Quantum Field Theory and the Standard Model by Schwartz M.D. z-lib.org · Problem 5.6

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

Physics — QFT · mandelstam-variables-electron-positron

Problem

In Section 5.3 we found that the

e^+e^- \to \mu^+\mu^-

cross section had the form

\frac{d\sigma}{d\Omega} = \frac{e^4}{64\pi^2 E_{\mathrm{CM}}^2}(1+\cos^2\theta)

in the center-of-mass frame. (a) Work out the Lorentz-invariant quantities

s = (p_{e^+} + p_{e^-})^2

t = (p_{\mu^-} - p_{e^-})^2

and

u = (p_{\mu^+} - p_{e^-})^2

in terms of

E_{\mathrm{CM}}

and

\cos\theta

(still assuming

m_\mu = m_e = 0

). (b) Derive a relationship between

s

t

and

u

. (c) Rewrite

\frac{d\sigma}{d\Omega}

in terms of

s

t

and

u

. (d) Now assume

m_\mu

and

m_e

are non-zero. Derive a relationship between

s

t

and

u

and the masses.