Isospin breaking decays as a diagnosis of the hadronic molecular structure of the $P_c(4457)$

The LHCb Collaboration announced the observation of three narrow structures consistent with hidden-charm pentaquark states. They are candidates of hadronic molecules formed of a pair of a charmed baryon and an anticharmed meson. Among them, the $P_c(4457)$ mass is consistent with earlier predictions of a $\Sigma_c\bar D^*$ molecule with $I=1/2$. We point out that if such a picture were true, one would have $\mathcal{B}(P_c(4457)\to J/\psi \Delta^+)/\mathcal{B}(P_c(4457)\to J/\psi p)$ at the level ranging from a few percent to about 30%. Such a large isospin breaking decay ratio is two to three orders of magnitude larger than that for normal hadron resonances. It is a unique feature of the $\Sigma_c\bar D^*$ molecular model, and can be checked by LHCb.

Since the isospin of the Σ c is 1 and that of theD * is 1/2, one can form I = 3/2 and I = 1/2 states out of them, In the Σ cD * molecular picture, the decays of the P c (4457) + into the J/ψp and J/ψ∆ + dominantly proceed through the Σ cD * loops with the intermediate states carrying different electric charges, as shown in Fig. 1. We denote the S-wave coupling constant for the P c (4457) + → Σ + cD * 0 vertex as g +,0 and that for the P c (4457) + → Σ ++ c D * − vertex as g ++,− . Assuming the P c (4457) to be a hadronic molecule generated from the I = 1/2 S-wave interaction between the Σ cD * pair, one gets from Eq. (3) Then from Fig. 1 one sees that in the isospin limit when all the masses in the same isospin multiplet are degenerate, the two loops exactly cancel with each other for the decay into the I = 3/2 final state J/ψ∆ + . The isospin splittings of the intermediate particles make the 3 Several interesting similarities between the P c (4450) and the X(3872), including the possibility of a sizeable isospin symmetry breaking, were discussed in Ref. [16]. 4 transition possible. In order to estimate the size of the isospin breaking effect, we make use of the method of Ref. [49] which was developed for the X(3872) (see also Refs. [50,51]).
The magnitudes of the three-momenta for the decays of the P c (4457) + into J/ψp and J/ψ∆ + are about 0.83 GeV and 0.52 GeV, respectively. They are much larger than the binding momenta which are 73 MeV and 124 MeV for the Σ + cD * 0 and Σ ++ c D * − , respectively (here the central values of all involved masses are used). Thus, these decays are shortdistance processes, and the decay rates would be determined by the wave function at the origin.
The wave function at the origin for the a two-body component (labeled by i) of a physical state with a mass M is given by where a Gaussian form factor with a cutoff Λ is introduced to regularize the ultraviolet divergence, and G Λ i is simply the nonrelativistic two-point scalar loop integral evaluated at the mass of the state. When M < m 1 + m 2 , it is given by where erf (x) is the error function.
Thus, for the P c (4457) + we have ψ Λ ++,− (r = 0) = g ++,− G Λ ++,− , and ψ Λ +,0 (r = 0) = g +,0 G Λ +,0 , for the Σ ++ c D * − and Σ + cD * 0 components, respectively. From Eq. In view that the ∆ resonances and the nucleons are in the same spin-flavor multiplet in the large N c limit (see, e.g., Ref. [52]), one gets the following relation for the decay amplitudes where the factor of √ 10 comes from the spin-flavor matrix elements worked out in Ref. [16] (see Eqs. (17,18) therein), and Eq. (4) has been used. From this equation, and taking into account the S-wave phase spaces for the decays of the P c (4457) + into the J/ψp and J/ψ∆ + , one can predict the isospin breaking ratio and the result is shown in Fig. 2 with the cutoff Λ in the region from 0.5 GeV to 1 GeV. One sees that the ratio ranges from a few percent to as large as 30% with the large uncertainty mainly from the uncertainty of the P c (4457) + mass. It is two to three orders of magnitude larger than the isospin breaking effects for the decays of normal hadron resonances. In order to see that, one notices that there are two sources of isospin breaking: the up and down quark mass difference, and the electromagnetic interactions (virtual photons). They give amplitudes of the order of (m d − m u )/Λ QCD and α, respectively, where Λ QCD is the 6 nonperturbative scale in quantum chromodynamics and α is the fine structure constant.
Both of them are of O(10 −2 ), and thus lead to a suppression for the branching fractions of O(10 −4 ). To give an example, the ratio of the branching fraction of the decay of an isoscalar state into another isoscalar and a π 0 over that into an isoscalar and an η is given by 2 π 0 η up to the phase space factor. The isospin breaking π 0 -η mixing angle is where the combinations of meson masses are constructed such that the virtual photon effects are cancelled out.
To summarize, in this paper we propose that the structure of the P c (4457) can be diagnosed using isospin breaking decays. If the P c (4457) + is an S-wave Σ cD * hadronic molecule with I = 1/2, which implies that it couples most strongly to the Σ cD * channels, then because its mass is closer to the Σ + cD * 0 threshold than to the Σ ++ c D * − one, one expects large isospin breaking effects in its decays. A quantitative estimate of the ratio B(P c (4457) + → J/ψ∆ + )/B(P c (4457) + → J/ψp) gives a value ranging from O(10 −2 ) to about 30%, where the large uncertainty comes mainly from the mass of the P c (4457) + . It is two to three orders of magnitude higher than the isospin breaking effects for the decays of normal hadron resonances. It is worthwhile to mention that the large isospin breaking effect is a key to unveiling the nature of the D * s0 (2317), whose isospin breaking decay width is about 100 keV [53][54][55][56][57] in the DK molecular picture and is one order of magnitude smaller [58,59] if it couples weakly to the DK (for detailed discussions, see Ref. [42]). Therefore, we suggest to search for the P c (4457) + (P c (4457) 0 ) in the J/ψ∆ + (J/ψ∆ 0 ) mode. Given the large ratio, it is feasible at the LHCb experiment.