Standard Model and Beyond · Problem 7.2

The Standard Electroweak Theory

Physics — Particle Physics · muon-decay-michel-parameter

Problem

(a) Suppose that the muon weak current were

V + A

rather than

V - A

, so that the interaction for

\mu

decay becomes

\mathcal{H} = \frac{G_F}{\sqrt{2}} \bar{\nu}_\mu \gamma_\mu \left(1 + \gamma^5\right) \mu \; \bar{e}\gamma^\mu \left(1 - \gamma^5\right) \nu_e

rather than (7.32). Calculate the electron energy distribution, and show that it corresponds to

\rho = 0

, where

\rho

is the Michel parameter defined in (7.55). Hint: the results for each part can be written down without additional trace calculations. (b) Calculate

\rho

for the (parity-conserving) vector interaction

\mathcal{H} = \sqrt{2}G_F \bar{\nu}_\mu \gamma_\mu \mu \; \bar{e}\gamma^\mu \nu_e,

where the

\sqrt{2}

is inserted to obtain the same overall rate. (c) Return to the Fermi

V - A

theory for unpolarized

\mu^- \to e^- \nu_\mu \bar{\nu}_e

. Give the energy distributions for the

\nu_\mu

and for the

\bar{\nu}_e

. In each case the momenta of the other two particles are integrated over. Give a simple helicity argument why one of them vanishes at the endpoint,

\epsilon = 1

, but not the other.