Abstract
We explore a general framework to treat coupled-channel systems in the presence of overlapping left- and right-hand cuts as well as anomalous thresholds. Such systems are studied in terms of a generalized potential, where we exploit the known analytic structure of $t$- and $u$-channel forces as the exchange masses approach their physical values. Given an approximate generalized potential the coupled-channel reaction amplitudes are defined in terms of nonlinear systems of integral equations. For large exchange masses, where there are no anomalous thresholds present, conventional $N/D$ methods are applicable to derive numerical solutions to the latter. At a formal level a generalization to the anomalous case is readily formulated by use of suitable contour integrations with amplitudes to be evaluated at complex energies. However, it is a considerable challenge to find numerical solutions to anomalous systems set up on a set of complex contours. By suitable deformations of left-hand and right-hand cut lines we manage to establish a framework of linear integral equations defined for real energies. Explicit expressions are derived for the driving terms that hold for an arbitrary number of channels. Our approach is illustrated in terms of schematic three-channel systems. It is demonstrated that despite the presence of anomalous thresholds the scattering amplitude can be represented in terms of three phase shifts and three inelasticity parameters, as one would expect from the coupled-channel unitarity condition.
Highlights
A reliable and systematic treatment of coupledchannel systems subject to strong interactions is still one of the remaining fundamental challenges of modern physics
Such systems are studied in terms of a generalized potential, where we exploit the known analytic structure of t- and u-channel forces as the exchange masses approach their physical values
Explicit expressions are derived for the driving terms that hold for an arbitrary number of channels
Summary
A reliable and systematic treatment of coupledchannel systems subject to strong interactions is still one of the remaining fundamental challenges of modern physics. In the previous works which attempted to deal with such systems the strategy was to perform an analytic continuation of an N=D ansatz for the reaction amplitudes in the external mass parameters so as to smoothly connect a normal system to an anomalous system This was studied for two-channel systems only [37,38]. The general framework has been known since the 1960s, it was only recently successfully integrated into an effective field theory approach based on the chiral Lagrangian [20,21,22,23,24,25,26] As it stands Eq (1) breaks down once a coupled-channel system involves unstable particles or anomalous thresholds arise.
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