How to Determine the Branch Points of Correlation Functions in Euclidean Space

Abstract

Two-point correlators represented by either perturbative or non-perturbative integral equations in Euclidean space are considered. In general, it is difficult to determine the analytic structure of arbitrary correlators analytically. When relying on numerical methods to evaluate the analytic structure, exact predictions of, e.g., branch point locations (i.e., the multiparticle threshold) provide a useful check. These branch point locations can be derived by Cutkosky's cut rules. However, originally they were derived in Minkowski space for propagators with real masses and are thus not directly applicable in Euclidean space and for propagators of a more general form. Following similar considerations that led Karplus et al., Landau and Cutkosky more than 50 years ago to the mass summation formula that became known as Cutkosky's cut rules, we show how the position of branch points can be derived analytically in Euclidean space from propagators of very general form.

The quark-gluon vertex in Landau gauge bound-state studies

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Gluon screening mass at finite temperature from the Landau gauge gluon propagator in lattice QCD

“…This is a nontrivial problem that would require, for example, the computation of the spectral density as discussed in [112][113][114]. The analytic structure of the gluon propagator has also been investigated within the Dyson-Schwinger approach in [115][116][117][118][119][120][121][122][123][124]. However, given the approximations involved in the calculation and the difficulty of numerical computation, the outcome of the Dyson-Schwinger equations requires an independent confirmation.…”

Deep reinforcement learning for complex evaluation of one-loop diagrams in quantum field theory

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“…With an adequate choice of coordinates, the four-fold integral over the Euclidean loop momentum can thus be * Electronic address: windisch@physics.wustl.edu † Electronic address: thomas.gallien@silicon-austria.com ‡ Electronic address: christopher.schwarzlmueller@siliconaustria.com reduced to a two-fold integral over the radial component of the loop momentum, as well as over the angle between the external momentum and the loop momentum, see Appendix A. As discussed on the basis of a very simple example in [9], the angular integral then produces branch cuts in the complex plane of the radial integration variable, and one has to deform the contour in order to avoid the cut, as well as poles that might be present as well. Such strategies have been applied in various situations, see e. g. [10][11][12][13][14][15][16][17][18].…”