Study of the $\eta$ to $\pi^{0}$ Ratio in Heavy-Ion Collisions

Photons are generally considered ideal probes to study the quark gluon plasma (QGP) created in heavy ion collisions [1], since they have a long mean free path and leave the collision volume without final state interactions. Of particular interest are low momentum or thermal photons with energies of up to several times the temperature of the QGP. The measurement of thermal photons has only recently been possible with the advance of the heavy ion programs at RHIC [2, 3, 4] and LHC [5].

One of the experimental key challenges for these measurements is to estimate and subtract photons from hadron decays that constitute the bulk of photons measured in experiments. The two major contributions of photons result from $\pi^{0}\rightarrow\gamma+\gamma$ and $\eta\rightarrow\gamma+\gamma$ decays. Precise knowledge of the parent $\pi^{0}$ and $\eta$ $p_{T}$ spectra is necessary to estimate the decay photon background. While spectra of pions from heavy ion collisions are well measured at RHIC and LHC, less data exists for $\eta$ spectra, in particular below $p_{T}$ of 2 GeV/$c$. Therefor experiments need to make assumptions how to model the $\eta$ spectra below 2 GeV/$c$, which leads to sizable systematic uncertainties. Frequently, experiments have based this extrapolation on the hypothesis of transverse mass $m_{T}$ scaling of meson spectra [3, 4, 5]. However, it is known since the late 1990’s [6] and was recently pointed out again [7] that $m_{T}$ scaling does not hold below 3 GeV for the $\eta$ meson.

In this paper we propose a new empirical approach to model the $\eta$ spectrum that is based on the universality of the $\eta/\pi^{0}$ ratio across collision systems, beam energies, and centrality selections in heavy ion collisions. With a good understanding of the $\eta/\pi^{0}$ ratio as function of transverse momentum $p_{T}$ and measured $\pi^{0}$ spectra, which are readily available for many collision systems, one can construct a more accurate $p_{T}$ distribution for $\eta$ mesons.

The paper is organized as follows. In the next section we elaborate more on the failure of $m_{T}$ scaling. In section III we will discuss two empirical fits and a Gaussian Process Regression (GPR) to describe the $\eta/\pi^{0}$ ratio for $p$+$p$ and $p$+A collisions, and document in section IV the universality of $\eta/\pi^{0}$ across different collision systems ($p$+$p$, $p$+A, A+A), energies, and collision centrality. In Section V, we estimate possible deviation from the universal trend at low $p_{T}$ due to radial flow in heavy ion collisions. We provide our result for $\eta/\pi^{0}$ for RHIC and LHC energies with systematic uncertainties in the final part.

For measurements of direct photons from heavy ion collisions, the photons from $\eta$ and heavier meson decays are frequently estimated using measured $\pi^{0}$ spectra in conjunction with the $m_{T}$ scaling hypothesis. A typical implementation of this method [8] starts with a fit to the $\pi^{0}$ spectra with a functional form like a modified Hagedorn function [9]:

with $M_{X}$ being the meson mass and $g(p_{T},M_{X})=\sqrt{p_{T}^{2}+m_{X}^{2}-m^{2}_{\pi}}$. In this implementation the spectra of the $\eta$ and heavier mass mesons follow the same distribution with respect to transverse mass $m_{T}\equiv\sqrt{m^{2}+p_{T}^{2}}$ as the $\pi^{0}$. The normalisation constant $A(M_{X})$ is the only free parameter, all other parameters are fixed by the fit to the $\pi^{0}$ data. $A(M_{X})$ is fitted to experimental data whenever such data exists.

Fig. 1 compiles available data of $\eta/\pi^{0}$ for $p$+$p$ [10, 11, 12, 13, 14] and $p$+A [6, 15] collisions. Also shown on the figure is the result of $m_{T}$ scaling for two different normalisation constants $A(M_{\eta})$ [12, 11, 16] and the expectation from a pythia-6 calculation from [17, 12]. While pythia and the $m_{T}$ scaling hypothesis agree well, a significant deviation from the data is seen at low $p_{T}$. This was originally discovered at the CERN SPS by CERES/TAPS [6] more than 20 years ago and recently confirmed by ALICE at the LHC [10]. Clearly the $m_{T}$ scaling hypothesis is not correct and should not be used to extrapolate meson spectra to low $p_{T}$ for systems where no data exists.

The quantitative agreement of the $\eta/\pi^{0}$ data shown in Fig. 1 is striking, consider the data covers more than 2 orders of magnitude in collision energy. In this section we will test different methods to obtain an empirical description of $\eta/\pi^{0}$. The first two methods (A,B) fit a functional shape of the ratio, while the third method (GPR) is a Gaussian Process Regression that does not assume a specific functional shape. All methods yield similar results below 10 GeV/$c$, at larger $p_{T}$ the deviations are sizable and we will include these deviations in our evaluation of systematic uncertainties.

In the previous section we established that the $\eta/\pi^{0}$ ratios measured in $p$+$p$ and $p$+A collisions are consistent with being constant at high $p_{T}$ with a value of $R^{\infty}=0.487\pm 0.024$ (Section III.B). Here we demonstrate that all available data from $p$+$p$, $p$+A, and A+B collisions listed in Table 1 are consistent with this $R^{\infty}$ value independent of the collision energy, collision system, or collisions centrality.

For this demonstration we adopt the functional form from Eq. 3 (empirical Fit B). The parameters are fixed using the simultaneous fit to the $p$+$p$ and $p$+A data to the following values: $a=-1.24$, $b=0.482$, $p_{0}=4.15$, $n=5.07$, and the composite parameter $R^{\infty}/A=2.28$. The final fit parameter $R^{\infty}$ is determined individually for each data set using the data shuffling method. For each data set we vary the points many times within their systematic uncertainties, as discussed in Appendix B, and create an ensemble of $R^{\infty}$ and $\sigma_{R^{\infty}}$ values. The mean of the $R^{\infty}$ ensemble is used as the measurement of $R^{\infty}$ for ${\eta/\pi^{0}}$ and the standard deviation is quoted as the systematic uncertainty. The mean of the $\sigma_{R^{\infty}}$ ensemble is quoted as the statistical uncertainty.

Fig. 4 shows the results as a function of the nucleon-nucleon center of mass energy $\sqrt{s_{NN}}$ for the minimum bias data samples of all collision systems. Also shown on the figure is the $R^{\infty}$ value obtained from the combined fit to the $p$+$p$ and $p$+A data sets using method B. Within uncertainties all data sets are consistent with this value and there is no evidence for a $\sqrt{s_{NN}}$ dependence of $R^{\infty}$.

For most publications of ${\eta/\pi^{0}}$ from heavy ion collisions, the data was also presented for centrality selected event classes. In order to include these in the comparison, we plot $R^{\infty}$ as a function of the number of produced particle $\evaluated{dN_{ch}/d{\eta}}_{\eta=0}$. The $d{N_{ch}/{d\eta}}$ values used are summarized in Tab. 2.

The results are given in Fig. 5. Again all values are consistent with a universal value within uncertainties. This analysis strongly suggest that $R^{\infty}$ does not depend on the collision systems, $\sqrt{s_{NN}}$, or the centrality of the collisions and that any apparent differences are likely due to systematic effects specific to individual data sets.

We have shown that $\eta/\pi^{0}$ can be described by one common function for all $p$+$p$ and $p$+A collisions over the measured $p_{T}$ range from 0.1 to 20 GeV/$c$. Furthermore, above $p_{T}$=5 GeV/$c$ the same function describes all data from heavy ion collisions. Whether this universal function also describes heavy ion data at lower $p_{T}$ can not be tested due to the absence of accurate experimental data. However, there are reasons to believe that this universality does not hold at low $p_{T}$.

Evidence for strong collective motion of the bulk of the produced particles has been observed in all high energy heavy ion collision. This motion is consistent with a Hubble like hydrodynamic expansion of the collision volume, with a linear velocity profile in radial direction. In this velocity profile heavier particles gain more momentum than lighter ones. Radial flow effectively depletes the particle yields at low $p_{T}$ and enhances them in an intermediate $p_{T}$ range, which is determined by the mass of the particle. For $p_{T}$ much larger than the particle’s mass radial flow becomes negligible. Fig. 6 shows the effect schematically by comparing $\pi,K$, and $p$ spectra at RHIC energies. The spectra shown are roughly to scale and consistent with experimental data from 200 GeV Au+Au collisions. They are normalized per particle of the corresponding type at mid rapidity.

Since the $\eta$ meson has about the same mass as the kaon, one would expect that in the momentum range from a few hundred MeV/$c$ to a few GeV/$c$ radial flow increases the yield of $\eta$ mesons significantly more than that of $\pi^{0}$. This in turn would increase the $\eta/\pi^{0}$ ratio in heavy ion collisions compared to that observed in $p$+$p$ and $p$+A collisions.

To quantify the size of the modification due to radial flow we will use a double ratio $R_{flow}$ defined as follows:

where we take advantage of the fact the momentum boost from radial flow is mostly determined by the particle mass and that $m_{K^{\pm}}\approx m_{\eta}$. Also charged pions are used instead of neutral pions, since $\pi^{\pm}$ and kaons are typically measured simultaneously with the same detector systems and thus most systematic uncertainties on the measurement cancel in the double ratio. The subscript $C_{i}$ refers to a specific collision system, energy and centrality selection.

Fig. 7 presents $R_{flow}$ for different centrality classes of Au+Au collisions at 200 GeV. The values were calculated from data published by PHENIX [29]. The data cover the $p_{T}$ range from 0.5 to 2 GeV/$c$ and GPR is used to extrapolate somewhat beyond the measured range. According to this estimate the $\eta/\pi^{0}$ ratio is enhanced in central collisions in a $p_{T}$ region from 0.4 to 3 GeV/$c$ with a maximum of about 25% near 1 GeV/c. The enhancement is reduced for more peripheral collisions and nearly vanishes for the 60-92% selection.

In Fig. 8 we depict the estimate for $R_{flow}$ for Pb+Pb data at 2.76 TeV calculated from $K$ and $\pi^{\pm}$ data measured by ALICE [30, 24]. Shown are results for a 0-10% centrality selections. Only statistical uncertainties are shown. The flow effect is significantly larger at LHC than at RHIC: the $p_{T}$ range affected is extended to 5-6 GeV/$c$, it reaches its maximum at higher $p_{T}$ around 3 GeV/$c$, and the maximum has increased to about 50%. All indicates that radial flow effects increase with beam energy, which is consistent with a higher initial pressure and a longer lifetime of the system at the LHC compared to RHIC.

ALICE also has published $\eta/\pi^{0}$ for Pb+Pb collisions at 2.76 TeV [24] down to 1 GeV/$c$, which can be used to verify the validity of the $R_{flow}$ estimate from $K/\pi$. For this we have divided Pb+Pb data by the universal $(\eta/\pi^{0})^{mc}_{pp}$ from Fig. 3. The result is also shown in Fig. 8, error bars represent the combine uncertainty of $(\eta/\pi^{0})^{mc}_{pp}$ and the statistical uncertainty of $(\eta/\pi^{0})_{PbPb}$. The ansatz that

is consistent with the data.

To construct an $\eta/\pi^{0}$ ratio for a specific collision system and centrality selection we modify the universal shape $(\eta/\pi^{0})^{mc}_{pp}$ determined from $p$+$p$ and $p$+A data (see Fig. 3 from section III) with $R_{flow}$ for the selected heavy ion sample:

Since $R_{flow}$ may be available only in a limited $p_{T}$ region, for example from 0.4 to 2 GeV/$c$ in Fig. 7, we propose the following procedure that can be applied to any A+B collisions system if $\pi^{\pm}$ and $K$ data are available for the $p_{T}$ range affected by radial flow. In the first step we create pseudo data for $\eta/\pi^{0}$ by multiplying $R_{flow}$ point-by-point with the $(\eta/\pi^{0})^{mc}_{pp}$ up to $p^{cut}_{T}$ where $R_{flow}(p_{T})\approx 1$. This range extends to 1.4 or 4.5 GeV/c for Au+Au at 200 GeV at 60-92% centrality and Pb+Pb at 2.76 TeV, respectively. To ensure that our flow estimate has the correct asymptotic behavior we add a second set of pseudo data with constant values of $\eta/\pi^{0}=0.487\pm 0.024$. These are added either above 4 GeV/c where all data sets can be described by a constant (see section IV) or above $1.6\times p_{T}^{cut}$, which ever is larger. The combine pseudo data are processed through a GPR to obtain a smooth curve. Finally, in order to account appropriately for the systematic uncertainties at high $p_{T}$ we merge the GPR describing the flow effect with $(\eta/\pi^{0})^{mc}_{pp}$ above $p_{T}^{cut}$. The uncertainty band at low $p_{T}$ is also taken to be whichever is larger.

In Fig. 10 the construction $\eta/\pi^{0}$ is presented step by step for three examples: 0-20%, 60-92% Au+Au at 200 GeV and 0-10% Pb+Pb at 2.76 TeV, in panels (a) to (c) respectively. The pseudo data generated are represented by points, which are then processed through a GPR resulting in the hashed green bands. They are contrasted with $(\eta/\pi^{0})^{mc}_{pp}$, the blue band, and merged with it above $p_{T}^{cut}$ to create the final green envelope representing our $\eta/\pi^{0}$ estimates. As discussed above the largest flow effect is observed for central Pb+Pb collisions at the LHC (panel (c)). For central Au+Au collison at RHIC (panel (a)) a much smaller effect is observed, and finally peripheral collisions of the same system are consistent with no flow effect (panel (b)).

These best estimates are compared to data in Fig. 9. For the comparison we selected data sets with similar charged particle densities, so that despite the difference in collision system, centrality or $\sqrt{s_{NN}}$ matter was created under similar conditions and evolved the same way with time. In all three cases our best estimates are consistent with the $\eta/\pi^{0}$ data.

We find a universal $p_{T}$ dependence of $\eta/\pi^{0}$ for all $p$+$p$ and $p$+A collisions independent of the center of mass energy from $\sqrt{s_{NN}}$=23 GeV to 8 TeV. We note that like originally discovered in [6], below 3 GeV/$c$ the universal ratio is significantly below $m_{T}$ scaling extrapolations from higher $p_{T}$.

That there is no $\sqrt{s_{NN}}$ dependence is surprising as the $p_{T}$ spectra of all particles vary strongly with $\sqrt{s_{NN}}$ and particle production from jet fragmentation becomes increasingly prevalent at higher energies. None-the-less there seems to be no impact on the relative yield at which $\eta$ and $\pi^{0}$ are produced. This may hint at a largely universal hadronisation process in which hadrons are always created under the same conditions, even if the underlying mechanism is considered different, for example bulk particle production or jet fragmentation.

For heavy ion collisions, $\eta/\pi^{0}$ has the same universal behavior at high $p_{T}$, independent of collision species, collision energy, or collision centrality. For lower $p_{T}$ we find evidence for modifications of the relative particle yields due to radial flow. One might speculate that the same universal harmonization process is at work but that hadrons are produced in a moving reference frame.

We have quantified the modification of the $\eta/\pi^{0}$ ratio due to radial flow using the double ratio $R_{AA}(K)/R_{AA}(\pi)$. This assumes that the change of the $p_{T}$ spectra depends entirely on the particle mass, but it does not make any assumptions about the similarity of $\eta$ and kaon spectra themselves. We note that our approach may overestimate the modification due to flow, since kaon production or generally strange quark production is enhanced in heavy ion collisions. In our estimate the modification increases with $\sqrt{s_{NN}}$. At 200 GeV at RHIC the maximum increase of $\eta/\pi^{0}$ is estimated to be 25% around 1 GeV/$c$, in contrast at 2.76 TeV at the LHC the maximum increase is nearly 50% and occurs at higher $p_{T}$ between 2 and 3 GeV/c.

With our original motivation in mind, which was to reduce systematic uncertainties on the measurement of direct photons, we proposed a new methodology to create $\eta/\pi^{0}$ ratios. This method is more accurate than frequently used extrapolations to lower $p_{T}$ based on $m_{T}$ scaling, and does not suffer from the frequent lack of statistics for $\eta$ measurements. Our approach can be applied to all systems for which $K/\pi^{\pm}$ is measured in the $p_{T}$ range affected by radial flow. The method does not require actual measurements of $\eta$ production for a given system.

We have tested this method for two specific collision systems. For Au+Au collisions at 200 GeV the deviations due to flow are found to be within $\pm$15% of the minimum bias values. Even for central collisions the $\eta/\pi^{0}$ underlying the estimate of photon from hadron decays used in direct photon measurements [2, 3, 4] is above what we propose. As a consequence direct photon yields have been slightly under estimated, though the differences are within quoted systematic uncertainties. For central Pb+Pb collisions at 2.76 TeV the flow modifications are larger and coincidentally bring $\eta/\pi^{0}$ much closer to the $m_{T}$ scaling assumption used in the measurement of direct photons published by ALICE [5].