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.