# Peskin & Schroeder Introduction to Quantum Field Theory, two-loop renormalization example

In Peskin & Schroeder's An Introduction to Quantum Field Theory, section 10.5, page 338, the book gives a two-loop renormalization example (in scalar $$\phi^4$$ theory).

Before we start the two-loops, let's us recall the renormalization condition in (10.19) For the second renormalization condition, my understanding for two-loop case is $$\text{One-loops + two-loops + counterterms}=-i\lambda.$$

However, the book in two-loop examples only considers the two loop case. The relevant two-loop feynman diagrams are given in (10.51) The value of last diagram in (10.51) is just a constant, which we can freely adjust to absorb any divergent terms that are independent of the external momenta.

1. Here, does the momentum independent divergence including double poles divergence? i.e. $$(\frac{1}{\epsilon})^2$$.

2. However, on page 339, the book refers which is the result from one-loop renormalization. In this case, the $$\delta_\lambda$$ is already specified. And we cannot freely adjust it to absorb momentum independent divergence. This is contradict with the sentence below (10.51).

So I am wondering what's wrong with my logic?

2. The 1-loop counterterms are of order $${\cal O}(\lambda^2)$$ while the 2-loop counterterms are of order $${\cal O}(\lambda^3)$$ in the perturbative formal power series for the coupling constant $$\lambda$$. [The order$$^1$$ is determined by the number of $$\times$$-vertices in the Feynman diagrams that a counterterm is designed to (partially) cancel.] The 1-loop and 2-loop counterterms are hence independent terms.
$$^1$$The counting is more complicated if the Feynman diagram contains $$\otimes$$-vertices. Since the counterterm $$\delta_{\lambda}$$ is a power series expansion in $$\lambda$$ (the lowest power is quadratic), the corresponding $$\otimes$$-vertex doesn't contain a definite power of $$\lambda$$.
• Thank you, that’s make sense! Could you please elaborate more on how we obtain this $\mathcal{O}(\lambda^2)$ and $\mathcal{O}(\lambda^3)$ counterterms? And they are independent? Mar 10 at 13:07