Near-threshold spectrum from a uniformized Mittag-Leffler expansion: Pole structure of the $Z(3900)$

We demonstrate how $S$-matrix poles manifest themselves as the physical spectrum near the upper threshold in the context of the two-channel uniformized Mittag-Leffler expansion, an expression written as a sum of pole terms (Mittag-Leffler expansion) under an appropriate variable where the $S$ matrix is made single-valued (uniformization). We show that the transition of the spectrum is continuous as a $S$-matrix pole moves across the boundaries of the complex energy Riemann sheets and that the physical spectrum peaks at or near the upper threshold when the $S$-matrix pole is positioned sufficiently close to it on the uniformized plane. There is no essential difference on which sheet the pole is positioned. What is important is the existence of a pole near the upper threshold and the distance between the pole and the physical region, not on which complex energy sheet the pole is positioned. We also point out that when the pole is close to the upper threshold, the complex pole does not have the usual meaning of the resonance. Neither the real part represents the peak energy, nor the imaginary part represents the half width. Subsequently, we try to understand the current status of $Z(3900)$ from the viewpoint of the uniformized Mittag-Leffler expansion reflecting in particular, Phys. Rev. Lett. 117, 242001 (2016) in which they concluded that $Z(3900)$ is not a conventional resonance but a threshold cusp. We point out that their results turn out to indicate the existence of $S$-matrix poles near the $\overline{D}{D}^{*}$ threshold, which is most likely the origin of the peak found in their calculation of the near-threshold spectrum. In order to support our argument, we set up a separable potential model which shares common behavior of poles near the $\overline{D}{D}^{*}$ threshold to the above-mentioned reference and show in our model that the structures near the $\overline{D}{D}^{*}$ threshold are indeed caused by these near-threshold poles.

Figure 1

The complex $z$ plane uniformizing the two-channel two-body $S$ matrix. The physical energy is represented by the red line. Below the lower threshold (labeled th1), the physical energy is on the imaginary axis. Between the lower and upper thresholds (labeled th2), the physical region is on the unit circle and then above the upper threshold on the real axis. The four domains specified by $[tt]$, $[bb]$, $[tb]$, and $[bt]$ correspond to four Riemann sheets of the complex energy, and ($+$), ($-$) show the imaginary part of the complex energy. For details refer to the main text.

Figure 2

Positions of the $S$-matrix poles, $A–F$, on the complex $z$ plane near the upper threshold (labeled th2). The red line represents the physical region.

Figure 3

The “normalized” pole-pair contributions, $f(z;z_p,{\varphi }_p)$, from poles $A–F$. Two cases are shown with different phases of residues, ${\varphi }_p={\varphi }_0$ and ${\varphi }_0-\pi /2$, where ${\varphi }_0$ is, respectively, chosen as ${\varphi }_0={\theta }_p-\pi /2$, $\mathrm{Arg}(z_p-1)-\pi /2$, and 0 for poles on the [$bt$], [$tb$], and [$bb$] sheet, so that the contribution, $f(z;z_p,{\varphi }_p)$, is maximized at physical energy $e_0$.

Figure 4

The behavior of $f(z;z_p,{\varphi }_p)$ on the unit circle, $z=\exp i\frac{\pi }{2}(1-t)$ (the red solid line, $0<t<1$, and the orange dashed line, $1<t<2$) and on the real axis, $z=t$ (the blue dotted line, $0<t<1$, and the red sold line, $1<t<2$) for typical cases of poles on the [$bt$], [$tb$], and [$bb$] sheets. The pole positions on the $z$ plane are $0.869+0.233i$ ([$bt$] sheet), $0.929-0.071i$ ([$tb$] sheet), and $1.300-0.100i$ ([$bb$] sheet), respectively.

Figure 5

The (locally) uniformized complex $z$ plane for the $\pi J/\psi -\rho {\eta }_c-\overline{D}{D}^{*}$ coupled-channel S matrix. $\pi J/\psi$, $\rho {\eta }_c$, and $\overline{D}{D}^{*}$ denote the corresponding thresholds on the physical energy, respectively. The $z$ plane is a two-sheeted Riemann surface connected by the two branch cuts running along the unit circle. Both the $S$-matrix poles given in Ref. [20], 1–5, and their conjugate poles $1^{*}–5^{*}$, (not given in Ref. [20]) are shown by filled and unfilled circles.

Figure 6

“Normalized” pole-pair contributions, $f(z;z_p,{\varphi }_p)$, from poles 2, $2^{*}$ and 3, $3^{*}$ for the cases where the phase of the residue is ${\varphi }_p=0$ (above left, above right) and ${\varphi }_p=-\pi /2$ (below left, below right).

Figure 7

$|{\mathcal{F}}_{11}{|}^2$(left) and $\mathrm{Im}[{\mathcal{F}}_{11}]$(right) when $\alpha =\gamma =1.7$, $\sqrt{{\mu }_1/{\mu }_2}=0.606$. $e$ is a dimensionless parameter given by $e=(E-{\mathrm{\Delta }}_1)/({\mathrm{\Delta }}_2-{\mathrm{\Delta }}_1)$. Individual contributions of poles $z_i$ and their conjugate pairs are shown by solid and dashed curves (red/blue), respectively.