Nonequilibrium photon production in partonic transport simulations

We discuss the implementation of leading-order photon production in nonequilibrium partonic transport simulations. In this framework photons are produced by microscopic scatterings, where we include the exact matrix elements of Compton scattering, quark-antiquark annihilation, and bremsstrahlung processes. We show how the hard-thermal loop inspired screening of propagators has to be modified such that the microscopic production rate agrees well with the analytically known resummed leading-order rate. We model the complete quark-gluon plasma phase of heavy-ion collisions by using the partonic transport approach called the Boltzmann approach to multiparton scatterings (BAMPS), which solves the ultrarelativistic Boltzmann equation with Monte Carlo methods. We show photon spectra and elliptic flow of photons from BAMPS and discuss nonequilibrium effects. Due to the slow quark chemical equilibration in BAMPS, the yield is lower than the results from other groups; in turn we see a strong effect from scatterings of energetic jet-like partons with the medium. This nonequilibrium photon production can dominate the thermal emission, such that the spectra are harder and the photonic elliptic flow of the quark-gluon plasma becomes negative.

Figure 1

The photon rate from Refs. [65, 66] compared with the rate obtained from the numerical solution of Eq. (B1) with matrix elements (8) and (7), using a Debye mass $\kappa m_{D,q}^2$. In the left panel the $\kappa$ is fixed. In right panel we keep $\kappa =2.45$ and fix the parameter $C_{\text{stat}}$ by integrating the Born matrix element with Boltzmann statistics. (a) The parameter $\kappa =2.45$ is tuned to make both integrated rates equal. (b) The Born matrix element integrated with Boltzmann statistics (green dotted line). Reducing this rate by $C_{\text{stat}}=0.84$ (orange dashed line), the total rate $R$ equals the Born rate with quantum statistics [which equals approximately the elastic HTL improved rate; see panel (a)].

Figure 2

For two values of $\kappa$ we compare the numerically obtained $2\leftrightarrow 2$ photon rate to the analytic expectation (obtained by using the method from Ref. [71]).

Figure 3

The two contributing vacuum diagrams we use for the numerical evaluation of the radiative photon rate. All internal propagators are screened by hand, and an overall factor $K_{\text{inel}}$ ensures the similarity to the AMY rate, as explained in Sec. 3b2. (a) $i{\mathcal{M}}_a$, (b) $i{\mathcal{M}}_b$.

Figure 4

An example of the photon self-energy and its cuts to obtain scattering matrix elements along the method from Ref. [73]. In diagram (a) the dashed line represents one possible cut, and the closed loops must be opened (“tic-ed”) to get a scattering matrix element. In (b) the dashed and double dashed lines are possible, topologically different cuts, generating Bremsstrahlung with on-shell gluons of the medium.

Figure 5

Here we show a sketch of an LPM interference effect: Due to a short quark mean-free path a subsequent radiation is suppressed. Note that this diagram is not used as shown here, we rather evaluate the quark-quark elastic mean-free path dynamically in BAMPS and compare it to the formation time of the photon. The photon is produced by the pure bremsstrahlung subdiagram.

Figure 6

Numerical results for the specific mean-free path for a quark corresponding to different reactions, depending on temperature and fugacity.

Figure 7

(a) The photon rate for bremsstrahlung and (b) the rate weighted by the photon energy from BAMPS compared to the full inelastic AMY result. The integral of the rate within $0.15<E/T<10$ is equal without a $K$ factor. (a) The equilibrium radiative photon rate. (b) The equilibrium radiative photon rate weighted by photon energy.

Figure 8

Dependence of the photon rate $R$ on the quark fugacity ${\lambda }_q$, and the comparison to the naive expectations $R\sim {\lambda }_q^{1,2}$ (solid, dotted line). The bremsstrahlung rate shows roughly a $R\sim {\lambda }_q^{1.36}$ dependence (dashed line, fit), whereas the $2\leftrightarrow 2$ photon production processes show a behavior $R\sim {\lambda }_q^{1.07}$.

Figure 9

Quark fugacity ${\lambda }_q$ scaling of Debye mass $m_D$, quark density $n$, average cross section at $T=0.4$ GeV, ${\sigma }_i\equiv 〈v_{\mathrm{rel}}{\sigma }_{\mathrm{tot},\mathrm{i}}\left(s\right)〉$ (where $i=1,2,3$ corresponds to the three different specific processes considered), and the specific inverse rate ${\lambda }_{\mathrm{mfp}}$.

Figure 10

Results for the qualitative understanding of elliptic flow of photons originating from flowing thermal background and nonthermal “jet”-like partons for the five scenarios explained in Sec. 4a. The thermal medium has a temperature of 0.4 GeV and, for simplicity, photons originate from $2\leftrightarrow 2$ processes only.

Figure 11

Spectra of photons which are produced by an incident jet particle with $E_{\text{jet}}=15$ GeV hitting a thermal bath. We show $2\leftrightarrow 2$ and $2\to 3$ contributions separately.

Figure 12

The $p_T$ spectrum of direct photons from the QGP phase of $\text{Au}+\text{Au}$ collisions at $\sqrt{s}=200$ GeV for $20\%–40\%$ most-central collisions. We show the elastic (magenta dotted) and inelastic (greed dashed) contribution from BAMPS as well as their sum (red solid) in comparison with a recent hydro result (yellow double dashed) from Ref. [25] and a result from off-shell transport PHSD [86].

Figure 13

Same as Fig. 12, but we switch on a running coupling for photon production (green dashed line).

Figure 14

The average quark and gluon fugacities over time in BAMPS for RHIC collisions at $\sqrt{(}s)=200$ GeV at $20\%–40\%$ centrality. Shown is the average over only the central cell and a tube of transverse radius 1.5 fm extending in spacetime rapidity $-0.5<{\eta }_s<0.5$.

Figure 15

Same as Fig. 14, but for artificially increased quark-antiquark production cross section ${\sigma }_{gg\to q\overline{q}}=K{\sigma }_{gg\to q\overline{q}}$, where $K=10$ (left panel) and $K=100$ (right panel). The photon spectra are mostly sensitive to the early phase, where the notion of fugacity (or temperature) can only be effective.

Figure 16

Same as Fig. 12, but here we change the chemical equilibration during the evolution by artificially increasing the $gg\to q\overline{q}$ cross section by a factor of 10 (magenta dotted line) and 100 (green dashed line).

Figure 17

The elliptic flow of photons from BAMPS for $\text{Au}+\text{Au}$ collisions at $\sqrt{s_{NN}}=200$ GeV and $20\%–40\%$ centrality. Shown are the sum and the elastic and inelastic contribution separately.

Figure 18

The exact photon bremsstrahlung matrix element is used to sample photons. Their momentum is given in $k_{\perp },q_{\perp },y,\varphi$ space; here we show several realizations (red dots) as an example. The green dashed curves represent the limits. The purple and blue dash-dotted lines show the limit from the LPM constraint for larger mean-free paths. The asymmetry in $y$ is forced by using only one fixed quark as the radiating one.

Figure 19

The differential cross section in the rapidity of the radiated photon for various mean-free paths.

Figure 20

The differential cross section in the transverse momentum of the radiated photon for various mean-free paths.