A sensitivity study of the primary correlators used to characterize chiral-magnetically-driven charge separation
aa r X i v : . [ nu c l - e x ] F e b A sensitivity study of the primary correlators used to characterizechiral-magnetically-driven charge separation
Niseem Magdy, ∗ Mao-Wu Nie,
2, 3
Guo-Liang Ma,
4, 5, † and Roy A. Lacey ‡ Department of Physics, University of Illinois at Chicago, Chicago, Illinois 60607, USA Institute of Frontier and Interdisciplinary Science,Shandong University, Qingdao, Shandong, 266237, China Key Laboratory of Particle Physics and Particle Irradiation,Ministry of Education, Shandong University, Qingdao, Shandong, 266237, China Key Laboratory of Nuclear Physics and Ion-beam Application (MOE),Institute of Modern Physics, Fudan University, Shanghai 200433, China Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China Depts. of Chemistry & Physics, Stony Brook University, Stony Brook, New York 11794, USA (Dated: February 20, 2020)A Multi-Phase Transport (AMPT) model is used to study the detection sensitivity of two of theprimary correlators – ∆ γ and R Ψ – employed to characterize charge separation induced by theChiral Magnetic Effect (CME). The study, performed relative to several event planes for differentinput “CME signals”, indicates a detection threshold for the fraction f CME = ∆ γ CME / ∆ γ , whichrenders the ∆ γ -correlator insensitive to values of the Fourier dipole coefficient a . . R Ψ correlator indicates concave-shaped distributions with inverse widths ( σ − Ψ2 ) that are linearlyproportional to a , and independent of the character of the event plane used for their extraction.The sensitivity of the R Ψ correlator to minimal CME-driven charge separation in the presence ofrealistic backgrounds, could aid better characterization of the CME in heavy-ion collisions. PACS numbers: 25.75.-q, 25.75.Gz, 25.75.Ld
Ion-Ion collisions at both the Relativistic Heavy IonCollider (RHIC) and the Large Hadron Collider (LHC)create hot expanding fireballs of quark-gluon plasma(QGP) in the background of a strong magnetic field [1–3].Topologically nontrivial sphaleron transitions [via the ax-ial anomaly] [4–6] can induce different densities of right-and left-handed quarks in the plasma fireballs, resultingin a quark electric current along the ~B -field. This phe-nomenon of the generation of a quark electric current( ~J Q ) in the presence of a magnetic field is termed thechiral magnetic effect (CME) [7, 8]: ~J Q = σ ~B, σ = µ Q π , (1)where, σ is the chiral magnetic conductivity, µ is thechiral chemical potential that quantifies the axial chargeasymmetry or imbalance between right- and left-handedquarks in the plasma, and Q is the electric charge [8–11].Full characterization of the CME, which manifestsexperimentally as the separation of electrical chargesalong the ~B -field [7, 8], can give fundamental insight onanomalous transport and the interplay of chiral symme-try restoration, axial anomaly and gluon topology in theQGP [12–16].Charge separation stems from the fact that the CMEpreferentially drives charged particles, originating fromthe same “P-odd domain”, along or opposite to the ~B -field depending on their charge. This separation can bequantified via measurements of the first P -odd sine term a , in the Fourier decomposition of the charged-particle azimuthal distribution [17]: dN ch dφ ∝ [1 + 2 X n v n cos( n ∆ φ ) + a n sin( n ∆ φ ) + ... ](2)where ∆ φ = φ − Ψ RP gives the particle azimuthal anglewith respect to the reaction plane (RP) angle, and v n and a n denote the coefficients of P -even and P -odd Fourierterms, respectively. A direct measurement of the P-oddcoefficients a n , is not possible due to the strict global P and CP symmetry of QCD. However, their fluctua-tion and/or variance ˜ a n = (cid:10) a n (cid:11) / can be measured withsuitable correlators.The CME-driven charge separation is small becauseonly a few particles from the same P -odd domain are cor-related. Moreover, both the initial axial charge and thetime evolution of the magnetic field (c.f. Eq. 1) are un-constrained theoretically, and it is uncertain whether aninitial CME-driven charge separation could survive thesignal-reducing effects of the reaction dynamics, and stillproduce a signal above the detection threshold. Besides,it is uncertain whether a charge separation that sur-vives the expansion dynamics would still be discerniblein the presence of the well-known background correla-tions which contribute and complicate the measurementof CME-driven charge separation [14, 18–22]. Thus, thecorrelators used to characterize the CME, not only needto suppress background-driven charge-dependent corre-lations, such as the ones from resonance decays, chargeordering in jets, etc., but should also be sensitive tosmall charge separation signals in the presence of thesebackgrounds. The latter requirement is especially im-portant for ongoing measurements [at RHIC] designedto detect the small signal difference between the Ru+Ruand Zr+Zr isobars [23].In this work we use the AMPT model [24] with varyingamounts of input charge separation ∆ S , characterized bythe partonic dipole term a , to study the detection sensi-tivity of the ∆ γ and the R Ψ (∆ S ) correlators. The modelis known to give a good representation of the experimen-tally measured particle yields, spectra, flow, etc.,[24–29].Therefore, it provides a realistic estimate of both themagnitude and the properties of the background-drivencharge separation one might encounter in the data setscollected at RHIC and the LHC.For these sensitivity tests, we simulated Au+Au col-lisions at √ s NN = 200 GeV with the new version ofthe AMPT model that incorporates string melting andlocal charge conservation. There are four primary in-gredients for each of these collisions: (i) an initial-state,(ii) a parton cascade phase, (iii) a hadronization phasein which partons are converted to hadrons, and (iv) ahadronic re-scattering phase prior to kinetic freeze-out.The initial-state mainly simulates the spatial and mo-mentum distributions of minijet partons from QCD hardprocesses and soft string excitations as encoded in theHIJING model [30, 31]. The parton cascade takes ac-count of the strong interactions among partons throughelastic partonic collisions controlled by a parton interac-tion cross section [32]. Hadronization, or the conversionfrom partonic to hadronic matter, is simulated via a co-alescence mechanism. Subsequent to hadronization, theART model is used to simulate baryon-baryon, baryon-meson and meson-meson interactions [33].A formal mechanism for the CME is not implementedin AMPT. However, modifications can be made to themodel to mimic CME-induced charge separation [34] byswitching the p y values of a fraction of the downwardmoving u ( ¯ d ) quarks with those of the upward moving ¯ u ( d ) quarks to produce a net charge-dipole separation inthe initial-state. Here, the x axis is along the directionof the impact parameter b , the z axis points along thebeam direction, and the y axis is perpendicular to the x and z directions, i.e , the direction of the proxy ~B -field.The strength of the proxy CME signal is regulated by thefraction f of the initial input charge separation [34, 35]: f = N +( − ) ↑ ( ↓ ) − N +( − ) ↓ ( ↑ ) N +( − ) ↑ ( ↓ ) + N +( − ) ↓ ( ↑ ) , f = 4 π a (3)where N is the number of a given species of quarks, “+ ′′ and “ − ′′ denote positive and negative charges, respec-tively, and ↑ and ↓ represent the directions along andopposite to that of the y axis. Eq. 3 also shows thatthe fraction f , is related to the P -odd dipole term a ,defined in Eq. 2. Note that this initial partonic chargeseparation a , is different from the final hadrons’ charge separation a , often referred to in the literature and im-plemented in other models. Cross-checks made with theAnomalous-Viscous Fluid Dynamics model [36, 37] sug-gests that the two are linearly proportional to a verygood approximation. Simulated events, generated for abroad set of f values, were analyzed with both the ∆ γ and the R Ψ (∆ S ) correlators, to evaluate their respectivesensitivity as discussed below quantitatively.The charge-dependent correlator, γ αβ [17] , has beenwidely used at RHIC [38–44] and the LHC [22, 45] inongoing attempts to identify and quantify CME-drivencharge separation: γ αβ = D cos (cid:0) φ ( ± ) α + φ ( ± ) β − (cid:1)E , ∆ γ = γ β − γ α , where φ α , φ β denote the azimuthal emission angles forlike-sign (++ , −− ) and unlike-sign (+ − ) particle pairs.The question as to whether the experimental measure-ments for ∆ γ indicate the CME, remain inconclusive be-cause of several known sources of background correlationsthat can account for most, if not all, of the measure-ments [14, 18–21].A recent embellishment to the ∆ γ correlator is the pro-posal to leverage the ratios of ∆ γ and elliptic flow ( v )measurements, obtained relative to the reaction plane(Ψ RP ) and the participant plane (Ψ PP ) r = ∆ γ (Ψ RP )∆ γ (Ψ PP ) , r = v (Ψ RP ) v (Ψ PP ) , (4)to simultaneously constrain the CME and background(Bkg) contributions to ∆ γ [46, 47]:∆ γ (Ψ PP ) = ∆ γ CME (Ψ PP ) + ∆ γ Bkg (Ψ PP ) , ∆ γ (Ψ RP ) = ∆ γ CME (Ψ RP ) + ∆ γ Bkg (Ψ RP ) , (5)and ∆ γ CME (Ψ PP ) = r × ∆ γ CME (Ψ RP ) , ∆ γ Bkg (Ψ RP ) = r × ∆ γ Bkg (Ψ PP ) , (6)where it is assumed that the CME is proportional to themagnetic field squared and the background (Bkg) is pro-portional to v . The fraction of the measured ∆ γ (Ψ PP ),attributable to the CME, can then be estimated as [46]; f CME = ∆ γ CME (Ψ PP ) / ∆ γ (Ψ PP ) = f /f , where f = r r − f = 1 r − . (7)The underlying idea behind the constraints expressedin Eqs. 4 - 7 is that the v -driven background is morestrongly correlated with Ψ PP [determined by the maxi-mal particle density in the elliptic azimuthal anisotropyand the beam axis], than with Ψ RP [determined by theimpact vector ~b and the beam direction]. By contrast,the ~B -field, which drives the CME, behaves oppositely– weaker correlation with Ψ PP and stronger correlationwith Ψ RP . We will employ this new method of leverag-ing the measurements of r and r to extract f CME fromAMPT events as discussed below.The operational details of the construction and theresponse of the R Ψ m (∆ S ) correlator is described inRefs. [48] and [49]. It is constructed for each event planeΨ m , as the ratio: R Ψ m (∆ S ) = C Ψ m (∆ S ) /C ⊥ Ψ m (∆ S ) , m = 2 , , (8)where C Ψ m (∆ S ) and C ⊥ Ψ m (∆ S ) are correlation functionsthat quantify charge separation ∆ S , parallel and per-pendicular (respectively) to the ~B -field. C Ψ (∆ S ) mea-sures both CME- and backgrond-driven charge separa-tion while C ⊥ Ψ (∆ S ) measures only background-drivencharge separation. The absence of a strong correlationbetween the orientation of the Ψ plane and the ~B -field, also renders C Ψ (∆ S ) and C ⊥ Ψ (∆ S ) insensitive toa CME-driven charge separation, but not to the back-ground, so it can give crucial additional insight on the rel-ative importance of background-driven and CME-drivencharge separation. However, they are not required forthe sensitivity studies presented in this work.The correlation functions used to quantify charge sep-aration parallel to the ~B -field, are constructed from theratio of two distributions [50]: C Ψ m (∆ S ) = N real (∆ S ) N Shuffled (∆ S ) , m = 2 , , (9)where N real (∆ S ) is the distribution over events, of chargeseparation relative to the Ψ m planes in each event:∆ S = (cid:10) S h + p (cid:11) − (cid:10) S h − n (cid:11) , (10)∆ S = p P sin( m ∆ ϕ m ) p − n P sin( m ∆ ϕ m ) n , (11)where n and p are the numbers of negatively- and posi-tively charged hadrons in an event, ∆ ϕ m = φ − Ψ m and φ is the azimuthal emission angle of the charged hadrons.The N Shuffled (∆ S ) distribution is similarly obtained fromthe same events, following random reassignment (shuf-fling) of the charge of each particle in an event. Thisprocedure ensures identical properties for the numeratorand the denominator in Eq. 9, except for the charge-dependent correlations which are of interest. The corre-lation functions C ⊥ Ψ m (∆ S ), that quantify charge separa-tion perpendicular to the ~B -field, are constructed withthe same procedure outlined for C Ψ m (∆ S ), but withΨ m replaced by Ψ m + π/m , to ensure that a possibleCME-driven charge separation does not contribute to C ⊥ Ψ m (∆ S ).The magnitude of the CME-driven charge separationis reflected in the width σ Ψ of the concave-shaped v p T (GeV/c) AMPT Au+Au 200 GeV10-50%
RPSPPP
FIG. 1. Comparison of the simulated v ( p T ) obtained in 10-50% Au+Au collisions ( √ s NN = 200 GeV) with Ψ RP , Ψ SP and Ψ PP , see text. distribution for R Ψ (∆ S ), which is also influenced byparticle number fluctuations and the resolution of Ψ .That is, stronger CME-driven signals lead to narrowerconcave-shaped distributions (smaller widths), which aremade broader by particle number fluctuations and poorerevent-plane resolutions. The influence of the particlenumber fluctuations can be minimized by scaling ∆ S bythe width σ ∆ Sh of the distribution for N shuffled (∆ S ) i.e. ,∆ S ′ = ∆ S/σ ∆ Sh . Similarly, the effects of the event ∆ γ a % AMPT Au+Au 200 GeV10-50% × -3 RPSPPP FIG. 2. Comparison of the simulated ∆ γ obtained in 10-50%Au+Au collisions ( √ s NN = 200 GeV) with respect to Ψ RP ,Ψ SP and Ψ PP , for several input charge separation fractionscharacterized by the P -odd dipole coefficient a . plane resolution can be accounted for by scaling ∆ S ′ bythe resolution factor δ Res , i.e. , ∆ S ′′ = ∆ S ′ × δ Res , where δ Res = σ Res × e (1 − σ Res ) and σ Res is the event plane res-olution [48].The 10 −
50% central AMPT events, generated for sev-eral input values of charge separation f (cf. Eq. 3),relative to the reaction- Ψ RP , spectator- Ψ SP and theparticipant plane Ψ PP , were analyzed to extract f CME via the ∆ γ correlator and σ Ψ via the R Ψ (∆ S ) correla-tor. Approximately 10 events were generated for eachvalue of f . The analyses included charged particles with -0.5 0 0.5 1 0 1 2 3 4 5 f x a % AMPT Au+Au 200 GeV10-50% f f f CME
FIG. 3. The dependence of f , f and f CME on differentinput charge separation characterized by the dipole coefficient a , see Eqs. 3 and 7. Results are shown for 10-50% centralAu+Au ( √ s NN = 200 GeV) AMPT events. | η | < . . < p T < c .To enhance the statistical significance of the measure-ments, the participant plane Ψ PP was determined withcharged hadrons in the range 2 . < η < . | η | < . PP . Representative results are summa-rized in Figs. 1 - 4.Figure 1 compares the v ( p T ) obtained with Ψ RP , Ψ SP and Ψ PP for 10-50% Au+Au collisions. It shows the ex-pected similarity between the results for Ψ RP and Ψ SP ,as well as larger values for Ψ PP that confirm the en-hanced fluctuations associated with the participant ge-ometry and consequently, the initial-state eccentricity ε .This difference is essential for the procedure outlined inEqs. 4 - 7.A similar comparison of the ∆ γ results for the threeplanes is given in Fig. 2. It shows that for a . γ values obtained with Ψ PP are larger than thoseobtained with Ψ RP and Ψ SP ; there is also little, if any,difference between the values obtained with Ψ RP and Ψ SP over the full range of the input a values. This lattertrend is to be expected since the fluctuation of Ψ SP aboutΨ RP is small. For a & γ values for Ψ RP andΨ SP become larger than the ones for Ψ PP (not shownin Fig. 2), consistent with a stronger influence from theproxy CME-driven charge separation.The extracted values of v and ∆ γ , with respect toΨ PP and Ψ SP were used to evaluate f , f and f CME fol-lowing the procedure outlined in Eqs. 4 - 7 [46]. Fig. 3summarizes the a dependence of f , f and f CME . It in-dicates a flat f , consistent with the expectation that the v fluctuations should be relatively insensitive to the in-troduction of small a signals. By contrast, f and f CME ,which are both negative for a . . a and become positive for a & . f CME suggests that for a . . f CME are invalid. Note that f CME is only 0.6even for a relatively large input signal of a = 4 . f CME (cf. Fig. 3) suggests a “turn-on” a value, below which,the modified ∆ γ correlator (cf. Eqs. 4 - 7) is unableto detect a CME-driven signal. This detection thresholdcould pose a significant limitation for CME detection andcharacterization with this correlator, because it is com-parable to, or larger than the magnitude of the CME-driven charge separation expected in actual experiments.Equally important is the fact that f CME ≤ . f CME measurements.The sensitivity of the R Ψ XX (∆ S ) (XX = RP , SP , PP)correlator to varying degrees of input CME-driven chargeseparation (characterized by a ) was studied using thesame AMPT events employed in the leveraged ∆ γ study.Figs. 4(a) - (e) show the R Ψ XX (∆ S ) correlator distribu-tions obtained for 10 −
50% central Au+Au collisions,relative to Ψ RP , Ψ SP and Ψ PP for several values of a asindicated. In each of these plots, ∆ S is scaled to accountfor the effects of number fluctuations and event planeresolution as outlined earlier and in Ref. [48].The concave-shaped distribution, apparent in eachpanel of Fig. 4 (b) - (e), confirms the input charge sepa-ration signal in each case; note the weakly convex-shapeddistribution for a = 0 in Fig. 4 (a). Note as well thatin contrast to the ∆ γ correlator, the R Ψ XX (∆ S ) distribu-tions are independent of the plane used to measure them,suggesting that they are less sensitive to the v drivenbackground and their associated fluctuations. The ap-parent decrease in the widths of these distributions with a , also confirm the expected trend.To quantify the implied signal strengths, we extractedthe width σ R Ψ2 of the R Ψ (∆ S ) distributions obtainedfor the respective values of a . Fig. 4(f) shows the in-verse widths σ − Ψ2 vs. a . They indicate an essentiallylinear dependence on a (note the dotted line fit). Here,it is noteworthy that for a . . σ R Ψ2 with good ac-curacy. These results suggests that the R Ψ m (∆ S ) corre-lator not only suppresses background, but is sensitive tovery small CME-driven charge separation in the presenceof such backgrounds.In summary, we have used both the R Ψ (∆ S ) correla-tor and an event-plane-leveraged version of the ∆ γ cor-relator to analyze AMPT events with varying degrees ofinput proxy CME signals. Our sensitivity study indi-cates a turn-on threshold for f CME = ∆ γ CME / ∆ γ , whichrenders the leveraged ∆ γ -correlator insensitive to inputsignals with a . . -3 -2 -1 0 1 2 3 (a) AMPT Au+Au10-50% a = 0.0% R Ψ ( ∆ S ″ ) RPSPPP (b) a = 2.0% ( 〈 p T 〉 ev ⁄ 〈 p T 〉 ) - 1 -3 -2 -1 0 1 2 3 (c) a = 2.5% (d) a = 3.0% R Ψ ( ∆ S ″ ) ∆ S ″ -3 -2 -1 0 1 2 3 (e) a = 4.0% ∆ S ″
0 1 2 3 4 5 0 0.1 0.2 0.3 (f) a % σ -1R Ψ FIG. 4. Comparison of the R Ψ (∆ S ) correlators obtained with respect to Ψ RP , Ψ SP and Ψ PP for several a values, as indicated,for 10-50% Au+Au collisions at √ s NN = 200 GeV (a) - (e). Panel (f) shows the a dependence of the inverse widths σ − Ψ2 ,extracted from the R Ψ (∆ S ) distributions; the dotted line represents a linear fit. strictions on its use to detect the CME. By contrast,the a -dependent R Ψ (∆ S ) correlators indicate inversewidths σ − Ψ2 , that are linearly dependent on a , and in-dependent of the character of the event plane (Ψ RP , Ψ SP or Ψ PP ) used for their extraction. These results not onlyhave implications for the interpretation of current andfuture f CME = ∆ γ CME / ∆ γ measurements; they furtherindicate that the R Ψ (∆ S ) correlator can provide robustquantification of minimal CME-driven charge separationin the presence of realistic backgrounds, that could aidcharacterization of the CME in RHIC and LHC collisions. ACKNOWLEDGMENTS
This research is supported by the US Department ofEnergy, Office of Science, Office of Nuclear Physics, undercontracts DE-FG02-87ER40331.A008 (RL), DE-FG02-94ER40865 (NM) and by the National Natural ScienceFoundation of China under Grants No. 11890714, No.11835002, No. 11961131011, and No. 11421505, the KeyResearch Program of the Chinese Academy of Sciencesunder Grant No. XDPB09. ∗ [email protected] † [email protected] ‡ [email protected][1] V. Skokov, A. Yu. Illarionov, and V. Toneev, “Es-timate of the magnetic field strength in heavy-ioncollisions,” Int. J. Mod. Phys.
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