Applying four functional derivatives to the functional equation gives the homogeneous equation for the renormalized <four-point one-particle-irreducible correlation function>:
$$
\boxed{\left[
\Lambda\partial_\Lambda+\beta_{m^2}\partial_{m^2}
+\sum_i\beta_i\partial_{g_i}-4\gamma_\phi
\right]
\Gamma_\Lambda^{(4)}(x_1,x_2,x_3,x_4;m^2,g_i)=0.}
$$
If the opposite convention for $\gamma_\phi$ is used, the last sign changes. The factor four counts the four external renormalized fields.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 304 / 2 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-304/2/c/solution