In-medium jet shape from energy collimation in parton showers: Comparison with CMS PbPb data at 2.76 TeV

0.33 MB·

by  Redamy Perez Ramos

In-medium jet shape from energy collimation in parton showers:
Comparison with CMS PbPb data at 2.76 TeV
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RedamyPe´rez-Ramos1
n
u DepartmentofPhysics,P.O.Box35,FI-40014UniversityofJyva¨skyla¨, Jyva¨skyla¨, Finland
J

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ThorstenRenk2
2

DepartmentofPhysics,P.O.Box35,FI-40014UniversityofJyva¨skyla¨, Jyva¨skyla¨, Finland,
]
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HelsinkiInstitute ofPhysics,P.O.Box64,FI-00014UniversityofHelsinki,Helsinki, Finland
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2
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1. Abstract: We present the medium-modified energy collimation in the leading-logarithmic approxima-
0 tion(LLA)andnext-to-leading-logarithmic approximation (NLLA)ofQCD.Asaconsequence ofmore
4
1 accurate kinematic considerations in the argument of the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi
v: (DGLAP) fragmentation functions (FFs) we find a new NLLA correction (αs) which accounts for
O
i the scaling violation of DGLAP FFs at small x. The jet shape is derived from the energy collimation
X
within the same approximations and we also compare our calculations for the energy collimation with
r
a
the event generators Pythia 6 and YaJEM for the first time in this paper. The modification of jets by
themediuminbothcases isimplemented byaltering theinfrared sector usingtheBorghini-Wiedemann
model. Theenergycollimationandjetshapesqualitativelydescribeaclearbroadening ofshowersinthe
medium, which isfurther supported byYaJEMinthe finalcomparison ofthejetshape withCMSPbPb
dataatcenter-of-mass energy 2.76TeV.Thecomparison ofthebiasedversusunbiased YaJEMjetshape
with the CMS data shows a more accurate agreement for biased showers and illustrates the importance
ofanaccurate simulationoftheexperimental jet-findingstrategy.
1e-mail:redamy.r.perez-ramos@jyu.fi
2e-mail:[email protected]
1 Introduction
Thephenomenonofjetquenchingwasfirstestablishedexperimentallythroughtheobservedsuppression
of high-p hadrons in nucleus-nucleus (A-A) collisions at the Relativistic Heavy Ion Collider (RHIC)
T
andtheLargeHadronCollider(LHC)[1–4]. Itwasthenconfirmedbyvariousothermeasurementswhere
highly virtual partons produced in hard processes in a medium showed modifications to the subsequent
evolution of a QCD shower, in particular softening and broadening of the resulting hadron distribution
whichleadstoareduction intheyieldofleading hadronsandjets[5–9].
In the vacuum, the production of highly virtual partons following the hard inelastic scattering of two
partons from the incoming protons (2 2 + X) is followed by the fragmentation into a spray of
→
hadrons which are observed in high-energy collider experiments. Theevolution of successive splittings
q(q¯) q(q¯)g,g gg andg qq¯(q,q¯andg label quark, antiquark andgluonrespectively) inside the
→ → →
parton shower prior to hadronization is well established and can be described by the DGLAPevolution
equations forfragmentation functions intheleading-logarithmic approximation (LLA)ofQCD[10–12]
or alternatively in terms of Monte Carlo (MC) formulations such as the PYSHOW algorithm [13,14].
In A-A collisions, partons produced in the hard inelastic scattering of two partons from nuclei (2
→
2+X)propagatethroughthehot/denseQCDmediaalsoproducedinsuchcollisionsandtheirbranching
pattern is changed by interacting with the color charges of the deconfined quark gluon plasma (QGP)
[15]. Asaconsequence, additional medium-induced soft gluon radiation isproduced inA-Acollisions,
which leads for instance to the modification of high-p hadroproduction [16–18]. At RHIC, the main
T
observables considered to probe this physics were the nuclear suppression factor of single inclusive
hadronsR [19,20]andthesuppressionfactorofhardback-to-backdihadroncorrelationsI [21,22].
AA AA
Morerecently, through theanalytical developments ofRefs.[23–25],itwasdemonstrated thatmedium-
induced soft gluon radiation is ruled by a condition of antiangular ordering over successive emissions
of such gluons, which oppositely to the condition of angular ordering in the vacuum (for a review see
Ref. [26]), leads to the jet broadening. However, further efforts are required in order to implement the
fulldescription ofcolordecoherence effectsinMonteCarloeventgenerators.
Inthispaper, weaimatdiscussing twodifferent observables currently relevantmainlyforLHCphysics:
the energy collimation and the jet shape of medium-modified showers. We use a QCD-inspired model
introduced by Borghini and Wiedemann (BW) [27] for the modification of the fragmentation functions
(FFs) by the medium where the medium evolution itself is described by a hydrodynamical evolution.
In the BW model, the DGLAPsplitting functions are enhanced in the infrared sector in order to mimic
themedium-induced softgluon radiation. Inpractice, the1/z dependence oftheQCDvacuum splitting
functions corresponding to the parton branchings q(q¯) q(q¯)g and g gg are altered by introducing
→ →
a parameter Ns = 1 + fmed (fmed 0) in the form Pa→bc(z) = Ns/z + (1), which is simple
≥ O
and mostly leads to analytical results. In Ref. [28], which appeared long after [27], Nayak derived in-
mediumexpressionsforthequarkandgluonDGLAPsplittingfunctionsinnonequilibrium(nonisotropic)
QCD at leading order in α as a function of arbitrary non-equilibrium distribution functions f (p~) and
s q
f (p~), where p~ is the momentum of the hard parton. The modification to the splitting functions turns
g
out to be quite similar to the one introduced in the BW model [27] by some prefactor depending on
f (p~)andf (p~), which affects both theinfrared and regular termsof theevolution kernels. Lateron, in
q g
1
Refs.[25,29],thesplitting functions weremodifiedbyanoverall factor whichdepends onthepertinent
medium parameters, such as the medium transport coefficient qˆ. Whether these modification prefactors
are related to those found in [28] may be an interesting issue of further investigation, but this is out of
the scope of this paper. Moreover, the BW model in Ref. [27] and the calculations in Ref. [29] show
thattheproduction ofsofthadrons asdescribed bytheFFsisenhanced atsmallenergy fraction xofthe
outgoing hadrons. Thesameresult wasalso found in Ref.[30], where afull resummation from large to
smallxwasperformed inthe sameframeofthevacuum Albino-Khiehl-Kramer parametrization ofFFs
(for a detailed review see Ref. [31]), which further motivates the use of the simple BW prescription in
thefollowing.
We start with the quantification of the jet energy collimation and a study of the jet broadening in gluon
and quark jets in LLA. The computation of the energy collimation was first performed analytically in
the vacuum [32] and subsequently modeled in the medium [33] by means of the inclusive spectrum of
partons provided by the medium-modified solution of DGLAP FFs at large x 1. For this purpose,
∼
a high-energy jet of half opening angle Θ , energy E and virtuality Q = EΘ produced in a nucleus-
0 0
nucleuscollision wasconsidered, followedbytheproduction ofoneconcentric sub-jetofopening angle
Θandtransversemomentumk⊥ = xEΘwherethebulkxE E ofthejetenergyiscontained [32,33].
∼
By definition, the smaller the angle Θ where the jet energy is concentrated, the higher the jet energy
collimation [32]. The half opening angle of the jet Θ should be fixed according to the jet definition
0
used bytheexperiment. Theenergy collimation canbethen determined bymaximizing thedistribution
ofpartonsD(x,EΘ ,EΘ),whichdominatesthehardfragmentation (x 1)ofthejetintoasub-jet, as
0
∼
discussed inRef.[33].
Inthispaper, wewillprovide amoreaccurate description oftheenergy collimation, whichaccounts for
the x dependence in the third argument of the FFs D(x,lnEΘ ,lnxEΘ). The account of the shift in
0
lnxleadstoasmallnext-to-leadinglogarithmicapproximation(NLLA)correctionoforder (α )which
s
O
decreases theenergycollimation atintermediate valuesofxasaconsequence ofthescalingviolation of
DGLAPFFs[34]. Ourfirst aim is indeed to make a comparison for the jet energy collimation between
the LLA, NLLA with Pythia 6 [13,14] and YaJEM [35] in the medium. Such studies have been done
for the vacuum case [36], but it is far from being evident that there are no additional differences in the
mediumandthisissueshould befurtherstudied.
Theintegrated jetshapeΨ(Θ;Θ )provides indeed ananalogous measurement ofhowwidelythetrans-
0
verseenergyofthejetisspread. Thisobservablewasfirststudiedinthevacuumin[36]andgeneralized
tothemediumthroughoutthecalculationspresentedin[37]and[38]intheframeworkoftheconeandk
t
jet reconstruction algorithms. By definition, Ψ(Θ;Θ ) determines the energy fraction [x Ψ(Θ;Θ )]
0 0
≡
of a jet of half opening angle Θ that falls into a sub-jet of half opening angle Θ for a fixed jet energy
0
E. In our framework, we extract for the first time the angular dependence of the sub-jet energy frac-
tion from the simple prescription provided by the LLA DGLAP energy collimation x = f(EΘ ,EΘ)
0
at fixed energy E; no other computation for this observable is known in the context of LLA DGLAP
evolution equations. Thus, the jet energy collimation is a LLA DGLAP observable; however it is not
measuredinaheavy-ions collisionscontext,butjetshapesare. Thatiswhy,inthecomparisonwithMC,
we do the jet shape analysis with Pythia 6 and YaJEM using the FastJet package [39,40] on the events
2
andcompareourresultswithCMSppandPbPbdataforcentralcollisions (0-10%)at2.76TeV[9]. The
BWprescriptionprovidesasimpletestcaseforthis,asitisanalyticallysolvableandeasilyimplemented
intheYaJEMcode,itwillbeshowntocapturethemainphysicsofadditional softgluonproduction and
jetbroadening through thisobservable.
Finally, for the purpose of a detailed comparison with data from the CMS experiment, the jet-shape
analysiswithYaJEMisperformedforbothPbPbandppcollisionsfollowingtheCMSanalysisprocedure
closely. Jets are reconstructed with the anti-k algorithm [39,40] with a resolution parameter R(
T
≡
Θ ) = 0.3. The clustering analysis is limited to charged particles with e > 1 GeV inside the jet cone
0 i
where E 100 GeV is required for a jet (i.e. E stands for the recovered jet energy inside the
rec rec
≥
cone)[9]. Theconditione > 1GeVremovesthesoftQCDmediumbackground whichmayblurthejet
i
fragmentation and jet-shape analysis. In order to illustrate the role of the bias caused by the jet-finding
procedureoutlinedabove,wecomparethebiasedjetshape(i.e. providedtheCMSjet-findingconditions
are fulfilled) with an unbiased jet shape (which is a purely theoretical quantity) for both PbPb and pp
CMSdata.
2 Theoretical framework
2.1 Description oftheprocess andkinematics
In Fig. 1, we consider the production of one gluon or quark (A = g,q,q¯) jet of total energy E and
openingangleΘ whichfragmentsintoasub-jetB ofenergyxEandopeningangleΘ(Θ Θ ),where
0 0
≤
xistheenergyfraction ofAcarriedbyB.
By definition, the virtualities of the jet A and the sub-jet B are Q = EΘ0 and k⊥ = EpΘ (Ep = xE)
respectively. Thevirtuality,alsoknownasthehardnessofthejet,determinesthephasespaceforradiation
andhencesetsthemaximaltransverse momentumofaparton inside thejet: k⊥ Q. Aminimalcutoff
≤
parameter Q0 can be introduced k⊥ Q0, such that the minimal angle reached by a parton inside the
≥
cascadeequalsΘ Q /xEΘ . Experimentally, thisphysicalpicturecorrespondstothecalorimetric
min 0 0
≥
Θ0 B Θ
DA
Ε A xE B
Figure 1: Fragmentation of a jet A of half opening angle Θ into a sub-jet B of half opening angle
0
Θ < Θ .
0
measurementoftheenergyfluxdepositedwithinagivensolidangle. Fromthepartonicpointofview,the
successive decaysofpartons inthecascade areordered ink⊥,i,orangles Θi duetotheLLAkinematics
3
forhard parton decays (x 1)ordue tothe QCDcoherence forsoft parton decays (x 1)[32]. Hard
∼ ≪
parton decays determine the bulk ofthejetenergy and areruled by theLLAkinematics, which leads to
DGLAPevolutionequations [41],whilesoftpartondecaysdeterminethebulkofthejetmultiplicityand
areruledbyQCDcoherence,whichleadsinsteadtothemodified-LLA(MLLA)evolutionequations[32].
The jet energy collimation is characterized by the large energy fraction x of the sub-jet where the bulk
ofthe jetenergy inside the givencone Θ < Θ 1isdeposited. Hence, theprobability for theenergy
0
≪
fractionxtobedepositedinaconeofapertureΘisrelatedtotheDGLAPinclusivespectrumofpartons
through theformula[42],
D (x,EΘ ,xEΘ) = DB(x,EΘ ,xEΘ), (1)
A 0 A 0
B=g,q
X
where the nature of partons B is not identified. In Eq. (1), the FFs DB(x,EΘ ,xEΘ) determine the
A 0
probability that a parton A produced at large p E in a high-energy collision fragments into a hard
T
∼
sub-jet B oftransverse momentum xEΘ,whichwewriteinthethirdargument oftheFF.Qualitatively,
Eq.(1)describestheevolutionofthejetAinthek⊥ rangexEΘ k⊥ EΘ0accordingtotheLLAk⊥
≤ ≤
ordering and hence, it determines the partonic skeleton of the sub-jet B before the hadronization takes
place.
As compared to the FF for the inclusive spectrum of partons where the third argument is set to EΘ for
hard partons x 1: DB(x,EΘ ,EΘ) [33], the formula (1) accounts for the energy fraction x of the
∼ A 0
sub-jet B in the FFs DB(x,EΘ ,xEΘ). We cannot compute DB(x,EΘ ,xEΘ) by using DGLAP
A 0 A 0
evolution equations because of the x dependence included in the third argument, but we can instead
expanditinpowersof“lnx”throughtheexponential operator,
DB(x,EΘ ,xEΘ) = elnx(∂/∂ln(EΘ))DB(x,EΘ ,EΘ) (2)
A 0 A 0
suchthat,
α (EΘ ) ∂DB
DB(x,EΘ ,xEΘ) = DB(x,∆ξ) lnx s 0 e4Ncβ0∆ξ A(x,∆ξ)+ (α2), (3)
A 0 A − 2π ∂∆ξ O s
where,
2
1 EΘ 2π
ξ(EΘ) = ln ln , ∆ξ = ξ(EΘ ) ξ(EΘ), α (EΘ ) = .
0 s 0
4Ncβ0 " (cid:18)ΛQCD(cid:19) # − 4Ncβ0ln ΛEΘ0
QCD
(cid:16) (cid:17)(4)
Hence,(1)canberewrittenintheform,
α (EΘ ) ∂DB
D (x,EΘ ,xEΘ)= DB(x,∆ξ) lnx s 0 e4Ncβ0∆ξ A(x,∆ξ) + (α2), (5)
A 0 A − 2π ∂∆ξ O s
B=g,q(cid:20) (cid:21)
X
where α is the QCD coupling constant, β is the first coefficient of the QCD β function given by
s 0
β = 1 11N 4T n , with N = 3, T = 1, n = 3 and Λ (= 300 MeV)is the mass scale
0 4Nc 3 c− 3 R f c R 2 f QCD
of QCD. The new correction (α ) is very small as x 1 and can be much larger for x 0.5. As
(cid:0) (cid:1) s
O → ≈
displayedinFig.1,theladderFeynmandiagramsleadingtoDGLAPevolutionequationsforDB(x,∆ξ)
A
4
should be iterated from the hardest virtuality Q = EΘ of the process to the lower sub-jet virtuality
0
k⊥ = EΘthroughthevariable∆ξ in(4). Thus,DB(x,∆ξ)describesthedistribution ofpartonsB with
A
transverse momentum k⊥ = EΘ contained inside the parton A, which fixesthe initial scale ofthe hard
process: Q = EΘ ;i.e. thevirtuality. Therefore, wecanestimate D (x,EΘ ,xEΘ)withthesolution
0 A 0
ofDGLAPevolution equations fortheFFsDB(x,∆ξ),whichappearontherhsofEq.(5).
A
2.2 Medium-modified DGLAP evolutionequations withtheBW model
TheDGLAPevolution equations for the splitting A[1] B[z]C[1 z](where z is theenergy fraction
→ −
ofonepartoninthesplitting) inthek⊥ rangeEΘ k⊥ EΘ0 takesthesimpleform[41],
≤ ≤
d α (EΘ) 1 dz x
s
D(x,EΘ ,EΘ) = P(z)D ,EΘ ,EΘ , (6)
0 0
dlnEΘ 4π z z
Zx (cid:16) (cid:17)
whereP(z)istheevolution“Hamiltonian”givenbytheregularizedsplittingfunctions[10–12]. Inorder
to account for the medium-induced soft gluon radiation in heavy-ion collisions, we make use of the
QCD-inspired model proposed in Ref. [27] which leads to a simple solution of the evolution equations
atx 1. Inthismodel,theinfraredpartsofthesplittingfunctions arearbitrarily enhanced bythefactor
∼
N = 1 + f , where f 0 accounts for medium-induced soft gluon radiation. The medium-
s med med
≥
modifiedsplitting functions inz spacearewrittenintheform[27],
N N
P (z) = 4N s + s +z(1 z) 2 , P (z) = 2T [z2+(1 z)2], (7a)
gg c z 1 z − − gq R −
(cid:20) (cid:20) − (cid:21)+ (cid:21)
2N 2N
s s
P (z) = 2C +z 2 , P (z) = 2C 1 z , (7b)
qg F z − qq F 1 z − −
(cid:18) (cid:19) (cid:18)(cid:20) − (cid:21)+ (cid:19)
1 1
wherethe[...] prescriptionisdefinedas dx[F(x)] g(x) dxF(x)[g(x) g(1)]. Thesolutions
+ 0 + ≡ 0 −
ofEq.(6)canmostconvenientlybeobtainedinMellinspace (j,EΘ ,EΘ)throughthetransformation
R R 0
D
1
(j,EΘ ,EΘ) = dxxj−1D(x,EΘ ,EΘ),
0 0
D
Z0
suchthattheconvolution (6)yields,
d 1
(j,EΘ ,EΘ) = (j) (j,EΘ ,EΘ), (j) = dzzj−1P(z). (8)
0 0
dlnEΘD P D P
Z0
Theadvantage ofthe Mellintransform canbeclearly seen inEq.(8). Theconvolution overz inEq.(6)
reduces to the product of the Mellin-transformed splitting functions (j) and the FFs (j,EΘ ,EΘ).
0
P D
Making use ofthe variables introduced in(4), (8)can bemore explicitly rewritten in thematrix form at
leadingorderLO:
(j,ξ) (j) 0 0 (j,ξ)
d DqNS Pqq DqNS
(j,ξ) = 0 (j) (j) (j,ξ) , (9)
dξ DqS   Pqq Pqg DqS 
(j,ξ) 0 (j) (j) (j,ξ)
 Dg   Pgq Pgg  Dg 
    
5
where and stand, respectively, for the flavor-nonsinglet and flavor-singlet quark distributions,
DqNS DqS
and (j)aretheMellintransformsoftheLOsplittingfunctions:
ik
P
N 1 N 1
s s
(j) = 4N N ψ(j +1)+N γ − −
Pgg − c s s E − j − j 1
(cid:20) − (cid:21)
11N 2n 8N (j2 +j+1)
+ c f + c , (10a)
3 − 3 j(j2 1)(j +2)
−
j2 +j +2
(j) = T , (10b)
Pgq R j(j +1)(j +2)
(2N 1)(j2 +j)+2
s
(j) = 2C − , (10c)
Pqg F j(j2 1)
−
N 1 2
s
(j) = C 4N ψ(j +1)+4N γ 4 − 3 . (10d)
Pqq − F s s E − j − − j(j +1)
(cid:20) (cid:21)
Thismethodallowsforthediagonalizationofthe“Hamiltonian”givenbytheset (j)withrespecttothe
P
“evolution-time” variable ξ t = ln(EΘ). In some limits at large and small x, analytical solutions of
∼
theequationscanbefoundthroughthismethod[41]butfornumericalcomputation,solvingtheequations
directly inxspace turns outtobe moreefficient than inverting the Mellintransform numerically. Thus,
atlargeenergy fraction x 1,orequivalently large j 1(whichweareinterested in),theexpressions
∼ ≫
fortheMellinrepresentation ofthesplitting functions (10a)-(10d)canbereducedto,
3 β
0
(j) 4C N lnj + γ , (j) 4N N lnj + γ , (11)
qq F s E gg c s E
P ≈ − 4N − P ≈ − N −
(cid:18) s (cid:19) (cid:18) s (cid:19)
wheretheasymptotic behavior ofthedigammafunction ψ(j +1) lnj isreplaced atj 1[41]. The
≈ ≫
off-diagonal matrixelementsvanishinthisapproximation: (j) = (j) = 0.
gq qg
P P
NotethattheN lnj dependenceinEq.(11)arisesfromthe[N /(1 z)] termsoftheDGLAPsplitting
s s +
−
functions, suchthat forhard partons z 1, theenhanced contribution ofthesoft 1 z 0component
∼ − ∼
[1/(1 z)] producesthesub-jetbroadeningwithinthisapproximation. Qualitatively,asaconsequence
+
−
′
of energy conservation, if soft gluon radiation is enhanced in the region Θ Θ Θ for a fixed jet
0
≤ ≤
energy E, the sub-jet energy (B, xE) should be smaller compared to its value in the vacuum and the
energy collimationshould thendecrease, i.e. Θshouldincrease.
Goingbacktoxspacerequires takingtheinverseMellintransform givenby
1
D(x,∆ξ) = djx−j (j,∆ξ), (12)
2πi D
ZC
where the contour C in the complex plane is parallel to the imaginary axis and lies to the right of all
singularities. Since weare interested inthe large j 1(x 1) approximation, weinsert Eq.(11)into
≫ ∼
Eq.(9). Afterintegrating theresult, wegetthemedium-modifieddistribution atlargex 1,
∼
exp[4C N ( 3 γ )∆ξ]
DA(x,∆ξ) (1 x)−1+4CANs∆ξ A s 4Ns − E . (13)
A ≃ − Γ(4C N ∆ξ)
A s
whereβ isreplacedby3/4(n = 3)inEq.(11). Thecorresponding resultinthevacuumforN = 1is
0 f s
giveninRef.[41].
6
Withinthisapproximation, thepartoninitiatingthejetAisidenticaltothatinitiating thesub-jetB = A,
C = N if A is a gluon and C = C = 4 if A is a quark. Indeed, the FF DA(x,∆ξ) in Eq. (13)
A c A F 3 A
describes the splittings g gg and q qg and as constructed, it neglects the others: g qq¯and
→ → →
q gq. Therefore, thesumoverB inEq.(5)disappears suchthat,
→
α (EΘ ) ∂DA
DA(x,EΘ ,xEΘ) = DA(x,∆ξ) lnx s 0 e4Ncβ0∆ξ A(x,∆ξ)+ (α2). (14)
A 0 A − 2π ∂∆ξ O s
In a more accurate solution of this problem which could only be achieved numerically, the whole sum
ofthe parton branchings givenby D (x,EΘ ,xEΘ) = Dq(x,EΘ ,xEΘ)+Dg(x,EΘ ,xEΘ)fora
q 0 q 0 q 0
quark jet and D (x,EΘ ,xEΘ) = Dg(x,EΘ ,xEΘ)+Dq(x,EΘ ,xEΘ) for a gluon jet, with the
g 0 g 0 g 0
full resummed contribution of the soft gluon/collinear logarithms arising from the N /z dependence of
s
thesplitting functions intheFOapproach ofDGLAPFFs[30,43].
2.3 Jetenergy collimation
As discussed in Ref. [41], the distribution (13) presents a certain maximum at some angle Θ where the
bulk of the jet energy is concentrated. The reason for this can be understood as follows: for ∆ξ 0,
→
or Θ Θ , almost all of the energy is contained inside the cone Θ [i.e. D δ(1 x)] and the
0 0
→ → −
probability distribution DA forx = 1shoulddecrease. ForΘdecreasing Θ Λ /E [noticethatthe
A 6 ≫ QCD
x dependence was reabsorbed on the pre-exponential term in Eq. (2)], the emission outside the cone Θ
growsandthefragmentation probabilitydecreases. Then,takingthefirstderivativeoverlnΘinEq.(14)
leadstotheNLLA(nottobeconfused withtheMLLA)equation forΘ:
3 α (EΘ )
ln(1 x)+ γ ψ(4C N ∆ξ) 1 4N β eb∆ξlnx s 0 =
E A s c 0
− 4N − − − 2π
(cid:20) s (cid:21)(cid:18) (cid:19)
2
α (EΘ ) 3
4C N e4Ncβ0∆ξlnx s 0 ln(1 x)+ γ ψ(4C N ∆ξ)
A s E A s
2π − 4N − −
(cid:20) s (cid:21)
α (EΘ )
4C N e4Ncβ0∆ξlnx s 0 ψ(1)(4C N ∆ξ), (15)
A s A s
− 2π
which is the main theoretical result of this section for medium N = 0 and also vacuum N = 0. We
s s
6
invert Eq.(15)numerically inorder to getthe NLLAjetenergy collimation Θ(x,E). InEq.(15), ψ(x)
isthedigammafunction andψ(1)(x) = dψ(x) isthepolygammafunctionofthefirstorder, whichisnew
dx
inthiscontext. Notethatthisisonecorrection; amorecompletesetofcorrections ofthesameordercan
bealsoaddedif,forinstance,oneconsiders thenext-to-leading-order corrections [44]totheapproached
splittingfunctions(11)inamorecumbersomeapproachofthisproblem. However,thistermgoesbeyond
DGLAPand corresponds to the so-called scaling violation in DGLAPfragmentation functions [34]. In
our framework, this correction slightly increases the available phase space from the hardest (B, x 1)
∼
to slightly softer partons (B, x 0.5) and is therefore expected to decrease the energy collimation or
∼
increase Θatintermediate x. Asexpected for harder partons lnx 0, the above equation (15)reduces
∼
tothesimplerone[33],
3
ψ(4C N ∆ξ)= ln(1 x)+ γ . (16)
A s E
− 4N −
s
7
Symbolically, the inversion of the NLLA (15) and LLA (16) can be written for quark (A = q,q¯) and
gluon(A = g)jetsinthesimpleform,
Θ EΘ −γA(x,Ns)
A 0
= . (17)
Θ Λ
0 (cid:18) QCD(cid:19)
Settinglnx 0inEq.(15),theLLAexpression forγ (x,N )issimplywrittenintheform[33]
A s
→
N β 3
γ (x,N )= 1 exp c 0 ψ−1 ln(1 x)+ γ , (18)
A s E
− −C N − 4N −
(cid:20) A s (cid:18) s (cid:19)(cid:21)
whereψ−1istheinverseofthedigammafunction. Theexponentγ (x,N )providesindeedthemedium-
A s
modifiedslopeoftheenergycollimationasafunctionofN forafixedvalueofthesub-jetenergyfraction
s
xandcanbeobtainednumericallyfromtheNLLAequation(15). InTable1,wedisplaythevaluesofthe
NLLAandLLAslopesforx = 0.5andx = 0.8,whichareinagreementwiththeLLA(NLLA)DGLAP
large sub-jet energy fraction x approximation where these predictions should be tested. As x 0, the
→
fixed-order(FO)approach oftheLLAfailstoprovideanyreliableresult.
NLLA,LLA x = 0.5 x = 0.8 NLLA,LLA x = 0.5 x = 0.8
γ (x,1.4) 0.37 0.26 γ (x,1) 0.54 0.38
g g
γ (x,1.4) 0.67 0.50 γ (x,1) 0.83 0.65
q q
Table1: NLLAandLLAvaluesoftheslopeγ (x,N )oftheenergycollimationforN = 1.4(medium)
A s s
andN = 1(vacuum).
s
Thenewequation (15)cannot berewrittenlikeEq.(17)butitcanbesolved numerically. FromTable1,
one may wonder why the NLLA (15) and LLA (16) slopes of the energy collimation are the same3.
Indeed, the coupling constant does not depend on the jet energy only, but rather on the product EΘ
≫
Λ throughthetermlnxe4Ncβ0∆ξα (EΘ ) lnxα (EΘ)inEq.(15). AsthejetenergyEincreases,
QCD s 0 s
∼
the sub-jet cone Θ decreases and α (EΘ) should remain roughly constant. Therefore, the NLLA and
s
LLAcurvesofthejetenergycollimationshouldstayapproximatelyparalleltoeachotherasymptotically.
WecanseeinbothcasesthatthenuclearsuppressionparameterN decreasestheslopeoftheenergycol-
s
limation, whichtranslates into increasing the rateof thejetbroadening asymptotically. In both medium
and vacuum γ > γ , which physically means that quark jets are more collimated than gluon jets. The
q g
sametrendsshouldbeconfirmedintheforthcominganalysisofthejetenergycollimationwiththeevent
generator YaJEM.
3 Comparison with YaJEM and QGP hydrodynamics
Inordertogaugetheimpactoftheapproximations madeinderivingtheresultsoftheprecedingsection,
wecomparethemwithresultsforjetenergycollimationobtainedinaMCformulationofthein-medium
3NoticethatTable1displaysindeed16valuesfortheslopes,butsuchnumbersareidenticalfortheNLLAandLLAenergy
collimation.
8
jet evolution. Within such a model, the parton initiating a jet A does not have to be identical to that
initiating the sub-ject B and hence the full set ofsplittings g gg and g qq is available fora gluon
→ →
jet. Inaddition, exactenergy-momentum conservation ateverysplitting vertexisenforced.
Invacuum, thePYSHOWalgorithm [13,14]isawell-tested numerical implementation ofQCDshower
simulations. For comparison with our analytic results, we use the Borghini-Wiedemann prescription
implemented withinthein-medium showercode[45].
3.1 The in-medium showergenerator YaJEM
YaJEM is based on the PYSHOW algorithm, to which it reduces in the limit of no medium effects. It
simulates theevolution ofaQCDshowerasaniterated seriesofsplittings ofaparent intotwodaughter
partons a bcwhere the energy of the daughters are obtained asE = zE and E = (1 z)E and
b a c a
→ −
thevirtualityofparentanddaughtersisorderedasQ Q ,Q . Thedecreasinghardvirtualityscaleof
a b c
≫
partonsprovidessplittingbysplittingthetransversephasespaceforradiation, andtheperturbative QCD
evolution terminates once the parton virtuality reaches a lower value Q = 1 GeV, at which point the
0
subsequent evolution isconsidered tobenonperturbative hadronization.
The probability distribution to split at given z is given by the same QCD splitting kernels and their
medium modification in the BW prescription, which we have used above, i.e. Eq. (7a) and Eq. (7b);
howeverintheexplicitkinematicsoftheMCshowerthesingularities forz 0orz 1areoutsideof
→ →
accessible phasespaceandno[...] regularization procedure isneeded.
+
We will refer to the implementation of the BW prescription for in-medium showers in the following as
YaJEM+BW (note that this corresponds to the FMED scenario described in Ref. [45]). This is distinct
from the default version of the code YaJEM, YaJEM-DE, which is tested against multiple observables
at both RHIC and LHC (see e.g. Refs. [46–48]) and is based on an explicit exchange of energy and
momentumbetweenjetandmediumratherthanamodificationofsplitting probabilities.
For a straightforward benchmark comparison with analytic results, a value of f can be chosen, the
med
parton shower can be computed and stopped at the partonic level or evolved using the Lund model to
the hadronic level, and then clustered using the anti-k algorithm and properties like collimation or jet
T
shapescanthenbeextracted.
In a MC treatment of the shower evolution, using a constant value of f to characterize the medium
med
is not needed and in fact not realistic once a comparison with data is desired. Following the procedure
in Ref. [45], the value of f is determined event by event by embedding the hard process into a
med
hydrodynamical medium[49]startingfromabinaryvertexwhichisat(x ,y )andfollowinganeikonal
0 0
trajectory ζ through themediumevaluating thelineintegral
f = K dζ[ǫ(ζ)]3/4(coshρ(ζ) sinhρ(ζ)cosψ). (19)
med f
−
Z
where ǫ is the local energy density of the hydrodynamical medium, ρ the local flow rapidity and ψ the
angle between the flow and the direction of parton propagation. Events are then generated for a large
numberofrandom(x ,y )sampledfromthetransverse overlapprofile
0 0
9