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Effects of HBT correlations on flow measurements
23 March 2000
Physics Letters B 477 Ž2000. 51–58
Effects of HBT correlations on flow measurements Phuong Mai Dinh a , Nicolas Borghini b, Jean-Yves Ollitrault
a
a
b
SerÕice de Physique Theorique, CE-Saclay, F-91191 Gif-sur-YÕette cedex, France ´ Laboratoire de Physique Theorique des Particules Elementaires, UniÕersite´ Pierre et Marie Curie, ´ ´ 4, place Jussieu, F-75252 Paris cedex 05, France Received 14 December 1999; received in revised form 21 January 2000; accepted 31 January 2000 Editor: W. Haxton
Abstract The methods currently used to measure collective flow in nucleus–nucleus collisions assume that the only azimuthal correlations between particles are those arising from their correlation with the reaction plane. However, quantum HBT correlations also produce short range azimuthal correlations between identical particles. This creates apparent azimuthal anisotropies of a few percent when pions are used to estimate the direction of the reaction plane. These should not be misinterpreted as originating from collective flow. In particular, we show that the peculiar behaviour of the directed and elliptic flow of pions observed by NA49 at low p T can be entirely understood in terms of HBT correlations. Such correlations also produce apparent higher Fourier harmonics Žof order n G 3. of the azimuthal distribution, with magnitudes of the order of 1%, which should be looked for in the data. q 2000 Published by Elsevier Science B.V. All rights reserved.
1. Introduction In a heavy ion collision, the azimuthal distribution of particles with respect to the direction of impact Žreaction plane. is not isotropic for non-central collisions. This phenomenon, referred to as collective flow, was first observed fifteen years ago at Bevalac w1x, and more recently at the higher AGS w2x and SPS w3x energies. Azimuthal anisotropies are very sensitive to nuclear matter properties w4,5x. It is therefore important to measure them accurately. Throughout this paper, we use the word ‘‘flow’’ in the restricted meaning of ‘‘azimuthal correlation between the directions of outgoing particles and the reaction plane’’. We do not consider radial flow w6x, which is usually measured for central collisions only. Flow measurements are done in three steps Žsee w7x for a recent review of the methods.: first, one
estimates the direction of the reaction plane event by event from the directions of the outgoing particles; then, one measures the azimuthal distribution of particles with respect to this estimated reaction plane; finally, one corrects this distribution for the statistical error in the reaction plane determination. In performing this analysis, one usually assumes that the only azimuthal correlations between particles result from their correlations with the reaction plane, i.e. from flow This implicit assumption is made, in particular, in the ‘‘subevent’’ method proposed by Danielewicz and Odyniec w8x in order to estimate the error in the reaction plane determination. This method is now used by most, if not all, heavy ion experiments. However, other sources of azimuthal correlations are known, which do not depend on the orientation of the reaction plane. For instance, there are quantum
0370-2693r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 1 8 1 - 7
P.M. Dinh et al.r Physics Letters B 477 (2000) 51–58
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correlations between identical particles, due to the Žanti.symmetry of the wave function: this is the so-called Hanbury-Brown and Twiss effect w9x, hereafter denoted by HBT Žsee w10,11x for reviews.. Azimuthal correlations due to the HBT effect have been studied recently in w12x. In the present paper, we show that if the standard flow analysis is performed, these correlations produce a spurious flow. This effect is important when pions are used to estimated the reaction plane, which is often the case at ultrarelativistic energies, in particular for the NA49 experiment at CERN w13x. We show that when these correlations are properly subtracted, the flow observables are considerably modified at low transverse momentum. In Section 2, we recall how the Fourier coefficients of the azimuthal distribution with respect to the reaction plane are extracted from the two-particle correlation function in the standard flow analysis. Then, in Section 3, we apply this procedure to the measured two-particle HBT correlations, and calculate the spurious flow arising from these correlations. Finally, in Section 4, we explain how to subtract HBT correlations in the flow analysis, and perform this subtraction on the NA49 data, using the HBT correlations measured by the same experiment. Conclusions are presented in Section 5.
respect to the reaction plane for spherical nuclei, ²sin n f : vanishes. Most of the time, because of limited statistics, Õn is averaged over p T andror y. The average value of ÕnŽ p T , y . over a domain D of the Ž p T , y . plane, corresponding to a detector, will be denoted by ÕnŽ D .. In practice, the published data are limited to the n s 1 Ždirected flow. and n s 2 Želliptic flow. coefficients. However, higher harmonics could reveal more detailed features of the f distribution w7x. Since the orientation of the reaction plane is not known a priori, Õn must be extracted from the azimuthal correlations between the produced particles. We introduce the two-particle distribution, which is generally written as dN 3
dN 3
d p1 d p 2
s
dN
3
d p1 d 3 p 2
c n Ž pT 1 , y1 , pT 2 , y 2 . ' ²cos n Ž f 1 y f 2 . : 1 y f2
s
d p1 d 3 p 2
H0
cos n f
dN
d3p 2p dN df d3p 0
df ,
Ž 1.
H
where the brackets denote an average value over many events. Since the system is symmetric with
df 1 df 2 .
Ž 3.
df 1 df 2
3
1
2p
3
dN 3
Õn Ž p T , y . ' ²cos n f : s
dN
.
HH d p d p
In nucleus–nucleus collisions, the determination of the reaction plane event by event allows in principle to measure the distribution of particles not only in transverse momentum p T and rapidity y, but also in azimuth f , where f is the azimuthal angle with respect to the reaction plane. The f distribution is conveniently characterized by its Fourier coefficients w14x
Ž 2.
where C Ž p 1 , p 2 . is the two-particle connected correlation function, which vanishes for independent particles. The Fourier coefficients of the relative azimuthal distribution are given by
HHcos n Ž f 2. Standard flow analysis
Ž 1 q C Ž p1 , p 2 . . ,
2
We denote the average value of c n over Ž p T 2 , y 2 . in the domain D by c nŽ p T 1 , y 1 , D ., and the average over both Ž p T 1 , y 1 . and Ž p T 2 , y 2 . by c nŽ D, D .. Using the decomposition Ž2., one can write c n as the sum of two terms: c n Ž p T 1 , y 1 , p T 2 , y 2 . s c nflow Ž p T 1 , y 1 , p T 2 , y 2 . q c nnon - flow Ž p T 1 , y 1 , p T 2 , y 2 . , Ž 4. where the first term is due to flow: c nflow Ž p T 1 , y 1 , p T 2 , y 2 . s Õn Ž p T 1 , y 1 . Õn Ž p T 2 , y 2 . ,
Ž 5.
P.M. Dinh et al.r Physics Letters B 477 (2000) 51–58
and the remaining term comes from two-particle correlations: c nnon - flow Ž pT 1 , y1 , pT 2 , y 2 . dN
HHcos n Ž f 1 y f 2 . C Ž p1 , p2 . d 3 p s
dN
HH d 3 p d 3 p 1
1
dN d 3 p2
df 1 df 2 .
Ž 6.
df 1 df 2 2
In writing Eq. Ž5., we have used the fact that ²sin n f 1 : s ²sin n f 2 : s 0 and neglected the correlation C Ž p 1 , p 2 . in the denominator. In the standard flow analysis, non-flow correlations are neglected w7,8x, with a few exceptions: the correlations due to momentum conservation are taken into account at intermediate energies w15x, and correlations between photons originating from p 0 decays were considered in w16x. The effect of non-flow correlations on flow observables is considered from a general point of view in w17x. In the remainder of this section, we assume that c nnon - flow s 0. Then, Õn can be calculated simply as a function of the measured correlation c n using Eq. Ž5., as we now show. Note, however, that Eq. Ž5. is invariant under a global change of sign: ÕnŽ p T , y . yÕnŽ p T , y .. Hence the sign of Õn cannot be determined from c n . It is fixed either by physical considerations or by an independent measurement. For instance, NA49 chooses the minus sign for the Õ 1 of charged pions, in order to make the Õ 1 of protons at forward rapidities come out positive w13x. Averaging Eq. Ž5. over Ž p T 1 , y 1 . and Ž p T 2 , y 2 . in the domain D, one obtains
™
(
Õn Ž D . s " c n Ž D , D . .
Ž 7.
This equation shows in particular that the average two-particle correlation c nŽ D, D . due to flow is positive. Finally, integrating Ž5. over Ž p T 2 , y 2 ., and using Ž7., one obtains the expression of Õn as a function of c n : Õn Ž p T 1 , y 1 . s "
c n Ž pT 1 , y1 , D .
(c Ž D , D .
.
Ž 8.
n
This formula serves as a basis for the standard flow analysis. Note that the actual experimental procedure is usually different: one first estimates, for a given
53
Fourier harmonic m, the azimuth of the reaction plane Žmodulo 2prm. by summing over many particles. Then one studies the correlation of another particle Žin order to remove autocorrelations. with respect to the estimated reaction plane. One can then measure the coefficient Õn with respect to this reaction plane if n is a multiple of m. In this paper, we consider only the case n s m. Both procedures give the same result, since they start from the same assumption Žthe only azimuthal correlations are from flow.. This equivalence was first pointed out in w18x.
3. Azimuthal correlations due to the HBT effect The HBT effect yields two-particle correlations, i.e. a non-zero C Ž p 1 , p 2 . in Eq. Ž2.. According to Eq. Ž6., this gives rise to an azimuthal correlation c nnon - flow , which contributes to the total, measured correlation c n in Eq. Ž4.. In particular, there will be a correlation between randomly chosen subevents when one particle of a HBT pair goes into each subevent. The contribution of HBT correlations to c nnon - flow will be denoted by c nHB T. In the following, we shall consider only pions. Since they are bosons, their correlation is positive, i.e. of the same sign as the correlation due to flow. Therefore, if one applies the standard flow analysis to HBT correlations alone, i.e. if one replaces c n by c nHBT in Eq. Ž8., they yield a spurious flow ÕnHB T, which we calculate in this section. First, let us estimate its order of magnitude. The HBT effect gives a correlation of order unity between two identical pions with momenta p 1 and p 2 if < p 2 y p 1 < Q "rR, where R is a typical HBT radius, corresponding to the size of the interaction region. From now on, we take " s 1. In practice, R ; 4 fm for a semi–peripheral Pb–Pb collision at 158 GeV per nucleon, so that 1rR ; 50 MeVrc is much smaller than the average transverse momentum, which is close to 400 MeVrc: the HBT effect correlates only pairs with low relative momenta. In particular, the azimuthal correlation due to the HBT effect is short-ranged: it is significant only if f 2 y f 1 Q 1rŽ RpT . ; 0.1. This localization in f implies a delocalization in n of the Fourier coefficients, which are expected to be roughly constant up to n Q RpT ; 10, as will be confirmed below.
P.M. Dinh et al.r Physics Letters B 477 (2000) 51–58
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For small n and Ž p T 1 , y 1 . in D, the order of magnitude of c nHB T Ž p T 1 , y 1 , D . is the fraction of particles in D whose momentum lies in a circle of radius 1rR centered at p 1. This fraction is of order Ž R 3 ² p T :2² mT :D y .y1 , where ² p T : and ² mT : are typical magnitudes of the transverse momentum and transverse mass Ž mT s p T2 q m2 , where m is the mass of the particle., respectively, while D y is the rapidity interval covered by the detector. Using Eq. Ž7., this gives a spurious flow of order
(
ÕnHB T
Ž D. ;
ž
1r2
1 R 3 ² p T :2² mT :D y
/
.
Ž 9.
The effect is therefore larger for the lightest particles, i.e. for pions. Taking R s 4 fm, ² p T : ; ² mT : ; 400 MeVrc and D y s 2, one obtains Õn Ž D . ; 3 %, which is of the same order of magnitude as the flow values measured at SPS. It is therefore a priori important to take HBT correlations into account in the flow analysis. We shall now turn to a more quantitative estimate of c nHB T. For this purpose, we use the standard gaussian parametrization of the correlation function Ž2. between two identical pions w19x: 2
2
2
2
2
2
C Ž p 1 , p 2 . s leyq s R s yq 0 R 0yq L R L .
Ž 10 .
One chooses a frame boosted along the collision axis in such a way that p 1 z q p 2 z s 0 Ž‘‘longitudinal comoving system’’, denoted by LCMS.. In this frame, q L , q0 and q s denote the projections of p 2 y p 1 along the collision axis, the direction of p 1 q p 2 and the third direction, respectively. The corresponding radii R L , R 0 and R s , as well as the parameter l Ž0 F l F 1., depend on p 1 q p 2 . We neglect this dependence in the following calculation. Note that the parametrization Ž10. is valid for central collisions, for which the pion source is azimuthally symmetric. Therefore the azimuthal correlations studied in this section have nothing to do with flow. Note also that we neglect Coulomb correlations, which should be taken into account in a more careful study. We hope that repulsive Coulomb correlations between like-sign pairs will be compensated, at least partially, by attractive correlations between opposite sign pairs. Since C Ž p 1 , p 2 . vanishes unless p 2 is very close to p 1 , we may replace dNrd 3 p 2 by dNrd 3 p 1 in the
numerator of Eq. Ž6., and then integrate over p 2 . As we have already said, q s , q0 and q L are the components of p 2 y p 1 in the LCMS, and one can equivalently integrate over q s , q0 and q L . In this frame, y 1 , 0 and one may also replace dNrd 3 p 1 by Ž1rmT 1 . dNrd 2 p T dy1. The resulting formula is boost 1 invariant and can also be used in the laboratory frame. The relative angle f 2 y f 1 can be expressed as a function of q s and q0 . If p T 1 4 1rR, then to a good approximation
f 2 y f 1 , q srpT 1 .
Ž 11 .
If p T 1 ; 1rR, Eq. Ž11. is no longer valid. We assume that R s , R 0 and use, instead of Ž11., the following relation: q s2 q q02 s p T2 1 q p T2 2 y 2 p T 1 p T 2 cos Ž f 2 y f 1 . .
Ž 12 . To calculate c nHB T Ž p T 1 , y 1 , D ., we insert Eqs. Ž10. and Ž11. in the numerator of Ž6. and integrate over Ž q s ,q0 ,q L .. The limits on q0 and q L are deduced from the limits on Ž p T 2 , y 2 ., using the following relations, valid if p T 1 4 1rR: q0 s pT 2 y pT 1 , q L s mT 1 Ž y 2 y y 1 . .
Ž 13 .
Since q s is independent of p T 2 and y 2 Žsee Eq. Ž11.., the integral over q s extends from y` to q`. Note that values of q0 and q L much larger than 1rR do not contribute to the correlation Ž10., so that one can extend the integrals over q0 and q L to "` as soon as the point Ž p T 1 , y 1 . lies in D and is not too close to the boundary of D. By too close, we mean within an interval 1rR 0 ; 50 MeVrc in p T or 1rŽ R L mT . ; 0.3 in y. One then obtains after integration c nHBT Ž p T 1 , y 1 , D . s
lp 3r2
ž
exp y
R s R0 RL 1
n2 4 p T2 1 R 2s
/
dN 2
=
mT 1 d p T1 dy 1 dN
HD d
2
p T 2 dy 2
.
d 2 p T 2 dy 2
Ž 14 .
P.M. Dinh et al.r Physics Letters B 477 (2000) 51–58
At low p T , Eq. Ž11. must be replaced by Eq. Ž12.. Then, one must do the following substitution in Eq. Ž14.:
ž
exp y
n2 4x
'p ™ / 2 xe
55
bution factorize, thereby neglecting the observed variation of ² p T : with rapidity w20x. The rapidity dependence of charged pions can be parametrized by w20x
yx 2 r2
2
dN s
x2
dy
x2
ž ž / ž //
= I ny 1 2
1
2
q I nq1 2
2
,
Ž 15 . where x s R s p T and Ik is the modified Bessel function of order k. Let us discuss our result Ž14.. First, the correlation depends on n only through the exponential factor, which suppresses c nHB T in the very low p T region p T 1 Q nr2 R s . For n smaller than R s² p T : , 10, the correlation depends weakly on n, as discussed above. Neglecting this n dependence, Ž14. reproduces the order of magnitude Ž9.. To see this, we normalize the particle distribution in D in order to get rid of the denominator in Ž14., and the numerator Ž 1r m T 1 .Ž dN r d 2 p T 1 dy 1 . is of order 1r² p T :2² mT :D y. However, Eq. Ž14. is more detailed, and shows in particular that the dependence of the correlation on p T 1 and y 1 follows that of the momentum distribution in the LCMS Žneglecting the mT and y dependence of HBT radii.. This is because the correlation c nHB T is proportional to the number of particles surrounding p 1 in phase space. Let us now present numerical estimates for a Pb–Pb collision at SPS. We assume for simplicity that the p T and y dependence of the particle distri-
s '2p
ž
exp y
2
Ž y y ² y :. 2s 2
/
,
Ž 16 .
with s s 1.4 and ² y : s 2.9. The normalized p T distribution is parametrized by em r T
dN d 2 pT
s
2p T Ž m q T .
ž
exp y
mT T
/
.
Ž 17 .
with T , 190 MeV w20x. This parametrization underestimates the number of low-pT pions. The values of R 0 , R s and R L used in our computations, taking into account that the collisions are semi-peripheral, are respectively 4 fm, 4 fm and 5 fm w22x. The correlation strength l is approximately 0.4 for pions w23x. Finally, we must define the domain D in Eq. Ž14.. It is natural to choose different rapidity windows for odd and even harmonics, because odd harmonics have opposite signs in the target and projectile rapidity region, by symmetry, and vanish at mid-rapidity Ž² y : s 2.9., while even harmonics are symmetric around mid-rapidity. Following the NA49 collaboration w21x, we take 4 - y - 6 and 0.05 - p T - 0.6 GeVrc for odd n, and 3.5 - y - 5 and 0.05 - p T - 2 GeVrc for even n. We assume that the particles in D are 85% pions w13x, half pq, half py. Then, for an identified charged pion Ža pq, say. with p T s p T 1 and y s y 1 , the right-hand side
Fig. 1. Apparent < ÕnHB T < from HBT correlations as a function of pT in MeVrc. Top: n s 1 Žsolid curve., n s 3 Žlong dashes., n s 5 Žshort dashes.. Bottom: n s 2 Žsolid curve., n s 4 Žlong dashes., n s 6 Žshort dashes.. The values are smaller for even n than for odd n because the rapidity intervals chosen to estimate the reaction plane differ.
P.M. Dinh et al.r Physics Letters B 477 (2000) 51–58
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of Eq. Ž14. must be multiplied by 0.85 = 0.5, which is the probability that a particle in D be also a pq. Substituting the correlation calculated from Eq. Ž14. in Eq. Ž8., one obtains the value of the spurious flow ÕnHB T Ž p T , y . due to the HBT effect. Fig. 1 displays ÕnHB T , integrated between 4 - y - 5 Žas are the NA49 data. as a function of p T . As expected, ÕnHB T depends on the order n only at low p T , where it vanishes due to the exponential factor in Eq. Ž14.. HBT correlations, which follow the momentum distribution, also vanish if p T is much larger than the average transverse momentum. Assuming that 1rR s < m,T, we find from Eq. Ž14. that the correlation is maximum at p T s p T max where p T max s
ž
2T mqT
1r4
/
(
nm 2 Rs
term from the measured correlation Žleft-hand side of Eq. Ž4., which will be denoted by c nmeasured in this section. to isolate the correlation due to flow. Then, the flow Õn can be calculated using Eq. Ž8., replacing in this equation c n by the corrected correlation c nflow s c nmeasured y c nHB T. In this section, we show the result of this modification on the directed and elliptic flow data published by NA49 for pions w13x. The published data do not give directly the twoparticle correlation c nmeasured , but rather the measured flow Õnmeasured . Since these analyses assume that the correlation factorizes according to Eq. Ž5., we can reconstruct the measured correlation as a function of the measured Õn . In particular, c nmeasured Ž p T 1 , y 1 , D .
, 60'n MeVrc,
s Õnmeasured Ž p T 1 , y 1 . Õnmeasured Ž D . .
Ž 18 . which reproduces approximately the maxima in Fig. 1. Although data on higher order harmonics are still unpublished, they were shown at the Quark Matter ’99 conference by the NA45 Collaboration w24x which reports values of Õ 3 and Õ4 of the same order as Õ 1 and Õ 2 , respectively, suggesting that most of the effect is due to HBT correlations. Similar results were found with NA49 data w25x. 4. Subtraction of HBT correlations Now that we have evaluated the contribution of HBT correlations to c nnon - flow , we can subtract this
Ž 19 .
We then perform the subtraction of HBT correlations in both the numerator and the denominator of Eq. Ž8.. The integrated flow values measured by NA49 are Õ 1measured Ž D . s y3.0 " 0.1% and Õ 2measured Ž D . s 3.0 " 0.1% w21x. After subtraction of HBT correlations, the coefficients are smaller by some 20%: Õ 1Ž D . s y2.5% and Õ 2 Ž D . s 2.6%. Fig. 2 displays the rapidity dependence of Õ 1 and Õ 2 at low transverse momentum, where the effect of HBT correlations is largest. Let us first comment on the uncorrected data. We note that Õ 1measured is zero below y - 4 Ži.e. outside D, where there are no HBT correlations. and jumps to a roughly constant value when y ) 4 Žwhere HBT correlations set in.. This gap disappears once HBT correlations are sub-
Fig. 2. Directed flow Õ1 and elliptic flow Õ 2 of pions, integrated over 50 - pT - 350 MeVrc, as a function of the rapidity, measured by NA49 Ž Õnme asured , dashed curves. and after subtraction of HBT correlations Ž Õn , full curves.. For clarity sake, experimental error bars are indicated for the corrected data only.
P.M. Dinh et al.r Physics Letters B 477 (2000) 51–58
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Fig. 3. Directed flow Õ1 and elliptic flow Õ 2 of pions, integrated between 4 - y - 5, as a function of the transverse momentum pT in MeVrc, measured by NA49 Ž Õnme asured , short dashes. and after subtraction of HBT correlations Ž Õn , full curves.. The long dashes show linear and quadratic fits at low pT for Õ1 and Õ 2 , respectively.
tracted, and the resulting values of Õ 1 are considerably smaller. The values of Õ 2 are also much smaller after correction, except near mid-rapidity. Fig. 3 displays the p T dependence of Õ1 and Õ 2 . The behaviour of ÕnŽ p T . is constrained at low p T : if the momentum distribution is regular at p T s 0, then ÕnŽ p T . must vanish like p Tn . One naturally expects this decrease to occur on a scale of the order of the average p T . This is what is observed for protons w13x. However, the uncorrected Õ 1measured and Õ 2measured for pions remain large far below 400 MeVrc. In order to explain this behaviour, one would need to invoke a specific phenomenon occurring at low p T . No such phenomenon is known. Even though resonance Žmostly D . decays are known to populate the low-pT pion spectrum, they are not expected to produce any spectacular increase in the flow. HBT correlations provide this low-pT scale, since they are important down to 1rR , 50 MeVrc. Once they are subtracted, the peculiar behaviour of the pion flow at low p T disappears. Õ1 and Õ 2 are now compatible with a variation of the type Õ 1 A p T and Õ 2 A p T2 , up to 400 MeVrc.
window used to estimate the reaction plane. Azimuthal correlations due to the HBT effect depend on p T and y like the momentum distribution in the LCMS, i.e. Ž1rmT . dNrdyd 2 p T , and depend weakly on the order of the harmonic n. The pion flow observed by NA49 has peculiar features at low p T : the rapidity dependence of Õ 1 is irregular, and both Õ 1 and Õ 2 remain large down to values of p T much smaller than the average transverse momentum, while they should decrease with p T as p T and p T2 , respectively. All these features disappear once HBT correlations are properly taken into account. Furthermore, we predict that HBT correlations should also produce spurious higher harmonics of the pion azimuthal distribution Ž Õn with n G 3. at low p T , weakly decreasing with n, with an average value of the order of 1%. The data on these higher harmonics should be published. This would provide a confirmation of the role played by HBT correlations. More generally, our study shows that although non-flow azimuthal correlations are neglected in most analyses, they may be significant.
Acknowledgements 5. Conclusions We have shown that the HBT effect produces correlations which can be misinterpreted as flow when pions are used to estimate the reaction plane. This effect is present only for pions, in the Ž p T , y .
We thank A.M. Poskanzer and S.A. Voloshin for detailed explanations concerning the NA49 flow analysis and useful comments, and J.-P. Blaizot for careful reading of the manuscript and helpful suggestions.
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P.M. Dinh et al.r Physics Letters B 477 (2000) 51–58
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Physics Letters B 477 Ž2000. 51–58
Effects of HBT correlations on flow measurements Phuong Mai Dinh a , Nicolas Borghini b, Jean-Yves Ollitrault
a
a
b
SerÕice de Physique Theorique, CE-Saclay, F-91191 Gif-sur-YÕette cedex, France ´ Laboratoire de Physique Theorique des Particules Elementaires, UniÕersite´ Pierre et Marie Curie, ´ ´ 4, place Jussieu, F-75252 Paris cedex 05, France Received 14 December 1999; received in revised form 21 January 2000; accepted 31 January 2000 Editor: W. Haxton
Abstract The methods currently used to measure collective flow in nucleus–nucleus collisions assume that the only azimuthal correlations between particles are those arising from their correlation with the reaction plane. However, quantum HBT correlations also produce short range azimuthal correlations between identical particles. This creates apparent azimuthal anisotropies of a few percent when pions are used to estimate the direction of the reaction plane. These should not be misinterpreted as originating from collective flow. In particular, we show that the peculiar behaviour of the directed and elliptic flow of pions observed by NA49 at low p T can be entirely understood in terms of HBT correlations. Such correlations also produce apparent higher Fourier harmonics Žof order n G 3. of the azimuthal distribution, with magnitudes of the order of 1%, which should be looked for in the data. q 2000 Published by Elsevier Science B.V. All rights reserved.
1. Introduction In a heavy ion collision, the azimuthal distribution of particles with respect to the direction of impact Žreaction plane. is not isotropic for non-central collisions. This phenomenon, referred to as collective flow, was first observed fifteen years ago at Bevalac w1x, and more recently at the higher AGS w2x and SPS w3x energies. Azimuthal anisotropies are very sensitive to nuclear matter properties w4,5x. It is therefore important to measure them accurately. Throughout this paper, we use the word ‘‘flow’’ in the restricted meaning of ‘‘azimuthal correlation between the directions of outgoing particles and the reaction plane’’. We do not consider radial flow w6x, which is usually measured for central collisions only. Flow measurements are done in three steps Žsee w7x for a recent review of the methods.: first, one
estimates the direction of the reaction plane event by event from the directions of the outgoing particles; then, one measures the azimuthal distribution of particles with respect to this estimated reaction plane; finally, one corrects this distribution for the statistical error in the reaction plane determination. In performing this analysis, one usually assumes that the only azimuthal correlations between particles result from their correlations with the reaction plane, i.e. from flow This implicit assumption is made, in particular, in the ‘‘subevent’’ method proposed by Danielewicz and Odyniec w8x in order to estimate the error in the reaction plane determination. This method is now used by most, if not all, heavy ion experiments. However, other sources of azimuthal correlations are known, which do not depend on the orientation of the reaction plane. For instance, there are quantum
0370-2693r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 1 8 1 - 7
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correlations between identical particles, due to the Žanti.symmetry of the wave function: this is the so-called Hanbury-Brown and Twiss effect w9x, hereafter denoted by HBT Žsee w10,11x for reviews.. Azimuthal correlations due to the HBT effect have been studied recently in w12x. In the present paper, we show that if the standard flow analysis is performed, these correlations produce a spurious flow. This effect is important when pions are used to estimated the reaction plane, which is often the case at ultrarelativistic energies, in particular for the NA49 experiment at CERN w13x. We show that when these correlations are properly subtracted, the flow observables are considerably modified at low transverse momentum. In Section 2, we recall how the Fourier coefficients of the azimuthal distribution with respect to the reaction plane are extracted from the two-particle correlation function in the standard flow analysis. Then, in Section 3, we apply this procedure to the measured two-particle HBT correlations, and calculate the spurious flow arising from these correlations. Finally, in Section 4, we explain how to subtract HBT correlations in the flow analysis, and perform this subtraction on the NA49 data, using the HBT correlations measured by the same experiment. Conclusions are presented in Section 5.
respect to the reaction plane for spherical nuclei, ²sin n f : vanishes. Most of the time, because of limited statistics, Õn is averaged over p T andror y. The average value of ÕnŽ p T , y . over a domain D of the Ž p T , y . plane, corresponding to a detector, will be denoted by ÕnŽ D .. In practice, the published data are limited to the n s 1 Ždirected flow. and n s 2 Želliptic flow. coefficients. However, higher harmonics could reveal more detailed features of the f distribution w7x. Since the orientation of the reaction plane is not known a priori, Õn must be extracted from the azimuthal correlations between the produced particles. We introduce the two-particle distribution, which is generally written as dN 3
dN 3
d p1 d p 2
s
dN
3
d p1 d 3 p 2
c n Ž pT 1 , y1 , pT 2 , y 2 . ' ²cos n Ž f 1 y f 2 . : 1 y f2
s
d p1 d 3 p 2
H0
cos n f
dN
d3p 2p dN df d3p 0
df ,
Ž 1.
H
where the brackets denote an average value over many events. Since the system is symmetric with
df 1 df 2 .
Ž 3.
df 1 df 2
3
1
2p
3
dN 3
Õn Ž p T , y . ' ²cos n f : s
dN
.
HH d p d p
In nucleus–nucleus collisions, the determination of the reaction plane event by event allows in principle to measure the distribution of particles not only in transverse momentum p T and rapidity y, but also in azimuth f , where f is the azimuthal angle with respect to the reaction plane. The f distribution is conveniently characterized by its Fourier coefficients w14x
Ž 2.
where C Ž p 1 , p 2 . is the two-particle connected correlation function, which vanishes for independent particles. The Fourier coefficients of the relative azimuthal distribution are given by
HHcos n Ž f 2. Standard flow analysis
Ž 1 q C Ž p1 , p 2 . . ,
2
We denote the average value of c n over Ž p T 2 , y 2 . in the domain D by c nŽ p T 1 , y 1 , D ., and the average over both Ž p T 1 , y 1 . and Ž p T 2 , y 2 . by c nŽ D, D .. Using the decomposition Ž2., one can write c n as the sum of two terms: c n Ž p T 1 , y 1 , p T 2 , y 2 . s c nflow Ž p T 1 , y 1 , p T 2 , y 2 . q c nnon - flow Ž p T 1 , y 1 , p T 2 , y 2 . , Ž 4. where the first term is due to flow: c nflow Ž p T 1 , y 1 , p T 2 , y 2 . s Õn Ž p T 1 , y 1 . Õn Ž p T 2 , y 2 . ,
Ž 5.
P.M. Dinh et al.r Physics Letters B 477 (2000) 51–58
and the remaining term comes from two-particle correlations: c nnon - flow Ž pT 1 , y1 , pT 2 , y 2 . dN
HHcos n Ž f 1 y f 2 . C Ž p1 , p2 . d 3 p s
dN
HH d 3 p d 3 p 1
1
dN d 3 p2
df 1 df 2 .
Ž 6.
df 1 df 2 2
In writing Eq. Ž5., we have used the fact that ²sin n f 1 : s ²sin n f 2 : s 0 and neglected the correlation C Ž p 1 , p 2 . in the denominator. In the standard flow analysis, non-flow correlations are neglected w7,8x, with a few exceptions: the correlations due to momentum conservation are taken into account at intermediate energies w15x, and correlations between photons originating from p 0 decays were considered in w16x. The effect of non-flow correlations on flow observables is considered from a general point of view in w17x. In the remainder of this section, we assume that c nnon - flow s 0. Then, Õn can be calculated simply as a function of the measured correlation c n using Eq. Ž5., as we now show. Note, however, that Eq. Ž5. is invariant under a global change of sign: ÕnŽ p T , y . yÕnŽ p T , y .. Hence the sign of Õn cannot be determined from c n . It is fixed either by physical considerations or by an independent measurement. For instance, NA49 chooses the minus sign for the Õ 1 of charged pions, in order to make the Õ 1 of protons at forward rapidities come out positive w13x. Averaging Eq. Ž5. over Ž p T 1 , y 1 . and Ž p T 2 , y 2 . in the domain D, one obtains
™
(
Õn Ž D . s " c n Ž D , D . .
Ž 7.
This equation shows in particular that the average two-particle correlation c nŽ D, D . due to flow is positive. Finally, integrating Ž5. over Ž p T 2 , y 2 ., and using Ž7., one obtains the expression of Õn as a function of c n : Õn Ž p T 1 , y 1 . s "
c n Ž pT 1 , y1 , D .
(c Ž D , D .
.
Ž 8.
n
This formula serves as a basis for the standard flow analysis. Note that the actual experimental procedure is usually different: one first estimates, for a given
53
Fourier harmonic m, the azimuth of the reaction plane Žmodulo 2prm. by summing over many particles. Then one studies the correlation of another particle Žin order to remove autocorrelations. with respect to the estimated reaction plane. One can then measure the coefficient Õn with respect to this reaction plane if n is a multiple of m. In this paper, we consider only the case n s m. Both procedures give the same result, since they start from the same assumption Žthe only azimuthal correlations are from flow.. This equivalence was first pointed out in w18x.
3. Azimuthal correlations due to the HBT effect The HBT effect yields two-particle correlations, i.e. a non-zero C Ž p 1 , p 2 . in Eq. Ž2.. According to Eq. Ž6., this gives rise to an azimuthal correlation c nnon - flow , which contributes to the total, measured correlation c n in Eq. Ž4.. In particular, there will be a correlation between randomly chosen subevents when one particle of a HBT pair goes into each subevent. The contribution of HBT correlations to c nnon - flow will be denoted by c nHB T. In the following, we shall consider only pions. Since they are bosons, their correlation is positive, i.e. of the same sign as the correlation due to flow. Therefore, if one applies the standard flow analysis to HBT correlations alone, i.e. if one replaces c n by c nHBT in Eq. Ž8., they yield a spurious flow ÕnHB T, which we calculate in this section. First, let us estimate its order of magnitude. The HBT effect gives a correlation of order unity between two identical pions with momenta p 1 and p 2 if < p 2 y p 1 < Q "rR, where R is a typical HBT radius, corresponding to the size of the interaction region. From now on, we take " s 1. In practice, R ; 4 fm for a semi–peripheral Pb–Pb collision at 158 GeV per nucleon, so that 1rR ; 50 MeVrc is much smaller than the average transverse momentum, which is close to 400 MeVrc: the HBT effect correlates only pairs with low relative momenta. In particular, the azimuthal correlation due to the HBT effect is short-ranged: it is significant only if f 2 y f 1 Q 1rŽ RpT . ; 0.1. This localization in f implies a delocalization in n of the Fourier coefficients, which are expected to be roughly constant up to n Q RpT ; 10, as will be confirmed below.
P.M. Dinh et al.r Physics Letters B 477 (2000) 51–58
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For small n and Ž p T 1 , y 1 . in D, the order of magnitude of c nHB T Ž p T 1 , y 1 , D . is the fraction of particles in D whose momentum lies in a circle of radius 1rR centered at p 1. This fraction is of order Ž R 3 ² p T :2² mT :D y .y1 , where ² p T : and ² mT : are typical magnitudes of the transverse momentum and transverse mass Ž mT s p T2 q m2 , where m is the mass of the particle., respectively, while D y is the rapidity interval covered by the detector. Using Eq. Ž7., this gives a spurious flow of order
(
ÕnHB T
Ž D. ;
ž
1r2
1 R 3 ² p T :2² mT :D y
/
.
Ž 9.
The effect is therefore larger for the lightest particles, i.e. for pions. Taking R s 4 fm, ² p T : ; ² mT : ; 400 MeVrc and D y s 2, one obtains Õn Ž D . ; 3 %, which is of the same order of magnitude as the flow values measured at SPS. It is therefore a priori important to take HBT correlations into account in the flow analysis. We shall now turn to a more quantitative estimate of c nHB T. For this purpose, we use the standard gaussian parametrization of the correlation function Ž2. between two identical pions w19x: 2
2
2
2
2
2
C Ž p 1 , p 2 . s leyq s R s yq 0 R 0yq L R L .
Ž 10 .
One chooses a frame boosted along the collision axis in such a way that p 1 z q p 2 z s 0 Ž‘‘longitudinal comoving system’’, denoted by LCMS.. In this frame, q L , q0 and q s denote the projections of p 2 y p 1 along the collision axis, the direction of p 1 q p 2 and the third direction, respectively. The corresponding radii R L , R 0 and R s , as well as the parameter l Ž0 F l F 1., depend on p 1 q p 2 . We neglect this dependence in the following calculation. Note that the parametrization Ž10. is valid for central collisions, for which the pion source is azimuthally symmetric. Therefore the azimuthal correlations studied in this section have nothing to do with flow. Note also that we neglect Coulomb correlations, which should be taken into account in a more careful study. We hope that repulsive Coulomb correlations between like-sign pairs will be compensated, at least partially, by attractive correlations between opposite sign pairs. Since C Ž p 1 , p 2 . vanishes unless p 2 is very close to p 1 , we may replace dNrd 3 p 2 by dNrd 3 p 1 in the
numerator of Eq. Ž6., and then integrate over p 2 . As we have already said, q s , q0 and q L are the components of p 2 y p 1 in the LCMS, and one can equivalently integrate over q s , q0 and q L . In this frame, y 1 , 0 and one may also replace dNrd 3 p 1 by Ž1rmT 1 . dNrd 2 p T dy1. The resulting formula is boost 1 invariant and can also be used in the laboratory frame. The relative angle f 2 y f 1 can be expressed as a function of q s and q0 . If p T 1 4 1rR, then to a good approximation
f 2 y f 1 , q srpT 1 .
Ž 11 .
If p T 1 ; 1rR, Eq. Ž11. is no longer valid. We assume that R s , R 0 and use, instead of Ž11., the following relation: q s2 q q02 s p T2 1 q p T2 2 y 2 p T 1 p T 2 cos Ž f 2 y f 1 . .
Ž 12 . To calculate c nHB T Ž p T 1 , y 1 , D ., we insert Eqs. Ž10. and Ž11. in the numerator of Ž6. and integrate over Ž q s ,q0 ,q L .. The limits on q0 and q L are deduced from the limits on Ž p T 2 , y 2 ., using the following relations, valid if p T 1 4 1rR: q0 s pT 2 y pT 1 , q L s mT 1 Ž y 2 y y 1 . .
Ž 13 .
Since q s is independent of p T 2 and y 2 Žsee Eq. Ž11.., the integral over q s extends from y` to q`. Note that values of q0 and q L much larger than 1rR do not contribute to the correlation Ž10., so that one can extend the integrals over q0 and q L to "` as soon as the point Ž p T 1 , y 1 . lies in D and is not too close to the boundary of D. By too close, we mean within an interval 1rR 0 ; 50 MeVrc in p T or 1rŽ R L mT . ; 0.3 in y. One then obtains after integration c nHBT Ž p T 1 , y 1 , D . s
lp 3r2
ž
exp y
R s R0 RL 1
n2 4 p T2 1 R 2s
/
dN 2
=
mT 1 d p T1 dy 1 dN
HD d
2
p T 2 dy 2
.
d 2 p T 2 dy 2
Ž 14 .
P.M. Dinh et al.r Physics Letters B 477 (2000) 51–58
At low p T , Eq. Ž11. must be replaced by Eq. Ž12.. Then, one must do the following substitution in Eq. Ž14.:
ž
exp y
n2 4x
'p ™ / 2 xe
55
bution factorize, thereby neglecting the observed variation of ² p T : with rapidity w20x. The rapidity dependence of charged pions can be parametrized by w20x
yx 2 r2
2
dN s
x2
dy
x2
ž ž / ž //
= I ny 1 2
1
2
q I nq1 2
2
,
Ž 15 . where x s R s p T and Ik is the modified Bessel function of order k. Let us discuss our result Ž14.. First, the correlation depends on n only through the exponential factor, which suppresses c nHB T in the very low p T region p T 1 Q nr2 R s . For n smaller than R s² p T : , 10, the correlation depends weakly on n, as discussed above. Neglecting this n dependence, Ž14. reproduces the order of magnitude Ž9.. To see this, we normalize the particle distribution in D in order to get rid of the denominator in Ž14., and the numerator Ž 1r m T 1 .Ž dN r d 2 p T 1 dy 1 . is of order 1r² p T :2² mT :D y. However, Eq. Ž14. is more detailed, and shows in particular that the dependence of the correlation on p T 1 and y 1 follows that of the momentum distribution in the LCMS Žneglecting the mT and y dependence of HBT radii.. This is because the correlation c nHB T is proportional to the number of particles surrounding p 1 in phase space. Let us now present numerical estimates for a Pb–Pb collision at SPS. We assume for simplicity that the p T and y dependence of the particle distri-
s '2p
ž
exp y
2
Ž y y ² y :. 2s 2
/
,
Ž 16 .
with s s 1.4 and ² y : s 2.9. The normalized p T distribution is parametrized by em r T
dN d 2 pT
s
2p T Ž m q T .
ž
exp y
mT T
/
.
Ž 17 .
with T , 190 MeV w20x. This parametrization underestimates the number of low-pT pions. The values of R 0 , R s and R L used in our computations, taking into account that the collisions are semi-peripheral, are respectively 4 fm, 4 fm and 5 fm w22x. The correlation strength l is approximately 0.4 for pions w23x. Finally, we must define the domain D in Eq. Ž14.. It is natural to choose different rapidity windows for odd and even harmonics, because odd harmonics have opposite signs in the target and projectile rapidity region, by symmetry, and vanish at mid-rapidity Ž² y : s 2.9., while even harmonics are symmetric around mid-rapidity. Following the NA49 collaboration w21x, we take 4 - y - 6 and 0.05 - p T - 0.6 GeVrc for odd n, and 3.5 - y - 5 and 0.05 - p T - 2 GeVrc for even n. We assume that the particles in D are 85% pions w13x, half pq, half py. Then, for an identified charged pion Ža pq, say. with p T s p T 1 and y s y 1 , the right-hand side
Fig. 1. Apparent < ÕnHB T < from HBT correlations as a function of pT in MeVrc. Top: n s 1 Žsolid curve., n s 3 Žlong dashes., n s 5 Žshort dashes.. Bottom: n s 2 Žsolid curve., n s 4 Žlong dashes., n s 6 Žshort dashes.. The values are smaller for even n than for odd n because the rapidity intervals chosen to estimate the reaction plane differ.
P.M. Dinh et al.r Physics Letters B 477 (2000) 51–58
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of Eq. Ž14. must be multiplied by 0.85 = 0.5, which is the probability that a particle in D be also a pq. Substituting the correlation calculated from Eq. Ž14. in Eq. Ž8., one obtains the value of the spurious flow ÕnHB T Ž p T , y . due to the HBT effect. Fig. 1 displays ÕnHB T , integrated between 4 - y - 5 Žas are the NA49 data. as a function of p T . As expected, ÕnHB T depends on the order n only at low p T , where it vanishes due to the exponential factor in Eq. Ž14.. HBT correlations, which follow the momentum distribution, also vanish if p T is much larger than the average transverse momentum. Assuming that 1rR s < m,T, we find from Eq. Ž14. that the correlation is maximum at p T s p T max where p T max s
ž
2T mqT
1r4
/
(
nm 2 Rs
term from the measured correlation Žleft-hand side of Eq. Ž4., which will be denoted by c nmeasured in this section. to isolate the correlation due to flow. Then, the flow Õn can be calculated using Eq. Ž8., replacing in this equation c n by the corrected correlation c nflow s c nmeasured y c nHB T. In this section, we show the result of this modification on the directed and elliptic flow data published by NA49 for pions w13x. The published data do not give directly the twoparticle correlation c nmeasured , but rather the measured flow Õnmeasured . Since these analyses assume that the correlation factorizes according to Eq. Ž5., we can reconstruct the measured correlation as a function of the measured Õn . In particular, c nmeasured Ž p T 1 , y 1 , D .
, 60'n MeVrc,
s Õnmeasured Ž p T 1 , y 1 . Õnmeasured Ž D . .
Ž 18 . which reproduces approximately the maxima in Fig. 1. Although data on higher order harmonics are still unpublished, they were shown at the Quark Matter ’99 conference by the NA45 Collaboration w24x which reports values of Õ 3 and Õ4 of the same order as Õ 1 and Õ 2 , respectively, suggesting that most of the effect is due to HBT correlations. Similar results were found with NA49 data w25x. 4. Subtraction of HBT correlations Now that we have evaluated the contribution of HBT correlations to c nnon - flow , we can subtract this
Ž 19 .
We then perform the subtraction of HBT correlations in both the numerator and the denominator of Eq. Ž8.. The integrated flow values measured by NA49 are Õ 1measured Ž D . s y3.0 " 0.1% and Õ 2measured Ž D . s 3.0 " 0.1% w21x. After subtraction of HBT correlations, the coefficients are smaller by some 20%: Õ 1Ž D . s y2.5% and Õ 2 Ž D . s 2.6%. Fig. 2 displays the rapidity dependence of Õ 1 and Õ 2 at low transverse momentum, where the effect of HBT correlations is largest. Let us first comment on the uncorrected data. We note that Õ 1measured is zero below y - 4 Ži.e. outside D, where there are no HBT correlations. and jumps to a roughly constant value when y ) 4 Žwhere HBT correlations set in.. This gap disappears once HBT correlations are sub-
Fig. 2. Directed flow Õ1 and elliptic flow Õ 2 of pions, integrated over 50 - pT - 350 MeVrc, as a function of the rapidity, measured by NA49 Ž Õnme asured , dashed curves. and after subtraction of HBT correlations Ž Õn , full curves.. For clarity sake, experimental error bars are indicated for the corrected data only.
P.M. Dinh et al.r Physics Letters B 477 (2000) 51–58
57
Fig. 3. Directed flow Õ1 and elliptic flow Õ 2 of pions, integrated between 4 - y - 5, as a function of the transverse momentum pT in MeVrc, measured by NA49 Ž Õnme asured , short dashes. and after subtraction of HBT correlations Ž Õn , full curves.. The long dashes show linear and quadratic fits at low pT for Õ1 and Õ 2 , respectively.
tracted, and the resulting values of Õ 1 are considerably smaller. The values of Õ 2 are also much smaller after correction, except near mid-rapidity. Fig. 3 displays the p T dependence of Õ1 and Õ 2 . The behaviour of ÕnŽ p T . is constrained at low p T : if the momentum distribution is regular at p T s 0, then ÕnŽ p T . must vanish like p Tn . One naturally expects this decrease to occur on a scale of the order of the average p T . This is what is observed for protons w13x. However, the uncorrected Õ 1measured and Õ 2measured for pions remain large far below 400 MeVrc. In order to explain this behaviour, one would need to invoke a specific phenomenon occurring at low p T . No such phenomenon is known. Even though resonance Žmostly D . decays are known to populate the low-pT pion spectrum, they are not expected to produce any spectacular increase in the flow. HBT correlations provide this low-pT scale, since they are important down to 1rR , 50 MeVrc. Once they are subtracted, the peculiar behaviour of the pion flow at low p T disappears. Õ1 and Õ 2 are now compatible with a variation of the type Õ 1 A p T and Õ 2 A p T2 , up to 400 MeVrc.
window used to estimate the reaction plane. Azimuthal correlations due to the HBT effect depend on p T and y like the momentum distribution in the LCMS, i.e. Ž1rmT . dNrdyd 2 p T , and depend weakly on the order of the harmonic n. The pion flow observed by NA49 has peculiar features at low p T : the rapidity dependence of Õ 1 is irregular, and both Õ 1 and Õ 2 remain large down to values of p T much smaller than the average transverse momentum, while they should decrease with p T as p T and p T2 , respectively. All these features disappear once HBT correlations are properly taken into account. Furthermore, we predict that HBT correlations should also produce spurious higher harmonics of the pion azimuthal distribution Ž Õn with n G 3. at low p T , weakly decreasing with n, with an average value of the order of 1%. The data on these higher harmonics should be published. This would provide a confirmation of the role played by HBT correlations. More generally, our study shows that although non-flow azimuthal correlations are neglected in most analyses, they may be significant.
Acknowledgements 5. Conclusions We have shown that the HBT effect produces correlations which can be misinterpreted as flow when pions are used to estimate the reaction plane. This effect is present only for pions, in the Ž p T , y .
We thank A.M. Poskanzer and S.A. Voloshin for detailed explanations concerning the NA49 flow analysis and useful comments, and J.-P. Blaizot for careful reading of the manuscript and helpful suggestions.
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