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I want to derive the basic equations for the rate of electron capture, taking into account the relativistic kinematics explicitly

$$p^{+}+e^{-}\rightarrow n^{0}+\nu_{e}$$

The idea is to treat electronic wave function like a particle-in-the-box, rather than use full blown atomic orbitals. But I want to do the full calculation over the kinematics, and compute the matrix elements, form factors, etc.

I am assuming one can derive this to using something like the differential cross section for 2+2 body scattering

$$1+2\rightarrow 3+4$$

$$d\sigma_{i,f}\sim\left(\dfrac{1}{64\pi^{2}s}\dfrac{|\vec{p}_{f}|}{|\vec{p}_{i}|}\big\vert\mathcal{M_{i,f}}\big\vert^{2}d\Omega\right)$$

(in the C.M. frame, $\mathbf{p_{e}}=-\mathbf{p_{p}}$)

and the rate is obtained by the integrating the differential cross section, to get something like

$$\Gamma=\big\Vert\Psi(0)\big\Vert\mathbf{v}\sigma$$

where $\sigma$ is the unpolarized cross section, and where $$\big\Vert\psi_{e}(0)\big\Vert$$ is the electronic density at the origin (the nuclear charge)

The matrix elements $M_{i,f}$ can be computed from the Lagrangian for the Weak interaction.

I am trying to get to an expression for the rate of capture which is something like

$$d\lambda_{ep}=\left(\dfrac{1}{2\pi}\right)^{2}\dfrac{\sum_{fi}\big\vert\mathcal{M}_{fi}\big\vert^{2}\big\Vert\psi_{e}(\mathbf{x})\big\Vert_{\mathbf{x}=0}}{16E_{p}E_{e}\;\big\vert\mathbf{k}\cdot(E{_n}\mathbf{k}-k^{0}\mathbf{p}_{n})\big\vert}k^{3}d\Omega_{k}$$

where $\mathbf{k}$ is the momentum 3-vector for the outbound neutrino $\nu_{e}$

Note that the rate calculations integrate over the solid angle of the neutrino

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