Near-threshold Spectrum from Uniformized Mittag-Leffler Expansion
-Pole Structure of $Z(3900)$-

In recent years, many resonance-like enhancements which are potential candidates of exotic hadrons, have been observed in hadronic spectra near the thresholds of hadronic channels Guo et al. (2018); Karliner et al. (2018). Typical examples are the $XYZ$-mesons, such as $X(3872)$, $Y(4260)$, $Z(3900)$ in the charmonium spectra Hosaka et al. (2016). There has been much discussion regarding the origins of these threshold enhancements, whether they are of resonant origin, such as tetra-quark, hadron molecular, or gluonic excitation states, or of kinematic origin, such as threshold cusps Graves-Morris (1967), or triangle singularities Guo et al. (2020). Clarification of such issues could possibly shed light on the complex nature of the strong interaction governing the dynamics of hadrons, and is of crucial importance in hadron physics.

For analyses of the hadronic spectra, model-independent expressions such as the Breit-Wigner formula Breit and Wigner (1936), Flatté formula or equivalently the two-channel Breit-Wigner formula Flatté (1976), or coupled-channel model calculations are most commonly used in practice. Though most of these practices are conducted under the parameterization of energy or channel-momentum, analysis in the coupled-channel uniformized parameterization, introduced by Ref. Kato (1965), and followed by Ref. Fujii and Kato (1969); Fujii and Fukugita (1975), can be particularly useful to clarify the interplay between the position of the pole on the unphysical sheets of the $S$-matrix Riemann surface and its contribution to physical quantities such as the cross section or invariant-mass distribution. In terms of the uniformized parameterization, a new method, the uniformized Mittag-Leffler expansion, was proposed and successfully applied to $\Lambda(1405)$ in Ref. Yamada and Morimatsu (2020, 2021), in which the S-matrix or the Green’s function is expressed as a simple sum of not only bound poles but also resonant poles.

The purpose of the present paper is two-folded. The first is to show how near-threshold poles of the S-matrix manifest themselves in the near-threshold spectrum. The second is to show that $Z(3900)$ can be naturally understood as a contribution of a set of poles near the $\bar{D}D^{*}$ threshold. In Sec. II, III, we will demonstrate the general behavior of contributions of near-threshold poles on various sheets of the Riemann surface, including contributions commonly regarded as “threshold cusps", based on the formalism of the Uniformized Mittag-Leffler expansion presented in Ref. Yamada and Morimatsu (2020, 2021). Then in the following section, (Sec. IV), we will discuss under the two-channel uniformization parameterization, a plausible interpreteation of the $Z(3900)$ enhancement, reflecting on the HAL QCD results obtained in Ref. Ikeda et al. (2016, 2018) combined with the pole symmetry condition of the $S$-matrix. We will also set up a separable potential model shraing common features as the HAL QCD spectrum, which support our interpretation.

Let us consider the spectrum of coupled double-channel two-body systems with threshold energies, $\varepsilon_{1}<\varepsilon_{2}$. The following findings of our analysis are valid in general, and are independent of the details of the system.

Observables of the two-body system such as scattering cross sections or invariant-mass distributions are given by the scattering $T$-matrix or the two-body Green’s function, which we generically write as ${\mathcal{A}}$. We assume that ${\mathcal{A}}$ has only singularities of poles and two unitarity cuts running from the thresholds, $\varepsilon_{1}$ and $\varepsilon_{2}$, when expressed as a function of the complex center-of-mass energy, $\sqrt{s}$.

Let us define a dimensionless parameter, $e$, by

where $e=0$ and $e=1$ at the lower and upper threshold, respectively. We call $e$, scaled “energy", because $e$ is real as long as the center-of-mass energy, $\sqrt{s}$, is real and is a monotonically increasing function of $\sqrt{s}$. Then, following Ref. Kato (1965), we introduce a complex dimensionless variable, $z$, by

where $e^{1/2}$ ($\left(e-1\right)^{1/2}$) is the scaled “momentum" in the lower (upper) channel. As a function of $z$, $e=\left(z+z^{-1}\right)^{2}/4$ and $\mathcal{A}$ is single-valued. Such a procedure where a function is made single-valued by introducing an appropriate parameterization is called uniformization Newton (1982). In this paper, for simplicity, we mainly deal with total scattering cross sections or missing-mass distributions, which are given by the imaginary part of the scattering $T$-matrix or the two-body Green’s function, Fetter and Walecka (1971); Bertsch and Tsai (1975); Morimatsu and Yazaki (1994), ${\rm Im}\,{\mathcal{A}}$.

The structure of the uniformized complex $z$ plane is shown in Fig. 1, where the physical energy is given by the red solid line. We also show the domains corresponding to the four complex energy Riemann sheets labeled by the convention following Ref. Pearce and Gibson (1989). For each channel momentum, $k_{i}$, $t$ and $b$ refer to the top and bottom half of the complex momentum plane with the argument, $0\leq\arg k_{i}<\pi$ and $\pi\leq\arg k_{i}<2\pi$, respectively. Then, four complex energy Riemann sheets are specified by a set of complex channel momenta as [$tt$], [$tb$], [$bt$] and [$bb$], where e.g. [$tb$] means the momentum in lower (upper) channel is $t$ ($b$). For later discussion we also show the regions where the imaginary part of the complex energy is positive and negative by $(+)$ and $(-)$, respectively.

In Ref. Yamada and Morimatsu (2020, 2021) it was shown that ${\mathcal{A}}$ can be expanded as a sum of pole terms in a form similar to that of Ref. Humblet and Rosenfeld (1961); Romo (1978); Bang et al. (1978); Berggren (1982) in the variable, $z$, (uniformized Mittag-Leffler expansion) as,

where $z_{n}$ and $r_{n}$ are, respectively, the position and residue of a pole in the variable, $z$. We define a real dimensionless function, $f(z;z_{p},\phi_{p})$, by

Then, ${\rm Im}\,{\mathcal{A}}$ is given as

where $r_{n}=|r_{n}|\exp(i\phi_{n})$. $f$ can be regarded as a ’normalized’ pole-contribution linearly contributing to $\mathcal{A}$ by a weight of the absolute value of the residue.

For later discussion, we give here a standard expression of the Breit-Wigner formula. If the amplitude is analytic in the square root of the center-of-mass energy squared, $\sqrt{s}$, the normalized contribution from a pole near the physical region to the physical amplitude is given by

Under the condition $\psi_{p}=0$, and the definition ${\rm Re}\sqrt{s}_{p}=\epsilon$ and ${\rm Im}\sqrt{s}_{p}=-\Gamma/2$, we obtain the standard Breit-Wigner form Breit and Wigner (1936),

In this section, we demonstrate how the near-threshold spectrum changes as a S-matrix pole moves from the [$bt$] sheet through the [$tb$] sheet to the [$bb$] sheet near the upper threshold. We choose six positions, $A$-$F$, of the S-matrix pole near the upper threshold shown in Fig. 2. Their position on the complex $z$-plane, $z_{p}$, and their scaled energy, $e_{p}$, are tabulated in Table. 1. We also show $e_{0}$, the scaled energy of the physical point nearest to the pole on the complex $z$ plane, and $\gamma_{0}$, the distance between the pole and the nearest physical point, defined by

where $r_{p}=|z_{p}|$, $\theta_{p}={\rm Arg}\,z_{p}$, $x_{p}={\rm Re}\,z_{p}$ and $y_{p}={\rm Im}\,z_{p}$.

The spectra of the contribution from poles, $A$-$F$, are shown respectively in Fig. 3. As can be observed, when the pole is located on the [$bt$] sheet, the peak shows up below the upper threshold energy. As the pole comes close to the [$tb$] sheet, the peak approaches the threshold. Then, when the pole is on the [$tb$] sheet, the peak remains at the upper threshold while changing the slopes below and above the threshold. Once the pole reaches the [$bb$] sheet, the peak position leaves the threshold, and increases again. Also, when the pole is close to the upper threshold, the complex pole energy does not have the usual meaning of the resonance. Neither the real part represents the peak energy, nor the imaginary part represents the width. Near the boundary of the [$bt$] and [$tb$] ([$tb$] and [$bb$]) sheets, ${\rm Re}\,e_{p}$ is larger (smaller) than 1 and ${\rm Im}\,e_{p}$ is very small, while the position of the peak is below (above) or equal to 1 and the width of the peak remains almost unchanged. Also, on the [$tb$] sheet ${\rm Im}\,e_{p}$ is positive. The position of the peak is given by the energy of the physical point nearest to the pole on the complex $z$-plane, and the width is given by the distance between the pole and the nearest physical point. Finally, and most importantly, the transition of the spectrum is continuous. There is no essential difference whether a pole is located on the [$bt$] sheet, [$tb$] sheet or [$bb$] sheet, contrary to the usual understanding that poles on the [$tb$] sheet are irrelevant. Therefore, if the observed spectrum is peaked about the upper threshold with experimental uncertainties, it would be difficult, in addition, non-essential to exactly determine which sheet the pole is really located. What is important is the existence of a pole near the upper threshold and how distant the pole is from the physical energy, which should be sufficient for us to know from experimental data.

Let us discuss the general behavior of the contribution from a near-threshold pole in comparison to the Breit-Wigner form. When $e>1$, $z$ is on the real axis and we parametrize $z$ and $z_{p}$ as $z=x$, $z_{p}=x_{p}+iy_{p}$, where $x$, $x_{p}$ and $y_{p}$ are real. Then, a pole contribution is given by

which is a Breit-Wigner form in the variable $x=z$ (real) with an additional phase. If we take the phase of the pole as, $\phi_{p}=0$, the contribution is given by,

which is peaked at $x=x_{p}$. Now let us consider the behavior when $0<e<1$, i.e., $z$ is on the unit circle. We can parametrize $z$ and $z_{p}$ as $z=\exp(i\theta)$, $z_{p}=r_{p}\exp(i\theta_{p})$, where $\theta$, $r_{p}$ and $\theta_{p}$ are real parameters. Then, a pole contribution becomes

similar to a Breit-Wigner form in the variable $\theta$ with a phase. If we take $\phi_{p}=\theta_{p}-\pi/2$, then,

which is peaked at $\theta=\theta_{p}$. Therefore, we can intuitively understand the behavior of $f(\sqrt{e}+\sqrt{e-1};z_{p},\phi_{p})$ in the whole energy region by combining Eq. (4) for $x>1$ and Eq. (5) for $0<\theta<\pi/2$, as shown in Fig. 4 for typical cases of poles on the [$bt$], [$tb$] and [$bb$] sheets. As a typical pole on the [$bt$] ([$bb$]) sheet, we take $A$ ($F$), while on the [$tb$] sheet we take a pole between $C$ and $D$. We have chosen the phase of the residue, $\phi_{p}$, such that the contribution is maximized at the physical energy nearest to the pole.

When the pole is close to the physical region and distant from the thresholds, in the region of ($0\ll\theta_{p}\ll\pi/2$) on the [$bt$] sheet or ($x_{p}\gg 1$) on the [$bb$] sheet, the contribution is of a Breit-Wigner form. The Breit-Wigner form along the unit circle with a peak at $\theta=\theta_{p}$ on the [$bt$] sheet and along the real axis with a peak at $x=x_{p}$ on [$bb$] sheet. When the pole is on the [$tb$] sheet and is close to the upper threshold, two Breit-Wigner forms overlap. Then, the contribution is given by connecting slopes of two Breit-Wigner forms and is peaked at the upper threshold, $x=1$ and $\theta=\pi/2$.

$Z(3900)$ is a candidate of exotic hadrons found by BESIII Ablikim et al. (2013, 2014, 2015); Liu (2015) and Belle Liu et al. (2013), which appears as a peak in both the $\pi J/\psi$ and $\bar{D}D^{*}$ invariant mass spectra in the reaction, $e^{+}e^{-}\to Y(4260)\to\pi\pi J/\psi$ and $\pi\bar{D}D^{*}$, and later confirmed by CLEO-c Xiao et al. (2013). Our ultimate goal is to analyze the experimental data for $Z(3900)$ by means of uniformized Mittag-Leffler expansion and draw a model independent conclusion on the pole structure of $Z(3900)$ properties, such as the mass and the width or on which sheet the S-matrix pole of $Z(3900)$ is located. Before carrying out the entire program, however, in this paper we confine our argument to addressing the possible origin of the $Z(3900)$ enhancement from the viewpoint of the uniformized Mittag-Leffler expansion.

A symmetry condition of S-matrix poles plays a vital role in the following argument. Let $S(\{k_{i}\})$ be the multi-channel S-matrix with $\{k_{i}\}$ as the set of channel momentum in all channels. Then, due to unitarity, the S-matrix has the symmetry,

From this relation, one can immediately conclude that if $\{k_{i}\}$ is a pole of S-matrix, so is $\{-k_{i}^{*}\}$, which will be referred to as the ’conjugate pole’ in the following. Then, if the complex scaled energy $e$ is a pole of S-matrix, so is $e^{*}$ on the same Riemann sheet specified by the set of complex channel momenta. In terms of the two-channel uniformization variable, $z$, the above symmetry is expressed as

and the conjugate pole of $z$ is $-z^{*}$ on the same Riemann sheet symmetric about the imaginary axis.

In subsection A, we reexamine the HAL QCD results including the poles conjugate to those given in Ref. Ikeda et al. (2016) from the symmetry of the S-matrix and calculating the pole contributions in the framework of the two-channel uniformized Mittag-Leffler expansion. There, we cannot determine the relative contributions of the poles in the spectrum since we do not know the residues of the poles in Ref. Ikeda et al. (2016). Then, in subsection B, we set up a separable potential model in which we have a pole near the $\bar{D}D^{*}$ threshold similar to Ref. Ikeda et al. (2016) and show that the contribution of such a pole really dominates the spectrum in the vicinity of the $\bar{D}D^{*}$ threshold.

In this paper we exhibited how S-matrix poles manifest themselves as the physical spectrum near the upper threshold, from the viewpoint of the uniformized Mittag-Leffler expansion, and argued that the enhancement of $Z$(3900) very close to the $\bar{D}D^{\ast}$-threshold most likely originates from a pole (a couple of poles) near the $\bar{D}D^{\ast}$-threshold.

After introducing the two-channel uniformized plane and the uniformized Mittag-Leffler Expansion, we numerically demonstrated how the spectrum changes as a S-matrix pole moves near the upper threshold across the borders of complex-energy Riemann sheets. As a pole moves from the [$bt$]-sheet, through the [$tb$]-sheet, to the [$bb$]-sheet, the contribution from the individual pole continuously transitioned, peaking at the energy of the physical point closest on the uniformized plane, with the width given by the corresponding distance. The continuous behavior of the transition implies that the identity on which sheet a pole is located, is not essentially important. The fundamental identities which characterize the contribution of a pole, is given by the residue of the pole, and the distance of the pole to the physical region on the uniformized plane. We also observed that the complex pole energy does not have the usual meaning of the resonance when the pole is close to the upper threshold. Neither the real part represents the peak energy, nor the imaginary part represents the half width. For example, the complex energy of a pole on the [$tb$] sheet in the near-threshold region of the upper threshold has a positive imaginary part.

Subsequently, we argued that $Z(3900)$ can be naturally understood as a contribution of a set of poles in the domain near the $\bar{D}D^{*}$ threshold. We showed that the HAL QCD results Ikeda et al. (2016, 2018) combined with the symmetry argument of the S-matrix indicate the existence of the S-matrix poles near the $\bar{D}D^{*}$ threshold, whose normalized contributions have narrow structures at the threshold. We further demonstrated such poles indeed exhibit a dominant enhancement in the spectrum by means of a separable two-channel non-relativistic potential model with a mono-polar form factor.

To further solidify our claim, it would be very meaningful to develop a parameterization which enables us to globally (entirely) unformize the three-channel $S$-matrix. Also we would like to fully exploit our uniformized Mittag-Leffler method to actual experimental observations of $Z(3900)$, such as the invariant-mass distributions of $Y\to\pi\pi J/\psi$ measured by Belle, and BESIII collaborations Ablikim et al. (2013, 2014, 2015); Liu (2015); Liu et al. (2013). Last but not least, we have not fully understood the nature of such a pole which is possibly responsible for the $Z(3900)$ enhancement and is located near the $\bar{D}D^{*}$ threshold on the complex energy sheet usually considered to be physically irrelevant. It is not clear how we should interpret such a pole because it would have a complex energy with positive imaginary part and increases in time while a resonance pole has a complex energy with negative imaginary part and decreases in time. It would be our future challenge to clarify the nature of such a pole, which would extend our understanding of the resonant phenomena in general.