Preamble

Prediction of States and Femtoscopic Study Ye Yan, Qi Huang, Qian Wu, Hong-Xia Huang, and Jia-Lun Ping

1 Department of Physics, Nanjing Normal University, Nanjing 210023, China 2 Changzhou College of Information Technology, Changzhou 213164, China 3 School of Physics, Nanjing University, Nanjing 210000, China 4 Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China

Inspired by recent research on the systems, we investigate the systems within the framework of the quark delocalization color screening model. Our result indicates that the nucleon- interaction is slightly stronger than the nucleon- interaction, implying a higher likelihood for the system to form bound states.

Dynamic calculations show that the systems with form bound states, whose binding energies are deeper than that of the system with . The scattering phase shifts and extracted scattering parameters also support the existence of bound states. Additionally, we discuss the behavior of the femtoscopic correlation function for the pairs for the first time. Building on the recent experimental progress on the correlation function, future femtoscopic investigations of the system in heavy-ion collisions will be particularly valuable for constraining baryon-antibaryon interactions.

Keywords

systems; femtoscopic correlation function; bound states; hadron-hadron interaction; scattering phase shifts

INTRODUCTION

The study of baryon-antibaryon bound states dates back to the proposal by Fermi and Yang [ ] to form the pion from a nucleon-antinucleon pair.

In the traditional one-boson- exchange theory of nucleon-nucleon interactions, it is shown that the nucleon-antinucleon system is more attractive than the nucleon-nucleon system due to the strong -exchange [ Therefore, possible bound states and resonances of nucleon- antinucleon systems have been proposed for many years. An extensive and comprehensive review of the possible bound states of was provided in Ref. [ More recently, the BESIII Collaboration reported the ob- servation of a new (1880) state in the line shape of the invariant mass spectrum [ ], which is considered as evidence for the existence of a proton-antiproton bound state. Many theoretical works have been sparked to study the system and the properties of (1880) ]. In addition to the possible proton-antiproton state, there has also been great progress in recent work related to ]. A narrow structure in the system near the mass threshold, named (2085) , is observed in the process with a

statistical significance exceeding 20 σ . Its spin and parity are 22

slightly favored to be J P = 1 + through an amplitude analy- 23

sis. Further theoretical results and discussions can be found

in Refs. [ 13 – 15 ]. Building on the significant progress in the 25

studies of nucleon-antinucleon and nucleon- systems, it is natural to explore whether bound states or resonance states can be formed between nucleon and other hyperons, or be- tween nucleon and other antihyperons.

In recent years, progress in understanding the strange dibaryon has renewed interest in dibaryon systems. The STAR Collaboration measured the correlation functions in Au+Au collisions at the Relativistic Heavy-Ion Collider (RHIC) [ ] and reported a positive scattering length for the interaction, which supports the hypothesis of a bound state. In addition, the ALICE Collaboration reported mea-

surements of the p - Ω correlation in pp collisions at √ s = 13 37

TeV at the Large Hadron Collider (LHC) [ Beyond

the p Ω system, femtoscopic techniques and correlation func- 39

tion studies have made significant progress, both experimen- 40

tally [ ] and theoretically [

The S = − 3 , I = 1 / 2 , J = 2 N Ω state was first pre- 42

dicted by J. T. Goldman et al. as a narrow resonance in a relativistic quark model [ ]. M. Oka also proposed the ex-

istence of a quasi-bound state with I ( J P ) = 1 / 2(2 + ) using 45

a constituent quark model [ ]. A lattice QCD study by the HAL QCD Collaboration reported that the state is a bound state at a pion mass of MeV [ ]. Later, the bound na- ture was also confirmed with nearly physical quark masses MeV and MeV) [ ]. Using the inter- actions obtained from ( 2 + 1 )-flavor lattice QCD simulations, K. Morita et al. studied the two-pair momentum correlation functions of the state in relativistic heavy-ion collisions to further investigate the existence of a bound state [ This state has also been confirmed to be a bound state in the frameworks of the chromomagnetic model [ ], QCD sum rules [ ] and other quark models [ Additionally, studies on the production of the systems can be found in Refs. [ analogy nucleon-nucleon nucleon- antinucleon systems, one may expect attractive interactions in both the channels.

If the state can be confirmed through further experimental measurements, we hope to observe an even stronger signal for the state in experiments.

Furthermore, the nucleon-antinucleon state

would annihilate very quickly in the ground state due to 66

the quark content of this system, making it challenging to provide a convincing theoretical confirmation of the nucleon- antinucleon bound state or resonance. In contrast, the

state cannot annihilate into the vacuum, as nucleon consists 70

of three ) quarks and consists of three quarks. In this context, the state is expected to be relatively stable and Supported by the National Natural Science Foundation of China (Nos. 12575088, 12305087, 11535005, and 11675080)

may serve as an ideal system for studying baryon-antibaryon interactions.

The copious production of anti-baryons in

high-energy colliders offers excellent opportunities to study 75

this type of spectrum. Clearly, the theoretical study of system is both interesting and necessary, as it can provide valuable insights for experimental searches of baryon-antibaryon bound states.

In our previous work, we studied the interactions and correlation functions based on the quark delocalization color

screening model (QDCSM) [ 72 ]. According to our calcula- 82

tions, the depletion of the correlation functions caused by bound state, which is not observed in the AL- ICE Collaboration’s measurements [ ], can be explained by

the contribution of the attractive J P = 1 + component in spin- 86

averaging. The QDCSM is a constituent quark model [ that introduces two key ingredients: first, quark delocaliza- tion, which accounts for orbital excitation by allowing quarks to delocalize from one cluster to another; second, the color

screening factor, which modifies the confinement interaction 91

between quarks in different cluster orbits. In the study of nucleon-nucleon and nucleon-hyperon interactions and the

properties of the deuteron, the mechanism of quark delocal- 94

ization and color screening plays a crucial role in generating 95

intermediate-range attraction [ ]. This model has also been used to investigate various dibaryon candidates, such as ], and others [ ]. It has been ex- tended to study baryon-antibaryon systems, including ]. Extending it to the system is a natural pro- gression. Therefore, we continue to investigate the tem within the framework of the QDCSM. In this work, the system is studied from three aspects: energy spectrum, scattering processes, and correlation functions.

This paper is organized as follows. A brief introduction 105

of the QDCSM is given in the next section. The correlation function and the inverse scattering method are introduced in , and Sec. , respectively. Sec. devotes to the numerical results and discussions. The summary is shown in the last section.

THEORETICAL FORMALISM

A. Quark delocalization color screening model 112

The details of the QDCSM employed in the present work can be found in Refs. [ 73 – 77 ]. Here, we present the salient features of the model. The model Hamiltonian is given by:

j>i =1 V qq ( r ij )

j =4 V q ¯ q ( r ij ) , (1)

j>i =4 V ¯ q ¯ q ( r ij ) +

where is the quark mass, is the momentum of the quark, is the center-of-mass kinetic energy. The dynam- ics of the hexaquark system is driven by two-body potentials, including color confinement ( ), perturbative one-gluon exchange interaction ( ), and dynamical chiral symmetry breaking (

V ( r ij ) = V CON ( r ij ) + V OGE ( r ij ) + V χ ( r ij ) . (2)

Here, a phenomenological color screening confinement po- tential ( V CON ) is used as:

occur in the same cluster

f ( r ij ) =

µqiqj occur in different cluster where are model parameters, and stands for the SU(3) color Gell-Mann matrices. Among them, the color screening parameter is determined by fitting the deuteron properties, nucleon-nucleon scattering phase shifts, and hyperon-nucleon scattering phase shifts, respectively, , and , sat- isfying the relation— ]. The one-gluon ex- change potential ( ) is written as:

V OGE ( r ij ) =1

where is the Pauli matrices and is the quark-gluon cou- pling constant. In order to cover the wide energy range from light to strange quarks, an effective scale-dependent quark- gluon coupling was introduced [

α s ( µ ) = α 0

Owing to the dynamical breaking of chiral symmetry, SU(3) Goldstone boson exchange interactions arise between the constituent light quarks , and . Accordingly, the chi- ral interaction is expressed as:

V χ ( r ij ) = V π ( r ij ) + V K ( r ij ) + V η ( r ij ) . (6)

Among them,

V π ( r ij ) = g 2 ch 4 π m 2 π 12 m i m j

V K ( r ij ) = g 2 ch 4 π m 2 K 12 m i m j

V η ( r ij ) = g 2 ch 4 π m 2 η 12 m i m j

where is the standard Yukawa function. The physical meson is considered by introducing the angle instead of the octet one. The are the SU(3) flavor Gell- Mann matrices. The values of are the masses of the SU(3) Goldstone bosons, which adopt the experimental values [ ]. The chiral coupling constant , is determined from the coupling constant through

g 2 ch 4 π = ( 3 5

Assuming that flavor SU(3) is an exact symmetry, it will only be broken by the different mass of the strange quark. The other symbols in the above expressions have their usual mean- ings.

As for in Eq. ( ), which represent the antiquark-antiquark ( ) and quark-antiquark ( ) interac- tions, for the antiquark we replace in Eqs. ( ) and ( ) with , and replace in Eqs. ( ) with . In this way, the forms of can be derived. It is noted that there

is no annihilation between quark and antiquark. The reason 127

is that the p ¯Ω state cannot annihilate to the vacuum due to the 128

different quark flavor contents of . All the parame- ters used in this work and the calculated baryon masses are listed in Table and Table , respectively. In quark model, the corresponding antibaryons have the same masses as their baryon partners.

Two parameter sets used in this work: = 0.54. In addition, quark delocalization was introduced to enlarge the model variational space to take into account the mutual distortion or the internal excitations of nucleons in the course of interaction. It is realized by specifying the single-particle orbital wave function of the QDCSM as a linear combination of left and right Gaussians, the single-particle orbital wave functions used in the ordinary quark cluster model ) = ( ) = (

N ( S i , ϵ ) = √

Masses of ground-state baryons (in MeV) with two param- eter sets.

Baryon 939 (939) 1232 (1232) 1124 (1118) 1238 (1224) 1360 (1358) 1374 (1365) 1496 (1499) 1642 (1654) justed one but determined variationally by the dynamics of the multiquark system itself. In this way, the multiquark sys- tem chooses its favorable configuration in the interacting pro-

cess. This mechanism has been used to explain the crossover 148

transition between the hadron phase and quark-gluon plasma Experimentally, the correlation function can be mea- sured based on:

C ( k ) = ξ ( k ) N same ( k ) N mixed ( k ) , (12)

where mixed represent the distributions of hadron-hadron pairs produced in the same and in differ- ent collisions, respectively, and denotes the corrections for experimental effects. In theoretical studies, the correla- tion function can be calculated using the Koonin Pratt (KP) formula [

where is the size parameter of the source. Thus, two important factors of the correlation function are included in Eq. ( ): the collision system, which is related to the source function , and the two-particle interaction, which is embedded in the relative wave function For a pair of non-identical particles, such as , assuming that only -wave part of the wave function is modified by the two-particle interaction, can be given by: where the spherical Bessel function represents the -wave part of the non-interacting wave function, and stands for the scattering wave function affected by the two- particle interaction. Substituting the relative wave function into the KP formula yields the correlation func- tion:

C p ¯Ω ( k ) = 1 + ∫ ∞

can be obtained by solving the Schrödinger equation, and a similar approach has been utilized in the fem- toscopic correlation analysis tool using the Schrödinger equa- tion [

where µ = m p m Ω / ( m p + m Ω ) is the reduced mass of the 157

system. Considering the case of the -wave, the wave function can be separated into a radial term and an angular term and expressed as:

ψ p ¯Ω ( r, θ, ϕ ) = R k ( r ) Y 0 0 ( θ, ϕ ) . (18)

Considering the interaction between a proton and an baryon, which includes both the strong interaction and the repulsive Coulomb interaction, the potential can be written

V ( r ) = V Strong ( r ) + V Coulomb ( r ) , (19)

where V Coulomb ( r ) = + α ℏ c/r , and α is the fine-structure con- 159

stant. The method to obtain the strong interaction potential Strong will be introduced in the next section.

Once the total interaction potential is determined, the radial Schrödinger equation can be solved:

where E = ℏ 2 k 2 / (2 µ ) and u k ( r ) = rR k ( r ) .

On this 162

basis, the correlation function for given spin-parity quantum numbers can be calculated through Eq. ( ). The calculation of the correlation functions described above is

based on obtaining the scattering wave functions by solv- 166

ing the Schrödinger equation in coordinate space [ ]. Additionally, the scattering wave functions can also be obtained by solving the Lippmann-Schwinger (Bethe- Salpeter) equation in momentum space [ ]. Fur- ther details on correlation functions for various systems can be found in the references mentioned above.

Additionally, for the S -wave p ¯Ω dibaryon system, the pos- sible spin-parity quantum numbers are J P = 1 − and 2 − , respectively. Since the experimentally measured correlation function is spin-averaged, the theoretically obtained correla- tion function should also consider the average over systems with different quantum numbers:

C p ¯Ω ( k ) = 3

8 C

J =1 p ¯Ω ( k ) + 5

8 C

J =2 p ¯Ω ( k ) . (21)

Gel’fand-Levitan-Marchenko method for inverse scattering problem To solve Eq. ( ), two-body interaction potential absolutely necessary. The QDCSM is actually a treatment on few-body problem, which means directly extracting a two- body interaction potential from it will not be so natu- ral. Hence, the QDCSM can be employed to investigate the scattering processes, from which the desired potential can be

obtained, since the hadronization is fully incorporated in the 181

model. The approach we adopted to extract the two-body equiva- lent potential is the GLM method, which is a very pow- erful tool in inverse scattering theory [ ]. It can provide us a systematic approach to reconstruct an equivalent potential from the scattering data of a specific process, which makes it as a very classical “inverse problem”. Thus, this method will give us another path to understand the nature of two-body in- teraction.

The key equation of the GLM method used in the work is the Marchenko equation [ ], which can be written in the -wave case in a integration equation form as:

r K ( r, s ) F ( s, r ′ ) d s = 0 . (22)

Here, the kernel function is the solution of the equa- tion to be determined, and is the inverse Fourier transformation of reflection coefficient as:

F ( r, r ′ ) = 1

i =1 M i e − κ i r e − κ i r ′ . (23)

The partial-wave scattering matrix S ( k ) is given by S ( k ) = 191

exp(2i , where is the scattering phase shift satis- fying . Here, repre- sent the scattering length and the effective range, respectively.

Additionally, is the number of bound states, denotes the wavenumber of the -th bound state, and is the norming

constant. Then, after solving Marchenko equation and ob-

taining K ( r, r ′ ) , the potential can be reconstructed as: 198

V ( r ) = − 2 d

There is one point we want to emphasize here. Generally, when there exists bound states, this method can not give us a fully determined potential but end up with a set of phase- equivalent potentials [ ]. However, if one fix all the

a unique way such as calculating from Jost solution, the ob- 204

tained potential will be unique for further calculation [ 98 , 99 ]. 205

By using this method, preparation for further calculation can be done, for a more comprehensive discussion on this method, one can refer to Refs. [ RESULTS AND DISCUSSION -wave systems with isospin , spin parity are investigated on the basis of the QD- CSM. In order to see whether or not there is any bound state, a dynamic calculation is performed as a first step. The resonat- ing group method (RGM) is employed to solve the bound- state problem. In this approach, the total wave function of the six-quark (three quarks and three antiquarks) system is con- structed as:

Ψ = A ϕ p ( ξ 1 ) ϕ ¯Ω ( ξ 2 ) χ ( R ) , (25)

where are the internal wave functions of the pro- ton and clusters, represents the relative motion wave function, and is the antisymmetrization operator account- ing for quark exchange effects. The RGM equation is derived by projecting the Schrödinger equation onto the cluster basis, leading to a coupled integro-differential equation:

where H and N are the Hamiltonian and normalization ker- 210

nels. The relative wave function is expanded with Gaussian basis functions, converting the integral equation into a generalized eigenvalue problem. Solving this equation provides the binding energies and wave functions of the states. This method has been widely and successfully applied to baryon–baryon interactions and multiquark systems [

The binding energies of the p ¯Ω systems with J P = 1 − and 218

J P = 2 − , denoted as E B , are listed in Table 3 [TABLE:3] . Here, E Theo th 219

represents the theoretical threshold, and represents the eigenvalue of the corresponding system. The calculation for systems does not involve channel coupling, because we limit our study to color-singlet sub-clusters consisting of three quarks and three quarks.

From Table 3 , we can find that both the J P = 1 − and 225

J P = 2 − p ¯Ω form bound states with the binding energies 226

about 10 MeV and 9 MeV, respectively. By contrast, in our previous work on the systems, the single channel calcu-

lation shows that neither the J P = 1 + nor J P = 2 + p Ω is 229

Calculated binding energies (MeV) of the systems for different spin-parity states with two parameter sets.

bound. After channel coupling, only the p Ω with J P = 2 + 230

forms a bound state with binding energy about 6 MeV. Ac- cording to the results obtained with the second parameter set,

the binding energies of the J P = 1 − and J P = 2 − p ¯Ω states 233

are about 18 MeV and 16 MeV, respectively. The correspond-

ing binding energy of the p Ω state with J P = 2 + after chan- 235

nel coupling is about 14 MeV. These numerical results indi- fore, considering that the attractive interaction is implied tions [ ], we look forward to the experimental progress on In order to further study the interaction between nucleon , we calculated the scattering phase shifts of the tems. The calculation is based on the well developed Kohn- Hulthen-Kato(KHK) variational method, the details of this method can be found in Refs. [ ]. The low-energy scat- tering phase shifts of the systems with two parameter sets

are shown in Fig. 1 [FIGURE:1] . For the p ¯Ω systems with both J P = 1 − 249

and J P = 2 − , the scattering phase shifts approach 180 ◦ when 250

MeV and rapidly decreases when increases, which indicates the existence of a bound state in these sys- tems. This conclusion is consistent with the the bound state calculation discussed earlier. Then, we can extract the scat- tering length and the effective range of the systems from the low-energy phase shifts obtained above by using the expansion:

k cotδ = − 1 a 0 + 1

where k is the momentum of the relative motion with k = 259 √ 2 µE c.m., µ is the reduced mass of two baryons, and E c.m. 260

is the incident energy; is the low-energy scattering phase shifts. And the binding energy can be calculated accord- ing to the following relation:

E ′ B = ℏ 2 α 2

where is the wave number which can be obtained from the relation [

r eff = 2

Note that this is another way to calculate the binding energy, therefore it is labeled . The scattering parameters of the systems, along with the binding energies obtained using the scattering parameters, are listed in Table

Calculated phase shifts of the systems for with two parameter sets.

Extracted scattering length , effective range , and binding energy of the systems with two parameter sets.

From Table , our results show that the scattering lengths

are positive for the p ¯Ω systems with J P = 1 − and 2 − , which 273

also confirms the existence of bound states. Besides, the broadly consistent with the numerical results shown in Ta- , which is obtained by the dynamic calculation. Ad-

ditionally, in the method of obtaining the binding energies 278

using scattering parameters, the binding energies of the two

J P = 2 + . 281

Moreover, by solving the inverse scattering problem, we in Eq. ( ). Before that, we can study the general properties well potential model. The correlation functions correspond- ing to different degrees of square well potentials are presented tions that influenced only by the repulsive Coulomb interac- tion. The dashed blue lines represent the correlation functions influenced only by square well potentials. We introduce three

square well potentials with width r 0 = 2 fm, weak attrac- 293

tion in panel(a) corresponding to V 0 = − 10 MeV, moder- 294

ate attraction in panel (b) corresponding to V 0 = − 28 MeV, 295

and relatively strong attraction in panel (c) corresponding to

V 0 = − 40 MeV. The dotted black lines represent the corre- 297

lation functions that influenced by both Coulomb interaction and square well potentials through Eq. ( ). Additionally, we

adopt a source size parameter R = 0 . 95 fm in Eq. ( 14 ), which 300

is the same value used in our previous correlation analy- sis [ ]. This value was originally extracted by the ALICE Collaboration [ ]. It should be emphasized that this source size is determined by the specific collision system and experimental conditions in heavy-ion collisions.

In Fig. , panels (a) and (b), the correlation function af-

fected only by the square potential is above unity in the low- 307

energy region, which is due to the attractive interaction. The difference is that the weak attraction is not enough to form a bound state, therefore the correlation function is always above

unity, while the moderate attraction forms a shallow bound 311

state. The existence of the bound state leads to the deple- tion of the correlation function, so there exists a part below

unity. After taking into account both the Coulomb interac- 314

tion and square well potentials, which mainly dominates the

low-energy region (0 < k < 25 MeV), the correlation func- 316

tion forms a peak-like structure. In panel (c), the correlation

function remains below unity for a relatively strong attrac- 318

tion. A discussion about this phenomenon can be found in Refs. [ ]. After considering the Coulomb interaction, one can see that the correlation function in the low-energy re- gion nearly coincides with the result obtained by considering only the Coulomb interaction. As the relative momentum increases, the correlation function closely matches the result obtained by considering only a relatively strong attraction.

After replacing the square well potentials with the effec- tive potentials obtained by solving the inverse scattering prob- lem using the GLM method, which is briefly introduced in , we can study the correlation function of the tems. As the two parameter sets give similar results, the dis- cussion of the correlation function is presented based on the first parameter set for conciseness. Both the effective poten-

tials of the p ¯Ω systems with J P = 1 − and 2 − are obtained. 333

The total correlation function is the superposition of the correlation functions corresponding to the two quantum num- bers according to Eq. ( ). Since the system forms bound states for both quantum numbers and the interactions are sim- ilar, we omit the comparison of the correlation functions for the two quantum numbers here. The total correlation func- tions calculated for different values of source size parameter are shown in Fig. . In addition, recent studies on the emis- sion source properties can be found in Refs. [ In Fig. , panels (a)–(e), the dashed gray lines and the dot- ted orange lines represent the correlation functions con- sidering only the Coulomb interaction and both the Coulomb interaction and strong interaction, respectively. In panel (f), the gray band stands for the correlation functions influenced only by the Coulomb interaction with size parameter rang- ing from 1.0 to 2.5 fm, while the other lines summarize the correlation functions shown in panels (a)-(e). According to our results, the change of the size parameter can greatly influence the correlation functions. An obvious feature is that as increases, the peak-like structure caused by the dif- ferent dominant regions of the Coulomb interaction and strong interaction gradually becomes less obvious and even- tually disappears. Since two bound states are obtained in our calculation, it is very important to verify this conclusion in the correlation functions. It can be seen that as the correlation function influenced only by the Coulomb interaction grad-

ually approaches unity, the depletion caused by the bound 360

R = 1 . 0 f m

R = 2 . 0 f m

states leads to the correlation function being below that of the Coulomb-only case.

While femtoscopic measurements in heavy-ion collisions provide important access to the interaction, such bound states may also be produced in other high-energy environ- ments capable of forming multi-strange baryon–antibaryon pairs, such as collisions, as well as high-energy fixed-target experiments, although the production probability may vary across systems. If a bound state is formed, it can decay through the weak decay of the , leading to final states such as , or through quark rearrangement into a three-meson final state,

R = 1 . 5 f m

R = 1 . 0 f m

R = 1 . 2 f m

R = 1 . 5 f m

R = 2 . 0 f m

R = 2 . 5 f m

which may provide complementary detection signatures be- yond femtoscopy in future searches.

In recent years, experimental data on correlation func- tions have increased rapidly [ ], providing unprece- dented insights into hadron-hadron interactions across a va-

riety of systems. Together with femtoscopic techniques, 378

these studies open up new possibilities for extracting low- energy scattering parameters that are difficult to access oth- erwise. On the theoretical side, continuous progress in lat- tice QCD, effective field theory, and quark-model-based ap- proaches has greatly enriched our understanding and offered valuable guidance for interpreting the experimental observa- Correlation functions for different square well potentials, with , where MeV in panel (a), MeV in panel (b), MeV in panel (c), and correlation functions with different values of source size parameter , where fm in panel (a), fm in panel (b), fm in panel (c), fm in panel (d), and fm in panel (e) are summarized in panel (f).

tions [ ]. The synergy between experimental measure- ments and theoretical developments will not only deepen our knowledge of the strong interaction, but also pave the way

for future explorations of exotic hadronic states and nuclear 388

physics [ ]. In this context, it is well established that nucleon-nucleon and hyperon-nucleon interactions provide the basis for the formation of nuclei and hypernuclei [ ], while antihyperon-antinucleon interactions give rise to anti-hypernuclei [ ]. An open question is whether antihyperon-nucleon interactions could also give rise to the formation of novel nuclear systems. As a continuation of the present study, we will further explore such systems based on the antihyperon-nucleon interactions obtained in this work.

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