Direct evaluation of the d=2 setting sun diagram with a priori complex external momenta disagrees with the Kallen-Lehmann spectral continuation, so complex momenta should be inserted only after performing the integral.
The Nielsen identities in screened theories
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One-loop explicit expressions are derived for the gluon Nielsen identity in the formalism of the screened massive expansion for Yang-Mills theory. The gauge-parameter-independence of the poles and residues is discussed in a strict perturbative context and, more generally, in extended resummation schemes. No exact formal proof was reached by the approximate resummation schemes, but some evidence is gathered in favor of an exact invariance of the phase, consistently with previous numerical studies.
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The setting sun diagram with complex external momenta
Direct evaluation of the d=2 setting sun diagram with a priori complex external momenta disagrees with the Kallen-Lehmann spectral continuation, so complex momenta should be inserted only after performing the integral.