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  Anomalous triangle vertex: divergencies and symmetry argument

+ 2 like - 0 dislike

Consider the triangle correlator of one axial-vector current $J_{\lambda 5}$ and two vector currents $J_{\mu}, J_{\nu}$ in a theory with a fermion with mass $m$:
\Gamma_{\lambda \mu\nu}(q,k,p) = F\bigg[\langle 0|T(J_{\lambda 5}(x)J_{\mu}(y)J_{\nu}(z) )|0\rangle \bigg]_{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
\Gamma_{\lambda\mu\nu}(q,k,p) = A(p,k,m)\epsilon_{\lambda \mu\nu\rho}k^{\rho}+ \epsilon_{\nu \lambda \sigma\rho}p^{\sigma}k^{\rho}(B(p,k,m)p_{\mu}+C(p,k,m)k_{\mu})\\ + \begin{pmatrix} k \leftrightarrow p \\ \mu \leftrightarrow \nu\end{pmatrix},
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 $\Gamma_{\lambda \mu\nu}$, 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 $p^{\mu}\Gamma_{\lambda \mu \nu} = 0$, or
\tag 1 A(p,k,m) = B(p,k,m)p^{2} +C(p,k,m)(p\cdot k)
Therefore one knows that $\Gamma_{\lambda \mu \nu}$ 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 $\Gamma_{\lambda \mu \nu}$ one has the general form
\Gamma_{\lambda \mu \nu}(0, k, -k) = D(k^{2},m^{2})\epsilon_{\lambda \mu\nu\sigma}k^{\sigma},
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

\Gamma_{\lambda \mu\nu}(0,k,-k) \equiv \lim_{p \to -k}\Gamma_{\lambda \mu\nu}(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?
asked Dec 9, 2017 in Theoretical Physics by NAME_XXX (1,060 points) [ revision history ]
edited Dec 9, 2017 by NAME_XXX

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