Coupled-channel system with anomalous thresholds and unitarity

A first nonperturbative and unitary treatment of multichannel systems with anomalous thresholds based on realistic potentials is presented. We consider the isospin one-half example system, with $D\pi$, $D\eta$, $D_s\overline{K}$, $D^{*}\pi$, $D^{*}\eta$, $D^{*}\overline{K}$ coupled channels in the $J^P=1^{-}$ partial wave, chosen such that various phenomena that come with the opening of an anomalous threshold can be illustrated in a stepwise procedure by a suitable variation of up, down and strange quark masses. We use a set of low-energy constants in the chiral Lagrangian that were adjusted to a large set of lattice QCD results. The six phase shifts and inelasticity parameters are presented for various choices of the pion mass. For a pion mass of 150 MeV there are no anomalous thresholds encountered. The small change from 150 MeV to 145 MeV pion mass causes a dramatic impact of the anomalous threshold on the phase shifts showing that our results are highly relevant for the extrapolation of lattice QCD calculations toward the physical pion mass.

Figure 1

The deformed left- and right-hand cut lines for the reaction amplitudes in (1). The crosses show the location of the normal threshold points. While the anomalous left-hand cut lines are shown with dashed red lines, the deformed right-hand cut lines are represented by blue solid lines. The filled and open red circles indicate the location of the anomalous threshold points and the return points respectively as explained in [20].

Figure 2

P-wave phase shifts with $I=1/2$ and $J^P=1^{-}$ quantum numbers. Results are shown for the pion mass $m_{\pi }=150\mathrm{MeV}$, where no anomalous threshold is encountered. Phase shifts are shown in blue solid lines, inelasticities in dashed red lines.

Figure 3

P-wave phase shifts with $I=1/2$ and $J^P=1^{-}$ quantum numbers. The left-hand panels show the phase shifts, the right-hand panels the inelasticity parameters for the $D\pi$ and $D^{*}\pi$ channels. Results are shown for two choices of the pion masses, $m_{\pi }=145\mathrm{MeV}$ and $m_{\pi }=150\mathrm{MeV}$.

Figure 4

P-wave phase shifts with $I=1/2$ and $J^P=1^{-}$ quantum numbers. The left-hand panels show the phase shifts, the right-hand panels the inelasticity parameters for the $D\pi$ and $D^{*}\pi$ channels. Results are shown for the physical pion mass.