Photon and dilepton production at the Facility for Antiproton and Ion Research and the beam energy scan program at the Relativistic Heavy-Ion Collider using coarse-grained microscopic transport simulations
PPhoton and dilepton production at the Facility for Antiproton and Ion Research andthe beam energy scan program at the Relativistic Heavy-Ion Collider usingcoarse-grained microscopic transport simulations
Stephan Endres, ∗ Hendrik van Hees, and Marcus Bleicher
Frankfurt Institute for Advanced Studies, Ruth-Moufang-Straße 1, D-60438 Frankfurt, Germany andInstitut f¨ur Theoretische Physik, Universit¨at Frankfurt,Max-von-Laue-Straße 1, D-60438 Frankfurt, Germany (Dated: May 2, 2016)We present calculations of dilepton and photon spectra for the energy range E lab = 2 − A GeVwhich will be available for the Compressed Baryonic Matter (CBM) experiment at the future Facil-ity for Anti-Proton and Ion Research (FAIR). The same energy regime will also be covered by phaseII of the Beam Energy Scan at the Relativistic Heavy-Ion Collider (RHIC-BES). Coarse-graineddynamics from microscopic transport calculations of the Ultra-relativistic Quantum Molecular Dy-namics (UrQMD) model is used to determine temperature and chemical potentials, which allows forthe use of dilepton and photon-emission rates from equilibrium quantum-field theory calculations.The results indicate that non-equilibrium effects, the presence of baryonic matter and the creationof a deconfined phase might show up in specific manners in the measurable dilepton invariant massspectra and in the photon transverse momentum spectra. However, as the many influences are diffi-cult to disentangle, we argue that the challenge for future measurements of electromagnetic probeswill be to provide a high precision with uncertainties much lower than in previous experiments. Fur-thermore, a systematic study of the whole energy range covered by CBM at FAIR and RHIC-BESis necessary to discriminate between different effects, which influence the spectra, and to identifypossible signatures of a phase transition.
PACS numbers: 25.75.Cj, 24.10.LxKeywords: Dilepton & photon production, Monte Carlo simulations
I. INTRODUCTION
A major goal of the study of heavy-ion collisions atrelativistic and ultra-relativistic collision energies is toexplore the properties of strongly interacting matter atfinite temperatures and densities [1, 2]. When two col-liding nuclei hit each other, the nuclear matter is com-pressed, and a large amount of energy is deposited in asmall spatial volume. This results in the creation of afireball of hot and dense matter [3, 4]. The fireball livesfor a time span of the order of several fm /c until the col-lective expansion of the matter has driven the stronglyinteracting system to a final state of freely streaming par-ticles.Today, almost the entire phase diagram governed byQuantum Chromodynamics (QCD) is accessible for ex-perimental exploration at various accelerator facilities.The temperature T and baryochemical potential µ B in-side the fireball are mainly determined by the energywhich is deposited in the nuclear collision; more precisely,the collision energy determines the trajectory of the sys-tem in the T − µ B plane of the QCD phase diagram.At the highest currently available energies at RHIC andLHC the reaction is dominated by high temperatures,significantly above the critical temperature T c , for whichthe creation of a deconfined state of quarks and gluonsis assumed. At the same time the baryochemical po- ∗ [email protected] tential is low or close to zero for the largest part of thefireball evolution. This situation is similar to the condi-tions which prevailed in the universe a short time afterthe big bang. On the other side, one finds a comple-mentary situation if considering heavy-ion collisions atlaboratory frame energies of the order of 1 A GeV. Hereonly moderate temperatures are obtained, insufficient tocreate a Quark-Gluon Plasma. However, the very highnet baryon densities or baryochemical potentials reachedin this case might provide valuable information aboutthose effects which are not mainly driven by tempera-ture but by the presence of compressed baryonic matter.This situation resembles the environments in (super)novaexplosions and neutron stars.To learn about the different regions of the phase di-agram one needs observables which do not only reflectthe diluted final state after the freeze-out of the systembut rather convey information about the entire fireballevolution. For this purpose electromagnetic probes, i.e.,photons and dileptons have been for long suggested asideal probes [5, 6]: Once produced, photons and dilep-tons only participate in electromagnetic and weak inter-actions for which the mean free paths are much longerthan the size and the lifetime of the fireball. Conse-quently, they can leave the zone of hot and dense matterundisturbed. Since electromagnetic probes are emitted ina large variety of processes over the whole lifetime of thefireball, the measured spectra reflect the time-integratedevolution of the thermodynamic properties of the system.While this allows to obtain convoluted information about a r X i v : . [ nu c l - t h ] M a y the properties of matter it also poses a serious challengefor the theoretical description. On the one hand, oneneeds to identify the relevant microscopic processes thatcontribute to dilepton and photon emission and to deter-mine the corresponding production rates. On the otherhand it is important to give a realistic description of thecomplete reaction dynamics.The intense experimental study of photon and dilep-ton production in the high-energy regime (at SPS [7–9], RHIC [10–13], and LHC [14] energies), but also forvery low collision energies as measured at SIS 18 andBEVALAC [15–18] in comparison to theoretical modelcalculations has significantly enhanced our knowledge ofthe reaction dynamics and the properties of matter in thehot and dense medium created in a heavy-ion reaction.The importance of partonic emission for the correct the-oretical description of the high-mass region of dileptoninvariant mass spectra and the high- p t photon spectrahas been pointed out [19–21] and the various differenthadronic contributions (especially for the photon pro-duction channels) could be identified [22–24]. Neverthe-less, the most important finding was the large influenceof the baryonic matter on the vector mesons’ spectralshape. Especially in the case of the ρ meson this causesa strong broadening of the spectral function with smallmass shifts [25–28]. This effect has been observed as anenhancement in the low-mass region of the dilepton in-variant mass spectra and also shows up as a strongerlow-momentum thermal photon yield. Note that the ρ broadening is most dominant at low collision energies,where one obtains the largest baryochemical potentials,but even at RHIC energies baryonic effects are by far notnegligible.However, there still remains an up to now unexploredenergy window between the E lab = 1 − A GeV dileptonmeasurements by the DLS and HADES Collaborationsand the CERES results for E lab = 40 A GeV. The futureCompressed Baryonic Matter (CBM) experiment at theFacility for Anti-Proton and Ion Research (FAIR) withthe SIS 100/300 accelerator provides the unique possi-bility to study heavy-ion collisions with beam energiesfrom 2 up to 35 A GeV and will therefore enable us to getan insight into exactly that regime of the phase diagramof highest baryon densities where no dilepton or photonmeasurements have been performed till now [29, 30]. Inaddition, also phase II of the Beam Energy Scan (BES)program at RHIC will allow to perform measurements infixed-target mode at lab-frame energies of 7.7, 9.1, 11.5,14.5 and 19 . A GeV, i.e., in the same collision energyrange as FAIR [31]. Further complementary investiga-tions are also planned for the NICA in Dubna [32].From a theoretical point of view, the handling of thisenergy range is quite challenging, as the transition froma purely hadronic fireball at low collision energies to thecreation of a partonic phase is expected here. Further-more, at the high baryochemical potentials which stilldominate the fireball at these energies, a first order phasetransition from a hadron gas to the QGP is assumed, in contrast to the situation at RHIC or the LHC where across-over is predicted by Lattice QCD calculations [33].Although transport models were applied successfullyto describe electromagnetic observables in heavy-ion col-lisions [34–36], they generally have some shortcomingswhen describing very hot and dense systems. In detail,problems include the following aspects: Firstly, while theBoltzmann approach works quite well for quasi-particlesof infinite lifetime, for broad resonances as the ρ me-son a correct description is challenging. Furthermore,in dense matter the intervals between scatterings be-come extremely short and will consequently modify thespectral characteristics of the single particles (collisionalbroadening). To describe the off-shell dynamics correctlya transport description with dynamical spectral functionsfollowing the description of Kadanoff and Baym [37] isrequired. However, a practical implementation of this iscurrently not possible. Secondly, in a dense medium notonly binary scatterings will occur but also multi-particleinteractions play a role, which is beyond the capabilitiesof the common transport models. And finally, the micro-scopic models usually concentrate on either the transportof hadrons or partons. However, modelling a transitionfrom an initially up-heating hadron gas to a deconfinedphase and the later particlization when the system coolsdown is extremely difficult to realize within a transportapproach.There have been several investigations over the lastyears on these aspects (see, e.g., Refs. [38–44]), but a fulltreatment of all these issues is still beyond the scope ofpresent investigations.On the other side, the short mean free paths of par-ticles in a medium might suggest to treat the reactionsfrom a macroscopic point-of-view. However, approachesas simple fireball expansion models [45] or hydrodynam-ics [46], which have been successfully applied for SPS,RHIC and LHC energies, also have their shortcomings inthe FAIR energy regime for three main reasons: Firstly,the separation of the fireball expansion from dynamicsof the initial projectile-target dynamics is not applica-ble; secondly, the often applied simplification to assumea 2+1-dimensional boost invariant geometry is not pos-sible; and finally, the time scale necessary for an approx-imate thermal equilibration of the fireball will be longerdue to the slower overall evolution of the reaction andthe lower temperatures reached.To avoid the disadvantages of both pictures the coarse-graining method has been developed, based on previ-ous studies [47], and was successfully applied to de-scribe dilepton production at SPS and SIS 18 energies[27, 28, 48]. The approach represents a combination ofthe microscopic picture from the underlying transportsimulations with the resulting description of the dynam-ics in terms of the macroscopic quantities temperatureand chemical potential. By averaging over many eventsone can extract the local energy and baryon densities ateach space-time point from the transport simulations anduse an equation of state to determine the correspondingtemperature and baryochemical potential. With this thecalculation method of thermal dilepton and photon emis-sion by application of full in-medium spectral functions isstraightforward, employing the rates available from equi-librium quantum field theory.In the present work the coarse-graining approach isused to calculate photon and dilepton spectra with focusthe FAIR energy regime, but naturally the results alsoserve as a theoretical prediction for the fixed-target mea-surements of the RHIC-BES since the prospected col-lision energies of both experimental programs overlap.Although the details of the future experimental set-upsare not yet determined, the results shall provide a generalbaseline calculation for the interpretation of the measure-ments to be conducted. Furthermore, it shall be investi-gated if and how one can obtain valuable information onthe properties of matter from the measured spectra anddiscriminate between several effects that might influencethe dilepton and photon results. In detail, we will concen-trate on the following three aspects: The modification ofthe thermal emission pattern by high baryochemical po-tentials, signals for a phase transition or the creation ofa deconfined phase and possible non-equilibrium effectson the thermal rates.This paper is structured as follows. In Section II thecoarse-graining approach will be presented. Thereafter,in Section III we will introduce the various microscopicsources for thermal emission of photons and dileptons andin short discuss the non-thermal cocktail contributions.In Section IV the results for the fireball evolution andthe photon and dilepton spectra at FAIR energies areshown. The results are used to systematically analyse inwhich way it might be possible to discriminate betweendifferent scenarios for the fireball evolution in Section V.We conclude the present work with a summary and anoutlook to subsequent investigations. II. THE MODEL
While microscopic transport models describe the reac-tion dynamics of a heavy-ion collision in terms of manydifferent degrees of freedom, the general idea of thecoarse-graining approach is that in principle only a veryreduced amount of the provided information is necessaryto account for the thermal production of electromagneticprobes. The microscopic information about all individ-ual particles and their specific properties – such as mass,charge and momentum – are ignored and the whole dy-namics is reduced to macroscopic quantities which are as-sumed to fully determine the local thermodynamic prop-erties: The energy and particle densities.The coarse-graining method combines two advantages:On the one hand the collision dynamics is still based onthe microscopic transport evolution and thereby gives avery nuanced picture of the entire collision evolution, onthe other hand the reduction to macroscopic state vari-ables enables an easy application of in-medium spectral functions from equilibrium quantum field theory calcula-tions.In the following the ingredients of the approach arepresented in detail.
A. Ultrarelativistic Quantum Molecular Dynamics
The underlying microscopic input for the presentcalculations stems from the Ultra-relativistic QuantumMolecular Dynamics (UrQMD) approach [49–51]. It isa non-equilibrium microscopic transport model based onthe principles of molecular dynamics [52, 53]. It consti-tutes an effective Monte Carlo solution to the relativis-tic Boltzmann equation and connects the propagation ofhadrons on covariant trajectories with a probabilistic de-scription of the hadron-hadron scattering processes. Toaccount for the quantum nature of the particles, eachhadron is represented by a Gaussian density distribution,and quantum statistical effects such as Pauli blocking areconsidered [54].The model includes all relevant mesonic and bary-onic resonances up to a mass of 2 . /c . Produc-tion of particles occurs via resonant scattering of par-ticles (e.g., N N → N ∆ or ππ → ρ ) or the decay ofhigher resonances, e.g., the process ∆ → πN . The in-dividual interaction and decay processes are describedin terms of measured and extrapolated hadronic cross-sections and branching ratios. For collision energiesabove √ s NN = 3 GeV also the excitation of strings ispossible. B. Coarse-graining of microscopic dynamics
Within the UrQMD model the particle distributionfunction f ( (cid:126)x, (cid:126)p, t ) is determined by the space and mo-mentum coordinates of all the different particles in thesystem at a certain time. However, due to the finitenumber h of hadronic particles involved and produced ina heavy-ion collision, one needs to take the average over alarge ensemble of events to obtain a smooth phase-spacedistribution of the form f ( (cid:126)x, (cid:126)p, t ) = (cid:42)(cid:88) h δ (3) ( (cid:126)x − (cid:126)x h ( t )) δ (3) ( (cid:126)p − (cid:126)p h ( t )) (cid:43) . (1)Note that this distribution is Lorentz invariant if all par-ticles are on the mass-shell, as provided in our case. Dueto the non-equilibrium nature of the model, one will ofcourse have to extract the particle distribution functionlocally. In the present approach, this is done by the useof a grid of small space-time cells where for each of thesecells we determine the (net-)baryon four-flow and theenergy-momentum tensor according to the relations j µ B = (cid:90) d p p µ p f B ( (cid:126)x, (cid:126)p, t ) , (2) T µν = (cid:90) d p p µ p ν p f ( (cid:126)x, (cid:126)p, t ) . (3)In practice, the integration is done by summing over the δ functions. As we use cells of finite size, we have δ (3) ( (cid:126)x − (cid:126)x h ( t )) = (cid:40) V if (cid:126)x h ( t ) ∈ ∆ V, O then becomes (cid:90) d p ˆ O ( (cid:126)x, (cid:126)p, t ) f ( (cid:126)x, (cid:126)p, t ) = 1∆ V (cid:42) (cid:126)x h ∈ ∆ V (cid:88) h ˆ O ( (cid:126)x, (cid:126)p, t ) (cid:43) . (5)Consequently, Eqns. (2) and (3) take the form T µν = 1∆ V (cid:42) N h ∈ ∆ V (cid:88) i =1 p µi · p νi p i (cid:43) ,j µ B = 1∆ V (cid:42) N B / ¯B ∈ ∆ V (cid:88) i =1 ± p µi p i (cid:43) . (6)Having obtained the baryon flow, we can boost each cellinto the rest frame as defined by Eckart [55], where j B µ is ( ρ B ,(cid:126) C. Non-equilibrium dynamics
While macroscopic models usually introduce thermaland chemical equilibrium as an ad-hoc assumption, mi-croscopic simulations – in the present case the UrQMDsimulations – are based on the description of singleparticle-particle interactions and non-equilibrium will bethe normal case. Consequently, we have to account forthese deviations from equilibrium in such a manner thatwe can reliably apply equilibrium spectral functions tocalculate the emission of photons and dileptons.In general it is difficult to really determine to whichdegree a system has reached equilibrium. Basically thereare two dominant effects, which may serve as indicatorsfor thermal and chemical equilibration: The momentum-space anisotropies and the appearance of meson-chemicalpotentials.
1. Thermal non-equilibrium
Regarding thermal equilibration, it was found in micro-scopic simulations that independent of the collision en-ergy the system needs a time of roughly 10 fm /c after the beginning of the heavy-ion collision until the transverseand longitudinal pressures are approximately equal [56].The pressure anisotropy stems from the initial strongcompression along the beam axis when the two nucleifirst hit and traverse each other. As thermal equilibriumrequires isotropy, one will obtain too high values for theenergy density in highly anisotropic cells. To obtain ef-fective quantities that account for the thermal proper-ties in the system we apply a description that explicitlyincludes the momentum-space anisotropies and in whichthe energy momentum-tensor is assumed to take the form[57, 58] T µν = ( ε + P ⊥ ) u µ u ν − P ⊥ g µν − ( P ⊥ − P (cid:107) ) v µ v ν . (7)where P ⊥ and P (cid:107) denote transverse or parallel pressurecomponents, respectively; u µ and v µ are the cell’s four-velocity and the four-vector of the beam direction. Theeffective energy density ε eff is obtained via the general-ized equation of state for a Boltzmann-like system of theform ε eff = εr ( x ) , (8)where the relaxation function r ( x ) and its derivative r (cid:48) ( x )are defined by r ( x ) = x − / (cid:16) x artanh √ − x √ − x (cid:17) for x ≤ x − / (cid:16) x arctan √ x − √ x − (cid:17) for x ≥ , (9)and x = ( P (cid:107) /P ⊥ ) / denotes the pressure anisotropy.As we have shown in our previous investigation at SPSenergies [27], with this description the effective energydensity deviates from the nominal one only for the veryinitial stage of the reaction, where the pressure compo-nents differ by orders of magnitude. Nevertheless, the ef-fective energy density ε eff allows to calculate meaningful T and µ B values for these cells. After the very initial col-lision phase the differences still exist, but have hardly anyinfluence on the energy density, so that we can assumethat these cells are in approximate local equilibrium.
2. Chemical non-equilibrium
Chemical equilibration is a more difficult problem, butone obvious deviation in microscopic models is the ap-pearance of meson chemical potentials, especially for thecase of pions as these are the most abundantly producedparticles. As the meson number is not a conserved quan-tity in strong interactions (in contrast to, e.g., the baryonnumber) meson chemical potentials can only show up ifthe system is out of chemical equilibrium. While pionchemical potentials are introduced in fireball models forthe stage after the chemical freeze-out to obtain the cor-rect final pion yields, in non-equilibrium transport mod-els they intrinsically appear in the early stages of thereaction when a large number of pions is produced inmany initial scattering processes [59]. At higher collisionenergies this mainly happens via string excitation. Thepion (and kaon) chemical potentials have a large influ-ence on the photon and dilepton production rates as anoverpopulation of pions increases the reactions in manyimportant channels, for example ππ → ρ → γ/γ ∗ [60].To implement the non-equilibrium effects in the calcu-lations, we extract the pion and kaon chemical potentialsin Boltzmann approximation as [61] µ π/K = T ln (cid:32) π ngT m K (cid:0) mT (cid:1) (cid:33) , (10)where n denotes the cell’s pion or kaon density and K theBessel function of the second kind. The degeneracy factor g is 3 in the case of pions and 2 for kaons. Note that theBoltzmann approximation is in order here, as the mesonsin the transport model also account for Boltzmann statis-tics and no Bose effects are implemented. However, whilefor a Bose gas the chemical potential is limited to the me-son’s mass, in principle one can get higher values for µ π or µ K here in rare cases. As such values are non-physical,we assume that the maximum values to be reached are140 MeV for µ π and 450 MeV for µ K . D. Equation of state
Once the rest-frame properties of each cell are deter-mined, an equation of state (EoS) is necessary to describethe thermodynamic system of the hot and dense matterin the cell under the given set of state variables, i.e., the(effective) local energy density and the local net densitiesof conserved charges (for the strong interactions consid-ered here the baryon number is the relevant quantity).For the present calculations we apply a hadron gas equa-tion of state (HG-EoS) that includes the same hadronicdegrees of freedom as the underlying transport model[62]. The EoS allows us to extract the temperature andbaryochemical potential for an equilibrated hadron gasat a given energy and baryon density. It is similar tothe result obtained for UrQMD calculations in a box inthe infinite time limit, when the system has settled to anequilibrated state.However, in the FAIR energy regime a purely hadronicdescription of the evolving hot and dense fireball will notbe sufficient. As the temperatures will exceed the crit-ical temperature T c , a transition from hadronic to par-tonic matter has to be implemented, and the dynamicevolution of the created Quark-Gluon Plasma has to beconsidered. On the other hand, it is necessary to keepthe EoS consistent with the underlying dynamics whichis purely hadronic. In our previous study at SPS energies[27], we supplemented the HG-EoS with a Lattice equa-tion of state [63] for temperatures above T c ≈
170 MeV,in line with the lattice results. In the range around thecritical temperature the results of the HG-EoS and theLattice EoS match very well for µ B ≈
0, while signifi- cantly higher temperatures are obtained with the latterif one reaches temperatures significantly above T c .However, this procedure is problematic for the presentstudy, as the transition from a hadronic to a partonicphase and back is assumed to take place at finite val-ues of µ B at FAIR energies, whereas the Lattice EoS isrestricted to vanishing chemical potential. To avoid dis-continuities in the evolution, we confine ourselves to theapplication of the HG-EoS, but with the assumption thatthe thermal emission from cells with a temperature above170 MeV stems from the QGP (i.e., we employ partonicemission rates). This should be in order, as the tempera-tures will not lie too much above the critical temperatureat the energies considered in the present work, where thedeviations from a full QCD-EoS explicitly including aphase transition are expected to be rather moderate.Nevertheless, we once again remind the reader thatthe underlying microscopic description is purely hadronicand it remains to be studied which consequences a phasetransition has at the microscopic level of the reactiondynamics. III. PHOTON & DILEPTON RATES
The mechanisms which contribute to the thermal emis-sion of photons and dileptons are the same. Any processthat can produce a real photon γ can also produce a vir-tual (massive) photon γ ∗ , decaying into a lepton pair.However, due to the different kinematic regimes probedby photons and dileptons, the importance of the singleprocesses varies. In the following the various sources ofthermal radiation considered in this work are presented.Determining quantity for the thermal emission of realphotons as well as virtual photos (i.e., dileptons) is theimaginary part of the retarded electromagnetic current-current correlation function Π ( ret ) em , to which the ratesare directly proportional. It represents a coherent sum-mation of the cuts of those Feynman diagrams which aredescribing the different processes contributing to thermal γ and γ ∗ emission, and therefore accounts for the photonor dilepton self-energy. In the rest frame, the thermalemission can be calculated according to [25]d N ll d x d q = − α L ( M ) π M f B ( q ; T ) × Im Π (ret)em ( M, (cid:126)q ; µ B , T ) , (11) q d N γ d x d q = − α em π f B ( q ; T ) × Im Π T, (ret)em ( q = | (cid:126)q | ; µ B , T ) . (12)Here L ( M ) is the lepton phase-space factor (which playsa significant role only for masses close to the threshold2 m l and is approximately one otherwise), f B the Bosedistribution function, and M the invariant mass of a lep-ton pair. Note that only the transverse polarization of thecurrent-current correlator enters for the photon rate, asthe longitudinal projection vanishes at the photon point,i.e., for M = 0. A. Thermal rates from hadronic matter
1. Vector meson spectral functions
In hadronic matter, all the spectral information of ahadron with certain quantum numbers is specified inIm Π (ret)em [64]. Assuming that Vector Meson Dominance(VMD) [65] is valid, the correlator can be directly relatedto the spectral functions of vector mesons. The impor-tant challenge for theoretical models is to consider themodifications of the particle’s self-energy inside a hot anddense medium. Different calculations of in-medium spec-tral functions exist [66–70], however not many of themfully consider the effects of temperature and finite chem-ical potential. We here apply a hadronic many-body cal-culation from thermal field theory for the spectral func-tions of the ρ and ω mesons [71–73], which has proven tosuccessfully describe photon and dilepton spectra fromSIS 18 to LHC energies [20, 22, 27, 28, 45, 74, 75]. Thecalculation of the different contributions to the ρ spectralfunction takes three different effects into account: Themodification due to the pion cloud, the direct scatteringof the ρ with baryons (nucleons as well as excited N ∗ and∆ ∗ resonances) and with mesons ( π, K, ρ, . . . )[76]. Whilethe pion cloud effects also contribute in the vacuum, thescattering processes only show up in the medium. Forthe ω , the situation is slightly more complex since thismeson basically constitutes a three-pion state. The vac-uum self-energy is represented by a combination of thedecays into ρ + π or three pions, respectively. Furtherthe inelastic absorption ωπ → ππ and the scattering pro-cesses with baryons as well as the pion (i.e. ωπ → b )are implemented [77]. In the same manner as in our pre-vious dilepton study at SIS 18 energy [28], we relinquisha treatment of thermal emission from the φ vector me-son here, for reason of the minor in-medium broadeningeffects observed for this hadron and its still low multi-plicities at least for lower FAIR energies. In the case ofvanishing invariant mass, i.e., for real photons, only the ρ vector meson will give a significant contribution. Forthe present calculation, we used the parametrization ofthe photon rates from the ρ as given in Eqs. (2)-(7) inRef. [78], while a more advanced parametrization is nec-essary for the dilepton rates from the ρ and ω , due totheir dependence on invariant mass and momenta [79].Note that presently the photon parametrization of the ρ contribution is limited to baryon chemical potentialslower than 400 MeV and momenta larger than 0 . /c .While we can easily neglect the lowest momentum re-gion in the present study, the restriction to low µ B is aproblem for the lowest collision energies considered here,where one expects values of µ B which significantly exceedthis range. The difference will be dominant at lower mo- menta, where the influence of baryonic effects is knownto be largest, while the effect of a finite chemical poten-tial is rather small at higher momenta [78]. However, forthe present work we can assume the photon contributionfrom the ρ meson as a lower limit with regard to thebaryonic effects.
2. Meson gas contributions
The contribution from vector mesons is not the onlyhadronic source of thermal emission. The mass regionabove the φ meson, i.e., for M > /c , is no longerdominated by well defined particles, but here one finds alarge number of overlapping broad resonances constitut-ing multi-meson (mainly four-pion) states, which have asignificant impact on the dilepton yield. We here apply adescription relying on model-independent predictions us-ing a low-temperature expansion in the chiral-reductionapproach [74].While the multi-meson effects only show up for highinvariant masses, i.e., in the time-like kinematic re-gion probed by dileptons, when going to real photonswith M → M . While baryonic bremsstrahlung processessuch as N N → N N γ and πN → πN γ are includedin the vector meson spectral functions, meson-mesonbremsstrahlung has to be added in the case of photonemission. The most dominant part will here come fromthe meson-meson scatterings ππ → ππγ and πK → πKγ ,for which we use the rates calculated within an effec-tive hadronic model [23] in form of the parametrizationgiven by Eqs. (8) and (9) in Ref. [78]. Note that thesebremsstrahlung processes are mainly contributing at lowmomenta, whereas they are rather subleading for photonsof higher energy.Besides the ππ bremsstrahlung, also other mesonicreactions contribute to the thermal photon produc-tion, such as strangeness bearing reactions and meson-exchange processes. In detail, these are πρ → πγ , πK ∗ → Kγ , πK → K ∗ γ , ρK → Kγ and K ∗ K → πγ .The corresponding thermal rates were calculated for ahot meson gas in Ref. [22], which are applied here to-gether with the respective form factors.Since the ω - t -channel exchange was found to give asignificant contribution to thermal photon spectra viathe πρ → πγ process, it has been recently argued thatalso other processes including a πρω vertex should bebe considered in the calculations, namely the πρ → γω , πω → γρ and ρω → γπ reactions for which the rates (in-cluding the form factors) are parametrized in AppendixB of Ref. [24]. Consequently, we also add these processeswhen calculating thermal photon spectra.
3. Influence of meson chemical potentials
As was already mentioned, we do not restrict our-selves to the consideration of emission from thermallyand chemically equilibrated matter, but also include non-equilibrium effects in form of finite pion and kaon chem-ical potentials µ π and µ K . It has been shown that theinfluence of a non-equilibrium distribution of the respec-tive mesons can be accounted for by introducing an ad-ditional fugacity factor z nM = π,K = exp (cid:16) nµ M T (cid:17) (13)in the thermal dilepton and photon rates in Eqs. (11) and(12). The exponent n depends on the difference in pionor kaon number N π/K between initial and final state ofthe process, i.e. n = N iπ/K − N fπ/K . Note that while thepion fugacity enters in most processes, the effects of afinite kaon chemical potential only play a role for the πK bremsstrahlung and π + K ∗ → π + γ , π + K → K ∗ + γ and K ∗ + K → π + γ photon production channels. For thedilepton channels considered here, µ K can be neglected.While for the single mesonic channels the initial andfinal state are always well defined, several different typesof processes are included in the ρ and ω spectral func-tions, especially processes with baryons. For the ρ notonly processes with an initial two-pion state of the type ππ → ρ → γ/γ ∗ are accounted for, but also reactionsincluding only one or no pion as ingoing particle (e.g., πN → ∆ → γN or N N → γN N ). However, as thecorrect fugacity depends on the initial pion number, onewould obtain different enhancements for each channel.But as the different processes interfere with each other itis difficult to determine the exact strength of each channeland consequently one might hardly be able to account forsome average enhancement factor. Instead we here ap-ply a fugacity factor z π which would be correct for pure ππ annihilation processes. This can be interpreted as anupper estimate of the influence which the meson chemi-cal potential might have on the thermal ρ emission rates.The same procedure is applied for the ω meson, wherewe assume a fugacity of z π . Note that while the multi-pion contribution also accounts for different initial states,they are each treated separately so that one can applythe correct fugacity factors here.A full list of all hadronic contributions considered forthe present calculation of thermal dilepton and photonemission, including the corresponding fugacity factorswhich account for the enhancement of the specific channeldue to the meson chemical potentials, is given in TableI. B. Quark-Gluon Plasma
For the thermal emission of electromagnetic probesfrom the Quark-Gluon Plasma one is again confronted
Type
Rates Fugacity Ref.Dilepton ρ (incl. baryon effects) ω (incl. baryon effects)Multi-Pion z π z π z π / z π / z π [73, 79][73, 79][45]Photon ρ (incl. baryon effects) ππ and πK Bremsstr. πρ → γππK ∗ → KγπK → K ∗ γρK → KγK ∗ K → πγπω → γρρω → γππρ → γω z π z π + 0 . z π z K z π z π z K z π z K z π z K z π z π — [73, 78][22, 78][22][22][22][22][22][24][24][24]TABLE I. Summary of the different dilepton contributionsconsidered in the present calculations. with the problem that different processes govern thedilepton production on the one side and photon emissionon the other. Consequently, one has to apply two differ-ent descriptions for thermal rates, which are presented inthe following.In case of photon emission from a partonic phase ofquarks and gluons the two main contributions stem fromquark-antiquark annihilation ( q ¯ q → gγ ) and Comptonscattering processes ( qg → qγ or ¯ qg → ¯ qγ ) [80]. However,it was shown that these processes are not sufficient to de-scribe the production mechanism correctly and that it isnecessary to (a) include Feynman diagrams accountingfor bremsstrahlung and inelastic annihilation processeswhich are enhanced due to near-collinear singularitiesand (b) to implement the Landau-Pomeranchuk-Migdaleffect [81, 82]. The results of a full calculation of the pho-ton emission, to leading order in α em and the QCD cou-pling g ( T ), was evaluated by Arnold, Moore and Yaffeand takes the following form [83]: q d R γ d q = − α em α s π T (cid:18) (cid:19) f B ( q ; T ) × (cid:20) ln (cid:18)(cid:114) πα s (cid:19) + 12 ln (cid:18) qT (cid:19) + C tot (cid:21) (14)with C tot = C ↔ (cid:16) qT (cid:17) + C annih (cid:16) qT (cid:17) + C brems (cid:16) qT (cid:17) , (15) α s ≈ π
27 ln(
T / . . (16)The functions C ↔ , C annih and C brems are approxi-mated by the phenomenological fits given in Eqs. (1.9)and (1.10) in Ref. [83]. Note that this calculation as-sumes the chemical potential to be vanishing. However,the overall effect of finite values of a quark chemical po-tential (i.e., non-equal numbers of quarks and anti-quarksin the QGP phase) is known to be rather small.In case of the thermal dilepton emission from the QGP,the leading order contribution is the electromagnetic an-nihilation of a quark and an anti-quark into a virtualphoton, q ¯ q → γ ∗ . This process is irrelevant in the light-cone limit for M →
0, as the annihilation of two massivequarks into a massless photon is kinematically forbidden.The pure perturbative quark-gluon plasma rate was cal-culated for the mentioned leading order process [84] asd R ll d p = α em π Tp f B ( p ; T ) (cid:88) q e q × ln ( x − + y ) ( x + + exp[ − µ q /T ])( x + + y ) ( x − + exp[ − µ q /T ]) (17)with x ± = exp[ − ( p ± p ) / T ] and y = exp[ − ( p + µ q ) /T ].The quark chemical potential µ q which shows up here isequal to µ B /
3. This calculation approximates the fullQCD results quite well at high energies, but for softprocesses of the order g s ( T ), i.e., for dileptons with lowmasses and momenta, the one-loop calculation is not suf-ficient and hard-thermal-loop (HTL) corrections to theresult as given in Eq.(17) have to be considered [85]. Itwas found that the rate for soft dileptons is then by or-ders of magnitude larger than the simple leading ordercalculation [26].Recent calculations from thermal lattice-QCD suggestan even stronger enhancement of the rates for low-massdileptons [86]. These results, which are applied in thepresent work, have been extrapolated to finite three-momenta by a fit to the leading-order pQCD rates suchthat the correlation function takes the form [26] − Im Π EM = C EM π M (cid:16) ˆ f ( q , q ; T ) + Q tot LAT ( M, q ) (cid:17) , (18)where Q totLAT ( M, q ) = 2 πα s T M (cid:18) M q (cid:19) × KF (cid:0) M (cid:1) ln (cid:20) . πα s q T (cid:21) (19)with a form factor F (cid:0) M (cid:1) = 4 T / (cid:0) T + M (cid:1) and afactor K = 2 to better fit the full lQCD rates. Notethat in contrast to the pQCD result the rate in Eqs. (18)and (19) is calculated for µ q = 0 only, as a calculationfor finite chemical potential is still beyond the currentlattice calculations. C. Hadronic decay contributons
While we restrict the calculation of photon yields to thethermal contribution, since all decay photons from long-lived hadronic resonances are usually subtracted from theexperimental results, a full description of the dileptonspectra requires to take also the non-thermal contribu-tions from the decay of pseudo-scalar and vector mesons into account. We here follow the same procedures as inour previous work for SIS 18 energies. In detail, we deter-mine the following non-thermal dilepton contributions:1. The Dalitz decays of the pseudo-scalar π and η mesons. To determine their contribution, we as-sume that each final state particle contributes witha weight of Γ M → e + e − / Γ tot .2. The direct decay of the φ meson into a lepton pair.As the lifetime of the φ is relatively short, we ap-ply a shining procedure which takes absorption andre-scattering processes inside the medium into ac-count.3. Finally, we restricted the calculation of thermaldileptons to those cells where the temperature islarger than 50 MeV, as otherwise a thermal de-scription becomes questionable. However, in prin-ciple one will of course also find ρ and ω mesonsat lower temperatures. To account for this, in thementioned cases we calculate a “freeze-out” contri-bution from the ρ and ω decays using the UrQMDresults for these mesons.For a more detailed description of the non-thermalhadronic contributions the reader is referred to Ref. [28].We refrain from an extensive reproduction of the proce-dure here, as the cocktail contributions will not play asignificant role in the present investigations. IV. RESULTS
In the following we present the results of calculationswith the coarse-graining approach for Au+Au collisionsin the energy range of E lab = 2 − A GeV. We re-strict the analysis to the 10% most central reactions,as the medium effects will be largest here. In termsof the microscopic UrQMD results, this roughly corre-sponds to an impact parameter range of b = 0 − . t = 0 . /c , and the size of the spatial gridis ∆ x = ∆ y = ∆ z = 0 . A. Reaction dynamics
As the dilepton and photon production is directly re-lated to the space-time evolution of the thermodynamicproperties of the system, it seems natural to start with astudy of the reaction dynamics obtained with the coarse-graining of UrQMD input.
Time t [fm/c]0 5 10 15 20 25 T e m p e r a t u r e T [ G e V ] Au+Au (0-10% central) central cell x=y=z=0
UrQMDSMASH p m /3 B m = q m T (a) Time t [fm/c]0 5 10 15 20 25 [ G e V ] B m B a r y o c h e m i ca l po t e n t i a l (b) FIG. 1. (Color online) Time evolution of (a) temperature T and (b) baryochemical potential µ B for the central cell of thecoarse-graining grid for different beam energies E lab = 2 − A GeV. The results are obtained for the 0-10% most centralcollisions in Au+Au reactions.
In Fig. 1 the time evolution of temperature and bary-ochemical potential in the central cell of the grid isdepicted for different beam energies. The evolutionshows a significant increase of the temperature maximafrom slightly above 100 MeV for 2 A GeV up to roughly225 MeV for top SIS 300 energy. While the temperatureis clearly below the critical temperature of 170 MeV forthe lower energies, the highest energies covered by FAIRcan probe also this deconfinement region of the phase di-agram. The thermal lifetime of the central cell, i.e., thetime for which it rests at temperatures above 50 MeV,increases slightly with increasing collision energy. This ismainly due to an earlier onset of thermalization after thefirst hadron-hadron collisions, which define the origin ofthe time axis. However, it is interesting that in the laterphase of the collision the temperature curves for all en-ergies show the same monotonous decrease and even lieon top of each other. A somewhat different behavior isobserved for the evolution of the baryochemical potential µ B . For all energies it shows a clear peak with values be-tween 700 and 900 MeV at the beginning of the collision,which is due to the high baryon densities reached in thecentral cell when the two nuclei first come into contact.Afterwards µ B decreases and then remains on a plateaulevel for a significant fraction of the reaction time for thelower energies, while one observes a slight increase forthe higher beam energies towards later times. If we ne-glect the peak in the early reaction stage, the chemicalpotential shows a clear decrease with increasing collisionenergy. While for 2 A GeV the baryochemical potentialremains around 900 MeV for the whole thermal lifetime, µ B is only 350 MeV at t = 5 fm for 35 A GeV and slowlyincreases up to 600 MeV after t = 20 fm.Note that the results shown in Fig. 1 are only for onesingle cell at the center of the collision. The evolution inother cells of the grid may differ largely in dependenceon their location (e.g., one finds in general lower temper- ature and chemical potential in more dilute peripheralcells). But yet it clearly depicts the influence of colli-sion energy on T and µ B . One finds two effects whengoing from the lowest to the highest FAIR energies: An increasing temperature combined with a decreasing bary-ochemical potential. This behavior is not specific for thecentral cell but reflected by the whole space-time evo-lution, as can be seen from Fig. 2. The two plots showthe thermal four-volume in dependence of temperature T (X-axis) and baryochemical potential µ B (Y-axis). Re-sults are shown for E lab = 4 A GeV (a) and 35 A GeV(b). We see that for the lower energy the largest partof the four volume is concentrated at values of the bary-ochemical potential between 500 and 800 MeV, while thetemperature remains below 160 MeV for all cells. In con-trast, for E lab = 35 A GeV the four-volume distributionextends to higher temperatures up to T = 240 MeV whileat the same time the distribution is shifted to lower bary-ochemical potentials especially for higher temperatureswhile the lower-temperature cells are mainly dominatedby high values of µ B . Interestingly, especially for thehigher collision energy of 35 AGeV one finds some cellsin a separate region with moderate to high temperatureand very low baryochemical potential µ B ≈
0. These cellsare mainly found in the more peripheral regions of thecollision, where the baryon density (and particle densityin general) is rather low and where nevertheless in somecases hadrons with large momenta are found, resulting inhigh energy density for these cells. However, comparedto the large overall total thermal volume the relevance ofthese low- µ B cells is negligible.It is important to bear in mind that the dilepton andphoton spectra will directly reflect the four-volume evo-lution in the T − µ B plane, as presented here. The resultsshow that at FAIR energies the region of the QCD phasediagram with temperatures above the critical tempera-ture and large µ B can be probed, in contrast to the situa-0 Temperature T [GeV] [ G e V ] B m B a r y o c h e m i ca l P o t e n t i a l -2 -1 = 4 AGeV lab Au+Au (0-10% central) E ] (a) Thermal 4-Volume [fm
Temperature T [GeV] [ G e V ] B m B a r y o c h e m i ca l P o t e n t i a l -4 -3 -2 -1
35 AGeV (b)
FIG. 2. (Color online) Thermal four-volume in units of fm from the coarse-grained transport calculations in dependence oftemperature T and baryochemical potential µ B . Results are shown for E lab = 4 A GeV (a) and 35 A GeV (b) in central Au+Aucollisions. tion at LHC or RHIC, where the transition from hadronicmatter to a deconfined phase is assumed to happen at µ B ≈
0. However, note that the results presented here areobtained with a purely hadronic equation of state whichdoes not include any effects of the phase transition itself.For an improvement the description one might need toimplement the transition properly, to account, e.g., forthe latent heat which would cause the cells to remain fora longer time at temperatures around T c . Neverthelessthe present results can serve as a lower limit baseline cal-culation, assuming that we have a smooth crossing fromhadronic to QGP emission. Significant deviations fromthis assumption might then show up in the photon and Temperature T [GeV]0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 > [ G e V ] / K pm < p m < > K m < Au+Au (0-10% centr.)
FIG. 3. (Color online) Average values of the pion chemicalpotential µ π (blue lines) and the kaon chemical potential µ K (green lines) in dependence on the cell temperature. Resultsare shown for central Au+Au collisions at three different col-lision energies, E lab = 4 ,
15 and 35 A GeV. dilepton spectra. We will discuss this later.As was pointed out before, the effects of chemical non-equilibrium show up in the form of finite meson chemicalpotentials for the π and K; and µ π and µ K can have asignificant effect on the population of several photon anddilepton production channels. The mean values of thepion chemical potential µ π and the kaon chemical poten-tial µ K in dependence on the cell’s temperature for dif-ferent collision energies are shown in Figure 3. Note thatthe results for the chemical potentials here are obtainedby averaging the values of µ π and µ K over all space-timecells with a specific temperature. The study indeed in-dicates that regarding the pion density the system willbe clearly out of equilibrium during the collision evolu-tion. The value of µ π increases with temperature, whichis not surprising since a large part of the pion produc-tion in the microscopic simulation takes place in initialscatterings and via string formation at the beginning ofthe reaction, when the system still heats up. At all tem-peratures one finds that the µ π decreases with increasingcollision energy, which may indicate a faster and strongerequilibration of the system if more energy is depositedin the system. In addition, for top SIS 300 energies theinitial emission is dominated by QGP radiation at tem-peratures above T c and consequently a larger fraction ofcells with T <
170 MeV is found later in the course ofthe fireball evolution, when the system is in a more equi-librated condition compared to the very beginning of thecollision. For the higher collision energies we get averagevalues up to µ π = 100 −
120 MeV around the critical tem-perature of 170 MeV. Note that for E lab = 4 A GeV themaximum temperature found in the evolution is around155 MeV, which explains the drop in the correspondingcurve around this temperature.In contrast to the large pion chemical potential, no1 ] [GeV/c ee M0 0.2 0.4 0.6 0.8 1 1.2 )] d N / d M [ / ( G e V / c -6 -4 -2
10 1 UrQMD ph fo w fo rf CG of UrQMD=0) p m Sum ( 0) „ p m Sum ( =0) p m ( r Thermal =0) p m ( w Thermal =0) p m Multi-pion (QGP (Lattice)
Au+Au (0-10% central) =2 AGeV lab
E (a) ] [GeV/c ee M0 0.2 0.4 0.6 0.8 1 1.2 )] d N / d M [ / ( G e V / c -6 -4 -2
10 1 (b) ] [GeV/c ee M0 0.2 0.4 0.6 0.8 1 1.2 )] d N / d M [ / ( G e V / c -6 -4 -2
10 1
15 AGeV (c) ] [GeV/c ee M0 0.2 0.4 0.6 0.8 1 1.2 )] d N / d M [ / ( G e V / c -6 -4 -2
10 1
35 AGeV (d)
FIG. 4. (Color online) Dilepton invariant mass spectra for Au+Au reactions at different energies E lab = 2 − A GeV within thecentrality class of 0-10% most central collisions. The resulting spectra include thermal contributions from the coarse-grainingof the microscopic simulations (CG of UrQMD) and the non-thermal contributions directly extracted from the transportcalculations (UrQMD). The hadronic thermal contributions are only shown for vanishing pion chemical potential, while thetotal yield is plotted for both cases, µ π = 0 and µ π (cid:54) = 0. such dominant off-equilibrium effect is observed for thekaons, where µ K ≈ π whereas the cross-section for kaon productionis rather low in the cases considered here (and espe-cially for the lower FAIR energies). Consequently, thekaon production is a slow process which seems to hap-pen synchronously with the equilibration of the systemwhile a large amount of pions is produced in the initialhard nucleon-nucleon scatterings before any equilibrationcould take place.The present results for the pion chemical potential arequite different from other model descriptions. For exam-ple, in fireball parametrizations the particle numbers arefixed at the chemical freeze-out of the system and con-sequently meson chemical potentials develop when thesystem cools down. However, in the fireball model this isjust an ad-hoc assumption, as such macroscopic models are based on a presumed equilibrium within the system.In contrast, the overpopulation of pions is an intrinsicresult stemming from the microscopic simulation in thecase of the coarse-graining approach. Nevertheless, thevery high pion chemical potentials in the temperatureregion close to the phase transition might be question-able, as one would assume that the transition from theQuark-Gluon Plasma to a hadronic phase should producea system where the mesons are in an equilibrium state. Afully satisfying description of the chemical off-equilibriumevolution is not feasible within the present approach andwould require a microscopic and dynamical descriptionof the phase transition and its underlying dynamics. B. Dilepton spectra
The dilepton invariant mass spectra in the low-massrange up to M e + e − = 1 . /c for four different beam2 ] [GeV/c ee M1.2 1.4 1.6 1.8 2 2.2 2.4 )] d N / d M [ / ( G e V / c -6 -5 -4 -3 =0) p m HG + QGP ( 0) „ p m HG + QGP ( =0) p m HG + enh. QGP (=0) p m pure HG ((a)8 AGeV Au+Au (0-10% central) ] [GeV/c ee M1.2 1.4 1.6 1.8 2 2.2 2.4 )] d N / d M [ / ( G e V / c -6 -5 -4 -3 (b)35 AGeV FIG. 5. (Color online) High-mass region of the dilepton M e + e − spectrum for central Au+Au collisions at E lab = 8 A GeV (a)and 35 A GeV (b), assuming different emission scenarios. The the baseline calculation assumes hadronic emission up to 170 MeVwith vanishing µ π and partonic emission for higher temperatures (full line). Besides the results with a five times enhancedemission around the critical temperature (dashed line) and for a pure hadron gas scenario with emission at all temperaturesfrom hadronic sources (dashed-triple-dotted line) are shown. Again, we also show the baseline result including finite pionchemical potential (dashed-dotted line). energies ( E lab = 2 , ,
15 and 35 A GeV) are presented inFig. 4. The comparison shows some interesting simi-larities and differences: While the very low masses up0 .
15 GeV /c are generally dominated by the Dalitz de-cays of neutral pions, the region beyond the Dalitz peakup to the pole masses of the ρ and ω mesons (i.e., ≈
770 MeV /c ) is dominated by a strong thermal ρ con-tribution. The thermal yield shows an absolute increasewith E lab , but its importance decreases relative to thenon-thermal η yield. This means that the thermal low-mass enhancement of the dilepton yield above a hadronicvacuum cocktail decreases with increasing collision en-ergy. This observation is explained by the decrease of thebaryon chemical potential at higher collision energies, ashas been mentioned in the previous section. In contrast,the increasing temperature leads to a significantly flat-ter shape of especially the ρ distribution in the invariantmass spectrum. While for low energies the thermal yielddecreases strongly when going to higher invariant masses,at the top FAIR energies this effect is less prominent andthe population of high masses is enhanced. This is can beclearly seen by the fact that the multi-pion yield showsa strong rise.While at the lowest of the four energies the whole sys-tem is well below the critical temperature T c , we knowfrom the temperature evolution in Fig. 1 (a) that theregion around T ≈
170 MeV from is reached E lab =6 − A GeV on. Consequently, in the dilepton invari-ant mass spectra of Fig. 4 the resulting QGP contribu-tion is very small at 8 A GeV, but even at 35 A GeV thepartonic yield is suppressed by roughly an order of mag-nitude compared to the leading contributions in the massrange up to 1 GeV /c .The hadronic thermal yields in Fig. 4 are shown for thecase of vanishing pion chemical potential. However, we also compare the result for µ π = 0 with the total yieldassuming finite values of µ π . One can see that chemicalnon-equilibrium can increase the overall dilepton yieldin the low-mass range up to a factor of two. For theregion above 1 GeV /c the effect can be even larger asthe fugacity factor enters the thermal rate with a powerof four for the multi-pion contribution. It is important tobear in mind that this result should rather be seen as anupper estimate, as the approximation µ ρ = 2 µ π is onlycorrect for the rate ππ → ρ , which represents only one ofthe many processes included in the ρ spectral function.Furthermore, as UrQMD has no intrinsic description ofthe phase transition, the pion chemical potential mightbe overestimated in vicinity of the critical temperature.Nevertheless, the results show that a deviation from pionequilibrium has a huge impact on the thermal dileptonrates.Considering possible signatures for a phase transitionand the creation of a deconfined phase, the low-massregion is rather unsuited due to the dominance of thehadronic cocktail contributions and hadronic thermalemission from the vector mesons. Consequently, it mightbe more instructive to explore the mass range above thepole mass of the φ , where one has a continuum dominatedby thermal radiation. In this region the hadronic cocktailcontributions can be neglected and thermal sources willdominate the spectrum. In previous works [27, 45] it hasbeen shown for SPS energies that the dilepton invariantmass spectrum at very high masses M l + l − > . /c could only be explained by including thermal radiationfrom the Quark-Gluon Plasma. In Fig. 5 the higher in-variant mass region for M e + e − > . /c is shown forthe two collision energies E lab = 8 and 35 A GeV. Herewe compare four different scenarios to study whether thehigh-mass invariant mass spectrum might help to iden-3 [GeV/c] t p0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ] [ / ( G e V / c ) t ) d N / dp t p p ( / -6 -5 -4 -3 -2 -1
10 110 Au+Au (0-10% centr.) < 0.2 GeV/c ee M (a) [GeV/c] t p0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ] [ / ( G e V / c ) t ) d N / dp t p p ( / -9 -8 -7 -6 -5 -4 -3 -2 -1
10 110 < 0.6 GeV/c ee [GeV/c] t p0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ] [ / ( G e V / c ) t ) d N / dp t p p ( / -9 -8 -7 -6 -5 -4 -3 -2 -1
10 110 < 0.9 GeV/c ee [GeV/c] t p0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ] [ / ( G e V / c ) t ) d N / dp t p p ( / -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1
10 110 > 1.1 GeV/c ee M(d)
FIG. 6. (Color online) Transverse-momentum spectra of the dilepton yield in central Au+Au reactions E lab = 4 A GeV (red)and 35 A GeV (blue). The results are shown for four different invariant mass bins: M e + e − < . GeV /c (a), 0 . < M e + e − < . /c (b), 0 . < M e + e − < . /c (c) and M e + e − > . /c (d). tify the creation of a Quark-Gluon Plasma. Besides thetwo standard scenarios (hadron gas + partonic emissionabove T c = 170 MeV) for (i) finite and (ii) vanishing µ π ,we include a scenario with (iii) a 5-times enhanced emis-sion from the partonic phase around the transition tem-perature to simulate the effect of a critical slowdown ofthe system due to a first order phase transition and, fi-nally, (iv) a pure hadron gas scenario, where we assumeall thermal radiation (also for T >
170 MeV) to stemfrom hadronic sources. For (iii) and (iv) µ π = 0 is as-sumed, too. The comparison shows that the spectralshape of the total yield is very similar for all scenarios,at both energies considered here. While the results fora purely hadronic scenario and including QGP emissionfrom temperatures above T c give quite the same resultswithin 10% deviation, also the artificially enhanced QGPemission does not significantly increase the overall yield.In contrast, one observes a very strong enhancement dueto a finite pion chemical potential, which shows up inour calculation by an overall increase by a factor of 5 at8 A GeV and still a factor of 2 at 35 A GeV. The resultsindicate that it will be difficult to draw unambiguous con-clusions from single measurements of the higher mass re- gion at a specific energy, as according to our calculationsa stronger QGP yield and less hadronic contribution canfinally result in the same overall dilepton spectrum. Fur-thermore, the non-equilibrium effects may lead to muchlarger modifications of the spectrum than caused by thedynamics of the phase transition.The transverse-momentum spectra, plotted for two dif-ferent energies in different invariant mass bins in Fig. 6underline the previous finding. Again the slopes of thecurves for hadronic and partonic emission are very similarfor high masses, especially for M e + e − > . T c . For this caseclear differences between the hadronic and the partonicemission are observed. Unfortunately, even at the top4 Temperature T [GeV]0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 ] - d N / dT [ G e V -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 M < 0.2 GeV/c M > 1.1 GeV/c
Au+Au @ 35 AGeV (0-10% centr.) (a)
Thermal Dileptons
Time t [fm/c] d N / d t [ / (f m / c )] -7 -6 -5 -4 -3 -2 -1 Au+Au (0-10% centr.)
Thermal Dileptons
35 AGeV2 AGeV
QGP (b)
FIG. 7. (Color online) (a) Temperature dependence of dilepton emission d N/ d T from thermal sources for central Au+Auat E lab = 35 AGeV, i.e., the lepton pairs directly extracted from the hadronic cocktail as calculated with UrQMD are notincluded. The results are shown for four different invariant mass bins: M e + e − < . /c , 0 . < M e + e − < . /c ,0 . < M e + e − < . /c and M e + e − > . /c ; (b) Time evolution d N/ d t of thermal dilepton emission for 2 (green)and 35 AGeV (blue). The dashed line shows the emission from the QGP for the top FAIR energy. SIS 300 energy only few cells reach temperature maximaabove 200 MeV and one will not see clear differences be-tween the partonic and hadronic emission patterns evenat high p t and high M e + e − .The reason for the duality showing up only at high in-variant masses (and momenta, respectively) is twofold:On the one hand, the low mass region is governed bythe vector mesons with their specific spectral shapesand the baryonic effects on them. This effect has beencalled the “duality mismatch” [25] since the hadronicrates show an increase for finite baryochemical poten-tials, while the partonic emission rates are quite insen-sitive with regard to µ q . On the other hand, while thespectra at low masses and momenta are populated bythermal emission at all temperatures, the production ofdileptons for masses above 1 GeV /c and for higher val-ues of p t is strongly suppressed at low temperatures. Thisis visible from Fig. 7 (a), where the temperature depen-dent dilepton yield from thermal sources for Au+Au at E lab = 35 A GeV is shown for different invariant-massbins. (Note that the non-thermal lepton pairs directlyextracted from the hadronic cocktail as calculated withUrQMD are not included here). The yields shown inthis plot represent the sum of the contributions fromall cells at a certain temperature. While for the lowestmass bin M e + e − < . /c the total thermal dilep-ton yield is built up by roughly equal fractions stemmingfrom the whole temperature range, with slight suppres-sion of emission from temperatures above T c , one cansee that the mass region above M e + e − = 1 . /c isdominated by emission from temperatures between 140to 220 MeV, which is exactly the assumed transition re-gion between hadronic and partonic emission. Dileptonemission at lower temperatures is strongly suppressed inthis mass range. Furthermore, one finds a smooth be-havior of the thermal emission in the highest mass bin, but at lower masses one observes a slight kink in therates at T c = 170 MeV. The finding indicates that forlower masses the partonic and hadronic rates do not per-fectly match, as was discussed above. Another observa-tion is the dominance of emission from the temperaturerange T = 100 −
140 MeV for the mass region from 0.6to 0 . /c . This result is in contrast to the generaltrend of a shift of the emission to higher temperatureswhen going to higher masses. However, the very massregion covers the pole masses of the ρ and ω meson. Asthe peak structures show a melting especially for finitebaryon densities, one will get the largest yields in thismass range from cells for which µ B ≈
0. A comparisonwith the T - µ B distribution of the cells at this energy inFig. 2 (b) shows that the largest fraction of cells for whichthe baryochemical potential is below 200 MeV lies exactlyin the temperature range from 100 to 140 MeV.Finally, it is instructing not only to look at the temper-ature but also at the time dependence of thermal dileptonemission as presented in Figure 7 (b). The total dilep-ton emission per timestep d N/ d t as sum from all cellsis shown for E lab = 2 and 35 AGeV. In principle, theresults reflect the findings from Fig. 1 and are similarto the temperature evolution depicted there. The sys-tem shows a faster heating for the top SIS 300 energywith higher temperatures, resulting in a larger numberof emitted dileptons; partonic emission from cells with T > T c is found for the first 5 −
10 fm/ c and only verysporadically thereafter. At lower energies the evolution isretarded and T remains below the critical temperature.However, in contrast to the slow heating of the system(which is simply due to the fact that the nuclei are mov-ing slower) the thermal emission drops much earlier for2 AGeV (compared to 35 AGeV) and only few cells withthermal emission are found for t >
30 fm/ c . The higherenergy deposited in the system with increasing E lab ob-5 Temperature T [GeV]0.1 0.12 0.14 0.16 0.18 0.2 0.22 ] - / dT [ G e V g d N -3 -2 -1
10 110 Au+Au (0-10% central) = 0 K m = p m | < 0.5 g |y > 1.0 GeV/c t p (a) UrQMDSMASH p m /3 B m = q m T2 AGeV 4 AGeV 6 AGeV 8 AGeV10 AGeV 15 AGeV 25 AGeV 35 AGeV [GeV] B m Baryochemical Potential ] - / dT [ G e V g d N -5 -4 -3 -2 -1
10 110 = 0 K m = p m | < 0.5 g |y (b) FIG. 8. (Color online) Thermal photon yield at mid-rapidity in dependence on (a) temperature, d N γ / d T , and (b) baryochemicalpotential, d N γ / d µ B , for central Au+Au collisions at different beam energies E lab = 2 − A GeV. The results are shown forthe case of vanishing µ π and µ K . viously results not only in higher initial temperatures,but also in an enhanced emission at later stages, as ittakes the system longer to cool down. Interestingly, forboth energies one finds some sparse cells with thermalemission even after 60-70 fm/c; however, their contribu-tion to the overall result is suppressed by 3-4 orders ofmagnitude compared to the early reaction stages. C. Photon spectra
While we have two kinematic variables (momentumand invariant mass) which can be probed for virtual pho-tons, real (i.e., massless) photons only carry a specificenergy. In this sense, dileptons are the more versatileprobes of the hot and dense medium and carry additionalinformation, especially regarding the spectral modifica-tions of the vector mesons. Nevertheless, the correct de-scription of the experimental photon spectra has beena major challenge for theory at SPS and RHIC ener-gies. In the kinematic limit M → µ B range of their applica-bility, it will be instructive to find out at the beginningunder which thermodynamic conditions the photons areemitted at FAIR. Fig. 8 shows the dependence of thermalphoton emission (at mid-rapidity) on temperature in theleft plot (a) and on baryochemical potential in the rightplot (b). As in Fig. 7, the yields are the sum of the ther-mal contributions from all cells with a certain tempera-ture or baryochemical potential, respectively. For bothresults we consider the case of vanishing meson chem- ical potentials. Note that for the temperature depen-dence we consider only the thermal emission at highertransverse momentum values p t > c , as here theduality between hadronic and partonic rates should beapproximately fulfilled, which is indeed visible from thecontinuous trend of the thermal photon emission around T c . One can see that especially for lower collision ener-gies the thermal emission is dominated by the cells whichreach the maximum temperature, whereas the curves be-come flatter at higher energies. Even at the top energyof 35 A GeV with maximum temperatures above 220 MeVstill a significant amount of emission also stems from thecells with temperatures around 100 MeV.Regarding the photon emission related to baryochemi-cal potential as presented in Fig. 8 (b), a clear energy de-pendent trend is visible: While at low energies the largestfraction of emission stems from cells with very high val-ues of µ B around 900 MeV, the emission weighted averagechemical potential drops continuously to 300–400 MeV at35 A GeV. However, the emission from cells with highervalues of the baryochemical potential is by far not negli-gible. In consequence, the findings once again underlinethat the strongest baryonic modifications of the spectralfunctions will be present at low energies. The results alsoshow that to fully account for the baryonic effects on thephoton emission the spectral functions should be able toreliably cover the whole µ B region from 0 to the nucleonmass (i.e. ≈
900 MeV). The presently used parametriza-tion will provide only a lower limit for the photon yield,especially for the lower collision energies.In Fig. 9(a)-(d) the transverse momentum spectra ofthermal photons for four different collision energies arepresented. As for the dilepton invariant mass spectra, weshow the results with (full black line) and without me-son chemical potentials (dashed black); once again thetwo calculations provide a lower and upper boundary forthe off-equilibrium influence on the thermal yields, re-spectively. Two observations can be made when com-6 [GeV/c] t p )] p ) [ / ( G e V E ( d N / d -8 -7 -6 -5 -4 -3 -2 -1
10 110 =0 /K p m „ /K p m SumMeson Gas w - r - p QGP Spect. Func. r Au+Au (0-10% central) =2 AGeV lab
E (a) | < 0.5 g |y [GeV/c] t p )] p ) [ / ( G e V E ( d N / d -8 -7 -6 -5 -4 -3 -2 -1
10 110 [GeV/c] t p )] p ) [ / ( G e V E ( d N / d -8 -7 -6 -5 -4 -3 -2 -1
10 110
15 AGeV (c) [GeV/c] t p )] p ) [ / ( G e V E ( d N / d -8 -7 -6 -5 -4 -3 -2 -1
10 110
35 AGeV (d)
FIG. 9. (Color online) Transverse-momentum spectra at mid-rapidity ( | y γ | < .
5) of the thermal photon yield for centralAu+Au reactions at E lab = 2 A GeV (a), 8 A GeV (b), 15 A GeV (c) and 35 A GeV (d). The total yields are plotted for bothcases µ π = 0 (full black line) and µ π (cid:54) = 0 (dashed line). The single hadronic contributions from the ρ spectral function (red,long dashed), the meson gas (green, dashed-double-dotted) and the π − ρ − ω complex (beige, dashed-dotted) are only shownfor vanishing pion chemical potential. The partonic contribution from the QGP is plotted as orange short-dashed line. paring the results for the different energies: An overallincrease of the photon p t -yield with increasing energyand, secondly, a simultaneous hardening of the spectra,i.e., one gets a stronger relative contribution for highermomenta. This is similar to the dilepton invariant-massspectra, where the yield in the higher mass region is sup-pressed for lower collision energies due to the lower over-all temperatures in the fireball. (A more explicit com-parison of the energy dependence of the results will beundertaken in Sec. IV D). Furthermore, one can see thatat all energies the contribution from the ρ meson dom-inates above the other hadronic contributions especiallyfor low p t , while the relative dominance of the ρ decreasesfor higher momenta. The contribution from the Quark-Gluon Plasma is visible for E lab = 8 A GeV and higherenergies, giving an increasing fraction of the overall yield.Note the similarity between the low-mass dilepton andphoton p t spectra for 35 A GeV: In both cases the slopeof the (virtual or real) photons emitted from the QGP stage is significantly harder than the contribution fromhadronic sources. Furthermore, looking only at the ρ and the partonic contribution, one finds that the firstis stronger for p t < /c , and the latter dominatesfor higher momenta - for both dileptons and photons.This behavior is expected as the real photon representsjust the M e + e − → µ π and µ K have an even morepronounced effect on the photon rates than on the dilep-ton rates, as several processes to be considered are verysensitive to an overpopulation of pions, it is remarkablethat the overall effect leaves the shape of the photon p t spectra mostly unchanged: The yields are enhanced bythe same factor at all transverse momenta. This is inter-esting as the effect of µ π/K (cid:54) = 0 on the different contribu-tions is varying in strength. For example, the yield fromthe π − ρ − ω system shows a much stronger enhancement7 [GeV] t p )] p ) [ / ( G e V E ( d N / d -6 -5 -4 -3 -2 -1 =0) /K p m Baryonic effects ( = 0 B m „ B m Sum r Au+Au (0-10%) | < 0.5 g |y (a) [GeV/c] t p )] p ) [ / ( G e V E ( d N / d -5 -4 -3 -2 -1 =0) /K p m Hadronic Scenario (
15 AGeV35 AGeV (x50)Sum r Meson Gas w - r - p HGHG+QGP (b)
FIG. 10. (Color online) Comparison of the transverse-momentum spectra at mid-rapidity ( | y γ | < .
5) of the thermal photonyield for central Au+Au reactions resulting from different emission scenarios. In (a) the effect of a finite baryon chemicalpotential µ B on the transverse momentum spectrum for the ρ contribution and the total yield is shown for 2 A GeV (lowerresults) and 35 A GeV (upper results), comparing the standard scenario with µ B (cid:54) = 0 (full lines) with the results for µ B = 0(dashed lines). Plot (b) shows the results for a purely hadronic scenario, i.e., for emission from the hadron gas also for T >
170 MeV and no partonic contribution. The single contributions are plotted as in Fig. 9; for comparison the total yield forthe standard result including hadronic + partonic emission is shown (black dashed). than the ρ contribution (compare Table I).But not only meson chemical potentials influence thephoton spectra, similar to the case of dileptons one ex-pects also an enhancement of the ρ contribution in thepresence of baryonic matter. In Fig. 10 (a) the effect ofa finite baryon chemical potential µ B on the transversemomentum spectrum for the ρ contribution and the to-tal yield is shown for 2 A GeV and 35 A GeV, comparingthe standard scenario with µ B (cid:54) = 0 (full lines) with theresults for µ B = 0 (dashed lines). The comparison showsthat especially at lower momenta the ρ contribution issignificantly increased for finite baryochemical potential,while this effect is less dominant at larger p t . Further-more the effect is stronger for lower collision energies,where one obtains larger average values of µ B . However,one should bear in mind that the parametrization forthe photon emission rates is limited to chemical poten-tials below 400 MeV, so that one can not fully accountfor the very large chemical potentials in this case. Inconsequence, one can expect an even larger enhancementin the experimental measurements than in the presentcalculation.To conclude the study of the different influences on thephoton spectra, we also consider whether the possible cre-ation of a deconfined phase has any effect on the thermalemission pattern. Plot (b) of Fig. 10 shows the resultsfor a purely hadronic scenario, i.e., for emission from thehadron gas also for T >
170 MeV and no partonic contri-bution. For comparison also the total yield for the stan-dard scenario including hadronic + partonic emission isshown. (For both cases the meson chemical potentials are assumed to be zero.) Again, as for the high-massdileptons (compare Fig. 5), the differences between thetwo scenarios are negligible, especially compared to theeffect of the meson and baryochemical potentials. Onlya very slight enhancement of the yield at low momentais obtained for the pure hadron gas scenario, reflectingthe different sensitivity of partonic and hadronic rates tofinite µ B . D. Excitation function of photon and dileptonyields
In the previous sections several differences and similar-ities between dilepton and photon spectra have alreadybeen discussed. However, it is instructing to do this inmore detail and to compare the energy dependence of theemission patterns for photons and dileptons. Consider-ing experimental measurements, an advantage of study-ing the excitation function of thermal yields might bethat the trends and results obtained hereby at differentenergies are more robust and less sensitive to errors ofmeasurement. It reflects the results of several differentmeasurements in contrast to single spectra at a specificenergy. For reason of comparison and because the bary-onic effects are strongest in this case, all results in thefollowing will be considered for mid-rapidity | y | < . p t and dilepton M e + e − spectra for eight different collision energies in therange E lab = 2 − A GeV are shown. It was already men-tioned that—besides the hadronic structures due to the8 [GeV/c] t p0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ] p ) [ / ( G e V / c ) ( d N / d y d p -7 -6 -5 -4 -3 -2 -1 Au+Au (0-10% central)
Thermal Photons = 0 K m = p m | < 0.5 g |y UrQMDSMASH p m /3 B m = q m T (a) ] [GeV/c ee M0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 )] d N / d y d M [ / ( G e V / c -6 -5 -4 -3 -2 Thermal Dileptons = 0 K m = p m | < 0.5 ee |y (b) FIG. 11. (Color online) Comparison of the overall photon transverse momentum spectra (a) and the thermal dilepton invariantmass spectra (b) at mid-rapidity | y | < . E lab = 2 − A GeV. The results are shown forvanishing meson chemical potentials µ π = µ K = 0. direct connection of the dilepton spectrum with the spec-tral function of the light vector mesons—the two spec-tra are strikingly similar. And also the change of thespectra with increasing collision energy is alike. At highmasses/momenta the yield shows a stronger increase with E lab than in the low-mass/ -momentum region. Morequantitatively, this can be seen in Fig. 12 (a), where therelative increase of the thermal photon and dilepton yieldfor different transverse momentum or invariant mass re-gions, respectively, is shown. The results are normalizedto 1 for E lab = 2 A GeV. One observes that the relativeincrease is stronger for high momenta and masses thanfor the lower p t or M e + e − bins. For example, the to-tal dilepton yield for masses below 300 MeV /c increasesonly by a factor of 2 from E lab = 2 to 8 A GeV and re-mains nearly constant thereafter, whereas in the highmass region above 1 GeV /c the yield increases by a fac-tor 300 when going to the top SIS 300 energy of 35 A GeV.A similar behavior is found for the photons, where theyield shows a more pronounced rise at high momenta. Aswas pointed out already before, one reason for this is thefact that much energy is needed to produce a dileptonat high masses or a photon with high momentum. Theirproduction is strongly suppressed at the rather moderatetemperatures obtained at lower collision energies. Notethat in general the overall increase in the photon spectrais slightly stronger than for the dilepton production. Thismight be due to the limitation of the photon parametriza-tion to temperatures above 100 MeV and baryochemicalpotentials below 400 MeV, which might somewhat un-derestimate the photon yield at the lowest collision en-ergies. Besides, one should keep in mind that in detailthe processes contributing to the thermal emission ratesfor dileptons and photons differ, which may also explainsome differences between the results.In contrast, we find that the fraction of the QGPyield compared to the total thermal emission is larger forthe high- M e + e − dileptons compared to the high- p t pho- tons. The first significant QGP contribution is found for E lab = 8 A GeV, and the fraction continuously increasesup to roughly 70% at 35 A GeV for dilepton masses above1 GeV /c and 50% for photon momenta over 1 GeV /c . Atlower masses or momenta, respectively, the hadronic con-tribution becomes more dominant. This is not surprising,as one could already conclude that we probe lower tem-peratures at lower masses and momenta (compare Fig. 7).Furthermore the baryonic influence increases here. In thedilepton spectra the direct connection to the vector me-son spectral functions makes a comparison between thephoton and dilepton results for low momenta and massesdifficult.Finally, we consider also the thermal photon yield inrelation to the number of (neutral) pions which are pro-duced in the heavy-ion collision, as presented in Fig. 13.This ratio is of theoretical and experimental interest: Forthe experimental study of photons, decays of neutral pi-ons are the major background in the analysis. On thetheoretical side, the number of pions gives an estimateof the freeze-out volume and is not sensitive to the de-tails of the reaction evolution. While the electromagneticemission takes place over the whole lifetime of the fireballand therefore reflects the evolution of the system in thephase diagram, the pion yield allows to scale cut trivialdependences. Previously, the thermal dilepton yield wasfound to scale with N / π if one compares different sys-tem sizes at SIS 18 energies [28]. However, the situationis more complex here as we consider a large range of ener-gies, which will be covered by FAIR. Several effects playa role, e.g., the lifetime of the fireball, the temperaturesand baryochemical potentials which are reached and thedifferent processes which contribute at different temper-atures. While in our study at SIS 18 energies only thesystem size was modified (and all the other parameterscould be assumed to remain quite constant, as one singleenergy was considered), in the present study only the sizeof the colliding nuclei is constant.9 [GeV] lab E = A G e V l a b E t h e r m a l N / t h e r m a l N
10 Thermal Dilepton Yield < 0.3 GeV/c ee M < 0.6 GeV/c ee > 1.0 GeV/c ee M Thermal Photon Yield < 1.0 GeV/c t t p (a) =0 K m = p m |y|<0.5, [GeV] lab E t h e r m a l N / QG P N Dilepton Yield
Fraction of QGP emission =0 K m = p m |y|<0.5, (b) < 0.3 GeV/c ee M < 0.6 GeV/c ee > 1.0 GeV/c ee M Photon Yield < 1.0 GeV/c t t p FIG. 12. (Color online) (a) Energy dependence of the thermal photon and dilepton yield in different invariant mass or transversemomentum bins, respectively. The yields are normalized to the result at E lab = 2 A GeV. (b) Fraction of thermal QGP emissionin relation to the total thermal yield of photons of dileptons in different M e + e − and photon- p t bins. All results in this figureare shown for mid-rapidity and vanishing meson chemical potentials. The investigation of the N γ /N π ratio is here combinedwith a comparison of the different scenarios for the condi-tions of thermal emission, which were already studied incase of the photon and dilepton spectra (see Figs. 5 and9). Varying E lab (and, in consequence, T and µ B ) mightresult in distinct excitation functions of the ratio N γ /N π for the various scenarios, in contrast to the spectra for onespecific energy where no unambiguous distinction waspossible. In Fig. 13, one can see a strong increase of thephoton to pion ratio for the lowest energies for both thelower p t range from 0.5 to 1 GeV/ c and the high trans-verse momentum region above 1 GeV/ c in case of thebaseline scenario with QGP emission for T >
170 MeVand µ π = 0. However, for higher collision energies above10 A GeV we still observe a further increasing ratio for thehigher momentum range, whereas at low p t the ratio re-mains relatively constant and even decreases for the high-est collision energy. One can understand the decreasingratio for lower p t reviewing again the energy dependenceof T and µ B as shown in Fig. 1. The rise of temperaturebecomes less intensive for higher collision energies, whilethe baryochemical potential decreases for higher collisionenergies, causing a less pronounced increase of the ther-mal yield (compare also Fig. 12). Besides, the effects dueto finite µ B are more pronounced at low momenta, ex-plaining the different trends for the two p t regions. In-cluding the finite meson chemical potentials, we observea strong increase of the ratio by factors 2-5 at all collisionenergies. The strongest effect in the present calculationis seen around 8 A GeV, so that a slight peak structurebuilds up. However, as mentioned already several times,this scenario can only be seen as an upper limit for thenon-equilibrium effects, most probably the increase willbe smaller.When comparing the two scenarios including enhancedQGP emission around T c on the one side and a pure hadronic scenario on the other side it is interesting thatboth cases lead to a similar result, namely an increase ofthe N γ /N π ratio at higher collision energies. The effectsshow up more dominantly at high momenta, as this re-gion is more sensitive with regard to emission from hightemperatures. Note, however, that there are also signif-icant differences between the two cases. The scenariowith enhanced QGP emission around T c shows the mostprominent increase at E lab = 8 − A GeV whereas thisenhancement becomes smaller again for higher energies.This can be explained by the fact that the relative frac-tion of emission from temperatures around 170-175 MeVis largest at those collision energies where the transitionto a partonic phase is just reached. At higher E lab thecorresponding higher temperatures may outshine any ef-fects from the transition region. On the contrary, theenhancement over the baseline scenario increases withenergy for the case of a pure hadron gas. However, forhighest collision energies the difference seems to remainstable or even to drop again.We remind again that for the experimental measure-ment the ratio of thermal photons from Fig. 13 is of im-portance, as almost all of the π mesons decay into a pho-ton. Therefore the vast majority will be decay photons,not stemming from direct (thermal or prompt) emissionprocesses. Their spectra have to be subtracted in experi-ment to draw conclusions about the direct photons fromthermal sources. This might be relatively difficult forthe lowest energies available at FAIR, as here the ratio issuppressed by up to an order of magnitude compared tothe higher collision energies.0 V. DISCRIMINATING DIFFERENTSCENARIOS
It has so far become clear that one can extract onlylimited information regarding the properties of the hotand dense fireball from single photon and dilepton spec-tra, as there are usually several different effects thatmight interfere and finally lead to the same invariant-mass or transverse-momentum yields. However, the pic-ture might be quite different if—in addition—the resultsin distinct mass or momentum regions, respectively, aresystematically compared for several collision energies. Inthis case one might be able to discriminate the hadronicand partonic effects from each other. The FAIR energyregime will be ideally suited for such a study, as thetransition from pure hadronic fireballs to the creationof a deconfined phase will take place somewhere around E lab = 6 − A GeV, as our results suggest. Neverthe-less, there is no single observable that seems to allowfor unambiguous conclusions on the details of the reac-tion evolution. On the contrary, it will still be necessary [GeV] lab E p t o t / N t h e r m a l g N -4 -3 -2 < 1.0 GeV/c t g t g p =0) /K p m HG + QGP ( 0) „ /K p m HG + QGP ( =0) /K p m HG + enh. QGP (=0) /K p m pure HG ( FIG. 13. (Color online) Energy dependence for the ratio ofthermal photon yield N thermal γ at mid-rapidity ( | y | < .
5) tothe overall number of neutral pions N π . The results areshown for two different regions of the photon transverse mo-menta, 0 . < p t < . /c (green) and p t > . c (red). The baseline calculation assumes hadronic emission upto 170 MeV with vanishing µ π/K and partonic emission forhigher temperatures (full line). In addition, the results witha five times enhanced emission around the critical tempera-ture (dashed-dotted line), for a pure hadron gas scenario withemission at all temperatures from hadronic sources (shortdashed line) and including meson chemical potentials (dashedline) are shown. to carefully compare theoretical calculations and experi-mental results.Based on the findings of the present work, one mayconsider the following scheme which might help to deter-mine the strength of the different effects on the thermalrates and discriminate between the contributions:1. The influence of baryonic matter leads to an en-hancement which is most dominant for low trans-verse momenta and low masses. In general, it steep-ens the p t slope of the overall yield. A large advan-tage is that today the spectral function of the ρ me-son is quite well known from previous experimen-tal and theoretical studies. Detailed and precisephoton (dilepton) measurements for low momenta(low masses) in the FAIR and RHIC-BES energyregime might give further constraints for the spec-tral function in the region of extremely high baryondensities and can, vice versa, help to see whetherthe models correctly describe the fireball evolutionin terms of µ B .2. In contrast to the baryonic effects on the emissionrates, non-equilibrium effects caused by finite pion(and kaon) chemical potentials will show up as en-hancement in the dilepton and photon spectra at allmasses and momenta, and will be visible at all col-lision energies. The effect should be slightly moredominant in the high invariant mass or high p t re-gion, as here the multi-meson contributions becomemore pronounced. Ideally, one can discriminate be-tween the µ B - and µ π -driven effects by comparingthe modification of the slope and the overall en-hancement in relation to baseline calculations.3. If the baryon and non-equilibrium effects are undercontrol, one might be able to find signals from thepartonic phase in the dilepton and photon spectrafor high p t and M e + e − . In general, the dilepton andphoton rates do not differ much from each otheraround T c , but effects such as a critical slowdownof the evolution might lead to an increased yieldfrom the Quark-Gluon Plasma. On the other handthe “duality mismatch” might lead to a relativedecrease of the yield, as hadronic rates are sensi-tive to finite baryon and meson chemical potentialswhile the QGP rates show hardly any modification.However, any effects connected to a phase transi-tion can only show up if the obtained temperaturesare large enough. Consequently, we would observesubsequent modifications of the spectra only for en-ergies larger than E lab = 6 − A GeV, in contrastto non-equilibrium and baryonic effects which alsoappear at lower temperatures. Significant differ-ences from calculations which only show up for thehigher energies might then indicate the creation ofa deconfined phase.4. Furthermore, these effects should be dominant inthe regions which are most sensitive to QGP forma-1tion: The region for M e + e − > p t in the photon respectivelylow-mass dilepton spectra (provided, it is possibleto get control over the hadronic decay background).Another advantage in these regions is that they arerelatively insensitive to the finite baryon chemicalpotential.The different issues are not easy to disentangle and sev-eral interdependencies exist. Another aspect, which isnot explicitly considered in our work but might furthercomplicate the situation, is the influence of different EoSon the thermal yields. In the present work we use aHadron-Gas EoS to provide consistency with the underly-ing microscopic model. However, previous investigationsin a transport + hydrodynamics hybrid model showedthat an MIT Bag Model EoS or a chiral EoS lead to dif-ferent evolutions in the hydrodynamic phase comparedto the HG-EoS, resulting in higher temperatures and,consequently, an increase of the emission rates. VI. CONCLUSIONS & OUTLOOK
We have presented photon and dilepton spectra for thecollision energy range E lab = 2 − A GeV, which willbe covered by the future FAIR facility (and, in parts, byphase II of the BES at RHIC). The calculations were per-formed using a coarse-graining approach with transportsimulations from the UrQMD model as input. In this ap-proach local particle and energy densities are extractedfrom an ensemble average of the microscopic transportcalculations, and an equation of state is used to calculatethe corresponding values for temperature and chemicalpotential. The local thermodynamic properties are thenused to determine the thermal emission rates.The resulting spectra show a strong influence of fi-nite baryochemical potentials and an enhancement dueto non-equilibrium effects caused by finite meson chem- ical potentials. Regarding the search for signals of thedeconfinement phase transition, there is no clear signalfrom which the creation of a partonic phase can be unam-biguously inferred. Similarly, the results suggest that it isalso hard to identify the type of the transition, whether itis a cross-over or a first-order phase transition, as effectsdue to a critical slowing down might be small comparedto other influences on the spectra. The main difficultyis the dual connection between hadronic and partonicemission rates in the transition region around the criti-cal temperature T c , resulting in very similar slopes in theinvariant-mass and transverse-momentum spectra.For a clarification of the open issues, experimentalinput is needed. Our results suggest that one needs veryprecise and detailed measurements, as different evolutionscenarios for the nuclear collisions are modifying thedilepton and photon spectra in a quite subtle manner.Systematic studies of several collision energies in thefuture FAIR energy range from E lab = 2 A GeV to35 A GeV are required to get more insights into thestructure of the phase diagram of QCD matter andespecially to find clues for the creation of a deconfinedphase. Besides the experimental efforts, it will besimilarly important to intensify the theoretical studies.
ACKNOWLEDGMENTS
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