A. Zee, Quantum Field Theory in a Nutshell · Problem IV.6.9

Field Theory Redux

Physics — QFT

Problem

Consider the theory defined in (12) with

\mu = 0

. Using the result of exercise IV.3.5 show that

V_{eff}(\phi) = \frac{1}{4}\lambda\phi^4 + \frac{1}{64\pi^2}(10\lambda^2 + 3e^4)\phi^4 \left[\log \frac{\phi^2}{M^2} - \frac{25}{6}\right] + \ldots

where

\phi^2 = \phi_1^2 + \phi_2^2

. This potential has a minimum away from

\phi = 0

and thus the gauge symmetry is spontaneously broken by quantum fluctuations. In chapter IV.3 we did not have the

e^4

term and argued that the minimum we got there was not to be trusted. But here we can balance the

\lambda\phi^4

against

e^4\phi^4 \log(\phi^2/M^2)

for

\lambda

of the same order of magnitude as

e^4

. The minimum can be trusted. Show that the spectrum of this theory consists of a massive scalar boson and a massive vector boson, with

\frac{m^2(scalar)}{m^2(vector)} = \frac{3}{2\pi} \frac{e^2}{4\pi}

. For help, see S. Coleman and E. Weinberg, Phys. Rev. D7: 1888, 1973.