Dilepton production at next-to-leading order and intermediate invariant-mass observables

The thermal QCD dilepton production rate is calculated at next-to-leading order in the strong coupling and at finite baryon chemical potential. The two-loop virtual photon self-energy is evaluated using finite temperature field theory and combined consistently with the self-energy in the Landau-Pomeranchuk-Migdal regime. We present new results for a dense baryonic plasma. The rates are then integrated using ($3+1$)-dimensional fluid-dynamical simulations calibrated to reproduce hadronic experimental results obtained at the BNL Relativistic Heavy Ion Collider (RHIC) at energies ranging from those of the Beam Energy Scan to $\sqrt{s_{NN}}=200$ GeV. We elaborate on the ability of dileptons to relay information about the plasma baryonic content and temperature.

Figure 1

Identified hadron yields at midrapidity for 0–10% $\mathrm{Au}+\mathrm{Au}$ collisions at eight beam energies from 7.7 to 200 GeV. The markers show the STAR measurements for net protons $p\text{−}\overline{p}$ (green square), pions ${\pi }^{+}$ (red cross), and kaons $K^{+}$ (blue circle), and the lines connect modeling results shown by black dots.

Figure 2

Hydrodynamic expansion trajectories in the phase diagram within three distinct spacetime rapidity windows $(-0.5,0.5), (0.5,1.0)$, and $(1.0,1.5)$ (from left to right curves) for 20–30% $\mathrm{Au}+\mathrm{Au}$ collisions at 7.7 GeV (dashed lines) and 19.6 GeV (solid lines). The two dotted lines correspond to ${\mu }_{\mathrm{B}}/T=2$ and 4, respectively. The dot-dashed line represents the chemical freeze-out line from Ref. [66].

Figure 3

Strict two-loop contributions to the photon self-energy in QCD. (The second and third diagrams are distinct for $\mu \ne 0$.)

Figure 4

Class of diagrams that contribute in the LPM regime. Gluons have HTL resummed propagators and quarks are “hard,” i.e., propagate with their asymptotic thermal mass.

Figure 5

Transverse and longitudinal spectral functions, with $k=2\pi T$ for a fixed value of the strong coupling ${\alpha }_s=0.3$. The red curves in each figure depict ${\rho }_{\mathrm{T},\mathrm{L}}$ for zero chemical potential. The solid lines depict ${\rho }_{\mathrm{L}}$ in panel (a) and ${\rho }_{\mathrm{T}}$ in panel (b). The family of blue curves indicates the effect of varying $\mu =0,\cdots ,4T$ in steps of $\mathrm{\Delta }\mu =\frac{1}{3}T$.

Figure 6

Thermal dilepton mass spectrum from a static source, for (a) ${\mu }_{\mathrm{B}}=0$ at several temperatures, and (b) for several values of ${\mu }_{\mathrm{B}}$ at $T=300$ MeV. The solid curves show the NLO rate with a fixed coupling ${\alpha }_s=0.3$ and the dotted lines depict the rate due to the LO spectral function. In both plots, $n_f=3$.

Figure 7

Fractional contribution of the longitudinal polarization to the total differential rate ($\mathrm{d}{\mathrm{\Gamma }}_{\mathrm{total}}=2\mathrm{d}{\mathrm{\Gamma }}_{\mathrm{T}}+\mathrm{d}{\mathrm{\Gamma }}_{\mathrm{L}}$) due to a point source Eq. (34). Note that for $M\to 0$ only the two transverse degrees of freedom contribute, whereas for large $M$ the two spectral functions become equal: ${\rho }_{\mathrm{T}}\approx {\rho }_{\mathrm{L}}$.

Figure 8

Evolution of dilepton yields with four different invariant masses as functions of (a) proper time (with ${\tau }_0$ subtracted) and (b) simulation temperature for 0–10% $\mathrm{Au}+\mathrm{Au}$ collisions at $\sqrt{s_{NN}}=19.6$ GeV.

Figure 9

Dilepton invariant mass spectra for 40–50% $\mathrm{Au}+\mathrm{Au}$ collisions at (a) $\sqrt{s_{NN}}=7.7$ GeV and (b) $\sqrt{s_{NN}}=19.6$ GeV presented in three scenarios: ($i$) Spectra within a rapidity window $\left|y\right|<1$ with ${\mu }_{\mathrm{B}}$-dependent emission rates (red solid line). ($ii$) Spectra within a rapidity window $\left|y\right|<1$ with ${\mu }_{\mathrm{B}}$ manually set to zero in the rate (green dotted line). (iii) Spectra at midrapidity $y=0$ with ${\mu }_{\mathrm{B}}$-dependent emission rates (blue dot-dashed line). The bottom panels display the ratios of the spectra with the first scenario as the denominator.

Figure 10

Effective temperature $T_{\mathrm{eff}}$ read off dilepton spectrum (“dilepton $T_{\mathrm{eff}}$”) emitted from fluid cells at different temperatures (“hydro $T$”) for $\mathrm{Au}+\mathrm{Au}$ collisions at $\sqrt{s_{NN}}=19.6$ GeV within three centrality classes. The two temperatures are exactly equal on the grey dashed line.

Figure 11

Effective temperature extraction from dilepton spectrum for $\mathrm{Au}+\mathrm{Au}$ collisions at (a) $\sqrt{s_{NN}}=27$ GeV and (b) $\sqrt{s_{NN}}=200$ GeV within three centrality classes. The straight lines represent the linear fit function in the mass window $1\le M\le 3$ GeV, with their inverse slope providing the effective temperature $T_{\mathrm{eff}}$. The shaded band indicates the uncertainties associated with the fitting procedure.

Figure 12

Effective temperature $T_{\mathrm{eff}}$ derived from dilepton spectra for $\mathrm{Au}+\mathrm{Au}$ collisions at eight beam energies from $\sqrt{s_{NN}}=7.7$ GeV (leftmost markers) to 200 GeV (rightmost markers). Horizontal coordinates of the markers represent $T_{\mathrm{eff}}$ from the minimum-bias collisions (0–80%), while vertical coordinates show $T_{\mathrm{eff}}$ within smaller centrality bins (0–10%, 10–20%$,...,$ 70–80%). Error bars denote associated uncertainties, and straight lines represent linear fits applied to the markers.

Figure 13

Integrated dilepton yields in the mass range $1\le M\le 3$ GeV, normalized by midrapidity charged hadron multiplicity ($\mathrm{d}{N}_{\mathrm{ch}}/\mathrm{d}y$), for $\mathrm{Au}+\mathrm{Au}$ collisions at eight beam energies spanning from $\sqrt{s_{NN}}=7.7$ GeV (leftmost markers) to 200 GeV (rightmost markers). Results are presented within centrality classes ranging from 0–10% (uppermost curve) to 70–80% (lowermost curve), in 10% increments. The dashed line represents results within centrality class 0–80%.

Figure 14

Correlation between normalized integrated dilepton yields and QGP lifetimes in $\mathrm{Au}+\mathrm{Au}$ collisions at eight beam energies, ranging from $\sqrt{s_{NN}}=7.7$ GeV (lowest curve) to 200 GeV (highest curve), across centrality classes from 0–10% (rightmost markers) to 70–80% (leftmost markers).

Figure 15

Double differential dilepton rate from Eq. (C1), as a function of $y-{\eta }_s$, assuming longitudinal boost invariance. The curves are shown for $M=\{1,2,3\}$ GeV with $T=0.25$ GeV and ${\mu }_B=0.1$ GeV and $u_{\perp }=0.3$.