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 $$\vec p_1=0,\,\, \vec p_2=-\vec p_3=\sqrt{(E-m)^2-4}\hat e_x$$

$$F^\infty=\int_{\vec k} \tilde\rho(k){\cal L}(k)$$ $$\tilde\rho(k)=\frac{H(k)\rho(k)}{2\omega_k}$$ $${\cal L}={\cal R}=\frac{1}{3}-2\omega_k{\cal M}_2^s(\vec k)\tilde\rho(\vec k)-\int_{\vec s} D^{(u,u)}(\vec k,\vec s)\tilde\rho(\vec s)$$ $$D^{(u,u)}(\vec k,\vec p)=-{\cal M}_2^s(\vec k)G^{\infty}(\vec k, \vec p){\cal M}_2^s(\vec p) -\int_{\vec s}\frac{1}{2\omega_s} {\cal M}_2^s(\vec k)G^{\infty}(\vec k, \vec s)D^{(u,u)}(\vec s,\vec p)$$

33.0.1 Parametrization of the resonance

from weinber we read