QUANTUM FIELD THEORY in .NET

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QUANTUM FIELD THEORY
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to all orders. Equation (12-147) may also be used to show that the corrections to the inverse propagator Ip, are transverse to all orders. A different regularization, investigated by Lee and Zinn-Justin, relies on the introduction of higher covariant derivatives in the initial lagrangian. This improves the large-momentum behavior of the gauge field and makes all diagrams with more than one loop finite. The one-loop diagrams must be regularized independently in a gauge-invariant way. The reader may carry out the renormalization program in the axial gauge; it is convenient to write the condition n A = 0 by using a Lagrange multiplier in the functional integral. Although the ghost fields are not really coupled to the gauge field, their introduction enables us to use a slightly modified Becchi-Rouet-Stora transformation and to derive a set of identities.
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12-4-4 Gauge Dependence of Green Functions
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Since at the very end of our computations we are supposed to check the gauge independence of the physical quantities, it is important to control the gauge dependence of Green functions. We observed that the transformation <5A = DA- 1 <5w shifts ff into ff + <5w and modifies the action Iff' of Eq. (12-133) according to
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(12-169)
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Consequently, an infinitesimal change of the function ff may be cancelled by a gauge transformation on the field A. The latter affects only the source term, and according to the equivalence theorem of Sec. 9-2-1 this should not modify physical quantities such as the S-matrix elements: exp [G.Hds-{J)]
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f2&(A) f~(A)
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det AfJiHfJiexp det AfJi exp
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{i fd x [~- ~ff2 + J o(A - DA-l~ff)]}
{i fd x CSP - ~(ff + ~ff)2 + J
(12-170) This discussion is quite formal at this point, since in the gauge theories studied so far the S matrix is not defined, due to severe infrared divergences.
In terms of the Becchi-Rouet-Stora transformation, the preceding property is reflected by the following structure of i5I:
NONABELIAN GAUGE FIELDS
This enables one to study the gauge dependence of proper functions and counterterms, for instance their dependence on the parameter A. As a typical result, one may show that the coupling constant renormalization Zg is A independent, at least in the minimal renormalization. With other prescriptions, this may be wrong. The last remark is a hint that the physical interpretation and observation of a nonabelian gauge coupling constant may be difficult.
12-4-5 Anomalies
We now reconsider a case of physical interest set aside at the end of Sec. 12-4-3. Let us assume that fermions are cOl,lpled to the gauge field through an axial current. We have seen in the previous chapter that anomalies may occur in the conservation (or quasiconservation) of such a current, as a consequence of the impossibility of regularizing the theory while preserving chiral symmetry. In the instances studied in Chap. 11, namely, quantum electrodynamics or the a model, this anomaly was acceptable and of physical interest to analyze the process n -4 2y. If the gauge field (abelian or not) is coupled to an anomalous current, the situation is drastically different. Slavnov-Taylor identities may become invalid and renormalizability is jeopardized. In theories where the gauge field remains massless, such as those considered so far, this would mean that all possible counterterms of dimension four would be required, spoiling the universality of the coupling renormalization. The issue is much more crucial when the symmetry is spontaneously broken. As we shall see in the next section, the gauge field becomes massive and renormalizability only results from the underlying gauge invariance. Anomalies are then harmful, and it is possible to devise models where they make the theory nonrenormalizable. It is therefore important to find a criterion to rule out their appearance. As in the previous chapter, the analysis may be restricted to one-loop diagrams. There exists a gauge-invariant regularization which preserves chiral invariance at higher orders (Sec. 12-4-3). Consider a gauge theory based on a compact group. In other words, we admit abelian factors. The lagrangian, including fermion fields,
(12-171) where ra stands for some combination of T a and Tays. It is easy to check that the anomaly of the axial current
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