Quantum Field Theory and the Standard Model by Schwartz M.D. z-lib.org · Problem 9.1

Scalar quantum electrodynamics — Schwartz M.D. z-lib.org

Physics — QFT · compton-scattering-scalar-qed

Problem

Compton scattering in scalar QED. ((a)) Calculate the tree-level matrix elements for

(\gamma\phi \to \gamma\phi)

. Show that the Ward identity is satisfied. ((b)) Calculate the cross section

\frac{d\sigma}{d\cos\theta}

for this process as a function of the incoming and outgoing polarizations,

\epsilon^{\rm in}_\mu

and

\epsilon^{\rm out}_\mu

, in the center-of-mass frame. ((c)) Evaluate

\frac{d\sigma}{d\cos\theta}

for

\epsilon^{\rm in}_\mu

polarized in the plane of the scattering, for each

\epsilon^{\rm out}_\mu

. ((d)) Evaluate

\frac{d\sigma}{d\cos\theta}

for

\epsilon^{\rm in}_\mu

polarized transverse to the plane of the scattering, for each

\epsilon^{\rm out}_\mu

. ((e)) Show that when you sum (c) and (d) you get the same thing as having replaced

(\epsilon^{\rm in}_\mu)^* \epsilon^{\rm in}_\nu

with

-g_{\mu\nu}

and

(\epsilon^{\rm out}_\mu)^* \epsilon^{\rm out}_\nu

with

-g_{\mu\nu}

. ((f)) Should this replacement work for any scattering calculation?