A. Zee, Quantum Field Theory in a Nutshell · Problem IV.3.5

Field Theory Redux

Physics — QFT

Problem

Consider the electrodynamics of a complex scalar field

\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + (\partial_\mu + ieA_\mu)\phi^\dagger (\partial^\mu - ieA^\mu)\phi + \mu^2\phi^\dagger\phi - \lambda(\phi^\dagger\phi)^2

In a universe suffused with the scalar field

\phi(x)

taking on the value

\phi

independent of

x

as in the text, the Lagrangian will contain a term

(e^2\phi^\dagger\phi)A_\mu A^\mu

so that the effective mass squared of the photon field becomes

M(\phi)^2 \equiv e^2\phi^\dagger\phi

. Show that its contribution to

V_{\text{eff}}(\phi)

has the form

\int \frac{d^4k}{(2\pi)^4} \log \frac{k^2 - M(\phi)^2}{k^2}

Compare with (14) and (26). [Hint: Use the Landau gauge to simplify the calculation.] If you need help, I strongly urge you to read S. Coleman and E. Weinberg, Phys. Rev. D7: 1883, 1973, a paragon in exposition.

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