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I'm following the book Quantum Field Theory and the Standard Model by Schwartz and I came to the rigorous non-perturbative proof of the Ward identity with path integrals via the Schwinger-Dyson equations in subsections 14.8.1-3. Since it is clear to me that the proof of the Ward-Takahaski identity is the "quantum version" of the Noether trick, I don't understand the passage from the Ward-Takahashi identity to the "standard" Ward identity. The latter can be thought of as a direct consequence of the photons being massless/without longitudinal polarization, but the proof followed by Schwartz does not seem to exclude the case of a massive vector boson. However this one breaks gauge invariance with its mass (or again equivalently it admit a longitudinal d.o.f.) so intuitively it would have no meaning that $p_\mu M^\mu=0$ for a generic amplitude $M^\mu$. Where am I wrong?

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    $\begingroup$ "the proof followed by Schwartz seems to do not exclude the case of a massive vector boson" --> Can you give more details here? What version of the Ward identity do you think holds for massive vector fields? In the usual formulation there is a (second class) constraint removing the time-like (not longitudinal) component, but no gauge symmetry or Ward identity. Also just as a friendly tip, it would help a lot to use MathJax (similar to LaTeX) to format the math in your question, here's a tutorial: math.meta.stackexchange.com/q/5020 $\endgroup$
    – Andrew
    Commented May 30, 2021 at 14:53
  • $\begingroup$ Thanks for the tip, I'm new so I hope you don't mind if in this topic I will keep writing with the normal text. For completeness, I'm referring to Schwartz 14.8.1 14.8.2 and 14.8.3. I think that the removing of timelike component holds for both Proca and Maxwell fields. For example, consider the process e(-)e(+)->u(-)u(+) at tree level: replacing kukv in the propagator of Proca and Maxwell will give me zero in both cases. But if I use the Ward identity in a process involving external vector bosons, when I replace the polarization with pu, I don't see where gauge invariance comes to kill puMu $\endgroup$ Commented May 30, 2021 at 15:04
  • $\begingroup$ Well it's a good idea to learn MathJax early and it's really just using LaTeX markdown as you type, so I do recommend using it, but I can follow what you are writing more or less. There is no gauge invariance for a massive gauge field (unless you use the Stuckelberg trick). The numerator of the propagator is $\eta_{\mu\nu}+p_\mu p_\nu / m^2$; if you contract this with $p^\mu$ you get $p_\nu - (m^2/m^2)p_\nu = 0$; this property follows from the second class constraint and is responsible for removing the time-like mode. $\endgroup$
    – Andrew
    Commented May 30, 2021 at 15:17
  • $\begingroup$ I will learn asap. I'm familiar with what you are saying, but then I would tell: take the Compton scattering (this means no internal vector propagators) but consider Proca instead of Maxwell and try to replace one external polarization with the momentum. Here we have no gauge invariance (intended as local U(1) symmetry of the lagrangian coming from having m=0 in A²), so there is no reason to discard the polarization which is proportional to the momentum. So why should even here puMu=0? Thanks for the help, much appreciating $\endgroup$ Commented May 30, 2021 at 15:22
  • $\begingroup$ So in this case all the vector propagators are on external legs. Then your question would be: what is the amplitude if there is a non-zero time-like component in the initial or final state? The answer is the amplitude is zero, since the external legs are on shell, and on shell $p_\mu A^\mu=0$ (so in the rest frame, $A^0=0$). I think it's possible you are assuming there is a property of the amplitudes is true, that isn't really true; there isn't a Ward identity like $p_\mu M^\mu$ on the amplitude itself, but there doesn't need to be, in the massive case. $\endgroup$
    – Andrew
    Commented May 30, 2021 at 15:30

2 Answers 2

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I'm just transferring some of what I wrote in the comments to an answer -- I may add more to this later.

There is no Ward identity for a massive spin-1 field; the massive and massless cases work differently.

For a massive photon, there exists a rest frame, so $p_\mu$ is timelike (on shell), so the fact that on shell $p_\mu A^\mu=0$ for a massive photon means that a timelike component is removed from external states. For internal lines, the numerator of the propagator is $\eta_{\mu\nu} + p_\mu p_\nu / m^2$; if you contract this with $p^\mu$ you get $p_\nu - (m^2/m^2) p_\nu=0$; this property follows from the second class constraint and is responsible for removing the time-like mode.

For a massless photon, there is no rest frame, and so on shell $p_\mu$ is null. Furthermore we need to remove two components from $A_\mu$ since there are only two polarizations. The Ward identity guarantees that the unphysical polarizations decouple from the other dofs, and so are never excited (so long as they are not present in external states) .

Another way to think of all this is in terms of first class and second class constraints (google "Dirac-Bergman quantization"). A first class constraint (gauge symmetry) removes two degrees of freedom, while a second class constraint (just a normal constraint) removes one. Massless electromagnetism has a first class constraint, Proca theory (without the Stuckelberg trick) has a second class constraint.

The story is different if you use the Stuckelberg trick in the massive case; then you introduce a new field, so naively you have 5 degrees of freedom (4 components of the vector field plus a scalar field). You also a new gauge symmetry, with an associated first class constraint. The first class constraint removes two degrees of freedom, and 5-2=3, which is the correct number of degrees of freedom for a massive spin-1 particle.

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There are two hierarchies of Ward-Takahashi identities (WTI):

  1. WTI for connected diagrams expressing charge conservation and derived via the Schwinger-Dyson (SD) equations from a global gauge symmetry.

  2. WTI for proper diagrams derived from BRST/local gauge symmetry. This implies (among other things) that the 2-point polarization tensor for the photon field is transverse in the $R_{\xi}$ gauge.

Both WTI hierarchies are in principle off-shell identities.

Already for QED with matter the pairing between the two WTI hierarchies is somewhat intricate, cf. e.g. this and this related Phys.SE posts, especially when one considers the roles of gauge-fixing conditions and renormalization.

Since Proca theory with matter fields has global but not local gauge symmetry, it only has the 1st WTI hierarchy.

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