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Doubly-heavy tetraquarks including
T{cc}[ˉqˉq′] ,T{bc}[ˉqˉq′] , andT{bb}[ˉqˉq′] have drawn considerable attention from theorists after the discovery of the first double-charm hadronΞ++cc [1]. The doubly-heavy tetraquarks can shed light on strong dynamics and greatly help us identify the nature of the exoticXYZ states or structures, e.g., cusps or true resonances. In addition, several studies of both heavy quark symmetry and lattice [2-8] suggest that some of them have masses below the threshold and thus can only decay weakly. If such weakly-decaying tetraquarks are found, they will provide evidence of compact diquarks as building blocks of hadronic matter. Among these doubly-heavy tetraquarks,T{cc}[ˉqˉq′] is the most promising one to be observed in experiments in the near future, because both its production rate and detection efficiency are the highest at the LHCb. In this work, we analyze the potential of findingT{cc}[ˉqˉq′] at the LHCb and also future Z-factories [9, 10], evaluating their production cross sections at the two kinds of facilities and proposing possible golden channels according to their decay properties.Decay properties of the doubly-heavy tetraquarks
T{cc}[ˉqˉq′] rely on whether their masses are above or below the thresholdsD0D∗+ andD∗D+s . If they are stable particles below the threshold, they will mainly decay radiatively (or even weakly according to [11]) and thus have a relatively long lifetime, which will considerably suppress the background and make the experimental search easier. In contrast, if they are unstable particles above the thresholds①, they will decay strongly and still have a good chance to be found at the LHCb using a method similar to that in [12]. Their masses have been calculated in many different procedures, such as QCD sum rule [13-17], lattice [18], Bethe-Salpeter equation approach [11], heavy quark symmetry [3, 19], and several QCD inspired models [2, 20-32]. The results are listed in Table 1 in the form of the mass differences ofT{cc}ˉnˉn′ andT{cc}ˉnˉs (n(′)=u or d) from their thresholdsD0D∗+ andD∗D+s , respectively. These results are far from conclusive because some of the calculations suggest they lie above the thresholds while the others are against it. As a result, the decay channels ofT{cc}[ˉqˉq′] are analyzed case by case.Table 1. Theoretical predictions on the mass differences of
T{cc}ˉnˉn′ andT{cc}ˉnˉs from their thresholdsD0D∗+ andD∗D+s , respectively, in units of MeV. The quarksn,n′ = u or d. For some works without explicit numerical results, we use "AT" and "BT" denoting "above threshold" and "below threshold," respectively.The rest of the paper is organized as follows. In section II, we study the production of
T{cc}[ˉqˉq′] at both the LHCb and future Z-factories. According to our estimation of the production cross sections, the LHCb has producedO(108) double-charm tetraquarks to date, and this number will increase by almost one order of magnitude by the end of Run 4 [33, 34]. It is also shown that a Tera-Z factory will produceO(106) double-charm tetraquarks in a very clean environment. In section III, possible decay channels ofT{cc}[ˉqˉq′] are discussed, in three cases with the double-charm tetraquarks either above or below theDD∗ andDDγ thresholds. For the LHCb, we propose some decay channels to search for the double-charm tetraquarks, because they have relatively large branching ratios, and more importantly, all of their final-state particles have high detection efficiencies, e.g.,T{cc}[ˉuˉd]→D0D∗+→ D0D0π+ andT{cc}[ˉuˉd]→D+K−π+ . In contrast, future Z-factories have an advantage over the LHCb in searching for other decay channels containing a photon, e.g.,T{cc}[ˉuˉd]→ D0D+γ . We conclude our study in section IV. -
Following the approach proposed in [35-37], we estimate the production rate of double-charm hadrons
Hcc at the LHCb and future Z-factories. The basic idea is as follows. First, the underlying processes for their production at the quark level arepp→ccˉcˉc+X ande+e−→Z→ ccˉcˉc at these two kinds of facilities, respectively. The two charm quarks must stay close enough to form acc diquark jet, which further fragments into different kinds of double-charm hadrons. The invariant mass of the two heavy quarksmQQ′ is used to parameterize their collinear level, and ifmQQ′ is smaller than some cut-off valueMQQ′(ΔM)≡mQ+mQ′+ΔM , we regard the two heavy quarks as aQQ′ diquark that will eventually produce a doubly-heavy hadron. In contrast, ifmQQ′ exceedsMQQ′(ΔM) , the two heavy quarks fragment separately. The parameterΔM can be determined by matching the partonicbˉc production simulation to the experimental measurements of theBc meson production cross sections [38], and it has been found by [35, 37] thatΔM={(2.0+0.5−0.4)GeV,forLHCb,(2.7+1.3−0.5)GeV,forZfactories.
(1) Next, we will discuss the production of the doubly-heavy hadrons at the LHCb and future Z-factories separately.
LHCb: For the analysis of
Hcc production at the LHC with√s=13 TeV, we generate105 pp→ccˉcˉc+X events via the Monte Carlo generator MadGraph5_aMC@NLO [39] at the next-to-leading-order level showered by Pythia8 [40, 41]. It is found that the total cross sectionσ(pp→ccˉcˉc+X)≈4.9×107 pb, and after applying the invariant mass cut in (1), we obtainσ(p+p→Hcc+X)=(3.1+1.7−0.7)×105 pb,
(2) in the transverse momentum range
4<pT<15 GeV and the pseudorapidity range2<η<4.5 .The double-charm hadrons
Hcc include both the double-charm baryonsΞ++cc(ccu) ,Ξ+cc(ccd) , andΩ−cc(ccs) and the double-charm tetraquarksT{cc}[ˉqˉq′] and their excited states. Therefore, we still need to know the relative fractions ofHcc to these hadrons. It is difficult to calculate these non-perturbative fragmentation ratios, while we can approximately borrow the corresponding ratios of a b-quark jet under the heavy quark - heavy diquark symmetry. Such ratios have been measured by the LHCb [42, 43] as[fΛ0bfd](pT)=(0.151±0.030)+exp[(−0.57±0.11)−(0.095±0.016)pT(GeV)],[fsfd](pT)=(0.263±0.008)+((−17.6±2.1)×10−4)⋅pT(GeV),
(3) which are
pT dependent. In addition, we assume that for states containing the same valence quarks, the ground state fraction is approximatelyrg=0.48±0.08 [44]. Then, convoluting thepT -distribution of thecc -diquark jets fetched from the simulated events with thepT -dependent fragmentation functions (3), we estimate the direct production cross sections of the double-charm tetraquarks to beσ(pp→T{cc}[ˉuˉd]+X)=(24+14−7) nb,σ(pp→T{cc}[ˉuˉs]+X)=σ(pp→T{cc}[ˉdˉs]+X)=(6.0+3.5−1.7) nb.
(4) The
pT distribution of thepp→T{cc}[ˉuˉd]+X events from the simulation is displayed in Fig. 1. In contrast, because almost all the excited states ofΞcc andΩcc decay into their ground states, we do not need to multiply therg ratio for their production cross section, which readsFigure 1. (color online) Projected
pT -dependence of the tetraquark production cross section inpp→T{cc}[ˉuˉd]+X at the LHC for√s=13 TeV with2<η<4.5 .σ(pp→Ξ++cc+X)=σ(pp→Ξ+cc+X)=(103+56−22) nb,σ(pp→Ω+cc+X)=(26+14−6) nb.
(5) The
Ξcc production cross section has been calculated in the framework of non-relativistic QCD asσ(Ξ++cc)= σ(Ξ+cc)≈62 nb [45], with the experimental cutspT> 4 GeV and|η|≤1.5 . To compare with this result, we reset the same cuts and findσ(Ξ++cc)=σ(Ξ+cc)≈100 nb. Taking into account the large uncertainties, they agree with each other. In addition, theΞ++cc production has been studied by the LHCb, with a relative production rate given as [46]σ(Ξ++cc)×B(Ξ++cc→Λ+cK−π+π+)σ(Λ+c)=(2.22±0.27±0.29)×10−4,
(6) choosing the cuts
4<pT<15 GeV and2<η<4.5 . From an LHCb measurement with√s=7 TeV [47], it can be extracted thatσ(Λ+c+Λ−c)≈38 μb with4<pT<8 GeV,2.0<η<4.5 (thepT>8 GeV contributions are negligible). By comparing the D meson production at 13 TeV [48] and 7 TeV [47], one expects that theΛc production rate with√s=13 TeV should exhibit an enhancement of approximately 50%, i.e.,σ(Λ+c+Λ−c)≈57 μb. AssumingB(Ξ++cc→Λ+cK−π+π+)∈ [5,20]% as suggested in [49], one can solve from (6) thatσ(Ξ++cc)∈[30,130] nb, which is also consistent with our result (5).The LHCb collaboration has collected approximately 9 fb
−1 of data up to now, which indicatesO(108) T{cc}[ˉqˉq′] particles. If reconstructed by proper decay channels, as we analyze in the next section, they have a good chance to be discovered in the near future.Z-factories: For future Z-factories which are designed to produce
1012 or more Z bosons, we also generate105 e+e−→Z→ccˉcˉc events via MadGraph5 [39] and Pythia8 [40, 41]. We find that the total cross sectionσ(e+e−→ccˉcˉc)≈120 pb, and after the diquark jet conditionmcc<Mcc(ΔM) is applied, the decay branching ratio of the Z boson into double-charm hadrons is found to beB(Z→Hcc+X)=(10.1+7.9−2.7)×10−5.
(7) Analogous to the analysis of the LHCb, we assume that the fragmentation fraction
f((cc)jet→T{cc}[ˉuˉd]+X)/ f((cc)jet→Ξcc+X) is close to two timesfΛb/(fBu+fBd)= 0.11±0.02 [50],fs/fd≈0.25 , and the ground-state fraction is stillrg=0.48±0.08 [44]. With these approximations, the branching ratios of the Z boson decays into double-charm tetraquarks are estimated to beB(Z→T{cc}[ˉuˉd]+X)=(4.1+3.4−1.5)×10−6,B(Z→T{cc}[ˉuˉs]+X)=B(Z→T{cc}[ˉdˉs]+X)=(1.0+0.8−0.4)×10−6.
(8) We also give the Z decay branching ratios to the ground-state double-charm baryons, including those secondarily decaying from their excited states,
B(Z→Ξ++cc+X)=B(Z→Ξ+cc+X)=(3.9+3.1−1.1)×10−5,B(Z→Ω+cc+X)=(1.0+0.8−0.3)×10−5.
(9) It can be seen that our estimation for the
Ξcc production rate at Z-factories is consistent with the NRQCD calculationB(Z→Ξcc+X)≈(1.7+1.3−0.6)×10−5 [51]. Therefore, it is expected that future Tera-Z factories will produceO(106) T{cc}[ˉqˉq′] particles. Owing to the clean background of electron-positron collisions, future Z-factories have the advantage of measuring some of the decay channels ofT{cc}[ˉqˉq′] containing neutral particles such as photons in their final states, while it is difficult for hadron colliders. More details are given in the next section. -
In addition to the production rates, the decaying processes are another important issue in the experimental searches for the double-charm tetraquarks. An example of the importance of decaying processes is the discovery of the first doubly charmed baryon. With the prediction of the most favorable decay channels of
Ξ++cc→Λ+cK−π+π+ andΞ+cπ+ [49], the LHCb collaboration observedΞ++cc via the above two modes [1, 52].The decaying properties depend on the masses of
T{cc}[ˉqˉq′] . Above theDD∗ thresholds, they will decay strongly. In contrast, between theDD∗ andDDγ thresholds, they will decay radiatively; below theDDγ thresholds, they will decay weakly. Since the theoretical predictions on the masses are very different from each other, as shown in Table 1, we will discuss the decay processes separately according to the above cases.In the case that the masses of double-charm tetraquarks are above the thresholds of
DD∗ , they are unstable and will decay strongly. The decaying processes includeT{cc}[ˉuˉd]→D0D∗+orD+D∗0 ,T{cc}[ˉuˉs]→D0D∗+sorD+sD∗0 , andT{cc}[ˉdˉs]→D+D∗+sorD+sD∗+ . The reconstructions ofD∗0 andD∗+s require the reconstructions of neutral particlesπ0 orγ , which will reduce the detection efficiencies significantly. Then, the favorable decay processes of the LHCb areT{cc}[ˉuˉd]→D0D∗+→D0D0π+,T{cc}[ˉdˉs]→D+sD∗+→D+sD0π+.
(10) If their masses are above the
DDγ thresholds but below theDD∗ thresholds, they will decay radiatively, such asT{cc}[ˉuˉd]→D0D+γ ,T{cc}[ˉuˉs]→D0D+sγ , andT{cc}[ˉdˉs]→D+D+sγ . These channels are expected to be better measured at future Z-factories.In the case of
T{cc}[ˉqˉq′] below theDDγ thresholds, they can only decay weakly and are thus stable particles. This is different from the exotic states discovered in experiments so far, which decay strongly. The scales of heavy quark weak decays, atO (GeV), are different from the ones of strong decays at the scale of hundreds of MeV. Thus, it will provide another way to distinguish the nature of hadrons to study the weak decays of such particles. Since they can only weakly decay, they must be long-lived particles and cannot be kinematical effects or coupled-channel effects. Due to large non-perturbative contributions, it is difficult to precisely predict the branching fractions of weak decays of charmed hadrons. Considering the Cabibbo-favored processes with charged particles in the final states, the most favorable processes of weak decays at the LHCb areT{cc}[ˉuˉd]→D+K−π+,T{cc}[ˉuˉs]→D+sK−π+,orD0π+,orD+K+K−,T{cc}[ˉdˉs]→D+K+K−π+,orD+π+.
(11) Considering a more abundant production of
T{cc}[ˉuˉd] compared to the ones containg a strange quark, we will discuss the possible signal events ofT{cc}[ˉuˉd] in experiments. If it decays strongly, it decays intoD0D∗+ withD∗+→ D0π+ ,D0→K−π+ , andK−π+π+π− . The total branching fraction would beO(10−2) . In case it is a stable particle, the branching fraction ofT{cc}[ˉuˉd]→D+K−π+ would be of the order of 10%. Considering the further decayD+→K−π+π+ , the total branching fraction of the weak decay ofT{cc}[ˉuˉd] would beO(10−2) as well. The order of the branching fractions ofT{cc}[ˉuˉd] decays is thus the same as that of the observedΞ++cc . Comparing with the production rates between double-charm tetraquarks and baryons and considering approximately2×103 events ofΞ++cc with the current LHCb data, the signal yields ofT{cc}[ˉuˉd] would beO(102) at LHCb, either for a strongly or weakly decayingT{cc}[ˉuˉd] . This number will reachO(103) at LHCb Run III. Therefore, it is expected that the double-charm tetraquark will be observed at the LHCb in the near future. Although the production rates are smaller at the future Z factories, it is also expected to be observed at the Tera-Z factories due to smaller backgrounds.
Discovery potentials of double-charm tetraquarks
- Received Date: 2021-07-29
- Available Online: 2021-10-15
Abstract: In this study, we investigate the discovery potential of double-charm tetraquarks