Origin of Temperature of Quark-Gluon Plasma in Heavy Ion Collisions

0.09 MB·English

by  Xiao-Ming Xu

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Origin of Temperature of Quark-Gluon Plasma in Heavy Ion Collisions
Xiao-MingXu∗
5
Department of Physics, Shanghai University,
1
Baoshan, Shanghai 200444, China
0 ∗E-mail: [email protected]
2

n
Initiallyproducedquark-gluonmatteratRHICandLHCdoesnothaveatemperature. A
a
J quark-gluonplasmahasahightemperature. Fromthisquark-gluonmattertothequark-
gluon plasma is the early thermalization or the rapid creation of temperature. Elastic
6
three-partonscatteringplaysakeyroleintheprocess. Thetemperatureoriginatesfrom
1
thetwo-partonscattering,thethree-partonscattering,thefour-partonscatteringandso

forthinquark-gluonmatter.
]
h
Keywords:Originoftemperature;Earlythermalization;Elasticthree-partonscattering;
t
- Transportequation,Quark-gluonmatter.
l
c
Temperature is common in daily life, and people believe that temperature exists
u
n no matter how cold matter is. In gas and liquid temperature reflects the existence
[ of thermal states where all particles randomly moveand obey isotropic momentum

1 distributions. The averagemomentum that a massless particle possesses is propor-
v tional to temperature. When temperature equals zero, the average momentum is
0
zero,andvice versa. However,there is one systemwhere the averagemomentum is
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9 large but particles do not obey isotropic momentum distributions, i.e., the system
3 doesnothavetemperature. Thesystemisquark-gluonmatterthatwasproducedin
0
initial gold-gold nuclear collisions performed at the Brookhaven National Labora-
.
1 toryRelativisticHeavyIonCollider(RHIC)inthelastfourteenyears. Intheparton
0
system the parton distribution functions in the direction of the incoming nucleus
5
1 beam are much largerthan those in the direction perpendicular to the beam direc-
: tion. The distributions areanisotropicin momentum space, anddo not correspond
v
i to a thermal state. Now we must wonder if this quark-gluon matter always has no
X
temperature. The answeris no, andthis matter is exactly a place for exploringthe
r
origin of temperature.
a
The reaction gg gg was considered by Shuryak1 to cause gluon matter to
→
thermalize. He assumed that every gluon suffers from scattering once in the same
period of time. In order to guarantee validity of the assumption, the cross section
for the gluon-gluon scattering must be very large (empirically be larger than 40
mb). This is not achievable! Eventhough a thermalization time of the order of 0.3
fm/c was obtained, the assumption makes the thermalization time unreliable. To
explore the kinetic equilibration of gluon momenta, inelastic scattering gg ggg
→
was taken into account in Refs.2,3. gg gg and gg ggg lead to a gluon-matter
→ →
thermalizationtimelargerthan1fm/c. Overpopulationisfoundtobeakeyfeature
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of the Glasma (far-from-equilibrium gluon matter), and can strengthen the kinetic
equilibration. But solutions of a transport equation with the elastic gg scattering
show that the thermalization time of the Glasma is also larger than 1 fm/c4.
However, the kinetic equilibration, i.e., the establishment of a thermal state
takes a time less than 1 fm/c, as concluded from the elliptic flow data of hadrons
produced in Au+Au collisionsat √sNN =130 GeV atRHIC5 andthe correspond-
ingexplanationofhydrodynamiccalculationswhichassumeearlythermalizationof
quark-gluonmatterandidealrelativisticfluidflowofthe quark-gluonplasma6,7. It
was pointed out by Kolb et al. that quark-gluon matter begins to have a temper-
ature at 0.6 fm/c from the moment when quark-gluon matter is created in initial
Au+Au collisions. The early thermalization thus became the first RHIC puzzle8.
Even for Pb+Pb collisions at √sNN = 2.76 TeV at the Large Hadron Collider
(LHC), a good explanation of the elliptic flow data of hadrons9 in hydrodynamic
calculations requires that quark-gluon matter begins to have a temperature at 0.6
fm/c10. Therefore,the early thermalization, i.e., the rapid equilibration also exists
at LHC.
Toexplaintheearlythermalizationortofindtheoriginoftemperatureofquark-
gluon plasma, we think about the prominent feature of initially produced quark-
gluonmatter: thepartonnumberdensityisashighas32fm−3. Atsuchahighnum-
ber density the occurrence probability of parton-parton-parton scattering is larger
than0.211. Thefeatureleadstothatthethree-partonscatteringisimportantines-
tablishingthermalstatesofquark-gluonmatter,i.e.,creatinganinitialtemperature
of quark-gluonplasma12,13. For example, gluon-gluon-gluonscattering was discov-
eredto dominatethe evolutionofgluonmatter towardsthermalequilibrium12. We
draw the conclusions from transport equations which include the elastic parton-
parton scattering14,15 and the elastic scattering of gluon-gluon-gluon12, gluon-
gluon-quark, gluon-gluon-antiquark13, gluon-quark-quark, gluon-quark-antiquark,
gluon-antiquark-antiquark16, quark-quark-quark17, quark-quark-antiquark,quark-
antiquark-antiquark11,andantiquark-antiquark-antiquark. Thetransportequation
for gluon matter is
∂f 1 d3p d3p d3p
∂tg1 +~v1·∇~~rfg1 =−2E1 Z (2π)322E2(2π)323E3(2π)324E4(2π)4
g
δ4(p1+p2 p3 p4) G gg→gg 2 [fg1fg2(1+fg3)(1+fg4)
× − − n 2 |M |
fg3fg4(1+fg1)(1+fg2)]+gQ( gu→gu 2 + gd→gd 2
− |M | |M |
+|Mgu¯→gu¯ |2 +|Mgd¯→gd¯ |2)[fg1fq2(1+fg3)(1−fq4)
f f (1+f )(1 f )]
g3 q4 g1 q2
− − }
1 d3p d3p d3p d3p d3p
2 3 4 5 6
−2E1 Z (2π)32E2(2π)32E3(2π)32E4(2π)32E5(2π)32E6
g2
(2π)4δ4(p1+p2+p3 p4 p5 p6) G ggg→ggg 2 [fg1fg2fg3
× − − − (cid:26)12 |M |
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(1+f )(1+f )(1+f ) f f f (1+f )(1+f )(1+f )]
g4 g5 g6 g4 g5 g6 g1 g2 g3
× −
g g
+ G2Q(|Mggu→ggu |2 +|Mggd→ggd |2 +|Mggu¯→ggu¯ |2 +|Mggd¯→ggd¯ |2)
[f f f (1+f )(1+f )(1 f ) f f f (1+f )(1+f )
g1 g2 q3 g4 g5 q6 g4 g5 q6 g1 g2
× − −
1 1
×(1−fq3)]+gQ2[4 |Mguu→guu |2 +2(|Mgud→gud |2 +|Mgdu→gdu |2)
1
+4 |Mgdd→gdd |2 +|Mguu¯→guu¯ |2 +|Mgud¯→gud¯ |2 +|Mgdu¯→gdu¯ |2
1 1
+|Mgdd¯→gdd¯ |2 +4 |Mgu¯u¯→gu¯u¯ |2 +2(|Mgu¯d¯→gu¯d¯ |2 +|Mgd¯u¯→gd¯u¯ |2)
1
+4 |Mgd¯d¯→gd¯d¯ |2][fg1fq2fq3(1+fg4)(1−fq5)(1−fq6)−fg4fq5fq6
(1+f )(1 f )(1 f )] ,
g1 q2 q3
× − − }
(1)
and the transport equation for up-quark matter is
∂f 1 d3p d3p d3p
∂qt1 +~v1·∇~~rfq1 =−2E1 Z (2π)322E2(2π)323E3(2π)324E4(2π)4
δ4(p1+p2 p3 p4) gG ug→ug 2 [fq1fg2(1 fq3)(1+fg4)
× − − |M | −
(cid:8) 1
fq3fg4(1 fq1)(1+fg2)]+gQ( uu→uu 2 + ud→ud 2
− − 2 |M | |M |
+|Muu¯→uu¯ |2 +|Mud¯→ud¯ |2)[fq1fq2(1−fq3)(1−fq4)
f f (1 f )(1 f )]
q3 q4 q1 q2
− − − }
1 d3p d3p d3p d3p d3p
2 3 4 5 6
−2E1 Z (2π)32E2(2π)32E3(2π)32E4(2π)32E5(2π)32E6
g2
(2π)4δ4(p1+p2+p3 p4 p5 p6) G ugg→ugg 2
× − − − (cid:26) 4 |M |
[f f f (1 f )(1+f )(1+f ) f f f (1 f )(1+f )
q1 g2 g3 q4 g5 g6 q4 g5 g6 q1 g2
× − − −
1
(1+fg3)]+gQgG( uug→uug 2 + udg→udg 2 + uu¯g→uu¯g 2
× 2 |M | |M | |M |
+|Mud¯g→ud¯g |2)[fq1fq2fg3(1−fq4)(1−fq5)(1+fg6)−fq4fq5fg6(1−fq1)
1 1
×(1−fq2)(1+fg3)]+gQ2[12 |Muuu→uuu |2 +4(|Muud→uud |2
1 1 1
+|Mudu→udu |2)+ 4 |Mudd→udd |2 +2 |Muuu¯→uuu¯ |2 +2 |Muud¯→uud¯ |2
1 1
+|Mudu¯→udu¯ |2 +|Mudd¯→udd¯ |2 +4 |Muu¯u¯→uu¯u¯ |2 +2(|Muu¯d¯→uu¯d¯ |2
1
+|Mud¯u¯→ud¯u¯ |2)+ 4 |Mud¯d¯→ud¯d¯ |2][fq1fq2fq3(1−fq4)(1−fq5)(1−fq6)
f f f (1 f )(1 f )(1 f )] ,
q4 q5 q6 q1 q2 q3
− − − − }
(2)
wheremasslesspartonshavethevelocityvector~v andthepositionvector~r;g and
1 G
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g are the spin-color degeneracy factors for the gluon and the quark, respectively;
Q
the distributionfunctions forthe gluon,the upquark,thedownquark,theupanti-
quark,andthedownantiquarkaredenotedbyfgi,fui,fdi,fu¯i,andfd¯i,respectively;
fui = fdi = fu¯i = fd¯i = fqi. Mijk→ijk of which the subscript, i, j, or k, stands for
a gluon, a quark, or an antiquark is the amplitude for the elastic parton-parton-
parton scattering. Squared amplitudes were calculated in perturbative QCD with
different number of Feynman diagrams for different types of scattering18. When
the squared four-momenta of gluon and/or quark propagators approach zero, the
divergences of the squared amplitudes for the elastic two-parton scattering and
the elastic three-parton scattering are encountered. The divergences are removed
while the propagators are regularized by a screening mass which is evaluated from
the distribution functions by a formula in Refs.19–21. It can be proved from the
transport equations that quark-gluonmatter in thermal nonequilibrium eventually
arrives at thermal equilibrium22. The larger the squared amplitudes, the shorter
the thermalization time of quark-gluonmatter23.
Every time the elastic two-parton scattering occurs, the two partons’ momenta
change but the scattering is limited to a plane. Due to the energy-momentumcon-
servation only two of the six momentum components of the two final partons are
free, i.e., angular distributions of the outgoing partons can be defined. The proba-
bility of producing two partons in an incoming parton direction is larger than the
oneinthedirectionperpendiculartotheincomingpartondirection. Ininitiallypro-
duced quark-gluonmatter more partons are in the gold-beamdirection than in the
directionperpendiculartothebeamdirection. Onthe onehandthecollisionoftwo
partons in the beam direction reduces partons in the beam direction and increases
partonsinthedirectionperpendiculartothebeamdirection,ontheotherhandthe
collisionoftwopartonsinthedirectionperpendiculartothebeamdirectionreduces
partons in the direction perpendicular to the beam direction and increases partons
in the beam direction. Since more partons are in the beam direction, the scatter-
ing reduces parton number in the beam direction and increases parton number in
the directionperpendiculartothebeamdirection. Partonmomentumdistributions
gradually become isotropic in such a manner via the elastic two-partonscattering.
In the elastic three-parton scattering the three initial partons are generally not
in a plane, so are the three final partons. Every time the elastic three-parton scat-
tering occurs, the three partons’ momenta change. Even restricted by the energy-
momentum conservation,five of the nine momentum components of the three final
partonsarefree. Thatthepartonmomentumdistributionsbecomeisotropicisthus
expected to be easy via the elastic three-parton scattering. Feynman diagrams ex-
hibit the following processes: a virtual parton breaks into three final partons, two
final partons, or one final parton and a virtual parton; a virtual parton combines
withaninitialpartontoformafinalparton;avirtualpartonscatterswithaninitial
parton to produce three final partons, two final partons, or one final parton and a
virtual parton; a virtual parton coming from the scattering of two initial partons
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does not stay in the plane of the two initial partons. Due to these processes the
three final partons are generally not in a plane.
Eventually, isotropic parton momentum distributions are established by the
three-partonscatteringandthetwo-partonscattering. Withanisotropicmomentum
distributions of partons in quark-gluonmatter created in central Au+Au collisions
at √sNN = 200 GeV at t = 0.2 fm/c24, the transport equations yield isotropic
momentum distributions of gluons first at t = 0.52 fm/c and of quarks first at
t=0.86fm/c. Thethermalizationtimesofgluonmatterandquarkmatterare0.32
fm/c and 0.66 fm/c, respectively. The temperature acquired is not close to zero,
and is much higher than the QCD critical temperature 0.175 GeV. Therefore, the
early thermalization is a consequence of the two-parton scattering and the three-
parton scattering. Since the four-parton scattering, the five-parton scattering, etc.
have certain occurrence probabilities11, they also contribute to the early thermal-
ization. The temperature of the quark-gluon plasma comes from the elastic and
inelastic two-parton scattering, the three-parton scattering, and the scattering of
more partons.
Acknowledgments
I thankX.-J.Qiuforfruitful discussions. This workwassupportedbythe National
Natural Science Foundation of China under Grant No. 11175111.
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