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eia e%a(p. Let's assume, for definiteness, we are in the vacuum with zero phase. Then, writing where x is a real field. Then
(4.2.19)
which form the group U(l). The infinitesimal transformations are 8ip = iSaip, 5(p* = —iSacp*. The current 3p = i(
(4.2.20)
is conserved, as a consequence of the Nother theorem (4.2.5). Solutions of the equation of motion (4.2.18) at g = 0 are plane waves cp = ae~ipx with p2 = m 2 ; they have j ^ = \a\22pfl. The standard relativistic normalization j ^ = 2pfl corresponds to \a\ = 1. A theory with many fields can have a more complicated symmetry. Let cp be a column vector of fields, and let a lagrangian be invariant with respect to transformations (p -> U(a)(p. (4.2.21) Here a is a set of parameters of a continuous group, and U(a) are matrices of its representation, in accordance to which the fields ip are transformed. Infinitesimal transformations have the form U(a) = 1 + i5aata, where ta are called generators of this representation. The currents (4.2.22)
172
4 Quantum electrodynamics
are conserved, as a consequence of the Nother theorem (4.2.5). If the group is non-abelian, the generators do not commute: [ta,tb] — if abctc, where fabc are called structure constants of the group. Generators are conventionally normalized as Trt a t b = \5ab. Let's now proceed to a theory of a real vector field A^{x). Its lagrangian 1 L = --F^F^ 4
m2 + —ApA^ L
F^v = d^,A
v
- dvA^
(4.2.23)
gives the equation of motion %FVV, + m2Afl = 0.
(4.2.24)
Applying d^ to it, we obtain (at m ^ 0) dpAp = 0.
(4.2.25)
Taking this into account, equation (4.2.24) gives (<92 + m2)A^ = 0.
(4.2.26)
Solutions of (4.2.24), or the equivalent equations (4.2.25), (4.2.26), are plane waves A^ = e^e~tpx withp 2 = m 2 , whose polarization vectors eM are orthogonal to pM. In the rest frame, the vector e has only spatial components, i.e. the theory describes a spin 1 particle. A massless vector field is a special case. Its lagrangian is invariant with respect to gauge transformations A^ —> A^ + dflf(x), where / is an arbitrary function. The equation of motion (4.2.24) dvFV[l = 0 does not imply (4.2.25). Is it possible to construct a theory invariant with respect to transformations (4.2.19), which are arbitrary functions of coordinates? The lagrangian (4.2.17) does not have such a gauge symmetry, because the derivative d^cp is not transformed to e^d^tp. Let's try to replace d^ip by the covariant derivative D^tp = {d^ — ieA^ip, and require that it is transformed like ip:
To this end, the field A^ should transform as A'^ = A^ + {l/e)d^,a. In addition to the replacement of d^ by D^, we also need a gauge invariant lagrangian of the field A^ itself. From what can we construct it? Let's consider the commutator [D^DU\ = —ieF^, where F^v = d^Au — d^A^. It is gauge invariant: {[D^.D^)' = e%ot[D^Dv]ip = [ D M , D j e i a ^ . We can construct a gauge invariant lagrangian —\F^ UF^U from the field strength tensor F^v. This is how the electromagnetic interaction is described, A^ being the electromagnetic field. Its quanta are massless vector particles — photons. It is also possible to construct a theory invariant with respect to local (gauge) transformations forming any group (4.2.21). We shall discuss such theories in § 5.1.
4.2 Fields
173
Spinor fields Spin \ particles (e.g., electrons) are described by spinor fields ip\ 4-component columns. Instead of using a hermitian-conjugated row -0"1", it is more convenient to use a Dirac-conjugated row ij) = V7"4"/?, where (3 = f3+ is a matrix. The Diracconjugate of a row ip is the column ij) = /J"1^"1", and that of a matrix F is the matrix F = /?~ 1 F + ^; for a scalar, it means complex conjugate. We have
and so on. Vectors are transformed under the action of Lorentz transformations as v^ — > L^vvv, and spinors as ij) —>• J7^, where [/ is a matrix, depending on the transformation parameters. Conjugated spinors are transformed as t/j —>• ij) E/. Dirac conjugation is defined to ensure that ij)ij) is a scalar: ipip —> ij) UUip = ilitj), hence UU = 1, i.e. ip —> IJJU~1. Dirac matrices 7M are introduced so that V;i7/i-'02 is transformed as a vector: A0I7MV;2 —> '0i^~17/x^V;2 — ^ixv^xlv^i^ i.e. C/~17/Xt/ = L^jjy. In this sense, 7M form a vector, when placed between ip and
form a tensor. Traces of products of 7-matrices are tensors, which don't change their form under the action of Lorentz transformations: I/
..
.7A =
therefore they can only be expressed via S^. Let's also require that 7^ = 7^. For any vector v, we shall use the notation i) = vM7M. Now we can write down the spinor field lagrangian: L = ijj(id - m)V>-
(4.2.27)
At first sight, it is not real: L* = ip(-i d - m ) ^ , but the difference L — L* = ) is inessential as a full divergence. Of course, one can write down a manifestly real lagrangian L = ij){i{d — d )/2 - m ) ^ , but it is equivalent to (4.2.27). Treating -0 and -0 as independent fields, we obtain the equation of motion by differentiating (4.2.27) in ij)\ (id - m)rp = 0. Differentiating in ip gives, of course, the conjugated equation ij)(—i d — m) = 0.
(4.2.28)
174
4 Quantum electrodynamics A plane wave ip = ue~ipx
is a solution of the Dirac equation (4.2.28) when (p - m)upa = 0,
(4.2.29)
where the index a numbers independent solutions at a given p. The conjugated equation has the form upa(p — m) = 0. Multiplying (4.2.29) by (p + m) from the left, we obtain (p2 — m2)upcr = 0. This equation should have nonzero solutions upa only at p2 = m 2 , i.e. the condition p2 = p^,pv{l^lv + 7i/7/n)/2 = p2 must hold for all p. This condition requires 7/^ + 7 ^ = 2 ^ .
(4.2.30)
This formula completely determines the algebraic properties of 7-matrices. The pseudoscalar matrix 75 is often used: 75 = -^e^palynviplG
= -Ho7i7273,
75 = !>
7s7M + 7/x7s = 0,
75 = - 7 5 . (4.2.31) The lagrangian (4.2.28) is invariant with respect to phase transformations %l> -> eza<0, %j) -¥ e~ lOLip. According to the Nother theorem, this fact leads to conservation of the current (4.2.32) U = ^7M^« We have agreed to normalize wave functions by the condition j ^ = 2pM, i.e. uj^u = 2pM.
(4.2.33)
Multiplying this equation by p^ and using the Dirac equation (4.2.29), we obtain the equivalent (at m ^ 0) form of the normalization condition: uu = 2m.
(4.2.34)
Quantum electrodynamics describes the interaction of point-like spin \ particles with electric charges ze (leptons and quarks) with photons. Here e is the elementary charge; electron, muon, and r-lepton have z — - 1 , all neutrinos have z = 0; u, c, ^-quarks have z = | , and d, 5, 6-quarks have z = - | . The fine structure constant a = e 2/(4?r) « 1/137 is usually used instead of e. The lagrangian of free spinor fields L o = Yli^ii^ ~ mi)'ll)i IS invariant with respect to many kinds of phase transformations. The [/(l)-group ipi —>• elZia^i plays a special role. Conservation of the electromagnetic current j M = H2i zi^il^i i s related to this symmetry. In quantum electrodynamics, this symmetry is made local by replacing d^ipi by D^i = (9M - ieziAy)^. As a result, the interaction term ej^A^ is added to LQ\ one also needs to add the free —^F^F^. electromagnetic field lagrangian Perturbative series in QED are asymptotic, not convergent. If they converged at small positive e, they would converge at small imaginary e, too. At an imaginary e, same sign charges attract each other. A configuration with many electrons in one region and many positrons in another region can have arbitrarily large negative energy. Vacuum is unstable with respect to tunnelling into such states, and perturbative series cannot converge. This argument is due to Dyson.
4.2 Fields
175
Working with Dirac matrices in REDUCE
It is possible to work with several independent sets of 7-matrices, related to different fermion lines. The matrix 7^, related to the line / , is denoted by g(f ,m). Indices and vectors are equivalent, therefore the matrix p, related to the same line, is denoted by g(f ,p). The product p^^q^v 1S denoted in short by g(f ,p,m,q,n). The matrix 75, related to the line / , is denoted by g(f ,a). The object a should be declared as vector a; it cannot be used as an ordinary vector. When evaluating a 7-matrix expression x, REDUCE automatically calculates Tr xj Tr 1 (where Tr 1 = 4). The traces Tr 7Ml ... 7Mn are invariant tensors, and can only be expressed via 8^v. Therefore, they don't vanish for even n only: input_case nil$ vector pl,p2,p3,p4,p5,p6; g ( f , p l ) ; g(f,pl,p2); g(f,pl, P 2,p3); g(f,pl, P 2,p3,p4); 0
0 P1-P2P3-P4 ~ P1-P3P2-P4 + P1-P4P2-P3
g(f,pl,p2,p3,p4,p5,p6); P1-P2P3-P4P5-P6 ~ P1-P2P3-P5P4-P6 + Pl-P2 P3-P6 P4-P5 ~ Pl-P3 P2-P4P5-P6 + P1-P3P2-P5P4-P6 ~ P1-P3P2-P6P4-P5 + P1-P4P2-P3P5-P6 ~ Pl-P4P2-PbP3-P6 + P1-P4P2-P6P3-P5 ~ P1-P5P2-P3P4-P6 + P1-P5P2-P4P3-P6 ~ Pl-Pt P2-P&P3-P4 + b — Pl'P6P2-P4P3'P5 + P1-P6P2-P5 P3-P4
Note that any pi may be vector or index; in the first case, the trace contains p, while in the second one it contains 7^. The traces Tr 757,^ .. .7Mn are invariant pseudotensors, and contain s^p^. Therefore, they don't vanish for even n only, starting from 4: vector a; g(f,a,pl,p2); g(f,a,pl,p2,p3,p4); 0 g(f,a,pl,p2,p3,p4,p5,p6); (
P4,P5,Pe) - Pi-P3 eps(p2,P4,P5,Pe) -h Pi-P*
Pi-Pb eps(p2,P3,P4,Pe) + Pi-Pe eps(p2,P3,P4,P5) + P2V P2-P4 eps(pi,p 3 ,p5,P6) + P2-PS eps(pi,p 3 ,p 4 ,p 6 ) - P2-P6 eps(pi,p 3 ,p 4 ,p 5 ) P3-P4 eps(pi,p2,P5,Pe) - P3-Pb eps(pi,p2,P4,Pe) + P3-P p 4 .p 5 eps(pi,p2,P3,Pe) - P4-P6 eps(pi,p 2 ,p 3 ,P5) + Ps-P
176
4 Quantum electrodynamics
The last trace is presented in some textbooks in a shorter form. In fourdimensional space, any tensor, antisymmetric over five or more indices, in particular e^vpvS^p, vanishes. Therefore, the form of expressions with e^pa and 6ap is not unique. Similar ambiguity arises in traces without 75, starting from ten 7-matrices, because the product of five 5^, antisymmetrized over five indices, vanishes. You can instruct REDUCE not to calculate traces of 7-matrix expressions, related to the line / , by saying nospur f;. The command spur f; returns it to the original mode of operation. It is better to perform 7-matrix calculations in the state nospur f; to say spur f; before a trace calculation, and nospur f; after it. Otherwise, surprises are possible, for example, g(f ,pl)*sub(p2=p3+p4,g(f ,p2)); 0 The function sub evaluates its argument; in the regime spur f; it obtains zero, which can hardly be intended by an author of this fragment. Calculations with 7-matrices at on gcd; sometimes lead to error messages. It is better to calculate all traces, and only after that to combine expressions for separate diagrams with on gcd;. Currently, only one line involved in an expression may be in the state nospur. Moreover, index contractions between a line in the states nospur and spur sometimes lead to CATASTROPHIC ERROR. This fact makes the possibility of using 7-matrices, related to several lines, not very useful. REDUCE performs some simplifications of 7-matrix expressions in the nospur state:
nospur f; g(f,pl,pl); Pi-Pi index m; g(f,m,m); g(f,m,pl,m); g(f,m,pl,p2,m); g(f,m,pl,p2,p3,m); 4
g(f ,m,pl,p2,p3,p4,m);
g(f,m,pl,p2,p3,p4,p5,m);
2
These examples lead one to the (correct) guess, that the expression 7 M ... 7M with an odd number of 7-matrices inside is equal to —2 times the product of the same 7-matrices in the opposite order. For an even number of 7-matrices, two terms appear: the product, in which the first matrix stands last, plus the same product in the opposite order (or vice versa, the last matrix may be moved to the first place). In the case of two 7-matrices, the result can be presented in a simpler form: as 4pi • p2>
4.2 Fields
177
The matrices 70, 71, 72, 73 anticommute with each other, and their squares are equal to 1, —1, - 1 , - 1 . Any product of these matrices reduces to the form in which each of these four matrices occurs 0 or 1 times. There are 16 such matrices. Let's denote them by 7^. Their squares are 7^ = SA = ± 1 . Any 7-matrix expression F can be expanded in these matrices: F = TAIA, where the contraction is defined as ^2A£^AryATraces of all 7^, except the unit matrix, are zero, therefore the coefficients TA = \ Tr 7AF. Hence we obtain the completeness relation: r = 7A|TY7Ar.
(4.2.35)
\^1A1B1CIn particular, the product is JAIB = C-ABCIC, where CABC = Its trace is \ Tr JA 1B — &AB ? where the diagonal elements SAA — £ A > so that 1A = SAB 7B_ Let's apply the completeness relation (4.2.35) to F = ^4^3* a n d sandwich it between IJJ1 . . . fo* l _
We have obtained a relation, expressing the product of 1-4 and 3-2 bilinear forms as a sum of products of 1-2 and 3-4 forms. Other formulae of such kind (called Fierz identities) are obtained by the substitution 1JJ2 —> 7BV;2> ^4 ~~^ 7A^4 :
Finally, _
_
where the Fierz coefficients are ,
1
_
_
1
JABCD = CEBCCEAD = T Tr
lAlDlBlC,
due to the completeness relation (4.2.35). The matrices JA axe components of the antisymmetric tensors 1, 7M, 7[M7^], 7[/i7^7p]? 7[/x7^7p7a] (there are no antisymmetric tensors with more indices in four-dimensional space). It is more convenient to use the dual pseudoscalar 75 (4.2.31) instead of the tensor 7[At7iy7p7a]5 and the dual pseudovector 7^75 instead of 7[^7i/7p]» The tensor 7^71/] is called 0 ^ ; its dual is \£nvp(jOpo — LLU. Any 7-matrix expression can be reduced to the canonical form 1 Ts = \ Tr F, TP = \ Tr 7 5 F , Yv^-Tv 4 4 4 1 1 ^Aii = 7 Tr 7/X75F, Fr#ii/ — 7 Tr cr^T. 4 4
7/x F,
(4.2.37)
178
4 Quantum electrodynamics
The factor \ is related to the fact that, in a contraction of two antisymmetric tensors, each term appears twice. Let's check the properties of JA with REDUCE: vector ml,m2,m3,nl,n2,n3; array gl(4),g2(4),g3(4),G(4); gl(0):=l$ gl(l):=g(f,a)$ gl(2):=g(f,ml)$ gl(3):=g(f,ml,a)$ gl(4):=(g(f,ml,nl)-g(f,nl,ml))/2$ for i:=0:4 do « g 2 ( i ) :=sub(ml=m2,nl=n2,gl(i)) ; g3(i) :=sub(ml=m3,nl=n3,gl(i))»; G(0):= n s n $ G(l):= u p"$ G(2):="vn$ G(3):= ua u$ G(4):="t"$ spur f; for il:=0:4 do if (r:=gl(il)) neq 0 then write G(il)," " , r ; s 1 for il:=0:4 do for i2:=il:4 do if (r:=gl(il)*g2(i2)) neq 0 then write G(il),G(i2)," " , r ; ss 1 PP 1 vv m,\.rfi2 CLCL
—
tt
— ml.m2nl.n2 + ml.n21n2.ni
7711.77T-2
Now let's find the coefficients
CABC,
which define the multiplication table of 7,4:
°/ 0 Multiplication table for i l : = 0 : 4 do for i2:=0:4 do for i3:=0:4 do if (r:=gl(il)* g 2(i2)*g3(i3)) neq 0 then write G(il),G(i2),G(i3)," " , r ; sss 1 spp 1 SVV 1712.1713 SOLO,
—
777-2*m-3
stt — m2.m3n2.n3 + m2.n3m3.n2 psp 1 pps 1 pVCL 777-2 « m 3
ptt — vsv m\ vpa —
4.2 Fields vvs vvt
mi.m2 — mi.m3m2.n3
179
-f mi.n3m2.m3
vap m\.m2 vat eps(mi, m 2 , 7713, n3) i vtv mi.m2m3.ri2 — mi.n2m2.m3 vta — eps(mi,m2,m 3 ,n2)i asa — m\.m3 apv mi.rri3 avp — mi.m 2 avt — eps(?ni,m2, ^3,^3) i aas — m\.m2 aat mi.m3m2.n3 — mi.n3m2.m3 atv eps(rai,ra2,m3, n2) i ata
— nii.m2m3.ri2 + mi.n2rn2.m3
tst
— mi.m3ni.n3 + mi.n3rn3.ni
tpt
— eps(rai,m3,?ii,n3)z
tw
— mi.m2m3.ni + mi.m3m2.ni
tva tav
eps(mi?m2,m3,ni) 2 — eps(rai,ra2,m3,ni) z
taa mi.m2m3.rii —
mi.m3m2.ni
tts
— mi.m2ni.n2 + mi.n2m2.ni
ttp
— eps(rai, ra2, ?^i, ri2)z
t t t mi.m2m3.nin2.n3
— mi.m2m3.n2ni.n3
mi.m3m2.n3ni.n2
+ mi.n2m2.m3ni.n3
mi.n3m2.m3ni.n2
+
— 7711-7713 m2.ni 722.713 +
— mi.n2m2.n3m3.ni —
mi.n3m2.nim3.n2
Fierz identities (4.2.36) are most often used for scalars. Let's find the Fierz coefficients, partially contracted in A and 5 within one of the classes 5, p, v, a, °/c Fierz coefficients index ml,nl; for il:=0:4 do for i2:=0:4 do for i3:=0:4 do if Cr:=gl(il)*g2(i2)*gl(il)*g3(i3)) neq 0 then write G(il),G(il),G(i2),G(i3)," ",r; ssss 1 sspp 1 SSVV
7712.7713
ssaa
— 7712.7713
sstt
— m2.m3n2.n3
+ m2.n3m3.n2
4 Quantum electrodynamics
180
ppss 1 pppp 1 ppvv
- m2.m3
ppaa m 2 •m 3 pptt - 772-2-772.3 72-2-713
+ 772-2-72.3 772-3.72-2
vvss 4 vvpp - 4 vvvv
-
2 7712-7723
vvaa
-
2 7712-7713
aass
- 4
aapp 4 -
aavv
2 7712-7723
aaaa
- 2 7712.77l3 — 12
ttss ttpp - 12 tttt 4( - m 2 . 7 7 i 3 n 2 . 7 i 3
+ 77i2.7i3m3.7i2)
In the weak interaction theory, Fierz identities for pseudoscalars are also used. To derive them, let's substitute JB —>• 7B757, Pseudoscalar Fierz coefficients for il:=0 step 2 until 4 do for i2:=0:4 do for i3:=0:4 do if (r:=gl(il)*g2(i2)*gl(il)*g(f f a)*g3(i3)) neq 0 then write G(il),G(il),G(i2),G(i3)," " , r ; sssp 1 ssps 1 ssva SSaV sstt
—
rri2.ms
7712-777,3 — e p s (7712,7713,712,713)2
vvsp 4 vvps VVVa
—
4
2 7712-7713
vvav 2m 2 .m3
ttsp ttps
- 12 - 12
tttt
— 4 eps(ra 2 ,m3,n 2 ,n 3 )z
The following procedure finds the coefficients in the expansion (4.2.37) of a 7-matrix in the basis matrices 1, 75, 7^, 7^75, a^. The indices \i and v are called ! 1 and ! 2, in order to minimize the risk of a name collision.
4.2 Fields
181
operator sig; for all f,m,n let sig(f,m,n)=(g(f,m,n)-g(f,n,m))/2; vector Ml,!!2; nospur !!; procedure Gexp(f,x); begin scalar y; y:=sub(f=!!,x); spur !!; y:={y,g(!!,a)*y,g(!!,!!l)*y,g(!!,a,!!l)*y,sig(M,!!2,!!l)/2*y}; nospur !!; return y end$ nospur f; °/0 several examples Gexp(f,g(f,pl,p2)); Gexp(f,g(f,pl,P2,p3));
W 2 ,o,o,o, ' ^ ^ - ' i ^ ' ^ } { 0 , 0 , \i-PiP2-P3-h-P2Pi-P3 + h'P3Pi-P2i eps(!i,pi,p2,p3)i, 0} Gexp(f,g(f,pl,P2,p3,p4));
{Pl-P2P3-P4 - PI-P3P2-P4 +P1-P4P2-P3, eps(pi,p 2 , P3,PA) i , 0, 0, ~ h-Pl }-2.P3P2-P4 + h-Pl }-2-P4P2-P3 ~ h-P2 h-P2 }'2-P3Pl'P4 ~ h-P2 h]
'2'P4Pl-P2 ~ h-P4 h'PlP2-P3 + W-P4
Gexp(f,g(f,a,pl,p2)); Gexp(f,g(f,a,pl, P 2,p3)); rn n n ~ eps(!x, !2,Pi,P2) ^ {0, pi.p 2 , 0 , 0 , ) { 0 , 0 , - eps(!i,pi,p 2 ,P3)^, ~h.piP2.P3 + h.P2Pi-P3-h-P3Pi.P2, 0} Gexp(f,g(f,a,pl,p2,p3,p4)); <eps(pUp2iP3,P4)i,
PI-P2P3-P4 ~ P1-P3P2-P4 + P1-P4P2-P3 , 0, 0,
0 ,p 2 ,p 3 ,p 4 ) + !2.p2 eps(!i,pi,p 3 ,p 4 ) ! 2 .p 3 eps(!i,pi,p2,P4) + !2.P4 eps(!i,pi,p 2 ,p 3 ) - px.p2 eps(!i, ! 2 ,p 3 ,p 4 ) Pi-Pz eps(!i,! 2 ,p 2 ,p 4 ) - pi.p4 eps(!i,!2,p2,p3) - p2.p3 eps(!i, ! 2 ,Pi,p 4 P2-P4 eps(!i,!2,pi,p3) - p3.p4 Gexp(f,g(f,pl)*sig(f,p2,p3)-sig(f,p2,p3)*g(f, { 0 , 0 , 2(-!i.p 2 Pi.P3 + !i*3Pi.P2), 0, 0} Gexp(f,sig(f,pl,p2)*sig(f,p3,p4)-sig(f,p3,p4)*sig(f,pl,p2)); {0, 0, 0, 0, - h-Pl h'P3P2-P4 + h-Pl ]'2'P
bye;
182
4 Quantum electrodynamics
The last two examples show that apa\ =
2 3
The matrix J7 of the infinitesimal Lorentz transformation ^M —> Vp+Sa^Vv must It satisfies C/~17/XC/ = 7M + C[7M>'Vrl^Vr = have the form [7 = 1 + cSa^a^. 7M + ^ ^ ^ 7 ^ . Taking into account the first equation in (4.2.38), we obtain c = | : J7 = 1 + ^Sa^cr^.
(4.2.39)
The second equation in (4.2.38) ensures the correct commutators of infinitesimal Lorentz transformations. For purely spatial rotations, they reduce to the commutators of the angular momentum operators (3.1.1).
4.3 Feynman diagrams Feynman rules Matrix elements iM of various processes are usually calculated in perturbation theory in interaction. To this end, the Feynman diagram technique is used. First, a set of diagrams is drawn for a process under consideration; expressions for contributions to the matrix element are produced from these diagrams, using some well-defined rules. We shall not derive the rules of the diagram technique here; if you are interested in the derivation, you can find it in standard textbooks. We shall only formulate these rules for the simplest example — the scalar field theory (4.2.15), and then discuss their generalization to more complicated theories. Propagation of a particle of some kind is depicted in a diagram by a line of the corresponding kind, and interaction of several particles by a vertex. External lines of a diagram correspond to real initial and final particles, which lie on their mass shells p2 = m2. Internal lines, which begin and end at the diagram's vertices, describe propagation of virtual particles; they are off their mass shells. Energy-momentum is conserved at every vertex. Integration d4p/(2?r)4 is performed over all momenta which are not fixed by these conservation laws. Each vertex gives the corresponding interaction constant, which is considered to be a small parameter of perturbation theory. Therefore, the leading contribution to the matrix element comes from diagrams with the minimum number of vertices. Such diagrams contain no loops, and are called tree diagrams. All their momenta are determined by the conservation laws. More complicated diagrams produce so-called radiative corrections. The first correction comes from one-loop diagrams. For their calculation, one needs to integrate over a single loop momentum. The second correction comes from two-loop diagrams with two integration momenta, and so on. In Chapters 4-6, we shall deal only with tree diagrams. Methods of calculation of one-loop radiative corrections will be considered in Chapter 7.
4.3 Feynman diagrams
183
In the scalar field theory (4.2.15), external lines contribute no factors to matrix elements. Internal lines contribute iG(p), where the propagator is
When integrating over momenta, the interpretation of the pole at p 2 = m2 is important. A stable particle should be considered as a limiting case of an unstable one with mass m — iT/2 at T —> +0, so that G(p) = l/{p2 - m2 + iO). For simplicity, let's assume that there is only the triple interaction — fcp3/31 in the theory (4.2.15). We shall formulate the rules of the diagram technique in two stages. In order to obtain the contribution of n-th order of perturbation theory in / to a matrix element zM, we start by drawing n three-leg vertices (vertices are labelled, and so are the legs of each vertex). Then we attach the external lines of the initial and final particles, and join the remaining legs by internal lines in all possible ways. Each vertex gives the factor —z//3!, and each internal line gives iG(p). Finally, we divide the resulting expression by n!. It is possible to simplify these rules considerably. There are n! diagrams which differ from each other by the labelling of vertices; their contributions are equal to each other, so that we can retain only one of them, and omit the factor 1/n!. Lines can be attached to a vertex in 3! ways, and this cancels the denominator in —z//3!. Therefore, one may draw diagrams without labelling from the very beginning. Vertices give the factor — z/, and there are no extra factors. However, this simplified approach yields a wrong result, if a diagram contains symmetric parts. Let's consider, for example, the diagram in Fig. 4.3a. For each way of attaching the external lines, there are only two ways of connecting the vertices by internal lines, instead of four. Therefore, the unlabelled diagram should be multiplied by the symmetry factor | . Similarly, interchanging the upper pair of vertices of the diagram in Fig. 4.36 with the lower one produces no new labelled diagram, and the factor ^ is required again. In general, if a diagram has n symmetric parts, it should be multiplied by the symmetry factor 1/n!.
a Fig. 4.3. Symmetric diagrams
Now let's formulate the generalization of the Feynman rules to the case of an arbitrary theory. Let there be n real fields <^. They may be scalars, components of vectors and spinors, etc. A complex field can always be considered as a pair of real ones. However, complex fields are so important that we shall formulate the rules for them separately. The lagrangian is separated into an unperturbed part Lo (which is quadratic in fields) and a perturbation. The particles' propagation is determined by the
184
4 Quantum electrodynamics
unperturbed lagrangian, which can always be rewritten in the form \ip%Mij{d)ipj (by adding a full divergence). The matrix M involves the differentiation operator d. In the momentum representation, d —> -ip. An internal line in a diagram gives the factor iGij, where the propagator is Gij = [M(-ip)]7*.
(4.3.2)
External lines correspond to wave functions of initial and final particles. Wave functions are normalized by the standard condition that there are 2e particles in a unit volume. The particles' interaction is determined by the perturbation. Each term in L of the form gi1...intyi1 - >Vin gives the vertex with n labelled legs, which corresponds to the factor ig^...^- If 9 involves the differentiation operator 9, acting on ipik, then it is replaced by —ip, where p is the momentum flowing into the vertex along the corresponding line. An unlabelled vertex can be obtained from the labelled one by summation over transpositions of same-kind legs. Let's check that these rules are self-consistent. Nothing forbids us to split the quadratic part of the lagrangian \ifiMij^pj into \^piMf^j and \ipi8Mijipj, and to consider the second term as a perturbation. Then the propagator will have the form GQ = MJ"1, and there will be the two-leg vertex i5M. The diagram sequence in Fig. 4.4 gives %G = iG0 + iGoiSMiGo + iGoiSMiGoiSMiGo
+ ... = t(M 0 + SM)~\
as expected. .-[-. >
Fig. 4.4. Diagrams for a propagator
Finally, let's formulate the rules for complex fields. Their propagation is depicted by directed lines. An incoming external line gives the corresponding wave function, and an outgoing one, the conjugated wave function. An unperturbed lagrangian has the form (p*Mij(d)
' • • VinWl ' ' ' Vjm
gives the vertex with m incoming and n outgoing legs. The operator 9, acting on (fjk, is replaced by — ip\ the one acting on tp*k, by ip. Let's consider a few examples. The scalar field lagrangian (4.2.15) can be converted to the form — \ip(d2 4- m2)(p — f(f3/S\ - g(p4/4\, which gives the propagator G(p) = l/(p 2 — ra2),the three-leg vertex —z/, and the four-leg one -ig. Note that the action (4.2.1) is dimensionless, hence a lagrangian has the dimensionality m 4 . Therefore, the field (p has the dimensionality m. The coupling
4-3 Feynman diagrams
185
constant / has the dimensionality ra, and g is dimensionless. Coupling constants with higher degrees of 4 are nonrenormalizable, i.e., in fact, undefined in perturbation theory. Our next example is the massive vector field (4.2.23). Its lagrangian is L o = ^fi[(d2 + rn2)5^v — d^dv\Av (up to a divergence), so that
M^-ip) = -(p2 - m^S The matrix D
(
is inverse to M^v in the sense MilvDv\ = —8^ u. An internal line of the vector field gives the factor - i D ^ p ) , and an external one gives the unit polarization vector eM (which is orthogonal to p^). If complex polarization vectors are used (e.g., circular), then an initial particle line gives eM, and a final one gives e^ The propagator D/J)U(p) behaves as a constant at p —» oo instead of l/p2. Due to this, theories with interacting massive vector fields are nonrenormalizable. Massless vector fields (e.g., the electromagnetic one) are an exception. The matrix M^ becomes singular at m —•» 0, and the propagator does not exist (this is related to the gauge ambiguity of the field A^). However, if A^ interacts with a conserved current j M , then the longitudinal part of the propagator ~ p^pu is inessential (this will be discussed in more detail later). Any propagator of the form 6
f
2
(4.3.4)
may be used. The term — ^(d^A^)2 may be added to the lagrangian without changing the physics of the theory. This gives the propagator P2.
(4 3 5)
The most convenient gauges are the Feynman one (a = 1) and the Landau one (a -> 0). The coupling constant e of a gauge interaction, which is introduced via D^ = d^ — iezA^, is dimensionless, and a theory with this interaction is renormalizable. Finally, let's consider the electron field (4.2.27). An internal line gives the factor z5(p), where the propagator is a/ \ S{p) =
1
p-m
P+m = -o2 2o •
p - m
, , (4.3.6)
186
4 Quantum electrodynamics
An external initial electron line gives up(T, and a final one gives upa. Note that the Dirac equation has negative energy solutions, and all these states are filled in the vacuum. A state which is not filled (hole) is a positron. Therefore, production of a positron with momentum p is annihilation of the negative-energy electron with momentum —p, and vice versa, annihilation of a positron with momentum p is production of the electron with momentum -p. Hence an initial positron line gives U-pa, and the final one gives U-pa. Spin | particles are fermions. Therefore, contributions of diagrams which differ from each other by interchanging lines of identical fermions in the initial or the final state have opposite signs. Diagrams with the same topology of fermion lines (they become identical, if boson lines are eliminated) have the same sign. In addition to this, each closed fermion loop gives the factor - 1 . Let's write down the summary of the diagram technique rules in quantum electrodynamics. Propagation of a fermion is depicted by a directed solid line, and that of an antifermion by a line directed to the opposite side. Propagation of a photon is depicted by an undirected wavy line (the photon coincides with its antiparticle). Interaction is depicted by a vertex. It can be emission or absorption of a photon by a fermion or an antifermion, production of a fermionantifermion pair by a photon, or annihilation of a fermion-antifermion pair into a photon. Factors which correspond to elements of a diagram in the expression for its contribution to iM are shown in Fig. 4.5. Dirac matrix factors, corresponding to each fermion line, are written down against the arrow direction, beginning from u and ending at u. A closed fermion loop produces a 7-matrix trace.
-=«-?*
p
p C
P>
•NAAAAAAAAA/
P> •WWWWVA. jJi 1/
Fig. 4.5. Feynman rules in quantum electrodynamics
For example, Fig. 4.6a shows the only tree diagram of the process e + e~ —>
4-3 Feynman diagrams
187
fi^fi . Its contribution to iM is U-p2(T2{—ie^^)Up
1(Tl
• (—iD^ u(pi
+ P2)) * uP3(T3(—ie /yl/)u-P4(T4.
(4.3.7)
Figs. 4.6b-d show some one-loop diagrams, producing radiative correction to the matrix element. Fig. 4.7 demonstrates the signs of several diagrams of the process e~e~ —» e~e~. e
PS/I/I1
"\\Pl
d
a F i g . 4 . 6 . Diagrams of e~*~e
Fig. 4.7. Signs of diagrams of e~e~ -> e~e~
The Furry theorem states that a fermion loop with an odd number of photon legs vanishes. It follows from the fact that an even number of photons cannot transform into an odd number (and vice versa) due to C-parity conservation. It is also easy to prove it diagrammatically: when the line direction in a fermion loop with n vertices is inverted, it is multiplied by (—l) n, and the sum of these two diagrams vanishes at an odd n. Feynman rules can be formulated directly for elements of S matrix: each vertex contains the energy-momentum conservation 5-function (2?r)4<5(p/ — p^, and integration over the momentum of each virtual particle GJ4P/(2TT)4 is performed. The single J-function reflecting conservation of the total energy-momentum always appears; after singling it out, iM remains (see (4.1.1)). Integrals remain only in the presence of loops. These rules in such a form can be easily generalized to the case of an external field A^{x). A photon line of the external field is considered incoming (with a momentum &), and gives the factor Al_L(k)d4k/(27r)4. Total energy and momentum are not conserved. In the case of a stationary field, A^{k) = 27T8(ko)AtJL(k). Then the energy is conserved: the factor 27rS(ef — Si) is singled out, and iM' remains (see (4.1.23)). When one calculates iM\ an external field line gives the factor AfJt(k)d3k/(27r)3. If there is only one such a photon in a diagram, then
188
4 Quantum electrodynamics
the momentum integral becomes trivial because of the momentum conservation ^-function, leaving us with A^{pf —pi). If in the external field rest frame (with the unit vector of the time axis n^) this field is electrostatic, then A^ = n^cp, where ip is the electrostatic potential. It satisfies the Poisson equation V2
* \(p+ k — m) — (p — m)\ J p + k-m L P~m = ize [S(p + k) - S(p)].
p
P Fig. 4.8. Photon insertion into an electron line
Fermion lines either form closed loops, or go through a diagram. Thanks to the Furry theorem, we need not consider inserting the additional photon to loops. Let's consider one of the open fermion lines in a diagram for M, and construct all the diagrams for M/i(A;), which are produced by inserting the additional photon to all propagators of this line (Fig. 4.9). Here, lower photon lines connect the
4-3 Feynman diagrams
189
selected fermion line with the rest of the diagram. All terms except the extreme ones cancel each other, and we arrive at the Ward identity MM(fc)*/* = *e [M(p + Jfe) - M(p)].
(4.3.8)
This is true for a single open fermion line: we have obtained the difference of the matrix element M, in which all momenta along this line are shifted by A;, and the original M. If there are several open fermion lines, the sum of such terms appears.
rrrn ~ rrrn}
m ri m n j Fig. 4.9. The Ward identity
Now let's see what happens if the ends of a fermion line are on-shell, and are multiplied by physical polarization spinors upa and up(T. Inserting the photon into the ends of an electron line gives (Fig. 4.10) If* 1 iS(p + k)izekup(7 = -ze-— s \(p + k - m) - (p - m) up
190
4 Quantum electrodynamics
P
P Fig. 4.10. Photon insertion into the ends of an electron line
In order to check for correctness of calculation of a matrix element or its square, it is useful to check the Ward identity for each external photon. It should hold not only for the complete sum of diagrams, but also for each subset of diagrams which is produced by inserting this photon into propagators of one of the open charged particle lines. Now we can easily understand why the longitudinal part of the photon propagator (4.3.4) does not influence matrix elements. Let's select a photon line, and consider the set of diagrams in which one of the ends of the selected photon line is fixed, and the other end is inserted into all possible places. The contribution of the longitudinal part of the propagator contains k^, at this end, and the sum of these contributions vanishes. Therefore, the sum of all diagrams is the same in any gauge (in particular, it does not depend on a), while the expressions for separate diagrams, of course, differ in different gauges. Standard algorithm Now we shall formulate the standard algorithm of calculation of decay widths and scattering cross sections. In some cases, other algorithms (such as the helicity amplitudes method) may appear more efficient; however, we have no possibility of considering them here. A matrix element has the form
where y?$, iff are wave functions of the initial and final particles. For spin 0, it is a scalar, normalized by a*a — 1; for spin 1, it is a vector e M, e^eM = —1; for spin \, it is a spinor upa, up(rup(T — 2m. The modulus squared is
\M\2 = TV A (J]
(4-3.10)
where the trace is taken over all spinor and vector indices. In the spinor case, Dirac conjugation is used instead of the complex one.
4-3 Feynman diagrams
191
In order.to obtain the probability of a process with unpolarized particles, one needs to calculate
where the sum is taken over polarizations of initial and final particles, and n^ is the number of spin states. Let's define the density matrices of unpolarized particles p? = ;jr Z>i
fc\ [nfpU .
(4.3.11)
If initial particles are polarized (in general, partially), then, instead of p?, one needs to use the density matrices pi =
™ ^
^
W
(4.3.12)
|M|2(0) Tr p Let's find the density matrix of an unpolarized electron 2po = Yla uv^v^i where the sum is taken over independent spin states. The only vector in the problem is p, hence is has the form 2p0 = ap + b. The Dirac equation (4.2.29) implies (p — m)2p0 = 0 (or 2po(p — m) = 0), hence b = ma, i.e. 2p0 = a(p -f m). The normalization condition Tr2p = Yla^p^up^ = ^m gi yes a = 1. Finally, 2po=p +
ra.
(4.3.13)
A partially polarized electron is characterized by the polarization pseudovector a, which is orthogonal to p: p^a^ = 0. In the electron's rest frame, aM has only spatial components, and is equal to the doubled average spin. It is a pseudovector, therefore it must appear in the density matrix p together with 75. The form p = p o (l + 0750) satisfies the Dirac equation (4.2.29) and its conjugate, because (1+C75a) commutes with p0. As follows from (4.2.39), a Dirac spinor transforms at an infinitesimal rotation in the rest frame as u -> (l + f 7o7s7 • 80) u; therefore the doubled average spin is a = u+70757?//(^+u) = T** PTST/T** PDue to the Lorentz invariance, the equation aM = Tr p757/x/Tr p holds in any frame. We obtain c = — 1, and finally \
(4.3.14)
192
4 Quantum electrodynamics
In considering processes involving positrons, we need to use negative energy wave functions xjj-po- — U-V(Te%vx. They satisfy the equation (p + m)u-p(T = 0 (or u-pa(p + m) = 0), and are normalized as U-p(T^^u-p(T = 2pM (or u_p(Tu-pcr = —2ra at m ^ 0). Their density matrix is 2po = (p - m),
P = g (P -
m
) ( X ~ 75a).
(4.3.15)
The density matrix of an unpolarized photon is 2pfxu = Ee^e*, where the sum is taken over two independent polarizations. The Ward identity allows us to use a simpler form. Let the photon fly along the z axis: k^ = e(e^ + efy. The basis vectors e^ and e 2 may be chosen as two independent transverse polarizations: 2pilv = e^el + e 2 e 2 . However, two nonphysical polarizations (longitudinal and timelike ones) may be included into this sum, too: 2pilv = e e
lb l + el>el + e^,el ~ e\e% ~ ~<W* ^n^s 1S D e c a u s e t h e Ward identity ensures
i.e. emission (or absorption) probabilities of the longitudinal photon and the timelike one cancel each other. It is most convenient to use the density matrix
2p% = - < W
( 4 - 3 - 16 )
The density matrix of a partially polarized photon p^v — eMe* = p*^ is defined up to p^ -> p^ + f^kj, + /^ifc^. With any choice of the basis vectors e^, e^, orthogonal to A;^, it can be cast into the form containing only components 1 and 2:
where & are called Stokes parameters. Its antisymmetric part describes circular polarization; £2 is the average helicity (pseudoscalar). Parameters £1, £2 describe linear polarization along axes 1 and 2, and along the diagonals; they depend on the choice of the basis vectors, but £2 = £jf + £f is invariant. By choosing the basis, one can diagonalize the symmetric part:
Thus, ps is characterized by the scalar £ — degree of linear polarization, and its direction e^. Let's formulate the algorithm for calculation of |M| 2 . First of all, we draw all the diagrams for M . Each term in \M\2 comes from some diagram in M and some diagram in M*. We draw a dashed line, and draw the diagram for M to the left of it in such a way that the final particles' lines look to the right. To the right of the dashed line, we draw the mirror image of the diagram for M*. We join the corresponding external lines of these two diagrams. This construction is called a squared diagram. If we have n diagrams for M, then there will be n2 squared diagrams for \M\2. For example, Fig. 4.11 shows the only tree diagram of the process e + e~ —> /i + /i~, and the corresponding squared diagram.
4-4 Scattering in an external field
193
Fig. 4.11. Diagram and squared diagram of the process e + e~ —>• /i + //~
On the left-hand side, elements of a diagram produce expressions according to the ordinary rules, while on the right-hand side they give conjugated expressions (Dirac conjugated, in the spinor case). A final particle's line, crossing the dashed line, produces rifp® (summation over polarizations); an initial particle's line gives p° (averaging). If there are n identical particles in the final state, then various transpositions of their momenta form a single state, which should be included once in the calculation of the total decay width or cross section. Instead of this, one may integrate over momenta independently, and then divide the result by the number of transpositions n!. There is a close relation between calculation of squared diagrams and of loop diagrams for a matrix element (the dispersion relations, which are not discussed in this book). Dividing a squared diagram with identical particles by n! corresponds to the symmetry factor in the loop diagram.
4.4 Scattering in an external field Electron scattering Electron scattering in an external field is depicted in Fig. 4.12. We shall start by calculating the lepton tensor TfjLl/(pi,p2) — the squared scattering matrix element, summed over polarizations of both the initial and the final electrons, with free photon polarization indices. This object is more general than we need now, but it can be used in many other problems, too. The lepton tensor contains e2 = 4TTCV; the factor a is not explicitly represented in the program. The factor 4 turns \ Tr (calculated by REDUCE) into Tr.
V Fig. 4.12. Scattering in an external field
For convenience, the operator V for the vertex and the operator r for the doubled density matrix are introduced. Then, the lepton tensor Tl is calculated.
194
4 Quantum electrodynamics
To this end, the statement for a l l . . . saveas . . . is used. It defines the substitution rule for a l l . . . l e t . . . = . . . , where the right-hand side is the value of the last evaluated expression (the value of ws). The expression s q r t ( k l . k l ) is used in the role of mass, so that Tl can be used for particles with different masses. After that, the program checks the Ward identity for T ^ . The program calculates the squared matrix element \M'\ 2 (§ 4.1), averaged over initial electron polarizations (this gives the factor \). The initial and final electron momenta are Pi, P2, and the 4-velocity of the field (and its source) rest frame is n. In this frame, the electron energy before and after the scattering are both equal to e. The variable t is t = q2, q = pi — p2. If the field is created by a point-like charge e, then its potential is A^ = —en^/t; in MM, the factor a2 is not explicitly represented. If the charge is not point-like, its squared form factor should be included. Using the formula (4.1.25), the program obtains
It is easy to obtain the cross section in the rest frame from the invariant one:
dCl
4(£2 — m 2 ) 2 sin4 f
4ra2v4 sin4 f
In the ultrarelativistic limit, da
a 2 p 2 cos 2 f
4e2 sin4 f
(4.4.3)
In this case, the electron cannot scatter backwards. Let's direct the z axis along the direction of motion of the initial electron. Projection of the orbital angular momentum onto this axis is zero. Therefore, the spin projection onto this axis cannot change, i.e. the electron helicity has to change its sign. It is forbidden for an ultrarelativistic electron. In the nonrelativistic limit, we recover the Rutherford formula ^ = ^ ! (444) dft 4m2?;4 s i n 4 f In the initial electron rest frame, the incident infinitely heavy field source transfers some energy to the electron. We shall denote it by mS. The cross section da/d5 is used for calculation of energy losses of a heavy charged particle, moving through matter containing electrons, which are (almost) at rest. The program obtains 2 da 2 22-(l-vv )S (A . K. ; -jj = TTOV . 4.4.5) 22 22 22 dS mvS
4-4 Scattering in an external
field
195
input_case nil$ °0/ Electron scattering in an external field 0/ /o
"""
™" "~
mass pl=m,p2=m,n=l; mshell pl,p2,n; °/0 pl,p2 - momenta let pl.p2=nT2-t/2,n.pl=E,n.p2=E; '/• n - 4-velocity of rest frame operator V; for all p let V(p)=g(f,p); % vertex vector kl,k2,el,e2; let k2.k2=kl.kl; nospur f; operator r; °/c doubled density matrix g(f,kl)+sqrt(kl.kl)$ for all kl saveas r(kl); spur f; operator Tl; */. lepton tensor - alpha is implied 4*4*pi*r(k2)*V(e2)*r(kl)*V(el);
16 7r(ei.e2 k\.k\ — ei.e2ki.k2 + ei.kie2.k2 + ei.k2e2.ki) for a l l kl,el,k2,e2 saveas Tl(kl,el,k2,e2); Tl(pl,p2-pl,p2,e2); °0/ Ward identity 0 °/0 averaged square of matrix element, alpha~2*formfactor~2 implied on gcd; MM:=Tl(pl,n,p2,n)*4*pi/(2*t~2)$
sigma:=MM/(16*pi*(E~2-nT2)); c l e a r MM; °/, d sigma / dt
+ t) a
2
2
'~ t (E - m 2 ) •/, rest frame, s=sin(theta/2) trf:=-4*(E^2-m^2)*s~2$ rf:=(E^2-m^2)/pi$ on factor; sig:=sub(t-trf,sigma)*rf; % d sigma / d Omega _ - (E2 s2 - E2 - m2 s2) S%9 ''~ 4(E + m)2{E - m)2s4 off factor; sub(m=0,sig); °0/ ultrarelativistic limit 4E2s4 Ev:=m/sqrt(l-v~2)$ °0/ v - velocity sig:=sub(E=Ev,sig)$ on factor; sig; off factor; (sv + l)(sv - l)(v + l ) ( i ; - 1) 4 m2 s 4 v 4 */, nonrelativistic limit procedure NonRel(x); begin scalar xl,x2,w,z; xl:=num(x); x2:=den(x); w:=l; while (z:=sub(v=0,xl))=0 do « xl:=xl/v; w:=w*v » ; xl:=z; while (z:=sub(v=0,x2))=0 do « x2:=x2/v; w:=w/v » ; x2:=z; return xl/x2*w end$
NonRel(sig); clear sig; 1 2 4 4 4m s v
196
4 Quantum electrodynamics
5J i n i t i a l electron rest frame, m*delta - energy transfer td:=-2*nT2*delta$ fd:=2*nT2$ sub(t=td,E=Ev,sigma)*fd; n(Sv2 -5 + 2) 52 m2 v2
clear sigma; off gcd; Polarization effects Let the initial electron be polarized. We want to find the cross section and the final electron polarization. To this end, we need the lepton tensor with polarizations Tfll/(pi,ai,p2,#2) (it contains doubled density matrices, so that at &1 = 0, a2 = 0 it reduces to the tensor obtained earlier). The initial and final electron polarizations are expanded into components: longitudinal, transverse in the scattering plane, and perpendicular to the scattering plane. In order to make finding corresponding unit vectors easier, the procedure ort is written, which constructs an orthonormalized system from a given list of vectors. Vectors in the list are given in the form v . l . The first one must be timelike. It is simply divided by its length, so that its square becomes 1. Components along already constructed basis vectors are subtracted from the following vectors, and the results are normalized so that their squares become —1. The variables yl, y2 are used to keep track of which vectors are timelike or spacelike. The procedure epol replaces polarization vectors by their expansions in these bases. The procedure pol calculates and prints the cross section, summed over final electron polarizations, and returns its polarization in the form of a three-component list. As you can see, the initial electron polarization does not influence the cross section, and the final electron polarization does not vanish only when the initial electron is polarized. The component of polarization, perpendicular to the scattering plane, does not change. For the other components, the program finds z
i
(cost? + v2sin2 f )z\ + y/l
2
-
-
-v2mwdz\
- 2 $
»
2
—y/1 — v2sini9z] + (costf -f v 2 sin %)zk
o
-,z 2
~~
•*•
1 _
2
2
• 2
(4.4.6)
*>
<§_
In the ultrarelativistic limit, z\ = z\^z\ — z\. The first equality reflects conservation of helicity of ultrarelativistic fermions in electromagnetic interactions. In the nonrelativistic limit, z\ = z\ cos d + z\ sin $, z\ = —z\ sin # + z\ cos $, i.e. the polarization vector is conserved, and its components are changed because of the direction of motion is rotated by the angle $. This is natural, because the spin-orbit interaction is absent in this limit. °/, polarization effects y vector al,a2; let kl.al=0,k2.a2=0,pl.al=0,p2.a2=0; nospur f; r(kl)*(l-g(f,a,al))$ for all kl,al saveas r(kl,al);
4-4 Scattering in an external
field
197
°/ 0 l e p t o n t e n s o r with p o l a r i z a t i o n spur f; 4 * 4 * p i * r ( k 2 , a 2 ) * V ( e 2 ) * r ( k l , a l ) * V ( e l ) ; 16 7T ( — a1.a2e1.e2fc1.fc1 -h a1.a2e1.e2fc1.fc2 - a1.a2e1.fc1e2.fc2 ~ a1.a2e1.fc2e2.fc1 H-a1.e1a2.e2fc1.fc1 — a1.e1a2.e2fc1.fc2 + a1.e1a2.fc1e2.fc2 + a1.e2a2.e1fc1.fc1 — a1.e2a2.e1fc1.fc2 + ai.e2a2.fciei.fc2 + a1.fc2a2.e1e2.fc1 + ai.fc 2 a2.e 2 ei.fcia1.fc2a2.fc1e1.e2 + ei.e2fci.fci - e1.e2fc1.fc2 + e1.fc1e2.fc2 + ei.fc2e2.fci — y/ki.ki
eps(ai, ei, e2,fci)i + y/ki.ki
eps(ai, ei, e2,fc2)i ~
y/ki.kx
eps(a 2 ,ei,e 2 ,fci)i + \/ f c i-*i eps(a 2 ,ei,e 2 ,fc2)*)
for all kl,al,el,k2,a2,e2 saveas Tl(kl,al,el,k2,a2,e2); Tl(pl,al,p2-pl,p2,a2,e2); % Ward identity 0 on gcd; MM:=Tl(pl,al,n,p2,a2,n)*4*pi/(4*t~2)$ sigma:=MM/(16*pi*(E~2-nT2)); clear MM; °/, 2*pi d sigma / dt d phi cr :=
t -f 4ai.na2.piE 2a1.p2a2.p1 4- 4 £ 2 + t))/(2t2(E2 - m 2 )) % some properties of abs let abs (m) =m,abs (s) =s, abs (M) =M, abs (v) =v; for all x let abs(sqrt(x))=sqrt(x); for all x,y let abs(x*y)=abs(x)*abs(y); for all x,y such that x neq 1 and remainder(y,x)=0 let abs(x+y)=abs(x)*abs(l+y/x); for all x,y,z such that remainder(z,x)=0 let abs (x*y+z) =abs (x) *abs (y+z/x); for all x,n such that evenp(n) let abs(x~n)=x~n; array epl(3) ,ep2(3); °/P basis polarization vectors °/0 normal to the scattering plane vector 1; et:=eps(n,pl,p2,l)$ index 1; x:=et*et$ remind 1; epl(3):=ep2(3):=et/sqrt(-x); clear x,et; ,o\
/ON
-2eps(f,n,pi,p 2 )
V
4m2t - t2
% Constructing orthonormalized system from a list of vectors 10 °/0 given in the form v,l; the first one must be timelike procedure ort(10); begin scalar 12,13,x,w,yl,y2; 12:=O; yl:=0; for each v in 10 do
198
4 Quantum electrodynamics
«
y2:=0; for each w in 12 do « x:=v*w; index 1; x:=x; remind 1; y2:=y2+l; v:=v-x*w*(if y2=yl then 1 else -1) x:=v~2; index 1; x:=x; remind 1; yl:=yl+l; v:=v/sqrt(x*(if yl=l then 1 else -1)); 12:=v.l2
return reverse(12) end$
l e t sqrt((-4*E~2*t+4*nT2*t-t"2)/(E~2-nT2)) = sqrt (-4*E~2*t+4*nr 2*t-t~2)/sqrt (E~2-irr2) ; x:=ort({pi.1,n.1,p2.1}); x := .p\ l.nm2 — l.pi E m ' JE2 — m2m ^
m2(l.nEt
- 2Lp1E2 + 2Lp1m2 - LPlt + 2Lp2E2 + 4m2t - t2(E2 - m2)
2Lp2m2) 1
epl(l):=~second(x)$ epl(2):=third(x)$ x:=ort({p2.1,n.l,pl.l})$ ep2(l):=-second(x)$ ep2(2):=-third(x)$ clear x; °/0 1 - longitudinal polarization, 2 - in the scattering plane °/0 Substituting ex (in the form v.l) for ax in x procedure epol(ax,ex,x); begin scalar xO,xl,el; xO:=sub(ax=O*ax,x); xl:=x-xO; index 1; el:=ex*ax.l; remind 1; index ax; xl:=xO+xl*el; remind ax; return xl end$ operator z; factor z; °/0 polarization components clear pl.al; sigma:=epol(al,for i:=l:3 sum z(pl,i)*epl(i),sigma)$ clear p2.a2; sigma:=epol(a2,for i:=l:3 sum z(p2,i)*ep2(i),sigma); a :=
- AE2m2 + E2t - m21) + 4
- 4E2m2 + E2t + m2t) +
4m 2 f - t2 2 \/-4E2t
+ 4m2t - t2z(pu 1) z(p2,2)
z(pul)z(p2,l)7r(4E4
- 4E2m2 + E2t + m2
n(4EA - 4E2m2 + E2t - m2t))/ (2t2(E4 - 2E2m2 + m4))
4-4 Scattering in an external field
clear sqrt((-4*E~2*t+4*nT2*t-t~2)/(E~2-m~2)); °/, polarization components have been substituted °/o Calculating polarization of the final fermion procedure pol(sig,n); begin scalar x,sigO; % cross section, summed over polarizations for all j let z(n,j)=0; write sig0:=2*sig; for all j clear z(n,j); % polarization of the final electron return for i:=l:3 collect « x:=df(sig,z(n,i)); for all j let z(n,j)=0; x:=2*x/sig0; for all j clear z(n,j); x end$ x:=pol(sigma,p2); clear sigma;
s
^° := *VP-+J)
x :=
{(2y/-4E2t z(Pl,l)(4E4 14
+ 4m2t - t2z(pu2)Em + - 4E2m2 + E2t + m2t))/
- 4E2m2 + 14
2 yj-AE'H
E2t-m2t),
- 4E2m2 + E2t + m2t)-
+ 4m2t - t2 z(pu
- AE2m2 + E2t -
l)Em)/
m2t),
let sqrt(l-s^2)=c,abs(E's2-m's2)=E's2-m"2,abs(v^2-l)=l-v"2; x:=for each w in x collect sub(t=trf,w); x := f -2z(p1,2)Ecms + z(pul){E2s2 ~ E2 + m2 s2) 2 2 2 2 2 I E s - E - m s ' z(pu2)(E2s2
- E2 + m2s2) + 2z(pul)Ecms 2 2 2 2 2 Es - E - ms ' Z^Pu' clear sqrt(l~s^2); °0/ rest frame; c = cos(theta/2) for each w in x collect sub(m=0,w); °o/ ultrarelativistic limit x:=for each w in x collect sub(E=Ev,w); °0/ v - velocity x :=
199
200
4 Quantum electrodynamics
f- 2 I
w-
« 2 + lz(pi,2) cs +
s2z>2 - 1
2 2 2 1)(- s z; + 2 s - 1)
2 2 2 2)(- -s t; + 2s -- 1) 4- 2V - • y
»- 1 «(Pi
2
-
for each w in x collect sub(v=0,w); °/o nonrelativistic limit {2z(pi,2)c5 + z(pi,l)(-2« 2 + 1), z(pu2)(-2s2 + 1) - 2s(pi,l)cs, *(pi,3)} clear x; Decay ijj -> e + e ~
A neutral vector meson (for example, ?/>) can transform into a virtual photon. The transition vertex has the form iM/e M , where M is the meson mass, and / is a constant with the dimensionality of mass, characterizing the meson. Therefore, these mesons can decay into e+e~ (Fig. 4.13). Summation over meson polarizations is performed using the formula ^e^e^ = p^p^/M2 — 8^u\ the first term does not contribute due to the Ward identity. The program calculates \M\ 2 and F (the factor a 2 / 2 is implied). The procedure ph2 from the package kpack is employed to find the two-particle phase space. The same formulae with the replacement / —> M describe the decay 7* —> e + e~.
92 Fig. 4.13. ip ->• e
The result is 1
47ra 2 / 2 (M 2 + 2m2)VM2 - 4m2 _ ira2f2v(d - v ~ 3M4 3m
(4 4 7)
' -
In the ultrarelativistic limit (relevant for ip —> e+e~, fi+fj,~),
5jf
(4.4.8)
In the nonrelativistic limit, r = ^ ^ . (4.4.9) m The threshold behaviour ~ v (4.4.9) is characteristic for production of a particle pair in the 5-wave: one may put v = 0 in the matrix element, while the phase space produces the factor v.
4-4 Scattering in an external
field
201
°/o Decay p s i -> e+e-
y#
in Mkpack.red"$ mass ql=m,q2=m; mshell ql,q2; let ql.q2=MT2/2-nT2; °/, M - psi mass °/o averaged square of matrix element, f(psi)~2 implied index 1; MM:=Tl(ql,l,-q2,l)*4*pi/(3*M~2)$ remind 1; Gamma:=MM/(2*M)*ph2(M,m,m); clear MM; 7, decay width
_ 4VM 2 - 4m 2 7r(M 2 + 2m 2 ) ~ 3M 4 sub(m=0, Gamma); °/c u l t r a r e l a t i v i s t i c limit 4?r 3M Mv: =2*m/sqrt (l-v~2) $ l e t sqrt(-l/(v~2-l))=sqrt(l-v~2)/(l-v~2); procedure dummy(x); x$ Gamma:= sub(M=Mv, Gamma) $ Gamma:=dummy (Gamma) ; °/« v - velocity :
V-v2
_
*~
+ IKV{-V2
4- 3)
3m
clear sqrt(-l/(v~2-l)) ; NonRel (Gamma); °/0 nonrelativistic limit 7TV
m
Polarization effects Let the meson have the density matrix p^ = £>%&]> Then the decay probability, summed over e + and e~ polarizations, has the form dT
4?r— = PijiASij -f BniUj) = T^e- n\
2
+ rj_(l - |e • n| 2 ),
(4.4.10)
where n is the unit vector in the direction of motion of e~. Averaging over polarizations gives
r=Q^i.
(4.4.U)
In order to find the two quantities Ty, F_L, which completely characterize the angular distribution, it is enough to consider e\\n and e_Ln. Let's perform the calculation in the specific basis no, ni, n2, n3, where no is the %j) velocity, n\ is the unit vector along the decay axis, and n2,3 are arbitrary. The vectors el and e2 mean e and e*, or the indices i and j , over which the result is contracted with pij. The program obtains _ 87ra/mVM
4m2
_ Sna2f2VM2 - 4m2 M2 ~
(
m
}
202
4 Quantum electrodynamics
In the ultrarelativistic limit, 11
'
±=
M
'
(4.4.13)
This follows frohi conservation of helicity of ultrarelativistic fermions: e + and e~ have opposite helicities, i.e. the same spin projections onto the decay axis. Projection of the orbital angular momentum onto this axis is zero. Hence the meson has spin projection ±1 onto this axis, i.e. its polarization is transverse. Iri the nonrelativistic limit, the angular distribution is isotropic:
% polarization effects clear ql.ql9q2.q29ql.q2; mass nO=l,nl-i,n2=i,n3=i; mshell nO,nl,n2,n3; °/« basis let nO.nl-0,nO.n2=0,nO.n3=0,nl.n2=0,nl.n3=0,n2.n3=0, eps(nO,nl,n2,n3)=l; ql:=M/2*iiO+sqrt(M~2/4-nT 2)*nl$ q2:=M/2*nO-sqrt(M~2/4-m~2)*nl$ vector el,e2,al,a2; let el.n0=0,e2.n0=0, al. nQ--*sqrt (l-4*nT 2/M~2) *al. nl, a2.nO= sqrt(l-4*nT2/M~2)*a2.nl; MM:=-Tl(ql,al,el,-q2,-a2,e2)*4*pi/(M~2)$ Gamma:-MM/(2*M)*ph2(M,m,m)$ clear MM; Garni:=sub(a2=0*a2,Gamma)$ GamO:=sub(al=0*al,Garni)$ GamL:=sub(el=nl,e2=nl,GamO); */, longitudinal
„
GamL
t
2 2 2 ^ SVM -_ 4m m 7r
l e t el.nl=0,e2.nl=0,el.e2=-l; GamT:=GamO; °0/ transverse ^ m 2\/M 2 - Am27T _ GamT S!e clear el.nl,e2.nl,el.e2; sub(m=0,GamL) ; sub(m=0,GamT); °/0 ultrarelativistic limit 0
~M
let sqrt(-l/(v"2-l))=sqrt(l-v*2)/(l-v*2); GamL: =sub (M=Mv, GamL) $ GamL: =dummy (GamL);
_ V~V2 -f l7T^(-i' 2 -f 1) m GamT:=sub(M=Mv,GamT)$ GamT:=dummy(GamT);
GamT := m
4-4 Scattering in an external
clear sqrt(-l/(v~2-l)); NonRel(GamL); NonRel(GamT); °/0 nonrelativistic
field
203
limit
7T V
m 7TV
m c l e a r GamO,GamL,GamT; Let's find the polarization of the final e~ in the decay of a polarized t/j. We are not going to consider correlations of e~ and e + polarizations, therefore we may put a 2 = 0. The polarization vector a\ is expressed via its longitudinal and transverse components. The choice of the axes 2 and 3 is arbitrary, therefore it is enough to consider the polarization along the axis 2 only. The electron is polarized only when the circular polarization a^ = ieijkPjk of the ?/>-meson is present (a = ie x e*)\ 211 = 11
.-!! - , 9 ,
1 — \v - e\z
^± =
,_ • H|9 •
(4.4.15)
1 — \v • e\z
ell:=second(ort({ql.l,n0.1})); °/ 0 longitudinal polarization 1 -Z.noAf2 + 4/.n 0 m 2 - \/M 2 2%/M2 el2:=second(ort({q2.1,n0.1}))$ °/ 0 longitudinal polarization 2 clear al.nO,a2.nO; Gaml:=epol(al,z(l,l)*ell+z(l,2)*n2.1,Gaml)$ x:=pol(Garni,1); clear Garni; 4\/M 2 — 4m 2 7r( —ei.e 2 M 2 - ei.nieo.niM 2 + 4ei.nie2.nim 2 ) x := /
- eps(ei,e 2 ,n 0 ,ni)M 2 h e1.n1e2.n1M2 -
2 eps(ei,e 2 ,no,n2 / ^^ .... , 2 2 + e1.n1e2.n1M — 4ei.nie2.nim ' / for each w in x collect sub(M=Mv,w); clear x; - eps(ei, e2, n 0 , ni) i eps(ei, e2, n 0 , n2) i ( - v2 ._ e1.n1e2.niv2 ' V - ^ 2 + l(ei.e 2 Now let's consider the inverse process e + e~ -> ip. Its cross section can be obtained from r ( ^ —> e + e~) by the formula (4.1.13). Using the freedom of choice of the axes 2 and 3, let's direct the axis 2 along the bisector of the transverse polarization vectors of e~ and e + . The cross section, summed over ip polarizations, is
M 2 + 2m2 + (M2 - 2m2)z\z2 - 2m2z\ • z2 a = 8n*a f \ " ^LS(y/i-M). M4y/M4 - Am2 2 2
(4.4.16)
204
4 Quantum electrodynamics
In the ultrarelativistic limit, • 3a2f2(l
4- ZnZ2)
:
-M).
(4.4.17)
This is natural, because only e~ and e + with opposite helicities annihilate. In the nonrelativistic limit,
This formula reflects the fact that the annihilation takes place only in the state with zero orbital angular momentum and the total spin 5 = 1. °/0 cross section, delta(sqrt(s)-M) implied sigma: =2*pi~2/(M~2/4-m~2)/4*Gamma$ clear Gamma; l e t cos(theta) "2=(1+cos(2*theta))/2, sin(theta)"2=(1-cos(2*theta))/2, cos(theta)*sin(theta)=sin(2*theta)/2; sigma:=epol(al, z(1,1)*ell+z(1,2)*(cos(theta)*n2.1+sin(theta)*n3.1),sigma)$ sigma:=epol(a2, z(2,1)*el2+z(2,2)*(cos(theta)*n2.1-sin(theta)*n3.1),sigma)$ clear el.nO,e2.nO; l e t el.e2=-nl.el*nl.e2-n2.el*n2.e2-n3.el*n3.e2; sigma:=sigma$ clear el.e2; index el,e2; sigma:=-el.e2*sigma; remind el,e2; a := ( - 16 z% 2) z{\, 2) cos(2tf) m2 7T3 + 8 *(2,1) z(l, 1) TT3 (M2 - 2 m2) + + 2m 2 ))/( V / M 2 - 4m 2 M 4 ) sub(m=0,sigma); 8TT 3 (M 2
)Z(1,1)TT 3 + 8TT3 let sqrt(~l/(v^2-l))=l/sqrt(l-v's2) ; sigma: =sub(M=Mv,sigma)$ sigma:=dummy(sigma); a :=
(y/-v2
+ 1^(2,2)^(1,2) cos(2i?)7T3(-?;4 + 2v2 7T3 (
-
clear sqrt(-l/(v~2-l)); NonRel(sigma); clear sigma; - z ( 2 , 2 ) ; g ( l , 2 ) COS(2^)TT 3 + ^(2,1) z(l, 1) TT3 + 3TT3
4-4 Scattering in an external
field
205
Problem. Derive the density matrix of the produced i/>. Demonstrate that in the ultrarelativistic limit
p =
The fact that only t/;-mesons with the spin projection ±1 onto the axis 1 are produced, i.e. the density matrix has only components in the 2-3 subspace, follows from the helicity conservation. Longitudinal polarizations of e~ and e + produce a circular meson polarization; transverse ones produce linear polarization along the axis 2 with the probability (1 + z1j_z^_)/2, and along the axis 3 with the probability (1 — Z-J_Z'J_)/2. This fact was used in § 3.4.
Pion scattering in an external field The vertex of charged pion (or any other spinless meson) interaction with an electromagnetic field has the form V^ = e[F{q2)p^ + F-(q2)qfJ], p = pi +£>2> q = pi —p2> The current conservation V^q^ = 0 leads to F-(q2) = 0. The meson tensor T^(pi,p2)-) similar to T ^ , is introduced in the program (the factor aF2 is implied in it). The cross section is da_ dt =
t2(e2 - m 2 )
'
In the rest frame, da _ a2F2p2e2 _ a2F2p2(1 - v2) 2 2 2 4 dQ, ~ 4(e - m ) sin f ~ 4m2v4 sin 4 f
(4.4.20)
In the ultrarelativistic limit, da
a2F2p2
(4.4.21)
In the nonrelativistic limit, the Rutherford formula (4.4.4) (with the additional factor F 2 ) is reproduced. % Pion s c a t t e r i n g in an e x t e r n a l f i e l d operator Vm; for all kl,k2,l let Vm(kl,k2,l)=(kl+k2).1; % vertex operator Tm; °/« meson tensor - alpha*formfactor~2 implied for all kl,el,k2,e2 let Tm(kl,el,k2,e2)= 4*pi*Vm(kl,k2,el)*Vm(kl,k2,e2); Tm(pl,p2-pl,p2,e2) ; °/c Ward identity 0 on gcd; MM:=Tm(pl,n,p2,n)*4*pi/t~2$
sigma: =MM/(16*pi*(E~2-nT2)); c l e a r MM; 7, d sigma / dt
206
4 Quantum electrodynamics
° :~ t2 (E2 - m2) sigma:=sub(t=trf,sigma)*rf$ on factor; sigma; '/, d sigma / d Omega m)2{E - m)2s4 off factor; sub(m=0,sigma); °0/ u l t r a r e l a t i v i s t i c limit 1 4E2s4 sigma: =sub(E=Ev,sigma); °/» v - v e l o c i t y 2 _ -v -f 1 # *~ Am2s4v4 NonRel(sigma); clear sigma; °/, nonrelativistic limit 1 4m2s4v4 Decay ip —•» TT+TT"
The decay via the strong interaction is suppressed because of the G-parity nonconservation, and the electromagnetic mechanism (Fig. 4.13) dominates. The same formulae, with the replacement / —> M, describe the decay 7* —> 7r+7r~. The program obtains r N _ 7ra2f2F2{M2 - 4m 2 ) 3 / 2 _ 7ra2f2F2v3(l - v2) " 6^ T ~ Ud4 In the ultrarelativistic limit,
'
-1 " (4.4.22)
At F = 1, this is four times smaller than the decay rate into e + e~ (4.4.8). In the nonrelativistic limit,
The ^-meson polarization vector e must appear in the matrix element M linearly. The only vector by which it can be multiplied is the 7r+-meson velocity v: M ~ e - v. Therefore, only Ty is nonzero. At v —> 0, the decay rate F contains v2 from | M | 2 , and v from the phase space, i.e. F ^ v3 (4.4.24). Such threshold behaviour is characteristic for all processes of pair production in the P-wave. °/0 Decay p s i -> p i + p i °/, f (psi)~2*formfactor~2 implied index 1; MM:=-Tm(ql,l,-q2,l)*4*pi/(3*M~2)$ remind 1; Gamma:=MM/(2*M)*ph2(M,m,m); c l e a r MM; •/. width
4-4 Scattering in an external
field
207
- 4m 2 7r(M 2 - 4m 2 )
_ VM
sub(m=O,Gamma);
70 u l t r a r e l a t i v i s t i c limit
7T
3M let sqrt(-l/(v~2-l))=sqrt(l-v~2)/(l-v~2) ; Gamma: =sub(M=Mv,Gamma)$ Gamma:=dummy(Gamma); °/ 0 v - velocity yj — V2 +
6m clear sqrt(-l/(v~2-l)) ; NonRel(Gamma) ; clear Gamma; °/0 nonrelativistic limit nv3
6m mass ee=i; mshell ee; l e t n0.ee=0; MM:=Tm(ql,ee,-q2,ee)*4*pi/M~2$ Gamma:=MM/(2*M)*ph2(M,m,m); clear MM; \JM2 — 4 m2 ee.n2 n (M2 — 4 m2) GamL:=sub(ee=(ql-q2)/sqrt(-(ql-q2). (ql-q2)) ,Gamma); °/ 0 longitudinal ^ y/M2 - 4m 2 7r(M 2 - 4m 2 ) r GamL := —~ M4 l e t ee.nl=0; GamT:=Gamma;
70 transverse
GamT := 0 clear ee.nl,Gamma,GamL,GamT; off gcd; Proton scattering in an external field The vertex of proton (or any other spin \ baryon) interaction with a photon has the form V^ = eT^F^iii, where T^ can be expanded in the basis 7-matrices. All structures involving p\ or p2 can be reduced to expressions with fewer 7-matrices, using commutation and the Dirac equation. As a result, three structures remain: V[i — (Pi +P2)/z, (7/x = (Pi — ^2)//, and 7^. The second one is forbidden by the current conservation V^q^ = 0. Therefore,
(4.4.25) (because afJiUqJ//(2m) = 7M — pfJL/(2m) when sandwiched in u 2 ...wi). In the first form, — F2 corresponds to electric charge (as in the pion case), and F\ + F2 to electric charge with the normal (Dirac) magnetic moment (as in the electron case). In the second form, Fi corresponds to electric charge with the normal magnetic moment, and F2 to the anomalous magnetic moment. In the third form, F\ corresponds to electric charge, and F\ + F2 to magnetic moment.
208
4 Quantum electrodynamics
As we shall see shortly, a different choice of form factors appears to be more convenient: J- /x = \*e - Jbm)
2
4m2
*~ -^m7/i
/
p2
2m
(4.4.26) because they do not interfere in the cross section. The program calculates the baryon tensor TpV(pi,j>2). With the choice of form factors (4.4.26), the interference term FeFm is absent in it. The cross section is
__/iufj
-m2) + t]Fl(t)_
w
(4427)
In the rest frame,
In the ultrarelativistic limit, only the magnetic form factor contributes:
da _ a2p2Fl cos2 f dtl 4 £ 2 sin 4 f '
(4.4.29)
while in the nonrelativistic limit, only the electric one: da _ o?p2Fl dtt - 4m2v4 sin4 |
(4.4.30)
(this cross section can be obtained from the Rutherford formula (4.4.4) by multiplying it by Fe2). °/o Proton scattering in an external field 0/
operator Vb; °/, vertex: g = Fm/Fe, Fe implied factor g; nospur f; vector p; p:=kl+k2$ (l-g)*2*sqrt(kl.kl)/p.p*p.l+g*g(f,l)$ for all kl,k2,l saveas Vb(kl,k2,l); clear p; spur f; operator Tb; % baryon tensor - alpha~2*Fe~2 implied 4*4*pi*r(k2)*Vb(kl,k2,e2)*r(kl)*Vb(kl,k2,el); (ei.e2ki.kl
- ei.e2ki.kl - e1.fe1e2.fe1fe1.fe1
ei.fe2e2.feifei-fe2— ei.^2 62-^2 fei-fei) +
4-4 Scattering in an external
field
209
\ 4- e\.k\e2-k2 + ei.k2e2.ki + e (ki.ki + fci.fo) for all kl,el,k2,e2 saveas Tb(kl,el,k2,e2); Tb(pl,p2-pl,p2,e2); °/, Ward identity 0 on gcd; MM:=Tb(pl,n,p2,n)*4*pi/(2*t~2)$ sigma:=MM/(16*pi*(E~2-nT2))$ clear MM; °/0 d sigma / dt sign:=num(sigma)$ sigd:=den(sigma)$
factor g; sign; remfac g; 7, numerator 2 2 2 2 2 g 7rt(-4E + 4m - t) + 16£ ra 7r on factor; sigd; off factor; clear sign,sigd; °/« denominator (E 4- m)(E - m) (Am2 - t) t2 sigma:=sub(t=trf, sigma) *rf$ °/0 d sigma / d Omega sign:=num(sigma)$ sigd:=den(sigma)$
factor g; sign; remfac g; °/0 numerator 2 2 4 2 4 2 2 2 2 2 4 2 g s (-E s + E + 2E m s - 2E m - m s + m4) + E2m2 on factor; sigd; off factor; clear sign,sigd; °/0 denominator 4 (£ 2 52 - m2 s2 + 77i2) (£ + m)2 (£7 - m)2 s4 sub(m=0,sigma); factor g; sigma:=sub(E=Ev,sigma); remfac g; _ g2s2v4(s2v2 - s2 - v2 + 1) + v4 - 2v2 + 1 a ''~ 4m2s4v4{s2v2 - v2 + 1) NonRel(sigma); clear sigma; off gcd; 1 4 m2 s4 v4 Problem. Demonstrate that the components of the final proton polarization in a polarized proton scattering are 2
zx-
_ az\ + bz\
,
2
z2-
V2 cog 2
_ -bz{ + az\ ,
t\
+
FeFmv2(l
sin2 0 (^cos^ - v2 cos2 ^
6 = y/l-v2sm0
- v2) sin2 ti ,
\F€ - Fmv2 cos2 -1 [F C (1 - v2) + Fmv2 sin2 -1 ,
c = [Fe2(l - t;2) + ±Flv4 sin In the case Fe = Fm, when the vertex (4.4.26) ^ 7^, this formula reduces to (4.4.6).
210
4 Quantum electrodynamics Decay 7* —> pp
The decay tp -> pp is mainly due to the strong interaction, therefore we shall consider the decay 7* —»pp. The program obtains
(4.4.31) The form factors are specially chosen in such a way that Fe describes the decay into pp with the spin projection 0, while Fm describes that with projection ±1. In the ultrarelativistic limit, r,|=0,
T± = 2na2FlM,
(4.4.32)
while in the nonrelativistic one r ( | = 27ra2F2mv,
T± = 27ra2F%lmv.
(4.4.33)
°/0 Decay gamma* -> p anti-p index 1; on gcd; MM:=Tb(ql,l,-q2,l)*4*pi/3$ remind 1; Gamma:=MM/(2*M)*ph2(M,m,m); clear MM; 7« width _ 4VM 2 - Am27r(M2g2 + 2m 2 ) :
3M 2
"
MM:=Tb(ql,ee,-q2,ee)*4*pi$ Garni:=-MM/(2*M)*ph2(M,m,m)$ clear MM; GamL: =sub(ee= (ql-q2)/sqrt(-(ql-q2) . (ql-q2)),Garni); % longitudinal „ T 8y/M2 - Am2m27r GamL := -— 5 M2 l e t ee.nl=0; GamT:=Gaml; clear e e . n l ; °/,transverse GamT := 2 \ / M 2 - 4m2g2ir sub(m=0,GamL) ; sub(m=0,GamT); °/ 0 u l t r a r e l a t i v i s t i c limit 0 2Mg27r let sqrt(-l/(v^2-l))=sqrt(l-v"2)/(l-v*2); GamL: =sub (M=Mv, GamL) $ GamL: =dummy (GamL) ;
GamL := 4 y —v 2 + lrrnrv let sqrt(-l/(v^2-l))=l/sqrt(l-v^2); GamT:=sub(M=Mv,GamT)$ GamT:=dummy(GamT); GamT
s=
*92m*v
V-v2 + 1
clear sqrt(-l/(v^2-l)) ; NonRel(GamL); NonRel(GamT); clear Gamma,Garni,GamL,GamT; off gcd; 4 7717TV
4 g2 miry
4-5 Scattering of charged particles
211
4.5 Scattering of charged particles This Section deals with the processes of charged particles scattering on each other. In most cases, the knowledge of the tensors T ^ , TjJ5, T^v suffices for calculation of squared diagrams. The code fragments of this Section continue the previous program. e/x —>> e\i, e + e~ -> /x + /i~ The simplest example is the scattering of different charged leptons on each other, such as e/j, -» e/i (Fig. 4.14). Any (or both) of them may be, in fact, antileptons. The same cross section is applicable to the scattering of charged leptons on quarks, it needs only to be multiplied by z2
Fig. 4.14. e/x -> e\x
The squared matrix element of Fig. 4.14 is equal to the contraction of two tensors T ^ , divided by t2 (from the photon propagators) and by 4 (averaging over initial polarizations). The cross section is calculated by the formula (4.1.21): da
2
]S
Tt
+ (s + t)2 - 4(mj + m\)8 + 2(m\ [s - (mi + m 2 ) 2 ][s - (mi - m 2 ) 2 ]* 2
(4.5.1)
sH2
,«t2—K)
It is not difficult to find da/dt in the second particle's rest frame, and to check that, when its mass is large, this cross section reduces to the one of the scattering in the coulomb field (4.4.1) (p = 1). If the energy, transferred to the second particle in its rest frame, is denoted m2#, then ! 2ei
dS
~
m2(e2-m2)62
At mi —> oo, this formula reduces to (4.4.5) (p = 1). °/« e - mu- -> e- mu0/
mass pl=ml,p2=m2,p3=ml,p4=m2; mshell pl,p2,p3,p4;
(4.5.2)
212
4 Quantum electrodynamics
l e t pl.p2Ks-ml~2-m2~2)/2,p3.p4=(s-ml~2-m2~2)/2, pl.p3=ml~2-t/2,p2.p4=m2~2-t/2, pl.p4Ks+t-ml~2-m2~2)/2,p2.p3=(s+t-ml~2-m2~2)/2; on gcd; index 11,12; MM:=Tl(pl,ll,p3,12)*Tl(p2,ll,p4,12)/4/t~2$ remind 11,12; on factor; sigma:=MM/(64*pi*JJ(s,ml,m2)~2); 7, d sigma / dt _ 2(2(m| - 2s) m\ + 2s2 + 2st + t2 + 2 (m? + 2 m2, - 2s)m\)-K (m2 4- 2 mi ra2 4- m?> - s) (m2 - 2 mi m2 + m?> — 5) £2 sub(ml=0,sigma); sub(m2=0,ws) ; off factor; 7, ml -> 0 m2 -> 0 2 {2 {ml - 2s)m22
+ 2s2 + 2 s £ + £ 2 ) TT
(m2 - 5 ) 2 1 2
2(2g 2 4- 2st + t2)7r sH2 7o rest frame 2 sr2:=m2~2+2*m2*E+ml~2$ sigma: =sub(s=sr2,sigma);
ml + 4Em2t + 2m?^2 + 2m?,* + t2)
( - m) sub(x=0,sub(m2=l/x,sigma)); % m2 -> infinity TT{4E2 2
+ Q
2
t (E - m2) tr2:=-2*m2^2*delta$ r2:=2*m2"2$ sigma:=sub(t=tr2,sigma)*r2; % d sigma / d delta
_ 7r(2ff2 - 2E8m2 + S2ml - 5m\ - 8m22) a ''~ 5 2 m 2 (^ 2 - m2) sub(x=O,sub(ml=l/x,sub(E=ml/sqrt(l-v~2),sigma))); 7o ml -> infinity TT{5V2 - 5
+ 2)
S2 m2 v2 clear sigma;
The squared matrix element of the process e +e~ —> /x+ji~ (Fig. 4.15) is obtained by the interchange s <-> t. This cross section is also applicable to the processes / + /~ -H- qq, when multiplied by z2 and the corresponding colour factors (see Chapter 6). The differential cross section is da
2
* =2TO
{s + t)2 + t2 - 4(mf + mpt + 2{m\ + ml)2
TU^i)
The total cross section is (4-5.4)
4-5 Scattering of charged particles
213
In the centre-of-mass frame,
— = j - 3 [s(s + 4m2 + 4m?>) + (5 - 4m2)(5 - 4m2) cos2 2
• — (l +
— 4m2, s—
h
(4.5.5)
2
cos tf).
mi,1712—>0 4 5
The angular distribution 1 + cos2 d in the ultrarelativistic limit follows from the helicity conservation.
Fig. 4.15. e+e~ -> ^+/i"
°/o e+ e- -> mu+ muMM:=sub(s=t,t=s,MM)$ sigma:=MM/(64*pi*JJ(s,ml,ml)^2)$ Mandel(ml,ml,m2,m2,s,tl,t2)$ sigt:=int(sigma,t,t2,tl)$ on factor; sigma; sigt; - 2 (2 {ml - 21) ml + 52 + 2 51 + 212 + 2 (m? + 2m% - 2t) m\) n (4 m 2 — 2
m| —
5) 5 3
4 m 2 5 - 4 m | 5 + 5 2 (2 m 2 +
sigO:=sub(ml=O,sigma); sigtO:=sub(ml=O,sigt); 2(2(m| - 2t)m\ + 52 H :=
•/. ml -> 0
-7
4 A/ - 4 m| 5 + s2 (2 m\ + 5) TT sub(m2=0,sig0); sub(m2=0,sigt0); clear sigO,sigtO; % ml, m2 -> 0 2(s 2
+ 2st
+ 2£2)TT
4TT
°/« centre-of-mass frame plcm: =sqrt(s/4-ml"2)$ p2cm: =sqrt(s/4-m2~2)$ tcm:=-plcm"2-p2cm"2+2*plcm*p2cm*c$ °/0 c = cos(theta) cm:=plcm*p2cm/pi$
214
4 Quantum electrodynamics
sigma: =sub(t=tcm,sigma)*cm;
7, d sigma / d Omega
+ s y - 4 mf + s (16 c 2 m 2 rn| - 4c2 mis - 4c2 m22s + c2 s2 + Amis + 4ml s + (4 (4 m? - s)s 3 ) sub(ml=O,sigma) ; sub(m2=0,ws) ; °/, ml -> 0, m2 -> 0 — 4 ml + s (4 m2 + 5 - (4 m 2 - 5) c2)
45 e?r —)> e-K, e + e ~ —>> 7r + 7r~
T h e differential cross section of en —> en is S 5
( + 0 ~~ rnl(2s + t) - 2m\s + (ml + m\)2 [ 5 -(m 1 -m 2 )2]^
d<7
(^
>^Tra^W^. 7721,7712—^0
56
If the lepton or meson mass is large, then it reduces to the cross section of the light particle in the coulomb field (4.4.1) or (4.4.19). °/o e p i -> e p i
I
index 11,12; MM:=Tl(pl,ll,p3,12)*Tta(p2,ll,p4,12)/2/t*2$ remind 11,12; on factor; sigma:=MM/(64*pi*JJ(s,ml,m2)~2); °0/d sigma / dt 4 ((ml + 2 ml - (2 5 + t)) ml + (m| - 2 5) m| + (5 + t) s J TT (m2 + 2 mi m2 + m2 — s) (ml - 2 mi ra2 4- m\ - s) t2 sub(ml=0,sigma); sub(m2=0,sigma); sub(ml=0,ws); 7, ml, m2 -> 0 4 ((ml - 2 5) ml + (5 + t) s) n (m2 - s)2t2 4 ( m 2 — 5 — t) 7T ( m 2 - 5) t2 4 (s + t) TT
sT2 off factor; sub(s=sr2,sigma); sub(x=0,sub(m2=l/x,ws)); 7o rest frame 2
4-5 Scattering of charged particles
215
ir(4E2m2 + 2Et + m2t) m2t2(E2 - m\) n(4E2 + t) t2 (E2 - m2) srl:=ml's2+2*ml*E+m2~2$ sub(s=srl,sigma); sub(x=O,sub(ml=l/x,ws)); °0/rest frame 1 n(4E2ml + 2Emxt + m\t) mlt2{E2 - m2) 4E2n t2 (E2 - m2) c l e a r sigma; The differential cross section e + e~ -> TT+TT" is d&
cy
dt
2 P 2 / ~KS + *) + TO?(2£ + s) + 2771^ — (ml + m^) 2
s3(s — 4m 2 )
^ 77li ,7712-^0
5
The total cross section is
In the centre-of-mass frame, a 2 F 2 (s) sin2 8s mi,m2Jo (4.5.9) The angular distribution sin2 $ in the case of ultrarelativistic electrons follows from the helicity conservation. da_
2 2
8 -{s-Am\) cos2 ^ (s - 4m|) 3 / 2 3 8s (s-4mf) 1 /2
°/0 e+ e > 7.
MM:=-sub(s=t,t=s,MM)/2$ sigma:=MM/(64*pi*JJ(s,ml,ml)"2 sigt:=int(sigma,t,t2,tl)$ on factor; sigma; sigt; 2 ((ml + 2m\ - (s + 2t))m\ + (m22 - 2t) m\ + (s + t) t) ir (4 m 2 - s) s3 ,lm2 — 4 ml s — 4 m\ s + s 2 (2 m 2 + s) (4 m2 s s
S)TT
3 (4ml ~ ) * sigO:=sub(ml=0,sigma) ; sigtO:=sub(ml=O,sigt) ; 7» ml -> 0 sig0 :=
- 2 ((ml - 21) m22 + (s + t) t) ir
216
4 Quantum electrodynamics — J— Amis
-f- s2 (4ml — S)TT
sub(m2=0,sig0) ; sub(m2=0,sigt0) ; °/ e ml, m2 -> 0 -2(s + t)nt 7T
clear sigO,sigtO,sigt; 70 centre-of-mass frame sigma: =sub (t=tcm, sigma) *cm;
y/ — 4 m \ -f s yj — ±m\ + s ((4m2 - s)c2 + s) (4m2 - s) a ''= 8 (4 m? - s)s3 sub(ml=0,sigma); sub(m2=0,sigma); sub(ml=0,ws);
V^V -4m?, + s (c -yfs~yf-4m2
+
l)(c: - 1)(<[ml -s)
2 2 +- s ((4 m - * ) c
1
8 (4 m? -
- (c + 1) (c - 1) 85 clear sigma; off factor; ep —> ep, e + e~
The differential cross section of ep —>> ep is ~dt ~" [s - (mi + m 2 ) 2 ][s - (m x - m 2 ) 2 ](4rn| - t)t2 ^)[52 + (s + £)2 - 4m?5 - 4m^(5 + *)
- m?,)2]
[s(s + t ) - m\(2s + t ) - 2m\s + {m\ (4.5.10) In the limit of large proton mass, it reduces to the cross section of electron scattering in the external field (4.4.1) with p = Fe. °/o ep -> ep let abs(ml)=ml,abs(m2)=m2; index 11,12; MM:=Tl(pl,H,p3,12)*Tb(p2,H,p4,12)/4/t^2$ remind 11,12; sigma: =MM/(64*pi*JJ(s,ml,m2)~2)$ °/, d sigma / dt sign:=num(sigma)$ sigd:=den(sigma)$
factor g; sign; sub (ml=0, sign); 2
°0/ numerator
2 g 7T t ( - 2 m\ + 4 m\ m\ + 4 m\ s - 2m\ + 4 m\ s + 4 m21 - 2 s2 -2st
- t 2) +
4.5 Scattering of charged particles 16 ml 7T (m\ + 2m\m\ 2g2"Kt(-2m\
217
— 2m\ s — m\t + m\ - 2 mis + s2 + st)
+ Am\s + 4m?,£ - 2 s 2 - 2 s t - t2) +
16m2 7r(m2 — 2 mis + s2 + st)
remfac g; clear sign; on factor; sigd; sub(ml=O,sigd); °/ 0 denominator (ml + 2 mi ra2 + m?> - s) (ml - 2 mi m2 + ml - s) (4 ml - t) t2 (4ml ~ t){m\ - s)2t2 off factor; clear sigd; sub(ml=0,m2=0,sigma) ; 7, ml, m2 -> 0 2g2n(2s2 + 1st + t 2 ) s^H2 sigma:=sub(s=sr2,sigma)$ % rest frame 2 sub(x=0,sub(m2=l/x,sigma)); clear sigma; °/« m2 -> infinity T T ( 4 E 2 + t)
t2 (E2 - m\) The differential cross section of e + e~ —> pp is da dt
s3(s
2-KOL2 — 4 m 2 ) (s — 4?TI 2 )
x {F^(s)s [(S + t)2 + t2- 4m\t - 4m22(s + *) + 2(m2 - m2)2] (4.5.11) + SF2(s)ml [-t(s + t) + ml(s + 2i) + 2m\t - (m? + m^ T h e t o t a l cross section is
°=
W
fg _j_ 2TTI i\F
(s^)s "4~ 2F
—^3?
'-
(S^TYI \
I
Is
4TTI
V J3l4-
( 4 - 5 - 12 )
In the centre-of-mass frame,
,,™+ 4.F (5)m^[8 - (s - 4ml) °s #]}\ ' ° ~ ™ 2
c
(4.5.13)
2
% e+ e- -> p anti-p •/ MM:=sub(s=t,t=s,MM)$ sigma:=MM/(64*pi*JJ(s,ml,ml)^2)$ s i g t : = i n t ( s i g m a , t , t 2 , t l ) $ sign:=num(sigma)$ sigd:=den(sigma)$ factor g; sign; sub (ml=0, sign) ; remfac g; clear sign; °/ 0 numerator 2 2 2 2g iTs(2m\ - 4m\ml - 4m\t + 2 m \ - 4m 2s - 4m 2t + s2 + 2st +
218
4 Quantum electrodynamics
2t2) + - st - t2) 2g2"Ks(2m\ - \m\s - Am\t + s2 -f 2st 4- 2t2) + + 2m\t - st - t2)
l§m\ir{-m\
on factor; sigd; sub(ml=O,sigd); clear sigd; (4 m? ~ s)(4ml - s) s3
7, denominator
-(Am22 ~ s)s4 sigt; sub(ml=O,sigt); off factor; - 4 ^/I6m?m| - 4 m? 5 - Am\s + s2 (#2 s + 2 m|) (2 m? + s) n 3 (4m? - s)s3 4 >J -Amis
+ s2 (^2 5 + 2 m|) ?r
sub(ml=0,m2=0,sigma) ; sub(ml=0,m2=0,sigt) ; */, ml, m2 -> 0 2# 2 7r(s 2 + 2st
35 clear sigt; % centre-of-mass frame sigma: =sub(t=tcm,sigma) *cm$ sign:=num(sigma)$ sigd:=den(sigma)$ factor g; factorize(sign); remfac g; clear sign; 7, numerator
{g2 s (4 c2 ml - c2 s - 4 m? - s) + 4 ml ( - 4 c2 ml + c2 s - s),
y - 4 r a § + s, y - 4 m ? + 5} on factor; sigd; off factor; clear sigd; 4 (4 m? - s)s
7o denominator
3
sub(ml=0,m2=0,sigma); clear sigma; °/0 ml, m2 -> 0 45
e e —tee,
e + e —> e + e
The electron-electron scattering is described by four squared diagrams (Fig. 4.16). The diagrams c, d are obtained from a, b by the interchange t «-» u. The direct diagram Fig. 4.16a is calculated by contraction of two tensors X^, as in e/x-scattering. The exchange diagram Fig. 4.166 has the opposite sign, and is calculated as a single long trace.
4-5 Scattering of charged particles
219
Fig. 4.16. e e -> e~e"
The electron mass is set to unity in the program. The simplest form of the differential cross section is obtained by the partial fraction decomposition: da _ 47rcr dt s(s — 4m 2 ) (s2 m->0
ira
s — Am2 \t t
u
2
S2t2U2
(4.5.14)
In the centre-of-mass frame (the Moller formula)
4m2V4
gm4
0
2 2 [4(1 + '..,2\2 ) - (3 + 6v2 - v4) sin2 t + ^4 sin4 ^1 . (4.5.15)
At v -C 1 it reduces to the nonrelativistic Mott formula. In the rest frame of one of the electrons, let's express the cross section via the incident electron energy my and the energy transfer mS. Using the partial fraction decomposition, we obtain da
d8
=
W
1 l*(l , ^ 2 1 (7- -*)2/ 7 - l [ V* 2
1 27-1/1 , ^ ill 7 - 1 V* 7 - 1 - * / J'
(4.5.16) Kinetic energies of two electrons after the scattering are m5 and m (7 — 1 — 8). The formula (4.5.16) is symmetric with respect to the interchange 8 «-> 7 — 1 — £, because the electrons are identical. If the recoil electron is defined as the one having the smaller energy, then <5max = (7 - l)/2.
220
4 Quantum electrodynamics
7, e- e- -> e- e0/
mass pl=l,p2=l,p3=l,p4=l; mshell pl,p2,p3,p4; l e t pl.p2=s/2-l,p3.p4=s/2-l, pl.p3=l-t/2,p2.p4=l-t/2, pl.p4=(s+t)/2-l,p2.p3=(s+t)/2-l; u:=4-s-t$ off gcd; index 11,12; MMl:=Tl(pl,ll,p3,12)*Tl(p2,ll,p4,12)/4/t^2; °/0 diagram a 32TT 2 (2S 2 + 2st
*,*,
- 8s + t2 + 8)
MM1 := ^ ^ MM2:=-4*(4*pi)"2*r(p4)*V(ll)*r(pl)*V(12)*r(p3)*V(ll)*r(p2)*V(12) /4/(t*u)$ remind 11,12; on factor; MM2; off factor; °/0 diagram c 2 - 32 (5 - 2) (s - 6) 7T (* - 4 + 5) £ on gcd; MM:=MM1+MM2$ clear MM1,MM2; MM:=MM+sub(t=u,MM)$ on factor; sigma:=MM/(64*pi*JJ(s,l, 1)^2); 7, d sigma / dt a :=
((s 3 + 2s 2 £ - 12s2 + 3s£ 2 - 12st + 52s + 2£3 - 12^2 -f 121 - 96) s + £4 - St3 + m 2 + 16* + 64)TT)/((S + t - 4) 2 (s - 4)s*2) sub(m=0,sub(s=s/nT2,t=t/nr2,sigma)/m"4); off factor; °/0 m -> 0 4(S2 + St + *2)27T
(s + 0 2 ^ 2 ^ 2 x:=pf(s*(s-4)*sigma/(4*pi),t)$
on factor; x; off factor; clear x;
(g - 2) 2 4(s-3) (s-2)2, -4(g-3) 1 2 ' (5 + t - 4) (5 - 4) ' (s + t - 4) ' (s - 4) t ' £2 J 7o centre-of-mass frame scm:=4/(l-v-2)$ tcm:=-2*v's2/(l-v^2)*(l-c)$ cm:=v"2/(l-v^2)/pi$ sig:=sub(s=scm,t=tcm,sigma)*cm$ 7, d sigma / d Omega let c~2=l-s~2; sig:=sig$ clear c^2; on factor; sig; off factor; clear sig;
-((s 2 ?; 4 + v4 - 6v2 - 3)s2 + 4(^ 2 + I)2) (v + 1) (t; - 1) •/. rest frame 2 sr2:=2*(l+E)$ tr2:=-2*delta$ r2:=2$ on factor; sigma:=sub(s=sr2,t=tr2,sigma)*r2; °/0 d sigma / d delta a :=
2(E4 - 2E36 - 2E3 + 3 ^ 2 ^ 2 + E2 - 2E63 + 3E6 + £4 4- 263 - S)TT (S + 1 - E)2 (E + 1) (E - 1) S2
4-5 Scattering of charged particles
221
off factor; x:=pf((E~2-l)*sigma/(2*pi),delta)$ on factor; x; off factor;
, -(2E-1) E2 -(2E-1) E2, * ' (E - S - 1) (E - 1) ' (E - 6 - I)2 ' (E - 1)6 ' T2"'
clear sigma,x;
T h e scattering e + e ~ -> e + e ~ (Fig. 4.17) is t h e u-channel of t h e same generalized reaction. In t h e centre-of-mass frame (the Bhabha formula) a2(l-
da d£l
16m 2 ?; 4 sin 4 %
(4.5.17)
+ (13 - 2v2 + v4)v4 sin 4 | - 4(1 + v2)v6 sin6 | + 4t;8 sin 8 ^ 1 . In t h e electron rest frame 2™
da_ __ _
2
~d8~~~(< x[7 2 (7-f I ) 2 - (27 2 +47 + 1 ) ( 7 + 1)5 + (37 2 + 67(4.5.18) In this case, up to the whole kinetic energy can be transferred, £ m a x = 7 — 1.
a Fig. 4.17. e + e~ -+ e + e" °/0 e+ e- -> e+ eMM:=sub(s=u,MM)$ on factor; 3
2
_ 4((5 4- 2s * - 4s
2
sigma:=MM/(64*pi*JJ(s,l,l)^2);
+ Zst2 + As + 2*3 - At) s + (t - 2) 2 * 2 )TT
:
^ ~ (s - A)sH2 sub(m=0,sub(s=s/m"2,t=t/m"2,sigma)/m"4); off factor; '/, m -> 0 4(s2 + st + t2)2n s4t2 °/0 centre-of-mass frame tcm:=-4*v~2/(l-v~2)*s2~2$ °/0 s2 = sin(theta/2) sig:=sub(s=scm,t=tcm,sigma)*cm$ on factor; sig; off factor; clear sig; °/0 d sigma / d Omega
s22v6 - 2s22v4
+
13s22v2
- 2v2 -
222
4 Quantum
7)s2v2
+ (v2 + l)2){v
+ l)(v
electrodynamics
1))/'(16s 4 2 v 4 )
-
% rest frame 2: d sigma / d delta factor delta; sigma:=sub(s=sr2,t=tr2,sigma)*r2; a \— (254TT - 453ETT 2STT(-2E
3
+ 262TT(SE2
- 6E
2
- 5E -
+ 6E + 4) + 1)+
+ 2E + l))/(<5 2 (£ 4 + 2 £ 3 - 2J57 - 1)) remfac delta; clear sigma; bye; 2E2TT(E2
Problem. Derive the angular distributions in the centre-of-mass frame in the reactions e + e~ —> /z"*"/i~, TT^TT"", pp with polarized e + , e~:
COS2 (e+e-
_
( e + e
£ _ ^
)
=
^
2
/
2
\ _ AA L _ **£\ s i n 2 t f c o s 2 1
3 / 2
4m \
1
Problem. Obtain the scattering cross section of polarized e and /Lt: da/dt = In the ultrarelativistic limit,
Problem. Consider polarization effects in e~e~ and e + e~ scattering.
4.6 Photon-electron scattering ej —>• e7
Scattering e7 -> e7 (the Compton effect) is depicted by four squared diagrams in Fig. 4.18. Electron virtualities are s — m2 and u - m2. Let's introduce the dimensionless variables x = (s — m2)/m2 and y = (m2 — u)/m2. The second pair of diagrams is obtained from the first pair by the interchange fei «-» — &2, i.e. s H u, or x f> —y.
4-6 Photon-electron scattering
223
a
Fig. 4.18.
In the program, the electron mass is set to unity; it is easy to reinstate the correct degree of m by dimensionality. The factor a2 is implied in the cross section. The factor | coming from the averaging over polarizations compensates for the factor 4, which is needed to transform \ Tr (which is calculated by REDUCE) to Tr. The package kpack from § 4.3 is used. The program obtains da
(T '==-
2TTQ;2 x3y — 4x2(y — 1) + xy(y2 + Ay — 8) -f 4^/2 TTQT
1
18x2 + 32x + 16
(.-c 2 -4a;-8)log(a;+l)
(4.6.1)
At low and high energies, the total cross section is a= 2na2 I" x -> oo : a = —r—2 loga; + mx \
O{x2)), 14/ 9\ logx- T 1 + 2 x V 4/
(4.6.2)
In the electron rest frame, x = 2u)\/m, y = 2a;2/m. The energy-momentum conservation law pi + &i = p2 4- A^2 gives (pi + hi - fc2)2 = p|? o r m 2 + 2mcJ 2mo;2 — 2UJILJ2(1 — cosi?) = m 2 , hence we have the Compton formula 1 - cos ^ m
(4.6.3)
224
4 Quantum electrodynamics
Substituting it into the cross section, we obtain the Klein-Nishina formula (£ = uj/m)
da _ a2 (l4-e + ^2)(l + cos 2 ^)-(l + 2QCcos^gcos^ dn ~ 2m2 (1 + f - £ cos tf)3 *
^
'
In the nonrelativistic limit, it reduces to the classical Thomson formula
£ £
+ «•*>•
(4-6.5)
input_case nil$ °/oe gamma -> e gamma 0/
mass pl=l,p2=l,kl=0,k2=0; mshell pl,p2,kl,k2; let pl.p2=l+(x-y)/2,kl.k2=(x-y)/2,°/, x=s-l y=l-u pl.kl=x/2,p2.k2=x/2,pl.k2=y/2,p2.kl=y/2; index m,n; operator V,r,S; °/« vertex, doubled density matrix, and propagator for all p let V(p)=g(f,p),r(p)=g(f,p)+l,S(p)=r(p)/(p.p-l); % diagram a MMl:=(4*pi)^2*r(p2)*V(m)*S(pl+kl)*V(n)*r(pl)*V(n)*S(pl+kl)*V(m); % diagram b MM2: = (4*pi)^2*r(p2)*V(m)*S(pl+kl)*V(n)*r(pl)*V(m)*S(pl-k2)*V(n); :=
S2,H-X
+
y-4)
xy
°/« averaged square of matrix element on gcd; MM:=MM1+MM2$ clear MM1,MM2; MM:=MM+sub(x=-y,y=-x,MM)$ in ll kpack.red"$ l e t abs(x)=x,abs(y)=y; sigma:=MM/(64*pi*JJ(l+x,l,0)"2) ; 7, d sigma / d y G
_ 2n(x3y - 4x2y + 4x2 + xy3 + Axy2 - Sxy + Ay2) ''~ x+y2
Mandel(l,0,0,l,l+x,ul,u2)$ for all x,y let log(x/y)=log(x)-log(y); sigt:=int(sigma,y,l-ul,l-u2)$ clear ul,u2; °/, total cross section on factor; df(sigt,log(l+x)); sub(log(l+x)=0,sigt); off factor; 2(x2 - 4x - 8)TT
73
(x3 + 18a;2 + 32x + 16) TT (x + l) 2 x 2
4-6 Photon-electron scattering
225
°/, low energies procedure binom(x,n); begin scalar s,u,i,j; s:=u:=i:=l; j:=n; while (u:=u*x*j/i) neq 0 do «s:=s+u;i:=i+l; j :=j-l»; return s end$ procedure logl(x); begin scalar s,u,i; s:=u:=x; i:=2; while (u:=-u*x) neq 0 do «s:=s+u/i;i:=i+l»; return s end$ procedure dummy(x); x$ sigl:=sigt*x~3*(l+x)~2$ weight x=l$ wtlevel 4$ on revpri; dummy(sub(log(l+x)=logl(x),sigl)*binom(x,-2))/x~3; clear x; 8TT(1 - x) "3 % high energies sigl:=sub(x=l/y,sigt)*(l+y)~2$ weight y=l$ wtlevel 2$ sub(log(l+y)=logl(y),sigl)*binom(y,-2);
ny (1 + lSy - 2 log(y) + 8 log(y)y) clear y,sigt,sigl,binom,logl; off revpri; % electron rest frame - d sigma / d Omega x:=2*omega$ y:=2*omega2$ sigma:=sigma*y"2/(4*pi);
omega2:=omega/(l+omega*(l-c))$ sigma:=sigma$ on factor; sigma; (c2u - cu2 - cu - c + 2a;2 + u) c - (u2 4- u + 1) 2{cu - u - I) 3 sub(omega=0,sigma); °o/ nonrelativistic limit C2
2 clear x,y,omega2,sigma; e+e~ The squared matrix element (Fig. 4.19) is obtained from the previous one by the interchange s <-> t. Let's introduce z = (m2 - t)/m2, x = s/m2. The most simple form of the differential cross section is obtained by the partial fraction decomposition. When calculating the total cross section, we should include the factor \ because the two final photons are identical. da
dz
87ra2
[1
1
m x(x — 4) [z + -,(x - z) 2
2
27ra2
rrr2
rc2+4x-8/l ;Ax I\zI
1 \
1
I +2 ~ x-zj
4 Quantum electrodynamics
226
In the centre-of-mass frame, 4m2v{l - v2 cos2
a =
+ 2 « 2 - 2r/4 - 2(1 - v2)v2 cos2 0 - v4 cos4 0], — v
(4.6.7)
In the nonrelativistic limit, a=
TTQT
(4.6.8)
the angular distribution is isotropic. In the ultrarelativistic limit, Trar m272
I Inn- O^/
(4.6.9)
the angular distribution has the forward and backward peaks with width d 1/7-
Fig. 4.19. e+e~ -+ 2 7
Differential cross section of the inverse process 77 -> e + e~ is obtained from (4.6.6), (4.6.7) by multiplying by the factor (s - 4m2)/s = v2. The total cross section should be additionally multiplied by 2, because the final particles are not identical. % e+ e~ ~> gamma gamma y sigma:=sub(x=-z,y=s-z,MM)/(64*pi*JJ(s,l,l)~2); °/0 d sigma / d z
4-6 Photon-electron scattering
227
a := - z)z + 4z - 4) (s - z)2 - {(z2 + 4 z - 4) z - {z2 + 4 z - 8) s) . ((s - z)2(s - A)sz2)
" off factor; c:=pf(sigma*s*(s-4)/(8*pi),z)$ on factor; c; off factor; clear c; rl ^2'
-{s2 + As - 8) 1 -(s2+ 4s-8) \ , r> 4 (5 - z) s ' (s - z)2 ' Ts~z ~£>
Mandel(l,l,0,0,s,tl,t2)$ let abs(s)=s; sigt:=int(sigma,z,l-t2,l-tl)/2$ % total cross section for all x,y clear log(x/y); let log(sqrt(s)*sqrt(s-4)+s)= log((sqrt(s)*sqrt(s-4)+s)/(-sqrt(s)*sqrt(s-4)+s)) +log(^sqrt(s)*sqrt(s-4)+s); factor log; sigt; 2 lo£( —%—/ V
s
V
s
) Trfs — 4— s
/
v
-j-
4 s — 8) ~h 2\fs v s — 4TT( — s — /
vv
\
4) /
2
s (s - 4)
clear t l , t 2 ; remfac log; clear log(sqrt(s)*sqrt(s-4)+s); °/0 centre-of-mass frame scm:=4/(l-v"2)$ zcm:=2*(l-v*c)/(l-v"2)$ cm:=v/pi/(l-v~2)$ on factor; sigma: =sub(s=scm,z=zcm,sigma)*cm; °/« d sigma / d Omega -(c4v4
- 2c2v4
+ 2c 2 v 2 + 2v4 - 2v2 - 1) (v + 1) (v - 1)
off factor; clear sigma; l e t abs(v)=v; sigt:=sub(s=scm,sigt)$ sigt:=dummy(sigt)$ l e t log(-4/(v-l))=log((l+v)/(l-v))+log(4/(l+v)); sigt:=sub(s=scm,sigt)$ clear log(-4/(v-l)); on factor; df(sigt,log((l+v)/(l-v))) ; (V4 - 3) (V + 1) (V - 1) 7T sub(log((l+v)/(l-v))=0,sigt); off factor; - (t>2 - 2) (v + 1) (t; - 1) 7T 2^ 7, nonrelativistic limit sub(v=0,sub(log((1+v)/(l~v))=2*v,sigt*v))/v; 7T
Jv
7o u l t r a r e l a t i v i s t i c limit sub(v=l,sub(log((l+v)/(l-v))=2*log(2*E),sigt/(l-v^2)))/E"2; 2E2
clear sigt;
228
4 Quantum electrodynamics
X gamma gamma —> e+ e~ JJ(s,l,l)~2/JJ(s,0,0)~2; sub(s=scm,ws); s - 4 s 2
v bye;
Polarization effects Let's consider polarization effects in the Compton scattering ej -> ej. We shall follow § 87 of the textbook [QED1]. Let the electron mass m = 1; it will be reintroduced in the final answers by dimensionality. The momenta are p + k = p1 -f fc'; x — s — 1, y = 1 — w, s = (p -f fc)2, u = (p - k')2. The matrix element squared is \M\2 = 64'7r z a^p pAt p^Trp ——p—
(4.6.10)
n3/z = - - — e ^ ^ K a q ^ ,
(4.6.11)
Q (£ +fc+ 1 ) xy x It is convenient to use the basis n 0 = —, ni = - , v
v
n
2
= — PJL, 2w
7
, q = k' -k=p-pf,
(
Xy
fe; + l )
2vw
P± = P - ^rK, P = p+p'; v = ^fx^y,
w = -^/xt/ — x + 2/. In this basis, \ \
p
—, ,
XV
j
, 1 / a ; + 2/ n P = (
a:-
v
k = - ( n 0 - ni), fc = - ( n 0 + ni). (4.6.12) The vectors n i 5 n 2 , n 3 form a right-handed system. The vectors n 2 , n 3 can be used as polarization vectors (in the scattering plane and orthogonal to it) of both photons. For the final photon, the vectors n 2 , n*3, kf form a right-handed system (in the K rest frame); therefore its density matrix is expressed via the Stokes parameters ££ in the standard way: juv, where £Q = 1 have been formally introduced, and (4.6.13) For the initial photon, the right-handed system is 712, — n~3, k. Therefore, \ J2j SjZjVjiMvi where So = S3 = 1, Si = S2 = - 1 .
4.6 Photon-electron scattering
229
The tensor Q^v in the ri2-n3 plane also can be expanded in cr-matrices: vi
Qk = -^QiivOkviii
k
w xy
, 7/x7o7^
|
v \
x
y
1
7^7o7/i ,
v
/
x
y (4.6.14)
Using the Dirac equation, we obtain K y,
K + y) — . _ _ (4.6.15) The conjugated tensor QMI/ = X)fc hQkO-ku^ because Qfc = 5kQk, a^ = akv^. The initial and final electron density matrices are pM = | ( p ^ + 1)(1 — &^). Let's introduce two bases eo=P,
ei=p
e' 0 =p',
ei=p;
2r fc, X 2
Q2 = -(«-2/)75,
Q3 = x-y-{x
w, N e2 = — ( n o - n i ) + n 2 , e 3 = n 3 ; ^ u; A;', e2 = (no + n i ) - n 2 , e3 = n 3 .
(46
*16)
Then a ^ = X)»Ci e i > where d is the longitudinal polarization, £2 is the transverse polarization in the scattering plane, and £3 is the transverse polarization perpendicular to this plane. Introducing formally Q = 1, we have
P{>) = \ E , Cl'Vf, where p£> = pW + 1, pW = - ^ O ^ O .
Finally, the cross section with the polarizations of all particles is da dydip
a2 mWy2
*-" U
V
kk'
3
n
^
^
'"
2
' (4 6 17)
1 3
3
' '
4
(the right-hand side depends on ip because the polarizations are denned relative to the scattering plane). The cross section summed over the final particles' polarizations is daa2 „ „
rr2' The final particles' polarizations are
& = | E ^ ° ^ . ^' = | E ^'C^r ij
(4-6.19)
The components F^-7 with i' ^ 0 and j ' / 0 describe the correlation of the final particles' polarizations.
230
4 Quantum electrodynamics
input_case nil$ ° 0/ Polarization
effects
7o basis and momenta mass nO=l,nl=i,n2=i,n3=i; mshell nO,nl,n2,n3; let nO.nl=0,nO.n2=0,nO.n3=0,nl.n2=0,nl.n3=0,n2.n3=0, eps(nO,nl,n2,n3)=-l; vector pl,p2,kl,k2,k; let v~2=x-y,w~2=x*y-x+y; pi:=((x+y)/2*nO+(x-y)/2*nl+w*n2)/v$ p2:=((x+y)/2*nO-(x-y)/2*nl+w*n2)/v$ kl:=v/2*(nO-nl)$ k2:=v/2*(nO+nl)$ k:=kl+k2$ °/0 electron density matrices and Q's nospur f; array rl(3),r2(3),Q(3); vector a; rl(0):=g(f,pl)+l$ rl(l):=-rl(O)*g(f,a,pl-2/x*kl)$ rl(2):=-rl(0)*g(f,a, w/x*(nO-nl)+n2)$ rl(3):=-rl(O)*g(f,a,n3)$ r2(0):=g(f,p2)+l$ r2(l):=-r2(0)*g(f,a,P2-2/x*k2)$ r2(2):=-r2(0)*g(f,a,-w/x*(nO+nl)-n2)$ r2(3):=-r2(0)*g(f,a,n3)$ Q(O):=x-y$ Q(l):=-i*(x-y)*g(f,a,k)/2$ Q(2):=-(x-y)*g(f,a)$ Q(3):=x-y-(x+y)*g(f,k)/2$ °/0 calculating traces spur f; array Sp(3,3,3,3); for i l : = 0 : 3 do for i2:=0:3 do for kl:=0:3 do for k2:=0:3 do Sp(il,i2,kl,k2):=r2(i2)*Q(kl)*rl(il)*Q(k2); clear r l , r 2 , Q ; clear pl,p2,kl,k2,k; clear n0.n0,n0.nl,n0.n2,n0.n3,nl.nl,nl.n2,nl.n3,n2.n2,n2.n3,n3.n3 , eps(nO,nl,n2,n3); clear nO,nl,n2,n3; °/« Pauli matrix traces array s ( 3 , 3 , 3 , 3 ) , d ( 3 , 3 ) , e p ( 3 , 3 , 3 ) ; for i : = l : 3 do d ( i , i ) : = l ; ep(l,2,3):=ep(2,3,l):=ep(3,l,2):=l$ ep(l,3,2):=ep(3,2,l):=ep(2,l,3):=-l$ s(0,0,0,0):=l$ for i : = l : 3 do s ( 0 , 0 , i , i ) : = s ( 0 , i , 0 , i ) : = s ( 0 , i , i , 0 ) : = for il:=l:3 do for i2:=l:3 do for i3:=l:3 do s(0,il,i2,i3):= s(il,0,i2,i3):=s(il,i2,0,i3):=s(il,i2,i3,0):=i*ep(il,i2,i3); for il:=l:3 do for i2:=l:3 do for i3:=l:3 do for i4:=l:3 do
clear d,ep; °/0 filling the array F array F(3,3,3,3),d(3); d(0):=d(3):=1$ d(l):=d(2):=-l$ for il:=0:3 do for i2:=0:3 do for jl:=0:3 do for j2:=0:3 do F(il,i2,jl,j2):=for kl:=0:3 sum for k2:=0:3 sum
4.6 Photon-electron scattering
231
d(jl)*d(k2)*s(j2,kl,jl,k2)*Sp(il,i2,kl,k2); clear Sp,d; clear v~2,w~2; % simplification for il:=0:3 do for i2:=0:3 do for jl:=0:3 do for j2:=0:3 do if F(il,i2,jl,j2) neq 0 then « n:=num(F(il,i2,jl,j2)); d:=den(F(il,i2,jl,J2)); while den(b:=n/(x-y))=l do n:=b*v~2; while den(b:=n/(x*y-x+y))=l do n:=b*w~2; while den(b:=d/(x-y))=l do d:=b*v~2; while den(b:=d/(x*y-x+y))=l do d:=b*w~2; F(il,i2,jl,j2):=n/d; clear n,d,b; °/« finding identical results max:=30$ array name(max),value(max); im:=-l$ for il:=0:3 do for jl:=0:3 do for i2:=0:3 do for j2:=0:3 do if F(il,i2,jl,j2) neq 0 then begin scalar i; i:=0; while i<=im do if (r:=F(il,i2,jl,j2)/value(i))=l or r=-l then « named) :=mkid(name(i),! ) ; if r=-l then name(i):=mkid(name(i),!-); name(i) :=mkid(named) ,il); name(i) :=mkid(name(i), jl); name(i) :=mkid(named) ,i2); name(i) :=mkid(name(i) , j2) ; L2.il.i2):=0; i:=im+2 else i:=i+l; if i=im+l then « im:=i; named) : = ! ; name(i) :=mkid(named) ,il); name(i) :=mkid(name(i), jl); name(i) :=mkid(named) ,i2); named) :=mkid(name(i), j2) ; value(i):=F(il,i2,jl,j2); F(il,i2,jl,j2):=0; i » end; clear F; on factor; for i:=0:im do «
write name(i); write value(i); » ;
0000 3333 3
x y - 4x2y + 4x2 + xy3 + 4xy2 - 8xy + 4y2 0003 0300 3033 3330 — 4v2 w2 0012 1200 2133 3321 {x y - 2 x + 2 y) (x + y) v2
232
4 Quantum electrodynamics 0022 0231 1133 -2200
-3102
-3311
-3011
-3201
-2v3wy 0101 3232 2(xy — 2x +
2y)xy
0132 0322 1130 -2203 3
2v wx 0202 3131 [x2 + y2)(xy
- 2x + 2y)
0210 1002 - 2331 - 3123 (x3y + x2y2 - 4x2y + 4x 2 + 4xy2 - 8xy + 4 x 0213 1302 - 2031 - 3120 — 4v4w2 x 0220 0223 -1031 - 1331 - 2002 - 2302 3110 3113 -2(xy - 2x + 2y)v3w x 0303 3030
2{x2y2 - 2x2y + 2x2 + 2xy2 - Axy + 2y2) 1010 2323 (x4 + x2 y2 - 4 x2 y + 4 x2 4- 4 x y2 — 8 x y + 4 y2) (x y — 2 x + 2 y) 1013 1310 2023 2320 - 2x + 2y)v2w2 1020 1323 -2010
-2313
1023 -2310 2{x2y + 4xy — 4x + 4y)v3w x~2 1111 2222 2(x3y2 - 2x3y + 2x3 - 2x2y + 2xy3 - 2xy2 + 2y3) x 1121 1222 -2111
-2212
— 2(xy — 2x + 2y) (x + y)t;w
4- 7 Positronium annihilation
233
1212 2121 x y — 4x3y + 4x 3 + x2 y3 — Ax2y + Axy3 — Axy2 + Ay3 4
X
1313 2020 2(x3y - 2x2y -h 2x2 HY2xy2 i52C>
- Axy + 2?/2)i ( x ?/
— 2x H
H2y)
—
4a;
bye; Problem. Derive the cross section of the two-photon annihilation of polarized e + e~ in the centre—of—mass frame. The total cross section differs from the unpolarized one by the factor 1 — z1 • z2, because the annihilation takes place only in the S = 0 state.
4.7 Positronium annihilation Estimates Positronium is the bound state of an electron and a positron. In the nonrelativistic approximation, it differs from the hydrogen atom only by replacement of the electron mass m by the reduced mass ra/2. In particular, binding energies and wave functions at the origin (of 5-wave states) are
Relativistic corrections to the energies (4.7.1) (the fine structure) are of the order ma 4 . Positronium states are characterized by the main quantum number n, the orbital angular momentum /, and the total spin s — 0 or 1; / and s are added, giving the total angular momentum j . As we have discussed in § 3.4, positronium states have definite parity P = (-1) / + 1 and charge parity C — (-l) z + s . Positronium with s = 0 is called parapositronium, and with s = 1, orthopositronium. States with C = +1 can annihilate into an even number of photons (most probably two), and with C = — 1, into an odd number (most probably three). The matrix element of annihilation into 2j contains e2, and the probability a2. In order to annihilate, the electron and the positron have to be at a distance of the order ~ 1/ra, which is small compared with the positronium radius ~ l/(ma). Therefore, the annihilation probability is proportional to the probability of being at such distances ~ 1^(0)|2/m3. Finally, it contains m by dimensionality: !
^
(4.7.2)
234
4 Quantum electrodynamics
The probability of annihilation into 37 is
These estimates are applicable to the states with / = 0, which have I/J(0) ^ 0. In general, the estimates involve if>(r) at r ~ 1/m; states with the angular momentum I have ijj(r) ~ rl, and hence ip(r) ~ (~) ip(0) ~ al/ipi=o(0): (4.7.4) The life time of the ground state (n = 1) of para- and orthopositronium is determined by the annihilation (4.7.2), (4.7.3). Excited states (n > 1) can decay into lower ones, emitting a photon. Therefore, it is necessary to understand which process is more probable. Probabilities of electric dipole transitions are FBI ~ d2u3 ~ ma 5 ,
(4.7.5)
where d ~ er is the dipole moment, r ~ l/(ma), UJ ~ ma2. Probabilities of electric quadrtipole transitions contain the extra factor (AT)2 ~ a 2 , and those of magnetic dipole ones contain v2 ~ a 2 : FE2 ~ TMi ~ ma7.
(4.7.6)
Ml transitions 5 -> 5 are an exception. They proceed with the spin flip; radial wave functions with different values of n are orthogonal to each other, and a nonvanishing matrix element emerges only from the relativistic correction ~ v ~ a: 5) ~ma9.
(4.7.7)
In this case, the transition with emission of two £ 1 photons is more probable: r 2 E i ~ d4uj5 - mas.
(4.7.8)
All states with n > 1, except 25, can undergo El-transitions (4.7.5). They are the most probable decay channel, except n5-states of parapositronium, which have the probability of annihilation into two photons (4.7.2) of the same order. 25 states of para- and orthopositronium mainly annihilate into 2j (4.7.2) and 37 (4.7.3), because the most probable transition is a two-photon one, and it is less probable (4.7.8). Annihilation probabilities of positronium 5-states F 2 7 , F3 7 can be obtained from the cross sections e + e~ -» 27, 37 at small relative velocities v. In the positron rest frame, incident electrons with velocity v in the velocity space element d3v form the flux v(dn/d3v)d3v. Multiplying it by the cross section of annihilation of e + e~ with the total spin 0 into two photons o"o(e+e~ -+ 27) and integrating over d3v, we obtain the annihilation rate F 2 7 . At small velocities, the
4- 7 Positronium annihilation
235
cross section behaves as 1/v, so that the integral gives simply n(wo) v-»o, where n = |-0(O)|2 is the total density of electrons at the point where the positron is situated. The cross section averaged over initial particles' spins cr(e+e~ —t 2j) is usually calculated. Only one spin state from four can annihilate into 27, therefore a = \CFQ. We arrive at the Pomeranchuk formula r 2 7 -4|V(0)| 2 (z; ( 7 2 7 )^o.
(4.7.9)
Similarly, three spin states from four can annihilate into 37, therefore a(e +e~ —>• 37) = fcr^e+e" ->• 37), and ^
(4.7.10)
In § 4.6, we have obtained a2l = 7ra2/(ra2t>), therefore m2
(4.7.11)
2 TT2 — In the next subsection, we shall derive the Ore-Powell formula cr37 == | (|(TT 3 2 9)a /(m v), and
e+e +
—> 37 at low energies
There are six diagrams of e e~ —> 37, which differ from each other by permutations of photons. Therefore, there are 36 squared diagrams. They can be obtained by permutations of photons from the six squared diagrams in Fig. 4.20, in which the order of photons in the left-hand side is fixed. The diagram Fig. 4.20c is transformed to Fig. 4.206 with interchanged photons by inverting the direction of the electron line (this operation does not change the trace). Similarly, the diagram Fig. 4.20e is transformed to Fig. 4.20d. Therefore, the diagrams Fig. 4.20c and e may be omitted, with doubling of the contributions of Fig. 4,20b and d. Let's set the electron mass to unity, and denote the photon energies by #1, x 2 , #3- The Dalitz plot is shown in Fig. 4.21; x\ + x2 + £3 = 2. In the program, 1 means the left-hand side, common to all squared diagrams of Fig. 4.20. Common factors, necessary for obtaining dT/(dxidxz), are also included in 1: 4 is the transition from \ Tr to Tr; \ comes from averaging over the electron and positron spins; e6 = (47ra)3; |(2TT) 3 is the factor from the phase space; 1/(4J) (where J = v is the invariant flux); ^^(O)!2?; comes from the Pomeranchuk formula (4.7.10) (the factor a3|'0(O)|2 is implied). The lefthand side 1 is multiplied by the right-hand sides of the diagrams of Fig. 4.20, traces are calculated, and the summation over permutations of x\, x 2, £3 is
236
4 Quantum electrodynamics
p-h\; .f-p+fcl
-p+k^ p
!\ P~h
P~h I'
Fig. 4.20. e+e~ -> 37
0
Fig. 4.21. Dalitz plot
4-7 Positronium annihilation
237
performed. The result can be written in the simple and symmetric form (the Ore-Powell formula) dT
_32c* 3 hK0)| 2 3
, .
^
, , .
^
t
, .
^ ,
ra2
During integration of this formula, each final state will occur 3! times, which differ from each other by permutations of three photons. Photons are identical, hence only one of them should be included, i.e. the integral should be divided by 3!. The program proceeds to calculate dT/dxi by integrating (4.7.13) in x2 from 1 - xi to 1 (Fig. 4.21). The indefinite integral, calculated by REDUCE, involves log(xl + x2 — 2), but we know that it should contain 2 — xl — x2 instead. The photon spectrum is dT dx
32 a3|V>(0)|2 9 m2x(2-x)2 x L 2 ( l * ) ( 8 ^ + 5^) I x(2 — x)
_x)
+ s
(4.7.14) Finally, the program calculates the total decay rate F. It is simpler not to integrate (4.7.14) in x, but to return to the formula (4.7.13), and to replace the expression in the brackets by the tripled first term, because the integration over the Dalitz plot is symmetric. At the second integration, dilogarithm (§ 1.4) dilog(x) = - [ ^-dy J y- i
(4.7.15)
i
appears. The REDUCE integrator cannot calculate such integrals. Therefore, we have to separate this term, and substitute the explicit result. The total decay
r
H
h
«
( l r 9 ) a
.
(4716)
input_case nil$ °/« e+e- -> 3 gamma at low energies y-
mass p=l,kl=0,k2=0,k3=0; mshell p,kl,k2,k3; vector el,e2,e3; let p.kl=xl,p.k2=x2,p.k3=x3, kl.k2=2*(l-x3),kl.k3=2*(l-x2),k2.k3=2*(l-xl); nospur f; operator r,S,V; for all p let r(p)=g(f,p)+l,S(p)=r(p)/(p.p-l),V(p)=g(f,p); l:=r(-p)*V(el)*S(kl-p)*V(e2)*S(p-k3)*V(e3)*r(p)*2/3$ •/, alpha~3*psi(0)~2 implied spur f; index el,e2,e3;
238
4 Quantum electrodynamics
Gal:=l*V(e3)*S(p-k3)*V(e2)*S(kl-p)*V(el)$ Ga2:=l*V(e2)*S(p-k2)*V(e3)*S(kl-p)*V(el)$ Ga3:=l*V(e2)*S(p-k2)*V(el)*S(k3-p)*V(e3)$ Ga4:=l*V(el)*S(p-kl)*V(e2)*S(k3-p)*V(e3)$ clear 1; for all p clear r(p),S(p),V(p); clear r,S,V; on gcd; Gamma:=Gal+2*Ga2+2*Ga3+Ga4$ clear Gal,Ga2,Ga3,Ga4; Gamma:=Gamma+sub(xl=x2,x2=x3,x3=xl,Gamma) +sub(xl=x3,x2=xl,x3=x2,Gamma)+sub(x2=x3,x3=x2,Gamma) +sub(xl=x3,x3=xl,Gamma)+sub(xl=x2,x2=xl,Gamma)$ x3:=2-xl-x2$ on factor; Gamma:=Gamma; off factor;
(64 {x\ + 2x\x2 - 4x? + ?>x\x\ - 9x\x2 + 7x\ A - 4^2 H" 7xl ~ 6 ^2 + 2))/(3(x2 - 2 c:—Gamma /(((1
_ 32 °/0 Photon spectrum factor log; Gamma: =int(Gamma,x2)/6$ 7, identical photons Gamma:=sub(log(xl+x2-2)=log(2-xl-x2),Gamma)$ Gamma:=sub(x2=l, Gamma) -sub(x2=l-xl, Gamma) $ 7, d Gamma / d xl on factor; df(Gamma,log(1-xl)); 64 (5x? - 12xi + 8)(xi - 1) 9(xi - 2)3xf sub(log(1-xl)=0,Gamma); off factor; - 3 2 (2x? - lx\ + 12xi - 8) 9(xi -2)2xi */, Total decay rate Gamma:=c/6*3*((l-x3)/(xl*x2))"2$ Gamma: =int (Gamma, x2) $ Gamma: =sub(x2=l, Gamma) -sub (x2=l-xl, Gamma) $ GO:=sub(log(l-xl)=0,Gamma)$ 7» without log(l-xl) Gl: = (Gamma-GO)/log(l-xl)$ 7, with log(l-xl) G2:=coeffn(Gl*xl~2,xl,l)$ % log(l-xl)/xl Gl:=Gl-G2/xl$ let dilog(l)=0,dilog(0)=pi~2/6; •/. dilogarithm Gamma: =int (GO, xl)+int(Gl*log( 1-xl), xl)-G2*dilog( 1-xl) $ V. caution at xl=0 Gamma:=sub(xl=l,Gamma)-sub(xl=0,sub(log(1-xl)=-xl,Gamma)); 16(TT2-9) r
bye;
=
9
4-7 Positronium annihilation
239
Positronium matrix elements There is an alternative approach, which allows one to calculate matrix elements of processes involving bound states, such as positronium. To be more general, let's consider a nonrelativistic bound state of a fermion with mass mi and an antifermion with mass m^ (for example, e~/i + ). They may be considered at rest relative to each other: pi = m ^ , P2 = m2v, v2 = 1. Summation over their spins can be performed with the use of the completeness relation (Fig. 4.22). Its first term corresponds to the pseudoscalar state (s = 0), and the second term, to the vector one (5 = 1). Summation is performed over three independent polarizations e, orthogonal to v. It may be extended to all four polarizations: e... e —> —7^ ... 7/x, because the timelike polarization e = v does not contribute.
Fig. 4.22. Spin completeness relation
Let's prove this relation, substituting the complete set of 7-matrices on the left and on the right: input_case nil$ mass v=l; mshell v; vector a,ul,u2,vl,v2,m; nospur f; rp:=g(f,v)+l$ rm:=g(f,v)-l$ ga:=g(f,a)$ gm:=g(f,m)$ array Gu(4),Gv(4); Gu(0):=l$ Gu(l):=ga$ Gu(2):=g(f,ul)$ Gu(3):=g(f,ul,a)$ Gu(4):=g(f,ul,u2)$ for i:=0:4 do Gv(i):=sub(ul=vl,u2=v2,Gu(i)); spur f; for i:=0:4 do for j:=0:4 do « x:=Gu(i)*rm*Gv(j)*rp; yu:=Gu(i)*rm*ga*rp; yv:=Gv(j)*rp*ga*rm; y:=-yu*yv/2; yu: =Gu(i) *rm*gm*rp; yv: =Gv(j ) *rp*gm*rm; index m; y:=y-yu*yv/2; remind m; if x neq y then « nospur f; write Gu(i), 11 \ G v ( j ) , n : H ,x," M,y; spur f; » »;
bye; A decay rate, summed (not averaged!) over initial polarizations, is equal to the product of the flux |V>(0)|2i> and the cross section ^ |M| 2 /(4J) d$, where the invariant flux is J = raira2v:
1v
1
32 E l ( )
(
W
)!
2
(4.7.17)
240
4 Quantum electrodynamics
In this sum, contributions of the pseudoscalar state (F = 275) and the vector one (F = e) are separated. We can rewrite it in the standard form Yl \M\ 2/(2m) d
V m
4mim2
T/,(0) TrM{p1 + mi)r(p2 - m 2 ).
(4.7.18)
Here F = 275 for the pseudoscalar state, and F = e for the vector one, where e is its polarization vector. The equality Tv = —-OF always holds, hence the equation (4.7.18) can be simplified: M = -^=^)TiM{p
+ m)F.
(4.7.19)
2 y 777,
However, in this form, the timelike polarization e = v may not be included. Note that in the ultrarelativistic case (p + ra)F —» ^75 for a pseudoscalar state, and pej. for a transversely polarized vector one. The longitudinal polarization vector is !ez -m2 where n is the timelike basis vector of the laboratory frame, and (p + m)e\\ =
=(^P ~" ^i 2 ^ + em — mph) —> p.
This variant of the formula (4.7.19) is used in QCD for calculating matrix elements of processes with ultrarelativistic mesons. Parapositronium In the program, n is the unit spacelike vector in the direction of motion of the first photon. The matrix element can be presented in the three-dimensional form (4.7.20) M = 8v^7ra^(0) e2 • e-1 x n. Polarizations of the produced photons are correlated: if the first one is linearly polarized, then the second one is linearly polarized in the orthogonal direction; if the first one is circularly polarized, then the second one is circularly polarized with the opposite angular momentum projection onto ft (i.e. with the same helicity). The effective vertex of the orthopositronium interaction with two photons can be introduced: 2 ^ 0 ^ ( 0 ) . (pF^F^,
(4.7.21)
where
W
(4.7.22)
4- 7 Positronium annihilation
241
input_case nil$ °/ 0 Parapositronium
yo
mass p=l,n=i; mshell p,n; l e t p.n=0; vector a , k l , k 2 , e l , e 2 , e 3 ; kl:=p+n$ k2:=p-n$ operator r,S,V; for a l l p l e t r(p)=g(f,p)+l,S(p)=r(p)/(p.p-l),V(p)=g(f,p); M:=2^(3/2)*i*4*pi*V(e2)*S(p-kl)*V(el)*r(p)*g(f,a)$ 7o alpha*psi(O) implied M:=M+sub(e3=e2,sub(e2=e1,sub(n=-n,eI=e3,M))); M := — 8\/2 eps(ei,e2,n,p)7r index e l ; M*kl.el; remind e l ; '/• Ward identity 0 index e l , e 2 ; Gamma:=M*M/4/(8*pi)/2; r := 4TT bye; This program, as well as some other ones, had to be changed in order to run under REDUCE 3.6. In earlier versions, sub(vl = v2,v2=vl,a) interchanged the vectors vl and v2 in a, just as in the case of scalar variables. REDUCE 3.6 ceases to do so, and a more complicated sub has to be used. Orth opositro nium The program calculates the matrix element of the orthopositronium decay into three photons. The effective interaction vertex can be introduced. Let A^ be the orthopositronium field. Then two structures of the interaction vertex are possible: T^F^F^F^ and TixvFvpFpaFa^ where T^ — d^Au — d^A^. Instead of the second one, T^F^FpGFpa may be used, where F^v = \epvpaFp
~
~
.
W \lvF \lvV pa? pv T ^fiu-Tfii/^ poFpO )'
M 7 9
V4- I'M)
All subsequent calculations are performed in the three-dimensional form. The matrix element is M =4(27ra)3/27/;(0)[e • ei(e 2 • e3 - e2 x n 2 • e3 x n 3 ) + e- ei x ni(e 2 * e*3 x n 3 + e3 • e*2 x n 2 ) + cycle].
^
Squaring it with contraction over ei, e2, e 3 gives the decay rate of polarized orthopositronium: dT
2
=16a 3
(l-|e-n2|2)+
/i x2 ^ ^ ^ (l-|e-n3n
(47 25)
-
242
4 Quantum electrodynamics
Averaging over e reproduces (4.7.13). Note that the decay rate (4.7.25) vanishes when positronium is linearly polarized, and all three photons fly along e. This is because the projection of the positronium angular momentum onto e is zero, and the sum of projections of the photon angular momenta ±1 ± 1 ± 1 cannot be zero. Spectrum-angular distribution of the photons has the form
Averaging it over
3\dx+
dx ) •
The spectrum of photons flying along e is rfT|| _16a 3 |V>(0)| 2 (l-a;)
x
-
It vanishes at x —> 1, because in this case all three photons have to fly along e. input_case n i l $ °/ 0 Orthopositronium
V.
mass p=l,kl=0,k2=0,k3=0; mshell p,kl,k2,k3; vector p,ee,el,e2,e3,e4,e5,k4,k5; operator r,S,V; for a l l p l e t r(p)=g(f,p)+l,S(p)=r(p)/(p.p-l),V(p)=g(f,p); M: = (8*pi)-*(3/2)*V(e3)*S(k3-p)*V(e2)*S(p-kl)*V(el)*r(p)*V(ee)$ M:=M + S ub(k4=k3,e4=e3,k5=kl,e5=el,sub(kl=k2,el=e2, sub(k2=k4,e2=e4,k3=k5,e3=e5,xl=x2,x2=x3,x3=xl,M))) +sub(k4=kl,e4=el,k5=k2,e5=e2,sub(kl=k3,el=e3, sub(k2=k4,e2=e4,k3=k5,e3=e5,xl=x3,x2=xl,x3=x2,M))) +sub(k4=kl,e4=el,sub(kl=k2,el=e2, sub(k2=k4,e2=e4,xl=x2,x2=xl,M))) +sub(k4=kl,e4=el,sub(kl=k3,el=e3, sub(k3=k4,e3=e4,xl=x3,x3=xl,M))) +sub(k4=k2,e4=e2,sub(k2=k3,e2=e3, sub(k3=k4,e3=e4,x2=x3,x3=x2,M)))$ let kl.k2=2*(l~x3),kl.k3=2*(l-x2),k2.k3=2*(l-xl), p.kl=x1,p.k2=x2,p.k3=x3, p.el=0,p.e2=0,p.e3=0,p.ee=O,k3.ee=-kl.ee-k2.ee, kl.el=0,k2.el=-k3.el,
4-7 Positronium annihilation
243
k2.e2=0,k3.e2=-kl.e2, k3.e3=0,kl.e3=-k2.e3; x3:=2-xl-x2$ on gcd; M:=M$ operator f,h; for all p,v,m,n let f(p,v,m,n)=p.m*v.n-p.n*v.m, h(p,v,m,n)=eps(p,v,m,n); index 11,12,ml,m2; Mf :=f(p,ee,ml,m2)*f(kl,el,ml,m2)*f(k2,e2,ll,12)*f(k3,e3,ll,12) +f(p,ee,ml,m2)*f(k2,e2,ml,m2)*f(kl,el,ll,12)*f(k3,e3,ll,12) +f(p,ee,ml,m2)*f(k3,e3,ml,m2)*f(kl,el,ll,12)*f(k2,e2,ll,12)$ Mh:=h(p,ee,ml,m2)*f(kl,el,ml,m2)*h(k2,e2,ll,12)*f(k3,e3,ll,12) +h(p,ee,ml,m2)*f(k2,e2,ml,m2)*h(kl,el,ll,12)*f(k3,e3,ll,12) +h(p,ee,ml,m2)*f(k3,e3,ml,m2)*h(kl,el,ll,12)*f(k2,e2,ll,12)$ c:=M/(Mf+Mh); •-2v/7TV/27r o •—
X\ X2 (Xi + X2 — 2) vecdim 3; mass nl=l,n2=l,n3=l; mshell nl,n2,n3; vector n4,n5; index 1; in Mvpack.red"$ M:=c*4*xl*x2*x3*(ee.el*(e2.e3-E3(1,e2,n2)*E3(1,e3,n3)) +E3(ee,el,nl)*(E3(e2,e3,n3)+E3(e3,e2,n2)))$ M:=sub(el=el-el.nl*nl,e2=e2-e2.n2*n2,e3=e3-e3.n3*n3,M)$ M:=M +sub(n4=n3,e4=e3,n5=nl,e5=e1,sub(nl=n2,el=e2, sub(n2=n4,e2=e4,n3=n5,e3=e5,M))) +sub(n4=nl,e4=e1,n5=n2,e5=e2,sub(nl=n3,eI=e3, sub(n2=n4,e2=e4,n3=n5,e3=e5,M)))$ vector ec; let ee.ec=l; index el,e2,e3; Gamma:=M*sub(ee=ec,M)/4/(4*(2*pi)~3) $ let nl.n2=(x3~2-xl~2-x2~2)/(2*xl*x2), nl.n3=(x2~2-xl~2-x3~2)/(2*xl*x3), n2.n3=(xl~2-x2~2-x3~2)/(2*x2*x3); let n3.ee=-(xl*nl.ee+x2*n2.ee)/x3,n3.ec=-(xl*nl.ec+x2*n2.ec)/x3; Gammai=Gamma$ c:=Gamma/ (((1-xl)/(x2*x3))^2*(1-nl.ee*nl.ec) +((l-x2)/(xl*x3))"2*(l-n2.ee*n2.ec) +((l-x3)/(xl*x2))^2*(l-n3.ee*n3.ec)); c := 16 GamL:=sub(ee=nl,ec=nl,Gamma)$ on factor; GamL; off factor; 7, first photon along ee 2x\ - 4x2 + 2) (x± + x2 - 1) (xi - I) (x2 - 1
(a?i + x2 - 2)2x\x\ GamL:=int(GamL,x2)/6$ GamL:=sub(log(xl+x2-2)=log(2-xl-x2),GamL)$
244
4 Quantum electrodynamics
GamL:=sub(x2=l,GamL)-sub(x2=l-xl,GamL)$ on factor; df(GamL,log(1-xl)); sub(log(l-xl)=0,GamL); off factor; -64(a:f - 6x1 + 18x1 ~
Ux
i
3 (an - 2fx\
+ 12) (a:i - 1)
128 (x\ - 3xi + 3)(xi - 1) 3 (an - 2 ) 2 x ? bye; Problem. Find the photon polarization in the decay of polarized orthopositronium.
5 Weak interactions
5.1 Electroweak interaction The weak interaction, together with the electromagnetic one, is described by the Weinberg-Salam theory. Its construction may be compared in significance with the creation of the Maxwell's electrodynamics which unified electricity, magnetism, and optics. The Weinberg-Salam theory is based on the combination of two ideas — gauge (Yang-Mills) fields and spontaneous symmetry breaking (the Higgs effect). It is described in detail, e.g., in the textbooks [W1-W3]. Gauge fields
We have seen that quantum electrodynamics is a gauge theory with local 17(1)symmetry. Theories with more complicated gauge groups can be constructed too (§ 4.2). In order to construct a lagrangian invariant with respect to local transformations
245
fbcaac)F Fub.
246
5 Weak interactions
This is called the adjoint representation; its generators are (ta)bc = —if cab. The commutation relation [ta,tb] = ifabctc in this representation follows from the Jacobi identity [ta, [tb,tc]) + [tb, [tc,ta]] + [tc, [ta,tb]] = 0 (§ 1.5).
It can be proved that gauge theories are renormalizable because the coupling constant g is dimensionless. Spontaneous symmetry breaking Let's consider a real scalar field ip with L = \{d^){d^) — V(ip). Here V((p) is the energy density of a constant field (p. At small 0 is allowed. Higher degrees of (p lead to nonrenormalizable theories. What if we take m2 < 0? In this case, the point (p = 0 will be a maximum of energy rather than a minimum. Two minima
Fig. 5.1. Spontaneous symmetry breaking
Thus the lagrangian has the symmetry (p —> —ip\ the theory has two degenerate vacua, each of which does not have this symmetry. They are transformed into each other by the action of the broken symmetry transformation. Such a situation is called spontaneous symmetry breaking (in the present example, it is a discrete symmetry that is broken). Many examples of spontaneous symmetry breaking are known from statistical physics, e.g. the choice of the magnetization direction in a ferromagnetic medium when it is being cooled below the Curie temperature. We should write cp = 77 -f x o r -r 7 + X depending on which vacuum we are in, where x describes the field fluctuations around the minimum of V. The potential (in the first case, for definiteness) is equal to Xr]2x2 + A77X3 + A%4/4; it describes particles with mass r)y/2\ having triple and quadruple interactions. Now let's consider an example of spontaneous breaking of a continuous symmetry, say U(l). Let the field ip be complex. The lagrangian L =
5.1 Electroweak interaction
247
{d,j,ip)*(d^(p) — V(
A4 A x V
This potential describes particles x with mass rjV2\, and massless particles ^, with a set of triple and quadruple interactions. The field ip corresponds to a shift along the line of minima, i.e. in the direction of action of the broken symmetry. In this direction the potential does not vary. Therefore, V does not contain a term ^ 2 , i.e. this particle is massless. This is the general situation: at a spontaneous breaking of a continuous symmetry, to every infinitesimal transform whose symmetry is broken (i.e. the vacuum is not invariant) there corresponds a direction in which V does not vary (degenerate vacua), and hence a massless scalar particle. This statement is called the Goldstone theorem, and the massless particles are Goldstone bosons. The Goldstone theorem is also applicable in statistical physics. Thus, to the spontaneous breaking of the spin rotation symmetry in a ferromagnetic medium there correspond magnons with energy tending to zero in the long wave limit. Higgs effect What happens if a gauge symmetry is spontaneously broken? Let's include the electromagnetic field in the above model: L = (D^)*(D^)
- - ( ^ V " V2)2 ~ -/vvF^,
D^ip = (0M - ieAJip.
Now the gauge invariance allows us to choose the phase of
- -F^F^
+ e2r/2AMAM + V2
y
This lagrangian describes photons A^ with mass \/2ery, and neutral scalar Higgs bosons x with mass v^2A?7, interacting with each other. The Goldstone boson has disappeared, but the photon has acquired mass, i.e. it now has not only two transverse polarizations but also the longitudinal one. The number of degrees of freedom has not changed.
248
5 Weak interactions
Including a gauge field mass using the Higgs effect does not spoil the renormalizability of a theory, because it is determined by the behaviour at high momenta where the spontaneous symmetry breaking is inessential. An analogue of the Higgs effect is known in statistical physics, too: in a superconducting medium, the photon acquires mass, which leads to a finite penetration depth of a magnetic field. Weinberg-Salam theory Weak interactions are not P- and C-invariant, and left and right particles are considered separately. Left leptons form weak isospin doublets
and right ones form singlets Z/R, IR (here / = e, /x, r, and v = ve, v^ vT). The weak hypercharge Y is defined by the formula z = y + T3, where T3 is the weak isospin projection (i.e. Y is twice the mean charge of a multiplet: Y\L = YVh — —1, Y\R = —2, Y VR = 0). Let's stress that the weak isospin and hypercharge have nothing to do with the ordinary ones known from strong interaction theory. There are the SU(2) gauge field W£ interacting with the weak isospin, and the U(l) gauge field B^ interacting with the weak hypercharge. For doublets, Dp — dp — igWp1^ — ig'B^, where ra are the Pauli matrices. Singlets don't interact with W£: D^ = 9M — ig'B^. Finally, there is a doublet of complex scalar fields
with Y = +1, and its antidoublet (p+ = (ip ,^°). They ensure the spontaneous symmetry breaking due to the potential V(
0)
5.1 Electroweak interaction
249
are the raising and lowering operators. Now we can write down the Weinberg-Salam theory lagrangian , p ^ L = L\ Lt = LiDL + IRIDIR + VRibvR,
where the field strength tensors are G^u = W^v H- geabcW^W^ (because [r a /2,rV2] = fefl6crc/2), W ^ = SMWJ? - ft,WJ, BMI/ = 9 ^ , - 8UB^. One cannot write a mass term for / or v because of the SU(2) symmetry, but one can write a Yukava interaction with
where x is a real field describing a neutral scalar particle (the Higgs boson). Let's start from ascertaining what masses the gaugefieldshave got. We can decompose Lw into quadratic terms r o _ _ _ w a wa 0
~~
4
B B
W + W~ — -W° W° W° W°
and interaction terms L'w. Mass terms come from the first term in L^. Let's calculate the derivative:
and hence
250
5 Weak interactions
We see that the fields W^ have acquired the mass mw = grj/y/2. Instead of B^ we introduce the rotated fields ZM, A^:
°r
= W° sin dw + B^ cos dw
\ B*= -Z^ sin flv
(5-1.2) where cosfiw — g/g-, sin^?^ = g'/g, g = \Jg2 H- g12 {tiw is called the Weinberg angle). Quadratic terms keep their form under the rotation: _
1 2
where Z^j, = dilZv — dvZ^
+
/*
_ f1
1 4
1 4
A^v = d^,Au — dyA^. Then we have 2
_
The field Z^ has acquired the mass mz = ~gr)/\/2, and the field A^ has remained massless — this is the photon. The second term in L^ is equal to —Xrj 2x2 — A77x3/\/2 — Ax 4 /8, from where we obtain mx — \/2Xrj. Now let's proceed to leptons. The terms with id in L\ give lidl + Vidis, where I = lL + lR, v = vL + VR, h,R = [(1 ± 7s)/2]/, i/L,fl = [(1 ± 7s)/2]^. The first term in L ^ is equal to
r, + - ^ j (lRlL Therefore, a lepton acquires the mass mi = / ^ . Similarly, the second term in L/^, equal to fv(r} 4- x/\/2)F^, gives the neutrino mass m^ = /^r/. Existence of nonzero neutrino masses is not proven, and we shall put j v — 0. In this case, right neutrinos interact with nothing, and one may forget about their existence. The other terms in L\ determine the electroweak interactions of leptons. The terms with Wjjf give
^
+VLW~lL) = mw
J-^
where the Fermi weak interaction constant
a.
o2
'
has been introduced for historical reasons. The remaining terms give
( W°
^Y
5.1 Electroweak interaction
251
where the summation runs over I/J = IL,R, ^L,R, and T3 = r 3 / 2 is the weak isospin projection. Substituting the physical fields A^ and ZM, we obtain - zsin2
^
Here z — Y/2 -f X3 is the electric charge, so that the first term is the usual electromagnetic interaction, and e = #cos$vpsin$ty. The second term can be rewritten as a sum over -0 = /, v as
where CA = i l for upper and lower components of the weak isospin doublets, and cy = CA — 4 sin2 # w Finally, let's consider the last part of the lagrangian — the gauge fields' selfinteractions L'w. Substituting G^v = W*v + geabcW*Wf; in L^,s and taking into account
abc ab c>
£
e '
= Sbb> Scc' - Sbc> Scb\ we obtain
Lw = -~eabcWa
2
WbWc - — (W aWa)2
4-
2
—W aWaWbWb
Now we calculate the sums over a, 6, c, taking account of e:0"^
= —i:
L'w = igW+W-W^ - igW°W;W+, + igW^W+
In the first term, we can move the derivative from W® to W^, omitting a full divergence, so that W® is singled out as a common factor in all cubic terms. Substituting the physical fields A^ and Z^ we obtain for these terms
The terms 2
- y [(W+Wp2 - (W+W+)(W~W-)] require no further manipulations. The remaining terms are equal to
- e2 [(W+W-)(AVAV) (W+A^W-Av)} [2(W+W-)(AVZU) - (W+AJiW-Z,,)
- {W~J.
Now let's discuss the inclusion of quarks. The difficulty here is that the primary quarks forming left doublets and right singlets,
252
5 Weak interactions
(i is the generation number) don't coincide with the physical quarks having definite masses. The Yukava coupling constants f$, f^J are matrices producing matrix mass terms. The mass matrices are diagonalized in the basis of physical quarks ui = (U+)ijUj, di = (D+)ijdj with the masses m ni , m^ (U, D are some unitary matrices). At the same time, the terms in the Lagrangian describing the electroweak interactions of quarks involve the primary quark fields u[, d[. The interactions with A^, Z^ have the form of sums over ip = u[, d[: ^ ij)... ijj. Substituting the physical fields u[ = UijUj, d[ = Dijdj, we obtain sums of the same form. But the interaction with W± gives y v ( diW^Ui + u^W di\ — 2_^ [di W^Ui + UiW a^ 1 , i
i
where d" = Vijdj, V = U+D. Of course, it could be equally well represented as
where u" — (V+)ijUj. The mixing (Kobayashi-Maskava) matrix Vij is a unitary 3 x 3 matrix; various parametrizations of it can be found in the literature. Let's summarize. The Weinberg-Salam theory lagrangian (5.1.1) contains the following parameters: the gauge coupling constants g, g'\ the Higgs coupling constant A and the vacuum average rj; the Yukava coupling constants fi, / ^ , f%. Instead of them, we shall use the elementary charge e and the Weinberg angle $w (g = e/sin$v^, g' = e/cos^\y); the Fermi constant G (rj = l/\/2\/2G) and the Higgs boson mass mx (A = \/2Gm^); the lepton and quark masses and the Kobayashi-Maskava mixing matrix Vij. The masses of W^ and Z° are not independent parameters:
A
^
(5.1.3)
Now we can obtain the Feynman rules in the Weinberg-Salam theory. Some of them coincide with the ones known from quantum electrodynamics. What's new are the W and Z propagators having the form (4.3.3) with the corresponding masses, and the Higgs propagator (4.3.1) with the mass mx. The vertices are presented in Fig. 5.2. Here Vi23 = <W2 (Pi -P2)M3 +<W 3 (Pa -Pz)in +*/*3A*I (P3 ~ Pi)^? w n e r e a ^ the momenta are considered incoming; T1234 = 2S At energies much less than mw,z, two vertices of fermion interactions with W or Z and a propagator between them are merged into an effective four-fermion x7/x( c y+ C A7S) vertex ^ [ ^ ] [ ^ - / ] 7 M ( l - f 7 5 ) x 7 / x (l+7 5 ) or i^j^cy+CA^) (Fig. 5.3), where the two 7-matrix expressions refer to the two fermion lines, and in the quark case the corresponding Vij factors are included.
5.1 Electroweak interaction
r
2 ie 2 —T sin v Jj£
253
= ie cot $w V\123
1234
= -ie cot iVT1234
1234
2
2
JKL
i e COt
X
= -3i\f2Gm
_V
= 2i\/2Gm
X.
= 2i\/2Gm
1^
= 2i\Jy/2G m2w
2
Fig. 5.2. Feynman rules in the Weinberg-Salam theory
a
b
Fig. 5.3. Feynman rules for low energy weak interactions
Classification of weak processes At high energies E > mw,z there is the single electroweak interaction characterized by the constant a. Somewhat conventionally, one can refer the processes of
254
5 Weak interactions
W and Z decay and production to weak interactions. W~ decays into l~V\, qq', where q = d, s, b;q' = u,c, and Z decays into /I, qq (§ 5.2). The production of a single W or Z can take place in the inverse processes: the resonance e + e~ —> Z is observed at e + e~ colliders; at pp colliding beams, the processes qq' —> W, qq —> Z take place. At e + e~ colliding beams, W are can only be produced in pairs: e+e~ —> W~*~W~ (§ 5.4). There are proposals about construction of je and 77 colliders; the reactions je~ -» W~u e and 77 —> W+W~ should take place on them (§ 5.4). The processes of Higgs production and decay may also be referred to here. But the mechanism of these processes essentially depends on the mass mx which is currently unknown. Therefore, we shall not consider these processes. At low energies E
I-,
,
1_
\
=AfJi - (uj^u - cfy^d) + - (uj^u -f 57Md - 257^5) , \_1 o J one can see that from the strong interaction point of view the photon behaves as a vector isovector (with 1% — 0) or isoscalar particle (like p° or a;0), and the isovector photon forms an isotriplet together with the vector W± (the isovector photon has G = +1, and the isoscalar one has G = -1). This allows one to obtain relations between weak and electromagnetic processes from the isospin symmetry. Thus, the r decays into nonstrange vector hadron systems are related to the cross sections e + e~ —> 7* —» hadrons with / = 1 (§ 5.2). Similarly, in the processes W~*~ —>• us, W~ —> su, W^ behaves as a vector or axial strange particle with I — \ and J 3 = ±\ (like K^), from where isospin relations in strangeness changing semileptonic decays follow. The vector nonstrange and strange W^, together with the isovector and isoscalar photons, are the members of a flavour SU(3) symmetry octet (like TT±, K±, TT0, and 77), which allows one to relate a wider range of weak and electromagnetic processes, though with a lower accuracy.
5.1 Electroweak interaction
255
Nonstrange particles semileptonic decays (n —» pe ve, TT —> [i v^) are caused by the interaction d —> uW~, W~ —» /~I7/ (there exist strangeness conserving decays of strange particles too, e.g. E~ —> Ae~Ve). Strange particle decays may be semileptonic (s —)• uW~, W~ —» l~T>i) or nonleptonic (5 —> uW~, W~ —•» du). It is evident that I" is produced in semileptonic decays of the particles containing s, while Z+ is produced in decays of those containing ~s. For example, K-mesons can decay into IV, nlV, 2TT/Z7; 2TT, 3TT. The neutral kaons have an interesting property: in the second order in the weak interaction, the transition K° «->> K can happen, leading to some very interesting phenomena (we shall not discuss them here). Semileptonic hyperon decays (A —> pl~V\ E~ -» nl~V\ E~ -> A/~z7, E°/~I7; S° —>• E+/~z7; Q~ —>• H°/~I7) have lower probabilities than nonleptonic ones (A —y pn~, ntr0; E + -> p7T°, n7r+; E~ -)- WK~\ S° -> ATT0; S~ -> ATT"; f2~ As we mentioned earlier, in strangeness changing semileptonic decays, W carries out the isospin | , which allows one to obtain relations between decay rates. In nonleptonic decays, W carries out the isospin | , and then transforms into a hadronic system with 7 = 1. The isospin change can be equal to \ or | . Amplitudes with A7 = -| are strongly enhanced as compared with ones with A7 = | , due to strong interactions dynamics (the reasons for it are not completely understood). Charmed hadrons' decays approximately reduce to free c-quark decays. Following this model, all charmed mesons and baryons should have the same life time, what does not quite fit the experiment. But it is sufficient for rough estimates. Semileptonic decays are subdivided into Cabbibo-allowed ones (c —> sl^v) and Cabbibo-suppressed ones (c —> dl+v). Nonleptonic decays are classified into Cabbibo-allowed ones (c —> sud), two kinds of suppressed ones (c —> sus, dud), and doubly suppressed ones (c —> dus). Decays of 6-quark containing hadrons are well described by the free quark decay model. It predominantly transforms into c, and with a lower probability into u. Semileptonic (b -> cl~v), Cabbibo-allowed (b -> cdu, esc) and suppressed (c -» csu, ede) nonleptonic decays can occur, as well as similar transitions into u. I?°-meson can transform into B in the second order in the weak interaction (like K°). Neutrinos can scatter at leptons and nucleons. At not too high energies, (quasi-) elastic scattering reactions like vN —> l~~N, vN dominate. At high energies and momentum transfers (the deep inelastic region), neutrinos scatter at practically free quarks. Heavy quark production can occur in such reactions.
5 Weak interactions
256
5.2 W and Z decays W decays Let's first define the lepton tensor (as in § 4.4) ml
/
\
G
(Fig. 5.4), which will be useful in many problems. Here OM = O^ — 7M(1 + 75) is the W interaction vertex, and 2p(p) = p + m.
v Fig. 5.4. Lepton tensor
input__case nil$ in llkpack.red",llpol.redll$ operator r,V,Tl; vector a,pi,p2,11,12; nospur f; g(f,ll)*(l+g(f,a))$ for all 11 saveas V(ll); % Vertex g(f ,pl)+sqrt(pl.pl)$ for all pi saveas r(pl); °/« Density matrix spur f; 4/sqrt(2)*V(ll)*r(pl)*V(12)*r(p2); 8(-h.l2Pi>P2 + h'Pih-P2 + I1.p2h.p1 + eps(/i,/2,ffi,P2)s)
V2
for a l l p i , 11,p2,12 saveas Tl(pl,ll,p2,12);
°/0 Lepton tensor
It is interesting that the lepton tensor, taking account of polarizations
can be expressed via T ^ p i , ^ ) - Specifically, 2p(p,a) = (p + m)(l — 75a) = (p — 77175 a) -h (ra — P75a), where the first bracket anticommutes with 75, and the second one commutes. If we retain the first bracket, then the factor (1 + 75) from OM will remain itself when transferred through it and through 7^, and we shall obtain (1 + 7 5 ) 2 = 2(1 + 75). On the other hand, if we retain the second bracket, we shall obtain (1 — 75) (1 + 75) = 0 instead. Thus, one may replace 2p(pi,ai) ->pi— 77117561. Moreover, one may omit 75, because (I+75) from OM stands to the left of 2p(pi, ai), and (1 + 75)75 = 1 +75: 2p(p±, ai) -» pi — m\a\. Similarly, 2p(p2,a2) -> p2 — m2d2. Therefore, ,p2 -
m2a2).
(5.2.1)
5.2 W and Z decays
257
Now it's easy to calculate the decay rates W —•» / z/j, qq' (where q is a charge — | quark, and q' is a charge -f | one) (Fig. 5.5). A dimensional estimate gives Tw ~ Gm\v ~ mwot. The program obtains r
_~—vv
m\+m\
M
2m
6V2TT L !
m m
,
i
,
{
(m^-m2,)21 2m
w
m
w
m
m
2
m
(5.2.2)
m m
l
w
(in the qq' case, the factor Nc\Vq>q\2 must be included).
Fig. 5.5. Decay W -* I v\
In the case of a W with polarization e*,
(m2 - ml)2'
11
77%
1 j_ = v
y
m
m
m^
m^
mm
l
m^r
m^
mi,m2-»o
omlm2
,
ml
—— 1 /I 4-
m
l
m^
-U
m
2
?rz^
9
m
l
m^
9 m2
m^
Gm
W
m w mi,m2->o 4v27r (5.2.3) where n is the decay axis. If fermion masses may be neglected, they behave like neutrinos, i.e. only negative helicity fermions and positive helicity antifermions interact with W. Therefore, the spin projection of the produced fermions onto the decay axis is — 1; the orbital momentum projection is zero, and hence the W spin projection onto the decay axis must be —1. Therefore, only Fj_ does not vanish. The polarization of a fermion produced in the unpolarized W decay is Ion-
258
5 Weak interactions
gitudinal, because there are no selected directions except n. Its degree is 1(5.2.4) YY)f*
m
w
<m°*
m
w
nm*'
im*1
nm^1 nm**
m
w
in accordance with the above reasoning. This program, as well as most of the programs in this Chapter, uses the procedure package kpack from § 4.1. For polarization calculations, it also uses the procedures ort and epol from § 4.4. Problem. Find the polarization of a fermion produced in the polarized W decay. 7, W decay 7.
mass p=l,pl=ml,p2=m2; mshell p,pl,p2; °/ W=1 0 M l e t p.pl=(l+ml"2-m2^2)/2,p.p2=(l+m2^2-ml^2)/2, pl.p2*(l-ml~2-m2~2)/2; index 11,12; MM:=-(p.ll*p.l2-11.12)*Tl(pl,ll,-p2,12)/3$ remind 11,12; Gamma:=MM/2*ph2(l,ml,m2); 1 G is assumed
r := {\Jm\ -2m\m\
- 2m\ + m\ - 2m\ + 1
(-mi + 2m\ml - m\ - m\ - m22 +
2))/(12V^TT)
l e t abs(ml"2~l)=l-ml^2; Gamma:=sub(m2=0,Gamma)$ on factor; Gamma; off factor; sub(ml=0,Gamma); (mj + 2) (mi + I) 2 (mi - I) 2
% Polarized W eW:FsecondCort({p.1,pi.1}))$ MM1:=-sub(1=11,eW)*sub(1=12,eW)*T1(pi,11,-p2,12)$ index 11,12; GamL:=MMl/2*ph2(l,ml,m2); 70 Along eW GamL := (ym\ — 2m 2 m 2 — 2 m2 + m\ — 2 m2 -f 1 (-m\ 4- 2m 2 m 2 + m2 - m\ + m 2 ))/(4\/2 7r) remind 11,12; GamL:=sub(m2=0,GamL)$ on factor; GamL; off factor; sub(ml=0,GamL); (mi + l) 2 (m x - l) 2 m 2 0 clear eW,MMl,GamL; off factor;
5.2 W and Z decays
259
mass n=i; mshell n; l e t n.p=0,n.pl=0,n.p2=0; GamT:=-Tl(pl,n,-p2,n)/2*ph2(l,ml,m2); °/0 Cross eW GamT :=
— 2 m 2 m 2 — 2 m 2 -f m\ — 2 m | + 1 (— m
yfm\
2
—
m\ -f 1)
4\/2TT
GamT:=sub(m2=0,GamT)$ on f a c t o r ; GamT; off f a c t o r ; sub(ml=0,GamT); (mi + I) 2 (mi - I) 2
4A/2TT
clear n.p,n.pl,n.p2; clear GamT; 7o Final particle polarization vector a l ; l e t pl.al=0,p.al=p2.al; el:=second(ort({pl.l,p2.1}))$ index 11,12; MMp:=(p.ll*p.l2-11.12)*Tl(pl-ml*al,ll,-p2,12)/3$ remind 11,12; clear p l . a l , p . a l ; Gamma:=MMp/2*ph2(l,ml,m2)$ l e t abs(ml)=ml; Gamma:=epol(al,z*el,Gamma)$ Pol:=df(Gamma,z)/sub(z=0,Gamma)$ on factor; num(Pol); off factor; den(Pol); (ml — ml — 2) (mi + ra2 + 1) (mx + ra2 - 1) (mi - m2 + 1) (mi — m2 —
l)
lm\ - 2 m\ ml - 2 m\ + m\ - 2 m\ + 1 (m* - 2 m\ m\ -f m2 + m\ + m^ - 2) Pol:=sub(m2=0,Pol)$ on factor; Pol; off factor; sub(ml=0,Pol); - (ml - 2) 1
ml
clear MMp,Gamma,Pol;
The t-quark is heavier than W. Therefore, it decays into qW+ (q — 6, s, d) (Fig. 5.6). For this process, \M\2 is obtained from the previous one by multiplying by | (from the spin averaging), and the decay rate is
r=
G\Vtq\2m*
ml
8V2TT
1 ~\
m o —>0
ml j
\
r —
mi - 2 — | - 2m;
1 + 2 mi 2 m
m$xr
(5.2.5)
260
5 Weak interactions
Fig. 5.6. Decay t -> qW+
7, t decay MM:=3/2*MM$ Gamma:=MM/(2*ml)*ph2(ml,l,m2) ;
T := (\Jm\ - 2m\m\-
2m\
m\ -
2m\
(-m} + 2 ml ml - m\ - m\ - m\ + 2))/(8 \/2raf TT) Gamma:=sub(m2=0,Gamma)$ on factor; Gamma; off factor; clear Gamma; (m 2 + 2) (mi + I) 2 (mi - I) 2
Decay r~ —> vTp~ This decay (Fig. 5.7) is very similar to the previous one, only now the virtual W~ transforms into p~ via the vertex mw \/ G/'\f2Vu&mp]'pe^ (cf. § 4.4). Therefore, the vertices produce the factor G 2 m 2 / 2 / 2 instead of
r=
T / |2 /rri 3 G 2 |\Vud\ ™> T Jp 16TT
i
m* mi
(5.2.6)
mi
A similar formula (with |Kt S|2) holds for the decay r
—> vTK* .
VT p
The decay r —> v rp is related to the reaction e + e —>• p° due to the isospin symmetry (§ 5.1). The formula (5.2.6) can be rewritten in the form xr
p
T(T~ -» u Te"i/e) 2
where xp = m p/m% (the decay rate r" Because the quark composition p° = contains
\2
J P
/i
,
r»
\ /-i
2\
m^ vTe~ve will be calculated in § 5.3). - dd)/y/2, its 7 interaction vertex
efP,
5.2 W and Z decays
261
and the cross section (§ 4.4) is
Therefore, 2
ff R o(m p
2
Tx)(l
+ 2x)(l - z) 2 dz.
(5.2.7)
The formula (5.2.7) is applicable to any vector nonstrange state with I — 1 in the r decay and the e+e~ annihilation, for example 2?r and 4TT, because they always can be considered as infinitely many p type resonances. The program finds the angular distributions of the longitudinally- and transversely-polarized />-mesons in the polarized r decay:
\
4 T T ^ aSZ
16?r 16TT
1 ( C
\
)
,
mj I m^ \
m* )
where C is the r polarization, and d is the angle between it and the p flight direction. 7« Decay tau -> nu rho - polarization effects m2:=0$ °0/ m tau = 1 m rho = ml rl:=sub(l=ll,el)*sub(l=12,el)$ °0/ longitudinal rho polarization r t :=pl .ll*pl .12/ml"2-11.12-rl$ °0/ transverse rho polarization l e t p.n=0,p2.n=-pl.n,pl.n=-c*PP(l,ml,0); index 11,12; °0/ n is the tau polarization °0/ Angular distribution of longitudinally polarized rho GamL:=Tl(p-n,ll,p2,12)*rl*ml^2/sqrt(2)/2/2*ph2(l,ml,0)$ °0/ Angular distribution of transversely polarized rho GamT:=Tl(p-n,ll,p2,12)*rt*ml^2/sqrt(2)/2/2*ph2(l,ml,0)$ remind 11,12; on factor; GamL; GamT; off factor; -I)2 1 6 7T
- ( c - l)(mi + l) 2 (m x - l) 2 m 2 8TT
clear m2,rl,rt,GamL,GamT; clear p.n,pl.n,p2.n; 7T decay The 7T~-meson can transform into a virtual W~. The vertex has the form fnPfj, (where fn is a constant with the dimensionality of mass), because the momentum pM is the only vector in the problem. Therefore, n~ decays into
262
5 Weak interactions
I v\, where I = e, [i (Fig. 5.8). Due to the angular momentum conservation, l~ has positive helicity, which is forbidden at mi = 0 (when the lepton behaves as a neutrino). Therefore, the matrix element M ~ uip( 1+75)^17 = ra/7Z/(l + 75)7;^ is proportional to mM, and a dimensional estimate gives T ~ G2mnf%mf. It shows that the decay n~ -± e~Ve is approximately {me/m^)2 times less probable than TT~ -» //~I7M. The program obtains
A similar formula (with |Ks| 2 ) holds for the decay K~~ -> l~Vi.
The matrix element squared of the decay r~ -> vTix~ (Fig. 5.9) is given by the same formula, and the decay rate is
(similarly for r~ —> i/TK~).
F i g . 5.9. Decay r
—>• z/r7r
•/. pi decay m2:=0$ MM:=-Tl(pl,p,-p2,p)/sqrt(2)$ Gamma:=MM/2*ph2(l,ml,0)$ % pi -> mu anti-nu on factor; Gamma; off factor; (rm + l)2(mi - I)2 mj 8TT
Gamma:=MM/(2*ml)*ph2(ml,1,0)$ % tau -> pi nu on factor; Gamma; off factor; - I)2 clear m2,MM,Gamma; bye;
5.2 W and Z decays
263
Z decays A dimensional estimate gives (Fig. 5.10) T(Z —> / / ) ~ G m | ~ ra^a, where / is a lepton or a quark. The Z interaction vertex is introduced in the program (cy is assumed in it, and c = CA/CV), and the lepton tensor for Z is calculated. The decay rate is Gm\ 24V2TT
1 - 4 - 171%
m%
(5.2.11)
mt —»0
(for quarks this should be multiplied by the number of colours iVc, see § 6.1). In the case of a polarized Z, the angular distribution is
r,, = *&. r. =
Gm%
U-4
A,
'1-4-
4
(5.2.12)
Fig. 5.10. Decay Z -> / / Problem. Find the polarization of Z decay products.
input_case nil$ in "kpack.red"$ °/, Z decay operator r,V,Tl; vector a,pl,p2,p,ll,12,n; nospur f; let p2.p2=pl.pl; g(f,pl)+sqrt(pl.pl)$ for all pi saveas r(pl); g(f ,ll)*(l+c*g(f,a))$ for all c,n saveas V(c,lD; °/0 Z vertex spur f; 4/(2*sqrt(2))*V(c,lD*r(pl)*V(c,12)*r(p2);
+ /1J2P1.P1 ~ i1.i2P1.p2c2 - i1.i2P1.p2 + i1.
264
5 Weak interactions
for a l l c , p l , l l , p 2 , 1 2 saveas T l ( c , p l , l l , p 2 , 1 2 ) ; °0/ Lepton tensor l e t p.p=l,pl.pl=ml~2,p.pl=l/2,p.p2=l/2,pl.p2=l/2-ml~2; index 11,12; MM:=-Tl(c,pl,ll,-p2,12)*(p.ll*p.l2-11.12)/3$ remind 11,12; Gamma:=MM/2*ph2(l,ml,ml); sub(ml=0,ws); _
y/4ml
+ l(-Ac2m\
+ c2 + 2m\
24V2 7T n:=pl-p2$ n:=n/sqrt(-n.n)$ GamL:=-Tl(c,pl,n,-p2,n)/2*ph2(l,ml,ml); sub(ml=O,GamL); °0/ Along GamL := 0
clear n; mass n=i; mshell n; l e t p.n=0,pl.n=0,p2.n=0; GamT:=-Tl(c,pl,n,-p2,n)/2*ph2(l,ml,ml); sub(ml=O,GamT); °0/ Cross
GamT := V
clear p.n,pl.n,p2.n; clear GamL,GamT; bye;
5.3 Weak decays ji decay
A dimensional estimate of the decay rate /x~ —>> p^e~Ve (Fig. 5.11) gives F ~ G 2 m^. The program obtains the simple formula for the matrix element squared
for any masses of all particles. A simpler derivation of this formula is presented in [Wl]. In the me = 0 approximation, the program obtains (xi = 2ei/m^)
dxedxi7 dxe~
967^^^
Xeh
- 1927T3-
This program, as well as other programs in this Section, uses the lepton tensor Tl calculated in § 5.2.
5.3 Weak decays
265
Pi Fig. 5.11. fj, decay
input_case nil$ in "kpack.red","lepten.red n $ °/0 mu decay X mass p=l,pl=ml,p2=m2,p3=m3; mshell p,pl,p2,p3; index 11,12; MM:=-Tl(p,12,pl,ll)*Tl(p2,ll,--p3,12)/2; MM := 64p.p3p1.p2 remind 11,12; '/• very simple result l e t p.pl=l-(x2+x3)/2,p.p2=x2/2,p.p3=x3/2, p2.p3=(-l+ml~2-m2~2-m3~2+x2+x3)/2; ml: =m2: =m3: =0$ °/0 massless case Ganma:=MM/2/(4*(2*pi)~3)/4; °/, d Gamma / dx2 dx3 _ s 3 ( - a?3 + l) 16 7T3 Gamma:=int(Gamma,x3,l-x2,l); % d Gamma / dx2 =
xj(-2x 2 + 3) 96 7T3
Gamma:=int(Gamma,x2,0,1);
7, Gamma
' 192 7T3 GammaO: =Gainma$ c l e a r Gamma; In order to consider the polarized \x decay, it is sufficient to replace p by p — ma in |M| 2 . Let's average \M\2 over the neutrino momenta pi, p 3 at a fixed
4 " " (n = p\ — P3, n2 — —q .
dT
IS
4*"w
12 V ""
2
). The program obtains G2ml
,r_
„
tp-J = — (q 6^ + 2qllqv)
a2 y
12
266
5 Weak interactions
In the zero mass approximation, the electron behaves like a neutrino, i.e. it is produced with negative helicity only. Therefore, at the maximum energy xe -> 1, when the neutrino and the antineutrino fly in parallel carrying out zero angular momentum projection, the electron prefers to fly opposite the muon spin (1 — cos$). At a small energy xe —>• 0, when the neutrino and the antineutrino fly to the opposite sides carrying out the unit angular momentum projection, the electron compensates the projection ~ and prefers to fly along the muon spin (1 + costf). % Polarized mu vector q,z; q:=pl+p3$ let p.z=0,p2.z=-x2/2*c,pl.z=-p2.z-p3.z; index 11,12; MMp:=64*(p-z).Il*p2.12*(q.q*11.12+2*q.ll*q.l2)/12$ remind 11,12; Gamma:=MMp/2/(4*(2*pi)~3)/4*x2;
__ xl(-2cx2 + c - 2x2 + 3) :
96 7T3
~
G a m m a : = i n t ( G a m m a , x 2 , 0 , 1 ) ; c l e a r MMp,Gamma;
r •= ~ c
+ 3
576 7T3
When one takes into account the nonzero electron mass, 9
rlr
Q6TT 3
axe
yDTr
V
;
m
[
T - ro/(—),
m2
f(x) -
»
v
;
2
6
e
\/
J V
In e rr? 2
A
m
8
M
1 - Sx + 8x - x - 24x4 logz.
(5.3.3)
Problem. Taking into account the nonzero mass, the electron is not completely polarized. Calculate its polarization. °/, taking the electron mass into account clear m2; Gamma:=MM/2/(4*(2*pi)^3)/4$ let abs(l-x2+m2~2)=l-x2+m2~2; Dalitz(1,m2,0,0,x2/2,E3m,E3p)$ Gamma:=int(Gamma,x3,2*E3m,2*E3p); clear E3m,E3p;
yj — 4 m\ + x\ ( - 3 m^ X2 + 4:1712 + 2x1 - 3X2) :=
96^ load_package algint$ Gamma:=int(Gamma,x2,2*m2,l+m2"2)$ for all x let log(-x)=log(x); for all x,y let log(x*y)=log(x)+log(y),log(x~y)=y*log(x); Gamma:=Gamma$ for all x clear log(-x); for all x,y clear log(x*y),log(x~y); Gamma/GammaO; clear Gamma,GammaO;
- 2 4 Iog(m2)m2 - m\\
%
\
These formulae refer to the decays r~ —> vTe~Ve, vTji~V^ too. In addition to the leptonic r decays, sernihadronic decays r -» v Tdu, vTsu occur. In the free quark approximation, rates of these decays differ from the leptonic ones by
5.3 Weak decays
267
the factor iVc|V^d| or iV c |V us | 2 (Nc is the number of colours, see § 6.1). This approximation is not very accurate because the r mass is not sufficiently large. The same formulae with the factor Nc and the corresponding \Vij\2 refer to the c and b quark decays. They are not very accurate in the c case. Problem. All decay products' masses may be nonzero. It is conventional to write down the decay rate in the form F = Tof(mi/m, m2/m, 7713/771). Because of the symmetry of \M\2 with respect to the interchange 1 «->• 2, f(xi,X2,X3) = f(x2,xi,xs). We have calculated /(0,#,0) (the formula (5.2.3)). In general, f(xi,X2,X3) contains elliptic integrals; when one of the decay products is massless, the integrals are calculable in elementary functions. Calculate /(0,#2,2:3) = /(2?2,0,2:3) and f(xi,X2,0).
Neutrino scattering The scattering v^e~ -» \i~v e (Fig. 5.12a) is another channel of the same reaction. A neutrino has only one polarization state, therefore it is superfluous to perform the polarization averaging (if one considers that right neutrinos exist but don't interact, then the averaging will give the factor \\ but then the incident neutrino should be considered completely polarized, which will give the factor 2). The program obtains
^ ^ at
4, , ^lZ
2!1Z4
n s — mj
TT
(5.3.4)
SS
The same formula multiplied by |V^ q | 2 describes the scattering v\q —>• l~q' (q is a charge — | quark, and q' is a charge | one).
Fig. 5.12. Neutrino scattering
The scattering vee reaction:
—^1/^/1
(Fig. 5.12&) is the ^-channel of the same
da = O2(t-ml)(t-ml) dt
7T
(s — ml)2
The same formula multiplied by \Vq>q\2 describes the scattering vee~ —> qq'. At s ^> m 2 , a dimensional estimate of these cross sections give a ~ G2s. In the first case (Fig. 5.13a), the angular momentum projection on both the initial and final particles' direction of motion is zero. The angular distribution is isotropic in the centre of mass frame, which is natural for a point-like interaction. In the second case (Fig. 5.136), the angular momentum projection on
268
5 Weak interactions
both directions is equal to unity. The probability amplitude that the angular momentum, equal to 1 and having the projection 1 on an axis, has at the same time the projection 1 on another axis directed at an angle $, is equal to sin2 T|. Therefore, the angular distribution has the form t2/s2 = sin4 f. The total cross section thus becomes three times smaller. One can understand this without considering the angular distribution, too: only one of three possible projections of the final particles' angular momentum on the collision axis is allowed.
Fig. 5.13. Helicity conservation at neutrino scattering Problem. Consider the scattering i ^ e " —> (Fig. 5.12a, c); vee~ -> Vee~ (Fig. 5.126, c).
vlle~, v\q —• v\q (Fig. 5.12c); v ee~ —• v ee~
°/0 nu-mu e -> mu nu-e l e t pl.p2=(s-m2~2)/2,p.p3=(s-l)/2,p.pl
, 2 7T [m^ —
r s)
Mandel (0,m2,1,0, s,tm,tp) $ sigma: =int (sigma,t,tp,tm) $ clear tm,tp; on factor; sigma; off factor; °/0 sigma (* - i) 2 7T S
°/0 e ant i-nu-e -> mu anti-nu-mu sigma:=sub(s=t,t=s,MM)/(64*pi*JJ(s,0,m2)~2)$ 7, d sigma / d t on factor; sigma; off factor;
- {ml - t){t - 1) (ml — s)2n Mandel(0,m2,1,0,s,tm,tp)$ sigma:=int(sigma,t,tp,tm)$ clear tm,tp; on factor; sigma; °/0 sigma ((2s + 1 ) 5 + (* + 2)m22)(s - I) 2 6TT5 3
bye; Problem. Consider the reaction e + e~ -> /i + /x~, qq taking into account both 7 and Z in the intermediate state. Find the charge asymmetry — the difference of the production probabilities of fj,^ and JJL~ (or of q and q) at an angle $.
5.3 Weak decays
269
Meson semileptonic decays Let's consider a decay of a pseudoscalar meson into a pseudoscalar meson and leptons, such as K~ -> 7r°rvh D~ -> K°l~Vu B~ -> D°l~Vi (Fig. 5.11). Only the vector current has the nonvanishing matrix element /x,
P = p + pu
q=P~Pi.
The program obtains
EM) - m2MEv +^(E°M-
EM)}
(5.3.6) where Ev = TU—EM—EI is the neutrino energy, and E^- = (m 2 +m 2 Vf -m 2 )/(2m) is the maximum meson energy. The integral in E\ can be calculated in a general form because q2 = m 2 + m2M — 2ITIEM > dV dEM
=
G2\Vqq,\2m2pM
(E°M-EM\2 )
fl [|PM (l + 2 ^ - ) + \^(™ mf m2-m2M
2
2
+ m2M + 2m£M)]
(5.3.7)
mf q2
where PM = V^h ~ m M *s t n e m e s o n momentum. If one may neglect the lepton mass, then / _ doesn't contribute, and
G2\Vqq,\2mf2
dT
^ 7 ^ : i = — J ^4TT3 3 — 2
dT
v2E» El ~ m ( £ « -
EM
>\ '
|
=
dEM
12TT3
In this case the virtual W is pure vector, and hence the decay occurs in the P-wave, which explains the threshold behaviour p z M .
input_case nil$ in "kpack.red","lepten.red"$ */. Ml -> M2 1 anti-nu o/
mass p=l,pl=ml,p2=m2,p3=0; mshell p , p l , p 2 , p 3 ; l e t p.pl=El,p.p2=E2,p.p3=l-El-E2,pi.p2=(l-ml~2-m2~2)/2-1+E1+E2,
270
5 Weak interactions
pl.p3=(l+m2-2-ml^2)/2-E2,p2.p3=(l+ml^2-m2^2)/2-El; °/ meson t r a n s i t i o n current - f+ assumed, f = f-/f+ 0 factor f; vector j ; j:=p+pl+f*(p-pl)$ MM:=Tl(p2,j,-p3,j)/sqrt(2)$ on gcd; Gamma:=MM/2/(32*pi~3); % d Gamma / d El d E2
r := (f2m\(2E1 "1 ^J TP
- ml + m\ - 1) + 2fm\(2Ex + 4E2 + m\ - m\ - 3) +
TP
d
TP
trm 2
Q ZT1
i
1 £} I7 | 2
Q
IT1
«~-, 2
i £* 77i
IOJC/IJC/2 ~ D-tSi 777-2 ~ o x^i + ID £L/2 — o £L/217l2 — ID H/
2
—
-__ 2 _~~ 2
m±m
2
m\ + 3 ml + 4)/(32?r3) let abs(2*El-ml^2+m2^2-l)=l-2*El+ml^2-m2^2; Dalitz(l,ml,m2,0,El,E2m,E2p); Gammal:=int(Gamma,E2,E2m,E2p)$ clear E2m,E2p; on factor; Gammal; off factor; °/0 d Gamma / d El 2 _
m2
(16Ef - 12E\f2m\ - SEfml - 4E\m\ 12E1f2ml - 12E1 fm\m22 + 12Exfm22 6 /
TTT-i ^ 2 ~~ ^ /
^ 2 + ^ / ^ H ^ 2 ~" ^ / ^ 2 ~~ ^ ^ 1 ^^2
[2J5?i - m\ + m22- I) 2 )/(96 {2E1 sub(m2=0,Gamma); sub(m2=0, Gammal); A TP TP O Z7 S 2 + m2 + 1 4-C/1XI/2 ~ Z JC/1 -
- ml
-I)
3
-3)
1
A/JE;2 -
m\
(El
-m\)
1 2 7T3
Problem. Find the final lepton polarization in this decay. Problem. Consider a semileptonic decay of a pseudoscalar meson into a vector one: introduce form factors and calculate the decay rate and the final meson polarization.
Decay TT~ —> 7r°e~77e The form factors of this decay are related to the pion electromagnetic form factors by the isospin symmetry. Due to the current conservation, /_ = 0. The meson TT~ = du can transform into TT°= (uu + dd)/y/2 due to the d —>• u transition (the charge 1) or due to the u -> d one (the charge - 1 ) . In the electromagnetic transition TT~ -> TT~, the charge i s - | - | = - l . Therefore, the weak form factor f+(q2) is obtained from the electromagnetic one f(q2) by multiplying by y/2. The form factor may be considered constant f(q2) = 1, because q2 is very small in this decay. The energy released in this decay is A = m^- — mno
5.3 Weak decays
271
mw —> oo. The power of A in the answer can be reconstructed by dimensionality. The program obtains
30TT3
(5.3.9) where T® = (A2—m 2)/(2m)is the maximum TT0 kinetic energy, q2 = A2—2m nT7n and v = yjl — m 2 /A 2 is the maximum electron velocity. The total rate can be obtained by integration of either the electron spectrum or the meson one. procedure NonRel(x); begin scalar a,b,c; let abs(d)=d; c:=sub(ml=l-d,m2=m2*d,El=l-d+d~2*Tl,E2=d*E2,x); a:=num(c); b:=den(c); while (c:=sub(d=0,a))=0 do a:=a/d; a:=c; while (c:=sub(d=0,b))=0 do b:=b/d; return a/c end$ Gamma:=NonRel(2*sub(f=0,Gamma)); _ AEl - AE2 - 2Ti - ml + 1 4TT 3
Gammal: =NonRel (2*sub (f=0, Gammal)) $ let abs(2*E2+ml~2-m2~2-l)=l-2*E2-ml~2+m2~2; Dalitz(l,m2,ml,0,E2,Elm,Elp)$ Tlm:=NonRel(Elm-ml)$ Tlp:=NonRel(Elp-ml)$ Gamma2:=int(Gamma,Tl,Tlm,Tlp)$ let abs(v)=v; on factor; Gammal:=sub(Tl=v~2/2,Gammal); Gamma2; off factor;
Gammai :=
(v2 + 3) ml - 2 (v + 1) {v - 1) v2) (ml + v2 - I)2 v
l - m\ (E2 -
12 (V + 1) 3 (U - 1)37T3
l)2E2
Gamma: =int (Gammal * v, v, 0, v) $ let Iog(-l)=0,log(v+l)=log((l+v)/(l-v))+log(v-l) ,v~2=l-rn2~2; Gamma; °/0 total decay rate v - 1 8 r a ^ + 4v bye;
120 7T3
Problem. Calculate the A/ra correction.
272
5 Weak interactions Neutron decay
dimensional estimate of the n —>> pe~Ve decay rate (Fig. 5.11) gives V 5 , where A = mn — mp
The terms fa and g
htf) = F2p{q2)2-*2niq2)-
htf) = Flptf)-Flntf),
In the neutron decay, q2 is small, and one may use the form factors at q2 = 0: the weak charge fi — 1 and the weak magnetism J2 — (9p — <7n)/(2ran) (gp and gn are the proton and the neutron anomalous magnetic moments). The effective pseudoscalar # 3 (so called because it yields the lepton mass and the point-like pseudoscalar interaction) is mainly determined by the pion exchange (Fig. 5.14), because the pion mass is small:
where the n —> pir~ transition vertex is denoted by #75. The characteristic scale at which #3 changes is m 2 , and it is much less than that of the other form factors. In the limit of the massless u- and d-quarks, the axial current is conserved: d^uy^d = —(m u + rrid)ud -> 0. The pions are massless in this limit: if one takes the matrix element between TT~ and the vacuum, one obtains f^m^ = (mu + md)<0\ud\7c~> —> 0. Applying the axial current conservation to the n —> p transition, we obtain ~2mNg1 + g3q2 = 0, or, substituting #3 at m^ — 0, we arrive at the Goldberger-Treiman relation for the axial charge =
2mN' All these formulae are rigorously applicable in the limit of the massless w, d quarks (and hence the massless pions) at q —> 0, and may be not very accurate in the real world.
wi TTY
n
P
Fig. 5.14. Effective pseudoscalar
5.3 Weak decays
273
The characteristic q is very small in the neutron decay, therefore one may neglect the weak magnetism and the effective pseudoscalar, and take the vector and the axial charge at q2 = 0. The form factors f2 and g% are more important in the //-capture process (see the problem). The program obtains
dT
rp
24TT3
dTp
I
q2
2g2(3A2 - AmpTp) -m2e(l dT
=
+ g\) (l -
G2\Vud\2(l + 3g2)y/E* - m2Ee(A - Ee)2 8TT3
dEe 3
240?r
?2) |Y
9m 2
m*\
L\
/
15
1+«"| J
(5:3.10) The shape of the electron spectrum, and hence the function of rae/A in the total decay rate, is the same as in the decay TT~ -> 7r°e~z7e. One can express the double differential rate via the angle between the electron momentum and the neutrino one: dT 167T 3
where ve = \/E2 — m1/Ee is the electron velocity. input_case nil$ in "kpack.red n,nlepten.red"$ °/o Neutron decay nospur f; g(f ,ll)*U+b*g(f,a))$ for all b,ll saveas V(b,ll); spur f; operator Tb; 4/sqrt(2)*V(b,lD*r(pl)*V(b,12)*r(p2)$ for all b,pl,ll,p2,12 saveas Tb(b,pl,ll,p2,12); 7, baryon tensor mass p=l,pl=ml,p2=m2,p3=0; mshell p,pl,p2,p3; let abs(ml)=ml; index 11,12; MM:=Tb(b,p,12,pl,ll)*Tl(p2,ll,-p3,12)/2$ let p.pl=El,p.p2=E2,p.p3=l-El-E2,pl.p2=(l-ml^2-m2^2)/2-l+El+E2, Gamma:=MM/2/(4*(2*pi)"3)/4; 7, d Gamma / d El d E2 •p .
(2Elb2 + 4E2b + 2E\ + 4E1E2b2 + SE1E2b + 4E1E2 - E1b2rn21 2E1b2m1 - E1b2ml - 3E1b2 - 2Eibml - 2E1bm22 - 6Exb - Ei -2E1ml - EYm\ - ZEX + ±E\b2 + \E\ - 2E2b2m\ - AE2b2 -
274
5 Weak interactions
4tE2bm\ - 4E2b - 2E2m\ - 4E2 - b2ml + b2ml + b2mxm\
- b 2 m1
-\-b2m\ + b2 4- 2bm\ + 2bm\ + 2 6 + m\ + m2 - m i m \ + mx + m^ + 1)/(32TT 3 )
procedure NonRel(x); begin scalar a,b,c; let abs(d)=d; c:=sub(ml=l-d,m2=m2*d,El=l-d+d^2*Tl,E2=d*E2,x); a:=num(c); b:=den(c); while (c:=sub(d=0,a))=0 do a:=a/d; a:=c; while (c:=sub(d=0,b))=0 do b:=b/d; return a/c end$ Gamma:=NonRel(Gamma);
_ 4fff b2 + 4 E | - AE2b2 - AE2 + 2Txb2 - 2TX + b2 m\ - b2 - m\ + 1 :
32 7T3
"
l e t abs(2*El-ml^2+m2^2-l)=l-2*El+ml"2-m2"2; Dalitz(l,ml,m2,0,El,E2m,E2p)$ E2m:=NonRel(E2m)$ E2p: =NonRel(E2p)$ Gammal:=int(Gamma,E2,E2m,E2p)$ l e t abs(2*E2+ml's2-m2^2-l)=l-2*E2-ml'N2+m2's2; Dalitz (1 ,m2 ,ml, 0, E2 ,Elm,Elp) $ Tlm:=NonRel(Elm-ml)$ Tlp:=NonRel(Elp-ml)$ Gamma2:=int(Gamma,Tl,Tlm,Tlp)$ let abs(v)=v; on factor; Gammal:=sub(Tl=v"2/2,Gammal); Gamma2; off factor; Gammai := ( - ( ( t ; 2 + 3 ) m 2 - 2(v + 1) (t; - 1 ) ^ 2 +
(m\v2 + 3m 2 + 4v4 - 10v2 + 6) b2) (m2 + v2 - l)2v)/(96{v
+ 1) 3(^ - 1)3TT3)
l - m\{E2 - I)2 (3b2 + 1)E2 8TT3
GammaT: =int (Gammai *v, v, 0, v) $ l e t log(-l)=0,log(v+l)=log((l+v)/(l-v))+log(v-l) ,v^2=l-m2^2; on factor; GammaT; off factor; °/0 t o t a l decay rate -(2(8 mj + 9 m2 - 2)v - 15 log( ~^_+11)) m|) (3 62 + 1) 960 7T3 Gamma: =sub (Tl= (E2"2-m2"'2+ (1-E2) ^2) /2+sqrt (E2^2-m2^2) * (1-E2) *c, Gamma) *sqrt (E2~2-m2~2) * (1-E2) $ on factor; sub(c=0,Gamma); df(Gamma,c)/sub(c=0,Gamma); off factor;
/~^~mJ(E2
- l)2(3fr2 + l)E2
1 6 7T3
- (E2 + m2) (E2 - m2) (b + 1) (b \ - rn2(Sb2 + 1) E2
5.3 Weak decays
275
In order to take the electron polarization into account, one should substitute Pe -> Pe — ra eae in |M| 2 ; this factor appears linearly. For the same reason, pv appears in \M\2 linearly too. In the approximation A/TTIN
°/, Polarized neutron vector ee; nospur f; r(pl)*(l-g(f,a,ee))$ for all pl,ee saveas r(pl,ee); spur f; let p.ee=0,pl.ee=-p2.ee~p3.ee,p2.ee=-sqrt(E2^2-m2^2)*ce, p3.ee=-(l-El-E2)*cn; MM:=4/sqrt(2)*V(b,lD*r(p,ee)*V(b,12)*r(pl)*Tl(p2,ll,-p3,12)/2$ Gamma:=MM/2/(4*(2*pi)"3)/4$ on gcd; Gamma:=NonRel(Gamma)$ on factor; Gamma:=sub(Tl=(E2^2-m2^2+(l-E2)"2)/2+sqrt(E2^2-m2^2)*(l-E2)*c,
Gamma)*sqrt(E2^2-m2^2)*(l-E2); F :=
l - m\b2c - 2 yjEl - m\ b2 ce - 2 yjE% - ml bee - yjE% - m\c 2
2b
cn - 3E2b2 - 2E2bcn - E2) (E2 - 1) 2 )/(16TT 3 )
Gamma:=int(Gamma,c,-l, 1)$ GammaO:=sub(ce=O,cn=O,Gamma); - sjE2 - m\ (E2 - I) 2 (3b2 + 1)E2 Gammao := — - — n » n
df(Gamma,ce)/GammaO;
Sir6 df(Gamma,en)/GammaO;
2 (E2 + 2m2) (E22 - 2m2) (b + 1) b - m (3 6 + \)E2 -2(6-1)6
bye;
362 + 1
Problem. Find the A/m correction. It can be noticeable in semileptonic hyperon decays with a large energy release.
276
5 Weak
interactions
Problem. Calculate the cross section /i p ^ ^ n at a small /J, velocity. Obtain the /i-capture rate from the /i-hydrogen atom ground state, and compare it with the fi decay rate. Problem. Express the cross section of vin -» l~p via the form factors. Problem. Consider the scattering vN —y vN.
Nonleptonic hyperon decays
A nonleptonic hyperon decay vertex (Fig. 5.15) has the form GVusmlu2(A + i, where A describes t h e 5 - w a v e decay, while B describes t h e P - w a v e one. T h e decay r a t e is r
G2\Vus\2mjpn
[(rm + m 2 ) 2 - mj] \A? + [(mt - m 2 ) 2 - mj] |gj 2 m?
8TT
7T
(5.3.13) where it is more convenient to use S = A and P = By/A2 — m2r/(2mi) in the nonrelativistic approximation. The pion angular distribution in a polarized hyperon decay has the form 1 — a£ • n^, where (53I4)
The final baryon polarization in an unpolarized hyperon decay is longitudinal, and is equal to a.
P
ZE$.
P
Pi Fig. 5.15. Nonleptonic hyperon decays input_case nil$ in "kpack.red","pol.red"$ °/0 Nonleptonic hyperon decays U mass p=l,pl=ml,p2=m2; mshell p,pl,p2; let p.pl=(l+ml~2-m2~2)/2,p.p2=(l-ml~2+m2~2)/2,
MM:=4*(l+b*g(f,a))*(g(f,p)+l)*(l-bc*g(f,a))*(g(f,pl)+ml)$ on gcd; Gamma:=MM/2*ph2(l,ml,m2); in
(\Jm\ - 2m\m\ - 2m\ + m\ - 2m\ + 1 l - 2bbcmi - bbcm\ + bbc + m2 + 2m1 - m\
5.3 Weak decays
277
procedure NonRel(x); begin scalar u,v,w; let abs(d)=d; w:=sub(ml=l-d,m2=d*m2,b=2*b/d/sqrt(l-m2~2), bc=2*bc/d/sqrt(l-m2~2) ,x); u:=num(w); v:=den(w); while (w:=sub(d=0,u))=0 do u:=u/d; u:=w; while (w:=sub(d=0,v))=0 do v:=v/d; return u/w end$ NonRel(Gamma);
\J -m\ + 1(6 6c + 1) 7T
% polarized hyperon vector ee; l e t p.ee=0,pl.ee = -p2.ee,p2.ee=-PP(l,ml,m2)*c; MM:=(l+b*g(f,a))*(g(f,p)+l)*(l-g(f,a,ee)) *(l-bc*g(f,a))*(g(f,pl)+ml)$ df(MM,c)/sub(c=O,MM); NonRel(ws); - yjm\ - 2 m\ m\ - 2 m\ + m\ - 2 m\ + 1 (b -f 6c) bbcm\
— 266cmi
—
bbcrrv^ -f 66c -f- ml -f 2mi — m | + 1
- (6 + 6c) 6 6c + 1 °/, final baryon polarization let abs(ml)=ml; ll:=-second(ort({pl.l,p.l}))$ index 1; pe:=p.l*ll$ remind 1; let pi.ee=0,p2.ee=p.ee,p.ee=z*pe; MM:=(l+b*g(f,a))*(g(f,p)+l) *(l-bc*g(f,a))*(g(f,pl)+ml)*(l-g(f,a,ee))$ df(MM,z)/sub(z=O,MM); NonRel(ws);
(6mi —
2bm\rri2
- 2 6 m \ + bm\ — 2bml + 6 + bcm\
-
2bcm\ + 6cm2 - 2 6cm^ + 6c)/ (\jm\ - 2m\m\ (bbcm\
— 266cmi
- 2m\ + m\ - 2m\ + 1 —
bbcrn\ -f 66c 4- m\ + 2 m i —
m\ + 1))
6c 66c + 1 bye; Problem. Find the final baryon polarization in a polarized hyperon decay.
278
5 Weak interactions 5.4 W production e+e" -> W+W~
The cross section of this process (Fig. 5.16) is (we have assumed mw = 1 to simplify the formula) TTC* 2
a = 9 6 s i n 4 $ i y s 3 ( s c o s 2 ' # w — 1) x { 24 \s(s2 + s + 4) - (s3 + 2s2 + 10s + 4) sin2 dw] log
s cos 2
—1
— 2—
2 |3s(21s - 20s + 32) - 16(8s3 + 352 + 16s + 6) sin2 ti L
4- 4(15s3 + 8s2 4- 160s + 96) sin
j
(5.4.1) All the programs in this Section were obtained by introducing some small improvements to the programs generated by the CompHEP system [HEP5]. Thus, in this case the middle loop is the same in Figs. 5.16a-c (up to the factors sinfiw and cosflw)- Of course, shorter programs could be written, but the presented programs use the memory highly economically and can be executed even on a PC. The economy is mostly achieved by the accurate treatment of tensor multiplications with the heavy vector particle propagators and density matrices. One should carefully study these tricks, because the lack of memory in complicated tensor contractions is by no means a rare situation. Problem. Investigate the angular distribution in this process.
input_case nil$ in "kpack.red"$ °/0 e+ e- -> W+ W-
0/
l e t cw~2=l-sw~2,e~2=4*pi; mZ:=l/cw$ mass kl=0,k2=0,pl=l,p2=l; mshell kl,k2,pl,p2; vector Il,12,nl,n2,n,nn,a; l e t kl.k2=s/2,pl.p2=s/2-l,kl.pl=(l-t)/2,k2.p2=(l-t)/2, kl.p2=(s+t-l)/2,k2.pl=(s+t-l)/2; operator r,V,VW,VZ,V3,D,DZ; for a l l p l e t r(p)=g(f,p),D(p)=l/p.p,DZ(p)=l/(p.p-mZ~2); for a l l 1 l e t V(l)=e*g(f,1), VW(l)=e/(2*sqrt(2)*sw)*g(f,l)*(l+g(f,a)), VZ(l)=e/(cw*sw)*g(f,l)*((l+g(f,a))/4-sw^2); for a l l pl,p2,p3,ll,12,13 let V3(pl,p2,p3,ll,12,13)= e*(11.12*(pl-p2).13+12.13*(p2-p3).11+13.11*(p3-pl).12); 7, diagram 1 vl:=V3(pl+p2,-pl,-p2,n2,ll,12)$ vr:=V3(pl+p2,-pl,-p2,nl,ll,12)$
279
5.4 W production
i
b
a
i
h I
d
+
Fig. 5.16. e e~ ->
f W+W~
vv:=sub(ll=pl,12=p2,vl)*sub(ll=pl,12=p2,vr)$ ul:=sub(12=p2,vl)$ ur:=sub(12=p2,vr)$ index 11; vv:=vv-ul*ur$ remind 11; ul:=sub(ll=pl,vl)$ ur:=sub(ll=pl,vr)$ index 12; vv:=vv-ul*ur$ clear ul,ur; index 11; vv:=vv+vl*vr$ remind 11,12; clear vl,vr; MM1:=r(kl)*V(nl)*r(k2)*V(n2)$ index nl,n2; MMl:=MMl*vv$ remind nl,n2; MMl:=MMl*D(kl+k2)*2$ % diagram 2 MM:=r(kl)*VZ(nl)*r(k2)*VZ(n2)$ n:=(pl+p2)/mZ$ MM2:=sub(nl=n,n2=n,MM)*sub(nl=N,n2=N,vv)$ ul:=sub(n2=n,MM)$ ur:=sub(n2=n,vv)$ index nl; MM2:=MM2-ul*ur$ remind nl; ul:=sub(nl=n,MM)$ ur:=sub(nl=n,vv)$
280
5 Weak interactions
index n2; MM2:=MM2-ul*ur$ clear ul,ur; index nl; MM2:=MM2+MM*vv$ remind nl,n2; clear MM; MM2:=MM2*DZ(kl+k2)~2*(cw/sw)~2$ °/, diagram 3 MM:=r(kl)*VZ(nl)*r(k2)*V(n2)$ ul:=sub(nl=n,MM)$ ur:=sub(nl=n,vv)$ index n2; MM3:=-ul*ur$ clear ul,ur; index nl; MM3:=MM3+MM*vv$ remind nl,n2; clear MM,vv; MM3:=2*sub(i=0,MM3)*D(kl+k2)*DZ(kl+k2)*cw/sw$ °/0 diagram 4 nospur f; MM:=VW(ll)*r(kl)*VW(ll)$ ul:=-sub(ll=pl,MM)$ index 11; ul:=ul+MM$ remind 11; MM:=VW(12)*r(k2)*VW(12)$ u2:=-sub(12=p2,MM)$ index 12; u2:=u2+MM$ remind 12; clear MM; spur f; MM4:=ul*r(kl-pl)*u2*r(kl-pl)$ clear ul,u2; MM4:=MM4*D(kl-pl)~2$ °/o diagram 5 vv:=V3(pl+p2,-pl,~p2,nn,ll,12)$ MM:=r(kl)*V(nn)*r(k2)*VW(12)*r(kl-pl)*VW(ll)$ ul:=sub(ll=pl,12=p2,MM)$ ur:=sub(ll=pl,12=p2,vv)$ index nn; MM5:=ul*ur$ remind nn; ul:=sub(12=p2,MM)$ ur:=sub(12=p2,vv)$ index nn,ll; MM5:=MM5-ul*ur$ remind nn,ll; ul:=sub(ll=pl,MM)$ ur:=sub(ll=pl,vv)$ index nn,12; MM5:=MM5-ul*ur$ clear ul,ur; index 11; MM5:=MM5+MM*vv$ remind nn,ll,12; clear MM; MM5:=-2*sub(i=0,MM5)*D(kl+k2)*D(kl-pl)$ °/0 diagram 6 MM:=r(kl)*VZ(nn)*r(k2)*VW(12)*r(kl-pl)*VW(ll)$ MM6:=-sub(nn=n,ll=pl,12=p2,MM)*sub(nn=n,ll=pl,12=p2,vv)$ ul:=sub(ll=pl,12=p2,MM)$ ur:=sub(ll=pl,12=p2,vv)$ index nn; MM6:=MM6+ul*ur$ remind nn; ul:=sub(nn=n,12=p2,MM)$ ur:=sub(nn=n,12=p2,vv) $ index 11; MM6:=MM6+ul*ur$ remind 11; ul:=sub(nn=n,1l=pl,MM)$ ur:=sub(nn=n,1l=pl,vv) $ index 12; MM6:=MM6+ul*ur$ remind 12; ul:=sub(12=p2,MM)$ ur:=sub(12=p2,vv)$ index nn,ll; MM6:=MM6-ul*ur$ remind nn,ll; ul:=sub(ll=pl,MM)$ ur:=sub(ll=pl,vv)$ index nn,12; MM6:=MM6-ul*ur$ remind nn,12; ul:=sub(nn=n,MM)$ ur:=sub(nn=n,vv)$ index 11,12; MM6:=MM6-ul*ur$ clear ul,ur; index nn; MM6:=MM6+MM*vv$ remind nn,ll,12; clear MM,vv; MM6:=-2*sub(i=0,MM6)*DZ(kl+k2)*D(kl-pl)*cw/sw$
5.4 W production
281
7c the sum on gcd; sigma:=(MMl+MM2+MM3+MM4+MM5+MM6)/(16*pi*s~2)$ clear MM1,MM2,MM3,MM4,MM5,MM6;
on factor; sigma; off factor; °/ 0 d sigma / d t
((s4sw4t
- 2s4sw2t
+ s4t + 3s3sw4t2
+ 2s3sw4t
+ s3 sw4-
6 s3 sw212 - 2 s3 sw21 - 2 s3 sw2 + 5 s2 sw4 £3 + 10 s2 sw41 - 8 s2 sw213 -12s2sw2t
+ 2525W2 + 355W4*4 + 2ssw4t3
+
27ssw4t2
-455W 2 * 4 - 45 5W2£3 - 4 5 5w;2^2 + 6ssw 2 £ + Asw4t4 + Ws 4sw4t2
- 10sw;2^4 + 24sw 2 * 3 - 18sw2*2 -f 4sw2t)s +
24(t - l ) 2 s ^ 4 t 2 + (3* 2 + l)s4) + (9^2 - 8 t + 4)(t - 1) 2 5 2 + (17t 3 - 20*2 + 121 - 8) 83>)K\/(16
(ssw2 - s +
l)2s4sw4t2)
let abs(s)=s; Mandel(0,0,l,l,s,tl,t2)$ sigma:=int(sigma,t,tl,t2)$ °0/ total cross section let log((sqrt (s)*sqrt(s-4)-s+2)/2) = log((s-2-sqrt(s)*sqrt(s-4))/(s-2+sqrt(s)*sqrt(s-4)))+ log((-sqrt(s)*sqrt(s-4)-s+2)/2); sigma:=sigma$ clear log((sqrt(s)*sqrt(s-4)-s+2)/2); for all x let log(x)=0; sigmaO:=sigma$ for all x clear log(x); sigma:=sigma-sigmaO$ on factor; sigma; sigmaO; off factor; (s3sw2 - s3 + 2s2sw2
- s2 + IQssw2 - 4 5 + 4sw2) 4(ssw2 — s + 1) s3 sw4
[yfsy/s —
4
((60s 2 sw 4 - 128s 2 sw 2 + 63s 2 + 32ssw 4 - 48ssw2 - 60s + 640sit; 4 256 sw2 + 96) s + 96(2sw + l)(2sw - 1) sw2) TT) /(96 (s sw2 - s + l ) 2 s 3 5W;
4
)
bye; e ~ 7 —)-
veW~
T h e cross section of this process (Fig. 5.17) is (x = 1 — t) ^ = ™ (2s ax 2 sin^ ^ v ^ s 3 ^ 2 2
3
- As2x + 3sx 2 - 25X + 25 - x3 + 2x 2 - 2x),
o- = — 7 ^ ? t4*3 + s2 + 2s-74 sin vws
,
(2s2 + s+l)logs] .
.
(0.4.zj
282
5 Weak interactions
7
-l£* 7 ^
\rj
'if %?} b
c Fig. 5.17. e~7 -» v eW~
input_case nil$ in "kpack.red"$ % e- gamma -> W- mi let cw~2=l-sw~2,e~2=4*pi; mass pl=0,p2=0,p3=0,p4=l; mshell pl,p2,p3,p4; vector 12,14,nl,n2,nn,a; let pl.p2=s/2,p3.p4=(s-l)/2,pl.P3=(x-l)/2,p2.p4=x/2, pi.p4=(1+s-x)/2,p2.p3=(s-x)12; operator r,V,VW,V3,D,DW; for all p let r(p)=g(f,p),D(p)=l/p.p,DW(p)=l/(p.p-l); for all 1 let V(l)=e*g(f,1), VW(l)=e/(2*sqrt(2)*sw)*g(f,l)*(l+g(f,a)); for all pl,p2,p3,H,12,13 let V3(pl,p2,p3,ll,12,13) = e*(11.12*(pl-p2).13+12.13*(p2-p3).11+13.11*(p3-pl).12); °/0 diagram 1 nospur f; MM:=VW(14)*r(p3)*VW(14)$ MMl:=-sub(14=p4,MM)$ index 14; MM1:=MM1+MM$ remind 14; clear MM; spur f; index 12; MMl:=MMl*r(p3+p4)*V(12)*r(pl)*V(12)*r(p3+p4)$ remind 12; MMl:=MMl*D(p3+p4)^2$ °/, diagram 2 vl:=V3(p2,-p4,pl-p3,12,14,nl)$ vr:=sub(nl=n2,vl)$ ul:=sub(14=p4,vl)$ ur:=sub(14=p4,vr)$ index 12; vv:=-ul*ur$ clear ul,ur; index 14; vv:=vv+vl*vr$ clear vr; remind 12,14;
54 W production
283
MM:=r(pl)*VW(n2)*r(p3)*VW(nl)$ nn:=pl-p3$ MM2:=sub(nl=nn,n2=nn,MM)*sub(nl=nn,n2=nn,vv)$ ul:=sub(n2=nn,MM)$ ur:=sub(n2=nn,vv)$ index n l ; MM2:=MM2-ul*ur$ remind n l ; ul:=sub(nl=nn,MM)$ ur:=sub(nl=nn,vv)$ index n2; MM2:=MM2-ul*ur$ c l e a r u l , u r ; index n l ; MM2:=MM2+MM*vv$ remind n l ; c l e a r MM,vv; MM2:=MM2*DW(nn)~2$ */. diagram 3 MM:=r(pl)*V(12)*r(p3+p4)*VW(14)*r(p3)*VW(nl)$ ul:=sub(14=p4,nl=nn,MM)$ ur:=sub(14=p4,nl=nn,vl)$ index 12; MM3:=ul*ur$ remind 12; ul:=sub(14=p4,MM)$ ur:=sub(14=p4,vl)$ index 12,nl; MM3:=MM3-ul*ur$ remind 12,nl; ul:=sub(nl=nn,MM)$ ur:=sub(nl=nn,vl) $ index 12,14; MM3:=MM3-ul*ur$ clear ul,ur; index nl; MM3:=MM3+MM*vl$ remind 12,14,nl; clear MM,vl; MM3:=-2*sub(i=0,MM3)*D(pl+p2)*DW(pl-p3)$ 7« the sum on gcd; sigma:=(MMl+MM2+MM3)/(16*pi*s~2) ; 7, d sigma / d x 7T (2 s3 - 4 s2 x + 3 s x2 - 2 s x + 2 s - x3 + 2 x2 - 2x) 2 s3 sw2 x2 clear MM1,MM2,MM3; let abs(s)=s,abs(s-l)=s-l; Mandel(0,0,0,l,s,tl,t2)$ sigma:=int(sigma,x,l-tl,l-t2)$ 7, total cross section factor log; sigma; 4 logO) 7T ( - 2 S2 - S - 1) + 7T (4 S3 -h S2 + 2 S - 7)
4 s3 sw2 bye;
77
_
The cross section of this process (Fig. 5.18) is (x = 2t + s — 2)
-
x2)2
(7 =
s+
(5.4.3)
5 Weak interactions
284
a
Fig. 5.18. 77->
input_case nil$ in "kpack.red"$ °/0 gamma gamma -> W+ W-
y#
let e~2=4*pi; mass kl=0,k2=0,pl=l,p2=l; mshell kl,k2,pl,p2; vector Il,12,ml,m2,nl,n2,nn; let kl.k2=s/2,pl.p2=s/2-l, kl.pl=(s-x)/4,k2.p2=(s-x)/4,kl.p2=(s+x)/4,k2.pl=(s+x)/4; operator V3,V4,D,DW; for all p let D(p)=l/p.p,DW(p)=l/(p.p-l); for all pl,p2,p3,ll,12,13 let V3(pl,p2,p3,ll,12,13)= e*(11.12*(pl-p2).13+12.13*(p2-p3).11+13.11*(p3-pl).12); for all 11,12,13,14 let V4(ll,12,13,14)= e~2*(2*11.12*13.14-11.13*12.14-11.14*12.13); % diagram 1 vl:=V4(ml,m2,ll,12)$ vr:=V4(ml,m2,ll,12)$ ul:=sub(ll=pl,12=p2,vl)$ u2:=sub(ll=pl,12=p2,vr)$ index ml,m2; MM1:=ul*u2$ remind ml,m2; clear ul,u2; ul:=sub(12=p2,vl)$ u2:=sub(12=p2,vr)$ index ml,m2,ll; MMl:=MMl-ul*u2$ remind ml,m2,ll; clear ul,u2; ul:=sub(ll=pl,vl)$ u2:=sub(ll=pl,vr)$ index ml,m2,12; MMl:=MMl-ul*u2$ clear ul,u2; index 11; MM1:=MMl+vl*vr$ clear vl; remind 11,12,ml,m2;
54 W production
285
°/o diagram 2 nn: = k2-p2$ vu:=V3(k2,-p2,-nn,m2,12,nl) $ vd:=V3(kl,-pl,nn,ml,ll,nl)$ vl:=-sub(nl=nn,vu)*sub(nl=nn,vd)$ index n l ; vl:=vl+vu*vd$ remind n l ; clear vu,vd; vl:=vl*DW(nn)$ ul:=sub(ll=pl,12=p2,vl)$ u2:=sub(ll=pl,12=p2,vr)$ index ml,m2; MM2:=ul*u2$ remind ml,m2; clear ul,u2; ul:=sub(12=p2,vl)$ u2:=sub(12=p2,vr)$ index ml,m2,ll; MM2:=MM2-ul*u2$ remind ml,m2,ll; clear ul,u2; ul:=sub(ll=pl,vl)$ u2:=sub(ll=pl,vr)$ index ml,m2,12; MM2:=MM2-ul*u2$ clear ul,u2; index 11; MM2:=MM2+vl*vr$ remind ml,m2,11,12; clear vr; MM2:=2*sub(i=0,MM2)$ MM2:=MM2+sub(x=-x,MM2)$ °/0 diagram 3 vr:=vl$ ul:=sub(ll=pl,12=p2,vl)$ u2:=sub(ll=pl,12=p2,vr)$ index ml,m2; MM3:=ul*u2$ remind ml,m2; clear ul,u2; ul:=sub(12=p2,vl)$ u2:=sub(12=p2,vr)$ index ml,m2,ll; MM3:=MM3-ul*u2$ remind ml,m2,ll; clear ul,u2; ul:=sub(ll=pl,vl)$ u2:=sub(ll=pl,vr)$ index ml,m2,12; MM3:=MM3-ul*u2$ clear ul,u2; index 11; MM3:=MM3+vl*vr$ remind ml,m2,11,12; clear vr; MM3:=MM3+sub(x=-x,MM3)$ % diagram 4 vector kk,mm; vr: =sub (kk=k2 ,mm=m2, sub (k2=kl ,m2=ml, sub (kl=kk ,ml=mm, x=-x, vl) ) ) $ ul:=sub(ll=pl,12=p2,vl)$ u2:=sub(ll=pl,12=p2,vr)$ index ml,m2; MM4:=ul*u2$ remind ml,m2; clear ul,u2; ul:=sub(12=p2,vl)$ u2:=sub(12=p2,vr)$ index ml,m2,ll; MM4:=MM4-ul*u2$ remind ml,m2,ll; clear ul,u2; ul:=sub(ll=pl,vl)$ u2:=sub(ll=pl,vr)$ index ml,m2,12; MM4:=MM4-ul*u2$ clear ul,u2; index 11; MM4:=MM4+vl*vr$ remind ml,m2,11,12; clear vl,vr; MM4:=2*sub(i=0,MM4)$ 7, the sum on gcd; sigma:=(MMl+MM2+MM3+MM4)/(128*pi*s~2); ° 0/d sigma / d x _ TT(19S 4 - 24s 3 + 1 0 s 2 x 2 + 96s 2 + 24sx2 + 3a:4) ~ s2(s4 - 2s2x2 + x4) let abs(s)=s; Mandel(0,0,l,l,s,tl,t2)$ sigma:=int(sigma,x,2*tl+s-2,2*t2+s-2)$ let log(-sqrt(s)*sqrt(s-4)-s)=log(sqrt(s)*sqrt(s-4)+s), log(sqrt(s)*sqrt(s-4)-s)=log(-sqrt(s)*sqrt(s-4)+s), log(-sqrt(s)*sqrt(s-4)+s)= log((-sqrt(s)*sqrt(s-4)+s)/(sqrt(s)*sqrt(s-4)+s)) +log(sqrt(s)*sqrt(s-4)+s); factor log; sigma; a
:
286
24
5 Weak interactions M - s + 2) + 2 v /i^/7^7^(-4 g 2 - 3a - 12)
bye; Problem. Investigate the polarization effects in the processes considered in this Section.
6 Quantum chromodynamics
6.1 Feynman diagrams in QCD Feynman rules This chapter is devoted to QCD. It contains examples of calculations in perturbation theory. There are several approaches which try to go beyond this framework (QCD sum rules, lattice simulation). We have no possibility of discussing them here. We only note that calculation methods used in QCD sum rules are based on the diagram technique, and are close to those used in perturbation theory. There exist several textbooks, entirely [QCD1-QCD4] or partially [HEP1HEP4] devoted to QCD. An excellent presentation of group theory, oriented towards applications to gauge theories, is contained in [QCD6]. The quark fields qa have a colour index. The number of colours is equal to 3, but we shall write down all formulae for an arbitrary number of colours Nc. The theory is symmetric with respect to the transformations q —>> Uq, where U is a unitary Nc x Arc matrix having detU = 1. These transformations form the group SU(NC). Infinitesimal transformations have the form £ 7 = 1 + iaata, where ta are called generators. It follows from U+U = H-za°(£o + — ta) = 1 that ta are hermitian matrices, and from detU = 1 + iaaTrta = 1 that Tr£ a = 0. The commutator of the generators is [£a, tb] = ifabctc, where fabc are called the structure constants. The generators are normalized by the condition Tr tatb =
¥ ab-
The colour symmetry is local. To ensure it, the ordinary derivative d^q in the free quark field lagrangian q(id — m)q is replaced by the covariant one D^q = (dy, — igAy)q, where Ay, — A^ta is the gluon field. The lagrangian of this field — \ TrFyvFyv is also added, where the gluon field strength is FyV = dy,Av — dvAy, — ig[Ay,^Au] = F^vta. Finally, the QCD lagrangian has the form
The quark propagator has the same form (4.3.6) as the electron one, the only difference is that the unit colour matrix is implied in it. The gluon propagator 287
288
6 Quantum chromodynamics
has the same form (4.3.5) as the photon one, with the additional factor Sab. The vertices contained in the lagrangian (6.1.1) (Fig. 6.1) are ig^^t™, gflmn[(pi — P2)vS\» + (P2 -PS)A<W + (P3 - P I ) A A L and -ig2frnnafrsa{5,ip5U(T - 8^5vp) ig2fmrafnsa(S^Spa -S^Spy) -ig2fmsafnra(5^S(7p -<WM- In fact, there are some subtleties in QCD Feynman rules, which are essential in diagrams with a gluon loop. We shall discuss them later.
Fig. 6.1. Feynman rules in QCD
Calculating colour factors with REDUCE
The problem of calculation of any QCD diagram consists of two subproblems: calculation of the colour factor, and that of the colourless diagram. When calculating colour factors, a quark-gluon vertex contributes the factor ta, and a three-gluon one, the factor ifabc. The remaining factors of vertices and propagators produce an expression for the colourless diagram. In doing so, one should not forget that, if the legs of a three-gluon vertex are enumerated, say, clockwise in the colour factor, they should be enumerated in the same order in the colourless part. A diagram with a four-gluon vertex has to be split into three diagrams. The colour factor of each of them is obtained by replacing the fourgluon vertex by two three-gluon ones; when calculating the colourless parts, the coefficients of the corresponding colour structures in the expression for the four-gluon vertex are taken. Colour factors can be most simply calculated graphically, using the Cvitanovic algorithm [QCD6]. The properties we already know are depicted in Fig. 6.2a-c. The definition of the three-gluon vertex ifabc via [ta,tb] is presented in Fig. 6.2d. Closing the quark legs to a gluon (Fig. 6.2e), we obtain the final expression for the three-gluon vertex (Fig. 6.3). The algorithm is based on an identity for a gluon exchange between two quark lines (Fig. 6.4a). The right-hand side can be expressed only via tensors invariant with respect to SU(NC). They are 5%, and the unit antisymmetric tensors e" 1 " 2 - 0 1 ^, eQia2...aNc- Any one of the last two tensors cannot occur in the right-hand side of Fig. 6.4a alone, because in this case the numbers of upper and lower indices would differ; and their product can be expressed via 5%. The most general form is shown in the right-hand side of Fig. 6.4a. In order to find
6.1 Feynman diagrams in QCD
289
—
/\A/\A/\A/W
v-v
Fig. 6.2. Properties of the generators ta
Fig. 6.3. Three-gluon vertex
the coefficients, let's first of all close the upper quark line (Fig. 6.46); we obtain b = l/Nc. After that, let's close it to a gluon (Fig. 6.4c); we obtain a= \. The final form of the identity is shown in Fig. 6.5. Now we can formulate the Cvitanovic algorithm. It consists of eliminating three-gluon vertices (Fig. 6.3) and internal gluon lines (Fig. 6.5). As a simplest example, let's find the number of gluons (Fig. 6.6). Several other problems are solved in Fig. 6.7. What is the difference of the Coulomb law for a heavy quark and antiquark in QCD from the ordinary QED case? If the quark-antiquark pair is in the colourless state, for example, it has been produced by a photon (not shown in Fig. 6.8a), then the diagram with the gluon exchange differs from the diagram without the exchange by the colour factor Cp = {N% - 1)/(2NC). Therefore, the quark and the antiquark attract each other following the Coulomb law with
290
6 Quantum
chromodynamics
-b
= a a
-b
= a
o
= 0
b
= a
x
1
JV,
2
Fig. 6.4. Derivation of an identity for a gluon exchange
3 C -kZ Fig. 6.5. The identity for a gluon exchange
N9 =
= 2
= 2
JV,
= JVC " I
Fig. 6.6. Counting gluons
6.1 Feynman diagrams in QCD
291
iVc2-l
iv.
2JNL
2
= 2 -=»—f
>
L >
^ >
< >
+ JV,
N,
k
= 8
—
*\A//^
V^^V
V ^
/
\A/^ 3 VV/ V V^^^
ZZZjSl 'VAAAAAAA/
Fig. 6.7. Examples of application of the Cvitanovic algorithm
292
6 Quantum chromodynamics
a —> Cpd-s' But if the quark-antiquark pair has the same colour as a gluon (Fig. 6.86), then the extra factor -1/(2NC) is negative; they repel each other following the Coulomb law with a —> —a s/(2Nc).
a
b
Fig. 6.8. Interaction of a quark-antiquark pair in the colour singlet and the octet state
More complicated colour factors can be calculated with REDUCE. To this end, one needs to use the program xCOLOR [QCD7] from the REDUCE network library. This program implements the Cvitanovic algorithm more efficiently than the older program COLOR [P4]. The number of quark colours is set by the command sudim. The quark-gluon vertex ta is denoted by qg(i, j ,a), where i and j are quark indices (matrix indices of ta), and a is a gluon index. The three-gluon vertex ifabc is denoted by g3(a,b,c). Let's consider several examples of colour factor calculations with xCOLOR (Fig. 6.9). input_case nil$ °/o Calculating colour factors load_package xcolor$ sudim Nc; q g ( i , j , a ) * q g ( j , i , a ) ; qg(i,j,a)*qg(j,k,b)*qg(k,l,a)*qg(l,i,b); Nc2 - 1 2 -Nc2 + I 4Nc qg(i,jJa)*qg(j,k,b)*qg(k,l,c)*qg(l,m,a)*qg(m,n,b)*qg(n,i,c); Nc4 - 1 SNc2 qg(i,j,a)*qg(j,k,b)*qg(k,i,c)*g3(c,b,a); Nc(Nc2 - 1) 4 qg(i,j,a)*qg(j,k,d)*qg(k,l,e)*qg(l,i,b)*g3(a,b,c)*g3(e,d,c); Nc2 (Nc2 - 1) 8 qg(i,j,a)*qg(j,k,d)*qg(k,l,e)*qg(l,i,b)*g3(a,e,c)*g3(b,d,c); 0
6.1 Feynman diagrams in QCD
293
Fig. 6.9. Colour factors
The first result easily follows from Fig. 6.7a, and the second one, from Fig. 6.76. Similarly, the fourth and the fifth results follow from Fig. 6.7c. The other answers are more difficult to obtain. Note that sometimes a colour factor vanishes. In such cases, there is no need to calculate a (possibly complicated) diagram. Here are several purely gluonic colour factors. g3(a,b,c)*g3(c,b,a); Nc(Nc2 - 1)
g3(c,b,a)*g3(a,d,e)*g3(e,b,f)*g3(f,d,c);
Nc2 (Nc2 - 1) 2 g3(c,b,a)*g3(a,e,d)*g3(d,f,h)*g3(h,b,i)*g3(i,e,g)*g3(g,f,c); 0
294
6 Quantum chromodynamics
g3(a,d,e)*g3(e,f,c)*g3(c,b,a)*g3(f,d,b); 2 2
Nc (Nc - 1) 2 g3(a,c,b)*g3(b,g,i)*g3(i,h,e)*g3(e,d,a)*g3(d,f,c)*g3(g,f,h); Nc3 (Nc2 - 1) 4 g3(a,c,b)*g3(b,g,i)*g3(i,h,e)*g3(e,d,a)*g3(h,f,c)*g3(g,f,d); 0 bye; The first result easily follows from Fig. 6.7o\ The gluon loop correction to the three-gluon vertex (Fig. 6.10) has the same colour structure as the bare vertex. We can find the coefficient from the fourth result (Fig. 6.10). This also explains the second and fifth results. The other answers are more difficult to obtain.
Fig. 6.10. Correction to the three-gluon vertex
6.2 e + e~ annihilation into hadrons Two-jet events In this Section, we shall consider one of the most important processes — e + e~ annihilation into hadrons. It has been studied in detail on e+e~ colliders. We shall not discuss the effects related to Z°, which are important at the energies E>mz. From the QCD point of view, this is the process e + e~ -» qq (Fig. 6.11a). At high energies, transformation of quarks into hadrons does not change the cross section. Quarks differ from muons only by their charges and by the colour, therefore e e > hadrons) = N ^ (6.2.1) R ^ (( + There are corrections to this cross section: interference of the diagram Fig. 6.11a with Fig. 6.116, and the process e+e~ —> qqg (Fig. 6.11c). We shall calculate these corrections in § 7.3, and shall obtain
^y
(6.2.2)
6.2 e+e
a
annihilation into hadrons
b
295
c +
Fig. 6.11. Diagrams of the process e e~ -» hadrons
It is convenient to calculate virtual photon decay widths 7* —>• X instead of cross sections e+e~ —» X. The virtual photon is partially polarized: if the e + e~ beams are unpolarized, it is polarized in the plane perpendicular to the beam axis; if they have complete transverse polarization, it is linearly polarized along the bisector between the e + and e~ polarization directions; if they have complete longitudinal polarization, then it is circularly polarized (see § 4.4). If a state X consists of two particle, there is one vector p in the problem. In this case, e^ej can be contracted with two structures Sij and PiPj. Everything is determined by two quantities Fy and Fj_, with e\\p and e±p. Longitudinal polarization yields no new information. If 7* is linearly polarized, TT-V
4TT—
dil
=
FII
"
cos2 a + Tj_ sin2 a,
where a is the angle between p and e. In the case of unpolarized e + e~ beams,
where $ is the angle between p and the beam axis. Averaging over directions gives F = (F|| +2Fj_)/3. If X is a pair of spin 0 particles, then only Fy is nonzero; if it is a pair of light fermions, then only F_L is nonzero. If a state X consists of three particles, then there are four symmetric structures (5^, puPij, ViiViji PuP2j + P2iPij) and one antisymmetric one (PuP2j — P2iPij)- Longitudinal polarization can produce a correlation of the normal to the event plane with the beam axis. A high energy quark, antiquark, or gluon created at e + e~ annihilation produces a jet of hadrons moving in approximately the same direction. Most e + e~ annihilation events have two jets (Fig. 6.11a); some have three jets (Fig. 6.116). The program starts by calculating the decay width F o of 7* —> qq (Fig. 6.12a) (without the correction of Fig. 6.126). The energy of each of e + and e~ is set to unity, i.e. 7* has energy equal to 2. The factor az^Nc is implied in the result. In what follows, the ratio F(X)/F 0 = a(X)/a0 will be calculated for various channels X. The program checks a±/a0 — | , cry /do = 0. The angular distribution of two-jet events is 4?r da
—-7^ CTQ
ail
=
1
7
4
296
6 Quantum chromodynamics
Fig. 6.12. Squared diagrams of 7* —> qq input_case nil$ % 2 jets
7. mass pq=0,pa=0; mshell pq,pa; % quark and antiquark momenta let pq.pa=2; index m; °/, alpha zq~2 Nc is implied GammaO:=-g(f ,m,pq,m,pa)*4/3*(4*pi)/4/(8*pi) $ procedure dummy(x); x$ % workaround for complete simplification GammaO:=dummy(GammaO); 2
Gammao := -
o °/o ee - polarization vector, ec - its conjugate % n - unit 3-dimensional vector along pq % scalar products involving it are 3-dimensional vector ee,ec,n; let ee.ec=-l,ee.pq=-ee.n,ee.pa=ee.n,ec.pq=-ec.n,ec.pa=ec.n; R:=g(f,ec,pq,ee,pa)*4*(4*pi)/4/(8*pi)/GammaO$ R:=dummy(R); 3( — ec.nee.n + 1) R:=
2 l e t ee.n=l,ec.n=l; R1:=R; °/ c along ee Rl := 0 l e t ee.n=0,ec.n=0; Rt:=R; °/«cross ee Rt:=3clear R,Rl»R"t,ee.n,ec.n,ee.pq,ee.pa,ec.pq,ec.pa,n,pq.pa; Three-jet events
The program proceeds to calculate F(7* —>• qqg)/To (Fig. 6.13). The quark, antiquark, and gluon energies are denoted by xq, x^ and xg. The Dalitz plot is presented in Fig. 6.14; xg = 2 - xq — xq. There are four squared diagrams; the diagrams of Fig. 6.13c, d are obtained from that of Fig. 6.13a, b by the interchange q «-»• q. The colour factor, divided by that of Fig. 6.12a, is Cp = (N% — 1)/(2NC). The three jets production cross section da 27T ( 1 - ;
(6.2.3)
is singular at the edges AB and AC of the Dalitz plot (Fig. 6.14), when the gluon is radiated along the quark or the antiquark, and especially at the corner
6.2 e^e
annihilation into hadrons
297
A, where the gluon is soft. The integral of (6.2.3) over the Dalitz plot diverges; as we shall see in § 7.3, this divergence is compensated by the divergence of the virtual correction of Fig. 6.126, producing the finite answer (6.2.2).
Fig. 6.13. Squared diagrams of 7*
QQ9
Fig. 6.14. Dalitz plot of 7* —> qqg
% 3 jets
7.
mass pg=0; mshell pg; °/0 gluon momentum index 1; °/0 xq, xa, xg - quark, antiquark, gluon energylet pg.pq=2*(l-xa),pg.pa=2*(l-xq),pq.pa=2*(l-xg); xg:=2-xq-xa$ operator S; for all p let S(p)=g(f,p)/p.p; R: = (g(f ,m)*S(pq+pg)*g(f ,l,pq,D*S(pq+pg)*g(f,m,pa) -g(f,m,pq,l)*S(pq+pg)*g(f,m,pa,l)*S(pa+pg)) *4/3*(4*pi)~2/4/(4*(2*pi)~3)/Gamma0$ °/o CF alphas is implied on factor; R:=R+sub(xq=xa,xa=xq,R); 7, adding symmetric diagrams
xa2 + xq2 2 (xa - 1) (xq - 1) n factor;
R := off
The program obtains (l/ao)(da/dxq) at Xq < 1 - <5. This condition can be formulated as tiqg > #o5 then S = xq(l — £g)sm 2 ($o/2), or as (pq -f pg)2 > [j,2, then S = /x 2 /4. The result is 1 da ac\ dxa
OLs_ 7
2TT
-
Xa
1-Xa
(6.2.4)
298
6 Quantum chrornodynamics
% distribution in xq at xa < 1-d
let log(-d)=-log(l/d),log(-l)=0; factor log; Rq:=int(R,xa,O,l-d); -2log(±)(xq2 + 1) + d2 - 4d + 3 Rq := 4TT (xq — 1) clear l o g ( - d ) , l o g ( - l ) ; clear Rq; A simple picture of a jet evolution emerges in the leading logarithmic approximation. The initial quark decays into quarks, antiquarks, and gluons (collectively called partons) with decreasing virtualities. Let's consider this cascade while virtualities exceed /i2. The initial quark virtuality can be arbitrary up to Q2. It is convenient to introduce
instead of ji2 (as we shall see in § 7.3, as depends on the characteristic virtuality /i 2 ). The initial quark distribution at £ = 0 (//2 = Q2) is wq(x) = 8(1 — x), where x is the fraction of the jet energy carried by the quark. At small A£, the probability of the quark decay into a quark and a gluon P(z) A£ should be taken into account: 1 + {1 z)2
±^ , Pq-+g(z) l — z
~ . (6.2.5) z Here z is the fraction of the initial quark energy carried by the final quark or gluon. When £ grows, further decays occur. In order to find the distributions wq(x), Wq(x), wg(x), one needs to solve the system of kinetic equations (called the Altarelli-Parisi equations) with the decay probabilities per unit A£: Pq_+q(z), Pq^g(z) (see (6.2.5)), and also Pg-+q(z), Pg->q(z), Pg^g(z). Infrared-stable quantities Probabilities of a soft gluon radiation, and also of parton decay into a nearly collinear parton pair, are not small. Therefore, in order to calculate quantities sensitive to these processes one needs to take higher orders of perturbation theory into account. Such quantities can also change during hadronization considerably. Quantities which don't change at radiation of a zero energy gluon, and at splitting of a parton into a collinear pair, are called infrared-stable. They may be calculated in the lowest order of perturbation theory. They are also only slightly changed at the transition from partons to jets of approximately collinear hadrons. For example, thrust is an infrared-stable characteristic of the event shape:
M
(6.2.6)
6.2 e+e annihilation into hadrons
299
Quantities which involve particle momenta, say, quadratically, are not infraredstable. The direction n at which the maximum in (6.2.6) is attained is called the jet axis. Thrust can be defined in a different though equivalent way:
£
T = max -
Pi
(6.2.7)
The maximum of the modulus of the momentum / \ = — Pi is taken over subdivisions of the set of particles into two subsets Ji, J^ (jets). The maximum in (6.2.7) is attained when all particles with n -p> 0 are referred to one jet, and all particles with n • p < 0 to another one. With this subdivision, the maximum in (6.2.6) is attained when the jet axis n is directed along Pi. Therefore, the definitions (6.2.6) and (6.2.7) are really equivalent. Thrust has its maximum value T = 1 in the case of an ideal two-jet event; it has its minimum value T = | in the case of a spherically-symmetric event with high multiplicity. For an ideal three-jet event, the maximum in (6.2.7) is attained when the most energetic jet is referred to Ji, and the other two to J2. The minimum value T = | possible in this case is attained when all three jets have equal energies and form the angles 2TT/3 with each other (the centre of the Dalitz plot of Fig. 6.14). Thrust has its maximum value T = 1 when two jets are collinear — at the edges of the Dalitz plot. Isolines T = const are the triangles in Fig. 6.14. The thrust of a two-jet qq event (Fig. 6.12a) equals 1, giving (l/ao)(da/dT) = 5(1 - T). We'll be interested in the distribution in T at T < 1. It is determined by three-jet events qqg (Fig. 6.13). Thrust is equal to the largest of the three energies xq, Xq, xg. The regions of the Dalitz plot where each of them is the largest are shown in Fig. 6.14. In order to obtain da/dT, one has to integrate the cross section (6.2.3) in each of these regions with the largest energy equal to T. In other words, this is the integral over the triangle — isoline of T in Fig. 6.14. The regions where the most energetic particle is q or q contribute equally, therefore it's enough to calculate one of them and double. The program obtains the result
(6.2.8)
°/0 Thrust Rt:=2*sub(xq=T,R)+sub(xq=2-xa-T,R)$ % quark or gluon has the largest energy l e t log(-T/2)=log(T/2),log(T-l)=log(l-T),log(l-2*T)=log(2*T-l); Rt:=int(Rt,xa,2*(l-T),T)$ clear log(-T/2),log(T-l),log(l-2*T); l e t log(2*T-l)=log((2*T-l)/(l-T))+log(l-T); Rt:=Rt;
300
6 Quantum chromodynamics 2 1og(-^|-zrp)(-3T2 + 3T - 2) + 3T(-3T 2 + 8T - 4)
R t
: =
2TTT(T
- 1)
clear log(2*T-l); clear Rt; The angular distribution of the energy flow dE/dfl, the correlation of the energy flows d£/(dCtid^), etc., are other examples of infrared-stable quantities. Here inclusive production cross sections are weighted with the particle energies and summed over particle kinds. Of course, the angular distribution is nonspherical only for a polarized 7*. The correlation contains three structures (5^-, nunij + n2iri2j, nun2j + ^22^1?); their coefficients are functions of the angle d between n\ and n^- We'll simplify the problem a little and calculate the correlation of the energy flows averaged over 7* polarizations. Two-jet qq events (Fig. 6.12a) contribute to the correlation only at $ = 0, 7T. We are interested in angle between these values where only three-jet qqg events (Fig. 6.13) contribute. Three cases are possible: the particles flying along ni, n
(6.2.9)
°/0 correlation of energy flows R:=2*sub(xq=xl,xa=2-xl-x2,R)+sub(xq=xl,xa=x2,R)$ °/0 q g and q anti-q x 2 : = ( l - x l ) / ( l - x l * s ) $ R:=R*xl*x2*df(x2,s)$ clear x2; R:=int(R,xl,0,l)$ l e t log(s-l)=log(l-s),log(-l)=0; on factor; R:=R; R := 1og(-(s - l ) ) s 2 - 12 1og(-(s - l))s + 6 1og(-(s - 1)) - 9s 2 + 6s) (2 s - 3)/(2(s - 1)TTS5) off factor; clear l o g ( s - l ) , l o g ( - l ) ; remfac log; clear R; Polarization effects The program proceeds to calculate F(7* —>• qqg)/T0 with the virtual photon having polarization e, and obtains
where n g , n-q are the unit vectors in the directions of motion of 0, q.
6.2 e + e
annihilation into hadrons
301
This allows one to obtain the angular distribution of the energy flow 47rd£ _ . 2 2 r — = in cos a + /j_ sin a (Jo eta 2 " for a linearly polarized 7*, or
for unpolarized e + e~ beams. It is characterized by two numbers /y = y I± = E_L/CT0. The energy flow averaged over directions (Ey + 2E_L)/3 is equal to the total cross section a multiplied by the total energy, which is equal to 2: (6-2-U)
4W •
Let's consider the leading terms (without as) first. Two-jet qq events (Fig. 6.12a) give 7y = 0. Either q or q with the energy equal to 1 can fly in a direction nLe with cross section |ao, therefore 7j_ = 3. Of course, this agrees with (6.2.11). Now let's proceed to the a s-correction. The program obtains the spectra of q, q, g, flying in a direction n||e*in three-jet qqg events (Fig. 6.13): 1 da q 0 dx uJbq
1 da UQ dx-q ujuq
0Ls
\n F
2 &n TT'
1 da &o UQ dXg uujbg
^
n
a
8
7T vi
1 ~Xg xq JUg
(6.2.12)
therefore j||=9CF^i.
(6.2.13)
The spectra for n_Le, similar to (6.2.12), diverge. The virtual correction to the qq cross section (Fig. 6.12&) also contributes in this case, and it diverges as well. Fortunately, we can use the energy conservation (6.2.11), taking into account the total cross section (6.2.2), and obtain I± = 3.
(6.2.14)
°/0 Polarization mass n=l,nq=l,na=l; °/« 3-dimensional vectors let ee.pg=-ee.(pq+pa),ec.pg=-ec.(pq+pa), ee.pq=-xq*ee.nq,ec.pq=-xq*ec.nq, ee.pa=-xa*ee.na,ec.pa=-xa*ec.na; R:=-(g(f,ec)*S(pq+pg)*g(f,l,pq,l)*S(pq+pg)*g(f,ee,pa) -g(f,ec,pq,l)*S(pq+pg)*g(f,ee,pa,l)*S(pa+pg)) *4*(4*pi)~2/4/(4*(2*pi)~3)/GammaO$ on gcd,factor; vector nn; °/0 workaround for a bug in vector sub R:=R+sub(nn=na,sub(na=nq,sub(xq=xa,xa=xq,nq=nn,R)));
302
6 Quantum chromodynamics — 3(ec.naee.naxa2 + ec.nqee.nqxq2 - xa2 - xq2)
R :=
4 (xa — 1) (xq — 1) TT off factor; mshell n,nq,na; let n.nq=(xa~2-xg~2-xq~2)/(2*xg*xq), n.na=(xq~2-xg~2-xa~2)/(2*xg*xa), nq.na=(xg~2-xq~2-xa~2)/(2*xq*xa); ql:=sub(ee=nq,ec=nq,R); % quark along ee
3 (xa + xq — 1) ql := nxq2 ql:=int(ql,xa,l-xq,l); ql:=int(ql*xq,xq,O,
clear xg; xq:=2-xa-xg$ gl:=sub(ee=n,ec=n,R); x m=
% gluon along ee
6(-xg + 1) nxg2
gl:=int(gl,xa,l-xg,l); gl:=int(gl*xg,xg,O,l); gl :=
6 ( - xg + 1)
nxg
Il:=2*ql+gl;
bye; Problem. The Higgs boson H can decay into gg via the vertex fHF£uF£u (via a heavy quark loop, Fig. 6.15). Calculate the decay probability H —>• gg, as well as H —» ggg and # —>• ggg. Obtain the parton decay probabilities Pg-+q(z), Pg_>q(z), and Pg-¥g (z) from these results.
^l/wv, Fig. 6.15. Decay H -+ gg
6.3 Charmonium decays
303
6.3 Charmonium decays Estimates In this Section, we shall consider decays of charmonium states cc (see § 3.4). The same formulae are applicable to bottomonium 66. We shall start from simple estimates. The dominant decay of ij) is ip —>> 3g —>• 3 jets (Fig. 6.16a). Its probability can be estimated just as in the case of orthopositronium (§ 4.7): 3$) - *\
(6.3.1)
maov
where v is the quark velocity inside ip (we may not assume v ~ as because the potential is not the coulomb one). The decays i\) —>• 2g + 7 —» 2 jets -f 7 (Fig. 6.166) *2
(6.3.2)
2
and tj) —> e + e~,/i + //~ (Fig. 6.16c) e+e~)=:
(6.3.3)
are also significant. The decay probability of %j) —> ^ —)• 2 jets (Fig. 6.16d) can be obtained from F(i0 -> e + e~) by multiplying it by R (6.2.1).
7
—fWVAA/
v a
Fig. 6.16. V decays T h e dominant decay of rjc is rjc -> 2g —>> 2 jets (Fig. 6.17a): F(r/C
(6.3.4)
mar.
•a:
The decay rjc —>• 27 (Fig. 6.176) can be estimated similarly, with as —> a. VAAAA/ WVAA/ VA^DT
a
6
c Fig. 6.17. 77c and x decays
<
d
KAAA/V
304
6 Quantum chromodynamics
The same decays exist for xo and X2* F(xo,2 -+2g)~c and similarly for xo,2 -> 27. In the case of xi, these decays are forbidden by the Landau theorem (two photons or gluons cannot have angular momentum equal to 1). It decays into qqg or 3g (Fig. 6.17c, d). But the estimate a^|V'0(O)| 2/ra4 for these decays is not quite correct, because the integral over the Dalitz plot logarithmically diverges (in the qqg case), and the estimate should include the large logarithm. In addition to this, all x states can decay into tpj: F E i ~ d2u3 ~
ram;4,
(6.3.6)
where d ~ e/(mv) is the electric dipole moment, and u ~ mv2. The dominant decay of i\)' is ip + 2p, where 2g transforms to 2?r or 77. It can be estimated similarly to the radiation of two El-photons (§ 4.4): T2g~d4uj5 ~maf2v6,
(6.3.7)
where d ~ g/(mv) is the chromoelectric dipole moment. In this formula, a!s is taken at a low virtuality (mv)2, and not ra2, and hence it is considerably larger than as in (6.3.1) and other estimates. In addition to this, the same decays (6.3.1-6.3.3) as in the case of ip exist, as well as the El-transitions to X7 (6.3.6) and the Ml-transition to rjcj: FMI ~ M2^3 ~ rnav6,
(6.3.8)
where \i ~ dv is the magnetic dipole moment. The dominant decays of rj'c are the hadronic transition to rjcgg (6.3.7), the annihilation into 2g (6.3.4) (also into 27), and the Ml-transition to ipj (6.3.8). Matrix elements of annihilation processes (in contrast to transitions) can be expressed via ^(0) (or V^(0) in the P-wave case), similarly to § 4.7. We only need to take colour into account. The formula of Fig. 6.5 from § 6.1 can be rewritten as the colour completeness relation (Fig. 6.18). A decay probability, summed over quark and antiquark colours, is equal to the sum of the singlet contribution (1/y/Nc) and the octet one (y/2ta). Therefore, the formula (4.7.18) becomes mi)r(p2-m2),
(6.3.9)
where F contains the unit colour matrix, and the trace is taken over Dirac and colour indices.
6.3 Charmonium decays
305
Fig. 6.18. Colour completeness relation: F = —?==, \/2ta
The vertex of a pseudoscalar meson transition into W or Z via the axial current (without the coupling constant) is usually represented by z/p^. Substituting F = «75 and M = 7^75 into (6.3.9), we obtain (6.3.10) Similarly, the vertex of a vector meson transition into 7, W, or Z via the vector current (without the coupling constant) is usually represented by m/e M . Substituting F = e and M = 7^ into (6.3.9), we obtain the same expression (6.3.10). ijj decays
Let's start from the simplest decays ip —> e + e~, ^+M~- They have been calculated in § 4.1. Substituting f^ (6.3.10) into (4.4.8), we obtain (6.3.11) where zc is the quark charge, and m^ is the meson mass. This result can be also derived from the Pomeranchuk formula (§ 4.7), using the cross section cc -> e + e~, which can be easily obtained from e + e~ -» /x+/i~ (§ 4.5). The decay width r(V> -> gg) differs from (6.3.11) by the factor R (6.2.1). The decay 'tp -> 3r^ is similar to the orthopositronium annihilation into 37 (4.7.12). Pairs of diagrams with the opposite directions of the quark line contribute equally to the matrix element, therefore it is obtained from the positronium one by the replacement a —> as and multiplying by the factor The colour factor is Tr(tatbtc + tctbta)/(2y/lQ. input_case nil$ % clour factor for psi -> 3g load_package xcolor$ sudim Nc; on factor; (qg(i,j,a)*qg(j,k,b)*qg(k,i,c)*qg(l,m,a)*qg(m,n,b)*qg(n,l,c) +qg(i,j,a)*qg(j,k,b)*qg(k,i,c)*qg(l,m,c)*qg(m,n,b)*qg(n,l,a))/2; (Nc + 2) (Nc + 1) {Nc - 1) (Nc - 2) 16iVc bye;
306
6 Quantum chromodynamics
Therefore, m
;
(6.3.12)
In the case of the decay ip —> 2# + 7, the colour factor is (iV2 — l)/4; a 3 is replaced by £ 2 aa 2 , and the identity factor 1/3! by 1/2!:
> 2g + 7) = £(*» - 9)^zlzlaall-^f.
(6.3.13)
The distribution in the gluon energies is given by the formula (4.7.13) (or (4.7.25) in the polarized case), and the gluon (or photon) spectrum by (4.7.14) (or (4.7.26-4.7.29) in the polarized case). The decay ip —> 3g produces a three-jet final state. The thrust distribution 1 dT
YdT
2
1
7r2-9T3(2-T)3
2T(1 - T)(8 - 12T + 5T2) log ( 2 ^ - ^ ^
+ (2-T)(2-T2)(3T-2)
(6.3.14) essentially differs from that for e +e~ -> hadrons off the I/J resonance (6.2.8). input^case nil$ 7o Thrust distribution in psi -> 3g R:=((l-xl)/(x2*x3))*2+((l-x2)/(xl*x3))*2+((l-x3)/(xl*x2))*2$ R:=sub(xl=T,x3=2-T-x2,R)$ R:=int(R,x2,2*(l-T),T)$ R:=sub(log(-T)=log(T),log(2*T-2)=log(2-2*T),R)$ R:=sub(log(2-2*T)=log((2-2*T)/T)+log(T),R)$ on factor; for all x let log(x)=0; Rl:=R; 7, term without log _ - 2 ( T 2 - 2)(3T - 2) 1: 2 3
~
(r - 2) r
for all x clear log(x); R-Rl; 4 ( 5 T 2 - 12T 4- 8)(r - 1) log(" 2( y" 1) )
°/, term with
log
(r - 2)3r2
off factor; clear R,R1; Let's find the angular distribution of the energy flow in the polarized decay _ 2 47rd£ T .2
T dn
=
"cos
a+
a
'
where | ( / | | + 2/x) = 2 due to the energy conservation. The program obtains
6.3 Charmonium decays
307
°/0 Angular distribution of energy flow for polarized psi 0/
R:=-(1-x)/x~2/(2-x)~2*((x~4+3*(2-x)~4)/x/(2-x)*log(l-x) +2*(x~2+3*(2-x)"2))$ % Now integrator needs help Rl:=sub(log(l-x)=0,R)$ R2:=(R-Rl)/log(l-x)$ Rl:=int(Rl,x)$ Rl:=sub(log(x-2)=log(2-x),R1)$ °/0 term without log l:=pf (R2,x)$ R3:=R4:=0$ °/0 partial fractions for each u in 1 do if sub(x=0,den(u))=0 then R3:=R3+u else R4:=R4+u; let dilog(l)=0,dilog(0)=pi~2/6,dilog(2)=-pi~2/12; °/0 dilogarithm 7o terms with denominators x~n c:=sub(x=0,sub(x=l/x,R3)/x)$ R3:=R3-c/x$ °/0 int(log(l-x)/x,x) = -dilog(l-x) R3:=int(R3*log(l-x),x)-c*dilog(l-x)$ °/0 terms with denominators (2-x)~n - change of variables x -> 2-x R4:=sub(x=2-x,R4)$ c:=sub(x=0,sub(x=l/x,R4)/x)$ R4:=R4-c/x$ •/. int(log(x-l)/x,x) = log(x)*log(x-l)+dilog(x) R4:=-(int(R4*log(x-l),x)+c*(log(x)*log(x-l)+dilog(x)))$ R4:=sub(x=2-x,R4)$ factor log,dilog; on gcd,rat; I1:=R1+R3+R4;
II := - log( - x + 2) log( - x + 1) + 2 1og(-x + l)(4x4 - 19x3 + 33x2 - 24x + 6) - dilog( - x + 2) + x2 (x2 — 4 x + 4) 3 dilog( - x + 1) +
2 (x2 - x - 3) x (x - 2)
clear Rl,R2,R3,R4,c,l; remfac log,dilog; off rat; df (Il,x)-R; clear R; 7, check 0
°/0 lower limit should be substituted with caution Il:=sub(x=l,Il)-sub(x=0,sub(log(l-x)=-x-x^2/2,Il))$ Il:=9/(pi~2-9)*Il; It:=3-11/2; % common factors T1 m
3(5TT2-48) 4(TT2 -
It :=
9)
9(TT2 - 8) 8 (TT2 - 9)
on rounded; II; It; °/« what's this numerically? 1.16261659074 2.41869170463 bye;
308
6 Quantum chromodynamics
Problem. Find the correlation of the energy flows in the decay ip —> Zg.
rjc decays
The decay widths r\c -> 2#, 27 can be obtained from the parapositronium annihilation (§ 4.7) (6.3.16) mi X decays
Until now, we have completely neglected the quark motion inside the meson, and used the matrix element (6.3.9), which contains ^(0). It vanishes in the P-wave case. For this reason, let's take into account small quark momenta in the linear approximation: v
hn
....
(6.3.17)
where ktp(O) = —iVip(Q) is assumed. In this approximation, k can be regarded as a purely spatial vector: pk = 0. The program calculates the decay matrix element x ~> 2#. In the three-dimensional form, it is equal to = 4nJ—
V ™c
Sabas
.18) \n > en • k ex - e2 + k • ei e- e2 + k • e2 e - e j ^ ( 0 ) . (6.3.
*-
-•
input_case nil$ 7o Matrix element of chi -> gg X
°/0 quark mass = 1; n - unit vector along the decay axis °/o k l , k2, e l , e2 - gluon momenta and polarizations mass p=l,n=i; mshell p,n; vector k,kl,k2,ee,el,e2,eO; l e t p.n=0,p.k=0,p.ee = 0,p.el=0,p.e2=0,n.el=0,n.e2=0; kl:=p+n$ k2:=p-n$ operator r , S ; for a l l p l e t r(p)=g(f,p)+l,S(p)=(g(f,p)+l)/(p.p-l); M:=g(f,e2)*S(p+k-kl)*g(f,el)*r(p+k)*g(f,ee)*r(-p+k)$ for a l l p clear r(p),S(p); clear r , S ; M:=M+sub(eO=e2,sub(e2=el,sub(n=-n,el=eO,M)))$ M:=sub(x=0,df (sub(k=x*k,M) ,x)) ; °/ o linear approximation M := 2 (ei.e2 ee.nk.n bye;
— ei.eee2-k — e\.kei>ee)
6.3 Charmonium decays
309
The formula (6.3.18) can be interpreted in a slightly different way: one may think that |V^(0)| is factored out, and k is the unit polarization vector of the orbital angular momentum. This matrix element has been obtained for a X state with the orbital angular momentum polarization k and the total spin polarization e fixed separately. Physical x states have definite values of the total angular momentum instead. Therefore, the matrix element (6.3.18) should be contracted with the corresponding angular momentum wave functions over the indices k, ee (these wave functions were considered in § 3.4). The tensor (6.3.18) is symmetric with respect to these indices, and hence its contraction with the antisymmetric wave function of Xi vanishes, in accordance with the Landau theorem. In the case of xo> the program obtains (6.3.19) Xo
(we have restored the power of the quark mass by dimensionality, and reexpressed the result via the meson mass). The decay matrix element of %2 with the angular momentum projection onto the decay axis ra = 0 happens to vanish. This fact was used in § 3.4 to obtain angular distributions. In the case of the projections ra = ±2, the gluon polarizations are fixed, so that there is no sum over the final particles' polarizations. Thus, only two %2 spin states of five can decay into 2g. The program obtains (6.3.20)
Of course, this result could also be derived using the formula for the sum over ra = ±2 states, derived in § 3.4. The decay widths xo,2 -> 27 can be obtained from (6.3.19), (6.3.20) by changing the charge factor a2 -» z*a2 and the colour trace \(N2 — 1) -» N 2: ^27) = mxo
f input_case nil$ V, chiO, chi2 decays
y.
vecdim 3; mass n=l; mshell n; vector k,ee,el,e2; let n.el=0,n.e2=0; M: =4*pi*sqrt(2/Nc)*(n.ee*n.k*el.e2+k.el*ee.e2+k.e2*ee.el)$ index k,ee; M0:=M*k.ee/sqrt(3); '/, chiO
(6.3.21)
310
6 Quantum chromodynamics
clear n.el,n.e2; M0:=sub(el=el-el.n*n,e2=e2-e2.n*n,M0)$ °/0 meson mass = 2; phase space = l/(8*pi); 2 identical gluons index el,e2; Gamma:=M0*M0*(Nc~2-1)/4/2/(8*pi); 3TT(NC2
-
1)
Nc remind e l , e 2 ; l e t n.el=0,n.e2=0; M2:=M*(k.ee-3*n.k*n.ee)/sqrt(6);
°/ 0 chi2 m=0
M2 := 0 remind k,ee; k:=ee$ l e t n.ee=0,ee.el=l,ee.e2=l; M2:=M; 70 chi2 m=2 M2 :=
/—
Gamma:=2/5*M2*M2*(Nc~2-1)/4/2/(8*pi); 4TT(NC2
-
1)
bye; Problem. Calculate dY/(dx\dx2) for the decays xi ~* QQ9i $9- Do the integrals over the Dalitz plot converge? If not, where are they cut off?
6.4 Scattering of quarks and gluons In this Section, we shall calculate scattering cross sections of partons (quarks, antiquarks, and gluons) on each other. The cross sections are averaged over colour and spin states of initial particles, and summed over states of final particles. These processes are observed in high energy hadron collisions, when partons from the initial hadrons scatter at a large angle, producing two jets. qq - » qq, qq -> qq
Let's start with the scattering of non-identical quarks qq1 —> qq1 (Fig. 6.19a). Its cross section differs from that of e~\±~ —> e~ fi~ (4.5.1) only by the change a —> as and the colour factor. It contains 1/N% from averaging over colours of two initial quarks, and the colour trace of Fig. 6.20a, which is equal to ^ Therefore, 2
s2 + u2 + 4(m 2 + ml2)t - 2(m 2 + m / 2 ) 2
s2t2
6.4 Scattering of quarks and gluons
a
b
311
c
Fig. 6.19. Diagrams of qq —> qq, qq —> qq
Fig. 6.20. Squared diagrams of qq —>• qq, qq -» qq
The process qq —> q'q' (Fig. 6.196) is the ^-channel of the same reaction (compare with e+e~ -* //+//" (4.5.3), (4.5.4)). Its colour factor (Fig. 6.206) is the same: da
4(m 2 + m'2)s - 2(m 2 + m / 2 ) 2
CF
m,m'-*0
> -T7-
iV c
CF7ra28 3s
(6.4.2) The scattering cross section of two identical quarks qq -> qq is a little more difficult to obtain. This process is described by two diagrams, Fig. 6.19a and c. There are four squared diagrams: two direct ones (like Fig. 6.20a), and two interference ones (like Fig. 6.20c). The diagrams in each pair are obtained from each other by the interchange t <-> u. The colour factor of the direct diagrams is the same as above; that of the interference diagrams contains the extra factor —1/N C. Therefore, we have to take the direct contribution and the interference one from the program listing in § 4.5, and combine them with the new factor. This results in da
CF
Nc s(s - Am2)
1 (s - 2m2)(s - 6m2) Nc tu
t2+u2)2 \ 2
As2t2u2
(6.4.3)
6 Quantum chromodynamics
312
The process qq —•» qq (Fig. 6.196, d) is the t-channel of the same reaction (compare with e+e~ —> e+e~, § 4.5). Its colour factors (Fig. 6.20a, d) are the same: da
~dT
2ira2
Nc s(s — 4m2) ( « - :
{
(u-2m2)(u-6m2)
1 iVc
si CF2TTO|
^ o Nc
s2
4sH2
(6.4.4) Q9 ->• Q9, qq ++ 99
In these processes, the difference between QCD and QED becomes essential. The diagrams of the process qg —>> qg are shown in Fig. 6.21. There the diagram with the three-gluon vertex appears. But an even more important difference is the fact that the Ward identity for the matrix element is more involved than in QED. Therefore, one may not sum over gluon polarizations including the time-like and the longitudinal ones, i.e. use 2pfJil/ = —8^. " • *
Fig. 6.21. Diagrams of qg -> qg
Let's derive this Ward identity. For the three-gluon vertex
(*3
-
we have (6.4.5) This leads to the identities for insertions into internal and external gluon lines (in the Feynman gauge), shown in Fig. 6.22a, b. Here, as in § 4.3, a large filled dot means that the gluon polarization index is contracted with its momentum k; a dot near a line means that its momentum is shifted byfc,as compared with the diagram without the gluon. The first terms of these identities are written as
6.4 Scattering of quarks and gluons
313
(colour structure) 0 (tensor structure), where the colour structure is z/ a i a 2 a 3 . In the last terms, we have introduced a new object — the ghost-gluon interaction vertex (Fig. 6.23). A dashed line represents propagation of a ghost — a fictious scalar massless particle having colour like a gluon. It is introduced to simplify summations over gluon polarizations.
I
I
#
r
-
|
Y
*
|- •r^/\/\J->- -*A/\/»
Y Fig. 6.22. Ward identities for gluon insertions into internal and external gluon lines
Fig. 6.23. Feynman rules for ghosts: propagator i ^ - , vertex
—gf
aia2b
p2n
Armed with this information, we can easily derive the Ward identity for the amplitude of qg —> qg (Fig. 6.24). The first three terms (whose colour structure is shown before ®) cancel; the fourth one vanishes due to the Ward identity for a quark-gluon vertex (§ 4.3). What's left is the last term — the ghost-quark scattering amplitude.
9 -
i
A Fig. 6.24. Ward identity for the amplitude of qg -> qg
Now we are ready to formulate the method of calculation of ^ (Fig. 6.25). One needs to use
±
\M\2
for summation over polarizations of each gluon, where a is an arbitrary vector with a2 = 0. Indeed, for any longitudinal vector v^ = xk^ + ya^ one has p^vvv =
6 Quantum chromodynamics
314
0. In Fig. 6.25, summation with p^u is denoted by a gluon line with symbol _L, multiplication by k^ by a filled dot, and multiplication by a^^ka) by an empty dot. A gluon line without symbol _L means summation over all four polarizations 2piiv = —Sfj,^. When p^v of the first gluon is expanded, the additional terms vanish, because eachfilleddot moves through M (due to Fig. 6.24) and produces 0, acting upon p^u. For the second gluon, each filled dot moves along the cycle, changing its sign when it crosses the dashed line (because 2ptJLl/ = —5^), and annihilates the empty dot, producing 1.
Fig. 6.25. Summing over gluon polarizations in the process qg —• qg
As a result, the sum over transverse polarizations of the gluons ^2± \M\2 is equal to the sum over all polarizations ^ \M\2 minus ^2 \M\2 for the ghost and antighost scattering. As a by-product, we have demonstrated that the result does not depend on the arbitrary vectors a of both gluons. Finally, let's turn to the calculation of the cross section of qg —>• qg. The matrix element (without uP2O2 and uPl(Tl) is (Fig. 6.21) • %S(pi + fci) - k2) •
Let's separate the symmetric colour structure and the antisymmetric one:
Kill = \ltai>
+
6.4 Scattering of quarks and gluons
315
(6.4.7)
These terms don't interfere when squared. The colour factors are input_case nil$ °/0 qg -> qg % load_package xcolor$ sudim Nc; on gcd; Ng:=Nc~2-l$ Cl:=qg(i,j,a)*qg(j,k,b)*qg(k,l,b)*qg(l,i,a)$ C2:=qg(i,j5a)*qg(j,k,b)*qg(k,l,a)*qg(l,i,b)$ Cs: = (Cl+C2)/(2*Nc*Ng); °/0 symmetric colour structure n C
Nc2 - 2
Ca: = (Cl-C2)/(2*Nc*Ng) ; °/0 antisymmetric colour structure
Ca:=\ off gcd; clear C1,C2;
The remaining factors in \M\2 = \ ^ \M\2 are mass pl=l,p2=l,kl=0,k2=0; mshell pl,p2,kl,k2; l e t pl.kl=(s-l)/2,p2.k2=(s-l)/2,pl.p2=l-t/2,kl.k2=-t/2, pl.k2=(l-u)/2,p2.kl=(l-u)/2; operator r,S,D,V; nospur f; for a l l p l e t r(p)=g(f,p)+l,D(p)=l/p.p,S(p)=(g(f,p)+l)/(p.p-l); for a l l ml,m2,m3,kl,k2,k3 let V(ml,m2,m3,kl,k2,k3)= 7« 3g vertex (k2-kl) .m3*ml.m2+(k3-k2) .ml*m2.m3+(kl-k3) .m2*m3.ml; vector ml,m2; index m3; % matrix element Al:=g(f,m2)*S(pl+kl)*g(f,ml)$ A2:=g(f,ml)*S(p2-kl)*g(f,m2)$ Ms:=Al+A2$ Ma:=~Al+A2-2*D(kl-k2)*g(f,m3)*V(ml,m2,m3,kl,-k2,k2-kl)$ °/o conjugated matrix element Al:=g(f,ml)*S(pl+kl)*g(f,m2)$ A2:=g(f ,m2)*S(p2-kl)*g(f,ml)$ Mcs:=Al+A2$ Mca:=-Al+A2-2*D(kl-k2)*g(f ,m3)*V(ml,m2,m3,kl,-k2,k2-kl)$ clear A1,A2; spur f; index ml,m2; : = (4*pi)^2*r(p2)*Ms*r(pl)*Mcs$ °/, symmetric structure : = (4*pi)~2*r(p2)*Ma*r(pl)*Mca$ 7, antisymmetric structure clear Ms,Ma,Mcs,Mca;
316
6 Quantum chromodynamics
Of course, the colour-symmetric term coincides with e'j —> ej (§ 4.6). We still need to subtract \ J2 \M\2 for the ghost and antighost scattering (Fig. 6.26). For example, the ghost scattering matrix element is
• (-t)D^(fci - k2) • igiX* = l^'^^lrf'
(6-4-8)
so that these terms are subtracted from the antisymmetric structure. MMh:=4*(4*pi)~2*D(kl-k2)~2* 7, ghost and antighost contributions (r(p2)*g(f,k2)*r(pl)*g(f,k2)+r(p2)*g(f,kl)*r(pl)*g(f,kl))$ u:=2-s-t$ on gcd; MMs:=MMs$ MMa:=MMa-MMh$ clear MMh,u;
F i g . 6 . 2 6 . Diagrams of qh —>• qh, qh —> qh (h is the ghost)
As a result, we have obtained \M\2 in the form CSAS -f CaAa. It can be rewritten as §(Ai - ^s) + i ^ ^ s - Looking at the listing, we discover that these two terms have identical numerators. Therefore, the cross section has a relatively compact form da, , nai [ 9 9 9 A -17 = —i m2)^2 I V + u2 + 6m2^ - 2m4 -{s — m2)(u-m2) dt (Q9 -> Q9) (s 2 2 dt (s1 mC )F I 1 9~ m _>o
sz
~N~C {s - m2){u - m2) [t*
Jyc su]
(6.4.9) In contrast with the QED case, the total cross section diverges, because the gluon has colour, and its long-range interaction with the quark produces the Rutherford behaviour da/dt ~ 1/t2. Cs:=2*(Cs+Ca); Ca:=8*Ca; Nc2 - I
Ca := 1 Sa:=(MMa-MMs)/(128*pi~2)$ Ss:=MMs/(32*pi~2)$ clear MMs,MMa; Sa:=sub(t=2-s-u,Sa*t^2)$ on factor; Sa; off factor; (s2u - s2 - 1 su + 3s + u3 - 7u2 + Uu)s - (u3 - Su2 + 6) (s ~l)(u1) on gcd; Ss:=-sub(t=2-s-u,Ss)/Sa$ on factor; Ss; off factor; 1 (s - 1) (u - 1)
6.4 Scattering of quarks and gluons
317
In the case of qq —>• gg, \M\2 differs from qg —> qg by the interchange t«-» s and changing the colour averaging factor l/(NcNg) -» l/N2: o
2
„2
n
4s 2 (t-m2)(u-m2)
4
2
dt
s(s-4m )[ x l —2 -
Nc(t-m2)(u-m2)
s
ra-»O
(6.4.10) When integrating this cross section, one should not forget that the two gluons are identical: -/-•=
2CFna2
--\
I"
yI
1+v
1 - v
1
3
2\
H- ^ f ((s2 + 4m25 - 8m4) log ^ ± ^ _ Nc \
(6.4.11)
1 — vv
where v = y/l — Am2/s is the quark velocity in the centre-of-mass frame. °/oq anti-q -> gg Sa:=sub(s=t,t=s,Sa)$ Ss:=sub(s=t,t=s,Ss) $ in "kpack.red"$ l e t abs(s)=s; Mandel(l,l,0,0,s,tm,tp)$ Sa:=int(Sa,t,tm,tp)/2$ Ss:=int(Ss,t,tm,tp)/2$ let sqrt(s-4)=sqrt(s)*v; Sa:=Sa$ Ss:=Ss$ clear sqrt(s-4); let log((-s*v-s)/2)=log(( s*v+s)/2), log(( s*v-s)/2)=log((-s*v+s)/2), log(( s*v+s)/2)=log((-s*v+s)/2)+log((l+v)/(l-v)); factor log; Sa:=Sa; Ss:=Ss; remfac log; 5a := (12 log( ~V ~
)( - u
2
+2 u - l ) +s v ( - s 2 u + s 2
21 u2 - 9u - 9))/(6(u - 1))
bye; The squared matrix element of the inverse process gg —>• qq can be obtained from qq —> gg by the substitution l/N2 —>• 1/iV2; the identity factor \ in the
6 Quantum chromodynamics
318
total cross section should be omitted: da at s2
Nc
^
~
4s 2
4.
2 m
~
2
(t - m )(u — m2)
(t-m2){u-m2) Nc tu\
4 log
-qq) =
1 — i;
\ + 5m2)v
iV c
1- v
(6.4.12)
99-* 99 This is the most difficult 2 -» 2 scattering process in QCD. We start from the derivation of the Ward identity for its amplitude (Fig. 6.27) (we consider all momenta as incoming, for the sake of symmetry) - fa3a4
a a a a M AtiM2M3M 1
2
3
3?P4,P3
PA) /
+
y +
+ (2
(6.4.13) where TMlMaAl3M4 = W2^^^^^ ~ ^IM4^2M 3 ) i s t n e coefficient of the selected colour structure in the four-gluon vertex. Using the identity (6.4.5), we obtain
^2
(Pi
(6.4.14)
(P3
^ ((P4 ~
(2 *+3)
(2
where ^ ( p ) = z^pM is the ghost-gluon vertex without the colour factor. The last terms cancel due to the Jacobi identity if^^b^a^b _ = ^a^b^^a^b ifa>ia,4bifa2azb^ a n d w e a r e l e f t w i t h t j i e W a r d identity shown in Fig. 6.28. The ghost propagator appears in this figure; it has been defined in Fig. 6.23.
6.4 Scattering of quarks and gluons
/J,2a2
319
A*3°3 A*2a2
p3
V2
PA
Pi
4.
r/*^
/i3a3
>4
/i 4 a 4
Fig. 6.27. Diagrams of gg
M' + V
to
Fig. 6.28. Ward identity for the amplitude of gg -> p^
We shall need the Ward identity for the amplitude of the gluon-ghost scattering, too. It is presented in Fig. 6.29; its derivation is left as an exercise.
Fig. 6.29. Ward identity for the amplitude of gh —t gh
Now we are ready to calculate the sum over transverse gluons' polarizations ^2± \M\2 (Fig. 6.30). Because of the large number of diagrams, notations are somewhat simplified as compared with Fig 6.25. Afilleddot moves, transforming a gluon line into a ghost one; when it crosses the dashed line, the sign is changed. Therefore, diagrams with one ghost loop have a minus sign, while those with two loops have a plus sign. We can combine 24 diagrams with four intermediate ghost lines into 6 diagrams, if we introduce the total ghost-[anti]ghost scattering amplitudes (Fig. 6.31). Diagrams which differ from each other by the interchange of ghost legs contribute with the opposite signs. Hence, ghosts are fermions. There are three colour structures in the matrix element. The diagrams for one of them are shown in Fig. 6.32. The term of the four-gluon vertex, which has the same colour structure, is meant in Fig. 6.32a. Two other structures are obtained by the interchanges 2 ++ 3 and 2 f> 4. Hence, there are 9 colour structures in the squared matrix element. They can be classified into direct (Fig. 6.33a) and interference (Fig. 6.336) ones. It's enough to calculate one direct contribution; the other ones are obtained by the interchanges 1, 2; 3, 4-+1, 3; 2, 4, and 1, 4; 2, 3, i.e. s <-• t and s «•» u (Fig. 6.33a). Similarly, it's enough to calculate one interference contribution; the other ones are obtained by the transpositions of s, £, and u (Fig. 6.336).
320
6 Quantum
NAA/
NAA/
/ \
NAA/
=
chromodynamics
V f
/ \
/ - > " t
/
=
NA^SZ
NA^/
NAA/
NAA/
/ \
NAA/ NAA/
y^y
NAA/
V f
< -
/ \
NA^V/
> -
^ _ 1 NAA/ 1 _ I f
\
NAA/
NAA/
NAA/
NAA/
NAA/
NAA/
< -
f
\
y^y
NA^S/
i ^
> -
<> -
/ > '
1 _ *"
NAA/
f
NA^/
< -
NAA/
NAA/
NAfcy
v±y
NAA/
)
NAA/
1
•
', NAA/
NAA/
- < -
y
\
\
v
i
y '
'/
- > - > - < -
'i
- < -
v f
> "
y
/
-<-<'0 - > ->-
NAA/
NAA/
NAA/
- > -
'i
->- ^ ->- ^ r - < - '/ -
«y
\
y
: > :
f
\
NAA/
1/ - < ^- > " ->-
vV }\
—
y .
NAA/
v
y" < -
NAA/
" < -
" > -
- < "
NAA/
NAA/
NAA/
- > - ^
y
1 _ 1 -VNA,
f
>
- > NAA/
y >
NAA/
- > - ^
v
_
- < -
y- > -
\
y
<-
- < - v - > -
<
>
y - < -
- > -
- < -
- < -
v
y
^
t
- < - y
< *
^<-
-
" < -
> -
- < -
- > -
- < -
" > -
NAA/
NAA/
NAA/
NAA/
NAA/
NAA/
- < -
- > "
NAA/
NAA/
NAA/
NAA/
NAA/
NAA/
- < -
- > "
NAA/
NAA/
NAA/
NAA/
NAA/
NAA/
- > "
" < -
" > -
" < -
NAA/
NAA/
- < -
- > -
NAA/
NAA/
- > -
- < -
NAA/
NAA/
- < -
- > -
- < -
- > "
> -
NAA/
- > -
NAA/
- < "
NAA/ NAA/
-
^
^
" > -
*
> "
< "
y- < -
> -
-
^
Fig. 6.30. Summing over gluon polarizations in the process gg —> gg
* > -
6.4 Scattering of quarks and gluons
7
>>•»
321
\z*"""* < \/'
ft " \ ft ^N ft * \ Fig. 6.31. Amplitude of hh -» hh
M
a
>•<>•<>•<>•<
VWW( l
\
/ '
\
/
'\ /
\
d Fig. 6.32. Diagrams for one of the colour structures
a
b
Fig. 6.33. Kinds of squared diagrams
Let's find the colour factors (Fig. 6.33). In the left-hand side of squared diagrams, we enumerate the legs of three-gluon vertices clockwise. We use the conjugate expressions in the right-hand side, therefore the legs are enumerated counterclockwise. The colour factor of interference diagrams appears to be equal to minus half of that of direct diagrams. input_case nil$ °/« gg -> gg load_package xcolor$ sudim Nc; Ng:=Nc"2-l$
322
6 Quantum chromodynamics
Cl:=g3(al,a2,bl)*g3(a3,a4,bl)*g3(al,a2,b2)* g3(a3,a4,b2)/Ng^2; _ Nc2 Cl '" Nc2 - I C2:=g3(a3,al,bl)*g3(a2,a4,bl)*g3(al,a2,b2)*g3(a3,a4,b2)/Ng's2; -Nc2 2 : ~ 2 {Nc2 - 1) Now let's proceed to the calculation of the tensor factors. There are four kinds of direct contributions in YL± |M| 2 , and four kinds of interference ones; all the other contributions are obtained by renamings (Fig. 6.34). It is implied that the term of the four-gluon vertex (Fig. 6.32a) is included in Fig. 6.34a, e. In the program, D(p) denotes the gluon propagator without the factor — i (which will cancel the factor i from the conjugate propagator in the right-hand side of a squared diagram). The ghost propagator has a different sign, and hence gives the additional minus sign. Every ghost loop gives a minus sign, too. In Fig. 6.346-d, symmetric diagrams are obtained by the transpositions A = (1 <-• 2), B = (3 ++ 4), and C = (1 <-> 3, 2 «-> 4) (the first two lead to t <-> u, the third one does not change the result). In Fig. 6.34/-ft, symmetric diagrams are obtained by the transpositions D = (2 «-» 3), E — (1 «-» 4), and F = (1 <-» 2, 3 <-* 4) (the first two lead to s <-> t, the third one does not change the result). The operator V with various numbers of parameters means the three-gluon vertex, the part of the four-gluon vertex with the selected colour structure, or the ghost-gluon vertex (all without colour factors, which have been taken into account separately). In the left-hand sides, legs of three-gluon vertices are enumerated clockwise. In the right-hand sides, the conjugate expressions are used; in the case of a three-gluon vertex, this means using outgoing momenta instead of incoming ones. Legs of a ghost-gluon vertex are enumerated in the order incoming ghost — outgoing ghost — gluon (Fig. 6.23). If this order is opposite to the one required by the colour structure (which has been taken into account already), then this vertex gives a minus sign. It is clear from the derivation in Fig. 6.30 that ghost-gluon vertices in the right-hand side of squared diagrams are not conjugate to the corresponding vertices in the left-hand side: they always contain the momentum of the outgoing ghost line. Including the necessary common factors, we finally arrive at the cross section da Nc 2ira2 ( tu su st\ * <»* -> 89) = c~F— (3 - y 2 ~ p ~ ~2) •
/C/I1_, (6.4.15)
operator V,D; for all pi let D(pl)=l/pl.pl; °/, propagator for all ml,pi let V(ml,pl)=i*pl.ml; °/0 ghost vertex for all ml,m2,m3,m4 let V(ml,m2,m3,m4)= °/c part of 4g vertex i*(ml.m3*m2.m4-ml.m4*m2.m3); 7, with one colour structure for all ml,m2,m3,pl,p2 let V(ml,m2,m3,pl,p2)= °/0 3g vertex i*((p2-pl) .m3*ml.m2-(pl+2*p2) .ml*m2.m3+(2*pl+p2) .m2*m3.ml) ; mass pl=0,p2=0,p3=0,p4=0; mshell pl,p2,p3,p4;
6.4 Scattering of quarks and gluons
>>AAA/£j
\
323
JjVAA/VC
<
4
a
b
2
2
+A+B+AB +C+AC+BC+ABC
+A+B+AB
/
d
e
+D+E+DE +F+DF+EF+DEF
2
+D+E+DE +F+DF+EF+DEF 9
f
h
Fig. 6.34. Squared diagrams
324
6 Quantum chromodynamics
l e t pl.p2=s/2,p3.p4=s/2,pl.p3=t/2,p2.p4=t/2,pl.p4=u/2,p2.p3=u/2; vector p5,p6; p5:=pl+p2$ p6:=p5$ u:=-s-t$ index ml,m2,m3,m4,m5,m6; A0; = (-i*V(ml,m2,m5,pl,p2)*V(m3,m4,m5,p3,p4)*D(p5)+V(ml,m2,m3,m4))* (i*V(ml,m2,m6,-pl,-p2)*V(m3,m4,m6,-p3,-p4)*D(p6)-V(ml,m2,m3,m4)); 9s 2 + 50s* + 50*2 A
Al:=V(m5,-p2)*V(m3,m4,m5,p3,p4)*V(m6,pl)*V(m3,m4,m6,-p3,-p4) *D(p5)~2$ Al:=2*(Al+sub(t=u,AD); _ s 2 - 10s* - 10*2 M
'-
72
'-
2?
A2:=-V(m2,p5)*V(m3,-p4)*V(m2,pl)*V(m3,p6)*D(p5)^2$ A2:=4*(A2+sub(t=u,A2)); A2 :== - 2 A3:=V(m5,-pl)*V(m5,-p4)*V(m6,p2)*V(m6,p3)*D(p5)^2$ A3:=2*(A3+sub(t=u,A3)); _ s2 + 2s* 4 2*2 As
A0:=A0+Al+A2+A3; clear A1,A2,A3; . 4(s 2 4 4s* 4 4*2) Ao := r s2 on gcd; AO:=AO+sub(s=t,t=s,AO)+sub(s=u,AO); off gcd; _ 4(4s 6 4 12s 5 * 4 15s4*2 4 10s 3 * 3 4 15s2*4 4 12s*5 4 4*6) 0 : ~ s2t2(s2 4 2s* 4 *2) p5:=pl+p3$ B0: = (-i*V(m3,ml,m5,p3,pl)*V(m2,m4,m5,p2,p4)*D(p5)+V(m3,ml,m2,m4))* (i*V(ml,m2,m6,-pl,-p2)*V(m3,m4,m6,-p3,-p4)*D(p6)--V(ml,m2,m3,m4)); 3 ( - 5 s 2 -^ 14s* - 5*2) JJ Q
:=
Bl;=-V(m3,p5)*V(m4,-p2)*D(p5)*V(m6,pl)*V(m3,m4,m6,-p3,-p4)*D(p6)$ Bl:=4*(Bl+sub(s=t,t=s,B1)); - (s2 4- t2) st
B2:=V(m2,-p4)*V(m3,p5)*D(p5)*V(m2,pl)*V(m3,p6)*D(p6)$ B2:=4*(B2+sub(s=t,t=s,B2)); B2 := 0 B3:=V(m5,-pl)*V(m5,-p4)*D(p5)*V(m6,p2)*V(m6,p3)*D(p6)$ B3:=B3+sub(s=t,t=s,B3); _ s2 + 2s* 4 *2 3 : ~ 2s* B0:=B0+Bl+B2+B3; clear B1,B2,B3; 2 2 —^ ) # 0 : = 4 ( - 2 s - 5s* - 2* st
6.4 Scattering of quarks and gluons
on gcd; B0:=2*(B0+sub(s=u,B0)+sub(t=u,B0)); £ 0 := -72 A0:=(A0-B0/2)*(4*pi)~2/4/(64*pi*s~2/4)$ clear BO; on factor; AO; off factor; 4 (g2 +
sf +
^3^ 4 2
^ (S + t)*S t bye;
325
7 Radiative corrections
7.1 Dimensional regularization and renormalization This chapter is devoted to the methods of calculation of one-loop corrections, based on dimensional regularization [RC1, RC2]. Divergences
Let's consider scalar field theory (4.2.16) L = \{d^){d,V)
- ^ V - ^
- | ^ 4 - |y> 5 .
(7.1.1)
Let's try to calculate the f2 correction to the propagator (Fig. 7.1) %G — iG$ + iGo(-iM)iGo, f2 r H4k J
j ^G(p
+ k)G{k).
(7.1.2)
This integral logarithmically diverges at large k, because both the numerator and the denominator contain k to the fourth power.
k p Fig. 7.1. One—loop correction to the propagator in the y?3 theory
If a coupling constant has the dimensionality of mass to a negative power (likeft),then, the higher the order of perturbation theory in this interaction, the higher the degree of divergence of momentum integrals. Such interactions are called nonrenormalizable. If a coupling constant is dimensionless (like g), then the degree of divergence does not depend on the order of perturbation theory — this is an exactly renormalizable interaction. Lastly, if a coupling constant 326
7.1 Dimensional regularization and renormalization
327
has the dimensionality of mass to a positive power (like / ) , then the interaction is called superrenormalizable; in this case, the degree of divergence decreases with the increase of the order of perturbation theory, and only a few diagrams diverge. In a renormalizable theory, only diagrams which describe corrections to vertices whose interaction constants in the lagrangian have non-negative dimensionalities can diverge. If coupling constants in a diagram are dimensionless, then this dimensionality is given by the momentum integral. At large momenta, masses in propagators can be neglected, and the dimensionality is equal to the degree of divergence. Fields and derivatives have positive dimensionalities, therefore there are only a few such vertices. In the case of scalar field theory (7.1.1), only four terms in the lagrangian have coupling constants of non-negative dimensionalities. They produce the two-leg vertices ip2 and -zra 2 , the three-leg vertex —if, and the four-leg vertex — ig. Therefore, in a renormalizable scalar field theory, only diagrams having two, three, or four legs can diverge. Strictly speaking, the term linear in ip should be added to this list. It produces a one-leg vertex; usually, it can be ignored. Let's denote a two-leg diagram (without the external propagators) by —iM(p 2). It can be expanded in a series M(p2) = M(0) + M'(0)p2 +... These terms are corrections to the vertices — im2, ip2, and vertices with more derivatives. If a diagram contains only vertices with the dimensionless coupling g, then M(0) diverges quadratically, and M'(0) logarithmically; all the other terms of the series converge. If a diagram contains two / vertices, then M(0) diverges logarithmically, and M(p2) — M(0) converges (as in (7.1.2)). Diagrams with four or more / vertices converge; an odd number of such vertices is impossible in a diagram with an even number of legs. A diagram with three external legs contains at least one / vertex (in general, an odd number of them). At zero external momenta, it is a correction to the / vertex, and diverges logarithmically. All the other terms of the momentum expansion converge. The same is true for diagrams with four legs, containing no / vertices. In summary, only a few types of diagrams can diverge in a renormalizable theory (where types of diagrams mean kinds and numbers of external legs). For each such type, divergences reside in a few first terms of expansions in external momenta. Types of divergences are in one-to-one correspondence with terms of the lagrangian, whose coupling constants have non-negative dimensionalities. Symmetries of a theory can strongly restrict a set of such terms. For example, if a scalar field theory is required to be be symmetric with respect to ip —>• —cp, then the term (p3 is forbidden. In contrast to this, if a theory contains a nonrenormalizable interaction (such as hep5), then any type of diagram becomes divergent at a sufficiently large order of perturbation theory in this interaction. We should not forget that all these considerations are based on the assumption that one may neglect masses in propagators at high momenta. This is not so in the case of massive vector fields (4.3.3). And though one can con-
328
7 Radiative corrections
struct fermion-boson, three- and four-boson interactions with non-negative dimensionalities of couplings, these theories are nonrenormalizable because of the non-decreasing propagator. Renormalization Divergences signal that we have not taken into account something essential. Specifically, we have forgotten that physical masses and charges don't coincide with parameters ra, / , #,..., appearing in the lagrangian, when we take radiative corrections into account. The correct problem statement is to calculate reaction amplitudes via physical measurable masses and charges. These expressions must be finite, if a theory makes sense. The procedure of re-expressing amplitudes via physical masses and charges by eliminating bare parameters of a lagrangian is called renormalization. It is physically necessary regardless of the problem of divergences, and must be applied always, even if there are no divergences at all, as, for example, in the theory of electron-phonon interaction in solid state physics. Let's start from scalar field theory
L = \(d,<po)(d^o) - !f
(7-1.3)
where quantities with the index 0 are called nonrenormalized or bare. Let's split every term in the lagrangian into two parts:
Here the renormalized field
7.1 Dimensional regularization and renormalization
329
The requirement of cancellation of divergences determines counterterms up to finite terms. They are fixed by exact definitions of physical masses and charges. These definitions (renormalization schemes) may vary. The most natural way to define the mass and the normalization of the field (p is to require the absence of corrections to the propagator on the mass shell. Diagrams for the propagator have the general structure shown in Fig. 7.2, where —iM(p) is the sum of one-particle-irreducible self-energy diagrams, i.e. two-leg diagrams (not including the external propagators), which cannot be separated into two disconnected parts by cutting a single line: iG = iG0 + iG0(-iM)iG0
+ iG0(-iM)iG0{-iM)iG0
+ ...
or p2 — m2 — M(p) Let's expand M(p) near p2 = m2: M(p) = A + B(p2 - m2) + M'(p). Then the propagator pole is at p2 = m2 + A. Defining the physical mass as the position of this pole, we must find the counterterm 6m2 from the equation A = 0. In this case, the propagator near the mass shell has the form 1
7
G=
2
2
2
p -m -ZM'{p y
1-JB'
and external legs of any diagram correspond to the wave function with the normalization factor Z1!2. The natural definition of the normalization of the physical field is Z = 1, so that the counterterm 5 is found from the equation B = 0. Let's stress that the on-shell renormalization is not the only possible scheme; for example, it is not applicable in QCD.
Fig. 7.2. Structure of diagrams for the propagator
In the on-shell renormalization scheme, self-energy insertions in external lines (Fig. 7.3) yield zero contribution. In the case B ^ 0, there appears an ambiguity: the term B(p2 — m 2 ), applied to the on-shell wave function, gives zero; but if it is combined with Go(p) = l/(p 2 — ^ 2 ) 5 it gives a nonzero contribution B. The correct result in the one-loop approximation is B/2. This corresponds to multiplying the wave function by Z 1 / 2 , This means that in the one-loop approximation (in schemes other than the on-shell one) self-energy corrections
330
7 Radiative corrections
to external lines should be included with the factor | . It is also possible to understand this in a different way. Let's consider our diagram as a part of a larger diagram, for which this line is internal. Then one half of the correction to this internal line should be attributed to our diagram, while the second half comes from the rest of this larger diagram.
Fig. 7*3. Self-energy correction to an external line
Physical charges / , g can be defined in a variety of ways. For example, one can require that some amplitudes at some fixed external momenta be exactly given by the tree approximation formulae, without any corrections. Regularization In order to have a possibility of calculating divergent diagrams and determining counterterms, we should first introduce a regularization — modify the theory in such a way that divergences are absent. In the limit of removing regularization, the original theory is restored. It is desirable that a regularization preserve as many symmetries of the theory as possible. Symmetries constrain possible forms of counterterms. The simplest regularization is cutting off momentum integrals. It is not convenient for calculations. The Pauli-Villars regularization is often used. It is inconvenient in the case of gauge theories, because it breaks the gauge invariance. The lattice regularization, in which space-time is replaced by a discrete lattice, is often used for numerical simulations. It can be done in a gauge-invariant manner, but breaks the Lorentz invariance. Dimensional regularization is most convenient for calculations in perturbation theory. It does not break the gauge or the Lorentz-invariance. In this scheme, expressions for diagrams are written down in space-time of an arbitrary dimension d. It must enter as a parameter; after that, it may be considered as an arbitrary complex quantity. The original theory is restored in the limit d —> 4; the dimension is usually written down as d = 4 — 2e. Divergences show up as 1/e poles. In d-dimensional space, the dimensionality of a lagrangian is
k
- ^v - {y - V N
)
Smm2 2o ^
5f
Sq
i
(7.1.5)
7.1 Dimensional regularization and renormalization
331
where Z = l + S, ZmZ = 1 + <Jm, 2 > Z 3 / 2 = 1 + Sf, ZgZ2 = 1 + Sg. When calculating quantities with integer dimensionalities, integrals over dk\ ) -
i l
_ 4 _ d d dk ( ) *
appear. We shall consider calculation of one-loop diagrams in dimensional regularization, using the integral (7.1.2) as an example. In d-dimensional space, it has the form M(p)
1
2 j \2*)d[{k+p)*-m*W-m*Y
(7B1 6)
'
Feynman parametrization The calculation starts by combining denominators using Feynman's formula 1
1
f
dx
o
which can be easily checked by a direct calculation in REDUCE. Differentiating it in a and 6, we can obtain from it l
anbm
~ r(n)r(m) J [xa + (1 - x)b]n+m' o
( • • )
In the case of three denominators, we can combine two of them using (7.1.7), and include the third one using (7.1.8):
J
o
o
y
[xya + {l-x)yb
+ {1 - y)c]3
+ Continuing in the same spirit, we obtain
1
= r(n) f 6(*i + *i + ---*n)dx1dx2.
...an
J
[xiai + x2a2 + ...
dxn
xnan\n
Differentiating this formula, we arrive at
«r< 2-«n" x-, —
m 2 -f . . . m n ) r(m 1n)r(m 2)...r(mn)
Xn . . . xnL o(xi + x2 + •.. xn)ax\dx2 • • • dxn *—. : [xiai + ^2(22 + . . . xnan
332
7 Radiative corrections Applying the formula (7.1.7) to the integral (7.1.6), we have
M{p) = 4 Then we shift the integration variable k' = k + xp to separate a complete square:
Calculating integrals in d dimensions One-loop integrals after combining denominators have the form
Strictly speaking, the denominator is k2 — a2 + z0, and the pole in the complex fco plane are at the points ±( v k2 + a2 — iO). We can turn the integration contour: kQ = ik0 (k0 varies from —oo to +oo) without crossing the poles. We are now in Euclidean space: k2 = —k 2, k2 = k% + k2. It is always convenient to calculate loop integrals in Euclidean space:
- «-!)«/ (iJ^gjB^.
(T.L13,
Now we shall proceed to spherical coordinates in d-dimensional euclidean space. A vector k^ can be expanded into the component kd = &cos$i, and the orthogonal component of length ksiwdi in the (d — 1)-dimensional subspace. Continuing this process, we obtain kd = A;cos$i, kd-i = ^sin^icos??2, kd-2 = A; sin $i sin $ 2 cos $3,
fei =fcsintfisin?!^• The angles #1, #2,. • • 5 $d-2 vary from 0 to TT; the last angle fld-i varies from 0 to 2TT. The volume element is ddk = kd~1dkdQd^ where the solid angle element is = dd\ • sin$id$2 * sind\ sin tf2^3 * • • • * sin#1 sin#2 • • • sin'dd^dfld-i-
7.1 Dimensional regularization and renormalization
333
The total solid angle is TV
TT
f • d-2 Qd = 2 J sin o The integral
i
f d-3 i y sin o
IT
. . .
/
2
can be found using the substitution x = sin2 #. Therefore, (7.1.14) This formula can also be derived in a different way. The integral
1 = j e~~k2ddk can be calculated in two ways. In Cartesian coordinates, +OO
1 = / • —oo
In spherical coordinates,
Comparing these two results, we again arrive at (7.1.14). The integral (7.1.13) in spherical coordinates has the form
nm
-i(
H
-n"+™ ^ d/2P-4-d (27r)^(d/2) J
ij
Tk2mld-^dk (fc2 +a 2)n-
Starting from this moment, it is not necessary to assume d to be integer. All integrals are understood as an analytical continuation from a region of d where they converge. The integral is made dimensionless by extracting the factor (a 2 ) d / 2 + m - n . Taking into account
7
J
xmdx _ r(n - m - l)r(m {x + 1)» " TH
334
7 Radiative corrections
(this can be derived using y = l / ( x + 1)), we finally arrive at j(-l)n+m^a2y/2+m-n^2-d/2 nm
~
r(d/2 + m)F(-d/2 + n - ro)
(4rc)W
r(n)r(d/2)
(7.1.15) As we shall see in a moment, results in the limit e —> 0 look simpler if we put
*
4n '
where 7 is the Euler constant. In this case, r nm —
1
The integral (7.1.15) with a2 = 0 converges at d > n — m, and equals zero. Therefore, for any n, in dimensional regularization
/ ( £ ) , <*•>*•- ft
(7xi7)
input_case nil$ °/, Calculation of one-loop integrals d:=4-2*E$ vecdim d; operator GO; °/0 G0(n) = gamma(l+n*E) 7, Gamma function procedure Gl(x); begin scalar y; y:=sub(E=0,x); return if fixp(y) then y>l then (x-l)*Gl(x-l) if else if y
7.1 Dimensional regularization and renormalization
335
°/ 0 Examples I10:=loop(l,0,a); I20:=loop(2,0,a); — GQ(1) ex(Egam)ai ho •= —TTT-
_ Gp(l) ex(Egam)i ~ 16aEEn2 I30:=loop(3,0,a); I31:=loop(3,l,a); — GQ(1) ex(Egam)i 20 :
ho •= /31
:=
32aEa7r2 G0(l)
ex(Egam)i(-E E
+ 2)
2
32a En As a rule, we are interested in the limit e —> 0. Then,
L
n=2
where AJ— 1
is the Riemann function. Its values at even arguments are expressed via powers of ?r: load_package specfn$ {zeta(2),zeta(4),zeta(6)}; ^ 6 ' 90 ' 945^ Here is a set of REDUCE procedures for expansion in e: procedure lnG(x); °/«log (gamma (1+x)) begin scalar s,u,n; s:=-gam*x; u:=-x; n:=2; while (u:=-u*x) neq 0 do « s:=s+zeta(n)/n*u; n:=n+l » ; return s end$ procedure exO(x); °0/ exp(x) begin scalar s,u,n; s : = l ; u : = l ; n : = l ; while (u:=u*x/n) neq 0 do « s:=s+u; n:=n+l » ; return s end$ procedure binom(x,n); ° 0/ (l+x)"n begin scalar s , u , i , j ; s : = l ; u : = l ; i:=n; j : = l ; while (u:=u*x*i/j) neq 0 do « s:=s+u; i : = i - l ; j : = j + l » ; return s end$ procedure dummy(x); x$
336
7 Radiative corrections
procedure exe(x); °/0 expansion in E begin scalar xn,xd,n; xn:=num(x); xd:=den(x); n:=0; while sub(E=0,xd)=0 do « xd:=xd/E; n:=n+l » ; weight E=l; wtlevel n; for a l l xM l e t G0(x!!)=ex(lnG(x!!*E)); for a l l xM l e t x! rE=ex(log(x! !)*E); for a l l x!!,nM l e t x!!~(n!!*E)=ex(n!!*log(x!!)*E); xn:=xn; xd:=xd; for a l l x!! clear G0(x!!),x!!~E; for a l l x!!,n!! clear x ! r ( n ! ! * E ) ; for a l l x!! let ex(x!!)=exO(x!!); xn:=xn; xd:=xd; for a l l x!! clear ex(x!!); clear E; xn:=xn/sub(E=0,xd); xd:=xd/sub(E=0,xd); weight E=l; xn:=dummy(xn*binom(xd-l,-l)); clear E; return xn/E~n end$ % Examples exe(HO); exe(I20); exe(I30); exe(I31); ai(- log(a)E + £7 + l) 16 En2 16 ETT2
32 an2 i ( - 2 log(a) E - E + 2) 32 ETT2
In particular, the mass operator (7.1.11), taking account of (7.1.16), takes the form
lrg1?
\dx-
o Let's calculate the integral at s = —p 2/m2 > 0 with REDUCE: r:=int(log(l+s*x*(l-x)),x); r := log(-sx2 + sx rr:=sub(x=l,r)-sub(x=0,r);
l 8 _ 4 ) - s)
7.1 Dimensional regularization and renormalization
337
°/o Unfortunately, REDUCE chooses a wrong branch. °/0 The correct one i s l e t atan((2*s*x-s)/(sqrt(s)*sqrt(-s-4) )) =sqrt(s+4)/sqrt(-s-4)/2 *log((sqrt (s)*sqrt (s+4) +2*s*x-s)/(sqrt (s)*sqrt (s+4)-2*s*x+s)) ; R:=r; R := (y/sVs + 4 log( log(-sa;
2
V
s
-
V
s
+ 4 — 2sx + s
+ sx + l)s —
) H- 2 log(-5x 2 + sx + l)sx-
4sx)/(2s)
clear atan((2*s*x-s)/(sqrt(s)*sqrt(-s-4))); df(R,x); ° 0/check log( — sx2 + sx + 1) let log((sqrt(s)*sqrt(s+4)-s)/(sqrt(s)*sqrt(s+4) +s)) = -log( (sqrt (s) *sqrt (s+4) +s)/(sqrt (s)*sqrt(s+4)-s)); RR:=sub(x=l,R)-sub(x=O,R); °/0 correct definite integral
s clear log((sqrt(s)*sqrt(s+4)-s)/(sqrt(s)*sqrt(s+4)+s)); % it can also be obtained from rr let atan(s/(sqrt(s)*sqrt(-s-4)))=sqrt(s+4)/sqrt(-s-4)/2 *log((sqrt(s)*sqrt(s+4)+s)/(sqrt(s)*sqrt(s+4)-s)); Rr:=rr$ clear atan(s/(sqrt(s)*sqrt(-s-4))); Rr-RR; % check 0 bye;
Adding the counterterm contributions, we obtain M(p2)=-
o2
--log-5--V v l + l/slog 2(4TT)2 \s ^ ^ ^ '"~*JT+T/i-l 2
2
)
(7.1.18)
2
+ Sm - S • {p - m ) .
where s = ~p2/(4m2). At p2 > 0, the function M(p2) is obtained by analytical continuation; it has a cut at p2 > 4m2. Equating its value and its derivative at p2 = rn2 to zero, we obtain the counterterms in the on-shell scheme:
f2 •=
(\ l 0. g m 2
f
U
2(47r) 2 m 2
(**
2
A *
The second one is finite, in agreement with the dimensional considerations discussed earlier.
338
7 Radiative corrections Tensors and 7-matrices in d dimensions
The tensor algebra in d-dimensional space is defined by the relation 6w = d.
(7.1.19)
Let's define the projector ^ . . . / i , , ^ . . . ^ as the tensor 8^,lVl . ..£/*„ i/n, antisymmetrized with respect to ii\... \in (or v\... vn). Its trace is equal to the number of linear independent antisymmetric tensors with n indices in d-dimensional space (Fig. 7.4, where the projector is denoted by a wavy line) = —d(d - 1)... (d - n + 1).
(7.1.20)
When d is integer, the projector, together with its trace, is equal to zero at n> d. In a space of a non-integer dimension, there exist antisymmetric tensors with arbitrarily many indices, because the trace (7.1.20) is nonzero for all n.
n
Fig. 7.4. Trace of the projector-antisymmetrizer with n indices
It is possible to define the Dirac matrix algebra, defined by the relation + 7I/7M = <W>
(7.1.21)
in a space of any even dimension d. As a consequence of (7.1.21), any product of 7-matrices reduces to the form in which each of the d matrices 7M appears 0 or 1 times. Therefore, the number of independent matrices is 2 d, and their dimension is 2 d / 2 : Trl = 2 d/2 . (7.1.22) In other words, any 7-matrix expression can be expanded in the basis of antisymmetric tensors rMi...M»=7[Mi---7M«]» (7.1.23) because symmetric parts are eliminated as a consequence of (7.1.21). The basis matrices (7.1.23) (denoted by vertices) obey the orthogonality (Fig. 7.5a) and completeness (Fig. 7.56) relations. In the first of them, n! is the number of terms in the trace which survive after the antisymmetrization. The relation of Fig. 7.5c follows from it. The completeness relation of Fig. 7.56 is checked by contracting it with arbitrary basis matrices from the left and from the right (Fig. 7.5d). Contracting it the other way (Fig. 7.5e), we again arrive at (7.1.22). The relation of Fig. 7.5/, as well as the Fierz identity (Fig. 7.5#), also follows from here.
7.1 Dimensional regularization and renormalization
339
n
n n
0™
d
in
>>rX> =Cnm
in
> O
Fig. 7.5. Properties of basis 7-matrices
In a space of an integer (even) dimension d, there exist a finite number of basis 7-matrices (7.1.23) with n ^ d. In a space of a non-integer dimension, there are infinitely many of them, so that the completeness relation of Fig. 7.56 is an infinite series. All the formulae for Tr7 Ml . . . 7 M n / Tr 1 (REDUCE calculates such expressions) look as usual (§ 4.2), only Tr 1 is now given by the formula (7.1.22).
340
7 Radiative corrections
Differences appear when there are index contractions. It is not difficult to derive from (7.1.21) that
(4 REDUCE produces slightly more complicated results: input_case n i l $ vector p l , p 2 , p 3 ; index m; vecdim d; nospur f; g(f,m,m); g(f,m,pl,m) ; g(f,m,pl,p2,m); g(f,m,pl,p2,p3,m); d -9(f,Pi)d
+
2g(f,Pl)
clear m,pl,p2,p3; There is a package CVIT [RC3], which implements the Cvitanovic algorithm [QCD6] of calculation of d-dimensional traces. It is based upon the Fierz identity, and is more efficient than the built-in algorithm of REDUCE for traces with index contractions. As a simple example, let's find several coefficients appearing in the Fierz identity (Fig. 7.bg): vector ml,m2,m3,nl,n2,n3$ gl:=g(f,ml)$ g2:=g(f,ml,m2)$ g3:=g(f,ml,m2,m3)$ al:=g(f,ml)$ a2:=(g(f,ml,m2)-g(f,m2,ml))/2$ a3: = (g(f ,ml,m2,m3)-g(f ,ml,m3,m2)+g(f ,m2,m3,ml)-g(f ,m2,ml,m3) +g(f ,m3,ml,m2)-g(f ,m3,m2,ml))/6$ hi: =sub (ml=nl, gl) $ h2: =sub (ml=nl ,m2=n2, g2) $ h3:=sub(ml=nl,m2=n2,m3=n3,g3)$ bl:=sub(ml=nl,al)$ b2:=sub(ml=nl,m2=n2,a2)$ b3:=sub(ml=nl,m2=n2>m3=n3,a3)$ index ml,m2,m3,nl,n2,n3$ spur f$ vecdim d$ load_package cvit$ on factor; al*bl*gl*hl; al*b2*gl*h2; al*b3*gl*h3;
- (d - 2) d - (d - 1) (d - 4) d (d - 1) (d a2*b2*g2*h2; (d2 - 9d+ (d - 1) (d -
2) (d - 6) d a2*b3*g2*h3; a3*b3*g3*h3; 16) (d - l)d 2) (d - 4) (d - 9) d
- (d2 - 15 d + 38) (d - 1) (d - 2) (d - 6) d bye;
7.1 Dimensional regularization and renormalization
341
The procedure aver (p, a), which averages an expression a over directions of a vector p, is presented here. Now it is more general than in § 3.4, and it can work with expressions containing 7-matrices in the nospur state. It returns the power of p. p in each term as an argument of the function f!. The procedure av applies the recurrence relation to a single term once. Replacement of an arbitrary single instance of p by v! is performed by substituting p+x! *v! for p and selecting terms linear in x!; an arbitrary term of the resulting expression is selected by the procedure term. This is not a very efficient method, but it's not too easy to invent a better one. index i M ; vector v!!; operator av! ! ,av!?,term! ! ,f! !; for a l l x,y l e t av!!(x/y)=av!!(x)/y,av!!(x+y)=av!!(x)+av!!(y), term!!(x/y)=term!!(x)/y,term!!(x+y)=term!!(x); procedure term(x); «
x:=term!!(x); for all x!! let term!!(x!!)=x!!; x:=x; for all x!! clear term!!(x!!); x
»$ procedure av(p,a); begin scalar n; a:=sub(p=x!!*p,a); n:=deg(a,x!!)/2; if not fixp(n) then return 0; a:=sub(x!!=l,a); if n=0 then return a; if n>l then more!!:=l; weight x!!=1; wtlevel 1; a:=sub(p=p+x!!*v!!,a); clear x!!; a:=sub(x!!=l,a)-sub(x!!=0,a); a:=term(a); weight x!!=l; a:=sub(p=p+x!!*v!!,a); clear x!!; a:=sub(x!!=l,a)-sub(x!!=0,a); index v!!; a:=a; remind i!!,v!!; a:=sub(v!!=i! ! ,a); index i!!; a:=x!!*a; for all n!! let x! !*f!!(n!!)=f!!(n!!+l)/2/(n+l-E); a:=a; for all n!! clear x!!*f!!(n!!); return a end$ procedure aver(p,a); for all n!! let p.p*f!!(n!!)=f!!(n!!+l); a:=f!!(0)*a; « for all n!! clear p.p*f!!(n!!); repeat « a:=av!!(a); more!!:=0; for all a!! let av!!(a!!)=av!?(a!!); a:=a; for all a!! clear av!!(a!!); for all a!! let av!?(a!!)=av(p,a!!); a:=a; for all a!! clear av!?(a!!); » until more!!=0; clear more!!; a
»$
342
7 Radiative corrections Calculating one-loop integrals with REDUCE
A package of procedures for automatization of calculation of one-loop diagrams is presented here. Names of non-local internal variables of this package end with ! or ?, in order to avoid conflicts with variable names in a program using it. The procedure Feyn(p,a,l) construct Feynman parametrization. In it, p is the loop integration momentum, a is the diagram numerator, and 1 is the list of denominators. In this list, every denominator of the form ((p + Pi)2 — mf)ni is represented by the list {pi,mi,ni}. Some elements of this list may be omitted, the default values are pi = 0, mi — 0, ni = 1. The only way to assign a zero vector to a vector variable in REDUCE is to use 0*p, where p is a vector (you can also use p-p). The Feynman parameters are denoted x(l), x(2),.. .Their number is stored in the global variable j ! . After shifting the integration momentum, the expression is averaged over p directions. vector p ! ! , p ! ? , p i ! ! ; operator x; for a l l n!! l e t abs(x(n!!))=x(n! ! ) ; procedure Feyn(p,a,l); begin scalar n , c , l i , m i , n i ; if 1=O then return 0; p!!:=0*p!?; j ! ! : = 0 ; repeat « j ! ! : = j ! ! + l ; l i : = f i r s t ( l ) ; l : = r e s t ( l ) ; if l i = O then pi!!:=0*p!? else « p i ! ! : =f i r s t ( l i ) ; l i : =rest ( l i ) » ; if l i = O then mi:=0 else «mi:=f i r s t ( l i ) ;li:=rest ( l i ) » ; if l i = O then ni:=l else n i : = f i r s t ( l i ) ; p!!:=p!!+x(j!!)*pi!!; c:=c+x(j!!)*(pi!!.pi!!-mi~2); n:=n+ni; if ni>l then a:=a*x(j ! ! ) ~ ( n i - l ) / G l ( n i ) » u n t i l 1=O; a:=Gl(n)*sub(p=p-p!!,a); a:=aver(p,a); c:=p!!.p!!-c; for a l l n!! l e t f! ! (n! !)=loop(n,n!!,c); a:=a; for a l l n!! clear f ! ! ( n ! ! ) ; x(j!!):=l-for j : = l : j ! ! - l sum x(j); a:=a; clear x ( j ! ! ) ; j ! ! : = j ! ! - l ; return a end$ Finally, the procedure Fint(a) integrates over the Feynman parameters. It calculates an integral over the last parameter (having the number j !), and decrements j !. Of course, it would be easy to do this in a loop until all integrals are calculated. But our less general procedure can be useful even in cases when only a few first integrals are calculable. Anybody can call it in a loop, if desired. The procedure tries to calculate each integral only once. The following method is used to acheive this: a linear operator int! is introduced, and parts of an expressions containing different integrals are separated by the function part with the proper output control settings. Generally speaking, this trick should not be used too widely, because the result can depend on other output control commands; moreover, output settings can change after calling such a
7.1 Dimensional regularization and renorrnalization
343
procedure, and this can be a surprise for a user considering this procedure as a black box. When calculating definite integrals, there appears a subtlety with logarithms whose arguments vanish at the boundaries. All such logarithms, irrespective of their origin, are treated by REDUCE as log(O), and can cancel each other, leading to incorrect results. The procedure treats such logarithms with care. operator i n t ! ! ; linear i n t ! ! ; procedure Fint(a); begin scalar l,xm,li,mi,c; if a=0 then return 0; if j!!=0 then return a; 1:=O; a:=int! ! (a,x(j ! !)); factor int!!; on div; c:=if part(a,0)=minus then «a:=-a;-l» else 1; while part(a,0)=plus do « li:=part(a,l); a:=a-li; l:=(c*li).l; if part(a,0)=minus then «a:=-a;c:=-c» »; l:=(c*a).l; remfac int!!; off div; a:=0; xm:=l-for j:=l:j!!-l sum x(j); repeat « li:=first(l); l:=rest(l); for all a!!,b!! let int!!(a!!,b!!)=1; mi:=li; for all a!!,bM clear int!!(a!!,b!!); li:=li/mi; for all a!!,b!! let int!!(a!!,b!!)=a!!; li:=li; for all a!!,b!! clear int!!(a!!,b!!);
for a l l a!! and a!!
such that den(a!!/x(j!!))=1 neq x(j!!)
let log(a!!)=log(a!!/x(j!!))+log(x(j!!)); for all a!! such that den(a!!/(xm-x(j!!)))=1 and a!! neq xm-x(j!!)
end$
l e t log(a!!)=log(a!!/(xm-x(j!!)))+log(xm-x(j!!)); li:=li; for a l l a!! such that den(a!!/x(j!!))=1 and a!! neq x(j!!) clear log(a!!); for a l l a!! such that den(a!!/(xm-x(j!!)))=1 and a!! neq xm-x(jM) clear log(a.U); li:=sub(x(j! !)=xm,li)-sub(x(j ! !)=0,li); a:=a+mi*li » u n t i l 1=O; j ! ! : = j ! ! - l ; return a
We performed Feynman parametrization of the whole diagrams, including numerators; then we shifted the integration momentum, and averaged over its directions. There is an alternative approach to calculation of one-loop integrals containing numerators: all possible tensor structures are written down with unknown coefficients, which are then found by solving a system of linear equations.
344
7 Radiative corrections
Such a reduction of tensor one-loop integrals to scalar ones is implemented in the package LERG-I [RC4]. A far more non-trivial problem of calculation of two-loop integrals, with two external legs, in the massless case, is solved in the package LOOPS [RC5]. 7.2 Radiative corrections in quantum electrodynamics The QED lagrangian (taking account of a gauge-fixing term (§ 4.3)) has the form
~ ^d» Al)2
L = 4oidil>o ~ mo?o*o - \KA
+ <wMVo-
(7.2.1)
et's re-express it via the renormalized quantities I/JQ = J1~£ZJ -0, A^ = ~£ Z1/2 A^, m0 = Zmm, a0 = Zaa, e0 = fl£Zee: L = fl-2£ \$idip - rmj)^ - -.F^F^ - —(d^) ip 6rmj>il>
2
FF
+ ^(d
(7.2.2) where Z^ = 1 + 5^,, ZA = 1 + 6A, ZmZ^ = 1 + J m , Z~XZA = l + <Ja, ZeZ^Z1/2 = l + 5e. The terms in the last line (counterterms) are considered as a perturbation and produce the vertices (Fig. 7.6) 1(6^ - Smm), ISA{PIIPV -p2^v) — i^V\LVv> and iSeCy^. They are tuned in such a way that z/>, A^ are the physical fields, m is the physical mass, and e is the physical charge. The on-shell renormalization is usually used in QED. The procedure package loop.red, described in § 7.1, is used in all calculations in this Section.
p
p
Fig. 7.6. Counterterm vertices in QED
Photon propagator It has the structure (Fig. 7.7): 2
where the polarization operator zll^^p) is the sum of one-particle-irreducible photon self-energy diagrams (which cannot be separated into two disconnected parts by cutting a single photon line). It follows from (7.2.3) that
ZVW = DIM + Dlp(p)Upa(p)Dai/(P), ((DXpip) ~ n w (p)) Dpv(p) = V ,
(7-2.4)
7.2 Radiative corrections in quantum electrodynamics where the inverse tensor (in the sense culated using the formula
345
= 8^) can be conveniently cal-
—» - l
_
A-l
2
Diagrams for U^ don't contain external off-shell charged particles. Hence, the Ward identity implies
= o. Therefore, (7.2.5)
For the non-perturbed photon propagator (4.3.5), we have
and finally
Thus, the interaction does not induce a photon mass, and does not influence the longitudinal part of the photon propagator. Therefore, the longitudinal counterterm is unnecessary: one can assume 5a = 0 to all orders, i.e. Za = ZA- As a definition of normalization of the physical field AM, we accept the requirement of the absence of corrections on the mass shell, i.e. 5A is found from the equation II(p2 = 0) = 0. (7.2.7)
-«\A/^
^\A/»-|-•V/V^
•
V A /
V
^NA/»-
Fig. 7.7. Structure of diagrams for the photon propagator
It is convenient to calculate the polarization operator using the formula
7 Radiative corrections
346
It is equal to (Fig. 7.8, the minus sign comes from the fermion loop) ill(p2) = — — — ^ / ( — ] TriejJ— r^ v ' (d-l)p2j \27r/d ^(k+p)2-m2 Calculation with REDUCE gives at p2 < 0 (s =
^iejj—z = ^ k2 - m2
-p2/4m2)
Using the condition (7.2.7), we finally arrive at
_
, a (I .
ZA = 1--
--log
2m2 (7.2.8)
a 3^
k p+k Fig. 7.8. One-loop diagram for the polarization operator input_case nil$ in "loop.red"$ °/, photon self energy on gcd; operator V,r; for all p let V(p)=g(f,p),r(p)=g(f,p)+m; vector p,k; index n; let p.p=-nT2*s; z:=-i*4*pi*4/(d-1)/p.p*ex(-E*log(2))* Feyn(k,V(n)*r(k+p)*V(n)*r(k),{{p,m},{O*p,m}})$ z:=Fint(exe(z))$ let atan(s/(sqrt(s)*sqrt(-s-4)))=sqrt(s+4)/sqrt(-s-4)/2 *log((sqrt(s+4)+sqrt(s))/(sqrt(s+4)-sqrt(s))); factor log; on div; 3*pi*z; - log(m 2 )
4 log(-
IS + 4 + y ~o-2 /
- 5 + 2) - log(2) + E - l
clear atan(s/(sqrt(s)*sqrt(-s-4))); remfac log;
7.2 Radiative corrections in quantum electrodynamics
347
Electron propagator It has the structure (Fig. 7.9): iS(p) = iS0(p) + *So(p)H)£(p)tSo(p) + «So(p)(»OS(p)»So(p)(-i)E(p)iS 0(p) + • • •,
(7.2.9)
\p) - E(p))S(p) = 1, where the mass operator —iS(p) is the sum of one-particle-irreducible electron self-energy diagrams (which cannot be separated into two disconnected parts by cutting a single electron line). It contains two structures: p2).
(7.2.10)
Taking into account So (p) = p — m (4.3.6), we obtain S(p) =
2
- (1 -
E2)2m2
Let's define the physical electron mass as the position of the pole of the propagator, then Sm is found from the condition Ei(m 2 ) + ^(m2) = 0. Then, near the mass shell p2 = m2, the propagator is S(p) =
1 22
1 - Ei(m )) -
p+ m + £'2(rn )) p2 - m2' 2
where prime means derivative in p2/m2. Let's define normalization of the physical field ip by the condition that its propagator coincides with the free one near the mass shell. Then 5^ is found from the condition Ei(m 2 ) + 2[E / 1 (m 2 ) + S / 2 (m 2 )] = 0 .
Fig. 7.9. Structure of diagrams for the electron propagator
Calculating the diagram of Fig. 7.10 with REDUCE, we obtain in the Feynman gauge (s = -p2/m2) (7.2.11)
348
7 Radiative corrections
On the mass shell,
ox
a
a /I
,
m2
The last two formulae cannot be derived from (7.2.11): after differentiation, there is a logarithmic singularity at s -» — 1. It is necessary to calculate the integrals over the Feynman parameter exactly in d-dimensional space, and then go to the limit e —> 0. The singularity appearing in this case is called infrared, and is related to the fact that the photon is massless.
k
; + ->-*>— P' p+fc
V
V
Fig. 7.10. One-loop diagram for the mass operator
Finally we obtain "~
(7.2.12)
The functions Si(j>2) and T,2{p2) contain infrared singularities 1/e. They should cancel when calculating any physically sensible quantity (like a total cross section). We shall consider an example of such a cancellation in the next Section. °/o electron self energy y# nospur f; l e t abs(m)=m; z:=-i*(4*pi)^2*Feyn(k,V(n)*r(k+p)*V(n),«p,m},O})$ remind n; spur f; zl:=z*V(n)/p.n$ z2:=z/m$ zla:=sub(s=-l,zl)$ z2a:=sub(s=-l,z2)$ °/« on mass shell zlb:=sub(s=-l,-df(zl,s))$ z2b:=sub(s=-l,-df(z2,s))$ 7, derivatives zl:=Fint(exe(zl))$ j ! ! : = l $ z2:=Fint(exe(z2))$ l e t log(m"2*s+m"2)=log(s+l)+log(m"2); zl:=zl$ z2:=z2$ clear log(m"2*s+m'v2) ; factor log; on div; z l ; z2; remfac log; log(m2) + log(s + 1) ( - 5 " 2 + 1) - E~l + s'1 - 1 - 4 log(m2) - 4 log(s + 1) (s-1 + 1) + 2(2E~1 + 3)
7.2 Radiative corrections in quantum electrodynamics
349
°/o Exact calculation of integrals in d dimensions operator f; zla:=zla*x(ir(2*E)*f(-2*E)$ z2a:=z2a*x(l)~(2*E)*f (-2*E)$ zlb:=zlb*x(l)~(l+2*E)*f(-l-2*E)$ z2b:=z2b*x(l)~(l+2*E)*f(-1-2*E)$ for a l l n l e t x(l)*f(n)=f(n+1); zla:=zla$ z2a:=z2a$ zlb:=zlb$ z2b:=z2b$ for a l l n clear x(l)*f(n); for a l l n l e t f(n)=l/(n+l); zla:=zla; z2a:=z2a; zlb:=zlb; z2b:=z2b; m-2EG0(l) ex(Egam)E-1 zla := 2E - 1 2E 2m- Go(l)ex(Egam){-2E-1 z2a := 2E - 1 2E ex(Egam)E-1 m- G0(l) zlb := 2E - 1 ex(Egam) (2E' 1 - 1) 2E - 1 for all n clear f(n); zla:=exe(zla); z2a:=exe(z2a); zlb:=exe(zlb) ; z2b:=exe(z2b); zla := 2 log(m) - E'1 - 2 z2b :=
m-2EG0(l)
z2a := 2 ( - 4 log(m) + 2E'1 + 3) zlb := - 2 log(m) + E" 1 + 2 z26 := 4 log(m) - 2£'~1 - 3 clear zl,z2,zla,z2a,zlb,z2b; Vertex function Let's define the vertex function ieT^ as the sum of one-particle-irreducible vertex diagrams (which cannot be be separated into two disconnected parts by cutting a single electron or photon line) (Fig. 7.11); F^ = 7M + A^, where AM starts from the one-loop diagram (Fig. 7.12). Generally speaking, FM contains 12 7-matrix structures, whose coefficients depend on three variables pf, p|? Q2 (q = P2 - P i ) . The Ward identity leads to
The same relation holds for the propagator of the bare field Z^S and the bare fields' vertex function Z~1F/X, therefore, in all orders of perturbation theory, Ze = Zj,.
(7.2.14)
7 Radiative corrections
350
Fig. 7.11. Structure of diagrams for the vertex
Fig. 7.12. One-loop diagram for the vertex function
At q -> 0, (7.2.13) implies *.(*,)-
o,
(7.2.15)
The program calculates the one-loop contribution to AM (Fig. 7.12) in the Feynman gauge. It is too bulky to print. The first Feynman parameter integral can be expressed via logarithms, and the second one via dilogarithms, but the general calculation is complicated. We shall restrict ourselves to the case when both the initial and the final
7.2 Radiative corrections in quantum electrodynamics
351
electron are on-shell and have physical polarizations. Then the vertex is expressed in terms of two form factors Fi(q2) and F2(q2) (4.4.26). They can be separated by calculating traces with specially tuned projectors. The form factor Fi at q2 / 0 contains an infrared divergence. We shall not calculate the Feynman parameter integral in d-dimensional space here (a similar, though simpler, example will be considered in the next Section). Let's restrict ourselves to calculating the ultraviolet divergence. According to (7.2.15), it is related to the ultraviolet divergence of Xa(p2). Calculation of F2 gives (t = -q2/m2)
-vr
(7 2 16)
--
In particular, F2(0) = Y
(7.2.17)
is the electron anomalous magnetic moment. °/ 0 vertex function 0/
vector pl,p2; index 1; nospur f; off gcd; off div; let pl.pl=-m~2*sl,p2.p2=--m~2*s2,pl.p2=m~2/2*(t-sl-s2); °/0 General vertex function (not printed to save space) z:=-i*(4*pi)~2* Feyn(k,V(l)*r(k+p2)*V(n)*r(k+pl)*V(l),{{pl,m},{p2,m},{}})$ sl:=-l$ s2:=-l$ % both initial and final electrons are on shell % Projectors to single out form factors operator xl,x2; vl:=-xl(2)/(2*m)*(pl+p2).n+(xl(l)+xl(2))*g(f,n)$ v2:=sub(xl=x2,vl)$ spur f; index n; vv:=vl*r(pl)*v2*r(p2)$ matrix mm(2,2),mv(2,l); for i:=l:2 do for j:=l:2 do mm(i,j):=df(vv,xl(i),x2(j)); nospur f; remind n; for i:=l:2 do mv(i,l):=df(vl,xl(i)); on gcd; mv:=mnr(-l)*mv$ Pfl:=mv(l,l); Pf2:=mv(2,l) ;
9\fin)'ffl't
r> p fl
Ph
~
~l" 4^(/,?7,) ?77. + An.piE — Qn.pi + An.p2E m3 (Et2 + 8Et + WE - t2 - 8t - 16)
— 6fi.p2
••=
+ 4g(f,n)m - n.piEt + n.pit - 2n.pi — n.p2Et + n.p2t -2n.p2))/(m3t(Et2
+ SEt + WE - t2 - St - 16))
clear mm,mv; clear vl,v2,vv;
352
7 Radiative corrections
% Form factors spur f; index n; off gcd;
zl:=r(pl)*Pfl*r(p2)*z$ z2:=r(pl)*Pf2*r(p2)*z$ clear Pfl,Pf2; on gcd; zl:=zl; z2:=z2;
ex(Egam) 2
(-x(2) E + x{2)2 + 2x(2)x(l)E2t
- Sx{2)x(l)Et - 2x{2)x(l)E +
x(2)x{l)t + 2x(2)ar(l) + x{2)Et + 2x{2)E - x{l)2E + x{l)2 + x(l)Et + 2x{l)E - Et - 2 # ) ) / ((z(2)2m2 + x(2)x(l)m2t + 2x(2)x(l)m2 +
x(l)2m2)EE
(x(2)2 + x(2)x(l)t
(4G 0 (l) ex(Egam) (x(2)2E - x(2)2 + 2x{2)x{l)E
- 2x(2)x(l) + x(2) + x{l)2 E - x{l)2
((x(2)2m2 + x{2)x(l)m2t + 2x(2)x(l)m2 + x(l)2 m2)E (x(2)2 + a;(2)a;(l)* °/0 Ultraviolet divergence of Fl zl:=exe(E*zl)$ off gcd; zl:=Fint(Fint(zl))$ on div; zl; 1 z2:=exe(z2); z0:=sub(t=0,z2)$ 7, F2 _ 4(-x(2) 2 - 2a;(2)g(l) + x(2) - x(l)2 + x(l)) Z2
''~
x(2)2 + x(2)x(l)t + 2x(2)a;(l) + x(l)2
j!!:=2$ z2:=Fint(z2)$ let atan((x(1)*t+2)/(sqrt(t)*sqrt(-t-4)*x(1))) = sqrt(t+4)/sqrt(-t-4)/2 *log((2/x(l)+t+sqrt(t)*sqrt(t+4)) /(2/x(1)+t-sqrt(t)*sqrt(t+4))), atan((t+2)/(sqrt(t)*sqrt(-t-4)))=sqrt(t+4)/sqrt(-t-4)/2 *log((2+t+sqrt(t)*sqrt(t+4))/(2+t-sqrt(t)*sqrt(t+4))); z2:=z2$ clear atan((x(l)*t+2)/(sqrt(t)*sqrt(-t-4)*x(l))), atan((t+2)/(sqrt(t)*sqrt(-t-4))); z2:=Fint(z2)$
7.2 Radiative corrections in quantum electrodynamics
353
l e t atan(t/ (sqrt (t) *sqrt (-t-4) )) =sqrt (t+4) /sqrt (-t-4) /2 *log( (sqrt (t+4) +sqrt (t) )/(sqrt (t+4)-sqrt (t))) , log( (-sqrt (t) *sqrt (t+4) -t-2) / (sqrt (t) *sqrt (t+4) -t-2) ) 2*log((sqrt (t+4) +sqrt ( t ) ) / ( s q r t (t+4)-sqrt ( t ) ) ) ; z2:=z2; °/0 F2 Z2
'
t(t + 4)
clear a t a n ( t / ( s q r t ( t ) * s q r t ( - t - 4 ) ) ) , log((-sqrt(t)*sqrt(t+4)-t-2)/(sqrt(t)*sqrt(t+4)-t-2)); °/o anomalous magnetic moment j!!:=2$ zO:=Fint(Fint(zO)); clear zl,z2,zO; zo := 2 There are no other divergences in QED, except the polarization operator, the mass operator, and the vertex function, because it is impossible to construct other terms in the lagrangian, whose coupling constants would have a nonnegative dimensionality. Axial anomaly We shall not discuss radiative corrections in the electroweak interaction theory in detail. We shall restrict ourselves to one important feature — the axial anomaly. Some vertices in this theory contain 75. In order to calculate diagrams in dimensional regularization, it is necessary to generalize 75 to d dimensions. The requirement 757^ + 7^75 = 0, combined with the standard algebraic properties of multiplication associativity and trace cyclicity, leads to a contradiction. Let's consider the following chain of equalities: =>dTr>y5 = 0
(75 anticommutativity was used in the second step, and trace cyclicity in the third one). We have learned that if d / 0, then TV75 = 0. We suppose that d 7^ 0, and continue: a7/3 = ~ Tr 757/x7a7/37/x = - ( d - 4) Tr 757,,7/3 = * ( d - 2) Tv 757*7/3 = 0. We have learned that if d ^ 2, then Tr 757a7/3 = 0. We suppose that d ^ 2, and continue:
= -(d - 8) Tr757 a 7 /3 7 7 75 => (d - 4) Tr 757a7/?777(5 = 0 (we have used 7At7a7/3777<57Ai — (d — &)lalplil5+ terms with fewer 7-matrices). We have learned that if d ^ 4, then Tr 757^707^,5 = 0, and this is not very good.
354
7 Radiative corrections
Supposing d ^ 4, we can show that (d—6) Tr757 a7ig777(57e7£ = 0, and so on. All traces vanish, if d is not an even integer number. Therefore, an anticommuting 75 is not usable in dimensional regularization. REDUCE quite sensibly refuses to work with g(f ,a) when vecdim is not 4. A way out of this difficulty is to split d-dimensional space into 4 - and (d — 4)-dimensional subspaces, and to define i
75 — Ty£/Xl£*2^3A t4 7/Xi7Ai2 7/X3 7/Li4
757/* + 7/^75 = 0,
=
*7()7l7374}
757M ~ 7/x75 = 0,
where the tensor s^liJL2^3^A lives in the 4-dimensional subspace. Indices /i refer to this subspace, while indices fi refer to the orthogonal (d — 4)-dimensional subspace. This prescription is inconvenient, because it spoils d-dimensional Lorentz invariance, but nobody knows a better consistent prescription. As an example, let's consider the one-loop contribution to the Z°77 vertex (Fig. 7.13). The vector Z° coupling does not contribute due to the Farry theorem. The Ward identity yields Af/iAiiMaPlMi
=
°>
M
MMiM2P2M2 = 0.
(7.2.19)
If the particles in the loop are massless, then, it seems, one can prove the axial Ward identity (pi + p 2 ) M = 0, (7.2.20) based on the equality 5(pi)(pi - p 2 ) 7 5 S ( p 2 ) = 7 5 S(p 2 ) + 5(pi) 7 5 + 2mS(p 1 ) 7 5 S(p 2 ).
(7.2.21)
But this identity is wrong!
v/ V\ *V\
c\v k
\v
v/> P^
V2 Mi *px
k
^
/i 2
Fig. 7.13. One-loop diagram for the Z°77 vertex
Let's calculate (7.2.20) for the diagram of Fig. 7.13 carefully:
= -9A92v f (^j + ( l o 2).
Tr
S(k+Pl)(p1+p2)lbS(k-p2)^2S(k)llMl
7.3 Radiative corrections in quantum chromodynamics
355
Taking into account p i + p 2 = (&+Pi) - (£-£2), (^-£2)75 = (£ + £-£2)75 = ~~75(& ~ k ~P2) = ~Jb{k — f>2 — 2fc) (where k and A; are the components of k in the physical and non-physical subspaces), we obtain 9A92y
2).
The first two traces vanish, because they contain only one of the two vectors pi, p2 (this is the basis of the naive proof of (7.2.20)). The last term contains k, and its numerator should vanish at d —>• 4. But the ultraviolet divergence of the integral can give 1/e, producing a finite result. The trace is equal to
(because we must retain both p\ and p2 in order to have a non-vanishing result, and jfc anticommutes with all physical subspace matrices, including pi j2 and 7A* 1 2) • T n e ^ wo diagrams contribute equally, and after combining the denominators we get I dxidx25(x1 + X2 - 1) f — J
Z Q2)3 '
Replacingfc2-> ^^fc 2 and calculating the integral, we arrive at the final result for the anomaly in the axial Ward identity: 2 M
^iti2
(Pi + P2)M = -g^£/iiM2C*ia2PlaiP2c*2 •
(7.2.22)
This contribution results from the ultraviolet divergence, and hence it does not depend on masses and external momenta. If masses of the particles in the loop are nonzero, the normal contribution (7.2.21) should be added. °/0 axial anomaly Fint(exe((d-4)/d*Feyn(k,k.k,{{0*k,a,3}}))); — i 3 2 7T2
bye;
An axial anomaly would spoil the renormalizability of electroweak theory. However, contributions of different particles in the loop compensate each other. Let's consider the first generation particles: ve, e, w, d. The coupling constant 9A is proportional to the weak isospin component T3 of the left particles, and gy is proportional to the charge z. The total contribution of the first generation (taking account of three quark colours) is | 0 2 - | ( - l ) 2 + 3 ^ (§) - 3 | (-§) = 0. The second generation (z/^, /i, c, s) and the third one (i/T, r, t, b) repeat the same pattern, and don't contribute, either. There are anomalies in several other three-boson vertices, in addition to Z°77; all of them are compensated within each generation. Thus, for the electroweak theory to be renormalizable, it is necessary to have a quark doublet corresponding to each lepton doublet.
356
7 Radiative corrections 7.3 Radiative corrections in quantum chromodynamics
We shall consider QCD with massless quarks. We shall need the renormalization factors of the quark field Z^ = 1+ £,/,, and of the gluon one ZA = 1 + #A- Ghosts, which were introduced in § 6.4, should be included not only when summing a squared matrix element over gluon polarizations, but also in loops. The reason for this is that these two cases are closely related (by dispersion relations, which we don't discuss here). Therefore, we shall also need the renormalization factor of the ghost field Zh = 1 + 5h. There are many vertices in QCD, determined by the single coupling constant 1/2
#o = P>EZgg. The quark-gluon vertex contains the factor ZgZ^Z^ , the ghostgluon one: ZgZhZj[ , the three-gluon one: ZgZj[ , and the four-gluon one: Z^Z\. Therefore, in contrast to the QED case, we shall simply define Zg = 1 + 5g. Then the counterterms for these vertices are equal, in the one-loop approximation, 5g + 5^ + \5A, &g + #/i + \&A, &g + §<^4? and 25g + 26 AIt is impossible to define the on-shell renormalization scheme in QCD. Therefore, we shall use the MS scheme, in which counterterms are chosen to cancel 1/e poles. This scheme contains a parameter — the normalization scale
/A
We shall do all calculations in the Feynman gauge; you are encouraged to generalize them to the case a ^ 1. All colour factors which we shall need have been found in § 6.1. The procedure package loop.red, described in § 7.1, is used in all calculations. Quark propagator
The quark mass operator (Fig. 7.10) differs from the electron one in QED (at m = 0) by the colour factor CF'
E(p) = -CF~P [ J - log -^ - l] - 6+p,
(7.3.1)
hence in MS Zi> = l-CF^--. input_case nil$ in u loop.red"$ °/o quark self energy
on gcd; operator V; for all p let V(p)=g(f,p); vector p,k,n; let p.p=-s; index m; °/0 CF alphas/(4*pi) assumed
z:=-i*(4*pi)~2*Fint(exe( Feyn(k,V(n)*V(m)*V(p+k)*V(m),«p},O})/p.n))$ on div; z; off div; log(a) - E-1 - 1
(7.3.2)
7.3 Radiative corrections in quantum chromodynamics
357
Gluon propagator The gluon polarization operator is given by the diagrams of Fig. 7.14. The quark contribution (Fig. 7.14a) differs from the QED case (at m = 0) by the factor n / / 2 , where nf is the number of kinds of quarks. The contribution of Fig. 7.14d vanishes, because it reduces to an integral of the form (7.1.17). The gluon contribution (Fig. 7.146) and the ghost one (Fig. 7.14c), taken separately, are not transverse. Their sum has the correct structure {p^Pu Adding the counterterm contribution (Fig. 7.14e), we obtain
n 2
>- £
/I
,
-p2
31\
CAr -5iV c ^--log^- + - j
(7.3.3)
(\ /I -2p 2 +2nf T - -- log log —=Hence in MS
(7.3.4) (usually the term with log 2 is also included in 5A). k
k
a
b
k
c
d
Fig. 7.14. One-loop diagrams for the gluon polarization operator
°/ 0 gluon self energy y
for a l l pl,p2,p3,ml,m2,m3 l e t V(ml,m2,m3,pl,p2,p3) = (p2-pl) .m3*ml.m2+(p3-p2) .ml*m2.m3+(pl-p3) .m2*m3.ml; °/0 Nf alphas/(4*pi) assumed zl:=i*(4*pi)^2*Fint(exe(Feyn(k, V(m)*V(k)*V(m)*V(p+k),{{p},{}})*2*ex(-E*log(2))/s/(d-l)))$ on div; zl; off div;
- | log(s) - I log(2) + \
+f
remind m; index 11,12; z2:=i*(4*pi)"2*Fint(exe( Feyn(k,V(m,ll,12,p,-p-k,k)*V(n,ll,12,-p,p+k,-k),-C{p},O})/2))$ on div; z2; off div; 19 , , N 19 ^.i 29 11 _ ,N — m.n log(s) s — — m.n E s — m.n s + — m.p n.p log{s) — LZ
LJI
y
o
358
7 Radiative corrections
11 _ _ ! 67 — m.pn.p E - — m.pn.p z3:=i*(4*pi)^2*Fint(exe(Feyn(k,(p+k).m*k.n,{{p},{}})))$ on div; z3; 1 1 2 1 — ra.n log(s) s — -~m.nE~1s — -m.ns — - m.pn.p log(s) + 12 12 9 6 1 ,-,-1 5 — m.pn.pE + —m.pn.p 6 18 z:=(z2+z3)/(p.m*p.n-p.p*m.n);
•=-I "«<•>-I*"-7
off div; clear zl,z2,z3; remind 11,12; Ghost propagator The ghost mass operator (Fig. 7.15) is
M(p2) = -Nc^p2
f -£ - log - £ - + 2j + ^ p 2 ,
(7.3.5)
hence the renormalization constant Z h = l + JVc — - .
(7.3.6)
k
Fig. 7.15. One-loop diagram for the ghost mass operator
°/ 0 ghost self energy z:=-i*(4*pi)^2*Fint(exe(Feyn(k,(p+k).k,{{p},{}})))$ on div; z; off div;
Quark-gluon vertex QCD three-leg vertex functions contain many spin structures, whose coefficients depend on three variables pf, p\, p\. All integrals can be expressed via dilogarithms, but calculations are very bulky. We shall restrict ourselves to calculation of divergent \je terms, which have the same structure as the bare vertex, and don't depend on external momenta.
7.3 Radiative corrections in quantum chromodynamics
359
The one-loop correction to the quark-gluon vertex is given by the diagrams in Fig. 7.16. The contribution of Fig. 7.16a differs from the QED case (at m = 0) by the colour factor —t a/(2Nc). The contribution of Fig. 7.16b has the colour factor Ncta. Adding also the counterterm contribution (Fig. 7.16c), we obtain
hence the coupling constant renormalization factor ^
(7.3.8)
c Fig. 7.16. One-loop diagrams for the quark-gluon vertex
°/0 quark-gluon vertex vector pl,p2; l e t pl.pl=-sl,p2.p2=-s2,pl.p2=(t-sl-s2)/2; nospur f; index n; z l : =V (n) *V (p2+k) *V (m) *V (pl+k) *V (n) $ zl:=i*(4*pir2/(2*Nc)*exe(E*Feyn(k,zl, {{pi}, {p2}, {}}))$ off gcd; zl:=Fint(Fint(zl)); on gcd; Zl
:=
2 No
remind n; index 11,12; z2:=V(12)*V(-k)*V(ll)*V(m,12,ll,p2-pl,-p2-k,pl+k)$ z2:=-i*(4*pi) A 2*Nc/2*exe(E*Feyn(k,z2,{{pi},{p2}, off gcd; z2:=Fint(Fint(z2)); on gcd; 3Ncg(f,m) Z2l= 2 remind 11,12; spur f; z:=(zl+z2)*V(n)/m.n; 3iVc2 - 1 z i—
2Nc
Ghost-gluon vertex The ghost-gluon vertex function gA^bc is given by the diagrams of Fig. 7.17. The colour factor is equal to -(Nc/2)fabc, and Aabc
=
fabCp
which gives (7.3.8) again.
360
7 Radiative corrections
Fig. 7.17. One-loop diagrams for the ghost-gluon vertex
% ghost-gluon vertex .p2*(p2+k) .m$ zl:=-i*(4*pi)^2*Nc/2*exe(E*Feyn(k,zl,{{pi},{p2},{}}))$ zl:=Fint(Fint(zl));
m.p2 Nc
*:=—§—
index 11,12; z2:=-k.ll*p2.12*V(m,12,ll,p2-pl,-p2-k,pl+k)$ z2:=-i*(4*pi)^2*Nc/2*exe(E*Feyn(k,z2,{{pi},{p2},{}}))$ z2:=Fint(Fint(z2));
3 m.p2 Nc z2:= 3 (zl+z2)/p2.m; Nc
bye;
Three-gluon vertex Calculation of the divergent part of the one-loop correction to the three-gluon vertex (Fig. 7.18) is not difficult, but somewhat lengthy, because of the large number of diagrams. It is left as an exercise; the result is
X
—
(7.3.10) which gives (7.3.8) again. This result could be reproduced once more from the one-loop correction to the four-gluon vertex.
J
/ \ ^
Fig. 7.18. One-loop diagrams for the three-gluon vertex
7.3 Radiative corrections in quantum chromodynamics
361
Asymptotic freedom The coupling constant #, as well as the normalizations of the quark, gluon, and ghost fields, depend on the normalization point /x. In principle, physical quantities may be calculated at any /x, and results taking account of all orders of perturbation theory must be the same. However, in practice, only a few first terms can be calculated. If /x2 is chosen far away from characteristic momenta p2 of the considered process, then terms of the perturbation series will contain powers of the large logarithm Iog(p 2 //x 2 ), and convergence of the series will be slow. On the contrary, if one chooses /x2 ~ p 2 , then there will be no large logarithms, and a few first terms will produce a good approximation to the exact result. The renormalized coupling #(/x) is related to the bare coupling go (which does not depend on /x) by the formula go = M £ ^P(A*)? where
2(4TT)V
Therefore,
In quantum chromodynamics
b = y Nc --nf>0
at nf < —N c.
Therefore, the coupling constant #(/i) decreases with the increase of fi (or with the decrease of characteristic distances). This is called asymptotic freedom. The equation (7.3.11) is usually written for as: odcte bo /x 2 —^ z = - —-a 2 Ofj, 4TT
or
d 1 b — - _ _ = — alogjLr asyn*) 4TT
, (7.3.12)
Its solution is
It can be also written as
where A2 is an integration constant. Of course, all these equations are valid only at large momenta (or small distances) /i 2 ^> A2, where as(n2)
362
7 Radiative corrections
quark confinement. Thus, in spite of the fact that the bare QCD lagrangian is characterized by the dimensionless coupling constant go, the physical constant which determines the strong interactions is the dimensionful parameter A. In the case of quantum electrodynamics, it is also not always convenient to use the on-shell renormalization scheme. It is good when characteristic momenta p2 ~ m2. At p2 ^> m2 the electron mass may be neglected, and the MS scheme with /I2 ~ p2 is more relevant. In this scheme, the dependence e(/i) is determined by the equation flefr) 4e3(/x) M
6n
3(4TT)2
(if there are several kinds of leptons, its right-hand side should be multiplied by their number). Its solution is (7-3.15) i.e. the charge increases with the increase of/i 2 (decreases with the increase of the characteristic distance). This is explained by screening of a bare charge by virtual e + e~ pairs. When the interaction becomes strong a(// 2) ~ 1, perturbation theory is no longer applicable. In reality, other particles and interactions become important much earlier than this breakdown of perturbation theory in QED. Correction to the cross section e + e~ —> hadrons As a specific example, we shall calculate the radiative correction to the total ) hadrons. We shall see how infrared divergences cancel in cross section e + e~ —> a total cross section. We are going to calculate R = F(7* —> hadrons)/Fo (where F o is the 7* —> qq decay rate in the tree approximation) in d-dimensional space: R = 1 + fi2 + 5* + (53,
(7.3.16)
where the one-loop correction to the 7* -> qq matrix element (Fig. 7.19) in the left-hand side and the right-hand side of the squared diagram of Fig. 7.20 produces S2 and 6%, and the 7* -» qqg decay in the tree approximation (Fig 7.21) produces 83. The program calculates the contribution of the one-loop vertex correction (Fig. 7.19a) to $2- The Feynman parameter integral is calculated in d dimensions using the formula x
2
3
F(3 + ai + a2 + a3)
(where £3 = 1 — x\ — #2), which can be derived by the substitution x 2 = x\z. The counterterm contribution (Fig. 7.196) cancels the ultraviolet divergence of
7.3 Radiative corrections in quantum chromodynamics
c
d
e
363
f
Fig. 7.19. Diagrams of 7* —>• qq in the one-loop approximation
Fig. 7.20. Squared diagram of 7* —> qq
Fig. 7.21. Squared diagrams of 7* —> qqg
Fig. 7.19a. We have not discussed renormalization of the quark-photon vertex in QCD yet, therefore the program separates this divergence and obtains *
r
otal 4TT
e
Of course, this counterterm can be obtained from the known ultraviolet diver-
364
7 Radiative corrections
gence of the QED vertex (§ 7.2) by multiplying it by the colour factor CF. The loop correction to the external line (Fig. 7.19c) vanishes, because at p2 — 0 it reduces to integrals of the form (7.1.17). This is a result of cancellation between the ultraviolet divergence and the infrared one: this contribution can be symbolically written as O7T \SUV
J
The counterterm contribution of Fig. 7.19d cancels the ultraviolet divergence contained in it. The contributions of Fig. 7.19e, / coincide with that of Fig. 7.19c, d. As a result, the counterterm contributions cancel each other, i.e. 82 is given by the contribution of Fig. 7.19a. However, all divergences in the total result are infrared. This correction acquires an imaginary part at p2 > 0 because of L = log(—p 2///2). The imaginary part cancels when adding the conjugated correction to the right-hand side of the squared diagram of Fig. 7.20. However, the term L2 has a remnant of this imaginary part in the form of TT2. input_case nil$ in "loop.red"$ % Radiative correction to e+ e- -> hadrons y# —^ 7, gamma* -> q anti-q mass pl=0,p2=0; mshell pl,p2; let pl.p2=s/2; index m,n; vector k; operator V,S; for all p let V(p)=g(f ,p) ,S(p)=g(f ,p)/p.p; % Matrix element squared in the tree approximation dO:=V(pl)*V(m)*V(-p2)*V(m); do := s(-E + 1) °/0 One-loop correction d2: =V (pi) *V (n) *V (pl+k) *V (m) *V (-p2+k) *V (n) *V (-p2) *V (m) /dO; , 2(-k.kEs + 4k.pxk.p2 + 2k.pis - 2k.p2s - s2) d2:= on gcd; '/. CF alphas/(4*pi) assumed d2:=-(4*pi)-2*i*Feyn(k,d2,{{pl},{-p2},O}); d2 := (2G O (1) ex(Egam) (2x(2)x(l)E2 - 3x(2)x(l)E + a;(2)a;(l) + x(2)E + x(\)E-E))/ ((-X(2)X(1)S)EX(2)X(1)E)
Fint(Fint(exe(E*d2))); 7, ultraviolet divergence 1 let (-s*x(l)*x(2))^E=ex(L*E)*x(l)^E*x(2)"E; d2:=d2$ clear (-s*x(l)*x(2))~E;
7.3 Radiative corrections in quantum chromodynamics
365
°/0 Calculation of the Feynman parameter integral operator f; d2:=d2*x(ir(l+E)*x(2K(l+E)*f(-l-E,-l-E)$ for all m,n let x(l)*f (m,n)=f (m+l,n) ,x(2)*f (m,n)=f (m,n+l) ; d2:=d2$ for all m,n clear x(l)*f(m,n),x(2)*f(m,n); for all m,n let f(m,n)=Gl(l+m)*Gl(l+n)/Gl(3+m+n); d2:=d2; t= 2 :
~
ex{Egam)(2E2 - E + 2) Go(-2) ex(EL)E2(2E - 1)
GQ(1)GQ(-1)2
for all m,n clear f(m,n); on div; d2:=exe(d2); % expansion in e d2 := 2E~1L - 3E~l - 2E~2 - L2 + 3L + -TT 2 - 8 6 9 /» Adding the conjugated contribution d2:=sub(L=L-i*pi,d2)$ d2:=d2+sub(i=-i,d2); off div; d2 := AE-XL - 6E-1 - 4E~2 - 2L2 + 6L + \ TT2 - 16 Before proceeding further, let's calculate the two-particle phase space for massless particles in d dimensions:
1 / s Yee*eT(l-e) \-2e The three-particle phase space is
4?r X 2-K8((y/s - pi - p2)2 ~P\-P\-
2pip2 COS tf).
Introducing pis2 = {•s/s/2)x\^' ) we obtain Xl
X
\
where xs = 2 — x\ — x2, and 128?r3 \fjL2J
(1 - 2e)r(l - 2e)
x [(1 - xi)(l - x2)(l - x3)]~£
dx\dx2.
366
7 Radiative corrections
The program obtains the 7* —> qqg decay rate, divided by Fo (this is the d-dimensional generalization of the formula (6.2.3)): _, OLS XQ^~X~a~
dR
dXqdXq
eX
Q
27T (1 - Xq)(l ~ Xg)
^
s x
I —=•
Integrating this expression using the substitution #1,2 = 1 — 2/1,2 and the formula (7.3.17), the program obtains the correction £3, which contains an infrared divergence. This divergence cancels when adding the two-particle contribution, and we are left with the finite result for the total cross section R=l
+ 3CF^.
(7.3.19)
% gamma* -> q anti-q g mass k=0; mshell k; l e t pl.p2=s/2*(xl+x2-l),pl.k=s/2*(l-x2),p2.k=s/2*(l-xl); off gcd; d31:=-V(pl)*V(n)*S(pl+k)*V(m)*V(-p2)*V(m)*S(pl+k)*V(n)$ d32:=-V(pl)*V(n)*S(pl+k)*V(m)*V(-p2)*V(n)*S(-p2-k)*V(m)$ d3:=d31+d32$ clear d31,d32; d3:=d3+sub(xl=x2,x2=xl,d3)$ on gcd; d3:=d3/dO$ Ph2:=s^(-E)*ex(gam*E)*G0(-l)/(8*pi*G0(-2)*(l-2*E))$ x3:=2-xl-x2$ Ph3:=s~(1-2*E)*ex(2*gam*E)/(128*pi^3*(1-2*E)*G0(-2) l e t s^E=ex(L*E); d3:=(4*pi)^2*d3*Ph3/Ph2; d3 := (2 ex(2Egam) {-Ex\
- 2Ex1x2 + 4Exi - Ex\ + 4^^ 2 - 4£ + x\ + x\))/
((xi + x2 - l)E(-x2
+ l) s (-a?i -h 1)EGO(-1) ex(EL + Egam)
(xix2 - xi - x2 + 1)) clear s~E; clear Ph2,Ph3; % Calculation of the integral d3:=sub(xl=l-yl,x2=l-y2,d3)$ for a l l l,m,n l e t yl*f(l,m,n)=f(1+1,m,n),y2*f(l,m,n)=f(l,m+l,n); d3:=d3$ for a l l l,m,n clear yl*f(l,m,n),y2*f(l,m,n); for a l l l,m,n l e t f(l,m,n)=Gl(l+l)*Gl(l+m)*Gl(l+n)/Gl(3+l+m+n); d3:=d3;
7.3 Radiative corrections in quantum chromodynamics 4G O (-1) 2 ex(2Egam)(-2E3 + 5 £ 2 - 6 £ + 2) 2 '' Go(-3) ex(EL + Egam)E (9E2 - 9E + 2) for a l l l,m,n clear f(l,m,n); on div; d3:=exe(d3); d:=d2+d3; off div; ds
7
A TTI—1
Gt3 . : = : — 4JC/
d := 3 clear d2,d3,d; bye;
7"
I
r> Tp—1
L T 0£/
I
A
771 — 2
-)- 4 £ /
I
O
T" 2
-\- Z L/
£J 7"
— 0 Ju — —71 O
367
References
References are grouped according to their topics, and are accompanied by short comments. Relevant volumes of the Course of Theoretical Physics by L. D. Landau and E. M. Lifshitz are recommended as the main source of information about various areas of physics; they require no comments. REDUCE language [Rl] A. C. Hearn, REDUCE User's Manual, version 3.6, (RAND Pub. CP78 Rev. 7/95, 1995). Definitive description of REDUCE, written by its author. In order to use REDUCE effectively, you should have the manual of the current version.
[R2] D. Stoutemayer, REDUCE Interactive Lessons.
Interactive lessons for beginners, contain many simple examples. They are distributed with REDUCE.
[R3] M. A. H. MacCallum and F. J. Wright, Algebraic Computing with REDUCE, (Oxford University Press, Oxford, 1991). Detailed textbook (version 3.3). Contains discussion of internal structure of REDUCE and symbolic mode programming.
[R4] G. Rayna, REDUCE: Software for Algebraic Computation, (Springer-Verlag, Berlin, 1987). Detailed textbook (version 3.2), with many examples.
[R5] D. Stauffer, F. W. Hehl, V. Winkelmann, and J. G. Zabolitzky, Computer Simulation and Computer Algebra, (Springer-Verlag, Berlin, 1993). Lectures for beginners, with examples.
[R6] N. MacDonald, REDUCE for Physicists, (Institute of Physics Pub., 1994). Good introduction for first-time users, with many examples from physics (unfortunately, they contain a lot of typos).
[R7] V. F. Edneral, A. P. Kryukov, and A. Ya. Rodionov, Language for Analytical Calculations REDUCE (in Russian), (Moscow University Press, Moscow, 1989). Detailed textbook (version 3.2).
[R8] W.-H. Steeb and D. Lewien, Algorithms and computations with REDUCE, (Bl-Wissenschaftsverlag, Mannheim, 1992). [R9] F. Brackx and D. Constales, Computer Algebra with Lisp and REDUCE, (Kluwer Academic Pub., Dordrecht, 1991). 368
References
369
[RIO] J. P. Fitch, Solving Algebraic Problems with REDUCE, J. Symb. Comp. 1 (1985) 211. Short introduction, with some examples from mathematics.
[Rll] A. P. Kryukov, A. Ya. Rodionov, A. Yu. Taranov, and E. M. Shablygin, Programming in R-Lisp (in Russian), (Radio and communications, Moscow, 1991). [R12] J. Marti, RLISP'88: an Evolutionary Approach to Program Design, (World Scientific, Singapore, 1993). These two books describe RLISP — the language in which REDUCE is written. If you want to write packages extending the capabilities of REDUCE, you should know this language.
[R13] H. Melenk, Symbolic mode primer, REDUCE network library.
Introduction to REDUCE internal structure and functioning, and symbolic mode programming.
Computer algebra [CA1] Computer Algebra, ed. B. Buchberger, G. E. Collins, and R. Loos, (Springer-Verlag, Berlin, 1985). Collection of reviews, mainly about algorithms for working with algebraic expressions; contains also short reviews of computer algebra systems and applications.
[CA2] J. H. Davenport, Y. Siret, and F. Tournier, Computer Algebra, (Academic Press, New York, 1993). Good computer algebra textbook, features both examples of applications and a sufficiently accessible discussion of basic algorithms. Contains a description of REDUCE (version 3.3).
[CA3] A. G. Akritas, Elements of Computer Algebra with Applications, (J. Wiley and Sons, Inc., New York, 1989). Textbook, mainly about algorithms.
[CA4] K. O. Geddes, S. Czapor, and G. Labahn, Algorithms for Computer Algebra, (Kluwer Academic Pub., Dordrecht, 1992). Fundamental textbook on computer algebra algorithms.
[CA5] D. Barton and J. P. Fitch, Applications of algebraic manipulation programs in physics, Rep. Progr. Phys. 35 (1972) 235. [CA6] V. P. Gerdt, O. V. Tarasov, and D. V. Shirkov, Analytic calculations on digital computers for applications in physics and mathematics, Sov. Physics Uspekhi 23 (1980) 59. These two early reviews were instrumental for wider application of computer algebra systems in physics.
Some additional packages [PI] H. Melenk, NUMERIC package, REDUCE network library. This package implements some basic methods of numerical mathematics. Calculations can be performed with arbitrarily high precision, though not very efficiently. Algorithms for standard cases are implemented; they should not be used in pathological situations. [P2] H. Melenk, GNUPLOT interface for REDUCE, REDUCE network library. REDUCE uses the well-known program GNUPLOT for drawing plots.
370
References
[P3] K. Mclsaac, PM — a REDUCE Pattern Matcher, REDUCE network library. An alternative pattern matching and substitution mechanism with wider possibilities than the standard one. Substitutions for functions with varying number of arguments are allowed. Unfortunately, it is too buggy.
[P4] A. P. Kryukov and A. Ya. Rodionov, Program COLOR for computing the group-theoretic weight of Feynman diagrams in non-abelian gauge theories, Comp. Phys. Comm. 48 (1988) 327. The package LETT, which allows substitutions for functions with varying number of arguments, has been written for calculation of colour factors. Unfortunately, it is no longer available in recent REDUCE versions.
[P5] A. C. Hearn, COMPACT: reduction of a polynomial in the presence of side relations, REDUCE network library. The package for simplification of a polynomial, when its variables obey one or several polynomial relations.
[P6] S. L. Kameny, The REDUCE roots finding package, REDUCE network library. Finds numerically all roots of a polynomial.
[P7] M. A. H. MacCallum, F. Wright, and A. Barnes, 0DES0LVE, REDUCE network library. The package for solving first order ordinary differential equations of simple kinds, and any order linear ordinary differential equations with constant coefficients.
[P8] A. Moussiaux and R. Mairesse, Convode: a REDUCE package for differential equations, Int. J. Mod. Phys. C4 (1993) 365. The package for solving some kinds of ordinary differential equations.
[P9] E. Feldmar and K. S. Kolbig, REDUCE procedures for the manipulation of generalized power series, Comp. Phys. Comm. 39 (1986) 267. A set of procedures for working with univariate series; powers may be positive and negative rational numbers.
[P10] A. P. Kryukov and E. M. Shablygin, A package for working with power series in REDUCE (in Russian), Preprint NPI MSU 88-005/26 (Moscow, 1988). [Pll] R. Schopf, A REDUCE package for manipulation of Taylor series, REDUCE network library. Packages for working with multivariate Taylor series.
[P12] A. Barnes and J. Padjet, Truncated power series, REDUCE network library. The package for univariate Laurent series, in which the maximum order of expansion can be changed after a calculation; it always obtains as many terms of intermediate series as necessary.
[P13] W. Kopf and W. Neun, FPS — A package for the automatic calculation of formal power series, REDUCE network library. [P14] S. L. Kameny, A REDUCE limits package, REDUCE network library. [PI5] F. Kako, The REDUCE sum package, REDUCE network library. [PI6] G. Stolting and W. Koepf, ZEILBERG: A package for indefinite and definite summation, REDUCE network library. Packages for calculation of sums and products.
[P17] C. Cannam, SPECFN: Special function package for REDUCE; V. S. Adamchik and W. Neun, ghyper, a package for simplification of generalized hypergeometric functions; meijerG, a package for simplification of Meijer's G function; REDUCE network library. Special functions packages.
[P18] M. Rebbek, A linear algebra package for REDUCE; A REDUCE package for the computation of several matrix normal forms; REDUCE network library. Packages for linear algebra.
References
371
[PI9] D. Harper, A vector algebra and calculus package for REDUCE, REDUCE network library. [P20] J. W. Eastwood, ORTHOVEC: a REDUCE program for 3-D vector analysis in orthogonal curvilinear coordinates, Comp. Phys. Comm. 47 (1987) 139; ORTHOVEC: version 2 of the REDUCE program for 3-D vector analysis in orthogonal curvilinear coordinates, 64 (1991) 121. Packages for three-dimensional vector algebra and calculus.
[P21] V. A. Ilyin, A. P. Kryukov, A. Ya. Rodionov, and A. Yu. Taranov, in Computer algebra in physical research, ed. D. V. Shirkov, V. A. Rostovtsev, and V. P. Gerdt (World Scientific, Singapore, 1990), p. 190; V. A. Ilyin and A. P. Kryukov, in Proc. ISSAC-91, ed. S. Watt (ACM Press, 1991), p. 224; in New computing techniques in physics research II, ed. D. Perret-Gallix (World Scientific, Singapore, 1992), p. 639; Comp. Phys. Comm. 96 (1996) 36. The package ATENSOR for working with expressions which contain tensors with symmetries, maybe obeying multiterm linear identities; they may be contracted over dummy indices. It can work only with expressions containing not too many indices, because the required memory and calculation time grow very quickly.
[P22] A. Ya. Rodionov and A. Yu. Taranov, in Lecture Notes in Computer Science 378 (Springer-Verlag, Berlin, 1989), p. 192; Programming 2 (1990) 91. The package RTENSOR, which solves the same problem, except multiterm indentities, and is more efficient. Unfortunately, it is no longer available in recent versions of REDUCE.
[P23] M. Warns, PHYS OP: A package for operator calculus in quantum theory, REDUCE network library. The package for working with noncommuting operators. It requires either the commutator or anticommutator of each pair of operators to be known.
[P24] A. Ya. Rodionov, in Computer algebra and its applications in theoretical
physics, JINR Dll-83-511 (Dubna, 1983); SIGSAM Bull 18 (1984) 16.
NCMP — an extensive and flexible package for working with noncommuting operators. Unfortunately, it is no longer available in recent versions of REDUCE.
[P25] B. Gates, GENTRAN user's manual, REDUCE version, (RAND Pub., 1991). The package for generation of numerical programs in Fortran, C, Pascal, and RATFOR.
[P26] J. A. van Hulzen, SCOPE 1.5: A source code optimization package for REDUCE, REDUCE network
library.
The package for optimization of numerical code by extracting common subexpressions.
[P27] R. Liska and R. Drska, A REDUCE-IM^X formula interface, REDUCE network library. A simple to use package for incorporating results obtained with REDUCE into a IATQEX article.
[P28] W. Antweiler, The IfeX-REDUCE interface, REDUCE network library. A similar package, mainly for plain TfeX, but can be used with IATJTJX, too. It can split long formulae into multiple lines. However, it is not so easy to use.
372
References
Mechanics [Ml] L. D. Landau and E. M. Lifshitz, Mechanics, (Pergamon Press, 1982). [M2] H. Goldstein, Classical Mechanics, (Addison-Wesley Pub., Inc., Reading, 1980). Fundamental textbooks on classical mechanics.
[M3] D. ter Haar, Elements of Hamiltonian Mechanics, (Pergamon Press, 1971). Brief textbook on classical mechanics, including the hamiltonian perturbation theory.
[M4] G. L. Kotkin and V. G. Serbo, Collection of Problems in Classical Mechanics, ed. D. ter Haar, (Pergamon Press, 1971). You are encouraged to solve problems from this excellent problem book with REDUCE.
Hydrodynamics [HD1] L. D. Landau and E. M. Lifshitz, Fluid Mechanics, (Pergamon Press, 1987). [HD2] H. Lamb, Hydrodynamics, (Cambridge University Press, Cambridge, 1975). [HD3] L. N. Sretensky, Theory of Wave Motions of Fluids (in Russian), (Nauka, Moscow, 1977). Very detailed treatment of waves in hydrodynamics.
[HD4] V. P. Gerdt, N. A. Kostov, and A. B. Shvachka, Preprint JINR E l l - 8 3 750 (Dubna, 1983). Water waves are found with the accuracy of the terms cubic in the amplitude, including the finite depth case.
Electrodynamics [EDI] L. D. Landau and E. M. Lifshitz, Classical Theory of Fields, (Pergamon Press, 1980). [ED2] L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, (Pergamon Press, 1984). [ED3] J. D. Jackson, Classical Electrodynamics, (J. Wiley and Sons, Inc., New York, 1975). Fundamental textbooks on classical electrodynamics.
[ED4] R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, vol. 2, (Addison-Wesley Pub., Inc., Reading, 1964). Throughout Chapters 4-7, we use notations for 4-vectors borrowed from this brilliant book.
[ED5] V. V. Batygin and I. N. Toptygin, Problems in Electrodynamics, ed. D. ter Haar, (Pion, London, 1978). You are encouraged to solve problems from this excellent problem book with REDUCE.
References
373
General relativity [GR1] C. W. Misner, K. S. Thorn, and J. A. Wheeler, Gravitation, (W. H. Freeman and Co., 1973). Fundamental textbook, written from the geometrical point of view. Modern mathematical techniques are explained.
[GR2] S. Weinberg, Gravitation and Cosmology, (J. Wiley and Sons, Inc., New York, 1972). Fundamental textbook, more physical than geometrical.
[GR3] R. Penrose and W. Rindler, Spinors and Space-Time, vol. 1, (Cambridge University Press, Cambridge, 1984). Useful modern approach to the tensor calculus — the abstract index method by Penrose. Spinor calculus is discussed in detail.
[GR4] A. P. Lightman, W. H. Press, R. H. Price, and S. A. Teukolsky, Problem Book in Relativity and Gravitation, (Princeton University Press, Princeton, 1975).
Quantum mechanics [QM1] L. D. Landau and E. M. Lifshitz, Quantum Mechanics, (Pergamon Press, 1981). [QM2] A. Messiah, Quantum Mechanics, (Halsted Press, 1963). Fundamental textbooks on quantum mechanics.
[QM3] D. ter Haar (ed.), Problems in Quantum Mechanics, (Pion, London, 1975). You are encouraged to solve problems from this excellent problem book with REDUCE.
[QM4] W.-H. Steeb, Quantum Mechanics using Computer Algebra, (World Scientific, Singapore, 1994). [QM5] S. I. Vinitsky and V. A. Rostovtsev, in Computer algebra and its applications in theoretical physics, JINR Dl 1-85-791 (Dubna, 1985); Preprint JINR Pll-87-303 (Dubna, 1987); A. G. Abrashkevich, S. I. Vinitsky, and V. A. Rostovtsev, Preprint JINR E4-88-404 (Dubna, 1988). Using the symmetry group SO(4,2) of the hydrogen atom.
High energy physics [HEP1] L. B. Okun, Particle Physics, (Gordon Breach Science Pub., 1985). A clearly written introduction to the subject.
[HEP2] F. Halzen and A. D. Martin, Quarks and Leptons, (J. Wiley and Sons, Inc., New York, 1984). Good introductory textbook.
[HEP3] M. B. Voloshin and K. A. Ter-Martirosyan, Theory of Gauge Interactions of Elementary Particles (in Russian), (Energoatomizdat, Moscow, 1984). [HEP4] J. F. Donoghue, E. Golowich, and B. R. Holstein, Dynamics of the Standard Model, (Cambridge University Press, Cambridge, 1992). Detailed treatment of all major parts of the Standard Model.
374
References
[HEP5] E. Boos et al, in New computing techniques in physics research, ed. D. Perret-Gallix and W. Wojcik (Editions de CNRS, 1990), p. 573; in New computing techniques in physics research II, ed. D. Perret-Gallix (World Scientific, Singapore, 1992), p. 665; L. Gladilin et al, in Proc. CHEP-92, ed. C. Verkerk and W. Wojcik, CERN 92-07 (1992), p. 573; A. Pukhov, in New computing techniques in physics research III, ed. K.H. Becks and D. Perret-Gallix (World Scientific, Singapore, 1993), p. 473; E. Boos et al, Int. J. Mod. Phys. C5 (1994) 615. CompHEP: a system for automatic calculation of cross sections and decay widths in the Standard Model and other similar models. For a given process, it generates all tree-level diagrams, and calculates the squared matrix element. It can write results in REDUCE-readable form. Alternatively, it can generate a REDUCE program for such calculation.
Quantum electrodynamics [QED1] V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics, (Pergamon Press, 1982). [QED2] A. I. Akhiezer and V. B. Berestetskii, Quantum Electrodynamics, (Interscience Pub., New York, 1965). [QED3] J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, (McGraw-Hill, Inc., 1964); Relativistic Quantum Fields, (McGraw-Hill, Inc., 1965).
Weak interactions [Wl] L. B. Okun, Leptons and Quarks, (North Holland, 1985). Very clear and concise textbook.
[W2] H. Georgi, Weak Interactions and Modern Particle jamin/Cummings, New York, 1984).
Theory,
(Ben-
Very clear textbook, with interesting original material.
[W3] E. D. Commins and P. H. Bucksbaum, Weak Interactions of Leptons and Quarks, (Cambridge University Press, Cambridge, 1983). Detailed textbook.
Quantum chromodynamics [QCD1] F. J. Yndurain, Quantum Chromodynamics, (Springer-Verlag, Berlin, 1993). [QCD2] Y. L. Dokshitzer, V. A. Khoze, and A. H. Mueller, Basics of Perturbative QCD, (Editions Frontieres, Gif sur Yvette, 1991). [QCD3] R. Field, Applications of Perturbative Quantum Chromodynamics, (Addison-Wesley Pub., Inc., Reading, 1989). [QCD4] T. Muta, Foundations of Quantum Chromodynamics, (World Scientific, Singapore, 1987).
References [QCD5] W. Greiner, Quantum 1994).
Chromodynamics,
375 (Springer-Verlag, Berlin,
Textbooks on QCD.
[QCD6] P. Cvitanovic, Group theory, p. 1, (Nordita, 1984). Excellent presentation of group theory with applications to gauge theories.
[QCD7] A. P. Kryukov, Program xCOLOR: User's manual, REDUCE network library. The package for calculating colour factors, based on the Cvitanovic algorithm [QCD6]; much more efficient than the older implementation [P4].
Radiative corrections [RC1] J. C. Collins, Renormalization, (Cambridge University Press, Cambridge, 1984). High-level textbook on renormalization theory, based on dimensional regularization.
[RC2] P. Pascual and R. Tarrach, QCD: renormalization for the practioner, Lecture notes in physics 194, (Springer-Verlag, Berlin, 1984). Renormalization of QCD in dimensional regularization.
[RC3] V. A. Ilyin, A. P. Kryukov, A. Ya. Rodionov, and A. Yu. Taranov, High speed Dirac algebra calculations in a space of arbitrary dimension by means of a computer algebra system, Comp. Phys. Comm. 57 (1989) 505. Implementation of the algorithm of trace calculations [QCD5], which is more efficient than the built-in algorithm of REDUCE in dimensional regularization.
[RC4] R. G. Stuart, Algebraic reduction of one-loop Feynman diagrams to scalar integrals, Comp. Phys. Comm. 48 (1988) 367; R. G. Stuart and A. GognoraT., Algebraic reduction of one-loop Feynman diagrams to scalar integrals II, Comp. Phys. Comm. 56 (1990) 337. Package LEGR-I for reducing tensor one-loop integrals to scalar ones.
[RC5] L. R. Surguladze and F. V. Tkachov, LOOPS: procedures for multiloop calculations in quantum field theory for the REDUCE system, Comp. Phys. Comm. 55 (1989) 205. Package for calculation of one- and two-loop massless propagator diagrams.
Index
r function, 52 7 matrices, 173, 338 75 matrix, 174, 353 C function, 54, 335 •0 function, 53 abstract index, 99 action, 168 adjoint representation, 245 Altarelli-Parisi equation, 298 angular momentum, 66, 110, 113, 154, 169 annihilation operators, 65, 122, 152 anomalous magnetic moment, 207, 272, 351 antisymmetric function, 20 tensor, 64, 145, 166 array, 27 assignment, 1, 37 asymptotic freedom, 361 series, 128, 171, 174 averaging over directions, 63, 149, 341 axial anomaly, 353 charge, 273 current, 272 baryon form factor, 207, 216, 272 basis, 100 natural, 102 Bernoulli integral, 89 Bessel functions, 53, 78 Bhabha formula, 221 Bianchi identity, 104 black hole, 107 block, 25 Bohr's correspondence principle, 129 bound variable, 2 Breit-Wigner formula, 162 Brillouin—Wigner perturbation theory, 119, 132 C, xi, 2, 74 C-parity, 139, 233 Cabbibo suppression, 254 chaotic motion, 87 charged current, 250 charmonium, 139, 303 Christoffel symbols, 102
classical oscillator, 82 perturbation theory, 82 Clebsch-Gordan coefficients, 113 colour completeness relation, 304 factor, 288, 305, 315, 322, 359 comment, 2 completeness relation colour, 304 spin, 239, 338 complex algebra, 18, 31 component, 100 Compton scattering, 222, 228 concrete index, 100 conditional statement, 26 confinement, 361 connection, 102 contraction, 63, 99, 156, 165 coordinates, 102 cylindrical, 62, 98, 110 spherical, 61, 98, 110, 332 correspondence principle, 129 coulomb field, 110, 154, 188, 194, 289 counterterm, 328, 344, 356, 363 covariant derivative, 102, 174, 245, 287 creation operators, 65, 122, 152 cross section, 158, 164 curl, 62, 88, 98 current, 169, 171, 174 axial, 272 charged, 250 neutral, 251 vector, 261, 269, 272 curvature, 103 Cvitanovic algorithm, 288, 340 cylindrical coordinates, 62, 98, 110 Dalitz plot, 159, 167, 235, 296, 304 debugging, 33 decay rate, 157 decomposition, 42 definite integration, 7 degenerate perturbation theory, 131 denominator, 40 density matrix, 191 electron, 191 photon, 192 dependent variable, 5, 43
377
378 detailed balance principle, 164 determinant, 57 differential equations, 46 numerical solution, 47 differentiation, 5 over vector, 63 dilogarithm, 54, 237, 307, 350, 358 dimensional regularization, 330 dipole, 112 Dirac equation, 173 matrices, 173 divergence, 62. 88, 98 infrared, 348, 351, 364 ultraviolet, 185, 326 dual tensor, 166, 177 dummy index, 63, 99, 156, 165 Dyson's argument, 174 effective pseudoscalar, 272 eigenvalue, 57 eigenvector, 57 Einstein tensor, 106 electric dipole transition, 141, 153, 234, 304 quadrupole transition, 234 electromagnetic field, 185, 250 interaction, 174, 186, 245 electron density matrix, 191 form factor, 351 propagator, 185, 347 elementary functions, 5, 22 energy flow, 300, 306 energy-momentum tensor, 168 equation, 44 differential, 46 linear, 45, 56 numerical solution, 45 polynomial, 44 error function, 54 Euler constant, 334 equation, 88 even functions, 20 exactly renormalizable theory, 326 exponent, 5 factorization, 41 Fermi constant, 250 Feynman diagrams, 182 gauge, 185, 347, 350, 356 parametrization, 331, 342 rules, 183, 186, 253, 288 field electromagnetic, 185, 250 gauge, 172, 245, 287 scalar, 170, 326, 328 spinor, 173 strength tensor, 172, 245, 249, 287 vector, 172, 185 Fierz identity, 177, 338 file, 67
Index form factor, 188, 194 baryon, 207, 216, 272 electron, 351 meson, 205, 214, 269 Fortran, xi, 69 Fourier series, 23, 83, 91 free variable, 2 Fridman metric, 110 function r, 52 C, 54, 335
Index interactions electromagnetic, 174, 186, 245 strong, 287 weak, 245, 253 invariant flux, 158, 161 matrix element, 157 irrotational flow, 89 isospin, 254, 260, 272 iteration, 92, 119, 132 Jacobi identity, 65, 104, 246, 318 Jensen's device, 40 jet, 295, 303 Jordan form, 60 Klein-Nishina formula, 224 Kobayashi-Maskava matrix, 252 label 25 lagrangian, 168 gauge field, 245 QCD, 287, 356 QED, 174, 344 scalar field, 170, 326 spinor field, 173 vector field, 172 Weinberg-Salam, 249 Laguerre polynomials, 55, 153 Landau gauge, 185 theorem, 304, 309 Laplacian, 62, 89, 98 LaTeX, 74 lattice regularization, 330 Legendre polynomials, 54, 76 leptonic decays, 254 limit, 51 line integral, 62 linear functions, 21 Lisp, xii, 2, 19 list, 27 local variable, 25 logarithm, 5, 22 loop diagrams, 182, 326, 328 Lorentz transformation, 173, 182, 330 magnetic dipole transition, 141, 234, 304 magnetic moment anomalous, 207, 272, 351 normal, 207 Mandelstam variables, 162, 167 manifold, 101 mass operator, 347, 356, 358 matrix, 56 7, 173, 338 75, 174, 353 determinant, 57 Dirac, 173, 338 eigenvalue, 57 eigenvector, 57 Jordan form, 60 Kobayashi-Maskava, 252 Pauli, 192, 228, 248 rank, 57 S, 156, 187 trace, 57 transposition, 57
379
Meijer function, 52 meson form factor, 205, 214, 269 metric tensor, 101, 105 minimization numerical, 46 Moller formula, 219 Mott formula, 219 multipole, 111, 141 natural basis, 102 neutral current, 251 noncommuting operators, 65 nondegenerate perturbation theory, 119 nonleptonic decays, 254 nonrenormalizable theory, 185, 326 normal magnetic moment, 207 normalization scale, 356, 361 Nother theorem, 168, 171, 174 numerator, 40 numerical calculations, 7, 69 integration, 8 minimization, 46 solution of differential equations, 47 solution of equations, 45 octupole, 112 odd functions, 20 one-particle-irreducible diagram, 329, 344, 347, 349 operator angular momentum, 66, 110, 113, 154, 169 annihilation, 65, 122, 152 creation, 65, 122, 152 noncommuting, 65 secular, 131 wave, 131 optimization, 71 Ore-Powell formula, 235, 237 orthopositronium, 233, 241 oscillator classical, 82 quantum, 118, 121, 152 P-parity, 139, 233 parapositronium, 233, 240 parity C, 139, 233 G, 254, 272 P, 139, 233 partial fractions, 42, 219, 225 Pascal, xi, xii, 2, 24, 25, 74 Pauli matrices, 192, 228, 248 Pauli-Villars regularization, 330 perturbation theory Brillouin-Wigner, 119, 132 classical, 82 degenerate, 131 hamiltonian, 88 in quantum field theory, 171, 182, 328 nondegenerate, 119 Rayleigh-Schrodinger, 120> 132 phase space, 157 three-particle, 160, 365 two-particle, 159, 365 photon density matrix, 192 propagator, 185, 344
380 plots, 8 Poisson brackets, 110 polarization operator, 344, 357 polynomial decomposition, 42 factorization, 41 Hermite, 55, 152 Laguerre, 55, 153 Legendre, 54, 76 Pomeranchuk formula, 235, 305 positron, 186 positronium, 233 PostScript, xiii, 14 power series, 30, 48 procedure, 29 recursive, 29, 123, 134 product. 27, 52 propagator, 183, 329 electron, 185, 347 ghost, 313, 358 gluon, 287, 357 photon, 185, 344 quark, 286, 356 scalar, 183, 329 spinor, 185 vector, 185 quadrupole, 112 quantum oscillator, 118, 121, 152 quark mixing, 252 radiative transitions, 138, 153 electric dipole, 141, 153, 234, 304 electric quadrupole, 234 magnetic dipole, 141, 234, 304 rank, 57 Rayleigh-Schrodinger perturbation theory, 120, 132 recurrence relation, 51, 77 recursive procedures, 29, 123, 134 substitutions, 77 regularization, 330 renormalizable theory, 185, 248, 326 renormalization, 328 MS, 356, 361 on-shell, 329, 337, 344 resonance, 83, 87, 161 Ricci tensor, 104 Riemann function, 54, 335 tensor, 103 riemannian space, 105 Rodrigues formula, 77 rotator, 133 Runge-Lenz vector, 110, 154 Rutherford formula, 194, 205, 208 5-matrix, 156, 187 scalar curvature, 106 field, 170, 326, 328 product, 61, 63, 101 propagator, 183, 329 scattering cross section, 158, 164 Schwarzshild metric, 107 secular operator, 131 self-energy, 329, 344, 356
Index semihadronic decays, 254 semileptonic decays, 254 series asymptotic, 128, 171, 174 Fourier, 23, 83, 911 power, 28, 48 Taylor, 28, 48 small variable, 4 special functions, 49 spherical coordinates, 61, 98, 110, 332 harmonics, 54, 153 spin completeness relation, 239, 338 spin-orbit interaction, 196 spinor, 101, 173 field, 173 propagator, 185 spontaneous symmetry breaking, 246 statement block, 26 conditional, 26 group, 26 loop, 27 stochastic motion, 87 Stokes parameters, 192, 228 strong interactions, 287 structure constants, 172, 245, 287 substitution, 17, 37 list, 32 recursive, 77 sum, 27, 51 superrenormalizable theory, 327 switch, 3 symmetric function, 20 tensor, 64, 145 symmetry, 168, 171, 174, 327 breaking, 246 factor, 183, 193 5C/(3), 254 tangent space, 101 Taylor series, 28, 48 tensor, 63, 64, 99, 338 antisymmetric, 64, 145, 166 component, 100 dual, 166, 177 Einstein, 106 energy-momentum, 168 field strength, 172, 245, 249, 287 metric, 101, 105 Ricci, 104 Riemann, 103 symmetric, 64, 145 unit, 63, 100, 156, 338 unit antisymmetric, 142, 150, 156, 165, 175 Thomson formula, 224 threshold behaviour, 200, 206, 269 thrust, 298, 306 torsion, 103 trace, 57 7-matrix, 175, 186 transposition, 57 tree diagrams, 182, 328 trigonometric functions, 5, 23 ultraviolet divergence, 185, 326
Index vertex, 182, 184 electron-photon, 186, 349 four-gluon, 288, 318, 356 ghost-gluon, 313, 356, 359 quark-gluon, 288, 356, 358 three-gluon, 288, 312, 356, 360 volume integral, 62
unit antisymmetric tensor, 142, 150, 156, 165, 175 unit tensor, 63, 100, 156, 338 vacuum, 246 variable bound, 2 dependent, 5, 43 free, 2 global, 25 local, 25 small, 4 vector 3-dimensional, 61 4-dimensional, 156 current, 261, 269, 272 field, 172, 185 product, 61 Runge-Lenz, 110, 154 symbolic, 63 velocity potential, 88
W-boson, 249, 256, 278 Ward identity, 188, 192, 194, 197, 200, 209, 312, 318, 345, 349, 354 water waves, 88 wave operator, 131 weak charge, 273 hypercharge, 248 interaction, 245, 253 isospin, 248 magnetism, 272 Weinberg angle, 250 Weinberg-Salam theory, 248, 355 Yang-Mills field, 245, 287 Z-boson, 250, 263
REDUCE language elements $, 3, 16 '/., 2 ., 27, 61, 63 •C, 28 }, 28 =,26 :=, 2, 37, 70 :=: (GENTRAN), 70 ::= (GENTRAN), 70 ::=: (GENTRAN), 70 >, 26 >=, 26 =>, 32 « , 25 » , 25 ',32 \ (RLFI), 74 _a RDEBUG 35 _c RDEBUG 35 RDEBUG 35 !larbconst (0DES0LVE), 46 a, 175, 354 abs, 5, 125, 197 accuracy (NUMERIC), 45 acinfo (SCOPE), 72 add_columns (LINALG), 59 add_rows (LINALG), 59 add_to_columns (LINALG), 59 add_to_rows (LINALG), 59 algint (package), 6 allfac (switch), 15 and, 26 antisymmetric, 20 append, 28 arbcomplex, 45 arbconst (ODESOLVE), 46
arbint, 45 arbreal, 45 arglength, 42 array, 27 atensor (package), 64 augment_columns (LINALG), 58 avec (AVECTOR), 61 avector (package), 61 band.matrix (LINALG), 58 begin, 25, 73 besselj (SPECFN), 53 br (RDEBUG), 35 clear, 2, 5, 17, 19, 39 clearrules, 32 clebsch_gordan (SPECFN), 118 coeff, 41 coeffn, 41 collect, 28 column_dim (LINALG), 58 compact (COMPACT), 43 compact (package), 43 cont, 67 convode (package), 47 coordinates (AVECTOR), 61 copy_into (LINALG), 60 cos, 5, 23 cosh, 95 cr!* GENTRAN), 70 curl AVECTOR), 62 cvit package), 340 cylindrical (AVECTOR), 62 declare (GENTRAN), 72 decompose, 42 def id (RLFI), 74 defindex (RLFI), 74
381
382 defint (package) 7 d e f l i n e i n t (AVECTOR), 62 deg, 40 delaydecs (GENTRAN), 71 delayopts (SCOPE), 72 delsq (AVECTOR), 62 den, 40 depend, 5, 62 det, 57 df, 5 diagonal (LINALG), 58 dilog (SPECFN), 54, 237, 307 div (AVECTOR), 62 div (switch), 15 do, 27, 29 e, 5, 7 echo (switch), 68 ei (SPECFN), 54 else, 26 end, 25, 67, 73 eps, 165, 175 erf (SPECFN), 54 even, 20 exp, 5 exp (switch), 3 extend (LINALG), 58 factor, 15 factor (switch), 41 factorial, 153 factorize, 41 first, 28 fix, 125 f ixp, 19, 26 for, 27 for all, 19, 31 for each, 27 fortlinelen!* (GENTRAN), 69 fps (FPS), 50 fps (package), 50 freeof, 92 g, 175 g3 (XCOLOR), 292 gamma (SPECFN), 52 gcd (switch), 3 gentran (GENTRAN), 69 gent ran (package), 69 gentranin (GENTRAN), 73 gentranlang!* (GENTRAN), 74 gentranout (GENTRAN), 70 gentranshut (GENTRAN), 70 getsystem (AVECTOR), 62 goto, 25 grad (AVECTOR), 62 hermitep (SPECFN), 55 hidden3d (plot), 14 hypergeometric (SPECFN2), 55 i, 18 if, 26 impart, 18 in, 28, 67 index, 63, 165 infinity, 50 input.case, 16
Index int, 6 integer, 25 join, 118 Jordan (NORMFORM), 60 jordan.block (LINALG), 58 laguerrep (SPECFN), 55, 153 lambert_w, 45 lasimp (RLFI), 74 latex (RLFI), 74 lcof, 40 legendrep (SPECFN), 54, 76 legr-i (package), 344 length 28 let, 17, 29, 33, 36, 37, 63 lett (package), 22 lhs, 44 limit, 51 limit!+, 51 limit!-, 51 linalg (package), 58 linear, 21 lineint (AVECTOR), 62 literal (GENTRAN), 70 load package, 6 log, 5, 22 loops (package), 344 lterm, 40 make identity (LINALG), 58 makedecs (GENTRAN), 71 makeopts (SCOPE), 72 mass, 166 mateigen, 57 matrix, 56 matrix.augment (LINALG), 58 matrix.stack (LINALG), 58 matrixp (LINALG), 58 max, 125 maxexpprintlen!* (GENTRAN), 69 mcd (switch), 3 min, 125 minor (LINALG), 59 mkid, 231 mshell, 166 mult.columns (LINALG), 59 mult.rows (LINALG), 59 nat (switch), 16 ncmp (package), 67 neq, 26 nero (switch), 106 nodepend, 5 noncom, 65 normform (package), 60 nospur, 176 not, 26 nullspace, 57 num, 40 num.int (NUMERIC), 8 num.min (NUMERIC), 46 num.odesolve (NUMERIC) 47 num.solve (NUMERIC), 45 numeric (package), 8, 45, 47 odd, 20 odesolve (ODESOLVE), 46
Index odesolve (package), 46 off, 3 on, 3 operator, 19, 31, 56, 65 or, 26 order, 14 ordp, 20 orthovec (package), 62 out, 67 output (plot), 14 part, 42 pascal (GENTRAN), 74 pause, 67 pf, 42 physop (package), 67 pi, 5, 7 plot, 8 pm (package), 22 points (plot), 14 precision, 7 pri (switch), 68 prod, 52 product, 27 ps (TPS), 48 pscompose (TPS), 49 psdepvar (TPS), 49 psexpansionpt (TPS), 49 psexplim (TPS), 48 psfunction (TPS), 49 psi (SPECFN), 53 psreverse (TPS), 50 psterm (TPS), 49 qg (XCOLOR), 292 rank, 57 rat (switch), 15 rationalize (switch), 18 rdebug (package), 33 real, 25 reduct, 40 remfac, 16 remind, 63, 165 repart, 18 repeat, 29 rest, 28 return, 25 reverse, 28 revpri (switch), 14 rhs, 44 rlf i (package), 74 root .multiplicities, 44 root_of, 44 rootacc (ROOTS), 45 roots (package), 45 roots (ROOTS), 45 rounded (switch) 7 row_dim (LINALG), 58 rtensor (package), 65 saveas, 194 scalar, 25 scalefactors (AVECTOR), 61 scope (package), 69 second, 28 set, 37
shut, 67 sin, 5, 23 sinh, 96 solve, 44 specfn (package), 52 specfn2 (package), 52 sphericalharmonicy (SPECFN), 54, 153 spur, 176 sqrt, 5 squarep (LINALG), 58 stack_rows (LINALG), 58 step, 27 sub, 17, 37 sub.matrix (LINALG), 59 such that, 19 sudim (XCOLOR), 292 sum, 27, 51 swap_columns (LINALG), 59 svap_rovs (LINALG) 59 symmetric, 20 system, 69 t, 67, 70 tclear (ATENSOR), 65 tensor (ATENSOR), 65 terminal (plot), 14 then, 26 third, 28 title (plot), 8 tp, 57 tps (package), 48 tr (RDEBUG), 33 trace, 57 tri (package), 76 trout (RDEBUG), 37 trrl RDEBUG), 36 trst RDEBUG), 34 tsym ATENSOR), 65 unbr (RDEBUG), 36 until, 30 untr (RDEBUG), 33 untrrl (RDEBUG), 36 untrst (RDEBUG), 35 vec (AVECTOR), 61 vecdim, 63, 340, 354 vector, 63, 165 verbatim (RLFI), 74 vmod (AVECTOR), 61 volintegral (AVECTOR), 62 volintorder (AVECTOR), 62 weight, 4 when, 32 where, 32 while, 29 write, 26, 67 ws, 1 wtlevel, 4 xcolor (package), 292 xlabel (plot), 8 ylabel (plot), 8 zeilberg (package), 52 zeta (SPECFN), 54, 335
383
384
Index Reactions
e+e~ "~, 139, 200, 295 7* 7* ->n+n~, 139, 206, 295 7* - * PP, 210 7* -> qq, 139, 295, 362 7* ~> 990, 296, 362, 366 W~ -> Z" 17, gg', 254, 257 Z - > / + / - , vv, qq, 254, 263 if -> ^ , 302 e+ee+e~ e+e" e+ee+e" e+e~ e+e" eV ^+ e "0e e~ e-
7*, 139, 205 Z, 254 —» 1 203 p, 254, 260 e+e-,221 -+,M+/I-, 139, 187, 212, 268, 305, 311 7T+7T-, 139, 215 pp, 217 gg, 139, 212, 268, 294, 305, 362 -+ qqg, 294, 296, 362, 366 _+ 7 7 ? 225, 235 -> 37, 235 -» W+W~, 254, 278
(e+e-)l 1 5 0 :+«(e+e
• 27, 233, 240 • 37, 234, 241
- e 7 , 222, 228 • Wv, 254, 281 77- • e + e - , 226 77 " v W+W~, 254, 283 e7* ->• e, 193 7T7* ->> 7T, 205 P7* -> p, 207 ee -> ee, 187, 218 e/x —> e/x, 211 en —> en, 214 ep ->> ep, 216 ix -> e ^ i / ^ , 254, 264 r -)> evevT, fiu^Vr 254, 266 r —>• c?ttz/, c?5z/, 254, 266 r ->• pz/, K*v, 260 r —>• 2nv, 4nis, 261 r -> nv, Kv, 262 c —y slv, dlv, 255 c —>• swd, sits, dud, dus, 255 b -» dI7, 255
CGFU, esc, C5u, cdc, 255
t —>•
bW, 259
•0 —> • e + e - , /x+M~, 141, 200, "0 —> qq, 303, 305 •0 —> 3tf, p£7, 303, 305 -0 —> 7T+7T-, 141, 206 ^ ' ->• X0,i,27, 141, 304 i\)' -> 77C7, 142, 304 •0' -> 'ifinn, tprj, 304 ^c — ¥ gg, 77, 303, 308 X0,2 -> p^, 77, 146, 304, 308 xi -•> 99P, 3#, 304, 310 Xo,i ,2 ->• V>7, 141, 304 Xo,i ' 2 ^TT+TT", 142 X0,l ,2 -» pp, 142
n -> Iv, 255, 261 Ti-.-• 7r°eZ/, 270 K -» ZI7, 255, 262
iC -> TTZF, 255, 269
K -> 27rfi7, 255 X — • 2?r, 3TT, 255 D —3 • Klv, B -)- D/F, 269 n -^ peF, 255, 272 A - > pez7, 255 A ^ Nn, 255, 276 E -> Aez/, nev, 255 E - > TVTT, 255, 276 2 -)• AeI7, Sez7, 255 S - ) • An, 255, 276 AK, En, 255, 276
n^
qq' --> go', 310 qq -* qq> 3 i i qq ^ n'a' 311 • 99, 312 > 99 317 qq • Z, 254 qq' -> W, 254 qg — »• qg, 312 99 ~> ^g, 317 > ^ , 318
-)> /xz/e, 267 eVe -+(&», 267
qv ->• q'l, qv, 267 Nv • -• NL Nv, 255, 276 pa -)> nz/, 276
e,
268
Codemist Ltd. h t t p : //www. b a t h . ac. uk/~mas jpf /Codemist /Codemist. html CSL REDUCE is available for the following computers: • Acorn Archimedes • Macintosh running System 7 • Power Macintosh running System 7.5 • PC 386, 486 or Pentium running MS-DOS, Windows 3.1 (with Win32s), Windows NT or Windows 95 • SUN4 running Sunos or Solaris • Silicon Graphics running Irix 4.0.5, 5.2 or 5.3 • IBM RS6000 running AIX • Dec Alpha running OSF/1 • Any computer with an ANSI C compiler (for example, MIPS, HP, Apollo) In some cases REDUCE 3.6 is not yet ready for distribution, and so we can supply the previous version pending the port. • The Professional REDUCE includes all sources of both REDUCE and CSL, instruction sheets, a printed manual and postage and packing. • The Personal REDUCE, only available for PC clones, Acorn Archimedes, Atari ST and Macintosh, is pre-built, and delivered with documentation in machine readable form, instruction sheets. It does not include a printed manual. • The Codemist REDUCE Manual is a 450 page single volume manual, incorporating the REDUCE manual, and the manuals for modules and libraries. Please note that one copy of the manual is included in the Professional REDUCE package. See the above URL for information about prices. Site licenses are available, email: [email protected]
Konrad-Zuse-Zentrum fur Informationstechnik Berlin http://www.zib-berlin.de/Symbolic/reduce/ Packages: Personal REDUCE: REDUCE 3.6 binary, PSL runtime system; no manual, no source, no compiler; available only for PCs (386, 486, DEC Alpha PC). Professional REDUCE: REDUCE 3.6 source and binary, REDUCE 3.6 documentation (1 printed manual, documents on tape/disk), PSL runtime system and compiler. full PSL: PSL source and binary, PSL documentation (on tape/disk). REDUCE 3.6 + full PSL.
The packages are available for the following system platforms: Group I: PC AT 386/486: DOS (^Windows), NT, OS/2, UNIX (SCO, Interactive), Linux; DEC Alpha PC (Windows NT). Group II: SUN SPARC (SunOs 4.x, Solaris); IBM RS6000 (AIX); HP 9000/300, 9000/700, 9000/800 (HP-UX); Silicon Graphics; DEC VAX (Ultrix and VMS), DecStation (Ultrix), Alpha (DEC Unix (OSF/1), VMS); Control Data 4000 series, Cyber 910; Convex C series; Data General AViiON; NeXT (UNIX); i860 (UNIX). Group III: CRAY X, Y, C90, T3D (UNICOS); Fujitsu 2400 (UXP/M). See the above URL for information about prices. Site licenses are available, email: neunOsc. zib-berlin. de
