Standard Model and Beyond · Problem 2.4

Lie Groups, Lie Algebras, and Symmetries

Physics — Particle Physics · pi-plus-pi-minus-scattering-amplitude

Problem

Consider the process

\pi^+(\vec{p}_1)\pi^-(\vec{p}_2) \to \pi^+(\vec{p}_3)\pi^-(\vec{p}_4)

, with

p_1 \neq p_3

and

p_2 \neq p_4

, in the theory of a complex scalar field with Lagrangian density given in (2.88) with

V_I = \frac{\lambda}{4}(\phi^\dagger\phi)^2.

As shown in Appendix B, the tree-level amplitude

M_{fi}

is given by

(2\pi)^4\delta^4(p_3+p_4-p_1-p_2)\,M_{fi} = \int d^4x\,\langle\vec{p}_3\vec{p}_4|{-iV_I(\phi_0(x),\phi_0^\dagger(x))}|\vec{p}_1\vec{p}_2\rangle.

Calculate this explicitly using the free-field expression for

\phi_0

, and show that

M_{fi} = -i\lambda