Equations of motion of pseudo-scalar and fermion fields

Question

Consider the Lagrangian of a massive pseudo-scalar and a fermion field, defined as $$\mathcal{L}=\bar{\psi}(i\gamma^\mu\partial_\mu-m)\psi-1/2(\partial_\mu\theta)(\partial^\mu\theta)-M^2\theta^2-ig\bar{\psi}\gamma^5\psi\theta,$$

where $g$ is the interaction constant between the fermion field and the pseudo-scalar field. The question is the following: writing the equations of motion for the two fields following the Euler-Lagrange equation gives us a system of three coupled equations which should (I guess) be solved perturbatively. Does anyone know of a source where I can follow the solution of those equations, or a source with a similar problem solved?

What I am also curious about is that if we calculate the Feynman diagrams with two and four external pseudo-scalar lines provides us with an effective potential $V(\theta)$ where the effective coupling constant takes into account the interaction with the pseudo-scalar field, so the effectively the two fields decouple. Then we can solve the equations of motion easily. Is that a valid method for finding those solutions?

Answer 1

If you redefine your pseudo-scalar $\theta$ as: $$\phi = i\gamma^5\theta$$ then the Lagrangian of yours would be exactly like the common Lagrangian of Higgs field $\phi$ with the typical Yukawa coupling, if you care to add in the gauge fields involved.

Therefore, you can reuse all the formula of your text book with regard to the Higgs calculation. Just replace $\phi$ with $i\gamma^5\theta$, that is all you need.

If the pseudo-scalar $\theta$ develops a non-zero VEV $\theta = \nu$, the fermion would acquire a pseudo-scalar mass: $$ m_{ps} = i\gamma^5 g\nu $$ The fun fact is that the pseudo-scalar mass accounts for the CKM-related CP violating phases such as $e^{i\delta_{13}}$ in the electroweak sector. Please see here for more details about the pseudo-scalar mass and CKM mixing phases.