Low-$p_{T}$ direct-photon production in Au$+$Au collisions at $\sqrt{s_{{}_{NN}}}=39$ and 62.4 GeV

Figure 1 presents the direct-photon $p_{T}$ spectra measured by PHENIX in Au$+$Au collisions in the 0%–20% centrality bin at $\sqrt{s_{{}_{NN}}}$ = 200 GeV, including data points from an analysis based on external conversions [35], internal conversions [34], and from calorimeter measurements [41]. Also shown are invariant yields of direct photons in $p$$+$$p$ collisions at 200 GeV from internal conversions [42, 34], calorimeter measurements [43, 44], a fit to the combined set of $p$$+$$p$ data, extrapolated below 1 GeV/$c$ [40, 45, 46, 47], and an $N_{\rm coll}$-scaled $p$$+$$p$ fit with $N_{\rm coll}=779.0$ [35].

The three techniques used for measuring direct photons deploy different detector systems within the PHENIX central arms³³3The PHENIX central arm acceptance is 0.7 units around midrapidity. Thus there is little difference between momentum and transverse momentum, so the terms will be used interchangeably in the following discussion. (see Ref. [48]) and various strategies to extract the direct photons from the decay-photon background include measuring:

• (i)

  photons that directly deposit energy into electromagnetic calorimeters. This is the method of choice to measure high momentum photons. At $p_{T}$ below a few GeV/$c$, the method suffers from significant background contamination from hadrons depositing energy in the calorimeter and the limited energy resolution [41].

• (ii)

  virtual photons that internally convert into $e^{+}e^{-}$ pairs and extrapolating their measured yield to zero mass. This technique was used for the original discovery of low-momentum direct photons at RHIC [34]. The pairs are measured in the mass region above the $\pi^{0}$ mass, which eliminates more than 90% of the hadron-decay-photon background. The extrapolation to zero mass requires the pair mass to be much smaller than the pair momentum, and thus limits the measurement to $p_{T}$ $>1$ GeV/$c$.

• (iii)

  photons that convert to $e^{+}e^{-}$ pairs in the detector material (”external conversion method”). This method gives access to a nearly background-free sample of photons down to $p_{T}$ below 1 GeV/$c$ [35].

The external-conversion method is used for the analysis presented here, which is the identical method used to analyze direct-photon production from 2010 Au$+$Au collisions at $\sqrt{s_{{}_{NN}}}$ = 200 GeV [35, 37]. Additional details can be found in Ref. [49]. The analysis proceeds in multiple steps. First established is $N_{\gamma}^{\rm incl}$, which is a sample of conversion photons measured in the PHENIX-detector acceptance. This is done in bins of conversion photon $p_{T}$. For a given $p_{T}$ selection, the $N_{\gamma}^{\rm incl}$ sample relates to the true number of photons $\gamma^{\rm incl}$ in that $p_{T}$ range as follows:

If direct photons are emitted from the collision system in a particular $p_{T}$ range, $R_{\gamma}$ will be larger than unity. The denominator in Eq. 4 can be obtained from the PHENIX hadron-decay generator exodus [50], based on the measured $\pi^{0}$ spectra. In the following sections, the determination of $N_{\gamma}^{\rm incl}$, $N_{\gamma}^{\pi^{0},\rm tag}$, $\langle\varepsilon_{\gamma}f\rangle$, and $\gamma^{\rm hadr}$/$\gamma^{\mbox{$\pi^{0}$}}$ will be discussed separately.

The 2010 data samples of $7.79\times 10^{7}$ (at 39 GeV) and $2.12{\times}10^{8}$ (at 62.4 GeV) MB Au$+$Au collisions were recorded with the two PHENIX central-arm spectrometers, each of which has an acceptance of $\pi/2$ in azimuthal angle and $|\eta|<$ 0.35 in pseudorapidity. For both collision energies, the MB data sets cover a range of 0%–86% of the interaction cross section. The data sample for 62.4 GeV is large enough so that two centrality classes (0%–20% central collisions, 20%–40% midcentral collisions) could be analyzed separately. The event centrality is categorized by the charge measured in the PHENIX beam-beam counters [51], which are located at a distance of 144 cm from the nominal interaction point in both beam directions, covering the pseudorapidity range of $3.1<|\eta|<3.9$ and $2\pi$ in azimuth.

The PHENIX central-arm drift chambers and pad chambers [52], located from 200 to 250 cm radially to the beam axis, are used to determine the trajectories and momenta of charged particles. The momenta are measured assuming the track originated at the event vertex (vtx) and traversed the full magnetic field. The tracks are identified as electrons or positrons by a combination of a minimum signal in the ring-imaging Čerenkov (RICH) detector [53] and a match of the track momentum with the energy measured in the electromagnetic calorimeter (EMCal) [54]. The RICH cut requires that a minimum of three RICH phototubes be matched to the charged track within a radius interval of 3.4 cm $<r<$ 8.4 cm at the expected ring location. For each electron candidate an associated energy measurement in the EMCal is required, with an energy/momentum ratio, E/$p$, greater than 0.5. Electrons and positrons are combined to $e^{+}e^{-}$ pairs and further selection cuts are applied to establish a clean sample of photon conversions. Most photon conversions occur in the readout boards and electronics at the back plane of the hadron blind detector (HBD) [55], located at a radius of $\approx$60 cm from the nominal beam axis. The relative thickness in terms of radiation length is equal to $X/X_{0}\approx 2.5\%$; all other material between the beam axis and the drift chamber is significantly thinner. Electrons and positrons from these conversions do not traverse the full magnetic field⁴⁴4A special field configuration was used in 2010 for the operation of the HBD. In this configuration there is a nearly field free region around the beam axis out to 60 cm. Thus the field integral missed by tracks from photon conversions in the HBD back plane is rather small.. Projecting the tracks back to the interaction point results in a small distortion of the reconstructed momenta, both in magnitude and in direction, which in turn results in an artificial opening angle of the $e^{+}e^{-}$ pair. This gives the pair an apparent mass ($M_{\rm vtx}$), which depends monotonically on the radial location of the conversion point and is approximately 0.0125 GeV/$c^{2}$ for conversions in the HBD back plane.

To select photon conversions in the HBD back plane, the track momenta are re-evaluated assuming the tracks originated at the HBD back plane. For $e^{+}e^{-}$ pair from conversions in the HBD back plane, a mass ($M_{\rm HBD}$) of below 0.005 GeV/$c^{2}$ is calculated with a distribution expected for an $e^{+}e^{-}$ pair of zero mass measured with the PHENIX-detector resolution. Figure 2 shows the correlation between the two different masses calculated for each pair. Photon conversions in the HBD back plane are clearly separated from $e^{+}e^{-}$ pairs from $\pi^{0}$ Dalitz decays, $\pi^{0}\rightarrow\gamma e^{+}e^{-}$, which populate a region $\mbox{$M_{\rm vtx}$}<0.005\ \mbox{GeV/$c^{2}$}$ and $M_{\rm HBD}$ around 0.012 GeV/$c^{2}$. The region between the $e^{+}e^{-}$ pairs from Dalitz decays and conversion in the HBD back plane is populated by conversions at radii smaller than 60 cm. To select a clean sample of photon conversions in the HBD back plane, $N_{\gamma}^{\rm incl}$, a two dimensional cut is applied: $\mbox{$M_{\rm HBD}$}<0.0045$  GeV/$c^{2}$ and $0.01<\mbox{$M_{\rm vtx}$}<0.015$ GeV/$c^{2}$. The purity of this photon sample was determined with a full Monte-Carlo simulation and is better than 99%. The sample sizes are $9.42\times 10^{4}$ and $3.28\times 10^{5}$, for 39 and 62.4 GeV, respectively.

Once the conversion-photon sample $N_{\gamma}^{\rm incl}$ is established, all $e^{+}e^{-}$ pairs in a given $p_{T}$ bin are combined with showers reconstructed in the EMCal in the same event and then the invariant mass is calculated. A minimum-energy cut of 0.4 GeV is applied to remove charged particles that leave a minimum-ionizing signal in the EMCal and further reduce the hadron contamination by applying a shower-shape cut. Figure 4(a) shows one example of the resulting mass distributions for a $p_{T}$ bin around 1 GeV/$c$ from the 62.4-GeV MB data set. The $\pi^{0}$ peak is clearly visible above a combinatorial background, which results from combining $e^{+}e^{-}$ pairs with all showers in the event, most of which are not correlated with the $e^{+}e^{-}$ pair.

A mixed-event technique is used to determine and subtract the mass distribution of these random combinations. In event mixing, all $e^{+}e^{-}$ pairs in a given event are combined with the EMCal showers from several other events. These other events are chosen to be in the same 10% centrality selection and within 1 cm of the interaction point of the event with the $e^{+}e^{-}$ pair. The ratio of the measured (foreground) mass distribution and mixed event (background) mass distribution is fitted with a 2nd-order polynomial, excluding the mass range $0.08<m_{ee\gamma}<0.19$ GeV/$c^{2}$, around the $\pi^{0}$ peak. Figure 4(b) shows the ratio and the fit, which is used to normalize the mixed event background distribution over the full mass range; the result is included in Fig. 4(b).

Figure 4(c) depicts the counts remaining after the mixed event background distribution is subtracted from the foreground distribution. The raw yield of tagged $\pi^{0}$ is calculated as the sum of all counts in mass window $0.11<m_{ee\gamma}<0.165\ \mbox{GeV/$c^{2}$}$. The counts in two side bands around the $\pi^{0}$ peak are evaluated to account for any possible remaining mismatch of the shape of the combinatorial background from mixed events compared to the true shape. These side bands are $0.035<m_{ee\gamma}<0.110$ GeV/$c^{2}$ and $0.165<m_{ee\gamma}<0.240$ GeV/$c^{2}$. The average counts per mass bin in the side-bands is subtracted from the raw tagged $\pi^{0}$ counts, the resulting counts are the number of tagged $\pi^{0}$, $N_{\gamma}^{\pi^{0},\rm tag}$ in the given $p_{T}$ bin.

Figure 4 shows both $N_{\gamma}^{\rm incl}$ and $N_{\gamma}^{\pi^{0},\rm tag}$ for 39 and 62.4 GeV MB Au$+$Au data. Figure 6 gives the ratios, $N_{\gamma}^{\rm incl}$/$N_{\gamma}^{\pi^{0},\rm tag}$.

The systematic uncertainties of the peak-extraction procedure were evaluated by choosing different-order polynomial function for the normalization and the various mass windows were varied in the procedure. It is found that $N_{\gamma}^{\pi^{0},\rm tag}$ changes by less than 8% and 5% for 39 and 62.4 GeV data, respectively. These systematic uncertainties are mostly uncorrelated between $p_{T}$ bins and thus are added in quadrature to the statistical uncertainties on $N_{\gamma}^{\pi^{0},\rm tag}$.

The conditional probability $\langle\varepsilon_{\gamma}f\rangle$, to tag an $e^{+}e^{-}$ pair that resulted from a conversion of a $\pi^{0}$ decay photon with the second decay photon, depends on the parent $\pi^{0}$ $p_{T}$ spectrum, the $\pi^{0}$ decay kinematics, the detector acceptance, and, the photon reconstruction efficiency. A Monte-Carlo method is used to calculate $\langle\varepsilon_{\gamma}f\rangle$. The method was developed for the direct-photon measurement from Au$+$Au collisions at $\sqrt{s_{{}_{NN}}}$ = 200 GeV, also recorded during 2010, as described in Ref. [35]. The calculation is done separately for MB and centrality selected Au$+$Au collisions at 39 and 62.4 GeV. Each calculation is based on an input $\pi^{0}$ spectrum that was measured for the same data sample [56].

Figure 6 shows the results for MB collisions. The conditional probability $\langle\varepsilon_{\gamma}f\rangle$ is small; it increases from approximately 10% to 20% over the $p_{T}$ range from 0.8 to 2.5 GeV/$c$. The visible difference between $\langle\varepsilon_{\gamma}f\rangle$ for 39 and 62.4 GeV is due to the $\sqrt{s}$ dependence of the $\pi^{0}$ $p_{T}$ spectra, which are much softer for the lower energies. Because $\langle\varepsilon_{\gamma}f\rangle$ is evaluated for a fixed $p_{T}$ range of the $e^{+}e^{-}$ pair, it is averaged over all possible $\pi^{0}$ $p_{T}$. Thus the value of $\langle\varepsilon_{\gamma}f\rangle$ at a fixed $e^{+}e^{-}$ pair $p_{T}$ is sensitive to the parent $\pi^{0}$ $p_{T}$ spectrum.

The EMCal acceptance contributes a multiplicative factor of 0.35 to $\langle\varepsilon_{\gamma}f\rangle$ at an $e^{+}e^{-}$ pair $p_{T}$ = 0.8 GeV/$c$, the factor increases to 0.45 at 2.5 GeV/$c$. This includes the geometrical dimension and the location of the EMCal sectors, the fiducial cuts around the sector boundaries and any dead areas in the EMCal. The minimum-energy cut of 0.4 GeV is the main contributor to the photon-reconstruction efficiency loss. This cut is equivalent to an asymmetry cut on the $\pi^{0}$ decay photons; the effect being largest at the lowest $\pi^{0}$ momenta that can contribute in a given $e^{+}e^{-}$ pair $p_{T}$ bin. With additional, but small, contributions from the shower-shape cut and the conversion of the second photon, the reconstruction efficiency rises from $\approx$0.3 to 0.45 over the $p_{T}$ range of 0.8 to 2.5 GeV/$c$.

Figure 6 also shows the systematic uncertainties on $\langle\varepsilon_{\gamma}f\rangle$, which are 8% and 5% for 39 and 62.4 GeV, respectively. The uncertainty of the energy calibration and the accuracy of the $\pi^{0}$ $p_{T}$ spectra are the two dominant sources of systematic uncertainties. A 2% change in the energy calibration, and with it a change of the actual energy cutoff, modifies $\langle\varepsilon_{\gamma}f\rangle$ by 3% to 4%. For 62.4 GeV, the measured $\pi^{0}$ $p_{T}$ spectra agree in shape within $\pm$10% with the charged-pion data from STAR [57]. Possible shape variations within this range translate into an uncertainty of 3% on $\langle\varepsilon_{\gamma}f\rangle$.

For 39 GeV, STAR has published charged-pion data up to 2 GeV/$c$ [58], these data agree in shape with the PHENIX $\pi^{0}$ data within $\pm$10%. However, due to the limited $p_{T}$ range, the systematic uncertainties on the shape of the $\pi^{0}$ $p_{T}$ spectrum were determined from the systematic uncertainties of the PHENIX measurement alone, which is less restrictive and, thus, results in a larger uncertainty.

The ratio of all photons from hadron decays to those from $\pi^{0}$ decays, $\mbox{$\gamma^{\rm hadr}$}/\gamma^{\pi^{0}}$ in the denominator of Eq. 3, is the final component that is needed to calculate $R_{\gamma}$. In addition to decays of $\pi^{0}$, decays of the $\eta$, $\omega$, and $\eta^{\prime}$ mesons contribute to $\gamma^{\rm hadr}$, with the $\eta$ decay being the largest contributor. Any other decays emit a negligible number of photons.

Photons from hadron decays are modeled based on the parent $p_{T}$ distributions. For each centrality class, the measured $\pi^{0}$ $p_{T}$ spectrum is used to generate $\pi^{0}$s, which are subsequently decayed to photons using the known branching ratios and decay kinematics. The decay photons from $\eta$, $\omega$ and $\eta^{\prime}$ are modeled similarly, with a parent $p_{T}$ distribution derived from the measured $\pi^{0}$ $p_{T}$ distributions, assuming ${\boldsymbol{m}}_{T}$ scaling (see Refs. [34, 59] for details)⁵⁵5Ref. [59] recently noted that using $m_{T}$ scaling overestimates the $\eta$ meson yield in $p$$+$$p$ collisions for $p_{T}$ below 2 GeV/$c$. The same work also shows that in Au$+$Au collisions at RHIC energies, this depletion is partially compensated by radial flow, which enhances the yield of $\eta$ in the same $p_{T}$ region. For this analysis, removing the $m_{T}$ scaling assumption, while including the effect of radial flow, will reduce the number of photons from hadron decays by $\approx$2% for $p_{T}{\approx}1$ GeV/$c$, where the change is the largest. Correspondingly the direct-photon yield would increase by 2%, which is within the systematic errors of 2.4% quoted on the contribution of $\gamma^{\rm hadr}$/$\gamma^{\rm\pi^{0}}$ to $R_{\gamma}$ and much smaller than the overall statistical ($>$7%) and systematic ($>$5%) uncertainties of the $R_{\gamma}$ measurement at $p_{T}$ of 1GeV/$c$. The normalization of photons from $\eta$, $\omega$, and $\eta^{\prime}$ is set to $\eta/\pi^{0}=0.46{\pm}0.06$, $\omega/\pi^{0}=0.9{\pm}0.06$ and $\eta^{\prime}/\pi^{0}=0.25{\pm}0.075$ all at $p_{T}=5$ GeV/$c$.

Figure 7 shows the $\mbox{$\gamma^{\rm hadr}$}/\gamma^{\pi^{0}}$ ratio. The ratio increases with $p_{T}$ and saturates at high $p_{T}$ between 1.22 and 1.23. There is no appreciable $\sqrt{s}$ dependence of $\mbox{$\gamma^{\rm hadr}$}/\gamma^{\pi^{0}}$. Following Ref. [35], the systematic uncertainties from $\mbox{$\gamma^{\rm hadr}$}/\gamma^{\pi^{0}}$ on $R_{\gamma}$ are estimated to be 2.4%.

After each factor in Eq. 4 is determined, $R_{\gamma}$ can be calculated. Figure 8 shows the results for all centrality classes. Despite the significant statistical and systematic uncertainties, the majority of the data points are above unity at a value around $\mbox{$R_{\gamma}$}\approx 1.2$. This indicates the presence of a direct-photon component of $\approx$20% relative to hadron-decay photons in Au$+$Au collisions at 39 and 62.4 GeV. There is no obvious $p_{T}$ dependence over the observed range; furthermore, the $\sqrt{s}$ and centrality dependence, if any, must be small.

To further analyze the data $R_{\gamma}$ is converted to a direct-photon $p_{T}$ spectrum $\gamma^{\rm dir}$ using the hadron-decay-photon spectra calculated in Sec. II.5:

Figure 9 presents the calculated direct-photon $p_{T}$ spectra. In addition to the systematic uncertainty on $R_{\gamma}$, the hadron-decay-photon spectra contribute $\approx$10% to the systematic uncertainties. These uncertainties cancel in $\mbox{$\gamma^{\rm hadr}$}/\gamma^{\mbox{$\pi^{0}$}}$, but need to be considered here. Each centrality and energy selection is compared to the expected prompt-photon contribution from hard-scattering processes based on perturbative-quantum-chomodynamics (pQCD) calculations from [10, 60]. Shown are the calculations at the scale $\mu=0.5\,p_{T}$, which were extrapolated down to $p_{T}=1$ GeV/$c$. The scale was selected as it typically gives a good description of prompt-photon measurements in $p$$+$$p$ collisions (see also Fig. 10). To represent hard scattering in Au$+$Au collisions, the calculation is multiplied with the nuclear-overlap function $T_{\rm AA}$ for the given event selection [61], assuming an inelastic $p$$+$$p$ cross sections of $\sigma_{\rm inel}=33.8$ mb at 39 GeV $\sigma_{\rm inel}=35.61$ mb at 62.4 GeV. Table 1 gives the values. Below 1.5 GeV/$c$, there is a clear enhancement of the data above the scaled pQCD calculation, consistent with the expectation of a significant thermal contribution.

To characterize the enhancement, the data is fitted with a falling exponential function given by

The data sets were fitted below a $p_{T}$ of 1.3 GeV/$c$, where statistics are sufficient. Table 2 summarizes the results, which are also shown in Fig. 9. Systematic uncertainties were obtained with the conservative assumption that the uncertainties are anticorrelatated over the observed $p_{T}$ range. All values are consistent with a common inverse slope $T_{\rm eff}$ of $\approx$0.170 GeV/$c$. For the MB and 0%–20% centrality Au$+$Au sample at 62.4 GeV, the data in the range from 0.9 to 2.1 GeV/$c$ is also fitted. The values are slightly above 0.24 GeV/$c$ and are larger than the value extracted for the lower-$p_{T}$ range. A possible increase of $T_{\rm eff}$ with $p_{T}$ is consistent with the values obtained from Au$+$Au at 200 GeV [35] and Pb$+$Pb at 2.76 TeV [39], which were fitted in the higher-$p_{T}$ range. See a more detailed discussion in the next section.

It was shown in Ref. [40] that the direct-photon yield from heavy ion collisions is approximately proportional to $(\mbox{$dN_{\rm ch}/d\eta$})^{\alpha}$ with common power $\alpha\approx 1.25$ across collision energies, systems, and centrality. Figure 10 presents the direct-photon yield normalized to $(\mbox{$dN_{\rm ch}/d\eta$})^{1.25}$ for a large range of data sets⁶⁶6The WA98 data are not shown here and in the following plots. The upper limits from WA98 for $p_{T}$ $<1.5$ GeV/$c$ are consistent with the lower end of the uncertainties of the PHENIX 62.4 GeV and 39 GeV data, but they do not significantly constrain the scaling behavior at low $p_{T}$. The STAR data are also not shown as the tension with the PHENIX data remains unresolved, while the multiple publications from PHENIX, based on different data sets and analysis methods, show self consistent results. If taken at face value, the STAR data do demonstrate a similar scaling behavior with $N_{\rm ch}$ for $p_{T}$ $<2$ GeV/$c$, but at a factor-3-lower direct-photon yield.. Panel (a) shows the data sets that are derived from the Au$+$Au measurements at 39 and 62.4 GeV shown in Fig. 9. Panel (b) presents PHENIX measurements from Au$+$Au [41, 34, 35] and Cu$+$Cu [47] collisions at $\sqrt{s_{{}_{NN}}}$ = 200 GeV. Panel (c) uses the ALICE measurement from Pb$+$Pb collisions at $\sqrt{s_{{}_{NN}}}$ = 2760 GeV [39]. All panels show pQCD calculations for $p$$+$$p$ collisions at the corresponding $\sqrt{s}$, extrapolated to $p_{T}$ = 1 GeV/$c$ at the scale $\mu=0.5~p_{T}$ [10, 60].

Table 3 gives the $dN_{\rm ch}/d\eta$ and $N_{\rm coll}$ values, which are are used to normalize the integrated yields and are obtained from published experimental data. The values for $p$$+$$p$ collisions at 62.4 are taken from Fig. 52.1 of Ref. [62], which was interpolated between UA5 data at $\sqrt{s}$ = 53 GeV [63] and 200 GeV [64]. The values for $p$$+$$p$ and heavy ion collisions from $\sqrt{s_{{}_{NN}}}$ = 7.7 GeV to 200 GeV are from PHENIX [61]; the values for 2760 GeV $p$$+$$p$ data are from ALICE [65]; and the values for Pb$+$Pb collision data at 2760 GeV are also from ALICE [66].

Figure 10(b) also gives a fit to the $p$$+$$p$ data at $\mbox{$\sqrt{s}$}=200$ GeV [40, 47] with the empirical form:

All three panels in Fig. 10 show that at a given $\sqrt{s_{{}_{NN}}}$ the normalized direct-photon yield from A$+$A collisions is independent of the collision centrality. This is true both for low and high $p_{T}$. Comparing the yield at $p_{T}$ below 3–4 GeV/$c$ across panels reveals that the yield is also remarkably independent of $\sqrt{s_{{}_{NN}}}$. Above $p_{T}$ of 4 to 5 GeV/$c$ the normalized yield does show the expected $\sqrt{s_{{}_{NN}}}$ dependence and is described by the pQCD calculations.

In the high-$p_{T}$ range, hard-scattering processes dominate direct-photon production, and these direct-photon contributions are not altered significantly by final-state effects. Different centrality selections show the same normalized yield, which reflects that empirically $\mbox{$N_{\rm coll}$}\propto\mbox{$dN_{\rm ch}/d\eta$}^{1.25}$ [40]. It remains surprising that within uncertainties the same scaling also holds at lower $p_{T}$ where direct-photon emission should be dominated by thermal radiation from the fireball. In the following sections, the similarity of the low-$p_{T}$ direct-photon spectra, both in shape and in normalized yield, is analyzed more quantitatively.

To better reveal the similarity of the low-$p_{T}$ direct-photon spectra across $\sqrt{s_{{}_{NN}}}$, the normalized yield from the most-central samples (0%–20%) for Pb$+$Pb at $\sqrt{s_{{}_{NN}}}$ = 2760 GeV, Au$+$Au at 200 GeV, and Au$+$Au at 62.4 GeV are superimposed on Fig. 11(a). Below 2.5 GeV/$c$, the data agree very well, even though they span almost two orders of magnitude in $\sqrt{s_{{}_{NN}}}$. As already suggested earlier by exponential fits to the 39 and 62.4 GeV data, the low-$p_{T}$ direct-photon spectra cannot be described by a single inverse slope, but seem consistent with an inverse slope that increases with $p_{T}$. Fitting all data shown in the $p_{T}$ range $\mbox{$p_{T}$}<1.3$ GeV/$c$ and $0.9<\mbox{$p_{T}$}<2.1$ GeV/$c$ results in inverse slopes of $T_{\rm eff}=0.174{\pm}0.018$ GeV/$c$ and $0.289{\pm}0.024$ GeV/$c$, respectively. Here the statistical and systematic uncertainties were added in quadrature in the fitting procedure. The fits are also shown in Fig. 11, where the dashed lines extrapolate the fits over the full $p_{T}$ range.

Figure 12 compares the inverse slopes from the common fit to the fits of the individual data sets. For $\sqrt{s_{{}_{NN}}}$ = 62.4 GeV, the values are from Table 2, for 200 GeV the data [34, 35] were fitted in the two $p_{T}$ ranges, and for 2760 GeV the value published in Ref. [66] is shown. For the lower-$p_{T}$ range a value for MB collisions at $\sqrt{s_{{}_{NN}}}$ = 39 GeV is also included.

Another way to illustrate the commonality of the spectra is to compare the ratio of the normalized yield divided by the extrapolated fit for $0.9<\mbox{$p_{T}$}<2.1$ GeV/$c$. The result is shown in Fig. 11(b). Within the uncertainties the ratios are consistent with unity over the fit range for all three $\sqrt{s_{{}_{NN}}}$. Below 1 GeV/$c$, where there is no data from $\sqrt{s_{{}_{NN}}}$ = 2760 GeV, the other two energies also agree very well.

The similarity of the spectra in the $p_{T}$ range up to $\approx$2 GeV/$c$ indicates that the source that emits these photons must be very similar, independent of $\sqrt{s_{{}_{NN}}}$, a finding that would be consistent with radiation from an expanding and cooling fireball evolving through the transition region from QGP to a hadron gas till kinetic freeze-out. This would naturally occur at the same temperature and similar expansion velocity, independent of the initial conditions created in the collisions.

Above 2 GeV/$c$, the normalized direct-photon yield becomes $\sqrt{s_{{}_{NN}}}$ dependent. The $\sqrt{s_{{}_{NN}}}$ = 200 GeV Au$+$Au data remain consistent with the exponential fit until $p_{T}{\approx}3$ GeV/$c$, where prompt-photon production from hard-scattering processes starts to dominate (see Fig. 10). In contrast, the Pb$+$Pb data from $\sqrt{s_{{}_{NN}}}$ = 2760 GeV begin to exceed the exponential $p_{T}{\approx}2$ GeV/$c$, while prompt-photon production only becomes the main photon source above 4 to 5 GeV/$c$, where the $N_{\rm coll}$-scaled pQCD calculation describes the heavy ion data well.

This leaves room for additional contributions to the direct-photon spectrum in the range from 2 to 5 GeV/$c$ beyond prompt-photon production, which are $\sqrt{s_{{}_{NN}}}$ dependent. Such contributions could reflect the increasing initial temperature that would be expected with increasing collision energy.

In this final section, the scaling behavior of the direct-photon yield with $(\mbox{$dN_{\rm ch}/d\eta$})^{\alpha}$ will be revisited. So far, a fixed value of $\alpha=1.25$ was used to calculate the normalized inclusive direct-photon yield. This value was obtained from the scaling relation $\mbox{$N_{\rm coll}$}{\propto}(\mbox{$dN_{\rm ch}/d\eta$})^{\alpha}$ [40]. Here, $\alpha$ will be determined from the direct-photon data itself as a function of $p_{T}$. For this purpose, the direct-photon $p_{T}$ spectra are integrated above a minimum $p_{T}$ value ($p_{T,{\rm min}}$) of 0.4 GeV/$c$, 1.0 GeV/$c$, 1.5 GeV/$c$, and 2.0 GeV/$c$. Panels (a) to (d) of Fig. 14 show the integrated yields as a function of $dN_{\rm ch}/d\eta$ for all data sets shown in Fig. 10. The systematic uncertainties, shown as boxes, give the uncertainty on the integrated yield and the uncertainty on $dN_{\rm ch}/d\eta$. The A$+$A data are compared to a band representing the integrated yields obtained from the fit to the $p$$+$$p$ data at $\sqrt{s}$ = 200 GeV, with the functional form given in Eq. 7, scaled by $N_{\rm coll}$. The width of the band is given by the uncertainties on the $p$$+$$p$ fit and on $N_{\rm coll}$, combined quadratically. Panels (b) to (c) also show the integrated yields from the $N_{\rm coll}$-scaled pQCD calculations for $\sqrt{s}$ = 200 and 2760 GeV.