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TRANSCRIPT
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
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
HBT Basics
Rout ≈Rside + vΔτ
Rlong
Rside
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
Blast-Wave Models
Rvz=z/τ
Vx=V(r/R)
Parameters: T, R, V , τ, Δτ
Schnedermann, Sollfrank and HeinzTomasik, Broniowski,Lisa and Retiere,…
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
NA49 vs. Therminator
3-dimensional details of blast-wave confirmed
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
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
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
HYDRO
Overall sizes depend on EoS & breakup criteria
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
Cascade/Boltzmann
More resonances -> softer -> biggerStrings -> softer -> bigger
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
HBT and the EoS: Dynamical Signature
Leads to: Rout/Rside >> 1
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
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)
⎡⎣⎢
⎤⎦⎥
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
HBT and EoS:
Entropy
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
Is Lattice Eq. of State excluded?
Range
Subrata Pal and S.P., PLB 2004
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
Total Entropy and the lattice EOS
• Final S consistent with lattice EOS (Crude)• Entropy moved from pions to baryons
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
The HBT puzzle
• Rside/Rout1 and τ10 fm/c suggest rapid expansion -> hard EoS
• Entropy suggests softer EoS
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
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
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
Extending Hydrodynamics
1. Longitudinal Acceleration2. Later times (Chemistry)3. Earlier times (Shear)4. In between (Hadronization dynamics)
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
No Longitudinal Acceleration
Rout =vtherm
dv/ dz ,dvdz
=1τ, τ =
Rout
vtherm
Bjorken
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
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
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
Later Times(Non-Equilibrium Chemistry
Below Tc) • Too many particles• Pion, kaon, baryon fugacities > 1• Smaller P/ • Less collective energy, more thermal
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
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()
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
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
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
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)
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
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
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
Intermediate Times(Bulk Viscosity in Mixed Region)
K.Paech and S.P., PRC 2006
P
P−B∇gv
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
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
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
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)
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
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)
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
How might this affect dynamics?
Txx
r
P
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
Tsunamis at RHIC
vwave = gh
See also nuclear doughnuts(Bauer and Bertsch, PRL '90)
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
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
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
HydrodynamicEvolution
L=2 GeV/fm3,c2
mixed=0.05
τ=1,3,5,7,9,11 fm/c
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
Changing EoS cmixed2 =0.05
cmixed2 =0.15
Large Rside
for L2 GeV/fm3
Bulk Viscosity in mixed phase strengthens
tsunami
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
Navie
r-Sto
kes
Dyn
am
ic Tij
Dyn
am
ic Field
s
Summary Slide
#1
Boltzm
an
n
?
Scott Pratt Extending Hydrodynamics and the HBT Puzzle Extending Hydrodynamics and the HBT Puzzle Michigan State University
Summary #2
HBT is only hope to disentagle • EoS• Early-Time Shear• Bulk Viscosity near Tc