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基于有效拉氏量方法, 本文研究了自旋宇称为
$J^P={{1}/{2}}^{-}$ 的单粲味五夸克态的产生. 本文根据强子可能的分子态图像, 分别以$ND_{\mathrm{s}}$ 或$ND^*_{\mathrm{s}}$ 不同的分子态构型, 讨论了$B_{\mathrm{s}}$ 介子产生单粲味五夸克态${c\bar suud}$ 和十重态重子$\bar \varDelta$ , 以及该五夸克态的两体强衰变过程. 通过复合粒子判据, 计算出与单粲味五夸克态${c\bar suud}$ 相关的强耦合常数. 借助于强子的有效拉氏量方法, 最终得到了单粲味五夸克态的产生分支比. 结果表明, 在单粲味五夸克态${c\bar suud}$ 为$ND_{\mathrm{s}}$ 的构型下, 具有Cabibbo允许的产生过程:$\bar B_{\mathrm{s}} \rightarrow P_{{\mathrm{c }}\bar{{\mathrm{s}}}} \bar \varDelta$ 的分支比可以达到$10^{-5}$ 量级, 而在$ND^*_{\mathrm{s}}$ 的构型下, 该过程的分支比仅为$10^{-8}$ 量级. 本文的研究结果可以为单粲味五夸克态的实验搜寻和深入研究提供参考, 并期望在将来的实验探测诸如LHCb, Belle II, BaBar等B工厂中得到验证.In this work, the authors use the effective Lagrangian method to investigate the production of singly charm pentaquark state with spin parity $J ^ P={1/2}^{-} $ . Based on the possible molecular state images of hadrons, the author discusses the production of singly charm pentaquark state${c\bar suud}$ and decuplet baryon$\bar \varDelta$ by$B_{\mathrm{s}}$ meson with different molecular state configurations of$ND_{\mathrm{s}} $ or$ND ^ * _{\mathrm{s}} $ . To determine the coupling between pentaquark and their constituents in the molecular scheme, the authors follow the Weinberg compositeness condition to estimate the self-energy diagram of the singly charmed pentaquark. Further study on the production of pentaquark from$B_{\mathrm{s}}$ meson can be propeled by computing the transition matrix elements, or the triangle diagrams, which can be careful divided into two part subprocess, one associated with weak transition can be represented into form factor and decay constant, another one related to strong coupling of hadrons can be described by effective Lagrangian. Selecting the scale parameter α (10–200 MeV) and binding energy ε (5, 20, 50 MeV), the authors can find the branching ratio of the production$\bar B_{\mathrm{s}} \to P_ {{\mathrm{c}}\bar {{\mathrm{s}}}}\bar \varDelta $ . Under the configuration of$ND_{\mathrm{s}}$ molecule, the branching ratio of the Cabibbo allowed process$\bar B_{\mathrm{s}} \rightarrow P_{{{\mathrm{c}} \bar{{\mathrm{s}}}}} \bar \varDelta$ can reach to order of$10^{-5}$ . Moreover, the production branching ratio of$ND^*_{\mathrm{s}}$ molecule is only at the order of$10^{-8}$ .A increasing scale parameter α can significantly improve the production branching ratio of the singly charm pentaquark. In addition, the binding energy and the coupling constants will also affect the magnitude of production. Therefore, considering the above factors, the production branching ratio of singly charm pentaquark in $B_{\mathrm{s}}$ decays have considerable results, which is worth experimental and theoretical research in the future. The findings of our work can provide a reference for the experimental search and study of singly charm pentaquark, and it is hoped that they will be verified in future experimental detections at B factories such as LHCb, Belle, and BaBar.
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Keywords:
- singly charm pentaquark /
- branching ratio /
- effective Lagrangian
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图 1 具有$ ND^{(*)}_{\mathrm{s}} $分子态构型的单粲味五夸克态的自能图
Fig. 1. Self-energy diagram of singly charm pentaquark as hadronic molecules $ ND^{(*)}_{\mathrm{s}} $.
图 2 $ \bar B_{\mathrm{s}} $介子产生单粲五夸克的三角图 (a), (b)具有$ ND_{\mathrm{s}} $分子态构型的单粲五夸克; (c), (d) 具有$ ND^*_{\mathrm{s}} $分子态构型的单粲五夸克
Fig. 2. Triangle diagrams of singly charm pentaquark produced by $ \bar B_{\mathrm{s}} $ meson: (a), (b) Singly charm pentaquark with $ ND_{\mathrm{s }}$ molecular state configuration; (c), (d) singly charm pentaquark with $ ND^*_{\mathrm{s}} $ molecular state configuration.
图 3 $ \bar B_{\mathrm{s}} $介子弱衰变过程的W发射图
Fig. 3. W emission diagram of $ \bar B_{\mathrm{s}} $ meson weak decay
图 4 $ \bar B_{\mathrm{s}} \xrightarrow[]{N} P_{{\mathrm{c}} \bar{{\mathrm{s}}}} \bar \varDelta $的分支比随参数α的变化曲线 (a) $ P_{{\mathrm{c}} \bar{{\mathrm{s}}}} $为$ ND_{\mathrm{s}} $分子态; (b) $ P_{{\mathrm{c}} \bar{{\mathrm{s}}}} $为$ N{D^*_{\mathrm{s}}} $分子态
Fig. 4. Branching ratios of $ \bar B_{\mathrm{s}} \xrightarrow[]{N} P_{{\mathrm{c}} \bar{{\mathrm{s}}}} \bar \varDelta $ vary with the parameter α: (a) $ P_{{\mathrm{c}} \bar{{\mathrm{s}}}} $ as hadronic molecule $ ND_{\mathrm{s}} $; (b) $ P_{{\mathrm{c}} \bar{{\mathrm{s}}}} $ as hadronic molecule $ ND^*_{\mathrm{s}} $
表 1 形状因子$ F_{1}(k^2) $, $ F_{2}(k^2) $和$ A_i(k^2) $(i = 1, 2, 3)的拟合展开参数$ a_i $和$ m_{{\mathrm{pole}}} $[34,35]
Table 1. Fitted parameters $ a_i $ and pole mass $ m_{{\mathrm{pole}}} $ of form factors $ F_{1}(k^2) $, $ F_{2}(k^2) $ and $ A_i(k^2) $(i = 1, 2, 3)[34,35].
参数 $ {\bar B_{\mathrm{s}}\to D} $ $ {\bar B_{\mathrm{s}}\to D^*} $ $ F_1(k_1) $ $ F_2(k_1) $ $ A_{0}(k_1) $ $ A_{1}(k_1) $ $ A_{2}(k_1) $ $ A_{3}(k_1) $ $ a_{0} $ $ 0.666 $ $ 0.666 $ $ 0.100 $ $ 0.105 $ $ 0.055 $ $ 0.059 $ $ a_{1} $ $ -0.206 $ $ -3.236 $ $ -0.180 $ $ -0.430 $ $ -0.010 $ $ -0.110 $ $ a_{2} $ $ -0.106 $ $ -0.075 $ $ -0.006 $ $ -0.100 $ $ -0.030 $ $ -0.250 $ $ a_{3} $ $ 0.00 $ $ -0.00 $ $ 0.00 $ $ -0.030 $ $ 0.060 $ $ -0.050 $ $ m_{{\mathrm{pole}}} $/GeV $ — $ $ — $ $ 6.335 $ $ 6.275 $ $ 6.745 $ $ 6.745 $ 
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表 2 单粲味五夸克态的产生分支比(α = 100 MeV)
Table 2. Production branching ratio of singly charm pentaquark state (α = 100 MeV).
分子态 产生道 分支比($ \times 10^{-6} $) $ \varepsilon $/MeV 5 20 50 $ ND_{\mathrm{s}} $ $ \bar B_{\mathrm{s}} \xrightarrow[]{N} P_{{\mathrm{c}} \bar{{\mathrm{s}}}} \bar \varDelta $ 29.40 31.37 24.51 $ \bar B_{\mathrm{s}}\xrightarrow[]{N} P_{{\mathrm{c}} \bar{{\mathrm{s}}}}(\to \varLambda_{\mathrm{c}} K) \bar \varDelta $ 0.223 0.194 0.137 $ ND^*_{\mathrm{s}} $ $ \bar B_{\mathrm{s}}\xrightarrow[]{N} P_{{\mathrm{c}} \bar{{\mathrm{s}}}} \bar \varDelta $ 0.055 0.408 1.570 $ \bar B_{\mathrm{s}}\xrightarrow[]{N} P_{{\mathrm{c}} \bar{{\mathrm{s}}}}(\to \varLambda_{\mathrm{c}} K) \bar \varDelta $ 0.0006 0.0041 0.0157 $ \bar B_{\mathrm{s}}\xrightarrow[]{N} P_{{\mathrm{c}} \bar{{\mathrm{s}}}}(\to \Sigma_{\mathrm{c}} K) \bar \varDelta $ 0.0004 0.0024 0.0072 $ \bar B_{\mathrm{s}}\xrightarrow[]{N} P_{{\mathrm{c}} \bar{{\mathrm{s}}}}(\to p D_{\mathrm{s}}) \bar \varDelta $ 0.0002 0.0015 0.0050
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[1] Aaij R, Advea B, Adinolfi M, et al. 2015 Phys. Rev. Lett. 115 072001
Google Scholar
[2] Aaij R, Abellán Beteta C, Adeva B, et al. 2019 Phys. Rev. Lett. 122 222001
Google Scholar
[3] Aaij R, Abellán Beteta C, Ackernley T, et al. 2021 Sci. Bull. 66 1278
Google Scholar
[4] Aaij R, Abdelmotteleb A S W, Abellán Beteta C, et al. 2022 Phys. Rev. Lett. 128 062001
Google Scholar
[5] Santopinto E, Giachino A 2017 Phys. Rev. D 96 014014
Google Scholar
[6] Deng C R, Ping J L, Huang H X, Wang F 2017 Phys. Rev. D 95 014031
Google Scholar
[7] Azizi K, Sarac Y, Sundu H 2023 Phys. Rev. D 107 014023
Google Scholar
[8] Chen R, Liu X, Li X Q, Zhu S L 2015 Phys. Rev. Lett. 115 132002
Google Scholar
[9] Guo F K, Meißner Ulf-G, Wang W, Yang Z 2015 Phys. Rev. D 92 071502
Google Scholar
[10] Branz T, Gutsche T, Lyubovitskij V E 2021 Phys. Rev. D 104 114028
Google Scholar
[11] Lebed R F, Martinez S R 2022 Phys. Rev. D 106 074007
Google Scholar
[12] Zhang Y, He G Z, Ye Q X, Y D C, Hua J, Wang Q 2024 Chin. Phys. Lett. 41 021301
Google Scholar
[13] Chen H X, Chen W Z, Shi L 2019 Phys. Rev. D 100 051501
Google Scholar
[14] Liu M Z, Pan Y W, Peng F Z, Sánchez Sánchez M, Geng L S, Hosaka A, Pavon V M 2019 Phys. Rev. Lett. 122 242001
Google Scholar
[15] Zhu J T, Kong S Y, He J 2023 Am. Phys. Soc. 107 034029
Google Scholar
[16] Wu Q, Chen D Y 2019 Phys. Rev. D 100 114002
Google Scholar
[17] Peng F Z, Yan M J, Sánchez Sánchez M, Valderrama M P 2021 Eur. Phys. J. C 81 666
Google Scholar
[18] Xiao C W, Wu J J, Zou B S 2021 Phys. Rev. D 103 054016
Google Scholar
[19] Lu J X, Liu M Z, Shi R X, Geng L S 2021 Phys. Rev. D 104 034022
Google Scholar
[20] Wu Q, Chen D Y, Ji R 2021 Chin. Phys. Lett. 38 071301
Google Scholar
[21] 叶全兴, 何广朝, 王倩 2023 72 201401
Google Scholar
Ye Q X, He G C, Wang Q 2023 Acta Phys. Sin. 72 201401
Google Scholar
[22] Shi P P, Baru Vadim, Guo F K, Hanhart C, Nefediev A 2024 Chin. Phys. Lett. 41 031301
Google Scholar
[23] Li N, Xing Y, Hu X H 2023 Eur. Phys. J. C 83 1013
Google Scholar
[24] Huang Y, Xiao C J, Lü Q F, Wang R, He J, Geng L S 2018 Phys. Rev. D 97 094013
Google Scholar
[25] Zhu H Q, Ma N N, Huang Y 2020 Eur. Phys. J. C 80 1184
Google Scholar
[26] Yan Y, Hu X H, Huang H X, Ping J L 2023 Phys. Rev. D 108 094045
Google Scholar
[27] Xin Q, Yang X, Wang Z G 2023 Int. J. Mod. Phys. A 38 2350123
Google Scholar
[28] Yan M J, Peng F Z, Pavon V M 2024 Phys. Rev. D 109 014023
Google Scholar
[29] Steven W 1963 Phys. Rev. 130 776
Google Scholar
[30] Tanja B, Thomas G, Valery E L 2009 Phys. Rev. D 79 014035
Google Scholar
[31] Xiao C J, Huang Y, Dong Y B, Geng L S, Chen D Y 2019 Phys. Rev. D 100 014022
Google Scholar
[32] Shen C W, Wu J J, Zou B S 2019 Phys. Rev. D 100 056006
Google Scholar
[33] Yalikun N, Zou B S 2022 Phys. Rev. D 105 094026
Google Scholar
[34] McLean E, Davies C T H, Koponen J, Lytle A T 2020 Phys. Rev. D 101 074513
Google Scholar
[35] Harrison J D, Christine T H 2022 Phys. Rev. D 105 094506
Google Scholar
[36] Heng H Y 1997 Phys. Rev. D 56 2799
Google Scholar
[37] Thomas G, Mikhail A I, Jürgen G K, et al. 2015 Phys. Rev. D 91 074001
Google Scholar
[38] Wu S M, Wang F, Zou B S 2023 Phys. Rev. C 108 045201
Google Scholar
[39] Li H N, Lu C D, Yu F S 2012 Phys. Rev. D 86 036012
Google Scholar
[40] Xing Y, Xing Z P 2019 Chin. Phys. C 43 073103
Google Scholar
[41] Xu Y J, Cui C Y, Liu Y L, Huang M Q 2020 Phys. Rev. D 102 034028
Google Scholar
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