Triple prompt $J/\psi$ hadroproduction as a hard probe of multiple-parton scatterings

We propose the process of triple prompt $J/\psi$ hadroproduction is a very clean hard probe of multiple-parton scatterings at high-energy hadron colliders, especially the least known triple-parton scattering. A first complete study is carried out by considering single-, double- and triple-parton scatterings coherently. Our calculation shows that it is a golden channel to probe double- and triple-parton scatterings as the single-parton scattering is strongly suppressed. The predictions of the (differential) cross sections in proton-proton collisions at the LHC and the future higher-energy hadron colliders are given. Our study shows that its measurement is already feasible with the existing data collected during the period of LHC Run2. A method is proposed to extract the triple-parton scattering contribution, and therefore it paves a way to study the possible triple-parton correlations in a proton.


I. INTRODUCTION
Multiparton scattering (MPI) physics at the high-energy hadron colliders, like Large Hadron Collider (LHC) at CERN and future proposed hadron colliders [1], is becoming increasingly important to study the new phenomenon and to search for beyond the Standard Model signatures as the fast increase of the parton-parton luminosity.As opposed to the leading MPI double-parton hard scattering (DPS), the measurements of the next-to-leading MPI triple-parton scattering (TPS) at the LHC are absent due to their more complicated final states and much fewer yields.Such rare processes, however, are possible to study with enough statistics at the high-luminosity phase of the LHC (HL-LHC).Similarly to the DPS case, the general factorization ansatz of MPI exists and perturbative QCD (pQCD) calculations are possible given the full knowledge of multiple-dimensional tomography of the proton.In practice, the phenomenological studies of MPI are either strongly model dependent or assuming no correlations between the multiple parton scatterings.We will use the latter approach here as a testable ground to study the multiple-parton correlations from MPI.If we assume zero correlations in MPI, a generic N-parton scattering (NPS) cross-section becomes [2] where the combinatorial factor m N !takes into account the indistinguishable final state symmetry.The effective cross section σ eff,N encodes all possible unknown parton tranverse profiles in the protons and should be determined by experiments.The DPS and TPS cases correspond to N = 2 and N = 3 in the above formula Eq. (1).From the pure geometrical consideration, Ref. [3] derives σ eff,3 = (0.82 ± 0.11) × σ eff,2 after a global survey of various parton transverse profiles.
Heavy quarkonia, bound states of heavy-flavored quarks, provide crucial insights of gluon-gluon and gluon-quark correlations in the proton by studying their associated production processes [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] covering a wide kinematic range.The values of σ eff,2 for DPS extracted from the quarkonium data are in general smaller than 10 mb as opposed to 15 mb from other final states at higher scales.However, we should bear in mind that it is still far from being conclusive in view of the remaining large uncertainties.Therefore, it is convincible to say that there is still much room to be improved in the future.
In contrast, the TPS theoretical studies on the market are only focusing on the open heavy-flavor productions [3,22] so far, which are motivated by their relatively large yields.Their complete study by including SPS and DPS is not available yet.We are trying to put triple-J/ψ hadroproduction as a possible TPS case study on the table.We will perform a first complete study by considering SPS, DPS and TPS simultaneously in the paper.

II. THEORETICAL FRAMEWORK
There are three scattering processes SPS, DPS and TPS entering into the calculations of the (differential) cross sections for triple J/ψ hadroproduction, where we have shown one typical Feynman diagram for each mode in Figure 1.As already mentioned, we will assume the zero correlations Eq.(1) in the different partonic scatterings to estimate DPS and TPS rates.In specific, we will use the following formula Therefore, we have to calculate three different SPS cross sections for one, two and three J/ψ production in protonproton collisions.The similar hybrid approach proposed in Refs.[6,9,13] to determine the matrix elements of one, two, and three prompt J/ψ SPS production entering into the computations of NPS contributions will be adopted here.The matrix elements for double and triple prompt J/ψ SPS productions are based on pQCD calculations in the non-relativistic QCD (NRQCD) factorization framework [23], while the single J/ψ hadroproduction is estimated by the data-driven approach in view of the poor understanding of its production mechanism and the availability of precision measurements.For multiple quarkonium SPS productions, its cross-section can be written as where the long-distance matrix elements (LDMEs) O Qi (n i ) follow the power counting of the velocity scaling rule.The leading Fock state Q Q[n] for S-wave quarkonium, with the assumption of the same order of magnitude in the shortdistance coefficients (SDCs) σ, shares the same quantum number J PC and color representation of the quarkonium.Depending on the kinematic region, the working assumption on the size of the SDCs may not always hold.A notorious example is the boosted single inclusive J/ψ production, which receives giant K factors from QCD radiative corrections and is dominated by the subleading Fock states, which can be seen in a recent discussion in Ref. [24].Therefore, one should always bear in mind to carefully check the working assumption case by case.It significantly complicates the corresponding quarkonium phenomenology studies.
The SPS cross sections for single and double J/ψ production have been extensively studied in the literature.One encounters the difficulties to understand single J/ψ production in the NRQCD framework, especially for the subleading color-octet channels.Given the availability of its precision measurements at the LHC covering a wide kinematic regime, we will use the data-driven approach to fit the matrix element of the single J/ψ production with the precise experimental data [6].On the other hand, the pQCD calculation of double J/ψ at leading-order (LO) in v 21 and next-to-leading order (NLO) in α s shows a fairly good agreement with the data when its DPS is small [10] and/or after subtracting the estimated DPS [6,11].In the present paper, we will use partial NLO result of double J/ψ SPS part by including infrared-safe real emission diagrams only, which shows a reasonable agreement with the complete NLO calculation [25].
We also performed the first calculation of the SPS cross-section for triple J/ψ production here.As opposed to double J/ψ production at hadron colliders, the LO SDC in α s at leading v 2 for single and triple J/ψ production must accompany a hard gluon in the final states within the pQCD framework.In other words, it is O( αs v 4 ) compared to their subleading Fock state channels.Therefore, the working assumption of the similar SDC size in different Fock states is worse in odd number J/ψ productions than in even number J/ψ productions.It has indeed concretely been checked in the single and double J/ψ production cases.Since triple J/ψ production is a very rare process and we are interested in its discovery potential at the LHC as well as the future colliders.We will not push J/ψ to the phase-space corners.Hence, we expect the leading 1 ] + g works quite well as long as the transverse momentum of J/ψ is not large.The situation should be quite similar to the single J/ψ production case [26].Even at LO, the process is already quite challenging on both sides of the scattering amplitude computations and the phase-space integrations.There are more than 2 • 10 4 Feynman diagrams to be tackled with.The computation is achieved here for the first time with the help of HELAC-Onia [27,28] due to the virtue of the recursion relations.

III. RESULTS
The numerical calculations for SPS, DPS, and TPS triple J/ψ production are performed in the HELAC-Onia framework.In pQCD parts for double and triple J/ψ yields, we take the charm mass to be 1. 5 GeV, and the central scale , where H T is the sum of the transverse masses of the final states.We will also independently vary the inclusive 2.0 < y J/ψ < 4. proton-proton colliders, where we have also included feeddown contributions from higher-excited quarkonia decay.
renormalization scale µ R and the factorization scale µ F by a factor of 2, i.e. µ R/F = ξ R/F µ 0 with ξ R/F = 0.5, 1, 2. It is conventionally used to estimate the missing higher order in α s , which is the dominant theoretical uncertainty.We choose the proton parton-distribution function as CT14NLO [29].In double and triple J/ψ cross sections, we have also included the feeddown contribution from the excited state ψ(2S).The corresponding LDMEs are estimated in a potential model via , where the squared wavefunctions at the origin are |R J/ψ (0)|2 = 0.81 GeV 3 and |R ψ(2S) (0)| 2 = 0.529 GeV 3 [30].For the single prompt J/ψ production cross section, we use the same ansatz of the averaged amplitude squared Eq.(1) in Ref. [31] and fit to the LHCb data measured at √ s = 7 and 8 TeV [32,33].The final fitted parameters in the ansatz are listed in the top row of Table 1 in Ref. [31].
The inclusive total cross sections as well as those in the LHCb forward rapidity acceptance 2.0 < y J/ψ < 4.5 and the ATLAS/CMS central rapidity acceptance |y J/ψ | < 2.4 are presented in Table I.We have multiplied the branching ratio of J/ψ into muon pairs in the cross-sections.Four different center-of-mass energies √ s = 13, 27, 75, 100 TeV are quoted to represent the LHC and the proposed higher-energy future hadron colliders [1].We have quoted two theoretical uncertainties in each SPS cross-section.The first one is the mentioned remaining renormalization and factorization scale uncertainty, while the second one is the error from the Monte Carlo integration since the phasespace integration is also very challenging.In the DPS cross sections, we only show the uncertainty from the scale variations, because their Monte Carlo errors are negligible.We do not show any theoretical error for TPS as the matrix element of the single J/ψ is determined by the very precise experimental data.In general, the SPS contributions are several orders of magnitude smaller than DPS and TPS cross sections as long as the unknown effective cross sections σ eff,2 and σ eff,3 are not significantly larger than the reference value 10 mb.Such a conclusion holds regardless of the center-of-mass energy √ s and the rapidity cuts on J/ψ.
A few comments on the integrated luminosities at the LHC and future hadron colliders are in order before we move to estimate the expected number of events.ATLAS and CMS experiments have collected around 150 fb −1 during the period of LHC Run2 at √ s = 13 TeV, and the corresponding number for LHCb experiment is 6 fb −1 .There will be two phases for HL-LHC runs [34]. 2 During the phase 1, LHCb aims to deliver 23 fb −1 and ATLAS and CMS aim to deliver 300 fb −1 .The integrated luminosity of LHCb (ATLAS/CMS) will increase to 300 fb −1 (3 ab −1 ) at the phase 2. The nominal integrated luminosities for the future hadron colliders (27 TeV high-energy LHC [34], 75 TeV Super proton-proton Collider [35] and 100 TeV Future Cicular Collider [36]) are in the range of 10 ab −1 to 20 ab −1 .After fixing σ eff,2 = σ eff,3 = 10 mb, we predict the numbers of triple J/ψ events to be 42 +108 −30 and 8 from DPS and TPS respectively with the data on tape recorded by the LHCb detector.These numbers will be 50 times higher at the end of the LHC life according to the targeted luminosity.On the other hand, we cannot directly use the numbers in Table I to estimate the number of events observed by the ATLAS and CMS experiments because of their large magnetic fields and their triggers on the low momentum muons.The lowest transverse momentum P T of J/ψ can be observed at these two detectors are not zero.For instance, the minimal P T of J/ψ in each event is from 4.5 GeV to 6.5 GeV in the CMS double J/ψ measurement [8].The cumulative distributions σ(P T > P T,min ) × Br 3 (J/ψ → µ + µ − ) are shown in Figure 2, where the SPS, DPS and TPS cross section are shown individually.σ(P T > P T,min ) is the cross section with the requirement of P T of each J/ψ candidate larger than P T,min .We have selected events by imposing the rapidity |y J/ψ | < 2.4 and used a Gaussian distribution with k T = 3 GeV to mimic the intrinsic k T smearing effect from the initial states.The TPS cross section decreases faster than the DPS cross section as P T,min increases.It is understood because TPS is typically a higher-twist effect than DPS.In other words, the former is more power suppressed at a higher scale than the latter one.The same argument should in principle be applied to the comparison between SPS and DPS because the latter is also more power suppressed than the former.It seems not the case in Figure 2 because of the caveat we have mentioned in the previous section.Like the case of single J/ψ production, the LO calculation in α s and in v 2 is not sufficient to account for the SPS yields at large P T .They might be strongly enhanced by higher-order QCD radiative corrections and the subleading color-octet channels in the same regime.However, given the substantial suppression of SPS compared to DPS and TPS, we do not expect the inclusion of the new channels will significantly change the total yields after summing of the three contributions when P T < ∼ 10 GeV.Three horizontal dashed lines in the figure are indicated for observing 100 events with the integrated luminosities 150 fb −1 (data on tape), 300 fb −1 (phase 1 of HL-LHC) and 3 ab −1 (phase 2 of HL-LHC) respectively.With 150 fb −1 , one is able to observe more than 100 selected events with P T > 5 GeV.Of course, the Monte Carlo simulations are necessary in order to properly take the realistic experimental conditions into account, for example the trigger, the reconstruction efficiency, and the combinatorial background.Such a study is beyond the scope of our paper.
In order to filter out the TPS events, a good observable is to use the minimal rapidity gap among the three J/ψ mesons.Such an observable has the virtue of insensitive to the intrinsic k T smearing as opposed to other observables like the azimuthal angles.The cumulative distributions σ(|∆y| > |∆y| min ) × Br 3 (J/ψ → µ + µ − ) can be found in Figure 3, where |∆y| is the minimal absolute rapidity difference among the three possible combinations of a J/ψ pair.Because none of the three J/ψ pairs is correlated in TPS, in contrast to SPS and DPS, it has higher possibility to generate an event with a large rapidity gap.Indeed, TPS contribution starts to be dominant when |∆y| > 1.The situation here is quite similar to the absolute rapidity difference in the double J/ψ production, which has been extensively used to extract DPS in the process.The minimal rapidity gap |∆y| can be readily used to determine TPS information in the triple J/ψ production.Thus, it paves the way to study the triple-parton correlations in a proton for the first time via the TPS triple J/ψ process.

IV. CONCLUSIONS
We have proposed to use triple prompt J/ψ production at the LHC and the future hadron colliders to improve our knowledge of the multiple-parton scattering physics.In particular, the TPS has never been observed yet by any experiment.The triple prompt J/ψ hadroproduction can be a very clean process to probe TPS and therefore the possible triple-parton correlations in a proton.We performed a first complete theoretical study of the process by including SPS, DPS and TPS contributions.Especially, we have accomplished the very challenging task of the pQCD calculation for triple J/ψ SPS production at O(α 7 s ), which involves more than 2 • 10 4 Feynman diagrams.Our calculation shows that it is a DPS and TPS dominant process, and therefore a golden channel to probe MPI.Although the process is rare, we have shown that the data on tape after LHC Run2 is already more than enough to measure this process.Finally, we also pointed out that the minimal rapidity gap among three J/ψs is a very useful observable to separate the TPS events from the DPS and SPS events.The horizontal dashed lines are the expected 100 events under targeted integrated luminosities.