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

Yi Jin  School of Physics and Technology, University of Jinan, Jinan 250022,
P. R. China  Shi-Yuan Li  School of Physics, Shandong University, Jinan
250100, P. R. China  Yan-Rui Liu  School of Physics, Shandong University,
Jinan 250100, P. R. China  Qin Qin  School of physics, Huazhong University of
Science and Technology, Wuhan 430074, China  Zong-Guo Si  School of Physics,
Shandong University, Jinan 250100, P. R. China  Fu-Sheng Yu  School of Nuclear
Science and Technology, and Frontiers Science Center for Rare Isotopes,
Lanzhou University, Lanzhou 730000, China  Lanzhou Center for Theoretical
Physics, and Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou
University, Lanzhou 730000, China  Center for High Energy Physics, Peking
University, Beijing 100871, China

###### Abstract

We suggest to study the production mechanism and some details of the
production properties to probe the structure of the $DD\pi$ resonance
$T_{cc}^{+}$ recently observed by the LHCb Collaboration. If the resonance is
produced as a four-quark state, one can find the corresponding finger prints
via measurements on some production properties that are the same as those of
$\Xi_{cc}$. On the other hand, if $T_{cc}^{+}$ is produced as a hadron
molecule, the measurement on the momentum correlation of $DD^{*}$ can be a
smoking gun to make the judgement.

##  1 Introduction

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:

$\displaystyle(q_{1}q_{3})_{\bar{3}}\otimes(\bar{q_{2}}\bar{q_{4}})_{3}\rightarrow
1$ (1)
$\displaystyle(q_{1}\bar{q}_{2})_{1~{}or~{}8}\otimes(q_{3}\bar{q}_{4})_{1~{}or~{}8}\rightarrow
1$ (2) $\displaystyle\cdot\cdot\cdot$

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.

##  2 Colour and baryon number fluctuation of preconfinement system and doubly
open charm hadron production

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

$(3_{1}\otimes 3_{2})\otimes(3_{1}^{*}\otimes 3_{2}^{*})=(3_{12}^{*}\oplus
6_{12})\otimes(3_{12}\oplus 6_{12}^{*})=(3_{12}^{*}\otimes
3_{12})\oplus(6_{12}\otimes 6_{12}^{*})\oplus\cdots,$ (3)

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

$\displaystyle(3_{1}\otimes 3_{2})\otimes 3_{1}^{*}\otimes
3_{2}^{*}=3_{12}^{*}\otimes 3_{1}^{*}\otimes 3_{2}^{*}\oplus\cdots,$ or
$\displaystyle 3_{1}\otimes 3_{2}\otimes(3_{1}^{*}\otimes
3_{2}^{*})=3_{1}\otimes 3_{2}\otimes 3_{12}\oplus\cdots.$ (4)

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.

##  3 Hadronization of cc diquark and finger prints of tetraquark in
experiment

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.

Figure 1: The $\Xi_{cc}$ production and the string formation for the
hadronization of the $(cc)\bar{c}\bar{c}$ system with the aid of quark
creation from the vacuum. Solid circles represent quarks, while hollow circles
represent antiquarks. The primary $\bar{c}\bar{c}$ connect to quarks
respectively via two strings. Figure 2: The $T_{cc}$ production and the string
formation for the hadronization of the $(cc)\bar{c}\bar{c}$ system with the
aid of quark creation from the vacuum. Solid circles represent quarks, while
hollow circles represent antiquarks. The primary $\bar{c}\bar{c}$ connect to
quarks respectively via two strings.

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.

Figure 3: Comparison of the $z$-distributions of the fragmentations (from the
$cc$ diquark) to $\Xi_{cc}$ (blue) and $T_{cc}$ (red).

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

$f(z)\propto\frac{1}{z(1-1/z-\epsilon_{Q}/(1-z))^{2}}\;,$ (5)

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.

Figure 4: 2D distribution onrapidity and $x_{T}=p_{T}/\sqrt{s*}$.

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

$f(z)\propto z^{-1}(1-z)^{a}exp(-bm_{\perp}^{2}/z),$ (6)

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.

##  4 Conclusion

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.

###### Acknowledgements.

We thank Prof. Wang Zhi-Gang for the discussion on application of QCD sum rule
in predicting the mass of $T_{cc}$, Prof. Jiang Jun for the discussion in some
aspects of the hadron molecule and Prof. Li Ning (Sun Yat-Sen Univ.) for the
nonrelativistic wave functions. This work is supported in part by National
Natural Science Foundation of China (grant Nos. 11635009, 11775130, 11775132,
11975112 and 12005068).

##  Appendix A $T_{cc}$ production as DD* molecule

In this appendix, as mentioned in the introduction, when we take ’$T_{cc}$’ as
the hadron molecule in production, we mean that such a production process
includes two factors if we can have a factorized formulation: First is the
productions of D and D* hadrons; Second is the description on how these two
hadrons D and D* combined to a ’hadron molecule’, which, if allowed, we also
call ’$T_{cc}$’.

For the second factor, there could be two possibilities. Corresponding to
these two ways, the requirement of the information to the first factor is
somehow also different. These two factorized formulations are investigated in
the following.

###  A.1 Non relativistic wave function formulation Jin:2016vjn ; lsy2016 ;
r2005lsy  and DD* momentum correlation measurement as smoking gun

If the molecule composed of DD* can be described by a non-relativistic wave
function (the $T_{cc}$ mass also got by solving the corresponding Schrödinger
function), then we can rely on this wave function to study the production of
the molecule of DD*, with the DD* produced from the hard interaction in the
collision, such as pp scattering at LHC. At last one gets a factorized
formulation.

The process $pp\to D+D*+X\to T(DD*)+X$ is illustrated in Fig. 5, and the
corresponding invariant amplitude is:

$\begin{array}[]{l l l}A_{inv}&=&\langle T(DD*),X|\hat{T}|pp\rangle\\\
&=&\frac{1}{\sqrt{\frac{{{m}_{D}}{{m}_{D*}}}{{{m}_{D}}+{{m}_{D*}}}}}\int\frac{d^{3}k}{(2\pi)^{3}}\Phi(\vec{k})\langle
D|\langle D*|\langle X|\hat{T}|pp\rangle\\\
&=&\frac{1}{\sqrt{\frac{{{m}_{D}}{{m}_{D*}}}{{{m}_{D}}+{{m}_{D*}}}}}\int\frac{d^{3}k}{(2\pi)^{3}}\Phi(\vec{k}){\mathbb{M}}(\vec{k}).\end{array}$
(7)

This formulation is valid in the rest frame of $T(DD*)$, the bound state of
the ingredient hadrons $D$ and $D*$. In the above equation, $\vec{k}$ is the
relative 3-momentum between $D$ and $D*$ in the rest frame of $T(DD*)$.
${\mathbb{M}}(\vec{k})$ is the invariant amplitude for the free (unbound) $D$
and $D*$ production. The factor
$\frac{1}{\sqrt{\frac{{{m}_{D}}{{m}_{D*}}}{{{m}_{D}}+{{m}_{D*}}}}}$ comes from
the normalization of the bound state to be $2E_{H}V$ just as a single
particle. The normalization for the wave function $\Phi(\vec{k})$ in momentum
space (Fourier transformation from the configuration space) is

$\int\frac{d^{3}k}{(2\pi)^{3}}|\Phi(\vec{k})|^{2}=1.$ (8)

From the above equation, it is obvious that if we know the analytical form of
the wave function as well as the free particle invariant amplitude, we can
calculate the amplitude of the bound state simply by integrating the relative
momentum $\vec{k}$. In practice, certain decomposition and simplification will
be taken for a concrete ${}^{2S+1}L_{J}$ state. The resulting formulations are
covariant.

Figure 5: Any hadron molecule $H(A,B)$ production process $p(p_{1})p(p_{2})\to
A(p_{A})+B(p_{B})+X\to H(A,B)(P_{H})+X$.

To get a factorized formulation, we should expand the $\Phi(\vec{k})$ around
$\vec{k}=0$. This is especially reasonable in the heavy hadron molecule case
since the typical relative momentum $\vec{k}$ is rather small compared to the
mass of the molecule. The binding energy is generally very small. For two
orbital angular momentum states L=0, 1, taking into account the parity of
these two wave functions, one respectively obtains

$\int\frac{d^{3}k}{(2\pi)^{3}}\Phi(\vec{k}){\mathbb{M}}(\vec{k})=\Psi(0){\mathbb{M}}(\vec{k})|_{\vec{k}=0}~{}~{}~{}(L=0),$
$\int\frac{d^{3}k}{(2\pi)^{3}}\Phi(\vec{k}){\mathbb{M}}_{\lambda}(\vec{k})=\Psi^{\prime}(0)\epsilon_{\lambda}^{\mu}\partial^{k}_{\mu}{\mathbb{M}}(\vec{k})|_{\vec{k}=0}~{}~{}~{}(L=1),$

to the leading contribution. Here $\Psi(0)$ or $\Psi^{\prime}(0)$ is the
configuration space wave function or divergence of the configuration space
wave function at origin. $\epsilon_{\lambda}^{\mu}$ is the polarization vector
of the vector hadron D*.

Since $T_{cc}$ is considered as $J^{P}=1^{+}$, one can assume the dominant
contribution is the S wave. In this case the production rate or cross section
can be calculated by these two factors: The absolute square of the wave
function at origin, and the distribution (differential cross section) with
respect to the relative momentum $\vec{k}$ extrapolated to $\vec{k}=0$.

The production rate of the P wave particles will be quite small as argued by
us in Jin:2016vjn  because
$\partial^{k}_{\mu}\mathbb{M}(\vec{k})|_{\vec{k}=0}$ is much more suppressed.
The reason for this is that $\mathbb{M}(\vec{k})$ is slowly varying on
$\vec{k}$. This can be checked by numerical study on the correlations
Jin:2016vjn . The following numerical result can show such a fact. In general,
because of the property of the spherical harmonics, typical of the the non-
relativistic (assuming central field) wave function, the L order partial wave
is relevant to the L order of moment of the function which convolute with the
L order spherical harmonics. So the L order of the derivative of the amplitude
enters. For slowly varying behaviour of the D D* correlation, the high order
of L partial wave contributions are generally suppressed in production for
this model. Needless to say, the high order L states/partial wave components
could be a good probe to the details of the correlation in the production.

The details of the molecule structure, e.g., the isospin effects, is relevant
to the absolute production rate. The reason is that different isospin
structure predicts different distribution near the origin. The weaker bound,
the smaller value near the origin, and the lower production rate predicted. In
Li:2012ss , it is predicted that only I=0 bound state exists. This indicates a
qualitatively direct judgement, i.e., to measure the isospin of the resonance,
to see whether isospin multi-states exist LHCb:2021auc .

To study the production of the hadron molecule, the other key point is to
study the relevant two hadron correlation in momentum space. The free pair
cross section $p(p_{1})p(p_{2})\to A(p_{A})+B(p_{B})+X$ can be expressed as:

$\displaystyle\frac{1}{N}\frac{dN}{d^{3}P_{H}d^{3}q}\propto\frac{1}{F}\sum_{j\neq
A,B}\int\prod\frac{d^{3}p_{j}}{(2\pi)^{3}2E_{j}}$
$\displaystyle\times\overline{|\hat{O}|^{2}}(p_{j},P_{H}=p_{A}+p_{B},q=p_{A}-p_{B})$
(9) $\displaystyle\times(2\pi)^{4}\delta^{(4)}(P_{intial}-\sum_{j\neq
A,B}p_{j}-p_{A}-p_{B}).$

Here the average is on various spin states, and the proper initial flux factor
$1/F$ and phase space integral are needed. $\hat{O}$ is the amplitude of
production of two free ingredient particles (with vanishing relative momentum
and proper angular momentum state). It is not possible to be calculated
directly with some effective quantum field theory/model when the initial state
is (anti) protons and $A$ and $B$ are hadrons or clusters. However, it can be
obtained with an event generator such as PYTHIA pythia  or equivalently
Shandong Quark Combination Model jin2010  etc., for the case that $A$ and $B$
are both on shell. It is the advantage that in the nonrelativistic framework
we employ, only the on shell case is considered, so that the numerical
calculation with event generator is plausible. The quantity of Eq. (A.1)
describes the two hadrons/clusters ($A$ and $B$) correlation in the phase
space. For the hadron case, by proper integral on components of $P_{H}$ and/or
$q$, the resulting correlations can be directly compared with data and serve
for tuning the parameters. This strongly implies that to study various hadron
molecule, careful measurement on the two (corresponding) hadron (momentum)
correlation in the corresponding scattering process is crucial, though such
study is in fact quite lack until now.

Since the special physical picture of the non-relativistic framework, it is
only valid in the rest frame of the two ingredient particles. One can define
the following covariant space-like relative momentum $\hat{q}$ as

$\hat{q}=(p_{A}-p_{B})-\frac{(p_{A}-p_{B})\cdot(p_{A}+p_{B})}{(p_{A}+p_{B})^{2}}(p_{A}+p_{B}).$
(10)

It is clear that in the rest frame of $A$ and $B$ ($H(A,B)$) where
$\vec{p}_{A}+\vec{p}_{B}=0$, $\hat{q}=(0,\vec{k})$ and the
$k=\sqrt{-\hat{q}^{2}}$ is exactly the absolute value of the 3-relative
momentum $|\vec{p}_{A}-\vec{p}_{B}|$.

Employing the event generator, one gets

$\frac{1}{N}\frac{dN}{d^{3}P_{H}d^{3}\hat{q}},\forall\hat{q},$ (11)

then extrapolates to the special case $k=0$. Numerically, one can take an
average around $k=0$ for the above quantity. Then we get, up to the kinematic
factors as for the covariant form,

$\displaystyle\frac{1}{F}\sum_{j\neq
A,B}\int\prod\frac{d^{3}p_{j}}{(2\pi)^{3}2E_{j}}\overline{|\hat{O}|^{2}}(p_{j},P_{H}=p_{A}+p_{B},k=0)$
$\displaystyle\times(2\pi)^{4}\delta^{(4)}(P_{intial}-\sum_{j\neq
A,B}p_{j}-P_{H}).$ (12)

There are several very basic facts supporting the extrapolation. First of all,
the amplitude and cross section are analytical in phase space. Any practical
generator should reproduce this property, and any ultraviolet divergence is
not present. Secondly, the study of strong interaction is complex because of
the SU(3) non-Abelian interaction, but its simulation has one simplicity: All
particles taking part in the strong interactions are massive, which eliminates
the infrared singularities.

Figure 6: The distribution of 3-relative momentum $k$ between DD* calculated
by PYTHIA.

From Figure 6 we can see, the distribution on the relative momentum between D
and D* is quite flat. The correlation quite near k=0 is generally not reliable
in generators because of lack of data to test. In general we can rather rely
on the average around the small value. However, a careful measurement on this
correlation surely plays the rôle of the smoking gun, i.e., if from experiment
we make sure this distribution on $\vec{k}$ is vanishing on the value
$\vec{k}=0$, such molecule will not possible to be produce from the
corresponding scattering process. As an illustration of the method rather than
the final conclusion, we estimate the production rate and cross section for a
S wave isospin-singlet DD* molecule Li:2012ss  based on the above calculation.
We use the wave function at origin value as ($0.1407GeV^{3/2}$ )Li:2012ss  and
the cross section of $pp\to c\bar{c}c\bar{c}+X$ as $3.6\times 10^{7}$ pb by
NLO calculation. The cross section $pp\to T_{DD^{*}}+X$ is around $3\times
10^{2}pb$, which is one order lower than that of the production rate of the
four-quark state Qin:2020zlg .

###  A.2 Combination matrix elelment formulation Li:2005hh ; r2021lsy

In high energies like LHC, the multi-scatterings and underlying events are
significant. It is possible that these two D’s are from different scattering
processes, or that one D/D* produced from the hard scattering flying by and to
combine with a D*/D from the underlying events. The Experiment has studied
some similarity with the DD pair with small invariant mass which is expected
significantly contributed from double scattering LHCb:2021auc . To calculate
the combination of a D/D* with the underlying D*/D, or with D*/D from another
scattering, we again need a factorized formulation. By the analogy to the
quark level combination Li:2005hh ; r2021lsy , we can conjecture that the
cross section to produce a molecule $T_{cc}$ with four momentum $(E,\vec{K})$
in pp scattering can be written as

$2E\frac{d{\sigma_{C}}}{d^{3}K}=\sum\limits_{ab}\int{dx_{1}dx_{2}}f^{a}_{1}(x_{1})f^{b}_{2}(x_{2})\frac{d\hat{\sigma}_{ab}}{d{\cal
I}}\frac{1}{\xi^{2}}\frac{(2\pi)^{2}}{(2M)^{2}}P(\xi_{l})\tilde{F}(\xi,\xi_{l})|_{\xi+\xi_{l}=1}.$
(13)

Here $f^{a}_{A}(x_{1})$ and $f^{b}_{B}(x_{2})$ are the initial parton
distributions with momentum factions $x_{1}$, $x_{2}$. $M$ is the mass of the
$T_{cc}$. $P(\xi_{l})$ is the distribution of the D/D* in underlying events or
the softer one from another ascattering and
$\tilde{F}(\xi,\xi_{l})|_{\xi+\xi_{l}=1}$ is the combination matrix element.
$d{\cal I}$ is the dimensionless invariant phase space for the ‘2-body’ final
state $D_{A}+X$ (Here $D_{A}$ refers one of the D/D* produced from the hard
interaction) where $X$ treated as one particle, with $\xi=E_{D_{A}}/E$. This
formula is also correct for higher order cross sections.

For this production mechanism, the $T_{cc}$ may behave like a single D/D* at
large transverse momentum. Since both the combination matrix element as well
as the underlying event distribution are yet unknown, we will not discuss the
numerical result of this combination process for hadron molecule production.

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