Abstract
The gauge principle is fundamental in formulating the Standard Model. Fermion–gauge-boson couplings are the inescapable consequence and the primary determining factor for observable phenomena. Vertices describing such couplings are simple in perturbation theory and yet the existence of strong-interaction bound-states guarantees that many phenomena within the Model are nonperturbative. It is therefore crucial to understand how dynamics dresses the vertices and thereby fundamentally alters the appearance of fermion–gauge-boson interactions. We consider the coupling of a dressed-fermion to an Abelian gauge boson, and describe a unified treatment and solution of the familiar longitudinal Ward–Green–Takahashi identity and its less well known transverse counterparts. Novel consequences for the dressed-fermion–gauge-boson vertex are exposed.
Highlights
Identities of the Ward-Green-Takahashi (WGT) type [1,2,3] have long been known and used in gauge theories
We consider the coupling of a dressed-fermion to an Abelian gauge boson, and describe a unified treatment and solution of the familiar longitudinal Ward-Green-Takahashi identity and its less well known transverse counterparts
Equation (1) is a nonperturbative consequence of gauge invariance in an Abelian theory and, following Ref. [4], it has been used extensively in the construction of models for the dressed-fermion–gauge-boson vertex
Summary
Identities of the Ward-Green-Takahashi (WGT) type [1,2,3] have long been known and used in gauge theories. The widely familiar forms provide constraints on the longitudinal part of n-point Schwinger functions; i.e., propagators and vertices. In an Abelian gauge theory the dressed-fermion–gaugeboson vertex, Γμ(k, p) in Fig. 1, satisfies qμiΓμ(k, p) = S−1(k) − S−1(p) ,. Equation (1) is a nonperturbative consequence of gauge invariance in an Abelian theory and, following Ref. [4], it has been used extensively in the construction of models for the dressed-fermion–gauge-boson vertex. (Renormalisation does not affect the form of the identities we consider, so we do not explicitly refer to it. A Euclidean metric is used : {γμ, γν } = 2δμν; γμ† = γμ; γ5 =
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