[QUE/QFT-04012] QFT-PROBLEM

$\newcommand{\Lsc}{\mathscr L}$

Consider electron proton scattering to be computed in the second quantized Schrodinger theory. The electrostatic interaction can be modeled as \[ V(\mathbf x_1-\mathbf x_2) = -(Ze^2)\int d\mathbf x_1\,d\mathbf x_2 \frac{\rho_e(\mathbf x_1) \rho_p(\mathbf x_2)}{|\mathbf x_1- \mathbf x_2|}\] This interaction suggests the following interaction term in the second quantized Schrodinger theory. \[ \Lsc_\text{int}=\int d\mathbf x_1d\mathbf x_2 \psi^*(\mathbf x_1)\psi(\mathbf x_1) V(\mathbf x_1-\mathbf x_2)\phi^*(\mathbf x_2)\phi(\mathbf x_2) \] where \(\psi ,\phi\) denote the electron and proton fields respectively, and \[V(\mathbf x_1-\mathbf x_2)=-\frac{(Ze^2)}{|\mathbf x_1- \mathbf x_2|}.\] Compute the electron proton scattering cross section in the center of mass frame. Use Yukawa potential \(V(\mathbf x_1-\mathbf x_2)=-V_0\dfrac{ e^{-\mu|\mathbf x_1-\mathbf x_2|}}{|\mathbf x_1- \mathbf x_2|}\) in \(\Lsc_\text{int}\). Show that in the limit \(\mu\to0\), the differential cross section coincides with the Rutherford scattering cross section.