# Interaction lagrangian neutral vector boson $Z_µ$ of mass $M_Z$ with Dirac fermion $ψ$

Let’s consider the following lagrangian density describing the interaction between a neutral vector boson $$Z_µ$$ of mass $$M_Z$$ with Dirac fermions $$ψ$$ $$\mathcal{L}=\mathcal{L_{free}}+\mathcal{L_{int}}=-\frac{1}{4}Z^{\mu\nu}Z_{\mu\nu}+\frac{1}{2}M_z^2Z^{\mu}Z_{\mu}+\bar{\psi}(i \displaystyle{\not} \partial -m)\psi+g\bar{\psi}\sigma_{\mu\nu}\psi Z^{\mu\nu} +d\bar{\psi}\sigma_{\mu\nu}\gamma_5\psi Z^{\mu\nu}$$ with $$\mathcal{L_{int}}=g\bar{\psi}\sigma_{\mu\nu}\psi Z^{\mu\nu}+d\bar{\psi}\sigma_{\mu\nu}\gamma_5\psi Z^{\mu\nu} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \sigma_{\mu\nu}=\frac{i}{2}[\gamma_{\mu},\gamma_{\nu}]$$ My problem is:

I would not have been able to write $$\mathcal{L_{int}}$$ starting only from $$\mathcal{L_{free}}$$.
Someone can link to a reference where it is schematized how to do or explain me.

Starting point:

I know that once all the generic terms have been written, the renormalizability condition applies, i.e. couplings $$M^{\alpha}$$ with $$\alpha \ge0$$. However, I do not know methodologically what procedure to follow to write all the terms.

• The first of the two interaction terms, the Pauli moment one, you also have in QED for photons, so you could find it in advanced QM books, like Sakurai's.... – Cosmas Zachos May 25 at 0:07
• Specifically, JJ Sakurai, Advanced QM, eqns (3.208)-(3.209) review the Gordon Decomposition of the Standard EM spinor current, isolating this, Pauli moment, term from the bland convective term. The Pauli moment term $F_{\mu\nu}\bar{\psi}\sigma^{\mu\nu}\psi$ summarize's the merman's magnetic interactions. – Cosmas Zachos Jun 1 at 15:13
• "summarizes the fermion's" is what the spell-checker monster should have started from... – Cosmas Zachos Jun 1 at 20:21