For a renormalized $n$-point one-particle-irreducible function, the <Callan-Symanzik equation> is
$$
\left(
M\frac{\partial}{\partial M}
+\beta(\lambda)\frac{\partial}{\partial\lambda}
+n\gamma_\phi(\lambda)
\right)\Gamma_R^{(n)}=0,
$$
with a convention-dependent sign on $\gamma_\phi$. At this order $\gamma_\phi=0$.

At a general Euclidean momentum scale $Q$, the one-loop four-point function contains
$$
\lambda_{\mathrm{eff}}(Q)
=\lambda(M)+\frac{3\lambda(M)^2}{32\pi^2}
\log\frac{Q^2}{M^2}+O(\lambda^3).
$$
Requiring a physical amplitude to be independent of the arbitrary subtraction scale gives
$$
0=M\frac d{dM}\lambda_{\mathrm{eff}}(Q)
=\beta(\lambda)-\frac{3\lambda^2}{16\pi^2}+O(\lambda^3),
$$
and therefore
$$
\beta(\lambda)=\frac{3\lambda^2}{16\pi^2}+O(\lambda^3).
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 304 / 2 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-304/2/c/solution