Momentum conservation and correlation analyses in heavy-ion collisions at ultrarelativistic energies

0.68 MB·English

by  Nicolas Borghini

BI-TP 2006/47
Momentum conservation and correlation analyses in heavy-ion collisions at
ultrarelativistic energies
∗
Nicolas Borghini
Fakult¨at fu¨r Physik, Universit¨at Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany
(Dated: February 9, 2008)
Global transverse-momentumconservation inducescorrelations betweenanynumberofparticles,
which contribute in particular to the two- and three-particle correlations measured in heavy-ion
collisions. These correlations are examined in detail, and their importance for studies of jets and
theirinteraction with themedium is discussed.
PACSnumbers: 25.75.Gz
I. INTRODUCTION philosophy of the cumulant study advocated in Ref. [12]
7 is to perform a measurement with minimal assumptions
0
0 The high-statistics data collected at the Brookhaven on the nature of the correlations, and later to compute
2 Relativistic Heavy-IonCollider (RHIC) have opened the the contributions (assumed to be additive) of the differ-
possibility of novel precision measurements aiming at ent sourcesof correlationto the three-particlecumulant.
n
characterizing the medium created in ultrarelativistic Incontrast,allothermethodsrelyontheassumption,be
a
J heavy-ion collisions. Among these new developments, it explicitly stated or not, that the measured correlation
3 significant effort is being devoted to extracting cor- consistsofajetsignal,whichvanishesawayenoughfrom
relations between pairs or triplets of high-momentum the jet, and an uncorrelated background.
2 particles from the data, to perform jet physics in the The purpose of this paper is to emphasize the role of
v high-multiplicity environment of a high-energy nucleus– theunavoidablecontributionofglobal transversemomen-
3 nucleus collision. Thus, the initial result from a study of tum conservationto the measuredcorrelation,either be-
9 azimuthal correlations between particles with transverse tween two or three particles. The realization that global
0
momenta 2 GeV/c < p < 6 GeV/c was the convincing momentumconservationcansignificantlyaffectmeasure-
2 T
1 demonstration that the back jet, at 180o from a high-pT ments in heavy-ion physics is not new. Thus early mea-
6 referenceparticle,issuppressed,inthe sensethatitdoes surements of collective anisotropic flow in nuclear colli-
0 not emerge above the background [1]. sions at intermediate and relativistic energies were cor-
h/ More recently, the focus has been to obtain a more rected [15] for that effect. Later, it was shown that mo-
mentum conservation biases the standard measurement
t quantitativedescriptionofthebehaviourintherecoilre-
l- gion away from a high-pT “trigger”particle. On the one of the first anisotropic-flowharmonic v1 at SPS [16] and
c anexplicitcorrectiontotheanalysiswasdevised[17]and
hand, various physical mechanisms have been proposed,
u
applied to the NA49 data [18]. Recently, the possibility
involving the interaction of the high-momentum parton
n
that the conservation of momentum and energy could
: initially emitted back-to-back to the trigger with the
v medium through which it propagates, that predict non- significantly affect the correlation functions measured in
i femtoscopy analyses was addressed in Ref. [19].
X trivial structures in this away-side region: shock-waves
alongaMachcone [2]generatedbythe energydeposited Here I shall argue that correlations at high transverse
r
a bythefast-movingparton[3,4];gluonbremsstrahlung[5] momentumareespeciallysensitivetotheconservationof
or Cherenkov-like radiation [6, 7, 8] by the parton; the total transverse momentum. The latter has several ef-
deflection of the back jet by the collective movement fects. On the one hand, it contributes to the cumulant
(“flow”) of the expanding medium [9]; or a path-length- measured in the technique of Ref. [12]. The specific de-
based selection of the particles emerging a random walk pendence on the particle momenta of the three-particle
through the medium [10]. cumulant from global transverse momentum conserva-
tion, which was already computed in Ref. [20], will thus
In parallel, severalanalysis techniques were developed
beinvestigatedinSec.II. Ontheotherhand,globalmo-
and are being applied to the experimental data, using
mentumconservationcorrelatesajettothe remainderof
two-[11]orthree-particle[12,13,14]correlations,todis-
the event, thereby invalidating the assumption that jet
tinguishbetweenthedifferentscenarios. Acommonneed
and background are uncorrelated. Some implications of
intheseapproachesisthenecessitytodealwiththeback-
this observation will be discussed in Sec. III.
ground properly. Thus, all measurements take into ac-
count the modulation of azimuthal correlations induced
byanisotropiccollectiveflow. However,the methodsdif-
II. MOMENTUM CONSERVATION AND
fer with respectto the other sourcesof correlations. The
CUMULANTS
In a nucleus–nucleus collision, all N emitted parti-
∗Electronicaddress: [email protected] cles are correlated together by the requirement that
2
their transverse momenta p add up to 0. As a con- in terms of distributions involving less than M parti-
Ti
sequence, the joint M-particle probability distribution cles [21], is finite. Using a generating function of the
f(p ,...,p ) of p , ..., p differs from the multiparticleprobabilitydistributionsandasaddle-point
Tj1 TjM Tj1 TjM
product of the single-particle probability distributions calculation,onecancomputethesuccessivecumulantsto
fan(pdT2j1)···Mf(pTjMN),.whIenrep1art≤icujl1ar<, t.h.e. c<umjMulan≤t Nof lMea-dpianrgtioclredecruminul1a/nNts[c2a0l]e.slTikheu1s,/NitMw−a1s.sAhoswsunmtihnagtththaet
≤ ≤
the M-particle distribution, which will be denoted by thesingle-particlep distributionisisotropic,oneinpar-
T
f (p ,...,p ) and corresponds to the part of the ticular finds that the two- and three-particle cumulants
c Tj1 TM1
joint probability distribution that cannot be expressed due to global transverse-momentumconservation read
2p p
fc(pT1,pT2) = − NT1p·2 T2, (1)
h Ti
2
f (p ,p ,p ) = (p p +p p +p p )
c T1 T2 T3 −N2 p2 T1· T2 T1· T3 T2· T3
h Ti
4
+ [(p p )(p p )+(p p )(p p )+(p p )(p p )], (2)
N2 p2 2 T1· T2 T1· T3 T1· T2 T2· T3 T1· T3 T2· T3
h Ti
where p2 denotes the averageover many particles and i.e., for practical purposes in high-p studies, always,
eventshofTtihe squared transverse momentum. since p p2 1/2 (as has been assuTmedin the second
T1 ≫h Ti
The meaning of the two-particle cumulant (1) is line of the above identity). For values of the associated-
clear and intuitive: the correlation is back-to-back and particle transverse momenta pT2= pT3 larger than the
stronger between particles with higher transverse mo- same pTmin, the three-particle cumulant fc also has a
menta. In plain words, given a high-p trigger-particle, minimum for
T
thereisalargerprobabilitytofindan“associated”parti-
cle awayfromitinazimuththancloseto it,just because cos∆ϕ12 =
of global transverse-momentumconservation. p2 p 2 p 2 2 12p 2 2p 2
ariTsihneg firnotmerpgrleotbaatliotnranosfvetrhsee-mthormeee-nptaurmticcloensceurmvautliaonnt, 12hpTT1piT21−hpT2T1i+s(cid:18)hpT2T1i−1(cid:19)+ hp2TTi1(cid:18)1+hpT2T2i(cid:19).
Eq.(2), is slightlymore involved,asit implies two terms  
with opposite signs. Yet, if one considers three parti- Onecancheckthatthe∆ϕ12 positionofthisminimumis
cles with transverse momenta significantly larger than an increasing function of pT2, which reaches a maximal
t“haettrrmacstitvrea”nstveermrse, imnotmheensteucomndhpli2Tnie1,/2d,oomnienaseteesstohvaetrtthhee vfoarlupeT∆2=ϕ12pT=1.arTchcoes(n1e/x3t+exhptr2Teim/6upmT21o)f≃th1e.2th3rreaed-p(a7r0t.i5cole)
“repulsive”one. Toillustratethe behaviourofthe three- cumulant in the range 0 ∆ϕ12 π only exists if the
≤ ≤
particlecumulant,IshallnowstudyEq.(2)bypromoting transverse momentum of the associated particles pT2 is
particle 1 to the role of “trigger particle,” with respect larger than
to which the azimuths of the other two are measured:
∆simϕ1p2lic≡ityϕ2th−atϕt1h,e∆“ϕa1s3so≡ciaϕte3d−pϕar1t.icIless”ha2llaansdsu3mehafvoer p′Tmin≡ 1h0pp2TTi1 hppT2T21i −1+shppT2T41i2 + 8hppT2T21i +1!,
equal transverse momenta p =p p , and use the
T2 T3 T1
≤
notationfc(pT1,pT2,pT3,∆ϕ12,∆ϕ13)forthe cumulant. whichis pT1/5whenthe transversemomentumpT1 of
≃
Consider first the symmetric case ∆ϕ13 =2π ∆ϕ12. the trigger particle is large. Then there is a second local
−
For symmetry reasons (parity and 2π-periodicity), the maximum, beyond that at ∆ϕ12 =0, for
three-particle cumulant fc(pT1,pT2,pT2,∆ϕ12, ∆ϕ12)
has trivial extrema at ∆ϕ12 = 0 and ∆ϕ12 = −π. The cos∆ϕ12 =
formeris a maximumfor values ofpT2=pT3 largerthan p2 p 2 p 2 2 12p 2 2p 2
a minimal value h Ti 1 T1 T1 1 + T1 1+ T2 .
12pT1pT2 −hp2Ti−s(cid:18)hp2Ti− (cid:19) hp2Ti(cid:18) hp2Ti(cid:19)
pTmin ≡ 1h0pp2TTi1 −hpp2TTi21 +1+shppT2T41i2 + 8hppT2T21i +1! TpTh2e,preoasicthioinngoaf tmhiinsimmaaxli∆mϕum12 d=ec2rπe/a3sews wheitnhpiTn2cr=eapsTin1g.
p2 This is actually a rather intuitive result: when the three
h Ti,
≃ 2pT1 particles have equal transverse momenta, the cumulant
3
10-4
2´10-5 2Π 5´10-5 2Π
0 3Π 0 3Π
€€€€€€€€€€ -5´10-5 €€€€€€€€€€
-2´10-5 2 2
00 Π 00 Π
Dj Dj
ΠΠ 13 ΠΠ 13
€€€€€€€€€€ Π €€€€€€€€€€ Π
22 ΠΠ €€€€€ 22 ΠΠ €€€€€
2 2
DDjj1122 33ΠΠ DDjj1122 33ΠΠ
€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€€€€€€€
22 2Π 0 22 2Π 0
FIG.1: (Coloronline)Three-particlecumulantduetoglobaltransverse-momentumconservationfc(pT1,pT2,pT3,∆ϕ12,∆ϕ13),
as a function of the relative angles ∆ϕ12, ∆ϕ13, assuming N = 8000 particles per event with a rms transverse momentum
hp2i1/2 =0.45 GeV/c. Left: p =3.2 GeV/c, p =p =1.2 GeV/c; right: p =6 GeV/c, p =p =1.2 GeV/c.
T T1 T2 T3 T1 T2 T3
′ ′
∆haϕs2a3 m=a∆xiϕm3u1m=a1t20tho.eEsyvmenmtueatrlliyc,ctohnefitghurreaet-ipoanrt∆icϕle12cu=- c(rhiogihcte)saprTe2di>splpayTemdinto∼illpuTst1r/a5te(tlehfet)diaffnedrenptTb2e<havpiTomurins
mulant fc(pT1,pT2,pT2,∆ϕ12, ∆ϕ12) has the already- discussedabove. Asanticipated,inthe firstcase,thecu-
−
mentioned extremum at ∆ϕ12 =π, which is a minimum mulant has a saddle point at ∆ϕ12 =∆ϕ13 =π and two
′
if pT2 >pTmin, a maximum otherwise. clearly separated maxima away from ∆ϕ12 = ∆ϕ13 = 0
Let me now study the three-particle cumulant fc in (modulo 2π) along the line ∆ϕ12 =2π ∆ϕ13; whereas
−
the specific case pT2 = pT3 and ∆ϕ13 = ∆ϕ12. Due in the second case, there are two maxima for values of
to parity and 2π-periodicity, it has two extrema at ∆ϕ12 =∆ϕ13 equal to 0 or π, and no further structure.
∆ϕ12 = 0 and ∆ϕ12 = π. The latter is always a max- The values of the total number of particles N = 8000
imum, irrespective of the value of pT2. That means andthermstransversemomentum p2 1/2 =450 MeV/c
that the point ∆ϕ12 = ∆ϕ13 = π is either a saddle adopted in Fig. 1 were chosen so ahsTtio mimic a central
point for the cumulant fc(pT1,pT2,pT2,∆ϕ12,∆ϕ13), if Au–AucollisionatRHIC.Suchnumbersresultinathree-
′ ′
p > p , or a local maximum if p p . The
T2 Tmin T2 ≤ Tmin particle cumulant from transverse-momentum conserva-
nature of the extremum at ∆ϕ12 = 0 depends on the tion of order 10−5 for transverse momenta of the trigger
value ofthe transversemomentumofthe associatedpar-
and associated particles of 3 and 1 GeV/c respectively.
ticle. If p ( p 2 2 p2 p )/2 — which is
p2 /2pT2 ≤ p2 T1/12−inhthTeire−gimTe1p p2 1/2, This is admittedly a small correlation, yet it is of the
≃ h Ti T1 ≪ hpTi T1 ≫ h Ti samesizeasthecontributionofanisotropiccollectiveflow
so that the case is most likely irrelevant for studies of
to the cumulant [12]. In addition, the correlation will
correlations at high transverse momentum — then the
be stronger in more peripheral collisions, since it scales
cumulanthasaminimum at∆ϕ12 =0. Onthe contrary, as the inverse squared multiplicity 1/N2, see Eq. (2).1
when p > ( p 2 2 p2 p )/2 the cumulant has
T2 T1− h Ti− T1 Similarly, the value of the correlation will be larger in
a maximum at ∆ϕ12 =0, and a minimum for smaller systems and at lower collision energies. In fact,
p
2p 2 p2 themultiplicityN thatentersEqs.(1-2)mightnotbethe
cos∆ϕ12 = T2−h Ti. total event multiplicity, but one could argue that differ-
− 2pT1pT2 entrapidityslicesaredecorrelated,sothattheconstraint
from transverse momentum conservation would actually
The position of this minimum grows with p , reaching
T2
∆ϕ12 = arccos(−1+hp2Ti/2pT21) for pT2= pT1. In the bvaeludersivoefnthbeycausmmualallnetromfuthlteipcloicrirteyl.atTiohnatfrwomoumldoamlseontguivme
case of a large transverse momentum p of the trigger
T1
conservation higher than those plotted in Fig. 1.
particle,thisminimumsitsat∆ϕ12 ≃π− hp2Ti/√2pT1,
close to the maximum at π.
p
Figure 1 shows the profile of the three-particle cu-
mulantduetoglobaltransverse-momentumconservation
1 Thermstransversemomentumhp2i1/2 mayalsodecreasewhen
fc(pT1,pT2,pT3,∆ϕ12,∆ϕ13),inthe caseofequaltrans- going tomoreperipheralcollisionsT,whichcontributes, although
verse momenta of the associated particles, p = p ,
T2 T3 much less than the drop in multiplicity, to the growth of the
largerthanthermstransversemomentum p2 1/2. Both cumulant.
h Ti
4
served.2 In other words, a jet and the remainder of the
“underlying” event are not uncorrelated, as is too often
assumed, but global transverse-momentum conservation
alone already constrains the momentum distribution of
the particles “outside the jet.”
A consequence of the existence of this correlation is
1.005 thatthemerenotionoftherebeing“anunderlyingevent
2Π over which the jet develops” is incorrect. The jet does
1
distort the event to which it belongs, it is not merely
0.995 3Π
€€€€€€€€€€ embedded in it as is done in many simulations. This
0.99 2
means that, at least in azimuth, there is no “region far
00 Π
ΠΠ Dj13 from the jet” where its influence would vanish, thereby
€€€€€€€€€€ Π allowing one to determine a correctly normalized “back-
22 ΠΠ €€€€€
2 ground” to the jet: even in the over-simplified case of a
DDjj1122 33ΠΠ one-particle jet and considering only two-particle corre-
€€€€€€€€€€€€€€€€€€€€
22 2Π 0 lations,the regionwhere the eventshape is not modified
bythe presenceofthejetisrestrictedtotwopointsonly,
at 90o away from it. Even in studies of correlations be-
FIG. 2: (Color online) Three-particle correlation due to tween“only”twoparticles,theapproachescurrentlyused
globaltransverse-momentumconservationvs.therelativean- todisentanglethe“jet”fromtheharmonicmodulationof
gles ∆ϕ12, ∆ϕ13, assuming N = 8000 particles per event azimuthal correlations due to anisotropic flow may thus
with a rms transverse momentum hp2i1/2 = 0.45 GeV/c,
T be inaccurate, since they ignore the modulation induced
for particles of transverse momentum p = 3.2 GeV/c and
T1 by transverse-momentumconservation. The effect ofthe
p =p =1.2 GeV/c, respectively.
T2 T3 latter will be even more important in three-particle cor-
relation studies.
In summary, I have shown that the conservation of
III. DISCUSSION
the total transverse momentum yields a correlation be-
tween pairs or triplets of high-p particles that is siz-
T
able—incentralAu–Au collisionsatRHIC energiesthe
Till now, I have mostly focussed on the cumulant (2)
resulting three-particle cumulant is of the same magni-
ofthe three-particlecorrelationdue toglobaltransverse-
tude as that due to anisotropic collective flow — and
momentum conservation. The three-particle correlation
thereforeshouldbeaccountedforproperlyinexperimen-
itself, which also involves two-body terms [controlled by
tal studies. This might be easily feasible in studies at
Eq. (1)], is much larger, as displayed in Fig. 2 for the
the cumulant level [12]: there one only has to consider
sameRHIC-inspiredvaluesofthemultiplicity,rmstrans-
the three-particle cumulant (2) — possibly generalized
versemomentum, andtransversemomentaofthe trigger
to include the known anisotropy of the p distribution
and associated particles as in the left panel of Fig. 1. T
(see Ref. [20]), especially in non-central collisions —, so
One sees that the main, quite intuitive effect of momen-
that the non-measured multiplicity N and rms trans-
tum conservation is to push the associated particles 2
verse momentum p2 1/2 enter the analysis only once.
and 3 away from the trigger particle 1. The resulting h Ti
The three-particle cumulant from transverse-momentum
“bump” at ∆ϕ12 = ∆ϕ13 = π is broader along the
conservation has a specific dependence on the values of
∆ϕ12 = 2π ∆ϕ13 axis than along the perpendicular
− the particle momenta, which was illustrated in special
∆ϕ12 = ∆ϕ13 axis. This is quite generic for the corre-
cases in Fig. 1. If the transverse momentum of the trig-
lation due to momentum conservation — configurations
ger particle is much larger (by a factor 5) than those
with the two associated particles on each side of the di-
∼
rection defined by p are more likely than those with of the associated particles, the cumulant has a simple
T1
shape with one peak on the trigger-particle side, and a
both particles on the same side —, while the symmetric
broader back-to-back bump, more elongated along the
shapeofthebumpacrossthe∆ϕ12 =∆ϕ13 axisinFig.2
reflects the specific choice p =p . ∆ϕ12 = 2π ∆ϕ13 axis than perpendicular to it (see
T2 T3 −
A most prominent feature of the profile, shown in
Fig. 2, of the three-particle correlation arising from the
momentum-conservation constraint, is the “dip” around 2 Surprisingly,thismutualexclusionofhightransverse-momentum
the origin ∆ϕ12 = ∆ϕ13 = 0. The meaning of this particlesseemstobeentirelydrivenbytheanticorrelationinthe
dip is transparent: if there is a high-p particle in the two-body term. On the contrary, unless the associated parti-
T
event, then its transversemomentum has to be balanced cles have transverse momenta smaller than hp2Ti1/2, the three-
by the others. Therefore, the probability of finding an- particle cumulant is maximal at ∆ϕ12 = ∆ϕ13 = 0: at that
level,allthreehigh-pT particlesaregroupedtogetherbytheat-
other high-p particle pointing into the same direction
T tractive term inthe second lineofEq. (2), ratherthan oriented
is smaller than if transverse momentum were not con- awayfromeachother.
5
Fig.1,rightpanel). However,ifonedecreasesthetrigger tween softer particles). In addition, the correlation in-
transversemomentumwhile keepingfixedthe associated duced by momentum conservation somehow blurs the
transverse momenta, then a two-bump structure devel- distinction between jet and background, since the lat-
ops along the ∆ϕ12 = 2π ∆ϕ13 axis on the side away ter is unquestionably different in the presence of the jet
−
from the trigger (Fig. 1, left). from what it would be in its absence. Before trying to
Takingproperlyintoaccounttheeffectoftotalmomen- identify definite structures, which require high-precision
tumconservationwillbemuchmoreinvolvedinanalyses measurements, in the recoil region of a high-p particle,
T
that rely on an assumed jet profile [11, 13, 14, 22]. In it is important to first determine precisely the reference
those, the effect of momentum conservation also has to pattern — including the effects of momentum conserva-
be considered at the two-body level, in the correlations tion and anisotropic collective flow — over which they
between “jet” and “background” or between two “back- would develop.
ground”particles(althoughthecorrelationissmallerbe-
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