On coupled-channel dynamics in the presence of anomalous thresholds

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.

Figure 1

Generic $t$- and $u$-channel exchange processes.

Figure 2

Left-hand cut line of an anomalous contribution to the generalized potential. The two crosses show the locations of the threshold points at $(m_a+M_a{)}^2$ and $(m_b+M_b{)}^2$ where we assume $a<b$ without loss of generality. The filled and open red circles indicate the location of the anomalous threshold point $({\mu }_{ab}^A{)}^2$ and a return point $({\mu }_{ab}^R{)}^2$ respectively.

Figure 3

Deformed left- and right-hand cut lines for the reaction amplitudes. The crosses show the locations of the threshold points at $(m_a+M_a{)}^2$. 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 locations of the anomalous threshold points ${\mu }_a^2$ and the return points ${\widehat{\mu }}_a^2$ respectively.

Figure 4

The generalized potential $U_{ab}^{\mathrm{tree}-\mathrm{level}}(s)$ (left panels) and $U_{ab}^{\mathrm{tree}-\mathrm{level}}(s-i\epsilon )$ (right panels) in the schematic model as defined in Eq. (15). Only nonvanishing elements are considered. Real and imaginary parts are shown with solid blue and dashed red lines respectively.

Figure 5

Reaction amplitudes $T_{ab}(s+i\epsilon )=T_{ba}(s+i\epsilon )$ in the schematic model of Eq. (15). Real and imaginary parts are shown with solid blue and dashed red lines respectively.

Figure 6

Phase shifts and inelasticity parameters of Eq. (16) in our schematic model (15).

Figure 7

The box contributions $U_{ab}^{\mathrm{box}}(s)$ (left panels) and $U_{ab}^{\mathrm{box}}(s-i\epsilon )$ (right panels) as defined in Eq. (19) with our model input (15). Only nonvanishing elements are considered. Real and imaginary parts are shown with solid blue and dashed red lines respectively.

Figure 8

The functions $D_{ab}(s+i\epsilon )$ of Eq. (23) in our schematic model (15). Real and imaginary parts are shown with solid blue and dashed red lines respectively.

Figure 9

The functions $B_{ab}(s+i\epsilon )$ of Eq. (24) in our schematic model (15) for $s>{\mu }_b^2$ or $s>(m_b+M_b{)}^2$. Real and imaginary parts are shown with solid blue and dashed red lines respectively.

Figure 10

The function ${\widehat{\varsigma }}_{ab}(s)$ of Eqs. (34) and (54) in our schematic model (15). Real and imaginary parts are shown with solid blue and dashed red lines respectively.

Figure 11

The nonvanishing functions $u_{n,ab}^L(s)$ of Eq. (44) for $n=1$, 2 in our schematic model (15). Real and imaginary parts are shown with solid blue and dashed red lines respectively.

Figure 12

The nonvanishing functions $U_{n,ab}^L(s)$ of Eq. (44) in our schematic model (15). Note the relation $U_{3,31}^L(s)=U_{1,31}^L(s)$. Real and imaginary parts are shown with solid blue and dashed red lines respectively.

Figure 13

The functions ${\widehat{N}}_{ab}(s)$ of Eqs. (65), (66), and (78) for $s>{\mu }_b^2$ or $s>(m_b+M_b{)}^2$ in our schematic model (15). Real and imaginary parts are shown with solid blue and dashed red lines respectively.

Figure 14

The functions ${\widehat{U}}_{ab}(s)$ of Eqs. (70) (74), and (77) for $s>{\mu }_b^2$ or $s>(m_b+M_b{)}^2$ in our schematic model (15). Real and imaginary parts are shown with solid blue and dashed red lines respectively.

Figure 15

The nonvanishing functions $H_{1,ab}^R(s)=\sum _{m=1}^3{\widehat{U}}_{1m}(s)u_m^R(s)$ for $s>{\mu }_b^2$ or $s>(m_b+M_b{)}^2$ in our schematic model (15). Real and imaginary parts are shown with solid blue and dashed red lines respectively.

Figure 16

The nonvanishing functions $H_{2,ab}^R(s)=\sum _{m=1}^3{\widehat{U}}_{2m}(s)u_m^R(s)$ for $s>{\mu }_b^2$ or $s>(m_b+M_b{)}^2$ in our schematic model (15). Real and imaginary parts are shown with solid blue and dashed red lines respectively.

Figure 17

The nonvanishing functions $H_{3,ab}^R(s)=\sum _{m=1}^3{\widehat{U}}_{3m}(s)u_m^R(s)$ for $s>{\mu }_b^2$ or $s>(m_b+M_b{)}^2$ in our schematic model (15). Real and imaginary parts are shown with solid blue and dashed red lines respectively.