Hybrid model calculations of direct photons in high-energy nuclear collisions

Direct photon emission in heavy-ion collisions is calculated within a relativistic micro $+$ macro hybrid model and compared to the microscopic transport model, the Ultra-relativistic Quantum Molecular Dynamics model. In the hybrid approach, the high-density part of the evolution is replaced by an ideal three-dimensional hydrodynamic calculation. This allows the effects of viscosity and full local thermalization, in comparison with the transport model of the ideal fluid dynamics, to be examined. The origin of high-$p_{\perp }$ photons as well as the impact of elementary high-$\sqrt{s}$ collisions is studied. The contribution of different production channels and nonthermal radiation to the spectrum of direct photons is further explored. Detailed comparisons to the measurements by the WA98 Collaboration are also undertaken.

Figure 1

UrQMD cross sections for ${\pi }^{+}{\pi }^{-}$ collisions as a function of center-of-mass energy. The resonant hadronic cross section [(red) dashed line], the cross section for the formation of strings [(blue) dotted line], and the cross section for hard scatterings via PYTHIA [(green) dash-dotted line] are shown. The peak at the $\rho$-meson pole mass has been cut out for better visibility.

Figure 2

Charged-particle spectra from proton-proton collisions at $E_{\mathrm{lab}}=158$ GeV calculated with UrQMD with (solid black line) and without [dashed (red) line] PYTHIA.

Figure 3

Temperature profiles after switching to fluid-dynamical description, calculated with the bag model (left) and hadron gas EOS (right). Along the lines, the temperature is constant, going from $T=50$ MeV at the outermost line to $T=250$ MeV at the innermost. Calculations have been done for Pb $+$ Pb collisions at $E_{\mathrm{lab}}=158A$ GeV with $b=0$ fm.

Figure 4

Cross sections for all included channels as a function of $\sqrt{s}$. For visibility, the cross sections for all processes $\pi \rho \to \gamma \pi$ are shown separately. They have been calculated with a $\rho$-mass $m_{\rho }=0.769$ GeV. The left plot shows the cross sections for $\pi \pi \to \gamma \rho$ both for fixed $\rho$-mass ($m_{\rho }=0.769$ GeV, labeled “$m_{\rho }$ fixed”) and for variable $\rho$-mass (labeled “$m_{\rho }$ Breit-Wigner”).

Figure 5

Comparisons between the rates from Ref. [57] (lines) and box calculations restricted to a $\pi ,\rho ,a_1$-system (points) with UrQMD at $T=150$ MeV.

Figure 6

UrQMD calculation. Contributions of the initial [$t<1.4$ fm, dash-dotted (green) line], intermediate [$1.4<t<13.25$ fm, dotted (blue) line], and final [$t>13.25$ fm, dashed (red) line] stages to the spectrum from pure cascade calculations (solid black line).

Figure 7

Hybrid model calculation with HG-EOS. Contributions of the initial [dash-dotted (green) line], intermediate [dotted (blue) line], and final [dashed (red) line] stages to the inclusive spectrum (solid black line). In the intermediate stage, the matter is described by hydrodynamics.

Figure 8

Hybrid model calculation with BM-EOS. Contributions to the initial [(green) dash-dotted line], intermediate [(blue) dotted line], and final [(red) dashed line] stages to the inclusive spectrum (solid black line). In the intermediate stage, the matter is described by hydrodynamics.

Figure 9

UrQMD calculation. Contributions of the different channels.

Figure 10

Hybrid-model calculation with HG-EOS. Contributions of the different channels.

Figure 11

Hybrid-model calculation with BM-EOS. Contributions of the different channels in hybrid calculation with BM-EOS.

Figure 12

Comparison of direct photon spectra from the hybrid model with HG-EOS and transport model, using only common channels.

Figure 13

Comparison of the direct photon spectra from all variations of the model to the experimental data of the WA98 Collaboration [8]. Calculations without intermediate hydrodynamic stage (pure cascade mode) are shown as (red) crosses, hybrid calculations with HG-EOS as a (blue) solid line, and BM calculations as the (dark green) dotted line. The data contain no photons from initial proton-proton collisions.

Figure 14

Comparison of the direct photon spectra from the model, with added pQCD photons from Ref. [57], to the experimental data by the WA98 Collaboration [8]. Calculations without intermediate hydrodynamic stage (pure cascade mode) are shown as (red) crosses, hybrid calculations with HG-EOS as a (blue) solid line, and BM calculations as a (dark green) dotted line.

Figure 15

UrQMD calculation. The $p_{\perp }$ spectra for all photons (solid black line), photons from boosted collisions [dotted (blue) line], and collisions with high center-of-mass energy [dashed (red) line]. The shaded gray area indicates the range of the $p_{\perp }$ region used for the solid black lines in Figs. 16 and 17.

Figure 16

UrQMD calculation. The number of photons as a function of the transverse center-of-mass momentum of the elementary collision, $q^{\mathrm{coll}}$: all photons [(blue) dotted line], photons that have high transverse momentum (black solid line), and photons from collisions with high center-of-mass energy [(red) dashed line]. The (light blue) shaded area indicates the range of the $q_{\perp }$ region used for the dotted (blue) lines in Figs. 15 and 17.

Figure 17

UrQMD calculation. The number of photons as a function of the center-of-mass energy of the elementary collision: all photons [(red) dashed line], photons that have high transverse momentum (black solid line), and photons from collisions with high transverse center-of-mass momentum. The (light red) shaded area shows the range of the $\sqrt{s}$ region used for the (red) dashed lines in Figs. 15 and 16.

Figure 18

UrQMD calculation. Average photon emission times as a function of the transverse momentum for all photons, $\pi \rho \to \gamma \pi$, $\pi \pi \to \gamma \rho$, and other processes [black solid, (red) dashed, (blue) dotted, and (green) dash-dotted lines, respectively]. The shaded areas correspond to the $p_{\perp }$ regions used for the curves shown in Fig. 19.

Figure 19

UrQMD calculation. Emission time distribution of photons from $\pi \rho$ scatterings for the different photon transverse-momentum regions indicated in Fig. 18. The vertical lines indicate the average emission time in the corresponding $p_{\perp }$ bin.

Figure 20

UrQMD calculation. Total yields of photons with (dashed line) and without (solid line) the contributions from colliding string ends.

Figure 21

UrQMD calculation. Photon spectra from collisions that produce a $\rho$-meson in the final state for the production of $\rho$-mesons at its pole mass (solid and dashed lines) and for the production of $\rho$-mesons according to a Breit-Wigner mass distribution (dotted and dash-dotted lines).

Figure 22

Comparison of the current model [(red) dashed line] to calculations from Dumitru et al. (Ref. [46], black solid line). For further comparision, we apply our model to an earlier UrQMD version [(blue) dotted line].