Two-body strong decays of the $P_{cs}(4338)$ as a hidden-charm pentaquark state via the QCD sum rules

Technical Summary: Two-Body Strong Decays of the $P_{cs}(4338)$ as a Hidden-Charm Pentaquark via QCD Sum Rules

Problem Statement
The LHCb collaboration observed a narrow resonance, $P_{cs}(4338)$, in the $J/\psi\Lambda$ system with a mass of $4338.2\pm 0.7\pm 0.4\,\mathrm{MeV}$ and a width of $7.0\pm 1.2\pm 1.3\,\mathrm{MeV}$. While its mass is close to the $\bar{D}\Xi_{c}$ threshold, suggesting a molecular state interpretation, mass alone is insufficient for unambiguous identification. Theoretical literature offers competing interpretations, including molecular states, diquark-diquark-antiquark pentaquark states, and threshold cusps. Previous studies using QCD sum rules have yielded inconsistent results regarding the decay widths and branching ratios of $P_{cs}(4338)$, often failing to rigorously account for higher resonances and continuum states in the three-point correlation functions. This work aims to resolve these ambiguities by calculating the hadronic coupling constants and partial decay widths of $P_{cs}(4338)$ assuming it is a compact $[us][dc]\bar{c}-[ds][uc]\bar{c}$ type pentaquark state with $J^{P}=\frac{1}{2}^{-}$.

Methodology
The authors employ the method of three-point QCD sum rules to study the strong decays $P_{cs}(4338) \to \eta_{c}\Lambda$, $\bar{D}_{s}\Lambda_{c}$, and $J/\psi\Lambda$.

1. Currents and Correlation Functions: The study utilizes a specific five-quark current of the diquark-diquark-antiquark type to interpolate the $P_{cs}(4338)$ state. Three-point correlation functions are constructed involving the pentaquark current and currents for the final state mesons ($\eta_{c}, \bar{D}_{s}, J/\psi$) and baryons ($\Lambda, \Lambda_{c}$).
2. Rigorous Quark-Hadron Duality: A critical methodological advancement in this work is the implementation of "rigorous quark-hadron duality." Unlike standard approaches that often neglect excited states, the authors introduce free parameters ($C$) to parameterize the transitions between the ground state and higher resonances/continuum states. This allows for the subtraction of contaminations from excited states, ensuring a more robust extraction of ground-state coupling constants.
3. Operator Product Expansion (OPE): On the QCD side, the OPE is carried out up to dimension 13 vacuum condensates, consistently including terms proportional to the strange quark mass ($m_s$) to account for $SU(3)$ flavor symmetry breaking. The spectral densities are calculated up to order $\mathcal{O}(\alpha_s)$.
4. Borel Transformation and Sum Rules: Double Borel transformations are applied to the variables $p^2$ and $q^2$. The authors optimize the subtraction constants $C$ (parameterized as linear functions of the Borel parameter $T^2$) to obtain stable "flat platforms" in the Borel windows, ensuring the reliability of the extracted hadronic coupling constants.

Key Contributions

• Application of Rigorous Duality: The paper applies a refined scheme for three-point QCD sum rules that explicitly subtracts contributions from higher resonances and continuum states via optimized free parameters, a method previously established by the authors for other exotic states but applied here to the $P_{cs}(4338)$ for the first time in this specific configuration.
• Comprehensive Decay Analysis: The study calculates the hadronic coupling constants for three distinct decay channels ($\eta_{c}\Lambda$, $\bar{D}_{s}\Lambda_{c}$, and $J/\psi\Lambda$) within a single consistent framework, rather than treating them in isolation.
• Comparison of Scenarios: The results are explicitly compared with previous calculations based on molecular state scenarios and other theoretical models (e.g., Born-Oppenheimer approximation), highlighting differences in predicted total widths and branching ratios.

Results
The numerical analysis yields the following hadronic coupling constants:

• $|G_{P\eta_{c}\Lambda}| = 0.30 \pm 0.05$
• $|G_{P\bar{D}_{s}\Lambda_{c}}| = 0.31 \pm 0.05$
• $G_{V} = 0.40 \pm 0.06$
• $G_{T} = 0.20 \pm 0.01$

Based on these couplings, the partial decay widths are determined as:

• $\Gamma(P_{cs}(4338) \to \eta_{c}\Lambda) = 2.49 \pm 0.83\,\mathrm{MeV}$
• $\Gamma(P_{cs}(4338) \to \bar{D}_{s}\Lambda_{c}) = 3.25 \pm 1.05\,\mathrm{MeV}$
• $\Gamma(P_{cs}(4338) \to J/\psi\Lambda) = 4.74 \pm 1.19\,\mathrm{MeV}$

Summing these channels, the total width is calculated to be $\Gamma(P_{cs}(4338)) = 10.48 \pm 1.79\,\mathrm{MeV}$. This result is consistent with the experimental value of $7.0 \pm 1.2 \pm 1.3\,\mathrm{MeV}$ reported by LHCb within uncertainties. The hierarchy of partial widths is found to be $\Gamma(J/\psi\Lambda) > \Gamma(\bar{D}_{s}\Lambda_{c}) > \Gamma(\eta_{c}\Lambda)$.

Significance and Claims
The authors claim that the calculated total width supports the assignment of $P_{cs}(4338)$ as a compact $[us][dc]\bar{c}-[ds][uc]\bar{c}$ type pentaquark state with $J^{P}=\frac{1}{2}^{-}$, rather than a molecular state. They note that while molecular scenarios can also reproduce the total width, the specific hierarchy of partial decay widths and the ratio $\Gamma(\bar{D}_{s}\Lambda_{c})/\Gamma(J/\psi\Lambda) \approx 0.69$ derived in this pentaquark scenario differ from those predicted in molecular models (e.g., a ratio of $\approx 0.18$ in a previous molecular study by the same authors). The paper concludes that precise future measurements of the total width and, crucially, the branching fractions of these decay channels are necessary to definitively distinguish between the pentaquark and molecular interpretations. The work reinforces the utility of rigorous quark-hadron duality in QCD sum rules for diagnosing the nature of exotic hadrons.