An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 9.1

Functional Methods — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · functional-methods-scalar-qed

Problem

Scalar QED. This problem concerns the theory of a complex scalar field

\phi

interacting with the electromagnetic field

A^\mu

. The Lagrangian is

\mathcal{L} = -\frac{1}{4}F_{\mu\nu}^2 + (D_\mu \phi)^*(D^\mu \phi) - m^2 \phi^* \phi,

where

D_\mu = \partial_\mu + ieA_\mu

is the usual gauge-covariant derivative. (a) Use the functional method of Section 9.2 to show that the propagator of the complex scalar field is the same as that of a real field:

\frac{------}{p} \quad = \quad \frac{i}{p^2 - m^2 + i\epsilon}.

Also derive the Feynman rules for the interactions between photons and scalar particles; you should find

\begin{array}{cc} p' \!\searrow & \\ & \!\!\sim\!\!\sim\!\! \mu = -ie(p+p')^\mu; \qquad \mu \quad \nu \\ p \!\nearrow & \qquad\qquad\qquad\qquad\quad = 2ie^2 g^{\mu\nu}. \end{array}

(b) Compute, to lowest order, the differential cross section for

e^+e^- \to \phi\phi^*

. Ignore the electron mass (but not the scalar particle's mass), and average over the electron and positron polarizations. Find the asymptotic angular dependence and total cross section. Compare your results to the corresponding formulae for

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

. (c) Compute the contribution of the charged scalar to the photon vacuum polarization, using dimensional regularization. Note that there are two diagrams. To put the answer into the expected form,

\Pi^{\mu\nu}(q^2) = (g^{\mu\nu}q^2 - q^\mu q^\nu)\Pi(q^2),

it is useful to add the two diagrams at the beginning, putting both terms over a common denominator before introducing a Feynman parameter. Show that, for

-q^2 \gg m^2

, the charged boson contribution to

\Pi(q^2)

is exactly

1/4

that of a virtual electron-positron pair.

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