Revisit the Chiral Magnetic Effect Expectation in Isobaric Collisions at the Relativistic Heavy Ion Collider

Abstract

Isobaric 96 44Ru+ 96 44Ru and 96 40Zr+ 96 40Zr collisions at √ snn = 200 GeV have been conducted at the Relativistic Heavy Ion Collider to circumvent the large flow-induced background in searching for the chiral magnetic effect (cme), predicted by the topological feature of quantum chromodynamics (QCD). Considering that the background in isobar collisions is approximately twice that in Au+Au collisions (due to the smaller multiplicity) and the cme signal is approximately half (due to the weaker magnetic field), we caution that the cme may not be detectable with the collected isobar data statistics, within ∼2σ significance, if the axial charge per entropy density (n5/s) and the QCD vacuum transition probability are system independent. This expectation is generally verified by the Anomalous-Viscous Fluid Dynamics (avfd) model. While our estimate provides an approximate “experimental” baseline, theoretical uncertainties on the cme remain large. The chiral magnetic effect (cme) in quantum chromodynamics (qcd) refers to charge separation along strong magnetic field caused by imbalanced numbers of left-handed and right-handed quarks due to interactions of chiral fermions with metastable topological domains [1, 2]. Such domains can form due to vacuum fluctuations in qcd, which might provide a mechanism for the large CP violation in the early universe, believed to be responsible for the matterantimatter asymmetry today [3]. The cme is being actively pursued in relativistic heavy ion collisions [4, 5, 6, 7, 8]. A common variable used to measure the cme-induced charge separation in those collisions is the azimuthal correlator [9], ∆γ = γos − γss with γ ≡ 〈〈cos(φα + φβ)〉〉, where φ is the particle azimuthal angle relative to the reaction plane (rp), and the subscript α and β indicate two particles with either opposite charge sign (os) or same charge sign (ss). The measurements of ∆γ in relativistic heavy ion collisions are contaminated by a major source of background, caused by correlations of particles from a common source (such as resonance decay) which itself is anisotropically distributed about the rp as quantified by the elliptic flow parameter v2 [10, 11, 12]. To circumvent the large flow background, many innovative approaches have been proposed [13, 14, 15, 16, 17]. In addition, isobaric 96 44Ru+ 96 44Ru and 96 40Zr+ 96 40Zr collisions were proposed [18] where the backgrounds are expected to be the same because of the same mass number ∗Corresponding author Email address: [email protected] (Fuqiang Wang) Preprint submitted to Elsevier August 3, 2021 ar X iv :2 10 3. 10 37 8v 3 [ nu cl -e x] 3 1 Ju l 2 02 1 of these isobars and the magnetic fields generated by these collisions (and hence the cme signals) are appreciably different because of the different atomic numbers. Isobar collisions at nucleon-nucleon center-of-mass energy of √ snn = 200 GeV have been conducted at the Relativistic Heavy Ion Collider (RHIC) [19] and approximately 2 × 109 good minimum bias collision events [20] were collected by the STAR experiment for each of the isobar collision systems. The CME signal is expected to be significant only in mid-central collisions and vanishes in peripheral and central collisions where the magnetic field disappears. A back-of-the-envelope estimation of the significance of cme measurement is as follows. Statistical uncertainty. A single pair quantity cos(φα+φβ) is an approximately uniform distribution between [−1, 1] with a root-mean-square of σ1 = 2/ √ 12. The typical charged hadron multiplicity measured in the STAR Time Projection Chamber (|η| . 1) in 20-60% centrality Au+Au collisions is of the order of NAuAu poi ≈ 320 [21], all of which are used in analysis as particles of interest (poi). Since multiplicity scales with A, the mass number of the colliding nuclei, N isobar poi ≈ 160 in isobar collisions of the same centrality range. Thus the width of the eventwise 〈cos(φα + φβ)〉 distribution is σN = √ 2σ1/ √ (N isobar poi /2)2 ≈ 0.01. Experimentally, the rp is assessed by the reconstructed event plane (ep) and corrected by the ep resolution factor Rep ≈ 0.5 1. The statistical uncertainty on the ∆γ difference between the isobar collision systems is therefore σ = √ 2σN/ √ N isobar evt /Rep ≈ 10−6 (where N isobar evt ≈ 109 is the number of events of each species in the 20-60% centrality range). Signal strength. To estimate the cme signal strength in isobar collisions, we use Au+Au collisions as a guide. The inclusive ∆γ measured in 20-60% central Au+Au collisions is on the order of ∆γAuAu ≈ 2 × 10−4 [23, 24]. Since it is dominated by the flow-induced background and thus approximately scales inversely with N (because the number of background correlation sources is likely proportional to N and ∆γ is a pair-wise quantity, while the v2 parameters are expected to be similar between isobar and Au+Au collisions), ∆γisobar ≈ 4× 10−4 is expected in isobar collisions. Suppose the cme signal fraction in isobar collisions is fcme ≡ ∆γcme/∆γ ∼ 10%, similar to what the Au+Au data [25, 26, 27] seem to indicate, then the cme signal difference between the isobar systems is on the order of ∆(∆γisobar cme ) ≈ 0.15 fcme∆γ ≈ 6×10−6 where the factor 0.15 comes from the magnetic field difference 2. The above back-of-the-envelope estimation suggests that the signal ∆(∆γisobar cme ) is 6 times the expected statistical uncertainty σ, i.e. a measurement of 6σ significance (systematic uncertainty is expected to be small [20]). This is consistent, within a factor of 2, with the more sophisticated estimate by Deng et al. [28] of 5σ with an assumed fcme = 1/3 and a factor of 5 smaller data volume, or 3.4σ using the numbers in our estimation. In those estimates the crucial input is the cme signal fraction fcme for isobar collisions. It has been generally expected that the cme fractions are similar in Au+Au and isobar collisions. However, there is no reason to expect so. In fact, as aforementioned, the background scales with 1/A, which yields a factor of 2 larger background in isobar collisions than in Au+Au. The magnetic field in heavy ion collisions is expected to scale as B ∝ A1/3 and since ∆γcme ∝ B2, it would possibly lead to a factor of 1.6 smaller signal in isobar collisions. In other words, based on simple reasoning, the natural expectation is rather fcme ∝ A5/3. As a net result, there could possibly be over a factor of 3 difference in the fcme between isobar and Au+Au collisions, smaller for the former. This suggests that our back-of-the-envelope estimate would give 2σ significance and the more sophisticated estimate [28] would give 1σ significance. Clearly, a more rigorous investigation is needed, for which we use the Anomalous-Viscous Fluid Dynamics (avfd) [31, 29]. The avfd model implements anomalous fluid dynamics to describe the evolution of fermion currents in the quarkgluon plasma created in relativistic heavy ion collisions [31, 29, 32]. The underlying bulk medium evolution is described by the VISH2+1 hydrodynamics [33]. The model integrates the anomalous fluid dynamics with the normal viscous hydrodynamics in the same framework. It incorporates ingredients including the initial conditions, the magnetic fields, and the viscous transport coefficients, and allows interplay between the evolution of the axial charge 1In experimental analysis, the ep is often reconstructed using the so-called subevent method, and the poi’s used for ∆γ are reduced in statistics. We use all particles in our estimation so that the estimated uncertainties are the best achievable. The ep in such analysis is often assessed by a single particle with the ep resolution being the particle v2. In this case the overall resolution effect on the final statistical uncertainty is given by √ Nv2 under small v2 approximation [22]. This factor is typically also of the order of 0.5. 2Although the atomic numbers differ by 10% between the isobars 96 44Ru and 96 40Zr, the expected ∆B/B difference is a bit smaller because of a slight radii difference in the charge density distributions [28, 29, 30]. Since ∆γcme ∝ B2 [2, 4], the effect is 2∆B/B. This uncertainty on ∆B/B does not significantly alter our estimates.