Interpretation of Axial Resonances in J/psi-phi at LHCb

We suggest that the J/psi phi structures observed by LHCb can be fitted in two tetraquak multiplets, the S-wave ground state and the first radial excitation, with composition [cs][cbar sbar]. When compared to the previously identified [cq][cbar qbar] multiplet, the observed masses agree with what expected for a multiplet with q -->s. We propose the X(4274), fitted by LHCb with a single 1^++ resonance, to correspond rather to two, almost degenerate, unresolved lines with J^PC = 0^++, 2^++. Masses of missing particles in the 1S and 2S multiplets are predicted.

As we shall see shortly, masses and mass differences lead to classify the lowest lying structures, X(4140) and X(4274), in the ground state (1S) multiplet, while the two heavier ones, X(4500) and X(4700) are attributed to the first radially excited (2S) multiplet.
Members of a tetraquark multiplet in S-wave differ for the arrangement of quark and antiquark spins and the spectrum is determined by spin-spin interactions, with couplings to be determined phenomenologically, as it happens for qq or qqq hadrons.
We denote by [cq] s=0,1 [cq ′ ]s =0,1 the S-wave tetraquarks with all possible spin quantum numbers. In the |s,s J basis we have the following states (we restrict to electri-cally neutral ones for simplicity 1 ) In the case of [cq][cq] states, X was identified with X(3872), and X (1,2) with Z(3900) and Z(4020), respectively [3]. It was shown in [5] that the ordering of the Z(3900) and Z(4020) masses could be simply explained with the hypohesis that the dominant spin-spin interactions in tetraquarks are those inside the diquark or the antidiquark. The ansatz explains why the Z state not degenerate with the X(3872) is the heaviest. In fact, under this hypothesis, the Hamiltonian simply counts the number of spin 1 in each diquark, and it is seen from (3) and (5) that X and X (1) have one spin 1 while X (2) has two spins 1 and therefore it is heavier.
A further (still untested) consequence is that all states originating from the |1, 1 configurations, namely X (2) , X ′ 0 and X 2 , should be degenerate in mass. Of course, the spin-spin couplings referring to different diquarks are not expected to vanish exactly, as indicated by the fact that X(3872) and Z(3900) are not exactly degenerate. Improving over the simple picture just described will however have to wait the identification of other members of the multiplet, to fix the subdominant spin-spin couplings 2 . It would be interesting to obtain information on spin-spin couplings from non-perturbative QCD methods, e.g. from lattice QCD studies like those in presented in [7][8][9].
It is not difficult to see that the spectrum of the Swave 1S ground states is characterised by two quantities, Fig. 1: the diquark mass, m [cq] (or m [cs] for J/ψ φ resonances), and the spin-spin interaction inside the diquark or the antidiquark, κ cq (κ cs ). The first radially excited, 2S-states are shifted up by a common quantity, the radial excitation energy, ∆E r , which is expected to be mildly dependent on the diquark mass [6]: we expect E r (cq) ∼ E r (cs). volved. A simple explanation of the dominance of inter-diquark interaction could be that diquarks and antidiquarks are at such relative distance in the hadron, as to suppress the overlap probability, unlike what happens, e.g., in the usual baryons. With this, we can predict all particles in the 1S and 2S multiplets. We predict the lower 0 ++ and higher 0 ++′ states to be at The lower 0 ++ state would not show up in the LHCb spectrum, being below the J/ψ φ threshold.
The higher mass 1S states, 0 ++ and 2 ++ , are close to the structure observed by LHCb at 4274 MeV. LHCb fits the 4274 structure with a single resonance and finds J P C = 1 ++ at 5σ [1]. This attribution is not compatible with the tetraquark model, which admits only one J P C = 1 ++ state. Rather, we would like to propose three alternative options for the structure at 4274 MeV 1. J P C = 0 ++ 2. J P C = 2 ++ 3. two unresolved, approximately degenerate, lines with J P C = 0 ++ and J P C = 2 ++ What we prefer in the third option is that, in that case, LHCb would have seen all the accessible C = +1, 1S states. A further experimental study of the structure at 4274 MeV, with respect to the three options presented above, would add valuable information.

I. CONCLUSIONS
In conclusion, the J/ψ φ structures observed by LHCb can be fitted in two tetraquak multiplets, the S-wave ground state and the first radial excitation. When compared to the previous [cq] [cq] multiplet, the observed masses agree well 3 with what expected for a multiplet with q → s.
The hypothesis is however inconsistent with the attribution of the X(4274) structure to a single 1 ++ resonance. Rather we propose this structure to correspond to two, almost degenerate, unresolved lines with J P C = 0 ++ , 2 ++ , an hypothesis which may not be in conflict with the present analysis. If this solution would be supported by a more detailed analysis, LHCb would have seen, in a single experiment, all possible 1S-wave states with C = +1 (since the lowest 0 ++ is predicted to be below threshold) and the beginning of the 2S multiplet.
In addition 1. Two 1 +− states should be observed very close in mass to X(4140) and X(4274) respectively.
2. Radial excitations with 1 +− quantum numbers should follow by the assignment of the observed X(4500) and X(4700) as the radial excitations of X(4140) and X(4274) respectively.
The discovery of C = +1 structures calls for an exploration of C = −1 channels and of other C = +1 channels, to survey different options of the heavy quark spin, S cc . Channels of choice could be C = −1 χ cJ φ (S cc = 1), η c φ (S cc = 0) C = +1 h c φ (S cc = 0) (18) An alternative view on C = −1 states is found in [11].
We thank A. Ali and S. Stone for interesting exchanges. ADP acknowledges fruitful collaboration with A. Esposito and A. Pilloni.