# A Systematic Study on Direct Photon Production from Central Heavy Ion Collisions

###### Abstract

We investigate the production of direct photons in central Au-Au collisions at the relativistic Heavy-Ion Collider (RHIC) at 200 GeV per nucleon, considering all possible sources. We treat thermal photons emitted from a quark-gluon plasma and from a hadron gas, based on a realistic thermodynamic expansion. Hard photons from elementary nucleon-nucleon scatterings are included: primordial elementary scatterings are certainly dominant at large transverse momenta, but also secondary photons from jet fragmentation and jet-photon conversion cannot be ignored. In both cases we study the effect of energy loss, and we also consider photons emitted from bremsstrahlung gluons via fragmentation.

###### pacs:

25.75.-q,12.38.Mh## I Introduction

The formation and observation of a quark-gluon plasma in heavy ion collisions is an important goal of modern nuclear physics. Even if a quasiequilibrated plasma is created for a brief time in the collisions, it is still a challenge to infer knowledge of the plasma from particle production. Among those proposed "probes" of the plasma are the directly produced real photons. Those photons interact only electromagnetically, and so their mean free paths are typically much larger than the transverse size of the hot dense matter created in the collisions. As a result, high-energy photons produced in the interior of the plasma usually pass through the surrounding matter without interaction, carrying information directly from wherever they were formed to the detector.

One expects to deduce properties of the highly excited matter, in particular it’s space-time evolution. Direct photons may provide these informations in two different ways: via so-called "thermal photons" emitted directly from hot and dense matter, and via secondary photons from initially produced jets. There are two kinds of such secondary photons: they may originate from the "fragmentation" of jets, or the "conversion" of jets into photons via interactions of jets with the partons of the equilibrated matter. Such secondary photons compete, however, with the photons directly produced in an initial hard nucleon-nucleon scattering, in the following referred to as "primordial NN scattering" contribution, so a carefully study of these photons is very important. In the following, we will shortly review these different photon sources.

Primordial NN scattering. The direct photon production via Compton scattering and quark-antiquark annihilation can be calculated in perturbation theory, using the usual parton distribution functions. In principle one should consider at this stage also higher order contributions, like bremsstrahlung of photons accompanying for example two-jet production in hards parton-parton scattering. However, we consider this as part of the so-called jet fragmentation (or bremsstrahlung) contribution, which will be affected by the thermalize matter, and which we will discuss separately.

Thermal photons. In high energy nuclear collisions, the density of secondary partons is so high that the quarks and gluons rescatter and eventually thermalize to form a bubble of hot quark-gluon plasma (QGP).The plasma expands and decreases its energy density so that a phase transition to hadronic gas (HG) phase appears. Thermal photons can be produced during the whole history of the evolution, from the QGP phase, the mixed phase, and from the pure HG phase. Photons from a thermal source are exponentially damped so that the contribution to very high is negligible. However, its contribution to low is dominant.

Jet-photon conversion. When jets pass through thermalized matter, they may interact. In case of the quark-gluon plasma, these interactions are elastic collisions between jets and deconfined partons. There is first of all quark-antiquark annihilation and quark-gluon Compton scattering, which both can produce a photon. These photons are affected by the plasma in two ways: obviously the plasma is needed to allow these interactions, but there is also a secondary effect, since the jets first of all lose energy in the plasma, before contributing to the photon production.

Jet fragmentation or bremsstrahlung photons. Photon production also occurs as higher order effect in purely partonic initial hard scatterings: at any stage of the evolution of a jet (final state parton emissions), there is a possibility of emitting photons. Also here the presence of a QGP will affect the results, since the jets lose energy during the fragmentation process. And not to forget the fragmentation contribution from induced gluon radiation in the plasma.

Our paper is organized as follows: in sections 2 and 3, we discuss thermal photon production; in section 4, we study hard photons from primordial NN scattering (leading order); in section 5 we study jet-photon conversion, including the modification due to jet energy loss; in section 6 we compute photon production from jet fragmentation, also referred to as bremsstrahlung’s photons, again considering the effect of energy loss; in section 7 we finally collect our results and compare with experimental data.

## Ii Photon emission rates

Thermal photon production is obtained by integrating the photon emission rate (number of reactions per unit time per unit volume which produce a photon ) over the space-time history of the expanding hot and dense matter. In this section we study the photon emission rates from different phases of the hot dense matter.

The spectral photon emissivity directly reflects the dynamics of real photon production reactions in thermalized matter. Commonly employed formalisms are finite-temperature field theory and kinetic theory. As systematically studied by Kapusta et al.Kapusta1991 (), the thermal emission rate of photons with energy and momentum from a small system (compared to the photon mean free path) is

(1) |

where is the retarded photon self-energy at finite temperature . This formula has been derived both perturbatively and nonperturbatively. It is valid to all orders in the strong interaction. If the photon self-energy is approximated by carrying out a loop expansion to some finite order, then the formulation of Eq. (1) is equivalent to relativistic kinetic theory, where the emission rate of photons with energy and momentum from a process of type 1+23+ reads

(2) | |||||

where the ’s are the Fermi-Dirac or Bose-Einstein distribution functions as appropriate. Eq.(2) is convenient if the scattering amplitude, , is evaluated in a perturbative expansion. Non-perturbative (model) calculations at low and intermediate energies, on the other hand, are more amenable to the correlator formulation, Eq.(1). In the hadronic medium, , can be directly related to vector-meson spectral functions within the vector dominance model (VDM). Instructive investigation on photon emission rates from both QGP and HG phases can be found in Kapusta1991 ()Rapp_rev_2004 ().

### ii.1 Quark-Gluon Plasma

In Kapusta1991 (), the thermal rate from a quark gluon plasma is computed using the kinetic theory formalism for the simplest two-to-two scattering diagrams such as the QCD Compton process and annihilation , see Fig.1. In case of a large photon energy, and with the energy of two initial partons being larger than the energy of the output photon (), the approximation is employed. Because the light quark masses are set to zero, an infrared cutoff must be placed on the four-momentum transfer. The infrared divergence is regulated by an infinite resummation of finite-temperature Feynman diagrams, following Braaten and Pisarski. This amounts to a careful treatment of a small part of phase space left out in the kinetic theory calculation by imposing the infrared cutoff. When the contributions from the two regions of phase space – below and above the cutoff – are added, the result is independent of the cutoff. This provides a parameterized thermal emission rate of photons with energy and momentum from an equilibrated QGP at temperature and zero net baryon density, for large values of , given as

(3) |

An additive “1" has been introduced in the argument of the logarithm to enable extrapolation to small Kapusta1991 ().

As noticed in Ref.Arenche_gelis () , Eq.(3) does not yet comprise the full result to leading order in the strong coupling constant . Due to collinear singularities, bremsstrahlung as well as pair annihilation graphs contribute at the same order as the resumed 22 processes, c.f. Fig.2. The full result, which also necessitates the incorporation of Landau-Pomeranchuk-Migdal (LPM) interference effects, as shown in Fig.3, has been computed in Ref.AMY2001 () as

with convenient parameterizations of the 3 functions as AMY2001 ()

(5) |

(6) | |||

We will employ the above formula, taking , and a temperature dependent running coupling constantKarsch1998 ()

(7) |

Effects from non-zero baryon density and from off-equilibrium are not included, the above rates can be considered to be a good approximation for the midrapidity region of Au+Au collisions at 200 AGeV. In Fig.7 we show photon emission rate from QGP. The dashed line represents the results obtained by Kapusta et alKapusta1991 (), summarized in eq.(3), the solid line refers to the calculations of AMYAMY2001 (), given in eq.(II.1), with additional contributions compared to Kapusta1991 (). We can see the full contribution to photon emission from a QGP is much higher than the one from partonic processes.

### ii.2 Hadronic Matter

Photons can also be produced in a hadronic phase, from several elementary interactions, see Fig.4. The dominant contribution comes from the reactions and ; The decay also contributes significantly. Interactions involving strange mesons or baryons can also produce photons, but these contributions are relatively small because of the phase space suppression due to their big masses. The situation of thermal photon radiation rates from a hadronic gas is uncertain, due to difficulties related to the strong coupling and the masses of hadrons. The study is usually carried out within effective Lagrangians. Constraints on the interaction vertices can, to a certain extent, be imposed by symmetry principles (, e.m. gauge and chiral invariance). Coupling constants are estimated by adjusting to measured decay branchings in the vacuum. Thus, for the temperature ranges relevant to practical applications, =100-200 MeV, the predicted emission rates are inevitably beset with significant uncertainties, and therefore a careful judgment of the latter becomes mandatory.

Investigations along these lines were initiated in Ref.Kapusta1991 (), where the photon self-energy has been computed to 2-loop order for a mesonic system consisting of sharp (zero width) -, - and -mesons (plus direct decays). In Ref.Xiong1992 () it was pointed out that scattering via resonance formation (or, equivalently, decay), c.f. Fig.5, constitutes an important contribution. This was followed up by a systematic treatmentSong1993 () of an interacting system to 2-loop order within the Massive Yang-Mills (MYM) framework of introducing axial-/vector mesons into a chiral Lagrangian, and, later, within the Hidden-Local Symmetry (HLS) approachHal98 (). The effect on in-medium vector and axial-vector meson masses is studied by Song and Fai Song1998 ().

We will use the results of the MYM calculation Rapp2004 (), where photon production from strangeness bearing mesons has been included as well as the axial meson as exchange particle for non-strange initial states. We show these rates in fig.6, and we list the corresponding parameterized rates, in unit of , with the photon energy () and the temperature () both in GeV:

(8) |

(9) | |||

(10) | |||

(11) | |||

(12) |

(13) |

(14) | |||

Parameterisations for and do not appear because their rates have been found to be negligible.

Hadrons are composite objects, so they may need vertex form factors to simulate finite hadronic size effect, in particular at high momentum transfer. How much will form factor influence the results? Form factors are a very delicate subject, especially when electromagnetism and its gauge invariance are involved. For those nonstrange reaction channels as originally studied by Kapusta et al. Kapusta1991 (), considering form factors provide a typical net suppression compared to the bare graphs by an appreciable factor of at photon energies around GeV. The reduction of the rate introduced by Rapp et al. Rapp2004 () in GeV region of photon energies amount to a factor of , confirming roughly Kapusta’s findings.

In ref. Arleo2004 (), for each hadronic vertex appearing in the amplitudes, it is proposed to take hadronic form factor for -channel meson -exchange according to

(15) |

with =1GeV and being the average momentum transfer , where is the photon energy and the mass of the hadron . In our calculation, we follow this procedure: The photon emission rate is taken to be the sum of all terms in eqs. (8-14), with each term multiplied by . The corresponding curve is shown in Fig.7 as the dashed-dotted line. The dotted line is the photon emission rate without considering hadronic form factors. The form factors of Eq.(15) makes the suppression stronger compared to Rapp2004 (); Kapusta1991 (). For example, at a photon energy of GeV, the suppression factor is about for -exchange contributions, and even for -exchange processes. The suppression are even larger for higher photon energies. So the hadronic emission rate is reduced considerably after including the hadronic form factor as in Eq.(15). The HG rate is therefore much smaller than QGP one (full line).

## Iii Thermal photons from the expanding hot and dense matter

The expanding thermalized matter is treated by employing three-dimensional hydrodynamics. We compute initial conditions at some given proper time , expressed via energy density , net flavor density , and collective velocity , by employing the EPOS model epos1 (); epos2 (). The hydrodynamic evolution is realized using SPheRIO SPHERIO (), which is a “Smoothed Particle Hydrodynamics” implementation, a method originally developed in astrophysics, and later adapted to relativistic heavy ion collisions . The three-dimensional hydrodynamics describes the space-time evolution of the hot dense matter created in heavy ion collisions, via the 3-velocity , the energy density , the entropy density and the baryon number density , as functions of the space-time position , with being the space-time rapidity, and the transverse coordinates. In this paper, we consider central AuAu collisions (10% most central events) at 200 AGeV. The corresponding results of a hydrodynamical evolutions is shown in Fig.8, where we plot the –dependence of at for different values of and . The initial time is . The solid lines and the corresponding dotted lines refer to and , respectively. The three horizontal dotted lines are the the energy densities GeV/fm and GeV/fm , limiting the mixed phase, and the freeze-out energy density GeV/fm..

The relation between energy density and temperature, , as used in SPheRIO is shown in Fig.9 (and tabulated for later use).

We define to be the fraction of matter in the QGP phase and as the corresponding fraction in the HG phase, at each space-time point . We have obviously , , if the energy density is bigger than , , , if the energy density is between and , and , , if the energy density is smaller than . In the mixed phase (, we have , with and being the entropy densities corresponding to and , is the total entropy, which is simply a linear function of , like . Then and are linear in as well, namely

(16) |

The photon emission rates discussed in the previous chapter are obtained in the local rest frame, so we should better write

(17) |

where quantities viewed in the local rest frame are decorated with the superscript . Bare symbols (without *) refer to the laboratory frame. We need the photon spectrum observed in the laboratory, which is given as

(18) | |||||

with

(19) |

where and are the photon emission rates from QGP phase and from HG phase as discussed in the previous chapter. The center-of-mass energy in eq.(18) is related to the photon momenta in the observer frame (appearing on the l.h.s. of Eq.(18)) as

(20) |

with

(21) |

where is the flow velocity at a given space-time point. The flow rapidity is roughly equal to the space-time rapidity (as in the Bjorken model). The radial dependence of the transverse velocity is shown in Fig.10, for central Au-Au collisions at 200 AGeV. The large transverse flows at large radii will boost thermal photons to higher regions.

In Fig.11 we present the transverse momentum spectra of thermal photons produced in the 10% most central Au-Au collisions at 200 AGeV with an initial time of =0.5 fm/c. The contributions from the two phases are presented separately: QGP phase (solid line) and HG phase (dashed line). We can see thermal contribution is dominated by photons from QGP phase, which is much bigger than the upper limit of the contribution from HG phase, , without consideration of hadron form factors (dotted line). The reason is simply that the QGP phase is "hotter" compared to the HG phase. Therefore, hadron form factors makes very little difference concerning the total thermal contribution, as seen in Fig.11, when comparing the complete results with form factor (full circles) and without form factor (empty squares).

## Iv Hard photons from primordial N-N scattering, including shadowing and isospin mixing

Here we present the leading order perturbative calculation of hard photon production, namely the hard photons from primordial N-N scattering. Higher order contributions are related to the production and propagation of jets, so we treat them later. The spectrum of hard photons from a collision between nuclei and nuclei is

(22) |

where and are parton distribution functions for nuclei and . We take MRST 2001 LO parton distributions for protonsMRST0201 (). Nuclear shadowing effects are taken into account by using EKS98 scale dependent nuclear ratios EKS98 (). The mixed isospin in nuclei with mass , neutron number and proton number is taken into account as

(23) |

The elementary cross sections after color sum and spin average are given as Owens1987 ()

(24) |

(25) |

with

(26) |

and MeV. The 4-momenta of the incoming particles (, ) and of the photon () in the center-of-mass are

(27) |

and

(28) |

The Mandelstam variables are then given as

(29) |

(30) |

(31) |

We set the factorization scale and renormalization scale to be . Due to the -function in Eq.(22), the -integration is trivial, leading to

(32) |

with . The condition requires with . So

(33) |

In Fig.12, we plot the corresponding spectrum, for central Au-Au collisions at 200 AGeV. Empty circles present PHENIX dataPHENIX data (). At high region, the contribution of primordial NN collisions is very close the experimental data.

## V Jet photon conversion, including jet energy loss

When jets propagate in the hot dense matter created in heavy ion collisions, they also interact with the matter and produce direct photons via the Compton process and the quark-antiquark annihilation process . We denote the momenta of the jet, the thermal parton, and the photon by , and , respectively. The leading order QCD Compton and annihilation cross sections are peaked in the forward and backward directions. In the laboratory frame we have , where is the temperature of the plasma. For high energy photons, i.e. , this implies that . That is why we call this process a conversion of a jet into a photon with similar momentum. The rate of photon production by annihilation and Compton scattering of jets in the medium can be approximated as WangCY ()Fries2005 ()

(34) |

with , , and where and are the strong and the electromagnetic couplings. Th subscript “” denotes all light quark and antiquark species with charge , and is the phase-space density of partons of flavor . It is worth emphasizing that the conversion property of the process is reflected in eq.(34) by the fact that the photon spectrum is directly proportional to the parton spectrum . Then at midrapidity the jet-photon conversion contribution to the direct photon production in the most 10% Au-Au collisions is gained via the integration over the space-time evolution of the hot dense matter in QGP phase:

(35) | |||

To get , the phase space densities of quarks and antiquarks, we have to fix the geometry of jet formation. The jet production from primordial N-N scattering is assumed to happen at the same proper time . Then at , the phase space distribution of partons of type is

(36) |

where in the impact parameter and and are thickness functions of nuclei and . The -function is motivated by the strong Lorentz contraction of the colliding nuclei. To have a simple form for a central - collision, i.e. most 10% central Au-Au collision, we take the approximation

(37) | |||

where is the radius of nuclei Au, and with . The momentum distribution is calculated as

(38) | |||

which is very similar to eq.(33). Here, the cross
sections of all possible partonic processes make
totally 127 termsOwens1987 (). =2 is used to take
into account higher order contributions to jet production in our calculation^{1}^{1}1 can be extracted from the ratio of experimental jet date
to leading order calculation. We find 2 via pp (p)
collisions at energy range from 27.4GeV to 630GeV. . The phase space distribution of jet satisfies the normalization

(39) |

where is the number of jets of type .

If we ignore the jet energy loss due to the interaction between jets and matter, then at any , the phase space distribution of jets is

(40) | |||||

where is the velocity of a jet, , and with being the space time rapidity.

If we consider the modification of the jet energy and its momentum due to the interaction between jets and hot dense matter, then the phase space distribution of jets should be replaced by

(41) |

with , and where is the energy loss of a jet propagates from the formation point to the jet-medium interaction point . In static matter, one has Baier97 ()

(42) |

with , and where the mean free path of the jet in the medium, and , are all temperature dependent quantities. The index “” refers to the jet type (quark or gluon). Due to the space-time evolution of temperature as , we have to replace in Eq.(42) by the integration along the jet’s trajectory in the QGP plasma. Light-flavour quarks and gluons are massless, with , and so we get

(43) |

The mean free path is given as

(44) |

(45) |

where the cross sections are given as GW94 () , with , and , and where and are the thermal parton densities.

Fig.13 shows the jet-photon conversion contribution to direct photon production in central Au-Au collisions at 200 AGeV. We show the results with and without energy loss, together with PHENIX dataPHENIX data (). We can see that considering energy loss can indeed suppress the jet-photon conversion contribution, by about a factor of three. Obviously, higher jet energies are needed to produce a photon with a given energy, in case of jet energy loss in the medium. The effect from different hydrodynamic initial times is also considered. However, results with =0.5 fm/c and 1 fm/c are indistinguishable. Our results also agree well with the earlier work by TurbideTurbide2005 (), although they use one dimensional hydrodynamics.

## Vi Bremsstrahlung photons, including jet energy loss

We discussed earlier leading order photon production from elementary NN collisions. However, also higher order diagrams give important contributions. Here, we discuss “bremsstrahlung photons” from jets, also referred to as “jet fragmentation”. We can treat the bremsstrahlung contribution via parton fragmentation functions being the probability for obtaining a photon from a jet , where the photon carries a fraction of the jet’s momentum. The effective fragmentation functions for obtaining photons from partons can be calculated. The leading order result is

(46) |

and

(47) |

where is the fractional charge of the quark . The photon fragmentation functions evolve with just as the usual hadronic fragmentation functions do, as a result of gluon bremsstrahlung and pair production. The resulting evolution equations are

(48) | |||

and

(49) | |||

where , , and are splitting functions appearing in the DGLAP equations. The parameterized solutions from OwensOwens1987 () are

(50) | |||

and

(51) |

The bremsstrahlung contribution to direct photon production is then

(52) |

with being the momentum carried by the jet before fragmentation, and . The jet cross section can be obtained from Eq.(38).

To take into account the energy loss of the jet in the QGP phase, the bremsstrahlung contribution to direct photon production is modified. We may use a modified fragmentation functionXNWANG04 (), given as

(53) | |||

where and are the rescaled momentum fractions carried by jet and the emitted gluons before fragmentation, and where and are the original fragmentation functions, given in Eqs.(50) and (51). So a parton has the probability to fragment directly without interacting with the medium, and the probability to interact with the medium before fragmentation. The ratio represents the number of scatterings, and a gluon of energy is emitted each time when a parton scatters with the medium.

As discussed earlier, the energy loss in the medium with a constant temperature is , where is the mean free path of jet in the medium and . Those quantities are temperature dependent. In our case, the temperature evolves with space and time. So similar to Eq.(43), we replace and by the corresponding mean values, namely

(54) |

(55) |

We take the mean energy loss per scattering as the energy carried by each emitted gluon.

In Fig.14, we present the bremsstrahlung contribution to direct photon production, with and without considering energy loss. We can see that the photon production is considerably suppressed due to the energy loss of the jets in the hot dense matter. The results from two different initial times =0.5 fm/c and 1 fm/c is indistinguishable.

## Vii results and discussion

In the following we are going to collect and discuss our results. In Fig.15 we compare the different non-thermal contributions to photon production in central Au-Au collisions at 200 AGeV. We show the contribution from primordial NN collisions (dashed line), bremsstrahlung (or fragmentation) photons (dotted line), and photons from jet-photon conversion (solid line). We have taken into account energy loss of jets in the hot and dense matter, both for bremsstrahlung photons and jet-photon conversion. We also plot PHENIX dataPHENIX data (), as a reference. Jet-photon conversion contributes the same magnitude as bremsstrahlung, but both are small compared to the hard photons from primordial NN collisions.

In Fig.16, the total contribution (solid line), including all non-thermal and thermal contributions, is compared with PHENIX data[PHENIX data] (empty circles). The thermal contribution from a hydrodynamic calculation with an initial time =0.5 fm/c is plotted (dotted line) as well as the results of a calculation with =1 fm/c (solid line with full circles). Smaller gives a somewhat broader distribution, but this effect is invisible in the total contribution. At very low , thermal production dominates the direct photon production. However, the thermal spectra decrease very fast with , and become negligible for 4GeV/c. The hydrodynamic configuration also affects the jet-photon conversion and the bremsstrahlung contribution. But this effect is very weak: taking different initial times within the range [0.5,1]fm/c make very little difference to the total contribution (invisible in Fig.16). The complete contribution, containing all the thermal and non-thermal contributions, follow quite well the PHENIX data.

It is useful to compare the AuAu results with proton-proton, and here it is convenient to study the nuclear modification factor, defined to be ratio of the nuclear spectrum to the proton-proton one, divided by the number of binary collisions. We use =880 for the 10% most central Au-Au collisions at 200 AGeV, treated in this paper. We compute the proton-proton differential cross section by using =40.83mb, and =2, and the unmodified photon fragmentation functions given in eq.(50, 51).

In Fig.17, we plot the different non-thermal contributions, namely the primordial NN contribution (dashed line), bremsstrahlung (dotted line), and jet-gamma conversion (solid line). The contributions from bremsstrahlung and jet-photon conversion are of the same magnitude, but one order smaller than that the primordial NN curve. We also show the rapidly falling thermal contribution(dotted dashed line). Above 5 GeV, only photons from primordial NN scattering contribute significantly. This is why we look more closely to this latter contribution.

In Fig.18, we show again the complete primordial NN contributions, including isospin mixing and shadowing. But we also show the results one would obtain by omitting isospin mixing (but considering shadowing), and by omitting shadowing (but considering isospin mixing). Obviously, both effect are crucial. In particular the isospin mixing is responsible for getting a nuclear modification factor of less than unity.

Finally we show in Fig.19 the complete calculation, for two different options of the initial time for the hydrodynamical evolution. The two curves are almost identical. As mentioned earlier, the total contribution to the nuclear modification factor is less than unity, due to isospin correction. However, the experimental data drop systematically below the theoretical curve, and it seems difficult with the processes discussed in this paper, to get the theoretical curve further down. So maybe some "new physics"? Before answering this question, one should not forget the large uncertainties of the experimental pp reference, which is a fit function passing between strongly fluctuating data points at large pt.

###### Acknowledgements.

This work is supported by the Natural Science Foundation of China under the project No. 10505010 and by MOE of China under project No. IRT0624. F.M.LIU thanks the IN2P3/CNRS and Subatech for their hospitality during her visit in Nantes.## References

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