Interpretation of structure in the di-$J/\psi$ spectrum

Structure in the di-$J/\psi$ mass spectrum observed by the LHCb experiment around 6.9 and 7.2 GeV is interpreted in terms of $J^{PC}=0^{++}$ resonances between a $cc$ diquark and a $\bar c \bar c$ antidiquark, using a recently confirmed string-junction picture to calculate tetraquark masses. The main peak around 6.9 GeV is likely dominated by the $0^{++}(2S)$ state, a radial excitation of the $cc$-$\bar c \bar c$ tetraquark, which we predict at $6.871\pm 0.025$ GeV. The dip around 6.75 GeV is ascribed to the opening of the S-wave di-$\chi_{c0}$ channel, while the dip around 7.2 GeV could be correlated with the opening of the di-$\eta_c(2S)$ channel. Description of the low-mass part of the di-$J/\psi$ structure appears to require a low-mass broad resonance consistent with a predicted $0^{++}(1S)$ state with $M_{rm inv} = 6191.5 \pm 25$ MeV. Implications for $bb \bar b \bar b$ tetraquarks are discussed.

In this paper we adopt the compact tetraquark point of view (see [20,21] for lists of related predictions) and point out a feature in the data which is characteristic of many processes. We note that the position of the dip roughly coincides with twice the mass of χ c0 (3415). If the major resonant di−J/ψ activity is in the J P C = 0 ++ channel, a pair of χ c0 (3415) charmonia can be produced in an S-wave as soon as M inv (di−J/ψ ) exceeds 6829 MeV. Unitarity then can induce a dip in the production channel. (See also [11].) In Section II we recall a number of instances in which the opening of an S-wave channel induces a dip in the production channel. We apply similar methods to the S-wave process J/ψ J/ψ → χ c0 χ c0 and J/ψ J/ψ → η c (2S)η c (2S) in Section III, discuss implications for cccc tetraquarks in Section IV and for bbbb tetraquarks in Section V, concluding in Section VI. An Appendix contains details of resonance fitting.

II Dips and cusps in S-wave production channels
Dips or cusps in the cross section for a number of S-wave processes occur when a new Swave threshold is crossed. Here we review several such cases. More details and references may be found in Ref. [22].
A ππ I = J = 0 amplitude at KK threshold The rapid drop in the magnitude of the I = 0 S-wave ππ scattering amplitude near a center-of-mass energy E cm ≃ 1 GeV is associated with the rapid passage of the elastic phase shift through 180 • . (See Ref. [23] for a recent parametrization.) This behavior is correlated with the opening of the KK threshold, forcing the I = J = 0 ππ amplitude to become highly inelastic [24]. It also reflects the effect of a narrow resonance f 0 (980) [25] coupling to both ππ and KK. For more details see [26,27]. A related discussion applies to the S-wave πη channel near the I = 1, J = 0 KK threshold [28].
B Cusp in π 0 π 0 spectrum at π + π − threshold The π 0 π 0 S-wave scattering amplitude is expected to have a cusp at π + π − threshold [29,30]. This behavior can be studied in the decay K + → π + π 0 π 0 , where the contribution from the π + π + π − intermediate state allows one to study the charge-exchange reaction π + π − → π 0 π 0 and thus to measure the ππ S-wave scattering length difference a 0 − a 2 [31]. The CERN NA48 Collaboration has performed such a measurement, finding results [32] in remarkable agreement with the prediction [31]. One can also study this effect in π + π − atoms [33]. C Hadron production by e + e − collisions around 4.26 GeV The value of R ≡ σ(e + e − → hadrons)/σ(e + e − → µ + µ − ) drops sharply just below threshold for production of D(1865) 0D 1 (2420) 0 + c.c. [34], which is the lowest-mass cc channel accessible in an S-wave from a virtual photon. If this behavior is not coincidental, the drop in R should be confined to the cc final states.

D Six-pion diffractive photoproduction
The diffractive photoproduction of 3π + 3π − leads to a spectrum with a pronounced dip near 1.9 GeV/c 2 [35,36]. This is just the threshold for production of a proton-antiproton pair in the 3 S 1 channel. This dip also occurs in the 3π + 3π − spectrum produced in radiative return in higher-energy e + e − collisions, i.e., in e + e − → γ 3π + 3π − , observed by the BaBar Collaboration at SLAC [37]. The feature can be reproduced by a 1 −− resonance with M = 1.91 ± 0.01 GeV/c 2 and width Γ = 37 ± 13 MeV interfering destructively with a broader 1 −− resonance at lower mass [35,36].

E Greater generality
The vanishing of an S-wave amplitude when its elastic phase shift goes through 180 • is not confined to particle physics. The Ramsauer-Townsend effect represents similar behavior in atomic physics [38]. Cusps in S-wave scattering cross sections occur at thresholds for any new channels [39,40]. Monochromatic neutrons may be produced by utilizing the vanishing absorption cross sections of neutrons of certain energies on specific nuclei [41].

F A cautionary note
Although the rapid passage of the I = J = 0 ππ phase shift through 180 • near KK threshold can be ascribed to the nearby f 0 (980) resonance, one cannot conclude that similar behavior in other of the above cases (or many more examined in [22]) is due to nearby poles in the scattering amplitude [40]. As in the case of diffractive six-pion production mentioned above, unitarity alone will cause a suppression of the input channel at the expense of the newly-open channel. The ability to fit the amplitude with a resonance does not guarantee its existence.

III Dips in M inv (di−J/ψ ) at di-charmonium thresholds
In Fig. 1 we show the spectrum of M inv (di−J/ψ ) [3] together with a fit to data in the range 6.2-7.5 GeV using the sum of three Breit-Wigner resonances with masses M i , widths Γ i , and parameters (normalizations) η i (i = 1, 2, 3). Signal normalization, background normalization, and background shape are described by parameters C i defined in the Appendix. The results of this fit are shown in Table I. The shapes of the peaks around 6.9 and 7.2 GeV suggest destructive interference between signal and background on the low-mass side of both peaks. The sudden rise following a dip is characteristic of an S-wave amplitude. Examples of this behavior were given in the previous Section. It was associated with the opening of a nearby threshold. In the case of the 6.9 GeV peak, we note that 2M(χ c0 ) = 6829 MeV, so we can ascribe the steep behavior between about 6750 and 6900 GeV as associated with opening of the di-χ c0 channel. The parameter η 2 < 1 indicates that the resonance with mass M 2 has a significant decay channel other than di−J/ψ . Figure 1: Spectrum of J/ψ pairs reported by the LHCb Experiment [3], together with our best fit to data (red line), as given in Table I   If the di-χ c0 channel is in an S-wave, as implied by its sudden onset, the S-wave behavior in the di-J/ψ channel requires the two J/ψ mesons to be in a state of J P C = 0 ++ . An initial state of two J/ψ mesons consists of two cc pairs, each in a 3 S 1 state. A χ c0 is a P -wave charmonium state with the quarks' spins coupled to 1 and spin coupled with L = 1 to give The final state with two 3 P 0 states in a relative S-wave can be reached from the initial state by orbital excitation of each spin-triplet state.
Detection of the presence of the two χ c0 states is challenging in view of the small branching fractions of χ c0 to observable final states. The only branching fractions of χ c0 that exceed a percent are given in Table II [25]. With sufficient mass resolution, one could combine the modes with all charged tracks to get an effective branching fraction of a bit above five percent, The total width of χ c0 is 10.8 ± 0.6 MeV. The experimental mass resolution in other LHCb analyses (see, e.g., [42,43]) is somewhat greater, and thus dominates the sensitivity to a signal. An explicit simulation would be helpful.
Similar behavior is apparent on the low-M inv shoulder of the peak at 7.2 GeV. The only nearby threshold is associated with a pair of η c (2S) mesons, with 2M[η c (2S)] = 7275 MeV. If this threshold plays an important role in the line shape of the peak, one should see decay products of two η c (2S) mesons on the high-M inv side of this peak. This, of course, is even more challenging than detecting a pair of χ c0 mesons. (Refs. [12,13] draw attention to the slightly lower Ξ ccΞcc threshold at 7242 MeV, which we shall discuss further at the end of Sec. IV.) We initially sought evidence for a di-η c (1S) threshold at 2M[η c (1S)] = 5968 MeV and inserted a corresponding pole below di−J/ψ threshold into our fitting amplitude. The expectation was that this would contribute a needed enhancement of the spectrum between M inv ≃ 6.2 and 6.6 GeV. The fitting program (see Appendix A) instead preferred a much higher-mass pole, as one sees for M 1 in Table I. However, the χ 2 for the fit is a very shallow function of M 1 (and several other parameters). In particular, the parameters in Table III are consistent with the prediction [20] M[T (cccc)] = 6191.5 ± 25 MeV for the lightest allcharm tetraquark. We shall explore the consequences of identifying M 1 with the mass of the 1S all-charm tetraquark.
Although we do not predict a tetraquark resonance near di-η c (1S) threshold, it might be worth examining channels that couple to a pair of η c (1S) to see if they exhibit cusps in S-wave amplitudes near M inv = 5968 MeV. Examples of such channels include DD and D * D * [8,14].

IV Implications for cccc tetraquarks
In Ref. [20], using a diquark-antidiquark picture, we predicted the ground state T (cccc) mass to be 6191.5 ±25 MeV. This error is taken to be twice that obtained when fitting nonexotic mesons and baryons in the string-junction picture (see also [12]), recently confirmed by the successful prediction of the mass of a T (csūd) tetraquark [44] and which we are assuming here [20]. This would be the 0 ++ (1S) state of the spin-1 color antitriplet diquark and the spin-1 color triplet antidiquark. The ingredients of the prediction included a term 2S = 2(165.1) MeV for two QCD string junctions, 2(M cc ) = 2(3204.1) MeV for the masses of two diquarks, an interpolated binding energy of the cc diquark with thecc antidiquark of -388.3 MeV, and a hyperfine term of -158.5 MeV. The predicted mass is just below 2M(J/ψ ) = 6194 MeV but above 2M(η c (1S) = 5968 MeV, so strong decay to a pair of η c (1S) is favored. Here and subsequently we use the latest Particle Data Group masses [25]. The above discussion is based on S-wave cc diquarks in a color 3 * state, with spin 1. There should also be states involving color 6 diquarks, with spin zero. There should be an additional spinless tetraquark made of a 6 in an S-wave state with a 6 * . Estimates, for example in Ref. [16], of its mass are not far from that of the 1S 3 * × 3 state, and the two may mix with one another.
The above estimate concerns the ground state 0 ++ mass. One estimates the ground state 2 ++ mass by noting that the hyperfine terms for a pair of spin-1 particles in states of J = 0, 1, 2 are in the ratio (1/2)[J(J + 1) − 4] = −2, −1, 1, so the hyperfine term for the lowest 2 ++ state is 79.3 MeV and the mass of the 2 ++ (1S) state is 6429.3 MeV, 237.8 MeV above the 0 ++ (1S) and well above 2M(J/ψ ) threshold. This 2 ++ state, if present in the data, could be contributing to the low-M inv di−J/ψ signal, allowing the 0 ++ component of the peak to lie at lower mass, more consistent with prediction. A spin-parity analysis should be able to detect whether there is any 2 ++ contribution to the amplitude.
The 1S-2S splittings of the charmonium and bottomonium systems are almost the same.  bb). They would be equal for a logarithmic interquark potential, providing a convenient interpolation between short-distance and long-distance QCD for these systems [45]. The   [9,13].) The 2 ++ (2S) state is then (0.4053)(237.8) = 96 MeV higher, at 6967 MeV. This state could also be contributing to the LHCb signal.
We have not discussed 1 ++ states of cc diquark andcc antidiquark decaying to a pair of J/ψ in an S-wave. Two identical spin-1 bosons in an S-wave are forbidden by Bose statistics to have total angular momentum J = 1.
The peak around 7200 MeV is in approximately the right place for a 3S state of (cc) 3 * (cc) 3 . The flavor threshold for charmonium lies just above the 2S level, while that for bottomonium lies just below the 4S level. As a system with reduced mass intermediate between that of charmonium and that of bottomonium, the di−J/ψ system can be expected to have a flavor threshold around the 3S level (see Fig. 1 of [50]). This estimate is based on the observation [51,52] that flavor threshold in a quarkonium system always occurs at a universal length of the QCD string connecting the two heavy constituents. Indeed, the first open-flavor state in which a QCD string connecting (cc) 3 * with (cc) 3 breaks is that in which a light qq pair is produced, giving Ξ ccΞcc with threshold 7242 MeV [12,13].

V Implications for bbbb tetraquarks
Some attention to the question of fully heavy tetraquarks was drawn by an unpublished report by the CMS Collaboration at CERN [46] of an exotic structure in the four-lepton channel at 18.4 ± 0.1 ± 0.2 GeV, an excess with a global significance of 3.6 σ. CMS reported 38 ± 7 events of Υ(1S) pairs produced with an integrated luminosity of 20.7 fb −1 at √ s = 8 TeV, each decaying to µ pairs [47]. There is no published confirmation of the structure [48,49], but in view of the di−J/ψ structure it is worth updating and extending the predictions of Ref. [20] for bbbb tetraquarks.
In Ref. [20] we predicted the ground state T (bbbb) mass to be 18826 ± 25 MeV, just above 2M[η b (1S)] = 18797 MeV, so its main decay will likely be to two η b -s. It would be the 0 ++ state of a color antitriplet spin-1 bb diquark and the corresponding antidiquark.
where S is the contribution of a QCD string junction, B (bb)(bb) is the binding energy between the bb diquark and thebb antidiquark, and ∆M HF is the hyperfine interaction between the  The radially excited 0 ++ (2S) bb-bb tetraquark at 19.434 ± 0.025 GeV is the bottom analogue of the 0 ++ (2S) excited cc-cc tetraquark at 6.871 ± 0.025 GeV, proposed here as the main component of the peak near 6.9 GeV reported by LHCb [3].
The predicted 0 ++ (2S) mass is large enough to imply a substantial partial width into a pair of Υ(1S). It lies below the χ b0 χ b0 threshold, which is 2(9859.44) = 19718.9 MeV, so its interference with the 0 ++ state will depend on the width of that state and should exhibit a different pattern from the T (cccc) case, where the χ c0 χ c0 threshold roughly coincides with the 0 ++ (2S) resonance mass. We should also keep in mind the Ξ bbΞbb threshold at 2(10162 ± 12) = 20324 ± 25 MeV, where we have used the prediction [53] M(Ξ bb ) = 10162 ± 12 MeV, in analogy with the Ξ ccΞcc threshold mentioned earlier.

VI Conclusions
We have interpreted the structure in the di−J/ψ mass spectrum observed by LHCb in terms of a diquark-antidiquark picture [20], with the predicted masses in Table IV. The irregular structure is seen to be due to the rapidly opening χ c0 χ c0 S-wave channel at 6829 MeV, interfering primarily with the 0 ++ 2S state. We have also updated and extended our prediction [20] for the tetraquark T (bbbb), with the results shown in Table V. The relative position of the 2χ b0 threshold with respect to the predicted 0 ++ (2S) state is different from that in the charm case, implying a structure in invariant mass of different shape.