Asymptotic symmetry

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Abstract
We discuss the deep Euclidean behavior of Green's functions in models in which symmetries are broken spontaneously or by "soft" operators. Our proof that the Green's functions approach their symmetric values in the deep Euclidean region uses the Callan-Symanzik equation in a way which avoids relying on order-by-order power counting for the scalar tadpole insertions. The vacuum expectation value of the scalar field appears in the Callan-Symanzik equation as an additional coupling constant; the solution of the equation introduces a momentum-dependent effective vacuum expectation value. We discuss theories with or without gauge invariance of the second kind.

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