Pathlength dependence of energy loss within in-medium showers

0.17 MB·English

by  Thorsten Renk

Pathlength dependence of energy loss within in-medium showers
∗
Thorsten Renk
Department of Physics, P.O. Box 35, FI-40014 University of Jyva¨skyla¨, Finland and
Helsinki Institute of Physics, P.O. Box 64, FI-00014 University of Helsinki, Finland
(Dated: January 12, 2011)
Studying the pathlength dependence of high PT hadron suppression in heavy-ion collisions by
measuringthedependenceofhardhadronproductionontheangleφwiththereactionplaneinnon-
central collisions has so far been one of the most successful tools in constraining the microscopical
picture of leading parton energy loss. With the imminent start of the LHC heavy-ion program
and the possibility of full jet reconstruction, the focus has shifted to models which are capable of
simulating full in-medium parton showers rather than tracing the leading parton only. Yet, on the
level of single inclusive hadron observables, such shower models need to reproduce the findings of
1 leading parton energy loss models, in particular they need to reproduce the correct reaction plane
1 angledependenceofhadronsuppression. Theaimofthisworkistoillustrateattheexampleofthe
0
in-medium shower code YaJEM (Yet another Jet Energy-loss Model) how pathlength dependence
2
arises in a MC shower formulation, how it relates to pathlength dependence of leading parton

n energy loss and how model results compare with data on the reaction plane dependence of the
a nuclearsuppression factor RAA(φ).
J

1 PACSnumbers: 25.75.-q,25.75.Gz
1

] I. INTRODUCTION Monte-Carlo (MC) models capable of describing the de-
h
velopment of a parton shower originating from a highly
p
virtual initial parton inside a medium, followed by
- The idea to use the modification of high P hadron
p T hadronizationintojets arebeing developed. Bothelastic
production due to the bulk medium produced in ultra-
e collisions [15] as well as medium-induced radiation [16–
h relativisticheavy-ioncollisionstostudypropertiesofthe
19] have been considered as mechanisms for the modifi-
[ medium is known as ’jet tomography’ [1–5]. However,
cation.
in order to gain any tomographical information about
2
the macroscopical medium density distribution and its
v
6 evolutionfrom measureddata in practice, it is necessary
1 to know first the microscopicalmechanism by which the
Monte-Carlo(MC)modelsarenotnaturalframeworks
1 medium modifies high P hadron spectra. A less ambi-
T to incorporate the physics of quantum coherence (which
4 tious approach is to determine under what assumptions
appears necessary to reproduce the pathlength depen-
.
0 a combination of macroscopical evolution in terms of a
dence of energy loss seen in the data), but it has been
1 bulkevolutionmodellikefluiddynamicsandmicroscopi-
demonstrated in [20] that an algorithm can be devised
0 calparton-mediuminteractionmodelcandescribealarge
which consistently reproduces known eikonal infinite en-
1 body of data in a systematic way [6–9]. So far, most of
: ergy analytical limits. Yet in the same work, it has also
v these studies have been carriedout for models which de-
been shown that including realistic effects such as finite
i scribe the medium modification as energy loss from the
X energy corrections, the pathlength dependence is effec-
leading shower parton prior to hadronization.
tively weaker than quadratic (see also [21]). This is in
r
a An emerging result is that the pathlength dependence manifestcontradictiontosystematicinvestigationswhich
of energy loss, accessible experimentally e.g. via the de- show that the data for R (φ) are marginally in agree-
AA
pendence of the nuclear suppression factor RAA(φ) on ment with a quadratic dependence and would be easier
angleof outgoinghadronswith the reactionplane φ pro- reproduced with a cubic dependence [13].
vides a crucial constraint for the microscopical physics.
In particular, the pathlength dependence cannot be lin-
ear in a constant medium (as characteristicfor elastic or
generally incoherent processes) [11, 12] but must be at Theaimofthispaperistoillustratetheconnectionbe-
least quadratic [13] characteristic of coherence, possibly tween the pathlength dependence of energy loss in lead-
even cubic [13, 14] as found in some versions of strongly ing parton models and the effect of pathlength on an
coupled QCD-like models. in-medium shower more clearly and to suggest a possi-
On the other hand, going beyond the capability to ble resolution for the apparent puzzle outlined above by
describe single inclusive hadron observables to full jets, suggesting that the pathlength dependence observed in
the data might be due to a yet different effect. For all
computations in the following, the radiative energy loss
scenario (RAD) of the MC in-medium shower code Ya-
∗Electronicaddress: thorsten.i.renk@jyu.fi JEM is used.
2
II. THE MODEL medium is increased by the ratio of their Casimir color
factors,3/4 =2.25. Themediumischaracterizedviathe
3
A. General description parameter qˆ(ζ) where a relation like
YaJEM is described in detail in [16, 17], here we just qˆ(ζ)=K·2·[ǫ(ζ)]3/4(coshρ(ζ)−sinhρ(ζ)cosψ) (4)
reproduce the essential details necessary to understand
how pathlength dependence arises within the model.
is assumed as the link to fluid dynamical parameters
In the absence of a medium, YaJEM utilizes the
such as the energy density ǫ or the local transverse flow
PHYSHOWalgorithm[22]inmomentumspacetomodel
rapidity ρ (at angle ψ with the parton trajectory).
the evolution from a highly virtual initial parton to a
One will observe that Eq. (1),(3) are not indepen-
showeroflowvirtualitypartonsandtheLundmodel[23]
dent — parton virtuality determines the parton forma-
to hadronize this shower. In this framework, the QCD
tiontime, but the formationtime inturndetermines the
evolution from a highly virtual state to a parton shower virtuality gain from the medium. If ∆Q2 ≪ Q2, holds,
is modelled as a series of branchings a → bc where the a a
i.e. the virtuality picked up from the medium is a cor-
virtuality scale decreases at each step. rection to the initial parton virtuality, we may add ∆Q2
In the presence of a medium, it is no longer sufficient a
tothe virtualityofpartonabeforedeterminingthe kine-
tomodelinmomentumspaceonlysincethemediumevo-
matics of the next branching. If the condition is not ful-
lution is caused by pressure gradients in spacetime and filled, the lifetime is determined by Q2 +∆Q2 and may
the density of matter felt by hard partons is a function a a
be significantly shortened by virtuality picked up from
ofpositionspace ratherthanmomentum space. In order
themedium. InthiscaseweiterateEqs.(1),(3)todeter-
to make a link between the shower evolution in momen- mine a self-consistent pair of (hτ i,∆Q2). This ensures
tum space and the medium density evolution in position a a
that on the level of averages,the lifetime is treated con-
space, YaJEM utilizes the uncertainty relation to com-
sistently with the virtuality picked up fromthe medium.
pute the average formation time of a virtual daughter
The actual lifetime is still determined by Eq. (2). This
parton b generated in the branching of a parent a as
consistency condition is fundamentally the same as the
algorithm presented in [20]. It corresponds to the limit
E E of a coherent summation of a large number of medium
b b
hτbi= Q2 − Q2. (1) interactionsduringtheformationtime(inYaJEM,virtu-
b a alitygainisasmoothfunctionofpathlength,ratherthan
where E denote parton energies and Q parton virtu- a sum over discrete interactions transferring virtuality)
i i
alities. YaJEMprovidesthe additionaloptionnotto use and hence represents LPM interference in this limit.
theaverageformationtimebutratherforeachbranching However,fortechnicalreasons,despitetheconsistency
an actual formation time given by an exponential distri- condition for medium-induced virtuality gain, the in-
bution medium shower simulated by YaJEM remains an ex-
pansion for thin media. The reason is that, as in the
PYSHOW algorithm, parton splitting no longer occurs
P(τ )=exp − τb (2) once a parton has reached a lower virtuality scale Q0
b
(cid:20) hτbi(cid:21) (where Q0 = O(1) GeV — to be more precise, for mas-
sive partons the limit is given by mq + mg with
eff eff
whichaveragestohτ i. Thisallowsto specifythe tem-
poral structure of thebshower, and combined with the mgeff = 21Q0 and mqeff = m2q + 14Q20 and mq the cur-
q
assumption that all shower partons sample the medium rent quark mass). Partons falling at any point of the
along the eikonaltrajectoryof the shower-initiating par- evolution below this scale are set on-shell and do not
ton, the complete space-time picture of the shower is longer branch. YaJEM inherits this condition also in a
fixed. medium. This limits the distance scale L for which
max
Intheradiativeenergylossscenario(RAD)ofYaJEM, an in-medium shower is evolved reliably parametrically
theparton-mediuminteractionismodelledbyamedium- to L ∼ E/Q2 where E is the energy of the shower-
max 0
inducedvirtualitygainduringthepartonformationtime initiating parton.
∆Q2 = τa0+τadζqˆ(ζ) (3) B. Pathlength dependence
a Zτa0
It should now be clear from where any dependence of
where the time τa is given by Eq. (2), the time τa0 partondistributionscreatedbyYaJEMonthein-medium
is known in the simulation as the endpoint of the pre- pathlength arise.
vious branching process and the integration dζ is along
the eikonal trajectory of the shower-initiating parton. If • The leading effect is given by Eq. (3) which states
the parton is a gluon, the virtuality transfer from the that the amount of medium-induced virtuality is
3
a monotonically increasing function of the path- C. Implications of a dynamical minimum virtuality
length. If qˆ is a constant, this states that the to-
tal amount of medium-induced virtuality (or the
Let us take some time to consider the implications of
number of scattering centers probed by a parton)
Eq.(5)insomemoredetail. Theobviousproblemisthat
is linearly proportional to the pathlength. Since
Q0 regulates both the endpoint for the vacuum and for
Eq. (3) holds for every parton in the shower, this
the in-medium parton shower evolution (which can’t be
would lead to a crescendo-like scenario [24] for the
disentangled on a branching-by-branching basis). While
virtuality transfer to the shower as a whole in the
lengthdependenceinthemediumcontributionisreason-
absence of finite energy or finite virtuality correc-
able, there is no justification for any length effect on the
tions (in practice however, such corrections turn
vacuum shower evolution. In contrast, in vacuum, Q0
out to dominate the dynamics [25]).
regulates the transition between the perturbative par-
tonicandnon-perturbativehadronicdomainoffragmen-
• The consistency condition between formation time
tation. This in turn is an ill-defined condition in a de-
and medium-induced virtuality forms a secondim-
confined medium, as hadrons cannot be formed in such
portant ingredient. It states that low Q2, i.e.
a medium.
collinear medium-induced radiationmay not occur
Eq.(5)canthereforenotbeusefulingeneralbutmust
after arbitrarily short formation times. When ap-
be seen as an approximationfor a situation in which the
plied to a single almost-on-shell parton propagat-
shower evolution is dominated by medium effects and
ing through a constant medium, it gives rise to a
in which the kinematics is such that the possibility of
quadraticpathlengthdependenceofmeanradiated
hadrons being formed in the medium is not a crucial is-
energyasin [20]. However,note that since YaJEM
sue. Thelatter conditionis readilyfulfilled whenconsid-
requires Q>Q0, this situation is strictly never re- ering single inclusive high P pion observables (as done
alized. To some degree, its effect is also obscured T
in this work), as the pion formation length (estimated
bythe randomizationofformationtimesEq.(2)as
as τ ∼E /m2) is far larger than the medium size. Note
will be demonstrated later. π π
thatthisisnotthecaseforthelowz partofthefragmen-
tation function or for the production of heavierhadrons.
• The minimum virtuality scale Q0 down to which
partons are evolved induces a spurious length de-
4
pendence whenever a pathlength L > L is 10
max
probed—forL≫Lmax (i.e. faroutside the valid- 103 vacuum (x10)
ity region of the model), all partons in the shower medium, ∆Q2 = 6 GeV2
are on-shell and no branching occurs for any den- 102
sityofthemedium. Thus,whenappliedoutsidethe
10
regionofvalidity,YaJEMleadstolessmodification z)
(
of showers than expected in reality. +-h 1
>
q -
D -1
• An interesting proposal,suggestedin [26, 27], is to 10
define the minimum virtuality scale by
-2
10
-3
10
Q0 = E/L (5)
0 0.2 0.4 0.6 0.8 1
p z
whereListheactualpathlengthashowerisevolved
in the medium and E the energy of the shower-
FIG. 1: (Color online) Response of the computed fragmen-
initiating parton, rather than leave it a constant.
tation function of a 20 GeV quark into charged hadrons to
If this is done, it replaces the spurious Q0 depen- choicesof theminimum virtualityscale Q20 from 0.5 GeV2 to
dence by a genuine scale dependence of the shower 2GeV2,bothforthevacuumcaseandforamedium-modified
evolutionwhich states that the medium cannotin- showerfortotalmediuminducedfirtuality∆Q2 =6GeV2 as
duce radiation below a scale Q0. We will test this typical for the majority of partons in RHIC high PT observ-
prescription in the following. ables within YaJEM.
In practical terms, it was found in [16, 17] that many It turns out that the first condition is also fulfilled for
observables (e.g. the fragmentation function) follow to typicalin-mediumpathlengthsasrelevantforcomputing
goodapproximationascalinglawwiththeintegratedto- observables. Fig.1 shows vacuum and medium-modified
tal virtuality gain from the medium, ∆Q2 = Ldξqˆ(ξ) fragmentation functions as bands indicating their re-
0
inside the region of validity. This implies thatRthe path- sponse to a change in Q20 from 0.5 GeV2 to 2 GeV2 for
length dependence of such observables is approximately typicalpartonpathsatRHICkinematics. Itisreadilyap-
linear in a constant medium. parent from the figure that the majority of the variation
4
withQ0 seeninthein-mediumcasecannotbeaccounted carryingaway80%oftheenergyofaparentquarkwould
for by the comparatively small effect of the scale change not be counted as large energy loss but as change in the
in vacuum, indicating that the pathlength condition in identity of the leading parton. No particular branching
thissituationindeeddominatesoverthescaledependence processcanbeattributeduniquelytovacuumormedium
of the transitionto the non-perturbative regime. Partof —differencesbetweenamedium-modifiedandavacuum
thereasonisthatQ0 doesnotonlydefinetheendpointof shower can only be defined at the level of averages over
theperturbativeshowerdescription,butalsothestarting many events. This implies that the baseline to compare
pointofthenon-perturbativeLundhadronizationmodel. with is not a single parton with a definite energy, but
Thus,tosomedegree,inthevacuumdecreasedshowering rather a vacuum shower with a statisticaldistribution of
iscompensatedbyanincreasedactionoftheLundmodel, hadrons at given momenta.
whereasnosuchcompensationoccursinthemediumcase As a way to overcome these difficulties and to make
(sincethe hadronizationisbyassumptionalwaystreated contactbetweenin-mediumshowerandenergylossmod-
as in vacuum). elling,in[17]c-quarksasshowerinitiatorshavebeencon-
Nevertheless, Eq. (5) has to be regarded as a first ap- sidered. Due to the deadcone effect, these have compar-
proximation. Ultimately,acriterionwhichalsotakesinto atively low vacuum shower activity and a hard fragmen-
accountthevirtuality∆Q2inducedbythemediumalong tationpattern,thusminimizingthedistortionofthevac-
the parton path would be superior. This can be under- uum baseline from a single parton with definite energy.
stoodinthelimitofsmallpathlengthL. Forapplications Moreover,since c quark generationin branchings g →cc
toobservables,∆Q2andLarenotindependentandsmall is mass suppressed, tagging the c quark almost always
L also implies small ∆Q2. In this case, the appropriate tags the shower initiator.
limit for Q0 is the vacuum default value Q0 = 1 GeV Energylossoftheleadingcquarkcanthenbededuced
rather than the value given by Eq. (5), as the assump- from the ansatz
tion that the medium effect dominates the dynamics of
the shower is not fulfilled when ∆Q2 is small. However,
one could think of an artificial problem in which ∆Q2 is dNmed dNvac ′ ′
(E)= d(∆E) (E )P(∆E)δ(E −E−∆E)
large while L is small, in which case the limit for small dEc Z dEc
pathlengths is not the vacuum. (6)
For the time being we will nevertheless explore the solving for the probability distribution of energy loss
consequences of the approximation Eq. (5) as it stands P(∆E). Note that this ansatz contains the rather dras-
rather than introduce a more general prescription which tic assumption that there is no parametric dependence
brings additional model dependence. This means that on the initial energy E. If we require P(∆E) to be a
the results are not reliable for small pathlengths where probability distribution, the assumption may imply that
the showerisnotdominatedbythe medium, butthe un- for some partons in the distribution dN/dEvac the en-
c
derlying assumption is justified by the vast majority of ergy loss ∆E is larger than their energy E in which case
partons encountered in the computation of experimen- they have to be considered lost to the medium.
tally relevant observables. In practice, Eq. (6) is discretized and cast it into the
form of a matrix equation
D. Medium-modified showers and energy loss
n
N (Ei)= K (Ei,∆Ej)P (∆Ej) (7)
i ij j
It is not straightforward for medium-modified shower Xj=1
models to make contact with leading parton energy loss
models. Typicallyinanenergylossmodellike[1–3]there where dN/dE is provided at m discrete values of E
c
isacleardistinctionbetweentheleadingpartonandradi- labelled N and P(∆E) is probed at n discrete values of
i
atedpartonsmadebythea prioriassumtionthattheen- ∆E labelled P . The kernel K is then the calculated
j ij
ergyoftheleadingpartonismuchlargerthantheenergy dN/dE for all pairs (Ei,∆Ej) where the energy loss
c
of radiation quanta. In this limit, the medium-induced acts as a shift of the distribution, i.e. dN/dEmed(E) =
c
energy loss is given by the sum of the energy of all ra- dN/dEvac(E +∆E). Eq. (7) can in principle be solved
c
diated quanta or equally by the energy difference of the for the vector P by inversion of K for m = n. How-
j ij
leading parton before and after the passage through the ever, in general this does not guarantee that the result
medium. A posteriori, one may however find (rare) pro- is a probability distribution. Especially in the face of
cesses in which the energy of a radiated parton is much statisticalerrorsandfinite numericalaccuracythedirect
largerthantheenergyoftheparent—theseareformally matrix inversion may permit negative P which have no
j
counted as large energy loss. probabilistic interpretation. Thus, a more promising so-
Inashowermodel,theleading(mostenergetic)parton lution which avoids the above problems is to let m > n
can be identified at all times, but the identity of that andfind the vector P which minimizes ||N−KP||2 sub-
parton may change easily during the evolution. Unlike ject to the constraints 0 ≤ P ≤ 1 and n P = 1.
i i=1 i
in the energy loss case, the radiation of a hard gluon This guarantees that the outcome can be Pinterpreted as
5
a probability distribution and since the system of equa- 100 GeV c quark, q = 5 GeV2 /fm
tions is overdetermined for m > n errors on individual
100
pointsR donothaveacriticalinfluence onthe outcome
i
any more. average formation time
Notethatthisansatzleadstoadditionalcomplications 80 randomized formation time
length dependent Q
incombinationwiththeprescriptiontosettheminimum 0
2
virtuality scale Q0 in the shower Eq. (5). This is due const. L
to two reasons: First, in essence for short paths, energy V] 60
e
losscannotbeextractedreliablybecausethescalechoice G
[
for the vacuum part becomes only irrelevant if the vac- ∆E 40
uumshowerisa smallcorrectiontothe medium-induced
shower as discussed above. Unfortunately, for truly long
paths the secondcomplicationbecomes apparent: unlike 20
in the light quark case, the charm mass m rather than
c
Eq. (5) determines the point at which the branching is
0
stopped. In order to avoid the second limit, we use very 0 5 10 15
L [fm]
energetic c quarks to extract energy loss, thus allowing
for a larger length scale to be probed.
For these reasons, the actual effect of Eq. (5) on the
FIG. 2: (Color online) The integrated mean energy loss ∆E
pathlength dependence of light parton energy loss is
as a function of pathlength extracted from YaJEM for a 100
only in a very limited way represented by evaluating GeV shower-initiating c quark propagating through a con-
Eq. (7), and all energy loss results shown for this pre- stant medium with qˆ = 5 GeV2/fm for different options
scriptionneedtobetreatedwiththeappropriateamount (seetext),comparedwith aquadraticaldependenceonpath-
of caution. However, this is largely a complication of length.
the way energy loss is determined from an in-medium
showerresult—thereisno compellingreasontosuspect
that Eq. (5) would be as problematic for the in-medium ity turn the result over to a constant again. This agrees
shower evolution result as such or in the computation of nicely with the scaling law identified in [17] and shows
observables. that an algorithm which in principle reproduces the an-
As a side remark, the fact that the mass of heavy alytical L2 dependence in the right limit may not show
quarks largely set the scale at which the shower evolu- this behaviour in practically relevant situations.
tion is terminated implies that there is no substantial Finally,thelastcaseshowstheeffectofavariablemin-
heavy-quark radiative energy loss in the kinematical re- imumvirtualityscaleQ0accordingtoEq.(5). Notethat,
gion of RHIC experiments in YaJEM. Without an ad- followingthepreviousdiscussion,certainlybeyond10fm
ditional mechanism like elastic energy loss, YaJEM thus the result is not characteristic for what would happen
fails (like other radiative energy loss models) to account in the light quark case as the charm quark mass rather
forthesuppressionofheavyquarksseeninsingleelectron than Q0 determines where the branching process is ter-
spectra. minated. For a light quark, the curve would rise more
strongly.
III. RESULTS
B. Pathlength dependence of the nuclear
A. Pathlength dependence of energy loss suppression factor RAA
In Fig. 2 the mean energy loss as a function of path- In order to illustrate how pathlength effects manifest
length extracted from YaJEM via Eq. (7) for a 100 GeV in observable quantities, we use a realistic 3+1d hydro-
charm quark as shower initiator is shown for different dynamicalevolutionmodel[28]forthe bulkmediumand
options. compute the dependence of the nuclear suppression fac-
In the first case, the formation time of virtual states tor R on the angle with the reaction plane in this
AA
Eq. (1) is not randomized by Eq. (2). The result is ini- background. The straightforward computation is de-
tially very well described by a quadratic length depen- scribed in detail in [16, 29]. It involves averaging the
dence until (around 3 fm) finite energy corrections be- medium-modified fragmentation function (MMFF) com-
come important and the limit of applicability is reached puted from YaJEM over all possible paths through the
and turn the curve over. medium, which are in turn determined by the distribu-
If randomization is taken into account the quadratic tion of binary collisions in the transverse plane and the
dependence becomes much blurred, and the net result orientationofthe partonmomentum vectorwithrespect
canbe describedwellbyalineargrowthwithpathlength to the reaction plane. This result is then folded with
untilfiniteenergycorrectionsandthelimitofapplicabil- a leading order perturbative Quantum Chromodynam-
6
ics (LO pQCD) calculation for the parton spectrum to R with increasingP is expected. The curvefor fixed
AA T
get the yield of high PT hadrons. Since YaJEM obtains Q0 has been obtained for a randomized formation time.
the MMFF at given partonic scale whereas LO pQCD The result computed for average formation time is not
computations require the fragmentation function to be shown,asit isindistinguishable giventhe averagingover
known at given hadronic scale, a scale matching proce- manydifferentin-medium pathlengthsas requiredinthe
dure as described in [29] has been used in the following. computation of R .
AA
Inallcomputations,thefactorK inEq.(4)isadjusted Note that the P dependence of R is different from
T AA
such that the measured R for 0-10% central Au-Au results shown in [16, 17]. The reason is that in these
AA
collisions at P = 6 GeV is reproduced by the calcula- worksthematchingprocedurebetweenhadronicandpar-
T
tion. Both the P dependence of R , the centrality tonic scale described in [29] has not been used, which
T AA
dependence and the dependence on the angle φ with the leads to an unrealistic decrease of R with P .
AA T
reactionplanefollowthenwithoutadditionalfreeparam-
eters. Note that the value of K depends onthe choice of 20 - 30 %
Q0 (as pointed out already in [17]).
PHENIX in plane
1
0 - 10 % PHENIX out of plane
fixed Q
0
0.8
PHENIX data length dependent Q
1 0
fixed Q
0
length dependent Q0 A0.6
0.8 A
R
0.4
A0.6
A
R
0.2
0.4
0
5 7.5 10 12.5 15
0.2
p [GeV]
T
0
5 10 15 20
p [GeV] FIG. 4: (Color online) The nuclear suppression factor RAA
T for 20-30% central 200 AGeV Au-Au collisions computed for
two different options (see text) in YaJEM, compared with
PHENIX data [10]. Shown are in-plane emission (solid) and
FIG. 3: (Color online) The nuclear suppression factor RAA out of plane emission (dashed).
for 0-10% central 200 AGeV Au-Au collisions computed for
two different options (see text) in YaJEM, compared with
PHENIX data [30]. The result for 20-30% centrality is shown in Fig. 4,
separated into in plane and out of plane emission, again
In Fig. 3 the result for the angular average of RAA is forafixedvalueofQ0andadynamicallycomputedvalue
shown in comparison with PHENIX data [30] for both a and compared with PHENIX data [10].
fixedvalueoftheminimumvirtualityscaleQ0 =0.7GeV Itis immediatelyapparentfromthe figurethatacom-
and the dynamical choice Q0 = E/L where L is the putation with fixed Q0 results in a spread between in-
length of the parton path in the pmedium. As discussed plane and out of plane emission much smaller than seen
in more detail in [13], there is no compelling reason to inthedata. Thisisconsistentwiththeobservationmade
identifytheCooper-Fryefreeze-outhypersurfaceinahy- earlierthattheeffectivepathlengthdependenceofenergy
drodynamical model with a sharp surface beyond which loss in YaJEM is approximately linear, and the spread
the medium interaction with partons is absent, and pre- found agrees with results from models in which the de-
scription to let the medium fade out gradually would be pendence was assumed to be linear [11, 12].
desirable. However, since the length dependence enters On the other hand, both spread and normalization of
under the squareroot,the dependence ondetails is suffi- the data are well reproduced by a dynamical choice for
ciently weak suchthat one may hope to geta reasonable Q0 whichintroducesforlightquarksandgluonsastrong
estimate of the effect. pathlength dependence of energy loss.
Asseenfromthefigure,bothcomputationsareableto The scenario with dynamically computed Q0 has thus
describe R well, however the P dependence is some- clearobservableconsequenceswhenobservinghadronsat
AA T
what different. This is not surprising, since the expres- higher energies, such as available at the LHC. Since the
sionforQ0 alsocontainsadependenceonthe initialpar- maximal in-medium pathlength is always limited by the
ton energy E which is proportional to the final hadron nuclear diameter times a factor of order one, Eq. (5) im-
PT. Generically, for the dynamical scale choice, a rise of plies that Q0 will grow with increasing energy, thus the
7
suppressioninducedbythemediumwilldecrease,leading perturbative hadronization description in terms of the
to a rise of R with increasing P . Somewhat unfor- Lundmodel,andthereissometransitionregioninwhich
AA T
tunately, at LHC kinematics, such a rise is expected in bothprescriptionsleadtosimilarresultssuchthatthefi-
variousmodelsfor anumber ofreasons,amongthemthe nal resultis not strongly influenced by the detailed scale
shapeofthepartonspectrumorthefactthattheMMFF choice. However within YaJEM, by construction (and
is, even for fixed Q0, energy dependent and reverts for due to a lack of insight into what physics hadronization
largeenergiestothevacuumresult[29]. However,therise inside the medium would entail), the medium only in-
induced by the dynamically computed Q0 occurs much fluences the partonic dynamics and there is no compen-
faster than any of these other effects. sation of e.g. a high scale choice leading to weak show-
In contrast, the ratio of computed Q0 at fixed energy ering by an increased action of the Lund model as in
fordifferentpathlengthsisaconstant. Whenconsidering the vacuum case. Thus, here Q0 more directly regulates
in-plane and out of plane emission for non-central colli- what part of the shower evolution happens in medium
sions,thisimpliesthatwhilethemean(angularaveraged) and what needs to be taken care off by vacuum dynam-
R changesrapidlywithenergy,thespreadbetweenin- ics later.
AA
plane and out of plane will be more robust and change If this result is confirmed by more detailed investiga-
much slower with energy (since asymptotically at high tions, it would imply the the idea to use R (φ) to dis-
AA
energies the MMFF reverts to the vacuum FF, eventu- tinguish betweenpQCD and stronglycoupled models on
ally the spread must vanish as well). Thus, the scenario the basis of quadratic vs. cubic pathlength dependence
proposedherecanbetestedbyanobservationofin-plane is largely a red herring, as the leading effect seen in the
vs. out of plane emission of hard hadrons in non-central data is something entirely different. However, since the
collisions for a sufficiently large range in P . Detailed scenario also predicts an energy dependence of the vir-
T
predictions for LHC kinematics will be postponed to a tuality scale, translating e.g. into a rise of R with
AA
subsequent publication, as they involve a significant nu- P , testing it should be comparatively easy with a large
T
merical effort. enoughkinematiclever-armase.g. theLHCexperiments
can provide.
In general, it seems that in the context of MC shower
IV. DISCUSSION modelswhichcanincorporatefiniteenergyeffectsandto
some degree also quantum interference, the question of
From the results of the previous section, it would ap- how pathlength dependence arises and how to define it
pearthatinthekinematicaldomaincurrentlyexperimen- fromobservablesbecomessignificantlymorecomplicated
tally accessible, imposing a global virtuality resolution than in the asymptotic limits of analytical models.
scale Eq. (5) down to which the medium can influence
the shower development has a much larger influence on
theobservablepathlengthdependenceofhighP hadron Acknowledgments
T
production than a local coherence condition imposed for
each branching which accounts for LPM interference. DiscussionswithAbhijitMajumder,KorinnaZappand
Note that Q0 acts in the presence of a medium quite Guangyou Qin are gratefully acknowledged. This work
different than in the vacuum case. In vacuum shower, was supported by an Academy Research Fellowship of
Q0 regulates the transition from a perturbative partonic T.R. from the Finnish Academy (Project 130472) and
description in terms of a shower evolution to a non- from Academy Project 133005.
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