Origin and limits of intermediate-mass dilepton thermometry

Abstract: We identify the physical origin and limits of intermediate-mass dilepton thermometry in relativistic heavy-ion collisions. Using a controlled expanding-fireball framework with thermal dilepton rates, we show that the local inverse-slope parameter of the invariant-mass spectrum follows, to percent-level accuracy, the emission-weighted harmonic mean of the temperatures contributing to the spectrum. As the thermal-stage initial temperature increases, the temperatures sampled by the radiation shift upward with the overall thermal scale, causing their harmonic mean to track the initial temperature closely. This provides the physical basis for the strong inverse-slope--initial-temperature correlation, while the resulting mapping remains nonuniversal: changes that only rescale the amount of radiation leave it unchanged, whereas changes in the cooling history or source composition redistribute the radiation among different temperatures and modify the response. The mapping is nevertheless nearly linear over the temperature range studied here. Because different invariant-mass windows weight the emission history differently, source scenarios with the same inverse-slope parameter in $1<M<3~{\rm GeV}$ develop different inverse slopes in harder mass windows. Intermediate-mass dilepton spectra therefore provide a quantitative but nonuniversal probe of the early thermal history, while measurements in multiple mass windows, when confronted with realistic calculations, can provide additional constraints on the thermal evolution and early electromagnetic source content.

• The paper clarifies the physical meaning of the inverse-slope parameter T used in intermediate-mass dilepton thermometry, demonstrating it measures the harmonic mean of spatial temperatures derived from thermal contributions to the dilepton spectrum.
• The study finds that the tracking of the mean initial temperature by T comes from the near-scaling of emission temperature distributions when normalized to their initial conditions.
• The thermometer's response varies by evolving conditions, with changes in cooling or source composition affecting the fitted value.

Motivation and scope

Intermediate-mass dileptons (IMR, $1\lesssim M\lesssim 3$ GeV) are attractive thermometers for the quark-gluon plasma (QGP) because their invariant-mass spectra are largely immune to collective-flow blue shifts and their Boltzmann mass suppression preferentially selects radiation from the hottest stages of a relativistic heavy-ion collision. A prior hydrodynamic analysis reported an approximately linear relation between the fitted inverse-slope parameter $T_{}$ of the IMR spectrum and an energy- and flow-weighted initial temperature $\langle T_{\rm in}\rangle$. That correlation, however, was empirical: it remained unclear what temperature scale $T_{}$ actually measures, why it tracks $\langle T_{\rm in}\rangle$, and which aspects of the evolution or electromagnetic source content can break the mapping. The paper by Du addresses precisely these questions using a controlled expanding-fireball framework rather than a full hydrodynamic simulation (2608.16102).

Controlled fireball setup

The framework is a cylindrically symmetric, boost-invariant fireball with a Gaussian transverse profile and Bjorken-like cooling, with baseline parameters $\tau_0=1~{\rm fm}/c$ and cooling exponent $c_{\rm eff}=1/3$. The thermal-stage initial temperature is defined on the initialization plane with the same $e\gamma$ weighting used in hydrodynamic calculations. Dilepton yields are computed from state-of-the-art thermal rates integrated over space-time and momentum ($0.2<p_T<4.5$ GeV), with emission restricted to $T\geq0.18$ GeV so that only the high-temperature contribution enters; no separate hadronic rate is included. A phenomenological pre-equilibrium component extending back to $T_{}$0 is added as a Born-motivated proxy: the finite-mass thermal rate is weighted by $T_{}$1, where $T_{}$2 parametrizes quark chemical undersaturation. The author is explicit that this early-source effective temperature is a prescription variable, not an equilibrium thermodynamic temperature, and that the proxy is not a microscopic description of pre-equilibrium dynamics.

The inverse-slope parameter $T_{}$3 is extracted by fitting $T_{}$4 over a chosen window $T_{}$5, defaulting to $T_{}$6 GeV.

What the inverse slope measures

The central analytic result decomposes the local logarithmic slope of the mass spectrum into two terms. Writing the temperature-resolved spectrum as $T_{}$7 — an exact rearrangement — one obtains

$T_{}$8

The first term is the emission-weighted mean inverse temperature, whose reciprocal defines the harmonic emission temperature $T_{}$9. Numerically, $\langle T_{\rm in}\rangle$0 follows $\langle T_{\rm in}\rangle$1 to within roughly $\langle T_{\rm in}\rangle$2 across the entire IMR (RMS differences of $\langle T_{\rm in}\rangle$3–$\langle T_{\rm in}\rangle$4) in all three scenarios tested: baseline thermal evolution, slower cooling ($\langle T_{\rm in}\rangle$5), and thermal plus early source. This establishes that the fitted IMR inverse slope is the emission-weighted harmonic mean of the temperatures contributing to the spectrum, not the initial temperature and not a simple arithmetic average.

The remaining question — why $\langle T_{\rm in}\rangle$6 tracks $\langle T_{\rm in}\rangle$7 — is answered by approximate scaling: when temperatures and masses are expressed in units of $\langle T_{\rm in}\rangle$8, the normalized emission-temperature distributions $\langle T_{\rm in}\rangle$9 become substantially more similar (a $T_{}$0 reduction in the mean pairwise Wasserstein distance at matched scaled mass $T_{}$1). If the scaled distributions collapsed exactly, perfect proportionality between $T_{}$2 and $T_{}$3 would follow identically. The collapse is incomplete, however, because fixed physical scales (the $T_{}$4 GeV emission cutoff, momentum acceptance) and the details of rate and evolution reshape $T_{}$5. Residual RMS deviations of the dimensionless response at matched $T_{}$6 are $T_{}$7–$T_{}$8 across scenarios. Notably, the author concedes that no simple universal mechanism enforcing near-linearity of the thermometer relation is identified; over the studied range the linearity emerges from the full coupled response.

Nonuniversality: cooling and source composition

Because $T_{}$9 depends only on the normalized distribution $\langle T_{\rm in}\rangle$0, changes that merely rescale the total yield leave it unchanged — confirmed explicitly by varying transverse size $\langle T_{\rm in}\rangle$1 and rescaled $\langle T_{\rm in}\rangle$2, which strongly alter the yield but not $\langle T_{\rm in}\rangle$3. This provides a clean interpretation of part of the robustness observed across centralities and beam energies in hydrodynamic studies.

Changes that redistribute radiation among temperatures do modify the thermometer. Quantitatively:

Scenario	Response coefficient $\langle T_{\rm in}\rangle$4	$\langle T_{\rm in}\rangle$5 at pivot
Baseline thermal	$\langle T_{\rm in}\rangle$6	$\langle T_{\rm in}\rangle$7 MeV
Slower cooling ($\langle T_{\rm in}\rangle$8)	$\langle T_{\rm in}\rangle$9	$\tau_0=1~{\rm fm}/c$0 MeV
Thermal + early source	$\tau_0=1~{\rm fm}/c$1	$\tau_0=1~{\rm fm}/c$2 MeV

Slower cooling shifts weight toward later, cooler emission and increasingly so at high $\tau_0=1~{\rm fm}/c$3, reducing $\tau_0=1~{\rm fm}/c$4. The chemically undersaturated early source raises $\tau_0=1~{\rm fm}/c$5 most strongly at low $\tau_0=1~{\rm fm}/c$6 (its window fraction falls from $\tau_0=1~{\rm fm}/c$7 to $\tau_0=1~{\rm fm}/c$8 along the scan), also flattening the relation. The author stresses that the direction and magnitude of this early-source effect are not universal properties of pre-equilibrium radiation but depend on how the early source evolves relative to the thermal medium.

Mass-window leverage

Since harder invariant-mass windows suppress cooler radiation more selectively, raising the lower boundary $\tau_0=1~{\rm fm}/c$9 strengthens the thermal response: $c_{\rm eff}=1/3$0 rises from $c_{\rm eff}=1/3$1 to $c_{\rm eff}=1/3$2, $c_{\rm eff}=1/3$3, and $c_{\rm eff}=1/3$4 as $c_{\rm eff}=1/3$5 moves to $c_{\rm eff}=1/3$6, $c_{\rm eff}=1/3$7, and $c_{\rm eff}=1/3$8 GeV. With the harder early source present, however, $c_{\rm eff}=1/3$9 rises much more weakly ($e\gamma$0), because the increasing relative weight of the early component counteracts the stronger thermal selectivity.

This produces a concrete degeneracy-breaking result. Two systems can share the identical broad-window value $e\gamma$1 MeV in $e\gamma$2 GeV while implying very different initial temperatures: a purely thermal system requires $e\gamma$3 MeV, whereas the mixed system achieves the same slope with only $e\gamma$4 MeV. In harder windows the degeneracy dissolves — the extracted slopes differ by $e\gamma$5 MeV in $e\gamma$6 GeV and $e\gamma$7 MeV in $e\gamma$8 GeV, tracking the rise of the early fraction from $e\gamma$9 to $0.2<p_T<4.5$0 and $0.2<p_T<4.5$1. Multi-window measurements confronted with realistic source calculations therefore carry genuine discriminating power beyond a single fitted slope.

Limitations and open questions

The paper is candid about its scope. The reduced fireball neglects transverse flow (justified by the flow-insensitivity of fully momentum-integrated mass spectra apart from residual finite-$0.2<p_T<4.5$2 effects) and uses a fixed transverse width, omitting an expansion timescale that could reshape $0.2<p_T<4.5$3. The early source is a Born-motivated fugacity-weighted proxy rather than a nonequilibrium calculation, so its inferred effect on the thermometer mapping should not be read as a prediction about realistic pre-equilibrium radiation. The near-linearity of the $0.2<p_T<4.5$4–$0.2<p_T<4.5$5 relation remains unexplained by any simple mechanism and may not persist outside the scanned range or in more complete dynamical treatments. Quantitative phenomenology will additionally require realistic heavy-flavor background treatment, and the extension across the Beam Energy Scan toward lower collision energies — with explicit nuclear-overlap heating and chemical equilibration — remains untested within this framework.

Conclusion

This work converts an empirical IMR thermometer into a physically interpreted one. The spectral inverse slope measures, to percent-level accuracy, the harmonic mean of emission temperatures; its strong tracking of $0.2<p_T<4.5$6 arises from approximate temperature rescaling of the emission distribution; and its nonuniversality is traced quantitatively to cooling history and source composition. The demonstrated degeneracy-breaking power of multiple mass windows gives experimental guidance: single-window extractions cannot disentangle thermal temperature from early-source hardness, but multi-window measurements constrained by realistic calculations can.