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QUANTUM FIELD TIIEORY
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where F I'V is the electromagnetic field strength. No result obtained previously is affected by this correction. The impossibility of maintaining for the regularized or renormalized theory all the Ward identities implied by the classical approximation justifies the name given to these anomalous Ward identities. Let us study some of their properties and consequences.
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11-5-3 General Properties
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The phenomenon analyzed in the framework of the (J model is common to all theories with fermions coupled to axial currents when we try to satisfy simultaneously axial and vector Ward identities. It was discovered as early as 1951 by Schwinger and was explicitly studied by Adler for electrodynamics and Bell and Jackiw for the (J model. In electrodynamics we want to verify the relation (11-214) which states that the failure of axial current conservation arises from the fermionic mass term. The axial current of Eq. (11-214) differs by a factor 2 from the one discussed above. A consequence of (11-214) would be a relation among Green functions, analogous to (11-205), of the form (Fig. 11-18)
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qPRl'vp= ie 2 qP = 2me 2
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d4 x d4 y ei(k,.x+k, y) Tjl' (x)j.(y)j5p(0)
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f d x d y ei(k,. x+k,' y) <Tjl'(x)jv(y)j5(0)
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(11-215)
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2mRI'v
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Let us introduce the proper functions for the axial current and the pseudo scalar density: (11-216)
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k2, v
Figure 11-18 Green function with two vector and one axial currents.
SYMMETRIES
where the subscript T means truncation of the fermion propagators. From (11-214) (11-217) According to our usual notations is(p) is the complete fermion propagator such that to lowest order S-l(p) = P- m. Equation (11-217) is the analog of the Ward identity for the vertex function (8-87). (11-218)
Equation (11-217) would imply a common multiplicative renormalization of mjs and normalization constant Zs = 1.
with re-
In fact, the identity (11-214) and its consequences, (11-215) and (11-217), are not verified perturbatively. They are modified by anomalies arising from the triangular diagram, as was the case for the (J model. The computation given in the previous section can readily be extended to the present case with the result
qPRl'vp = 2mRI'v
+ - excl'vpukfk'2 n
(11-219)
as the corrected version of (11-215) to one-loop order. In operator form the anomaly reads (11-220) Similarly, (11-217) is modified as
(p~ - pl')n(p',p) = - 2imr s(p', p)
+ YsS-l(p) + S-l(p')yS + i 4n F(p',p)
(11-221)
y- p' x)
F(P',p)
d4 x d4 x' ei(p"
<T!/J(y)ljI(x)cI'VpupvFPU(O)T
As a consequence of (11-221) evaluated at zero momentum, 2mjs has still a renormalization constant equal to unity. The same is no longer true for j~. The breaking of chiral invariance now involves a hard component epvpaFPvFpa.
The structure of anomalies in a renormalizable theory with fermion fields endowed with an internal symmetry coupled to vector, axial, scalar, and pseudoscalar fields may be analyzed along the same lines. The general conclusion is that only diagrams containing fermion loops coupled to vector and axial currents lead to anomalies. Axial currents must occur in odd numbers along the loop. Moreover, any loop containing a scalar or pseudo scalar coupling may be eliminated by an adequate subtraction. If we insist on preserving the normal Ward identities for vector currents, the diagrams of Fig. 11-19 lead to anomalies for the axial ones. As in the triangular case, the manipulation performed when trying to verify the identity al'j~ = 2mjs involve divergent integrals up to the pentagon diagram.
QUANTUM FIELD THEORY
Figure 11-19 One-fermion-loop diagrams leading to anomalies in the axial Ward identities.
Let "Yp. and d'p. stand for the vector and axial fields considered as matrices acting on the internal symmetry indices of fermions in such a way that the interaction lagrangian reads (11-222) The corresponding currents
J p.---", T u"Yp.
off'
(11-223)
are also matrices in the internal symmetry indices. The vector Ward identities are by hypothesis the normal ones:
Op.Jp.(x)
j(x)
(11-224)
while the axial current anomalies take the form
V V '" 5 - . _1_ 1 up. J"( X ) - J5 (x) - 4n 2 Bp.v P,,(lFp.v F P" + 17 FA FA "4 p.v P"
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