How Wide is the Transition to Deconfinement?

0.42 MB·English

by  Adrian Dumitru

BNL-94342-2010-JA, KUNS-2310, NIKHEF 2010-040, RBRC-871
How Wide is the Transition to Deconfinement?
Adrian Dumitrua,b, Yun Guoc,d, Yoshimasa Hidakae, Christiaan P. Korthals Altesf,g, and Robert D. Pisarskih
aDepartment of Natural Sciences, Baruch College,
17 Lexington Avenue, New York, NY 10010, USA
bRIKEN/BNL Research Center,
Brookhaven National Laboratory, Upton, NY 11973, USA
cDepartment of Physics, Brandon University,
Brandon, Manitoba, R7A 6A9 Canada
dWinnipeg Institute for Theoretical Physics,
Winnipeg, Manitoba, Canada
eDepartment of Physics, Kyoto University,
Sakyo-ku, Kyoto 606-8502, Japan
fCentre Physique Th´eorique au CNRS Case 907,
Campus de Luminy F-13288 Marseille, France
gNikhef Theory Group, Science Park 105,
1
1098 XG Amsterdam, The Netherlands
1
0 hDepartment of Physics,
2 Brookhaven National Laboratory, Upton, NY 11973

n
(Dated: January 20, 2011)
a
J Pure SU(3) glue theories exhibit a deconfining phase transition at a nonzero temperature, T .
c
9 Using lattice measurements of the pressure, we develop a simple matrix model to describe the
1 transition region, when T ≥ T . This model, which involves three parameters, is used to compute
c
the behavior of the ’t Hooft loop. There is a Higgs phase in this region, where off diagonal color

] modes are heavy, and diagonal modes are light. Lattice measurements of the latter suggests that
h
thetransitionregionisnarrow,extendingonlytoabout∼1.2T . Thisisinstarkcontrasttolattice
p c
measurementsoftherenormalizedPolyakovloop,whichindicatesamuchwiderwidth. Thepossible
-
p implications for the differences in heavy ion collisions between RHIC and the LHC are discussed.
e
h PACSnumbers: 12.38.Mh,12.38.Gc,25.75.Nq
[

2
Heavy ion collisions at the Relativistic Heavy Ion Col- coupling is moderate even down to the transition tem-
v
lider (RHIC) have demonstrated a rich variety of unex- perature, T [8]. We use an elementary matrix model,
0 c
pected behavior [1]. Notably, in peripheral collisions the involving three parameters, to parametrize the behav-
2
8 elliptical flow can only be described by nearly ideal hy- ior of the deconfining phase transition. A version of
3 drodynamics, with a very small ratio between the shear this model with one parameter was first proposed by
. viscosity, η, and the entropy density, s. The differences Meisinger, Miller, and Ogilvie [9]. Similar models arise
1
1 betweencollisionsatRHIC,andthosewhichwillsoonbe for theories in which one (or more) spatial directions are
0 observed soon at the Large Hadron Collider (LHC), will of femtoscale size [10–13].
1 be especially interesting: does the nearly ideal hydrody-
The parameters of the model are fixed from lattice
: namic behavior, observed at RHIC, persist at the much
v measurements of the pressure [14–17]. It then predicts
i higher energies of the LHC? how the ’t Hooft loop [18–22] changes with temperature
X
One approach to deconfinement exploits the analogy near T , which we compare to the results of lattice sim-
r c
a to N = 4 supersymmetric gauge theories: using the ulations [23, 24]. Further, the model predicts that for a
AdS/CFTcorrespondence, suchtheoriesarecomputable range of temperatures above T , there is a Higgs phase,
c
analyticallyinthelimitofinfinitecoupling,foraninfinite wherecorrelationfunctionsofelectricfieldsareamixture
number of colors [2]. By introducing a potential for the ofheavyandlightmodes,fromfieldswhichareoffdiago-
dilation field, the behavior of the entropy density near nal,anddiagonal,incolor,respectively. Thismayhelpto
the deconfining phase transition, at a temperature T , understand the results of lattice simulations [15, 25–27],
c
can be fit from measurements on the lattice [3–6]. While which are otherwise somewhat puzzling.
the entropy density, s, decreases strongly as T → T+
c The most direct prediction of our model is for the ex-
because it is related to Hawking radiation, in AdS/CFT
pectation value of the Polyakov loop. For the pure glue
models the ratio η/s remains completely independent of
SU(3)theory,latticesimulationsfindthatthe(renormal-
temperature. This suggests that like RHIC, that colli-
ized) Polyakov loop vanishes below T , jumps to ∼ 0.4
sionsattheLHCshouldalsobedescribedbynearlyideal c
at T+, and then rises with T, until it is approximately
hydrodynamics; see, also, Ref. [7]. c
constantabove∼4.0T [28–30]. Thisrepresentsconfine-
c
In this work we consider a very different approach to ment below T , a complete Quark Gluon Plasma (QGP)
c
the deconfining phase transition. It assumes that the athightemperature, anda“semi”-QGPinbetween[31–
2
34]. Physically, there is no ionization of color in the con- physical quantities. Indeed, the grand canonical ensem-
fined phase, total ionization in the complete QGP, and ble is characterized by a fixed value for the chemical po-
only partial ionization in the semi-QGP [34]. (While we tential, and not by an integral over µ’s.
discuss a purely gluonic plasma, we adopt the common Nevertheless, precisely this mechanism arises for the
term QGP.) deconfining phase transition in a SU(N) gauge theory.
Theprincipalthrustofthispaperisthatfromindirect Consider the expansion about a background field for the
measurements on the lattice, we suggest that the width time-like component of the vector potential,
of the semi-QGP is much narrower than indicated by
present results for the renormalized Polyakov loop: not (cid:0)Acl(cid:1) = 2πT q δab ; (3)
up to ∼4.0T , but only ∼1.2T . We do not understand 0 ab g a
c c
this discrepancy in detail, but suggest a possible reason
later. This discrepancy is the reason why, having fit the a and b are colors indices, running from 1...N. For
parameters of our model from the pressure, we compute nonzeroqa’s,thisbackgroundfieldactslikeanimaginary
both the ’t Hooft loop and gluon masses. chemicalpotentialforthediagonalelementsofthegauge
group. Integration over the q ’s arises from imposing
While we treat the pure glue theory, our model can a
Gauss’ law for those elements of the gauge group [19].
be extended to QCD, with dynamical quarks [17]. It is
reasonable to assume that in QCD, the semi-QGP is like This background field generates a non-trivial expecta-
that of the pure glue theory, relatively narrow. We thus tion for the Polyakov loop, (cid:96), which is the color trace of
concludebydiscussingthepossiblephenomenologicalim- the thermal Wilson line, L:
plications of our results for heavy ion collisions.
(cid:32) (cid:33)
How confinement arises in our model can be under- 1 (cid:90) 1/T
(cid:96)= trL ; L=Pexp ig A dτ ; (4)
stoodbyanalogy. Forabosonicfield,withenergyE and N 0
0
chemical potential µ, the Bose-Einstein statistical distri-
bution function is P represents path ordering, T is the temperature, and τ
the imaginary time, τ :0→1/T.
1 1
n(E,q)= = . (1) ThelatticedemonstratesthatnearT ,theexpectation
c
e(E−µ)/T −1 eE/T−2πiq−1
valueofthePolyakovloopisnotnearone, anddecreases
asthetemperaturedoes. Insucharegion,theeigenvalues
Instead of taking µ to be real, as in ordinary thermody-
of the logarithm of the thermal Wilson line are nonzero.
namicalsystems,forthepurposesoftheanalogywetake
TakinganansatzsuchasEq. (3)isthesimplestpossible
it to be purely imaginary, and define µ=2πT iq, where
way to model this. We do not attempt to derive the
q is real.
distribution of these eigenvalues, but to guess that from
Doing so, n(E,q) is clearly a periodic function of q,
lattice results.
invariant under q →q+1. Thus we can choose to define
Since the gauge potential A is an element of SU(N),
q to lie within the range from −1 to +1. 0
2 2 (cid:80)N q = 0, modulo one, there are N −1 independent
Now assume that we integrate over q, with a distri- a=1 a
q ’s. At infinite N, the q ’s form a continuum, and the
bution which is flat in q. Expand for large energy, so a a
example of Eq. (1) is exact; see, e.g., computations on
that the first term is the Boltzmann statistical distribu-
a femtosphere at N = ∞ [13]. For two colors, we can
tion function. Given the assumed distribution in q, the
choose the eigenvalues to be q = −q ; for three, q =
integral of this term vanishes, 1 2 1
−q , and q =0.
2 3
(cid:90) +1/2 In the presence of the background field of Eq. (3), a
e−E/T e2πiq dq =0. (2) potential for the q ’s is generated at one loop order [18–
a
−1/2 22],
Indeed, we can expand the Bose-Einstein distribution
function term by term in powers of Boltzmann factors, V (q ) = 2π2T4 (cid:88)N q2 (1−|q |)2−(N2−1)π2T4 .
e−E/T+2πiq [13]; doing so, the integral over each and ev- pt a 3 ab ab 45
a,b=1
ery term obviously vanishes. The same is true for the (5)
Fermi-Dirac distribution function as well. where q =q −q , defined modulo one. The minimum
ab a b
Thus a flat distribution in q represents the confined is at q = 0, where −V (0) is the pressure for an ideal
a pt
phase. To represent a phase with partial deconfinement, gas of gluons.
one integrates over a limited region, say q : −q0 → +q0, The potential Vpt(qa) enters in computations of the ’t
with q0 < 21. Complete deconfinement occurs when one Hooft loop. It is useful to consider deconfinement as a
integratesoveradistributionwhichisadelta-functionin type of spin system. A pure SU(N) gauge theory has
q. N degenerate vacua, where the thermal Wilson line L
Thisexampleappearssomewhatartificial. Foragiven equals one of the N roots of unity,
q, the statistical distribution functions are complex val-
ued, and so only integrals over q can possibly represent L=e2πij/N 1, (6)
3
j = 0...(N −1). The usual vacuum, with j = 0 and one q :
ab
L = 1, corresponds to all q = 0. A Z(N) vacua with
a
j =1andL=e2πi/N1correspondstoN−1qa’s=1/N, V (q )=T2 T2 (cid:88)N (c |q |(1−|q |)
and the remaining element =−1+1/N. non a c 1 ab ab
a,b=1
At high temperature in the complete QGP, the the-
ory lies in one spin state, which we can choose to be
(cid:17)
j = 0. One can compute tunneling between two degen- +c q2 (1−|q |)2+c . (7)
2 ab ab 3
erate vacua by constructing a box which is long in one
spatial direction, with j = 0 at one end, and j = 1 at The model of Ref. [9] involves terms ∼ c and c ; these
1 3
the other. An interface between the two ordered states are kept in fixed ratio, given by the second Bernoulli
forms in the center of the box, with the interface tension polynomial. Instead,weallowc andc tovaryindepen-
1 3
between the two computable semi-classically, using the dently. This helps to avoid a pathology of the model of
potential V (q ) [18–22]. This interface is equivalent to Ref. ([9]), where the pressure is negative at T .
pt a c
a’tHooftloopwhichwrapsaroundthecenterofthebox We also introduce a term ∼ c q2 (1 − q )2; this is
2 ab ab
[20]. proportional to the perturbative term in Eq. (5), and is
related to the fourth Bernoulli polynomial.
As the temperature decreases and T approaches T ,
c We take all of the non-perturbative terms to be ∼T2,
domainswithj (cid:54)=0formandgrowinsize. Theybecome
since lattice simulations indicate that in the pure glue
increasingly probable, until at T− and below, as a spin
c theory, the leading corrections to terms ∼ T4 are ∼ T2
system the vacuum is completely disordered, a sum over
[9, 10, 33, 37]. There is obviously no fundamental rea-
many spin domains.
son why other terms, such as those ∼ 1, could not also
We want to add terms to the effective potential which appear.
model the transition to deconfinement. We could add When the q ’s develop an expectation value, this rep-
a
perturbative corrections to V (q ), which have been resentssymmetrybreakingforanadjointscalarfield,A ,
pt a 0
computedto∼g3[21],butinvariablytheygiveq =0(or coupled to an SU(N) gauge field, the A ’s [33]. As an
a i
a Z(N) equivalent state) as the vacuum. With a com- adjoint scalar, though, there is no strict order parameter
plete theory, such as the monopole model of Liao and which distinguishes between the symmetric and broken
Shuryak [7], this potential could be computed directly. phases. Thus there need not be a phase transition in go-
ingfromthesymmetricphase,thecompleteQGP,tothe
We adopt a more modest approach, attempting to
“broken” phase, the semi-QGP.
guess the form of the non-perturbative potential. We fit
If there were such a phase transition, it would repre-
thecoefficientswhichentertolatticeresultsforthepres-
sent a second transition, above T , separate from that
sure, and then use it to compute other quantities. The c
for deconfinement. While possible, in a pure SU(N)
advantageofourapproachisthatwecancomputequan-
gauge theory lattice simulations only find evidence for
tities not just in, but near thermal equilibrium. Such
one phase transition, at T [14–17]. To avoid a phase
quantities, like the shear viscosity [34], are much harder c
transition between the complete and semi-QGP, it is es-
to extract on the lattice.
sential that the non-perturbative potential has a term
Since the Polyakov loop is an order parameter for de- which is linear in the q ’s. Assume that the effective po-
a
confinement,anaturalguessisthatthenon-perturbative tential only involved terms such as ∼ qn (1−q )n for
ab ab
potential involves Z(N) invariant elements of the Lie n ≥ 2. For small q , these are of quadratic or higher
a
group. Thefirstsuchtermistheadjointloop[12,31–36]. order in the q ’s, and of necessity, there would then be a
a
Instead, following the authors of Ref. [9], and computa- phase transition when the q ’s developed a nonzero ex-
a
tions of the ’t Hooft loop [18–22], we write a potential pectation value. This transition might be of either first
which is a polynomial in the q ’s. or second order, but there would be a phase transition.
a
When c (cid:54)= 0, though, a term linear in the q ’s ensures
There are several symmetries which any potential of 1 a
that there is no such phase transition. Instead, even for
the q ’s must satisfy. It must be periodic in each q ,
a a high temperature, there is always a small but non-zero
with q →q +1. It must also be invariant under Z(N)
a a expectation value for the q ’s, < q >∼ 1/T2; that is,
transformations,whereN−1oftheq ’sshiftby1/N,and a a
a the theory is always in a Higgsed phase. As we shall see
the last element, by −1+1/N. Lastly, if we interchange
however, this point is somewhat academic. For the pa-
the ordering of the q ’s, we can change q → q =
a ab ba rameters relevant to two and three colors, the region in
−q . These symmetries can be satisfied by constructing
ab which Higgsing matters is very narrow.
a potential as a function of q (1−q ).
ab ab We remark that effective theories on the lattice often
We can still form an infinite number of terms by tying exhibitphaseswithbrokensymmetry[38]. Thenecessity
together the color indices in different ways; see, e.g., the of such a broken phase near T does not seem to have
c
examplesattwo[21]andthree[13]looporder. Weadopt been appreciated previously, though.
the simplest approach, and take terms like those which Todeterminetheparametersofthemodelwecompare
ariseatonelooporder,Eq. (5),whichinvolveasumover tolatticemeasurementsofthepressure. Forthreecolors,
4
thisisillustratedinFig. (1);fortwocolors,inFig. (2). If where
p(T) is the pressure, and e(T) the energy density, then a
moresensitivetestofthefitisalsotoplottheinteraction ξ(g2)=1−0.16459g2(T).
measure, ∆=e−3p. Thus in each figure we plot p/T4,
e/T4, and ∆/T4, both from the lattice, from Ref. ([14]) The factors involving Tc/T are special to the semi-QGP,
so that as T (cid:29) T , the result reduces to that in the
for two colors, and from Ref. ([15]) for three colors. c
complete QGP [18]. The function ξ(g2) is the correction
The parameters of the fit are
∼g2 in the complete QGP; in plotting, we take g2(2πT)
c =−.41488; c =−5.45957; c =0.21954. (8) [8].
1 2 3
The’tHooftloopvanishesatT ,asexpectedforasec-
c
for three colors, and ond order phase transition. From universality, the result
in the Ising model is σ(T)∼(T −T )2ν, with 2ν ∼1.26;
c
c1 =−0.30267; c2 =−5.97440; c3 =0.18341. (9) lattice results in a gauge theory [23] find 2ν ∼1.32 [23].
Our result, 2ν =1.5, is not too far off, as expected for a
for two colors.
meanfieldtheory. Wenote,however,thatbecauseofthe
While our model appears to involve three parameters,
term in the denominator, that the numerical agreement
this is misleading. One parameter fixes the critical tem-
isn’t close. This is presumably related to the unphys-
perature, T . A second is chosen so that the pressure
c ical behavior of the energy density near T , mentioned
c
vanishesatT . Thuswereallyhaveonlyonefreeparam-
c previously.
eter, which is tuned to fit the behavior of the pressure
Forthreecolors,inthesemi-QGPthevacuaisalongλ
3
near T .
c (using the Gell-Mann notation), while the path for the
For two colors, our model exhibits unphysical behav-
’t Hooft loop depends upon a change in λ . The path
8
ior, as the energy density is negative below ∼ 1% of T .
c was determined numerically, and lies along both λ and
3
This might be corrected by adding further terms in the
λ . Theactionofthetunnelingpathwasalsodetermined
8
nonperturbativepotential,suchashigherBernoullipoly-
numerically,andtheresultforthe’tHooftloopforthree
nomials.
colors is illustrated in Fig. (3). (For N = 2, we take
In any case, since we fix the pressure to vanish at T ,
within our approximations the confined phase has vanc- Tc/ΛM¯S = 1.31; for N = 3, 1.14. For the same value of
ishing pressure. How to match to a more realistic de- Tc/ΛM¯S =1.31, the results unexpectedly coincide.)
Including ξ(g2), the semi-classical computation of the
scription of the confined phase is an important problem,
’t Hooft loop in the complete QGP agrees with lattice
which we defer for now.
simulationsabove∼4.0T ;belowthattemperature,they
c
Given the effective Lagrangian, it is then straightfor-
agree with the result in the semi-QGP [24]. To obtain
wardtocomputethe’tHooftloop. InthecompleteQGP,
agreement,however,itisnecessarytoincludethecorrec-
the potential includes only the perturbative potential,
tion ξ(g2); this is computed in the complete QGP, which
V (q ),Eq. (5);inthesemi-QGP,itisasumofthisand
pt a isincorrect. Twothingsarerequiredtocomputeξ(g2)in
the non-perturbative potential, V (q ).
non a the semi-QGP. First, the potential for constant q needs
a
For two colors, as q =−q there is only one indepen-
2 1 to be computed to two loop order, expanding about the
dent direction, and it is direct to compute the tunneling
full potential, V (q )+V (q ). Second, corrections to
pt a non a
path, and its associated action, analytically. The result
onelooporderneedtobecomputedforthekineticterm.
for the ’t Hooft loop is
In the complete QGP this brings in new functions, the
ψ(q ) [18]. Other functions could arise in the semi-QGP.
4π2T2 (1−(T /T)2)3/2 a
σ(T)= ξ(g2) c , (10) Fornow,wedefertheseinvolvedcomputations; sincethe
3(cid:112)6g2(T) 1−0.908(Tc/T)2
1.5
3
2.5 1
2
1.5 0.5
1
0
0.5
p/T4 e/3T4 (cid:54)/T4
0 p/T4 e/3T4 (cid:54)/T4 Latt. p/T4 Latt. e/3T4 Latt. (cid:54)/T4
-0.5 1Latt. p/T4 1.5 Latt. e/3T4 2 Latt .2 (cid:54).5/T4 3 -0.5 1 1.5 T / 2TC 2.5 3
T / T
C
FIG. 2: Comparison of lattice results for SU(2) pure gauge
FIG. 1: Comparison of lattice results for SU(3) pure gauge theory to the model, for the pressure, energy density, and
tothemodel,forthepressure,energydensity,andinteraction interaction measure.
measure.
5
corrections ∼g2 are large, ∼50%, our results should be Theshiftintheenergy,p =2πTn→2πT(n+q −q ),is
0 a b
considered as tentative. becauseweareexpandingaboutabackgroundfield. The
Besides the ’t Hooft loop, which is an interface ten- backgroundfieldAcl actsuponquantumfluctuationslike
0
sion for an order-order interface at T ≥T , the interface an adjoint Higgs field. Because the field is diagonal in
c
tensionfortheorder-disorderinterface,atT ,isalsocom- color, Eq. (3), diagonalfieldsdonotfeelthebackground
c
putable in our model. This only exists for a first order field. Thusfordiagonalfields,theonlymasstheydevelop
transition; for three colors, is the Debye mass, m . This is of order ∼ gT times a
D
function of the q ’s [34]. In contrast, off diagonal fields
T2 a
σ =0.0258012 c . (11) have non-trivial commutators with a diagonal field, and
dis (cid:112)
g2 so they develop “masses” which are large, ∼ 2πT(n+
q −q ).
a b
It is necessary to compute the corrections ∼ g2 before
We illustrate this in Fig. (4) for three colors. The
comparing to lattice data, though.
massesofthetwodiagonalgluonsareequal,anddecrease
The parameters for three colors, Eq. (8), and two col- as T → T+. There are two types of off-diagonal gluons:
ors, Eq. (9), are similar; the difference is commensurate c
four with |a−b| = 1, and two with |a−b| = 2. The
with a dependence on ∼ 1/N2, with the coefficient of
splitting of the masses is evident only close to T , for
c
order one. We have then assumed that the parameters
T <1.2T .
c
for three colors are close to those for higher N. We find
We do not plot lattice data, because it is somewhat
reasonable agreement for the interaction measure to lat-
contradictory. Latticemeasurementsofagaugeinvariant
tice results [16]. When N ≥ 4, there is more than one
quantity,thetwopointfunctionbetweenPolyakovloops,
’t Hooft loop. Lattice simulations find that they obey
showsthattheassociatedmassdecreasesasT →T+[15].
Casimirscalingtogoodapproximation[24]. Wehavenot c
Incontrast,thetwopointfunctionofgluonsindicatethat
checked this explicitly, but suspect that in our model, ’t
the gauge dependent mass increases as T → T+ [27].
Hooft loops respect Casimir scaling. c
Clearly it would be best to reanalyze the lattice data
The most novel prediction of our model is that there
with a Higgsed propagator in the effective theory, Eq.
is a Higgs effect in the semi-QGP. This was noted first
(13), with its characteristic combination of modes whose
in Ref. [33], and in theories at a femtoscale [12]. To
masses both increase and decrease.
understand it, consider the quantum fluctuations about
The static, spatial gluon fields, the A , also undergo a
the background field of Eq. (3): i
Higgs effect. This happens as well in a monopole gas [7].
(cid:10)(Aqu) ((cid:126)x) (Aqu) (0)(cid:11)∼(cid:90) d3p eip(cid:126)·(cid:126)x (cid:88)+∞ ∆ ouWr mehodaveel:ntohtedcisocmuspsuetdattihoenmoofstthoebPviooluysakaopvplliocoapti.onWoef
0 ab 0 ba (2π)3 00
n=−∞ plotthisquantity,andthelatticeresults,forthreecolors
(12) in Fig. (5). A direct comparison of the two is somewhat
where ∆00 is the quantum propagator misleading. We have not computed perturbative correc-
tions to the Polyakov loop, which enter at ∼ g3 [39].
e−ip0τ
∆ = ; p =2πT (n+q −q ) . This contribution is positive, and will increase the re-
00 (p(cid:126))2+p2+m2(q) 0 a b
0 D sult. Nevertheless, while the two coincide at Tc — which
(13)
is presumably coincidence — they immediately diverge
from one another. From Fig. (5), in our model the loop
quickly goes up to a constant value by ∼ 1.2T ; this is
c
2.5
) 2
1
N- 1.5 1.8
( 1.6 a=b
2/T / (cid:109) 0. 51 mmLooddaeettlli,,c SSeG UUdK((a23Ata)) m / gT 011. ..8241 ||aa--bb||==12
0 0.6
1 1.5 2 2.5 3 3.5 4 4.5 5 0.4
T / TC 0.2
0
1 1.2 1.4 1.6 1.8 2
FIG.3: The’tHooftloopforSU(3)puregaugetheory: lattice T / TC
data from Ref. [24], and GKA, Giovannangeli and Korthals
Altes, Ref. [21], the semi-classical computation in the com-
plete QGP, including corrections of ∼ g2. In our model we FIG. 4: Gluon masses, mab/(gT), for SU(3) pure gauge the-
show results for two and three colors, assuming that the cor- ory: diagonal gluons, with a = b, are light; while there are
rectionsof∼g2 areidenticalinthecompleteandsemi-QGP; twooffdiagonalgluonswithheavymasses,for|a−b|=1and
see text. |a−b|=2.
6
very different from lattice measurements, for which it is system to probe the q-dependence of the potential in a
not constant until a much higher temperature, ∼ 4.0T nontrivial way. In the present model, these are deter-
c
[28–30]. mined by the coefficients for c and c . (As a constant
1 2
If our model is correct, why does the value of the term, the value of c doesn’t matter.) In this way the
3
Polyakovloop,computedfromourmodel,differsosignif- lattice measurements of the ’t Hooft loop [23, 24] are an
icantly from lattice measurements of the (renormalized) absolutely essential constraint on our model.
Polyakov loop? There is an ambiguity associated with In contrast, near T all of the parameters of the model
c
the renormalized Polyakov loop, from the zero point en- matter, contribute to both the behavior of the pressure
ergy. InRef. [34]wearguedthatperturbatively,thezero and the value of the q ’s. By comparison, consider the
a
point energy vanishes for a straight Polyakov loop. This model of Ref. ([9]), where c = 0. We take two colors,
2
argument fails for a “smeared” loop (see, e.g., the ap- sincethentheexpressionsarealgebraicallysimple. From
pendix of Ref. [29]). If so, then the effects of smearing Ref. ([9]), with q =q =−q ,
1 2
must be very dramatic.
 (cid:115) 
We comment that a similar rapid growth in the 1 (cid:18)T (cid:19)2
Polyakov loop is found in solutions of the Schwinger- qc2=0 = 41− 1− Tc  . (14)
Dyson equations [40]. Our results do not coincide nu-
merically, though.
When c = 0, though, the interaction measure is much
To understand our results better, consider first the 2
broader than indicated by the lattice data.
limit of intermediate temperature: say above ∼ 1.5T ,
c In the present model, with c ,c (cid:54)=0,
and up to ∼ 4.0T . In this limit the q ’s are small. 1 2
c a
As noted before, for small q the dominant terms are (cid:32) (cid:115) (cid:33)
1 2c
∼ c1T2q in the nonperturbative potential, Eq. (7), and q = 1− 1+ 1 . (15)
that ∼T4q2 in the perturbative potential, Eq. (5). Bal- 4 c2+(2π2/3)(T/Tc)2
ancing these two gives (cid:104)q (cid:105) ∼ −c /π2(T /T)2; for three
a 1 c
colors, (cid:104)q (cid:105) ∼ 0.04(T /T)2. Thus while the theory is The values of c1 and c2 are dictated by fitting the pres-
a c
sure, or more sensitively, the interaction measure. By
nominally always in a “Higgsed” phase, as a practical
making c increasingly negative from c = 0, one finds
matterthiseffectisnumericallyminiscule. Further,since 2 2
thatthepeakintheinteractionmeasurebecomesincreas-
the loop involves the cosine of the q ’s, asymptotically
a
ingly sharp. Since the loop vanishes at T for two colors,
the deviation of the loop from unity is even smaller: c
1−(cid:96)∼q2 ∼(T /T)4. c1 is then adjusted so that q =1/4 at T =Tc.
c
Tuning c and c in this way, one finds that as the
At intermediate temperatures, what is the origin of 2 1
the term ∼ T2 in the pressure; or equivalently, the be- peak in the interaction measure sharpens, that the re-
havior of the interaction measure, ∆/T4 ∼1/T2? If the gion in which q is nonzero narrows as well. This is due
q ∼ 1/T2, for small q terms ∼ T4q2 and ∼ T2q are of to the particular values of our fit: near Tc, the term
order one. Then the only contribution to a term ∼ T2 ∼ c2Tc2T2 is not only large, but approximately cancels
the similar term, with coefficient (2π2/3)T4, in the per-
in the pressure is from the q-independent terms in the
turbative potential. Requiring q = 1/4 at T = T fixes
nonperturbative potential. In our model, this is a single c
term, ∼c T2T2. c2+2π2/3=−2c1,sothat|c1|issmall,|c1|(cid:28)|c2|. From
3 c Eq. (15), the combination c /(c +(2π2/3)(T /T)2) en-
On the other hand, for intermediate temperatures, 1 2 c
ters into q, and implies that it is much sharper than the
even if the pressure does not probe the other terms in
corresponding factor, (T/T )2 for c = 0, Eq. (14). A
the potential, the ’t Hooft loop does. By measuring a c 2
similar cancellation happens for three colors.
’t Hooft loop, the altered boundary conditions force the
We suggest that this reflects real physics: the sharp-
nessoftheinteractionmeasurereflectsthenarrownessof
the region in which the loop deviates significantly from
1 unity. For the model with c2 = 0, Eq. (14), q al-
waysexceedsthecorrespondingvalueinourmodel, with
0.8
c (cid:54)= 0. This is also seen for T (cid:29) T : when c = 0,
2 c 2
0.6 q ∼0.125(T /T)2; with c (cid:54)=0, q ∼0.011(T /T)2.
c 2 c
0.4 It is also worth contrasting our results with those in
0.2 l(T) a Polyakov loop model. Consider a theory which only
Latt. l(T) involves the Polyakov loop of Eq. (4),
0
1 1.5 2 2.5 3
T / T (cid:18) b 1 (cid:19)
C V ((cid:96))= − 2|(cid:96)|2+ (|(cid:96)|2)2 b T4 ; (16)
eff 2 4 4
FIG. 5: The Polyakov loop for a SU(3) pure gauge theory see, e.g., Eq. (2) of Ref. [32], and Polyakov Nambu-
from lattice simulations [30] and in our model. Jona-Lasino (PNJL) models [35]. For three colors, the
Z(3) symmetry also allows a cubic term, ∼ (cid:96)3 +((cid:96)∗)3,
7
but its addition would only complicate the algebra, and continuation of the thermal Wilson line from imaginary
not our qualitative conclusion. There is no cubic term to real time.
for two colors. √ TocomparewithQCDitisnecessarytoincludetheef-
The minimum of the potential is (cid:96) = b , which we fectsofdynamicalquarks. Itwillbeespeciallyinteresting
0 2
choose to be real. As it is related to the pressure of toseeif, uponaddingtheeffectsofquarkstothepertur-
an ideal gas of gluons, we assume that the coefficient b bative potential, V (q ), whether the thermodynamics
4 pt a
is independent of the temperature, and that only b , or [17],andtheDebyemass,arereproducedusingthesame
2
equivalently (cid:96) , depends upon T. The pressure and the parameters for the non-perturbative potential, V (q ),
0 non a
interaction measure are then in the pure glue theory. (With dynamical quarks, the ’t
Hooft loop does not exist as an order parameter.)
p e−3p ∂(cid:96)
=(cid:96)4 ; = (cid:96)3 T 0 . (17) Without detailed computation, we assume that a nar-
p 0 4p 0 ∂T
ideal ideal row width for the semi-QGP in the pure glue theory im-
plies the same for QCD. We thus conclude with some
Considerfirstintermediatetemperature,wheretheex-
speculations for the phenomenology of heavy ion colli-
pectation value of the loop is near one. To obtain a
sions.
term ∼ T2 in the pressure, in a loop model it is nec-
IfRHICprobestosometemperatureintheQGP,then
essary to assume that the loop deviates from one as
1−(cid:96) ∼ (T /T)2 +.... This is contrast to our matrix LHC may probe to a temperature approximately ∼50%
0 c
model, where the deviation of the loop is ∼ 1/T4, but higher. If the AdS/CFT correspondence holds for QCD,
there is still a term ∼T2 in the pressure. thenresultsattheLHCmustmimicthoseatRHIC.With
the present analysis, the picture is rather more compli-
Near T , from Eq. (17) a peak in the interaction mea-
c
cated.
sure corresponds to a rapid change in (cid:96) . This is similar
0
to what we find in a matrix model. For (cid:96) to decrease as We assume, for the sake of argument, that RHIC
√
T → T+, so must b in Eq. (16). This is proportional probesonlytoatemperatureinthesemi-QGP,verynear
c 2
tothemassofthe(cid:96)field,andsothe(cid:96)massdecreases,like Tc. Then the LHC begins at a temperature well in the
thatofthediagonalmodesinthematrixmodel,Fig. (4); completeQGP.Anyconclusionsaretemperedbythefact
it is in contrast to the masses of the off-diagonal modes, that even if the LHC starts at a higher temperature, as
which increase. it cools it must traverse through the semi-QGP.
In Ref. [32] and other loop models [35], in order to fit In the semi-QGP, the ratio of η/s decreases as the
thepressurethetemperaturedependenceofthepressure, squareofthePolyakovloopasT →Tc+;thisistrueboth
b must have a complicated form. In our matrix model, in the pure glue theory, and with dynamical quarks [34].
2
the coefficients are just ∼ T4, Eq. (5), and ∼ T2, Eq. Conversely, then, η/s increases as the temperature goes
(7). In a mean field theory such as this, simplicity is a up from Tc. This is in sharp contrast to models based
virtue. upon the AdS/CFT correspondence, where η/s is con-
Lastly, the splitting of gluon masses near T is special stant [3–5]. Unfortunately, a computation beyond lead-
c
toamatrixmodel,asaHiggseffectfortheadjointscalar ing logarithmic order is required to compute the precise
A field. See,e.g.,theloopmodelofRef. [10],wherethe dependence of η/s with temperature.
0
splitting of masses does not occur. If the shear viscosity increases strongly from Tc, and
Our analysis is a preliminary first step. In deriv- the system is in thermal equilibrium, then an increased
ing our results, we balance the perturbative potential, shearviscosityshouldleadtoanincreaseinparticlemul-
V (q ),againstthenon-perturbativepotential,V (q ). tiplicity, and a decrease in the elliptical flow, over the
pt a non a
Inpowersofg2,theperturbativepotentialisoforderone, results expected from a (nearly) ideal gas. If the shear
so implicitly we have assumed that the non-perturbative viscosity increases significantly, though, a hydrodynamic
is as well. Since the non-perturbative potential repre- description could easily break down.
sents a resummation of effects to all orders, this is a It is also possible that the temperature dependence
strong assumption. Nevertheless, it allows us to envis- of η/s is weak; if so, then the particle multiplicity and
agecomputingtohigherordering2. Correctionsatleast elliptical flow at LHC should be similar to that expected
to order ∼ g2 and ∼ g3 are needed in order to make by an extrapolation from the results at RHIC. There are
a serious comparison to lattice data. This also requires then other ways to probe the effects of the semi-QGP.
preciselatticedata,closetothecontinuumlimit,notjust Consider, for example, energy loss, which is controlled
for the pressure, but also for the ’t Hooft loop and gluon by a parameter qˆ. In the complete QGP, qˆ ∼ T3, or
masses. equivalently, the entropy density, s. In kinetic theory,
Thereareseveralformalquestionsraisedbyouranaly- qˆ/s and s/η are each proportional to a cross section,
sis. Theparametersofeffectivetheoriescanbecomputed so one expects that a minimum in η/s corresponds to
fromlatticesimulations[36]; doingsoforelementsofthe a maximum in the energy loss, qˆ/s [7, 41]. Following the
Liealgebra,insteadofforelementsoftheLiegroup,may methods of Ref. [34], the energy loss of a quark can be
be useful. It is also necessary to extend the analysis of computed in the semi-QGP; as T →T+, it vanishes lin-
c
Hard Thermal Loops in the complete QGP to the semi- early in the Polyakov loop. Thus in the semi-QGP, both
QGP. This is equivalent to understanding the analytic η/s and qˆdecrease as T →T+; the difference from Refs.
c
8
[7, 41] is because the kinetic theory for the semi-QGP is tities such as the ’t Hooft loop and screening masses.
in the presence of a background A field. As the tem- This can then be tested against predictions from the
0
perature increases from T , then, excluding the obvious AdS/CFT correspondence [2–6] and other models [7].
c
dependence upon the entropy, the energy loss is larger
in the complete QGP than in the semi-QGP. As with
the shear viscosity, determining the precise dependence
upon temperature requires computation beyond leading
Acknowledgments
logarithmic order.
There is also a qualitatively new phenomenon in the
semi-QGP: besides energy loss, the propagation of a col- We thank A. Bazavov, P. de Forcrand, O. Kaczmarek,
ored field is suppressed by the background A field [34]. F.Karsch,J.Liao,M.Pepe,P.Petreczky,andE.Shuryak
0
Thissuppressionisuniversal,independentofthemassor for discussions and comments. The research of A.D.
momentumofthecoloredfield. Acompleteanalysisneed was supported by the U.S. Department of Energy under
incorporate this universal suppression as well as energy contract#DE-FG02-09ER41620,andbyPSC-CUNYre-
loss. searchgrant63382-0041;ofY.G.,inpartbytheNatural
Lastly, we note that given the modified propagator of Sciences and Engineering Research Council of Canada;
the semi-QGP, Eq. (13), there are also significant modi- of Y.H., by the Grant-in-Aid for the Global COE Pro-
ficationstotheheavyquarkpotential[42]. Thiscanalso gram “The Next Generation of Physics, Spun from Uni-
be compared to lattice data, which we defer for now. versality and Emergence” from the Ministry of Educa-
Intheend,ourspeculationswillsoonberenderedmoot tion, Culture, Sports, Science and Technology (MEXT)
by the wealth of results which will flow from heavy ion of Japan; of R.D.P., by the U.S. Department of En-
collisions at the LHC. The present approach is based ergyundercontract#DE-AC02-98CH10886;R.D.P.also
upon constructing an effective theory from the results thankstheAlexandervonHumboldtFoundationfortheir
of lattice simulations; not just of the pressure, but quan- support.
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