Azimuthal asymmetry of direct photons in intermediate energy heavy-ion collisions

Hard photon emitted from energetic heavy ion collisions is of very interesting since it does not experience the late-stage nuclear interaction, therefore it is useful to explore the early-stage information of matter phase. In this work, we have presented a first calculation of azimuthal asymmetry, characterized by directed transverse flow parameter $F$ and elliptic asymmetry coefficient $v_2$, for proton-neutron bremsstrahlung hard photons in intermediate energy heavy-ion collisions. The positive $F$ and negative $v_2$ of direct photons are illustrated and they seem to be anti-correlated to the corresponding free proton's flow.

The properties of nuclear matter at different temperatures or densities, especially the derivation of the Equation-of-State (EOS) of nuclear matter, are one of the foremost challenges of modern heavy-ion physics. Since heavy ion collisions provide up to now the unique means to form and investigate hot and dense nuclear matter in the laboratory, many experimental and theoretical efforts are under way towards this direction. Because of their relatively high emission rates, nucleons, mesons, light ions and intermediate mass fragments, produced and emitted in the reactions, are conveniently used to obtain information on the reaction dynamics of energetic heavy ion collisions. However, these probes interact strongly with the nuclear medium such that the information they convey may bring a blurred image of their source. Fortunately, energetic photons offer an attractive alternative to the hadronic probes [1]. Photons interacting only weakly through the electromagnetic force with the nuclear medium are not subjected to distortions by the final state (neither Coulomb nor strong) interactions. They therefore deliver an undistorted picture of the emitting source. For hard photons, defined as γrays with energies above 30MeV in this paper, many experimental facts supported by model calculations [1,2,3] indicate that in intermediate energy heavy-ion collisions they are mainly emitted during the first instants of the reaction in incoherent proton-neutron bremsstrahlung collisions, p + n → p + n + γ, occurring within the participant zone. This part of hard photons are called as direct photon. Direct hard photons have thus been exploited to probe the pre-equilibrium conditions pre-vailing in the initial high-density phase of the reaction [4,5]. Aside from the dominant production of hard photons in first-chance p-n collisions, a significant hard-photon production in a later stage of heavy-ion reactions, called as thermal photons, are also predicted by the Boltzmann-Uehling-Uhlenbeck (BUU ) theory [6,7]. These thermal photons are emitted from a nearly thermalized source and still originate from bremsstrahlung production by individual p-n collisions, which was also confirmed by the experiments at last decade [8,9].
In this work, we take the BUU transport model improved by Bauer [10]. The isospin dependence was incorporated into the model through the initialization and the nuclear mean field. The nuclear mean field U including isospin symmetry terms is parameterized as where ρ 0 is the normal nuclear matter density; ρ, ρ n , and ρ p are the nucleon, neutron and proton densities, respectively; τ z equals 1 or -1 for neutrons and protons, respectively; The coefficients a, b and σ are parameters for nuclear equation of state. C sym is the symmetry energy strength due to the density difference of neutrons and protons in nuclear medium, which is important for asymmetry nuclear matter (here C sym = 32 MeV is used), but it is trivial for the symmetric system studied in the present work.
For the calculation of the elementary double-differential hard photon production cross sections on the basis of individual proton-neutron bremsstrahlung, the hardsphere collision was adopted from Ref. [11], and modified as in Ref. [12] to allow for energy conservation. The double differential probability is given by (2) Here R is the radius of the sphere, α is the fine structure constant, β i and β f are the initial and final velocity of the proton in the proton-neutron center of mass system, and θ γ is the angle between incident proton direction and photon emitting direction. More details for the model can be found in Ref. [10].
In this paper, we simulate the reaction of 40 Ca + 40 Ca collisions at 30MeV /nucleon, and use the EOS with the compressibility K of 235 MeV (a = -218 MeV , b = 164 MeV , σ = 4/3) for the nuclear mean field U. As a first attempt to extract the photon's azimuthal asymmetry, we only take the semi-central events (40 − 60%) as an example in this Letter.
In Fig. 1 we show the time evolution of production rate of bremsstrahlung hard photons as well as the time evolution of system densities, including both maximum (closed circles) and average density (open circles). We found that hard-photon production is sensitive to the density oscillations of both the maximum and the average density during the whole reaction evolution. When the density of collision system increases, that is in the compression stage, the system produces more hard photons. In contrary, when the system expends, the hard photon production decreases. Actually, the density oscillations of the colliding heavy ions systems can be observed in the experiments via hard-photon interferometry measurements [13,6]. Apparently, hard photons are mostly produced at the early stage of the reaction. Combining the time evolution of the nuclear density, we know that this part of hard photons are dominantly emitted from the stage of the first compression and expansion of the system. Thereafter we call these photons, emitted before the time of the first maximum expansion of the system (t = 80 f m/c in this reaction), as direct photons (on the left side of blue dashed line in Fig. 1(a)). It is also coincident with the definition of direct photons above. And we call the residual hard photons produced in the later stage as thermal photons (on the right side of blue dashed line in Fig. 1(a)). So in the simulation, we can identify the produced photon as direct or thermal photon by the emitting time. Because of the sensitivity to the density oscillations of colliding system, hard photon may be sensitive to the nuclear incompressibility [6,7].
It is well known that collective flow is an important observable in heavy ion collisions and it can bring some essential information of the nuclear matter, such as the nuclear equation of state [14,15,16,17,18,19,20,21,22,23]. Anisotropic flow is defined as the different n − th harmonic coefficient v n of the Fourier expansion for the particle invariant azimuthal distribution [15]: where φ is the azimuthal angle between the transverse momentum of the particle and the reaction plane. Note that the z-axis is defined as the direction along the beam and the impact parameter axis is labelled as x-axis. Anisotropic flows generally depend on both particle transverse momentum and rapidity, and for a given rapidity the anisotropic flows at transverse momen- where · · · denotes average over the azimuthal distribution of particles with transverse momentum p t , p x and p y are projections of particle transverse momentum in and perpendicular to the reaction plane, respectively. The first harmonic coefficient v 1 is called directed flow parameter. The second harmonic coefficient v 2 is called the elliptic flow parameter v 2 , which measures the eccentricity of the particle distribution in the momentum space.
In relativistic heavy-ion collisions azimuthal asymmetry of hard photons have been recently reported in the experiments and theoretical calculations [24,25,26,27]. It shows a very useful tool to explore the properties of hot dense matter. However, so far there is still neither experimental data nor theoretical prediction on the azimuthal asymmetry of hard photons in intermediate energy heavy ion collisions. Does the direct photon also exist azimuthal asymmetry so that it leads to non-zero directed transverse flow or elliptic asymmetry parameters in the intermediate energy range? Moreover we know that direct photons mostly originate from bremsstrahlung produced in individual proton-neutron collisions, and free nucleons are also emitted from nucleon-nucleon collisions. Does the azimuthal asymmetry of the direct photons correlate with the one of free nucleons? To answer the above question, we focus on the azimuthal asymmetry analysis for both photons and protons in this Letter. tion are emitted from a more thermalized system, they prefer more isotropic emission (i.e. the vanishing "flow" parameters) than direct ones produced in the pre-equilibrium stage. Thereafter we only consider direct photons to discuss the azimuthal asymmetry results. For protons, we take the values of flows when the system has been already in the freeze-out time at 180 f m/c.
The directed transverse flow parameter at mid-rapidity can be also defined by the slope: , where (y) c.m. is the rapidity of particles in the center of mass and p x is the mean in-plane transverse momentum of photons or protons in a given rapidity region. In Fig. 3(a) and (b), we show p x plotted versus the c.m. rapidity y c.m. for direct photons (a) as well as p x plotted versus the reduced c.m. rapidity (y/y beam ) c.m. for free protons (b). The errors shown are only statistical. A good linearity was seen in the mid-rapidity region (−0.5, 0.5) and the slope of a linear fit can be defined as the directed transverse flow parameter. The extracted value of the directed transverse flow of direct photons is about +3.7 MeV /c, and that of free protons is about −12.4 MeV /c. Thus direct photons do exist the directed transverse  asymmetry even though the absolute value is smaller than the proton's flow, and its sign is just opposite to that of free protons.
As Eq. 3 shows, elliptic flow is defined as the second harmonic coefficient v 2 of an azimuthal Fourier expansion of the particle invariant distribution. In order to extract the value of elliptic asymmetry coefficient v 2 and reduce the error of fits, we fit the azimuthal distribution to the 4th order Fourier expansion. Shown in Fig. 3(c) and (d), direct photons demonstrate out-of-plane enhancement and the v 2 is about −2.7%. Whereas, for free protons, azimuthal distribution displays the preferential in-plane emission and the v 2 is about +7.2%. Furthermore, we can extract the transverse momentum dependence of the elliptic asymmetry coefficient v 2 . Fig. 4 shows v 2 of direct photons (a) and free protons (b) as a function of transverse mo- To explain the above anti-correlation of anisotropic emission between direct photons and free protons, we should note that direct photons originate from the individual proton-neutron collisions. As Eq. 2 shows, we can roughly consider that in the individual proton-neutron center of mass system, in directions perpendicular to incident proton velocity, i.e. θ γ = π/2, the probability of hard photon production is much larger than that in the parallel direction, i.e. θ γ = 0, which is in agreement with the theoretical calculations and the experiments [29,30], that causes hard photon preferential emission perpendicular to the motion plane of corresponding nucleons. As a whole, the azimuthal anisotropy of hard photons is shifted by a phase of π/2 with respect to that associated with the anisotropy of nucleons, leading to the opposite signs of the values of F and v 2 between them. Consequently, azimuthal anisotropic emission of hard photon and free nucleon are anti-correlated, presenting the opposite behavior.
In conclusion, we have presented a first calculation of azimuthal asymmetry, both directed and elliptic asymmetry, for direct photons produced by proton-neutron bremsstrahlung from intermediate energy heavy-ion collisions. It was, for the first time, presented that in the intermediate energy heavy-ion collisions the proton-neutron bremsstrahlung hard photon shows non-zero directed transverse flow parameter and elliptic asymmetry coefficient which have opposite sign to the corresponding free proton flow parameters. The time evolutions of azimuthal parameters v 1 and v 2 of hard photons exhibit rich structures as the density oscillation of the system during the pre-equilibrium and thermalization stage of reaction system. Therefore direct photons can server for a good probe to nuclear matter properties. Considering that hard photons are dominantly produced by individual neutron-proton bremsstrahlung, so they are sensitive to the in-medium neutron-proton cross section, but not to the in-medium proton-proton or neutron-neutron cross section, that can be advantaged in the isospin dependent study of in-medium nucleon-nucleon cross section by direct photons. Of course, systematic studies of the influences from equation of state, in-medium nucleon-nucleon cross section, impact parameter and incident energy etc on the azimuthal asymmetry of direct photon should be carried out. The progress along this line is underway.