Consider the triangle correlator of one axial-vector current Jλ5 and two vector currents Jμ,Jν in a theory with a fermion with mass m:
Γλμν(q,k,p)=F[⟨0|T(Jλ5(x)Jμ(y)Jν(z))|0⟩]q,k,p,
where F[] means the Fourier transformation. The momentum q is q=p+k. By using the perturbation theory, it can be related to triangle diagram with massive fermion running in the loop.
Adler proposes to use tensor decomposition
Γλμν(q,k,p)=A(p,k,m)ϵλμνρkρ+ϵνλσρpσkρ(B(p,k,m)pμ+C(p,k,m)kμ)+(k↔pμ↔ν),
following from lorentz covariance, parity symmetry and bose symmetry. This expansion has historical name Rosenberg expansion.
The ''form-factors'' B,C have the dimension -2 in energy units, while A has the dimension 0; this follows from the dimensionality of Γλμν, which is 1. Therefore, since the form-factors are obtained by the loop integration, A is potentially logarithmically divergent. However, actually it can be shown that it is finite assuming the vector-like current conservation. Really, one has pμΓλμν=0, or
A(p,k,m)=B(p,k,m)p2+C(p,k,m)(p⋅k)
Therefore one knows that Γλμν is free from any divergences without any calculations, since now potentially diverged A is expressed in terms of finite B,C with finite pre-factors.
Let's now assume particular case p=−k which corresponds to q=0. For the vertex Γλμν one has the general form
Γλμν(0,k,−k)=D(k2,m2)ϵλμνσkσ,
where D is dimension 0 and a priori contains logarithmic singularity.
My questions are the following:
- is it possible to use the identity (1) for the case p=−k by considering first the general case of unrelated k,p and then by taking the limit
Γλμν(0,k,−k)≡limp→−kΓλμν(q,k,p)?
- if the answer on the previous question is no, is there any other way to ensure that D contains or doesn't contain logarithmic divergence without direct computations of the loop integral?