Colour and baryon number fluctuation of preconfinement system in production process and $T_{cc}$ structure

The quark model predicts quark-antiquark and three-quark hadrons, i.e., mesons and baryons. In these hadrons, quarks are bound by the strong interactions and they are in colour singlet. However, a cluster of more (anti)quarks can also be in colour singlet. It is possible for this kind of colour-singlet clusters with modest mass to be a ’real’ particle. These belong to the exotic hadrons, among whom the four-quark states ($qq\bar{q}\bar{q}$, tetraquark) and five-quark states ($qqqq\bar{q}$, pentaquark) with at least one heavy quark ($c,~{}b$) have been observed. Almost all the four- and five-quark states are observed to be produced from decays of heavier hadrons (e.g., bottom hadrons) rather than promptly produced from multi-production processes in experiments. From the theoretical aspects, this fact is understood that their production rate generally is quite small in multiproduction processes because of the unitarity constraint Han:2009jw ; Jin:2016vjn as well as the modest mass of the preconfinement clusters Li:2019vrc .

All kinds of the multiquark state hadrons have one common property that the bound (anti)quarks inside can be grouped into several clusters, with each cluster possibly in colour-singlet. Hence, these hadrons could just be hadron molecules rather than real multiquark states. However, the ways of grouping these (anti)quarks in a multiquark state hadron are not unique, as it is simply known from the group theory that the reduction ways for a direct product of several representations are not unique. Furthermore, these clusters in one hadron need not necessarily be in colour-singlet, since the only requirement is that the whole set of these clusters are in colour-singlet. For example, the system $q_{1}\bar{q_{2}}q_{3}\bar{q_{4}}$ (the constituents of a four quark state) can be decomposed/clustered in the following ways:

Here we just mention that such group theory analysis is applicable to the quark states as well as the quark field operators pire . In the above example, only the second case, when these two $q\bar{q}$ pairs are both in colour-singlet, it seems possible to be considered as a hadron molecule. However dynamically, the colour interactions in the system via exchanging gluons can change the colour state of each individual cluster, so each kind of grouping/reduction way seems having no special physical reasons. Such an ambiguity, which has been considered in many hadronization and decay processes as “colour recombination/rearrangement” our1 ; our2 ; gsj , obstacles the possibility to consider the multiquark hadron in a unique and uniform way, while leads to the possibility of introducing some phenomenological duality. Namely, even we consider the production of multiquark hadron as “hadron molecule” formation, the subsequent colour interactions in the system can eventually transit this “molecule” into a “real” multiquark hadron, at least by some probability — et vice verse.

However, this ambiguity just gives us a clue: even though we can not determine the structure from its static property via the model of a multiquark state or a hadron molecule (in case that both kinds of models explain well the data of the static properties such as masses and decay widths), we can try to explore a ’production definition’ via the study of the production mechanism, especially in the multi-production processes. This means to investigate the production mechanism for the specific hadron to see whether it is produced like a multiquark state or like a hadron molecule Han:2009jw . In this paper we investigate the $T_{cc}$ structure via its production mechanism. The $T_{cc}$ state has been long studied theoretically and has recently been observed as a narrow resonance in the $DD\pi$ spectrum by the LHCb collaboration LHCb:2021vvq ; LHCb:2021auc (for theoretical review, see Liu:2019zoy and Refs. therein). Its mass is just below the $D^{*+}D^{0}$ threshold and its quantum numbers are consistent with $I(J^{P})=0(1^{+})$. The static properties of this resonance have been well predicted by both assumptions as tetraquark or molecule Wang:2017uld ; Wang:2017dtg ; Li:2012ss though some potential models of multiquark state fail to give the mass jbc . Since the observation of this $T_{cc}$ state, discussions of double-heavy four quark states in both molecular configuration Li:2021zbw ; Meng:2021jnw ; Chen:2021vhg ; Ling:2021bir ; Feijoo:2021ppq ; Yan:2021wdl ; Dai:2021wxi ; Wang:2021yld ; Xin:2021wcr ; Huang:2021urd ; Fleming:2021wmk ; Ren:2021dsi ; Chen:2021tnn and tetraquark picture Agaev:2021vur ; Weng:2021hje ; Guo:2021yws ; Azizi:2021aib are performed to explain the static and/or decay properties. However, $T_{cc}$ can be taken as an excellent example for the study of the production mechanism (rather than via decay or static properties) to gain insight on the hadronic structure, and to shed light on the confinement mechanism. It is also special comparing to other exotic ones like the XYZ particles, which will be explained in details later. The analysis of this paper serves to answer: What is it, a multiquark bound state ($cc\bar{q}\bar{q}$) or a hadron molecule ($DD^{*}$)? What have been observed and what to be observed for this purpose in the production process? The main body of our paper emphasizes the finger prints of its production as a four-quark state. As a comparison, we also investigate its production as a molecule by calculating the production rate, and propose methods to make judgement. This content is in the appendix because of large uncertainties.

First of all, before paying any attention to its structure, one should notice that this production process is characteristic of the fact that two heavy quark pairs $cc\bar{c}\bar{c}$ are produced. This is the very reason why the $T_{cc}$ is an outstanding example for the study of the production mechanism to gain insight on its structure. The production of two heavy quark pairs is a perturbative QCD process, so the interface between perturbative and nonperturbative QCD plays an important rôle in the $T_{cc}$ production, i.e., if $T_{cc}$ is a tetraquark rather than molecule of two $D$’s, the colour connection of the $cc$ quark pair must be a special and nontrivial one. This colour connection just shares exact similarity in the production mechanism of the $\Xi_{cc}$, a baryon, which can not be produced as a hadron molecule. Therefore, once the production mechanism of $T_{cc}$ and $\Xi_{cc}$ is proved to be the same by various observables, one can determine that $T_{cc}$ is produced as $\Xi_{cc}$ at the quark level rather than at the hadron level. This is the key point of this paper. As a mater of fact, $T_{cc}$ has been investigated from this viewpoint and suggested to be measured in various processes, e.g., $e^{+}e^{-}$ annihilation in various energies Jin:2013bra ; Jin:2014nva ; Jin:2015mla , $\Upsilon$ decay Li:2020ggh , as well as hadronic collisions Qin:2020zlg . Besides the production mechanism, some models on static property of the $T_{cc}$ also is fixed by the experimental data of $\Xi_{cc}$ Karliner:2017qjm . The recent discovery of the resonance is from the $pp$ collision at LHC and agrees well with the suggestion by Qin, Shen and Yu Qin:2020zlg , in which a full analogy to $\Xi_{cc}$ is employed as the key input for the prediction. This strongly motivates the investigation of this paper. In the next section (Sec. 2), we investigate the special colour connection for $\Xi_{cc}$ and $T_{cc}$ production. Then in section 3 we study the hadronization and demonstrate the finger print observables for $T_{cc}$ as four-quark state. The conclusion is in section 4.

As is mentioned above, if the $T_{cc}$ is produced as a four-quark state, there must be a $cc$ cluster produced from the primary hard interaction. In high energy processes such as proton-proton collisions at the LHC, the hard scattering produces the $cc$ cluster, or more explicitly the $cc\bar{c}\bar{c}$, which transfer into possible preconfinement systems before the final hadronization process, in which $T_{cc}$ or other handrons are formed. Those preconfinement systems with a specific colour connection favour the baryon ($\Xi_{cc}$) and $T_{cc}$ production. The corresponding phenomena in the production will play the rôle of the finger prints of the four-quark state structure of $T_{cc}$.

The concept of preconfinement is introduced by Amati and Veneziano AV . In our case, the production of the charm quark system $cc\bar{c}\bar{c}$ can be calculated by pertubative QCD. In general, it is accompanied by gluon radiation and light quark pair production. The dominant contribution can be described in a parton evolution picture, which is embedded in, e.g., parton shower models. The preconfined system is considered as the end of this evolution, which is typically of the formation of colour-singlet clusters with modest masses independent on the center-of-mass energy of the scattering process. Each colour-singlet cluster will independently hadronize into some hadrons. The preconfinement cluster formation has many evidences such as the local parton hadron duality (for details, see Li:2019vrc and refs. therein).

It has long been argued that the colour structure of the preconfined system is not unique our2 ; gsj . It has also been recognized that different colour structures will lead to different non-trivial baryon number distribution of the colour-singlet clusters, which is referred as the baryon number fluctuation of the preconfined system Li:2019vrc . Some phenomena such as the baryon number enhancement (with saturation) has been investigated in Li:2019vrc , which is also an important feature of the $T_{cc}$ production as a four-quark state. These issues of the preconfinement can be easily understood from the analysis on the colour connection of the most simple colour-singlet four quark system $c\bar{c}c\bar{c}$ produced from some perturbative processes (e.g., Jin:2013bra ).

The $cc\bar{c}\bar{c}$ colour space can be decomposed as Eqs. (1) and (2) in the introduction section or in other ways, but we investigate the specific one

where $3_{12}$ ($3_{12}^{*}$) denotes the triplet (anti-triplet) representation of the $SU_{c}(3)$ group and $6_{12}$ ($6_{12}^{*}$) denotes the sextet (anti-sextet) representation of the $SU_{c}(3)$ group. When two (anti)quarks in the color state $3^{*}(3)$ attract each other and their invariant mass is sufficiently small, such a cluster can be considered as a(n) (anti)diquark Jin:2013bra . For the case in which only one of the pair has a small invariant mass, the color configuration can be better written as

The above equation indicates that after the perturbative QCD evolution for the whole system, it is possible to form clusters, like ’ccq + (other quark anti-quark pairs) + gluons’, with cc in the 3*. This is a colour singlet cluster with a non-zero baryon number in the preconfinement system. It can enhance baryon production, especially for the $\Xi_{cc}$, as displayed in Fig 1. On the other hand, one can see from Fig. 2 that this colour structure can also lead to $T_{cc}$ production, if $T_{cc}$ is indeed a four-quark state. After the $T_{cc}$ formed, there can be a remnant diquark, which again can enhance the baryon production as Fig. 1.

From the above analysis, such a specific colour and baryon number fluctuation of the preconfinement system enhances both $\Xi_{cc}$ and $T_{cc}$. Et vice verse, the doubly charm baryon or tetraquark play the role of the ’trigger’ to probe this special colour connection. In brief, the $T_{cc}$ and $\Xi_{cc}$ production requires the above physical mechanism happen on the interface between PQCD and NPQCD. Therefore, the corresponding unique signals can be taken as the fingerprints for judging that the resonance structure is a true tetraquark rather than a molecule. The details of the hadronization and the signals are investigated in the next section.

The preconfinement and subsequent hadronization are ‘branching’ processes via the creation of quarks from the vacuum by the strong interactions within the system. The created quarks and the primary quarks are combined into color-singlet clusters and then hadrons at last. In Eq. (4), the colour configuration as a whole is like a ‘big antibaryon’ ($3^{*}3^{*}3^{*}$) but with a large invariant mass. For example, in Figs. 1, 2, The diquark $cc$ must combine with a quark $q$ (or antidiquark) to form a color-singlet system, $\Xi_{cc}$/$T_{cc}$. To balance the quantum numbers of color and flavour, an antiquark/diquark must be created simultaneously. To branch them further, more quark pairs and diquark pairs will be created from the vacuum via the interactions among the quark system. Such a cascade process will proceed until the end of time, when most of the ‘inner energy’ of the entire system is transformed into the kinematical energies and masses of the produced hadrons. In this process some clusters with non-zero baryon number produce because of some diquark pair creation. Each of the two primary $\bar{c}$’s combines with a created quark or antidiquark to respectively hadronize into two open charmed hadrons. To quantitively describe such a nonperturbative hadronization procedure, we adopt a concrete hadronization model, the Lund string model string , which is realized by PYTHIA/JETSET pythia . For the configurations considered here, the above process is straightforward, except that for each step, we must assign special quantum numbers for each specific kind of hadrons according to their production rates.

In the case of $\Xi_{cc}$ production, the complementary antiquark can produce an antibaryon by combining with an antidiquark, and then the balancing diquark is broken by the interactions with the remaining system and then each becomes connected to the two primary $\bar{c}$’s to form two strings. The resultant hadronization can be described by the conventional string-fragmentation picture. This procedure is illustrated in FIG.1. In the case of $T_{cc}$ production, if the complementary diquark is broken and proceeds in the same manner described above, the procedure can be described by FIG.2. However, it is also possible that the new created diquark is not broken, and then the hadronization proceeds like the $\Xi_{cc}$ production shown in Fig.1. In this scenario, the baryon production is also enhanced. Therefore, there is baryon enhancement in both the $\Xi_{cc}$ and $T_{cc}$ production, and this similarity is expected to be observed. We hereby propose an observable to measure this effect. First find a cluster including a $\Xi_{cc}$ or $T_{cc}$ hadron and other particles with relative small momenta, and then measure the invariant mass of this cluster $M_{T(\Xi)}+\Delta$ ($M_{T(\Xi)}$ is mass of $T_{cc}(\Xi_{cc})$) and the total baryon number $N_{B}$ (algebric sum or arithmetic sum). The $N_{B}$ as a function of $\Delta$ is thus obtained. The value of $N_{B}$ should be significantly larger than the averaged baryon production rate in various high energy scattering processes, which is around 10 per cent with respect to all hadrons.

The next observable that we propose to measure on the similarity between the $\Xi_{cc}$ and $T_{cc}$ is their kinematic spectra. The fragmentation of the heavy diquark can be described by the Peterson formula pet83

where $z$ is defined by $p_{+}^{\text{hadron}}/p_{+}^{\text{cluster}}$ with $p_{+}$ being the sum of the energy and the momentum projected in the moving direction of the cluster, and the free parameter $\epsilon_{Q}$ is expected to scale between flavours as $\epsilon_{Q}\propto 1/m_{Q}^{2}$. In practice, different $\epsilon_{Q}$ values inversely proportional to the hadron masses squares are used for $T_{cc}$ and $\Xi_{cc}$. The results for $f(z)$ of $T_{cc}$ and $\Xi_{cc}$ are displayed in FIG. 3 to show the difference. It can be seen that for reasonable kinematics region, the differences between these two hadron can be neglected. Therefore, this similarity in the line shape of the momentum spectra with respect to those of $\Xi_{cc}$ is another feature to judge $T_{cc}$ as a four-quark state, which can be examined in experiments.

To further prove this physical picture, we propose that the light hadrons produced adjointly with $\Xi_{cc}$/$T_{cc}$ can provide additional clues. One notices that the two-string structure in FIG. 1&2 is typical for this kind of production processes. The reason is that however the $cc$ diquark fragments, to balance the colour, the subsequent hadronization of the remnant except the $cc$ diquark cluster can only happen in two (or more) colour-singlet clusters rather than only one. Thus, one can measure this special property, the string effect, which was suggested by the Lund group in 1980s string . In experiments, one should get quite similar kinematic distributions such as those of the energy, momentum, etc., for $\Xi_{cc}$ and $T_{cc}$. To visualize this effect, a simulation is performed. The fragmentation of the complementary (anti) quark is handled by the fragmentation function employed by the LUND group string

where $a$ and $b$ are free parameters. In our simulation program, we take $a=0.3$ GeV^-2 and $b=0.58$ GeV^-2, as used in PYTHIA pythia . The fragmentation of the strings can be referred to the classical book about the Lund model by B. Anderson string and the PYTHIA manual pythia . With the simulated events, we first have to reasonably define a cluster, whose total transverse momentum is zero at the jet level. In practice, we approximately use the doubly heavy hadron momentum as jet momentum, i.e., the Peterson function taken as a $\delta$ function. In this scenario, we collect hadrons around it to balance the transverse momentum of $\Xi_{cc}$/$T_{cc}$ to obtain the cluster of hadrons (with invariant mass $\sqrt{s*}$) to be investigated. Then, we should Lorentz transform this cluster to its rest frame. Take the direction opposite to the $\Xi_{cc}$/$T_{cc}$ momentum as the $z$ axis direction, then we calculate the rapidities and re-scaled transverse momenta $x_{T}\equiv p_{T}/\sqrt{s*}$ of the hadrons in the two string system to get the distribution, which is shown in FIG. 4.

The above suggested ’similarity’ study is very plausible since the experiment LHCb:2021auc has attempted to make comparisons between the $T_{cc}^{+}$ signal and the $DD$ pair production, which is expected to be contributed significantly from double scattering. Nevertheless, more data are required. Furthermore, one should give more definite confirmation on the the new discovered resonance, as well as on $\Xi_{cc}$. To eliminate some kinematic uncertainty, future measurements by other kinematic region, such as the CMS collaboration in the central rapidity region, are valuable.

Except for the similarity between the kinematics distributions of the $\Xi_{cc}$ and $T_{cc}$ production processes, their total production rates can also be a helpful observable to distinguish the nature of $T_{cc}$. If $T_{cc}$ is a four-quark state, both $\Xi_{cc}$ and $T_{cc}$ are generated from the fragmentation of the heavy diquark $cc$ and are managed by Eq. (5). In this scenario, Qin, Shen and Yu Qin:2020zlg estimated the production ratio of $T_{cc}$ which was assumed to be a isospin singlet state. The key point is that, taking into account all these similarities, they gave a reasonable conjecture of the production ratio, $T_{cc}$/ $\Xi_{cc}$ $\sim$ 1/4. It is quite significant that with this estimation, they have predicted the signal yield which agrees well with the data LHCb:2021vvq ; LHCb:2021auc . It was also estimated in lsy2016 that the diquark versus quark fragmentation ratio as $\lambda/2$ ($\lambda\sim 0.3$), which is a result without taking the isospin counting. The present observed one is composed with the $ud$ antidiquark, and it is thus 2 times of $uu$ or $dd$ case. Consequently, an estimation around the range 1/3-1/5 is generally reasonable and adoptable, consistent with Qin:2020zlg . In contrast, if $T_{cc}$ is regarded as a molecule state, the production rate would be smaller by one order of magnitude, which is disfavored by the current data LHCb:2021vvq ; LHCb:2021auc . The details can be found in Appendix A.

In the end, it is valuable to examine whether there exists a further resonance, here denoting as $T^{\prime}_{cc}$, which has a different isospin from $T_{cc}$. The relative production rate of $T_{cc}/T^{\prime}_{cc}$ is a good observable to decide whether $T_{cc}$ has the four-quark state structure. The reason is as follows. Consider the $cc$ as a $3^{*}$ anti heavy ’quark’, then $T_{cc}$ and $T^{\prime}_{cc}$ are anomalous to heavy anti-baryons from the colour aspect. By such a viewpoint, their production ratio follows the mass and production rate rules observed from the experiment Belle:2017caf for heavy baryons, dependent on the isopspin structures. More explicitly, the mass splitting and production rate between $T_{cc}$ and $T^{\prime}_{cc}$ should be similar to $\Lambda_{c}$ and $\Sigma_{c}$, with the production rate of $\Sigma_{c}$ is smaller than that of $\Lambda_{c}$ by one order of magnitude. If the present observed $T_{cc}$ is isospin singlet and the $\Lambda_{c}$-like one, then a scan to the larger mass region with more data by LHCb can find a isospin-1 resonance $T^{\prime}_{cc}$ with a lower production rate. If, on the contrary, the observed $T_{cc}$ is the $\Sigma_{c}$-like one, then a scan to the smaller mass region will find such a isospin-0 $T^{\prime}_{cc}$ with a much larger production rate. Now the experiment LHCb:2021auc favours a $\Lambda_{c}$-like $T_{cc}$, since there is not another significant resonance observed and the signal yield is consistent with the estimation of Qin:2020zlg . Based one the above discussion, the production rate is a perfect evidence for the four-quark state production mechanism.

The consistence between the theoretical analysis on the $T_{cc}$ production by Qin, Shen and Yu Qin:2020zlg and the data LHCb:2021vvq ; LHCb:2021auc strongly favours the newly discovered resonance $T_{cc}$ is produced as a real four-quark state. We in this paper clarify the production mechanism, and provide further experimental observables to look for the finger prints of the $T_{cc}$ as a tetraquark. The observables include: the similarity of $T_{cc}$ and $\Xi_{cc}$ in the kinematics spectra, in the baryon number enhancement, in the string effect, etc. Moreover, measurements of the production cross section of $T_{cc}$ and of the production ratio between $T_{cc}$ to a different isospin state $T^{\prime}_{cc}$ (if it exists) would also be very helpful. With these measurements, we will have a good chance to confirm or deny the $T_{cc}$ produced as a four-quark state.