s c o t t pratprat michiganmichigan s t t e univrsityunivrsity t h btbt puzzle puzzle z a n d e x t...

S C O T T

PRA

T

MICHIGAN

S T T E

UNIV

RSITY

T HBT

PUZZLE

A N D

E X T E N D I N G

HY

RODYNAMICS

Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University

OUTLINE

• Remnants of the HBT Puzzle• Extending Hydrodynamics

• Tsunamis


HBT Basics

Rout ≈Rside + vΔτ

Rlong

Rside


Blast-Wave Models

Rvz=z/τ

Vx=V(r/R)

Parameters: T, R, V , τ, Δτ

Schnedermann, Sollfrank and HeinzTomasik, Broniowski,Lisa and Retiere,…


NA49 vs. Therminator

3-dimensional details of blast-wave confirmed


Blast-Wave and the HBT Puzzle

Parameters:T 110 MeVR 13 fmV 0.7c τ 10 fm/c Δτ 3 fm/c

Similar Conclusions:Blast Waves (Lisa-Retiere, Tomasik)TherminatorBuda-Lund

Requires Kp spectra

Rapid expansionsudden disintegration


HYDRO

Overall sizes depend on EoS & breakup criteria


Cascade/Boltzmann

More resonances -> softer -> biggerStrings -> softer -> bigger


HBT and the EoS: Dynamical Signature

Leads to: Rout/Rside >> 1


HBT and EoS: Entropy

f (rp)≡

d3r[ f(rp,

rr )]2∫

d3r f(rp,

rr )∫

=π 3/2

(2S+1)dN / d3p

RoutRlongRside

Sizes and spectra give phase space density

Phase space density gives Entropy

dS

dy≅2π ptdpt E∫

dNd3p

52−32

ln(2)−ln[ f (pt)] ±123/2 f (pt)

⎡⎣⎢

⎤⎦⎥


HBT and EoS:

Entropy


Is Lattice Eq. of State excluded?

Range

Subrata Pal and S.P., PLB 2004


Total Entropy and the lattice EOS

• Final S consistent with lattice EOS (Crude)• Entropy moved from pions to baryons


The HBT puzzle

• Rside/Rout1 and τ10 fm/c suggest rapid expansion -> hard EoS

• Entropy suggests softer EoS


Explanations to the HBT Puzzle

• Refraction (Cramer-Miller, 2005)

- Requires extreme assumptions

• Surface Emission (Heiselberg, 2002)

- What happened to energy in center?

• Initial transverse velocity (Sinyukov, 2007)

- Cause ???

• Super-Cooling (Csorgo&Csernai, hep-th/9312230)

Needed: Explanation with consistent dynamical model


Extending Hydrodynamics

1. Longitudinal Acceleration2. Later times (Chemistry)3. Earlier times (Shear)4. In between (Hadronization dynamics)


No Longitudinal Acceleration

Rout =vtherm

dv/ dz ,dvdz

=1τ, τ =

Rout

vtherm

Bjorken


Analytic 1-D Solution(S.P. PRC 2007)

dvdz =

2xD(x)τ

,

x≡σ y (1−cs2 ) / 2cs

2

Accelerationless models underestimate lifetime by

10%

Bjorken

cs2 =0.2

τ 0 =0.5 fm/c


Later Times(Non-Equilibrium Chemistry

Below Tc) • Too many particles• Pion, kaon, baryon fugacities > 1• Smaller P/ • Less collective energy, more thermal


Early Times (Shear)

In rest frame:

Tαβ =

0 0 00 Txx 0 00 0 Tyy 00 0 0 Tzz

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

∂ux∂τ= −

∂Txx / ∂x

ε +Txx

Always true

Perfect Hydro:

Viscous Hydro:

Txx =Tyy =Tzz =P()

Txx ≠Tyy ≠Tzz ≠P()


QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Navier-Stokes

Tzz =P −η(2∂zvz −∂xvx −∂yvy)−B∇gv

Shear

Bulk

Txx >Tzz• at early times• Boosts early acceleration (D.Teaney,)

• Increases collective energy vs. thermal energy


Early times

• For η=s/4π, Tzz becomes negative for τ0.6 fm/c• Eq.s of Motions for Tij

• Saturate Anisotropy

Tij(target) =P(1+ Aij

(target))

A(target) =Amax tanh ΔT (NS)

AmaxP( )

ddτ

A=−(A−A(target))τ (I.S.)

saturatesanisotropy

relaxation time(Israel-Stewart)


Anisotropy at early times

Classical Longitudinal Fields: Txx =Tyy =Tzz =−

CGC (Krasnitz, Nara &Venugopalan) : Txx =Tyy ≈ / 2Tzz ≈0

Collisionless Boltzmann:Txx =Tyy = / 2

Tzz =0


Intermediate Times(Bulk Viscosity in Mixed Region)

K.Paech and S.P., PRC 2006

P

P−B∇gv


Sigma Field out of Equilibrium

∂2σ∂t2

+ Γ∂σ

∂t+m2 (σ − σ equil ) = R(t)

Langevin "force"

md2x

dt2+γ dx

dt+ k(x−xequil ) =R(t)

kδx=−γ&xequil

δσ =−

Γm2

&σ equil

δσ can blow up at phase transition!

B =∂P∂σ

Γsmσ

2

∂σeq

∂s


Example: Linear Sigma Model

H =λ2

4σ 2 − fπ

2 +mπ2 / λ2( )

2−hqσ +quarks(T,m=gσ )

1st order when

g>3.55

K.Paech & A.Dumitru, PLB 623, 200 (2005)


Example: Linear Sigma

Model

For g=3.4, Txx -> 0

For big effects:Should solve Sigma Eq.s of Motion

K.Paech & A.Dumitru, PLB 623, 200 (2005)


How might this affect dynamics?

Txx

r

P


Tsunamis at RHIC

vwave = gh

See also nuclear doughnuts(Bauer and Bertsch, PRL '90)


Modeling Tsunamis at RHIC with Hydrodynamics

Pcs2 =dP

d

H

L

cs2 =0.3

cs2 =constant

Increasing L or decreasing c2mixed -> stronger

tsunami

Resonance Gas


HydrodynamicEvolution

L=2 GeV/fm3,c2

mixed=0.05

τ=1,3,5,7,9,11 fm/c


Changing EoS cmixed2 =0.05

cmixed2 =0.15

Large Rside

for L2 GeV/fm3

Bulk Viscosity in mixed phase strengthens

tsunami


Navie

r-Sto

kes

Dyn

am

ic Tij

Dyn

am

ic Field

s

Summary Slide

#1

Boltzm

an

n

?


Summary #2

HBT is only hope to disentagle • EoS• Early-Time Shear• Bulk Viscosity near Tc

s c o t t pratprat michiganmichigan s t t e univrsityunivrsity t h btbt puzzle puzzle z a n d e x t...

Documents

hbt puzzleparameters

lattice eos crudeentropy

fmc eq

lattice eosfinal s consistent

s of motions

sigma eq

collective energy

s of motionk