# Hydrodynamic scaling and analytically solvable models

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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, /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

/s~Nc

2

dip(crossover)

/s~0.1

2

Resonances?Hagedorn

2

/s~ ~Ln(T)

At T0 Tc speed of sound experiences a dip (not to 0,as its a cross-over,buta dip). Above Tc, /s N0c , below Tc, /s N2c . We should expect...

life

phase

InitialT

Initial

Phase 2

Phase 1

Data across1/2s , A,Npart

Transition/threshold

Sdydy

dN dN (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, /s, T0(y), 0(y),freezeout,... ) Need3D viscous hydro to investigate interplay between: EoS,/s,,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!

/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 gunlink between theory and experiment.

Perhaps, but I would not give up just yet!

By decreasing energyTinitial,final decreases, B increasesLifetime increases Flow etc has more time to developPhases change Intensive parameters change (/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 /(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 ?

Buckingams 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 (1, 2, ..., nk) = 0

wherei =

xii ,

i = 0

Widely applied within hydrodynamics in the 19th century: Knudsensnumber, Reynolds number, Rayleighs number, etc.

Since we are varying a whole slew of experimental (y, pT , Npart,s, A) And

theoretical (T, , , s, q, 0, life) 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,,vn

/ s,Cs,...

Heavy ionspecificdimensionless number "O"

life

phase

InitialT

Initial

Phase 2

Phase 1

Data across1/2s , A,Npart

Transition/threshold

Sdydy

dN dN (Or , ,...)

(Intensive quantities)

,T,,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

/s

Initial flow

The shortest course possible on hydro I:EvolutionThe 5 energy momentum conservation equations

T = 0

have 10 unknowns. They can be closed by assuming approximate isotropy

T = (p+ )uu + pg + u + u

And thermodynamic equations for p, , in terms of .

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

dpTdyd=

pdf(pu, T )

If is the locus of constant T , parametrized by t(x, y, z, T ) then

d = d

dx

d

dy

d

dz

In this formalism

vn =

cos(n)dN

dpTdydd

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 n

Freeze-out isothermal Cooper-Frye

The basic idea Conformal invariance of the solution constrains flow to be,in addition to the usual Bjorken

u 2xL2 + 2 + x2

, uz z

t

plugging this into the Relativistic Navier-Stokes equation gives yousomething you can solve

ENS = T4NS =

1

4C4

(cosh )8/3

[

1 +09C

(sinh )3 2F1

(3

2,7

6,5

2; sinh2

)]4

where

sinh = L2 2 + x2

2LNB: issues at 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 ( s 0s )

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

L

L or x

(

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 (n) , y (

x a2

x

)

sin ()

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, a

2/x3)(Again, early freezeout )

To first order in a/L (i.e., n 1 ) we get

E C4

4/3(2L)8/3

(L2 + x2)8/3

(

1 02C

(L2 + x22L

)2/3)4

[

1 4n(

1 +02C

(L2 + x22L

)2/3)(

2LxL2 + x2

)n

cosn

]

,

Deformation breaks down at L

this can be solved for an expression of an isothermal surface, ready forfreeze-out

T 3 =C3(2L)2

(L2 + x2)2

(

1 02C

(L2 + x22L

)2/3)3

[

1 3n(

1 +02C

(L2 + x22L

)2/3)(

2LxL2 + x2

)n

cosn

]

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

pTdpTdyd=

dp exp

(

up

T

)(

1 +pp

2(e+ P )T 2(p)

)

where

(p) = 1, = (g uu) u, = T 3dxdT

dxdT

dxdT

and

dN

dy=

dpTpTdf(~p), pT =

dpTp2Tdf(~p), vn =

dpTpTdf(~p) cos (2n)

we can analytically map

L, T, n,

s, B dN

dy, pT , vn

After quite a bit of Algebra... (2)3 dNdY pTdpTdp J1 + J2 + J3 .

J1 = 4mTK1(mT/T ) 0

dxx0

I0(z) (1 )+

0In(z)n cosnp +(1

)pT2T

[(u

ux

)In1(z) +

(u +

ux

)In+1(z)

]n cosnp

J2 = 4pTK0(mT/T ) 0

dxx0

0x

I1(z)(1

)+

0x

0I n(z)n cosnp

+(1)

0

xpT2T

((u

ux

)I n1(z) +

(u +

ux

)I n+1(z)

)+ x

I n(z)

n cosnp

J3 = 4pTK0(mT/T ) 0

dx0n2z In(z)(1 )n cosnp J3n cosnp .

where Jn = Jn0 + Jnn z pTu0T =2xpT (2L)

5

TB3(L2+x2)3 (1 ) .,

Expanding linearly in n and pT/(TB3), In(x) xn/2nn!

J01 = 4mTK1(mT/T )16L3

B3

{1 x

2max64L2

(6 +

m2T2T2

K3K1K1

p2T

T2

)},

J02 = 4K0(mT/T )215L3p2T

TB9

{121

640

(12 +

m2TT2

K2K0K0

p2T

T2

)},

J1 = 4mTT K1(mT/T )

(3n)(4n)

926nL3pnTB3(n+1)Tn1

(n1){

2(3n+2)4n+1

n8(3n1)

(6n + 6 +

m2T2T2

K3K1K1

p2T

T2

)

J2 = 4K0(mT/T )(3n)(4n)

926nL3pnTB3(n+1)Tn1

2n{

6n26n54n+1

(6n210n+1)48(3n1)

(6n +

m2TT2

K2K0K0

p2T

T2

)},

J3 = 4K0(mT/T )(3n)(4n)

926nL3pnTB3(n+1)Tn1

2n{1 (4n1)48(3n1)

(6n +

m2TT2

K2K0K0

p2T

T2

)},

Low pT vn(pT/(TB

3) 1), but B 1

vn(pT )

n=

9(n 1)32

(3n)

(4n)

(64pTB3T

)n [2(3n+ 2)

4n+ 1 n

8(3n 1)

(

6n+ 9 +2mTT

p2T

T 2

)]

The v2 and Knudsen number for this solution:

vn

O((pT

T

)n)

(1K) , K s

(L

)2/3

A bit different from Gomebaud et al, Lacey et al(vnn nTR

)Sensitivity

to form of solution , Interplay of L,

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

d cos (1 tf cos() exp [ (E vT ()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) ez2z

exp(pTT

2x(2L)5

B3(x2 + L2)3

(1 ))

.

The x-integral can be evaluated by doing the saddle point at x = L/

5.

The result is

vn(pT ) n2

pTTu0 = n

500pT27TB3

(5

3

)n1(

n 1 27200

n

)

.

but jet contamination likely. Experimental opportunity to see how scalingofvn(pT ) changes with n, pT [email protected] pT , [email protected] pT . NB: High, low w.r.t. TSize/Lifetime 1

The role of bulk viscosityPlugging in the 14-moment correction of the distribution function

f bulk

feq=

12T 2

m2

[

12 +8

Tup

+1

T 2(up

)2]u

T

S,

and assuming early time u 1/ , we carry these terms to be

vbulkn 81

128

(128

B3

)nn2(n 1)(3n)

(4n)2

(n

2

)((3n+ 2)2

4(4n+ 1)

x2maxL2

3n3n 1

)B2

CSn ,

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

m2bulk ,

Now we fix K,C,B in terms of bulk obvservablesThese are dominated by soft regions, so can calculate

dN

dY=

1

(2)2

dpTpT (J01 + J

02 )

4C3

pT (dN

dY

)1

pTdpTdN

dY dpT 3T

4=

3CB

8L

Therefore

C (dN

dY

)1/3

,1

B3 1pT 3L3

dN

dY.

As for azimuthal coefficients, these are

(vn(pT )

n

)1/n

pTA

3/2 pT 4

dN

dY(1n) ,

(vnn

)1/n

1A

3/2 pT 3

dN

dY(1n) ,

Note that vn pnT robust against assumptions we made, should survive forrealistic scenarios where the knudsen number is

B2

C

S ApT

2

dN/dY

S, , A L2, A3/2 Npart

NB: this is a bit different from Bhalerao et al , as well as GT,1310.3529

v22

f() (const.O ())

Plugging in some more empirical formulae

dN

dY Npart(

s) , pT F

(

1

N2/3part

dN

dY

)

F(

N1/3part(

s)

)

,

where 0.15 in AA collisions and 0.1 in pA and pp collisions, andF is a rising function of its argument, we get

(vnn

)1/n

(s)G

(

N1/3part(

s)

)

(1n) , H(

N1/3part(

s)

)

S,

where G(x) = F3(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)/

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)/

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 (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, 1S dNdyThis 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 )

n

)1/n

pTA

3/2 pT 4

dN

dY(1n) ,

(vnn

)1/n

1A

3/2 pT 3

dN

dY(1n) ,

but since 1AdNdy , it is enough to naively extrapolate from B

2 O (1)to B2 O (L/). Extra A1/2 power enough for scaling but Need Realistichydrodynamics to test this extrapolation

pT

BRAHMS,0907.4742v2

nuclex/0608033PHENIXAuAu,CuCu

Low energy scan, STAR 1206.5528

v2(pT ) constant (at least at high pT ).Definitely not dependent on pT as in

(vn(pT )

n

)1/n

pTA

3/2 pT 4

dN

dY(1 n)

unless 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 n , 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 A3/2 for early freezeout vn(pT ) pT 1 dNdy Given a constant /s , 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 n and 2nLets put in two eccentricities

v2n(pT ) pT

2TB3

(10

3

)3(

5

3

)2n1

(2n1)2n+1

2

( pT2TB3

)2(10

3

)6(

5

3

)2n2

(n1)2

For integrated v2 it becomes

v2n v2n2n +O(n22n)

2n

Can be tested by finding v3 in terms of centrality

More generally

v2n(pT ) pT

2TB3

(10

3

)3(

5

3

)2n1

(2n1)2n+1

2

( pT2TB3

)2(10

3

)6(

5

3

)2n2

(n1)2

together with the definition of the two-particle correlation function

dN

dpT1dpT2d(1 2)

n

vn (pT1) vn (pT2) cos (n (1 2))

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/dypT vn

=

... ... ...

... ... ...

... ... ...

/s,cs,,...

TinitialLn

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 n , run jets through it and calculate vn . Now increase R whilemantaining n constant.

vnn

tomo

SurfaceV olume

0, vnn

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, (T)1 1

When we expand any jet energy loss model, f (pT/T, T ) around

T/pT , (T)1

The ABC-model!

dE

dx= paT b c +O

(T

pT,1

T

)

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 QCD) 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)/

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)/

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

cn...

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.