Table Of ContentShear viscosity of pion gas due to ρππ and σππ interactions
Sabyasachi Ghosh,1,∗ Gasta˜o Krein,1,† and Sourav Sarkar2,‡
1Instituto de F´ısica Te´orica, Universidade Estadual Paulista,
Rua Dr. Bento Teobaldo Ferraz, 271 - Bloco II, 01140-070 S˜ao Paulo, SP, Brazil
2Theoretical Physics Division, Variable Energy Cyclotron Centre, 1/AF Bidhannagar, Kolkata 700064, India
We haveevaluated theshear viscosity of pion gas taking intoaccount its scattering with thelow
mass resonances, σ and ρ during propagation in the medium. The thermal width (or collisional
rate)ofthepionsiscalculatedfromπσ andπρloopdiagramsusingeffectiveinteractionsinthereal
timeformulation offinitetemperaturefieldtheory. Averysmallvalueofshearviscositybyentropy
densityratio (η/s), close totheKSSbound,is obtained which approximately matchestherange of
valuesof η/s used by Niemi et al. [25] in orderto fit the RHICdata of elliptic flow.
4 PACSnumbers: 25.75.Ag,25.75.-q,21.65.-f,11.10.Wx,51.20.+d
1
0
2 I. INTRODUCTION both Kubo and KT approaches - without unitarization,
n η decreases with T. Again in Ref. [31], it was shown
a In order to explain the elliptic flow parameter, v , ex- that a KT approach leads to an η of pionic medium
J tractedfromdatacollectedatthe RelativisticHeav2yIon that increases with T when a phenomenological inter-
1 Collider (RHIC) [1–7], hydrodynamical calculations [8– action used, while a decreasing function of T is obtained
2 12]aswellassometransportcalculations[13–16]suggest when using χPT in that same approach. An increas-
that the matter produced in the collisions is likely to ing trend of η with T has also been observed by Mitra
]
h havea verysmallratio ofshearviscosityto entropyden- et. al. [39, 40], who have incorporated a medium de-
-t sity, η/s. Recent studies [17–22] have shown that η/s pendent π − π cross-section in the transport equation
l may reach a minimum in the vicinity of a phase tran- for a pion gas. They also found a significant effect of
c
u sition - for earlier studies, see e.g. Ref. [23]. In this a temperature dependent pionic chemical potential [40].
n context, the smallness of this minimum value with re- Again, the question of magnitude of η is also an unset-
[ spect to its lower bound, η/s = 1/4π, commonly known tled issue. For example, near the critical temperature,
v1 aAsgathine KfroSmS btohuendrec[2e4n]t, wasosrukmoesf pNairetmiciuleatrasli.gn[2ifi5c],antchee. Tinc ≃Re0fs.1.7[530G,e3V9,,R4e4f]s,.[η31=,380].0p0r2ed−ict0.a0n03η ≈Ge0V.030;1aGnedVi3n;
2 transversemomentump dependenceonellipticflowpa- Refs. [26, 27], η =0.4 GeV3.
T
9 rameter extracted from RHIC data is highly sensitive to
3 the temperature dependence of η/s in hadronic matter,
5 and is almost independent of the viscosity in the QGP From these considerations, it is evident that the issue
1. phase. Thisresultattributesextraimportancetothemi- of the temperature dependence of hadronic shear viscos-
0 croscopic calculations of viscosity of hadronic matter in ity is still a matter of debate and warrants further in-
4 recent years [26–41], though these investigations began vestigation. Motivated by this, we have calculated η of
1 some time ago [43, 44]. a pion gas using an effective Lagrangian for ππσ and
:
v Calculations basedon kinetic theory (KT) approaches ππρ interactions which may be treated as an alternative
i in Refs. [35, 39, 43, 44] predict a shear viscosity η of way to describe π π cross sections up to the √s = 1
X −
pionic matter that increases with T, whereas using a GeV [39, 40] beside unitarization technique [30]. Using
ar Kubo approach, Lang et al. [38] predict η to decrease real-time thermal field theory we have calculated the in-
with T. For the interaction of pions in the medium, medium pion correlator to obtain the thermal width, a
Lang et al. [38] used lowest order chiral perturbation necessary ingredient to calculate η. We have also esti-
theory (χPT), which describes well experimental data mated the temperature dependence of the shear viscos-
on π π cross sections up to center-of-mass energies ity to entropy density ratio η/s of the pionic gas and
of √s−= 0.500 GeV. For higher energies, resonances, compared our results to others of the recent literature.
particularly σ and ρ, become important and iteration Although the hadronic matter that is formed in heavy-
of the amplitude (unitarization) is necessary to describe ioncollisionsatRHICiscomprisedofmorehadronsthan
data. In the χPT approach, σ and ρ resonancesin π π pions only, our study nevertheless is of relevance to the
scattering can be generated dynamically under un−ita- real situation as, at least in the central rapidity region,
rization. Fernandez-Fraile et al. [30] showed that un- pions are the dominantcomponentof the hadronic fluid.
der unitarization, χPT predicts η increasing with T in
In the next Section, we present the formalism used to
∗ [email protected] evaluatethe shearviscosityofapiongas. Ournumerical
† [email protected] resultsarepresentedinSec.IIIandinSec.IV,wepresent
‡ [email protected] the summary and conclusions.
2
II. FORMALISM For clarity of presentation, we start considering the
correlator in the narrow-width approximation, in which
Let us start with the standard expression of the shear the widths of the σ and ρ resonances are neglected. At
viscosity for pion gas: one-loop order - see Fig. 1 - one can write:
β d3kk6 Π1π1(k)=Π1π1(k,σ)+Π1π1(k,ρ), (5)
η = 10π2 Z Γ (k,T)ω2 n(ωk)[1+n(ωk)], (1) with
π k
d4l
where Π11(k,u)= i L(k,l)D11(l,m )D11(u,m ),
π − Z (2π)4 l u
1 (6)
n(ω )= , (2)
k eβωk 1 for eachloop(πσ orπρ), where ml =mπ, u=k l,and
− m =m for the πσ loop and m =m for the π−ρ loop;
u σ u ρ
is the Bose-Einstein distribution function for a temper- the propagatorsD11(l) are given by:
ature T = 1/β, with ω = (k2 +m2)1/2, and Γ (k,T)
k π π
is the thermal width of π mesons in hadronic matter at 1
D11(l)= − +2πin(ω)δ(l2 m2), (7)
temperature T. We note that this expression can be de- l2 m2+iη l − l
rivedeitherwiththeKuboformalism[45]usingretarded − l
correlator of the energy-momentum tensor, or with a where n(ω ) is the Bose-Einstein distribution given in
l
kinetic approach using the Boltzmann equation in the Eq. (2); and
relaxation-time approximation[46]. In both approaches,
to evaluate Γ (k,T) one needs the interactions of the g2m2
π L(k,l)= σ σ, (8)
pions in medium. Here, we pursue the use of retarded − 4
correlators.
for the πσ loop, and
As mentioned previously, from the lowest order χPT,
tahgereeesmtiemnattwedithπt−heπecxrpoessrimseecntitoanl dinatfareuepsptaoctehiescweenltleirn- L(k,l)= gρ2 k2 k2 m2 +l2 l2 m2
of-mass energy √s=0.5 GeV. Beyond this value of √s, −m2ρ n (cid:0) − ρ(cid:1) (cid:0) − ρ(cid:1)
the σ and ρ resonances play an essential role to explain 2 (k l) m2+k2l2 , (9)
the data. On unitarization, the σ and ρ resonances are − (cid:2) · ρ (cid:3)o
generated dynamically [30] in the amplitude. An alter-
for the πρ loop.
native way, which we follow in the present paper, is to
incorporatethese resonancesby using the effective inter-
action for ππσ and ππρ interactions:
g
=g ρ π ∂µπ+ σm π πσ, (3)
L ρ µ· × 2 σ ·
where the coupling constants g and g are fixed from
ρ σ
their experimental decay widths. We use this effective
Lagrangian to calculate the contributions of the πρ and
πσ loops to the self-energy of π meson at finite temper-
ature. The contributions coming from the interactions
of the pions in medium, which are the relevant ones for
Γ (k,T) in Eq. (1), can be obtained from the imaginary
π
part of the retarded pion correlator ΠR(k) evaluated at
π
theπ-mesonpole,k =(k =ω ,k). Inreal-timethermal
0 k
fieldtheory,thisrelationshipcanbeexpressedas[47,48]: FIG. 1. One-loop self-energy diagram of pion.
1
Γ (k,T)= ImΠR(k)
π −mπ π |k0=ωk Using Eq. (7) in Eq. (6), one can perform the l0 inte-
grationand,fromtherelationbetweenImΠR andImΠ11
βk 1
= tanh 0 ImΠ11(k) . (4) in Eq. (4), one obtains:
− (cid:18) 2 (cid:19)m π |k0=ωk
π
d3l
ImΠR(k,u)= L(k,l) 1+n(ω )+n(ω ) δ(k ω ω ) n(ω ) n(ω ) δ(k ω +ω )
π Z 32π2ωlωun |l0=ωlh(cid:0) l u (cid:1) 0− l− u − (cid:0) l − u (cid:1) 0− l u i
+L(k,l) (n(ω ) n(ω ))δ(k +ω ω ) 1+n(ω )+n(ω ) δ(k +ω +ω ) . (10)
|l0=−ωlh l − u 0 l− u − (cid:0) l u (cid:1) 0 l u io
3
TheDiracdeltafunctionsprovidebranchcutsinthek - In the integration limits, m± = m 2Γ0, with Γ0 =
0 u u ± u σ
axis,identifyingthedifferentkinematicregionswherethe Γ (M =m )andΓ0 =Γ (M =m ). InviewofEq.(5),
σ σ ρ ρ ρ
imaginary part of the pion self-energy acquires non-zero the total pionic width is the sum
values. Therelevanttermforthein-mediumdecaywidth
is the one proportionalto n(ωl) n(ωu), whichis due to Γπ(k,T)=Γπ(k,T,ρ)+Γπ(k,T,σ). (19)
−
the interactionsof in-medium pions only andvanishes in
A quantity closely related to the thermal width is the
vacuum. Therelevantbranchcut,theLandaucut,isthe
region [k2+(m m )2]1/2 k [k2+(m m )2]1/2; mean free path:
u π 0 u π
− − ≤ ≤ −
it gives:
k
1 ω− λπ(k,T)= ω Γ|(|k,T). (20)
Γnw(k,T,u)= dωL(ω) k π
π 16π km Z
| | π ω+ Phenomenologically,analysis of this quantity is interest-
ingforgettingfurtherinsightinthepropagationofpions
[n(ω) n(ω +ω)], (11)
k
× − in medium; in particular, it allows to know the values
where the superscript nw indicates that this expression
oftypicalpionmomenta that are responsiblefor dissipa-
is obtained in the narrow-widthapproximation, and
tion in medium, as we shall discuss in the next section.
R2 Onthetheoreticalside,thisquantityisinterestingas[52]
ω± = 2m2π (−ωk±|k|W), (12) λdeπp≡en1d/eΓncπeinoftλhe cphriorvaildleismiint,simghπt=on0;effaescstsucdhu,ethteomexπ-
π
with R2 =2m2 m2 and W = 1 4m4/R4 1/2, and plicit chiral symmetry breaking [38].
π− u − π
(cid:0) (cid:1)
L(ω)=L(k =ω ,k,l = ω, l = ω2 m2). (13)
0 k 0 − | | − π III. RESULTS AND DISCUSSION
p
The physical interpretation of the Landau cut contri-
butions is straightforward [49]. During propagation of
Let us first consider the separate contributions of the
π+,itmaydisappearbyabsorbingathermalizedπ−from
πρ and πσ loops to the imaginary part of the pion self-
the medium to create a thermalized ρ0 or σ. Again the
energyasafunctionoftheinvariantmassm2 =k2 k2
π+ may appearby absorbinga thermalizedρ0 or σ from 0−| |
for fixed values of temperature, T = 0.150 GeV, and
the medium as well as by emitting a thermalized π−.
three-momentum, k =0.300GeV-resultsareshownin
nl(1+nu) and nu(1+nl) are the corresponding statis- Fig.2. Wehaveuse|d|herethefollowingsetofparameters:
tical probabilities of the forward and inverse scattering
m =0.140GeV,m =0.770GeV,Γ0 =0.150GeV,and
respectively. By subtracting them, one gets the factor π ρ ρ
g =6. The parameters for the σ resonance are those of
(n n ) in Eq. (11). ρ
l− u Set 1 in Table I.
Next,totakeintoaccountthewidthsoftheresonances,
In Fig. 2, the dashed lines clearly indicate the sharp-
weusethe spectralrepresentationsoftheσ andρpropa-
ends of the Landau cuts at m = m m = 0.630 GeV
gatorsinEq.(6)-seee.g. Refs.[50,51]. Thisresultsina ρ− π
for the πρ loop (upper panel) and at m = m m =
folding of the narrow-width expression for Γπ(k,T,mu): 0.250 GeV for the πσ loop (lower panel). Thσes−e shπarp
1 (m+u)2 endsturnintosmoothfalloffsatlargevaluesofm dueto
Γπ(k,T,mu)= Nu Z(m−u)2 dM2ρu(M)Γnπw(k,T;M), othneanfocledsi.ngThwiisthlatrhgee-smpeecfftreacltfudnocetsionnostoaffftehcetσΓa(nkd,Tρ)reass-
(14) π
this quantity is calculated at m=m . However, folding
whereΓnw(k,T;M)isthenarrow-widthexpressiongiven π
does affect Γ (k,T) via a large effect induced by the πσ
in Eq. (11), with m replaced by M; ρ (M) is the spec- π
u u channel; at m = m , folding decreases the contribution
tral density: π
ofthe πσ loopby50%ascomparedtothe corresponding
1 1 contribution in the narrow-width approximation. This
ρ (M)= Im − , (15)
u π (cid:20)M2 m2 +iMΓ (M)(cid:21) does not come as a surprise, as the σ resonance has a
− u u largewidth,whilethe widthoftheρ isnotaslarge. One
and N is the normalization
u should also notice that numerically, the contribution of
(m+u)2 the ρ resonance to Γπ is one order of magnitude larger
Nu =Z(m−u)2 dM2 ρu(M). (16) tshhaanlltsheeesohnoerftrloy,mthtihsedσoelosonpotatmmea=ntmhaπt. oHnoewceavnern,eagslewcet
Γ (M), u=σ,ρ, are the spectral widths of the mesons: the σ resonance altogether.
u
3g2m2 4m2 1/2
Γ (M)= σ σ 1 π , (17) Next, we consider the momentum dependence of ther-
σ 32πM (cid:18) − M2 (cid:19) mal width and of the mean free path for a fixed temper-
ature. Results are shown in Fig. 3. First of all, one sees
Γ (M)= gρ2M 1 4m2π 3/2. (18) that the effects of folding are not big when considering
ρ 48π (cid:18) − M2 (cid:19) thejointcontributionsoftheπρandπσloops-thisisdue
4
0.25
πρloop 0.3
0.2 0.25
0.15 GeV) 0.2 with folding
0.1 Γ (π0.15 without folding
V) 0.1 T=0.150 GeV
Π/m (Geππ0.005 0.005
m0.04 T=0.150 GeV
-I k=0.300 GeV 60
with folding
0.02 without folding λ (fm)π40
20
πσloop
00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 00 0.4 0.8 1.2 1.6 2
m (GeV) k (GeV)
FIG. 2. The imaginary part of pion self-energy from πρ (up- FIG.3. Momentum dependenceofthethermalwidth (upper
per panel) and πσ (lower panel) loops as function of the in- panel) and of the mean free path (lower panel) for a fixed
variant massm=pk02−|k|2 for fixedvaluesoftemperature value of temperature, T = 0.150 GeV. Parameters are the
T = 0.150 GeV and three-momentum |k| = 0.300 GeV. The same as in Fig. 2.
vertical dotted line indicates theon-shell valuem=mπ. Pa-
rameters are: mπ = 0.140 GeV, mρ = 0.770 GeV, Γ0ρ =
0.150 GeV, gρ = 6 and Set 1 in Table I for parameters of σ 0.4
resonance. without folding
0.3
with folding
GeV) 0.2 k=0.300 GeV
Γ (π
tothecombinedfactsthatthewidthofρhasonlyamild 0.1
effectandthedominanceoftheπρloopovertheπσ loop.
0
Onealsoseesthatthevalueofλ isverybigformomenta
π
0.100 GeV k 0.300 GeV, but for k 0.400 GeV 60
≤ | | ≤ | | ≥
the value of mean free path varies very little, reaching
m)
an average value of λπ ≃ 25 fm. In a typical relativis- λ (fπ40
tic heavy ion collision at RHIC, the size of the hadronic
20
systems produced after freeze-out varies between 20 fm
and 40 fm. Therefore, scattering processes with center 0
0.12 0.125 0.13 0.135 0.14 0.145 0.15 0.155 0.16 0.165 0.17
of mass momenta larger than k =0.400 GeV are those T (GeV)
| |
responsiblefordissipationinthemedium,atleastforthe
chosen temperature T =0.15 GeV. FIG.4. Temperaturedependenceofthethermalwidth(upper
panel)andofthemeanfreepath(lowerpanel)forafixedvalue
of momentum |k| = 0.300 GeV. Parameters are the same as
In Fig. 4 we present results for the temperature de-
in Fig. 2.
pendence of the thermal width (upper panel) and of the
meanfreepath(lowerpanel)forafixedvalueofmomen-
tum k = 0.300 GeV. Clearly, folding does not affect
| |
much the temperature dependence of these quantities; perature dependence of η. Interestingly we see that the
the reason for this is the same as for their momentum πρ and πσ contributions play a complementary role in
dependence: the dominance of the contribution of the η to be non-divergent in the higher (T > 0.100 GeV)
πρ loop over that from πσ loop. The figure also shows and lower (T <0.100 GeV) temperature regions respec-
that only temperatures larger than T = 0.120 GeV give tively. The lesson here is that consideration of both res-
a mean free path smaller than the typical size of the onancesinπ π scatteringisstrictlynecessarytoobtain
−
hadronic system produced in a typical heavy ion colli- a smooth, non divergent η for temperatures below the
sion at RHIC. critical temperature, T 0.175 GeV. Moreover,though
c
≃
η at very low temperatures (T < 0.020 GeV) tends to
become very large in the narrow-width approximation
Of course, the viscosity of the pion gas is determined
not only by the value of Γ (or λ ), which is given ba- (upper panel),this trenddisappearsaftertakingintoac-
π π
count the the widths of the resonances (lower panel).
sically by the π π interaction; it depends also on the
−
momentum distribution of the in-medium pions, which
is determined by the temperature in the Bose-Einstein We have compared our results with the earlier results
distribution. InFig.5wepresenttheresultsforthetem- in Kubo approach by Fernandez-Fraile et al. [30] and
5
0.0015 0.003
without folding
tot with folding
πσloop
0.001 πρloop RReeff..[[3308]]
Ref.[39]
0.0005 0.002
3GeV) 0 3GeV)
η ( η (
0.001
0.001
0.0005
0 0
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
T (GeV) T (GeV)
FIG. 5. Temperature dependence of η from the πσ (dashed FIG. 6. Resultsof η vs T obtained in this work compared to
lines)andπρ(dottedlines)loops. Thelowerandupperpanels some other results.
respectively show theresults with and without folding.
TABLE I. The mass mσ (in GeV) and vacuum width Γ0σ (in
GeV)oftheσresonancetakenfromRefs.[55–57],fromwhich
Lang et al. [38], along with previous results obtained by
thecorresponding coupling constants gσ are extracted.
someofus[39]inaKT approach. Inthe KT approaches
of Refs. [35, 39, 43, 44], the predicted η is a monotoni-
cally increasing function of temperature in the tempera- mσ Γ0σ gσ
ture range 0.100 GeV < T < 0.175 GeV and vanishing
baryon chemical potential (µ = 0). The results of Lang
Set 1 (BES) [55] 0.390 0.282 5.82
et al. [38] obtained with the Kubo approach indicate an
η decreasing in that same temperature range. Similar
Set 2 (E791) [56] 0.489 0.338 5.73
trends are obtained by Fernadez-Fraile et al. [30] with
the Kubo-approach without unitarization of Γ, but the Set 3 (PDGmin) [57] 0.400 0.400 6.85
trend is reversed when dynamically generated (through
unitarization) ρ and σ resonances come into play. Our Set 4 (PDGmax) [57] 0.550 0.700 7.03
calculations,basedonaneffectiveLagrangiantakinginto
account the low-mass σ and ρ resonances, found a simi-
lar trend of an increasing η with T for T > 0.100 GeV,
although smaller in magnitude and slope, lending sup- In the calculation of the entropy density,
port to other calculations which take into account those
resonances. d3k k2
s=3β ω + n(ω ), (21)
Z (2π)3 (cid:18) k 3ω (cid:19) k
k
Now we concentrate on the sensitivity of our predic-
tions associated with phenomenological uncertainty of where ωk = k2+m∗π2(k,T), we have explored the ef-
q
theparametersoftheσ resonance. Theresultspresented fect of the loops in the real part of ΠR on the effec-
π
above have been obtained by choosing (arbitrarily) the tivepionmass,m∗ = m2 +ReΠR(k =ω ,k,T). The
π π π 0 k
parameters of Set 1 shown in Table I. Although long- variation of m∗ withpT for two different values of k is
π
standing controversies about the properties of this res- shownin the lowerpanelofFig. (8). As canbe seen, the
onance seem to be settling to a consensus [53], recent effect is not big, at most 15% for the highest values of
literature[54]stillshowsconflictingvaluesforthoseprop- momentum and temperature. The effect of this change
erties, as one can see in Table I. We have explored the in the pion mass on the entropy density is marginal and
impactofthedifferentvaluesfortheσparameters;there- can be safely neglected.
sultsareshowninFig.7. Ascanbeseen,allsetspredictη
tobesmall,althoughparametersetswithsmallerwidths
The dependence of the ratio η/s on T is shown in the
predictsmallerη’satlowtemperatures;forT >0.1GeV,
upper panel of Fig. (8). Our results respect the KSS
all sets predict essentially the same result.
bound η/s 1/4π, as indicated by the dotted line. We
≤
recallthatNiemietal.[25]intheirinvestigationofv (p )
2 T
Finally, we estimate the temperature dependence of of RHIC data, have used an η/s(T) from Ref. [33] that
shearviscositytoentropydensityratioη/sinourmodel. is in the same range of our results shown in the figure.
6
IV. SUMMARY
Set 1
Set 2
0.0006 Set 3 We have calculated the shear viscosity of a pion gas
Set 4
at finite temperature taking into account the low mass
resonancesσ andρonpionpropagationinmedium. The
3V)0.0004 thermal width Γπ is calculated from one-loop pion self-
Ge energy at finite temperature in the framework of real-
η ( time thermal field theory. We have evaluated the contri-
butions of πσ and πρ loops to the pion self-energy with
0.0002 the help of an effective Lagrangianfor the σππ and ρππ
interactions. To take into account the widths of σ and ρ
resonances, we have folded the zero-widths self-energies
withtheirspectralfunctions. Wehaveseenacomplemen-
0
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 tary role played by the πσ and πρ loops in producing a
T (GeV)
smooth temperature dependence for η.
FIG.7. Thebandof uncertaintyofη in thelow temperature We have also explored the impact of uncertainties in
domain for different sets of mσ, Γ(mσ) and gσ from Table I. the parameters of the σ resonance on our results. Using
therangeofσ mass(m =0.400 0.550GeV)andwidth
σ
−
(Γ = 0.400 0.700 GeV) from the latest PDG compi-
σ
−
0.5 lation [57], we have obtained smaller values for η at low
0.4 with folding temperaturesthanthosewhenusingtheearlierPDGval-
without folding ues[58],(m =0.400 1.200GeVandΓ =0.600 1.00
0.3 σ − σ −
η/s GeV). For temperatures largerthan 0.1 GeV, all param-
0.2 eter sets give essentially the same value for η
KSS bound
0.1 Our estimated temperature dependence for the ratio
η/srespectstheKSSboundη/s 1/4π,andcomesvery
0 ≤
closetotheboundfortemperaturesnearthecriticaltem-
0.16
k=0
perature T = 175 MeV. It agrees with the results of
V) 0.155 k=0.300 GeV Refs. [32,3c3]. Fromthe recentworkby Niemi et al.[25],
(m*Geπ0.01.4155 tshuechelslimptailcl flvoawluepsaroafmηe/tse(rTv)2(foPrT)haodfrRonHiIcCmdaatttaerp.reTfehres
0.14 results seem to provide experimental justification to the
microscopic calculations of shear viscosity which include
0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17
T (GeV) σππ and ρππ interactions, as the one performed in the
present work.
FIG. 8. Upperpanel : η/s vs T and the KSS bound (dotted
line). Lowerpanel: Dependenceoftheeffectivepionmassm∗
π
with temperature T at two different values of three momen-
ACKNOWLEDGMENTS
tum |k|.
Work partially financed by Funda¸ca˜o de Amparo `a
Pesquisa do Estado de Sa˜o Paulo - FAPESP, Grant
This alsolends supporttothe validity ofthe physicalin- Nos. 2009/50180-0 (G.K.), 2012/16766-0 (S.G.), and
put our phenomenological analysis, in that the σ and ρ 2013/01907-0(G.K.); Conselho Nacional de Desenvolvi-
resonances play a decisive role in the dissipation proper- mento Cient´ıfico e Tecnol´ogico - CNPq, Grant No.
ties of the pion gas. 305894/2009-9(G.K.).
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