Srednicki — Quantum Field Theory · Problem 27.1

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · renormalization-group-mass-running

Problem

Suppose that we have a theory with

\begin{aligned} \beta(\alpha) &= b_1\alpha^2 + O(\alpha^3) \,, \tag{27.27} \\ \gamma_m(\alpha) &= c_1\alpha + O(\alpha^2) \,. \end{aligned}

Neglecting the higher-order terms, show that

m(\mu_2) = \left[\frac{\alpha(\mu_2)}{\alpha(\mu_1)}\right]^{c_1/b_1} m(\mu_1) \,. \tag{27.29}