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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$\overline \Delta$ by$B_s$ meson with different molecular state configurations of$ND_s $ or$ND ^ * _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$B_s$ 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 \sim200 $ MeV) and binding energy ε($5,20,50 $ MeV), the authors can find the branching ratio of the production$\overline B_s \to P_ {c\bar {s}}\overline \Delta $ . Under the configuration of$ND_s$ molecule, the branching ratio of the Cabibbo allowed process$\overline B_s \rightarrow P_{c \bar{s}} \overline \Delta$ can reach to order of$10^{-5}$ . Moreover, the production branching ratio of$ND^*_s$ 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 $B_s$ 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^{(*)}_s $分子态构型的单粲味五夸克态的自能图
Figure 1. The self-energy diagram of singly charm pentaquark as hadronic molecules $ ND^{(*)}_s $.
图 2 $ \overline B_s $介子产生单粲五夸克的三角图. (a-b) 具有$ ND_s $分子态构型的单粲五夸克; (c-d) 具有$ ND^*_s $分子态构型的单粲五夸克
Figure 2. The triangle diagrams of singly charm pentaquark produced by $ \overline B_s $ meson. (a-b) singly charm pentaquark with $ ND_s $ molecular state configuration; (c-d) singly charm pentaquark with $ ND^*_s $ molecular state configuration.
图 3 $ \overline B_s $介子弱衰变过程的W发射图
Figure 3. The W emission diagram of $ \overline B_s $ meson weak decay
图 4 $ \overline B_s \xrightarrow[]{N} P_{c \bar{s}} \overline \Delta $的分支比随参数α的变化曲线 (a) $ P_{c \bar{s}} $为$ ND_s $分子态; (b) $ P_{c \bar{s}} $为$ N{D^*_s} $分子态
Figure 4. The branching ratios of $ \overline B_s \xrightarrow[]{N} P_{c \bar{s}} \overline \Delta $ vary with the parameter α: (a) $ P_{c \bar{s}} $ as hadronic molecule $ ND_s $; (b) $ P_{c \bar{s}} $ as hadronic molecule $ ND^*_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_{pole} $[33,34]
Table 1. The fitted parameters $ a_i $ and pole mass $ m_{pole} $ of form factors $ F_{1}(k^2) $, $ F_{2}(k^2) $ and $ A_i(k^2) $(i = 1, 2, 3).
参数 $ {\overline B_s\to D} $ $ {\overline B_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_{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. The 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_s $ $ \overline B_s \xrightarrow[]{N} P_{c \bar{s}} \overline \Delta $ 29.40 31.37 24.51 $ \overline B_s\xrightarrow[]{N} P_{c \bar{s}}(\to \Lambda_c K) \overline \Delta $ $ 0.223 $ 0.194 0.137 $ ND^*_s $ $ \overline B_s\xrightarrow[]{N} P_{c \bar{s}} \overline \Delta $ 0.055 $ 0.408 $ 1.570 $ \overline B_s\xrightarrow[]{N} P_{c \bar{s}}(\to \Lambda_c K) \overline \Delta $ $ 0.0006 $ 0.0041 0.0157 $ \overline B_s\xrightarrow[]{N} P_{c \bar{s}}(\to \Sigma_c K) \overline \Delta $ $ 0.0004 $ 0.0024 0.0072 $ \overline B_s\xrightarrow[]{N} P_{c \bar{s}}(\to p D_s) \overline \Delta $ 0.0002 0.0015 0.0050
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