This document discusses using an exactly solvable hydrodynamic model to study scaling behavior near a phase transition in heavy ion collisions. It proposes using a deformed Gubser solution with azimuthal asymmetries parameterized by dimensionless εn to model initial inhomogeneities. Analytical expressions are derived relating εn, which characterizes the initial geometry, to final state particle distributions like dN/dy and vn. This allows studying how observables change with transport coefficients and other parameters across a phase transition through their scaling behavior.
Hydrodynamic scaling and analytically solvable models
1. Hydrodynamic scaling in an exactly solvable model
Based on 1407.5952 with Yoshitaka Hatta,Bowen Xiao, Jorge
Noronha
G.Torrieri
2. 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 (onset
of deconfinement): How do opacity, η/s , EoS etc. change at that point?
Hydrodynamics can be used as a tool to connect statistical physics (more
or less understood) to particle distributions
3. 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 λ
2
/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 ∼ N0
c , below Tc, η/s ∼ N2
c . We should expect...
4. life
phase
Initial
T
Initial µ
Phase 2
Phase 1
Data across
1/2
s , 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 via
a kink in the speed of sound to the sQGP regime.
6. 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,... ) Need
3D viscous hydro to investigate interplay between: EoS,η/s,τΠ,transverse
initial conditions,longitudinal initial conditions,pre-existing flow,freeze-out
dynamics, jet showers in-medium, fragmentation outside the medium .... .
No jump clearly seen! In which parameters is the phase transition hiding?
7. The problem!
η/s
Equation of state
Rapidity dependence
Initial flow
"With enough
parameters
you can fit..."
vn from ALICE
fits well with a
NAIVE
model with
5 parameters
We understand the equation of state and hopefully the viscosity from first
principles. But initial conditions and their dependence in energy, and
transport coefficients, and jets, and freezeout... Even when you are trying
to fit lots of data simultaneusly, a model with many correlated parameters
can describe nealry any physical system
8. Some people think that this will always be with us
The system we are studying is so complicated that models with lots of
parameters 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!
9. • By decreasing energy
Tinitial,final decreases, µB increases
Lifetime increases Flow etc has more time to develop
Phases 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 decreases
Gradients 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 1
S
dN
dy ?
10. 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
where
πi = xλi
i , λi = 0
Widely applied within hydrodynamics in the 19th century: Knudsen’s
number, 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, ˆq, τ0, τlife) parameters it would be nice to represent
heavy ion observables this way
11. This is how hydrodynamics
was done in the 19th
century!!!!
The idea:
when you have a pipe
and you make it
twice as big
does your variable
of interest grow asn
2 ? What is n?
12. s,A,Npart,y
dN/dy,<pT>,vn
η/s,Cs,...
Heavy ion−specific
dimensionless
number "O"
life
phase
Initial
T
Initial µ
Phase 2
Phase 1
Data across
1/2
s , 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 does
vn, pT , ... change? Given enough variable conditions, a scaling dimensionless
number makes it straight-forward to look for scaling violations
13. “And the theorist says.... Consider a spherical elephant in a vacuum”
η/s
Initial flow
14. The shortest course possible on hydro I:Evolution
The 5 energy momentum conservation equations
∂µTµν
= 0
have 10 unknowns. They can be closed by assuming approximate isotropy
Tµν
= (p + ρ)uµ
uν
+ pgµν
+ η∆µναβ∂α
uβ
+ ζ∆µνα
α ∂βuβ
And thermodynamic equations for p, η, ζ in terms ofρ .
Once closed these equations can be integrated from initial conditions
15. The shortest course possible on hydro I:Freezeout
At a critical condition (here critical T ) the fluid has to convert into particles.
Energy-momentum and entropy conservation, plus ”fast” conversion, force
the Cooper-Frye formula
E
dN
d3p
=
1
pT
dN
dpT dydφ
= pµ
dΣµf(pµ
uµ, T)
If Σµ is the locus of constant T , parametrized by t(x, y, z, T) then
dΣµ = ǫµαβγ dΣα
dx
dΣβ
dy
dΣγ
dz
In this formalism
vn = cos(nφ)
dN
dpT dydφ
dφ
16. A ”semi-realistic” but solvable model: A deformed Gubser solution
Gubser flow includes
Viscosity , finite Knudsen number
Transverse flow with ”Conformal” setup
We add
Inhomogeneities parametrized by dimensionless ǫn
Freeze-out isothermal Cooper-Frye
17. 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 you
something you can solve
ENS = λT4
NS =
1
τ4
λC4
(cosh ρ)8/3
1 +
η0
9λC
(sinh ρ)3
2F1
3
2
,
7
6
,
5
2
; − sinh2
ρ
4
where
sinh ρ = −
L2
− τ2
+ x2
⊥
2Lτ
NB: issues at ρ ≪ −1 (negative temperature!) Physically this reflects
implicit non-causality of NS limit, see 1307.6130 (Noronha et al) to fix this
18. Not (yet!) the real world:
• Strictly conformal EoS (s ∼ T3
, e ∼ T4
) 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 limitation
of the solution ansatz!
19. 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 a
solution 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 )
20. To first order in a/L (i.e., ǫn ≪ 1 ) we get
E ≈
λC4
τ4/3
(2L)8/3
(L2 + x2
⊥)8/3
1 −
η0
2λC
L2
+ x2
⊥
2Lτ
2/3 4
× 1 − 4ǫn 1 +
η0
2λC
L2
+ x2
⊥
2Lτ
2/3
2Lx⊥
L2 + x2
⊥
n
cos nφ ,
Deformation breaks down at τ ≃ L
21. this can be solved for an expression of an isothermal surface, ready for
freeze-out
T3
=
C3
(2L)2
τ(L2 + x2
⊥)2
1 −
η0
2λC
L2
+ x2
⊥
2Lτ
2/3 3
×
1 − 3ǫn 1 +
η0
2λC
L2
+ x2
⊥
2Lτ
2/3
2Lx⊥
L2 + x2
⊥
n
cos nφ ≡
C3
B3
(2L)3
,
C: overall multiplicity. B Lifetime of the system
NB: Need B ≫ 1, so lifetime ≪ L, “early” freezeout w.r.t. size. .
22. Now we are set
f(p) =
dN
pT dpT dydφ
= dσµpµ
exp −
uµpµ
T
1 +
Πµν
pµpν
2(e + P)T2
χ(p)
where
χ(p) = 1, πµν
= (gµα
− uµ
uα
) ∂αuν
, σµ = T3
ǫµναβ
dxν
dT
dxα
dT
dxβ
dT
and
dN
dy
= dpT pT dφf(p), pT = dpT p2
T dφf(p), vn = dpT pT dφf(p) cos (2nφ)
we can analytically map
L, T, ǫn,
η
s
, B ⇔
dN
dy
, pT , vn
24. Expanding linearly in ǫn and pT /(TB3
), In(x) ∼ xn
/2n
n!
J0
1 = 4πmT K1(mT /T )16L3
B3 1 −
κx2
⊥max
64L2 6 +
m2
T
2T 2
K3−K1
K1
−
p2
T
T 2 ,
J0
2 = 4πK0(mT /T )
215L3p2
T
T B9
1
21 − κ
640 12 +
m2
T
T 2
K2−K0
K0
−
p2
T
T 2 ,
δJ1 = 4π
mT
T K1(mT /T )Γ(3n)
Γ(4n)
9·26nL3pn
T
B3(n+1)T n−1
×(n−1) 2(3n+2)
4n+1 − nκ
8(3n−1) 6n + 6 +
m2
T
2T 2
K3−K1
K1
−
p2
T
T 2
δJ2 = 4πK0(mT /T )Γ(3n)
Γ(4n)
9·26nL3pn
T
B3(n+1)T n−1
×2n 6n2−6n−5
4n+1 − (6n2−10n+1)κ
48(3n−1) 6n +
m2
T
T 2
K2−K0
K0
−
p2
T
T 2 ,
δJ3 = 4πK0(mT /T )Γ(3n)
Γ(4n)
9·26nL3pn
T
B3(n+1)T n−1
×2n 1 − (4n−1)κ
48(3n−1) 6n +
m2
T
T 2
K2−K0
K0
−
p2
T
T 2 ,
25. Low pT vn pT /(TB3
) ≪ 1 , but B ≫ 1
vn(pT )
ǫn
=
9(n − 1)
32
Γ(3n)
Γ(4n)
64pT
B3T
n
2(3n + 2)
4n + 1
−
nκ
8(3n − 1)
6n + 9 +
2mT
T
−
p2
T
T2
The v2 and “Knudsen number” for this solution:
vn
ǫ
∼ O
pT
T
n
(1 − K) , K ∼
η
s
L
τ
2/3
A bit different from Gomebaud et al, Lacey et al vn
n ∼ n
T R Sensitivity
to form of solution , Interplay of L, τ
NB: vn(pT ) ∼ pn
T phenomenologically important general prediction
(Depends on azimuthal integral, independent of approxuimations!
26. vn ∼ pn
T : A robust prediction
All it requires is that
vn ∼ dφ cos φ (1 − tf cos(φ) exp [γ (E − vT (φ)pT )]) ∼ In O
pT
T
∼
pT
T
n
This is much more robust than the assumptions of Gubser flow
27. A large momentum region, pT ≫ TB3
is also possible,
In(z) ≈
ez
√
2πz
∼ exp
pT
T
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 ) ≈
ǫn
2
pT
T
δu∗
⊥0 = ǫn
500pT
27TB3
√
5
3
n−1
n − 1 −
27κ
200
n .
but jet contamination likely. Experimental opportunity to see how scaling
ofvn(pT ) changes with n, pT
∼ pn
T @low pT , ∼ pT @High pT . NB: High, low w.r.t. T×Size/Lifetime≫ 1
28. The role of bulk viscosity
Plugging in the 14-moment correction of the distribution function
δfbulk
feq
=
12T2
m2
12 +
8
T
uµpµ
+
1
T2
(uµpµ
)2 ∇µuµ
T
ζ
S
,
and assuming early time ∂µuµ
∼ 1/τ , we carry these terms to be
δvbulk
n ≈
81
128
128
B3
n
n2
(n − 1)Γ(3n)
Γ(4n)
Γ2 n
2
(3n + 2)2
4(4n + 1)
x2
max
L2
−
3n
3n − 1
B2
ζ
CS
ǫn ,
Shear and bulk viscosity compete with terms which may be of opposite sign
and non-trivial contribution, Confirming the numerical work of Noronha-
Hostler et al
vn
videal
n
− 1 ∼ ±n2 T2
m2
κbulk
,
29. Now we fix K, C, B in terms of bulk obvservables
These are dominated by soft regions, so can calculate
dN
dY
=
1
(2π)2
dpT pT (J0
1 + J0
2 ) ≈
4C3
π
pT ≡
dN
dY
−1
pT dpT
dN
dY dpT
≈
3πT
4
=
3πCB
8L
Therefore
C ∼
dN
dY
1/3
,
1
B3
∼
1
pT
3L3
dN
dY
.
30. As for azimuthal coefficients, these are
vn(pT )
ǫn
1/n
∼
pT
A
3/2
⊥ pT
4
dN
dY
(1−nκ) ,
vn
ǫn
1/n
∼
1
A
3/2
⊥ pT
3
dN
dY
(1−nκ) ,
Note that vn ∼ pn
T robust against assumptions we made, should survive for
realistic 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
v2
ǫ2
∼ f(τ) (const. − O (κ))
31. Plugging in some more empirical formulae
dN
dY
∼ Npart(
√
s)γ
, pT ∼ F
1
N
2/3
part
dN
dY
∼ F N
1/3
part(
√
s)γ
,
where γ ≈ 0.15 in AA collisions and γ ≈ 0.1 in pA and pp collisions, and
F is a rising function of its argument, we get
vn
ǫn
1/n
∼ (
√
s)γ
G N
1/3
part(
√
s)γ
(1−nκ) , κ ∼ H N
1/3
part(
√
s)γ η
S
,
where G(x) = F−3
(x) and H(x) = F2
(x)/x.
32. Flow... the experimental situation
0 10 20 30 40 50 60
pT
-0.5
0
0.5
1
1.5
2
2.5
3
v2
(pT
)/<v2
>
0-10%
10-20%
20-30%
30-40%
40-50%
0 5 10 15 20
pT
(GeV)
-2
0
2
4
6
v2
(pT
)/<v2
>
CMS 0-5%
60-70%
PHENIX 0-10%
50-60%
BRAHMS,NPA 830, 43C (2009)
pT
CMS
1204.1850
CMS
1204.1409
PHENIX PRL98, 162301 (2007)
PHENIX
PRL98:162301,2007
CMS
PRL109 (2012) 022301
NPA830 (2009)
PHOBOS
STAR 1206.5528
Here is what we know experimentally
v2 ≃ ǫ(b, A)F(pT ), v2 ≃ dpT F(pT )f pT , pT y,A,b,
√
s
F(pT ) universal for all energies , f(pT ) tracks mean momentum, ∼ 1
S
dN
dy
This is an experimental statement, as good as the error bars. Very different
from our scaling!
33. knew
this:
for years
and we
Wrong power w.r.t.
vn(pT )
ǫn
1/n
∼
pT
A
3/2
⊥ pT
4
dN
dY
(1−nκ) ,
vn
ǫn
1/n
∼
1
A
3/2
⊥ pT
3
dN
dY
(1−nκ) ,
but since κ ∼ 1
A⊥
dN
dy , it is enough to “naively extrapolate” from B2
∼ O (1)
to B2
∼ O (L/τ). Extra A1/2
power enough for scaling but Need Realistic
hydrodynamics to test this extrapolation
35. LHC vn(pT ) data allows us to test vn ∼ pn
T
a 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 4
pT
(GeV)
0
0,5
1
1,5
2
Ratio
v3
(pT
)/v2
(pT
)
v4
(pT
)/v2
(pT
)
v5
(pT
)/v2
(pT
)
0 1 2 3 4
pT
(GeV)
0
0,1
0,2
0,3
0,4
0,5
Ratio
v3
(pT
)/v2
(pT
)
v4
(pT
)/v2
(pT
)
v5
(pT
)/v2
(pT
)
0 1 2 3 4
pT
(GeV)
0
0,5
1
1,5
2
Ratio
v3
(pT
)/v2
(pT
)
v4
(pT
)/v2
(pT
)
v5
(pT
)/v2
(pT
)
Data from ALICE
1105.3865
36. PRL107 032301 (2011)
vn from ALICE
eccentricities from Glauber
model
vn actually fit quite well with Glauber model ǫn , but see my intro... this is
not how one checks this model is realistic
37. What we learned
• A simplified exactly solvable model incorporating vn yields some very
simple scaling patters
– vn(pT ) ∼ pn
T
– vn ∼ A
−3/2
⊥ for early freezeout
– vn(pT ) ∼ pT
−1 dN
dy
– Given a constant η/s , κ ∼ A⊥ pT
2
(dN
dy )−1
– ...
• These scaling patters Can be compared to experiment! provided different
system sizes, energies, rapidities compared! . This way no free
parameters!
What else can we do?
38. More detailed correlations... Mixing between ǫn and ǫ2n
Lets put in two eccentricities
v2n(pT ) ≈
pT
2TB3
10
3
3 √
5
3
2n−1
(2n−1)ǫ2n+
1
2
pT
2TB3
2 10
3
6 √
5
3
2n−2
(n−1)2
For integrated v2 it becomes
v2n → v2n
ǫ2n + O(n2
ǫ2
n)
ǫ2n
Can be tested by finding v3 in terms of centrality
39. More generally
v2n(pT ) ≈
pT
2TB3
10
3
3 √
5
3
2n−1
(2n−1)ǫ2n+
1
2
pT
2TB3
2 10
3
6 √
5
3
2n−2
(n−1)2
together with the definition of the two-particle correlation function
dN
dpT 1dpT 2d(φ1 − φ2)
∼
n
vn (pT 1) vn (pT 2) cos (n (φ1 − φ2))
Predicts a systematic rotation of the reaction plane that can be compared
with data
40. A hydrodynamic outlook
Calculate the same things we had with realistic hydro simulations
• Long life
• Realistic transverse initial conditions
dN/dy
pT
vn
=
... ... ...
... ... ...
... ... ...
η/s,cs,τπ,...
×
Tinitial
L
ǫn
→Npart,A,
√
s
Finding a scaling variable ≡ finding a basis to diagonalize this
41. Should hydrodynamic scaling persist in tomographic regime? NO!
Take, as an initial condition, an elliptical distribution of opaque matter at
a given ǫn , run jets through it and calculate vn . Now increase R while
mantaining ǫn constant.
vn
ǫn tomo
→
Surface
V olume
→ 0,
vn
ǫn 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 of
Same multiplicity!
42. Can we investigate this both quantitatively and generally?
When we study a jet traversing in the medium, we assume
• Fragments outside the medium phadron
T ∼ f(pparton
T )
• 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, Tτ) around
T/pT , (Tτ)−1
43. The ABC-model!
dE
dx
= κpa
Tb
τc
+ 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
44. Embed ABC model in Gubser solution
And calculate v2(pT ≫ ΛQCD) as a function of pT , L, T .
0 10 20 30 40 50 60
pT
-0.5
0
0.5
1
1.5
2
2.5
3
v2
(pT
)/<v2
>
0-10%
10-20%
20-30%
30-40%
40-50%
0 5 10 15 20
pT
(GeV)
-2
0
2
4
6
v2
(pT
)/<v2
>
CMS 0-5%
60-70%
PHENIX 0-10%
50-60%
CMS
1204.1850
CMS
1204.1409
PHENIX PRL98, 162301 (2007)
v2 at low and high pT look remarkably similar.
45. Conclusions: heavy ions beyond fitting
Choose observable O and your favorite theory, try to determine a, b, c, ...
O ≃ La dN
dy
b
ǫc
n...
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.