Standard Model and Beyond · Problem 3.16

Gauge Theories

Physics — Particle Physics · isospin-invariant-meson-nucleon-cross-sections

Problem

The effective (non-renormalizable) interaction of a hypothetical doublet of mesons

\mathcal{A}

with the nucleons is assumed to take the isospin-invariant form

\mathcal{L}_{\mathcal{A}N} = \lambda(\bar{\psi}\vec{\tau}\psi) \cdot (\mathcal{A}^\dagger \vec{\tau} \mathcal{A}),

where

\psi = \begin{pmatrix} \psi_p \\ \psi_n \end{pmatrix}, \quad \mathcal{A} = \begin{pmatrix} \mathcal{A}^+ \\ \mathcal{A}^0 \end{pmatrix},

and

\tau^i

are the Pauli matrices. Calculate the tree-level spin-averaged CM cross sections for

\mathcal{A}^+ p \to \mathcal{A}^+ p

\mathcal{A}^0 p \to \mathcal{A}^0 p

, and

\mathcal{A}^+ n \to \mathcal{A}^0 p

. Assume that

m_p = m_n

and

m_{\mathcal{A}^+} = m_{\mathcal{A}^0}