Srednicki — Quantum Field Theory · Problem 88.5

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · standard-model-lepton-sector

Problem

a) Write down the term in

\mathcal{L}_{\rm eff}

that is relevant for

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

. Express your answer in the form

\mathcal{L}_{\rm eff} = \tfrac{1}{\sqrt{2}}G_{\rm F}\overline{N}\gamma^\mu(1-\gamma_5)N\,\overline{\mathcal{E}}\gamma_\mu(C_{\rm V}-C_{\rm A}\gamma_5)\mathcal{E} \,, \tag{88.43}

where

N

is the muon neutrino field, and determine the values of

C_{\rm V}

and

C_{\rm A}

. b) Repeat part (a) for

\nu_e e^- \to \nu_e e^-

. c) Compute

\langle|\mathcal{T}|^2\rangle

as a function of the Mandelstam variables, the electron mass, and

C_{\rm V}

and

C_{\rm A}