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A few years ago, I studied Fujikawa's method of deriving the chiral anomaly (section 13.2.1 in Nakahara's book), i.e. the equation \begin{equation}\tag{1} \partial_\mu J^\mu=\frac{1}{16\pi^2}\operatorname{tr}\epsilon^{ijkl}F_{ij}F_{kl}. \end{equation} I recently started to wonder about its consequences for physics and I found some lecture notes online (which do not seem to be available any more) that stated that \begin{equation}\tag{2} \langle\pi^0|\gamma\gamma\rangle\propto\langle\Omega|\partial_\mu J^\mu|\gamma\gamma\rangle, \end{equation} meaining that the decay rate for $\pi^0\to\gamma\gamma$ is related to the anomalous divergence. Since I was studying mathematics, I learned the necessary mathematical tools to understand $(1)$ (e.g. Riemannian manifolds, Clifford modules, Dirac operators, Berezin integrals), but I have no background in QFT (apart from a QM lecture) and hence I have no understanding of $(2)$. I looked into some standard QFT books that discuss the chiral anomaly, but I only found derivations of $(1)$, not $(2)$.

That being said, the literature on QFT is very vast and I probably just looked in the wrong places. I hope that someone can either provide a list of key-words and references to get a quick understanding of $(2)$, or even explain $(2)$ right away, ideally using mathematical language. To give you an idea of what I am looking for, here are two questions that I have:

  • Is it correct that $\langle\pi^0|\gamma\gamma\rangle$ is called a transition amplitude and if yes, what is the mathematical definition?
  • The bracket notation suggests that $\langle\Omega|\partial_\mu J^\mu|\gamma\gamma\rangle$ is an inner product. If yes, what is the Hilbert space and what are the definitions of the two states and the operator $\partial_\mu J^\mu$?
Qmechanic
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Filippo
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1 Answers1

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I am not sure which parts of your QFT books you have left unread...

The QFT machinery serves to compute amplitudes of particle transitions, like decays, in this case $\pi^0\to \gamma \gamma$. If you appreciate the quark model, then, you appreciate the fundamental relation connecting quarks (in J) and hadrons, in this case Goldstone pions, $$ \langle \Omega| J^{a ~\mu} (x)|\pi^b(p)\rangle =-ip^\mu f_\pi \delta^{ab} e^{-ip\cdot x}, \tag {19.88} $$ where I have tagged the formula to the standard textbook on An introduction to Quantum Field Theory by M Peskin & D Schroeder, and the pion decay constant $f_\pi\sim 130$MeV summarizes the hadronization of the u and d quarks in the anomalous current J in (1). Taking the divergence of this, you get $$ \langle \Omega| \partial_\mu J^{a ~\mu} |\pi^0(p)\rangle \propto m^2_{\pi^0} f_\pi \leadsto \\ \langle \Omega| \partial_\mu J^{a ~\mu} |\pi^0\rangle \langle \pi^0|\gamma \gamma\rangle \implies (2) , $$
whence (19.113,118,119) in the same text, but you may use other books, like the one by M Schwartz, which even hide/place your (1) inside the gauged terms of the effective Wess-Zumino term summarizing flavor chiral anomalies, much more elegantly. (30.13,14)

  • In any case, the transition amp of a decay process like the one you are contemplating is a Dirac bra-ket, a complex number amounting to a dot product, as you guessed. The Hilbert space is that of observable particles, photons and pions, but, as per the fundamental trick/cheat of particle physics, also quarks!

That is to say (a very long story indeed!) the axial current J is a two-headed monster, obeying similar relations, the very heart of the hadroniztion magic: $$ J^\mu \sim \bar u \gamma^\mu \gamma^5 u - \bar d \gamma^\mu \gamma^5 d, $$ $$ J^\mu \sim f_\pi \partial^\mu \pi^0 + ... $$ The ellipsis denotes higher order terms in pseudoscalar pseudogoldstons, depending on the parameterizations in the effective Lagrangian of low energy QCD. There are dozens of questions on the subject on this site.

Cosmas Zachos
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