Impact of medium anisotropy on quarkonium dissociation and regeneration

Abstract

Quarkonium production in ultra-relativistic collisions plays a crucial role in probing the existence of hot QCD matter. This study explores quarkonia states dissociation and regeneration in the hot QCD medium while considering momentum anisotropy. The net quarkonia decay width ($\Gamma _{D}$) arises from two essential processes: collisional damping and gluonic dissociation. The quarkonia regeneration includes the transition from octet to singlet states within the anisotropic medium. Our study utilizes a medium-modified potential that incorporates anisotropy via particle distribution functions. This modified potential gives rise to collisional damping for quarkonia due to the surrounding medium, as well as the transition of quarkonia from singlet to octet states due to interactions with gluons. Furthermore, we employ the detailed balance approach to investigate the regeneration of quarkonia within this medium. Our comprehensive analysis spans various temperature settings, transverse momentum values, and anisotropic strengths. Notably, we find that, in addition to medium temperatures and heavy quark transverse momentum, anisotropy significantly influences the dissociation and regeneration of various quarkonia states.

1 Introduction

The relativistic heavy-ion collision (HIC) experiments at the relativistic heavy ion collider (RHIC) and the large hadron collider (LHC) offered an opportunity to explore the properties of hot dense QCD matter known as the quark gluon plasma (QGP) [1,2,3,4,5,6,7]. The transient nature of this phase, resulting from the confinement properties of quantum chromodynamics (QCD), necessitates the study of discernible signatures and probes. One of the most prominent and analyzed signatures is the suppression of heavy quarkonia ($q-{\bar{q}}$) due to their interactions with the QGP medium [8, 9]. The pioneering work on the dissolution of various quarkonia states, attributed to color screening within the finite-temperature QGP medium, was first undertaken by Matsui and Satz [8]. Since then, a substantial body of literature has contributed critical refinements to this research [10,11,12,13,14,15,16,17,18,19,20]. Interestingly, an expected suppression of $J/\psi$ was observed at the super proton synchrotron (SPS) at CERN [21]. However, this suppression did not exhibit a corresponding increase with higher beam energies at RHIC [22]. To address this puzzle, the concept of recombination of $c-{\bar{c}}$ pairs into $J/\psi$ within the QGP medium was introduced. Consequently, recombination processes play an important role in understanding the observed number of $J/\psi$ particles in experiments [23, 24].

In HIC, the quarkonia, characterized by their considerable mass, predominantly emerge during the initial stages directly following the collisions. The production and evolution of quarkonia can be categorized into two distinct processes: (i) perturbative production: the generation of quark and anti-quark ($q {{{\bar{q}}}}$) pair through nucleon-nucleon collisions constitutes the perturbative aspect of quarkonium production [25]. (ii) Non-perturbative evolution: the subsequent formation and development of bound-state from these $q {{{\bar{q}}}}$ pairs are governed by non-perturbative QCD phenomena. This dual nature of quarkonia allows us to explore both perturbative and non-perturbative QCD effects. Theoretical frameworks grounded in QCD, such as non-relativistic QCD (NRQCD), potential NRQCD/ (pNRQCD) [26,27,28,29] and fragmentation approaches [30, 31], are widely employed to investigate quarkonium production.

It is reported that the stability of quarkonium states can be compromised by various QGP effects such as color screening [8, 20], collisional damping, and gluonic dissociation [32,33,34,35], etc. The static potential between $q-{\bar{q}}$ pair placed in a QCD medium consists of two parts; the first part represents the real part of the potential, and the other part of the potential is the imaginary part. The collisional damping arises due to the imaginary part of the potential between $q-{\bar{q}}$ pairs in the time limit, $t\rightarrow \infty$. It represents the thermal decay width induced by collisional (Landau) damping of the low-frequency gauge fields that mediate interactions between heavy $q-{\bar{q}}$ pair. [19]. The gluonic dissociation occurs when quarkonia absorb soft gluons (E1 gluons), leading to a transition from a color singlet state to a color octet state-an unbound quark-antiquark pair [16, 36, 37]. The mechanism governing quarkonia production in the color singlet state has been explored in Refs. [38,39,40]. Furthermore, the regeneration of quarkonia within the QGP medium is also possible and offers two primary avenues: (i) Uncorrelated $q-{\bar{q}}$ recombination: Free flowing q and ${\bar{q}}$ produced within the medium, through various different processes, have the potential to merge into open mesons or hidden quarkonia through the recombination of q and ${\bar{q}}$. The production of quarkonia through the recombination of q and ${\bar{q}}$ is considered as regeneration of quarkonia through uncorrelated recombination of quark and anti-quark [41,42,43,44,45,46,47,48,49,50]. It is particularly significant for charmonium states like $J/\psi , \chi _{c}, \psi ^{'}$, etc., at LHC energies due to the relatively higher production of c and ${\bar{c}}$ in HICs compared to b and ${\bar{b}}$ quarks. One may use the coalescence mechanism to calculate the formation of quarkonia from such quarks and anti-quarks. (ii) Correlated $q-{\bar{q}}$ recombination: this process reverses the gluonic dissociation, which produces a color octet state. The color octet state represents an unbound configuration of $q-{\bar{q}}$ pairs, and an octet potential is introduced to account for such configurations [51], establishing the spatial correlation between $q-{\bar{q}}$ pairs. The transition of these pairs from a color octet to a color singlet state is regarded as quarkonia regeneration, attributed to the correlated nature of the $q-{\bar{q}}$ pair. In this particular situation, it is worth mentioning that besides the charmonia, there is also the possibility for the regeneration of bottomonia via the de-excitation of correlated $b-{\bar{b}}$ pairs [35, 52]. It is to be noted that the regeneration of quarkonia may happen at any point in time, given that the medium temperature should be less than the dissociation temperature of quarkonia.

Following the UMQS model [20, 35, 52,53,54], centered on the formulation of gluonic dissociation and collisional damping within an isotropic medium, the current study extends its scope to accommodate momentum anisotropy. Several researchers have delved into the dissociation of quarkonia, exploring diverse facets, including momentum anisotropy. However, the prospect of recombination in the context of an anisotropic QGP medium remains largely unexplored. This aspect is important because a significant increase in the production of $c{\bar{c}}$ and higher $R_{AA}$ for $J/\psi$ at low momentum in experiments has been noticed at LHC. These findings strongly suggest that quarkonium, like $J/\psi$, might be created through recombination either at the boundary between phases or during the QGP phase [55]. Although the decay width has been investigated previously by some of us [56], our study represents the inaugural attempt to devise a methodology for investigating this recombination process while considering momentum anisotropy. This anisotropy remains prevalent throughout all stages, particularly in the context of non-central collisions. Consequently, our objective is to evaluate the decay width and regeneration for $J/\psi$ and $\Upsilon$(1 S) within the QGP medium, considering the presence of momentum anisotropy. The exploration of momentum anisotropy has been a focal point in various contexts, as evidenced by studies such as [57,58,59,60,61,62,63,64,65]. We adopt an approach inspired by Refs. [66,67,68], where anisotropy is introduced at the level of the distribution function by stretching and squeezing it along one direction. Subsequently, we derive the medium-modified Cornell potential, considering anisotropic dielectric permittivity. This modification yields the potential as a complex quantity consisting of finite real and imaginary parts [19, 69,70,71,72,73,74]. Furthermore, along with the anisotropic effects, we have also observed the influence of relativistic Doppler shift on both, i.e., the particle’s net decay width ($\Gamma _{\textrm{D}}$) and recombination reactivity ($\Gamma _{\textrm{F}}$) which is responsible for quarkonia regeneration. Transverse momentum ($p_{\textrm{T}}$) and temperature (T) dependent $\Gamma _{\textrm{D}}$, $\Gamma _{\textrm{F}}$ are obtained for $J/\psi$, $\Upsilon$(1 S) for anisotropic strength, $\zeta = 0.2, 0.4$ and T = 200, 300, 400 MeV. This study provides a comprehensive method to estimate the net quarkonia suppression. It can be extended to obtain the survival probability of quarkonia in ultra-relativistic collisions at the RHIC and LHC energies. However, it is beyond the scope of the current analysis; we leave it for future investigations.

The manuscript is structured as follows: Sect. 2 provides the formalism for the dissociation and correlated recombination of various quarkonia states in the presence of momentum anisotropy. Section 3 is dedicated to presenting our results and engaging in discussions. Finally, in Sect. 4, we summarize our findings and contemplate future aspects of the current work. Throughout the text, we employ natural units with $c=k_B=\hbar =1$. Three vectors are in bold typeface, while four are in regular font. The center dot denotes the four-vector scalar product, with the metric tensor specified as $g_{\mu \nu }={\text{ diag }}(1,-1,-1,-1)$.

2 Formalism

The interaction between a ($q-{\bar{q}}$) pair is characterized by the vacuum Cornell potential, which includes both a Coulombic and a string component [75, 76]. This potential is expressed as:

$$\begin{aligned} \text {V(r)} = -\frac{\alpha _s}{r}+\sigma r. \end{aligned}$$

(1)

Here, r represents the radius of the quarkonia state under investigation, $\alpha _s$ denotes the strong coupling constant, and $\sigma$ represents the string tension. To account for medium effects, we modify this potential using the dielectric permittivity, $\epsilon (k)$, which considers the presence of momentum anisotropy. In Fourier space, this modification is formulated as:

$$\begin{aligned} \grave{V}(k)=\frac{{\bar{\text{ V }}}(k)}{\epsilon (k)}, \end{aligned}$$

(2)

where ${\bar{\text {V}}}(k)$ is the Fourier transform of $\text {V(r)}$ and is obtained as:

$$\begin{aligned} {\bar{\text {V}}}(k)=-\sqrt{\frac{2}{\pi }}\bigg (\frac{\alpha _s}{k^2}+2\frac{\sigma }{k^4}\bigg ). \end{aligned}$$

(3)

Here, the transition from the hadronic phase to the QGP is assumed to be a crossover [77], ensuring that the string tension does not abruptly disappear at or near the critical temperature, $T_c$. As mentioned earlier, some of us have derived the medium-modified potential in the presence of anisotropy in Ref. [56]; we briefly provide the necessary steps for completeness in the upcoming sections.

2.1 Dielectric permittivity in the anisotropic QGP medium

The anisotropy in this formalism is introduced through the medium’s particle distribution functions, represented as $f_{g}(p)$ and $f_{q/{{{\bar{q}}}}}(p)$:

$$\begin{aligned} f_{g}(p) = \frac{1}{e^{\beta E_g} - 1}, ~~~ f_{q/{{{\bar{q}}}}}(p) = \frac{1}{e^{\beta E_p} + 1}. \end{aligned}$$

(4)

Here, $\beta = 1/T$ (T is the temperature of the system in the units of energy). For the gluons, $E_{g}=|\textbf{p}|$ and $E_{q} = \sqrt{|\textbf{p}|^2+m_q^2}$ for the quarks/antiquarks and $m_q$ denotes their mass. $\textbf{p}$ represents the medium quarks and gluons momentum (not to mix up with heavy quark momentum later). We adopt the method utilized in these Refs. [66,67,68], where anisotropic distribution functions are derived from isotropic ones through rescaling in one direction in momentum space:

$$\begin{aligned} f({\textbf{p}})\rightarrow f_{\zeta }({\varvec{p}}) = C_{\zeta }~f(\sqrt{\textbf{p}^{2} + \zeta (\textbf{p}\cdot {\varvec{\hat{n}}})^{2}}). \end{aligned}$$

(5)

Here, ${\varvec{{\hat{n}}}}$ is a unit vector (${\varvec{\hat{n}}}^{2} = 1$) representing the direction of anisotropy, and $\zeta$ quantifies the anisotropic strength in the medium. It describes the degree of squeezing ($\zeta > 0$, oblate form) or stretching ($-1<\zeta <0$, prolate form) along the ${\varvec{\hat{n}}}$ direction. Normalizing the Debye mass $m_D$, defined as [67]:

$$\begin{aligned} m^{2}_D= & 4\pi \alpha _s \Bigg (-2N_c \int \frac{d^3 p}{(2\pi )^3} \partial _p f_g(p)\nonumber \\ & - N_f \int \frac{d^3 p}{(2\pi )^3} \partial _p \left( f_q(p)+f_{\bar{q}}(p)\right) \Bigg ). \end{aligned}$$

(6)

In both isotropic and anisotropic media, yields the normalization constant $C_{\zeta }$:

$$\begin{aligned} C_{\zeta }= {\left\{ \begin{array}{ll} \frac{\sqrt{|\zeta |}}{\tanh ^{-1}\sqrt{|\zeta |}}& \text {if }~~ -1\le \zeta <0\\ \frac{\sqrt{\zeta }}{\tan ^{-1}\sqrt{\zeta }} & \text {if }~~ \zeta \ge 0. \end{array}\right. } \end{aligned}$$

(7)

In the limit of small $\zeta$, this expression approximates to:

$$\begin{aligned} C_{\zeta }= {\left\{ \begin{array}{ll} 1-\frac{\zeta }{3} +O\left( \zeta ^{\frac{3}{2}}\right) & \text {if }~~ -1\le \zeta <0\\ 1+\frac{\zeta }{3} +O\left( \zeta ^{\frac{3}{2}}\right) & \text {if }~~ \zeta \ge 0. \end{array}\right. } \end{aligned}$$

(8)

The inclusion of $C_\zeta$ serves the purpose of averting the sudden loss of anisotropy as the momentum of the medium particles aligns perpendicularly to the anisotropic direction, particularly as $\theta _n$ approaches $\pi /2$. This scenario compromises the physical integrity of the situation. On the contrary, maintaining anisotropy is assured through $C_{\zeta }$ as long as $\zeta \ne 0$. In the context of the current situation, where $\zeta$ is small, the value of $C_\zeta$ approximates 1.1. This adjustment subtly enhances the fidelity of the physical representation in the model. It was initially set to unity in Ref. [66]. However, in a subsequent study by Romatschke and Strickland [78], it was normalized based on the anisotropic number density to the isotropic one, yielding $C_\zeta = \sqrt{1+\zeta }$ an alternative expression for the same.

Before calculating the dielectric permittivity for the anisotropic QGP medium, it is essential to know that perturbative theory at $T>0$ faces infrared singularities and gauge-dependent results due to the incomplete nature of perturbative expansion at these temperatures. However, HTL resummation [79], semi-classical transport theory, or many-particle kinetic theory yield consistent results up to one-loop order. These methods lead to the same expression for the gluon self-energy, $\Pi ^{\mu \nu }$, which in turn affects the medium’s dielectric permittivity [80]. In the static limit, the dielectric permittivity, $\epsilon ^{-1}(\textbf{k})$ can be derived from the temporal component of the gluon propagator considering the Coulomb gauge within the linear response theory as [56, 81]:

$$\begin{aligned} \epsilon ^{-1}(\textbf{k}) = -\lim _{\omega \rightarrow 0}k^2\Delta ^{00}(\omega ,\textbf{k}). \end{aligned}$$

(9)

The temporal components of the real and imaginary parts of the gluon propagator are given as follows:

$$\begin{aligned} \Re [\Delta ^{00}({\omega = 0,\textbf{k}})]= & \frac{-1}{k^2+m_{D}^2}-\zeta \Bigg (\frac{1}{3(k^2+m_{D}^2)}\nonumber \\ & -\frac{m_{D}^2(3\cos {2\theta _n}-1)}{6 (k^2+m_{D}^2)^2}\Bigg ), \end{aligned}$$

(10)

$$\begin{aligned} \Im [\Delta ^{00}({\omega = 0,\textbf{k}})]= & \pi ~ T~ m_{D}^2\Bigg (\frac{-1}{k(k^2+m_{D}^2)^2} \nonumber \\ & + \zeta \Bigg (\frac{-1}{3k(k^2+m_{D}^2)^2}+\frac{3\sin ^{2}{\theta _n}}{4k(k^2+m_{D}^2)^2} \nonumber \\ & - \frac{2m_{D}^2\big (3\sin ^{2}({\theta _n})-1\big )}{3k(k^2+m_{D}^2)^3}\Bigg )\Bigg ). \end{aligned}$$

(11)

These expressions contain direction and strength to account for the anisotropy in the medium. Now, using Eqs. (10) and (11) in Eq. (9), respectively the real and imaginary parts of $\epsilon (\textbf{k})$ can be obtained as,

$$\begin{aligned} \Re [\epsilon ^{-1}(\textbf{k})]= & \frac{k^2}{k^2+m_{D}^2}+k^2\zeta \Bigg (\frac{1}{3(k^2+m_{D}^2)}\nonumber \\ & - \frac{m_{D}^2(3\cos {2\theta _n}-1)}{6(k^2+m_{D}^2)^2}\Bigg ), \end{aligned}$$

(12)

and

$$\begin{aligned} \Im [\epsilon ^{-1}(\textbf{k})]= & \pi T m_{D}^2\Bigg (\frac{k^2}{k(k^2+m_{D}^2)^2}-\zeta k^2\Bigg (\frac{-1}{3k(k^2+m_{D}^2)^2}\nonumber \\ & + \frac{3\sin ^{2}{\theta _n}}{4k(k^2+m_{D}^2)^2}-\frac{2m_{D}^2 \big (3\sin ^{2}({\theta _n})-1\big )}{3k(k^2+m_{D}^2)^3}\Bigg )\Bigg ).\nonumber \\ \end{aligned}$$

(13)

It is to note that at $\zeta \rightarrow 0$, and $T\rightarrow 0$, the real part $\Re [\epsilon ^{-1}(\textbf{k})]$ goes to unity, and the imaginary part $\Im [\epsilon ^{-1}(\textbf{k})]$ vanishes, and we get back the vacuum Cornell potential.

2.2 Modified Cornell potential in the anisotropic medium

The medium-modified Cornell potential can be obtained using $\epsilon (k)$ in Eq. (2). The updated potential in the coordinate space can be found by doing an inverse Fourier transform and is given as,

$$\begin{aligned} V(r)= \int \frac{d^3\textbf{k}}{(2\pi )^{3/2}}(e^{i\textbf{k} \cdot \textbf{r}}-1)\grave{V}(k). \end{aligned}$$

(14)

Since the medium permittivity is a complex quantity, the updated potential also contains the real and imaginary parts. Employing Eq. (12) in Eq. (2), the real part of the potential is obtained as,

$$\begin{aligned} \Re [V]= & \alpha _s m_D \left( -\frac{e^{-\grave{r}}}{\grave{r}}-1\right) +\frac{\sigma }{ m_D} \Bigg (\frac{2 e^{-\grave{r}}}{\grave{r}}-\frac{2}{\grave{r}}+2\Bigg )\nonumber \\ & +\alpha _s m_D \zeta \Bigg [-\frac{3 \cos (2 \theta _r )}{2 \grave{r}^3} -\frac{1}{2 \grave{r}^3}+\frac{1}{6}+e^{-\grave{r}} \Bigg \{\frac{1}{2 \grave{r}^3}\nonumber \\ & +\frac{1}{2 \grave{r}^2}+\Bigg (\frac{3}{2 \grave{r}^3} +\frac{3}{2 \grave{r}^2}+\frac{3}{4 \grave{r}}+\frac{1}{4}\Bigg ) \cos (2 \theta _r )+\frac{1}{4 \grave{r}}\nonumber \\ & -\frac{1}{12}\Bigg \}\Bigg ]+\frac{\zeta ~ \sigma }{m_D} \Bigg [\left( \frac{6}{\grave{r}^3}-\frac{1}{2 \grave{r}}\right) \cos (2 \theta _r )+\frac{2}{\grave{r}^3}\nonumber \\ & -\frac{5}{6 \grave{r}}+\frac{1}{3}+e^{-\grave{r}} \Bigg \{ -\frac{2}{\grave{r}^3}-\frac{2}{\grave{r}^2}+\Bigg (-\frac{6}{\grave{r}^3}-\frac{6}{\grave{r}^2}\nonumber \\ & -\frac{5}{2 \grave{r}}-\frac{1}{2}\Bigg ) \cos (2 \theta _r )-\frac{1}{6 \grave{r}}+\frac{1}{6}\Bigg \}\Bigg ], \end{aligned}$$

(15)

where $\grave{r} = r m_D$. Assuming the limit, $\grave{r}\ll 1$, Eq. (15) becomes,

$$\begin{aligned} \Re [V]= & \frac{ \grave{r}~ \sigma }{m_D}\left( 1+\frac{\zeta }{3}\right) -\frac{\alpha _s~ m_D}{\grave{r}} \bigg (1+\frac{\grave{r}^2}{2}\nonumber \\ & +\zeta \left( \frac{1}{3}+\frac{\grave{r}^2}{16}\left( \frac{1}{3}+ \cos \left( 2 \theta _r\right) \right) \right) \bigg ), \end{aligned}$$

(16)

here, $\sigma = 0.192$ $\hbox {GeV}^{2}$. The coupling constant is considered as follows;

$$\begin{aligned} \alpha _{s}(\Lambda ) = \frac{6\pi }{(11N_c - 2N_f) \ln \left( \frac{\Lambda }{\Lambda _{MS}}\right) }, \end{aligned}$$

(17)

with $\Lambda _{MS}$ = 0.176 GeV, $N_f = 3$ and $N_c = 3$. Here, the renormalization scale $\Lambda$ under HTL limit is considered $2 \pi T$ [82]. Now, using Eq. (13), the imaginary part of the potential is found to be:

$$\begin{aligned} & \Im [V]=\frac{\alpha _s ~T ~m^2_D}{2~\pi }\int d^3\textbf{k}(e^{i\textbf{k} \cdot \textbf{r}}-1) \frac{1}{k}\Bigg [\frac{-1}{(k^2+m_D^2)^2}\nonumber \\ & +\zeta \Bigg [\frac{-1}{3(k^2+m_D^2)^2}+\frac{3\sin ^2\theta _n}{2(k^2+m_D^2)^2} -\frac{4 m_D^2(\sin ^2\theta _n-\frac{1}{3})}{(k^2+m_D^2)^3}\Bigg ]\Bigg ]\nonumber \\ & +\frac{\sigma T m^2_D}{\pi }\int d^3\textbf{k}(e^{i\textbf{k} \cdot \textbf{r}}-1) \frac{1}{k^3}\Bigg [\frac{-1}{(k^2+m_D^2)^2}\nonumber \\ & +\zeta \Bigg [\frac{-1}{3(k^2+m_D^2)^2}+\frac{3\sin ^2\theta _n}{2(k^2+m_D^2)^2} -\frac{4 m_D^2(\sin ^2\theta _n-\frac{1}{3})}{(k^2+m_D^2)^3}\Bigg ]\Bigg ].\nonumber \\ \end{aligned}$$

(18)

The analytical solutions of Eq. (18) can be obtained in the limit $\grave{r}\ll 1$ as,

$$\begin{aligned} \Im [V]= & \frac{\alpha _s ~ \grave{r}^2~ T}{3} \Bigg \{\frac{ \zeta }{60} (7-9 \cos 2 \theta _r)-1\Bigg \}\log \left( \frac{1}{\grave{r}}\right) \nonumber \\ & +\frac{\grave{r}^4 ~\sigma ~ T}{m_D^2}\Bigg \{\frac{\zeta }{35} \left( \frac{1}{9}-\frac{1}{4} \cos 2 \theta _r \right) \nonumber \\ & -\frac{1}{30}\Bigg \}\log \left( \frac{1}{\grave{r}}\right) . \end{aligned}$$

(19)

It is noteworthy that obtaining an analytical solution for the complete imaginary potential described in Eq. (18) is unattainable. Nevertheless, one can numerically solve it under certain reasonable assumptions.

2.3 Collisional damping

Collisional damping is an inherent characteristic of the complex potential, and as mentioned above, quarkonia potential becomes complex within the medium; the imaginary part of the potential gradually reduces the force fields that bind $q-{\bar{q}}$ together. Consequently, quarkonia gets dissociated within the QGP medium, and the decay width ($\Gamma _{d,nl}$) corresponding to collisional damping can be obtained by performing a first-order perturbation calculation. The calculation of $\Gamma _{d,nl}$ involves folding the imaginary part of the potential with the radial wavefunction [20]:

$$\begin{aligned} \Gamma _{d,nl}(\tau ,p_{T}) = \int g_{nl}(r)^{\dagger } \Im [V] g_{nl}(r) \, dr. \end{aligned}$$

(20)

Here, $g_{nl}(r)$ represents the singlet wavefunction corresponding to the specific quarkonia state being investigated, where n and l are the quantum numbers with their usual meanings. For simplicity, some of the authors have chosen the Hydrogen atom wavefunctions [56, 81, 83]. However, we have obtained these wavefunctions by solving the Schrödinger equation for various $q-{\bar{q}}$ states. As the potential is anisotropic, the particle’s decay width also contains the respective effect.

2.4 Gluonic dissociation

The gluon-induced excitation from color singlet to color octet state is defined as “gluonic dissociation.” The cross-section corresponding to the process is obtained as given by [53]:

$$\begin{aligned} & \sigma _{gd,nl}(E_g) = \frac{\pi ^2\alpha _s^u E_g}{N_c^2}\sqrt{\frac{{m_{Q}}}{E_g + E_{nl}}}\nonumber \\ & \quad \times \left( \frac{l|J_{nl}^{q,l-1}|^2 + (l+1)|J_{nl}^{q,l+1}|^2}{2l+1} \right) , \end{aligned}$$

(21)

where, $E_{nl}$ is energy eigenvalues corresponding to $g_{nl}(r)$, $\alpha _{s}^{u}$ is coupling constant, scaled as $\alpha _{s}^{u} = \alpha _{s}(\alpha _{s}m_{Q}^{2}/2)$ and $m_{Q}$ is the mass of the heavy quark. The $J_{nl}^{ql^{'}}$ is the probability density obtained by using the singlet $g^*_{nl}(r)$ and octet $h_{ql'}(r)$ wavefunctions as follows,

$$\begin{aligned} J_{nl}^{ql'} = \int _0^\infty r\; g^*_{nl}(r)\;h_{ql'}(r)dr. \end{aligned}$$

(22)

Here, $h_{ql'}(r)$ is the wavefunction of the color octet state of quarkonia. It is obtained by solving the Schrödinger equation with the octet potential $V_{8} = \alpha _{eff}/8r$. The effective coupling constant, $\alpha _{eff}$ is defined at soft scale $\alpha _{s}^{s} = \alpha _{s} (m_{Q} \alpha _{s}/2)$, given as $\alpha _{eff} = \frac{4}{3}\alpha _{s}^{s}$. The value of q here, is determined using conservation of energy, $q = \sqrt{m_{Q}(E_{g}+E_{nl})}$. Next, we obtained the mean of the gluonic dissociation cross-section to obtain $\Gamma _{gd,nl}$ of a quarkonia state moving with speed v. To do so, we thermal averaged over the modified Bose–Einstein distribution function for gluons in the rest frame of quarkonia [35]. Thus, gluonic dissociation is found as,

$$\begin{aligned} & \Gamma _{gd,nl}(\tau ,p_{T},b) \nonumber \\ & \quad = \frac{g_d}{4\pi ^2} \int _{0}^{\infty } \int _{0}^{\pi } \frac{dp_g\,d\theta \,\sin \theta \,p_g^2 \sigma _{gd,nl}(E_g)}{e^{ \{\frac{\gamma E_{g}}{T_{eff}}(1 + v\cos \theta )\}} - 1}, \end{aligned}$$

(23)

where $\gamma$ is a Lorentz factor and $\theta$ is the angle between v and incoming gluon with energy $E_{g}$. $p_T$ is the transverse momentum of the quarkonia and $g_d = 16$ is the number of gluonic degrees of freedom.

2.5 Doppler Shift

As a massive particle, the quarkonia may not experience the same temperature as the surrounding medium. The relativistic Doppler shift induced by the relative velocity ($v_{r}$) between the medium and heavy meson causes an angle-dependent effective temperature ($T_{eff}$), which is expressed as detailed in Refs. [35, 84]. As a consequence, the Doppler effect, a blue-shifted $T_{eff}$, is expected in the forward direction and a red-shift in the backward direction. In the blue-shifted region $T_{eff} > T$, however, the blues-shifted is confined to very smaller angles [84]. While the red-shifted region dominates over the blue-shifted at all relative velocities $v_{r} > 0$; i.e., $T_{eff} < T$ [85]. As the red-shift grows with increasing velocity, effective temperature decreases further for particles with high $p_{T}$ values. The $T_{eff}$ is defined as;

$$\begin{aligned} T_{eff}(\theta ,|v_{r}|) = \frac{T\;\sqrt{1 - |v_{r}|^{2}}}{1 - |v_{r}|\;\cos \theta }, \end{aligned}$$

(24)

here $\theta$ is the angle between $v_{r}$ and incoming light partons. To calculate the relative velocity, $v_{r}$, we have taken medium velocity, $v_{m} = 0.7c$, and quarkonia velocity, $v_{nl} = p_{T}/E_{T}$. Here $p_{T}$ is transverse momentum of quarkonia and $E_{T} = \sqrt{p_{T}^{2} + M_{nl}^{2}}$ is its transverse energy, $M_{nl}$ is the mass of corresponding quarkonium state. We have averaged Eq. (24) over the solid angle and obtained the average effective temperature given by:

$$\begin{aligned} T_{eff}(\tau ,b,p_{T}) = T(\tau ,b)\;\frac{\sqrt{1 - |v_{r}|^{2}}}{2\;|v_{r}|}\;\ln \Bigg [\;\frac{1 + |v_{r}|}{1 - |v_{r}|}\Bigg ]. \end{aligned}$$

(25)

After incorporating the $T_{eff}$ correction, we have obtained the total dissociation decay width by taking the sum over the Eqs. (20) and (23) correspond to the collisional damping and the gluonic dissociation, respectively.

$$\begin{aligned} \Gamma _{D,nl} = \Gamma _{d,nl} + \Gamma _{gd,nl}. \end{aligned}$$

(26)

Next, we shall discuss the regeneration of the bound state in the presence of the anisotropic QGP medium.

2.6 Regeneration of quarkonia in the anisotropic QGP medium

The regeneration via correlated $q-{\bar{q}}$ pairs is considered through the de-excitation of the octet state to a singlet state via emitting a gluon. The recombination cross-section $\sigma _{f,nl}$ is obtained from gluonic dissociation cross-section $\sigma _{gd,nl}$ using the detailed balance approach [20],

$$\begin{aligned} \sigma _{f,nl} = \frac{48}{36}\sigma _{gd,nl} \frac{(s-M_{nl}^{2})^{2}}{s(s-4\;m_{Q}^{2})}. \end{aligned}$$

(27)

Here, s is the Mandelstam variable, $s = ({\varvec{p}_{q} + p_{{\bar{q}}}})^{2}$, where $\textbf{p}_{q}$ and $\textbf{p}_{{\bar{q}}}$ are four momentum of q and ${\bar{q}}$, respectively. The recombination factor, $\Gamma _{F,nl}$ is obtained by taking the thermal average of the product of recombination cross-section and relative velocity $v_{rel}$ between q and ${\bar{q}}$ as [50]:

$$\begin{aligned} \Gamma _{F,nl}=<\sigma _{f,nl}\;v_{rel}>_{p_{q}}, \end{aligned}$$

(28)

rewriting Eq. (28) as,

$$\begin{aligned} \Gamma _{F,nl} = \frac{\int _{p_{q,min}}^{p_{q,max}}\int _{p_{{\bar{q}},min}}^{p_{{\bar{q}},max}} dp_{q}\; dp_{{\bar{q}}}\; p_{q}^{2}\;p_{{\bar{q}}}^{2}\; f_{q}\;f_{{\bar{q}}}\;\sigma _{f,nl}\;v_{rel} }{\int _{p_{q,min}}^{p_{q,max}}\int _{p_{{\bar{q}},min}}^{p_{{\bar{q}},max}} dp_{q}\; dp_{{\bar{q}}}\; p_{q}^{2}\;p_{{\bar{q}}}^{2}\; f_{q}\;f_{{\bar{q}}}}, \nonumber \\ \end{aligned}$$

(29)

where, $p_{q}$ and $p_{{\bar{q}}}$ are respectively, 3-momentum of heavy quark and heavy anti-quark under investigation. The relative velocity of $q-{\bar{q}}$ pair in the medium is given as,

$$\begin{aligned} v_{rel} = \sqrt{\frac{( {\varvec{p}_{q}^{\mu }\; p_{{\bar{q}} \mu }})^{2}-m_{Q}^{4}}{p_{q}^{2}\;p_{{\bar{q}}}^{2} + m_{Q}^{2}(p_{q}^{2} + p_{{\bar{q}}}^{2} + m_{Q}^{2})}}. \end{aligned}$$

(30)

The $f_{q,{\bar{q}}}$ is the modified Fermi-Dirac distribution function of heavy quark and heavy anti-quark given as,

$$\begin{aligned} f_{q,{\bar{q}}} = \lambda _{q,{\bar{q}}}/(e^{E_{q,{\bar{q}}}/T_{eff}} + 1). \end{aligned}$$

(31)

Here $E_{q,{\bar{q}}} = \sqrt{p_{q,{\bar{q}}}^{2} + m_{Q}^{2}}$ is the energy of heavy quark and heavy anti-quark, in medium and $\lambda _{q,{\bar{q}}}$ is their respective fugacity terms. It should be emphasized that while these fugacity terms do not affect the integration outlined in Eq. (29). They effectively cancel out from both the numerator and denominator, rendering them unnecessary for the present analysis. Nevertheless, they are included here for the sake of thoroughness and completeness. Next, there is an open discussion on accounting for the Bose Stimulation factor for outgoing gluon in the quarkonia regeneration through the octet-to-singlet transition or gluonic de-excitation. It is argued that the Bose stimulation factor has a minimal impact on the regeneration rate as it merely affects it by less than 10% [86]. Since the present work investigates quarkonia regeneration within an anisotropic medium, the Bose stimulation factor is omitted in Eq. (29) to reduce the complexity in the formulation. Next, we shall discuss the results obtained in this analysis.

3 Results and discussion

In this study, we investigate how the anisotropic properties of the medium impact the decay width ($\Gamma _{D}$) and recombination reactivity ($\Gamma _{F}$) of $J/\psi$ and $\Upsilon$(1 S) with masses 3.1 and 9.46 GeV, respectively. We determine $\Gamma _{D}$ and $\Gamma _{F}$ using the expressions presented in Eqs. (26) and  (29), respectively. We thoroughly analyze how these quantities depend on the degree of medium anisotropy, $\zeta$. Additionally, we examine the relativistic Doppler effect, which arises from particle motion and leads to changes in the effective temperature of the particles in relation to the medium. We quantify this effect using the parameter $T_{eff}$ included through the color map in all the plots of $\Gamma _D$. Our results demonstrate the dependence of $\Gamma _{D}$ on both $T_{eff}$ and transverse momentum ($p_{T}$). We omit the detailed discussion of the $T_{eff}$ dependence for $\Gamma _{F}$, as it is difficult to show effective temperature for the quark-antiquark pair as depending upon their velocity $q-{\bar{q}}$ may have different values of $T_{eff}$. However, the effective temperature for $q-{\bar{q}}$ is considered while calculating the $\Gamma _{F}$. Our analysis considers three distinct temperature values, T = 200, 300, and 400 MeV, and two values for anisotropic parameter $\zeta$ = 0.2 and 0.4. These specific temperature and anisotropy choices are made to elucidate the qualitative impact of medium anisotropy on the dissociation of quarkonia within the QGP medium.

Fig. 1

The net decay width, $\Gamma _{D}$ for $J/\psi$ as a function of transverse momentum, $p_{T}$ is demonstrated for the anisotropic parameter, $\zeta$ = 0, 0.2, 0.4 and medium temperature, T = 200, 300, 400 MeV. The $\Gamma _D$ with effective temperature, $T_{eff}$ is shown through a color map corresponding to the individual values of T

Figure 1 shows the change in $\Gamma _D$ for $J/\psi$ with respect to $p_T$ and $T_{eff}$, at different values of T and $\zeta$. The $\Gamma _D$ increases with increasing $\zeta$ from 0 to 0.4 and further grows with respect to the change in the medium temperature from 200 MeV to 400 MeV. At low-$p_{T}$ and large $T_{eff}$, the impact of anisotropy on $\Gamma _D$ is distinguishable, while with increasing $p_{T}$, this difference diminishes ($T_{eff}$ also decreases). Initially, at $p_{T}<3$ GeV, there is a slight increase in the $\Gamma _D$ for all $\zeta$ values. This rising trend is a consequence of the $T_{eff}$, as $T_{eff}$ increases up to $p_{T}<3$ GeV, and after that, it decreases with increasing $p_{T}$, and as a result, $\Gamma _D$ also decreases for $p_{T}>3$ GeV. It suggests that the particle dissociation rate will be high if the particle temperature is large and comparable to medium temperature. The quarkonia with high-$p_{T}$ quickly traverses through the medium; therefore, high-$p_{T}$ $J/\psi$ feels less heat, and survival of such particle becomes more probable.

Fig. 2

The net decay width, $\Gamma _{D}$ for $\Upsilon$(1 S) as a function of transverse momentum, $p_{T}$ and effective temperature, $T_{eff}$ is demonstrated for the anisotropic parameter, $\zeta$ = 0, 0.2, 0.4 and medium temperature, T = 200, 300, 400 MeV. The $\Gamma _D$ with effective temperature, $T_{eff}$ is shown through a color map corresponding to the individual values of T

The decay width of $\Upsilon$(1 S) is shown in Fig. 2; it follows the same reasoning as Fig. 1 except the difference in $\Gamma _D$ corresponding to $\zeta$ values remains consistent throughout $p_{T} = 30$ GeV at T = 200 and 300 MeV. While at T = 400 MeV, $\Gamma _D$ is almost constant up to $T_{eff}\approx$360 MeV, then a sudden change is observed, further decreasing the $\Gamma _D$. This anomaly in decay width for $\Upsilon$(1 S) at T = 400 MeV comes because the change in the $T_{eff}$ is almost negligible up to $p_{T}\sim 20$ GeV, and as a consequence, change in $\Gamma _D$ becomes stagnant at $p_{T}<20$ GeV. It is to be noted that the difference in the analyses of $J/\psi$ and $\Upsilon$(1 S) relies on their masses. As a heavy particle, the $\Upsilon$(1 S) dissociates at higher temperatures than $J/\psi$. Various articles have studied collisional damping, including the anisotropic medium, from different perspectives. However, in most of these analyses, the aspect of gluonic dissociation is missing. Figures 1 and 2 contain both the collisional damping and gluonic dissociation considering the anisotropy in the QGP medium.

In Fig. 3, the variation of recombination reactivity, $\Gamma _F$ of $J/\Psi$, is shown with respect to $p_T$ at different anisotropic strength and temperature values. It is found that $\Gamma _F$ has a non-trivial dependence on the three i.e., $p_T$, T and $\zeta$. Regeneration of $J/\psi$ is almost independent of medium anisotropy at T = 200 MeV. Small dependence on anisotropy can be observed in the intermediate $p_{T}$ ($5<p_{T}<17$ GeV) at T = 300 MeV. It depicts that an anisotropic medium is less favorable to $J/\psi$ regeneration than an isotropic medium. As anisotropy induces or supports $J/\psi$ dissociation, it is obvious that it would restrict the particle regeneration, though its impact is negligible. Regeneration increases with increasing $p_{T}$, except at mid-$p_{T}$ where particle interactions and large medium temperature constrain the $J/\psi$ to form. While at $p_{T}>17$ GeV $c-{\bar{c}}$ pair feels a lower temperature than the medium, and as they move at a high speed, they easily traverse through the medium. Consequently, we get a linear increase in regeneration reactivity with $p_{T}$. As we move to a very high temperature, T= 400 MeV, the probability of recombination suppresses to a very narrow range of $p_T$. Here, $\Gamma _F$ showed a non-monotonic behavior, as medium temperature T = 400 MeV is larger than the dissociation temperature of $J/\psi$ ($T_{D}\sim$350 MeV); therefore a large number of $c-{\bar{c}}$ pair are expected, and as the interaction probability is large at mid-$p_{T}$, most of the $c-{\bar{c}}$ recombine to form $J/\psi$. Hence, a maximum recombination probability is attained at $p_{T}\sim 8-10$ GeV for the isotropic case. At $p_{T}>15$ GeV for T = 400 MeV, the order of recombination becomes almost equivalent to the T = 200 and 300 MeV, following a similar explanation as the previous one corresponding to high-$p_{T}$. Moreover, regeneration due to un-correlated $q-{\bar{q}}$ pair dominates at low-$p_{T}$ and decreases rapidly at high-$p_{T}$. Thus, charmonia regeneration due to correlated $c-{\bar{c}}$ pair is contrary to the regeneration due to un-correlated $q-{\bar{q}}$ pair.

Fig. 3

The recombination reactivity, $\Gamma _{F}$ for $J/\psi$ is demonstrated in the semilog scale as a function of transverse momentum, $p_T$ with medium anisotropy parameter, $\zeta$ = 0, 0.2, 0.4 for T = 200, 300, 400 MeV

Fig. 4

The recombination reactivity, $\Gamma _{F}$ for $\Upsilon$(1 S) is demonstrated in the semilog scale as a function of transverse momentum, $p_T$ with medium anisotropy parameter, $\zeta$ = 0, 0.2, 0.4 for T = 200, 300, 400 MeV

The regeneration of $\Upsilon$(1 S) is depicted in Fig. 4, there is a slight increase in $\Gamma _F$ magnitude at T = 300 MeV than 200 MeV temperature. However, it follows the same trend at both temperature values. The effect of anisotropy on $\Upsilon$(1 S) regeneration is visible, though it decreases with increasing temperature. As shown in the figure, at T = 200 and 300 MeV, $\Upsilon$(1 S) regeneration is dominated for the isotropic medium, which is equivalent to $J/\psi$ as shown in Fig. 3. Initially, $\Gamma _F$ increases because of the abundance of $b-{\bar{b}}$ octet states at mid-$p_{T}$ range but at high-$p_{T}$, this abundance decreases for low temperatures, and so does the $\Gamma _F$. The physics of $\Upsilon$(1 S) regeneration probability at T = 400 MeV is as follows: since gluonic dissociation increases with the increase in temperature, it produces a significant number of $b-{\bar{b}}$ octet states for temperature more than 300 MeV. Such that the de-excitation of $b-{\bar{b}}$ octet states to $\Upsilon (1S)$ enhances the regeneration of $\Upsilon (1S)$ at T = 400 MeV, which can be seen in Fig. 4, where the value of $\Gamma _{F}$ is higher at T = 400 MeV as compared with T = 300 MeV. It also shows that high temperature produces enough $b-{\bar{b}}$ pair even at high-$p_{T}$ and therefore, we get an increasing pattern in $\Upsilon$(1 S) regeneration with $p_{T}$. As gluonic excitation decreases at high-$p_{T}$, so the de-excitation of $b-{\bar{b}}$ octet state into $\Upsilon (1S)$ becomes more probable with increasing $p_{T}$ at T = 400. For isotropic case ($\zeta = 0$), $\Gamma _{F}$ decreases between $p_{T}$ 1 to 10 GeV, because as it is shown in Fig. 2 the dissociation $\Upsilon$(1 S) is more favorable at this $p_{T}$ range. Meanwhile, $\Gamma _{F}$ for $\Upsilon$(1 S) always has a rising trend for the anisotropic case at T = 400 MeV. It seems that at high temperatures where anisotropy increases the decay width but also favors the $\Upsilon$(1 S) to regenerate. Due to high dissociation temperature, $\Upsilon$(1 S) can regenerate even at T = 400 MeV or more, and therefore, $\Gamma _{F}$ for $\Upsilon$(1 S) is almost two times larger than $J/\psi$.

4 Summary

This research investigates the modification in quarkonia potential induced by momentum anisotropy within the anisotropic hot QCD medium. This anisotropic medium intricately influences the gluon distribution, leading to significant changes in quarkonia dissociation and regeneration phenomena. We comprehensively examine the interplay of dissociation mechanisms, including gluonic dissociation and collisional damping, considering various strengths of the medium anisotropy through the parameter, $\zeta$. Moreover, we meticulously explore the role of the relativistic Doppler effect (RDE) on quarkonia dissociation by considering the effective temperature, $T_{eff}$.

Our findings reveal distinct effects of medium anisotropy and RDE on the dissociation and regeneration of $J/\psi$ and $\Upsilon$(1 S). Notably, $\Upsilon$(1 S) dissociation is enhanced in the presence of anisotropy at all temperature levels, while $J/\psi$ is more profoundly affected at higher temperatures and lower transverse momentum ($p_{T}$) values. The influence of anisotropy on the regeneration of $J/\psi$ is marginal, whereas $\Upsilon$(1 S) regeneration is particularly dependent on the degree of medium anisotropy. We have observed that medium anisotropy increases and supports particle dissociation by reducing its binding strength within the QGP medium as it alters the particle’s potential. Therefore, if the medium is anisotropic, then quarkonia states become more vulnerable and can have a higher decay width than an isotropic medium.

The primary objective of this study was to elucidate the effects of medium anisotropy at the recombination process at various temperatures. These insights can serve as corrections for future investigations into the total survival probability of bottomonium and charmonium in heavy-ion collisions at facilities such as RHIC at Brookhaven National Laboratory (BNL) and LHC at the European Organization for Nuclear Research (CERN). As the future upgrades at the LHC focus on the measurement of heavy flavors (along with some others) with increased low-$p_{T}$ reach and vertexing efficiencies close to the interaction point, this study will help the theoretical model tuning to better understand quarkonia production/decay dynamics at the RHIC and LHC energies.

Lastly, an outstanding article investigated the regeneration of bottomonia through an open quantum systems approach [87], and we duly recognize the sophistication of this method. However, our current approach diverges slightly, and we have yet to derive the survival probability, as it demands additional models beyond our present scope. Consequently, it proves challenging to furnish a direct comparison with the open system approach. Thus, we defer this aspect to future extensions of our work, enabling a thorough comparison with the open system approach.