# Parity conserving QFT

I was reading Peskin's QFT text and in Chapter 6 (Radiative Corrections), there is a line in section 6.2, page 185 which I am quoting below. The passage is about the $$\Gamma^{\mu}(p^{'},p)$$ and its dependence on possible objects that appear in the Feynman rules. The dependence should be on $$p^{'},p,\gamma^{\mu}$$ and constants like $$m, e, etc$$ according to the text. Additionally, the book reads:

"The only other object that could appear in any theory is $$\epsilon^{\mu\nu\rho\sigma}$$ (or equivalently $$\gamma^{5}$$); but this is forbidden in any parity-conserving theory."

My question is two-fold:

1. Why is the appearance of $$\epsilon^{\mu\nu\rho\sigma}$$ forbidden for parity to be conserved? Is it because the antisymmetricity of $$\epsilon$$ causes a change in sign under parity transformation?

2. How is Peskin relating $$\epsilon$$ to $$\gamma^{5}$$?

Starting from Adam and Eve, the space reflection operator (see for example Bjorken & Drell, Relativistic Quantum Field, Vol. I, p. 24) is $$P = \gamma^0$$ except for an irrilevant phase; this means that while the bilinear $$j^{\mu} = \bar\psi\gamma^{\mu}\psi$$ is invariant under parity transformation, the bilinear $$j^{\mu}_A = \bar\psi\gamma^{\mu}\gamma^5\psi$$ is certainly not. Since the phenomenological request of $$P$$-symmetry for the EM current implies the total absence of $$\gamma^5$$ in the QED interaction term $$L_{int} = -e\bar\psi\gamma^{\mu}A_{\mu}\psi,$$ where $$A_{\mu}$$ is the EM four-potential, Gell-Mann & Low's formula (ibidem, Vol. II, chap. 17) then guarantees us the absence of $$\gamma^5$$ also from the form factor $$\Gamma^{\mu}$$. It is important to notice that this last object can't depend "directly" from a four-index tensor such as $$\epsilon$$, as Peskin seems to state; the only information it can carry around about parity violation is contained in $$\gamma^5$$.
The completely antisymmetric tensor $$\epsilon^{\mu\nu\alpha\beta}$$ appears only in the S-matrix squared element (properly contracted in all is indexes), and emerges related to $$\gamma^5$$ when you take the trace; as a matter of fact, up to a normalization, $$Tr(\gamma^{\mu}\gamma^{\nu}\gamma^{\alpha}\gamma^{\beta}\gamma^{5}) = 4i\epsilon^{\mu\nu\alpha\beta}.$$ This accounts for Peskin's supposed equivalence between the presence of $$\gamma^5$$ in $$\Gamma^{\mu}$$, which implies $$P$$-symmetry violation, and the presence of $$\epsilon^{\mu\nu\alpha\beta}$$ in your final result, the S-matrix squared element.