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Advances in High Energy Physics
Volume 2013 (2013), Article ID 176578, 41 pages
http://dx.doi.org/10.1155/2013/176578
Review Article

Electromagnetic Radiations from Heavy Ion Collision

1HENPP Division, Saha Institute of Nuclear Physics, 1/AF Bidhan Nagar, Kolkata 700064, India
2Theoretical Physics Division, Variable Energy Cyclotron Centre, 1/AF Bidhan Nagar, Kolkata 700064, India
3Ctr. for Astroparticle Physics and Space Bose Institute, 5/EN Bidhan Nagar, Kolkata 700091, India

Received 15 April 2013; Accepted 5 July 2013

Academic Editor: Jan E. Alam

Copyright © 2013 Payal Mohanty et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this review, we have discussed the different sources of photons and dileptons produced in heavy ion collision (HIC). The transverse momentum () spectra of photons for different collision energies are analyzed with a view of extracting the thermal properties of the system formed in HIC. We showed the effect of viscosity on spectra of produced thermal photons. The dilepton productions from hot hadrons are considered including the spectral change of light vector mesons in the thermal bath. We have analyzed the and invariant mass () spectra of dileptons for different collision energies too. As the individual spectra are constrained by certain unambiguous hydrodynamical inputs, so we evaluated the ratio of photon to dilepton spectra, , to overcome those quantities. We argue that the variation of the radial velocity extracted from with is indicative of a phase transition from the initially produced partons to hadrons. In the calculations of interferometry involving dilepton pairs, it is argued that the nonmonotonic variation of HBT radii with invariant mass of the lepton pairs signals the formation of quark gluon plasma in HIC. Elliptic flow () of dilepton is also studied at  TeV for 30–40% centrality using the hydrodynamical model.

1. Introduction

The main objective of relativistic heavy ion collisions is to study the transient phase, that is, quark gluon plasma (QGP) which is believed to permeate the early universe a few microseconds after the Big Bang. Collision between nuclei at ultrarelativistic energies produces charged particles either in hadronic or in partonic state depending on the collision energy. Interaction among these charged particles produces electromagnetic (EM) radiation [19]. However, hadrons being strongly interacting objects give snapshot of evolution only from the freezeout surface. So they have hardly any information about the interior of the plasma. Whereas EM radiations, for example, the thermal photons and dileptons, are expected to provide an accurate information about the initial condition and the history of evolution of the plasma. This is possible since photons and dileptons interact only through the EM interaction. The EM interaction strength is small compared to that of strong interaction () and thus dominates the dynamics of nuclear collision processes. Therefore, its mean free path () is larger than the size of the system. Because of their negligible final-state interactions with the hadronic environment, once produced it brings the electromagnetic particles about to escape unscathed carrying the clean information of all stages of the collision. The EM radiations produce all stages of collision process which contribute to the measured photon spectra; in principle, a careful analysis may be useful to uncover the whole space-time history of nuclear collision. Hence EM radiations—real and the virtual photons (dilepton)—are considered as efficient probes to study dynamical evolution of the matter formed in relativistic heavy ion collision. However, as they are emitted continuously, they sense in fact the entire space-time history of the reaction. This expectation has led to an intense and concerted efforts toward the identification of various sources of such radiations. While initially these signals were treated as thermometer of the dense medium created, but later on recent calculations suggest it might serve as chronometer [10] and flow-meter [1116] of HIC.

The review is organized as follows. In Section 2, we start with possible sources of photons and dileptons that were produced in HIC. We have discussed the formalism of static emission rate of photons and dileptons in Section 3. To get total yield, we need concept of hydrodynamics. So, we briefly outlined relativistic hydrodynamics in Section 4 which takes care of the evolution. In Section 5 we have presented the thermal emission rate of photons from QGP (Section 5.1) and hadronic matter (Section 5.2) which is used to produce the results, and total invariant yield of direct photon for different collision energy is shown in Section 5.4. The effect of viscosity on the transverse momentum () spectra of photon is discussed in Section 5.5. Similarly, the details of the emission rate of dileptons from QGP and hot hadrons are given in Sections 6.1 and 6.2, respectively. Using these rates, the results of and invariant mass () spectra of dileptons are presented in Section 6.4. In Section 7 the radial flow is extracted by simultaneous use of spectra of photons and dileptons and ratio of the spectra, and is conferred. The correlation function for dilepton has been calculated and HBT radii are extracted as function of in Section 8. We have also evaluated the dilepton in Section 9 taking into account the medium effect on the spectral function of the vector mesons. Finally, we have summarized the work in Section 10.

2. Various Sources of EM Radiations

As argued previously that EM radiations emerge out copiously from all stages of collision, so, in order to proceed, it is useful to identify various sources of photons and dileptons produced in the HIC. So the “inclusive” photon spectrum coming from such collision in usual sense can be defined as the unbiased photon spectrum observed in pp, pA, or AA collision. This spectrum is built up from a cocktail of various components.

Depending on their origin, there are two different types of sources which are “direct photons” and “photons from decay of hadrons.” The term “direct photons” is meant for those photons and dileptons which produce directly from collision between the particles. One can subdivide this broad category of “direct photons” into “prompt photons,” “preequilibrium photons” and “thermal photons,” depending on their origin. On the other hand, the decay photons do not come directly from the collision, rather from the decay of hadrons.

2.1. Transverse Momentum () Dependence of EM Radiations

The EM spectra provided by the experimentalist are mingled with various sources of photons and dileptons and it is difficult to distinguish different sources experimentally. However, real interest lies in the thermal photons and dileptons since it is expected to render information about the initial condition and the history of evolution of the plasma while it cools and hadronizes. Thus, theoretical models are used with great advantage to identify these sources of photons and their relative importance and characteristics in the spectrum.

Depending on the process through which photons/dileptons produce, they are categorized as follows.(1)Prompt: the EM radiations produced by hard scattering of the partons inside the nucleons of incoming nuclei in the initial stage of collision, before the thermalization sets in, are known as prompt photons and dileptons (Drell Yan). This contribution may be evaluated by using pQCD.(2)Preequilibrium: the preequilibrium photons and dileptons are produced in the preequilibrium stage, that is, before the thermalization sets in the system. In such scenario the contribution from preequilibrium stage will be very small and hence neglected.(3)Thermal: EM radiations which are emitted from the thermalized systems of quarks and gluons or hadronic gas.(4)Decay: after the freezeout of the fireball, photons and dileptons are also produced from the decays of long-lived (compared to strong interaction time scale) hadrons and known as “photons from decay.

Out of different sources, the thermal photons and dileptons are privileged as they carry information about the formation of QGP. As indicated in Figure 1, the hard photons dominate the high part of the invariant momentum spectra, and decay photon populates the low part and rest over thermal contribution shines in the intermediate domain of the spectra ~1–3 GeV. And the calculations based on theory infer that the photons and lepton pairs from hadronic matter dominate the spectrum at lower (~1-2 GeV) whereas photons and dileptons form QGP dominate in the intermediate range, that is,  GeV (depending on the models) [17]. This small window may help in learning the properties of QGP. Thus one has to subtract out the nonthermal sources to understand the properties of the QGP. However, it is not possible experimentally to distinguish between different sources. Thus, theoretical models and calculations can be used to great advantage to identify different sources of direct photons and their relative importance and characteristics in the spectrum. The hard photons and dileptons are well understood in the framework of pQCD, and decay contributions can be filtered out experimentally using different subtraction methods, like invariant mass analysis, mixed event analysis, internal conversion method, and so forth.

176578.fig.001
Figure 1: Schematic diagram of different sources of photon production in heavy ion collision as function of transverse momentum ().

The invariant momentum distribution of photons and dileptons produce from a thermal source depends on the temperature () of the source through the thermal phase space distributions of the participants of the reaction that produces the photons and dileptons [18]. As a result the spectra of thermal photons and dileptons reflect the temperature of the source through the phase space factor (). Hence ideally the photons with intermediate values (~2-3 GeV, depending on the value of initial temperature) reflect the properties of QGP (realized when , is the transition temperature). Therefore, one should look into the spectra for these values of for the detection of QGP. However, for an expanding system the situation is far more complex. The thermal phase space factor changes by flow; for example, the transverse kick received by low photons due to flow originating from the low temperature hadronic phase (realized when ) populates the high part of the spectra [19]. As a consequence the intermediate or the high part of the spectra contains contributions from both QGP and hadrons. Thus, it is not an easy task to disentangle the photons coming from pure partonic phase. Thus photons appear to be a more restrictive probe since they are characterized only by their momentum whereas the dileptons have two kinematic variable, and invariant mass () to play with. A soft photon (low ) in one frame of reference can be hard (high ) in another frame, whereas the integrated invariant mass distribution of dileptons is independent of any frame. In addition to it the spectra are affected by the flow; however, the integrated spectra remain unaltered by the flow in the system. Also in the spectra of dileptons, again in spectra, the dileptons from QGP dominates over its hadronic counterpart above the peak. All these suggests that a judicious choice of and windows will be very useful to characterize the QGP and hadronic phase separately.

2.2. Invariant Mass () Dependence of EM Radiations

Being massive, dileptons make situation different from photons. They have two kinematic variables— and . As argued before, the spectra are affected by the flow; however, the -integrated spectra remain unaltered by the flow in the system. It should be mentioned here that for below peak and above peak dileptons from QGP dominates over its hadronic counterpart (assuming the contributions from hadronic cocktails are subtracted out) if the medium effect of spectral function of the low mass vector mesons are not taken into account. However, the spectral function of low mass vector mesons (mainly ) may shift toward lower invariant mass region due to nonzero temperature and density effects. As a consequence the contributions from the decays of mesons to lepton pairs could populate the low window and may dominate over the contributions from the QGP phase [5, 8, 20]. All these suggest that the invariant mass distribution of dilepton can be used as a clock for HIC, and a judicious choice of and windows will be very useful to characterize the flow in QGP and hadronic phase.

The measured dilepton spectra can be divided into several phases. Depending on the invariant mass of the emitted dileptons, it can be classified into three distinct regimes (discussed below [5]), and a schematic diagram of dilepton mass distribution is shown in Figure 2.

176578.fig.002
Figure 2: Expected different sources of dilepton production in HIC as function of invariant mass [5].

(i) High mass region (HMR):

The HMR region corresponds to early preequilibrium phase (), where the lepton pairs are produced with large invariant mass ( GeV) and the dominant contributions are from the hard scattering between the partons, like Drell Yan annihilation [21, 22]. The final abundance of the heavy quarkonia and their contribution to the spectrum is suppressed due to the Debye screening and as a result the bound states are dissolved.

(ii) Intermediate mass region (IMR):

Thermalization is achieved in the system after a time scale (). In this domain, the dileptons from the QGP are produced via quark-antiquark annihilation. In this regime, due to higher temperature the continuum radiation from QGP dominates the dilepton mass spectrum and thus this region is important for the detection of QGP. The decays of “open charm” mesons, that is, pairwise produced mesons [23] followed by semileptonic decays, contribute largely in this domain of . Although an enhanced charm production is interesting in itself—probably related to the very early collision states—it may easily mask the thermal plasma signal. To somewhat lesser extent, this also holds true for the lower-mass tail of Drell-Yan production [21, 22]. If the heavy quark does not get thermalized, then their contribution may be estimated from pp collision data with the inclusion of nuclear effects like shadowing, and so forth, and they do not contribute to the flow also [24].

(iii) Low mass region (LMR):

With subsequent expansion and cooling, the QGP converts into a hot hadron gas at the transition temperature, . At later stages, the dileptons are preferentially radiated from hot hadron gas from the decay of (light) vector meson, such as the , , and . The low domain of the lepton pairs is dominated by the decays of . Medium modification of will change the yield in this domain of . The change of spectral function is connected with the chiral symmetry in the bath; therefore, the measurement of low lepton pairs has great importance to study the chiral symmetry restoration [25, 26] at high temperature and density. Thus the invariant mass of the lepton pair directly reflects the mass distribution of the light vector mesons. This explains the distinguished role that vector mesons in conjunction with their in-medium modifications play for dilepton measurements in HIC.

So far, we have discussed the different sources of photons and dileptons. As QGP is expected to form in the HIC experiments, the basic intention of the present study is to study the properties of QGP. Therefore, we have emphasized more on the study of thermal photons and dileptons in this review, as they may provide information to understand the formation and unique properties of the novel matter.

The emission of thermal photons and dileptons coming from HIC consists of two important segments: (1)firstly, static emission rate () which takes care of the basic interactions in respective phases (QGP or hadronic phase),(2)Secondly, the space-time integration over four volume () which takes care of the evolution of the thermal matter created in HIC. As the EM radiations produced from each space-time point of the evolving matter, we need the concept of relativistic hydrodynamics (described in Section 4) for understanding the evolution.

3. Formulation of Thermal Emission Rate of EM Radiations

The importance of the electromagnetic probes for the study of thermodynamic state of the evolving matter was first proposed by Feinberg in 1976 [27]. Feinberg showed that the emission rates can be related to the electromagnetic current-current correlation function in a thermalized system.

3.1. Dilepton Emission Rate from Thermal Medium

Let us consider an initial state which goes to a final state producing a lepton pair with momenta and , respectively. The dilepton multiplicity thermally averaged over initial states is given by [4, 28] where in which is the lepton field operator and is the electromagnetic current and . Following [1, 4, 8] this expression can be put in the form where the factor is of the order of unity for electrons, being the invariant mass of the pair, and the electromagnetic (e.m.) current correlator is defined by Here is the electromagnetic current and indicates ensemble average. The rate given by (6) is to leading order in electromagnetic interactions but exact to all orders in the strong coupling encoded in the current correlator . The in the denominator indicates the exchange of a single virtual photon and the Bose distribution implies the thermal weight of the source. We can also express the dilepton rate in terms of a photon spectral function . Using the relation [4], in (6), we have where .

3.2. Photon Emission Rate from Thermal Medium

The photon emission rate is calculated in the similar way to that of dilepton rate. The photon emission rate differs from the dilepton rate in the following way: the factor appearing in the dilepton rate (in (9)) which is nothing but the product of electromagnetic vertex , the leptonic current involving Dirac spinors, and the square of the photon propagator should be replaced by the factor . And the phase space factor should be replaced by . Then the photon emission rate becomes The above emission rate is correct up to order in electromagnetic interaction but exact, in principle, to all order in strong interaction. However, for all practical purposes, one is able to evaluate up to a finite order of loop expansion. Now it is clear from the above results that to evaluate photon and dilepton emission rate from a thermal system we need to evaluate the imaginary part of the photon self-energy. The Cutkosky rules at finite temperature or the thermal cutting rules [2932] give a systematic procedure to calculate the imaginary part of a Feynman diagram. The Cutkosky rule expresses the imaginary part of the -loop amplitude in terms of physical amplitude of lower order ( loop or lower). This is shown schematically in Figure 3. When the imaginary part of the self-energy is calculated up to and including order loops where satisfies , then one obtains the photon emission rate for the reaction particles particles +, and the above formalism becomes equivalent to the relativistic kinetic theory formalism [2, 3].

176578.fig.003
Figure 3: Optical Theorem in Quantum Field Theory.
3.3. Emission Rate Using Relativistic Kinetic Theory Formalism

According to relativistic kinetic theory formulation, the production of -type particles from the reaction of type is given as follows: where is the overall degeneracy for the reaction under consideration, is the square of the invariant amplitude for the process under consideration, , , and are the three momentum, energy, and thermal distribution functions (Fermi-Dirac or Bose-Einstein) of the incoming and outgoing particles “.”

The transverse momentum () distribution of photons from a reaction of the type: taking place in a thermal bath at a temperature, is given by [2, 3]: Using the Mandelstam variables we can write the differential photon production rate as [33] where

In a similar way the dilepton emission rate for a reaction can be obtained as where is the appropriate occupation probability for bosons or fermions.

4. Relativistic Hydrodynamics

To evaluate the photon and dilepton production from HIC we need to convolute the static rate over space-time integration. Thus, we need to know hydrodynamics which takes care of the evolution of the matter. In this section, we briefly discuss the relativistic hydrodynamics for an ideal as well as viscous medium formed in HIC. Ideally, one cannot describe heavy ion experimental data from the first principle, that is, quantum chromodynamics (QCD) due to its complexity which mainly arises from nonlinearity of interactions of gluons, strong coupling, dynamical many body system, and color confinement. One promising strategy to connect the first principle with phenomena is to introduce hydrodynamics as a phenomenological theory. Relativistic hydrodynamics [18, 3443] plays an important role for an expanding system where pressure, temperature, and so forth vary with space and time. It is assumed that, due to intense rescatterings among the produced secondaries, the system reaches a state of local thermal equilibrium and then the evolution of the system is described by relativistic fluid dynamics. To describe the space-time evolution of such expanding system during the collision, the prescription of relativistic hydrodynamics is essential which assumes the system to be in local thermodynamic equilibrium, which means that pressure and temperature are not constant but rather are the function of space and time. This prescription is valid in the regime where the mean-free path in this “thermalised” system is much smaller than the characteristic dimensions of the system ; that is, .

4.1. Basic Equations of Ideal Hydrodynamics

The space-time evolution of the pressure, energy density, particle densities, and the local fluid velocities is controlled by energy momentum conservation equations from hydrodynamics. The basic equations of relativistic hydrodynamics which result from applying constraints of energy-momentum conservations relevant for heavy ion collision at relativistic energies are expressed in where is the energy-momentum tensor of fluid element, and in its local rest frame it is given by

Local rest frame is the frame in which the velocity of the fluid element is zero. In such a frame the becomes diagonal since the energy flux of the fluid and the momentum density turns to be zero. In absence of any dissipative processes the component becomes the energy density and since is the th component of force acting on the surface element which according to Pascal’s law is isotropic and perpendicular to the surface. is the pressure of the fluid element in the local rest frame. Isotropy implies that the energy flux and the momentum density vanish in the rest frame of fluid. In addition, it implies that the pressure tensor is proportional to the identity matrix, that is, , where is the thermodynamic pressure.

By doing a proper Lorentz transformation, the energy-momentum tensor in a moving frame, where the fluid moves with an arbitrary four-velocity, where , is given in where is the Mankowski metric tensor and is the fluid 4-velocity referred to as “collectivity” of the system which can be defined as with and , where is the velocity of fluid element. In the above equation, the and are the energy density and pressure, respectively, in the fluid rest frame, and both are functions of space time coordinate .

Apart from the energy-momentum conservation, a fluid may contain several conserved charges, such as total electric charge, and net baryon number. The conserved charges obey the following continuity equation given in (19): is the conserved net baryonic current and is baryon number density. For the present work the net baryon number is assumed to be negligible small, so (16) is the only relevant equation to deal with. In addition to it, the total entropy of an inviscid fluid is conserved throughout (). If we define the entropy current: , then the conservation of entropy results in [44].

4.2. Basic Equations of Viscous Hydrodynamics

In the above discussion we considered an idealized situation of a perfect fluid with no internal friction or energy dissipation. But in practice most of the times we have to deal with a system of imperfect fluid in which the density, pressure, and fluid velocity changes over a distance of the order of mean-free path. Such presence of a space-time gradient of those thermodynamic quantities results in modifying the energy momentum tensor and the conserved current to the first-order gradient of these quantities:

One thing should be mentioned here that for a relativistic fluid it is necessary to specify whether is the velocity of energy transport or velocity of particle transport. In the approach of Landau and Lifshitz, is taken to be the velocity of energy transport and so vanishes in a comoving frame. In the approach of Eckart, is taken to be the velocity of particle transport and so in a comoving frame. The second approach is adopted here to obtain the following assumptions. The modification in the energy momentum tensor and conserved current is such that in a comoving frame: With these assumptions we need to construct to quantify the dissipative processes within the system. This has to be done in such a way that the rate of entropy production per unit volume is positive, which is again required from second law of thermodynamics. To accomplish this task some guidelines are to be followed.(1)The thermodynamic quantities , , and vary slightly over the mean-free path of the particles within the fluid; that is, the system is only very slightly away from equilibrium. So the dissipative term in energy momentum tensor must be a linear combination of space-time derivatives of , , , and so forth.(2)Only the space time derivative of and can occur in because if derivative of , , or appeared in , then would contain pressure or density gradient, with velocity or temperature gradient and these products are not always positive for all fluid configurations.

The entropy production rate comes out to be

From the condition that for all fluid configuration we obtain where is the coefficient of shear viscosity, is coefficient of bulk viscosity, and is thermal conductivity.

Generalizing this expression comes out to be [45] Here we have ignored the terms related to thermal conductivity since we are not showing any effect of that on any observables. We have defined , where is the projection operator.

For the present study, the evaluation of matter from QGP (initial) to the hadronic system (final) via an intermediate quark-hadron transition is studied by applying relativistic hydrodynamics.

4.3. Space-Time Evolution

Hydrodynamics is a general framework to describe the space-time evolution of locally thermalized matter for a given equation of state (EoS). The basic ingredients required to solve the ideal hydrodynamic equations are EoS and initial conditions. As the system expands from its initial state, the mean-free path between particles within the system increases. At certain stage, the mean-free path becomes comparable to the system size, and then the hydrodynamic description breaks down and the phase space distribution of the particle gets fixed by the temperature of the system at this stage. This stage of evolution is called freezeout state and the corresponding temperature of the system is called thermal freezeout temperature (). The hydrodynamic evolution stops at the freezeout point.

4.3.1. Initial Condition

The initial conditions are crucial to the description of space-time evolution. Initial conditions in hydrodynamics may be constrained in the following ways to reproduce the measured final multiplicity. We assume that the system reaches equilibration at a time (called initial thermalization time) after the collision. The can be related to the measured hadronic multiplicity () by the following relation [46]: where is the radius of the system, is the Riemann zeta function, and , is the degeneracy of quarks and gluons in QGP, = number of colors, = number of flavors. The factor “7/8” originates from the difference between the Bose-Einstein and the Fermi-Dirac statistics. depends on the centrality through the multiplicity, . The value of for various beam energies and centralities can be obtained directly from experiment or calculated using the following relation [47]: where is the multiplicity per unit rapidity measured in pp collisions: , the fraction of is due to “hard” processes, with the remaining fraction being “soft” processes. The multiplicity in nuclear collision has then two components: “soft,” which is proportional to number of participants, and “hard,” which is proportional to number of binary collision, .

After the initial thermalization time, , the system can be treated hydrodynamically. The initial conditions to solve the hydrodynamic equations are given through the energy density and velocity profile: where is the initial energy density which is related to initial (), is the nuclear radius, and is the diffusion parameter taken as 0.5 fm.

4.3.2. Equation of State (EoS)

The set of hydrodynamic equations are not closed by itself; the number of unknown variable exceeds the number of equations by one. Thus a functional relation between any two variables is required so that the system become deterministic. The most natural course is to look for such relation between the pressure and the energy density . Under the assumption of local thermal equilibrium, this functional relation between , , and is the EoS: which expresses the pressure as function of energy density, , and baryon density, . This can be obtained by exploiting numerical lattice QCD simulation [48].

Different EoSs (corresponding to QGP vis-a-vis that of hadronic matter) will govern the hydrodynamic flow quite differently. It is thus imperative to understand in what respects the two EoSs differ and how they affect the evolution in space and time. The role of the EoS in governing the hydrodynamic flow lies in the fact that the velocity of sound, sets an intrinsic scale in hydrodynamic evolution. One can thus write simple parametric form of the EoS: , for baryon-free system which is relevant for the present study.

4.3.3. Freeze-Out Criteria

The expansion persists as long as the fluid particles interact. At sufficiently longer when it is comparable to system size the particles decouple to behave as free particles which is called “freeze-out” stage. This freeze-out scenario is characterized by a system temperature which is of the order of pion mass and defines a space-time surface which serves as the boundary of the hydrodynamical flow [49].

5. Emission of Thermal Photons from Heavy Ion Collision

The thermal photons emerge just after the system thermalizes () from both QGP due to partonic interactions and hot hadrons due to interactions among the hadrons. Now with the formalism discussed in Section 3, the production of thermal photons from QGP and hot hadronic gas is given in Sections 5.1 and 5.2, respectively. And using the hydrodynamic equations, we have convoluted these static rates by space-time integration (discussed in Section 4) and obtained the total invariant yield of photon for different collision energies. The space-time integration is constrained to the hydrodynamical inputs which has been discussed elaborately in this section.

5.1. Photons Emission from Quark Gluon Plasma

The contribution from QGP to the spectrum of thermal photons due to annihilation () and Compton () processes has been calculated in [17, 50] using hard thermal loop (HTL) approximation [51, 52]. The rate of hard photon emission is then obtained as [17] where is the strong coupling constant. Later, it was shown that photons from the processes [53]: ,  , , and contribute in the same order as Compton and annihilation processes (shown in Figure 4). The complete calculation of emission rate from QGP to order has been performed by resuming ladder diagrams in the effective theory [54, 55]. In the present work this rate has been used. The temperature dependence of the strong coupling, , has been taken from [56].

fig4
Figure 4: Partonic processes for production of photons.
5.2. Photons Emission from Hot Hadronic Gas

For the photon spectra from hadronic phase we consider an exhaustive set of hadronic reactions and the radiative decay of higher resonance states [33, 57, 58].

To evaluate the photon emission rate from a hadronic gas we model the system as consisting of , , , and . The relevant vertices for the reactions and and the decay are obtained from the following Lagrangian [57] (see Figure 5): where is the Maxwell field tensor and is the hadronic part of the electromagnetic current given by with .

176578.fig.005
Figure 5: Photon-producing reactions and decays in hadronic gas.

For the sake of completeness we have also considered the photon production due to the reactions , , and the decay using the following interaction: The last term in the above Lagrangian is written down on the basis of vector meson dominance (VMD) [60, 61]. To evaluate the photon spectra, we have taken the relevant amplitudes for the abovementioned interactions from [33, 57]. The effects of hadronic form factors [62] have also been incorporated in the present calculation. The reactions involving strange mesons: , , , and [62, 63] have also been incorporated in the present work. Contributions from other decays, such as , , , , and , have been found to be small [63] for  GeV.

With all photon-producing hadronic reaction, the static thermal emission rate of photons for hadronic phase has been evaluated [17, 33, 54, 55, 57, 62]. The reaction involving mesons has dominant contribution. The rate at low photon energy is dominated by reaction with in final state, because these reactions are endothermic with most of the available energy going into rho mass. At high photon energy reactions with the in initial state are dominant because these reactions are exothermic; most of the rho mass is available for the production of high energy photons. Similar remarks can be made concerning reactions involving mesons, but as the value of is smaller, thus so are the rates. All the isospin combinations for the above processes have properly been implemented.

5.3. Total Invariant Momentum Spectra of Thermal Photons

In this section we evaluate photon spectrum from a dynamically evolving system. The evolution of the system is governed by relativistic hydrodynamic. The photon production from an expanding system can be calculated by convoluting the static thermal emission rate with the expansion dynamics, which can be expressed as follows: where the is the four volume. The energy, , appearing in (33) should be replaced by for a system expanding with space-time-dependent four-velocity . Under the assumption of cylindrical symmetry and longitudinal boost invariance, can be written as where is the radial velocity, and, therefore, for the present calculations, For massless photon the factor can be obtained by replacing in (35) by . For the system produced in QGP phase reverts to hot hadronic gas at a temperature . Thermal equilibrium may be maintained in the hadronic phase until the mean-free path remains comparable to the system size. The term “” is the static rate of photon production where stands for quark matter (QM), mixed phase () (in a 1st-order phase transition scenario), and hadronic matter (HM), respectively. The dependence of the photon and dilepton spectra originating from an expanding system is predominantly determined by the thermal factor . The total momentum distribution can be obtained by summing the contribution from QM and HM, where the distribution for both the phases can be obtained by choosing the phase space appropriately.

The integration has been performed by using relativistic hydrodynamics with longitudinal boost invariance [41] and cylindrical symmetry [64] along with the inputs (given in Table 1) as the initial conditions for SPS and RHIC energies.

tab1
Table 1: The values of various parameters—thermalization time (), initial temperature (), and hadronic multiplicity (the value of for various beam energies and centralities is calculated from (26))—used in the present calculations.

To estimate for RHIC, we have taken and at  GeV. It should be mentioned here that the values of (through and in (26)) and hence the (through in (25)) depend on the centrality of the collisions. For SPS, is taken from experimental data [65]. We use the EoS obtained from the lattice QCD calculations by the MILC collaboration [66]. We consider kinetic freeze-out temperature,  MeV for all the hadrons. The ratios of various hadrons measured experimentally at different indicate that the system formed in heavy ion collisions chemically decouple at which is higher than which can be determined by the transverse spectra of hadrons [67, 68]. Therefore, the system remains out of chemical equilibrium from to . The deviation of the system from the chemical equilibrium is taken into account by introducing chemical potential for each hadronic species. The chemical nonequilibration affects the yields through the phase space factors of the hadrons which in turn affects the productions of the EM probes. The value of the chemical potential has been taken into account following [69].

5.4. Results and Discussion on Distributions of Photons

For comparison with direct photon spectra as extracted from HIC two further ingredients are required. With all the ingredients we have reproduced the spectra of direct photon for both SPS and RHIC energies. The prompt photons are normally estimated by using perturbative QCD. However, to minimize the theoretical model dependence here, we use the available experimental data from p-p collisions to estimate the hard photon and normalized it to A-A data with for different centrality; that is, the photon production from A-A collision and p-p collision are related to the following relation: where is taken for the corresponding experiments and the the typical ( 41 mb for RHIC and 30 mb for SPS).

5.4.1. Photon Spectrum for WA98 Collaboration

The WA98 photon spectra from Pb+Pb collisions are measured at  GeV. However, no data at this collision energy is available for pp interactions. Therefore, prompt photons for p+p collision at  GeV have been used [70] to estimate the hard contributions for nuclear collisions at  GeV. Appropriate scaling [65] has been used to obtain the results at  GeV. For the Pb+Pb collisions the result has been appropriately scaled by the number of collisions at this energy (this is shown in Figure 6 as prompt photons). The high part of the WA98 data is reproduced by the prompt contributions reasonably well. At low the hard contributions underestimate the data indicating the presence of a thermal source. The thermal photons with initial temperature  MeV along with the prompt contributions explain the WA98 data well (Figure 6), with the inclusion of nonzero chemical potentials for all hadronic species considered [25, 26, 69, 71, 72]. In some of the previous works [7378] the effect of chemical freezeout is ignored. As a result either a higher value of or a substantial reduction of hadronic masses in the medium was required [73]. In the present work, the data has been reproduced without any such effects.

176578.fig.006
Figure 6: Transverse momentum spectra of photon at SPS energy for Pb+Pb collision at midrapidity [12].

5.4.2. Photon Spectrum for PHENIX Collaboration

In Figure 7, transverse momentum spectra of photons at RHIC energy for Au-Au collision for three different centralities (0–20%, 20–40%, and min. bias.) at midrapidityi shown, where the red tangles are the direct photon data measured by PHENIX collaboration [79] from Au-Au collision at  GeV, blue-dashed line is the contribution of the prompt photons and the black solid line is thermal + prompt photons. For the prompt photon contribution at  GeV, we have used the available experimental data from pp collision and normalized it to Au-Au data with for different centrality [80] (using (36)). At low the prompt photons underestimate the data indicating the presence of a possible thermal source. The thermal photons along with the prompt contributions explain the data [79] from Au-Au collisions at  GeV reasonably well. The reproduction of data is satisfactory (Figure 7) for all the centralities with the initial temperature shown in Table 1 [81].

176578.fig.007
Figure 7: Transverse momentum spectra of photons at RHIC energy for Au-Au collision for different centralities at midrapidity [12].
5.4.3. Photon Spectrum for ALICE Collaboration

The direct photon spectra from Pb+Pb collisions are measured at  TeV for 0–40% centrality by ALICE collaboration. However, no data at this collision energy is available for pp interactions. Therefore, prompt photons from p+p collision at  TeV have been used to estimate the hard contributions for nuclear collisions at  TeV by using the scaling (with ) procedure used in [65]. For the Pb+Pb collisions the result has been scaled up by the number of collisions at this energy (this is shown in Figure 8 as prompt photons). The high part of the data is reproduced by the prompt contributions reasonably well. At low the hard contributions underestimate the data indicating the presence of a possible thermal source.

176578.fig.008
Figure 8: Transverse momentum spectra of photons at LHC energy for Pb-Pb collision for different 0–40% centrality at midrapidity [59].

The thermal photons with initial temperature ~553 MeV along with the prompt contributions explain the data well (Figure 8), with the inclusion of nonzero chemical potentials for all hadronic species considered [69] (see also [71, 72]).

It is well known that transverse momentum spectra of photons act as a thermometer of the interior of the plasma. The inverse slope of the thermal distribution is a measure of the average (over evolution) effective (containing flow) temperature of the system. We have extracted the average effective temperature () from the thermal distributions of photons at different collision energies—that is, for SPS, RHIC, and LHC energies. Figure 9 shows the variation of with multiplicity for different collision energies. To minimize the centrality dependence of the results the is normalized by . The results clearly indicate a significant rise in the average () while going from SPS to RHIC to LHC. The values of for different collision energies are given in Table 1. Since photons are emitted from each space time point of the system, therefore, the measured slope of the spectra represents the average effective temperature of the system.

176578.fig.009
Figure 9: The variation of with the increase in multiplicity for different collision energies.

The quantity, ), is proportional to the entropy density. Therefore, , the average effective statistical degeneracy, a quantity which changes drastically if the colour degrees of freedoms deconfined; that is, if a phase transition takes place in the system. We find that the entropy density () at LHC increases by almost compared to RHIC, and there is an enhancement of at RHIC compared to SPS. However, part of this increase is due to the increase in the temperature and part is due to increase in degeneracy. To estimate the increase in the degeneracy we normalize the quantity by . Therefore, we estimate from the analysis of the experimental data and found that there is a increase in this quantity from SPS to RHIC and increase from RHIC to LHC.

5.5. Total Invariant Momentum Spectra of Thermal Photons in Viscous Medium

Effects of viscosity on the transverse momentum distribution of photons were earlier considered in [82, 83] and recently the interest in this field is renewed [8486]. The measured photon spectra () are the yield obtained after performing the space time integration over the entire evolution history—from the initial state to the freezeout point using (33). Beyond a certain threshold in collision energy the system is expected to be formed in QGP phase which will inevitably make a transition to the hadronic matter later. The measured spectra contain contributions from both QGP and hadronic phases. Therefore, it becomes imperative to estimate the photon emission with viscous effects from QGP as well as hadrons and identify a kinematic window where photons from QGP dominate. While in some of the earlier works [8486] contributions from hadrons were ignored, in others [82, 83] the effects of dissipation on the phase space factors were omitted. In the present work we study the effects of viscosity on the thermal photon spectra originating from QGP and hadronic matter and argue that photons can be used as a very useful tool to estimate and hence characterize the matter.

Equation (12) can be simplified to the following form (see the appendix) [87]: The effects of viscosity on the photon spectra resulting from HIC enter through two main factors: (i) the modification of the phase space factor due to the deviation of the system from equilibrium and (ii) the space time evolution of the matter governed by dissipative hydrodynamics. One more important issue deserves to be mentioned here. Normally, the initial temperature () and the thermalization time () are constrained by the measured hadron multiplicity (). This approach is valid for a system where there is no viscous loss and the time reversal symmetry is valid. However, for a viscous system the entropy at the freezeout point (which is proportional to the multiplicity) contains the initially produced entropy as well as the entropy produced during the space time evolution due to nonzero shear and bulk viscosity. Therefore, the amount of entropy generated during the evolution has to be subtracted from the total entropy at the freezeout point, and the remaining part which is produced initially should be used to estimate the initial temperature. Therefore, for a given (which is associated with the freezeout point) and the magnitude of will be lower in case of viscous dynamics compared to ideal flow.

5.5.1. Viscous Correction to the Distribution Function

We assume that the system is slightly away from equilibrium which relaxes back to equilibrium through dissipative processes. Here we briefly recall the main considerations leading to the commonly used form for the first viscous correction, , to the phase space factor, , defined as follows [88]: where is the equilibrium distribution function of “th” particle, , , , being the four-velocity of the fluid. The coefficients and can be determined in the following way. Substituting in the expression for stress-energy tensor we get where is the energy momentum tensor for ideal fluid. From general considerations [44] the dissipative part can be written as Equating the part containing from (38) with (40), and can be expressed in terms of the coefficients of shear () and bulk () viscosity, respectively, in terms of which the phase space distribution for the system can be written as

For a boost invariant expansion in dimension this can be simplified to get where where is the -component of the momentum in the fluid comoving frame. The phase space distribution with viscous correction (42) thus enters the production rate of photon through (37).

5.5.2. Viscous Correction to the Expansion Dynamics

As mentioned before the distribution of thermal photons is obtained by integrating the emission rate over the evolution history of the expanding fluid. Relativistic viscous hydrodynamics can be used as a tool for the space-time dynamics of the fluid.

For a dimensional boost invariant expansion [41] the evolution equation, , can be written as [89] where is the pressure and is the energy density. We assume that the baryonic chemical potential is small in the central rapidity region for RHIC/LHC collision energies. Therefore, the equation corresponding to the net baryon number conservation need not be considered in these situations.

We assume that the system achieves thermal equilibrium at a time after the collision at an initial temperature . With this initial condition and equation of state (EoS) the solution of (44) can be written as [82] where , , and .

Equation (45) dictates the cooling of the QGP phase from its initial state to the transition temperature, , at a time, , when the QGP phase ends.

In a first-order phase transition scenario, the pure QGP phase is followed by a coexistence phase of QGP and hadrons. The energy density, shear, and bulk viscosities in the mixed phase can be written in terms of the corresponding quantities of the quark and hadronic phases at temperature as follows [82]: where indicates the fraction of the quark (hadronic) matter in the mixed phase at a proper time . We have , , , , is the bag constant, () denote statistical degeneracy for the QGP (hadronic) phase. In the mixed phase the temperature remains constant but the energy density varies with time as the conversion of QGP to hadrons continues. This time variation is executed through . Substituting (46) in (44) and solving for we get [82] where , , , and . Equation (47) indicates how the fraction of QGP in the coexistence phase evolves with time.

The variation of with in the hadronic phase can be obtained by solving (44) with the boundary condition and , where is the (proper) time at which the mixed phase ends; that is, when the conversion of QGP to hadronic matter is completed, Similar to QGP, has been used for hadronic phase. For a vanishing bulk viscosity () the cooling of the QGP is dictated by Similarly the time variation of temperature in the hadronic phase is given by In a realistic scenario the value of may be different for QGP [9094] and hadronic phases [9598]. However, in the present work we take the same value of both for QGP and hadronic matter as shown in Table 2.

tab2
Table 2: The values of various and parameters—initial temperature (), starting time of mixed phase (), and ending time of mixed phase ()—used in the present calculations.

5.5.3. Results and Discussion on Viscous Effect on Distributions of Photons

In case of an ideal fluid, the conservation of entropy implies that the rapidity density is a constant of motion for the isoentropic expansion [41]. In such circumstances, the experimentally observed (final) multiplicity, , may be related to a combination of the initial temperature and the initial time as . Assuming an appropriate value of (taken to be ~0.6 fm/c in the present case), one can estimate .

For dissipative systems, such an estimate is obviously inapplicable. Generation of entropy during the evolution invalidates the role of as a constant of motion. Moreover, the irreversibility arising out of dissipative effects implies that estimation of the initial temperature from the final rapidity density is no longer a trivial task. We can, nevertheless, relate the experimental to the freezeout temperature, , and the freezeout time, , by the relation: where is the radius of the colliding nuclei (we consider collision for simplicity) and is a constant ~3.6 for massless bosons.

To estimate the initial temperature for the dissipative fluid we follow the following algorithm. We treat as a parameter; for each , we let the system evolve forward in time under the condition of dissipative fluid dynamics (44) till a given freezeout temperature is reached. Thus is determined. We then compute at this instant of time from (51) and compare it with the experimental . The value of for which the calculated matches the experimental number is taken to be the value of the initial temperature. Once is determined, the evolution of the system from the initial to the freezeout stage is determined by (45), (47), and (48).

In Figure 10 we display the variation of temperature with proper time. It is clear from the results shown in the inset (Figure 10) that initial temperature for system which evolves with nonzero viscous effects is lower compared to the ideal case for a fixed . Because of a nonviscous isentropic evolution scenario the multiplicity (measured at the freezeout point) is fixed by the initial entropy. However, for a viscous evolution scenario the generation of entropy due to dissipative effects contributes to the multiplicity. Therefore, for a given multiplicity (which is proportional to the entropy) at the freezeout point one requires lower initial entropy; hence, initial temperature will be lower. It is also seen (Figure 10) that the cooling of the system is slower for viscous dynamics because of the extra heat generated during the evolution.

176578.fig.0010
Figure 10: Variation of temperature with proper time for different phases for various values of the shear viscosities. Inset shows the effect of viscosity on the cooling of the QGP phase (in an amplified scale) for different values of .

In this section we present the shift in the distribution of the photons due to viscous effects. The integrand in (33) is a Lorentz scalar; consequently the Lorenz transformation of the integrand from the laboratory to the comoving frame of the fluid can be effected by just transforming the argument; that is, the energy of the photon () in the laboratory frame should be replaced by in the comoving frame of the fluid, where is the four momenta of the photon.

The results presented here are obtained with vanishing bulk viscosity. The effects of viscosity enter into the photon spectra through the phase space factor as well as through the space time evolution. We would like to examine these two effects separately. For convenience we define two scenarios: (i)the effects of viscosity on the phase space factor are included () in (42), but the viscous effects on the evolution are neglected ( ) in (44),(ii)the effects of are taken into account in the phase space factors as well as in the evolution dynamics.

The space time-integrated photon yield originating from the QGP in scenario (i) is displayed in Figure 11. Note that the value of the initial temperatures for the results displayed in Figure 11 is the same (for all ) because the viscous effects on the evolution are ignored in scenario (i). The viscous effects on the distribution of the photons are distinctly visible. The higher values of make the spectra flatter through the dependence of the correction, .

176578.fig.0011
Figure 11: Transverse momentum distribution of thermal photons from QGP for various values of in the scenario (i).

Next we assess the effects of viscosity on photon spectra for scenario (ii). In Figure 12 we depict the photon spectra for various values of . In this scenario the value of is lower for higher for reasons described above. As a result the enhancement in the photon production due to change in phase space factor, , is partially compensated by the reduction in for nonzero , which is clearly seen in the results displayed in Figures 11 and 12.

176578.fig.0012
Figure 12: Transverse momentum distribution of thermal photons from QGP for various values of in the scenario (ii).

In Figures 13 and 14 we exhibit results for the hadronic phase for scenarios (i) and (ii), respectively. The effects of dissipation on the distribution of photons from hadronic phase are qualitatively similar to the QGP phase; that is, the effect is more prominent in scenario (i) than in (ii). It is also clearly seen that the effects of viscosity though the effect is stronger in the QGP phase than in the hadronic phase. It is expected that the observed shift in the photon spectra due to viscous effects may be detected in future high precision experiments.

176578.fig.0013
Figure 13: Transverse momentum distribution of photons from thermal hadrons for various values of in the scenario (i).
176578.fig.0014
Figure 14: Transverse momentum distribution of photons from thermal hadrons for various values of in the scenario (ii).

Finally in Figures 15 and 16 we plot the spectra of photons for the entire life time of the thermal system; that is, the photon yield is obtained by summing up contributions from QGP, mixed and hadronic phases for different values of for scenario (i) and (ii), respectively. The effect of viscosity for the scenario (i) is stronger than (ii).

176578.fig.0015
Figure 15: Transverse momentum distribution of thermal photons from the entire evolution history of the system for various values of in the scenario (i).
176578.fig.0016
Figure 16: Transverse momentum distribution of thermal photons from the entire evolution history of the system for various values of in the scenario (ii).

6. Emission of Thermal Dileptons from Heavy Ion Collision

Unlike real photon, dilepton is massive. Thus dilepton has two kinematic variables, invariant mass and transverse momentum (). Again, the spectra are affected due to flow, whereas the -integrated spectra remain unaltered by flow. By tuning these two parameters, different stages of expanding fireball can be understood. Dileptons having large and high are emitted early from the hot zone of the system. On the other hand, those having lower and produced at later stage of the fireball when the temperature is low. Because of an additional variable, the invariant pair mass , dileptons have the advantage over real photons [99].

The production of thermal dileptons from QGP (Section 6.1) and hot hadronic gas (Section 6.2) is described below.

6.1. Dileptons Emission from QGP

In the QGP, where quarks and gluons are the relevant degrees of freedom, the can be directly evaluated by writing the electromagnetic current in terms of quarks of flavor , that is, . Confining to the leading order contribution we obtain The rate in this case corresponds to dilepton production due to process . The static thermal emission rate of dilepton from QM is given by () [100, 101] (also [102, 103]), where is the charge of the quark and .

6.2. Dileptons Emission from Hot Hadronic Gas

To obtain the rate of dilepton production from hadronic interactions it is convenient to break up the quark current into parts with definite isospin: where and denote iso-vector and iso-scalar currents and the dots denote currents comprising of quarks with strangeness and heavier flavors. These currents couple to individual hadrons as well as multiparticle states with the same quantum numbers and are usually labeled by the lightest meson in the corresponding channel [104]. We thus identify the isovector and isoscalar currents with the and mesons, respectively. Defining the correlator of these currents analogously as in (7), we can write The correlator of vector-isovector currents has in fact been measured [105, 106] in vacuum along with the axial-vector correlator by studying decays into even and odd number of pions. The former is found to be dominated at lower energies by the prominent peak of the meson followed by a continuum at high energies. The axial correlator, on the other hand, is characterized by the broad hump of the . The distinctly different shape in the two spectral densities is an experimental signature of the fact that chiral symmetry of QCD is dynamically broken by the ground state [107]. It is expected that this symmetry may be restored at high temperature and/or density and will be signaled by a complete overlap of the vector and axial-vector correlators [17].

In the medium, both the pole and the continuum structure of the correlation function gets modified [8, 108]. We will first evaluate the modification of the pole part due to the self-energy of vector mesons in the following. Using vector meson dominance the isovector and scalar currents are written in terms of dynamical field operators for the mesons allowing us to express the correlation function in terms of the exact (full) propagators or the interacting spectral functions of the vector mesons in the medium. To reach that goal we have to specify the coupling of the currents to the corresponding vector fields. For this purpose we write, in the narrow width approximation [104], where denotes the resonance in a particular channel and is the corresponding polarization vector. The coupling constants are obtained from the partial decay widths into through the relation yielding = 0.156 GeV, 0.046 GeV, and 0.079 GeV for , , and , respectively. Equation (56) suggests the operator relations: where denotes the field operator for the meson. So using the above relations connecting currents to fields (so-called field-current identity), the current commutator becomes where are the spectral functions of corresponding vector meson resonances and is the diagonal element of the thermal propagator matrix. The form of the diagonal element of the exact thermal propagator matrix for the spin 1 particle is given by where

The imaginary part is then put in (59) and then in (6) to arrive at the dilepton emission rate: where, for example, is given by the sum running overall meson loops and baryon loops . Here is the diagonal element of vector meson ( and ) self-energy at finite temperature and density which is also a matrix in the real-time formalism. We have taken [109] and , , , , , , and [110] for meson whereas for meson, (with folding), and , , , , , and [111] are taken. These self-energy graphs are diagrammatically represented in Figure 17.

176578.fig.0017
Figure 17: One-loop Feynman diagrams for or self-energy involving mesons (first figure) and baryons (second and third figures). stands for the or in the external line. In the internal lines of meson loops, , , , and for whereas for . For the baryonic loops, and indicate, respectively, nucleon and baryonic internal lines.

The general expression of for meson loop (representing the first diagram of Figure 17) is given by [109, 111] where ’s are Bose-Einstein distribution functions for the internal meson lines and ’s are their on-shell energies. In the above expression denote the values of for , , , and , respectively. The corresponding expression for the baryon loop (second diagram of Figure 17) is given by [110, 111] where ’s are Fermi-Dirac distribution functions for the internal baryon (antibaryon) lines. Here, denote the values of for , , , , respectively. The expression for the third diagram of Figure 17 can be obtained by changing the sign of the external momentum in (65).

The numerical results for the and meson self-energy are, respectively, shown in Figures 18(a) and 18(b). The individual contribution from the meson and baryon loops is also shown for two values of the baryon chemical potential. For both and mesons, the small positive contribution from the baryon loops to the real part is partly compensated by the negative contributions from the meson loops which can be clearly seen in the lower panels of Figure 18.

fig18
Figure 18: The imaginary (upper) and real (lower) part of self-energy function of (a) and (b) at different chemical potential ( or ). The contribution coming from meson, baryon loops, and their sum is shown.

We now use these self-energy functions in the expression for the exact propagator (60) to obtain an explicit results of in-medium spectral functions for and meson. In view of the fact that the and peaks are close to each other, it is worthwhile to compare their relative spectral strengths below their nominal masses. This is shown in Figure 19 for two values of the chemical potential. The characteristic and thresholds for the and in the vacuum case are also visible. At fixed temperature and density, the contribution is lower than but of comparable magnitude below their nominal masses. However, the fact that the is suppressed by a factor ~10 () compared to the in the dilepton emission rate makes a quantitative study of the difficulty. In the above expressions the meson () and baryon resonances () have been treated in the narrow width approximation. These have then been folded with the width of the resonances as shown in [110].

176578.fig.0019
Figure 19: The spectral function seen in comparison with the .

Thus, the dilepton emission rate in the present scenario actually boils down to the evaluation of the self-energy graphs of and as a function of , , temperature , and net baryon density (). Using those functions in (63) we can get a numerical estimation of dilepton static rates. With all the ingredients discussed previously, we have calculated the static emission rate of dilepton from QGP and hadronic matter. The emission rate from both the phases is plotted in Figure 20 for a given temperature of 175 MeV and baryonic chemical potential of 30 MeV. We observe significant enhancement in the dilepton yield in the mass region below the pole compared to vacuum. This rate has been used in the analysis of the dimuon spectra obtained from In-In collisions at 17.3 GeV at CERN SPS [112, 113] (discussed in Section 6.4.1). The calculations show a reasonable agreement with the invariant mass spectra for different ranges as well as the spectra for different bins.

176578.fig.0020
Figure 20: The dilepton emission rate from different sources at  MeV and  MeV.

As indicated earlier, coupling of the hadronic current to multiparticle states gives rise to a continuum structure in the current correlation function . Following [104], we take a parameterized form for this contribution and augment the dilepton emission rate with where with = 1.3, 1.1 GeV for , , and for both and . The continuum contribution for the contains an additional factor of .

6.3. Invariant Mass and Momentum Spectra of Dileptons

The total invariant transverse momentum distribution of thermal dileptons () is obtained as follows: In a similar manner, the invariant transverse mass distribution of thermal dileptons () can be obtained by integrating static emission rate over certain window and by convoluting that by four volume and expressed as follows: The limits for integration over and can be fixed judiciously to detect contributions either from quark matter or hadronic matter. Experimental measurements [79, 114, 115] are available for different window.

6.4. Results and Discussion on and Distributions of Dileptons

Thus far we have discussed the dilepton emission rate for a given temperature. In HIC the dilepton yield is obtained by convoluting the static emission rate over space and time.

6.4.1. Dileptons at SPS Energy

With all these ingredients the and spectra of dileptons measured by NA60 collaboration at SPS energy are reproduced as follows.

Finally, we have obtained the dimuon yield () in In-In collisions at SPS at a center of mass energy of 17.3 GeV. The initial energy density is taken as 4.5 GeV/fm3 corresponding to a thermalisation time  fm, the QGP to hadronic matter transition temperature  MeV, and the freezeout temperature  MeV (fixed from the slope of the hadronic spectra measured by the NA60 Collaboration) has been taken to compare the data measured by NA60 collaboration. In Figure 21 we have shown the invariant mass spectra for different transverse momentum () windows calculated for 17.3 GeV energies.

176578.fig.0021
Figure 21: Dilepton invariant mass spectra for different bins compared with the NA60 data.

The theoretical curves agree quite well with the experimental data [114, 115] for all the ranges. The strong enhancement in the low domain is clearly due to the large broadening of the in the thermal medium which comes entirely from the Landau cut in the self-energy diagrams. In the last panel, the blue-dashed line curve is the result of a previous calculation [113] where the self-energy due to baryons has been evaluated following the approach of [116]. In the present work [112] we have included an exhaustive set of baryon loops using the real-time thermal field theoretical approach where we have employed the full relativistic baryon propagators in which baryons and anti-baryons appear on an equal footing. This [112] is seen to be in better agreement with the experimental data [114, 115] than [113] in the range  GeV.

Apart from the spectra, we have also evaluated the transverse mass spectra of dimuon pairs at SPS energy measured by NA60 collaboration [114, 115].

The results are compared with the data obtained by NA60 collaborations [114, 115, 117, 118] at SPS energy (Figure 22). Theoretical results contain contributions from the thermal decays of light vector mesons (, , and ) and also from the decays of vector mesons at the freezeout [10, 49] of the system has also been considered. The nonmonotonic variation of the effective slope parameter extracted from the spectra of the lepton pair with evaluated within the ambit of the present model [113] reproduces the NA60 [114, 115] results reasonably well.

176578.fig.0022
Figure 22: Transverse mass spectra of dimuons in In+In collisions at SPS energy. Solid lines denote the theoretical results [12].
6.4.2. Dileptons at RHIC Energy

For Au+Au collisions at  GeV, we have evaluated the dilepton spectra for different invariant mass bins with the initial condition (min bias) shown in Table 1 and lattice QCD equation of state. The results are displayed in Figure 23. The slopes of the experimental data on distribution of lepton pairs for different invariant mass windows measured by the PHENIX collaboration [119, 120] could be reproduced well with the same initial condition that reproduces photon spectra [79]. In fact, the reproduction of data for the mass bins and does not need any normalization factors (Figure 23). For lower mass windows slopes are reproduced well but fail to reproduce the absolute normalization. Therefore, it should be clarified here that the theoretical results shown in Figure 22 for lower mass windows (to be precise for , and ) contain arbitrary normalization constant.

176578.fig.0023
Figure 23: Transverse momentum spectra of dileptons for different invariant mass windows for minimum bias Au-Au collisions at RHIC energy [12].

Assuming hard (i.e., ) and soft collisions for initial entropy production the value of turns out to be about 2.43 at  GeV. For RHIC energy, we take  MeV with initial time  fm/c which acts as inputs to the hydrodynamic evolution.

For studying thermal dileptons at the RHIC energy (as well as the LHC energy) we have included the vacuum spectral function of meson because its mass appears at a boundary between quark and hadronic sources of dileptons.

We begin by plotting the space-time integrated invariant mass spectra of dileptons. In Figure 24 we plot the yield of lepton pairs from the hadronic matter (HM), evaluated with and without the modified spectral function for RHIC energy. The enhancement in the region  GeV is purely a medium effect and is a contribution from the Landau cut of the meson and baryon loops. In contrast, the vacuum spectral function naturally starts from the threshold coming from the unity in the unitary cut contribution. The (small) kink at 0.42 GeV in this curve is due to the threshold for production. The enhancement in the yield due to medium effects is ~ for around 400 MeV.

176578.fig.0024
Figure 24: Invariant mass distribution of dileptons from hadronic matter (HM) for modified and unmodified meson for RHIC energy.

In Figure 25, we have shown the dependence of the yield from the two phases on the EoS. Dilepton radiation from hadronic phase outshines the emission from quark matter for up to mass. Since the internal loops of self-energy contain and interactions, we ignore the four pion annihilation process [121] to avoid double counting. The contributions from quark matter phase dominate over its hadronic counter part for both the EoS for beyond peak. This fact may be used to extract various properties, that is, average flow, temperature, and so forth of quark matter and hadronic matter by selecting windows judiciously. The dilepton yield from hadronic matter is observed to be larger when the HRG EoS is employed in comparison with LQCD. This can be understood in terms of the velocity of sound evaluated at constant entropy) which controls the rate of expansion. For EoS of the type (a) in the QGP phase which is larger than the value of the corresponding quantity for EoS of the type (b). Therefore, the rate of expansion in the scenario (b) is comparatively slower, allowing the QGP to emit lepton pairs for a longer time resulting in greater yield for LQCD EoS. In contrast, for the EoS (a), the lower value of for the hadronic phase results in a slower cooling and hence a larger yield. Also shown for comparison is the yield from the decays of mesons at the freezeout for the two types of EoS used. The yield from this source is much smaller and we will not consider it any further.

176578.fig.0025
Figure 25: Freeze-out, QGP, and total contribution for EoS (a) HRG (dash-double dotted, dotted, and dash-dotted lines) and EoS (b) LQCD (bold dashed, dashed, and solid lines) at RHIC energy.
6.5. Dileptons at LHC Energy

At LHC the measured values of for  GeV, 2.36 TeV, and 7 TeV are 3.02, 3.77, and 6.01, respectively [122]. The value at  TeV is obtained by interpolating the above experimental data mentioned above. Assuming in (26) we obtain in Pb+Pb collision for 0–10% centrality. For  fm/c we get  MeV.

The invariant mass spectra of lepton pairs are displayed for LHC initial conditions in Figure 26. Although the results are qualitatively similar to RHIC, quantitatively the yield at LHC is larger by an order of magnitude, primarily because of the large initial temperature. This enhancement is also seen in the transverse mass distributions of the lepton pairs at LHC.

176578.fig.0026
Figure 26: QGP and total contribution for EoS (a) HRG (dotted and dash-dotted lines) and EoS (b) LQCD (dashed and solid lines) at LHC energy.

All the results presented above for photon and dilepton production are reproduced using hydrodynamical model with cylindrical symmetry [64] and boost invariance along the longitudinal direction [41]. The two approximations, such as cylindrical symmetry and boost invariance, remain intact at very high and in central collision. So our results will not differ much even if hydrodynamics is used at very high and in central collision. In [123], the transverse momentum spectra of photons and of photons are calculated at RHIC energy using hydrodynamical model. Recently, in [124, 125], a realistic hydrodynamical model is established which can be used further to calculate the dilepton production, higher harmonics of flow of dileptons, interferometry with dileptons, and many more.

7. Radial Flow of Thermal Photons and Dileptons

The average magnitude of radial flow can be extracted from the transverse mass spectra spectra of the hadrons only at freezeout surface. However, hadrons being strongly interacting objects can bring the information of the state of the system when it is too dilute to support collectivity; that is, the parameters of collectivity extracted from the hadronic spectra are limited to the evolution stage where the collectivity ceases to exist. These collective parameters have hardly any information about the interior of the matter. On the other hand, electromagnetic (EM) probes; that is, photons and dileptons are produced and emitted [18] from each space time point. Therefore, estimating radial flow from the EM probes will shed light on the time evolution of the collectivity in the system.

The calculations of EM probes from thermal sources depend on the parameters such as , , , , are EoS, which are not known uniquely. These abovementioned uncertainties have been used in the evaluation of individual single spectra of photon as well as dilepton. In order to overcome the uncertainties and minimize the dependence of thermal sources on these parameters, the importance of the ratio of the transverse momentum spectra of photon to dilepton () has been emphasized in the present study (see [12, 16, 126, 127]), where the uncertainities are canceled out partially. It may be mentioned here that in the limit of the lepton pairs (virtual photons) emerge as real photons. Therefore, the evaluation of the ratio of the spectra of photons to dileptons for various invariant mass bins along with a judicious choice of the and windows will be very useful to extract the properties of QGP as well as those of hadronic phase. This will be demonstrated in the present work by analyzing WA98 and PHENIX photons (results are shown in Section 5.4) and NA60 and PHENIX dilepton (results are shown in Section 6.4) spectra.

The spectra of photon and dilepton can be parametrized as follows: where is the blue-shifted effective temperature for massless photons and is the effective temperature for massive dileptons. is the average temperature and is the average radial flow of the system. The can be obtained by parameterizing the spectra of photons and dileptons (see Sections 5.4 and 6.4), respectively, with the expressed form of (70). The ratio, , for different windows (Figure 27) can be parametrized as follows: with different values of for different invariant mass windows. The argument of the exponential in (71) can be written as [126]

fig27
Figure 27: Variation thermal photon to dilepton ratio, , with for different invariant mass windows at SPS energy (a) and RHIC energy (b) (see text).

As mentioned before some of the uncertainties prevailing in the individual spectra may be removed by taking the ratio, , of the distribution of thermal photon to dileptons. In the absence of experimental data for both photon and dilepton from the same colliding system for SPS energies, we have calculated the ratio for Pb+Pb system, where the initial condition and the EoS are constrained by the measured WA98 photon spectra. The results are displayed in Figure 27.

Also we evaluate the ratio of the thermal photon to dilepton spectra constrained by the experimental data from Au+Au collisions measured by PHENIX collaboration. The results for the thermal ratio, , displayed in Figure 27(b) are constrained by the experimental data on the single-photon and -dilepton spectra. The behavior of with for different invariant mass windows which is extracted from the available data is similar to the theoretical results obtained in [16, 126]. It is observed that the ratio decreases sharply and reaches a plateau beyond  GeV.

This behavior of as a function of can be understood as follows: (i) for , and consequently giving rise to a plateau at large . The height of the plateau is sensitive to the initial temperature of the system [16, 126] and (ii) for , indicating a decrease of the ratio with (at low ) as observed in Figure 27.

7.1. Variation of Radial Flow with Average Temperature

For a given and , (72) can be written as