Quark and gluon distribution functions in a viscous quark-gluon plasma medium and dilepton production via q\bar{q}– annihilation

Quark and gluon distribution functions in a viscous quark-gluon plasma medium and dilepton production via – annihilation

Vinod Chandra vchandra@iitgn.ac.in, $^b$: The work was started at the Department of Physics, IISER Bhopal, India    V. Sreekanth sreekanth@cts.iisc.ernet.in Indian Institute of Technology Gandhinagar, 382424, Gujarat, India Centre for High Energy Physics, Indian Institute of Science, Bangalore 560012, India
July 15, 2019

Viscous modifications to the thermal distributions of quark-antiquarks and gluons have been studied in a quasi-particle description of the quark-gluon-plasma medium created in relativistic heavy-ion collision experiments. The model is described in terms of quasi-partons that encode the hot QCD medium effects in their respective effective fugacities. Both shear and bulk viscosities have been taken in to account in the analysis and the modifications to thermal distributions have been obtained by modifying the the energy momentum tensor in view of the non-trivial dispersion relations for the gluons and quarks. The interactions encoded in the equation of state induce significant modifications to the thermal distributions. As an implication, dilepton production rate in the annihilation process has been investigated. The equation of state is found to have significant impact on the dilepton production rate along with the viscosities.

PACS: 25.75.-q; 24.85.+p; 05.20.Dd; 12.38.Mh

Keywords: Viscous quark-gluon plasma; Quasi-particle model; Dilepton production; Equation of state

I Introduction

There are strong indications from relativistic heavy-ion collider experiments (RHIC) at BNL concerning the creation of strongly coupled quark-gluon-plasma(QGP) expt () that possess near perfectly fluidity. These observations on the QGP are mainly corroborated by two of the most striking finding of the RHIC, viz., the large elliptic flow shown by QGP, and the large jet quenching expt () at RHIC. The former led to the near perfect fluid picture and latter indicated towards the strongly coupled picture of the QGP. Preliminary results from heavy-ion collisions at the LHC alice (); alice1 () reconfirm similar picture of the QGP. There are interesting possibilities for observing the other higher order flow parameters (dipolar and triangular etc.) at LHC, that are crucial for the quantitative understanding of collectivity and the viscous coefficients of the QGP higher_flow1 (); higher_flow2 ().

The strongly coupled picture of the QGP is seen to be consistent with the lattice simulations of the QCD equation of state (EoS) leos_lat (); cheng (); leos1_lat (). The EoS is an important quantity that plays crucial role in deciding the bulk and transport properties of the QGP. Therefore, it need to be implemented in an appropriate way as a model for the equilibrium state of the QGP while investigating its properties within the framework of semi-classical transport theory. Furthermore, the form of local thermal distribution functions that describe the hydrodynamic expansion of the QGP liquid must contain the effect from the realistic EoS. This sets the motivation for the present investigations. For the temperatures higher than the QCD transition temperature, , this issue can be addressed by adopting the quasi-particle approaches quasi_eff (); pesh (); polya (); casto (). The way to couple the non-trivial dispersion (single particle energy) for the effective degrees of freedom of the QGP in those quasi-particle approaches to the transport theory, is the modification in the definition of the  dusling (); chandra_2012 ().

On the other hand, transport coefficient of the QGP (shear viscosity, and bulk viscosity ) are essential to understand and characterizes its liquid state, and the hydrodynamic evolution in heavy-ion collisions. A tiny value of (: entropy density) can be associated with the near perfect fluid picture and the strongly coupled nature of the QGP provided that the is relatively smaller. Theoretical investigations suggest that this is true for the temperatures not very close to where bulk viscosity is large kharzeev (); chandra_bulk (); chandra_2012 (). Several phenomenological and theoretical investigations do suggest that the QGP indeed possess a very tiny value of the  shrvis (); bmuller (); chandra_eta ().

Moreover, in certain situations, the temperature behavior of the and/or lead to cavitation and it may cause the hydrodynamic evolution of the QGP to stop before the freeze out is actually reached rajagopal (); Bhatt:2010cy (); Bhatt:2011kr (); sinha (). Both the bulk and shear viscosities play vital role in deciding the observed properties of final state hadrons in the RHIC heinz_visco (). Furthermore, these transport coefficients have significant impact on the important phenomena such as heavy quark transport  hq (), photon and dilepton production in heavy-ion collisions  Bhatt:2009zg (); Bhatt:2010cy (); Bhatt:2011kx (); dusling1 (); visco_dilep (); recent_lep (); Bhalerao:2013aha (). All these investigations calls for an appropriate modeling of viscous modified thermal distribution functions of quarks and gluons in the QGP medium. Importantly, such modifications naturally encode hot QCD medium effects through the QGP EoS (described in terms of the quasi-particle approaches at high temperature).

The present analysis is devoted to obtain the viscous modified thermal distributions for quarks and gluons in the QGP medium, within the framework of transport theory, coupling it with a recently proposed effective fugacity quasi-particle model chandra_quasi (). As an implication of these distribution functions, the dilepton production rate via annihilation process is analyzed, and significant modifications are obtained, as compared to those obtained by considering a viscous quark-gluon medium with non-modified particle distribution functions. The ideal QCD/QGP EoS refers to the system of ultra-relativistic non-interacting gas of quarks-antiquarks and gluons (the Stefan-Boltzmann limit of hot QCD).

The paper is organized as follows. Sec. II deals with a recently proposed quasi-particle description of hot QCD in terms of effective quasi-parton distribution functions along with how it modifies the kinetic theory definition of the energy momentum tensor. Furthermore, the modifications to the thermal distributions of the quasi-particles (quasi-gluons, and quasi-quarks) in the presence of dissipation that is induced by shear and bulk viscosity of the QGP, are obtained by coupling the kinetic theory with the hydrodynamic description of the QGP. In Sec. III, dilepton production rate is investigated employing these viscous modified thermal distribution functions, and interesting observation are discussed. Sec. IV articulates the conclusions and future directions.

Ii Viscous modification to quark and gluon thermal distribution functions

The determination of transport properties of any fluid is subject to the matter of moving away from equilibrium followed by adopting either the transport theory approach or equivalently the field theory approach utilizing the well known Green-Kubo formulae green-kubo (). Once these transport coefficients such as shear and bulk viscosities are known, it is pertinent to ask what kind of modifications are induced to the momentum distributions of the fluid degrees of freedom

Now, to obtain the modified distribution function of quarks and gluons which describe the viscous QGP, firstly we need an appropriate modeling of the equilibrium state of the QGP in terms of its degrees of freedom. To that end, we employ a recently proposed quasi-particle description of the QGP chandra_quasi () as a model for its equilibrium state. This is followed by the linear perturbation induced in terms of shear and bulk viscous effects adopting the quadratic ansatz dusling () (quadratic in terms of momentum dependence). To obtain, viscous corrections to the momentum distribution of quarks-antiquarks and gluon that constitute the QGP, kinetic theory expression for the energy momentum tensor, needs to be equated with its hydrodynamic decomposition in the presence of viscosities. Let us first briefly review the quasi-particle model followed by the obtained from this model.

ii.1 The quasi-particle description of hot QCD

Let us now discuss the quasi-particle understanding of hot QCD medium effects employed in the present analysis, recently proposed by Chandra and Ravishankar chandra_quasi (). This description has been developed in the context of the recent (2+1)-lattice QCD equation of state (lQCD EoS) cheng () at physical quark masses. There are more recent lattice results with the improved actions and more refined lattices leos1_lat (), for which we need to re-visit the model with specific set of lattice data specially to define the effective gluonic degrees of freedom. This is beyond the scope of the present analysis. Henceforth, we will stick with the one set of lattice data utilized in the model chandra_quasi ().

The model initiates with an ansatz that the lQCD EoS can be interpreted in terms of non-interacting quasi-partons having effective fugacities, , which encode all the interaction effects, where denotes the effective gluon fugacity, and , denotes the effective quark-fugacity respectively chandra_quasi (). In this approach, the hot QCD medium is divided in to two sectors, viz., the effective gluonic sector, and the matter sector (light quark sector, and strange quark sector). The former refers to the contribution of gluonic action to the pressure which also involves contributions from the internal fermion lines. On the other hand, latter involve interactions among quark, anti-quarks, as well as their interactions with gluons. The ansatz can be translated to the form of the equilibrium distribution functions, (this notation will be useful later while writing the transport equation in both the sector in compact notations) as follows,


where for gluons and light quarks, and for strange quarks ( denotes the mass of the strange quark) and in the natural units. The minus sign is for gluons and plus sign is for quark-antiquarks. The quarks and antiquarks possess the same distribution functions since we are working at the zero baryon chemical potential.The determination of achieved by fixing the temperature dependence of the effective fugacities and from the QGP EoS which in our case is the lQCD EoS (for details we refer the reader to chandra_quasi ()). Effective fugacity in our quasi-particle model should not be confused with the presence of any chemical potential. It does not indicate the presence of any conserved quantity in the medium. Its physical significance is described below.

It is worth emphasizing that the effective fugacity is not merely a temperature dependent parameter which encodes the hot QCD medium effects. It is very interesting and physically significant, and can be understood in terms of effective number density of quasi-particles in hot QCD medium, and equivalently in terms of an effective Virial expansion chandra_quasi (). Interestingly, its physical significance reflects in the modified dispersion relation both in the gluonic and matter sector by looking at the thermodynamic relation of energy density . One thus finds that the effective fugacities modify the single quasi-parton energy as follows,


This leads to the new energy dispersions for gluons (), light-quarks/antiquarks () and strange quark-antiquarks, (). These dispersion relations can be explicated as follows. The second term in the right-hand side of Eq. (II.1), is like the gap in the energy-spectrum due to the presence of quasi-particle excitations. This makes the model more in the spirit of the Landau’s theory of Fermi -liquids. A detailed discussions regarding the interpretation and physical significance of , and is discussed at a length in chandra_quasi (); chandra_eta (). Note, that the quasi-particle model is reliable for the temperatures that are higher than , hence in this situation, the effects induced by the strange quark mass can be neglected (strange quark mass is around in EoS and the model is reliable beyond ). In this case, we can describe the hot QCD EoS as a system with the effective gluons () and effective quark-antiquarks () as the degrees of freedom. The effective fugacity model has further been employed to study the anisotropic hot QCD matter and quarkonia dissociation chandra_aniso () and to study the heavy-quark drag/diffusion coefficients in the QGP medium hq (), leading to significant impact of the realistic QGP equation of sate on both these important phenomena.

Note that there are other quasi-particle descriptions of hot QCD medium effects, viz., the effective mass models pesh (), effective mass models with gluon condensate casto (), quasi-particle models with Polyakov loop polya (), along with our effective fugacity model. Our model is fundamentally distinct from these models and the differences are discussed at a length in chandra_quasi (). The major difference between our model and the effective mass models is in their philosophy itself. The effective fugacities in our model are not the effective masses. However, these can be interpreted as effective mass in some limiting case (chandra_quasi (). Another substantive difference, between the two approaches can be seen in terms of group velocity, which is not the same in two approaches In the effective mass approaches the group velocity of quasi-particles depends upon the thermal mass parameter . In contrast, in our model the description does not touch the . One can alternatively interpret the effective fugacities in terms of effective mass, (: effective coupling) as, . The effective coupling, comes out to be less than unity for (both in gluonic and quark sector) chandra_quasi ().

ii.2 Modification to the thermal distributions

Shear and bulk viscosities are essential to understand space-time evolution of the QGP during its hydrodynamic expansion. Physically, shear viscosity accounts for the entropy production during the anisotropic expansion of the system maintaining its volume constant, on the other hand bulk viscosity accounts for the entropy production while the volume of the system changes at constant rate (isotropic expansion). Since these transport coefficients are related to the non-equilibrium properties of the fluid, this requires to go beyond the equilibrium modeling of the fluid within linear response theory.

The general linear response (Chapman-Enskog) formalism assumes a small perturbation of the thermal equilibrium distribution (considering the small perturbation around the equilibrium distributions of the quark-antiquarks and gluons) as:




denote the local thermal equilibrium distribution function in Eq. (5) in the absence of viscous effects. The quantity, is the linear perturbation which encodes the viscous effects as described below. Here, stands for quasi-gluons and for quasi-quarks (we have also neglected the mass of the strange quark which is justified at high temperature), is the 4-velocity of the fluid and . The isotropic distribution, reduced to in the local rest frame of the fluid (LRF).

Now, using , the linear perturbation can be expressed as bmuller ():


Here, plus is for gluons and minus for the quark-antiquarks. The perturbation (combined notation for quarks and gluons) can be thought of as a change in the argument of as ( bmuller (), and can be thought of as a local fugacity factor leading to following form of the near-equilibrium distributions:


Note that Eq. (5) is obtained by expanding Eq. (II.2) and keeping only the linear term in the perturbation, . Next, we discuss the energy-momentum tensor for the QGP fluid obtained from these distribution functions that is essential for determining the form of in terms of shear and the bulk viscosities.

ii.2.1 Energy-momentum tensor

To obtain viscous modifications to the quark-antiquarks and gluon distribution functions, we need to couple the fluid dynamic description of the QGP to the kinetic theory description. At the point of freeze-out in heavy-ion collisions, the fluid dynamic description of QGP should smoothly change to the particle description (hardonization). This is understood in terms of the matching of energy momentum tensor, in these two descriptions. The in the QGP phase is the energy-stress tensor encoding shear and bulk viscous effects, should match with the kinetic theory expression (in terms of ) at freeze-out. To achieve the continuity of of the , in our case, the kinetic theory definition of needs to be revised in a way that it must capture the hot QCD medium effects in terms of non-trivial dispersion relations and the effective fugacities. We further have to satisfy the the Landau-Lifshitz (LL) conditions landau () discussed below.

The energy density and the pressure can be obtained in terms of quasi-gluons and quasi-quarks in our quasi-particle model chandra_quasi () as,


We use the notation for gluonic degrees of freedom , ( in the present case).

In kinetic theory is obtained from the single particle momentum distributions as,


It is emphasized in chandra_2012 (), the above expression of can not simply be utilized in the present case, since, it does not capture the non-trivial dispersions of quasi-particles. In other words, the thermodynamic consistency condition is not satisfied with this expression of yielding incorrect expressions for energy density and the pressure.

This issue has recently been addressed in chandra_2012 () by arguing for a modified form of the , in a similar spirit as it is done in the effective mass quasi-particle models dusling ():


One can clearly realize the presence of the factors, and , in the expression for in Eq. (9) that are the part of the modified dispersions. The second term in the right-hand side of Eq. (9) ensures the correct expression for the pressure, and the third term ensures the correct expression for the energy density, and hence the definition of incorporates the thermodynamic consistency condition correctly. In view of the reliability of the quasi-particle descriptions of hot QCD for temperature beyond the QCD transition temperature, we may ignore the strange quark mass effects. In this case the QGP can be described by massless quasi-gluons, and massless quasi-quarks having non-trivial dispersion relations. Therefore, in Eq. (9), , and summation is over the gluons and quarks. On the other hand, the effective mass quasi-particle models, include the terms containing temperature derivative of the effective mass in the modified  dusling (). It is to be noted that the effective mass models are fundamentally different from the effective fugacity model chandra_2012 () employed here. The differences can be understood in terms of the modified dispersions in the two cases. The major difference can be realized in terms of non-changing particle velocities in the effective fugacity model, in contrast, to the effective mass models. Moreover, the effective fugacity model can be understood in term of charge renormalization in the hot QCD, on the other hand the effective mass quasi-particle models are motivated by mass renormalization in the hot QCD medium.

To realize the LL conditions ( and ), we can resolve the as


From Eq. (9),


which gets the modifications from the EoS leads to right expressions for the energy density and pressure following the LL condition. The form of , here is based on the quadratic ansatz (see Eq.(13)) which also follow the LL condition (since , , ).

On the other hand, the fluid dynamic definition of in the presence of shear and bulk viscous effects is given as,


where , and are the shear and bulk part of the viscous stress tensor.

The form of the perturbations to the thermal distributions of gluons and quarks can be obtained in terms of the and by relating the two definitions (kinetic theory and fluid dynamic) of the . The two definitions can be matched through the following quadratic ansatz for  dusling (),


where the coefficients and are obtained by the matching of the two definitions of in the local rest frame of the fluid (LRF). This follows from the fact that shear and bulk viscosities are Lorentz invariant quantities and can conveniently be obtained in the LRF of the fluid. The factor is introduced for convenience, since for the QGP in RHIC, we consider viscosities scaled with entropy density (). While matching hydrodynamic and kinetic theory descriptions one should ensure the Landau-Lifshitz matching conditions. The modified form of the in Eq.(12) ensures LL-matching conditions in the temperature range where our quasi-particle model is valid chandra_2012 ().

Next, utilizing the notations in Eq. (5), and matching right-hand sides of Eq. (9) and Eq. (13) in the LRF, we obtain,


Here, and are contracted and summed over. Note that shear and bulk viscous part of the stress in gluonic and quark-sector are distinguished by their respective transport coefficients (their total value is obtained by adding up appropriately the gluonic and quark contributions). The fluid velocity fields for quark-antiquark and gluonic degree of freedom are assumed to be same in the QGP fluid. The integral over the momentum in the above equations can be expressed as in dusling (): (the subscripts and are used to distinguish the gluonic and the matter sector), where


Now, from Eq. (II.2.1) in the gluonic sector,


and in the matter sector,


The viscous modified thermal distributions of gluons and quarks in the QGP in terms of ,


As mentioned earlier, .

Let us discuss the validity of the above expression of the viscous modified thermal distributions. The validity criterion is simply (near equilibrium condition). In other words, for the validity of our formalism, the viscous corrections ( and ) must induce small corrections to the equilibrium distribution of the gluons and quarks. This translates to the condition,


Next, we consider a case, where the integral displayed in Eq. (II.2.1) can be solved analytically. In the limit (high temperature limit), we can obtain analytic expressions for and as,


The function appearing in Eq. (II.2.1) is having the series representation, (convergence of the series is subject to the condition that ). The Stefan-Boltzmann (SB) limit (employment of ideal QGP EoS) is obtained only asymptotically (by putting ) in right-hand side of Eq. (II.2.1). It can easily be seen that and are of the order of unity in the case of ideal EoS. This is also realized in dusling (). To see the difference in these two cases, we plot the quantities and , defined as:


for the ideal QGP EoS, and lQCD EoS (temperature dependence of and are taken from Ref. chandra_quasi ()) in Fig. 1. Here, we use the identities , and to obtain and in the case of ideal EoS.

Figure 1: Behavior of , and as function of . The temperature dependence of the and are taken from Ref. chandra_quasi (). Clearly the modifications induced by the EoS are quite significant even at higher temperatures as compared to the ideal QGP EoS.

Clearly and , will approach to their SB limit that is asymptotically. The interaction effects are significant even at . Therefore, one can not simply ignore these effects while obtaining the viscous modified forms of the thermal distributions of gluons and quarks in the QGP medium. This crucial observation has been realized in the case of effective mass quasi-particle model in dusling (). Next, we shall investigate the significance of such viscous modified thermal distributions of gluons and quarks in the context of dilepton production.

Iii Effects of the EoS and viscosities on dilepton production via annihilation

The dilepton production in the QGP medium has dominant contributions from the annihilation process via the mechanism, . The kinetic theory expression for the dilepton production rate for a given dilepton mass and momentum is given by Vogt (),


where the 4-momenta are of quark and anti-quark respectively with , if one neglects the quark masses. The quantity is the invariant mass of the intermediate virtual photon. Here, is the degeneracy factor, and is the thermal dilepton production cross section. Here, is the 4- momentum of the dileptons.

The quantity is the quark (anti-quark) distribution function in thermal equilibrium, (this form is in view of the effective quasi-particle model based on realistic QGP EoS. In the case of ideal QGP EoS the factor will be replaced by unity, as done in most of works on dilepton production in the QGP medium in the literature. As we shall argue that the EoS effects are quite significant even if we take high temperature limit of quark (anti-quark) distribution functions. Recall from the previous section that the realistic EoS strongly influence the viscous modified portion of the thermal distributions of gluons, and quarks (anti-quarks). In the present analysis we are interested in the invariant masses that are larger compared to the temperature, . In this limit, we can take the high temperature limit of quark (antiquark) equilibrium thermal distribution functions (replacing Fermi-Dirac distribution with classical Maxwell-Boltzmann distribution in the case of Ideal QGP EoS) as


where . The form will remain the same for the quarks and antiquarks since the baryon chemical potential is zero here. It is straightforward to observe from Eq. (22) that the effects coming from the EoS are of the order (this quantity is quite significant even at ). In other words, the dilepton production rate is modulated by a factor . Let us now proceed to explore the impact of EoS and the viscous modifications to the dilepton production rate.

Next, we employ the result obtained in Eq. (18) for the viscous modified quark (antiquark) distribution function , and take its high temperature limit, and analyze shear and bulk viscous contributions one by one. In this limit, the viscous modified quark (anti-quark) distribution functions become,

Note that the first term in the above equation accounts for the equilibrium part of the quark (anti-quark) thermal distribution, the second encodes the shear viscous effects, and the third one encodes the bulk viscous effects.

Now, the effects of viscosities on the production rate of dileptons, we employ Eq. (III) to Eq. (22), and rewrite the dilepton production rate in the component form as,


The notations and are introduced since , involve them as the first order transport coefficient in their definitions. These three terms in Eq. (25) have already been computed for the Ideal QGP EoS in Bhatt:2011kx (), and straight-forward to compute in our case (difference are there in the definition of the distribution functions). The first term is given by the following integral,


This integral is well known in the literature Vogt () in the case of . Since is independent of the of the momentum of the particles, so the integral can be evaluated in the same way as Vogt (),


The modification to rate due to the shear viscosity (at first order , where is the Navier-Stokes tensor (, , ) can be obtained from the following equation,


Following the analysis of dusling1 (), we obtain the following expression for the shear viscous correction of the rate,


Now, the third term which is the correction to the rate due to the bulk viscosity (at first order, , where is the expansion rate of the fluid) can be evaluated from the following expression,


This integral can be evaluated using the analysis of sree_33 () as,

The full expression for the rate displayed in Eq. (26) can be obtained by combining Eq. (27-III). These expressions reduces to the those obtained in Bhatt:2011kx () (the expressions obtained by employing the ideal QCD EoS) by substituting (in this case as already described in dusling ()).

Figure 2: Behavior of as function of is shown along with its SB limit (). The temperature dependence of the effective quark fugacity, is taken from Ref. chandra_quasi ().

If we ignore the viscous corrections, it is obvious that the EoS induced modifications appear as a factor, . On the other hand, the shear and bulk viscous corrections to dilepton production rate gets a factor of (whose SB limit is ), as a modification from the EoS. We have plotted both of these factors, employing the quasi-particle model for (2+1)-flavor QCD chandra_quasi () in Fig. 2 and Fig. 3. On looking at the temperature behavior of both these factors, we can safely say that all the three terms in the dilepton rate in Eqs. (27), (29), and (III) get significant modifications from the QGP EoS. From Figs. 2 and 3, both , and approach their respective SB limit only asymptotically.

Figure 3: Behavior of (modification factor to the viscous contribution to the dilepton rate induced by the realistic EoS) as function of is shown along with its SB limit (). The temperature dependence of the effective quark fugacity, is taken from Ref. chandra_quasi (). Here, we assume that and are the phenomenological parameters for the QGP, and assumed to be same for realistic and ideal QGP EoSs.

Let us discuss the interesting observations that can be made out, based on the results of the dilepton production rate obtained in the viscous environment and the realistic EoS. Since the quantities, and are the phenomenological numbers, hence they can safely assumed to be same in case of the ideal and the realistic equations of state. Therefore, the role of the viscous corrections in both the cases will be qualitatively similar. However, the quantitative differences are mainly induced by the EoS. The realization that the dilepton production is sensitive to the EoS has been also been seen in the recent work of Deng et. al deng ().

Finally, the EoS induces significant modifications to the viscous modified thermal distribution functions. These modifications play significant role in the dilepton production rate in the RHIC. The rate is suppressed significantly as compared to that obtained by employing the ideal EoS (the modifications are of the order of in the absence of the viscosities). Let us now proceed to the quantitative understanding of these effects from the realistic EoS and the viscosities.

We choose the value of (the number from the from AdS-CFT (KSS bound) kovtun ()). On the other hand, the bulk viscosity of QGP can taken according to the studies from strongly interacting gauge theories kanti () as,


Here, is the speed of the sound square. The temperature dependence of is dictated by and the speed of sound in the QGP phase.

Iv Dilepton spectra from heavy-ion collisions

We now study the effect of modified gluon and quark, antiquark distribution functions and viscosity on thermal dilepton spectra produced from the QGP in the heavy-ion collision experiments. Evolution of the fireball is modeled using relativistic hydrodynamics. In this qualitative study, we use one dimensional boost-invariant scaling flow to analyze the system Bjorken:1982qr (). We choose the parametrization cosh and sinh for the coordinates, with the proper time and space-time rapidity . With fluid four-velocity expressed as , we can write down the equation governing the longitudinal expansion of the plasma as


where we have neglected the effect of viscosity on the expansion as the significant contribution to particle production comes from the viscous modified rates Bhatt:2011kx (). In order to close the system we use the lQCD EoS cheng (). With critical temperature MeV, we take the initial conditions relevant to the RHIC energies: fm/c and MeV in our calculation. By numerically solving the energy equation Eq.(33), we obtain the temperature profile . We note that for fm/c system reaches .

Once we obtain the temperature profile, particle spectra can be calculated by integrating the viscous modified particle production rates over the space-time history of the collisions


Where within the Bjorken model, volume element is given by , with representing the radius of the nucleus used for the collision (for ). Here and are the initial and final values of proper time that we are interested (i.e. duration of the QGP phase in present analysis). The production rates calculated in Section III need to be modified while considering a longitudinally expanding system. This is done by replacing of Eq.(23) with in rate expressions of Eq.(25). With four-momentum of the dilepton parametrized as = , where = , we get . The other factors appearing in the rate Eqs.(29) and (III) are now given as


We use temperature dependent and constant value of as prescribed in the previous section for our calculations. All results are presented for case only.

Figure 4: Thermal dilepton yield from viscous quark-gluon plasma at RHIC energies for different dilepton invariant masses using the modified distribution functions. Dotted curves represent the rates calculated ignoring the EoS modifications on the thermal distribution functions.

We plot the transverse momentum spectra of thermal dileptons produced from the viscous QGP in Fig. 4 for invariant masses GeV. In order to understand the impact of the EoS effects to the dilepton production through the modified distribution functions, we compare the results in a case where such effects are ignored (). Note that in both these situations, we choose the lQCD EoS for the purpose of hydrodynamical evolution of the temperature. Solid lines represent the rate while considering the modified distribution functions and dotted lines while ignoring such effects.

We observe significant modifications to the spectra while using the lQCD EoS through the modified distribution functions for the quark-antiquarks. From the curves, it is clear that effect of these terms is to suppress the particle spectra. Significant suppression is observed for all the values of transverse momentum e.g.; at GeV, suppression is about 89 for GeV, which reduces to about 79 for GeV and about 47 for GeV.

We also observe that suppression of low particles is strong indicating that effect of the modifications is more dominant in the later stages of collision, when system is near . The high particles, produced predominantly during the early stages of the evolution of the system, are also affected by these modifications albeit less compared to the low region. This behavior can be understood by observing the behavior of the EoS induced modifications to the dilepton rate in Fig 3. Since these terms are getting multiplied with the dilepton rates employing ideal EoS, they suppress the spectra at low .

Figure 5: Transverse momentum spectra for the thermal dileptons (with invariant mass GeV) from the viscous quark-gluon plasma produced in RHIC energies. Effect of viscosities is highlighted.

Next, we look into the behavior of various dissipative terms in the dilepton production. In Fig. 5, we plot dilepton yield as a function of transverse momentum of the pair for the invariant mass GeV. In this plot we show the effect of various viscosity terms in the total rate. Firstly, we plot the spectra without considering the viscous effects in the calculations ( case). As emphasized earlier, the presence of terms in the expression, as we saw from Eq. [27], will lead to the overall suppression of the spectra. One can see that inclusion of the the effect of bulk viscosity () has only marginal effect. As observed before in Bhatt:2011kx (), at high , effect of bulk viscosity is to suppresses the spectra, for e.g.; at GeV suppression is about 7. Next, we consider only the effect of shear viscosity ( case) in the spectra. Shear viscosity significantly enhances the particle production and its effect becomes stronger as increases. This can be understood from the presence of the first (positive) term on the right hand side of Eq.(35). At GeV enhancement of spectra due to shear viscosity is around 177. Since the shear viscosity coefficient in the entire temperature regime we are interested, its effect is expected to be more dominant. As before Bhatt:2009zg (); Bhatt:2011kx (), we observe that even the lowest value of shear viscosity has significant effect on the spectra. Consequently, when we consider fully viscous case (), spectra gets highly enhanced due to shear viscosity, albeit the presence of bulk viscosity suppression. For instance, at GeV the total enhancement of the spectra is 170.

It is worth emphasizing that the main source of thermal dileptons in the QGP medium is the quark-anti-quark annihilation processes considered here. There are other higher order processes that can also contribute to the thermal dilepton production Altherr:1992th (); Thoma:1997dk (). Such higher order processes are not considered in the present analysis. It may further be noted that the thermal dileptons from the annihilation process is dominant in the regime of intermediate invariant mass GeV and transverse momentum of the pair, in that range vanHees:2007ma (); Akiba:2009es (). We intend to extend our present studies by incorporating higher order contributions in view of the quasi-particle description in the near future.

V Conclusions

In conclusion, the form of viscous modified thermal distribution functions for quasi-quarks and quasi-gluons are obtained in the QGP medium by systematically employing the realistic EoS for the QGP (the lQCD EoS). As an implication, the impact of them is demonstrated on the dilepton production via annihilation in RHIC. The EoS also induces significant modifications to the viscous modified thermal distributions of the gluons and quark-antiquarks that constitute the QGP. The effects are equally significant in deciding the dilepton production rate in the viscous QGP medium. In particular, even in the high temperature regime, where the hot QCD medium effects are weaker, the realistic EoS and viscosities play crucial role.

Finally, coupling the present analysis to the relativistic second order viscous hydrodynamic evolution of the QGP and impact of the temperature dependence of the shear and bulk viscosities on the dilepton production rate will be matters of future investigation.


VC acknowledges financial support from the Department of Science and Technology, Govt. of India [Grant:DST/IFA-13/PH-55]. We sincerely thank F. Becattini and V. Ravishankar for fruitful discussions and encouragement. We are highly indebted to the people of India for their generous support for the research in basic sciences.


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