Heavy quark potential and LQCD based quark condensate at finite magnetic field

In the present work, we have studied heavy quarkonia potential in hot and magnetized quark gluon plasma. Inverse magnetic catalysis (IMC) effect is incorporated within the system through the magnetic field modified Debye mass by modifying the effective quark masses. We have obtained the real and imaginary part of the heavy quark potential in this new scenario. After the evaluation of the binding energy and the decay width we comment about the dissociation temperatures of the heavy quarkonia in presence of magnetic field.


Mumbai, Maharashtra 400094, India
In the present work, we have studied heavy quarkonia potential in hot and magnetized quark gluon plasma.Inverse magnetic catalysis (IMC) effect is incorporated within the system through the magnetic field modified Debye mass by modifying the effective quark masses.We have obtained the real and imaginary part of the heavy quark potential in this new scenario.After the evaluation of the binding energy and the decay width we comment about the dissociation temperatures of the heavy quarkonia in presence of magnetic field.

I. INTRODUCTION
A lot of information is being provided by the ongoing relativistic heavy-ion collisions (HIC) in respect of deconfined state of matter, i.e.Quark Gluon Plasma (QGP).QGP at sufficiently high temperature behaves like a weakly interacting gas of quarks and gluons, which can also be studied perturbatively by using hard thermal loop (HTL) resummation [1][2][3][4] techniques, apart from the first principle lattice QCD estimations.Depending on the noncentrality, HIC can also produce a very strong magnetic field in the direction perpendicular to the reaction plane [5,6].At the RHIC energies, the estimated strength of the magnetic field is around B ∼ m 2 π ≡ 10 18 Gauss whereas at the LHC the estimated strength is around B ∼ 15m 2 π ≡ 1.5 × 10 19 Gauss [5,6], where m π is the pion mass.The issue -whether this large initial magnetic field will decay very fast [7] or slow [8,9] is still probably an open problem.Studies have also shown that such an extensive magnetic field might even have survived from the very early stages of universe [10,11].These possibilities provide the opportunity to revisit the entire QGP phenomenology in presence of magnetic field, and in the present work we have attempted the same with heavy quark phenomenology.
According to recent lattice quantum chromodynamic (LQCD) calculation [12,13], nonzero QCD vaccum at finite temperature and magnetic field can face both magnetic catalysis (MC) and inverse magnetic catalysis (IMC).MC shows the low temperature enhancements in the values of quark condensates with increasing magnetic field and extensively studied through lattice QCD and effective models.On the other hand, IMC shows a decreasing behaviour of the condensates with increasing magnetic field close to the transition temperature.IMC was first discovered by lattice QCD simulations using physical values of pion mass and for the light quarks.Since then there have been many efforts to implement IMC in the effective models.In our present work, we have implemented this important IMC effect in our calculation through the constituent quark mass generated by the lattice QCD simulations, which incorporates the complex and non-monotonic temperature and magnetic field dependence within the quark condensates.
One of the useful probe of the QGP formation is heavy quarkonium which is a bound state of Q Q pair [14,15].After the discovery of J/ψ ( a bound state of cc), [16,17], in 1974, a large number of excellent articles have been published that proposed several essential refinements in the study of heavy quark potential.The first work to study the quarkonia at finite temperatures using potential models have been done by Karsch, Mehr, and Satz [18].Subsequently another pioneering work to study the dissociation of quarkonia due to the color screening in the deconfined medium with finite temperature, was carried out by Matsui and Satz [19].In recent years various studies have been executed to see the impact of magnetic field on heavy quark phenomenology [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36].Specifically speaking, the effects of the external magnetic field on the quarkonia production have been discussed in Refs.[23,28].In Ref. [32], the authors have studied the directed flow of charm quarks which is considered as an efficient probe to characterize the evolving magnetic field produced in ultrarelativistic HIC.In Ref. [29] the authors have investigated QCD sum rules in calculation of the mass of heavy mesons to estimate the modification of the charged B meson mass, (m B ), in the presence of an external Abelian magnetic field.The momentum diffusion coefficients of heavy quarks, in a strong magnetic field within the lowest Landau level (LLL) approximation, along the directions parallel and perpendicular to B, at the leading order in QCD coupling constant have been computed within and beyond the static limit of the heavy quarks, respectively in Refs.[31] and [36].
The physics about the fate of quarkonia at zero temperature can be understood with the help of non-relativistic potential models.Masses of heavy quarks (m Q ) are much larger than QCD scale (Λ QCD ) and velocity of the quarks in the bound state is small, v 1 [37].Hence, to understand the binding effects in quarkonia generally one uses the Cornell potential which belongs to the family of the non-relativistic potential models [38] and can be derived directly from QCD using the effective field theories (EFTs) [37][38][39].Refs.[33,35] has studied the effect of a strong magnetic field on the heavy quark complex potential with the lowest Landau level (LLL) approximation.In this regard, present work has moved beyond LLL estimation and considered all Landau level summation, which is valid for the entire range of magnetic field from weak to strong.This is one of the new ingredients of the present work.The main goal of the present work though is to incorporate the IMC effect in the heavy quark potential through the effective quark masses.These two components are mainly introduced within the standard formalism of heavy quark potential [41][42][43][44][45], which is the sum of both Coulomb and string terms [40].
The paper is organized as follows.In section II, we will discuss the basic formalism of our present work which includes discussions about the real and imaginary parts of the heavy quark potential, decay width, binding energy and the Debye screening mass.In section III we will show our results for the same as well as find out the dissociation temperatures and discuss their magnetic field dependence.Finally in section IV, we shall conclude the present work.

II. FORMALISM
In this section, we have described the entire formalism required for our current study.In the first subsection (II A), we have addressed the standard framework of the heavy quark potential in presence of an external magnetic field, and established the connection with the gluon propagator through the dielectric permittivity.In the process, a temperature and magnetic field dependent Debye mass enters into the heavy quark potential through the gluon propagator.This Debye mass is calculated from semi-classical transport theory, whose magneto-thermodynamical phase space can be obtained by projecting the temperature and magnetic field dependent condensates from the lattice quantum chromodynamics (LQCD) calculation, which incorporate the effects of both MC and IMC.This part is discussed in details in subsection (II B), which is the main motive our present study.Next in subsection (II C), we have discussed about the framework of thermal width and binding energy of heavy quarkonia (e.g.J/ψ or Υ) from imaginary and real part of heavy quark potential respectively.
A. In-medium heavy quark potential in presence of magnetic field Let us start our discussion with the full Cornell potential [38,46], that contains the Coulombic as well as the string part given as, Here, r is the effective radius of the corresponding quarkonia state, α is the strong coupling constant given by (α = ) and σ is the string tension.The Fourier transform of V(r) is The assumption given in Ref. [41] has been followed which says that, the in-medium modification can be obtained in the Fourier space by dividing the heavy-quark potential from the medium dielectric permittivity, (k, T, eB) as, where (k, T, eB) can be obtained from the static limit of the longitudinal part of gluon self-energy [47].So, information of temperature T and magnetic field eB (in M 2 dimension) are entered through this medium dielectric permittivity (k, T, eB).By making the inverse Fourier transform, we can obtain the modified potential at finite T and eB as, Next, the dielectric permittivity can be obtained in the static limit, in the Fourier space, from the temporal component of the propagator (∆ 00 ) as [1,47], Now, to obtain the real part of the inter-quark potential in the static limit, the temporal component of real part of the retarded propagator in the Fourier space is demanded, which is given as The imaginary part of the same can be derived from the imaginary part of the temporal component of symmetric propagator in the static limit which is given as [48] Im Thus in the short-distance limit (r 1), the sum of Coulomb and string term gives the real and imaginary part of the potential in terms of modified coordinate space r = rm D given as [45,49,50] and respectively.Reader can notice that the information of T and B are entering through Debye mass m D = m D (T, B), whose mathematical derivation is addressed in next subsection.

B. Debye mass in presence of magnetic field
Debye screening mass is an important observable in the context of heavy ion collisions which also acts as a QGP signature through heavy quarkonia (i.e.J/Ψ and Υ) suppression.Debye screening mass can be directly evaluated from the temporal component of the gluon self energy tensor (Π 00 (p 0 , p)) by employing the static limit (| p| = 0, p 0 → 0) through a perturbative order by order evaluation.On the other hand one can also determine the Debye screening mass through the semi-classical transport theory by using the relation [45,51,52] where g 2 s ≡ 4πα s is the strong coupling constant, C q/g are the Casimir constants for quarks and gluons and f q/g = 1 e βE ±1 are their respective distribution functions -Fermi-Dirac (FD) for quark and Bose-Einstein (BE) for gluons.For an ideal non interacting QGP medium, one can readily trace back the leading order hard thermal loop (HTL) expression of the Debye mass, i.e.
There are several investigations on the Debye screening masses of the QGP as a function of the magnetic field from the temporal component of the gluon polarization in perturbative QCD (pQCD) calculation [33,34,[53][54][55][56][57], whose equivalence anatomy in semi-classical transport theory has been shown by Ref. [58].The gluonic distribution function remains unchanged in presence of an external anisotropic magnetic field along the z direction ( B = B ẑ), whereas the quark distribution function gets modified to: where the Landau quantized dispersion relation reads as . being the number of Landau levels and q f = + 2 3 , − 1 3 being the fractional charge of the u and d quarks respectively.Again, in a magnetized medium the phase space quantization [59][60][61] can be represented as Incorporating these, the expression for the Debye screening mass m D for a magnetized QGP medium becomes, Now, in the earlier studies of the Debye screening mass in a magnetized medium [33,34,[53][54][55][56][57][58], one of the important aspect of QCD at finite magnetic field have not been considered.It is Inverse Magnetic Catalysis (IMC) phenomenon near quark-hadron phase transition temperature, recently revealed from lattice QCD (LQCD) calculations [12,13].Their observed phenomenon and connected physics can be described as follows.We know that QCD vacuum or quark condensate in vacuum is non-zero, for which current quark mass m f ∼ 5 − 10 MeV get an effective constituent quark mass M f (more than 300 MeV) and fusion of three constituent quarks can able justify the origin of nucleon mass (∼ 939 MeV).Now, in presence of magnetic field, condensate value in vacuum or T = 0 enhances, which is known as magnetic catalysis (MC) phenomenon.LQCD calculations [12,13] found this MC effect within low T domain but near transition T range, its inverse nature is observed, which is called as IMC.To capture this detailed MC and IMC aspects of QCD through T and B dependent quark condensate and constituent quark mass M f (T, B) in the present work, we have redefined the dispersion relation in a magnetized medium as where M f (T, eB) represents the effective constituent mass in terms of LQCD [12,13] predicted normalized quark condensate q q f (T, eB), which varies from 1 to 0 during the transition from hadronic to quark temperature regions for eB = 0 case.Corresponding constituent quark mass will vary from M f (T = 0, eB = 0) to m f (T, eB = 0) and owing to effective QCD model, we can build a connecting relation between M f (T, eB) and LQCD data q q f (T, eB) as Using the LQCD data of q q f (T, eB) from Ref. [12] and using those in Eq. ( 14 With this modified Lattice QCD inspired dispersion relation we have evaluated the Debye screening mass as where we have considered g s (T ) -temperature dependent one loop running coupling [62]:

C. Thermal Width and Binding Energy
Next, we focus on other relevant quantities like thermal width or dissociation rate, binding energy in the context of heavy quark potential.Their working formulas are described below.
Let us first discuss about thermal width or dissociation rate Γ, which can be formulated from the imaginary part of potential Im V (r, T, B), discussed in Sec.(II A).In fact, the quantity Im V (r, T, B) provide more detailed structure of dissociation rate along r, T and B axes.Now, if we know the r-profile of quarkonia wave function Ψ(r), then the thermal width Γ can be computed as [45] Γ(T, B) where imaginary part of the potential is folded by the unperturbed (1S) Coulomb wave function, owing to the first-order perturbation theory.Here we take Ψ(r) as the Coulombic wave function for ground state (1s,corresponding to n = 1 (J/ψ and Υ)) given as where, r B =2/(α s m Q ) is the Bohr radius of the quarkonia system.By solving Eq.17, we have [45] Γ(T, Here also, one can see that the T and B dependent profile of Γ is coming through m D (T, B).
Following Ref. [41], we can consider simple Culombic type real part of heavy quark potential and then by solving its corresponding Schrödinger equation, we can get the eigenvalues for charmonium and bottomonium system as where m Q is the mass of the heavy quark c and b respectively.Now its ground state eigen value for n = 1 can be considered as binding energies of lowest possible charmonium and bottomonium states i.e.J/ψ and Υ respectively.Similar to thermal width Γ(T, B), binding energy E n=1 (T, B) of J/ψ and Υ will carry magnetic field dependent information through Debye mass m D (T, B).

III. RESULTS
In this section we will show and discuss about our results corresponding to heavy quark potential in a magnetized medium incorporating IMC based quark condensate.In the present work we have used N c = 3, N f = 2 and Λ MS = 0.176 GeV [63], the string tension, σ = 0.184 GeV 2 [65] and the value of α s from Equation 16.
In the formalism part, we already discussed about the Fig. 1, which represent graphically T and eB profile of constituent quark mass, based on LQCD quark condensate data.So, we  15), then we can identify the T and eB dependent mathematical components.Sources of T are FD distribution function f l q (T ) and coupling constant g s (T ), which provide increasing and decreasing T profile.Hence, due to their collective role, we notice that m D (T ) decreases first and then increase.On the other hand, we can notice a straight forward m D ∝ eB relation in Eq. ( 15).However, right panel of Fig. (2) shows a deviation from the proportional relation, which is because a non-trivial eB dependency is entering through the eB dependent constituent quark mass M (eB), located in FD distribution function.Also we notice that the effect of magnetic field is stronger at a lower temperature and becomes weaker at a higher temperature.Now, if we compare with earlier m D (T, B) calculations, done by Refs.[33,34,[53][54][55][56][57][58], then one can find that the new aspect of present work is the adoption of IMC based constituent quark mass.Though quark condensate and constituent quark mass are the good quantities, where we can see the IMC effect near transition temperature, but for the quantity like Debye mass m D (T, B), it is not zoomed in as it becomes fade due to integration of thermal distribution.For thermodynamical quantities like pressure, energy density or transport coefficients like electrical conductivity [64], we get similar fact.So we will continue to present our final results as a IMC based outcome instead of zooming the actual modification, which Within the T = 0.080-0.160GeV, a mild suppression of Debye mass is noticed due to considering the IMC based constituent quark mass M f (T, B) in place of current quark This suppression of Debye mass is connected with the magnetized quark condensate in non-perturbative QCD (non-pQCD) domain, obtained from LQCD calculation [12,13], which reveals IMC effect near transition temperature.So, this effect will propagate to other quantities of heavy quark phenomenology, as discussed next.
We had plotted the variation of the real part of the potential with the separation distance (r) between the Q Q pair for different values of magnetic field (eB= 0.2 GeV 2 and 0.4 GeV 2 ) at T = 100 MeV (left) and T = 200 MeV (right) in Figure 3 and compared it with the absence of magnetic field i.e eB = 0. From the figure 3 we find that the screening is increasing with the increase in magnetic field and temperature.One can notice that the exponential decay with distance become more at a higher temperature (T = 200 MeV in right panel) as compared to lower temperature (T = 100 MeV in left panel).By shifting zero to non-zero magnetic field and increasing its values, similar dominancy in exponential behavior is noticed.It is because the exponential decay term is controlled by Debye mass, which increases with T as well as eB.So increasing of screening with T and eB indicates loosely bound of quarkonium state at high T and eB.
Similar to the real-part of the potential we have plotted the imaginary part of the potential with the separation distance (r) for different values of magnetic field (eB= 0.2 GeV 2 and 0.4 GeV 2 ) at T = 100 MeV (left) and T = 200 MeV (right) in Fig. 4 and compared it with the absence of magnetic field i.e eB = 0.As we can see from the figure that magnitude of the imaginary part of the potential increases with the increase in magnetic field and hence it provides more contribution to the thermal width obtained from the imaginary part of the potential.We also find that the magnitude of imaginary part of potential is more at a higher temperature (T = 200 MeV) as compared to a lower temperature (T = 100 MeV) for a given eB and r.This observation is again coming from the fact that Debye mass which is a function of temperature and magnetic field is found to be increased with temperature and magnetic field.So, grossly we notice that real and imaginary part of heavy quark potential are modified with T and eB due to non-trivial profile of LQCD based condensate and their modifications approach towards more screening and dissociation with increasing of T and B. It indicates a possibility of low dissociation temperature due to magnetic field, which we will explicitely see latter.
Next, let us integrate the r dependence of imaginary and real potential through Coulombtype probability distribution function and proceed to compute thermal width or dissociation probability Γ(T, B) by using Eq. ( 18) and binding energy using the Eq.(19).Imaginary and real part of heavy quark potential are their respective sources, whose T and eB profiles are mainly coming from m D (T, eB).Let us first discuss about binding energy (BE), plotted in Fig. 5. Its left panel shows the binding energies of J/ψ as a function of T for various values of magnetic field (eB=0.2GeV 2 and 0.4 GeV 2 ) and compared it with the results, considering current quark mass (red line).We observe that binding energy is decreasing as the temperature and magnetic field both are increasing.In the right panel of Fig. 5 we have plotted the binding energies of Υ as a function of T for various values of magnetic field (eB=0.2GeV 2 and 0.4 GeV 2 ) and compared it with without magnetic field (eB = 0).The similar behaviour has been observed for Υ also, except that the value of binding energy for J/Ψ is lower as compared to the value for Υ, which is due to their mass difference.We have taken (charmonium) J/ψ and (bottomonium) Υ masses as m c = 1.5 GeV and m b = 4.5 GeV respectively.
Next, let us come to the estimation of thermal width or dissociation probability, given in Eq. ( 18), which is obtained by substituting Eq. (( 8)) in Eq. (( 17)).So Eq. (( 8)) represents a detailed tomography of dissociations of quarkonia state, while Eq.(( 17)) present its integrated values.Now this heavy quark bound states, called quarkonia, can be dissociate completely, when thermal width will be greater than twice of the the binding energy i.e.Γ ≥ 2BE [66].Now, if one plots twice the binding energy (2BE) along with the thermal width (Γ) against T -axis, then their intersection point can be considered as dissociation temperature (T d ).We have done this in fig.6 and in fig.7 for ground state of charmonium and bottomonium respectively with different values of magnetic field.
If we analyse the fig.6 and fig.7 we can find that the thermal width is increasing with the increase in magnetic field.Also we can observe that the width for the J/ψ is much  as magnetic field increases.This fact is exactly similar with the reduction of quark-hadron phase transition temperature due to magnetic field, which is connected with IMC aspect of QCD.If QCD follow MC near transition temperature, then magnetic field will push the location of transition temperature towards the higher values.However, reduction of dissociation temperature with magnetic field can not be linked with IMC as constituent quark mass has very mild impact on Debye mass and heavy quark phenomenology.In other word, the fact of reduction of dissociation temperature with magnetic field remain same for both MC and IMC.This conclusion can be established from Figs. ( 6) and ( 7), where we notice that T d remain almost same for IMC-based constituent quark mass (black line) and massless quark mass case (red line).

IV. CONCLUSIONS
We have revisited the medium modified heavy quark potential at finite magnetic field.This is done by obtaining the real and imaginary parts of the resummed gluon propagator, which in turn gives the real and imaginary parts of the dielectric permittivity.A temperature and magnetic field dependent Debye mass, capturing the information of inverse magnetic catalysis, is entering into the gluon propagator.Now the real and imaginary parts of the dielectric permittivity will be used to evaluate the real and imaginary parts of the complex heavy quark potential.We notice that the Debye screening mass increases with temperature and magnetic field, so Debye exponential part of real part of potential become more dominant, which interpret more screening and favoring the dissociation process.With respect to earlier works, present work has incorporated two new ingredients -inverse magnetic catalysis information and all Landau level summations.
The real part of the potential is used in the time-independent Schrödinger equation for the radial wave function to obtain the binding energy of heavy quarkonia, whereas the imaginary part is used to calculate the thermal width.We observe that the binding energies of J/ψ and Υ are decreasing and their thermal widths are increasing as the temperature and magnetic field both are increasing.These decreasing of binding energy and increasing of thermal width of heavy quarkonia will push its dissociation probability.By plotting the twice of binding energy along with the thermal width, one can obtain the dissociation temperature as a point of their intersection.We noticed that dissociation temperature of heavy quarkonia can be reduced due to magnetic field.

FIG. 1 :
FIG. 1: Variation of the constituent quark masses (M u and M d ) with temperature for different values of magnetic field eB = 0.2 (solid line), 0.4 (dashed line), 0.6 (dotted line) GeV 2 in left panel and with magnetic field for different values of T = 100 (solid line), 150 (dash line) MeV in right panel.Black and Red curves denotes respectively u and d quarks.
), we have plotted constituent quark mass M f =u,d against T -axis (left panel) and eB-axis (right panel) in Fig. (1).Here, reader can notice that constitute quark mass follow MC effect in low temperature domain and IMC effect near transition temperature domain.Enhancement of constituent quark mass by increasing values of eB in low T -axis is noticed in left panel of Fig. (1) as well as in its right panel, we notice an increasing constituent mass curve with eB-axis at T = 100 MeV.This enhancement of constituent quark mass with magnetic field in low temperature domain is proportionally mapping the MC effect of quark condensate.On the other hand, reduction of constituent quark mass by increasing values of eB near transition temperature is noticed in left panel of Fig.(1) as well as in its right panel, we notice that constituent mass curve at T = 150 MeV first increases then decreases with eB.This reduction of constituent quark mass with magnetic field near quark-hadron phase transition temperature domain is proportionally mapping the IMC effect of quark condensate.

FIG. 2 :
FIG. 2: Variation of the Debye mass (m D ) with temperature for different values of magnetic fields eB = 0.2 (black solid line), 0.4 (dash line), 0.6 (dotted line) GeV 2 in left panel and with magnetic field for different values of temperatures T = 100 (black solid line), 200 (dash line) MeV in right panel.Red solid line in left panel denotes for massless quark case.

FIG. 3 :
FIG. 3: Variation of the real part of potential with separation distance r between Q Q for three different values of the magnetic field and with fixed temperature T = 100 MeV (left) and T = 200 MeV (right).

FIG. 4 :
FIG. 4: Variation of the imaginary part of potential with separation distance r between Q Q for various values of magnetic field T = 100 MeV (left) and T = 200 MeV (right).
FIG. 7: Variation of Γ, 2BE(E b ) with Temperature with different values of magnetic field(eB) for Υ larger than the Υ, because charmonium states are larger in size and smaller in masses as compared to bottomonium states which are smaller in size and larger in masses and hence will get dissociated at higher temperatures.Interestingly, we see that Γ increases and BE decreases with magnetic field, which results in the early dissociation of quarkonium states.Our findings of the dissociation temperatures from intersection points in graphs are enlisted in table I.We notice that dissociation temperature of quarkonia states decreases

TABLE I :
The dissociation temperature(T D ) for the quarkonia states (in units of MeV), when Γ= 2BE