Full jet in quark-gluon plasma with hydrodynamic medium response

3.1 MB·

by  Yasuki Tachibana

Full jet in quark-gluon plasma with hydrodynamic medium response
Yasuki Tachibana,1,∗ Ning-Bo Chang,2,1,† and Guang-You Qin1,‡
1Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics (MOE),
Central China Normal University, Wuhan, Hubei, 430079, China
2Institute of Theoretical Physics, Xinyang Normal University, Xinyang, Henan 464000, China
(Dated: January 30, 2017)
We study the nuclear modifications of full jets and their structures in relativistic heavy-ion colli-
sions including the effect of hydrodynamic medium response to jet quenching. To study the evolu-
tionsofthefulljetshowerandthetraversedmediumwithenergyandmomentumexchangesbetween
them, we formulate a coupled jet-fluid model consisting of a set of jet transport equations and rel-
ativistic hydrodynamics equations with source terms. In our model, the full jet shower interacts
with the medium and gets modified via collisional and radiative processes during the propagation.
Meanwhile, the energy and momentum are deposited from the jet shower to the medium and then
evolve with the medium hydrodynamically. The full jet defined by a cone size in the final state
7
1 includes the jet shower and the particles produced from jet-induced flow. We apply our model to
0 calculate the full jet energy loss and the nuclear modifications of jet rate and shape in Pb+Pb
2 collisions at 2.76A TeV. It is found that the inclusion of jet-induced flow contribution leads to
strongerjet-conesizedependenceforjetenergylossandjetsuppression. Jet-inducedflowalsohasa
n
significant contribution to jet shape function and dominates at large angles away from the jet axis.
a
J
7 (cid:113)
I. INTRODUCTION R = ∆η2+∆φ2 [19–23]. Furthermore, the measure-
2 p p
ments of substructures inside the full jets have provided

] In relativistic heavy-ion collisions at the Relativistic us more detailed information on jet quenching [24–30].
h
Heavy Ion Collider (RHIC) and the Large Hadron Col- Motivated by the detailed measurements, a lot of theo-
t
- lider (LHC), the deconfined state of quarks and gluons, reticalefforthasbeendevotedtothestudyofjet-medium
l
c namely quark-gluon plasma (QGP), has been created. interaction and the medium effect on the full jets with
u One of the remarkable properties of the produced QGP showering structures in relativistic heavy-ion collisions
n is the strong collective motion which has been well de- [31–64]
[
scribedbyrelativistichydrodynamicswiththeextremely The interactions between the energetic partons in the

1 smallviscositytoentropyratio[1–9]. Thehydrodynamic jet shower and the QGP medium constituents exchange
v behavior of the QGP has been confirmed by the large energies and momenta between them. For the QGP
1 anisotropicflowsasmeasuredbyexperiments, whichim- fluid, the jet shower is a bunch of fast-moving energy-
5
plies strong interactions among the QGP constituents. momentum deposition sources, which are supposed to
9
7 Inadditiontothecollectivephenomena,onecanstudy excite the medium fluid, and induce flows propagating
0 the novel features of the QGP as strongly interacting with the jet as a collective medium response [54, 65–93].
. matterviajetquenching[10–18]. Inhigh-energynucleus- The jet-induced flow carries the energy and momentum
1
nucleus collisions, the energetic jet showering partons deposited by the jet and enhances the hadron emission
0
7 may be produced in hard processes at very early time. from the medium around the direction of the jet axis.
1 During their propagation through the QGP, the jet par- Some of these enhanced hadrons are detected as part of
: tonsinteractwiththemediumconstituentsviacollisional the jet together with the fragments from the jet shower
v
and radiative processes, which change the momenta of and affect the final state full jet energy and structures
i
X the shower partons in the hard jet. The phenomena re- [54, 60, 62]. Since the structure of the jet-induced flow
r latedtothemodificationofthejetsandtheirinnerstruc- is characterized by the bulk properties of the QGP, e.g.,
a ture by the QGP medium effect are commonly referred soundvelocity,andviscosities,thedetailedinvestigations
to as jet quenching in a broad sense. of the contribution of the collective medium response to
InrecentexperimentsofPb+PbcollisionsattheLHC, the jet structure may provide us not only the precise in-
detailed measurements of fully reconstructed jets with terpretation of the experimental data, but also unique
very high transverse momentum have become available, opportunities to study the bulk properties of the QGP
thanks to the high center of mass energy. One of the through the jet events in relativistic heavy-ion collisions.
most general consequence of jet quenching seen in the
In this work, we study the nuclear modification of full
experiments is the suppression of jet rates due to the en-
jet structures in the QGP medium including the con-
ergy loss of the full jets with a given cone of the size
tributions from the hydrodynamic medium response to
jet-deposited energy and momentum. We employ a cou-
pled full jet shower and QGP fluid model which is com-
∗ [email protected] posed of a set of transport equations for the jet shower
† [email protected] evolution and the hydrodynamic equations with source
‡ [email protected] terms for the QGP medium evolution. The transport
2
equationsdescribetheevolutionofthethree-dimensional gluon), ω is its energy, and k is its transverse momen-
i i⊥
momentum distributions of partons in the jet shower tumwithrespecttothejetaxis. Thetransportequations
[57], including the collisional energy loss, the transverse have the following generic form [57]:
momentum broadening, and medium-induced partonic
(cid:18) (cid:19)
d ∂ 1
splittings for all partons within the showering jet. The f (ω ,k2 ,t)= eˆ + qˆ∇2 f (ω ,k2 ,t)
space-timeevolutionoftheQGPmediumisdescribedby dt j j j⊥ j∂ωj 4 j k⊥ j j j⊥
(t3io+n1s)w-diitmhesnosuiorncealtreerlmatsiv[i8s9t,ic9i3d]e.alThhyedrsooudrycneamteircmesquaac-- +(cid:88)(cid:90) dω dk2 dΓ˜i→j(ωj,kj2⊥|ωi,ki2⊥)f (ω ,k2 ,t)
i i⊥ dω dk2 dt i i i⊥
count for the transfer of the deposited energy and mo- i j j⊥
mcoennstturumctferdomwitthhethjeetevsohlovwinegrdtoisttrhibeuQtiGonPsoflfutidh,eapnadrtoanres −(cid:88)(cid:90) dωidki2⊥dΓ˜j→id(ωωi,dkki22⊥|dωtj,kj2⊥)fj(ωj,kj2⊥,t).(2)
in the jet shower obtained as solutions of the jet trans- i i i⊥
portequations. Basedonourcoupledjet-fluidmodel,we
The first two terms on the right hand side in Eq. (2)
perform simulations of jet events in Pb+Pb collisions at
describe the energy-momentum changes of the jet par-
2.76A TeV, investigate the flow induced in the medium
tons via scatterings with the medium constituents. The
as a hydrodynamic response to jet quenching, and study
first term accounts for the energy-momentum changes in
how the jet-induced flow affects the full jet structures in
the longitudinal direction with respect to the jet axis;
the final state. Our study shows that the contribution
such contribution is called collisional energy loss and its
of theparticles originating fromthe jet-induced flows in-
strength is determined by the longitudinal momentum
creasesthejet-conesizedependenceofthefulljetenergy
loss rate eˆ = d(cid:104)E(cid:105)/dt. The momentum changes in the
loss. It is also found that jet-induced flow has a signif-
transversedirection,whoseeffectiscalledtransversemo-
icant contribution to the final state full jet shape, and
mentum broadening, is taken into account through the
dominates jet shape function at very large angles away
second term, with the exchange rate of transverse mo-
from the jet direction.
mentum squared qˆ = d(cid:104)∆p2(cid:105)/dt [94–96]. The last two
Thepaperisorganizedasfollows. InSec.II,wepresent ⊥
terms are the contributions from the medium-induced
the formulation of our coupled full jet shower and QGP
partonicsplittingprocesses,includingthegaintermcom-
fluid model used in this work. In Sec. III, we present
ing from the radiation of parton j with ω and k from
and discuss the results from the simulations of jet events j j⊥
the parton i with ω and k and the loss term for the
in Pb+Pb collisions at 2.76A TeV. We will focus on the i i⊥
radiation of parton i from parton j. It should be noted
effect of the hydrodynamic medium response to the jet
thatthetransportequations(2)arecoupledtoeachother
quenching on the final state full jet observables. Sec-
through the terms for the medium-induced parton split-
tion IV is devoted to the summary and concluding re-
tings, i.e., the gain term for the process i → j in the
marks of this work.
transport equation for parton j also appears as a loss
terminthetransportequationforpartoni. Themedium-
induced parton splittings change the number of partons
II. THE COUPLED JET-FLUID MODEL
and the internal momentum distribution of the jet, but
donotchangethetotalenergyandmomentuminsidethe
A. Jet Shower Evolution in Medium
full jet (for sufficiently large cone size) because they are
conserved in the splitting processes.
Jets are collimated clusters of the particles originating For the rates of the medium-induced partonic split-
from high-pT partons produced in early stage partonic tings, we employ the results from higher-twist jet energy
hard scatterings. The produced high-pT parton succes- loss formalism [97, 98]:
sively radiates partons and develops a shower of partons
due to its high virtuality. In relativistic heavy-ion colli- dΓi→j(ωj,kj2⊥|Ei) = 2αsqˆ xPi→j(x)sin2(cid:18)t−ti(cid:19).(3)
sions, the interactions between the propagating jet and dω dk2 dt π g ω k4 2τ
j j⊥ j j⊥ f
theconstituentsoftheQGPmedium,includingcollsional
and radiative processes, change the momenta of the jet Here, P (x = ω /E ) is the vacuum splitting function
i→j j i
shower partons, and thus modify the energy as well as for the process i → j with ω the energy of the radi-
j
the structure of the full jet defined by a cone size. ated parton and E the energy of the parent parton,
i
In this work, to describe the time-evolution of the τ = 2E x(1−x)/k2 is the formation time of the ra-
f i j⊥
jet shower structure driven by the interaction with the diated parton with k the transverse momentum with
j⊥
medium, we employ a set of coupled transport equations respecttothepropagationdirectionoftheparentparton,
fortheenergyandtransversemomentumdistributionsof andt istheproductiontimeoftheparentparton. Toob-
i
the partons contained in the jet shower, tain the splitting rates of the form dΓ˜i→j(ωj,kj2⊥|ωi,ki2⊥) in
fi(ωi,ki2⊥,t)= dNdi(ωωiid,kki2i2⊥⊥,t), (1) Eisqn.e(c2e)s,stahreym(sueletipRleicfa.t[i5o7n]wfoitrhmthoereJadceotbdaωiilajsdn)2.kJjI⊥n=dtt(cid:12)(cid:12)(cid:12)h∂∂ekki2cj2j⊥a⊥l(cid:12)(cid:12)(cid:12)-
where the index i denotes the parton species (quark or culation, we put a constraint that the medium-induced
3
radiations are allowed only for the partons whose forma- tions∂ Tµν =0,whereTµν istheenergy-momentum
µ QGP QGP
tion times are achieved (t>τ ). tensor of the QGP fluid. This equation represents
f
In our model, the strength of the each contribution the energy-momentum conservation only in the fluid.
of the medium modification in Eq. (2) is controlled by However, in the case that jets propagate through the
two transport coefficients: eˆ for collisional energy loss, QGP, the QGP and the jets exchange their energies
andqˆfortransversemomentumbroadeningandmedium- andmomentathroughthescatteringsbetweentheircon-
induced partonic splittings. Assuming that the medium stituents. Therefore,theenergy-momentumconservation
is in almost local thermal equilibrium, the fluctuation- is satisfied not for the QGP only, but for the combined
dissipation theorem can relate these two transport co- system of the QGP and the jets:
efficients to each other: qˆ = 4Teˆ [99, 100]. Also since
(cid:104) (cid:105)
qˆ for quarks and gluons are related by a color factor, ∂ Tµν (x)+Tµν(x) =0. (5)
µ QGP jet
qˆ /qˆ =C /C ,onlyonetransportcoefficient(wechoose
g q A F
qˆ for quark) determines the sizes of all the medium ef- Here Tµν is the energy-momentum tensor of the jet
q jet
fects in Eq. (2). In this work, we use qˆ of the form, shower. Assuming that the energy and momentum de-
q
positedbythejetsarequicklythermalized,wemodelthe
(cid:20)T(x)(cid:21)3 p·u(x) QGP as an ideal fluid in local equilibrium whose energy-
qˆ (x)=qˆ . (4)
q q,0 T p momentum tensor can be decomposed as:
0 0
where T0 = T (τ =τ0,x=0,y =0,ηs =0) is the initial TQµGνP =(ε+p)uµuν −pηµν, (6)
temperatureatthecenteroftheQGPmediumproduced
in central Pb+Pb collisions at 2.76A TeV at the LHC, where ε is the energy density, p is the pressure, uµ is the
pµ is the four-momentum of the propagating parton, u flow four-velocity, and ηµν = diag(1,−1,−1,−1) is the
µ
is the flow four-velocity of the medium, and qˆ is the Minkowski metric. If we define the source terms by
q,0
exchangerateoftransversemomentumsquaredforquark
Jν(x)=−∂ Tµν(x), (7)
in a static QGP medium at T . The factor p·u/p is µ jet
0 0
introduced to account for the flow effect in a non-static Equation (5) becomes the hydrodynamic equations with
medium [101]. source terms:
We solve the transport equations (2) numerically to
describe the jet evolution in terms of the energy and ∂ Tµν (x)=Jν(x). (8)
µ QGP
transverse momentum distributions of the partons con-
In the numerical hydrodynamic calculations for the
tained in the jet shower. The initial condition for full
jet shower is constructed from Pythiasimulation [102]. medium evolution in the relativistic heavy-ion collisions,
we use the relativistic τ-η coordinates which is conve-
The jet shower evolves according to the transport equa- s
nient to describe the longitudinal dynamics. To do so,
tions during the propagation through the QGP medium
we employ the equations in which the partial derivative
with the local temperature above T =160 MeV. We in-
c
intheEq.(8)arethereplacedwiththecovariantderiva-
troduceaminimumenergyω forthepartonsinthejet
cut
tive in the relativistic τ-η coordinates:
shower. Whentheenergyofthepartonsinthejetshower s
becomesbelow ωcut duringtheirevolution, theyarecon- D Tµ¯ν¯ =Jν¯, (9)
sideredtobeabsorbedinthemedium. Weputtheenergy µ¯ QGP
andmomentumoftheabsorbedpartonsintothemedium where µ¯ and ν¯ are the suffixes for the components in the
together with the collisional energy loss and transverse τ-η coordinate system: xµ¯ = (τ,x,y,η ). Equation (9)
s s
momentum broadening terms in Eq. (2). The same cut- isnumericallysolvedtodescribethespace-timeevolution
off energy is also used for the medium-induced partonic oftheQGPmedium. Usingthisframework,thecollective
splittings,i.e,onlypartonswithenergyaboveω canbe responsetothejetquenchingintheexpandingQGPfluid
cut
radiated; this is to take into account some effect for the can be properly described.
balancebetweenradiationandabsorption. Inthisstudy, Toclosethesystemofhydrodynamicequations(8),an
we set qˆ = 1.7 GeV2/fm and ω = 1 GeV. These equation of state is necessary. Here we employ the equa-
q,0 cut
values provide a good description to the nuclear mod- tion of state from the lattice QCD calculation [104]. As
ification factor R for inclusive jet spectrum in most the fluid expands, the QGP fluid cools down and finally
AA
central Pb+Pb collisions at 2.76A TeV measured by the turnsintoahadronicmatteraccordingtotheequationof
ATLAS, ALICE, and CMS Collaborations [19, 20, 23]. state. We keep using Eq. (8) to describe the space-time
Our value of qˆ is also consistent with the one obtained evolution of the hadronic matter until the temperature
q,0
by JET Collaboration [103]. drops to the freeze-out temperature T .
FO
B. Hydrodynamic Equations with Source Terms C. The Source Terms
The conventional way to describe the space-time evo- ThesourcetermsinEq.(8)describethetransferofthe
lution of the QGP is to use the hydrodynamic equa- energy-momentum between the jet and the QGP fluid.
4
(cid:112)
Here we construct the source terms from the evolving Hererisdefinedasr = (η −η )2+(φ(cid:48)−φ )2,with
s jet jet
parton distributions in the jet shower obtained as the φ(cid:48) =arctan[(y−yjet)/(x−xjet)] denoting the azimuthal
0 0
solutions of the transport equation (2). We recall the anglewithrespecttotheinitialpositionofthejetcenter
kinetic definition of the energy-momentum tensor of the xjet,η andφ thepseudorapidityandazimuthalangle
0 jet jet
jet shower [105]: ofthejet,andxjet(t)thepositionofthejetcenteratthe
time t. Using Eqs. (13), (14) and (15), we can construct
Tµν(x)=(cid:88)(cid:90) d3kjkνkµf (k ,x,t), (10) thesourcetermsfromtheenergyandtransversemomen-
jet ω j j j j
j tum distributions of the partons inside the jet shower.
j
To obtain the source terms in the relativistic τ-η coor-
s
where fj(kj,x,t) is the phase-space distribution of the dinates for Eq. (9), we perform Lorentz transformation:
parton j in the jet shower. Referring to Eq. (7), the
source term can be written as: dPν¯
Jν¯(τ,x,y,η )=− jet
Jν(x)=−ddtPdj3νext =−(cid:88)(cid:90) dω3kjjkjνkjµ∂µfj(kj,x,t),(11) s =Λν¯τJdµτ(dxx)d=yd−ηsΛν¯ dPjµet , (16)
j µ µdtd3x
where Pν is the total four-momentum of the jet shower
jet where
dPν
and d3jxet is its 3-dimensional space density. As men-  coshηs 0 0 −sinhηs
tioned above, there are three types of processes, i.e., the
0 1 0 0
collisional energy loss, the transverse momentum broad- Λνµ¯ = 0 0 1 0 . (17)
ening, and the medium-induced partonic splitting, con- −1 sinhη 0 0 1 coshη
tributingtothederivative ofthe phase-spacepartondis- τ s τ s
tribution in Eq. (11). However, since the total energy
and momentum in the jet shower are conserved in the
D. Initial Condition of the Medium
splitting processes, their contribution turns out to van-
ishbytheintegrationandsummationinEq.(11). There-
We assume that the QGP medium is locally thermal-
fore, the energy and momentum are exchanged between
izedatthepropertimeτ =τ (setasτ =0.6fm/cinour
the jet and the QGP fluid through the processes of first 0 0
calculation), and then apply relativistic hydrodynamics
two terms on the right hand side of Eq. (2), i.e., the
to describe its space-time evolution. We set the initial
collisional energy loss and transverse momentum broad-
entropy density distribution in the transverse plane at
ening. We also have to estimate the position of each
midrapidity η =0 as:
energeticpartoninthefulljet. Forapartonwithenergy s
ω and momentum k at time t, its position is estimated C (cid:20)(1−α) (cid:21)
j j s (x )= nb (x )+αnb (x ) ,(18)
as: x = xjet + (k /ω )t, where xjet is the production T ⊥ τ 2 part ⊥ coll ⊥
0 j j 0 0
point of the jet. Then the source term is obtained as:
wherenb andnb arethenumberdensitiesofnucleon-
Jν(x)=−(cid:88)(cid:90)d3kjkjνdfj(dktj,t)(cid:12)(cid:12)(cid:12)(cid:12) δ(3)(cid:18)x−xj0et−ωkjt(cid:19).(12) neruactleedonfrcboolimlnartyhecooplpalritstiicoanlsGalnadubpearrtmicoipdaetlinwgitnhuicmlepoancstgpena--
j col. j
rameter b, respectively. The parameters C = 41.4 and
Here df (k,t)/dt| is the part of the time derivative α = 0.08 are chosen for Pb+Pb collision at 2.76A TeV
i col.
of the momentum distribution corresponding to the first by fitting the centrality dependence of the multiplicity
and second terms on the right hand side in Eq. (2): atmidrapidityfromtheALICECollaboration[106,107].
Inη direction,weinitializethemediumprofileusingthe
dfj(cid:0)ω,k⊥2,t(cid:1)(cid:12)(cid:12)(cid:12) =(cid:18)eˆ ∂ +1qˆ∇2 (cid:19)f(ω ,k2 ,t). (13) folloswing function form:
dt (cid:12)(cid:12)col. j∂ωj 4 j k⊥ j j j⊥ (cid:34) (|η |−η /2)2 (cid:16) η (cid:17)(cid:35)
H(η )=exp − s flat θ |η |− flat .(19)
Considering the rotational symmetry for the distribu- s 2σ2 s 2
η
tions of the shower partons along the jet axis and using
the collimated shower approximation (r = θ ≈ sinθ = ThefunctionH(ηs)hasashapeconsistingofaflatregion
k⊥), the source term can be written as: aroundthemid-rapidity,liketheBjorkenscalingsolution
ω [108], and two halves of a Gaussian connected smoothly
Jν(x)≈−2π1rt3(cid:0)xν−xνjet,0(cid:1)ddEtdjert(cid:12)(cid:12)(cid:12)(cid:12) δ(cid:16)|x−xj0et|−t(cid:17),(14) troamtheteervsacηufluamt =at3b.8otahnedndσsηo=f t3h.e2flaarterfietgtieodn.toThreepprao--
col. duce the pseudorapidity distribution for the multiplicity
where for central Pb+Pb collisions at 2.76A TeV. Finally, the
(cid:16) (cid:17)(cid:12) full 3-dimensional profile of the initial entropy density is
ddEtdjert(cid:12)(cid:12)(cid:12)(cid:12) =(cid:88)(cid:90)dωdkj2⊥ωjdfj ωjd,tkj2⊥,t (cid:12)(cid:12)(cid:12)(cid:12) δ(cid:18)r− kωj⊥(cid:19).(15) given as:
col. j (cid:12)col. j s(τ0,x⊥,ηs)=sT(x⊥)H(ηs). (20)
5
We further assume that there is no initial flow at τ =τ obtain the final state full jet. For the particles in-
0
in the transverse direction, which means that the radial cluded in Eq. (23), we impose the transverse momenta
expansion of the medium is caused solely by the initial cut ptrk,hyd > 1 GeV/c for inclusive jet analyses, and
T
pressuregradientincorporatedintheinitialprofilecondi- ptrk,hyd >0.5GeV/cfordijetanalysesfollowingthemea-
tionofEq. (20). Fortheflowvelocityinthelongitudinal T
surements by CMS Collaboration [24, 28].
direction, the space-time rapidity component is initially
set to zero: uηs(τ =τ0)=0 [108].
In this work, we employ the smooth averaged initial
III. SIMULATIONS AND RESULTS
profile of the medium obtained from the optical Glauber
model,whichdoesnotincludetheinitialgeometricalfluc-
tuation of the nucleons and their internal structures in In this work, we initialize the jet production points in
the colliding heavy ions. This initial state fluctuations the transverse plane ηs = 0 according to the distribu-
in principle can affect both the jet evolution and the tions of the binary nucleon-nucleon collisions which are
mediumresponsetojetquenching;wewouldliketoleave calculatedbyusingtheGlaubermodel[111]. Jetspectra
theevent-by-eventstudiesincludingtheinitialstatefluc- and the momentum distributions of the shower partons
tuations as a future work. inside the full jets are obtained via Pythiasimulation
[102] with FastJet package [112] used for the full jet
reconstruction. Hard jets are assumed to be created at
E. Freeze-out τ = 0, and travel freely until the thermalization proper
time of the QGP, τ = 0.6 fm/c. Then the jet shower
startstointeractwiththeQGPandevolvesaccordingto
Thehydrodynamicmediumresponsetotheenergyand
the transport equations (2). The jet-medium interaction
momentumdepositedfromthejetswillinduceadditional
is turned off when the local temperature of the medium
flowsintheQGPmedium. Thesejet-inducedflowsprop-
is below T =160 MeV. The interaction in the hadronic
agateandenhancehadronemissionsindirectionsaround c
matterisusuallysmallcomparedtotheQGPphase,and
the jet axes. Some of the hadrons produced from jet-
is neglected in this work.
inducedflowsremaininthejetconeafterthebackground
subtraction and are counted as part of the full jets. To The medium profile at initial time τ =τ0 is calculated
obtainthemomentumdistributionsofparticlesproduced by employing Eq. (20) with the impact parameter b=0
fromthemedium, weusetheCooper-Fryeformula[109]: (central collisions). The evolution of the medium is gov-
erned by the ideal hydrodynamic equations with source
E dNi = gi (cid:90) pµidσµ(x) , (21) terms (8). As the system expands and cools down, it
id3pi (2π)3 Σ exp[pµiuµ(x)/T(x)]∓1 transits from QGP phase to hadronic phase, and finally
the freeze-out occurs. The momentum distributions of
where g is the degeneracy, ∓ corresponds to Bose or
i hadrons produced from the medium are calculated via
Fermi distribution for particle species i, and Σ is the
the Cooper-Frye formula (21). After the subtraction of
freeze-out hypersurface. The freeze-out is chosen to oc-
the background (without jet), the remaining part (23)
cur at a fixed temperature T . Here a typical value
FO contributes to final state full jets.
T = 140MeV (e.g., Ref. [110]) is used. For the sake
FO Hereafterwecallthepartofjetdescribedbythetrans-
of simplicity, when calculating the particle spectra via
port equations (2) as shower part, and that coming from
theCooper-Fryeformula,weconsideronlyonespeciesof
the jet-induced flow via freeze-out as hydro part. In this
bosons with the same mass as the charged pions. The
study, we mainly focus on the contribution of the hydro
contributions from other hadron specicies are taken into
partofthefulljet. Detailedstudiesofthecontributionof
account by replacing g by the effective degrees of free-
i each medium modification effect on the jet shower part
dom d defined as follows:
eff can be found in Ref. [57]. Also since we perform the
(cid:113) simulation independently for each single jet shower, the
e (T)= deff(T)(cid:90) d3p m2π± +p2 ,(22) effect of possible interference between the flows induced
lat (2π)3 exp(cid:104)(cid:113)m2 +p2/T(cid:105)−1 by multiple jet showers in one event is not included.
π±
where e is the energy density obtained from lattice
lat
QCD calculations [104]. A. Flow Induced by Full Jets
Thecontributionfromthejet-inducedmediumflowto
thefulljetisobtainedbyremovingthecontributionfrom
Figure1showsthesnapshotsoftheenergydensitydis-
the background medium without jet propagation:
tributionofthemediuminthetransverseplaneatη =0
s
dN dN(cid:12)(cid:12) dN(cid:12)(cid:12) at different proper times for an example event. For this
∆ = (cid:12) − (cid:12) . (23) event, the single jet with the initial pjet = 150 GeV/c
d3p d3p(cid:12) d3p(cid:12) T
w/jet w/ojet is produced at (xjet,yjet) = (0 fm,6.54 fm) and travels
0 0
The contribution ∆dN/d3p is added to that of the jet in the direction of φ = 5π/8. The upper panels show
p
shower calculated from the transport equations (2) to the whole energy density of the medium, and the lower
6
����� ����� �����
τ � 3 fm/c e��GeV/fm3� τ � 7.5 fm/c e��GeV/fm3� τ � 12 fm/c e��GeV/fm3�
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��� ������ ��� �� ��� ���������
�� ��� �� �� ����
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��� ���� ��� ���� ��� ���������
��� �� ��� �� ��� ��
������ �� �� �� ������ ������ �� �� �� ������ ������ �� �� �� ������
x��fm� x��fm� x��fm�
����� ����� �����
τ � 3 fm/c Δe��GeV/fm3� τ � 7.5 fm/c Δe��GeV/fm3� τ � 12 fm/c Δe��GeV/fm3�
��� ��� ��� ���� ��� ����
��� �� ��� ���� ��� �����
m� �� ���� m� �� �������� m� �� ����������
f �� f �� ���� f ��
�y� �� ���� �y� �� �� �y� �� �������
����
��� �� ��� ���� ��� �����
��� �� ��� ���� ��� �����
������ �� �� �� ������ ������ �� �� �� ������ ������ �� �� �� ������
x��fm� x��fm� x��fm�
FIG. 1. (Color online) Energy density distribution of the medium in the transverse plane at midrapidity η = 0 at different
s
propertimesτ =3.0,7.5,and12fm/c. Thejetwiththeinitialpjet =150GeV/cisproducedat(xjet,yjet)=(0fm,6.54fm)and
T 0 0
propagatesinthedirectionofφ =5π/8. Theupperpanels(a-1),(a-2),and(a-3)showthewholemediumenergydensity,and
p
thelowerpanels(b-1),(b-2),and(b-3)showthemediumenergydensitysubtractedbythatinthecasewithoutjetpropagation.
panels show the energy density after the subtraction of response to the jet-deposited energy and momentum,
the energy density in the events without jet propaga- e.g., the Mach cone can be smeared by the finite shear
tion. From these figures, we can see that the V-shaped viscosity [79, 83, 87, 88]. In our study, we assume the
wave fronts (shown by higher energy density region) are instantaneous thermalization of the energy and momen-
induced by the jet propagation, and develop with time tumdepositedbythejet;thefiniterelaxationtimeeffects
in the medium. This V-shaped wave front is the Mach may be included in the source terms [79, 118] (note that
cone [65–67], a conical shock wave that appears as an the smearing due to the finite grid size in the hydrody-
interference of sound waves caused by an object moving namic simulation mimics some relaxation effect). Since
faster than the medium sound velocity. Here the highly the relaxation times for the deposited energy and mo-
collimatedjetshowerdepositsitsenergyandmomentum mentum are closely related to the transport coefficients
and induces a Mach cone whose vertex is the center of of the QGP, and the inclusion of such effects would pro-
thejet[80,81]. ThiswavefrontoftheMachconecarries vide further information on the QGP’s properties, which
theenergyandmomentum,propagatesoutwardandalso we would like to leave as a future work.
causes the lower energy density region behind the wave
front. During the propagation, the Mach cone and the
radial flow of the medium are pushed and distorted by B. Full Jet Energy Loss and Suppression
eachother. OnecanseethattheMachconeisasymmet-
rically deformed in this example because the jet travels
In our framework, the final full jets are contributed
through the off-central path in the medium.
from two parts: jet shower part and hydrodynamics re-
In this work, we neglect the effect of the finite small sponse part. The shower part of the jet loses energy
shear viscosity of the QGP and model the medium cre- due to three mechanisms: the collisional energy loss and
ated in relativistic heavy-ion collisions as an ideal (non- the absorption of the soft partons by the medium, the
viscous) fluid. The finite viscosities are important for transverse momentum broadening which kicks the par-
more precise description of the medium evolution and tons out of the jet cone, and the medium-induced radia-
thecollectiveanisotropicflowsobservedinthefinalstates tionoutsidethejetcone. Thehydropartofthejetcomes
[6, 113–117]. It can also affect the shape of the medium from the lostenergyand momentum from the jet shower
7
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q^q,0= 1.7�GeV2/fm q^q,0= 1.7�GeV2/fm
��� ωpTtrcku, th y=d >1. 01�.0G�eGVe/Vc/c ��� ωpTtrcku, th y=d >1. 01�.0G�eGVe/Vc/c
) )
c c
/ /
V V
Ge��� Ge���
( (
� �
�pjetT��� �pjetT���
Δ Δ
� � � �
R=0.2�������������� R=0.3��������������
� �
�� RR==00..32����������������������� �� RR==00..63�����������������������
� �
R=0.3�������� R=0.6��������
� �
R=0.4�������������� R=0.9��������������
� �
R=0.4�������� R=0.9��������
�� ��
��� ���� ���� ���� ���� ��� ���� ���� ���� ����
pjet (GeV/c) pjet (GeV/c)
T�init T�init
FIG. 2. (Color online) Total p loss for jets in central Pb+Pb collisions at 2.76A TeV as a function of initial jet p for the
T T
cone sizes (left panel) R = 0.2, 0.3, and 0.4 and (right panel) R = 0.3, 0.6, and 0.9. The solid lines for jets with hydro part,
and the dashed lines for jets without hydro part.
which thermalize into the medium and induce connical The effect of full jet energy loss in the relativistic
flow; some of the energy is still inside the jet cone. Thus heavy-ion collisions can be quantified by the measure-
the hydro part will partially compensate the energy loss ments of nuclear modification factor R for single in-
AA
experienced by the jet shower part. Here we study the clusive jet spectrum, defined as:
effect of jet-induced medium flow on full jet energy loss
and full jet suppression. R = 1 d2NjAetA/dηpdpjTet, (24)
Figure 2 shows the mean value of the total energy AA (cid:104)Ncoll(cid:105) d2Njpept/dηpdpjTet
(transverse momentum) loss for inclusive jets with and
without the hydro part contribution as a function of ini- where (cid:104)N (cid:105) is the number of binary nucleon-nucleon
coll
tial full jet transverse momentum. The left panel shows collisionsaveragedovereventsinagivencentralityclass,
the results for the jet-cone sizes R = 0.2, 0.3, and 0.4, NAA is the number of jets in nucleus-nucleus collisions,
jet
and the right shows for R = 0.3, 0.6, and 0.9. One can and Npp is that in p+p collisions. One important re-
jet
see the general feature that the amount of the energy sult of jet energy loss is that jet p spectrum in nucleus-
T
loss increases with increasing initial jet transverse mo- nucleuscollisionsisshiftedtolowerpjet comparedtothat
T
mentum while the fractional energy loss decreases. The in p+p collisions. Since the jet spectrum is a steeply de-
total pjet loss for the full jets with the inclusion of the creasing function of pjet, jet R will become smaller
T T AA
hydro part contribution is smaller than that without the than unity in high-pjet region.
contribution from the medium response. For jets with T
Figure 3 shows the nuclear modification factor R
the cone size R = 0.3, about 10 % of the lost pjTet from for single inclusive jets as a function of pjet for differeAnAt
the jet shower part is recovered by the hydro part. T
jet cone sizes: the left panel for the jet-cone sizes R =
We can also see the jet cone size dependence of jet en- 0.2, 0.3, and 0.4, and the right for R = 0.3, 0.6, and
ergylossfromFigure2. Fortheshowerpartwithoutthe 0.9. We also compare the results with and without the
hydro part contribution, the jet cone size dependence is inclusion of the contribution from the jet-induced flow.
rather weak. This is due to the reason that the shower Wefindthatwithoutthehydropartcontribution,thejet
partofthejetisquitecollimated,i.e.,mostoftheenergy cone size dependence for jet R is very week, which is
AA
intheshowerpartiscoveredbyanarrowjetcone,there- consistent with the weak dependence for jet energy loss
fore,jetenergydoesnotchangemuchwithincreasingjet as seen in Figure 2. The inclusion of the contribution
conesizes. Onthecontrary,jet-inducedflowevolveswith from jet-induced flow decreases the total energy loss and
medium,diffuses,andcanspreadquitewidelyaroundjet thus increase the value of R ; it also increases the jet-
AA
axis. As a result, the jet cone size dependence becomes conesizedependenceofR . Ourresultsarecomparable
AA
much strongerwhenaddingthehydropart contribution. withCMSmeasurementswiththejet-conesizesR=0.2,
8
���� ��
����������������������������������22..7766 TTeeVV���� ������ �����������������2.76 TeV�� ���
qq^^qq,,00== 11..77��GGeeVV22//ffmm q^q,0= 1.7�GeV2/fm
�������� ωωccuutt == 11..00��GGeeVV//cc ���� ωcut = 1.0�GeV/c
ppttrrkk,, hhyydd >> 11..00��GGeeVV//cc ptrk, hyd > 1.0�GeV/c
TT T
�������� ����
AA A
AA A
RR R
�������� ����
�
R=0.3������0-5 %�
�������� RRR===000...432���������������������000---555 %%%��� RRRRRR======000000......443322������������������������������������������������������������������������ ���� RRRRRR======000000......996633������������������������������������������������������������������������
���� ��
������ �������� �������� �������� �������� ��� ���� ���� ���� ����
ppjjeett ((GGeeVV//cc)) pjet (GeV/c)
TT T
FIG. 3. (Color online) Nuclear modification factor R for inclusive jet spectrum in central Pb+Pb collisions at 2.76A TeV
AA
with jet cone of the sizes (a) R=0.2, 0.3, and 0.4, and (b) R=0.3, 0.6, and 0.9. The solid lines are the results for inclusive
jets with hydro part, and the dashed lines for without hydro part. The turquoise plus markers, the magenta circles, and the
orange triangles show the experimental data taken from CMS Collaboration [23] for the jet-cone sizes R = 0.2, 0.3, and 0.4,
respectively. The colored shaded boxes indicate the systematic uncertainties of the same colored data points.
0.3, and 0.4, which show relatively small jet cone size themediumeffectonthejetshapefunctionmoreclearly,
dependence (but with large error bars). the nuclear modification factor for the jet shape func-
tionRρ (r)=ρ (r)/ρ (r)isshownintherightpanel
AA AA pp
of Figure 4. We can see that our results (both with
C. Full Jet Shape Function andwithoutthecontributionfromthehydrodynamicre-
sponse part) show similar nuclear modification pattern
for the jet shape function to the experimental data from
One of the advantages of studying fully reconstructed
CMSCollaboration,i.e.,littlechangeforsmallr,adipat
jetsinrelativisticheavy-ioncollisionsisthatonemayin-
r ∼ 0.1 and an enhancement at large r. In other words,
vestigatenotonlythefulljetenergylossandsuppression,
the inner hard core of the jet is more collimated while
butalsotheirinternalstructureswhichprovideusthede-
the tail (the outer soft part) of the jet is broadened, in
tailedinformationonhowtheenergyisdistributedinside
central Pb+Pb collisions compared to pp collisions.
the full jets and how the energy distribution is modified
by the interaction with the QCD medium. Jet shape
The medium modification feature for the shower part
function describes how the energy inside (and outside)
of the full jet has been extensively studied in Ref.
the full jets is distributed in the radial direction (trans-
[57] which shows that the collisional energy loss and
verse to the jet axis) and is defined as follows:
the thermalization of the soft shower partons (into the
ρ (r)= 1 (cid:88)(cid:34) 1 (cid:80)trk∈(r−δr/2,r+δr/2)ptTrk(cid:35), (25) mhaerddiucmor)e,mwahkeilethtehejettrannasrvreorwseermwoimthenmtuomrebcroolalidmenatinedg
jet Njet jet pjTet δr andmedium-inducedradiationtransporttheenergyfrom
the inner to the outer sides of the jet and broaden the
(cid:112)
where r = (η −η )2+(φ −φ )2 is the radial dis- tail of the jet shape function. After the inclusion of the
p jet p jet
tanceofthejetconstituentsfromthejetaxis,δristhebin contributionfromjet-inducedmediumflow,thejetshape
size, and the sum is taken over all constituents (tracks) function at small r is not modified much, but for large
of the full jets in the bin at r. r region (r > 0.2-0.25), there is a significant enhance-
The left panel of Figure 4 shows our result for the mentofthejetbroadeningeffect. Thisseemstobequite
jet shape function inside the jet cone for inclusive jets natural considering the jet cone size dependence of full
with pjet > 100 GeV/c and R = 0.3 in central Pb+Pb jet energy loss as seen in Figure 2, i.e., the energy loss
T
collisions and in p+p collisions, compared to the exper- from the shower part of the jet induces conical flow and
imental data from CMS Collaboration [24, 28]. To see medium excitation which evolve with the medium and
9
����
��� �����������2.76 TeV� �����������������2.76 TeV��
pTjet > 100�GeV/c��R= 0.3 ���� pTjet > 100�GeV/c��R= 0.3
�����������0-10 %� ���� q^q,0= 1.7�GeV2/fm��ωcut = 1.0�GeV/c
���������������������������������������� (r)pp ���� pTtrk, hyd > 1.0�GeV/c
(r) ����PYTHIA� r)/ρ ���� ����������0�-�1�0 �%����
ρ (b ������
��� bP ����
P
ρ
q^q,0= 1.7�GeV2/fm ��
ωcut = 1.0�GeV/c ����
pTtrk, hyd > 1.0�GeV/c
����
�� ����� ���� ����� ���� ����� ���� �� ����� ���� ����� ���� ����� ����
r r
FIG. 4. (Color online) Left panel: Jet shape function for inclusive jets with pjet > 100 GeV/c in central Pb+Pb and p+p
T
collisionsat2.76A TeV. Rightpanel: Nuclearmodificationfactorforjetshapefunctionforinclusivejetswithpjet >100GeV/c
T
in central Pb+Pb collisions. The solid and dashed lines are results for jets with and without hydro part. The dotted line is
Pythiasimulation, and the turquoise triangles are data for p+p collisions (left panel). The black circles are data for Pb+Pb
collisions with the shaded boxes for the systematic uncertainties. Data are taken from CMS Collaboration [24].
broadening effect, we show in Figure 5 the jet shape
���
function ρ(r) for inclusive jets with an extended radial
�����������2.76 TeV
distance 0 < r < 1. The trigger p threshold for the
pjet > 100�GeV/c��R= 0.3 T
��� T inclusive full jets is set to be pjTet > 100 GeV/c. Here
�����������0-10 %� we still use pjet defined by the jet-cone size R = 0.3
�������� T
������������������� as the normalization factor for the jet shape function
��� ������������� at r > R = 0.3, to be consistent with the experimental
) ������������ results from CMS Collaboration [24, 28]. The red solid
ρ(r ����PYTHIA� line shows the result for jets with both shower and hy-
���� dro parts, the orange dashed dotted solid line shows the
contribution from the hydro part, and the blue dashed
line shows the result for jet without hydro part. The
���� green dotted line shows the result from Pythia simula-
q^q,0= 1.7�GeV2/fm��ωcut = 1.0�GeV/c tion. Aswecansee,theshowerpartofjetshapefunction
pTtrk, hyd > 1.0�GeV/c is a deep falling function of r, while the energy (momen-
���� tum)fromthehydrodynamicresponsepartisaquiteflat
�� ���� ���� ���� ���� ��
distribution in a wide range of r. This is because the en-
r
ergylossfromtheshowerpartiscarriedawaybythejet-
inducedflowwhichevolveswiththemediumanddiffuses
FIG. 5. (Color online) Jet shape function for single inclusive tolargedistances[89]. ComparedtoPythiasimulation,
jets with pjet >100 GeV/c in central Pb+Pb and p+p colli- the broadening of the shower part of the jet continues to
T
sionsat2.76A TeV. Thesolidanddashedlinesareresultsfor large r by the transverse momentum kicks and medium-
jets with and without hydro part, and the dash-dotted line
induced radiation, but the contribution from the hydro
shows the pure hydro part contribution. The dotted line is
parttothejetshapefunctionisquiteflatandfinallydom-
Pythiasimulation, and the turquoise triangles are data for
inates over the shower part in the region with r >0.5.
p+pcollisions. TheblackcirclesaredataforPb+Pbcollisions
withtheshadedboxesindicatingthesystematicuncertainties.
CMS Collaboration has recently measured the jet
Data are taken from CMS Collaboration [24].
shape functions with a wide range of r (up to r = 1)
for both leading and subleading jets in asymmetric dijet
events in Pb+Pb collisions at 2.76A TeV [24]. We also
diffuse to larger angles with respect to the jet axis. perform the calculation for the jet shape functions in di-
To see more clearly the contribution from the hydro- jet events, and the comparison with CMS data is shown
dynamic response part (jet-induced medium flow) to jet in Figure 6. In the calculation, we chose dijet events
10
��� ���
��������������� 2.76 TeV��R= 0.3 ������������������ 2.76 TeV��R= 0.3
pjet> 120�GeV/c ����������0-30 %� pjet> 120�GeV/c ����������0-30 %�
T,1 T,1
��� pjet> 50�GeV/c ������������������� ��� pjet> 50�GeV/c �������������������
T,2 T,2
Δφ1,2> 5π/6 ������������� Δφ1,2> 5π/6 �������������
������������ ������������
��� ����PYTHIA� ��� ����PYTHIA�
) )
r r
( (
ρ ρ
���� ����
���� q^q,0= 1.7�GeV2/fm ���� q^q,0= 1.7�GeV2/fm
ωcut = 1.0�GeV/c ωcut = 1.0�GeV/c
pTtrk, hyd > 0.5�GeV/c pTtrk, hyd > 0.5�GeV/c
���� ����
�� ���� ���� ���� ���� �� �� ���� ���� ���� ���� ��
r r
FIG.6. (Coloronline)Jetshapefunctionfor(a)leadingand(b)subleadingjetsindijeteventsincentralPb+Pbcollisionsand
inp+pcollisionsat2.76A TeV. Theleadingjetpjet >120GeV/c,thesubleadingjetwithpjet >50GeV/c,andtheazimuthal
T,1 T,2
angle between the leading and subleading jets ∆ϕ > 5π/6. The solid and dashed lines are results with and without hydro
1,2
partcontributionincentralPb+Pbcollisions,respectively. Thedasheddottedlineshowsthepurehydropartcontribution. The
dotted line is result for p+p collisions obtained from Pythia. The black circles are data in central (0-30%) Pb+Pb collisions
from CMS Collaboration [28], with the shaded boxes indicating the systematic uncertainties.
withtheleadingjetwithpjet >120GeV/c,thesublead- show that the hydrodynamic medium response to jet-
T,1
ing jet with pjet > 50 GeV/c, and the azimuthal angle medium interaction plays an important role in the study
T,2 of fully reconstructed jets, especially at large r region.
between the leading and subleading jets ∆ϕ > 5π/6.
1,2 This means that the jet shape function at large angles
In the figure, we do not show the jet shape function for
with respect to the jet direction provides a good oppor-
p+p collisions from CMS [24] since the data contain the
tunity to study the hydrodynamic medium response to
contamination from the underlying event (and therefore
jet quenching.
are quite different at large r region as compared to the
jet shape function obtained from Pythia simulation).
In Pb+Pb collisions, such background effect is supposed
to be small at large r, since the jet shape function is
IV. SUMMARY
dominated by the shower part at small r and by the jet-
induced medium flow at large r (see below).
In this work, we have studied the nuclear modifica-
From Figure 6(a), we can see that the jet shape func- tions of full jet structures in relativistic heavy-ion col-
tion for leading jets in central Pb+Pb collisions is quite lisions with the inclusion of the contribution from the
similar to that for inclusive jets shown in Figure 5. The mediumexitationsinducedbythepropagatinghardjets.
showerpartdominatesthejetshapefunctionatrelatively We have formulated a model which consists of a set of
small r and is broadened by the medium effect starting transport equations to describe the full jet shower evolu-
from r = 0.2−0.3, while the hydro part starts to dom- tioninmediumandrelativisticidealhydrodynamicequa-
inate the jet shape function at large r region (r > 0.5). tions with source terms to describe the dynamical evo-
Our full result on jet shape function with the contribu- lution of the QGP medium. The transport equations
tions from both shower and hydro parts reproduces the control the evolutions of energy and transverse momen-
experimental result quite well throughout the entire r tum distributions of the shower partons within the full
range(uptor =1). ForsubleadingjetsasshowninFig- jet. Thecontributionfromthemomentumexchangewith
ure 6(b), we can see that the jet shape function is much the medium via scatterings with medium constituents is
broaderthanthatofleadingjetsduetolargerjet-medium taken into account by collisional energy loss and trans-
interaction for subleading jets. As a result, the shower verse momentum broadening terms. The partonic split-
partofthejetshapefunctioniswiderandthehydropart ting terms account for the contribution of the medium-
contribution is also larger and more widely distributed induced radiations; the rates for the induced splittings
than that in leading jets. Our full result also provides a were taken from higher-twist jet energy loss formalism.
good description of the jet shape function for subleading The local temperature and flow velocity of the medium
jetsexceptthemiddlerregion. Theaboveresultsclearly are embedded in the jet quenching parameter qˆ which