 # Joint Physics Analysis Center JPAC acknowledges support from the U.S. Department of Energy, the U.S. National Science Foundation and the Comunidad de Madrid

# $P_c(4312)^+$

The animations show the behavior of the poles in the two amplitudes cases described in [Fer19a] . In case A, it is possible to identify a cluster of virtual state poles across the II and IV sheet above the $\Sigma_c^+ \bar{D}^0$ threshold. If we use the customary definition of mass and width, $M_P = \text{Re } \sqrt{s_p}$, $\Gamma_P = -2\text{Im } \sqrt{s_p}$ the main cluster has $M_P = 4319.7 \pm 1.6\text{ MeV}$, $\Gamma_P = -0.8 \pm 2.4\text{ MeV}$, where positive or negative values of the width correspond to II or IV sheet poles, respectively. To establish the nature of this singularity, we track down the movement of the poles as the coupling between the two channels is reduced. By taking $m_{12} \to 0$, we can see how the cluster moves over to the upper side of the IV sheet and ends up on the real axis below the $\Sigma_c^+ \bar{D}^0$ threshold. The fraction of poles that reach the real axis from the lower side of the II sheet is $0.7\%$ only, and thus not significant. This result reinforces the interpretation of the pole as an unbound virtual state, meaning that the binding between the $\Sigma_c^+$ and the $\bar D^0$ is insufficient to form a molecule. In case B, the poles on the II sheet accumulate in two clusters. The one closer to threshold has $M_P = 4319.8 \pm 1.5\text{~MeV}$ and $\Gamma_P = 9.2 \pm 2.9\text{MeV}$, and it is the one responsible for the $P_c(4312)^+$ signal. As we did for case A, we study the motion of the poles as the channels decouple. The lighter cluster migrates onto the IV sheet (see the two animations), where it hits the zero of $T_{11}(s)$, and annihilates when $m_{12} = 0$. Since the pole does not survive the decoupling, it is entirely due to the interaction between the two channels, and its motion to the furthest unphysical sheet also suggest a virtual state nature for the $P_c(4312)^+$ as in case A. The other poles, which are located further away from the $\Sigma_c^+ \bar{D}^0$ threshold can also be interpreted. There are two clusters of poles on the III sheet, one far above the $\Sigma_c^+ \bar{D}^0$ threshold, the other below. The former could correspond to a resonance with standard Breit-Wigner lineshape as it appears to originate from the broad bump in the mass spectrum centered at $\sqrt{s} \sim 4.37\text{ GeV}$. As $m_{12} \to 0$, the two III sheet clusters move close to each other, and the heavier one disappears, multiplied by the amplitude zeros when $m_{12} = 0$. In this uncoupled limit, only one pole per channel is left. Furthermore, as the channels close, which is achieved by replacing $ik_i \to \lambda i k_i$ and letting $\lambda \to 0$, the poles move onto the real axis. It is worth noting that the fit chooses almost identical values for the ratios $m_{11}/c_{11}\simeq m_{22}/c_{22}$. This ratio determines the independent positions of the bare poles on the real axis in the two uncoupled channels and being equal, suggests existence of a single compact pentaquark.  We also performed a study of the three-channel case, including the $\Sigma_c^{++}D^-$ threshold. To simplify the approach we work in the scattering length approximation (as in case A) and in the isospin limit for the fitting parameters. The result of the fit and the pole positions are shown here. We find a single pole close to the $\Sigma_c^+ \bar{D}^0$ threshold on the II sheet. No other pole appears close to the physical axis above the $\Sigma_c^{++}D^-$ threshold. When the couplings between the channels are reduced, the pole quickly moves far to the left (on different sheets depending on the specific solution considered, as shown here), and cannot interpreted as a physical state. We therefore conclude that the $P_c(4312)^+$ signal could be a result of a complicated interplay of thresholds and feeble $\Sigma_c \bar D$ interactions.   ## References

[Fer19a]
C. Fern´ndez-Ramírez, A. Pilloni, M. Albaladejo, A. Jackura, V. Mathieu, M. Mikhasenko, J. Silva-Castro, and A. Szczepaniak, Interpretation of the LHCb Pc(4312) Signal,'' arXiv:1904.10021 [hep-ph]