Scaling properties of background- and chiral-magnetically-driven charge separation: evidence for detection of the chiral magnetic effect in heavy ion collisions

The scaling properties of the $R_{\Psi_2}(\Delta S)$ correlator and the $\Delta\gamma$ correlator are used to investigate a possible chiral-magnetically-driven (CME) charge separation in $p$+Au, $d$+Au, Ru+Ru, Zr+Zr, and Au+Au collisions at $\sqrt s_{\mathrm{NN}}=200$~GeV, and in $p$+Pb ($\sqrt s_{\mathrm{NN}}=5.02$~TeV) and Pb+Pb collisions at $\sqrt s_{\mathrm{NN}}=5.02$ and $2.76$~TeV. The results for $p$+Au, $d$+Au, $p$+Pb, and Pb+Pb collisions, show the $1/{\rm N_{ch}}$ scaling for background-driven charge separation. However, the results for Au+Au, Ru+Ru, and Zr+Zr collisions show scaling violations which indicate a CME contribution in the presence of a large background. In mid-central collisions, the CME accounts for approximately 27\% of the signal + background in Au+Au and roughly a factor of two smaller for Ru+Ru and Zr+Zr, which show similar magnitudes.

Metastable domains of gluon fields with non-trivial topological configurations can form in the magnetized chiral relativistic quark-gluon plasma (QGP) [1] produced in collisions at RHIC and the LHC.The colliding ions generate the magnetic field ( B) at early times [2].The interaction of chiral quarks with the gluon fields can drive a chiral imbalance resulting in an electric current J V = N c e B 2π 2 µ A , along the B-field, i.e., perpendicular to the reaction plane; N c is the color factor, and µ A is the axial chemical potential that quantifies the imbalance between right-and left-handed quarks.The resulting final-state charge separation, termed the chiral magnetic effect (CME) [1], is of great experimental and theoretical interest.However, its experimental characterization has been hampered by significant background.
The charge separation can be quantified via the P-odd sine term a 1 , in the Fourier decomposition of the charged-particle azimuthal distribution [3]: where ∆φ = φ − Ψ RP gives the particle azimuthal angle with respect to the reaction plane (RP) angle, and v n and a n denote the coefficients of the P-even and P-odd Fourier terms, respectively.A direct measurement of a 1 , is not possible due to the strict global P and CP symmetry of QCD.However, their fluctuation and/or variance ã1 = a 2 1 1/2 can be measured with charge-sensitive correlators such as the γ-correlator [3] and the R Ψ 2 (∆S ) correlator [4].
The γ-correlator measures charge separation as: where Ψ 2 is the azimuthal angle of the 2 nd -order event plane which fluctuates about the RP, φ denote the particle azimuthal emission angles, α, β denote the electric charge (+) or (−) and SS and OS represent same-sign (++, −−) and opposite-sign (+ −) charges.Measurements of the quotient ∆γ/v 2 with the 2 nd -order anisotropy coefficient v 2 , are usually employed to aid quantification of the background-driven charge separation.
The R Ψ 2 (∆S ) correlator measures charge separation relative to Ψ 2 via the ratio: , where C Ψ 2 (∆S ) and C ⊥ Ψ 2 (∆S ) are correlation functions that quantify charge separation ∆S , approximately parallel and perpendicular (respectively) to the B-field.The charge-shuffling procedure used to construct the correlation functions ensures identical properties for their numerator and denominator, except for the charge-dependent correlations, which are of interest [4].C Ψ 2 (∆S ) measures both CME-and backgrounddriven charge separation while C ⊥ Ψ 2 (∆S ) measures only the background.After correcting the R Ψ 2 (∆S ) distributions for the effects of particle-number fluctuations and the event-plane resolution, their inverse variance σ −2 R Ψ 2 are used to quantify the charge separation [4].
In this work, we use model simulations to chart the scaling properties of σ −2 R Ψ 2 and ∆γ/v 2 for the background and signal + background, respectively, in A+A collisions.We then leverage these scaling properties to identify and characterize a possible CME-driven charge separation using previously published data for p+Au, d+Au, Ru+Ru, Zr+Zr and Au+Au collisions at RHIC [5][6][7][8][9][10], and p+Pb and Pb+Pb collisions at the LHC [11][12][13][14].
Figure 1 shows the results for σ −2 and ∆γ/v 2 obtained with the AVFD and Hijing models for Au+Au collisions.Note that these models emphasize different sources for the charge-dependent non-flow background; the initial axial charge density n 5 /s and the degree of local charge conservation (LCC) regulate the magnitude of the CME-and backgrounddriven charge separation in the AVFD model.The solid triangles in Fig. 1 show that the background scales as 1/N ch -the expected trend for the charge-dependent non-flow correlations.By contrast, the signal (Sig.)+ background values (solid diamonds) indicate positive deviations from the background scaling [16,17].This dependence can be represented as; Note that for c ∼ 0 the 1/N ch scaling for the background is retrieved, as demonstrated with the AVFD model in Fig. 1.
The scaling violation gives a direct signature of the CME-driven contributions to the charge separation (Figs. 1 (a) and (c)).
It can be quantified via the fraction: and ∆γ/v 2 measurements for these collisions can be leveraged with 1/N ch scaling to obtain a quantitative estimate of the background over the entire centrality span (cf.Fig. 1).Here, an important proviso is to experimentally establish that the background in p(d)+A and A+A collisions scale over the full centrality span.The v 2 and ∆γ values reported for p+Au, d+Au, Ru+Ru, Zr+Zr and Au+Au collisions at RHIC [5][6][7][8][9][10], and p+Pb and Pb+Pb collisions at the LHC [12][13][14][15] were used to investigate the scaling properties of ∆γ/v 2 .Fig. 2 shows the results for p+Pb and Pb+Pb collisions at √ s NN = 5.02 TeV.They indicate that ∆γ/v 2 essentially scales as 1/N ch (c ≈ 0), suggesting negligible CME contributions in these collisions.They also confirm that the combined sources of background (LCC, resonances, back-to-back jets, ...), which should be substantial, especially for p+Pb, scale as 1/N ch .Note as well that the CME contribution is negligible in p(d)+A collisions because of significant reductions in B, and the sizable de-correlation between the event plane and the B-field [12].Thus, the scaling patterns of ∆γ/v 2 for these systems' sizable backgrounds give a direct experimental constraint on the validity of 1/N ch scaling of the background.The scaling results for Au+Au collisions at √ s NN = 200 GeV are shown in Fig. 3.The 1/N ch scaling apparent for d+Au collisions (Fig. 3 (a)) confirms the expectation that the CME is negligible in these collisions.It also confirms that the combined sources of background (LCC, resonances, back-to-back jets, ...), which could be substantial in d+Au collisions, show 1/N ch scaling.In contrast to d+Au, the results for Au+Au (Fig. 3(b)) show visible indications of a violation (c > 0) to the 1/N ch scaling observed for background-driven charge separation in p(d)+A collisions.Similar violations were observed for Ru+Ru and Zr+Zr [17].The scaling violation is similar to that observed for signal + background in Figs. 1 (a) and (c), suggesting an unambiguous non-negligible CME contribution to the measured ∆γ/v 2 in Au+Au, Ru+Ru, and Zr+Zr collisions.The estimates of the background for all three systems are obtained by leveraging the ∆γ/v 2 measurements for peripheral and central collisions with 1/N ch scaling [17].Here, it is noteworthy that the simulated results from the AVFD and HIJING models, as well as the measurements presented in Figs. 2 and 3(a), provide strong constraints that the combined sources of background, scale as 1/N ch over the full centrality span.The background estimates were used to extract f CME values for Au+Au, (Fig. 3 (c)) Ru+Ru and Zr+Zr collisions respectively.They indicate non-negligible f CME values that vary with centrality.In mid-central collisions, f CME ∼ 27% for Au+Au collisions, which is roughly a factor of two larger than the values for Ru+Ru and Zr+Zr.Within the uncertainties, no significant difference between the values for Ru+Ru and Zr+Zr was observed, suggesting that ∆γ/v 2 is sensitive to CME-driven charge separation in Ru+Ru and Zr+Zr collisions but may be insensitive to the signal difference between them [17].
In summary, the scaling properties of the R Ψ 2 (∆S ) and the ∆γ correlators have been used to characterize the CME in several colliding systems at RHIC and the LHC.The results for p+Au and d+Au collisions at

Figure 1 .
Figure 1.σ −2 R Ψ 2 vs. 1/N ch (a) and ∆γ/v 2 vs. 1/N ch (b) and (c) for simulated Au+Au collisions at √ s NN = 200 GeV.The results for σ −2 R Ψ 2 and ∆γ/v 2 from the AVFD model are shown for background and for signal + background as indicated.The Hijing model results are only shown for the background.The dashed lines represent linear fits to the background values.

2 EPJ
for the small values of n 5 /s indicated in Fig. 1.Here, a, b and c are parameters; c characterizes the degree of the scaling violation.Web of Conferences 276, 06005 (2023) https://doi.org/10.1051/epjconf/202327606005SQM 2022

R Ψ 2 ( 2 R Ψ 2 and
S ig.+ Bkg.)].The scaling patterns in Fig. 1 suggest that the observation of 1/N ch scaling for the experimental σ −∆γ/v 2 measurements would strongly indicate background-driven charge separation with little room for a CME contribution.However, observing a violation of this 1/N ch scaling would indicate the CMEdriven contribution.Figs. 1 (a) and (c) also indicate comparable background and signal + background σ −2 R Ψ 2 and ∆γ/v 2 values in central and peripheral collisions, suggesting that the background dominates over that of the CME-driven contributions in these collisions.Note the reduction of B in central collisions and the enhanced de-correlation between the event plane and the B-field in peripheral collisions.Since the background dominates in central and peripheral collisions, the σ −2 R Ψ 2

Figure 3 .
Figure 3. ∆γ/v 2 vs. 1/N ch [(a) and (b)] and f CME vs. centrality (c) for d+Au and Au+Au collisions at √ s NN = 200 GeV.The dotted and dashed lines indicate an estimate of the background contributions.The data are taken from Refs.[7, 9? ].
√ s NN = 200 GeV and p+Pb ( √ s NN = 5.02 TeV) and Pb+Pb collisions at √ s NN = 5.02 and 2.76 TeV, scales as 1/N ch consistent with backgrounddriven charge separation.However, the results for Au+Au, Ru+Ru and Zr+Zr collisions ( √ s NN = 200 GeV) show scaling violations which indicate a CME-driven contribution in the presence of significant background.In mid-central collisions, f CME ∼ 27% for Au+Au collisions and approximately a factor of two smaller in Ru+Ru and Zr+Zr collisions but with similar magnitudes for the two isobars.