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

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 $4376$ $\rm{MeV}$.