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0(a:) , A»{x) ^ A^x) + -d»a(x)
,
(4)
while a gauge field mass term (i.e., a term quadratic in the fields A^) would not be gauge invariant and cannot be added to £ if the U(l) gauge symmetry has to be preserved. Indeed, the Lagrangian in Eq. (3) can still describe the physics of a massive gauge boson, provided the potential V(*0^O). The occurrence of a non trivial minimum, or, better, of a non trivial degeneracy of minima only depends on
264
L. Reina
the sign of the /J,2 parameter in V(<j>). For /i2>0 there is a unique minimum at 0*0 = 0, while for /u2 < 0 the potential develops a degeneracy of minima satisfying the equation 0*0 = —/i2/(2A). This is illustrated in Fig. 1, where the potential V{) is plotted as a function of the real and imaginary parts of the field 0 = 0i + lV>2- In the case of a unique minimum at 0*0 = 0 the
~15 \
V V ' ^ i r ' y J/ X -15 10 X / 15
10 15
Figure 1. The potential V($) (> =
Lagrangian in Eq. (3) describes the physics of a massless vector boson (e.g. the photon, in electrodynamics, with g = — e) interacting with a massive charged scalar particle. On the other hand, something completely different takes place when \j? < 0. Choosing the ground state of the theory to be a particular 0 among the many satisfying the equation of the minimum, and expanding the potential in the vicinity of the chosen minimum, transforms the Lagrangian in such a way that the original gauge symmetry is now hidden or spontaneously broken, and new interesting features emerge. To be more specific, let's pick the following 0o minimum (along the direction of the real part of 0, as traditional) and shift the 0 field accordingly: 1/2
~
2A
—
0(x) = 0 o + - ^ ( 0 i ( a ; ) + i 0 2 ( a ; ) ) .
(5)
The Lagrangian in Eq. (3) can then be rearranged as follows: 1 „II.I/„
•-iF^Fp,
.
1
+J
2
5
V ^ ^ + i ( ^ 0 i ) 2 + M 2 ^ 1 + ; (d>14>2)2+gvAMd>l
massive vector field
./;.
.
.
1 ,nii
.
N2
.
2
,2
massive scalar field
.
1
Goldstone boson
(6)
and now contains the correct terms to describe a massive vector field A11 with mass m2A =g2v2 (originating from the kinetic term of Cj,), a massive
Higgs Boson Physics 265
real scalar fieldi and 2fieldsirrelevant to this discussion. The gauge symmetry of the theory allows us to make the particle content more transparent. Indeed, if we parameterize the complex scalar field <j> as:
^)
= f
7T ( V + F(:C)) ^
j=(v + H(x)),
(7)
the x degree of freedom can be rotated away, as indicated in Eq. (7), by enforcing the U(l) gauge invariance of the original Lagrangian. With this gauge choice, known as unitary gauge or unitarity gauge, the Lagrangian becomes: L = LA + ^-A»A„
+ l- (d^Hd^H + 2^H2)
+ ...
(8)
which unambiguously describes the dynamics of a massive vector boson A11 of mass m2A =g2v2, and a massive real scalar field of mass m2H = —2fi2 = 2Aw2, the Higgs field. It is interesting to note that the total counting of degrees of freedom (d.o.f.) before the original U(l) symmetry is spontaneously broken and after the breaking has occurred is the same. Indeed, one goes from a theory with one massless vector field (two d.o.f.) and one complex scalar field (two d.o.f.) to a theory with one massive vector field (three d.o.f.) and one real scalar field (one d.o.f.), for a total of four d.o.f. in both cases. This is what is colorfully described by saying that each gauge boson has eaten up one scalar degree of freedom, becoming massive. We can now easily generalize the previous discussion to the case of a non-abelian Yang-Mills theory. LA in Eq. (3) now becomes: LA = \Fa^Fl„
with F^ = d^Aau - dvA% + gfabcA\Al
,
(9)
where the latin indices are group indices and fabc are the structure constants of the Lie Algebra associated to the non abelian gauge symmetry Lie group, defined by the commutation relations of the Lie Algebra generators ta: [ta, tb]—ifabctc. Let us also generalize the scalar Lagrangian to include several scalar fields
- V{) = ^
+ ±tf
,
where the sum over the index i is understood and D^ = <9M — igtaAa.
(10) The
266
L. Reina
Lagrangian of Eq. (3) is invariant under a non-abelian gauge transformation of the form: 0 i ( i ) - ( l + ta°(a:)ta)ii^ ,
(11)
Al{x) -> Al(x) + -gd^aa(x) + rbcA^(x)ac(x)
.
When y? < 0 the potential develops a degeneracy of minima described by the minimum condition: 0 2 = <^ = -fx2/X, which only fixes the magnitude of the vector <j>0. By arbitrarily choosing the direction of <j>0, the degeneracy is removed. The Lagrangian can be expanded in a neighborhood of the chosen minimum and mass terms for the gauge vector bosons can be introduced as in the abelian case, i.e.: \{D^t)2
—» 4>m_^0
. . . +±g2(ta)i(tb4>)iA*Ab» + ... _
1 g2{ta(f)o)i{tb(f)o)i 2s v
AaAb»
+
(12) _
,
Upon diagonalization of the mass matrix rr?ab in Eq. (12), all gauge vector bosons A^ for which ta(f)o ^ 0 become massive, and to each of them corresponds a Goldstone particle, i.e. an unphysical massless particle like the x field °f the abelian example. The remaining scalar degrees of freedom become massive, and correspond to the Higgs field H of the abelian example. The Higgs mechanism can be very elegantly generalized to the case of a quantum field theory when the theory is quantized via the path integral method a . In this context, the quantum analog of the potential V() is the effective potential Veff(
eff(tfci)
= -yji^[ci] for (pd(x) = constant = ipcl ,
(13)
where VT is the space-time extent of the functional integration and 4>d {x) is the vacuum expectation value of the field configuration 4>(x): cf>cl(x) =.
(14)
a Here I assume some familiarity with path integral quantization and the properties of various generating functionals introduced in that context, as I did while giving these lectures. The detailed explanation of the formalism used would take us too far away from our main track
Higgs Boson Physics
267
The stable quantum states of the theory are defined by the variational condition: = 0 (Xpcl
-^
^—Veff(ipcl)
= 0,
(15)
Pcl=
which identifies in particular the states of minimum energy of the theory, i.e. the stable vacuum states. A system with spontaneous symmetry breaking has several minima, all with the same energy. Specifying one of them, as in the classical case, breaks the original symmetry on the vacuum. The relation between the classical and quantum case is made even more transparent by the perturbative form of the effective potential. Indeed, Veff(ipci) can be organized as a loop expansion and calculated systematically order by order in h: Veff(ifci) = V(ipd) + loop effects ,
(16)
with the lowest order being the classical potential in Eq. (2). Quantum corrections to Veff(ipci) affect some of the properties of the potential and therefore have to be taken into account in more sophisticated studies of the Higgs mechanism for a spontaneously broken quantum gauge theory. We will see how this can be important in Section 2.3 when we discuss how the mass of the SM Higgs boson is related to the energy scale at which we expect new physics effect to become relevant in the SM. Finally, let us observe that at the quantum level the choice of gauge becomes a delicate issue. For example, in the unitarity gauge of Eq. (7) the particle content of the theory becomes transparent but the propagator of a massive vector field A^ turns out to be: n
'"'(&) = - r 2 2 - K " - - T > ( 17 ) A z k - ml V rnz and has a problematic ultra-violet behavior, which makes more difficult to consistently define and calculate ultraviolet-stable scattering amplitudes and cross sections. Indeed, for the very purpose of studying the renormalizability of quantum field theories with spontaneous symmetry breaking, the so called renormalizable or renormalizability gauges (R^ gauges) are introduced. If we consider the abelian Yang-Mills theory of Eqs. (l)-(3), the renormalizable gauge choice is implemented by quantizing with a gauge condition G of the form: G = ^ ( ^
+ W2)
,
(18)
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L. Reina
in the generating functional Z\J} = C [DADfaDfaexp
f(C-^G2)
i
«(£
(19)
where C is an overall factor independent of the fields, £ is an arbitrary parameter, and a is the gauge transformation parameter in Eq. (4). After having reduced the determinant in Eq. (19) to an integration over ghost fields (c and c), the gauge plus scalar fields Lagrangian looks like:
£ - l-G2 + Cghost = - ^ (-g^d2 + (l - 0 d»d" - M V") Av 2
;(^i)
^ < ^ + ^ 2 )
2 2 + c -d - CM
2
- | M
2
^
i+
(20)
such that: (A»(k)A»(-k))
k»kv 2
k
—m
k2 (fa(k)fa(-k))
+ k2
2 A
-it — £jr?A \
(21)
m\
= (c(k)c(-k))
k»k? k2
=
k2 — £m2A
where the vector field propagator has now a safe ultraviolet behavior. Moreover we notice that the fa propagator has the same denominator of the longitudinal component of the gauge vector boson propagator. This shows in a more formal way the relation between the fa degree of freedom and the longitudinal component of the massive vector field A11, upon spontaneous symmetry breaking. 2.2. The Higgs sector of the Standard
Model
The Standard Model is a spontaneously broken Yang-Mills theory based on the SU(2)i, x U(l)y non-abelian symmetry group 4 ' 5 . The Higgs mechanism is implemented in the Standard Model by introducing a complex scalar field >, doublet of SU(2) with hypercharge Y$ = 1/2, (22)
with Lagrangian
^HC'WZV-V^-A^W ,
(23)
Higgs Boson Physics
269
where D^ = {d^-igA^ -ig'Y^B^), and ra = aa/2 (for a= 1,2,3) are the SU(2) Lie Algebra generators, proportional to the Pauli matrix aa. The gauge symmetry of the Lagrangian is broken to U{l)em when a particular vacuum expectation value is chosen, e.g.: (0) =
72U)
with
v=
^2<0-A>°)-
\~*~)
(24
^
Upon spontaneous symmetry breaking the kinetic term in Eq. (23) gives origin to the SM gauge boson mass terms. Indeed, specializing Eq. (12) to the present case, and using Eq. (24), one gets: (D^D^
— • • • + 1(0 v) {gAfr* + g'B») (
— • • • + J T fcW + 92(AD2 + (~9Al + 9'B^] +••• (25) One recognizes in Eq. (25) the mass terms for the charged gauge bosons
W± = -±=(Al ± A\)
Mw=gV-,
-^
(26)
and for the neutral gauge boson Z®:
Z° = J—figAl
MZ = ^Tg^V-
- g'B») -^
,
(27)
l V9 + 9 while the orthogonal linear combination of A^ and BM remains massless and corresponds to the photon field (A^):
A
» = -rrr^^'Al+
9B
^ —> MA = ° >
(28
)
the gauge boson of the residual U(l)em gauge symmetry. The content of the scalar sector of the theory becomes more transparent if one works in the unitary gauge and eliminate the unphysical degrees of freedom using gauge invariance. In analogy to what we wrote for the abelian case in Eq. (7), this amounts to parametrize and rotate the
0
\
su(2)
„ ,
1 /
0
\
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L. Reina
after which the scalar potential in Eq. (23) becomes 1 A = _ 1 2 I-H rrA = -^-MlH \MHH* (30) -XH* A lH nl 4 2 ~" V 2"'"" 4' Three degrees of freedom, the x a ( a ; ) Goldstone bosons, have been reabsorbed into the longitudinal components of the W^ and Z° weak gauge bosons. One real scalar field remains, the Higgs boson H, with mass
£ 0 = v2H2
XvH6
M]j = —2/LA2 = 2Xv2 and self-couplings: H.
H.
\
H=_
3 f
M|
H
= - 3 i M?
X
H'
H"
"•H
Furthermore, some of the terms that we omitted in Eq. (25), the terms linear in the gauge bosons Wj^ and Z°, define the coupling of the SM Higgs boson to the weak gauge fields:
Ml
= 2i*$-
We notice that the couplings of the Higgs boson to the gauge fields are proportional to their mass. Therefore H does not couple to the photon at tree level. It is important, however, to observe that couplings that are absent at tree level may be induced at higher order in the gauge couplings by loop corrections. Particularly relevant to the SM Higgs boson phenomenology that will be discussed in Section 3 are the couplings of the SM Higgs boson to pairs of photons, and to a photon and a Z^ weak boson:
Y.Z H.
lAAA/ y
Higgs Boson Physics
271
as well as the coupling to pairs of gluons, when the SM Lagrangian is extended through the QCD Lagrangian to include also the strong interactions:
mnnp s
irtRRp
g
The analytical expressions for the # 7 7 , H^Z, and Hgg one-loop vertices are more involved and will be given in Section 3.1. As far as the Higgs boson tree level couplings go, we observe that they are all expressed in terms of just two parameters, either A and /i appearing in the scalar potential of Cj, (see Eq. 23)) or, equivalently, MH and v, the Higgs boson mass and the scalar field vacuum expectation value. Since v is measured in muon decay to be v= (y/2GF)~1/2 = 246 GeV, the physics of the SM Higgs boson is actually just function of its mass MHThe Standard Model gauge symmetry also forbids explicit mass terms for the fermionic degrees of freedom of the Lagrangian. The fermion mass terms are then generated via gauge invariant renormalizable Yukawa couplings to the scalar field (f>: Cyukawa c
= -TVQifu'n
- ^j QlLdR - ^
ULlR + h.C.
(31)
2
where (f> — —ia*, and Tf (/ = u, d, I) are matrices of couplings arbitrarily introduced to realize the Yukawa coupling between the field <j> and the fermionic fields of the SM. QlL and L\ (where i = 1,2,3 is a generation index) represent quark and lepton left handed doublets of SU(2)L, while ulR, dlR and lR are the corresponding right handed singlets. When the scalar fields acquires a non zero vacuum expectation value through spontaneous symmetry breaking, each fermionic degree of freedom coupled to <j> develops a mass term with mass parameter
where the process of diagonalization from the current eigenstates in Eq. (31) to the corresponding mass eigenstates is understood, and Tf are therefore the elements of the diagonalized Yukawa matrices corresponding to a given fermion / . The Yukawa couplings of the / fermion to the Higgs boson (yf) is proportional to Tf.
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L. Reina
V___.H =-i^f. = -i^
= -iyf
As long as the origin of fermion masses is not better understood in some more general context beyond the Standard Model, the Yukawa couplings y/ represent free parameter of the SM Lagrangian. The mechanism through which fermion masses are generated in the Standard Model, although related to the mechanism of spontaneous symmetry breaking, requires therefore further assumptions and involves a larger degree of arbitrariness as compared to the gauge boson sector of the theory. 2.3. Theoretical constraints boson mass
on the Standard
Model
Higgs
Several issues arising in the scalar sector of the Standard Model link the mass of the Higgs boson to the energy scale where the validity of the Standard Model is expected to fail. Below that scale, the Standard Model is the extremely successful effective field theory that emerges from the electroweak precision tests of the last decades. Above that scale, the Standard Model has to be embedded into some more general theory that gives origin to a wealth of new physics phenomena. From this point of view, the Higgs sector of the Standard Model contains actually two parameters, the Higgs mass (MJJ) and the scale of new physics (A). In this Section we will review the most important theoretical constraints that are imposed on the mass of the Standard Model Higgs boson by the consistency of the theory up to a given energy scale A. In particular we will touch on issues of unitarity, triviality, vacuum stability, fine tuning and, finally, electroweak precision measurements. 2.3.1. Unitarity The scattering amplitudes for longitudinal gauge bosons (Vx,Vj, —> VLVL, where V = W±,Z°) grow as the square of the Higgs boson mass. This is easy to calculate using the electroweak equivalence theorem 4 ' 5 , valid in the high energy limit (i.e. for energies s = Q2^>My), according to which the scattering amplitudes for longitudinal gauge bosons can be expressed in
Higgs Boson Physics
273
terms of the scattering amplitudes for the corresponding Goldstone bosons, i.e.:
Ml
= (iTi-irAiw1... ujn - u,1... O + o s
A(v[ ...vE^v£...vn
(33) where we have indicated by UJ1 the Goldstone boson associated to the longitudinal component of the gauge boson V1. For instance, in the high energy limit, the scattering amplitude for W^W^ —> W^W^ satisfies: A{W+W£
-> W£W£)
/ M2 \ +Ol-^-j
= A(UJ+LO- -> w+w-)
(34)
where .,
4- -
+ -s
v
Ml
/
S
t
v2 \ s - M%
'
(35)
t - Mfj
Using a partial wave decomposition, we can also write A as: oo
A =16TT ^(21
+ l)Pi(cos6)ai
,
(36)
1=0
where ai is the spin I partial wave and Pi(cos6) are the Legendre polynomials. In terms of partial wave amplitudes ai, the scattering cross section corresponding to A can be calculated to be:
< r = is^ £ ( 2 i + l)|a,|2 ,
(37)
t=t
where we have used the orthogonality of the Legendre polynomials. Using the optical theorem, we can impose the unitarity constraint by writing that: 1 fi
.°°_
s
i
+ l)\ai\2
a=—^2(2l
= -Im[A(e
= 0)} ,
(38)
s
i=o
where A{9 = 0) indicates the scattering amplitude in the forward direction. This implies that: \ai\2 = Re(at)2 +lm(ai)2
= Im(oj)
—•* |Re(aj)| < ^ -
(39)
Via Eq. (39), different a; amplitudes can than provide constraints on MHAs an example, let us consider the J = 0 partial wave amplitude ao for the W£W£ —> W£W£ scattering we introduced above: 1 '° a0 = —— / Adt loirs
M2 167T1I
2
r2 4-
MH2
r 1 *-M%
M2
(
-^'°<
1+
s \1
«j)J (40)
274
L. Reina
In the high energy limit (Mjj < s), ag reduces to: ao
•
<41>
MH < 870 GeV .
(42)
^
- 8 ^
from which, using Eq. (39), one gets:
Other more constraining relations can be obtained from different longitudinal gauge boson scattering amplitudes. For instance, considering the coupled channels like W^W^ —> Z^ZL, one can lower the bound to: MH < 710 GeV .
(43)
Taking a different point of view, we can observe that if there is no Higgs boson, or equivalently if M\ » s, Eq. (39) gives indications on the critical scale yfsl above which new physics should be expected. Indeed, considering again W£W£ —> W^W^ scattering, we see that: a 0 (o; + w
-^w+a; ) —> " 3 2 ^ 2 '
( 44 )
from which, using Eq. (39), we get: V ^ < 1.8 TeV .
(45)
Using more constraining channels the bound can be reduced to: v ^ < 1.2 TeV .
(46)
This is very suggestive: it tells us that new physics ought to be found around 1-2 TeV, i.e. exactly in the range of energies that will be explored by the Tevatron and the Large Hadron Collider. 2.3.2. Triviality and vacuum stability The argument of triviality in a A>4 theory goes as follows. The dependence of the quartic coupling A on the energy scale (Q) is regulated by the renormalization group equation
This equation states that the quartic coupling A decreases for small energies and increases for large energies. Therefore, in the low energy regime the coupling vanishes and the theory becomes trivial, i.e. non-interactive.
Higgs Boson Physics
275
In the large energy regime, on the other hand, the theory becomes nonperturbative, since A grows, and it can remain perturbative only if A is set to zero, i.e. only if the theory is made trivial. The situation in the Standard Model is more complicated, since the running of A is governed by more interactions. Including the lowest orders in all the relevant couplings, we can write the equation for the running of X(Q) with the energy scale as follows: 32TT2^ =24A
2
-(35'2+952-24yt2)A+^'4
+
^ V + ^ 4 - 2 4 y t 4 + . . . (48)
where i = ln(Q 2 /Qo) is the logarithm of the ratio of the energy scale and some reference scale Qo square, yt=mt/v is the top-quark Yukawa coupling, and the dots indicate the presence of higher order terms that have been omitted. We see that when MH becomes large, A also increases (since MH = 2At>2) and the first term in Eq. (48) dominates. The evolution equation for A can then be easily solved and gives:
KQ) =
A(Qo) a
,„,N •
(49)
1 -&A(Q 0 )ln(g)
When the energy scale Q grows, the denominator in Eq. (49) may vanish, in which case X(Q) hits a pole, becomes infinite, and a triviality condition needs to be imposed. This is avoided imposing that the denominator in Eq. (49) never vanishes, i.e. that X(Q) is always finite and 1/A(Q) > 0. This condition gives an explicit upper bound on MH'M
8TTV
H < „, , A ^ . (50) 31°g(&) obtained from Eq. (49) by setting Q = A, the scale of new physics, and Qo = v, the electroweak scale. On the other hand, for small MH , i.e. for small A, the last term in Eq. (48) dominates and the evolution of X(Q) looks like: A A
( )=^)-i^
2 l
°s(^)
•
(51)
To assure the stability of the vacuum state of the theory we need to require that A(A) > 0 and this gives a lower bound for MH' A(A)>0
—
2 ,, /A2 f2 _ 2,v Mj* > ^ log ( ^ - j
.
(52)
More accurate analyses include higher order quantum correction in the scalar potential and use a 2-loop renormalization group improved effective
276
L. Reina
potential, V e //, whose nature and meaning has been briefly sketched in Section 2.1.
2.3.3. Indirect bounds from electroweak precision measurements Once a Higgs field is introduced in the Standard Model, its virtual excitations contribute to several physical observables, from the mass of the W boson, to various leptonic and hadronic asymmetries, to many other electroweak observables that are usually considered in precision tests of the Standard Model. Since the Higgs boson mass is the only parameter in the Standard Model that is not directly determined either theoretically or experimentally, it can be extracted indirectly from precision fits of all the measured electroweak observables, within the fit uncertainty. This is actually one of the most important results that can be obtained from precision tests of the Standard Model and greatly illustrates the predictivity of the Standard Model itself. All available studies can be found on the LEP Electroweak Working Group and on the LEP Higgs Working Group Web pages 19 ' 20 as well as in their main publications 23 . 22 . 21 . 25 . 24 . An excellent recent series of lectures on the subject of Precision Electroweak Physics is also available from a previous TASI school 27 . The correlation between the Higgs boson mass MH , the W boson mass M\y, the top-quark mass mt, and the precision data is illustrated in Figs. 2 and 3. Apart from the impressive agreement existing between the indirect determination of My/ and mt and their experimental measurements we see in Fig. 2 that the 68% CL contours from LEP, SLD, and Tevatron measurements select a SM Higgs boson mass region roughly below 200 GeV. Therefore, assuming no physics beyond the Standard Model at the weak scale, all available electroweak precision data are consistent with a light Higgs boson. The actual value of MH emerging from the electroweak precision fits strongly depends on theoretical predictions of physical observables that include different orders of strong and electroweak corrections. As an example, in Fig. 2 the magenta arrow shows how the yellow band would move for one standard deviation variation in the QED fine-structure constant a(m2z). It also depends on the fit input parameters. As we see in Fig. 3, MH grows for larger mt. The sensitivity of the indirect bound on MH to mt is clearly visible both in Fig. 2 and in Fig. 4, where you can find the famous blue band plot. In both figures, we compare the results of the Winter 2005 and Summer 2005 electroweak precision fits. As far as the Higgs boson mass goes,
Higgs Boson Physics i
,
i
i
|
i
i
i
277 i
— LEP1 and 3LD LEP2 and Tevatron (prel. 80.5-
80.5
68% CL
>
> Q)
S
M0
CD
80.4
& 80.4<: E
g E
80.3-
80.3
: y
!•:... i - J M
150
200
150
1
t M
t
. .
•••'.
•/• .
175
200
m, [GeV] Figure 2. Comparison of the indirect measurements of Mw a n d mt (LEP I+SLD data) (solid contour) and the direct measurement (pp colliders and LEP II data) (dashed contour). In both cases the 68% CL contours are plotted. Also shown is the SM relationship for these masses as a function of the Higgs boson mass, m,ji. The arrow labeled A a shows the variation of this relation if a(M^) is varied by one standard deviation. The left hand side plot is from Ref. 2 2 and corresponds to the Winter 2005 situation (see Eq. 53), the right hand side plot is from the Ref. 2 3 and corresponds to the Summer 2005 situation (see Eq. 54). The comparison between Winter and Summer 2005 is shown in the right hand side plot.
the main change between Winter and Summer 2005 has been the value of the top-quark mass. We go from: (MH \MH<
= ll7t%
GeV
251 GeV (95% CL)
for m t = 178. ± 4 . 3 GeV
(53)
in Winter 2005 22 , to: MH = 91113 GeV MH < 186 - 219 GeV (95% CL)
for mt = 172.7 ± 2 . 9 GeV , (54)
in Summer 2005 23 . We see in Fig. 2 that the overlap between the direct and indirect determination of MH is greatly reduced when the value of mt decreases and at the same time the minimum of the Ax 2 band in Fig. 4 considerably shifts. While in the first case the electroweak precision fits are still largely compatible with the direct searches at LEP II that have placed a 95% CL lower bound on MH at: MH > H4.4 GeV ,
(55)
278
L. Reina
• High Q except m t 68% CL
200 > CD
O z~ 180
160
Excluded 10
10 m H [GeV]
10
Figure 3. The 68% confidence level contour in mt and MH for the fit to all data except the direct measurement of mt, indicated by the shaded horizontal band of ±1
in the second case a large region of the A\2 band in Fig. 4, in particular the region about the minimum, is already excluded, and values of MH very close to the experimental lower bound seem to be favored. It is fair to conclude that the issue of constraining MH from electroweak precision fits is open to controversies and, at a closer look, emerges as a not clear cut statement. With this respect, Fig. 5 illustrates the sensitivity of a few selected electroweak observables to the Higgs boson mass as well as the preferred range for the SM Higgs boson mass as determined from all electroweak observables . One can observe that Mw and the leptonic asymmetries prefer a lighter Higgs boson, while ApCB and the NuTeV determination of sin 9w prefer a heavier Higgs boson. A certain tension is still present in the data. We could just think that things will progressively adjust and, after the discovery of a light Higgs boson at either the Tevatron or the LHC, this will result in yet another amazing success of the Standard
Higgs Boson Physics
m H [GeV]
279
m H [GeV]
Figure 4. A x 2 = X2 — x\xin vs- MH curve. The line is the result of the fit using all electroweak data; the band represents an estimate of the theoretical error due to missing higher order corrections. The vertical band shows the 95% CL exclusion limit on MH from direct searches. The solid and dashed curves are derived using different evaluations of A a l ^ A f f ) . The dotted curve includes low Q2 data. The left hand side plot is from Ref. and corresponds to the Winter 2005 situation (see Eq. 53), the right hand side plot is from the Ref. 2 3 and corresponds to the Summer 2005 situation (see Eq. 54). Note the different horizontal scale in the two plots.
Model. Or, one can interpret the situation depicted in Fig. 5 as an unavoidable indication of the presence of new physics beyond the Standard Model. Indeed, since the data compatible with a lighter Higgs boson are very solid, one could either interpret the data compatible with a larger value of MH as an indication of new physics beyond the Standard Model, or one could drop them as wrong, and still, the Higgs boson mass would turn out to be so small not to be compatible anymore with the Standard Model, signaling once more the presence of new physics.
2.3.4. Fine-tuning One aspect of the Higgs sector of the Standard Model that is traditionally perceived as problematic is that higher order corrections to the Higgs boson mass parameter square contain quadratic ultraviolet divergences. This is expected in a \<j)A theory and it does not pose a renormalizability problem, since a A>4 theory is renormalizable. However, although per se renormalizable, these quadratic divergences leave the inelegant feature that the Higgs
280
L. Reina
-• a
1
had
RJ> A,(P,)
R° A o,b M
fb
M
fb
\ A,(SLD) sin 2 e^(Q fb ) m
w*
Q„(Cs) sin2e
Bis( e _ e T
sin26w(vN) 9L(VN)
gR
'preliminary
10
10 M H [GeV]
10
Figure 5. Preferred range for the SM Higgs boson mass MH as determined from various electroweak observables. The shaded band shows the overall constraint on the mass of the Higgs boson as derived from the full data set. From Ref. 2 2 .
boson renormalized mass square has to result from the adjusted or finetuned balance between a bare Higgs boson mass square and a counterterm that is proportional to the ultraviolet cutoff square. If the physical Higgs mass has to live at the electroweak scale, this can cause a fine-tuning of several orders of magnitude when the scale of new physics A (the ultraviolet cutoff of the Standard Model interpreted as an effective low energy theory) is well above the electroweak scale. Ultimately this is related to a symmetry principle, or better to the absence of a symmetry principle. Indeed, setting to zero the mass of the scalar fields in the Lagrangian of the Standard Model does not restore any symmetry to the model. Hence, the mass of the scalar fields are not protected against large corrections.
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Models of new physics beyond the Standard Model should address this fine-tuning problem and propose a more satisfactory mechanism to obtain the mass of the Higgs particle(s) around the electroweak scale. Supersymmetric models, for instance, have the remarkable feature that fermionic and bosonic degrees of freedom conspire to cancel the Higgs mass quadratic loop divergence, when the symmetry is exact. Other non supersymmetric models, like little Higgs models, address the problem differently, by interpreting the Higgs boson as a Goldstone boson of some global approximate symmetry. In both cases the Higgs mass turns out to be proportional to some small deviation from an exact symmetry principle, and therefore intrinsically small. As suggested in Ref. 28 , the no fine-tuning condition in the Standard Model can be softened and translated into a maximum amount of allowed fine-tuning, that can be directly related to the scale of new physics. As derived in Section 2.1, upon spontaneous breaking of the electroweak symmetry, the SM Higgs boson mass at tree level is given by Mjj — — 2/x2, where JJL2 is the coefficient of the quadratic term in the scalar potential. Higher order corrections to M ^ can therefore be calculated as loop corrections to /x2, i.e. by studying how the effective potential in Eq. (16) and its minimum condition are modified by loop corrections. If we interpret the Standard Model as the electroweak scale effective limit of a more general theory living at a high scale A, then the most general form of fx2 including all loop corrections is: oo
2
2
2
/2 = M + A E
C
"(A*) l o g n (A/Q) ,
(56)
n=0
where Q is the renormahzation scale, Aj are a set of input parameters (couplings) and the cn coefficients can be deduced from the calculation of the effective potential at each loop order. As noted originally by Veltman, there would be no fine-tuning problem if the coefficient of A2 in Eq. (56) were zero, i.e. if the loop corrections to fi2 had to vanish. This condition, known as Veltman condition, is usually over constraining, since the number of independent c n (set to zero by the Veltman condition) can be larger than the number of inputs A,. However the Veltman condition can be relaxed, by requiring that only the sum of a finite number of terms in the coefficient of A2 is zero, i.e. requiring that:
Y, c n (A i )log n (A/M H )=0 , o
(57)
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L. Reina
where the renormalization scale fx has been arbitrarily set to MH and the order n has been set to nmax, fixed by the required order of loop in the calculation of Veff. This is based on the fact that higher orders in n come from higher loop effects and are therefore suppressed by powers of (167r 2 ) -1 . Limiting n to nmax, Eq. (57) can now have a solution. Indeed, if the scale of new physics A is not too far from the electroweak scale, then the Veltman condition in Eq. (57) can be softened even more by requiring that: n
max
2
X^c n (A i )log"(A/M H )<^ .
(58)
o This condition determines a value of Amax such that for A < Amax the stability of the electroweak scale does not require any dramatic cancellation in p?. In other words, for A < Amax the renormalization of the SM Higgs boson mass does not require any fine-tuning. As an example, for nmax = 0, c0 = (327r 2 w 2 )- 1 3(2M^ + M | 4- MH - 4mt2), and the stability of the electroweak scale is assured up to A of the order of 4TTV ~ 2 TeV. For nmax = 1 the maximum A is pushed up to A ~ 15 TeV and for nmax = 2 up to A ~ 50 TeV. So, just going up to 2-loops assures us that we can consider the SM Higgs sector free of fine-tuning up to scales that are well beyond where we would hope to soon discover new physics. For each value of nmax, and for each corresponding Amax, MH becomes a function of the cutoff A, and the amount of fine-tuning allowed in the theory limits the region in the (A, MH) plane allowed to MH(A). This is well represented in Fig. 6, where also the constraint from the conditions of unitarity (see Section 2.3.1), triviality (see Section 2.3.2), vacuum stability (see Section 2.3.2) and electroweak precision fits (see Section 2.3.3) are summarized. Finally, the main lesson we take away from this plot is that if a Higgs boson is discovered new physics is just around the corner and should manifest itself at the LHC. 2.4.
The Higgs sector of the minimal Standard Model
supersymmetric
In the supersymmetric extension of the Standard Model, the electroweak symmetry is spontaneously broken via the Higgs mechanism introducing two complex scalar SU(2)L doublets. The dynamics of the Higgs mechanism goes pretty much unchanged with respect to the Standard Model case, although the form of the scalar potential is more complex and its minimization more involved. As a result, the W± and Z° weak gauge bosons
Higgs Boson Physics
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10
283
10
A (TeV) Figure 6. The SM Higgs boson mass Mn as a function of the scale of new physics A, with all the constraints derived from unitarity, triviality, vacuum stability, electroweak precision fits, and the requirement of a limited fine-tuning. The empty region is consistent with all the constraints and less than 1 part in 10 fine-tuning. From Ref. 2 8 .
acquire masses that depend on the parameterization of the supersymmetric model at hand. At the same time, fermion masses are generated by coupling the two scalar doublets to the fermions via Yukawa interactions. A supersymmetric model is therefore a natural reference to compare the Standard Model to, since it is a theoretically sound extension of the Standard Model, still fundamentally based on the same electroweak symmetry breaking mechanism. Far from being a simple generalization of the SM Higgs sector, the scalar sector of a supersymmetric model can be theoretically more satisfactory because: (i) spontaneous symmetry breaking is radiatively induced (i.e. the sign of the quadratic term in the Higgs potential is driven from positive to negative) mainly by the evolution of the top-quark Yukawa coupling from the scale of supersymmetry-breaking to the electroweak scale, and (ii) higher order corrections to the Higgs mass do not contain quadratic
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L. Reina
divergences, since they cancel when the contribution of both scalars and their super-partners is considered (see Section 2.3.4). At the same time, the fact of having a supersymmetric theory and two scalar doublets modifies the phenomenological properties of the supersymmetric physical scalar fields dramatically. In this Section we will review only the most important properties of the Higgs sector of the MSSM, so that in Section 3 we can compare the physics of the SM Higgs boson to that of the MSSM Higgs bosons. I will start by recalling some general properties of a Two Higgs Doublet Model in Section 2.4.1, and I will then specify the discussion to the case of the MSSM in Section 2.4.2. In Sections 2.4.3 and 2.4.4 I will review the form of the couplings of the MSSM Higgs bosons to the SM gauge bosons and fermions, including the impact of the most important supersymmetric higher order corrections. A thorough introduction to Supersymmetry and the Minimal Supersymmetric Standard Model has been given during this school by Prof. H. Haber to whose lectures I refer 2 9 . 2.4.1. About two Higgs doublet models The most popular and simplest extension of the Standard Model is obtained by considering a scalar sector made of two instead of one complex scalar doublets. These models, dubbed Two Higgs Doublet Models (2HDM), have a richer spectrum of physical scalar fields. Indeed, after spontaneous symmetry breaking, only three of the eight original scalar degrees of freedom (corresponding to two complex doublet) are reabsorbed in transforming the originally massless vector bosons into massive ones. The remaining five degrees of freedom correspond to physical degrees of freedom in the form of: two neutral scalar, one neutral pseudoscalar, and two charged scalar fields. At the same time, having multiple scalar doublets in the Yukawa Lagrangian (see Eq. (31)) allows for scalar flavor changing neutral current. Indeed, when generalized to the case of two scalar doublet2, Eq. (31) becomes (quark case only):
cYUkaWa = - £
T
ii,kQi*k'CuR - E
T
UQi*k4i+h-c-.
(59)
fc=l,2 fc=l,2
where each pair of fermions (i,j) couple to a linear combination of the scalar fields 0 1 and (j)2. When, upon spontaneous symmetry breaking, the fields (p1 and
($fc) = J L
for
A; = 1,2 ,
(60)
Higgs Boson Physics
285
the reparameterization of Cyukawa of Eq. (59) in the vicinity of the minimum of the scalar potential, with <3>fc = $'fe + vk (for k = 1, 2), gives: Cvukawa = - « L X ) r"i,fc ~M u « " ^ 5 3 r«,fc ^ k "
d
fl + h.c. + FC couplings ,
k v
'
"
*
'
(61) where the fermion mass matrices M4" and Mfj are now proportional to a linear combination of the vacuum expectation values of2. The diagonalization of M," and M^ does not imply the diagonalization of the couplings of the (j)'k fields to the fermions, and Flavor Changing (FC) couplings arise. This is perceived as a problem in view of the absence of experimental evidence to support neutral flavor changing effects. If present, these effects have to be tiny in most processes involving in particular the first two generations of quarks, and a safer way to build a 2HDM is to forbid them all together at the Lagrangian level. This is traditionally done by requiring either that u-type and d-type quarks couple to the same doublet (Model I) or that u-type quarks couple to one scalar doublet while d-type quarks to the other (Model II). Indeed, these two different realization of a 2HDM can be justified by enforcing on Cyukawa the following ad hoc discrete symmetry: $ i _» _ $ i
a n d
$2 _, $2
d* - • - d * and v? -> ±v?
{
'
The case in which FC scalar neutral current are not forbidden (Model III) has also been studied in detail. In this case both up and down-type quarks can couple to both scalar doublets, and strict constraints have to be imposed on the FC scalar couplings in particular between the first two generations of quarks. 2HDMs have indeed a very rich phenomenology that has been extensively studied. In these lectures, however, we will only compare the SM Higgs boson phenomenology to the phenomenology of the Higgs bosons of the MSSM, a particular kind of 2HDM that we will illustrate in the following Sections. 2.4.2. The MSSM Higgs sector: introduction The Higgs sector of the MSSM is actually a Model II 2HDM. It contains two complex SU(2)L scalar doublets:
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L. Reina
with opposite hypercharge (Y = ±1), as needed to make the theory anomaly-free13. $1 couples to the up-type and2 to the down-type quarks respectively. Correspondingly, the Higgs part of the superpotential can be written as: VH = (|M|2 + m ? ) | $ 1 | 2 + ( H 2 + m 2 ) | $ 2 | 2 - / i B e y ^ * ^ + /i.e.)
+ ^ ± ^ ( | $ 1 | 2 - | $ 2 | 2 ) 2 + ^|$t$2|2
f
in which we can identify three different contributions
(64) 29,7
:
fi) the so called D terms, containing the quartic scalar interactions, which for the Higgs fields $1 and $2 correspond to:
^ ± » 2 (!*!!»-|* a |») a + ^ | * t $ a | a , with g and g' the gauge couplings of SU(2)L tively; (ii) the so called F terms, corresponding to: |M|2(|*I|2 + |*2|2)
(65)
and C/(l)y respec-
;
(66)
(Hi) the soft SUSY-breaking scalar Higgs mass and bilinear terms, corresponding to: m 2 | $ i | 2 + m 2 . | $ 2 | 2 - / i £ e i j ( $ i ^ 2 + /i.c.) .
(67)
Overall, the scalar potential in Eq. (64) depends on three independent combinations of parameters, |/i| 2 + m 2 , |/x| 2 + TO2, and piB. One basic difference with respect to the SM case is that the quartic coupling has been replaced by gauge couplings. This reduced arbitrariness will play an important role in the following. Upon spontaneous symmetry breaking, the neutral components of $1 and $2 acquire vacuum expectation values
b
Another reason for the choice of a 2HDM is that in a supersymmetric model the superpotential should be expressed just in terms of superfields, not their conjugates. So, one needs to introduce two doublets to give mass to fermion fields of opposite weak isospin. The second doublet plays the role of 4>c in the Standard Model (see Eq. (31)), where tj>c has opposite hypercharge and weak isospin with respect to 4>.
Higgs Boson Physics
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and the Higgs mechanism proceed as in the Standard Model except that now one starts with eight degrees of freedom, corresponding to the two complex doublets $1 and <3>2- Three degrees of freedom are absorbed in making the W * and the Z° massive. The W mass is chosen to be: M{y = g2(v2 +v2)/4 — g2v2/4, and this fixes the normalization of v\ and v2, leaving only two independent parameters to describe the entire MSSM Higgs sector. The remaining five degrees of freedom are physical and correspond to two neutral scalar fields h° = -(V2Re4% - v2) sin a + (v^Re^? - v{) cos a
(69)
H° = (\/2Re02 - v2) cos a + (v^Re^? - vx) sin a , one neutral pseudoscalar field A0 = V2 (Im<^ sin /3 + Im0? cos 0) ,
(70)
and two charged scalar fields tf ± = 4>± sin /3 + (t>f cos P ,
(71)
where a and /3 are mixing angles, and tan/3 = v\jv2. At tree level, the masses of the scalar and pseudoscalar degrees of freedom satisfy the following relations: M2H± ^Ml Mlh
+ M2,
= \[M2A
,
+ M2±
(72) ((Mi + M2Z)2 - AM2ZM\ cos2 2/3) 1 / 2 ) ,
making it natural to pick M& and tan /3 as the two independent parameters of the Higgs sector. Eq. (72) provides the famous tree level upper bound on the mass of one of the neutral scalar Higgs bosons, h°: Ml < M2Z cos 2P<M2Z
,
(73)
which already contradicts the current experimental lower bound set by LEP II: Mh > 93.0 GeV 26 . The contradiction is lifted by including higher order radiative corrections to the Higgs spectrum, in particular by calculating higher order corrections to the neutral scalar mass matrix. Over the past few years a huge effort has been dedicated to the calculation of the full oneloop corrections and of several leading and sub-leading sets of two-loop corrections, including resummation of leading and sub-leading logarithms via appropriate renormalization group equation (RGE) methods. A detailed discussion of this topic can be found in some recent reviews 8>30'31 and in the original literature referenced therein. For the purpose of these lectures,
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L. Reina
let us just observe that, qualitatively, the impact of radiative corrections on MJ^ax can be seen by just including the leading two-loop corrections proportional to y\, the square of the top-quark Yukawa coupling, and applying RGE techniques to resum the leading orders of logarithms. In this case, the upper bound on the light neutral scalar in Eq. (73) is modified as follows:
i
*?
(74)
12M|yj
where Mf = (M-2 + M? )/2 is the average of the two top-squark masses, mt is the running top-quark mass (to account for the leading two-loop QCD corrections), and Xt is the top-squark mixing parameter defined by the top-squark mass matrix: M
m D o,Q + t + i -T "H T- ^L mtXt t
rntXt ••"<•"<M%t + m\ + DlR
I
/ 7 rN
with Xt = At — // cot /3 (At being one of the top-squark soft SUSY breaking trilinear coupling), DlL — (1/2 — 2/3sin^w)A4"|cos2/3, and D^ = 2/3sin 9wM^zcos2(i. Fig. 7 illustrates the behavior of Mh as a function of tan/3, in the case of minimal and maximal mixing. For large tan/3 a plateau (i.e. an upper bound) is clearly reached. The green bands represent the variation of Mh as a function of mt when mt = 175 ± 5 GeV. If top-squark mixing is maximal, the upper bound on Mh is approximately Mrnax ^ 1 3 5 G e V c . T h e behavior of both Mh,H and MH± as a function of MA and tan/3 is summarized in Fig. 8, always for the case of maximal mixing. It is interesting to notice that for all values of MA and tan/3 the MH > M™ax. Also we observe that, in the limit of large tan/3, i) for MA < M%iax: Mh ^ MA and MH ~ M^ax, while ii) for MA > M^ax: rmax MH ^ MA and Mh ^ M^ 2.4.3. MSSM Higgs boson couplings to electroweak gauge bosons The Higgs boson couplings to the electroweak gauge bosons are obtained from the kinetic term of the scalar Lagrangian, in strict analogy to what we have explicitly seen in the case of the SM Higgs boson. Here, we would like to recall the form of the HiVV and HiHjV couplings (for : 144 GeV for mt = 178 GeV.
Higgs Boson Physics
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1
-
|
' '
< <<< |
289
|
......
maximal miyir;
120
~
1 rn.:i:.. % _i^ HL*i"-;i:iu
100
; : - •
/
•
Mt = 1 7 5 ± 5 GeV MEUSY =
60
mA
=
1
TeV
— ; -
fi = - 2 0 0 GeV
-
1
,
,
1 , , , ,1
1
-
5
20
50
10
tan f> Figure 7. The mass of the light neutral scalar Higgs boson, h°, as a function of tan/3, in t h e minimal mixing and maximal mixing scenario. T h e green bands are obtained by varying the top-quark mass in the mt = 175 ± 5 GeV range. The plot is built by fixing MA = 1 TeV and MsuSY = MQ =MV = MD = 1 TeV. From Ref. 8 .
Hi = hQ,H°,A°,H±, and V = W±,Z°) that are most important in order to understand the main features of the MSSM plots that will be shown in Section 3. First of all, the couplings of the neutral scalar Higgs bosons to both W± and Z° can be written as: 9hvv - 9vMv sin(/3 - a)g^
, gHvv = 9vMv cos(j3 - a)g^
,
(76)
where gv-2Mv/v, while the A°VV and H^V couplings vanish because of CP-invariance. As in the SM case, since the photon is massless, there are no tree level 7 7 ^ and ryZ°Hi couplings. Moreover, in the neutral Higgs sector, only the h°A°Z° and H°A°Z° couplings are allowed and given by: 9hAZ =
n___n
2 cos
8\Y
(ph-VAT
,
9HAZ = -
_ „ -{PH-PAY 2 COS 9 w
,
(77)
290
L. Reina 250
1
1
1
1
1
1
1
1
*5 200 O
H* ( H ( h (
1
1
i s
I
y\i—-7
/,^
#f
30
•i/'
— "
'if
Sv / 30
X
£sr ^sr
X
X
'-/
^sfsS
—
• —^——^— s /'
'
/ /
' / '"' 100 ,•
1
. —
^y^S _.—^> S
CO
"
1
1
) ) )
S Q0
1
maximal mixing fj, = - 2 0 0 GeV Msugy = 1 TeV
.
60 150
1
1
/
•
.' ^ • ""
, r - r
I
'
/y
30
."
"
-
tan 0 = 3
<~ ' I
,
1
i
i
i
•
•
100
I
i
mA
-
.
200
150
250
(GeV)
Figure 8. The mass of the light (h°) and heavy (H°) neutral scalar Higgs bosons, and of the charged scalar Higgs boson (H^) as a function of the neutral pseudoscalar mass MA, for two different values of tan/3 (tan/3 = 3,30). The top-quark mass is fixed to mt = 174.3 GeV and MsuSY = MQ =M\J = MQ = 1 TeV. The maximal mixing scenario is chosen. From Ref. 8 .
where all momenta are incoming. We also have several HiHjV involving the charge Higgs boson, namely: 9H+H-Z
—
2 cos 9\y -ie(pH+
9H+H-
cos 20W(PH+
-PH-Y
couplings
(78)
-PH-Y
9H*hw± = Ti-^cos(p-a)iph-pH^Y
.
9HTHW±
= ±i-sm((3
.
9HTAW±
= ^(PA-PH±Y
- a)(pH-PHTY •
At this stage it is interesting to introduce the so called decoupling limit, i.e. the limit of MA > Mz, and to analyze how masses and couplings behave in this particular limit. MH± in Eq. (72) is unchanged, while Mh,H become: Mh ~ M^ax
and MH ~ MA + M\ sin2 2/? .
(79)
Higgs Boson Physics
291
Moreover, as one can derive from the diagonalization of the neutral scalar Higgs boson mass matrix: 0^m
^ cos (0-a)=
Ml{M\-Mp _ Ml) Ml{Ml
M\*M% M | sin2 4/3 - > 4M1 •
(80)
From the previous equations we then deduce that, in the decoupling limit, the only light Higgs boson is h° with mass Mh — M™ax, while MH — MH± ~ MA » Mz, and because cos(/3 - a) —> 0 (sin(/3 - a) —> 1)), the couplings of /i° to the gauge bosons tend to the SM Higgs boson limit. This is to say that, in the decoupling limit, the light MSSM Higgs boson will be hardly distinguishable from the SM Higgs boson. Finally, we need to remember that the tree level couplings may be modified by radiative corrections involving both loops of SM and MSSM particles, among which loops of third generation quarks and squarks dominate. The very same radiative corrections that modify the Higgs boson mass matrix, thereby changing the definition of the mass eigenstates, also affect the couplings of the corrected mass eigenstates to the gauge bosons. This can be reabsorbed into the definition of a renormalized mixing angle a or a radiatively corrected value for cos(/3 — a) (sm(/3 — a)). Using the notation of Ref. 8 , the radiatively corrected cos(/3 — a) can be written as: cos(/3 — a) = K
MJ sin 4/3 2M\
Ml \M\
+U
(81)
where y - - , , SM2n-SMl2 + 2 M | cos 2/3
5M\2 Ml sin 2/3 '
l
'
and SMij are the radiative corrections to the corresponding elements of the CP-even Higgs squared-mass matrix (see Ref. 8 ) . It is interesting to notice that on top of the traditional decoupling limit introduced above (MA S> MZ), there is now also the possibility that cos(/? — a) —> 0 if K —> 0, and this happens independently of the value of MA2.4.4. MSSM Higgs boson couplings to fermions As anticipated, $ i and $2 have Yukawa-type couplings to the up-type and down-type components of all SU(2)L fermion doublets. For example, the Yukawa Lagrangian corresponding to the third generation of quarks reads: ^Yukawa = ~ht [tR(j)°tL - iRfbL] - hb \bR<jP2bL - bR4>2tL] + h.c. (83)
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L. Reina
Upon spontaneous symmetry breaking Cyukawa provides both the corresponding quark masses: vi vsinp mt =
v2 , vcosP and mb = hb—j= = hb—-j=-
and the corresponding Higgs-quark couplings: cos (y. 9h.tt = -^—fiVt = [sin(/3 - Oi) + cot 0 cos(/3 - a)] yt , sin fD sin OL 9hbb = ~^w}2/ & = t s i n (0 - a) - tan/3cos(/? - a)] yb ,
,
(84)
(85)
sin ex 9mi = ^—aVt = [cos(/8 - a) - cot/3sin(/3 - a)] j / t , sin ^o COS CK
3/J&5 = -—-oVb = [cos(/3 - a) + tan/3sin(/3 - a)] yb , 9Ati= cot Pyt
, gAbb = ta,npyb
,
9H±tb = n>M lmt c o t 0(1 - 75) + " i b tan /?(1 + 75)] , where yg = m g /w (for q = t,b) are the SM couplings. It is interesting to notice that in the MA 3> Mz decoupling limit, as expected, all the couplings in Eq. (85) reduce to the SM limit, i.e. all H°, A0, and H± couplings vanish, while the couplings of the light neutral Higgs boson, h°, reduce to the corresponding SM Higgs boson couplings. The Higgs boson-fermion couplings are also modified directly by oneloop radiative corrections (squarks-gluino loops for quarks couplings and slepton-neutralino loops for lepton couplings). A detailed discussion can be found in Ref. 8 ' 7 and in the literature referenced therein. Of particular relevance are the corrections to the couplings of the third quark generation. These can be parameterized at the Lagrangian level by writing the radiatively corrected effective Yukawa Lagrangian as: Ceff Yukawa
(hb + 6hb)bRQJL&2 + (ht +
8ht)iRQm
(86)
K - AhttRQm*kfok* - AhbbAU.h~nk(f,k* + h.c RQ L$l*
where we notice that radiative corrections induce a small coupling between $1 and down-type fields and between $2 and up-type fields. Moreover the tree level relation between hb, ht, mb and mt are modified as follows: hbv ( 5hb A/ibtan/A hbv mb = —= cos/3 1 + -—- + = -^= cos P(l + Ab) , (87) \ hb hb J yj2 hV2 tv • o (t , Sht1 , A M a n / 3 \ htv mt = —7= -7=smp[l \7 == -—7= > s i n: / 3 ( l + A t ) SID p 1 + + -- +1
Higgs Boson Physics
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where the leading corrections are proportional to Ahb and t u r n out to also be t a n / 3 enhanced. On the other hand, the couplings between Higgs mass eigenstates and third generation quarks given in Eq. (85) are corrected as follows: cos a: 9hti = ——aVt snip 9hbb —
_
sin a cos/3
1 Afet (cot /3 + t a n a) 1 + A t ht
1
(^ - A b ) (1 + c o t a c o t / 3 ) Vb 1 + ^ V 1 + Ab \ fib
sin a 9 ml = —.—oVt snip cos a
1-
1
1+
9 Abb = t a n / 3 y 6
/ 5hb
1 + A 6 V—/ib 1
9Ati = cot 0yt
9H±tb
1 Aht (cot (3 — cot a ) 1 + A t /it
A/it
1 + At ht 1+
» \ /i /i\ Ab I (1 — tan a cot p)
(cot /3 + t a n /3)
1
^ - A (l + A 6 ) s i n 2 / 3 V ^6
9 2y/2M>w
TO(
mb t a n /?
1+
cot /3
Aht 1 + At
ht
t
(cot /3 + t a n /?) (1 + 75)
^ - A (1 + A 6 ) sin2 0 V hb
6
(1 " 75)
where the last coupling is given in the approximation of small isospin breaking effects, since interactions of this kind have been neglected in the Lagrangian of Eq. (86).
3 . P h e n o m e n o l o g y of t h e H i g g s b o s o n 3 . 1 . Standard
Model
Higgs
boson
decay
branching
ratios
In this Section we approach the physics of the SM Higgs boson by considering its branching ratios for various decay modes. In Section 2.2 we have derived the SM Higgs couplings to gauge bosons and fermions. Therefore we know t h a t , at the tree level, the SM Higgs boson can decay into pairs of electroweak gauge bosons (H —> W+W~,ZZ), and into pairs of quarks and leptons (H —> QQ,l+l~); while at one-loop it can also decay into two photons (H —> 7 7 ) , two gluons (H —> gg), or a 7 Z pair (H —> yZ). Fig. 9 represents all the decay branching ratios of the SM Higgs boson as functions of its mass MH- T h e SM Higgs boson total width, sum of all the partial widths r ( i l —> XX), is represented in Fig. 10.
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L. Reina
••••.
0.1 ~ TT—-/_
1
T
-
BR(ff)
V \
"' /
I
r"^
.•-;
cc —r~^. / \ \ 0.01
"I
1
77-.'"' / **V | \
-
\ ^
•
0.001 SS
-^L
<
\
\
" " ~ ^ - - ~ • * * < - / '
~~~~'~~~
w-*~^\ 0.0001 100
130
i \ 160
V 200
S
N X
UH
I / 300
> ^
i
500
700
"•••-,.
1000
[GeV]
Figure 9. SM Higgs decay branching ratios as a function of MH- The blue curves represent tree-level decays into electroweak gauge bosons, the red curves tree level decays into quarks and leptons, the green curves one-loop decays. From Ref. 6 .
1000
T(H) [GeV] 100
10
0.1
0.01
0.001
100
130
j 160
i_ 200
300
500
700
1000
MH [GeV] Figure 10.
SM Higgs total decay width as a function of MH- From Ref.
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Fig. 9 shows that a light Higgs boson (MH < 130 - 140 GeV) behaves very differently from a heavy Higgs boson (MH > 130 — 140 GeV). Indeed, a light SM Higgs boson mainly decays into a bb pair, followed hierarchically by all other pairs of lighter fermions. Loop-induced decays also play a role in this region. H —> gg is dominant among them, and it is actually larger than many tree level decays. Unfortunately, this decay mode is almost useless, in particular at hadron colliders, because of background limitations. Among radiative decays, H —> 77 is tiny, but it is actually phenomenologically very important because the two photon signal can be seen over large hadronic backgrounds. On the other hand, for larger Higgs masses, the decays to W+W~ and ZZ dominates. All decays into fermions or loop-induced decays are suppressed, except H —> tt for Higgs masses above the tt production threshold. There is an intermediate region, around MH ^ 160 GeV, i.e. below the W+W~ and ZZ threshold, where the decays into WW* and ZZ* (when one of the two gauge bosons is off-shell) become important. These are indeed three-body decays of the Higgs boson that start to dominate over the H —> bb two-body decay mode when the largeness of the HWW or HZZ couplings compensate for their phase space suppression d . The different decay pattern of a light vs a heavy Higgs boson influences the role played, in each mass region, by different Higgs production processes at hadron and lepton colliders. The curves in Fig. 9 are obtained by including all available QCD and electroweak (EW) radiative corrections. Indeed, the problem of computing the relevant orders of QCD and EW corrections for Higgs decays has been thoroughly explored and the results are nowadays available in public codes like HDECAY 32 , which has been used to produce Fig. 9. Indeed it would be more accurate to represent each curve as a band, obtained by varying the parameters that enters both at tree level and in particular through loop corrections within their uncertainties. This is shown, for a light and intermediate mass Higgs boson, in Fig. 11 where each band has been obtained including the uncertainty from the quark masses and from the strong coupling constant. In the following we will briefly review the various SM Higgs decay channels. Giving a schematic but complete list of all available radiative corrections goes beyond the purpose of these lectures. Therefore we will only discuss those aspects that can be useful as a general background. In pard
Actually, even four-body decays, corresponding to H —> W*W*, Z* Z* may become important in the intermediate mass region and are indeed accounted for in Fig. 9.
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100
110
120
130
140
150
160
MH [GeV] Figure 11. SM Higgs boson decay branching ratios in the low and intermediate Higgs boson mass range including the uncertainty from the quark masses nit = 178 ± 4 . 3 GeV, m(, = 4.88±0.07 GeV, and m c = 1.64±0.07 GeV, as well as from as(Mz) = 0.1172±0.002. From Ref. 6 .
ticular I will comment on the general structure of radiative corrections to Higgs decay and I will add more details on QCD corrections to H —> QQ (Q = heavy quark). For a detailed review of QCD correction in Higgs decays we refer the reader to Ref. 33 . Ref. 6 also contain an excellent summary of both QCD and EW radiative corrections to Higgs decays.
3.1.1. General properties of radiative corrections to Higgs decays All Higgs boson decay rates are modified by both EW and QCD radiative corrections. QCD corrections are particularly important for H —> QQ decays, where they mainly amount to a redefinition of the Yukawa coupling by shifting the mass parameter in it from the pole mass value to the running mass value, and for H —> gg. EW corrections can be further separated into: i) corrections due to fermion loops, ii) corrections due to the Higgs boson self-interaction, and Hi) other EW corrections. Both corrections of type (ii) and (Hi) are in general very small if not for large Higgs boson masses, i.e. for MH » Mw- On the other hand, corrections of type (i) are very important over the entire Higgs mass range, and are particularly relevant for MH < 2mt> where the top-quark loop corrections play a leading role.
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297
Indeed, for MR >C 2m t , the dominant corrections for both Higgs decays into fermion and gauge bosons come from the top-quark contribution to the renormalization of the Higgs wave function and vacuum expectation value. Several higher order radiative corrections to Higgs decays have been calculated in the large mt limit, specifically in the limit when MH -C 2mtResults can then be derived applying some very powerful low energy theorems. The idea is that, for an on-shell Higgs field (p2H = MH), the limit of small masses (M# -C 2m t ) is equivalent to a pn —» 0 limit, in which case the Higgs couplings to the fermion fields can be simply obtained by substituting
m°->m°(l + ^ )
,
(89)
in the (bare) Yukawa Lagrangian, for each massive particle i. In Eq. (89) H° is a constant field and the upper zero indices indicate that all formal manipulations are done on bare quantities. This induces a simple relation between the bare matrix element for a process with (X —> Y + H) and without (X —> Y) a Higgs field, namely Pi
1
lim A{X -> Y + H) = -jr Y
m°i--^A{X
-> Y) .
(90)
When the theory is renormalized, the only actual difference is that the derivative operation in Eq. (90) needs to be modified as follows o 9 mi
rrii
d
dm°^TT^dm~
(91)
where j m i is the mass anomalous dimension of fermion / j . This accounts for the fact that the renormalized Higgs-fermion Yukawa coupling is determined through the Z 0 limit above). The theorem summarized by Eq. (90) is valid also when higher order radiative corrections are included. Therefore, outstanding applications of Eq. (90) include the determination of the one-loop Hgg and .f/77 vertices from the gluon or photon self-energies, as well as the calculation of several orders of their QCD and EW radiative corrections. Indeed, in the mt —> 00 limit, the loop-induced H 7 7 and Hgg interactions can be seen as effective vertices derived from an effective Lagrangian of the form:
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Ceff =
^^"^^^f(1 +0{as)) '
(92)
where F^a)^v is the field strength tensor of QED (for the # 7 7 vertex) or QCD (for the Hgg vertex). The calculation of higher order corrections to the H —> 77 and H —> gg decays is then reduced by one order of loops! Since these vertices start as one-loop effects, the calculation of the first order of corrections would already be a strenuous task, and any higher order effect would be a formidable challenge. Thanks to the low energy theorem results sketched above, QCD NNLO corrections have indeed been calculated. 3.1.2. Higgs boson decays to gauge bosons: H —> W+W ±
The tree level decay rate for H —* VV (V = W ,Z)
v{H vv)
- =Wt5v{i-Tv+\Tv)pv'
,ZZ
can be written as:
(93)
where /3V = y/1 -TV, TV = 4M^/MH, and Sw,z = 2,1. Below the W+W~ and ZZ threshold, the SM Higgs boson can still decay via three (or four) body decays mediated by WW* (W*W*) or ZZ* (Z*Z*) intermediate states. As we can see from Fig. 9, the off-shell decays H —> WW* and H —> ZZ* are relevant in the intermediate mass region around MH — 160 GeV, where they compete and overcome the H —> bb decay mode. The decay rates for H -> VV* - • Vfjj (V = W±,Z) are given by:
r( ™.,.|^(^),
(M)
where sw =sin 0w is the sine of the Weinberg angle and the function F(x) is given by
F(x) - - ( 1 - x2) (fx2 - y + i ) - 3 (1 - 6x2 + Ax4) \n{x) o
+ 3
1 - 8x2 + 20a;4
V^-1
(2>x2 - 1 \ arCC0S
l-^^j •
(95)
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299
3.1.3. Higgs boson decays to fermions: H —> QQ,l+l The tree level decay rate for H —> / / ( / = Q,l, Q =quark, I =lepton) can be written as:
T(H - //) = 9zE^Nfm^f
,
(96)
where 0f = v/T^^ry, 77 = 4m2f/M%, and (iV c )'' Q = 1,3. QCD corrections dominate over other radiative corrections and they modify the rate as follows: T(H
= 3G4^"m2Q{MH)(3l
-> QQ)QCD
[AQCD
+ At]
,
(97)
where At represents specifically QCD corrections involving a top-quark loop. AQCD and At have been calculated up to three loops and are given by: AQCD
= 1 + 5 . 6 7 ^ ^ + (35.94 - 1MNF) ( ^ ^ Y + (98) (164.14 - 25.77NF + 0.26N2N F)
At =
, as{MH) IT
".(^M^.E^+l^^) mt
y
M2H
where as(M{j) and TTIQ{MH) are the renormalized running QCD coupling and quark mass in the MS scheme. It is important to notice that using the MS running mass in the overall Yukawa coupling square of Eq. (97)is very important in Higgs decays, since it reabsorbs most of the QCD corrections, including large logarithms of the form l n ( M ^ / m g ) . Indeed, for a generic scale fi, ffiQ{n) is given at leading order by: •">0
m
Q
(^o=m
Q
(m
Q
)(^M_)
to
mQ(mQ) 1 - — I n
(99)
2 Q.
where bo and 70 are the first coefficients of the (3 and 7 functions of QCD, while at higher orders it reads:
^M-^(nH,)/^W
(100)
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where, from renormalization group techniques, the function f(x) is of the form:
fix)
25 N "
[1 + 1.0142: + ...]
for mc
,
/23 \ " f{x) = i—xj [1 + 1.175a: + ...]
for mb<\i<mt
,
(101)
12
4
f(x)=(-x)
[1 + 1.398a:+...]
for n>mt
.
As we can see from Eqs. (100) and (101), by using the MS running mass, leading and subleading logarithms up to the order of the calculation are actually resummed at all orders in as. The overall mass factor coming from the quark Yukawa coupling square is actually the only place where we want to employ a running mass. For quarks like the b quark this could indeed have a large impact, since, in going from /j, ~ MH to (j, ~ mb, rfin(fi) varies by almost a factor of two, making therefore almost a factor of four at the rate level. All other mass corrections, in the matrix element and phase space entering the calculation of the H —> QQ decay rate, can in first approximation be safely neglected. 3.1.4. Loop induced Higgs boson decays: H —> 7 7 , 7 ^ , As seen in Section 2.2, the .#77 and HjZ couplings are induced at one loop via both a fermion loop and a W-loop. At the lowest order the decay rate for H —> 77 can be written as: 2
T(H -> 77) =
GFa2Mfj 128\/27r 3
(102) f
where N[ = 1,3 (for / = l,q respectively), Qf is the charge of the / fermion species, r/ = 4m?/Mjj, the function / ( r ) is defined as: [ arcsin2 -7=
T >1 (103)
1-
n^T
T<1 ,
- VK
and the form factors A J and A^ are given by: ^
= 2T[1 + ( 1 - T ) / ( T ) ]
,
^ ( r ) = - [ 2 + 3r + 3 r ( 2 - r ) / ( r ) ]
(104)
Higgs Boson Physics 301
On the other hand, the decay rate for H —> 7 Z is given by:
nH-^lZ)-
G%M^aM%
f M
^
2
V
\^-MT)
J2Aj(Tf,\f)+A^(TW,\W)
,
/ (105) where n = 4M?/M% and A* = 4 M ? / M § (i = / , W ) , and the form factors Af(r, A) and A ^ ( r , A) are given by: A?{r,\) = 2NfQ^-2Q/Sin2°w)
[h(r,X)-I2(r,X))
( 1 + - J t a n 2 6W
A ^ ( T , A) = cos 9W \
6+H
, (106)
/i(r,A)
+ 4(3-tan26»vy)/2(r,A)| ,
(107)
where TV"/ and Q / are defined after Eq. (102), and l( is the weak isospin of the / fermion species. Moreover: /i(r,A)
TA
2(r-A)
+ . / 2 A L [2/ ( r ) - /(A)] + 7-^To[ff(r) - (A)] , 2 2(r-A)
A)
^(r,A) = - ^ - [ / ( r ) - / ( A ) ]
(108)
2(T-A)'
and \ / T — 1 arcsin -4=
5(r) I
2
l + v/T^r In 1 3
L " i-x/T ^
T > 1 (109)
T< 1
while / ( r ) is defined in Eq. (103). Q C D and E W corrections to b o t h Y{H -> 7 7 ) and T{H —> 7 Z ) are p r e t t y small and for their explicit expression we refer the interested reader to the literature 3 3 ' 6 . As far as H —> gg is concerned, this decay can only be induced by a fermion loop, and therefore its rate, a t the lowest order, can be written as: T(# -
gg)
GFa2sMl 36\/27r 3
i£<w
(110)
where Tq = Arn^/M%, / ( T ) is defined in Eq.(103) and the form factor A^ (r) is given in Eq. (106). Q C D corrections t o H —> gg have been calculated u p to NNLO in the mt —> 00 limit, as explained in Section 3.1.1. At NLO the expression of the corrected rate is remarkably simple r r(H^gg(g),qqg)=rLO(H^gg)
a(iVi)l l + E(rQ)-^—
,
(111)
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where ri/
£(
s
Minimi
^
-
95
33 - 2NF , ( u2 \ + l 0 g
7 nr
T - 6^ — 6 —
Kk)
•
, , (U2)
When compared with the fully massive NLO calculation (available in this case), the two calculations display an impressive 10% agreement, as illustrated in Fig. 12, even in regions where the light Higgs approximation is not justified. This is actually due to the presence of large constant factors in the first order of QCD corrections. We also observe that the first order of 0.8
T(H -)• gg)
0.75
^EH
0.7 0.65 0.6 0.55
full corrections infinite m( case
0.5 0.45 0.4 0.35
1000
DO
MH [GeV] Figure 12. The QCD correction factor for the partial width T(H —> gg) as a function of the Higgs boson mass, in the full massive case with rat = 178 GeV (dotted line) and in the heavy-top-quark limit (solid line). The strong coupling constant is normalized at as(Mz) = 0.118. From Ref. 6 .
QCD corrections has quite a large impact on the lowest order cross section, amounting to more than 50% of TLO on average. This has been indeed the main reason to prompt for a NNLO QCD calculation of T(H —> gg). The result, obtained in the heavy-top approximation, has shown that NNLO QCD corrections amount to only 20% of the NLO cross section, therefore pointing to a convergence of the T(H —> gg) perturbative series. We will refer to this discussion when dealing with the gg —> H production mode, since its cross section can be easily related to T(H —> gg).
Higgs Boson Physics
3.2. MSSM
Higgs boson branching
303
ratios
The decay patterns of the MSSM Higgs bosons are many and diverse, depending on the specific choice of supersymmetric parameters. In particular they depend on the choice of MA and tan/3, which parameterize the MSSM Higgs sector, and they are clearly sensitive to the choice of other supersymmetric masses (gluino masses, squark masses, etc.) since this determines the possibility for the MSSM Higgs bosons to decay into pairs of supersymmetric particles and for the radiative induced decay channels (h°,H° —> <7,77,7.£) to receive supersymmetric loop contributions. In order to be more specific, let us assume that all supersymmetric masses are large enough to prevent the decay of the MSSM Higgs bosons into pairs of supersymmetric particles (a good choice could be M-g = MQ = Mu = MD = 1 TeV). Then, we only need to examine the decays into SM particles and compare with the decay patterns of a SM
100 mh :
ISO
140
(GeV) bb
m„ '
160 1B0 (GeV)
200
200
500 •n»
(GeV)
I " " i""i""r"i""r
bb ton 0=30 • T*T"
\ V 1
ss
•
sa
i v :
/' /
+ M M
W + W~'
100
, , , ,120 /i m „ (GeV)
¥
**7'
1 '140
160 1B0 m „ (GeV)
zz : 200
300 m„
600 (GeV)
Figure 13. Branching ratios for the h° and H° MSSM Higgs bosons, for tan (5 — 3, 30. The range of MH corresponds to M A = 9 0 G e V - 1 TeV, in the MSSM scenario discussed in the text, with maximal top-squark mixing. The vertical line in the left hand side plots indicates the upper bound on Mh, which, for the given scenario is M™ax = 115 GeV (tan/3 = 3) or M™ax = 125.9 GeV (tan/3 = 30). From Ref. 8 .
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Higgs boson to identify any interesting difference. Prom the study of the MSSM Higgs boson couplings in Sections 2.4.3 and 2.4.4, we expect that: i) in the decoupling regime, when MA S> MZ, the properties of the h° neutral Higgs boson are very much the same as the SM Higgs boson; while away from the decoupling limit ii) the decay rates of h° and H° to electroweak gauge bosons are suppressed with respect to the SM case, in particular for large Higgs masses {H°), Hi) the A0 -> VV (V = W±,Z°) decays are absent, iv) the decay rates of h° and H° to T+T~ and bb are enhanced for large tan/3, v) even for not too large values of tan/3, due to ii) above, the h°,H° —> T+T~ and h°,H° —> bb decay are large up to the tt threshold, when the decay H° —> tt becomes dominant, vi) for the charged Higgs boson, the decay H+ —> T+VT dominates over H+ —> tb below the tb threshold, and vice versa above it. As far as QCD and EW radiative corrections go, what we have seen in Sections 3.1.2-3.1.4 for the SM case applies to the corresponding MSSM decays too. Moreover, the truly MSSM corrections discussed in Sections 2.4.3 and 2.4.4 need to be taken into account and are included in Figs.13 and 14.
3.3. Direct bounds on both SM and MSSM
Higgs
bosons
LEP2 has searched for a SM Higgs at center of mass energies between 189 and 209 GeV. In this regime, a SM Higgs boson is produced mainly through Higgs boson strahlung from Z gauge bosons, e+e~ —> Z* —> HZ, and to a lesser extent through WW and ZZ gauge boson fusion, e + e ~ —> WW, ZZ —> Hveue, He+e~ (see Fig. 15). Once produced, it decays mainly into bb pairs, and more rarely into T+T~ pairs. The four LEP2 experiments have been looking for: i) a four jet final state (H —> bb, Z —> qq), ii) a missing energy final state (H —> bb, Z —> vv), Hi) a leptonic final state (H —> bb, Z —> l+l~) and iv) a specific r-lepton final state (H —* bb, Z —> T+T~ plus H —> T+T~, Z —> qq). The absence of any statistical significant signal has set a 95% CL lower bound on the SM Higgs boson at MHsM > 114.4 GeV . LEP2 has also looked for the light scalar (h°) and pseudoscalar (A0) MSSM neutral Higgs bosons. In the decoupling regime, when A0 is very heavy and h° behaves like a SM Higgs bosons, only h° can be observed and the same bounds established for the SM Higgs boson apply. The bound can however be lowered when TUA is lighter. In that case, h° and A0 can also be pair produced through e+e" -> Z —> h°A° (see Fig. 15). Combining
Higgs Boson Physics
^r^-
-J t t
:
—r
T*V
tb / \
r
1
T+T~ bb
r
--,. - - . ^ Z h
5
i""r M T" tun 0=3
i' • " i "T"
tb
TJ
tan 0=3
Zh,,
_ / c b
-
:
\ /
''
cs 'I
r-
9S
/
-
1
. cc
/
m 10"
r •
.
30
. . I 200
300
M,l, 500
1 W»h
^ T * C
~
• - - .
'"-
\ \\
us
10
1
S,
\Y.
200
m A (GeV)
: 1
W>A
MIMMI,,.,!,
700
:
-._
It
n 10-3
305
.
1 , ...1 ,I,,,,L,,I,„ 300 500 700 1000
m „ . (GeV)
10° bb
tan 0=30 '
T + T"
r r
Tt
-
93 ~--„
r
=
--._ I
M .
. . 1 .. 200
. , I . J . J .. . . i , . . . ! , . . , ! . . . " ! : . : . ^ , 300
500
700
1000
100
m „ (GeV)
200
300 500 m H . (GeV)
700
1000
Figure 14. Branching ratios for the A0 and H+ MSSM Higgs bosons, for tan (3 = 3, 30. The range of MH± corresponds to MA = 90 GeV— 1 TeV, in the MSSM scenario discussed in the text, with maximal top-squark mixing. From Ref. 8 .
ve,e
.h°,H°
vc,e Figure 15.
SM and MSSM neutral Higgs boson production channels at LEP2.
the different production channels one can derive plots like those shown in Fig. 16, where the excluded (Mh,tan/3) and (MA, tan (3) regions of the MSSM parameter space are shown. The LEP2 collaborations 26 have been able to set the following bounds at 95% CL: MKA > 93.0 GeV , obtained in the limit when cos(/3 — a) ~ 1 (anti-decoupling regime) and for large tan/3. The plots in Fig. 16 have been obtained in the maximal mixing
L.
306
Reina
i
CO.
CO.
a
a
10
"
|mh„-max|
|
10
••;
I •. I i-l -I I- I I I '
\
1
•
1 •.. I . I • !•;. 1 1 I 1
Hi -i In.i,
0
2U
•.!•!.
-rU
W
M
lUU 120 140
0
200
CQ.
C
C3
.1
400
m.c (GeV/c2)
mho (GeV/c2)
CQ.
C
:"V.-'.'ML-!ii!!
\
V.
CS
'/]
10
\t-
i, •
10
&
r
0
" N --.
/
••1 - 1 rtically ••. 1 • c e s s i b l e
1 •• h
1
. . . . i . . . i . . • i^^^fts. " 20 40 60 80 100 120 140
m h0 (GeV/c2)
:•!
:
•••. 1 1 1-
200
400
m.o (GeV/c2)
Figure 16. 95% CL exclusion limits for MSSM Higgs parameters from LEP2: (Mh,tan/3) (left) and (MA,tan/3) (right). Both the maximal and no-mixing scenarios are illustrated, for Ms = 1 TeV and mt = 179.3 GeV. The dashed lines indicate the boundaries that are excluded on the basis of Monte Carlo simulations in the absence of a signal. From Ref. 7 .
scenario (explained in Section 2.4.2). For no-mixing, the corresponding plots would exclude a much larger region of the MSSM parameter space. Finally, the LEP collaborations have looked for the production of the MSSM charged Higgs boson in the associated production channel: e+e~ —> 7, Z* —+ H+H~ 34 . An absolute lower bound of MH± > 79.3 GeV has been set by the ALEPH collaboration, and slightly lower values have been obtained by the other LEP collaborations.
Higgs Boson Physics
3.4. Higgs boson studies
at the Tevatron
307
and at the LHC
The parton level processes through which a SM Higgs boson can be produced at hadron colliders are illustrated in Figs. 17 and 18. q.q
z,w
nnnnnn
"•H
g Tnnnn: Figure 17.
q.q
Leading Higgs production processes at hadron colliders: gg —• H, qq —> ggif,
a n d g<j - • TO, Z-ff.
. t,b
<% <7> tf>
tf>
Tpsm^
,$
g Tyffinn
Ttfmr1
^f,b
. t,b
. t,b
*.
V -- H • t,b
- "
>*
t,b
Figure 18. Higgs production with heavy quarks: sample of Feynman diagrams illustrating the two corresponding parton level processes qq, gg —»tiH, bbH. Analogous diagrams with the Higgs boson leg attached to the remaining top(bottom)-quark legs are understood.
Figures 19 and 20 summarize the cross sections for all these production modes as functions of the SM Higgs boson mass, at the Tevatron (center of mass energy: "^5=1.96 TeV) and at the LHC (center of mass energy: y / s = 14 TeV). These figures have been recently produced during the TeV4LHC workshop 35 , and contain all known orders of QCD corrections as well as the most up to date input parameters. We postpone further details about QCD corrections till Section 4, while we comment here about some general phenomenological aspects of hadronic Higgs production. The leading production mode is gluon-gluon fusion, gg —> H (see first diagram in Fig. 17). In spite of being a loop induced process, it is greatly enhanced by the top-quark loop. For light and intermediate mass Higgs
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SM Higgs production 10-
a [ft]
10 :
10 -
100
180
200 mh [GeV]
Figure 19. Cross sections for SM Higgs boson production processes at the Tevatron, Run II ( v ^ = 1 . 9 6 TeV). From Ref. 3 5 .
bosons, however, the very large cross section of this process has to compete against a very large hadronic background, since the Higgs boson mainly decays to 66 pairs, and there is no other non-hadronic probe that can help distinguishing this mode from the overall hadronic activity in the detector. To beat the background, one has often to employ subleading if not rare Higgs decay modes, like H —* 77, and this dilutes the large cross section. For larger Higgs masses, above the ZZ threshold, on the other hand, gluongluon fusion together with H —•> ZZ produces a very distinctive signal, and make this mode a "gold-plated mode" for detection. For this reason, gg —> H plays a fundamental role at the LHC over the entire Higgs boson mass range, but is of very limited use at the Tevatron, where it can only be considered for Higgs boson masses very close to the upper reach of the machine (MH ~ 200 GeV). Weak boson fusion (qq —> qqH, see second diagram in Fig. 17) and the associated production with weak gauge bosons (qq —> WH, ZH, see third diagram in Fig. 17) have also fairly large cross sections, of different relative size at the Tevatron and at the LHC. qq —> WH, ZH is particularly important at the Tevatron, where only a relatively light Higgs boson (MH <
Higgs Boson Physics
309
SM Higgs production 10-
'
= L-
o[fb]
L
LHCE "
-
10'
-—•-
:- --.'.
10 :
-
E*^X ^v\ -
10
' '
^ - . . ^
^ \ u q -^ Wh
-
bb -> h ^ ^ § N T ^ \
'.j H r*J Q
:
--
:~--~^ "=S...^^~^
;
qb - > qth
"
'
- TeV4LHC Higgs working group 100
200
-=i^S^__
• qq ->
: -
^-b^-~^~-J-^~~~-H^^
r^-->-~T<~-~~ 300
400
T~~~~,—~i^T~ 500 m„ [GeV]
Figure 20. Cross sections for SM Higgs boson production processes at the LHC (y/s = U TeV). From Ref. 3 5 .
200 GeV) will be accessible. In this mass region, gg —» H, H —-> 77 is too small and qq —> qqH is suppressed (because the initial state is pp). On the other hand, qq —> qqH becomes instrumental at the LHC (pp initial state) for low and-Intermediate mass SM Higgs bosons, where its characteristic final state configuration, with two very forward jets, has been shown to greatly help in disentangling this signal from the hadronic background, using different Higgs decay channels. Finally, the production of a SM Higgs boson with heavy quarks, in the two channels qq,gg —» QQH (with Q = t,b, see Fig. 18), is sub-leading at both the Tevatron and the LHC, but has a great physics potential. The associated production with ti pairs is probably too small to be seen at the Tevatron, given the expected luminosities, but will play a very important role for a light SM Higgs boson (MH < 130 - 140 GeV) at the LHC, where enough statistics will be available to fully exploit the spectacular signature of a tiH, H —> bb final state. Moreover, at the LHC, the associated production of a Higgs boson with top quarks will offer a direct handle on the top-quark Yukawa coupling (see Section 3.4.2). On the other hand, the production of a SM Higgs boson with bb pairs is tiny, since the SM bottom-
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quark Yukawa coupling is suppressed by the bottom-quark mass. Therefore, the bbH, H —» bb channel is the ideal candidate to provide evidence of new physics, in particular of extension of the SM, like supersymmetric models, where the bottom-quark Yukawa coupling to one or more Higgs bosons is enhanced (e.g., by large tan/? in the MSSM). bbH production is kinematically well within the reach of the Tevatron, RUN II. First studies from both CDF 38 and D0 39 have already translated the absence of a bbh°,H°,A° signal into an upper bound on the tan/? parameter of the MSSM. Were a signal observed, bbH could actually provide the first piece of evidence for new physics from RUN II.
3.4.1. Searching for a SM Higgs boson at the Tevatron and the LHC Discovering a Higgs boson during RUN II of the Tevatron is definitely among the most important goal of this collider. It will be challenging and mainly luminosity limited, but recent studies have confirmed that RUN II can push the 95% CL exclusion limit much farer than LEP2 and also shoot for a 3a or 5
combined CDF/DO thresholds
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Higgs mass (GeV/c ) Figure 21. Integrated luminosity required for each experiment at the Tevatron, Run II, to exclude a SM Higgs boson at 95% CL or to observe it at the 3CT or 5
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The plot in Fig. 21 shows the integrated luminosity that was originally estimated to be necessary to reach the 95% CL exclusion limit, the 3<x, and the 5cr discovery levels. It is given for a SM Higgs boson mass up to 200 GeV, that is to be considered as the highest Higgs boson mass reachable by RUN II. The curves have been obtained mainly by using the associated production with weak gauge bosons, qq —> VH (V = W±,Z°), with H -> 66 and H —> W+W~, over the entire Higgs boson mass range, and gg —• H with H —> ZZ in the upper mass region. As discussed in the introduction to Section 3.4, this can be understood in terms of production cross sections (see Fig. 19) and decay branching ratios (see Fig. 9) over the MH = 115 - 200 GeV mass range. From Fig. 21 we see that with, e.g., 10 fb" 1 of integrated luminosity RUN II will be able to put a 95% CL exclusion limit on a SM Higgs boson of mass up to 180 GeV, while it could claim a 3CT discovery of a SM Higgs boson with mass up to 125 GeV. A 5CT discovery of a SM Higgs boson up to 130 GeV, i.e. in the region immediately above the LEP2 lower bound, seemed to require 30 f b - 1 of integrated luminosity, well beyond what is currently expected for RUN II.
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More recently, a new sensitivity study has appeared l s , where the low mass region only has been revisited and new luminosity curves have been drawn, as shown in Fig. 22. Mainly using the qq —> VH (V = W±,Z°) production mode, it appears that new analyses techniques will allow to obtain better results with less integrated luminosity. A 3cr discovery of a SM Higgs boson with mass up to 125 GeV will now require only about 5 ftr1, while 10 fb" 1 could allow a 5130 - 140 GeV) mass region, as becomes evident by simultaneously looking at both production cross sections (see Fig. 20) and decay branching ratios (see Fig. 9) over the entire 115 — 1000 GeV SM Higgs boson mass range. In the region of MH < 130 - 140 GeV the SM Higgs boson at the LHC will be searched mainly in the following channels: gg^H,H^rr,W+W-,ZZ , qq^qqH,
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These have been the modes used by both ATLAS and CMS to provide us with the discovery reach illustrated in Figs. 23 and 24. The ATLAS plots give the signal significance for a total integrated luminosity of 100 f b - 1 (upper plot) and of 30 fb _ 1 (lower plot). The high luminosity (upper) plot belongs to the original ATLAS technical design report 12 , and the weak boson fusion channels had not been studied in detail at that time. The lower luminosity (lower) plot is taken from a more updated study 36 , and the weak boson fusion channels have been included in the low mass region, up to about MH — 200 GeV, where they play an instrumental role towards discovery. Other instrumental channels in the low mass region are the inclusive Higgs production with H —> 77 and, below MH = 130 GeV, tiH production with H —> bb. In the high mass region, the inclusive production
Higgs Boson Physics
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H -» yy + WH,ttH(H -> yy) ttH (H -» bb) H -> ZZ(*> -> 41 H -> WW1*' -> lvlv H -» ZZ -» llvv H -> WW -> lvjj Total significance
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F i g u r e 2 3 . Significance for t h e S M Higgs b o s o n d i s c o v e r y in v a r i o u s d e t e c t i o n c h a n n e l s as a f u n c t i o n of Mu- T h e u p p e r plot is for 100 f b - 1 of d a t a a n d w i t h n o qqH c h a n n e l i n c l u d e d , t h e lower p l o t for 30 f b - 1 a n d w i t h t h e qqH c h a n n e l i n c l u d e d over t h e m a s s r a n g e MH < 2 0 0 G e V . R e s u l t s a r e from t h e A T L A S c o l l a b o r a t i o n 3 6 .
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CMS
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with H —> ZZ, WW dominates, although CMS has found a substantial contribution coming from weak gauge boson fusion with H —> ZZ, WW. 3.4.2. Studies of a SM Higgs boson If a Higgs boson signal is established, the LHC will have the capacity of measuring several of its properties at some level of accuracy. In particular, it will be able to measure its mass, width, and couplings. At the same time, the charge and color quantum numbers of the newly discovered particle will be established by detecting a single production-decay channel; while a precise determination of its spin and parity will probably require more statistics than available at the LHC and will have to wait for a high energy Linear Collider to be established (see Section 3.5). The two plots in Fig. 25 show the precision with which both the mass and width of a SM-like Higgs boson will be determined by ATLAS and CMS combining 300 fb _ 1 of data per experiment. We see that below M # ~ 400 GeV the Higgs mass can be determined with a precision of about 0.1%, through H -» ZZ -> 41, complemented by H,WH,ttH(H -> 77) and
Higgs Boson Physics Experimental precision on the SM Higgs moss
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Figure 25. Expected precision in the measurement of a SM-like Higgs boson mass (MH) and width (TH) combining 300 f b - 1 of data per experiment from ATLAS and CMS. From Ref. 4 0 .
tiH(H —» bb) in the low mass region. Above MH — 400 GeV the accuracy deteriorates for the smaller statistics available, although precisions of the order of 1% can still be obtained. We also see that the Higgs width above M # ~ 2 0 0 GeV will be entirely determined through H -> ZZ —> 4/, while below MH — 200 GeV it is too small to be resolved experimentally and can only be determined indirectly, as we will discuss in the following. Finally, many studies in recent years have pointed to the fact that the LHC, under minimal theoretical assumptions, will have the potential to measure several Higgs boson couplings with an accuracy in the 10-30% range. The proposed strategy 41 consists of measuring the production-decay channels listed in Eqs. (113) and (114) for a light (MH < 130 - 140 GeV) or heavy (MH > 130 — 140 GeV) Higgs boson respectively, and combine them to extract individual partial widths or ratios of partial widths. Indeed, if a given production-decay channel is observed, one can write that the experimentally measured product of production cross section times decay branching ratio corresponds, in the narrow width approximation, to the following expression: (<JP(H)BI(H
-» dd))exp =
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where Tp and Yd are the partial widths associated with the production and decay channels respectively, while V is the Higgs boson total width. The coefficient Gf(H)/Tf can be calculated, while Tp, Ta, and T is what needs
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to be determined. To each production-decay channel one can therefore associate a measurable observable
zf = ^
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,
where p and d label the production and decay channels respectively. ZJf' is obtained from the experimental measurement of (ap(H)Bi(H —> dd))exp, normalized by the theoretically calculable coefficient aPh (H)/Tph. A signal in the (p, d) channel will measure Z^', and therefore the product of Higgs couplings Upi/l, since Tp ~ y2p and Td ~ y\. Combining many different (p, d) channels, a system of equations of the form of Eq. (116) is obtained. Ratios of partial widths Ti/Fj, and therefore ratios of Higgs couplings, can then be derived in a model independent way, e.g.:
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relation between yw and yz, while the right hand plots also assume a SMlike relation between yb and yT. The more the assumptions, the better the accuracy with which the considered couplings can be determined, and the more model dependence is introduced in the coupling determination. More sophisticated analyses have appeared in recent studies 16 - 43 . Overall, we can however conclude that the LHC has a great potential of giving a first fairly precise indication of the nature of the couplings of a Higgs boson candidate, although under some (well justified) model assumptions. In particular, in the specific case of the top-quark Yukawa coupling, the LHC will be for a long time the only machine to be able to measure it with enough precision, since the measurement of yt in e + e~ —> ttH at a
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•y/i = 500 GeV Linear Collider is statistically very limited, as we will see in Section 3.5. 3.4.3. Searching for a MSSM Higgs boson at the Tevatron and the LHC Most of the characteristics of the MSSM Higgs couplings that determine the pattern of decays reviewed in Section 3.2 also affect the mechanism of
Higgs Boson Physics
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production of the MSSM Higgs bosons. In particular: • for MA 3> Mz, the so called decoupling limit, —> h° —> HSM, while —> MA ~ MH and g{A:H)bb » 9HSMbb .
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tral MSSM Higgs boson production rates at the Tevatron and at the LHC is given in Figs. 29 and 30, for two values of tan/3, in the maximal mixing scenario (see Section 2.4.2). For the Tevatron the mass range is limited to the range kinematically accessible while for the LHC the entire Higgs mass range up to 1 TeV is covered. It is important to notice how for large tan /3 (tan/3 = 30 in the plots of Figs. 29 and 30) the production of both scalar and pseudoscalar neutral Higgs boson with bottom quarks becomes dominant. In particular the inclusive production (denoted in the Figs. 29 and 30 as bb —> cp°, forbb(jP) is right below gluon-gluon fusion but above all other production modes. More details on exclusive vs inclusive production of a Higgs boson with bottom quarks will be given in Section 4. We also notice the subleading role played by vector boson fusion production (qq —> qqcj)0) due to the suppression (or absence in the case of
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a(pp->h/H+X> [pb] Vs = 2 TeV M , . 175 GeV CTEQ4
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A0) of themt — rrib. When MH± < mt — mb, -ff± is mainly produced via the decay of a top or anti-top in it production, i.e. via pp,pp —> ti with t —> bH+ or t —> bH~. On the other hand, when MH± > mt — nib, • ^ ± is mainly produced through pp —> ibH+,tbH~, as well as through the tree level production modes qq —> H+H~ and bb —> W±HT, and at one loop through the associated modes gg —> H+H~ and gg —> W±HT. The overall cross section for the Tevatron and the LHC is illustrated in Figs. 31 and 32 as a function of the charged Higgs boson mass, for different values of tan/3. The threshold behavior at M^ ~ mt — rrib is clearly visible.
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The evidence or the absence of evidence for an MSSM Higgs boson at the Tevatron or at the LHC will place definite bounds on the parameter space of the MSSM. The reach of the Tevatron in the (M,i,tan/3) plane is illustrated in Fig. 33 for different integrated luminosities. The shaded regions are spanned by using only the qq —> V(f>° (<j>° —> bb) channel (withbbcjP ((f>0 -* bb) channel (with 4>° = h°,H°,A°). The region below the black solid line is excluded by searches at LEP2. Fig. 33 shows that, although discovery may require integrated luminosities that are beyond the reach of RUN2 of the Tevatron, with 5 fb _ 1 CDF and D0 will be able to exclude (in the maximal mixing scenario) almost all the parameter space of the MSSM at 95% C.L., a pretty impressive result by itself! Both CDF and D0 have indeed already presented results from searches conducted in the pp —> bJxfP (
bb) channel with three or four 6-quark jets tagged in the final state 39 . The most recent results, from D0, are illustrated in Fig. 34, where we can see that, depending on MA, values of tan/3 as low as tan/3 = 50 have already been excluded. The LHC 5cr discovery reach in the (MA, tan/3) parameter space is illustrated in Fig. 35. Thanks to the high luminosity available and to the complementarity of various production and decay modes, the entire (MA, tan/3) parameter space can be covered, up to MA of the order of 1 TeV. This gives us the exciting perspective that the LHC will be able to either discover or completely rule out the existence of an MSSM Higgs boson!
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3.5. Higgs boson studies
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As we all know, an e+e~ collider provides a very clean environment, with relatively simple signatures and very favorable signal to background ratios. Therefore, one of the most important roles that a high energy e + e~ collider (to which we will refer as International Linear Collider (ILC) or simply Linear Collider (LC)) will play is to unambiguously identify any new particle discovered at the Tevatron or at the LHC, through a thorough program of precision measurements. This is true in Higgs boson physics as well, where we expect the mass, width, spin, and couplings of any Higgs boson candidates to be determined at the few percent level. In this Section I will mainly focus on a SM Higgs boson, since this is enough to illustrate the role played by a LC in Higgs physics, and since the true impact on the study of the MSSM parameter space will be shaped by the discoveries occured by the time a LC is built. First of all, the most important SM Higgs boson production processes in e + e~ collisions are illustrated in Figs. 36 and 37 and are: i) e+e~ —» ZH, the Higgs strahlung or associated production with Z gauge bosons, ii) e+e~ —> HuP, the W+W~ fusion production, and e + e _ —> He+e~, the ZZ fusion production, Hi) e+e~ —> tiH, the associated production with a ti pair. In addition, in the MSSM we also have: iv) e+e~ -> h0A°,H°A0 and
Higgs Boson Physics
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e+e~ —> H+H~, the pair production of two Higgs bosons, either neutral or charged, as illustrated in Fig. 38. The cross sections of these processes as functions of the corresponding Higgs masses are illustrated in Figs. 39 and 40 for various center of mass energies (\/i). e+e~ —> ZH and e + e~ —* Hvv
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Figure 40. MSSM Higgs boson production cross sections in e+e collisions, for two choices of tan/3: tan/3 = 3 (solid) tan/3 = 30 (dotted), at ^ / i = 3 5 0 GeV, for h° and H°, and at v ^ = 8 0 0 GeV, for A0 and H±. From Ref. 1 8 .
are the leading production modes for a SM Higgs bosons. Their relative size varies with the center of mass energy, since a(e+e~ —> ZH) scales as 1/s (s-channel process), while a(e+e~ —> Hvv) scales as log(s) (^-channel process). e+e~ —• Hvv always dominates over e+e~ —> He+e~ by almost one order of magnitude. In the MSSM, e+e~ —> Hvv plays a lesser role, due to the suppression of the VVH coupling (V = W±,Z), but the h°A°, H°A° and H+H~ pair production modes become important. tiH production is always very rare, in particular at center of mass energies around 500 GeV or lower, but it plays a really important role at higher energies, around 800 GeV-1 TeV, for the determination of the top-quark Yukawa coupling, as we will discuss later. Other rare production modes that could play an important role in determining some Higgs boson properties are the double Higgs boson production modes: e+e~ —» HHZ and e+e~ —> HHvv With a LC running at energies between 350 GeV and 1 TeV, one or more Higgs bosons can be observed over the entire mass spectrum and all its properties can be precisely studied. Reconstructing the recoiling l+l~ mass (for l = e,fi) in e + e _ —> HZ —> Hl+l~, where the Z is monoenergetic, allows an excellent and model independent determination of the Higgs boson mass. Z —> qq decays can also be used and actually provide a very large statistics. Accuracies of the order of 50-80 MeV can be obtained, depending on the center of mass energy and the Higgs boson mass. The spin and parity of the Higgs boson candidate (expected to be Jp = 0 + ) can be determined in several ways, among others: i) from the onset of cr(e + e~ —>
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Vs(GeV) Figure 41. The e+e —> ZH cross section energy dependence near threshold for MH120 GeV and spin J p = 0 + , l - , 2 + . From Ref. 1 8 .
ZH), since the energy dependence near threshold strongly depend on the Jp quantum number of the radiated H (see Fig. 41); ii) from the angular distribution of H and Z in e + e~ —•> ZH —> 4 / ; Hi) from the differential cross section in e+e ttH. Finally, a high energy LC will measure the Higgs boson couplings to unprecedented precision and in a model independent way. Thanks to the precise knowledge of the initial state energy configuration (once the initial state radiation, or beam strahlung, has been properly taken into account), one can indeed measure both a(e+e~ —> HZ —> Hl+l~) and a{e^ W*W*vv —> Hvv), reconstructing the mass recoiling against the l+l~ or uv pair, and from there determine in a model independent way the isolated HZZ and HWW couplings6. This is probably the most important intrinsic difference between measuring the Higgs couplings at a lepton versus a hadron collider. At a lepton collider the Higgs couplings to the weak gauge bosons, i.e. the Higgs couplings associated to the production mode (yp of Section 3.4.2), can be isolated in a model independent way. Any other coupling can then be also determined in a model independent e
We notice that the two production modes e + e - —• HZ —> Hvv and e + e _ —> W*W*vv —> Hvv can be well isolated, since their distribution in the invariant vv mass are very distinctive. See Ref. 1 8 for further details.
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way, measuring the individual Br{H —> XX) in e+e~ —> HZ followed by H —> XX. Several recent studies have confirmed the possibility of determining Higgs couplings to both gauge boson and fermions within a few percent (2-5%). For instance, for a Higgs boson of MH = 120 GeV, the bottom-quark Yukawa coupling, yb, could be determined within 2%, the r one, yT, within 5%, and the charm quark one, yc within 6% due to the larger error on the charm quark mass. This will test in a very stringent way the proportionality of the Higgs couplings to the mass of the interacting fermion. Even the indirect coupling of the Higgs boson to a pair of gluons, arising at the one-loop level (see Section. 2.2), will be determined with a precision of 4-5%. This will allow an indirect check of the top-quark Yukawa coupling for a SM Higgs boson (when the top-quark loop dominates), or will result in some anomalous coupling if new physics contributes in the loop. In this second case, probably, other Higgs coupling will show anomalous behaviors. The attainable precisions at a LC running at \fs = 350 GeV, and with a 500 f b - 1 of data, are summarized in Fig. 42. A LC will then be the ideal machine to discover small new physics effects. For instance, it could play a fundamental role in distinguishing the h° Higgs boson of the MSSM from the SM Higgs boson, in regions of the MSSM parameter space close to the decoupling limit. With this respect,
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MSSM prediction: 200 GeV < m < 400 GeV
400 GeV <
I
600 GeV <
LC 1 cj (with fusion) LC 95% CL (with fusion) \
LC 1a (w/o fusion) LC 95% CL (w/o fusion)
9w / 9w( SM )
StAJSM)
Figure 43. Higgs couplings determination at a LC (TESLA in this case) with 500 f b _ 1 of data, for M}j = 120 GeV. The labels with fusion and w/o fusion refers to WW-fusion inclusion/exclusion. From Ref. 1 8 .
Fig. 43 shows two combined fits: the first one combines the ratio between the bottom-quark Yukawa coupling to h° and its SM expected value and the ratio between the h° coupling to W bosons and its SM expected value; the second one combines the same ratios for the bottom-quark and r-lepton Yukawa couplings. Different contours illustrates the precision with which the correlation between the combined ratios of couplings can be measured. On the same plots we see what the ratios would look like in the MSSM for different ranges of MA- A definite distinction between SM and MSSM Higgs bosons can clearly be established. The only problem in completing a full study of the Higgs boson couplings is the determination of the top-quark Yukawa coupling, and of the Higgs self couplings. The top-quark Yukawa coupling is indirectly determined by measuring the tiH cross section, when the Z contribution is under control. This cross section is very small at -y/i = 500 GeV, and peaks around A/S = 800 GeV (see Fig. 44) Recent studies show that with a LC operating at y / i = 500 GeV the top-quark Yukawa coupling, yt, for a MH = 120 GeV Higgs boson, will probably be determined only at the 25% precision level, while with a LC operating at ^/s = 800 GeV precisions as high as 5 - 6% becomes available. An updated summary of the existing studies can be seen in Fig. 45. The initial phase of a high energy LC will therefore not be able to give us probably the most important Yukawa coupling, and with this respect the role of the LHC becomes crucial, since, as we have seen
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e V -> tth
Vs (GeV) Figure 44. Cross section for e + e ~ —• ttH, in the SM, as a function of the center of mass energy y/s, for two different values of the Higgs boson mass.
in Section 3.4.2, the LHC can obtain yt within 10-15% accuracy, although with some intrinsic model dependence. The production of Higgs boson pairs is also very rare (see Fig. 46), and the measurement of the Higgs boson self-couplings will probably have to wait for a very high energy Linear Collider, like the CLIC collider, a multi-TeV e+e~ machine being studied at CERN. Finally, the total width of a SM-like Higgs boson can be determined in a model independent way by using any well measured branching ratio. For example, one can use that T = T(H -> WW*)/Br{H -> WW*), where Br(H —> WW") is measured directly and T(H —> WW*) can be calculated from the direct determination of the HWW coupling. 4. Highlights of theoretical calculations in Higgs boson physics I would like to conclude these lectures by reviewing some important theoretical results that have recently been obtained in the calculation of Higgs boson physics observables, i.e. total and differential cross sections. Having to limit my discussion to a selection of topics, I prefer to focus on hadron colliders, since the discovery of a Higgs particle very much depends on our
Higgs Boson Physics
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Vs = 500 GeV
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ability to provide precise theoretical predictions for hadronic cross sections. In this context I will concentrate on a few processes for which some outstanding progress has been made recently. The cross section for pp and pp collisions to produce a final state containing a Higgs bosons {H) can be schematically written as: a(pp,pp^H
+ X) = Y^ f dx1dx2fp(x1)fp'p(x2)a(ij ij
^ H + X) , (118)
J
where the partonic cross section &(ij —> H + X) is convoluted with the Parton Distribution Functions (PDF) of partons i and j . ff'p(x) denotes indeed the PDF of parton i into a proton (or anti-proton) and can be interpreted as the probability of finding parton i into a proton (or anti-proton) with a fraction x of its longitudinal momentum. Both the partonic cross section and the parton distribution functions are calculated perturbatively. At hadron colliders, the most important effects arise from strong interactions, and it is therefore mandatory to have the QCD perturbative expansion of cr(pp,pp —> H + X) under control. At each order in the perturbative expansion, the calculation of both a(ij —> H+X) and ff'p(x) contains ultraviolet divergences that are subtracted through a standard renormalization procedure. This, at each finite order, leaves a dependence on the renormalization scale, [1R. In the same way, when the PDF's are defined, a factorization scale HF is introduced in the calculation of ff'p(x). The dependence on both HR and /J,F is indicative of the residual theoretical uncertainty present at a given perturbative order, and should improve the higher the order of QCD corrections that are taken into account. Indeed, it is well known that the theoretical predictions for most Higgs production hadronic cross sections at lowest or leading order (LO) are affected by a very large renormalization and factorization scale dependence. In general, at least the next-to-leading order (NLO) of corrections need to be calculated and this should stabilize or improve the theoretical prediction for the cross section, making the residual theoretical uncertainty comparable or smaller than the corresponding experimental precision. In some cases, as we will see, even next-to-next-to-leading order (NNLO) QCD corrections are necessary to obtain reliable theoretical predictions. To be more specific, the NLO cross section for pp,pp —> H + X can in full generality be written as: a
ppL°p
=
Yl /
dx
ldx2Jrf{xi,
HRHFF?'P(X2, HRHFV%LO(x1,X2, VR, VF) , (119)
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where we have made explicit the dependence on both renormalization and factorization scale. T^ denote the NLO PDF's, while cr^LO is the parton level cross section calculated at NLO as: ~LO a sx^NLO -NLO (120)
+ 4TT ^y
where d&ij represents the 0(as) NLO
6*%
real and virtual corrections: virt
*
real
(121)
In a similar way, and with due differences, we could write cr^pNp^°. A lot of theoretical effort has gone in recent years in the calculation of SajjLO and 5afjNLO for several Higgs production processes. I have collected in Table 1 all the existing work on higher order QCD corrections to Higgs production modes. My apologies for any omission! The result of this effort can be naively summarized by investigating the residual renormalization (/J.R) and factorization ([IF) scale dependence in each SM Higgs boson production mode. This is illustrated in Fig. 47 for the case of the LHC. For each production mode the scale H = /J.R = fip has been varied according to the corresponding original literature as indicated in Fig. 47, such that the scale Table 1.
Existing QCD corrections for various SM Higgs production processes.
process
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S.Dawson, NPB 359 (1991), A.Djouadi, M.Spira, P.Zerwas, PLB 264 (1991) C.J.Glosser et al., JHEP 0212 (2002); V.Ravindran et al, NPB 634 (2002) D. de Florian et al, PRL 82 (1999) R.Harlander, W.Kilgore, PRL 88 (2002) (NNLO) C.Anastasiou, K.Melnikov, NPB 646 (2002) (NNLO) V.Ravindran et al., NPB 665 (2003) (NNLO) S.Catani et al. JHEP 0307 (2003) (NNLL), G.Bozzi et al., PLB 564 (2003),hep-ph/0508068 T.Han, S.Willenbrock, PLB 273 (1991) O.Brien, A.Djouadi, R.Harlander, PLB 579 (2004) (NNLO) T.Han, G.Valencia, S.Willenbrock, PRL 69 (1992) T.Figy, C.Oleari, D.Zeppenfeld, PRD 68 (2003)
qq, gg - • ttH
W.Beenakker et al, PRL 87 (2001), NPB 653 (2003) S.Dawson et al., PRL 87 (2001), PRD 65 (2002), PRD 67,68 (2003)
QQ, 99 -> bbH
S.Dittmaier, M.Kramer, M.Spira, PRD 70 (2004) S.Dawson et al, PRD 69 (2004), PRL 94 (2005) J.Campbell et al, PRD 67 (2003)
gb(b) -> b(b)H bb^
H
D.A.Dicus et al PRD 59 (1999); C.Balasz et al, PRD 60 (1999). R.Harlander, W.Kilgore, PRD 68 (2003) (NNLO)
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•
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Figure 47. Residual renormalization and factorization scale dependence (/U = fj.fi = HF) of the SM Higgs boson production cross section, when all available orders of QCD corrections are included. The scale fi is varied according to the original work present in the literature, and can therefore be slightly different case by case.
interval can be different case by case. Since different production modes have very different cross sections, we have separated them into two plots, containing the leading and sub-leading production modes respectively. The bands in Fig. 47 are in no way indicative of the overall theoretical error, since they do not include systematic errors coming from PDFs and other input parameters. Moreover the effect of setting HR ^ \±F needs and has been investigated case by case, but it is not included in Fig. 47. Nevertheless Fig. 47 gives us a qualitative idea of the perturbative stability of the existing theoretical predictions for the SM Higgs boson production cross sections. Overall the existing theoretical predictions are in good control. In a parallel series of lectures given at this school 44 you have been exposed to the complexity of higher order QCD calculations, and to the variety of techniques that have been developed to perform them. I will not then directly proceed and comment about the results of some higher order calculations in Higgs physics. In particular, I would like to report about: i) the calculation of gg —> H at NNLO of QCD, a pioneer effort that has provided for the first time a reliable theoretical results for the most important Higgs production mode at hadron colliders; ii) the calculation oipp,pp —> ttH, bbH at NLO of QCD, a challenging task, due to the many massive degrees of freedom involved, that has provided for the first time reliable theoretical predictions for the cross sections of these two physically very important production modes. Both i) and ii) rely on the development of several innovative techniques that have allowed the successful completion
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of both calculations. Given the degree of technicalities involved, I will not review them in detail, but only point to the kind of difficulties that had to be faced. The interested reader can find all necessary technical details in the original literature, listed in Table 1 and in the bibliography. 4.1. gg ->• H at
NNLO
g innnnn s innrtro Figure 48.
The gg —> H production process at lowest order.
Most of the basic ideas that motivate the techniques used in the NNLO calculation of the cross section for the gg —> H production process have been already introduced in Section 3.1.4, where we discussed the H —> gg loopinduced decay. In particular we know that in the SM, the main contribution to gg —> H comes form the top-quark loop (see Fig. 48) since: GFas(fi)2 °LO — 288\/27r
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H gives origin to a ggH effective
As we saw in Section 3.1.4, one can work in the infinite top-quark mass limit and reduce the one-loop Hgg vertex to a tree level effective vertex, derived from an effective Lagrangian of the form: Ceff = %-C(aa)Ga>»G° 4v
,
(123)
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where the coefficient C(as), can be written as:
including NLO and NNLO QCD corrections,
C{as) = - — O
1 + Cl
h C2 TV
7T
(124)
+ •• \ TV I
NLO and NNLO QCD corrections to gg —> H can then be calculated as corrections to the effective Hgg vertex, and the complexity of the calculation is reduced by one order of loops. The NLO order of QCD corrections has actually been calculated both with and without taking the infinite top-quark mass limit. The comparison between the exact and approximate calculation shows an impressive agreement at the level of the total cross section, and, in particular, at the level of the if-factor, i.e. the ratio between NLO and LO total cross sections {K = CTNLO/VLO), as illustrated in Fig. 50. It is indeed expected that
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900
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Figure 50. The NLO cross section for gg —> H as a function of MH- The two curves represent the results of the exact calculation (solid) and of the infinite top-quark mass limit calculation (dashed), where the NLO cross section has been obtained as the product of the K-factor (K = CTNLO/VLO) calculated in the rat —> oo limit times the LO cross section. From Ref. 1 0 .
methods like the infinite top quark mass limit may not reproduce the correct kinematic distributions of a given process at higher order in QCD, but are very reliable at the level of the total cross section, in particular when
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the cross section receives large momentum independent contribution at the first order of QCD corrections. As for the H —> gg decay process, the NLO corrections to gg —> H are very large, changing the LO cross section by more than 50%. Since the gg —> H is the leading Higgs boson production mode at hadron colliders, it has been clear for quite a while that a NNLO calculation was needed in order to understand the behavior of the perturbatively calculated cross section, and if possible, in order to stabilize its theoretical prediction. Recently the NNLO corrections to the total cross section have been calculated using the infinite top-quark mass limit (see Table 1). The calculation of the NNLO QCD corrections involves then 2-loop diagrams like the ones shown in Fig. 51, instead of the original 3-loop diagrams (a quite formidable task!). Moreover, thanks to the 2 —» 1 kinematic of the gg —> H
/& » - — H # %$ Figure 51.
Pmrorofi)-— H Is #
Two-loop diagrams that enter the NNLO QCD corrections to gg —• H.
process, the cross section has in one case be calculate in the so called soft limit, i.e. as an expansion in the parameter x = Mjj/s about x=l, where s is the partonic center of mass energy (see paper by Harlander and Kilgore in Table 1). The n-th term in the expansion of OY,- of Eq. (118):
*« = E(v) B *S°'
^
n>0
can then be written in the soft limit (x —+ 1) as follows: 2n-l
^=a^6(l-x)+^bi
oo 2n —1
"ln f c (l-a
fe=0 purely soft t e r m s
+
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(126) where we have made explicit the origin of different terms in the expansion. The NNLO cross section is then obtained by calculating the coefficients a^, (2\
(2)
b), , and clk , for / > 0 and k = 0,..., 3. In Fig. 52 we see the convergence behavior of the expansion in Eq. (126). Just adding the first few terms
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K^Cpp^H+X)
VT= 14 TeV
120 140 160 180 200 220 240 260 280 300 M H [GeV] Figure 52. if-factor for gg —+ H at the LHC (y/s = 14 TeV), calculated adding progressively more terms in the expansion of Eq. (126). From Harlander and Kilgore as given in Table 1.
i{)2a(pp^H+X)[pb]
Vs~= 14 TeV
100 120 140 160 180 200 220 240 260 280 300 M H [GeV] Figure 53. Cross section for gg —• H at the LHC ( v / s = 1 4 TeV), calculated at LO, NLO and NNLO of QCD corrections, as a function of MH, for HF=HR = MH/4- From Harlander and Kilgore in Table 1.
provides a remarkably stable .^-factor. The results shown in Fig. 52 have been indeed confirmed by a full calculation 51 , where no soft approximation has been used.
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The results of the NNLO calculation 50,51 are illustrated in Figs. 53 and 54. In Fig. 53 we can observe the convergence of the perturbative calculation of a(gg —> H), since the difference between NLO and NNLO is much smaller than the original difference between LO and NLO. This is further confirmed in Fig. 54, where we see that the uncertainty band of the NNLO cross section overlaps with the corresponding NLO band. Therefore the NNLO term in the perturbative expansion only modify the NLO cross section within its NLO theoretical uncertainty. This is precisely what one would expect from a good convergence behavior. Moreover, the narrower NNLO bands in Fig. 54 shows that the NNLO result is pretty stable with respect to the variation of both renormalization and factorization scales.
o(pp ->H+X) [pb]
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o(pp^H+X) [pb]
Vs"=14TeV
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ss-s~—_
NNLO - - NLO LO 100 120 140 160 180 200 220 240 260 280 300 100 120 140 160 180 200 220 240 260 280 300 MH [GeV] MH [GeV] Figure 54. Residual renormalization/factorization scale dependence of the LO, NLO, and NNLO cross section for gg —> H, at the Tevatron (^/s = 2 TeV) and at the LHC ( V ^ = 1 4 TeV), as a function of MH- The bands are obtained by varying HR=HF by a factor of 2 about the central value HF = HR = MH/4- From Ref. 5 0 .
Figure 55. NNLL and NNLO cross sections for Higgs boson production via gluon-gluon fusion at both the Tevatron and the LHC. From Ref. 5 4
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This has actually been checked thoroughly in the original papers, by varying both /J,R and fip independently over a range broader than the one used in Fig. 54. The NNLO cross section for gg —> H has been further improved by Catani et al. 54 by resumming up to the next-to-next-to leading order of soft logarithms. Using the techniques explained in their papers, they have been able to obtain the theoretical results shown in Figs. 55 and 56 for the total
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and differential cross sections respectively. In particular, we see from Fig. 55 that the NNLO and NNLL results nicely overlap within their uncertainty bands, obtained from the residual renormalization and factorization scale dependence. The residual theoretical uncertainty of the NNLO+NNLL results has been estimated to be 10% from perturbative origin plus 10% from the use of NLO PDF's instead of NNLO PDF's. Moreover, in Fig. 56 we see how the resummation of NNL crucially modify the shape of the Higgs boson transverse momentum distribution at low transverse momentum (qr), where the soft ln(M^/g^) are large and change the behavior of the perturbative expansion in as 56 .
4.2. pp,pp
—• ttH
and pp,pp
—• bbH at NLO
The associated production of a Higgs boson with heavy quark pairs, PPiPP ~^ ttH and bbH, has been for a while the only Higgs production process for which the NLO of QCD corrections had not been calculated. Given the relevance of both production modes to Higgs physics (see discussion in Section 3.4) and the large renormalization and factorization scale dependence of the LO cross sections (see, e.g., the LO curves in Figs. 60 and 63), a full NLO calculation was mandatory. This has been completed in the papers by Beenakker et al. 57 ' 58 and Dawson et al. 59>60>61>62 listed in Table 1 and we will briefly report about their most important results in this Section. The NLO calculation of pp,pp —» QQH (for Q = b,t) presents several challenges, since it has to deal with a 2 —> 3 process that involves all massive particles in the final state. At tree level, QQH proceeds through the QQ: 99 -* QQH parton level processes illustrated in Fig. 57 for Q = t. As expected, the qq —» QQH dominates for large fraction of the parton longitudinal momentum x, while gg —> QQH dominates at small x. This translate into the fact that the parton level cross section for ttH production is dominated by qq —> ttH at the Tevatron, and by gg —> ttH at the LHC, while bbH production is always dominated by gg —> bbH. The 0(as) virtual corrections include up to pentagon diagrams, such that the problem of calculating scalar and tensor integrals with up to five denominators, several of which massive, has to be faced. Most integrals have to be calculated analytically since both ultraviolet (UV) and infrared (IR) singularities have to be extracted. The 0(as) real corrections involve factoring out IR (soft and collinear) divergences from a 2 - * 4 phase space with several massive particles. Samples of Feynman diagrams corresponding to the 0{as) vir-
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tual and real corrections are illustrated in Figs. 58 and 59. Several new methods and algorithms have been used by the two collaborations that have calculated the NLO cross section for pp, pp —> QQH, and we refer to their papers for all technical details. In the following we will comment separately on ttH and bbH NLO results. For ttH production, the most important outcome of the full NLO calculation is illustrated in Fig. 60, where the renormalization (fiR) and factorization scale (/UF) dependence of the LO and NLO total inclusive cross section is presented, for a SM Higgs boson mass of MH = 120 GeV, at both
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Figure 58. Example of 0{as) virtual corrections to gg —• tt/f: pentagon diagrams. The circled crosses denote all possible insertion of the final Higgs boson leg. All t-channel diagrams have corresponding u channel diagrams, where the two initial state gluon legs are crossed. Analogous diagrams with t —• b contribute to the 0(as) virtual corrections to gg —> bbH.
g«
s nrffirmrcrc
%>w< r^
<
nmsm
q,q(k)
q,q(k)
¥ g innnnnn-—®— i
Figure 59. Examples of 0(as) real corrections to gg —•> t t # . The circled crosses denote all possible insertion of the final Higgs boson leg. Analogous diagrams with t —> b contribute to the 0(cts) real corrections to gg —» bb/f.
the Tevatron and the LHC. We note that the factorization and renormalization scales have been set equal in the plots of Fig. 60, [J,R = /J,F = M, while in the original work both scales have been first varied independently to verify that HR = /j,p = \x is not a particular point at which both scale dependences accidentally mutually cancel. It is evident from the plots of Fig. 60 that the NLO total cross section sensitivity to ^R and ^F is drastically reduced with respect to the LO cross section. Indeed, the residual systematic error coming from scale dependence is at NLO reduced to about 15%, as opposed to more than 100% of the LO cross section. The NLO predictions for ttH production can now be confidently used to interface with experimental analyses.
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0.8
1
2
4
oz
0.5
1
2
4
Figure 60. Dependence of
Let us now consider bbH production f . Naively, one would expect the calculation of bbH production at NLO to follow that of ttH, with the universal replacement of the top-quark mass with the bottom-quark mass, mt «-* fnbHowever, the theoretical prediction of bbH production at hadron colliders involves a few subtle issues not encountered in the calculation of ttH production. Indeed, both from the experimental and theoretical standpoint, it is important to distinguish between inclusive and exclusive bbH production. More specifically, the production of a Higgs boson with a pair of b quarks can be detected via: (i) a fully exclusive measurement, where both b jets are observed, (ii) a fully inclusive measurement, where no b jet is observed, or (Hi) a semi-inclusive measurement, where at least one b jet is observed. Experimentally, b quarks are identified or tagged by imposing selection cuts on their transverse momentum and their angular direction with respect to the beam axis or pseudorapidity. Inclusive modes have larger cross sections, but also larger background, such that more exclusive modes are often preferred experimentally. Moreover, only the exclusive and semi-inclusive modes are unambiguously proportional to the bottom-quark Yukawa coupling. Theoretically, different approaches may be adopted depending on the fact that a final state b quark is either treated inclusively (untagged) or exclusively (tagged). Indeed, when a final state b quark is not identified through some selection cuts, the corresponding integration over its phase space, in particular over its transverse momentum, gives rise to logarithms of the form: For an updated review see Ref.
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A . - * ( $ ) ,
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(127)
where m\, and [in represent the lower and upper bounds of the integration over the transverse momentum of the final state b quark, HH is typically of O(MH) and therefore, due to the smallness of the bottom-quark mass, these logarithms can be quite large. Additionally, the same logarithms appear at every order in the perturbative expansion of the cross section in as, due to recursive gluon emission from internal bottom-quark lines. If the logarithms are large, the convergence of the perturbative expansion of the cross section could be severely hindered and it can be advisable to reorganize the expansion in powers of a"A™, further resumming various orders of logarithms via renormalization group techniques 8 . Currently, there are two approaches to calculating the inclusive and semi-inclusive cross sections for Higgs production with bottom quarks. Working under certain kinematic approximations, and adopting the socalled five-flavor-number scheme (5FNS), the collinear logarithms, A;,, can be factored out and resummed by introducing a bottom-quark Parton Distribution Function (PDF) 63 > 64 ' 65 . This approach restructures the calculation to be an expansion in both as and A^"1. At tree level, the semi-inclusive production is then described by the process bg —> bH illustrated in Fig. 61, while the fully inclusive production process becomes bb —* H, illustrated in Fig. 62. Alternatively, working with no kinematic approximations, and adopting the so-called four-flavor-number scheme (4FNS), one can compute the cross section for pp,pp —> bbH at fixed order in QCD with no special treatment of the collinear logarithms, considering just the parton level processes qq, gg —* bbH illustrated in Fig. 57 (with t —> b) and their radiative corrections. The fully exclusive bbH production cross section can only be computed in the 4FNS framework. As far as the inclusive and semi-inclusive production cross sections go, the comparison between the 4FNS and 5FNS needs to consider QCD corrections beyond the LO, in order to work with stable results. Indeed, the two calculation schemes represent different perturbative expansion of the same physical cross section, and therefore should agree at sufficiently high order. The discussion to follow is based on the s
T h e logarithms mentioned here also appear in the tiH calculation but, since /i// is typically of the order of mt, the logarithms are small and the convergence of the perturbative expansion in as is preserved.
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9 onnnri—• Figure 61.
Tree level Feynman diagram for bg —> bH in the 5FNS.
b Figure 62.
'
Tree level Feynman diagram for bb —> H in the 5FNS.
NLO calculation of qq, gg -> bbH 68 - 69 and bg ^ bH 72 , and on the NNLO calculation of bb —> H 73 . It should be noted that our discussion for the production of a scalar Higgs boson with bottom quarks applies equally well to the production of a pseudoscalar Higgs boson. In fact, if one neglects the bottom-quark mass in the calculation of the NLO corrections, the predictions for bbA0 is identical to those for bbh°(H°) upon rescaling of the Yukawa couplings (see Section 2.4.4). On the other hand, for massive b quarks, the situation becomes more complicated due to the 75 matrix appearing in the bbA° Yukawa coupling. The 75 Dirac matrix is intrinsically a four-dimensional object and care must be taken in its treatment when regularizing the calculation in dimensional regularization (d ^ 4). However, bottom-quark mass effects 2
are expected to be small, 0{j&), h
_
and predictions for bbh°, upon rescaling _
of the Yukawa coupling, provide good estimates of bbA° production even in the massive 6-quark case. In the following we will present results for the exclusive, semi-inclusive, and inclusive cross sections separately. The fully exclusive bbH NLO total cross section is illustrated in Figs. 63 and 64, both for the Tevatron and for the LHC. Both 6 quarks in the final state are identified by the cuts explicitly given in Fig. 63, which have been chosen to closely mimic experimental searches. The curves in Fig. 63 show the dependence of the LO and NLO exclusive bbH cross section from both renormalization and factorization scales (set equal in these plots, i.e. Mfi = MF = M)- The two sets of curves represent the case in which the bottom-quark mass in the bottom-quark Yukawa coupling is renormalized
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0.6
\
\\
Vs=2 TeV Mh=120GeV C„=mb+Mh/2
Vs=14TeV Mh=120 GeV
CT
NLO,OS
CT
NLO,MS
^NLO.OS °NLO,MS
- s s s Ho=mb+Mh/2
°LO,OS
°LO,OS
S ""
°LO,MS S
pTb>20 GeV
N
h|<2
°LO,MS
pTb>20 GeV N<2.5
— 0.04 3
y^^ 1 MA*Q
Figure 63.
Scale dependence of the LO and NLO cross sections for pp —• bbH (Tevatron)
and pp —• bbH (LHC), for MH = 120 GeV. The curves labeled OLO,OS
use the OS renormalization scheme, while the curves labeled ULO.MS the MS renormalization scheme. From Ref. 6 9 .
and O'ATLO.OS
and CAfio.MS
us
e
in the on-shell scheme (OS, blue curves) or in the modified Minimal Subtraction scheme (MS, red curves). As we have already observed in Section 3.1.3 when we considered the decays of a Higgs boson into quark pairs, the bottom-quark renormalized mass varies substantially when the renormalization scale fi is varied from scales of the order of nib to scales of the order of MH, and therefore it is important to know how the large logarithms that determine the running of rrif, are treated in the different renormalization schemes. This is particularly true for the factor mb that appears in the bottom-quark Yukawa coupling, yb = rrib/v, since the cross section depends quadratically on yb- It is much less relevant for the cross section kinematic dependence on m j , coming from the amplitude square or from the integration over the final state phase space. For this reason, the OS and MS labels in Fig. 63 refer to the cases in which only the bottom-quark mass in yb is renormalized one way or the other. The kinematic 6-quark mass is always taken to be the pole mass. Looking at the curves in Fig. 63 we learn that, as expected, the NLO set of QCD corrections stabilizes the total cross section, drastically reducing the scale dependence of the LO cross section. Moreover we see that there is a non negligible dependence on the bottom-quark mass renormalization scheme. This dependence is intrinsic of any perturbative calculation, and should decrease the more orders are added in the perturbative expansion of a given physical observable. However, from the behavior of the residual scale dependence, it is possible to estimate which renormalization scheme provides a better perturbative ex-
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10'
\/s=2 TeV >T">20 GeV
H=m b+ M,/2 s
10
Vs=14TeV pT">20 GeV M<2.5 H=m b +M l /2
10*
|t||<2
10'
~
10
°
^»-.
Q.10"' o
tf
• »MSSM,tanp=10 • — • MSSM,tanp=20 •---•MSSM.tanMO
1Qo
< '
10"' io-a
t
100
•
•
120
M„° (GeV)
^*^_^
10"" 10""
.
10-
'
•
' '
"'-»-. --. '" ••... . ^
• »SM ~~~-»-_ • «MSSM,tanM0 •---•MSSM,tanp=20 • • MSSM,tan0=4O
* <
m"6
•
110
?
i!10 ! o
'
130
0
200
400
600
800
MH° (GeV)
Figure 64. NLO cross section for pp —> bbH (Tevatron) and pp —> bbH (LHC) as a function of MH, in the SM and in the MSSM w i t h t a n / 3 = 10,20 and 40. For the Tevatron the production process considered is pp —> bbh° with Mha = 100,110, 120, and 130 GeV, while for the LHC it is pp -> bbH° with MHo = 120, 200,400, 600, and 800 GeV. From Ref. 6 9 .
pansion. In the case illustrated in Fig. 63, both OS and MS NLO cross sections show a well defined plateau region where the cross section is very mildly dependent on the scale \i. Nevertheless, the MS cross section overall performs better, since it has a more regular behavior also at small scales. As a result, MS is often the preferred choice in all processes that depend on the bottom-quark Yukawa coupling. Finally, Fig. 64 shows the dependence of the total cross section from the Higgs boson mass, in both the SM (solid black curve) and the MSSM (dashed colored curves, corresponding to different values of tan (3), over a significant MH range for both the Tevatron and the LHC. The quadratic growth of the MSSM cross sections with tan/J is evident, and this graphically confirms the possibility of finding evidence of new physics already at the Tevatron, if an MSSM-like 2HDM with large tan/3 is realized in nature (see Sec. 3.4.3). Let us now turn to the semi-inclusive and inclusive bbH cross sections. For both production modes, much effort has been spent recently in understanding the difference between 4FNS and 5FNS in order to assess the reliability of the existing theoretical results 67 ' 68 ' 71,70 . First of all, let us briefly review the idea behind the introduction of a bottom-quark PDF, which naturally leads to the 5FNS framework. For the purpose of illustration, consider the prototype case depicted in Fig. 65: one of the final state bottom quarks is directly originating from the g —> bb splitting of one of the initial state gluons, while the shaded blob represents all the possible non collinear configurations of the remaining particles. In the mj —> 0 limit, the
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349
Pb
g nnnnftnnn—*— g
nrcftn
Figure 65. Tree level Feynman diagrams for gg —• bbH illustrating the almost collinear emission of a bottom quark (upper leg).
g —> bb configuration gives origin to collinear singularities, when the two b quarks are emitted in the same direction of the splitting gluon. In the case of bbH production, the singularities appear in the phT —> 0 phase space region, where phT is the transverse momentum of the upper leg bottom quark in Fig. 65. If we take rrn, ^ 0, these singularities are regulated by the non zero 6-quark mass, leaving behind collinear logarithms Ab of the form given in Eq. (127). The contribution to the total partonic cross section from this diagram can be written as: An avgg^bbH
1
d3Pb
1 d3Pl
3
1 d3pH
1 Y^,,
(2)2El2E2 (2TT) 2Eb (2TT) 2Eh (2TT)3 2EH ^ •(2TT)4*<4>
3
(qi +q2-Pb-Ph-
PH)
l
,2
l
99^bbH\
,
(128)
where we have used the momentum notation of Fig. 65, such that E\%2 are the energies of the initial gluons while Eb 5 H are the energies of the final state particles. The amplitude for this process is denoted by Mgq-^bbHParameterizing the 6-quark propagator momentum as: k^ = zq? + (3n» + k^ ,
(129)
where z is the fraction of the initial gluon momentum carried by the b quark, and k± is transverse to both qf and nM, for an arbitrary vector nM, with k^k±^ = — kj., one can explicitly show (after some Dirac algebra manipulations) that:
\Mgg^bbH\2 ~ g2sCF~
( 1 ^ ) Pqg(z)\Mgl^lH|2
,
(130)
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where Pqg(z) is the Altarelli-Parisi splitting function for g —> qq: P,9 = \[z2 + (l-z)2}
,
(131)
CF =_(N2 - l)/2/N (for N = 3 colors), and Mgb^bh is the amplitude for gb —> bH. The previous equation is approximate, i.e. it has been obtained by neglecting higher powers of kj,, keeping only the terms that would give the most singular or leading contribution upon integration over the collinear 6-quark phase space: d3pb
1 _ 3
1
dzdk\ 2
(2ir) 2Eb
16TT (1-Z)
'
where we have used that in the small kr limit the transverse part of the four momentum of the outgoing quark, p^ coincides with kx- Inserting Eqs. (130) and (132) into Eq. (128) one finds: dtgg^bhH - -jfdz^-Pqg{z)dag-b^-bH
.
(133)
The integration over k\, with lower bound ml and upper bound [i2H, gives origin to the collinear logarithm Ab introduced in Eq. (127). Moreover, when one convolutes with the gluon PDFs, g(xi,/j,) (for i= 1,2), of the two initial gluons to obtain the hadronic cross section, a bottom-quark PDF of the form:
b(x, HF) = ^~f^^b
I ^Pqg •
/
*
<
( -. ) g(y, HF) )
(134)
•
naturally appears. In Eq. (134) fip represents the factorization scale. The collinear Ab logarithms are factored out and then resummed in the bottomquark PDF when the factorization scale is set to ^F—^H- Indeed, Eq. (134) gives the bottom-quark parton density at the lowest order in as, while the leading a"Ab logarithms are resummed via the DGLAP equation upon evolution: d
dlogn
as(jjL)
b(x, /i) = ^
n
t ^Pqg Jx y
(-) g(y, \yj
M)
.
(135)
The 5FNS approach is therefore based on the approximation that the outgoing b quarks are at small transverse momentum, since this is the region of phase space that is emphasized by the kr expansion. The incoming b
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partons are given zero momentum at leading order, and acquire transverse momentum at higher order. With the use of a fr-quark PDF, the 5FNS effectively reorders the perturbative expansion to be one in as and AjJ" . To see how this works, let us consider the perturbative expansion of the inclusive process bb —> H (Fig. 62) which, according to what we just saw, is intrinsically of order a2 A2. At NLO, the virtual and real corrections to the tree level process make contributions of 0(a^A2). However, at NLO, we must also consider the contribution from bg —> bH where the final state b is at high transverse momentum. This process makes a contribution of order «jAf, and is, thus, a correction of 0(A^1) to the tree level cross section. Similarly, at NNLO, besides the myriad of radiative corrections of 0(a4sA2), we must also include the contribution from the process gg —> bbH, where both b and b are at high pr- The contribution from these diagrams are of order a2, and are, thus, 0(A^2) (or NNLO) corrections to the tree level process bb —> H 65 ' 66 . The above discussion for bb —> h also applies to the perturbative expansion of bg —> bH. In this case, the tree level process is of order a2 A& and the contribution from gg —+ bbH is a NLO correction of
O(A^)
72
.
The comparison between 4FNS and 5FNS has been initially performed in the SM 67>68>71>70. This has been crucial to understand several important issues. However, since the production of a Higgs boson with bottom quarks will only be physically interesting if the bottom-quark Yukawa coupling is enhanced beyond its SM value, we will present the 4FNS vs 5FNS comparison in the MSSM, with tan/3 = 40 7 4 . Figs. 66 and 67 show the comparison for the inclusive and semi-inclusive total cross sections respectively, at both the Tevatron and the LHC. They both give the cross section as a function of the Higgs boson mass, chosen to be h° at the Tevatron and H° at the LHC. In the inclusive case, Fig. 66 illustrates both the NLO and NNLO predictions. Both at LO and NLO (NNLO) the band are obtained by varying the renormalization and factorization scales independently as explained in the figure captions. In both the inclusive and semi-inclusive case there is good agreement between the 4FNS and 5FNS results within their respective scale uncertainties, although the 5FNS tends to always give slightly higher results. It is actually very satisfactory that the NLO (and NNLO) calculations for the semi-inclusive and inclusive bbH production agree within their systematic errors. Indeed, if all perturbative orders were to be considered, the two approaches would produce the very same outcome. The truncation of the perturbative series at a given order gives origin to discrepancies, because
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CTNLO [pb] 100 r \ i $••--.._
Tevatron, yft = 1.96 TeV 1
LHC,
O"NL0 [ p b ]
1 1 1 1 NLO, 4FNS, gg,qq-> (bb)ha —f_ NLO, 5FNS, bb->h" - - * NNLO 5FNS, 66 -» A°—x -
14 TeV
NLO,
4FNS, gg,qq -> {bb)H
NLO,
5FNS, bb -> H"
--*-.
NNLO, 5FNS, 6 6 ^ f f ° - - « - •
r"-"::"::;:-'-'-"-V - - ' - H ' " - * '* MSSM, tan/? = 40 l
l
i
^ \x; l
i
250
Mho [GeV]
300
MHo [GeV]
Figure 66. Total cross sections for pp,pp —> (bb)h (h = h°,H°) in the MSSM with no bottom-quark jet identified in t h e final state in the 4FNS (at NLO) and 5FNS (at NLO and NNLO) as a function of the light and heavy MSSM Higgs boson masses, at both the Tevatron and the LHC. The error bands have been obtained by varying the renormalization (fj,r) and factorization {fXf) scales separately between /Uo/4 and fj,o (with H0=mi, + Mf//2) in the 4FNS, while keeping // r = M^ and varying Hf between O.lMjj and 0.7MH in the 5FNS (see Ref. 7 3 for details). From Ref. 7 4 .
0NLO [ p b ]
Tevatron, Js = 1.96 TeV
10
1
:
LHC, 1
v ^ = 14 TeV
' NL6, gg,qk^b(t)Ha !+__ : N L O M C F M , gb-ibH"._X:__ 0.2/io < (' < IM> Ho = mi + MHt>/2
K
: '
: MSSM, tan /9 = 40 t
150
Mho [GeV]
i
200
i
250
^ ^ ^ ^ > = ^ _ i
i
i
i
300
350
400
450
'
500
MHo [GeV]
Figure 67. Total NLO cross section in the MSSM for pp,pp —> b(b)h production at the Tevatron and the LHC as a function of Mho ^ o . We varied fj,r and fif independently to obtain the uncertainty bands, as explained in the text. The solid curves correspond to the 4FNS, the dashed curves to the 5FNS. The error bands have been obtained by varying the renormalization (/j,r) and factorization (fJ-f) scales separately between 0.2/io and no (with no=mb + Af/f/2). From Ref. 7 4 .
the 4FNS and 5FNS perturbative series are differently ordered. However, we see that considering the first (second) order of corrections already brings the agreement between the two schemes within the respective theoretical uncertainties. The results shown in Figs. 67 and 66 represent indeed a major advancement, since the comparisons existing in the literature before
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Refs. 67>68>71>70 showed a pronounced disagreement. This was mainly due to the absence of the (now available) NLO results for qq, gg —> bbH. Acknowledgments I would like to t h a n k the organizers of TASI 2004 for inviting me to lecture and for providing such a stimulating atmosphere for b o t h students and lecturers. I am most thankful to Chris B. Jackson and Fernando FebresCordero for carefully reading this manuscript and providing me with several valuable comments. This work was supported in p a r t by the U.S. Department of Energy under grant DE-FG02-97ER41022. References 1. F. Englert and R. Brout, Phys. Rev. Lett. 13, 321 (1964). 2. P. W. Higgs, Phys. Rev. Lett. 13, 508 (1964). 3. G. S. Guralnik, C. R. Hagen and T. W. B. Kibble, Phys. Rev. Lett. 13, 585 (1964). 4. M. E. Peskin and D. V. Schroeder, "An Introduction to quantum field theory", Harper Collins Publishers (June 1, 1995) 5. S. Weinberg, "The Quantum theory of fields. Vol. 2: Modern Applications", Cambridge University Press (August 13, 1996) 6. A. Djouadi, "The anatomy of electro-weak symmetry breaking. I: The Higgs boson in the standard model", arXiv:hep-ph/0503172. 7. A. Djouadi, "The anatomy of electro-weak symmetry breaking. II: The Higgs bosons in the minimal supersymmetric model", arXiv:hep-ph/0503173. 8. M. Carena and H. E. Haber, "Higgs boson theory and phenomenology", Prog. Part. Nucl. Phys. 50, 63 (2003) [arXiv:hep-ph/0208209]. 9. S. Dawson, "Introduction to the physics of Higgs bosons", lectures given at TASI 94, arXiv:hep-ph/9411325. 10. S. Dawson, "Introduction to electroweak symmetry breaking", lectures given at the 1998 ICTP Summer School in High-Energy Physics and Cosmology, arXiv:hep-ph/9901280. 11. CMS Collaboration, Technical Design Report, CERN/LHCC/94-38. 12. ATLAS Collaboration, Technical Design Report, Vol. II, Detector and physics performance, CERN/LHCC/99-15. 13. M. Carena et al. [Higgs Working Group Collaboration], arXiv:hepph/0010338. 14. D. Cavalli et al., arXiv:hep-ph/0203056. 15. L. Babukhadia et al. [CDF and DO Working Group Members], FERMILABPUB-03-320-E 16. K. A. Assamagan et al. [Higgs Working Group Collaboration], arXiv:hepph/0406152.
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73. R. V. Harlander and W. B. Kilgore, Phys. Rev. D 68, 013001 (2003) [arXiv:hep-ph/0304035]. 74. S. Dawson, C. B. Jackson, L. Reina and D. Wackeroth, arXiv:hepph/0508293, to appear in Mod. Phys. Lett. A.
HEIDI SCHELLMAN
L E A R N H A D R O N COLLIDER PHYSICS IN 3 DAYS
HEIDI SCHELLMAN Dept. of Physics and Astronomy, Northwestern University Evanston, IL USA 60208 A brief overview of experimental hadron collider physics for theoretical physics students, illustrated with recent results from the D0 and CDF experiments at the Tevatron.
1. Introduction and acknowledgements Because these lectures are meant to be educational rather than cutting edge, I have gone into more detail than usual. Such details are usually only available in doctoral dissertations. In particular I have relied on the D 0 Dissertations of Levan Babukhadia 12 , Robert Snihur 5 , Juan Estrada 9 and Florencia Canelli 10 . I would also like to thank the organizers and participants at TASI04. Their questions shaped the direction of these lectures and raised many other questions I'm still working on. 2. The characters in the story These lectures describe physics done at hadron colliders, which have had four historical phases. (1) During the 1970's protons were collided with protons at a center of mass energy of around 60 GeV at the CERN ISR. Supersymmetry didn't really exist yet as a prediction and it was not discovered at the ISR. (2) During the 1980's the UA1 and UA2 experiments at CERN took data on proton-anti-proton collisions at y/s = 540 GeV. The W and Z bosons were discovered by these experiments. They were shut down in order to run the LEP e~e+ collider during the 90's. They wiped out many popular models but failed to find Supersymmetry, despite theoretical expectations that it would be there. 359
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(3) During the late 80's through the mid 90's the CDF and D 0 experiments ran at the Fermilab Tevatron, colliding protons and antiprotons at an energy of yfs = 1800 GeV. The major discovery was the top quark. Precision measurements of the top and W masses were also performed. CDF installed a silicon vertex tracker which enhanced their top signal and started a program in B physics at hadron colliders. CDF and D 0 shut down for 5 years in the late 90's for a detector and accelerator upgrade. During this time, the CDF silicon tracker was replaced with an improved version and the D 0 experiment added a solenoidal magnetic field and a silicon tracker and now has similar capabilities to CDF. The accelerator upgrade led to an increased center of mass energy of ,/s = 1960 GeV and peak luminosities of 10 3 2 cm _ 2 sec _ 1 . The cross section for inelastic proton anti-proton scatters at 1960 GeV is 60 mb = 6 x 10 26 cm~ 2 sec _1 which works out to 4.2 million interactions per second at the peak luminosity. The beams only cross 1.7M times/second so at peak luminosity, D 0 and CDF are now seeing two or more inelastic scatters per beam crossing. D 0 and CDF have already logged close to a factor of four more data than in the previous run and hope to multiply the current sample by another factor of 10 before the LHC turns on. Supersymmetry has not been found at the Tevatron, despite expectations, and many more interesting models have been ruled out. (4) In the late 90's LEP was shut down so that CERN could finish construction of the Large Hadron Collider (LHC), scheduled to start running in 2007. This will be a proton-proton, not proton-antiproton machine at -^s = 14,000 GeV, 7 times the Tevatron energy. The LHC will have 4 detectors, two of which, CMS and Atlas, are general purpose detectors optimized for high p± physics. The machine luminosity is expected to be 10 33 - 10 34 cm~ 2 sec'" 1 with close to 20 inelastic scatters per beam crossing at high luminosity. Discovery of Supersymmetry is expected in the first months of LHC operation.
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3. The technical details 3.1.
Luminosity
Luminosity measures the flux of particles capable of creating a reaction of interest. The number N0bserved of events observed in an experiment is N«bserved
—
^process
* ^detection
*
/ L,uZ
~T~ ^background
\*-J
where the observable aprocess is the cross section for the process and should not depend on the experimental details, ^detection is the probability that a signal event will be observed in a given detector, / Cdt is the Integrated Luminosity and Nbackground are events from other processes that got counted incorrectly. At colliders, the luminosity depends on both the beam intensities and the beam densities. N N£ = / ^Vv Z L
(2)
where / is the frequency with which beam bunches cross (1.7 MHz at the Tevatron), Np is the number of protons/bunch, Np is the number of antiprotons/bunch and ax and cry are the gaussian sizes of the beam. See http://www-bd.fnal.gov/notifyservlet/www for the real-time numbers for the Tevatron. Typical beam sizes at hadron colliders are 20-100 /itms and typical instantaneous luminosities are 5 x 10 31 cm~ 2 sec - 1 . For integrated luminosities, we normally use inverse pico-barns (1 pico-barn - 1 = 10 36 cm~ 2 ) as a unit. During a typical running week, which has around 200,000 seconds of beam in it, 7-10 pb~ of luminosity will be delivered to each of the Fermilab experiments. This means that if a particle has a production cross section of 100 fb, one a week will be produced (but probably not detected) at the Tevatron. 3.2. Overview
of collider
detectors
Particle detectors at colliders have evolved to be pretty similar - the technologies used in each component differ but they all have the same basic layout. Starting at the interaction point, there is • a tracking volume with almost no material and a high magnetic field. This is used to measure the trajectory of charged particles
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with high precision. It normally has an inner, high resolution section built of silicon to detect the decays of short lived particles and an outer tracker made of less expensive materials and optimized for momentum measurement. • a 'calorimeter' made of very heavy material which absorbs and detects almost all strongly and electromagnetically interacting particles. It is normally divided in to a high Z electromagnetic part and a cheaper outside hadronic part. • a muon detection system, which measures the momentum of any muons which make it through the calorimeter. These different pieces are illustrated in the picture of the D 0 detector (Figure 1). More detail on these components is given below.
3.3. Collider
physics
basics
As theorists, you think of processes as one or two incoming fundamental particles interacting to form an interesting final state. In e~e+ physics this is a good approximation, but in hadron colliders it is an approximation and it turns out, a bad one. Most of my examples will be from proton anti-proton collisions at the Tevatron (since that's what I know) but I will include comments on the LHC which will collide protons with protons. The problem is that your fundamental incoming partons, quarks and gluons, are delivered in protons and antiprotons. The hard collision of interest only occurs when partons with the right quantum numbers happen to have the right center of mass energy to make the desired final state. Most of the time, the hard collision involves partons with the wrong quantum numbers or the wrong energy and all you get, from your point of view, is junk. The longitudinal momentum distribution of the desired partons in the proton is described by Parton Distribution Functions or PDF's, which can be determined from other processes. But these are probabilities, not certainties. As a result, in a given proton-antiproton reaction you do not know the longitudinal momentum of the initial state although you can predict the distribution of such momenta for an ensemble of events. Figure 4 shows typical parton distribution functions for important partons such as u quarks and gluons. The total cross section for a parton of type 1 and a parton of type 2 to scatter is the integral over the probability of finding those partons in the proton to begin (the PDF) with times the hard scattering matrix element.
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—
o
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E
Figure 1. Side view of the D 0 detector. The innermost area contains a silicon vertex detector surrounded radially by a scintillating fiber magnetic tracker. The intermediate region is the calorimeter, shown in more detail in the next figure, and the outermost system detects muons which have escaped from the calorimeter.
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Figure 2. One quadrant of the D 0 calorimeter illustrating the segmentation in pseudorapidity.
In the following, the hatted quantities refer to the hard parton scatter while the unhatted quantities are for the proton/ antiproton system. Since we don't know the longitudinal momentum for the initial state, we should use cylindrical coordinates. Unless the protons are polarized, the cross section should be symmetric in azimuth so the relevant variables are Pll and px- The parton distributions can be written as fi(x;n)
(3)
where i is the parton flavor, x = pparton/pProton is the fraction of the proton momentum carried by the parton and fx is an appropriate hard scattering scale for the interaction. In the absence of strong interactions between the quarks in the proton, the PDF's would be just a function of x but interactions introduce a log^i dependence. a(p + p-> X) =
a(l + 2->X;n)fi(xi;n)f2(x2:,fi)dx1dx2
(4)
<7 is the quark scattering cross section, it depends on the scale n but (in principle) the observable cross section a does not. In practice, one guesses that n is the hard scattering momentum scale Q, which is often assumed
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Anti-Proton
Proton
Figure 3. What is really going on in a hadron collider, partons collide and a mess of target remnants and scattered particles ensues. In this event a u from the proton and u from the anti-proton have produced a final state with many particles, mainly concentrated in two jets.
to be the mass (*c) of the final state object or the transverse momentum of the final state particles. For a detailed discussion you might wish to look at the CTEQ Handbook of Perturbative QCD 2 or other QCD texts. The parton center of mass energy is: S = XXX2S
=
XiX2{2Pbeam)2
(5)
and the momentum of the parton center of mass is: pz(cm) = (xi p±(cm) ~ 0
-x2)Pbiteam
(6) (7)
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Figure 5 shows the first order hard scattering diagrams for proton antiproton scattering. Figure 6 shows the typical x and Q ranges for different experiments. Figure 4 shows typical parton distribution functions at collider energies. If you look at the parton probabilities in Figure 4, you note that gluons are the most probable partons, except at the highest momentum fractions, and in fact, the cross section at very low p± is dominated by gluon-gluon and quark-gluon scattering via the t channel. 3.4. The final state Quarks and gluons do not appear as particles in the final state, instead
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x XX x XX x Figure 5. First order diagrams for proton-anti-proton scattering. If one assumes that time runs bottom to top (the theorist's convention), the first column indicate exchange in the t channel, the second s channel exchange, the third the u channel and the 4th is a special QCD diagram.
they fragment into 'jets' of reasonably long-lived hadronic particles such as 7r+, ir~, 7T°, K+, K~, KL, KS, rj, rf, p, Ti etc. The 7r°, TJ decay quickly into photons. This jet of particles generally follows the path of the original quark or gluon but there are important problems in making that identification. These include: • The final state particles are color neutral, while the quarks and gluons are not. This means that there is a color connection between the final state partons (and the remnants of the proton and antiproton in most cases).
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Kinematic coverage of various experiments.
• Higher order diagrams cause jet splitting. Some models of jet production take only the leading order hard diagram and then do fragmentation of those jets using parton shower models while others attempt to include higher order hard scattering diagrams and then fragment them. The degree to which fragmentation is handled in the original matrix element or in the fragmentation model is a rapidly evolving art form. • Algorithms for finding the jets vary - generally fast algorithms such as cones are hard to map onto theoretical observables while algorithms which are theoretically robust, such as the kperp algorithm, are time consuming and prone to experimental biases. For a nice recent review see reference 7 . • The final state particles deposit energy in the detector in different ways - finding and summing the energy can be quite difficult.
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Figure 7 shows the production of a jet when a quark is knocked out of a proton (by a neutrino). In step a), the quark is knocked out but remains connected to the proton by its color charge. At some point the energy in the color field becomes so high that it is energetically favorable to produce a quark anti-quark pair b) which can neutralize some of the color field. In c) the color neutral objects have 'hadronized' to form real observable particles, in this case a neutron and two kaons.
n Figure 7. Illustration of fragmentation when a quark is knocked out of a proton, for example in a neutrino interaction.
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3.5.
Kinematics
You may, personally, be interested in Higgsino production but the total proton anti-proton scattering cross section is dominated by t channel exchange of a gluon. Because the backgrounds are dominated by t channel processes, which have factors of 1/i oc s i n - 2 | in the matrix element it was realized early on that the polar angle 6 was a lousy variable for describing what one actually sees in most produced events, even though most interesting interactions involve s channel quark anti-quark annihilation. Instead of the polar angle, the rapidity, y, is used.
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The big deal about rapidity is that: • Differences in rapidity Ay are Lorentz invariant for boosts along the z (or rapidity) axis. You can verify this for yourself. Because of this, Lorentz Invariant Phase Space can be written as
H
= dWydpl = 2Trdydpl
(10)
• and if you go to a frame where the rapidity of a final state object is 0, it has a polar angle of J and small variations in y are 5y « 56 + O{50f
(11)
equivalent to small variations in the polar angle 0. This means that one can define 'jet's of hadrons in y — cj> space and achieve results similar to those one would get at 90° in 9 — <j) space. The rapidity of a particle of mass M has kinematic limits set by the total energy available for that particle. ;»-'M
(12)
r\cr log ^§ M
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For example, at the Tevatron, Z° bosons will have rapidities of less than 3, while top quarks will be less than 2.3.
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Spherical D
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Figure 8. Illustration of collider coordinates. The sphere at top left has lines drawn at rapidity intervals. The cylinder on the right is the same space after the transformation to rapidity space 6 —> n. The bottom left diagram shows the cylinder being unrolled to make an r) — > grid. Particles can then be plotted in T)
The actual rapidity distributions are determined by the product of parton distributions, which determines the longitudinal momentum distributions of interactions, but empirically, the rapidity distribution for soft processes is closely approximated by a constant distribution per unit rapidity within the kinematics limits. For massless particles (which are a good approximation for the decay products of almost anything in a collider) , the rapidity y reduces to the pseudo-rapidity:
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Figure 9. A very high energy event in the D 0 detector. The boxes represent calorimeter towers, with blue boxes having higher energy than green ones, the purple line are charged tracks, the brown cylinders are momentum vectors for the jets. The momentum vector scale is set very low, the two largest jets extend far outside the picture.
Tj = - l o g ( t a n - )
(14)
Figure 1 shows a side view of the D 0 detector at Fermilab. Figure 2 shows a quadrant of the calorimeter and illustrates the segmentation in pseudo-rapidity. Collider detectors are designed so that each detector element covers the same area in rj - <> / space. For example, at D 0 the detector elements are 0.1 x 0.1 in size. The utility of plotting things in -q - <j> ~ Pperp space is illustrated when one looks at real data. Figure 9 shows a normal space view of the objects detected in a very high energy parton scatter. The initial state partons
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carried more than half of the proton's momentum and scattered at around 90 degrees. Figure 10 shows a lego plot in rj — (f> — p± coordinates of the energy flow in the final state. Figure 8 illustrates the different coordinate systems and their relationship.
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Figure 10. A lego plot of the same event. T h e calorimeter transverse momentum is plotted in JJ —coordinates. The two high energy jets are very visible. The strange patterns for |f}| > 3 are due to energy deposited by the beam particle remnants and a change in segmentation from 0.1 units to 0.2 in the far forward region.
4. E x a m p l e 1: J e t p r o d u c t i o n Our first example will be the simple partonic scattering illustrated in Figure 5, where two initial state partons scatter into two or more final state partons. These results have been published in references 8 , 13 . Levan Babukhadia's thesis 12 contains a full description of the methods used. The observed final state will consist of 2 or more jets. Because the major diagrams have matrix elements that go like l/t or 1/s which act like \/p\ and the parton distributions approximately go as l/;r(l — x)n, the jet spectrum falls off very rapidly with transverse momentum.
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Figure 11. Illustration of a photon interacting destructively in some heavy material. The photon interacts with a nucleus, pair produces an electron positron pair. The typical length scale is XQ. Those electrons then bremsstrahlung photons, again over a scale of XQ, which then pair produce and the process continues. All of the energy ends up as ionization caused by the electrons passing through the material which can then be detected. A typical length for such a shower is L = XQ log E/EC where Ec ~ 700MeV/(Z + 1) is the critical energy. See chapter 26 and 27 of the PDG 4 for a discussion.
4.1.
Calorimetry
The jets of particles are a mix of hadrons and electrons, muons and photons from decays. The energy of these particles are measured in a calorimeter, a detector designed to destructively measure the total energy of particles which enter it. 4.1.1. Electromagnetic calorimeter Calorimeters normally consist of two sections, one optimized to have high Z and detect electromagnetic energy, and another optimized to be both dense and thick which lies behind the electromagnetic calorimeter and detects any
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hadronic particles which make it through the electromagnetic part. The typical scale for electromagnetic showers is the radiation length Xo (1/e absorption length) which ranges between 14 meters in liquid hydrogen to 0.32 cm in Uranium. The best material for dense detectors is Platinum at 0.305 cm but it's a bit pricey. Thanks to the cold war, depleted Uranium is much cheaper. Figure 11 attempts to explain photon detection. Electron detection is almost identical - a photon shower is just two electron showers superposed in some sense. The decays T(9460), J/^(3100) and Z° to two electrons provide a very accurate calibration for electromagnetic energy at the 0.1% level. Figure 12 shows a simulation of 10 GeV electrons and photons hitting the CMS Electromagnetic Calorimeter which consists of Pb-W glass blocks.
3m*e~,gamnw,e+ 10G0V
Figure 12. Simulation of the interactions of 10 GeV photons and electrons in the CMS electromagnetic calorimeter.
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Sampling Calorimeter Uranium Liquid Argon
JC+
Figure 13. The shower induced by a charged pion. The length scale is much longer and the range of interactions at each stage is much more complex than in the electromagnetic case. In fact, if particles are moving slow enough they can undergo weak decays and muons and neutrinos can be produced which leave the calorimeter undetected. In this case, a sampling calorimeter is illustrated where liquid argon lies between uranium plates. The charged particles in the shower ionize the Argon and the electrons from the ionization are swept up by high voltage across the gap. Ionization in the uranium is not detected and a 'sampling' correction is applied for this loss.
4.1.2. Hadron calorimetry Unfortunately, a hadronic jet is not all electromagnetic and in addition to pions and kaons, contains neutrinos and muons from weak decays, which don't interact enough to deposit their full energy. Hadrons such as pions, kaons, protons and neutrons do deposit energy but over a much longer distance scale, the interaction length Aj, which ranges from around 600 cm in liquid hydrogen to 10 cm in Uranium. Since the hadronic showers are generally much longer than electromagnetic showers, hadron calorimeters are generally put behind electromagnetic calorimeters and build of cheaper materials. Figure 13 shows a hadronic shower, it looks similar to an electron shower but the particles involved are more diverse and the length scales are
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much longer. Generally, hadronic calorimeters need to be 6-10 interaction lengths thick with the limiting factor being cost and mass. Figure 14, from the PDG 4 Chapter 27, shows measurements of the energy containment of hadron calorimeters vs energy and thickness. For hadronic calorimeters, which need to have lots of material, both for thickness and because other large parts of the detector are inside them, cheaper materials such as Fe or Cu are normally used with Liquid Argon or scintillator readout. Such calorimeters are 'sampling' calorimeters. ~rT T T -
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4.1.3. Sampling vs. non-sampling Fi gure 13 also illustrates 'sampling'. Calorimeters can be made 100% active by using glasses such as Cs-I or Pb-W or liquid Xe. But 'sampling' calorimeters, in which an passive heavy material such as Uranium is interleaved with an active material, such as Argon or scintillating plastic, are much less expensive and usually used for large detectors. In these detectors a 'sampling' correction must be made for the fraction of energy lost in the passive material. The losses in the passive material cause statistical fluctuations in the amount of energy detected and substantially degrade
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the resolution of the detector. However, by appropriate choice of sampling fractions as a function of depth, the response of the calorimeter to photons (which interact early) and hadrons (which interact late) can be compensated, leading to smaller errors for jets with mixed electromagnetic and hadronic energies. detector type
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4.1.4. Hadron energy calibration Nature does not provide very many jet calibration lines. For example, the W and Z bosons decay to jets most of the time but the dijet cross section shows no appreciable enhancement around the mass peaks both due to poor resolution and a factor of ( — ) 2 in electroweak cross sections relative to the QCD cross section. In future, when statistics are higher, the hadronic decays of W"s in top decays should become a promising calibration point with much less background. Jet scales are found either by a combination of Monte Carlo simulation and test beam measurements for individual particles or by in situ measurement of transverse momentum balance between photons and jets from the QCD/QED process qG —+ q7- After enormous effort, errors of the order of 3% on the energy scale can be achieved. n is a 94 page article describing the procedure used by DO. 3% sounds good until you remember that the jet spectrum is falling very quickly as a function of the jet transverse momentum. This 3% error on the x (transverse momentum) axis quickly becomes a 30-50% error on the y (cross section) axis when the spectrum is falling fastest.
4.2.
Result
Figures 15 and 16 summarizes the results from a measurement done at D 0 in the late 90's. It is described in great detail in Levan Babukhadia's thesis 12 . The DO result and a similar measurement from CDF have been published 8 ' 1 3 . Figure 15 shows the p± spectrum of jets for several rapidity bins. Figure 16 the p± spectrum normalized to different theoretical predictions. The main source of variation in theoretical predictions is the input parton density functions, in particular, the gluon content at high x which is not well constrained by other experiments. The uncertainty on the measurement is completely dominated by the jet energy scale. What the plot does not show is that error is very highly correlated from point to point. One can note that the MRSTjg PDF set does match the data more closely and a full statistical analysis indicates that the better match is in fact significant. This has been interpreted as the presence of a larger gluon distribution at high x than had previously been believed. In particular, it indicates that a large fraction of the momentum of the proton can end up in a single gluon.
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5. Simulations and neutrinos Now that we've seen the QCD cross sections, we might want to look at how the others compare. Figure 17 shows the event rates for various processes in hadron-hadron collisions. All other processes have much lower rates than the QCD rates discussed above and have to contend with huge backgrounds from simple QCD scatters in which the final state particles mimic something more interesting.
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For example, a W boson can decay to an electron and a neutrino, each with an energy of around 40 GeV in the W center of mass frame. This leads to electrons and neutrinos with transverse momenta in the range 0-40 GeV, depending on the decay angle relative to the beam axis. The total
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cross section for this process at the Tevatron is around 2.6 nb once you take into account the branching fraction to electrons. The QCD cross section for producing 2 partons with transverse momenta of 25-40 GeV or above is roughly 2-10 /ubarn or roughly a factor of a thousand higher. If your detector has a 0.1% chance of calling a QCD jet an electron, and messing up its transverse energy, you may be looking at a background as large as your signal. We have better ways of detecting W which will be described later, but one has to be very careful in detector design to measure rare processes in the presence of such large backgrounds. 5.1. QCD backgrounds
to rarer
processes
There are several ways in which a normal quark can fragment in a way which looks like an electron, muon or neutrino in a detector. • Jet fluctuations - a jet of cm energy E > 40 resulting from a quark or gluon will have an average number Nchg ~ 7.71og 10 (£yi0 GeV) + 1.318 charged particles, mainly charged pions, and around Nneu ~ Nchg/2 neutral pions in it. The neutral pions decay to two photons and hence look electromagnetic. Statistical fluctuations a can lead to Nchg being much smaller than the average, or the relative neutral and pion ratios changing radically in a small fraction of events. A small fraction of jets can either be intrinsically highly electromagnetic and electron like or have a single very energetic charged pion and little else. • Decay in flight - pions and kaons in the jet can decay in flight, mainly to muons. • Dalitz decays and Photon interactions in the detector. A n° can decay to two photons and one can interact in the detector to form an electron-positron pair. • Charm and B decay - a small fraction of jets contain a b or c quark which carries most of the jet energy, the heavy quark can decay semileptonically, producing a muon or electron which carries a large fraction of the jet energy. • Charge exchange reactions. A charged pion can turn into a neutral pion through a quasi-elastic charge exchange interaction in the calorimeter. This will look identical to an electron, with a charged particle pointing at an electromagnetic shower. a
Which are not Poisson in N due to charge and flavor correlations, but even broader.
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None of these are common - but when one has QCD rates that are thousands of times higher than the signal you are interested in, they become important.
5.2. Discussion
of
simulations
One way of studying and understanding the backgrounds is to simulate your process, the likely QCD backgrounds and the detector. I'm not going to talk about detector simulations and will concentrate on the methods used as inputs to the detector. These simulations take the process you are interested in and produce real particles. These simulations differ in two ways from an analytical calculation of a cross section. First, if you are going to do a detector simulation later on, your physics generator needs to generate real events with real 4-vectors, not just amplitudes or rates. And you get much faster convergence if all events have positive weights and preferably the same weight. Just generating events flat in phase space and then using your analytical calculation to assign a weight to each event is not going to work very well. Second, nobody knows how to do a perturbative QCD calculation that produces real particles like pions - it would have to be very high order and still could not handle the non-perturbative hadronization phase. Instead we rely on very useful codes such as PYTHIA 1 7 and HERWIG 1 9 which use parton showering or string models to convert partons into particles. These programs manage to simulate many of the interference effects you expect when you have colored particles radiating by using concepts such as angular ordering. Over the past 20 years, the parameters of the models have been tuned to match data from e~e+, lepton-hadron and hadron-hadron scattering.
6. Neutrino detection at colliders So far we've discussed the detection of electrons, photons and jets. Most interesting high px. processes at colliders involve neutrinos or other noninteracting particles (such has hypothetical Lightest Supersymmetric Particle or LSSP). But even high energy neutrinos have a probability of only 10~ 10 of interacting in the typical hadron collider detector. While they cannot be detected, some of their parameters can be estimated by calculating the missing p± in the event.
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As noted earlier, we do not know the longitudinal momentum of the scattering quarks but we do have a pretty good idea what their transverse momenta are, close to zero. In principle, if you can measure the transverse -> (*)
momentum of every scattered particle p± any non-interacting particle will be:
, the transverse momentum of
2L =-X>~l W
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i
The magnitude of this variable is very frequently, and incorrectly, referred to as the missing Transverse Energy, $± or MET. Which is reasonable in the case of a neutrino but just plain wrong in the case of a 70 GeV LSSP or two neutrinos. The missing transverse momentum estimate only works if the p± and direction of all scattered particles except the missing particle are detected. This requires a 'Hermetic' detector which covers almost all of 4n solid angle with active components. Such coverage is very difficult and expensive to achieve. Detectors such as D 0 and CDF have active calorimetry down to angles of order 3° from the beam axis or rapidities out to ±4 and are able to achieve ^ resolutions of order 5 GeV/c. However there is a potential for very large fluctuations in the missing momentum, for example if a jet fluctuates to be very electromagnetic in a calorimeter which responds differently to hadrons and electrons, exactly the kind of events which fake real electrons. For events with one missing neutrino from a semi-leptonic decay, such as W boson production, the neutrino reconstruction is almost unambiguous. This is illustrated in Figure 18. However, di-boson production, leptonic top decays and almost any supersymmetric signal have multiple neutral particles in the final state and the missing momentum method can only yield the sum of the missing particles transverse momenta.
7. Example 2: The top quark at the Tevatron 7.1. Standard
Model top
production
Pairs of top quarks are produced either by quark-antiquark annihilation or gluon-gluon fusion. At the Tevatron, the cross section for this process is believed to be around 7 pb 2 2 . Single top can also be produced by electroweak diagrams with the exchange of a W but this process has not yet been observed.
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Proton antiProton collision
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K Figure 18. Illustration of W production and decay. The first 3 frames show the longitudinal view. The initial state consists of two quarks with different momenta, so the hard scatter is moving relative to the lab frame. T h e W is produced with a small recoil and then decays to a neutrino and an electron. The inset frame shows the view along the beam axis. The neutrino's p j_ balances the transverse momenta of the recoil and the electron, but there is no information about the longitudinal component of the neutrino momentum.
The D 0 and CDF collaborations have recorded an integrated luminosity of over 400 p b - 1 each in run 1 and 2 combined. This means that the number of top-antitop pairs produced per experiment is:
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w+ or u,x
W" Figure 19. Top anti-top production followed by the decays t —> b W + , t —» bW~ one W decaying leptonically and the other hadronically.
with
However, at the same time, the total QCD cross section for inelastic scatters is 60 mb. Ninei = J Cdtainei = 60mb * 400pb _ 1 = 2.4 x 10 13
(17)
So only one in 10 10 interactions is a top event. If there are not additional quark species and the CKM matrix is unitary, top is believed to decay .997 of the time to b + W with the W then decaying to e, /z, r + v or to u, c -f- d, s, b. Figure 19 illustrates this process. Both kinds of W decay mode are problematic. In the case of leptonic decays, the neutrino leaves the detector without being detected while the hadronic modes look just like the 10 10 QCD background events. Figure 20 shows the different W^W^ - decay signatures predicted for Standard Model top decays. 7.2.
Backgrounds
Typical energies for the leptons and jets produced in top decay are - y 1 or 40-100 GeV and the total energy flowing transverse to the beam direction can be expected to be greater than 200 GeV. The major backgrounds to top production depend on the final state. For the all hadronic final state, the background is QCD going to 6 jets. For the semi-leptonic final states,
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W+W" decay signatures
TV+ X
21% Figure 20. Fraction of tt events with different W~^~W~ decay signatures, T decays are lumped together. Note that decay modes involving e or ji~ are a small fraction of the total.
the QCD/EW process W+jets mimics a top event, or one of the jets from a true QCD event can mimic an lepton and hence a W. The decays with two leptons have much lower backgrounds but are very rare and harder to reconstruct because they have two neutrinos in the final state. For now I am going to concentrate on the semi-leptonic decays which have the signature t —> b + Iv and t —> b + q1q2 (and vice versa). The experimental signature is 4 jets, two of which contain bs while the other two should reconstruct to close to the W mass. The other W decays to a lepton £ and missing transverse energy from the neutrino. One of the major technologies that has made top detection much easier is tagging of b-jets. QCD backgrounds mainly consist of jets initiated by
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X(cm) Figure 21. A top-antitop candidate from CDF viewed along the beam axis. Two separated vertices are found which indicate the presence of b-jets from top decay.
light quarks while top events should have 2 jets containing b quarks. Multilayer silicon vertex detector with resolutions in the 10-20 /jm range allow measurement of the decay length of long lived particles in jets, the presence of a long lived particle is a very strong signature for a b jet and hence a top quark. Figure 21 shows the region within 5 mm of a likely top interaction from CDF which shows two b tags. Figure 22 shows the transverse (xy) decay length distance L2 for a set of events from the CDF experiment which have passed all other top cuts. All of the backgrounds have lower L2 values than expected for top events. So we have several experimental handles, the large transverse energy, the presence of a high pj_ lepton, missing transverse momentum from the
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Figure 22. Distribution of the transverse decay length for vertices in b jet candidates from CDF compared to backgrounds.
neutrino and the b-jets. A combination of all of these can now get a very clean top signal. Figure 23 shows the number of jets found in CDF events with an identified leptonic W decay from and at least one tagged b jet. For total jet numbers of 3 and 4, the top signal becomes significant. The CDF collaboration have recently submitted their new measurement of the top cross section in the dilepton channel at 1960 GeV to PRL 2 3 . It is atl = 7.0 ± 2A{stat.) ± l.6{syst.) ± 0A(lum.)
7.3. Top
(18)
mass
The top mass is one of the most important parameters in the Standard model, because the top quark is heavy enough to have a significant influence on electroweak observables through virtual diagrams. These virtual
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diagrams lead to correlations between the masses of the W and t and the mass of the Higgs particle. New measurements of the top quark mass from DO's first run were recently published in Nature 25 and CDF have reported but not published results from their most recent running. The interesting part is that improved analysis techniques, better background rejection and calibration improvements have moved the central value to from 174 to 178 ± 4.5 GeV 24 . The effect of this changes is illustrated in Figure 24. The solid green line shows the combined W and top mass measurements from the Tevatron after addition of the more precise D 0 measurement. The dashed line shows the previous values. Figure 25 shows the effect of the change in the central value for the top mass on estimates of the Higgs mass. 7.4. Extraction
of the top
mass
The top mass can be measured with surprising precision - mainly because it is so high compared to the QCD scale of 1 GeV that strong interaction
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effects do not dominate, as they do in the determination of the other quark masses. The basic method goes as follows. We have a top event with one semileptonic and 1 hadronic decay. We know 5 momenta (those of the lepton, 2 jets from W decay and 2 jets from b-decay) and the transverse portion of the neutrino momentum. We also assume that the lepton and neutrino are massless and can estimate the 'masses' of the jets. We can now start applying constraints. First, the non-b jets come from a W and should have the appropriate mass.
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400 Higgs Boson Mass [GeV/c ] Figure 25. Predictions [24] for the Standard Model Higgs mass. The dark blue dashed parabola shows the new best results with the improved top mass while the black parabola shows the results previous to 2004. The best estimate for the Higgs mass has risen substantially from 96 to 117 G e V / c 2
Second, the lepton and neutrino should also reconstruct to the W mass. Here X\\ is the unknown longitudinal component of the neutrino momentum. (19) m2w = {Ee + Eu)2 - ^ l , e p t o n + ^ ± »? ~ (P\\e +
X
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This is a quadratic equation with two solutions for the neutrino mo mentum, and hence the W momentum.
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We can then impose the remaining constraint, that the masses of the two tops must be equal and extract a top mass. If detectors were perfect, that would be it but generally, top events have only 1 or zero tagged b jets and the charge of the b jet is not known so it cannot be automatically associated with the W+ or W~. In general 12-14 combinations of objects must be considered and their consistency with the top anti-top hypothesis evaluated. In addition, b decays are weak decays and very likely to include neutrinos, which makes the energy determination for b jets different than that for ordinary jets.
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Unfortunately, there are also a lot of background events out there as well which cannot be arbitrarily thrown out of the event sample. The D 0 collaboration very recently published 25 a determination of the top mass using likelihood methods. These methods have an advantage over the techniques used in previous analyses as configurations and events with smaller errors are given greater weight. The data sample used is from the previous run, where vertex tagging was not available to clean up the data sample or indicate which jets contained b quarks. Each combination of particles in each event had its log likelihood of being top or background calculated by comparison to a theoretical model as a function of the top mass. Figure 26 shows this distribution for a single event with a high
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likelihood for being top. This plot is the sum over all of the combinations. is almost certainly background. Figure 28 shows the log likelihood distribution for the ensemble of 71 events and the variation of likelihood as a function of the assumed mass near the peak. The optimal value after correction is mt = 180.1 ± 5.3 GeV/c 2 . This value is expected to improve greatly once the run II data are well understood as both D 0 and CDF have larger statistics and much better
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b tagging. The dominant error remains the hadronic energy scale, which should also improve with better top signals as the W bosons in hadronic top decays are indeed the pure calibration line for hadronic energy scales we've needed for a decade. 8. Example 3: looking for the Higgs at Hadron colliders or where is it anyways? 8.1. Standard
Model Higgs production
and
decay
The Higgs is an excellent example of the difference between production and detection of rare signals. Because the Higgs couples to mass, Standard Model Higgs production generally involves t, W, Z either through loops or direct production. At the Tevatron, low mass Higgs bosons are produced via GG —> top loop —> H (Figure 29) and associated production (Figure 30 in which a quark and anti-quark produce a W* which decays to WH. Figure 31 illustrates the relative production rates for these processes 27 ' 26 at the Tevatron. The Higgs also likes to decay into the highest mass particles possible. Figure 32 shows the decay modes for low mass Higgs, below WW threshold, the Higgs has to decay to bb quarks. So for a Higgs just above the current limit of 114 GeV set by LEP, one would expect a Standard Model cross section for the process qq —> H —> bb of 1 picobarn. During its best week in 2004, the Tevatron recorded over 20 inverse picobarns of data - or enough to produce 20 115 GeV Higgs going into bb. Even if the Higgs mass were 200 GeV, the cross section for GG —» toploop —> H —> WW is still over 0.1
Figure 29.
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picobarn so one would expect to see 2 or 3 in a good week. So why haven't you heard of the discovery or a new limit? The reason is backgrounds, both due to particle misidentification and real physics processes which have the same final state. In this case, it's likely that real physics is the problem. The total cross section for producing b or b quarks at the Tevatron was recently measured by the CDF collaboration 30 in the central (rapidity < 1) region, it is around 25 micro barns, or around 1/3,000 of the total proton anti-proton cross section. The cross section for producing bb is 1/2 that, but both CDF and D 0 use twice the rapidity range for Higgs analysis as they did for this measurement, so the two factors
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cancel. This means, at typical luminosities, that 300-500 bb pairs are being produced per second, or 10,000,000 times the rate of H —> bb Most such b pairs are highly correlated and have a low invariant mass, but even 1 in a million is enough to swamp the Higgs signal. 8.2. Triggers
and
detection
There are also instrumental problems in detecting the bb channel. Over 1 million proton-anti-proton interactions occur per second at the Tevatron, but only around 100 can be recorded due to limits on CPU and bandwidth. (Even so the Tevatron fills tapes faster than Fox News does). The rest of the events are eliminated by a multi-level trigger, which tries to distinguish interesting physics (in this case 2 b quarks) from backgrounds (light quark QCD). CDF has a B physics trigger which relies on detection of the B decay length in the silicon tracker. The D 0 B triggers rely more on the detection of muons in semi-leptonic B decay. Figure 33 illustrates the flow of data in the CDF trigger 31 . The Level 1 trigger makes a decision about the usefulness of an event (high energy, has a muon) in 5.5 microseconds. That crude decision takes the raw rate of around 1.5 Million events/second down to around 25,000/second b . A Level 2 trigger, which does fast tracking in the silicon detector and detects the separated vertex, reduces the data by a further factor of 100 to around 300/second which are then reduced to 100/second by running a full reconstruction program in Level 3. At each stage, real 6's are lost and fake ones can slip through. One can estimate the probability of a simple bb event with a large invariant mass surviving by considering the process qq —> Z —> bb which has a cross section of around 15 nanobarns, about half way between the raw bb cross section and the Higgs. Figure 34 shows a previous analysis of Z —> bb from the CDF experiment . This was done with an earlier, less powerful, version of the new trigger and vertex detectors. The Z° is the slight enhancement above background on the falling edge of the background. The signal was 91 ± 30 ± 19 events over a background of 250 observed in 110 inverse picobarns of data. This implies that the cross section for observing Z —> bb was around 1 picobarn, where the cross section for producing Z —* bb is around 1.5 nanobarns. CDF were only able to detect and identify 1/1,500 of the events with very 30
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large backgrounds. One should contrast this with the process Z —> ee which has 1/5 the production cross section. 3,000 events with less than 1% background were observed in a data sample of similar size. For Z —> ee the detection probability is of order 10% after triggering and holes in the detector are taken into account. Since those data were taken, CDF has added the separated vertex trigger, which should raise the Z —> bb signal substantially. But even with a good trigger, they won't see the 1 Higgs/week from GG —> bb the Standard Model suggests in our detectors.
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8.3. The
solution
There is a solution, look for rarer final states such as Wbb from associated production which have much lower backgrounds and higher trigger efficiencies. In particular, the presence of the W eliminates the need to trigger on a b quark. Figure 35 shows the data and background/signal sources for Wbb production at the D 0 detector which was recently submitted to PRL 33 . The dominant backgrounds are real W+ continuum bb and top production. The limit set on Higgs production is around 10 times the expected Standard
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Model cross section. The Tevatron Higgs Sensitivity study 26 done in 2003, indicated that integrated luminosities of 5,000-10,000 inverse picobarns will be necessary to see a Standard Model Higgs at the Tevatron. 9. Conclusions I've discussed how particle detectors work at hadron colliders and the signatures for old and new physics. I've emphasized the problems in extracting rare signals from a very large background. I'll end with some advice for the aspiring theorist who wishes to have his/her ideas tested in the next 20 years. • Rate alone cannot guarantee that a process you predict will be detectable. • The key is special signatures - final state electrons, muons, taus, heavy quarks, which are harder for QCD processes to mimic.
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H. Schellman • To find truly rare processes, you probably need to use multiple special signatures. • Get experimentalists interested in your physics - so they don't throw it out at Level 1 in their trigger.
References 1. T. Sjostrand, P. Eden, C. Friberg, L. Lonnblad, G. Miu, S. Mrenna and E. Norrbin, Computer Physics Commun. 135 (2001) 238. and hep-ph/0108264. 2. http://www.phys.psu.edU/wcteq/handbook/vl.l/handbook.pdf 3. R. Schwitters, "Jet Production in High Energy Hadron Collisions", Proceedings of the 11th Slac Summer Institute, P. McDonough editor, SLAC Report 267, 1984. 4. Review of Particle Properties - K. Hagiwara et al., Phys. Rev. D66, 010001 (2002) 5. R. M. Snihur, "Subjet Multiplicity Of Quark And Gluon Jets Reconstructed With The K(T) Algorithm In P Anti-P Collisions,", Northwestern University Doctoral Dissertation, FERMILAB-THESIS-2000-25, 6. D. Denegri, Standard Model physics at the LHC (pp collisions), 27 November 1990, Proceedings of the Large Hadron Collider Workshop, Aachen, 4-9 October 1990, Vol 1,11,111, CERN 90-10. 7. Jay Dittman, Talk given at the Jet/Met workshop at Fermilab, January 28, 2004. http://agenda.cern.ch/fullAgenda.php?ida=a04329 . 8. B. Abbott et al. [DO Collaboration], "Inclusive jet production in p antip collisions," Phys. Rev. Lett. 86, 1707 (2001) [arXiv:hep-ex/0011036], V. M. Abazov et al. [DO Collaboration], "Multiple jet production at low transverse energies in p anti-p collisions at s**(l/2) = 1.8-TeV," Phys. Rev. D 67, 052001 (2003) [arXiv:hep-ex/0207046] 9. J. C. Estrada Vigil, "Maximal Use Of Kinematic Information For The Extraction Of The Mass Of The Top Quark In Single-Lepton T Anti-T Events At DO," Univ. of Rochester Doctoral Dissertation, FERMILAB-THESIS2001-07, 10. M. F. Canelli, "Helicity of the W boson in single-lepton t anti-t events," FERMILAB-THESIS-2003-22 11. B. Abbott et al. [DO Collaboration], "Determination of the absolute jet energy scale in the DO calorimeters," Nucl. Instrum. Meth. A 424, 352 (1999) [arXiv:hep-ex/9805009]. 12. L. R. Babukhadia, "Rapidity Dependence Of The Single Inclusive Jet CrossSection In Proton - Anti-Proton Collisions At The Center-Of-Mass Energy Of 18-Tev With The DO Detector,", University of Arizona Doctoral Dissertation, FERMILAB-THESIS-1999-03, 13. T. Affolder et al. [CDF Collaboration], Phys. Rev. D 64, 032001 (2001) [Erratum-ibid. D 65, 039903 (2002)] [arXiv:hep-ph/0102074]. 14. S. Frixione and B. R. Webber, "Matching NLO QCD computations and parton shower simulations," JHEP 0206, 029 (2002) [arXiv:hep-
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15.
16.
17.
18.
19.
20. 21. 22.
23.
24. 25. 26.
27. 28.
29. 30.
ph/0204244].S. Prixione and B. R. Webber, "The MC@NLO 2.3 event generator," arXiv:hep-ph/0402116. M. Kramer and D. E. Soper, "Next-to-leading order QCD calculations with parton showers. I: Collinear singularities," Phys. Rev. D 69, 054019 (2004) [arXiv:hep-ph/0306222]. S. Mrenna and P. Richardson, "Matching matrix elements and parton showers with HERWIG and PYTHIA," JHEP 0405, 040 (2004) [arXiv:hepph/0312274]. T. Sjostrand, L. Lonnblad, S. Mrenna and P. Skands, "PYTHIA 6.3: Physics and manual," arXiv:hep-ph/0308153. T. Sjostrand, P. Eden, C. Friberg, L. Lonnblad, G. Miu, S. Mrenna and E. Norrbin, "High-energy-physics event generation with PYTHIA 6.1," Comput. Phys. Commun. 135, 238 (2001) [arXiv:hep-ph/0010017]. T. Affolder et al. [CDF Collaboration], "Charged particle multiplicity in jets in p anti-p collisions at s**(l/2) = 1.8-TeV," Phys. Rev. Lett. 87, 211804 (2001). G. Corcella et al, "HERWIG 6: An event generator for hadron emission reactions with interfering JHEP 0101, 010 (2001) [arXiv:hep-ph/0011363]. G. Marchesini, B.R. Webber, G. Abbiendi, I.G. Knowles, M.H. Seymour and L. Stanco, Computer Phys. Commun. 67 (1992) 465. Jason Nielsen, talk at the 2004 Lake Louise Winter Institute; http://www-cdf.fnal.gov/physics/top/RunIIWjets/webpages/ secvtx/public.html M. Cacciari, S. Frixione, M. L. Mangano, P. Nason and G. Ridolfi, "The t anti-t cross-section at 1.8-TeV and 1.96-TeV: A study of the systematics due to parton densities and scale dependence," JHEP 0404, 068 (2004) [arXiv:hep-ph/0303085]. D. Acosta et al. [CDF Collaboration], "Measurement of the t anti-t production cross section in p anti-p collisions at s**(l/2) = 1.96-TeV using dilepton events," arXiv:hep-ex/0404036. Lep Electroweak Working group and Tevatron Electroweak working group, http://lepewwg.web.cern.ch/LEPEWWG/, http://tevewwg.fnal.gov/i V. M. Abazov et al. [DO Collaboration], "A precision measurement of the mass of the top quark," Nature 429, 638 (2004). L. Babukhadia et al. [CDF and DO Working Group Members Collaboration], "Results of the Tevatron Higgs sensitivity study," FERMILAB-PUB-03-320E T. Han and S. Willenbrock, "QCD correction to the p p -> W H and Z H total cross-sections," Phys. Lett. B 273, 167 (1991). A. Djouadi, J. Kalinowski and M. Spira, "HDECAY: A program for Higgs boson decays in the standard model and its supersymmetric extension," Comput. Phys. Commun. 108, 56 (1998) [arXiv:hep-ph/9704448]. M. Spira, "Higgs boson production and decay at the Tevatron," arXiv:hepph/9810289. P. J. Bussey [CDF Collaboration], "Charm and beauty production at CDF," arXiv:hep-ex/0408020. M. Cacciari, S. Frixione, M. L. Mangano, P. Nason
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and G. Ridolfi, "QCD analysis of first b cross section data at 1.96-TeV," JHEP 0407, 033 (2004) [arXiv:hep-ph/0312132]. 31. S. Baroiant et al. [CDF-II Collaboration], "The CDF-II tau physics program: Triggers, tau ID and preliminary results," arXiv:hep-ex/0312020. 32. T. Dorigo [CDF collaboration], "Observation of Z decays to b quark pairs at the Tevatron collider," arXiv:hep-ex/9806022. 33. V. M. Abazov et al. [DO Collaboration], arXiv:hep-ex/0410062.
TAO HAN
COLLIDER PHENOMENOLOGY: BASIC KNOWLEDGE A N D TECHNIQUES*
TAO HAN Department of Physics, 1150 University University of Wisconsin Madison, WI 53706, USA than@physics. wise, edu
Avenue
These lectures are meant to provide an introductory presentation on the basic knowledge and techniques for collider physics. Special efforts have been made for those theorists who need to know some experimental issues in collider environments, and for those experimenters who would like to know more about theoretical considerations in searching for new signals at colliders.
Contents 1
Introduction
408
2
High energy colliders: Our powerful tools 2.1 Collider parameters 2.2 e+e~ colliders 2.2.1 Production cross sections for standard model processes 2.2.2 Resonant production 2.2.3 Effective photon approximation 2.2.4 Beam polarization 2.3 Hadron colliders 2.3.1 Hard scattering of partons 2.3.2 Production cross sections for standard model processes
410 410 413 414 416 417 418 419 420 422
3
Collider detectors: Our electronic eyes 3.1 Particle detector at colliders
423 423
*I will maintain an updated version of these lectures at http://pheno.physics.wisc.edu/ M,han/collider-2005.pdf. 407
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3.2 3.3 3.4 4
What do particles look like in a detector More on measurements Triggering
Uncover new dynamics at colliders 4.1 Kinematics at hadron colliders 4.2 s-channel singularity: Resonance signals 4.2.1 The invariant mass variable 4.2.2 The transverse mass variable 4.2.3 The cluster transverse mass variable 4.3 i-channel enhancement: Vector boson fusion 4.4 Forward-backward asymmetry 4.5 Be prepared for more involved inclusive signatures
423 426 429 430 431 433 433 436 438 439 441 443
A Relativistic kinematics and phase space A.l Relativistic kinematics A.2 One-particle final state A.3 Two-body kinematics A.4 Three-body kinematics A.5 Recursion relation for the phase space element
445 445 446 447 448 450
B Breit-Wigner resonance and the narrow w i d t h approximation
451
1. I n t r o d u c t i o n For the past several decades, high energy accelerators and colliders have been our primary tools for discovering new particles and for testing our theory of fundamental interactions. W i t h the expectation of the Large Hadron Collider (LHC) in mission in 2007, and the escalated preparation for the International Linear Collider (ILC), we will be fully exploring the physics at the electroweak scale and beyond the s t a n d a r d model (SM) of the strong and electroweak interactions in the next twenty years. New exciting discoveries are highly anticipated t h a t will shed light on the mechanism for electroweak symmetry breaking, fermion mass generation and their mixings, on new fundamental symmetries such as Supersymmetry (SUSY) and grand unification of forces (GUTs), even on probing the existence of extra spatial dimensions or low-scale string effects, and on related cosmological implications such as particle dark matter, baryon and C P asymmetries of the Universe, a n d dark energy as well. Collider phenomenology plays a pivotal role in building the bridge between theory and experiments. On the one hand, one would like to decode the theoretical models and to exhibit their experimentally observable consequences. On the other hand, one needs to interpret the d a t a from exper-
Collider Phenomenology:
Basic Knowledge and Techniques
409
iments and to understand their profound implications. Phenomenologists working in this exciting era would naturally need to acquaint both fields, the more the better. These lectures are aimed for particle physicists who need to know the basics in collider phenomenology, both experimental issues and theoretical approaches. Special efforts have been made for those theorists who need to know some realistic experimental issues at high-energy collider environments, and those experimenters who would like to know more about theoretical considerations in searching for generic new signals. In preparing these lectures, I had set up a humble goal. I would not advocate a specific theoretical model currently popular or of my favorite; nor would I summarize the "new physics reach" in a model-dependent parameter space at the LHC and ILC; nor would I get into a sophisticated level of experimental simulations of detector effects. The goal of these lectures is to present the basic knowledge in collider physics including experimental concerns, and to discuss generic techniques for collider phenomenology hopefully in a pedagogical manner. In Sec. 2, we first present basic collider parameters relevant to our future phenomenological considerations. We then separately discuss e+e~ linear colliders and hadron colliders for the calculational framework, and for physics expectations within the SM. In Sec. 3, we discuss issues for particle detection — what do the elementary particles in the SM theory look like in a realistic detector? — which are necessary knowledge but have been often overlooked by theory students. We also illustrate what parameters of a detector and what measurements should be important for a phenomenologist to pay attention to. Somewhat more theoretical topics are presented in Sec. 4, where I emphasize a few important kinematical observables and suggest how to develop your own skills to uncover fundamental dynamics from experimentally accessible kinematics. If I had more time to lecture or to write, this would be the section that I'd like to significantly expand. Some useful technical details are listed in a few Appendices. The readers are supposed to be familiar with the standard model of the strong and electroweak interactions, for which I refer to Scott Willenbrock's lectures [1] and some standard texts [2]. I also casually touch upon topics in theories such as SUSY, extra dimensions, and new electroweak symmetry breaking scenarios, for which I refer the readers to the lectures by Howie Haber [3,4] on SUSY, and some recent texts [5], Raman Sundrum [6] and Csaba Csaki [7] on physics with extra-dimensions. For more extensive experimental issues, I refer to Heidi Schellman's lectures [8]. The breath and
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depth covered in these lectures are obviously very limited. For the readers who need more theoretical knowledge on collider phenomenology, there are excellent text books [9,10] as references. As for experimental issues, one may find a text [11] very useful, or consult with the Technical Design Reports (TDR) from various detector collaborations [12-14]. 2. High energy colliders: Our powerful tools 2.1. Collider
parameters
In the collisions of two particles of masses mi andTO2and momenta pi and p-2, the total energy squared in the center-of-momentum frame (cm.) can be expressed in terms of a Lorentz-invariant Mandelstam variable (for more details, see Appendix A) s = {p1+p2)2=\K
(Ei + E2)2 2
in the c m . frame pi +p~2 = 0,
2
TO? + ra\ + 2{ElE2 - pi • pi).
In high energy collisions of our current interest, the beam particles are ultrarelativistic and the momenta are typically much larger than their masses. The total c m . energy of the two-particle system can thus be approximated as _ ~
i- ^ { 2E\ fa 2E2 in the c m . frame pi + P2 = 0, ~ \ \J2E\m2 in the fixed target frame p 2 = 0.
while the kinetic energy of the system is T « E\ in the fixed-target frame p~2 = 0, and T = 0 in the c m . frame pi + p*2 = 0. We see that only in the c m . frame, there will be no kinetic motion of the system, and the beam energies are maximumly converted to reach a higher threshold. This is the designing principle for colliders like LEP I, LEP II and LHC at CERN; the SLC at SLAC; and the Tevatron at the Fermi National Accelerator Laboraroty. Their c m . energies are listed in Tables 1 and 2, respectively. The limiting factor to the collider energy is the energy loss during the acceleration, known as the synchrotron radiation. For a circular machine of radius R, the energy loss per revolution is [4,11]
where E is beam energy,TOthe particle mass (thus E/m is the relativistic 7 factor). It becomes clear that an accelerator is more efficient for a larger radius or a more massive particle.
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Table 1. Some e + e ~ colliders and their important parameters: c m . energy, instantaneous peak luminosity, relative beam energy spread, bunch crossing frequency, longitudinal beam polarization, and the total length of the collider. The parameters are mainly from PDG [4], ILC working group reports [14], and a recent CLIC report [15]. Colliders
5E/E
/
^ i (GeV) (GeV)
C (cm~2s-1)
LEP I SLC LEP II
Mz ~ 100 ~210
2.4 x 10 3 1 2.5 x 10 3 0 10 32
~ 0.1% 0.12% ~ 0.1%
45 0.12 45
ILC CLIC
(TeV) 0.5-1 3-5
2.5 x 10 3 4 ~ 10 3 5
0.1% 0.35%
(MHz) 3 1500
polar.
L (km)
55% 80%
26.7 2.9 26.7
80, 60% 80, 60%
14-33 33-53
(kHz)
Table 2. Some hadron colliders and their important parameters [4]: c m . energy, instantaneous peak luminosity, relative beam energy spread, bunch crossing frequency, number of particles per bunch, and the total length of the collider. For reference, the cancelled SSC and a recently discussed future VLHC [16] are also listed. Colliders
v^ (TeV)
C (cm-2s-1)
5E/E
/ (MHz)
#/bunch (10 1 0 )
L (km)
Tevatron HERA
1.96 0.314
9 x 10"5 0.1,0.02%
2.5 10
p: 27, p: 7.5 e: 3, p: 7
6.28 6.34
LHC SSC VLHC
14 40 40-170
2.1 x 10 3 2 1.4 x 10 3 1 10 3 4 10 3 3 2 x 10 3 4
0.01% 5.5 x 1 0 " 5 4.4 x 1 0 - 4
40 60 53
10.5 0.8 2.6
26.66 87 233
In e+e~ annihilations, the c m . energy may be fully converted into reaching the physics threshold. In hadronic collisions, only a fraction of the total c m . energy is carried by the fundamental degrees of freedom, the quarks and gluons (called partons). For instance, the Tevatron, with the highest c m . energy available today, may reach an effective parton-level energy of a few hundred GeV; while the LHC will enhance it to multi-TeV. Another important parameter for a collider is the instantaneous luminosity, the number of particles passing each other per unit time through unit transverse area at the interaction point. In reality, the particle beams usually come in bunches, as roughly illustrated in Figure 1. If there are n\ particles in each bunch in beam 1 and n-i in each bunch in beam 2, then the collider luminosity scales as C oc
fniri2/a,
(2)
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T. Han
Colliding beam "l .•:;•)
c ° ; .
"2
<•;;•; —»•
• « — o>s;so
t°
?=1// Figure 1.
Colliding beams with a bunch crossing frequency / .
where / is beam crossing frequency and a the transverse profile of the beams. The instantaneous luminosity is usually given in units of c m - 2 s _ 1 . The reaction rate, that is the number of scattering events per unit time, is directly proportional to the luminosity and is given by a R(s) = a(s)C,
(3)
where <j(s) is defined to be the total scattering cross section. Though the units of cross sections are conventionally taken as cm 2 , these units are much too big to use for sub-atomic particle scattering, and thus more suitable units, called a barn, are introduced 1 cm2 = 1024 barn = 10 27 mb = 10 30 fib = 10 33 nb = 1036 pb = 10 39 fb = 1042 ab. It may also be convenient to use these units for the luminosity accordingly like 1 cm" 2 s" 1 = 10" 3 3 n b - 1 s _ 1 . In fact, it is often quite relevant to ask a year long accumulation of the luminosity, or an integrated luminosity over time. It is therefore useful to remember a collider's luminosity in the units' 3 10 33 c m - 2 s" 1 = 1 n b - 1 s _ 1 w 10 fb _ 1 /year. In practice, the instantaneous luminosity has some spread around the peak energy \fs, written as dL/dr with T = s/s where s is the c m . energy squared with which the reaction actually occurs. The more general form for Eq. (3) is R(s) = £ I dr— a
a(s).
(4)
T h e r e will be another factor e < 1 on the right-hand side, which represents the detection efficiency. b Approximately 1 year « 7r x 10 7 s. It is common that a collider only operates about 1/7T of the time a year, so it is customary to take 1 year —• 10 7 s.
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With the normalization J dL/dr dr = 1, then C is the peak instantaneous luminosity. The energy spectrum of the luminosity often can be parameterized by a Gaussian distribution with an energy spread as given by 5E (~ y/s — %/I) in Tables 1 and 2. For most of the purposes, the energy spread is much smaller than other energy scales of interest, so that the luminosity spectrum is well approximated by 5(1 — T). Thus, Eq. (3) is valid after the proper convolution. The only exception would be for resonant production with a physical width narrower than the energy spread. We will discuss this case briefly in the next e+e" collider section. While the luminosity is a machine characteristics, the cross section is determined by the fundamental interaction properties of the particles in the initial and final states. Determining the reaction cross section and studying the scattering properties as a function of energy, momentum, and angular variables will be of ultimate importance to uncover new dynamics at higher energy thresholds. The electrons and protons are good candidates for serving as the colliding beams. They are electrically charged so that they can be accelerated by electric field, and are stable so that they can be put in a storage ring for reuse to increase luminosity. In Table 1, we list the important machine parameters for some e+e~ colliders as well as some future machines. In Table 2, we list the important machine parameters for some colliders. Electron and proton colliders are complementary in many aspects for physics exploration, as we will discuss below. 2.2. e + e -
colliders
The collisions between electrons and positrons have some major advantages. For instance, • The e + e~ interaction is well understood within the standard model electroweak theory. The SM processes are predictable without large uncertainties, and the total event rates and shapes are easily manageable in the collider environments. • The system of an electron and a positron has zero charge, zero lepton number etc., so that it is suitable to create new particles after e + e~ annihilation. • With symmetric beams between the electrons and positrons, the laboratory frame is the same as the c m . frame, so that the total c m . energy is fully exploited to reach the highest possible physics threshold.
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T. Han
• With well-understood beam properties, the scattering kinematics is well-constrained. • It is possible to achieve high degrees of beam polarizations, so that chiral couplings and other asymmetries can be effectively explored. One disadvantage is the limiting factor due to the large synchrotron radiation as given in Eq. (1). The rather light mass of the electrons limits the available c m . energy for an e+e~ collider. Also, a multi-hundred GeV e+e~ collider will have to be made a linear accelerator [14]. This in turn becomes a major challenge for achieving a high luminosity when a storage ring is not utilized. When performing realistic simulations for high energy e + e~, e~e~~ reactions at high luminosities, the beamstrahlung effects on the luminosity and the c m . energy become substantial and should not be overlooked. Another disadvantage for e+e~~ collisions is that they predominantly couple to a vector (spin 1) state in s-channel, so that the resonant production of a spin-0 state (Higgs-like) is highly suppressed. For a higher spin state, such as spin-2, the resonant production will have to go through a higher partial wave. 2.2.1. Production cross sections for standard model processes For the production of two-particle a, b and for unpolarized beams so that the azimuthal angle can be trivially integrated out (see Appendix A.3), the differential cross section as a function of the scattering polar angle in the c m . frame is given by
where /? = A 1 / 2 (l,m 2 a /s,m1/s) is the speed factor for the out-going particles, 131-MI2 i s t n e scattering matrix element squared, summed and averaged over unobserved quantum numbers like color and spins. It is quite common that one needs to consider a fermion pair production + e~e —> / / . For most of the situations, the scattering matrix element can be casted into a V ± A chiral structure of the form (sometimes with the help of Fierz transformations) M = -Qa0 s
[ve+(P2h>1Paue-{Pl)}
0/(«ih/W/(<72)],
(6)
'
where a,/3 = L,R are the chiral indices, PLR = (1 T 7s)/2, and Qap are the chiral bilinear couplings governed by the underlying physics of the
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Basic Knowledge and Techniques
415
T. Han 1 0 °
fc
l
i i i I i i i I I i i i i
I i i I '
I i i i i ;
SM processes at LC
b
1(F
0
200
400
600
Vs
(GeV)
800
1000
Figure 2. Scattering cross sections versus c m . energy for the SM processes in e + e _ collisioins. The Higgs boson mass has been taken as 120 GeV.
interactions with the intermediate propagating fields. With this structure, the scattering matrix element squared can be conveniently expressed as
J2\M\2 = ^ [(\QLL\2 + \QRR\2) uiUj + (\QLR\2 + \QRL\2) Utj + 2Re(Q*LLQLR + QRRQRL)mfmfs}
,
(7)
where U = t - mf = (pi - qi)2 - m2 and w, = u - m2 = (pi - q2)2 - m2. Exercise: Eq. ( 6 ) .
Derive Eq. (7) by e x p l i c i t c a l c u l a t i o n s from
Figure 2 shows the cross sections for various SM processes in e+e~ collisions versus the c m . energies. The simplest reaction is the QED process
416 T. Han e+e
__> -y* _» /U+^t
anc[
it s cross section is given by
/ + * + -\ 47ra2 + + a(e e -> 7 -> /i /x ) = <rpt = — — .
.. (8)
OS
In fact, <7P£ « 100 fb/(-^/i/TeV) 2 has become standard units to measure the size of cross sections in e+e~ collisions. However, at energies near the EW scale, the SM Z boson resonant production dominates the cross section, seen as the sharp peek slightly below 100 GeV. Above the resonance, cross sections scale asymptotically as 1/s, like the s-channel processes typically do. This is even true for scattering with t, u-channel diagrams at finite scattering angles. The only exceptions are the processes induced by collinear radiations of gauge bosons off fermions, where the total cross section receives a logarithmic enhancement over the fermion energy. For a massive gauge boson fusion process,
a
(9)
~ik^W
To have a quantitative feeling, we should know the sizes of typical cross sections at a 500 GeV ILC a(W+W~)
« 20apt w 8 pb;
a(ZZ) « a(tt) « apt « 400 fb; a(ZH)
W a(WW
a(WWZ)
->H)KS
apt/4 « 100 fb;
« 0.1 apt w 40 fb.
Now let us treat these two important cases in more details. 2.2.2. Resonant production The resonant production for a single particle of mass My, total width IV, and spin j at c m . energy y/s is a{e
e
->V^x)
=
{s-MlY
+ TlMl
M£'
(10)
where r ( V —> e+e~) and T(V —> X) are the partial decay widths for V to decay to the initial and final sates, respectively. This is the BreitWigner resonance to be discussed in Eq. (B.l) of Appendix B. For an ideal monochromatic luminosity spectrum, or the energy spread of the machine much smaller than the physical width IV, the above equation is valid. This is how the Z resonant production cross section was calculated in Fig. 2 as
Collider Phenomenology:
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a function of the c m . energy yfs, and how the Z line-shape was measured by the energy-scan in the LEP I and SLC experiments. E x e r c i s e : Verify Eq. (10) by assuming a generic v e c t o r production and decay. It can occur that the energy spectrum of the luminosity is broader than the narrow resonant width. One could take the narrow-width approximation as given in Eq. (B.l) and thus the cross section is a(e+e-
-* V -* X) =
4TT 2 (2J
+ l ) r ( V -> e+e-)BF(V MlL v
-> X) dL_ dr l s = M v
(11) where dL/dr\s=M2 presents the contribution of the luminosity at the resonant mass region . In other complicated cases when neither approximation applies between 6E and IV, the more general convolution as in Eq. (4) may be needed. For a discussion, see e.g. Ref. 17. E x e r c i s e : Derive Eq. (11) by applying Eq. (4) with the narrow width approximation as i n Eq. ( B . l ) . The resonant production is related to the s-channel singularity in the 5-matrix for an on-shell particle propagation. It is the most important mechanism for discovering new particles in high energy collider experiments. We will explore in great detail the kinematical features in Sec. 4. 2.2.3. Effective photon approximation A qualitatively different process is initiated from gauge boson radiation, typically off fermions. The simplest case is the photon radiation off an electron. For an electron of energy E, the probability of finding a collinear photon of energy xE is given by „
p
,
N
a 1 + (1 - x)2 ,
E2
w*)=^-V- Lln ^'
(12)
which is known as the Weizsacker-Williams spectrum. We see that the electron mass enters the log to regularize the collinear singularity and \/x leads to the infrared behavior of the photon. These dominant features are a result of a ^-channel singularity for the photon. This distribution can be obtained by calculating the splitting process as depicted in Fig. 3, for
418
T. Han
Figure 3.
Illustrative Feynman diagram for effective photon approximation.
e a —> e X. The dominant contribution is induced by the coUinear photon and thus can be expressed as a{e~a -> e~X) w / dx P^/e{x) cr(^a -> X).
(13)
This is also called the effective photon approximation. This picture of the photon probability distribution in Eq. (13) essentially treats the photons as initial state to induce the reaction. It is also valid for other photon spectrum. It has been proposed recently to produce much harder photon spectrum based on the back-scattering laser techniques [18] to construct a "photon collider". There have been dedicated workshops to study the physics opportunities for e + e~ linear colliders operating in such a photon collider mode, but we will not discuss the details further here. A similar picture may be envisioned for the radiation of massive gauge bosons off the energetic fermions, for example the electroweak gauge bosons V = W±,Z. This is often called the Effective W-Approximation [19,20]. Although the coUinear radiation would not be a good approximation until reaching very high energies y/s 3> My, it is instructive to consider the qualitative features, which we will defer to Sec. 4.3 for detailed discussions. 2.2.4. Beam polarization One of the merits for an e + e~ linear collider is the possible high polarization for both beams, as indicated in Table 1. Consider first the longitudinal polarization along the beam line direction. Denote the average e± beam polarization by P±, with P± = - 1 purely left-handed and + 1 purely right-
Collider Phenomenology:
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419
handed. Then the polarized squared matrix element can be constructed [21] based on the helicity amplitudes Mae_ae+ ^2\M\2 = \[(l-
P-)(l ~ P+)\M-\2
+ (1 - ^ f )(1 +
P+)\M-+\2
+(1 + P f ( l - P ^ ) | M + - | 2 + (1 + P f )(1 + p £ ) | M + + | 2 ] . (14) Since the electroweak interactions of the SM and beyond are chiral, it is important to notice that contributions from certain helicity amplitudes can be suppressed or enhanced by properly choosing the beam polarizations. Furthermore, it is even possible to produce transversely polarized beams with the help of a spin-rotator. If the beams present average polarizations with respect to a specific direction perpendicular to the beam line direction, - 1 < P£ < 1, then there will be one additional term in Eq. (14) (in the limit me —> 0), ^ 2 PlP$
Re(M-+M*+_).
The transverse polarization is particularly important when the interactions under consideration produce an asymmetry in azimuthal angle, such as the effect of CP violation. For a comprehensive count on physics potential for the beam polarization, see a recent study in Ref. 22. 2.3. Hadron
colliders
Protons are composite particles, made of "partons" of quark and gluons. The quarks and gluons are the fundamental degrees of freedom to participate in strong reactions at high energies according to QCD [23]. The proton is much heavier than the electron. These lead to important differences between a hadron collider and an e+e~ collider. • Due to the heavier mass of the proton, hadron colliders can provide much higher c m . energies in head-on collisions. • Higher luminosity can be achieved also, by making use of the storage ring for recycle of protons and antiprotons. • Protons participate in strong interactions and thus hadronic reactions yield large cross sections. The total cross section for a proton-proton scattering can be estimated by dimensional analysis to be about 100 mb, with weak energy-dependence.
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T. Han
'Hard" Scattering outgoing parton
proton underlying event •*
*~~\ A t - - ^
*• underlying event
outgoing parton Figure 4.
An illustrative event in hadronic collisions.
• At higher energies, there are many possible channels open up resonant productions for different charge and spin states, induced by the initial parton combinations such as qq, qg, and gg. As discussed in the last section for gauge boson radiation, there are also contributions like initial state WW, ZZ and WZ fusion. The compositeness and the strong interactions of the protons on the other hand can be disadvantageous in certain aspect, as we will see soon. An interesting event for a high-energy hadronic scattering may be illustrated by Fig. 4. 2.3.1. Hard scattering of partons Thanks to the QCD factorization theorem, which states that the cross sections for high energy hadronic reactions with a large momentum transfer can be factorized into a parton-level "hard scattering" convoluted with the parton "distribution functions". For scattering of two hadrons A and B to produce a final state F of our interest, the cross section can be formally written as a sum over the sub-process cross sections from the contributing partons a{AB-*F
X) = J2 [dx1dx2Pa/A(x1,Q2)Pb/B(x2,Q2)v(ab->F), a.b •*
(15)
Collider Phenomenology:
X
Basic Knowledge and Techniques
421
X
Figure 5. Parton momentum distributions versus their energy fraction x at two different factorization scales, from CTEQ-5.
where X is the inclusive scattering remnant, and Q2 is the factorization scale (or the typical momentum transfer) in the hard scattering process, much larger than A Q C D RS (200 MeV) 2 . The parton-level hard scattering cross section can be calculated perturbatively in QCD, while the parton distribution functions parameterize the non-perturbative aspect and can be only obtained by some ansatz and by fitting the data. For more discussions, the readers are referred to George Sterman's lectures [23] on QCD, or the excellent text [10] on thesse topics. Since the QCD parton model plays a pivotal role in understanding hadron collisions and uncovering new phenomena at high energies, we plot in Fig. 2.3.1 the parton momentum distributions versus the energy fractions x, taking CTEQ-5 as a representative [24]. For comparison, we have chosen the QCD factorization scale to be Q 2 =10 GeV2 and 104 GeV 2 in these two panels, respectively. Several general features are important to note for future discussions. The valence quarks well as the gluons carry a large momentum fraction, typically x ~ 0.08 - 0.3. The "sea quarks" (u = us s, c, b) have small x, and are significantly enhanced at higher Q2. Both of these features lead to important collider consequences. First of all, heavy objects near the energy threshold are more likely produced via valence quarks. Second, higher energy processes (comparing to the mass scale of the parton-level subprocess) are more dominantly mediated via sea quarks and gluons.
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T. Han
0 10
09 o
08 0? n6
B
si I
B o
x
3 o
t ft ft )2
II o *i
u
)"1
>
i-2
E om
(TeV)
10*
Figure 6. Scattering cross sections versus c m . energy for the SM processes in pp collisioins. The Higgs boson mass has been taken as 120 GeV.
2.3.2. Production cross sections for standard model processes In Figure 2.3.1, we show the integrated cross sections for various typical processes in the SM versus c m . energy of a pp hadron collider in units of mb. The scale on the right-hand side gives the event rate for an instantaneous luminosity 1034 c m - 2 s - 1 , a canonical value for the LHC. We have indicated the energies at the Tevatron and the LHC by the vertical dashed lines. First of all, we have plotted the pp total cross section as the line on the top. It is known that the cross section increases with the c m . energy [25]. Unitarity argument implies that it can only increase as a power of In s. An empirical scaling relation s 0 0 9 gives a good fit to the measurements upto date, and has been used here. All integrated cross sections in hadronic collisions increase with the c m . energy due to the larger parton densities at higher energies. The jet-inclusive cross section o"(jets) is given by the blue line. The reason the cross section falls is due to our choice of an energy-dependent cut on the jet's transverse momuntum. The bb pair production is also sensitively
Collider Phenomenology:
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423
dependent upon the transverse momentum cutoff since the mass rrib is vanishingly small comparing to the collider energies and thus the integrated cross section presents the familiar collinear singularity in the forward scattering region. The production at the leading order is dominantly via the gluon-initiated process gg —> bb, and is of the order of 1 /xb at the LHC energy (with a cutoff pr > 30 GeV). The top-quark production is again dominated by the gluon fusion, leading to about 90% of the total events. The rate of the leading order prediction is about 700 pb or about 7 Hz with a canonical luminosity, and higher order QCD corrections are known to be substantial [26]. It is thus justifiable to call the LHC a "top-quark factory". We also see that the leading Higgs boson production mechanism is also via the gluon fusion, yielding about 30 pb. QCD corrections again are very large, increasing the LO cross section by a significant factor [27]. Another interesting production channel is the gauge-boson fusion VV —> h, that is about factor of 5 smaller than the inclusive gg —> h in rate, and the QCD correction is very modest [28]. Most of the particles produced in high-energy collisions are unstable. One would need very sophisticated modern detector complex and electronic system to record the events for further analyses. We now briefly discuss the basic components for collider detectors. 3. Collider detectors: Our electronic eyes Accelerators and colliders are our powerful tools to produce scattering events at high energies. Detectors are our "e-eyes" to record and identify the useful events to reveal the nature of fundamental interactions. 3.1. Particle
detector
at
colliders
The particle detection is based on its interactions with matter of which the detectors are made. A modern particle detector is an electronic complex beyond the traditional particle detection techniques, which typically consists of a secondary displaced vertex detector/charge-tracking system, electromagnetic calorimetry, hadronic calorimetry and a muon chamber, etc. A simplified layout is shown in Fig. 7. 3.2. What do particles
look like in a
detector
As theorists, we mostly deal with the fundamental degrees of freedom in our SM Lagrangian, namely the quarks, leptons, gauge bosons etc. in our
424
T. Han
hadronic calorimeter ^••;;;.:;E-CAL:-
. : ;;•..:
tracking (in B field)
beam
RZl&S&£&SgS3$&8gS&SJI£^^
pipe
vertex detector
\
\
\
muori chambers Figure 7.
Modern multi-purpose detector at colliders.
calculations. The truth is that most of them are not the particles directly "seen" in the detectors. Heavy particles like Z, W, t will promptly decay to leptons and quarks, with a lifetime 1/r ~ 1/(2 GeV) « 3.3 x 10~ 25 s. Other quarks will fragment into color-singlet hadrons due to QCD confinement at a time scale of th ~ 1/AQCD « 1/(200 MeV) « 3.3 x 10~ 24 s. The individual hadrons from fragmentation may even behave rather differently in the detector, depending on their interactions with matter and their life times. Stable paricles such as p, p, e^, 7 will show up in the detector as energy deposit in hadronic and electromagnetic calorimeters or charge tracks in the tracking system. In Fig. 8, we indicate what particles may leave what signatures in certain components of the detector. In order to have better understanding for the particle observation, let us recall the decay length of an unstable particle (/? c r ) 7 « (300 ^ m ) ( _ I — - ) 7, (16) 10 where r is the particle's proper lifetime and 7 = E/m is the relativist s factor. We now can comment on how particles may show up in a detector. d
Collider Phenomenology:
Tracking chamber
Electromagnetic Hadron ;:AIO: ::::i?re;' caloiirnerer
Innermost Lay ex Figure 8.
Basic Knowledge and Techniques
425
Muon chamber
———$• .,. Outermost Layer
Particle signatures left in the detector components.
• Quasi-stable: fast-moving particles of a life-time r > 1 0 - 1 0 s will still interact in the detector in a similar way. Those include the weak-decay particles like the neutral hadrons n, A, K^, ... and charged particles fj^, -K±,K±, ... • Short-lived resonances: particles undergoing a decay of typical electromagnetic or strong strength, such as TV°, p0,±... and very massive particles like Z, W±, t, (H...), will decay "instantaneously". They can be only "seen" from their decay products and hopefully via a reconstructed resonance. • displaced vertex: particles of a life-time r ~ 10~ 12 s, such as B0^, D0^, T1*1, may travel a distinguishable distance (CT ~ 100 /xm.) before decaying into charged tracks, and thus result in a displaced secondary vertex, as shown in Fig. 9, where the decay length between the two vertices is denoted by L. As an interesting and important case, Kg with CT ~ 2.7 cm also often results in a secondary vertex via its decay to 7r+7r_. • Things not "seen": those that do not participate in electromagnetic nor strong interactions, but long-lived as least like the quasi-stable particles, will escape from detection by the detector, such as the neutrinos v and neutralinos x° in SUSY theories, etc.
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T. Han
prompt tracks
^z
Figure 9. An illustrative event leading to a displaced secondary vertex.
Now coming back to the elementary particles in the SM, we illustrate their behavior in Table 3. A check indicates an appearance in that component, a cross means no, a check-cross is partially yes. Other symbols are self-explained. 3.3. More on
measurements
It is informative to discuss in a bit more detail a few main components for the particle detection. We hope to indicate how and how well the energy, momentum, and other properties of particles can be measured. When needed, we will freely take either the ATLAS or the CMS detector as an example for the purpose of illustration. Vertexing: Normally, at least two charged tracks are needed to reconstruct a secondary decay vertex, as illustrated in Fig. 9. Note that if the decaying particle moves too fast, then the decay products will be collimated, with a typical angle 0 « 7 _ 1 = m/E. The impact parameter do as in Fig. 9 can be approximated as do « Lxy0. The impact parameter is crucial to
Collider Phenomenology: Table 3.
Basic Knowledge and Techniques
What the elementary particles in the SM look like in detectors.
Leptons e± r±
Vetexing x x Vx X
Quarks u, d, s
Tracking p P sf X
x
ECAL E v/ e±
HCAL
V
Muon Cham.
X
X
V
P
h±;
3h± X
X
V
V
X
h's h's 6 + 2 jets
X
b->B t-^bW± Gauge bosons 7 9
V
V
e± e±
W ± - • SL±v
427
b
-J
x
x
£
X
X
x
V
V
X
x x x
p x/ p
e* V e±
V 2 jets
X
X
M±
(«*)
V
V
2 jets
X
X
determine the displaced vertex. For instance, the ATLAS detector [12] has the resolution parameterized by Ad0 = 11
73
(pT/GeV) x/ihTe
(/xro),
(17)
where the notation © implies a sum in quadrature. It is possible to resolve a secondary vertex along the longitudinal direction alone, which is particularly important if there will be only one charged track observed. In this case, the resolution is typically worse and it can be approximated [12] as Az 0 = 87 ©
115 (p r /GeV) Vsin 3 9
(fim).
(18)
Tracking: Tracking chamber determines the trajectories of traversing charged particles as well as their electromagnetic energy loss dE/dx. The rapidity coverage is 1^1*2.5,
(19)
for both ATLAS and CMS. When combined with a magnetic field (2 T for ATLAS and 4 T for CMS), the system can be used to measure a charged particle momentum. The curvature of the trajectory is inversely proportional to the particle momentum
428
T. Han 1
QB = - oc -=—, (20) P P where Q is the particle's electric charge and B the external magnetic field. Therefore, knowing B and assmuing a (unity) charge, the momentum p can be determined. The energy-loss measurement dE/dx for heavy charged particles may be used for particle identification. For instance, the Bethe-Bloch formula for the energy loss by excitation and ionization gives a scaling quadratically with the particle charge and inversely with the speed K
dE
2
a (Q\
/on (21)
^ UJ '
independent of the charged particle mass. The mass can thus be deduced from p and /?. However, if we allow a most general case for a particle of arbitrary m, Q, then an additional measurement (such as /? from a Cerenkov counter or time-of-flight measurements) would be needed to fully determine the particle identity. From the relation Eq. (20), the momentum resolution based on a curvature measurement can be generically expressed as apT © b,
(22)
PT
For instance, the ATLAS [12] (CMS [13]) detector has the resolution parameterized by a = 36% T e V - 1 (15% TeV - 1 ), b = 1.3%/V^8 (0.5%). In particular, the momentum resolution for very high energy muons about pT « 1 TeV in the central region can reach 10% (6%) for ATLAS (CMS). Good curvature resolution for highly energetic particle's tracks is important for the charge determination. ECAL: High-energy electrons and photons often lead to dramatic cascade electromagnetic showers due to bremsstrahlung and pair production. The number of particles created increase exponentially with the depth of the medium. Since the incident energy to be measured by the electromagnetic calorimetry (ECAL) is proportional to the maximum number of particles created, the energy resolution is characterized by 1/vN, often parameterized by =
E
,
^/E/GeV
© 6,
'
(23) V
^
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429
where a is determined by the Gaussian error and b the response for cracks. For ATLAS (CMS), a = 10% (5%), b = 0.4% (0.55%). The coverage in the rapidity range can reach
toe,7l*3
(24)
or slightly over for both ATLAS and CMS. HCAL: Similar to the ECAL, showers of subsequent hadrons can be developed from the high-energy incident hadrons. An HCAL is to measure the hadronic energy, and the Gaussian error again is parameterized as AE -
=
a e
7 I
f o
, .
(25)
For ATLAS (CMS), a = 80% (100%), b = 15% (5%). The rapidity coverage by the forward hadronic calorimeter can reach \Vh\ « 5
(26)
for both ATLAS and CMS.
3.4.
Triggering
So far, we have ignored one very important issue: data acquisition and triggering. Consider pp collisions at the LHC energies, the hadronic total cross section is of the order about 100 mb, and the event rate at the designed luminosity (10 34 c m - 2 s _ 1 ) will be about 1 GHz (compare with the clock speed of your fast computer processor). A typical event will take about one megabytes of space. It is therefore impossible for the detector electronic system to record the complex events of such a high frequency. Furthermore, the physical processes of our interest occur at a rate of 10~ 6 lower or more. Thus, one will have to be very selective in recording events of our interest. In contrast, there will be no such problems at e+e~ colliders due to the much lower reaction rate. Trigger is the decision-making process using a desired temporal and spatial correlation in the detector signals. It is provided by examining the properties of the physical process as appeared in the detector. Modern detectors for hadron colliders such as CDF, DO at the Tevatron and ATLAS, CMS at the LHC typically adopt three levels of triggering. At the LHC experiments, Level-1 triggering brings the event rate down to the order of 105 Hz; Level-2 to about 10 3 Hz; and Level-3 finally to about 100 Hz to tape.
430
T. Han Table 4. Level-1 trigger thresholds in r]-pT coverage for the ATLAS [12] detector. Entries are for a luminosity of 10 3 3 c m - 2 s _ 1 (10 3 4 c m - 2 s _ 1 in parentheses). Objects fj, inclusive e/photon inclusive Two e's or two photons 1-jet inclusive 3 jets 4 jets r/hadrons Jets+#T
ATLAS vT (GeV) 2.4 6 (20) 2.5 17 (26) 2.5 12 (15) 3.2 180 (290) 3.2 75 (130) 3.2 55 (90) 2.5 43 (65) 4.9 100 3.2, 4.9 50,50 (100,100) V
There are many means to design a trigger, such as particle identification, multiplicity, kinematics, and event topology etc. Modern detectors at colliders usually can trigger on muons by a muon chamber, electrons/photons as electromagnetic objects, "r/hadrons and jets as hadronic objects, global energy sum and missing transverse energy, and some combinations of the above. Relevant to collider phenomenology is to know what particles may be detected in what kinematical region usually in pr-V coverage, as the detector's acceptance for triggering purposes. Table 4 summarizes Level-1 trigger thresholds for ATLAS for the commonly observed objects and some useful combinations. Inversely, if we find some trigger designs inadequate for certain physics needs, such as a well-motivated new physics signal with exotic characteristics or unusual kinematics, it is the responsibility of our phenomenologists to communicate with our experimental colleagues to propose new trigger designs. 4. Uncover new dynamics at colliders Instead of summarizing which new physics scenario can be covered by which collider to what extent, I would like to discussion a few examples for observing signals to illustrate the basic techniques and the use of kinematics. The guiding principles are simple: maximally and optimally make use of the experimentally accessible observables to uncover new particles and to probe their interactions. In designing the observables, one will need to concern their theoretical properties, like under Lorentz transformation, charge and C, P, T discrete symmetries etc., as well as their experimental feasibil-
Collider Phenomenology:
Basic Knowledge and Techniques
431
ity, like particle identification, detector acceptance and resolutions etc. I hope that this serves the purpose to stimulate reader's creativity to cleverly exploit kinematics to reveal new dynamics in collider experiments. 4 . 1 . Kinematics
at hadron
colliders
In performing parton model calculations for hadronic collisions like in Eq. (15), the partonic c m . frame is not the same as the hadronic c m . frame, e. g. the lab frame for the collider. Consider a collision between two hadrons of A and B of four-momenta PA = (EA, 0,0,p A ) and Pg = (EA, 0,0, — pA) in the lab frame. The two partons participating the subprocess have momenta p\ = X\PA and p2 = x2PB. The parton system thus moves in the lab frame with a four-momentum Pom = [(xi + x2)EA,
0, 0, ( n - x2)pA\
or with a speed pcm = {%i — x2)/(x\
{EA « pA),
(27)
+ x2), or with a rapidity
Vcm = Jin—-
(28)
Denote the total hadronic c m . energy by S = 4EA and the partonic c m . energy by s, we have S = TS,
T
— x\x2 — —.
(29)
The parton energy fractions are thus given by x
\,2 = VT e±y™.
(30)
One always encounters the integration over the energy fractions as in Eq. (15). With this variable change, one has /•l
rl
/ dxi / •'To
JTO/XI
rl
dx2=
p— ^ l n r
dr J TO
dycm.
(31)
J^lnr
The variable r characterizes the (invariant) mass of the reaction, with r 0 = rn^es/S and mres is the threshold for the parton level final state (sum over the masses in the final state); while ycm specifies the longitudinal boost of the partonic c m . frame with respect to the lab frame. It turns out that the T — Vcm variables are better for numerical evaluations, in particular with a resonance as we will see in a later section. Consider a final state particle of momentum pM = (E,p) in the lab frame. Since the c m . frame of the two colliding partons is a priori undetermined with respect to the lab frame, the scattering polar angle 6 in these two
432
T. Han
frames is not a good observable to describe theory and the experiment. It would be thus more desirable to seek for kinematical variables that are invariant under unknown longitudinal boosts. Transverse momentum and the azimuthal angle: Since the ambiguous motion between the parton c m . frame and the hadron lab frame is along the longitudinal beam direction (z), variables involving only the transverse components are invaraint under longitudinal boosts. It is thus convenient, in contrast to Eqs. (A.6) and (A.9) of Appendix A in the spherical coordinate, to write the phase space element in the cylindrical coordinate as d3p dpz dpz — = dpxdpy— = pTdpTd—,
(32)
where <j> is the azimuthal angle about the z axis, and PT=\JPl+V2y=Psin6
(33)
is the transverse momentum. It is obvious that both pT andare boostinvariant, so is dpz/E. E x e r c i s e : Prove t h a t dpz/E boost-invariant.
is longitudinally
Rapidity and pseudo-rapidity: The rapidity of a particle of momentum pM is defined to be y= ln
(34)
2 E-^p-z-
E x e r c i s e : With t h e i n t r o d u c t i o n of r a p i d i t y y, show t h a t a p a r t i c l e four-momentum can be r e w r i t t e n as pV = (ET cosh y, pT sin <j>, pT cos, ET sinh y),
ET = -Jp\ + m2.
The phase space element then can be expressed as d?p —— = pTdpTd(j) dy = ETdETdcj) dy. E
(36)
Consider the rapidity in a boosted frame (say the parton c m . frame), and perform the Lorentz transformation as in Eq. (A.4) of Appendix A, , y
1
E'+p'z
=2l»w^7z
1
( 1 - # ) ) ( £ + Pz)
= 2ln(i+e0)(E-pz)=y-yo-
,~7, (37)
(35)
Collider Phenomenology:
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433
In the massless limit, E w |p|, so that 1, l + cos0 0 . . 2/-^olni ^ = lncot-=7?, (38) 2 1 — cos 0 2 where rj is the pseudo-rapidity, which has one-to-one correspondence with the scattering polar angle 7r > 0 > 0 for — oo < ry < oo. Since y as well as 77 is additive under longitudinal boosts as seen in Eq. (37), the rapidity difference Ay = y2 - y\ = y'2 - y{ is invariant in the two frames. Thus the shape of rapidity distributions da/dy in the two frames would remain the same if the boost is by a constant velocity. In realistic hadronic collisions, the boost of course varies on an event-by-event basis according to Eq. (28) and the distribution is generally smeared. The lego plot: It should be clear by now that it is desirable to use the kinematical variables (pT, rj,(j>) to describe events in hadronic collisions. In collider experiments, most often, electromagnet and hadronic calorimeters provide the energy measurements for (essentially) massless particles, such as e^, 7, and light quark or gluon jets. Thus = Ecos6 = E cosh" 1 n.
ET=pT
(39)
A commonly adopted presentation for an electromagnetic or hadronic event is on an 77—2.
(40)
As a quantitative illustration, for two objects back-to-back in the central region, typically AT? < A~ n. Another important consequence for the introduction of separation is that it provides a practical definition of a hadronic jet, and AR specifies the cone size of a jet formed by multiple hadrons within AR. 4.2. s-channel
singularity:
Resonance
signals
4.2.1. The invariant mass variable Searching for a resonant signal in the s-channel has been the most effective way of discovering new particles. Consider an unstable particle V produced by a + b and decaying to 1 + 2 +... + n. For a weakly coupled particle Ty <
434
T. Hon
Figure 10. A CDF di-jet event on a lego plot in the r) —plane. The height presents the transverse energy scale, and the two colors (blue and pink) indicate the energy deposit in the two calorimeters (ECAL and HCAL).
My, according to the Breit-Wigner resonance Eq. (B.l), the amplitude develops a kinematical peak near the pole mass value at (Pa+P 6 ) 2 = ( X > )
2
«M2.
(41)
i
This is called the invariant mass, and is the most effective observable for discovering a resonance if either the initial momenta or the final momenta can be fully reconstructed. As a simple example of a two-body decay, consider Z —• e + e~, Wee = (Pe+ + Pe~ f
» 2 p e + • pe-
« 2Ee+ Ee-~ ( 1 - COS 0 e + e - ) « M | ,
(42)
which is invariant in any Lorentz frame, and leads to Ee « Mz/2 in the Z-iest frame. Figure 11 shows the peak in the e+e~~ invariant mass spectrum at Mz, indicating the resonant Z production observed by the DO collaboration [29] at the Tevatron collider. Now let us examine the transverse momentum variable of a daughter particle peT = pe sin 9*, where 9* is the polar angle in the partonic
Collider Phenomenology:
Basic Knowledge and Techniques
90
100
110
435
120
m(ee) (GeV) Figure 11.
The resonant signal for a Z boson via Z —> e+e
at the DO detector.
c m . frame. For a two-body final state kinematics, we thus have da dpeT
4p,eT sy/l - AplT/s
da dcos6*'
(43)
The integrand is singular at p\r = s/4, but it is integrable. Exercise:
Verify t h i s equation for Drell-Yan production of
Combining with the Breit-Wigner resonance, we obtain da dmledplT
TZMZ " (ml - M\Y + Y\M\
m^I
1 - Ap'iT/mle
da dcosO* '
(44)
We see that the mass peak of the resonance leads to an enhanced distribution near peT = Mz/2. This is called the Jacobian peak. This feature is present for any two-body kinematics with a fixed subprocess c m . energy. E x e r c i s e : While t h e i n v a r i a n t mass d i s t r i b u t i o n da/dme+eis unaffected by the motion of the produced Z boson, show t h a t
436 T. Han the dcr/dpeT distribution for a moving Z with a momentum pz is changed with respect to a Z at rest at the leading order of
Pz=pz/Ez. It is straightforward to generalize the invariant mass variable to multibody system. Consider a slightly more complicated signal of a Higgs decay H -> ZXZ2 -> e+e~ n+yT.
(45)
Obviously, besides the two Z resonant decays, the four charged leptons should reconstruct the mass of the parent Higgs boson 4
"& = ( X > ) 2 = 2(Ml+ P;Jl .p Za )
(46)
i
= (Ee+ + Ee- + £ M + + Ep- f - (pe+ + Pe- + P»+ + P»- ?•
(47)
4.2.2. The transverse mass variable As another example of a two-body decay, consider W —> ev. The invariant mass of the leptonic system is m\v = (Ee + Eu)2 - (peT + PUT? - (pez + Puz)2•
(48)
The neutrino cannot be directly observed by the detector and only its transverse momentum can be inferred by the imbalancing of the observed momenta, T/>T — ~ /_,PT(observed),
(49)
called missing transverse momentum, identified as j>r = PvT- Missing transverse energy is similarly defined, and ET = Ev. The invariant mass variable thus cannot be generally reconstructed. We would get the correct value of mev if we could evaluate it in a frame in which the missing neutrino has no longitudinal motion pvz = 0; but this is impractical. Instead, one may consider to ignore the (unknown) longitudinal motion of the leptonic system (or the W boson) all together, and define a transverse mass of the system [30] r r & r = (EeT + EvTf
- (peT + PUT?
« 2peT-PvT « 2EeTET
(50)
(1 - COSev),
where (f>eu is the opening angle between the electron and the neutrino in the transverse plane. When a W boson is produced with no transverse motion, Eer — ET = mei/T/2. It is easy to see that the transverse mass
CM 2500 o >
o 2000
Collider Phenomenology:
Basic Knowledge and Techniques
CDF Run II 72.0 pb" 1
• W-^(iv Data 0Sum O W ^ i v MC | mZJy^iu. MC ©W-ycv MC E3QCD
CM
11500
437
LU
1000
500
80
100
120 140
MT (GeV/c2) Figure 12. T h e transverse mass peak for a W —• iw signal observed at CDF.
variable is invariant under longitudinal boosts, and it reaches the maximum meuT = mel/, for pez = pvz, so that there is no longitudinal motion for the electron and the neutrino when boosting to the W^-rest frame. In general, (51)
0 < mei/T < me
The Breit-Wigner resonance at mev = Mw naturally leads to a kinematical peak near meUT « Mw again due to the Jacobian factor da
dmlv dm*
oc
TWMW
K „ - MWY + TwMlw
mf
1
ri
'\f ' 2
(52) l
ev,T
In the narrow width approximation, meUT is cut off sharply at Mw- In practice, the distribution extends beyond Mw because of the finite width Tw This is shown in Fig. 12 as observed by the CDF collaboration [31] in the channel W —> \xv.
438
T. Han
E x e r c i s e : While the i n v a r i a n t mass d i s t r i b u t i o n d(j/dmeu i s unaffected by t h e motion of t h e produced W boson, show t h a t the da/dmel/T d i s t r i b u t i o n for a moving W with a momentum pw i s changed with r e s p e c t t o a W a t r e s t a t the n e x t - t o leading order of j3w = pw/EwCompare t h i s conclusion with t h a t obtained for da/dpeT. 4.2.3. The cluster transverse mass variable A transverse mass variable with more than two-body final state is less trivial to generalize than the invariant mass variable as given in Eq. (47). This is mainly due to the fact that for a system with more than one missing neutrino, even their transverse momenta pViT cannot be individually reconstructed in general, rather, only one value fx is experimentally determined, which is identified as the vector sum of all missing neutrino momenta. Thus the choice of the transverse mass variable depends on our knowledge about the rest of the kinematics, on how to cluster the other momenta, in particular realizing the intermediate resonant particles. H —> \V\W2 —> qi<72 ev: As the first example of the transverse mass variable with multi-body final state, let us consider a possible Higgs decay mode to WW with one W subsequently decaying to qxfa and the other to ev. Since there is only one missing neutrino, one may construct the transverse mass variable in a straightforward manner according to the kinematics for the two on-shell W bosons MT,WW
=
{ET,W! + ET,W2)
= (yJPTjj
+
M
W
- (PT,W! + ^PT,ev
+
+PT,W2) M
w)
~ (PTJje
+ fTf
•
Note that if the decaying Higgs boson has no transverse motion (like being produced via gg fusion), then the last term vanishes fa = —PTjjeHowever, this simple construction would not be suitable for a light Higgs boson when one of the W bosons (or both) is far-off shell. The above expression can thus be revised as AfrVw = {\jphi
+ m% +
\IPT,O>
+m^r)
" (Pr,«e
+ir)2-
Alternatively, one can consider to combine the observed two jets and a lepton together into a cluster, and treat the missing neutrino separately.
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One has M 2
= (^Phje+Mfje
T WW
+
fa)
~ (pTJje +
fa?•
Which one of those My variables is the most suitable choice for the signal depends on which leads to the best Higgs mass reconstruction and a better signal-to-background ratio after simulations. H —> Z1Z2 —* e+e~ vv: In searching for this signal, we define the cluster transverse mass based on our knowledge about the Z resonances [32], MT,ZZ
= (ET,Zl
+ETtz2)2
- (pr.z,
+PT,z2f
If the parent particle (H) is produced with no transverse motion, then Mr,zz*2v/^ie+e_+M|. E x e r c i s e : Consider how t o r e v i s e the above MT,ZZ c o n s t r u c t i o n i f mH < 2Mz i n order t o b e t t e r r e f l e c t the Higgs resonance. H -+W1W2
-*l\v\
^2^2:
As the last example, we consider a complicated case in which the two neutrinos come from two different decays. The two missing neutrinos do not present a clear structure, and thus one simple choice may be to cluster the two charged leptons together [33] M
c,ww
= (y/Pr,ee + Ma + fa) - (#r,« +
fa)2-
( 53 )
It was argued that since fr ~ ~PT,U, thus one should have, on average, ET,W ~ ET,U- This leads to a different construction [34] M
T,WW « ^\JPT,U + Mu-
(54)
These two choices are shown in Fig. 13, for mH = 170 GeV at the Tevatron, along with the SM WW background [35]. 4.3. t-channel
enhancement:
Vector boson
fusion
A qualitatively different process is initiated from gauge boson radiation, typically off energetic fermions, as introduced in Sec. 2.2.3. Consider a
440
T. Han
0.08 0.07
_ - a)
0.06
r
0.05 0.04
-_
0.12
: b 0.1
vv v
0.08
' r j
1
0.06
I.
0.03
i\
— _ -
0.04
i r
0.02 0.01
.-'
! I r ,
0
0.02
j "
J ; •
300
0
;L
"
I
.:7a£
200
M c (GeV)
MT (GeV)
Figure 13. Normalized distributions -^jj^ for mH = 170 GeV (histogram) and the leading WW background (shaded) at the Tevatron for (a) MT in Eq. (54) and (b) Mc in Eq. (53).
fermion / of energy E, the probability of finding a (nearly) collinear gauge boson V of energy xE and transverse momentum pT (with respect to pf) is approximated by [19,20] v/f (x,pT) pL
(„ „2) (x,p T
gv + g\ l + ( l - z ) 2 8TT 2 9V+9A
4TT 2
a 1-g
x
PT
(j?T + (1 - x)Ml {p2T +
2\2' (l-x)M*)
{l-x)MlY,
(55) (56)
which T (L) denotes the transverse (longitudinal) polarization of the massive gauge boson. In the massless limit of the gauge boson, Eq. (55) reporduces the Weizsacker-Williams spectrum as in Eq. (12), after integration of p\ resulting in the logarithmic enhancement over the fermion beam energy. In fact, the kernel of this distribution is the same as the quark splitting function q -» qg* [10,23]. The scattering cross section can thus be formally expressed as a{fa -> f'X)
w f dx dp2T Pv/f{x,p2T)
a(Va -> X).
(57)
This is also called the Effective W Approximation. Although this may not be a good approximation until the parton energy reaches E 3> My, it is quite important to explore the general features of the t-channel behavior. First of all, due to the non-zero mass of the gauge boson, there is no more collinear singularity. The typical value of the transverse momentum
Collider Phenomenology:
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441
(gauge boson or jet) is pT ~ \/l - x My & Mw. Since x prefers to be low reflecting the infrared behavior, the jet energy (1 — x)E tends to be high. These observations provide the arguments for "forward jet-tagging" for massive gauge boson radiation processes: a highly energetic companion jet with pT < My/12 and a scatteirng polar angle of a few degrees [36]. Furthermore, it is very interesting to note the qualitative difference between PT and PL for the pT dependence (or equivalently the angular dependence) of the outgoing fermion. For pT < Mw, PT is further suppressed with respect to PL; while for pT > Mw, PT is enhanced instead in the central scattering region. This was the original design for a forward jet-tagging and a "central jet-vetoing" [37] to enhance the longitudinal gauge boson fusion with respect to the transverse gauge boson fusion in the search for strong WW scattering signals [38]. It has been further realized that the t-channel electroweak gauge boson mediations undergo color singlet exchanges, and thus do not involve significant QCD radiation. Consequently, there will be little hadronic activities connecting the two parton currents. This further justifies the central jet-vetoing, and is developed into a "mini-jet vetoing" to further separate the gauge boson fusion processes from the large SM backgrounds in particular those with QCD radiation in the central region [39].
4.4. Forward-backward
asymmetry
The precision test of universal chiral couplings of the gauge bosons Z°, W^ to SM fermions is among the most crucial experimental confirmation for the validity of the SM. Similar probe would be needed to comprehend any new vector bosons once they are discovered in future collider experiments. The forward-backward asymmetry, actually the parity-violation along the beam direction, is very sensitive to the chiral structure of the vector boson to fermion couplings. Consider a parton-level process for a vector boson production and decay ii _» V _> / / ,
(58)
where the initial state ii = e~e+, qq. Let us now parameterize the coupling vertex of a vector boson VM to an arbitrary fermion pair / by L,R
ij^drPr.
(59)
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T. Han
Then the parton-level forward-backward asymmetry is defined as NF-NB
u
^*
s
AWA^
3 =
4
A
(60)
^'
where NF {NB) is the number of events with the final-state fermion momentum pf in the forward (backward) direction denned in the parton c m . frame relative to the initial-state fermion pi. The asymmetry A/ is given in terms of the chiral couplings as 2 S Af = (rt) - i9 R? (9fL)2 + (gfR)2' The formulation so far is perfectly feasible in e+e~~ collisions. However, it becomes more involved when applied to hadron colliders AB —> VX —> ffX. The first problem is the mismatch between the parton c m . frame (where the scattering angle is defined to calculate the asymmetry) and the lab frame (where the final-state fermion momentum is actually measured). This can be resolved if we are able to fully determine the final state momenta pf, pf. We thus construct the vector boson momentum Pv = Pf + Pf, (62) and then boost pf back to the y-rest frame, presumably the parton c m . frame. The second problem is the ambiguity of the p^ direction: both a quark and an anti-quark as initial beams can come from either hadrons A or B, making the determination of pi impossible in general. Fortunately, one can resolve this ambiguity to a good approximation. This has something to do with our understanding for quark parton distributions in hadrons. We first recognize that for a heavy vector boson production, the parton energy fraction is relatively large x ~ My/^/s, and thus the contributions from valance quarks dominate, recall Fig. 2.3.1. Consider the case at the Tevatron for pp collisions, we can thus safely choose the beam direction of the protons (with more quarks) as Pi. As for the LHC with pp collisions, this discrimination of p versus p is lost. However, we note that for uu, dd annihilations, valance quarks (u, d) carry much larger an x fraction than the anti-quark in a proton. We thus take the quark momentum direction Pi along with the boosted direction as reconstructed in Eq. (62), recall the boost relation Eq. (28). With those clarifications, we can now define the hadronic level asymmetry at the LHC [40] A
LHC
fdXl
^g=u,dAFfB
(Pq(xi)Pg(x2)
fdx1Eq=UtdiaiC(Pq(x1)Pq(x2)
- P9(Xl)Pq{x2))
+
BJgnfri -
X2)
Pll(x1)Pq(x2)) (63)
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443
where Pq{x\) is the parton distribution function for quark q in the proton with momentum fraction x\, evaluated at Q2 = Mv. The momentum fraction x^ is related to x\ by the condition x\x% = Mv/s in the narrowwidth approximation. Only u and d quarks contribute to the numerator since we explicitly take the quark and antiquark PDFs to be identical for the sea quarks; all flavors contribute to the denominator. Some recent explorations on the hadronic level asymmetry for various theoretical models have been presented in Ref. [41]. 4.5. Be prepared for more involved
inclusive
signatures
The previous sections presented some basic techniques and general considerations for seeking for new particles and interactions. They are applicable to many new physics searches. Prominent examples include: • Drell-Yan type of new particle production in s-channel [41-47]: Z> _+ £+£-, W+W~; ZH -> ZH;
WH ->
W -> Iv,
W±Z;
W±H;
p?£->tt, W+W-;
tb,W±Z;
heavy KK gravitons —> £+£~, 77,...; single q, £ via R parity violation. • ^-channel gauge boson fusion processes [41,44,48-50]: W+W~,
ZZ, W±Z^H,
p0^,
light SUSY partners;
++
W+W+ -* H ; W+b -> T. However, Nature may be trickier to us. Certain class of experimental signals for new physics at hadron collider environments may be way more complex than the simple examples illustrated above. The following possible scenarios may make the new physics identification difficult: • A new heavy particle may undergo a complicated cascade decay, so that it is impossible to reconstruct its mass, charge etc. For example, think about a typical gluino decay [51] in SUSY theories g-^qq^q
q'x+ -» Q «' X°W+ -> q q' x° e+u.
• New particles involving electroweak interactions often yield weakly coupled particles in the final state, resulting in missing transverse momentum or energy, making it difficult for reconstructing
444
T. Han
the kinematics. Examples of resulting in missing energies include neutrinos in the SM, neutralino LSP in SUSY theories [52], light Kaluza-Klein gravitons in large extra dimension models [53], and the lightest stable particles in other theories like in universal extra dimensions (UED) [54] and little Higgs (LH) model with a T-parity [55] etc. • Many new particles may be produced only in pair due to a conserved quantum number, such as the R-parity in SUSY, KK-parity in UED, and T-parity in LH, leading to a smaller production rate due to phase space suppression and more involved kinematics. For the same reason, their decays will most commonly yield a final state with missing energy. The signal production and the decay products are lack of characteristics. On the other hand, one may consider to take the advantage of those less common situations when identifying new physics signatures beyond the standard model. Possible considerations include: • Substantial missing transverse energy is an important hint for new physics beyond the SM, since ftr in the SM mainly comes from the limited and predictable sources of W,Z,t decays, along with potential poor measurements of jets. • High multiplicity of isolated high pT particles, such as multiple charged leptons and jets, may indicate the production and decay of new heavy particles, rather than from higher order SM processes. • Heavy flavor enrichment is again another important feature for new physics, since many classes of new physics have enhanced couplings with heavy flavor fermions, such as H —>, bb, T+T~; H+ —»tb, T+V; H -> xH; i -+ x+6, x°i; Pre % -> ** e t c Clever kinematical variables may still be utilized, such as the lepton momentum and invariant mass endpoints as a result of certain unique kinematics in SUSY decays [56]. We are always encouraged to invent more effective observables for new signal searches and measurements of the model parameters. When searching for these difficult signals in hadron collider environments, it is likely that we have to mainly deal with event-counting above the SM expectation, without "smoking gun" signatures. Thus it is of foremost importance to fully understand the SM background processes, both for total production rates and for the shapes of kinematical distributions.
Collider Phenomenology:
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445
This should be recognized as a serious challenge to theorists working on collider phenomenology, in order to be in a good position for discovering new physics in hadron collider experiments. To conclude these lectures, I would like to say that it is highly anticipated that the next generation of collider experiments at the LHC and ILC will reveal exciting new physics beyond the currently successful standard model. Young physicists should be well prepared for understanding the rich but complex data from the experiments in connection to our theoretical expectation and imagination, and thus contributing to the major discovery. Acknowledgments I would like to thank the TASI-2004 organizers, John Terning, Carlos Wagner, and Dieter Zeppenfeld for inviting me to lecture, K.T. Mahanthappa for his arrangements and hospitality, and the participating students for stimulating discussions. I would also like to express my gratitude to Yongsheng Gao, Yibin Pan, Heidi Schellman, Wesley Smith, Weimin Yao, for their help in preparing the sections related to the experimental issues. This work was supported in part by the U. S. Department of Energy under contract No. DE-FG02-95ER40896, and by the Wisconsin Alumni Research Foundation. A. Relativistic kinematics and phase space A . l . Relativistic
kinematics
Consider a particle of rest mass m and momenta p moving in a frame O. We denote its four-momentum p = pM = {E,p). The Lorentz invariant m2 defines the on-mass-shell condition p>lptl = E2-p-p
= m2.
(A.l)
Its velocity in units of c is 0=-
= ^ C
(-13<1);
and
7
= (i _ / ? * ) - *
Hi
=
*:.
(A.2)
m
Consider another frame O' that is moving with respect to O along the z direction (without losing generality). It is sufficient to specify the Lorentz transformation between the two frames by either the relative velocity (/3o) of the moving O' or its rapidity 1 .
1 +
/SO
y°=oln:i W I i - Po
/
v
(-00 < j/o < oo).
/ A ON
(A.3)
446
T. Han
For instance, for the four-momentum vector
'E'\
=
yzJ
/70
-TbAA (E
v - 7 0 A 70
_ /coshj/o
) \PZ
— sinhy 0 \
V-sinhj/o cosh y0
(E
J \pz J '
These transformations are particularly useful when we need to boost the momentum of a decay product from the parent rest frame (O1) to the parent moving frame (O). In this case, the relative velocity is given by the velocity of the decaying particle 0 = parent /Eparent _ The Lorentz-invariant phase space element for an n-particle final state can be written as :1
(2TT)3 2Ei'
(
5j
The 8A imposes the constraint on the phase space by the four-momentum conservation of the initial state total momentum P. Each final state particle satisfies an on-shell condition pf = mf, and the total c m . energy squared i S S = p2 = ( E r = i f t ) 2 _ A.2. One-particle
final
state
Most straightforwardly, we have the phase space element for one-particle final state dPS1 = (2TT) ~S4(P-Pl)
=7r|pi|dni(J3(P-pi),
(A.6)
here and henceforth, we adopt a notation " = " to indicate that certain lessconcerned variables have been integrated out at this stage. For instance, the variable E\ has been integrated out in the last step of Eq. (A.6), which leads to a trivial (but important) relation Eym = yfs in the c m . frame. Making use of the identity
£E = Jd*p6(p2-m2),
(A.7)
we can rewrite the phase space element as dPS! =
= - 6(1 - ^ L ) . (A.8) s Vs We will call the coefficient of the phase-space element "phase-space volume" , after integrating out all the variables. Here it is 27T for one-particle final state in our convention. 2TT S(S
-m\)
Collider Phenomenology:
A.3. Two-body
Basic Knowledge and Techniques 447
kinematics
For a two-particle final state with the momenta p\, P2 respectively, the Lorentz-invariant phase space element is given by dPS2
= -(2^s - iA ^2 (47T)
2
{p pi P2)
- ~ mm
r
dn
i = 77"^ ^ ^ dcosM^i(47T) 2
-y/S
(A.9)
y/S
Two-body phase space element is dimensionless, and thus no dimensionful variables unfixed. That is to say that the two-body phase space weight is constant and the magnitudes of the energy-momentum of the two particles are fully determined by the four-momentum conservation. It is important to note that the particle energy spectrum is monochromatic. Specifically, in the c m . frame L-cn»i_L-cmi_Al/2^m^2) lPl M P 2 I " %fs '
s+m\-m22 2^/i '
wcm El
_ s+ffl| - m? >2^i '
FCm E
where the "two-body kinematic function" is defined as X(x,y, z) = (x-y-z)2-4yz
= x2+y2 + z2-2xy-2xz-2yz,
(A.10)
which is symmetric under interchange of any two variables. While the momentum magnitude is the same for the two daughter particles in the parent-rest frame, the more massive the particle is, the larger its energy is. The only variables are the angles for the momentum orientation. We rescale the integration variables dcos9\ = 2dx\ and d\ = 2-ndxi to the range 0 < x\, X2 < 1, and thus dPS2
=
±lX1/2(l,r^,^-)dx1dx2. 47T 2
\
S
(A.ll)
SJ
It is convenient to do so in order to see the phase-space volume and to implement Monte Carlo simulations. The phase-space volume of the two-body is scaled down with respect to that of the one-particle by a factor sdPSt"
(4n)f
(A 12)
'
Roughly speaking, the phase-space volume with each additional final state particle (properly normalized by the dimensonful unit s) scales down by this similar factor. It is interesting to note that it is just like the scaling factor with each additional loop integral.
448
T. Han
It is quite useful to express the two-body kinematics by a set of Lorentzinvariant variables. Consider a 2 —> 2 scattering process pa +Pb —• Pi + P2> the Mandelstam variables are defined as
* = {Pa+Pbf = (pi +P 2 ) 2 = C . * = (Pa - P i ) 2 = (Pb-P2)2 u={jpa-
=m2a + m\-
2{EaE1 -PaPl
cos6»ol), (A.13)
= (Pb - Pi) 2 = m2a + m\ - 2(£ a £2 - PaP2 cos0 a2 ).
p2f
The two-body phase space can be thus written as td
dPS2 = - ^
xl/2(t
t)
2T-V
12
2
(47r) s X '
(l,ml/s,mi/s)
Exercise: Assume that ma = m\ and m\t=m2.
i=-2p M
2
(A.14)
Show that
m(l-cos0;i),
= -2pL(i+cosfl: 1 ) +
K
;
m i ) 2
,
where p c m = A 1 / 2 (s,mi,m2)/2^/i i s the momentum magnitude i n the c m . frame. This leads t o t —> 0 i n t h e c o l l i n e a r l i m i t . E x e r c i s e : A p a r t i c l e of mass M decays t o two p a r t i c l e s i s o t r o p i c a l l y i n i t s r e s t frame. What does the momentum d i s t r i b u t i o n look l i k e i n a frame i n which t h e p a r t i c l e i s moving with a speed /3Z? Compare t h e r e s u l t with your e x p e c t a t i o n for t h e shape change for a basket b a l l . A.4.
Three-body
kinematics
For a three-particle final state with the momenta p i , p2, ps respectively, the Lorentz-invariant phase space element is given by „r DC
l
d3pi d3p2
x4/D
. |fc|2d|ft|dni
|pf3)'
I
3
(2TT) 2 £ I
(4TT) ,2
2
m23
V
m
23
dVL2
(A. 15)
™2
= T A T X1'2 (l, ^f-, ^4-) (4TT) 3
d3p3
m
23/
2|pi| dE1 dx2dx3dx4dx5.
(A.16)
Collider Phenomenology:
1.0
£
Basic Knowledge and Techniques
—i—i—i—i—I—i—i—r
449
I ' ' ' •
0.8
a
0.6
-•J
'S
0.4
CO
W
0.2 -
_l
0.0
I
I
I
I
l_J
I
10
I
L_J
20
I
I
l_J
I
30
I
I
I
I
40
I
I
I
L
50
60
E, = K.+m, (GeV) Figure 14. Three-body phase space weight as a function of Ei (i = 1, 2, 3) for y/~s = 100 GeV, mi,2,3 = 10,20,30 GeV, respectively.
The angular scaling variables are dcos#i,2 — 2dx2,i, and dcj>i^ = 2-Kdxz$ in the range 0 < £2,3,4,5 < 1- Unlike the two-body phase space, the particle energy spectrum is not monochromatic. The maximum value (the endpoint) for particle 1 in the c m . frame is s + ml - (m 2 + m 3 ) 2
•pmax
^max or \p 1
I I
A 1 / 2 (a,mj,(m2 + m 3 ) 2 ) (A.17)
It is in fact more intuitive to work out the end-point for the kinetic energy instead - recall that this is how a direct neutrino mass bound is obtained by examining /?-decay processes [4], Kmax
=
Emax
_
^
( - y / i - m i - 7712 - r7i 3 )( v / s - mi + m 2 4- m 3 ) 2y/E (A.18)
450
T. Han
P\
P i "
P
P
Figure 15.
Pn-l
1
P
Illustration for the recursion relation for an n-body kinematics.
Practically in Monte Carlo simulations, once Efm is generated between mi to E™ax, then all the other variables are determined
IPTI2
= \vT+PT? = (ED2 - ml
ml3 = s-2^sEr
+ ml
\pf\ = \pf\
A 1 / 2 (mj 3 ,mj,mj) 2ra 23
along with the four randomly generated angular variables. To see the non-monochromaticity of the energies in three-body kinematics, in Fig. 14, we plot the three-body phase space weight dPSd as a function of Ei (i = 1,2,3). For defmiteness, we choose ^/s = 100 GeV and TO i,2,3 = 10, 20, 30 GeV, respectively. It is in arbitrary units, but scaled to dimensionless by dividing E™ax^/s. We see broad spectra for energy distributions. Naturally, the more massive a particle is, the more energetic (energy and momentum) it is, but narrower for the energy spread. However, its kinetic energy Ki = Ei — rrii is smaller for larger ra^.
A . 5 . Recursion
relation for the phase space
dPSn(P;Pl,...,Pn)
=
element
dPSn-i(P; i
dPS2(pn-l,n;Pn-l,Pn)
2
Tr^
•
(A.19)
ZTT
This recursion relation is particularly useful when we can write the intermediate mass integral for a resonant state.
Collider Phenomenology:
Basic Knowledge and Techniques 451
B. Breit-Wigner resonance and the narrow width approximation The propagator contribution of an unstable particle of mass M and total width Ty is written as 1_ ^ 2 , ^ 2 ^ 2 , (B.1) (s-M^+TlM*' This is the Breit-Wigner Resonance. Consider a very general case of a virtual particle V* in an intermediate state, *(*) = ,„
a^bV*^bpip2.
(B.2)
An integral over the virtual mass can be obtained by the reduction formula in the last section. Together with kinematical considerations, the resonant integral reads ax 2
r{m™
) =(ma-mb)2
dml
(B.3)
J(rr /(mjli")2 = (mi+m2)2
The integral is rather singular near the resonance. Thus a variable change is effective for the practical purpose, tanfl=
m ;-5, TvMv
(B.4) y
resulting in a flat integrand over 6 (ml - Mv)2 + T2VMV
y(mri„)2
where 9 = t a n _ 1 ( m 2 - Mv)/TvMy.
Jemin
TVMV'
max
then 9 -> -7T, 9 approximation:
j
(B.6)
-> 0. This is the condition for the narrow-width 1
T^>
'
In the limit
(mi+m2) + r v < M y « ; m 0 - r v , min
[
7r
» r 2 \ 9 , -n2 ^ 2 ~ T. , ,
^ ( W * - My).
(B.7)
v v K J (m 2 - My)2 + TvMy TvMy * ' Exercise: Consider a three-body decay of a top quark, t —> bW* —> b ev. Making use of the phase space recursion relation and the narrow width approximation for the intermediate W boson, show that the partial decay width of the top quark can be expressed as
T(t -> bW* -> b ev) w T{t -> bW) • BR(W -> ev).
(B.8)
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T. Han
References 1. S. Willenbrock, Symmetries of the Standard Model, these TASI lectures, arXiv:hep-ph/0410370, and references therein. 2. C. Quigg, Gauge Theories of the Strong, Weak, and Electromagnetic Interactions, Addison-Wesley Publishing Company, Inc. (1983); T.-P. Cheng and L.-F. Li, Gauge Theory of Elementary Particle Physics, Oxford University press (1984); J. Donoghue, E. Golowich, B.R. Holstein, Dynamics of the Standard Model, Cambridge University Press (1992). 3. H. Haber, Practical Super symmetry, these TASI lectures. Also see his review in Reference [4]. 4. Particle Data Group, Review of Particle Physics, Phys. Lett. B592, 1 (2004). 5. M. Drees, R. M. Godbole, and Theory and Phenomenology of Sparticles, World Scientific Publisher (2004). 6. R. Sundrum, Introduction to Extra Dimensions, these TASI lectures. 7. C. Csaki, Higgsless Electroweak Symmetry Breaking, these TASI lectures. Also, see his previous TASI lectures, arXiv:hep-ph/0404096. 8. H. Schellman, these TASI lectures. 9. V.D. Barger and R.J.N. Phillips, Collider Physics, Addision-Wesley Publishing Company (1987). 10. R.K. Ellis, W.J. Stirling, and B.R. Webber, QCD and Collider Physics, Cambridge University Press (1996). 11. R. Fernow, Introduction to Experimental Particle Physics, Cambridge University Press (1990). 12. ATLAS Technical Design Report, CERN-LHCC-94-43. 13. CMS Technical Design Report, CERN-LHCC-94-38. 14. J.A. Aguilar-Saavedra et al., ECFA/DESY LC Physics Working Group Collaboration, DESY 2001-011 and arXiv:hep-ph/0106315; T. Abe et al., American LC Working Group Collaboration, arXiv:hep-ex/0106055; K. Abe et al, ACFA LC Working Group Collaboration, arXiv:hep-ex/0109166. 15. CLIC Working Group, Physics at the CLIC Multi-TeV Linear Collider, arXiv:hep-ph/0412251. 16. M. Blaskiewicz et al.. VLHC Accelerator Physics, FERMILAB-TM-2158. 17. V.D. Barger, M.S. Berger, J.F. Gunion, and T. Han, Phys. Rept. 286, 1 (1997). 18. I.F. Ginzburg, G.L. Kotkin, S.L. Panfil, V.G. Serbo, and V.I. Telnov, Nucl. Instr. and Methods in Phys. Res. 219, 5 (1984). 19. S. Dawson, Nucl. Phys. B249, 42 (1985); M. Chanowitz and M.K. Gailard, Phys. Lett. B142, 85 (1984). 20. G. Kane, W.W. Repko, and W.B. Rolnick, Phys. Lett. B148, 367 (1984). 21. K. Hagawara and D. Zeppenfeld, Nucl. Phys. B313, 560 (1989). 22. G. Moortgat-Pick et al. arXiv:hep-ph/0507011. 23. G. Sterman, QCD and Jets, these TASI lectures. arxive:hep-ph/0412013. 24. H. L. Lai et al. [CTEQ Collaboration], Eur. Phys. J. C12, 375 (2000). 25. H. Cheng and T.T. Wu, Phys. Rev. Lett. 24, 1456 (1970). 26. E. Laenen, J. Smith, and W.L. van Neerven, Phys. Lett. B321, 254 (1994);
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R. SEKHAR CHIVUKULA
• J-v^lW*
ELIZABETH H. SIMMONS
D Y N A M I C A L ELECTROWEAK S Y M M E T R Y B R E A K I N G
R. SEKHAR CHIVUKULA 1 AND ELIZABETH H. SIMMONS 2 Department of Physics and Astronomy Michigan State University East Lansing, MI 48824 USA sekhar@msu. edu esimmons@msu. edu
We provide a brief description of the material covered during the lectures on dynamical electroweak symmetry breaking and a guide to further reading.
1. Lecture themes and overview These lectures explored the idea that the physics of electroweak symmetry breaking arises from novel strong dynamics. We started from the familiar strong-interaction dynamics of QCD, introduced simple dynamical electroweak symmetry breaking models whose strong dynamics imitate those of QCD, analyzed the limitations of these models, and finally moved on to theories whose dynamics differ from those of QCD in intriguing and useful ways. Because strong-dynamics models predict new states at accessible energies, experimental constraints have played a large role in the development of these theories. Accordingly, we discussed a number of specific classes of models that have been created to meet specific experimental challenges and examine their phenomenology. Some may ask why it is necessary to keep creating and studying new kinds of models, why the Standard Model and its supersymmetric extension^) do not suffice. The answer, quite simply, is that we lack experimental evidence as to the mechanism of electroweak symmetry breaking. Theoretical arguments about which general kinds of mechanisms are plausible leave us with too many open possibilities, each with its own strengths and weaknesses. It is often pointed out that we need experimental data to winnow the options. At the same time, we must think carefully about what conclusions one will be able to draw from the data. For example, if a collider experiment 457
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Simmons
produces a new scalar state decaying to bb, how will we know whether it is a standard model Higgs if we have not understood the characteristics of other candidates such as a supersymmetric Higgs, a technipion, or a top-pion? Likewise, if we find a color-octet of new heavy gauge bosons, how will we distinguish whether these are colorons, Kaluza-Klein modes of gluons, or colored extended-technicolor bosons, unless we understand the phenomenology of the models that contain such states? Studying specific models helps us to understand the strengths and limitations of general theoretical ideas. It gives us a vantage from which to suggest promising directions for experiments. And it enables us to construct frameworks that can help us decipher the implications of experimental data. But a good model-builder should always recall Aristotle's words: "It is the mark of an educated mind to be able to entertain a thought without accepting it." 2. A guide to further reading The TASI-2004 lectures on dynamical symmetry breaking drew heavily on reviews that we have previously published. The first is the most recent and comprehensive, and focuses exclusively on dynamical electroweak symmetry breaking - this is the place to start for a pedagogical introduction to the subject, and also contains a complete set of references to the original literature. The second and third reviews, based on lectures we gave at TASI-2000, place the material on dynamical electroweak symmetry breaking in slightly different contexts - in the context of compositeness and topquark physics, respectively. The last review provides a survey of current experimental constraints on these models. (1) C. T. Hill and E. H. Simmons, "Strong Dynamics and Electroweak Symmetry Breaking," Phys. Rept. 381, 235 (2003) [Erratum-ibid. 390, 553 (2004)] [arXiv:hep-ph/0203079]. (2) R. S. Chivukula, "Technicolor and Compositeness," TASI-2000 [arXiv:hep-ph/0011264]. (3) E. H. Simmons, "Top Physics," TASI-2000 [arXiv:hep-ph/0011244]. (4) R. S. Chivukula, M. Narain and J. Womersley, "Dynamical Electroweak Symmetry Breaking," in S. Eidelman et al. [Particle Data Group], "Review of Particle Physics," Phys. Lett. B 592, 1 (2004). The last lecture included a brief introduction to the most complete attempt to construct a model of extended technicolor, and incorporate or-
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Breaking
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dinary fermions with a quasi-realistic mass matrix. This work is not extensively described in the references above: the model(s) are built on the ideas introduced in the first reference given below, and are thoroughly analyzed in the second and third. (1) T. Appelquist and J. Terning, "An Extended Technicolor Model," Phys. Rev. D 50, 2116 (1994) [arXiv:hep-ph/9311320]. (2) T. Appelquist, M. Piai and R. Shrock, "Fermion Masses and Mixing in Extended Technicolor Models," Phys. Rev. D 69, 015002 (2004) [arXiv:hep-ph/0308061]. (3) T. Appelquist, N. Christensen, M. Piai and R. Shrock, "FlavorChanging Processes in Extended Technicolor," Phys. Rev. D 70, 093010 (2004) [arXiv:hep-ph/0409035].
M. SCHMALTZ
LITTLE HIGGS GOES TO TASI
M. SCHMALTZ* Physics Department, Boston University 590 Commonwealth Avenue, Boston, MA 02215, USA schmaltz@bu. edu
These TASI 2004 lecture notes provide a pedagogical introduction to Little Higgs models. The "Simplest Little Higgs" is used wherever explicit examples are given. Precision electroweak constraints and collider phenomenology as well as T-parity are briefly discussed.
1. I n t r o d u c t i o n A few years before the start of the LHC program, electroweak symmetry breaking remains poorly understood. T h e detailed quantitative fit of Standard Model predictions to precision experiments at the weak scale strongly suggests t h a t electroweak symmetry is broken by one or more weakly coupled Higgs doublets. However, fundamental scalar particles suffer from a radiative instability t o their masses, leading us to expect additional structure (such as compositeness, supersymmetry, little Higgs, ...) near the weak scale. Interestingly, we can t u r n this problem into a prediction for the LHC. T h e argument goes as follows: Let us assume t h a t precision electroweak d a t a are indeed telling us t h a t there are no new particles beyond the Standard Model (with the exception of possible additional Higgs doublets) with masses at or below the weak scale. T h e n physics at the weak scale may be described by an "effective S t a n d a r d Model" which has the particle content of the Standard Model and in which possible new physics is parametrized by higher dimensional operators suppressed by the new physics scale A > TeV. All renormalizable couplings are as in the S t a n d a r d Model. If there are additional Higgs fields then more complicated Higgs self-couplings as *Work partially supported by DOE grant DE-FG02-90ER-40560, DOE-OJI Award DEFG02-91ER40676 and an Alfred P. Sloan Research Fellowship. 463
464
M.
Schmaltz
well as Yukawa couplings are possible. Since no Higgs particles have been discovered so far, the effects of additional Higgs fields can be parametrized by effective operators for the Standard Model fields. The higher dimensional operators can be categorized by the symmetries which they break. The relevant symmetries are baryon and lepton number (B and L), CP and flavor symmetries, and custodial SU(2) symmetry. The wealth of indirect experimental data can then be translated into bounds on the scale suppressing the operators 1 > 2,3 ' 4 . Examples of such operators and the resulting bounds are summarized in Table 1. Table 1. Lower bounds on the scale which suppresses higher dimensional operators that violate approximate symmetries of the Standard Model. broken symmetry
operators
scale A
B, L
(QQQL)/A2
10 1 3 TeV
flavor (1,2
nd
flavor (2,3
rd
2
family), CP family)
(dsds)/A
1000 TeV 2
TO(j(SCTM„F'"'fr)/A
50 TeV
custodial SU(2)
(h^D^/A2
5 TeV
none (S-parameter)
(D2h^D2h)/A2
5 TeV
The bounds imply that physics at the TeV scale has to conserve B and L, flavor and CP to a very high accuracy, and that violations of custodial symmetry and contributions to the S-parameter should also be small. The question then becomes if it is possible to add new physics at the TeV scale to the SM which stabilizes the Higgs mass but does not violate the above bounds. To understand the requirements on this new physics better we must look at the source of the Higgs mass instability. The three most dangerous radiative corrections to the Higgs mass in the Standard Model come from one-loop diagrams with top quarks, SU(2) gauge bosons, and the Higgs itself running in the loop (Figure 1). All other diagrams give smaller contributions because they involve small coupling constants. Assuming that the Standard Model remains valid up to a cut-off scale A the three diagrams give -^A2A2op
~ - ( 2 TeV) 2
SU(2) gauge boson loops
^g2A2gauge
~ (700 GeV) 2
Higgs loop
j^X2A%iggs
~ (500 GeV) 2
top loop
The Higgs soft mass includes the sum of these contributions and a tree level mass-squared parameter.
Little Higgs Goes to TASI
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Figure 1. The most significant guadratically divergent contributions to the Higgs mass in the Standard Model.
In order for this to produce an expectation value for the Higgs of order the weak scale without fine tuning to more than 10%, the cut-off must satisfy Atop<2TeV
Agauge<5TeV
A „ i g g s < 10 TeV
(1)
We see that the Standard Model with a cut-off near the maximum attainable energy at the Tevatron (~ 1 TeV) is natural, and we should not be surprised that we have not observed any new physics. However, the Standard Model with a cutoff of order the LHC energy would be fine tuned and we expect to see new physics at the LHC. More specifically, we expect new physics which cuts off the diverging top loop at or below 2 TeV. In a weakly coupled theory this implies that there are new particles with masses at or below 2 TeV. These particles must couple to the Higgs, and enable us to write a one-loop quadratically divergent diagram which cancels the contribution from the top loop. In order for this cancellation to be natural, the new particles must be related to the top quark by a symmetry which implies that the new particles have similar quantum numbers to top quarks. Thus naturalness arguments predict a new multiplet of particles with mass below 2 TeV which carry color and are therefore easily produced at the LHC. In supersymmetry these new particles are of course the top squarks. The contributions from SU(2) gauge loops must also be canceled by new particles which are related to the Standard Model SU{2) gauge bosons by a symmetry. The masses of these particles must be at or below 5 TeV for the cancellation to be natural. Similarly, the Higgs loop requires new particles related to the Higgs at 10 TeV. We summarize the upper bounds on new particle masses which we obtain from naturalness in Table 2. Given the center of mass energy of the LHC of 14 TeV these predictions are very exciting, and encourage us to explore different possibilities for what the new particles could be.
466
M.
Schmaltz Table 2. Predictions for maximum masses of "partner" particles. Standard Model loop
maximum partner mass
top
2 TeV
weak bosons
5 TeV
Higgs
10 TeV
One example of new particles near the TeV scale which can appear in loops to cancel quadratic divergences are the superpartners predicted in the Minimal Supersymmetric Standard Model 5 . There the top quark loop is canceled by a corresponding loop with stop squarks. Supersymmetry also predicts the necessary relationship between top and stop coupling constants. Furthermore, the two diagrams are each proportional to A 2 , the cut-off used to regulate the two divergences. In general, the cut-offs for the two diagrams need not be the same, and therefore the divergences from two diagrams do not cancel. However, if a supersymmetric cut-off is used, then the A's for the Standard Model particles and their superpartners are the same. Another possibility is that the Higgs is a composite resonance at the TeV scale as in technicolor 6 or composite Higgs models 7 ' 8 ' 9 . Or extra dimensions might be lurking at the TeV scale with possible new mechanisms to stabilize the Higgs mass 10 . Here, we explore another possibility, that the Higgs is a pseudo-NambuGoldstone boson as suggested in Refs. 11 and 12. This idea was recently revived by Arkani-Hamed, Cohen and Georgi who also constructed the first successful "little Higgs" model 13 , and thereby started an industry of "little model building" 1 4 " 3 3 . 2. Nambu-Goldstone bosons Nambu-Goldstone bosons (NGBs) arise whenever a continuous global symmetry is spontaneously broken. If the symmetry is exact, the NGBs are exactly massless and have only derivative couplings. U{1) example: Consider for example a theory with a single complex scalar field <j> with potential V = V((f>*4>). The kinetic energy term d^d^cj) and the potential are invariant under the U(l) symmetry transformation > - >
ela
(2)
Little Higgs Goes to TASI
467
If the minimum of the potential is not at the origin but at some distance / away as in the famous "wine bottle" or "Mexican hat" potential (Figure 2), then the U(l) symmetry is spontaneously broken in the vacuum. We
Figure 2. The "Mexican hat" potential for €>. The black dot represents the expectation value f, r is the radial mode and 9 the Nambu-Goldstone boson.
vacuum
expand the field for small fluctuations around the vacuum expectation value (VEV)(x) = ^=(f
+ r(x))eie^f
(3)
where / is the VEV of r, r(x) is the massive "radial mode" and 9(x) is the NGB. The factor of l/-\/2 ensures canonical kinetic terms for the real fields r and 9. The radial field r is invariant under the Z7(l) symmetry transformation Eq. (2) whereas the NGB field 9 shifts 9-^9
+a
(4)
under U(l) transformations. We say that the U(l) symmetry is "nonlinearly" realized. We may now imagine integrating out the massive field r and writing the resulting effective Lagrangian for the NGB 9(x). 9 cannot have a mass or any potential, because the shift symmetry forbids all nonderivative couplings of 9. Non-Abelian examples: In the generalization to spontaneously broken nonAbelian symmetries we find one NGB for every broken symmetry generator. For example, we may break SU(N) —> SU(N — 1) with a VEV for a single fundamental 0 of SU(N). The number of broken generators is the total number of generators of SU(N) minus the number of unbroken generators,
468
M. Schmaltz
i.e. [TV2 - 1] - [(TV - l ) 2 - 1] = 2TV - 1
(5)
T h e NGBs are conveniently parametrized by writing 7Ti
/ exp\-
•
U-
(°\
\
}
TTiV-1 •K*N-1
= eiw/f4>0
'• 0
(6)
\fJ
TTo/v^/
The field no is real whereas the the fields n\ • • • 7TJV_I are complex. The last equality defines a convenient short-hand notation which we will employ whenever the precise form of n and 4>o ls clear from the context. Another example of symmetry breaking and NGBs which has found applications in little Higgs model building is SU(N) -> SO(N) .
(7)
Here the number of NGBs is the number of fields in the adjoint of SU(N) minus the number of fields in the adjoint of SO(N) (antisymmetric tensor), i.e. [TV2 - 1]
N(N-l)
N(N+1)
1
(8)
For even N we also have
SU{N) - • SP(N)
(9)
and the number of NGBs is the number of fields in the adjoint of SU(N) minus the number of fields in the adjoint of SP(N) (symmetric tensor), i.e. [Nz - 1]
N(N+1)
N(N-l)
2
2
-1
(10)
Finally, for SU(N) x SU(N) -> SU(N)
(11)
the number of NGBs is 2[/V2 - 1] - [N2 - 1] = iV2 - 1 . (12) In this last case the symmetry breaking is achieved by a VEV which transforms as a bi-fundamental under the two SU(N) symmetries. Denoting transformation matrices of the two SU(N) as L and R respectively we have 4>^L(j)B)
.
(13)
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469
The symmetry breaking VEV is proportional to the unit matrix A <0>=0O = /
OX ••.
(14)
This VEV is left invariant under "vector" transformations for which L = R=U (15)
<j>o — > U 4>o U^ = 4>o ,
all other symmetry generators (the "axial" generators) are broken and correspond to NGBs which can be parametrized as 60e
iir/f — f g W /
(16)
where rr is a Hermitian traceless matrix with N2 — 1 independent components. 2.1. How do NGBs
transform?
We now show how NGBs transform under the broken and unbroken symmetries in the example of SU(N) —> SU(N — 1) which is often denoted in more mathematical notation as SU(N)/SU(N — 1). The NGBs are parametrized as <> / = en4>o as in Eq. (6). Let's consider first transformations under the unbroken SU(N — 1). Then we have UN.
(UN-!
e™ U]N_{) UN-! fa = e ^ - i ' ^ - i )
fo
(17)
where in the second equality we used the fact that the symmetry breaking 4>0 is invariant under the unbroken UN-I transformations. Therefore the NGBs transform in the usual "linear" way under SU(N— 1) transformations ir —> UN-mUlf^. Explicitly, in the case of SU(N)/SU(N - 1) the unbroken SU(N — 1) transformations are (UN-IO
UN-i
V o
(18)
i
The single real NGB transforms as a singlet whereas the TV - 1 complex NGBs transform as 0
0 UN-i
7ft
0
uJV-1
UN-iTT
(19) n7rt77t U N-1
470
M.
Schmaltz
where we used a vector notation 7? to represent the N — 1 complex NGBs as a column vector. We see that 7? —* UN-ITT, i.e. 7? transforms in the fundamental representation of SU(N-l). Under the broken symmetry transformations we have U e™ >o = exp {i[
_t
) } exp
{'(*:)}•
; . / 0 77' = exp <j i ( _,,f ) }• UN-i(a,n)
^^{^Vo)}*
(20)
where in the second equality we used the fact that any SU(N) transformation can be written as a product of a transformation in the coset SU(N)/SU(N) times an SU{N - 1) transformation 34 . The UN-i(a,Tr) transformation which depends on a and TT leaves <po invariant and can therefore be removed. Equation (20) defines the transformed field 7?' which in general - is a complicated function of a and 7?. To linear order the transformation is simple 7? -> 7?' = 7? + <3
(21)
which shows that the NGBs shift under the non-linearly realized symmetry transformations. As in the U(l) case, the shift symmetry ensures that NGBs can only have derivative interactions. 2.2. Effective
Lagrangian
for
NGBs
Our goal for this section is to write the most general allowed effective Lagrangian for only the massless NGB fields, and respecting the full SU(N) symmetry. This is where the utility of the exponentiated fields <j> becomes obvious: while the full SU(N) transformations on the 7r's are complicated, the 0's transform very simply. To get the low energy effective Lagrangian we expand in powers of c^/A and write the most general possible SU(N) invariant function of= elv^(j)o at every order. With no derivatives we can form two basic gauge invariants objects <$4> = f2 and eai---aN(f)ai(f)a2 • • • 4>aN = 0. Thus the most general invariant contribution to the potential is simply a constant. You can convince yourself that the most general term that can be written at quadratic order is a constant times \d^
+ O(04)
(22)
Little Higgs Goes to TASI
471
where we normalized the coefficient of the second order term such that the -K fields have canonical kinetic terms. Note that the second order term expanded to higher order in the 7r fields contains interactions which determine the scattering of arbitrary numbers of 7r's at low energies in terms of the single parameter / . 3. Constructing a little Higgs model Now that we know how to write a Lagrangian for NGBs we would like to use this knowledge to write a model where the Higgs is a NGB. The explicit model we are going to construct in the remainder of this section is the "simplest little Higgs" 19>20>29. For example, consider the symmetry breaking pattern SU{3)/SU(2) with NGBs TT =
|
-r//2
0
0
-77/2
h
(23)
/it
V, Note that h is a doublet under the unbroken SU(2) as required for the Standard Model and it is an NGB, it shifts under "broken" 5(7(3) transformations. r\ is an SU(2) singlet which we will ignore for simplicity in most of the following. To see what interactions we get for h from the Lagrangian we expand = exp
f\tfoJ\f.
}C
+i
1
2? tfh +
(24)
and therefore
m
0
|aMfe[2/itfc
= \d»h\2 + — 2
+
T
+'
(25)
which contains the Higgs kinetic term as well as interactions suppressed by the symmetry breaking scale / . Since the Lagrangian contains nonrenormalizable interactions, it can only be an effective low-energy description of physics. To determine the cut-off A at which the theory becomes strongly coupled we can compute a loop and ask at which scale it becomes as as important as a corresponding tree level diagram. The simplest example is quadratically divergent one-loop contribution to the kinetic term which stems from contracting h)h in the second term in Eq. (25) into a loop. Cutting the divergence off at A we find a renormalization of the kinetic term proportional to 1 2
A2
f 167T2
and therefore A S 47r/.
(26)
472
M.
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Summarizing, we now have a theory which produces a "Higgs" doublet transforming under an exactly preserved (global) SU(2). This "Higgs" is a NGB and therefore exactly massless. It has non-renormalizable interactions suppressed by the scale / which become strongly coupled at A = Airf. Because of the shift symmetry no diagrams, divergent or not, can give rise to a mass for h. Anticipating that we are going to use this theory as a model for the Higgs in the Standard Model we summarize the relevant scales in Figure 3. On the other hand, this theory is still vary far from
A lOTeV--
A
ITeV—
f
100 G e V ~ Figure 3.
Mweak
Energy scales of a typical Little Higgs theory.
what we want. An NGB can only have derivative interactions, i.e. no gauge interactions, no Yukawa couplings and no quartic potential. Any of these interactions explicitly break the shift symmetry h —> h + const. In the following subsections we discuss how to add these interactions without re-introducing quadratic divergences. 3.1. Gauge
interactions
Let us try to introduce the SU{2) gauge interactions for h (we ignore hypercharge for the moment, it will be easy to add later). To do so we simply follow our nose and see where it leads us. We will arrive at the right answer after a few unsuccessful attempts. First attempt: Let's simply add couplings to 5(7(2) gauge bosons in the usual way, i.e. in addition to the Lagrangian Eq. (25) we add the term
IffW^I2
(27)
and another term with one derivative and one SU{2) gauge boson W^ as required by gauge invariance. These terms allow us to write Feynman
Little Higgs Goes to TASI
(a) Figure 4.
473
(b)
Quadratically divergent gauge loop contributions
to the Higgs mass.
diagrams with quadratic divergences (Figure 4) which contribute to the "Higgs" mass
(28)
jL^k
Note that these diagrams are exactly the quadratically divergent Standard Model gauge loops which we set out to cancel. We apparently gained nothing, we started with a theory in which the Higgs was protected by a non-linearly realized SU (3) symmetry (under which h shifts) but then we added the term Eq. (27) which completely and explicitly breaks the symmetry. Of course, we necessarily have to break the shift symmetry in order to generate gauge interactions for h but we must break the symmetry in a subtler way to avoid quadratic divergences in the Higgs mass. Second attempt: Let's write more Sf/(3) symmetric looking expressions and add the coupling |5(^
0
)^|
2
(29)
where W^ contains the three SU(2) gauge bosons. (Really, we write |-DM0|2 where the covariant derivative involves only SU(2) gauge bosons. The twogauge-boson-coupling is then Eq. (29)). This still allows a quadratically divergent contribution to the Higgs mass. The diagram is the same as before except with external
(30)
where the projection matrix diag(l, 1,0) arises from summing over the three SU(2) gauge bosons running in the loop. Not surprisingly, we got the same answer as before because we added the same interactions, just using a fancier notation.
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Third attempt: Let us preserve SU(3) by gauging the full SU(3) symmetry, i.e. by adding ID^]2 where now the covariant derivative contains the 8 gauge bosons of SU (3). Again we can write the same quadratically divergent diagram and find
^
16TT2
A2/2
(31)
which has no dependence on the Higgs field. The quadratic divergence only contributes a constant term to the vacuum energy but no Higgs mass! Unfortunately, we have also lost the "Higgs"! The NGBs are "eaten" by the heavy SU(3) gauge bosons corresponding to the broken generators, i.e. they become the longitudinal components of the gauge bosons. We have now exhausted all possible ways of adding SU(2) gauge interactions to our simple toy model for h. The lesson is that we can avoid the quadratically divergent contribution to the Higgs mass by writing SU(3) invariant gauge interactions, the problem that remains is that our "Higgs" was eaten. But this is easy to fix. Fourth attempt (successful): We use two copies of NGBs fa andfaand add SU(3) invariant covariant derivatives for both. We expect no quadratic divergence for either of the NGBs but only one linear combination will be eaten. To see how this works in detail we parametrize ^=ei7ri/f(f)
^2=ei,r2//(/)
(32)
where - for simplicity - we assumed identical symmetry breaking scales / i = $i = / , and we also assumed that the VEVs for fa andfaare aligned. We write the Lagrangian C=\DM2 + \D»fa?
(33)
The two interaction terms allow writing two quadratically divergent oneloop diagrams similar to the one's of the previous attempt (Figure 5.a) which give
^
{4>\fa + 44») = jf^A 2 (/2 + f)
(34)
i.e. no potential or mass term for any of the NGBs. However only one linear combination of m and 7T2 is eaten as there is only one set of hungry massive SU(3) gauge bosons.
Little Higgs Goes to TASI
b.)
A2
475
^2
Figure 5. a.) Quadratically divergent gauge loop contributions which do not to the Higgs potential, b.) log-divergent contribution to the Higgs mass.
contribute
A simple way to understand this result is to notice that each of the two diagrams only involves one of the <j> fields, therefore the diagrams are the same as in the theory with only onein which all NGBs are eaten, therefore none can get a potential. This reasoning changes once we consider diagrams involving both
^ l o g ( ^ ) \4>W
(35)
which does depend on the "Higgs" field but is not quadratically divergent. To calculate the Higgs dependence we choose a convenient parametrization
*1=ea^ (/)}**{*(* ")}(/)
(36)
* , = c a 9 , Kfct f c )} c ^{- i ( f c t")}(/)
(^
The field k can be removed by an 5(7(3) gauge transformation, it corresponds to the "eaten" NGBs, h cannot simultaneously be removed from <j>i and cf>2, it is physical. In the following we will work in the unitary gauge for SU(3) where k has been rotated away. Then we have
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*"<°'>-{-7C/)}0) = f2 - 2h]h + •••
(38)
and we see that Eq. (35) contains a mass for h g4/(16n2)\og r
e
( ^ ) f2
~
ual
•^uieafe f° 5 q to the SU(2) gauge coupling and / ~ TeV. To summarize, the theory of two complex triplets which both break SU(3) —> SU(2) automatically produces a "Higgs" doublet pseudo-NGB which does not receive quadratically divergent contributions to it's mass. There are log-divergent and finite contributions and from these the natural size for the "Higgs" mass is //47r ~ Mw 3.2. Symmetry
argument,
collective
breaking
Let us understand the absence of a quadratic divergence to the mass of h using symmetries. The lesson we learn is valuable as it generalizes to other couplings, it provides a general recipe for constructing little Higgs theories. Without gauge interactions, our theory would consist of two non-linear sigma models corresponding to the spontaneous breaking of 5/7(3) to 517(2), the coset [SU(3)/'SU(2)]2. There are 10 spontaneously broken generators and therefore 10 NGBs. The gauge couplings explicitly break some of the global symmetries. For example, the two gauge boson - two scalar coupling C ~ IgA^fal2 + \gA^2\2
(39)
breaks the two previously independent 51/(3) symmetries to the diagonal (gauged) SU(3). Thus only one of the spontaneously broken symmetries is exact, and therefore only one set of exact NGBs arises, the eaten ones. The other linear combination, corresponding to the explicitly broken axial 5£/(3), gets a potential from loops. However, as we saw before, there is no quadratically divergent contribution to the potential. This is easy to understand by considering the symmetries left invariant by each of the terms in Eq. (39) separately. Imagine setting the gauge coupling of fa to zero, then the Lagrangian has 2 independent 5(7(3) symmetries, one acting on fa (and A^) and the other acting on fa. Thus we now have two spontaneously broken SU{2>) symmetries and therefore 10 exact NGBs (5 of which are eaten). Similarly, if
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the gauge coupling of 4>i ls s e t to zero, there are again two spontaneously broken Sf/(3)'s. Only in the presence of gauge couplings for both <j>i and $2 are the two S[/(3) symmetries explicitly broken to one 5C/(3) and only then can the "Higgs" h develop a potential. Therefore any diagram which contributes to the Higgs mass must involve the gauge couplings for both 4>i and 02- But there are no quadratically divergent one-loop diagrams involving both couplings. This is the general mechanism employed by "little Higgs" theories 13 : The "little Higgs" is a pseudo-Nambu-Goldstone boson of a spontaneously broken symmetry. This symmetry is also explicitly broken but only "collectively", i.e. the symmetry is broken when two or more couplings in the Lagrangian are non-vanishing. Setting any one of these couplings to zero restores the symmetry and therefore the masslessness of the "little Higgs". We now know how to construct a theory with a naturally light scalar doublet coupling to SU(2) gauge bosons. To turn this into an extension of the Standard Model we still need i. Yukawa couplings, ii. hypercharge and color, and Hi. a Higgs potential with a quartic coupling. 3.3. Top Yukawa
coupling
The numerically most significant quadratic divergence stems from top quark loops. Thus the cancellation of the quadratic divergence associated with the top Yukawa is the most important. Let us construct it explicitly. The crucial trick is to introduce SU(3) symmetries into the Yukawa couplings which are only broken collectively. First, we enlarge the quark doublets into triplets * = (t,b,T) transforming under the SU(3) gauge symmetry. The quark singlets remain the same tc and bc except that we also need to add a Dirac partner T° for T. Note that we are using a notation in which all quark fields are left-handed Weyl spinors, and the Standard Model Yukawa couplings are of the form h^Qtc. Let us change notation slightly to reflect the fact that tc and Tc mix and call them t\ and t\. We can now write two couplings which both look like they contribute to the top Yukawa couplinga Cyuk = Ai(/>l*ii + A 2 4**2
(40)
To see what couplings for the Higgs arise we substitute the parametrization Eq. (37) and expand in powers of h. For simplicity, let us also set Ai = a
We do not write the couplings 4>\^t2 a n t ^ ^2**2 a s t n e v w o u ' d reintroduce quadratic divergences. They can be forbidden by global U(1) symmetries and are therefore not generated by loops.
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M. Schmaltz
A2 = A/-\/2. This will reduce the number of terms we encounter because it preserves a parity 1 *-» 2, but the main points here are independent of this choice. We find
r
X
fT(tc2 + t\) + i^Q(tc2 - t{) - jjrfhT(tc2 + t\) +
\/2 = \f(l--^rfh)TTc
+ \rfQtc + ---
(41)
where in the second line we have redefined fields Tc = (t2 + if )/\/2 and tc = i(t2 — t1)/y/2. We find a top Yukawa coupling and identify A = At. The Dirac fermion T, Tc has a mass \tf and a coupling to two Higgs fields with coupling constant A t /(2/). The couplings and masses are related by the underlying SU(3) symmetries. To see how the new fermion and it's
tc
Figure 6. The quadratically divergent contribution is canceled by the T loop.
to the Higgs mass from the top loop
couplings to the Higgs cancel the quadratic divergence from the top quark loop we compute the fermion loops including interactions to order A2. The two relevant diagrams (Figure 6) give
ife
A2fl h +
^
l e ^ ( 1 " 7 ? } A2 + °{h4) = const"+ °{hi)
(42)
The quadratically divergent contribution to the Higgs mass from the top and T loops cancel ! b While this computation allowed us to see explicitly that the quadratic divergence from t and T cancel, the absence of a quadratic divergence to the Higgs mass is much more naturally understood by analyzing the symmetries b I n order for the two cut-offs for the two loops to be identical, the new physics at the cut-off must respect the SU(3) symmetries. This is analogous to the situation in SUSY where the boson-fermion cancellation also relies on a supersymmetric regulator/cutoff.
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479
of the Lagrangian for the 0, fields, Eq. (40). First note that the Yukawa coupling Lagrangian preserves one 517(3) symmetry, the gauge symmetry. The term proportional to Ai forces symmetry transformations of\ and $ to be aligned and the term proportional to A2 also forces i can now rotate independently. Thus with either of the Aj we expect two sets of NGBs. One linear combination is eaten but the other is the "little Higgs". To understand radiative stability of this result we observe that a contribution to the Higgs potential can only come from a diagram which involves both Aj. The lowest order fermion diagram which involves both Aj is the loop shown in Figure 7, it is proportional to |AiA212- You can easily convince yourself
Figure 7. A log divergent contribution proportional to [Ai A212-
to the Higgs mass from the top and T loops
that you cannot draw a diagram which contributes to the Higgs potential and is proportional to only a single power of Ai A2. This also follows from an argument using "spurious" symmetries: assign t\ charge 1 under a 17(1) 1 symmetry while all other fields are neutral. The symmetry is broken by the Yukawa coupling Ai, but we can formally restore it by assigning the "spurion" Ai charge -1. Any effective operators which may be generated by loops must be invariant under this symmetry. In particular, operators which contribute to the Higgs potential and do not contain the fermion field t\ can depend on the spurion Ai only through |Ai| 2 . Of course, the same argument shows that the dependence on A2 is through jA212 only. A contribution to the Higgs potential requires both couplings \\ and A2 to appear and therefore the potential is proportional to |AiA2I2, i.e. at least four coupling constants. But a one-loop diagram with 4 coupling constants
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can at most be logarithmically divergent, and therefore does not destabilize the Higgs mass. In the explicit formulae above, we assumed - for simplicity - that / i = /2 = / and X1 = X2 = \t/V2. In the general case we find ™T =
/*?/? +A|/f
(43)
At=A1A2^M
(44)
rriT
Note that the top Yukawa coupling goes to zero as either of the A, is taken to zero as anticipated from the SU(3) symmetry arguments. Furthermore note that the mass of the heavy T quark can be significantly lower than the larger of the two ft if the corresponding A; is smaller than 1. This is a nice feature because it will allow us to take the heavy gauge boson's masses large (£ few TeV as required by the precision electroweak constraints) while keeping the T mass near a TeV. Keeping the T mass as low as possible is desirable because the quadratic divergence of the top loop in the Standard Model is cut off at the scale of the mass of T.
3.4. Other Yukawa
couplings
The other up-type Yukawa couplings may be added in exactly the same way. We enlarge the SU(2) quark doublets into triplets because of the gauged SU(3). Then we add two sets of Yukawa couplings which couple the triplets to fa and fa and quark singlets q\ and q%. In the Standard Model, Yukawa couplings for down type quarks arise from a different operator where the SU(2) indices of the Higgs doublet and the quark doublets are contracted using an epsilon tensor (or, equivalently, the conjugate Higgs field hc = zo^/i*) is used. Before explicitly constructing this operator from the quark and fa fields note that even the bottom Yukawa coupling is too small to give a significant contribution to the Higgs mass. The quadratically divergent one loop diagram in the Standard Model yields ^ A
2
« (30 GeV) 2 .
(45)
Therefore, we need not pay attention to symmetries and collective breaking when constructing the down type Yukawa couplings. The Standard Model Yukawa is Xb^ijhiQj
bc
(46)
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481
To obtain the epsilon contraction from an ST/(3) invariant operator we write
j eeiMi^oV ijk
(47)
Note that the e^-fe contraction breaks both SU(3) symmetries (acting on the two scalar triplets (j>\ and 02) to the diagonal and therefore this operator does lead to a quadratic divergence. But the quadratic divergence is harmless because of the smallness of the bottom Yukawa coupling. 3.5. Color and
hypercharge
Color is added by simply adding SU (3) color indices where we expect them from the Standard Model. ST/(3) color commutes with all the symmetry arguments given above, therefore nothing significant changes. hypercharge is slightly more complicated. The VEVsi oc (0,0,1) break the St/(3) wea k gauge group to SU(2), i.e. no t/(l) hypercharge candidate is left. Therefore, we gauge an additional U(l)x- In order for the hypercharge of the Higgs to come out correctly, we assign the SU(3) x U(l)x quantum numbers
(48)
The combination of generators which is unbroken by
where
v"3
T8 = ~
I
2^3 ^
1
|
(49)
2J
and X is the generator corresponding to U(l)x- This uniquely fixes the U(l)x charges of all quarks and leptons once their SU(3) transformation properties are chosen. For example, the covariant derivative acting on >, is D^^d^-^igxA^
+ igAf^^
(50)
Note that the U(l)x generator commutes with SC/(3), and the U(l)x gauge interactions do not change any of the symmetry arguments which we used to show that the Higgs does not receive quadratic divergences to its mass. There are now three neutral gauge bosons corresponding to the generators T3,T&,X. These gauge bosons mix, the mass eigenstates are the photon, Z and a Z' which leads to interesting modifications to predictions for precision electroweak measurements.
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3.6. Quartic Higgs
coupling
To generate a quartic Higgs coupling we want to write a potential V((j>i, fa) that i. contains no mass at order / for the Higgs, ii. contains a quartic coupling, Hi. preserves the "collective" symmetry breaking of the 5f/(3)'s: i.e. the quartic coupling is generated by at least two couplings in V and if one sets either one of them to zero the Higgs becomes an exact NGB. This last property is what guarantees radiative stability, no A2 contributions to the Higgs mass. Writing down a potential which satisfies these properties appears to be impossible for the pure SU(3) model (if you can figure out how to do it please let me know, and write a paper about it). To see why it is not straightforward, note that fa fa is the only non-trivial gauge invariant which can be formed from fa and fa. {fa fa = const = fafa and t^kfafafa = 0.) But the fa fa invariant is a bad starting point because it breaks the two SU{3)'s to the diagonal, and it is not surprising that generic functions of fa fa always contain a mass as well as a quartic. For example,
\fa~f2-h)h+-{h)h)2
+ ---
(51)
so that
-pU(42) n ~f4~ fWh + (rfh)2 + •••
(52)
By dialing the coefficient of this operator we can either get a small enough mass term or a large enough quartic coupling but not both. Of course, we could try to tune two terms with different powers n such that the mass terms cancel between them but that tuning is not radiatively stable. There are two different solutions to the problem in the literature. Both require enlarging the model and symmetry structure. One solution, due to Kaplan and Schmaltz 20 , involves enlarging the gauge symmetry to SU(4) and introduce four (f> fields which transform as a 4 of SU(4). The fourfields break SU(4) -» SU{2), yielding 4 SU{2) doublets. Two of them are eaten, the other two are "little Higgs" fields with a quartic potential similar to the quartic potential in SUSY. The other solution, due to Skiba and Terning 22 , keeps the SU{3) gauge symmetry the same but enlarges the global SU{3)2 symmetry to SU{3)3 which is then embedded in a SU(9). The larger symmetry also leads to two "little" Higgs doublets for which a quartic coupling can be written. Both of these solutions spoil some of the simplicity of the SU{3) model but they allow a large quartic coupling for the Higgs fields with natural electroweak
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symmetry breaking. I refer you to the original papers for details on these models. A third option 29 is to simply add a potential with a very small coefficient. The resulting quartic coupling is then also very small but radiative corrections from the top loop as in the MSSM give a contributions which can raise the Higgs mass above the experimental bound of 114 GeV. Explicitly,
h
K>-. hi
Figure 8.
The top loop contribution
to the Higgs quartic
coupling.
below the T mass the cancellations in the top sector no longer occur and the diagram in Figure 8 gives a contribution to the quartic
which is too small by itself but does give successful electroweak symmetry breaking when combined with a small tree level contribution. Since the tree level term also contributes to the soft mass for the Higgs a moderate amount of tuning (~10%) is required. While this is not completely satisfactory it is better than most other models of electroweak symmetry breaking and certainly better than the MSSM with gauge coupling unification which requires tuning at the few % level or worse. 3.7. The simplest
little
Higgs
This section summarizes the construction and salient features of the "simplest little Higgs" 29 , the SU(3) model in which the Higgs quartic coupling is predominantly generated from the top loop. In the phenomenology section, we will use the model as an example to discuss typical experimental signatures and constraints.
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M.
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The model has an S£/(3)coior x 5rC/(3)weak x £/(l)x gauge group with each of the three generations of quark and lepton fields transforming as *Q = (3,3)I
* L = (l,3)_j
c
ec=(l,l)!
d = (3,l)i 2 x « c = (3,l)-|
nc = ( l , l ) 0
(54)
The triplets ^Q and \&jr, contain the quark and lepton doublets, the (chargeconjugated) singlets are the fields uc,dc,ec,nc.c The SU(3)W x U(l)x symmetry breaking stems from expectation values for fa =<j>2 = ( l , 3 ) _ i / 3 . The Lagrangian of the model contains the usual kinetic terms Ckin ~
+ ••• + [D^l2
*IPVQ
+ •••
(55)
Yukawa couplings + \%4*QUC2 + jfafa^Q
Cyuk ~ K^QUl
+ \n\^Lnc+jfafa^Lec
dC
+ h.c.
(56)
and the tree level Higgs potential arises from £Pot ~fJ.2<j>\fa+h.c.
(57)
We substitute the parametrization for the NGBs ! = e ^
0
, fc = e - « ° *
0
(58)
/ 2 = /i 2 + / 2 2 -
(59)
where © =-7=7 + 7
0°
and
and solve for the spectrum of massive particles. The numerical values provided below correspond to the example point / i = 0 . 5 TeV and fc = 2 c
This fermion content is anomalous under the extended electroweak gauge group. The anomaly must be canceled by additional fermions which can be as heavy as A. It is also possible to change the charge assignments such that anomalies cancel among the fields ^Q,^L,uc,dc,ec,nc alone 29 > 35 > 36 .
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TeV (/ ~ 2 TeV). In addition to the Standard Model particles we have 0.9 TeV
heavy gauge bosons <
•1.1 TeV 2
\/(Ai/i) + (A2/2)2~lTeV
top partner
A2/2~0.7TeV
up, charm partners
300 GeV
real scalar
(60)
4. Phenomenology 4.1. Direct production
of little
partners
Precision electroweak constraints (which we will discuss in the next section) force the masses of the new states to be at or above 1 TeV, and therefore probably out of reach of the Tevatron. LHC ("Little Higgs Collider") little Higgs phenomenology is very exciting [37-44]. All new states may be within reach and give interesting signatures.
i' # Events/bin
(a)
(b)
Figure 9. a.) Lepton pairs from s-channel Z' production, mass distribution of lepton pairs.
MZ' b.) bump in the
invariant
little Z (Z'): Heavy neutral gauge bosons would be produced in the schannel in quark-antiquark collisions with large rates. An easy signature comes from the decay of the Z' to pairs of highly energetic leptons (Figure 9.a). A plot of the invariant mass of the lepton pairs should show a clear bump at the mass of the Z' (Figure 9.b). The reach in this channel is about 5-7 TeV 3 7 . little W ( W ) : The heavy SU{2) doublet gauge bosons W = (W° , W+ ) T are less straightforward to see. The reason is that to lowest
486
M.
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order in an expansion in mw/ f they couple an 5f/(2) wea k doublet fermion of the Standard Model Q to a heavy partner fermion as can be seen explicitly from the gauge couplings of the triplet fermions
gWui)
(61)
Thus W gauge bosons would always have to be produced in association with heavy "little quarks" (£/). This picture is not quite correct because we ignored mixing effects. In fact, little quarks mix with the light quarks Uh,eavy
(62)
= u- f
which results in v/f suppressed couplings of pairs of light quarks to W. For example, a charged W+ may be produced (Figure 10.a) and then decay into a light quark and a little quark (Figure 10.b) which then decays further as in Figure 10.d. However the rates are expected to be small because the W' is heavy and because of the v/f in the coupling.
w'
U (b)
(a)
Figure 10.
a.) W
production,
b.) U production,
c.) W
decay d.) U decay.
The signatures quoted here for little W's are unique to "simple little Higgs" models. Models based on product gauge groups such as \SU(2) x U(l)]2 have little W's which transform as triplets under SU(2)weak which can couple directly to pairs of light quarks. They are produced with larger
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487
cross sections and also contribute more strongly to precision electroweak observables. little quarks (U): Heavy U quarks may be pair produced directly via their coupling to gluons. However, because of mixing there is also the possibility of single U production as shown in Figure 10.c. The produced £/'s would subsequently decay into a light quark and a Standard Model gauge boson (U —> u + Z , [/ —» d + W). The rate is suppressed by v2/f2 from the coupling in the production cross section and the reach is estimated to be of the order 2-3 TeV. 4.2. Precision
electroweak
constraints
As we already showed in the Introduction, indirect constrains from precision electroweak (PEW) measurements on any new Physics in the TeV energy regime are very severe. Little Higgs models are no exception and we must check that there are significant regions of parameter space in which the model is not ruled out by PEW, and where the little Higgs mechanism solves the Higgs naturalness problem. In practice, the most significant effects arise from the exchange of the new neutral gauge boson, the Z'. The SU(3) model has three neutral gauge bosons which mix (in the SU(3) charge eigenstate basis: A3, As, Ax)• The mass eigenstates are the photon, Z, and a Z'. To understand all possible effects from the Z' we must work out it couplings. From the gauge couplings of the fermions ipU/Dip and scalars \D^(j)\2 we find the mass and couplings. Neglecting order one group theory factors, these couplings are \g2, f22, si 7 / 7 / j ' ^ghWuh -+
i
" ' ^iptp
g2v2Z'^
-> gZ'^U^ip
(63)
The Feynman diagrams corresponding to these three Z' interactions are shown in Figure 11. Integrating out the Z' at tree level by solving it's classical equations of motion we find the following effective Lagrangian :eff „
C 0 t r y ^ ), 2 VT - , y
^){h^D ^ h), +- r 2{^i - V T1 . , ^ >1
-
+
r
^
2 jh^D^h) ^ ;
( g 4 )
P These three operators correspond to new four fermion operators, modifications of the Z couplings to fermions, and modifications of the Z mass, respectively. In terms of Feynman diagrams they arise as shown in Figure 12. The four fermion operators are most strongly constrained by limits from
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Figure 11.
a.) Z' coupling to fermions,
b.) Z-Z'
Figure 12. Integrating out Z' to obtain a.) four fermion couplings c.) shift in Z mass.
mixing c.) Z'
operators,
mass.
b.)
shift in Z
LEP II and from atomic parity violation 2 ' 45 . Modifications of the Z couplings are constrained by Z-pole data such as Rb and A^B. Finally, shifts in the Z mass change the mass splitting between the Z and the W from the Standard Model prediction, deviations from the Standard Model value are usually parametrized with the p or T parameters. These constraints generally imply / > 2 — 4 TeV. A global fit of various little Higgs theories using a general operator analysis and the Han-Skiba precision electroweak matrix is currently being performed 4 ' 4 6 . For more details regarding precision electroweak constraints on little Higgs models see Refs. 47-53. 4.3. Precision
electroweak,
fine-tuning,
and
T-parity
The result of PEW is that the scale of the new particles in most little Higgs models is somewhat higher than 1 TeV. This implies some fine tuning as quantum corrections to the Higgs (mass) 2 parameter are directly proportional to the new particle masses. In particular, we would really like the mass of the top partner to be at or below 1 TeV, not 2-4 TeV. Since the naturalness problem we set out to solve involved fine tuning of order 1%,
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it is somewhat disappointing to find that fine-tuning is still on the order of 10%. Can we do better? The answer is "yes". To understand the problem better, note that the problematic contributions to PEW all stem from tree level exchange of little partners (as in Figure 12). However, naturalness only requires the new particles to appear in loops in order to cancel quadratic divergences. Thus a solution would be to forbid all tree level couplings of little partners to Standard Model particles while keeping the loops. But how can this be enforced in a natural way? To see how it might work, note that the MSSM does not appear to have this problem. In the MSSM superpartner masses can be well below 1 TeV without causing problems with PEW. Why is that? The answer is R-parity. R-parity has a reputation of being the ugliest part of the MSSM but with regards to PEW R-parity is really the MSSM's best part. Recall that R parity is defined such that all Standard Model particles are R-parity even and all superpartners are odd. If R-parity is exact then all interactions must involve an even number of superpartners. Interactions of single superpartners with Standard Model fields are forbidden. Therefore contributions from superpartners to processes in which all external states are Standard Model particles are loop suppressed as shown in Figure 13. Thus contributions to PEW from superpartners suppressed by l/(167r 2 ) mw/^susY which is sufficiently small even for MSUSY < 1 TeV.
(a)
/
Figure 13.
(b)
a.) forbidden by R-parity,
,
b.) superpartner
loop, allowed.
The equivalent of R-parity in the case of the little Higgs models has been dubbed T-parity by Cheng and Low 24>27>30. Under T-parity all Standard Model fields are even whereas little partners are odd. Just like in the case of supersymmetry, this forbids the dangerous contributions to PEW while allowing the cancellation of quadratic divergences. The difficult part, of course, is to construct a model in which T-parity can be consistently imposed. It appears that this is impossible to do with any of the "simple
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group" models, such as our favorite SU(3) model. However, it can be done with all the other little Higgs models. In models with T-parity the masses of little partners can be smaller t h a n 1 TeV. T h e phenomenology of such a model was recently discussed in Ref. 54. Acknowledgements I wish to t h a n k the TASI organizers for inviting me lecture a t this exciting summer school. I particularly enjoyed the enthusiasm of this year's students. References 1. S. Eidelman et al. [Particle Data Group], "Review of particle physics," Phys. Lett. B 592, 1 (2004). 2. t. L. E. Group, t. S. Electroweak and H. F. Groups [OPAL Collaboration], "A combination of preliminary electroweak measurements and constraints on the [arXiv:hep-ex/0412015]. 3. G. D'Ambrosio, G. F. Giudice, G. Isidori and A. Strumia, "Minimal flavour violation: An effective field theory approach," Nucl. Phys. B 645, 155 (2002) [arXiv:hep-ph/0207036]. 4. Z. Han and W. Skiba, "Effective theory analysis of precision electroweak data," arXiv:hep-ph/0412166. 5. TASI 2004: H. Haber, "Introduction to Supersymmetry"; M. Luty, "Supersymmetry Breaking". 6. TASI 2004: S. Chivukula and E. Simmons "Strong Interaction Models". 7. D. B. Kaplan and H. Georgi, "SU(2) X U(l) Breaking By Vacuum Misalignment," Phys. Lett. B 136, 183 (1984). 8. D. B. Kaplan, H. Georgi and S. Dimopoulos, "Composite Higgs Scalars," Phys. Lett. B 136, 187 (1984). 9. H. Georgi and D. B. Kaplan, "Composite Higgs And Custodial SU(2)," Phys. Lett. B 145, 216 (1984). 10. TASI 2004: R. Sundrum, "Introduction to Extra Dimensions"; C. Csaki, "Electroweak in Extra Dimensions"; G. Kribs, "Phenomenology in Extra Dimensions". 11. H. Georgi and A. Pais, "Calculability And Naturalness In Gauge Theories," Phys. Rev. D 10, 539 (1974). 12. H. Georgi and A. Pais, "Vacuum Symmetry And The Pseudogoldstone Phenomenon," Phys. Rev. D 12, 508 (1975). 13. N. Arkani-Hamed, A. G. Cohen and H. Georgi, "Electroweak symmetry breaking from dimensional deconstruction," Phys. Lett. B 513, 232 (2001) [arXiv:hep-ph/0105239]. 14. N. Arkani-Hamed, A. G. Cohen, T. Gregoire and J. G. Wacker, "Phenomenology of electroweak symmetry breaking from theory space," JHEP 0208, 020 (2002) [arXiv:hep-ph/0202089].
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15. N. Arkani-Hamed, A. G. Cohen, E. Katz, A. E. Nelson, T. Gregoire and J. G. Wacker, "The minimal moose for a little Higgs," JHEP 0208, 021 (2002) [arXiv:hep-ph/0206020]. 16. N. Arkani-Hamed, A. G. Cohen, E. Katz and A. E. Nelson, "The littlest Higgs," JHEP 0207, 034 (2002) [arXiv:hep-ph/0206021]. 17. T. Gregoire and J. G. Wacker, "Mooses, topology and Higgs," JHEP 0208, 019 (2002) [arXiv:hep-ph/0206023]. 18. I. Low, W. Skiba and D. Smith, "Little Higgses from an antisymmetric condensate," Phys. Rev. D 66, 072001 (2002) [arXiv:hep-ph/0207243]. 19. M. Schmaltz, "Physics beyond the standard model (Theory): Introducing the little Higgs," Nucl. Phys. Proc. Suppl. 117, 40 (2003) [arXivihepph/0210415]. 20. D. E. Kaplan and M. Schmaltz, "The little Higgs from a simple group," JHEP 0310, 039 (2003) [arXiv:hep-ph/0302049]. 21. S. Chang and J. G. Wacker, "Little Higgs and custodial SU(2)," Phys. Rev. D 69, 035002 (2004) [arXiv:hep-ph/0303001]. 22. W. Skiba and J. Terning, "A simple model of two little Higgses," Phys. Rev. D 68, 075001 (2003) [arXiv:hep-ph/0305302]. 23. S. Chang, "A 'littlest Higgs' model with custodial SU(2) symmetry," JHEP 0312, 057 (2003) [arXiv:hep-ph/0306034]. 24. H. C. Cheng and I. Low, "TeV symmetry and the little hierarchy problem," JHEP 0309, 051 (2003) [arXiv:hep-ph/0308199]. 25. E. Katz, J. y. Lee, A. E. Nelson and D. G. E. Walker, "A composite little Higgs model," arXiv:hep-ph/0312287. 26. A. Birkedal, Z. Chacko and M. K. Gaillard, "Little supersymmetry and the supersymmetric little hierarchy problem," JHEP 0410, 036 (2004) [arXiv:hep-ph/0404197]. 27. H. C. Cheng and I. Low, "Little hierarchy, little Higgses, and a little symmetry," JHEP 0408, 061 (2004) [arXiv:hep-ph/0405243]. 28. D. E. Kaplan, M. Schmaltz and W. Skiba, "Little Higgses and turtles," Phys. Rev. D 70, 075009 (2004) [arXiv:hep-ph/0405257]. 29. M. Schmaltz, "The simplest little Higgs," JHEP 0408, 056 (2004) [arXiv:hepph/0407143]. 30. I. Low, "T parity and the littlest Higgs," JHEP 0410, 067 (2004) [arXiv:hepph/0409025]. 31. K. Agashe, R. Contino and A. Pomarol, "The minimal composite Higgs model," arXiv:hep-ph/0412089. 32. P. Batra and D. E. Kaplan, "Perturbative, non-supersymmetric completions of the little Higgs," arXiv:hep-ph/0412267. 33. J. Thaler and I. Yavin, "The littlest Higgs in anti-de Sitter space," arXiv:hepph/0501036. 34. C. G. Callan, S. R. Coleman, J. Wess and B. Zumino, "Structure Of Phenomenological Lagrangians. 2," Phys. Rev. 177, 2247 (1969). 35. O. C. W. Kong, "A completed chiral fermionic sector model with little Higgs," arXiv:hep-ph/0307250.
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36. O. C. W. Kong, "Little Higgs model completed with a chiral fermionic sector," Phys. Rev. D 70, 075021 (2004) [arXiv:hep-ph/0409238]. 37. G. Azuelos et al., "Exploring little Higgs models with ATLAS at the LHC," arXiv:hep-ph/0402037. 38. J. L. Hewett, F. J. Petriello and T. G. Rizzo, "Constraining the littlest Higgs. ((U))," JHEP 0310, 062 (2003) [arXiv:hep-ph/0211218]. 39. G. Burdman, M. Perelstein and A. Pierce, "Collider tests of the little Higgs model," Phys. Rev. Lett. 90, 241802 (2003) [Erratum-ibid. 92, 049903 (2004)] [arXiv:hep-ph/0212228]. 40. T. Han, H. E. Logan, B. McElrath and L. T. Wang, "Phenomenology of the little Higgs model," Phys. Rev. D 67, 095004 (2003) [arXiv:hep-ph/0301040]. 41. M. Perelstein, M. E. Peskin and A. Pierce, "Top quarks and electroweak symmetry breaking in little Higgs models," Phys. Rev. D 69, 075002 (2004) [arXiv:hep-ph/0310039]. 42. T. Han, H. E. Logan, B. McElrath and L. T. Wang, "Loop induced decays of the little Higgs: H -+ g g, gamma gamma," Phys. Lett. B 563, 191 (2003) [Erratum-ibid. B 603, 257 (2004)] [arXiv:hep-ph/0302188]. 43. H. E. Logan, "The littlest Higgs boson at a photon collider," Phys. Rev. D 70, 115003 (2004) [arXiv:hep-ph/0405072]. 44. W. Kilian, D. Rainwater and J. Reuter, "Pseudo-axions in little Higgs models," Phys. Rev. D 71, 015008 (2005) [arXiv:hep-ph/0411213]. 45. C. S. Wood, S. C. Bennett, D. Cho, B. P. Masterson, J. L. Roberts, C. E. Tanner and C. E. Wieman, "Measurement of parity nonconservation and an anapole moment in cesium," Science 275, 1759 (1997). 46. Z. Han and W. Skiba, to appear. 47. C. Csaki, J. Hubisz, G. D. Kribs, P. Meade and J. Terning, "Big corrections from a little Higgs," Phys. Rev. D 67, 115002 (2003) [arXiv:hep-ph/0211124]. 48. C. Csaki, J. Hubisz, G. D. Kribs, P. Meade and J. Terning, "Variations of little Higgs models and their electroweak constraints," Phys. Rev. D 68, 035009 (2003) [arXiv:hep-ph/0303236]. 49. T. Gregoire, D. R. Smith and J. G. Wacker, "What precision electroweak physics says about the SU(6)/Sp(6) little Higgs," Phys. Rev. D 69, 115008 (2004) [arXiv:hep-ph/0305275]. 50. M. C. Chen and S. Dawson, "One-loop radiative corrections to the rho parameter in the littlest Higgs model," Phys. Rev. D 70, 015003 (2004) [arXiv:hepph/0311032]. 51. R. Casalbuoni, A. Deandrea and M. Oertel, "Little Higgs models and precision electroweak data," JHEP 0402, 032 (2004) [arXiv:hep-ph/0311038]. 52. C. Kilic and R. Mahbubani, "Precision electroweak observables in the minimal moose little Higgs model," JHEP 0407, 013 (2004) [arXivihepph/0312053]. 53. W. Kilian and J. Reuter, "The low-energy structure of little Higgs models," Phys. Rev. D 70, 015004 (2004) [arXiv:hep-ph/0311095]. 54. J. Hubisz and P. Meade, "Phenomenology of the littlest Higgs with T-parity," arXiv:hep-ph/0411264.
MARKUS A. LUTY
SUPERSYMMETRY BREAKING
MARKUS A. LUTY Physics Department University of Maryland College Park, MD 20742 [email protected]
These lectures give an introduction to the problem of finding a realistic and natural extension of the standard model based on spontaneously broken supersymmetry. Topics discussed at some length include the effective field theory paradigm, coupling constants as superfleld spurions, gauge mediated supersymmetry breaking, and anomaly mediated supersymmetry breaking, including an extensive introduction to supergravity relevant for phenomenology.
Contents 1
Introduction
497
2
Effective field theory and naturalness
499
2.1 2.2 2.3 2.4 2.5 2.6 2.7
499 501 502 503 504 504 505
3
Matching in a toy model Relevant, irrelevant, and marginal operators Naturally small parameters and spurions Fine tuning in a toy model Fine tuning versus quadratic divergences Fine tuning versus small parameters To tune or not to tune?
Model-building boot camp 3.1 The standard model 3.2 The GIM mechanism 3.3 Accidental symmetries 3.4 Neutrino masses 3.5 Extending the standard model 495
507 507 510 510 511 512
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M. A. Luty
4
T h e minimal supersymmetric standard model 4.1 Superfields and couplings 4.2 R parity
512 512 515
5
Soft 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10
516 516 517 519 525 527 528 529 530 531 534
6
Spontaneous S U S Y breaking 6.1 F-type breaking of SUSY 6.2 D-type breaking of SUSY 6.3 Generalities of tree-level SUSY breaking 6.4 SUSY breaking in the observable sector 6.5 The messenger paradigm
535 536 539 540 542 544
7
Hidden sector S U S Y breaking 7.1 The 'minimal SUGRA' ansatz
544 549
8
Gauge mediated S U S Y breaking 8.1 Gauge messengers 8.2 The gauge mediated spectrum 8.3 Phenomenology of gauge mediation 8.4 Gravitino cosmology
549 549 552 557 559
9
' N e e d - t o - k n o w ' supergravity 9.1 SUSY breaking in SUGRA: Polonyi model 9.2 'No scale' SUSY breaking 9.3 The SUGRA potential
559 566 568 569
S U S Y breaking Coupling constants as superfields Superfield couplings in perturbation theory Soft SUSY breaking from superfield couplings Renormalization group equations for soft SUSY breaking Soft SUSY breaking in the MSSM The SUSY flavor problem The fi problem The SUSY CP problem The SUSY fine-tuning problem The next-to-minimal supersymmetric standard model
10 A n o m a l y mediated S U S Y breaking 10.1 The fj, problem in anomaly mediation 10.2 Anomaly-mediated phenomenology
571 576 576
Supersymmetry Breaking 497 10.3 Naturalness of anomaly mediation
577
11 Gaugino mediation
578
12 No conclusion
579
1. Introduction Our present understanding of particle physics is based on effective quantum field theory. Quantum field theory is the inevitable result of combining quantum mechanics and special relativity, the two great scientific revolutions of the early twentieth century. An effective quantum field theory is one that includes only the degrees of freedom that are kinematically accessible in a particular class of experiments. Presently, the highest energies probed in accelerator experiments are in the 100 GeV range, and the standard model is an effective quantum field theory that describes all physical phenomena at energies of order 100 GeV and below. The standard model contains a Higgs scalar that has not been observed as of this writing. Therefore, the minimal effective theory that describes the present data does not contain a Higgs scalar. This effective theory allows for the possibility that the dynamics that breaks electroweak symmetry does not involve elementary scalars (as in 'technicolor' theories, for example). This effective theory necessarily breaks down at energies of order a TeV, and therefore new physics must appear below a TeV. This is precisely the energy range that will be explored by the LHC starting in 2007-2008, which is therefore all but guaranteed to discover the interactions that give rise to electroweak symmetry breaking, whether or not it involves a Higgs boson. If the physics that breaks electroweak symmetry does not involve particles with masses below a TeV, then it must be strongly coupled at a TeV. Contrary to what is sometimes stated, precision electroweak experiments have not ruled out theories of this kind. For example, if we estimate the size of the S and T parameters assuming that the electroweak symmetry breaking sector is strongly coupled at a TeV with no large or small parameters, we obtain AS~-, IT
AT~-!-,
(1)
4lT
which are near the current experimental limits. The strongly-coupled models that are ruled out are those that contain iV ;g> 1 degrees of freedom at the TeV scale; in these models the estimates for S and T above are multiplied by N. Another difficulty with building models of strongly-coupled
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electroweak symmetry breaking is incorporating quark mixing without large flavor-changing neutral current effects. However, given our profound ignorance of strongly-coupled quantum field theory, it may be prudent to keep an open mind. Fortunately, the LHC will tell us the answer soon. The subject of these lectures is weakly-coupled supersymmetry ('SUSY'). As you have heard in previous lectures, there are several hints that SUSY is correct. First, the simplest supersymmetric grand unified models predict a precise relation among the three gauge couplings of the standard model that is in excellent agreement with observation. Second, the best fit to precision electroweak data is obtained with a Higgs boson with a mass close to the experimental lower limit of 114 GeV from LEP. Such a light Higgs boson arises automatically in SUSY. Finally, SUSY naturally contains a viable cold dark matter candidate. If SUSY is in fact discovered at the LHC, it will be the culmination of decades of work by many hands, starting with general theoretical investigations of spacetime symmetries and the construction of supersymmetric quantum field theories, to the realization that SUSY can solve the hierarchy problem and the construction of realistic models of broken supersymmetry. It will be an intellectual triumph comparable to general relativity, another physical theory that fundamentally changed our view of space and time, and which was also proposed based on very general theoretical considerations and later spectacularly verified by experiment. If SUSY is realized in nature, it must be broken. In this case, the pattern of SUSY breaking can give us a great deal of information about physics at much higher energy scales. There are a number of theoretically well-motivated mechanisms for SUSY breaking, each of which give distinct patterns of SUSY breaking that can be experimentally probed at the weak scale. If nature is supersymmetric at the weak scale, then the experimental program in particle physics after the LHC turns on will be largely the study of superpartners. These are exciting prospects, but we are not there yet. There is at present no direct experimental evidence for superpartners, or a light Higgs boson. Indirect experimental constraints place strong constraints on the simplest SUSY models, requiring either accidental cancelations (fine tuning) or additional non-minimal structure in the theory. It is premature to say that these constraints rule out the idea of SUSY, but they must be addressed by any serious proposal for weak scale SUSY. These lectures will review both the progress that has been made in constructing realistic models of SUSY breaking, and the problems faced
Supersymmetry
Breaking
499
by SUSY in general, and these models in particular. It is my hope that these lectures will challenge and inspire—rather than discourage—the next generation of particle physicists. 2. Effective field theory and naturalness If we take seriously the idea that the standard model is an effective field theory, then the coupling constants of the standard model are not to be viewed as fundamental parameters. Rather they are to be thought of as effective couplings determined by a more fundamental theory. How do we know there is a more fundamental theory? For one thing, we hope that there is a more fundamental theory that explains the ~ 20 free parameters of the standard model (mostly masses and mixings). One piece of evidence for a simpler fundamental theory comes from the fact that the standard model gauge couplings approximately unify at a scale MGUT ~ 10 16 GeV. This is evidence that M G U T is a scale of new physics described by a more fundamental theory. Another evidence for a new scale is the fact that gravity becomes strongly interacting at the scale Mp ~ 1019 GeV, and we expect new physics at that scale. Finally, as we review below, the recent observation of neutrino mixing suggests the existence of another scale in physics of order 10 15 GeV. These scales are so large that we cannot ever hope to probe them directly in accelerator experiments. The best that we can do is to understand how the effective couplings that we can measure are determined by the more fundamental theory. 2.1. Matching
in a toy
model
Let us consider a simple example that shows how effective couplings are determined from an underlying theory. Consider a renormalizable theory consisting of a real scalar h coupled to a Dirac fermion field tp: C = ipi$il) + \(dhf
-\M2h2-^-hA
+ yhi>ilj.
(2)
This theory has a discrete chiral symmetry i> >-> 75^,
h
H->
-h,
(3)
that forbids a fermion mass term, since -ipip — t > —*!/>?/>. For processes with energy E
500
M. A. Luty
We determine the effective theory by matching to the fundamental theory. Let us consider fermion scattering. In the fundamental theory, we have at tree level + crossed,
(4)
\ while in the effective theory the scattering comes from an effective 4-fermion coupling:
(5)
Matching is simply demanding that these two expressions agree order by order in an expansion in 1/M. This corresponds to expanding the scalar propagator in inverse powers of the large mass: 1 p2 - M2
-w-w
+
°^M^-
(6)
Equivalently, we can solve the classical equations of motion for h in the fundamental theory order by order in 1/M2: h =
i
yipi/j-
nh-±h3
JLipip-JLn(w)
(7)
+ o(i/M6).
(8)
Substituting this into the lagrangain, we obtain the effective Lagrangian at tree level
Ceff
y
^H +2i^H^r M2Vrry
y
4 - 2M -~Mum)^Oi\iM%
(9)
This effective Lagrangian will give the same results as the Lagrangian Eq. (2) for all tree level processes, up to corrections of order 1/M 6 . We can continue the matching procedure to include loop corrections. The new feature that arises here is the presence of UV divergences in both the fundamental and effective theories. However, the matching ensures that the results in the effective theory are finite, and this determines the counterterms that are required in the effective theory. The result is that the renormalized couplings in the effective theory are a power series in 1/M even at loop level.
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Exercise 1: Carry out the one-loop matching of the four fermion vertex using a momentum space cutoff regulator in both the fundamental and the effective theory. If we write the effective Lagrangian as £ eff = jriM + G(^)2
+ 0(1/M4),
(10)
show that G
y2 |\ 2M 2 1
+
ci« 2 A 2 y2 + 1 6 ^ 1 6 ^ l
(
, A M +
C 2 l n
C 3
(11)
where A is the momentum space cutoff and cy^fi are numbers of order 1. Note that all the non-analytic behavior of the loop amplitudes cancels in the matching computation. Compute the fermion-fermion scattering amplitude at one loop in the effective theory and show that it is independent of A at one-loop order.
What is the value of A that is appropriate for a matching calculation? Renormalization theory tells us that the physics is insensitive to the value of the cutoff, and so it does not matter. To get complete cutoff sensitivity one should take the cutoff to infinity, in which case the matching gives a relation among renormalized couplings in the fundamental and effective theory. Note that the couplings in the effective theory are determined by dimensional analysis in the scale M, provided that the dimensionless couplings y and A in the fundamental theory are order 1. If the scale M is much larger than the energy scale E that is being probed, then the effects of the higher-order effects in the 1/M 2 expansion are very small. 2.2. Relevant,
irrelevant,
and marginal
operators
The features we have seen in the example above are very general. A general effective Lagrangian is denned by the particle (field) content and the symmetries. An effective Lagrangian in principle contains an infinite number of operators, but only a finite number of them are important at a given order in the expansion in 1/M 2 , where M is the scale of new physics. It is therefore useful to classify the terms in the effective Lagrangian according to their dimension. An operator of dimension d will have coefficient
A£eff
~ w^°d-
(12)
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M. A. Luty
Inserting this term as a first-order perturbation to a given amplitude, we obtain an expansion of the form A ~ Ao 1 +
E d-i
(13)
Md-4
where E is the kinematic scale of the physical process. We see that the effects of operators with dimension d > 4 decrease at low energies, and these are called 'irrelevant' operators. The effects of operators with d = 4 are independent of energy, and these operators are called 'marginal.' Finally, the effects of operators with d < 4 increase with energy, and these operators are called 'relevant.' Loop effects change the scaling behavior of couplings. For weakly coupled theories, the most important change is that dimensionless couplings run logarithmically. The effects of quantum corrections on scaling behavior can be much more dramatic in strongly-coupled conformal field theories. See the lectures by Ann Nelson at this school. For irrelevant and marginal operators, it makes sense for their coefficient to be fixed by dimensional analysis at the scale M. But if a relevant operator has a coefficient fixed by dimensional analysis at the scale M, then its effects are not small below the scale M. In this case, the simple picture based on dimensional analysis cannot be correct. 2.3. Naturally
small parameters
and
spurious
In the toy model considered above, there is a relevant operator that could be added to the effective Lagrangian, namely a fermion mass term AC = miprp.
(14)
However, this was forbidden by the chiral symmetry Eq. (3), and so it is natural to omit this term. As we now discuss, this symmetry also means that it is natural for the mass to be nonzero but very small, i.e. m
A
16^CmlnM'
(15)
where c ~ 1. The size of the loop correction is controlled byTOrather than M because an insertion of m is required to break the symmetry.
Supersymmetry
Breaking
503
This can be formalized in the following way. We can say that the mass parameter m transforms under the discrete symmetry Eq. (3) as m
H->
— m.
(16)
What this means is that if we view m as a parameter, all expressions must depend on m in such a way that the chiral symmetry including the transformation Eq. (16) is a good symmetry. We can think of m as a field, and the numerical value of m as a vacuum expectation value for the field. We say that m is a 'spurion field.' This spurion analysis immediately tells us that quantities that are even under the chiral symmetry will depend only on even powers of m, while quantities that are odd will depend on odd powers of m. This kind of spurion analysis will be very useful when we consider SUSY breaking. 2.4. Fine tuning in a toy model To illustrate the problem with relevant operators that are not forbidden by any symmetry, let us consider another example of a light real scalar field <> / coupled to a heavy Dirac fermion &: A A C = ±{d(f>)2 - \m24>2 - -4> + ^i$^ - M * # + # * * . C 4!
(17)
Note that the mass operator (f>2 has d = 2, and is therefore relevant. The mass of the scalar cannot be forbidden by any obvious symmetry, but we simply assume that m? is chosen to make the scalar lighter than the fermion. We can describe processes with energies E
2 ,
y
16?r2
ClA
2
+ c2m2 In - + c3M2 +
0(M4/A2)
(18)
where A is the cutoff used. If we use dimensional regularization in 4 — e dimensions and minimal subtraction, then we obtain m2eS=m2
+^ [ ^ m
2
+ c3M2 + 0(e)}.
(19)
In either case, we can write this in terms of the renormalized mass m2(/j, = M): m 2 ff ( M = M ) = m 2 ( M = M) + | | ^ M 2 .
(20)
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M. A. Luty
In this expression, the cutoff dependence has disappeared, but the dependence on the (renormalized) mass M remains. This shows that if we want to make the scalar light compared to the scale M, we must tune the renormalized couplings in the fundamental theory so that there is a cancellation between the terms on the right-hand side of Eq. (20). The accuracy of this fine tuning is of order y2m2/(16ir2M2). There is no obvious symmetry or principle that makes the scalar naturally light in this model.
2.5. Fine tuning versus quadratic
divergences
We see that we must fine-tune parameters whenever the low-energy effective Lagrangian contains a relevant operator that cannot be forbidden by symmetries. The naturalness problem for scalar mass parameters is often said to be a consequence of the fact that scalar mass parameters are quadratically divergent in the UV. We have emphasized above that the fine-tuning can be formulated in terms of renormalized quantities, and has nothing to do with the regulator used. (In particular, we have seen in the example above that fine tuning can be present in dimensional regularization, where there are no quadratic divergences.) The naturalness problem is simply the fact that relevant operators that are not forbidden by symmetries are generally sensitive to heavy physical thresholds in the theory. Although fine-tuning can be formulated without reference to UV divergences, there is a close connection that is worth commenting on. We can view a regulator for UV divergences as a UV modification of the theory that makes it finite. The dependence on the cutoff A can therefore be viewed as dependence on a new heavy threshold.
2.6. Fine tuning versus small
parameters
Not all small parameters are finely tuned. We have seen in the first toy model above that a small fermion mass is not fine tuned, because there is an additional symmetry that results when the mass goes to zero. This automatically ensures that the radiative corrections to the fermion mass are small. In this case, we say that the small parameter is 'protected by a symmetry.' There is another general mechanism by which a parameter in an effective Lagrangian can be naturally small, and that is if two sectors of the theory completely decouple as the parameter is taken to zero. As an example, consider a theory of two real scalars with Lagrangian
Supersymmetry
Breaking
505
C = \{d^f - \m\fi + \(902)2 - \ml4>\ - ±4>\ (21) We have forbidden terms that with odd powers of the scalar fields with discrete symmetries 01 ,-> - 0 1 .
02 •-> 02,
(22)
and ?1 l - K p i ,
(23)
This symmetry also forbids mixing terms of the form 0102- If we take re —> 0, the theory becomes the sum of two 'superselection sectors,' i.e. two theories that are not coupled to each other. It is therefore natural to take re <^ Ai,A2- It is also easy to see that any radiative correction to the coupling re is proportional to re itself. In this case, we say that the parameter re is small because of an 'approximate superselection rule.' Approximate superselection rules explain why it is natural for the electromagnetic coupling to be weaker than the strong coupling. If we take the electromagnetic coupling to zero, the theory splits into superselection sectors, consisting of QCD and a free photon. Approximate superselection rules also explain why it is natural for gravity to be much weaker than the standard model gauge interactions. a
2.7. To tune or not to tune? Is fine tuning really a problem? If we want to explain the effective couplings of the standard model in terms of a more fundamental underlying theory, then it is at least disturbing that the underlying couplings must be adjusted to fantastic accuracy in order to reproduce even the qualitative features of the low-energy theory. A fine tuned theory is like finding a pencil balancing on its tip: it is possible that it arises by accident, but one suspects that there is a stabilizing force. a
Gravity couples to all forms of matter with universal strength, and therefore sets the
ultimate limit on how decoupled two approximate superselection sectors can be.
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M. A. Luty
Recently, the possibility that the standard model may be fine tuned has received renewed attention, motivated in part by the fact that there is another grave naturalness problem: the cosmological constant problem. In general relativity, there is an additional relevant operator that must be added to the Lagrangian, namely the unit operator: A£ c c = - A 4 .
(24)
This can be thought of as a constant vacuum energy, which is not observable in the absence of gravity. In the presence of gravity, Eq. (24) is covariantized to A£cc = -\/det(3^)A4,
(25)
where gM„ is the metric field. This coupling means that vacuum energy couples to gravity. The term Eq. (24) gives rise to the infamous cosmological constant term in Einstein's equations, which gives rise to a nonzero spacetime curvature at a length scale
In order to explain the present universe, L must be at least of order the size of the present Hubble horizon, I/Hubbie ~ 10 32 cm ~ 1 0 - 4 2 G e V - 1 . This requires A < 10~ 3 eV. This is an enormous problem, because loops of particles with mass M give rise to a correction to the vacuum energy of order AC ~ ~ M I67H
4
.
(27)
For M ~ Mz ~ 100 GeV, this is too large by 54 orders of magnitude! No one has ever found a symmetry that can cancel this contribution to the required accuracy. This problem has prompted some physicists to consider the possibility that there could be a kind of anthropic selection process at work in nature. The idea is that there are in some sense many universes with different values of the effective couplings, and we live in one of the few that are compatible with our existence. If the cosmological constant were much larger than A ~ 10~ 3 eV, then structure could not form in the universe [1].
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The fact that cosmological observations favor a value of the cosmological constant in this range has given added impetus to this line of thinking. Also, string theory appears to have a large number of possible ground states, as required for anthropic considerations to operate. For recent discussions, see e.g. Refs. [2]. There is another possibility to save naturalness as we know it, advocated in Ref. [3]. The point is that although we have tested the standard model to energies of order 100 GeV, we have only tested gravity to a much lower energy scale, or much longer distances. The shortest distances probed in present-day gravitational force experiments are presently of order 0.1 mm, corresponding to an energy scale of 10~ 3 eV. If we assume that new gravitational physics comes in at the 0.1 mm scale, the small value of the cosmological constant may be natural. Note that this approach also predicts that the cosmological constant should be nonzero and close to its experimental value. The difficulty with this approach is that it is not known how to modify Einstein gravity in a consistent way to cut off the contributions to the cosmological constant. However, given our ignorance of UV completions of gravity, we should perhaps keep an open mind. See Ref. [4] for a recent idea along these lines. These are interesting ideas, and worth pursuing. But in these lectures I will assume that naturalness is a good guide to non-gravitational physics at least.
3. Model-building boot camp We are now ready to start building effective field theory models. If we believe in the naturalness principle articulated in the previous section, then the models should be defined by specifying the particle content and the symmetries of the theory. Then we should write down all possible couplings consistent with the symmetries.
3.1. The standard
model
Let us apply these ideas to the standard model. The standard model is defined to be a theory with gauge group SU(3)C x SU(2)W
x U(1)Y.
(28)
The fermions of the standard model can be written in terms of 2-component
508
M. A. Luty
Weyl spinor fields as b Q'~(3,2)+i, (Wcr~(3,l)_f, (dT~(3,l)+i, 2/~(l,2)_
(29)
2 '
(ecr~(l,l)+1, where i = 1,2,3 is a generation index. In addition, the model contains a single scalar multiplet tf~(l,2)+i.
(30)
According to the ideas above, we must now write the most general interactions allowed by the symmetries. The most important interactions are the marginal and relevant ones. The marginal interactions include kinetic terms for the Higgs field, the fermion fields, and the gauge fields: ^kinetic = (D^H^D^H
+ QliafDpQi
+ •••-
{B^B^
+ •••
(31)
Note that these include the gauge self interactions. Also marginal is the quartic interaction for the Higgs A£ q u a r t i c = -\{&H?
(32)
and Yukawa interactions: A^Yukawa = ( y u ) « Q < H ( « C ) J ' + ( j / d J i j Q ' ^ W + (Veh^H
V )
j
•
(33)
Note that the Yukawa interactions are the only interactions that break a SU (3) 5 global symmetry that would otherwise act on the generation indices of the fermion fields. This means that the Yukawa interactions can be naturally small without any fine tuning. This is reassuring, since it means that the small electron Yukawa coupling ye ~ 10~ 5 is perfectly natural. Finally, the marginal interactions include 'vacuum angle' terms for each of the gauge groups: £vacuumangle = ^ B ^ B ^
+ ^
tv(W^WMV)
+ ^
t
v ( G ^ G ^ ) ,
(34) b
We use the spinor conventions of Wess and Bagger [5], which have become conventional
in the SUSY literature. See e.g. Ref. [6] for a pedagogical introduction.
Supersymmetry
Breaking
509
where B M " = \t^vpaBpo, etc. These terms break CP, and are therefore very interesting. These terms are total derivatives, e.g. B^B„U
= d^K^,
K" = \t'""mAvFpa.
(35)
This is enough to ensure that they do not give physical effects to all orders in perturbation theory. They can give non-perturbative effects with parametric dependence ~ e1^9 , but these are completely negligible for the SU(2)w x U(l)y terms, since these gauge couplings are never strong. The strong vacuum angle gives rise to CP-violating non-perturbative effects in QCD, most importantly the electric dipole moment of the neutron. Experimental bounds on the neutron electric dipole moment require 0 3 < 10 10 . Explaining this small number is the 'strong CP problem.' There are a number of proposals to solve the strong CP problem. For example, there may be a spontaneously broken Peccei-Quinn symmetry [7] leading to an axion [8], or there may be special flavor structure at high scales that ensures that the determinant of the quark masses is real [9]. There is one relevant interaction that is allowed, namely a mass term for the Higgs field: ^relevant = -mjjH^
H.
(36)
Note that mass terms for the fermions such as Lec are not gauge singlets, and therefore forbidden by gauge symmetry. The Higgs mass parameter cannot be forbidden by any obvious symmetry, and therefore must be fine tuned in order to be light compared to heavy thresholds such as the GUT scale. For example, in GUT models there are massive gauge bosons with masses of order M Q U T that couple to the Higgs with strength g, where g is the unified gauge coupling. These will contribute to the effective Higgs mass below the GUT scale Am 2 , ~ ^ g p i ~ 10 30 GeV 2
(37)
for M Q U T ~ 1016 GeV. In order to get a Higgs mass of order 100 GeV we must fine tune to one part in 10 26 ! We can turn this around and ask what is the largest mass threshold that is naturally compatible with the existence of a light Higgs boson. The top quark couples to the Higgs with coupling strength yt ~ 1, and top quark loops give a quadratically divergent contribution to the Higgs mass.
510
M. A. Luty
Assuming that this is cut off by a new threshold at the scale M, we find a contribution to the Higgs mass of order Am
^~T6^'
(38)
which is naturally small for M < 1 TeV. We get a similar estimate for M from loops involving SU{2) x U{1) gauge bosons. So the standard model is natural as an effective field theory only if there is new physics at or below a TeV. This is the principal motivation for the Large Hadron Collider (LHC) at CERN, which will start operation in 2007-2008 with a center of mass energy of 14 TeV. It is expected that the LHC will discover the mechanism of electroweak symmetry breaking and the new physics that makes it natural. 3.2. The GIM
mechanism
One very important feature of the standard model is that it violates flavor in just the right way. The quark mass matrices are proportional to the up-type and down-type Yukawa couplings. Diagonalizing the quark mass matrices requires that we perform independent unitary transformations on the two components of the quark doublet Qj. This gives rise to the CKM mixing matrix, which appears in the interactions of the mass eigenstate quarks with the W± ('charged currents'). Crucially, the interactions with the photon and the Z ('neutral currents') are automatically diagonal in the mass basis. This naturally explains the phenomenology of flavor-changing decays observed in nature, including the 'GIM suppression' of flavor changing neutral current processes such as K°-K° mixing. For our purposes, what is important is that this comes about because the quark Yukawa couplings are the only source of flavor violation in the standard model. If there were other couplings that violated quark flavor, these would not naturally be diagonal in the same basis that diagonalized the quark masses, and would in general lead to additional flavor violation. A simple example of this is a general model with 2 Higgs doublets, in which there are twice as many Yukawa coupling matrices. 3.3. Accidental
symmetries
It is noteworthy that the standard model was completely defined by its particle content gauge symmetries. In particular, we did not have to impose any additional symmetries to suppress unwanted interactions. If we
Supersymmetry
Breaking
511
look back at the terms we wrote down, we see that all of the relevant and marginal interactions are actually invariant under some additional global symmetries. One of these is baryon number, a U(l) symmetry with charges
B(Q) = l
B(u<) = B(d<) = - i ,
B(L) = B(ec) = B{H) = 0. Another symmetry is lepton number, another U{\) symmetry with charges L{Q) = L{uc) = L{dc) = 0, c
L(e ) = - 1 ,
L{L) = + 1 , (40)
L{H) = 0.
These symmetries can be broken by higher-dimension operators. For example, the lowest-dimension operators that violate baryon number are dimension 6: A £ ~ ~QQQL
+ -l^ucucdcec,
(41)
where the color indices are contracted using the SU(3)c invariant antisymmetric tensor. Consistency with the experimental limit on the proton lifetime of 10 33 yr gives a bound M > 1022 GeV. Although this is larger than the Planck mass, these couplings also violate flavor symmetries, and it seems reasonable that whatever explains the small values of the light Yukawa couplings can suppresses these operators. A very appealing consequence of this is that if the standard model is valid up to a high scale M, then the proton is automatically long-lived, without having to assume that baryon number is an exact or approximate symmetry of the fundamental theory. Baryon number emerges as an 'accidental symmetry' in the sense that the other symmetries of the model (in this case gauge symmetries) do not allow any relevant or marginal interactions that violate the symmetry. 3.4. Neutrino
masses
Lepton number can be violated by the dimension 5 operator A £ ~ ±(LH)(LH).
(42)
When the Higgs gets a VEV, these gives rise to Majorana masses for the neutrinos of order v2 m,,-—. (43)
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M. A. Luty
In order to get neutrino masses in the interesting range mv ~ 10~ 2 eV for solar and atmospheric neutrino mixing, we require M ~ 10 15 GeV, remarkably close to the GUT scale. The interaction Eq. (42) also has a nontrivial flavor structure, so the actual scale of new physics depends on the nature of flavor violation in the fundamental theory, like the baryon number violating interactions considered above. The experimental discovery of neutrino masses has been heralded as the discovery of physics beyond the standard model, but it can also be viewed as a triumph of the standard model. The standard model predicts that neutrino masses (if present) are naturally small, since they can only arise from an irrelevant operator. We can view the discovery of neutrino masses as evidence for the existence of a new scale in physics. This is analogous to the discovery of weak (3 decay, which can be described by an effective 4-fermion interaction with coupling strength Gp ~ 1/(100 GeV) 2 . (Therefore, Fermi was doing effective quantum field theory in the 1930's!)
3.5. Extending
the standard
model
The steps in constructing an extension of the standard model are the same ones we followed in constructing the standard model above. The model should be defined by its particle content and symmetries. We then write down all couplings allowed by these principles. The goal is to find an extension of the standard model that cures the naturalness problem, but preserves the successes of the standard model described above.
4. The minimal supersymmetric standard model We now apply the ideas of the previous section to constructing a supersymmetric extension of the standard model. The motivation for this is that supersymmetry can naturally explain why a scalar is light. This because unbroken SUSY fixes scalar and fermion masses to be the same. Since fermion masses can be protected by chiral symmetries, the same chiral symmetries will also protect the masses of the scalar superpartners.
4.1. Superfields
and
couplings
To construct a supersymmetric extension of the standard model, we simply embed all fermions of the standard model into chiral superfields, and all
Supersymmetry
Breaking
513
gauge fields into vector superfields. The chiral superfields are therefore Q% ~ ( 3 , 2 )+*'
{ucy ~ ( 3 , D-i, {Dcy ~ ( 3 , !)+*' V ~(1,
2)-i,
(Ecy ~ ( 1 , l ) + i , where i = 1,2,3 is a generation index. The Higgs scalar fields are also in chiral superfields. If there is a single Higgs multiplet, the fermionic partners of the Higgs scalars will give rise to gauge anomalies. The minimual model is therefore one with two Higgs chiral superfields with conjugate quantum numbers Hu~(l,2)+i,
(45) ffd~(l,2)_i.
The next step is to write the most general allowed couplings between these fields. Let us begin with the relevant interactions. These are A£reievant = U29[nHuHd
+ KiUHu} + h.c.
(46)
where fi and Kj have dimensions of mass. Right away, we have some explaining to do. We see that SUSY allows us to write a supersymmetric mass for the Higgs, as well as a term that mixes the Higgs with the lepton doublets. (Note that L and Hd have the same gauge quantum numbers, so the distinction between them is only a naming convention up to now.) The terms K, can be forbidden by lepton number symmetry, defined by L(Q) = L{UC) = L{DC) = 0, L(L) = +l,
L{EC) = -1, (47)
L(HU) = L(Hd) = 0. The '/x term' can be forbidden by a U(l) 'Peccei-Quinn' c symmetry with charges C
A similar symmetry plays a role in the solution of the strong CP problem by axions,
as first discussed by Peccei and Quinn.
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M. A. Luty
P(HU) = P(Hd) = + 1, (48)
P(Q) = P(UC) = P(DC) = P(L) = P(EC) = - i .
There are many other symmetries that we could invent to control these terms. The motivation for the particular symmetries given here is that they are not violated by Yukawa interactions (see below). The important point is that the relevant terms in Eq. (46) can be naturally zero or small due to additional symmetries. The marginal interactions include kinetic terms for all the gauge and chiral superfields, which we write schematically as Adnetic ~ fdH [Q\evQi
+ •••] + ( fd2eWaWa
+ ••• + h.c.)
(49)
Note that the kinetic terms have been chosen to be diagonal in the flavor indices. It also includes the Yukawa couplings ^Yukawa = [#0
(VD)*,®Hd{Dcy
[(jftOy^iWT' +
J
(50) + (yE)ijLiHd(Ecy]
+h.c.
Note that the Yukawa couplings are invariant under both the lepton number symmetry Eq. (47) and the Peccei-Quinn symmetry Eq. (48). There are also additional Yukawa-like interactions d 9[(\LQD)ijkLlQj(D°)
^dangerous =
+ (\LLE)ijkLlLJ
J
(Ec)
(51) c
c
c k
+ (XuDDhk(u y(D y(D ) ]
+h.c.
Once again, we have some explaining to do. These couplings violate lepton and baryon number symmetries, and therefore give rise to proton decay and other processes that are not observed unless these couplings are small. The couplings XLQD and \LLE violate lepton number, and XLQD and XUDD
violate baryon number, defined by B(Q) = +l
B(UC) = B(DC) = - i , (52)
B(L) = B(EC) = B(HU) = B{Hd) = 0. Imposing these symmetries therefore suppresses these terms.
Supersymmetry
4.2. R
Breaking
515
parity
Another type of possible symmetry in SUSY theories acts differently on different components of the same supermultiplet. For example, we can define a U(1)R symmetry acting on chiral and vector superfields as $(9) ^ eiR*a$(8e-ia),
V{6) i-> y(6»e" iQ ),
(53)
where i?$ is the 'R charge' of the chiral superfield $. From this definition, we see that the R charges of the scalar and fermion fields in $ are R(4>) = R*,
R(ip) = R$ - 1,
(54)
while the R charge of a gaugino field is + 1 . In order for a supersymmetric Lagrangian C = fd46K+(
jd29W
+ h.c.^
(55)
to be invariant, we require R(K) = 0 and R(W) = +2. Note that if we define a U(1)R transformation in the MSSM where all chiral superfields have R = | , this is automatically preserved by all renormalizable couplings except the fi term. Another R symmetry in the MSSM is a discrete symmetry called 'R parity.' It can be defined by $(0) _• ± $ ( - 0 ) ,
(56)
where the sign is — 1 for Q, Uc, Dc, L, and Ec, and + 1 for Hu, Hd- The idea is that the observed matter fermions have R parity + 1 , while their scalar partners have R parity — 1. Note that the gauginos also have R parity — 1, so all superpartners have odd R parity. R parity ensures that superpartners are produced in pairs, and that the lightest R parity odd particle is absolutely stable. R parity is sufficient to forbid all of the dangerous relevant and marginal interactions in Eqs. (46) and (51). (In fact, the couplings in Eq. (51) are often called lR parity violating operators.') Conversely, if these operators are forbidden by another symmetry, such as B and L conservation, then R parity emerges as an accidental symmetry. Unbroken R parity is often taken as part of the definition of the MSSM, but it is worth keeping in mind that R parity may only be an accidental and/or approximate symmetry. Note that unlike the standard model, the MSSM requires that we impose certain exact or approximate global symmetries in addition to the gauge symmetries and particle content. In this sense, we have given up the
516
M. A. Luty
attractive automatic explanation of the suppression of baryon and lepton number violation in the standard model. 5. Soft S U S Y breaking We now begin our discussion of supersymmetry breaking. It is obvious that the world is not exactly supersymmetric, since SUSY predicts the existence of superpartners with the same mass and quantum numbers as existing particles. Once SUSY is broken the masses of the superpartners can be different from the observed particles, and must be larger than 100 GeV or so to have avoided detection in accelerator experiments performed so far. As we have already seen above, new physics at or below TeV is required in any solution of the naturalness problem. In SUSY the new physics is superpartners, and therefore these must be at or below the TeV scale, and can be discovered at LHC. If superpartners are discovered, the most important question in particle physics will be to understand the pattern of SUSY breaking. It is no exaggeration to say that SUSY phenomenology is SUSY breaking phenomenology. A simple way to break SUSY is to break it explicitly in the effective Lagrangian. If we do this, we would like to ensure that the breaking terms do not introduce power-law sensitivity to heavy thresholds (i.e. 'quadratic divergences'), which must be canceled by fine-tuning. SUSY breaking terms with this feature are called 'soft breaking terms.' This way of breaking SUSY may be ad hoc, but it does realize the goal of constructing a natural extension of the standard model. Also, we will see below that if SUSY is spontaneously broken at high scales, the effective theory below the SUSY breaking scale is a softly broken SUSY theory.
5.1. Coupling
constants
as
superfields
To discuss soft breaking we will use a tool that will be very useful to us throughout these lectures. This is the idea of coupling constants as superfields. Note that if a superfield $ has a nonzero value ($), it does not break SUSY as long as Qa($)
= Qa($) = 0,
(57)
where Q° = M£-<«0&dli
(58)
Supersymmetry
Breaking
517
is the SUSY generator. In particular, a constant nonzero value of the lowest component of a superfield does not break SUSY. We can therefore view the coupling constants that appear in a SUSY theory as superfields with only their lowest components nonzero. For example, in the Wess-Zumino model fd4eZ&$+
/d20(iM$2 + iA$3)+h.c.
(59)
we can view Z = Z^ as a real superfield and M and A as chiral superfields. 5.2. Superfield
couplings
in perturbation
theory
This simple idea is very useful for understanding the structure of loop corrections. For example, in the model defined by Eq. (59) the 1PI effective action at one loop contains the divergent term
Ar IPI
/•
d4ez
1
A2
A
267H
\i
+ -^r-^ In — + finite
(60)
where A
z3/2
(61)
is the physical Yukawa coupling. By direct calculation, we find that there are no further divergences at one loop. In particular there are no corrections to the superpotential, even though these are allowed by dimensional analysis. The absence of corrections to the superpotential holds to all orders in perturbation theory. We now show that this can be understood very easily if we view the couplings as superfields. Note that a one-loop correction the superpotential in the IPI effective action would have the form AT
IPI
- /
*>£*$+**.
(62)
Note that we treat A as a chiral superfield, and therefore A* cannot appear in the IPI superpotential. That is, the superpotential must be a holomorphic function of the couplings as well as the chiral superfields. For this argument, it is important that the couplings can be treated as superfields even in the fully regulated theory. In this theory, this can be easily done by using a higher derivative regulator: 6>Z$f / '
! + §'!• + •
(63)
518
M. A. Luty
where the cutoff A is a real superfield. For example, the scalar propagator is modified p2
(64)
p2-p4/A2'
This makes loops of Q fields UV convergent.d Because A is a real superfield, it cannot appear in the superpotential, immediately ruling out divergent corrections like Eq. (62). What about finite contributions? This holomorphy of the superpotential allows us to easily show that these are also absent to all orders in perturbation theory. We consider a U{1) x U{1)R symmetry with charges given below: U(l) $ M A
U(1)R
+1
0 2 2
-2 -3
(65)
Note that we are treating the couplings as spurions that transform nontrivially under these symmetries. The most general 1PI superpotential is therefore a function of the neutral (and dimensionless) ratio A $ 3 / M $ 2 :
Ar IPI
/>*«*2/(f
(66)
-f-h.c.
Expanding this in powers of A we obtain
ArIPI
/<*
"M 2 $ + M $ 2 + A$ 3 + — $ 4 + A M
+ h.c.
(67)
Only the $ 2 and $ 3 terms can be present, since the higher order terms are singular in the limit A —> 0 or M —> 0. The conclusion is that the superpotential is not corrected in this theory. We can use the non-renormalization of the superpotential to understand the structure of the renormalization group (RG) equations for this theory. Since there is only wavefunction renormalization, the physical couplings A d
A £3/2'
T h i s regulator also introduces a ghost, i.e.
w
M
(68)
a state with wrong-sign kinetic term at
p 2 = A 2 . However, this decouples when we take the limit A —+ oo and does not cause any difficulties.
Supersymmetry
Breaking
519
run only because of the running of Z. We therefore have (exactly) dX
o ,
dM
"
, .
Demanding that the one loop 1PI effective action Eq. (60) is independent of /tx, we obtain the anomalous dimension d\nZ _ \X\2 7 = M -d[l r — = -T7Z2 167T
( 7 °)
which summarizes the renormalization of the theory. 5.3. Soft SUSY
breaking from superfield
couplings
We can include SUSY breaking terms in the Lagrangian by allowing the superfield couplings to have nonzero higher components. For example, the Wess-Zumino model above, we can write Z->l M^M
x^x
+ {82B + h.c.) + 0262C, + 82FM,
(71)
+ e2Fx.
Working out the potential by integrating out the auxiliary fields, we find V = VSVSY + m V 0 + [\AU + \Axc^ + h.c] ,
(72)
where VSvSY = \Mt+±Xtf\2
(73)
is the supersymmetric potential, and m2 = -C +\B\2 =
-{\nZ]e252,
AM = -2(FM - BM) = -2{M}62, AX = -2(FX-^BX)
(74)
= -2[X}92,
where M = M/Z is the physical mass. Now let us consider the divergence structure of this theory including the SUSY breaking terms. The analysis in the previous subsection showed that in the supersymmetric case, the only divergence is in the wavefunction renormalization, given to one loop by Eq. (60). When we turn on higher components of the superfield couplings, this divergent contribution is given
520
M. A. Luty
by the same expression, but now it contains SUSY breaking from the superfield couplings. We see that the renormalization of SUSY breaking terms that can be written as higher components of superfield couplings is completely fixed by the renormalization of the couplings in the SUSY limit. We will explore the consequences of this in the following subsections. Are there any additional divergences in the presence of SUSY breaking that are not present in the SUSY limit? These can arise from couplings that vanish identically in the SUSY limit. We can get such couplings by taking the total superspace integral of a chiral quantity: ATIPI
= fd49 [ai$ + \a2§2}
+ h.c.
(75)
The couplings a\ and a2 are renormalizable by power counting, but are total derivatives if ct\ and a2 have only the lowest component nonvanishing (which is why we did not include them in the original Lagrangian). However, if OL\ and a2 depend on superfield couplings with higher components, they can have nontrivial effects. For example, (76) dA6ai<& = [ai]e2g2(j) + • / As above, we can understand the possible counterterms by considering the U(l) and U(1)R symmetries defined in Eq. (65). Under these, the couplings a\ and a2 transform as U(l) «1
a2
-1 -2
U(1)R
0 0
(77)
From these symmetries, we can see that the diagram
allows a logarithmically divergent contribution AM* ai
A
o h i —. I67H fj,
(78)
Note that a counterterm of the form a\ is allowed only if the field $ is a singlet. The following exercise illustrates the danger of this kind of divergence.
Supersymmetry
Breaking
521
Exercise 2: Consider a model of two superfields 5 and X with Lagrangian C = fd49 [ZSS^S +
ZXX^X] (79)
2
2
2
+ fd 9 (±\SX
+ \MX )
+ h.c.
Show that in the SUSY limit (X) = 0 while (S) is undetermined. The theory therefore has a space of vacua parameterized by S. Now break SUSY by turning on soft masses m | , m2x
M .
(80)
y
Similar reasoning shows that
<8i
-~iS^
>
because there cannot be a UV divergence that is singular in the limit M^O. Another kind of term that can appear in the effective Lagrangian once SUSY is broken involves higher SUSY derivatives of the superfield couplings, such as D2X = -4FX,
D2D2Z
= 16C.
(82)
However, it is not hard to check that all such terms have positive mass dimension, and therefore cannot be UV divergent by simple power counting.
Exercise 3: Carry out an operator analysis to show that there are no additional divergent counterterms involving SUSY derivatives of the couplings in the Wess-Zumino model above. Note that D& vanishes on chiral superfields, and that D2X is chiral for any superfield X.
We conclude that in the model Eq. (59) all SUSY breaking terms that can be parameterized by a nonzero higher component of coupling constant
522
M. A.
Luty
superfields are soft, in the sense that they do not lead to any quadratic divergences. Our argument has been rather abstract, and it is worth pointing out how the the absence of quadratic divergences comes about in explicit calculations. In component calculations, the quadratic divergences cancel between graphs involving loops of fermions and bosons. For example, the contributions to the counterterm for the scalar mass term in the model denned by Eq. (59) are given by (after analytic continuation to Euclidean momenta) ^ T O b o s o n loop
=
+ | A | 2
d*kE 1 2 2 / (2TT)4 fc| + ( | M | + m ) ' (83)
A
fdikE
2
^ ' ' ^ f e r m i o n loop
'2| A | 2 . / ( 2 T T ) 4
trl fc| +
|M|2'
where tr 1 = 2 is the trace over the Weyl fermion indices. We see that the quadratically divergent part cancels.
Exercise 4: Check the signs and combinatoric factors in Eq. (83).
The superfield analysis can be easily extended to include gauge fields. To fix our superfield conventions, we write a U(l) gauge superfield as V = Vf =
26><7MMM + • • • + 92d2D
(84)
and write the field strength as Wa = -\D2DaV.
(85)
We then have fd29 WaWa = -2F"UF^
+ • • • + 4D 2
(86)
+ 0 f £ty + • • • ,
(87)
and fdA9 &ev<$> = (D^D^
whereis the scalar component of the chiral superfield $ , and D^ = d„- iA„.
(88)
Supersymmetry
Breaking
523
We therefore write the action of scalar QED as [d40Z[&ev$
C=
&e-v$}
+
(89) 4- fd2e1-WaWa
+ h.c.,
where T
=
1^ 2g>
iQ 16^
a2m\ e
"~f
^
is a chiral superneld that contains the gauge coupling g, vacuum angle G, and gaugino mass m\. The gauge kinetic term is jd26
T
-WaWa + h.c. = -J^F^F^
+ ••• ,
(91)
so the canonically normalized gauge field is
( 92 )
K = 2V
A general renormalizable SUSY theory can be written in superfields as
C = Jdi9Ql(Zev*T*)\Qb + j,P9T±W«WaA + fd26W(Q)
+ h.c.
(93)
+ h.c,
where a, b,... are field indices (including both gauge and flavor indices), A,B,... are gauge generator indices, and the superpotential W is a cubic function of the fields. W = naQa + \MabQaQb
+ \\abcQaQbQc-
(94)
Exercise 5: Check that the D term potential in the general theory above is given by (for Z = 1) 2 A
524
M. A. Luty
Using the same arguments as above, we can see that turning on nonzero higher components of these fields breaks SUSY softly, in the sense that there are no quadratic divergences in the theory. If there are singlets, then there may be logarithmic divergences of the type found in Eq. (76) that may destabilize the desired vacuum. Is this the most general soft SUSY breaking? We can answer this question by again using higher components of superneld couplings to break SUSY. Any term that breaks SUSY can be written in this way. Consider for example the term AC = JdA9 CD^ia^d^D^
= 2i\C)ewri<M«
+ •••
(96)
changes the coefficient of the fermion kinetic term relative to the scalar kinetic term, and therefore gives rise to SUSY breaking perturbations in the physical couplings of the canonically normalized fermion relative to the scalar. The superneld coupling ( has mass dimension —2, and at one loop we find divergent contributions of the form A r 1 P I ~ jd*6 ^ A
2
*t$ „ &
A 2 < ^ + ... .
E
(97)
We see that this gives rise to a quadratically divergent contribution to the scalar mass. In terms of diagrams, the couplings of the scalar and fermion in Eq. (83) are no longer equal, and the quadratic divergences no longer cancel. A more subtle example is AC = fd49 ± C $ + $ 2 + h.c. = i f C V ^ V + h.c. + • • • .
(98)
The superneld coupling C has mass dimension — 1, and at one loop we find divergent contributions of the form AriPi ~ Jd*e 3 ^ 2 A 2 $ + h.c. ~ & f
A2cf> + h.c. + • • • .
(99)
We see that this term is not soft in general. However, the quadratic divergence Eq. (99) is absent if there are no singlets in the theory, so that the tadpole is not allowed. For example, a term of the form AC = fd46 | C $ { $ ! + h.c.
(100)
is soft if there is a U{\) symmetry with charges Q($i) = 2Q($2)- Although SUSY breaking terms of the form Eq. (98) are soft, they are usually neglected. We will see that they are not naturally generated in the more fundamental theories of SUSY breaking that we consider.
Supersymmetry
Breaking
525
The result is therefore that the most general soft SUSY breaking terms are precisely those that can be written as higher components of superfield couplings in the Lagrangian, plus possible ' C terms, of the form Eq. (98). 5.4. Renormalization breaking
group equations
for soft
SUSY
The fact that the divergences in this theory are controlled completely by the divergences in the SUSY limit gives rise to nontrivial relations between the renormalization of SUSY and SUSY breaking couplings. For example, the soft scalar mass is given by m 2 = -[\nZ}92§2.
(101)
Exercise 6: Check that this formula is correct, even for the case where the lowest component of Z is nonzero. Therefore, dm2
, ,
,,„„s
M — = -[7W2
(102)
1 Z3
167T 2
1 16TT 2
(\Ax\2
+ •••
(103)
+ 3m2) + ---
(104)
e2e2
where
We see that the RG equations for the SUSY breaking parameters are completely determined in terms of the supersymmetric ones. We can easily extend these results to gauge theories. The one loop renormalization group equation for a U (1) gauge theory is ^
=
"16^'
(106)
where Q is the U(l) charge matrix of chiral superfields. For a non-abelian gauge theory with gauge group SU(N) and F fundamentals, the one-loop beta function is
526
M. A. Luty
3N-F
dr
Note that this immediately implies the RG equation for the gaugino mass
<%(?)-<>•
«
There is a subtlety in the gauge coupling superfield beyond one loop. It is not hard to see that the RG equation Eq. (107) for the gauge coupling superfield has no corrections to all orders in perturbation theory. This follows from the fact that r is a chiral superfield, and therefore the beta function for r must be a holomorphic function of r: ^=0(r).
(109)
The RG equation for the real part of r must be independent of the vacuum angle 6 , which is a total derivative and therefore irrelevant in perturbation theory: # - = 0. almr
(110)
But because /3 is holomorphic, this implies that
(m,
|-o.
i.e. P is a constant. (It can be similarly shown that /? is independent of Yukawa couplings by considering U(l) charges under which the Yukawa couplings are charged spurions.) We conclude that holomorphy implies that the one-loop RG equation Eq. (107) is in fact valid to all orders in perturbation theory. This does not contradict the fact that the physical gauge coupling does run at two loops and beyond because the physical gauge coupling differs from the gauge coupling defined by Re(r) beyond one loop. The gauge coupling defined by the lowest component of r is often called the holomorphic gauge coupling to distinguish it from the physical gauge coupling. The physical gauge coupling is the lowest component of a real superfield defined by #
= r + Tt +
N F — j zl n i Z - —,hxZ 8ir SIT-1
+ 0(l/R)
(112)
Supersymmetry
Breaking
527
where the terms of order 1/R and higher are scheme dependent and R=^—+(e2l^^+h.c^+..-. "phys
\
"phys
(113)
/
Differentiating this expression, we obtain the famous expression for the beta function first written down in Ref. [11] 1
M w
^-F-Fj
87r2
;
i -^ ^2 ffphys + ^ p h y s )
where dlnZ
mw
7=/
(U5)
^T"
For a complete discussion with many applications, see Ref. [10]. 5.5. Soft SUSY
breaking in the
MSSM
We now apply these results to the MSSM. We assume R parity (or equivalent symmetry) so that the 'R parity violating' terms in Eq. (51) and the Hd~L mixing terms in Eq. (46) are absent. The most general soft terms are as follows. There are gaugino masses for the SU{2>)c x SU(2)w x U(l)y gauginos: A£gaUgino = -M1A1A1 - M2A2A2 - M3A3A3 + h.c.
(116)
These can be thought of as arising from the 02 component of the gauge coupling superfield as in Eq. (90).e There are scalar masses for all scalars that can be thought of as arising from the 0202 component of the kinetic coefficients: A £ s c a i a r = -m2HuHlHu
-
m2HdH]dHd
-(mlyjQl&-(ml)ijU?Uci
+ --..
(117)
Here we are using standard notation where the scalar components of matter fields (those with even R parity) are denoted by a tilde, while the scalar components of the Higgs superfields are given the same name as the superfield itself. There are also 'A' and '£?' terms that can be thought of as e
Beyond one loop, the physical gaugino mass is defined by the 62 component of the real
gauge superfield Eq. (112). See Ref. [10].
528
M. A. Luty
arising from higher components of superfield couplings, or 92 components of kinetic coefficients: ACB = -BnHuHd
+ h.c.
A£A = -(Au^j&HuU^
(118) + ••• + h.c.
(119)
The names of these couplings have become traditional. Finally, there are cubic interactions arising from the ' C terms of the form Eq. (98): ACp = {Cu^Q'Hlu^
+ ••• + h.c.
(120)
These are soft because they do not involve singlet fields. These are usually neglected, and we will see that they do not arise in any of the models of SUSY breaking that we consider. For an interesting possible application of these terms, see Ref. [12]. What is the status of the MSSM with general soft breaking terms? First of all, we should note that there are enough terms allowed to give masses to all of the unobserved superpartners. This is obvious for the gauginos and squarks and sleptons. For the Higgs sector this requires some work. See e.g. the lectures by H. Haber at this school. This must be counted as a success. On the other hand, there are an enormous number of parameters in the theory once SUSY is broken, about 100 even if we use our freedom to make field redefinitions to reduce the number of independent parameters. 5.6. The SUSY
flavor
problem
To make matters worse, many of the soft SUSY breaking parameters have nontrivial flavor structure. This means that they will in general give an additional source of flavor mixing that is not diagonal in the basis where the quark masses are diagonal. This is the SUSY flavor problem. Because the new flavor violation must be small, the scalar masses must be dominantly flavor-independent, e.g.
(™§Yi=™lPi
+ (A™2Q)ii>
( 121 )
with Am2
+ (AAu)y,
(122)
Supersymmetry
Breaking
529
The flavor-violating parts of the scalar masses and A terms are constrained to be small by observational constraints on flavor-changing neutral current processes. The most constraining process is K°-K° mixing. In the standard model, this arises from the famous box diagram u,c,t s
< \ <
v-+
d
9
W
16TT 2
(123)
(vtsvtdy
u,c,t In the MSSM, there are additional diagrams from squark loops. We can treat the flavor-violating soft masses Am 2 as insertions. This gives a new contribution from box diagrams involving superpartners, e.g.
u,c,t
w d
-<^d 2 \2
94
*< W
167T
(A™~j)
m
2
l
(124)
<-.
ii,c,t
Because the standard model contribution does a good job in accounting for the observed rate, we must demand that the SUSY contribution is not larger. This gives the bound
Am?: sd
m
^m^VtsVtd^
10"
Q
500 GeV
(125)
We see that the squark masses have to be very nearly flavor diagonal in order to avoid flavor-changing processes that are much larger than what is observed. There are many similar constraints coming from various processes. See Ref. [13] for a comprehensive discussion. Because flavor symmetry is broken in the Yukawa couplings, it cannot explain why the squark masses are nearly diagonal. This is the SUSY flavor problem. 5.7. The n
problem
In order to give the Higgsino a mass in the MSSM, we need a term of the form A£=
/
d29 fiHuHd + h.c.
(126)
530
M. A. Luty
If n » 100 GeV, the Higgs multiplet has a large supersymmetric mass and electroweak symmetry cannot be broken. If fx < 100 GeV the Higgsino is lighter than M^/M2. The parameter /j, must therefore be of order 100 GeV, that is, the same size as the other SUSY breaking parameters. Explaining why the \x term is the same size as the soft SUSY breaking parameters is the so-called '[i problem.' Since \i contributes to the mass term of the Higgs scalars, we cannot explain the weak scale without explaining the origin of the [i term. It is not easy to understand why [i is the same size as the soft SUSY breaking parameters discussed above because it breaks a different set of symmetries. For example, it does not break SUSY, but it does break the Peccei-Quinn symmetry defined in Eq. (48). 5.8. The SUSY
CP
problem
Since the MSSM with SUSY breaking has many new parameters, it is not surprising that there are new CP violating phases in the soft SUSY breaking parameters that cannot be rotated away. For example, there is a contribution to the strong CP phase from the phase of the gluino mass: 6 Q C D = 6 3 - argdet(m u ) + argdet(m d ) - 3arg(M 3 ).
(127)
Here 63 is the coefficient of tr(F A " / F M ^) in the QCD Lagrangian, and the other terms are the phases in the masses of strongly coupled fermions. Only the combination 0 Q C D is physically observable. Bounds on the neutron electric dipole moment require
e Q cD £ icr 10 .
(128)
Explaining this small number is the 'strong CP problem.' The strong CP problem must be solved somehow, whether or not nature is supersymmetric. For example, there may be a spontaneously broken Peccei-Quinn symmetry [7] leading to an axion [8], or there may be special flavor structure at high scales that ensures that the determinant of the quark masses is real [9], These mechanisms work just as well with or without SUSY, although in the latter case, we must independently insure that the gluino mass does not have a phase that upsets the mechanism. There is also an interesting proposal for solving the strong CP problem that works only in SUSY models [14]. Phases in the squark masses can also give rise to electric dipole moments for quarks and leptons. Demanding that the quark eletric dipole moments
Supersymmetry
Breaking
531
do not give rise to a too-large neutron electric dipole gives the constraint ImAm%
/
rrif,
\2
-sr^Ksio&v) •
(129
>
Even if the SUSY breaking violates CP, this can be satisfied if the phases are accidentally somewhat smaller than unity. There are several constraints that are similarly strong. See Ref. [13] for a review. The 'SUSY CP problem' is the problem of explaining why these phases are small. It does not involve very large or small numbers, and is therefore not clearly a serious problem. 5.9. The SUSY
fine-tuning
problem
In the MSSM the Higgs potential is constrained by softly broken SUSY to have the form Vniggs - (m2Hu + \fi\2)\Hu\2(m2Hd + \fi\2)\Hd\2 - B^{HuHd + h.c.) (130) + \{9l + 922)(\HU\2 - \Hd\2)2 + y22\HlHd\2. In particular, the quartic terms are completely determined, since they arise entirely from the D term potential from the SU{2)w x U(l)y gauge multiples Even though the quadratic terms are free parameters, we obtain an upper bound on the physical Higgs masses. The basic reason is that the Higgs potential has the form Vmsss~m2HH2
+ g2H4,
(131)
so the Higgs VEV is determined to be v~(H)~—.
(132)
This implies that we need m # ~ gv ~ Mz, and so the physical Higgs mass is also of order Mz- This can be viewed as a consequence of eliminating the mass term ran in favor of the VEV. We might imagine that we can get larger physical higgs masses because there are several independent quadratic terms, but famously this is not the case for the neutral CP-even Higgs boson h°. Computing the physical mass of h° from Eq. (130) we obtain a bound f mho < Mz\cos2p\. f
(133)
This assumes that the CP-odd neutral scalar A0 is heavier than h°. If this is not the
case, then the bound is even stronger.
532
M. A. Luty
Here tan / ? = — ,
(134)
Vd
where (HU) = ( ; " ) ,
(Hd) = ( ° ) ,
(135)
with t; = \ / ^ + ~ ^ I = 174 GeV. A Higgs lighter than Mz was ruled out by LEP I, and the current limit from LEP II is m^o > 114 GeV. (It is ironic that in the standard model, the Higgs mass is in a sense too large, while in SUSY it is too small!) Because this bound holds independently of the Higgs quadratic terms, we need additional contributions to the Higgs quartic couplings in order to raise the mass of the ft0. In the MSSM, these corrections can come only from loop effects. We can compute the loop contribution to the Higgs potential systematically by integrating out the particles heavier than the ft0. Since the heavy particles have masses and couplings that break SUSY, the effective field theory below their masses is non-supersymmetric. It therefore contains non-supersymmetric quartic couplings such as AVHiggs = \AXu(HtHu)2
+ \&\d{H\Hd)2
+ ••• .
(136)
(Another way to say this is that SUSY predicts relations among the most general allowed quartic couplings, which are violated at loop level.) The largest loop contribution comes from the particles with the largest couplings to the Higgs, which are the gauge particles and the top and stop quarks. g Numerically, the top contribution dominates and gives
AA tt = 4 l n «. Air2
(137)
mf
The log can be thought of as the result of RG running between the stop and the top mass scale. The precise argument of the logarithm is chosen to include the finite 1-loop matching corrections. The large size of the coefficient can be understood from the fact that it is enchanced by a color factor. This contributes to the physical mass of the ft0 AmJ, = i±vl2 47r g
u
sin4 0 In
mi
'-^h..
(i38)
If tan f3 ;> 50, then j/t ~ 1 in order to explain the observed bottom quark mass. In this
case, the sbottom can also give a significant contribution.
Supersymmetry
Breaking
533
For numerical estimates, an important correction to the value of yt comes from the QCD corrections to the physical top quark mass: mt = ytvu 1 + - ^ - 4-
(139)
Again, the large coefficient of the 1-loop correction can be understood as color enhancement. For a top quark mass of 175 GeV, we have ytvu ~ 165 GeV. The correction Eq. (137) is often written with the substitution yt —> mt/vu, but it is the Yukawa coupling and not the mass that enters directly into the diagram. The loop contribution to the Higgs quartic coupling Eq. (137) grows logaritmically with the stop mass, so we can try to get a large Higgs mass by increasing the stop mass. However, there is a heavy price to pay for this: the Higgs mass parameter m2Hu also gets loop contributions that grow quadratically with the stop mass. In fact, this contribution is also logarithmically divergent, and therefore has a logarithmic sensitivity to the scale ASUSY- These logs can be summed using the RG equations
If the logarithm is not large, we have
Amk--gWlnA
(141)
where we have assumed a common stop mass for simplicity. It should not be surprising that the Higgs mass is quadratically sensitive to SUSY breaking mass scales. This is just a particular manifestation of the fine-tuning problem for light scalars. We argued on very general grounds that naturalness of a light Higgs requires new physics below a TeV. Now that we have a specific model with a specific type of new physics, we can put in the numbers and make this more precise. To satisfy the current experimental bounds on the Higgs mass, we require Am2h0 > (114 GeV) 2 -M\
= (69 GeV) 2 .
(142)
We are assuming tan/3, which enhances the Higgs mass. We therefore require the stop mass to be f27r 2 Am? 0 l mt- % mt exp j g ^ J ~ 620 GeV.
(143)
To quantify how much fine tuning is involved, we note that the general scaling of Eq. (132) tells us that the natural size of the Higgs mass parameter is
534
M. A. Luty
of order Mz- An approximate measure of the fine tuning in the Lagrangian is therefore Ami,, 3y 2 m? A tuning = - — p - ~ / ' « In — .
, 144
This is approximately 16 for mt- ~ 620 GeV and A ~ 100 TeV. This means that the positive contribution to the quadratic Higgs mass terms must cancel the large negative contribution from the loop correction to an accuracy of at least 1 part in 16 to get acceptable electroweak symmetry breaking (see Eq. (130)). The full 1-loop corrections and the largest two loop corrections to the quartic of order yf and y\g\ increase the required stop mass above 1 TeV for small At, so this analysis actually underestimates the amound of fine tuning. This is not very much fine tuning if the parameters are at their current experimental limits, but note that the amount of fine tuning grows exponentially with the experimental bound on the Higgs mass (see Eq. (143)). This sensitivity also means that the precise amount of fine tuning is sensitive to other corrections. At present, we cannot say that there is anything clearly wrong with the MSSM, but this may be taken as a hint that the MSSM is not a completely natural solution to the fine-tuning problem of the standard model. 5.10. The next-to-minimal
super symmetric
standard
model
We can take the SUSY fine tuning problem as a motivation to go beyond the MSSM. A simple and well-motivated extension of the MSSM is obtained by adding a singlet chiral superfield S to the theory. This is relevant for the SUSY fine tuning problem because a superpotential term AW = \SHuHd
(145)
gives a new contribution to the potential that is quartic in the Higgs fields: AV=\Fs\2
= \\\2\HuHd\2.
(146)
This can raise the mass of the lightest neutral Higgs scalar at tree level. Another motivation for this model is that we can naturally obtain nonzero values for (S) and (Fs) of order the other SUSY breaking parameters. This gives weak scale fi and B/J, terms, solving the pt problem.
Supersymmetry
Breaking
535
The superpotential Eq. (145) preserves a Peccei-Quinn symmetry with P(S) = —2 (see Eq. (48)). If this is broken spontaneously, it gives rise to a weak-scale axion, which is ruled out. This problem is easily solved by adding the superpotential AWNMSSM
+ ±kS3
= \SHuHd
(147)
to the MSSM. This model is called the 'next-to-minimal supersymmetric standard model,' or NMSSM. In principle, the lightest Higgs mass can be arbitrarily large by choosing the couplings A and k large, but there is a constraint if we impose the condition that the coupling A remains perturbative up to the GUT scale. This is because the RG evolution %
=
+
8 ^
+
-
(148)
drives A larger in the UV, and A will diverge at a finite scale if its value at the weak scale is too large. This gives a bound of approximately 150 GeV on the lightest Higgs mass in the NMSSM if we require that the theory be perturbative up to the GUT scale MQUT — 2 X 10 16 GeV, as suggested by the success of gauge coupling unification. (Even in the standard model, requiring perturbativity of the quartic Higgs coupling up to the GUT scale gives an upper bound of 170 GeV on the Higgs mass.) Extending the model further does not relax this bound. There are several ways to avoid these bounds. One possibility is that the coupling A indeed becomes strongly coupled, so that some or all of S, Hu, or Hd are composite above the strong interaction scale. This need not interfere with perturbative gauge coupling unification, since the gauge couplings naturally remain weak going through a strong threshold. (For example, the electromagnetic coupling gets only a small renormalization going through the QCD threshold.) For examples of this kind of model, see Refs. [15]. The NMSSM phenomenology differs from the MSSM phenomenology mainly in that scalar component of S mixes with the neutral Higgs, and can therefore change the signals of the lightest scalar. Also, the new singlet fermion mixes with the neutralinos. 6. S p o n t a n e o u s S U S Y b r e a k i n g There are good reasons to be dissatisfied with the softly broken MSSM. It has (9(100) unexplained parameters, and correspondingly no understanding of flavor conservation, one of the great successes of the standard model.
536
M. A. Luty
To address this question, it is natural to consider models in which SUSY is broken spontaneously, with the hope that the many SUSY breaking parameters will be naturally explained in terms of a simpler underlying theory. That is, we look for models with a SUSY invariant Lagrangian in which the ground state breaks SUSY. A famous condition for SUSY breaking comes from the fact that the Hamiltonian can be written in terms of the supercharges as H=\
[Q1Q1 + Q1Q1 + Q2Q2 + Q2Q2] > 0.
(149)
This shows that the energy of any state is positive or zero, and also that a state is SUSY invariant Qa\n)=0
(150)
if and only if its energy is exactly zero ff|fi)=0.
(151)
The vacuum energy is therefore an order parameter for SUSY breaking. A VEV for a superfield S can also act as an order parameter for breaking SUSY. SUSY is broken if Qa(S} = 0
oxQa(S)=0.
(152)
This occurs if higher 0 components of (S) are nonzero, for example ($) = 62F, 6.1. F-type
breaking of
(V) = 9292D.
(153)
SUSY
The simplest examples of spontaneous breaking of SUSY are O'Raifeartaigh models. The idea behind this class of models is very simple. Consider a theory of chiral superfields with Lagrangian of the form JCO-R
= jdH
QlQa + (jd26
W(Q) + h.c.) .
(154)
In order for there to be a vacuum with unbroken SUSY, we must have
°-<*>-(*£)•
(155)
If there are N fields Qa, this gives Af conditions, which generically have a solution. However, for special choices of W, there may be no solution, and SUSY is spontaneously broken. The order parameter is the F component of
Supersymmetry
Breaking
537
a chiral superfield, and this type of SUSY breaking is called 'F-type SUSY breaking.' The simplest example of this mechanism is a model with a single superfield S with superpotential W =
(156)
KS.
However, this theory is trivial, as is easily seen in components. It consists of a free chiral multiplet with a constant poential V=\K\2.
(157)
This formally breaks SUSY, but there are no boson fermion splittings in the model. Suppose however that we add higher-dimension operators in the J d48 terms:
£ = fd4e f(S\ S)+(
fd29 KS + h.c. J .
(158)
This model is often called the 'Polonyi model.' The higher order terms in / may arise from integrating out heavy particles at scale M, in which case we expect f = S*S+-^(&S)2
+ 0(S*/M%
(159)
where c ~ 1. The potential terms can be computed from £ = fsisF^F
+ (KF + h.c.) + • • • ,
(160)
where we use the abbreviation
fsis =
d2f
asms-
(161)
This gives
V =•££-.
(162)
J SiS
We see that this has a nontrivial minimum if / s t s n a s a maximum. For a potential of the form Eq. (159), we obtain (5) = 0 for negative c. Expanding about this minimum, we find a scalar mass
Iclkl 2
(163)
SUSY is broken because the fermion component of S remains massless
538
M. A. Luty
We can write a renormalizable model of F-type SUSY breaking following the original idea of O'Raifeartaigh. One simple model contains 3 chiral superfields Si, S2, and X, with superpotential W = \XiSiX2
+ \X2S2{X2-v2).
(164)
Note that dW
h y 2
dSi
2 x
""
9W
'
i
w v 2
(165)
ds2
Since these cannot both vanish, SUSY is spontaneously broken. Since dW dX
(X1S1 + X2S2)X,
(166)
the scalar potential is V = \\Xi\2\X\4
+ \\X2\2\X2
- v2\2 + \X1S1 + A 2 5 2 | 2 | X | 2 .
(167)
Extremizing this potential with respect to Si and 52, it is easy to see that ( A i 5 i + A 2 5 2 ) = 0 or
(X) = 0.
(168)
Extremizing with respect to X gives (X2)
IA2I2 2 |Ai| + | A 2 | 2 V
or
2
{X) = 0.
(169)
The value of the vacuum energy is
(V)
4 ['YH = <
NW ,
2(|A1|2 + |A 2 | 2 ) 1
for (X) = 0, (170) 4
for (X) ^ 0.
It is easy to see that the vacuum energy is minimized at (X) =^ 0 provided that IA21 > |Ai|. In this case (Ai5i + A252) = 0, but one linear combination of Si and 52 is completely unconstrained.
Exercise 7: Show that in an arbitrary O'Raifeartaigh model there is always one linear combination of the superfields that is unconstrained by minimizing the potential at tree level.
Supersymmetry
Breaking
539
Working out the scalar and fermion masses at the minimum of the potential, we find that all scalar and fermion masses are of order Xv except for the scalar and fermion components of the superfield that is orthogonal to the linear combination X\Si + X2S2- This suggests that we can integrate out all the fields except one chiral superfield, and write the effective Lagrangian below the scale Xv as an effective theory of a single chiral superfield. This can be formally justified by noting that (Fx) = 0,
(171) 2
Xi(Fsi) + X2(FS2) = ||Ax| X
t2
2
2
2
+ ||A 2 | (Xt - J ) = 0,
(172)
so that X = 0,
AiSi + X2S2 = 0
(173)
are superfield constraints that can be used to define the light degrees of freedom in the effective theory. The massless chiral multiplet can therefore be parameterized by £2 (for example), which gives the effective Lagrangian CeS = fdA8 1 + j ^ L slS2 + ( fd26 [~lX2v2} S2 + h.c.) .
(174)
This has precisely the form of the Polonyi model considered above! In this sense, O'Raifeartaigh models reduce to Polonyi models at low energies. Note that in the O'Raifeartaigh model, the corrections to the effective kinetic function / come from integrating out massive fields at the scale Xv, and is therefore fully calculable. Computing the Coleman-Weinberg potential in this model, one finds that the minimum is at (S2) = 0. 6.2. D-type
breaking of
SUSY
We can also break SUSY with an order parameter that is the highest component of a gauge superfield: (V) = 6W(D).
(175)
The simplest way works for only for a U{\) gauge field, and consists of adding a 'Fayet-Iliopoulos term' A £ F I = fd4e^V:
(176)
where £ is a coupling with mass dimension +2. This is is gauge invariant because under a gauge transformation SV — fi + fi^, where Q is a chiral superfield, so under gauge transformations
540
M. A. Luty
(5(A£Fi) = fdA6£{£1 + fif) = total derivative.
(177)
However, notice that this term is not gauge invariant if the coupling is a superfield. This means that this term cannot be generated by a more fundamental theory in which the couplings are superfields, such as a SUSY gauge theory. As we will discuss below, this also means that this term is not allowed by supergravity. We will therefore not discuss these terms further. It is possible to get (D) ^ 0 if SUSY and gauge symmetry are broken at the same time. For example, consider a theory with superfields Q, Q, X, and X, with U(l) gauge charges ± 1 and ±2 respectively, and one singlet S. The superpotential W = \\{XQ2
+ XQ2) + \'S(QQ
- v2)
(178)
breaks SUSY and gauge symmetry at the same time, and gives rise to a nonzero value of (D). As this example illustrates, models of this type are necessarily somewhat complicated. An interesting open question is whether one can naturally get (D) 3> (F) without Fayet-Iliopoulos terms. 6.3. Generalities
of tree-level
SUSY
breaking
Let us consider a tree-level SUSY Lagrangian of the form given in Eq. (93). We are interested in tree-level SUSY breaking, so we take Z = 1 and all couplings to be SUSY preserving. The potential can be written as V = FaF£ + \DADA
(179)
where F* = WU
=
™ ,
DA = 9AQi(TA)\Qb-
Fl = ^
(180)
(Note that we have absorbed a factor of the gauge coupling into the definition of DA-) SUSY is spontaneously broken if (Fa) and/or (DA) are nonzero. We first show that if SUSY is spontaneously broken, there is a massless fermion in the spectrum, called the 'Goldstino.' The fermion mass terms in this notation are Aermionmass = ~\V^'abtf
+ V
^
D
^
+ h.C,
(181)
where
"*• - w-
(182)
Supersymmetry
Breaking
541
etc., and we use the notation Wab = {Wab) in the remainder of this subsection. We can write the fermion masses in a matrix notation as Aermionman = * M i
/ 2
* + h.C,
(183)
where * = (f)
,
(184)
-V2gBDBa C , ) •
(185)
and Wab <»={-V2DAb
M
We now claim that the fermion d e s t i n e = {_DB/^)
(186)
is massless. (Note that if SUSY is unbroken, *Goidstino is trivial.) This follows from M1/2*Goldstino = (
WaabW^b + DBDBa /Fir, „,+(, ° ) V2D AbWV>
•
(187)
The upper component vanishes because 0
= 757^ = WabW^b + DBDBa.
(188)
The lower component vanishes due to the gauge invariance of the superpotential: 0 = 6W = Wa5Qa = Wa{TA)abQb = —WaD%
(189)
9A
The existence of the Goldstino when SUSY is spontaneously broken is analogous to the existence of a Goldstone boson when a global symmetry is spontaneously broken. Its existence can also be established on general grounds, without referring to any Lagrangian. For a very clear discussion, see Ref. [16]. Let us continue our discussion of the masses with the scalar masses. Combining the scalar fields into a vector
•-(£)•
(190)
we can write the scalar masses as £scalarmass = - * t M 0 2 $ ,
(191)
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M. A. Luty
where M2 = 0
(W^Wd, V
+ D\DAb
+ DaAbDA
W^abcWc + DADbA WacW^b + DAaDbA + DbAaDA
WabcW^ + DAaDAb
(192) Finally, the gauge boson masses are *-gaugemass
=
^A
M^A^,
(\\)6)
with (Mf)AB
= 2DaADBa.
(194)
From these formulas, we can read off the fact that the 'supertrace' of the mass matrix vanishes: str(M 2 ) = tr(M02) - 2 tr(M\/2Ml/2)
+ 3 ta^Af?) = 0.
(195)
The supertrace is the sum of the squared masses of the particles, counting spin multiplicities, with fermions contributing with opposite sign as bosons. The vanishing of the supertrace puts strong constraints on how SUSY is broken, as we will see. 6.4. SUSY
breaking in the observable
sector
We have seen if SUSY makes electroweak symmetry breaking natural, superpartner masses must be below ~ 1 TeV, and to explain their nonobservation they must have masses greater than ~ 100 GeV. The superpartner masses must therefore be at the weak scale, and a good model of SUSY breaking should explain why this is so. An obvious thing to try is to break SUSY and electroweak symmetry at tree level by some extended Higgs sector. That is, we imagine a renormalizable extension of the MSSM to include extra fields and interactions that break SUSY and electroweak symmetry at tree level. In such a model, superpartner masses are nonzero because of direct couplings to the fields that break SUSY. We can see that this is difficult from the supertrace constraint discussed in the previous subsection. First of all, note that the mass matrix has a block-diagonal form, with each block corresponding to states that do not mix with the states in the other blocks. (For example, colored particles to do not mix with color singlets.) Unless there are heavy fermions in every block containing observed quarks and leptons, the supertrace constraint
Supersymmetry
Breaking
543
immediately implies that the scalar superpartners must be lighter than the heaviest observed fermion, which is a phenomenological disaster. It can be shown that even if we allow for the possibility of heavy particles, there are always light scalar color triplets (squarks) lighter than either mu or md, the lightest quark masses. Let us therefore restrict attention to the color triplet part of the mass matrix Eq. (192), which does not mix with anything else. Because color is unbroken we have (Wa) = <W ta ) = (DAa) = (DaA) = 0
(196)
when a is a color triplet index. The color triplet part of the scalar mass matrix is therefore _ (M\/2M1/2 °-{
+ {DlbDA) A
A
f
M1/2Ml/2
+
\ (DAaDA))-
(197)
where Aab =
(yV\abcWc^
( 19 g)
The idea is to think of MQ as a Hamiltonian, with the lightest mass eigenvalue equal to the ground state energy. We can use the standard variational method from quantum mechanics to estimate the ground state energy. For any 'state vector' $o, we have —Q—r > smallest eigenvalue. (199) $o$o To choose $ 0 , note that the D term contribution (DAbDA) is proportional to charges, and is therefore negative at least one of the fields U, Uc, D, and Dc in each generation (in a mass basis). Suppose for concreteness it is U that gives a negative result. We then define *o = ( It ) -
(200)
where 0 (201)
w is a unit vector in the direction of the first generation U field, with (DaAbDA)4>b = -\\\4>a.
(202)
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M. A. Luty
Then $jM 0 2 $o = \4>]{M\/2M1/2 + {DAgATA)+ h.c,
(203)
where we have used the fact that 4>A(j) vanishes, since there are no mass terms of this form UU allowed. Therefore
$JM 0 2 $ 0 _ , < $J$o
~ |A|.
(204)
This implies that the matrix M§ has at least one eigenvalue lighter than the quark mass m 2 . This is clearly ruled out. If the quark with the negative eigenvalue is down-type, we have an eigenvalue less than m2d, which is also ruled out. The basic problem is that D-type masses are proportional to charges, and therefore have both positive and negative signs, while the '.B-type' masses parameterized by the off-diagonal A terms tend to split the eigenvalues, and therefore cannot raise the lowest eigenvalue. 6.5. The messenger
paradigm
To make a viable model of SUSY breaking, we need either large loop corrections, or non-renormalizable terms in the Kahler potential. SUSY breaking from either of these sources is suppressed, by loop factors and/or by high mass scales. This means that the theory must contain a sector in which SUSY is broken at a scale much larger than the weak scale. This large primordial SUSY breaking will then communicated to the standard model fields through 'messenger' interactions. This gives us a way of thinking about the SUSY flavor problem. Since the interactions of the messengers with the standard model fields determine the pattern of SUSY breaking in the visible sector, a natural way to avoid additional flavor violation in the MSSM is if the messenger interactions do not violate flavor symmetries, i.e. are 'flavor blind.' We will see that this paradigm gives rise to successful models of SUSY breaking. 7. Hidden sector SUSY breaking An obvious candidate for the messenger of SUSY breaking is gravity. From a particle physics perspective, the unique low-energy effective theory of
Supersymmetry
Breaking
545
gravity is general relativity.11 Its consistency requires that gravity couples to matter through the stress-energy tensor, which is the origin of the equivalence principle. Because gravity couples to all forms of energy, it necessarily couples the SUSY breaking sector with the visible sector, even if there are no other interactions between the two sectors. All that is required for gravity to be the messenger of SUSY breaking is that there are no other stronger interactions between the two sectors. In this case, we refer to the SUSY breaking sector as the 'hidden sector.' The fact that gravity couples to the stress-energy tensor also means that general relativity is flavor-blind. (Different flavors have different masses, and in this sense couple differently to gravity. But what we want is that there be no additional flavor violation beyond that of the Yukawa couplings.) The difficulty with this in practice is the fact that general relativity is only an effective theory, and requires UV completion above the Planck scale (if not at lower scales). It is far from clear that the fundamental theory of gravity can have flavor symmetries that guarantee that the UV couplings of gravity are flavor-blind. In fact, there are strong hints from what little is known about the UV theory of gravity to suggest that the fundamental theory of gravity is unlikely to respect global symmetries such as flavor. One of these hints comes from studies of black holes, where Hawking radiation appears to be incapable of radiating away any global charge that was thrown into the black hole when it was formed. The final stages of black hole evaporation occur when the mass of the black hole becomes of order the Planck mass, and what happens there is not understood. However, requiring the conservation of global quantum numbers appears to require a large number of charged states at the Planck scale (corresponding to all the different possible charges of the initial black hole) which seems unlikely. Another hint comes from string theory, the only known candidate for a fundamental theory of gravity. String theory does not appear to allow exact global symmetries, although the full space of string theory vacua is still poorly understood. Prom a low-energy point of view, we can parameterize the most general effects of the unknown physics at the Planck scale by higher dimension operators suppressed by powers of the Planck scale Mp. Of particular interest to us are operators that connect the fields in the hidden sector This assumes that gravity is mediated by a spin-2 boson, and also assumes locality and Lorentz invariance. The cosmological constant problem has motivated attempts to relax the assumption of locality [17] or Lorentz invariance [18].
546
M. A. Luty
with those in the observable sector. We assume that SUSY is broken in the hidden sector by the F component of a field X, and without loss of generality we shift the field X so that (Fx) ± 0,
(205)
(X) = 0.
We can then write the most general interactions between X and the visible sector fields:
A£ = Jd*0{&£l-XiXQiQ + + +
jkXHuHd+mx'XHuHd+hx}
MP
_£i_ de XW?Wla MP 2
+
^XQ'H^Uy
(206)
+ h.c. +
When we substitute the SUSY breaking VEV (Fx), we find that this generates all the soft SUSY breaking terms of the MSSM (other than the ' C terms of Eq. (98)) with the size of all SUSY breaking masses of order AfeusY Taking M S U S Y
(Fx)
(207)
~ (10 11 GeV) 2 .
(208)
TeV gives (Fx) ~
MPMSUSY
(The scale 10 11 GeV is often called the 'intermediate scale.') Note that the Id and B/J, terms are generated by the terms with coefficients b and b' in Eq. (206). This gives a very simple solution to the l(j, problem', as first pointed out by Ref. [19]. Note these terms are only allowed if the SUSY breaking field X is a singlet, as are the terms with couplings si,... that give rise to gaugino masses. It is easy to see why SUSY breaking terms are suppressed in this approach. For example, 'C terms' are generated by operators of the form vfy
(fiO—s-QHlV* / P
+ h.c,
(209)
Supersymmetry
Breaking
547
which give rise to SUSY breaking trilinear couplings of order M | U S Y / M p -C MSUSY- 'Hard' SUSY breaking is also small. For example, a fermion kinetic term arises from /
yty dA9 -j^- DaQa^d^Q\
(210)
which gives AZ ~ Mf U S Y /Mp < 1. It is striking that simply writing all possible terms connecting the hidden sector to the visible sector suppressed by powers of a single large scale gives all required SUSY breaking terms (including \x and B\x terms), all of the same order.
Exercise 8: Write the leading additional SUSY breaking allowed in the NMSSM coupled to a hidden sector. Does this automatically give rise to all allowed SUSY breaking of order MSUSY, as in the MSSM? Can we impose symmetries so that all required SUSY breaking is generated with size MSUSY?
The difficulty with this approach is that the soft masses and A terms can violate flavor. The A terms arise from the terms with coefficients a^-, and we can imagine forbidding these by symmetries acting on the field X. However, the soft scalar masses are generated by the operators with coefficients zlj, which are invariant under all symmetries. Unless there are flavor symmetries at the Planck scale, there appears to be no reason for these coefficients to be flavor-diagonal. This is the flavor problem of hidden sector models of SUSY breaking. One way to avoid the flavor problem is to assume that there is a gauged flavor symmetry at the Planck scale. The existence of gauge symmetries (as opposed to global symmetries) is compatible with what is known about string theory and black hole physics. A gauge symmetry must be free of anomalies, but extra fermions can always be added to cancel the anomalies, and once the flavor symmetries are broken, all these extra fermions can in principle become massive. The flavor gauge symmetry must be broken at a high scale to avoid dangerous flavor-changing neutral currents. The flavor symmetry may also be discrete. For an example of this kind of model, see Ref. [20].
548
M. A. Luty
Exercise 9: To illustrate some of the issues involved in models with flavor symmetries, consider the following model. Assume that the model at the Planck scale preserves an SU(3)5 flavor symmetry that acts separately on the generation indices of the five multiplets Q, Uc, Dc, L, and Ec. Find additional particle content that can make the SU(3)5 flavor symmetry free of gauge anomalies, such that the additional particles can get masses below the SU(3)5 that do not violate standard model gauge symmetry. Suppose further that these symmetries are spontaneously broken by the VEVs of scalar fields Y that have the same quantum numbers as the Yukawa couplings. (For example, there are one or more 'up-type' fields (Yu)ij, where the i is a SU(3)Q and j is a SU(3)u index.) The Yukawa couplings are of order V~TT--
(211)
y
v MP ' The hierarchy of VEVs for different components of Y gives rise to the hierarchy for Yukawa couplings. Show that in this model, the off-diagonal squark masses have size
~^-y\
(212)
m
Q
where y is an appropriate Yukawa coupling. Show that this is sufficient to suppress dangerous flavor-changing neutral currents.
The model in the example above is complicated, and contains many particles. The top quark Yukawa coupling is yt ~ 1, which means that the expansion in powers of (Y)/Mp is marginal. (Since loop effects are suppressed by powers of l/(167r 2 ), the expansion really only breaks down for yt ~ 47r, so this is not fatal.) However, all these features arise because the model is an ambitious attempt to explain the origin of the Yukawa couplings. Since the Yukawa couplings presumably have some explanation, and no simple one is known, this may not be a drawback of this approach. We should note that this is not the way flavor arises in conventional string compactifications, in which there are no flavor symmetries at the string scale (see Ref. [21] for a review). On the other hand, string theory has not had any real success so far in explaining the observed features of our
Supersymmetry
Breaking
549
world. Maybe string theory is right, but we have not found the appropriate string vacuum. Given the richness of string theory, it might be right and we might never know. In this situation it is worth keeping an open mind about mechanisms that do not fit into string theory in an obvious way. 7.1. The 'minimal
SUGRA'
ansatz
An Ansatz that has been extensively analyzed in the literature is to assume that the couplings that give rise to scalar masses are equal to a universal value at /u = Mp: (ZQYJ = {ZL)1J
= ••• = z05lj,
zHu = zHd=z0
(213)
This is called 'minimal SUGRA' for historical reasons. One feature of this Ansatz is that the up-type Higgs mass runs negative because of the large top Yukawa coupling. This is called 'radiative symmetry breaking.' We have argued above that if there is no flavor symmetry at the Planck scale, this Ansatz is not natural. Nonetheless, there is an extensive literature on this, so you should at least know what it is.
Exercise 10: In this Ansatz, we must run the z couplings down from the scale Mp down to the mass of the field X. Below this scale, we match onto a theory where X is replaced by its SUSY breaking VEV. Show that if we include only standard model fields in the loops, this procedure is equivalent to running universal scalar masses down from the Planck scale.
8. Gauge mediated S U S Y breaking Following the messenger paradigm for SUSY breaking, another natural flavor-blind messenger to consider are the standard model gauge interactions themselves. We have seen that tree-level SUSY breaking in the visible sector has severe difficulties, so we look at loop effects. 8.1. Gauge
messengers
A simple and predictive framework is to assume that SUSY breaking is communicated to the standard model via heavy chiral supermultiplets that are charged under the standard model gauge symmetries. If the masses
550
M. A. Luty
of these messenger fields are not exactly supersymmetric, integrating them out will give rise to SUSY breaking in the visible sector. A simple example that illustrates how this kind of SUSY breaking can arise is an O'Raifeartaigh-type model with superpotential W = ±XxSiX2 + iA 2 5 2 (X 2 - v2) + A 3 S 3 $$ (214) + (M + AS 2 )$$. Here S* (i = 1,2,3) and X are singlet fields, and <3> and l> are charged under the standard model. This model breaks SUSY, and for appropriate choices of the parameters, the minimum occurs for (F 52 ) ~ Xv2
(215)
and {Si), {S3), {$), ($) = 0. This means that <E> and $ effectively have a SUSY breaking mass term, with a superfield mass parameter M = M + \{Si)+92{Fs2).
(216)
As long as the fields in this sector do not have any direct couplings to the MSSM fields, the leading effects on the standard model will come from loop graphs involving the fields <E> and $, which depend on their mass Eq. (216). These fields are therefore the messengers of SUSY breaking. The messenger mass parameters are the only parameters in this model that will have an observable effect on physics in the visible sector. The only unsatisfactory feature of this model is that the mass terms M and v2 are put in by hand. There is a large literature on models that break SUSY dynamically, in which the SUSY breaking messenger mass arises from dimensional transmutation (see Ref. [22] for a review). However, since the messenger mass parameters are the only parameters in the SUSY breaking sector that have observable effects in the visible sector, we will be content to assume that a fully satisfactory model exists and work out the consequences. We therefore simply assume that there are charged messengers with SUSY breaking mass A£mess = [d20 M<M> + h.c.
(217)
with chiral superfield mass parameter M = M + 92F.
(218)
The messengers must be in a vector-like representation of the standard model gauge group to allow this mass. In order to keep the successful unification of couplings in the MSSM, we can take the messengers in complete
Supersymmetry
Breaking
551
SU(5) multiplets, such as 5 8 5. After integrating out the auxiliary fields, the scalar masses are AAness - -\M\2{^4>
+ ft$) + ( F # + h.c).
(219)
We can rephase <j> and 4> to make F real, in which case the scalar mass eigenstates are (<£ ±)/\/2, with masses \M\2 ± F. We see that stability of the vacuum (>) = (<j>) = 0 requires F < \M\2. The fermion masses are unaffected by F, and so the fermion mass is \M\. When we integrate out the messengers at loop level, the resulting lowenergy effective theory breaks SUSY. At one loop, we get a gaugino mass from the diagram
m
— ~ r,^jC\>^^
~ if^lr
(220)
and at two loops, we get a scalar mass from diagrams like
TO2
scalar
S
(
\l6lT2 )
F M
(221)
These are the leading terms in an expansion in powers of F. (The corrections are suppressed by powers of F/M2.) The estimates give the right order of magnitude even if F ~ M2. One immediate consequence of this is that the masses of the scalars and gauginos are of the same order, which is important for getting a realistic and natural model of SUSY breaking. Also, note that the scalar masses depend only on the gauge quantum numbers of the scalars, and are therefore flavor-blind. This gives a natural solution to the SUSY flavor problem. Furthermore, the spectrum is determined by just a few parameters, so this is a highly predictive framework for SUSY breaking. This mechanism is called 'gauge mediation' of SUSY breaking, for obvious reasons.
Exercise 1 1 : Show that the spurion F/M has the right U(1)R charge to give rise to a gaugino mass. Use symmetry properties and dimensional analysis to show that the leading contribution to the scalar mass for small F must be proportional to \F/M\2. Also, show that the subleading terms for small F give rise to fractional corrections of order \F/M2\2 to Eqs. (220) and (221).
552
M. A.
Luty
If we want the superpartner masses to of order 100 GeV or more, we need to have F/M ~ 5-50 TeV. However, the mass scale M can be quite large, keeping this ratio fixed. An upper bound on M is obtained by requiring that the gauge mediated contributions to scalar masses to be sufficiently larger than the contributions from Planck suppressed higher-dimension operators. If SUSY is broken primordially by the VEV of a chiral superfield {X) = 02FO,
(222)
then no symmetry can forbid operators of the form AC ~ fd4e -^X^XQ^Q.
(223)
These will in general give rise to flavor-violating scalar masses of order
Demanding that Am^. is small enough to avoid FCNC's gives the bound W
VF~o
/ m ~ \ 3/2 (—£-= • ,500 GeVy
( 22 5)
Note that the primodial SUSY breaking scale F0 need not be the same as the scale F in the messenger mass. In particular, it is possible to have Fo 3> F if SUSY breaking is communicated weakly to the messengers. However, we must have FQ > F, so the largest possible value of M is obtained when F0 ~ F:
( 50blM TO
8.2. The gauge mediated
-
\
3
'
(226)
spectrum
We now turn to the calculation of the gauge mediated spectrum. Even at the qualitative level, it is crucial to know the signs of the squark and slepton mass-squared terms. If any of these are negative, the theory does not have a minimum that preserves color and electomagetism, which is certainly ruled out! We will compute the induced SUSY breaking masses using an elegant method due to Giudice and Rattazzi [23] that makes essential use of superfield couplings. We treat the messenger mass M as a chiral superfield M = M + 62F.
(227)
Supersymmetry
Breaking
553
This reduces the problem to how the superfield couplings in the effective theory below the scale M depend on M. The leading dependence for large M is given by the RG, making the calculation of the loop diagrams very simple. For example, the value standard-model gauge coupling at a scale li < M can be obtained from the one-loop RG equation: 1 2
g (n)
1 5'
2
b' 2
(A)
8TT
M A
b , 2
8TT
n
,nnn.
M'
Here g' is the gauge coupling in the theory above the scale M, and b' is the beta function coefficient in this theory, while g and b are the corresponding quantities in the effective theory below the scale M. We started the running at an arbitrary scale A > M. For a non-abelian group, b-b'
= N,
(229)
where N is the number of messengers that get mass at the scale M if the messengers are in the fundamental representation. Note that N is always positive. The idea of Giudice and Rattazzi is to extend the formula Eq. (228) to superfield couplings. We therefore have
. M ^ A H j ^ l ^ +J l ^ A
(230)
Here T is the chiral superfield containing the (holomorphic) gauge coupling. Both sides of this equation are now well-defined chiral superfields. We can then compute the gaugino mass just by taking the higher 9 components of both sides:
N 16TT2
F W
(231)
Note that we have assumed that the only SUSY breaking is contained in M. In particular, the couplings at the cutoff A have no higher 9 components, which means that SUSY is unbroken in the fundamental theory above the scale M. In components, this would have been a finite one-loop computation, but this method reduces it to a simple RG calculation. Note that the result Eq. (231) includes the running from the matching scale M down to scales JJL < M. We can find the value of the gaugino mass at the matching scale M by expanding about /x = M. We illustrate this
554
M. A. Luty
method here because it is very useful for the scalar masses to be discussed below. For the gaugino mass, we write T{H)
= T{M) +
dr din/J
in H=M
£ + °('°2£)-
(232)
When we take the 92 component of both sides, the terms of order ln 2 (/i/At) do not contribute in the limit fi —> M. We then have
iM)}02 +
lim [T(/X)]92 fl—yM
dr dliifi
M
H=M
(233)
We then compute
-(M)}6
F dr{M) M d\nM - F b'
F M
dr'{M) d\nM
~ M 8TT2 '
(234)
Note that in our expansion the UV couplings are held fixed, which is why the result is proportional to the beta function in the theory above the scale M. Putting this together, we obtain V-bF M
W
T
< *
=
(235)
8«* M>
in agreement with Eq. (231). This method is even more powerful when used to compute scalar masses. These are extracted from the wavefunction coefficient Z via m2 = -[lnZ]e2S2.
(236)
Here Z is a real superfield, so it depends on M via the real superfield In M -»In |M | = In \M\ + \ (62 j - + h.c.) .
(237)
Expanding about fi = M, we have \imJ\nZ^)}e2S2
= ( l n Z ( M ) W + (\rr(M)]e> [ i n - ^ ] _2 + h.c.) (238)
1 d-f
:2 _M/
i-f-(M) fln
M-
92e2
where 7(M)
d\nZ dln/i
(239)
Supersymmetry
Breaking
555
is the anomalous dimension in the effective theory below the scale M. As in the calculation of the gaugino mass, we must perform the expansion keeping the UV cutoff fixed, which means that we must expand in M in the fundamental theory. We therefore have F_ ~M
[lnZ(M)W =
2 /
o
\ 2
d
d\nM
lnZ'(M)
dj (M), dlnfi
F_ M
(240)
where
VM =
dlnZ' dln/x
(241)
is the anomalous dimension in the theory above the scale M. Similarly,
[7(M)]e2
1 F d 7(3'(M)) 2 M d In M
^S<M)":(M)'
(242)
where gt (<^) denotes the dimensionless couplings of the theory below (above) the scale M, and ft ( $ ) are the corresponding beta functions, e.g. Pi
dgi dln/x'
(243)
Putting it all together, we obtain m2(M)
-lirn [lnZ(n)}e2§2 /x—*M
F M
dg[
dgi
dg{
(244)
Here all anomalous dimensions are evaluated at /J = M. This shows that the gauge mediated scalar mass at the threshold is a simple function of the anomalous dimensions of the theory. From this formula, we see that the scalar masses arise at two loops, since both 7 and (3 start at one loop. Note that we have performed a two-loop finite matching calculation using only the RG equations. We see that the threshold corrections are determined completely by the anomalous dimensions and beta functions of the theory. This is another illustration of the power of superfield couplings. Squarks and sleptons do not couple directly to the messengers, so they have 7' = 7 at one loop. (This means that 7' is the same function of the
556
M. A. Luty
couplings g' as 7 is of the couplings g.) In this case, the expression for the scalar mass simplifies further: F_ (245) Pi). M The one-loop RG for a kinetic coefficient (of a quark field, say) from a gauge loop is m2(M)
dlnZ
c
,
0
where c is the quadratic Casimir of the field. For a fundamental representation of an SU(N) gauge group, c = (TV2 - 1)/(2N). Putting this in, we obtain m2(fi = M) =
„4
(247)
;2cN.
(16TT 2 ) 2
Note that the scalar masses are positive at the matching scale fi = \M\, which is certainly a good starting point for a realistic model. RG evolution down to the weak scale can make the up-type Higgs mass run negative (due to the large top Yukawa coupling), triggering electroweak symmetry breaking. Using the same techniques, we can see that lim \\aZ{u)}e2 =
1 F -(V - 7 ) .
(248)
Again, for particles that do not couple directly to the messengers 7' = 7 at one loop, and so we do not get A terms at one loop. (This is also obvious from the fact that there are no one-loop diagrams that could give an A term.) Direct couplings of the quarks and leptons to the messengers violate flavor symmetries, but the Higgs can have nontrivial couplings to the messengers. Some of the consequences of this are explored in Refs. [24]. It is important to remember that the results above are only the leading result in an expansion in powers of F/M2. In the effective theory below the messenger scale M, these additional terms are parameterized by terms with additional SUSY covariant derivatives, such as D2M M2
QfQ
(249) / ' Unlike the leading terms computed above, these terms are not related to the dimensionless couplings of the low-energy theory, and therefore require an independent calculation. This calculation has been performed in Refs. [25]. A£,eff
e
QiQ
M3
Supersymmetry
Breaking
557
The result is that the scalar mass in particular is very insensitive to corrections unless F is very near |M| 2 . 8.3. Phenomenology
of gauge
mediation
We now mention briefly some highlights of the phenomenology of gaugemediated SUSY breaking. For more detail, see Ref. [26] and references therein. First, note that because superpartner masses are controlled by gauge couplings, colored states will be much heavier than uncolored states. In particular, the ratio of stop masses to right-handed slepton masses is of order m
%
i -.v/3 4 - 1 0 . Si
(250)
(The factor of \/3 comes from the color factor c ~ 3 in Eq. (247).) The experimental bound miR > 99 GeV therefore implies rat~ <^ 980 GeV, which implies a sizable fine-tuning for electroweak symmetry breaking. The actual value of rri£ can be smaller, but fine-tuning is a concern in gauge mediated SUSY breaking. As always, this fine-tuning is more severe if SUSY is broken at high scales. Another important feature of gauge-mediated SUSY breaking is that the gravitino is generally the LSP. In standard scenarios for SUSY breaking, the gravitino gets a mass
ra
^=7&~io°Gev(K?£y2'
(25i>
where FQ is the primordial scale of SUSY breaking. (We will review the origin of this formula below when we discuss supergravity.) The bound Eq. (226) implies that m 3 / 2
558
M. A. Luty
spin | field is part of the supergravity multiplet, and therefore couples to matter with strength suppressed by powers of 1/Mp. On the other hand, the 'eaten' Goldstino couples to matter with strength determined by the priomordial SUSY breaking scale Fo. Since F0
+ h.c. + 0(G2),
(252)
where the supercurrent is J£ = O M M O a 0 ^ a - ^ ( A A ^ O a i W
(253)
This can be used to compute the decays of the other superpartners into Goldstinos. Since y/F~o is much larger that superpartner masses, the heavy superpartners will decay rapidly to the next-to-lightest superpartner (NLSP), which will then decay more slowly into Goldstinos. (We are assuming that R parity or a similar symmetry prevents rapid decays of the NLSP.) The phenomenology therefore depends on the value of Fo and the identity of the NLSP. If the NLSP is Bino, its dominant decay is
If the NLSP is the right-handed stau (the lightest of the right-handed sleptons because of mixing effects), its dominant decay is
Depending on the value of Fo, the NLSP can have a visible decay length:
1
" 7 ~ 1
For y/Fo < 10~ 6 GeV this decay is inside the detector. Even for y/Fo ~ 100 TeV (the smallest allowed value) this gives a displaced vertex small enough to be seen in a silicon vertex detector. Measurement of the decay length of the NLSP therefore gives direct information about the scale of primordial SUSY breaking!
Supersymmetry
8.4. Gravitino
Breaking
559
cosmology
We now make some brief remarks on gravitino cosmology in gauge-mediated models with R parity. The gravitino is stable in these models, and can therefore contribute to the energy density of the universe today. If the gravitino has a thermal abundance early in the universe, it freezes out at temperatures of order its mass. The relic abundance today is of order
^"(UFSV)"
<257)
where Q is the fraction of critical density contributed by the gravitino. (See Ref. [27] for a discussion of this standard calculation.) In order to avoid overdosing the universe we need fi3/2 < 1, or \fFo < 106 GeV. Note that this implies that the gravitino decays inside the detector in collider experiments! It is still possible to have y/Fo > 106 GeV if the primordial gravitino abundance is diluted by inflation with a low reheat temperature or by significant late-time entropy production. 9. 'Need-to-know' supergravity We now switch gears to a more formal subject: supergravity (SUGRA). SUGRA is the supersymmetric generalization of Einstein gravity, and as such unquestionably has a fundamental place in a supersymmetric world. However, the gravitational force is so weak that it is generally unimportant for particle physics experiments, so we start by explaining the motivation for a particle phenomenologist to learn about SUGRA. One reason has already emerged in our discussion of gauge-mediated SUSY breaking. Namely, the superpartner of the spin-2 graviton is a the spin-1 gravitino. Its interactions with ordinary matter are suppressed by powers of the Planck scale, but it can have interesting (or dangerous) cosmological effects. Another motivation is the cosmological constant problem, which is clearly a gravitational effect. The cosmological constant can be naturally zero in the limit of unbroken SUSY, but the cosmological constant problem comes back when SUSY is broken. There is at present no convincing solution to the cosmological constant problem, but SUSY is the only known symmetry that can explain why the cosmological constant smaller than the Planck scale, and may therefore play a role in the eventual solution. However, the primary motivation for studying SUGRA in these lectures is that SUGRA can be the messenger of SUSY breaking. In section 7 we
560
M. A. Luty
already considered gravity as the messenger of SUSY breaking. However, we really only considered the effect of integrating out heavy physics at the scale Mp. We did not include the effects of the SUGRA fields, which are light! In particular, we want to explore the idea that SUSY breaking can be communicated to the visible sector via the VEV of an auxiliary field in the SUGRA multiplet. This auxiliary field is in some ways analogous to a D field in SUSY gauge theory: if (D) ^ 0, it will give rise to SUSY breaking for fields that are charged under the gauge group.1 Since gravity couples universally, all fields are 'charged' under SUGRA, and we might expect that this gives rise to flavor-blind SUSY breaking. This idea can be made to work, but it turns out to be rather subtle and we will have to develop some formal machinery before we can get to the physics. Let us begin. There are several different formalisms for SUGRA, all of which are related by field redefinitions and give equivalent physical results. The simplest formulation for our purposes is the tensor calculus approach. The basic idea of this approach is to write off-shell supermultiplets as a collection of component fields, and to define the usual superfield operations directly on this collection of components. For example, a chiral multiplet is written as $ = (<^, j , a , F),
(258)
and products of chiral superfields are defined by $ ! $ 2 = (fafc,
4>l1p2a +lF2 + 0 2 ^ 1 + Iplfo)-
(259)
SUSY invariants are defined by taking the highest components of superfields, e.g. fd26$
= F.
(260)
For chiral and real superfields without SUGRA, this is just a rewriting of the usual rules for combining superfields. In the tensor calculus approach, matter and gauge supermultiplets are coupled to SUGRA by 'covariantizing' the rules for combining superfields and forming SUSY invariants. The minimal off-shell SUGRA multiplet is ( e / , VMa, £ „ , F^),
(261)
'One might wonder why we don't try to couple an additional f/(l) to the MSSM and break SUSY by (D) ^ 0. One obstacle to building a model of this kind is that the scalar masses are proportional to U(l) charges, which must occur with both signs in order to cancel gauge anomalies. Nonetheless, this approach may be viable [28].
Supersymmetry
Breaking
561
where eMa is the 4-bein, ^ p Q is the gravitino field, and B^ and F<j, are vector and scalar auxiliary fields, respectively. In the tensor calculus approach, the SUGRA fields are included by suitably covariantizing the usual rules for multiplying supermultiplets and taking their highest components to define SUSY invariants. In particular, the 4-bein e M a is coupled according to the standard rules from general relativity.J For SUSY breaking, interested in (F^,) ^ 0, since a VEV for B^ would break Lorentz invariance. The dependence on F$ is governed by supercovariance, and is closely related to a local (gauged) conformal invariance of the theory. To understand this local conformal invariance, let us see how it can be introduced in Einstein gravity without SUSY. There we can write a theory in a way that is invariant under local scale transformations by introducing an additional real scalar. The additional gauge symmetry can be used to gauge away the scalar, so this theory is equivalent to ordinary Einstein gravity. We first introduce local scale transformations acting on the metric as <7M„(a;) >-> n2(x)g^(x).
(262)
It is easy to see that the usual Einstein kinetic term is not invariant:
T%~g-ld9 => r%~r*v R^^dT + T2 => R^^R^
+ o(dn), + O{d0),
(263) (264)
where O(0Vl) denotes terms with derivatives acting on fl. Thererfore, R = g^R^
^ n~2R + 0(dSl),
T ^ ^ Q
4
^ ,
(265) (266)
and we see that the Einstein Lagrangian y/—gR is not invariant. (Alternatively, it is clear that the Einstein Lagrangian is not conformally invariant because it has a dimensionful coefficient proportional to Mp.) There is a 4-derivative action that is invariant under local scale transformations, but there is no obvious way to make sense out of theories whose leading kinetic term has 4 derivatives. To make the Lagrangian invariant, we introduce a real scalar rj transforming under local scale transformations as •q{x) .-» n~2(x)r]{x). J
(267)
For a clear introduction to the 4-bein formalism of general relativity, see Ref. [29] or
Ref. [21].
562
M. A. Luty
We can then write an invariant action in terms of the invariant 'metric' 9nv = V 9nv • fd^xy/^R
= jdAx^g~
[rj2R - 6(dri)2} .
(268) (269)
The signs are such that if the kinetic term for gravity has the right sign for positive energy, the kinetic term for the scalar has the 'wrong' sign (negative energy). Usually, a 'wrong' sign kinetic term means that the theory has a catastrophic instability to creation of negative energy modes. However, this is not a disaster in this case, because rj is not a physical degree of freedom: it can be gauged away. In fact, this theory is equivalent to Einstein gravity, as we can easily see by chosing the gauge r]{x) -> MP.
(270)
Eq. (270) is a good gauge choice as long as r\ is everywhere nonzero, which is good enough for perturbative expansions. k We have seen that we can rewrite Einstein gravity as a theory with an extra gauge symmetry (local scale invariance) and an extra scalar field. In fact, scale invariance implies invariance under an extended set of symmetries, the so-called conformal symmetries.1 The scalar field r/ is called the 'conformal compensator.' The same trick is useful in writing the SUGRA Lagrangian. This approach is called the superconformal tensor calculus. The full group of symmetries is the superconformal tranformations, which includes scale transformations and a U(l)u symmetry."1 One writes a theory that is invariant under local superconformal transformations, based on the superconformal supergravity multiplet
(eMa, Wa, Bp, RJ.
(271)
k
It may be disturbing that the strength of the gravitational coupling is apparently determined by a gauge choice. However, one must remember that only ratios of scales have physical significance. An operator of dimension d in the lagragian will be proportional to •q4~d by scale invariance, so the value of r\ just sets the overall scale. 'For an introduction, see Ref. [30]. m T h e appearance of local scale symmetry can be understood in a deductive way in the superfleld formulation of SUGRA. For a discussion of superfield SUGRA at the level of these lectures, see Refs. [31].
Supersymmetry
Breaking
563
Here B^ and i?M are vector auxiliary fields. To break the superconformal symmetry dowm to super-Poincare symmetry, one introduces a superconformal compensator supermultiplet, which is a chiral multiplet 4> = (r,, X , F*).
(272)
The real part of the scalar complex field rj plays the same role as the real scalar field called rj above. The dimension and R charge of are dO) = 1,
R(<j>) = I
(273)
Note the dimension is the same as the real scalar (f> of Eq. (269), and the R charge is such that the superpotential J d29(f)3 is U(1)R invariant. To see that the theory with the compensator is equivalent to ordinary nonconformal SUGRA, one makes the gauge choice 0 - » ( l , 0, .FV),
-R M ->0.
(274)
This discussion has been very sketchy. For more details, see Ref. [32]. The utility of all this formalism is that the couplings of the superfield <j>, and hence F^, are completely fixed by superconformal invariance. To determine the couplings of <j> it is sufficient to keep track of scale transformations and U(1)R transformations, which are determined by the dimension d and the R charge. The basic rule is that the Lagrangian C has d{C) — 4 and R(C) = 0. For a SUSY Lagrangian of the form £ = fd4e f+(
fd26 W + h . c ]
(275)
this means that d(f) = 2,
R(f) = 0,
(276)
d{W) = 3,
R(W) = +2.
(277)
It is convenient to choose all chiral and vector matter multiplets to have d = 0 and R = 0. (It may appear strange to choose d = 0 for matter fields, but we will see that we can make field redefinitions so that d coincides with the usual mass dimension.) For a Lagrangian of the form Eq. (275), this implies in particular d(f) = 0, so it is not superconformally invariant. To make it invaraint, we use the conformal compensator. To convariantized kinetic and superpotential terms for a Lagrangian of the form Eq. (275) are
564
M. A.
Luty
then
jtotftf
= e [/|94 + (f\92 • F\ +h.c.) + /I • (FlFf + 6R(g) + irifal*) + fermions (278)
d203W = e W\62 + W\ • (3Ftf, + ipip) + fermions , (279) / where e = det(e M a ) is the determinant of the 4-bein and the terms involving the gravitino ip have been written only schematically. Note that the constant term in / contains a kinetic term for the 4-bein and gravitino field, and a constant term in W contains a mass term for the gravitino. We have not written terms involving the matter fermions, since we are interested in SUSY breaking. For a gauge field with field strength Wa, note that d(WaWa) = 3 and R(WaWa) = 2, so there is no 4> dependence in the standard gauge kinetic terms: 0S(Q)WaWa + h.c. (280) / ' The sum of Eqs. (278), (279), and (280) is the most general Lagrangian terms with 2 or fewer derivatives, coupled to supergravity. Now we are (finally) ready to make our first main point. Consider a theory with no dimensionful couplings. The Lagrangian can be written schematically as A£ =
C
Id'e-
QievQ
+
H?
;WaWa
A3^i3 6
Q
It appears that this has nontrivial couplings between fields, the field redefinition
+ h.c.
and the matter (282)
Q = (pQ dependence completely:
removes the
di0QlevQ
C •
(281)
+
fd2e(\wawa
+g ;
+ h.c.
(283)
/
Note thatand Q are both chiral superfields, so Q is also chiral. Note also that this redefinition implies that the superconformal dimension of Q coincides with its canonical dimension, i.e. d(Q) = 1. In the case of the MSSM, the only dimensionful parameter is the fi term. Coupling this to SUGRA and using the canonical 'hatted' fields defined in
Supersymmetry
Breaking
565
Eq. (282), the SUSY breaking part of the Lagrangian would be (assuming <**> ^ 0) A £ S U S Y break = / x ( F ^ ) J f u f f d + h.C.
(284)
That is, SUSY is broken only by a B\i term. This does not give rise to a realistic model (e.g. the squarks and sleptons are much lighter than the Higgs). However, we will see that there are important loop effects that can make this form of SUSY breaking realistic. The fact that F^ decouples from a conformally invariant Lagrangian at tree level would seem to imply that there are no supergravity corrections to the potential at tree level. However, this is not quite correct, because the kinetic term for scalars includes a non-minimal coupling to gravity: fd494>^QtQ
= e
[\dQ\2 + \\Q\2R{g) + • • • ] .
(285)
This coupling means that the scalar fields in general mix with gravity. We can eliminate the non-standard scalar couplings by a field redefinition. For a component Lagrangian of the form C = V=9 [{M$ + f{Q))R{g) - V(Q)]
(286)
we can redefine the metric g»v = &§»„,
(287)
to obtain £ = ^ f t
4
[fl-2(M2 + f(Q))R(g)
- V(Q) + 0(df/M2)]
, (288)
where the omitted terms involve derivatives acting on / . Choosing
"2" s?fW'
<289)
we obtain C =
MlR{g) - V(Q) + 0(8f/M})
,
(290)
where
*m = wrmmw-
(291)
We have eliminated the non-standard couplings to gravity at the price of multiplicatively changing the scalar potential. This choice of definition of metric is often called 'Einstein frame.' The additional terms involving derivatives acting on / mean that the new Lagrangian does not contain
566
M. A. Luty
canonically normalized scalar fields in general. Further field redefinitions of the scalar fields can make these canonical. When all this is done, the expression for the potential for the canonically normalized scalar fields with no non-standard couplings to gravity is more complicated. If we go to Einstein frame in supergravity, the connection to scale invariance is obscured. For purposes of understanding SUSY breaking, it is often better not to go to Einstein frame, as we will see. 9.1. SUSY
breaking in SUGRA:
Polonyi
model
In this subsection and the following two, we consider SUSY breaking in the presence of SUGRA. Readers who are willing to take it for granted that (F^) will be nonzero in the presence of SUSY breaking can skip to section 10 below. The Polonyi model is the simplest model of SUSY breaking, and we now consider what happens when it is coupled to SUGRA. The Lagrangian is
£ = jdAetf[-?,Ml + f{X,X^)] + ( [d20(j)3[c + KX}+h.c. J .
(292)
If the dimensionful parameters c and K are small in units of Mp we expect the SUGRA corrections to the scalar potential to be small perturbations. In the absence of gravity, the scalar potential is V = j ^ ,
(293)
Jxix where fx^x — d2f/dX^dX. We assume that the Kahler function / is nontrivial so that this potential has a nontrivial minimum in which SUSY is broken. Since the superpotential is the most general linear function of X, we can shift the field so the minimum occurs at (X) = 0. To find the SUGRA corrections, we write out the terms without derivatives in the Lagrangian C = F j P V ( - 3 M | + / ) + (FxFlfx
(294)
fx\xFxFx
+
+ 3i^(c +
+ h.c.)
KFX
+ KX)+
h.c.
+ h.c. + • • •
(295)
Supersymmetry
Breaking
567
Integrating out Fx, we obtain
N2
l/x|
3M$ + /
/xtx
/xtx
+
«/.x t
3(C+KX)
F]F A
Frf, + h.c
(296)
/xt X
If the dimensionful couplings in the Polonyi sector are small compared to Mp, we can approximate the coefficient of F\FJ, by - 3 M p and write 3|c| 2
V = xtx
Ml
X
1 +
/.x t
(297)
3/xtx.
Note that the SUGRA corrections to the potential are negative definite. (This is related to the 'wrong-sign' kinetic term for the conformal compensator.) This is a crucial property that allows us to cancel the positive vacuum energy due to SUSY breaking and obtain a ground state with vanishing cosmological constant. This requires that we tune 3(/xtx) (Note that since c oc Mp, it is a good approximation to neglect the terms proportional to K/C in Eq. (297).) Because the Kahler function / is renormalized, this is not stable under radiative corrections. Note that if the fluctuations of X about (X) = 0 are canonically normalized, then ( / x t x ) ~ 1 and c ~ nMp. In this vacuum, SUSY is broken by the auxiliary fields (Fx) = -71?—T,
(/xtx)'
f
{F } =
*
Ml
CFx> MP
(299) (300)
up to corrections suppressed by powers of 1/Mp, and where we have assumed ( / x t x ) ~ 1 in the last relation. If we include the gravitino couplings, we find that there is also a gravitino mass (W) (301) = (Fl). MP MP This is the origin of the formulas for the gravitino mass used in the section on gauge mediated SUSY breaking. We see that if SUSY is broken below the Planck scale by the usual O'Raifeartaigh (or Polonyi) mechanism, the only effect of SUGRA is to m3/2
568
M. A. Luty
allow the fine-tuning of the cosmological constant, and to generate a nonzero VEV for the auxiliary field F^. 9.2.
'No scale' SUSY
breaking
The above analysis assumed that SUSY is broken in the absence of SUGRA. There is another possibility for SUSY breaking that can only occur in the presence of SUGRA: 'no scale' SUSY breaking. To see how this works, consider a model of a single chiral superfield T with Lagrangian C = fd49 >*(T + T f ) + ( fd29 <j>3c + h.c") ,
(302)
where c is a constant. Note that T has been chosen to have dimensions of mass-squared. This choice of Lagrangian is rather arbitrary, and in fact it is not radiatively stable. We will address this below, but let us start by understanding this simple Lagrangian. First, note that in the absence of gravity, the Kahler term would be a total derivative. The field T acquires a kinetic term only by mixing with gravity, so this is only a healthy theory in the presence of gravity. The terms with no derivatives are
£ = F ^ ( T + r+) + (FJFr + h.c.) + 3F^c + h.c. + • • •
(303)
Note that we have not included the Kinetic term for gravity, since this can be absorbed into a shift of T. In order to get the right strength for gravity, we need (T) = - § M | .
(304)
Varying with respect to FT tells us that F$ = 0, and hence the potential vanishes identically. In particular, this means that the cosmological constant vanishes. On the other hand, the gravitino mass is nonzero: m 3 / 2 = ^L
= -£-.
(305)
Thus, SUSY is broken with vanishing cosmological constant! This kind of SUSY breaking is called 'no scale' SUGRA for historical reasons. However, the fact that the potential vanishes identically also means that the scalar field T is completely undetermined. Also, the form of the Lagrangian Eq. (302) is not preserved by radiative corrections.
Supersymmetry
Breaking
569
Suppose therefore that we add a nontrivial Kahler corrections to the Lagrangian above: AC = ftiPetftAfiT,^).
(306)
Such corrections will in any case be induced radiatively, and may play a role in the stabilization of T. Let us treat A / perturbatively, and ask what are the conditions that we get a vacuum that is 'close' to the one found above. It is easy to work out that the scalar potential to first order in A / : AV = - | c | 2 A / T t r -
(307)
Therefore, if fotT has a local maximum, the theory will have a stable minimum where (T) is given by Eq. (304), as required. In this vacuum, we have (to first order in A / ) (FT) = -3c\
(308)
/AfTiTFT\ \ T + Tt
(F4>)
_
ct(A/Ttr) 3M' P2
,30gs
Note if A / is small, we can make (F^) as small as we want. This kind of vacuum can be thought of as 'almost no scale' SUGRA. For a more complete discussion see Ref. [33]. 9.3. The SUGRA
potential
We now consider the SUGRA corrections to the scalar potential. This is important to make contact between the approach to SUGRA taken here, which emphasizes the auxiliary fields, and more conventional treatments which use the SUGRA corrections to the potential as a starting point. We consider a 2-derivative Lagrangian of the form C = fd49 tf<j>f + f fd29
(310)
where / and W are functions of some matter fields Qa. In order to get the right kinetic term for gravity, we require (/) = -3Ml
(311)
The terms in the Lagrangian with no derivatives are £ = F^f
+ [F\faFa
+ h.c.) +
+ 3Fj,W + WaFa+h.c,
fabFlFa (312)
570
M. A. Luty
where /„ = df/dQa, find
fa = df/dQl,
etc. Solving for the auxiliary fields, we
Fl = -(Ma" „+
f
1
f
wb-^w
(3W + faF*),
(313) (314)
where ( / _ 1 ) a b is the matrix inverse of _ f b * i fb fa b = fa" ~ jfaf-
(315)
Integrating out the auxiliary fields, we find after some algebra (316) As discussed above, this is not the potential in Einstein frame. To make the gravity kinetic term canoncial, we define the Einstein frame metric
/ The potential in Einstein frame is then
t-(*f)\.
(318,
Note also that the function / is not what is called the Kahler potential in the SUGRA literature. The Kahler potnetial is related to / by f =-3M$e-K/3Mk
(319)
With these relations, Eq. (318) reduces to the standard expression for the SUGRA potential (see e.g. Ref. [5]). Let us apply these results to find the conditions for a vacuum that preserves SUSY and has a vanishing cosmological constant. To preserve SUSY it is sufficient for all auxiliary fields to vanish in the vacuum. The condition (F%) = 0 is equivalent to Wa - 3Wfa/f = 0 provided that (fab) is a non-singular matrix. This in turn is equivalent to the condition that W/f3 is stationary. The condition (F!) = 0 then imposes the additional requirement that (W) = 0. Combining these, we see that the conditions for a SUSY vacuum with vanishing cosmological constant are W stationary,
(W) = 0.
(320)
Supersymmetry
Breaking
571
Note that we can always impose (W) — 0 by adding a constant to the superpotential. This shows that the SUSY-preserving vacua in the presence of SUGRA with vanishing cosmological constant are in one-to-one correspondence with the SUSY vacua in the absence of SUGRA. In particular, the SUGRA corrections to the potential cannot turn a SUSY preserving vacuum into a SUSY breaking one.
10. Anomaly mediated S U S Y breaking Now we finally have enough machinery to discuss SUGRA as the messenger of SUSY breaking. We therefore assume that the only source of SUSY breaking comes from a non-vanishing value of (F^). This can be viewed as a SUGRA background in which we are calculating. Let us consider a SUSY model with no dimensionful couplings in the SUSY limit. (The NMSSM is such a model, as is the MSSM if we omit the /J, term. We will see that the fx term in the MSSM will have to be treated differently.) As we have seen in the previous section, the fact that the there are no dimensionful couplings means that at tree level there is no SUSY breaking felt by the matter fields. However, at loop level scale invariance is broken by the running of the couplings. We therefore expect SUSY breaking related to the conformal anomaly. This is called 'anomaly mediated SUSY breaking' (AMSB). The original papers are Refs. [34,35]. To see how this works in a very concrete way, note that the presence of UV divergences requires us to regulate the theory, and the regulator necessarily introduces a mass scale, and therefore breaks conformal symmetry. For example, in a Wess-Zumino model, we can use higher-derivative terms to regulate the theory:
C = jd*8 Z0& (l + ^ )
Q + (^j#9 I Q3 + h.c.) ,
(321)
where • is the covariant second derivative operator. We have directly written the Lagrangian in terms of the canonical 'hatted' fields defined by Eq. (282). The factors of (j> are required because the operator • has d = 2 and R = 0. The D / A term modifies the propagator of the component fields in Q. For example, setting (F$) = 0 for the moment, the scalar propagator is modified & ~* P 2 - P 4 / A 2 '
(322)
572
M. A. Luty
This makes loops of Q fields UV convergent." Reintroducing (F^) ^ 0 introduces a small splitting between the regulated scalars and fermions required by the coupling to the SUGRA background, but the loop diagrams are still finite. Note that we are only regulating loops of Q fields, not SUGRA loops. SUGRA loops are suppressed by additional powers of Mp, and are therefore much smaller than loop effects from dimensionless couplings. We now compute standard model loop corrections in the SUGRA background. We do this (once again) by treating the couplings as superfields. At one loop, the Q^Q term in the 1PI effective action is logathimically divergent:
w
^
l+ ^ln^+finite 267H
fi
(323)
where A is the physical Yukawa coupling. Note thedependence in Eq. (323) is required by conformal invariance. There is no <j> dependence in the 'finite' part of the amplitude because it is by definition independent of A as A —> co. The divergence (A dependence) must be cancelled by absorbing it into the 'bare' coupling ZQ, but we cannot absorb the <j> dependence into ZQ because we want all SUSY breaking to be due to the nontrivial SUGRA background. (If the superfield ZQ had some nonzero SUSY breaking components, this would be a theory with SUSY broken in the fundamental theory.) This means that there is some SUSY breaking left over in the finite part after we subtract the divergence. We can view this as the replacement l n M ^ l n ^ r = l n / i - i ( 6 » 2 F < / , + h.c).
(324)
The fact that this substitution parameterizes all the SUSY breaking is clearly more general than than this example. The cutoff A and the renormalization scale fi always appear in the combination A//z, and for any real superfield coupling the correct substitution is A —> A\(j>\, implying Eq. (324). Note also that Eq. (324) holds to all orders in perturbation theory. We can therefore compute SUSY breaking in real superfield couplings by
W
=
"2^dW?
"This regulator also introduces a ghost, i.e.
(325)
a state with wrong-sign kinetic term at
p2 = A 2 . However, this decouples when we take the limit A —• co and does not cause any difficulties.
Supersymmetry
Breaking
573
For example, the gaugino mass computed from the real superfield gauge coupling is mx = -g2[R}92 = P^F+.
(326)
We also have [lnZ] 92 = - i
7
^,
(327)
which gives rise to nonzero A terms. Finally, we have soft masses m2 = - ( l n Z W = - i ^ |
2
^ . dln/x'
(328)
Eqs. (326), (327), and (328) are exact in the sense that they hold to all orders in perturbation theory. They hold at each renormalization scale, and therefore define the 'AMSB renormalization group trajectory.' As in gauge mediation, the dominant source of SUSY breaking comes from gauge loops, and therefore the scalar masses are flavor-blind and the model solves the SUSY flavor problem. Also as in gauge mediation, gaugino masses arise at one loop and scalar mass-squared parameters at two loops, so all SUSY breaking masses are of the same order: m
*~l&-
m
^(i^¥
(329)
To get SUSY breaking masses of order 100 GeV, we need (F^) ~ 10 TeV. It is very interesting that the model gives the entire superpartner spectrum in terms of a single new parameter (F^), which just sets the overall scale of the superpartner masses. Let us check the crucial sign of the scalar masses. For fields with only gauge interactions, we have 7 ~ +g2 and therefore m2~-2/3g|
(330)
We see that if the gauge group is asymptotically free {j3g < 0) the scalar mass-squared parameter is positive. Unfortunately in the MSSM, the SU(2)w and U(1)Y gauge groups are not asymptotically free, so sleptons have negative mass-squared. Adding more fields can only make this worse. We cannot live on the AMSB renormalization trajectory. Nonetheless, it is possible to have realistic SUSY breaking from AMSB. To understand this, let us consider the effect of massive thresholds in AMSB. Note that the formulas Eqs. (326), (327), and (328) are claimed to hold independently of the details of the high energy theory, in particular
574
M. A. Luty
the nature of ultrahigh energy thresholds {e.g. at the GUT scale). Let us see how this works. Consider some new chiral superfields P and P that transform as a vectorlike representation of the standard model gauge group, and which have a large supersymmetric mass term A £ = fd29 MtpPP + h.c.
(331)
Note that we have included the superconformal compensator in a normalization where the fields P and P have canonical kinetic terms. Because the fields P and P are charged, the gauge beta functions will have different values above and below the scale M, so the SUSY breaking masses above and below the scale M are different. To understand this, note that the P threshold is not supersymmetric because of thedependence. Because of this, there is a gauge-mediated threshold correction at the scale M, with £
= **.
(332)
For example, for gaugino masses, the threshold correction is (see Eq. (231)) Am A = ^ i V
(333)
Adding this to the AMSB value above the threshold, we find that the gaugino mass below the threshold is precisely on the AMSB trajectory below the threshold. Another way to understand this point is to consider again the superfield couplings. The holomorphic gauge coupling superfield T below the threshold M is given by T{H)
=
T0
M
b'
2
167T
^
b
,
fj,
+ —-5- In — + ——r In —: +
A
T^ln^
2
167T +
M
i6^ln^-
(334)
where b and b' are the beta function coefficients below and above the scale M, respectively. (The notation is the same as in section 8.) We see that the <> / dependence induced by A and M exactly cancel in the contribution from above the scale M. The gaugino mass below the scale M is correctly given by the substitution fj, —> fi/<j>, just as if the threshold did not exist. We can think of M as the new cutoff. The fact that AMSB is independent of thresholds is very striking, and makes the theory very predictive. Unfortunately, we have seen that it is too predictive, and is ruled out by negative slepton mass-squared parameters.
Supersymmetry
Breaking
575
This discussion however suggests a way out. If a heavy threshold is not supersymmetric, the cancelation discussed above no longer occurs, and we can be on a different RG trajectory below the threshold. If the F/M of the threshold is much larger than FA, we have gauge mediation, and if it is much smaller it is irrelevant. The only interesting case is where F/M ~ FA, and we would like this to occur naturally. A very simple class of models where this occurs was first discussed in Ref. [36]. Consider a theory with a singlet S in addition to the vector like fields P and P, with superpotential terms
sn
XSPP +
AC =!dH
(M4>) n - 3
+ h.c.
(335)
The potential for S is V =
MUnz
S_ M
2(n-l)
+
<"- 3 )is)"§ +1 "-
(336)
3 (FA) n(n - 1) M
(337)
Minimizing the potential we find
M From this we see that (Fs) (S)
-
!
<
*
*
>
•
(338)
For n > 3 and M > (JF^), Eq. (337) implies that (FA) « (S) < M, while (Fs)/(S) ~ (FA). Because the coefficient in Eq. (338) is different from unity, the theory will not be on the AMSB trajectory below the threshold. These theories can be viewed as a combination of gauge and anomaly mediation. We can obtain a realistic SUSY breaking spectrum in this way (in particular, the slepton masses can be positive). See Ref. [36] for more details. There are also other ways proposed in the literature to make AMSB realistic. One class of models is similar to the proposal discussed above, in that they have an additional 'gauge mediated' contribution to SUSY breaking that is naturally the same size as the AMSB contribution [37]. For a very different approach, see Ref. [38].
576
M. A. Luty
10.1. The \x problem
in anomaly
mediation
The n problem is more severe in AMSB because we cannot simply add a conventional fi term of the form A £ = fd20 n4>HuHd + h.c.
(339)
The reason is that the explicit breaking of conformal invariance means that B/J. = {F^fi,
(340)
which is far too large for (F^) ~ 10 TeV. One possibility is the NMSSM, discussed in subsection 5.10. This has no dimensionful couplings, and therefore this problem is absent. The effective JJ, term arises from a VEV for the singlet (S), which is ultimately triggered by AMSB itself. Given the fact that this Another possibility was pointed out in Ref. [34]. If there is a chiral superfield X with a shift symmetry X — i » X + constant, then the [i term can arise from an operator of the form A £ = fd46^jj(X
+ X^)HuHd
+ h.c.
(341)
Assuming (X) = 0, we have AC = fd29 ^HuHd
+ h.c.
(342)
i.e. we generate a ji term with no B/j, term. The B/J term can come from AMSB in this model. Yet another possibility to generate the [i term from the VEV of a singlet is described in Ref. [36]. 10.2. Anomaly-mediated
phenomenology
Since the theory cannot be on the AMSB trajectory at low energies, the low-energy phenomenology depends on how these problems are resolved. Discussions can be found in the original papers, quoted above. We do want to point out, however, that these theories share the finetuning problem of gauge mediated SUSY breaking, since scalar masses arise from 2-loop gauge diagrams, and therefore TO
2
Ncg43
£- ~ ^ . m in 91
(343)
Supersymmetry
10.3. Naturalness
of anomaly
Breaking
577
mediation
So far we have not addressed the question of whether it is natural for the theory to be on the AMSB trajectory. What we would like is to have a theory that breaks SUSY spontaneously in a hidden sector in such a way that the breaking is communicated to the observable sector dominantly through the SUGRA conformal compensator. As we have seen in subsections 9.1 and 9.2, spontaneous SUSY breaking in SUGRA generally gives (F,) < A
(344)
where Fo is the primordial SUSY breaking scale. Generally, FQ = (Fx), where X is some chiral superfield. In this case, we expect the effective theory to contain operators of the form A £ ~ fd49~X^XQ^Q
(345)
where Q are standard model fields. As already discussed in section 7, these couplings have the quantum numbers of a product of kinetic terms, and cannot be forbidden by any symmetries. We therefore expect them to be present in any UV completion of the theory at the Planck scale. This gives rise to scalar masses of order
(Fx)2 Ml
m2 ~ ^ - ,
(346)
which is much larger than the AMSB value. (Furthermore, there is no reason for the term Eq. (345) to conserve flavor, so we expect the masses to give rise to FCNC's.) If we consider all the other possible terms coupling the visible and the hidden sector suppressed by powers of Mp (see Eq. (206)), we find that they can all naturally be absent due to symmetries. Therefore, the viability of anomaly mediation depends on whether there are sensible models where the couplings Eq. (345) are naturally absent. A simple rationale for this was given in Ref. [34]. The idea is that the hidden and visible sectors are localized on 'branes' in extra dimensions. 0 That is, the standard model matter and gauge interactions are localized on the visible brane, and the SUSY breaking sector is localized on the hidden °We will not go into details here, but will just state the main ideas. For an introduction into many of the technical and conceptual issues in theories with extra dimensions and branes, see the lectures by Raman Sundrum at this school.
578
M. A. Luty
brane. In fact, this type of setup naturally occurs in string theory, e.g. in the setup of Ref. [39]. In the higher-dimensional theory, interactions like Eq. (345) are fobidden because X and Q are localized on different branes, so the interaction is not local. Fields that propagate in the bulk can give rise to interactions between X and Q, so we must check whether interactions like Eq. (345) are generated in the 3 + 1 dimensional theory below the compactification scale i ? - 1 , where R is the distance between the visible and hidden branes. If the scale of new physics is M (e.g. the string scale), then for R ~^> M - 1 the propagator of a massive field (e.g. and excited string state) connecting the visible and hidden branes is suppressed by the Yukawa factor e~MR <s£ 1. (Since the suppression factor is exponential, R > fewx M _ 1 is sufficient in practice.) Therefore, operators like Eq. (345) are not generated by the exchange of massive states. This leaves only the effect of fields that are light compared to the compactification scale. Only supergravity must propagate in the bulk, so the minimalthe minimal light fields in the bulk are the minimal 5D SUGRA multiplet. It was shown in Ref. [40] that this does not generate terms of the form Eq. (345). For details, see Refs. [34,40]. In this setup, the SUSY breaking sector is 'more hidden' than in conventional hidden sector models, and is sometimes referred to as a 'sequestered sector.' Another way to make this natural is to replace the extra dimension in the setup above with a conformal field theory via the AdS/CFT correspondence. For a discussion of this 'conformal sequestering,' see Ref. [41]. 11. Gaugino mediation The final model we will mention (very briefly) is gaugino mediation. Like anomaly mediation, this can be motivated by an extra-dimensional setup. This time we assume that the standard-model gauge fields propagate in the bulk, while the matter fields are localized on the visible brane. In this case, the gauginos can get a mass from contact terms on the hidden brane of the form AC ~ fd29 ^-WaWa
+ h.c,
(347)
where X is the hidden sector field that breaks SUSY. In this type of model, the gaugino gets a mass at tree level, while the visible matter fields get a mass only at one-loop order. This means that the gaugino masses are much larger than scalar masses at the compactification scale, but the RG between this scale and the weak scale generates scalar masses of order the
Supersymmetry
Breaking
579
gaugino masses. This scenario is called 'gaugino mediated SUSY breaking' for obvious reasons. The original papers are Refs. [42]. Note that gaugino mediation shares the fine-tuning problem of gaugeand anomaly-mediation. In a GUT model, we expect that the gaugino masses are unified at the GUT scale: MI(MGUT) ^ M 2 ( M G U T ) ^ M3(MGUT).
(348)
(Even if there are GUT breaking splittings, we expect Mi ~ Af2 ~ M 3 , which is sufficient for our argument.) Since the quantity Mi/gf is RG invariant at one loop, at the weak scale we have
Ml^Ml^M*.
(349)
Si 92 9s The scalar masses are generated from the RG equation
^ Using the fact that m\/g2
- ~f<
(3»)
is RG invariant, we have the solution
™ 2 (M) = j where we have assumed that
[fl 4 M - 94(MGVT)] TO2(MGUT)
m m
?
lR
AT
(^
,
(351)
*C m2(n). We therefore have „4
Ncgi 9x
(352)
just as in gauge mediation and anomaly mediation. 12. N o conclusion There is much more to say, but I will stop here. I hope that I have introduced some of the problems and issues with SUSY breaking, as well as introducing some ideas that may point in the right direction. I hope that some of the readers will be inspired by these lectures to go beyond them. References 1. S. Weinberg, "Anthropic Bound On The Cosmological Constant," Phys. Rev. Lett. 59, 2607 (1987). 2. L. Susskind, "The anthropic landscape of string theory," arXiv:hepth/0302219; N. Arkani-Hamed, S. Dimopoulos and S. Kachru, "Predictive landscapes and new physics at a TeV," arXiv:hep-th/0501082.
580
M. A. Luty
3. R. Sundrum, "Towards an effective particle-string resolution of the cosmological constant problem," JHEP 9907, 001 (1999) [arXiv:hep-ph/9708329]. 4. D. E. Kaplan and R. Sundrum, arXiv:hep-th/0505265. 5. J. Wess and J. Bagger, "Supersymmetry and supergravity," Princeton (1992). 6. S. P. Martin, "A supersymmetry primer," arXiv:hep-ph/9709356. 7. R. D. Peccei and H. R. Quinn, "CP Conservation In The Presence Of Instantons," Phys. Rev. Lett. 38, 1440 (1977). 8. S. Weinberg, "A New Light Boson?," Phys. Rev. Lett. 40, 223 (1978). F. Wilczek, "Problem Of Strong P And T Invariance In The Presence Of Instantons," Phys. Rev. Lett. 40, 279 (1978). 9. A. E. Nelson, "Naturally Weak CP Violation," Phys. Lett. B 136, 387 (1984); S. M. Barr, "Solving The Strong CP Problem Without The Peccei-Quinn Symmetry," Phys. Rev. Lett. 53, 329 (1984). 10. N. Arkani-Hamed, G. F. Giudice, M. A. Luty and R. Rattazzi, "Supersymmetry-breaking loops from analytic continuation into superspace," Phys. Rev. D 58, 115005 (1998) [arXiv:hep-ph/9803290]. 11. M. A. Shifman and A. I. Vainshtein, "On holomorphic dependence and infrared effects in supersymmetric gauge theories," Nucl. Phys. B 359, 571 (1991). 12. F. Borzumati, G. R. Farrar, N. Polonsky and S. Thomas, "Soft Yukawa couplings in supersymmetric theories," Nucl. Phys. B 555, 53 (1999) [arXivihepph/9902443]. 13. F. Gabbiani, E. Gabrielli, A. Masiero and L. Silvestrini, "A complete analysis of FCNC and CP constraints in general SUSY extensions of the standard model," Nucl. Phys. B 477, 321 (1996) [arXiv:hep-ph/9604387]. 14. G. Hiller and M. Schmaltz, "Solving the strong CP problem with supersymmetry," Phys. Lett. B 514, 263 (2001) [arXiv:hep-ph/0105254]. 15. R. Harnik, G. D. Kribs, D. T. Larson and H. Murayama, "The minimal supersymmetric fat Higgs model," Phys. Rev. D 70, 015002 (2004) [arXiv:hepph/0311349]; A. Birkedal, Z. Chacko and Y. Nomura, "Relaxing the upper bound on the mass of the lightest supersymmetric Higgs boson," Phys. Rev. D 71, 015006 (2005) [arXiv:hep-ph/0408329]. 16. E. Witten, "Constraints On Supersymmetry Breaking," Nucl. Phys. B 202, 253 (1982). 17. N. Arkani-Hamed, S. Dimopoulos, G. Dvali and G. Gabadadze, "Non-local modification of gravity and the cosmological constant problem," arXiv:hepth/0209227. 18. N. Arkani-Hamed, H. C. Cheng, M. A. Luty and S. Mukohyama, "Ghost
Supersymmetry Breaking 581
19. 20.
21. 22. 23.
24.
25.
condensation and a consistent infrared modification of gravity," JHEP 0405, 074 (2004) [arXiv:hep-th/0312099]. G. F. Giudice and A. Masiero, "A Natural Solution To The Mu Problem In Supergravity Theories," Phys. Lett. B 206, 480 (1988). R. Barbieri, G. R. Dvali and L. J. Hall, "Predictions From A U(2) Flavour Symmetry In Supersymmetric Theories," Phys. Lett. B 377, 76 (1996) [arXiv:hep-ph/9512388]. M. B. Green, J. H. Schwarz and E. Witten, "Superstring Theory. Vol. 2: Loop Amplitudes, Anomalies And Phenomenology," Cambridge (1987). Y. Shadmi and Y. Shirman, "Dynamical supersymmetry breaking," Rev. Mod. Phys. 72, 25 (2000) [arXiv:hep-th/9907225]. G. F. Giudice and R. Rattazzi, "Extracting supersymmetry-breaking effects from wave-function renormalization," Nucl. Phys. B 511, 25 (1998) [arXiv:hep-ph/9706540]. M. Dine, Y. Nir and Y. Shirman, "Variations on minimal gauge mediated supersymmetry breaking," Phys. Rev. D 55, 1501 (1997) [arXivrhepph/9607397]; Z. Chacko and E. Ponton, "Yukawa deflected gauge mediation," Phys. Rev. D 66, 095004 (2002) [arXiv:hep-ph/0112190]. S. Dimopoulos, G. F. Giudice and A. Pomarol, "Dark matter in theories of gauge-mediated supersymmetry breaking," Phys. Lett. B 389, 37 (1996) [arXiv:hep-ph/9607225]; S. P. Martin, "Generalized messengers of supersymmetry breaking and the sparticle mass spectrum," Phys. Rev. D 55, 3177 (1997) [arXiv:hep-ph/9608224].
26. G. F. Giudice and R. Rattazzi, "Theories with gauge-mediated supersymmetry breaking," Phys. Rept. 322, 419 (1999) [arXiv:hep-ph/9801271]. 27. E. W. Kolb and M. S. Turner, "The Early Universe," Addison-Wesley (1990). 28. B. A. Dobrescu, "B — L mediated supersymmetry breaking," Phys. Lett. B 403, 285 (1997) [arXiv:hep-ph/9703390]. 29. S. Weinberg, "Gravitation and Cosmology," Academic Press (1972). 30. P. H. Ginsparg, "Applied Conformal Field Theory," arXiv:hep-th/9108028. 31. W. D. Linch, M. A. Luty and J. Phillips, "Five dimensional supergravity in N = 1 superspace," Phys. Rev. D 68, 025008 (2003) [arXiv:hep-th/0209060]; T. Gregoire, M. D. Schwartz and Y. Shadmi, "Massive supergravity and deconstruction," JHEP 0407, 029 (2004) [arXiv:hep-th/0403224]. 32. E. Cremmer, S. Ferrara, L. Girardello and A. Van Proeyen, "Yang-Mills Theories With Local Supersymmetry: Lagrangian, Transformation Laws And Superhiggs Effect," Nucl. Phys. B 212, 413 (1983). 33. M. A. Luty and N. Okada, "Almost no-scale supergravity," JHEP 0304, 050 (2003) [arXiv:hep-th/0209178].
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34. L. Randall and R. Sundrum, "Out of this world supersymmetry breaking," Nucl. Phys. B 557, 79 (1999) [arXiv:hep-th/9810155]. 35. G. F. Giudice, M. A. Luty, H. Murayama and R. Rattazzi, "Gaugino mass without singlets," JHEP 9812, 027 (1998) [arXiv:hep-ph/9810442]. 36. A. Pomarol and R. Rattazzi, "Sparticle masses from the superconformal anomaly," JHEP 9905, 013 (1999) [arXiv:hep-ph/9903448]. 37. D. E. Kaplan and G. D. Kribs, "Gaugino-assisted anomaly mediation," JHEP 0009, 048 (2000) [arXiv:hep-ph/0009195]; Z. Chacko and M. A. Luty, "Realistic anomaly mediation with bulk gauge fields," JHEP 0205, 047 (2002) [arXiv:hep-ph/0112172]; R. Sundrum, "'Gaugomaly' mediated SUSY breaking and conformal sequestering," Phys. Rev. D 71, 085003 (2005) [arXiv:hepth/0406012]. 38. N. Arkani-Hamed, D. E. Kaplan, H. Murayama and Y. Nomura, "Viable ultraviolet-insensitive supersymmetry breaking," JHEP 0102, 041 (2001) [arXiv:hep-ph/0012103]. 39. P. Horava and E. Witten, "Eleven-Dimensional Supergravity on a Manifold with Boundary," Nucl. Phys. B 475, 94 (1996) [arXiv:hep-th/9603142]. 40. M. A. Luty and R. Sundrum, "Radius stabilization and anomaly-mediated supersymmetry breaking," Phys. Rev. D 62, 035008 (2000) [arXiv:hepth/9910202]. 41. M. A. Luty and R. Sundrum, "Supersymmetry breaking and composite extra dimensions," Phys. Rev. D 65, 066004 (2002) [arXiv:hep-th/0105137]. 42. D. E. Kaplan, G. D. Kribs and M. Schmaltz, "Supersymmetry breaking through transparent extra dimensions," Phys. Rev. D 62, 035010 (2000) [arXiv:hep-ph/9911293]. 43. Z. Chacko, M. A. Luty, A. E. Nelson and E. Ponton, "Gaugino mediated supersymmetry breaking," JHEP 0001, 003 (2000) [arXiv:hep-ph/9911323].
•^j\>'
RAMAN SUNDRUM
TO T H E FIFTH D I M E N S I O N A N D B A C K
RAMAN SUNDRUM Department of Physics and Astronomy The Johns Hopkins University 3400 North Charles Street Baltimore, MD 21218, USA [email protected]. edu
Introductory lectures on Extra Dimensions delivered at TASI 2004.
1. Introduction There are several significant motivations for studying field theory in higher dimensions: (i) We are explorers of spacetime structure, and extra spatial dimensions give rise to one of the few possible extensions of relativistic spacetime symmetry, (ii) Extra dimensions are required in string theory as the price for taming the bad high energy behavior of quantum gravity within a weakly coupled framework, (iii) Extra dimensions give rise to qualitatively interesting mechanisms within effective field theory that may play key roles in our understanding of Nature, (iv) Extra dimensions can be a type of "emergent" phenomenon, as best illustrated by the famous AdS/CFT correspondence. These lectures are intended to provide an introduction, not to the many attempts at realistic extra-dimensional model-building, but rather to central qualitative extra-dimensional mechanisms. It is of course hoped that by learning these mechanisms in their simplest and most isolated forms, the reader is well-equipped to work through more realistic incarnations and combinations, in the literature, or better yet, as products of their own invention. (Indeed, to really digest these lectures, the reader must use them to understand some particle physics models and issues. The other TASI lectures are a good place to start.) When any of the examples in the lectures yields a cartoon of the real world, or a cartoon solution to real world problems, I point this out. 585
586
R.
Sundrum
The lectures are organized as follows. Section 2 gives the basic language for dealing with field theory in the presence of extra dimensions, "compactified" in order to hide them at low energies. It is also shown how particles of different spins in four dimensions can be unified within a single higherdimensional field. Section 3 illustrates the "chirality problem" associated with fermions in higher dimensions. Section 4 illustrates the emergence of light scalars from higher dimensional theories without fundamental scalars, computes quantum corrections to the scalar mass (potential), and assesses how natural these light scalars are. Section 5 describes how extra dimensional boundaries (and boundary conditions) can be derived from extra dimensional spaces without boundary, by the procedure of "orbifolding". It is shown how the chirality problem can thereby be solved. The localization of some fields to the boundaries is illustrated. Section 6 describes the matching of the higher dimensional couplings to the effective four-dimensional long-distance couplings. In Section 7, the issue of non-renormalizability of higher-dimensional field theory is discussed and the scale at which a UV completion is required is identified. Higher-dimensional General Relativity is discussed in Section 8, in partiucular the emergence of extra gauge fields at low energies as well as scalar "radion" fields (or "moduli") describing massless fluctuations in the extra-dimensional geometry. Section 9 illustrates how moduli may be stabilized to fix the extra-dimensional geometry at low energies. Section 10 describes the unsolved Cosmological Constant Problem as well as the less problematic issue of having a higher-dimensional cosmological constant. Section 10 shows that a higher dimensional cosmological constant leads to "warped" compactifications, as well as the phenomenon of "gravity localization". Section 11 shows that strongly warped compactifications naturally lead to hierarchies in the mass scales appearing in the low energy effective four-dimensional description. Section 12 shows that when warped hierarchies are used to generate the Planck/weak-scale hierarchy, the extra-dimensional graviton excitations are much more strongly coupled to matter than the massless graviton of nature, making them observable at colliders. Section 13 shows how flavor hierarchies and flavor protection can arise naturally in warped compactification, following from a study of higher-dimensional fermions. Section 14 studies features of gauge theory, including the emergence of light scalars, in warped compactifications. The TASI lectures of Ref. 1 and Ref. 2, and the Cargese lectures of Ref. 3, while overlapping with the present lectures, also contain complementary topics and discussion. The central qualitative omissions in the
To the Fifth Dimension
and Back
587
present lectures are supersymmetry, which can combine with extra dimensions in interesting ways (see the TASI lectures of Refs. 1 and 4), a longer discussion of the connection of extra dimensions to string theory 5 ' 6 , a discussion of fluctuating "branes" (see Refs. 1 and 3), and the (very illuminating) AdS/CFT correspondence between some warped extra-dimensional theories and some purely four-dimensional theories with strong dynamics 7 ' 8 ' 9 . Phenomenologically, there is no discussion of the "Large Extra Dimensions" scenario 10, although these lectures will equip the reader to easily understand it. The references included are meant to be useful and to act as gateways to the broader literature. They are not intended to be a complete set. I have taken moderate pains to get incidental numbers right in the notes, but I am fallible. I have taken greater pains to ensure that important numbers, such as exponents, are correct. 2. Compactification and spin unification Let us start by considering 517(2) Yang-Mills (YM) theory in five-dimensional (5D) Minkowski spacetime, a in particular all dimensions being infinite in size,
S = Tvfd4xfdx5
L±FMNFMN\
= Trjd4xjdx5 I-IF^F^
- \F^FA
,
(2.1)
where M,N = 0,1,2,3,5 are 5D indices, while n, v = 0,1,2,3 are 4D indices. We use matrix notation for SU{2) so that the gauge field is AM = A%]Ta, where ra are the isospin Pauli matrices. We will study this theory in an axial gauge, A$ = 0. To see that this is a legitimate gauge, imagine that AM is in a general gauge and consider a gauge transformation, A'M =
-SI^DMQ,
fi(x",x5)
€ SU(2),
(2.2)
where g is the gauge coupling. It is always possible to find n(x,xs), such that A'5 = 0. Ex. Check that this happens for fi(:r, X5) = Ve%9^° dx5A5(x,x'5)^ w j l e r e V represents the path-ordering of the exponential. a
O u r metric signature convention throughout these lectures is (+ — . . . —).
588
R.
Sundrum
A a-M
x5
Figure 1.
=R(j)
A 5D hypercylindrical spacetime.
Ex. Check that in this gauge,
S = Tv JdAx jdx5
I-IF^F""
+ \(d5A„)2\ •
(2.3)
Let us now compactify the fifth dimension to a circle, so that X5 = Rep, where R is the radius of the circle and
A^A)
= 4 0 ) ( z ) + J2(A^(x)ein*
+h.c).
(2.4)
71=1
But now we can no longer go to axial gauge; in general our Q. above will not be 27r-periodic. The best we can do is go to an "almost axial" gauge where A§ is) = A5 (x), where the action can be written
To the Fifth Dimension and Back 589
= 2*RTrJ*x{ - \(d,A^ - dvAff + \{d,A^f
(2.5)
showing that the 5D theory is equivalent to a 4D theory with an infinite tower of 4D fields, with masses, m}n = n2/R2. This rewriting of 5D compactified physics is called the Kaluza-Klein (KK) decomposition. Ex. Show that if AM is in a general gauge it can be brought to almost axial gauge via the (periodic) gauge transformation n(x,(j>)
=J>ei9 tf dtf RAs(x,
^
(2.6)
Note that the sum over n of the fields in any interaction term must be zero since this is just conservation of fifth dimensional momentum, where for convenience we define the complex conjugate modes, A}? , to be the modes corresponding to —n. In this way a spacetime symmetry and conservation law appears as an internal symmetry in the 4D KK decomposition, with internal charges, n. Since all of the n ^ O modes have 4D masses, we can write a 4D effective theory valid below \/R involving just the light A^ modes. Tree level matching yields SeVE~JnRTrJd*x{-\FWF^ + \(n,A^y
(2.7)
The leading (renormalizable) non-linear interactions follow entirely from the 4D gauge invariance which survives almost axial gauge fixing. We have a theory of a 4D gauge field and gauge-charged 4D scalar, unified in their higher-dimensional origins. This unification is hidden along with the extra dimension at low energies, but for E » l/R the tell-tale "Kaluza-Klein" (KK) excitations A^ are accessible, and the full story can be reconstructed in principle. Ex. Check that almost axial gauge is preserved by 4D gauge transformations, $l(x) (independent of >). Our results are summarized in Fig. 2.
590
R.
Sundrum
miD
A
complex field charged under 5th momentum
n = 2 4D effective theory
n= 1
T
n= 0 J =0 Figure 2.
J =1
4D KK spectrum of 5D gauge field.
3. 5D fermions and the chirality problem To proceed we need a representation of the 5D Clifford algebra, {TM, TN} 2I]MN- This is straightforwardly provided by T M = 7M>
r
5 =
(3.1)
~in/5,
where the 7's are the familiar 4D Dirac matrices. Therefore, 5D fermions are necessarily 4-component spinors. We decompose them as
vM*,0)= J2 *£° (*)*'"*•
(3.2)
Plugging this into the 5D Dirac action gives S* = [d4x[dx5y(iDMTM f^xfdx^iiD^
- m)¥ - TO)* - * 7 5 a 5 * + igVAs-ysV
(3.3)
oo
/
d4x Y, y{n)(h>*d)i-m-i^5)¥n)
+ 0($A<S>).
To the Fifth Dimension
and Back
591
miD
i/R 3/R 2/R l/R m Left Figure 3.
Right
4D KK spectrum of 5D fermion.
We see that we get a tower of 4D Dirac fermions labelled by integer n (no longer positive), with physical masses, n2 ™2phys = ™2 + jp-
(3-4)
For small m, this is illustrated in Fig. 3. These fermions are coupled to the gauge field KK tower, again with all interactions conserving 5D momentum, the sum over n of all 4D fields in an interaction adding up to zero. At low energies, E <§; 1/.R, we can again get a 4D effective action for the light n = 0 modes, S^
=
2nR
fd4x [^°\i^D^
- m)*<°> + i<7*^7540)*(0)} - (3-5)
where the covariant derivative contains only the gauge field A^(0)'. Note that we also have a Yukawa coupling to the 4D scalar, A^', of the same strength as the gauge coupling, so-called gauge-Yukawa unification. The idea that the Higgs particle may originate from extra-dimensional components of gauge fields was first discussed in Refs. 11.
592
R.
Sundrum
A™
jT^x scalar
^ * Figure 4.
Some quantum corrections to 4D scalar self energy.
An unattractive feature in this cartoon of the real world, emerging below 1/.R, is that the necessity of having Dirac 4-component spinor representations of 5D Lorentz invariance has resulted in having 4-component nonchiral 4D fermion zero-modes. The Standard Model is however famously a theory of chiral Weyl 2-component fermions. Even as a cartoon this looks worrying. This general problem in theories of higher dimensions is called the "chirality problem" and we will return to deal with it later.
4. Light scalar quantum corrections Given that light scalars are unnatural in (non-supersymmetric) quantum field theories, it is rather surprising to see a massless 4D scalar, A$ , emerge from higher dimensions. Of course, we should consider quantum corrections to our classical story and see what happens to the scalar mass. From a purely 4D effective field theory viewpoint we would expect there to be large divergent corrections to the scalar mass coming from its gauge and Yukawa couplings, from diagrams such as Fig. 4, ^scalar ~ J ^
A
"V '
^'^
suggesting that the scalar is naturally very heavy. But from the 5D viewpoint A5 is massless because it is part of a 5D gauge field, whose mass is protected by 5D gauge invariance. So the question is which viewpoint is correct? To find out let us first compute the 1-fermion-loop effective potential for A5 12 . For this purpose we treat a = gA^ as completely constant, and A^ = 0. Then,
S* = 2TTR fd4xJ2^n\x)
[i$-m-i(j:-a)
75] ^{n)(x),
(4.2)
To the Fifth Dimension
and Back
593
where (4.3)
= 7MdM. Since a is constant,
5 =27TR
*
(n)(p) m
V' -l G _ a ) 751¥n){p)-
IWJ^ £ *
(4 4)
-
After Wick rotating, this gives ofB = _ X£
•2T
*(n)(p) [* + *m + g - a) 75] *(n)(p).
(4-5)
n
Integrating out the fermions by straightforward Gaussian Grassman integration,
Vef =
ri det V+ im + (lj> ~ a) 7s ] fd4p trln I +im + (i-«) = exp V
75
(4.6)
<W(2TT)4
From now on, I will simplify slightly by considering a U(l) gauge group rather than SU(2). All subtleties will come from finite R, so we focus on d4P
dVeff e OR
d4p (2TT)4
n
tr tr
~ l Rv2 j 7 ,0 5 j+im+(%
n ^
- a ) 75
^ - i m + (-g - a) 75 7 5
p2 +
(%-a)2+m2_
d4p 4n(n — a) 2 (27TJ4 p + (n - a) 2 + TO2 4"i(27T
(4.7)
where we have gone to R = 1 units in the last line. Naively, this integal and sum over n is quintically divergent! So let us carefully regulate the calculation by adding Pauli-Villars fields, in a 5D gauge-invariant manner. These fields have the same quantum numbers as \&, but have cut-off size masses m —> Auv, some with Bose rather than Fermi statistics. Thereby,
dVeS dR
4
?i(27T) U
2
4n(n — a) + (n — a)2 + TO'
+
.regulator\ terms
(4g)
The regulator terms resemble the physical term except for having cutoff size masses and with signs (determined by the statistics of the regulator
594
R.
Sundrum
b. iy/p +A*jV
•
i\Jp2-\-m2
•
2
-3
-2
-1
0
2
2
-is/p2+K\JV
b
'l
b
•
— iyjp +m
Figure 5.
C
•
-
Contour for turning KK sum into integral.
field) chosen in such a way that the entire expression converges. The big trick for doing the sum on n is to replace it by a contour integral, dVen OR
dZ
Hki \
4z(z - a)
1 e2niz
_l
yp2
+
(z _ a)2
m2
+
+ Reg.
(4.9)
where the contour is shown in Fig. 5, following from the simple poles of the factor l/(e2niz — 1) and from the residue theorem. The semi-circles at infinity needed to have a closed contour are irrelevant because the integrand vanishes rapidly enough there, precisely because of the addition of the regulator terms. We can deform the contour to that shown in Fig. 6 without encountering any singularities of the integrand, so that by the residue theorem, dVeeff OR
a + i^Jp2 +m" J (2TT)
4
D 27ria e -27ri/p
2
+m 2 _ -^
iVl?
m'
+ -,2-Kia 2nyJp +m 2
e
2
_ -^
+ Reg.
(4.10)
To the Fifth Dimension
/p*+A^
iy/p2-\-m2
-h;
h
(0 0 0 0
-i\/p2+r,
Figure 6.
595
-Ci
1
-fc
'a 'l
-1
and Back
2
3
-C2
Deformed contour for KK sum.
We can also write this as SVeff
OR
= 47
7w
V^+ m* — la
o2viae-2iry/p2+m2
_ -y
y/p2 + m2 + ia 2
+m2
e2viae2iiy/\p
-i
2
02Triae-2-Ky/p
2
+ WP
2
+m
— ia)
+m2 2
_ 2
o2-Kiae-2-ny/p +m
-y _ y
"(vP 2 +m2 — ia) + Reg.
(4.11)
where we have just added and subtracted the same quantity in the last two terms (not counting the regulator terms). Note that the overbraced terms cancel out, leaving
aVeff dR
=
f d*P J (2irY
\Jp2 + m2 — ia -2iriaRe2TrR^/p2+m2
( VP2 +i~n2 — ia)
_ y
+ C.C.
+ Reg.
where we have put back R explicitly, by dimensional analysis.
(4.12)
596
R.
Sundrum
Now let us integrate with respect to R,
VeS = f - ^
{ - 4Reln ( l -
-4TTR f \ / p 2 +m2
e-
2
^\/p2+™V"afl)
- iaj
\ +
Reg.
+ irrelevant const.
(4.13)
In the R —» oo limit, V^F must be independent of a since certainly all potential terms for gauge fields vanish by gauge invariance as usual. This yields the identity, Veff —> - 4?r# / - ^ J (\/p 2 + m 2 - ia) + Reg. = \R , R-*OO
J
(4.14)
\^r
where A is a constant independent of a and R. Ex. Directly show the cancellation of a-dependence in the right hand side of eq. (4.14) by carefully writing out the regulator terms. Using this identity in eq. (4.13) yields
VeS = AR-4
f-~^
Rein ( l - e - 2 ^ v V + m 2 e 2 ™ f l ) + Reg. (4.15)
This formula has some remarkable properties. The first term is indeed highly cutoff dependent, but it does not depend on a. The integrand of the second term behaves as e ~ 2 7 r R p for large p and therefore the p integrals converge. The regulator terms are suppressed by e - 2 7 r f l A uv f ac tors and can be completely neglected for Auv-R 3> 1 (or more formally, for Auv —* oo). We therefore drop the a-dependent regulator terms from now on. Finally, combining complex exponentials we arrive at our final result,
VeS = AR - 2 J - ^
In (l +
e-^V^
-2e-2nRVp2+m\os
(2nRgA{°})
J,
(4.16)
which is illustrated in Fig. 7 . For small A5 , this can be approximated,
To the Fifth Dimension
and Back
RgAf
True vacuum Figure 7. Quantum effective potential for A5 but it is periodic.)
VeS ~ AR +
fd4p [ 4
i(27T) \
1 _
41n
597
. (The potential is not exactly sinusoidal,
2
e-2irRy/p
+m2
2
c,-2nR^/p
-
+m2
(2wRgA
+ (2*R9A y
+m2Y
-2TvRy/p2+m2
o)
6 (l -
(4.17)
2
e-2nRy/p
2
e-2irR^/p
+m2\
-4TrRy/p2+m2
^ _ _
2 +
f± _
2
£~2nRy/p^+ri
"}
We see immediately that the vacuum has non-vanishing (A(5 '), ,(0)
~Rg
for
(4.18)
m
Let us now return from considering U(l) gauge group back to SU(2). Nothing much changes as far as the \I> loop contribution we have just considered (vll(0) is just to be replaced by \A(5'\ = \/tiAl, where the trace is 5 over gauged isospin) but now there are also diagrams involving gauge loops which contribute to the A 5 °' effective potential. See Fig. 8. By similar methodology, these give a contribution illustrated in Fig. 9. We see that there is a competition now between the contribution from gauge loops which prefers a vacuum at A\' = 0 versus the fermion loops
598
R.
Sundrum
i(n)
Figure 8.
Examples of gauge loop contribution to As potential.
veS A
RgAf
Figure 9.
Result of gauge loop contribution to As potential.
which prefer a vacuum at \A§'\ ^ 0. But clearly if we include sufficiently many identical species of \&, their contribution must dominate, and |(0), |(0) 1-^5 I ~ V(-^s)- Since A§' is an isovector, a non-zero expectation necessarily breaks the gauge group SU(2) down to U(l). One can think of this as a caricature of electroweak symmetry breaking where the preserved U(l) is electromagnetism and A§ ' is the Higgs field! We refer to it as "radiative symmetry breaking" (also the "Hosotani mechanism" 12 ) because it is a loop effect that sculpted out the symmetry breaking potential. In this symmetry breaking vacuum or Higgs phase, we can easily estimate the physical mass spectrum,
To the Fifth Dimension and Back 599 m7(o) = 0
"V±<°> ~ ^
_1_
J_
TTl^(o)
-R2 m - • 0 -R
mKK ~ -„ " W "
~ 3 ^ 3
•
(4-19)
Now this is certainly an interesting story theoretically, but it is surely dangerous to imagine anything like this happening in the real world because we are predicting TTIKK ~ fnwi a n d such light KK states should already have been seen. However, there is a simple way to make the KK scale significantly larger than mw, by making
^« i
4)><-5-Note that for small A\
(4-20)
we have
VeS = V*s-loop + Vg
uge loop
-
~
AR
small a
+ [ci - c2(m)N] (Ra)2 + [c3 + c4{m)N] (Raf
,
(4.21)
where the c's are order one and positive, and C2,C4 depend on the 5D fermion mass m, and N is the number of species of fermions. Now let us tune m to achieve - c i + c2(m)N = e
(4.22)
from which it follows that there is a local minimum of the effective potential (a possibly cosmologically stable, false vacuum) with AT5 ~ ^ • gR
(4.23)
This yields the hierarchy, mw±
~ — ~ \/emKK K
•
(4.24)
600
R.
Sundrum
S+-* 4f=TV
Identify +0 with —
0= 0 x 5 =0 Figure 10.
XS=-KR
Orbifolding the circle to an interval.
5. Orbifolds and chirality If we ask whether our results thusfar could be extended to a realistic model of nature, with the standard model as a low energy limit, we encounter some big problems, not just problems of detail: a) The previously mentioned chirality problem. b) Yukawa couplings of the standard model vary greatly. Our low energy fermion modes seem to have Yukawa couplings equal to their gauge coupling, a reasonable cartoon of the top quark but not of other real world fermions. A very simple way of solving (a) is to replace the fifth dimensional circle by an interval. The two spaces can be technically related by realizing the interval as an "orbifold" of the circle. This is illustrated in Fig. 10, where the points on the two hemispheres of the circle are identified. Mathematically, we identify the points at <j> or X5 with —or — X5. In this way the physical interval extends a length irR, half the circumference of our original circle. This identification is possible if we also assign a "parity" transformation to all the fields, which is respected by the dynamics (i.e. the action). The action we have considered above has such a parity, given by +A»
P(x6)
P(4°>) = _4°>
P(#L) = + * L P(*R) = _ $ R ,
(5.1)
precisely when the 5D fermion mass vanishes, m = 0. We consider this case for now. Ex. Check that the action is invariant under this parity transformation. With such a parity transformation we continue to pretend to live on a circle, but with all fields satisfying $(x*,-xs)
=
P($){x",xB).
(5.2)
To the Fifth Dimension
and Back
601
That is, the degrees of freedom for x5 < 0 are merely a reflection of degrees of freedom for xs > 0, they have no independent existence. Of course we also require circular periodicity, $ ( ^ , 0 + 27r) = $(^,>).
(5.3)
These conditions specify "orbifold boundary conditions" on the interval, derived from the the circle, which of course has no boundary. We can write out the mode decompositions (in almost axial gauge) for all the fields subject to orbifold boundary conditions, oo
AM(x,4>) = ^4")(x)cos(n>) n=0
A 5 (x,>)=0
Lost "Higgs"!
oo
n=0 oo
*R(i,
) = Y, * R } (x) sin(n^)
Lost ^
!
(5.4)
n=l
One unfortunate consequence we see is that A$ has no modes, in particular orbifolding has eliminated our candidate Higgs! The good consequence is for the chirality problem, in that the massless right-handed fermion is eliminated, only the massless left handed fermion mode is left. The low energy effective theory below 1/R is just
S«E=X2*RJ*X{-\
(Fjg))2+ 9?UDrf*™y
(5.5)
With SU(2) gauge group, if \& is an isodoublet (so that ^ is an isodoublet), the only possible gauge invariant mass term for the light mode,
*LJ,*4e«e a/9 ,
(5-6)
vanishes by fermi statistics. Therefore we apparently have a chiral effective gauge theory below 1/R. Unfortunately this theory is afflicted by a subtle non-perturbative "Witten anomaly", so the theory is really unphysical. However, if we consider * to be in the isospin 3/2 representation, we again get a chiral gauge theory, but now not anomalous in any way. Having seen that the chirality problem is soluble, we need to recover our Higgs field. (For discussion of related mechanisms and further references
602
R. Sundrum
see the TASI review of Ref. 13.) To do this we must enlarge our starting gauge group, from SU(2) 2* 50(3)
(5.7)
to 50(4). Gauge fields are conveniently thought of as anti-symmetric matrices, A1^, in the fundmental gauge indices i,j = 1,2,3,4. For simplicity we choose fermions in the fundamental representation, \P l . The action, 5 = tvjd4xjdx5{
- \F^F^
+ \(d5A^)2
+ !(ZV40))a
+ * t D M 7 " * - ^ 5 d 5 - * + igViAZW-YsVj},
(5.8)
is invariant under the orbifold parity given by P{A$) = +A* P(AJ?) = -A*
P(*i!) = +*£
P(Af?) = -Aj? P(At)
P(*£) = -H
= +At
P(*A) = -Erf* P(H)
=
+*A,
where i,j = 1, 2,3. Ex. Check by mode decomposition that this leaves 4D massless fields, Ati0),
A&°\
*fr°\
*^(0),
(5.10)
that is, a 4D 50(3) gauge field, a 4D Higgs triplet of 50(3), a left-handed fermion triplet of 50(3), and a right-handed singlet of 50(3). This illustrates how (orbifold) boundary conditions on extra dimensions can break the gauge group of the bulk of the extra dimensions. The lowenergy effective theory is given by 5 eff
= i 2nR jtfxl
£<Jj
J
I.
- ^)P"(°> + i(^44(0))2 4
+ ig ( n ° U ^ ^
2
+ ^ A l
m
^ )
}• (5-11)
This contains 4D 50(3) gauge theory with two different representations of Weyl fermions Yukawa-coupled to a Higgs field. This again bears some resemblence to the standard model if we think of the fermion as the left and right handed "top" quark. But what of the second problem we identified, (b), that the standard model contains some fermions with much smaller Yukawa couplings than gauge coupling? Such fermions can arise by realizing them very differently in the higher-dimensional set-up. The simplest example is illustrated in Fig. 11, where beyond the fields we have thusfar
To the Fifth Dimension
and Back
603
AM,$
I
= TT
> = 0 x5 = 0 Figure 11.
X5 = TTR
Orbifolded higher dimensional spacetime (with boundaries).
considered, which live in the "bulk" of the 5D spacetime, there is a 4D Weyl fermion precisely confined to one of the 4D boundaries of the 5D spacetime, say cf> = 7r. It can couple to the gauge field evaluated at the boundary if it carries some non-trivial representation, say triplet. This represents a second way in which the chirality problem can be solved, localization to a physical 4D subspace or "3-brane" (a "p"-brane has p spatial dimensions plus time), in this case the boundary of our 5D spacetime. The new fermion has action,
- / *
x Xl(x)
i d ^
•gA*{x,= ir) XJAx) •
(5.12)
At low energies, E
604
R.
Sundrum
6. Matching 5D to 4D couplings Let us study how effective 4D couplings at low energies emerge from the starting 5D couplings. Returning to pure SU(2) Yang-Mills on an extradimensional circle, we get a low-energy 4D theory,
S4eff ^
0
2*RJ*x {-\F$F^
+ I(JDM4 >)2
j.
(6.1)
The fields are clearly not canonically normalized, even though the 5D theory we started with was canonically normalized. We can wavefunction renormalize the 4D effective fields to canonical form,
tp = 4 0 ) V^R,
A„ = A™ V2^R ,
(6.2)
and see what has happened to the couplings,
54eff = 2nRJdix{ - \ (dpAlW - dvA^
- ig5eabc Ab^
A^f
+i(^4 0) -%4 0) 4 0) ) 2 } = Jd*x{ - \ (d»K - duA° +
i-^e^AlA^j
K^-v§i^) 2 }-
(6 3)
-
Prom this we read off the effective 4D gauge coupling, 54eff
= -§=
-
(6.4)
Ex. Check that this is dimensionally correct, that 4D gauge couplings are dimensionless while 5D gauge couplings having units of l / ^ m a s s . For experimentally measured gauge couplings, roughly order one, we require g5 ~ 0(V2nR).
(6.5)
7. 5D non-renormalizability Now, having couplings with negative mass dimension is the classic sign of non-renormalizability, and as you can easily check it happens rather readily in higher dimensional quantum field theory. There are various beliefs about non-renormalizable theories:
To the Fifth Dimension
and Back
605
a) A non-renormalizable quantum field theory is an unmitigated disaster. Throw the theory away at once. Only a few people still hold to this incorrect viewpoint. b) A non-renormalizable quantum field theory can only be used classically, for example in General Relativity where GNewton has negative mass dimension. All quantum corrections give nonsense. This incorrect view is held by a surprisingly large number of people. c) The truth (what I believe): Non-renormalizable theories with couplings, K, with negative mass dimension can make sense as effective field theories, working pertubatively in powers of the dimensionless small parameter, /c(Energy) n , where —n is the mass dimension of K. TO any fixed order in this expansion, one in fact has all the advantages of renormalizable quantum field theory. There are even meaningful finite quantum computations one can perform. In fact we have just done one in computing the A5 quantum effective potential. But of course there is a price: the whole procedure breaks down once the formal small parameter is no longer small, E ~ 1/K 1 /™. At higher energies the effective field theory is useless and must be replaced by a more fundamental and better behaved description of the dynamics. Ex. Learn (non-renormalizable) effective field theory at the systematic technical level as well as a way of thinking. A good place to start is the chiral Lagrangian discussion of soft pions in Ref. 15. In more detail, perturbative expansions in effective field theory will have expansion parameters, K(Energy)", divided by extra numerical factors such as 2's or 7r's. These factors are parametrically order one, but enough of them can be quantitatively significant. These factors can be estimated from considerations of phase space. I will just put these factors in correctly without explanation. Ex. Learn the art of naive dimensional analysis, including how to estimate the 2's and 7r's (for some discussion in the extra-dimensional context see Ref. 16). Use this in your work on extra dimensions. Our findings so far are summarized in Fig. 12. The non-renormalizable effective field theory of 5D gauge theory breaks down when the formal small parameter, Eg%, gets large, that is we can define a maximum cutoff on its validity, Ayv, max ~ 167r2/c?§. The 5D effective theory cannot hold above this scale and must be replaced by a more fundamental theory. Let us say this happens at Ayv < Auv, max- From here down to l/R we have 5D effective field theory, and below l/R we have 4D effective field theory.
606
R.
Sundrum
E EFT cannot make sense . A u V , max ~
16TT2 8TT ,— ~ "FT-n
AUV 5D EFT
— ~ mKK ~ mw ~ my
4DEFT ?Tl"Hiees" ~ —. Sg
Figure 12.
95
=~
V32^W
54 —
ATTR
Natural scales of the 5D gauge theory.
We found it interesting that a Higgs-like candidate emerged from 5D gauge fields because it suggested a way of keeping the 4D scalar naturally light, namely by identifying it as part of a higher-dimensional vector field. But given that Auv < Auv, max r^J ^($TT/(-^74))J and 4D gauge couplings are measured to be not much smaller than one, we must ask how well this extra-dimensional picture is doing at addressing the naturalness problem of the Higgs. In Fig. 13 we make the comparison with purely 4D field theory with a UV cutoff imposed. We see that in the purely 4D scenario one naturally predicts a weak scale v ~ 0(Auv/47r), while with the 5D regime included, v ~ 0(g4:Auy/8n), which is parametrically better. Numerically, it does not look like much improvement, and indeed one can do better, but we do not pursue that here.
To the Fifth Dimension
2 fll + AH:i g g s ~
mB.
,
Figure 13.
/
87T
87TU
A u V max ~ - ^ - 2 ~ -R94
Auv / g\ \l
mw ~ 54« ~ —
uv
+
~ —;
V^Higgs
l
\
.Jf^k. V ~
A2
167T 2
H
^7T V ^ ' g g
. , ^ Auv s
607
5D
4D Standard Model
SQMmH
and Back
h1> — 47T
v
94
94 * ^ — - A u v max 87T
Natural relationship between W mass and cutoff.
8. Gravity-gauge unification and the radion The non-renormalizable effective field theory of quantum General Relativity, whose small expansion parameter is E2G Newton, is expected to be "UV-completed" by string theory, just as the Fermi theory of weak interactions is completed by the renormalizable electroweak theory. One price of calculable string theories is that they predict the. existence of extra dimensions! You do not have to ask for extra dimensions, they are forced on you as the price of maintaining the stringy gauge symmetries needed to tame the higher spins in the theory. In fact the extra dimensions are required in order to cancel quantum anomalies in the stringy gauge symmetries 5 . However, string theory has been notoriously unpredictive about the size and shape of the extra dimensions, so we must gedanken experiment, or really experiment, with the possibilities. Since string theory contains a regime of (higher-dimensional) General Relativity, the size and shape of the extra dimensions is dynamical as is all of spacetime. This raises several interesting issues, which we study below. Let us begin by returning to 5D spacetime with circular extra dimension. The 5D gravitational field is given by the metric function on this spacetime, in terms of which we can write infinitesimal distances in arbitrary coordinates,
608
R.
Sundrum
ds2 =
GMN(X)dXMdXN
= GMN(X(X'))
dX'M
j ^ r
= GM>N>{X')dX'M'dX'N'
dX'N
j ^ r
.
(8.1)
From this, we deduce how the metric field transforms under 5D general coordinate transformations (the gauge symmetry of 5D General Relativity) in order to maintain the invariance of physical distances, BXM GM'N'(X')
r)XN
= G W ( X ( X ' ) ) ^ M 7 ^ ^
•
(8.2)
Expanding around flat (cylindrical) spacetime, GMN(X)
= VMN + hMN(X)
,
(8.3)
and working to leading order in HMN and small coordinate transformations, X,M
=
XM
_
£
M
( X )
f
( g 4 )
we see that the general coordinate transformation of the metric reduces to ^MTVPO
= hMN(X) + dM£N +
0N£M
•
(8.5)
Again, we can go to an almost axial gauge of the form,
V(s^)
= h<$(x) + £
(hftWe1"* + c.c.) ,
(8.6)
n=l
where we use coordinates X = {x»,4>)-
(8.7)
Ex. Show how this is to be done using the linearized transformation of the metric. Since the 5D Einstein action, D
5D Einstein — / « -<\
/ '
g£)
,
written in terms of the 5D Ricci scalar curvature TZ contains only terms with derivatives, the h\] must be massless 4D fields, without any potential i ( 0 N) (x) (
To the Fifth Dimension
and Back
609
m4D
5DGR 4/iJ 3/R 2/R l/R 4DGR 0 J =0 Figure 14.
J=\
J = 2
KK 4D spectrum of 5D metric decomposition.
at all. The h^N have n/R masses as usual. These facts are illustrated in Fig. 14. Of course once one goes beyond the linearized approximation, all these fields are interacting. The interacting massless vector field, h(J(x), must therefore have a protective gauge symmetry. Ex. Check that in almost axial gauge there is a residual unfixed symmetry,
h'll5(X) = h^(x) + dfle5(x).
(8.9)
Because of this gauge invariance, inherited from 5D general coordinate invariance, the quadratic terms involving h( s:(x) in the low energy 4D effective action must be just the Maxwell action. This is part of the original Kaluza-Klein idea, 4D gauge fields and 4D General Relativity (protected by the residual 4D general coordinate invariance) both emerge from a unifed 5D action upon compactification. The 4D effective action is tightly con-
610
R.
Sundrum
strained by the protective symmetries to be of Maxwell-Einstein type, as can be checked by plugging in the massless modes as usual. It is easiest to just check this at quadratic order in the action and then use the residual symmetries. Since the vector field gauge symmetry corresponds to shifts in the extra dimensional circle (and because there is precisely one emergent vector field), the emergent gauge group is U{1). Charged matter must correspond to states which transform under such extra-dimensional translations. These are precisely the KK excitations which carry non-zero extra-dimensional momentum. It is intriguing to see how "internal" gauge groups can emerge from the symmetries of the extra-dimensional geometry. One might wonder whether more complex and realistic gauge structure, as well as chiral 4D massless charged matter, can emerge from extra-dimensional geometry. While higher dimensional set-ups go in this direction, it is in general a very difficult game. However, in one way of analysing heterotic string theory realistic gauge structure does emerge from several KK U(l) gauge fields with special stringy enhancements to non-abelian form 6 . In this sense the old KaluzaKlein program of unification of gauge theory (just electromagnetism in the early days) and gravity is alive and well. There is one more 4D massless state which does not follow from a residual symmetry, and to that extent is unexpected at first sight, namely h55(x). To repeat, it has no potential because the 5D action has only derivative terms, and this mode only has ^-derivatives, being independent of the extra dimension by gauge-fixing. This 4D scalar can therefore have any VEV, (h$)=t.
(8.10)
To see what this VEV means physically, note that for the simple VEVs,
the VEV of the 5D geometry is given by (ds2) = r)iivdxp,dxv - (1 + £)dx5dx5 = •q^dxTdx" - (1 + £)R2d(f>2 ,
(8.12)
which is just the geometry of an undeformed (flat) cylinder with physical radius y/\ + £ R. Thus, fluctuations in the scalar field correspond to a dynamical radius for the extra dimension, as illustrated in Fig. 15. The quantum of this scalar is therefore referred to as the "radion".
To the Fifth Dimension
Figure 15.
and Back
611
Fluctuations of dynamical radius (radion).
We are used to scalar fields without potentials in the form of Goldstone bosons. But in these cases, any VEV in field space is equivalent to any other by the associated symmetry. The radion field space is an example of a "moduli space" of physically inequivalent vacua (differing radii are physically quite different obviously). The radion is an example of a "modulus". The string theories that have been studied start life with very large moduli spaces of inequivalent vacua. Thus even a unique theory can lose predictivity in the maze of physically inequivalent vacua Nature has to choose from. The good news is that we see a cartoon of the 4D ingredients needed for the real world emerging from our very simple unified 5D example: 4D scalars, 4D gauge theory and 4D gravity. If one throws in supersymmetry, another key ingredient of superstring theory, one must necessarily have fermions as well. The size and shape of the extra dimensions, and the low energy 4D world they produce, is determined by the moduli VEVs. Small corrections to the vanishing potential of moduli space can actually favor one or a discrete set of vacua, yielding greater predictivity (in string theory right now "greater" does not mean great). Let us continue with our 5D cylinder to see a simple example of such small corrections, in this case from quantum effects related to the famous Casimir effect.
612
R.
Sundrum
9. M o d u l u s stabilization Let us add a 5D fermion to the (gravitating) cylinder. Our old effective potential calculation from fermion loops, now dropping A5 since there are no 5D gauge fields, is given by VeB ~ KR - 4 / ^ J L l n ( i _ e - 2 ^ V ^ W ) .
(9.1)
In the 5D general relativistic context we should consider the 1-fermionloop induced effective potential for the dynamical radius VEV, R. Recall, A contains all UV divergences in this calculation. The renormalized value of A is frequently tuned to zero (more on this later) so that crudely, Vreea ~ e-2nRm / d4p ~ m*e-2irRm . (9.2) m»i Jo Now, let us assume that there is also a 5D scalar boson as well and add its quantum-loop contribution to the R effective potential. In complete analogy to our fermion loop computation, except for a sign due to the different statistics, Ves ~ m V 2 * * " 1 - / x V 2 * * " .
(9.3)
Upon minimizing the effective potential with respect to the dynamical radius, fl-5/^ (9.4) v ; \nj 27r(m-/x) That is the corrections have selected a particular radius from the original moduli space! The general conclusion is that in (non-supersymmetric) extra dimensions small corrections can generate an effective potential for moduli which stabilizes the size and shape of the extra dimensions. This radius can readily be moderately larger than the fundamental length scale of the cutoff of non-renormalizable higher-dimensional effective field theory. In the present example, this is achieved by taking the matter masses,TO,/X, to be moderately lighter than the mass scale of the cutoff. If the present example is orbifolded so that the extra dimension is an interval, one can also add a constant "potential" to be localized to one of the boundaries. Since such a localized term is insensitive to R, it will just contribute an .R-independent constant to the effective potential. Such a constant seems irrelevant from the point of view of the radion, but it e2nR(m-»)^(™\\
To the Fifth Dimension
and Back
613
is important when the effective dynamics of 4D General Relativity is also taken into account, as we discuss in the next section. 10. The cosmological constant (problem) The dominant interactions in 4D General Relativity at long distances include not just the Einstein action but also a cosmological constant term,
In flat space quantum matter generates zero-point energies or vacuum energy, corrected by interactions in all possible ways, but we usually throw this away as a physically irrelevant field-independent constant in the effective action. But when coupled to gravity, all vacuum energy from matter and radiation contributes to the cosmological term. If we trust the standard model up to TeV energies, one naturally expects contributions to A of order at least TeV 4 . Crudely, in the presence of such a term Einstein's Equations read K~GN\,
(10.2)
where the left-hand side is a measure of the curvature of spacetime. In particular, flat Minkowski spacetime is not a solution to Einstein's equations for A ^ 0. If A ~ TeV 4 , the radius of curvature of spacetime would be 1 —=
VTl
MPI ~ —=• ~
V\
MPI 7T ~ 1 m m ! 2
TeV
(10.3)
'
This would be tremendously at odds with even everyday experience, where space appears to satisfy Euclid's postulates to excellent approximation. We therefore have to assume all the varied contributions to the cosmological term cancel to very high precision for some mysterious reason. We will assume this from now on to make contact with the observed universe. This unnatural feature is called the Cosmological Constant Problem 17 . In the modulus stabilization example of the last section, the value of Vefi at its minimum represents (in general, a contribution to) the infrared 4D cosmological constant. As pointed out at the end of that section, in the orbifolded version one can also add an R-independent constant to Veff, originating from a boundary-localized constant potential. By fine-tuning the value of this constant we can ensure that the minimal Veff vanishes, that is, the 4D effective cosmological constant is zero (or very small). The ugliness of having to do this tuning reflects the unresolved cosmological constant problem.
614
R.
Sundrum
Just like 4D General Relativity, 5D General Relativity can also have a cosmological term,
S = Jd5xVG^-Ay
(10.4)
Now what we have learned in gauge theory is generally true: 4D effective couplings derive from 5D couplings, but they are not the same thing. In particular, our tuning of the 4D cosmological term, A, to be very nearly zero does not indicate that A is near zero (in fact we will see this explicitly below). The most natural thing is to therefore assume that A ^ 0 and see what happens. In our 5D cylinder example, let us treat A as a small perturbation, where at zeroth order we have a cylindrical solution to Einstein's equations, (ds 2 ) = nlivdx'idxv
- R2phys dtf .
(10.5)
Therefore the cosmological term in the 5D action becomes S B fd4xf
d4>R(-k) = fd4x(-2nRA)
,
(10.6)
which looks precisely like one of our old contributions to the radion (R) effective potential. In fact we saw that there were divergent renormalizations of this A from 5D matter loops, again suggesting that the renormalized A should naturally be significant. Once we decide that A is not very small, we really should consider its effects in determining the 5D geometry. It cannot be treated perturbatively, we must resolve Einstein's equations in the presence of the 5D cosmological constant. The same is true for any boundary-localized potential terms coupled to 5D gravity. 11. Warped compactiflcation We will study the simplest model where we include the higher dimensional cosmological term, known as the Randall-Sundrum I (RSI) model 18>19. It is also a good prototype of more complex constructions. Rather than continue with the cylindrical spacetime we return to the orbifolded variant. The 5D gravitational action of the "bulk" of the spacetime is still given by
Shu^ = fdixJ ^ v ^ j ^ - A J .
(11.1)
To the Fifth Dimension
and Back
615
We can assign the orbifold parities as +1 for G^ and G55 and —1 for GMs, which is respected by the action. In almost axial gauge then, G^5 (X) = 0 by the parity. Therefore in the orbifolded set-up we automatically have no KK massless gauge boson emerging in the 4D effective theory. Of course there can be even more dimensions in non-minimal set-ups from which such gauge bosons might emerge. Here, when we need gauge bosons we will just add them at the 5D level. The radion does remain after orbifolding since G55 is parity even. The orbifolded set-up also has "branes" or boundaries of 5D spacetime, namely the 4D subspaces at <j> = 0, n. In general there can be 4D actions localized to these branes for the 5D fields. In fact such actions must be there since they are renormalized by 5D loops 20 . The leading terms in these brane actions are "tensions", which look like localized cosmological terms, SbraneM = ~ j ^ X y f ^ T ^
,
(11.2)
where 9${x)
= Glu,(x,= 0)
9$(x)=GfU,(x,
= n),
(11.3)
and the T ^ are constant tensions. The induced 4D metrics define distances along the branes, for example, dsfx) = G^ix,
4> = 0)dx^dxl/ ,
(11.4)
since dip = 0 along the brane. Since we are looking for solutions to Einstein's equations that might fit the vacuum of the real world, let us try the ansatz that the 5D metric should respect at least 4D Poincare invariance, ds2 = e-2aWrilu,dxitdxv
- R2d
(11.5)
Here, ?7M„ is the 4D Minkowski metric, and we have chosen the extradimensional coordinate to be proportional to proper distance. The prefactor to 77^1/ is written as an exponential as a convenient convention and is called the "warp factor". Its potential (^-dependence means that the higher-dimensional geometry cannot be defined as a product geometry of 4D Minkowski space and some purely extra-dimensional geometry, but rather all the dimensions are entangled. Plugging this ansatz into the equations
616
R.
Sundrum
Figure 16.
Warp factor solution satisfying orbifold boundary condition.
of motion following from our bulk plus brane actions, one finds ,2 6a"
A 4M53
6k'
T(l)
T(2)
(11.6) where we define a 5D Planck scale,
Mi
(11.7)
2G%D
The only consistent solution to these equations, satisfying periodicity in <j> and the orbifold parity is illustrated in Fig. 16. But even this solution only exists if the kinks have the right size to reproduce the ^-functions in the equations of motion. This requires the relationships between brane tensions and bulk cosmological constant given by r(i)
=
_r(2)
=
24/cM53 .
(11.8)
Thus the vacuum metric solution is given by ds2
=
e-^^ri^dafdx" -2ky
ri^dx^dx"
-i?2#2, -dy2,
0
<6
0 < y < irR .
(11.9)
y = R
This geometry is a slice of AdSs, 5D anti-de Sitter spacetime, which is the maximally symmetric 5D spacetime of negative curvature (just as a sphere is the maximally symmetric space (not spacetime) of positive curvature).
To the Fifth Dimension
and Back
617
The forms of the massless 4D modes, the 4D graviton and radion, are easy to guess: ds{0) = e-2kR(x)'t'gi°J(x)dx^dx'y
~ R(x)2d
0 < <j> < n .
(11.10)
That is we promote the radius, which thusfar has been an integration constant of our vacuum solution, into an x-dependent field, and rj^ into a dynamical 4-metric, g\lJ(x). There is a slick proof that these fluctuations indeed do correspond to the 4D massless modes, that indeed they have no potential at all classically. Plugging the above ansatz into our full action must give a vanishing effective 4D cosmological term since we found a 4D Poincare invariant solution. Now a 4D potential for g^J and R can be computed for ^-independent fields. A constant linear coordinate transformation can take any non-singular symmetric matrix, gjiJ, to the form 77M„. Therefore by residual 4D general coordinate invariance of the action, plugging ^-independent gliJ,R must also give zero. Since the effective lagrangian is the negative of the effective potential for constant fields, the effective potential for gp,J,R clearly vanishes. So we have identified the zero-modes and the radion is a modulus again. Radius stabilization can be readily achieved much as in the unwarped case we studied earlier for the cylindrical spacetime, by adding suitable 5D matter. The simplest such implementation is the classical GoldbergerWise mechanism 2 1 using a 5D scalar, naturally allowing a moderately large "radius", k{R)~O{10),
(11.11)
where k (A) is taken not too much smaller than the 5D effective field theory cutoff. We could also proceed with stabilization by quantum corrections from massive matter as we studied earlier in the unwarped case. We will not pursue these in detail here, just assume some such stabilization, the warping has mild impact on the stabilization. Ex. Work through Goldberger-Wise stabilization 2 1 . After radion stabilization, and neglecting the small backreaction of the stabilizing physics on the vacuum metric solution, we have only the 4D metric zero mode, ds^
= e-2k(-R^g
- (R)2d<j>2, 0 < <j> < ir .
(11.12)
The 4D low-energy effective action for the 4D metric zero mode is obtained by plugging the above ansatz into the fundamental action. The 5D curvature term contains 2-derivative terms, which by 4D Poincare invariance of the vacuum solution must be either two ^-derivative acting on the
618
R. Sundrum
zero mode or two <j> derivatives acting on the warp factor. From the 4D point of view all 4> derivatives just contribute to potential terms, which we saw above all cancelled for the 4D graviton zero-mode. So we focus on pairs of ^-derivatives. Plugging in the zero-mode ansatz, a typical term in the fundamental action has the schematic form, jdixf
G]?S3
dcj)VGGxxGxxGxxdxGxxdxGxx
(11.13)
= fd*xrd4>Re-2k^y^^g^xxg^''g^''dxg^dxg^
,
which allows us to count powers of the warp factor easily. The residual 4D general coordinate invariance then gives a unique form for the 4D effective action of the graviton zero-mode, SAD«S
= -^D fdAx (J*dcf>Re-2kRA =
2kGW
(1
~ e'2kVR)
V^)K(4D)lgM}
/ ^ V ^ ^ W ^ ]
•
(11-14)
That is, at low energies we obtain 4D General Relativity. We easily deduce the 4D effective Newton's constant, ig„/o4£)eff_ 167T °N ~
1
%k GN _ e-2k*R
mien (U-15)
•
Remarkably, even when we take the decompactified limit, R —> oo, this effective coupling stays finite, ^
= ^ ( l - e -
2
^ ) ^ ^ .
(11.16)
This is a result of the fact that the zero-mode is localized in the vicinity of the brane at <j> = 0. This localization mechanism for gravity is sometimes known as the Randall-Sundrum II (RS2) mechanism. Because of gravity localization, the brane at <j> = 0 is usually called the "Planck brane". All these classical derivations are considered to take place as the leading approximation of non-renormalizable quantum effective field theory. The control parameter is in general given by
(^-i7)
5«i in particular,
*£„(*££!«! ^
* «l.
Ml
M5
Ml
(IMS)
To the Fifth Dimension
H
y
Figure 17.
619
H(x)
Warp factor (profile of 4D graviton)
y= 0
and Back
•
= ITR
Localization of 4D graviton and Higgs.
In practice we usually take all the scales within an order of magnitude or two of the 4D Planck scale, k & M5 & M4Pl ~ 10 18 GeV .
(11.19)
12. Warped hierarchy Consider the set-up of Fig. 17, where we have a warped compactification with localized 4D gravity, and now we add a 4D Higgs field by hand (not from As's for simplicity), 6-function localized to the opposite brane (<j) = n), with 4D brane-localized action, Smggs = Jd4xV=m^{9Zd^d„H
- X(\H\2 - v20)2} ,
(12.1)
620
R. Sundrum
where 9™(x) = Gttv(x, = *) = e-2k"R9$(x)
,
(12.2)
gives the induced 4D geometry of the brane, and where the second line gives the low-energy approximation where only the gravity zero-mode can propagate. In this approximation the Higgs action becomes
SH = Jd^x^^)
{e-2k«Rg^d^d„H - e-4fc-«A(|#|2 - v2)2} ,
(12.3) where the warp factor appears like a conventional constant wavefunction renormalization of the Higgs. Canonical normalization of the Higgs is achieved by the field redefinition, e-knRH
_^ H
(12_4)
;
giving a canonical action,
- A (\H\2 - e ^ * v ^ \ .
SH = JcPxy/^wlgfodprfdvH
(12.5)
The miracle is that the "bare" weak scale parameter for the Higgs, VQ, receives a large warp factor renormalization reducing it to the physical weak scale Higgs VEV, v = e-k7rRv0
•
(12.6)
Even if vo is very large, Planckian in size, for kR ~ 0(10) one can easily accomodate the physical weak scale, v ~ 250 GeV. Because of this large warping down of UV scales, this brane is sometimes called the IR brane, or in the present context, the "TeV brane". How does this magic trick work? Consider again the vacuum solution for the 5D metric, ^vacuum = e ^ * " V * ^ "
~ V
•
(12"7)
In any extra-dimensional locale, y ~ yo, we can approximate this metric, dslc
~ e-^v^dx^dx" = j]ilvdxildxv
- dy2 - dy2 ,
(12.8)
which looks just like a piece of 5D Minkowski spacetime provided we define x = e-kyox,
m4D = ekvomAD
.
(12.9)
To the Fifth Dimension
Regulator fields for 5D GR mass Afrav Af
av
and Back
621
Pauli-Villars
Mpi
for Higgs mass A5H - Mpi
~ e - ^ M ]PI
~ e- f e ^ M p i
Planck brane
IR brane
Figure 18. Profiles of 5D regulator fields for gravity and 4D regulator for Higgs.
Without warping the translation from 5D to 4D masses is straightforward, so for physics localized in the vicinity of yo, we have the translation, fhiD ~ m5D
=>•
m4D = e~kyom5D
.
(12.10)
For the case of the Higgs, yo = *R,
(12-11)
so v = e-knRv0
.
(12.12)
Warped hierarchies are radiatively stable, essentially because even (general coordinate invariant) cutoff scales get warped down near the IR brane, as illustrated in Fig. 18. 13. K K gravitons at colliders Let us begin by removing the IR brane to infinity, R —> oo, and expand the general 5D metric (in almost axial gauge) about the vacuum solution, ds2 = f e " 2 * ' " 1 ^ + e-k^/2h^(x,y)}
dx»dxv - dy2 .
(13.1)
622
R. Sundrum
The parametrization of fluctuations with the extra exponential is purely for later convenience. It is also convenient to switch to an extra-dimensional coordinate, z = sgn(y)
ek\y\
_ i
.
(13.2)
Plugging the general 5D metric into the fundamental action, and expanding to quadratic order in small fluctuations about the vacuum gives, +00
G%DS = jd^xjdz^h^U^^h^
- \h^v^r°HQMhpa\
,
(13.3)
—00
where the first term contains an operator made of two ^-derivatives while the second term contains an operator made from y-derivatives and nonderivative terms, specified by HQM
1 = --d?
+ VQM(z),
1 ^A-2
V
QM(z)^+mzl
3k
+ 1)2-YS(z).
(13.4)
The unspecified index structures are exactly such that, if HQM is only a constant and if the fields are purely 4D fields, and if we get rid of the z integral, this action would just be that of a free massive 4D spin-2 field, with mass-squared given by the constant HQM- Of course, HQM is a hermitian operator in the extra dimension, so we should really try to diagonalize the operator. The eigenvalues will then be the 4D mass-squareds of the tower of gravitational KK modes. Note that this eigenvalue problem is analogous to finding the energy eigenvalues of a ID non-relativistic quantum mechanics problem with "unit" mass and a "volcano potential",
^^WTIF-T^'
(13 5)
-
as illustrated in Fig. 19. By inspection the analog time-independent Schrodinger equation should have precisely one bound state (which of course must therefore be noneother than our localized zero-mode) and a continuum of waves (asymptoting to sinusoidal waves far away from the volcano) with positive "EQM" > 0. Now let us put back in the fact that R < 00. In the z coordinates, l^maxl ~ ek*R/k
.
(13.6)
While this does little to the bound state mode, it will quantize the previously continuum modes, with " £ Q M " ~ (n/2 m a x ) 2 .
(13.7)
To the Fifth Dimension
and Back
623
z
Figure 19. "Volcano" potential of analog quantum mechanics problem describing KK graviton fluctuations.
Prom this we deduce that the zero mode graviton is accompanied by KK spin-2 excitations with masses given by m&K ~ (nke-k"R)2
.
(13.8)
Even though all the input scales of the fundamental 5D set-up are Planckian, given that the warp factor renormalized the weak scale to its observed size, we get an estimate for KK gravitons, mKK £ nMpie-kvR
~ TeV !
(13.9)
That is, these states are accessible to TeV scale colliders. Of course it is not good enough to be kinematically accessible, to be seen a new particle must have an appreciable coupling to ordinary matter. If ordinary matter couples to KK gravitons with the same strength as the usual graviton, these states would be completely invisible as a practical matter. Fortunately this is not the case. To be concrete let us assume that all matter, including the Higgs is localized on the TeV brane. They can only couple to the graviton KK modes via their extra-dimensional profiles evaluated on the TeV brane. This determines the relative strength of coupling of different gravity KK modes to brane localized matter since the fundamental coupling constant, the 5D Newton constant, is the same for all the KK modes. The relative strength is easy to estimate. A typical lowlying KK gravity excitation will have roughly a plane-wave wave-function (for the analog non-relativistic quantum mechanics problem) in most of the z space, so that its normalized value on the IR brane is roughly of order 1/'\JZmax ~ Vke~knR/2. On the other hand, the bound state 4D massless
624
R.
Sundrum
graviton mode has the normalized analog wavefunction evaluated on the IR brane given by ~ y/ke~3k7rR^2. Ex. Check this last fact carefully accounting for the factor of 3/2 in the exponent arising from the precise definition of the analog problem in terms of the 5D metric ansatz. Thus the amplitude for a KK excited graviton to couple to brane matter is ~ ekvR larger than the amplitude for the massless graviton. Now the massless graviton of course couples with 4D Planck-suppressed ~ l/(10 18 GeV) strength which is why such couplings are invisible at TeV scale colliders. But since we are taking ekwR ~ Mpi/TeV, KK gravitons will couple to matter with only 1/TeV suppression, that is order one amplitudes at TeV colliders 22 ! In this scenario these exotic spin-2 states should be visible at upcoming colliders such as the LHC. Similar mass splitting, wavefunctions and coupling strengths to IR brane matter hold for KK excitations of any field (with any spin) one considers in the 5D bulk. 14. Warped (fermionic) bulk matter A useful and pioneering reference for this section and the next is Ref. 23. Let us consider 5D fermions in the warped context subject to the orbifold boundary conditions, P(*L) = +*L P ( * R ) = -tf R .
(14.1)
Using the proper-distance y coordinate ranging from — irR to irR, we consider a bulk action of the form +TTR 4
S* = fd x
f dy e~4kM * (i TMDM
- m sgn(y)) tf .
(14.2)
—irR
In general, the covariant derivative for fermions cannot be written directly in terms of the metric alone, but requires a compatible "spin-connnection". We will just consider the RSI vacuuum metric for which the compatible covariant derivatives are given by £>M = d,, - -T 5 r M A; sgn(y) D5 = dy.
(14.3)
The exponential is just the measure factor arising from the root of the metric determinant as usual, but it appears that the mass term has explicit
To the Fifth Dimension
and Back
625
y dependence in order to be compatible with the orbifold parity symmetry. This is not really cheating, such a mass term can be thought of as arising from a Yukawa coupling to a orbifold-parity-odd 5D scalar field, whose fluctuations are very massive but whose VEV is non-zero and proportional to sgn(y). The curved space YN are given in terms of the usual flat space Dirac matrices by T„ = e^^-y^ T5 = -if5, T" = e + f c ^7 M , T 5 = ry 5 . Decomposing the action in 4D notation, and making the convenient field redefinition, tf = e+3'2kM&,
(?x dy*U0 -f*,f«i{
S*=
+e
-k\y\
~k 2
sgn(j/)75 - J5dy - msgn(y)
* .
(14.4) Despite the 5D mass parameter, which we will from now on dimensionlessly parametrize as Cs
(145)
J '
there are 4D massless chiral fermion zero-modes. We clearly get a zeromode from the equation of motion if the mode is annihilated by the 4D Dirac operator, so that
k
( ? " C) Sgn(?/) ~ l5dy] * = ° '
(14 6)
-
We see two possible chiral solutions to this equation. The left-handed one, $L(z,i/)=*L ( 0 ) (:E)e ( i " c ) f c | 1 ' 1 ,
(14.7)
satisfies the orbifold parity condition and is therefore physical. The righthanded one, * R ( ^ y ) = *R ( 0 ) (z)e ( ^ + c ) f c | y | ,
(14.8)
does not satisfy being parity-odd, and therefore is inadmissible. The parity therefore gives us a chiral 4D massless left-handed zero-mode. While the 5D mass parameter, c, does not affect the existence of 4D chiral modes, they clearly influence their profile in the extra dimension, as illustrated in Fig. 20. If the Higgs is considered to be localized on the IR brane still, 4D Yukawa couplings with two species of chiral zero modes coming from bulk fermions with mass parameters ca,Cb will be given by AD Yukawa ^ j i _ couplings
Ca)k7rR
e{\
- cb)knR
x
5D Yukawa couplings
Thus even without large hierarchies at the 5D level, hierarchical effective Yukawa couplings are naturally generated. In the real world, we can identify light fermions as chiral modes arising from bulk fermions with c > 1/2,
626
R.
Sundrum
H Yukawa couplings
Planck , brane Figure 20.
/
IR brane
Fermion zero-mode profiles for different 5D fermion masses.
and heavy fermions with chiral modes arising from bulk fermions with c < 1/2. Therefore light fermion profiles are suppressed at the IR brane. This suppresses their wave-function overlap with low-lying KK excitations of all bulk fields, thereby suppressing a host of dangerous KK-mediated effects. This is the central part of an automatic GIM mechanism suppressing flavor-changing neutral currents. On the other hand one can predict that the heavy top quark in this scenario should display significant non-standard corrections to its couplings. 15. Warped bulk Y M Of course if fermionic matter lives in the 5D bulk so must the gauge fields. The 5D YM action in background curved space is given by S = -\
fd4xfd
We will consider the RS warped vacuum background.
.
(15.1)
To the Fifth Dimension
and Back
627
We begin by considering the orbifold boundary conditions given by P{A„) = +AM P(A5) = -A5 .
(15.2)
In almost axial gauge, the single expected A5 mode is eliminated by the orbifold conditions, but A^ does give rise to a zero-mode. Ex. Show that All(x,)=AW(x)
(15.3)
is the zero-mode 4D gauge boson and that if coupled to matter (or if nonabelian), the effective 4D gauge coupling is given by
<15 4)
" = 7§I'
Note the form of the zero-mode and the 4D coupling look exactly as they did without warping. Gauge theory is very different from gravity in this regard. In particular the warped explanation for "why gravity is weak" (the zero-mode graviton of course) does not simultaneously greatly weaken the strength of gauge forces, which is obviously a good thing. The other interesting example to consider is given by orbifold conditions, P(A,)
=
-A,
P(A5) = +A5 ,
(15.5)
so that now there is a 4D scalar zero-mode but not a vector zero-mode. (Of course, as illustrated earlier in flat spacetime, for non-abelian gauge group different gauge fields corresponding to different generators can be treated with different orbifold boundary conditions. We will not consider that level of complexity here.) To isolate the scalar zero-mode in the warped background we first look at all terms in the action containing A5, S B \ Jd^xjdcjiRe-^ne1^ 3 \ fd4xfd(P R e~2k^R
(d5A, -
d,A5f
[{O^As)2 - 2 d5A^A5]
.
(15.6)
The usual almost axial gauge does not disentangle the vector-scalar mixing in warped spacetime and is therefore not convenient. But if we try As{x,<j>) = Af\x)e+2kWR,
(15.7)
the mixing term vanishes (as you can check by doing an integration by parts with respect to the <j> derivative). The 4D effective action for the zero-mode
628
R. Sundrum
scalar below the mass of the lightest KK excitations ( ~ ke given by
5eff4D = \jd±xjd4>Re+*kWR
kvR
) is then
(5,40))2
— 7T
= Jdx-^Ji(d»A°)
•
(15 8)
-
As the zero-mode ansatz shows such massless scalars are localized (but not J-function localized) near the IR brane and therefore makes a good Higgs candidate 24 ! One can ask what the advantage is of having the Higgs realized as the fifth component of a gauge field as opposed to as a fundamental IR-branelocalized scalar. The answer is that in the former case radiative corrections to the scalar mass-squared are effectively cut off at ~ TBKK ~ ke~klxR while in the latter case the cutoff is the "warped down" fundamental cutoff A\jye~k7rR, which is parametrically larger. This is a generalization of the flat spacetime result we derived in detail earlier where the cutoff on the scalar mass was effectively TTIKK ~ 1/R. One can repeat the kind of calculations we did at the beginning of these lectures which carve out an A$ effective potential, but now in the warped context, to build realistic warped models of the Higgs and electroweak symmetry breaking. The central considerations needed to achieve realism are worked out in Ref. 25. Combining these with the Higgs realized as an A$ was done in Ref. 26. 16. Last words Well, I have just gone through the simplest examples of many of the interesting mechanisms connected with extra dimensions. They can appear together in interesting combinations. When this happens there is greater complexity and the probability for error is multiplied. I have found that in the arena of warped compactifications, the qualitative insight gained from the AdS/CFT connection between such compactifications and strongly coupled 4D dynamics, has saved me time and time again from errors. It's like "checking units" as an undergraduate, in principle it's not necessary, but in practice indispensible. But while checking units is dull, the AdS/CFT connection is pure magic! It also allows you to design more efficient methods of analysing your models quantitatively. You can learn to think in terms of this connection by starting with Refs. 7 and 9.
To the Fifth Dimension and Back 629 Acknowledgements T h e a u t h o r is supported by N S F grant P420-D36-2043-4350. He is grateful to D m i t r y Belyaev for assistance with the figures and equations, and for catching several errors. References 1. C. Csaki, arXiv:hep-ph/0404096. 2. I. Z. Rothstein, arXiv:hep-ph/0308266. 3. R. Rattazzi, Cargese lectures in "Particle Physics and Cosmology: The Interface", eds. D. Kazakov and G. Smadja, Springer 2005 4. M. A. Luty, arXiv:hep-th/0509029. 5. H. Ooguri and Z. Yin, arXiv:hep-th/9612254. 6. J. Polchinski, "String Theory" Vol. 1, Ch.8, Cambridge University Press (1998). 7. J. M. Maldacena, arXiv:hep-th/0309246. 8. O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y. Oz, Phys. Rept. 323, 183 (2000) [arXiv:hep-th/990511lj. 9. N. Arkani-Hamed, M. Porrati and L. Randall, JHEP 0108, 017 (2001) [arXiv:hep-th/0012148]; R. Rattazzi and A. Zaffaroni, JHEP 0104, 021 (2001) [arXiv:hep-th/0012248]. 10. N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B 429, 263 (1998) [arXiv:hep-ph/9803315]. 11. N. S. Manton, "A New Six-Dimensional Approach To The Weinberg-Salam Model," Nucl. Phys. B 158, 141 (1979); D. B. Fairlie, Phys. Lett. B 82, 97 (1979); J. Phys. G 5, L55 (1979); P. Forgacs and N. S. Manton, Commun. Math. Phys. 72, 15 (1980); S. Randjbar-Daemi, A. Salam and J. A. Strathdee, Nucl. Phys. B 214, 491 (1983). 12. Y. Hosotani, Phys. Lett. B 126, 309 (1983); Annals Phys. 190, 233 (1989). 13. M. Quiros, arXiv:hep-ph/0302189. 14. C. A. Scrucca, M. Serone and L. Silvestrini, Nucl. Phys. B 669, 128 (2003) [arXiv:hep-ph/0304220]. 15. H. Georgi, "Weak Interactions and Modern Particle Theory", Benjamin/Cummings, Menlo Park (1984). 16. Z. Chacko, M. A. Luty and E. Ponton, JHEP 0007, 036 (2000) [arXiv:hepph/9909248]. 17. S. Weinberg, Rev. Mod. Phys. 61, 1 (1989). 18. L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999) [arXiv:hepph/9905221]. 19. L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 4690 (1999) [arXiv:hepth/9906064]. 20. H. Georgi, A. K. Grant and G. Hailu, Phys. Lett. B 506, 207 (2001) [arXiv:hep-ph/0012379]. 21. W. D. Goldberger and M. B. Wise, Phys. Rev. Lett. 83, 4922 (1999) [arXiv:hep-ph/9907447].
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22. H. Davoudiasl, J. L. Hewett and T. G. Rizzo, Phys. Rev. Lett. 84, 2080 (2000) [arXiv:hep-ph/9909255]. 23. T. Gherghetta and A. Pomarol, Nucl. Phys. B 586, 141 (2000) [arXiv:hepph/0003129]. 24. R. Contino, Y. Nomura and A. Pomarol, Nucl. Phys. B 671, 148 (2003) [arXiv:hep-ph/0306259]. 25. K. Agashe, A. Delgado, M. J. May and R. Sundrum, JHEP 0308, 050 (2003) [arXiv:hep-ph/0308036]. 26. K. Agashe, R. Contino and A. Pomarol, Nucl. Phys. B 719, 165 (2005) [arXiv:hep-ph/0412089].
. * * • *
GRAHAM D. KRIBS
P H E N O M E N O L O G Y OF E X T R A D I M E N S I O N S
G R A H A M D. KRIBS Department of Physics and Institute of Theoretical University of Oregon, Eugene, OR 97403 kribs@uoregon. edu
Science
The phenomenology of large, warped, and universal extra dimensions are reviewed. Characteristic signals are emphasized over an extensive survey. This is the writeup of lectures given at the Theoretical Advanced Study Institute in 2004.
1. Introduction The most exciting development in physics beyond the Standard Model in the past ten years is the phenomenology of extra dimensions. A cursory glance at the SLAC Spires "all time high" citation count confirms this crude statement. As of the close of the summer of 2005, the original papers on large [1], warped [2], and universal [3] extra dimensions have garnered nearly 4000 citations 1 among them. Extra dimensions have been around for a long time. Kaluza and Klein postulated a fifth dimension to unify electromagnetism with gravity. A closer look at this old idea reveals both its promise and problems. Imagine a Universe with five dimensional gravity, in which one dimension is compactified on a circle with circumference L. The Einstein-Hilbert action is
S = I' d5xMlR(b)
(1)
where R^ is the Ricci scalar for the five dimensional spacetime. The five dimensional graviton KMN contains five physical components that are decomposed on R4 ® S1 at the massless level as
A5„ 0 ) 1
Nearly 4000 papers have cited one or more these three papers. 633
Kl)
634
G. D. Kribs
where h^ is the four dimensional graviton, A^ is a massless vector field, andis a massless scalar field. The action reduces to
J dAx (M*L)RW + \F^F^ + \d^d^
(3)
comprising four dimensional gravity plus a gauge field with coupling strength g2 = (M^L)-1 as well as a massless scalar field (with only gravitational couplings). The remarkable finding that gauge theory could arise from a higher dimensional spacetime suitably compactified has been a tantalizing hint of how to unify gravity with the other gauge forces. As it stands, however, the original Kaluza-Klein proposal suffers from three problems: (1) there is a gravitationally coupled scalar field; (2) the gauge field strength is order one only when L~x ~ M„; (3) fermions are not chiral in five dimensions, leading to fermion "doubling" at the massless level. Orbifolding the compactified spacetime on S1 /Z^ doesn't help, since the same operation that projects out the fermion doubles also projects out the massless gauge field. Where the original hope of Kaluza-Klein's idea fails, string theory takes over and I refer you to other TASI lectures and books to give you the past and present scoop on string theory (for starters, try Ref. [4]). These lectures, instead, concentrate on what the world would be like if some of or all of the fields we know and love live in extra dimensions. There is some overlap between these lectures and those of Sundrum [5] and Csaki (with Hubisz and Meade) [6], however, I believe you will find that my perspective on this (and the direction given to me by the TASI 2004 organizers) is somewhat different. Hopefully it is useful! There is one issue that I think is useful to dispense with right away, namely: Why study extra dimensions? In light of deconstruction [7, 8], one is tempted to believe everything can be studied from a purely four dimensional view. This is certainly true of gauge theory. Does this mean we discard extra dimensions and just consider product gauge theories? Here it useful to consider the point of view of Hill, Pokorski, Wang [8], in which they sought an effective theory of the Kaluza-Klein modes of an extra dimension. They emphasized that imposing a lattice cutoff on the extra dimensional space was tantamount to writing a product gauge theory with particular relationships among all of the couplings and masses. Perhaps an analogy to gauge theory is useful here. Imagine that you know nothing of gauge theory
Phenomenology
of Extra Dimensions
635
and just go out and measure couplings of fermions to gluons and gluons to themselves (3-point and 4-point couplings). Gradually, through careful measurement you would find that the couplings are all related, up to certain overall constants (later identified as group theory factors dependent on the representation of the fermions). These relationships would be curious, but certainly would not prevent you from writing down the low energy effective theory of the couplings of these particles. Eventually, once the couplings are established to be the same up to some experimental accuracy, you would discard the effective theory of totally separate couplings and instead just write down the QCD Lagrangian. Analogously, once the couplings and masses of Kaluza-Klein modes are measured to sufficient accuracy, one will likely cease to characterize this as "a product gauge theory with relationships among the couplings" and instead simply begin saying one has "discovered an extra dimension". All of this is true for gauge theory and gravity. However, deconstructing gravity has proved more elusive, due to various issues of strong coupling that appear inevitable. I refer you to several papers [9] for discussions of this fascinating topic. Suffice to say, it is much more straightforward to understand the low energy effective theory of a compactified "physical" extra dimension rather than an extra dimension built out of multiple general coordinate invariances. Since the focus of these lectures is on phenomenology, this view is efficient, simple, and prudent.
2. Large extra dimensions The renaissance of extra dimensions began with the Arkani-Hamed, Dimopoulos, and Dvali (ADD) proposal [1,10] to lower the scale of quantum gravity to a TeV by localizing the SM to a 3+1 dimensional surface or "brane" in a higher dimensional spacetime. The extra dimensions are compactified into a large volume that effectively dilutes the strength of gravity from the fundamental scale (the TeV scale) to the Planck scale. A sketch of the setup is shown in Fig. 1. The idea that the quantum gravity scale could be lowered while the SM remain on a brane was motivated by earlier results in string theory. In particular, it was realized in string theory that the quantum gravity scale could be lowered from the Planck scale to the GUT scale [11] Others also pursued extra dimensions opening up between the TeV scale to the GUT scale [12]. In this section, however, I will concentrate solely on the ADD model and discuss several of its important phenomenological implications.
636
G. D. Kribs
extra compactified dimensions
4-d spacetime
Figure 1.
Sketch of the large extra dimension ADD model worldview.
First, let's be explicit about the assumptions. The ADD model consists of • n extra dimensions, each compactified with radius r (taken to be the same size for each dimension) on a torus with volume Vn = (2irr)n. • All SM fields (matter, Higgs, gauge fields) are localized to a 3-brane ("SM brane") in the bulk ("gravity only") spacetime. • Bulk and boundary spacetime is flat, i.e., the bulk and boundary cosmological constants vanish. • The SM 3-brane is "stiff"; the fluctuations of the brane surface itself in the higher dimensional spacetime can be ignored (or, more technically, the brane fluctuations have masses of order the cutoff scale) The action for this model divides into two pieces: S = £b u lk + Sbrane
(4)
where we are assuming for the moment that there is only one SM brane on which all of the SM fields live. Concentrate on the bulk action first, which
Phenomenology
of Extra Dimensions
637
is just the Einstein-Hilbert action for 4 + n dimensional gravity: 2 Sbuik = - / d4+nx v 7 + ^ M ; + 2 . R ( 4 + n )
(5)
We obviously integrate over all spacetime coordinates; hence the Lagrangian has mass dimension D = 4 + n. The higher dimensional Ricci curvature scalar i?( 4 + n ), formed in the usual way from two derivatives acting on the metric, having mass dimension 2. This determines the mass dimension of the coefficient of this highly relevant operator, namely n + 2. The line element of the bulk is ds2=g^+Nn)dxMdxN
(6)
where we use capital letters, M,N = 0 . . . (4 + n — 1), as the indices for the bulk spacetime. Assumption (2) of the ADD model is that spacetime is flat, so we can write expand gMN about flat spacetime including fluctuations. For the moment, let's only consider 4-d metric fluctuations, h^. Then the line element is ds2 = (T)»V + h^) dx»dxv - r2dn\n)(n)
(7)
where dfi(n) are n-dimensional toroidal coordinates. Given this factorization of the metric, it is easy to show that
vV 4 + n ) =r n VsW ^(4+n)
=
R(4)
(8) (Q)
where g^ and R^ implicitly depend on /iM„. Now rewrite the higher dimensional action in terms of 4-d modes by "integrating out" the extra dimensions: 5 b u l k = - M ^ + 2 J di+nx^/g^+n)R{A+n) = -M?+2
J dAx [dn(n)rny/g~WRW
= -M?+2{2irr)n
f dAx\fg~WR^
(10) (n) (12)
2 This is my definition of M*, and there are plenty other others out there absorbing various factors of 2 and TT. One can argue that this one has the most intuitive physical interpretation [13], and thus is the one that ought to be used. But, having not stumbled onto this until four years after ADD's original paper, the other definitions are unlikely to go away.
638 G. D. Kribs The last line is the action for 4-d gravity. Matching the coefficient of the above action with the Planck scale and one obtains the famous result M& = M?+2(2wr)n
(13)
in which our measured Planck scale is determined by the fundamental scale of quantum gravity, M*, and the volume of the extra dimensions, Vn = (27rr)n. What does this result mean physically? The weakness of 4-d longdistance gravity is fundamentally to due the graviton being spread rather thin across the extra dimensions with a small intersection with the SMbrane. At long distances, gravity behaves exactly as it does in 4-d by construction, since we have integrated out the extra dimensions and matched this action* to the usual 4-d action. Close to the length scale of the extra dimensions, however, the gravitational potential changes and one expects to see macroscopic changes in the strength of the force of gravity. 2.1. Deviations
from Newtonian
gravity
Classically, gravitational potential changes at short distance. The Newtonian potential between two bodies of mass m\ and mi is , '^N n+n V(r') = { i mim 2 -GN —
r
< r v(14)
r'>r
where r' represents the distance separating the objects, not to be confused with the size of the extra dimensions, r. This leads to a vital question: How well is gravity measured? The answer is, for short distances, rather poorly compared with all the other forces. With hindsight, we really should not be surprised since gravity is so weak in comparison to the other forces. Nevertheless, it was the genius of ADD to exploit this fact in constructing their model. There have for years been crazy ideas on the potential modification of gravity at small but macroscopic distances, from fifth force shenanigans to light scalar moduli from certain string theories. There is some history here, and since each particular idea has a somewhat different functional dependence of the strength of gravity as a function of distance, experimentalists simplified all this by parameterizing deviations in Newton's law as ^ )
= _
G
W ^ ( l
+ ac-VA)
(15)
Phenomenology
1 Q 10
rm
1—i—i i i 11 I I
of Extra Dimensions
1—i—i i i 11 I I
Lamoreaux
108
639
1—i—i I I ii i
excluded region
Stanford
106
moduli
10* 2 extra dimensions scenario
102 10° vacuum energy scenario
10"2
Eot-Wash
10-I
i
10- e 2
i
i i 11111
5
10- 5 2
j
5
10"* 2 X [m]
i
I
5
i
10" 3 2
i
' 111
5
II
10-2
Figure 2. 95%-confidence-level constraints on ISL-violating Yukawa interactions with 1 /im < A < 1 cm. The heavy curves give experimental upper limits. (Fig. 5 from Ref. [14].)
The Yukawa form of the correction to Newton's law roughly corresponds to the exchange of virtual bosons of mass 1/A, and a = 1 corresponds to gravitational strength. Fig. 2 shows how well gravity is tested at macroscopic distances of direct relevance to the ADD model. You have all seen this graph many times, since it was showcased on the poster for this TASI school! This is rather busy graph showing several experimental results as well as theoretical predictions for deviations in assorted models. For us, what is relevant are the experimental bounds (i.e., ignore all of the non-capitalized identifications on the graph). Five experimental results are plotted that provide the strongest constraint on deviations from Newton's law for various ranges of distances and strengths of forces. For gravitational strength deviations, relevant to
640
G. D. Kribs
models of extra dimensions, the strongest constraint comes from the EotWash experiment that is consistent with Newtonian gravity down to about 200 microns [15]. These are remarkable experiments, and I encourage you to read the very nice review by Adelberger, Heckel and Nelson [14] for more complete details on how the experiments are done and what they imply for the assorted models that predict deviations. How big is the deviation in the ADD model? The changeover from four dimensional gravity to higher dimensional gravity in Eq. (14) implies that once objects are brought to distances smaller than r apart, they begin to experience gravity increasing in strength proportional to l/rn+1. Gravity becomes far stronger at short distances. Since experiment has not found any (unambiguous) deviations from the 1/r force law, the best we can do is to constrain the parameters of the ADD model using these null results. For the ADD model, it's obvious that a ~ 1, A ~ r is where an 0(1) deviation from Newtonian gravity is expected. Doing this a bit more carefully (for example, see [14]), one obtains A= r
(16)
a = |(2n)
(17)
where In sums over the number of KK gravitons with the same mass, and the 4/3 factor results from summing over all polarizations of the massive KK graviton. Numerically, the size of this correction depends on the volume of the extra dimensions and the size of the fundamental Planck scale M*. Let's take the lowest value that we could possibly imagine, namely M* ~ 1 TeV. This choice "solves" the hierarchy problem by lowering the cutoff scale of the Standard Model to 1 TeV. There are many implications of this, particularly if cutoff scale effects violated the global symmetries of the Standard Model such as flavor, baryon number, lepton number, etc. This is certainly what an effective field theorist should expect happens, and so ADD is immediately faced with serious problems. Let's not forget, however, that every other "solution" to the hierarchy problem also faces the same problems, i.e., excess violation of SM global symmetries. (For an amusing comparison of ADD to supersymmetry, see [16].) Just as there are fixes to these model-induced problems in supersymmetry, there are also some ingenious fixes for extra dimensions, and I'll mention a few at the end of this section and in the third lecture.
Phenomenology
of Extra Dimensions
641
For now, let's just calculate. Assuming equal-sized extra dimensions, we can trivially solve for the radius of n extra dimensions from Eq. (13),
2TT \M?+2J
V
;
Setting M* = 1 TeV, here is table of the distance scale where one expects order one deviations from Newtonian gravity: number of extra dimensions n= 1 n = 2 n = 3
r ~ 10 12 m ~10"3 m ~10-8 m
n = 6
~10_u m
Clearly, one extra dimension n = 1 with M* = 1 TeV is totally ruled out by solar system tests of Newtonian gravity. It is amusing to see how well gravity really is measured at these distances. This is shown in Fig. 3, where estimates of the bounds on new Yukawa-like forces are shown from a diverse set of experimental techniques across the distance scales in the figure. As an aside, notice that for distances several orders of magnitude longer than the solar system, deviations from Newtonian gravity are not well constrained. Indeed, astronomers have in fact measured very significant deviations from Newtonian gravity on galactic distance scales: the famous mismatch between the observed rotation curves with what one would expect from Newtonian gravity given the luminous mass distribution. This is of course one motivation for dark matter; for reviews, see Refs. [17, 18, 19]. For two extra dimensions, the predicted deviation from Newtonian gravity occurs at r ~ 1 mm. In 1998, when ADD wrote the first paper on their model, the best experimental limit on gravitational strength forces happened also to be at about 1 mm! Subsequent experiments, however, have found no deviation down to 200 microns, and thus rules out two extra dimensions with a quantum gravity scale of M* = 1 TeV. For three or more extra dimensions, the predicted deviation from Newtonian gravity occurs at considerably smaller distances, less than ten nanometers. The experimental constraints on new forces at these distances are extremely weak, as shown in Fig. 4. The good news is that ADD with M* = 1 TeV and n > 3 is not ruled by these experiments! The bad news is that the experiments
642
G. D. Kribs
planetary I
I 2
10~
I
I
10°
I
10
I 2
I
I
10*
I
J 6
10
I 8
10
I
10
I 10
I
10
I 12
L
10u
X [ml Figure 3. 95%-confldence-level constraints on ISL-violating Yukawa interactions with A > 1 cm. The LLR constraint is based on the anomalous perigee precession; the remaining constraints are based on Keplerian tests. (Fig. 4 from Ref. [14].)
are so far from testing gravitational strength interactions that it is hopeless to attempt to observe the change from 1/r to l/r™ +1 directly.3 2.2. Dynamics
of the higher dimensional
graviton
Let's generalize the line element for arbitrary metric fluctuations about a flat background bulk spacetime ds = gMNdx MJ„N dx ( VMN -\ 3
(19) VFTTThMN
)
MJ„N M
dx dx
(20)
There may be "auxiliary" effects associated with extra dimensions that lead to observable deviations from Newtonian gravity, such as additional scalar moduli that couple stronger than gravity. Some of these effects are shown in Fig. 2, and I refer you to Ref. [14] for details.
Phenomenology
1030
1
1
1 1 1 i 111
i
i
-
i
of Extra Dimensions
i i i 111
i
i
643
i i i 11
i
: 10,27 : Si
- ^\vdW
-
10
1021 -
EderthX^
-
1018
1015
_
Mohideen
—
Casimir
1012 109
-
-
Lamoreaux i
10"
Nv
9
i
2
i
i i i 111
5
i
10"
8
i
2
1
5
i
7
10~
i
2
i i i i ?
i
5
10" 6
X [ml Figure 4. Constraints on ISL-violating Yukawa interactions with l n m < A < l^im. (Fig. 9 from Ref. [14].)
where now HMN is the higher dimensional graviton. The coefficient is chosen for convenience to lead to canonical normalization for the graviton. Insert the metric expansion into the higher dimensional equations of motion GAB
= RAB
-
n
,
QABR •
2+ n""""
(21)
M*2+"
and one obtains [20] M?/2+1GAB
= UhAB - dAdchCB ,c
, „
- VABOh'c +
~ dBdchCA
+ dAdBhGc
oCaC
VABdudvhcD
(22)
T
AB
M
(23)
Define the higher dimensional coordinates as XM = {xnWi) =
{x0,xi,x2,x3;yi,...yn)
(24)
644
G. D. Kribs
where compactification of the extra dimensions implies the y's are periodic Vi —> Vi + 27rr. Upon imposing these periodic boundary conditions, the expansion of the higher dimensional graviton in terms of 4-d Kaluza-Klein fields is OO
OO
!.("»)/
hAB(x;y)= 22 •" 12
ixr
\
e
( 25 )
mn = — oo
where Mm> is a shorthand for /l(™i>™2.-"»n)> The SM is confined to a 3-brane within this bulk spacetime, so that TAB{x^)
= rfArfBTilv^\y)
(26)
taking the SM brane to be located at y = 0. The physics content of this equation is important: • We are neglecting brane fluctuations. • We are not bothering about 5{) functions. Any would-be singularities are smoothed out by replacing 5( )'s by e~v M *. The brane has a width of at least 1/M*. This distinction can be neglected so long as we work in an energy regime where -y/s <S M*. Now, plug in the KK expansion into Einstein's equations
G&H*) = f (h®,h%\h$>) = - g |
(27)
*P1
G%{x)=
...
=0
(28)
G%\x) =
•••
= 0
(29)
where the precise forms of / ( ) are not particularly illuminating (and can be found in [20]). Rewrite Einstein's equations in terms of propagating (i.e., physical) and non-propagating (i.e., redundancies associated with general coordinate invariance) degrees of freedom:
(• + n2)G^> (n
Mpi
+
-Tiiv + ( ^ 2 + 7W J 3 ft2
)"n=2Mp^
(n + ni 22) )V#> ^n)=0 - 22 \
c(") (n + h )S%>=0
(30) (31)
(32) (33)
Phenomenology
of Extra Dimensions
645
where Eqs. (30)-(33) correspond to KK gravitons, KK radions 4 , KK graviphotons, and KK scalars, respectively. The notation
E in* Ir
(34)
was also used. These excitations are coming from the decomposition of the higher dimensional metric fluctuations ' hu h\fN
h(i5 h^G • • • \ ^55 ^56
=
(35)
he, /
Notice that the vectors and most scalars from this decomposition do not couple to SM brane-localized fields. The gravitons and radions do! Let's do an example of this in 5-d. The five dimensional metric fluctuation decomposition is "-iiv ri/j,5
(36)
naively has KK models /i(0)
h{l)
h(0)
h{1)
«/,(°)
(37)
,(D and However, a massive graviton in 4-d has five polarizations: h p,v' eats hi (f)^; the latter are the longitudinal components. It's now appropriate to go through the degree of freedom counting for gravitons. As a warmup, let's begin with gauge theory. A general gauge field AM in D dimensions has D real components. One can always choose a gauge, such as Coulomb gauge, dMA
M
0
(38)
reducing the number of independent components to D — 1. One can then do a gauge transformation on AM'i-M
4
AM
+ dMX
O n e "KK radion" for each KK level for n > 1.
(39)
646
G. D. Kribs
for some real function X. This gauge transformation leaves the kinetic term invariant FMN
-> [dM (AN + dNX) - dN (AM + dMx)\
(40)
= FMN
(41)
since the OM^NX terms drop out. A massless, on-shell .D-dimensional gauge field has therefore D — 2 independent components. A mass term for the gauge field, however, is famously known not to be gauge invariant since \m2AMAM
-
±m2 [AMAM
+ 2AMdMX
+ dMXdMX]
•
(42)
The Higgs mechanism reinterprets the last term as a kinetic term for a scalar (Goldstone) field X and the mixing term represents the gauge boson/scalar mixing that gives rise to a gauge boson mass. A massive, on-shell gauge field in D dimensions therefore has D — 1 independent components. The story for the graviton is entirely analogous. A D-dimensional graviton is a D x D real symmetric matrix with two indices, and therefore D(D + l ) / 2 components. We can first choose a gauge, such as the harmonic gauge,
dMh% = \dNhZ
(43)
which reduces the number of independent components by D since Eq. (43) represents D independent constraints. One can then do a general coordinate transformation on HMNtlMN —> h-MN + 9M^N + ON^M
(44)
for some real vector function ejv. This gauge transformation leaves the kinetic term for gravity invariant. To show this, one needs the graviton kinetic term, namely the Einstein-Hilbert action expanded to leading order in the graviton fluctuation HMN [20] R = -^hABOhAB
+ ^hinh%
- hABdAdBhcc
+ hABdAdchcB
. (45)
I leave it as an exercise to verify that R is invariant under the general coordinate transformation Eq. (44). A massless, on-shell .D-dimensional graviton has therefore D(D - 3)/2 independent components. For D = 4 we obtain two real components, consistent with our expectations. A mass term for the graviton field, however, is not gauge invariant with respect to
Phenomenology
of Extra Dimensions
647
general coordinate invariance. The Fierz-Pauli graviton mass term in D dimensions is \m2
(hMNhMN
- h%hNN)
(46)
picks up non-zero contributions that include terms like h,MNdMeN. Reinterpreted as a Higgs mechanism for gravity, this graviton-vector mixing is entirely analogous to the vector-scalar mixing we found above. The vector is itself composed of a Goldstone (massless) vector with a Goldstone scalar, contributing a total D — 1 components to the graviton. A massive graviton in D dimensions therefore contains D(D — l ) / 2 — 1 components, for example which evaluates to 5 components for a 4-d graviton, matching our expectations. A nice description of how massive gravitons absorbs vectors, the issues surrounding massive gravity (vDVZ discontinuity [21], etc.), and what this means for a deconstruction of gravity can be found in Ref. [9]. The potentially dangerous mode is the scalar degree of freedom that couples to the energy momentum tensor: the radion. The radion is a conformally-coupled scalar, and thus couples to explicit conformal violation in the Standard Model, for example T£ ~ M&W+W1- + mfff + ... .
(47)
We have implicitly assumed that a stabilization mechanism is in place to fix the size of the extra dimensions and thus give the radion a mass sufficiently heavy so as to not modify gravity in experimentally unacceptable ways. This is obviously a highly model-dependent statement, and several groups have explored constraints on radion couplings in large extra dimension scenarios (for example, see Refs. [22, 23, 24]). At this point, what I have done is to show you that the effects of large extra dimensions (suitably stabilized) is reduced to the problem of determining the effects of the KK modes of the graviton. Given the results thus far, it is straightforward to derive the Feynman rules. I'll simply sketch the well known procedure for obtaining the interactions of the KK models with matter. The graviton couples to the energy momentum tensor, which we obtain from the SM action by
648
G. D. Kribs
For example, the QED part of the energy-momentum tensor is T
,?K ED
= 4> ( 7 , A +
7^M) 0 - J
+ \eQ4> h ^
u
{dl^>dudv'4n^) (j>
+ 7 v A M ) + FpxF* + ^F^Fpx
(49)
The Feynman rules follow directly. A few of them are shown in Fig. 5. Given the Feynman rules, we are now ready to do phenomenology! 2.3. Scales and graviton
counting
KK gravitons have a mass k/r so that the mass splittings between KK gravitons is 2/r,
Am~l
= = 2 2KM* . M{ *^ -^ ''
.
(50)
r Plugging in some numbers to give some feeling for the size 0.003 eV n =2 Am ~ i 0.1 MeV n =4 for M* = 1 TeV. 0.05 GeV n =6
(51)
Obviously this is very small! For illustration, the KK mass spectrum of gravitons for n = 2 extra dimensions is shown in Fig. 6. It is convenient to replace the sum over modes with an integral over the density of states. Fig. 6 graphically shows that the continuum rapidly becomes an excellent approximation, so long as experiments are not sensitive to the mass splitting. (This is certainly true for M* near the TeV scale and the number of extra dimensions is, say, n < 6.) The number of states dN in an n-dimensional spatial volume having momenta between |fe| to \k\ +dk is dN = Sn^\k\n~ldk where Sn-\ states is
27r"/ 2
5n_! = — — r(n/2)
(52)
is the area of an n-sphere. Using m = \k\/r, the density of dN = Sn-1rnmn-1dm
(53)
so that the differential cross section for inclusive graviton production becomes ^ - 5 n - i , „ M f' n+22 m " - * % dtdm " (27r)"M* dt
(54)
Phenomenology
of Extra Dimensions
649
f(kl)
G.iiv
Gliv
4MPI
MPI
[WM„ + Wv
[W^ra/3 + Wvlia0]
Gfw
2Mpi
ffS(*i
eQ
[7M ^ Q
+ 7^a]
Gnu
i£-fabcK(kuk2,k3)^af37 Mpi
$(**)
9CM
Figure 5. Some of the Feynman rules connecting gravitons to SM fields, from Ref. [20]. Here Wpi, = (fci + £2)^71/ and the other kinematical functions W^Ja„ and K(ki, A)2, kz)nua0~t c a n D e found in [20]. Rules (b) and (c) are present for all SM groups; rule (d) occurs for non-Abelian groups (gluons shown).
where t = (j>\ — P3)2; da/dt is the differential cross section for a single graviton of mass m; and I have substituted for r using Eq. (50).
650
G. D. Kribs
100/r
mass
10/r
1/r
Figure 6. The mass spectrum of the KK gravitons is shown for n = 2. Notice that the density of KK states fills in the energy axis quite rapidly, allowing us to very accurately replace the discrete set of KK states with a continuum.
This general formula can applied to any specific process involving gravitons. For example, consider real graviton production in association with photons, / / —> jG. The differential cross section to produce a particular graviton is
dt
VJ
'
;
lWfsM^
where a is the photon coupling, Qf is the electric charge of the fermion, Nf is number of colors, and s is the center of mass energy. Using the Feynman rules given above, the kinematical function F\(x,y) can be computed, and is given in the Appendix of Ref. [20]. Notice that the cross section for producing a single graviton is suppressed by the 4-d Planck scale, as you
Phenomenology
of Extra Dimensions
651
would expect. But upon integrating over the huge density of states between 1/r to the energy of the process s, one obtains da
™ lt-i
n\
aQ2
f
1
o
rnn~l
(t
m2\
where the dependence on the 4-d Planck scale cancels! This of course had to happen, since we could have just as easily done the same calculation in D dimensions, where the only scale in the problem is the fundamental quantum gravity scale, M*, which is the coupling of the D-dimensional graviton! The total cross section is obtained by integrating over all angles. To give you a feeling for the size of this signal, consider the process where the initial state particles / / = e+e~ from a 1 TeV center-of-mass energy collider (such a linear collider that is under active consideration by the high energy physics community). The result is shown in Fig. 7. There are several things to glean from the figure. Holding the energy of the incident particles fixed (as a partonic collider does for you for free), you can see that more extra dimensions generically means a smaller signal; i.e., contrast the n = 2 curve with the n = 5 curve. This is easy to understand: As the number of dimensions increases, 1/r increases, and hence the density of graviton states per unit energy interval increases dramatically. Holding y/s/Mi, fixed, then, implies that the integrated density of states between 1/r to y/s always decreases as the number of dimensions increase. Hence, all other things considered equal, signals associated with graviton emission will always be harder to see as the number of extra dimensions increases. Second, notice in Fig. 7 that the estimated size of the SM background is rather substantial, and one really has to get rather lucky with M* awfully close to y/s to get a signal at a TeV e+e~ collider. Hadron colliders can do much better. This subject has received enormous attention (the first few papers that performed calculations for hadron colliders are [20,25-27]). As just one example, consider the basic partonic diagrams for the LHC that lead to graviton emission in association with one colored parton that becomes a jet in the detector. The basic subprocesses include qg —> qG (which gives the largest contribution), qq —> gG, and gg —> gG. Since missing energy by itself leads to no signal at a hadron collider detector, these processes allow for "tagging" of large missing energy using the monojet signal. The resulting hadronic collider process is thus pp —> jet + fir- For a sufficiently high enough cut on the jet energy, the background from one jet plus Z —> Vv can be sufficiently reduced to weed
652
G. D. Kribs -r
i
i
i
|
i
i
i
i
'
1
'
1
1
+
10*
1*
10 1
_
=
~
\
^
>v
\
"~*
\a
v^s N^
1
Vs = 1 TeV
;
^ ^ - C ^ ^ .
—
:
-\^^
_ " ^ v
1 0 " 1 E E7 < 450 GeV -
10"
_
---^SM Bkgd 90% Pol-
v— — ^ v _
»
1
^
^s.
I
+
1
SM Bkgd
V
10°
1
rvv\\ V\\\ •
+ ^ t
e e"
1
—
v.
E T7 > 300 GeV i
i
i
i
1
E ^
i
i
.
1 1
1
1
i^k
i
-
v
i
i
i
MD [TeV] Figure 7. Total e+e —» 7 + nothing cross-section at a 1 TeV center-of-mass energy e + e " collider. Here M , = (27r)- n /( 2 + n >M D ~ (0.4 -> 0.25)M D for n (called <5 in the figure) between (2 —> 6). The signal from graviton production is presented as solid lines for various numbers of extra dimension (n = 2, 3, 4, 5). The Standard Model background for unpolarized beams is given by the upper dash-dotted line, and the background with 90% polarization is given by the lower dash-dotted line. The signal and background are computed with the requirement E-, < 450 GeV in order to eliminate the -yZ —• 7P1/ contribution to the background. The dashed line is the Standard Model background subtracted signal from a representative dimension-6 operator. (Fig. 2 from Ref. [20].)
out the signal. Doing the calculation in detail [20] one finds the cut on the jet energy must be in the several hundred GeV to TeV range. As an illustration of the size of this process in comparison to background, Fig. 8 shows the hadronic production cross section at leading order as a function of the lowered quantum gravity scale. Another interesting signal is virtual graviton exchange. Here one now must sum over all KK gravitons exchanged, so that the amplitude contains A
1 M|,
(57)
s — mKK
s(«-2)/2
MJ?n+2
(58)
Phenomenology
1Q—2
Li
i
I
2
i
i
i
i
I
i
i
i
i
4
I
i
of Extra Dimensions
i
6
i
i
I
8
i
i
i
653
L.
10
MD [TeV] Figure 8. The total jet + nothing cross-section versus Mp at the LHC integrated for all Er.jet > 1 TeV with the requirement that |?jjet| < 3-0- Here again M* = ( 2 7 r ) - n / ( 2 + n > M D ~ (0.4 - • 0.25)M D for n (called <S in the figure) between (2 -> 4). The Standard Model background is the dash-dotted line, and the signal is plotted as solid and dashed lines for n = 2 and 4 extra dimensions. The "a" ("b") lines are constructed by integrating the cross-section over s < M^ (all s).
where like before, the sum over the KK modes removes the 1/Mpi suppression in favor of 1/M* suppression.5 This result, unlike the real graviton emission in association with photons discussed above, has certain theoretical ambiguities. The central issue is the positive power of y/s in the numerator. This means that this process is ostensibly diverging as one approaches M*. This is analogous to what happens to the amplitude of gauge boson scattering in the Standard Model without a Higgs boson. Unlike the SM, however, we don't know what regulates quantum gravity (even if there are extra dimensions), and so this amplitude could well be affected by the UV physics that smoothes out quantum gravity (strings at a TeV!). In effective field theory, this UV dependence corresponds to higher dimensional operators suppressed by the cutoff scale, i.e., the quantum gravity scale. Hence, 5
For the case n = 2, the amplitude should be multiplied by
Ins/fj.2.
654
G. D. Kribs
another effect of the ADD model is to look for effects of these higher dimensional operators. At dimension-8, one can write the effective operator corresponding to virtual graviton exchange at tree-level
where the coefficient c ~ 4ir corresponds to the NDA estimate of the size of this operator at strong coupling. At dimension-6, graviton loops can also induce new four-fermion operators, of the form
7/7/ M?
(60)
where again c ~ 4TT at strong coupling. The effects of these operators correspond to what is usually called "compositeness" in older literature, and indeed the same analysis applies. Large extra dimensions are simply one realization of these operators. The precise constraints depend on the assumption of strong coupling (the A-K coefficient) and the particular operators in question, but one finds numbers of order 1.5 TeV for Eq. (59) and of order 15 TeV for Eq. (60) (see 2005 update of PDG [28]). 2.4.
Astrophysics
Collider experiments can probe large extra dimensions by integrating over a large number of graviton modes. To get stronger bounds one must either increase the energy of the collider or increase the luminosity. The existence of light gravitons, however, allows a different window on this physics: namely, thermal systems that are hot enough to produce graviton KK modes and large enough to produce enough of them to have an effect on the astrophysical system. Specifically, consider astrophysical systems whose temperature is T > mKK .
(61)
We can estimate the rate of thermal graviton production by multiplying the coupling of each graviton by the number of modes accessible, 1 Tn rate of graviton production oc —s- (Tr)n ~ ——rn . Mf,x MT +
(62)
To find the best astrophysical bound, we want the hottest astrophysical system in the Universe. The system must, however, be well enough understood via ordinary SM physics so that we can use it as testing laboratory. This is system is a supernova, and SN1987A in particular.
Phenomenology
of Extra Dimensions
655
SN1987A is a core collapse type II supernova that went off in our sister galaxy, the Large Magellanic Cloud, emitting a huge amount of energy mostly into neutrinos. The neutrinos appeared after the stellar core of the dying star got hot enough so that protons and electrons could combine into neutrons via the weak interaction. Several neutrino events were recorded by underground detectors on Earth, including Kamiokande in Japan and 1MB in the USA. The time extent of the neutrino burst was several seconds, suggesting the supernova remained hot enough for the inverse reaction p + e~ —> n + ve to proceed on a macroscopic time scale. The temperature of SN1987A is estimated to be T ~ 50 ± 20 MeV. SN1987A has been used to constrain all sorts of non-standard physics (for example, see Ref. [29]). Here what is of most interest are the constraints on axions that have rather weak couplings to matter. The basis for constraints on axions is that too large a coupling leads to too much axion emission from the supernova that has the effect of providing a means to more rapidly cool the supernova. In this case, the time extent of neutrino observations limits the total amount of cooling, and thus the strength of the axion coupling. Similarly, graviton emission can cause excess cooling of a supernova, and this is what we want to work out now. The situation is a bit different between axions and gravitons, since axions are derivatively coupled. The relevant reaction for gravitons is N + N -> N + N + G
(63)
where strong interaction effects (through pion exchange) are unsuppressed while the nucleons N themselves are still yet non-relativistic. The graviton coupling to non-relativistic matter is Pi T"" = (m , ) (64) KPiPiPj/mJ where the transverse-traceless part couples to PiPj/m ~ T, the temperature of the non-relativistic plasma. This causes an extra suppression T2/M+ for the cross section to produce gravitons compared with axions. The thermally averaged cross section is roughly [10]
h^T^
where
rp \ n+2
<™>~(30mb)^—J
(65)
During SN collapse, roughly 10 53 erg are released in a few seconds. To use this to place a bound on graviton emission, we simply require that the
656
G. D. Kribs
graviton luminosity is less than 10 53 erg/s ~ (10 16 GeV) 2 . The graviton luminosity is 2
/
T
\ n+2
LG ~ M core -Z- (30 mb) I — J
(66)
where TIN is the nucleon number density in the core and p is the mass density. This calculation was done carefully in [30], who found the constraints r 50 TeV MD > I 4 TeV [ 1 TeV
n =2 n =3 n = 4
(67)
where M/j = (27r) n /( n+2 )M*. There is no bound for n > 4 since the mass splitting between the gravitons is between a few to tens of MeV, leaving a rather small range of KK gravitons that can be emitted without Boltzmann suppression. These bounds are significant for several reasons. Perhaps the most important conclusion that can be drawn from these results becomes manifest if we translate the bounds on M* into upper bounds on the size of the extra dimensions:
{
1 0 - 4 mm 10~ 7 mm
n =2 n = 3
(68)
10" 8 mm n =4 Hence, the scale where gravitational strength deviations from Newton's law are guaranteed to be present from large extra dimension models is far smaller than the present experimental bound (about 0.2 mm) and indeed much smaller than future experiments are likely able to probe (at gravitational strength). There is good lesson here. New physics can appear in myriad experimental situations, and one must consider all of them to obtain the best bounds on the parameters of a new physics model. Of course this is not to suggest that continued experiments probing gravitational deviations is futile, but it does mean that a deviation attributed to KK gravitons would be (apparently) inconsistent with graviton emission from SN1987A. 2.5.
Cosmology
There is another constraint that I want to discuss that concerns excess cooling of another big astrophysical system: the entire Universe! The source of cooling is the same as for supernovae, namely graviton emission. In 5-d
Phenomenology
of Extra Dimensions
657
language, a 5-d graviton can be emitted into the bulk with a coupling that is suppressed by just 1/M*. In 4-d language, the probability to emit some KK mode goes as 1/M™+2 while the decay of a given mode goes as 1/Mp\. This means for high enough temperatures in the early universe, graviton emission becomes large, and this depletes the energy of the coupled plasma causing excess cooling that would be observed by large differences in BBN. The decay rate for a graviton into two photons is
rG^7 = s ^ r
(69)
which corresponds to a decay time of
(70)
'-("Mw)'
Hence, once a graviton is produced it decouples from the thermal plasma and does not decay for a long, long time. To extract a bound on large extra dimension models, let's compare the ordinary Hubble expansion rate to that of cooling by gravitons. Cooling by Hubble expansion is roughly dp ~-32Tp~-3—TP (71) ~dl expansion PI whereas cooling by graviton emission goes as dp dt
rpn
(72)
w*
evaporation
These rates are equal at the "normalcy" temperature, which is easily found by equating the above rates, and one obtains f 10 MeV
n =2
T* = lO^Vr MeV = { : 10 GeV
.
(73)
n = 6
The normalcy temperature is is the maximum reheat temperature of the Universe such that cooling by ordinary Hubble expansion dominates. The good news is that this temperature is above the temperature of BBN (about 1 MeV), and so we do not expect BBN predictions to be modified. The bad news is that we generally have thought that the Universe was far hotter than tens of MeV to tens of GeV, for example to generate weakly interacting dark matter, baryogenesis, inflation, etc. All of these phenomena need new mechanisms that operate at low temperatures. See for example Ref. [31] for a discussion of some of these issues.
658
G. D. Kribs
O ASCA (Gendrcau 1995) HEAO (LED) (Gruber 1992) A HEAO (MED) (Kinzeret.aL 1996) + SMM (Watanabe 1997) * APOLLO (Trombka 1997) •
COMPTEL(Kappadalhet.al.l996) SAS-2 (Thompson & Fichtel 1982)
•
EGRET (Srcckumar el.al 1997)
vr 10 ioJ Photon Energy (keV) Figure 9. Multiwavelength spectrum from X-rays to 7-rays including the revised ICT upper limits from the Apollo experiment (Trombka 1997). The thick solid line indicates the sum of all the components. (Fig. 6 from Ref. [32].)
2.6. Relic
photons
Even if the Universe is reheated to a temperature that is below the normalcy temperature, many light, long-lived gravitons are produced. By themselves, the relic KK gravitons are not a nuisance, but their decay products may well be. The KK graviton decay rate into photons was given above in Eq. (69), and we see that some significant fraction of the KK gravitons (ones lighter than about 5 MeV) produced in the early Universe will have decayed by now. This excess source of keV to MeV photons contributes to the diffuse cosmic 7-ray background. Measuring the diffuse high energy photon spectrum is an ongoing enterprise. The region of interest of us has been covered by the Comptel 7-ray observations, shown (in conjunction with measurements throughout the high energy photon spectrum) in Fig. 9. Since BBN requires the Universe be reheated to at least about 1 MeV, we can obtain the best bound by requiring that the diffuse photon flux does not exceed the diffuse 7-ray observations assuming the normalcy temperature T* = 1 MeV.
Phenomenology
of Extra Dimensions
659
Ref. [33] worked this out, obtaining dn 7
/rn = an{E)
~d~E
/TeV\n+2 ( M )
1 MeV cm 2 s sr
(74)
V 1V1* /
T, = l MeV
where the coefficient was found to be a 2 (4 MeV) ~ 104
(75)
a 2 (4 MeV) ~ 0.4
(76)
for n = 2,3 extra dimensions evaluated at a photon energy of 4 MeV. Comparing this result to the data allows us to extract a new bound on the quantum gravity scale MD>
{
rilO->350TeV 5^14TeV
n = 2 n = 3
,„_. (7?)
where the first (second) number corresponds to restricting the normalcy temperature to be 1 (2.2) MeV. Regardless, this is clearly the strongest bound we have seen on the scale of quantum gravity from experiment for 2 and 3 extra dimensions. No significant bound is obtained for n > 3 dimensions. 3. Warped extra dimensions Prom Sundrum's lectures [5] and Csaki's lectures (with Hubisz and Meade) [6] you are already well versed on warped extra dimensions. Their emphasis, however, is a bit different from the charge of these lectures. Both Sundrum and Csaki were largely interested in warped spacetimes in which some or all of the Standard Model fields propagate in the bulk. This is done for all sorts of reasons that they have expertly explained in their lectures. In this lecture, I want to focus on the original proposal of Randall and Sundrum (RS) [2] in which only gravity exists in the warped extra dimension while the SM is confined to a 3-brane whose dimensionful parameters are scaled to the TeV scale. This is the historical approach, which may seem a bit dated by now, however I see this as a minimalist approach to general topic of warped extra dimensions: Virtually everything that I describe below applies to these model variants involving warped extra dimensions. That is, there are graviton resonances in the composite unification model [34] as well as the warped Higgsless model [35] as well as effects of radius stabilization, even if they are not of primary interest in those scenarios.
660
G. D. Kribs
bulk
Vpianck
Planck
-A
VTW
TeV Brane
Brane
V=0 Figure 10.
y=b
Sketch of the warped extra dimension RS model.
Without further ado, let's plunge into the original RS model. The RS model is 5-d theory compactified on on S1 /Z% orbifold, with bulk and boundary cosmological constants that precisely balance to give a stable 4-d low energy effective theory with vanishing 4-d cosmological constant. The basic setup is sketched in Fig. 10. The background spacetime metric is taken to be ds2 = e-^rj^dxKdx"
- dy2
(78)
where the metric is not as trivial as in the ADD model. Specifically, the y dependence that enters the metric as e _fe ' !/ ' that is known as the "warp factor". The absolute value of y is taken because the extra dimension is compactified on an orbifold that identifies y <-> — y. We will see the physical significance of the warp factor shortly. The action for the model is S = Sbulk + SWnck + S T W
(79)
in which
Sbulk=-Jd5x^(MlsR-A) Splanck = 'TeV
d aV^PlanckVplanck
/ d4x^/gTey
(Vrev + SM Lagrangian)
(80) (81) (82)
where gpianck and
Phenomenology
of Extra Dimensions
661
Vpianck = -Vrev = 12A;MRS
(83)
A = -fcVp lanck .
(84)
in terms of the AdS curvature k and the fundamental quantum gravity scale M R S . 6 This solution balances bulk curvature with boundary brane tensions (4-d cosmological constants). There is one troubling fact we see already, namely the brane tension of the TeV brane is negative. One simple way to see the effects of this is to phase rotate the entire TeV brane action, S —* —S, that shifts the negative sign to be in front of the kinetic terms of brane fields. Negative kinetic terms (for scalars, anyway) are well known to signal a instability in the theory in which the kinetic terms can grow (negative) arbitrarily large. We will ignore this issue here, and assume that the UV completion of the RS model stabilizes the negative tension brane (in field theory see e.g. Ref. [36] and in string theory see e.g. Ref. [37]). Examine the SM action:
SSM = Jd4x^g^
[g^v(D^DvH
- X(H^H - v2)2 + ...]
(85)
Now insert the induced metric evaluated on the TeV brane, (gTev)iiu = e~2kbVn" a n d o n e obtains fd4x
- A (A^H - ( e - f c V ) 2 ) 2 +
rr(DpH)\DvH)
(86)
in terms of the canonically normalized fields H = e -kbH
4„ =
A e"-kb M-p,
-3kb/2 / • / = e~
(87) (88) (89)
The central result is that the warp factor can be rescaled away from all of the dimensionless terms in the SM (at tree-level) by field redefinitions. The only dimensionful operator, the Higgs (mass) 2 , gets physically rescaled. We can define a new Higgs vacuum expectation value (VEV) that absorbs the warp factor v = e~kbv 6
(90)
1 use M R S for the quantum gravity scale in RS to be distinguished from M* in the ADD model.
662
G. D. Kribs
which we see can be exponentially smaller in the canonically normalized basis. The RS model presupposes that we take all fundamental mass parameters to be 0(Mp\), including v ~ O.lMpi. Then for a suitably large large enough slice of AdS space relative to the curvature size, kb ~ 35, we obtain v ~ 0.1e- fc6 M P1 ~ 0.1 TeV
(91)
This is the key result that got everyone excited about the RS model! The interpretation is that all dimensionful parameters on the TeV brane are "warped" down to TeV scale assuming only that the size of the AdS space is parametrically larger than the inverse of the AdS curvature. What are the sizes of fundamental parameters? The 4-d effective Planck scale can be obtained by integrating out the extra dimension to extract the coefficient of
j dhx^f^)e-2k^R^
(92)
which is
^ p i = Mis [V~+b e~2k^dy = Mk(iJy=-b
e~2kb) .
(93)
K
Clearly, given a tiny warp factor e~kb, all fundamental parameters can be of order the 4-d Planck scale, MRS ~ k ~ Mp\. 3.1. Metric
fluctuations
The generic ansatz for metric fluctuations about the RS background is ds2 = e-2kMgiM/dx»dxu
+ A^dx^dy - b2dy2 .
(94)
RS is by definition an AdS spacetime on an S1 jZ-i orbifold, and thus the zero mode of the graviphoton, A^ , is absent. Furthermore, we saw from ADD that in 5-d there are only graviton KK excitations since the wouldbe graviphoton and graviscalar KK excitations are completely absorbed into the longitudinal components of the massive KK gravitons. The mass spectrum of RS is thus exceedingly simple: the massless graviton h^J, the massive KK graviton excitations h^J and a single real scalar field <j>, called the radion. Consider first just the tensor excitations [38,39]. This is easiest to consider when the RS metric in written in the the conformal frame ds2 = e~Mz)
(v^ + h^(x,
z)dx»dxv
- dz2)
(95)
Phenomenology
of Extra Dimensions
663
with e Mz)
~
=
{l+\lzl)2
> A(z)=2log(k\z\
+ l)
(96)
where the z <-> y relationship between the two coordinates is simply l + k\z\ We seek linearized fluctuations about the background that satisfy SGMN
= K STMN
•
(98)
First let's fix the gauge, the "RS gauge", in which the fluctuations satisfy hfi = d^h^ = 0. Expanding out GMN, keeping the leading order terms for h^v, one obtains -\dRdRh^
+ ^dRAdRh^
=0.
(99)
This can be written in a somewhat cleaner way by rescaling the graviton perturbation by h^v = e3A/4hlil/ and then the linearized Einstein equations become 1 R R ^dnh^ + ^d AdRA-\d dRA V = 0. (100) Now separate variables h^ix.z)
where D/iM„ = m2h^
— h^(x)(j)(z)
(101)
and <j)(y) satisfies the 1-D Schrodinger-like equation -d2z4> + V{z)(j> = m2
(102)
where the potential can be read off from Eq. (100) to be V(z)
= YQA'2
- \A"
(103)
and primes denote derivatives with respect to z. Substituting for A(z) using Eq. (96), one obtains
^
) =
T(iTJw-TTfeM* (z)
(104)
where V(z) is the famous "Volcano potential" that falls off as 1/z2 far from the UV brane with a single S() function at the caldera implying there is a single bound state (the massless 4-d graviton). One can solve for the zero mode wavefunction
<=•
4>{V) = e~3/4klyl
(105)
664 G. D. Kribs and thus we find that the overlap of the graviton with the TeV brane is exponentially suppressed. This is why gravity appears so weak to us in the RS model. Notice that there were no small parameters needed to obtain this result, beyond the requirement of the size of the extra dimension being a factor of 35 times the inverse of the AdS curvature. To obtain the wavefunctions and masses of the KK modes, we first must impose boundary conditions on the 5-d graviton wavefunction: dz
= -§*
(106) Z — Z\JV
k 2k\z\ + l
(107)
where zuv = 0 and ZIR = e /k are the locations of the Planck and TeV branes in the conformal coordinates. Inserting the expression for A(z) into Eq. (102), we obtain
-did +
15 k2 4 {k\z\ + l)2
m2(j}
•
(108)
The solution to this PDE can be cast in terms of a sum over Bessel functions 4>(z) = {kz + I)1'2 [amY2[m(z + 1/fc)] + bmJ2[m(z + 1/k)]]
(109)
where the boundary conditions Eqs. (106),(107) completely determine the wavefunction coefficients am,bm. The mass of the Kaluza-Klein graviton modes is easily found rrij = Xjke-kb where
(110)
the roots of the Bessel function J\ (XJ ) = 0. Roughly,
Xj
~
<
3.8 7.0 10.2 16.5
j = 1 J = 2 J =3
(111)
j=4
where the modes are obviously not evenly spaced. Furthermore, the quantity ke~kb is roughly the TeV scale for a TeV brane, and thus these gravitons have masses of order the TeV scale! This is a central prediction of the RS model: distinct spin-2 resonances with a Kaluza-Klein spectrum that is spaced according to the roots of the first Bessel function. What is the strength of the coupling of these gravitons? This where there is a big distinction between the zero mode and the Kaluza-Klein
Phenomenology
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665
modes of the graviton. The zero mode graviton has a 1/Mpi strength coupling, since it's wavefunction is peaked near the Planck brane, while the Kaluza-Klein modes have 1/TeV strength couplings since their wavefunctions are peaked near the TeV brane. This is easy to see by simply evaluating ^( Z )l'='m „ ekb
(112)
{z)\z=zw
that is exponentially enhanced by the (inverse) warp factor. Hence, the full action for the graviton fluctuations interacting with matter on the TeV brane is 1 1 °° £TeV = _ - i - r ^ / i f S - ——TUT^ Y hiuJ • (113) kb M Afpi ^ Mpie^-J n=1
3.2. Phenomenology
of Kaluza-Klein
gravitons
We found that the Kaluza-Klein excitations of the graviton have O(TeV) masses with 0(1/TeV) strength couplings. This means that they will behave like new spin-2 resonances that can be produced and observed individually. In particular, Kaluza-Klein gravitons will decay on a timescale of order 1/TeV. This we can easily estimate on dimensional grounds to be TYl TG nsM
(H4) ~ (MPle-^ where USM is the number of SM particles into which the graviton could decay. There are two parameters that determine the graviton production cross sections and decay rates: the warp factor and the AdS curvature. The warp factor can be equivalently replaced by the mass of the first graviton resonance. At the LHC, the .s-channel exchange of gravitons leads to new resonances, analogous to the search for new Z' gauge bosons. The subprocesses for this include qq, gg —> G^ —> £+£" which provides relatively clean leptonic events as well as qq,gg —> G^ —> qq, gg which leads to pairs of jets that will reconstruct the graviton mass. Using these parton processes to calculate the rate of graviton production, one can calculate a constraint on the warped Planck scale as a function of the AdS curvature. This is shown in Fig. 11, where the warped Planck scale is related to the first graviton mass by Eq. (110). The LHC is thus able to probe to several TeV.
At a hypothetical e + e~ collider, the broad resonances of s-channel KK graviton exchange are clearly visible (if the energy of the collider is high
666
G. D. Kribs 1.00
0.50
0.20 ^
0.10 0.05
0.02 0.01
>'
I
>
I
L
2
3
4
5
6
7
m i (TeV) Figure 11. Exclusion regions for resonance production of the first KK graviton excitation at the the LHC. The dashed, solid curves correspond to 10, 100 f b - 1 . The excluded region lies above and to the left of the curves. (Fig. 1 from Ref. [40].)
enough), allowing a detailed study of the graviton properties. This is shown in Fig. 12, where an example of graviton production and decay is shown for the process e + e~ —» /J + /i~ at a rather optimistic mass for the first KK resonance. In any case, the total cross section plot is rather striking!
3.3. Radius
stabilization
As it stands, the radion mass in the the RS model vanishes. This is because the brane tensions were tuned to balance the bulk cosmological constant. As there is no energy cost to increase or decrease the size of the extra dimension, radial excitations are uninhibited. This is a disaster, since massless scalar fields which couple with gravitational strength or stronger are ruled out. However, there is a simple modification of the original RS proposal in which dynamics can be used to stabilize the extra dimension. Goldberger and Wise realized that adding a bulk scalar field with an expectation value that had a profile, namely vhu\k(y), would be sufficient [41]. Roughly speaking, the bulk kinetic term wants to maximize the size of the extra dimension
Phenomenology
of Extra Dimensions
667
Z~l—I—l—I—l—l—l—I-1—l—l—I—l—|—I—l—I—I—I—I—I—l—I—I—l—I—I—t~z
250
500
750
1000
1250
1500
Vs (GeV) Figure 12. The cross section for e+e~ —> fi+ \x~ including the exchange of a tower of KK gravitons, taking the mass of the first mode to be 600 GeV, as a function of y/s. From top to bottom the curves correspond to k/Mp\ = 1.0, 0.7, 0.5, 0.3, 0.2, 0.1. (Fig. 2 from Ref. [40].)
whereas the bulk mass term wants to minimize the size. The balance between these two forces results in a stabilized warped extra dimension. A complete analysis of radius stabilization is a rather intricate affair. The original Goldberger-Wise paper [41] provides a very clear account of the problem and their solution. There are, however, a few subtleties concerning how they implemented their solution. Specifically, a naive ansatz for the radion field was used which ignores both the radion wavefunction and the backreaction of the stabilizing scalar field on the metric. This was remedied in Ref. [42] (see also Ref. [43]), were we combined the metric ansatz of Ref. [44] that solves Einstein's equations, with a bulk scalar potential that includes backreaction effects that was given by Ref. [45]. Rather than repeating the analysis in these lectures, let me simply quote the main results here and refer interested readers to the original literature for details on the derivation. The main properties of the radion relevant to phenomenology are its mass and couplings. The radion mass is given by m 0 ~ e^==Mpie- fcb Vkb
(115)
668
G. D. Kribs
where e is parameter that characterizes the size of the backreaction on the metric due to the bulk scalar field VEV profile. In Ref. [42], our analysis computed the mass of the radion treating the backreaction as a perturbation, and thus Eq. (115) is valid for e < 1. The key result here is that the radion mass is of order the warped Planck scale, i.e., the TeV scale, but parametrically smaller by a factor l/y/kb. This is unlike the KK gravitons that we found above, where the first KK mode had a mass of ~ Ake~kb. This suggests that the radion is the lightest mode in the RS model, and thus possibly the most important excitation to study at colliders. The couplings of the radion to matter are also of vital importance. They were first given in [46,47] and confirmed using the formalism described above in [42]. At leading order in the radion field, the radion couplings are -Tji (116) V6Mpie-kb M where Tfi is the trace of the energy-momentum tensor of the Standard Model. This coupling is precisely the same as a conformally coupled scalar field, which is not surprising given the holographic interpretation of the RS model [48, 49, 50]. There is a fascinating story here, based on the AdS/CFT, concerning the duality between the 5-d RS model and a 4-d strongly-coupled conformal field theory. This would take me well beyond the scope of these lectures, and so I refer you to some excellent TASI lectures for details [51] In any case, a phenomenologically successful RS model must have dynamics that stabilize the extra dimension and give the radion a mass, and this can be seen in either the 5-d AdS picture or the 4-d CFT dual picture. 3.4. Radion
phenomenology
Given that the radion mass, Eq. (115), is of order (slightly smaller than) the warped Planck scale, and the couplings of the radion are dimension-5 operators suppressed by the warped Planck scale, Eq. (116), the radion clearly has observable effects at high energy colliders. The mass of the radion is effectively a free parameter m^,, which can be bounded from above by the warped Planck scale A = y/QMPle-kb
(117)
and bounded from below by naturalness, namely quadratically divergent radiative corrections to its mass. Roughly, then, we anticipate ^<m^
(118)
Phenomenology
of Extra Dimensions
669
which corresponds to perhaps tens of GeV up to a TeV or so assuming A ~ 1 TeV. The conformal couplings of the radion are straightforward to work out. The Standard Model in an unbroken electroweak vacuum, i.e., without a Higgs boson, is classically scale invariant and so T^ vanishes classically. However, once a Higgs boson is introduced and electroweak symmetry is broken, the radion couples at tree-level to the conformal violation, namely, everywhere the Higgs VEV enters (as well as the Higgs (mass) 2 term). Scale invariance of the SM is broken at the quantum level, since for example coupling constants change with energy scale under the renormalization group. The radion also couples to this (loop-level) breaking of scale invariance. Thus, the tree-level couplings of the radion are identical to the Higgs boson of the SM, except that the coupling is universally scaled by a factor v/A. At loop-level, the scaling factor v/A remains, but the coefficients change since the Higgs boson is not a conformally coupled scalar! The radion couplings are shown along with the couplings of the SM Higgs boson are shown for comparison in Fig. 13. There are several comments to make about these couplings. First, the coefficient v/A is expected to be of order 1/10 for A ~ TeV scale. This drops out of branching ratios, and thus at tree-level the radion's branching ratios are the same as (tree-level) Higgs branching ratios. This suggests it is rather hard to tell the radion and the Higgs apart! The definitive measure is total width: again, at tree-level we would expect T^ = jjThUnfortunately, measuring the total width of the Higgs boson or radion directly is somewhere between hard to really hard to measure. How hard it is depends on the mass of the scalar in question, and thus its width, and what instruments are at our disposal. Obviously s-channel production at a lepton collider would be ideal, measuring the width in the same way that LEP measured the Z width. The Z, however, has an 0(1) coupling with the colliding leptons, whereas the Higgs and radion couple (at tree-level) with only Yukawa strength. This means an e+e~ collider is hopeless: the cross section is simply too small. A muon collider is much more promising, if such a machine could actually be made to work. At loop-level there are differences in the branching ratios, and fortunately the loop effects associated with the radion couplings are significant. In particular, the radion coupling to gluons has a strength that far exceeds that of the Higgs, and this has several important consequences. One is that the production cross section for radions at hadron colliders is enhanced by roughly a factor of b\ = (7) 2 , the QCD beta function coefficient. This nearly
670
G. D. Kribs
radion couplings
Higgs couplings
.2M$y
w9n
2?rA
{PlP2V»v ~ Pl„P2v) 9> "
,cjT.(~ i)aa ( %
2-KV
, WWl^-PipP*")
Figure 13. The leading order radion couplings to several fields in the Standard Model are shown. At tree-level, radion couplings are identical to Higgs boson couplings up to an overall factor v/A. At loop-level, the conformal couplings of the radion are manifest, for example, in the proportionality t o the beta function coefficients illustrated by the coupling to gluons.
compensates for the v2 / A 2 suppression factor from the conformal coupling, and leads to radion production through gluon fusion t h a t is comparable to t h a t of Higgs production. T h e production cross section for radion production a t the LHC and the Tevatron is shown in Fig. 14 as a function of the radion mass. T h e second effect of the large coupling t o gluons is t h a t if the radion mass is less t h a n about 2Mw, the only open decay modes are <j> —>bb and loop-level decays into gluons or photons. In the S t a n d a r d Model, the Higgs branching ratio to gluons never dominates for any range of the Higgs boson
Phenomenology
of Extra Dimensions
671
m, (GeV) Figure 14. Production cross sections versus the mass of the radion for pp —•(gg fusion), pp —> qq'4> (WW,ZZ fusion), pp -» W<j>, pp -» Z «>. (Fig. 3(b) from Ref. [52].)
mass. For radions, however, the far larger coupling to gluons dominates over the small b Yukawa coupling causing the radion to decay dominantly to gluons for the radion mass range 20 GeV < m^ < 2MW- This is a strikingly different signal as compared with a Higgs boson in the same mass range. It is, in fact, a far more challenging signal to find since the two gluons become two jets, and the LHC is overwhelmed with multi-jet events with modest transverse momentum. Branching ratios for the radion into different Standard Model final states were calculated in [52,53] are shown in Fig. 15. 4. Universal extra dimensions 4.1.
Motivation
We have seen that both large and warped extra dimensions have the potential to lower the cutoff scale of the SM to the TeV scale. In ADD, we
672
G. D. Kribs
Figure 15. Branching ratios of the radion versus the radion mass. Here we have used mh = 150 GeV. (Fig. 2 from Ref. [52].)
saw that the fundamental quantum gravity scale is the TeV scale, and so this is automatically the cutoff scale of the SM. In RS, this is a bit more subtle, but the end result is the same. This is easy to see by writing the full effective theory for the SM plus all higher dimensional operators on the TeV brane with natural size, i.e., of order the 4-d Planck scale. After the extra dimension is stabilized, the warp factor appears everywhere there is a dimensionful scale. This causes the Higgs (mass) 2 to warp down to the TeV scale, and simultaneously causes the higher dimensional operators proportional to 1/M£, to scale "up" to 1/TeV™. In both cases, the cutoff scale is set ultimately by quantum gravity. It is well known lore that quantum gravity generically violates global symmetries (e.g., black holes with Hawking evaporation). Hence, we expect that the presence of higher dimensional operators proportional 1/TeV" to violate the global symmetries of the SM. The global symmetries of the SM protect against an awful lot of curious phenomena, including B and L violation
Phenomenology
of Extra Dimensions
673
(together or separately), non-GIM flavor symmetry violation, excessive CP violation, custodial SU(2) violation, etc., which has relied on having a large energy desert between the weak scale and the cutoff scale. Experimental limits on the absence of these phenomena correspond to raising the appropriate cutoff scale of these operators high enough so that these phenomena do not occur. For instance, proton decay in the SM occurs through
among other dimension-6 B and L violating operators. Very roughly, this operator leads to a proton lifetime x ~ ^
(120)
ml that must be longer than about 10 33 yrs based on the super-Kamiokande bounds [54]. Converting this lifetime into a bound on the scale M suppressing the operator one finds M > 1016 GeV. Examples of other global symmetries of concern are: • Lepton number via the dim-5 operator M leading to way too large Majorana neutrino masses. Flavor-changing neutral current (FCNC) operators, such as • dsds M2
(121)
(122)
leading to excessive KQ <-> KQ mixing. • Baryon number violating operators, such as QQuudd M5 leading to neutron-anti-neutron oscillations [55].
(123)
Some of these operators, such as the one leading to KQ <-> KQ mixing, cannot be simply forbidden by exact symmetries since this mixing has been observed experimentally and successfully explained by GIM-suppressed flavor violation in the SM. There are numerous proposals for solving these problems within the contexts of the ADD model and the RS model. A small set of examples include physical separation of fermions [56], discrete symmetries [1,57], fermions in
674
G. D. Kribs
the bulk [5], and so on. Unfortunately, I don't have the time, space, or energy to review these ideas here. Instead, I simply want to emphasize that the ADD and RS models are incomplete as originally proposed, and require mechanisms to explain why these processes are small. Universal extra dimensions have the potential to solve some of these problems as well as provide interesting "what if" scenarios that can be tested in experiments. 4.2. UED: The
model(s)
Universal Extra Dimensions are models in which all of the SM fields live in 4 + n dimensions with the n extra dimensions taken to be fiat and compact. This basic idea has a long history; for some of the early and earlier work see [12]. In this lecture I will primarily discuss the original proposal [3] and then touch on several of its numerous spin-offs: a solution to the proton decay problem [58]; a rationale for three generations [59]; a test-bed for a scenario with experimental signatures that have great similarity with (versions of) supersymmetry [60, 61, 62, 63]; and a dark matter candidate [60,64]. Promoting the full SM to extra dimensions seems like a crazy idea for several reasons. First, the spin-1/2 representations of the Poincare group in higher dimensions generically have more degrees of freedom and differing restrictions based on chirality properties and anomalies (see Ref. [65] for a nice review). This means that fermions are generically non-chiral (with respect to our four dimensions). A simple example of this is that in five dimensions, -y5 becomes part of the group structure, and so no chiral projection operators can be constructed to reduce what are intrinsically four-dimensional fermion representations to chiral two-dimensional representations. This problem is solved by "orbifolding", i.e., compactifying on surfaces with endpoints. In five dimensions, the only choice is S1/Z2, which identifies opposite sides of a circle to create a line segment with two endpoints. In six and higher dimensions, there are many more surfaces to compactify on; the one that is more interesting for this discussion is T2/Z2The next problem is that gauge couplings are dimensionful. Given the higher dimensional gauge field action
S= J dA+nxFMNFMN
(124)
(where M, N are the higher dimensional indices running from 0 to 3 + n), one deduces that the canonical dimension of the gauge fields is (2 + n)/2. This means the gauge couplings have dimension —n/2. Gauge field couplings in higher dimensions become analogous to graviton couplings in
Phenomenology
of Extra Dimensions
675
four dimensions, and this means these effective theories have a cutoff of order the scale of the coupling. Here I have been loose with "of order"; a more complete accounting of the relationship between the cutoff and the compactification scale can be found in Ref. [66]. I should warn you, however, that my own partially substantiated hunch is that counting 47r's in higher dimensional calculations to estimate the cutoff scale may be even more subtle. This is because if one matches these higher dimensional theories with four dimensional product gauge theories via deconstruction, it appears that the 4TT counting may be better estimated using just four dimensional naive dimensional analysis. Anyways, for "few extra dimensional" theories, the difference is a rather innocuous 0(1) number, and so will not really affect the discussion below. Lastly, as good phenomenologists we will push the compactification scale to the lowest possible value that is not excluded by experiment. Putting gauge fields, fermions, and Higgs bosons in extra dimensions means there is a tower of KK excitations for all of these fields. Given that we have not seen excited massive resonances of KK photons or gluons, while colliders have probed up to the few hundred GeV scale, we can expect that 1/R > hundreds of GeV. We'll be much more precise below. 4.3. Three
generations
Dobrescu and Poppitz [59] found a very interesting result of promoting the SM into six dimensions. Six dimensions happens to be the most interesting numbers of dimensions due to existence of chiral fermions, as occurs for even numbers of dimensions, and also additional anomaly cancellation constraints, in particular due to the gravitational anomaly [67] that exists in two, six, ten, . . . dimensions. The anomalies that exist in six dimensions can be classified as "irreducible" gauge anomalies, "reducible" gauge anomalies, or pure or mixed gravitational anomalies. Here "reducible" gauge anomalies correspond to most of the SM ones involving U(l)y and SU(2)r,. Dobrescu and Poppitz argue that we do not need to care about anomalies associated with symmetries that are spontaneously broken. One way to rationalize this argument is that since the electroweak breaking scale and the compactification scale is roughly the same for UED, any effects associated with anomalies of these spontaneously broken gauge symmetries can be absorbed by cutoff scale effects. Alternatively, they also point out that the reducible gauge anomalies can be canceled by a higher dimensional version of the Green-Schwarz mechanism, see Ref. [59] for details.
676
G. D. Kribs
The irreducible gauge anomalies are those associated with SU(3)C and C/(l) em . Since colored particles and electrically charged particles are vectorlike, the only non-trivial anomaly comes from U(l)em[SU(3)c]3 (in six dimensions, anomalies correspond to box diagrams connecting four gauge fields together). The pure and mixed gravitational anomalies arise with respect to the 6-d chirality assignments of the six dimensional SM fermions. Fermions can take on either chirality, using r7 • / - ±/±
(125)
where the f± are the 6-d chiral fermions (each chirality is a four-component spinor) analogous to the more familiar /L,R 4-d chiral fermions (where each chirality is a two-component spinor). Requiring the irreducible gauge anomaly to cancel combined with the pure and mixed gravitational anomalies leads to one of four possible 6-d chiral assignments Q+,u-,d-,L+,e-,N-
(126)
Q+,u-,d.,L-,e+,N+
(127)
where the other two assignments simply flip + <-> — for all fermion species. Already, an interesting result is that a gauge-neutral fermion, N, is required to exist so that the pure and mixed gravitational anomaly is canceled. This requirement is curiously similar to an analogous phenomena that occurs with gauged flavor symmetry extensions of the (4-d) SM [68]. Finally, there are the global gauge anomalies. These are higher dimensional analogues of the Witten anomaly for SU(2) [69]. Global gauge anomalies potentially exist for SU(3), SU(2), and Gi in six dimensions. In UED, SU(3)C is vector-like and so automatically cancels. SU(2)L, however, requires n ( 2 + ) - n ( 2 _ ) = 0 mod 6
(128)
where n(2±) corresponds to the number of doublets with 6-d chirality ± . For one generation, n(Q) = 3, n(L) = ± 1 implying n(+) - n ( - ) = 2 or 4. For ng generations, this becomes n(Q) = 3ng
(129)
n(L) = ±ng
(130)
n ( 2 + ) - n ( 2 _ ) = 2ng or 4ng
(131)
and we see that the global gauge anomaly is canceled with ng = 3 (mod 3) generations!
Phenomenology
4.4. Proton
of Extra Dimensions
677
decay
We already remarked on the potential problems with operators that lead to proton decay in models with a low cutoff scale. Ref. [58] pointed out that part of the global symmetry associated with the extra dimensional coordinates can be utilized to restrict the forms of higher dimensional operators that are allowed. In particular, in 6-d the Poincare symmetry 50(1,5) —> 50(1,3) x (7(1)45 where the [7(1)457 corresponds to rotations between the fourth and fifth (extra dimensional) coordinates. Chiral 6-d fermions decompose under 4-d SU(2)L chirality as± = 4>±L + ±R
(132)
defined via the projection operators P±L =PTn=\{lT
£45)
(133)
where S Q/3 /2 are the generators of the spin-1/2 representations of 50(1,5). The 4-d chiral fermions have ^7(1)45 charges given by the eigenvalues of E 4 5 /2: T l / 2 for±L and ±1/2 for ±R. This implies the U(l)i5 charge assignmentsforthe SM fields: fermion Q+L U-R, d-R
L+L e-R, N-R
{7(1)45 charge -1/2 -1/2 -1/2 -1/2
U(1)B
1/3 1/3 0 0
Baryon number violation requires three quark fields, but obviously no combination of three quarks is invariant under £7(1)45. To obtain a A S = 1 operator, therefore, we need three lepton fields to make the operator [7(1)45invariant. The lowest dimensional operator 8 occurs at dimension-17 On -
(L+Ld-R)3Hi p j
(134)
After integrating over the extra dimensions, the 4-d low energy effective theory contains the dim-9 operator -^^{VLdR^LdR)2 7
(135)
"45" is a label for one U(l), not to be confused with forty-five U(l)'s. There are several operators at dim-16 involving the singlet N, but for conciseness I will only consider the lowest dimensional operator involving SM fields. 8
678
G. D. Kribs
leading to proton decay via the 5-body process such as p Putting in appropriate coefficients and form factors, one obtains [58]
Hence, for 1/R of order the weak scale with a cutoff scale A > 5/R, the 6-d UED model based on T2/Z2 is completely safe from proton decay. One can show more generally that the sum rule 3 A S ± AL = 0 mod 8
(137)
is satisfied for all of the zero-mode fields. This forbids: proton decay with less than 6 fermions; AB = 2, AL = 0 baryon-number violating interactions leading to neutron-anti-neutron oscillations; and A B = 0, AL = 2 leptonnumber violating interactions leading to Majorana neutrino masses. Hence, many of the most dangerous violations of the SM global symmetries are forbidden or sufficiently suppressed. 4.5. UED: The model Having piqued your interest in UED by argument for three generations as well as naturally allowing a low cutoff scale, let's now delve into the UED model and its phenomenology. The action for the SM in higher dimensions is
W'
y
+ iQTMDMQ
1
T7> jpMN •Z^J-TMN-r
2 2
+ iuTMDMu
+
idTMDMd
+ QXuuia2H* + QXddH + ^Higgs + leptons + ..
(138)
where gauge interactions, Yukawa interactions, and Higgs interactions are all bulk interactions. These couplings are thus dimensionful, since this is a higher dimensional theory. In particular, it must be stressed that there are no S(y) functions present. The UED model, by definition, has no tree-level brane-localized fields or interactions. Demanding that all fields and interactions are bulk interactions, with no S() functions, has one extremely important consequence. To see this, let's
Phenomenology
of Extra Dimensions
679
first decompose a (4 + n)-dimensional gauge field into its 4-d Kaluza-Klein (KK) components: Afi{x,y)
Af\x)
=
• j » )
(x) cos
jiyi +... R
+jnyn
(139)
We could continue this discussion in an arbitrary number of dimensions, but for simplicity let's concentrate on just one extra dimension. There is no loss of generality to the basic argument I am about to present by specializing to 5-d. Indeed, numerous papers that have been written about UED have concentrated on the 5-d version, so this sets us up nicely to discuss this body of work. In most cases, it is more complicated but nevertheless straightforward to extend the 5-d discussions into 6-d to preserve the properties that we found in the first two subsections. Back to the significance of the absence of S( )-functions. This is best understood with an example. Consider two distinct 5-d theories: one contains bulk fermions F, the other contains boundary fermions / (localized at y — 0), while both are coupled to a bulk gauge field AM- The theory with boundary fermions has an action d*xdyfTMDMf6(y)
(140)
/ •
that on KK expansion becomes /
^xdyfT"
4°> + V25>W>COB
m
fS(y)
(141)
where there is an overall constant as well as an additive set of interactions with the fifth component of the gauge field (an additional scalar field) that I'm not bothering about here. Integrate out the fifth dimension, assumed to be on the interval S1 /Z%, jf
dy cos (?jL\ 6(y) = 1
(142)
and one is left with the 4-d Lagrangian
f d^xfT*1 4°>+V2 5>W>
/
(143)
where the 4-d boundary fermions couple to all of the KK modes with the same strength.
680
G. D. Kribs
Now contrast this to what happens in the theory with only bulk fermions. The action / '
(144)
is KK expanded into
/
d4xdyF{0)T"
4°) + y/2Y,Af cos
i?(°)
(145)
where a KK expansion of the bulk fermions and gauge field were done but only keeping the zero modes of the bulk fermions for this discussion. Integrate out the fifth dimension O
P'KR
dy cos where this integral vanishes for all j except j Lagrangian
0
(146)
0. One is left with the 4-d
d4xFi0)T»M0)FW (147) / where the 4-d zero mode fermions couple only to the zero mode of the gauge field! Generalizing to the kth KK mode of one of the bulk fermions interacting with the j t h KK mode of the gauge field, the integral Eq. (146) becomes 2 TTR
r-nR
r-(sM*)
' ' " > c o s ( ^ ) =Sjk
(148)
leading to the non-zero 4-d interactions
d*x F ( f c ) FM ( f c ) .F ( 0 ) + FPwWrTM ( f c ) . F ( f c ) (149) /• also shown diagrammatically in Fig. 16. The presence of interactions with an even number of same-level KK modes is due precisely to the absence of 6() functions. The S() functions are sources for brane-localized interactions which completely break translation invariance in the fifth dimension. The absence of 5() functions implies that a discrete remnant of translation invariance survives compactification: KK number conservation. To obtain 4-d chiral fermions, UED is compactified on an orbifold, and this introduces fixed points on which interactions that break KK number conservation could exist. Generically, KK number conservation is broken to a subgroup called KK parity [3] by brane-localized interactions that
Phenomenology
A
(0)
A
\AA7V
(fc)
wvv F(o/
i?(o)
F(0)
681
F<°\
j?(fc)
iP(O)
of Extra Dimensions
(a)
(b)
(c)
Figure 16. Example of interactions that are allowed [(a) and (b)] and are not allowed [(c)] by KK number (or KK parity) conservation.
can arise radiatively [70]. The size of the one-loop brane-localized corrections for UED have been explicitly calculated in Ref. [60], which we'll discuss more below. Nevertheless, KK parity remains unbroken so long as no explicit KK parity violating interactions are added to the orbifold fixed points. In other words, KK parity is technically natural, in that the symmetry structure is enhanced when coefficients of these bare would-be KK parity violating interactions are taken to zero. This is entirely analogous to i?-parity in supersymmetric models. KK parity can be written succinctly as PKK = ( — l) fc for the kth KK mode. This implies • The lightest level-one KK mode is stable. • Odd level KK modes can only be produced in pairs. • Direct couplings to even KK modes are occur through branelocalized, loop-suppressed interactions. We'll now discuss these implications for the phenomenology of the UED model. 4.6. Corrections
to electroweak
precision
observables
The typical problem with additional gauge bosons that couple to light fermions is that can give large contributions to electroweak precision observables. Consider the quintessential observable, the Z-width. Given measurements of Mz, GF, and aem(Mz), the Z width can be calculated at tree-level. New contributions to the width potentially arise from the exchange of heavier gauge bosons, but such contributions do not exist in UED models since KK parity forbids tree-level couplings of Z^ with the fermion zero modes as well as the Z^Z^H^H^ four-point coupling.
682
G. D. Kribs
Through one-loop interactions, however, there are calculable corrections 9 to the electroweak precision observables. Using the parameterization given by Peskin and Takeuchi [71], the contributions to S and T were calculated in Ref. [3]. They found T = ^Dj
(Tj + T* + TY)
(150)
3
where the sum is over all modes up to the cutoff scale of the D-dimensional theory, and Dj is the density of states at each level j . The individual contributions are 3
Ancos26w
aTh3 =
a
i
5m
l +
2
4ncos 6w mt
aT
6M? 7M\Al.
w
12M?
(152)
4
(153)
~ o 0 9^2 ' 3
2 2
v
8ir v Mf ' Using the experimental values for the SM parameters, the T parameter is roughly T=,0.76££^.
(154)
J
3
A similar calculation can be done for S, ScOM^Dj^
(155) 3
3
where the contribution to the isospin-breaking parameter T is two orders of magnitude larger than the isospin-preserving parameter S. There is no large contribution to S because the heavy KK quarks acquire their mass dominantly from the vector-like contribution arising from compactification. Note that the sum over states for these electroweak parameters is convergent log divergent power divergent
D = 5 for
D =6 D >6.
Contemplating UED models with D > 6 therefore appear somewhat problematic, since even the calculable contribution to the electroweak precision observables diverges. Distinguished from cutoff scale contributions, discussed below.
Phenomenology of Extra Dimensions
683
In any case, using the calculable corrections we can find a lower bound on the inverse radius of the extra dimensions. Ref. [3] required the moderately loose constraint T < 0.4 which leads to l_ > f 300 GeV R ~ \ 500 GeV
D = 5 D = 6.
. [
. '
These bounds are probably a bit too low given the latest electroweak fits [28], but in any case the bound is in the several hundred GeV range. These bounds on UED dimensions should be contrasted with those that result from extra dimensions that are not universal, i.e., SM gauge bosons living in higher dimensions with SM fermions localized to 4-d. For example, Ref. [72] found constraints on the inverse size of an extra dimension of this type is in the several TeV range. 4.7. UED cutoff
scale
We have alluded to the fact that UED models are effective theories of extra dimensions with a cutoff scale. What is the cutoff scale? Since gauge couplings in extra dimensional theories are dimensionful, a rough guess is 47T
A - ^ l /:n aD
(157)
where ao is the D-dimensional gauge coupling and I have been excessively naive about my NDA counting. Matching this D-dimensional gauge coupling to a 4-d coupling of the SM, 1
(irR)n
2 - - „ 2 -
(158)
9D
we obtain . „ A.R
4 f 30 for D = 5 77-^{,r,r r, n (159) v a 1 /" 1 10 for D = 6 ' which is roughly what is expected. A 4-d version of the same calculation, which is arguably better defined, sums over the number of KK particles running in loops to determine the scale of strong coupling. In this way of counting, since the number of KK modes is proportional to n2, we would expect Ae-dR — i/A 5 _ d i?, and thus A 6 _ d i? ~ 5. These are the typical numbers given for the cutoff scales of the 5-d and 6-d theories. Cutoff scales that are only about one order of magnitude above the compactification scale may be problematic in other ways. While proton decay and lepton number violating operators can be suppressed or eliminated in
684
G. D. Kribs
six dimensions, given a cutoff only a factor of 5 above the compactification scale one ought to be concerned about other higher dimensional operators that violate flavor symmetries or custodial SU(2). Also, the cutoff scale may be even lower than the above estimates suggest. Ref. [73] found that requiring scattering amplitudes satisfy the unitarity bound results in rather low estimates for the scale where strong coupling appears, only a small factor above the compactification scale. 4.8. UED in 5-d: The
spectrum
For the remainder of the discussion, I want to focus on the spectrum of the 5-d version of UED. Fortunately, the spectrum depends linearly on 1/R and only logarithmically on AR, which we will leave as a free parameter varied in some reasonable range. We will also ignore the effects of higher dimensional operators suppressed by the cutoff scale. What we will do is to examine more closely the spectrum of UED and the implications for collider searches and for the possibility of having a dark matter candidate. I'll briefly mention the ways in which a UED dark matter candidate could be detected, emphasizing the difference from a typical supersymmetric candidate. The spectrum of the 5-d UED model consists of all of the particles of the SM and their Kaluza-Klein excitations. Let's focus on the first KK level. At tree-level, the masses of the KK particles are simply ™\K
= #2 +
m
SM
(160)
where TTISM is the mass of level-zero (Standard Model) particle. This suggests a high degree of degeneracy for the KK excitations of light SM particles, but this degeneracy is not preserved beyond tree-level. Indeed, in Ref. [61] it was realized that radiative corrections to the KK particle masses are often much larger than the tree-level SM contribution. There are two classes of calculable radiative corrections. One arises from diagrams involving bulk loops, namely particles that traverse around the circle (or actually from one side to the other, in an orbifold). The loops are non-contractable, with finite extent in the extra dimension, implying they give finite corrections to the masses. The second class of corrections involves brane-localized kinetic terms that appear on the boundaries of the orbifold. These corrections are logarithmically sensitive to the cutoff scale of the theory. The generic form of this correction to the Lagrangian is
Phenomenology
of Extra Dimensions
685
5L=(S(y)+S(y-nR))^\n~ x \F+i$F+ + 5{d5F^)F+ + 5(F+{d5F„)]
(161)
where F+ and F- are the components of a bulk four-component spinor corresponding to any of the SM fermions. These corrections are necessarily logarithmically sensitive to the cutoff scale. The shifts in the KK masses resulting from these two classes of radiative corrections are [60]: 2
9>
l6n2R2 2
9
f-39C(3)
n2
7T2
3
\
2
/-5C(3)
167T2i?2 V 2 S(m2g(n)) (5(m Q( n))
7T2
167T2i?2 n („ 9 . 27 o 1 (6 53 2 + f 92 + \d'2) 167r2i? n
6{muln))
16ffa*
6(md(n))
16TT 2 #
5(mLM)
n /27 , . 9 l6ir2R \~8
<5(me(n))
15n2 In AR
1" Ail
(6*3 + 2 , 0 In A *
(egj + l-g'^ In AR
(f* 2+ H lnA *
H %9'2^AR 16ir2R 2
(162)
All of the non-colored KK excitation masses are within about 10% of m7(i) up to moderately high values of the cutoff scale (AR « 30). The strongly interacting particles are somewhat heavier, up to perhaps 20% — 25% above 1/.R. Fig. 17 gives an example of this spectrum for a compactification size relevant to upcoming collider experiments. I should emphasize that there are several assumptions built into this radiatively-corrected spectrum. One is that the matching contributions to the brane-localized kinetic terms are assumed to be zero when evaluated at the cutoff scale. This leads to a correction that should be compared against a log-enhanced correction. However, since the log is relatively small, of order log 30 ~ 3.4, the finite contribution could easily compete or dominate
686
G. D. Kribs
650
650
600 -
600
550 -
550
>
500 Figure 17. One-loop corrected mass spectrum of the first KK level for R AR = 20 and mh = 120 GeV. (Fig. 1 from Ref. [61].)
1
500
= 500 GeV,
over this correction. Also, the spectrum assumes that there are no branelocalized quadratically-divergent contributions to the Higgs (mass) 2 . Nevertheless, it is intriguing that the spectrum is so qualitatively similar to a moderately degenerate supersymmetric spectrum. The spin of the KK excitations is of course equal to the spin of the corresponding SM (zero mode) field, whereas superpartners have spin that differ by 1/2 from their SM counterparts. Unfortunately, measurements of the spin of newly discovered heavy particles at hadron colliders is not easy. This had led to suggestions that a KK spectrum could easily be mistaken for a degenerate supersymmetric spectrum [61]. Another similarity to supersymmetry is that UED possesses an exact auxiliary discrete symmetry that (if exact) forces pair production of the lightest level-one KK excitations and prevents the lightest level-one KK excitation from decaying into SM particles. The latter property implies that a stable particle exists in the spectrum, potentially a dark matter candidate.
Phenomenology
10l
•'
'
of Extra Dimensions
687
1
- (a)
AR=20 tree—level, any n
•
•
10- 1 \ \
S.
Ob
-
one-loop:
N •
\
10-2 -
\
\
^v
\
•
n=l -
\ \ \
5-
\\
\ 2 \3
\
-
^-^ ^ \
•
V
10-3 ',
^,
i
•
200
400
600 R"1 (GeV)
Figure 18. The effective Weinberg angle 6^ lightest level-n KK mode, yW = cosO^B^
800
1000
that determines the gauge content of the - s i n e ^ W 3 ^ ) . (Fig. 5 from Ref. [60].)
If the spectrum of the level-one KK excitations follows precisely that of Eqs. (162), then the lightest KK particle is the KK photon. Saying "KK photon" is somewhat misleading, however, since the neutral gauge boson mass matrix in the (B^n\ W3^) basis
R2
•
Sm2B(,
., + iff'V
b'92V
2
\g'92v2 W 4- dmiy<™>
hW
(163)
depends on both the tree-level contributions (proportional to v2) and the radiative corrections. The effective mixing angle, sin 2 0\j/ for the n t h mode is much smaller than the Weinberg angle, shown in Fig. 18. Clearly, once l/R > 500 GeV combined with AR ~ 20, 7 ( 1 ) ~ B ( 1 ) to within a percent, and so for all subsequent purposes we can consider them equivalent. From now on I'll just use B^ to refer to the lightest level-one KK excitation of the hypercharge gauge boson.
688
G. D. Kribs
B
/
VWVAJ
B^
t / B W V W
w Wl) (1
/
•
/
£?(D
(a) B
(i)
Figure 19. Relevant annihilation and scattering processes for (a) supersymmetry and (b),(c) UED. The supersymmetric annihilation diagram (a) is s-wave suppressed by a factor m'j/mB(i), whereas the UED diagram (b) is unsuppressed. Observable annihilation in the Sun occurs through diagram (b) with / = v. Annihilation in the galactic neighborhood to positrons occurs through diagram (b) with / = t. Scattering off nuclei occurs via diagram (c) with / = q, suitably "dressed" into a proton or neutron.
4.9. UED dark
matter
It is amusing that B^> happens to be the candidate for dark matter in UED models. The close analogy with supersymmetry would seem to continue here, since the supersymmetric partner to the hypercharge gauge boson, the Bino (B), is the typical candidate for supersymmetric dark matter. But, this is where the similarity ends. In supersymmetry, Bino annihilation typically proceeds through sfermion exchange, shown in Fig. 19(a). Since the Binos are Majorana fermions, Fermi statistics requires that they have their spins oppositely directed when prepared in an initial s-wave. This means that a chirality flip of the fermions is required, and thus a mass insertion in the diagram. This causes the cross section to be suppressed by a factor rni/m2-. In UED, B^ annihilation also proceeds through KK fermion annihilation shown in Fig. 19(b), but because the incoming states are bosons, there is no s-wave suppression. This means that the mass range for B^1' dark matter is significantly higher for UED dark matter as compared with Bino
Phenomenology
of Extra Dimensions
689
supersymmetric dark matter. We can estimate the thermally-averaged annihilation cross section by assuming that only diagrams of type (b) shown in Fig. 19 are present. Given the KK spectrum above, Eqs. (162), the radiative correction to level-one KK fermions is typically at the few to tens of percent level. A reasonable approximation is to assume that the mass of the exchanged KK particle f^ is roughly degenerate with B^. The cross section is then simply l/m^ ( 1 ) times the coupling factors, which is just g\ Ei(>1) 4 - The final result is
(av) = oJ59[
(164)
N
v ' 3247rm| (1) where this cross section with numerical factors was worked out in Ref. [64]. If you take a rough estimate of the relic density given in Kolb and Turner [74]
Qh2
„ 0.77 x 10-3
W
(av) and then plug in the thermally-averaged cross section above, one obtains
n*-0-1^)
(166)
which is accurate to within about 15% of the numerical results given in Ref. [64]. A plot of the relic density is shown in Fig. 20, along with several additional curves that represent including coannihilation with one to three generations of the level-one KK excitations of the right-handed leptons. An interesting outcome of Ref. [64]'s analysis is that coannihilation 10 with light KK leptons caused an increase in the effective annihilation cross section and thus a decrease in the mass range of the KK particle. This happens because the additional KK particles, when close enough in mass to B^\ have small annihilation and coannihilation cross sections, freeze out later, causing them to act as additional components to dark matter. These RH KK leptons decay into B^ after their mutual interactions are too slow compared with the expansion rate, and thus they decay into B^\ boosting the B^ relic density, or equivalently lowering the mass range of B^ when the relic density is held fixed. This result is, however, unique to right-handed KK leptons. As shown by two groups Ref. [76,77], coannihilation with left-handed KK leptons, KK See Ref. [75] for a nice general discussion of the effects of coannihilation.
690
G. D. Kribs
0.6
-
«
"/ >."• /
Overt" Insure Limit
*
-V
/ .* .• #** / / / »« / /r *,**~--^ * »c * *.» »_- »* ».# / /
0.4
/
a °-3 c
*,• --
.
•
»
»- / * •V
i
*:
•
/
/
.••
/
0.5
*
/
A /
-
X /
•s * /
: / / /
0.2 .-
.'
*.••
»
*
A
«••
• V n h 2 = 0.16 ± 0.04;
0.1
I-»«W*II
0.2
i
0.4
0.6
i
'
0.8
i
1 i i i 1 i i i 1
iii
I I !
1
1.6
1.8
1.2
1.4
mKK (TeV) Figure 20. Prediction for QBih2. The solid line is the case for B1 alone, and the dashed and dotted lines correspond to the case in which there are one (three) flavors of nearly degenerate e^. For each case, the black curves (upper of each pair) denote the case A = (rriNLKP — mLKp)/mLKP = 0.01 and the red curves (lower of each pair) A = 0.05. Note that the "favorable range" of Qh2 for dark matter is now out-of-date; W M A P suggests Qh2 ~ 0.1, just below the bottom of the shaded band. (Fig. 3 from Ref. [64].)
quarks, KK gluons, etc., all have cross sections that are larger than B^ annihilation, causing the total effective cross section to go up. Holding the relic density fixed, this implies the mass range of B^1' must also increase. If the KK quarks and KK gluon are below about 10% of the mass of B^\ these coannihilation effects can cause the mass range for B^1' to go up to the few TeV range. On face value, such a small separation between the mass of B^ and the strongly interacting level-one KK particles is not expected from the radiative corrections to the masses of the first KK level computed in [60]. However, if the cutoff scale is not much larger than the
Phenomenology
of Extra Dimensions
691
compactification scale, and thus matching corrections are comparable in size while opposite in sign to compensate, the level-one KK spectrum could be much more degenerate. 4.10. Direct detection
of KK dark
matter
KK dark matter can also be detected by the usual methods, namely direct detection by scattering off nuclei, as well as indirect detection through annihilation in the Sun to neutrinos or annihilation in our galactic neighborhood to positrons. First consider the direct detection of B^ dark matter. Dark matter particles are currently non-relativistic, with velocity v ~ 10~ 3 . For weak scale dark matter, the recoil energy from scattering off nuclei is far less than for scattering off electrons, and thus one need only consider elastic scattering off nucleons and nuclei. At the quark level, B^ scattering goes through KK quarks, such as shown in Fig. 19(c). The amplitudes and cross sections for the quark level processes are easy to calculate, but then these processes must be convoluted with structure functions for nucleons and nuclei. The interactions divide into spin-dependent and spin-independent parts [78]. Higgs exchange contributes to scalar couplings, while KK quark exchange contributes to both. In Refs. [79,80] both spin-independent and spin-dependent cross sections were calculated and shown in Fig. 21. This figure assumes all level-one KK quarks are degenerate with that is different from m R (i). Projected sensitivities of near future experiments are also shown in Fig. 21. For scattering off individual nucleons, scalar cross sections are suppressed relative to spin-dependent ones by ~ mp/mB{i). However, this effect is compensated in large nuclei where spin-independent rates are enhanced by ~ A2. In the case of bosonic KK dark matter, the latter effect dominates, and the spin-independent experiments have the best prospects for detection with sensitivity to mB(i) far above current limits. 4.11. Indirect
detection
of KK dark
matter
Weakly interacting dark matter particles are expected to become gravitationally trapped in large bodies, such as the Sun, and annihilate into neutrinos or other particles that decay into neutrinos. The calculation of the flux of neutrinos from WIMP annihilation in the Sun has been explored in some detail, particularly the case of neutralino dark matter (for reviews,
692
G. D. Kribs
0
200
400 mBi
600 800 (GeV)
1000
1200
Figure 21. Predicted spin-dependent proton cross sections (dark-shaded, blue), along with the projected sensitivity of a 100 kg NAIAD array; and predicted spin-independent proton cross sections (light-shaded, red), along with the current EDELWEISS sensitivity, and projected sensitivities of CDMS, GENIUS, and CRESST. The predictions are for rrih = 120 GeV and 0.01 < r = (m ? i — m B (i) )/»nB(i) < 0.5, with contours for specific intermediate r labeled. (Fig. 1 from Ref. [79].)
see Refs. [17,19]). The basic idea is to begin with the relatively well-known local dark matter density from the galactic rotation data, compute the interaction cross section of the WIMPs with nuclei in the Sun, compare the capture rate with the annihilation rate to determine if these processes are in equilibrium, and then compute the flux of neutrinos that result from this rate of WIMP capture and annihilation. There is a huge detector at the south pole that has instrumented a large area of antarctic ice by stringing detectors down deep holes. The first version of this experiment had an effective area of 0.1 km 2 (called AMANDA) that is now in progress towards expansion to 1 km 2 (called IceCube). The calculation of the annihilation rate in the Sun involves similar scattering processes as we found for direct detection, except that now the dominant process is simply scattering off protons. The new ingredient is to
Phenomenology of Extra Dimensions
693
determine when the capture rate equilibrates with the annihilation rate, which is determined by the mass of the (core of the) Sun, the dark matter density and (relative) velocity, as well as the microscopic scattering cross section. Since the WIMPs in UED are rather heavy, it is not surprising that the elastic scattering is not particularly large, and a detailed calculation [81] shows that B^ dark matter just barely comes into equilibrium after 4.5 billion years. This gives the maximal neutrino signal emitted from the Sun. The actual outgoing flux depends on the annihilation fraction directly into neutrinos, as well as indirectly through decays. Muon neutrinos are the main actors, since at the energies relevant to B^ annihilation, neutrino telescopes only observe muon tracks generated in charged-current interactions. In Ref. [81] the heavier level-one KK modes were approximated to have about the same mass, but this mass was taken to be slightly larger than mB(D by a fraction
typically about 0.1 - 0.2 given the spectrum from Eqs. (162). Using the neutrino energy spectrum, the event rate expected at an existing or future neutrino telescope can be calculated. This is shown in Fig. 22 for a detector with an effective area of 1 km . Combining the spectrum determined by the one-loop radiative corrections with a relic density appropriate for dark matter, the expectation is to get between a few to tens of events per year at the IceCube detector. Finally, there are speculations that KK dark matter annihilation in the galactic halo might account for the positron excess, see Ref. [82] for details. 5. Conclusions In these lectures I have showed how the phenomenology of extra dimensions is very rich, should Nature choose to following one or more of the ideas discussed in this review. I have tried to give a overview of what I perceive to be the main characteristic signals of the specific extra dimensional models that I discussed. Nevertheless, there are several related models and a host of other aspects to extra dimensions that I did not have the time or space to review. This remains a very active field of investigation with new ideas continually developed. How likely is any given extra dimensional proposal? This is not a question that has any scientific answer, even though physicists try hard to
694
G. D. Kribs
10^
1
1
I
I ""-I
1
1
I"
10*
CM
I
10 J -t->
Pi >
10 l
\0.3 10 - 1
i
•
400
_•_!_
•
600
800
HILKP
v
s
1000
1200
(GeV)
Figure 22. The number of events per year in a detector with effective area equal to one square kilometer. Contours are shown for r„i = 0.1, 0.2, and 0.3. The r„i = 0.3 is q
R
q
R
shown merely for comparison, since this mass ratio is larger than would be expected from the one-loop radiative correction calculations of the KK mode masses. The relic density of the B W ' s lies within the range £lB(i) h2 = 0.16 ± 0.04 for the solid sections of each line. Matching to the W M A P data, in which flg^h2 ~ 0.1 is preferred, corresponds roughly to the left-hand side dash-to-solid transition for each curve. (Fig. 2 from [81].)
quantify their qualitative instincts. This much can be said with relative certainty: In all cases the cutoff scale of the Standard Model is drastically lowered from the Planck scale to near the TeV scale. Since the lore of quantum gravity is that all global symmetries are broken by Planck scale effects, naturalness suggests cutoff-scale suppressed higher dimensional operators should appear at the 1/TeV level with order one coefficients. If this were true, all of these models would be ruled out immediately by B and L violating operators, operators leading to FCNC, and operators modifying precision electroweak observables. As model builders, we must conclude that either the cutoff scale (i.e., the quantum gravity scale) is larger or there are mechanisms to suppress or eliminate these dangerous effects. Some of
Phenomenology
of Extra Dimensions
695
these remarkably creative mechanisms were discussed or referenced in the preceding sections. As phenomenologists, however, we are blissfully free to assume t h a t these operators are suppressed by an unspecified mechanism or simply tuned to be small, and then we have the opportunity to probe the physics of these scenarios directly in colliders and indirectly in all sorts of ways from astrophysics to table-top experiments. During the lectures, I repeatedly emphasized t h a t while extra dimensions are interesting in themselves, the real take home lesson is to understand how to t u r n "great model A" into "predictions 1,2,3" and compare with "experiments X,Y,Z". E x t r a dimensions provide a fascinating and exciting set of models to illustrate precisely this important exercise. As we approach the next energy frontier with the LHC and future experiments, I hoped to instill some of the techniques to t u r n new physics ideas into a testable theory. Good luck developing your own great ideas and turning t h e m into calculable phenomenology!
Acknowledgments I t h a n k J o h n Terning, Carlos Wagner, and Dieter Zeppenfeld, the organizers of the 2004 "Physics in D > 4" TASI, for the invitation to present these lectures and for putting together a fabulous program. I particularly t h a n k J o h n Terning for persistent nagging emails which ensured the writeup of these lectures was finally completed. I am grateful to K. T. M a h a n t h a p p a for his hospitality. I also t h a n k the many TASI participants for their insightful questions and comments. This work was supported in part by the Department of Energy grant number DE-FG02-96ER40969.
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CSABA GSAKI
ELECTROWEAK SYMMETRY BREAKING FROM EXTRA DIMENSIONS
CSABA CSAKI, JAY HUBISZ AND PATRICK MEADE Institute for High Energy Phenomenology Newman Laboratory of Elementary Particle Physics Cornell University, Ithaca, NY 14853, USA
This is a pedagogical introduction into the possible uses and effects of extra dimensions in electroweak (TeV scale) physics, and in particular to models of electroweak symmetry breaking via boundary conditions ("higgsless models"). It is self contained: all the aspects of extra dimensional and electroweak physics used here are reviewed, before we apply these concepts to higgsless models. In the first lecture gauge theories in an extra dimension and on an interval are discussed. In the second lecture we describe the basic structure of higgsless models, while in the third lecture we discuss fermions in extra dimensions and the inclusion of fermions into higgsless models. The final lecture is devoted to the issue of electroweak precision observables in theories beyond the standard model and its applications to extra dimensional theories and in particular the higgsless models.
Contents
1
Introduction
704
2
Gauge theories in an extra dimension and on an interval 2.1 A scalar field on an interval 2.2 Pure gauge theories on an interval: gauge fixing and BC's 2.3 Gauge theories with boundary scalars 2.4 Orbifold or interval?
705 706 710 713 716
3
Higgsless m o d e l s of electroweak s y m m e t r y breaking 3.1 Large energy behavior of scattering amplitudes 3.2 Naive Higgsless toy model 3.3 Custodial SU(2) and flat space Higgsless model 3.4 The AdS/CFT correspondence 3.5 The warped space Higgsless model
720 721 725 726 731 735
703
704 C. Csdki, J. Hubisz & P. Meade Fermions in extra dimensions 4.1 Brief summary of fermions in D = 4 4.2 Fermions in a fiat extra dimension 4.3 Boundary conditions for fermions in 5D 4.4 Examples and a simple application 4.5 Fermions in warped space 4.6 Fermion masses in the Higgsless model
739 740 742 743 746 748 751
Electroweak precision observables for general BSM and extra dimensional theories 754 5.1 EW observables and the effective Lagrangian point of view . . . . 754 5.2 Electroweak precision and extra dimensions 762 5.3 Electroweak precision and Higgsless models 766 Conclusions
772
1. Introduction Theories with extra dimensions have again become very popular over the past ten years. The reason why is that it was realized that extra dimensions could actually play an active role in the physics of the TeV scales (rather than being irrelevant up to the Planck scale). The first proposal along these lines were theories with large extra dimensions [1], where one would explain the discrepancy of the weak and Planck scale via the presence of large extra dimensions diluting the strength of gravitational interactions. The second main wave of excitement was brought on by the Randall-Sundrum models [2,3], where it was understood that the geometry of the extra dimensions could actually lead to novel approaches to the hierarchy problem or even to 4D gravity. Many of the aspects of these models have been reviewed in a previous TASI lecture [4] (and other excellent introductions to these topics include lectures at this TASI given by Graham Kribs [5] and Raman Sundrum [6], and can also be found in [7]). The aim of these lectures is to emphasize less (compared to [4]) the gravitational aspects of extra dimensional theories, but rather the use of such models for electroweak physics. Some knowledge of the material in [4] could be useful for reading these notes, however we have attempted to present a self-contained series of lectures focusing on gauge theories in extra dimensions. In the first lecture we discuss generalities about extra dimensional gauge theories. We mostly focus on the issue of how to assign a consistent set of boundary conditions (BC's) after proper gauge fixing. We also comment on the relation between the orbifold and the interval approaches to describing an extra dimension with a boundary. In the second lecture we
Electroweak Symmetry
Breaking from Extra Dimensions
705
apply the tools from the first lecture towards building an extra dimensional model where electroweak symmetry is broken via BC's (rather than by a scalar higgs). Such models will be referred to as higgsless models, and will serve throughout these notes as the canonical example of applying the various concepts discussed. In order to find a (close to) realistic higgsless model we need to review the basics of the AdS/CFT correspondence for warped extra dimensions. The third lecture deals with fermions in extra dimensions. After a brief review of fermions in general we show how one generically introduces fermions into extra dimensional models and how theories with boundaries can render such models chiral. We then show how these tools can be applied to higgsless models to obtain a realistic fermion mass spectrum. Finally, we discuss the issue of how to calculate corrections to electroweak precision observables in theories beyond the standard model. We give a general effective field theory approach applicable to any model, and show how the Peskin-Takeuchi S,T,U formalism fits into it. We then show how these parameters can be generically calculated in an extra dimensional model, and evaluate it for the higgsless theory.
2. Gauge theories in an extra dimension and on an interval In this first lecture we will be discussing the structure of a gauge theory in a single extra dimension. For now we will be assuming that there is no non-trivial gravitational background, that is the extra dimension is flat. However we still need to discuss what the geometry of the extra dimension is. There are three distinct possibilities: • Totally infinite extra dimension like the other 3+1 dimensions • The extra dimension is a half-line, that is infinite in one direction • Finite extra dimension: this could either be a circle or an interval For now we will not be dealing with the very interesting possibility of a half-infinite extra dimension, which could be phenomenologically viable in the case of localized gravity (the so-called RS2 model [3]). In most interesting cases one has to deal with a compactified extra dimension. The first question we will be discussing in detail is how to define a field theory with an extra dimension classically. When the space is infinite, one usually requires that the fields vanish when the coordinates go to infinity, that is ip —> 0 as r —> oo
(2.1)
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However when the space is finite, one does not necessarily need to impose ip —> 0 as the boundary conditions (BC's). What will be the possible BC's then? The two possibilities as mentioned above for the finite extra dimensions are • circle (= an interval with the ends identified): in this case the boundary conditions for the fields are clear, ^(2irR) = *(0). • interval (= the ends are not identified) In either case, we can just start from the action principle and see what BC's one can impose that are consistent with the action principle. As the simplest example let us first discuss the case of a single scalar field in an extra dimension [8,9]. 2.1.
A scalar field on an
interval
We start with a bulk action for the scalar field P'KR
d4x I
'1
M, (2.2) dy, :d JM ) / where we have assumed that the interval runs between 0 and TTR. The coordinates are labelled by M = 0,1,2,3,5, while Greek letters /i, v,... will denote our usual four dimensions 0,1,2,3. We will also assume throughout these lectures that the signature of the metric is given by
/I
\ -1
9MN=
(2-3)
—1
-1
-1/
V
We will for simplicity first assume that there is no term added on the boundary of the interval. Let us apply the variational principle to this theory: SS-
/-f(
dM<j>dM5-
%S*
'
dy
-
(2.4)
Separating out the ordinary 4D coordinates from the fifth coordinate (and integrating by parts in the ordinary 4D coordinates, where we apply the usual requirements that the fields vanish for large distances) we get d4x /
dy -d^4>S
- ^-6<j> - dy4>dy54>
(2.5)
Electroweak Symmetry
Breaking from Extra Dimensions
707
Since we have not yet decided what boundary conditions one wants to impose we will have to keep the boundary terms when integrating by parts in the fifth coordinate y: -KR
SS=
f dAx r
-dMdMtt> - ? v
I dAxdv
(2.6)
To ensure that the variational principle is obeyed, we need 5S = 0, but since this consists of a bulk and a boundary piece we require: • The bulk equation of motion (EOM) <9M<9M> = - f ^ as usual • The boundary variation needs to also vanish. This implies that one needs to choose the BC such that dy4>5(j)\bound = 0.
(2.7)
We will be calling a boundary condition natural, if it is obtained by letting the boundary variation of the field 54>\bOUnd to be arbitrary. In this case the natural BC would be dy4> = 0 - a flat or Neumann BC. But at this stage this is not the only possibility: one could also satisfy (2.7) by imposing 6(j)\bound = 0 which would follow from the 4>\bound = 0 Dirichlet BC. Thus we get two possible BC's for a scalar field on an interval with no boundary terms: • Neumann BC dy4>\ = 0 • Dirichlet BC <j>\ = 0 However, we would only like to allow the natural boundary conditions in the theory since these are the ones that will not lead to explicit (hard) symmetry breaking once more complicated fields like gauge fields are allowed. Thus in order to still allow the Dirichlet BC one needs to reinterpret that as the natural BC for a theory with additional terms in the Lagrangian added on the boundary. The simplest possibility is to add a mass term to modify the Lagrangian as S = Sbuik - j dAx±Mf2\y=0 - jd4x^M^\y=1TR.
(2. 8)
These will give an additional contribution to the boundary variation of the action, which will now given by: SSbound = ~ ft4>(d+ Ml4>)\v=^R + f dAx5 (dy - Ml4>)\y=0.
(2.9)
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Thus the natural BC's will be given by dv(f> + Ml4> = 0 a t y = irR, dv4>- M?= Q a t y = 0.
(2.10)
Clearly, for Mi —» oo we will recover the Dirichlet BC's in the limit. This is the way we will always understand the Dirichlet BC's: we will interpret them as the case with infinitely large boundary induced mass terms for the fields. Let us now consider what happens in the case [10] when we add a kinetic term on the boundary (which we will also be calling branes throughout these lectures) for. This is a somewhat tricky question that had many people confused for a while. For simplicity let us set the mass parameters on the branes to zero, and take as the action
S = Sbulk + J dtx^d^tly^.
(2-11)
Note that the boundary term had to be added with a definite sign, that is we assume that the arbitrary mass parameter M is positive. This is in accordance with our expectations that kinetic terms have to have positive signs if one wants to avoid ghostlike states. For simplicity we have only added a kinetic term on one of the branes, but of course we could easily repeat the following analysis for the second brane as well. The boundary variation at y = 0 will be modified to
<55|0 = Jd4xS
(2.12)
Thus the natural BC will be given by: dy4> = ^ D 4 0 .
(2.13)
Using the bulk equation of motion (in the presence of no bulk potential) \3z4> = D40 — 4>" = 0 we could also write this BC as $' = jjwhere one usually assumes that the 4D modes <j>n have the x dependence
n(y)etPn'x, where p„ = m^ is the nth KK mass eigenvalue. Using this form the BC will be given by:
^=j^
= -§* = - ^ .
(2-14)
In either form this BC is quite peculiar: it depends on the actual mass eigenvalue in the final form, or involves second derivatives in the first form.
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Breaking from Extra Dimensions
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This could be dangerous, since from the theory of differential equations we know that usually BC's that only involve first derivatives are the ones that will automatically lead to a hermitian differential operator on an interval. The usual reason is that the second derivative operator d2/dy2 is hermitian if the scalar product rirR
(/,)=/
r(v)g(v)
(2-15)
Jo
obeys the relation
i l ) = (—
( / > X29^ ) = ([ - ^ / , 3 ) . dy ' ~ dy
(2-16)
These two terms can be easily transformed into each other using two integrations by parts up to two boundary terms:
(/
' ^9)
=
Jo* riy)~d¥9{y)
= C
=
~ C r'{y)9'{y) + v*M9'iv)]oR
r"(y)g(y) + [r{y)9\y)\ZR - [/*'\y)g{y)YoR•
(2.17)
Jo
Thus we can see that if the boundary condition for the functions on which we define this scalar product is of the form / ' | o , 7 r « = Oif\o,nR
(2-18)
then the two boundary terms will cancel each other and the operator d2 /dy2 is hermitian, and the desired properties (completeness, real eigenvalues) will automatically follow. However, the boundary condition / " = 1/Mf is not of this form, and the second derivative operator is naively not hermitian. This is indeed the case, however, one can choose a different definition of the scalar product on which the above boundary condition will nevertheless ensure the hermiticity of the second derivative operator. To find what this scalar product should be, let us try to prove the orthogonality of two distinct eigenfunctions of the second derivative operator. Let / and g be two eigenfunctions of the second derivative operator / " = Xff and g" — Xgg, then
f
Jo
nR
rg"dy = Xg r r(y)g(y)dy Jo
i
r"(y)9(y)dy + r'g\o-g'r\o.
(2.19)
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C. Csdki, J. Hubisz & P. Meade
Using the boundary condition / " = Mf that
(I
= A / / , so / ' = A / / M / we find -.
•nR
(A/ - A9) ( /
f(y)g(y)dy
+ — fg\0 I = 0.
(2.20)
What we find is the combination that is orthogonal, which means that this is the combination that one should call the scalar product in the presence of a non-trivial boundary kinetic term. Thus the scalar product in this case should be defined by
(f,g) = f
f(y)g(y)dy+^fg\0
(2.21)
This definition still satisfies all the properties that a scalar product should satisfy, and with this definition the second derivative operator will now be hermitian. This is easy to check:
(f,9") = f
fg"dy + ^fg"\o = -f
f'g'dy-fg'\0 + jjfg\o ( 2 - 22 )
= J f'gdy + f'g\0 - fg'\o + ^ffW
Due to the BC g" = Mg' the last two boundary terms cancel, and the first can be rewritten as jjf"g\o, and so the full expression really equals (/",)• So we have shown that there really is no problem with a theory with brane kinetic terms added. However, one needs to be careful when using a KK decomposition: the proper scalar product needs to be used when one is trying to use the orthogonality and the completeness of the wavefunctions. For example, the completeness relation will be given by
Y,9n{x)gn{y) = 5{x - y ) - jjS(x)J2gn(o)gn(y)2.2. Pure gauge theories BC's
on an interval:
gauge fixing
(2.23) and
Next we will consider a pure gauge theory in an extra dimension [9,11,12]. A gauge field in 5D AM contains a 4D gauge field A^ and a 4D scalar A5. The 4D vector will contain a whole KK tower of massive gauge bosons, however as we will see below the KK tower of the A5 will be eaten by the massive gauge fields and (except for a possible zero mode) will be nonphysical. That this is what happens can be guessed from the fact that the Lagrangian contains a mixing term between the gauge fields and the scalar,
Electroweak Symmetry
Breaking from Extra Dimensions
711
reminiscent of the usual 4D Higgs mechanism. The Lagrangian is given by the usual form
S = jd5x{-\FaMNFMNa)
= Jd5x(-l-F^F^a
- \Fa^F^a),
(2.24)
where the field strength is given by the usual expression FjfaN = 8M A% 3N AaM + g5fabc AbMAcN. gs is the 5D gauge coupling, which has mass dimension —1/2, thus the theory is non-renormalizable, so it has to be considered as a low-energy effective theory valid below a cutoff scale, that we will be calculating later on. To determine the gauge fixing term, let us consider the mixing term between the scalar and the gauge fields:
J dAX ^
a
dy - \F^F^
\quadrauc
= f d4x f
dy~ hd^
- d5Aa^)(d>"A5 a - d5A» a)
= IdAx f
dy- i(3MAg3M5 a + d5Ald5A» a - 2d5Aa^A5a).
(2.25)
Thus the mixing term that needs to be cancelled is given by rirR
/
dsAad»Aha.
(2.26)
Jo Integrating by parts we find /•TTR
- / Jo
WAldsAl + [d^AlY^.
(2.27)
The bulk term can be cancelled by adding a gauge fixing term of the form SGF = j # x £
_l(dMA^-£a5Ag)2.
(2.28)
This term is chosen such that the A5 independent piece agrees with the usual Lorentz gauge fixing term, and such that the cross term exactly cancels the mixing term from (2.27). Thus within R^ gauge, which is what we have defined, the propagator for the 4D gauge fields will be the usual ones. Varying the full action we then obtain the bulk equations of motion and
712
C. Csdki, J. Hubisz & P. Meade
the possible BC's. After integrating by parts we find: dSbulk
!*•£
(dMFMva
- dvdbAl)8Aav
- g5fabcFMubAcM
+
-dvd°Aaa
+ {d°F%5 - g5fabcF>5Ac°d5d„Aa°
-
^Al)5A% (2.29)
The bulk equations of motion will be that the coefficients of 6A° and 6A^ in the above equation vanish. We can see, that the Ag field has a term £<9f Ag in its equation. This will imply that if the wave function is not flat (e.g. the KK mode is not massless), then the field is not physical (since in the unitary gauge ^ - t o o this field will have an infinite effective 4D mass and decouples). This shows that as mentioned above, the scalar KK tower of A1 will be completely unphysical due to the 5D Higgs mechanism, except perhaps for a zero mode for A%. Whether or not there is a zero mode depends on the BC for the A§ field. We will see later how to interpret A$ zero modes. In order to eliminate the boundary mixing term in (2.27), we also need to add a boundary gauge fixing term with an a priori unrelated boundary gauge fixing coefficient Abound± &ow^)2|0,.fi,
(2.30)
where the — sign is for y = 0 and the + for y = nR. variations are then given by:
The boundary
-^r— Abound
I' dAx{d^a J
(05A»a +
-J—d^AntA-l^R Abound
+ (£d5Aa5
± ZboundAfiSAtlovR.
(2.31)
The natural boundary conditions in an arbitrary gauge £, Abound are given by d5A»a + - J — 3 „ 0 M " ° = 0, ZdhAl±ZboundAl=Q.
(2.32)
Abound
This simplifies quite a bit if we go to the unitary gauge on the boundary given by Abound —> 00. In this case we are left with the simple set of boundary conditions d6A>M = 0, Aab=Q.
(2.33)
This is the boundary condition that one usually imposed for gauge fields in the absence of any boundary terms. Note, that again we could have chosen
Electroweak Symmetry
Breaking from Extra Dimensions
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some non-natural boundary conditions, where instead of requiring that the boundary variation be arbitrary we would require the boundary variation itself (and thus some of the fields on the boundary) to be vanishing. It turns out that these boundary conditions would lead to a hard (explicit) breaking of gauge invariance, and thus we will not consider them in the following discussion any further. We will see below how these simple BC's will be modified if one adds scalar fields on the branes. 2.3. Gauge theories
with boundary
scalars
Let us now consider the case when scalar fields that develop VEV's are added on the boundary [9,12]. For simplicity we will be considering a U(l) theory, but it can be straightforwardly generalized to more complicated groups. The localized Lagrangians for the two complex scalar fields $ ; will be the usual ones for a Higgs field in 4D, and the subscripts i = 1,2 correspond to the two boundaries:
ID^I 2 -^!^! 2
A
k)2-
(2-34)
These boundary terms will induce non-vanishing VEV's and we parameterize the Higgs as usual as a physical Higgs and a Goldstone (pion): hi)ei*''Vi.
$i = -j=(vi +
(2.35)
We can now expand again the action to quadratic order in the fields to find the expression
Cw=
£
dy(-±F*l/ + ±(dIJA5)2-dtlA5d5A'*
+ \{d^f +
2
\(d,h2)
-Xivlhl
+
^d^x-v^f v=o
- \\2v\h\
+ \(d^2
-
2
v2A„)
(2.36) V=irR
Repeating the procedure in the previous section and integrating by parts, we find that the following bulk and boundary gauge fixing terms are necessary in order to eliminate all the mixing terms between the gauge field and the scalars As,^: CGF=
~ j *
dy{d»A" - d5A5)2
- -^(V^+^ITTI-AS))
2
y=0
lzr(d^+^2(v27r2
2
+ A5))
(2.37) y—irR
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C. Csdki, J. Hubisz & P. Meade
With this gauge fixing the gauge field will be decoupled from the rest of the fields, and its bulk action will be given by /
d5x-Au 2
l
M
,2 c>2\ » " >~ (dz - d$)rfv - (1 - -)d»dv Av.
(2.38)
The KK decomposition can then be obtained by writing the field as Afl(x,y)
= ell(p)eiP-xf(z),
(2.39)
where eM(p) is the polarization vector and f(z) is the wave function. For a given mode we assume that p2 = m2^. The equation of motion satisfied by the wave function will simply be (d2y+m2n)fn(y)=0,
(2.40)
which will be linear combinations of sine and cosine functions. The natural boundary condition (analogous to the previous case) will now be modified to dyA^
T
v2h2A^ = 0.
(2.41)
Thus we can see, that introducing a boundary scalar field will modify the BC (just like for the case of the simple bulk scalar). In the limit when vt —> oo we simply obtain a Dirichlet BC for the gauge field. In this limit of a Dirichlet boundary condition, the fields ftj,7Tj will clearly decouple from the gauge field, since they are non-vanishing only where the gauge field itself vanishes. Thus their effect will be to repel the wave function of the gauge field from the brane, and to make the gauge field massive. However, as we will see later even in the limit Vi —> oo the mass of the gauge field will not diverge, but rather it will be given by the radius of the extra dimension. Finally, let us consider what will happen to the scalar fields and their BC's. The physical Higgs hi does not have any mixing term with any of the other fields, so it will have its own equation of motion on the branes. Since its mass is determined by the parameter Aj which does not appear anywhere else in the theory, we could just make this scalar arbitrarily heavy and decouple it without influencing any of the other fields. Turning to the fields A5 and iti the bulk equation of motion of As will still be given by d2yA5 + ^A5
= 0.
(2.42)
Here m2 is the mass of a scalar state that could live in a combination of As and the 7r*'s. The boundary equation of motions for the 7Tj's will give a
Electroweak Symmetry
Breaking from Extra Dimensions
715
relation between these fields and the boundary values of A5:
(1^--vl\n1+v1A5\y=0 (j^-vi\Tr2
= 0,
+ V2A5\y=TTR = 0.
(2.43)
Finally, requiring that the boundary variation for arbitrary field variations still vanishes (combined with the above two equations) will give the BC's for the field A5: ^ 5 -
? m 2 / C i
AA + I
_ ^
rl1^
5
| , = 0 - 0 ,
a^5|y^fl
= 0.
(2.44)
We can see that when one of the VEV's vt is turned on, in the unitary gauge £ —> 00 the boundary condition for the A5 field will change from Dirichlet to Neumann BC: dyAs = 0. In the unitary gauge it is also clear that all the massive modes are again non-physical, since they will provide the longitudinal components for the massive KK tower of the gauge fields. However, now it may be possible for physical zero modes in the scalar fields to exist. Without boundary scalars, the BC for A$ is Dirichlet, and no nontrivial zero mode may exist. This basically means that there are just enough many modes in A5 to provide a longitudinal mode for every massive KK state in the gauge sector, but no more. When one adds additional scalars on the boundary, some combination of the A$'s and 7r's may remain uneaten. If we turn on the VEV for a scalar on both ends (in the non-ablian case for the same direction), then the A5 will obey a Neumann BC on both ends and there will be a physical zero mode. As we will see in the AdS/CFT interpretation this will correspond to a physical pseudo-Goldstone boson in the theory. In this case the wave function of the A5 is simply flat (which obeys both the bulk eom's and the BC's), and from (2.43) we find that the boundary scalars will be given by 7Tj = A^/vi. In the limit v^ —> 00 the boundary scalars will vanish as expected and decouple from the theory. If a direction is higgsed only on one of the boundaries (that is v\ ^ 0 but V2 — 0) then there will be no physical scalars in the spectrum. This is the situation we will be using most of the time in these lectures, thus in most cases we can simply set all scalar fields to zero and safely decouple them from the gauge fields in the v —> 00 limit.
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C. Csdki, J. Hubisz & P. Meade
2.4. Orbifold or
interval?
The more traditional way of introducing BC's in theories with extra dimensions is via the procedure [13] known as "orbifolding". By orbifolding we mean a set of identifications of a geometric manifold which will reduce the fundamental domain of the theory. In the case of a single extra dimension, the most general orbifolding can be described as follows. Let us first start with an infinite extra dimension, an infinite line R, parametrized by y, —oo < y < oo. One can obtain a circle S1 from the line by the identification y —> y + 2irR, which we is usually referred to as modding out the infinite line by the translation T, M. —> S1 = R/T. This way we obtain the circle. Another discrete symmetry that we could use to mod out the line is a Z2 reflection which takes y —> —y. Clearly, under this reflection the line is mapped to the half-line R1 —> R1 jZi- If we apply both discrete projections at the same time, we get the orbifold S1 /Z2. This orbifold is nothing else but the line segment between 0 and TTR. Let us now see how the fields tp{y) that are defined on the original infinite line K will behave under these projections, that is what kind of BC's they will obey. The fields at the identified points have to be equal, except if there is a (global or local) symmetry of the Lagrangian. In that case, the fields at the identified points don't have to be exactly equal, but merely equal up to a symmetry transformation, since in that case the fields are still physically equal. Thus, under translations and reflection the fields behave as T(2wR)
(2.45) (2.46)
where T and Z are matrices in the field space corresponding to some symmetry transformation of the action. This means that we have made the field identifications
(2.47) (2.48)
Again, Z and T have to be symmetries of the action. However, Z and T are not completely arbitrary, but they have to satisfy a consistency condition. We can easily find what this consistency condition is by considering an arbitrary point at location y within the fundamental domain 0 and 2nR, apply first a reflection around 0, 2(0), and then a translation by 2irR, which
Electroweak Symmetry
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717
will take y to 2-KR — y. However, there is another way of connecting these two points using the translations and the reflections: we can first translate y backwards by 2TVR, which takes y —> y — 2TTR, and then reflect around y = 0, which will also take the point into 2TTR - y. This means that the translation and reflection satisfy the relation: T(2TTR)Z(0)
= 2(0)r- 1 (2 7 ri?).
(2.49)
When implemented on the fields ip this means that we need to have the relation TZ = ZT~1,or
ZTZ = T~1
(2.50)
which is the consistency condition that the field transformations Z and T have to satisfy. As we have seen, the reflection Z is a Z2 symmetry, and so Z2 = 1. T is not a Z2 transformation, so T 2 ^ 1. However, for non-trivial T's T ^ l (T is sometimes called the Scherk-Schwarz-twist) one can always introduce a combination of T and Z which together act like another Z2 reflection. We can take the combined transformation T(2TTR)Z(0). This combined transformation takes any point y into 2TTR — y. That means, that it is actually a reflection around nR, since if y = nR — x, then the combined transformation takes it to nR + x, so x —> —x. So this is a Zi reflection. And using the consistency condition (2.50) we see that for the combined field transformation Z' = TZ Z'2 = (TZ)2 = (TZ^ZT-1)
= 1,
(2.51)
so indeed the action of the transformation on the fields is also acting as another Z^ symmetry. Thus we have seen that the description of a generic S1 /Z2 orbifold with non-trivial Scherk-Schwarz twists can be given as two non-trivial Z^ reflections Z and Z', one which acts around y = 0 and the other around y — irR. These two reflections do not necessarily commute with each other. A simple geometric picture to visualize the two reflections is to extend the domain to a circle of circumference 4TTR, with the two reflections acting around y = 0,2TTR for Z and TTR^TTR for Z'. One can either use this picture with the fields living over the full circle, or just living on the fundamental domain between y = 0 and 2irR. The two pictures are equivalent. So what we find is that in the case of an S1 jZ-i orbifold a field tp(y) will live on the fundamental domain 0 < y < TTR, and will either have positive
718
C. Csdki, J. Hubisz & P. Meade
2?rR
3TIR
Figure 1. The action of the two Z2 reflections in the extended circle picture. The fundamental domain of the S1 /Z2 orbifold is just the interval between 0 and TTR, and the theory can be equivalently formulated on this line segment as well.
or negative parities under the two Z2 symmetries, which means that it will either have Dirichlet or Neumann BC's at the two boundaries f\o,7vR
= 0 Or dy(p\0,irR
= 0.
(2.52)
Let us assume that the Z2 's are a subgroup of the original symmetries of the theory*, thus in the case of gauge symmetry breaking they need to be a subgroup of the gauge symmetries. Since it is a Z^ subgroup it is necessarily abelian, so it is a subgroup of the Cartan subalgebra. This means that the symmetry with which we are orbifolding commutes at least with the Cartan subalgebra (since it is a subgroup of the Cartan subalgebra itself), so it will never reduce the rank of the gauge group. This would imply that the interval approach would give more general BC's than the orbifolds. However, one can of course in addition consider orbifolds with fields localized at the fixed points. With these you can also reproduce the more general BC's that arise as natural ones from the interval approach. However, these are more awkward to deal with since one has to often deal with fields that have discontinuities (jumps) at the fixed points. So while the two approaches are nominally equivalent, the interval approach is by far easier to deal with when the BC's are complicated. The interval approach "The other possibility would be to use a discrete symmetry of the generators that can not be expressed as the action of another generator known as an outer automorphism. There are few examples of such orbifolds which can indeed reduce the rank but will not be considered in these lectures.
Electroweak Symmetry
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719
is also more advantageous since it allows for a dynamical explanation of the BC's. One usual application [14,15] of orbifold theories for example is to break an SU(5) GUT symmetry to the SM subgroup SU(3)xSU(2)xU(l), without a Higgs field, and thus avoiding the doublet-triplet splitting problem. The doublet-triplet splitting problem is the question of why in an SU(5) GUT theory (usually a supersymmetric one) the Higgs doublets are light (of the order of the weak scale) while triplet that is necessary to make it a complete SU(5) multiplet has to be heavy (of the order of the GUT scale) in order to obtain unification of couplings (and to avoid proton decay in SUSY models). In an orbifold theory one can simply assume that the gauge fields (which form an adjoint 24 of SU(5)) have the following Zi parities at one of the boundary: / a
A
—>
/
\
+
V-
a
A
—>
J
\
u
+
(2.53)
-/
If we take the point of view that the orbifold is the fundamental object, then one never has to mention the doublet triplet splitting problem. However a different starting point could be an SU(5) theory on an interval, with some boundary conditions (that are caused by some dynamics of the fields on the boundary). To obtain the BC's used in the orbifold picture one could for example consider the SU(5) theory with an adjoint scalar on the boundary, which has a VEV /!
\ (2.54)
<£> = «
V
9 /
and then the limit v —> oo will give the orbifold BC's from (2.53) for the gauge fields. In order to also solve the doublet triplet splitting problem in this picture, one needs to require that the Higgs (which is a 5 of SU(5)) has Zi parity at the same boundary: / (2.55)
H = 5
\T,
720
C. Csaki, J. Hubisz & P. Meade
If this is also coming from some dynamics in the interval picture, then one would have to assume the presence of a boundary Lagrangian H*Y,H + mH*H,
(2.56)
where in order to obtain the above parities one needs to assume that —3/2v + m = 0. This condition is equivalent to the usual fine tuning solution in supersymmetric GUTs. So from the interval point of view the doublet-triplet splitting problem would mainifest itself in the following way: the 3-2-1 invariant boundary conditions for the Higgs field are dyH3 + m3H3 = 0, dyH2 + m2H2 = 0.
(2.57) 14
In order to obtain doublet-triplet splitting we need m2/mz ~ 10~ , and thus the set of BCs solving the doublet-triplet splitting problem is equivalent to the usual fine-tuning problem of SUSY GUTs. Thus in this picture the doublet-triplet splitting problem would not really be resolved, but rather just hidden behind the question of what dynamics will cause these fields to have the above BC's. Whether or not the doublet-triplet splitting problem is really resolved then depends on how the extra dimension is really emerging: if some string theory compactification naturally yields the S1/Z2 orbifold as its vacuum state with the necessary parities then the doublet-triplet splitting problem would be indeed resolved. If however the BC's are due to some boundary dynamics as discussed above the problem would reemerge. Thus one can not really decide which interpretation is the right one purely from the low-energy effective theory, but some knowledge of the UV theory would be necessary. 3. Higgsless models of electroweak symmetry breaking We have shown above how to find the BC's for a general gauge theory in an extra dimension. We would like to use now this knowledge to construct a model of electroweak symmetry breaking, where the electroweak symmetry is broken by boundary conditions, and without the presence of a physical scalar Higgs boson in the theory. First we want to show how the presence of extra dimensions can postpone the unitarity violation scale in a theory with massive gauge bosons but without a Higgs scalar [9,16]. Then we will show how to find the simplest model with a massive W and Z bosons without a scalar Higgs from an extra dimensional model with a flat extra dimension. We will see that in this model the prediction for the ratio of the W/Z mass is far from the SM value, which is due to the absence of a custodial SU(2) symmetry protecting this ratio [17]. We will show that
Electroweak Symmetry
-Ufa)
Breaking from Extra Dimensions
721
(") r-J
(« Figure 2. Elastic scattering of longitudinal modes of KK gauge bosons, n + n —> n + n, with the gauge index structure a + b —+ c + d.
such a global symmetry indeed predicts the right W/Z mass ratio and then following the suggestion of [17] use the AdS/CFT correspondence to build an extra dimensional model incorporating custodial SU(2) [18,19]. For other aspects of higgsless models see [20-42]. 3.1. Large energy behavior of scattering
amplitudes
Our aim is to build a higgsless model of electroweak symmetry breaking using BC breaking in extra dimensions. However, usually there is a problem in theories with massive gauge bosons without a higgs scalar: the scattering amplitude of longitudinal gauge bosons will grow with the energy and violate unitarity at a low scale. What we would like to first understand is what happens to this unitarity bound in a theory with extra dimensions. For simplicity we will be focusing on the elastic scattering of the longitudinal modes of the nth KK mode (see fig. 2). The kinematics of this process are determined by the longitudinal polarization vectors and the incoming and outgoing momenta:
_ M M
p™ =
{
M'M
^p_ \p\> (E,0,0,±y/E*=M*)
p° u t = {E,±y/E*
- M2.sin0,O,±V£ 2 - A^costf).
(3.1)
The diagrams that can contribute to this scattering amplitude in a theory with massive gauge bosons (but no scalar Higgs) are given in Fig. 3 (where the ^-dependence can be estimated from e ~ E,p^ ~ E and a propagator ~ E~2). This way we find that the amplitude could grow as quickly as E4, and then for E » Mw can expand the amplitude in decreasing powers of
722
C. Csdki, J. Hubisz & P. Meade
W
W
X w
w
Figure 3. The tree-level diagrams contributing to the scattering of massive longitudinal gauge bosons in the SM without a Higgs.
a -, (n)
(n)
, c
a -. (n)
•*x JK b' n-- )/ \
(-P- d
P
+
\.
° T_>)
+
iw |J
r (n)
d
(nKo c -J?
P \,
p
(n)<-
s channel exchange
+
b
, c
/"*
6 ^ (n)
contact interaction a -, (n)
(n)
p-^L
(") s
c
A
(*)i,r P" p T
T->5~
6 ^(n)
(n)1 d
t channel exchange
7? (n) t rf
u channel exchange
Figure 4. The four diagrams contributing at tree level to the elastic scattering amplitude of the nth KK mode.
E as A = A^ ^ - + A^ ^ + A(°> + O ( *£2 Mi Ml VE
(3.2)
In the SM (and any theory where the gauge kinetic terms form the gauge invariant combination F 2 ) the A^ term automatically vanishes, while A^ is only cancelled after taking the Higgs exchange diagrams into account. In the case of a theory with an extra dimension with BC breaking of the gauge symmetry there are no Higgs exchange diagrams, however one needs to sum up the exchanges of all KK modes, as in Fig. 4. As a result we will find the following expression for the terms in the amplitudes that grow with energy: A^
=
i (gLnn-Ystink)
+2(3 - cos2 e)facefbde)
( / ^ / ^ ( S + 6 COS B - COS2 0)
,
(3.3)
Electroweak Symmetry
Breaking from Extra Dimensions
723
In order for the term A^ to vanish it is enough to ensure that the following sum rule among the coupling of the various KK modes is satisfied: "nnnn
/ j9nnk'
V' /
Assuming A^ = 0 we get
X-3$>LfcMj * r
k abe cde fcde 2 $ feabe f ).
.f - s i n '
(3.5)
Here gnnnn is the quartic self-coupling of the nth massive gauge field, while gnnk is the cubic coupling between the KK modes. In theories with extra dimensions these are of course related to the extra dimensional wave functions fn(y) of the various modes as 9mnk
=
9mnki =95
95jdyfm(y)fn(y)fk(y), dyfm(y)fn(y)fk(y)fi(y).
(3.6)
The most important point about the amplitudes in (3.3-3.5) is that they only depend on an overall kinematic factor multiplied by an overall expression of the couplings. Assuming that the relation (3.4) holds we can find a sum rule that ensures the vanishing of the A^ term: 9nnnnMl = -J29nnkMl k
(3.7)
Amazingly, higher dimensional gauge invariance will ensure that both of these sum rules are satisfied as long as the breaking of the gauge symmetry is spontaneous. For example, it is easy to show the first sum rule via the completeness of the wave functions fn{y)'-
[* dytiiy) = J2 [R(iy Cdz fi(y)ffc)fk(y)fk(z), (3-8) Jo Jo k Jo and using the completeness relation ^My)fk(z)
= 8(y-z),
(3.9)
k
we can see that the two sides will indeed agree. One can similarly show that the second sum rule will also be satisfied if the boundary conditions
724
C. Csdki, J. Hubisz & P. Meade
are natural ones (as defined in Section 2) and all terms in the Lagrangian (including boundary terms) are gauge invariant. What we see from the above analysis is that in any gauge invariant extra dimensional theory the terms in the amplitude that grow with the energy will cancel. However, this will not automatically mean that the theory itself is unitary. The reason is that there are two additional worries: even if A^ and A^ vanish A^ could be too large and spoil unitarity. This is what happens in the SM if the Higgs mass is too large. In the extra dimensional case what this would mean is that the extra KK modes would make the scattering amplitude flatten out to a constant value. However if the KK modes themselves are too heavy then this flattening out will happen too late when the amplitude already violates unitarity. The other issue is that in a theory with extra dimensions there are infinitely many KK modes and thus as the scattering energy grows one should not only worry about the elastic channel, but the ever growing number of possible inelastic final states. The full analysis taking into account both effects has been performed in [34], where it was shown that after taking into account the opening up of the inelastic channels the scattering amplitude will grow linearly with energy, and will always violate unitarity at some energy scale. This is a consequence of the intrinsic non-renormalizability of the higher dimensional gauge theory. It was found in [34] that the unitarity violation scale due to the linear growth of the scattering amplitude is equal (up to a small numerical factor of order 2 — 4) to the cutoff scale of the 5D theory obtained from naive dimensional analysis (NDA). This cutoff scale can be estimated in the following way. The one-loop amplitude in 5D is proportional to the 5D loop factor - ^ -3
(3.10)
24TT '
The dimensionless quantity obtained from this loop factor is (3.11)
24TT3'
where E is the scattering energy. The cutoff scale can be obtained by calculating the energy scale at which this loop factor will become order one (that is the scale at which the loop and tree-level contributions become comparable). Prom this we get ANDA
=
2
-^--
(3-12)
Electroweak Symmetry
Breaking from Extra Dimensions
725
We can express this scale using the matching of the higher dimensional and the lower dimensional gauge couplings. In the simplest theories this is usually given by g25=nRg2,
(3.13)
where TtR is the length of the interval, and g\ is the effective 4D gauge coupling. So the final expression of the cutoff scale can be given as 24TT2
ANDA
= -2K.
(3.14)
g4n We will see that in the Higgsless models \/R will be replaced by M^/MKK, where Mw is the physical W mass, and MKK is the mass of the first KK mode beyond the W. Thus the cutoff scale will indeed be lower if the mass of the KK mode used for unitarization is higher. However, this AJVDA could be significantly higher than the cutoff scale in the SM without a Higgs, which is around 1.8 TeV. We will come back to a more detailed discussion of h-NDA in higgsless models at the end of this section. 3.2. Naive
Higgsless
toy
model
Now that we have convinced ourselves that one can use KK gauge bosons to delay the unitarity violation scale basically up to the cutoff scale of the higher dimensional gauge theory, we should start looking for a model that actually has these properties and resembles the SM. It should have a massless photon, a massive charged gauge boson to be identified with the W and a somewhat heavier neutral gauge boson to be identified with the Z. Most importantly, we need to have the correct SM mass ratio (at tree-level) M2 ^w=cos2ew Af|
n2 = ^ 2- - , 2, g + g'
(3.15)
where g is the S U ( 2 ) L gauge coupling and g' the U(l)y gauge coupling of the SM. We would like to use BC's to achieve this. This seems to be very hard at first sight, since we need to somehow get a theory where the masses of the KK modes are related to the gauge couplings. Usually the KK masses are simply integer or half-integer multiples of 1/R. For example, if we look at a very naive toy model with an SU(2) gauge group in the bulk, we could consider the following BC's for the various gauge directions: dyAl = 0 at y dyA]f 2
=
0,TTR,
= 0 at y = 0
A]; = 0a,ty = nR.
(3.16)
726
C. Csdki, J. Hubisz & P. Meade
With these BC's the wave functions for the various gauge fields will be for A3 ny J3 \y) = cos R (3.17) while3 for the 1,2 directions Am Jl,2
\y) = cos
(2m + l)y -~— 2R (3.18)
The mass spectrum then is A3 -*•
TOn
=
1 2
n
°>R k
'I'"'
m,+
1
_3_ (3.19) '"" ~ i? ' 2R'2R' This spectrum somewhat resembles that of the SM in the sense that there is a massless gauge boson that can be identified with the 7, a pair of charged massive gauge bosons that can be identified with the W±, and a massive neutral gauge boson that can be identified with the Z. However, we can see that the mass ratio of the W and Z is given by Mz = 2, (3.20) Mw and another problem is that the first KK mode of the W,Z is given by ay Al>2^7
77,
m
Mz> Mw n (3.21) Mz Mw Thus, besides getting the totally wrong W/Z mass ratio there would also be additional KK states at masses of order 250 GeV, which is phenomenologically unacceptable. We will see that both of these problems can be resolved by going to a warped higgsless model with custodial SU(2). 3.3. Custodial
SU(2)
and flat space Higgsless
model
We have seen above that a major question in building a realistic higgsless model is how to ensure that the W/Z mass ratio agrees with the tree-level result. Let us first understand why the tree result in the SM is given by M2 r
2
cos 6wMl
The electro weak symmetry in the SM is broken by the Higgs scalar H, which transforms as a 2 i under SU(2) L x U(l)y. The Higgs potential is given by V(H) = -n2H^H
+ X(H^H)2.
(3.23)
Electroweak Symmetry
Breaking from Extra Dimensions
111
This potential is only a function of H^H, which can also be written as H*H = hl + hl + hl + hi
(3.24)
where the Higgs doublet has been written in terms of its real and imaginary components as H=
(hl+ifl2 \h3 + ih4
(3.25)
We can see from (3.24) that the Higgs potential actually has a bigger global symmetry than S U ( 2 ) L X U(l)y: it is invariant under the full SO(4) rotation of the four independent real fields in the Higgs doublet. The SO(4) group is actually not a simple group, but rather equivalent to S U ( 2 ) L X S U ( 2 ) K . The origin of the S U ( 2 ) H symmetry can also be understood as follows. A doublet of SU(2) is a pseudo-real representation, which means that the complex conjugate of the doublet is equivalent to the doublet itself. The way this manifests itself is if we consider the doublet to be a field with a lower SU(2) index Hi, then the complex conjugate would automatically have an upper index. However, using the SU(2) epsilon extras can be lowered again, and so Hi and i€ij(H*y transform in the same way. This means that in addition to an SU(2)x, acting on the usual SU(2) index, there is another SU(2) symmetry that mixes H with eH*. To make this more intuitive, we could write a 2 by 2 matrix as (HeH*),
(3.26)
and then the ordinary S U ( 2 ) L would act from the left, and the additional S U ( 2 ) H would be a global symmetry acting on this matrix from the right. Once the Higgs scalar gets a VEV, it will break the SO (4) global symmetry to an SO(3) subgroup. In the SU(2) language this means that S U ( 2 ) L X S U ( 2 ) . R is broken to the diagonal subgroup SU(2)r>. This is most easily seen from the representation in (3.26) since then we have a matrix whose VEV is given by diag(n,w), and obviously leaves the diagonal subgroup unbroken. The claim is that once such an S\](2)D subgroup (which is usually referred to as the custodial SU(2) symmetry of the Higgs potential) is left unbroken during electroweak symmetry breaking, it is guaranteed that the p-parameter will come out to be one at tree level. Let us quickly prove that this is indeed the case. The generic description of the global symmetry breaking pattern S U ( 2 ) L X S U ( 2 ) H —> SU(2)£> can be achieved via the non-linear a-model which will describe the physics of the 3 Goldstonemodes appearing in this symmetry breaking. In this description the possible
728
C. Csdki, J. Hubisz & P. Meade
massive Higgs modes are integrated out. This model will give all the consequences of the global symmetries. In this model the Goldstone fields are represented by a 2 by 2 unitary matrix S, which is given in terms of the Goldstone modes na as S = ei^,
(3.27)
and transforms under SU(2).£,xSU(2)fl as £ —> UiEXJ^. One can think of the SU(2)i,x U(l)y electroweak symmetries as a subgroup of SU(2)iX SU(2) K , with U(l)y C SU(2)i*. This gauging of a subgroup of the global symmetries will explicitly break some of the global symmetries, but this is easily incorporated into the non-linear cr-model description. (Note, that in the presence of fermions the global symmetry needs to be enlarged to SU(2) L x SU(2) fl x U ( 1 ) B - L , and then U(l)y C SU(2) f l xU(l) B _ L , where U ( 1 ) B - L is the baryon number minus lepton number symmetry.) The covariant derivative is then given by D» = d„ - ig^-A"
- i9-B„.
(3.28)
The leading kinetic term for the pions is given by / 2 Tr(D, 1 E t )(£>''E).
(3.29)
Expanding this expression in the pion fluctuations will result in mass terms for the gauge fields M^ = ^f,
Mi=(g2+f2)/2,
(3.30)
and thus M2W _ g2 M*z-g*+g»-
(3 31)
-
Note, that in the above derivation the only information that has been used was the global symmetry breaking pattern SU(2)iXSU(2)/j —> SU(2)£>, with the appropriate subgroup being gauged. Once this symmetry breaking pattern is established, it is guaranteed that the tree-level prediction for the p-parameter will be 1. From the above discussion it is clear that in order to find a higgsless model with the correct W/Z mass ratio one needs to find an extra dimensional model that has the custodial SU(2) symmetry incorporated [17]. Once such a construction is found, the gauge boson mass ratio will automatically be the right one. Therefore we need to somehow involve SU(2)fl in the construction. The simplest possibility is to put an
Electroweak Symmetry
Breaking from Extra Dimensions
729
entire SU(2)LxSU(2) f i xU(l)s-i gauge group in the bulk of an extra dimension [9]. In order to mimic the symmetry breaking pattern in the SM most closely, we assume that on one of the branes the symmetry breaking is SU(2) L xSU(2) K -> SU(2) D , with U(1)B-L unbroken. On the other boundary one needs to reduce the bulk gauge symmetry to that of the SM, and thus have a symmetry breaking pattern SU(2)flxU(l)s-L —> U(l)y, which is illustrated in Fig. 5. Z=7lR
z=0
custodial SU(2) broken
SU(2) L XSU(2^XU(1) B _ L
custodial SU(2) obeyed .
SU(2)RX W\-Z
Figure 5.
U(DY
SU(2)LX SU(2) r
SU(2^
The symmetry breaking structure of the flat space higgsless toy model.
We denote by A^a, Aj^a and BM the gauge bosons of SU{2)R, SU(2)L and U(1)B-L respectively; g$L and g^n are the the gauge couplings of the two 517(2)'s and g$, the gauge coupling of the U(1)B-LI n order to obtain the desired BC's as discussed above we need to follow the procedure laid out in the first lecture. We assume that there is a boundary Higgs on the left brane in the representation ( l , 2 ) i under S U ( 2 ) L X S U ( 2 ) H X U ( 1 ) B _ L , which will break S\J(2)RX\J(1)B_L to U(l)y. We could also use the more conventional triplet representation under S U ( 2 ) K which will allow us to get neutrino masses later on. On the right brane we assume that there is a bi-doublet higgs in the representation (2, 2)o which breaks the electroweak symmetry as in the SM: SU(2) L xSU(2) f i -+SU(2) D . We will then take all the Higgs VEV's to infinity in order to decouple the boundary scalars from the theory, and impose the natural boundary conditions as described in the first lecture. The BC's we will arrive at then are: at y = nR
'dz(gBRBll
+ g!iA*3) = 0 , C V 1 ^ = 0 ,
(3.32)
. 95#M - gsRAf at y — 0 :
.4f'2=0,
= 0,
dz(g5RAf[a + g5LA*a) = 0, 0, Q*hALa 95RA£ Ra dzBn
0.
(3.33)
730 C. Csdki, J. Hubisz & P. Meade The BC's for the A5 and B5 components will be the opposite of the 4D gauge fields as usual, i.e. all Dirichlet conditions should be replaced by Neumann and vice versa. The next step to determine the mass spectrum is to find the right KK decomposition of this model. First of all, none of the A5 and B5 components have a flat BC on both ends. This means that there will be no zero mode in these fields, and as we have seen all the massive scalars are unphysical, since they are just gauge artifacts (supplying the longitudinal components of the massive KK towers). So we will not need to discuss the modes in these fields. The main point to observe about the KK decomposition of the gauge fields is that the BC's will mix up the states in the various components. This will imply that a single 4D mode will live in several different 5D fields. Since in the bulk there is no mixing, and we are discussing at the moment a flat 5D background, the wave functions will be of the form fk(z) oc a cos iV/fc z -\- b sin Ad^ z. If we make the simplifying assumption that g$L = 95R, then the KK decomposition will be somewhat simpler than the most generic one, and given by (we denote by A^'R± the linear combinations -j^(AL'R1 ^iAL'R2)):
B„(x,y)
55 ao7M (*) + 9 E hk fc=i
C0
<MkV)
Z^ (x),
(3.34)
00
Ajl3(x,y) A*3(x,y) = ^CfeCosfM^O, - irR)) Wjf^ix),
(3.37)
k=l 00
A^(x,y)
= J2 c* <xya(M?(y + TTR)) W^tf)
•
(3-38)
fc=i The coefficients and the masses are then determined by imposing the BC's on this KK decomposition. The resulting mass spectrum that we find is the following. The spectrum is made up of a massless photon, the gauge boson associated with the unbroken U(1)Q symmetry, and some towers of massive charged and neutral gauge bosons, W^ and Z^ respectively. The masses of the W ± ' s are given by Mf = ^ i ,
k = 1,2...
(3.39)
Electroweak Symmetry
Breaking from Extra Dimensions
731
while for Z's there are two towers of neutral gauge bosons with masses
Aff = ( _ M 0 + - £ ) , fc = l , 2 . . .
(3.41)
where M 0 = ^ arctan ^ / l + 2g2/g%. Note that l/(4i?) < M 0 < l/(2i?) and thus the Z"s are heavier than the Z's (Aff > Aff). We also get that the lightest Z is heavier than the lightest W (Aff > Af^), in agreement with the SM spectrum. However, the mass ratio of W/Z is given by
M\
(3.42)
16
and hence the p parameter is H
M& M\ cos2 9W
1.10 .
(3.43)
Thus the mass ratio is close to the SM value, however the ten percent deviation is still huge compared to the experimental precision. The reason for this deviation is that while the bulk and the right brane are symmetric under custodial SU(2), the left brane is not, and the KK wave functions do have a significant component around the left brane, which will give rise to the large deviation from p = 1. Thus one needs to find a way of making sure that the KK modes of the gauge fields do not very much feel that left brane, but are repelled from there, and only the lightest (almost zero modes) 7, Z, W± will have a large overlap with the left brane. 3.4. The AdS/CFT
correspondence
We have seen above that one would need to modify the flat space setup such that the KK modes get pushed away from the left brane without breaking any new symmetries. There are two possible ways that this can be done (and in fact we will see that these two are basically equivalent to each other). One possibility is to simply add a large brane kinetic term for the gauge fields on the left brane where custodial SU(2) is violated [20]. The effect of this will be exactly to push away the heavy KK modes (since as we have seen in the first lecture the BC does depend on the eigenvalue). This way the custodial SU(2) can be approximately restored in the KK sector of the theory. The second possibility which we will be pursuing here is to use the AdS/CFT correspondence. This has been first pointed out in [17].
732
C. Csdki, J. Hubisz & P. Meade
It has been realized by Maldacena [43] in 1997 that certain string theories on an anti-de Sitter (AdS) background are actually equivalent to some 4D conformal field theories. The crucial ingredient was that the conformal group SO(2,4) is equivalent to the isometries of 5D AdS space, whose metric is given by ds2 = f^\
{y^dx^dx" - dz2^j
(3.44)
where 0 < z < oo is the radial AdS coordinate. Basically the point is that besides the usual 5D Poincare transformations this metric has an additional rescaling invariance z —> az,:r M —> ax^. Using several checks Maldacena was able to gather convincing evidence that J\f = 4 supersymmetric Yang Mills theory in a certain limit is equivalent to type IIB string theory on AdSs x S5. This field theory is automatically conformal due to the number of supercharges. One crucial observation is that the coordinate along the AdS direction actually corresponds to an energy scale in the CFT. This is quite clear from the above mentioned rescaling invariance. Rescaling z implies rescaling x. But rescaling x means changing the energy scale in the CFT. So this will imply that the z —* 0 region corresponds to the most energetic sector of the CFT E —> oo, while the z —* oo corresponds to low energies E —> 0. The other important observation of this equivalence is that the field theory has a large global symmetry S U ( 4 ) H . The way this is realized in the string theory side is that SO(6)=SU(4) is the isometry of the 5D sphere S5. Thus there will be massless gauge boson corresponding to this SU(4) global symmetry in the AdSs theory. What this seems to suggest is that the AdSs bulk is really supplying the modes of the CFT itself, while the global symmetries of the CFT will manifest themselves in gauge fields appearing in the 5D theory. One final crucial ingredient needed for us is how this correspondence will be modified if the AdS space is not infinite, but we are considering only a finite interval (slice) of 5D AdS space. In this case we clearly do not have the full conformal invariance, since the appearance of the boundaries of the slice of AdS will explicitly break it. One way of interpreting the appearance of a boundary close to z = 0 (usually called the UV brane or Planck brane since it corresponds to high energies) is that the field theory has an explicit cutoff corresponding to this energy scale. If the cutoff is at z = R, the field theory will have a UV cutoff A = -jj. The interpretation of the other boundary (usually referred to as IR brane or TeV brane) is trickier. It has been argued in [44,45] that the proper interpretation of such an IR brane is that the CFT spontaneously
Electroweak Symmetry
Breaking from Extra Dimensions
733
breaks the conformal invariance at low energies. The location of the IR brane will supply an IR cutoff. For those familiar with the phenomenology of the Randall-Sundrum models, this can be explained by realizing that the KK spectrum of the fields will be localized on this IR brane. In the presence of the IR brane there will also be a discrete spectrum for these KK modes. These discrete KK modes can be thought of as the composites (bound states) formed by the CFT after it breaks conformality and becomes strongly interacting (confining). The analogy could be a theory that is very slowly running (its /?-function is very close to zero), then after a long period of slow running ("walking") the theory will become strongly interacting, and the theory will confine (if it is QCD-like) and form bound states. What happens then to the gauge fields if they are in the bulk of a finite slice of the AdS space? It depends on their BC's. The main difference between the full AdS and the case of a slice is that while a gauge field zero mode is not normalizable in an infinite AdS space, it will become normalizable in the case of a slice. So this means that if there is a zero mode present, then one would need to identify this as a weakly gauged global symmetry of the CFT. Whether the gauge field in the finite slice actually has a zero mode or not, depends on its BC's. If it has Dirichlet BC on the UV brane, then the zero mode will pick up a mass of the order of the scale at the UV brane (that is proportional to the UV cutoff), so it will be totally eliminated from the theory. Thus in this case even in a finite slice the symmetry should be thought of as a global symmetry only. However, if the BC on the UV brane is Neumann, while on the IR brane is Dirichlet, then the zero mode picks up a mass of the order of the IR cutoff (the confinement scale, the KK scale of the other resonances), so the way this should be interpreted is that the CFT became strongly interacting, and that breaking of conformality also resulted in breaking the weakly gauged global symmetry spontaneously. Later on it was also realized that perhaps supersymmetry may not be necessary for such a correspondence to exist. So let us summarize the rules laid out above for the AdS/CFT correspondence (at least the ones relevant for model building) in Table 1. Using the rules of the correspondence found in Table 1 we can now relatively easily find the theory that we are after. We want a theory that has an SU(2)L x SU(2)R x U(1)B-L global symmetry, with the SU(2)L x U(1)Y subgroup weakly gauged, and broken by BC's on the IR brane. To have the full global symmetry, we need to take SU(2)L x SU(2)R x U(1)B-L in the bulk of AdSs. To make sure that we do not get unwanted gauge fields at low energies, we need to break SU(2)R X U(1)B-L to U(l)y on
734
C. Csdki, J. Hubisz & P. Meade Table 1. Relevant rules for model building using the A d S / C F T correspondence. Bulk of AdS Coordinate (z) along AdS Appearance of UV brane
CFT Energy scale in C F T C F T has a cutoff conformal symmetry broken spontaneously by C F T composites of C F T Elementary fields coupled to C F T C F T has a global symmetry Global symmetry not gauged
Appearance of IR brane KK modes localized on IR brane Modes on the UV brane Gauge fields in bulk Bulk gauge symmetry broken on UV brane Bulk gauge symmetry unbroken on UV brane Higgs on IR brane
Global symmetry weakly gauged C F T becoming strong produces composite Higgs Strong dynamics that breaks C F T also breaks gauge symmetry
Bulk gauge symmetry broken on IR brane by BC's
the UV brane, which we will do by BC's as in the flat case. Finally, the boundary conditions on the TeV brane break SU(2)i x SU(2)R to SU(2)D, thus providing for electroweak symmetry breaking. This setup is illustrated in Fig. 6. Note, that it is practically identical to the flat space toy model considered before, except that the theory is in AdS space.
SU(2) L X SU(2}x U(1) B L
AdS.
SU(2)RX U d t -
Figure 6.
U(1)Y
SU(2)LX S U ( 2 ) r SU(2^
The symmetry breaking structure of the warped higgsless model.
Electroweak Symmetry
3.5. The warped space Higgsless
Breaking from Extra Dimensions
735
model
Before coming to the detailed prediction of the mass spectrum of the warped space higgsless model outlined above, let us first briefly discuss how to deal with a gauge theory in an AdS background [46,47]. We will be considering a 5D gauge theory in the fixed gravitational background v (ri^dx^dx" dzl)2} ds2 = ( - ) (n, - dz iV dx^dx
(3.45)
where z is on the interval [R, R'}. We will not be considering gravitational fluctuations, that is we are assuming that the Planck scale is sent to infinity, while the background is frozen to be the one given above. In RS-type models R is typically ~ l/Mpi and R' ~ TeV - 1 . For higgsless models we will see later on what the optimal choice for these scales are. The action for a gauge theory on a fixed background will be given by d5x^j
TPa
TPa
^ MP
NQ
(3.46)
Putting in the metric (3.45) we find
w; ?
S
lR
lpa
2
4 >"
2
(3.47)
^
To get the right gauge fixing term, we have to repeat the procedure from the first section. The mixing term between Ap, and As is given by f dAx f
rR'
r
D
= - f d4x f
dz-d^A"
rR'
dzd^d5
' (-A " it 5 ) •
(3.48)
In the second equality we have integrated by parts and neglected a boundary term (which is necessary for determining the BC's, however these will not change due to the presence of the warping so the BC's derived for the flat space model will be applicable here as well). This implies that the gauge fixing term necessary in the warped case is given by
/^r
dz—:~ (3.49) 2$ z Due to the chosen BC's the Ac, fields will have no zero modes they will all again become massive gauge artifacts and can be eliminated in the unitary gauge. The quadratic piece of the action for the gauge fields will be then given by S
9f
wr
A
R
1
dz
A
Au z 2
M
^-4*(T*
#
-s«
R
d2
V"
1
d^d" Av. (3.50)
736
C. Csdki, J. Hubisz & P. Meade
As before, we go to 4D momentum space by writing A,j.(x,z) = e ti(p)f(z)eip'x. The equation of motion for the wave function f(z) will then become (p2 = M2): -M2
- zdb ( -zdb
f(z) = 0.
(3.51)
Equivalently it can be written as 1 / " - - / ' + M 2 / = 0. z
(3.52)
This will lead to a Bessel equation for g(z) after the substitution f{z) = zg(z): g" + -g' + (M2 - -^)g = 0,
(3.53)
which is a Bessel equation of order 1. The solution is of the form f(z) = z (AJr (qkz) + BK2 (qkz)).
(3.54)
The BC's corresponding to the symmetry breaking pattern discussed above for the warped higgsless model are identical to the ones for the flat space case [18]: f d^gsnBp + § 5 < 3 ) = 0 dzAL^ at z = R : < \gsBv-gBRA«3 = 0,
= 0, Af- 2 = 0, (3.55)
Again the BC's for the As's are the opposite to that of the corresponding combination of the AD gauge fields, and these BC's can be thought of as arising from Higgses on each brane in the large VEV limit. Using the above Bessel functions the KK mode expansion is given by the solutions to this equation which are of the form ^lA\z)
= z(akA)J1{qkz)
+ b(kA)Y1(qkz))
,
(3.57)
where A labels the corresponding gauge boson. Due to the mixing of the various gauge groups, the KK decomposition is slightly complicated but it
Electroweak Symmetry
Breaking from Extra Dimensions
737
is obtained by simply enforcing the BC's: (again assuming g^L = 9SR) oo
B^x,z) = gsao-y^x) + ^2^\z)ZJje\x),
(3.58)
fc=i oo
+ Y,il>lL3)(z)ZlJ')(x),
A™(x,z)=gsa
(3-59)
fc=i oo
A™ (x,z) =~g5aolll(x)+ Y,4m\z)4k)(x)>
( 3 " 6 °)
fc = l OO
AJ1±{x,z)^y£4L±\z)W^±{x),
(3.61)
fc = l OO
( 3 - 62 )
±
A2 (X,Z) = Y,4R±\Z)W{JC)±(*)fc=i
Here 7(x) is the 4D photon, which has a flat wavefunction due to the unbroken U(1)Q symmetry, and Wp, (x) and Zp \x) are the KK towers of the massive W and Z gauge bosons, the lowest of which are supposed to correspond to the observed W and Z. To leading order in 1/R and for log (R'/R) 3> 1, the lightest solution for this equation for the mass of the W±,s is
"""•i&zm-
(363)
Note, that this result does not depend on the 5D gauge coupling, but only on the scales R, R'. Taking R = 1(T 19 GeV" 1 will fix R' = 2 • 10" 3 GeV" 1 . The lowest mass of the Z tower is approximately given by M
g\ + 2g§
1_
2 Z = gl ! 2 +, gl -J ni?' ^„,R^( 3 - 64 ) log(f)' If the SM fermions are localized on the Planck brane then the leading order expression for the effective 4D couplings will be given by (see Section 5 for more details)
i 2
9
R
^ (£) 9B
* = M*(*)(a + g)-
9 thus the 4D Weinberg angle will be given by
<3 65)
'
738
C. Csdki, J. Hubisz & P. Meade
We can see that to leading order the SM expression for the W/Z mass ratio is reproduced in this theory as expected. In fact the full structure of the SM coupling is reproduced at leading order in l/log(R'/R), which implies that at the leading log level there is no S-parameter either. An S'-parameter in this language would have manifested itself in an overall shift of the coupling of the Z compared to its SM value evaluated from the W and 7 couplings, which are absent at this order of approximation. The corrections to the SM relations will appear in the next order of the log expansion. Since log [ ^ - j ~ O(10), this correction could still be too large to match the precision electroweak data. We will be discussing the issue of electroweak precision observables in the last lecture. The KK masses of the W (and the Z bosons as well due to custodial SU(2) symmetry) will be given approximately by mWn = ^(n+-)
— ,n=l,2,....
(3.67)
We can see that the ratio between the physical W mass and the first KK mode is given by raw m' ww
°
4 3n
1
(3.68)
°" \/Mf)'
We can see that warping will achieve two desirable properties: it will enforce custodial SU(2) and thus automatically generate the correct W/Z mass ratio, but it will also push up the masses of the KK resonances of the W and Z. This will imply that we can get a theory where the W', Z' bosons are not so light that they would already be excluded by the LEP or the Tevatron experiments. Finally, we can return to the issue of perturbative unitarity in these models. In the flat space case we have seen that the scale of unitarity violation is basically given by the NDA cutoff scale (3.12). However, in a warped extra dimension all scales will be dependent on the location along the extra dimension, so the lowest cutoff scale that one has is at the IR brane given by AN DA
24TI-3 R
a"-57.
(3-69)
Using our expressions for the 4D couplings and the W and W masses above we can see that [34,40] ANDA
~ -JM^
•
(3 70)
-
Electroweak Symmetry
Breaking from Extra Dimensions
739
From the formula above, it is clear that the heavier the resonance, the lower the scale where perturbative unitarity is violated. This also gives a rough estimate, valid up to a numerical coefficient, of the actual scale of non-perturbative physics. An explicit calculation of the scattering amplitude, including inelastic channels, shows that this is indeed the case and the numerical factor is found to be roughly 1/4 [34]. Since the ratio of the W to the first KK mode mass squared is of order M2 - ^ -
= 0(l/log(R!/R))
,
(3.71)
raising the value of R (corresponding to lowering the 5D UV scale) will significantly increase the NDA cutoff. With R chosen to be the inverse Planck scale, the first KK resonance appears around 1.2 TeV, but for larger values of R this scale can be safely reduced down below a TeV.
4. Fermions in extra dimensions Since the early 1980's, there was a well known issue with allowing fermions to propagate in the bulk of an extra dimension. This problem arises from the spin-1/2 representations of the Lorentz group in higher dimensions. The principle issue is that the irreducible representations of the Lorentz group in higher dimensional spaces are not necessarily chiral from the 4D point of view. This means that a low energy effective theory derived from this higher dimensional theory would not, in general, contain chiral fermions. However, the SM does contain chiral fermions, and so models with bulk fermions appeared to be doomed. It was realized in the 80's, however, that it is possible to obtain chiral fermions in models with extra dimensions via orbifolding. Our first objective will be to outline how this is possible. We cover cases where the background geometry of the extra dimension is either flat or warped [48-52], and give some explicit examples of interesting models with bulk fermions [53]. The standard orbifold method of producing chiral modes is generalized to include arbitrary fermionic boundary conditions. In Higgsless models, the boundary condition techinique is used to generate the entire spectrum of SM fermion masses through boundary conditions [21]. We also discuss a simpler model of obtaining fermions masses through localization methods. It is interesting to note that extra dimensions have provided an alternative framework to possibly resolve the flavor hierarchy problem of the standard model.
740
C. Csdki, J. Hubisz & P. Meade
4.1. Brief summary
of fermions
in D — 4
Before we begin our discussions of fermions in higher dimensions, we first review the basic properties of 4D fermions [54]. We follow the spinor conventions given in [55]. We use the chiral representation for the Dirac 7 matrices: 7 " = ( ; / ; )
-
d
7
5
=(;_°.)
,=0,1,2,3,
(4.1)
where
(4.2)
where the A matrices are such that they leave the Minkowski inner products invariant: ^g'ttvy"/
= x»9lM,y"
(4.3)
Spinors are a different type of representation of the Lorentz group. To start of the discussion of spin-1/2 representations, we note that in 4 dimensions, the 2-dimensional complex special linear group, 5L(2,C), can be shown to be a covering space for the Lorentz group. This equivalence is similar to the mapping of the special unitary group, SU(2), onto the rotation group 50(3). To see this more explicitly, consider the following parametrization of a Lorentz four-vector xM x" -* [x] = x°-
IX
xiai = IX
x° + x',3
(4.4)
where [x] has the following properties: [x] = [z] t and det[x] = x^xv'g^
= x^x^.
(4.5)
Now take an arbitrary matrix A e SL(2, C). Such a matrix is a general 2 x 2 complex matrix with unit determinant. Under a rotation by A, [x] -
[x]A = A[x)tf
(4.6)
Finally note that [X]A = [X\A> an< ^ t n a t detfa;]^ = det[x]. These all correspond precisely to the properties of the inner product under a general Lorentz transformation. Thus, for some A^, [X]A = [AAZ].
(4.7)
Thus the mapping A —> A^ is a homomorphism of the Lorentz group.
Electroweak Symmetry
Breaking from Extra Dimensions
741
The group SL(2, C) is isomorphic to the product group SU(2) x SU(2), with the transformation parameters for the £[/(2)'s being complex, but related by complex conjugation. The imaginary component of the transformation parameters is associated with the non-compact directions of the Lorentz group (the boosts) while the rotations are associated with the real part of these parameters. Because of this isomorphism, we can express representations of SX(2,C) in terms of their breakdown under the 517(2) subgroups. The two SU(2) indices are represented as dotted and un-dotted. The two most simple (non-trivial) irreducible representations of SL(2,C) can then be written as xa a n d '4,a- To introduce a notation, these are the (1/2,0) and (0,1/2) representations of the Lorentz group, respectively. These are the familiar left and right handed Weyl spinors. Any representation of the Lorentz group can then be written in terms of the transformation laws under the two complexified SU(2) subgroups. Such a representation is labelled (m, n) where m is the number of un-dotted indices that the representation has, while n is the number of dotted indices. As a side note, we mention that if we require that a physical theory be invariant under a parity transformation, then we require that it contain special combinations of fundamental representations. Parity exchanges left and right handedness, or in terms of the labelling (m, n) of a representation, exchanges m, and n. For a theory to be invariant under parity, it must contain representations in the form of direct sums (m,n)+(n,m). In the case of Weyl spinors, the lowest representation that is invariant under parity is the (l/2,0)+(0,1/2). This representation is the familiar Dirac spinor. In the index notation that we have discussed, the rules for the indices are quite simple. Complex conjugation, exchanges dotted and undotted indices. The metrics on the dotted and un-dotted spaces which raise and lower indices are given by the anti-symmetric tensors eap and e&g. In terms of the SL(2,C) notation, the a matrices exchange representations between the dotted and undotted spaces:
d
(4.8)
This follows from the transformation properties of \x\. With these conventions, spinors can be combined into invariants which have the following property: Xa^Peal3
= XaTpa = ipaXa-
(4-9)
There are two minus signs that cancel each other in switching the ordering of the two spinors. One is from changing the order of the Grassman vari-
742
C. Csdki, J. Hubisz & P. Meade
ables making up the spinors, and the other is from permuting the indices in the totally anti-symmetric tensor, tap. Because the spinor sums can be interchanged in this way, in the proceeding sections we will frequently drop the spinor indices completely: xa'tPa = X^P- Note, however, that Xa4>a = - X Q ^ Q -
4.2. Fermions
in a flat extra
dimension
In 5D, the Clifford algebra includes, in addition to the four dimensional Dirac algebra, a 7 5 . This 7 5 is precisely the parity transformation that we discussed in the previous section. This means that in 5D the simplest irreducible represention will break up under the 4D subgroup of the full 5D Poincare algebra as a (0, l / 2 ) + ( l / 2 , 0 ) . That is, the simplest irreducible representation in 5D is a Dirac spinor, rather than a Weyl spinor. This is expressing the fact that bulk fermions are not chiral, as mentioned in the introduction to this lecture. It is not possible to start only with 2 component spinors, as can be done in 4D theories. As a warmup to working in more general compactified spaces, we consider the minimal 5D Lagrangian for a bulk spinor field which is propagating in a flat extra dimension with the topology of an interval::
= /**(j(*:r M a M *-5 M *r A 1 *)-m** i.
(4.io)
The field \I/ decomposes under the 4D Lorentz subgroup into two Weyl spinors:
* =
TJ
•
(4-H)
In finding the consistent boundary conditions for these Weyl fermions, it is useful to express the Lagrangian (4.10) in terms of the 4D Weyl spinors: 5 = Jd5x
{-ixa^d^x
~ ilH^drf
+ \ (V%X - xA^P) + m(V>x + x4>)) ,
(4.12) <—> —> <— where <95 = 85 — 85. We note that in 4D theories, the terms with the left acting derivatives are generally integrated by parts, so that all derivatives act to the right. However, since we are working here in a compact space with boundaries, the integration by parts produces boundary terms which can not be neglected.
Electroweak Symmetry
Breaking from Extra Dimensions
743
The bulk equations of motion for the 4D Weyl spinors which result from the variation of this 5D Lagrangian are: -ia^dpX -ia^d^ 4.3. Boundary
conditions
~ dsi> + mi> = 0, + d5X + mX = 0. for fermions
(4.13)
in 5D
Our goal now is to find what the possible consistent boundary conditions are. We consider a consistent boundary condition to be one which satisfies the action principle. Naively, one might think that there are two independent spinors, \ a n d V>, and that one would require two independent boundary conditions for each spinor. However, because the bulk equations of motion are only first order, there is only one integration constant. So for the Dirac pair, (x, 4>), there is only one boundary condition f{\, ip) = 0 at each boundary, where / is some function of the spinors and their conjugates. The form of / together with the bulk equations of motion in Eq. (4.13) then determines all of the arbitrary coefficients in the complete solution to the spinor equation of motion on the interval. We would now like to see what the restrictions are on the function / , so now let us examine the variations that include the derivatives acting along the extra dimension: ss =
/ \ (5^X
+ ^5x
~ S*^
~* ^ )
( 4 - 14 )
To get the boundary equations of motion, we need to integrate by parts so that there are no derivatives acting on the variations of the fields left over. However, this procedure results in residual boundary terms given by ^bound
=
J
d5x
r_^, x
+
^
x
+
6
^
_ ^
L
( 4 1 5 )
The most general boundary conditions which satisfy the action principle then are given by the solutions to -61>X + # X + 6x$ ~ xH = 0.
(4.16)
As a simple example, consider the case when the spinors are proportional to each other on the boundaries: tp = aX-
(4.17)
The variations of the spinors are then related by Sip = a5x
(4.18)
744
C. Csdki, J. Hubisz & P. Meade
Plugging these relations into (4.16), we find that it simplifies to aSxX ~
OL8\X
= 0,
(4.19)
and the variation of the action on the boundaries does indeed vanish. This is only one of many boundary conditions which satisfy the action principle. The most general solution consistent with the Lorentz symmetries of the interval is given by V>„ = M%xp + Na^-
(4.20)
We have put the Weyl indices back in to show the structure of the operators M and N. These operators can contain derivatives along the extra dimension. This most general set of boundary conditions is further restricted by additional symmetries such as gauge symmetries that are allowed on the boundaries. For example, if a fermion is transforming under a complex representation of a gauge group, then the operator M@ must vanish. This is because the spinors x a n d V* transform under conjugate representations, thus the spinors cannot consistently be proportional to each other. If the fields are in real representations of the gauge group, such as the adjoint, such boundary conditions are allowed. Let us consider a simple boundary condition: take the spinor tp, and set it equal to zero on both boundaries. The resulting boundary condition for the other Weyl spinor x, which comes from the bulk equation of motion, is (d5+m)X\o,L
= 0.
(4.21)
Solving the equations of motion with these boundary conditions results in a zero mode for x-, but not for if>. That is, the low energy theory is a chiral theory, which has been obtained from an inherently non-chiral 5D theory. By appropriately choosing the boundary conditions, one can get a chiral effective theory from a geometry which would naively not allow chiral modes. It is useful to have in mind a physical picture which could result in this type of boundary condition. For this purpose, we can consider an infinite extra dimension where there is a finite interval where the bulk Dirac mass is vanishing, but outside of which the mass is either positive and infinite, or negative and infinite. Then a constant mass m is added. This is shown pictorially in Figure 7. In the case where the sign of the mass is opposite on either end of the interval, after solving the bulk equations in the entire bulk space, we get the boundary condition above at the points y — 0 and
Electroweak Symmetry
y=0 •
Breaking from Extra Dimensions
745
y=L m = oo
i
m=0
m — -oo Figure 7. A fermion mass profile in an infinite extra dimension which leads to a 4 dimensional chiral zero mode. The resulting zero mode wave function is flat and finite in the interval 0 < y < L. The wave function for this zero mode is vanishing at all other points in the extra dimension.
y = L that resulted in a zero mode for the Weyl spinor, \. This mass profile is a discretized version of the boundary wall localized chiral fermion approach [56,57]. In the case where the Dirac mass is the same sign on either end of the interval 0 < y < L, the fermions are again localized, however the boundary conditions are x\y=o — 0 and ip\y=L — 0. In this case, no zero mode results, and the lowest lying KK resonance has a mass of order 1/L. To show that the orbifold picture does not easily give all possibilities, let us consider this same example in the orbifold language. In the orbifold setup, the boundary conditions are determined by imposing Z2 parity (y —> —y) symmetry on the spinors, where the I/J spinor is odd, and the x spinor is even. The parity transformation of one spinor is then determined by the other, since the action contains terms of the form ipd$x- If V' i s 0( id under the Z2 then x must be even, since 85 —> —85 under the parity transform. The situation becomes complicated, however, if one wants to give a bulk Dirac mass to the 5D fermion: Mtjjx- This term is not allowed under the Z2 symmetry unless the bulk mass term is given a transformation law under the Zi as well. This means that the bulk mass term must undergo a discrete jump at the orbifold fixed points [58,59] (those points that are stationary under the parity transformation). In the interval picture, there are no such issues. The vanishing of the boundary and bulk action variation give the requirements that ip = 0, and (85 + m)x = 0 at the endpoints.
746
C. Csdki, J. Hubisz & P. Meade
4.4. Examples
and a simple
application
We begin this section with a discussion of the KK-decomposition of the 5D fermion fields, which we then apply to an application which provides an interesting solution to the fermion mass hierarchy problem, the Kaplan-Tait model [53]. This approach utilized the boundary wall fermion localization method [56]. As with gauge and scalar fields, there will be, in the 4D effective theory, a tower of massive Dirac fields that arise from solving the full 5D spectrum. These fermions will obey the 4D Dirac equation, which, when broken into Weyl spinors, is given by: -za^x(n) + m n
n
^ =0
{n)
-U7"3^< > + mnX
=0
(4.22)
The 5D spinors \ a n d ip can be written as a sum of products of the 4D Dirac fermions with 5D wavefunctions: X = Y,9n(y)Xn(x),
(4.23)
n
^ = £/n(i/)$n(aO.
(4.24)
n
Substituting this decomposition into the 5D bulk equations of motions gives the following g'n + m9n-mnfn fn-mfn
= 0,
(4.25)
+ mngn = 0.
(4.26)
The standard approach to solving this system of equations is to combine the two first order equations into two second order wave equations: & + (m2n - m2)gn = 0, f» + (ml-m*)fn = 0.
(4.27) (4.28)
The solution is simply a sum of sines and cosines, with coefficients that are determined by reimposing the first order equations, and imposing the boundary conditions. In the Kaplan-Tait model, there is a Higgs field which is confined to one boundary of an extra dimension, and there are gauge fields which are propagating in the bulk. Assign the bulk fermions the boundary conditions where ip\o,L = 0, and (d<j + m)x\o,L = 0. The main question concerns the
Electroweak Symmetry
Breaking from Extra Dimensions
747
zero mode solutions. Take the first order equations (4.26), and set the 4D mass eigenvalue to zero. The resulting equations are 9n + m9n = 0 / ; - mfn = 0.
(4.29)
The solutions are simply exponentials. The solution which obeys the boundary condition V|O,L = 0 is / 0 = 0 and go{y) = goe~my. The wave-function then either exponentially grows or decays, depending on whether the bulk mass term is positive or negative. The constant go is determined by the choice of normalization for the fermion wave function. To obtain a 4D theory in which the zero mode has the canonical normalization, we impose that
which has the solution
L
gl{y)dy = 1 ,
(4.30)
so = ^rz^r-
(4-3D
Now we can propose that all of the Yukawa couplings of the bulk fermions to the Higgs on the boundary are of order one, and try to find what the masses are for different bulk Dirac masses. The Yukawa couplings in the 5D theory are given by XuLuRHQL6(y
- L) -> XnLx°QHX^S{y
- L)
(4.32)
where the expression on the right leaves out all modes except the zero mode left from the solution above. The effective Yukawa coupling in the 4D picture involves the wave function evaluated at the boundary where the Higgs is located, and is expressed as (for example): Xu
=
A^ VmQmuLi y/2 y/(l - e~2mQL) (1 -
e-(mq+n,u)L-
(4-33)
2m L e- * )
This turns out to be an interesting solution to the fermion mass hierarchy. For all parameters of order one, it is possible to get a very wide spectrum of fermion masses. This is due to the exponential dependence of the wave functions on the bulk masses. For a small range of bulk Dirac masses that are all 0(1), one can easily lift the zero modes. The exponential dependance of the effective 4D Yukawa coupling on the bulk Dirac mass implies that this small range can give the fermions 4D masses that span the observed standard model spectrum. A graphical representation of how this works is given in Figure 8.
748
C. Csdki, J. Hubisz & P. Meade(on brane
Figure 8. This figure, from [53], displays how the fermion mass hierarchy is achieved through localization of the chiral zero modes. The lighter quarks are localized away from y = 0, while the third generation is peaked on the y = 0 brane, so that it has a sizable overlap with the Higgs VEV.
4.5. Fermions
in warped
space
As shown in earlier parts of these lectures, the tools of warped spacetime could fix some of the phenomenological problems of flat extra dimensions, and we would now like to see whether we can replicate the standard model spectrum of fermions in AdS setups. The first complication is that we need to know the form of the covariant derivative acting on fermions in this curved spacetime. To this end, we need an object constructed from the metric which lives in the spin 1/2 representation of the Lorentz group. In very rough terms, this object is the square root of the metric, g. In index notation, we write down the metric in terms of "vielbeins," or in the case of 5 dimensions, a "funfbein": gMN
=
eMvabeN
(4.34)
The Dirac algebra is written in curved space in terms of the flat space Dirac matrices as
TM = ef 7 a
(4.35)
To write down the covariant derivative that can act on fermions, we use a spin connection, w^J: DM = 9M +
1 n^h^ab, 2''
(4.36)
Electroweak Symmetry
Breaking from Extra Dimensions
749
where aab = \"1{alb\- The spin connection can be expressed in terms of the funfbeins as [60]
< = \9RPe[^d{Me^ + J f l f l V S e £ 4 V p ] < W
(4.37)
When the background geometry is given by AdS space, the metric (in conformal coordinates) is given by ds2 = (-}
{dx^dxv-rf" - dz2) .
(4.38)
One can show that eaM = ( f ) 6^, D^ip = (<9M + 7^75 J J ) >, and D5ip = d^. Proving that the spin connection terms cancel each other in this manner is left as an exercise for the reader. Written in terms of the two component Weyl spinors, the AdS action will be S=
Id5x(^\
[-ix^d^x-^^d^+^Kx-xA^P)
+ -z^X + X$)),
(4-39)
where the coefficients c — mR, and m is the bulk Dirac mass term for the 4-component Dirac spinor. In AdS space, the bulk equations of motion are [48]: -ia^d^x -ia^d^
- dd + — ^ = 0, z + d5X + ^ X = 0.
(4.40) (4.41)
These have some subtle yet important features. The terms in the equations of motion that contain the bulk mass, c, are dependent on the extra dimensional coordinate, z. The z dependent terms 2/z play an important role in determining the localization of any potential zero modes. As before, we perform the KK decomposition. Everything from flat space carries over, except that the bulk equations of motion for the wave functions are different. X = ^29n(z)Xn{x)
and $ = ^
n
fn(z)$n(x),
(4.42)
n
where the 4D spinors Xn and T/>„ satisfy the usual 4D Dirac equation with mass mn: -io^d^Xn
+ rnni)n = 0 and - ia^d^n
+ mnxn
= 0.
(4.43)
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C. Csdki, J. Hubisz & P. Meade
The bulk equations then become ordinary (coupled) differential equations of first order for the wavefunctions / „ and gn: In + mngn - ~~fn / 9n -
= 0,
(4.44)
9n = 0.
(4.45)
c-2 m
n9n
+
For a zero mode, if the boundary conditions were to allow its presence, these bulk equations are already decoupled and are thus easy to solve, leading to: /o = Co ( j j )
,
(4.46)
9O = A
'
(4 47)
°\R)
-
where AQ and Co are two normalization constants of mass dimension 1/2. For the massive modes, just as with the flat space scenario, the first order differential equations can be uncoupled by combining them to get second order equations: /n " -X + (m2n £-U'n
c
^^)fn
2
+ (rn n-£tS=*)gn
= 0,
(4.48)
= 0.
(4.49)
The solutions are now linear combinations of Bessel functions, as opposed to sin and cos functions: gn{z) = zi (AnJc+i(mnz) fn(z)
= zi (cnJc_i(mnz)
+ BnYc+i(mnz)J + DnYc_i(mnz)^j
(4.50) .
(4.51)
The first order bulk equations of motion (4.44)-(4.45) further impose that An = Cn and Bn = Dn.
(4.52)
The remaining undetermined coefRcients are determined by the boundary conditions, and the wave function normalization. As in the flat space Kaplan-Tait model, the bulk mass determines the localization of the fermions. For instance, let us take 9o(y) = A > ( ! ) 2 ~ \
/ = 0.
(4.53)
This solution corresponds to the boundary condition where V'lfi.fl' = 0. The coefficient AQ is determined by the normalization condition
Electroweak Symmetry
Breaking from Extra Dimensions
751
To understand from these equations where the fermions are localized, we study the behavior of this integral as we vary the limits of integration. For the boundary conditions that we are studying right now, if we send R' to infinity, we see that the integral remains convergent if c > 1/2, and the fermion is then localized on the UV brane. If we send R to zero, the integral is convergent if c < 1/2, and the fermion is localized on the IR brane. The value of the Dirac mass determines whether the fermion is localized towards the UV or IR branes. We note that the opposite choice of boundary conditions that yields a zero mode (X\R,R' ~ 0) results in a zero mode solution for ip with localization at the UV brane when c < —1/2, and at the IR brane when c > - 1 / 2 . The interesting feature in the warped case is that the localization transition occurs not when the bulk mass passes through zero, but at points where \c\ = 1/2. This is due to the curvature effects of the extra dimension. 4.6. Fermion
masses
in the Higgsless
model
Recall the gauge symmetries of the Higgsless model in warped space shown in figure 6. The fermions in this model can not be completely localized on the UV or IR branes. If they were on the IR brane, the up-type and downtype quarks could not have any mass splitting as the theory is non-chiral at z = R\ and if they were on the UV brane, the theory would be completely chiral, and the zero modes could not be lifted. The fermions must then live in the bulk, and feel the breakings of both branes. The quantum numbers of the fermions are given in Table 2. The preliminary boundary conditions Table 2. These are the quantum numbers of the bulk fermions under the bulk left-right symmetric Higgsless model. SU(2)L
n W*
$ '
a. (0.
SU(2)R
U(1)B-L
•
1
1/6
1
•
1/6
•
1
-1/2
1
•
-1/2
752
C. Csdki, J. Hubisz & P. Meade
for the fermions that give the zero modes that we desire are given by
(XuL\
+ + 4>UR
XdL
++
+ +
.
(4.55)
XdR
\$dn)
++
Where the + and — refer to whether we give those spinors Neumann or Dirichlet boundary conditions, respectively. These boundary conditions give massless chiral modes that match the fermion content of the standard model. However, the UL, di, UR, and d,R are all massless at this stage, and we need to lift the zero modes to achieve the standard model mass spectrum. While simply giving certain boundary conditions for the fermions will enable us to lift these zero modes, in the following discussion, we talk about boundary operators, and the boundary conditions that these operators induce. There are some subtleties in dealing with boundary operators for fermions. These arise from the fact that the fields themselves are not always continuous in the presence of a boundary operator. This is due to the fact that the equations of motion for fermions are first order. The most straightforward approach is to enforce the boundary conditions that give the zero modes as shown in Eq. (4.55) on the real boundary at z = R, R' while the boundary operators are added on a fictitious brane a distance e away from it. The distance between the fictitious brane and the physical one is taken to be e. The new boundary condition is then obtained by taking the distance e to be small. This physical picture is quite helpful in understanding what the different boundary conditions will do. To lift a doublet, we can give Dirac masses on the TeV brane which mix the SU(2)i and SU(2)R multiplets. This is possible because the theory on the IR brane is non-chiral. The boundary conditions that this Dirac mass gives on the IR brane are ^L = -MDR'ipR
and
XR = MDR'XL
(4.56)
At this stage, the up and down-type quarks, and the charged leptons and neutrinos are degenerate in mass. Mass splittings must be acquired on the UV brane, where the theory is chiral. For leptons and neutrinos, this can be accomplished simply by adding a Majorana mass for the neutrinos on the UV brane. The neutrino mass is suppressed by a type of see-saw mechanism while the lepton masses are unaffected. As a complete example of the boundary operator prescription, we consider adding a Majorana mass for the VR on the UV brane. This is displayed
Electroweak Symmetry V|/=0
Breaking from Extra Dimensions
753
8(z-R-e)RMxx
e
TeV
Planck
Figure 9. To give a physical picture to the boundary conditions used to lift a zero mode, a 4D majorana mass is added a slight distance from the boundary. The equations of motion are then solved, and the limit e —• 0 is taken.
in Figure 9. Going through the procedure described above, the new equation of motion is given by MR2
-xS(z-R-e) = 0 (4.57) z z Integrating over the discontinuity, we obtain a jump condition for the spinor -iad^x ~ dbi> H
—ip +
mRn+e
MRX\
(4.58)
R+e-
With the condition that ip\R = 0 and the small e limit, we have a new effective boundary condition ^R+e
= MRX\n+e
(4.59)
The general solutions to the bulk equations of motion are given by: XL,R = z5/2 [AL,RJ1/2+CLR(mz) 5/2
$L,R = z
[ALiRJ_1/2+CLR{mz)
+
BLiRJ_1/2_CLR(mz)]
- BLtRJ1/2_CLR(mz)]
.
(4.60)
The coefficients are fixed by the boundary conditions given in Eqs. (4.56) and (4.59), and by imposing canonical normalizations for the KK modes in the 4D effective theory. The resulting mass spectrum for the lifted zero modes can be modified by changing the cr,tR parameters, and the strengths of the boundary terms Mu, and M. Accomplishing the quark splittings is slightly more complicated, and involves adding induced kinetic terms on the boundaries, or mixings with
754
C. Csdki, J. Hubisz & P. Meade
localized fermions. However, the splittings of the quarks can be achieved as well. The final result is a spectrum which agrees with the standard model. It is possible to give the lighter fermions the right masses while all of them are strongly localized on the UV brane. This is ideal, since the distortion of the gauge boson wave functions towards the IR brane due to the warp factor will shift effective couplings to the fermion currents. The third generation of quarks poses a difficulty, however. To get a large enough mass for the top quark, the (t, b)i doublet must be localized towards the IR brane. This poses some difficulty for matching on to the measured value of the Zbb coupling [12]. This will be discussed further in the next lecture. 5. Electroweak precision observables for general B S M and extra dimensional theories This lecture dealing with electroweak (EW) observables in extra dimensional theories will be split into two main lines of discussion. The first part of this lecture will be dealing with how to calculate electroweak precision observables from the effective Lagrangian point of view. This part of the lecture will be completely independent of extra dimensions and can be applied to any weakly coupled model of beyond the SM (BSM) physics. After this self contained introduction to EW observables in BSM physics we will move on to analyzing where generic corrections to SM EW observables come from in extra dimensional models. The focus will then be shifted to the particular cases of EW observables in RS models and the Higgsless models. The Higgsless models will be discussed in detail at the end of this section showing how they can both be ruled out from EW observables in their original incarnations; as well as how these models can be made consistent with EW fits when the original models are modified. 5.1. EW observables view
and the effective
Lagrangian
point of
There have already been a series of lectures on EW precision tests by James Wells during this TASI [61], however this lecture will give a slightly different point of view that we find very useful in practice. We will follow closely here the analysis done by Burgess et. al. in [62] but will attempt to give a mostly self contained presentation. There is a very generic program that can be applied to electroweak precision observables, and can be divided in the following way.
Electroweak Symmetry
Breaking from Extra Dimensions
755
• First one needs to write down the most general effective Lagrangian allowed by the symmetries that are unbroken at the relevant energy scale (for instance for LEP I electroweak precision observables this would be Mz)• The second step after writing down the most general Lagrangian is to fix the coefficients for this effective Lagrangian for whatever model you are interested in analyzing. • Once this work has been done the third and final step is to calculate the expressions for the desired observables in terms of the input parameters and the effective Lagrangian coefficients. The desired observables for the particular model in question, Omodel, are best usually to be expressed as Omod,il = OSM + SOmodel. The reason for expressing the observables for a particular model in this form is that the corrections SOmodel are usually small (or you better hope so if this is your model you are studying). One can then include the SM loop contributions into O S M and only deal with the tree-level SOmodel. This is not always the case as there are cases of BSM physics where the tree-level contribution vanishes and loop level contributions must be calculated, for instance as in [63], however it can be successfully applied to a whole host of models. At this point once one has calculated the electroweak observables in the model of interest one simply needs to perform a fit to the experimental data and find the allowed regions of parameter space for a particular model. At this point several comments are in order after laying out such a short and yet powerful program for testing new BSM physics possibilities. The first is that at no point in this program have we made any reference to "oblique" corrections or the S, T, U parameters [64] or the ei, €2, £3 parameters [65]. Some students may have heard of these various parameterizations and associated their existence as the only calculations necessary for EW precision tests (EWPT). This is obviously not the case from the prescription we have laid out so far, yet there will be a large set of "universal" corrections that can be included in oblique corrections. The choice of parametrization is a matter of conventions and how you want to choose to express large sets of common corrections. For those who want more information on the common parameterizations we refer the reader to some original papers as well as useful reviews [61,64-70]. We will begin now with taking the Lagrangian at the energy scale of the Z mass, mz, with the top quark and the Higgs integrated out along with any other BSM particles as well. The general form of the gauge boson Lagrangian will be
756
C. Csdki, J. Hubisz & P. Meade
=
^eff
C-SM{e-i) + £new,
(5.1)
where CsM{e-%) is the ordinary SM Lagrangian with SM loop effects taken into account. The couplings of SM Lagrangian gj do not take their usual numerical values, because the new physics can contribute and has not been included. For example f^ = j^g will not necessarily hold due to the effects of Cnew which must be taken into account. The form of £new for the gauge boson sector up to dimension 4 is
Cnew = -\h^v
- jW^W^ - jZ^Z'"' + I ^ Z " "
-wm^W+W11-
-^mz2Z^.
(5.2)
The reason the gauge fields A,W and Z are hatted is because these are not canonically normalized fields. There was no trick to the procedure of what to write down in (5.2), we simply wrote down all dimension < 4 terms that are allowed by the remaining symmetries of the theory. FMJ, and Z^v are the usual Abelian field strengths, while the unbroken U{1)EM forces W„v = D^W,, - DVW^
(5.3)
D„WV = d^Wv + ieAuW„.
(5.4)
where
Let us assume for now that the rest of the SM Lagrangian in the gauge sector is unchanged, that is C-EM =
-ey^fiYQifjAn, i
Ccc = -^-7=
E
ViJa»PLfjW+
+ c.c,
CNC = -zrArswcw
V Ji-f [T3iPL - QiS2w] UK^
(5-5)
If (5.5) holds we have 6 parameters A, B, C, G, w, and z however not all of them are physically observable. The reason for this is that field redefinitions of W£, Bfj, and the Higgs scalarcan absorb 3 of the six parameters. Therefore in the end only 3 combinations of these parameters will appear in observables. Conventionally these three physical combinations are called
Electroweak Symmetry
Breaking from Extra Dimensions
757
the familiar S, T, and U parameters, which can be defined in terms of the parameters in (5.2) as aS = As2wc2w{A -C-
^ ' ^ G ) , CwS\y
aT = w — z, 4 w
aU = As (A -s 4- +s 4 ^ "s — °)
(5-6)
w w w Let us now begin to take into account the effect of the new physics by rescaling the fields to get canonically normalized gauge kinetic terms. Assuming the parameters A, B, C, G, w, and z are small we can go to canonically normalized kinetic terms with the following rescaling A. A, = (1 -•^)A,
K = (1 -
+ GZ,
-fw.
z» = (i ->
(5.7)
,
This rescaling will bring the Lagrangian into the form Ceff =
- \ F ^ F ^
- \w^W»v
- \z^Z^
- (1 + w -
B)m~w2W+W»-
_I(1 + z - C)rriz2ZliZ^ - e(l - ^ / i V Q i M , i
Sw
'-m(! - f )Y,ViJa'lPLfiW+ + c.c
6
swcw
(x - ? ) 5 1 /*^ [ T ^ " ^ 4 / + <2iSwcwG] /
The Lagrangian (5.8) depends on 3 input parameters e, §w, and m z (neglecting the CKM elements). However we would like to trade these input parameters for well measured quantities a, MZ,GF that take on their usual SM values. This will give us a relationship between the tilded parameters e, §w,inz, and the measured observables CC,MZ,GF, which can be further used to express any other observables in terms of. Starting with a general model as we have expressed so far, from (5.8) we have the relationship Ana = e 2 (l - A).
(5.9)
758
C. Csdki, J. Hubisz & P. Meade
In the SM this relationship takes the form Ana = e 2 , thus equating the two will lead us to a relationship between e and e, g = C
( 1+ ^ ) '
(5 10)
'
Similarly defining M% = m | ( l + z - C)
(5.11)
allows us to write the obvious relationship m2z = M2(l-z
+ C).
(5.12)
We next turn our attention to Gp, Fermi's constant, which is measured via /u-decay. The relevant diagram is shown in Figure 10. Integrating out the W boson generates an effective four fermion operator whose dimensionful coupling constant is given by GF- In the SM the expression defining GF is
7 1 ~ %s2wc2wm2z' There will be two effects that alter this relationship from the introduction of the new operators BSM. The first is the modified charged current interactions in (5.8) and the second will be the shift in the W propagator which comes from a shift in the pole mass which can be read off from (5.8) as well. With these modifications from the effective Lagrangian we have the relation GF e 2 (l - B) e 2 (l - w) (5.14) 2 2 2 y/2 %s wc wm z{\ + w-B) %s2wc2wm2z From this relationship and taking (5.13) as the definition of Sw we can express Sy^ as _ 2 w — sw 1 +
s- 2
(5.15) + z) 2 ™ 2 (A-C-w w sw We have now fixed part of the Lagrangian based on our chosen input parameters and by construction c
Cz =
-^M2Z^
CEM =-e^fi-fQifiA^
(5.16)
i
However now we can express other observables in terms of our chosen input observables and find the predictions for the effects of the new physics. For
Electroweak Symmetry
Breaking from Extra Dimensions
759
instance the term in the Lagrangian governing the mass of the W was of the form -(! + « ; - B)fh2wW+W»= - ( 1 + w - B)c2wrh2zW+W»(5.17) -w where we have used the relationship rhw = rhzcw since these refer to the tilded (SM only) parameters which continue to take their natural relations. We have previously fixed the relationship of the tilded parameters to the input observables in (5.15) and (5.12) therefore we can re-express (5.17) as Cw = -(l + w-B)(l-z
w
-w
M2c2wW+W^ -M2c2w
y
+ C)
l-B+C+w-z
-(A-C-w
>w
c
+ z)
'w
s
w
-{A-C-w
+ z)
w (5.18)
If we now change to the S,T,U parametrization given in (5.6) then the prediction for the physical mass of the W as a function of the new physics is given by aS
c2waT
aU
(5.19) 2 *w) We can also analyze how other sectors of the Lagrangian are changed by our choice of input parameters and what the effects are for other observables. In the charged current sector the Lagrangian now takes the form (Mw)phys
C-cc
—
{MW)SM
\f2s-w \
1-
1+
X ^ / ^ L / J
2
+r
2
2
2(c-w v
- s2
+ 4s
(A-C-w
A-B•w
+ z)
>w
W++/i.e.
(5.20)
r
U - • »
Figure 10.
The process in the SM which contributes to Gp.
760
C. Csdki, J. Hubisz & P. Meade
which can be rewritten with the help of S, T, and U as _
r
, V
y/2sw
aS 4(c^-s^)
J2 Vafi-fPLfiW^
(?waT 2(c2w-s2w)
all 8s^,
+ h.c.
(5.21)
This now allows us to define a shifted coupling for charged current interactions as hij = hfjM + 5hij, where hfjM = Vij and the shift from new physics is
which can be read off from (5.21). The neutral current sector is also changed to the form CNC =
—(1
S\vcW
2
Ql
Sw
'
+
+ ^)E/^
P^Z,-
2
aS uo 4(c w - S ^ ) w °w)
c
a WcLshaT W
2
v
fiZfj,.
°w
(5.23)
With this form of the neutral current we can define the couplings gi, = gfM + 6gL and gu = g^4 + 5gR analogously to the previously defined h^, where nSM
_
„SM bJR
_ —
rp
n
2
2 n —Wl^W'
and Xn
_
QT
„SM
n
/
dgiL,R - -y9iL,R - Qi I jra
aS
c2ws2waT\
i r r - „2 _ ,2 J •
(5-24)
With these definitions we can tackle other observables such as the left right asymmetry at the Z-pole A
=
r ( z -> ILIL) - v{z -> /fl/fl)
Electroweak Symmetry
Breaking from Extra Dimensions
761
The left-right asymmetry can be rewritten in terms of the neutral current couplings denned in (5.24) as A LR
A
„2'2 „2 A„SMnnaSM A „am ivi 9eL ~ 9eR ASM , ^9 eh 9eR /„SMX„ 9eL nSMfin A = „2 , -.2 = LR + , SM* i SM^2^9eR ~ 9eL 9eL + 9eR \9eL + 9eR ) An nSM ASM / nSM nSM nq a o ASM , ^9 eh 99eR eR nSM\ A SM ({„SM SM __ 9el nSM\ I I LR +, / s ^9eh 2 , _SM2\o.\9eR M
(5 e s r+9?r? [9eR
cwswaT\ A
_
' W2W - *2w)
r>1 >
9eR
9eL
^526^
s2
This program can be carried out for all other relevant observables at the Z-pole and similarly one will get expressions that depend only upon the oblique parameters S, T, and U. This conclusion should be expected if we recall the form of the "new" physics BSM that we introduced in (5.2). All corrections that we assumed in (5.2) appeared in the gauge boson sector only. Sometimes this will be the case for a given model, sometimes it won't. The most common exceptions to new physics appearing only obliquely are: • exchange of heavy gauge bosons (KK modes!) gives an additional contribution to \i decay as well as additional four fermion operators • mixing of heavy and light gauge bosons could give a non-oblique contribution to the shift of Zff, Wfifj couplings • non universal fermion wave functions in extra dimensions can also lead to the shift of Zff, Wfifj couplings while all of these have examples in extra dimensions there are analogues in four dimensional models as well. To account for non-oblique corrections the program is no different in practice than the one we have just discussed, the only difference being the form of the effective Lagrangian (5.2). Instead of only including new operators in the gauge sector alone one could include shifts in other sectors of the model up to whatever dimension of operator was desired for accuracy. This program has been carried out in [62] up to and including dimension five operators and we refer the interested reader there for more details. Of course including all operators up to dimension five will not necessarily yield all the interesting physics since new four fermion operators are dimension six. However, this process can be straightforwardly extended to include a more general effective Lagrangian up to dimension six. The interested reader may also find useful the extended (beyond S,T, and U) "universal" parametrization by Barbieri et. al. [26] which can account for new four fermion operators.
762
C. Csdki, J. Hubisz & P. Meade
Before we move on to investigating the specific effects of extra dimensions on EW precision observables we will further discuss oblique corrections in the Peskin-Takeuchi S, T, and U formalism since it is the most widely used. In the beginning of the effective Lagrangian approach (5.2) we started off with the parameters A, B, C, G, w, and z and then expressed these six parameters in terms of three physically measurable combinations S, T, and U. The Peskin-Takeuchi formalism calls the parameters that we used in (5.2) by slightly different names,
- ^j^F^Z^
+ IIWW(0)W+W»- + H £ | M Z M Z M .
(5.27)
These are clearly the same parameters as what we used in (5.2), their alternative names come from the reference to the SM loop calculations in the Peskin-Takeuchi paper [64]. These parameters simply represent shifts in the propagators of the gauge boson. Using this language and defining
n w ( o ) = 5 2 n n (o) u'ww = 52n'n(o) i W o ) = (ff2 + '2)n33(o) n'zz(o) = (g2 + g'2W33(o) - 2s2wn'3Q(0) + s4wWQQ(o)) n;7(o) = e2n^Q(o) n; z (o) = S f l , (n^(o)- s ^n' Q Q )
(5.28)
we can express the S, T, and U parameters in terms of these II as
s = i67r(n'33(o)-n^(o)) T= , b
47T c
2 A/f2 (nu(o)-n 33 (o))
W WlvlZ
C/ = 1 6 r r ( n i 1 ( 0 ) - n 3 3 ( 0 ) ) .
(5.29)
We have now summarized a basic program that has been applied previously to a wide range of models from little Higgs models [71] to extra dimensional ones [72,73] and we hope the reader can apply it to any model of their own interest. 5.2. Electroweak
precision
and extra
dimensions
With the formalism set up in Section 5.1 we may now move forward with showing in detail how we implement this for extra dimensional models. For our first example let us assume that the fermions will be localized at some given point. An important point we should make is that matching
Electroweak Symmetry
Breaking from Extra Dimensions
763
between 5D and 4D couplings is a convention. However when choosing your convention you want to make sure that your expressions are the simplest such that it minimizes the amount of work for you to do. For instance in a simple fiat extra dimension the matching condition between 4D and 5D gauge couplings usually is -2 = —, and -^ = -^.
(5.30)
However if fermions are localized at one point, for instance y — 0, the coupling of fermions to gauge bosons is not necessarily given by g and g' but rather the AD effective Lagrangian is of the form C D - J ^F^F^dy
+ 55^(0)7^^(0)^(0).
(5.31)
If we canonically normalize the gauge field A^ this in turn shifts the interaction term to the form Cint = - ^ ( 0 ) 7 ^ ( 0 ) ^ ( 0 ) ,
(5.32)
where the coupling will be given by # ^ ( 0 ) if we match the gauge coupling as in (5.30). To ensure that all corrections in this example will be oblique requires that we need to pick the wave function at the location of the fermions to be 1 (if we use this simple matching relationship). We note here that this is not always possible in the case that the fermions are localized at different points in the extra dimension. In this case the choice of the wave function will then tell us how to completely calculate the S, T, and U parameters. The effective wave function for the light modes will be - \ j \^(z)\2dzF^F^
- l - J \^z{z)\2dz
- \ J \^w(z)\2dzW+uW^+ / dzldzipz^Z^Z11
+J
Z^Z^
dz\dzipw\2W+W-»
+ any mass from Higgs terms
(5.33)
Equation (5.33) simply comes from the generic KK decomposition into 4D fields. Beginning with (after setting A5 = 0 as previously discussed) -\FMNFMNdz
= —JF^F^dz
- \ f(d5A»)2dz
(5.34)
and writing A^z)
= A° {x)ij)0A(z) + KKmoAes
(5.35)
764
C. Csdki, J. Hubisz & P. Meade
we get contributions to the Lagrangian of the form
-IJdzWAiztfidnAlW-dvAXx))2
\Jdz\d^A(z)\2Al(x)A°^x).
+
(5.36) As one can see from the last term in (5.36) a non-flat wave function V ' A W will contribute to a AD mass term! With these, the expression for the various IPs will be <7 2 n u (0) = J"
\dzijjw(z)\2dz
(g2 + 5,2)n33(o) = J \dMz)\2 dz 1 - g2n'n(0) l-(g2
= J \i/>w{z)\2dz
+ g'2)U'33(0) = J \i>z(z)\2dz.
(5.37)
Now it is simply a matter of substituting these expressions into (5.29) to get the S, T, and U parameters. For our next example we will look at the Randall-Sundrum model with all the fermions as well as the Higgs on the TeV brane, but with the gauge fields propagating in the bulk (this is not the most interesting case but probably the simplest to actually calculate [72]). Starting with the metric ds2
- dz2)
-{ri^dx^dx"
(5.38)
the form of the action will be S$D
— I d4x / J JR
dz — z
_ 1 ( l_
J_
— H—-n; +
4U +
e
)
2g2
MN
TIT-ZMNZ
FMNF
^6(z-R')^W+W
MN
4{9t+9?)
M-
rM
-5(z-R')-ZMZ
(5.39)
Solving for the wave functions, the expressions for the masses and wave functions of the W and Z are given by: M2w
- 1
4RlogR'/R
^(0)
1 +
(0)
,,(0)
1>
9%
Mw
^>(Mw^Mz).
R2v2 R'2 R '2
„,
M2z —
2z2\og-
R
1 g25+g'52 AR\ogR'/R 2 l + 2Rz'\og
R2v2 R'2 R! — R (5.40)
Electroweak Symmetry
Breaking from Extra Dimensions
765
The normalization of the wave functions is chosen such that ip(R') = 1 (where the fermions are assumed to be localized). We note that the Higgs being on the TeV brane will not allow a flat wave function for the W and Z in the bulk, which will give rise to non-zero S, T, and U parameters. Now we want to pick our matching conditions so we can find the relevant ITs. We pick the simplest matching conditions 1
RlogR'/R
2
9
ff§
1
RlogR'/R
2
9
9?
(5.41)
which leads to the following expressions ~v2U2R'\ogR'/R-2R2
n'„(0) = n y o ) =
M2 M2 n n ( 0 ) = - - ^ n ' u ( 0 ) and n 3 3 (0) = —f%M-
(5-43)
With these IPs we can easily calculate the S, T, and U parameters as we did before and we find that S = -4wfR2logR'/R,
T=--?—f2R2, U = 0, (5.44) 2cw where / is the SM Higgs VEV. We see from (5.44) that the contributions to S and T are quite sizeable. However in this case these are not the only corrections one must take into account. The KK modes of the gauge bosons localized on the TeV brane will generate large corrections to 4-fermi operators, in particular \i decay and the definition of Gp. We can introduce a new parameter V that measures the effect of the 4-fermi operators on the definition of GF • The effect of the ordinary W boson does not give the full GF but we will call it Gp,w which is given by 4V2GF,w = -n
•
(5-45)
4r + nn(o) The expression for the full GF defines the parameter V as GF = GF,w(l + V)
(5.46)
where V captures the effect of the additional KK modes of the gauge bosons. If we recall from Figure 10, the process that defines GF, we see that we need the full contribution to the W propagator. This means that what we really need to do is calculate the full 5D zero momentum brane to brane
766
C. Csdki, J. Hubisz & P. Meade
propagator Aw(q = 0, R', R'). We can calculate V in terms of A(0, R', R') and the W contribution (5.45) which gives V = -(AW(0,R>,R')
+ l
f
T
^ j
(£+n
l l (
0)).
(5.47)
For the RS model the W propagator will yield the result that
V=|W2log^.
(5.48)
With the S, T, and U results (5.44) and the V parameter (5.48) we can now express all SM Z-pole observables and find a bound on R'. In this particular example one finds that -kj > 11 TeV [72]. Of course this iis not the most interesting case since we would prefer to have the fermions in the bulk (mostly localized to the Planck brane). However even in this case there is still a strong constraint on the model from the T parameter. The strong constraint from the T parameter in the original RS models with the SM gauge symmetries are due to the absence of a custodial SU(2) symmetry. The solution to this is to put another SU(2)R gauge symmetry in the bulk of RS that is broken on the Planck brane [17]. By the AdS/CFT correspondence this is equivalent to adding a global custodial symmetry to the RS model as discussed in Section 3. 5.3. Electroweak
precision
and Higgsless
models
We have seen from the previous Section 5.2 that to leading order in log R'/R that S = T = 0. However the first correction to the oblique parameters is 0(lo R,/R), which is relatively large and one must check whether this is compatible with experimental results. We first analyze the Higgsless model discussed in Section 3 with the addition of Planck brane kinetic terms L W 2
A ^
4
+
+ g5W™)3
Jn^T^^SLB^ 4
(5.49)
9$R "+" 95
and TeV brane kinetic terms
E. R
+T
l M
^ w^ -
T
-7Bl» + 7 - 2
? -2
(9SLW^
+
g6LW")
(5.50)
We perform the matching calculation and carry out the program that we have been discussing and find the following approximate expressions for S
Electroweak Symmetry
Breaking from Extra Dimensions
767
and T to leading order in r, 2 5
6?r
2
Iog f- i + % 1 + RlosrR,/R
- -
V
3#
951,
(5.51)
TwO. However M^ is given by M,w
~
2 1 r 2 >& 1+ 1+ mog R,/RR' logR>/R
05*
g26R+glLr+R\ogR'/R,
\;
(5.52) The experimental constraints from the oblique parameters are S = -0.13 ±0.10 T = -0.17 ±0.11 U = 0.22 ± 0.12
(5.53)
for a 117 GeV Higgs with 1 a error bars given [74]. However the values are also correlated in the usual S-T plot as shown in Figure 11. In the Higgsless
Oblique Parameters constraints on gauge boson self-energies
i
1 ' i i i
asymmetries
—- M »
~ —-~ Vscalterina
i i 1
i
-
,k
<
:
:
<
i a l l : M H = 117 GeV :
i
Figure 11.
all: M H = 1000 GeV •
S-T Plot from PDG [74]
models for g^i = g$R and no induced kinetic terms the results for S and T are 5=1.15
T =0
obviously completely excluding this form of the model [24].
(5.54)
768
C. Csdki, J. Hubisz & P. Meade
This isn't the end of the story since there is a solution [40] to the 5 problem which has additional beneficial side-effects. It has been known for a long time in Randall-Sundrum (RS) models with a Higgs that the effective S parameter is large and negative [72] if the fermions are localized on the TeV brane as originally proposed. When the fermions are localized on the Planck brane the contribution to S is positive, and so for some intermediate localization the S parameter vanishes, as first pointed out for RS models by Agashe et al. [17]. The reason for this is fairly simple. Since the W and Z wavefunctions are approximately flat, and the gauge KK mode wavefunctions are orthogonal to them, when the fermion wavefunctions are also approximately flat the overlap of a gauge KK mode with two fermions will approximately vanish. Since it is the coupling of the gauge KK modes to the fermions that induces a shift in the S parameter, for approximately flat fermion wavefunctions the S parameter must be small. Note that not only does reducing the coupling to gauge KK modes reduce the S parameter, it also weakens the experimental constraints on the existence of light KK modes. This case of delocalized bulk fermions is not covered by the no-go theorem of [26], since there it was assumed that the fermions are localized on the Planck brane. In order to quantify these statements, it is sufficient to consider a toy model where all the three families of fermions are massless and have a universal delocalized profile in the bulk. Before showing some numerical results, it is useful to understand the analytical behavior of S in interesting limits. For fermions almost localized on the Planck brane, it is possible to expand the result for the 5-parameter in powers of (R/R')2°L~l -C 1. The leading terms, also expanding in powers of 1/ log, are:
and U « T « 0. The above formula is actually valid for 1/2 < CL < 3/2. For CL > 3/2 the corrections are of order (R'/R)2 and numerically negligible. As we can see, as soon as the fermion wave function starts leaking into the bulk, S decreases. Another interesting limit is when the profile is almost fiat, CL « 1/2. In this case, the leading contributions to S are: S = —
^
(l + (2cL - 1) log f
r log % V
K
+ O ((2cL - l ) 2 ) ) .
/
(5.56)
Electroweak Symmetry
Breaking from Extra Dimensions
769
In the fiat limit CL = 1/2, S is already suppressed by a factor of 3 with respect to the Planck brane localization case. Moreover, the leading terms cancel out for: 1 1 (5.57) 0.487 . CL 2 2 1ogf For CL < 1/2, S becomes large and negative and, in the limit of TeV brane localized fermions (cr,
1-
~cfh-2cL while, in the limit CL T U -
2cL
'
(5.58)
-oo: 2TT
g2 log
- ^ ( l + t a n 2 ^ ) wO.5 ,
8?r tan 2 6w 1 g log f- 2 + tan 2 0W cL 2
(5.59)
R
(5.60)
Figure 12. Plots of the oblique parameters as function of the bulk mass of the reference fermion. T h e values on the right correspond to localization on the Planck brane. S vanishes for c = 0.487.
In Fig. 12 we show the numerical results for the oblique parameters as function of CL- We can see that, after vanishing for CL « 1/2, S becomes negative and large, while T and U remain smaller. With R chosen to be the inverse Planck scale, the first KK resonance appears around 1.2 TeV, but for larger values of R this scale can be safely reduced down below a TeV. Such resonances will be weakly coupled to almost flat fermions and can easily avoid the strong bounds from direct searches at LEP or Tevatron. If we are imagining that the AdS space is a dual description of an approximate conformal field theory (CFT), then l/R is the scale where the CFT is no
770
C. Csdki, J. Hubisz & P. Meade
longer approximately conformal and perhaps becomes asymptotically free. Thus it is quite reasonable that the scale 1/R would be much smaller than the Planck scale.
Figure 13. In the left plot we show the contour plots of A N D A (solid blue lines) and M z ( ! ) (dashed red lines) in the parameter space ci,-R. The shaded region is excluded by direct searches of light Z' at LEP. In the center, the contours of S (red), for | S | = 0.25 (solid) and 0.5 (dashed) and T (blue), for \T\ = 0.1 (dotted), 0.3 (solid) and 0.5 (dashed), as function of C£ and R are shown. On the right, contours for the generic suppression of fermion couplings to the first resonance with respect to the SM value can be seen. The region for CL, allowed by 5 , is between 0.43 ± 0.5, where the couplings are suppressed at least by a factor of 10.
In Fig. 13 we have plotted the value of the NDA scale (3.12) as well as the mass of the first resonance in the (c^ — R) plane. Increasing R also affects the oblique corrections. However, while it is always possible to reduce S by delocalizing the fermions, T increases and puts a limit on how far R can be raised. One can also see from Fig. 13 that in the region where |5| < 0.25, the coupling of the first resonance with the light fermions is generically suppressed to less than 10% of the SM value. This means that the LEP bound of 2 TeV for SM-like Z' is also decreased by a factor of 10 at least (the correction to the differential cross section is roughly proportional to g2 jM\,). In the end, values of R as large as 10~ 7 G e V - 1 are allowed, where the resonance masses are around 600 GeV. So, even if, following the analysis of [34], we take into account a factor of roughly 1/4 in the NDA scale, we see that the appearance of strong coupling regime can be delayed up to 10 TeV. At the LHC it will be very difficult to probe WW scattering above 3 TeV. The major challenge facing Higgsless models is the incorporation of the third family of quarks. There is a tension [17,23] in obtaining a large top quark mass without deviating from the observed bottom couplings with the Z. It can be seen in the following way. The top quark mass is proportional both to the Dirac mixing Mp on the TeV brane and the overall scale of the
Electroweak Symmetry
Breaking from Extra Dimensions
771
extra dimension set by 1/R'. For cL ~ 0.5 (or larger) it is in fact impossible to obtain a heavy enough top quark mass (at least for g5R = gsL)- The reason is that for MpR' » 1 the light mode mass saturates at 2 R'2 log % which gives for this case rnt0p < \f2M\y. Thus one needs to localize the top and the bottom quarks closer to the TeV brane. However, even in this case a sizable Dirac mass term on the TeV brane is needed to obtain a heavy enough top quark. The consequence of this mass term is the boundary condition for the bottom quarks XbR = MDR'XbL-
(5.62)
This implies that if MDR' ~ 1 then the left handed bottom quark has a sizable component also living in an SU(2)R multiplet, which however has a coupling to the Z that is different from the SM value. Thus there will be a large deviation in the Zbi/bh- Note, that the same deviation will not appear in the ZbRl>R coupling, since the extra kinetic term introduced on the Planck brane to split top and bottom will imply that the right handed b lives mostly in the induced fermion on the Planck brane which has the correct coupling to the Z. The only way of getting around this problem would be to raise the value of 1/R', and thus lower the necessary mixing on the TeV brane needed to obtain a heavy top quark. One way of raising the value of 1/R' is by increasing the ratio g5R/gsi (at the price of also making the gauge KK modes heavier and thus the theory more strongly coupled). Another possibility for rasing the value of 1/R' is to separate the physics responsible for electroweak symmetry breaking from that responsible for the generation of the top mass. In technicolor models this is usually achieved by introducing a new strong interaction called topcolor. In the extra dimensional setup this would correspond to adding two separate AdSs bulks, which meet at the Planck brane [12]. One bulk would then be mostly responsible for electroweak symmetry breaking, the other for generating the top mass. The details of such models have been worked out in [12]. The main consequences of such models would be the necessary appearance of an isotriplet pseudo-Goldstone boson called the top-pion, and depending on the detailed implementation of the model there could also be a scalar particle (called the top-Higgs) appearing. This top-Higgs would however not be playing a major role in the unitarization of the gauge boson scattering amplitudes, but rather serve as the source for the top-mass only.
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C. Csdki, J. Hubisz & P. Meade
6. Conclusions We have attempted to give an introduction to the uses of extra dimensions for electroweak physics. Clearly there are many other very interesting models in this field besides the higgsless theories, which we were not able to cover in these lectures. Here is a partial list of the most prominent models of new electroweak phenomenology from extra dimensions not covered here: • The Randall-Sundrum model with custodial SU(2) [17], which can be thought of as the simplest implementation of the composite Higgs idea of Georgi and Kaplan. • The hierarchy problem could also be possibly solved if the Higgs was secretly an extra dimensional component (A5) of the gauge field. These theories are usually referred to as models with "gaugehiggs unification" [75] and were also the original inspirations for the little Higgs models of [76]. • Most recently [77], the above two approaches have been combined to get a model with gauge-higgs unification in warped space, with the A5 component localized on the TeV brane. This way the Higgs would be a composite pseudo-Goldstone boson, explaining its lightness compared to the TeV scale. To date these are probably the most successful models of electroweak phenomenology from extra dimensions, which may even incorporate a successful unification of the gauge couplings [78]. We can only guess that there must be many other interesting models of the TeV scale that no one has thought of yet. We are all hoping that the guessing will end abruptly about 2-3 years from now, and at TASI 2009 we will already be lecturing about the definitive theory of electroweak symmetry breaking (and start a new guessing game of what lies beyond)... Acknowledgments We thank K.T. Mahanthappa, John Terning, Carlos Wagner and Dieter Zeppenfeld for organizing a stimulating TASI. C.C. thanks Giacomo Cacciapaglia, Christophe Grojean, Hitoshi Murayama, Luigi Pilo, Matt Reece, Yuri Shirman and John Terning for collaborations which form much of the basis of these lectures. We also thank Giacomo Cacciapaglia, Cristophe Grojean, and Matt Reece for comments on the manuscript. This research is supported in part by the DOE OJI grant DE-FG02-01ER41206 and in part by the NSF grants PHY-0139738 and PHY-0098631.
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KEITH A. OLIVE
ASTROPARTICLE PHYSICS
KEITH A. OLIVE* William I. Fine Theoretical Physics Institute, School of Physics and University of Minnesota, Minneapolis, MN 55455 USA [email protected]
Astronomy
Selected topics in Astroparticle Physics including the CMB, dark matter, BBN, and the variations of fundamental couplings are discussed.
1. I n t r o d u c t i o n T h e background for all of the topics to be discussed in these lectures is the Big bang model. T h e observed homogeneity and isotropy enable us to describe the overall geometry and evolution of the Universe in terms of two cosmological parameters accounting for the spatial curvature and the overall expansion (or contraction) of the Universe. These two quantities appear in the most general expression for a space-time metric which has a (3D) maximally symmetric subspace of a 4D space-time, known as the Robertson-Walker metric: ds2 = dt2 -
R2(t)
dr2 kr2
+ r 2 (d0 2 +sin 2 0# 2 )
(1)
where R(t) is the cosmological scale factor and k is the curvature constant. By rescaling the radial coordinate, we can choose k to take only the discrete values + 1 , —1, or 0 corresponding to closed, open, or spatially flat geometries. T h e cosmological equations of motion are derived from equations ? V " o ^ f t = 87rG N T Ml/ + Ag^
Einstein's
(2)
"This work was supported in part by DOE grant DE-FG02-94ER40823 at Minnesota. 779
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K. A. Olive
where A is the cosmological constant. It is common to assume that the matter content of the Universe is a perfect fluid, for which Tiiv = ~V9p.v + (p + p) u^uv
(3)
where gM„ is the space-time metric described by (1), p is the isotropic pressure, p is the energy density and u = (1,0,0,0) is the velocity vector for the isotropic fluid in co-moving coordinates. With the perfect fluid source, Einstein's equations lead to the Friedmann-Lemaitre equations R2
2
H
=
&
STTGNP =
^
k
- - &
A +
3
(4)
and R
A
4TTGN
(P + 3p) (5) R 3 3 where H(t) is the Hubble parameter. Energy conservation via T^" = 0, leads to a third useful equation [which can also be derived from Eqs. (4) and (5)] p=-3H(p
+ p)
(6)
The Friedmann equation can be rewritten as (Q-l)ff2 = A
(7)
so that k = 0,+l,—1 corresponds to fl = l,fl > 1 and Q, < 1. However, the value of Q appearing in Eq. (7) represents the sum Q. = Qm + J1A of contributions from the matter density (fl m ) and the cosmological constant (fiA = A/3ff 2 ). 2. The C M B There has been a great deal of progress in the last several years concerning the determination of both Qm and Q\. Cosmic Microwave Background (CMB) anisotropy experiments have been able to determine the curvature (i.e. the sum of Clm and fl\) to within a few percent, while observations of type la supernovae at high redshift provide information on a (nearly) orthogonal combination of the two density parameters. The CMB is of course deeply rooted in the development and verification of the big bang model and big bang nucleosynthesis (BBN) 1 . Indeed, it was the formulation of BBN that led to the prediction of the microwave background. The argument is rather simple. BBN requires temperatures greater
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than 100 keV, which according to the standard model time-temperature relation, ts^MeV — 2.4/A/JV, where N is the number of relativistic degrees of freedom at temperature T, and corresponds to timescales less than about 200 s. The typical cross section for the first link in the nucleosynthetic chain is av{p + n ^ D + 7 ) ~ 5 x 10- 2 0 cm 3 /s
(8)
This implies that it was necessary to achieve a density „ „ _ L r, 10 1 7 cm- 3 (9) crvt for nucleosynthesis to begin. The density in baryons today is known approximately from the density of visible matter to be UB0 ~ 10~ 7 c m - 3 and since we know that that the density n scales as R~3 ~ T 3 , the temperature today must be To = (nBo/n)l/3TBBN ~ 10K (10) thus linking two of the most important tests of the big bang theory. An enormous amount of cosmological information is encoded in the angular expansion of the CMB temperature T{9,) = Y,airnYem{9, ).
(11)
Im
The monopole term characterizes the mean background temperature of T 7 = 2.725 ± 0.001 K as determined by COBE 2 , whereas the dipole term can be associated with the Doppler shift produced by our peculiar motion with respect to the CMB. In contrast, the higher order multipoles, are directly related to energy density perturbations in the early Universe. When compared with theoretical models, the higher order anisotropics can be used to constrain several key cosmological parameters. In the context of simple adiabatic cold dark matter (CDM) models, there are nine of these: the cold dark matter density, Q,xh2; the baryon density, £lsh2; the curvature characterized by Ototai; the hubble parameter, h; the optical depth, r; the spectral indices of scalar and tensor perturbations, ns and nt; the ratio of tensor to scalar perturbations, r; and the overall amplitude of fluctuations, Q. Microwave background anisotropy measurements have made tremendous advances in the last few years. The power spectrum 3,4 ' 5 ' 6,7,8 ' 9,10 has been measured relatively accurately out to multipole moments corresponding to £ ~ 2000. A compilation of recent data is shown in Fig. 1 n , where the power in at each £ is given by (2£ + l)Cg/(An), and Cf =< |a^ m | 2 >.
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K. A. Olive
i
r
i
6000
i
. i
Ta.4 0 0 0
•
i
r
i
^
" I - I
CJ^
i
i
i
i
DCBI
*
*ACBAR
i
—
i
- i + i -~*»2 0 0 0
4I T 1,T ]
-i
~ —
J J
X
i—i
V' ' i l l '
I I
n
5' i
•WMAP
i 5
CM
0J
Angular Scale 10/ . . , , 1 1 i i
1
1
1
1
1
500
1
i
1 < 1000 Multipole / 1
i i
I
1
i
*
1500
2000
Figure 1. The power in the microwave background anisotropy spectrum as measured by W M A P 9 , CBI 6 , and ACBAR 1 0 . Taken from Ref. 11.
As indicated above, the details of this spectrum enable one to make accurate predictions of a large number of fundamental cosmological parameters. The results of the WMAP data (with other information concerning the power spectrum) is shown in Table 1. For details see Ref. 9.
On ft2 nB/i h
2
WMAPext + 2dFGRS + Lyman a running
W M A P alone power-law
WMAPext + 2dFGRS power-law
0.14 ± 0 . 0 2 0.024 ± 0.001
0.134 ± 0 . 0 0 6 0.023 ± 0.001
0.72 ± 0 . 0 5
0.93 ± 0 . 0 3 0.17 ± 0 . 0 6
ns
0.99 ± 0.04
0.73 ± 0 . 0 3 0.97 ± 0 . 0 3
T
n 1icfi+0.076 °- 66-0.071
0.148_oon
,,,-+0.008 u.±oo_0 009 n
0.0224 ± 0.0009 n71+0.04
U-'i-0.03
Of particular interest to us here is the CMB determination of the total density, Qtot, as well as the matter density Clm. There is strong evidence that the Universe is flat or very close to it. The best constraint on fitotai is 1.02 ±0.02. Furthermore, the matter density is significantly larger than the
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baryon density implying the existence of cold dark matter and the baryon density, as we will see below, is consistent with the BBN production of D/H and its abundance in quasar absorption systems. The apparent discrepancy between the CMB value of Q to t and fim, though not conclusive on its own, is a sign that a contribution from the vacuum energy density or cosmological constant, is also required. The preferred region in the flm — Q\ plane is shown in Fig. 2 under four different assumptions 9 .
0.2
Figure 2.
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Two-dimensional confidence regions in the ( Q M , ^ A ) plane 9 .
The presence or absence of a cosmological constant is a long standing problem in cosmology. We know that the cosmological term is at most a factor of a few times larger than the current mass density. Thus from Eq. (4), we see that the dimensionless combination, GATA <> 1Q~121. Nevertheless, even a small non-zero value for A could greatly affect the future history of the Universe: allowing open Universes to recollapse (if A < 0), or closed Universes to expand forever (if A > 0 and sufficiently large). When the SN l a results 12 are included (see the last panel of Fig. 2) we are led to a seemingly conclusive picture. The Universe is nearly flat with fitot ^ 1. However the density in matter makes up only 23% of this total, with the remainder in a cosmological constant or some other form of dark energy.
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3. Dark matter 3.1. Observational
evidence
Direct observational evidence for dark matter is found from a variety of sources. On the scale of galactic halos, the observed flatness of the rotation curves of spiral galaxies is a clear indicator for dark matter. There is also evidence for dark matter in elliptical galaxies, as well as clusters of galaxies coming from the X-ray observations of these objects. Also, direct evidence has been obtained through the study of gravitational lenses. For example, assuming that galaxies are in virial equilibrium, one expects that one can relate the mass at a given distance r, from the center of a galaxy to its rotational velocity by M(r) oc v2r/GN
(12)
13 14
The rotational velocity, v, is measured ' by observing 21 cm emission lines in HI regions (neutral hydrogen) beyond the point where most of the light in the galaxy ceases. A subset of a compilation 15 of nearly 1000 rotation curves of spiral galaxies is shown in Fig. 3. The subset shown is restricted to a narrow range in brightness, but is characteristic for a wide range of spiral galaxies. Shown is the rotational velocity as a function of r in units of the optical radius. If the bulk of the mass is associated with light, then beyond the point where most of the light stops, M would be constant and v2 oc 1/r. This is not the case, as the rotation curves appear to be flat, i.e., v ~ constant outside the core of the galaxy. This implies that M oc r beyond the point where the light stops. This is one of the strongest pieces of evidence for the existence of dark matter on galactic scales. Velocity measurements indicate dark matter in elliptical galaxies as well 16 . For a more complete discussion see Ref. 17. 3.2.
Theory
Theoretically, there is no lack of support for the dark matter hypothesis. The standard big bang model including inflation almost requires fitot = 1 18 . This can be seen from the following simple solution to the curvature problem. The unfortunate fact that at present we do not even know whether ft is larger or smaller than one, indicates that we do not know the sign of the curvature term further implying that it is subdominant in Eq. (4) k
8wG
W2<—P
(13)
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785
<M,> = -21.2 I I I I | I I I I | I I I I | I I I I | I
HIIIIII-
1 . . .i . i II
i
Figure 3. Synthetic rotation curve 1 5 for galaxies with (M) = - 2 1 . 2 . The dotted curve shows the disk contribution, whereas the dashed curve shows the halo contribution.
In an adiabatically expanding Universe, i? ~ T l where T is the temperature of the thermal photon background. Therefore the quantity k
8TTG
„
,
7*2* < 31? < 2 X 10
.o
(14)
is dimensionless and constant in the standard model. This is known as the curvature problem and can be resolved by a period of inflation. Before inflation, let us write R = Ri, T = Ti and R ~ T " 1 . During inflation, R ~ T~1 ~ eHt, where H is constant. After inflation, R = Rf ^> Rt but T = Tf = TR <^ Ti where TR is the temperature to which the Universe reheats.^ Thus R^T and k -> 0 is not constant. But from Eqs. (7) and (14) if fc —> 0 then Cl —> 1, and since typical inflationary models contain much more expansion than is necessary, 17 becomes exponentially close to one. The existence of non-baryonic dark matter can be immediately inferred from the determination of the cosmological parameters through the microwave background anisotropy as described above. If flmh2 ~ 0.13 and £lsh2 ~ 0.02, then the difference must be dark matter which contributes to the total density fluuh2 ~ 0.11. In addition, because the amplitude of fluctuations is relatively small, dark matter is necessary to have sufficient time to grow primordial perturbations into galaxies (for a more complete discussion see Ref. 17).
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K. A. Olive
Candidates
3.3.1. Baryons Accepting the dark matter hypothesis, the first choice for a candidate should be something we know to exist, baryons. Though baryonic dark matter can not be the whole story if Qm > 0.1, the identity of the dark matter in galactic halos, which appear to contribute at the level of Q, ~ 0.05, remains an important question needing to be resolved. A baryon density of this magnitude is not excluded by nucleosynthesis. Indeed we know some of the baryons are dark since Cl < 0.01 in the disk of the galaxy. It is interesting to note that until recently, there seemed to be some difficulty in reconciling the baryon budget of the Universe. By counting the visible contribution to CI in stellar populations and the X-ray producing hot gas, Persic and Salucci19 found only SlviS ~ 0.003. A subsequent accounting by Fukugita, Hogan and Peebles 20 found slightly more (fl ~ 0.02) by including the contribution from plasmas in groups and clusters. At high redshift on the other hand, all of the baryons can be accounted for. The observed opacity of the Ly a forest in QSO absorption spectra requires a large baryon density consistent with the determinations by the CMB and BBN 21 . In galactic halos, however, it is quite difficult to hide large amounts of baryonic matter. Sites for halo baryons that have been discussed include Hydrogen (frozen, cold or hot gas), low mass stars/Jupiters, remnants of massive stars such as white dwarfs, neutron stars or black holes. In almost every case, a serious theoretical or observational problem is encountered 22 .
3.3.2. Neutrinos Light neutrinos (m < 30eV) are a long-time standard when it comes to non-baryonic dark matter 23 . Light neutrinos are, however, ruled out as a dominant form of dark matter because they produce too much large scale structure 24 . Because the smallest non-linear structures have mass scale Mj « 3 x 10 1 8 M Q /m 2 (eV) and the typical galactic mass scale is ~ 1O 12 M 0 , galaxies must fragment out of the larger pancake-like objects. The problem with such a scenario is that galaxies form late 25 ' 26 (z < 1) whereas quasars and galaxies are seen out to redshifts z > 6. The neutrino decoupling scale of (9(1) MeV has an important consequence on the final relic density of massive neutrinos. Neutrinos more massive than 1 MeV will begin to annihilate prior to decoupling, and while
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MeV
GeV
Physics
787
TeV Illy
Figure 4. Summary plot 2 8 of the relic density of Dirac neutrinos (solid) including a possible neutrino asymmetry of r\u = 5 x 1 0 - 1 1 (dotted).
in equilibrium, their number density will become exponentially suppressed. Lighter neutrinos decouple as radiation on the other hand, and hence do not experience the suppression due to annihilation. Therefore, the calculations of the number density of light (m„ < 1 MeV) and heavy (m„ > 1 MeV) neutrinos differ substantially. The energy of density of light neutrinos with mu < 1 MeV can be expressed at late times as pv = mvYvn^ where Yv = n^/n^ is the number density of v's relative to the density of photons, which today is 411 photons per cm 3 . It is easy to show that in an adiabatically expanding universe Yv = 3/11. This suppression is a result of the e+e~ annihilation which occurs after neutrino decoupling and heats the photon bath relative to the neutrinos. Imposing the constraint Slvh2 <> 0.13, translates into a strong constraint (upper bound) on Majorana neutrino masses 27 : "Hot = ^
m
"
~
1 2 e V
'
(15)
where the sum runs over neutrino mass eigenstates. The limit for Dirac neutrinos depends on the interactions of the right-handed states. The limit (15) and the corresponding initial rise in VLvh2 as a function of mv is displayed in the Figure 4. Combining the rapidly improving data on key cosmological parameters with the better statistics from large redshift surveys has made it possible
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to go a step forward along this path. It is now possible to set stringent limits on the light neutrino mass density ttuh2, and hence on the neutrino mass based on the power spectrum of the Ly a forest 29 , m t o t < 5.5 eV, and the limit is even stronger if the total matter density, flm is less than 0.5. Adding additional observation constraints from the CMB and galaxy clusters drops this limit 30 to 4.2 eV. This limit has recently been improved by the 2dF Galaxy redshift31 survey by comparing the derived power spectrum of fluctuations with structure formation models. Focussing on the the presently favoured ACDM model, the neutrino mass bound becomes m t o t < 1-8 eV for Clm < 0.5. When even more constraints such as HST Key project data, supernovae type la data, and BBN are included 32 the limit can be pushed to m to t < 0.9 eV. With WMAP data, an upper limit of mtot < 0.7 eV has been derived 9 . The calculation of the relic density for neutrinos more massive than ~ 1 MeV, is substantially more involved. The relic density is now determined by the freeze-out of neutrino annihilations which occur at T ^ m„, after annihilations have begun to seriously reduce their number density 33 . For particles which annihilate through approximate weak scale interactions, annihilations freeze out when T ~ m x / 2 0 . Roughly, the solution to the Boltzmann equation, which tracks the neutrino abundance, goes as Yv ~ / ~ {m{av)ann)~l and hence Qvh2 ~ 2 2 {w)ann~ , so that parametrically Quh ~ 1/m . As a result, the constraint on CI now leads to a lower bound 33,34 ' 35 on the neutrino mass, of about m„ j> 3 — 7 GeV, depending on whether it is a Dirac or Majorana neutrino. This bound and the corresponding downward trend Clvh2 ~ 1/m2 can again be seen in Figure 4. The result of a more detailed calculation is shown in Figure 5 35 for the case of a Dirac neutrino. The two curves show the slight sensitivity on the temperature scale associated with the quarkhadron transition. The result for a Majorana mass neutrino is qualitatively similar. Indeed, any particle with roughly weak scale cross-sections will tend to give an interesting value of flh2 ~ 1. The deep drop in Q,vh2, visible in Figure 4 at around m„ = M^/2, is due to a very strong annihilation cross section at Z-boson pole. For yet higher neutrino masses the Z-annihilation channel cross section drops as ~ 1/m 2 , leading to a brief period of an increasing trend in Q,vh2. However, for mv > mw the cross section regains its parametric form {o-v)ann ~ m 2 due to the opening up of a new annihilation channel to W-boson pairs 36 , and the density drops again as Clvh2 ~ 1/m 2 . The tree level ^-channel cross section breaks the unitarity at around C(few) TeV 3 7 however, and
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Figure 5. The relic density of heavy Dirac neutrinos due to annihilations 3 5 . The curves are labeled by the assumed quark-hadron phase transition temperature in MeV.
the full cross section must be bound by the unitarity limit 38 . This behaves again as Y/m2,, whereby Q,vh2 has to start increasing again, until it becomes too large again at 200-400 TeV 38 ' 37 . If neutrinos are Dirac particles, and have a nonzero asymmetry the relic density could be governed by the asymmetry rather than by the annihilation cross section. Indeed, it is easy to see that the neutrino mass density corresponding to the asymmetry r)u = (n„ — n^/n^ is given by 39 p = mvr)vn~i, which implies nvh2 ~ 0.004 rjvW (m„/GeV).
(16)
where ?y„io = l O 1 0 ^ . The behaviour of the energy density of neutrinos with an asymmetry is shown by the dotted line in the Figure 4. In the figure, we have assumed an asymmetry of r)v ~ 5 x 1 0 - 1 1 for neutrinos with standard weak interaction strength. Based on the leptonic and invisible width of the Z boson, experiments at LEP have determined that the number of neutrinos is N„ = 2.994±0.012 4 0 . Conversely, any new physics must fit within these brackets, and thus LEP excludes additional neutrinos (with standard weak interactions) with masses mu < 45 GeV. Combined with the limits displayed in Figures 4 and 5, we see that the mass density of ordinary heavy neutrinos is bound to be very small, Vt^h? < 0.001 for masses mu > 45 GeV up to mv ~ 0(100)
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TeV. Lab constraints for Dirac neutrinos are available 41 , excluding neutrinos with masses between 10 GeV and 4.7 TeV. This is significant, since it precludes the possibility of neutrino dark matter based on an asymmetry between v and v 39 . 3.3.3. Axions Due to space limitations, the discussion of this candidate will be very brief. Axions are pseudo-Goldstone bosons which arise in solving the strong CP problem 42 ' 43 via a global U(l) Peccei-Quinn symmetry. The invisible axion 43 is associated with the flat direction of the spontaneously broken PQ symmetry. Because the PQ symmetry is also explicitly broken (the CP violating OFF coupling is not PQ invariant) the axion picks up a small mass similar to pion picking up a mass when chiral symmetry is broken. We can expect that ma ~ m^f^/fa where fa, the axion decay constant, is the vacuum expectation value of the PQ current and can be taken to be quite large. If we write the axion field as a — fa6, near the minimum, the potential produced by QCD instanton effects looks like V ~ m 2 # 2 / 2 . The axion equations of motion lead to a relatively stable oscillating solution. The energy density stored in the oscillations exceeds the critical density 44 unless fa < 1012 GeV. Axions may also be emitted stars and supernova 45 . In supernovae, axions are produced via nucleon-nucleon bremsstrahlung with a coupling gAN ocTOJV/fa- As was noted above the cosmological density limit requires fa ^ 1012 GeV. Axion emission from red giants imply 46 fa > 10 10 GeV (though this limit depends on an adjustable axion-electron coupling), the supernova limit requires 47 fa > 2 x 1 0 n GeV for naive quark model couplings of the axion to nucleons. Thus only a narrow window exists for the axion as a viable dark matter candidate. 4. Supersymmetric dark matter For the remaining discussion of dark matter, I will restrict my attention to supersymmetry and in particular, the minimal supersymmetric standard model (MSSM) with R-parity conservation. R-parity is necessary if one wants to forbid all new baryon and lepton number violating interactions at the weak scale. If R-parity, which distinguishes between "normal" matter and the supersymmetric partners and can be defined in terms of baryon, lepton and spin as R = (_i) 3 B + L + 2 S ' ) i s unbroken, there is at least one supersymmetric particle (the lightest supersymmetric particle or
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LSP) which must be stable. Thus, the minimal model contains the fewest number of new particles and interactions necessary to make a consistent theory. There are very strong constraints, however, forbidding the existence of stable or long lived particles which are not color and electrically neutral 48 . Strong and electromagnetically interacting LSPs would become bound with normal matter forming anomalously heavy isotopes. Indeed, there are very strong upper limits on the abundances, relative to hydrogen, of nuclear isotopes 49 , n/nH < 10" 1 5 to 1(T 29 for 1 GeV £ m < 1 TeV. A strongly interacting stable relic is expected to have an abundance n/nu ^ 10~ 10 with a higher abundance for charged particles. There are relatively few supersymmetric candidates which are not colored and are electrically neutral. The sneutrino 50 is one possibility, but in the MSSM, it has been excluded as a dark matter candidate by direct 41 and indirect 51 searches. In fact, one can set an accelerator based limit on the sneutrino mass from neutrino counting, mp ^ 44.7 GeV 52 . In this case, the direct relic searches in underground low-background experiments require mp J> 20 TeV 41 . Another possibility is the gravitino which is probably the most difficult to exclude. I will concentrate on the remaining possibility in the MSSM, namely the neutralinos but will return to the case of gravitino dark matter as well.
4.1.
Parameters
The most general version of the MSSM, despite its minimality in particles and interactions contains well over a hundred new parameters. The study of such a model would be untenable were it not for some (well motivated) assumptions. These have to do with the parameters associated with supersymmetry breaking. It is often assumed that, at some unification scale, all of the gaugino masses receive a common mass, m ^ - The gaugino masses at the weak scale are determined by running a set of renormalization group equations. Similarly, one often assumes that all scalars receive a common mass, mo, at the GUT scale. These too are run down to the weak scale. The remaining supersymmetry breaking parameters are the trilinear mass terms, AQ, which I will also assume are unified at the GUT scale, and the bilinear mass term B. There are, in addition, two physical CP violating phases which will not be considered here. Finally, there is the Higgs mixing mass parameter, /ix, and since there are two Higgs doublets in the MSSM, there are two vacuum expectation values. One combination of these is re-
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lated to the Z mass, and therefore is not a free parameter, while the other combination, the ratio of the two vevs, tan/3, is free. The natural boundary conditions at the GUT scale for the MSSM would include // and B in addition to rrii/ 2 , mo, and AQ. In this case, upon running the RGEs down to a low energy scale and minimizing the Higgs potential, one would predict the values of Mz, tan (3 (in addition to all of the sparticle masses). Since Mz is known, it is more useful to analyze supersymmetric models where Mz is input rather than output. It is also common to treat tan P as an input parameter. This can be done at the expense of shifting \i (up to a sign) and B from inputs to outputs. This model is often referred to as the constrained MSSM or CMSSM. Once these parameters are set, the entire spectrum of sparticle masses at the weak scale can be calculated.
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RG evolution of the mass parameters in the CMSSM.
In Fig. 6, an example of the running of the mass parameters in the CMSSM is shown. Here, we have chosen m 1 / 2 = 250 GeV, m 0 = 100 GeV, tan/3 = 3, AQ = 0, and /t < 0. Indeed, it is rather amazing that from so few input parameters, all of the masses of the supersymmetric particles can be determined. The characteristic features that one sees in the figure, are for example, that the colored sparticles are typically the heaviest in the
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spectrum. This is due to the large positive correction to the masses due to «3 in the RGE's. Also, one finds that the B (the partner of the U(l)y gauge boson), is typically the lightest sparticle. But most importantly, notice that one of the Higgs mass 2 , goes negative triggering electroweak symmetry breaking 53 . (The negative sign in the figure refers to the sign of the mass 2 , even though it is the mass of the sparticles which are depicted.) 4.2.
Neutralinos
There are four neutralinos, each of which is a linear combination of the R = — 1 neutral fermions48: the wino W3, the partner of the 3rd component of the SU(2)L gauge boson; the bino, B; and the two neutral Higgsinos, Hi and B.2- Assuming gaugino mass universality at the GUT scale, the identity and mass of the LSP are determined by the gaugino mass m ^ 2 , At, and tan/3. In general, neutralinos can be expressed as a linear combination X = aB + PW3 + 1Hl + 5H2
(17)
The solution for the coefficients a, /3,7 and S for neutralinos that make up the LSP can be found by diagonalizing the mass matrix (
M2
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^
^ /W3\
fl£
B
(W3,B,H°,H%) \
V2
\/2
32^2
-giV2
m
^ _.
n
(18)
I
where M\{M2) is a soft supersymmetry breaking term giving mass to the U(l) (SU(2)) gaugino(s). In a unified theory M\ = M-x = mi/2 at the unification scale (at the weak scale, Mi ~ l^p-A^). As one can see, the coefficients a,/3,7, and 5 depend only on m ^ 2 , /«, and tan/3. In the CMSSM, the solutions for [i generally lead to a lightest neutralino which is very nearly a pure B. 4.3. The relic
density
The relic abundance of LSP's is determined by solving the Boltzmann equation for the LSP number density in an expanding Universe. The technique 35 used is similar to that for computing the relic abundance of massive neutrinos 33 with the appropriate substitution of the cross section. The relic density depends on additional parameters in the MSSM beyond an TO I/2)MJ d tan/3. These include the sfermion masses, m? and the Higgs
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pseudo-scalar mass, 7714, derived from m 0 (and mx/2). To determine the relic density it is necessary to obtain the general annihilation cross-section for neutralinos. In much of the parameter space of interest, the LSP is a bino and the annihilation proceeds mainly through sfermion exchange. Because of the p-wave suppression associated with Majorana fermions, the s-wave part of the annihilation cross-section is suppressed by the outgoing fermion masses. This means that it is necessary to expand the cross-section to include p-wave corrections which can be expressed as a term proportional to the temperature if neutralinos are in equilibrium. Unless the neutralino mass happens to lie near near a pole, such as mx ~ m z / 2 or m/t/2, in which case there are large contributions to the annihilation through direct s-channel resonance exchange, the dominant contribution to the BB annihilation cross section comes from crossed t-channel sfermion exchange. Annihilations in the early Universe continue until the annihilation rate r ~ avnx drops below the expansion rate. The final neutralino relic density expressed as a fraction of the critical energy density can be written as 4 8
^'••*10-u(£)VkTK)
where (Tx/T.y)3 accounts for the subsequent reheating of the photon temperature with respect to x, due to the annihilations of particles with mass m < Xfmx 54 and Xf = Tf/mx is proportional to the freeze-out temperature. The coefficients a and b are related to the partial wave expansion of the cross-section, av = a + bx + Eq. (19 ) results in a very good approximation to the relic density expect near s-channel annihilation poles, thresholds and in regions where the LSP is nearly degenerate with the next lightest supersymmetric particle 55 . 4.4.
The CMSSM
after
WMAP
For a given value of tan/3, ^40> and sgn(/j,), the resulting regions of acceptable relic density and which satisfy the phenomenological constraints can be displayed on the mi/2 — mo plane. In Fig. 7a, the light shaded region corresponds to that portion of the CMSSM plane with tan/3 = 10, AQ = 0, and [i > 0 such that the computed relic density yields 0.1 < Q.xh2 < 0.3. At relatively low values of m i / 2 and mo, there is a large 'bulk' region which tapers off as my2 is increased. At higher values of mo, annihilation cross sections are too small to maintain an acceptable relic density and il.xh2 > 0.3. Although sfermion masses are also enhanced at large my2
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(due to RGE running), co-annihilation processes between the LSP and the next lightest sparticle (in this case the fj) enhance the annihilation cross section and reduce the relic density. This occurs when the LSP and NLSP are nearly degenerate in mass. The dark shaded region has m ^ < mx and is excluded. Neglecting coannihilations, one would find an upper bound of ~ 450 GeV on my2, corresponding to an upper bound of roughly 200 GeV on m g . The effect of coannihilations is to create an allowed band about 2550 GeV wide in m 0 for m x / 2 <> 1400 GeV, which tracks above the m,fx = mx contour 56 . tan p = 1», ^ > 0
tan p = 10, n > 0
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Figure 7. The (m 1 / 2 ,mo) planes for (a) tan/3 = 10 and y, > 0, assuming A0 = 0,mt = 175 GeV and mb{mb)fE = 4.25 GeV. The near-vertical (red) dot-dashed lines are the contours mh =114 GeV, and the near-vertical (black) dashed line is the contour m ± = 104 GeV. Also shown by the dot-dashed curve in the lower left is the corner excluded by the LEP bound of mi > 99 GeV. The medium (dark green) shaded region is excluded byb-> s'y, and the light (turquoise) shaded area is the cosmologically preferred regions with 0.1 < Qxh2 < 0.3. In the dark (brick red) shaded region, the LSP is the charged fi. The region allowed by the E821 measurement of aM at the S-cr level, is shaded (pink) and bounded by solid black lines, with dashed lines indicating the 1-a ranges. In (b), the relic density is restricted to the range 0.094 < Qxh2 < 0.129.
Also shown in Fig. 7a are the relevant phenomenological constraints. These include the limit on the chargino mass: mx± > 104 GeV 57 , on the selectron mass: m~e > 99 GeV 58 and on the Higgs mass: mh > 114 GeV 5 9 . The former two constrain m i / 2 and m0 directly via the sparticle masses, and the latter indirectly via the sensitivity of radiative corrections to the Higgs mass to the sparticle masses, principally mi 6 . FeynHiggs 60 is used
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for the calculation of m/,. The Higgs limit imposes important constraints principally on m i / 2 particularly at low tan/3. Another constraint is the requirement that the branching ratio for b —> sy is consistent with the experimental measurements 61 . These measurements agree with the Standard Model, and therefore provide bounds on MSSM particles 62 ' 63 , such as the chargino and charged Higgs masses, in particular. Typically, the b —> S'y constraint is more important for \i < 0, but it is also relevant for \i > 0, particularly when tan /3 is large. The constraint imposed by measurements of b —> sj also excludes small values of m\/2- Finally, there are regions of the (mi/2, mo) plane that are favoured by the BNL measurement 64 of 0 and values of tan/3 from 5 to 55, in steps A(tan/3) = 5. We notice immediately that the strips are considerably narrower than the spacing between them, though any intermediate point in the (mi/ 2 ,mo) plane would be compatible with some intermediate value of
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As in Fig. 7 for tan/3 = 50.
tan f3. The right (left) ends of the strips correspond to the maximal (minimal) allowed values of m i / 2 and hence m x . The lower bounds on mi/2 are due to the Higgs mass constraint for tan /? < 23, but are determined by the b —> S7 constraint for higher values of tan f3. Finally, there is one additional region of acceptable relic density known as the focus-point region 71 , which is found at very high values of mo. An example showing this region is found in Fig. 10, plotted for tan/3 = 10, (j, > 0, and m t = 175 TeV. As m 0 is increased, the solution for /i at low energies as determined by the electroweak symmetry breaking conditions eventually begins to drop. When /x <; m x /2, the composition of the LSP gains a strong Higgsino component and as such the relic density begins to drop precipitously. These effects are both shown in Fig. 11 where the value of fi and ilh2 are plotted as a function of m 0 for fixed m i / 2 = 300 GeV and tan/3 = 10. As m 0 is increased further, there are no longer any solutions for \i. This occurs in the shaded region in the upper left corner of Fig. 10. Fig. 11 also exemplifies the degree of fine tuning associated with the focus-point region. While the position of the focus-point region in the mo,mi/2 plane is not overly sensitive to supersymmetric parameters, it is highly sensitive to the top quark Yukawa coupling which contributes to the evolution of /J, 72 ' 73 . As one can see in the figure, a change in mt of 3 GeV produces a shift of about 2.5 TeV in m0. Note that the position of the focus-point region is also highly sensitive to the value of Ao/rriQ. In Fig. 11, A0 = 0 was chosen. For A0/m0 = 0.5, the focus point shifts from 2.5 to 4.5 TeV and moves to larger mo as Ao/m0 is increased.
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o
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m 1/2 (GeV) Figure 9. The strips display the regions of the (mi/ 2 ,mo) plane that are compatible with 0.094 < Qxh2 < 0.129 and the laboratory constraints for n > 0 and tan/3 = 5,10,15,20,25,30,35,40,45,50,55. The parts of the strips compatible with gM - 2 at the 2-a level have darker shading.
4.5. A likelihood
analysis
of the
CMSSM
In displaying acceptable regions of cosmological density in the mo,m 1 / 2 plane, it has been assumed that the input parameters are known with perfect accuracy so that the relic density can be calculated precisely. While all of the beyond the standard model parameters are completely unknown and therefore carry no formal uncertainties, standard model parameters such as the top and bottom Yukawa couplings are known but do carry significant uncertainties. The optimal way to combine the various constraints (both phenomenological and cosmological) is via a likelihood analysis. When performing such an analysis, in addition to the formal experimental errors, it is also essential to take into account theoretical errors, which introduce systematic uncertainties that are frequently non-negligible. Recently, we have preformed an extensive likelihood analysis of the CMSSM 74 . The interpretation of the combined Higgs likelihood, Lexp, in the (mi/2,mo) plane depends on uncertainties in the theoretical calculation of mh. These include the experimental error in mt and (particularly at large tan/3) m,b, and theoretical uncertainties associated with higher-order
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m1/2(GeV) Figure 10. As in Fig. 7a, where the range in rno is extended to 5 TeV. In the shaded region at very high mo, there are no solutions for /J. which respect the low energy electroweak symmetry breaking conditions.
corrections to mj,. Our default assumptions are that rnt = 175 ± 5 GeV for the pole mass, and rnb - 4.25 ± 0.25 GeV for the running MS mass evaluated at rnb itself. The theoretical uncertainty in rrih,
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Typically, we find that (dmh/dmt) ~ 0.5, so that ath is roughly 2-3 GeV. The combined experimental likelihood, £ e ip, from direct searches at LEP 2 and a global electroweak fit is then convolved with a theoretical likelihood (taken as a Gaussian) with uncertainty given by ath from (20) above. Thus, we define the total Higgs likelihood function, Ch, as Ch(mh) = -j£— 2TT
j dm'h Cexp(m'h)
c
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(21)
ath
where Af is a factor that normalizes the experimental likelihood distribution. In addition to the Higgs likelihood function, we have included the likelihood function based on b —• 57. While the likelihood function based on the measurements of the anomalous magnetic moment of the muon was considered in 7 4 , it will not be discussed here.
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tan (3 = 10, m 1 / 2 = 300 GeV, \L > 0
1 o>
D
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m 0 (GeV) Figure 11. The value of fj, as a function of mo for fixed m\/2 = 300 GeV and tan j3 — 10 for two choices of mt as indicated. The scale on the right gives the value of Qh2. The curves corresponding to this is scale rise sharply at low mo to values much larger than 1. For mt — 175 GeV and mo ~ 2500 GeV, the value of Clh2 drops to acceptable values when fi becomes small. When the mt = 178 GeV, Clh2 drops at mo ~ 5000 GeV.
Finally, in calculating the likelihood of the CDM density, we take into account the contribution of the uncertainties in mt,b- We will see that the theoretical uncertainty plays a very significant role in this analysis. The likelihood for Q,h2 is therefore, v 1 -Qhz -(Uhl Y/2az -n/i2
(22)
27T0-
where a2 = a2xp + afh, with aexp taken from the WMAP 9 result and a2h from (20), replacing m i by Q,h2. The total likelihood function is computed by combining all the components described above: Ctot = Ch x £bs~i x Caxh2{xCail)
(23)
The likelihood function in the CMSSM can be considered a function of two variables, Ctot(^1/2,^0), where m i / 2 and m 0 are the unified GUT-scale gaugino and scalar masses respectively. Using a fully normalized likelihood function Ctot obtained by combining both signs of /J, for each value of tan /?, we can determine the regions in the (m 1 / 2 ,m 0 ) planes which correspond to specific CLs as shown in Fig. 12.
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The darkest (blue), intermediate (red) and lightest (green) shaded regions are, respectively, those where the likelihood is above 68%, above 90%, and above 95%.
tan|3=10, n>0
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Figure 12. Contours of the likelihood at the 68%, 90% and 95% levels for tan/3 = 10, Ao = 0 and fi > 0 (left panel) or [i < 0 (right panel), calculated using information of mi,, 6 —* sf and CICDM^2 and the current uncertainties in mt and mi,.
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The bulk region is more apparent in the right panel of Fig. 12 for n > 0 than it would be if the experimental error in mt and the theoretical error in rrih were neglected. Fig. 13 complements the previous figures by showing the likelihood functions as they would appear if there were no uncertainty in mt, keeping the other inputs the same. We see that, in this case, both the coannihilation and focus-point strips rise above the 68% CL. 4.6. Beyond
the
CMSSM
The results of the CMSSM described in the previous sections are based heavily on the assumptions of universality of the supersymmetry breaking parameters. One of the simplest generalizations of this model relaxes the assumption of universality of the Higgs soft masses and is known as the NUHM 75 In this case, the input parameters include fj, and TUA, in addition to the standard CMSSM inputs. In order to switch fj, and TUA from outputs to inputs, the two soft Higgs masses, mi,m2 can no longer be set equal to mo and instead are calculated from the electroweak symmetry breaking conditions. The NUHM parameter space was recently analyzed 75 and a sample of the results are shown in Fig. 14. In the left panel of Fig. 14, we see a m ^ 2 , m o plane with a relative low value of /U. In this case, an allowed region is found when the LSP contains a non-negligible Higgsino component which moderates the relic density independent of mo- To the right of this region, the relic density is too small. In the right panel, we see an example of the TUA,^ plane. The crosses correspond to CMSSM points. In this single pane, we see examples of acceptable cosmological regions corresponding to the bulk region, coannihilation region and s-channel annihilation through the Higgs pseudo scalar. Rather than relax the CMSSM, it is in fact possible to further constrain the model. While the CMSSM models described above are certainly mSUGRA inspired, minimal supergravity models can be argued to be still more predictive. In the simplest version of the theory 76 where supersymmetry is broken in a hidden sector, the universal trilinear soft supersymmetrybreaking terms are A = (3 — V3)mo and bilinear soft supersymmetrybreaking term is B = (2 — V3)mo, i.e., a special case of a general relation between B and A, Bo = Ao — moGiven a relation between BQ and Ao, we can no longer use the standard CMSSM boundary conditions, in which m i / 2 , mo, A0, tan/3, and sgn(ii) are input at the GUT scale with fi and B determined by the electroweak symmetry breaking condition. Now, one is forced to input Bo and instead tan /?
Astroparticle Physics tan(i = 3 5 . \x = 400 GeV, mx = 700 GeV
"•
""»
m1/2(GeV)
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tanP = 10, m , / 2 = ?0<>, m„ = 100
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Figure 14. a) The NUHM (m 1 / 2 ,m 0 ) plane for tan/3 = 35, (a) \i = 400 GeV and mA = 700 GeV b)the NUHM (jx, mA) plane for tan0 = 10, mo = 100 GeV and m 1 / 2 = 300 GeV, with A0 = 0. The (red) dot-dashed lines are the contours mh = 114 GeV, and the near-vertical (black) dashed lines are the contours m%± = 103.5 GeV. The dark (black) dot-dashed lines indicate the GUT stability constraint. Only the areas inside these curves (small n) ore allowed by this constraint. The light (turquoise) shaded areas are the cosmologically preferred regions with 0.1 < ttxh2 ^ °-3- Tfle darker (blue) portion of this region corresponds to the WMAP densities. The dark (brick red) shaded regions is excluded because the stau is the LSP, and the lighter (yellow) shaded regions is excluded because the LSP is a sneutrino. The medium, (green) shaded region is excluded byb—> ay. The regions allowed by the g - 2 constraint are shaded (pink) and bounded by solid black lines. The solid (blue) curves correspond to mx = m.A/2.
is calculated from the minimization of the Higgs potential 77 . In Fig. 15, the contours of tan/3 (solid blue lines) in the (mi/2,m0) planes for two values of A = A0/m0, B = B0/m0 = A - 1 and the sign of /x are displayed 77 . In panel (a) of Fig. 15, we see that the Higgs constraint combined with the relic density requires tan/3 £, 11, whilst the relic density also enforces tan/3 <, 20. For a given point in the mx/2 - m0 plane, the calculated value of tan/3 increases as A increases. This is seen in panel (b) of Fig. 15, when A = 2.0, close to its maximal value for /i > 0, the tan/3 contours turn over towards smaller rn1/2, and only relatively large values 25
Detectability
Direct detection techniques rely on an ample neutralino-nucleon scattering cross-section. In Fig. 16, we display the allowed ranges of the spin-
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Figure 15. Examples o/(m1//2,'Wo) planes with contours of tan 0 superposed, for fi > 0 and (a) the simplest Polonyi model with A = 3 — \/3, B = A — 1 and (b) A = 2.0, B = A — 1. In each panel, we show the regions excluded by the LEP lower limits on MSSM particles, those ruled out by b —* s-y decay (medium green shading), and those excluded because the LSP would be charged (dark red shading). The region favoured by the WMAP range has light turquoise shading. The region suggested by g^ — 2 is medium (pink) shaded.
independent cross sections in the NUHM when we sample randomly tan/3 as well as the other NUHM parameters 78 . The raggedness of the boundaries of the shaded regions reflects the finite sample size. The dark shaded regions includes all sample points after the constraints discussed above (including the relic density constraint) have been applied. In a random sample, one often hits points which are are perfectly acceptable at low energy scales but when the parameters are run to high energies approaching the GUT scale, one or several of the sparticles mass squared runs negative. This has been referred to as the GUT constraint here. The medium shaded region embodies those points after the GUT constraint has been applied. After incorporating all the cuts, including that motivated by g^ — 2, we find that the light shaded region where the scalar cross section has the range 10 ~ 6 pb ^ &si ^ 10~ 10 pb, with somewhat larger (smaller) values being possible in exceptional cases. The results from this analysis 78 for the scattering cross section in the NUHM (which by definition includes all CMSSM results) are compared with the previous CDMS 79 and Edelweiss80 bounds as well as the recent CDMSII results 81 in Fig. 16. While previous experimental sensitivities were not strong enough to probe predictions of the NUHM, the current CDMSII
Astroparticle Physics 805
0
100
200
300
400
500
600
700
m Y (GeV) Figure 16. Ranges of the spin-independent cross section in the NUHM. The ranges allowed by the cuts on fixh2; mh and b -* sy have dark shading, those still allowed by the GUT stability cut have medium shading, and those still allowed after applying all the cuts including
bound has begun to exclude realistic models and it is expected that these bounds improve by a factor of about 20. See ref. 8 3 for updated direct detection calculations in the MSSM.
5. Big bang nucleosynthesis The standard model 84 of big bang nucleosynthesis (BBN) is based on the relatively simple idea of including an extended nuclear network into a homogeneous and isotropic cosmology. Apart from the input nuclear cross sections, the theory contains only a single parameter, namely the baryonto-photon ratio, rj. Other factors, such as the uncertainties in reaction rates, and the neutron mean-life can be treated by standard statistical and
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Monte Carlo techniques 85 ' 86 ' 87 - 88 - 89 . The theory then allows one to make predictions (with well-defined uncertainties) of the abundances of the light elements, D, 3 He, 4 He, and 7 Li. 5.1.
Theory
Conditions for the synthesis of the light elements were attained in the early Universe at temperatures T J> 1 MeV. In the early Universe, the energy density was dominated by radiation with
from the contributions of photons, electrons and positrons, and N„ neutrino flavors (at higher temperatures, other particle degrees of freedom should be included as well). At these temperatures, weak interaction rates were in equilibrium. In particular, the processes n + e+ <-> p + ve n + ve <-> p + e~ n <-> p + e~ + Ve
(25)
fix the ratio of number densities of neutrons to protons. At T ;§> 1 MeV, (n/p) ~ 1. The weak interactions do not remain in equilibrium at lower temperatures. Freeze-out occurs when the weak interaction rate, Twk ~ G%T5 falls below the expansion rate which is given by the Hubble parameter, H ~ VUFp ~ T2/MP, where MP = 1/y/G^ ~ 1.2 x 10 19 GeV. The /3-interactions in eq. (25) freeze-out at about 0.8 MeV. As the temperature falls and approaches the point where the weak interaction rates are no longer fast enough to maintain equilibrium, the neutron to proton ratio is given approximately by the Boltzmann factor, (n/p) ~ e ~ A m / T ~ 1/6, where Am is the neutron-proton mass difference. After freeze-out, free neutron decays drop the ratio slightly to about 1/7 before nucleosynthesis begins. The nucleosynthesis chain begins with the formation of deuterium by the process, p + n —> D + 7. However, because of the large number of photons relative to nucleons, r]^1 = TI^/UB ~ 10 10 , deuterium production is delayed past the point where the temperature has fallen below the deuterium binding energy, EB = 2.2 MeV (the average photon energy in a blackbody is E7 ~ 2.IT). This is because there are many photons in the exponential tail of the photon energy distribution with energies E > EB
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despite the fact that the temperature or Ey is less than EB- The degree to which deuterium production is delayed can be found by comparing the qualitative expressions for the deuterium production and destruction rates, (26)
rigay
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When the quantity r / ~ 1 e x p ( - £ B / T ) ~ 1, the rate for deuterium destruction (D + 7 —> p + n) finally falls below the deuterium production rate and the nuclear chain begins at a temperature T ~ O.lMeV. The dominant product of big bang nucleosynthesis is 4 He and its abundance is very sensitive to the (n/p) ratio 2(n/p) [1 + (n/p)}
0.25
(27)
i.e., an abundance of close to 25% by mass. Lesser amounts of the other light elements are produced: D and 3 He at the level of about I O - 5 by number, and 7 Li at the level of 1 0 - 1 0 by number.
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Recently the input nuclear data have been carefully reassessed 87 ' 88 ' 89 ' 90 ' 91 , leading to improved precision in the abundance predictions. The NACRE collaboration presented an updated nuclear compilation 90 . For example, notable improvements include a reduction in the uncertainty in the rate for 3 He(n,p)T from 10% 86 to 3.5% and for T(a,7) 7 Li from ~ 23 - 30% 86 to ~ 4%. Since then, new data and techniques have become available, motivating new compilations. Within the last year, several new BBN compilations have been presented 92 ' 93 ' 94 . The resulting elemental abundances predicted by standard BBN are shown in Fig. 17 as a function of r? 88 . The left plot shows the abundance of He by mass, Y, and the abundances of the other three isotopes by number. The curves indicate the central predictions from BBN, while the bands correspond to the uncertainty in the predicted abundances. This theoretical uncertainty is shown explicitly in the right panel as a function of 77. In the standard model with Nv = 3, the only free parameter is the density of baryons which sets the rates of the strong reactions. Thus, any abundance measurement determines 77, while additional measurements overconstrain the theory and thereby provide a consistency check. BBN has thus historically been the premier means of determining the cosmic baryon density. With the increased precision of microwave background anisotropy measurements, it is now possible to use the the CMB to independently determine the baryon density. The WMAP value for fig ft2 = 0.0224 translates into 7710 = 6.14 ±0.25
(28)
With 77 fixed by the CMB, precision comparisons to the observations can now be attempted 95 . 5.2. Light element
observations
and comparison
with 3
4
theory
BBN theory predicts the universal abundances of D, He, He, and 7 Li, which are essentially determined by t ~ 180 s. Abundances are however observed at much later epochs, after stellar nucleosynthesis has commenced. The ejected remains of this stellar processing can alter the light element abundances from their primordial values, and produce heavy elements such as C, N, O, and Fe ("metals"). Thus one seeks astrophysical sites with low metal abundances, in order to measure light element abundances which are closer to primordial. For all of the light elements, systematic errors
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are an important and often dominant limitation to the precision of derived primordial abundances. 5.2.1. D/H In recent years, high-resolution spectra have revealed the presence of D in high-redshift, low-metallicity quasar absorption systems (QAS), via its isotope-shifted Lyman-a absorption. These are the first measurements of light element abundances at cosmological distances. It is believed that there are no astrophysical sources of deuterium 96 , so any measurement of D/H provides a lower limit to primordial D/H and thus an upper limit on 77; for example, the local interstellar value of D/H=(1.5 ± 0.1) x 1 0 - 5 citelin requires that ryio < 9. In fact, local interstellar D may have been depleted by a factor of 2 or more due to stellar processing; however, for the highredshift systems, conventional models of galactic nucleosynthesis (chemical evolution) do not predict significant D/H depletion 98 . The five most precise observations of deuterium 99,100 ' 101 ' 102 in QAS give D/H = (2.78 ± 0.29) x 1 0 - 5 , where the error is statistical only. These are shown in Fig. 18 along with some other recent measurements 103 ' 104 ' 105 . Inspection of the data shown in the figure clearly indicates the need for concern over systematic errors. We thus conservatively bracket the observed values with a range D/H = 2 — 5 x 10~ s which corresponds to a range in ?7io of 4 - 8 which easily brackets the CMB determined value. Using the WMAP value for the baryon density (28) the primordial D/H abundance is predicted to be 8 8 ' 9 2 : (D/H) p = 2.55i°; 21 x 10" 5
(29)
As one can see, this value is in very good agreement with the observational value. 5.3.
4
He
We observe 4 He in clouds of ionized hydrogen (HII regions), the most metalpoor of which are in dwarf galaxies. There is now a large body of data on 4 He and CNO in these systems 106 . Of the modern 4 He determinations, the work of Pagel et al. 1 0 7 established the analysis techniques that were soon to follow108. Their value of Yp = 0.228 ± 0.005 was significantly lower than that of a sample of 45 low metallicity HII regions, observed and analyzed in a uniform manner 106 , with a derived value of Yp = 0.244 ± 0.002. An
810
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[Si/H] Figure 18.
D / H abundances shown as a function of [Si/H].
analysis based on the combined available data as well as unpublished data yielded an intermediate value of 0.238 ± 0.002 with an estimated systematic uncertainty of 0.005 1 0 9 . An extended data set including 89 HII regions obtained Yp = 0.2429 ± 0.0009 u o . However, the recommended value is based on the much smaller subset of 7 HII regions, finding Yp = 0.2421 ± 0.0021. 4 He abundance determinations depend on a number of physical parameters associated with the HII region in addition to the overall intensity of the He emission line. These include, the temperature, electron density, optical depth and degree of underlying absorption. A self-consistent analysis may use multiple 4 He emission lines to determine the He abundance, the electron density and the optical depth. In 106 , five He lines were used, underlying He absorption was assumed to be negligible and used temperatures based on OIII data. The question of systematic uncertainties was addressed in some detail in Ref. 111. It was shown that there exist severe degeneracies inherent in the self-consistent method, particularly when the effects of underlying ab-
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sorption are taken into account. The results of a Monte-Carlo reanalysis 112 of NCG 346 113 is shown in Fig. 19. In the left panel, solutions for the 4 He abundance and electron density are shown (symbols are described in the caption). In the right panel, a similar plot with the 4 He abundance and the equivalent width for underlying absorption is shown. As one can see, solutions with no absorption and high density are often indistinguishable (i.e., in a statistical sense they are equally well represented by the data) from solutions with underlying absorption and a lower density. In the latter case, the He abundance is systematically higher. These degeneracies are markedly apparent when the data is analyzed using Monte-Carlo methods which generate statistically viable representations of the observations as shown in Fig. 19. When this is done, not only are the He abundances found to be higher, but the uncertainties are also found to be significantly larger than in a direct self-consistent approach. 'II111
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Figure 19. Results of modeling of 6 He I line observations of NGC 346 1 1 3 . The solid lines show the original derived values and the dashed lines show the 1 a errors on those values. The solid circles (with error bars) show the results of the x 2 minimization solution (with calculated errors) 1 1 2 . The small points show the results of Monte Carlo realizations of the original input spectrum. The solid squares (with error bars) show the means and dispersions of the output values for the \ 2 minimization solutions of the Monte Carlo realizations.
Recently a careful study of the systematic uncertainties in 4 He, particularly the role of underlying absorption has been performed using a subset of the highest quality from the data of Izotov and Thuan 1 0 6 . All of the physical parameters listed above including the 4 He abundance were determined self-consistently with Monte Carlo methods 111 . Note that the 4 He abundances are systematically higher, and the uncertainties are several times larger than quoted in 106 . In fact this study has shown that the determined
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K. A. Olive
value of Yp is highly sensitive to the method of analysis used. The result is shown in Fig. 20 together with a comparison of the previous result. The extrapolated 4 He abundance was determined to be Yp = 0.2495 ± 0.0092. The value of r\ corresponding to this abundance is r/io = G ^ t ^ 8 and clearly overlaps with T\CMB- Conservatively, it would be difficult at this time to exclude any value of Yp inside the range 0.232 - 0.258. At the WMAP value for j], the 4 He abundance is predicted to be 8 8 ' 9 2 : Yp = 0.2485 ± 0.0005
(30)
This value is considerably higher than any prior determination of the primordial 4 He abundance, it is in excellent agreement with the most recent analysis of the 4 He abundance 112 . Note also that the large uncertainty ascribed to this value indicates that the while 4 He is certainly consistent with the WMAP determination of the baryon density, it does not provide for a highly discriminatory test of the theory at this time.
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5.4.
7
Li/iJ
The systems best suited for Li observations are metal-poor halo stars in our Galaxy. Observations have long shown 114 that Li does not vary significantly in Pop II stars with metallicities < 1/30 of solar — the "Spite plateau". Recent precision data suggest a small but significant correlation between
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Li and Fe 115 which can be understood as the result of Li production from Galactic cosmic rays 116 . Extrapolating to zero metallicity one arrives at a primordial value 117 Li/H|„ = (1.23±g;?|) x 1(T 10 . Figure 21 shows the different Li components for a model with ( 7 Li/H) p = 1.23 x 10~ 10 . The linear slope produced by the model is independent of the input primordial value. The model of ref. 118 includes in addition to primordial 7 Li, lithium produced in galactic cosmic ray nucleosynthesis (primarily a + a fusion), and 7 Li produced by the z/-process during type II supernovae. As one can see, these processes are not sufficient to reproduce the population I abundance of 7 Li, and additional production sources are needed. Recent data 1 1 9 with temperatures based on Ha lines (considered to give systematically high temperatures) yields 7 Li/H = (2.19 ± 0.28) x 10~ 10 . These results are based on a globular cluster sample (NGC 6397). This result is consistent with previous Li measurements of the same cluster which gave 7 Li/H = (1.91±0.44) x 1(T 10 12° and 7 Li/H = (1.69±0.27) x 10" 1 0 m . A related study (also of globular cluster stars) gives 7 Li/H = (2.29 ±0.94) x i g - 1 0 122
The 7 Li abundance based on the WMAP baryon density is predicted to be - : 88 92
814
K. A. Olive 7
Li/H = 4.26J:{J;SJ
x
!0~ 1 0
(31)
This value is in clear contradiction with most estimates of the primordial Li abundance. It is a factor of ~ 3 higher than the value observed in most halo stars, and just about 0.2 dex over the globular cluster value. 5.5.
Concordance
In Fig. 22, we show the direct comparison between the BBN predicted abundances given in eqs. (29), (30), and (31), using the WMAP value of 7710 = 6.25 ± 0.25 with the observations 123 . As one can see, there is very good agreement between theory and observation for both D/H and 4 He. Of course, in the case of 4 He, concordance is almost guaranteed by the large errors associated to the observed abundance. In contrast, as was just noted above, there is a marked discrepancy in the case of 7 Li.
10 B xD/H
Yp
10 lo **Li/H
Figure 22. Primordial light element abundances as predicted by BBN and W M A P (dark shaded regions) 1 2 3 . Different observational assessments of primordial abundances are plotted as follows: (a) the light shaded region shows D / H = (2.78 ± 0.29) X 10~ 5 ; (b) the light shaded region shows Yp = 0.249 ± 0.009; (c) the light shaded region shows 7 L i / H = 1 . 2 3 l ° ; ^ x 1 0 - 1 0 , while the dashed curve shows 7 L i / H = (2.19±0.28) x 1 0 " 1 0 .
The quoted value for the 7 Li abundance assumes that the Li abundance in the stellar sample reflects the initial abundance at the birth of the star. However, an important source of systematic uncertainty comes from the possible depletion of Li over the > 10 Gyr age of the Pop II stars. The atmospheric Li abundance will suffer depletion if the outer layers of the stars have been transported deep enough into the interior, and/or mixed
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with material from the hot interior; this may occur due to convection, rotational mixing, or diffusion. Standard stellar evolution models predict Li depletion factors which are very small (<0.05 dex) in very metal-poor turnoff stars 124 . However, there is no reason to believe that such simple models incorporate all effects which lead to depletion such as rotationallyinduced mixing and/or diffusion. Current estimates for possible depletion factors are in the range ~ 0.2-0.4 dex 125 . As noted above, this data sample 115 shows a negligible intrinsic spread in Li leading to the conclusion that depletion in these stars is as low as 0.1 dex. Another important source for potential systematic uncertainty stems from the fact that the Li abundance is not directly observed but rather, inferred from an absorption line strength and a model stellar atmosphere. Its determination depends on a set of physical parameters and a modeldependent analysis of a stellar spectrum. Among these parameters, are the metallicity characterized by the iron abundance (though this is a small effect), the surface gravity which for hot stars can lead to an underestimate of up to 0.09 dex if log g is overestimated by 0.5, though this effect is negligible in cooler stars. Typical uncertainties in logg are ±0.1 — 0.3. The most important source for error is the surface temperature. Effective-temperature calibrations for stellar atmospheres can differ by up to 150-200 K, with higher temperatures resulting in estimated Li abundances which are higher by ~ 0.08 dex per 100 K. Thus accounting for a difference of 0.5 dex between BBN and the observations, would require a serious offset of the stellar parameters. While there has been a recent analysis 126 which does support higher temperatures, the consequences of the higher temperatures on the inferred abundances of related elements such as Be, B, and O observed in the same stars is somewhat negative 127 . Finally a potential source for systematic uncertainty lies in the BBN calculation of the 7 Li abundance. As one can see from Fig. 17, the predictions for 7 Li carry the largest uncertainty of the 4 light elements which stem from uncertainties in the nuclear rates. The effect of changing the yields of certain BBN reactions was recently considered by Coc et al. 91 . In particular, they concentrated on the set of cross sections which affect 7 Li and are poorly determined both experimentally and theoretically. In many cases however, the required change in cross section far exceeded any reasonable uncertainty. Nevertheless, it may be possible that certain cross sections have been poorly determined. In 9 1 , it was found for example, that an increase of either the 7Li(d, n)2 4 He or 7 Be(d,p)2 4 He reactions by a factor of 100 would reduce the 7 Li abundance by a factor of about 3.
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The possibility of systematic errors in the 3 He(a, 7) 7 Be reaction, which is the only important 7 Li production channel in BBN, was considered in detail in 128 . The absolute value of the cross section for this key reaction is known relatively poorly both experimentally and theoretically. However, the agreement between the standard solar model and solar neutrino data thus provides additional constraints on variations in this cross section. Using the standard solar model of Bahcall 129 , and recent solar neutrino data 130 , one can exclude systematic variations of the magnitude needed to resolve the BBN 7 Li problem at the > 95% CL 128 . Thus the "nuclear fix" to the 7 Li BBN problem is unlikely. Finally, we turn to 3 He. Here, the only observations available are in the solar system and (high-metallicity) HII regions in our Galaxy 131 . This makes inference of the primordial abundance difficult, a problem compounded by the fact that stellar nucleosynthesis models for 3 He are in conflict with observations 132 . Consequently, it is not appropriate to use 3 He as a cosmological probe 133 ; instead, one might hope to turn the problem around and constrain stellar astrophysics using the predicted primordial 3 He abundance 134 . For completeness, we note that the 3 He abundance is predicted to be: 3
He/H = 9.28i°yJ4 x 1(T 6
(32)
at the WMAP value of rj. 6. Constraints on decaying particles and gravitino dark m a t t e r from B B N As an example of constraints on particle properties from BBN, I will concentrate here on life-time and abundance limits on decaying particles as it ties in well with the previous discussion on supersymmetric dark matter. There are of course many other constraints on particle properties which can be derived from BBN, most notably the limit on the number of relativistic degrees of freedom. For a recent update on these limits, see 123 . Because there is good overall agreement between the theoretical predictions of the light element abundances and their observational determination, any departure from the standard model (or either particle physics, cosmology, or BBN) generally leads to serious inconsistencies among the element abundances. Gravitinos have long been known to be potentially problematic in cosmology135. If gravitinos are present with equilibrium number densities,
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we can write their energy density as /3C(3)\ A3/2 = "l3/2«3/2 = W13/2 ( —5- J TI/2
( 33 )
where today one expects that the gravitino temperature T3/2 is reduced relative to the photon temperature due to the annihilations of particles dating back to the Planck time 54 . Typically one can expect the gravitino abundance Y3/2 = n3/2/n^ ~ (T 3 / 2 /T 7 ) 3 ~ 10~ 2 . Then for tt3/2h2 < 1, we obtain the limit that 7713/2 £j 1 keV. Of course, the above mass limit assumes a stable gravitino, the problem persists however, even if the gravitino decays, since its gravitational decay rate is very slow. Gravitinos decay when their decay rate, r 3 / 2 — m3/2/Mp, becomes comparable to the expansion rate of the Universe (which becomes dominated by the mass density of gravitinos), H ~ m3',2T3j2/Mp. Thus decays occur at Td ~ m3',2/Mp' heated" to a temperature
. After the decay, the Universe is "re-
TR ~ p(Td)1/4
~ m^l/Mp'2
(34)
The Universe must reheat sufficiently so that big bang nucleosynthesis occurs in a standard radiation dominated Universe. For TR J> 1 MeV, we must require m 3 / 2 S; 20 TeV. This large value threatens the solution of the hierarchy problem. Inflation could alleviate the gravitino problem by diluting the gravitino abundance to safe levels 136 . If gravitinos satisfy the noninflationary bounds, then their reproduction after inflation is never a problem. For gravitinos with mass of order 100 GeV, dilution without over-regeneration will also solve the problem, but there are several factors one must contend with in order to be cosmologically safe. Gravitino decay products can also upset the successful predictions of Big Bang nucleosynthesis, and decays into LSPs (if R-parity is conserved) can also yield too large a mass density in the now-decoupled LSPs 48 . For unstable gravitinos, the most restrictive bound on their number density comes form the photo-destruction of the light elements produced during nucleosynthesis 137 ' 138 ' 139 . Here, we will consider electromagnetic decays, meaning that the decays inject electromagnetic radiation into the early universe. If the decaying particle is abundant enough or massive enough, the injection of electromagnetic radiation can photo-erode the light elements created during primordial nucleosynthesis. The theories we have in mind are generally supersymmetric, in which the gravitino and neutralino are the next-to-lightest and lightest
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supersymmetric particles, respectively (or vice versa), but the constraints hold for any decay producing electromagnetic radiation. We thus constrain the abundance of such a particle given its mean lifetime TX • The abundance is constrained through the parameter C,x = mx^x/n-y The BBN limits in the (Cx,Tx) plane is shown in Fig. 23 1 4 0 . The constraint placed by the 4 He abundance comes from its lower limit, as this scenario destroys 4 He. Shown are the limits assuming Ymin = 0.232 and 0.227 138.123>!40 The area above these curves are excluded. The deuterium lines correspond to the contours (1.3 or 2.2) x 1 0 - 5 < D/H < 5.3 x 10~ 5 . The first of the lower bounds is the higher line to the left of the cleft, and represents the very conservative lower limit on D/H assumed in 138 . The range 1.3 - 5.3 x l O - 5 effectively brackets all recent observations of D/H in quasar absorption systems as discussed above. The second of the lower bounds is the lower line on the left side and represents the 2-a lower limit in the best set of D/H observations. The upper bound is the line to the right of the cleft. A priori, there is also a narrow strip at larger C,x and TX where the D/H ratio also falls within the acceptable range but this is excluded by the observed 4 He abundance.
-15
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10
11
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L o g [Xx (sec)] Figure 23. The constraints imposed by the astrophysical observations of4 He (red lines), D/H (green lines), 6 L i (yellow line), 6Li/7Li (blue lines), 7Li (blue band) and 3He (black lines).
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The constraint imposed by the 6 Li abundance is shown 138 as a solid yellow line in Fig. 23. Also shown, as solid blue lines, are two contours representing upper limits on the 6 Li/ 7 Li ratio: 6 Li/ 7 Li < 0.07 or 0.15. The lower number was used in 138 and represented the upper limit available at the time, which was essentially based on multiple observations of a single star. The most recent data 1 4 1 includes observations of several stars. The Li isotope ratio for most metal-poor stars in the sample is as high as 0.15, and we display that upper limit here 140 . The main effect of this constraint is to disallow a region in the near-vertical cleft between the upper and lower limits on D/H, as seen in Fig. 23. The blue shaded band in Fig. 23 corresponds to a 7 Li abundance of 0.9 x l O - 1 0 < 7 Li/H < (2 or 3) x 10~ 10 , with the 7 Li abundance decreasing as (x increases and the intensity of the shading changing at the intermediate value. In 138 , only the lower bound was used due the existing discrepancy between the primordial and observationally determined values. It is apparent that 7 Li abundances in the lower part of the range are possible only high in the Deuterium cleft, and even then only if one uses the recent and more relaxed limit on the 6 Li/ 7 Li ratio. Values of the 7 Li abundance in the upper part of the range are possible, however, even if one uses the more stringent constraint on 6 Li/ 7 Li. In this case, the allowed region of parameter space would also extend to lower T\ , if one could tolerate values of D/H between 1.3 and 2.2 x 10" 5 . Finally, we show the impact of the 3 He constraint 139,140 . Since Deuterium is more fragile than 3 He, whose abundance is thought to have remained roughly constant since primordial nucleosynthesis when comparing the BBN value to it proto-solar abundance, one would expect, in principle, the 3 He/D ratio to have been increased by stellar processing. Since D is totally destroyed in stars, the ratio of 3 He/D can only increase in time or remain constant if 3 He is also completely destroyed in stars. The present or proto-solar value of 3 He/D can therefore be used to set an upper limit on the primordial value. Fig. 23 displays the upper limits 3 He/D < 1 or 2 as solid black lines. Above these contours, the value of 3 He/D increases very rapidly, and points high in the Deuterium cleft of Fig. 23 have absurdly high values of 3 He/D, exceeding the limit by an order of magnitude or more. The previous upper limit on rjx 138 corresponded to the constraint mxnx/n^ < 5.0 x 10~ 12 GeV for TX = 108 s. The weaker (stronger)
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version of the 3 He constraint adopted corresponds 140 to mx—
< 2.0(0.8) x 10" 1 2 GeV
(35)
= 108 s. Returning to the case of a decaying gravitino, recall that thermal reactions are estimated to produce an abundance of gravitinos given by 142>138:
for
TX
i - i i .. (
TR
Assuming that m 3 / 2 = 100 GeV and TX = 108 s, and imposing the constraints (35), we now find TR < (0.8 - 2.8) x 107 GeV, ((0.3 - 1.1) x 107 GeV)
(37)
for the weaker (stronger) version of the 3 He constraint. Finally, we consider the possibility that gravitinos are stable and the LSP 1 4 3 ' 1 4 4 . In this case, in the CMSSM, the next lightest supersymmetric particle (NSP) is either the neutralino or the stau. In Fig. 24, we fix the ratio of supersymmetric Higgs vacuum expectation values tan/3 = 10 (left panel), and tan/3 = 57 (right panel), and assume m 3 / 2 = 100 GeV. In each panel of Fig. 24, we display accelerator, astrophysical and cosmological constraints in the corresponding ( m x ^ m o ) planes as discussed above for the CMSSM, concentrating on the regions to the right of the near-vertical black lines, where the gravitino is the LSP. The NSP is the r lepton below the (red) dotted line. Below and to the right of the upper (purple) dashed lines, the density of relic gravitinos produced in the decays of other supersymmetric particles is always below the WMAP upper limit: Q3/2h2 < 0.129. The code used in 138 , when combined with the observational constraints used in 138 , yielded the astrophysical constraint represented by the dashed grey-green lines in both panels of Fig. 24 and did not include the constraint due to 3 He/D. These constraints on the CMSSM parameter plane were computed in 144 . For each point in the ( m ^ 2 , m o ) , the relic density of either \ o r T is computed and (x is determined using flxh2 = 3.9 x 107 GeV Oc- When X — f, Qx is reduced by a factor of 0.3, as only 30% of stau decays result in electromagnetic showers which affect the element abundances at these lifetimes. In addition, at each point, the lifetime of the NSP is computed. Then for each TX, the limit on £x is found from the results shown in Fig. 23. The region to the right of this curve where r = Cx/Cx""' < 1 i s allowed.
Astro-particle Physics m3/2 = 100 GeV, tan f = 10, n > 0
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m 3 / 2 = 100GeV,tanP = 5 7 , p.>0
Ji'V 400-
Sam-
/
£1
M
•••
/'Si™
•
200-
'W 100-
*s /
\
'
1
r
<1
' I Mil
m^GeV)
1000
mm (GeV)
Figure 24. The ( m ^ ) "K>) planes for fj, > 0, m 3 / 2 = 100 GeV and (a) tan 13 = 10 (b) tan/8 = 57. We restrict our attention to the regions between the solid black lines, where the gravitino is the LSP and the NSP lifetime exceeds 104 s. In each panel, the nearvertical dashed black (dash-dotted red) line is the constraint mx± > 104 GeV (m>, > 114 GeV), the upper (purple) dashed line is the constraint Q3/2h2 < 0.129, and the light green shaded region is that where the NSP would have had 0.094 < Clh2 < 0.129 if it had not decayed. The solid red (dashed grey-green) line is the region now (previously) allowed by the light-element abundances: r < 1 as described in the text. The red (blue) shaded region is that where the 7Li abundance could have been improved by NSP decays, but which is now excluded by the 3 He (D) constraint.
The astrophysical constraints obtained with the newer abundance limits 140 yields the solid red lines in Fig. 24. The examples where TX and Cx for the NSP decays fall within the ranges shown by the blue band of Fig. 23, and hence are suitable for modifying the 7 Li abundance 145,146 , are shown as red and blue shaded regions in each panel of Fig. 24. If we had been able to allow a Deuterium abundance as low as D/H ~ (1 - 2) x 10 ~ 5 , the blue shaded region would have been able to resolve the Li discrepancy in the context of the CMSSM with gravitino dark matter. The blue region that we now regard as excluded by the lower limit on D/H, which is stronger than that used in 138 , extends to large m 1 / 2 . The red shaded region, which is consistent even with this limit on D/H, but yields very large 3 He/D. Fig. 24 show as solid red lines the additional restrictions these constraints impose on the (mi/ 2 ,mo) planes 140 . 7. The variation of fundamental constants There has been considerable interest of late in the possible variation of the fundamental constants. The construction of theories with variable
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"constants" is straightforward. Consider for example a gravitational Lagrangian which contains the term £ ~R,
(38)
whereis some scalar field and R is the Einstein curvature scalar. The gravitational constant is determined if the dynamics of the theory fix the expectation value of the scalar field so that GN
m
= idw) •
Similarly a coupling in the Lagrangian of a scalar to the Maxwell term F2, fixes the fine-structure constant
Indeed, gravitational theories of the Jordan-Brans-Dicke type do contain the possibility for a time-varying gravitational constant. However, these theories can always be re-expressed such that GN is fixed and other mass scales in the theory become time dependent (i.e., dependent on the scalar field). For example, the JBD action can be written as /
d4x,/g 0R ~ T 5 M ^ M ( A + A +
(41)
£m(lpmatter,guv)
where a; is a number which characterizes the degree of departure from general relativity (GR is recovered as w —> oo), A is the cosmological constant, and the matter action for electromagnetism and a single massive fermion can be written as Cm = - — F 2 - * # > * - T O * * .
(42)
Written this way, if the scalar field <j> evolves, then GN does as well. In another conformal frame, the JBD action can be rewritten as /
dtxy/g
1
2'
2
S2
4e2
d>2
(43) In this frame, Newton's constant is constant, but the fermion mass (after * is rescaled) varies as >-1/2 and the cosmological constant varies as l/>2. The physics described by either of these two actions is identical and the two frames can not be distinguished as the measurable dimensionless quantity Gm2 oc (j)"1 in both frames. While the fine-structure constant remains constant in this construction, it is straight forward to consider theories
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where it is not. In what follows, I will restrict attention to variations in the fine-structure constant. In any unified theory in which the gauge fields have a common origin, variations in the fine structure constant will be accompanied by similar variations in the other gauge couplings 147 (see also, 1 4 8 ). In other words, variations of the gauge coupling at the unified scale will induce variations in all of the gauge couplings at the low energy scale. It is easy to see that the running of the strong coupling constant has dramatic consequences for the low energy hadronic parameters, including the masses of nucleons 147 . Indeed the masses are determined by the QCD scale, A, which is related to the ultraviolet scale, M\jy, by dimensional transmutation:
where 63 is a usual renormalization group coefficient that depends on the number of massless degrees of freedom, running in the loop. Clearly, changes in gs will induce (exponentially) large changes in A: AA = 2TT Aas(Muv) Aas{Muv) [ A 9as{Muv) as(Muv) ^ as{Muv) ' ' where for illustrative purposes we took the beta function of QCD with three fermions. On the other hand, the electromagnetic coupling a never experiences significant running from Muv to A and thus AA/A » Aa/a. A more elaborate treatment of the renormalization group equations above Mz 149 leads to the result that is in perfect agreement with 147 : AA „ A« , x -30 . 46 v A a ' In addition, we expect that not only the gauge couplings will vary, but all Yukawa couplings are expected to vary as well. In 147 , the string motivated dependence was found to be ^ = ** (47) h av where au is the gauge coupling at the unification scale and h is the Yukawa coupling at the same scale. However in theories in which the electroweak scale is derived by dimensional transmutation, changes in the Yukawa couplings (particularly the top Yukawa) leads to exponentially large changes in the Higgs vev. In such theories, the Higgs expectation value corresponds to the renormalization point and is given qualitatively by v ~ Mpexp(-27rc/a t ) (48)
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where c is a constant of order 1, and at = h2/An. Thus small changes in ht will induce large changes in v. For c ~ \i% ~ 1, AT;
— ~ 80 V
Aan
„ , (49)
OL\J
This dependence gets translated into a variation in all low energy particle masses. In short, once we allow a. to vary, virtually all masses and couplings are expected to vary as well, typically much more strongly than the variation induced by the Coulomb interaction alone. Unfortunately, it is very hard to make a quantitative prediction for A«/« simply because we do not know exactly how the dimensional transmutation happens in the Higgs sector, and the answer will depend, for example, on such things as the dilaton dependence of the supersymmetry breaking parameters. This uncertainty is characterized in Eq. (48) by the parameter c. For the purpose of the present discussion it is reasonable to assume that Av/v is comparable but not exactly equal to AA/A. That is, although they are both O(10 — 100)Aa/a, their difference |AA/A — Av/v\ is of the same order of magnitude which we will take as ~ 50Aa/a. Much of the recent excitement over the possibility of a time variation in the fine structure constant stems from a series of recent observations of quasar absorption systems and a detailed fit of positions of the absorption lines for several heavy elements using the "many-multiplet" method 150 ' 151 . A related though less sensitive method for testing the variability of a, is the alkali doublet method, which neatly describes the physics involved. Absorption clouds are prevalent along the lines of sight towards distant, high redshift quasars. As such, the quasar acts as a bright source, and the absorption features seen in these clouds reflect their chemical abundances. Consider an absorption feature in a doublet system involving for example, S1/2 —> -P3/2 a n d S\/2 —> -Pi/2 transitions. While the overall wavelength position of the doublet is a measure of the redshift of the absorption cloud, the separation of the two lines is a measure of the fine structure constant. This is easily seen by recalling the energy splitting due to the spin-orbit coupling, AE ~ -4^S
• L ~ me8,
~
~ e4 ~ a2.
(50)
Since the line splitting AA/A ~ AE/E, the relative change in the line splitting is directly proportional to Aa/a. The many multiplet method compares transitions from different multiplets and different atoms and utilizes the effects of relativistic corrections on the spectra. The alkali doublet
Astroparticle Physics 825
method 152 has been applied to quasar absorption spectra, but the sensitivity of the method only limits the variation in a within an of order 10 ~ 5 . Similarly, at present, considerations based on OIII emission line systems 153 are also only able to set limits on the variation of a at the level of 10 . In contrast, the many multiplet method based on the relativistic corrections to atomic transitions using several transition lines from several elemental species allows for sensitivities which approach the level of IQ-Q 150,151,154 r p ^ m e ^ n o ( j compares the line shifts of elements which are particularly sensitive to changes in a with those that are not. At relatively low redshift (z < 1.8), the method relies on the comparison of Fe lines to Mg lines. At higher redshift, the comparison is mainly between Fe and Si. At all redshifts, other elemental transitions are also included in the analysis. Indeed, when this method is applied to a set of Keck/Hires data, a statistically significant trend for a variation in a was reported: Aa/a = (-0.54 ± 0.12) x 10~ 5 over a redshift range 0.5 < z < 3.0. The minus sign indicates a smaller value of a in the past. More recent observations taken at VLT/UVES using the many multiplet method have not been able to duplicate the previous result 154,155 . The use of Fe lines in 155 on a single absorber found Aa/a = (0.01 ± 0.17) x 10~ 5 . However, since the previous result relied on a statistical average of over 100 absorbers, it is not clear that these two results are in contradiction. In 154 , the use of Mg and Fe lines in a set of 23 high signal-to-noise systems yielded the result Aa/a = (—0.06 ± 0.06) x 1 0 - 5 and therefore represents a more significant disagreement and can be used to set very stringent limits on the possible variation in a. There exist various sensitive experimental checks that constrain the variation of coupling constants (see e.g., 1 5 6 ) . Limits can be derived from cosmology (from both big bang nucleosynthesis and the microwave background), the Oklo reactor, long-lived isotopes found in meteoritic samples, and atomic clock measurements. The most far-reaching limit (in time) on the variation of a comes from BBN. The limit is primarily due to the limit on 4 He. Changes in the fine structure constant affect directly the neutron-proton mass difference which can be expressed as ATUN ~ aah.QCD + bv, where AQCD ~ 0(100) MeV
is the mass scale associated with strong interactions, v ~ 0(100) GeV determines the weak scale, and a and b are numbers which fix the final contribution to Am;v to be —0.8 MeV and 2.1 MeV, respectively. From the previous discussion on BBN, changes in a directly induce changes in ATUN, which affects the neutron to proton ratio. The relatively good agreement
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between theory and observation, |AY/Y| < 3.5% allows one to set a limit \Aa/a\ < 0.06 (AY/Y scales with Aa/a) 157,i47,i23_ S i n c e t h i s l i m i t i s applied over the age of the Universe, we obtain a limit on the rate of change \a/a\ <> 4x 1 0 - 1 2 y r _ 1 over the last 13 Gyr. When coupled variations of the couplings are considered, the above bound is improved by about 2 orders of magnitude to Aa/a <> 10~ 4 as confirmed in a numerical calculation 158 . One can also derive cosmological bounds based on the microwave background. Changes in the fine-structure constant lead directly to changes in the hydrogen binding energy, Ef,. As the Universe expands, its radiation cools to a temperature, T&&c, at which protons and electrons can combine to form neutral hydrogen atoms, allowing the photons to decouple and free stream. Measurements of the microwave background can determine this temperature to reasonably high accuracy (a few percent) 9 . At decoupling i~l~l exp(—Eb/Tdec) ~ I- Thus, changes in a of at most a few percent can be tolerated over the time scale associated with decoupling (a redshift of z~1100)159. Interesting constraints on the variation of a can be obtained from the Oklo phenomenon concerning the operation of a natural reactor in a rich uranium deposit in Gabon approximately two billion years ago. The observed isotopic abundance distribution at Oklo can be related to the cross section for neutron capture on 149 Sm 1 6 °. This cross section depends sensitively on the neutron resonance energy Er for radiative capture by 149 Sm into an excited state of 150 Sm. The observed isotopic ratios only allow a small shift of |AJ5 r | < Er from the present value of Er = 0.0973 eV. This then constrains the possible variations in the energy difference between the excited state of 150 Sm and the ground state of 149 Sm over the last two billion years. A contribution to this energy difference comes from the Coulomb energy Ec = (3/5)(e2/ro)Z2/A1/3 (r0 = 1.2 fm) for a nucleus with Z protons and (A — Z) neutrons. This contribution clearly scales with a and is -E c ( 150 Sm) - £ c ( 1 4 9 S m ) = 1.16(a/a 0 ) MeV, where a0 is the present value of a. Considering the time variation of a alone, |A.E r | ~ 1.16|Aa/a| MeV and a limit \Aa/a\ <> 10~ 7 can be obtained 160 . However, if all fundamental couplings are allowed to vary interdependently, a much more stringent limit | A a / a | < (1 - 5) x 1 0 - 1 0 may be obtained 161 . Bounds on the variation of the fundamental couplings can also be obtained from our knowledge of the lifetimes of certain long-lived nuclei. In particular, it is possible to use precise meteoritic data to constrain nuclear decay rates back to the time of solar system formation (about 4.6 Gyr ago). Thus, we can derive a constraint on possible variations at a redshift z ~ 0.45
Astroparticle Physics 827
bordering the range (z = 0.5-3.0) over which such variations are claimed to be observed. The pioneering study on the effect of variations of fundamental constants on radioactive decay rates was performed by Peebles and Dicke and by Dyson 162 . The /3-decay rate, A^, depends on some power n of the energy Qg released during the decay, Xg oc Qjg. A contribution to Qg again comes from the Coulomb energy Ec oc a. Isotopes with the lowest Qg are typically most sensitive to changes in a as AXg/Xg = n(AQg/Qg) is large for small Qg. The isotope with the smallest Qg (2.66 ± 0.02 keV) is 187 Re, which decays into 1 8 7 0s. If some radioactive 187 Re was incorporated into a meteorite formed in the early solar system, the present abundance of 1 8 7 0 s in the meteorite is ( 187 Os) 0 = ( 187 Os)j + ( 1 8 7 Re) 0 [exp(Ai 8 7 t a )-l], where the subscripts "i" and "0" denote the initial and present abundances, respectively, A187 is Xg for 187 Re, and ta is the age of the meteorite. The above correlation between the present meteoritic abundances of 1 8 7 0s (daughter) and 187 Re (parent) can be generalized to other daughter-parent pairs. All these correlations can be used to derive the product of the relevant decay rate and the meteoritic age. Using the decay rates of 238 U and 235 U from laboratory measurements, the correlations for the 2 0 6 Pb- 2 3 8 U and 2 0 7 Pb235 U pairs give a precise age of ta = 4.558 Gyr for angrite meteorites 163 . This determination of ta has the advantage that the decay rates of 238 U and 2 3 5 U, and hence ta, are rather insensitive to the variation of fundamental couplings 162 . The above age for angrite meteorites allows for a precise determination of Ai87 from the correlation for the 1 8 7 0s- 1 8 7 Re pair in iron meteorites formed within 5 Myr of the angrite meteorites 164 . Comparing this value of Ai87, which covers the decay over the past 4.6 Gyr, with the present value from a laboratory measurement 165 limits the possible variation of a to Aa/a = (8 ± 8) x 10~ 7 166 . Once again, if all fundamental couplings are allowed to vary interdependently, a more stringent limit Aa/a = (2.7 ± 2.7) x 10" 8 may be obtained. Finally, there are a number of present-day laboratory limits on the variability of the fine-structure constant using two kinds of atomic clocks: one based on hyperfme transitions involving changes only in the total spin of electrons and the nucleus and the other based on electronic transitions involving changes in the spatial wavefunction of electrons. The electronic transition frequency ve\ depends on a relativistic correction Frei(a), which is a function of a and is different for different atoms. Relative to uei, the hyperfme transition frequency Vhf has an extra dependence on {Unuci/HB)&2, where finuci is the magnetic moment of the relevant nucleus ar.d /is is the atomic Bohr magneton. For atoms A and
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K. A. Olive
B , Vhf,A/vhf,B
OC (Hnucl,A/^nucl,B)FreltA(a)/Frei:B{a)
a n d Vhf,A/Vel,B
OC
2
(UnudI'PB)& Frei,A(oi)/Frel>B(a). If only the variation of a is considered, this can be tested by comparing Vhf,A with Vhf,B or Vhf,A with vei,B over a period of time. Three recent experiments have led to marked improvement in the limit on the variation of a: Aa/a < 6 x 10~ 15 from comparing hyperfine transitions in 8 7 Rb and 133 Cs over a period of about 4 years 167 , Aa/a < 4 x 1 0 - 1 5 from comparing an electric quadrupole transition in i99jjg+ t o t n e g r o u n c i- S tate hyperfine transition in 133 Cs over a 3 year period 168 , and Aa/a = (1.1 ± 2.3) x 10" 1 5 from comparing the 1S-2S transition in atomic hydrogen to the hyperfine transition in 133 Cs over a 4 year period 169 . If both a and Unud/l^B are allowed to vary, then constraints on these two distinct variations can be obtained by combining the latter two experiments, which give Aa/a = (—0.9 ±4.2) x 10~ 15 or a/a ^ 10~ 15 y r
- l 169
A summary of the constraints on a is found in Fig. 25, taken from Ref. 170. The result found in 154 and in the statistically dominant subsample of 74 out of the 128 low redshift absorbers used in 151 are sensitive to the assumed isotopic abundance ratio of Mg. In both analyses, a solar ratio of 24 Mg: 25 Mg: 26 Mg = 79:10:11 was adopted. However, the resulting shift in a is very sensitive to this ratio. Furthermore, it is commonly assumed that the heavy Mg isotopes are absent in low metallicity environments characteristic of quasar absorption systems. Indeed, had the analyses assumed only pure 24 Mg is present in the quasar absorbers, a much more significant result would have been obtained. The Keck/Hires data 1 5 1 would have yielded Aa/a = (-0.98 ±0.13) x 10~ 5 for the low redshift subsample and Aa/a = (-0.36 ± 0.06) x 10" 5 for the VLT/UVES data 1 5 4 . The sensitivity to the Mg isotopic ratio has led to a new interpretation of the many multiplet results 171 . The apparent variation in a in the Fe-Mg systems can be explained by the early nucleosynthesis of 25,26 Mg. The heavy Mg isotopes are efficiently produced in intermediate mass stars, particular in stars with masses 4-6 times the mass of the sun, when He and H are burning in shells outside the C and O core. There may even be evidence for enhanced populations of intermediate mass stars at very low metallicity. Recall the dispersion seen in D/H observations in quasar absorption systems as seen in Fig. 18. Is there a real dispersion in D/H in these high redshift systems? The data may show an inverse correlation of D/H abundance with Si 100>102. This may be an artifact of poorly determined
Astroparticle
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0.5
Figure 25. Constraints on the rate of variation (da/dt)/ct as a function of the fractional "look-back" time (to — t ) / t o , where to « 13 Gyr is the present age of the Universe. The results shown are taken from data on atomic clocks 1 6 8 , considerations of the Oklo phenomenon 1 6 0 , meteoritic data on 1 8 7 R e decay 1 6 4 > 1 6 6 ; and many-multiplet ( M M 1 1 5 1 , MM2 1 5 4 , MM3 1 5 5 ) and alkali-doublet (AD 1 5 2 ) analyses of quasar absorption spectra. For convenience, the results for MM1, MM2, MM3, and AD are shown at the mean redshift for the data used and then converted to (to — t)/to- Note that the result for MM1 actually covers (t 0 — t)/to = 0.37-0.84. Except for this result, all others are consistent with no time variation of a.
Si abundances, or (as yet unknown) systematics affecting the D/H determination in high-column density (damped Lyman-a, hereafter DLA) or low-column density (Lyman limit systems) absorbers. On the other hand, if the correlation is real it would indicate that chemical evolution processes have occurred in these systems and that some processing of D/H must have occurred even at high redshift. It is interesting to speculate 172 that the possible high redshift destruction of D/H is real and related to the chemical evolutionary history of high red shift systems. For example, these observations could be signatures of an early population of intermediate-mass stars characterized by an initial mass function different from that of the solar neighborhood. An example of such an IMF is shown in Fig. 26 173 . There are a number of immediate consequences of an IMF of the type shown in Fig. 26. In addition to the destruction of D/H at low metallicity, one expects observable C and N enhancements in high redshift absorption
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0
5
10
15 20 25 Stellar Mass (M )
30
35
40
Figure 26. An IMF with an early enhancement of intermediate mass stars
systems. In addition, one also expects an enhancement of the heavy Mg isotopes, 2 5 , 2 6 M g , which may account 1 7 1 ' 1 7 3 for the apparent variation of the fine-structure constant in quasar absorption systems. In this sense, the many multiplet method can be used to trace the chemical history of primitive absorption clouds 1 7 1 . This hypothesis will be tested by future observations and examinations of correlation among other heavy elements produced in intermediate mass s t a r s 1 7 4 . Acknowledgments I would like to t h a n k T. Ashenfelter, M. Casse, R. Cyburt, J. Ellis, T. Falk, B. Fields, K. Kainulainen, G. Mathews, M. Pospelov, Y. Qian, Y. Santoso, J. Silk, E. Skillman, V. Spanos, M. Srednicki, and E. Vangioni for recent (and enjoyable) collaborations on BBN. This work was partially supported by D O E grant DE-FG02-94ER-40823. References 1. R. A. Alpher and R. C. Herman, Phys. Rev. 74, 1737 (1948); Phys. Rev. 75, 1089 (1949). 2. J. C. Mather, et al., Ap.J. 512 (1999) 511. 3. A. T. Lee et al, Ap. J. 561 (2001) LI [arXiv:astro-ph/0104459]; R. Stompor et al., Ap.J. 561 (2001) L7 [arXiv:astro-ph/0105062].
Astro-particle Physics 831 4. C. B. Netterfield et al. [Boomerang Collaboration], Ap. J. 571 (2002) 604 [arXiv:astro-ph/0104460]; J. e. Ruhl et al., Astrophys. J. 599 (2003) 786 [arXiv:astro-ph/0212229]. 5. N. W. Halverson et al. MNRAS 568 (2002) 38 [arXiv.astro-ph/0104489]; C. Pryke, N. W. Halverson, E. M. Leitch, J. Kovac, J. E. Carlstrom, W. L. Holzapfel and M. Dragovan, Ap. J. 568 (2002) 46 [arXiv:astroph/0104490]. 6. S. Padin, et al. Ap.J. 549 (2001) LI; T.J. Pearson et al. Ap.J. 591 (2003) 556. 7. A. Rubino-Martin et al., Mon. Not. Roy. Astron. Soc. 341 (2003) 1084 [arXiv:astro-ph/0205367]. 8. A. Benoit et al. [the Archeops Collaboration], Astron. Astrophys. 399 (2003) L25 [arXiv:astro-ph/0210306]. 9. C. L. Bennett et al, Astrophys. J. Suppl. 148 (2003) 1; D. N. Spergel et al, Astrophys. J. Suppl. 148 (2003) 175. 10. C. 1. Kuo et al. [ACBAR collaboration], Astrophys. J. 600 (2004) 32 [arXiv:astro-ph/0212289]. 11. D. Scott and G. F. Smoot, Phys. Lett. B 592 (2004) 1. 12. A. G. Riess et a/., A. J. 116 (1998) 1009; S. Perlmutter et al, Ap. J. 517 (1999) 565; A. G. Riess et al, Ap. J. 560 (2001) 49. 13. S. M. Faber and J. J. Gallagher, Ann. Rev. Astron. Astrophys. 17 (1979) 135. 14. A. Bosma, Ap. J. 86 (1981) 1825; V. C. Rubin, W. K. Ford and N. Thonnard, Ap. J. 238 (1980) 471; V. C. Rubin, D. Burstein, W. K. Ford and N. Thonnard, Ap. J. 289 (1985) 81; T. S. Van Albada and R. Sancisi, Phil. Trans. R. Soc. Land. A320 (1986) 447. 15. M. Persic and P. Salucci, Ap. J. Supp. 99 (1995) 501; M. Persic, P. Salucci, and F. Stel, MNRAS 281 (1996) 27P. 16. R. P. Saglia et al., Ap. J. 403 (1993) 567; C. M. Carollo et al, Ap. J. 411 (1995) 25; A. Bordello, P. Salucci, and L. Danese, MNRAS 341 (2003) 1109; A. Dekel, F. Stoehr, G. A. Mamon, T. J. Cox and J. R. Primack, arXiv:astro-ph/0501622. 17. K. A. Olive, arXiv:astro-ph/0301505. 18. For reviews see: A. D. Linde, Particle Physics And Inflationary Cosmology Harwood (1990); K. A. Olive, Phys. Rep. 190 (1990) 307; D. H. Lyth and A. Riotto, Phys. Rept. 314 (1999) 1 [arXiv:hep-ph/9807278]. 19. M. Persic and P. Salucci, MNRAS 258 (1992) 14p. 20. M. Fukugita, C. J. Hogan, and P. J. E. Peebles, Ap. J. 503 (1998) 518. 21. M. G. Haehnelt, P. Madau, R. Kudritzki, and F. Haardt, Ap. J. 549 (2001) L151; D. Reimers, Sp. Sci. Rev. 100 (2002) 89. 22. D. J. Hegyi and K. A. Olive, Phys. Lett. 126B (1983) 28; Ap. J. 303 (1986) 56. 23. D. N. Schramm and G. Steigman, Ap. J. 243 (1981) 1. 24. S. D. M. White, C. S. Frenk and M. Davis, Ap. J. 274 (1983) 61. 25. C. S. Frenk, S. D. M. White and M.Davis, Ap. J. 271 (1983) 417.
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STUDENT SEMINARS* Friday, J u n e 1 1 , 7 p m Brian Murray: Quintessence Model of Dark Energy Monday, J u n e 14, 3:45 p m Kathryn Zurek: New Matter Effects in Neutrino Oscillation Experiments Artonio De Felice: BD- Baryogenesis Monday, J u n e 14, 7 p m Jeremy Jones: Electroweak Instanton Mediated Neutrino Junhai Kang: EW Baryogenesis
Interactions
Tuesday, J u n e 15, 7 p m David Morrissey: Cosmology of the MSSM with a Minimal Singlet Thomas Underwood: Resonant Leptogenesis Thursday, J u n e 17, 7 p m Leanne Dufry: Axion Dark Matter Roshan Foadi: Higgsless EWSB from Theory Space Friday, J u n e 18, 7 p m Jay Hubisz: Fermion Masses in Models without a Higgs Monday, J u n e 2 1 , 3:45 p m Fernando Febres-Cordero: Higher order Calculation in QCD and Background Process in Higgs Production Ting Wang: Implications of B —> cj>Ks Result for Flavor Models Monday, J u n e 2 1 , 7 p m Ben Lillie: Higgsless EW SB-Precision EW and Unitarity KC Kong: Collider Phenomenology of UED Tuesday, J u n e 22, 3:45 p m Tsedenbaljor Enkhbat: Femion Mass and Flavor Symmetry Saif Rayyan: Phenomenological Consequences of TeV Sterile Neutrinos 'Organized by Jay Hubisz and Patrick Meade 841
"RH"
842
Student
Seminars
Tuesday, June 22, 7 pm Suharyo Sumowidagdo: Top Quark to Tau lepton at DO Matt Dorsten: Sum Rule Bounds on B —> D Form Factors Wednesday, June 23, 7 pm Sonny Mantry: B-physics and SCET Masaoki Kusunoki: Production of X(3870) from B Decays by the Coalescence of D mesons Thursday, June 24, 7 pm Gil Paz: Precision Measurement of Vub Zhenyu Han: Family Unification on an Orbifold Friday, June 25, 7 pm Neil Christensen: Dynamical EWSB Monday, June 28, 3:45 pm Chrislopher Lee: SCET and Jet Physics Seung Lee: Factorization in SCET Monday, June 28, 7 pm Ines Cavero: Casimir Effect on Dielectric Cylinders Rahul Malhotra: Discovering MSSM Higgs Bosons Produced with a b Quark Tuesday, June 29, 3:45 pm Cyril Anoka: Anomaly Mediation Wednesday, June 30, 3:45 p m Linda Carpenter: UV Insensitive Anomaly Mediation Wednesday, June 30, 7 pm Andrew Blechman: Shining Mechanism of SUSY Breaking I.W. Kim: Fayet-Illiopoulos Terms in 5D SUGRA Thursday, July 1, 3:45 pm Irene Hidalgo: Fine Tuning in Little Higgs Daniel 1,arson: The Fat Higgs
Student Seminars
Thursday, July 1, 7 pm Patrick Meade: Evidence for a New Phase of SUSY Gauge Theories Can Kilic: The New Fat Higgs
843
Symposium in Honor of
Saturday June 26, 2004 (Room Cj-125 (Physics (Department University of Colorado at
Dinner at Kittredge
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S T U D E N T PARTICIPANTS Xavier Amador CINVESTAV Departamento de Fisica Ave. Instituto Politecnico Nacional 2508 Colonia Zacatenco, Mexico D.F. MEXICO David Anderson College of William and Mary Department of Physics P.O. Box 8795 Williamsburg, VA 23187-8795
Linda Carpenter Johns Hopkins University Department of Physics 3400 N.Charles Street Baltimore, MD 21218 Ines Cavero-Pelaez University of Oklahoma Department of Physics Sz Astronomy 440 W. Brooks Street Norman, OK 73019
Cyril Anoka Oklahoma State University Department of Physics 145 Physical Sciences Stillwater, OK 74078
Neil Christiansen Stony Brook University C.N. Yang Institute for Theoretical Physics Stony Brook, NY 11794-3840
Juan Barranco-Monarca CINVESTAV Departamento de Fisica Apdo. Postal 14-740 Mexico, DF 07000 MEXICO
Justin Conroy College of William and Mary Department of Physics P.O. Box 8795 Williamsburg, VA 23187-8795
Andrew Blechman Johns Hopkins University Bloomsberg Center for Physics 3701 San Martin Drive Baltimore, MD 21218
Shahida Dar University of Delaware Department of Physics & Astronomy Newark, DE 19716
Piyabut Burikham University of Wisconsin Department of Physics 1150 University Ave Madison, WI 53706
Chad Davis Stanford University Physics Department 382 Via Pueblo Mall Stanford, CA 94305-4060 847
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Student
Participants
Antonio De Felice Syracuse University Physics Department 201 Physics Bldg. Syracuse, NY 13244
Peter Graham Stanford University Physics Department 382 Via Pueblo Mall Stanford, CA 94305-4060
Matthew Dorsten California Institute of Technology MC 452-48 1200 E. California Blvd. Pasadena,CA 91125
Zhenyu Han Yale University Physics Department 217 Prospect St. P.O. Box 208120 New Haven, CT 06520-8120
Leanne Duffy University of Florida Department of Physics P.O. Box 118440 Gainesville, FL 32611-8440 [email protected]
Motomichi Harada Ohio State University Department of Physics 174 W. 18th Avenue Columbus, OH 43210
Tsedenbaljir Enkhbat Oklahoma State University Department of Physics 145 Physical Sciences Stillwater, OK 74078-3072 Fernando Febres-Cordero Florida State University Department of Physics 315 Keen Building Tallahassee, FL 32306-4350 Roshan Foadi Michigan State University Department of Physics &; Astronomy Biomedical k. Physical Sciences Bldg. East Lansing, MI 48824
Irene Hidalgo Universidad Autonoma de Madrid Facultad De Ciencias Instituto De Fisica Teorica Modulo C-XVI 3 Planta. Cantoblanco, Madrid, 28049 SPAIN Jordan Hovdebo University of Waterloo Department of Physics Waterloo, ON N2L 3G1 CANADA Jay Hubisz Cornell University Department of Physics 109 Clark Hall Ithaca, NY 14853-2501
Student Participants
Marcin Jankiewicz Vanderbilt University Physics & Astronomy Department 1807 Station B Nashville, TN 37235
Masaoki Kusunoki Ohio State University Department of Physics 174 West 18th Avenue Columbus, OH 43210
Jeremy Jones University of Arizona Physics Department 1118 E. 4th Street Tucson, AZ. 85721
Daniel Larson Lawrence Berkeley National Lab Theoretical Physics Group One Cyclotron Rd., 50A5104 Berkeley, CA 94720-8162
Junhai Kang University of Pennsylvania Physics Department, DRL Bldg. 209 S. 33rd St. Philadelphia, PA 19104
Christopher Lee California Institute of Technology Physics Department MC 452-48 Pasadena, CA 91125-4800
Can Kilic Harvard University Physics Department Jefferson Physical Laboratory 17 Oxford Street Cambridge, MA 02138 Ian-Woo Kim KAIST Department of Physics Guseongdong Yuseonggu Daejeon, 305-701 REPUBLIC OF KOREA Kyoungchul Kong University of Florida Physics Department P.O. Box 118440 Gainesville, FL 32611
Hye-Sung Lee University of Wisconsin Department of Physics 1150 University Avenue Madison, WI 53706 Keith Lee MIT Center for Theoretical Physics Room 6-405A 77 Massachusetts Avenue Cambridge, MA 02139-4307 Seung Lee Cornell University Department Physics 109 Clark Hall Ithaca, NY
849
850
Student
Participants
Michael Lennek University of Arizona Department of Physics 1118 E. 4th Street Tucson, AZ 85721
Patrick Meade Cornell University Department of Physics 109 Clark Hall Ithaca, NY 14853-2501
Ben Lillie SLAC MS81 2575 Sand Hill Road Menlo Park, CA 94025
Arjun Menon University of Chicago Department of Physics 5640S.Ellis Ave. Chicago, IL 60637
Zhaofeng Liu University of Colorado Department of Physics UCB 390 Boulder, CO 80309
Rahul Malhotra University of Texas at Austin Department of Physics Center for Particle Physics 1 University Station C 1600 Austin, TX 78712-0264
David Morrissey University of Chicago Department of Physics, RI 388 5640 S. Ellis Avenue Chicago, IL 60637 Brian Murray University of Oregon Institute of Theoretical Science 5203 University of Oregon Eugene, OR 97403-5203
Sonny Mantry MIT Center for Theoretical Physics Cambridge, MA 02139
Azar Mustafayev Florida State University Department of Physics 513 Keen Building Tallahassee, FL 32306
Thomas McElmurry University of Illinois Department of Physics 1110 West Green Street Urbana, IL 61801-3080
Minjoon Park Enrico Fermi Institute Rm388 5640 S. Ellis Ave. Chicago, IL 60637
Student Participants
Gil Paz Cornell University Particle Physics Theory Group 405 Newman Laboratory Ithaca, NY 14853 Parul Rastogi University of Maryland, College Park Department of Physics College Park, MD 20742 Saifuddin Rayyan Virginia Tech Physics Department Robeson Hall Blacksburg, VA 24061-0435 Andreas Ross University of Massachusetts Department of Physics 1126 Lederle Graduate Research Tower Amherst, MA 01003-4525 Leslie Schradin Ohio State University Department of Physics 1012 Smith Laboratory 174 West 18th Avenue Columbus, OH 43210 Marc Schreiber Stanford Linear Accelerator Center MS 81 2575 Sand Hill Road Menlo Park, CA
Vedat Senoguz University of Delaware Department of Physics & Astronomy 223 Sharp Lab Newark, DE 19716 Suharyo Sumowidagdo Fermilab Florida State University Group MS352, DO Experiment Batavia, IL 60510-0500 Thomas Underwood University of Manchester Department of Physics & Astronomy Theoretical Physics Group Brunswick Street Manchester, M13 9PL UNITED KINGDOM Ting Wang University of Michigan Department of Physics Randall Laboratory 500 E. University Ave. Ann Arbor, MI 48109 Kathryn Zurek University of Washington Department of Physics Box 351560 Seattle, WA 98195
851
LECTURERS R. Sekhar Chivukula Michigan State University Physics & Astronomy Department 3243 Biomedical Physical Sciences East Lansing, MI 48824-1116 [email protected] Csaba Csaki Cornell University Physics Department Ithaca, NY 14853 csaki@mail. Ins. comell. edu
Graham Kribs Institute for Advanced Study School of Natural Sciences Princeton, NJ 08540 [email protected] Markus Luty University of Maryland Physics Department College Park, MD 20742-4111 mluty ©physics. umd. edu
Andre de Gouvea Northwestern University Department of Physics & Astronomy 2145 Sheridan Road Evanston, IL 60208-3112 [email protected]
Ann Nelson University of Washington Physics Department Box 351560 Seattle, WA 98195-1560 [email protected]
Howard Haber University of California Physics Department Santa Cruz, CA 95064 [email protected]
Matthias Neubert Cornell University Physics Department Ithaca, NY 14853 [email protected]
Tao Han University of Wisconsin Physics Department 1150 University Ave. Madison, WI 53706 [email protected]
Yasunori Nomura LBNL Theoretical Physics Group Berkeley, CA 94720-7300 [email protected]
Nemanja Kaloper University of California Physics Department Davis, CA 95616 kaloper@physics .ucdavis. edu
Keith Olive University of Minnesota Physics Department Minneapolis, MN 55455 [email protected] 853
854
Lecturers
Laura Reina Florida State University Physics Department 315 Keen Building Tallahassee, FL 32306-4350 [email protected] Heidi Schellman Northwestern University Physics Department Evanston, IL 60208-3112 hschellman@northwestern. edu Martin Schmaltz Boston University Physics Department Boston, MA 02215 [email protected] Elizabeth Simmons Michigan State University Physics Department 3221 Biomedical Physical Sciences East Lansing, MJ 48824-1116 esimmons@msu .edu George Sterman Stony Brook University C.N. Yang Institute for Theoretical Physics Stony Brook, NY 11794-3840 sterman@insti. physics. suny sb. edu
Raman Sundrum Johns Hopkins University Physics Department Baltimore, MD 21218 [email protected] Doreen Wackeroth SUNY at Buffalo Physics Department 239 Fronczak Hall Buffalo, NY 14260 dow@ubpheno. physics. buffalo. edu James Wells University of Michigan Physics Department Randall Laboratory 500 E. University Ave. Ann Arbor, MI 48109-1120 [email protected] Scott Willenbrock University of Illinois Physics Department 1110 W. Green Street Urbana,IL 61801 [email protected]
DIRECTORS KT Mahanthappa University of Colorado Department of Physics UCB 390 Boulder, CO 80309 [email protected]
Carlos Wagner Argonne National Laboratory HEP Theory Division 9700 South Cass Ave. Argonne,IL 60439-4815 c wagner @hep. anl. gov
John Trning University of California Department of Physics One Shields Ave. Davis, CA 95616 [email protected]
Dieter Zeppenfeld Institute for Theoretical Physics University of Karlsruhe Wolfgang-Gaede-Str. 1 76131 Karlsruhe Germany [email protected]
LOCAL ORGANIZING C O M M I T T E E Shanta deAlwis University of Colorado Department of Physics UCB 390 Boulder, CO 80309 dealwis@pizero .Colorado. edu
Anna Hasenfratz University of Colorado Department of Physics UCB 390 Boulder, CO 80309 anna@eot vos. color ado. edu
Tom DeGrand University of Colorado Department of Physics UCB 390 Boulder, CO 80309 [email protected]
KT Mahanthappa University of Colorado Department of Physics UCB 390 Boulder, CO 80309 ktm@pizero. Colorado. edu
856
HABER
KALOPER
NELSON
NOMURA
WACKEROTH
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Proceedings of the Theoretical Advanced Study Institute in Elementary Particle Physics This book contains write-ups of lectures from a summer school for advanced graduate students in elementary particle physics. In the first lecture, Scott Willenbrock gives an overview of the standard model of particle physics. This is followed by reviews of specific areas of standard model physics: precision electroweak analysis by James Wells, quantum chromodynamics and jets by George Sterman, and heavy quark effective field by Matthias Neubert. Developments in neutrino physics are discussed by Andre de Gouvea and the theory behind the Higgs boson is addressed by Laura Reina. Collider phenomenology from both experimental and theoretical perspectives are highlighted by Heidi Schellman and Tao Han. A brief survey of dynamical electroweak symmetry breaking is provided by R Sekhar Chivukula and Elizabeth H Simmons. Martin Schmaltz covers the recent proposals for "little" Higgs theories. Markus Luty describes what is needed to make supersymmetric theories realistic by breaking supersymmetry. There is an entire series of lectures by Raman Sundrum, Graham Kribs, and Csaba Csaki on extra dimensions. Finally, Keith Olive completes the book with a review of astrophysics.
IN
I ISBN 981-256-809-3
torld Scientific YEARS 0 1
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www.worldscientitic.com