2.1 PYTHIA decay process

PYTHIA is an event generator widely used in high-energy particle and nuclear physics communities [20]. In this study, PYTHIA6.319 was used to decay π⁰ and η mesons into photons through both two-photon and Dalitz decay channels for both p+p and Au+Au collisions. The pT and rapidity distributions of π⁰ and η mesons are obtained from experimental measurements. Information on daughter particles was stored for further analysis.

2.2 p+p Collisions at $\sqrt{s}$ = 200 GeV

The four-momenta of parent particles (π⁰ and η) are input to PYTHIA for decay into photons. The measured π⁰ cross-section at mid-rapidity (|η|<0.35) from the PHENIX collaboration [21-23] and calculated π⁰ ((π⁺+π^-)/2) cross section at mid rapidity (|y|<0.1 and |y|<0.5) from the STAR collaboration [24-28] during p + p collisions at $\sqrt{s}$ = 200 GeV are shown in Fig. 1 (a). The η meson cross-section was measured at mid-rapidity by both STAR (|y|<0.5 and 0<η<1) and PHENIX (|η|<0.35) collaborations [21, 29-31], as shown in Fig. 1 (a). Their cross-sections as a function of pT, assumed to have a similar shape in different mid-rapidity intervals, are parameterized using the Tsallis statistics

Fig. 1.Full-screen View

(a) Cross-sections as a function of pT for π⁰, (π⁺+π^-)/2, and η mesons at mid-rapidity in p+p collisions at $\sqrt{s}$ = 200 GeV measured using STAR and PHENIX [21-27, 29-31]. Error bars indicate point-to-point uncertainties. Solid lines represent the Tsallis function fits. The error bands depict parameterization uncertainties. (b) Parameterization of dN/dy as a function of rapidity for π⁰ and η mesons [32, 33]. (c) Direct photon cross-section as a function of pT measured by PHENIX [34-36] along with a power-law fit at mid-rapidity in p + p collisions at $\sqrt{s}$ = 200 GeV. Error bars denote point-to-point uncertainties. The error band shows the parameterization uncertainty. (d) Inclusive photon cross-section as a function of pT at mid-rapidity (|y|<0.5) in p + p collisions at $\sqrt{s}$ = 200 GeV. Different photon sources are also shown. Bands represent uncertainties, including those from parameterization and branching ratios. A global uncertainty of 8% is not shown. Owing to the large y-axis scale, the error bands are difficult to see given the small relative errors. The same is shown in the following figures.

where $m_{\text{T}}=\sqrt{(p_{\text{T}}^2+m^2)}$, m is the particle mass, and Cn, q, and T are free parameters. The fitting results, shown as solid lines in Fig. 1 (a), are used as inputs. The input rapidity distributions of π⁰ and η are parameterized based on Landau hydrodynamics, with a Gaussian-like function ${{\displaystyle \cosh }}^{-2}(\frac{3y}{4{\sigma }_{\text{Landau}}(1-\frac{y^2}{2\sqrt{s}/m})})$, where ${\sigma }_{\text{Landau}}=\sqrt{\ln (\sqrt{s}/(2m_{\text{N}}))}$, $\sqrt{s}$ is the nucleon-nucleon center of mass energy, m is the particle mass, and m_N is a nucleon mass [32, 33]. These are shown in Fig. 1 (b).

The parent particle production is isotropic in azimuth, and therefore, their azimuthal angle (ϕ) distributions are sampled uniformly within the 0∼2π range. With the input distributions extracted, the pT distributions of photons from π⁰ and η decay are obtained using PYTHIA and normalized based on the branching ratio of π⁰/η→e⁺e^-γ/γγ and π⁰/η dN/dy. The pT distributions of the decayed photons are shown in Fig. 1 (d).

In addition, the direct photon contribution is based on the PHENIX measurement of the direct photon cross-section as a function of pT during 200 GeV p + p collisions at mid-rapidity (|η|<0.35), fitted using a power-law function $a{(1+\frac{p_{\text{T}}^2}{b})}^c$, where a, b, and c are free parameters [34-36]. The data points and the fit curve are shown in Fig. 1 (c). The inclusive photon cross-section is obtained by summing π⁰ and η meson decayed and direct photons, as shown in Fig. 1 (d).

2.3 Au+Au Collisions at $\sqrt{s_{\text{N}N}}$ = 200 GeV

The inclusive photon invariant yields at mid-rapidity in Au + Au collisions at $\sqrt{s_{\text{N}N}}$ = 200 GeV are obtained using a similar approach as for p + p collisions described in Section 3.1. One notable difference is that the π⁰ and η production yields and pT distributions are modified in the QGP mainly owing to the collective flow and paton energy loss. In addition to the contributions from the initial hard scatterings and jet fragmentation to the direct photon production, the thermal photon radiation, mainly from the partonic phase of the medium created during heavy-ion collisions, is also included in this category.

The invariant yields of π⁰ as a function of pT, inferred from the STAR measurement ((π⁺+π^-)/2, |y| < 0.5) [37] and directly measured through the PHENIX experiment (|y| < 0.35) [38, 39], are shown in Fig. 2 (a) for Au + Au collisions of different centrality classes. Good agreements are seen among the different measurements in the overlapping kinematic range. Because of in-medium modifications of these yields compared to those in p + p collisions [40], the modified Tsallis function is used to fit the pT spectra:

Fig. 2.Full-screen View

(a) Invariant yields of π⁰ and (π⁺+π^-)/2 as a function of pT at mid-rapidity for different centrality classes in Au + Au collisions at $\sqrt{s_{\text{N}N}}$ = 200 GeV measured by STAR and PHENIX [37-39]. Error bars indicate point-to-point uncertainties. Solid lines represent modified Tsallis function fits. The error bands depict parameterization uncertainties. (b) Invariant yields of η mesons as a function of pT at mid-rapidity, shown as solid symbols, for 0%–10%, 10%–20%, 20%–40%, and 40%–60% Au + Au collisions at $\sqrt{s_{\text{N}N}}$ = 200 GeV as measured by PHENIX [41]. Error bars on the data points denote point-to-point uncertainties. The shaded bands represent parameterized η meson yields in various centrality classes based on mT scaling of parameterized π⁰ pT spectra and the measured η to π⁰ yield ratio. The spread of the shaded bands arises from the uncertainties in parameterizing the measured π⁰ yields, as shown in the left panel, and the error in the η to π⁰ yield ratio.

where A₁, β, p₁, n₁, A₂, B, p₂, q₀, and n₂ are free parameters. This function takes into account the thermal production and collective effects at a low pT and the parton energy loss effect at a high pT [40]. We found that this function can describe well the π⁰ spectra with pT_th = 7 GeV/c. The fitting results are also shown in Fig. 2 (a) as solid curves. The error bands arise from point-to-point uncertainties in the data. The rapidity distribution for π⁰ is assumed to be the same as that in the p + p collisions, as shown in Fig. 1 (b).

Because of the limited precision and kinematic coverage by the available measurements of η meson invariant yields, the pT spectrum shape is obtained by utilizing the mT scaling method [19], that is, taking the parameterized pT spectrum shape for π⁰ (Fig. 2 (a)) and replacing its pT with $\sqrt{p_{\text{T}}^2+m_{\eta }^2-m_{{\pi }^0}^2}$. To obtain the absolute normalization, the yield ratios of η to π⁰ from measurements [31] are 0.4±0.04(stat)±0.02(sys), with no strong pT or centrality dependencies observed. This ratio is used to fix the normalization for η meson-invariant yields in all centrality bins, shown as the solid bands in Fig. 2 (b). Also shown in Fig. 2 (b) are the measurements of η meson-invariant yields at mid-rapidity (|η| < 0.35) [41], which are seen to agree spectra quite well with the parameterized pT. As a consistent check, this method is also applied to p + p collisions, and the result from mT scaling for η meson is seen to agree with the experimental measurement down to 1 GeV/c. Similar to the case of π⁰, the rapidity distribution of the η meson is also assumed to be the same as that during p + p collisions, as shown in Fig. 1 (b).

Invariant yields of direct photons at mid-rapidity (|y|<1) in different centralities of 200 GeV Au+Au collisions were measured using the STAR experiment [42], as shown in Fig. 3 (a). They are fitted with an exponential function plus the fit to the corresponding distribution in p + p collisions scaled by N_coll [43], that is, Ae^-pT/T+N_coll × App(1+pT²/b)^-n, where A, T, App, b, and n are free parameters, and N_coll is the number of binary nucleon–nucleon collisions in different Au + Au centrality classes. The first term is used to describe the thermal photon radiation, whereas the second term is motivated by the fact that photons do not interact strongly with the QGP, and thus do not exhibit a spectrum modification compared to the p + p collisions. To obtain the direct photon yields in 0%–10% and 10%–20% centralities that are not measured, the following extrapolation procedure [25] is used:

Fig. 3.Full-screen View

(a) Invariant yields of direct photons as a function of pT at mid-rapidity (|y|<1) for different centrality classes in Au + Au collisions at $\sqrt{s_{\text{N}N}}$ = 200 GeV as measured using STAR [42]. Error bars indicate point-to-point uncertainties. Solid lines represent fits to the data points with an exponential function plus the fit to the corresponding distribution measured during p + p collisions scaled by N_coll. Error bands depict uncertainties in the parameterization and N_coll. (b) Direct photon dN/dy/⟨0.5N_part⟩ as a function of N_part along with a fit utilizing the second-order polynomial function. Error bars denote point-to-point uncertainties, and the error band shows the parameterization uncertainty. (c) Direct photon invariant yields as a function of pT from measurements and extrapolation for different centrality classes of Au+Au collisions. For the 0%–10% and 10%–20% centrality classes, the error bands show uncertainties from extrapolated inclusive direct photon yields, N_part values, and parameterization of the direct photon pT spectrum in the 0%–20% centrality bin.

• Obtain dN/dy/⟨0.5N_part⟩ as a function of N_part, where dN/dy is the integrated direct photon yield based on the fit to the measured spectrum, and N_part is the number of participating nucleons in each centrality class.

• Fit dN/dy/⟨0.5N_part⟩ versus N_part with a second-order polynomial function, and extrapolate the inclusive direct photon yields in desired centrality bins. Such a distribution and the fit are shown in Fig. 3 (b).

• The shapes of the invariant yields for direct photons in 0%–10% and 10%–20% centrality bins are taken to be the same as that of the 0%–20% centrality class, as shown in Fig. 3 (a). Normalization is based on the extrapolated inclusive direct photon yields.

The resulting invariant yields of direct photons in 0%–10% and 10%–20% centrality bins are shown in Fig. 3 (c) as dash-dotted curves.

With all ingredients in hand, invariant yields of inclusive photons are obtained with contributions from π⁰ and η two-body and Dalitz decays, and direct photon production. These are shown as solid curves in Fig. 4 with different panels corresponding to different centrality classes. The dashed and dotted lines represent individual sources. The integrated IPT yields per unit rapidity at mid-rapidity for different centrality classes are summarized in Table 1, along with those from each component. The contribution from π⁰ two-body decay is the dominant source of photon production at low pT, whereas the contribution from direct photon production increases with increasing pT and overtakes that from π⁰ two-body decay at high pT. In all cases, the neutral meson Dalitz decays constitute less than 1% of inclusive photons within the entire kinematic range.

Fig. 4.Full-screen View

Invariant yields of inclusive photons as a function of pT, along with those from different sources, at mid-rapidity (|y|<0.5) for different centrality classes in Au + Au collisions at $\sqrt{s_{\text{N}N}}$ = 200 GeV. Error bands on the solid curves include uncertainties from the parameterized π⁰, η, and direct photon yields as well as the branching ratios of π⁰ and η decaying into photons.