Chapter 33 integral eq
\[ {\cal M}_{3s}(E)=\frac{\sum_i{\cal L(\vec p_i)} {\cal R(\vec p_i)}}{1/K_{df} + F^\infty} \] where the sum over the momenta of the three particles \(\vec p_i=\). The momenta \(\vec p_i\) have to satisfy the on-shell condition \[ E=\sqrt{m+\vec p_1^2}+\sqrt{m+\vec p_2^2}+\sqrt{m+\vec p_3^2} \] thus a valid choice is \[ \vec p_1=0,\,\, \vec p_2=-\vec p_3=\sqrt{(E-m)^2-4}\hat e_x \]
\[ F^\infty=\int_{\vec k} \tilde\rho(k){\cal L}(k) \] \[ \tilde\rho(k)=\frac{H(k)\rho(k)}{2\omega_k} \] \[ {\cal L}={\cal R}=\frac{1}{3}-2\omega_k{\cal M}_2^s(\vec k)\tilde\rho(\vec k)-\int_{\vec s} D^{(u,u)}(\vec k,\vec s)\tilde\rho(\vec s) \] \[ D^{(u,u)}(\vec k,\vec p)=-{\cal M}_2^s(\vec k)G^{\infty}(\vec k, \vec p){\cal M}_2^s(\vec p) -\int_{\vec s}\frac{1}{2\omega_s} {\cal M}_2^s(\vec k)G^{\infty}(\vec k, \vec s)D^{(u,u)}(\vec s,\vec p) \]
