The Interaction of Moving $\mathbf{Q\bar{Q}}$ and QQq in the Thermal Plasma
Xuan Liu Sheng Lin Xun Chen





School of Nuclear Science and Technology, University of South China, Hengyang 421001, China

School of Nuclear Science and Technology, University of South China, Hengyang 421001, China

chenxun@usc.edu.cn
School of Nuclear Science and Technology, University of South China, Hengyang 421001, China
Key Laboratory of Quark and Lepton Physics (MOE), Central China Normal University, Wuhan 430079,China
§ ABSTRACT 
    The strength of the force between heavy quarks is studied for heavy quarkonium (QQ) and doubly heavy baryons (QQq) at finite temperature and rapidity using the gauge/gravity duality in this paper. The strength of the interaction is defined as an effective running coupling α from lattice. By considering the QQ and QQq moving through the thermal quark-gluon plasma, we find that the interaction of heavy quarks for QQ is small and remains almost constant when L<0.2 fm. The strength of the interaction for QQq is always less than that for QQ and approaches a constant when L<0.4 fm. The α and the maximum of effective running coupling α̃ for both QQ and QQq decrease with increasing temperature or rapidity. Through comparison, we find that QQq is more sensitive to changes in temperature and rapidity, and the interaction force between quarks is always less than that of QQ, indicating that QQ is more stable than QQq in the presence of temperature and rapidity.



The Interaction of Moving 𝐐𝐐̅ and QQq in the Thermal Plasma
    Xun Chen
    September 9, 2024
===========================================================





§ INTRODUCTION

The running coupling constant serves as a crucial metric for measuring the strength of the strong interaction between quarks, and the force between quarks is essential for our understanding of QCD and the QCD phase transition. When quarks are confined within hadrons, the force between them increases with the separation distance until the string breaks, resulting in the creation of new quark-antiquark pairs, thereby preventing the isolation of free quarks. QCD predicts that deconfinement occurs under extreme conditions, where color screening happens at long distances between quarks, leading to the emergence of relatively free quarks <cit.>. A system composed of such deconfined particles is referred to as a Quark-Gluon Plasma (QGP) <cit.>. Moreover, the property of asymptotic freedom indicates that quarks behave as free particles at very high energy scales. These phenomena can be attempted to be explained through the running of the coupling constant. Additionally, the running coupling constant plays a key role in many important physical processes such as jet quenching, heavy quarkonium production, and Higgs particle production <cit.>. Lattice QCD defined an effective running coupling <cit.>,

    α_Q Q̅(r)=1/C_F r^2 ∂ E(r)/∂ r,

to study the force between the static quark and antiquark. C_F=4 / 3 is the Casimir operator in the fundamental representation of SU(3) and E is the static energy of QQ.

Creating QGP through heavy-ion collisions allows for the simulation of the extreme conditions of high temperature and rapid expansion present in the early universe <cit.>. This helps us to understand the evolution of the early universe and the properties of QCD. Heavy quarkonium is an important probe for studying conditions of extreme high temperature and rapid expansion. Moreover, in the recent LHCb experiment at CERN, researchers have discovered a new particle known as Ξ_cc^++ <cit.>. It is composed of two heavy quarks and one light quark, and its discovery has greatly increased interest in the study of doubly heavy baryon. The running coupling constant has been extensively studied in the Refs. <cit.>, as discussed through lattice calculations at finite temperatures in the Refs. <cit.>. It is well known that the QGP has been rapidly expanding since its inception. Therefore, rapidity is an unavoidable factor to consider when discussing the effective running coupling of particles. This paper attempts to reveal more information about the QGP by contrasting heavy quarkonium and doubly heavy baryon at finite temperatures and rapidities. This can aid in our understanding of particle transport in the QGP and the plasma's effect on particle properties. Moreover, the interaction forces between quarks in the QGP can also reflect the state of the QGP to a certain extent, such as determining whether it is a strongly coupled QGP (sQGP) or a weakly coupled QGP (wQGP) <cit.>.

Lattice gauge theory remains the fundamental tool for studying non-perturbative phenomena in QCD, yet its application to doubly heavy baryon has been relatively limited <cit.>. Gauge/gravity duality offers a new avenue for probing strongly coupled gauge theories. Originally,  Maldacena <cit.> proposed the gauge/gravity duality for conformal field theories, but it was subsequently extended to encompass theories akin to QCD, thereby establishing to some extent a linkage between string theory and heavy-ion collisions <cit.>. Research on moving heavy quarkonium can be found in lattice <cit.>, effective field theory <cit.>, perturbative QCD <cit.>,  S matrix <cit.>, and holographic QCD <cit.>.  Moreover, the multi-quark potential obtained through effective string model is in good agreement with lattice results <cit.>.

In this paper, we primarily investigate the effective running coupling of heavy quarkonium and doubly heavy baryon at finite temperature and rapidity. The rest of the paper is organized as follows: in the Sec. <ref>, we determine the holographic parameters by fitting their lattice potentials, and present the state diagram of particles in the T-η plane. In the Sec. <ref>, we discuss the effective running coupling of heavy quarkonium and doubly heavy baryon, including the effective running coupling with distance, and with temperature and rapidity, as well as the impact of rapidity on the temperature dependence of the effective coupling constant. Additionally, we examine the effects of temperature and rapidity on their screening distances. In the Sec. <ref>, we provide a summary of this paper.



§ PRELIMINARIES

The effective string holographic models have been proposed by Andreev recently. These models can not only describe the potential of heavy quarkonium <cit.>, but also describe the potential of exotic hadrons <cit.>. The purpose of this study is to reveal the properties of QQ and QQq based on the effective string model by investigating the effective running coupling and the screening distance of their motion in a thermal medium. First, we present the metric

    ds^2   =w(r)  (-f(r)dt^2+dx^2+1/f(r)dr^2 )+e^-𝐬r^2g_ab^(5)dω^adω^b,

where

    w(r)   =e^𝐬r^2R^2/r^2 ,
     f(r)   =1-r^4/r_h^4.


The metric signifies a deformation of the Euclidean AdS_5×S_5 space, controlled by a single parameter 𝐬 and with a radius R = 1. In this work, 𝐬 is determined to be 0.41 GeV^2 by fitting the lattice potential of heavy quarkonium. Therefore, the metric is composed of an AdS_5 space and a five-dimensional compact space X with coordinates ω^a. The function f(r) is the blackening factor, which decreases within the interval [0, r_h], where r_h represents the black hole horizon (brane). Additionally, when hadrons are confined, an imaginary wall exists at r_w on the r-axis <cit.>. The Hawking temperature T associated with the black hole is given by:

    T=1/4π | df/dr |_r=r_h=1/π r_h.


In this paper, we consider a particle moving at a rapidity
η along the x_3 direction in a thermal medium at temperature T. It is known that the particle is also subject to drag force in the medium, but this is not the main focus of our discussion <cit.>. We can assume that the particle is at rest, while the thermal medium moves relative to it at rapidity η, which can be considered as a "thermal wind" blowing past the particle in the x_3 direction <cit.>. Through a Lorentz transformation, we can provide a new background metric:

    ds^2=w(r)   (-g_1(r)dt^2
     -2sinh(η)cosh(η)(1-g_1(r)/g_2(r))dx_3dt
       +g_3(r)dx_3^2+dx_1^2+dx_2^2+g_2(r)/g_1(r)dr^2)+e^-𝐬r^2g_ab^(5)dω^adω^b,

where

    g_1(r)   =f(r)cosh^2(η)-sinh^2(η),
    g_2(r)   =g_1(r)/f(r),
    g_3(r)   =cosh^2(η)-f(r)sinh^2(η).


The Nambu-Goto action of a string is

    S_NG = - g ∫ dξ^0 dξ^1√(-  g_ab),

where the g_ab is an induced metric (ξ^0, ξ^1) are worldsheet coordinates, and g is related to the string tension.
Based on the AdS/CFT correspondence, we know that the vertex corresponds to a five brane <cit.>, the baryon vertex action is S_vert = τ_5∫ d^6ξ√(γ^(6)), where τ_5 is the brane tension and ξ^i are the world-volume coordinates. Since the brane is wrapped on the compact space X, it appears point-like in AdS_5. We choose a static gauge where ξ^0 = t and ξ^a = θ^a, with θ^a being the coordinates on X. Consequently, the action is:

    S_vert=τ_v∫ dte^-2𝐬r^2/r√(g_1(r)),

where τ_v is a dimensionless parameter defined by τ_v=τ_5Rvol(X) and vol(X) is a volume of X. Finally, we consider the light quarks at the endpoints of the string as a tachyon field, which is coupled to the worldsheet boundary through S_q=∫ dτ𝐞T, where T(x, r) is a scalar field that describes open string tachyon, τ is a coordinate on the boundary, and 𝐞 is the boundary metric <cit.>. We consider only the case where T(x, r)=T_0 and the worldsheet boundary is a line in the t direction, in which case the action can be written as:

    S_q=m∫ dte^𝐬r^2/2/r√(g_1(r)),

This action represents a particle of mass T_0 at rest, with a medium at temperature T moving past it at a rapidity η.



 §.§ The potential and effective running coupling of heavy quarkonium


In this paper, we investigate the scenario where the direction of motion is perpendicular to the heavy quark pair, with the heavy quark pair located at x_1 and the rapidity η along the x_3 direction. For brevity, x will be used to denote x_1 in the following. Then we choose the static gauge ξ^0=t, ξ^1=x, and the action of QQ can be written as:


    S_QQ   =g_QQ∫_0^t∫_-L/2^L/2 w(r)√(g_1(r)+g_2(r)(∂_xr)^2) dx 
       =g_QQt∫_-L/2^L/2 w(r)√(g_1(r)+g_2(r)(∂_xr)^2) dx,


where the g_QQ = 0.176. And the boundary condition of r(x) is

    r(± L/2)=0, r(0)=r_0, (∂_xr|_r=r_0)^2=0.

By substituting into the Euler-Lagrange equation, we can obtain:

    ∂_xr=√(w(r)^2g_1(r)^2-g_1(r)w(r_0)^2g_1(r_0)/w(r_0)^2g_1(r_0)g_2(r)).

The distance between heavy quark and anti-quark is:

    L=2∫_0^r_0∂_rx  dr,

where ∂_rx=∂ x/∂ r=1/∂_xr.
By E=S/t and after normalizing, we can obtain the potential of QQ,

    E_QQ=2g_QQ(∫_0^r_0(w(r)√(g_1(r)(∂_rx)^2+g_2(r))-1/r^2)dr-1/r_0).

We consider the pattern of QQ̅ string breaking as

    QQ⟶Qq+Qq.

It is easy to know that

    E_break= E_Qq+E_Qq=2g_QQ(∫_0^r_q(w(r)√(g_2(r))-1/r^2)dr-1/r_q+ne^sr_q^2/2/r_q√(g_1(r_q))).

At T=0, η=0, we obtain E_break=1.057 GeV which means that string breaking occurs for the QQ̅ at E_QQ=1.057 GeV, resulting in a breaking distance of L_break=1.23 fm. We present a comparison between the potential obtained by the model and the lattice potential in Fig. <ref>.

Therefore, the effective running coupling of QQ from lattice QCD is given by  <cit.>

    α_QQ=3L^2/4dE_QQ/dL.

Then we present its basic behavior in Fig. <ref>. The effective coupling constant of QQ is very small and increases only slightly at small scales when L < 0.2 fm, which can explain the phenomenon of asymptotic freedom exhibited by quarks at short distances. At larger scales, the obvious increase of the effective coupling constant with distance also corresponds to our understanding of the strong interaction force. We discuss in detail its effective running coupling properties in moving thermal media in the subsequent sections.





 §.§ The potential and effective running coupling of doubly heavy baryon


The double heavy baryon consists of two heavy quarks and one light quark, which introduces a baryon vertex. When considering a string configuration for the ground state of QQq, it is natural due to symmetry to place the light quark between the two heavy quarks <cit.>. The QQq potential depends only on the distance L between the heavy quarks. In the Ref. <cit.>, the QQq potential is also fitted to a function similar to the Cornell potential:

    E_QQq=σ_effL-A_eff/L+C_eff,

where the first term represents the confinement potential, and the second term comes from one-gluon exchange <cit.>. Due to the octet nature of gluons, the factor 4/3 is included in A_eff.
As the distance between the heavy quark pair increases, its string configuration will change, as shown in Fig. <ref>. And the three configurations from left to right are called: small L, intermediate L, and large L. In the small L configuration, the action is composed of three strings, the baryon vertex, and the light quark. After r_v=r_q, it transitions to the intermediate L configuration, where the position of the light quark coincides with the baryon vertex, and the action is constituted by two strings, the baryon vertex, and the light quark. When the string at the baryon vertex transitions from a "convex" to a "concave" shape, it becomes the large L configuration. At this configuration, the composition of the action is consistent with the intermediate L. Therefore, the total action at small L is:

    S=∑_i=1^3 S_NG^(i)+S_vert+S_q.

Then we choose the static gauge where ξ^0=t, ξ^1=r, and the boundary conditions of x(r) is:

    x(0)=± L/2, x(r_v)=x(r_q)=0, 
        (∂_rx)^2=^2(α),   r=r_v
    
          (∂_rx)^2=0,   r∈ (r_v,r_q].

And we obtain the total action as

    S=g_QQqt   (2∫_0^r_v w(r)√(g_1(r)(∂_rx)^2+g_2(r)) dr+ ∫_r_v^r_q w(r)√(g_2(r))dr
       +3ke^-2𝐬r^2/r√(g_1(r))+ne^𝐬r^2/2/r√(g_1(r))),

where k=τ_v/3g_QQq, n=m/g_QQq. By substituting the first term of the action into the Euler-Lagrange equation, we obtain

    ∂_rx=√(w(r_v)^2g_1(r_v)^2g_2(r)/w(r)^2g_1(r)^2(g_1(r_v)+g_2(r_v)tan^2(α))-g_1(r)w(r_v)^2g_1(r_v)^2).

Furthermore, there must be a balance of forces at the light quark and the baryon vertex, with the light quark site satisfying f_q+e_3'=0, and the vertex site satisfying f_v+e_3+e_1+e_2=0. These forces are obtained by taking the variation of their action:

    e_1   =g_QQqw(r_v)(-√(g_1(r_v)f(r_v)/f(r_v)+tan^2α), -√(g_1(r_v)/f(r_v)^2^2(α)+f(r_v))),
    
        e_2   =g_QQqw(r_v)(√(g_1(r_v)f(r_v)/f(r_v)+tan^2α), -√(g_1(r_v)/f(r_v)^2^2(α)+f(r_v))),
    
        e_3   =g_QQqw(r_v)(0, √(g_2(r_v))),
    
        e_3'   =g_QQqw(r_q)(0, -√(g_2(r_q))),
    
        f_q   =(0, -g_QQqn∂_r_q(e^𝐬r_q^2/2/r_q√(g_1(r_q)))),
    
        f_v   =(0, -3g_QQqk∂_r_v(e^-2𝐬r_v^2/r_v√(g_1(r_v)))),

it is evident that when T and η are fixed, the force at the light quark site depends only on r_q, while the force at the vertex involves only two unknowns, r_v and α. Therefore, we can determine the value of r_q and the function of α in terms of r_v, and from these, we can derive the potential of small L. The potential, with the divergent terms eliminated through normalization, is represented as:

    E_small=g_QQq   (2∫_0^r_v(w(r)√(g_1(r)(∂_rx)^2+g_2(r))-1/r^2)dr-2/r_v+∫_r_v^r_q w(r)√(g_2(r))dr
       +3ke^-2𝐬r_v^2/r_v√(g_1(r_v))+ne^𝐬r_q^2/2/r_q√(g_1(r_q)))+c.





At intermediate L, the total action is given by

    S=∑_i=1^2 S_NG^(i)+S_vert+S_q,

and the boundary conditions of x(r) become

    x(0)=± L/2, x(r_v)=0, (∂_rx|_r=r_v)^2=^2(α).

And only the forces at the vertices need to be considered: f_v+f_q+e_1+e_2=0, Since r_q = r_v, we only need to replace r_q with r_v in the force. Similarly, we can obtain its potential at intermediate L as:

    E_intermediate=g_QQq   (2∫_0^r_v(w(r)√(g_1(r)(∂_rx)^2+g_2(r))-1/r^2)dr-2/r_v
       +3ke^-2𝐬r_v^2/r_v√(g_1(r_v))+ne^𝐬r_v^2/2/r_v√(g_1(r_v)))+c.

The distance between heavy quark pairs for small L and intermediate L is calculated using the following function

    L=2∫_0^r_v∂_rx  dr.


As Fig. <ref> shows the large L is special because the string has two additional smooth inflection points. For the string configuration at large L, we choose a new metric ξ^0=t, ξ^1=x, and now the boundary condition of r(x) is

    r(± L/2)=0, r(0)=r_v,  
        (∂_xr)^2=tan^2(α),   r=r_v
    
          (∂_xr)^2=0,   r=r_0.

For configurations at large L, the form of the force balance equation at the vertex remains consistent with that at intermediate L but introduces an additional unknown r_0, which can be determined through the properties of the first integral

    ℋ|_r=r_0   =w(r_0)√(g_1(r_0)),
    ℋ|_r=r_v   =w(r_v)g_1(r_v)/√(g_1(r_v)+g_2(r_v)tan^2(α)),
    
        w(r_0)√(g_1(r_0))   =w(r_v)g_1(r_v)/√(g_1(r_v)+g_2(r_v)tan^2(α)).

Furthermore we obtain

    ∂_xr=√(w(r)^2g_1(r)^2-g_1(r)w(r_0)^2g_1(r_0)/w(r_0)^2g_1(r_0)g_2(r)).

For convenience in the calculation, we present the potential after equivalent transformation and normalization as follows

    E_large=g_QQq   (2∫_0^r_v(w(r)√(g_1(r)(∂_rx)^2+g_2(r))-1/r^2)dr+2∫_r_v^r_0 w(r)√(g_1(r)(∂_rx)^2+g_2(r))dr
       -2/r_v
        +3ke^-2𝐬r_v^2/r√(g_1(r_v))+ne^𝐬r_v^2/2/r_v√(g_1(r_v)))+c.

The distance at large L should be

    L=2(∫_0^r_v∂_rx dr+∫_r_v^r_0∂_rx dr).

Clearly, the complete potential is pieced together from three potential functions, but we do not need to focus on which configuration it belongs to. Therefore, in the following text, the potential of QQq will be collectively referred to as E_QQq.

The parameters determined through fitting the lattice points for the QQq potential are: 𝐬=0.41 GeV^2, k=-0.321, n=2.941, g_QQq=0.082, c=0.73 GeV. The potential of QQq is shown in Fig. <ref>. The last point for small L is (r_v, L, E) = (1.1600, 0.3250, 1.0365), the starting point for intermediate L is (r_v, L, E) = (1.1602, 0.3251, 1.0366), and its last point is (r_v, L, E) = (1.4132, 0.7174, 1.2282). The starting point for large L is (r_v, L, E) = (1.4150, 0.7233, 1.2310), with no sudden change in the data.



Based on the model of heavy quarkonium in Refs. <cit.>, similarly fitted to the Cornell potential for the QQq, we can also determine the effective running coupling,

    α_QQq = 3L^2/4dE_QQq/dL,

to study the force between heavy quarks. It is the inter-two-quark potential in baryons which effectively includes the light-quark effects <cit.>. Moreover, it can be proven that the physical significance of Eq. (<ref>) and Eq. (<ref>) is the same, as both represent the effective coupling strength between heavy quarks. Thus, the effective running coupling of QQq is a function of the separation distance between the two heavy quarks as shown in Fig. <ref>. It can be seen that at small scales, QQq also exhibits asymptotic freedom behavior, and its range (L< 0.4 fm) is broader compared to QQ. Moreover, the effective running coupling of QQq is always smaller than that of QQ and is almost half of it. This is very close to the relationship between A_QQq and A_QQ fitted in the lattice <cit.>, which is due to the presence of the light quark reducing the interquark force <cit.>.






§ NUMERICAL RESULTS AND DISCUSSION


Before we proceed with the discussion, we need to establish the critical points (critical temperatures at different rapidities). At low temperatures and rapidities, QQ is confined, an imaginary wall r_w. The potential of the confined QQ increases with distance, but when it rises to a certain value, string breaking occurs, exciting light quark and anti-quark from the vacuum, with quarks always remaining confined within a hadron. At high temperatures and rapidities, QQ becomes deconfined, and the imaginary wall disappears. At this point, there is a maximum quark separation distance. When this distance is exceeded, the QQ no longer forms a U-shaped string configuration but instead consists of two straight string segments extending from the boundary to the horizon <cit.>. At this point, the potential also reaches its maximum, indicating that the QQ is screened at this point. This distance is known as the screening length. Considering string breaking or screening as a characteristic to distinguish between confinement and deconfinement, we can obtain the critical point of QQ.

The critical points for QQq can be determined similarly, however, it possesses an additional property. At high temperature and/or rapidity QQq can dissociate at a certain distance. However, At extreme high temperature and/or rapidity the QQq can not exit judged from the force balance. Base on the previous discussion, we can draw out the state diagrams of QQ and QQq on the (T, η) plane as in Fig. <ref>. The detailed discussions about the state diagram have already been completed in our previous work <cit.>.





 §.§ Discussion about heavy quarkonium


Next, we discuss the temperature dependence of the effective running coupling of QQ. From Fig. <ref>, it can be seen that critical temperature T_c=0.1345 GeV when η=0. We calculate the effective running coupling in the range of T/T_c ∈ [0, 3] as shown in Fig. <ref>. When T/T_c ∈ [0, 1], QQ is in a confined state and the effect of temperature on the effective running coupling is relatively small. Therefore, our discussion of effective running coupling primarily concentrates on the deconfined state when T/T_c > 1. Additionally, the string breaking can happen when T/T_c < 1. The detailed discussion of string breaking at finite temperature and rapidity can be found in our previous works <cit.>.




From the left of Fig.<ref>, it can be observed that the higher the temperature, the smaller the effective running coupling, and at small scales, the effective running coupling's dependence on temperature is minor. This is understandable due to the asymptotic freedom of quarks at small scales. Additionally, the effect of temperature on effective running coupling primarily manifests in the maximum effective coupling constant α̃_QQ (T, η), with the maximum effective coupling constant decreasing as the temperature increases, and the maximum coupling distance L_α̃_QQ reduces. The impact of rapidity on the effective running coupling is similar to that of temperature, as shown in the right of Fig. <ref>. We choose a lower temperature, T=0.1412 GeV (when η=0, T=1.05T_c), to observe a comprehensive process of the influence of rapidity on the effective running coupling. It can be seen that the greater the rapidity, the smaller the effective running coupling, and the smaller the maximum effective coupling constant. Furthermore, it is noteworthy that the screening distance L_screen for QQ is greater than the maximum coupling distance
L_α̃_QQ. Here we provide the definition: △ L= L_screen-L_α̃_QQ.

The maximum effective coupling constant as a function of T/T_c(η=0) and η/η_c(T=0.1 GeV) is shown in Fig. <ref>. From the left graph, it is apparent that when T/T_c < 1.5, the maximum effective coupling constant rapidly decreases to a lower level as the temperature increases, and becomes relatively flat for T/T_c > 3. The rapidity dependence of the maximum effective coupling constant is qualitatively similar to that of temperature; however, as a function of rapidity, it exhibits a more gradual decline and a broader range (η/η_c <2) in the falling region. We present the screening distance and the maximum coupling distance in Fig. <ref>. The screening distance decreases with increasing temperature or rapidity, while the slope also decreases with increasing temperature or rapidity. The screening distance as a function of T/T_c behaves similarly to that observed in the Ref. <cit.>.  The △ L increases with temperature and becomes conspicuously flat, and as a function of η/η_c, it displays a declining trend for (η/η_c >3.5).







Next, we select three sets of data from the critical points in Fig. <ref> for comparison: (η, T_c)=(0, 0.1345), (0.6, 0.123), (1.2, 0.0985). From Fig. <ref> and Fig. <ref>, it can be seen that although they both represent the function of the maximum effective coupling constant or screening distance with respect to T/T_c, there are slight differences under different rapidities. Specifically, the larger the rapidity, the smaller the maximum effective coupling constant and screening distance. This suggests that the larger the rapidity, the stronger the dependence of the maximum effective coupling constant and screening distance on temperature.









 §.§ Discussion about doubly heavy baryon

As before, we first present the temperature dependence of the effective running coupling for QQq as shown in the left of Fig. <ref>, where η=0, T_c=0.1245 GeV. The rapidity dependence of the effective running coupling is shown in the right of Fig. <ref>, where T=0.1245 GeV. Their temperature and rapidity values reach up to the maximum values for which QQq can exist, T ∈ [0, 0.1435] (η=0), η∈ [0, 0.751] (T=0.1245 GeV). As shown in the two figures, the temperature and rapidity dependence of the effective running coupling for QQq is essentially consistent with that of QQ. The higher the temperature and rapidity, the lower the effective running coupling curve. Furthermore, the distance at which QQq reaches maximum coupling is almost identical to the screening distance.



We focus on the maximum effective coupling constant of QQq, with the maximum effective coupling constant of QQq in relation to QQ displayed as the function of T/T_c and η/η_c in Fig. <ref>. And the screening distances for QQq and QQ is shown in Fig. <ref>. The graphs indicate that the maximum coupling constant of QQq quickly reduces with an increase in temperature or rapidity, and it consistently remains much lower than that of effective QQ, showing a greater sensitivity to both temperature and rapidity. However, QQq is subject to a limit from the maximum (T, η), and its graph ends as the maximum effective coupling constant begins to level off. Similarly, the screening distance for QQq is invariably shorter than that of QQ, but it diminishes more quickly. Additionally, since QQq is always within the descending region, its trend of reduction with rapidity is consistently less intense than with temperature.








§ SUMMARY


In this work, we initially determined the parameters of our model by fitting the lattice potentials for QQ and QQq. Subsequently, we calculated their the strength of the interaction which is defined as effective running coupling from the lattice. We discovered that, as a function of the distance between quarks, the effective coupling constant of QQ is always noticeably larger than that of QQq, and the rate of increase with distance is more pronounced for QQ than for QQq. Furthermore, the effective coupling constant for QQ is extremely low and flat within the range of L < 0.2 fm, whereas for QQq, this occurs approximately within L < 0.4 fm. We interpret this as a manifestation of asymptotic freedom in the behavior of the effective coupling constants. Additionally, their effective coupling constants, as functions of distance, decrease with an increase in temperature or rapidity, which is particularly evident in the effective coupling constants at long distances, especially in the maximum effective coupling constant α̃. The effective coupling constants for small scales remain almost unchanged with variations at finite temperature or rapidity.

Based on the holographic model, we have conducted a detailed analysis of the effective running coupling constant at finite temperature and rapidity in this paper. Through the above discussion, we can infer that the QQq system is less stable than the QQ̅ system in the presence of temperature and rapidity. Furthermore, the study of the effective running coupling for the QQq system at finite magnetic fields and under rotation can also be conducted in the future work.


§ ACKNOWLEDGMENTS

This work is supported by the NSFC under Grant Nos. 12405154 and Open Fund for Key Laboratories of the Ministry of Education under Grants Nos. QLPL2024P01.



§ REFERENCES


