# Radiative corrections to $V-A$ and $V+A$ currents

In Appendix A of the following reference, p.128,

Masaru Doi, Tsuneyuki Kotani, Eiichi Takasugi, Double Beta Decay and Majorana Neutrino, Progress of Theoretical Physics Supplement, Volume 83, March 1985, Pages 1–175, https://doi.org/10.1143/PTPS.83.1

The radiative correction to arbitrary order of perturbation to the effective tree level $$V\pm A$$ quark current is given as, \begin{align} \overline{u}\gamma^\mu(1\mp\gamma^5)d = \overline{u}(p_1)\Gamma^{\rho\nu ... ..\lambda}\gamma^\mu(1\mp\gamma^5)\Gamma'_{\rho\nu ... ..\lambda}d(p_2), \tag{1}\label{eqn1} \end{align} where $$u$$ and $$d$$ are up and down quarks. $$\Gamma^{\rho\nu ... ..\lambda}$$ consists of $$\gamma^\mu$$, $$\gamma^\nu$$, ...... $$\gamma^\lambda$$, $$\gamma^\mu p_{1\mu}$$, $$\gamma^\mu p_{2\mu}$$, masses of quarks and coupling constants. How do we get the r.h.s of \eqref{eqn1} from l.h.s by applying radiative correction?

• You are misreading/conflating two formulas there, (A.2.19) with (A.2.20). The latter is every possible radiative correction to the former, your left hand side. It just preserves charge, color (0), and Lorentz invariance. Write down just one correction, say an exchange of a photon, or a gluon, to see the point. Commented Apr 17, 2021 at 14:01
• Ahh... I see what you mean, thanks a lot! Commented Apr 20, 2021 at 2:58