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
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Transcript of Hydrodynamic scaling and analytically solvable models

  • 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.