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.
High Precision Statistical Landau Gauge Lattice Gluon Propagator Computation vs. the Gribov-Zwanziger approach
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abstract
The compatibility of the results from the Gribov-Zwanziger tree level prediction and lattice simulations, using large statistical ensembles, for the Landau gauge gluon propagator are investigated, thereby complementing earlier work using small-scale statistics. Our results show that the data is well described by the tree level estimate only up to momenta $p \lesssim 1$ GeV while clearly favoring the so-called Refined Gribov-Zwanziger scenario, implying particular relations between certain possible $d=2$ condensates. We also provide a global fit of the lattice data which interpolates between the above scenario at low momenta and the usual continuum one-loop renormalization improved perturbation theory after introducing an infrared log-regularizing term.
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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.