An Explanation for Heavy Quark Energy Loss Puzzle by Flow Effects

0.16 MB·English

by  Luan Cheng

An Explanation for Heavy Quark Energy Loss Puzzle by Flow Effects
Luan Chenga,b and Enke Wanga,b
aInstitute of Particle Physics, Huazhong Normal University, Wuhan 430079, China
bKey Laboratory of Quark & Lepton Physics (Huazhong Normal University), Ministry of Education, Wuhan 430079, China
Based on our new model potential in the presence of collective flow describing the interaction
of the hard jet with scattering centers, an explanation for heavy quark energy loss puzzle is given
by collective flow effects. It is shown that the collective flow changes the pole of propagator which
lead to reduce significantly the dead cone and change the LPM destructive interference comparing
to that in the static medium. Consider the collective flow with velocity vz along jet direction, the
heavyquarkenergylossisnearlythesameaslightquarkintheregionof0.065<vz <0.13. Itturns
outthattheaveragevz ofrescatteringcentersisaround0.08,thedifferencesoftheeffectiveaverage
9 energylossamongcharm,bottomandlightquarksareverylittlefromafull3Didealhydrodynamic
0 simulationfor0-10%centraleventsofAu-AucollisionatRHICenergybyincludingcollectiveeffects.
0 The flow dependence of the quark energy loss implies that the light and heavy quark have almost
2 thesame suppression of high pT hadron spectrum in high-energy heavy-ion collisions.

g
PACSnumbers: 12.38.Mh,24.85.+p;25.75.Bh;25.75.-q
u
A
Introduction — The formation and observation of of collective flow in perturbative Quantum Chromody-
6
1 quark-gluonplasma(QGP)arethemostimportantgoals namics(pQCD)andgiveanexplanationforheavyquark
in high energy heavy ion physics. Jet quenching [1] or energylosspuzzleintheframeworkofjetquenchingthe-

] suppression of large transverse momentum p hadrons ory. Based on our new potential model with collective
h T
caused by the energy loss of a propagating parton in a flow, we find that the flow effects reduces significantly
t
- dense medium, has become a powerful toolfor the study the dead cone of heavy quark jets. We are led to the
l
c of properties of QGP. Lightquark and gluonjet quench- conclusion that the decrease of the dead cone increases
u
ingobservedviaπ,η suppressioninCu+CuandAu+Au the heavy quark energy loss. Using the full 3D ideal
n
collisions at √s=62 200GeV at the relativistic heavy hydrodynamic simulations [17], we obtained the average
[ −
ion collider (RHIC) has been remarkably consisted thus flow velocity of rescattering centers and effective aver-

2 farwithpredictions[2,3,4,5]fromjetquenchingtheory. age energy loss of heavy quark jets. It turns out that
v It is believed that a good test of QGP formation is the the effective average energy loss of heavy quark jets is
8
opencharmsuppression,whichcanbemeasuredatRHIC nearly the same as the light quark jets, this agrees with
1
by comparingthe p distributions of single electronpro- the experimental data that the suppression of electrons
9 T
0 duction in d+Au and Au+Au collisions. It is estimated for heavy quarks is similar to light hadrons.
. that the heavy quark mass leads to a kinematical ”dead Flow Effect on Gluon Radiation for Heavy Quark
8
cone” effect that reduces significantly the induced radia- Jet — In our previous work [16] we model the jet
0
tiveenergylossofheavyquarks[6,7,8,9]basedonthejet interactions in the QGP in the presence of collective
9
0 quenching picture. However, recent non-photonic single flow as random color screened potentials Aνi(qi) =
: electrondata[10,11,12,13,14]showsthatamuchlarger (Vflow(q ),Aflow(q )),
v i i i i
suppression of electrons, which is similar to the suppres-
i
X sionoflighthadrons,thanpredictedwasobservedinthe Vflow(q ) = 2πδ(q0 vq)e−iq·xiv˜(q)T (j)T (i), (1)
ar pisTin=cl4u−de8dGtoeVexrpelgaiionn.thLeatneornt-hpehoctoollnisicionsianlgeleneerlgeyctlroosns Aifilow(qii) = 2πδ(qii0−−v··q)ve−iq·xiv˜(q)aTa(j)aTa(i),(2)
data, but this still doesn’t yield a consistent description.
where v˜(q) = 4πα /(q2 (v q)2+µ2), q = (q0,q) is
Recently the ADS/CFT approachhas been used for this s − · i i
the momentum transfer, µ the Debye screening mass, v
question[15]. At presentthe heavy quark energy loss be-
the flow velocity, T (j) and T (i) the color matrices for
a a
comesapuzzletochallengethescenarioofjetquenching.
thejetandtargetpartonatpositionx ,respectively. The
i
The currently available studies suffer from one cru- collectiveflowchangesthe color-electricfield,producesa
cialdrawback,the medium inducedradiativeenergyloss color-magneticfieldandleadstonon-zeroenergytransfer
should be computed in a expanding QCD medium with q0 =v q from the target parton to the jet which differs
i ·
collective flow instead of a static medium. The medium with Gyulassy-Wang’s static potential model [1].
produced in nucleus-nucleus collisions at RHIC equili- Here we investigate the rescattering-inducedgluon ra-
brates efficiently and builds up a flow field. This flow diation of heavy quark jet by considering the flow effect
changes the color-electric field and produces a color- resultingfromthemovingpartontarget. Wewillworkin
magnetic field [16], and will surely change the parton the framework of opacity expansion developed by Gyu-
energy loss. In this letter, based on our previous new lassy, L´evai and Vitev (GLV)[18] and Wiedemann[19].
modelpotential[16]withcollectiveflow,wewillreporta The opacity is defined as the mean number of collisions
firststudyoftheheavyquarkenergylossinthepresence in the medium. In the framework of the opacity expan-
2
sion, the firstorder opacitycontributionis dominantbe- where z = z z , C and C are the Casimirs of jet
10 1 0 R A
−
causethe higherordercorrectionscontributelittle tothe in fundamental representation in d dimension and the
R
radiative energy loss as shown by GLV [18]. target parton in adjoint representation in d dimension,
A
Inordertocomputethemedium-inducedradiativeen- respectively.
ergy loss to the first order in the opacity expansion,
k2 (k q )2
we need to obtain the amplitudes for self-quenching in ω = ⊥ , ω = ⊥− ⊥ , (10)
0 1
2ω 2ω
the vacuum, and for single rescatteringand double Born
scattering with the same target in the QCD medium in m2g+x2M2 vzxM2
ω = , ω = , (11)
m M
the presence of collective flow. Consider a hard heavy 2ω (1 v )ω
z
−
quark jet with mass M and initial energy E produced k
at x0 = (z0,x0⊥). It interacts with the target parton H1 = k2 +m2⊥+x2M2 , (12)
at x = (z ,x ) with flow velocity v by exchanging ⊥ g
1 1 1⊥ k
gluonwithfour-momentumq,radiatesagluonwithfour- H = ⊥ , (13)
momentum k and polarization ǫ(k), and emerges with 2 k2⊥+m2g+x2M2− 12−vvzzxM2
final four-momentump. In Ref. [6], it is shownthat glu- k q
ons in the medium can be be approximated as massive C1 = (k q )2⊥+−m⊥2+x2M2 , (14)
transverse plasmons with mass m µ/√2. Then p, ⊥− ⊥ g
g
≈ k q
k and polarization ǫ(k) can be written in the light-cone C = ⊥− ⊥ . (15)
components, 2 (k⊥−q⊥)2+m2g+x2M2− 12−vvzzxM2
k2+m2 Different from the static medium case, the single and
k = [2ω, ⊥ g,k ], (3)
⊥ double Born scattering amplitudes depends on the col-
2ω
ǫ k lective flow of the medium. Our results agree with the
⊥ ⊥
ǫ(k) = [0,2 xE·+ ,ǫ⊥], (4) GLV results [6] in static medium at zero flow velocity.
In calculating radiation amplitude in Eqs.(8) and (9),
p2+M2
p = [(1 x)E++2vq, ⊥ ,p ], (5) compared to the static medium case we can see that the
− · (1 x)E++2vq ⊥
collective flow changes the poles of the propagator as
− ·
where ω =xE, E+ =2E µ. follows: the poles of the quark propagators are shifted
At zeroth order in opac≫ity, the jet has no interaction by a flow velocity-dependent term ωM and then times a
withthetargetparton,weobtainthesameradiationam- flow factor 1/(1 vz). The pole shift leads to that the
−
plitude as that in the static medium in Ref. [6], mass-dependent energy difference ωm in Ref. [6] shifts
to ω ω as shown in Eqs. (8) and (9). For the pro-
m M
k ǫ −
R(0) =2igT ⊥· ⊥ , (6) cesses that heavy quark jet interacts directly with the
ak2⊥+m2g+x2M2 target partons, the contribution of shifted poles of both
quark propagators cancels each other, the net contribu-
which indicates the depletion of radiation in the ”dead
tion keeps the same radiation amplitude as that in the
cone” [8] at angles
static medium except the phase factor, as shown by H
1
θ <θc1 = m2g+x2M2/(xE). (7) and C1 terms in Eqs. (8) and (9). From H1 and C1 in
q Eqs. (12) and(14)we canseeeasilythat these processes
Based on our new potential with the collective flow of have the dead cone θ in Eq. (7) which is same as that
c1
thequark-gluonmediuminEqs.(1)and(2),assumingthe inthestaticmedium. Fortheprocessesthatheavyquark
flowvelocity v 1,theradiationamplitudesassociated jet interacts with target partons via triple-gluon vertex,
| |≪
withasinglescatteringandthedoubleBornrescattering the contribution of shifted pole of the quark propaga-
with two scattering centers in the contact limit can be tor can’t cancel the contribution coming from the pole
expressed as of gluon propagator, the flow effects changes the radia-
tion amplitude as shown by H and C terms in Eqs.
R(S) =2ig(cid:16)C2(eiω10−−vωz1z10−eiω0+1ω−mv−zωMz10)[Tc,Ta] (8) and (9). From H2 and C2 i2n Eqs. (123) and (15) we
+H1eiω0+1ω−mv−zωMz10TcTa+H1(1−vz)2(e−i1ω−Mvzz10 cdaencrseeaesecsledaeralydtchoanteiθnc1thaesse processes the collective flow
−eiω0+1ω−mv−zωMz10)TaTc(cid:17)·ǫ⊥(1−vz), (8) θc2 =qm2g+x2M2−(2vzxM2)/(1−vz)/(xE). (16)
R(D) =2igTc CA(1+vz)2H2(eiω0+1ω−mv−zωMz10 1) Compared to the dead cone term M2/E2 coming from
(cid:16) 2 − the mass effects, the dead cone change 2v M2/[(1
+CR(1 vz)2H1(eiω0+1ω−mv−zωMz10 e−i1ω−Mvzz10) vz)xE2] coming from the flow effects is propzortional t−o
2 − −
1/x = E/ω, this term deduces significantly the mass-
CAC1(eiω0+1ω−mv−zωMz10 eiω0−1ω−1v−zωMz10) effect dead cone for high energy heavy quark jet. The
− 2 −
real dead cone with collective flow effects should be
−C2RH1eiω0+1ω−mv−zωMz10(cid:17)·ǫ⊥(1−vz)2, (9) θc =min[θc1,θc2]=θc2 ≥0. (17)
3

5.5
1.0 c quark 5.0 blig qhut aqrukark
b quark c quark
0.8 /E=0.2 4.5 E/=20
flowstatic cc 0.6 E/34..50
3.0
0.4 vz=0.065~0.13
2.5
0.2 2.0
1.5
0.0
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.05 0.10 0.15
FIG.1: Theratioofdeadconevzinthemediumwithandwith- FIG.2: Theenergylossofcharvmz ,bottomandlightquarksas
out flowasfunction offlowvelocity in thejet direction when function of flow velocity in the jet direction when E/µ=20,
ω/E =0.2 L/lg =5.
Fig.1 is the ratio between the dead cone in the medium in the presence of collective flow leads to that the ra-
with flow and that in the static medium as function of diated gluon formation time in some interference pro-
flow velocity vz as ω/E = 0.2. It shows that the dead cesses changes by replacing ωm with ωm−ωM as shown
cone decrease to zero rapidly with increasing v . The in the first term in Eq. (18), so that the flow effects
z
dead cone of bottom quark reduces more strongly than result in different gluon formation time for different in-
that of charm quark because of the heavier quark mass. terference processes. From Eq. (18) we see clearly three
From the radiation amplitude in Eqs.(8) and (9), the different formation time, τf1 = (1 vz)/(ω1+ωm), τf2 =
−
medium induced radiation probability to the first order (1 vz)/(ω1+ωm ωM), τf3 = (1 vz)/ωM. The gluon
− − −
in opacity can be expressed as formation factor should be averaged over the longitudi-
naltargetprofilewhichisdefinedas = dzρ(z) .
dP(1) C2 N dx d2k⊥ d2q⊥ ω We take the target distribution as ahn··e·xiponRential Ga·u·s·-
= P( )
dω 8πdAdRA⊥ Z x Z (2π)2 Z (2π)2 E sian form ρ(z)=exp( z/Le)/Le with Le =L/2.
−
As shown in Ref. [6], the scale of the hard scattering
[(1 v )2v2(q )] Tr R(S) 2+2Re R(0)†R(D)
− z ⊥ D h| | (cid:16) (cid:17)iE can be taken as Q2 = 4E2, the kinematic limits of the
α C L dx gluon’stransversemomentumareµ2 k2 4ωE. From
s R dk2 d2q P(x)v¯(q )2(1 v )2 ≤ ⊥ ≤
≈ 2π l Z x Z ⊥Z ⊥ | ⊥ | − z Eq. (18)wegettheinducedenergylosstothe firstorder
g
of the opacity as ∆E = dωωdP(1)/dω. Its numerical
(cid:16)(2C22−H1·C1−H1·H2)DRe(1−eiω1+1ω−mv−zωMz10)E results are shown in Fig.R2 for charm, bottom and light
quarks as E/µ = 20, L/l = 5. At zero flow velocity,
+(H21−H1·C1)DRe(1−eiω11−+vωzmz10)E the lightquarkenergylossgislargerthancharmandbot-
+2vzH1 C2 Re e−i1ω−Mvzz10(1 eiω11−+vωzmz10) tmoamssqueffarekctesnoerfghyelaovsysbqeucaaruksejeotfsthaes dsheaodwncoinnefRroefm. t[h6e]
· D (cid:16) − (cid:17)E
+2vz(H1·C2−H21)DRe(1−eiω11−+vωzmz10)E ilnossthiencsrteaatiscesmweditihumin.crHeoawsienvgert,hteheflohweavveyloqcuitayrkbeencaerugsye
−2vzH21DRe(1−eiω11−+vωzmz10)E(cid:17), (18) othfethmeeddieucmreawseithofcothlleecdtievaedflcoown.eTbhyethheeavflioewr tehffeecqtusarink
mass is, the more rapidly the heavy quark energy loss
whereC , N,LandA are,respectively,the Casmierin
2 ⊥ increases. The light quark energy loss decreases with in-
fundamental representation, the number, the thickness,
creasing the flow velocity as shown in our previous work
and the transverse area of the targets, l is the mean-
g [16]. In the region of 0.065<v <0.13, the energy loss of
free path of the gluon, α = g2/4π the strong coupling z
s light quark is a bit less than that of bottom quark but a
constant,P (x) P(x)/x=[1+(1 x)2]/xthesplitting
gq ≡ − bit larger than that of charm quark. In this region the
functionforq gq. v¯(q )2isdefinedasthenormalized
→ | ⊥ | average heavy and light quark energy loss is possible to
distribution of momentum transfer from the scattering
be nearly the same.
centers as shown in Ref. [16]. The opacity factor L/l
g ForA-Acollisionsatimpactparameterb,withrespect
reflectingthemeannumberofrescatteringinthemedium
tocollisionnumberthe averagev ofrescatteringcenters
z
arises from the sum over the N distinct targets.
intheQGPmediumbeforefreezeoutcanbeexpressedas
The gluon formation factor inside Re( ) in Eq.(18)
···
reflects the destructive interference of the non-Abelian d2r dφ dτ v cosφ+v sinφ ρσt t
LPM effect [20]. Comparing to the radiated gluon for- hvzi= R R 2π R d2(cid:0)rx dτρσt ty (cid:1) A B , (19)
mation time τstatic = 1/(ω +ω ) for heavy quark jet A B
f 1 m R R
goesthroughthestaticmediuminRef.[6],fromthegluon where φ is the angle between jet and x axis, v =
x
formationfactorinEq.(18)weseethattheradiatedgluon v (r+nτ b/2),v =v (r+nτ b/2)aretheflowve-
x y y
| − | | − |
formation time in the medium with collective flow con- locity of the expanding elliptic medium along the minor
tracts by a factor 1 v because of the moving of the andmajorsemi-axes,nistheunitvectoralongjetdirec-
z
−
target parton. The pole shift of the quark propagator tion. Partondensityρ=ρ(τ,b,r+nτ),nuclearthickness
4

9
bitlargerthanthatofcharmquark. Thedifferenceofthe
b quark
8 light quark effective average energy loss among three quarks is very
c quark
7 little,whichimpliesthatthelightandheavyquarkshave
V)
Ge 6 almostthesamesuppressionofhighp hadronspectrum
E>( T
< 5 for Au-Au collisions at RHIC energy.
4 Conclusion — In summary, based on new potential in
3 our previous work, we studied the flow effects on energy
2 loss of the heavy quark jets in the quark-gluon medium
withcollectiveflowandgiveanexplanationfortheheavy
1
10 20 30 40 50 60
quarkenergylosspuzzle. Thecollectiveflowchangesthe
E(GeV)
FIG. 3: The effective average energy loss of charm, bottom pole of the quark propagators, which lead to reduce sig-
andlightquarksasfunctionofjetenergyE inthepresenceof
nificantly the dead cone of radiative gluon and increase
collective flow for 0−10% central events of Au-Au collisions
obviously the energy loss of heavy quark jets. The col-
at RHICenergy.
lective flow also affect the gluon formation time because
functions t = t (r), t = t (r b). Cross section ofthemovingofthetargetpartonandtheshiftingofthe
A A B B
| | | − |
σ = C 2πα2/µ2 (C = 1 for qg and 9/4 for gg scatter- pole of the quark propagators. It has been shown that
a s a
ing)obtainedinpQCD[5,21],µ=g T(τ)attemperature in the region of 0.065 < v < 0.13, the light and heavy
s z
T(τ). Using the data from a full 3D ideal hydrodynamic quarkenergylossarenearlythesame. For0 10%central
−
simulations[17],weobtain v =0.08for0 10%central events of Au-Au collisions at RHIC energy, by using the
z
h i −
events of Au-Au collisions at RHIC energy, v is inside data from 3D ideal hydrodynamic simulations we obtain
z
h i
the region of 0.065<v <0.13 as shown in Fig. 2. thattheaveragevelocityofrescatteringcentersalongthe
z
The effective averageenergy loss of partonjet for A-A jet direction v =0.08,the difference of the effective av-
z
h i
collisions can be written as erage energy loss among the charm, bottom and light
quarks is very little. Our results shall have implications
∆E = d2r d2πφ dτ dωωdPdω(1)tAtB . (20) for comparisons between theory and experiment in the
h i R R dR2r Rdτt t future.
A B
R R We thank Fuqiang Wang, Xin-Nian Wang for help-
TheopacityL/l indP(1)/dωinEq.(18)canbeexpressed ful comments and Xiao-Fang Chen for helpful discus-
g
as L/l = dτσ(τ)ρ(τ,b,r+nτ). Fig.3 is the effective sions. ThisworkwassupportedbyNSFCofChinaunder
g
averageeneRrgylossofcharm,bottomandlightquarksas ProjectsNo. 10825523,No. 10635020andNo. 10875052,
functionofjetenergyE inthe presenceofcollectiveflow by MOE of China under Projects No. IRT0624, by
for0 10%centraleventsofAu-AucollisionsatRHICen- MOST of China under Project No. 2008CB317106;
−
ergy. It is shownthat the effective averageenergylossof and by MOE and SAFEA of China under Project No.
lightquarkis abit lessthanthatofbottomquark,but a PITDU-B08033.
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