Hydrodynamic scaling in an exactly solvable model
Based on 1407.5952 with Yoshitaka Hatta,Bowen Xiao, Jorge
Noronha
G.Torrieri
What we think we know
High pT distributions determined by tomography in dense matter
Low pT distributions determined by hydrodynamics
Missing: A connection of this to a change in the degrees of freedom (onsetof deconfinement): How do opacity, η/s , EoS etc. change at that point?
Hydrodynamics can be used as a tool to connect statistical physics (moreor less understood) to particle distributions
A phase transition and/or a cross-over implies scaling violations
η/s~Nc
2
dip(crossover)
η/s~0.1
−2
Resonances?Hagedorn
η λ2
/s~ ~Ln(T)
At T0 ≃ Tc speed of sound experiences a dip (not to 0,as its a cross-over,but
a dip). Above Tc, η/s ∼ N0c , below Tc, η/s ∼ N2
c . We should expect...
life
phase
InitialT
Initial µ
Phase 2
Phase 1
Data across1/2s , A,Npart
Transition/threshold
Sdydy
dN dN<N> (Or , ,...)
(Intensive quantities)
v2
An change in v2 as the system goes from the viscous hadron gas regime viaa kink in the speed of sound to the sQGP regime.
140
145
150
155
160
165
170
175
180
185
190
1 10 100 1000
T [M
eV
]
A
p-p C-C Si-Si Pb-Pb
√ sNN = 17.2 GeV
Not a hit of this is seen! Why?
140
145
150
155
160
165
170
175
180
185
190
1 10 100 1000
T [
MeV
]
A
p-p C-C Si-Si Pb-Pb
√ sNN = 17.2 GeV
lots of correlated parameters (Qs, η/s, T0(y), µ0(y),freezeout,... ) Need3D viscous hydro to investigate interplay between: EoS,η/s,τΠ,transverseinitial conditions,longitudinal initial conditions,pre-existing flow,freeze-outdynamics, jet showers in-medium, fragmentation outside the medium .... .No jump clearly seen! In which parameters is the phase transition hiding?
The problem!
η/s
Equation of state
Rapidity dependence
Initial flow
"With enoughparametersyou can fit..."
vn from ALICEfits well with a
NAIVEmodel with5 parameters
We understand the equation of state and hopefully the viscosity from firstprinciples. But initial conditions and their dependence in energy, andtransport coefficients, and jets, and freezeout... Even when you are tryingto fit lots of data simultaneusly, a model with many correlated parameterscan describe nealry any physical system
Some people think that this will always be with usThe system we are studying is so complicated that models with lots ofparameters will always be necessary and well never have a “smoking gun”link between theory and experiment.
Perhaps, but I would not give up just yet!
• By decreasing energy
Tinitial,final decreases, µB increasesLifetime increases Flow etc has more time to developPhases change Intensive parameters change (η/s,,opacity, EoS )Boost-invariance breaks down (regions at different rapidities talk)
• By decreasing system size (pA at high√s is an extreme example)
Tfinal increases, Lifetime decreasesGradients go up , driving up Knudsen number lmfp/R ≃ η/(sTR)
Thermalization/medium “turns off”
• By varying rapidity Initial density decreases (Phase changes? )(pA also effectively more ”forward” than AA at central rapidity)
All these need to be compared against intensive variable 1SdNdy ?
Buckingam’s theorem (How to do hydro, circa 19th century)Any quantitative law of nature expressible as a formula
f(x1, x2, ..., xn) = 0
can be expressed as a dimensionless formula
F (π1, π2, ..., πn−k) = 0
whereπi =
∏
xλii ,
∑
λi = 0
Widely applied within hydrodynamics in the 19th century: Knudsen’snumber, Reynolds number, Rayleigh’s number, etc.
Since we are varying a whole slew of experimental (y, pT , Npart,√s, A) And
theoretical (T, µ, η, s, q̂, τ0, τlife) parameters it would be nice to representheavy ion observables this way
This is how hydrodynamicswas done in the 19thcentury!!!!
The idea:when you have a pipeand you make ittwice as bigdoes your variableof interest grow as
n
2 ? What is n?
s,A,Npart,y
dN/dy,<pT>,vn
η/ s,Cs,...
Heavy ion−specificdimensionless number "O"
life
phase
InitialT
Initial µ
Phase 2
Phase 1
Data across1/2s , A,Npart
Transition/threshold
Sdydy
dN dN<N> (Or , ,...)
(Intensive quantities)
<O>
µ,εT,<R>,life,
So, when you double size (or initial temperature, or whatever) how doesvn, pT , ... change? Given enough variable conditions, a scaling dimensionlessnumber makes it straight-forward to look for scaling violations
“And the theorist says.... Consider a spherical elephant in a vacuum”
η/s
Initial flow
The shortest course possible on hydro I:EvolutionThe 5 energy momentum conservation equations
∂µTµν = 0
have 10 unknowns. They can be closed by assuming approximate isotropy
Tµν = (p+ ρ)uµuν + pgµν + η∆µναβ∂αuβ + ζ∆µνα
α ∂βuβ
And thermodynamic equations for p, η, ζ in terms ofρ .
Once closed these equations can be integrated from initial conditions
The shortest course possible on hydro I:FreezeoutAt a critical condition (here critical T ) the fluid has to convert into particles.Energy-momentum and entropy conservation, plus ”fast” conversion, forcethe Cooper-Frye formula
EdN
d3p=
1
pT
dN
dpTdydφ=
∫
pµdΣµf(pµuµ, T )
If Σµ is the locus of constant T , parametrized by t(x, y, z, T ) then
dΣµ = ǫµαβγdΣα
dx
dΣβ
dy
dΣγ
dz
In this formalism
vn =
∫
cos(nφ)dN
dpTdydφdφ
A ”semi-realistic” but solvable model: A deformed Gubser solutionGubser flow includes
Viscosity , finite Knudsen number
Transverse flow with ”Conformal” setup
We add
Inhomogeneities parametrized by dimensionless ǫn
Freeze-out isothermal Cooper-Frye
The basic idea Conformal invariance of the solution constrains flow to be,in addition to the usual Bjorken
u⊥ ∼ 2τx⊥
L2 + τ2 + x2⊥, uz ∼
z
t
plugging this into the Relativistic Navier-Stokes equation gives yousomething you can solve
ENS = λT 4NS =
1
τ4λC4
(cosh ρ)8/3
[
1 +η09λC
(sinh ρ)3 2F1
(3
2,7
6,5
2;− sinh2 ρ
)]4
where
sinh ρ = −L2 − τ2 + x2⊥
2LτNB: issues at ρ ≪ −1 (negative temperature!) Physically this reflectsimplicit non-causality of NS limit, see 1307.6130 (Noronha et al) to fix this
Not (yet!) the real world:
• Strictly conformal EoS (s ∼ T 3, e ∼ T 4 ) and viscosity (η ∼ s ≡ η0s )
• Azimuthally symmetric
• Transversely much more uniform than your “average” Glauber
• “Small times”, or temperature becomes negative (Israel-Stewart needed).Temperature becomes negative (i.e., the solution becomes unphysical)for
τL
L or x⊥≫( η
sC
)3/2
Where C is an overall normalization constant ∼ dN/dy . NB limitationof the solution ansatz!
Azimuthal asymmetries: The Zhukovsky transform
x →(
x⊥ +a2
x⊥
)
cos (nφ) , y →(
x⊥ − a2
x⊥
)
sin (φ)
In two dimensions this is a conformal transformation, so it transforms asolution into a solution up to a calculable rescaling up to a volume rescaling.This can be neglected to O
(a2/x2
⊥, τa2/x3
⊥)(Again, early freezeout )
To first order in a/L (i.e., ǫn ≪ 1 ) we get
E ≈ λC4
τ4/3(2L)8/3
(L2 + x2⊥)
8/3
(
1− η02λC
(L2 + x2
⊥2Lτ
)2/3)4
×[
1− 4ǫn
(
1 +η02λC
(L2 + x2
⊥2Lτ
)2/3)(
2Lx⊥
L2 + x2⊥
)n
cosnφ
]
,
Deformation breaks down at τ ≃ L
this can be solved for an expression of an isothermal surface, ready forfreeze-out
T 3 =C3(2L)2
τ(L2 + x2⊥)
2
(
1− η02λC
(L2 + x2
⊥2Lτ
)2/3)3
×
[
1− 3ǫn
(
1 +η02λC
(L2 + x2
⊥2Lτ
)2/3)(
2Lx⊥
L2 + x2⊥
)n
cosnφ
]
≡ C3B3
(2L)3,
C: overall multiplicity. B Lifetime of the systemNB: Need B ≫ 1, so lifetime ≪ L, “early” freezeout w.r.t. size. .
Now we are set
f(~p) =dN
pTdpTdydφ=
∫
dσµpµ exp
(
−uµpµ
T
)(
1 +Πµνpµpν
2(e+ P )T 2χ(p)
)
where
χ(p) = 1, πµν = (gµα − uµuα) ∂αuν, σµ = T 3ǫµναβ
dxν
dT
dxα
dT
dxβ
dT
and
dN
dy=
∫
dpTpTdφf(~p), 〈pT 〉 =∫
dpTp2Tdφf(~p), vn =
∫
dpTpTdφf(~p) cos (2nφ)
we can analytically map
L, T, ǫn,η
s, B ⇔ dN
dy, 〈pT 〉 , vn
After quite a bit of Algebra... (2π)3 dNdY pTdpTdφp
≡ J1 + J2 + J3 .
J1 = 4πmTK1(mT/T )∫ ∞0
dx⊥x⊥τ0
I0(z) (1 − βπ)+
δττ0In(z)ǫn cosnφp +(1−
βπ)pT2T
[(δu⊥ −
δuφx⊥
)In−1(z) +
(δu⊥ +
δuφx⊥
)In+1(z)
]ǫn cosnφp
J2 = −4πpTK0(mT/T )∫ ∞0
dx⊥x⊥τ0
∂τ0∂x⊥
I1(z)(1 − β̃π
)+
∂τ0∂x⊥
δττ0I ′n(z)ǫn cosnφp
+(1−β̃π)
∂τ0
∂x⊥pT2T
((δu⊥ −
δuφx⊥
)I ′n−1(z) +
(δu⊥ +
δuφx⊥
)I ′n+1(z)
)+ ∂δτ
∂x⊥I ′n(z)
ǫn cosnφp
J3 = −4πpTK0(mT/T )∫ ∞0
dx⊥τ0n2δτz In(z)(1− β̃π)ǫn cosnφp ≡ δJ3ǫn cosnφp .
where Jn = Jn0 + δJnǫn z ≡ pTu⊥0T = 2x⊥pT (2L)5
TB3(L2+x2⊥)3
(1− α) .,
Expanding linearly in ǫn and pT/(TB3), In(x) ∼ xn/2nn!
J01 = 4πmTK1(mT/T )16L
3
B3
{1 − κx2⊥max
64L2
(6 +
m2T
2T2K3−K1
K1− p2T
T2
)},
J02 = 4πK0(mT/T )
215L3p2TTB9
{121 − κ
640
(12 +
m2T
T2K2−K0
K0− p2T
T2
)},
δJ1 = 4πmTT K1(mT/T )Γ(3n)Γ(4n)
9·26nL3pnT
B3(n+1)Tn−1×(n−1)
{2(3n+2)4n+1 − nκ
8(3n−1)
(6n + 6 +
m2T
2T2K3−K1
K1− p2T
T2
)
δJ2 = 4πK0(mT/T )Γ(3n)Γ(4n)
9·26nL3pnT
B3(n+1)Tn−1×2n
{6n2−6n−5
4n+1 − (6n2−10n+1)κ48(3n−1)
(6n +
m2T
T2K2−K0
K0− p2T
T2
)},
δJ3 = 4πK0(mT/T )Γ(3n)Γ(4n)
9·26nL3pnT
B3(n+1)Tn−1×2n
{1 − (4n−1)κ
48(3n−1)
(6n +
m2T
T2K2−K0
K0− p2T
T2
)},
Low pT vn(pT/(TB
3) ≪ 1), but B ≫ 1
vn(pT )
ǫn=
9(n− 1)
32
Γ(3n)
Γ(4n)
(64pTB3T
)n [2(3n+ 2)
4n+ 1− nκ
8(3n− 1)
(
6n+ 9 +2mT
T− p2T
T 2
)]
The v2 and “Knudsen number” for this solution:
vnǫ
∼ O((pT
T
)n)
(1−K) , K ∼ η
s
(L
τ
)2/3
A bit different from Gomebaud et al, Lacey et al(vnn ∼ n
TR
)Sensitivity
to form of solution , Interplay of L, τ
NB: vn(pT ) ∼ pnT phenomenologically important general prediction(Depends on azimuthal integral, independent of approxuimations!
vn ∼ pnT : A robust predictionAll it requires is that
vn ∼∫
dφ cosφ (1− tf cos(φ) exp [γ (E − vT (φ)pT )]) ∼ In
(
O(pTT
))
∼(pTT
)n
This is much more robust than the assumptions of Gubser flow
A large momentum region, pT ≫ TB3 is also possible,
In(z) ≈ez√2πz
∼ exp
(pTT
2x⊥(2L)5
B3(x2⊥ + L2)3
(1− α)
)
.
The x⊥-integral can be evaluated by doing the saddle point at x∗⊥ = L/
√5.
The result is
vn(pT ) ≈ǫn2
pTTδu∗
⊥0 = ǫn500pT27TB3
(√5
3
)n−1(
n− 1− 27κ
200n
)
.
but jet contamination likely. Experimental opportunity to see how scalingofvn(pT ) changes with n, pT∼ pnT@low pT , ∼ pT@High pT . NB: High, low w.r.t. T×Size/Lifetime≫ 1
The role of bulk viscosityPlugging in the 14-moment correction of the distribution function
δf bulk
feq=
12T 2
m2
[
12 +8
Tuµp
µ +1
T 2(uµp
µ)2]∇µu
µ
T
ζ
S,
and assuming early time ∂µuµ ∼ 1/τ , we carry these terms to be
δvbulkn ≈ 81
128
(128
B3
)nn2(n− 1)Γ(3n)
Γ(4n)Γ2(n
2
)((3n+ 2)2
4(4n+ 1)
x2max
L2− 3n
3n− 1
)B2ζ
CSǫn ,
Shear and bulk viscosity compete with terms which may be of opposite signand non-trivial contribution, Confirming the numerical work of Noronha-Hostler et al
vnvidealn
− 1 ∼ ±n2T2
m2κbulk ,
Now we fix K,C,B in terms of bulk obvservablesThese are dominated by soft regions, so can calculate
dN
dY=
1
(2π)2
∫
dpTpT (J01 + J0
2 ) ≈4C3
π
〈pT 〉 ≡(dN
dY
)−1 ∫
pTdpTdN
dY dpT≈ 3πT
4=
3πCB
8L
Therefore
C ∼(dN
dY
)1/3
,1
B3∼ 1
〈pT 〉3L3
dN
dY.
As for azimuthal coefficients, these are
(vn(pT )
ǫn
)1/n
∼ pT
A3/2⊥ 〈pT 〉4
dN
dY(1−nκ) ,
(vnǫn
)1/n
∼ 1
A3/2⊥ 〈pT 〉3
dN
dY(1−nκ) ,
Note that vn ∼ pnT robust against assumptions we made, should survive forrealistic scenarios where the “knudsen number” is
κ ∼ B2
C
η
S∼ A⊥〈pT 〉2
dN/dY
η
S, , A⊥ ∼ L2, A
3/2⊥ ∼ Npart
NB: this is a bit different from Bhalerao et al , as well as GT,1310.3529
v2ǫ2
∼ f(τ) (const.−O (κ))
Plugging in some more empirical formulae
dN
dY∼ Npart(
√s)γ , 〈pT 〉 ∼ F
(
1
N2/3part
dN
dY
)
∼ F(
N1/3part(
√s)γ)
,
where γ ≈ 0.15 in AA collisions and γ ≈ 0.1 in pA and pp collisions, andF is a rising function of its argument, we get
(vnǫn
)1/n
∼ (√s)γG
(
N1/3part(
√s)γ)
(1−nκ) , κ ∼ H(
N1/3part(
√s)γ) η
S,
where G(x) = F−3(x) and H(x) = F 2(x)/x.
Flow... the experimental situation
0 10 20 30 40 50 60p
T
-0.5
0
0.5
1
1.5
2
2.5
3
v2(p
T)/
<v
2>
0-10%10-20%20-30%30-40%40-50%
0 5 10 15 20
pT (GeV)
-2
0
2
4
6
v2(p
T)/
<v
2>
CMS 0-5%60-70%PHENIX 0-10%50-60%
BRAHMS,NPA 830, 43C (2009)
pT
CMS1204.1850
CMS1204.1409
PHENIX PRL98, 162301 (2007)
PHENIXPRL98:162301,2007
CMSPRL109 (2012) 022301
NPA830 (2009)PHOBOS
STAR 1206.5528
Here is what we know experimentally
v2 ≃ ǫ(b,A)F (pT ), 〈v2〉 ≃∫
dpTF (pT )f(
pT , 〈pT 〉y,A,b,√s
)
F (pT ) universal for all energies , f(pT ) tracks mean momentum, ∼ 1SdNdy
This is an experimental statement, as good as the error bars. Very differentfrom our scaling!
knewthis:for years
and we
Wrong power w.r.t.
(vn(pT )
ǫn
)1/n
∼ pT
A3/2⊥ 〈pT 〉4
dN
dY(1−nκ) ,
(vnǫn
)1/n
∼ 1
A3/2⊥ 〈pT 〉3
dN
dY(1−nκ) ,
but since κ ∼ 1A⊥
dNdy , it is enough to “naively extrapolate” from B2 ∼ O (1)
to B2 ∼ O (L/τ). Extra A1/2 power enough for scaling but Need Realistichydrodynamics to test this extrapolation
pT
BRAHMS,0907.4742v2
nucl−ex/0608033PHENIXAu−Au,Cu−Cu
Low energy scan, STAR 1206.5528
v2(pT ) constant (at least at high pT ).Definitely not dependent on 〈pT 〉 as in
(vn(pT )
ǫn
)1/n
∼ pT
A3/2⊥ 〈pT 〉4
dN
dY(1− nκ)
unless κ depends funnily on dN/dy . Problem also with realistic calculations.
LHC vn(pT ) data allows us to test vn ∼ pnTa robust prediction, based on In ≃ (z/2)n/n! , independent of lifetime.
Not bad, not ideal! Can experimentalists constrain this further?
0 1 2 3 4p
T (GeV)
0
0,5
1
1,5
2
Rat
io
v3(p
T)/v
2(p
T)
v4(p
T)/v
2(p
T)
v5(p
T)/v
2(p
T)
0 1 2 3 4p
T (GeV)
0
0,1
0,2
0,3
0,4
0,5
Rat
io
v3(p
T)/v
2(p
T)
v4(p
T)/v
2(p
T)
v5(p
T)/v
2(p
T)
0 1 2 3 4p
T (GeV)
0
0,5
1
1,5
2
Rat
io
v3(p
T)/v
2(p
T)
v4(p
T)/v
2(p
T)
v5(p
T)/v
2(p
T)
Data from ALICE1105.3865
PRL107 032301 (2011)vn from ALICE
eccentricities from Glaubermodel
vn actually fit quite well with Glauber model ǫn , but see my intro... this isnot how one checks this model is realistic
What we learned
• A simplified exactly solvable model incorporating vn yields some verysimple scaling patters
– vn(pT ) ∼ pnT– vn ∼ A
−3/2⊥ for early freezeout
– vn(pT ) ∼ 〈pT 〉−1 dNdy
– Given a constant η/s , κ ∼ A⊥ 〈pT 〉2 (dNdy )−1
– ...
• These scaling patters Can be compared to experiment! provided differentsystem sizes, energies, rapidities compared! . This way no freeparameters!
What else can we do?
More detailed correlations... Mixing between ǫn and ǫ2nLets put in two eccentricities
v2n(pT ) ≈pT
2TB3
(10
3
)3(√
5
3
)2n−1
(2n−1)ǫ2n+1
2
( pT2TB3
)2(10
3
)6(√
5
3
)2n−2
(n−1)2
For integrated v2 it becomes
v2n → v2nǫ2n +O(n2ǫ2n)
ǫ2n
Can be tested by finding v3 in terms of centrality
More generally
v2n(pT ) ≈pT
2TB3
(10
3
)3(√
5
3
)2n−1
(2n−1)ǫ2n+1
2
( pT2TB3
)2(10
3
)6(√
5
3
)2n−2
(n−1)2
together with the definition of the two-particle correlation function
dN
dpT1dpT2d(φ1 − φ2)∼∑
n
vn (pT1) vn (pT2) cos (n (φ1 − φ2))
Predicts a systematic rotation of the reaction plane that can be comparedwith data
A hydrodynamic outlookCalculate the same things we had with realistic hydro simulations
• Long life
• Realistic transverse initial conditions
dN/dy〈pT 〉vn
=
... ... ...
... ... ...
... ... ...
︸ ︷︷ ︸
η/s,cs,τπ,...
×
Tinitial
Lǫn
︸ ︷︷ ︸
→Npart,A,√s
Finding a scaling variable ≡ finding a basis to diagonalize this
Should hydrodynamic scaling persist in tomographic regime? NO!
Take, as an initial condition, an elliptical distribution of opaque matter ata given ǫn , run jets through it and calculate vn . Now increase R whilemantaining ǫn constant.
vnǫn
∣∣∣∣tomo
→ Surface
V olume→ 0,
vnǫn
∣∣∣∣hydro
→ constant
Role of “size” totally different in tomo vs hydro regime .Probe by comparing vn in Cu-Cu vs Au-Au, Pb-Pb vs Ar-Ar collisions ofSame multiplicity!
Can we investigate this both quantitatively and generally?
When we study a jet traversing in the medium, we assume
• Fragments outside the medium phadronT ∼ f(ppartonT )
• Comes from a high-energy parton, T/pT ≪ 1
• Travels in an extended hot medium, (Tτ)−1 ≪ 1
When we expand any jet energy loss model, f (pT/T, T τ) around
T/pT , (Tτ)−1
The ABC-model!
dE
dx= κpaT bτ c +O
(T
pT,1
Tτ
)
A phenomenological way of keeping track of every jet energy loss model:
c = 0 Bethe Heitler
c = 1 LPM
c > 2 AdS/CFT “falling string”
Conformal invariance, weakly or strongly coupled, implies a+ b− c = 2
Embed ABC model in Gubser solutionAnd calculate v2(pT ≫ ΛQCD) as a function of pT , L, T .
0 10 20 30 40 50 60p
T
-0.5
0
0.5
1
1.5
2
2.5
3
v2(p
T)/
<v
2>
0-10%10-20%20-30%30-40%40-50%
0 5 10 15 20
pT (GeV)
-2
0
2
4
6
v2(p
T)/
<v
2>
CMS 0-5%60-70%PHENIX 0-10%50-60%
CMS1204.1850
CMS1204.1409
PHENIX PRL98, 162301 (2007)
v2 at low and high pT look remarkably similar.
Conclusions: heavy ions beyond fitting
Choose observable O and your favorite theory, try to determine a, b, c, ...
O ≃ La
(dN
dy
)b
ǫcn...
compare a,b,c with all experimental data
We did this with a highly simplified analytically solvable hydro model .Calculations fro ”real” hydro and tomography also possible.
Top Related