An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 12.1

The Renormalization Group — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · renormalization-group-beta-functions

Problem

Beta functions in Yukawa theory. In the pseudoscalar Yukawa theory studied in Problem 10.2, with masses set to zero,

\mathcal{L} = \frac{1}{2}(\partial_\mu \phi)^2 - \frac{\lambda}{4!}\phi^4 + \bar{\psi}(i\partial\!\!\!/)\psi - ig\bar{\psi}\gamma^5\psi\phi,

compute the Callan-Symanzik

\beta

functions for

\lambda

and

g

\beta_\lambda(\lambda, g), \qquad \beta_g(\lambda, g),

to leading order in coupling constants, assuming that

\lambda

and

g^2

are of the same order. Sketch the coupling constant flows in the

\lambda

g

plane.