I am uncomfy with the calculation of the neutral pion’s decay rate via the triangle anomaly diagram, which gets touted as evidence of three colors. The calculation invokes PCAC in the guise of the occult Goldberger-Treiman relation, which says that ${{f}_{\pi }}$ (the charged pion’s proportionality to the divergence of the axial current) is inversely related to ${{g}_{\pi NN}}$ (the pion’s effective coupling to a nucleon line). The charged pion’s decay amplitude is directly proportional to ${{f}_{\pi }}$by definition, but the neutral pion’s is found to be inversely proportional to it.

Since I think of pions as $Q\bar{Q}$ bound states that could in principle be described by a wave function, I would intuitively have expected both decay amplitudes to be directly proportional to aspects of the elusive wave function. The axial current operator is strictly local, so the charged pion’s decay amplitude should depend on the value of the wave function at zero separation. But what about the triangle diagram? One would have to project the wave function onto something, but what? The quark propagator between the two electromagnetic vertices suggests something extended, on the order of the Compton wavelength of a quark, whatever that might mean for confined quarks. (I’ve never seen a colored quark, and I never hope to see one, but I can tell you anyhow, I’d rather see than free one.) What is the numerical success of the G.-T. relation telling us about the pion’s (and maybe the nucleon’s) wave function?

  • $\begingroup$ Related. The accepted answer is more "modern", in that it reads off the chiral anomaly from the WZW term containing it. But you wanted something more "direct"... $\endgroup$ – Cosmas Zachos Feb 17 '18 at 22:04

OK, sir, here is a minimal inelegant answer, not matching the elegance of the question. But, as MGM reportedly told BR in the GOR paper, "Let us leave it like this, Bruno, and any well-meaning, intelligent reader will know exactly what we mean!" Was he right? But, given the minimal facts, I will desist from any and all interpretations and pion wavefunctions... It seems to me you might be seeking to compare tapples and tangelos.

Anyway, I will follow M Schwartz's modern text § 30.1.2 & 30.1.1, which, perversely, pays homage to the old fashioned Steinberger calculation, but uses effective Lagrangians--the modern angle! And I'll be cavalier... won't even bother with how the π gets a mass via the explicit χSBreaking σ term, $f_\pi m_\pi ^2 \sigma$ , which shifts the σ-model vacuum, covered best, I feel, in Itzykson & Zuber § 11-4. So I'll just keep the asymmetric π mass term out of sight.

OK, now, the effective Lagrangian that has the full SU(2)×SU(2) symmetry, with the axial part realized nonlinearly, is something like the following, where I'm cavalier with conventions, normalizations, etc... $$ {\cal L}= \frac{1}{2} \partial \vec{\pi}\cdot \partial \vec{\pi} + g f_{\pi} \partial_\mu \vec{\pi} \cdot \vec{W}^\mu +...+\bar{\psi}(i \displaystyle{\not} \partial -m)\psi + 2i \frac{m}{f_\pi} \vec{\pi} \cdot \bar{\psi}\gamma^5 \vec{\tau}\psi +.... $$

m is the induced mass of the fermion (nucleon, massive constituent quark, it doesn't matter, except for factors of 3 and such...) It incorporates Goldberger-Treiman, which the moderns consider unfortunately complicated and indirect. The point is this σ-model lagrangian is chirally invariant, with the axials realized just right, and, crucially, you actually need these fermions to compute the triangle diagram.

I have skipped all implied expressions involving the σ, but, of course, its v.e.v. is $f_\pi$, the order parameter; so that the SSBroken axial currents $f_\pi \partial_\mu \vec{\pi}+... $ shift the $\vec{\pi}$s by $f_\pi$,
$$\delta_\vec{\theta_A} \vec{\pi}=f_\pi \vec{\theta_A}+\sigma' \vec{\theta_A},\\ \delta_\vec{\theta_A}\sigma= - \vec{\theta_A}\cdot \vec{\pi},\\ \delta_\vec{\theta_A}\psi=i\vec{\theta_A} \cdot \vec{\tau} \gamma^5\psi~ ;$$ so the Yukawa term variation cancels the fermion mass variation, etc. It's all been pre-packaged in with a minimum of intermediate parameters!

The order parameter appears in the denominator of the Yukawa to balance off the shift of the pion in this procedure: large χSB needs weaker Yukawa coupling to yield the same fermion mass! (Subjective subtext: the π owes nothing to the fermions--they owe it!).

The point, however, is that, behold! The coupling to the charged W is visibly $f_\pi$, coming from the covariant derivatives of the nonlinear Gürsey chiral lagrangian, ("half the axial currents") and cares not about any fermions--they might as well be absent, as far as it is concerned; supplants GOR; the left piece of its axial current is simply gauged. Leading, ultimately, to the charged pion rate $\Gamma \propto G_F^2 f_\pi^2 m_\pi$...,

Whereas one need do the Steinberger triangle fermion loop to get the neutral pion decay, which Schwartz covers, yielding the dreaded $\Gamma\propto \alpha^2 m_\pi^3/f_\pi^2 $.

At a minimum, one might inspect this effective action, then, to draw deep conclusions on the meaning of $f_\pi$ and pion wavefunctions, but the brute chiral realization of the symmetry just dictates its emergence like so.

Charged pion to W couples with $f_\pi$, but neutral pion has to go through heavier fermion coupling, which goes like the inverse of that. In my mind, this sounds like a cautionary tale to not think of pions as bound states $Q\bar Q$, but as just the agent of the fermions (nucleons, massive constituent quarks, etc...) getting their mass. A theory without fermions here would be a fine chirally symmetric theory--even though the neutral pion would then decay via a Z. But this is all mental picturing...

As $f_\pi$ increases and pions pop in and out of the seized vacuum more copiously, the charged pion is more likely to disappear in favor of a virtual W; but needs couple to fermions more weakly to give them a fixed large mass... it's easier to achieve. And then the intermediating fermion triangle loop will be less effective...

  • A teaching moment aside. Today's students might be rolling their eyes at all this, as their teachers most probably drag them immediately to the effective action fragment $\frac{e^2 N_c}{48 \pi^2 f_\pi}\pi^0 F\cdot \tilde{F}$, eqn (21), hiding in the celebrated WZW chiral anomaly containing term, where no quarks appear, but $N_c$ is still (rather breathtakingly!) present for consistency. It's gotten by consistency, not modeling, and $f_\pi$ is there for dimensional reasons--there is no other way. The effective Lagrangian maven would be uncomfortable with any other arrangement, beyond this one, instead! But, "If they can get you asking the wrong questions, they don't have to worry about answers." (Thomas Pynchon, "Gravity's Rainbow").
  • $\begingroup$ A whole nother language. It's Hellenistic Greek to an Egyptian from the age of pyramids like myself. Now where is that Rosetta stone? I'm only 2000 years behind the times. $\endgroup$ – Bert Barrois Feb 19 '18 at 18:53
  • $\begingroup$ Ach, ach, Rip Van, maybe a perusal of M Schwartz's text cited... Surely it couldn't hurt... especially read under a pyramid.... $\endgroup$ – Cosmas Zachos Feb 19 '18 at 19:35

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