Abstract
We employ a scalar model to exemplify the use of contour deformations when solving Lorentz-invariant integral equations for scattering amplitudes. In particular, we calculate the onshell 2 -> 2 scattering amplitude for the scalar system. The integrals produce branch cuts in the complex plane of the integrand which prohibit a naive Euclidean integration path. By employing contour deformations, we can also access the kinematical regions associated with the scattering amplitude in Minkowski space. We show that in principle a homogeneous Bethe-Salpeter equation, together with analytic continuation methods such as the Resonances-via-Pad\'e method, is sufficient to determine the resonance pole locations on the second Riemann sheet. However, the scalar model investigated here does not produce resonance poles above threshold but instead virtual states on the real axis of the second sheet, which pose difficulties for analytic continuation methods. To address this, we calculate the scattering amplitude on the second sheet directly using the two-body unitarity relation which follows from the scattering equation.
Highlights
The study of resonances is a central task in the nonperturbative treatment of quantum field theories
We show that in principle a homogeneous Bethe-Salpeter equation, together with analytic continuation methods such as the Resonances-via-Pademethod, is sufficient to determine the resonance pole locations on the second Riemann sheet
We focus on the methodological aspects, namely how to calculate scattering amplitudes in the kinematical domains where contour deformations become necessary, and how that information can be used to extract the resonance information on higher Riemann sheets
Summary
The study of resonances is a central task in the nonperturbative treatment of quantum field theories. To extract the properties of resonances it is necessary to access unphysical Riemann sheets, whereas straightforward numerical calculations are restricted to the first sheet only Both issues are typical obstacles in the functional approach of Dyson-Schwinger equations (DSEs) and BSEs, where one determines quark and gluon correlation functions and solves BSEs to arrive at hadronic observables. We focus on the methodological aspects, namely how to calculate scattering amplitudes in the kinematical domains where contour deformations become necessary (which is usually referred to as “going to Minkowski space”), and how that information can be used to extract the resonance information on higher Riemann sheets. Technical details on contour deformations are relegated to the Appendix
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