Tutorial on translating particle physics effective Lagrangians to conventional atomic physics and quantum chemistry operators

Occasionally we have to carry out calculations with some effective Lagrangians
supplied by our particle physics friends (possibly related to new physics
beyond the standard model). For example, we could be given a Lagrangian density

\mathcal{L}^{\prime}=g~\phi\bar{\psi}\psi,


where \phi is some scalar field, \psi is the Dirac field (electrons) and g is a coupling constant. The Dirac equation that is conventionally used in atomic physics reads

 H_{D} =c\mathbf{\alpha\cdot p}+\beta m_{e}c^{2}+V^{\prime}.

(I suppress interactions of electrons with each other and with the nucleus). Given \mathcal{L}^{\prime} what is that extra operator V^{\prime} that I would have to add to my Dirac Hamiltonian? I consistently derive V^{\prime} in this tutorial (pdf).

For the impatient, the result is \begin{equation}
V^{\prime}\psi=-\gamma_{0}\left( \frac{\partial\mathcal{L}^{\prime}}
{\partial\bar{\psi}}-\partial_{\mu}\left( \frac{\partial\mathcal{L}^{\prime}
}{\partial\left( \partial_{\mu}\bar{\psi}\right) }\right) \right) .
\end{equation} Applications to axions and "Higgs portal" interactions are also covered in the tutorial (pdf).