Remarks on possible local parity violation in heavy ion collisions
aa r X i v : . [ nu c l - t h ] D ec Remarks on possible local parityviolation in heavy ion collisions
Adam Bzdak a,b , Volker Koch a , and Jinfeng Liao aa Lawrence Berkeley National Laboratory, 1 Cyclotron RoadMS70R0319, Berkeley, CA 94720, USA ∗ b Institute of Nuclear Physics, Polish Academy of SciencesRadzikowskiego 152, 31-342 Krakow, PolandAugust 1, 2018
Abstract
In this note we discuss some observations concerning the possible localparity violation in heavy ion collisions recently announced by the STAR Col-laboration. Our results can be summarized as follows (i) the measured corre-lations for same charge pairs are mainly in-plain and not out of plane, (ii) ifthere is a parity violating component it is large and, surprisingly, of the samemagnitude as the background, and (iii) the observed dependence of the signalon the transverse momentum ( p t ) is consistent with a soft boost in p t andthus in line with expectations from the proposed chiral magnetic effect.PACS numbers: 25.75.-q, 25.75.Gz, 11.30.ErKeywords: local parity violation, chiral magnetic effect ∗ e-mails: [email protected], [email protected], [email protected] Introduction
Recently the STAR collaboration announced [1] the results on possible local parityviolation in heavy ion collisions. In Ref. [2, 3] it was argued that in the hot densematter created in heavy ion collisions local, instanton or sphaleron, transitions toQCD vacua with different topological charge may result in metastable domains,where parity is locally violated.In this paper we will solely concentrate on an analysis of the experimental re-sults. We will neither attempt to provide alternative explanations for the observedcorrelations, such as e.g. given in Ref. [4] nor will we discuss the likelihood that theproposed effect may occur in a heavy ion collision. For detailed discussion of theunderlying mechanism and the latest theoretical review of this problem we refer thereader to Ref. [3].The phenomenon due to local parity violation, which is of relevance for thediscussion here, is the so called chiral magnetic effect [2, 3]. It leads to the separationof negatively and positively charged particles along the system’s angular momentum(or equivalently the direction of the magnetic field) into two hemispheres separatedby the reaction plane. As a result, the system exhibits a electric current along thedirection of the angular momentum, and thus breaks parity locally in a given event.However, since instanton (sphaleron) and anti-instanton (anti-sphaleron) transitionsoccur equally likely, the chiral magnetic current is either aligned or anti-alignedwith the angular momentum. As a result, the expectation value of any parity oddobservable, such as D ~j CM ~I E vanishes. Here ~j CM is the chiral-magnetic current and ~I is the angular momentum. Consequently, a direct measurement of parity violationeven in a small subsystem is impossible. However, one may attempt to identifythe existence of these parity violation domains by studying the fluctuations or thevariance of a parity-odd observable. Since the variance of a parity-odd observableis parity even, in principle other, genuinely parity-even, effects may contribute, andone needs to separate those carefully before being able to draw any conclusionsabout the existence of local, parity violating domains.In Ref. [5] Voloshin proposed a method to measure the variance of a parity oddobservables. He suggested to measure the following correlator h cos( φ α + φ β − RP ) i ,where Ψ RP , φ α and φ β denote the azimuthal angles of the reaction plane and pro-duced charged particles respectively, see Fig. 1.As we will discuss in more detail in Section 2 this rather involved correlationfunction has the advantage that correlations which are independent of the reactionplane do not contribute. As a result, a large fraction of the expected backgroundshould cancel. Recently the STAR collaboration has reported the measurement ofthe above correlation function [1], both integrated over the entire acceptance as well2 φ α φ β Figure 1: The transverse plain in a collision of two heavy ions. Ψ RP , φ α and φ β denote the azimuthal angles of the reaction plane and produced charged particles,respectively.as differential in transverse momentum and pseudo-rapidity.This paper is organized as follows. In the following section we will analyzethe integrated STAR result and will suggest additional measurements necessary tofurther clarify the situation. In the subsequent section, we will concentrate on the p t differential results and explore to which extend they are consistent with the expectedsoft phenomena due to the chiral magnetic effect. In Ref. [1] the details of the STAR measurement are given. Among other thingsSTAR shows the results for h cos( φ α − φ β ) i and for h cos( φ α + φ β − φ c ) i , where φ α,β,c are the azimuthal angles of the produced charged particles. The paper gives reason-able arguments that h cos( φ α + φ β − φ c ) i = h cos( φ α + φ β − RP ) i v ,c , (1)where Ψ RP is the angle of the reaction plane, and v ,c characterizes the ellipticanisotropy for the particle with angle φ c .For the rest of the discussion we will assume that the relation (1) is correct. Asa consequence, we will work in a frame where the reaction plane is defined by the x − z coordinates and where the y direction is perpendicular to the reaction plane.In other word we work in a frame where Ψ RP = 0, see Fig. 1. Furthermore, since h cos( φ α − φ β ) i is independent of the direction of the reaction plane , it will be thesame also in the frame where the reaction plane is specified e.g., Ψ RP = 0. Thus Indeed h cos( φ α − φ β ) i ≡ h cos([ φ α − Ψ RP ] − [ φ β − Ψ RP ]) i h cos( φ α − φ β ) i = h cos( φ α ) cos( φ β ) i + h sin( φ α ) sin( φ β ) i , h cos( φ α + φ β ) i = h cos( φ α ) cos( φ β ) i − h sin( φ α ) sin( φ β ) i . (2)STAR has measured both these correlation functions for same sign, (+ , +) , ( − , − ),and opposite sign, (+ , − ), pairs of charged particles. Qualitatively the data for Au + Au collisions can be characterized as follows. • For same sign pairs: h cos( φ α + φ β ) i same ≃ h cos( φ α − φ β ) i same < . (3)Using Eq. (2) this implies h sin( φ α ) sin( φ β ) i same ≃ , h cos( φ α ) cos( φ β ) i same < . (4) • For opposite sign pairs we find that h cos( φ α + φ β ) i opposite ≃ h cos( φ α − φ β ) i opposite > . (5)Again, using Eq. (2), this means h sin( φ α ) sin( φ β ) i opposite ≃ h cos( φ α ) cos( φ β ) i opposite > . (6)The actual data decomposed into the above components are shown in Fig. 2.The fact that for same charge pairs the sinus-term in Eq. (4) (see Fig. 2) is es-sentially zero whereas the cosine term is finite, tells us that the observed correlationsare actually in plane rather than out of plane. This is contrary to the expectationfrom the chiral magnetic effect, which results in same charge correlation out of plane.In addition, since the cosine term is negative, the in-plane correlations are strongerfor back-to-back pairs than for small angle pairs. Second, we see that for oppositecharge pairs the in- and out-of-plane correlations are virtually identical. This ishard to comprehend, given that there is a sizable elliptic flow in these collisions. Atpresent, there is no simple explanation for neither of these observations. However,they may be explained by a cluster model, which requires several, not unreasonable,assumptions [4].One may ask if there is room for a parity violating component if for the samesign h sin( φ α ) sin( φ β ) i same ≃
0, i.e. the signal is in-plane rather than out of plane.Following the argument of Ref. [1, 5], we can always write h sin( φ α ) sin( φ β ) i same = B out + P, (7)4 C o rr e l a ti on x < Sin( φ α ) Sin( φ β ) >< Cos( φ α ) Cos( φ β ) >Same Charge70 60 50 40 30 20 10 0Centrality %00.511.52 C o rr e l a ti on x < Sin( φ α ) Sin( φ β ) >< Cos( φ α ) Cos( φ β ) >Opposite Charge Figure 2: Correlations in-plane h cos( φ α ) cos( φ β ) i and out of plane h sin( φ α ) sin( φ β ) i for same and opposite charge pairs in Au + Au collisions. As can be seen thecorrelations for same charge pairs are mainly in-plane.where P is the part of the correlation which is caused be the parity violation (at thisstage we do not claim that P = 0) and B out represents all other contributions bycorrelations projected on the direction perpendicular to the reaction plane. Denotingthe correlations in-plane h cos( φ α ) cos( φ β ) i same by B in we obtain: h cos( φ α + φ β ) i same = [ B in − B out ] − P, h cos( φ α − φ β ) i same = [ B in + B out ] + P. (8)The advantage of h cos( φ α + φ β ) i is obvious. The background is B in − B out ,meaning that all correlations that do not depend on the reaction plane orientation5ancel. The STAR collaboration studied many known sources of reaction planedependent correlations and all effects produce B in − B out which is much smallerthan the observed signal. We note, however, that at present the background isnot understood since none of the present models is able to explain the value of h cos( φ α − φ β ) i .Following the above argument, however, immediately implies that (using Eq. (4)and Eq. (7)) P ≃ − B out ≃ − B in , (9)i.e., the parity violating effect has to be precisely of the same magnitude as all other,standard correlations. This relation is quite an unexpected coincidence. It meansthat the parity signal is quite strong, and consequently should also be visible in h cos( φ α − φ β ) i same if the background is well understood.In our view, it is mandatory to explore if the relation, Eq. (9), is just a co-incidence or an indication of potential problems with the present interpretation ofthe data. To answer this question it is essential to analyze the correlation function h cos( φ α − φ β ) i same differentially in transverse momentum and pseudo-rapidity as ithas been already done for h cos( φ α + φ β ) i same . Should relation (9) persist also forthe differential correlations, one would have to conclude that the proposed parityviolating effect is not seen in the data. The STAR collaboration has also presented [1] the measurement of h cos( φ α + φ β ) i in mid-central Au + Au collisions as a function of p + = ( p t,α + p t,β ) / p − = | p t,α − p t,β | , where p t,α and p t,β are the absolute values of the particles momenta.Qualitatively the data can be characterized as follows. • For same sign pairs in the range 0 < p + , p − < . h cos( φ α + φ β ) i p + , same ∝ p + h cos( φ α + φ β ) i p − , same ≃ const. (10) • For opposite sign pairs the signal vs p + and p − is consistent with zero.One would expect [1, 2] that the parity violating signal should be a soft, low p t phenomenon. Thus the observed increase of the signal for same sign pairs with p + seems to be inconsistent with the chiral magnetic effect. As we will show such a In the present Section we are only interested in the p t dependence of the signal, not in theoverall normalization. p t dynamics.Indeed, by definition h cos( φ α + φ β ) i = N corr N all , (11)where N corr is the number of correlated pairs [via cos( φ α + φ β )] and N all is thenumber of all pairs. The latter can be easily approximated by [ p + = ( p t,α + p t,β ) / p − = | p t,α − p t,β | ]: N all ( p + ) ∝ Z d p t,α d p t,β exp (cid:16) − p t,α T (cid:17) exp (cid:16) − p t,β T (cid:17) δ (2 p + − [ p t,α + p t,β ]) ∝ p e − p + /T (12)and N all ( p − ) ∝ Z d p t,α d p t,β exp (cid:16) − p t,α T (cid:17) exp (cid:16) − p t,β T (cid:17) δ ( p − − | p t,α − p t,β | ) ∝ T e − ( p − /T ) ( p − + T ) , (13)where in the following calculations we take T = 0 .
22 GeV. The calculated distributions of all pairs vs ( p t,α + p t,β ) / | p t,α − p t,β | arepresented in Fig. 3. It is worth noticing that both functions are concentrated in thesmall p t region, reflecting typical thermal distributions for p − and p + . Due to thesoft nature of chiral magnetic effect, one expects that the distributions in p − and p + for the correlated particles should not differ much from the underlying thermaldistributions. This is indeed the case as we will demonstrate next.In order to estimate the distribution of correlated same sign pairs it is sufficientto multiply Eq. (10) by the expressions (12) and (13), respectively. Consequentlywe obtain N corr ( p − ) ∝ N all ( p − ) ,N corr ( p + ) ∝ p + N all ( p + ) . (14)As can be seen the dependence of the number of correlated same pairs vs | p t,α − p t,β | is identical to the dependence of all pairs presented in Fig. 3. Clearlythe signal is concentrated in the low p t region and indeed is unchanged from a ther-mal distribution. In Fig. 4 the dependence of the number of same sign pairs vs( p t,α + p t,β ) / p t and the shape is roughly similar. The momentum shift required by the It corresponds to the average transverse momentum of the pions h p t i = 0 .
45 GeV. .0 0.5 1.0 1.5 2.00.00.51.01.52.0 H p t , Α + p t , Β L(cid:144) @ GeV D den s i t y o f a ll pa i r s N all È p t , Α - p t , Β È @
GeV D den s i t y o f a ll pa i r s N all Figure 3: The distributions of all charge pairs vs. ( p t,α + p t,β ) / | p t,α − p t,β | ,respectively.data is δp + ≃
150 MeV which could conceivably be due to the large magnetic field,although it is somewhat on the high end of what one would naively expect fromelectromagnetic phenomena.
In this note we have discussed several aspects of the recent measurement of possiblelocal parity violation in Au + Au collisions by the STAR Collaboration. We madethe following three observations:(i) For particles with the same charge STAR sees large negative correlations in-plain h cos( φ α ) cos( φ β ) i same and very small correlations out of plain h sin( φ α ) sin( φ β ) i same .For opposite sign correlations in-plane and out-plain are both positive and of thesame magnitude. 8 .0 0.5 1.0 1.5 2.00.00.51.01.52.0 H p t , Α + p t , Β L(cid:144) @ GeV D den s i t y o f pa i r s N corr N all Figure 4: Distribution of all pairs (solid line) compared with the distribution ofsame sign correlated pairs (dashed line). Both functions are concentrated in the low p t region.(ii) If there is indeed a parity violating component in the STAR data it has to beof the same magnitude as all other, “trivial’ correlations projected on the directionperpendicular to the reaction plane. This may be a pure coincidence or an indica-tion that the present interpretation of the data as a signal for local parity violationneeds to be revised. To investigate this problem in more detail we need differen-tial distribution (vs pseudo-rapidity or transverse momenta) of h cos( φ α + φ β ) i and h cos( φ α − φ β ) i at the same time.(iii) We have also argued that the distribution of the number of correlated pairsis concentrated in the low p t region i.e. p t < h cos( φ α − φ β ) i differential in transverse momentum and pseudo-rapidity is absolutely essential to further distinguish between trivial correlations andthose due to the chiral magnetic effect. Acknowledgments
We thank D. Kharzeev, L. McLerran, S. Voloshin, and F. Wang for useful discus-sions. This work was supported in part by the Director, Office of Energy Research,Office of High Energy and Nuclear Physics, Divisions of Nuclear Physics, of theU.S. Department of Energy under Contract No. DE-AC02-05CH11231 and by thePolish Ministry of Science and Higher Education, grant No. N202 125437, N202 03492/0918. A.B. also acknowledges support from the Foundation for Polish Science(KOLUMB program).