Jet Quenching: the medium modification of the single and double fragmentation functions

0.18 MB·English

by  A. Majumder

Jet Quenching: the medium modification of the
single and double fragmentation functions
5
0
0
A. Majumder
2
NuclearScienceDivision,LawrenceBerkeleyNationalLaboratory
n 1Cyclotronroad,Berkeley,CA94720
a
J
Abstract. The physics ofthe quenching of hardjets indense matter is briefly
1
reviewed. This is presented within the framework of the partonic medium
1
modificationofthefragmentationfunctions. Modificationsinbothdeeplyinelastic

scattering(DIS)offlargenucleiandhigh-energyheavy-ioncollisionsarepresented.
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v
9
2 PACSnumbers: 12.38.Mh,11.10.Wx,25.75.Dw
0 The quenching of jets and the modification of jet structure has emerged as a
1
new diagnostic tool for the study of the partonic properties of dense matter [1]. The
0
modification includes, not only a suppression of inclusive spectra of leading hadrons
5
0 commonlyreferredtoasjetquenching,butcanbeextendedtoincludethemodification
/ of many particle observables. The simplest of these are the two-hadron correlations
h
within the jet cone. Such one and two particle observables have been measured both
t
-
in DIS [2] and high-energy heavy-ion collisions [3, 4].
l
c ThesingleparticleinclusivespectruminDISoffanucleusversusthatinanucleon
u
(or deuterium) target demonstrates an increasing suppression as the momentum
n
: fraction z of the detected hadron with respect to the initial partonic energy is
v
increased. This suppressionata givenz hasalsobeen foundto increasequadratically
i
X withthe sizeofthe nucleartarget. Forthe twohadroncorrelation,aconditionalratio
r is measured: the distribution of associated hadrons given that there already exists a
a
trigger hadron. This is measured in the case of a large nucleus and then divided by
thesameratioindeuterium. Thetwo-hadroncorrelationis,however,foundtobevery
slightly suppressed, and almost independent of the nuclear size [2].
In high energy Au + Au collisions, the single inclusive spectra are suppressed
comparedtothatinp+pcollisions. Whatispresentedis the differentialcrosssection
in Au+Au collisions divided by the binary scaled differential cross section in p+p
collisions. This ratio is denoted as R . Beyond a transverse momentum p of
AA T
3 GeV for π0’s and 6 GeV for chargedhadrons,this suppressionis almost a constant,
independent of p (at a √s = 200 GeV). The suppression is seen to increase with
T
centrality with R assuming a value of barely one fifth for the most central events.
AA
Insharpcontrasttothis,thenearsidetwoparticlecorrelationismoderatelyenhanced
in central Au+Au collisions relative to that in p+p [3, 4].
Theoretically,n-particleobservables(wheren 1)injetfragmentationandtheir
≥
mediummodificationcanbestudiedthroughn-hadronfragmentationfunctionswhich
can be defined as the expectation values of partonic field operators on n-hadron
states. These n-hadron fragmentation functions are non-perturbative and involve
long distance processes. However,they may be factorized from the hard perturbative
Jet Quenching 2
processes and their evolution with momentum scale may be systematically studied
in perturbative Quantum Chromo-Dynamics (pQCD) [5]. Such an evolution is very
similar to the well known case of n=1 hadron fragmentation functions [6].
In these proceedings, the medium modification of fragmentation functions in
DIS off nuclei will be sketched within the framework of generalized factorization
and twist expansion [7]. The results are then extended to the case of parton
propagation in heavy-ion collisions. Alternative methods for calculating the single
inclusivedistributionsinheavy-ioncollisions,basedonascreenedpotentialscattering
model, have been investigated by many authors [8, 9, 10]. The advantage of the
twist formalism [7] lies, not merely in the ease of its applicability to DIS and heavy-
ion collisions, but also in the presence of a single unknown normalization constant.
This is set by fitting one experimental data point at one set of kinematic variables.
With this parameterdetermined, one maypredict the variationofthe modifiedsingle
hadron fragmentation functions [7, 11], as well as the medium modification of two-
hadron correlations. Such behaviour in both DIS off nuclei and heavy-ion collisions
are presented in comparison with experimental data.
Applying factorization to hadron production in single jet events in DIS off a
nucleus, e(L ) + A(p) e(L ) + h (p ) + ... + X, one can obtain the n-hadron
1 2 1 1
→
semi-inclusive cross section as,
dσh1... α2 1 dWµν
E DIS = L , (1)
L2d3L dz ... 2πsQ4 µνdz ...
2 1 1
where the ellipsis indicates the possible presence of more than one detected hadron.
The semi-inclusive tensor at leading twist has the simple form,
dWµν
= dxfA(x,Q2)Hµν(x,p,q)Dh1...(z ,...,Q2). (2)
dz ... Z q q 1
1 X
q
In the above, Dh1...(z ...) is the n-hadron fragmentationfunction, L is the leptonic
q 1 µν
tensor, the factor Hµν represents the hard part of quark scattering with a virtual
photon which carries a four-momentum q = [−Q2/2q−,q−,~0⊥] and fqA(x,Q2) is the
quark distribution in the nucleus which has a total momentum A[p+,0,~0⊥]. The
− −
hadron momentum fractions, z = p /q are defined with respect to the initial
n n
−
momentum q of the fragmenting quark.
Atnext-to-leadingtwist,thedihadronsemi-inclusivetensorreceivescontributions
from multiple scattering of the struck quark off soft gluons inside the nucleus with
inducedgluonradiation. Onecanreorganizethetotalcontribution(leadingandnext-
to-leading twist) into a product of the effective quark distribution in a nucleus, the
hard part of photon-quark scattering Hµν and a modified n-hadron fragmentation
function D˜h1...(z ,...). The calculation of the modified dihadron fragmentation
q 1
function at next-to-leading twist in a nucleus [12] proceeds in similar fashion to that
forthe modifiedsinglehadronfragmentationfunctions[7]. Theresultsofthe medium
modification of the single hadron fragmentation functions are shown in the left plot
ofFig.1 incomparisonwithexperimentaldata. Theexperimentalpointatthe lowest
z in Nitrogen is used to fit the over all normalization constant. The variation with z
and with nuclear size is a predictionof the calculationand shows excellentagreement
with the experimental results.
With no additional parameters, one can predict the nuclear modification of
dihadron fragmentation functions within the same kinematics. This is shown in the
right hand plot of Fig. 1, where results are presented for DIS off nitrogen.
Jet Quenching 3
1.1 D) 1.2
1.05 o
1 ati Nevt.(z1,z2, N)/Nevt.(z1, N)/(Ratio D)
R 1.1 HERMES data
0.95 /(5 Q2=2.15−2.58 GeV2
0.9 >0.1 n =21.4−16.4 GeV
0.85 )]z 1
1
0.8 (z
N
0.75 D
)/ 0.9
0.7 2
z
0.65 z,1
(
0.60.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 [DN0.80 0.1 0.2 0.3 0.4 0.5
z
2
Figure 1. Results of the medium modification of the single (left plot, from
Ref. [11]) and associated (right plot, from Ref. [12]) hadron distribution, in a
cold nuclear medium versus its momentum fraction. The ratio of the respective
quantityinthenucleus[Nitrogen(N)orKrypton(Kr)]withthatindeuterium(D)
aretheHERMESdatapoints. Intheplotforassociatedhadrons,themomentum
fractionoftheleadinghadronz1 isintegratedoverallallowedvaluesabove0.5.
Inhigh-energyheavy-ion(orp+pandp+A)collisions,jetsarealwaysproducedin
back-to-back pairs. Correlationsof two high-p hadrons in azimuthal angle generally
T
have two Gaussian peaks [3, 4]. Relative to the triggered hadron, away-side hadrons
come from the fragmentation of the away-side jet and are related to single hadron
fragmentation functions. On the other hand, near-side hadrons come from the
fragmentationofthesamejetasthetriggeredhadronandthereforetheintegralofthe
near side Gaussian peaks are related to dihadron fragmentation functions.
To extend the study of the medium modification of the fragmentation functions
to heavy-ion collisions, one also has to include the effect of thermal gluon absorption
[13]suchthatthe effective energydependence ofthe energylosswill be differentfrom
the DIS case. Such a procedure,appliedto the study ofthe modification ofthe single
hadronfragmentationfunctionssuccessfullydescribesthequenchingofsingleinclusive
hadron spectra, their azimuthal anisotropyand the suppressionof the away-sidehigh
p hadron correlations [14]. The result for the suppression of the single inclusive
T
spectra as a function of the centrality of the collision is shown in the left side plot
in Fig. 2. The ratio R for π0’s (PHENIX [4]) and charged hadrons (STAR [3]) is
AA
plotted. Theoverallnormalizationconstantisfittedatagivenvalueofp forthemost
T
central event. The variation of the suppression with p and centrality is a prediction
T
and shows very good agreementwith the data. The overall gluon density is assumed
to vary with centrality as the number of participants. Due to space constraints,
we will skip the discussion of the suppression of the away side spectra and baryon
meson differences. We refer the readerinstead to Ref. [14] for this and further details
regarding the figure.
The change of the near-side correlation due to the modification of dihadron
fragmentationfunctionsinheavy-ioncollisionscanbesimilarlycalculated. Foragiven
value of ptrig of the triggered hadron, one can calculate the average initial jet energy
T
E . Because of trigger bias and parton energy loss, E in heavy-ion collisions is
T T
h i h i
generally larger than that in p+p collisions for a fixed ptrig [14].
T
Usingtheoverallparameter,determinedinthesingleinclusivemeasurement,one
Jet Quenching 4
calculates the modified dihadron distributions. The ratio of such associated hadron
distributionsinAu+Auversusp+pcollisions,referredtoasI [3],isplottedasthe
AA
solidlineintherighthandplotofFig.2togetherwiththeSTARdata[3],asafunction
of the number of participant nucleons. We also present data from PHENIX [4] which
accounts for correlations at a lower ptrig. In central Au+Au collisions, triggering on
T
ahighp hadronbiasestowardalargerinitialjetenergyandthereforesmallerz and
T 1
z . This leads to an enhancement in I due to the shape of dihadron fragmentation
2 AA
functions[5]. TheenhancementincreaseswithN becauseofincreasedtotalenergy
part
loss. In the most peripheral collisions, the effect of smaller energy loss is countered
by the Cronineffect due to initialstate multiple scatteringthatbiasestowardsmaller
E relative to p+p collisions. As a result, the associated hadron distribution is
T
h i
slightly suppressed. For higher ptrig, the suppression will diminish because of the
T
disappearence of the Cronin effect (See Ref. [12] for further discussions).
11 4
STAR p =4−6 GeV, p =2−4 GeV
trig assoc
PHENIX p =2.5−4 GeV, p =1.7−2.5 GeV
trig assoc
p = 5 GeV, p = 2−4 GeV
trig assoc
1100--11 3 ppttrriigg== 88 GGeeVV,, ppaassssoocc== 32.−24− 6G.4eV GeV
11
A2
A
I
--11
1100
1
11
0
--11 0 100 200 300 400
1100 N
222 444 666 888 111000 222 444 666 888 111000 part
Figure2. Calculatedmediummodificationofsingle(leftplot,fromRef.[14])and
associated(rightplot,fromRef.[12])hadrondistributionfromjetfragmentation
inAu+Aucollisionsat√s=200GeV.TheleftplotshowsvariationofRAA vs.
pT fordifferentcentralities. TherightplotshowsvariationofIAA vs. centrality
fordifferenttriggerandassociated pT ascomparedtoexperimental data[3,4].
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