Varying the chiral magnetic effect relative to flow in a single nucleus-nucleus collision
We propose a novel method to search for the chiral magnetic effect (cme) in heavy ion collisions. We argue that the relative strength of the magnetic field (mainly from spectator protons and responsible for the cme) with respect to the reaction plane and the participant plane is opposite to that of the elliptic flow background arising from the fluctuating participant geometry. This opposite behavior in a single collision system, hence with small systematic uncertainties, can be exploited to extract the possible cme signal from the flow background. The method is applied to the existing data at rhic, the outcome of which is discussed.
pacs:25.75.-q, 25.75.Gz, 25.75.Ld
Introduction. One of the fundamental properties of quantum chromodynamics (QCD) is the creation of topological gluon fields from vacuum fluctuations in local domains Lee and Wick (1974); Morley and Schmidt (1985); Kharzeev et al. (1998). The topological charge () refers to the change from right-handed to left-handed quarks () and vice versa () resulted from interactions of quarks with those gluon fields under the approximate chiral () symmetry restoration Kharzeev et al. (2008). The resultant chirality imbalance can lead to an electric current, or charge separation (cs), along a strong magnetic field (), a phenomenon called the chiral magnetic effect (cme) Fukushima et al. (2008). This local parity violating phenomenon is not specific to QCD but a subject of interest for a wide range of physics communities Kharzeev et al. (2016). Similar phenomena have been observed in magnetized relativistic matter in three-dimensional Dirac and Weyl materials Li et al. (2016); Lv et al. (2015); Huang et al. (2015).
Both conditions for the cme–the -symmetry and a strong –may be met in relativistic heavy ion collisions (hic) Kharzeev et al. (2016). At high energies, a nucleus-nucleus collision can be considered as being composed of participant nucleons in an overlap interaction zone and the rest spectator nucleons passing by continuing into the beam line. The spectator protons produce the majority of . Its direction may be assessed by the sidewards flow () of the spectators which lies, on average, in the reaction plane (rp) spanned by the impact parameter and beam directions Reisdorf and Ritter (1997); Poskanzer and Voloshin (1998). The sign of the net is, however, random due to its fluctuating nature; consequently the cs dipole direction is random Kharzeev et al. (2008). As a result the cs can only be measured by charge correlations. A commonly used observable is the three-point azimuthal correlator Voloshin (2004), , where and are the azimuths of two charged particles, and is that of the rp. Because of charge-independent backgrounds, such as correlations from global momentum conservation, the correlator difference between opposite-sign (os) and same-sign (ss) pairs, , is used. is ambiguous between a back-to-back pair perpendicular to the rp (potential cme signal) and an aligned pair in the rp (e.g. from resonance decay). It is therefore contaminated by backgrounds, which arise from the coupling between particle correlations with the elliptic flow anisotropy () Voloshin (2004); Wang (2010); Bzdak et al. (2010); Schlichting and Pratt (2011); Wang and Zhao (2017) and are hence proportional to .
The search for the cme is one of the most active research in hic at the Relativistic Heavy Ion Collider (rhic) and the Large Hadron Collider (lhc) Abelev et al. (2009a, 2010, 2013a); Adamczyk et al. (2013); Adamczyk et al. (2014a, b); Khachatryan et al. (2017); Sirunyan et al. (2017); Acharya et al. (2017). A finite signal is observed Abelev et al. (2009a, 2010); Adamczyk et al. (2014a); Adamczyk et al. (2013); Abelev et al. (2013a), but how much background contamination is not yet settled. There have been many attempts to gauge, reduce or eliminate the flow backgrounds, by event-by-event dependence Adamczyk et al. (2014b), event-shape engineering Sirunyan et al. (2017); Acharya et al. (2017), comparing to small-system collisions Khachatryan et al. (2017); (for the STAR Collaboration) (2017); Sirunyan et al. (2017), invariant mass study Zhao et al. (2017), and by new observables Ajitanand et al. (2011); Magdy et al. (2017). The lhc data seem to suggest that the cme signal is small and consistent with zero Sirunyan et al. (2017); Acharya et al. (2017), while the situation at rhic is less clear Kharzeev et al. (2016).
To better gauge background contributions, isobaric Ru+Ru (RuRu) and Zr+Zr (ZrZr) collisions have been proposed Voloshin (2010) and planned at rhic in 2018. Their QCD backgrounds are expected to be almost the same because of the same mass number, whereas the atomic numbers, hence , differ by 10%. These expectations are qualitatively confirmed by studies Deng et al. (2016) with Woods-Saxon (ws) nuclear densities; the cme signal over background could be improved by a factor of seven in comparative measurements of RuRu and ZrZr collisions than each of them individually. A recent study by us Xu et al. (2017) has shown, however, that there could exist large uncertainties on the differences in both eccentricity () and due to nuclear density deviations from ws. As a result, the isobaric collisions may not provide a clear-cut answer to the existence or the lack of the cme.
In what follows, we argue that one has, in a single collision system, all and even better advantages of significant and minimal differences of the comparative isobaric collisions, and with the benefit of minimal theoretical and experimental uncertainties.
General idea. Due to fluctuations, the second harmonic participant plane (pp) azimuthal angle () is not necessarily aligned with the rp’s () Alver et al. (2007). The eccentricity of the transverse overlap geometry is by definition . The overlap geometry averaged over many events is an ellipse with its short axis being along the rp; its eccentricity is and
The factorization, which is assumed in Eq. (1), holds when, at a given centrality, the magnitude does not vary with the fluctuation around .
is mainly produced by spectator protons. Their positions fluctuate; the direction, , is not always perpendicular to the rp Bzdak and Skokov (2012); Deng and Huang (2012); Bloczynski et al. (2013). The cme-induced cs is along the direction Kharzeev et al. (2008); when measured perpendicular to a direction , its relevant strength is proportional to Deng et al. (2016). Because the position fluctuations of participant nucleons and spectator protons are independent, and fluctuate independently about . This yields
Because contains contributions also from participant protons, the factorization is only approximate.
It is convenient to define a relative difference,
where and are the measurements of quantity with respect to and (or described below), respectively. Those in and are
The upper panel of Fig. 1 shows and calculated by a Monte Carlo Glauber (mcg) model Xu et al. (2014); Zhu et al. (2017) for AuAu, CuCu, RuRu, ZrZr collisions at rhic and PbPb collisions at the lhc. The centrality percentage is determined from the impact parameter () in mcg. The nucleon-nucleon cross sections are set to be 42 and 64 mb for rhic and lhc, respectively. The larger lhc value is responsible for the relatively larger fluctuations seen in PbPb compared to AuAu. Spherical nuclei with the ws as well as the energy density-functional (edf) calculated distributions Xu et al. (2017) are used. The uncertainties due to nuclear deformation are small (not shown). The results are compared to the corresponding : the Eq. (1) approximation is good and that of Eq. (3) is excellent.
The pp is not experimentally measured, nor is . As a proxy for pp, the event plane (ep) is often reconstructed from final-state particle momenta. is measured by the ep method with a correction for the ep resolution () Poskanzer and Voloshin (1998), , or almost equivalently, by two-particle correlations, . can also be obtained with respect to the rp, . Although a theoretical concept, the rp may be assessed by Zero-Degree Calorimeters (zdc) measuring spectator neutrons Reisdorf and Ritter (1997); Abelev et al. (2013b); Adamczyk et al. (2017). Similar to Eq. (1),
where is given by
Similar to , one can obtain , and a similar relationship as Eq. (3),
Similarly, the relative differences in and are
The lower panel of Fig. 1 shows A Multi-Phase Transport (ampt) Lin and Ko (2002); Lin et al. (2005) simulation results of and , compared to . Again, good agreements are found. Note that the ampt centrality is determined from the midrapidity () final-state charged particle multiplicity Xu et al. (2017). The mcg and ampt results cannot be readily compared quantitatively because the former involves pp while the latter uses ep as it would be in experiments. Although ampt employs mcg as its initial geometry, the subsequent parton-parton scatterings in ampt, currently using the same parameter setting for the rhic and lhc energies, are important for the final-state ep determination. In addition, other distinctions exist, such as the nuclear shadowing effect and the Gaussian implementation of the nucleon-nucleon cross-section, which yield different predictions for the eccentricity (hence flow harmonics) and its fluctuations Zhu et al. (2017); Xu et al. (2016). Nevertheless, the general features are similar between the mcg and ampt results. Both show the opposite behavior of and , which approximately equal to .
The commonly used variable contains, in addition to the cme it is designed for, -induced background,
can be measured with respect to (using the 1st order event plane by the zdc) and (2nd order event plane via final-state particles). If is proportional to and to , then
Here can be considered as the relative cme signal to background contribution,
If the experimental measurement equals to (i.e. scales like ), then cme contribution is zero; if (i.e. scales like ), then background is close to zero and all would be cme; and if , then background and cme contributions are of similar magnitudes. The cme signal fractions with respect to rp and ep are, respectively,
Apply to data. The quantities and , and consequently and , are mainly determined by fluctuations. The smaller the collision system, the smaller the and the larger the values as shown in Fig. 1. Being defined in a single nucleus-nucleus collision, they are insensitive to many details, such as the structure functions of the colliding nuclei. This is in contrast to comparisons between two isobaric collision systems where large theoretical uncertainties are present Xu et al. (2017). There have been tremendous progresses over the past decade in our understanding of the nuclear collision geometry and fluctuations. The mcg and ampt calculations of these quantities are therefore on a rather firm ground.
Experimentally, can be assessed by measurements. cannot but may be approximated by , as demonstrated by the mcg and ampt calculations. Table 1 shows the measured in 200 GeV AuAu collisions by STAR via the zdc at beam rapidities () Wang (2006) and the forward time projection chamber (ftpc) (i.e. ) at forward/backward rapidities () Voloshin (2007), together with those via the midrapidity tpc ep () and the two- and four-particle cumulants (, ). The relative difference () between and is smaller in magnitude than from mcg and from ampt; moreover, may already be on the too-large side as it is larger than for some of the centralities whereas the opposite is expected because of a smaller nonflow contribution to . These may suggest that may not measure the purely relative to the rp, but a mixture of rp and pp. This is possible because, for instance, the zdc could intercept not only spectator neutrons but also those having suffered only small-angle elastic scatterings.
Table 1 also lists the correlator measurements by STAR with respect to Abelev et al. (2009a, 2010); Adamczyk et al. (2013) and Adamczyk et al. (2013). Although from zdc may not strictly measure the rp, our general formulism is still valid, and one can in principle extract the cme signal from those measurements. Many of the experimental systematics related to event and track quality cuts cancel in their relative difference . The remaining major systematic uncertainty comes from those in the determinations of the rp and ep resolutions or the Abelev et al. (2009a, 2010). In the STAR measurement Abelev et al. (2009a, 2010), the Voloshin (2007) was used and the systematic uncertainty was taken to be half the difference between and . In the later STAR measurement Adamczyk et al. (2013), the uncertainty is taken to be the difference between and , perceived to be physically equal, but shown not to be the case by the present work. Below we use the later, higher statistics data Adamczyk et al. (2013) but the earlier systematic uncertainty estimation Abelev et al. (2009a, 2010). The systematic uncertainty was not estimated on Adamczyk et al. (2013), though statistical uncertainties are large and likely dominate.
We average the and measurements over the centrality range 20-60%, weighted by (because the is a pair-wise average quantity). We extract the cme to bkg ratio by Eq. (14), replacing with and assuming and . We vary the “true” over the wide range between and , and at each the is replaced by (i.e. three-particle correlator over ). The fraction is obtained and shown in Fig 2 by the thick curve as a function of the “true” . The gray area is the uncertainty, , determined by the statistical uncertainty in the measurements. The vertical lines indicate the various measured values. At present the data precision does not allow a meaningful constraint on ; the limitation comes from the measurement which has an order of magnitude larger statistical error than that of . With ten-fold increase in statistics, the constraint would be the dashed curves. This is clearly where the future experimental emphasis should be placed: larger AuAu data samples are being analyzed and more AuAu statistics are to be accumulated; zdc upgrade is ongoing in the CMS experiment at the lhc; fixed target experiments at the SPS may be another viable venue where all spectator nucleons are measured in the zdc allowing possibly a better determination of .
Summary. Elliptic flow () develops in relativistic heavy ion collisions from the anisotropic overlap geometry of the participant nucleons. The participant plane azimuthal angle (), due to fluctuations, does not necessarily coincide with the reaction plane’s (). With respect to , is stronger than that with respect to . This has been known for over a decade. The magnetic field () is, on the other hand, produced mainly by spectator protons and its direction fluctuates nominally about , not . Therefore, with respect to is weaker than with respect to . This has so far not been well appreciated. We have verified these with MC Glauber (mcg) calculations and A Multi-Phase Transport (ampt) model simulations of AuAu, CuCu, RuRu, ZrZr, and PbPb collisions. One can effectively “change” in a single nucleus-nucleus collision and, at the same time, “change” in the opposite direction; the change is significant, as large as 20% in each direction in AuAu collisions. We demonstrate that this opposite behavior in a single collision system, thus with small systematic uncertainties, can be exploited to effectively disentangle the possible chiral magnetic effect (cme) from the -induced background in three-point correlator () measurements. We argue that the comparative measurements of with respect to and in the same collision system is superior to isobaric collisions where large systematics persist. We have applied this novel idea to experimental data, however, due to the poor statistical precision of the data, no conclusion can presently be drawn regarding the possible cme strength. This calls for future efforts to accumulate data statistics and to improve capabilities of Zero-Degree Calorimeters.
Acknowledgments. This work was supported in part by the National Natural Science Foundation of China under Grants No. 11647306, U1732138, 11505056 and 11605054, 11628508, and US Department of Energy Grant No. DE-SC0012910.
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