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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.

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    $\begingroup$ 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! $\endgroup$ – G. Smith Mar 24 at 5:47
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    $\begingroup$ Are you familiar with the actual $\partial_\mu A_\nu$, the Pauli moment term multiplying that very bilinear? $\endgroup$ – Cosmas Zachos Mar 25 at 18:32
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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.

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

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  • $\begingroup$ Okay, thank you! $\endgroup$ – Étienne Bézout Mar 27 at 15:28

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