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I already know that QCD is renormalizable in several gauges, including the $\xi$ gauge and the background field gauge. That is, the divergence of the quantum effective action is limited by symmetry, and as a result the divergence can be removed by adjusting the coefficients of the original action. However, I feel it odd that the proof of the renormalizability of QCD or some gauge theories as I know it depends on certain gauge fixing conditions. This is because, according to the philosophy of the FP method, a gauge fixing is basically a way to remove the huge non-physical redundancy of gauge degrees of freedom, which in that sense is not an essential component of the theory, and thus these proofs only mean "that the theory is renormalizable in that specific gauge. Therefore, I would like to ask two questions:

  1. When one show the renormalizability of QCD for example , a particular gauge is chosen. Why is such a "specific gauge dependent" proof widely used? In other word, why is such a gauge dependent proof essential?

  2. Is there a proof of renormalizability that does not depend on gauge fixing conditions? In particular, is the theory renormalizable no matter how strange gauge fixing conditions are imposed? (Intuitively, I don't think that's true.)

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While OP is right that morally going from one gauge-fixing condition to another is a flow along gauge-orbits, it can in practice be difficult to check that the theory is independent of the gauge-fixing condition: Gauge-fixing in the path-integral may be only partial, and in a quantum average sense, there could be Gribov ambiguities, etc.

Moreover, various objects, such as propagators and vertex-correlators, which we use in the renormalization, explicitly depend on the gauge-fixing condition.

At the formal perturbative level, the Batalin-Vilkovisky formalism has been employed to show renormalization and independence of gauge-fixing, see e.g. Ref. 1.

References:

  1. S. Weinberg, Quantum Theory of Fields, Vol. 2, 1996; Chapters 15 + 17.
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