Energy loss predicts no $v_2$ in small systems

Abstract

We present high-$p_T$ $R_{AB}$ and $v_2$ from a perturbative quantum chromodynamics-based energy loss model that includes event-by-event hydrodynamic evolution of the medium and small system size corrections to the energy loss. The model is calibrated on, and describes well, large system $R_{AA}$ and $v_2$ experimental data. The extrapolation of our model to $\mathrm{Ne}+\mathrm{Ne}$ and $\mathrm{O}+\mathrm{O}$ agrees quantitatively with recent experimental measurements of $R_{AA}$. Surprisingly, at high-$p_T$ our energy loss model predicts $v_2\approx0$ for all symmetric and asymmetric small systems when extracted using either hard-hard or hard-soft two-particle correlations. We argue that all energy loss models will in general predict $v_2\approx0$ when extracted using hard-soft correlations, which is the usual experimental method for measuring anisotropy in hadronic collisions, due to a generic geometric decorrelation between the hard and soft sector participant planes.

Figures (5)

• Figure 1: The left and middle columns compare energy loss model predictions for $\mathrm{Pb}+\mathrm{Pb}$$R_{AA}$ and $v_2\{\text{SP}\}$ at $\sqrt{s_{NN}}=5.02$ TeV, respectively, as functions of $p_T$ across multiple centrality classes, with data from ATLAS ATLAS:2022kquATLAS:2018ezv, ALICE ALICE:2018vuuALICE:2018rtz, and CMS CMS:2016xefCMS:2017xgk. The right column presents comparisons to small-system data: the top panels show $R_{AB}$ for $\mathrm{Ne}+\mathrm{Ne}$ (first) at $\sqrt{s_{NN}}=5.36$ TeV CMS:2026qef, $\mathrm{O}+\mathrm{O}$ (second) at $\sqrt{s_{NN}}=5.36$ TeV CMS:2025btaALICE:2025oop, and for $p+\mathrm{Pb}$ (third) at $\sqrt{s_{NN}}=5.02$ TeV CMS:2016xefALICE:2018vuuATLAS:2022kqu, while the bottom panel shows $v_2\{\text{SP}\}$ predictions for $p+\mathrm{Pb}$ compared to ATLAS ATLAS:2019vcm (0-5%, $\sqrt{s_{NN}}=8.16$ TeV, $v_2\{2\}$), ALICE ALICE:2022cwa (0-10%, $\sqrt{s_{NN}}=5.02$ TeV, $v_2\{2\}$), and CMS CMS:2025kzg ($185 \leq N_{\text{trk }}^{\text{offline }}<250\approx 0-5\%$, $\sqrt{s_{NN}}=8.16$ TeV, $v_2\{4\}$, four-subevent method). Experimental data are shown with statistical (bars) and systematic (boxes) uncertainties, while theory results are shown in red with statistical (bars) and systematic (bands) uncertainties, along with central values (lines). Diamonds, triangles, and circles denote ATLAS, ALICE, and CMS data, respectively. Bands around unity are experimental normalization uncertainties. Normalization uncertainties for $R_{p\mathrm{Pb}}$ are $4.6\%,~ 3.9\%, \text{ and } 2.3\%$ for ATLAS ATLAS:2022kqu, ALICE ALICE:2018vuu, and CMS CMS:2016xef, respectively; the $R_{p\mathrm{Pb}}$ uncertainties are not shown for visual purposes.
• Figure 2: Left (Right): Model predictions for $R_{AB}$ (top), $v_2^{\text{hard}}$ (middle), $v_2\{\text{SP}\}$ (bottom) as a function of $p_T$ (centrality) for $\mathrm{Pb}+\mathrm{Pb}$ (gray), $\mathrm{Ne}+\mathrm{Ne}$ (green), $\mathrm{O}+\mathrm{O}$ (red), and $p+\mathrm{Pb}$ (blue) collision systems. The left panel shows model predictions for the 20-30% centrality class and the right panel shows model predictions at a fixed $p_T$ of 15 GeV. The $v_2^{\text{hard }}$ is calculated from \ref{['vnHardEq']} and the $v_2\{\text{SP}\}$ is calculated from \ref{['vnSPDefEQ']}. Results are shown with statistical uncertainties (bars), systematic uncertainties (bands), and central values (lines).
• Figure 3: Left: Hydrodynamic temperature profiles at the formation time of the medium $\tau_0=0.4$ fm/c for $\mathrm{Pb}+\mathrm{Pb}$ (top), $\mathrm{Ne}+\mathrm{Ne}$ (second), $\mathrm{O}+\mathrm{O}$ (third), and $p+\mathrm{Pb}$ (bottom) collision systems at 20-30% centrality. The dot-dashed lines over the temperature profiles are plotted for visual purposes and are calculated as the integral of the temperature profile from the origin in the azimuthal direction $\phi$. Center (Right): Dashed lines show the calculated $R_{AB}(\phi)/R_{AB}$ for $\alpha_s=0.3$ ($\alpha_s=0.45$) plotted on a polar axis; the curves are calculated using the corresponding hydrodynamic events from the left panel. The solid lines are $1+v_2^{\text{hard}}\cos(2(\phi-\psi_2^{\text{hard }}))$, where $v_2^{\text{hard}}$ and $\psi_2^{\text{hard }}$ are calculated from the energy-loss model using the corresponding hydrodynamic events from the left panel (see \ref{['RAAFourierEquation']} and \ref{['psi_nEQ']} for further details). For visual purposes, all figures are rotated so that the $\psi_2^{\text{hard }}$ angles align with the horizontal axis. All results from the energy-loss model are shown here with $|\mathbf{k}|_{\rm max}$ multiplier = 1 and with $p_T$ = 15 GeV.
• Figure 4: Probability distributions of $\text{folded }(\psi_2^{\text{hard }}-\psi_2^{\text{soft }})$ for $\mathrm{Pb}+\mathrm{Pb}$ (top), $\mathrm{Ne}+\mathrm{Ne}$ (second row), $\mathrm{O}+\mathrm{O}$ (third row), and $p+\mathrm{Pb}$ (bottom) collision systems in the 20-30% centrality class. For visual purposes, in the $\mathrm{Pb}+\mathrm{Pb}$ panel, we plot a normal distribution $N(\bar{X},\sigma^2)$ of mean $\bar{X} = 0.03$ and standard deviation $\sigma = 0.35$; in the $\mathrm{Ne}+\mathrm{Ne}$, $\mathrm{O}+\mathrm{O}$ and $p+\mathrm{Pb}$ panels, we plot the constant distribution of $1/\pi$. All distributions are generated with $\alpha_s$ = 0.3, $p_T$ = 15 GeV, and with $|\mathbf{k}|_{\rm max}$ multiplier = 1. Statistical uncertainties are shown in each bin with error bars.
• Figure 5: Dependence on the effective strong coupling $\alpha_s$ of the event-averaged $\cos(2[\psi_2^{\text{hard}}-\psi_2^{\text{soft}}])$ (top), $v_2^{\text{hard}}$ (middle), and $v_2\{\text{SP}\}$ (bottom) for $\mathrm{Pb}+\mathrm{Pb}$ (gray), $\mathrm{Ne}+\mathrm{Ne}$ (green), $\mathrm{O}+\mathrm{O}$ (red), and $p+\mathrm{Pb}$ (blue) collision systems. All results are shown for 20-30% centrality at fixed $p_T=15$ GeV with $|\mathbf{k}|_{\rm max}$ multiplier set to 1. Statistical uncertainties are indicated by error bars.