Implications of the isobar run results
for chiral magnetic effect in heavy ion collisions

Introduction.— Chiral magnetic effect (CME) is a macroscopic transport phenomenon induced by quantum anomaly in chiral matter. In the presence of an external magnetic field $\mathbf{B}$ and a chiral imbalance, the CME amounts to the generation of an electric current $\mathbf{J}$ along $\mathbf{B}$:

where $\sigma_{5}$ is the chiral magnetic conductivity that is proportional to the chiral chemical potential parameterizing the chirality imbalance between the left- and right-handed chiral fermions Kharzeev (2006); Kharzeev et al. (2008); Fukushima et al. (2008). The CME has an impact on the physics of high-density QCD matter Kharzeev et al. (1998, 2002); Kharzeev (2006); Kharzeev and Zhitnitsky (2007); Kharzeev et al. (2008); Fukushima et al. (2008), condensed matter physics Son and Spivak (2013); Zyuzin and Burkov (2012); Basar et al. (2014); Li et al. (2016); Huang et al. (2015), astrophysics Grabowska et al. (2015); Masada et al. (2018); Yamamoto (2016), cosmology Tashiro et al. (2012); Vilenkin and Leahy (1982); Vilenkin (1980), plasma physics Akamatsu and Yamamoto (2013); Hirono et al. (2016); Gorbar et al. (2016), and quantum information Kharzeev and Li (2019); Shevchenko (2013); for reviews, see e.g. Kharzeev and Liao (2021); Kharzeev et al. (2016); Kharzeev (2014); Fukushima (2019); Hattori and Huang (2017); Gao et al. (2020); Burkov (2015); Armitage et al. (2018); Shovkovy (2021).

Relativistic heavy ion collisions provide a unique opportunity to create and study a quark-gluon plasma (QGP) at a temperature of over a trillion degrees. In QGP, the fluctuations of quark chirality imbalance are generated through the topological fluctuations of gluon fields. Moreover, the QGP produced in heavy ion collisions experiences an extremely strong magnetic field Kharzeev et al. (2008) created mostly by the fast-moving spectator protons. Thus the CME is expected to occur in the produced QGP Kharzeev (2006), and may lead to a detectable signal in these collisions Voloshin (2004). The observation of CME in heavy ion collisions would establish the presence of topological fluctuations in a non-Abelian plasma, which represent a crucial ingredient of the baryon asymmetry generation in the Early Universe.

Extensive experimental efforts have been made by STAR, ALICE and CMS Collaborations to look for CME in collisions at both the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) Abelev et al. (2009, 2010); Acharya et al. (2020, 2018); Khachatryan et al. (2017), see reviews Zhao and Wang (2019); Li and Wang (2020); Bzdak et al. (2020). The search has proved to be challenging due to a relatively small signal masked by a strong background contamination Voloshin (2004); Wang (2010); Pratt (2010); Bzdak et al. (2010, 2011), see e.g. discussions in Zhao and Wang (2019); Li and Wang (2020); Bzdak et al. (2020, 2013); Choudhury et al. (2022); Christakoglou et al. (2021).

To disentangle the signal driven by magnetic field (in addition to topological fluctuations) and the background driven by the collective flow determined by the collision geometry, it has been proposed to perform a measurement of CME observables in RuRu and ZrZr isobar collisions Voloshin (2010); Koch et al. (2017). The motivation for this measurement was that the similar size and shape of the colliding nuclei would lead to a nearly identical background, whereas the difference in electric charge of Ru and Zr nuclei would result in a difference in the created magnetic field, and thus in a difference in the observed CME signal.

In 2018 the STAR Collaboration performed the corresponding measurements at RHIC. A careful blind analysis was carried out subsequently Adam et al. (2021), with the first set of data released in September 2021 Abdallah et al. (2022a). STAR results are inconsistent with the “pre-defined” criteria for the CME, i.e. the criteria based on the assumption that the backgrounds in RuRu and ZrZr collisions are identical. Namely, the ratios of the CME observables measured in RuRu and ZrZr collisions are smaller than 1, whereas a stronger magnetic field in RuRu system would apparently make this ratio bigger than 1 in the presence of CME. The problem with this result however is that if the CME is absent, the ratios of these observables would have to be equal to 1, and not be smaller than 1. Indeed, none of the theory models predicted the ration smaller than 1, so this experimental result begs for an explanation.

The examination of STAR data Abdallah et al. (2022a) shows the key to understanding this puzzle is the observed difference between the gross properties of hadron production in RuRu and ZrZr collisions that stem from the difference in the shape and size of Ru and Zr nuclei. This observed difference in the multiplicity distributions and the collective flow invalidates the “pre-defined” criteria for the presence of CME, and clearly indicates the need for a post-blinding re-analysis of STAR data. Only after such an analysis is performed, one will be able to draw conclusions about the presence or absence of CME in the data. In this Letter, we address this issue by combining insights from theoretical simulations based on the event-by-event anomalous-viscous fluid dynamics (EBE-AVFD) framework Shi et al. (2020, 2018); Jiang et al. (2018); An et al. (2022); Shen et al. (2016) with the analysis of STAR data.

Correlation observables.— In heavy ion collisions, the CME leads to a charge separation along the magnetic field which is approximately perpendicular to the reaction plane Bloczynski et al. (2013). Such a charge separation can be measured via charge-dependent azimuthal correlations Voloshin (2004); Abelev et al. (2009, 2010), with the most commonly used $\Delta\gamma$ and $\Delta\delta$ observables defined as:

In the above, $\phi_{1,2}$ are azimuthal angles of the charged hadron pairs while $\Psi_{2}$ is the event-plane angle. The “OS-SS” means the difference between the opposite-sign hadron pairs (i.e. the pairs of hadrons with opposite electric charges) and same-sign pairs. Other observables have also been developed and used for experimental analysis Abdallah et al. (2022a); Choudhury et al. (2022), such as $\Delta\gamma$ comparison between reaction and event plane Xu et al. (2018a); Voloshin (2018), $\Delta\gamma$ invariant mass dependence Zhao et al. (2019), R-correlator Magdy et al. (2018), signed balance function Tang (2020), event-shape engineering Acharya et al. (2018); Wen et al. (2018) and others.

The CME signal induces the parity-odd harmonic in the azimuthal angle distribution of charged hadrons Kharzeev (2006):

Therefore it contributes to the above observables as $\Delta\gamma_{cme}=+2a_{1}^{2}$ and $\Delta\delta_{cme}=-2a_{1}^{2}$ which are thus proportional to the square of the magnetic field strength.

The main challenge in the experimental search of CME is the background correlations that dominate the observables. The identified backgrounds are local charge conservation at hydrodynamic freeze-out and resonance decays. Their contributions to the observables scale approximately as $\Delta\gamma_{bkg}\propto+\frac{v_{2}}{N_{ch}}$ and $\Delta\delta_{bkg}\propto+\frac{1}{N_{ch}}$ where $N_{ch}$ is the charged particle multiplicity and $v_{2}$ is the elliptic flow coefficient. One may also consider the observable $\Delta{\bar{\gamma}}\equiv\Delta\gamma/v_{2}$ for which $\Delta{\bar{\gamma}}_{bkg}\propto+\frac{1}{N_{ch}}$. Such scaling behaviors were found to approximately hold in model simulations. For more detailed discussions on signals and backgrounds see e.g. Choudhury et al. (2022).

The isobar collision experiment in principle allows to separate the signal and background contributions. In the idealized scenario, the two systems would be identical in their bulk properties (such as multiplicity and collective flow), which would result in the identical background contributions to the $\Delta\gamma$ and $\Delta\delta$ correlators. Therefore in the case of pure background, with no CME present, the measured isobar ratios would be $\Delta\gamma^{Ru}/\Delta\gamma^{Zr}=1$ and $\Delta\delta^{Ru}/\Delta\delta^{Zr}=1$. The case of a finite CME contribution would imply $\Delta\gamma^{Ru}/\Delta\gamma^{Zr}>1$ and $\Delta\delta^{Ru}/\Delta\delta^{Zr}<1$. These are the “pre-defined criteria” used in the STAR blind analysis Abdallah et al. (2022a); Adam et al. (2021). However, as clearly shown by the STAR data, the bulk properties of hadrons produced in the two isobar systems are not identical. For example, in the same centrality class, the hadron multiplicities differ at a few percent level – this difference is extremely important in the search for the $\sim 1\%$ CME effect. This situation requires a more careful isobar comparison with a proper baseline identification.

Isobar multiplicity comparison.— As discussed above, the event multiplicity plays a key role in the background correlations and it is important to first examine the multiplicity difference between the two isobar pairs. While the Ru and Zr nuclei have an equal number of nucleons, the geometric distributions of protons and neutrons within these nuclei have a non-negligible difference Shi et al. (2019); Xu et al. (2018b); Hammelmann et al. (2020). This difference translates into the difference in the initial conditions (e.g. the participant and the binary collision densities), which in turn affects the subsequent bulk evolution and leads to the observed discrepancy in multiplicity.

First, we will show that by adopting suitable parameters (like charge radius, neuron skin and harmonic deformation coefficients) for Ru and Zr nuclear distributions, one can reasonably reproduce the observed multiplicity difference. In Fig. 1 (upper panel), we show the ratio for the multiplicity distribution $P(N_{ch})$ between RuRu and ZrZr from simulations with several choices for the nuclear parameters Zhang and Jia (2022); Abdallah et al. (2022a); Xu et al. (2021). The STAR data are also shown for comparison, along with vertical bars indicating centrality class definition by STAR analysis. The simulation results compare well with data for the $(20\sim 50)\%$ centrality class which will be the focus of our analysis. Considerable deviations occur in the very central and peripheral regions where fluctuations and uncertainties of both simulations and data become large. In Fig. 1 (lower panel) for the ratio of average multiplicity between RuRu and ZrZr in the same centrality class as defined by the STAR analysis, one sees nice agreement between simulation results and experimental data. With such multiplicity difference quantitatively accounted, one can expect the simulations to provide useful and realistic baseline resulting from the background correlations.

Understanding the measured correlations.— Given the isobar multiplicity difference, a reasonable way to compare the correlators is to take into account the expected scaling behavior of background correlations by examining the following re-scaled correlators: $N_{ch}\times\Delta\gamma/v_{2}$ and $N_{ch}\times\Delta\delta$. Using STAR data from Abdallah et al. (2022a); STAR Collaboration (2022), we plot these re-scaled observables for RuRu and ZrZr in Fig. 2 (upper panels) as well as the RuRu/ZrZr ratios (lower panels).

Let us discuss the RuRu/ZrZr ratios shown in the lower panels of Fig. 2. If the background correlations were to scale exactly as $v_{2}/N_{ch}$, then the pure-background baseline for both $N_{ch}\times\Delta\gamma/v_{2}$ and $N_{ch}\times\Delta\delta$ ratios would be unity. (As a caveat, the non-flow effect has the potential of shifting this ratio at about $1\%$ level and requires careful scrutiny Feng et al. (2022).) A nonzero CME contribution on top of the baseline would then lead to $R_{\gamma}\equiv[N_{ch}\times\Delta\gamma/v_{2}]_{RuRu/ZrZr}>1$ and $R_{\delta}\equiv[N_{ch}\times\Delta\delta]_{RuRu/ZrZr}<1$. As one can see from Fig. 2, the ratio for the scaled $\gamma$ correlator is around unity for very central collisions and gradually increases to a value well above one towards the relatively more peripheral region.

This trend is consistent with a nonzero CME contribution that should increase with growing magnetic field strength from central to peripheral collisions. On the other hand, the ratio for the scaled $\delta$ correlator also increases from unity in very central collisions to be above unity in more peripheral collisions, while a nonzero CME contribution would have decreased this ratio to be below unity. This apparent “inconsistency” between the $\gamma$ and $\delta$ trends requires a closer scrutiny of the background behavior.

To resolve this issue, we have used our $(20\sim 50)\%$ simulation events in the pure background case to verify the assumption about the $1/N_{ch}$ background scaling. It turns out that the scaling is not exact and a non-negligible additional dependence on the average transverse momentum $\langle p_{T}\rangle$ can be identified, especially in the $\delta$ correlator. The physical origin of this effect is due to the initial fluctuations that could induce a spread of radial flow “push” and thus a spread of $\langle p_{T}\rangle$ even for events with similar multiplicity. Stronger radial flow (i.e. larger $\langle p_{T}\rangle$) would lead to a stronger angular collimation of correlated charged hadron pairs, and thus to the enhancement of background correlations Pratt (2010).

To demonstrate the impact of this effect on the $\delta$ and $\gamma$ correlators, we bin the $(20\sim 50)\%$ simulation events based on $\langle p_{T}\rangle$ and compute the corresponding correlators in each bin. The results, plotted in Fig. 3, clearly show a linear increase of $N_{ch}\times\Delta\delta$ with $\langle p_{T}\rangle$. The $N_{ch}\times\Delta\gamma/v_{2}$, on the other hand, appears to be relatively insensitive to the $\langle p_{T}\rangle$. We also note that hydrodynamic simulations performed in  Nijs and van der Schee (2021) and in our calculations demonstrate that the RuRu events have a larger $\langle p_{T}\rangle$ than ZrZr events in the same centrality class.

Our findings suggest that while unity is a suitable baseline ratio of the $N_{ch}\times\Delta\gamma/v_{2}$ correlator, flow-induced corrections need to be taken into account for the baseline ratio of $N_{ch}\times\Delta\delta$. Since RuRu system has a larger multiplicity than ZrZr system in the same centrality class, the scaled $\delta$ correlator would have a relative enhancement in the RuRu system due to a slightly larger radial flow “push”. To quantify this correction, we have evaluated the baseline ratio from pure background case to be $1.033$ by using the $(20\sim 50)\%$ AVFD simulation events for the isobar pairs. In short, our analysis of the baseline ratios can be summarized as $R_{\gamma}^{baseline}=1$ and $R_{\delta}^{baseline}=1.033$ for $(20\sim 50)\%$ collisions, shown as blue lines in Fig. 2 (lower panels). Comparing with the corresponding STAR data, $R_{\gamma}^{STAR}=1.011>R_{\gamma}^{baseline}$ and $R_{\delta}^{STAR}=1.028<R_{\delta}^{baseline}$, shown as red bands in Fig. 2 (lower panels), we conclude that both the scaled $\gamma$ and $\delta$ correlators are consistent with a nonzero CME contribution.

Extracting CME signal fraction.— Given the indication of a nonzero CME signal from our analysis above, we now make an attempt to extract the CME signal fraction from both the $\gamma$ and $\delta$ correlators based on the available information for $(20\sim 50)\%$ centrality from the STAR data as well as from the simulation results.

Let us first examine the $\Delta{\bar{\gamma}}$ correlator. Assuming that the CME signal fraction is $f_{s}$, we split the correlator measured in RuRu collisions into a signal and background contributions as follows:

where the subscripts “s” and “b” denote the signal and background components, respectively. Since the ZrZr collisions are expected to possess a weaker magnetic field, and thus relatively smaller signal and larger background, we then re-scale these quantities from RuRu to ZrZr collisions as follows:

Therefore, the total $\Delta\bar{\gamma}$ correlator for ZrZr is:

which means the $\Delta\bar{\gamma}$-ratio between the isobars is

Let us rewrite the ratio as $R\equiv\frac{1}{1+\lambda_{R}}$ (or equivalently $\lambda_{R}=1/R-1$) and then we can express the $f_{s}$ as

Under the naïve assumption of identical backgrounds in RuRu and ZrZr systems, as used in the STAR predefined criteria, the pure background case would correspond to $\lambda_{b}=\lambda_{R}=f_{s}=0$, while a nonzero signal would correspond to $\lambda_{b}=0$, $\lambda_{R}<0$ and $f_{s}>0$. This assumption is however invalided by the data, as already discussed above.

The estimates based on experimental data and simulations suggest instead the following values: (a) $R\simeq 0.9641\pm 0.0037$ or $\lambda_{R}\simeq+(0.0372\pm 0.0040)$ directly from measurements; (b) $\lambda_{b}$, dominated by the multiplicity difference, is estimated according to the ratio of $\langle N^{-1}\rangle$ and $\lambda_{b}\simeq+0.0508$; (c) The isobar signal ratio is dictated by the magnetic fields, for which the ratio is determined from simulations to be $\lambda_{s}\simeq+(0.15\pm 0.05)$ Shi et al. (2019) . See Supplemental Material A Sup for details of how these values are obtained or estimated. Putting these inputs together, one arrives at the following estimate for the CME signal fraction in the measured $\Delta\bar{\gamma}$ correlator:

We note that this value is consistent with the isobar analysis results from the event-plane/spectator-plane contrast method Abdallah et al. (2022a). To make the outcome of this analysis more transparent, we illustrate it in Fig. 4.

Summary.— To summarize, we have combined the insights from theoretical simulations with the analysis of STAR experimental data to understand the implications of isobar collisions for the chiral magnetic effect. First, we have shown that the measured multiplicity difference between the RuRu and ZrZr systems, which plays a key role for establishing the background baseline, can be successfully described by simulations with suitable initial nuclear structure inputs. Furthermore, we have identified the radial flow “push” as an important contributor to background correlations, in addition to the multiplicity and elliptic flow. Quantitatively accounting for these two effects on the backgrounds has allowed us a calibration of the appropriate baselines for both the scaled $\gamma$ and $\delta$ correlators. Compared with the experimental data, we conclude that the correlation measurements could be consistent with a finite CME signal contribution, estimated at a level of about $(6.8\pm 2.6)\%$ fraction as illustrated in Fig. 4. Such a fraction is obtained by assuming that the non-CME background of $\Delta\bar{\gamma}$ is inversely proportional to multiplicity. There is however non-flow effect Feng et al. (2022) that could make nontrivial contributions to background ratio, the influence of which clearly deserves future investigations (— see Supplemental Material A Sup for more discussions).

Let us discuss possible future measurements that can further help to establish or rule out the presence of CME signal in heavy ion collisions. Given that the radial flow push is found to bear impact on the calibration of backgrounds, it would be very useful to measure and compare the average transverse momentum of charged particles between the isobar systems. Another useful approach for the isobar comparison is to apply identical event selection criteria for multiplicity, $v_{2}$ and $\langle p_{T}\rangle$ and then compare the subset of isobar events that are ensured to have identical background correlations Shi et al. (2020, 2019). Both the participant-plane/spectator-plane contrast method Xu et al. (2018a); Voloshin (2018) and the event-shape engineering approach Milton et al. (2021) have the potential of maximizing the signal/background contrast, extracting the signal fraction and allowing a verification of expected signal scaling between isobar pairs. Beyond the $\gamma$ and $\delta$ correlators, it would be interesting to understand the background scaling and their implications for the interpretation of isobar comparison in e.g. the R-correlator Lacey et al. (2022) as well as the signed balance function Tang (2020). Finally, it is of great importance to achieve a coherent understanding of both isobar and AuAu systems Feng et al. (2021). Recent STAR measurements in AuAu collisions based on the participant-plane/spectator-plane contrast method, while suffering from a limited statistics, do suggest a nonzero CME fraction of $(6.3\sim 14.7)\%$ Abdallah et al. (2022b). Future high precision measurements based on the anticipated large sample of AuAu events together with the ongoing post-blinding analysis of the extensive isobar data set will hopefully allow to improve the statistical significance of these results and allow to make a conclusive statement on the presence or absence of CME in heavy ion collisions.