Standard Model and Beyond · Problem 4.11

The Strong Interactions and QCD

Physics — Particle Physics · su2-model-fermion-doublet

Problem

Extend the

SU(2)

model on pages 149--151 (with

\mu^2 < 0

) by the addition of a non-chiral fermion doublet

\psi = \begin{pmatrix} \psi^+ \\ \psi^0 \end{pmatrix}

with mass

m_\psi

. (a) Find the interaction vertices for

\psi^+ \to \psi^0 A^+

\psi^+ \to \psi^+ A^0

, and

A^0 \to A^+ A^-

. (b) Write the tree-level amplitude for

\psi^+(p_1)\psi^-(p_2) \to A^+(p_3)A^-(p_4)

in the

R_\xi

gauge. Show that it is independent of

\xi

. (c) Show that the individual diagrams (for the longitudinal polarizations) grow

\propto s

for

s \gg M_A^2, m_\psi^2

, but that the leading term cancels in the full amplitude. (The implications for unitarity and renormalizability will be discussed in Chapter 7.1.)