Standard Model and Beyond · Problem 2.7

Lie Groups, Lie Algebras, and Symmetries

Physics — Particle Physics · massive-photon-pion-qed

Problem

Consider charged pion QED with a massive photon, i.e., add the term

\frac{1}{2}M_A^2 A_\mu A^\mu

to the Lagrangian density in (2.133) (with

V_I = 0

). Assume

M_A > 2m

. (The photon mass term is not gauge invariant, but the model still makes sense at tree level.) (a) Calculate the decay rate for

\gamma \to \pi^+\pi^-

in the photon rest frame at tree level for an unpolarized massive photon. (b) Calculate the

\pi^+

angular distribution

d\Gamma/d\cos\theta

for a polarized photon, where

\theta

is the angle between the photon polarization direction in the rest frame and the

\pi^+

direction. (c) Show that one recovers the result in (a) when

d\Gamma/d\cos\theta

is integrated over

\cos\theta