An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 10.4

Systematics of Renormalization — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · phi4-asymptotic-behavior

Problem

**10.4 Asymptotic behavior of diagrams in

\phi^4

theory.** Compute the leading terms in the

S

-matrix element for boson-boson scattering in

\phi^4

theory in the limit

s \to \infty

t

fixed. Ignore all masses on internal lines, and keep external masses nonzero only as infrared regulators where these are needed. Show that

i\mathcal{M}(s,t) \sim -i\lambda - i\frac{\lambda^2}{(4\pi)^2}\log s - i\frac{5\lambda^3}{2(4\pi)^4}\log^2 s + \cdots.

Notice that ignoring the internal masses allows some pleasing simplifications of the Feynman parameter integrals.