ANOMALIES in .NET

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11-5 ANOMALIES
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Up to now we have only encountered examples where renormalization did not conflict with the symmetries apparent at the classical level. In this section we discuss situations of great interest where the opposite is true, and new features are
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QUANTUM FIELD THEORY
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uncovered by renormalization. In Chap. 13 similar phenomena will appear when dealing with scale invariance. We introduce this subject with an apparent paradox in the application of current algebra.
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11-5-1 The nO _ 2y Decay and Current Algebra
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When reviewing the successes of current algebra in Sec. 11-3 we carefully omitted what seems at first a failure in its application to the nO decay in two photons (Fig. 11-15). Let us now study this process. The amplitude may be written
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!7 = !7(qZ) Iq2=m~ = lim [;~[;2 TI'v(q)
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q2-+m;
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(11-183) (11-184)
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We have exhibited the time-ordered product of two electromagnetic currents coupled to the two photons (kb [;1 and kz, [;z) and of the pion field n(x). Here q = k1 + kz and the photons are on-shell kr = k~ = O. In the spirit of PCAC (11-117) we are led to replace the neutral pion field by the divergence of the axial current, omitting the isospin index 3. Therefore, (11-185) with (11-186)
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To extract the derivative acting on the axial current from the time-ordering symbol we have used the fact that the commutator b(Xo - yO) [JI'(x), Ao(y)] vanishes for the quantum numbers entering (11-186). Let us now write the most general decomposition of the tensor Tl'vp(k b k z). This involves taking into account a negative parity from the axial current, a symmetry in the combined exchange (kb /1) +-+(kz, v) from the Bose statistics of photons, and the transversity to k~ and k"Z as a result of the electromagnetic current conservation (11-187)
q --k 2 , 2
Figure 11-15
nO .....
2)' decay.
SYMMETRIES
With this information we may write the following expression when
T!,vp(kb k 2) = c!'w;tkTk'2qp Tl (q2)
kr = k~ = 0 :
(11-188)
+ (c!'pqt k 2v -
cvpqrk 1 !')kTk'2 T2(q2)
+ [(C!'pqtklV Consequently,
CvPqtk2!')kTk'2 - c!'vpq(kT - k~)kl k 2]T3 (q2)
(11-189) We evaluate these amplitudes to leading order in electromagnetism. Since no strongly interacting zero-mass particle is around (m; is considered to be different from zero) it follows from (11-189) that
.:1(0) = 0 .:1(m;) is suppressed:
(11-190)
For the soft pion theory to be valid, this should be interpreted as meaning that (11-191) The dominant part of the nO -+ 2y is forbidden according to this observation of Sutherland and of Veltman. Fortunately this conclusion turns out to be incorrect due to the effects of renormalization. We stress that this does not question the validity of the extrapolation from (11-190) to (11-191) but really means that the statements (11-189) or (11-190) have to be modified. A method of approaching the problem is to perform a perturbative calculation within a given model. The interpretation of the result will be that we have to modify the PCAC relation (11-117) in the presence of electromagnetic interactions.
11-5-2 Axial Anomaly in the
Model
To be specific we use the a model with fermions. For simplicity we keep only one Fermi field of charge + 1 (the proton) and two mesons nO and a. From Sec. 11-4-1 the lagrangian is
= i/f[i - m + g(a + inys)]'" + i(on)2 + i(oa)2 - ~" n 2 - ~q a 2
(11-192)
To lowest order m= -gv (11-193) The proton is the only charged particle to contribute to the current; therefore
i/fy!'", (11-194)
The axial current
552 QUANTUM FIELD THEORY
Figure 11-16 One-loop diagrams for
decay.
has a divergence formally given by
81'AI' =
m;J"n
j,,= -v
(11-195)
The nO-decay amplitude computed to the lowest one-loop order is (Fig. 11-16)
TJJ)(k 1, k 2) + Tv~)(k2' k 1) (11-196) 4 T(1)(k k) _ 2 d p tr [(~ + P- /<.1 + m)yl'(~ + p + m)ys(p + m)yv] I'V 1, 2 - ge (2n)4 [(q + P _ k1 _ m 2] [(q + p _ m 2] (p2 _ m2) Tl'v
The trace in the numerator is equal to 4imcl'vpakfM (with remaining integral is convergent:
C0123 =
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