Electromagnetic Radiations from Heavy Ion Collision

Abstract

In this review, we have discussed the different sources of photons and dileptons produced in heavy ion collision (HIC). The transverse momentum (𝑝_𝑇) 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 𝑝_𝑇 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 𝑝_𝑇 and invariant mass (𝑀) 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, 𝑅_em, to overcome those quantities. We argue that the variation of the radial velocity extracted from 𝑅_em with 𝑀 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 (𝑣₂) of dilepton is also studied at √𝑠_𝑁𝑁=2.76 TeV for 30–40% centrality using the (2+1)𝑑 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 [1–9]. 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 (𝛼≪𝛼_𝑠) and thus dominates the dynamics of nuclear collision processes. Therefore, its mean free path (𝜆=1/𝑛𝜎) 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 [11–16] 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 (𝑝_𝑇) 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 𝑝_𝑇 and invariant mass (𝑀) spectra of dileptons are presented in Section 6.4. In Section 7 the radial flow is extracted by simultaneous use of 𝑝_𝑇 spectra of photons and dileptons and ratio of the spectra, and 𝑅_em is conferred. The correlation function for dilepton has been calculated and HBT radii are extracted as function of 𝑀 in Section 8. We have also evaluated the dilepton 𝑣₂ 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 (𝑝_𝑇) 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 𝑝_𝑇 part of the invariant momentum spectra, and decay photon populates the low 𝑝_𝑇 part and rest over thermal contribution shines in the intermediate domain of the 𝑝_𝑇 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 𝑝_𝑇 (~1-2 GeV) whereas photons and dileptons form QGP dominate in the intermediate 𝑝_𝑇 range, that is, 𝑝_𝑇∼2-3 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.

Figure 1 

Schematic diagram of different sources of photon production in heavy ion collision as function of transverse momentum (𝑝𝑇).

The invariant momentum distribution of photons and dileptons produce from a thermal source depends on the temperature (𝑇) 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 𝑝_𝑇 spectra of thermal photons and dileptons reflect the temperature of the source through the phase space factor (𝑒^−𝐸/𝑇). Hence ideally the photons with intermediate 𝑝_𝑇 values (~2-3 GeV, depending on the value of initial temperature) reflect the properties of QGP (realized when 𝑇>𝑇_𝑐, 𝑇_𝑐 is the transition temperature). Therefore, one should look into the 𝑝_𝑇 spectra for these values of 𝑝_𝑇 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 𝑝_𝑇 photons due to flow originating from the low temperature hadronic phase (realized when 𝑇<𝑇_𝑐) populates the high 𝑝_𝑇 part of the spectra [19]. As a consequence the intermediate or the high 𝑝_𝑇 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, 𝑝_𝑇 and invariant mass (𝑀) to play with. A soft photon (low 𝑝_𝑇) in one frame of reference can be hard (high 𝑝_𝑇) in another frame, whereas the 𝑝_𝑇 integrated invariant mass distribution of dileptons is independent of any frame. In addition to it the 𝑝_𝑇 spectra are affected by the flow; however, the 𝑝_𝑇 integrated 𝑀 spectra remain unaltered by the flow in the system. Also in the 𝑀 spectra of dileptons, again in 𝑀 spectra, the dileptons from QGP dominates over its hadronic counterpart above the 𝜙 peak. All these suggests that a judicious choice of 𝑝_𝑇 and 𝑀 windows will be very useful to characterize the QGP and hadronic phase separately.

2.2. Invariant Mass (𝑀) Dependence of EM Radiations

Being massive, dileptons make situation different from photons. They have two kinematic variables—𝑝_𝑇 and 𝑀. As argued before, the 𝑝_𝑇 spectra are affected by the flow; however, the 𝑝_𝑇-integrated 𝑀 spectra remain unaltered by the flow in the system. It should be mentioned here that for 𝑀 below 𝜌 peak and above 𝜙 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 𝜌) may shift toward lower invariant mass region due to nonzero temperature and density effects. As a consequence the contributions from the decays of 𝜌 mesons to lepton pairs could populate the low 𝑀 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 𝑝_𝑇 and 𝑀 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.

Figure 2 

Expected different sources of dilepton production in HIC as function of invariant mass [5].

(i) High mass region (HMR): 𝑀≥𝑀_𝐽/Ψ(=3.1GeV),𝑝_𝑇∼3--5GeV. (1)

The HMR region corresponds to early preequilibrium phase (𝜏<𝜏_𝑖), where the lepton pairs are produced with large invariant mass (𝑀>3 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 (𝐽/Ψ,Υ) 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): 𝑀_𝜙≤𝑀≤𝑀_𝐽/Ψ,𝑝_𝑇∼1--3GeV. (2)

Thermalization is achieved in the system after a time scale (>𝜏_𝑖). 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 𝐷𝐷 mesons [23] followed by semileptonic decays, contribute largely in this domain of 𝑀. 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): 𝑀≤𝑀_𝜙(=1.02GeV),𝑝_𝑇<1GeV. (3)

With subsequent expansion and cooling, the QGP converts into a hot hadron gas at the transition temperature, 𝑇_𝑐. At later stages, the dileptons are preferentially radiated from hot hadron gas from the decay of (light) vector meson, such as the 𝜌, 𝜔, and 𝜙. The low 𝑀 domain of the lepton pairs is dominated by the decays of 𝜌. Medium modification of 𝜌 will change the yield in this domain of 𝑀. The change of 𝜌 spectral function is connected with the chiral symmetry in the bath; therefore, the measurement of low 𝑀 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: 𝑑𝑁𝑑⁴𝑝=𝑑𝑅𝑑⁴𝑝𝑑⁴𝑥, (4)(1)firstly, static emission rate (𝑑𝑅/𝑑⁴𝑝) which takes care of the basic interactions in respective phases (QGP or hadronic phase),(2)Secondly, the space-time integration over four volume (𝑑⁴𝑥) 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 |𝐼⟩ which goes to a final state |𝐹⟩ producing a lepton pair 𝑙⁺𝑙⁻ with momenta 𝑝₁ and 𝑝₂, respectively. The dilepton multiplicity thermally averaged over initial states is given by [4, 28] 𝑁=_𝐼𝐹𝐹,𝑙⁺𝑙⁻𝑒^{𝑖∫ℒint𝑑4𝑥}𝐼²×𝑒^{−𝛽𝐸𝐼}𝑍𝑑³𝑝₁(2𝜋)³2𝐸₁𝑑³𝑝₂(2𝜋)³2𝐸₂, (5) where ℒ_int=𝑒𝜓_𝑙(𝑥)𝛾_𝜇𝜓_𝑙(𝑥)𝐴^𝜇(𝑥)+𝑒𝐽_𝜇(𝑥)𝐴^𝜇(𝑥) in which 𝜓_𝑙(𝑥) is the lepton field operator and 𝐽_𝜇(𝑥) is the electromagnetic current and 𝑍=Tr[𝑒^−𝛽𝐻]. Following [1, 4, 8] this expression can be put in the form 𝑑𝑁𝑑⁴𝑞𝑑⁴𝑥𝛼=−²6𝜋³𝑞²𝐿𝑀²𝑓_𝐵𝐸𝑞₀𝑔^𝜇𝜈𝑊_𝜇𝜈𝑞₀,𝑞, (6) where the factor 𝐿(𝑀²)=(1+2𝑚²_𝑙/𝑀²)(1−4𝑚²_𝑙/𝑀²)^1/2 is of the order of unity for electrons, 𝑀(=𝑞²) being the invariant mass of the pair, and the electromagnetic (e.m.) current correlator 𝑊_𝜇𝜈 is defined by 𝑊_𝜇𝜈𝑞₀,𝑞=𝑑⁴𝑥𝑒^{𝑖𝑞⋅𝑥}𝐽^em_𝜇(𝑥),𝐽^em_𝜈(0). (7) Here 𝐽^em_𝜇(𝑥) 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 𝑊_𝜇𝜈. The 𝑞² 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 𝐴_𝛾^𝜇𝜈. Using the relation [4], 4𝜋𝛼𝑊_𝜇𝜈=2𝜋𝑞²𝑔_𝜇𝛼−𝑞_𝜇𝑞_𝛼𝐴_𝛾^𝛼𝛽−𝑞𝑞²𝑔_𝛽𝜈−𝑞_𝛽𝑞_𝜈 (8) in (6), we have 𝑑𝑁𝑑⁴𝑞𝑑⁴𝑥𝛼=−(⋯)𝐿^𝜇𝜈𝑞⁴𝑓_𝐵𝐸𝑞₀ImΠ^em_𝜇𝜈𝑞₀, (9) where 𝐴_𝜇𝜈(𝑞₀,𝑞)=ImΠ^em_𝜇𝜈(𝑞₀).

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 𝑒²𝐿_𝜇𝜈/𝑞⁴ appearing in the dilepton rate (in (9)) which is nothing but the product of electromagnetic vertex 𝛾^∗→𝑙⁺𝑙⁻, the leptonic current involving Dirac spinors, and the square of the photon propagator should be replaced by the factor ∑𝜖^𝜇𝜖^𝜈∗(=−𝑔^𝜇𝜈). And the phase space factor 𝑑³𝑝₁/(2𝜋)³𝐸₁𝑑³𝑝₂/(2𝜋)³𝐸₂ should be replaced by 𝑑³𝑝/(2𝜋)³𝐸. Then the photon emission rate becomes 𝑞₀𝑑𝑅𝑑³𝑞=𝛼2𝜋³𝑔^𝜇𝜈𝑓_𝐵𝐸𝑞₀ImΠ^em_𝜇𝜈. (10) The above emission rate is correct up to order 𝑒² 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 [29–32] give a systematic procedure to calculate the imaginary part of a Feynman diagram. The Cutkosky rule expresses the imaginary part of the 𝑛-loop amplitude in terms of physical amplitude of lower order (𝑛−1 loop or lower). This is shown schematically in Figure 3. When the imaginary part of the self-energy is calculated up to and including 𝐿 order loops where 𝐿 satisfies 𝑥+𝑦<𝐿+1, then one obtains the photon emission rate for the reaction 𝑥 particles →𝑦 particles +𝛾, and the above formalism becomes equivalent to the relativistic kinetic theory formalism [2, 3].

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 𝑖-type particles from the reaction of type 1(𝑝₁)+2(𝑝₂)→3(𝑝₃)+4(𝑝) is given as follows: ℛ_𝑖𝑑=𝒩³𝑝₁(2𝜋)³2𝐸₁𝑑³𝑝₂(2𝜋)³2𝐸₂𝑑³𝑝₃(2𝜋)³2𝐸₃𝑑³𝑝(2𝜋)³×()2𝐸2𝜋⁴𝛿⁽⁴⁾𝑝₁+𝑝₂−𝑝₃−𝑝|ℳ|²×𝑓₁𝑓₂1±𝑓₃1±𝑓₄, (11) where 𝒩 is the overall degeneracy for the reaction under consideration, |ℳ|² is the square of the invariant amplitude for the process under consideration, 𝑝_𝑖, 𝐸_𝑖, and 𝑓_𝑖(𝐸_𝑖) are the three momentum, energy, and thermal distribution functions (Fermi-Dirac or Bose-Einstein) of the incoming and outgoing particles “𝑖.”

The transverse momentum (𝑝_𝑇) distribution of photons from a reaction of the type: 1+2→3+𝛾 taking place in a thermal bath at a temperature, 𝑇 is given by [2, 3]: 𝐸𝑑𝑅𝑑³𝑝=𝒩2(2𝜋)⁸𝑑³𝑝₁2𝐸₁𝑑³𝑝₂2𝐸₂𝑑³𝑝₃2𝐸₃𝑓₁𝑓₂1±𝑓₃×𝛿⁽⁴⁾𝑝₁+𝑝₂−𝑝₃−𝑝|ℳ|². (12) Using the Mandelstam variables (𝑠,𝑡,𝑢) we can write the differential photon production rate as [33] 𝐸𝑑𝑅𝑑³𝑝=𝒩16(2𝜋)⁷𝐸^∞_{(𝑚1+𝑚2)}²𝑑𝑠^𝑡max_𝑡min𝑑𝑡|ℳ(𝑠,𝑡,𝑢)|²×𝑑𝐸₁𝑑𝐸₂𝑓𝐸₁𝑓𝐸₂1+𝑓𝐸₃𝑎𝐸²₂+2𝑏𝐸₂,+𝑐 (13) where 𝑎=−𝑠+𝑡−𝑚²₂−𝑚²₃²,𝑏=𝐸₁𝑠+𝑡−𝑚²₂−𝑚²₃𝑚²₂−𝑡+𝐸𝑠+𝑡−𝑚²₂−𝑚²₃𝑠−𝑚²₁−𝑚²₂−2𝑚²₁𝑚²₂−𝑡,𝑐=−𝐸²₁𝑚²₂−𝑡²−𝐸²𝑠−𝑚²₁−𝑚²₂²−4𝑚²₁𝑚²₂+𝑚²₂𝑠+𝑡−𝑚²₂−𝑚²₃²+𝑚²₁𝑚²₂−𝑡²−2𝐸₁𝐸×2𝑚²₂𝑠+𝑡−𝑚²₂−𝑚²₃−𝑚²₂−𝑡𝑠−𝑚²₁−𝑚²₂−𝑠+𝑡−𝑚²₂−𝑚²₃𝑚²₂−𝑡𝑠−𝑚²₁−𝑚²₂,𝐸_1min=𝑠+𝑡−𝑚²₂−𝑚²₃+4𝐸𝐸𝑚²₁𝑠+𝑡−𝑚²₂−𝑚²₃,𝐸_2min=𝐸𝑚²₂𝑚²₂+𝑚−𝑡²²−𝑡,𝐸4𝐸_2max𝑏=−𝑎+𝑏²−𝑎𝑐𝑎. (14)

In a similar way the dilepton emission rate for a reaction 𝑎𝑎→𝑙⁺𝑙⁻ can be obtained as 𝑑𝑅𝑑⁴𝑝𝑑=𝒩³𝑝_𝑎2𝐸_𝑎(2𝜋)³𝑑³𝑝_𝑎2𝐸_𝑎(2𝜋)³𝑑³𝑝₁2𝐸₁(2𝜋)³𝑑³𝑝₂2𝐸₂(2𝜋)³×(2𝜋)⁴𝛿⁽⁴⁾𝑝_𝑎+𝑝_𝑎−𝑝₁−𝑝₂𝛿⁽⁴⁾×𝑝−𝑝_𝑎−𝑝_𝑎|ℳ|²_{𝑎𝑎→𝑙}⁺_𝑙⁻𝑓𝑝_𝑎𝑓𝑝_𝑎, (15) where 𝑓(𝑝_𝑎) 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, 34–43] 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 𝜆 is much smaller than the characteristic dimensions of the system (𝐿); that is, 𝐿≥𝜆.

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 𝜕_𝜇𝑇^𝜇𝜈=0, (16) where 𝑇^𝜇𝜈 is the energy-momentum tensor of fluid element, and in its local rest frame it is given by 𝑇₀^𝜇𝜈=diag[𝜖(𝑥),−𝑃(𝑥),−𝑃(𝑥),−𝑃(𝑥)]. (17)

Local rest frame is the frame in which the velocity of the fluid element is zero. In such a frame the 𝑇^𝜇𝜈 becomes diagonal since the energy flux of the fluid 𝑇^𝑖0 and the momentum density 𝑇^0𝑗 turns to be zero. In absence of any dissipative processes the 𝑇⁰⁰ component becomes the energy density 𝜖 and 𝑇^𝑖𝑗=𝑃𝛿^𝑖𝑗 since 𝑇^𝑖𝑗𝑑𝑆_𝑗 is the 𝑖th component of force acting on the surface element which according to Pascal’s law is isotropic and perpendicular to the surface. 𝑃 is the pressure of the fluid element in the local rest frame. Isotropy implies that the energy flux 𝑇_𝑖0 and the momentum density 𝑇_0𝑗 vanish in the rest frame of fluid. In addition, it implies that the pressure tensor is proportional to the identity matrix, that is, 𝑇_𝑖𝑗=𝑃𝛿_𝑖𝑗, where 𝑃 is the thermodynamic pressure.

By doing a proper Lorentz transformation, the energy-momentum tensor 𝑇^𝜇𝜈 in a moving frame, where the fluid moves with an arbitrary four-velocity, 𝑢^𝜇=𝛾(1,𝑣) where 𝛾=(1−𝑣²)^−1/2, is given in 𝑇^𝜇𝜈=[𝜖(𝑥)+𝑃(𝑥)]𝑢^𝜇𝑢^𝜈−𝑃(𝑥)𝑔^𝜇𝜈, (18) where 𝑔^𝜇𝜈≡diag(1,−1,−1,−1) is the Mankowski metric tensor and 𝑢^𝜇 is the fluid 4-velocity referred to as “collectivity” of the system which can be defined as 𝑢^𝜇=𝛾(1,𝑣) with 𝛾=1/√1−𝑣² and 𝑢^𝜇𝑢_𝜇=1, where 𝑣 is the velocity of fluid element. In the above equation, the 𝜖 and 𝑃 are the energy density and pressure, respectively, in the fluid rest frame, and both are functions of space time coordinate 𝑥^𝜇.

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): 𝜕_𝜇𝑁^𝜇_𝐵=0. (19)𝑁^𝜇_𝐵=𝑛_𝐵𝑢^𝜇 is the conserved net baryonic current and 𝑛_𝐵 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 (𝑆=constant). If we define the entropy current: 𝑠^𝜇=𝑠𝑢^𝜇, then the conservation of entropy results in 𝜕_𝜇𝑠^𝜇=0 [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 𝑇^𝜇𝜈 and the conserved current 𝑁^𝜇 to the first-order gradient of these quantities: 𝑇^𝜇𝜈=(𝜖+𝑃)𝑢^𝜇𝑢^𝜈−𝑃𝑔^𝜇𝜈+Δ𝑇^𝜇𝜈,𝑁^𝜇=𝑛𝑢^𝜇+Δ𝑁^𝜇. (20)

One thing should be mentioned here that for a relativistic fluid it is necessary to specify whether 𝑢^𝜇 is the velocity of energy transport or velocity of particle transport. In the approach of Landau and Lifshitz, 𝑢^𝜇 is taken to be the velocity of energy transport and so 𝑇^𝑖0 vanishes in a comoving frame. In the approach of Eckart, 𝑢^𝜇 is taken to be the velocity of particle transport and so 𝑁^𝑖=𝑛𝑢^𝑖=0 in a comoving frame. The second approach is adopted here to obtain the following assumptions. The modification in the energy momentum tensor Δ𝑇^𝜇𝜈 and conserved current Δ𝑁^𝜇 is such that in a comoving frame: 𝑢_𝜇𝑢_𝜈Δ𝑇^𝜇𝜈=0,Δ𝑁^𝜇=0. (21) With these assumptions we need to construct Δ𝑇^𝜇𝜈 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 𝜖, 𝑃, and 𝑢^𝜇 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 Δ𝑇^𝜇𝜈 must be a linear combination of space-time derivatives of 𝜖, 𝑃, 𝑢^𝜇, and so forth.(2)Only the space time derivative of 𝑇 and 𝑢^𝜇 can occur in Δ𝑇^𝜇𝜈 because if derivative of 𝜖, 𝑃, or 𝑛 appeared in Δ𝑇^𝜇𝜈, then 𝜕_𝜇𝑆^𝜇 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 𝜕𝑆^𝜇𝜕𝑥^𝜇=1𝑇𝑢̇_𝑖−1𝑇²𝜕𝑇𝜕𝑥^𝑖Δ𝑇^𝑖0+1𝑇𝜕_𝑗𝑢_𝑖Δ𝑇^𝑖𝑗. (22)

From the condition that 𝜕_𝜇𝑠^𝜇≥0 for all fluid configuration we obtain Δ𝑇^𝑖𝑗=𝜂𝜕𝑢^𝑖𝜕𝑥^𝑗+𝜕𝑢^𝑗𝜕𝑥^𝑖−23∇⋅𝑢𝛿^𝑖𝑗+𝜁∇⋅𝑢𝛿^𝑖𝑗,Δ𝑇^𝑖0=−𝜒𝜕𝑇𝜕𝑥^𝑖+𝜁𝜕𝑢^𝑖,𝜕𝑡Δ𝑇⁰⁰=0, (23) where 𝜂 is the coefficient of shear viscosity, 𝜁 is coefficient of bulk viscosity, and 𝜒 is thermal conductivity.

Generalizing this expression Δ^𝜇𝜈 comes out to be [45] Δ𝑇^𝜇𝜈=𝜂∇^𝜇𝑢^𝜈+∇^𝜈𝑢^𝜇−23Δ^𝜇𝜈∇_𝜌𝑢^𝜌+𝜁Δ^𝜇𝜈∇_𝜌𝑢^𝜌. (24) 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 ∇^𝜇=Δ^𝜇𝜈𝜕_𝜈, where Δ^𝜇𝜈=𝑔^𝜇𝜈−𝑢^𝜇𝑢𝜈 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 (𝑇_𝑓). 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 𝜏_𝑖 (called initial thermalization time) after the collision. The 𝑇_𝑖 can be related to the measured hadronic multiplicity (𝑑𝑁/𝑑𝑦) by the following relation [46]: 𝑇³_𝑖𝜏_𝑖≈2𝜋⁴(3)145𝜁4𝑎_eff1𝜋𝑅²_𝐴𝑑𝑁,𝑑𝑦 (25) where 𝑅_𝐴∼1.1𝑁^1/3_part is the radius of the system, 𝜁(3) is the Riemann zeta function, and 𝑎_eff=𝜋²𝑔_eff/90, 𝑔_eff(=2×8+(7/8)×2×2×𝑁_𝑐×𝑁_𝐹) is the degeneracy of quarks and gluons in QGP, 𝑁_𝑐 = number of colors, 𝑁_𝐹 = number of flavors. The factor “7/8” originates from the difference between the Bose-Einstein and the Fermi-Dirac statistics. 𝑇_𝑖 depends on the centrality through the multiplicity, 𝑑𝑁/𝑑𝑦. The value of 𝑑𝑁/𝑑𝑦 for various beam energies and centralities can be obtained directly from experiment or calculated using the following relation [47]: 𝑑𝑁=()𝑛𝑑𝑦1−𝑥_pp𝑁_part2+𝑥𝑛_pp𝑁_coll, (26) where 𝑛_pp is the multiplicity per unit rapidity measured in pp collisions: 𝑛_pp=2.5−0.25ln(𝑠)+0.023ln²(𝑠), the fraction 𝑥 of 𝑛_pp is due to “hard” processes, with the remaining fraction (1−𝑥) being “soft” processes. The multiplicity in nuclear collision has then two components: “soft,” which is proportional to number of participants, 𝑁_part and “hard,” which is proportional to number of binary collision, 𝑁_coll.

After the initial thermalization time, 𝜏_𝑖, the system can be treated hydrodynamically. The initial conditions to solve the hydrodynamic equations are given through the energy density and velocity profile: 𝜖𝜏_𝑖=𝜖,𝑟⁰1+exp𝑟−𝑅_𝐴,𝑣𝜏/𝛿_𝑖,𝑟=0, (27) where 𝜖₀ is the initial energy density which is related to initial (𝑇_𝑖), 𝑅_𝐴 is the nuclear radius, and 𝛿 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 𝑃 and the energy density 𝜖. Under the assumption of local thermal equilibrium, this functional relation between 𝑃, 𝜖, and 𝑛_𝐵 is the EoS: 𝑃=𝑃𝜖,𝑛_𝐵 (28) which expresses the pressure as function of energy density, 𝜖, and baryon density, 𝑛_𝐵. 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, 𝑐²_𝑠=(𝜕𝑃/𝜕𝜖) sets an intrinsic scale in hydrodynamic evolution. One can thus write simple parametric form of the EoS: 𝑃=𝑐²_𝑠𝜖, 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 𝜆 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 𝑇_𝑓 which is of the order of pion mass and defines a space-time surface 𝑇(𝑥,𝑡)=𝑇_𝑓 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 (𝜏>𝜏_𝑖) 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 (𝑞𝑞→𝑔𝛾) and Compton (𝑞(𝑞)𝑔→𝑞(𝑞)𝛾) 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] 𝐸𝑑𝑅^QGP_𝛾𝑑³𝑞=_𝑓𝑒_𝑓𝛼𝛼_𝑠2𝜋²𝑇²𝑒^−𝐸/𝑇ln2.912𝐸𝑔²_𝑠𝑇, (29) where 𝛼_𝑠 is the strong coupling constant. Later, it was shown that photons from the processes [53]: 𝑔𝑞→𝑔𝑞𝛾,  𝑞𝑞→𝑞𝑞𝛾, 𝑞𝑞𝑞→𝑞𝛾, and 𝑔𝑞𝑞→𝑔𝛾 contribute in the same order 𝑂(𝛼𝛼_𝑠) as Compton and annihilation processes (shown in Figure 4). The complete calculation of emission rate from QGP to order 𝛼_𝑠 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, 𝛼_𝑠, has been taken from [56].

(a)

(b)

(c)

(d)

(e)

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 𝜋, 𝜌, 𝜔, and 𝜂. The relevant vertices for the reactions 𝜋𝜋→𝜌𝛾 and 𝜋𝜌→𝜋𝛾 and the decay 𝜌→𝜋𝜋𝛾 are obtained from the following Lagrangian [57] (see Figure 5): ℒ=−𝑔_𝜌𝜋𝜋𝜌^𝜇⋅𝜋×𝜕_𝜇𝜋−𝑒𝐽^𝜇𝐴_𝜇+𝑒2𝐹^𝜇𝜈𝜌_𝜇×𝜌_𝜈3, (30) where 𝐹_𝜇𝜈=𝜕_𝜇𝐴_𝜈−𝜕_𝜈𝐴_𝜇 is the Maxwell field tensor and 𝐽^𝜇 is the hadronic part of the electromagnetic current given by 𝐽^𝜇=𝜌_𝜈×𝐵^𝜈𝜇₃+𝜋×𝜕^𝜇𝜋+𝑔_𝜌𝜋𝜋𝜋×𝜌^𝜇₃ (31) with 𝐵_𝜇𝜈=𝜕_𝜇𝜌_𝜈−𝜕_𝜈𝜌_𝜇−𝑔_𝜌𝜋𝜋(𝜌_𝜇×𝜌_𝜈).

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 𝜋𝜂→𝜋𝛾, 𝜋𝜋→𝜂𝛾, and the decay 𝜔→𝜋𝛾 using the following interaction: 𝑔ℒ=^𝜌𝜌𝜂𝑚_𝜂𝜖_{𝜇𝜈𝛼𝛽}𝜕^𝜇𝜌^𝜈𝜕^𝛼𝜌^𝛽𝜂+𝑔_𝜔𝜌𝜋𝑚_𝜋𝜖_{𝜇𝜈𝛼𝛽}𝜕^𝜇𝜔^𝜈𝜕^𝛼𝜌^𝛽𝜋+em²_𝜌𝑔_𝜌𝜋𝜋𝐴_𝜇𝜌^𝜇. (32) 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: 𝜋𝐾^∗→𝐾𝛾, 𝜋𝐾→𝐾^∗𝛾, 𝜌𝐾→𝐾𝛾, and 𝐾𝐾^∗→𝜋𝛾 [62, 63] have also been incorporated in the present work. Contributions from other decays, such as 𝐾^∗(892)→𝐾𝛾, 𝜙→𝜂𝛾, 𝑏₁(1235)→𝜋𝛾, 𝑎₂(1320)→𝜋𝛾, and 𝐾₁(1270)→𝜋𝛾, have been found to be small [63] for 𝑝_𝑇>1 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 𝜌 mesons has dominant contribution. The rate at low photon energy is dominated by reaction with 𝜌 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 𝜌 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 𝜂 mesons, but as the value of 𝑔_𝜌𝜌𝜂 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: 𝑑𝑁_𝛾𝑑²𝑝_𝑇𝑑𝑦=_𝑖𝑖𝑑𝑅_𝛾𝑑²𝑝_𝑇𝐸𝑑𝑦^∗,𝑇_𝑖𝑑⁴𝑥, (33) where the 𝑑⁴𝑥 is the four volume. The energy, 𝐸^∗, appearing in (33) should be replaced by 𝑢^𝜇𝑝_𝜇 for a system expanding with space-time-dependent four-velocity 𝑢^𝜇. Under the assumption of cylindrical symmetry and longitudinal boost invariance, 𝑢^𝜇 can be written as 𝑢=𝛾_𝑇()𝑡𝜏,𝑟𝜏,𝑣_𝑟(),𝑧𝜏,𝑟𝜏=𝛾_𝑇𝑀_𝑇cosh𝜂,𝑢_𝑥,𝑢_𝑦,𝑀_𝑇sinh𝜂=𝛾_𝑇𝑀_𝑇cosh𝜂,𝑣_𝑟cos𝜙,𝑣_𝑟sin𝜙,𝑀_𝑇sinh𝜂, (34) where 𝑣_𝑟(𝜏,𝑟) is the radial velocity, 𝛾_𝑟(𝜏,𝑟)=(1−𝑣_𝑟(𝜏,𝑟))^−1/2 and, therefore, for the present calculations, 𝑢^𝜇𝑝_𝜇=𝛾_𝑟𝑀_𝑇cosh𝑦−𝜂−𝑣_𝑟𝑝_𝑇cos𝜙. (35) For massless photon the factor 𝑢^𝜇𝑝_𝜇 can be obtained by replacing 𝑀_𝑇 in (35) by 𝑝_𝑇. For the system produced in QGP phase reverts to hot hadronic gas at a temperature 𝑇∼𝑇_𝑐. Thermal equilibrium may be maintained in the hadronic phase until the mean-free path remains comparable to the system size. The term “(𝑑𝑅/𝑑²𝑝_𝑇𝑑𝑦)_𝑖=[(⋯)𝑓_𝐵𝐸]” is the static rate of photon production where 𝑖 stands for quark matter (QM), mixed phase (𝑀) (in a 1st-order phase transition scenario), and hadronic matter (HM), respectively. The 𝑝_𝑇 dependence of the photon and dilepton spectra originating from an expanding system is predominantly determined by the thermal factor 𝑓_𝐵𝐸. 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 𝑑⁴𝑥 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.

Table 1 

The values of various parameters—thermalization time (𝜏_𝑖), initial temperature (𝑇_𝑖), and hadronic multiplicity 𝑑𝑁/𝑑𝑦 (the value of 𝑑𝑁/𝑑𝑦 for various beam energies and centralities is calculated from (26))—used in the present calculations.

To estimate 𝑑𝑁/𝑑𝑦 for RHIC, we have taken 𝑑𝑛_pp/𝑑𝑦=2.43 and 𝑥=0.1 at √𝑠_𝑁𝑁=200 GeV. It should be mentioned here that the values of 𝑑𝑁/𝑑𝑦 (through 𝑁_part and 𝑁_coll in (26)) and hence the 𝑇_𝑖 (through 𝑑𝑁/𝑑𝑦 in (25)) depend on the centrality of the collisions. For SPS, 𝑑𝑁/𝑑𝑦 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, 𝑇_𝑓=140 MeV for all the hadrons. The ratios of various hadrons measured experimentally at different √𝑠_𝑁𝑁 indicate that the system formed in heavy ion collisions chemically decouple at 𝑇_ch which is higher than 𝑇_𝑓 which can be determined by the transverse spectra of hadrons [67, 68]. Therefore, the system remains out of chemical equilibrium from 𝑇_ch to 𝑇_𝑓. 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 𝑝_𝑇 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 𝑝_𝑇 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 𝑇_AA(𝑏) for different centrality; that is, the photon production from A-A collision and p-p collision are related to the following relation: 𝑑𝑁^AA𝑑²𝑝_𝑇=𝑁𝑑𝑦^coll(𝑏)𝜎^pp_𝑖𝑛𝑑𝜎^𝑁𝑁𝑑²𝑝_𝑇𝑑𝑦=𝑇_AA(𝑏)𝑑𝜎^𝑁𝑁𝑑²𝑝_𝑇,𝑑𝑦 (36) where 𝑁_coll(𝑏) is taken for the corresponding experiments and the the typical 𝜎^pp_𝑖𝑛 (𝜎^pp_𝑖𝑛 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 √𝑠_𝑁𝑁=17.3 GeV. However, no data at this collision energy is available for pp interactions. Therefore, prompt photons for p+p collision at √𝑠_𝑁𝑁=19.4 GeV have been used [70] to estimate the hard contributions for nuclear collisions at √𝑠_𝑁𝑁=17.3 GeV. Appropriate scaling [65] has been used to obtain the results at √𝑠_𝑁𝑁=17.3 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 𝑝_𝑇 part of the WA98 data is reproduced by the prompt contributions reasonably well. At low 𝑝_𝑇 the hard contributions underestimate the data indicating the presence of a thermal source. The thermal photons with initial temperature =200 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 [73–78] the effect of chemical freezeout is ignored. As a result either a higher value of 𝑇_𝑖 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.

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 √𝑠_𝑁𝑁=200 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 √𝑠_𝑁𝑁=200 GeV, we have used the available experimental data from pp collision and normalized it to Au-Au data with 𝑇_AA(𝑏) for different centrality [80] (using (36)). At low 𝑝_𝑇 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 √𝑠_𝑁𝑁=200 GeV reasonably well. The reproduction of data is satisfactory (Figure 7) for all the centralities with the initial temperature shown in Table 1 [81].

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 √𝑠_𝑁𝑁=2.76 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 √𝑠_𝑁𝑁=7 TeV have been used to estimate the hard contributions for nuclear collisions at √𝑠_𝑁𝑁=2.76 TeV by using the scaling (with √𝑠_𝑁𝑁) 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 𝑝_𝑇 part of the data is reproduced by the prompt contributions reasonably well. At low 𝑝_𝑇 the hard contributions underestimate the data indicating the presence of a possible thermal source.

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 (∼⟨𝑝_𝑇⟩) from the thermal distributions of photons at different collision energies—that is, for SPS, RHIC, and LHC energies. Figure 9 shows the variation of ⟨𝑝_𝑇⟩ with multiplicity for different collision energies. To minimize the centrality dependence of the results the 𝑑𝑁_ch/𝑑𝜂 is normalized by 𝑁_part. The results clearly indicate a significant rise in the average 𝑝_𝑇 (⟨𝑝_𝑇⟩) while going from SPS to RHIC to LHC. The values of ⟨𝑝_𝑇⟩ 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 𝑝_𝑇 spectra represents the average effective temperature of the system.

Figure 9 

The variation of ⟨𝑝𝑇⟩ with the increase in multiplicity for different collision energies.

The quantity, 𝜌^av_eff(=1/𝑁_part𝑑𝑁_ch/𝑑𝜂), is proportional to the entropy density. Therefore, 𝜌^av_eff/⟨𝑝_𝑇⟩³∝𝑔^av_eff, 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 (𝑠∼𝑔_eff𝑇³) at LHC increases by almost 96% compared to RHIC, and there is an enhancement of 46% 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 𝜌^av_eff by ⟨𝑝_𝑇⟩³. Therefore, we estimate 𝜌^av_eff/⟨𝑝_𝑇⟩³ from the analysis of the experimental data and found that there is a 15% increase in this quantity from SPS to RHIC and 35% 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 [84–86]. The measured photon spectra (𝑑𝑁/𝑑²𝑝_𝑇𝑑𝑦) 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 [84–86] 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 𝜂/𝑠 and hence characterize the matter.

Equation (12) can be simplified to the following form (see the appendix) [87]: 𝑑𝑅𝑑²𝑝_𝑇=𝒩𝑑𝑦16(2𝜋)⁸𝑝_1𝑇𝑑𝑝_1𝑇𝑑𝑝_2𝑇𝑑𝜙₁𝑑𝑦₁𝑑𝑦₂×𝑓₁𝑝₁𝑓₂𝑝₂1±𝑓₃𝑝₃×|ℳ|²×𝑝_1𝑇sin𝜙₁−𝜙₂+𝑝_𝑇sin𝜙_2𝜙⁻¹_2=𝜙02. (37) 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 (𝑇_𝑖) and the thermalization time (𝜏_𝑖) are constrained by the measured hadron multiplicity (𝑑𝑁/𝑑𝑦). 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 𝑑𝑁/𝑑𝑦 (which is associated with the freezeout point) and 𝜏_𝑖 the magnitude of 𝑇_𝑖 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, 𝛿𝑓, to the phase space factor, 𝑓, defined as follows [88]: 𝑓_𝑖𝑝=𝑓_𝑖01+𝛿𝑓_𝑖=𝑓_𝑖0𝑝1+^𝛼𝑝^𝛽2𝑇³𝐶∇_𝛼𝑢_𝛽+𝐴Δ_𝛼𝛽∇⋅𝑢, (38) where 𝑓_𝑖0 is the equilibrium distribution function of “𝑖th” particle, ⟨∇_𝛼𝑢_𝛽⟩≡∇_𝛼𝑢_𝛽+∇_𝛽𝑢_𝛼−(2/3)Δ_𝛼𝛽∇_𝛾𝑢^𝛾, Δ_𝛼𝛽=𝑔_𝛼𝛽−𝑢_𝛼𝑢_𝛽, ∇_𝛼=(𝑔_𝛼𝛽−𝑢_𝛼𝑢_𝛽)𝜕^𝛽, 𝑢_𝜇 being the four-velocity of the fluid. The coefficients 𝐶 and 𝐴 can be determined in the following way. Substituting 𝑓 in the expression for stress-energy tensor 𝑇^𝜇𝜈 we get 𝑇^𝜇𝜈=𝑑³𝑝(2𝜋)³𝐸𝑝^𝜇𝑝^𝜈𝑓₀1+𝛿𝑓=𝑇₀^𝜇𝜈+Δ𝑇^𝜇𝜈, (39) where 𝑇₀^𝜇𝜈=(𝜖+𝑃)𝑢^𝜇𝑢^𝜈−𝑔^𝜇𝜈𝑃 is the energy momentum tensor for ideal fluid. From general considerations [44] the dissipative part can be written as Δ𝑇^𝜇𝜈=𝜂∇^𝜇𝑢^𝜈+𝜁Δ^𝜇𝜈∇⋅𝑢. (40) Equating the part containing 𝛿𝑓 from (38) with (40), 𝐶 and 𝐴 can be expressed in terms of the coefficients of shear (𝜂) and bulk (𝜁) viscosity, respectively, in terms of which the phase space distribution for the system can be written as 𝑓=𝑓₀1+𝜂/𝑠2𝑇³𝑝^𝛼𝑝^𝛽∇_𝛼𝑢_𝛽−𝜁/𝑠5𝑇³𝑝^𝛼𝑝^𝛽Δ_𝛼𝛽∇⋅𝑢. (41)

For a boost invariant expansion in (1+1) dimension this can be simplified to get 𝑓=𝑓₀1+𝛿𝑓_𝜂−𝛿𝑓_𝜁, (42) where 𝛿𝑓_𝜂=𝜂/𝑠3𝑇³𝜏𝑝²_𝑇−2𝑝^′_𝑧²,𝛿𝑓_𝜁=𝜁/𝑠5𝑇³𝜏𝑝²_𝑇+𝑝^′_𝑧², (43) where 𝑝^′_𝑧=𝑚_𝑇sinh(𝑦−𝜂) is the 𝑧-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 𝑝_𝑇 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 (1+1) dimensional boost invariant expansion [41] the evolution equation, 𝜕_𝜇𝑇^𝜇𝜈=0, can be written as [89] 𝑑𝜖+𝑑𝜏𝜖+𝑃𝜏=(4/3)𝜂+𝜁𝜏², (44) where 𝑃 is the pressure and 𝜖 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 𝜏_𝑖 after the collision at an initial temperature 𝑇_𝑖. With this initial condition and equation of state (EoS) 𝑃=𝜖/3 the solution of (44) can be written as [82] 𝑇=𝑇_𝑖𝜏_𝑖𝜏^1/3+𝐴_𝑄8𝑎_𝑄𝜏_𝑖𝜏_𝑖𝜏^1/3−𝜏_𝑖𝜏, (45) where 𝐴_𝑄=((4/3)𝜂_𝑄0+𝜁_𝑄0), 𝜂_𝑄0=𝜂_𝑄/𝑇³=4𝑎_𝑄(𝜂/𝑠)_𝑄, and 𝜁_𝑄0=𝜁_𝑄/𝑇³=4𝑎_𝑄(𝜁/𝑠)_𝑄.

Equation (45) dictates the cooling of the QGP phase from its initial state to the transition temperature, 𝑇_𝑐, at a time, 𝜏_𝑞, 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 𝑇_𝑐 as follows [82]: 𝜖_𝑀(𝜏)=𝑓_𝑄𝜖_𝑄𝑇_𝑐+1−𝑓_𝑄(𝜏)𝜖_𝐻𝑇_𝑐,𝜂_𝑀(𝜏)=𝑓_𝑄𝜂_𝑄𝑇_𝑐+1−𝑓_𝑄(𝜏)𝜂_𝐻𝑇_𝑐,𝜁_𝑀(𝜏)=𝑓_𝑄𝜁_𝑄𝑇_𝑐+1−𝑓_𝑄(𝜏)𝜁_𝐻𝑇_𝑐, (46) where 𝑓_𝑄(𝜏)(𝑓_𝐻(𝜏)) indicates the fraction of the quark (hadronic) matter in the mixed phase at a proper time 𝜏. We have 𝜖_𝑄(𝑇_𝑐)=3𝑎_𝑄𝑇⁴_𝑐+𝐵, 𝜖_𝐻(𝑇_𝑐)=3𝑎_𝐻𝑇⁴_𝑐, 𝑎_𝑄=𝑔_𝑄𝜋²/90, 𝑎_𝐻=𝑔_𝐻𝜋²/90, 𝐵 is the bag constant, 𝑔_𝑄 (𝑔_𝐻) 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 𝑓_𝑄(𝜏). Substituting (46) in (44) and solving for 𝑓_𝑄(𝜏) we get [82] 𝑓_𝑄=𝑒^−𝑏/𝜏𝜏^𝜏′_𝜏𝑄𝑐𝑒^𝑏/𝜏′𝜏^′−𝑎𝑒^𝑏/𝜏′𝑑𝜏^′+𝜏_𝑄𝜏𝑒^{(𝑏/𝜏𝑄−𝑏/𝜏)}, (47) where 𝑎=4𝜖_𝐻/(3Δ𝜖), 𝑏=[4(𝜂_𝑄−𝜂_𝐻)/3+2(𝜁_𝑄−𝜁_𝐻)]/Δ𝜖, 𝑐=((4/3)𝜂_𝐻+2𝜁_𝐻)/Δ𝜖, and Δ𝜖=𝜖_𝑄−𝜖_𝐻. Equation (47) indicates how the fraction of QGP in the coexistence phase evolves with time.

The variation of 𝑇 with 𝜏 in the hadronic phase can be obtained by solving (44) with the boundary condition 𝑇=𝑇_𝑐 and 𝜏=𝜏_𝐻, where 𝜏_𝐻 is the (proper) time at which the mixed phase ends; that is, when the conversion of QGP to hadronic matter is completed, 𝑇=𝑇_𝑐𝜏_𝐻𝜏^1/3+𝐴_𝐻8𝑎_𝐻𝜏_𝐻𝜏_𝐻𝜏^1/3−𝜏_𝐻𝜏. (48) Similar to QGP, 𝑃=𝜖/3 has been used for hadronic phase. For a vanishing bulk viscosity (𝜁=0) the cooling of the QGP is dictated by 𝑇=𝑇_𝑖𝜏_𝑖𝜏^1/3+23𝜏_𝑖𝜂𝑠_𝑄𝜏_𝑖𝜏^1/3−𝜏_𝑖𝜏. (49) Similarly the time variation of temperature in the hadronic phase is given by 𝑇=𝑇_𝑐𝜏_𝐻𝜏^1/3+23𝜏_𝐻𝜂𝑠_𝐻𝜏_𝐻𝜏^1/3−𝜏_𝐻𝜏. (50) In a realistic scenario the value of 𝜂/𝑠 may be different for QGP [90–94] and hadronic phases [95–98]. However, in the present work we take the same value of 𝜂/𝑠 both for QGP and hadronic matter as shown in Table 2.

Table 2 

The values of various 𝜂/𝑠 and parameters—initial temperature (𝑇_𝑖), starting time of mixed phase (𝜏_𝑞), and ending time of mixed phase (𝜏_ℎ)—used in the present calculations.

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

In case of an ideal fluid, the conservation of entropy implies that the rapidity density 𝑑𝑁/𝑑𝑦 is a constant of motion for the isoentropic expansion [41]. In such circumstances, the experimentally observed (final) multiplicity, 𝑑𝑁/𝑑𝑦, may be related to a combination of the initial temperature 𝑇_𝑖 and the initial time 𝜏_𝑖 as 𝑇³_𝑖𝜏_𝑖. Assuming an appropriate value of 𝜏_𝑖 (taken to be ~0.6 fm/c in the present case), one can estimate 𝑇_𝑖.

For dissipative systems, such an estimate is obviously inapplicable. Generation of entropy during the evolution invalidates the role of 𝑑𝑁/𝑑𝑦 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 𝑑𝑁/𝑑𝑦 to the freezeout temperature, 𝑇_𝑓, and the freezeout time, 𝜏_𝑓, by the relation: 𝑑𝑁=𝑑𝑦𝜋𝑅²_𝐴4𝑎_𝐻𝑇³_𝑓𝜏_𝑓𝜅, (51) where 𝑅_𝐴 is the radius of the colliding nuclei (we consider AA collision for simplicity) and 𝜅 is a constant ~3.6 for massless bosons.

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

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 𝜂/𝑠.

In this section we present the shift in the 𝑝_𝑇 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 (𝐸=𝑝_𝑇cosh(𝑦)) in the laboratory frame should be replaced by 𝑢_𝜇𝑝^𝜇 in the comoving frame of the fluid, where 𝑝^𝜇 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 (𝛿𝑓_𝜂≠0) in (42), but the viscous effects on the evolution are neglected (𝜂=0 ) in (44),(ii)the effects of 𝜂≠0 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 𝜂/𝑠) because the viscous effects on the evolution are ignored in scenario (i). The viscous effects on the 𝑝_𝑇 distribution of the photons are distinctly visible. The higher values of 𝜂/𝑠 make the spectra flatter through the 𝑝_𝑇 dependence of the correction, 𝛿𝑓_𝜂.

Figure 11 

Transverse momentum distribution of thermal photons from QGP for various values of 𝜂/𝑠 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 𝜂/𝑠. In this scenario the value of 𝑇_𝑖 is lower for higher 𝜂/𝑠 for reasons described above. As a result the enhancement in the photon production due to change in phase space factor, 𝛿𝑓_𝜂, is partially compensated by the reduction in 𝑇_𝑖 for nonzero 𝜂, which is clearly seen in the results displayed in Figures 11 and 12.

Figure 12 

Transverse momentum distribution of thermal photons from QGP for various values of 𝜂/𝑠 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 𝑝_𝑇 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.

Figure 13 

Transverse momentum distribution of photons from thermal hadrons for various values of 𝜂/𝑠 in the scenario (i).

Figure 14 

Transverse momentum distribution of photons from thermal hadrons for various values of 𝜂/𝑠 in the scenario (ii).

Finally in Figures 15 and 16 we plot the 𝑝_𝑇 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 𝜂/𝑠 for scenario (i) and (ii), respectively. The effect of viscosity for the scenario (i) is stronger than (ii).

Figure 15 

Transverse momentum distribution of thermal photons from the entire evolution history of the system for various values of 𝜂/𝑠 in the scenario (i).

Figure 16 

Transverse momentum distribution of thermal photons from the entire evolution history of the system for various values of 𝜂/𝑠 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 (𝑀) and transverse momentum (𝑝_𝑇). Again, the 𝑝_𝑇 spectra are affected due to flow, whereas the 𝑝_𝑇-integrated 𝑀 spectra remain unaltered by flow. By tuning these two parameters, different stages of expanding fireball can be understood. Dileptons having large 𝑀 and high 𝑝_𝑇 are emitted early from the hot zone of the system. On the other hand, those having lower 𝑀 and 𝑝_𝑇 produced at later stage of the fireball when the temperature is low. Because of an additional variable, the invariant pair mass 𝑀, 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 𝑊_𝜇𝜈 can be directly evaluated by writing the electromagnetic current in terms of quarks of flavor 𝑓, that is, 𝐽^em_𝜇=∑_𝑓𝑒_𝑓𝜓_𝑓𝛾_𝜇𝜓_𝑓. Confining to the leading order contribution we obtain 𝑔^𝜇𝜈𝑊_𝜇𝜈=−3𝑞²2𝜋_𝑓𝑒²_𝑓1−4𝑚²_𝑞𝑞². (52) The rate in this case corresponds to dilepton production due to process 𝑞𝑞→𝛾^∗→𝑙⁺𝑙⁻. The static thermal emission rate of dilepton from QM is given by (𝑞𝑞→𝛾^∗→𝑙⁺𝑙⁻) [100, 101] (also [102, 103]), 𝑑𝑅_𝑙⁺_𝑙⁻𝑑⁴𝑝𝛼=−²4𝜋⁴𝐿𝑀²𝑓_𝐵𝐸𝑓𝑒²_𝑓1+2𝑇𝑝𝑛ln⁺𝑛₋, (53) where 𝑒_𝑓 is the charge of the quark and 𝑛_±=1/(𝑒^{(𝑝0±|𝑝|/2𝑇)+1}).

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 𝐽^ℎ_𝜇 into parts with definite isospin: 𝐽^ℎ_𝜇=12𝑢𝛾_𝜇𝑢−𝑑𝛾_𝜇𝑑+16𝑢𝛾_𝜇𝑢+𝑑𝛾_𝜇𝑑+⋯=𝐽^𝑉_𝜇+𝐽^𝑆_𝜇+⋯=𝐽^𝜌_𝜇+𝐽^𝜔_𝜇3+⋯, (54) where 𝑉 and 𝑆 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 𝜌 and 𝜔 mesons, respectively. Defining the correlator of these currents 𝑊^{𝜌,𝜔,𝜙}_𝜇𝜈 analogously as in (7), we can write 𝑊_𝜇𝜈=𝑊^𝜌_𝜇𝜈+𝑊^𝜔_𝜇𝜈9+⋯. (55) The correlator of vector-isovector currents 𝑊^𝜌_𝜇𝜈 has in fact been measured [105, 106] in vacuum along with the axial-vector correlator by studying 𝜏 decays into even and odd number of pions. The former is found to be dominated at lower energies by the prominent peak of the 𝜌 meson followed by a continuum at high energies. The axial correlator, on the other hand, is characterized by the broad hump of the 𝑎₁. 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], 0𝐽^em_𝜇(0)𝑅=𝐹_𝑅𝑚_𝑅𝜖_𝜇, (56) where 𝑅 denotes the resonance in a particular channel and 𝜖_𝜇 is the corresponding polarization vector. The coupling constants 𝐹_𝑅 are obtained from the partial decay widths into 𝑒⁺𝑒⁻ through the relation 𝐹²_𝑅=3𝑚_𝑅Γ_{𝑅→𝑒+𝑒−}4𝜋𝛼² (57) yielding 𝐹_𝑅 = 0.156 GeV, 0.046 GeV, and 0.079 GeV for 𝜌, 𝜔, and 𝜙, respectively. Equation (56) suggests the operator relations: 𝐽^𝜌_𝜇(𝑥)=𝐹_𝜌𝑚_𝜌𝑉^𝜌_𝜇(𝑥),𝐽^𝜔_𝜇(𝑥)=3𝐹_𝜔𝑚_𝜔𝑉^𝜔_𝜇(𝑥),andsoforth, (58) where 𝑉_𝜇^𝜌(𝜔)(𝑥) denotes the field operator for the 𝜌(𝜔) meson. So using the above relations connecting currents to fields (so-called field-current identity), the current commutator becomes 𝑊_𝜇𝜈=_{𝑅=𝜌,𝜔,…}𝐹²_𝑅𝑚²_𝑅𝑑⁴𝑥𝑒^{𝑖𝑞⋅𝑥}𝑉^𝑅_𝜇(𝑥),𝑉^𝑅_𝜈=_{𝑅=𝜌,𝜔,…}𝐹²_𝑅𝑚²_𝑅𝐴^𝑅_𝜇𝜈𝑞₀,𝑞=2𝜖𝑞_{0𝑅=𝜌,𝜔,…}𝐹²_𝑅𝑚²_𝑅Im𝐷^𝑅_𝜇𝜈𝑞₀,𝑞, (59) where 𝐴^𝑅_𝜇𝜈 are the spectral functions of corresponding vector meson resonances (𝑅) and 𝐷^𝑅_𝜇𝜈 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 𝐷^𝑅_𝜇𝜈𝑞=𝐺^𝑅_𝜇𝜈−𝑞_𝜇𝑞_𝜈𝑞²𝑚²_𝑅, (60) where 𝐺^𝑅_𝜇𝜈𝑃=−^𝜇𝜈𝑞²−𝑚²_𝑅−Π^𝑅_𝑡𝑞−𝑄_𝜇𝜈/𝑞²𝑞²−𝑚²_𝑅−𝑞²Π^𝑅_𝑙𝑞. (61)

The imaginary part is then put in (59) and then in (6) to arrive at the dilepton emission rate: 𝑑𝑁𝑑⁴𝑞𝑑⁴𝑥=𝛼²𝜋³𝑞²𝐿𝑞²𝑓_𝐵𝐸𝑞₀×𝐹²_𝜌𝑚²_𝜌𝐴_𝜌𝑞₀,𝑞+𝐹²_𝜔𝑚²_𝜔𝐴_𝜔𝑞₀,𝑞+⋯, (62) where, for example, 𝐴_𝜌(=−𝑔^𝜇𝜈Im𝐷^𝜌_𝜇𝜈/3) is given by 𝐴_𝜌1=−32∑ImΠ^𝑅_𝑡𝑞²−𝑚²_𝜌−∑ReΠ^𝑅_𝑡²+∑ImΠ^𝑅_𝑡²+𝑞²∑ImΠ^𝑅_𝑙𝑞²−𝑚²_𝜌−𝑞²∑ReΠ^𝑅_𝑙²+𝑞⁴∑ImΠ^𝑅_𝑙², (63) the sum running overall meson loops 𝜋𝐻 and baryon loops 𝑁𝐵. Here Π^𝑅_𝑡,𝑙 is the diagonal element of vector meson (𝑅=𝜌 and 𝜔) self-energy at finite temperature and density which is also a matrix in the real-time formalism. We have taken 𝐻=𝜋,𝜔,𝑎₁,ℎ₁ [109] and 𝐵=𝑁(940), 𝑁^∗(1520), 𝑁^∗(1650), 𝑁^∗(1700), 𝑁^∗(1720), Δ(1230), and Δ^∗(1620) [110] for 𝜌 meson whereas for 𝜔 meson, 𝐻=𝜌 (with 𝜋𝜋 folding), and 𝐵=𝑁(940), 𝑁^∗(1440), 𝑁^∗(1520), 𝑁^∗(1535), 𝑁^∗(1650), and 𝑁^∗(1720) [111] are taken. These self-energy graphs are diagrammatically represented in Figure 17.

Figure 17 

One-loop Feynman diagrams for 𝜌 or 𝜔 self-energy involving mesons (first figure) and baryons (second and third figures). 𝑉 stands for the 𝜌 or 𝜔 in the external line. In the internal lines of meson loops, 𝐻=𝜋, 𝜔, 𝑎1, and ℎ1 for 𝑉=𝜌 whereas 𝐻=𝜌 for 𝑉=𝜔. For the baryonic loops, 𝑁 and 𝐵 indicate, respectively, nucleon and baryonic internal lines.

The general expression of Π^𝑅_𝑡,𝑙 for meson loop 𝜋𝐻 (representing the first diagram of Figure 17) is given by [109, 111] Π^𝑅_𝑡,𝑙𝑞=𝑑³𝑘(2𝜋)³14𝜔_𝜋𝜔_𝐻×1+𝑛^𝜋𝐿¹_𝑡,𝑙+𝑛^𝐻𝐿³_𝑡,𝑙𝑞₀−𝜔_𝜋−𝜔_𝐻+𝑖𝜂𝜖𝑞₀+−𝑛^𝜋𝐿¹_𝑡,𝑙+𝑛^𝐻𝐿⁴_𝑡,𝑙𝑞₀−𝜔_𝜋+𝜔_𝐻+𝑖𝜂𝜖𝑞₀+𝑛^𝜋𝐿²_𝑡,𝑙−𝑛^𝐻𝐿³_𝑡,𝑙𝑞₀+𝜔_𝜋−𝜔_𝐻+𝑖𝜂𝜖𝑞₀+−𝑛^𝜋𝐿²_𝑡,𝑙−1+𝑛^𝐻𝐿⁴_𝑡,𝑙𝑞₀+𝜔_𝜋+𝜔_𝐻+𝑖𝜂𝜖𝑞₀, (64) where 𝑛’s are Bose-Einstein distribution functions for the internal meson lines and 𝜔’s are their on-shell energies. In the above expression 𝐿^𝑖=1,…,4_𝑡,𝑙 denote the values of 𝐿_𝑡,𝑙(𝑘₀) for 𝑘₀=𝜔_𝜋, −𝜔_𝜋, 𝑞₀−𝜔_𝐻, and 𝑞₀+𝜔_𝐻, respectively. The corresponding expression for the baryon loop 𝑁𝐵 (second diagram of Figure 17) is given by [110, 111] Π^𝑅_𝑡,𝑙𝑞=𝑑³𝑘(2𝜋)³14𝜔_𝑁𝜔_𝐵×1−𝑛^𝑁₊𝐿¹_𝑡,𝑙−𝑛^𝐵₋𝐿³_𝑡,𝑙𝑞₀−𝜔_𝑁−𝜔_𝐵+𝑖𝜂𝜖𝑞₀+𝑛^𝑁₊𝐿¹_𝑡,𝑙−𝑛^𝐵₊𝐿⁴_𝑡,𝑙𝑞₀−𝜔_𝑁+𝜔_𝐵+𝑖𝜂𝜖𝑞₀+−𝑛^𝑁₋𝐿²_𝑡,𝑙+𝑛^𝐵₋𝐿³_𝑡,𝑙𝑞₀+𝜔_𝑁−𝜔_𝐵+𝑖𝜂𝜖𝑞₀+𝑛^𝑁₋𝐿²_𝑡,𝑙+−1+𝑛^𝐵₊𝐿⁴_𝑡,𝑙𝑞₀+𝜔_𝑁+𝜔_𝐵+𝑖𝜂𝜖𝑞₀, (65) where 𝑛₊₍₋₎’s are Fermi-Dirac distribution functions for the internal baryon (antibaryon) lines. Here, 𝐿^𝑖=1,…,4_𝑡,𝑙 denote the values of 𝐿_𝑡,𝑙(𝑘₀) for 𝑘₀=𝜔_𝑁, −𝜔_𝑁, 𝑞₀−𝜔_𝐵, 𝑞₀+𝜔_𝐵, respectively. The expression for the third diagram of Figure 17 can be obtained by changing the sign of the external momentum 𝑞 in (65).

The numerical results for the 𝜌 and 𝜔 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 𝜌 and 𝜔 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.

(a)

(b)

Figure 18 

The imaginary (upper) and real (lower) part of self-energy function of 𝜌 (a) and 𝜔 (b) at different chemical potential (𝜇 or 𝜇𝐵). 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 𝜌 and 𝜔 meson. In view of the fact that the 𝜌 and 𝜔 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 2𝜋 and 3𝜋 thresholds for the 𝜌 and 𝜔 in the vacuum case are also visible. At fixed temperature and density, the 𝜔 contribution is lower than 𝜌 but of comparable magnitude below their nominal masses. However, the fact that the 𝜔 is suppressed by a factor ~10 (≃𝐹²_𝜌/𝐹²_𝜔) compared to the 𝜌 in the dilepton emission rate makes a quantitative study of the 𝜔 difficulty. In the above expressions the meson (𝐻) and baryon resonances (𝐵) have been treated in the narrow width approximation. These have then been folded with the width of the resonances as shown in [110].

Figure 19 

The 𝜔 spectral function seen in comparison with the 𝜌.

Thus, the dilepton emission rate in the present scenario actually boils down to the evaluation of the self-energy graphs of 𝜌 and 𝜔 as a function of 𝑞₀, 𝑞, temperature (𝑇), and net baryon density (𝜌_𝐵). 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 𝜌 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 𝑝_𝑇 ranges as well as the 𝑀_𝑇 spectra for different 𝑀 bins.

Figure 20 

The dilepton emission rate from different sources at 𝑇=175 MeV and 𝜇𝐵=30 MeV.

As indicated earlier, coupling of the hadronic current to multiparticle states gives rise to a continuum structure in the current correlation function 𝑊^𝜇𝜈. Following [104], we take a parameterized form for this contribution and augment the dilepton emission rate with 𝑑𝑁𝑑⁴𝑞𝑑⁴𝑥=𝛼²𝜋³𝐿𝑞²𝑓_𝐵𝐸𝑞_{0𝑉=𝜌,𝜔}𝐴^cont_𝑉, (66) where 𝐴^cont_𝜌=1𝛼8𝜋1+^𝑠𝜋11+exp𝜔₀−𝑞₀/𝛿 (67) with 𝜔₀ = 1.3, 1.1 GeV for 𝜌, 𝜔, and 𝛿=0.2 for both 𝜌 and 𝜔. The continuum contribution for the 𝜔 contains an additional factor of 1/9.

6.3. Invariant Mass and Momentum Spectra of Dileptons

The total invariant transverse momentum distribution of thermal dileptons (𝑙⁺𝑙⁻) is obtained as follows: 𝑑²𝑁_𝑙⁺_𝑙⁻𝑑²𝑝_𝑇𝑑𝑦=_{𝑖=𝑄,𝑀,𝐻𝑖}𝑑𝑅_𝑙⁺_𝑙⁻𝑑²𝑝_𝑇𝑑𝑦𝑑𝑀²_𝑖𝑀𝑑𝑀𝑑⁴𝑥. (68) In a similar manner, the invariant transverse mass distribution of thermal dileptons (𝑙⁺𝑙⁻) can be obtained by integrating static emission rate over certain 𝑝_𝑇 window and by convoluting that by four volume and expressed as follows: 𝑑²𝑁_𝑙⁺_𝑙⁻2𝑀𝑑𝑀𝑑𝑦=_{𝑖=𝑄,𝑀,𝐻𝑖}𝑑𝑅_𝑙⁺_𝑙⁻𝑑²𝑝_𝑇𝑑𝑦𝑑𝑀²_𝑖𝑝_𝑇𝑑𝑝_𝑇𝑑⁴𝑥. (69) The limits for integration over 𝑝_𝑇 and 𝑀 can be fixed judiciously to detect contributions either from quark matter or hadronic matter. Experimental measurements [79, 114, 115] are available for different 𝑀 window.

6.4. Results and Discussion on 𝑝_𝑇 and 𝑀 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 𝑀_𝑇 and 𝑀 spectra of dileptons measured by NA60 collaboration at SPS energy are reproduced as follows.

Finally, we have obtained the dimuon yield (𝑑𝑁/𝑑𝑀) 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 𝜏_𝑖=0.7 fm, the QGP to hadronic matter transition temperature 𝑇_𝑐=175 MeV, and the freezeout temperature 𝑇_𝑓=120 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 (𝑝_𝑇) windows calculated for 17.3 GeV energies.

Figure 21 

Dilepton invariant mass spectra for different 𝑝𝑇 bins compared with the NA60 data.

The theoretical curves agree quite well with the experimental data [114, 115] for all the 𝑝_𝑇 ranges. The strong enhancement in the low 𝑀 domain is clearly due to the large broadening of the 𝜌 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 0.35≤𝑀≤0.65 GeV.

Apart from the 𝑀 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 (𝜌, 𝜔, and 𝜙) 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 𝑀_𝑇 spectra of the lepton pair with ⟨𝑀⟩ evaluated within the ambit of the present model [113] reproduces the NA60 [114, 115] results reasonably well.

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 √𝑠_𝑁𝑁=200 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 𝑝_𝑇 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 0.5<𝑀(GeV)<0.75 and 0.81<𝑀(GeV)<0.99 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 0.1<𝑀(GeV)<0.2, 0.2<𝑀(GeV)<0.3 and 0.3<𝑀(GeV)<0.5) contain arbitrary normalization constant.

Figure 23 

Transverse momentum spectra of dileptons for different invariant mass windows for minimum bias Au-Au collisions at RHIC energy [12].

Assuming 10% hard (i.e., 𝑥=0.10) and 90% soft collisions for initial entropy production the value of 𝑑𝑁^ch_pp/𝑑𝑦 turns out to be about 2.43 at √𝑠=200 GeV. For RHIC energy, we take 𝑇_𝑖=320 MeV with initial time 𝜏_𝑖=0.2 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 𝜙 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 𝜌 spectral function for RHIC energy. The enhancement in the region 0.1≤𝑀≤0.7 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 2𝑚_𝜋 threshold coming from the unity in the unitary cut contribution. The (small) kink at 0.42 GeV in this curve is due to the 3𝑚_𝜋 threshold for 𝜔 production. The enhancement in the yield due to medium effects is ~20 for 𝑀 around 400 MeV.

Figure 24 

Invariant mass distribution of dileptons from hadronic matter (HM) for modified and unmodified 𝜌 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 𝑀 up to 𝜙 mass. Since the internal loops of 𝜌 self-energy contain 𝑎_1𝜋 and 𝜔_𝜋 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 𝑀 beyond 𝜙 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 𝑀 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 𝑐²_𝑠(=𝑑𝑃/𝑑𝜖 evaluated at constant entropy) which controls the rate of expansion. For EoS of the type (a) 𝑐²_𝑠∼1/3 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 𝑐²_𝑠 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 𝜌 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.

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 𝑑𝑁^ch_pp/𝑑𝑦 for √𝑠_𝑁𝑁=900 GeV, 2.36 TeV, and 7 TeV are 3.02, 3.77, and 6.01, respectively [122]. The value 𝑑𝑁^ch_pp/𝑑𝑦 at √𝑠_𝑁𝑁=5.25 TeV is obtained by interpolating the above experimental data mentioned above. Assuming 𝑥=0.2 in (26) we obtain 𝑑𝑁/𝑑𝑦=2607 in Pb+Pb collision for 0–10% centrality. For 𝜏_𝑖=0.1 fm/c we get 𝑇_𝑖=756 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.

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 𝑝_𝑇 and in central collision. So our results will not differ much even if (3+1) hydrodynamics is used at very high 𝑝_𝑇 and in central collision. In [123], the transverse momentum spectra of photons and 𝑣₂ of photons are calculated at RHIC energy using (3+1) hydrodynamical model. Recently, in [124, 125], a realistic (3+1) 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 𝑚_𝑇(=𝑝²_𝑇+𝑚²_ℎ) 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 [1–8] 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 𝑇_𝑖, 𝜏_𝑖, 𝑇_ch, 𝑇_𝑓, 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 (𝑅_em) 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 𝑀→0 the lepton pairs (virtual photons) emerge as real photons. Therefore, the evaluation of the ratio of the 𝑝_𝑇 spectra of photons to dileptons for various invariant mass bins along with a judicious choice of the 𝑝_𝑇 and 𝑀 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 𝑝_𝑇 spectra of photon and dilepton can be parametrized as follows: 𝑑𝑁𝑑²𝑝_𝑇𝑑𝑦_𝛾=𝐴₁1𝑝_𝑇^𝐵1exp−𝑐₁𝑝_𝑇;𝑐₁=1𝑇_eff1,𝑑𝑁𝑑²𝑝_𝑇𝑑𝑦_𝑙+𝑙−=𝐴₂1𝑀_𝑇^𝐵2exp−𝑐₂𝑀_𝑇;𝑐₂=1𝑇_eff2, (70) where 𝑇_eff1=𝑇_av(1+𝑣_𝑟)/(1−𝑣_𝑟) is the blue-shifted effective temperature for massless photons and 𝑇_eff2=𝑇_av+𝑀𝑣²_𝑟 is the effective temperature for massive dileptons. 𝑇_av is the average temperature and 𝑣_𝑟 is the average radial flow of the system. The 𝑇_eff1,2 can be obtained by parameterizing the 𝑝_𝑇 spectra of photons and dileptons (see Sections 5.4 and 6.4), respectively, with the expressed form of (70). The ratio, 𝑅_em, for different 𝑀 windows (Figure 27) can be parametrized as follows: 𝑅_em𝑀=𝐴^𝑇𝑝_𝑇^𝐵exp−𝑐𝑀_𝑇−𝑝_𝑇1;𝑐=𝑇_eff (71) with different values of 𝑇_eff for different invariant mass windows. The argument of the exponential in (71) can be written as [126] 𝑀_𝑇−𝑝_𝑇𝑇_eff=𝑀_𝑇𝑇_eff2−𝑝_𝑇𝑇_eff1=𝑀_𝑇𝑇_av+𝑀𝑣²_𝑟−𝑝_𝑇𝑇_av1+𝑣_𝑟/1−𝑣_𝑟. (72)

(a)

(b)

Figure 27 

Variation thermal photon to dilepton ratio, 𝑅em, with 𝑝𝑇 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, 𝑅_em, of the 𝑝_𝑇 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 𝑅_em 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, 𝑅_em, displayed in Figure 27(b) are constrained by the experimental data on the single-photon and -dilepton spectra. The behavior of 𝑅_em with 𝑝_𝑇 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 𝑝_𝑇>1.5 GeV.

This behavior of 𝑅_em as a function of 𝑝_𝑇 can be understood as follows: (i) for 𝑝_𝑇≫𝑀, 𝑀_𝑇∼𝑝_𝑇 and consequently 𝑅_em∼𝐴 giving rise to a plateau at large 𝑝_𝑇. The height of the plateau is sensitive to the initial temperature of the system [16, 126] and (ii) for 𝑝_𝑇<𝑀, 𝑅_em∼exp(−𝑝_𝑇/𝑇_eff)/𝑝^𝐵_𝑇 indicating a decrease of the ratio with 𝑝_𝑇 (at low 𝑝_𝑇) as observed in Figure 27.

7.1. Variation of Radial Flow with Average Temperature

For a given 𝑝_𝑇 and 𝑀, (72) can be written as 𝑣_𝑟=𝑓(𝑇_av). The 𝑇_eff’s are obtained from the ratio of individual spectra of photon and dilepton (by parametrising the 𝑅_em in Figure 27 using (71)). Thus we obtained the variation of radial flow with average temperature (𝑣_𝑟(𝑇_av)) for SPS (a) and RHIC (b) that has been depicted in Figure 28. It is clear from Figure 28 that the magnitude of the flow is larger in case of RHIC than SPS because of the higher initial pressure. Because of the larger initial pressure and QGP life time the radial velocity for QGP at RHIC is larger compared to SPS.

(a)

(b)

Figure 28 

The variation of radial flow velocity with average temperature of the system for ⟨𝑀⟩=0.75 GeV and 1.2 GeV at SPS energy (a) and for ⟨𝑀⟩=0.625 GeV and 0.9 GeV at RHIC energy (b).

The 𝑇_eff obtained from the parametrization of ratio at SPS energy is 263 MeV and 243 MeV for 𝑀=0.75 and 1.2 GeV, respectively. The average flow velocity 𝑣_𝑟 versus 𝑇_av has been displayed for 𝑀=0.75 GeV and 1.2 GeV in Figure 28(a). The hadronic matter (QGP) dominates the 𝑀∼0.75(1.2) GeV region. Therefore, these two mass windows are selected to extract the flow parameters for the respective phases. The 𝑣_𝑟 increases with decreasing 𝑇_av (increase in time) and reaches its maximum when the temperature of the system is minimum; that is, when the system attains 𝑇_𝑓, the freezeout temperature. Therefore, the variation of 𝑣_𝑟 with 𝑇_av may be treated as to show how the flow develops in the system. The 𝑣_𝑟 is larger in the hadronic phase because the velocity of sound in this phase is smaller, which makes the expansion slower, as a consequence system lives longer—allowing the flow to fully develop. On the other hand, 𝑣_𝑟 is smaller in the QGP phase because it has smaller life time where the flow is only partially developed. In Figure 28(b) the variation of average transverse velocity with average temperature for RHIC initial conditions is depicted.

7.2. Variation of Radial Flow with Invariant Mass

Obtaining 𝑇_eff1 and 𝑇_eff2 from the individual spectra and eliminating 𝑇_av one gets the variation of 𝑣_𝑟 with 𝑀. Figure 29(a) shows the variation of 𝑣_𝑟 with 𝑀 for SPS conditions. The radial flow velocity increases with invariant mass 𝑀 up to 𝑀=𝑀_𝜌 and then drops. How can we understand this behavior? From the invariant mass spectra, it is well known that the low 𝑀 (below 𝜌 mass) and high 𝑀 (above 𝜙 peak) pairs originate from a partonic source [16]. The collectivity (or flow) does not develop fully in the QGP because of the small life time of this phase, which means that the radial velocity in QGP will be smaller for both low and high 𝑀. Whereas the lepton pairs with mass around 𝜌-peak mainly originate from a hadronic source (at a late stage of the evolution of system) are largely affected by the flow resulting in higher values of flow velocity. In summary, the value of 𝑣_𝑟 for 𝑀 below and above the 𝜌-peak is small but around the 𝜌 peak is large—with the resulting behavior displayed in Figure 29. Similar nonmonotonic behavior is observed in case of elliptic flow of photon as a function of 𝑝_𝑇 [128]. The variation of 𝑣_𝑟 with 𝑀 in RHIC (Figure 29(b)) is similar to SPS though the values of 𝑣_𝑟 at RHIC are larger than those of SPS as expected due to higher initial pressure.

(a)

(b)

Figure 29 

The variation of radial flow with invariant mass pairs for SPS (a) and RHIC (b) energies.

It is shown that simultaneous measurements of photon and dilepton spectra in HIC will enable us to quantify the evolution of the average radial flow velocity for the system, and the nature of the variation of radial flow with invariant mass indicates the formation of partonic phase at SPS and RHIC energy. The stronger radial flow at RHIC compared to SPS is due to higher initial energy densities and a longer lifetime of the reaction zone.

8. Dilepton Interferometry

The two-particle intensity interferometry, commonly known as Hanbury Brown Twiss (HBT) interferometry [129], is considered as one of the efficient methods to extract the information of space-time structure of the fireball formed in HIC. The utility of the intensity interferometry with dileptons [14, 15] for extracting fireball properties is as follows. As EM radiation produces from each stages of HIC, it retains information of the fireball at each space-time point. In contrast to correlation studies with hadrons which give information of the system when the system has frozen out, two-particle intensity interferometry using lepton pairs [14, 15], or photon [130, 131], can provide the information on the history of evolution of hot matter efficiently because EM probes do not rescatter after its production. As argued previously, photons appear to be more restrictive compared to dilepton, and we have attempted to do the correlation calculations with dileptons. So that with judicious choice of 𝑝_𝑇 and 𝑀 windows we can get the spatial as well as temporal information of QGP and the hadronic phases separately by making use of correlations between two dilepton pairs.

8.1. Formalism and Equation of Bose-Einstein Correlation Function (BECF)

As interferometry of the dilepton pairs actually reflects correlations between two virtual photons, the analysis then concentrates on computing the Bose-Einstein correlation function (BECF) for two identical particles defined as 𝐶₂𝑝₁,𝑝₂=𝑃₂𝑝₁,𝑝₂𝑃₁𝑝₁𝑃₁𝑝₂, (73) where 𝑝_𝑖 is the three momenta of the particle 𝑖, and 𝑃₁(𝑝_𝑖) and 𝑃₂(𝑝₁,𝑝₂) represent the one- and two-particle inclusive lepton pair spectra, respectively, and is expressed as follows: 𝑃₁𝑝=𝑑⁴(),𝑃𝑥𝜔𝑥,𝐾₂𝑝₁,𝑝₂=𝑃₁𝑝₁𝑃₁𝑝₂+𝜆3𝑑⁴𝑥₁𝑑⁴𝑥₂𝜔𝑥₁𝑥,𝐾×𝜔₂,𝐾cosΔ𝑥^𝜇𝑞_𝜇, (74) where 𝐾=(𝑝₁+𝑝₂)/2, 𝑞_𝜇=𝑝_1𝜇−𝑝_2𝜇=𝑞_𝜇, Δ𝑥_𝜇=𝑥_1𝜇−𝑥_2𝜇, 𝑥_𝑖𝜇 and 𝑝_𝑖𝜇 are four coordinates for position and momentum variables, respectively, and 𝜔(𝑥,𝐾) is the source function related to the thermal emission rate of lepton pairs per unit four volume, expressed as as follows: 𝜔(𝑥,𝐾)=^𝑀22_𝑀21𝑑𝑀²𝑑𝑅𝑑𝑀²𝑑²𝐾_𝑇.𝑑𝑦 (75) With further simplification, the 𝐶₂ can be redefined as 𝐶₂𝑝₁,𝑝₂𝜆=1+3∫𝑑⁴𝑥𝜔(𝑥,𝐾)cos(Δ𝛼)²𝑃₁𝑝₁𝑃₁𝑝₂+∫𝑑⁴𝑥𝜔(𝑥,𝐾)sin(Δ𝛼)²𝑃₁𝑝₁𝑃₁𝑝₂, (76)

where Δ𝛼=𝛼₁−𝛼₂, 𝛼_𝑖=𝜏𝑀_𝑖𝑇cosh(𝑦_𝑖−𝜂)−𝑟𝑝_𝑖𝑇cos(𝜃−𝜓_𝑖), 𝑀_𝑖𝑇=𝑝²_𝑖𝑇+𝑀² is the transverse mass, 𝑦_𝑖 is the rapidity, and 𝜓_𝑖’s are the angles made by 𝑝_𝑖𝑇 with the 𝑥-axis.

The inclusion of the spin of the virtual photon will reduce the value of 𝐶₂−1 by 1/3. The correlation functions can be evaluated for different average mass windows, ⟨𝑀⟩(≡𝑀_𝑙⁺_𝑙⁻)=(𝑀₁+𝑀₂)/2. The leading order process through which lepton pairs are produced in QGP is 𝑞𝑞→𝑙⁺𝑙⁻ [100, 101]. For the low 𝑀 dilepton production from the hadronic phase the decays of the light vector mesons 𝜌, 𝜔, and 𝜙 have been considered including the continuum [1, 5, 7, 8, 104]. Since the continuum part of the vector meson spectral functions is included in the current work, the processes like four pions annihilations [121] are excluded to avoid double counting.

For the space time the evolution of the system relativistic hydrodynamical model with cylindrical symmetry [64] and boost invariance along the longitudinal direction [41] has been used. The values of the parameters required for space-time evolution are displayed in Table 3. With all these ingredients we evaluate the correlation function 𝐶₂ for 0–5% Au+Au collisions centrality for RHIC at √𝑠_𝑁𝑁=200 GeV [132] and Pb+Pb collisions at for LHC at √𝑠_𝑁𝑁=2.76 TeV [133] for different invariant mass windows as a function of 𝑞_side and 𝑞_out which are related to transverse momenta of individual pair [134, 135]. By choosing appropriate phase space for the QGP and hadron gas and performing the space time integration using the initial condition tabulated in Table 3, the 𝐶₂ for different phase has been evaluated. We have evaluated the 𝐶₂ for ⟨𝑀⟩= 0.3, 0.5, 0.7, 1.2, 1.6, and 2.5 GeV. In Figure 30 the results for only three values of ⟨𝑀⟩ corresponding to low and high mass which are expected to be dominated by radiations from QGP (⟨𝑀⟩∼1.6 GeV) and hadronic phase (⟨𝑀⟩∼0.77 GeV), respectively, are displayed.

Table 3 

Values of the various parameters used in the relativistic hydrodynamical calculations.

(a)

(b)

Figure 30 

Correlation function for dilepton pairs as a function of 𝑞side ((a), for 𝑝1𝑇=𝑝2𝑇=2 GeV and 𝜓2=0) and 𝑞out ((b), for 𝜓1=𝜓2=0 and 𝑝1𝑇=2 GeV) for three values of ⟨𝑀⟩ [14, 15]. The solid lines show the parameterization of 𝐶2 using (77).

In Figure 30, we plot the 𝐶₂ as a function of 𝑞_side and 𝑞_out for RHIC initial conditions as tabulated in Table 3. A clear difference of dilepton pair mass dependence of the BEC studied as a function of 𝑞_side is observed for the contributions from different 𝑀 domains. The differences are however small when BEC is studied as a function of 𝑞_out.

8.2. Source Dimension

The source dimensions can be obtained by parameterizing the calculated correlation function of the dilepton pairs with the empirical (Gaussian) form: 𝐶₂𝑞,𝐾=1+𝜆exp−𝑅²_𝑖(𝐾)𝑞²_𝑖, (77) where 𝑖 stands for side, out, and long. Thus 𝑅_side, 𝑅_out, and 𝑅_long appearing in (77) are commonly referred to as HBT radii, which is measure of Gaussian widths of source size and can be expressed as follows: 𝑅²_side(𝐾)=𝑦²,𝑅²_out(𝐾)=𝑥−𝑣_𝑟𝑡̃²,𝑅²_long(𝐾)=𝑧−𝑣_𝑧𝑡̃². (78)

The 𝑞_out, 𝑞_side, and 𝑞_long can be expressed in terms of individual particle momenta as [136] 𝑞_side=𝑞_𝑇−𝑞_out𝐾_𝑇𝐾_𝑇=2𝑝_1𝑇𝑝_2𝑇1−cos²𝜓₁−𝜓₂𝑓𝑘_1𝑇,𝑘_2𝑇,𝑞_out=𝑞_𝑇⋅𝐾_𝑇𝐾_𝑇=𝑝²_1𝑇−𝑝²_2𝑇𝑓𝑘_1𝑇,𝑘_2𝑇,𝑞_long=𝑝_1𝑧−𝑝_2𝑧=𝑝_1𝑇sinh𝑦₁−𝑝_2𝑇sinh𝑦₂, (79) where 𝑓(𝑘_1𝑇,𝑘_2𝑇)=𝑝²_1𝑇+𝑝²_2𝑇+2𝑝_1𝑇𝑝_2𝑇cos(𝜓₁−𝜓₂) and 𝜆 (in this review, 𝜆=1/3) represents the degree of chaotic of the source. The deviation of 𝜆 from 1/3 will indicate the presence of nonthermal sources. A representative fit to the correlation functions is shown in Figure 30 (solid lines). While the radius (𝑅_side) corresponding to 𝑞_side is closely related to the transverse size of the system and considerably affected by the collectivity, the radius (𝑅_out) corresponding to 𝑞_out measures both the transverse size and duration of particle emission [136–139]. The extracted 𝑅_side and 𝑅_out for different ⟨𝑀⟩ are shown in Figures 31(a) and 31(b), respectively.

(a)

(b)

Figure 31 

(a) 𝑅side is evaluated with 𝑝1𝑇=𝑝2𝑇=2 GeV and 𝜓2=0 as a function of ⟨𝑀⟩ for RHIC energy and (b) 𝑅out is evaluated with 𝜓1=𝜓2=0 and 𝑝1𝑇=2 as a function of ⟨𝑀⟩ for RHIC [14, 15].

8.2.1. Variation of 𝑅_side and 𝑅_out with ⟨𝑀⟩

𝑅_side. The variation of 𝑅_side for QGP, hadronic and QGP + hadronic phase, is obtained from the respective 𝐶₂ in that phase with an appropriate selection of phase space in space-time integration. Figure 31(a) shows nonmonotonic dependence of 𝑅_side on 𝑀, starting from a value close to QGP value (indicated by the dashed line); it drops with increase in 𝑀 finally again approaching the QGP value for ⟨𝑀⟩>𝑚_𝜙. It can be shown that 𝑅_side∼1/(1+𝐸_collective/𝐸_thermal) [134, 135]. In the absence of radial flow, 𝑅_side is independent of 𝑞_side. With the radial expansion of the system a rarefaction wave moves toward the center of the cylindrical geometry; as a consequence the radial size of the emission zone decreases with time. Therefore, the size of the emission zone is larger at early times and smaller at late time. The high ⟨𝑀⟩ regions are dominated by the early partonic phase where the collective flow has not been developed fully; that is, the ratio of collective-to-thermal energy is small, and hence the source has larger 𝑅_side. In contrast, the lepton pairs with 𝑀∼𝑚_𝜌 are emitted from the late hadronic phase where the size of the emission zone is smaller due to larger collective flow giving rise to a smaller 𝑅_side. The ratio of collective to thermal energy for such cases is quite large, which is reflected as a dip in the variation of 𝑅_side with ⟨𝑀⟩ around the 𝜌-mass region (Figure 31(a)). Thus the variation of 𝑅_side with 𝑀 can be used as an efficient tool to measure the collectivity in various phases of matter. The dip in 𝑅_side at ⟨𝑀⟩∼𝑚_𝜌 is due to the contribution dominantly from the hadronic phase. We observe that, by keeping the 𝜌 and 𝜔 contributions and setting radial velocity, 𝑣_𝑟=0, the dip in 𝑅_side vanishes, confirming the fact that the dip is caused by the radial flow of the hadronic matter. Therefore, the value of 𝑅_side at ⟨𝑀⟩∼𝑚_𝜌 may be used to estimate the average 𝑣_𝑟 in the hadronic phase.

𝑅_out. The 𝑅_out probes both the transverse dimension and the duration of emission, and unlike 𝑅_side it does not remain constant even in the absence of radial flow. As a result its variation with 𝑀 is complicated. The values 𝑅_out for different phases are obtained in a similar fashion as followed for obtaining the 𝑅_side values for the different phases. The large 𝑀 regions are populated by lepton pairs from early partonic phase where the effect of flow is small and the duration of emission is also small—resulting in smaller values of 𝑅_out. For lepton pair from 𝑀∼𝑚_𝜌 the flow is large which could have resulted in a dip as in 𝑅_side in this 𝑀 region. However, 𝑅_out probes the duration of emission too which is large for hadronic phase because the expansion is slower in this phase for the EoS used in the present work. The velocity of sound which controls the rate of expansion and hence the duration of the phase has larger value in hadronic phase than in the partonic phase. Thus resulting in the larger 𝑅_out in the hadronic phase than that in partonic phase, the larger duration compensates the reduction of 𝑅_out due to flow in the hadronic phase resulting is a bump in 𝑅_out in this region of 𝑀 (Figure 31(b)). Again the duration of particle emission from both the phases obviously is larger than that from the individual phases.

Both 𝑅_side and 𝑅_out approach QGP values for ⟨𝑀⟩∼2.5 GeV implying dominant contributions from partonic phase.

8.3. Comparison of HBT Radii with Different Collision Energies

Now we study the sensitivity of the HBT radii on the different collision energy. The 𝑅_side and 𝑅_out extracted from the 𝐶₂’s evaluated for 0–5% centrality in Au+Au collisions for RHIC at √𝑠_𝑁𝑁=200 GeV [132] and Pb+Pb collisions for LHC at √𝑠_𝑁𝑁=2.76 TeV [133] for different invariant mass windows as a function of 𝑞_side and 𝑞_out are shown in Figure 32. The change of 𝑅_side with ⟨𝑀⟩ for RHIC and LHC is qualitatively similar but quantitatively different. The smaller values of 𝑅_side for LHC are due to the larger radial expansion which can be understood from the fact that the quantity 𝐸_collective/𝐸_thermal is larger at LHC than RHIC. So, the dip in the 𝑅_side variation at LHC is below than that at RHIC confirming a larger flow at LHC than RHIC. As the 𝑅_out probes both the transverse size and the duration of emission, from the previous discussion in Section 8.2.1, the larger duration compensates the reduction of 𝑅_out due to that flow resulting is a bump in 𝑅_out for 𝑀∼𝑚_𝜌. Though the duration of particle emission is more at LHC compared to RHIC (shown in Figure 36), the larger flow (corresponds to smaller size) at LHC [126] than that of RHIC compensates other factor (like duration of emission) which has an enhancing effect on 𝑅_out. So the value 𝑅_out at LHC is smaller than that of RHIC.

(a)

(b)

Figure 32 

𝑅side (a) is evaluated with 𝑝1𝑇=𝑝2𝑇=1 GeV and 𝜓2=0 and 𝑅out (b) is evaluated with 𝑝1𝑇=1 GeV and 𝜓1=𝜓2=0 as a function of ⟨𝑀⟩ for RHIC (dashed line) and LHC (solid line) energies [14, 15].

8.4. Radial Flow from HBT Radii

According to the discussion given in the Section 8.2, 𝑅_side is independent of 𝑞_side in the absence of radial flow. 𝑅_side is related to radial flow as follows: 𝑅_side(𝑀)=𝒦𝑝_𝑇(𝑀;𝑃)^𝑇(𝑀)=𝑇_av+𝑀𝑣²_𝑟. (80)

The values of 𝑅_side(𝑀) are obtained for different ⟨𝑀⟩ windows (shown in Figure 31). The higher mass, that is, ⟨𝑀⟩=2.5 GeV, corresponds to the initial stage of collision where the flow is not developed fully. So assume for ⟨𝑀⟩=2.5, 𝑣_𝑟=0, and 𝑇=𝑇_𝑖 the value of 𝒦=𝑇_𝑖×𝑅_side|_{⟨𝑀⟩=2.5}. Once we know the value of 𝒦, we can calculate the ⟨𝑝_𝑇(𝑀)⟩(=𝒦/𝑅_side(𝑀)) The variation of ⟨𝑝_𝑇⟩ with ⟨𝑀⟩ has been displayed in Figure 33.

Figure 33 

Variation of ⟨𝑝𝑇⟩ as function of 𝑀 (see (80)).

The high ⟨𝑀⟩ regions are dominated by the early partonic phase where the collective flow has not been developed fully and hence show smaller ⟨𝑝_𝑇⟩. In contrast, due to larger collective flow for the lepton pairs with 𝑀∼𝑚_𝜌, emitted from the late hadronic phase, ⟨𝑝_𝑇⟩ is larger. The larger value of ⟨𝑝_𝑇⟩ around the 𝜌-mass region is due to the contribution of large flow in the hadronic phase. Thus the variation of 𝑅_side with 𝑀 (Figure 31) can be used as an efficient tool to measure the collectivity in various phases of matter.

8.5. Sensitivity of HBT Radii on 𝑝_𝑖𝑇

In this section, the sensitivity of the HBT radii for different values of the individual transverse momentum of the pairs is described. In Figure 34, the variation of 𝑅_side and 𝑅_out with ⟨𝑀⟩ is shown for 𝑝_1𝑇= 1 and 2 GeV. The lepton pairs coming from higher 𝑝_𝑇 and high mass region enable us to quantify the size of hotter zone. As mentioned before, the 𝑝_𝑇 contains the effect of flow as well as thermal motion. Hence the larger 𝑅_side at 𝑀∼𝑀_𝜌 for 𝑝_𝑇=2 GeV is associated with longer flow and hence smaller source size. The observed bump in 𝑅_out (Figure 34(b)) is resulted from the fact that it contains both the size of the system as well as the duration of dilepton emission as discussed earlier.

(a)

(b)

Figure 34 

(a) Shows 𝑅side as a function of ⟨𝑀⟩ which is evaluated with 𝑝1𝑇=𝑝2𝑇= 1 and 2 GeV and 𝜓2=0 and similarly (b) shows 𝑅out as a function of for 𝜓1=𝜓2=0 and 𝑝1𝑇=1 and 2 GeV ⟨𝑀⟩. The 𝑝1𝑇=1 and 2 GeV results are shown as dashed line and solid line, respectively.

8.6. Duration of Particle Emission from HBT Radii

The HBT radii, 𝑅_out and 𝑅_side, provide the information of average source size. However, in the ratio, 𝑅_out/𝑅_side, some of the uncertainties associated with the space time evolution get canceled out. The quantity, 𝑅_out/𝑅_side gives the duration of particle emission [134, 135, 140, 141] for various domains of 𝑀. The difference between 𝑅²_side(𝐾) and 𝑅²_out(𝐾) at nonzero in 𝐾 is then only due to the explicit 𝐾 dependence in (78), that is, the term 𝑣_𝑟⟨𝑡²⟩. This implies that the explicit 𝐾 dependence dominates if the emission duration is sufficiently large or if the position-momentum correlations in the source are sufficiently weak: 𝑅²_diff=𝑅²_out(𝐾)−𝑅²_side(𝐾)=𝑣_𝑟𝑡². (81)

In this case, the difference between these two HBT radius parameters gives direct access to the average emission duration ⟨𝑡²⟩ of the source and allows to partially disentangle the spatial and temporal information contained in (78).

Figure 35 shows the 𝑅_out/𝑅_side and the difference 𝑅²_out−𝑅²_side as a function of ⟨𝑀⟩ for Au+Au collisions at √𝑠_𝑁𝑁=200 GeV. Both show a nonmonotonic dependence on ⟨𝑀⟩. The smaller values of both the quantities, particularly at high mass region, reflect the contributions from the early partonic phase of the system. The peak around 𝜌-meson mass reflects dominance of the contribution from hadronic phase as discussed before. Figure 36 shows a comparative study of the above two quantities (the ratio and the difference of 𝑅_out and 𝑅_side) for RHIC and LHC energies. They reflect a larger life time of thermal system for LHC than RHIC.

(a)

(b)

Figure 35 

The ratio 𝑅out/𝑅side and the difference 𝑅2out−𝑅2side as a function of ⟨𝑀⟩ [14, 15].

(a)

(b)

Figure 36 

The ratio 𝑅out/𝑅side (a) and the difference 𝑅2out−𝑅2side (b) as a function of ⟨𝑀⟩ for RHIC (dashed line) and LHC (solid line) energies are shown.

9. Elliptic Flow of Thermal Dileptons

It has been argued that the anisotropic momentum distribution of the hadrons can bring the information on the interaction of the dense phase of the system [142] despite the fact that the hadrons are emitted from the freezeout surfaces when the system is too dilute to support collectivity. Therefore, a suitable dynamical model is required to extrapolate the final hadronic spectra backward in time to get the information about the early dense phase. Such an extrapolation is not required for lepton pairs because they are emitted from the entire space-time volume of the system. Therefore, the 𝑣₂ of lepton pairs provides information of the hot and dense phase directly. The 𝑣₂ of dileptons can also be used to to test the validity and efficiency of the extrapolation required for hadronic 𝑣₂. The 𝑣₂ of real photons and dileptons [128, 143–146] has been evaluated for RHIC energies and shown that it can be used as effective probes to extract the properties of the partonic plasma. The sensitivity of the 𝑣₂ of lepton pairs on EoS has been elaborated in [145] for RHIC collision conditions. The lepton pairs are produced from each space time point of the system and hence the study of 𝑣₂ of lepton pairs will shed light on the time evolution of collectivity in the system [12, 147]. The radial flow alters the shape of the 𝑝_𝑇 spectra of dileptons; it kicks the low 𝑝_𝑇 pairs to the higher 𝑝_𝑇 domain, making the spectra flatter. Therefore, the presence of large radial flow may diminish the magnitude of 𝑣₂ at low 𝑝_𝑇 [37, 38], and this effect will be larger when the radial flow is large, that is, in the hadronic phase which corresponds to lepton pairs with 𝑀∼𝑚_𝜌.

9.1. Formalism of Elliptic Flow of Dilepton

The elliptic flow of dilepton, 𝑣₂, can be defined as 𝑣₂𝑝_𝑇==∑,𝑀cos2𝜙_{𝑖=𝑄,𝐻}∫cos2𝜙𝑑𝑁^𝛾∗/𝑑²𝑝_𝑇𝑑𝑀²𝑑𝑦_𝑦=0𝑖𝑑𝜙∑_{𝑖=𝑄,𝐻}∫𝑑𝑁^𝛾∗/𝑑²𝑝_𝑇𝑑𝑀²𝑑𝑦_𝑦=0𝑖,𝑑𝜙 (82) where the ∑ stands for summation over quark matter (QM) and hadronic matter (HM) phases. The quantity 𝑑𝑁/𝑑²𝑝_𝑇𝑑𝑀²𝑑𝑦|_𝑦=0 appearing in (82) can be obtained from the dilepton production per unit four volume, 𝑑𝑁/𝑑⁴𝑝𝑑⁴𝑥 in a thermalized medium by integrating over the space-time evolution of the system. The 𝑑𝑁/𝑑⁴𝑝𝑑⁴𝑥 for lepton pairs for QGP and hadrons are discussed in Sections 6.1 and 6.2, respectively (see [13, 109] for details).

To evaluate 𝑣₂ from (82) one needs to integrate the fixed temperature production rate given by (9) over the space time evolution of the system—from the initial QGP phase to the final hadronic freezeout state through a phase transition in the intermediate stage. The space-time evolution is done over the 4-volume, which is defined as 𝑑⁴𝑥(=𝜏𝑑𝜏𝑑𝑥𝑑𝑦𝑑𝜂) are expressed in terms of 𝑥^𝜇=(𝜏,𝑥,𝑦,𝜂). We assume that the matter is formed in QGP phase with zero net baryon density in Pb+Pb collision at √𝑠_𝑁𝑁=2.76 TeV. The energy of the lepton pair (𝑝₀) should be replaced by its value in the comoving frame of the expanding system which is given by [𝑝⋅𝑢=𝛾_𝑇(𝑀_𝑇cosh(𝑦−𝜂)−𝑣_𝑥𝑝_𝑇cos𝜙−𝑣_𝑦𝑝_𝑇sin𝜙)], 𝑝_𝜇=(𝑀_𝑇cosh𝑦,𝑝_𝑇cos𝜙,𝑝_𝑇sin𝜙,𝑀_𝑇sinh𝑦), and 𝑢^𝜇=𝛾_𝑇(cosh𝜂,𝑣_𝑥,𝑣_𝑦,sinh𝜂). The EoS required to close the hydrodynamic equations is constructed by complementing Wuppertal-Budapest lattice simulation [48] with a hadron resonance gas comprising all the hadronic resonances up to mass of 2.5 GeV [148, 149]. The necessary initial conditions to solve the hydrodynamic equations are 𝑇_𝑖=456 MeV, the value of the temperature corresponding to the maximum of the initial energy profile for 30–40% centrality at √𝑠_𝑁𝑁=2.76 TeV, with 𝜏_𝑖=0.6 fm/c, the thermalization time. The transition temperature, 𝑇_𝑐, for quark hadron conversion is taken as 175 MeV. The system is assumed to get out of chemical equilibrium at 𝑇=𝑇_ch=170 MeV [69]. The kinetic freezeout temperature 𝑇_𝐹=130 MeV is fixed from the 𝑝_𝑇 spectra of the produced hadrons at the same collision energy of Pb+Pb system. The EoS and the values of the parameters mentioned above are constrained by the 𝑝_𝑇 spectra (for 0–5% centrality) and elliptic flow (for 10–50% centrality) of charged hadrons [148] measured by ALICE collaboration [150, 151].

9.2. Results and Discussion

In Figure 37 we depict the constant temperature contours corresponding to 𝑇_𝑐=175 MeV and 𝑇_𝑓=130 MeV in the 𝜏-𝑥 plane (at zero abscissa) indicating the boundaries for the QM and HM phases, respectively.

Figure 37 

Constant temperature contours denoting space-time boundaries of the QGP and hadronic phase.

The life time of the QM phase ~6 fm/c and the duration of the HM are ~6–12 fm/c. Throughout this work by early and late will approximately mean the duration of the QM and HM, respectively.

With all the ingredients mentioned above we evaluate the 𝑝_𝑇 integrated 𝑀 distribution of lepton pairs originating from QM and HM (with and without medium effects on the spectral functions of 𝜌 and 𝜔). The results are displayed in Figure 38 for the initial conditions and centrality mentioned above. We observe that for 𝑀>𝑀_𝜙 the QM contributions dominate. For 𝑀_𝜌≤𝑀≤𝑀_𝜙 the HM shines brighter than QM. For 𝑀<𝑀_𝜌, the HM (solid line) over shines the QM due to the enhanced contributions primarily from the medium-induced broadening of 𝜌 spectral function. However, the contributions from QM and HM become comparable in this region of 𝑀 if the medium effects on 𝜌 spectral function are ignored (dotted line). Therefore, the results depicted in Figure 38 indicate that a suitable choice of 𝑀 window will enable us to unravel the contributions from a particular phase (QM or HM).

Figure 38 

Invariant mass distribution of lepton pairs from quark matter and hadronic matter [13].

To further quantify these issues we evaluate the following quantity: ∫𝐹=^′𝑑𝑁/𝑑⁴𝑥𝑑²𝑝_𝑇𝑑𝑀²𝑑𝑦𝑑𝑥𝑑𝑦𝑑𝜂𝜏𝑑𝜏𝑑²𝑝_𝑇𝑑𝑀²∫𝑑𝑁/𝑑⁴𝑥𝑑²𝑝_𝑇𝑑𝑀²𝑑𝑦𝑑𝑥𝑑𝑦𝑑𝜂𝜏𝑑𝜏𝑑²𝑝_𝑇𝑑𝑀², (83) where the 𝑀 integration in both the numerator and denominator is performed for selective 𝑀 windows from 𝑀₁ to 𝑀₂ with mean 𝑀 defined as ⟨𝑀⟩=(𝑀₁+𝑀₂)/2. The prime in ∫^′ in the numerator indicates that the 𝜏 integration in the numerator is done from 𝜏₁=𝜏_𝑖 to 𝜏₂=𝜏_𝑖+Δ𝜏 with progressive increment of Δ𝜏, while in the denominator the integration is done over the entire lifetime of the system. In Figure 39, 𝐹 is plotted against 𝜏_av(=(𝜏₁+𝜏₂)/2). The results substantiate the fact that pairs with high ⟨𝑀⟩∼2.5 GeV originate from QM (𝜏_av≤6 fm/c, QGP phase) and pairs with ⟨𝑀⟩∼0.77 GeV mostly emanate from the HM phase (𝜏_av≥6 fm/c). The change in the properties of 𝜌 due to its interaction with thermal hadrons in the bath is also visible through 𝐹 evaluated for ⟨𝑀⟩∼0.3 GeV with and without medium effects. This clearly indicates that the ⟨𝑀⟩ distribution of lepton pairs can be exploited to extract collectivity of different phases of the evolving matter.

Figure 39 

Fractional contribution of lepton pairs for various invariant mass windows as a function of average proper time (see text for details) [13].

Figure 40(a) shows the differential elliptic flow, 𝑣₂(𝑝_𝑇), of dileptons arising from various ⟨𝑀⟩ domains in quark matter. Similarly Figure 40(b) shows the differential elliptic flow, 𝑣₂(𝑝_𝑇), of dileptons arising from various ⟨𝑀⟩ domains from hadronic matter. The individual 𝑣₂ for QM and HM is obtained by doing an integration over specific invariant masses (𝑀) window as well as space time integration over the regime where 𝑇_𝑐<𝑇(𝜏,𝑥,𝑦)<𝑇_𝑖 and 𝑇_𝑓<𝑇(𝜏,𝑥,𝑦)<𝑇_𝑐, respectively. The 𝑣₂ is small at low 𝑝_𝑇 and gradually increases and attains large value around 𝑝_𝑇∼2-3 GeV/c. Also there is clear mass ordering that has been observed for 𝑣₂(𝑝_𝑇) for QM; that is, 𝑣₂ decreases with increase in 𝑀. This is because dileptons come from high 𝑀 region, 𝑀>𝑀_𝜙, come mostly from hot partonic phase where the fluid velocity is not strong to support the collectivity but the spatial eccentricity of the source is large. On the other hand dileptons that come from low 𝑀 region, 𝑀 below 𝜙 peak dominantly come from late hadronic matter where the collectivity is strong and the spatial asymmetry dissolve into momentum asymmetry.

(a)

(b)

Figure 40 

Elliptic flow of quark matter (a) and hadronic matter (b) as a function of 𝑝𝑇 for various mass windows.

Figure 41 shows the differential elliptic flow, 𝑣₂(𝑝_𝑇) of dileptons arising from various ⟨𝑀⟩ domains. We observe that, for ⟨𝑀⟩=2.5 GeV, 𝑣₂ is small for the entire 𝑝_𝑇 range because these pairs arise from the early epoch (see Figure 39) when the flow is not developed entirely. However, the 𝑣₂ is large for ⟨𝑀⟩=0.77 GeV as these pairs originate predominantly from the late hadronic phase when the flow is fully developed.

(a)

(b)

Figure 41 

Total elliptic flow as function of 𝑝𝑇 for various mass windows.

It is also interesting to note that the medium-induced enhancement of 𝜌 spectral function provides a visible modification in 𝑣₂ for dileptons below 𝜌 peak. Figure 42 shows the comparison between 𝑣₂(𝑝_𝑇) of dilepton at ⟨𝑀⟩=300 MeV with and without medium effects.

Figure 42 

The figure displays the effect of the broadening of 𝜌 spectral function on the elliptic flow for ⟨𝑀⟩=300 MeV.

In Figure 43 we depict the variation of 𝑅_𝑄 with 𝑝_𝑇 for ⟨𝑀⟩=0.3 GeV (line with solid circle) 0.77 GeV (solid line) and 2.5 GeV (line with open circle). The quantity 𝑅_𝑄 and 𝑅_𝐻 is defined as 𝑅_𝑄=𝑣^QM₂𝑣^QM₂+𝑣^HM₂,𝑅_𝐻=𝑣^HM₂𝑣^QM₂+𝑣^HM₂, (84) where 𝑣^QM₂ and 𝑣^HM₂ are the elliptic flow of QM and HM phases, respectively. The results clearly illustrate that 𝑣₂ of lepton pairs in the large ⟨𝑀⟩(=2.5GeV) domain (open circle in Figure 43) originates from QM for the entire 𝑝_𝑇 range considered here. The value of 𝑅_𝑄 is large in this domain because of the large (negligibly small) contributions from QM (HM) phase. 𝑓_QM is large here. It is also clear that the contribution from QM phase to the elliptic flow for ⟨𝑀⟩(=0.77GeV) is very small (solid line in Figure 43). The value of 𝑅_𝐻 for ⟨𝑀⟩=0.77 GeV is large (not shown in the figure). The 𝑣₂ at the (late) hadronic phase (either at 𝜌 or 𝜙 peak) is larger than the (early) QGP phase (say at ⟨𝑀⟩=2.5 GeV) for the entire 𝑝_𝑇 range considered here. Therefore, the 𝑝_𝑇-integrated values of 𝑣₂ should also retain this character of 𝑣₂ at the corresponding values of ⟨𝑀⟩. It is also important to note that the differential elliptic flow, 𝑣₂(𝑝_𝑇), obtained here at LHC is larger than the values obtained at RHIC [143–145] for all the invariant mass windows. The value of 𝑅_𝐻 for ⟨𝑀⟩=0.77 GeV is large (not shown in the figure).

Figure 43 

It shows the variation of 𝑅𝑄 (see text) with 𝑝𝑇 for ⟨𝑀⟩=0.3 GeV, 0.77 GeV and 2.5 GeV.

The 𝑣₂ at the HM phase (either at 𝜌 or 𝜙 peak) is larger than its value in the QGP phase (say at ⟨𝑀⟩=2.5 GeV) for the entire 𝑝_𝑇-range considered here. Therefore, the 𝑝_𝑇 integrated values of 𝑣₂ should also retain this character at the corresponding values of ⟨𝑀⟩, which is clearly observed in Figure 44 which displays the variation of 𝑣₂(⟨𝑀⟩) with ⟨𝑀⟩. The 𝑣₂(∝𝜖_𝑝) of QM is small because of the small pressure gradient in the QGP phase. The 𝑣₂ resulting from hadronic phase has a peak around 𝜌 pole indicating the full development of the flow in the HM phase. For ⟨𝑀⟩>𝑚_𝜙 the 𝑣₂ obtained from the combined phases approachs the value corresponding to the 𝑣₂ for QGP. Therefore, measurement of 𝑣₂ for large ⟨𝑀⟩ will bring information of the QGP phase at the earliest time of the evolution. It is important to note that the 𝑝_𝑇-integrated 𝑣₂(⟨𝑀⟩) of lepton pairs with ⟨𝑀⟩∼𝑚_𝜋, 𝑚_𝐾 is close to the hadronic 𝑣^𝜋₂ and 𝑣^𝐾₂ (symbol ∗ in Figure 44) if the thermal effects on 𝜌 properties are included. Exclusion of medium effects gives lower 𝑣₂ for lepton pairs compared to hadrons. The fact that the 𝑣₂ of the (penetrating) lepton pairs is similar in magnitude to the 𝑣₂ of hadrons for (⟨𝑀⟩∼𝑚_𝜋, 𝑚_𝐾, 𝑚_proton, etc.) ascertains that the anisotropic momentum distribution of hadrons carries the information of the HM phase with duration ~6–12 fm/c [13]. We also observe that the variation of 𝑣₂(⟨𝑀⟩) with ⟨𝑀⟩ has a structure similar to 𝑑𝑁/𝑑𝑀 versus 𝑀. As indicated by (82) we can write 𝑣₂(⟨𝑀⟩)∼∑_𝑖=QM,HM𝑣^𝑖₂×𝑓_𝑖, where 𝑓_𝑖 is the fraction of QM or HM from various space-time regions. The structure of 𝑑𝑁/𝑑𝑀 is reflected in 𝑣₂(⟨𝑀⟩) through 𝑓_𝑖. We find that the magnitude of 𝑣₂(⟨𝑀⟩) at LHC is larger than its value at RHIC [143–145].

Figure 44 

(Color online) Variation of dilepton elliptic flow as function of ⟨𝑀⟩ for QM, HM (with and without medium effects), and for the entire evolution. The symbol ∗ indicates the value of 𝑣2 for hadrons, for example, 𝜋, kaon, proton, and 𝜙.

10. Summary

In this review, we have attempted to summarize the photon and dilepton spectra measured at different beam energies by different experimental collaborations which have been analyzed to understand the evaluation of collectivity in the system. The initial conditions of the evolving matter required to calculate the photon and dilepton spectra have been constrained to reproduce the measured multiplicity in these collisions. The EoS, the other crucial input to the calculations, has been taken from lattice QCD calculations. The deviation of the hadronic phase from chemical equilibrium is taken into account by introducing nonzero chemical potential for each hadronic species. For dilepton emission from hot hadrons, the spectral function of 𝜌, 𝜔 at finite temperature and 𝜙 in vacuum is considered. We have also observed the dissipative effects on the photon spectra by taking into account the viscous contribution to phase space factors of all the participating partons/hadrons in the photon production rate as well as to the expansion dynamics with finite shear viscosity.

It is shown that simultaneous measurements of photon and dilepton spectra in heavy ion collisions will enable us to quantify the evolution of the average radial flow velocity for the system, and the nature of the variation of radial flow with invariant mass will indicate the formation of partonic phase. All this study suggests with judicious choice of 𝑝_𝑇 and 𝑀 window will infer information of partonic and hadronic phase separately.

In this work, we present a new proposal for carrying out an experimental measurement of dilepton interferometry both for RHIC and LHC. We establish through a hydrodynamical model-based space-time evolution the promise of such a dilepton interferometry analysis that will be useful to understand the properties of the partonic phase. We have evaluated the correlation function, 𝐶₂, for two dilepton pairs for various invariant mass domains and extracted the HBT radii, that is 𝑅_side and 𝑅_out as a function of 𝑀. These HBT radii show a nonmonotonic dependence on the invariant mass, reflecting the evolution of collective flow in the system which can be considered as a signal of the QGP formation in heavy ion collisions. The 𝑀 dependence of the 𝑅_out/𝑅_side and 𝑅²_out−𝑅²_side which can be experimentally measured could be used to characterize the source properties at various instances of the evolution.

We have evaluated the 𝑣₂ of dileptons originating from the Pb+Pb collisions at √𝑠_𝑁𝑁=2.76 TeV for 30–40% centrality. Our study shows that 𝑣₂(𝑀) provides useful information on the collective motion of the evolving QCD matter formed in high energy heavy-ion collisions. The present work indicates that experimental observation of the reduction of 𝑣₂(𝑀) with increasing 𝑀 beyond 𝜙 mass would reflect the presence of small momentum space anisotropy through small collective motion in the partonic phase. We observe that 𝑣₂(⟨𝑀⟩) of the penetrating probe (lepton pairs) for ⟨𝑀⟩=𝑚_𝜋 and 𝑚_𝐾 is similar to the hadronic 𝑣^𝜋₂ and 𝑣^𝐾₂ when the medium-induced change in the 𝜌 spectral function is included in evaluating the dilepton spectra. Since the medium effects are large during the dense phase of the system, therefore, this validates the statement that the hadronic 𝑣₂ carries the information of the early dense phase of the collisions. Our study also establishes the fact that the invariant mass dependence of dilepton 𝑣₂ can in principle act as a clock for the space time evolution of the system formed in HIC.

Some comments on effect of magnetic field on photon and dilepton production are in order here. The magnetic contribution is significant when photons propagate perpendicular to magnetic field or in reaction plane in noncentral collision [152]. Recently, Tuchin has implemented the magnetic field effect on the photon as well as dilepton production which is summarized here. Recently in [153, 154], it has been argued that photons, in particular thermal and direct photons from earliest times of QGP, will be affected by the magnetic field. In dilepton sector, the magnetic contribution is important at lower electron energies (below ~0.5 GeV at midrapidity at RHIC). In fact, it becomes dominant source of dileptons at lower electron energies. For LHC energies, the effect of magnetic field on dilepton should be taken seriously because 40% of energy density of the QGP resides in the strong magnetic field (~10¹⁸ Gauss [152]), which can be generated in heavy ion collision at LHC. Whereas in RHIC energy this fraction is within 3% so can be safely ignored. Some more realistic calculation and hydrodynamic model with magnetic field are required to conclude anything in future.

Appendix

Phase Space

In this appendix we derive (37) from (12). The photon production rate from the process, 1+2→3+𝛾 is given by 𝐸𝑑𝑅𝑑³𝑝=12𝒩(2𝜋)⁸×𝑑³𝑝₁2𝐸₁𝑑³𝑝₂2𝐸₂𝑑³𝑝₃2𝐸₁𝑓₁𝐸₁𝑓₂𝐸₂×1±𝑓₃𝐸₃|𝑀|²𝛿×𝑝₁+𝑝₂−𝑝₃−𝑝. (A.1) Performing the 𝑑³𝑝₃ integration using the delta function and using 𝑑³𝑝/𝐸=𝑝_𝑇𝑑𝑝_𝑇𝑑𝑦𝑑𝜙 we get 𝐸𝑑𝑅𝑑³𝑝=1𝒩16(2𝜋)⁸𝑝_1𝑇𝑑𝑝_1𝑇𝑑𝑦₁𝑑𝜙₁𝑝_2𝑇𝑑𝑝_2𝑇𝑑𝑦₂𝑑𝜙₂×1𝐸₃𝑓₁𝐸₁𝑓₂𝐸₂1±𝑓₃𝐸₃×|𝑀|²𝛿𝐸₁+𝐸₂−𝐸₃−𝐸, (A.2) where 𝜙₁ and 𝜙₂ are the angles made by the transverse momenta of first and second particles with the transverse momentum of the emitted photon. The momentum conservation along the 𝑧-direction: 𝑝_3𝑧=𝑝_1𝑧+𝑝_2𝑧−𝑝_𝑧 can be written in terms of rapidity as 𝑚_3𝑇sinh𝑦₃=𝑚_1𝑇sinh𝑦₁+𝑚_2𝑇sinh𝑦₂−𝑝_𝑇sinh𝑦. (A.3) Now the energy, 𝐸₃, can be written as 𝐸₃=𝑚_3𝑇cosh𝑦₃=𝑚²_3𝑇+𝑚²_3𝑇sinh²𝑦₃. (A.4) Substituting (A.3) in (A.4) we get 𝐸₃=𝑚_1𝑇sinh𝑦₁+𝑚_2𝑇sinh𝑦₂−𝑝_𝑇sinh𝑦²+𝑚²_3𝑇^1/2. (A.5) Considering the energy conservation (𝐸₃=𝐸₁+𝐸₂−𝐸) and writing the energies in terms of rapidity (𝐸_𝑖=𝑚_𝑖𝑇cosh𝑦_𝑖) we get 𝐸₃=𝑚_1𝑇cosh𝑦₁+𝑚_2𝑇cosh𝑦₂−𝑝_𝑇cosh𝑦. (A.6) Equating (A.5) and (A.6) we have 𝑚_3𝑇=𝑚²_1𝑇+𝑚²_2𝑇+𝑝²_𝑇+2𝑚_1𝑇𝑚_2𝑇cosh𝑦₁−𝑦₂−2𝑚_1𝑇𝑝_𝑇cosh𝑦₁−𝑦−2𝑚_2𝑇𝑝_𝑇cosh𝑦₂−𝑦^1/2. (A.7)

However, we also have 𝑚_3𝑇=𝑝²_3𝑇+𝑚²₃^1/2=𝑝_1𝑇+𝑝_2𝑇−𝑝_𝑇²+𝑚²₃^1/2=𝑝²_1𝑇+𝑝²_2𝑇+𝑝²_𝑇+2𝑝_1𝑇𝑝_2𝑇cos𝜙₁₂−2𝑝_𝑇𝑝_1𝑇cos𝜙₁−2𝑝_𝑇𝑝_2𝑇cos𝜙₂+𝑚²₃^1/2, (A.8) where cos𝜙₁₂=cos𝜙₁cos𝜙₂+sin𝜙₁sin𝜙₂. (A.9) Equating (A.7) with (A.8) leads to the expression: 𝑝_1𝑇cos𝜙₁−𝑝_𝑇cos𝜙₂+𝑝_1𝑇sin𝜙₁sin𝜙₂=12𝑝_2𝑇𝑚²₁+𝑚²₂−𝑚²₃+2𝑚_1𝑇𝑚_2𝑇cosh𝑦₁−𝑦₂−2𝑚_1𝑇𝑝_𝑇cosh𝑦₁−𝑦−2𝑚_2𝑇𝑝_𝑇cosh𝑦₂−𝑦+2𝑝_𝑇𝑝_1𝑇cos𝜙₁. (A.10) Solving (A.10) for 𝜙₂ one gets 𝜙⁰₂=tan⁻¹𝑝_1𝑇sin𝜙₁𝑝_1𝑇cos𝜙₁−𝑝_𝑇−cos⁻¹𝐻2𝑅𝑝_2𝑇, (A.11) where 𝑅=𝑝²_1𝑇+𝑝²_𝑇−2𝑝_1𝑇𝑝_𝑇cos𝜙₁,𝐻=𝑚²₁+𝑚²₂−𝑚²₃+2𝑚_1𝑇𝑚_2𝑇cosh𝑦₁−𝑦₂−2𝑚_1𝑇𝑝_𝑇cosh𝑦₁−𝑦−2𝑚_2𝑇𝑝_𝑇cosh𝑦₂−𝑦+2𝑝_𝑇𝑝_1𝑇cos𝜙₁. (A.12)(A.13) Now we express the argument of the delta function in (A.2) as function of 𝜙₂ as 𝑓𝜙₂=𝐸₁+𝐸₂−𝐸₃−𝐸=𝑚_1𝑇cosh𝑦₁+𝑚_2𝑇cosh𝑦₂−𝑝_𝑇−𝑚cosh𝑦_23𝑇+𝑚_1𝑇sinh𝑦₁+𝑚_2𝑇sinh𝑦₂−𝑝_𝑇sinh𝑦^21/2, (A.14) and performing the 𝜙₂ integration in (A.2) we get 𝐸𝑑𝑅𝑑³𝑝=1𝒩16(2𝜋)^8∞0𝑝_1𝑇𝑑𝑝_1𝑇^∞₀𝑑𝑝_2𝑇×^∞_−∞𝑑𝑦₁^∞_−∞𝑑𝑦₂₀^2𝜋𝑑𝜙₁×𝑓₁𝐸₁𝑓₂𝐸₂1±𝑓₃𝐸₃×|𝑀|²𝑝_1𝑇sin𝜙₁−𝜙₂+𝑝_𝑇sin𝜙_2𝜙02 (A.15) with the constraint |𝐻/2𝑅𝑝_2𝑇|≤1 originating from |cos(𝜙)|≤1.

Acknowledgments

The authors thank Sourav Sarkar and Jan E. Alam for their major collaborative contribution in these works presented here and also for comments and suggestions to complete the paper. Payal Mohanty and Sabyasachi Ghosh thank Victor Roy, Santosh K. Das, Jajati K. Nayak, Bedangadas Mohanty, and Asis K. Chaudhuri for their assistance in the work presented in the review.