Bjorken flow and symmetry
principles
Steve Gubser, Princeton University
CERN-CKC TH Institute on Numerical Holography
December 15, 2014
ptContents
1
Introduction
3
2
Symmetry principles
5
3
Bjorken flow
7
4
Conformal deformation
8
5
Complex deformation
14
6
Match to AdS/CFT numerics
18
7
Complex and conformal deformations
22
8
Non-relativistic boost invariance
25
9
Conclusions
28
10 Backup material
29
Bjorken flow and symmetries, 12-15-14
1.
3
S. Gubser
Introduction
I want to modify Bjorken flow so that it has finite extent in both transverse and
longitudinal directions.
• This would make it a better starting point for phenomenology.
• I don’t want to recreate Landau-Khalatnikov flow.
• Instead let’s see if we can massage Bjorken’s symmetry principles.
Conformal deformations of Bjorken’s symmetries lead to finite transverse extent.
Can be used in hydro [Gubser 1006.0006, Gubser-Yarom 0902.4062, Marrochio-Noronha-DenicolLuzum-Jeon-Gale 1307.6130], for BK equation [Gubser 1102.4040], and in relativistic Boltzmann dynamics [Denicol-Heinz-Martinez-Noronha-Strickland 1408.5646].
Complex deformations of Bjorken’s symmetries lead to finite longitudinal extent
[Gubser 1210.4181]. Find regions with no Landau frame and a fairly good match to
AdS/CFT numerics [Gubser-van der Schee 1410.7408].
AdS/CFT treatment of collisions motivates conformal construction, but main calculations are in hydro.
Bjorken flow and symmetries, 12-15-14
S. Gubser
4
14
12
210 MeV
A partial preview of results:
Conformal deformations have
boost invariance and radial flow.
Τ HfmcL
10
8
130 MeV
6
4
2
30 MeV
0
2
4
6
8
10
12
14
x¦ HfmL
• Complex deformations have
yF = η/2
at τ = t3
• Flow is only approximately
e
gl
t3
as
gl
e
ik
l
a−
x3
m
m
a−
l
as
t
ik
Bjorken−like
Landau−like
full stopping
rapidity dependence but no
transverse structure.
hydrodynamical.
Bjorken flow and symmetries, 12-15-14
2.
S. Gubser
5
Symmetry principles
TH1L in transverse plane
SO(3, 1) acts on R3,1 as follows:
• Translations T(µ) = ∂µ
• Spatial rotations R(ij) = xi∂j − xj ∂i
• Boosts B(i) = t∂i − xi∂t
Conformal group SO(4, 2) has five additional generators:
• Dilatations D = t∂t + xi∂i = xµ∂µ
• Special conformal transformations
K(µ) = xν xν ∂µ − 2xµxν ∂ν
xµ = (t, xi)
xµ = (−t, xi)
(1)
KH1L at t = x3 = 0
Bjorken flow and symmetries, 12-15-14
S. Gubser
6
Interesting subgroups of SO(3, 1):
modified beamline boost
transverse planar symmetry
SO(1, 1)C
ISO(2)
generated by B(3) + t3 T(3)
generated by T(1) , T(2) , and R(12)
(2)
and of SO(4, 2):
“5d little group”
SO(3)q
generated by W(1) , W(2) , and R(12)
(3)
where 1/q is a length (say 4 fm) and
W(i) = T(i) − q 2K(i)
(i = 1, 2)
(4)
Ordinary Bjorken flow has SO(1, 1) × ISO(2) symmetry.
Conformal deformation is based on SO(1, 1) × SO(3)q .
Complex deformation is based on SO(1, 1)C × ISO(2) with t3 imaginary.
I always assume a Z2 symmetry under x3 → −x3 : colliding “nuclei” are identical.
Bjorken flow and symmetries, 12-15-14
3.
S. Gubser
7
Bjorken flow
Bjorken flow is simple because it respects a four-parameter symmetry group,
SO(1, 1) × ISO(2).
q
= (τ )
where
τ = t2 − x23
(5)
because τ is the unique combination of the xµ invariant under SO(1, 1) × ISO(2).
The velocity field is also determined by symmetry:
uµ = − p
∂µ τ
.
∂µτ ∂ µτ
(6)
Dynamics only enter into determining the form of (τ ): ∇µ Tµν = 0 where
Tµν = uµuν + (ηµν + uµuν ) ,
3
leads to = 0 /τ 4/3 .
Many calculations to follow are variations on (5)-(7).
(7)
Bjorken flow and symmetries, 12-15-14
4.
S. Gubser
8
Conformal deformation
AdS/CFT motivation: A pointlike massless particle at a depth 1/q in AdS5 has
SO(3)q symmetry: This is the 5-d little group.
Head-on collisions in AdS5
preserve SO(3)q .
R
Boost-invariance is not part
of this story.
Collisions of such particles
can make a black hole, and
some interesting
SO(3)q
multiplicity estimates can
be developed based on
H3
black hole entropy
3,1
x3
x 1,2
z=
S2
S1
C
z
1
q
SO(3)q
H3
[Gubser-Pufu-Yarom 0902.4062]
With a UV cutoff at Λ ∼ 2 GeV, find for central collisions Nch ≥ 16,800 @
√
2.76 TeV and 22,200 @ 5.5 TeV. Generally, Nch ∼ sN N 1/3 with the cutoff.
Bjorken flow and symmetries, 12-15-14
µ
The combination of x
invariant under
SO(1, 1) × SO(3)q is
1 − q 2τ 2 + q 2x2⊥
g=
,
2qτ
S. Gubser
9
v¦
0.14
0.12
0.10
0.08
1q = 4.
1q = 4.5
1q = 5.
1q = 5.5
1q = 6.
1q = 6.5
1q = 7.
0.06
and the “symmetrical”
velocity profile is
determined by
∂µ g
uµ = p
.
∂µg∂ µg
[Gubser 1006.0006]
0.04
v¦ = tanh
0.02
1
2
3
x¦
50
4
5
6
ux⊥
2q 2τ x⊥
v⊥ ≡
=
.
−uτ
1 + q 2(τ 2 + x2⊥)
7
x¦ HfmL
(8)
A natural choice for q is rms radius of colliding nucleus, so about 4 fm.
Figure shows v⊥ at τ = 0.6 fm/c for different choices of q ; cf. v⊥ = tanh x50⊥ from
a phenomenological study.
Bjorken flow and symmetries, 12-15-14
S. Gubser
10
Because we have four continuous symmetries, we can uniquely determine from
inviscid hydro:
ˆ(g)
ˆ0
(2q)8/3
= 4 = 4/3
.
τ
τ [1 + 2q 2(τ 2 + x2⊥) + q 2(τ 2 − x2⊥)2]4/3
(9)
14
12
210 MeV
Plotted is T ≡ (/11)1/4 .
Normalization of
temperature is based on
Τ HfmcL
10
8
130 MeV
dNcharged
dη
≈ 5000 .
6
dS/dη ≈ 7.5
4
2
30 MeV
0
2
4
6
x¦ HfmL
8
10
12
14
Bjorken flow and symmetries, 12-15-14
S. Gubser
11
Conformal deformation is wonderfully simplified by mapping future wedge of R3,1
into a contracting half-wedge of dS3 × R [Gubser-Yarom 1012.1314]:
(10)
ds2R3,1 = τ 2 −dρ2 + cosh2 ρ (dθ2 + sin2 θdφ2) + dη 2
X0
Τ�2.5
Τ= Τ= Τ
0.
0. =2
5
9
.5
t
Τ�0.9
−→
Τ�0.5
x3
X1
X3
ρ = − sinh−1 g
η = pseudorapidity
tan θ =
2qx⊥
q 2τ 2 − q 2x2⊥
φ = beamline azimuthal angle
(11)
Bjorken flow and symmetries, 12-15-14
S. Gubser
12
Let hatted quantities refer to dS3 × R, where “expansion” is just a stationary fluid.
∂ x̂ν
uµ = τ µ ûν
∂x
ˆ
τ4
(12)
entropy conserving
(13)
=
Then find ûρ = −1, ûθ = ûφ = ûη = 0, and
ˆ = ˆ0(cosh ρ)−8/3
Viscous corrections are accessible: uµ is unchanged,
d 3/4
2
cosh ρ = viscous heating
dρ
(14)
Bjorken flow and symmetries, 12-15-14
S. Gubser
13
HShaded region is not part of R3,1 L
2
An important instability at early
times is against non-zero uη .
Kinetic freez
eo
0 Chemical fre ut, T = 110
MeV
ezeou
t, T =
QGP
Ρ
Flow can be shown to be stable
against small perturbations in
realm where Navier-Stokes
applies (τ not too small).
-2
Τ = 1 fmc
Τ = 0.6 fmc
Τ = 0.3 fmc
170
Τ=¥
, x
¦ =¥
MeV
Thermalization
-4 Total overlap, Τ = 0.
07 fm
c
ΝΗ instability, Ρ = -4.4
x¦ = 7 fm
-6
`
T = 0, Ρ = -6
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Θ
Perturbations initialized via Glauber Monte Carlo at early τ propagate to a phenomenologically acceptable power spectrum, suppressed by viscous factor Pk =
2 η k2 t
e− 3 s T [Shuryak-Staig 1106.3243 and refs therein].
Bjorken flow and symmetries, 12-15-14
5.
14
S. Gubser
Complex deformation
Deforming ISO(2) → SO(3)q made the collision finite in the transverse direction.
Deforming
SO(1, 1) → SO(1, 1)C
with t3 imaginary makes the
collision finite in the beamline
direction.
But what could imaginary t3
mean?
Energy density, Poynting vector, and beamline pressure
match in SO(1, 1)C -invariant can be made to match
fairly well to results of AdS/CFT numerics.
Modified boost generator b = B(3) + t3 T(3) commutes with ISO(2).
p
τ C ≡ (t + t3)2 − x23 is the SO(1, 1)C × ISO(2) invariant.
Bjorken flow and symmetries, 12-15-14
S. Gubser
15
We glibly define [Gubser 1210.4181]
∂µτ C
= −p C µ C
∂µτ ∂ τ
C
C
C
C C C
Tµν = uµ uν + (ηµν + uC
µ uν )
3
C
C = C0 4/3
(τ )
uC
µ
(15)
C
∇µTµν
= 0 is automatic because we’re just translating Bjorken flow.
But all quantities that are supposed to be real have become complex!
No problem, specify C
0 > 0 and define
C
Tµν = Re eiθ Tµν
But how to choose θ?
Clearly, ∇µ Tµν = 0.
(16)
Bjorken flow and symmetries, 12-15-14
S. Gubser
16
Positive energy condition: If we require Tµν ξ µ ξ ν ≥ 0 for null or timelike ξ µ inside
future light-cone, it follows that θ = π/3.
Tµν generally does not obey
hydrodynamic ansatz.
We can still define local
4-velocity by
Tµν uν = −Luµ
with L > 0.
��t3 ��20
��t3 ��7.4
��t3 ��2.7
��t3 ��1.0
��t3 ��0.37
��t3 ��0.14
��t3 ��0.050
yF
6
yF �Η
4
2
�5
5
�2
�a�
�4
Define yF through
uµ = (cosh yF , 0, 0, sinh yF )
�6
Τ4�3 ΕL
in lab frame.
0.6
Then yF ≈ η at mid-rapidities, but dyF /dη ≈ 1/2 at forward rapidities.
0.4
�b�
Η
0.6
Bjorken flow and symmetries, 12-15-14
S. Gubser
17
In local frame where uµ = (1, 0, 0, 0), find
T µν = diag{−L, p⊥,�b�
p⊥, pL}
0.4
0.2
(17)
@
I
@
6
@
Landau frame
energy density
�5
p3L �ΕL
5
transverse longitudinal
pressure
pressure
Η
0.2
��t3 ��20
��t3 ��7.4
��t3 ��2.7
��t3 ��1.0
�5
Η
5
�0.2
�0.4
�0.6
�0.8
�1.0
�c�
Then p⊥ 6= pL is a
measure of how far
we are from inviscid
hydro.
In forward region,
pL ≈ −L, similar to
glasma.
Bjorken flow and symmetries, 12-15-14
6.
18
S. Gubser
Match to AdS/CFT numerics
Following numerics of van der Schee and collaborators, consider collisions of shocks
of the form
Nc2 µ3
2
2
(18)
T±±(z±) = 2 √
e−z±/2w ,
2π
2πw
where z± = t ± z , w is the width of the sheets and Nc2 µ3 is proportional to energy
per unit area.
We work in thin shock limit, µw 1.
• E = T 00 now in the lab frame.
• Positive energy condition is
violated near the light cone.
• Maybe we should try fitting to
SO(1, 1)C-invariant flow with
t3 ∼ i/µ.
• θ and t3 will be adjusted to
optimize the fit [Gubser-van der Schee
1410.7408]
Bjorken flow and symmetries, 12-15-14
S. Gubser
19
Instead of the inviscid result = 0 /τ 4/3 , we start with
2 4 N Λ
= c 2
2π
1
2η0
1
−
+
(Λτ )4/3 (Λτ )2 (Λτ )8/3
10 2 6 ln 2 − 17
√
η +
3 0
36 3
,
(19)
where η0 = √2133/4 (corresponding to η/s = 1/4π ), and plug into the complete 2nd
order hydrodynamical stress tensor:
µν
µν
µν
T µν = T(0)
+ T(1)
+ T(2)
µν
T(1)
= −2ησ
µν
Uniformly replacing τ → τ C
µν
leads to conserved TC as before,
and we finally extract
T µν = Re eiθ TCµν .
Λ fixed from late t at z = 0; find
θ = −0.425 and
µt3 = 0.080 + 0.318i.
µν
T(0)
= uµuν + P µν
3
η
µν
T(2)
∼ (∂σ + σ 2)
T
(20)
Bjorken flow and symmetries, 12-15-14
S. Gubser
20
Some details of the fit:
• We exclude region
τ < τmin = 0.75/µ.
• Curve τ = τmin is shown in
solid black.
√2 2
p
t −τ
dz
δE 2 + δPL2 + δS 2.
• We compute δ(t) = √ 2 min
2
R
−
t −τmin
• We minimize ∆/µ3 = δ(1.5/µ) + δ(3.0/µ).
The fit has some arbitrariness, but the main message is clear: SO(1, 1)C gives a
pretty good fit, including regions with negative E , and the fit improves with time.
Maybe the system is evolving toward a state with some SO(1, 1)C symmetry?
Bjorken flow and symmetries, 12-15-14
S. Gubser
21
Tried to study linear perturbations, but success was only so-so:
x±C
u ≡ u ± u = √ (1 ± ν)
g
±
t
z
T = T0(g)(1 + σ)
(21)
g = xC + x−C = (τ C)2 .
(22)
where we defined
x±C = t + t3 ± x3
SO(1, 1)C symmetry of background implies that we can use the ansatz
σ = (x+C)nΣ(g)
ν = (x+C)nN(g) .
(23)
µν
Then ∇µ TC = 0 implies
n
n
Σ + Σ+ N+ (viscous) = 0
2
6
0
n
n 1
N + Σ+
+
N+ (viscous) = 0 . (24)
2
2 3
0
Using n = 1 − 2i improved the overall fit by 25% at expense of four extra real
parameters (amplitude and phase of perturbations).
Description (23) of perturbations is overcomplete.
Bjorken flow and symmetries, 12-15-14
7.
S. Gubser
22
Complex and conformal deformations
How about combining the complex deformation of boost invariance,
B(3) → C(3) = B(3) + t3T(3) ,
aka
t → t + t3 ,
with the conformal deformation of transverse translation invariance,
T(i) → W(i) = T(i) − q 2K(i)
(i = 1, 2) ?
Seek an invariant function g and also a function h of weight 1, i.e.
αX
(∂µξ µ)X = 0
for
ξ ∈ {W(1), W(2), R(12), C(3)} ,
4
with αg = 0 and αh = 1. “By inspection,” find
ξ µ∂µX = −
j12
g=
j2
j1 = i1 − 4q 2t3t
h=
1
j1
where
j2 = i2 + 2t3t(i1 − 2q 2t3t)
i1 = 1 + q 2(−t2 + x21 + x22 + x23)
i2 = t2 − x23 .
(25)
Bjorken flow and symmetries, 12-15-14
S. Gubser
23
µν
Inviscid hydro ansatz plus ∇µ TC = 0 leads to
h4g 2C
0
C
=
[(g +
4q 2)2
+ (1 + (g +
2/3
4q 2)t23)]
and require C
0 > 0.
(26)
Minor technical complication: the phase of [...] varies quite a lot, so to keep [...]2/3
from having discontinuities we have to continue as needed to the appropriate sheet
of the Riemann surface of z 2/3 .
Just to have a nice example
to look at, consider:
Tµν
C
0
t3
q
C
= Re eiπ/3Tµν
=1
=i
= 1/5 .
(27)
eπi/3 gives T 00 > 0 inside
|z| < t lightcone for
t<
∼ 1/q .
Bjorken flow and symmetries, 12-15-14
24
S. Gubser
• Start with a compressed pancake, evolve to a combination of longitudinal and
radial flow.
• Eventually, T 00 < 0 at ~x = 0! Different choice of phase tends to introduce
more negative energy density regions inside |z| < t.
Bjorken flow and symmetries, 12-15-14
8.
S. Gubser
25
Non-relativistic boost invariance
Non-relativistic systems in 1 + 1 dimensions can have Galilean symmetry:
[H, P ] = [H, M ] = 0
[H, B] = −iP
[M, P ] = [M, B] = 0
[B, P ] = iM .
(28)
For example, {H, B, P, M } can act on a complex wave-function ψ(t, x) as
H = i∂t
P = −i∂x
B = it∂x + mx
M =m
(29)
where we impose the equation of motion
i∂tψ = −
1 2
∂ ψ + V (|ψ|)ψ .
2m x
(30)
Obvious adaptation of Bjorken flow is Bψ = 0. Arrive without much trouble at
ψ=
From NLSE
:
ψ0
t
e
1
+iθ(t)
2
imx2
2t
.
YH
H
H
(31)
From Bψ = 0
Bjorken flow and symmetries, 12-15-14
S. Gubser
26
Complex deformation:
Pass to Cψ = 0 where C = B + it∗ P ; in other words, t → t − it∗ , taking t∗ ∈ R.
Free Schrodinger works fine:
imx2
1
2(t−it
ψ=√
e ∗)
t − it∗
spreading Gaussian, from Quantum 101
But NLSE doesn’t generally have a solution of the form ψ = f (t)e
imx2
2(t−it∗ )
.
(32)
Bjorken flow and symmetries, 12-15-14
S. Gubser
27
One more deformation:
Pass to Dψ = 0 where D = B + a1 H − x0 M . For simplicity, set m = 1/2 = a1 .
For free Schrodinger equation, find “Airy beams” [Berry-Balazs ’79]
2i 3
ψx0 (t, x) = eitx− 3 t Ai(x − x0 − t2) ,
hψx1 |ψx2 i = δ(x1 − x2) . (33)
For NLSE, and for x0 = 0, arrive at
2i 3
ψ(t, x) = eitx− 3 t f (x − t2)
(34)
• All such wave forms propagate
non-dispersively.
• NLSE solutions discovered
recently in optics literature
[Kaminer-Segev-Christodoulides ’11]
• Quite a range of systems should
exhibit similar phenomena since
it’s all based on symmetries.
E.g. non-local ψ -ψ interactions.
Bjorken flow and symmetries, 12-15-14
9.
28
S. Gubser
Conclusions
• Could we choose any function g(xµ), call it the “invariant,” and make some flow
from it, say with uµ ∝ ∂µ g ?
No: It’s crucial that g is constrained by symmetries compatible with ∇µ Tµν = 0.
• Status of complex deformation is tricky because dynamics is only pseudo-hydrodynamical.
And yet: Approximate match to AdS/CFT numerics is striking enough to suggest that there’s something right about this funny construction.
• Conformal & complex deformation may at least provide a convenient analytical
starting point for phenomenology.
Outlook: Could we match onto hydro? Freeze-out? Is it best to think of 1/q as
size of nucleus or transverse size of a participant?
• Non-relativistic Bjorken flow lets us recover some interesting features of quantum mechanics.
Possible generalizations: NLSE, many-body systems, non-linear optics....
Bjorken flow and symmetries, 12-15-14
10.
29
S. Gubser
Backup material
Perturb conformal deformation by perturbing static plasma in dS3 × R:
T̂ = T̂b(1 + δ)
ûµ = (−1, νi, νη ) ,
(35)
Split into S 2 -scalar (sound plus η shear) and S 2 -vector (shear) modes:
δ = δ(ρ)S(θ, φ)eikη η
νi = νs(ρ)∂iS(θ, φ)eikη η + νv (ρ)Vi(θ, φ)eikη η
νη = νη (ρ)S(θ, φ)eikη η .
(36)
S(θ, φ) is a Y`m. Vi(θ, φ) is a vector spherical harmonic.
νv0 (ρ) = −Γv (ρ)νv (ρ)
(37)
where primes denote d/dρ and [Gubser-Yarom 0902.4062]
Γv =
1 H20
4 Tb
2
tanh
ρ
+
tanh4 ρ
9 Tb0
3 Tb Tb0
H0 2
2
−
−64
+
18`(`
+
1)
+
9k
+
(16
+
9k
)
cosh
2ρ
sech2 ρ tanh ρ .
η
η
36Tb0
(38)
Here H0 = η̂/T̂ 3 . Γv > 0 for reasonable values of parameters and τ not too early.
Bjorken flow and symmetries, 12-15-14
S. Gubser
30
SO(3)q can also be used to constrain initial conditions as captured by LO BK:
∂S(r1 , r2 ; Y )
αs Nc
=
∂Y
2π 2
Z
d2 z
|r1 − r2 |2
[S(r1 , z; Y )S(z, r2 ; Y ) − S(r1 , r2 ; Y )] .
|r1 − z|2 |r2 − z|2
(39)
SO(3)q -invariant ansatz is [Gubser 1102.4040]
where
S(r1, r2; Y ) = Sq (dq (r1, r2); Y ) .
(40)
|r1 − r2|
dq (r1, r2) ≡ p
,
(1 + q 2|r1|2)(1 + q 2|r2|2)
(41)
Because
dq (b, b + δr) ≈
δr
,
1 + q 2 b2
(42)
saturation scale defined through S(b, b + δr; Y ) ≈ Ŝ(Qs (b; Y )δr) gives
Qmax
(Y )
Qs(b; Y ) = s 2 2
1+q b
as previously suggested in [Iancu-McLerran hep-ph/0701276].
(43)