Coarse-graining approach for dilepton production at energies available at the CERN Super Proton Synchrotron

Coarse-grained output from transport calculations is used to determine thermal dilepton emission rates by applying medium-modified spectral functions from thermal quantum field theoretical models. By averaging over an ensemble of events generated with the Ultrarelativistic Quantum Molecular Dynamics (UrQMD) transport model, we extract the local thermodynamic properties at each time step of the calculation. With an equation of state the temperature $T$ and chemical potential ${\mu }_{\mathrm{B}}$ can be determined. The approach goes beyond simplified fireball models of the bulk-medium evolution by treating the full (3+1)-dimensional expansion of the system with realistic time and density profiles. For the calculation of thermal dilepton rates we use the in-medium spectral function of the $\rho$ meson developed by Rapp and Wambach and consider thermal quark-gluon plasma (QGP) and multipion contributions as well. The approach is applied to the evaluation of dimuon production in In+In collisions at top CERN Super Proton Synchrotron (SPS) energy. Comparison to the experimental results of the NA60 experiment shows good agreement of this ansatz. We find that the experimentally observed low-mass dilepton excess in the mass region from 0.2 to 0.6 GeV can be explained by a broadening of the $\rho$ spectral function with a small mass shift. In contrast, the intermediate-mass region $(M>1.5$ GeV) is dominated by a contribution from the quark-gluon plasma. These findings agree with previous calculations with fireball parametrizations. This agreement, in spite of differences in the reaction dynamics between both approaches, indicates that the time-integrated dilepton spectra are not very sensitive to details of the space-time evolution of the collision.

Figure 1

Comparison of the two equations of state used in our model: the hadron gas EoS [51] with UrQMD-like degrees of freedom (black line) and the lattice EoS [55] to model the deconfined phase (blue line). We show the temperature dependence on the energy density (in units of normal nuclear-matter density, ${\varepsilon }_0)$. While for the lattice EoS ${\mu }_{\mathrm{B}}=0$ intrinsically, we set the chemical potential to zero as well for the hadron gas EoS for reasons of comparison.

Figure 2

Longitudinal and transverse pressure for the central cell, as well as the anisotropy parameter $x={(P_{\parallel }/P_{\perp })}^{3/4}$ and the relaxation function $r\left(x\right)$ as defined in Eq. (14).

Figure 3

Transverse (a) and longitudinal (b) profiles of the energy density $\varepsilon$ and the baryon density ${\rho }_{\mathrm{B}}$. The left plot shows the dependence of $\varepsilon$ and ${\rho }_{\mathrm{B}}$ on the position along the $x$ axis, with $y$ and $z$ coordinates fixed to 0. The right plot shows the dependence along the $z$ axis (i.e., along the beam axis) with $x=y=0$. The results are presented in units of the normal nuclear-matter densities, ${\varepsilon }_0$ and ${\rho }_0$.

Figure 4

Left panel (a): Time evolution of the baryon density ${\rho }_{\mathrm{B}}$ (short dashed) and energy density $\varepsilon$ (long dashed) for the cell at the center of the coarse-graining grid $\left(x=y=z=0\right)$. The results are given in units of the ground-state densities ${\varepsilon }_0$ and ${\rho }_0$. Right panel (b): Time evolution of the temperature $T$ (red dash-dotted), baryon chemical potential ${\mu }_{\mathrm{B}}$ (green short dashed) and the pion chemical potential ${\mu }_{\pi }$ (blue dotted) in the central cell. The thin grey line indicates the transition from the kattice EoS to the hadron gas EoS at the transition temperature of $T=170\text{MeV}$.

Figure 5

Invariant mass spectra of the dimuon excess yield in In+In collisions at a beam energy of $158A\text{GeV}$, for the low-mass region up 1.5 GeV (a) and the intermediate-mass regime up to $2.8\text{GeV}$ (b). We show the contributions of the in-medium $\rho$ emission according to the Rapp-Wambach spectral function [56] (blue short dashed), the contribution from the quark-gluon plasma, i.e., $q\overline{q}$ annihilation, according to lattice rates [32, 67] (green dashed) and the emission from multipion reactions, taking vector–axial-vector mixing into account [15] (orange dash-dotted). Additionally a nonthermal transport contribution for the $\rho$ is included in the yield (dark blue dash-dotted). Only left plot: For comparison the thermal $\rho$ without any baryonic effects, i.e., for ${\rho }_{\text{eff}}=0$, is shown (violet dash-double-dotted) together with the yield from pure perturbative $q\overline{q}$-annihilation rates (green dotted). The results are compared to the experimental data from the NA60 Collaboration [11, 12, 71].

Figure 6

Transverse-mass $\left(m_t-M\right)$ spectra in four different mass bins of the dimuon-excess yield in In+In collisions at a beam energy of $158A\text{GeV}$. We have the mass ranges $0.2<M<0.4\text{GeV}$ (a), $0.4<M<0.6\text{GeV}$ (b), $0.6<M<0.9\text{GeV}$ (c), and the highest mass bin with $1.0<M<1.4\text{GeV}$ (d). The different contributions are the same as in Fig. 5. The results are compared to the experimental data from the NA60 Collaboration [72].

Figure 7

The invariant mass spectra of the dimuon excess yield in In+In collisions at a beam energy of $158A\text{GeV}$, for the low-mass region as in the left plot of Fig. 5, but for 6 different $p_t$ bins ranging from $p_t=0$ to $1.2\text{GeV}$. The different contributions are the same as in Figs. 5 and 6. The results are compared to the experimental data from the NA60 Collaboration [11].

Figure 8

Same as in Fig. 7, but for higher $p_t$ bins ranging from $p_t=1.2$ to $2.4\text{GeV}$. The different contributions are the same as in Figs. 5 and 6. The results are compared to the experimental data from the NA60 Collaboration [11].