Hadronic and partonic sources of direct photons in relativistic heavy-ion collisions

The direct photon spectra and flow ($v_2, v_3$) in heavy-ion collisions at CERN Super Proton Synchrotron, BNL Relativistic Heavy Ion Collider, and CERN Large Hadron Collider energies are investigated within a relativistic transport approach incorporating both hadronic and partonic phases, the parton-hadron-string dynamics (PHSD). In the present work, four extensions are introduced compared to our previous calculations: (i) going beyond the soft-photon approximation (SPA) in the calculation of the bremsstrahlung processes $\mathrm{meson}+\mathrm{meson}\to \mathrm{meson}+\mathrm{meson}+\gamma$, (ii) quantifying the suppression owing to the Landau-Pomeranchuk-Migdal (LPM) coherence effect, (iii) adding the additional channels $V+N\to N+\gamma$ and $\mathrm{\Delta }\to N+\gamma$, and (iv) providing PHSD calculations for $\mathrm{Pb}+\mathrm{Pb}$ collisions at $\sqrt{s_{NN}}=2.76\text{TeV}$. The first issue extends the applicability of the bremsstrahlung calculations to higher photon energies to understand the relevant sources in the region $p_T=0.5–1.5\text{GeV}$, while the LPM correction turns out to be important for $p_T<0.4\text{GeV}$ in the partonic phase. The results suggest that a large elliptic flow $v_2$ of the direct photons signals a significant contribution of photons produced in interactions of secondary mesons and baryons in the late (hadronic) stage of the heavy-ion collision. To further differentiate the origin of the direct photon azimuthal asymmetry (late hadron interactions vs electromagnetic fields in the initial stage), we provide predictions for the photon spectra, elliptic flow, and triangular flow $v_3\left(p_T\right)$ of direct photons at different centralities to be tested by the experimental measurements at the LHC energies. Additionally, we illustrate the magnitude of the photon production in the partonic and hadronic phases as functions of time and local energy density. Finally, the “cocktail” method for an estimation of the background photon elliptic flow, which is widely used in the experimental works, is supported by the calculations within the PHSD transport approach.

Figure 1

Cross section of $\mathrm{pion}+\mathrm{pion}$ elastic scattering within two effective models are compared to the experimental data from Refs. [67, 68]: the exchange of two mesonic resonances, scalar $\sigma$ and vector $\rho$ (blue dashed line), and the exchange of three resonances $\sigma , \rho$, and the tensor resonance $f_2\left(1270\right)$ of the particle data booklet [63] (red solid line). The green dotted line shows the constant ${\sigma }_{\text{el}}=10$ mb for comparison.

Figure 2

Angular differential cross section for $\mathrm{pion}+\mathrm{pion}$ elastic scattering within two effective models: the exchange of two mesonic resonances, scalar $\sigma$ and vector $\rho$ (blue dashed line), and the exchange of three resonances, $\sigma , \rho$, and the tensor particle $f_2\left(1270\right)$ (red solid line with symbols and gray dashed line). The red solid line with star symbols shows the model with the full momentum dependence of the $f_2$ propagator, while the gray dashed line is obtained neglecting the momentum dependence of the $f_2$ propagator. The green dashed line shows the constant and isotropic ${\sigma }_{\text{el}}=10$ mb for comparison.

Figure 3

Feynman diagrams for photon production in the reaction $\pi +\pi \to \pi +\pi +\gamma$ in the OBE model. Time goes from left to right. For identical pions, e.g., ${\pi }^{+}+{\pi }^{+}$, the $u$-channel diagrams have to be added.

Figure 4

Cross section for the production of a photon with energy $q_0=0.005\text{GeV}$ in the process $\pi +\pi ->\pi +\pi +\gamma$ within the following models: the exact OBE cross section within the effective model taking into account scalar, vector, and tensor interactions via the exchange of $\sigma , \rho$, and $f_2\left(1270\right)$ mesons (red line with star symbols) and the SPA to this model (blue dotted line); the OBE result within the model taking into account only the scalar and vector interactions via the exchange of $\sigma$ and $\rho$ mesons (blue dashed line) and the SPA to this model (cyan dash-dot-dotted line).

Figure 5

Invariant rate of the bremsstrahlung photon production from an equilibrated pion gas at a temperature of $T=200$ MeV and pion chemical potential ${\mu }_{\pi }=0$ as calculated in the OBE model with three resonance exchange within the SPA (red dashed line). The black solid line “Haglin:2004” from Ref. [71] is shown for comparison.

Figure 6

Invariant rate of bremsstrahlung photons produced from an equilibrated pion gas at $T=150$ MeV and ${\mu }_{\pi }=40$ MeV versus the photon energy $q_0$. The inset shows the same quantity for the range of photon energies $q_0=0.1–0.4\text{GeV}$. The calculations have been performed within the following models: (1) OBE model beyond the SPA (red solid line with star symbols); (2) OBE model within the SPA (blue dotted line); (3) OBE model within the improved SPA (black short-dashed line) (the invariant energy $s_2$ of the on-shell $\pi +\pi$ elastic process is not equal to the total invariant energy of the process $s$: $s_2=s-q_0\sqrt{s}$); (4) the SPA with the constant isotropic elastic cross section of ${\sigma }_{\text{el}}=10$ using the formula (30) (green dashed line). The cyan solid line “Liu and Rapp:2007” from Ref. [65] is shown for comparison.

Figure 7

The distribution in the invariant collision energy $\sqrt{s}$ for the elastic scattering of mesons on mesons (a) and mesons on baryons (b) and for the elastic scattering of partons (c) in the course of a $\mathrm{Au}+\mathrm{Au}$ collision at $\sqrt{s_{NN}}=200\text{GeV}$ for $b=7$ fm as calculated within the PHSD approach. Here $V\equiv (\rho ,\omega ,\varphi )$; $K$ denotes all the strange mesons $K\equiv (K,\overline{K},K^{*},{\overline{K}}^{*})$ and $B$ stands for the baryons $B=(p,n,\mathrm{\Delta },\cdots )$.

Figure 8

Invariant rate of photons produced from the sQGP consisting of massive broad quasiparticle quarks and gluons (red solid line). The leading-order pQCD rate (blue dashed line) from Ref. [78] is shown for comparison.

Figure 9

Incoherent invariant photon production rate from the sQGP consisting of massive broad quasiparticle quarks and gluons (red solid line) scaled by $8{\pi }^3/\left(3T^2\right)$ to match the electric conductivity for $q_0\to 0$ [cf. Eq. (37)]. The blue dashed line and the magenta dotted line show the coherent rates with the two assumptions for the average time between the collisions $\tau$, from the DQPM model and from the AdS/CFT correspondence.

Figure 10

Comparison of the PHSD calculations to the data of the WA98 Collaboration from Ref. [96]. In comparison to the original HSD study [41], (i) the $\mathrm{meson}+\mathrm{baryon}$ bremsstrahlung (blue dash-dotted line), $\mathrm{\Delta }$ decays (black short-dashed line), and the photons from QGP (green line with round symbols) are added and (ii) the $\mathrm{meson}+\mathrm{meson}$ bremsstrahlung is now calculated beyond the SPA (magenta dashed line). The black line with diamond symbols labeled as “other” includes $\omega$-, ${\eta }^{\prime }$-, $\varphi$-, and $a_1$-meson decays and binary channels $\pi +\rho /\pi \to \pi /\rho +\gamma$ and $N+V\to N+\gamma$.

Figure 11

PHSD results for the spectrum of direct photons produced in minimum bias $\mathrm{Au}+\mathrm{Au}$ collisions at $\sqrt{s_{NN}}=200\text{GeV}$ as a function of the transverse momentum $p_T$ at midrapidity $\left|y\right|<0.5$. The data of the PHENIX Collaboration are taken from Refs. [30, 31]. For the individual lines, see the legend in the figure.

Figure 12

Contribution of the photon production in the two-to-two $\rho +\mathrm{nucleon}$ interaction (orange dashed lines) to the total direct photon spectra (red lines) at the top RHIC energy for different centralities. The dominant sources are the photons from the QGP and from the hadronic two-to-three bremsstrahlung processes. The PHENIX data are from Refs. [30, 31].

Figure 13

(a) Yield of photons with the transverse momentum $p_T=0.5\text{GeV}$ at midrapidity produced in 0%–20% most central $\mathrm{Au}+\mathrm{Au}$ collisions as functions of the approximate local “temperature” (the fourth-root of the energy density) from the PHSD from meson-meson bremsstrahlung (dash-dotted lines) and gluon Compton scattering (solid lines). (b) Same as in the top panel for photons with the transverse momentum $p_T=1.5\text{GeV}$.

Figure 14

Yield of direct photons at midrapidity in $\mathrm{Pb}+\mathrm{Pb}$ collisions at the invariant energy $\sqrt{s_{NN}}=2.76\text{TeV}$ for 0%–40% centrality within the PHSD in comparison to the preliminary data from the ALICE Collaboration [2].

Figure 15

Transverse-momentum spectra of direct photons at midrapidity in $\mathrm{Pb}+\mathrm{Pb}$ collisions at the invariant energy $\sqrt{s_{NN}}=2.76\text{TeV}$ for the three centrality bins—0%–20%, 20%–40%, and 40%–80%—as predicted within the PHSD. The black symbols show the very recently available data from the ALICE Collaboration [98]. See the legend for the individual contributions.

Figure 16

Elliptic flow $v_2$ versus transverse momentum $p_T$ for the direct photons produced in minimum bias $\mathrm{Au}+\mathrm{Au}$ collisions at $\sqrt{s_{NN}}=200\text{GeV}$ calculated within the PHSD (solid red line); the blue band reflects the uncertainty in the modeling of the cross sections for the individual channels. The data of the PHENIX Collaboration are from Ref. [1].

Figure 17

Rate of photon production in the QGP versus time for 0%–20% $\mathrm{Au}+\mathrm{Au}$ collisions at $\sqrt{s_{NN}}=200\text{GeV}$ from the PHSD. The inset shows the time evolution of the elliptic flow of photons from the partonic sources.

Figure 18

Elliptic flow $v_2$ versus transverse momentum $p_T$ for the charged particles produced in 20%–30% central $\mathrm{Pb}+\mathrm{Pb}$ collisions at $\sqrt{s_{NN}}=2.76\text{TeV}$ from the PHSD (solid red line) in comparison to the data from the ALICE Collaboration [103]; the blue band reflects the statistical uncertainty of the PHSD calculations.

Figure 19

Elliptic flow $v_2$ versus transverse momentum $p_T$ for the inclusive photons produced in 0%–40% central $\mathrm{Pb}+\mathrm{Pb}$ collisions at $\sqrt{s_{NN}}=2.76\text{TeV}$ from the PHSD (solid red line) in comparison to the data from the ALICE Collaboration [33]; the blue error band reflects the finite statistics and the uncertainty in the modeling of the cross sections for the individual channels.

Figure 20

Elliptic flow $v_2$ versus transverse momentum $p_T$ for the direct photons produced in 0%–40% central $\mathrm{Pb}+\mathrm{Pb}$ collisions at $\sqrt{s_{NN}}=2.76\text{TeV}$ from the PHSD (solid red line) in comparison to the preliminary data from the ALICE Collaboration [33]; the blue error band is dominated by the uncertainty in the modeling of the cross sections for the individual channels.

Figure 21

Centrality dependence of the direct photon $v_2$ for $\mathrm{Au}+\mathrm{Au}$ collisions at $\sqrt{s_{NN}}=200\text{GeV}$ for different centralities (see legend); the data from the PHENIX Collaboration [31, 107] are compared to the earlier PHSD predictions from Ref. [7].

Figure 22

Elliptic flow $v_2$ of direct photons from PHSD versus transverse momentum $p_T$ produced in $\mathrm{Pb}+\mathrm{Pb}$ collisions at $\sqrt{s_{NN}}=2.76\text{TeV}$ for different centrality classes (see legend).

Figure 23

Triangular flow $v_3$ versus transverse momentum $p_T$ for the direct photons produced in $\mathrm{Au}+\mathrm{Au}$ collisions at $\sqrt{s_{NN}}=200\text{GeV}$ in three centrality classes (see legends). The PHSD results are shown by the solid red lines in comparison to the data of the PHENIX Collaboration (black symbols) taken from Refs. [107, 112].

Figure 24

Triangular flow $v_3$ versus transverse momentum $p_T$ for the direct photons produced in different centrality classes for $\mathrm{Pb}+\mathrm{Pb}$ collisions at $\sqrt{s_{NN}}=2.76\text{TeV}$ from the PHSD (see legend); the blue band reflects the uncertainty in the modeling of the cross sections for the individual channels and give a measure of the present level of accuracy.