Establishing the heavy quark spin and light flavor molecular multiplets of the $X(3872)$, ${Z}_{c}(3900)$, and $X(3960)$

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Establishing the heavy quark spin and light flavor molecular multiplets of the $X (3872)$ , $Z_{c} (3900)$ , and $X (3960)$

Teng Ji, Xiang-Kun Dong, Miguel Albaladejo, Meng-Lin Du, Feng-Kun Guo, and Juan Nieves

Phys. Rev. D 106, 094002 – Published 3 November 2022

Abstract

Recently, the LHCb Collaboration reported a near-threshold enhancement $X (3960)$ in the $D_{s}^{+} D_{s}^{-}$ invariant mass distribution. We show that the data can be well described by either a bound or a virtual state below the $D_{s}^{+} D_{s}^{-}$ threshold. The mass given by the pole position is ( $3928 \pm 3$ ) MeV. Using this mass and the existing information on the $X (3872)$ and $Z_{c} (3900)$ resonances, a complete spectrum of the $S$ -wave hadronic molecules formed by a pair of ground state charmed and anticharmed mesons is established. Thus, pole positions of the partners of the $X (3872)$ , $Z_{c} (3900)$ , and the newly observed $D_{s}^{+} D_{s}^{-}$ state are predicted. Calculations have been carried out at the leading order of nonrelativistic effective field theory and considering both heavy quark spin and light flavor SU(3) symmetries, though conservative errors from the breaking of these symmetries are provided.

Received 21 July 2022
Accepted 12 October 2022

DOI:https://doi.org/10.1103/PhysRevD.106.094002

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Heavy quark effective theory Strong interaction

Exotic mesons

Particles & Fields

Authors & Affiliations

Teng Ji ^1,2,*, Xiang-Kun Dong ^1,2,†, Miguel Albaladejo ^3,‡, Meng-Lin Du ^3,4,§, Feng-Kun Guo ^1,2,∥, and Juan Nieves ^3,¶

¹CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
²School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
³Instituto de Física Corpuscular (centro mixto CSIC-UV), Institutos de Investigación de Paterna, Apartado 22085, 46071 Valencia, Spain
⁴School of Physics, University of Electronic Science and Technology of China, Chengdu 611731, China

^*jiteng@itp.ac.cn
^†dongxiangkun@itp.ac.cn
^‡Miguel.Albaladejo@ific.uv.es
^§du.menglin@ific.uv.es
^∥fkguo@itp.ac.cn
^¶jmnieves@ific.uv.es

Article Text

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Issue

Vol. 106, Iss. 9 — 1 November 2022

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Images

Figure 1
Bottom: fitted line shapes of the $D_{s}^{+} D_{s}^{-}$ invariant mass distribution in the $B^{+} \to D_{s}^{+} D_{s}^{-} K^{+}$ reaction. Data taken from Ref. [56]. The shaded area on the right, starting at $m_{D_{s}^{+} D_{s}^{-}} = 4.12 GeV$ , is the region excluded in all fits. The solid blue line and green histogram stand for the best fit ( $χ^{2} / d . o . f . = 1.40$ ) to the ten nonzero data points included in the nonshaded area in the plot (from 3.92 to 4.12 GeV), which leads to a virtual state pole. The dash-dotted red line is obtained considering only the nine points from 3.94 GeV ( $χ^{2} / d . o . f . = 0.83$ ). The band around the solid line and the highlighted area in the top of each histogram shows the uncertainty obtained Monte Carlo propagating the errors from the fit to the data. The curves obtained from fits leading to a bound state pole, as discussed in the text, are almost indistinguishable from those shown here and have an almost identical $χ^{2} / d . o . f .$ . Top: pull of the data for the fit, defined as the difference between the heights of the experimental and theoretical histograms divided by the error of the former.
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Figure 2
Complete $H \bar{H}$ molecular spectrum obtained for both the S-I (top) and S-II (bottom) schemes and with a cutoff $Λ = 0.5 GeV$ . Quantum numbers $(I) J^{P C}$ are specified at the bottom of the plots. The filled rectangle bands, green or orange for bound or virtual states, respectively, cover the range of the pole positions given in Tables 2, 3, 4, 5 (second sets of errors). Thresholds are marked by dotted horizontal lines. The band closest to, but below, the threshold would correspond to an $H \bar{H}$ hadronic molecule, with quantum numbers indicated at the bottom. In the cases where the range of the pole position lies in both the bound and virtual regions, we show the two filled shorter bands. The input channels are signaled by a blue rectangle. The band width for the $X (3872)$ is multiplied by a factor of 10 for a better illustration. Note that, in some cases, the error for the pole position is so large that it exceeds the validity region of the NREFT treatment, and hence the bands fade away.
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Figure 3
SU(3) content of the pseudoscalar-vector $J^{P C} = 1^{+ -}$ molecular sector, deduced from the $[\bar{3}] \otimes [3] = [1] \oplus [8]$ reduction. See text for details.
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Figure 4
Pole-position histograms obtained from a $10^{4}$ MC sampling of the S-I LECs, including both the uncertainty from the physical inputs discussed in the text and that produced by HQSS and SU(3) flavor symmetry breaking effects. The cutoff has been fixed to $Λ = 0.5 GeV$ . See text for details.
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