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 $\gamma^{5}$?

Thanks in advance


1 Answer 1


Peskin's statement is a little confusing, so let me turn your questions around.
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.


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