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What's the fundamental difference between the way a gauge boson gets coupled to a matter field, preferably a Fermionic field and the way a Goldstone boson gets coupled to the matter field ?

In electron-photon (a U(1) boson) interaction we write the coupling as $A_\mu J^\mu$. Is it true for any fermionic field getting coupled to any (Abelian or non-Abelian) gauge field ? Why can't it involve derivatives? Can we write down a generic expression for the coupling of a Goldstone boson (such as phonon) with an electron? One thing I'm sure of is, it can't escape the Adler's rule, but what are the other constraints?

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Take a look at WP:Gauge theory and WP:Goldstone boson. In point of fact, there is little "fundamental" difference, and the signatures you are proposing are not it. Basically, the Higgs mechanism tells you the two couplings are magically primed to correspond to each other, so, in the Higgs mechanism, the second is subsumed/absorbed into the first.

First, for your detailed, specific questions: In gauge theory, the minimal coupling prescription $\partial_\mu \mapsto \partial_\mu -i e A_\mu$ dictates that the leading gauge coupling (in e) starts with the $eA_\mu J^\mu$ you mention. That's where the story ends for fermions, when their action is only linear in derivatives, as in the standard kinetic term thereof.

But, as you seem to be aware, for bosons, the second derivative introduces further couplings bilinear in the gauge fields, e.g. the "seagull" terms for gauged scalar fields. And on and on....

Worse yet, if renormalizability is not at issue, you may have different couplings, such as the Pauli term, $F^{\mu \nu} \overline{\psi} \sigma_{\mu \nu}\psi$ where each factor is separately gauge invariant, even for fermions. You may, of course, introduce as many derivatives as desired, provided your gauge fields enter through minimal coupling designed to ensure gauge invariance.

Now, Goldstone bosons $\theta$ are always the leading (lowest order) term in the currents corresponding to the symmetry broken charges, $J_\mu \sim v \partial_\mu \theta+ ...$, $v$ being the ("pion") decay constant, which is how such currents are conserved, given a spontaneous broken symmetry transformation which shifts the Goldstone by a constant! (Its defining feature, essentially).

So, the leading behavior of the spontaneous broken current is "poised" to be gauged into gauge fields in the Higgs mechanism, and vanish in the unitary gauge. Indeed, for many theories, the heart of the interaction is of the form $(J_g ^\mu + J_\psi ^\mu+...)(J_{g ~\mu} + J_{\psi ~\mu}+...) $, so, then, yes, the leading interaction is, indeed, of the form $v\partial_\mu \theta J_\psi^\mu+...$ which reminds you of the gauge coupling, and, upon a gauge transformation, produces the same increment as the standard "current times the gradient of the gauge parameter" increment of the gauge minimal coupling term.

Moreover, as you pointed out, the magic of breaking up Wigner-Weyl mode multiplets into Nambu mode reduced arrays ensures the derivative couplings you see dominating the current above, and producing Adler zeros. Now, I have to invoke "magic": it turns out this type of coupling is universal, provable by abstract arguments, but not always manifest in the way the fields have been parameterized originally in the theory, and, oftentimes, in such suboptimal languages, Adler zeroes are "detected" by magic cancellations, notably in tree amplitudes!!!

Normally even simple groups broken, like U(1)s, lead to messy reduced arrays, and it might be easier for you to actually look at the possible/prospective terms in the action, observe their behavior under a gauge transformation, $\delta J_\mu \sim v\partial _\mu \epsilon (x) +...$, and convince yourself of the parity of terms in having their variations cancel... a handful of cancellations in-the-hand are worth a thousand words!

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