Isospin breaking decays as a diagnosis of the hadronic molecular structure of the $P_c(4457)$
Feng-Kun Guo Hao-Jie Jing Ulf-G. Meißner Shuntaru Sakai




[E-mail address: ]fkguo@itp.ac.cn


[E-mail address: ]jinghaojie@itp.ac.cn

[E-mail address: ]meissner@hiskp.uni-bonn.de

[E-mail address: ]shsakai@itp.ac.cn



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 Σ_cD̅^* molecule with I=1/2. We point out that if such a picture were true, one would have ℬ(P_c(4457)→ J/ψΔ^+)/ℬ(P_c(4457)→ J/ψ 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  molecular model, and can be checked by LHCb.





Isospin breaking decays as a diagnosis of the hadronic molecular structure of the 
    Shuntaro Sakai
    December 30, 2023
==================================================================================








Four years after the discovery of the hidden-charm pentaquark-like states P_c(4380) and P_c(4450) <cit.>, with the full set of Run-1 and Run-2 data the LHCb Collaboration announced  the observation of more structures consistent with hidden-charm pentaquark states with masses and widths given by <cit.>

    M_   = 4311.9±0.7^+6.8_-0.6 ,    Γ_   = 9.8±2.7^+3.7_-4.5 , 
    
     M_   = 4440.3±1.3^+4.1_-4.7 ,    Γ_   = 20.6±4.9^+8.7_-10.1 ,
    
     M_   = 4457.3±0.6^+4.1_-1.7 ,    Γ_   = 6.4±2.0^+5.7_-1.9 .

That is, the P_c(4450) reported earlier is split into two peaks corresponding to the  and the , and the small spike (sticking out in a single bin) at slightly above 4.3 GeV in the 2015 measurement is resolved into a pronounced peak with a 7.3σ significance. Pentaquark states with hidden-charm as hadronic molecules of a pair of a charmed baryon and an anticharmed meson were predicted to exist in this mass region prior to the LHCb observations <cit.>. In particular, the masses of the  and  are in a remarkable agreement with the predictions for the isospin I=1/2 Σ_cD̅ (J^P=1/2^-) and Σ_cD̅^* (J^P=1/2^- or 3/2^-) S-wave bound states in Ref. <cit.> where a coupled-channel formalism with the vector-meson exchange potential is used. 
The first observation in Ref. <cit.> inspired a flood of models for the P_c structures, such as the baryon–meson
molecules <cit.>, compact pentaquark 
states <cit.>  and
baryocharmonia <cit.>, while the importance of triangle singularities, in particularly for the P_c(4450), has
also been discussed <cit.>.[It could be that the triangle singularities enhance the production of the P_c states at around 4.45 GeV.] Reviews of these models can be found in Refs. <cit.>. Of particular interest here is the interpretation of the P_c(4450) as an I=1/2  molecular with  J^P=3/2^- <cit.> (see also early predictions in Ref. <cit.>), which is adopted as the interpretation for the  in Refs. <cit.>. We notice that such an interpretation will lead to large isospin breaking effects in the decays. We have the following nearby Σ_cD̅^* thresholds:[As noticed in Ref. <cit.>, the mass of the  coincides with the Λ_c(2595)^+D̅^0 threshold, 4457.09±0.28 MeV. ]

    M_Σ_c^+ + M_D̅^*0 = 4459.8±0.4 ,      
        M_Σ_c^++ + M_D^*- = 4464.23±0.15 .

Thus, the binding energy of the  with respect to the  threshold, 2.5^+1.8_-4.2 MeV, is sizably smaller than that with respect to the  threshold, 6.9^+1.8_-4.1 MeV. As a result, one would expect sizeable isospin breaking effects in the decays, similar to the case of the X(3872) which decays with comparable rates into the I=0 J/ψπ^+π^-π^0 and I=1 J/ψπ^+π^-  final states,[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. <cit.>.] though with a much more modest magnitude as will be shown in the following.

Since the isospin of the Σ_c is 1 and that of the D̅^* is 1/2, one can form I=3/2 and I=1/2 states out of them,

    | ; I=1/2,I_3=1/2⟩   = √(2/3)| ⟩ - 1/√(3)| ⟩, 
    | ; I=3/2,I_3=1/2⟩   = 1/√(3)| ⟩ + √(2/3)| ⟩.

In the  molecular picture, the decays of the  into the  and  dominantly proceed through the  loops with the intermediate states carrying different electric charges, as shown in Fig. <ref>.



We denote the S-wave coupling constant for the → vertex as g_+,0 and that for the  → vertex as g_++,-. Assuming the  to be a hadronic molecule generated from the I=1/2 S-wave interaction between the  pair, 
from the Lippmann-Schwinger equation (LSE), we have 

    T^-1_1/2 = V_1/2^-1-  G_^Λ = 0

when the energy equals to the mass of the P_c. Here, T_1/2 is the I=1/2  scattering T-matrix, V_1/2 is the corresponding nonrelativistic potential, and G_^Λ is the  two-body Green's function whose form is irrelevant here (it will be given below when it is used). In G_^Λ, the isospin averaged masses for the Σ_c and D̅^* should be used. Now let us switch on isospin breaking and consider the two-channel (,) nonrelativistic system. Because the products of couplings are the residues of the T-matrix elements, i.e., g_++,-^2=Res T_++,-→ ++,- and g_++,- g_+,0=Res T_++,-→ +,0, we get from the two-channel LSE the following ratio (the energy is at the P_c mass),

    g_++,-/g_+,0 = 2V_1/2 + V_3/2 - 3 V_1/2 V_3/2 G_+,0^Λ/-√(2)(V_1/2-V_3/2),

where V_3/2 is the potential for the I=3/2  scattering, G_+,0^Λ is the two-body Green's function for , and we have used Eq. (<ref>) to express the particle-basis potentials in terms of the isospin-basis ones V_1/2 and V_3/2. From Eq. (<ref>), we get V_1/2G_+,0^Λ = 1 - V_1/2(G_+,0^Λ - G_^Λ), where the second term is an isospin breaking effect and is much smaller than 1. Therefore, Eq. (<ref>) becomes

    g_++,-≃ -√(2) g_+,0 .

Then from Fig. <ref> 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 . The isospin splittings of the intermediate particles make the transition possible. In order to estimate the size of the isospin breaking effect, we make use of the method of Ref. <cit.> which was developed for the X(3872) (see also Refs. <cit.>).

The magnitudes of the three-momenta for the decays of the  into  and  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  and , respectively (here the central values of all involved masses are used).
Thus, these decays are short-distance processes, and the decay rates would be determined by the wave function at the origin.

The wave function at the origin for a two-body component (labeled by i) of a physical state with a mass M  is given by

    ψ_i(r= 0) =  ∫d^3q⃗/(2π)^3⟨q⃗ |ψ⟩ = -2μ_i ∫d^3q⃗/(2π)^3⟨q⃗ |V̂_i| ψ⟩/γ_i^2 + q⃗^ 2,

where we have used the Schrödinger equation (q⃗^ 2/(2μ_i) + V̂_i )|ψ⟩ = (M-m_1-m_2)|ψ⟩, and the binding momentum is defined as γ_i = √(2μ_i(m_1+m_2-M)),  with m_1,2 the masses of the constituents and μ_i=m_1 m_2/(m_1+m_2) the reduced mass. Since the physical state is nearby the threshold, one can approximate the S-wave vertex form factor ⟨q⃗ |V̂_i| ψ⟩ by the coupling constant g_i. Then, one gets

    ψ^Λ_i(r= 0) = -2μ_i g_i ∫d^3q⃗/(2π)^3exp(-2 q⃗^ 2/Λ^2)/γ_i^2 + q⃗^ 2≡ g_i G^Λ_i,

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 

    G^Λ_i = - μ_i Λ/(2π)^3/2 - μ_iγ_i/2π e^2γ_i^2 /Λ^2[ erf(√(2)γ_i /Λ) -1 ],

where erf (x) is the error function.

Thus, for the  we have

    ψ^Λ_++,-(r=0) = g_++,-G^Λ_++,- ,   and  ψ^Λ_+,0(r=0) = g_+,0G^Λ_+,0 ,

for the  and  components, respectively. 
From Eq. (<ref>), the isospin I=1/2 and I=3/2 components are 

    ψ^Λ_1/2 (r=0)     = √(2/3)ψ^Λ_++,-(r=0) - 1/√(3)ψ^Λ_+,0(r=0), 
    ψ^Λ_3/2 (r=0)     = 1/√(3)ψ^Λ_++,-(r=0) + √(2/3)ψ^Λ_+,0(r=0) .


In view that the Δ resonances and the nucleons are in the same spin-flavor multiplet in the large N_c limit (see, e.g., Ref. <cit.>), one gets the following relation for the decay amplitudes

    |(→)|/|(→)|≃√(10)| ψ^Λ_3/2(r=0) /ψ^Λ_1/2(r=0)| = 2√(5) | G^Λ_++,- - G^Λ_+,0/ 2G^Λ_++,- + G^Λ_+,0|,
 
where the factor of √(10) comes from the spin-flavor matrix elements worked out in Ref. <cit.> (see Eqs. (17,18) therein), and Eq. (<ref>) has been used. We have further assumed that V_3/2 is much smaller than V_1/2 so that we can neglect contribution from the isospin breaking effect in Eq. (<ref>) here. This is plausible in the molecular picture as the I=1/2 interaction needs to be strong to produce the P_c(4457) as a  bound state.[For the X(3872) in the DD̅^*+c.c. hadronic molecular picture, the I=1 potential is indeed much weaker than the I=0 one, see, e.g., Ref. <cit.>.] From this equation,
and taking into account the S-wave phase spaces for the decays of the  into the  and , 
one can predict the isospin breaking ratio

    R_Δ^+/p≡(→)/(→),

and the result is shown in Fig. <ref> 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  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 nonperturbative scale in quantum chromodynamics and α is the fine structure constant. Both of them are of 10^-2, and thus lead to a suppression for the branching fractions of 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 the same isoscalar and an η is given by ϵ_π^0η^2 up to the phase space factor. The isospin breaking π^0-η mixing angle is 

    ϵ_π^0η = √(3)/2m_d-m_u/2m_s - m_u-m_d≃√(3)/2 M_K^0^2 - M_K^±^2 - M_π^0^2 + M_π^±^2 / M_K^0^2 + M_K^±^2 - M_π^0^2 - M_π^±^2≃ 0.01  ,

where the combinations of meson masses are constructed such that the virtual photon effects are canceled out. 

To summarize, in this paper we propose that the structure of the  can be diagnosed using isospin breaking decays. If the  is an S-wave  hadronic molecule with I=1/2, which implies that it couples most strongly to the  channels, then because its mass is closer to the  threshold than to the  one, one expects large isospin breaking effects in its decays. 
A quantitative estimate of the ratio ℬ(→)/ℬ(→) gives a value ranging from 10^-2 to about 30%, where the large uncertainty comes mainly from the mass of the . 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 <cit.> in the DK molecular picture and is one order of magnitude smaller <cit.> if it couples weakly to the DK (for detailed discussions, see Ref. <cit.>). 
Therefore, we suggest to search for the  (^0) in the (J/ψΔ^0) mode. Given the large ratio, it is feasible at the LHCb experiment.







F.-K. G. is grateful for the hospitality of the Helmholtz-Institut für Strahlen- und       Kernphysik (HISKP) where this work was done.
This work is supported in part by the National Natural Science Foundation of China (NSFC) and  the Deutsche Forschungsgemeinschaft (DFG) through the funds provided to the Sino-German Collaborative Research Center “Symmetries and the Emergence of Structure in QCD"  (NSFC Grant No. 11621131001, DFG Grant No. TRR110), by the NSFC under Grant No. 11747601 and No. 11835015, by the Chinese Academy of Sciences (CAS) under Grant No. QYZDB-SSW-SYS013 and No. XDPB09, by
the CAS Center for Excellence in Particle Physics (CCEPP),  by the CAS President’s International Fellowship Initiative (PIFI) (Grant No. 2018DM0034 and No. 2019PM0108), and by the VolkswagenStiftung (Grant No. 93562).







