Srednicki — Quantum Field Theory · Problem 11.4

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · three-scalar-field-scattering-amplitudes

Problem

Consider a theory of three real scalar fields (

A

B

, and

C

) with

\begin{aligned} \mathcal{L} &= -\tfrac{1}{2}\partial^\mu A\partial_\mu A - \tfrac{1}{2}m_A^2 A^2 \\ &\quad - \tfrac{1}{2}\partial^\mu B\partial_\mu B - \tfrac{1}{2}m_B^2 B^2 \\ &\quad - \tfrac{1}{2}\partial^\mu C\partial_\mu C - \tfrac{1}{2}m_C^2 C^2 \\ &\quad + gABC . \tag{11.57} \end{aligned}

Write down the tree-level scattering amplitude (given by the sum of the contributing tree diagrams) for each of the following processes:

\begin{aligned} AA &\to AA , \\ AA &\to AB , \\ AA &\to BB , \\ AA &\to BC , \\ AB &\to AB , \\ AB &\to AC . \tag{11.58} \end{aligned}

Your answers should take the form

\mathcal{T} = g^2\left[\frac{c_s}{m_s^2 - s} + \frac{c_t}{m_t^2 - t} + \frac{c_u}{m_u^2 - u}\right], \tag{11.59}

where, in each case, each

c_i

is a positive integer, and each

m_i^2

m_A^2

m_B^2

m_C^2

. Hint:

\mathcal{T}

may be zero for some processes.