Viscosities of Quark-Gluon Plasmas

18 Pages·0.22 MB·

by  H. Heiselberg

Viscosities of Quark-Gluon Plasmas
H. Heiselberg
Nuclear Science Div., MS 70A-3307, Lawrence Berkeley Laboratory, Berkeley, CA 94720, USA
The quark and gluon viscosities are calculated in quark-gluon plasmas to leading orders in the
couplingconstantbyincludingscreening. ForweaklyinteractionQCDandQEDplasmasdynamical
screening of transverse interactions and Debye screening of longitudinal interactions controls the
infrared divergences. For strongly interacting plasmas other screening mechanisms taken from lat-
tice calculations are employed. By solving theBoltzmann equation for quarksand gluons including
screeningtheviscosityiscalculatedtoleadingordersinthecouplingconstant. Theleadinglogarith-
4 micorderiscalculated exactlybyafullvariational treatment. Thenexttoleadingordersarefound
9 to be very important for sizable coupling constants as those relevant for the transport properties
9 relevant for quark-gluon plasmas created in relativistic heavy ion collisions and theearly universe.
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I. INTRODUCTION transport relaxation rates for viscous processes, momen-
tumstopping,thermalandelectricalconductionhavethe
Transportandrelaxationpropertiesofquarkandgluon same dependence as (1) on the QED fine structure con-
(QCD) plasmas are importantin a number of a different stant α [5].
contexts. Theydeterminethetimethatittakesaquark- Thedependence ofthetransportratesonthecoupling
gluonplasmaformedinaheavy-ioncollisiontoapproach constants is very sensitive to the screening. Besides the
equilibrium, and they are of interestin astrophysicalsit- factor α2s from the matrix element squared of the quark
uations such as the early universe, and possibly neutron and gluon interactions, the very singular QCD interac-
stars. tions for small momentum transfers lead to a logarithm,
The basic difficulty in calculatingtransportproperties ln(qmax/qmin), of the maximum and minimum momen-
ofsuchplasmas,as wellasofrelativisticelectron-photon tum transfers. The typical particle momenta limits the
(QED) plasmas, is the singular nature of the long-range maximum momentum transfer, qmax ∼ T, and Deby
interactions between constituents, which leads to diver- and dynamical screening leads to effective screening for
gences in scattering cross sections similar to those for small momentum transfers of order qmin ∼ qD ∼ g
Rutherford scattering. This makes the problem of fun- This gives the leading logarithmic order in the coupling
damental methodological interest, in addition to its pos- constant, ln(T/qD) ∼ ln(1/αs), to the transport rates
sible applications. The first approaches to describe the (1).
transport properties of quark-gluon plasmas employed Thecalculationsin[4,5]werebriefanddealtonlywith
the relaxation time approximation [1,2,3] for the colli- the leading logarithmic order in the coupling constan
sionterm. Thisapproximationsimplifiesthe collisionin- with a given ansatz for the distribution function. Here,
tegral enormously and transport coefficients are related more detailed calculations of the quark and gluon vis-
directly to the relaxation time. The latter is typically cositiesinthehightemperaturequark-gluonplasmasare
estimated from a characteristic cross section times the presented. The leading logarithmic order is calculated
density of scatterers. In Refs. [2,3] the divergent part of exactly by a variational method and the next to leading
the total cross section at small momentum transfers was order-theα2s termin(1)-iscalculatedaswell. Because
assumedtobescreenedatmomentumtransferslessthan αs is not exponentially small, the next to leading order
the Debye momentum. However, Debye screening influ- is important in many realistic physical situations as rel-
encesonlythelongitudinal(electric)partoftheQEDand ativistic heavyioncollisionsandthe earlyuniverse. Fur-
QCD interactions, and the transverse (magnetic) part is thermorewhenthe Debyescreeninglengthis largerthan
unscreened in the static limit. the interparticle screening, which occur when αs∼>0.1 as
Recentlyithasbeenshownthatthephysicsresponsible weshallseebelow,Debyeanddynamicalscreeningbreaks
for cutting off transverse interactions at small momenta down. Insteadlatticegaugecalculationshavefoundthat
is dynamical screening [4]. This effect is due to Landau quark-gluon plasmas seem to develop a constant screen-
damping of the exchanged gluons or photons. Within ingmass,mpl ≃1.1T,fortemperaturesT∼>2−3Tcand
perturbative QCD and QED rigorous analytical calcula- isimportanttoseewhateffectsthisalternativescreening
tions of transport coefficients to leading order have been mechanism has in strongly interacting plasmas.
made for temperatures high [4,5] as well as low [6] com- WeshallfirstdescribeinsectionIIthetransporttheory
pared with the chemical potentials of the constituents. we use, namely the Boltzmannequation, andthe screen-
Transport processes depend on a characteristic relax- ing oflongrangeQCDandQEDinteractions. Insection
ation time, τ , of the particular transport process con- III, we describe the process of shear flow and the varia-
tr
sidered. For example, in high temperature plasma the tionalcalculationnecessaryinordertofindthe viscosit
viscosities, η = w τ /5, of particle type i are propor- InsectionIVwethenevaluatethe collisiontermtolead-
i i η,i
tional to the characteristic times for viscous relaxation, ing logarithmic order with a simplifying ansatz for the
τ ∼ τ , which were first calculated in [4] to leading trial function and refer to Appendix A for a full and ex-
η,i tr
order in the coupling constant. More generally one finds actvariationalcalculation. InsectionV we calculate the
that the typical transport relaxation rates, that deter- viscosity to higher orders in the coupling constant and
mines momentum stopping, thermal and viscous relax- discuss strongly interacting plasmas. Finally, in section
ation, is in a weakly interacting QCD plasma VI we give a summary and discuss generalizationsof the
methods developed here to other transport coefficients.
1
∝α2ln(1/α )T +O(α2). (1)
τ s s s
tr
II. TRANSPORT THEORY
wheretheexpansionisintermsofthefinestructurecon-
stant α = g2/4π. The coefficients of proportionality to
s Transport processes are most easily described by the
the leading order in α (in the following called the lead-
s Boltzmann equation
ing logarithmic order) has been calculated analytically
for a number of transport processes in high temperature ∂
( +v ·∇ +F·∇ )n = 2πν |M|2
plasmas [4,5]. Likewise in a QED plasma the typical ∂t 1 r p1 1 2
234
X
2
×[n n (1±n )(1±n )−(1±n )(1±n )n n ] 9 1 (1−µ2)cosφ 2
1 2 3 4 1 2 3 4 |M |2= g4 − . (7)
×δ(ǫ +ǫ −ǫ −ǫ )δ , (2) gg 8 q2+Π q2−ω2+Π
1 2 3 4 p1+p2;p3+p4 (cid:12) L T(cid:12)
(cid:12) (cid:12)
The interactions ar(cid:12)e modified by inclusion of(cid:12) the gluon,
whereǫiistheenergyandpithemomentumofthequasi- orphoton,selfener(cid:12)gies,Π andΠ [8](seeal(cid:12)soFig. (1
particles,Fsomeforceactingonthe quasi-particles,and L T
Intherandom-phaseapproximationthepolarizationsare
therighthandsideof(2)isthecollisionterm. n (p )are
i i given in the long wavelength limit (q ≪T) by
the Fermi and Bose quasi-particle distribution functions
for quarks and gluons and the signs ± include stimu- µ µ+1
Π (q,ω)=q2 1− ln , (8)
lated emission and Pauli blocking. The spin and color L D 2 µ−1
(cid:20) (cid:18) (cid:19)(cid:21)
statistical factor ν2 is 16 for gluons and 12Nf for quarks µ2 µ(1−µ2) µ+1
andantiquarkswithN flavors. |M|2 is the squaredma- Π (q,ω)=q2 + ln , (9)
f T D 2 4 µ−1
trix element for the scatteringprocess 12→34,summed (cid:20) (cid:18) (cid:19)(cid:21)
over final states and averaged over initial states. It is whereµ=ω/qandq =1/λ istheDebyewavenumber.
D D
related to the Lorentz-invariantmatrix element |M|2 by In a weakly-interacting high temperature QCD plasma,
|M|2 = |M|2/(16ǫ ǫ ǫ ǫ ). For gluon-gluon scattering [8,9]
1 2 3 4
[7] (see Fig. (1))
q2 =4π(1+N /6)α T2, (10)
D f s
9 us st ut whereα =g2/4πisthefinestructureconstantforstrong
|M |2 = g4 3− − − , (3) s
gg 4 t2 u2 s2 interactions,thefactor(1+Nf/6)isthesumofcontribu-
(cid:18) (cid:19)
tions from gluon screening, the “1,” and from light mass
where s, t, and u are the usual Mandelstam variables. quarks, of number of flavors,N . In a high temperature
f
In Eq. (3) the double counting of final states has been QED plasma, q2 = 4παT2/3, where α is the QED fine
D
corrected for by inserting a factor 1/2. For quark-gluon structure constant.
scattering One should keep in mind that the self energies of (8
and (9) are only valid in the long wavelength limit, i.e.,
1 4
|M |2 =g4(u2+s2) − , (4) for q ≪ T. When q ∼ T other contributions of order
gq t2 9us α qT enter(see,e.g.,[8])whichmaybegaugedependen
(cid:18) (cid:19) s
[14]. However, as long as α is small all contributions
s
and for scattering of two different quark flavors
from the self energies can be ignored in the gluon prop-
agator when q ∼ T because the matrix element squared
4 u2+s2
|M |2 = g4 . (5) already carry the order α2.
q1q2 9 t2 s
Inthe abovederivationswe haveconsistentlyassumed
The matrix element for scattering of the same quark fla- that the screening was provided in RPA by the gluon
vors or quark-antiquark scattering is different at large selfenergieswhichgivetheDebyeanddynamicalscreen-
momentum transfer but the same as (5) at small mo- ing of longitudinal and transverse interactions respec-
mentum transfers. tively. Both effects provide a natural effective cutoff
The t−2 and u−2 singularities in Eq. (3-5) lead to momentum transfer less than qmin ∼ qD. These pertur-
divergingtransportcrosssectionsandthereforevanishing bativeideasmust,however,breakdownwhenthescreen-
transport coefficients. Including screening, it was shown ing length becomes as short as the interparticle spac-
in [4,5,6] that finite transport coefficients are obtained. ing, i.e. qD ∼ T, or in terms of the coupling constan
In fact, the leading contribution to transportcoefficients αs∼<(4π(1+Nf/6))−1 ∼0.1accordingtoEq. (10). Inlat-
comesfromthesesingularities. Inthet=ω2−q2channel ticegaugecalculationsofquark-gluonplasmaabovetwice
the singularityoccursforsmallmomentumqandenergy the temperature of the phase transition, Tc ≃ 180MeV,
ω transfers (see Fig. (1)). one finds strongnonperturbativeeffects inthe plasmaso
For small momentum transfer, q ≪ǫ1,ǫ2 ∼ T, energy that the typical screening mass is mpl ∼ 1.1T [12]. One
conservation implies that ω =ǫ −ǫ ≃v ·q=−v ·q may argue [13] that perturbation theory still applies for
1 3 1 2
where v =p . Therefore the velocity projections trans- large momentum transfers so that the matrix elemen
i i
aregivenbythesimpleFeynmantreediagrams,butthat
versetoqhavelengths|v |=|v |= 1−µ2,where
1,T 2,T
µ=ω/q. Consequentlyv ·v =(1−µ2)cosφ,where perturbationtheorydoesnotapplyforsmallmomentum
φistheanglebetweenv 1,Tand2,vT . Forpq ≪T wethus transfersoforderq∼<qD andthatoneshouldratherinsert
1,T 2,T
the effective cutoff found by lattice gauge calculations
have
ΠL ≃ΠT ≃mpl, αs∼>0.1. (11)
s≃−u≃2p p (1−cosθ )
1 2 12
Thephenomenologicalscreeningmassof(11)providesus
≃2p p (1−µ)(1−cosφ), (6)
1 2 withamethodtoextendourcalculationsoftransportco-
efficientstolargervaluesforα anditcanbecombinedto
and the interactions splits into longitudinal and trans- s
theDebyeanddynamicalscreeninginweaklyinteracting
verse ones, [8]
quark-gluon plasmas.
3
III. THE VISCOSITY value η. Equation (19 is therefore convenient for varia-
tional treatment, which will be carried out in Appendix
With screeningincludedinthe interactionwecannow A.
proceedtocalculatetransportpropertiesastheviscosity. To find the viscosity we must solve the integral equa-
In the presence of a small shear flow, u(y), in the x- tion (16 for which we have to evaluate
direction we obtain from the Boltzmann equation
hΦ| I |Φi=2πν |M|2n n (1±n )(1±n )
2 1 2 3 4
∂n ∂u
p1xv1y 1 x =2πν2 |M|2 p1X,p2,q
∂ǫ1 ∂y (Φ +Φ −Φ −Φ )2
234 1 2 3 4
X × δ(ǫ +ǫ −ǫ −ǫ ). (20)
1 2 3 4
× [n n (1±n )(1±n )−(1±n )(1±n )n n ] 4
1 2 3 4 1 2 3 4
× δ(ǫ1+ǫ2−ǫ3−ǫ4)δp1+p2;p3+p4. (12) Momentum conservation requires that p3 = p1+q and
p = p −q where q is the momentum transfer. Intro-
For small u we can furthermore linearize the quasi- 4 2
ducinganauxilliaryintegraloverenergytransfers,ω,the
particle distribution function
delta-function in energy can be written
∂n ∂u
ni =nLiE + ∂ǫ Φi ∂yx , (13) δ(ǫ +ǫ −ǫ −ǫ )= dω p3 δ(cosθ −µ− t )
p 1 2 3 4 1
p q 2p q
Z 1 1
where the local equilibrium distribution function is p t
4
× δ(cosθ −µ+ ), (21)
2
p q 2p q
nLE =(exp[(ǫ −u·p )/T]∓1)−1, (14) 2 2
i i i
where θ is the polar angle between q and p and θ
1 1 2
and Φ is an unknown function that represents the devi-
i the corresponding one between q and p (see Fig. 2).
2
ations from local equilibrium. By symmetry Φ has to be
Consequently, we find
on the form
1 ∞ q
Φ=pˆ pˆ f(p/T), (15) hΦ|I|Φi= dq dω
x y 4π8T Z0 Z−q
where now the function f must be determined from the ∞
× dp p2n (p )(1±n (p +ω))
Boltzmann equation. Inserting (13 in the Boltzmann 1 1 1 1 1 1
equation we find Z(q−ω)/2
∞
× dp p2n (p )(1±n (p −ω))
∂n 2 2 2 2 2 2
p1xv1y 1 =2πν2 |M|2[n1n2(1±n3)(1±n4) Z(q+ω)/2
∂ǫ
1 234 2π dφ
X × |M|2(Φ +Φ −Φ −Φ )2. (22)
−(1±n1)(1±n2)n3n4] Z0 2π 1 2 3 4
×δ(ǫ +ǫ −ǫ −ǫ )δ
1 2 3 4 p1+p2;p3+p4 This integral equation for Φ has been solved in a few
×(Φ +Φ −Φ −Φ ). (16)
1 2 3 4 cases under simplifying circumstances. F.ex., in Fermi
liquids the sharp Fermi surface restricts all particle mo-
Itisveryconvenienttodefineascalarproductoftworeal
menta near the Fermi surface and with a simplified form
functions by:
for the scattering matrix element techniques have been
∂n developedto calculate anumber of transportcoefficien
hψ |ψ i=−ν ψ (p)ψ (p) . (17)
1 2 2 1 2 ∂ǫ exactly[10]. FortheQCDandQEDplasmastheverysin-
p p
X gularinteractioncan,oncescreened,beexploitedsince
Thus Eq. (16 may be written on the form |Xi = I|Φi allowsanexpansionatsmallmomentumtransfers. Thus
where |Xi = p v and I is the integral operator acting ananalyticalcalculationofthe transportcoefficientscan
x y
on Φ. The viscosity is given in terms of Φ [10] and can becarriedoutatleasttoleadinglogarithmicorderinthe
now be written as coupling constant [4,5,6].
∂n
η =−ν p v Φ =hX|Φi. (18)
2 x y∂ǫ p IV. VISCOSITY TO LEADING LOGARITHMIC
p p
X ORDER
Equivalently, the viscosity is given from (16 as
In Ref. [4], through solution of the Boltzmann kinetic
hX|Φi2
η = . (19) equation, the first viscosity of a quark-gluonplasma was
hΦ|I|Φi derivedtoleadinglogarithmicorderintheQCDcoupling
strength. We will in the following give a more thorough
Since h.|.i defines an inner product, the quantity
and exact derivation of the quark and gluon viscosit
hX|Ψi2/hΨ|I|Ψi is minimal for Ψ=Φ with the minimal
4
The total viscosity, to leading order, is an additive sum 28π3 qmax 1
hΦ|I|Φi= α2T3 q3dq dµ
of the gluon and quark viscosities, η =ηg +ηq. 45 s Z0 Z0
The leading logarithmic order comes from small mo-
1 1/2
mentum transfers because of the very singular matrix [ + ].
|q2+Π (µ)|2 |q2+Π (µ)/(1−µ2)|2
element (7) dominates. For small q the kinematics sim- L T
(26)
plify enormously and, as we will now show, the integrals
separate allowing almost analytical calculations. First,
This integral is discussed in detail in appendix B. For
we can set the lower limits on the p and p integrals
1 2 the longitudinal interactions Π ≃ q2 due to Deby
to zero, however, then replacing the upper limit on q by L D
screening and the leading term is a logarithm of the
the natural cutoff from the distribution functions which
ratio of maximum to minimum momentum transfer,
is q ∼T. Thus we find from (22)
max ln(q /q ) ∼ ln(T/q ). Likewise for the transverse
max min D
∞ interactions Π ≃ i(π/4)µq2 due to Landau damp-
hΦ|I|Φi= 2π18T dp1p21n1(1±n1) ing and the deTpendence on µD= ω/q provides sufficien
Z0 screening to render the integral finite and the leading
∞
× dp p2n (1±n ) term is the same logarithm as for the longitudinal inter-
2 2 2 2
Z0 actions. Whereas the details of the screening are unim-
qmax 1 dµ 2π dφ portant for the leading logarithmic order, they are im-
× qdq
portant for the higher orders and they are calculated in
Z0 Z−1 2 Z0 2π detail in appendix B. The final result is thus to leading
×|M|2(Φ +Φ −Φ −Φ )2. (23)
1 2 3 4 logarithmic order
to leading logarithmic order 27π3
The solution to the integral equation or equivalently hΦ|I|Φi= 15 α2sln(T/qD)T3, (27)
the variational calculation of (19) is quite technical and
is for that reason given in Appendix A. A much simpler SincehΦ|Xi=(64ξ(5)/π2)T3,wefindfrom(19)and(27
calculationistomakethestandardassumptioninviscous
processes, i.e., to take the trial function as 2515ξ(5)2 T3
η =
gg π7 α2ln(T/q )
f(p/T)=(p/T)2. (24) s D
T3
≃0.342 , (28)
As will be shown in Appendix A this turns out to be a α2sln(1/αs)
very good approximation. It is accurate to more than
99% for reasons also explained in the appendix. f can to leading logarithmic order in αs =g2/4π.
be defined up to any constant which cancels in (19) and Toobtainthe fullgluonviscositywemustaddscatter-
therefore never enters in the viscosity. ing on quarks and antiquarks which is calculated analo-
The quantity (Φ +Φ −Φ −Φ )2 can be averaged gously and only has a few factors different. Firstly, from
1 2 3 4
over x- and y-directions while keeping µ and φ fixed. (3) and (4) we see that the matrix element squared
This corresponds to keeping the relative positions of the a factor 4/9 smaller. Secondly, the statistical factor
threevectorsq,p,andp′ fixedrelativetoeachotherand ν2 = 12Nf instead of 16. Thirdly, in integrating over
rotatingthis systemoverthe three Eulerangles(see also the factor (p21+p22) in Eq. (25) we note that the distri-
Appendix A). Consequently, we obtain bution function, n2, in Eq. (23) is now a fermion one.
Consequently, the p and p integrations give a factor
1 2
q2 (1/2+7/8)/2lessfor gluon-quarkcollisionsascompared
h(Φ +Φ −Φ −Φ )2i= [3(p −p )2
1 2 3 4 15T4 2 1 to gluon-gluon collisions and we find
+(qˆ·(p2−p1))2] η =(η−1+η−1)−1 = ηgg . (29)
q2 g gg gq 1+11N /48
= [(3+µ2)(p2+p2) f
15T4 1 2
−2p p (4µ2+3(1−µ2)cosφ)]. (25) In [4] the slightly different result ηg = ηgg/(1+Nf/6)
1 2
was obtained.
Theintegralsoverp andp in(23)areelementary. Next The quark viscosity can be obtained analogously to
1 2
we perform the integrations or averages over µ and φ the gluon one. The quark viscosity due to collisions on
requiredin(23). Wenoteinpassingthatthetermin(25) quarks only, ηqq, deviates from ηgg by a factor (4/9)2 in
proportionalto p p vanishes andthat µ2 effectively can the matrix elements and differences in having Fermi and
1 2
bereplacedby1/3(seeAppendixA).Letusfirstconsider Bose integrals. By comparing to (19,18,20)we find
the case of gluon-gluon scattering inserting |M |2 from
gg (15/16)2 5236
(3). We thus find η =η =η . (30)
qq gg(4/9)2(7/8)(1/2) gg 287
5
Note that the statistical factors ν cancel in η and η . quark-quark scattering. The function Q and the effec-
gg qq
Includingquarkscatteringsongluonsleadtosimilarfac- tive maximum and minimum momentum transfer, q
max
torsininhΦ|I|Φi,namelyafactor(9/4)fromthematrix and q , are given in Appendix B. In weakly interact-
min
element, a factor 16/12N from statistics, and a factor ing plasmas, where the screening is provided by Deby
f
(8/7+2)/2 from Bose instead of Fermi integrals. Thus anddynamicalscreening,the functionQis basicallyjust
a logarithm of the ratio of the maximum and minimum
ηqq 1+11Nf/48 momentum transfer, i.e.,
η = ≃2.2 N η , (31)
q f g
1+33/7N 1+7N /33
f f
q q
which for Nq = 2 results in ηq = 4.4ηg, a quark viscos- Q(cid:18)qmmainx(cid:19)=ln(cid:18)qmmainx(cid:19) , αs∼<0.1 (33)
ity that is larger than the gluon one partly because the
gluons generally interact stronger than the quarks and By numerical integration we find that the distribution
partlybecauseofdifferencesbetweenBoseandFermidis- functions leads to an effective cutoff of q(gg) ∼3T. This
max
tribution functions. is because the distribution functions are weighted with
several powers of particle momenta and thus contribute
themostforp≃3T. Theeffectivecutoffisslightlylarger
V. VISCOSITY TO HIGHER ORDERS IN αS for quark-gluon and quark-quark scattering because the
Fermi distribution functions emphasize large momenta
The leading logarithmic order dominates at extremely thantheBoseones. Debyeanddynamicalscreeningleads
high temperatures, where the running coupling constant to qmin ≃ 1.26qD as described in Eq. (B6) and so from
is small,but it is insufficient atlowertemperatures. The (B8)
next to leading order correction to the viscous rate in
the coupling constant is of order α2s. It may be signifi- Q qm(gagx) =ln 0.44 , αs∼<0.1. (34)
cantbecause the leading logarithmis a slowlyincreasing qmin! (cid:18)αs(1+Nf/6)(cid:19)
function. In the derivation of the leading logarithmic
order, Eq. (27), we have been very cavalier with any The numerical factor inside the logarithm, which gives
factors entering in the logarithm, which are of order α2. the order α2, is discussed in more detail in Appendix B.
s s
It was only argued that the leading logarithmic order In the other limit, qD∼>T or equivalently αs∼>0.1, per-
ln(q /q ) ∼ ln(T/q ) because q and q were turbative ideas breaks down and we assume an effectiv
max min D max min
of order ∼ T and ∼ qD respectively. Finally, if thermal screening mass taken from lattice calculations, qmin
quark-gluonplasmasare createdinrelativistic heavyion 1.1T,asdescribedbyEq. (11). Thuswefind(see(B10
collisionsatCERNandRHICenergies,thetemperatures
achieved will probably be below a GeV. We can thus Q(qm(gagx)/qmin)=0.626, αs∼>0.1, (35)
estimate the interaction strength from the running cou-
and similarly for quark-gluon and gluon-gluon scatter-
pling constant α ≃ 6π/(33−2N )ln(T/Λ) which, with
Λ≃150MeVandsT∼<1GeV,givesfαs∼>0.4. Forsuchlarge ing Q(qm(gaq)x/qmin) = 0.819 and Q(qm(qaq)x/qmin) = 1.024
respectively.
coupling constants Debye and dynamicalscreening is re-
Adding gluon-gluon and gluon-quark scatterings w
placed by an effective screening mass, m , as discussed
pl
obtain the gluon viscosities
above which will affect the viscosity considerably.
To calculate the viscosity to order α2s exactly, the 5- 2515ξ(5)2 T3
dimensional integral of (22) must be evaluated numeri- ηg = π7 α2
cally and at the same time a variationalcalculationof Φ s
must be performed. This is a very difficult task and we q(gg) 11N q(gq)
[Q max + fQ max ]−1, (36)
shall instead use the information obtained in the previ- q 48 q
D ! D !
ous section, that the trial function f ∝p2 is expected to
be an extremely good approximation. With that ansatz which extends Eq. (37) to higher orders. In weakly in-
forthetrialfunction,itisthenstraightforwardtocalcu- teracting plasmas (36) reduces to
latetheintegralof(21)numericallyandfindtheviscosity
to order α2 for the given screening mechanism. The 5- T3 0.44
s η ≃0.342 [ln
dimensionalnumericalevaluationofthecollisionintegral g α2 α (1+N /6)
s (cid:18) s f (cid:19)
of(21)isacomplicatedfunctionofthecouplingconstant.
11N 0.72
It is convenient to write it in terms of the function Q + f ln ]−1,αs∼<0.1 (37)
48 α (1+N /6)
(cid:18) s f (cid:19)
hΦ|I|Φi = 27π3α2Q qm(gagx) T3, (32) to leading orders in αs. In strongly interacting plasmas
gg 15 s qmin! we obtain by inserting (B10) in (32)
twhheeraentahloegionudsexdegfignrietifoernsstoapgplluieosn-gtoluognlusocna-tqtuearirnkgabnudt ηg ≃0.55Tα23 1+1.31114N8f −1 , αs∼>0.1 (38)
s (cid:20) (cid:21)
6
InFig. (3)weshowthegluonviscositywiththevarious sincetheleadingorderisγ(g) =3α ln(1/α )asexplained
p s s
assumptions for screening. With dash-dotted curve the in[16]. Fortheviscosity,however,vertexcorrectionscan
result of Eq. (38) assuming a constant screening mass, beignoredsincetheextraverticesaddsafactorα2. Even
s
m =1.1T, is shown. With dashed curve the numerical though integration over soft momenta may cancel a fac-
pl
result assuming Debye and dynamical screening of Eqs. tor α the result is still of higher order in the coupling
s
(8) and (9) is shown. For αs∼<0.05 it is given by Eq. constant.
(37) to a good approximation whereas for αs∼>0.05 the Writing each of the viscosities ηi (i=q,g) in terms
result of Eq. (B4) is better. The final viscosity shown the viscous relaxation time, τ , as
ηi
by full curve is obtained by combining the two limits,
i.e., applying Debye and dynamical screening in weakly ηi =wiτηi/5, (42)
interacting plasmas when qD∼<T or equivalently αs∼<0.1 where w = (32π2/45)T4 and w = (N 7π2/15)T4 are
but an effective screening mass m = 1.1T as given by g q f
pl the gluon and quark enthalpies respectively, we obtain
Eq. (11) when αs∼>0.1. This corresponds to chosing the the viscous relaxation rate for gluons
smallest value of the viscosities as seen in Fig. (3), i.e.,
the two limits of Eqs. (37) and (38). 1 π9 T
Similarly, adding quark-quark and quark-gluon scat- =
τ 3353ξ(5)2 α2
terings we find the quark viscosity η,g s
q(gg) 11N q(gg)
ηq = 5323361ξ1(π57)2NfTα23 ×"Q qmmainx!+ 48fQ qmmainx!#, α2sT, (43)
s
q(gq) 7N q(qq) and quarks and antiquarks
[Q max + fQ max ]−1, (39)
qD ! 33 qD ! 1 11π9237 T
=
τ 3755ξ(5)2 α2
η,q s
which in weakly interacting plasmas gives
q(gq) 7N q(qq)
× Q max + fQ max , α2T . (44)
T3 0.72 q 33 q s
η ≃0.752N [ln " min! min!#
q fα2 α (1+N /6)
+7Nf sln (cid:18) s1.15 f ]−(cid:19)1,αs∼<0.1, (40) Tτ−h1e)vairsecotuhsusrevlaerxyatsiiomniltaimrteos,thτηe,gc,oτrrηe,qspaonnddτiηng=v1is/c(oτsη−i,tg1ies
33 (cid:18)αs(1+Nf/6)(cid:19) wηh,qen divided by a factor of T4. The curves on Figs.
and in the strongly interacting plasmas (4) and (5) therefore applies to the viscous relaxation
times (times temperature) as well when divided a factor
ηq ≃0.92NfTα23 1+1.2573N3f −1 ,αs∼>0.1. (41) oafcc∼or1d.i4nagntdo∼Eq0..9(24N2)f.forgluonsandquarksrespectively
s (cid:20) (cid:21)
In weakly interacting plasma the viscous rates can b
Thequarkviscosityincreaseswiththenumberofquark approximated by
flavors,N , whereas the gluon viscosity decreasesas can
f
be seen in Fig. (4) and (5), where the viscosities are 1 0.44
≃4.11α2[ln
shown for two and three flavors respectively. The total τ s α (1+N /6)
η,g (cid:18) s f (cid:19)
viscosity of a quark-gluon plasma, η = η +η , is domi-
g q 11N 0.72
nated by the quark viscosity. + f ln ], αs∼<0.1 (45)
48 α (1+N /6)
Fromthedefinitionoftheviscosityintermsofthe col- (cid:18) s f (cid:19)
lisionintegral(18)and(20),whichonlycontainspositive and
quantities,itfollowstriviallythattheviscosityispositive
1 0.72
asisaphysicalnecessity. TheresultingviscositiesofEqs. ≃1.27α2[ln
(36) and(39) arepositive quantities whereasthe αs∼<0.1 τη,q s (cid:18)αs(1+Nf/6)(cid:19)
expansions of Eqs. (37) and (40) are not when extended 7N 1.15
to the region αs∼>0.5. This explains the results found in + 33f ln α (1+N /6) ], αs∼<0.1, (46)
[11], where it was claimed that estimates of the next to (cid:18) s f (cid:19)
leading order α2 could lead to a negative viscosity. to leading orders in α .
s s
Contributions from vertex corrections should also be
considered. In fact for the calculation of the quasi-
particle damping rate, γ , Braaten and Pisarski [15] VI. SUMMARY
p
found that vertex corrections contributed to leading or-
der γ(g) ≃ 6.6α for zero gluon momenta, p. Vertex BysolvingtheBoltzmannequationforquarksandglu-
p=0 s
correctionsdoalsocontributetoorderα forlargequasi- ons the viscosities in quark-gluon plasmas were calcu-
s
particle momenta,p≫gT,but they canherebe ignored latedtoleadingordersinthecouplingconstant. Inclusion
7
of dynamical screening of transverse interactions, which ent for thermal dissipation processes as described in [6
controls the infrared divergences in QED and QCD, is because the transport of energy introduce dependences
essential for obtaining finite transport coefficients in the on ω which also is present in the transverse screening,
weakly interacting plasmas. The solution of the trans- Π (ω/q).
T
port process was extended to strongly interacting plas- All the transport processes discussed above depend
mas by assuming an effective screening mass of order only on momentum scales from the typical particle
m = 1.1T, as found in lattice calculations, when the momentum, q ∼ T down to the Debye screening
pl max
Debye screening length became larger than the interpar- wavenumber q ∼ q ∼ gT which also is the momen-
min D
ticle distance or when αs∼>0.1. The Boltzmann equation tum scale for dynamical screening. There is, however,
was solved exactly to leading logarithmic order numeri- a shorter momentum scale of order the magnetic mass,
cally but the result only differed by less than a percent m ∼ g2T, at which perturbative ideas of the quark-
mag
fromananalyticalresultobtainedby asimple ansatzfor gluon plasma fails [17]. As shown in [16] the quark and
the deviation from local equilibrium, Φ ∝ p p . The gluon quasiparticle decay rates depend on this infrared
x y
next to leading orders was also calculated and found to cutoff, m . Furthermore, recent studies [19] find that
mag
be very important for the transport properties relevant the color diffusion and conductivity also depend on this
for quark-gluonplasmas created in relativistic heavy ion cutoffandthereforetherateofcolorrelaxationisafactor
collisions and the early universe. For αs∼>0.1 we find 1/αs larger than Eq. (1).
ηi = Ci,1T3/α2sln(Ci,2/αs) whereas for αs∼>0.1 we find
η =C T3/α2 with coefficients C given above.
i i,3 s i,j
The viscosityin degenerateplasmas ofquarks,i.e., for ACKNOWLEDGMENTS
T ≪ µ was calculated in [6]. Several differences were
q
found. In the high temperature quark-gluon plasma the This work was supported by DOE grant No. DE-
chemicalpotentialcanbeignoredandthe transportpro- AC03-76SF00098, NSF grant No. PHY 89-21025 and
cesses depend on two momentum scales only, namely T the Danish Natural Science Research Council. Discus-
and qD ∼ gT. In degenerate quark matter three mo- sionswithGordonBaymandChrisPethickaregratefully
mentum scales enter, namely µq, T, and qD ∼ gµq, and acknowledged.
the transport process depends considerably on which of
q and T is the larger. In fact for T ≪ q transverse
D D
interactions turn out to be dominant in contrast to the APPENDIX A: EXACT VARIATIONAL
high temperature quark-gluon plasma where transverse CALCULATION TO LEADING LOGARITHMIC
and longitudinal interactions contribute by similar mag- ORDER
nitude. Furthermore, the existence of a relative sharp
Fermi surface allows an almost analytical calculation of InthisappendixwesolvetheBoltzmannequationand
both the leading (logarithmic) order as well as the next find the deviation from local equilibrium, Φ, by a varia-
order α2s. tional treatment of Eq. (18).
Thetechniquesforcalculatingtheviscositiestoleading For a general function Φ=pˆ pˆ f(p/T) we have
x y
orders in the coupling constants can be applied to other
transport coefficients as well. The leading logarithmic Φ +Φ −Φ −Φ =pˆ pˆ f(p )+pˆ pˆ f(p )
1 2 3 4 1,x 1,y 1 2,x 2,y 2
orders to momentum stopping, electrical conductivities
−pˆ pˆ f(p )−pˆ pˆ f(p )
3,x 3,y 3 4,x 4,y 4
and thermal dissipation in QCD and QED plasmas have
=−(qˆ pˆ +qˆ pˆ )f(p)−µpˆ pˆ f (p)
been estimated with simple ansa¨tze for the distribution x y y x x y 1
functions in [5]. Based on the experience with the vis- +(qˆxpˆ′y +qˆypˆ′x)f(p′)+µpˆ′xpˆ′yf1(p′), (A1)
cosity studied here, we do not expect the leading loga-
rithmic order for these transport coefficients to decrease where we have changed notation to p = p1 + q/2
bymuchwhenafullvariationalcalculationisperformed. p3−q/2 and p′ =p2−q/2 = p4+q/2. We have used
The next to leading logarithmic order for these trans- that the energy conserving δ-functions of (21) implies
port coefficients can also be estimated with the experi- pˆqˆ =pˆ′pˆ =µ. Furthermore, have defined the function
ence obtained above for the viscosity. Good estimates
f (p)=p3d(f/p2)/dp=pf′−2f, (A2)
are obtained if one in (B5) replaces q by the average 1
max
particle momenta entering the collision integral for the
that vanishes when f ∝ p2 which was the case for the
relevant transport process and q by ∼q .
min D ansatz used in section V.
A few transport coefficients are, however, different.
For small momentum transfer the matrix element (3
The second viscosity ζ is zero for a gas of massless rela-
depends onlyonenergyandmomentumtransferandthe
tivisticparticles[1]andonecannotdefineathermalcon-
azimuthalangleφ. Theδ-functionsof(21)takingcare
ductivity in a plasma of zero baryon number. One can,
energyconservationfixesthe polaranglesθ andθ with
however,considerthermaldissipationprocesses[5]where 1 2
respect to q. Thus all the angular integrals for fixed
the leading orders also can be calculated with the above
and φ reduces to rotating the three vectors q, p and p
methods. Theeffectivesoftcutoffwill,however,bediffer-
8
overallEulerangleskeepingthemfixedrelativelytoeach hΦ|Xi= 8 fp3 −∂n dp, (A13)
other. Only (Φ +Φ −Φ −Φ )2 depends on the Euler 15π2 ∂ǫ
1 2 3 4 Z (cid:18) p(cid:19)
angles and the integration or averaging over the three
we find from (18)
Eulerian angles, while keeping the relative positions of
the vectors q, p and p′ fixed, i.e. keeping µ and φ fixed, ∞ 2
gives 15 T3 fn′x3dx
η = ∞(cid:18)Z0 (cid:19) , (A14)
(qˆxpˆy+qˆypˆx)2 =(qˆxpˆ′y +qˆypˆ′x)2 = 3+15µ2 , (A3) 8πg4ln(T/qD) Z0 (f2+ 61x2f′2)n′dx
wherex=p/T. Asmentionedabove,thefunctionf(x)
1 determined by minimizing (A14). A functional variation
(pˆxpˆy)2 =(pˆ′xpˆ′y)2 = 15, (A4) withrespecttof resultsinasecondorderinhomogeneous
differential equation for f
2
(qˆxpˆy +qˆypˆx)(pˆxpˆyµ)= 15µ2, (A5) f′′+(2 + n′′)f′− 6 f =−C˜x, (A15)
(qˆ pˆ′ +qˆ pˆ′)(pˆ′pˆ′µ)= 2 µ2, (A6) x n′ x2
x y y x x y 15 wheren′′/n′ =−(1+2n). C˜isanarbitraryconstantthat,
1
(qˆ pˆ +qˆ pˆ )(qˆ pˆ′ +qˆ pˆ′)= (3pˆpˆ′+µ2), (A7) by rescaling f, can be chosen as C˜ =2 for convenience.
x y y x x y y x 15
Forx≫1wecanapproximaten≃0andsowefindthe
solution to (A15), that does not increase exponentially
(qˆ pˆ +qˆ pˆ )(qˆ′pˆ′)=(qˆ′pˆ′ +qˆ′pˆ′)(qˆ pˆ ) for x→∞, to be
x y y x y x x y y x y x
µ2 f(x)=x2, x≫1. (A16)
= (3pˆpˆ′−1), (A8)
15 For x≪1 we canapproximaten≃1/xandthe solution
to (A15) that is finite at the origin is
(qˆypˆx)(pˆ′xpˆ′yµ2)= µ302(3(pˆpˆ′)2−1), (A9) f(x)=x3(C− 15lnx), 0≤x≪1, (A17)
where pˆ = p/ǫ and pˆ′ = p′/ǫ′. Since we assume that whereC ≃0.7isaconstantthatcanonlybe determined
p p
the plasma temperature is much larger than any of the by finding the full solution to (A15) and matching it to
particle masses, the particles are relativistic and pˆ, pˆ′ (A17). This is done by a numerical Runge-Kutta inte-
andqˆ areunit vectors. The vectorproductofpˆ andpˆ is grationandtheresultisshowninFig. (6). Theviscosit
most useful in terms of µ and φ (see Fig. (2) isnowfoundbyinsertingf in(A14). Theexactvaluefor
η thusobtainedisonly0.523%lessthantheapproximate
pˆpˆ′ =µ2+(1−µ2)cosφ. (A10) value, ηgg, of Eq. (38). Since the exact value is a vari-
ational minimum, it has to be smaller than that of (28
Next we integrate over µ and φ. The µ integration is only slightly less because f ≃ x2 for large as just as
averagesµ2 to 1/3 whereasthe φ integrationis weighted theansatzof(24)andf ismainlysampledovervalues
by a factor (1−cosφ)2 from the matrix elements. Thus x=p/T ≫1becausetheintegralsoverp andp in(23
1 2
we find that (A7-A9) vanishes whereby all combinations have powers ∼ p4 to ∼ p5 times n (1+n ). In Ref. [4
p p
mixing p1 and p2 very conveniently disappear. After av- a variational calculation with trial functions f(p) ∝
eraging over both Euler angles and µ and φ we obtain lead to a minimal viscosity for ν = 2.104. This result
close to the quadratic power of (A16) but tends slightly
q2 1
h(Φ +Φ −Φ −Φ )2i= (10f2+f2+4ff ) towards the asymptotic form of (A17) (see also Fig. (6
1 2 3 4 p215 1 1 It has almost the same slope and curvature as the exact
q22 1 solution around p = 5T (note that the absolute value
= (f2+ p2f′2). (A11)
p25 6 unimportantsinceitcancelsintheviscosity). Thecorre-
sponding viscositywas 0.364%smallerthan thatof (28
Let us first consider the pure gluon plasma for which i.e., in between the exact result and the ansatz f ∝x2
(23) gives The above analysis was restricted to a pure glue
plasma. As mentioned above the distribution functions
8π
hΦ|I|Φi= g4ln(T/q )T3 are weighted with several powers of momentum and w
D
15 donotfindmuchdifferencebetweenfermionsandbosons.
∞
× f2+ 1p2f′2 −∂n dp, (A12) Thereforethedeviationfromlocalequilibriumforquarks
6 ∂ǫ willnotbemuchdifferentfromgluonsandwecanbecon-
Z0 (cid:20) (cid:21)(cid:18) p(cid:19)
fident that the ansatz, Φ ∝ p p , of Eq. (24) will be
x y
where n = (exp(p/T)− 1)−1 is the gluon distribution good approximation for quarks as well accurate within
function. Since less than a percent.
9
APPENDIX B: SOFT AND HARD q =1.26q , (B7)
min D
CONTRIBUTIONS
because the additional terms in Π lead to some addi-
L,T
tionalscreeningbesidestheDebyescreeningandLandau
The essential contribution to hΦ|I|Φi is the integral
damping of (B2) and (B3). This effective cutoff is de-
terminedbythe screeningonlyandisthereforethe same
q 1 1 qmax 1
Q max = dµ q3dq[ forgluon-gluon,quark-gluonandquark-quarkscattering.
(cid:18)qmin(cid:19) 3Z−1 Z0 |q2+ΠL(µ)|2 Whereas qmin may serveas an effective “cutoff” of small
1/2 momentum transfers interactions, it is not a parameter
+ ]. (B1)
|q2+Π (µ)/(1−µ2)|2 put in by hand as discussed in [20]. Contrarily, it
T
caused and determined by Debye and dynamical screen-
For dimensionalreasons the function Q canonly depend ing.
ontheratioofq tothemomentumscale,q ,which If the transverseinteractionsare assumedto be Deby
max min
is provided by the screening. For Debye and dynami- screened like the longitudinal ones, i.e., Π = q2, then
T D
cal screening q ∼ q whereas lattice calculations of theresultwouldhavebeenq =q exp(0.5)=1.65q
min D min D
strongly interacting plasmas give q ∼m =1.1T. ThisisbecausedynamicalscreeningofEq. (B3)islessef-
min pl
As described in connection with screening non- fectivethantheDebyescreeningof(B2)andthusresults
perturbativeeffectsbecomeimportantwhenqD∼>T which in a smaller qmin.
correspondstoαs∼<0.1. Weshalltreatthetwolimitssep- It is convenient to express the results in terms of α
arately starting with the weakly interaction plasmas for In weakly interacting plasmas we find
which the gluonself energies,Π (µ), aregivenby Eqs.
L,T q q
(8) and (9). It is straight forward to calculate Q nu- Q max ≃ln max =
q 1.26q
merically and the result will be given below, but let us (cid:18) min(cid:19) (cid:18) D(cid:19)
fibrusttiomnatkoetahsisiminptleegarnaallcyatnicableeostbitmaainteed. Tbyheinmcaluindicnogntthrie- = 12ln 4πα(qm(1ax+/TN)2/6) , αs∼<0.1. (B8)
(cid:18) s f (cid:19)
leading terms in the self energies (8,9)
The upper effective cutoff q is provided by the
max
Π (q,ω)≃q2 , (B2) quark and gluon distribution functions as discussed in
L D
π connection with Eq. (22) and it therefore varies some-
ΠT(q,ω)≃i4µqD2 . (B3) whatwithparticletype. BecauseBosedistributionfunc-
tions emphasizesmallermomentathanFermiones, q
max
Thus we find for (B1) is larger for quarks. We find q(gg) =3.0T, q(gq) =3.8
max max
(qq)
and q = 4.8T for gluon-gluon, quark-gluon, and
q 1 q2 q2 max
Q max = [ln 1+ max − max quark-quark scattering respectively. The lower effectiv
(cid:18)qmin(cid:19) 3 (cid:18) qD2 (cid:19) qD2 +qm2ax cutoff qmin is, however, the same for the three cases b
1 q4 4 causeitonlydependsonthescreeninginthegluonprop-
+ ln 1+ max( )2
4 q4 π agator. Furthermore,we findthatthe extraterms inthe
(cid:18) D (cid:19) matrix elements of (3), (4) and (5) besides the t−2 part
2q2 π q2
+ maxArctg D ]. (B4) donotcontributemuchsincetheyhavevaryingsignsand
π qD2 (cid:18)4qm2ax(cid:19) turn out to be partially cancelling. Thus the constan
within the logarithms of (37) and (40) just reflects the
Expanding in the limit q ≫ q or equivalently for
max D different q for gluon-gluon, gluon-quark and quark-
small α we obtain the leading orders up to α2 in the max
s s quark scattering.
coupling constant
Lacking screening of transverse interactions in the
static limit, it has often been assumed that some mech-
q q
max max
Q ≃ln , q ≫q , (B5) anism like Debye screening might lead to screening
max D
q q
(cid:18) min(cid:19) (cid:18) min(cid:19) transverse interactions as well, i.e. mpl = qD. Recently
lattice gauge calculations of QCD plasmas have found
where
effective screening masses of order m ≃ 1.1T near the
pl
1 4 phase transition point, T ≃180MeV. In both cases it
q =q exp 1−ln ≃1.13q . (B6) c
min D 6 π D thus assumed that
(cid:26) (cid:18) (cid:19)(cid:27)
Π =Π /(1−µ2)=m (B9)
The two terms in (B5) corresponding to lnq and L T pl
max
lnqmin are often referred to as “hard” and “soft” con- in (B1) which leads to
tributions in the literature [11].
Anumericalevaluationof(B1)withΠL,T givenbyEqs. Q qmax = 1 ln 1+ qm2ax − qm2ax .
(8)and(9)insteadof(B2)and(B3)givesaslightlylarger q 2 m2 m2 +q2
(cid:18) min(cid:19) " pl ! pl max#
value for the effective minimum momentum transfer
(B10)
10