Universal Scaling in Direct-Photon Production

• Universal scaling of direct-photon production is the empirical observation that low-pT yields follow a power-law relation with charged-particle density.
• Experiments use internal and external conversion techniques to reveal a super-linear scaling with an exponent of about 1.2–1.25 across various collision systems and energies.
• The scaling results constrain theoretical models by linking thermal radiation and pre-equilibrium dynamics to a common underlying photon emission mechanism.

Universal scaling of direct-photon production denotes the empirical observation that low-$p_T$ direct-photon yields from heavy-ion collisions, when organized by the charged-particle pseudorapidity density at midrapidity, follow a common power law,

$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$

with $\alpha$ near $1.2$–$1.25$, across Au+Au, Cu+Cu, and Pb+Pb systems and across beam energies from $39$ GeV to $2.76$ TeV. In PHENIX measurements this scaling is accompanied by a large low-$p_T$ yield, a large azimuthal anisotropy relative to the reaction plane, and similar scaled spectral shapes up to about $p_T\sim 2$ GeV/$c$; prompt high-$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$0 photons, by contrast, follow the expected binary-collision scaling (Esha, 2022).

1. Experimental observable and discovery context

Direct photons are photons not coming from hadronic decays such as $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$1 and $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$2. Because they escape the medium essentially unmodified, they provide an integral probe of the space-time evolution of the fireball. PHENIX measured low-momentum direct photons in $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$3, $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$4Au, $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$5Au, Cu+Cu, and Au+Au collisions at RHIC, using internal conversions in some cases and external conversions in detector material such as the HBD/VTX in others (Khachatryan, 2018).

The central experimental observations are twofold. First, large A+A systems show a significant excess over scaled $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$6 baselines at low $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$7. Second, the integrated low-$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$8 direct-photon yield from large systems follows a universal scaling as a function of charged-particle multiplicity, $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$9, with $\alpha$0. PHENIX summarized this as a discovery that the photon production yield increases faster than the charged-particle multiplicity (Khachatryan, 2018).

This empirical organization of the data was not limited to a single collision energy. PHENIX reported consistency across Au+Au at $\alpha$1, $\alpha$2, and $\alpha$3 GeV, Cu+Cu at $\alpha$4 GeV, and broader comparisons including Pb+Pb at the LHC. The multiplicity variable $\alpha$5 was used because direct comparison by $\alpha$6 or $\alpha$7 is useful only at a fixed beam energy, whereas multiplicity allows comparisons across different $\alpha$8 (Khachatryan, 2018).

2. Multiplicity scaling in large systems

The phenomenological core of universal scaling is the power-law relation

$\alpha$9

In the PHENIX compilation of low-$1.2$0 direct photons from large systems, the fitted exponent is $1.2$1. The abstract statement that this means “the photon production yield increases faster than the charged-particle multiplicity” is a direct consequence of $1.2$2 (Khachatryan, 2018).

A later PHENIX study of Au+Au collisions at $1.2$3 and $1.2$4 GeV sharpened the scaling statement. It found that a universal scaling is observed when the direct-photon $1.2$5 spectra for different center-of-mass energies and for different centrality selections at $1.2$6 GeV are scaled with $1.2$7 for $1.2$8. This scaling also holds for earlier PHENIX Au+Au data at $1.2$9 GeV and for ALICE Pb+Pb spectra at $1.25$0 TeV. The scaling power $1.25$1 seems to be independent of $1.25$2, center-of-mass energy, and collision centrality (Abdulameer et al., 2022).

PHENIX subsequently summarized the same pattern in terms of system size thresholds and global systematics. For systems with $1.25$3–30, a large direct-photon yield and a large azimuthal anisotropy are observed, and the yield follows

$1.25$4

According to that study, the scaling holds across A+A systems spanning almost two orders of magnitude in system size, and it is also consistent with the scaled $1.25$5 trend, though the heavy-ion yields are much larger, by about an order of magnitude. Importantly, the observed scaling is reported to be independent of collision system size, beam energy, and, within uncertainties, $1.25$6 over the measured range from about $1.25$7 to $1.25$8 GeV/$1.25$9 (Esha, 2022).

A related centrality formulation predates the multiplicity-collapse plots. In Au+Au at $39$0 GeV, PHENIX parameterized the integrated excess yield as

$39$1

with exponents around $39$2–$39$3, and a simultaneous fit giving $39$4. The paper also quoted $39$5 as a very similar value for the centrality scaling of the excess yield. This established that the yield grows faster than linear in $39$6 but slower than $39$7 (Adare et al., 2014).

3. Spectral shapes, inverse slopes, and anisotropy

Universal scaling does not imply that the direct-photon spectrum is characterized by a single temperature-like parameter over the full measured $39$8 range. PHENIX emphasized that, after subtracting the prompt component, the inverse slope for the $39$9 range from $2.76$0–$2.76$1 GeV/$2.76$2 is $2.76$3 MeV/$2.76$4, but increases to about $2.76$5 MeV/$2.76$6 for the range from $2.76$7 to $2.76$8 GeV/$2.76$9. Within the experimental uncertainty, there is no indication of a system size dependence of the inverse slope (Esha, 2022).

The lower-energy Au+Au measurements at $p_T$0 and $p_T$1 GeV show the same curvature in another form. The spectra have a local inverse slope $p_T$2 increasing with $p_T$3, from $p_T$4 GeV/$p_T$5 in the range $p_T$6 GeV/$p_T$7 to $p_T$8 GeV/$p_T$9 for $p_T\sim 2$0 GeV/$p_T\sim 2$1. The spectra from different collision energies have a similar shape up to $p_T\sim 2$2 of $p_T\sim 2$3 GeV/$p_T\sim 2$4 (Abdulameer et al., 2022).

Earlier PHENIX work at $p_T\sim 2$5 GeV, reaching down to $p_T\sim 2$6 GeV/$p_T\sim 2$7, found that the excess yield above the $p_T\sim 2$8-scaled $p_T\sim 2$9 baseline is well described by an exponential distribution with an inverse slope of about $c$0 MeV/$c$1 in the $c$2 range from $c$3–$c$4 GeV/$c$5. A major result was that, once the prompt component is removed, the shape of the excess spectrum is consistent across centrality bins within the experimental uncertainties; the centrality dependence appears mainly in the normalization, not in the $c$6-shape (Adare et al., 2014).

The anisotropy measurement is central to the scaling discussion because PHENIX observes a large azimuthal anisotropy with respect to the reaction plane together with the large low-$c$7 yield. Thermal radiation from the quark-gluon plasma is qualitatively the natural explanation for the large low-$c$8 yield and the strong anisotropy, since photons escape without strong final-state interactions and therefore carry information about the medium’s evolution. However, the combination of large yield and large anisotropy remains difficult to reconcile with naive expectations for early thermal photons (Esha, 2022).

Independent STAR data reinforce the same low-$c$9 excess pattern. In Au+Au at $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$00 GeV, STAR found that for $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$01 GeV/$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$02 the direct-photon yield lies clearly above the $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$03-scaled $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$04 reference, while for $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$05 GeV/$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$06 it becomes consistent with the scaled $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$07 expectation. STAR compared the spectrum with a model by Ralf Rapp and collaborators including QGP thermal radiation, hadronic gas radiation, in-medium $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$08 contributions, and primordial hard-scattering photons, and found consistency within uncertainties (Yang, 2014).

4. Baselines, small systems, and prompt–nonprompt separation

The universality claim for low-$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$09 photons is inseparable from the prompt baseline. At high $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$10, direct photons are dominated by prompt hard processes, and scaling with $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$11 is expected because $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$12. In the PHENIX scaling compilations, the slopes in the high-$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$13 integrated-yield plot are similar to those at low $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$14, but the physical interpretation there is straightforward: prompt production follows binary-collision scaling (Khachatryan, 2018).

The cold-nuclear-matter benchmark is supplied by $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$15Au collisions. PHENIX measured direct photons at $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$16 GeV over $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$17 GeV/$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$18 using nearly-real virtual photons and real photons, and found that the invariant yield over the scaled $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$19 cross section is consistent with unity. Theoretical calculations assuming standard cold nuclear matter effects describe the data well for the entire $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$20 range. The paper concluded that the large enhancement of direct photons observed in Au+Au collisions for $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$21 GeV/$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$22 is due to a source other than the initial-state nuclear effects (Adare et al., 2012).

Small systems occupy an intermediate position. PHENIX reported a non-zero excess over the scaled $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$23 yield in central $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$24Au collisions, observed within systematic uncertainties and at roughly the $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$25 level, while $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$26Au serves as a binary-scaled baseline. The low-$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$27 region in the integrated-yield plots shows a trend that may connect smoothly toward the large-system trend. The PHENIX interpretation was suggestive of a possible “thermal transition region or point” between small and large systems (Khachatryan, 2018).

A more differential decomposition was introduced by PHENIX under the term nonprompt direct photons. In Au+Au at $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$28 GeV, the prompt component was estimated as the $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$29-scaled $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$30 direct-photon yield, with

$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$31

An excess of direct photons above prompt-photon production from hard-scattering processes is observed for $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$32 GeV/$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$33. For the nonprompt component, the spectrum has an increasing inverse slope from approximately $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$34 to $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$35 GeV/$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$36 with increasing $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$37, and the $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$38-integrated nonprompt direct-photon yields follow a power-law scaling behavior as a function of collision-system size with $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$39 consistent with $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$40 and no apparent $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$41 dependence (Acharya et al., 2022).

The $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$42 baseline at the LHC does not show a corresponding low-$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$43 enhancement. ALICE measured inclusive and direct photon production at mid-rapidity in $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$44 collisions at $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$45 and $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$46 TeV and found no statistically significant direct-photon signal below $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$47 GeV/$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$48 at either energy; the combined $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$49 results are consistent with unity throughout the measured low- and intermediate-$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$50 range, and above $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$51 GeV/$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$52 $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$53 is at least one standard deviation above unity and consistent with next-to-leading order pQCD expectations (Collaboration, 2018).

5. Geometrical scaling, saturation variables, and other notions of universality

The multiplicity scaling observed by PHENIX motivated attempts to connect direct-photon data to saturation-based geometrical scaling. One line of analysis showed that low-$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$54 direct-photon spectra from Au+Au, Cu+Cu, and Pb+Pb systems, for transverse momenta up to about $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$55–$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$56 GeV/$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$57, are approximately universal once divided by $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$58 with experimental $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$59. The same spectra also exhibit geometrical scaling when expressed as a universal function of

$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$60

or, in the heavy-ion form,

$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$61

with $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$62, $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$63, and $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$64. Under power-law assumptions for the scaling functions, the derived estimates $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$65 and $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$66 overshoot the experimental value by about $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$67 on average (Khachatryan et al., 2019).

A related phenomenology tested geometrical scaling directly by participant number and centrality. Using

$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$68

the authors showed that direct-photon data from PHENIX Au+Au at $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$69 GeV and ALICE Pb+Pb at $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$70 TeV are broadly consistent with geometrical scaling, with the ratios of scaled spectra closest to unity for $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$71. The paper emphasized that the evidence for a precise universal energy-scaling exponent $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$72 is not strong enough yet because the available data come from different experiments with different rapidity intervals and centrality binning (Praszalowicz, 2018).

An earlier saturation-based analysis made the same point in a simpler form. It argued that the inclusive spectrum of photons in the transverse momentum range $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$73 satisfies geometric scaling when written as

$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$74

with

$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$75

$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$76 GeV and $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$77–$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$78. The same scaling form was reported to work across $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$79, $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$80, AuAu, and PbPb data from RHIC to the LHC, while scaling at SPS energy is only approximate at low $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$81 and clearly breaks above about $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$82 GeV/$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$83 (Klein-Bösing et al., 2014).

A distinct use of universality appears in QCD kinetic theory for pre-equilibrium photons. In a boost-invariant, transversely homogeneous setup, photon emission during chemical equilibration can be written in terms of universal scaling functions controlled by $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$84, $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$85, and $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$86, with

$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$87

and

$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$88

These scaling functions were proposed as practical tools for realistic predictions and were implemented into KoMPoST (Garcia-Montero et al., 2024).

6. Physical interpretation, misconceptions, and unresolved tensions

The dominant interpretation of the low-$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$89 excess is that it reflects radiation from the hot and dense medium created in heavy-ion collisions. In PHENIX summaries, the excess likely reflects substantial thermal radiation from a hot, dense medium in large A+A systems, while the hint of excess in central $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$90Au raises the possibility that even small systems may occasionally reach conditions sufficient for some thermal-like photon emission. The fact that the yield scales faster than multiplicity hints that photon production is sensitive not just to the number of produced particles, but also to the space-time volume, temperature history, and lifetime of the emitting medium (Khachatryan, 2018).

At the same time, the scaling results do not imply that a single microscopic source or a single exponential describes the full spectrum. PHENIX stressed that the nonprompt spectrum is not a single constant-slope exponential over all $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$91; instead, the inverse slope increases from roughly $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$92 MeV to about $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$93 MeV as $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$94 moves from around $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$95 to $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$96 GeV/$$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$97. This behavior is consistent with a time-integrated emission history in which the earliest, hottest phases contribute more strongly at higher $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$98. A plausible implication is that universal multiplicity scaling can coexist with a changing relative weight of QGP, hadron-gas, and pre-equilibrium emission (Esha, 2022).

A common misconception is that universal scaling means simple binary-collision scaling everywhere. The data show the opposite. Hard photons at high $$\frac{dN_\gamma}{dy} \propto \left(\frac{dN_{ch}}{d\eta}\right)^\alpha,$$99 follow $\alpha$00 scaling, but low-$\alpha$01 photons do not: they lie above the $\alpha$02-scaled $\alpha$03 baseline in A+A collisions, and in large systems the integrated yield follows a super-linear multiplicity law instead (Yang, 2014).

Another misconception is that the universality of the scaling already amounts to a quantitative theoretical solution. PHENIX repeatedly emphasized that the measured yield is often larger than predicted, particularly below $\alpha$04 GeV/$\alpha$05, where one comparison to a hybrid model with thermal plus pre-equilibrium emission falls short by a factor of about $\alpha$06–$\alpha$07. The calculations do capture the general spectral shape reasonably well and indicate that pre-equilibrium radiation may become dominant above $\alpha$08 GeV/$\alpha$09, but they do not yet quantitatively reproduce the full observed yield and anisotropy. The observed scaling, inverse slopes, and large $\alpha$10-like anisotropy therefore remain among the most constraining observables for models of thermal radiation, pre-equilibrium dynamics, and the space-time evolution of the quark-gluon plasma (Esha, 2022).

The broader significance of universal scaling is that it suggests a remarkably simple systematics underlying photon production in a very complex dynamical environment. The observed similarity of low-$\alpha$11 direct-photon production from $\alpha$12 to $\alpha$13 GeV suggests a common source of direct photons for the different collision energies and event centrality selections, and suggests a comparable space-time evolution of direct-photon emission. This suggests a stable organizing principle for direct-photon phenomenology, even though a quantitative first-principles description remains incomplete (Abdulameer et al., 2022).