# Two photon vertex from tensor fermion bilinear

The fermion bilinears are $$\bar{\Psi}\Psi$$, $$\bar{\Psi}\gamma⁵\Psi$$, $$\bar{\Psi}\gamma^\mu\Psi$$, $$\bar{\Psi}\gamma^\mu\gamma^5\Psi$$ and $$\bar{\Psi}\sigma^{\mu\nu}\Psi$$, where $$\sigma^{\mu\nu} = (i/2)[\gamma^\mu,\gamma^\nu]$$. I understand the interpretation of the first four, and how we might build Lagrangians from them.

For example, the interaction Lagrangian in QED looks like (up to a constant) $$\bar{\Psi}\gamma^\mu\Psi A_\mu$$, where $$A_\mu$$ is the photon field. But how about the second term? Would it make sense to add a term like e.g. $$\bar{\Psi}\sigma^{\mu\nu}\Psi A_\mu A_\nu$$ as an interaction term to a Lagrangian? Obviously this is fine w.r.t. Lorentz invariance, but does it correspond to any actual physical process in nature? Basically it corresponds to a 2 fermion + 2 photon vertex.

• Note: $\bar{\Psi}\sigma^{\mu\nu}\Psi A_\mu A_\nu$ is zero because $\sigma^{\mu\nu}$ is antisymmetric in $\mu$ and $\nu$ while $A_\mu A_\nu$ is symmetric. Thus it is not a possible interaction term. One photon only per vertex! – G. Smith Mar 24 '19 at 5:47
• Are you familiar with the actual $\partial_\mu A_\nu$, the Pauli moment term multiplying that very bilinear? – Cosmas Zachos Mar 25 '19 at 18:32

To maintain gauge invariance, we would need to consider something like $$(D_\mu\psi)^\dagger\gamma^0\sigma^{\mu\nu}D_\nu\psi$$ instead, where $$D_\mu=\partial_\mu+iA_\mu$$.

With or without the derivative parts, such a term has mass-dimension $$5$$ (from $$3/2$$ for each factor of $$\psi$$ and $$1$$ for each factor of $$A_\mu$$), so it is "irrelevant" in the sense of Wilson renormalization; it makes the Lagrangian non-renormalizable. That's okay, but the effects of such a term would tend to be suppressed at low energies.

Terms that have been proposed to account for non-zero neutrino masses are also "irrelevant" in this sense, as are terms added to account for gravity. As expected for a non-renormalizable terms, both of those observed effects are small. As far as I know, no process requiring a term of the form $$(D_\mu\psi)^\dagger\gamma^0\sigma^{\mu\nu}D_\nu\psi$$ has been observed.

• Thanks! You're right about the gauge invariance, of course. I just wrote the interaction part of the Lagrangian. – Étienne Bézout Mar 24 '19 at 0:47

Because of anti-symmetric property of $$\sigma^{\mu\nu}$$ OP's two-photon term $$\bar{\Psi}\sigma^{\mu\nu}\Psi A_\mu A_\nu$$ would vanish identically.

On the other hand, the gauge-invariant mass-dimension 5 term $$(D_\mu\psi)^\dagger\gamma^0\sigma^{\mu\nu}D_\nu\psi$$ suggested by @Chiral Anomaly can be transformed into (via partial integration of the action, and throwing away the surface term), $$\bar{\psi}\sigma^{\mu\nu}F_{\mu\nu}\psi,$$ where $$F_{\mu\nu} = \frac{1}{2} (\partial_\mu A_\nu - \partial_\nu A_\mu).$$ Again, there is no two-photon vertex term.

The fundamental reason is that the electromagnetic interaction is Abelian. In contrast, for non-Abelian strong interactions, $$\sigma^{\mu\nu}F^{G}_{\mu\nu}$$ would entail two-gluon vertices, where $$\sigma^{\mu\nu}G^a_\mu G^b_\nu$$ can be non-zero, provided gauge matrices $$T_a$$ and $$T_b$$ do not commute.

• Okay, thank you! – Étienne Bézout Mar 27 '19 at 15:28