How to reveal the exotic nature of the $P_c(4450)$

The LHCb Collaboration announced two pentaquark-like structures in the $J/\psi p$ invariant mass distribution. We show that the current information on the narrow structure at 4.45 GeV is compatible with kinematical effects of the rescattering from ${\chi }_{c1}p$ to $J/\psi p$: First, it is located exactly at the ${\chi }_{c1}p$ threshold. Second, the mass of the four-star well-established $\mathrm{\Lambda }(1890)$ is such that a leading Landau singularity from a triangle diagram can coincidentally appear at the ${\chi }_{c1}p$ threshold, and third, there is a narrow structure at the ${\chi }_{c1}p$ threshold but not at the ${\chi }_{c0}p$ and ${\chi }_{c2}p$ thresholds. In order to check whether that structure corresponds to a real exotic resonance, one can measure the process ${\mathrm{\Lambda }}_b^0\to K^{-}{\chi }_{c1}p$. If the $P_c(4450)$ structure exists in the ${\chi }_{c1}p$ invariant mass distribution as well, then the structure cannot be just a kinematical effect but is a real resonance; otherwise, one cannot conclude that $P_c(4450)$ is another exotic hadron. In addition, it is also worthwhile to measure the decay $\mathrm{\Upsilon }(1S)\to J/\psi p\overline{p}$: a narrow structure at 4.45 GeV but not at the ${\chi }_{c0}p$ and ${\chi }_{c2}p$ thresholds would exclude the possibility of a pure kinematical effect.

Figure 1

Two-point and three-point loops for the mechanism of the ${\chi }_{c1}p\to J/\psi p$ rescattering in the decay ${\mathrm{\Lambda }}_b^0\to K^{-}J/\psi p$.

Figure 2

Fit to the real and imaginary parts of the $P_c(4450)$ amplitude shown in Fig. 9 of Ref. [10] with Eq. (6). The blue curve represents the best fit. It is counterclockwise with increasing $J\psi p$ invariant mass from 4.41 GeV to 4.49 GeV, the same range as for the LHCb diagram.

Figure 3

Absolute values of amplitudes in arbitrary units: Panel (a) is for the amplitude in Eq. (6) fitted to the Argand plot; (b) is for the triangle loop integral with the ${\chi }_{c1}p$ vertex in a $P$ wave. In (b), we assume the $\mathrm{\Lambda }(1890)$ with a mass of 1.89 GeV is exchanged in the triangle diagram. The solid, dashed and dotted lines correspond to a width of the $\mathrm{\Lambda }(1890)$ of 10, 60 and 100 MeV, in order.

Figure 4

Left panel: Motion of the two triangle singularities in the complex plane of $\sqrt{s}=M_{{\chi }_{c1}p}=M_{J/\psi p}$ with respect to changing the mass of the exchanged ${\mathrm{\Lambda }}^{*}$ baryon (several values are labeled in the plot in units of GeV). In order to distinguish the trajectories from the real axis, we put a small imaginary part, $-5\mathrm{MeV}$ corresponding to a width of 10 MeV, to $M_{{\mathrm{\Lambda }}^{*}}$. Only the part between the two filled triangles, labeled as A and B, has a large impact on the physical amplitude. The thick solid straight line represents the unitary cut starting from the ${\chi }_{c1}p$ threshold. Right panel: The corresponding Dalitz plot which shows the region between A and B.