Quantum Field Theory and the Standard Model by Schwartz M.D. z-lib.org · Problem 5.3

Cross sections and decay rates — Schwartz M.D. z-lib.org

Physics — QFT · muon-decay-rate

Problem

A muon decays to an electron, an electron neutrino and a muon neutrino,

\mu^- \to e^- \nu_\mu \nu_e

. The matrix element for this process, ignoring the electron and neutrino masses, is given by

|\mathcal{M}|^2 = 32 G_F^2 (m^2 - 2mE)mE

, where

m

is the mass of the muon and

E

is the energy of the outgoing

\nu_e

G_F = 1.166 \times 10^{-5}\,\mathrm{GeV}^{-2}

is the Fermi constant. (a) Perform the integral over

d\Pi_{\mathrm{LIPS}}

to show that the decay rate is

\Gamma = \frac{G_F^2 m^5}{192\pi^3}. \tag{5.55}

(b) Compare your result to the observed values

m = 106\,\mathrm{MeV}

and

\tau = \Gamma^{-1} = 2.20\,\mu\mathrm{s}

. How big is the discrepancy as a percentage? What might account for the discrepancy?