Universality of the Diffusion Wake from Stopped and Punch-Through Jets in Heavy-Ion Collisions

1.1 MB·English

by  Barbara Betz

Universality of the Diffusion Wake from Stopped and Punch-Through Jets in
Heavy-Ion Collisions
Barbara Betz,1 Jorge Noronha,2 Giorgio Torrieri,3 Miklos Gyulassy,2,3 Igor Mishustin,3 and Dirk H. Rischke1,3
1Institut fu¨r Theoretische Physik, Johann Wolfgang Goethe-Universit¨at, Frankfurt am Main, Germany
2Department of Physics, Columbia University, New York, 10027, USA
3Frankfurt Institute for Advanced Studies (FIAS), Frankfurt am Main, Germany
We solve (3+1)–dimensional ideal hydrodynamical equations with source terms that describe
punch-throughand fully stoppedjetsin ordertocomparetheirfinalaway-sideangularcorrelations
inastaticmedium. Forfullystoppedjets,thebackreactionofthemediumisdescribedbyasimple
Bethe–Bloch-like model which leads to an explosive burst of energy and momentum (Bragg peak)
close to the end of the jet’s evolution through the medium. Surprisingly enough, we find that the
medium’s response and the corresponding away-side angular correlations are largely insensitive to
9 whether the jet punches through or stops inside the medium. This result is also independent of
0 whethermomentumdeposition islongitudinal (asgenerally occursinpQCDenergy lossmodels) or
0 transverse (as the Bethe–Bloch formula implies). The existence of the diffusion wake is therefore
2 shown to be universal to all scenarios where momentum as well as energy is deposited into the
n medium, which can readily be understood in ideal hydrodynamics through vorticity conservation.
a The particle yield coming from the strong forward moving diffusion wake that is formed in the
J wakeofbothpunch-throughandstoppedjetslargely overwhelmstheirweakMachconesignalafter
freeze-out.
9

PACSnumbers: 13.90.+i,25.75.Bh,25.75.Gz
]
h
t
-
I. INTRODUCTION model is needed [23, 26, 29, 30, 31, 32, 33, 34, 35].
l
c
u In general, supersonic probes that shoot through a
n One of the major discoveries found at the Relativis- fluid can deposit energy and momentum in the medium
[ tic Heavy Ion Collider (RHIC) was the suppression of in such a way that collective excitations such as Mach

highlyenergeticparticlesincentralA+Acollisions[1,2]. cones and diffusion wakes are formed [36]. These struc-
3
v Jets are assumed to be created in the early stage of tures have indeed been found [37, 38] in the wake of a
1 a heavy-ion collision where they interact with the hot supersonic heavy quark that travels through an =
N
0 and dense nuclear matter and serve as a hard probe for 4 Supersymmetric Yang-Mills (SYM) thermal plasma
4 the created medium [3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. [39, 40, 41]. The validity of a hydrodynamic description
4 Two- and three-particle correlations of intermediate–p⊥ of the supersonic heavy quark wake was studied in Refs.
.
2 particles provide an important test of the medium re- [42, 43, 44, 45, 46] in the framework of the Anti-de Sit-
1 sponse to the details of the jet quenching dynamics and ter/ConformalFieldTheorycorrespondence(AdS/CFT)
8 they show a re-appearance of a broad or double-peaked [47,48]. Theangularcorrelationscreatedbyheavyquark
0
structure in the away-side of jet angular correlations jets in AdS/CFT have recently been computed in Ref.
:
v [13, 14, 15, 16, 17, 18]. [49,50,51]andcompared[52]totheresultsobtainedbya
Xi The observation of strong elliptic flow in non-central punch-throughheavyquark jet describedby the Neufeld
Au+Aucollisionsconsistentwithfluiddynamicalpredic- et. al. [53, 54, 55] chromo-viscous hydrodynamic model,
r
a tions [19, 20] suggests that a thermalized medium that which is formulated within perturbative quantum chro-
evolves hydrodynamically is created in these collisions. modynamics (pQCD).
Moreover,sincetheaveragemomentumofparticlesemit- Ingeneral,afastmovingparton(whichcouldbealight
tedontheaway-sideapproachesthevalueofthethermal- quark/gluonoraheavyquark)willloseacertainamount
ized medium with decreasing impact parameter [15], the of its energy and momentum along its path through the
energy lost by the jet should quickly thermalize. Thus, mediumandthendecelerate. Thus,thefateoftheparton
the disturbance caused by the jet may also be described jet strongly depends on its initial energy: if the parton
hydrodynamically. hasenoughenergyitcanpunchthroughthemediumand
Recent interest in Mach-like conical di-jet correlations fragmentinthe vacuum(punch-throughjet) oritcanbe
isbasedonsuggestions[21,22,23,24,25,26]thatamea- severely quenched until it becomes part of the thermal
surementofthedependence ontheconeangleassociated bath (stopped jet). Of course, the amount of initial en-
with a supersonic jet moving with velocity v could pro- ergy required for the parton to punch through depends
vide via Mach’s law (cosφ =c /v) a constraint on the on the properties of the medium (a very large energy
M s
average speed of sound in the strongly coupled Quark- loss per unit length dE/dx means that most of the jets
Gluon Plasma (sQGP) [27, 28]. For a quantitative com- will be quenched while only a few would have enough
parison to RHIC data, a detailed model of both energy energy to leave the plasma). In this paper we solve the
and momentum deposition coupled to a relativistic fluid (3+1)–dimensional ideal hydrodynamical equations [56]
2
T0 = 200 MeV, v=0.999 (a) 1.4
4 (a) ddMM//ddtt == 00..0705 GGeeVV//ffmm (c) v=00..795909
3 240 1.2 dM/dt = 1.50 GeV/fm 0.580
2 230 1
1
y [fm]- 10 221200 T [MeV] φCF() 00..68
-2
-3 200 0.4
-4 190 0.2
-5 -4 -3 -2 -1 0 1
x [fm] 0
(b) (d)
(b) 1
4
3 240 u.] 0.8
12 230 φdy [a. 0.6
y [fm]- 10 221200 T [MeV] dE/d 0.4
-2 0.2
200
-3
-4 190 0
-5 -4 -3 -2 -1 0 1 π 3π/2 2ππ 3π/2 2π
x [fm] φ [rad] φ [rad]
FIG. 1: (Color online) Temperature pattern and flow ve-
locity profile (arrows) after a hydrodynamical evolution of FIG. 2: (Color online) The left panels show the normalized
t=4.5/vjet fm, assuming (a) an energy loss rate of dE/dt= angular distribution created by a punch-through jet at mid-
1.5 GeV/fm for a vanishing momentum loss rate and (b) rapidity with a fixed energy loss of dE/dt = 1.5 GeV/fm
an energy and momentum loss rate of dE/dt = dM/dt = and different momentum loss rates. The jet moves at a con-
1.5 GeV/fm for a punch-through jet moving with a constant stant velocity vjet = 0.999 through the medium. The right
velocityofvjet =0.999alongthex–axisthroughastaticback- panels show the angular distributions associated with jets
ground plasma with temperature T0 = 200 MeV. The jet is where dE/dt = 1.5 GeV/fm and vanishing momentum loss
sitting at the origin of the coordinates at the time of freeze- (dM/dt = 0). Here, the jets move with different veloci-
out. ties through the medium: vjet = 0.999 (black), vjet = 0.75
(blue), and vjet = 0.58 (magenta). In the upper panels, an
isochronous Cooper–Frye freeze-out at p⊥ = 5 GeV is used
with source terms that describe the two scenarios in or- while in the lower panels we employed the bulk flow freeze-
outprocedure[52]. ThearrowsindicatetheangleoftheMach
der to compare the final away-side angular correlations
cone as computed via Mach’s law.
produced by a punch-through and a fully stopped jet in
a static medium with background temperature T . We
0
would like to point out that the wake formed by fully describe how a jet deposits energy and momentum in
stopped jets has not yet been studied using hydrody- (3+1)–dimensional hydrodynamics and how we extract
namics. observables for the Mach cone created by the away-side
For simplicity, our medium is a gas of massless SU(3) jet. In Sec. III we present our results for punch-through,
gluons in which p = e/3, where p and e are the pres- and in Sec. IV for completely stopped jets. A summary
sureandtheenergydensity,respectively. Anisochronous concludes this paper in Sec. V.
Cooper–Frye (CF) [57] freeze-out procedure is employed We use natural units and the Minkowskimetric gµν =
in order to obtain the angular distribution of particles diag(+, , , ). LorentzindicesaredenotedwithGreek
− − −
associated with the away-side jet. We use a simplified letters µ,ν =0,...,3. In our system of coordinates, the
Bethe–Blochmodel[58]to showthat the explosiveburst beam axis is aligned with the z–direction and the as-
of energy and momentum [known as the Bragg peak sociated jet moves along the x–direction with velocity
[59, 60, 61, 62]] deposited by a fully quenched jet im- ~v =vxˆ.
mediately before it thermalizes does not stop the dif-
fusion wake and, thus, no new structures in the away-
side of angular correlation functions can be found. This II. JETS IN IDEAL HYDRODYNAMICS
explosive release of energy before complete stopping is
a general phenomenon that has been employed, for in- Energetic back-to-back jets produced in the early
stance, in applications of particle beams for cancer ther- stages of a heavy-ion collision transverse to the beam
apy [63, 64, 65]. axis can travel through the sQGP and deposit energy
This paper is organized as follows. In Sec. II we and momentum along their path in a way that depends
3
onthephysicsbehindtheinteractionbetweenthejetand case, the momentum distribution for associated (mass-
theunderlyingmedium. Inthecasewhereoneofthejets less) particles pµ = (p⊥,p⊥cos(π φ),p⊥sin(π φ)) at
− −
is produced near the surface (trigger jet), the other su- mid-rapidity y =0 is computed via
personic away-side jet moves through the medium and
dN
excites a Mach wave as well as a diffusion wake. The ass = dΣ pµ[f (uµ,pµ,T) f ] . (4)
µ 0 eq
resulting angular correlation with respect to the away- p⊥dp⊥dydφ(cid:12)y=0 ZΣ −
(cid:12)
side jet axis is then expected to lead to an enhancement (cid:12)
Here, φ is the azimuthal angle between the emitted par-
of associated hadrons at the characteristic Mach angle
ticle and the trigger, p⊥ is the transverse momentum,
[23, 24, 25, 26, 30].
f = exp[ uµ(t,~x)p /T(t,~x)] the local Boltzmann equi-
Inidealhydrodynamics,theenergy–momentumtensor 0 − µ
librium distribution, and feq ≡ f|uµ=0,T=T0 denotes the
isotropic background yield. We checked that our results
Tµν =(e+p)uµuν pgµν , (1)
− donotchangesignificantlyifweuseaBose–Einsteindis-
tribution instead of the Boltzmann distribution. The
is locally conserved, i.e.,
background temperature is set to T = 0.2 GeV. Fol-
0
∂ Tµν =0, (2) lowing Refs. [23, 26, 33, 50], we perform an isochronous
µ
freeze-outwheredΣµ =d3~x(1,0,0,0)anddefinethe an-
where uµ = γ(1,~v) is the flow 4-velocity and γ = gular function
(1 ~v2)−1/2. We takethe net baryondensity to be iden-
tic−ally zero in this study. Here, we only consider a static CF(φ)= 1 dNass(φ) , (5)
medium. More realistic initial conditions involving an Nmaxp⊥dp⊥dydφ(cid:12)y=0
(cid:12)
expanding medium will be consideredin a further study. (cid:12)
where the constant N is used to normalize the plots.
max
Oncethejetisincludedinthesystemtheconservation
We would like to remark that in the associated p⊥–
equationschange. Weassumethattheenergylostbythe
rangeofinterestacoalescence/recombinationhadroniza-
jet thermalizes and gives rise to a source term Sν in the
tion scenario [67, 68, 69, 70, 71] may be more appro-
energy–momentum conservation equations
priate than CF freeze-out. However, we expect that the
main features of the away-side angular correlations ob-
∂ Tµν =Sν . (3)
µ tainedusing CFhadronizationarerobustenoughto sur-
vive other hadronization schemes.
Thus, one has to solve Eq. (3) numerically in order to
The other freeze-out prescription (called bulk flow
determine the time evolution of the medium which was
freeze-out) used in the present paper was introduced in
disturbed by the moving jet. The source term that cor-
Ref. [52]. The main assumption behind the bulk flow
rectly depicts the interaction of the jet with the sQGP
freeze-out is that all the particles inside a given small
isunknownfromfirstprinciples,althoughrecentcalcula-
sub-volume of the fluid will be emitted in the same di-
tions in AdS/CFT [42, 43, 44] and pQCD [54] have shed
rection as the averagelocal energy flow
somelightonthisproblem. WhilepQCDiscertainlythe
correct description in the hard-momentum region where d
jetsareproduced(Q T ),inthesoftpartoftheprocess E = d3~x (~x)δ[φ Φ(~x)] δ[y Y(~x)] . (6)
≫ 0 dφdy Z E − −
(Q T ) non-perturbative effects may become relevant.
0
∼
Inthispaper,weomitthenear-sidecorrelationsassoci- Here, φ is again the azimuthal angle between the de-
ated with the trigger jet and assume that the away-side tected particle and the trigger jet and y is the parti-
jet travels through the medium according to a source cle rapidity. Only the y = 0 yield is considered. The
term that depends on the jet velocity profile which shall cells are selected according to their local azimuthal an-
be discussed below for the case of punch-through and gle Φ(~x) = arctan[ y(~x)/ x(~x)] and rapidity Y(~x) =
P P
stopped jets. Artanh[ z(~x)/ (~x)]. The local momentum density of
The away-side jet is implemented in the beginning of the cell iPs T0i(~xE) = i(~x), while its local energy density
the hydrodynamical evolution at x = 4.5 fm, and its in the lab frame is P(~x) = T00(~x). The δ–functions are
− E
motion is followed until it reaches x = 0. For a jet implemented using a Gaussian representation as in Ref.
moving with a constant velocity v this happens at [52]. Due to energy and momentum conservation, this
jet
t =4.5/v fm. quantity should be conservedafter freeze-out. Note that
f jet
We use two different methods to obtain the away-side Eq. (6) is not restricted to a certain p⊥ and does not in-
angularcorrelations. IntheCFmethod[57],thefluidve- clude the thermal smearing that is always present in the
locity uµ(t ,~x) and temperature T(t ,~x) fields are con- CF freeze-out.
f f
vertedintofreeparticlesatafreeze-outsurfaceΣatcon-
stanttimet . Inprinciple,onehastoensurethatenergy
f
and momentum are conservedduring the freeze-out pro- III. PUNCH-THROUGH JETS
cedure [66]. However,the associatedcorrectionsare zero
if the equation of state is the same before and after the Inthissectionweconsiderajetmovingwithauniform
freeze-out, as it is assumed in the present study. In this velocity v = 0.999 through the medium. The source
jet
4
term is given by
1.4
Sν = τfdτdMνδ(4) xµ xµ (τ) , (7) 4 T0 = 200 MeV, v0=0.999 (a) 1.2 dML/dtd=E0/.d2t5= ddMMTT//ddtt
wcihateerde τwfit−h τtihτZeidejnetotedesvτotluhteiophnr.op−eWretjeitfmuerthiinerteravsaslumasesoa- y [fm]-- 210123 222220123400000 T [MeV] φCF() 000 ...1468
constant energy and momentum loss rate dMν/dτ = --43 190 0.2
(dE/dτ,dM~/dτ)alongthe trajectoryofthe jetxµjet(τ)= -5 -4 -3x -[f2m]-1 0 1 0
xµ +uµ τ. In non-covariant notation, this source term 1
0 jet (b)
has thSeνf(otr,m~x) = (√d2E1πσd)M3 exp(cid:26)−[~x−2~xσje2t(t)]2(cid:27) y [fm]-- 2101234 222212340000 T [MeV] φdE/ddy [a.u.] 0000....2468
, ,0,0 , (8) -3 200
× (cid:18) dt dt (cid:19) 0
-4 190 π 3π/2 2π
-5 -4 -3 -2 -1 0 1 φ [rad]
x [fm]
where~x describes the locationof the jet, ~x is the posi-
jet
tiononthecomputationalgrid,andσ =0.3. Thesystem
plasma+jetevolves accordingto Eq.(3) until the freeze- FIG. 3: (Color online) Left panel: Temperature pattern and
out time t =4.5/v fm is reached. flow velocity profile (arrows) after a hydrodynamical evolu-
f jet
The temperature and flow velocity profiles created by tion of t = 4.5/vjet fm, assuming an energy loss rate of
dE/dt=dM/dt=1.5GeV/fmfor(a)fulltransversemomen-
a punch-through jet with a constant energy loss rate of
tumdepositionand(b)longitudinalaswellastransversemo-
dE/dt = 1.5 GeV/fm and vanishing momentum deposi-
mentum deposition with a ratio of dML/dt = 0.25 dMT/dt.
tion are shown in Fig. 1 (a). In Fig. 1 (b) the jet has
Right panel: The normalized angular distribution created
lost the same amount of energy and momentum and in
by a punch-through jet at mid-rapidity for the two above
this case one can clearly see that the space-time region mentioned transverse momentum deposition scenarios. In
close to the jet, where the temperature disturbance is the upper panel, an isochronous Cooper–Frye freeze-out at
the largest, is bigger than in the pure energy deposition p⊥ = 5 GeV is used while in the lower panel the bulk flow
scenario. The creationof a diffusion wake behind the jet freeze-out procedure [52] is employed. The arrows indicate
in the case of equal energy and momentum deposition theideal Mach cone angle.
is clearly visible, which is indicated by the strong flow
observed in the forward direction (at φ=π).
NoteinFig.2(a)thatforthepunch-throughjetdepo- is reduced. The associatedparticle distribution fromthe
sitionscenariowithequalenergyandmomentumlossone CF freeze-out in Fig. 2 (a) reveals a peak at φ = π for
6
always obtains a peak in the associated jet direction af- pure energy deposition (solid black line), however, the
ter performing the freeze-outusing the two prescriptions openingangleisshiftedtoavaluesmallerthantheMach
described in Sec. II. However, the energy flow distribu- cone angle due to thermal smearing [33].
tion in Fig. 2 (b) displays an additional small peak at In Figs. 2 (c,d) we consider dM/dt = 0 jets that
the Mach cone angle indicated by the arrow. This Mach movethroughthemediumwithdifferentvelocitiesv =
jet
signal cannot be seen in the Cooper–Frye freeze-out be- 0.999,0.75,and0.58. NoteinFig.2(d)thatthepeakpo-
causeofthermalsmearing[23, 33,50, 52]andthe strong sition changes in the bulk flow distribution according to
influence ofthe diffusion wake,whichleads to the strong theexpectedMachconeangles(indicatedbythearrows).
peak around φ π in the bulk energy flow distribution. However, due to the strong bow shock created by a jet
∼
However,giventhat the exact formof the sourceterm moving at a slightly supersonic velocity of vjet = 0.58,
in the sQGPis unknown,one may wantto exploreother there is a strong contribution in the forwarddirection in
energy–momentum deposition scenarios where the jet this case and the peak position is shifted from the ex-
deposits more energy than momentum along its path. pected value. In the CF freeze-out shown in Fig. 2 (c),
While this may seem unlikely, such a situation cannot the peak from the Mach cone can again be seen for the
be ruled out. Thus, for the sake of completeness, we jetmovingnearlyatthespeedoflight(vjet =0.999),but
additionally consider in Fig. 2 (a) the case where the for slower jets thermal smearing again leads to a broad
jet source term is described by a fixed energy loss of distributionpeakedinthedirectionoftheassociatedjet.
dE/dt=1.5GeV/fmanddifferentmomentumlossrates.
InthebulkflowdistributioninFig.2(b),onecanseethat It is apparently surprising that the above mentioned
the peak at the Mach cone angle is more pronouncedfor results are independent of whether the momentum de-
smallermomentumlosswhilethecontributionofthedif- posited by the particle is in the longitudinal (along the
fusion wake (indicated by the peak in forwarddirection) motion of the jet) or transversal (perpendicular) direc-
5
tion. RepeatingthecalculationshowninFig.1including 5 5
vjet
transverse momentum deposition dE/dt
4 4
dE/dt
m]
Sν(t,~x)∝((ddMMdTTM//dLdtt/))dcstoinsϕϕ  , (9) v [c]jet 23 23 dE/dt [GeV/f
1 1
where ϕ is the latitude angle in the y z plane with re-
spect to the jet motion and the magni−tude of Sν(t,~x) is
0 0
the same as Eq. (7), shows that transverse momentum 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
t [fm/c]
depositionwillnotalterthe resultspresentedinthis sec-
tion(seeFig.3). Alongitudinaldiffusionwakestillforms
FIG. 4: (Color online) The jet velocity vjet(t) (solid black
duringthefluidevolutionstage,anditscontributionwill line) and energy deposition rate dE(t)/dt (dashed blue line)
still dominate the resulting angular inter-particle corre- accordingtoEq.(12). Theinitialjetvelocityandenergyloss
lations though a peak occurs around the expected Mach rate are vjet =0.999 and a≃−1.3607 GeV/fm, respectively.
cone angle in the CF freeze-out.
The reason is that transverse momentum deposition
will force the fluid around the jet to expand, and the and the identities dE/dt = vjetdM/dt and dM/dyjet =
empty space left will create a shock wave in the longitu- mcoshyjet (as well as vjet = tanhyjet), one can rewrite
dinal direction that behaves much like a diffusion wake. Eq. (10) as
In terms of ideal hydrodynamics, this universality of the
m
diffusion wake can be understood in the context of vor- t(yjet) = [sinhyjet sinhy0
a −
ticity conservationsince momentum deposition, whether
1 1
transverse or longitudinal, will add vorticity to the sys- arccos +arccos ,(11)
− coshy coshy (cid:21)
tem. This vorticity will always end up behaving as a jet 0
diffusion wake [72]. In the next section, we demonstrate
where y is the jet’s initial rapidity. The equation
0
thatthese resultsarelargelyindependentofwhether the
above can be used to determine the time-dependent ve-
jet is fully quenched or survives as a hard trigger.
locity v (t). The initial velocity is taken to be v =
jet 0
Artanhy = 0.999. The mass of the moving parton is
0
taken to be of the order of the constituent quark mass
IV. STOPPED JETS
m = 0.3 GeV. Moreover, the initial energy loss rate
a 1.3607GeV/fmisdeterminedbyimposingthatthe
Intheprevioussectionweconsideredauniformlymov- ≃−
jetstopsafter∆x=4.5fm(asintheprevioussectionfor
ing jet that deposited energy and/or momentum in the
a jet with v =0.999). Thus, the jet location as well as
jet
medium at a constant rate. However, due to its inter-
the energy and momentum deposition can be calculated
action with the plasma, the jet will decelerate and its
as a function of time via the following equations
energyand/ormomentumlosswillchange. Thus,thede-
m
celerationroughlyrepresentstheresponseofthemedium. x (t) = x (0)+ (2 v2 )γ (2 v2)γ ,
In general, a decelerating jet should have a peak in the jet jet a − jet jet− − 0 0
(cid:2) (cid:3)
energy loss rate because the interaction cross section in- dE 1 dM 1
= a , =a , (12)
creasesasthe parton’senergydecreases. Inother words, dt v dt v2
jet jet
when the particle’s velocity goes to zero there appears a
peak in dE/dx knownas the Braggpeak [59]. The ques- whichcanbeusedtodeterminethecorrespondingsource
tion to be considered in this section is whether this en- term for the energy-momentum conservation equations.
ergy deposition scenario might be able to somehow stop The change of the jet velocity vjet(t) and energy deposi-
the diffusion wake and, thus, change the angular distri- tiondE(t)/dtaredisplayedinFig.4. Thestrongincrease
butions shown in Fig. 2. The source term in this case is of energy deposition shortly before the jet is completely
still given by Eq. (8) and, according to the Bethe–Bloch stopped corresponds to the well-known Bragg peak [59].
formalism [59, 60, 61, 62], one assumes that Themaindifferencebetweentheansatzdescribedhere
andtheBethe–Blochequationisthatthemomentumde-
dE(t) 1 position is longitudinal (parallel to the motion of the
=a , (10)
dt v (t) jet) rather than transverse (perpendicular to the mo-
jet
tion of the jet). According to most pQCD calcula-
which shows that when the jet decelerates the energy tions, this is true in the limit of an infinite energy jet
loss rate increasesand has a peak as v 0. Note that [3, 4, 5, 6, 7, 8, 9, 10, 11, 12], but it is expected to break
jet
→
here dE/dt is the energy lost by the jet, which is the down in the vicinity of the Braggpeak where the jet en-
negative of the energy given to the plasma. Using this ergy is comparable to the energy of a thermal particle.
ansatzforthevelocitydependenceoftheenergylossrate However,aswedemonstratedintheprevioussection,the
6
FIG. 5: (Color online) Temperature pattern and flow velocity profile (arrows) after a hydrodynamical evolution of t=4.5 fm
(left panel), t = 6.5 fm (middle panel) and t = 8.5 fm (right panel) for a jet that decelerates according to the Bethe–Bloch
formula and stops after ∆x=4.5 fm. The jet’s initial velocity is vjet =0.999. In the upper panel a vanishing momentum loss
rate is assumed while in the lower panel the momentum loss is related to theenergy loss byEq. (12).
freeze-outphenomenologyisratherinsensitivetowhether both Mach cones and diffusion wakes are similarly af-
the momentum deposition is transverse or longitudinal. fected [23, 26, 36].
Fig. 5 displays the temperature and flow velocity pro-
The angular distribution associated with the deceler-
files of a jet that stops after ∆x = 4.5 fm, with an en-
ating jet (which stops after ∆x=4.5 fm), shown in Fig.
ergylossaccordingtoEq.(10)andvanishingmomentum
6,isdeterminedaccordingtothetwofreeze-outprescrip-
deposition (upper panel) as well as an energy and mo-
tions describedinSec.II. Whenthe energyandmomen-
mentum deposition following Eq. (12) (lower panel). In
tumlossratesaredeterminedbyEq.(12)(magentaline),
the left panel the medium decouples immediately after
both freeze-outproceduresdisplay a feature discussedin
t=4.5fmwhenthejetisstoppedwhileinthemiddleand
the previous section for the case of punch-through jets:
right panel the decoupling takes place after t = 6.5 fm
the formation of a strong diffusion wake which leads to
and t=8.5 fm, respectively.
a strong peak in the associated jet direction. The re-
Comparingthisresultto Fig.1leadsto theconclusion
sults after the isochronous CF freeze-out are shown in
thatthediffusionwakeispresentindependentofwhether
the upper panel of Fig. 6. As in Fig. 5, the medium
the jet is quenched or survives until freeze-out. In the
decouples after t = 4.5 fm (left panel), t = 6.5 fm (mid-
former case, however, the diffusion wake is only weakly
dle panel) and t = 8.5 fm (right panel). Only the pure
sensitive to the duration of the subsequent evolution of
energy deposition scenario produces a peak at an angle
the system.
close to the Mach angle [see Fig. 6 (a)] which is smeared
Within ideal hydrodynamics this can be understood out thermally for larger decoupling times [cf. Fig. 6 (b)
via vorticity conservation. The vorticity-dominated dif- and(c)]. Onthe otherhand,thebulkenergyflowfreeze-
fusion wake will always be there in the ideal fluid, out displayed (lower panel) shows in all cases a peak at
whetherthesourceofvorticityhasbeenquenchedornot. the Mach cone angle. Note that in this case the peak
The only way this vorticity can disappear is via viscous becomes more pronounced when dM/dt = 0. While the
dissipation. While a (3+1)–dimensional viscous hydro- Mach cone signal increases with the decay time, the sig-
dynamic calculation is needed to quantify the effects of nal is still smaller than the forwardyield of the diffusion
this dissipation, linearized hydrodynamics predicts that wake.
7
1.4
(a) dM/dt = 0.00 GeV/fm (c) (e)
dM/dt = 1/v dE/dx
1.2
1.4
1.2
1 1
0.8
φCF() 00..68 000 ...0246π 3π/2 2π
0.4
0.2
0
(b) (d) (f)
1
1
φE/ddy [a.u.] 00..68 0000 ....02468π 3π/2 2π
d 0.4

0.2
0
π 3π/2 2π π 3π/2 2π π 3π/2 2π
φ [rad] φ [rad] φ [rad]
FIG. 6: (Color online) The normalized angular distribution generated by a decelerating jet (cf. also Fig. 5) at mid-rapidity is
shown(upperpanel)accordingtoanisochronousCooper–Fryefreeze-outatp⊥ =5GeVforajetthatstopsafter∆x=4.5fm
and a hydrodynamical evolution of t = 4.5 fm (left panel), t = 6.5 fm (middle panel) and t = 8.5 fm (right panel). The
corresponding bulk flow pattern [52] is shown in the lower panel. The solid black line in all plots depicts the pure energy
deposition case while the dashed magenta line corresponds to the energy and momentum deposition scenario given by Eq.
(12). The arrows indicate the angle of the Mach cone as computed via Mach’s law. The inserts repeat Fig. 2 (a) and (b) for
comparison.
V. SUMMARY The same features also appear when different jet ve-
locities are considered. In the bulk energy flow distribu-
tion the peaks occur nearly at about the expected Mach
In this paper we compared the away-side angular cor-
cone angles (for slow velocities they are shifted due to
relationsatmid-rapidityassociatedwithuniformlymov-
the creation of a bow shock), but in the CF freeze-out
ing jets and also decelerating jets in a static medium. In
distributionthese peaks only occurat largejet velocities
general, a fast moving parton will lose a certain amount
(see Fig. 2). This result is consistent with the angular
of its energy and momentum along its path through the
correlations obtained for a punch-through heavy quark
mediumandthusdecelerate. Therefore,dependingonits
jet [52] described by the Neufeld et. al. pQCD source
energythejetwilleitherpunchthroughthemediumand
term [53, 54, 55].
fragment in the vacuum or it will be severely quenched
until it cannot be distinguished from the other thermal The diffusion wake created behind the jet dominates
partons in the plasma. the freeze-out distribution for a jet moving through a
Ourresults confirmpreviousstudies [23,26, 33]where static medium, even in case of large transverse momen-
a similarsourceterm[see Eq.(7)]was usedto showthat tum deposition (see Fig. 3) and independent of whether
the diffusion wake created by these jets leads to a sin- the jet has enough energy to punch through (see Fig. 1)
gle peak in the away-side of the associated di-hadron themediumornot(Fig.5). Assumingthatthejetdecel-
correlations that overwhelms the weak Mach signal af- erates according to the Bethe–Bloch formalism, see Eq.
ter isochronousCF freeze-outunless the total amountof (10), we checkedwhether the largeamountofenergyde-
momentum loss experienced by the jet is much smaller positedaroundthestoppingpoint(thewell-knownBragg
than the corresponding energy loss. However, according peak)canblockthediffusionwakeandthusalterthean-
to the bulk energy flow the peak always occurs at the gular correlations. However, our results show that no
expected Mach cone angle but the diffusion wake still significantdifferencesoccurbetweenthe away-sideangu-
leadstoa largepeakinthe associatedjetdirectionwhen lar correlations associated with punch-through jets and
dM/dt=0 (see Fig. 2). deceleratingjetsdescribedwithintheBethe–Blochmodel
6
8
(compare Fig. 2 and Fig. 6). Clearly, it would be inter- conical correlations in an expanding three-dimensional
esting to study other models that describe decelerating ideal fluid with a realistic equation of state [73] compat-
jets in strongly-coupled plasmas. However, the simple ible with current lattice results [74] is currently under
Bethe–Bloch model used here displays the main qualita- investigation.
tive features relevant for the hydrodynamic wake associ-
ated with decelerating jets. The path lengths of both
types of jets were taken to be the same. A different
scenario in which the light jets are almost immediately
Acknowledgments
stopped in the medium while the heavy quark jets are
still able to punch through may lead to different angular
correlations. Such an analysis is left for a future study. TheauthorsthankI.Pshenichnov,L.Satarov,M.Ble-
We would like to underline that the formation of a icher, C. Greiner, P. Rau, J. Reinhardt, J. Steinheimer
diffusion wake that trails the supersonic jet is a generic and H. St¨ocker for helpful discussions and D. Dietrich
phenomenon [36] and, thus, its phenomenological conse- for carefully reading the manuscript. The work of B.B.
quences must be investigated and not simply neglected. is supported by BMBF and by the Helmholtz Research
Our results indicate that the diffusion wake is univer- SchoolH-QM. J.N. and M.G. acknowledgesupport from
sal to all scenarios where momentum as well as energy DOE under Grant No. DE-FG02-93ER40764. M.G. also
is deposited to the medium, independent of whether the acknowledges sabbatical support from DFG, ITP, and
jet stops or is quenched. However, one can expect that FIAS at Goethe University. G.T. is grateful to LOEWE
the strong forward moving column of fluid represented forthe financialsupportgivenandthanksthe Alexander
by the diffusion wake can be considerably distorted in von Humboldt foundation as well as Goethe University
an expanding medium by the presence of a large radial for support. I.M. acknowledges support from the grant
flow. The interplay between radial flow and away-side NS-3004.2008.2(Russia).
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