Second order hydrodynamics based on effective kinetic theory and electromagnetic signals from QGP

The experiments at Relativistic Heavy Ion Collider (RHIC) and at Large Hadron Collider (LHC) suggest the existence of strongly coupled quark-gluon matter at extreme high temperature and density Adams et al. (2005); Adcox et al. (2005); Arsene et al. (2005); Back et al. (2005). These experiments provide opportunity to inspect the features of the hot nuclear matter, QGP which is believed to have existed in the primordial universe. Theoretical and experimental investigations of properties of the hot and dense QGP is being pursued vigorously by the community Jaiswal et al. (2021). Analysis of the experimental data from the heavy-ion collision experiments imply that the QGP has a near perfect fluid nature with extreme low value of shear viscosity to entropy ratio, $\eta/s=1/4\pi$ Kovtun et al. (2005). This surprisingly low value of viscosity has evoked interest in the application of relativistic dissipative hydrodynamics to heavy ion collisions Romatschke and Romatschke (2019).

The evolution of viscous QGP has to be modelled using higher order relativistic viscous hydrodynamical theories, since first order Navier-Stokes theory exhibits acausal behaviour and numerical instabilities Hiscock and Lindblom (1983, 1985); Geroch and Lindblom (1990). Recently, several investigations are being pursued towards the development of causal first order hydrodynamic theory Bemfica et al. (2018); Hoult and Kovtun (2020); Biswas et al. (2022). When it comes to second order theories, there is no unique prescription and there exist several successful formalisms. The earliest attempts in this direction were by Muller, Israel and Stewart MÃŒller (1967); Israel and Stewart (1979). Development of second order causal theories is an active field of research and several new formalisms have been proposed and studied in the context of heavy ion collisions Jaiswal and Roy (2016); Romatschke and Romatschke (2019). Second order viscous hydrodynamics developed in Ref. Bhadury et al. (2021) is a recently proposed formalism within the effective fugacity quasiparticle model (EQPM) for the hot QCD medium Chandra and Ravishankar (2009, 2011). The EQPM prescription incorporates the thermal QCD medium interaction effects into a system of quasipartons by considering thermal modifications to the phase-space distribution functions in terms of effective quark and gluon fugacities. The dissipative hydrodynamic evolution equations have been determined by employing the effective covariant kinetic theory developed for EQPM Mitra and Chandra (2018). The non-equilibrium corrections to the distribution functions have been estimated in this hydrodynamic framework by the Chapman-Enskog (CE) expansion in relaxation time approximation (RTA) Jaiswal (2013a, b). Moreover, a quasiparticle model such as EQPM distinguishes the quark and gluon sector in the hot QCD medium and can be used to study thermal dilepton and photon emission from QGP Chandra and Sreekanth (2015).

Properties of the QCD matter created in high energy heavy ion collisions can be studied by analysing the different signals emitted, such as thermal dileptons and photons. As these particles interact only electromagnetically, they can easily decouple from the strong nuclear matter and reach the detectors without further interaction with other particles. Effect of viscosities has consequences on the thermal dilepton and photon spectra from heavy ion collisions. The role of shear viscosity on thermal particle production using causal dissipative hydrodynamics has been investigated in Refs. Dusling and Lin (2008); Dusling (2010); Bhatt and Sreekanth (2010); Bhatt et al. (2012); Chaudhuri and Sinha (2011); Vujanovic et al. (2014, 2018). The influence of bulk viscosity on the expansion of QGP and signals emanating from it has been studied in Refs. Torrieri et al. (2008); Fries et al. (2008); Rajagopal and Tripuraneni (2010); Bhatt et al. (2010, 2012); Paquet et al. (2016). In certain situations, the inclusion of dissipation into the heavy ion collision scenario may induce a phenomenon called cavitation which causes the hydrodynamic description to be invalid before the freeze-out and thereby affecting the signals Rajagopal and Tripuraneni (2010); Bhatt et al. (2011). Recently, thermal particle production has been investigated by employing the concept of hydrodynamic attractors for temperature evolution Coquet et al. (2021); Naik et al. (2021a, b). The EQPM prescription has been employed to study several observables from heavy ion collisions. Impact of chromo-Weibel instability on thermal dilepton emission has been analysed by the authors of Ref. Chandra and Sreekanth (2017) within EQPM. The effect due to collisional contributions of thermal QCD medium has also been investigated on thermal dileptons Naik et al. (2022) and heavy quark transport Prakash et al. (2021), using EQPM. However, in these studies employing the EQPM, ideal hydrodynamics was used to model the evolution of the system.

In the present analysis, we proceed to implement the causal second order hydrodynamic framework within EQPM to study the evolution of QGP and thermal particle production from heavy ion collisions. We analyze the evolution for different temperature dependent relaxation times. The thermal particle emission rates have to be calculated by proper modelling of the momentum distribution functions incorporating the viscous effects. The non-equilibrium part of distribution function is generally determined by the 14-moment Grad’s method or the CE expansion, out of these, the CE type expansion of non-equilibrium distribution function is shown to be well behaved even up to second order in gradients Bhalerao et al. (2014). In this work, we intend to calculate thermal dilepton and photon production rates using the viscous modified quasiparticle thermal distribution functions determined by the CE method in RTA. The particle emission spectra is dependent on the temperature profile of QGP and is found by modelling the expansion of QGP with relativistic dissipative hydrodynamics under one-dimensional ($1-$D) boost invariant Björken expansion.

The paper is structured as follows. In section II, we review the EQPM description and the formalism used to derive the second order viscous hydrodynamics. The causal hydrodynamic evolution equations of the expanding QGP medium are prescribed within the 1-D boost invariant Björken flow. Then, we analyze the evolution of shear stress and bulk viscous pressure for different relaxation times. Section III is devoted to the calculation of thermal dilepton and photon production rates in the presence of viscous modified momentum distribution function. In section IV, thermal particle spectra is computed in 1-D Björken expansion and we present the results of our analysis in section V. Section VI details conclusions and future outlook.

Notations and conventions: We are working with natural units $i.e.,\,c=\hbar=k_{B}=1$. The Minkowski metric is taken as $\eta^{\mu\nu}=\textrm{diag}(1,-1,-1,-1)$. The fluid four-velocity is denoted as $u^{\mu}$, which is normalized as $u^{\mu}u_{\mu}=1$ and in the local rest frame (LRF) of the fluid, $u^{\mu}=(1,0,0,0)$. The quantity $\Delta^{\mu\nu}=\eta^{\mu\nu}-u^{\mu}u^{\nu}$ represents the projection operator orthogonal to $u^{\mu}$ and $\nabla^{\mu}=\Delta^{\mu\nu}\partial_{\nu}$. Also, $\Delta_{\alpha\beta}^{\mu\nu}\equiv\frac{1}{2}(\Delta_{\alpha}^{\mu}\Delta_{\beta}^{\nu}+\Delta_{\beta}^{\mu}\Delta_{\alpha}^{\nu})-\frac{1}{3}\Delta^{\mu\nu}\Delta_{\alpha\beta}$ is the traceless symmetric projection operator orthogonal to $u^{\mu}$.

In this section, we present the formalism to estimate the second order relativistic viscous hydrodynamic equations within the EQPM description of QCD medium as derived in Ref. Bhadury et al. (2021). The EQPM maps the thermal QCD medium effects through temperature dependent effective fugacity parameters of constituent noninteracting quasiparticles. Within this model, the equilibrium momentum distribution functions of the quasiparticles are given by Chandra and Ravishankar (2009),

$$f_{k}^{0}=\frac{z_{k}\exp[-\beta(u_{\mu}p_{k}^{\mu})]}{1\pm z_{k}\exp[-\beta(u_{\mu}p_{k}^{\mu})]}.$$

(1)

Here, $k\equiv(q,g)$ represent the quarks and gluons respectively and $z_{q},z_{g}$ denote the effective quark, gluon fugacities which capture the QCD medium interactions. The model divides the hot QCD medium into two sectors : (i) the effective gluonic sector, which describes the contribution of gluonic action to the pressure as well as the contribution due to the internal fermion lines, (ii) the matter sector, which represents the interactions between quarks and antiquarks, along with their interactions with gluons. The form of $z_{g}$ and $z_{q}$ is determined by equating the expression for pressure obtained within EQPM with the lattice data (see Ref. Chandra and Ravishankar (2011) for detailed discussion). Here, the ($2+1$) flavor lattice QCD equation of state Cheng et al. (2008); Borsanyi et al. (2014) is considered. In Eq. (1), $p_{k}^{\mu}=(E_{k},\vec{p}_{k})$ represent the particle four-momenta of quarks and gluons and $\beta=1/T$ is the inverse of temperature. The quasiparticle momenta ($\tilde{p}_{k}^{\mu}$) and bare particle momenta ($p_{k}^{\mu}$) are related through the dispersion relation:

$$\tilde{p}_{g,q}^{\mu}=p_{g,q}^{\mu}+\delta\omega_{g,q}u^{\mu};\,\,\,\,\,\,\,\,\,\,\,\delta\omega_{g,q}=T^{2}\partial_{T}\ln(z_{g,q}),$$

(2)

where $\delta\omega_{g,q}$ denotes the modified part of the dispersion relation. From Eq. (2), the single particle energy of the quasiparticles is given by

$$\tilde{p}_{g,q}^{0}\equiv\omega_{g,q}=E_{g,q}+\delta\omega_{g,q}.$$

(3)

In order to obtain the viscous hydrodynamic equations, one need to quantify the dissipative corrections of the system. The non-equilibrium corrections to the phase space distribution functions are estimated by considering the effective Boltzmann equation within EQPM. The form of relativistic transport equation in RTA for the collision term is obtained as Mitra and Chandra (2018)

$$\tilde{p}_{k}^{\mu}\partial_{\mu}f_{k}^{0}(x,\tilde{p}_{k})+F_{k}^{\mu}\partial_{\mu}^{(p)}f_{k}^{0}=-\frac{\delta f_{k}}{\tau_{R}}\omega_{k},$$

(4)

where $\tau_{R}$ is the relaxation time and $\delta f_{k}$ denotes the non-equilibrium part of the distribution function. The force term is defined from energy-momentum and particle flow conservation and is given by $F_{k}^{\mu}=-\partial_{\nu}(\delta\omega_{k}u^{\nu}u^{\mu})$ Mitra and Chandra (2018).

Using the Landau definition : $u_{\nu}T^{\mu\nu}=\epsilon u^{\mu}$ Landau and Lifshitz (1987), the energy-momentum tensor can be decomposed in terms of hydrodynamic degrees of freedom as

$$T^{\mu\nu}=\epsilon u^{\mu}u^{\nu}-P\Delta^{\mu\nu}+\tau^{\mu\nu},$$

(5)

where $\epsilon$ and $P$ denote the equilibrium energy density and pressure of the system respectively. $\tau^{\mu\nu}=\pi^{\mu\nu}-\Pi\Delta^{\mu\nu}$ is the dissipative current, with $\pi^{\mu\nu}$ being the traceless part. The quantities $\pi^{\mu\nu}$ and $\Pi$ represent the shear stress tensor and bulk viscous pressure respectively. The evolution equations for $\epsilon$ and $u^{\mu}$ can be obtained by projecting the energy-momentum conservation equation $\partial_{\mu}T^{\mu\nu}=0$ along and orthogonal to $u^{\mu}$ and are given by

	$\displaystyle\dot{\epsilon}+(\epsilon+P+\Pi)\Theta-\pi^{\mu\nu}\sigma_{\mu\nu}$	$\displaystyle=$	$\displaystyle 0,$		(6)
	$\displaystyle(\epsilon+P+\Pi)\dot{u}^{\alpha}-\nabla^{\alpha}(P+\Pi)+\Delta_{\nu}^{\alpha}\partial_{\mu}\pi^{\mu\nu}$	$\displaystyle=$	$\displaystyle 0.$		(7)

Here, $\Theta\equiv\partial^{\mu}u_{\mu}$ is the four-divergence of fluid velocity, $\sigma^{\mu\nu}\equiv\Delta_{\alpha\beta}^{\mu\nu}\nabla^{\alpha}u^{\beta}$ represents the shear stress tensor and $\dot{A}\equiv u^{\mu}\partial_{\mu}A$ is the co-moving derivative of $A$. The derivatives of $\beta$ can be calculated from the above equations

	$\displaystyle\dot{\beta}$	$\displaystyle=$	$\displaystyle\beta c_{s}^{2}\left(\Theta+\frac{\Pi\Theta-\pi^{\mu\nu}\sigma_{\mu\nu}}{\epsilon+P}\right),$		(8)
	$\displaystyle\nabla^{\alpha}\beta$	$\displaystyle=$	$\displaystyle-\beta\left(\dot{u}^{\alpha}+\frac{\Pi\dot{u}^{\alpha}-\nabla^{\alpha}\Pi+\Delta_{\nu}^{\alpha}\partial_{\mu}\pi^{\mu\nu}}{\epsilon+P}\right),$		(9)

where $c_{s}^{2}=dP/d\epsilon$ denotes the speed of sound squared.

Now, the form of energy-momentum tensor in terms of quasiparticle four-momenta can be written as

	$\displaystyle T^{\mu\nu}(x)$	$\displaystyle=$	$\displaystyle\sum_{k}g_{k}\int d\tilde{P}_{k}\tilde{p}_{k}^{\mu}\tilde{p}_{k}^{\nu}f_{k}(x,\tilde{p}_{k})$		(10)
			$\displaystyle+\sum_{k}g_{k}\delta\omega_{k}\int d\tilde{P}_{k}\frac{<\tilde{p}_{k}^{\mu}\tilde{p}_{k}^{\nu}>}{E_{k}}f_{k}(x,\tilde{p}_{k}),$

with $g_{k}$ being the degeneracy factor of quasiparticles. Here, $<\tilde{p}_{k}^{\mu}\tilde{p}_{k}^{\nu}>\equiv\frac{1}{2}(\Delta_{\alpha}^{\mu}\Delta_{\beta}^{\nu}+\Delta_{\beta}^{\mu}\Delta_{\alpha}^{\nu})\tilde{p}_{k}^{\alpha}\tilde{p}_{k}^{\beta}$ and $d\tilde{P}_{k}\equiv\frac{d^{3}|\vec{p}_{k}|}{(2\pi)^{3}\omega_{k}}$ represents the phase-space factor. Using Eqs. (5) and (10), the expressions for shear stress tensor and bulk viscous pressure are obtained as Mitra and Chandra (2018)

	$\displaystyle\pi^{\mu\nu}$	$\displaystyle=$	$\displaystyle\sum_{k}g_{k}\Delta_{\alpha\beta}^{\mu\nu}\int d\tilde{P}_{k}\tilde{p}_{k}^{\alpha}\tilde{p}_{k}^{\beta}\delta f_{k}$		(11)
			$\displaystyle+\sum_{k}g_{k}\delta\omega_{k}\Delta_{\alpha\beta}^{\mu\nu}\int d\tilde{P}_{k}\tilde{p}_{k}^{\alpha}\tilde{p}_{k}^{\beta}\frac{\delta f_{k}}{E_{k}},$
	$\displaystyle\Pi$	$\displaystyle=$	$\displaystyle-\frac{1}{3}\sum_{k}g_{k}\Delta_{\alpha\beta}\int d\tilde{P}_{k}\tilde{p}_{k}^{\alpha}\tilde{p}_{k}^{\beta}\delta f_{k}$		(12)
			$\displaystyle-\frac{1}{3}\sum_{k}g_{k}\delta\omega_{k}\Delta_{\alpha\beta}\int d\tilde{P}_{k}\tilde{p}_{k}^{\alpha}\tilde{p}_{k}^{\beta}\frac{\delta f_{k}}{E_{k}}.$

The transport coefficients of the system can be determined once we know the form of $\delta f_{k}$. In Ref. Bhadury et al. (2021), viscous correction is calculated by employing the Chapman-Enskog like iterative solution of the Boltzmann equation in RTA. Within the effective kinetic theory considered, the non-equilibrium corrections to the quark (antiquark) distribution function up to first order in gradients is then obtained as

	$\displaystyle\delta f_{q}$	$\displaystyle=$	$\displaystyle\tau_{R}\bigg{[}\tilde{p}_{q}^{\mu}\partial_{\mu}\beta+\frac{\beta\,\tilde{p}_{q}^{\mu}\,\tilde{p}_{q}^{\nu}}{u\!\cdot\!\tilde{p}_{q}}\partial_{\mu}u_{\nu}-\beta\Theta(\delta\omega_{q})$		(13)
			$\displaystyle-\beta\dot{\beta}\left(\frac{\partial(\delta\omega_{q})}{\partial\beta}\right)\bigg{]}f_{q}^{0}\bar{f}_{q}^{0},$

where $\bar{f}_{q}^{0}=1-f_{q}^{0}$. By employing the above equation in Eqs. (11) and (12), and considering $\tau_{R}$ to be independent of momenta, the following relations are obtained:

$\displaystyle\pi^{\mu\nu}=2\tau_{R}\beta_{\pi}\sigma^{\mu\nu},\,\,\,\,\,\,\,\,\,\,\Pi=-\tau_{R}\beta_{\Pi}\Theta.$

(14)

We note that, as a result of RTA, there will be only a single time-scale for both shear and bulk relaxation times. First order transport coefficients within the effective covariant kinetic theory can be determined by comparing the above equations with that of the Navier-Stokes equation

$$\eta=\tau_{R}\beta_{\pi},\,\,\,\,\,\,\,\,\,\zeta=\tau_{R}\beta_{\Pi}.$$

(15)

Here, $\eta$ and $\zeta$ are the coefficients of shear and bulk viscosities respectively. The quantities $\beta_{\pi},\beta_{\Pi}$ appearing in the above equations are determined in terms of thermodynamic integrals and are given as

	$\displaystyle\beta_{\pi}$	$\displaystyle=$	$\displaystyle\beta\sum_{k}\bigg{[}\tilde{J}^{(1)}_{k~{}42}+\delta\omega_{k}\tilde{L}^{(1)}_{k~{}42}\bigg{]},$		(16)
	$\displaystyle\beta_{\Pi}$	$\displaystyle=$	$\displaystyle\beta\sum_{k}\bigg{[}c_{s}^{2}\bigg{(}\tilde{J}_{k~{}31}^{(0)}+\delta\omega_{k}\tilde{L}_{k~{}31}^{(0)}$
			$\displaystyle-\frac{\partial(\delta\omega_{k})}{\partial\beta}\left(\tilde{J}_{k~{}21}^{(0)}+\delta\omega_{k}\tilde{L}_{k~{}21}^{(0)}\right)\bigg{)}$
			$\displaystyle+\frac{5}{3}\bigg{(}\tilde{J}_{k~{}42}^{(1)}+\delta\omega_{k}\tilde{L}_{k~{}42}^{(1)}\bigg{)}-\delta\omega_{k}\left(\tilde{J}_{k~{}21}^{(0)}+\delta\omega_{k}\tilde{L}_{k~{}21}^{(0)}\right)\bigg{]}.$

The form of thermodynamic integrals, $\tilde{J}^{(r)}_{k~{}nm}$ and $\tilde{L}^{(r)}_{k~{}nm}$ appearing in the above equations are given in Appendix A. Following the methodology of Ref. Bhadury et al. (2021), the evolution equations for shear stress tensor and bulk viscous pressure are obtained as

	$\displaystyle\dot{\pi}^{\langle\mu\nu\rangle}+\frac{\pi^{\mu\nu}}{\tau_{R}}$	$\displaystyle=$	$\displaystyle 2\beta_{\pi}\sigma^{\mu\nu}+2\pi_{\phi}^{\langle\mu}\omega^{\nu\rangle\phi}-\delta_{\pi\pi}\pi^{\mu\nu}\theta$		(18)
			$\displaystyle-\tau_{\pi\pi}\pi_{\phi}^{\langle\mu}\sigma^{\nu\rangle\phi}+\lambda_{\pi\Pi}\Pi\sigma^{\mu\nu},$
	$\displaystyle\dot{\Pi}+\frac{\Pi}{\tau_{R}}$	$\displaystyle=$	$\displaystyle-\beta_{\Pi}\theta-\delta_{\Pi\Pi}\Pi\theta+\lambda_{\Pi\pi}\pi^{\mu\nu}\sigma_{\mu\nu}.$		(19)

Here, $\omega^{\mu\nu}=\frac{1}{2}(\nabla^{\mu}u^{\nu}-\nabla^{\nu}u^{\mu})$ denotes the vorticity tensor. The second order transport coefficients in Eqs. (18) and (19) are obtained in terms of different thermodynamic integrals and are shown in Appendix A.

Dissipation in the system influences the particle production rates in two ways: firstly through the hydrodynamic evolution of the system and secondly via non-equilibrium corrections to the single particle distribution functions. In the previous section, we have analyzed the impact of viscosity on the evolution of QGP. Now, we incorporate the effect of viscosities on thermal dilepton and photon production rates through viscous modified distribution functions upto first order in gradients. The major source of thermal dileptons in QGP medium is from the $q\bar{q}$-annihilation, $q\bar{q}\rightarrow\gamma^{*}\rightarrow l^{+}l^{-}$. From relativistic kinetic theory, the rate of dilepton production for this process, within the EQPM can be written as

	$\displaystyle\frac{dN}{d^{4}xd^{4}p}$	$\displaystyle=$	$\displaystyle\iint\frac{d^{3}\vec{p}_{1}}{(2\pi)^{3}}\frac{d^{3}\vec{p}_{2}}{(2\pi)^{3}}\,\frac{M_{\textrm{eff}}^{2}\,g^{2}\,\sigma(M_{\textrm{eff}}^{2})}{2\omega_{1}\omega_{2}}$		(27)
			$\displaystyle\times f_{q}(\vec{p}_{1})f_{q}(\vec{p}_{2})\delta^{4}(\tilde{p}-\tilde{p}_{1}-\tilde{p}_{2}).$

Here, $\tilde{p}_{1,2}=(\omega_{1,2},\vec{p}_{1,2})$ is the four-momentum of the quark, anti-quark respectively, with $\omega_{1,2}$ being the corresponding modified single particle energy as given by Eq. (3). When quark (anti-quark) masses are neglected, we can write $\omega_{1,2}\approx|\vec{p}|$. Four-momentum of the dilepton is given as $\tilde{p}=(\omega_{0}=\omega_{1}+\omega_{2},\vec{p}=\vec{p}_{1}+\vec{p}_{2})$. The quantity $M_{\textrm{eff}}^{2}=(\omega_{1}+\omega_{2})^{2}-(\vec{p}_{1}+\vec{p}_{2})^{2}$ represents the modified effective mass of the virtual photon in the interacting QCD medium. Keeping the terms up to linear order in $\delta\omega_{q}$, we get Naik et al. (2021a)

$\displaystyle M_{\textrm{eff}}^{2}$

$\displaystyle\approx$

$\displaystyle M^{2}\left(1+\frac{4\,\delta\omega_{q}\,(E_{1}+E_{2})}{M^{2}}\right),$

(28)

where $M^{2}$ is the invariant mass of dilepton in the ultrarelativistic limit, $z_{q}\rightarrow 1$. In Eq. (27), the term $\sigma(M_{\textrm{eff}}^{2})$ represents the cross-section for the $q\bar{q}$-annihilation process and $g$ is the degeneracy factor. In the Born approximation, we obtain $M_{\textrm{eff}}^{2}\,g^{2}\,\sigma(M_{\textrm{eff}}^{2})=\frac{80\pi}{9}\alpha_{e}^{2}$ Alam et al. (1996) for $N_{f}=2$ and $N_{c}=3$, with $\alpha_{e}$ being the electromagnetic coupling constant.

We note that, in Eq. (27), $f_{q}(\vec{p})\equiv f_{q}^{0}+f_{q}^{0}\bar{f}_{q}^{0}\delta f_{q}$ represents the quark (anti-quark) momentum distribution function away from equilibrium, with the form of equilibrium distribution function $f_{q}^{0}$ as given by Eq. (1). The viscous modification to the distribution function (Eq. (13)) can be rewritten in terms of first order gradients of hydrodynamic quantities by employing Eq. (14) and the form of $\delta f_{q}$ thus obtained is given below:

	$\displaystyle\delta f_{q}$	$\displaystyle=$	$\displaystyle\delta f_{\pi}+\delta f_{\Pi}$		(29)
		$\displaystyle=$	$\displaystyle\frac{\beta}{2\beta_{\pi}(u\cdot\tilde{p})}\tilde{p}^{\mu}\tilde{p}^{\nu}\pi_{\mu\nu}+\frac{\beta\Pi}{\beta_{\Pi}}\Big{[}\xi_{1}-\xi_{2}(u\cdot\tilde{p})\Big{]},$

where

	$\displaystyle\xi_{1}$	$\displaystyle=$	$\displaystyle\beta c_{s}^{2}\frac{\partial\delta\omega_{q}}{\partial\beta}+\delta\omega_{q},$
	$\displaystyle\xi_{2}$	$\displaystyle=$	$\displaystyle\left(c_{s}^{2}-\frac{1}{3}\right)+\frac{\delta\omega_{q}}{3(u\cdot\tilde{p})^{2}}\left[2(u\cdot\tilde{p})-\delta\omega_{q}\right].$		(30)

We intend to calculate the spectra of dileptons with large invariant mass i.e., $M>>T$. Hence we can approximate the distribution function with the Maxwell-Boltzmann one, $f(\vec{p})\approx z_{q}e^{-\omega/T}$. Keeping the terms up to second order in momenta, we write the total dilepton production rate as Bhatt et al. (2012)

$$\frac{dN}{d^{4}xd^{4}p}=\frac{dN^{(0)}}{d^{4}xd^{4}p}+\frac{dN^{(\pi)}}{d^{4}xd^{4}p}+\frac{dN^{(\Pi)}}{d^{4}xd^{4}p}.$$

(31)

The ideal part of the rate, within the EQPM is obtained as Chandra and Sreekanth (2015); Naik et al. (2021a)

	$\displaystyle\frac{dN^{(0)}}{d^{4}xd^{4}p}$	$\displaystyle=$	$\displaystyle z_{q}^{2}\int\frac{d^{3}\vec{p}_{1}}{(2\pi)^{6}}\,\frac{M_{\textrm{eff}}^{2}\,g^{2}\,\sigma(M_{\textrm{eff}}^{2})}{2\omega_{1}\omega_{2}}\,e^{-(\omega_{1}+\omega_{2})/T}$		(32)
			$\displaystyle\times\delta(\omega_{0}-\omega_{1}-\omega_{2})$
		$\displaystyle=$	$\displaystyle\frac{z_{q}^{2}}{2}\frac{M_{\textrm{eff}}^{2}\,g^{2}\,\sigma(M_{\textrm{eff}}^{2})}{(2\pi)^{5}}e^{-\omega_{0}/T}.$

The contribution to the thermal dilepton rate due to shear viscosity can be written as

	$\displaystyle\frac{dN^{(\pi)}}{d^{4}xd^{4}p}$	$\displaystyle=$	$\displaystyle z_{q}^{2}\iint\frac{d^{3}\vec{p}_{1}}{(2\pi)^{3}}\frac{d^{3}\vec{p}_{2}}{(2\pi)^{3}}\,e^{-(\omega_{1}+\omega_{2})/T}$		(33)
			$\displaystyle\times\left[\frac{\beta}{\beta_{\pi}(u\cdot\tilde{p})}\right]\frac{M_{\textrm{eff}}^{2}\,g^{2}\,\sigma(M_{\textrm{eff}}^{2})}{2\omega_{1}\omega_{2}}$
			$\displaystyle\times\delta^{4}(\tilde{p}-\tilde{p}_{1}-\tilde{p}_{2})\,\,\tilde{p}_{1}^{\mu}\tilde{p}_{1}^{\nu}\pi_{\mu\nu}.$

We note that, shear viscous correction to distribution function, $\delta f_{\pi}$ is qualitatively similar to the non-equilibrium correction obtained in Refs. Bhalerao et al. (2013); Naik et al. (2021a, b). Hence, by following the analysis of the above Refs., we obtain the contribution to the dilepton rate due to shear viscosity as

	$\displaystyle\frac{dN^{(\pi)}}{d^{4}xd^{4}p}$	$\displaystyle=$	$\displaystyle\frac{dN^{(0)}}{d^{4}xd^{4}p}\Bigg{\{}\frac{\beta}{\beta_{\pi}}\frac{1}{2|\vec{p}|^{5}}\Bigg{[}\frac{(u\cdot\tilde{p})|\vec{p}|}{2}\left(2|\vec{p}|^{2}-3M_{\textrm{eff}}^{2}\right)$		(34)
			$\displaystyle+\frac{3}{4}M_{\textrm{eff}}^{4}\ln\left(\frac{(u\cdot\tilde{p})+|\vec{p}|}{(u\cdot\tilde{p})-|\vec{p}|}\right)\Bigg{]}\tilde{p}^{\mu}\tilde{p}^{\nu}\pi_{\mu\nu}\Bigg{\}},$

where $|\vec{p}|=\sqrt{(u\cdot\tilde{p})^{2}-M_{\textrm{eff}}^{2}}$. Following the same analysis, we calculate the contribution to the dilepton rate due to bulk viscosity as

	$\displaystyle\frac{dN^{(\Pi)}}{d^{4}xd^{4}p}$	$\displaystyle=$	$\displaystyle\frac{dN^{(0)}}{d^{4}xd^{4}p}\frac{2\beta\Pi}{\beta_{\Pi}}\Bigg{\{}\beta c_{s}^{2}\frac{\partial\delta\omega_{q}}{\partial\beta}-\frac{2}{3}\delta\omega_{q}-\left(c_{s}^{2}-\frac{1}{3}\right)\frac{(u\cdot\tilde{p})}{2}$		(35)
			$\displaystyle+\frac{\delta\omega_{q}^{2}}{3}\frac{1}{2|\vec{p}|^{5}}\Bigg{[}\frac{(u\cdot\tilde{p})|\vec{p}|}{2}\left(2|\vec{p}|^{2}-3M_{\textrm{eff}}^{2}\right)$
			$\displaystyle+\frac{3}{4}M_{\textrm{eff}}^{4}\ln\left(\frac{(u\cdot\tilde{p})+|\vec{p}|}{(u\cdot\tilde{p})-|\vec{p}|}\right)\Bigg{]}\Bigg{\}}.$

We now determine the modification to the thermal photon emission rate due to shear and bulk viscosities. We consider two major sources of thermal photons: Compton scattering, $q(\bar{q})g\rightarrow q(\bar{q})\gamma$ and $q\bar{q}$-annihilation, $q\bar{q}\rightarrow g\gamma$. We emphasize that, in the present work we only examine the hard contributions of thermal photon emission. Following Bhatt et al. (2010); Dusling (2010); Wong (1995), the total photon rate in the presence of dissipation within EQPM can be written as

	$\displaystyle\omega_{0}\frac{dN_{\gamma}}{d^{3}pd^{4}x}$	$\displaystyle=$	$\displaystyle\frac{5}{9}\frac{\alpha_{e}\alpha_{s}}{2\pi^{2}}T^{2}f_{q}(\vec{p})\left[\ln\left(\frac{12\,(u\cdot\tilde{p})}{g^{2}T}\right)\!+\!\frac{C_{\textrm{ann}}\!+\!C_{\textrm{Comp}}}{2}\right]$		(36)
		$\displaystyle=$	$\displaystyle\omega_{0}\frac{dN_{\gamma}^{(0)}}{d^{3}pd^{4}x}+\omega_{0}\frac{dN_{\gamma}^{(\pi)}}{d^{3}pd^{4}x}+\omega_{0}\frac{dN_{\gamma}^{(\Pi)}}{d^{3}pd^{4}x},$

where the constants take the values $C_{\textrm{ann}}=-1.91613$, $C_{\textrm{Comp}}=-0.41613$. Here $\alpha_{s}$ denotes the strong coupling constant and $g=\sqrt{4\pi\alpha_{s}}$. The ideal contribution to the photon rate takes the following form:

$$\omega_{0}\frac{dN_{\gamma}^{(0)}}{d^{3}pd^{4}x}=\frac{5}{9}\frac{\alpha_{e}\alpha_{s}}{2\pi^{2}}T^{2}z_{q}^{2}e^{-(u\cdot\tilde{p})/T}\ln\left(\frac{3.7388\,(u\cdot\tilde{p})}{g^{2}T}\right).$$

(37)

Viscous contributions to photon rate are obtained as

	$\displaystyle\omega_{0}\frac{dN_{\gamma}^{(\pi)}}{d^{3}pd^{4}x}$	$\displaystyle=$	$\displaystyle\omega_{0}\frac{dN_{\gamma}^{(0)}}{d^{3}pd^{4}x}\left\{\frac{\beta}{2\beta_{\pi}(u\cdot\tilde{p})}\right\},$		(38)
	$\displaystyle\omega_{0}\frac{dN_{\gamma}^{(\Pi)}}{d^{3}pd^{4}x}$	$\displaystyle=$	$\displaystyle\omega_{0}\frac{dN_{\gamma}^{(0)}}{d^{3}pd^{4}x}\left\{\frac{\beta\Pi}{\beta_{\Pi}}\Big{[}\xi_{1}-\xi_{2}(u\cdot\tilde{p})\Big{]}\right\},$		(39)

where $\xi_{1}$ and $\xi_{2}$ are defined in Eq. (III).

Next, in order to do a comparative study with the standard hydrodynamic results, we calculate the thermal dilepton and photon emission rates by employing the viscous modified distribution function of the form: $f=f_{0}(1+\delta\phi)$, where $f_{0}=e^{-E/T}$. The non-equilibrium correction is obtained from the $14-$moment Grad’s method and is shown below Dusling and Teaney (2008) :

$\displaystyle\delta\phi=\frac{p^{\mu}p^{\nu}}{2(\epsilon+P)T^{2}}\left(\pi_{\mu\nu}+\frac{2}{5}\Pi\Delta_{\mu\nu}\right).$

(40)

The ideal part of the dilepton rate, in the absence of viscosity is well known and is given by Vogt (2007)

$$\frac{dN_{0}}{d^{4}xd^{4}p}=\frac{1}{2}\frac{M^{2}\,g^{2}\,\sigma(M^{2})}{(2\pi)^{5}}e^{-(u\cdot p)/T}.$$

(41)

The contribution to the dilepton rate due to shear and bulk viscous terms in $\delta\phi$ are calculated as Dusling and Lin (2008); Bhatt et al. (2012)

	$\displaystyle\frac{dN_{\pi}}{d^{4}xd^{4}p}$	$\displaystyle=$	$\displaystyle\frac{2}{3}\left(\frac{p^{\mu}p^{\nu}}{2sT^{3}}\pi_{\mu\nu}\right)\frac{dN_{0}}{d^{4}xd^{4}p},$		(42)
	$\displaystyle\frac{dN_{\Pi}}{d^{4}xd^{4}p}$	$\displaystyle=$	$\displaystyle\frac{2}{5sT^{3}}\left(\frac{M^{2}}{12}g^{\alpha\beta}-\frac{1}{3}p^{\alpha}p^{\beta}\right)\Delta_{\alpha\beta}\Pi\frac{dN_{0}}{d^{4}xd^{4}p},$

respectively. Similarly, the ideal and viscous contributions to thermal photon rate are obtained as given below Dusling (2010); Bhatt et al. (2010) :

	$\displaystyle E\frac{dN_{0}}{d^{3}pd^{4}x}$	$\displaystyle=$	$\displaystyle\frac{5}{9}\frac{\alpha_{e}\alpha_{s}}{2\pi^{2}}T^{2}e^{-(u\cdot p)/T}\ln\left(\frac{3.7388\,(u\cdot p)}{g^{2}T}\right),$		(44)
	$\displaystyle E\frac{dN_{\pi}}{d^{3}pd^{4}x}$	$\displaystyle=$	$\displaystyle E\frac{dN_{0}}{d^{3}pd^{4}x}\left\{\frac{p^{\mu}p^{\nu}\pi_{\mu\nu}}{2(\epsilon+P)T^{2}}\right\},$		(45)
	$\displaystyle E\frac{dN_{\Pi}}{d^{3}pd^{4}x}$	$\displaystyle=$	$\displaystyle E\frac{dN_{0}}{d^{3}pd^{4}x}\left\{\frac{2}{5}\Pi\frac{p^{\mu}p^{\nu}\Delta_{\mu\nu}}{2(\epsilon+P)T^{2}}\right\}.$		(46)

In the next section, we proceed to calculate the thermal dilepton and photon spectra from heavy-ion collisions by convoluting the rate expressions over the space-time evolution of the collisions along with the temperature profile and viscous evolution of the QGP obtained in section II.1.

Thermal dileptons and photons are produced from a thermalized QGP throughout its evolution. We calculate the emission spectra of these particles over the entire evolution by considering Björken’s $1-$D model Bjorken (1983). Within the model, four-dimensional volume element in the Minkowski space gets modified as $d^{4}x=A_{\bot}\tau d\tau d\eta_{s}$. Here, $A_{\bot}=\pi R_{A}^{2}$ is the transverse area of the collision, with $R_{A}=1.2\,A^{1/3}$ fm being the radius of the colliding nuclei (for Au, $A=197$). Now, we write the total thermal dilepton and photon yields in terms of transverse momentum $p_{T}$, invariant mass $M$ and rapidity $y$ of the particle produced

	$\displaystyle\frac{dN}{dM^{2}d^{2}p_{T}dy}$	$\displaystyle=$	$\displaystyle A_{\bot}\,\int_{\tau_{0}}^{\tau_{f}}d\tau\,\tau\int_{-\infty}^{\infty}d\eta_{s}\,\chi(T,\eta_{s})\left(\frac{1}{2}\frac{dN}{d^{4}xd^{4}p}\right)$
	$\displaystyle\frac{dN}{d^{2}p_{T}dy}$	$\displaystyle=$	$\displaystyle A_{\bot}\,\int_{\tau_{0}}^{\tau_{f}}d\tau\,\tau\int_{-\infty}^{\infty}d\eta_{s}\,\left(\omega\frac{dN}{d^{3}pd^{4}x}\right),$		(47)

where the factor $\chi(T,\eta_{s})=\left[1+\frac{2}{m_{T}}\cosh(y-\eta_{s})\delta\omega_{q}\right]$ and $\tau_{f}$ is the time taken by the fireball to cool down to the critical temperature $T_{c}$. Four-momentum of the dilepton can be parametrized as $\tilde{p}^{\mu}=(M_{T}\cosh y,p_{T}\cos\phi_{p},p_{T}\sin\phi_{p},M_{T}\sinh y)$, where $M_{T}=\sqrt{p_{T}^{2}+M_{\textrm{eff}}^{2}}$ is the medium modified transverse mass of dilepton. Now, for an expanding QGP within Björken flow, we evaluate the factors appearing in particle rate as

	$\displaystyle(u\cdot\tilde{p})$	$\displaystyle=M_{T}\cosh(y-\eta_{s}),$		(48)
	$\displaystyle\tilde{p}^{\mu}\tilde{p}^{\nu}\pi_{\mu\nu}$	$\displaystyle=\pi\left[\frac{p_{T}^{2}}{2}-M_{T}^{2}\sinh^{2}(y-\eta_{s})\right],$		(49)
	$\displaystyle\tilde{p}^{\mu}\tilde{p}^{\nu}\Delta_{\mu\nu}$	$\displaystyle=-p_{T}^{2}-m_{T}^{2}\sinh^{2}(y-\eta_{s}).$		(50)

By noting the above expressions, we write the ideal contribution to thermal dilepton yields as

	$\displaystyle\frac{dN^{(0)}}{dM^{2}d^{2}p_{T}dy}$	$\displaystyle=$	$\displaystyle\mathcal{Q}\int_{\tau_{0}}^{\tau_{f}}d\tau\,\tau z_{q}^{2}\,\int_{-\infty}^{\infty}d\eta_{s}\,\chi(T,\eta_{s})$		(51)
			$\displaystyle\times\textrm{Exp}\left(-\frac{M_{T}}{T}\cosh(y-\eta_{s})\right).$

The viscous contributions to the dilepton yields are obtained as

	$\displaystyle\frac{dN^{(\pi)}}{dM^{2}d^{2}p_{T}dy}$	$\displaystyle=$	$\displaystyle\mathcal{Q}\int_{\tau_{0}}^{\tau_{f}}d\tau\,\tau z_{q}^{2}\,\frac{\beta\pi}{4\beta_{\pi}}\int_{-\infty}^{\infty}d\eta_{s}\,\chi(T,\eta_{s})\textrm{Exp}\left(-\frac{M_{T}}{T}\cosh(y-\eta_{s})\right)\frac{\left[p_{T}^{2}/2-M_{T}^{2}\sinh^{2}(y-\eta_{s})\right]}{[M_{T}^{2}\cosh^{2}(y-\eta_{s})-M_{\textrm{eff}}^{2}]^{5/2}}$		(52)
			$\displaystyle\times\Bigg{\{}M_{T}\cosh(y-\eta_{s})\sqrt{M_{T}^{2}\cosh^{2}(y-\eta_{s})-M_{\textrm{eff}}^{2}}\Big{(}2M_{T}^{2}\cosh^{2}(y-\eta_{s})-5M_{\textrm{eff}}^{2}\Big{)}$
			$\displaystyle+\frac{3}{2}M_{\textrm{eff}}^{4}\ln\left(\frac{M_{T}\cosh(y-\eta_{s})+\sqrt{M_{T}^{2}\cosh^{2}(y-\eta_{s})-M_{\textrm{eff}}^{2}}}{M_{T}\cosh(y-\eta_{s})-\sqrt{M_{T}^{2}\cosh^{2}(y-\eta_{s})-M_{\textrm{eff}}^{2}}}\right)\Bigg{\}},$
	$\displaystyle\frac{dN^{(\Pi)}}{dM^{2}d^{2}p_{T}dy}$	$\displaystyle=$	$\displaystyle\mathcal{Q}\int_{\tau_{0}}^{\tau_{f}}d\tau\,\tau z_{q}^{2}\frac{2\beta\Pi}{\beta_{\Pi}}\int_{-\infty}^{\infty}d\eta_{s}\,\chi(T,\eta_{s})\textrm{Exp}\left(-\frac{M_{T}}{T}\cosh(y-\eta_{s})\right)$		(53)
			$\displaystyle\times\Bigg{\{}\beta c_{s}^{2}\frac{\partial\delta\omega_{q}}{\partial\beta}-\frac{2}{3}\delta\omega_{q}-\frac{1}{2}\left(c_{s}^{2}-\frac{1}{3}\right)M_{T}\cosh(y-\eta_{s})+\frac{\delta\omega_{q}^{2}}{3}\frac{1}{4[M_{T}^{2}\cosh^{2}(y-\eta_{s})-M_{\textrm{eff}}^{2}]^{5/2}}$
			$\displaystyle\times\Bigg{[}M_{T}\cosh(y-\eta_{s})\sqrt{M_{T}^{2}\cosh^{2}(y-\eta_{s})-M_{\textrm{eff}}^{2}}\Big{(}2M_{T}^{2}\cosh^{2}(y-\eta_{s})-5M_{\textrm{eff}}^{2}\Big{)}$
			$\displaystyle+\frac{3}{2}M_{\textrm{eff}}^{4}\ln\left(\frac{M_{T}\cosh(y-\eta_{s})+\sqrt{M_{T}^{2}\cosh^{2}(y-\eta_{s})-M_{\textrm{eff}}^{2}}}{M_{T}\cosh(y-\eta_{s})-\sqrt{M_{T}^{2}\cosh^{2}(y-\eta_{s})-M_{\textrm{eff}}^{2}}}\right)\Bigg{]}\Bigg{\}},$

where $\mathcal{Q}=\frac{A_{\bot}}{4(2\pi)^{5}}\frac{80\pi}{9}\alpha_{e}^{2}$. From Eq. (31), we note the total dilepton yield to be

$$\frac{dN}{dM^{2}d^{2}p_{T}dy}=\frac{dN^{(0)}}{dM^{2}d^{2}p_{T}dy}+\frac{dN^{(\pi)}}{dM^{2}d^{2}p_{T}dy}+\frac{dN^{(\Pi)}}{dM^{2}d^{2}p_{T}dy}.$$

The thermal photon yield is obtained as

$$\frac{dN}{d^{2}p_{T}dy}=\frac{dN^{(0)}}{d^{2}p_{T}dy}+\frac{dN^{(\pi)}}{d^{2}p_{T}dy}+\frac{dN^{(\Pi)}}{d^{2}p_{T}dy}.$$

(54)

Noting the photon energy to be $(u\cdot\tilde{p})=p_{T}\cosh(y-\eta_{s})$, the ideal part of photon yield is obtained as

	$\displaystyle\frac{dN^{(0)}}{d^{2}p_{T}dy}$	$\displaystyle=$	$\displaystyle\mathcal{C}\int_{\tau_{0}}^{\tau_{f}}d\tau\,\tau z_{q}^{2}T^{2}\int_{-\infty}^{\infty}d\eta_{s}\textrm{Exp}\left(-\frac{p_{T}}{T}\cosh(y-\eta_{s})\right)$		(55)
			$\displaystyle\times\ln\left(\frac{3.7388\,p_{T}\cosh(y-\eta_{s})}{g^{2}T}\right).$

Viscous contributions to photon yield are obtained as

	$\displaystyle\frac{dN^{(\pi)}}{d^{2}p_{T}dy}$	$\displaystyle=$	$\displaystyle\mathcal{C}\int_{\tau_{0}}^{\tau_{f}}d\tau\,\tau z_{q}^{2}T^{2}\int_{-\infty}^{\infty}d\eta_{s}\textrm{Exp}\left(-\frac{p_{T}}{T}\cosh(y-\eta_{s})\right)$		(56)
			$\displaystyle\times\frac{\beta}{2\beta_{\pi}}\frac{1}{p_{T}\cosh(y-\eta_{s})}\ln\left(\frac{3.7388\,p_{T}\cosh(y-\eta_{s})}{g^{2}T}\right)$
			$\displaystyle\times\pi\left[\frac{p_{T}^{2}}{2}-M_{T}^{2}\sinh^{2}(y-\eta_{s})\right]$
	$\displaystyle\frac{dN^{(\Pi)}}{d^{2}p_{T}dy}$	$\displaystyle=$	$\displaystyle\mathcal{C}\,\int_{\tau_{0}}^{\tau_{f}}d\tau\,\tau z_{q}^{2}T^{2}\int_{-\infty}^{\infty}d\eta_{s}\textrm{Exp}\left(-\frac{p_{T}}{T}\cosh(y-\eta_{s})\right)$		(57)
			$\displaystyle\times\frac{\beta\Pi}{\beta_{\Pi}}\left[\xi_{1}-\xi_{2}p_{T}\cosh(y-\eta_{s})\right]$
			$\displaystyle\times\ln\left(\frac{3.7388\,p_{T}\cosh(y-\eta_{s})}{g^{2}T}\right),$

where $\mathcal{C}=\frac{5A_{\bot}}{9}\frac{\alpha_{e}\alpha_{s}}{2\pi^{2}}$.