On the mechanism of $T_{4c}(6900)$ tetraquark production

We discuss the production mechanism of a new state, a putative fully charm tetraquark, observed recently by the LHCb at M = 6.9 GeV in the $J/\psi J/\psi$ channel. Both single parton scattering (SPS) and double parton scattering (DPS) mechanisms are considered. We calculate the distribution in the invariant mass of the four-quark system $M_{4c}$ for SPS and DPS production of $c c \bar c \bar c$ in the $k_t$-factorization approach with modern unintegrated gluon distribution functions (UGDFs). The so-calculated contribution of DPS is almost two orders of magnitude larger than the SPS one, but the tetraquark formation mechanism is unknown at present. Imposing a mass window around the resonance position we calculate the corresponding distribution in $p_{t,4c}$ -- the potential tetraquark transverse momentum. The cross section for the $J/\psi J/\psi$ continuum is calculated in addition, again including SPS (box diagrams) and DPS contributions which are of similar size. The formation probability is estimated trying to reproduce the LHCb signal-to-background ratio. The calculation of the SPS $g g \to T_{4c}(6900)$ fusion mechanism is performed in the $k_T$-factorization approach assuming different spin scenarios ($0^+$ and $0^-$). The $0^+$ assignment is preferred over the $0^-$ one by the comparison of the transverse momentum distribution of signal and background with the LHCb preliminary data assuming the SPS mechanism dominance. There is no reliable approach for the DPS formation mechanism of tetraquarks at present as this is a complicated multi-body problem.


I. INTRODUCTION
The potential existence of tetraquarks was discussed a decade after the quark model was formulated [1]. Although the potential territory of tetraquarks is large there is an ongoing debate regarding their identification. The conjectured light tetraquarks, like e.g. f 0 (980), are ambiguous, because often a competitive interpretation as a hadronic molecule (KK for f 0 (980)) or as a more generic coupled-channel/threshold phenomenon is possible.
Much attention has recently been paid to possibly exotic hadrons containing heavy quarks, where a plethora of newly discovered states await their definitive theoretical understanding/interpretation [2][3][4][5].
The recent observation by the LHCb collaboration [6] of a sharp peak in the di-J/ψ channel at M = 6.9 GeV seems to strongly suggest the presence of a fully charm tetraquark, consisting of cccc.
During the last years a number of theoretical models for the spectroscopy of tetraquarks were developed and are waiting for experimental verification. The most popular approach treats the fully heavy (cccc, bbbb or ccbb) tetraquarks as a bound system of a color antitriplet diquark and color triplet antidiquark. In early works the diquarks were treated as structureless objects, and the interaction between diquark and antidiquark is then constructed in analogy to that between heavy quarks developed earlier for quarkonia [7][8][9][10][11]. The second color configuration of a sextuplet diquark and antisextuplet antidiquark is most often neglected. Reservations regarding the diquark approach from the point of view of more rigorous approaches to the few-body problem have been raised in [12].
Assuming the enhancement observed by LHCb is indeed caused by a new state, models suggest that it is rather an excited state. There is no clue at present on its spin and parity. Clearly, higher statistics studies which may give model-independent answer [13] are required in future. Different models predict slightly different quantum numbers. For example the nonrelativistic potential quark model (NRPQM) [14] predicts the state to be J PC = 0 −+ or 1 −+ . A similar result is obtained in the framework of a QCD sum rule approach [15]. A different pattern is obtained in the relativized quark model with quarkquark, antiquark-antiquark and quark-antiquark interactions [16]. In this approach the 6.9 GeV state can be a radial excitation with J PC = 0 ++ , 2 ++ . A caveat regarding the interpretation of the LHCb result is in order: in [17] the authors describe the di-J/ψ invariant mass distribution by a non-resonant rescattering of quarkonium pairs. The peaks are then associated with threshold enhancements for different channels.
With this reservation in mind, we will in the following assume that indeed an (excited) tetraquark state has been observed. Different J PC combinations are possible in general [9,10,15,16], the details depend on the method used for the the four-body systems.
The decays of the fully heavy tetraquarks was discussed e.g. in [15,18]. On the other hand the production mechanism of the T 4c tetraquarks is terra incognita. There are only a few papers [19,20] on the production of the ground-state fully charm tetraquark. The production cross section for T 4c in [20] was estimated to be one order of magnitude smaller than that for X(3872) (assumed to be a tetraquark) production which was measured by the CMS collaboration, the production mechanism of X(3872) however itself is under debate.
What is the mechanism of the fusion of four charm quarks/antiquarks is not clear at the moment. Some time ago a large cross section for cccc production at the LHC due to doubleparton scattering (DPS) mechanism was predicted in [21]. The SPS mechanism was also considered but its contribution to the cccc production is much smaller [22][23][24][25]. The prediction of Ref. [21] was verified by the LHCb collaboration by observing many sameflavor D mesons [26]. The calculation in the k T -factorization approach explained many correlation observables for double D meson production [23][24][25].
Recently, within k T -factorization, also the gluon-gluon fusion mechansism of production of pseudoscalar [39] and scalar [40] charmonia was studied. Here we shall consider the mechanism of gluon-gluon fusion for the fully charm tetraquark production. In this letter we shall discuss only 0 + and 0 − scenarios where the formalism was tested.

II. CROSS SECTION FOR SIGNAL AND BACKGROUND
In this section we discuss several issues related to the production of the T 4c (6900) tetraquark.
After many years of investigation there is no agreement on production mechanism even for quarkonia, pure QQ states. For C = +1 quarkonia rather color singlet mechanism dominates [39]. How big is color octet contribution is not quite clear at present.
The reaction mechanism for C = + 1 tetraquark production (the LHCb case) can be categorized as: (a) cccc are produced in color singlet state, (b) cccc are produced in color octet state and extra emission(s) of soft gluon(s) is(are) necessary to bring the cccc system to color singlet state relevant for the tetraquark hadron.

A. pp → cccc cross section
In this subsection we wish to calculate the cross section for four charm quark/antiquark production. In particular, we wish to calculate distribution in invariant mass of the four charm quarks/antiquarks in the region of low invariant masses. In particular, such a cross section in the mass window arround the mass of the tetraquark can be compared to the cross section for the tetraquark which at present can be only estimated with poor precision. In Fig.1 we show the dominant reaction mechanisms: SPS type (left diagram) and DPS type (right diagram).
In the present study both the SPS and the DPS contributions are calculated in the framework of k T -factorization [41][42][43][44]. According to this approach the SPS cross section for pp → cccc X reaction can be written as In the formula above F g (x, k 2 t , µ 2 ) is the unintegrated or transverse momentum dependent gluon distribution function (gluon uPDF). The uPDF depends on longitudinal momentum fraction x, transverse momentum squared k 2 t of the partons entering the hard c c c c p 1 process, and in general also on a (factorization) scale of the hard process µ 2 . The elementary cross section in Eq. (2.1) can be written as: (2.2) where E l and p l are energies and momenta of final state charm quarks. The matrix element takes into account that both gluons entering the hard process are off-shell with virtualities k 2 1 = −k 2 1t and k 2 2 = −k 2 2t . In numerical calculations we limit ourselves to the dominant gluon-gluon fusion channel of the 2 → 4 type parton-level mechanism. We checked numerically that the channel induced by the qq-annihilation can be safely neglected in the kinematical region under consideration here.
A formal theory of multiple-parton scattering (see e.g. Refs. [45,46]) is rather well established but still not fully applicable for phenomenological studies. In general, the DPS cross sections can be expressed in terms of the double parton distribution functions (dPDFs). However, the currently available models of the dPDFs are still rather at a preliminary stage. So far they are formulated only for gluon or for valence quarks and only in a leading-order framework which is for sure not sufficient for many processes, especially when heavy quark production is considered.
In general, the DPS cross sections can be expressed in terms of the double parton distribution functions (dPDFs) (see e.g. Refs. [45,46]). However, the currently available models of the dPDFs are still rather at a preliminary stage. Therefore, in phenomenological studies one usually follows the assumption of the factorization of the DPS cross section. Within the factorized ansatz, the dPDFs are taken in the following form: is the dPDF and f i (x i , µ) are the standard single PDFs for the two generic partons in the same proton. The factor θ(1 − x 1 − x 2 ) ensures that the sum of the two parton momenta does not exceed 1.
According to the above, the differential cross section for pp → cccc X reaction within the DPS mechanism can be then expressed as follows: where ξ 1 and ξ 2 stand for generic phase space kinematical variables for the first and second scattering, respectively. The combinatorial factor m is equal 0.5 for the cccc case.
Here, the dσ SPS (pp → cc X) ingredient cross sections are also calculated with the off-shell initial state partons. The effective cross section σ eff provides normalization of the DPS cross section and can be roughly interpreted as a measure of the transverse correlation of the two partons inside the hadrons. The longitudinal parton-parton correlations are far less important when the energy of the collision is increased, due to the increase in the parton multiplicity. For small-x partons and for low and intermediate scales the possible longitudinal correlations can be safely neglected (see e.g. Ref. [47]). In this paper we use world-average value of σ eff = 15 mb provided by several experiments at Tevatron [48][49][50] and LHC [51][52][53][54][55]. Future experiments may verify this value and establish a systematics.
The numerical calculations for both the SPS and the DPS mechanisms are performed with the help of KaTie [56], which is a complete Monte Carlo parton-level event generator for hadron scattering processes. It can can be applied to any arbitrary processes within the Standard Model, for several final-state particles, and for any initial partonic state with on-shell or off-shell partons. We use µ 2 = ∑ 4 i=1 m 2 it /4 as the renormalization/factorization scale, where m it 's are the transverse masses of the outgoing charm quarks. We take running α s at next-to-leading order (NLO) and charm quark mass m c = 1.5 GeV. Uncertainties related to the choice of the parameters were discussed very recently in Ref. [57] and will be not considered here. We use the Kimber-Martin-Ryskin (KMR) [58,59] unintegrated distributions for gluon calculated from the MMHT2014nlo PDFs [60]. The above choices are kept the same also in the case of double-parton scattering calculation except of the scales.
Having calculated differential cross section for cccc-system production one can obtain the cross section for T 4c (6900) tetraquark within the framework of color evaporation model (CEM) [61,62]. The cccc → T 4c (6900) transition can be written as follows: where F T 4c is the probability of the cccc → T 4c transition which is unknown and could be fitted to a future experimental data, M T 4c = 6.9 GeV is the mass of T 4c tetraquark and M 4c is the invariant mass of the cccc-system. In the numerical calculations we take ∆M = 100 MeV.
In Fig.2 we show the invariant mass distribution calculated for SPS (solid line) and DPS (dashed line) contribution. In this calculation we take limitation on rapidity of the cccc system 2 < Y < 4.5 relevant for the LHCb apparatus. Clearly the cross section for DPS is much larger than the cross section for SPS in the vicinity of the tetraquark position. This does not mean that the tetraquark is produced mainly in the DPS mechanism. The underlying production mechanism is complicated as it involves many-body correlations and four-body wave function. Furthermore the production mechanism for DPS is (must be) different than for SPS. In particular, the quarks and antiquarks produced in the DPS mechanism may be less space-time correlated than those from the SPS mechanism. Now we wish to visualize the p t,4c distribution in a very narrow window of M 4c in the sourrounding of the tetraquark mass. Such a distribution is shown in Fig.3. This calculation requires large statistics of the experimental sample. Of course in general p t,4c  it is not p t,T 4c but must be closely related. It would be so in a bit naive coalescence or color evaporation model.
In the present calculations we have summed over all possible color states of outgoing c quarks andc antiquarks. It would be of interest to make the color study in a future. Then, having selected the wave function of the teraquark (requires selecting a model), one could select final states more relevant for the tetraquark production. We leave such a J/ψ study for a future work.

B. pp → J/ψJ/ψ background
It is of interest to calculate also background to the J/ψJ/ψ final state used in the LHCb experiment. There are two dominant mechanisms shown in Fig.4. The normalized cross section within the LHCb acceptance was measured [63].
Here we have considered only the dominant mechanisms. There are some other mechansims like gluon exchange [32] or χ c (J 1 )χ c (J 2 ) contributions [64] which are important for the ATLAS or CMS kinematics (large M J/ψJ/ψ , large ∆y) but negligible for the LHCb kinematics relevant for production of the tetraquark.
As far as DPS is concerned we parametrize 1 the single J/ψ production in terms of a simple color evaporation model based on k T -factorization approach [65]. This approach is simple enough and can be nicely adjusted to the experimental data [65].
The σ e f f is relatively well known and is about 15 mb [66]. We estimate the precision of the DPS calculation at 30 % level.
In Fig.5 we show distribution in M J/ψJ/ψ for the two mechanisms shown in Fig.4. We see that in the vicinity of the tetraquark mass the SPS contribution is similar as the DPS one so both of them must be included in the evaluation of the background.
As for cccc production in the previous subsection we wish to show distribution in p t,J/ψJ/ψ for the narrow window of invariant mass arround the tetraquark mass. Such a distribution is shown in Fig.6. The distributions for the background here can be compared to the distribution of the signal from Fig.3 after multiplying the latter by a factor 10 −4 for the DPS and 10 −2 for the SPS contributions. The p t dependence of the background and the so-obtained signal have similar magnitude and the shape.

production, examples of the spin-parity assignment
Finally we consider the calculation of the SPS-type signal as a fusion of two (off-shell) gluons for two different spin-parity assignments: 0 + and 0 − of the tetraquark. The corresponding diagram is shown in Fig.7.
In the following we use the formalism worked out recently for the inclusive production of pseudoscalar [39] and scalar [40] quarkonia. The off-shell gluon fusion cross sections will be proportional to a form-factor, which depends on the virtualities of gluons, with X = (M 4 + 2(Q 2 1 + Q 2 2 )M 2 + (Q 2 1 − Q 2 2 ) 2 )/4. Note, that for the 0 + assignment we use only the TT coupling, as in analogy with [40] we expect the LL contribution to be smaller. In our calculation for the tetraquark production we also use the KMR UGDFs.
The g ggT 4c coupling constants are in both cases roughly adjusted to get the signal-tobackground ratio of the order of 1. In our calculation here we use the nonfactorizable where Q 2 1 and Q 2 2 are gluon virtualities and vary corresponding form factor parameter Λ. For the fully charm tetraquark one may expect naively Λ ∼ m T 4c or Λ ∼ 4m c . We will use also a smaller value having in mind uncertainty related to the tetraquark wave function.
Since the ratio of signal-to-background improves with transverse momentum of the tetraquark [6] and knowing relatively well the behaviour of the SPS and DPS background (see Fig.6) we can conclude that the 0 − assignment is disfavoured by the LHCb experimental results.
Here we have considered only 0 + and 0 − (C = +1) spin-parity assignments. Other assigments (1 + . 2 + , etc.) should (will be) be considered in a future. In fact the peak observed by the LHCb does not need to be a single spin but a mixture of different spins [14,16]. This is the main argument that we do not consider interference of the resonance and continuum at the present stage.

III. CONCLUSIONS
In the present letter we have considered several aspects related to the production of T 4c (6900) tetraquark (called signal) observed recently by the LHCb collaboration in the J/ψJ/ψ channel and the J/ψJ/ψ background. Both for the signal and the background the SPS and DPS mechanisms have been considered.
The background distributions can in our opinion be reliably calculated. It is not the case for the signal. In the naive coalescence model we have to adjust a normalization factor C responsible for the formation probability P T 4c and decay branching fraction Br(T 4c (6900) → J/ψJ/ψ) 2 . In the moment the formation probability cannot be calculated from first principles. In our opinion the branching fraction is a simpler issue but also goes beyond the scope of the present letter where we try to explore the general situation. Thus in the moment the product of the two unknowns can be roughly adjusted to the current signal-to-background ratio. We get C = 10 −4 for the DPS and C = 10 −2 for the SPS production of cccc. We have not considered the mixed scenario in which both SPS and DPS mechanisms contribute.
We have considered also more explicitly the SPS mechanism of the resonance production via gluon-gluon fusion in the k T -factorization approach with modern UGDFs. Also in this case the normalization, related to the underlying formation process and/or wave function of the tetraquark and the decay branching fraction T 4c → J/ψJ/ψ must be adjusted to the experimental signal-to-background ratio. In this study we have considered two examples of the 0 + and 0 − assignment. The current data seem to exclude the 0 − assignment as the final result contradicts qualitatively to the transverse momentum dependence of the signal-to-background ratio as observed by the LHCb collaboration.
In this letter we have only set the general approach leading to a better understanding the tetraquark production. We expect that in the future more states will be observed by the LHCb collaboration and disentagling spins and parities will be easier. In addition, one could study the angular correlation of the J/ψ mesons in the tetraquark rest frame. This requires a better statistics available in run 3. Then a model independent analysis [13] will be possible.
In a future one could try to search for the T 4c (6900) tetraquark production also in the pp → ppJ/ψJ/ψ exclusive reaction or in AA → AAJ/ψJ/ψ ultraperipheral collisions. The corresponding studies will be done elsewhere.