Structure and production mechanism of the enigmatic $X(3872)$ in high-energy hadronic reactions

We calculate the total cross section and transverse momentum distributions for the production of enigmatic $\chi_{c,1}(3872)$ (or X(3872)) assuming different scenarios: $c \bar c$ state and $D^{0*} {\bar D}^0 + D^0 {\bar D}^{0*}$ molecule. The derivative of the $c \bar c$ wave function needed in the first scenario is taken from a potential model calculations. Compared to earlier calculation of molecular state we include not only single parton scattering (SPS) but also double parton scattering (DPS) contributions. The latter one seems to give smaller contribution than the SPS one. The upper limit for the DPS production of $\chi_{c,1}(3872)$ is much below the CMS data. We compare results of our calculations with existing experimental data of CMS, ATLAS and LHCb collaborations. Reasonable cross sections can be obtained in either $c \bar c$ or molecular $D {\bar D}^*$ scenarios for $X(3872)$. Also a hybrid scenario is not excluded.


I. INTRODUCTION
The X(3872) state was discovered already some time ago by the Belle collaboration [1]. Since then its existence has been confirmed in several other processes and numerous theoretical studies have been performed, see for example the review articles [2][3][4]. There is at present agreement that the X(3872) has the axial vector quantum numbers J PC = 1 ++ , and accordingly the state is named as χ c1 (3872) [5].
The internal structure of X(3872) stays rather enigmatic. While its quantum numbers are not exotic -it could indeed be a quarkonium cc state, e.g. a radial excitation of the χ c1 , there are strong arguments for a non-cc scenario, manifested e.g. by the violation of isospin in its decays [5].
More importantly, the mass of X is very close to the DD * threshold. It is therefore rather popular to consider X(3872) as very weakly bound state of the DD * system -a hadronic molecule, see the review [6]. A tetraquark scenario was considered in [7]. The cc quarkonium scenario, where X(3872) is the χ c1 (2P) state has been advocated in [8]. Other approaches treat the X(3872) as a cc bound state in the meson-meson continuum taking into account coupling of cc and meson-meson channels [9][10][11]. The possible mixture of quarkonium and molecule/virtual state is considered in [12] and found to be consistent with current data.
There is a debate in the literature [16][17][18][19], whether the rather large production rate at large p T allows one to exclude the molecular scenario. With few exceptions [20][21][22] the discussion in the literature is limited to estimates of orders of magnitude.
In this paper we shall consider two scenarios of prompt X(3872) production, which are not mutually exclusive and in fact both can contribute depending on the structure of X(3872). Both scenarios have in common that the production is initiated by the production of a cc pair in a hard process.
In the first scenario we shall consider that the X(3872) is a pure cc state, the first radial excitation of χ c1 . The corresponding wave function and its derivative were calculated in potential models e.g. in [23].
In the second scenario, where the X(3872) is treated as a weakly bound s-wave state in the DD * + D * D -system, we exploit the connection between the low-energy scattering amplitude in the continuum and at the bound state pole below threshold, well known from effective range theory. In this work we follow [17] and give an estimate of a p Tdependent upper bound for X(3872) production in the molecule scenario.
As far as the hard production mechanism is concerned, we employ the k T -factorization framework [24][25][26]. For the quarkonium scenario, the dominant mechanism of C = +1 is probably color singlet two virtual gluon fusion. The production of χ c0 , χ c1 , χ c2 production was considered e.g. in [27][28][29]. Recently we have shown that the transverse momentum distribution of η c measured by the LHCb [30] can be nicely described as g * g * → η c fusion within k T -factorization approach [31].
For the molecular scenario, in addition to the single-parton scattering (SPS) mechanism of fusion of two off-shell gluons g * g * → cc, we will also consider production through the double-parton scattering (DPS) mode.

II. FORMALISM
In Fig.1 we show two generic Feynman diagrams for X(3872) quarkonium production in proton-proton collision via gluon-gluon fusion: for the quarkonium (left) and molecule (right). These diagrams illustrate the situation adequate for the k T -factorization calculations used in the present paper.
The inclusive cross section for X(3872)-production via the 2 → 1 gluon-gluon fusion (2.1) Here the matrix element squared for the fusion of two off-shell gluons into the 3 P 1 color singlet cc charmonium is (see e.g. [27,32] for a derivation): where φ is the azimuthal angle between q T 1 , q T 2 . The momentum fractions of gluons are fixed as The derivative of the radial quarkonium wave function at the origin is taken for the first radial p-wave excitation from Ref. [23], |R ′ (0)| 2 = 0.1767 GeV 5 .
The unintegrated gluon parton distribution functions (gluon uPDFs) are normalized such, that the collinear glue is obtained from The hard scale is taken to be always µ F = m T , the transverse mass of the X(3872). In order to estimate the production cross section for the molecule we also start from a hard production of a cc-pair, which we then hadronize into a DD * + h.c system using a prescription given below.
The parton-level differential cross section for the cc production, formally at leadingorder, reads: where M off−shell g * g * →QQ is the off-shell matrix element for the hard subprocess [24], we use its implementation from [33].
Here, one keeps exact kinematics from the very beginning and additional hard dynamics coming from transverse momenta of incident partons. Explicit treatment of the transverse momenta makes the approach very efficient in studies of correlation observables. The two-dimensional Dirac delta function assures momentum conservation. The gluon uPDFs must be evaluated at longitudinal momentum fractions: where m Ti = p 2 Ti + m 2 c is the quark/antiquark transverse mass. In the present analysis we employ the heavy c-quark approximation and assume that three-momenta in the pp-cm frame are equal: This approximation could be relaxed in future. The hadronization is then included only via fragmentation branching fractions: The first number is from [34] while the other numbers are from [35].
The cross section for cc production are then multiplied by and, when comparing to experimental data by the relevant branching fraction i.e.
by BR(X → J/ψπ + π − ) for the case of CMS and LHCb, and by BR(X → The branching fractions are taken from [5]. The factor 1 2 is related to the factor 1 √ 2 in the definition of the molecular wave function, In our calculation, we control the dependence on the relative momentum of quark and antiquark in the rest frame of the pair: where M cc is invariant mass of the cc system and m c is the quark mass. In order to obtain an upper bound for the molecule production cross section, we should integrate the DD * cross section over the relative momentum k DD * rel up to a cutoff k DD max [17]. We will instead impose a cutoff k max on the relative momentum k rel . Within our kinematics the latter will be similar to k DD * rel , but somewhat larger. In reality, for larger p T X ≈ p T 1 + p T 2 , we have k DD < k rel . We therefore estimate, that k max = 0.2 GeV corresponds roughly to k DD max ≈ 0.13 GeV. A better approximation would be to add simultaneous c → D, c →D fragmentation to our Monte Carlo code, which however means at least two more integrations. We do not consider here any model of the DD * wave function. What is the appropriate choice for k DD * max was a matter of discussion in the literature. In Ref. [16] it was suggested, that k DD * max should be of the order of the binding momentum k X = 2µε X , where µ is the reduced mass of the DD * -system, and ε X is the binding energy of X(3872). This would lead to a very small value for k DD * max , similar to k DD * max = 35 MeV used in [16]. However problems with this estimate have been pointed out in [17][18][19]. As has been argued in these works, the integral should be extended rather to a scale k DD * max ∼ m π . In our choice of k max , we follow this latter prescription.
In the following for illustration we shall therefore assume k max = 0.2 GeV. The calculation for the SPS molecular scenario is done using the VEGAS algorithm for Monte-Carlo integration [36].
We also include double parton scattering contributions (see Fig.2). The corresponding cross section is calculated in the so-called factorized ansatz as: Above the differential distributions of the first and second parton scattering dσ dy i d 2 p T i are calculated in the k T -factorization approach as explained above. In the following we take σ eff = 15 mb as in [37]. The differential distributions (in p T of the X(3872) or y diff = y 1 − y 2 , etc.) are obtained by binning in the appropriate variable. We include all possible fusion combinations leading to X(3872): This leads to the multiplication factor two times bigger than for the SPS contribution (see Eq.(2.12)).

III. RESULTS
In this section we shall show our results for recent CMS [13] ATLAS [14] and LHCb [15] data. The CMS data is for √ s = 7 TeV and -1.2 < y X < 1.2, the ATLAS data for √ s = 8 TeV, -0.75 < y X < 0.75 and the LHCb data for √ s = 13 TeV, 2 < y X < 4.5. In all cases the X(3872) was measured in the J/ψπ + π − channel. We have used a number of different
Similar shapes in k rel are obtained for the different windows of p Tcc .
To visualize this better we show in the right panel of Fig.6 k −2 rel dσ/dk rel . As expected, the so-obtained distributions closely follow phase-space, and are almost flat in a broad range of k rel . Therefore, the cross section has essentially the phase-space behaviour which implies the strong, cubic dependence ∝ k 3 max on the upper limit k max of the k relintegration.
The calculation in the whole phase space p T1 ∈ (0, 20) GeV and p T2 ∈ (0, 20) GeV leads to fluctuations at large p Tcc > 20 GeV. This can be understood as due to steep dependence of the cross section on p T1 , p T2 , y 1 , y 2 , φ. Only a narrow range in the p T1 ⊗ p T2 space with p T1 ≈ p T2 fulfills the condition k rel < k max = 0.2 GeV.
We remind the reader that this value of k max is imposed on the cc final state, and would correspond to a smaller k DD * max ∼ 0.14 GeV for the DD * mesons. Due to the behaviour shown in Eq.(3.1), the cross section for k max = 0.14 GeV or k max = 0.1 GeV imposed on the cc state would go down by a factor three or eight, repsectively.
The distribution in relative azimuthal angle between cc in the pp-frame with the cut k rel < 0.2 GeV is shown in Fig.7. This is rather a steep distribution around φ = 0 o . This is not a typical region of the phase space. Recall, that in the leading-order collinear approach φ = 180 o . It is obvious that the region of φ ≈ 0 cannot be obtained easily in the collinear approach. As discussed in [33] the k T -factorization approach gives a relatively good description of D 0D0 correlations in this region of the phase space.
Having understood the kinematics of the X(3872) production we can improve the description of transverse momentum distribution of X(3872). The rather strong correlation p T1 ≈ p T2 allows to perform VEGAS calculation simultaneously limiting to p T1 , p T2 > p Tmin . We have also verified that Therefore imposing lower limits on p T1 > p Tmin and p T2 > p Tmin means also lower limit on p Tcc > 2p Tmin . The solid line in Fig.3 shows result of such a calculation. The fluctuations are gone. Combining the two calculations at say p Tcc = 15 GeV gives a smooth result everywhere.

C. Hybrid approach
Still another option is to assume that X is the combination: Such a state could be called hybrid. Since the cross section for production of conventional and molecular X(3872) are rather similar, it is difficult to select one combination of α and β. For example in Fig.8 we show also such a result for α 2 = β 2 = 0.5 (dash-dotted line).
The corresponding result is similar to the result for α 2 = 1, β 2 = 0 (pure cc state) and α 2 = 0, β 2 = 1 (pure molecular state) calculated for the KMR UGDF. The fifty-fifty mixture gives quite good representation of all the experimental data.
In summary, it is very difficult at present to definitely conclude what is the mechanism of the X(3872) production. The same is true for the wave function of the X(3872) meson (see Eq.(3.3)). At present we cannot answer the question whether X(3872) is of conventional, molecular or hybrid type. More work is definitely needed in future.

IV. CONCLUSIONS
We have performed the calculation of X(3872) production at the LHC energies. We have performed two independent calculations: one within nonrelativistic QCD approach assuming pure cc state and second assuming a coalescence of D andD * orD and D * , consistent with molecular state assumption. The first calculation requires usage of derivative of the cc wave function. In the present analysis we have used the wave function obtained in [23]. The resulting cross section was calculated within the k T -factorization approach with a few unintegrated gluon distributions. In the second approach first the hard production of a cc pair is calculated. Next a simple hadronization is performed giving a correlation distribution of D and D * mesons. Imposing limitations (upper limit) on relative momenta of D andD * we get a p T -dependent upper limit of the cross section for D-D * orD-D * fusion (coalescence). We compare, for the first time, to all available experimental data on the p T -dependent cross section. The both, quite different, scenarios give the cross section of the right order of magnitude, very similar to the CMS, ATLAS and LHCb experimental data.
Also a mixture of both mechanism leads to correct description of the CMS, ATLAS and LHCb data. Our study within k T -factorization approach shows that all the options are in the game.