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In standard QFT, each term in the perturbative expansion for a gauge theory is not necessarily gauge-invariant. Only the whole sum of Feynman diagrams is guaranteed so.

However, at least for QED, there have been attempts to do the perturbative expansion solely in terms of gauge invariant quantities, cf. "QED : A Proof of Renormalizability" by Joel Feldman etc...

I wonder if there have been any attempt for the non-Abelian Yang-Mills theory yet?

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A formulation of a large class of non-Abelian gauge theories of Yang-Mills-Higgs type in terms of manifestly gauge-invariant quantities is developed in "Higgs phenomenon without symmetry breaking order parameter" by Fröhlich, Morchio and Strocchi. They explicitly derive the relations between the perturbative correlation functions in terms of gauge-invariant fields and the standard perturbative correlation functions in terms of gauge-variant fields.

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