Anisotropic Behavior of S-Wave and P-Wave States of Heavy Quarkonia at Finite Magnetic Field

Lal, Manohar; Solanki, Siddhartha; Sharma, Rishabh; Agotiya, Vineet Kumar

doi:https://doi.org/10.1155/2023/6922729

Advances in High Energy Physics

On this page

Abstract Introduction Results and Discussion Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 6922729 | https://doi.org/10.1155/2023/6922729

Anisotropic Behavior of S-Wave and P-Wave States of Heavy Quarkonia at Finite Magnetic Field

Manohar Lal,¹Siddhartha Solanki,¹Rishabh Sharma,¹and Vineet Kumar Agotiya¹

Academic Editor: Ming Liu

Received20 Jan 2023

Revised13 Mar 2023

Accepted27 May 2023

Published15 Jun 2023

Abstract

We studied the effect of momentum space anisotropy on heavy quarkonium states using an extended magnetized effective fugacity quasiparticle model (EQPM). Both the real and imaginary part of the potential has been modified through the dielectric function by including the anisotropic parameter . The real part of the medium-modified potential becomes more attractive in the presence of the anisotropy and constant magnetic field. The binding energy of the 1S, 2S, and 1P quarkonium states including anisotropy effects for both the oblate and the isotropic case were studied. We find that the binding energy of states becomes stronger in the presence of anisotropy. However, the magnetic field is found to reduce the binding energy. The thermal width of the charmonium and bottomonium states has been studied at constant magnetic field for isotropic and prolate cases. The effect of magnetic field on the mass spectra of the 1P state for the oblate case was also examined. The dissociation temperature for the 1S, 2S, and 1P states of charmonium and bottomonium has been determined to be higher for the oblate case with respect to the isotropic case.

1. Introduction

Quarkonium states are sensitive to several important features of the quark-gluon plasma (QGP), including Landau damping and energy loss as mentioned in ref. [1–3]. Based on the experimental observations, quarkonium suppression, among the other signatures, is regarded as the clear probe of the QGP [4–8]. In the five decades since the discovery of the in 1974 [9, 10], quarkonium dissociation due to color screening in the deconfined medium suggested by Matsui and Satz [11] has become a pioneering research area in the particle physics. Several studies such as [12–17] discuss the important refinements essential to the study of quarkonium in the thermal hot quantum chromodynamics (QCD) medium. The physics of the heavy quarkonia, a strongly interacting matter, in the last two decades has great interest in the presence of the magnetic field upto the scale square mass of the pion or even larger details of which can be found in ref. [18]. Such kind of the studies are very relevant to the highly dense astrophysical compact bodies like the magnetars [19] and also useful for the study of the cosmological aspects [20]. Besides of these facts, the main motivation to study the effect of magnetic field in the heavy quarkonia was triggered by the fact that the order of the magnitude of this field, e.g., at Relativistic Heavy Ion Collider (RHIC) and at Large Hadron Collider (LHC) during the lead-lead collision, could be produced in the laboratory when the two heavy ions, traveling nearly equal to the speed of light, colliding with each other at zero impact parameter in the colliding region [21–25]. It is believed that such a large magnitude of the magnetic field produces at very early stages of the universe shortly after the big bang. However, it is not certain how long these generated magnetic field and to what extent survives in the thermalisation process of the QGP formation.

Various theoretical model-based studies, as well as the lattice quantum chromodynamics (LQCD) prediction, discuss several phenomena effecting the properties of the quarkonia in the presence of the background magnetic field. However, the effects related to the magnetic field are of particular interest because of the fact that the heavy quarkonia are very sensitive to the earlier condition of the universe. Studies such as [26–33] briefly explained the quarkonium spectra and the production rates in the magnetic field regime. Since QCD with nonzero magnetic field does not have a sign problem, one can obtain the QCD phase diagram in the plane using Monte Carlo calculations of the LQCD as the first ab initio principle. Since, we know that the magnetic field generally causes the momentum anisotropy which is responsible for the instability in the Yang-Mills fields. Hence, this momentum anisotropy has played a vital role in the evolution of the QCD medium. When the charged particle placed in the strong magnetic field, the energy associated with circular motion of the charged particle, because of Lorentz force, is discretized (quantized). These quantized energy levels, due to the magnetic field effect, are known as Landau levels. However, in the presence of the strong magnetic field , only lowest Landau levels are populated . This indicates the importance of the lowest Landau level dynamics.

In the present case, we are working under strong magnetic field using the effective fugacity quasiparticle model to study the anisotropic behavior of the heavy quarkonia. One of the remarkable reasons to include the momentum anisotropy is that the QGP produced in the noncentral collision does not possess isotropy. Moreover, momentum anisotropy is present at each and every stages of the heavy ion collisions. This fact triggered most to study the anisotropic effect on the quarkonium properties in magnetic field regime. Several authors [34–41] studied various observables of the quark-gluon plasma by considering the momentum anisotropy. Following the ref. [42–44], the anisotropy has been introduced at the levels of distribution function. The gluon self-energy is used to obtained gluon propagator and in turn to determine the dielectric permittivity in the presence of the anisotropy.

The present manuscript is organized in the following manner.

Quasiparticle Debye screening mass in the presence of magnetic field has been briefly discussed in Section 2.1. The quark-antiquark potential in the anisotropic medium is described in Section 2.2. The effect of momentum space anisotropy on the binding energy, dissociation temperature, thermal width, and the mass spectra of the quarkonium states in the presence of magnetic field has been briefly studied in Section 3. We discuss the results of the present work in Section 4. Finally, we conclude our work in Section 5.

2. Model Setup

2.1. Effective Quasiparticle Model Extension in Magnetic Field and Debye Screening Mass

In the quasiparticle description, the system of the interacting particles is supposed to be noninteracting or weakly interacting by means of the effective fugacity [45] or with the effective mass [46, 47]. Nambu-Jona-Lasinio (NJL) and Ploylov Nambu-Jona-Lasinio (PNJL) quasiparticle models [48], self-consistent quasiparticle model [49–51], etc. include the effective masses. Here, we considered the effective fugacity quasiparticle model (EQPM), in the presence of magnetic field, which interprets the QCD EoS as noninteracting quasipartons with effective fugacity parameter for gluons and for quarks encoding all the interacting effects taking place in the medium. The distribution function for quasigluons and the quasiquarks/quasiantiquarks [52] is given as

It is noted that for the gluons, whereas for quarks/antiquarks, where “” denotes the Landau levels. In high energy physics, . Therefore, . One can look the physical significance of the effective fugacity in the following dispersion relation:

The first term in the above equations (Equations (3) and (4)) represents the collective excitation of the quasigluons and quasiquarks (quasipartons). Thus, it is infer that the effective fugacity and describes the hot QCD medium effects. The effective fugacity like in the other quasiparticle model modifies the energy tensor as discussed in [53]. Now, the extended version of the effective fugacity quasiparticle model in the presence of magnetic field requires the modification of the dispersion relation as defined in the above Equations (2) and (3) by the relativistic discretized Landau levels. Thus, in view of this, the quark/antiquark distribution function can be obtained as below:

The effect of the magnetic field is taken along the -axis. Since the plasma contains both the charged and the quasineutral particles, hence, it shows collective behavior. The Debye mass is an important quantity to describe the screening of the color forces in the hot QCD medium. The Debye screening mass can be defined as the ability of the plasma to shield out the electric potential applied to it. In the studies [54–57], detailed definition of the Debye mass can be found. To determine the Debye mass, in terms of the magnetic field, we start from the gluon self-energy as below:

According to the [58], gluon self-energy modified as

Thus, Debye mass for quarks using the distribution function defined by Equation (5) is given below as

Since the magnetic field has no effect on the gluons, therefore, the gluonic contribution to the Debye mass will remain unchanged. In other words, the distribution function for the gluons with or without magnetic field will remain intact. For perturbative QCD, in the presence of magnetic field, the Debye screening mass can be derived with the application of kinetic theory approach. Both these approaches provide the similar results for the Debye mass in the presence of the magnetic field. So, the Debye mass for the and will be as follows:

The Debye mass for the ideal EoS [] representing noninteracting quarks and gluons becomes

is two loop coupling constant depending upon the temperature [59] and is given below: where denotes the number of flavor which is in our case and is the QCD renormalization scale.

2.2. Quark-Antiquark Potential in the Anisotropic Medium

The solution of the Schrödinger equation (SE), although a century has passed, is still an important tool for both the physicists and chemists. The SE played a vital role to obtain not only the energy spectrum of the diatomic and polyatomic molecule but also the spectrum of the heavy quarkonium system. The solution of the SE for the different potentials, as found in [60–63], has been obtained using generalized Boopp’s shift method and standard perturbation theory. In the present work, medium-modified potential [64] has been used to investigate the properties of the heavy quarkonia. The Cornell potential having both coulombic as well as the string part [64] is given by

To modify this static potential (Equation (12)), we use the Fourier transformation. In the above equation, and are the coupling constant and the string tension, respectively. Here, we take the two loop coupling constant depending upon on the temperature. The value of the string constant has been taken . “r” is the effective radius of the respective quarkonium states. The reason behind the modification of the potential is that the string tension does not vanish at or near the transition temperature , and transition is just “crossover” from hadronic to quark-gluon plasma (QGP). Since the heavy ion collisions are noncentral, the spatial anisotropy generates at the very early stages. As the QGP expands or evolves with time, different pressure gradient arises which are responsible for mapping the spatial anisotropy to the momentum anisotropy. In the present formalism, anisotropy has been introduced at the particle distribution level. Following the studies [43, 44, 65], the isotropic function has been employed to determine the anisotropic distribution function given below: where denotes the effective fugacity quasiparticle isotropic distribution function [66–68]; is the unit vector representing the direction of momentum anisotropy. The parameter represents the anisotropy of the medium. For isotropic case , for oblate form, and for prolate form, the parameter in the direction lies in the ranges and , respectively. The effects of different equations of state (EoS) enter through the Debye mass (). So, to make Debye mass intact from the effects of anisotropy, present in the medium, we use the following normalization constant . With this normalization constant , the Debye mass will remain the same for both the isotropic and anisotropic cases [65]. Therefore, the normalization constant, , is written as

Equation (14) can be simplified as

For small anisotropic effect, can be written as or simply in the small limit, we have

Following the assumption given by [14, 69, 70], the potential of the dissipative anisotropic medium has been modified in the Fourier transform by dividing it with medium dielectric permittivity, :

Taking the inverse of Fourier transform defined above, the in medium/corrected potential reads off:

is the Fourier transform of defined by Equation (12) and given as

Thus, to obtain the modified form of the potential, it is necessary to calculate the dielectric permittivity , and this can be done by two methods: (I) using the gluon self-energy in finite temperature QCD [71, 72] and (II) using semiclassical transport theory (many particle kinetic theory upto one loop order) [44, 73, 74]. By using the above mentioned methods, one can obtain the gluon self-energy which in turn provides static gluon propagator as given below:

Now, from the temporal component of the gluon propagators [75], the dielectric tensor in the Fourier space can be written as below: where represents the static limit of the “00” component of the gluon propagators in the Coulomb gauge. Also, Equation (22), according to linear response theory, provides the relation between the dielectric permittivity and the . The real and the imaginary part of the dielectric tensor obtained from the real part of the retard propagator and imaginary part of symmetric propagators, respectively [76], in the static limit, are given below by where

is the angle between the particle momentum and anisotropy direction . is the angle between and . The azimuthal and the polar angle lie between and . The term denotes the quasiparticle Debye mass which depends on the temperature and magnetic field and is briefly described in Section 2.1.

As the limit, real part of the potential goes to unity, and when , the imaginary part becomes zero. With these limits, the modified form of the potential simply reduces to the Cornell potential. Now, by substituting the real part of the dielectric tensor defined by Equation (23) in Equation (19), the real part of the interquark potential can be written as below:

After separating the coulombic () and string () term from the above equation, the real part of the potential will look like

Similarly, by putting the imaginary part of the dielectric tensor defined by Equation (24) in Equation (19), the imaginary part of complex potential will become

Again, separating the coulombic () and the string () part of above Equation (28), the imaginary potential can be rewritten as

3. Properties of Heavy Quarkonia

3.1. Binding Energy of the Different Quarkonium States

Real binding energies of the heavy quarkonium can be obtained by solving the Schrödinger equation with the first-order perturbation in anisotropy parameter, , as done in [75, 77, 78]. With this, the real binding energy () becomes

This is valid only for the ground and first excited states of the charmonium and bottomonium, i.e., , , , and . But in order to find the binding energies of 1P states of the charmonium and bottomonium, the correction term (potential and the kinetic energy) must be added to the binding energy of the and . These correction terms have been obtained by using the variational treatment method [79–83], in which the total energy consists of the kinetic energy correction and most importantly the correction added to the spin-dependent potential which makes the and (both are first excited state) degenerate and hence obeys the Pauli’s exclusion principle. The correction energy term, as found in the [81, 82, 84], is given below:

Therefore, to evaluate the binding energy of the 1P states of heavy quarkonia, we add up this correction energy term to the binding energy of the as defined in the above equation. Hence, we have

This implies

In the present work, the masses of 1P state as charmonium () and bottomonium () have been taken, and details of which can be found in [85] and references therein.

3.2. Dissociation of Quarkonium States in the Presence of Anisotropy and Strong Magnetic Field

Once we obtained the binding energies of the quarkonium states, it is customary to study the dissociation pattern of the quarkonium states when their binding energies become zero. But in the ref. [86], the authors have argued that it is not essential to have zero binding energy to dissociate the states, but when the binding energy is less than the temperature , a state is weakly bound, and hence, it is destroyed by the thermal fluctuations. The authors in [75, 86, 87] propose another condition for the dissociation of the quarkonium states and that is or , where is the thermal width of the respective state. Thus, there are two major criteria used to determine the dissociation temperature (). The upper bound and the lower bound of the dissociation temperature, as one can found in [88] and reference therein, using thermal effects can be obtained by the following conditions:

Equation (34) is applicable for the 1S and 2S states of the heavy quarkonia. The dissociation temperature for the 1P states of charmonium and bottomonium has been calculated using the following relation:

3.3. Thermal Width of 1S State of Charmonium and Bottomonium

Since the imaginary part of the potential gives rise to the thermal width which in turn also used to calculate the dissociation point of the quarkonium states. The thermal width of the quarkonium states can be obtained by using following ansatz: where is the coulombic wave function. The coulombic wave function for ground state (1S, corresponding to ( and )) is given as where denotes the Bohr radius of the quarkonium system. Now, from Equation (36), we have

Solving the above equation, we get the thermal width for 1S-state as below:

It is important to note that in ref. [75], while considering up to leading logarithmic order of imaginary potential, the authors too have taken the width up to the leading logarithmic. Thus, the dissociation width for 1S-state would be of the form:

3.4. Mass Spectra of the Different Quarkonium States in Anisotropic Hot QCD Medium at Finite Magnetic Field

The mass spectra of the different quarkonium states in the presence of magnetic field along with the effect of anisotropic parameter can be obtained by the relation:

But in the current work, we have calculated the mass spectra of only 1P state of the heavy quarkonia by using the following relation:

Hence, using Equations (30) and (33) in Equation (42), we have

Where is the mass of the heavy quarkonia, is the binding energy of the and , and is the energy correction/mass gap correction obtained using variational treatment method.

4. Results and Discussion

Heavy quarkonium properties have been investigated, by means of in medium modification to the Cornell potential (perturbative as well as nonperturbative), using extended quasiparticle approach in the presence of strong magnetic field limit . Since at the early stages of the ultrarelativistic heavy ion collisions (URHIC), the anisotropy arises in the beam direction as the system expands. At , the string term makes the real potential more attractive compared to the case when the potential is modified using coulombic part only. This means that the respective quarkonium states become more bound with both the coulombic and string part in comparison to the case when coulombic part is modified alone. Here, in the present work, we have consider the weak anisotropy for the oblate case, , and isotropic case, , with the fixed value of the critical temperature . The variation of the real potential with the distance ( in ) is shown in Figure 1 at and temperature for oblate and isotropic case. For the isotropic case, , we have the same variation for both parallel and perpendicular case. This is because of the fact that the system is expanding longitudinally. On the other hand, for the oblate case , the real potential has a lower value for the parallel case () in comparison to the perpendicular case ().

Figures 2–7 show the variation of the binding energy of , , , , , and with the temperature at finite values of the magnetic field for both the isotropic case (a) and oblate case (b), respectively. From all these figures, it has been deduced that the binding energy of all the quarkonium states 1S (, ), 2S (, ), and 1P (, ) decreases with the temperature. Also, as we go from lower to higher magnetic field values, the binding energy also has lower values as can be seen from Figures 2–7. However, it is noticed that binding energy of all the above mentioned states has higher values for the oblate case in competition to the isotropic case. In other words, the anisotropy seems to be an additional handle to decipher the properties of the quarkonium states. In anisotropic medium, the binding energy of the pair gets stronger with increase in the anisotropy. This is due to the fact that the Debye screening mass in anisotropic medium is always much lower compared to the isotropic one. Hence, quarkonium states are strongly bound.

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

Figure 8 shows the variation of the thermal width of charmonium () (a) and bottomonium () (b) at for the isotropic case and oblate case. It has been noticed from Figure 8 that there is an increase in the thermal width with the temperature for both cases. Although, thermal width has lower value for the oblate case in competition to the isotropic case. It is also noticed that the thermal width of the upsilon () is much smaller than the (). This is due to the fact the bottomonium states are smaller in size and larger in masses than the charmonium states and hence get dissociated at higher temperatures. Mass spectra of the and have been shown in Figure 9. There is a decreasing pattern of the mass spectra of and with the temperature at at finite magnetic fields. Mass spectra of these states at (oblate case) are found to be very close to the particle data group 2018 [89].

(a)

(b)

(a)

(b)

The dissociation temperatures for the 1S (, ), 2S (, ), and 1P (, ) have been given in Tables 1–4. Lower bound of dissociation temperatures for different states is shown in Tables 1 and 3 for the isotropic case. Tables 2 and 4 show the different values of dissociation temperatures for the oblate case . In general, the dissociation temperatures decrease with the magnetic field. It is pertinent to mention here that for the oblate case, dissociation temperatures have found to be higher in comparison to the isotropic case as seen from the tables.

5. Conclusions

The dissociation process of the heavy quarkonium states 1S (, ), 2S (, ), and 1P (, ) in the anisotropic medium at finite magnetic field using extended quasiparticle model has been investigated. The real part of potential becomes more attractive in anisotropic hot QCD medium compared to isotropic case at constant magnetic field. The binding energy of the different quarkonium states decreases with the temperature as well as with magnetic field for both the isotropic and oblate case. However, the binding energy has higher values for the oblate case compared to isotropic case (). From Figures 2–7, it is deduced that the binding energies of the 2S states (, ) are smaller than the binding energies of 1S (, ) and 1P (, ) states for both the isotropic and oblate case in the presence of magnetic field. The dissociation temperatures reduce as we increase the magnetic field for both the isotropic and oblate case. It is also found that the dissociation temperature for all the 1S, 2S, and 1P states has higher values for the oblate case compared to the case when . It is also noted here that the dissociation temperature of 1S and 1P states of charmonium and bottomonium are higher than the 2S states. This is because of the fact that the and states are highly unstable or loosely bound states. The hierarchical order of dissociation temperatures () for the different quarkonium states is . Thermal width is found to increase with the temperature at constant magnetic field . The thermal width also increases with indicating quicker dissociation of the states. In the future, this work will be extended to calculate the survival probability or the nuclear modification factor of different quarkonium states with respect to transverse momentum, centrality, and rapidity which is the key point to quantify the various properties of the medium produced during heavy ion collisions (HIC) at LHC and RHIC.

Data Availability

The data statement will remain intact as the data shall be provided by the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

Acknowledgments

VKA acknowledges the Science and Engineering Research Board (SERB) project no. EEQ/2018/000181 New Delhi for providing the financial support. We record our sincere gratitude to the people of India for their generous support for the research in basic sciences.

References

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, vol. 8, Heinemann, Butterworth, 2nd edition, 1984.
View at: Publisher Site
M. C. Chu and T. Matsui, “Dynamic Debye screening for a heavy-quark-antiquark pair traversing a quark-gluon plasma,” Physical Review D, vol. 39, no. 7, pp. 1892–1897, 1989.
View at: Publisher Site | Google Scholar
Y. Koike, “Energy loss of high energy partons traversing a quark-gluon plasma,” AIP Conference Proceedings, vol. 243, p. 916, 1992.
View at: Google Scholar
L. McLerran, “The physics of the quark-gluon plasma,” Reviews of Modern Physics, vol. 58, no. 4, pp. 1021–1064, 1986.
View at: Publisher Site | Google Scholar
K. Adcox, S. S. Adler, S. Afanasiev et al., “Formation of dense partonic matter in relativistic nucleus-nucleus collisions at RHIC: experimental evaluation by the PHENIX Collaboration,” Nuclear Physics A, vol. 757, no. 1-2, pp. 184–283, 2005.
View at: Publisher Site | Google Scholar
B. B. Back, M. D. Baker, M. Ballintijn et al., “The PHOBOS perspective on discoveries at RHIC,” Nuclear Physics A, vol. 757, no. 1-2, pp. 28–101, 2005.
View at: Publisher Site | Google Scholar
I. Arsene, I. G. Bearden, D. Beavis et al., “Quark-gluon plasma and color glass condensate at RHIC? The perspective from the BRAHMS experiment,” Nuclear Physics A, vol. 757, no. 1-2, pp. 1–27, 2005.
View at: Publisher Site | Google Scholar
J. Adams, M. M. Aggarwal, Z. Ahammed et al., “Experimental and theoretical challenges in the search for the quark-gluon plasma: the STAR Collaboration's critical assessment of the evidence from RHIC collisions,” Nuclear Physics A, vol. 757, no. 1-2, pp. 102–183, 2005.
View at: Publisher Site | Google Scholar
J. J. Aubert, U. Becker, P. J. Biggs et al., “Experimental observation of a heavy particleJ,” Physical review focus, vol. 33, no. 23, pp. 1404–1406, 1974.
View at: Publisher Site | Google Scholar
J. E. Augustin, A. M. Boyarski, M. Breidenbach et al., “Discovery of a narrow resonance ine+e−annihilation,” Physical Review Letters, vol. 33, no. 23, pp. 1406–1408, 1974.
View at: Publisher Site | Google Scholar
T. Matsui and H. Satz, “J/ψ suppression by quark-gluon plasma formation,” Physics Letters B, vol. 178, no. 4, pp. 416–422, 1986.
View at: Publisher Site | Google Scholar
A. Mocsy and P. Petreczky, “Heavy quarkonia survival in potential model,” Europian Physical Journal C, vol. 43, no. 1-4, pp. 77–80, 2005.
View at: Publisher Site | Google Scholar
A. Mocsy and P. Petreczky, “Quarkonia correlators above deconfinement,” Physical Review D, vol. 73, no. 7, article 074007, 2006.
View at: Publisher Site | Google Scholar
V. K. Agotiya, V. Chandra, and B. K. Patra, “Dissociation of quarkonium in a hot QCD medium: modification of the interquark potential,” Physical Review C, vol. 80, no. 2, article 025210, 2009.
View at: Publisher Site | Google Scholar
B. K. Patra, V. K. Agotiya, and V. Chandra, “Charmonium suppression in the presence of dissipative forces in a strongly-coupled quark-gluon plasma,” Europian Physical Journal C, vol. 67, no. 3-4, pp. 465–477, 2010.
View at: Publisher Site | Google Scholar
S. Datta, F. Karsch, P. Petreczky, and I. Wetzorke, “Behavior of charmonium systems after deconfinement,” Physical Review D, vol. 69, no. 9, article 094507, 2004.
View at: Publisher Site | Google Scholar
N. Brambilla, J. Ghiglieri, P. Petreczky, and A. Vairo, “Static quark-antiquark pairs at finite temperature,” Physical Review D, vol. 78, no. 1, article 014017, 2008.
View at: Publisher Site | Google Scholar
D. Kharzeev, K. Landsteiner, A. Schmitt, and H. U. Yee, “Strongly Interacting Matter in Magnetic Fields,” in Lecture Notes Physics, vol. 871, pp. 1–11, Springer, Berlin Heidelberg, 2013.
View at: Publisher Site | Google Scholar
R. C. Duncan and C. Thompson, “Formation of very strongly magnetized neutron stars: implications for gamma-ray bursts,” Astrophysical Journal, vol. 392, pp. L9–L13, 1992.
View at: Publisher Site | Google Scholar
D. Grasso and H. R. Rubinstein, “Magnetic fields in the early universe,” Physics Reports, vol. 348, no. 3, pp. 163–266, 2001, [astro-ph/0009061].
View at: Publisher Site | Google Scholar
V. Skokov, A. Y. Illarionov, and V. Toneev, “Estimate of the magnetic field strength in heavy-ion collisions,” International Journal of Modern Physics A, vol. 24, no. 31, pp. 5925–5932, 2009.
View at: Publisher Site | Google Scholar
V. Voronyuk, V. D. Toneev, W. Cassing, E. L. Bratkovskaya, V. P. Konchakovski, and S. A. Voloshin, “Electromagnetic field evolution in relativistic heavy-ion collisions,” Physical Review C, vol. 83, no. 5, article 054911, 2011, [arXiv:1103.4239 [nucl-th]].
View at: Publisher Site | Google Scholar
A. Bzdak and V. Skokov, “Event-by-event fluctuations of magnetic and electric fields in heavy ion collisions,” Physics Letters B, vol. 710, no. 1, pp. 171–174, 2012.
View at: Publisher Site | Google Scholar
W. T. Deng and X. G. Huang, “Event-by-event generation of electromagnetic fields in heavy-ion collisions,” Physical Review C, vol. 85, no. 4, article 044907, 2012.
View at: Publisher Site | Google Scholar
K. Tuchin, “Particle production in strong electromagnetic fields in relativistic heavy-ion collisions,” Advances in High Energy Physics, vol. 2013, Article ID 490495, 34 pages, 2013.
View at: Publisher Site | Google Scholar
C. S. Machado, F. S. Navarra, E. G. de Oliveira, J. Noronha, and M. Strickland, “Heavy quarkonium production in a strong magnetic field,” Physical Review D, vol. 88, no. 3, 2013.
View at: Publisher Site | Google Scholar
C. S. Machado, S. I. Finazzo, R. D. Matheus, and J. Noronha, “Modification of theBmeson mass in a magnetic field from QCD sum rules,” Physical Review D, vol. 89, no. 7, 2014.
View at: Publisher Site | Google Scholar
J. Alford and M. Strickland, “Charmonia and bottomonia in a magnetic field,” Physical Review D, vol. 88, no. 10, 2013.
View at: Publisher Site | Google Scholar
D. Dudal and T. G. Mertens, “Melting of charmonium in a magnetic field from an effective AdS/QCD model,” Physical Review D, vol. 91, no. 8, 2015.
View at: Publisher Site | Google Scholar
S. Cho, K. Hattori, S. H. Lee, K. Morita, and S. Ozaki, “QCD sum rules for magnetically induced mixing between η_c and J/ψ,” Physical review focus, vol. 113, no. 17, article 172301, 2014.
View at: Publisher Site | Google Scholar
S. Cho, K. Hattori, S. H. Lee, K. Morita, and S. Ozaki, “Charmonium spectroscopy in strong magnetic fields by QCD sum rules: S-wave ground states,” Physical Review D, vol. 91, no. 4, article 045025, 2015, [arXiv:1411.7675 [hep-ph]].
View at: Publisher Site | Google Scholar
P. Filip, “Heavy flavor mesons in strong magnetic fields,” 2013, https://arxiv.org/abs/1504.07008.
View at: Google Scholar
X. Guo, S. Shi, N. Xu, Z. Xu, and P. Zhuang, “Magnetic field effect on charmonium production in high energy nuclear collisions,” Physical Letters B, vol. 751, pp. 215–219, 2015.
View at: Publisher Site | Google Scholar
P. Romatschke, “Momentum broadening in an anisotropic plasma,” Physical Review C, vol. 75, no. 1, article 014901, 2007.
View at: Publisher Site | Google Scholar
P. Romatschke and M. Strickland, “Collisional energy loss of a heavy quark in an anisotropic quark-gluon plasma,” Physical Review D, vol. 71, no. 12, article 125008, 2005.
View at: Publisher Site | Google Scholar
B. Schenke and M. Strickland, “Photon production from an anisotropic quark-gluon plasma,” Physical Review D, vol. 76, no. 2, article 025023, 2007.
View at: Publisher Site | Google Scholar
M. Martinez and M. Strickland, “Measuring quark-gluon-plasma thermalization time with dileptons,” Physical review focus, vol. 100, no. 10, article 102301, 2008.
View at: Publisher Site | Google Scholar
M. Martinez and M. Strickland, “Pre-equilibrium dilepton production from an anisotropic quark-gluon plasma,” Physical Review C, vol. 78, no. 3, article 034917, 2008.
View at: Publisher Site | Google Scholar
A. Dumitru, Y. Nara, B. Schenke, and M. Strickland, “Jet broadening in unstable non-Abelian plasmas,” Physical Review C, vol. 78, no. 2, article 024909, 2008.
View at: Publisher Site | Google Scholar
R. Baier and Y. Mehtar-Tani, “Jet quenching and broadening: the transport coefficient q̂ in an anisotropic plasma,” Physical Review C, vol. 78, no. 6, article 064906, 2008.
View at: Publisher Site | Google Scholar
A. Dumitru, Y. Guo, and M. Strickland, “The heavy-quark potential in an anisotropic plasma,” Physics Letters B, vol. 662, no. 1, pp. 37–42, 2008.
View at: Publisher Site | Google Scholar
Y. Burnier, M. Laine, and M. Vepsäläinen, “Quarkonium dissociation in the presence of a small momentum space anisotropy,” Physics Letters B, vol. 678, no. 1, pp. 86–89, 2009.
View at: Publisher Site | Google Scholar
P. Romatschke and M. Strickland, “Collective modes of an anisotropic quark-gluon plasma,” Physical Review D, vol. 68, no. 3, article 036004, 2003.
View at: Publisher Site | Google Scholar
M. Y. Jamal, S. Mitra, and V. Chandra, “Collective excitations of hot QCD medium in a quasiparticle description,” Physical Review D, vol. 95, no. 9, article 094022, 2017.
View at: Publisher Site | Google Scholar
V. Chandra and V. Ravishanker, “Quasi-particle model for lattice QCD: quark-gluon plasma in heavy ion collisions,” European Physics Journal C, vol. 64, no. 1, pp. 63–72, 2009.
View at: Publisher Site | Google Scholar
V. Goloviznin and H. Satz, “The refractive properties of the gluon plasma in SU(2) gauge theory,” Zeitschrift für Physik C Particles and Fields, vol. 57, pp. 671–675, 1993.
View at: Google Scholar
A. Peshier, B. Kampfer, O. P. Pavlenko, and G. Soff, “Massive quasiparticle model of the SU (3) gluon plasma,” Physical Review D, vol. 54, no. 3, pp. 2399–2402, 1996.
View at: Publisher Site | Google Scholar
A. Dumitru and R. D. Pisarski, “Degrees of freedom and the deconfining phase transition,” Physics Letters B, vol. 525, no. 1-2, pp. 95–100, 2002.
View at: Publisher Site | Google Scholar
V. M. Bannur, “Self-consistent quasiparticle model for quark-gluon plasma,” Physical Review C, vol. 75, article 044905, 2006.
View at: Publisher Site | Google Scholar
V. M. Bannur, “Comments on quasiparticle models of quark-gluon plasma,” Physics Letters B, vol. 647, no. 4, pp. 271–274, 2007.
View at: Publisher Site | Google Scholar
V. M. Bannur, “Quasi-particle model for QGP with nonzero densities,” Journal of High Energy Physics, vol. 2007, no. 9, p. 046, 2007.
View at: Publisher Site | Google Scholar
M. Kurian and V. Chandra, “Bulk viscosity of a hot QCD medium in a strong magnetic field within the relaxation-time approximation,” Physical Review D, vol. 97, no. 11, article 116008, 2018.
View at: Publisher Site | Google Scholar
V. Chandra, “Transport properties of anisotropically expanding quark-gluon plasma within a quasiparticle model,” Physical Review D, vol. 86, no. 11, article 114008, 2012.
View at: Publisher Site | Google Scholar
A. Rebhan, “Non-abelian Debye mass at next-to-leading order,” Physical Review D, vol. 48, no. 9, pp. R3967–R3970, 1993.
View at: Publisher Site | Google Scholar
E. Shuryak, “Theory of hadron plasma,” Journal of Experimental and Theoretical Physics, vol. 47, p. 212, 1978.
View at: Google Scholar
S. Mrowczynski, “Topics in the transport theory of quark–gluon plasma,” Physics of Particles and Nuclei, vol. 30, no. 4, p. 419, 1999.
View at: Publisher Site | Google Scholar
K. Yagi, T. Hatsuda, and Y. Miake, “Quark-gluon plasma: from big bang to little bang,” Particles and Nuclei Physics Cosmology, vol. 23, pp. 1–446, 2005.
View at: Google Scholar
J. JanKowski, D. Blaschke, and O. Kaczmarek, “Debye masses at the QCD transition in the PNJL model,” 2015, https://arxiv.org/abs/1508.05322.
View at: Google Scholar
M. Laine and Y. Schroder, “Two-loop QCD gauge coupling at high temperatures,” Journal of High Energy Physics, vol. 2005, no. 3, 2005.
View at: Publisher Site | Google Scholar
A. Maireche, “Heavy-light mesons in the symmetries of extended nonrelativistic quark model,” Yanbu Journal of Engineering and Science, vol. 17, no. 1, p. 1441H, 2019.
View at: Publisher Site | Google Scholar
A. Maireche, “The impact of deformed space-phase parameters into HAs and HLM systems with the improved Hulthen plus Hellmann potentials model in the presence of temperature-dependent confined Coulomb potential within the framework of DSE,” Revista Mexicana de Física, vol. 68, no. 5 Sep-Oct, pp. 1–21, 2022.
View at: Publisher Site | Google Scholar
A. Maireche, “Analytical expressions to energy eigenvalues of the hydrogenic atoms and the heavy light mesons in the framework of 3D-NCPS symmetries using the generalized Bopp’s shift method,” Bulgarian Journal of Physics, vol. 49, no. 3, pp. 239–265, 2022.
View at: Publisher Site | Google Scholar
A. Maireche, “A new nonrelativistic investigation for interactions in one-electron atoms with modified vibrational-rotational analysis of supersingular plus quadratic potential: extended quantum mechanics,” Journal of nano- and electronics physics, vol. 8, article 04076, 2016.
View at: Google Scholar
E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane, and T. M. Yan, “Charmonium: comparison with experiment,” Physical Review D, vol. 21, no. 1, pp. 203–233, 1980.
View at: Publisher Site | Google Scholar
M. E. Carrington, K. Deja, and S. Mrowczynski, “Plasmons in anisotropic quark-gluon plasma,” Physical Review C, vol. 90, no. 3, article 034913, 2014.
View at: Publisher Site | Google Scholar
V. Chandra, R. Kumar, and V. Ravishankar, “Hot QCD equations of state and relativistic heavy ion collisions,” Physical Review C, vol. 76, no. 5, article 054909, 2007, [Erratum: Phys. Rev. C, vol. 76, pp. 069904, 2007.
View at: Publisher Site | Google Scholar
V. Chandra, A. Ranjan, and V. Ravishankar, “On the chromo-electric permittivity and Debye screening in hot QCD,” The European Physical Journal A, vol. 40, no. 1, pp. 109–117, 2009.
View at: Publisher Site | Google Scholar
V. Chandra and V. Ravishankar, “Quasiparticle description of (2+1)-flavor lattice QCD equation of state,” Physical Review D, vol. 84, no. 7, article 074013, 2011.
View at: Publisher Site | Google Scholar
S. Solanki, M. Lal, R. Sharma, and V. K. Agotiya, “Study of quarkonium properties using SUSYQM method with baryonic chemical potential,” International journal of modern physics A, Particles and fields, gravitation, cosmology, nuclear physics, vol. 37, no. 31n32, article 2250196, 2022.
View at: Publisher Site | Google Scholar
V. K. Agotiya, V. Chandra, M. Y. Jamal, and I. Nilima, “Dissociation of heavy quarkonium in hot QCD medium in a quasiparticle model,” Physical Review D, vol. 94, no. 9, article 094006, 2016.
View at: Publisher Site | Google Scholar
M. H. Thoma, “New developments and applications of thermal field theory,” 2000, https://arxiv.org/abs/hep-ph/0010164.
View at: Google Scholar
E. Braaten and R. D. Pisarski, “Soft amplitudes in hot gauge theories: a general analysis,” Nuclear Physics B, vol. 337, no. 3, pp. 569–634, 1990.
View at: Publisher Site | Google Scholar
J. P. Blaizot and E. Iancu, “Soft collective excitations in hot gauge theories,” Nuclear Physics B, vol. 417, no. 3, pp. 608–673, 1994.
View at: Publisher Site | Google Scholar
P. F. Kelly, Q. Liu, C. Lucchesi, and C. Manuel, “Classical transport theory and hard thermal loops in the quark-gluon plasma,” Physical Review D, vol. 50, no. 6, pp. 4209–4218, 1994.
View at: Publisher Site | Google Scholar
L. Thakur, U. Kakade, and B. K. Patra, “Dissociation of quarkonium in a complex potential,” Physical Review D, vol. 89, no. 9, article 094020, 2014.
View at: Publisher Site | Google Scholar
M. Y. Jamal, I. Nilima, V. Chandra, and V. K. Agotiya, “Dissociation of heavy quarkonia in an anisotropic hot QCD medium in a quasiparticle model,” Physical Review D, vol. 97, no. 9, article 094033, 2018.
View at: Publisher Site | Google Scholar
M. Margotta, K. McCarty, C. McGahan, M. Strickland, and D. Y. Elorriaga, “Quarkonium states in a complex-valued potential,” Physical Review D, vol. 83, no. 10, article 105019, 2011.
View at: Publisher Site | Google Scholar
M. Strickland and D. Bazow, “Thermal bottomonium suppression at RHIC and LHC,” Nuclear Physics A, vol. 879, pp. 25–58, 2012.
View at: Publisher Site | Google Scholar
G. Hardekopf, J. Sucher, G. Hardekopf, and J. Sucher, “Relativistic wave equations in momentum space,” Physical Review A, vol. 30, no. 2, pp. 703–711, 1984.
View at: Publisher Site | Google Scholar
J. W. Darewych and M. Horbatsch, “The relativistic two-body coulomb system,” Journal of Physics B: Atomic, Molecular and Optical Physics, vol. 22, no. 7, pp. 973–986, 1989.
View at: Publisher Site | Google Scholar
J. W. Darewych and M. Horbatsch, “A variational treatment of the relativistic two-fermion bound-state system in quantum electrodynamics,” Journal of Physics B: Atomic, Molecular and Optical Physics, vol. 23, no. 3, pp. 337–354, 1990.
View at: Publisher Site | Google Scholar
J. W. Darewych, M. Horbatsch, and R. Kounik, “Variational basis-state expansion calculation of the mass gap in scalar field theory,” Physical Review D, vol. 39, no. 2, pp. 499–502, 1989.
View at: Publisher Site | Google Scholar
L. Kluberg, H. Satz, L. Kluberg, and H. Satz, “Color deconfinement and charmonium production in nuclear collisions,” 2009, https://arxiv.org/abs/0901.3831.
View at: Google Scholar
A. Jakovac, P. Petreczky, K. Petrov, and A. Velytsky, “Quarkonium correlators and spectral functions at zero and finite temperature,” Physical Review D, vol. 75, no. 1, article 014506, 2007.
View at: Publisher Site | Google Scholar
I. Nilima and V. K. Agotiya, “Bottomonium suppression in nucleus-nucleus collisions using effective fugacity quasi-particle model,” Advances in High Energy Physics, vol. 2018, Article ID 8965413, 12 pages, 2018.
View at: Publisher Site | Google Scholar
A. Mocsy and P. Petreczky, “Color screening melts quarkonium,” Physical Review Letters, vol. 99, no. 21, article 211602, 2007.
View at: Publisher Site | Google Scholar
D. Kharzeev, L. D. McLerran, and H. Satz, “Non-perturbative quarkonium dissociation in hadronic matter,” Physics Letters B, vol. 356, no. 2-3, pp. 349–353, 1995.
View at: Publisher Site | Google Scholar
M. Lal, S. Solanki, R. Sharma, and V. K. Agotiya, “Melting of quarkonia in strong magnetic field,” Indian Journal of Pure & Applied Physics (IJPAP), vol. 60, no. 6, pp. 475–481, 2022.
View at: Publisher Site | Google Scholar
M. Tanabashi, K. Hagiwara, K. Hikasa et al., “Review of particle physics,” Physical Review D, vol. 98, no. 3, article 030001, 2018.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2023 Manohar Lal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

PDF Download Citation

Download other formats

Order printed copies

Views

362

Downloads

306

Citations