Theory & Phenomenology of Sparticles

Vwz{z) = Oa^eA^x) + ee ex(x) + ee ex(x) + l-ee eeD(x).

(4.34)

A vector superfield has mass dimension zero and can be exponentiated. We summarize the three main points learnt above. • A vector superfield obeys the reality condition and has a vanishing mass dimension. • An abelian "supergauge transformation", simply adding a real combination of chiral superfields, can transform a vector superfield to the Wess-Zumino gauge where it has only the gauge boson, gaugino and auxiliary D field components. • [V]D = f d46V(x,6,S), occuring in the Lagrangian density, would yield both a supersymmetric and supergauge invariant action. 4

The interactions of V are taken to keep physics invariant under this transformation.

63

4.4. Vector Superfields

We can identify the LHS of (4.34) as a gauge superfield in the Wess-Zumino gauge. Among its components, A^ can be designated as a gauge field with three independent real field components (there is a gauge restriction, say d^A^ = 0) and X(x) the corresponding gaugino (cf §3.3), which is a two-component complex spinor with four real independent fermionic fields. D{x) is an auxiliary field since it is not a dynamical field like A^(x) and X(x); it does not have a kinetic energy term in C and can be eliminated from the system through the equations of motion. In terms of particle helicity components, one has two helicities of the massless gauge boson and two of the gaugino and hence the gauge supermultiplet of Table 3.1. However, the above "gauge-fixing" breaks manifest supersymmetry since the relations C(x) = £(x) = M(x) = 0 cannot be maintained under a supersymmetry transformations. Nonetheless, after every supersymmetry transformation, the Wess-Zumino gauge can be restored by a field-dependent compensating gauge transformation. One interesting property of the Wess-Zumino gauge is the ease with which powers of V can be computed. Thus, from (4.34), 1 Vwz = nM 99A^A^ Vwz = 0 V n > 3 . (4.35) 2L Moreover, exp Vwz

=

1 + Vwz + rVjy Z

= i + 0CT"&4M + 99 ex + ee ex + x-ee ee (D + \A»AA

.

(4.36)

For later utility, we provide a couple of additional superfield products involving VwzFirst, consider the triple product between a chiral superfield, its hermitian conjugate and a vector superfield in the Wess-Zumino gauge. The result is

$tvwz$j = etfeA^j

+ 4=00(&X%A<# + y/ifixtfh) v2

+±=96(-eo»liApi + l

+ -ee Mipfttj -yfi^Xli

y/iOXfth)

- 2iA»v;\dAh

- &*%A

- V2*XQ .

(4.37)

Next, let us write a quadruple product: $\VwzVwz®i

= \ee

~99A»A^i

.

(4.38)

Given the properties of V(z), one can view it as the supersymmetric generalization of an abelian gauge potential. The natural question arises: what is the field strength that remains invariant under an abelian supergauge transformation? For an arbitrary V(z), not necessarily in the Wess-Zumino gauge, one can construct left and right chiral spinorial field-strength superfields: WA = -jDV

VAV ,

(4.39a)

4. Free Superfields in Superspace

64

WA = -\VV

VAV

(4.39b)

as superfield generalizations of the abelian field-strength tensor. While V (as well as $, $t) is even under the Z2-grading [cf.(3.4)], W and W are odd. Moreover, the following observations on (4.39) can be made readily. • Since the product of any 3 P ' s or P ' s vanishes, VBWA = 0 and VBW A = 0 in conformity with the claim that WA and WA are indeed left and right chiral superfields respectively. • Both W and W are seen to have the mass dimension of 3/2. • WA and WA are invariant under the supergauge transformation V -> V + tA - tA*. This follows as a consequence of A and A* being left chiral and right chiral superfields respectively with VAA — 0 = T>A^ and the facts that 5 VhVB VAA = VB\vB, DA] A=[f>B, B

B

V VB VAtf = V [VB,VA]+tf _

A

V.

VBA = 0 ,

= [VB,VA]+VBtf

=0

A

• VAW = VAWA, which can be derived immediately on applying the covariant chiral derivatives to (4.39) and using (4.11k). Since W^(W,j) is a left- (right-) chiral superfield, it is convenient to use the variable y\x — xii _ jflo-Mfl (jjn = xv _|. iQo^ff) to calculate its component expansion. Moreover, since either of them is supergauge invariant, we need work only in the Wess-Zumino gauge. We take (4.34), substitute xM = y^ + iOa^O = y^ — iOa^O, Taylor-expand and make use of (3.19k) to get + 00 0\(y) + 00 0X(y) +l-6660 {D(y) + id^A^y)}

,

(4.40a)

Vwz(y, 9,9) = 0°>16All{y) + 00 0\(y) + 00 0\{y) +l-0006 {D(y) - id^A^y)}

,

(4.40b)

VWz(y, e, 0) = 6^0 A^y)

Now we can utilize the forms of the covariant derivatives V and V written in terms of y and y in (4.17) and (4.18) respectively. Thus, employing (4.39a), we can write WA = -\v{y)V^V^Vwz(y,

0,0) .

(4.41)

5 Prom (4.11b), [VB,T>A]+ is proportional to a^dy. which commutes with VB. Similarly [T>B,T>A]+ commutes with VB.

4.4. Vector Superfields

65

The evaluation of the component expansion of the RHS of (4.41) is straightforward though tedious. First, one needs to compute VAVWz{y,9,9). The result can be given in terms of the gauge field strength FM„ = d^A^ — d^A^.

V{X]Vwz(y, 9,9) = o^jjPA^y) + 29A9X(y) + 8§XA(y) +ee{5ABD(y)-(onABFM}eB +i66

66{a^X{y)}A.

The application of -\f>(v)f>W leads to WA = XA(y) + D(y)6A - (a^e^F^y)

+ i99a^d^X^(y).

(4.42)

Similar manipulations in the (y, 8,6) superspace yield WA = XA(y) + D(y)§A - e^a^F^y)

- iM {d^X(y)^}A

.

(4.43)

Thus WA, WA contain only the component fields A, D and FM„; each has a total of 4 + 1 + 3 = 8 real components. WAWA also has to be a left chiral superfield in the light of the discussion in §4.2. In fact, one finds after some algebra that WAWA

= X(y)X(y) + 29{D(y)X(y) + +eelD\y)

+ 2iX{y)a^X{y)

a^X(y)FliV(y)} -

^Flw(y)F^(y)

-lFia,(y)F^{y)\,

(4.44)

where F^ is the dual field strength \tilvpaFp<'. It follows similarly that WAWA

= X(y)X(y) + {2D(y)X(y) + +9elD\y)

- 2idfX{y)o»X{y)

2X(y)a^F^(y)}0 -

^F^F^y)

+ \F,M^v{y)Y

(4.45)

An important and interesting consequence of (4.44) and (4.45) is the following relation: i [WAWA + WAWA]

p

= \D\X) - ^F,v(x)F^(x)

+ iA(zK[c>M] A » •

( 4 - 46 )

4. Free Superfields in Superspace

66

Here [d^] is as defined after (4.28) and in the last step we have utilized the evident fact that, in the F-terms of WAWA (W^W ), the coordinates y*1 (y1') can be replaced by x1*. Since the last two RHS terms stand respectively for the kinetic energy (in a Lagrangian density) of an abelian gauge boson and of a Majorana fermion (gaugino), (4.46) can serve as the kinetic energy term for a vector superfield. We conclude by noting an interesting identity, namely

f^(W*VAV

+

WAV*V)B = f^(W*WA

+

WAWX .

(447)

The integrands in (4.47) are equal modulo surface terms. Before ending this section, we want to comment on the quantization of various classical superfields that we have introduced above. One can construct a supersymmetric quantum field theory in component language by postulating canonical (anti-) commutation relations for the physical (fermionic) bosonic component fields. This is most easily accomplished after the auxiliary fields have been removed through their equations of constraint. Alternatively, one could try to construct quantum superfields 'from scratch'. The starting point of such a construction would be to define an infinitesimal supersymmetry transformation of a quantum superfield T not by 5J7 = i(eQ + lQ)T but rather by [4.4]

In view of the well-known problems, with which the canonical quantization of gauge theories is beset, we shall not pursue this approach any further in this book. Instead, in Ch.6, we shall discuss the extension of the functional integral quantization formalism to classical superfields and show how it leads to the notion of supergraphs.

4.5

M a t t e r Parity and imparity

In §3.1 we already introduced it-symmetry as a global [/(l)-invariance of the supersymmetry algebra. Under phase rotations of the Grassmann coordinates 6 —> elv0 and 0 —» e~%v6, from (3.26f, 3.26g) one has that QA -»• e»RQAe-*R QA ->• eivRQAe-i,pR

= e*>QA , = eivQA.

This means that 6, 6, Q and Q have the i?-charges 1 , - 1 , - 1 and 1 respectively. We can define an ^-transformation on a general left chiral superfield as $ —> $', where $'(x, eitpe, e~i,p6) = ei,fiR*${x, 9,8) ,

(4.48)

i.e. the .R-charge of $ is i?$. Correspondingly, $* —> $'* where &(x, eive, e~iipe) = e~i,fiR*&(x, 6,6) ,

(4.49)

so that the .R-charge of $ t is - i?$. The consistency of the ^-transformations of both sides of (4.19) would moreover dictate the following .R-charges for the component fields: R(4>) = i?$ ,

(4.50a)

4.5. Matter Parity and R-Parity

R(0 = -R(0

67

= R*-i,

(4.50b)

R(F) = J2* - 2 .

(4.50c)

For products of chiral superfields, the J?-charges will add algebraically. Turning to a vector superfield, we can see that its reality requires a vanishing .R-charge, i.e. R(V) = 0. A comparison with (4.34) makes it evident that R(AJ

=0,

(4.51a)

R{\) = -R(X) = 1 ,

(4.51b)

R(D) = 0 .

(4.51c)

Eqs. (4.51) are true in any gauge, as can be checked by comparing with (4.30). In a general gauge [cf. (4.32)] we have the additional i?-charge values R{C)

= 0,

R(Z) = -R(0 = -i, R(M)

= -R(M*)

= -2 .

There has been much speculation in the literature on whether i?-invariance can be a symmetry of nature. This possibility arises since the kinetic energy term J d&z —$ or $ —> $ under 9 —t —6. The corresponding scalar component then has either negative or positive i?-parity while Rp for its fermionic partner is correspondingly positive or negative. These two options (i.e. i?$ = ± 1 or 0) pertain to the 6 A discrete symmetry, which is the invariant remnant of some gauge symmetry that gets destroyed by anomalies, is called a discrete gauge symmetry. It has been argued [4.7] that quantum gravity effects destroy not only all global symmetries but also all discrete symmetries except those that are discrete gauge symmetries.

68

4. Free Superfields in Superspace

cases when <£ is matterlike or quantalike. Here we are calling a chiral superfield matterlike if its fermionic component can be identified as a known elementary particle such as a quark or lepton; the corresponding scalar will then be a sparticle such as a squark or slepton. In contrast, we call a chiral superfield quantalike if its scalar component is identifiable as an elementary particle like a Higgs boson; now the corresponding fermion is a sparticle like a higgsino. In this way, particles always turn out to have positive R-parity whereas the same for a sparticle is always negative (see Table 4.1). This property of i?$ can be ensured in the Standard Model by identifying it with 3(B - L) where B = baryon number and L = lepton number associated with $. Thus we can say that the matter parity Mp of a superfield is (_1)3(B-£)_ Q n e -IS foen led, with S being the spin, to the following formulae [4.8] Supermultiplet Particle Sparticle Particle Sparticle Particle Sparticle Particle

Name Quark q Squark q Lepton / Slepton I Higgs h_ Higgsino h Gauge boson g

Spin 1/2 0 1/2 0 0 1/2 1

Sparticle

Gaugino g

1/2

Rp

+ + + +

Superfield nature Chiral, matterlike, odd matter parity Chiral, matterlike, odd matter parity Chiral, quantalike, even matter parity Vector

Table 4.1. Information on four classes of supermultiplets

Mp = (-l) 3 ^-*-) , Rp = (_l)3(B-t)+2S

(4.52a) >

(4

52b)

the latter being for the .R-parity of any Standard Model particle and its sparticle partner(s) in a supermultiplet. Since L is always zero or an integer, 3(B — L) in the RHS of (4.52) can also be replaced by 3B — L or 3B + L. Note that the .R-parity of a state containing several particles and sparticles is the product of the individual /^-parities. Evidently, for the particles and sparticles under discussion, matter parity conservation and Rp conservation in any interaction vertex (that conserves angular momentum) are the same thing since the product of (—l) 25 factors for all the particles involved in the vertex must equal + 1 . In case Rp is exactly conserved, a starting assumption of the Minimal Supersymmetric Standard Model (Ch.8), the Lightest Supersymmetric Particle (LSP), being the lightest i?-odd state in nature, has to be stable. As will be elaborated in Ch.16, this has rather fundamental implications for cosmology in that the LSP becomes a very attractive candidate for cold dark matter, supposedly pervading the Universe. An additional consequence of Rp invariance is that sparticle production processes must produce them in even numbers (usually a pair) while every sparticle other than the LSP will eventually decay into particles plus an odd number of LSPs (usually one). Each LSP, being weakly interacting like neutrinos, will escape detection, leading to a mismatch in the total momentum. Such a situation in particle collision at high energies will lead to spectacular effects in the laboratory like events with a large missing transverse energy or I$T, as will be explored in Ch.15.

References

69

i?-parity need not be an exact symmetry of nature, though such may indeed turn out to be the case. This question has been investigated [4.9] in connection with high scale physics. The issue is whether or not an exact discrete symmetry can survive at low energies from of the discrete symmetries of the manifold on which a higher dimensional fundamental theory, such as superstring theory, needs to be compactified or as the protected subgroup of a gauge symmetry destroyed [4.10] by gravitational anomalies. If so, can this discrete symmetry be Rp or is it something like (—1)3B? These investigations have been inconclusive [4.9, 4.10] so far - the answer depending on the specific manifold chosen. There is also a rich phenomenology of models with Rp violating interactions about which we shall have more to say in Ch.14. We summarize this discussion of i?-parity as follows. • Rp is a discrete Z2 symmetry which equals (-1) 3 ( B - L )+ 2 S = (_i)3B-z,+2S for Standard Model particles and their superpartners and is positive for particles but negative for sparticles; it is related to the matter parity Mp = (—l)R of the corresponding superfield. • The matter parity of a vector superfield V is always positive. For a chiral superfield $ it is positive or negative depending on whether $ is quantalike or matterlike. • Rp has been conjectured to be an exact symmetry of nature in which case the LSP is absolutely stable. This fact has deep cosmological and spectacular laboratory consequences. However, there are models with Rp violation.

References [4.1] D.V. Volkov and V.P. Akulov, Phys. Lett. B46 (1973) 109. S. Samuel and J. Wess, Nucl. Phys. B226 (1983) 289; B233 (1984) 488. S. Deser and B. Zumino, Phys. Rev. Lett. 38 (1977) 1433. [4.2] A. Salam and J. Strathdee, Nucl. Phys. B76 (1974) 477; Phys. Rev. D l l (1975) 1521. [4.3] J. Wess and B. Zumino, Nucl. Phys. B78 (1974) 1. J. Wess and J. Bagger, op. cit., Bibl. P.P. Srivistava, op. cit., Bibl. [4.4] S. Weinberg # 3 , op. cit., Bibl. [4.5] G.R. Farrar and S. Weinberg, Phys. Rev. D27 (1983) 2732. [4.6] A. Salam and J. Strathdee, Nucl. Phys. B87 (1975) 85. P. Fayet, Nucl. Phys. B90 (1975) 104. G.R. Farrar and P. Fayet, Phys. Lett. B76 (1978) 575. [4.7] L.M. Krauss and F. Wilczek, Phys. Rev. Lett. 62 (1989) 1221. T. Banks, Nucl. Phys. B323 (1989) 90. [4.8] S. Weinberg, Phys. Rev. D26 (1982) 287. N. Sakai and T. Yanagida, Nucl. Phys. B197 (1982) 533.

70

4. Free Superfields in Superspace

[4.9] M.C. Bento, L.J. Hall and G.G. Ross Nucl. Phys. B292 (1987) 400. L.E. Ibanez and G.G. Ross, Phys. Lett. B260 (1991) 291. L.E. Ibanez, Nucl. Phys. B368 (1992) 3. A.E. Faraggi, Phys. Lett. B398 (1995) 97. [4.10] T. Banks and M. Dine, Phys. Rev. D45 (1992) 424.

Chapter 5 INTERACTING SUPERFIELDS 5.1

System of Interacting Chiral Superfields

For a supersymmetric action, the Lagrangian density - describing a set of interacting superfields - can at most change under a supersymmetry transformation into itself plus a total spacetime derivative. As explained in §4.1, that is precisely how the highest (£)-) component of a general superfield transforms. So does the F-component of a chiral superfield. The construction of a supersymmetric Lagrangian density is thus a straightforward exercise. The D-term needs to be isolated from a product of superfields that contains such a component. If some products are chiral superfields, their F-terms are also eligible. In general, a renormalizable Lagrangian density admits only terms with mass dimension not more than four [5.1]. A general renormalizable supersymmetric Lorentz invariant Lagrangian density, involving only polynomials of left chiral superfields $i (i referring to the type with repeated indices being summed), is therefore C=

$t$,j +[w(* i ) + &.c.]F,

W($i) = h&i + -rriij^i^j + -fij^i^j^k

(5.1a) .

(5.1b)

The first term in the RHS of (5.1a) is the D-term of a vector superfield, namely $}$j. The second is the F-term of a chiral superfield (plus its hermitian conjugate) defined as a polynomial of <£; (upto cubic terms so as to retain renormalizability) and called the superpotential W. The couplings my and / „ * are symmetric in their indices. Though the explicit form of W appears in (5.1b), we shall try to keep a general W($i) in what follows. Invariance under supersymmetry dictates that W be an analytic function of $*; in particular, it cannot involve $ | . Similarly, W t involves the right chiral $ | and not $ j . This property of the superpotential in any interacting chiral superfield system is known as holomorphy. Note that (5.1a) can be understood with either y or x (cf. §4.3) in the argument of each <£ since it is invariant1 under the map x —> y or y ->• x. In other words, / d6z$ = J d4y and this integral has no ^-dependence. Similarly, f d6z& = f diyd?8& has no ^-dependence. 'This is true for any D- or F- term since B8G(y) = 66G(x) and 66 66H{y) = 66 66H(x) V functions G and H.

71

5. Interacting Superfields

72

We only need to remember that $(x,9,6) supersymmetric action is

= $(y,0) and &(x,6,6)

= &(y,0).

Finally, the

I d8z [$J$ 4 + W($i)6W(6) + Wt($<)* (2) (0)

S=

We shall find it notationally convenient to use [5.2] two subspaces 4>i and 4>l = cf>* for the complex scalar field and define: (5.2a)

dW W(4>) =

(5.2b)

d2W

(5.2c)

Wj(i) =

(5.2d)

d3W

(5.2e)

dQidQjdQk 03vyt

Wjlc(4>)

(5.2f)

a$|a$t.a$t

In each RHS of (5.2) the symbol | on the right is as defined after (4.23). The superfield products in the RHS of (5.1) can be decomposed into their components in the manner shown in §4.4. In particular, one can make use of (4.26), (4.27) and (4.28). Then, ignoring total spacetime derivatives, one can rewrite the Lagrangian density in terms of component fields as

^[dtfi

+ Vtid^

+ FtFi

hk + miki + -fijki<j>j 1 Fk - -&£,- (mi:j + fijk(t>k) + h.c.

(5.3)

(5.3) makes an "off-shell" description of the system since it includes the auxiliary fields F* which do not possess kinetic energy terms. It is an instructive but straightforward exercise to see that J d^xCfa), constructed from the above, is invariant under infinitesimal supersymmetry transformations (4.25) applied to each field of type i. Indeed, the usual Noether procedure goes through. This means that one can define a conserved spinorial supercurrent density K^A and its conjugate K* such that 2

eK'i + eK'i = 2

ec

J26xd^X)

In precise terms, K^ = -v / 2(
U for the system of (5.3)

5.1. System of Interacting Chiral Superfields

73

with the summation covering all fields X present and 5C = d^L^ being a total derivative. Finally, the supercharge QA = J d3x K0A and its conjugate QA = J d3x R£, constructed from these currents, can be checked to consistently obey SX = i(eQ + eQ)X . The verification of these statements is left as an exercise for the reader. Let us just remark that the concept of a four spinor supercurrent density a

\K»A)

goes beyond this particular set of interacting chiral superfields and is generally valid for any supersymmetric system. Returning to (5.3), the auxiliary fields can be eliminated by using the equations of constraint 0 = dC/dF* and 0 = dC/dFi which respectively yield F{ = -W{4>) = -h* - m\$

- \ttikW

,

F* = -Wi{(f>) = -hi - rriijcpj - -^fnk^jfa

(5.4a) •

(5.4b)

Utilizing (5.4), (5.3) can be recast in the "on-shell" form, i.e. purely in terms of propagating dynamical fields, as £ = i^[d^i

+ Pfidrfi

- Q&&VI*>(0) + h-c)

Vtfi, # ) = Wi{)W{4>) = F*F{ .

- V(i> <%) -

( 5 - 5a ) (5.5b)

Thus Wy ((/>), the double superfield derivative of the superpotential evaluated with nonzero scalar fields only, contains both the fermion mass terms and the Yukawa interaction terms. Indeed, one can write . £vukawa

1/..

d2W

- ~2 {^d^M'j

\ +

)'

Turning to the scalars, we find that the potential V(4>i, 0|) is the sum of the modulus squares of all the auxiliary fields. If they exist, configurations with F, = 0 Vi correspond to absolute minima of the potential. Such configurations define the supersymmetric ground state, cf. §2.2. If no such solution exists, supersymmetry is broken spontaneously - as will be discussed in §7.4. We will now discuss the mass terms in the Lagrangian density (5.5a) with fa = ((pi) and develop a formalism in which squared masses of the bosons and fermions get related to various scalar derivatives of the superpotential. Of course, for exact supersymmetry (as discussed so far), all supermultiplet members are degenerate in mass. However, this degeneracy gets split in any realistic broken supersymmetric case. Later, when we come

74

5. Interacting Superfields

to that, this formalism will prove useful. The coefficient of ——£i£j in C, namely Wij((cf>)), measured in these configurations, yields the fermion mass matrix: mFij = Wy((0)) .

(5.6)

(5.6) follows from the fermionic equations of motion which are i^dyjii - WijiMj

= 0,

ia^d^i - W'iffij = 0 . In discussing scalar masses, we can employ the notation d2V d4>idj

d2V d2V

= Vij ,

(5.7a)

= y\

(5.7b)

= Vj — yz

d2V

dpty--

=

(5.7c) >

yy

(5Jd)

•

The scalar mass terms (SMT) in the Lagrangian density can then be written as £sMT = -\(
(5.8a)

V2*i=[v'. v>)

•

(5 8b)

-

Employing (5.5b) and (5.6) in (5.8b), we have the sum of squared masses of all scalars Y2 mo and that of all fermions Y2 m\n m * n e c m r a l supermultiplets related by

= Tr[(mFmF) + (mFmF)] = 2 ^ m ? / 2 . (5.9) Thus follows the supertrace mass sum rule [5.3] 1/2

STr m 2 = ^2(-\fJ(2J

+ \)m2j = 0 .

(5.10)

j=o

In other words, the spin weighted graded trace of the squared mass matrix, taken over all chiral supermultiplets vanishes. In fact, we will derive below a stronger result, namely that (5.10) is true over each chiral supermultiplet.

5.1. System of Interacting Chiral Superfields

In detail, the diagonal blocks of ™ ly (V"j)*=M = {mFmF)ih

are

75

given by (V; 1)^=^) = (mFmF)ij

,

while the off-diagonal elements can be written as = W«*(W)W*«0», ^>«0)*) = W i j *«0»VP*«0» .

Vam)

For a supersymmetric ground state, V is at its absolute minimum with a vanishing (i^), i.e. U

—

W*((0)) = 0 = W ((<£)). In this configuration the off-diagonal blocks in w | vanish. In fact, we have m 0 m 2 = f K f)ii ~Sij V 0 {mFmF)ij. so that (5.10) is indeed true for mass eigenstates, (chiral) supermultiplet by supermultiplet. Indeed, at the moment, this is just the evident supersymmetric fact that fermions and bosons in each chiral supermultiplet are degenerate in mass. Strictly, in this situation, the content of the sum rule (5.10) is empty 3 . However, even if (F*) ^ 0 (as happens with supersymmetry broken spontaneously, a la O'Raifeartaigh, Ch.7) and those off-diagonal elements in JJ}|y are nonvanishing, the supertrace sum rule (5.10) will be shown to be still valid. We have given a somewhat elaborate discussion of scalar and fermion masses here to lay the foundation for the derivation of that result, which will come in Ch.7. Wess-Zumino model A celebrated special case is the Wess-Zumino model [5.4] which consists of only one chiral superfield $. We impose time reversal invariance so that all couplings are real, but do not insist on the invariance of matter parity (cf. §4.5). For this specific case, the action can be written as

S=fdsz&$

+ (h$ + ^m$2 + t&\s(V(g)

T h e corresponding "on-shell" Lagrangian density in component form is

c = i^id,]i-4>*d24> - \m(tf + tf) - {m* + ^)-\h 3 The relative factor of 2 in front of the fermionic contribution to

+ mcf>+^2\2.

(5.11)

(5.10) comes from the following fact: whereas l are counted as two separate scalar states, & is counted as a single two component fermion state. It is important to realize that this way of counting does not correspond to the number of degrees of freedom described by those fields.

76

5. Interacting Superfields

This Lagrangian density makes it evident that the Wess-Zumino model is the supersymmetric generalization of the theory of a massive scalar field with upto quartic self interactions. The equations of motion following from (5.11) are ia'dpt - m£ = f£4> ,

(5.12a)

ia»d^-mt

(5.12b)

= f(;
(d2 + m2)(j) = -mh - { ( £ £ + 2 m | 0 | 2 + m2 + f\<j>\2<j>* + 2h*) .

(5.12c)

In the limit of vanishing h and / , (5.12a) and (5.12b) combine to yield, cf §3.1 and (3.6) ff, the free Majorana equation in the Weyl representation while (5.12c) yields the free KleinGordon equation respectively: (ty5 M - m)AM = 0 , AM = ( $ J ,

(5.13a)

(d2+m2)4> = 0.

(5.13b)

Eqs. (5.13) describe free fields with identical masses corresponding to a Majorana spin half fermion and two spinless bosons which make up (cf §3.3) the on-shell Wess-Zumino supermultiplet. Of the spinless bosons, the linear combination (4> + 4>*)/V2 corresponds to the scalar while the orthogonal ((f> — *)/iy/2 stands for the pseudoscalar. With two WessZumino superfields, $+ and $_, say, one could make a Dirac fermion field ip with left chiral component tpL = £ + and right chiral component ipR = £_ as well as complex left and right chiral sfermion scalar fields (j>+ and 4>*_ respectively. This will become clearer in §5.3 when we take up SQED. Even in the interacting case, we can rewrite (5.11) as C = 9M
5.2

-

7S)AM
-

{AM(1

+

75)AMr-

(5.14)

Abelian Gauge Interactions

A U(l) gauge transformation of the chiral superfields $;, $J can be denned by $J = e-2i9tiA{z)$i, $n

= $ t e a 9 t J At(,) )

VAA = 0 , VAK]

= 0

(5.15a) (5

15b)

Here g is the gauge coupling, tt is a real number which is the U(l) charge of $,, A is a complex function of z = (x, 9,6) specifying the local gauge transformation and there is no sum on i. The requirements of vanishing chiral covariant derivatives have been imposed on A and A* to maintain the chiral nature of the superfields "3>j and $ j under gauge transformations. A sufficient condition for these is for A to be function of y only (cf. 4.16a). A and A* can be thought of as chiral superfunctions in superspace. The factor 2 appears in the exponents

5.2 Abelian Gauge Interactions

77

of (5.15) because the imaginary part of the scalar component of the superfield ih is minus half the usual gauge transformation function introduced in an abelian gauge field theory: Ap —»• Afj, — 29M Qm
(5.16)

29 iV

and by generalizing the said K.E. term to [$Je ' $;]£). The introduction of the exponent does not destroy renormalizability, as can be simply seen by going to the Wess-Zumino gauge; from (4.35) (Vwz)n vanishes for n > 3. In addition, the D-component of V being gauge invariant, we can add that multiplied by a real constant r\. We can also introduce, following (4.47), the kinetic energy term for the vector superfield in terms of the gauge invariant supersymmetric field strengths W and W of (4.39). The full Lagrangian density for a system of self-interacting chiral superfields $, coupling to an abelian gauge vector superfield V reads

H

V A W 4 + W^WMl

+

JF

|$teW$. + vy] L

+[w($i)+

h.c.]F ,

(5.17)

JO

where the superpotential W($i) has to respect gauge invariance. The corresponding action is

S=

fd8z

(^\e29tiV^i

+ r,v)

+

f 66z l^WAWA

+ W(*0

+

f S6zf^WAWA

+

WH$t)\.

In component form and with the superpotential of §5.1, the Lagrangian density of (5.17) is given by use of (4.36), (4.37), (4.38) and (4.46) to be C = l-D2 + nD- (IteiWiM

i v " + h.c\

+ iXo-xd^X + iZiO-x^i - yfigtiQ&h

+ | A^i

+ h.c.) + gt^D

| 2 - F?Fi .

(5.18)

In (5.18) Aj^ is the gauge covariant derivative d^ + igUA^ and we have discarded a total derivative. We see that the coefficient 2 in the exponents of (5.15) has been taken care of by factors of 1/2 generated in (4-37) and (4-38). Moreover, the real field D acts as another auxiliary field in addition to the complex Ft given by (5.4). The corresponding equation of constraint dC/dD = 0 yields D = -gtftifa ~ V • (5-19) (5.11) is now generalized to C =

z & ^ A ^ + |A iM &| 2 - J F ^ F ^ + a ^ [ d p ] A -^gUiXUi

+ h.c.) - Q&fcW«(0) + h.c^j - Vfa,
(5.20a)

78

5. Interacting Superfields V(<j>i,*) = FiF*+^D2

,

(5.20b)

where Fi remains If the abelian gauge symmetry group is a product of several C/(l) factors, one uses the index a to refer to a particular factor. Now Da = —rf — ga4>*(ta4>)i, a not summed in the second term, whereas the linear D-term in the Lagrangian density is rjaDa with repeated a summed as usual. Moreover, the gauge covariant derivative Aiu is da + ig"(ttAu)a while the gauge kinetic energy terms and the supersymmetric gaugino-scalar-chiral fermion (g-s-cf) interaction respectively are -Cgauge = —.F;vF^a Cg.s.cf

+ i\ao»[du\\a

,

= - V V [(UX)%4>i + h-c] .

(5.21a) (5.21b)

The scalar potential generalizes to V(
M> 9$

(5.22a)

Da = -r? a - 5>*(t°0)i •

(5.22b)

Our remaining discussion in this section addresses this more general situation. First, though, let us remark on the ground state configuration with <j> = (4>). The absolute minimum of V may yield a supersymmetric vacuum (cf. §2.2) with (F,) = 0 = {Da). (ia())j may not vanish, but any nonzero value of it would imply a spontaneous breakdown of the Abelian U(l) symmetry corresponding to a and a mass (via the Higgs mechanism) for the concerned gauge boson. Let us consider this more general situation and direct our attention to the mass terms of the vector bosons and their superpartners. We employ the subspace notation, introduced in §5.1, to investigate the gauge invariance of the superpotential W. Now Dai = dDald4>i = ~ga{ta4>)i and Df = dDa/d(f>i = -ga(ta(j>y, a not summed. The vector-boson mass term (VMT) in the Lagrangian density, referred above, may now be written as CVMT = \m2vabAauA»b

,

m2vab = {D°Dbi + DaiD%=w

(5.23a) .

(5.23b)

Furthermore, considering an infinitesimal gauge transformation of the superpotential, we have Wj()Dai{) = -F*Daj = 0 . (5.24) Differentiation with respect to 4>i yields Wyfo)D a ''(0) - FfDZJ = 0 .

(5.25)

5.2 Abelian Gauge Interactions

79

For a supersymmetric ground state, (Fj) vanishes and (5.25) becomes V^«tf»Z>««0» = 0 .

(5.26)

It follows that, for ($) ^ 0 (otherwise (Dai) also vanishes), the matrix (AOS- = W*()VM<0>) aj

(5.27) 2

has a null eigenvalue on the eigenvector D (((/>)); thus det M = 0. A similar statement holds for M2 j = WikdMW'({$)). We now discuss the masses of the members of the matter supermultiplets. Once again, though they are mass degenerate in exact supersymmetry, we wish to develop a formalism like that in §5.1 which can be easily generalized to the case of supersymmetry breaking. Concerning scalar masses, one needs to be somewhat more general. For scalars with abelian gauge interactions, (5.8b) is still true. However, on using (5.22a), one now obtains rather different expressions for elements of the scalar mass squared matrix: m

z

~Sy

—

ai [WWYj +• D D] <•• • • •' y a DfD i

J

-

J

Dai-Daj [WWV + DfDv)^' -

I

(5

^

OR1)

S

where we have used the supersymmetric vacuum condition that Wk (()) = W ((«/>)) = Da({4>)) = 0. Now it is easy to see that (5.28) has a vanishing eigenvalue with eigenvector

i.e. for each broken symmetry component (labelled by a), there is a massless scalar which is the Goldstone mode that gets eliminated by the Higgs mechanism. Turning to fermion masses, (5.6) needs to be generalized to include the mixing between gauginos and chiral fermions. The latter arises from the gaugino-scalar-chiral fermion interaction (5.21b) as soon as the scalar develops a VEV. Identifying the fermion mass terms (FMT) from (5.20a) and (5.21b), we can now write

UMT = -yfigrWtSi)

- IteiWAtt)) + h-c.

= -l&WFH^+h-c-,

(5-29)

where a is summed and

W.-1-&, -1CfLr Thus iv(mjfm/)] T r ( ™ ^ ) '

= [(WW)ij + 2DaiDaj

+ 2D?DaJ]
= [{Wp?)i + 2 D?D*i + 2 D*D3]^W.

(5.31a) (5.31b)

80

5. Interacting Superfields

Finally, from (5.23), (5.28) and (5.31), we obtain

Tr SL-Tr(m«t V2F+

^ F ^ F ) i +3 ^ v a a = °

(5.32)

(5.32) is now the more general statement of the supertrace mass sum rule (5.10), as applied to the supermultiplet containing a massive vector boson. Note that here a chiral supermultiplet and a massless vector supermultiplet combine to form a massive one with members that are mass degenerate in the limit of exact supersymmetry. Once again, the vanishing of the supertrace is not only valid in exact supersymmetry but survives F-type spontaneous supersymmetry breaking, as will be shown in Ch.7. The latter makes (5.32) even more nontrivial.

5.3

Supersymmetric Quantum Electrodynamics (SQED)

The spectrum of the supersymmetric generalization of QED includes the electron which is a four component Dirac fermion. It can be thought of as a complex linear combination of two degenerate Weyl fermion fields. This obliges us to start with two left chiral superfields $+, <£_ with respective charges (multiplied by the QED coupling strength) q, —q and the gauge transformations $'+ = e -2i 9 A(;) $+ ^

#_

= e+2iqA(z)Q_

_

(533)

Here $ + contains the left chiral component of the fermion field and <J>_ that of the antifermion field, i.e. $ 1 contains the right chiral fermion field. The vector superfield, of course, remains the same and obeys the transformation (5.16). Now (5.17) gets specialized (with a real mass M and setting 77 = 0 for definiteness) to C =

wAwA + wAwA

2 v

+ [&+e « $

+ + $ + _e- 29V, $_

+ M [$+$_ + h.c.]F .

(5.34)

The corresponding action is = J dsz &+e2qV$+

+ &_e-2(>v$-+6W{e)(hvAWA

+ 6i2\e)(hvAwA

+ M+$}

+ M&+&_

(5.34) is manifestly gauge invariant. Furthermore, if each of the above chiral superfields is assigned a matter parity (§4.5) of value —1 and the vector superfield a value + 1 , the Lagrangian density contains no term that is odd under matter parity. In the light of the discussion in §4.5, this means that the Lagrangian conserves i?-parity. On using the equations of constraint, in place of (5.4) and (5.19), we have (5.35a) Z? = - 9 ( | ^ | 2 - | 0 _ | 2 ) .

(5.35b)

5.3. Supersymmetric Quantum Electrodynamics

(SQED)

81

Finally, the "on-shell" Lagrangian density for supersymmetric quantum electrodynamics is CSQED

= -\Fia,F'u'

+ i\a»dll\ +

iZ+a»Alt+

+ t £ - a " A p £ . + | A ^ + | 2 + | A ^ _ | 2 - M 2 (\<j>+\2 + |_|2) -M(^-

+ ^-)-^(|^+|2-|0_|2)2

-V2q [X ( f + 0 + - £_0_) + A (£+<j>*+ - £-0*)] ,

(5.36)

with AM = dft. + iqAp. Next, we can define the four component Dirac fermion and gaugino fields ip(x), \M(X) by [cf. (3.20) and (3.24)] 4>a = ( ^ )

, 4>„ = {e

f+s) , XMa = ( ^ )

, Xm = (AB

The definitions are consistent with V> = tp^j^,, XM — ^ 7 ^ with 7 ^ = I

As) • J in the Weyl

representation and Vl = ( ^ ?-A) = (£+A £-) a n d AMa = ( \ i A-4). By using (3.19a), (3.28c) and (3.29c), the gauge invariant fermion K.E. terms in (5.36) can be rewritten modulo total spacetime derivatives as |A
+

l

-Xa^X + »£+a"A„£+ + t£_a"A„£_ = ^ W ^ A

M

+ i ^ A ^ ,

the invariance of which under the transformation ip(x) —> ip'{x) = e~'qA^ip(x) Furthermore, by use of (3.30), (5.36) can be recast as a whole: CSQED

= - JV "

+ ^u-fd^Xu

+ iirfdrrfr - qfrfW

~

is transparent.

M^

+ | ^ + | 2 + | 9 M 0 _ | 2 - M 2 ( | 0 + | 2 + |^_| 2 ) +q2A2(\ct>+\2 + |0_| 2 ) - 2iqA»l[dll]4>+

+2M"0*[a^_-^(|0+|2-|0_|2)2 -V2q((p*+XMijjL

+ ipLXM<j>+ - <j>-\Mij)R - ipRXM(f>t) ,

(5.37)

with ipLtR = | ( 1 T J5)IJJ and [dj as defined after (4.28). (5.37) describes our supersymmetric quantum electrodynamic system. We shall refer to the supersymmetric partner of a fermionic particle as a sfermion and that of a bosonic particle as a bosino. The spectrum of the matter sector of this theory thus consists of a Dirac fermion (particle / and its antiparticle / ) as described by the Dirac field ip, the left chiral sfermion fi (plus its antiparticle fi) as described by the complex scalar field + (plus

5. Interacting Superfields

82

0+) and the right chiral sfermion fR (plus its antiparticle fR) as described by the complex scalar field 4>*_ (plus 0_); f,fi and / # all carry charge q and their conjugates have charge —q. In the gauge sector the spectrum consists of the photon 7 as described by the vector field AM and the photino 7 as described by the four component Majorana field XM- The supermultiplet structure is as shown below:

£)• (tl). (£)• ( £ ) • (? The fermion and the photon are to be regarded as ordinary particles with positive i?-parity (cf. §4.5). But their superpartner sfermions (left as well as right chiral) and the photino are to be regarded as sparticles with negative Rp. The conservation of Rp is automatic in SQED and dictates that sparticles appear in a pair at every vertex. The vertex Feynman rules, following from the interaction part of ICSQED, are shown in Fig. 5.1 with the vertex factors, to be inserted between spinors, given explicitly. N.B. Here PL = | ( 1 — 75), PR = | ( 1 + 75) and the vertex factors are to be written between final state Dirac spinors u, v and initial state ones u, v. In this chapter we do not put arrows on any Majorana fermion line. Therefore, one is free to choose either u o r i i for a Majorana fermion linking to an initial state Dirac fermion and similarly u or v for one linking to a Dirac fermion in the final state. A more definitive formulation of rules for Majorana fermions will be given in Ch.9. Fig.5.1

—iq^f"

T

L,R\

^

y

P \

~iV2qPL

L

')r^J^/\A/\/\/^^

r

f-+fLl

.••• In

V '4m

— iq(pi + P;)n

Chiral indices I m n p L L L L Ft R R i t L R L R

7/"/m~/n"/p

-2iq2 -2iq2 +iq2

vertex

5.3. Supersymmetric Quantum Electrodynamics (SQED)

83 Fig.5.1 contd.

« \ L

2ig2»V 'V"

Legend f Dirac fermion ' V W W ^ '"V/V/X/V/V*

photon photino left sfermion right sfermion

Fig.5.1. Feynman rules for vertices in SQED.

The four main features of SQED to remember are: • Two left chiral superfields with opposite electric charges are needed to describe the supersymmetric quantum electrodynamics of a Dirac fermion. • A sfermion, appearing at a vertex, has a specific chirality. Chirality is conserved in all vertices. Multisfermion vertices involve two or four sfermions, grouped in chiral pairs. • Rp conservation is automatic in SQED and sparticles appear in a pair (or quartet) at any vertex. • The coupling strength of the fermion-chiral sfermion-photon vertex is \/2 times that of the gauge coupling strength and the strengths of the four sfermion vertices are also proportional to the square of the latter. Indeed, these features are also true of the two gauge theories described later in this chapter.

5. Interacting Superfields

84

5.4

Nonabelian Gauge Interactions

To start with, let us confine ourselves to a simple gauge group G with a universal gauge coupling constant g since the extension to a semisimple group or a product group with U(l) factors is straightforward. The nonabelian gauge transformations on a left chiral superfield $, belonging to the unitary irreducible representation 11 of G, are nonabelian generalizations of (5.15). They can be described as $'7 = | e - 2 ^ A a ( 2 ) T a l

* y = *S

$ j

,+2igAa' {z)Ta

^ pAAaTa

= 0 )

DAAa*Ta = 0 .

(5 3 8 a )

(5.38b)

JI

Here Aa(x,6,9) are the complex chiral superfunctions specifying the gauge transformation and Ta are the hermitian generators of G in the space of the representation 1Z whose basis indices are denoted by subscripts / , J. Unless otherwise specified, we work in a Cartesian basis. Thus [T°,

T 6j =

UabcTc.

TJ. TaTb

= T^gab

^

(539)

where tabc are the completely antisymmetric real structure constants and T(TZ) is the representation constant of TZ. For the adjoint representation, T is a positive constant k which is the quadratic Casimir C2(G) defined by tabctdbc = C2(G)5ad. More generally, (TaTa)u - C2(n)5ij. It is convenient to define A „ = 2gkaTfj,

Vtj = 2gVaTfj .

(5.40)

In (5.40) Va(z) are the vector superfields transforming in the adjoint representation of G. Now the extension of (5.18) to a nonabelian gauge invariant and supersymmetric Lagrangian density is simple provided the gauge transformation of V is generalized from (5.16) to ev' = e - * A t e V A , e-v

= e-iAe-veitf

(5.41a) (5 41b)

The transformations (5.41a) and (5.41b) are inverses of each other and imply that T r ^ e ^ ] is gauge invariant. The transformations (5.41) are, in fact, valid for any representation of the generators provided they have a consistent solution V = 2gV'aTa. It is sufficient for us to consider infinitesimal gauge transformations neglecting 0(A 2 , A*A, AA*, A t2 ) terms. Then, writing V — V + 6V and using the Baker-Campbell-Hausdorff formula [5.5], we have eV,=

V + i{A-A<)+'-[V,A

+

A<}+±[V,[V,A-A<}}+-

{V = V " _ y = i ( A - A t ) + i[V,A + A,] + i [ V , [ V , A - A t ] ] + . . . .

^ (5.42b)

5.4 Nonabelian Gauge Interactions

85

In terms of the superfields V°, (5.42) works out to yia _ya

=

+

^

A

«_

A«t)

_ gfobcyb^c

+

^

_ l^abc^deybyd^e o

_

A

^

+

_

_ ^

^

As before, we choose to work in the Wess-Zumino gauge of the superfields Va where triple products of V's or those with a higher number of V-factors vanish. Since, in this gauge, a vector superfield has no ^-independent term, cf. (4.34), the ^-independent component of A" must be real. A general gauge superfunction A" will lead to a V'° which is not in the WZ gauge, though an appropriate supergauge transformation can transform it to that gauge. Suppose one insists on both V and V'a being in the WZ gauge. Then, if VaVbVc vanishes, one must consistently choose A° in a way such that VlaVlbV'c is zero. This means not only that ( A a - A a +)(A 6 - A 6 t )(A c - Act) and Va{Ab- A 6 t )(A c - Act) vanish, but also that the third and successive terms in the RHS of (5.43) consequently vanish, leading to the infinitesimal transformation of a vector superfield obeying the WZ condition: 6Vwz = i(A-tf)+*-[Vwz,A

+

tf].

(5.44)

The definitions (4.39) of the abelian supersymmetric field strengths WA and WA need to be extended to the nonabelian case. The generalized definitions are WA = -\vf>e-vVAev,

(5.45a)

WA = --VVevVAe~v. (5.45b) 4 Unlike the gauge invariant abelian field strengths, these transform nontrivially (covariantly) under gauge transformations, i.e. WA —> W'A, with = -lvVe-iAe-veiA*VAe-iA*eveiA = --e-iAVVe-vVAeveiA . (5.46) 4 4 In the second step of (5.46), use has been made of the chiral properties of A and A*, i.e. 2)A = 0 = VA^. These two relations enable one to commute e~tA through VV as well as e tAt through T>A and write the gauge transformed field strengths as W

W'A = e~iAWAeiA

,

(5.47a)

W'A = eiAWAe~iA

.

(5.47b)

Expanding in powers of V, one has e~vVAev

= VAV + \ [VAV, V] + \{[VAV, V], V] + 0(V 4 ) + • • • .

(5.48)

In the WZ gauge only the first two RHS terms in (5.48) survive and (5.45a) becomes WA = -\VV

VAVwz - \vV[VAVwz,

Vwz\ .

(5.49)

86

5. Interacting Superfields

In analogy with V in (5.40), we can define W% by WA = 2gWaATa .

(5.50)

(5.49) and (5.50) lead to the expression W% = -\VV[VAV&Z

+

igtabc(VAV^z)V^z]

(5.51)

On expanding the superfield WA in terms of its components, one obtains the nonabelian generalization of (4.42), namely W% = XaA(y) + Da(y)eA - (a^9)AF^(y)

+ ieda^A„\aB

(y) ,

(5.52)

with FZ^d.Al-dvAl-gt^A^Al,

(5.53a)

abc AAb a»c ^Xc*^ AMA * = <^A * - gt-+ ai

(5

aJ

53b)

The nonabelian component expansion of W? can be made in a similar fashion. The final Lagrangian density is a nonabelian generalization of (5.17). It can be written as C =

1 Tr 16g2k

[w^wu +

WAW

+

*, T (e v )«*>l

+[W(*i)

+ h.c.]F ,

(5.54)

where k = T(TZ) and W($i) is a gauge invariant superpotential. We have reverted to the use of i as a general type index which covers the gauge representation space index / and can include other types such as generations. Note that the first RHS term above is simply j(W>VoW^ + W^WAa)p- Component expansion plus the use of the equation of constraint dC/dDa = 0 allow (5.54) to be rewritten as £

=

i^aIMA]^i

+ (A^jy(A0k(l)k)--F^F^a

1.

-y/2g(\aSJ?jj + h.c.)-V{i,<%)

+

iXa^A,X

2 & W J M

+

hx

-

(5.55)

with4 (5.56a) V(cj>i,cj>*) = FiF* + Fi =

dW

m

lDaDa

Da = -gfiTtfr

(5.56b) (5.56c)

If we generalize the above discussion to include gauge groups that are products of factor groups (including C/(l) factors), G = r L ^ " ' w e nee( ^ to put a superscript a to the gauge 4

The reader can verify that the conserved spinorial supercurrent density for the system of (5.55) is

5.5. Supersymmetric Quantum Chromodynamics (SQCD)

87

coupling and make it ga and characterize T° by Ta(-a^, a referring to the factor. Moreover, if we use the index h to refer generically to the generators of abelian factor groups, we can admit the term r\hDh in the Lagrangian density of (5.55). Now we have £>a(a) = -rf -galT^a) - T£a, a not summed. For a supersymmetric vacuum, the rest of the discussion on masses proceeds identically to that of §5.2. Thus (5.28) and (5.31) hold here also and the derivation of the supertrace mass sum rule (5.32) follows exactly as in the abelian case.

5.5

Supersymmetric Quantum Chromodynamics (SQCD)

Let us take one quark flavor transforming as an SU(3) color triplet and eight gluons transforming as the adjoint color octet. In order to supersymmetrize this uniflavor QCD theory, we proceed in analogy with electrodynamics and introduce two chiral superfields $+ and $_ transforming as an SU(3) color triplet and antitriplet respectively: «V = e - 2 ^ T ° A ° $ + $'_ = e

2 i

f

(5 57a)

^°AV ,

(5.57b)

where gs is the QCD coupling, the generators T° equal one-half times the Gell-Mann SU(3) A-matrices in their fundamental representation and T° are the corresponding complex conjugate matrices. Since T° are hermitian, $T = ^e219'7""^^. Thus the mass term W($+, $_) = M&+

(5.58)

qualifies to be a color invariant superpotential term and the full scalar potential at the tree level then is

V&+, .) = M2(j>l
(5.59)

a

Proceeding as in the SQED case, one can write the onshell Lagrangian density in terms of the four component Dirac field ip as CSQCD

= - j C ^

0

+ ^A a M 7n^ ac + ^ o6c 4)A c M + |A^ + | 2

+(A"£. y(A^*_) + t^7"AMV - Mfal> - M2(4>U+ + 4>U-) - v / 2 < 7 s ( ^ T 0 A ^ + + 4>lXaMTail>L - ct>T_~XaMTa^R -

-\a2 X > + T > + - ^-f°0_) 2 ,

^XaMTa4>*_)

(5.60)

5. Interacting Superfields

88

where A p = dli + ig,TaAall, with (Ta)bc = -itabc in the adjoint representation, where tabc are the structure constants for SU(3) in the standard form. The hermitian conjugation and the transposition in (5.60) are in color space. We can now use (5.60) to describe our uniflavor supersymmetric quantum chromodynamics. The quark q (as well as the antiquark q) is described by the color triplet Dirac field ip, the left squark q^ (together with its antiparticle q~L) is denoted by the color triplet complex field (j>+ (plus cj)*+) while the similar field
-ig,Ta(pi+pf)ll

-iV2gsTaPL

N?L,R

t

+iV2gsTaPR

-gst""^

5.5. Supersymmetric

Quantum Chromodynamics

(SQCD)

89 Fig.5.2 contd.

- gstabc

[(r - p)„r]pil + (p-

q)pt]liv

+ (q - r) M r? vp ]

p+q+r=0

dbe

Ws PbeVde(jlpprl»p)+*uxt {luaVvp

~ •niwVlxr) + t

t

{•nV.Vr\f>a -

n^wa)

ig2s[Ta T 6 ]+»V

Color indices

/* % \

^N Chiral

% /

\ %

Q M

N

L L

L L

R R R

P

QQ-QM-qN-qp vertex

•92s

-i-^-i^MP^NQ

+ &MQ&NP)

•92s

R

indices 9s L

R

L

R

1

i y {5MP5NQ

-

^MQ&NP)

Legend quark

'W\AA/

I3 •>

N

gluon gluino left squark

• >

right squark a rpa 1 \a Fig.5.2. Feynman rules for SQCD vertices with 55 T = ±X 5

Here Aa are the Gell-Mann SU(3) matrices. Thus now T^LT^P = \{SMP5NL

— \&ML5NP) •

5. Interacting Superfields

90

We can end this discussion of SQCD with the following comments. • SQCD vertices include those that are straightforward generalizations of SQED ones. • The intrinsically nonabelian triple gluon coupling of QCD has a supersymmetric analog in the gluino-gluon-gluino coupling carrying the same strength. • The quadruple gluon coupling does not have such a supersymmetric analog; this is easy to see in terms of mass dimensions since any such analog would mean a term in the Lagrangian density with a dimension exceeding four. • The quark-squark-gluino vertices, as given above, are in pure SQCD in which quark/ squark mass and interaction eigenstates are identical. When supersymmetry is broken, differences in general arise between quark and squark flavor rotations effecting mass diagonalization. In consequence, there are some additional mixing factors in these vertices which will be detailed in Ch.9.

5.6

Supersymmetric chiral gauge theory (S^GT)

We now discuss the supersymmetric generalization of a chiral gauge theory based on the SU(2)i x U(l)y gauge group of the electroweak theory of Glashow, Weinberg and Salam. For simplicity, we restrict ourselves to one fermionic generation without worrying about ABJ anomalies [5.6] and consider the purely massless gauge theory without the Higgs mechanism. A more realistic description with spontaneous symmetry breakdown will follow in Ch.8 when we discuss the Minimal Supersymmetric Standard Model. Here we start with one left chiral SU{2)L doublet (up type and down type) and two SU{2)L singlet left chiral superfields:

* + = ( J ^ ) ; * 0 -,*d-.

(5.6i)

Let the SU(2)L and f/(l)y gauge (vector) superfields be V (a = 1,2,3) and VY respectively. The gauge transformations of the superfields in (5.61) are straightforward applications of (5.15) and (5.38). We do not give them explicitly but simply specify the SU(2)L and U(l)y gauge couplings to be g2 and gy respectively. The matter kinetic energy term can now be written, following the discussions given in §5.2 and §5.4, as •$t_ e (»VT- + 9 y v^y) $ + + ^ $ t _ e 9 , v - y $ r _ J } (5 62) r=u,d

where r a are the Pauli matrices and Y is the hypercharge operator. The electric charge is Q = (T3 + Y)/2. We shall use YL and YrR to denote the hypercharge eigenvalues for <3>+ and %._. The gauge kinetic energy term, following from (5.17) and (5.54) similarly is WYAW\

1

Y^rrYA

+WA'AW

+ F

4

•aA

WaAWaAA "•" + vyWA' AW

(5.63)

The sum of (5.62) and (5.63) leads us to the full Lagrangian density. This can be written in component notation by using (5.22) and (5.55). Let us use a more specialized notation in

5.6. Supersymmetric Chiral Gauge Theory (SxGT)

91

the gauge sector, namely W^, W^, A (the vector sign covering Cartesian subscripts 1,2,3) to denote A^, F£v, XaM respectively in the SU(2)L factor and BM, B^, A0 to denote A^, F/iv, XM respectively in the U(1)Y part. Furthermore, we use A 2 / and A* for the respective SU(2)L x U(1)Y and U{l)y gauge covariant derivatives:

A;;1 = d„ + ig2Wli • £ + igYB„- ,

Now we can write the Lagrangian density of this theory as CSXGT

= -\w»„ • W^ -

\B^B^

+ l-Xol»dllX0 + iJl-f^L

+%-(xrd/x +i Y

- g2lr

^K^R

x W, • 1) + |A 2 1 ^+| 2

r—u,d

+ Y |A^ r _| 2 - \&{&\*+)2 - \gY(^YL^+4>+ r—u,d \ Y ^l^r-l 2 ) 2 - V2g2{<j>lX-^L+^

• U+)

r=u,d

Vlgy [-y [4>+^oipL + 4>LX0+) \ Y 2

Y

.

rR ( # - ^ * 0 + 4>T-1}1>R) ] •

(5.64)

r=u,d

The relative minus sign between the terms in (5.64) containing Yj, and those with YR occurs since, while Yj, is the hypercharge of +, YR is the hypercharge of 4>*_. As done with SQED and SQCD earlier, the vertex Feynman rules for this S%GT can also be given from (5.64). This time we use the fields }L = ( r

J > II — f 7

I , fuR, fdR, fuR, fdR

(5.65)

in place of ipL, 4>+, ipUR, ipdR, 0£_, *d- respectively. Charged gauge boson and gaugino fields respectively are W* = (W? =F iW^)/y/2 and A* = (A\ =p i\2)/\/2. The weak isospin up, down fermionic particles will be designated / , with q = u, d respectively. The electroweak vector boson particles will be described by the symbols W*, W 3 and B, while the sparticles, corresponding to the fields A*, A3 and A0, will be referred to as w±, w3 and B respectively. As before, states without (with) tilde will have positive (negative) J?-parity. The vertices are given below in Fig. 5.3 in terms of the chiral projection matrices P^R = (1 T 7 S ) / 2 . Note that no arrows have been put on Majorana fermion lines here. Once again, we draw attention to an extra factor of 2 for the w 3 -w ± -w :F and w ± -w :F -w 3 vertices.

5. Interacting Superfields Fig.5.3

• 92

q = u

D

92 -I—^P,PL

q = d i^--y„PL

q = u, d

- ^ 7 M ( ^ L +

W W

*92 [(r - p)„f?PM + (p - q)PViw + (q- r)fj,r)vp) p+q+r=0

Wi = W+ , W2 =

W+ ^92\^r)p.pr)va

~ VptoVvp ~

r

}p.vr1pa)

>v3 = w - , vv4 = w W i = W + , >V 2 = W ~

n>3 = w 3 , vv4 = w 3

92 i \ i-7^(Pi+Pf)p.

-iS&CZVupVva

- Vp.aVvp - Vp-vVpcr)

5.6. Supersymmetric

Chiral Gauge Theory

(SxGT)

Kh pw\ru\ns\ii

w>

>A

q= d

i-^{pi+Pf)n

Pf \f",< '9\r\nj\J\/\r>

B

i/L

^

^ WVWWN|1

•i9-fYL{Pi+P})„

q = u, d

- i y Y,fl(pi + Pf)u,

B

/

V A.

K/

/ B

q

ffy&

= u,q' = d,>Vi = W -

q = d,q' = u , W i = W +

" ^

V2

V

*X/

,$•%•

A rip, 2

/ V

\

2

^v

/^

W = W 3 , q = u,d W = B,q = u,d

,.••

V

J ^

.

/

V

^

V

q= u

i^f-Y^

q= d

- iY ^2 y ^

99,

iy»? M „ i^Yl^v

5. Interacting Supernelds Fig.5.3 contd. On

q = u,d

9

i|y;,iv

q = u, q' = d, w = w+

-ig2PL

q = d, q' = u, w = w~

-igiPi,

w= w ,q = u

—i—/=PL v2

w = w, q = d

i—pPh v2

w = B,q = u,d

q = u,d

-i^YLPL

i^j=YqRPR

±*027/i

Ti92lii

q

= q' = u,d

--{9l

q = u,q' = d -\{9l

+ 9\Yl) + 92yYl)

5.6. Supersymmetric Chiral Gauge Theory (S\GT)

95 Fig.5.3 contd.

V"" \&k

v

?

'\

,--

VR

q = q' = u,d

--gyY?R

q = u, q' = d

-gyYuRYdR

-i^yLyg-fi

X

Legend

>—

f 'wol/wr

Dirac fermion SU{2)L

ry A A I A A-*=>

x U{l)y gauge boson

charged 5t/(2)i gauginos neutral SU(2)i and l 7 (l)y gauginos

3

W ,B left sfermion right sfermion

Fig.5.3. Vertex Feynman rules in S\GT We end with the following comments. • One needs to be careful about a factor of 2 in going from (5.64) to the vertices in so far as Majorana fermions are concerned. • Mixed SU(2)L and U(1)Y vertices involve the coupling product gYg2 as well as the squared sum g\ + gYY£. This is a characteristic of quartic vertices with complex scalars which transform nontrivially under the two gauge groups.

96

References • Since chiral symmetry is unbroken here, there exist no couplings which can take left chiral (s)fermions to right chiral ones or vice versa. Indeed, in the nonsupersymmetric case, there are no couplings involving both /& and JR. But, in the scalar sector of S^GT, there is a mixed chiral scalar vertex, namely the last quartic one shown above, arising from the — \{DY)2 term. However, its structure ensures 'chirality' conservation even for sfermions.

References [5.1] S. Coleman, op. cit., Bibl. [5.2] S. Dimopoulos and H. Georgi, Nucl. Phys. B193 (1981) 150. [5.3] S. Ferrara, L. Girardello and F. Palumbo, Phys. Rev. D20 (1979) 403. [5.4] J. Wess and B. Zumino, Phys. Lett. B49 (1974) 52. J. Iliopoulos and B. Zumino, Nucl. Phys. B76 (1974) 310. J. Wess and J. Bagger, op. cit, Bibl. [5.5] W. Miller, op. cit, Bibl. [5.6] S.B. Treiman, R. Jackiw and D. Gross, op. cit., Bibl.

Chapter 6 SUPERSPACE PERTURBATION THEORY AND SUPERGRAPHS 6.1

Nonrenormalization of Superpotential Terms

One of the striking properties of any supersymmetric theory pertains to the superpotential which is controlled by the nonrenormalization theorem [6.1]. That theorem has the conseqence that the superpotential does not receive any renormalization to any order1 in perturbation theory. This means that the wave function renormalizations of the superfields appearing in superpotential terms must match oppositely with the renormalizations of the coefficient masses and couplings in those terms. If the former are regarded as independent, the latter will be dependent on them. Thus, for renormalized perturbation calculations, no counterterms need to be added to those terms in the Lagrangian density which are derived from the superpotential. To one loop order, the nonrenormalization theorem has been illustrated in a toy supersymmetric model in §1.3. A technical point, that was made (cf. ftnt.8, Ch.l) in connection with this illustration, was the need to avoid using the customary dimensional regularization [6.3] procedure but to employ modified dimensional reduction or DR [6.4]. Here the subtraction procedure is as in the MS scheme (cf. §1.3) and the momentum integrals, with propagator factors in the denominator, are evaluated in d — 2UJ dimensions before eventually taking the limit w -> 2. But the 7-algebra and the P-algebra in the numerator are done strictly in four dimensions since it is only there that the numbers of fermions and bosons match in the case of a supersymmetric system. The components of a gauge field A^, with 2u> < fj, < 4, act as scalar fields ("e-scalars") in the adjoint representation of the gauge group, but a variant of the DR scheme exists2 where they completely decouple from physical degrees of freedom. Even the DR scheme in superspace is not completely free of trouble; there are problems [6.4] when the number of loops gets large (> 4), though they may not cause any practical difficulties. This, however, is the most convenient supersymmetric regularization scheme available so far and we shall adopt it. A more important point, earlier alluded to in Ch.l, is the following. The nonrenormaliza1

Certain deviations can arise in higher loops [6.2] when massless fields are present, but we ignore such subtleties here. 2 I. Jack, D.R.T. Jones, S.P. Martin, M.T. Vaughn and Y. Yamada, loc. cit., Ref. [6.4].

97

98

6. Superspace Perturbation Theory and Supergraphs

tion, mentioned above, is the key argument behind the perturbative stability of the hierarchy between the electroweak scale (~ 102 GeV) and a high unification scale (~ 1016 GeV) or the scale where gravity becomes strong (~ 1019 GeV) in a supersymmetric theory. Once this type of a hierarchy is fixed at the tree level, it does not get destabilized by perturbative radiative corrections. In order to understand the origin of this result, one needs to employ supergraphs. The latter are also needed to efficiently perform renormalization group calculations in supersymmetric field theories, which will be required later. Therefore, in this chapter, we aim to introduce supergraph techniques with super-Feynman rules of perturbation theory in superspace. We have to develop, for this purpose, a formalism for the quantization of superfields. To that end, we first need to generalize functional methods to superspace. This is done below by largely following the two works cited in Ref. [6.5].

6.2

Functional Methods in Superspace

Points 1 and 2 in superspace can be represented by supercoordinates Z\ = (xi,6i,Si) and z2 — (x2,62,62) respectively. We shall always use the notations d8z = d/kxdi9, d6z = dixd26 and d6z = dixd20, introduced in Ch.4. Referring back to (4.4i,j), we can define a fermionic subspace J-function and describe its properties 3 : 6l2

= 5(4)(01 _ 02) = s^(e1 - e2)6^(e1 - e2) = (ex - e2){px - e2) fa - e2)fa - e2), (e.ia) / •

d%6l2 = l,

(1 = 1,2),

(6.1b)

—didi dmdm8i2 = 1 (l,m = 1 or 2, not summed) . (6.1c) 16 It follows from (4.4) as well as from (4.9 - 4.11), by ignoring surface terms, that - under a spacetime integral J d4x - the operator V{Di Vmf>m behaves like d[di dmdm, l,m not summed. (6.1a) and (6.1c) then imply that —V{Di VmVm5l2 = 1 , lo ^8l2V{Di

(6.2a)

VmVm612 = 512 ,

(6.2b)

I, m not being summed. The symbol '= in (6.2a) means that the equality is valid only under a spacetime integral J d4x; multiplication by a general superfunction from the right, but not from the left, is permitted. No such restriction is necessary for (6.2b). Take a general superfield J-(z) = T{x, 9,6) which spans all of superspace, e.g. a vector superfield V. (Chiral superfields, which span only parts of superspace are therefore not covered here; they will be treated in the next paragraph). A functional derivative of J-{z) can be defined with the following properties: 5T{2) &F(1)

Sw (an - x2)612 = 6^ (Zl -

Z2)

.

(6.3a)

3 The subscript I or m here standing for 1,2, refer to two points in superspace and should not be confused with the spinorial components 1,2 of 9 which are denoted by A, B or with those of 6, as denoted by A, B .

6.2. Functional Methods in Superspace

/

^fg

= l.

99

(6.3b)

The functional products between two superfields J~i(z), J-2{z) and more specifically between a superfield J7 and its source J are given by ?yT2 = J dh^iz^iz), f dszF(z)J(z)

T-J=

(6.4a) (6.4b)

respectively. We now turn to chiral superfields $(x, 9,6) = $(y, 9) and &(x, 9,9) = &{y, 9) with y, y as defined just before (4.16). It is clear from the action (cf. §5.1) of any theory, incorporating chiral superfields, that the latter do not span all of superspace. Instead of connecting to the full superspace integral J dsz, they span the parts picked by a factor 5^(9) or 5^(9). The equivalents of (6.3b) are now , 6 . **(2)

/

^ ^ i y =1 > J6

/

,

( 6 - 5a )

^(2)

~Z2mij = 1 •

( 6 - 5b )

while those of (6.4a) are 4 $ r $ 2 = / d&z^x^2

,

(6.6a)

$\4>\ = f d6z\$l .

(6.6b)

For a general supersymmetric action, the superpotential term, which is a chiral superfield, will have this limitation. In contrast, the kinetic energy and other terms, that allow being clubbed under a vector superfield, connect to the full superspace and are therefore not covered here; they can be treated functionally by using (6.3) or (6.4). In order to overcome this difficulty and develop a common formalism, we introduce a left chiral superspace projector n + and a right chiral superspace projector II_, defined by n+ = - ^ 5 ^ ™> .

(6.7a)

U_ = - ^ V V V V ,

(6.7b)

where d2 = d^d^ is the d'Alambertian in spacetime and the nonlocal operator 1/d2 is defined as in (4.14e,f). The choice of the forms of II ± is motivated by (4.14a,b) which also imply that

n + $ = $, n_$t = $ f . 4

$ i $ 2 = $2'$i and ${-$2 = $2'$i> assuming that the $'s are even under the ^-grading mentioned in §3.1.

100

6. Superspace Perturbation Theory and Supergraphs

Evidently, acting on a general superfield T, II + yields a left chiral superfield since VAU+Jr : 0 while n _ yields a right chiral one since P ^ I L T = 0. Finally, n

+

=

1 V --1vv v v v v v v v= 1 1 6 9 2 ) 1 vVV ~\m VV VV VT) = .— 1^

v

v=n +

'

(6,8a)

2

ni = I —r™ ) vv vv vv vv = n_ ,

(6.8b)

where we have used (4.11J,m). Moreover, n + n _ = I I _ n + = 0. Thus n + satisfies all the requisite properties of a left chiral projector and n _ satisfies those of a right chiral one. One can also introduce a third projector _ VAVVVA n

°=

VAVVVA

892 =

8d2

>

(6-9)

cf. (4.11k). The set of projectors IIj = (II + ,IIo,n_), i = 1,2,3, may be seen to satisfy the relations 3

5 3 ^ = 1, 11^ = 5^ .

(6.10)

We can now rewrite (6.6a,b) as

r . r R vvvv r R vv ( vv 6 *!42 = J d Z^U+^2 = -Jtfz^— —^ = -J Sz— (*1^*2 where we have used the facts that VA$i — 0 = VA$\. Now one can employ (4.12d) and footnote 4 to rewrite the above equations as $ r $ 2 = j d S ^ g f c j = j dsz (~^x) $2 ,

*J4J = Jd^l^l

= fd*z ( ^ * t ) *t ,

(6.11a)

(6.llb)

though one still has $ | - $ 2 = f d8z$\<&2- T h e shifting of t h e nonlocal operators VV/d2, 2 VV/d in (6.11) from one chiral superfield t o t h e other (by applying I I ± on t h e latter instead) is a useful trick. Now, returning t o (6.5a) and comparing with (4.12c), we see t h a t the former, as well as (6.5b), is solved by the following equations for t h e integrands: <5<S>(2) = --VlVl5^\zl-z2), <5$(1) ~ 4

(6.12a)

g^-IlWC^-.,).

(6.12b)

6.3. Functional Formulation of Superfield Theory

101

Eqs.(6.12) are the left and right chiral superfield equivalents of (6.3a). We may note here some consequences of the definitions (4.9) and (4.10) of V and V and of <5(8)(zi - z2) as ip ( 1-X2) (2TT)- 4 / dVr ' * <$i2, namely V1A6^(Zl-z2) VtSP\zi-Z2)

6.3

= ~V2AS^(Zl-z2)

,

(6.13a)

= ~V^8\Zl-z2),

(6.13b)

V1VlS^(zl-z2)

= V2V25^(Zl-z2)

,

(6.13c)

V1V15^{z1-z2)

= V2V26W{Zl-z2)

.

(6.13d)

Functional Formulation of Superfield Theory

We start with a general superfield (cf. §4.1) T% chiral superfields $, $* and a generic action S = f d8zK0(F,$,$f)

+ ( f d6zW($)

+ h.c^j .

(6.14)

In (6.14) KQ is a vector superfield containing all the kinetic energy as well as other allowed interaction terms, while W($) is the superpotential. The generating superfunctional - with source superfields j , J matching onto T, $ respectively - is given by the following functional integral over superfields: Z[j,J,jl]

= N f[dF}[d$}[d&}expi{S+j-Jr+J-$+jt-&)

,

(6.15)

where TV is the normalization constant. Let us use a square bracket ] to the right to mean an evaluation at 0 = J — j . We can then write the m-point function corresponding to (6.15) as 6mZ[j,J,Jl] dj(l) • -Sj{n)SJ{n + 1) • -6J{k)5Jt(fc + 1) • -<5Jt(m) = im{Q.\TF(l).

.JF(n)$(n + 1) • •$(*)$*(* + 1) • - ^ ( m ) ^ ) .

The connected Green's function (^"(1) • • •J'(n)$(n+1) • ••$(k)&(k+l) in terms of W\j,J,J*] = -i]nZ\j,J,Jt]

• • • $ t ( m ) ) c is given

by 6mW[j,J,Jt] 5j(l) • • • 6j{n)5J(n + 1) • • • 6J(k)6Jl(k m

= i (F{\) • • • F{n)${n + ! ) • • • ${k)&(k

+ 1) • • • 6Jt(m) + 1) • • • <E>t(m))c .

(6.16)

The "classical" superfields $, & and T, which are solutions to the equations of motion, relate to W via

102

6. Superspace Perturbation Theory and Supergraphs

These relations can be inverted to obtain expressions for J, J* and j in terms of $, & and T. The substitution of those in W, plus the explicit subtraction of the j-J7 + J $ + «/*•$* piece, lead to [6.5] the effective action r ( $ , $ t , . F ) which is the quantum analog of the classical action S[3>]: r ( $ , $ t , ^ ) = V K [ j ( $ , $ t , J - ) , J ( $ , J - ) , J t ( $ t , J T ) ] -j.f-j.$-

./t.$t .

(6.17)

Just as W generates the m-point connected Green's function, cf. (6.16), T generates one particle irreducible supergraphs Tm(l,- • • ,m) V positive integral m. If the classical action (6.14) splits additively into a free part 5° and an interacting part Sint- as S = S° + Sint- , then (6.15) can be rewritten as

ZM,jt ] = exp(tf- [!£,!£, I ^ z ^ j . j t ]

(6.18)

with Z0\j, J, ^} = No f[dF][d$}[d&} zxpi(S°+j-F+

J-$+Jt-^) ,

W0[j, J, Jt] = -i In Z0\j, J, Jt] ,

(6.19a) (6.19b)

No being the normalization and Wo being the generating functional for free, connected Green's functions. Eq. (6.18) can be inverted [6.5] and rewritten as

Zo[j,J,Ji}=exp(-iSh

1.1 1JL I_i_ zy.iJ1] i S j ' i 5 J' i (5 Jt

The perturbative procedure goes as follows. Free, connected propagators of the theory are first obtained by explicitly evaluating the integrals in Wo[j, J, J*] and then substituting that expression in place of W in the LHS of (6.16) for m = 2. The exponent in the RHS of (6.18) is then expanded in a perturbation series and the Feynman factors obtained order by order. This can be seen more concretely in the explicit case of the chiral superfield, as illustrated below. Chiral Superfield: connected free Green's functions The main technically nontrivial point in developing the superperturbation expansion for a chiral superfield is the need to keep using equations such as (6.11a) or (6.11b) in converting the subspace integral Jd6z, as in (6.6a), or Jd6z, as in (6.6b), to the full superspace one, namely J dsz. Thus the free part of a chiral superfield generating functional, corresponding to that a Wess-Zumino model (cf. §5.1), with vanishing couplings h = f = 0, is Z0[J, Jt] = No Ad$][d$ + ] exp? j * 1 - * + (J-$ + ^ m $ - $ + hxA j .

6.3. Functional Formulation of Superfield Theory

103

It can be recast as Zo[J,ji]

jd8zl^(z)^(z)

= N0J[d$][d&}&Vi m^u

,X>X>

+,

.

1 . .VV

+ r,

,

j^(z)'^-^(z)

1 , . .VV /

=

8

N0 f[d$][d&]expi

fd zl^-($(z)

T+/

.

mW

&(z))

1 mW

J1P~

+ ($(z) $*(*))

49s"

k*

f

(*)

J(*) (6.20)

• g ^ In the last step of (6.20), we have employed a complex two dimensional linear space of chiral superfields spanned by the row vector ($(z) &(z)) and the corresponding column vector. We can now use the identity /

dwdwt expz

£(w w t ) 0 ( W t U ( w

= const, exp

W t)f»

-sMs-fa

(6.21)

where Q is any symmetric 2 x 2 matrix with nonzero determinant acting in the complex two dimensional linear space spanned by the variable vectors w, w* and the fixed vectors y, t y For /mW

o=

\

mVV

acting between the row and column vectors of chiral superfields, as in (6.20), we have o-i =

mVV '4(3 2 + m 2 ) 2 1 + m 2VV2 VV2 16d (d' + m )

1 +

m2VV VV 16d2(d2 + m2) mW 2 4(5a22 +, m4N )

That O - i O = O O - i = / can be verified by means of (4.11 l,m). Now the use of (6.21) in (6.20) enables us to rewrite the latter as

Zo[J,J^}

= Z0[0,0]exP\-l~

J d8z (^J(z)

g ^ t o )

S-1

/J

6. Superspace Perturbation Theory and Supergraphs

104

After straightforward algebra this equation can be simplified to

+

J_ CE2. rt\ 16 V 9

2

vv 2

J_ ("BE T\ ™> / t l \ 16 V d2

2

J 9 +m

) d2 + m2

\y

The number of covariant derivatives in each of the terms of the RHS exponent above can be reduced. Observe that J, (d2 + m2)~lJ and VV(d2 + m 2 ) - 1 Jt are all left chiral superfields while each of j t , (d2 + m 2 )"* Jt and VV{d2 + m^J is right chiral. The shifting of the nonlocal operators VV/d2, W/d2 from the first superfield factor to the second, as per (6.11), in each RHS term above and the use of (4.14) lead to the following expression for Z0: (

/mDV

Z0[J, Jt] = Z0[0,0] exp | - \ j d*z (J(z) J t ( z ) ) _ 2 L 2_ d +m

I I

_2

^

2

4d _1

mVV 4d2 J

J(z) J*(z)

We can then write Z0[J, Jt] =

Zo[0,0] exp | - i J dszdsz' (J(z) S(z))

=

Zo[0,0]exp|-i(Jjt).AF.^tH,

(

mVV

^

AF(z - z') (

fify (6.22a)

\

n^W"')4a2

<6"22b>

/

One can try to draw a direct analogy between (6.22a) and the corresponding expression [6.6] for the generating functional of free connected Green's functions in a scalar quantum field theory (QFT). That would suggest that AF should be the chiral superfield propagator matrix, i.e. iAp should be the matrix for the connected two point Green's functions in the ($(z) &(z)) linear space. Unfortunately, such is not the case. Unlike in nonsuper6 J (2) superfields J, Jt are chiral. Therefore, 5J(2)/5J(1) and symmetric scalar QFT, the source \t>lV15^{zl-z2) , (6.23a) 5J{1) ~ 4— z%), are given by <5Jt(2)/5Jt(l), instead of being just 5^(zi <5jt(2) <5Jt(l)

"

-i2?l2V(8)(*1_Zi)-

(6 23b)

'

Now, on using (6.22) in (6.16), we see that the only nonzero connected free particle Green's functions are

W1) *. (2))c =

-, * S U

5J(l)(5Jt(2)

-

« **»!>>(„ - „) ,

16 d^ + rn2

(6.24a)

6.3. Functional Formulation of Superfield Theory

WWOcB-t-.15J(1)6J{2) *™ ($t(l)$t(2))c

=

{zi Z2h

-AWT^6

_ *^ ^ W f c

105

~

_

Z2)

.

(6 24b)

-

(6.24c)

Note that, in deriving (6.24b,c), use has been made of (4.11^,m) and (6.13c,d). These chiral superfield propagators, as described by (6.24), are originally due to Salam and Strathdee [6.7]. In comparison with (6.22b), the corresponding matrix form of the above chiral superpropagators in the ($(z) &(z')) linear space is Afs(z - z') where

_ _ Vv vv \ 4 (6.25) Ii_ i mVV_ _ » 8W(z-z') F 4(d* + m*) \ VV VV mVV / 4 and not the A F of (6.22b). By setting #1,2 and #1,2 in (6.24a) to zero, one can replace V1A, V* in its RHS by 8IA, &i respectively. Making use of (6.1c), one is then able to obtain the free connected scalar two point function: ASS

_

(0(1)
(6.26)

The corresponding Green's functions with higher components in the superfields $1,2 can be extracted by the application of appropriate covariant derivatives on (6.24a). For instance, suppose we apply VIAV2B> u s e (6-la) and then put #1]2 = 0 = #1,2, utilizing (4.23b) and (4.24b). We are then left with

(UDU2))c = -i^^1*??™^

- «)

9X2 + m 2

(6.27)

where we have used (4.lie) in the last step of (6.27). As an immediate demonstration of the power of the Green's functions (6.24), we can state the following result [6.8]: Every L-loop (L > 1) in a supergraph, containing either only left chiral or only right chiral vertices, must vanish5. It is sufficient to prove the above in the left chiral case. Note that such a loop will involve only left chiral internal lines and will therefore be proportional to a product of superpropagators of the type of (6.24b), namely: ($(l)$(2)) c ($(2)$(3)) c • • • <$(*)*(l))c •

(6-28)

We can now employ the superspace variables y, 9, y, 9 of §4.3 and the fact that $ = $(y, 9), $t = ¥{y,d). Thus, on using (4.17b), (6.24b) becomes

6

In particular, the one loop tadpole for $ or $t is zero.

6. Superspace Perturbation Theory and Supergraphs

106

where (4.4j) and (4.3) have been used in putting did\5^{6\ — 02) = 4. On substituting the last displayed equation into the expression (6.28), one obtains a string of ^-function products in 9, namely 6^(0! - 02)<5(2)(02 - 03) • • • <5(2) (
5^(0i - 02)5{2)(9i - 03) • • • 5^(61 - ek)5®(ek - 0X)

=

0,

since, for any function / , 5^{01-02)}{61) = 6^{0i-02)f{92) and moreover 6^{0)8^{-0) = 0 as a consequence of (4.4j). It follows that, for a system of only chiral superfields, nontrivial supergraphs must involve both left and right chiral vertices. We can summarize the important physics points brought out in this section as follows. • The effective action r ( $ , $ t , ^ r ) of (6.17) in superspace generates one particle irreducible supergraphs. • The connected two point functions of chiral superfields are given in the linear space by iAFs(z — z') of (6.25).

($(z)&(z1))

• Every L-loop (L > 1) in a supergraph, containing either only left chiral or only right chiral vertices, vanishes.

6.4

GRS Feynman Rules for the Wess-Zumino Model

Let us consider the Wess-Zumino model which was introduced in §5.1. We wish to develop for it a formalism of supergraphs with superfield lines and vertices in superspace in analogy with Feynman graphs. We first return to (6.18) and note that factors of

^^'^J = '/" 4 a : £ i n t (^'4) generate vertices at superspace points zk. Specifically, Cmt, contains functional derivatives which act on more such derivatives to the right and eventually on Z0 itself. Each such derivative connects an existing propagator to zk in a way that every new vertex is properly linked with the requisite propagators. These functional derivatives constitute the key operations. So let us refocus our attention on the properties of chiral superfield functional derivatives (6.12), as compared with those of something like a vector superfield V, namely (6.3a). The latter are evidently more convenient since they span the entire superspace, whereas the partial spanning of superspace by the former leads to complications. Specifically, the matrix form of the chiral superpropagator is Afs rather than the suggestive Ap of (6.22b). Difficulties increase when one turns to interaction vertices. An interaction, such as f dsz\&V$, yields the vertex contribution i\ J d8z • • • to a transition amplitude. In contrast, the Wess-Zumino

f

cubic self interaction (cf. §5.1) — J d6z$3 contributes if J d6z- • •, whereas its hermitian

6.4. GRS Feynman Rules for the Wess-Zumino Model

107

conjugate contributes if* J d6z • ••. As we saw at the end of the last section, both left and right chiral vertices will be present in a nontrivial supergraph with chiral lines. The former WD VD vertex can be recast as if J dsz—^ • • • and the latter as if* f d8z—-^ • • • by using (4.14e,f). But the nonlocal operators and the asymmetric way of treating different vertices differently make life hard. All of this can be avoided: in effect, (1) AF can be made the superpropagator matrix and (2) every vertex contribution can be given as i times the coupling strength times the full superspace integral / dsz ••-. What we need to do is adopt a modified set of conventions. These are due to Grisaru, Rocek and Siegel (GRS) [6.9] and lead to an improved set of super-Feynman rules. The "problem" with the original convention of Salam and Strathdee can be traced back to the occurrence of the operator factor — -WD and — -WD in the RHS of (6.23a) and (6.23b) respectively. Suppose we agree to drop each such factor from those functional derivatives, modify the corresponding superpropagator accordingly, but agree to insert back a — -DD (—-WD) factor on each internal left chiral (right chiral) superfield line attached to a vertex. Then we ought to get the same amplitude as before. Thus, in this new improved convention, the chiral superpropagators - now marked with primes - can be rewritten "as though derived functionally from (6.22) with SJ(2)/SJ(1) = 6Ji(2)/6J^(1) = S^(Zl - z2)," so long as additional chiral derivative factors get inserted on internal lines at the time of writing the interaction vertex. The new GRS-improved chiral superpropagators are (*(l)&W)'c = -QY~^6{8)^

~ *) »

($(1)$(2)) c

' = l y f f m / ' ^ - Z2) '

( 6 " 29a )

(6 29b)

-

corresponding to the matrix operator form iAF of (6.22b). This AF is sometimes written [6.5] as AFRS to distinguish it from AFS. An extra modification is, however, called for in the insertion of such — -WD (or —-WD) factors on internal left (right) chiral superfield lines when three of them meet at a vertex describing a cubic term of the superpotential, say in an interacting Wess-Zumino model (cf. §5.1) with h = 0, / ^ 0: 6 3 Siint [$] = ^ J d z$ + h.c.

(6.30)

In this situation (a) none, (b) one, (c) two or (d) all the three of the superfield lines could be internal. The corresponding number of insertions would be (a) zero, (b) zero, (c) one or (d) two (i.e. one less than the number of internal lines) respectively. We demonstrate this below explicitly for case (d), i.e. when three new chiral superpropagators are created at the superspace point z. At the lowest order, this part of the supergraph is generated from

•a,

108

6. Superspace Perturbation Theory and Supergraphs

Let us use the "old" functional derivative formula (6.23a) first, generate the corresponding — -VV

factors and then associate them with appropriate internal superfield lines. We can

keep using (6.13c,d) and 6J(zk)/6J(z) becomes

=

-(1/4)PP<5(8)(ZA;

- z). The above expression then

fc=l

-iS^(z-zk)J^zk))z0[0,0] We need to convert the Jd6z- • • integration into a Jd8z• • • one and can do so by means of (4.12c), provided any one of the three I — -VV

I factors is absorbed. It is simplest when we

select the first one, for instance, in which case we have

%Jd&z **wh* (*T^ ( 8 ) ( Z -Zl)j{zi) -i6i&){z ~Zi)Ji{Zk)) • -i5W(z-zk)f(zkj)z0[0,0]

It is now evident that we need not include a factor — -VV for one of the three propagators at the vertex so long as we do so for each of the other two and include a contribution if J d8z • • • from the vertex. Thus the procedure would be to insert a factor of — T - ^^ each for two of the three left chiral internal lines in a cubic Wess-Zumino vertex: the rule for case (d). If any of the lines is an external leg, no such factor is to be inserted and in this way we also understand the rules for cases (a), (b) and (c) enumerated earlier. For the hermitian conjugate vertex, the same rules are valid, except that the insertion factor is —-VV.

In

case of the A$^V$ vertex, with both chiral superfields corresponding to internal lines, the factor to be inserted would be iX J dsz—VV

VV- • •; once again, if any of the chiral lines

is external, the corresponding insertion factor, i.e. —-VV or —-VV, is to be dropped. Of course, we need to mark each particular internal line on which the factor is inserted. GRS super-Feynman rules in momentum space The standard practice is to use the modified super-Feynman rules in momentum space and it would be useful to first discuss some preliminaries in that space. We define T{p, 6,9)

6.4. GRS Feynman Rules for the Wess-Zumino Model

109

as the Fourier transform of the superfield6 F{x, 8,6): d4p

?{x,0,G) = (2TT)< j e-%vxF(p,6,9) We further adopt the convention that J-{j>) corresponds to a superfield line carrying a four momentum p into a vertex so that 9M on ^(p) becomes —ip^ip) • Turning to the action of covariant derivatives in momentum space, we can define VpAf(p) = J
,

(6.31a)

F(p)

(6.31b)

T{p) ,

(6.31c)

= (3A - o^jjPpjFip) = [-dA

+ ega^Ap^j

= (-dA+9Ba»BAPli)

Vp'AT{p) = I d^xJpxVAT{x)

= (BA - a»ABeBp^

J-(p) .

(6.31d)

Furthermore, VA,t>\

T{p) = 2a.

.p,T(p)

which agrees with (4.11b) and our convention of J-{p) as an annihilating superfield (denoted in Fig. 6.1 by an inward ordinary arrow) with d^ = — ip^. According to (6.31), in momentum space, for a superfield line flowing with four momentum ft1 (denoted by a bold arrow) into a superspace point with coordinates (x\.,d\,8\) and labelled 1 (Fig. 6.1), VPA can be replaced w-p

%

-w

v'l '2A = d2A - o*J${-Pli) V\A =

dXA-o-»JZp»

Fig.6.1. Flow into point 1 and out of point 2 in superspace. by d\A — a^OiPp. On the other hand, for a superfield created at superspace point 2 with coordinates (^2,^2,^2) (denoted by an outward ordinary arrow) but with p" flowing out, consistency demands that the covariant derivative be taken as d2A — cr^6B(—pu) and labeled T>2A. This is why we have shown the bold arrow there as inward with label —p. When there is a propagator between 1 and 2, the multiplying factor <512 ensures that §1 — 92, whereas 6 N.B. Since T can be either a left chiral superfield $ or its right chiral conjugate $t, *t(—p) is the conjugate of $(p) in momentum space.

6. Superspace Perturbation Theory and Supergraphs

110

(6.1a) implies that diA^u = —d-iA&vi- Thus the validity of (6.13a) is retained in momentum space via V\A6l2

=

-V£612

V[A612

= ~V;l5n.

,

(6.32a) (6.32b)

There is also the super-Leibniz rule VVA(TQ) = {VA?)Q±F{p£qG)

,

V\{TQ) = (V\T)Q ± T{t>YQ)

,

where the +, - signs correspond to T', Q being superfields/superfunctions that are even, odd under the Z2 grading of §3.1. Moreover, f d*8{VAF)g = T f dA6T{V-Aqg) ,

I tfioipffig = =F I d^Tiv^g). These results generalize further to products of three or more superfields/superfunctions. For instance, if % is a third member of the latter category, we can write

I
(6.33a)

f d49{vg^)gn

(6.33b)

+ f d^Tpfflu

+ f d46Tg{t>q*H) = o,

provided qi + q2 + q$ = 0 and T,g,H are all even under the Z2 grading. Next, we turn to superfield propagators in momentum space. We need to take Fourier transforms with respect to the spatial coordinates of both points 1 and 2 multiplying by one set of delta functions of 5^(xi — :r2). In the modified convention [6.9], the momentum space superpropagators are [ d*x1dkX2eW*1-**Wi\x1 [ fx^x^r^-'Wix! J [ dix1d4x2ei^-^S^(x1

- z 2 X$(l)$t(2))' 0 = - f i ^

,

m - x 2 )($(l)$(2))' c = ~ 2 2AV{V1512) 4 j r y r — mr)

- x2){&(l)&(2))'c

= 1

TO

(6.34a) ,

(Pffig 1 2 ) .

(6.34b) (6.34c)

We can now list below the super-Feynman rules for computing the contribution to the effective action from a particular supergraph in the Wess-Zumino model. Recall that for any line, internal or external, an ordinary arrow keeps track of annihilation or creation (i.e. into a vertex for $ and out of a vertex for $+) while, wherever needed, a bold arrow is put to denote momentum flow. The direction of the internal four momentum Pint is to be fixed as

6.4. GRS Feynman Rules for the Wess-Zumino Model

111

per the convention of Fig. 6.1. One further comment can be made regarding the momentum labels of covarant derivatives appearing in supergraphs. Apart from the sign (which is controlled by the direction of momentum flow), the momentum in the covariant derivative can always be taken as the momentum of the superfunction/superpropagator on which it acts. This property can be maintained even after partial integration through a judicious use of the super-Leibniz rule. Finally, the different factors associated with various parts of a supergraph, with the action of Vp or Vp as given in (6.31), can be enumerated as follows:

D

-, for the superpropagator

L-™2'

- - - -

-x

%Tfl

-(VPmtT>PiM6i2), for the superpropagator iffi

„

— —

%4Mt -

PiM Pint ™2) (T> V 6i2), for the superpropagator

p_nt

2

m

x

m c

^

4 • / d ^vert with each vertex;

an overall vertex factor of if times a factor of —-f>Pi»tfp>int

( o r — pp.ntpPmA

eacn7

for n — 1 of the n left (or right) chiral internal lines, each marked by a double bar perpendicular to the line, which meet at a $ 3 vertex (denoted by a closed circle) or a $t 3 vertex (denoted by an open circle for which the vertex factor is if*), but no such factor should be put for any external line; J (27T)4

with each independent loop;

/ IT m T4 |(^ 7 r ) 4 ^^ 4 H/lPext)l for the external momenta; Pext

•

$(PextAert)

ext Or

$+(

~Pextinvert) f° r a n external left chiral superfield line respectively entering or leaving a vertex with four momentum pext and the reverse for a right chiral superfield; • usual symmetry and combinatorial factors; • since the above rules yield a supergraph corresponding to the quantum version of iSmt[$], an overall multiplicative factor of —i is needed to obtain the effective action. An immediate convenient consequence of these ("improved") GRS super-Feynman rules is in the following fact. In a graph with v vertices one can readily integrate out 6,0 for v — 1 of them. We shall have a greater appreciation of the power of the above rules when we do explicit loop supergraph calculations later. 7

Under the Grassmann integrations, generated by vertices, the order of these T>V (or T>V terms) turns out not to matter. However, pmt. is always defined as the momentum flowing into the vertex.

112

6. Superspace Perturbation Theory and Supergraphs

6.5

Feynman Rules for Nonabelian Supergauge Theories

Let us first start with the pure super Yang-Mills theory. We shall take its action from (5.54) but will rescale the gauge coupling strength g —> y/2g and the vector superfields V" —>• —y=Va, so that V (cf. 5.40), W etc, are unchanged. The original normalization was V2 useful in making contact with the component content of the theory, but the present one is more convenient for perturbative superfield computations [6.8, 6.9]. Thus we write the super-YM action as SSYM = ^ I ^ T r ( / d6zWW + j d6zWw\

.

(6.35)

Each chiral superspace integral in the RHS of (6.35) is real8 and hence equal to the other on account of the constraints on WA, WA. The definitions (5.45a,b), as well as the chiral properties of WA, WA, can now be utilized to convert the partial superspace integrals f d6z, Jd6z in (6.35) into the full fd8z by means of (4.12c,d). This leads to SSYM = ^ ^ J d8zTr (WAe-vVAev).

(6.36)

On using (5.45) and (5.48), we can expand the RHS of (6.36) in powers of V: SSYM

=

^

^

f ^ T r j VVAVVVAV

- (f>f>VAV) [VAV, V}-^

- i (Vt>VAV) [{VAV,V},V]

[DAV, V] VV [VAV, V]

+ 0(V5) I.

(6.37)

In writing (6.37), we have used the formulae (4.12a,b). Since V = 2gVaTa (cf. 5.40), each power of V brings in one power of the gauge coupling strength g. The super Yang-Mills action has now acquired the form SSYM = SfYM + 0(g), SSYM

" *^e P a r t ^ S

s™

=

a t

(6.38)

^s quadratic in V - being given by

647fc / ^ ^ (VVAf)f>VAV)

= ^TkJdh

Tr(VU0d2V)

,

(6.39)

where Ho is as in (6.9). In terms of vector superfields Va, S{SYM = \J' 8

d*zVaYl0d2Va .

We are here ignoring FF terms (cf. §4.4) which, in fact, cancel out in (6.35).

(6.40)

6.5. Feynman Rules for Nonabelian Supergauge Theories

113

(6.40) makes it explicitly clear that the SgYM is actually of zeroth order in g, so that the remainder in the RHS of (6.38) can be identified as being O(g). Gauge fixing and superpropagator In defining propagators of gauge fields, one requires [6.6] the addition of a gauge fixing term to the Yang-Mills action. A common choice is -{Sag2},)-1

fdixnt(&,Altff'Av)

,

(6.41)

a being the gauge parameter and A^ being 2gA^Ta. The supersymmetric generalization [6.8] of this term, covering gaugino fields as well, is (cf. 6.55 below for justification) SoF

= - 6 4 ^ / ^ I™™™) = -Y^k Id&zTr (V mVV]+v)

. (6.42)

The second step in (6.42) follows from the trace property and partial integration. We shall justify this form of SQF when we discuss the Faddeev-Popov superdeterminant later in this section. The sum of (6.39) and (6.42) can be recast, by means of (4.11f,g,k), as S$u + SGF = ^j
+ ±(\-^V[VD,VT>]+v}.

(6.43)

The complications of fourth order derivatives in the RHS of (6.43) are avoided in the supersymmetrized Feynman gauge {sFg) with a = 1. Henceforth this is our chosen gauge for the rest of this discussion. Thus, with repeated superscripts understood as summed, we have

S$M

+ SSGFF9 = g^jfc / d8^(V92V)

= \ J fzV'&V*.

(6.44)

In order to derive the super Yang-Mills propagator, it is sufficient to retain only that part of the action which is quadratic in V, together with the gauge fixing term. The free field generating superfunctional, which pertains to a vanishing value of g, can be written as Zo[j] = Zo[0] j[dV) e x p i ( S $ M + SGF + ja-Va)

.

(6.45)

On substituting (6.44) into (6.45) and doing the functional integration in V°, one derives the final result for Wo[j] = —i In Z0[j]:

Wo[jy9 = -lJdhf±f.

(6.46)

The two point connected Green's function obtains easily from (6.46):

(V(l)V(2)>^ = -%?$%)

= i ^ ( 8 ) ^ - *> •

(6-47)

6. Superspace Perturbation Theory and Supergraphs

114

In momentum space (6.47) becomes G(p)sF9 = -\5a%2

.

(6.48)

The overall sign in the RHS of (6.48) is opposite to that of the chiral superpropagator, as expected. We can diagrammatically represent it as a

J

b

i

.

Unitarity and ghosts The gauge fixing conditions need to be introduced into the functional integral defining Z[j] via ^-functional constraints. In this respect, however, there is a nontrivial generalization from ordinary Yang-Mills theory to the super-YM one. We need to first quickly review the former case [6.6]. A gauge transformation there involves one gauge parametric function of spacetime, namely ui(x), constructed out of real gauge functions uja{x): LO{X) = 2gu>a(x)T", defined in analogy with A^ = 2gA^(x)Ta. An infinitesimal gauge transformation is now given by

SA^A^A^d^

+

^A^w}.

There is, in consequence, one gauge fixing function F(A^) whose vanishing is ensured by the insertion of the ^-functional <5[F(A^)] under the functional integral [oL4p]. Returning to super-YM field theory, however, we see that here the gauge transformation (5.41) of the vector superfield involves two chiral superfunctions: the left chiral A(z) and the right chiral A* (2). Hence two gauge fixing superfunctions, one left chiral and the other right chiral, need to be introduced. These are chosen to be [6.8] )C(V) = ~VVV(z) K)(V) = -l-VVV

+ f(z), + f\z)

(6.49a) ,

(6.49b)

where f{z) and /*(z) are external left chiral and right chiral source superfields respectively, with f(z) = 2gfa(z)Ta . (6.50) The two (5-functionals, that are needed to ensure the vanishing of the lefthand sides of (6.49), are thus <5(/C[V]) and ^(/C^V]). Note furthermore that /C has unit mass dimension though that of V is zero. One can now introduce a Faddeev-Popov superdeterminant A.Fp in analogy with that in ordinary Yang-Mills field theory [6.9]. The free field generating superfunctional can actually be written as a functional integral over the vector superfield V as follows:

Z0[j] = Z[0] J[dV] exp [i ( S $ M + f • Va)] AFP8(K[V])5(tf[V\)

.

(6.51)

6.5. Feynman Rules for Nonabelian Supergauge Theories

115

In (6.51), S^M is that of (6.39) and (6.40). Moreover, we have introduced the measure [dV] oc n o [dV°], while the (5-functionals are given by 6[f - \VVV] oc UaS[fa - \VVVa] etc. Here each product Il 0 extends over the dimension of the gauge group and proportionality constants are immaterial. The Faddeev-Popov superdeterminant is defined by = f[dA+][dA]6[K(Vx)]6[tf(Vx)]

AFP

,

(6.52)

with Vx as the vector superfield that is gauge transformed (cf. 5.43) away from V via the chiral superfunctions A(z) and A*(2). Since, in (6.51), App multiplies the two <5-functionals, we need only its specific value A'FP when K\V] and K?[V] vanish. Suppose V represents a solution to the equations -VVV and -VVV

= /*.

=f

For the purpose of (6.51), the only contributions to A'FP from the

functional integrations over A, A* in (6.52) evidently arise when A, At are quite close to zero. This means that we can use (5.42b) and keep only linear terms in A, A*. In other words, we can take

SVX = VX-V~H{V)A

The linear operations H{V),H*{V) H(V)A fr(V)tf

(6.53)

on A, At in (6.52) are defined (cf. 5.41b) by

= iA+l[V,A] =

+ £P{V)A* .

+ ±\y,[V,A]]

+ --- ,

-tA* + l-[V, At] - ±[V, [V, At]] + • • • .

(6.54a) (6.54b)

Thus, we can rewrite (6.52) as A'pP

=

f[dA][dAi]6\-^Vv{H(V)A ~W{H(V)A

+

+

H*(V)Al}

H*(V)A*}

Finally, since the RHS of this equation is independent of the external chiral superfields f(z), /t(x), one can - as in ordinary YM theory [6.6] - average over the latter by integrating the RHS of (6.51) functionally over J[df][dp] with exp[-i(4ag2k)-lrTr{p(z)f{z)}] as the weight factor9. Such a procedure gets rid of the <5-functionals from the RHS of (6.51) and results in the introduction of the additional factor exp ISGF where ScF =

- i — f d8z Tr(VVVVVV) 64a<72K J

.

(6.55)

This justifies the first step of (6.42) since we have shown (6.55) to be the supersymmetric generalization of (6.41). Going to the super-Feynman gauge with a = 1, we now have the 9

Note that /[d/tjfd/] exp [i(4ag2k) 1 Tr {P(z)f(z)}] contribute to any of the n-point functions.

is just a field independent constant which does not

6. Superspace Perturbation Theory and Supergraphs

116

full Yang-Mills generating functional as Z\j] = N f[dV]exp[i(

SsYM + SsGFFg +

f-Va) A',FP

)

(6.56)

N being the normalization. For the restoration of unitarity, ghost superfields need to be introduced as supersymmetric generalizations of Faddeev-Popov ghost fields in quantized Yang-Mills theory [6.6]. Let us briefly review the latter. One needs there to introduce one complex pair of anticommuting spin zero Grassmann ghost fields ca(x), c^a(x), for each gauge field component AaJx), to tackle a single gauge fixing ^-functional <5[F(A^)] involving the gauge parametric function LU(X). Indeed, one defines c(x) = 2gca(x)Ta and replaces the Faddeev-Popov determinant by (cf. 2.9) del '5F dui

f[dc^}[dc\ exp

<&{*{%'«

Since there are two <5-functionals in the supersymmetric generalization, we need to introduce two pairs of hermitian conjugate chiral superfields, each of unit mass dimension, which are odd under the Z2-grading (cf. §3.1). Let us denote these by C(z), C^(z) and C'(z), C«{z) with C{z) = 2gCa{z)Ta etc. and VAC = VkC = VA& = VAC* = 0. Now A' F P in (6.56) can be written as A'FP

=

f{dC^}[dC'}[dC^][dC} o '

S

OU

(6.57)

with the ghost contribution to the action SQH being given by10 [6.7]

-

>GH

—

' [/^M^").SW^{(>").SS/

Ag2k 1 4g2k

yA(-i«)Tv{ C w\» & }

+./^(-I O T )TV{(^)_ I & | ^}\ On substituting SVX from (6.53) into the RHS above, we obtain SGH = ^JdszTr 10

[C{H(V)(-iC)

+ H\V)(i&)}

+ h.c' .

The evaluation of C,C* or of C',C'^ at the values iA, -iA+ is due to the fact that, to linear order, 5VX ~ i(A — At), cf. (6.53,4). N.B. Twice the imaginary part of the scalar component of A equals —ui.

6.5. Feynman Rules for Nonabelian Supergauge Theories

117

The substitution of H(V), H^(V) from (6.53) enables us to write the ghost contribution to the action (upto quadratic terms in V) as

Sgjf = ^L- f d8zTr{&'C + &C + \(C + &')[V, C - Cf] + ^(C' + C^')[V,[V,C + C^]]}

(6.58)

N.B. The purely left chiral terms C'C and the purely right chiral terms C^C'^ have vanished under J d8z ••-. By utilizing (6.57) and (6.58), the generating superfunctional of (6.56) can now be rewritten, with j = 2gjaTa, as r

Z[j] = N STOT

+

t

i[STOT+^7Tr(j-V)}

Wk

[dV][dC*][dC^[dC*)[dC]e

,

— SSYM + S^fp + SQH •

(6.59a) (6.59b)

Let So be the interaction free quadratic part of the total action STOT and let us define Sjnt. = STOT — So as the interaction part. As in (6.18), we can then write ~r-i f n I"1 S 1 S 1 S 1 1 1\ „ ,. , t, t l Z[j) = exp ^ 5 i n , ^ - , 7 - , - - , —v —^ j Z0[j, „ , „, „t, vX^=^=o

.60) , (6.

where Zo[j, rf, r,, V'\ rf] = j[dV][dC*}[dC'}[d&}[dC} exp i (s0[V, C, C, C'\ C+]

J. +7^rTT{j-V 2

+ W-C' + V-C + h.c.)} )

(6.61)

Ag k

with j-V = Jd8zj(z)V(z),

r)-C = Jd*zri(z)C{z),

j-C =

jd*zrf{z)C{z)

and similarly for the hermitian conjugates of the last two dot products. In (6.60) and (6.61), we have introduced chiral source superfields 77', r}'\ r\ and if, with r) = 2gi]aTa etc., which are odd under the Z2-grading (cf. §3.1). They obey variational relations analogous to (6.12), namely

%®> - %$-'(-&*>) as contrasted with that obeyed by the source superfields

^ 4 = ^(8)(^i-^).

W*->•

<•»»*>

ja(x):

(6.63)

6. Superspace Perturbation Theory and Supergraphs

118

Utilizing (6.37), we can now write11 (5 0 ) s F s

=

- i - /" d*zlk{\vd2V

+ C'^C + &C)

,

(6.64a)

(Sini.)sF9 = g^g I dszTrl ±(VVVAV)[V, VAV] + (C + C*)[V, C - C+] +\{C + &')[V, [V, C + C+]] - h v , VAV)VV[V, O

-±(f>VVAV)[V,

61

VAV\

[V, VAV}} + 0(g5) 1.

(6.64b)

The evaluation of the RHS of (6.61), by substituting (6.64a), involves the use of standard functional integration methods and yields the following generalization of (6.46):

W0\j, rf, v, if, J]'F' = J d*z {-\ja^f

+ VVa^Va + VU^V"^J •

(6.65)

Thus, alongside the Yang-Mills superpropagator, we have two sets of chiral ghost superpropagators which can be treated with the methodology of §6.4. The only point of difference is that the ghost superfields have negative metric, no mass terms and propagate not freely but only within loops and have just the two superpropagators (C(l)C" t (2))c and {C'{l)C^{2))c which can be written in the GRS-improved form. Thus, in momentum space, the ghost superpropagators appear as shown below.


- 4 --2sz ™ • 2

p

2

p2

A comparison between the different terms of (6.65) shows that the ghost superpropagator has a sign opposite to that of a massless vector superpropagator. However, since ghosts always propagate within loops and SQH is bilinear in ghost superfields, their superpropagators must occur in pairs so that this sign is inconsequential. Coming to interactions, the self interacting V3 and V1 vertices, as well as the V-ghost-ghost and l /2 -ghost-ghost vertices, are included in (5j n t.) sFs . The explicit forms can be read off from (6.64b). Matter fields The above considerations can be readily extended to a more general supersymmetric nonabelian gauge theory comprising both gauge and matter fields. The latter are contained in vector and chiral superfields respectively, as described by the Lagrangian density of (5.54). "Whereas in Ch.5 we had fixed the supergauge to the WZ gauge and kept the ordinary gauge degree of freedom, here we have fixed the latter to the supersymmetrized Feynman gauge and kept the former free.

6.5. Feynman Rules for Nonabelian Supergauge Theories

119

Indeed, the total action STOT now consists of super-YM, gauge fixing, ghost and matter parts, i.e. STOT

—

SSYM

+ SSGJ +

SMAT

(6.66)

d6z W($i) + h.c.

(6.67)

SGH

+

In (6.66) SMAT = / dsz

/

*T(e

)y**

+

I

cf. (5.54), while the rest of the RHS remains as in the matterless case. We are once again using i,j as type indices which include /, J in the space of the gauge group representation 1Z to which $ belongs. The generating superfunctional now is simply that of (6.15) with T,3 specified to be Va,ja. Furthermore, S in (6.15) needs to be replaced by the total action STOT of (6.66). For the sake of being definite, let us suppose W($j) is given by (5.1b) with rriij — mSij. Then m can be identified as a common mass parameter of the chiral superfields. The total action STOT now splits easily between the free part So and the interaction part Sj nt . In the super-Feynman gauge S0=

f dsz

(h/ad2Va

+ CaCa + C^cA

+ $\$i

+ l- (m f cPz $& + h.c.") (6:68a)

sFg

Si„t. = (SiDt.)

s

v

+ J d z $+ [(e ) y - Siil *,• + i (f

6

d z fij&Q&k

+ h.c.) ,

(6.68b)

with {SfnlM)sF9 as given by (6.64b). Of course, the last RHS terms in (6.68a,b) must be gauge singlets. It is now straightforward to derive the superpropagators and the vertex rules for supergraphs in this theory. They are just a combination of the rules for chiral superfield theory and super-YM theory. Consider the lowest order gauge-matter-matter vertex, with one type of matter and the interaction (*,V)

2g fdsz\(Ta)ij$jVa

+

(6.69)

In (6.69) we have restricted the type index i to only the 7^-space basis index / and the ellipsis stands for terms of order g2 and higher order corrections, which will not be needed for the explicit calculations done later in this chapter. One puts in, for this vertex, a factor of 2ig J divert times an insertion of --V Pint V Pint or --T>p'ntVPint for each left (or right) chiral internal line, shown by an ordinary arrow flowing into (or out of) the gauge vertex, but not for such an external line. Similarly, to order g2, there is a &VV$ vertex and so on12. Ghost superpropagators, which only enter into loops, involving gauge-ghost-ghost or gauge-gaugeghost-ghost vertices, are to be treated as massless left or right chiral superpropagators in the improved GRS framework described in §6.4. One point of difference is that, since ghost chiral superfields are odd under the Z2-grading of §3.1, there needs to be an additional -1 factor for each ghost superloop. Finally, one should not forget to put in a V(pext,Overt) or y(—Pext, Overt)foran external vector superfield line entering or leaving a vertex. In the Wess-Zumino gauge one need consider only these two vertices.

120

6.6

6. Superspace Perturbation Theory and Supergraphs

Sample One Loop Supergraph Calculations

One loop chiral superpropagator We shall illustrate the utility of the Feynman rules, derived in the two previous sections, with several one loop calculations. First, let us calculate the one loop correction to the superpropagator (6.34a) in the Wess-Zumino model. Recall (cf. Ch.5) that the action of this model is given by 'wz

f d8z&$ +£•( f d6z$3 + h.c.

The concerned supergraph appears in Fig. 6.2, 1 and 2 being the $ and $* ends respectively. The closed circle represents the $ 3 vertex and the open circle stands for the $ t 3 vertex. The contribution to the effective action, after setting i 4 = 1 for the product of the i factors from the two vertices and the two propagators, reads

^wf^e>*H-pA

-^»i

(p — k)2 — m 2

k Fig.6.2. One loop correction to the superpropagator of (6.34a) in momentum space. the overall 1/2 being the combinatoric factor arising from (3 x 3 x 2)/(3!3!). The internal four momentum k flows into vertex 2 and out of vertex 1; so, as per our convention (cf. ftnt. 7), the momentum label on the covariant derivative at 2 is A: while the same on the anticovariant derivative at 1 is -k. On replacing X>ffc by f>\ (cf. 6.32b) and then 13 using (6.2b), the above expression simplifies and can be written in terms of the Passarino-Veltman (cf. §1.3) function B0 {p2,m2,m2) [6.10] as

- ^ J ^ £ 0 ( p 2 , m 2 , m 2 ) J dW(-p,0)(p,0) , cftk /

1

(6.70a)

1

(2^k2-m2(P-k)2-m2 = *>&>«) • (6-70b) 2 2 2 It is evident from (6.70b) that B0(p ,m ,m ) of (6.70a) has a logarithmic ultraviolet divergence which needs to be regulated by the dimensional reduction procedure, as explained in 'Recall that all the 7-matrix algebra, and thus the D-algebra, must be done in four dimensional space.

6.6. Sample One Loop Supergraph Calculations

121

§1.3 and §6.1. An evaluation, along the lines discussed there, yields [6.8, 6.10] the following expression B0(p2,m2,m2)

= -,A . h finite terms , (47r)^ 2 — ui as u) —> 2. (Recall that the number of spacetime dimensions is d = 2u). Therefore, the divergent contribution to the effective action from the one loop superpropagator, corresponding to (6.34a), is given by \f\2..2u)-i

5|(oneloop)=

( 4

r

^(^_

M

y^t

(6.71)

$ ;

where we have introduced the renormahzation scale /i in order to ensure that the RHS of (6.71) has a vanishing mass dimension for 2w ^ 4. We end this calculation with two remarks. (1) Were we to evaluate the one loop correction to (6.34b) or (6.34c), we would have needed the corresponding superpropagator in the internal lines of the loop. The loop then would be a purely chiral loop, thereby vanishing on account of the result proved at the end of §6.3. (2) Suppose the final $* line were missing from Fig. 6.2, i.e. one was computing the one loop $-tadpole supergraph. The result of the computation would be similar to the RHS of (6.70a), but with one factor of if and one propagator less and with the $*(—p, 6) factor missing. Hence the answer would be proportional to J d46$(p, 9) which vanishes explicitly demonstrating (cf. ftnt. 5) the absence of a <5>-tadpole. p-k

k Fig.6.3. Gauge superfield contribution at one loop to the superpropagator of (6.34a). As another exercise, let us calculate the one loop gauge superfield correction to the superpropagator of (6.34a) in the Wess-Zumino model incorporated within a super-YM theory. The action of this theory is given by (6.66) and (6.67) where the superpotential is as discussed in §5.1 and SSYM is as given in (6.35). The particular contribution to the effective action shown in Fig. 6.3 can be written as

{

->^-{->

4

k*-m*)TLJ*j{P'ei)

•

By manipulations, similar to those detailed for the one loop correction to the chiral superpropagator, the above expression can be simplified to

4g2C2(n) J j^B0(p2,0,

m2) f d*0*}(-p, 0)*,(p, 9) .

122

6. Superspace Perturbation Theory and Supergraphs

The relative minus sign here, in comparison with the contribution of Fig. 6.2, is due to the sign difference between the (VV) and ( $ $ ^ superpropagators. Moreover, C2{TV) is the quadratic Casimir constant of the representation 11 of the gauge group to which the <3>'s belong: (TaTa)u = C2(TZ)SU . (6.72) The expression for B0(p2,0, m 2 ), in the limit UJ —>• 0, is (cf. 6.70b): B0(p2,0, m 2 ) = i J

d*k

1 1 1_ x1 _ ^fc)2 k,_ 29 _' m_20 -= _- , (44 ? r_) 2„ 2n _ \ w. + finite terms

in the DR scheme. Comparing B0(p2,0,m2) with Bo(p2,m2,m2) of (6.70b), we see that any difference is only in the finite piece. Thus the one loop contribution to the effective action from the present divergence is given by

One loop vector superpropagator As the next example, consider the one loop correction to the vector superpropagator (VaVb), due to internal $-lines, as illustrated in the supergraph of Fig.6.4. The corresponding contribution to the effective action reads

-4ig2 T(n)\6">> J

^Y^yd%d%V«{-p,62A) (

\

-nP+k^p+k 1 2

4 (

(

T~.-p-ksp.-p-k I

\

l

J

4

_ ) ^ ( _ ) ^ _ ^ _ )

(p+k)2

- m2 I

V>(p,0M

,

(6.74)

where T(1Z) is the representation constant (cf. 5.39) of TZ and an extra factor of 1/2 has been inserted because the term in the effective action contains V2. We can first replace T>\ by -f>2k and Vip~h by -Vv^k in the expression (6.74), cf. (6.32). This makes all the covariant derivatives act at point 2 and in partial integrations Vb(p, di,0i) can always be factored out. k

p+k Fig.6.4. One loop internal chiral superfield contribution to the vector superpropagator.

6.6. Sample One Loop Supergraph Calculations

123

Now one can partially integrate the V%+kf>%+k factor off the left most 5 i 2 by utilizing the super-Leibniz rule of §6.4 while ensuring that the momentum label on any covariant derivative remains (upto a sign) that of the superfunction it is acting on. Thus (6.74) changes to

+2f>-Pva(-p,e2,e2) k k V~2 k
(vp+kvp+kj-^-

t>okVzk "2 2 fc2-m2

+v\-vM){v?kv?k{p V2kV2k

V2kV2k

+

k»_m2

p-*p2-*_*iL

V"^,eu0i).

(6-75)

In going from (6.74) to (6.75), identities like (6.33b) have been repeatedly used with 2k5i2/(k2 — m 2 ). Similarly, the second term can be simplified, by utilizing the commutator (cf. 4.11h) [VA'~k ,V2kV2k) = -4k"oABV2k and also (3.18a), to

sv^v-i-pM)

(^+k^+\p + ^_m2) k^Bd (*>»* vt^wr^)

•

We are now in a position to partially integrate the V2+ T>p+ factor in each of the terms of (6.75) by sequential use of the super-Leibniz rule (6.33). Furthermore, we can employ the trick of manipulating with partial integrations of covariant derivatives by the use of relations like14 0 = 5l2&l2 = 612VA6l2 = 5l2VB5l2 = S12VV512 = 5nt>VS12 = S12VA Vt>S12 = 5\2VB VV612 etc. and the most effective 5\2W W5i2 = 16<512 till the product of two 5i2's reduces to a single 8\2 in every term. Consequently, the #i-integration can be performed and we are left with

\ppVp VPVP + WD^o-^V*,

+ 16k2} Vb{p, 9, §) .

(6.76)

In going from (6.75) to (6.76), identities such as (6.33a) with qi = —p, q2 = p + k and
The basic point here is that anything less than two Vs and two V's between two <5i2's yields zero.

124

6. Superspace Perturbation Theory and Supergraphs

be rewritten by means of the relation VV VV = VAVDVA + MVa^Vd^, which follows by contracting V^ with both sides of (4.11h). Thus (6.76) now appears in the notation of (6.9) as -ilTCR)

8

{

}

f

^

** 4

^

4

1

2

2

2

VI

J (2TT) (2^) [(p + k) - m } k -w?

n 9 9)

l P

' ' ''

- 8 n 0 p 2 + 4(p + 2k)llV>d
(6.77)

Suppose now that we are interested only in the infinite part of the contribution of Fig. 6.4 to the effective action. Then we can set the mass m, associated with the chiral superfield, to zero in (6.77). We are now able to use the following formulae in the DR scheme [6.4,6.5]:

f d?"k f d2"k _ f d^k 2 2 J (p + k) ~J k ~J (p+k)*k2{P

+ 2k)l1

in isolating the infinite part of (6.77) as

92T(TZ) J JJilLtfeBoWV'i-p,

9)noP2Va(p, 9) ,

(6.78)

where B0(p2) is the m1>2 -*• 0 limit of B0{p2,m\,m\) introduced in (6.70b). There are two more contributions to the vector superpropagator at one loop. The first comes from internal V-lines effected by a V3 vertex at either end (Fig. 6.5). (The V-loop from a quartic V-vertex does not have a divergent part in dimensional reduction). The calculation for this involves lengthy algebra. There are six terms at each vertex from six

Fig.6.5. One loop contribution from V internal lines to the vector superpropagator. possible contractions in (64g2k)-lTr f d8zV'D'DAV[V,VAV], cf.(6.64b), producing thirty six terms in total. We know that, in the string of covariant derivatives, there must be at least four involving the product of VV and VV between the two <J12's to produce a nonzero result. Owing to the symmetry between the two lines and the two vertices, there are, in fact, only seven separate contributions with at least four covariant derivatives in the form mentioned above. The long algebra leads to the following result [6.9] for the infinite part:

g2C2(G) J -0^d*6Bo(P2)va(-p,

e, S) ( - | n 0 + \n+ + i n _ ) P2va(P, e, e),

(6.79)

where n ± are the same as in (6.7). Recall from the discussion in §5.4 that C2(G) is the quadratic Casimir of the gauge group G. Finally, there is the ghost loop contribution of Fig. 6.6 which is the only divergent ghost contribution at one loop. The infinite part of this can be evaluated, by methods similar to those outlined earlier, as

~C2(G)

J J?lLdl6B0tf)Va(-p,

8,9)p2Va(p, 9,9) .

(6.80)

6.7. The Nonrenormalization Theorem

125

Adding (6.78), (6.79) and (6.80) and utilizing the result 1 = n 0 +11+ + EL, the infinite part of the net one loop contribution to the vector superpropagator can be rewritten as -g2 [3C2(G) - T(K)] J ^ d ^ B ^ V ^ - p , 6,9)n0p2Va(p,

9,9) .

Once again, cf. (6.70b), Bo(p2) =

-

^

+

finite

part

'

2

Taking into account the change of sign in going from p in momentum space to d2 in configuration space, the divergent contribution to the effective action from the one loop vector superpropagator is S£(one loop) = - J | - ^

{3C2(G) - T(Tl)} - J d* zVaIl0d2Va.

(6.81)

We end this section with a remark. If we rescale the definitions of the gauge coupling and gauge superfield back (cf.§6.5), i.e. g —> —7=9 and Va —> \/2Va, so as to recover the standard V2 normalizations introduced in Ch.4, (6.81) remains unchanged. Indeed, we shall now revert back to those standard normalizations for g and Va and use them henceforth.

Fig.6.6. Ghost contribution to the one loop vector superpropagator.

6.7

The Nonrenormalization Theorem

The one loop contributions to the two point vertex functions, calculated in the previous section, have one thing in common. They finally involve only one integral f d49 • • • of an integrand that is a local function of the external superfields15 and their covariant derivatives in the 9,9 subspace of superspace. This integrand also has a factor J d*k • • • of a nonpolynomial function of the internal loop four momentum k. When higher loops are taken into account, the latter form may change to multiple integrals of this sort, involving several internal loop four momenta ki or equivalently several spacetime coordinates y, or several Feynman parameters a*. But they can all be expressed as a single integral J d^9 • • • of an integrand with the property mentioned above. This is the content of the nonrenormalization theorem [6.1], the technical version of which is enunicated below. Any perturbative quantum contribution to the effective action is expressible as a single integral f d49 • • • over the entire 9,9 subspace of superspace. These, in turn, are polynomials in 0,8.

6. Superspace Perturbation Theory and Supergraphs

126

In other words, the above contribution must have the form r = Y, I I / rfxidteGnixu • • •, * „ ) # ( $ , $ f , V, £>^$, VA&, VAV, •••). n

(6.82)

»=1 J

In (6.82) the Gn's are translationally invariant functions on Minkowski spacetime and the Fj are local functions of possible external superfields $, $*, V and their covariant derivatives. Proof. This results in a simple and direct way from the super-Feynman rules given in the earlier sections. Consider the 9, ^-structure of an L-loop supergraph. We recall that each propagator - linking the zth and jth vertices, say - yields <5y while each vertex (ith, say) yields / d49i • • • as well as X>'s and/or V's acting on the <$y factors of the propagators. Suppose we take up a fixed irreducible loop containing n vertices and n propagators. Let us focus on one particular propagator connecting the iith and i 2 th vertices. We can partially integrate any V's or V's, acting on the 5^ factor, onto other propagators or external fields/sources. Once these partial integrations are done, a residual naked 5ili2 remains which can be "killed" by doing the / di9i integration. After the latter, all i\ indices have to be replaced by i2 at all other points in the supergraph, i.e. the propagator gets contracted in the 9,#-subspace to one point at the i2 vertex. We can continue this procedure n — 1 times, i.e. for all but one of the propagators. The loop has now shrunk to contain only two vertices (1 and 2, say), with its 8, ^-structure given by the expression J J f di9Ed%d%SnVl E

•••Vl---V1---Vl---512.

•*

In this expression all vertices external to the loop are covered by the label E. Note that the appearance of fewer than two V's and two Z>'s - between two <$i2's in this, or any other earlier occurring expression, will make the whole supergraph vanish on account of relations described earlier. On the other hand, more than two factors of V's or V's can always be reduced to two such factors by the V-, P-algebra. Finally, for the remaining two V's and two V's, we can use the result S12VAVfVlcVlD612

= AeA^eCD5l2

(6.83)

to eliminate one of the <5i2's and perform the dl92 with the remaining one. The entire loop, in consequence, gets contracted to a single point in the 9, ^-subspace. We can now continue this procedure, loop by loop, till we are left with an expression involving loop as well as external four momenta, external superfields, covariant derivatives acting on external superfields and one final / dA9 • • • vertex integration, i.e. an expression like that in the RHS of (6.82). QED.

Corollary. All vacuum supergraphs vanish. Proof. If a supergraph has no external legs, at the end of the procedure outlined above, we are left with an expression C f d^9 where C represents one or more integrals with loop four momenta and there are no 9, ^-dependent terms inside the integral. This is evidently

6.7. The Nonrenormalization

Theorem

127

zero. In consequence, the generating superfunctional of (6.15) is naturally normalized by Z[0,0,0] = 1. A somewhat more pointed statement can be made as follows. The quantum contribution to the absolute minimum of the effective potential for classically supersymmetric configurations vanishes in perturbation theory. Proof. The said configuration must have zero vacuum energy at the classical level. This was already demonstrated earlier and will be discussed in more detail in Ch.7. Indeed, all the auxiliary F- and D-fields, whose absolute squares add to make the classical potential (cf. 5.56b), must individually vanish in this configuration. The perturbative quantum contribution to the effective potential, which is the momentum independent part of the effective action, can only have the following form

J ofzF (($), <©$), ($+), 0D$t>, (V), (W), • • •), where F is some function of the superfields and their covariant derivatives in that configuration. However, in classically supersymmetric configurations, the auxiliary fields vanish while spinorial fields can never have nonzero VEVs. This means that all the VEVs in the argument of F are independent of 6,0 and so is F. As a result, the integral vanishes, and the statement follows. Of course, nonvanishing quantum corrections to the effective potential do exist away from the minimum. This statement has an important implication. The perturbatively quantum corrected effective potential will be at the same absolute minimum of field configurations as it is for a classically supersymmetric theory. Specifically, any degeneracies at that minimum will not be removed by these corrections. This means that vacuum configurations, which were classically different, will remain different after quantum fluctuations are perturbatively switched on. Moreover, if supersymmetry is unbroken at the tree level, it will not be broken perturbatively by quantum corrections. The latter cannot shift the minimum of the potential away from zero where it already is located classically. Only nonperturbative quantum effects may be able to achieve such a shift. Remark on the nonrenormalization theorem Each term in the RHS of (6.82) involves the full integral / di0 • • • in 6, 0-subspace but not the subintegrals / d20 • • • or / d29 • • • separately. This means that counterterms of the form f d6zW($i) or f d6zW($\), where W($i) is a local analytic function of chiral superfields $i and W' its hermitian conjugate, cannot arise in perturbation theory. The holomorphy property of the superpotential protects it from any perturbative renormalization and is the basis of the following alternative and generally more practical statement of the nonrenormalization theorem that can be made for any supersymmetric field theory: No superpotential term gets renormalized in perturbation theory16. Nonperturbative renormalization of a superpotential term is, however, possible; cf. §7.6.

6. Superspace Perturbation Theory and Supergraphs

128

More specifically, counterterms - added to the Lagrangian density after renormalization cannot have the structure of superpotential terms. There will, of course, be wave function renormalizations of the chiral superfields. Thus renormalizations of couplings and masses in each term of the superpotential must balance the corresponding chiral wave function renormalizations. The latter, being dimensionless, can at most have a logarithmic dependence on the mass scale of the theory. Therefore, the same must be true of the former.

Fig.6.7. A two loop contribution to the ($$) superpropagator. In this context, a clarification is perhaps necessary. A purely left or right chiral contribution to the effective action involving / d6z • • • or J d6z • • • can arise with nonlocal combinations of chiral superfields, the nonlocality being in coordinate space17. For instance, the two loop supergraph of Fig. 6.7 contributes the following expression as a higher order correction to the superpropagator of (6.34b):

™/(^£(PV**(-P,*)*(P.*) = -mJ-0^^p-d^(-P,e)^(p,9) . Here B(p2) is a Lorentz invariant function of p that satisfies dB/dp2 = 0 for p2 —> 0 and (4.14e) has been used to derive the equality. Superpotential terms are, however, local analytic functions and remain unaffected. We focus on two important aspects and consequences of the nonrenormalization theorem. • No term in the superpotential can be perturbatively renormalized; wave function renormalizations of chiral superfields appearing in any term must be cancelled by the renormalization of the coefficient coupling/mass. • Given a classically supersymmetric configuration, there are no perturbative quantum corrections to the absolute minimum of the corresponding effective potential.

6.8

One Loop Infinities and

(3-,7-functions

We return to the classical Lagrangian density of a supersymmetric nonabelian gauge theory, as described by (5.54). Let us, for the moment, suppress the general type index i. We are then left with a generic chiral superfield $ along with the gauge superfield V. Let us also 17

This is simply because J d 6 z / ( $ ) = — \ f dszVD/d2f($) for any analytic function / ( $ ) and similarly / d?zf(&) = —j f d?zf)'D/d2f(&), cf. (4.14e,f). Nonlocality in coordinate space shows up as a nonpolynomial factor in momentum space.

6.8. One Loop Infinities and /3-,7-Functions

129

specify the superpotential W($) to be that of the Wess-Zumino model. The classical action, cf. (5.54), is

Tr ( J d6zWW + f d6zWW J + j f d6z (im$$

+ ^ $ $ $ J + h.c. j . (6.84)

After renormalization, we can put subscripts R on all superfields and parameters in (6.84), referring to renormahzed versions of the same. But then there will be additional counterterms for the gauge covariant kinetic energy terms in the RHS of (6.84), though not for the V-independent curly bracketted set of terms which come from the superpotential. We shall therefore have a counterterm 18 for each chiral superfield $ which is AZ J dsz$ReVn$R Tr ( / d*zWRWR + / d6zWRWR ) .

and one for each gauge superfield V that is A Z y

Defining Z = 1 + AZ and Zy = 1 4- AZy, the action can be cast in terms of renormahzed superfields $R, VR and the renormahzed parameters mR,fR,gR as S

= Z Jd8z$Rev*$R

+ ZVj^Tr(Jd6zWRWR

+ M cfz (±mR$R$R

+ ^<$>R$R$R)

+j +h.c.J.


In (6.85) WRA and WR are renormahzed spinorial supersymmetric field strengths with WRA

= ~VVe-v«VAev"

W$

= ~-VVeVRVAeVR 4

, .

A comparison between (6.84) and (6.85) leads to relations between renormahzed and unrenormalized superfields, namely

ZliHR,

(6.86a)

va = 4 /2 V!,

(6.86b)

$ =

as well as between renormahzed and unrenormalized parameters, i.e. V = VR=>gVa

=

gRVZ,

m$$

=

mR$R$R

A$$$

=

\R$R$R$R

(6.87a) ,

(6.87b) .

(6.87c)

18 The fact that renormalizations constants AZ and AZV are only logarithmically divergent can be shown by an analysis of the superficial degree of divergence of the concerned one loop supergraphs [6.8].

130

6. Superspace Perturbation Theory and Supergraphs

Relations (6.87) indeed follow directly from the nonrenormalization theorem and supersymmetric gauge invariance19. We can formally introduce renormalization constants Zm, Zf and Zg for the mass and the two coupling strengths / and g respectively:

m = ZmmR ,

(6.88a)

/

(6.88b)

= ZffR , •

(6.88c)

1,

(6.89a)

= 1.

(6.89b)

=

(6.89c)

9 =

Z99R

ZSZ3'2

=

It follows from (6.86-6.88) that

ZmZ y

7l/2

1.

Finally then, there are only two independent renormalization constants: Z and Zv. Let us now consider the formally divergent expressions for the infinite parts of these renormalization constants. That for AZ can be obtained easily. Expand the factor eVR in the covariant chiral superfield kinetic energy term in (6.84) as 1 + VR + -VR H and treat the contribution from the linear VR term as the lowest order perturbative gauge interaction on par with the trilinear self coupling term. Now (6.71) and (6.73) already tell us what the divergent one loop contributions from these two couplings are. To one loop, this is all there is for the divergent part. The one loop counterterm AZ^1' J d8z
1

AZW

2

= 'it

<

i/|2

- V<7a(fc)} fz^

(6-9°)

•

One can now display the one loop expression for the wavefunction renormalization constant Z1^2(fi) of (6.86a), including the dependence [6.6] on the renormalization mass scale (cf. §1.3) fi, as Zl/2(/x)W = 1

~ 3 2 ^
( 6 - 91 )

• 2

2

i.e. the one loop wavefunction renormalization constant vanishes if | / | = 4g C2(Ti). The anomalous dimension of the chiral superfield 7 = lim fidIn Z1^2(/j.)/d/i can be immediately w—»2 19

A caveat is called for here. Counterterms in quantized gauge theories generally become noninvariant under classical gauge transformations of the original action because of gauge fixing. However, they have to obey the Slavnov-Taylor identities [6.6] which ensure gauge invariance of the effective action. This is manifestly attainable in the background field method [6.5]. For supersymmetric nonabelian gauge theories, the viability of the background field method was demonstrated by Grisaru, Rocek and Siegel [6.9], i.e. they formulated it as a framework within which the effective action is invariant under the initial gauge transformations. It is in this framework, i.e. within the background field method, that relations like (6.87a) and (6.89c) can be shown to be valid.

6.8. One Loop Infinities and /3-,7-Functions

131

computed to be 7 (1) = 3 ^ 3 {l/l 2 - 4ff 2 C 2 (^)} = - 1 lim(4 - 2u,)AZ« .

(6.92)

The superscript (1) in (6.90), (6.91) and (6.92) refers to the fact that the RHS has been calculated only upto one loop. Turning to Zv, let us refer back to (6.40) and rescale Va —> y/2Va. The counterterm AZy Jd8zV^U0d2V^ must be chosen to cancel the divergence of (6.81). This allows us to deduce that „2..2w-4

A 4 1 ' = 1 + 8 / 2 ( 4 _ 2 a ; ) (3C 2 (G) - T(K)} ,

(6.93a)

4 / 2 M ( 1 ) = 1 + i § ^ [3C2(G) - T{K)} f - ^

.

(6.93b)

It follows from (6.89c) and (6.93b) that ZMm

= 1 - i f ^ s [ 3 ^ ( G ) - T(tt)] ^ - ^

.

(6.94)

If we make use of (6.88c) now, we can derive for this supersymmetric nonabelian gauge theory the /3-function20 which is the derivative of the gauge coupling strength with respect to the logarithm of the renormalization mass scale \i. Suppose the lowest order nontrivial (one loop) part of the /3-function is written as (3^(g). Then, from (6.88c), we can write ^)=^9^9^Z;^^.

(6.95)

Now (6.94) and (6.95) imply that P{1)(9) = - j f ^

[3C2(G) - nil)].

(6.96)

Let us consider the more general case with a type index i. Eq. (6.86b) remains unchanged, but (6.86a) generalizes to *< = {Zll2)w
Again, (6.88c) remains unaffected while (6.88a) and (6.88b) get changed to mij

= {Zm)ijtiljl(mR)iiji

,

(6.98a)

fijk

= (Zf)ijk,i'j'k'(fR)i'j'k'

(6.98b)

"See §6.9 for a more elaborate discussion of the /3-function.

6. Superspace Perturbation Theory and Supergraphs

132

respectively. Furthermore, (6.89c) remains the same but (6.89a) and (6.89b) do not. The extended versions of the latter respectively are {Zf)ijk,i'j'k'(Z1/2)i>lJ' {Z1/2)j>j" {Z1/2)k,k» =

6 ( 6u"6jj"6kk"

+ 6u"5jk"skj"

+ sij"6a"skk"

6

ij"5jk"5kj"

+

(6'99a)

+*ik" V V + <W' V V J > (ZM W (Z1/2)^ (Zl/2)jlf = i fe« %< + V V ) .

(6.99b)

Finally, the generalizations of (6.92) and (6.95) respectively are 7£>

= ^

WZ1,2)$

= 3^5 | / ^ / i « - V ] [ > ( 7 ^ j ,

(6.100a)

,3

/? (1) (
ff

167T 2

3C 2 (G)-2 T (^i)

(6.100b)

We can also write the /3-functions for quantities in the superpotential, namely the mass my and coupling strength fijk as follows: /?(m)y = n—{mR)%j(n)

= 'yu-mi'j + -yjrmir

P(f)ijk = P-Q-{fR)ijk{ll) = lii'fi'jk + Ijffij'k

,

(6.101a)

+ Ikk'fijk' •

(6.101b)

Before ending this discussion, we wish to comment on the gauge (in-)dependence that can arise when calculating /3-functions in a supersymmetric theory with gauge interactions. Consider the coupling coefficient of a superpotential term that is a product of three different left chiral superfields. Let the fermionic component of one of these (F x , say) be combined with the charge conjugate of that of another (F2, say) to form a Dirac fermion / . If <j> is the scalar component of the third, called $, the Yukawa interaction hfijucj) + n c - w m then be derived from the said superpotential term. The /?-function of the coupling strength h of this Yukawa interaction should be obtainable from the wavefunction renormalizations of the superfields Fi$ and $ on account of the nonrenormalization theorem. Despite contributions from gauge interactions in the loop, /3(h) is gauge invariant to one loop. This, however, does not mean the gauge invariance of the wavefunction renormalizations of the component fields of the left chiral superfields. Only in a manifestly supersymmetric gauge, such as the sFg (cf. §6.5), will the wavefunction renormalizations of the scalar and fermionic components be the same; in such a gauge the sum of all proper vertex corrections will vanish. However, in such a gauge there will generally be loops involving the auxiliary components of the vector

6.9. Renormalization Group Evolution

133

superfield, in addition to those involving gauge bosons and gauginos. The story is different with a choice such as the Wess-Zumino supergauge, followed by the i?£-gauge for the gauge boson propagator. Here the auxiliary components of the vector superfield have been gauged away, but such a gauge fixing procedure violates manifest supersymmetry. In this case the gauge dependence of the divergent part of the proper vertex part of the one (gauge boson) loop correction to the Yukawa interaction will need to cancel that of the fermionic wavefunction renormalization. In other words, it is no longer possible here to compute 0(h) from the wavefunction renormalizations of the component fields alone. Moreover, those wavefunction renormalizations will in general be different for the different component fields of a single superfield. However, the one loop fi(h), even if calculated with this gauge choice, will come out the same as calculated in the sFg from the supersymmetric wavefunction renormalization constants. To conclude this section, let us summarize its contents: • The calculation of the /^-function of the gauge coupling in a super-YM theory bears some resemblances to the corresponding calculation in ordinary YM theory; owing to Ward identities, the /3-function can be computed solely from vacuum polarization (super-) graphs, though the numerical coefficients will differ in the two cases. • For /3-functions of Yukawa and scalar four point couplings, there is a qualitative difference between theories with and without supersymmetry. While, for the former, all /3-functions of superpotential couplings orginate from the wave function renormalizations of the chiral superfields and are computable solely from two point functions, such is generally not the case for the latter. • All superpotential parameters renormalize multiplicatively, the multiplication being in a matrix sense (cf. 6.101), in the presence of nondiagonal wave function renormalization. Since scalar self couplings in a supersymmetric theory are related to either superpotential (Yukawa) or gauge couplings, this statement applies to them, too, unlike21 in nonsupersymmetric theories.

6.9

Renormalization Group Evolution

The /?- and 7-functions, derived above, control the renormalization group evolution (RGE) of any supersymmetric system. A treatment of RGE can be found in most modern books [6.6] on quantum field theory. We need therefore give only a brief introduction here into its basic methodology - always with a supersymmetric field theory at the back of our mind. Our approach would be somewhat intuitive rather than rigorous. The main assumption is that we are dealing with a problem containing only two basic scales that matter: an "external" energy or momentum scale Q and the renormalization scale \i. The specific implication is that all masses, pertinent to the problem, are much smaller than Q and fi. 21 For instance, the /^-function of the coefficient A of the scalar quartic coupling of the Standard Model receives [6.11] a contribution proportional to the fourth power of the top Yukawa coupling, independent of A.

134

6. Superspace Perturbation Theory and Supergraphs

Such a situation can always be ensured by integrating out fields with masses which are larger than those two scales. Of course, matrix elements or connected Green's functions, describing physical processes, cannot really depend on fi; the latter is a pure artifice which gets introduced in the process of removing loop divergences from quantum field theory. The parameters of the Lagrangian such as couplings and masses, that appear in expressions for concerned matrix elements or Green's functions, do nonetheless acquire ju-dependence through radiative corrections. In physical amplitudes this dependence gets cancelled by explicit /i-dependent terms emerging from loops. This cancellation is quantitatively described by the renormalization group equation. To one loop order, a connected Green's function F can be schematically written as

r = 1£[xi(rir Ci + 2_]dijXj(n)

I In

1- finite terms J

(6.102)

Here {Xi(fj,)} are a set of renormalized Lagrangian parameters while C; and d^ are coefficients describing tree level and one loop contributions respectively. On renormalization, the logarithmic divergences in the latter generate logarithmic Q-dependent terms. These must involve In Q/fJ, since, by assumption, Q and JJ, are the only dimensional quantities available. Two important consequences of (6.102) can be highlighted right away. First, it allows the deduction of the /i-dependence of Xi from the requirement that dT /dji vanish; in a renormalizable theory the answer is guaranteed to be the same for all processes, i.e. for all connected Green's functions V. Second, the solution to the resulting differential equation automatically resums all leading logarithmic corrections of the form {Xi In Q//j)m, Vm. Moreover, it is easy to see from (6.102) that (most) one loop corrections to T can be absorbed into the "running parameters" Xi{Q) by the choice22 /J, = Q. The resummation of leading loop corrections is now included, by simply plugging the running parameters Xi(Q), taken at the external momentum scale Q, into the tree level expression for T. Suppose we introduce the evolution variable t as t = \n— , (6.103) Mo Ho being a fixed but arbitrary reference mass. The latter does not contribute to dt = d\xj y,. Now the beta-function for the specific parameter Xi: defined by fi—Xt

= /3(Xi) ,

(6.104)

controls [6.6] the i-evolution of X,

^

= «W).

(6-105)

This dependence can be computed from any matrix element or connected Green's function to any loop order. In practice, the complexity of the calculation is reduced by finding the 22 Setting fi = Q exactly, one removes only the logarithmic corrections. The "finite terms" / can be absorbed by setting fj, = Qe?, but this evidently requires a calculation of these terms which vary from process to process.

6.9. Renormalization Group Evolution

135

simplest Green's function which contains the required information. The resulting expressions for the running parameters can then be used to describe the Q-dependence of far more complicated matrix elements. The RGE equations constitute a very powerful tool to study the dependence of physical quantities on the external scale Q. They have the general form of a loop expanded series: p{Xi)

= pW(Xi) + /3W(Xi) + --. (8TT2)2

j,k

j,k,t

(6.106) In (6.106) (87r2) n is the numerical denominator associated with the nth loop term which is proportional to the coefficient (b£ )jke~ carrying n + 1 indices. The first RHS term in (6.106) with the coefficient (bx )jk, which is a bilinear in X, is the lowest order nontrivial term coming from one loop. This is the term that we shall be concerned with since we work only upto one loop and neglect higher ones. If we compare (6.106) with the /?-functions, calculated to one loop order in §6.8, we notice something interesting. Since the first RHS term in that equation is quadratic in X, the square of the gauge coupling strength g2, rather than g itself, is the right choice for our X. Thus, upto one loop, we can use (6.100b), (6.105) and (6.106) to write the evolution equation for g2 in a supersymmetric nonabelian gauge theory as

s 2

* iH'v

-3C2{G) + Y,Ti(n)

(6.107)

Let us recall that C2(G) is the quadratic Casimir of the gauge group G, being N for SU(N), and Ti(TZ) is the representation constant of the representation TZ of the gauge group according to which the left chiral superfield of type i in the loop transforms. For comparison, the equation corresponding to (6.107) in the same nonabelian gauge theory, but without supersymmetry and sparticles, is [6.6]:

dt

~c2(G) + lJ2T>W + ^ £ W )

(6.108)

where i now refers to the ith chiral fermion and a to the ath complex scalar in the loop. It is easy to see how (6.108) is related to (6.107). Suppose we start with the former, impose supersymmetry and introduce the requisite sparticles into the system. First, each cartesian gauge boson component in the loop will need the addition of one gaugino. The latter counts as a single chiral fermion field in the adjoint representation with T{TV) = C^iG), since its two chirality states are related through the Majorana condition. One then obtains (-11/3 + 2/3)C 2 (G) = -3C 2 (G) as the coefficient of # \ yielding the first RHS term of (6.107). Next, each chiral fermion will now have an accompanying complex sfermion in the loop. For each i (and with a = i), this would yield £ ( 2 / 3 + l/3)Tj(ft) = £ 7 i ( f t ) as

136

6. Superspace Perturbation Theory and Supergraphs

the coefficient of g4, viz. the second RHS term of (6.107). Of course, no third RHS term is needed now. Finally, turning to the masses and Yukawa couplings of a supersymmetric theory, (6.101a) and (6.101b) translate to the evolution equations dt dt

3

Jijk

—

juirriiij + jjj'rriiji ,

(6.109a)

lii'fi'jk + Jjj'fifk + lkk'fijk1 ,

(6.109b)

the 7's being given to the one loop order by (6.100a). Eqs. (6.107) and (6.109) will be of considerable utility to us in Chs. 11-13.

References [6.1] J. Wess and B. Zumino, Phys. Lett. 49B (1974) 52. J. Iliopoulos and B. Zumino, Nucl. Phys. B76 (1974) 310. S. Ferrara, J. Iliopoulos and B. Zumino, Nucl. Phys. B77 (1974) 41. B. Zumino, Nucl. Phys. B89 (1975) 535. M. Grisaru, M. Rocek and W. Siegel, Nucl. Phys. B159 (1979) 429. [6.2] P.S. Howe and P. West, Phys. Lett. 227B (1989) 379. I. Jack and D.R.T. Jones, ibid 258B (1991) 382. [6.3] G. 't Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189. C. Bollini and J. Giambiagi, Nuov. Cim. 12B (1972) 20. [6.4] W. Siegel, Phys. Lett. 84B (1979) 193; 94B (1980) 37. D.M. Capper, D.R.T. Jones and P. van Nieuwenhuizen, Nucl. Phys. 167 (1980) 479. L.V. Avdeev, G.V. Chochia and A.A. Vladimirov, Phys. Lett. 105B (1981) 272. L.V. Avdeev and A.A. Vladimirov, Nucl. Phys. B219 (1983) 262. I. Jack, D.R.T. Jones, S.P. Martin, M.T. Vaughn and Y. Yamada, Phys. Rev. D50, 5481 (1994). I. Jack and D.R.T. Jones in (G.L. Kane, ed.) Perspectives on Supersymmetry, op. cit., Bibl., pl49. [6.5] P. West, op. cit, Bibl. J. Wess and J. Bagger, op. cit., Bibl. [6.6] M.E. Peskin and D.V. Schroeder, op. cit., Bibl. A. Zee, op. cit., Bibl. [6.7] A. Salam and J. Strathdee, Nucl. Phys. B86 (1975) 142. [6.8] I.L. Buchbinder and S.M. Kuzenko, op. cit., Bibl. [6.9] M. Grisaru, M. Rocek and W. Siegel, loc. cit, Ref. [6.1]. [6.10] M. Drees, K. Hagiwara and A. Yamada, Phys. Rev. D45 (1992) 1725, cf. Appendix. [6.11] M. Sher, Phys. Lett. B317 (1993) 159; ibid. 331 (1994) 448.

Chapter 7 GENERAL ASPECTS OF SUPERSYMMETRY BREAKING 7.1

Initial Remarks

Sparticles would be degenerate in mass with particles in an exactly supersymmetric world and would have the same abundance. Such is not the case in reality; no sparticle has yet been observed. In fact, experiment already tells us that (1) the superpartner of the 0.511 MeV mass electron has to have a mass at least of the order of 100 GeV and (2) the superpartner of the notionally massless gluon is obliged (in the minimal theory) to weigh more than about 200 GeV. Any supersymmetry, present in nature, must be broken severely indeed! All global continuous symmetries can be broken [7.1] in broadly two different ways - (1) explicitly in the Heisenberg-Wigner mode or (2) spontaneously in the Nambu-Goldstone mode. (A combination is also possible). In the former a "small" part of the Lagrangian breaks a symmetry of the remaining larger part, but the vacuum is left invariant; multiplets are split in mass but are otherwise kept intact. In the latter the Lagrangian is unchanged, but the vacuum does not remain symmetric and one or more massless Goldstone particles will occur; the multiplet structure is destroyed. If the symmetry in question is bosonic, the Goldstone particles will have spin zero. When one has a combination of both modes, the Goldstone particles acquire masses and are called pseudo-Goldstone particles. In QCD the breaking of flavor SU(n) by different quark masses is an example of (1), while the violation of the chiral symmetry SU(n) x SU(n) —> SU(n), for massless quarks, illustrates (2) with pseudoscalar mesons as massless Goldstone bosons. When quark masses are switched on, one has a combination of (1) and (2); consequently, the pseudoscalar mesons become massive pseudoGoldstone bosons. We shall explore both possibilities for global iV=l supersymmetry, taking up spontaneous violation first. The supersymmetry algebra is assumed to be unaffected in either case. 137

138

7.2

7. General Aspects of Supersymmetry Breaking

Spontaneous Supersymmetry Breaking : Some Generalities

Supersymmetry can be broken spontaneously, but there are specific constraints. We came across some such restrictions in §2.2 during our discussions of supersymmetric quantum mechanics. These generally carry over mutatis mutandis to field theory. In particular, the conclusion regarding the ground state - namely its energy being zero for exact supersymmetry but turning positive under spontaneous breaking - is valid for the vacuum in a supersymmetric field theory. So is the positive semidefiniteness property of energy for any state. We already know from (3.26a) that [QA,QB\+

KBP^-

=

This can be inverted, via the trace relation Tr(crM0-") — 2rfv, to yield \^A[QA,QB}+-

P, =

Thus the Hamiltonian H = P0 can be written as H

= \[QuQi]+

+ \[Q^Qi]+-

(7-i)

Since QA is the hermitian conjugate of QA, it follows from (7.1) that

(a\H\a) = l-J2H 0 « » | 2 + l
(7.2)

n

for any state \a). The positivity of the energy of any eigenstate of the Hamiltonian with a nonzero energy is evident from (7.2) and is inevitable so long as the supersymmetry algebra is intact. A supersymmetric vacuum state \fl) is defined by the condition that it remains invariant under any supersymmetry transformation. For an infinitesimal transformation, the condition requires 8\Q) = i (eAQA + eAQA) |fi> = 0 .

(7.3)

Since eA, eA are arbitrary Grassmann parameters, (7.3) is equivalent to the statement that a supersymmetric vacuum is annihilated by the supersymmetry generators QA, QA, i.e. Qa\ty = 0 or, in two component form, QA\n) = 0 ,

(7.4a)

QA\ty = 0 .

(7.4b)

It follows then from (7.2) that such a vacuum must have zero energy. Consider the scalar sector of the theory described generically by a complex field ), including quantum corrections. The supersymmetric vacuum must correspond to a

7.2. Spontaneous Supersymmetry Breaking: Some Generalities

139

global minimum of V() with a zero value as shown in Fig. 7.1a. On the other hand, supersymmetry is spontaneously broken if and only if any one of the generators QA, QA does not annihilate the vacuum state, thereby forcing that state - via (7.2) - to have a positive definite energy. In other words, such a violation occurs if and only if the said global minimum has a positive value. This situation is described by Fig. 7.1b. These situations may be contrasted with the corresponding ones in the spontaneous breakdown of an ordinary global internal symmetry as discussed below.

(a)

(b)

Fig.7.1. Vacua in supersymmetric theories: (a) exactly supersymmetric case, (b) spontaneously broken case. In general, in a field theory with a global internal symmetry, the zero point of the energy is not well defined (the null point energy is usually a divergent integral). In case one exercises the choice to make the ground state energy zero, one subtracts any nonzero constants from the Hamiltonian density #oo and appropriately normal orders the field dependent terms so that the VEV of (9oo vanishes. However, this does not have to be the case and the ground state energy could very well have been nonzero. Not so for global supersymmetry, where the ground state in the unbroken case must necessarily have a vanishing energy. For instance, the famous Mexican hat potential in the complex 0-plane, as shown in Fig. 7.2a, spontaneously violates the global U(l) internal symmetry 0 —>• eltp<j) but preserves supersymmetry. In contrast, the configuration shown in Fig. 7.2b leads to the spontaneous breakdown of both the said internal symmetry and supersymmetry.

(a)

(b)

Fig.7.2. Mexican hat supersymmetric potential (a) without and (b) with spontaneous supersymmetry breaking. A bosonic continuous internal symmetry, which is unbroken at the tree level, is usually stable under small1 loop corrections; only the existence of some kind of degeneracy in the potential can lead to a (dynamical) symmetry breakdown - arbitrarily induced by such tiny perturbations. A particularly well known example is the Coleman-Weinberg potential [7.2]. Here the vanishing of the curvature near the origin at the tree level leads to an 0(a) one loop violation of the global U(l) symmetry, a being (47r)_1 times the square of a 1

Of course, 'large' quantum corrections can always break a classical bosonic symmetry.

140

7. General Aspects of Supersymmetry Breaking

small coupling strength. Similarly, there are examples with special degeneracies [7.3] where a global symmetry, that is spontaneously broken at the tree level, gets restored by loop corrections, i.e. the curvature of the potential becomes positive everywhere instead of being negative over a finite region. Without such degeneracies, though, this type of restoration is not possible. But in a supersymmetric system, once supersymmetry is spontaneously broken at the tree level (i.e. V^ > 0), it can never be restored (cf. [7.4]) for any range of couplings. Also, given unbroken supersymmetry at the tree level and even with any amount of classical degeneracy, a potential is immune to supersymmetry breaking by perturbative quantum corrections on account of the corollary to the nonrenormalization theorem (§6.7). It has been shown, however, that strong nonperturbative effects can lead to a dynamical breakdown of supersymmetry in a classically supersymmetric field theory; an explicit example will be given in §7.6. Vacuum expectation values We know that the spontaneous breakdown of a bosonic internal symmetry arises from [7.1] a nonzero VEV acquired by a scalar or pseudoscalar operator which transforms nontrivially under the symmetry. Similarly, the spontaneous violation of global supersymmetry can be expected to arise from a nonzero VEV accruing to some operator that changes under a supersymmetry transformation. Consider first a chiral superfield $ with component fields , £ and F . Their infinitesimal supersymmetry transformations are given by (4.25). Examining the right hand sides of (4.25a-c), we can see that £ cannot acquire a nonzero VEV without compromising Lorentz invariance. Furthermore, the VEVs of dM, d^t, must be zero since the vacuum carries a vanishing four momentum. Only the auxiliary component F and hence S£ can have this property. Thus

l(n|F|n>| = A ^ o will lead to a spontaneous violation of supersymmetry. Since F has mass dimension 2, As can be identified as a mass scale, namely the scale of supersymmetry breaking. This type of spontaneous supersymmetry breaking is called F-type breaking or the O'Raifeartaigh mechanism [7.5]. Turning to a vector superfield in the Wess-Zumino gauge, with component fields A^, X and D, we can again apply the criterion of the preservation of Lorentz invariance and make use of the fact that the vacuum has no four momentum. Then it follows that only the auxiliary component D could have a nonzero VEV: |
7.3. The Goldstino

141

• The necessary and sufficient condition for SSB is the failure of the supercharge to annihilate the vacuum. • The energy of the ground state is always positive after SSB. • Two kinds of SSB have been outlined: F-type or D-type with a nonzero VEV of an auxiliary F or (abelian) D field respectively.

7.3

The goldstino

Just as the spontaneous violation of any ordinary bosonic symmetry leads to [7.1] a massless spinless Goldstone boson, that of global supersymmetry generates [7.6] a massless spin half Majorana fermion called the goldstino2. We had introduced in §5.1 the two component spinorial supercurrent density K^A and its conjugate K* for a system of interacting chiral superfields. Let us now define a general four component Majorana fermionic supercurrent density K% as the four spinor \ m x ) = ( K%x) K

*W ~ {R^ix) J '

with the space integral of its timelike component being the supercharge Qa. Thus

Qa = jdzxK°a{x)

= Qca ,

(7.5a)

d.K^x) = 0 . (7.5b) Of course, supercurrent conservation (7.5b) holds only if supersymmetry is not broken explicitly. Since Qa has mass dimension 1/2, that of K% is 7/2. As with bosonic global symmetries, the supersymmetry algebra (3.12) involving Qa and the Poincare generators can be promoted to an algebra of the corresponding densities with the addition of [7.7] Schwinger terms (S.T.). The latter involve derivatives of delta functions and hence drop out of VEVs. Thus, corresponding to (3.26a), we have [KW,QB]+

= 2VuAB&"W + S.T. ,

with 61"' as the stress energy tensor, i.e. P^ = J d3x6oll(x). 0^{x) = \a»AB

[QA, K%(x)\

+

(7.6)

(7.6) can be recast as + S.T.

(7.7)

Relations that are similar to (7.6) and (7.7) can easily be written with the anticommutator between QA and K^{x). 2

This terminology deviates from our usual custom of employing the suffix -ino to the name of a known boson in order to denote its superpartner. Here the goldstino is not the superpartner of a Goldstone boson, but is itself a Goldstone fermion. However, we choose to use this term because of its prevalent occurrence in the literature. Any superpartner of a Goldstone boson will instead be explicitly described as such. See the discussion at the end of this section.

142

7. General Aspects of Supersymmetry Breaking

(7.7) makes two things quite clear. The VEV of 9flv(x) vanishes when QA (and correspondingly QA) annihilates |fi), i.e. when the vacuum is supersymmetric. Also, the same VEV acquires a nonzero value when, on account of spontaneous supersymmetry breakdown, <5a|fi) / 0. In this case we may write (Q\0^(x)\Q)=E}lr,llv,

(7.8)

where EQ is an energy scale associated with the vacuum state. Taking VEVs of either side in (7.7) and using (7.8), we have E ^

= \^AB(a\[QA,K^B}+\^).

(7.9)

(7.9) demonstrates the reverse of the result derived earlier, namely that whenever the vacuum has a nonzero (necessarily positive) energy, it fails to be annihilated by the supercharge, i.e. QA\ty 7^ 0, Q" 4 ^) / 0 and supersymmetry has been spontaneously broken. Since, according to (3.26c), the supercharge commutes with the Hamiltonian, it is conserved as a consequence of Heisenberg's equation, i.e. ^

= 0. (7.10) at Therefore, for any local fermionic operator A(x') taken at the spacetime point x'^ = {t',x'), we can write

^(n\[Qa(t),A(x')} +\n) = o. dt'

(7.ii)

Let us insert a complete set of states y~J n )( n li which are eigenstates of the operator P M , n

between the two operators in (7.11). Then the replacement of Qa by Jd3xK°(x) use of the four translational properties (with a;0 = t) of the operators K°a(x) =

e"pK°a(0)e-^p,

A(x') =

eix'pA{0)e~ix'p,

and the

followed by a differentiation with respect to time, yield the result EnEJ^(pn)[(n\K°a((,)\n)(n\A(0me-iE^-^ -(ft|A(0)|n)(n|#0(0)|ft)e i£ »('- f ')] = 0 ,

(7.12)

with En defined as the energy of the state \n). Eq. (7.12) has a very important consequence. Because of the different time dependences of the two coefficients in the square bracketted expression, the latter can vanish V t, t' if and only if En5^(pn)(n\K°a(0)\n)(n\A(0)\a) =0. (7.13) Hence, if 3 some \n) for which (r2|A'°|n) ^ 0, the validity of (7.13) would require the condition

En6^(pn) = S^(pn)^pfTK

=0

7.3. The Goldstino

143

where mn is the "mass" \JE\ — p\ of the state \n). However, this implies that m„ = 0 .

(7.14)

In other words, the state \n), which must be a fermion since it admits a nonzero matrix element (fi|ifo|n), has to be massless on account of (7.14). Note that such a state must exist once supersymmetry is spontaneously broken, since (f2|Q0 7^ 0 =>• (ii\K^ ^ 0 and 3 some state \n) for which (Q|/iT°|n) ^ 0; the operator A can always be chosen in a way such that (ft|j4(0)|n) is nonzero. This state then is the goldstino. Because of (3.12b) and the commutation relation [J*,Qa] = 0 following from it, the state Q a |^) must have the same four momentum as |Q) and can be regarded as another degenerate and fermionic ground state comprising |f2) and a goldstino of vanishing energy and momentum. There is another (perhaps more physical) way to see the existence of the goldstino. Let us return to four component notation and start from (3.12a). Eq. (7.9) can be recast, with K% taken at the origin, as follows.

*hr

= l(n\[Qa,K£(o)}+\n)(Y)ba = \ J d'x dp(n\TK^x)K^o)\n)(Y)ba = ±JdSp{n\TKttx)K£(0)\n)(Y)i»-

(7.15)

We have used the result dp9(x°) = 6p0S(x°) in the second step of (7.15), whereas in the last step Gauss' theorem has been used, employing a spacelike three dimensional surface S at spatial infinity. For a finite surface integral in (7.15), the two point supercurrent correlator needs to fall off as \x|~3 for large spatial distances |x|. Since this falloff behavior can come only from the three dimensional Fourier transform of a massless spin half propagator, the existence of an intermediate massless spin half state, viz. the goldstino |A9), is established. From the fact that it contributes through the nonzero matrix element (fi|J!rMO|A9), it is evidently a Majorana fermion. In fact, one can define a dimensional goldstino constant f\ in the limit of vanishing three momentum via (2^) 3 / 2 (fi|^ a (0)|A 9 (o,6)) = / A ( 7 M U ,

(7.16)

where b is the free spinor index of the goldstino state in (7.16), defined at zero four momentum. If the goldstino had a nonzero four momentum p, there would be a term proportional to p^Sab in the RHS of (7.16); but that part would lead to a spatial fall off of the correlator of (7.15) faster than \x\~3 causing a vanishing contribution to En- The only nonvanishing contribution to En comes from an intermediate |A9) at zero four momentum and then (7.15) and (7.16) imply that El = h(7-17)

144

7. General Aspects of Supersymmetry Breaking

The matrix element of the spinorial supercurrent density K^, between bosonic and fermionic states in the same supermultiplet is of considerable importance. Let the bosonic (fermionic) state \b) (|/)) carry the four momentum pb (pf) with p/ = pb — q. One can then write (2Tr)3(b(pb)\K,a(0)\f(Pf))

= F1(q2)(1,1p)a + F2(q2)(pb+pf)^a

+ Fz{q2)q^a

,

where tpa is the four spinor from | / ) and Fi,2,3(<72) are form factors. If there is a spontaneous breakdown of supersymmetry, the above matrix element will have a goldstino pole occuring in the form factor F\ on account of (7.16). Suppose hfb\ is defined as the strength of the Xg-f-b Yukawa coupling. One can then write

the ellipsis standing for terms which behave smoothly when q —> 0. On contracting the matrix element above with q^ and taking the limit q —¥ 0, we can use the conservation law (7.5b) to write 2 2 hfbxf\ where mj 0, i.e. when supersymmetry breaking disappears. A related interesting issue emerges when | / ) and \b), as defined in the above paragraph, do not describe single particles but represent multiparticle states. In such a situation one can replace hfb\ by A4fb\, the amplitude describing a goldstino scattering process f + \g—>b or / —>• b + Xg. Now the matrix element of the vanishing four divergence d^K^ — 0 between (6| and | / ) implies that MfbxhF^q2)

+ q • (pb+pf)F2(q2)

+ q2F3(q2) = 0 .

On letting q —> 0, we derive the result that the amplitude M.fb\ vanishes in the limit of a 'soft' goldstino. This is sometimes called the soft goldstino theorem and has important physical consequences. For instance, no leptonic or semileptonic weak decay amplitude vanishes in the limit when the neutrino four momentum goes to zero. That proves, therefore, that the neutrino is not a goldstino. Let us conclude this section with the following comments: • Not every massless fermion in a supersymmetric theory need be a goldstino, a gaugino being a counterexample. In particular, one should distinguish between the goldstino and the superpartner of a Goldstone boson (cf. ftnt.' 1). • An exactly supersymmetric theory may be obliged to have a massless fermionic superpartner of a Goldstone boson. For instance, take a theory with a spontaneously broken global U{1) symmetry and containing one massless Goldstone boson. If the system is supersymmetrized, the Goldstone particle must necessarily acquire a massless fermionic partner. The latter is not a goldstino since the system remains exactly invariant under supersymmetry.

7.4. Model of F- Type Supersymmetry

Breaking

145

• The feature characterizing a goldstino is not so much its massless fermionic nature but rather its coupling (7.16) to the spinorial supercurrent density K%.

7.4

Model of F-type Supersymmetry Breaking

As explained in §7.2, an F-type spontaneous supersymmetry breaking is characterized by the nonzero VEV of some auxiliary F-component of a chiral superfield. A field theory exhibiting supersymmetry, broken by an F-type mechanism, needs to have the following feature. It must admit a solution of the equations of motion with some Ft ^ 0. If one insists (and this may not be strictly necessary) that supersymmetry be broken in the global minimum of the scalar potential V(0), one needs to require in addition that the choice Fj = 0, Vj, does not represent a solution of these equations. In the following discussion we shall assume that such is indeed the case. An immediate corollary of this is the fact that, in a system of interacting 1 1 chiral superfields3 $j, any superpotential of the form W = - m ^ $,., + -fijk$i$j$k without a linear $ term will fail to produce F-type supersymmetry breaking. In such a case F* = —dW/d$i | can always be made zero with the choice of all expectation values (0,) = 0. The latter correspond to a perfectly realizable configuration which is a global minimum of V(0). In order to effect an F-type supersymmetry breaking, we need to have a linear term in the superpotential. Take the Wess-Zumino model of one self interacting chiral superfield (cf §5.1). Replacing / by A, we get F* = -h - m<j> - ^ 0 2 .

(7.18)

Now the condition F = 0 can always be achieved by the choice 0± = - ^ ± i ( m 2 - 2 / * A ) 1 / 2 (7.19) A A and evidently it does correspond to the vanishing minimum of V = |F| 2 . Hence this model is also incapable of attaining the spontaneous breakdown of supersymmetry. One needs at least three chiral superfields $123 (012 3 with £ 12 3 and F123), as was noted by O'Raifeartaigh [7.5]. In O'Raifeartaigh's model the superpotential is chosen to be W($i, $2, $3) = m$ 2 $3 + A$i($g - /x2) ,

(7.20)

where fi is a fixed mass parameter. For simplicity, we assume that m, A and /x are all real. The equations of constraint yield the relations

3

F* = -A(0 2 - M2) ,

(7.21a)

F* = - m 0 3 ,

(7.21b)

F3* = - m 0 2 - 2A0!03 .

(7.21c)

If there are gauge interactions present, we assume here the $; are gauge singlets.

146

7. General Aspects of Supersymmetry Breaking

There is no consistent set of solutions to (7.21a-c) which can make all the F's vanish simultaneously, i.e. some F is always nonzero everywhere in the ^-space including the minimum of V{
Fig.7.3. Two regions in the parameter space of O'Raifearteigh's model. The scalar potential in O'Raifeartaigh's model is given by V(4>u 02, 2 + 2A01<^3|2 ,

(7.22)

with A, (i and m as real parameters. The minima of this potential are different in two distinct regions (Fig. 7.3) of the fx2, m2 parametric plane: (A) \i2 < m2/2X2 and (B) fi2 > m2/2X2. The difference in behavior can be easily seen under the simplifying assumption that (\) = 0.

AV

r\

xV

Region A

Region B

Fig.7.4. Scalar potential of O'Raifeartaigh's model near 3, (fa) has to be real and, at (i) = (z). For region A, we have one global minimum of the potential with Vm,„ = A2//4 > 0, {2) = {fa) = 0 and (i) is undertermined means that the minimum has a

7.4. Model of F-Type Supersymmetry

Breaking

147

"flat direction"4 along (i). Since the nonzero value of Vmi„ comes only from |(.Fi)|2, £1 the fermionic component of $1 - has to be the goldstino5. Its masslessness is shown below. Generalizing from (5.3) and using (5.5a), the fermion mass terms (FMT) in a Lagrangian density of interacting chiral superfields, with some scalar fields acquiring VEVs, can be written as C-FMT -

-2mijtitj

~ 7;fijk(k)£i£j + h.c.

(7.23)

For the present model (7.23) becomes CFMT = - m & 6 , - 2A(01>&£3 + h.c.

(7.24)

(7.24) is consistent with £i being the massless goldstino while £2 and £3 combine to yield a massive state. Suppose we choose (
with £-FMT

= —mtpip .

(7-25)

Turning our attention to the bosonic spectrum for the case of the parametric region A, we can isolate the quadratic terms in boson fields from (7.22) and identify the boson mass terms in the Lagrangian density as CBMT

=

A V 2 ( ^ + ^ 2 ) - m 2 ( | ^ 2 | 2 + |
=

-m2\2\2 ~\{m2-

2 A V ) A2

-i(m2 + 2AV)B2.

(7.26)

In (7.26) we have introduced real scalar fields A, B defined as A = (3 + %)/\/2, B = {4>z — %)/V2i. This equation implies a bosonic spectrum consisting of a massless scalar field 0i, a complex scalar field & of mass \m\ and two real scalar fields A,B of masses mA = | m | v / l - 2 A V / ^ 2 = y/ma-2X/^, mB = \m\y/l + 2AV/™ 2 = y/m2 + 2\A2t. The physical significance of the above mass spectrum is clear. The complex scalar 4>z has a mass \m\ in the supersymmetric limit As —>• 0. However, under supersymmetry breaking, it splits into two real scalars lying symmetrically in mass on either side of \m\ with a difference in squared mass 62 = 4AA2. In detail, this is caused by the operator A [ ( $ I ) $ 2 ] F = X(Fi)(A2 — B2). Moreover, we have the sum rule (5.10), namely STr m 2 = 2m2 + m\ + m2B - 4ml

=

°

( 7 - 27 )

4 "Flat directions" are noncompact lines and surfaces in the space of scalar fields along which the scalar potential vanishes. The present flat direction is an accidental feature of the classical potential and gets removed by quantum corrections. 5 This fact, namely that the goldstino is the fermionic component of the superfield whose auxiliary field develops a VEV, is also true of D-type supersymmetry breaking, cf. §7.5.

148

7. General Aspects of Supersymmetry Breaking

in this case with m as the mass of the complex scalar field 2 and m^, (= m) as the mass of the Dirac field tp. Thus the real scalar components of the chiral superfield $ 3 get split in mass from their fermionic superpartner, one becoming heavier and the other lighter. Their fermionic superpartners, together with those of the complex scalar field fa, make up the Dirac fermion field ip. It is interesting to see how this simple model leads us to quite important conclusions which will be pertinent when we discuss supersymmetry breaking for the MSSM in the hidden sector in Ch.12. First, all the physical states of the model can be made very massive by choosing m to be very large without affecting either the scale of supersymmetry breaking As or the mass splitting characterized by 5. Second, if A <S 1, the 'mass splitting' 6 will be much smaller than the supersymmetry breaking scale As. Thus all the three scales m, As and 5 in this model can be very different from one another. Finally, the masses of the physical states of the model obey (7.27). The graded trace of the mass squared operator over a supermultiplet vanishes at the tree level in the O'Raifeartaigh model. This sum rule is, however, subject to modifications involving any explicit supersymmetry breaking terms. It also changes under D-type supersymmetry breaking which is treated in §7.5. We would also like to clarify, in the context of F-type supersymmetry breaking, the interrelation between the auxiliary field VEV (F), the spinorial supercurrent KA and the (two component) goldstino field A9 which is its superpartner. Referring to ftnt. 3 of Ch.5, we can characterize the second RHS term in K\ as the contribution from (F) and Xg. Identifying, in particular, £ with Xg and utilizing (5.4a), we can then rewrite the supercurrent conservation condition (7.5b) as 0 = d^Kl = iV2(F) (a^Xg)A

+ dVA + • • •.

(7.28)

In (7.28) k\ generically represents the contribution to the supercurrent from all other supermultiplets (cf. ftnt. 3, Ch.5), i.e. -{a"a"a^Xa)AF^p - ig(\T?j4>j){a*\a)A , (7.29) where &, 4>i n o w represent components of other chiral superfields. Finally, the ellipsis in (7.28) stands for the contribution to the divergence of the supercurrent from the remaining members of the goldstino supermultiplet. As a result of (7.28), we can write an effective Lagrangian density involving the goldstino and its interactions with all other fermionic and bosonic fields. This reads kfX = -y/2{cva»Zi)A(Uiij)* ~ iV2(a%)Wj

-

l

Cg = iXgand»]Xg + _ L ^ A s d „ * " + h.c.) .

(7.30)

The appearance of the inverse of a VEV ( F ) _ 1 as a coupling strength in (7.30) is a characteristic feature of the interactions of Goldstone particles (bosons or fermions) with others. Eqs. (7.29) and (7.30) do show that the goldstino field has Xg£i
7.5. Model of D- Type Supersymmetry Breaking

7.5

149

Model of D-type Supersymmetry Breaking

A D-type spontaneous supersymmetry breaking is effected by the nonzero value of the VEV of the auxialiary D-component of a vector superfield (cf. §7.2). The simplest model exhibiting this mechanism, due to Fayet and Iliopoulos [7.6], is a supersymmetric U(l) gauge theory with a single chiral superfield of charge q in which the auxiliary component D(x) of the vector superfield V develops a nonzero VEV. Tracing back to (4.8h) and using £ = A, we see the consequent implication that |(ee)-1(fi|6<5A(a;)|Q>| = \(D(x))\ = A2 # 0 .

(7.31)

(7.31) means that the gaugino field has a nonzero variation in the vacuum configuration. Moreover taking {F) to be zero, it is evident from (5.20b) that Vmin = \K

(7.32)

is indeed positive, as required for spontaneous supersymmetry breakdown. In this model (5.19) becomes D = - V - }. If the sign of rjq is negative, it becomes possible through a nonzero VEV of

) = \{ri + q\\2)2

(7.34)

requires () = 0 for minimization and V

•

=

-r?2

i.e. supersymmetry does get violated spontaneously while the U{1) symmetry remains intact. However, the scalar component (f> of the chiral superfield $ becomes massive, in consequence, with the mass m = VOV i (7.35) while its fermionic superpartner £ remains massless and is the goldstino. In other words, the members of the supermultiplet are split in mass. Moreover, this means that 2_^ m o longer equals 2^2ml/2y

no

unlike in (5.9). In fact, the sum rule (5.10) has changed to STrrn2

= 2qri=-2q(D)

.

(7.36)

Gauge invariance has, however, been maintained so that both the gauge boson, described by An, and the gaugino, described by A, remain massless. One problem with this U(l) gauge model, however, is that it suffers from the occurrence of an ABJ anomaly [7.7]. Suppose we

150

7. General Aspects of Supersymmetry Breaking

then consider SQED (cf. §5.3) which, with two chiral superfields, is anomaly free. Here the introduction of the Fayet-Iliopoulos term simply changes V((p) to ^

+

, ^ ) = i[r? + ^ + | 2 - | 0 _ | 2 ) ] 2 .

(7.37)

But now the potential of (7.37) vanishes at its minimum independently of the sign of qr). The latter merely determines whether it is + or 0_ which develops a VEV. This example illustrates the difficulty [7.9] of implementing the Fayet-Iliopoulos mechanism in an anomaly free gauge theory. In order to produce a nonzero (D) in a field theory of the latter kind, one needs to introduce both extra fields and extra mass terms in the superpotential. We shall not go into this discussion any further but only refer the reader to Ref. [7.9] for details.

7.6

Dynamical Model of Supersymmetry Breaking

There is a model [7.10] constructed to achieve supersymmetry breaking via an interplay between perturbative and nonperturbative effects. It consists of a supersymmetric nonabelian gauge theory (cf. §5.4) with the semisimple gauge group SU(3) x SU(2), i.e. eleven gauge superfield components, characterized by gauge coupling strengths (33,52) and containing left chiral superfields Q(3,2), £7(3,1), £>(3,1) and $(1,2) in the matter sector6. The bold pair of integers in the parenthesis of each chiral superfield refers to the dimensions of the representations of SU(3), SU(2) that the superfield transforms as. Let us also assume an additional global U(l) symmetry, called H, and take the corresponding //-charges of Q, U, D and $ respectively to be 1, - 4 , 2 and —3. Now there is only one possible polynomial form for the superpotential of the model which can be written as WP = \{Q-$D).

(7.38)

In (7.38) A is a coupling constant, the subscript p in the LHS stands for the designation 'perturbative', Q-$ = £ABQA$B with A,B being SU(2) indices and the brackets imply an SU(3) index contraction. Had (7.38) been the only contribution to the superpotential, all VEVs would vanish and supersymmetry would remain unbroken. However, SU(3) instantons [7.11] generate [7.10] a contribution (that is of a nonpolynomial form in the chiral superfields) whose nonperturbative origin we undescore with the subscript np:

w

-=mh(m-

(739)

Here A3 is the mass scale at which the 5C/(3) coupling strength g3 becomes strong, though the SU(2) gauge coupling strength g2 is assumed to remain in the weak coupling regime. Thus the nonperturbative effects are determined by the dynamics of the 577(3) gauge group 6

We are using notation similar to that of quark and antiquark superfields of the Minimal Supersymmetric Standard Model, to be described in Ch.8, because we later use an anology with multiflavored SQCD.

7.6. Dynamical Model of Supersymmetry Breaking

151

alone. The actual derivation [7.12] of (7.39) makes use of instanton calculus and is beyond the scope of this book, but we provide below a qualitative symmetry argument justifying the form of the RHS. By assumption, 5(7(3) instantons dominate and determine Wnp. One can then draw an analogy with multiflavored supersymmetric quantum chromodynamics with Nc (= 3) colors and Nf (= 2) flavors. The effective action of this kind of a theory is greatly constrained by its full flavor related global symmetry which happens [7.10] to be 5(7(/V/)L x SU(Nf)R x (7(1)v x U(l)n x U(1)R. The subscripts L, R and V for the three factor groups refer respectively to left chiral, right chiral and vectorial representations while the U(l)R factor is a consequence of supersymmetry (cf. Ch.3). Anomaly cancellation requires [7.10] each chiral superfield to carry an iZ-charge (Nc — Nf)/Nf (= 1/2). The only product combination of chiral superfields, that is invariant under transformations of the complete gauge and flavor group is (QU)-(QD), i.e. that appearing in the right hand denominator of (7.39). However, the i?-charge of this combination is 2 while that of the superpotential, on the grounds that f d26W must be Rinvariant, is —2. Hence the only allowed contribution to W is proportional to [(QC7)-(Q5)]-1, while the power of A3 in the numerator follows from the requirement that W must have a mass dimension of 3. Moreover, the power of A3 has to be positive in order to facilitate a smooth transition to the weak coupling regime which corresponds to the limit A3 —> 0. The unit coefficient in the RHS of (7.39) cannot be determined by this argument and has to be computed by instanton calculus [7.10]. There is also a caveat. The perturbative superpotential in (7.38) breaks 5 ( 7 ( 2 ) R explicitly. For the above argument to go through, we need to assume that the magnitude of A is small enough not to change the nonperturbative behavior of the theory. Eq. (7.39) clearly violates the nonrenormalization theorem of §6.7. However, the latter only applies to perturbative contributions to W. Perturbation theory in the trivial vacuum cannot [7.11] describe instanton effects which are entirely nonperturbative in origin. Hence (7.39) is outside the purview of the said theorem. A more important feature of the RHS of (7.39) is that it is singular in the vacuum configuration for vanishing VEVs. This signals the breakdown of 5(7(3) perturbation theory in the infrared region where the perturbative part of W vanishes. Conversely, the contribution (7.39) by itself would drive some (or all) VEVs to infinite values, but there the perturbative contribution (7.38) diverges. A simple scaling argument then shows [7.10] that the complete scalar potential, computed from W = Wp + Wnp, has a minimum where all fields have VEVs of order v ~ AzX^1^7. All of these VEVs point in a direction in field space along which the entire set of D-term contributions to the scalar potential vanishes. The system is characterized by the spontaneous breakdown, not only of supersymmetry, but also of the 5(7(3) x 5(7(2) symmetry. The latter is completely destroyed, i.e. every generator fails to annihilate the vacuum. The Higgs mechanism becomes operative via the VEVs. In consequence, all eleven gauge supermultiplets become massive, eight with masses ~ gzv and three with ~ g?y. The VEVs also generate masses for eleven of the fourteen chiral superfields; only the (7(3,1) are left massless on account of its nonappearance in (7.38). Let us understand the presence of the massless fermions contained in the latter in terms of symmetries. There is a massless goldstino (as always when supersymmetry is spontaneously broken) plus the superpartner of a massless "axion" associated with the

152

7. General Aspects of Supersymmetry Breaking

spontaneous breakdown of the global axial [7(1) symmetry. There is an additional massless fermion associated with the composite superfield X = (UQ)-$ which is charged under U(1)H• The latter is required so that the 't Hooft anomaly matching condition [7.13] is satisfied. The supersymmetry violating VEVs of F-components are of order Xv2, i.e. (F) ~ A 5 / 7 ^ while the mass splitting between members of a given supermultiplet is characterized by 6 ~ Xv. If the gauge coupling strengths are of order unity and yet |A|