Constraining the hidden-charm pentaquark predictions and discriminating the $P_c(4440)$ and $P_c(4457)$ spins through the effective range expansion

TL;DR

This paper uses effective field theory and the effective range expansion to analyze the nature of hidden-charm pentaquarks, constraining their properties and distinguishing their spins, with implications for understanding their molecular structure.

Contribution

It introduces a systematic EFT approach to connect scattering parameters with pentaquark properties, providing constraints on their molecular interpretations and spin assignments.

Findings

Positive effective ranges of natural size support molecular interpretations.

Predicted spin assignments for $P_c(4440)$ and $P_c(4457)$ based on effective range analysis.

Estimated mass of the broad $P_c(4380)$ resonance as approximately 4376 MeV.

Abstract

The Weinberg compositeness criterion dictates that a pure shallow bound state is characterized by a large scattering length $a_0\gg\mathcal{O}(1/\beta)$ and a positive effective range $r_0$ that naturally scales to the size of $\mathcal{O}(1/\beta)$, where $1/\beta$ signifies the interaction range. In constructing the contact-range effective field theory (EFT) up to the next-to-leading order to describe the pentaquarks $P_c(4312)$, $P_c(4440)$, and $P_c(4457)$ observed by the LHCb collaboration in 2019, we match the effective range $r_0$ at single-channel situation for these pentaquarks with the low-energy couplings within the EFT framework. Three different schemes are used to connect the couplings with the effective range. We find positive effective ranges $r_0$ of the natural size of $\mathcal{O}(1/\beta)$ for the spin configurations $J^P=\frac{3}{2}^-$ for $P_c(4440)$ and $J^P=\frac{1}{2}^-$ for $P_c(4457)$ within the molecular $\bar{D}^* \Sigma_c$ description. Additionally, predictions from the power counting for low-energy couplings or Wilsonian coefficients suggest that, under heavy quark spin symmetry, the broad $P_c(4380)$ resonance, discovered by the LHCb collaboration in 2015, when considered as part of the single-channel $\bar{D}^{(*)} \Sigma_c^{(*)}$ molecular system alongside $P_c(4312)$, $P_c(4440)$, and $P_c(4457)$, has a mass of approximately $4376$ $\rm{MeV}$.