Quantum Field Theory and the Standard Model by Schwartz M.D. z-lib.org · Problem 5.1

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

Physics — QFT · differential-cross-section-rest-frame

Problem

Show that the differential cross section for

2 \to 2

scattering with

p_i^\mu + p_A^\mu \to p_f^\mu + p_B^\mu

in the rest frame of particle

A

can be written as

\frac{d\sigma}{d\Omega} = \frac{1}{64\pi^2 m_A} \left[ E_B + E_f \left(1 - \frac{|\vec{p}_i|}{|\vec{p}_f|}\cos\theta\right) \right]^{-1} \frac{|\vec{p}_f|}{|\vec{p}_i|} |\mathcal{M}|^2, \tag{5.54}

where

\theta

is the angle between

\vec{p}_i

and

\vec{p}_f

E_B = \sqrt{(\vec{p}_f - \vec{p}_i)^2 + m_B^2}

and

E_f = \sqrt{\vec{p}_f^2 + m_f^2}