Prompt $J/\psi$ production in associated with top quark pair at the LHC

In this work, we investigate the prompt $J/\psi$ production in associated with top quark pair to leading order in the nonrelativistic QCD factorization formalism at the LHC with $\sqrt{s} =13$ TeV. In addition to the contribution from direct $J/\psi$ production, we also include the indirect contribution from the directly produced heavier charmmonia $\chi_{cJ}$ and $\psi^\prime$. We present the numerical results for the total and differential cross sections and find that the $\sideset{^3}{^{(8)}_1}{\mathop{{S}}}$ states give the dominant contributions. The prompt $t\bar t J/\psi$ signatures at the LHC are analyzed in the tetralepton channel $pp\to (t\to W^+(\ell^+\nu)b) (\bar t \to W^-(\ell^- \bar \nu)\bar b) (J/\psi\to\mu^+\mu^-)$ and trilepton channel $pp\to (t\to W(q q^\prime)b) ( t \to W(\ell \nu) b) (J/\psi\to\mu^+\mu^-)$, with the $J/\psi$ mesons decaying into muon pair, and the top quarks decaying leptonically or hadronically. We find that $t\bar t J/\psi$ proudction can be potentially detected at the LHC, whose measurement is useful to test the heavy quarkonium production mechanism.

In this work, we investigate the prompt J/ψ production in associated with top quark pair to leading order in the nonrelativistic QCD factorization formalism at the LHC with √ s = 13 TeV. In addition to the contribution from direct J/ψ production, we also include the indirect contribution from the directly produced heavier charmmonia χcJ and ψ . We present the numerical results for the total and differential cross sections and find that the 3 S states give the dominant contributions. The prompt ttJ/ψ signatures at the LHC are analyzed in the tetralepton channel pp → (t → W + ( + ν)b)(t → W − ( −ν )b)(J/ψ → µ + µ − ) and trilepton channel pp → (t → W (qq )b)(t → W ( ν)b)(J/ψ → µ + µ − ), with the J/ψ mesons decaying into muon pair, and the top quarks decaying leptonically or hadronically. We find that ttJ/ψ proudction can be potentially detected at the LHC, whose measurement is useful to test the heavy quarkonium production mechanism.

I. INTRODUCTION
The nonrelativistic QCD (NRQCD) [1] provides a rigorous theory to study heavy-quarkonium physics, where the production cross section and decay rates can be divided into short-distance and long-distance parts. The short-distance coefficients are process-dependent which can be calculated perturbatively in QCD as expansions in the strong-coupling constant α s . The long-distance matrix elements (LDMEs) are process-independent and univeral which are governed by nonperturbative QCD dynamics and can be extracted from experiments. The relative importance of the LDMEs can be estimated by means of the velocity-scaling rules [2]; i.e., the LDMEs can be classified with a definite power of the realative velocity v of heavy quarks in the bound state within the limit v 1. As a result, the theoretical predictions can be experessed as a double series in α s and v.
Top quark as the heaviest particle in the standard model (SM), play an important role in deciphering the fundamental interactions, and its studies are currently driven the Large Hadron Collider (LHC) experiments. At the LHC, top quarks produced in pairs and associated with bosons through strong interactions, have started to be accessible, like for example ttγ [25][26][27], ttW/Z [28][29][30][31], and ttH [32,33]. With the running of the LHC, more tt associated produciton channels may can be discovered, such as tt production in associated with heavy quarkonium which is helpful to understand the heavy quarkonium production mechanism and deepen our understanding of the strong interaction.
This paper is organized as follows. In Sec. II, we describe the details of the calculation strategies. In Sec. III, we present the numerical results of the total and differential cross sections for ttJ/ψ at the produciton level. In Sec IV, we analyse ttJ/ψ signatures at the LHC with the decay of the top quarks and J/ψ. Discussion and summary are given in Sec. V.

II. THE DETAILS OF THE CALCULATION
In this section, we present the details of the calculation for the process pp → tt + J/ψ of prompt J/ψ production associated with top pair at the LHC in the NRQCD factorization formalism to LO. The cross section for the direct production of charmonium state Q assiciated with top quark pair pp → tt + Q can be expressed as (1) Here O Q [n] is the LDME describing the hadronization of the cc[n] pair into the observable quarkonium state Q, n denotes the cc Fock states contributing at LO in v, which is specified for Q = J/ψ, ψ , χ cJ in Tab. I, with J = 0, 1, 2. From spin symmetry of the heavy quark, the LDMEs at LO in v have the multiplicity relations x, µ f ) represent the distribution function at the scale µ f of gluon which carries the momentum fraction x of the proton A or B. Theσ(gg → tt + cc[n]) decribes the short-distance cross section for the partonic process g(p 1 ) + g(p 2 ) → t(p 3 ) + t(p 4 ) + cc[n](p 5 ) of a cc pair in a Fock state n, and is calculated from the amplitudes using certain projectors onto the usual QCD amplitudes for open cc production. The amplitude for gg → tt + cc[n] involves 36 Feynman diagrams, which are drawn in Fig. 1 representatively.
The LO short-distance cross section can be obtained by performing the integration over the phase space expressed as below,σ The summation is taken over the spins and colors of initial and final states, and the bar over the summation denotes averaging over the spins and colors of initial partons. The Mandelstam variableŝ = (p 1 + p 2 ) 2 , is the partonic centerof-mass system energy. N col and N pol refer to the numbers of color and polarization state of cc[n]. The three-body phase space element can be defined as A refers to the QCD amplitude with amputated heavy-quark spinors. In the notations of Ref. [35]: where the lower index q denotes the momentum of the charm-quark in the cc rest frame. Π 0/1 are spin projectors onto the spin singlet and spin triplet states. C 1/8 are color projectors onto the color-singlet and color-octet states. ε α and ε αβ represent the polarization vector and tensor of the cc states, respectively. c c g g (14) g g t t c c c c g (15) g g t t c c g g c (16) g g t t c c g g g Figure 1. Respresentative diagrams at LO for the partonic process gg → tt + cc[n].

III. THE RESULTS AT THE THE PRODUCTION LEVEL
In this section, we present the numerical results for the production of prompt J/ψ mesons in associated with top pair at the LHC with √ s = 13 TeV. The masses of the charm quark and top quark are taken as m c = 1.5 GeV and m t = 172 GeV. We use the CTEQ6L1 parton distribution functions [36] with a one-loop running α s in the LO

calculations. Our default choice of factorization scale is
c is the charmina Q transverse mass , and p Q T is the charmina Q transverse momentum. We apply the cut to the final charmonia, which is defined as the "basic cut" hereafter. We use FeynArts [37] to generate Feynman diagrams and amplitudes for the partonic process gg → tt+cc[n]. Then the corresponding amplitudes are reduced with FeynCalc [38] and FeynCalcFormLink [39]. At last the numerical calculations are obtained with FormCalc [40].
We adopt the LDMEs for direct J/ψ production from [41,42] as The LDMEs for direct ψ and χ cJ production and are choosen from [43] and [44,45] as and with N c = 3. In our calculations, the relations of LDMEs with conventions of Bodwin-Braaten-Lepage have been considered [35]. In Tab. II, we present the cross section of the direct production of the charmina J/ψ, ψ , χ cJ associated with top quark pair at the 13 TeV LHC. We list the contribution from each cc Fock state, respectively. It can be seen that the contribution from the Fock state 3 S (8)  1 is dominant for all the direct production of charmina J/ψ, ψ , χ cJ in associated with tt at the 13 TeV LHC. Especially for J/ψ (χ c2 ), 3 S (8)  1 state can contribute about 714 (701) fb which is nearly 96.3% (96.2%)of the corresponding total cross section. The summation of cross section for the processes pp → ttχ cJ with J = 0, 1, 2, can reach 1350 fb, which will provide abundant and fascinating studies of phenomenology at the LHC [46].
We summarize the direct and indirect contributions from the radiative or hadronic decays of heavier charmonium states for the prompt J/ψ production in Tab. III. The cross sections of the four residual indirect production channels can be obtained approximately by multiplying the cross sections of the respective intermediate directly produced charmonium states with their decay braching ratios to J/ψ mesons: We can see that the total prompt ttJ/ψ produciton rates at the 13 TeV LHC can reach more than 1100 fb, of which the indirect contribution accounts for about 34%. The indirect contribution mainly comes from the pp → ttχ c2 and pp → ttχ c1 channels which can account for nearly 77% of the total indirect contribution. The indirect contribution from χ c0 is less than 1%, which can be neglected.  In Fig. 2, we present the distributions of the final J/ψ transverse momentum p J/ψ T and the rapidity y J/ψ for the direct production process pp → tt + J/ψ at the 13 TeV LHC. For camparison, we also show the Fock states 3 S  The transverse momentum distributions for the direct production charmina Q in the channels pp → tt + Q are shown in the Fig. 3 on the left. We also plot the p J/ψ T spectra from the indirect contributions in the Fig. 3 on the right. We can find that, at 13 TeV LHC, the p T distribution for the direct production χ c2 and χ c1 are larger in the lower p T region than J/ψ. The behavior of the spectra for the direct and indirect contributions are similar; and the indirect contribution from χ c0 is much less than other channels in the whole plotted p J/ψ T region.

IV. ANALYSES OF ttJ/ψ SIGNATURES AT THE LHC
In this section, we analyse the prompt J/ψ production associated with top quark pair signatures at the LHC. We focus on the J/ψ decaying into a pair of opposite-sign (OS) muons in ttJ/ψ production 1 , which presents as irreducible background the ttµ + µ − production. The events for the background pp → ttµ + µ − are simulated with Madgraph [48]. The J/ψ → µ + µ − decay is implemented with the narrow width approximation (NWA) method, and event weights are rescaled by the branching ratio BR(J/ψ → µ + µ − ) = 5.961%, which is taken from [34].

A. Stable top quarks
We begin our discussion without including top quark decays. For the final muons of the signal and background processes, the following cuts are applied: where p µ ± T and η µ ± are the transverse momentum and pseudorapidity of the final muon and m(µ + µ − ) denotes the invariant mass of the final muon pair. Using the cuts for J/ψ probed in the ATLAS experiment, additional cuts are also applied to the final reconstructed J/ψ mesons [8,9]: where p J/ψ T and y J/ψ are the transverse momentum rapidity of the J/ψ mesons reconstructed from the muon pair. In Fig. 4, we display the normalized distributions of the transverse momentum of the final top (p t T ) and the final reconstructed J/ψ (p J/ψ T ) and the top quark pair invariant masses M tt for the prompt ttJ/ψ(→ µ + µ − ) and the background ttµ + µ − . We can see that the signal decrease faster than the background with increasing p the significance can be maximized based on the cuts (12) and (13). In Tab. IV, we present the total cross sections for tt associated with the direct and indirect production J/ψ mesons including J/ψ mesons decay to muon pair, and the SM background pp → ttµ + µ − , after each cut above at 13 TeV LHC, In the last column, the corresponding prompt significance S = σ signal √ L/ √ σ bkg is also given with the luminosity L=100 fb −1 . After all the selection cuts, we can see that with L=100 fb −1 the total number of events for the prompt ttJ/ψ production with the subsequent decay J/ψ → µ + µ − can account for more than 600 and the corresponding significance can reach more than 100. The reason for the decline of the significance after the cut (13) is that the contributions from the lower p J/ψ T region dominate, which can be seen in the Fig. 2. Therefore, we suggest that the p J/ψ T should be measured as lower as the experiment can to get larger significance and more signal events.

B. Unstable top quarks
Then we continue our discussion including top quark decays. For both the signal and background, the top quarks also decay using NWA method with BR(t → W b) = 100%; both W bosons, resulting from the top-quark decays, are required to decay leptonically or hadronically with Br(W → eν) = 10.71%, Br(W → µν) = 10.63% and Br(W → qq )= 67.41% [34].
1. ttJ/ψ production in tetralepton channel at the LHC First, we consider that all the top quarks leptonically decay, which will bring out the tetralepton signatures with the process pp All the corresponding possible tetralepton channels are listed in Tab.V. Events should contain two pairs of opposite-sign leptons, and at least one pair must be mouns. Table V. List of the tetralepton channels for ttJ/ψ production.
For the channels listed in Tab.V, all the the final leptons in the signal and background events satisfy the following cuts: (15) in the following analyses according to the Sec.IV A, where the J/ψ mesons are reconstructed from one pair of opposite charge mouns. In case the event includes more than one such moun pair, the pair with an invariant mass closet to the nominal value of m J/ψ is attributed to J/ψ mesons and selected to reconstruct the J/ψ meson candidate. The two remaning leptons are considered as top quark decay candidates. The efficiency b we used to correctly tag a b-quark jet is approximately 77%, as determined for b-jets with in simulated tt events [49]. In this paper, only b-jets statisfy the cuts (16)  Table VI. The cross sections (in unit of fb) for the signal from the direct and prompt J/ψ production and background in the tetralepton channels at the 13 TeV LHC. The significances with L = 100 fb −1 are also given.
In Tab. VI, we list the total cross section for the signal of direct and indirect J/ψ production and the SM background in the tetralepton channels after considering the cuts (15) and (16) and the b-tagging efficiency. We also list the prompt production J/ψ significance in the last column with the luminosity L = 100 fb −1 . We can see that for the prompt J/ψ production in all the tetralepton channels the significances are all more than 5 with L = 100 fb −1 . At the future HL-LHC with L = 3000 fb −1 , prompt J/ψ production in associated with top quark pair can cumulate about more than 300 events in the tetralepton channel.
2. ttJ/ψ production in trilepton channel at the LHC In this part, we focus on the production rate of ttJ/ψ events measured for the final state with three leptons in the process pp → (t → W (qq )b)(t → W ( ν)b)(J/ψ → µ + µ − ), where one top quark decays hadronically and the other leptonically. We list all the corresponding possible trilepton channels in Tab.VII. Events should contain three leptons, and at least one pair of opposite-sign mouns. We also use the cut (15) for the final leptons and J/ψ, cut (16) for final b-jets and the b-tagging efficiency b = 77%. In the 3µ2b + / E T channel, the opposite-sign moun pair with an invariant mass closet to m J/ψ is selected to reconstruct the J/ψ meson candidate. In Tab. VII, we summarize the contributions for the total cross section from the signal of J/ψ produced directly and indirectly after the selection cuts and the b-tagging efficiency, respectively. The corresponding irreducible background production rates and the prompt J/ψ significances with L = 100 fb −1 are also list there. We can see that the direct contribution for the prompt J/ψ production in the trilepton channel at 13 TeV LHC can account for about 0.53 fb, and the indirect contribution is half of the direct. Table VII. List of the trilepton channels for ttJ/ψ production. The symbols b and ν denote a b-quark or antiquark and neutrino or antinutrino, respectively, with charge conjugation implied.  3   Table VIII. The cross sections (in unit of fb) for the signal from the direct and prompt J/ψ production and background in the trilepton channels at the 13 TeV LHC. The significances with L = 100 fb −1 are also given.

V. DISCUSSION AND SUMMARY
In this paper, we investigate the prompt J/ψ production in associated with top quark pair to LO in the NRQCD factorization formalism at the 13 TeV LHC. The prompt J/ψ candidates can be produced directly and indirectly. The contributions for the indirect J/ψ production come from radiative decays of χ cJ → J/ψ + γ or hadronic decays of ψ → J/ψ + X. We present the total and differential cross section for the direct and indirect J/ψ production with the basic cut p Q T > 3 GeV. We find that the prompt production rates can account for more than 1100 fb and the Fock state 3 S gives the dominate contribution to the total and differential cross section. Then we present the analyses of the prompt ttJ/ψ signatures at the LHC. We consider the J/ψ decaying into a pair of muons, and take the pp → ttµ + µ − process as the irreducible background. We begin our studies with stable top quarks, and implement the kinematic cuts for the final reconstructed J/ψ mesons with p J/ψ T > 8.5 GeV and |y J/ψ | < 2.1, which were used in the ATLAS experiments before. Through the investigation of the normalized distributions of the p t T , p J/ψ T and M tt , we can get the maximum significance for the prompt J/ψ production with addition cuts of p J/ψ T < 30 GeV. In this situation, we can get more than 600 prompt ttJ/ψ events at 13 TeV LHC with L = 100 fb −1 via the decay channel J/ψ → µ + µ − .
Furtherly, we consider the top quarks decay leptonically or hadronically, and anlysis prompt ttJ/ψ production in tetralepton and trilepton channel at the LHC with the processes pp → (t → W + ( + ν)b)(t → W − ( −ν )b)(J/ψ → µ + µ − ) and pp → (t → W (qq)b)(t → W ( ν)b)(J/ψ → µ + µ − ). In the tetralepton channels, we find that the prompt significance can all be more than 5 with L = 100 fb −1 , and at the future HL-LHC with L = 3000 fb −1 , the total number of events can account for more than 300. In the trilepton channels, the prompt production rates for ttJ/ψ at 13 TeV LHC can cumulate about 0.53 fb. We can find that the ttJ/ψ production at the LHC have the potential to be detected. The measurement of the production J/ψ in associated with top quark pair is useful to investigate the production mechanism of the heavy quarkonium and deepen our understanding about the strong interaction.