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

TL;DR

This paper studies prompt $J/\psi$ production associated with top quark pairs at the 13 TeV LHC, including direct and indirect contributions, and explores its potential detection to test heavy quarkonium production mechanisms.

Contribution

It provides the first detailed calculation of prompt $J/\psi$ production with top pairs at NLO in the nonrelativistic QCD framework, including indirect contributions from heavier charmonia.

Findings

$tar t J/\psi$ production can be potentially observed at the LHC.

The $ ext{S}^{(8)}$ states dominate the production.

Results can test heavy quarkonium production models.

Abstract

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.

Figures

Tables

Table 1. Table 1: Values of k 𝑘 k in the velocity-scaling rule ⟨ 𝒪 𝒬 ​ [ n ] ⟩ ∝ v k proportional-to delimited-⟨⟩ superscript 𝒪 𝒬 delimited-[] 𝑛 superscript 𝑣 𝑘 \langle{\cal O^{Q}}[n]\rangle\propto v^{k} for the leading c ​ c ¯ 𝑐 ¯ 𝑐 c\bar{c} Fock states n 𝑛 n pertinent to 𝒬 𝒬 \cal Q .

k	$J/\psi$, ${\psi }^{\prime }$	${\chi }_{c,J}$
3	${}_{}{}^{3}S_1^{\left(1\right)}$	–
5	–	${}_{}{}^{3}P_J^{\left(1\right)}$,${}_{}{}^{3}S_1^{\left(8\right)}$
7	${}_{}{}^{1}S_0^{\left(8\right)}$, ${}_{}{}^{3}S_1^{\left(8\right)}$, ${}_{}{}^{3}P_J^{\left(8\right)}$	–

Table 2. Table 2: Total cross section (in unit of fb) of the direct charmonium production process p ​ p → t ​ t ¯ + 𝒬 → 𝑝 𝑝 𝑡 ¯ 𝑡 𝒬 pp\to t\bar{t}+{\cal Q} at the 13 TeV LHC with basic cut.

Fock state	${}_{}{}^3𝑆_1^{\left(1\right)}$	${}_{}{}^1𝑆_0^{\left(8\right)}$	${}_{}{}^3𝑆_1^{\left(8\right)}$	${}_{}{}^3𝑃_J^{\left(8\right)}$	${}_{}{}^3𝑃_J^{\left(1\right)}$	Total
$\sigma \left(pp\to t\overline{t}+J/\psi \right)$	0.125	9.57	714.08	17.25	–	741.03
$\sigma \left(pp\to t\overline{t}+{\psi }^{\prime }\right)$	0.077	4.79	127.51	4.93	–	137.31
$\sigma \left(pp\to t\overline{t}+{\chi }_{c2}\right)$	–	–	701.33	–	27.67	729.00
$\sigma \left(pp\to t\overline{t}+{\chi }_{c1}\right)$	–	–	420.80	–	43.95	464.75
$\sigma \left(pp\to t\overline{t}+{\chi }_{c0}\right)$	–	–	140.27	–	16.23	156.50

Table 3. Table 3: Total cross section for J / ψ 𝐽 𝜓 J/\psi production from direct and indirect contributions at 13 TeV LHC with basic cut.

Source	From ${\psi }^{\prime }$	From ${\chi }_{c2}$	From ${\chi }_{c1}$	From ${\chi }_{c0}$	Indirect	Direct	Prompt
$\sigma$ (fb)	84.31	138.51	159.41	2.19	384.42	741.03	1125.45

Table 4. Table 4: The cross sections (in unit of fb) for the signal from the direct and prompt J / ψ 𝐽 𝜓 J/\psi and background after each cut at the 13 TeV LHC. The significance is given with L=100 fb -1 .

	Direct			Indirect			Background	$𝒮$
		From $\psi \left(2S\right)$	From ${\chi }_{c2}$	From ${\chi }_{c1}$	From ${\chi }_{c0}$	Total Indirect
Cut (12)	12.56	1.38	2.37	2.65	0.036	6.44	0.82	209.8
Cut (13)	5.62	0.62	1.05	1.145	0.015	2.83	0.74	98.2
Cut (14)	4.25	0.47	0.79	0.86	0.01	2.13	0.34	109.4

Table 5. Table 5: List of the tetralepton channels for t ​ t ¯ ​ J / ψ 𝑡 ¯ 𝑡 𝐽 𝜓 t\bar{t}J/\psi production.

$t\overline{t}$ decay	$J/\psi$ decay	Channel
$\left(e^{+}\nu b\right)\left(e^{-}\overline{\nu }\overline{b}\right)$	${\mu }^{+}{\mu }^{-}$	$2\mu 2e2b+{\mathrm{E̸}}_T$
$\left(e^{+}\nu b\right)\left({\mu }^{-}\overline{\nu }\overline{b}\right)$	${\mu }^{+}{\mu }^{-}$	$3\mu 1e2b+{\mathrm{E̸}}_T$
$\left({\mu }^{+}\nu b\right)\left(e^{-}\overline{\nu }\overline{b}\right)$	${\mu }^{+}{\mu }^{-}$	$3\mu 1e2b+{\mathrm{E̸}}_T$
$\left({\mu }^{+}\nu b\right)\left({\mu }^{-}\overline{\nu }\overline{b}\right)$	${\mu }^{+}{\mu }^{-}$	$4\mu 2b+{\mathrm{E̸}}_T$

Table 6. Table 6: The cross sections (in unit of fb) for the signal from the direct and prompt J / ψ 𝐽 𝜓 J/\psi production and background in the tetralepton channels at the 13 TeV LHC. The significances with L = 100 fb -1 are also given.

	Direct			Indirect			Background	$𝒮$
		From $\psi \left(2S\right)$	From ${\chi }_{c2}$	From ${\chi }_{c1}$	From ${\chi }_{c0}$	Total Indirect
$2\mu 2e2b+{\mathrm{E̸}}_T$	0.019	0.0021	0.0035	0.0038	0.00005	0.0095	0.0015	7.4
$3\mu 1e2b+{\mathrm{E̸}}_T$	0.038	0.0042	0.0071	0.0077	0.00010	0.019	0.0032	10.1
$4\mu 2b+{\mathrm{E̸}}_T$	0.019	0.0021	0.0035	0.0038	0.00005	0.0095	0.0016	7.1
Combined	0.076	0.0084	0.0141	0.0153	0.00020	0.038	0.0063	14.4

Table 7. Table 7: List of the trilepton channels for t ​ t ¯ ​ J / ψ 𝑡 ¯ 𝑡 𝐽 𝜓 t\bar{t}J/\psi production. The symbols b 𝑏 b and ν 𝜈 \nu denote a b-quark or antiquark and neutrino or antinutrino, respectively, with charge conjugation implied.

$t\overline{t}$ decay	$J/\psi$ decay	Channel
$\left(q{q}^{\prime }b\right)\left(e\nu b\right)$	${\mu }^{+}{\mu }^{-}$	$2\mu 1e2b+{\mathrm{E̸}}_T$
$\left(q{q}^{\prime }b\right)\left(\mu \nu b\right)$	${\mu }^{+}{\mu }^{-}$	$3\mu 2b+{\mathrm{E̸}}_T$

Table 8. Table 8: The cross sections (in unit of fb) for the signal from the direct and prompt J / ψ 𝐽 𝜓 J/\psi production and background in the trilepton channels at the 13 TeV LHC. The significances with L = 100 fb -1 are also given.

	Direct			Indirect			Background	$𝒮$
		From $\psi \left(2S\right)$	From ${\chi }_{c2}$	From ${\chi }_{c1}$	From ${\chi }_{c0}$	Total Indirect
$2\mu 1e2b+{\mathrm{E̸}}_T$	0.266	0.0292	0.050	0.054	0.00073	0.134	0.020	28.3
$3\mu 2b+{\mathrm{E̸}}_T$	0.264	0.0289	0.049	0.053	0.00073	0.132	0.021	27.3
Combined	0.530	0.0581	0.099	0.107	0.00146	0.266	0.041	39.3

Equations

$$\sigma (pp\to t\bar{t}+\mathcal{Q})=\int d{x}_{1}d{x}_2\sum _n⟨{\mathcal{O}}^{\mathcal{Q}}[n]⟩\hat{\sigma }(gg\to t\bar{t}+c\bar{c}[n])[G_{g/A}(x_1,{\mu }_f)G_{g/B}(x_2,{\mu }_f)].$$

$$⟨{\mathcal{O}}^{J/\psi ,{\psi }^{\prime }}[\text{sideset}{}^3{}_J^{(8)}P]⟩$$

$$⟨{\mathcal{O}}^{{\chi }_{cJ}}[\text{sideset}{}^3{}_1^{(8)}S]⟩$$

$$⟨{\mathcal{O}}^{{\chi }_{cJ}}[\text{sideset}{}^3{}_J^{(1)}P]⟩$$

$$\hat{\sigma }(gg\to t\bar{t}+c\bar{c}[n])=\frac{(2\pi {)}^4}{2\hat{s}{N}_{col}{N}_{pol}}\int \overline{\sum }|{\mathcal{A}}_{Q\bar{Q}[n]}{|}^{2}d{\Omega }_3.$$

$$d{\Omega }_3={\delta }^{(4)}\left(p_1+p_2-\sum _{i=3}^5{p}_i\right)\prod _{j=3}^5\frac{d^3{\vec{p}}_j}{(2\pi {)}^{3}2E_j}.$$

$${\mathcal{A}}_{Q\bar{Q}[{}^1{S}_0^{(1/8)}]}$$

$${\mathcal{A}}_{Q\bar{Q}{[}^3{S}_1^{(1/8)}]}$$

$${\mathcal{A}}_{Q\bar{Q}{[}^1{P}_1^{(1/8)}]}$$

$${\mathcal{A}}_{Q\bar{Q}{[}^3{P}_J^{(1/8)}]}$$

$$p_T^{\mathcal{Q}}>3\mathrm{GeV}$$

$$⟨{\mathcal{O}}^{J/\psi }[\text{sideset}{}^3{}_1^{(1)}S]⟩$$

$$⟨{\mathcal{O}}^{J/\psi }[\text{sideset}{}^1{}_0^{(8)}S]⟩$$

$$⟨{\mathcal{O}}^{J/\psi }[\text{sideset}{}^3{}_1^{(8)}S]⟩$$

$$⟨{\mathcal{O}}^{J/\psi }[\text{sideset}{}^3{}_0^{(8)}P]⟩$$

$$⟨{\mathcal{O}}^{{\psi }^{\prime }}[\text{sideset}{}^3{}_1^{(1)}S]⟩$$

$$⟨{\mathcal{O}}^{{\psi }^{\prime }}[\text{sideset}{}^1{}_0^{(8)}S]⟩$$

$$⟨{\mathcal{O}}^{{\psi }^{\prime }}[\text{sideset}{}^3{}_1^{(8)}S]⟩$$

$$⟨{\mathcal{O}}^{{\psi }^{\prime }}[\text{sideset}{}^3{}_0^{(8)}P]⟩$$

$$⟨{\mathcal{O}}^{{\chi }_{c0}}[\text{sideset}{}^3{}_1^{(8)}S]⟩$$

$$⟨{\mathcal{O}}^{{\chi }_{c0}}[\text{sideset}{}^3{}_0^{(1)}P]⟩$$

$${\sigma }^{\mathrm{indirect}}(\mathrm{From}{\psi }^{\prime })=\sigma (pp\to t\bar{t}+{\psi }^{\prime })\times \mathrm{Br}({\psi }^{\prime }\to J/\psi +X)$$

$${\sigma }^{\mathrm{indirect}}(\mathrm{From}{\chi }_{cJ})=\sigma (pp\to t\bar{t}+{\chi }_{cJ})\times \mathrm{Br}({\chi }_{cJ}\to J/\psi +\gamma )$$

$$p_T^{{\mu }^{\pm }}>2.5\mathrm{GeV},|{\eta }_{{\mu }^{\pm }}|<2.3,$$

$$|m({\mu }^{+}{\mu }^{-})-m_{J/\psi }|<0.5\mathrm{GeV},$$

$$p_T^{J/\psi }>8.5\mathrm{GeV},|y_{J/\psi }|<2.1,$$

$$p_T^{J/\psi }<30\mathrm{GeV},$$

$$p_T^{{\ell }^{\pm }}>2.5\mathrm{GeV},|{\eta }_{{\ell }^{\pm }}|<2.3,$$

$$|m({\mu }^{+}{\mu }^{-})-m_{J/\psi }|<0.5\mathrm{GeV},$$

$$8.5\mathrm{GeV}<p_T^{J/\psi }<30\mathrm{GeV},|y_{J/\psi }|<2.1$$

$$p_T^b>20\mathrm{GeV},|{\eta }_b|<2.5$$

Full text

I Introduction

The nonrelativistic QCD (NRQCD) Bodwin:1994jh 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 $\alpha_{s}$. 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 Lepage:1992tx ; 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 $v\ll 1$. As a result, the theoretical predictions can be experessed as a double series in $\alpha_{s}$ and $v$.

The charmonium $J/\psi$ associated-production channels are very important to test the heavy quarkonium physics. At the hadron colliders, such as LHC and Tevatron, there already exist data for $J/\psi+J/\psi$ by LHCb Aaij:2011yc ; Aaij:2016bqq , D0 Abazov:2014qba , CMS Khachatryan:2014iia and ATLAS Aaboud:2016fzt , $J/\psi+Z$ by ATLAS Aad:2014kba , $J/\psi+W$ by ATLAS Aad:2014rua , $J/\psi$ + charm Aaij:2012dz by LHCb. With the future projected LHC luminosities, more data can be available. On the theoretical side, many efforts have been made for the associated-hadroproduction channels, $J/\psi+J/\psi$ Sun:2014gca ; Likhoded:2016zmk , $J/\psi+Z$ Kniehl:2002wd ; Mao:2011kf ; Gong:2012ah , $J/\psi+W$ Kniehl:2002wd ; Li:2010hc , $J/\psi+\gamma$ Li:2008ym ; Lansberg:2009db ; Li:2014ava , $J/\psi$ + charm Baranov:2006dh ; Artoisenet:1900zz ; Qiao:2002nd , $J/\psi$ + bottom Gang:2012js , where some can be know up to next-to-leading-oder (NLO) accuracy for the short-distance coefficient.

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 $t\bar{t}\gamma$ Aad:2015uwa ; Aaboud:2017era ; Sirunyan:2017iyh , $t\bar{t}W/Z$ Aad:2015eua ; Aaboud:2016xve ; Khachatryan:2015sha ; Sirunyan:2017uzs , and $t\bar{t}H$ Aaboud:2018urx ; Sirunyan:2018hoz . With the running of the LHC, more $t\bar{t}$ associated produciton channels may can be discovered, such as $t\bar{t}$ production in associated with heavy quarkonium which is helpful to understand the heavy quarkonium production mechanism and deepen our understanding of the strong interaction.

In this paper, we plan to study the prompt $J/\psi$ + $t\bar{t}$ production at the LHC in the NRQCD factorization formalism to leading-order (LO). Prompt $J/\psi$ candidates that can also be produced indirectly via radiative or hadronic decays of heavier charmonium states, such as $\chi_{cJ}\to J/\psi+\gamma$ and $\psi^{\prime}\to J/\psi+X$, are not distinguished from directly-produced $J/\psi$. The respective decay branching ratios are Tanabashi:2018oca ${\rm Br}(\chi_{c0}\to J/\psi+\gamma)=(1.40\pm 0.05)\%$, ${\rm Br}(\chi_{c1}\to J/\psi+\gamma)=(34.3\pm 1.0)\%$, ${\rm Br}(\chi_{c2}\to J/\psi+\gamma)=(19.0\pm 0.5)\%$, and ${\rm Br}(\psi^{\prime}\to J/\psi+X)=(61.4\pm 0.6)\%$. Contributions to the total $t\bar{t}J/\psi$ production rate can also come from the process of $t\bar{t}+b$-hadrons with weak decays of $b\to J/\psi+X$, which can be separated easily in the detectors. Therefore, we don’t consider the $J/\psi$ production through $b$-hadron decay here.

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 $t\bar{t}J/\psi$ at the produciton level. In Sec IV, we analyse $t\bar{t}J/\psi$ signatures at the LHC with the decay of the top quarks and $J/\psi$. 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 $pp\to t\bar{t}+J/\psi$ of prompt $J/\psi$ 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 $\cal Q$ assiciated with top quark pair $pp\to t\bar{t}+{\cal Q}$ can be expressed as

[TABLE]

Here $\langle{\cal O}^{{\cal Q}}[n]\rangle$ is the LDME describing the hadronization of the $c\bar{c}[n]$ pair into the observable quarkonium state ${\cal Q}$, $n$ denotes the $c\bar{c}$ Fock states contributing at LO in $v$, which is specified for ${\cal Q}=J/\psi,\,\psi^{\prime},\,\chi_{cJ}$ in Tab. 1, with $J=0,1,2$. From spin symmetry of the heavy quark, the LDMEs at LO in $v$ have the multiplicity relations

[TABLE]

$G_{g/A,B}(x,\mu_{f})$ represent the distribution function at the scale $\mu_{f}$ of gluon which carries the momentum fraction $x$ of the proton $A$ or $B$.

The $\hat{\sigma}(gg\to t\bar{t}+c\bar{c}[n])$ decribes the short-distance cross section for the partonic process $g(p_{1})+g(p_{2})\to t(p_{3})+\bar{t}(p_{4})+c\bar{c}[n](p_{5})$ of a $c\bar{c}$ pair in a Fock state $n$, and is calculated from the amplitudes using certain projectors onto the usual QCD amplitudes for open $c\bar{c}$ production. The amplitude for $gg\to t\bar{t}+c\bar{c}[n]$ 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,

[TABLE]

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 variable $\hat{s}=(p_{1}+p_{2})^{2}$, is the partonic center-of-mass system energy. $N_{col}$ and $N_{pol}$ refer to the numbers of color and polarization state of $c\bar{c}[n]$. The three-body phase space element can be defined as

[TABLE]

${\cal A}$ refers to the QCD amplitude with amputated heavy-quark spinors. In the notations of Ref. Petrelli:1997ge :

[TABLE]

where the lower index $q$ denotes the momentum of the charm-quark in the $c\bar{c}$ rest frame. $\Pi_{0/1}$ are spin projectors onto the spin singlet and spin triplet states. ${\cal C}_{1/8}$ are color projectors onto the color-singlet and color-octet states. ${\varepsilon}_{\alpha}$ and ${\varepsilon}_{\alpha\beta}$ represent the polarization vector and tensor of the $c\bar{c}$ states, respectively.

III The results at the the production level

In this section, we present the numerical results for the production of prompt $J/\psi$ mesons in associated with top pair at the LHC with $\sqrt{s}=13$ 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 Pumplin:2002vw with a one-loop running $\alpha_{s}$ in the LO calculations. Our default choice of factorization scale is $\mu_{f}=m_{T}^{\cal Q}$, where $m_{T}^{\cal Q}=\sqrt{(p_{T}^{\cal Q})^{2}+4m_{c}^{2}}$ is the charmina ${\cal Q}$ transverse mass , and $p_{T}^{\cal Q}$ is the charmina ${\cal Q}$ transverse momentum. We apply the cut

[TABLE]

to the final charmonia, which is defined as the “basic cut” hereafter. We use FeynArts Hahn:2000kx to generate Feynman diagrams and amplitudes for the partonic process $gg\to t\bar{t}+c\bar{c}[n]$. Then the corresponding amplitudes are reduced with FeynCalc Shtabovenko:2016sxi and FeynCalcFormLink Feng:2012tk . At last the numerical calculations are obtained with FormCalc Hahn:1998yk .

We adopt the LDMEs for direct $J/\psi$ production from Fleming:1997fq ; Kniehl:2001tk as

[TABLE]

The LDMEs for direct $\psi^{\prime}$ and $\chi_{cJ}$ production and are choosen from Kniehl:2006qq and Eichten:1995ch ; Ma:2010vd as

[TABLE]

and

[TABLE]

with $N_{c}=3$. In our calculations, the relations of LDMEs with conventions of Bodwin-Braaten-Lepage have been considered Petrelli:1997ge .

In Tab. 2, we present the cross section of the direct production of the charmina $J/\psi$, $\psi^{\prime}$, $\chi_{cJ}$ associated with top quark pair at the 13 TeV LHC. We list the contribution from each $c\bar{c}$ Fock state, respectively. It can be seen that the contribution from the Fock state $\sideset{{}^{3}}{{}^{(8)}_{1}}{\mathop{{S}}}$ is dominant for all the direct production of charmina $J/\psi$, $\psi^{\prime}$, $\chi_{cJ}$ in associated with $t\bar{t}$ at the 13 TeV LHC. Especially for $J/\psi$ ($\chi_{c2}$), $\sideset{{}^{3}}{{}^{(8)}_{1}}{\mathop{{S}}}$ 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 $pp\to t\bar{t}\chi_{cJ}$ with $J=0,1,2$, can reach 1350 fb, which will provide abundant and fascinating studies of phenomenology at the LHC nextwork .

We summarize the direct and indirect contributions from the radiative or hadronic decays of heavier charmonium states for the prompt $J/\psi$ production in Tab. 3. 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 $J/\psi$ mesons:

[TABLE]

We can see that the total prompt $t\bar{t}J/\psi$ 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 $pp\to t\bar{t}\chi_{c2}$ and $pp\to t\bar{t}\chi_{c1}$ channels which can account for nearly 77% of the total indirect contribution. The indirect contribution from $\chi_{c0}$ is less than 1%, which can be neglected.

In Fig. 2, we present the distributions of the final $J/\psi$ transverse momentum $p_{T}^{J/\psi}$ and the rapidity $y_{J/\psi}$ for the direct production process $pp\to t\bar{t}+J/\psi$ at the 13 TeV LHC. For camparison, we also show the Fock states $\sideset{{}^{3}}{{}^{(1)}_{1}}{\mathop{{S}}}$, $\sideset{{}^{1}}{{}^{(8)}_{0}}{\mathop{{S}}}$, $\sideset{{}^{3}}{{}^{(8)}_{1}}{\mathop{{S}}}$ and $\sideset{{}^{3}}{{}^{(8)}_{J}}{\mathop{{P}}}$ there. We can see that the $J/\psi$ spectrum of all the Fock states steadily decrease with the increment of $p_{T}^{J/\psi}$. In the range of $3\,{\rm GeV}\leq p_{T}^{J/\psi}\leq 40\,{\rm GeV}$, the $d\sigma/dp_{T}^{J/\psi}$ is in the range of [0.91, 154.18] fb/GeV. We can see that the $\sideset{{}^{3}}{{}^{(8)}_{1}}{\mathop{{S}}}$ state gives the dominant contribution in the whole plotted region, and the curves of total and $\sideset{{}^{3}}{{}^{(8)}_{1}}{\mathop{{S}}}$ state contribution for the distribution are almost overlap. For the $y_{J/\psi}$ distributions, at the center region $y_{J/\psi}=0$, the contributions from $\sideset{{}^{3}}{{}^{(1)}_{1}}{\mathop{{S}}}$ and $\sideset{{}^{3}}{{}^{(8)}_{1}}{\mathop{{S}}}$ states reach their maximum, while the contributions from $\sideset{{}^{1}}{{}^{(8)}_{0}}{\mathop{{S}}}$ and $\sideset{{}^{3}}{{}^{(8)}_{J}}{\mathop{{P}}}$ states reach their minimum.

The transverse momentum distributions for the direct production charmina ${\cal Q}$ in the channels $pp\to t\bar{t}+{\cal Q}$ are shown in the Fig. 3 on the left. We also plot the $p_{T}^{J/\psi}$ 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 $\chi_{c2}$ and $\chi_{c1}$ are larger in the lower $p_{T}$ region than $J/\psi$. The behavior of the spectra for the direct and indirect contributions are similar; and the indirect contribution from $\chi_{c0}$ is much less than other channels in the whole plotted $p_{T}^{J/\psi}$ region.

IV Analyses of $t\bar{t}J/\psi$ signatures at the LHC

In this section, we analyse the prompt $J/\psi$ production associated with top quark pair signatures at the LHC. We focus on the $J/\psi$ decaying into a pair of opposite-sign (OS) muons in $t\bar{t}J/\psi$ production 111The $J/\psi\to e^{+}e^{-}$ channel is not used because the ATLAS detector has poor signal effciency for the $J/\psi$ relevant momentum range Bertsche:2016skf . , which presents as irreducible background the $t\bar{t}\mu^{+}\mu^{-}$ production. The events for the background $pp\to t\bar{t}\mu^{+}\mu^{-}$ are simulated with Madgraph Alwall:2014hca . The $J/\psi\to\mu^{+}\mu^{-}$ decay is implemented with the narrow width approximation (NWA) method, and event weights are rescaled by the branching ratio BR$(J/\psi\to\mu^{+}\mu^{-})=5.961\%$, which is taken from Tanabashi:2018oca .

IV.1 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:

[TABLE]

where $p_{T}^{\mu^{\pm}}$ and $\eta_{\mu^{\pm}}$ are the transverse momentum and pseudorapidity of the final muon and $m(\mu^{+}\mu^{-})$ denotes the invariant mass of the final muon pair. Using the cuts for $J/\psi$ probed in the ATLAS experiment, additional cuts are also applied to the final reconstructed $J/\psi$ mesons Aad:2014rua ; Aad:2014kba :

[TABLE]

where $p_{T}^{J/\psi}$ and $y_{J/\psi}$ are the transverse momentum rapidity of the $J/\psi$ 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 $J/\psi$ ($p_{T}^{J/\psi}$) and the top quark pair invariant masses $M_{t\bar{t}}$ for the prompt $t\bar{t}J/\psi(\to\mu^{+}\mu^{-})$ and the background $t\bar{t}\mu^{+}\mu^{-}$. We can see that the signal decrease faster than the background with increasing $p_{T}^{J/\psi}$, $M_{t\bar{t}}$ and $p_{T}^{t}$. In order to increase the significance ${\cal S}=S/\sqrt{B}$, where $S$ ($B$) is the number of signal (background) events, we investigate the $p_{T}^{J/\psi}$, $M_{t\bar{t}}$ and $p_{T}^{t}$ cuts effects, and find that when

[TABLE]

the significance can be maximized based on the cuts (12) and (13).

In Tab. 4, we present the total cross sections for $t\bar{t}$ associated with the direct and indirect production $J/\psi$ mesons including $J/\psi$ mesons decay to muon pair, and the SM background $pp\to t\bar{t}\mu^{+}\mu^{-}$, after each cut above at 13 TeV LHC, In the last column, the corresponding prompt significance ${\cal S}={\sigma^{\rm signal}}\sqrt{\rm L}/\sqrt{\sigma^{\rm bkg}}$ 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 $t\bar{t}J/\psi$ production with the subsequent decay $J/\psi\to\mu^{+}\mu^{-}$ 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 $p_{T}^{J/\psi}$ region dominate, which can be seen in the Fig. 2. Therefore, we suggest that the $p_{T}^{J/\psi}$ should be measured as lower as the experiment can to get larger significance and more signal events.

IV.2 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($t\to Wb$) = 100%; both $W$ bosons, resulting from the top-quark decays, are required to decay leptonically or hadronically with Br($W\to e\nu$) = 10.71%, Br($W\to\mu\nu$) = 10.63% and Br($W\to qq^{\prime}$)= 67.41% Tanabashi:2018oca .

IV.2.1 $t\bar{t}J/\psi$ 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 $pp\to(t\to W^{+}(\ell^{+}\nu)b)(\bar{t}\to W^{-}(\ell^{-}\bar{\nu})\bar{b})(J/\psi\to\mu^{+}\mu^{-})$. All the corresponding possible tetralepton channels are listed in Tab.5. Events should contain two pairs of opposite-sign leptons, and at least one pair must be mouns.

For the channels listed in Tab.5, all the the final leptons in the signal and background events satisfy the following cuts:

[TABLE]

in the following analyses according to the Sec.IV.1, where the $J/\psi$ 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 $m_{J/\psi}$ is attributed to $J/\psi$ mesons and selected to reconstruct the $J/\psi$ meson candidate. The two remaning leptons are considered as top quark decay candidates. The efficiency $\epsilon_{b}$ we used to correctly tag a $b$-quark jet is approximately 77%, as determined for b-jets with

[TABLE]

in simulated $t\bar{t}$ events btag . In this paper, only b-jets statisfy the cuts (16) are selected.

In Tab. 6, we list the total cross section for the signal of direct and indirect $J/\psi$ 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 $J/\psi$ significance in the last column with the luminosity L = 100 fb*-1*. We can see that for the prompt $J/\psi$ 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 $J/\psi$ production in associated with top quark pair can cumulate about more than 300 events in the tetralepton channel.

IV.2.2 $t\bar{t}J/\psi$ production in trilepton channel at the LHC

In this part, we focus on the production rate of $t\bar{t}J/\psi$ events measured for the final state with three leptons in the process $pp\to(t\to W(qq^{\prime})b)(t\to W(\ell\nu)b)(J/\psi\to\mu^{+}\mu^{-})$, where one top quark decays hadronically and the other leptonically. We list all the corresponding possible trilepton channels in Tab.7. 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 $J/\psi$, cut (16) for final $b$-jets and the b-tagging efficiency $\epsilon_{b}=77\%$. In the $3\mu 2b+{\not{E}}_{T}$ channel, the opposite-sign moun pair with an invariant mass closet to $m_{J/\psi}$ is selected to reconstruct the $J/\psi$ meson candidate. In Tab. 7, we summarize the contributions for the total cross section from the signal of $J/\psi$ produced directly and indirectly after the selection cuts and the b-tagging efficiency, respectively. The corresponding irreducible background production rates and the prompt $J/\psi$ significances with L = 100 fb*-1* are also list there. We can see that the direct contribution for the prompt $J/\psi$ 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.

V Discussion and summary

In this paper, we investigate the prompt $J/\psi$ production in associated with top quark pair to LO in the NRQCD factorization formalism at the 13 TeV LHC. The prompt $J/\psi$ candidates can be produced directly and indirectly. The contributions for the indirect $J/\psi$ production come from radiative decays of $\chi_{cJ}\to J/\psi+\gamma$ or hadronic decays of $\psi^{\prime}\to J/\psi+X$. We present the total and differential cross section for the direct and indirect $J/\psi$ production with the basic cut $p_{T}^{\cal Q}>3$ GeV. We find that the prompt production rates can account for more than 1100 fb and the Fock state $\sideset{{}^{3}}{{}^{(8)}_{1}}{\mathop{{S}}}$ gives the dominate contribution to the total and differential cross section.

Then we present the analyses of the prompt $t\bar{t}J/\psi$ signatures at the LHC. We consider the $J/\psi$ decaying into a pair of muons, and take the $pp\to t\bar{t}\mu^{+}\mu^{-}$ process as the irreducible background. We begin our studies with stable top quarks, and implement the kinematic cuts for the final reconstructed $J/\psi$ mesons with $p_{T}^{J/\psi}>8.5$ GeV and $|y_{J/\psi}|<2.1$, which were used in the ATLAS experiments before. Through the investigation of the normalized distributions of the $p_{T}^{t}$, $p_{T}^{J/\psi}$ and $M_{t\bar{t}}$, we can get the maximum significance for the prompt $J/\psi$ production with addition cuts of $p_{T}^{J/\psi}<30$ GeV. In this situation, we can get more than 600 prompt $t\bar{t}J/\psi$ events at 13 TeV LHC with L = 100 fb*-1* via the decay channel $J/\psi\to\mu^{+}\mu^{-}$.

Furtherly, we consider the top quarks decay leptonically or hadronically, and anlysis prompt $t\bar{t}J/\psi$ production in tetralepton and trilepton channel at the LHC with the processes $pp\to(t\to W^{+}(\ell^{+}\nu)b)(\bar{t}\to W^{-}(\ell^{-}\bar{\nu})\bar{b})(J/\psi\to\mu^{+}\mu^{-})$ and $pp\to(t\to W(q\bar{q})b)(t\to W(\ell\nu)b)(J/\psi\to\mu^{+}\mu^{-})$. 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 $t\bar{t}J/\psi$ at 13 TeV LHC can cumulate about 0.53 fb. We can find that the $t\bar{t}J/\psi$ production at the LHC have the potential to be detected. The measurement of the production $J/\psi$ 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.

Acknowledgements.

We thank Jinmian Li and Ze-Bo Tang for helpful discussions. This work was supported in part by the National Natural Science Foundation of China (Grants No. 11805001, No. 11305001, No. 11575002, No. 11675033, and No. 11747317) and the Key Research Foundation of the Education Ministry of Anhui Province of China (Grant No. KJ2017A032).