Quarkonium dissociation by anisotropy in a strongly coupled plasma - - PowerPoint PPT Presentation
Quarkonium dissociation by anisotropy in a strongly coupled plasma Mariano Chernicoff DAMTP, University of Cambridge Based on 1208.2672 (and 1202.3696) in collaboration with D. Fernandez, D. Mateos and D. Trancanelli
Based on 1208.2672 (and 1202.3696) in collaboration with
For the creation of the quark gluon plasma 400 nucleons go in 8000 hadrons are produced Energy of CM 200 GeV 1-10 GeV per hadron
Collision Far from equilibrium Anisotropic hydrodynamics Isotropic hydrodynamics
P? 6= Pk P? = Pk
longitudinal ¡direc>on ¡ transverse ¡direc>on ¡
z ≡ x, y ≡
[Romantschke ¡et ¡al; ¡Mrowczynski ¡et ¡al.] ¡
τiso . 1fm The time evolution of the quark gluon plasma
Things we know about the quark gluon plasma After a very short period of time the system is in thermal equilibrium
∼ 200 − 400MeV
Things we know about the quark gluon plasma After a very short period of time the system is in thermal equilibrium
∼ 200 − 400MeV
It is a strongly coupled system ( Perturbative methods are inapplicable and lattice QCD is not well suited to study real-time dynamics. ideal fluid + small )
η/s
Things we know about the quark gluon plasma After a very short period of time the system is in thermal equilibrium
∼ 200 − 400MeV
It is a strongly coupled system ( Perturbative methods are inapplicable and lattice QCD is not well suited to study real-time dynamics. ideal fluid + small )
η/s
Some observables are sensitive to the presence of an anisotropy
[Dumitru ¡et ¡al.; ¡Philipsen ¡et ¡al.] ¡ ¡
J/Ψ
[Dumitru ¡et ¡al.; ¡Mehtar-‑Tani; ¡Romantschke ¡et ¡al.] ¡
Quarkonium refers to charm-anticharm mesons ( , , , ... )
J/Ψ Ψ0 χc
and bottom-antibottom mesons ( , , ... )
Υ Υ0
1s ¡state ¡
About quarkonium in heavy ion collisions:
About quarkonium in heavy ion collisions: Quarkonium refers to charm-anticharm mesons ( , , , ... )
J/Ψ Ψ0 χc
and bottom-antibottom mesons ( , , ... )
Υ Υ0
dissociation temperature that is higher than the deconfinement temperature ( lattice predicts: ). mesons survive as bound states in a hot medium up to some
J/Ψ Tc Td Td(J/Ψ) ' 2Tc
RHIC data: - suppression in nucleus-nucleus collisions when compared to proton-proton collisions. dissociation temperature that is higher than the deconfinement temperature ( lattice predicts: ).
About quarkonium in heavy ion collisions: Quarkonium refers to charm-anticharm mesons ( , , , ... )
J/Ψ Ψ0 χc
and bottom-antibottom mesons ( , , ... )
Υ Υ0
mesons survive as bound states in a hot medium up to some
J/Ψ J/Ψ Tc Td Td(J/Ψ) ' 2Tc
mesons are screened in the quark gluon plasma
J/Ψ
They might be moving with significant transverse momentum through the hot medium, what is the effect of such “wind”? It is important to understand how they are screened by the QGP What is the effect of the anisotropy? We will use the AdS/CFT correspondence to address this questions
λ ≡ L4 l4
s
= g2
YMNc
u = 0
N = 4 SYM AdS5 × S5
Type IIB on
z xy u = ∞ ds2 = L2 u2 ⇥ − dt2 + d~ x2 + du2⇤
z xy u = 0 u = uh
T 6= 0
at finite temperature Schwarzschild AdS black hole
N = 4 SYM
ds2 = L2 u2 ⇥ − f(u)dt2 + d~ x2 + du2 f(u) ⇤ Note: we will refer to this metric as the isotropic metric f(u) = 1 − u4 u4
h
with
External quark
Fundamental string
u = 0 u = uh
Quark ¡
T 6= 0
codifies the profile of the gluonic field
horizon corresponds to an infinitely massive quark.
u = 0
meson (bound state) = U-shaped string
u = 0 u = uh
T 6= 0
` `
meson (bound state) moving at constant velocity
U-shaped string moving at constant velocity
u = 0 u = uh
T 6= 0
meson (bound state) moving at constant velocity
U-shaped string moving at constant velocity
u = 0 u = uh
T 6= 0
But we are interested in studying an anisotropic strongly coupled plasma How can we do that using the AdS/CFT correspondence?
The gauge theory that we will consider is a deformation of N = 4 SYM
S = SN =4 + Z ✓(~ x)TrF ∧ F with
T 6= 0
dimensions ¡of ¡energy ¡
✓(~ x) = 2⇡nD7z
z xy u = 0 u = uh
T 6= 0
S = SN =4 + Z ✓(~ x)TrF ∧ F The gauge theory that we will consider is a deformation of .
N = 4 SYM
with
χ = az
τ = θ 2π + 4πi g2
YM
= χ + ie−φ ✓(~ x) = 2⇡nD7z The field dual to the -term is the axion
The axion is magnetically sourced by D7-branes
z xy u = 0 u = uh
T 6= 0
S = SN =4 + Z ✓(~ x)TrF ∧ F The gauge theory that we will consider is a deformation of N = 4 SYM
with t x y z u S5 Nc D3 nD7 D7 x x x x x x x x ✓(~ x) = 2⇡nD7z
χ = az
The field dual to the -term is the axion
z xy u = 0 u = uh
T 6= 0
S = SN =4 + Z ✓(~ x)TrF ∧ F The gauge theory that we will consider is a deformation of N = 4 SYM
with
χ = az
a = λnD7 4πNc ✓(~ x) = 2⇡nD7z and
Density ¡of ¡D7-‑branes ¡ distributed ¡along ¡z ¡ is ¡a ¡measure ¡of ¡the ¡anisotropy ¡
The field dual to the -term is the axion
z xy u = 0 u = uh
T 6= 0
S = SN =4 + Z ✓(~ x)TrF ∧ F The gauge theory that we will consider is a deformation of N = 4 SYM
with
χ = az
✓(~ x) = 2⇡nD7z Finally, putting all these ingredients together, and solving the eom... The field dual to the -term is the axion
ds2 = L2 u2 h − F(u)B(u)dt2 + dx2 + dy2 + H(u)dz2 + du2 F(u) i
The anisotropic metric is
[Mateos ¡and ¡Trancanelli] ¡
0. 0.2 0.4 0.6 0.8 1.
1 2 3 4 5 6
1 2 3 4 5 6 0. 0.2 0.4 0.6 0.8 1.
0.2 0.4 0.6 0.8 1. 1.2
0.2 0.4 0.6 0.8 1. 1.2
H H F F B B φ φ u/uh u/uh
χ(z) = az φ ≡ φ(u)
and
The anisotropic metric is
[Mateos ¡and ¡Trancanelli] ¡
RG flow between AdS (UV) and Lifshitz type (IR) ds2 = L2 u2 h − F(u)B(u)dt2 + dx2 + dy2 + H(u)dz2 + du2 F(u) i The entropy density interpolates between
T a s ⇠ T 3 T ⌧ a s ⇠ a1/3T 8/3
and Regular on and outside the horizon There is an analytical expression for the near-boundary behavior
such that for it is energetically favorable for the pair The screening length is define as the separation between a
to be bound (unbound).
such that for it is energetically favorable for the pair The screening length is define as the separation between a
to be bound (unbound).
We will determine by comparing the action of the pair to
the action of the unbound system; i.e.
The screening length is the maximum value of for which is positive
(In the Euclidean version, this criterion corresponds to determining which configuration has the lowest free energy)
∆S(`) = S(`) − Sunb
u = 0 u = uh
θ Given the rotational symmetry in the xy-plane, the most general case is to consider the dipole in the xz-plane.
z x
Choosing the static gauge , , and the string embedding:
τ = t σ = u
Z(u) = z(u) cos θ X(u) = x(u) sin θ
and
The action for the U-shaped string takes the form S = − L2 2πα0 2 Z dt Z umax du 1 u2 q B(1 + FH cos2 θz02 + F sin2 θx02)
The action for the U-shaped string takes the form S = − L2 2πα0 2 Z dt Z umax du 1 u2 q B(1 + FH cos2 θz02 + F sin2 θx02) Two conserved momenta and associated to translation
Πz
invariance in the x, z direction. Then, the on-shell action can be written as
Πx
S = − L2 2πα0 2 Z dt Z umax du 1 u2 B √ FH p FBH − u4(Π2
z + HΠ2 x)
where the turning point is determined from the condition
umax
x0(umax) = z0(umax) → ∞
The action for the U-shaped string takes the form S = − L2 2πα0 2 Z dt Z umax du 1 u2 q B(1 + FH cos2 θz02 + F sin2 θx02) Two conserved momenta and associated to translation
Πz
invariance in the x, z direction. Then, the on-shell action can be written as
Πx
S = − L2 2πα0 2 Z dt Z umax du 1 u2 B √ FH p FBH − u4(Π2
z + HΠ2 x)
which in terms of the momenta reads FBH − u4(Π2
z + HΠ2 x)|umax = 0
umax ≡ umax(a, T, Πi)
From the boundary conditions we obtain the relation between the
Πz Πx
momenta , and the quark-antiquark separation `
l 2 = Z umax duX0 = Z umax duZ0
From the boundary conditions we obtain the relation between the
Πz Πx
momenta , and the quark-antiquark separation Finally, to determine , we need to subtract from the U-shaped string action, that of the unbound pair (i.e. two straight strings)
Ls
Sunb = − L2 2πα0 2 Z dt Z uh du √ B u2 The UV divergences associated to integrating all the way to the boundary cancel out in the difference, and there are no IR divergences.
l 2 = Z umax duX0 = Z umax duZ0
Ls ' 0.24 T
[Rey ¡et ¡al; ¡Brandhuber ¡et ¡al] ¡
Ls ' 0.24(π2N 2
c
2s )1/3 The isotropic result ( )
a = 0 H → 0, B → 1, F → f
at ¡constant ¡entropy ¡ density ¡
Ls ' 0.24 T
[Rey ¡et ¡al; ¡Brandhuber ¡et ¡al] ¡
Ls ' 0.24(π2N 2
c
2s )1/3 The isotropic result ( )
a = 0 H → 0, B → 1, F → f
The anisotropic results:
5 10 15 20 25 30 0.5 0.6 0.7 0.8 0.9 1.0
Ls/Liso(T) a/T
5 10 15 20 25 30 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
Ls/Liso(s) aN 2/3
c
/s1/3
θ = π/2 θ = 0 θ = π/2 θ = 0
We will first consider the ultra-relativistic limit. There are at least two important reasons for doing so: It is relevant for the experiments. It can be understood analytically. For the isotropic case:
Ls(T, v) ∼ (1 − v2)1/4
[Liu ¡et ¡al.] ¡
: direction of the wind (velocity) with respect to the anisotropic direction
θv θ, φ : parametrized the orientation
Our set up:
z x y θv θ φ q ¯ q
Dipole rest frame
As in the static case, to determine the screening length we need to compare the actions of a bound and an unbound quark-antiquark pair.
As in the static case, to determine the screening length we need to compare the actions of a bound and an unbound quark-antiquark pair. The unbound action is that of a string moving with constant velocity Let us summarize the important steps:
[MCh, ¡Fernandez, ¡Mateos ¡and ¡Trancanelli] ¡
As in the static case, to determine the screening length we need to compare the actions of a bound and an unbound quark-antiquark pair. The unbound action is that of a string moving with constant velocity The position of the turning point is now Let us summarize the important steps:
[MCh, ¡Fernandez, ¡Mateos ¡and ¡Trancanelli] ¡
It is easy to check that for a fixed separation of the string endpoints, Then the dynamics of the string can be determined using the near boundary expansion of the metric. umax ≡ umax(a, T, Πi, v)
lim
v→1 umax → 0
As in the static case, to determine the screening length we need to compare the actions of a bound and an unbound quark-antiquark pair. The unbound action is that of a string moving with constant velocity The position of the turning point is now Let us summarize the important steps:
[MCh, ¡Fernandez, ¡Mateos ¡and ¡Trancanelli] ¡
It is easy to check that for a fixed separation of the string endpoints, The analytical expression for the near boundary metric
[Mateos ¡et ¡al.] ¡
T - independent F, B, H ∼ 1 + Wi(a)u2 + Gi(a, T)u4 + O(u6) umax ≡ umax(a, T, Πi, v)
lim
v→1 umax → 0
We want to compare the two actions in the ultra-relativistic limit and see how they scale with
(1 − v2)
We want to compare the two actions in the ultra-relativistic limit and see how they scale with
(1 − v2)
After some algebra:
motion outside the transverse plane motion within the transverse plane
∆S(l, v) ∼ (1 − v2)−1/2 × (finite integral) ∆S(l, v) ∼ (1 − v2)−1/4 × (finite integral) I(a, θv, Πi, O(u6)) I(a, T, θv, Πi, O(u6))
And finally, using the boundary conditions, we obtain how the screening length scales in the ultra-relativistic limit: We want to compare the two actions in the ultra-relativistic limit and see how they scale with
(1 − v2)
After some algebra:
motion outside the transverse plane motion within the transverse plane
∆S(l, v) ∼ (1 − v2)−1/2 × (finite integral) ∆S(l, v) ∼ (1 − v2)−1/4 × (finite integral) Ls ⇠ (1 v2)1/2 ⇥ I(a, Πi, O(u6)) if θv 6= π/2 (1 v2)1/4 ⇥ J (a, T, Πi, O(u6)) if θv = π/2
A glimpse of the numerical results: Wind along “z” and dipole along “x”
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Ls/Liso(T)
5 10 15 20 25 30 0.0 0.2 0.4 0.6 0.8 1.0
Ls/Liso(T) a/T
v
a/T = 744 a/T = 12.2 v = 0.9995 v = 0.25
vanishes as (1 − v2)1/4 in the limit v → 1
Ls/Liso
5 10 15 20 25 30 0.6 0.7 0.8 0.9 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
A glimpse of the numerical results: Wind along “x” and dipole along “z”
Ls/Liso(T) Ls/Liso(T) a/T
v
a/T = 744 a/T = 12.2 v = 0.9995 v = 0.25
approaches a finite, non-zero value as v → 1
Ls/Liso
Another interesting limit to consider: a/T 1
The proper velocity along “z” of a point on the string at some
vproper(u) = vz s H(u) F(u)B(u) increases from to , more steeply as increases, a/T
H(u) u = 0 u = uh F(u)B(u) has the opposite behavior. ¡
Maximum value of beyond which becomes superluminal
umax vproper
Another interesting limit to consider: a/T 1 Two observations:
The proper velocity along “z” of a point on the string at some
vproper(u) = vz s H(u) F(u)B(u) increases from to , more steeply as increases, a/T
H(u) u = 0 u = uh F(u)B(u) has the opposite behavior. ¡
Maximum value of beyond which becomes superluminal
umax vproper
decreases as increases.
umax
a/T We can show that for ,
vz 6= 0
Another interesting limit to consider: a/T 1 Two observations:
lim
a/T 1 umax → 0
More over, Use near boundary metric to study Ls
For a fixed value of vz 6= 0
Ls ∼ a−1
It is straight forward to check that T independent as in the ultra-relativistic
case
Another interesting limit to consider: a/T 1
For a fixed value of vz 6= 0
Ls ∼ a−1
T = 0 a meson will dissociate for sufficiently large anisotropy ¡
adiss
It is straight forward to check that The limit can be understood as at fixed , or
a/T 1 a → ∞ T
as at fixed .
T → 0 a
Another interesting limit to consider: a/T 1
For a fixed value of vz 6= 0
Ls ∼ a−1
The limit can be understood as at fixed , or
a/T 1 a → ∞ T
as at fixed .
T → 0 a T = 0 a meson will dissociate for sufficiently large anisotropy ¡
adiss
For motion within the transverse plane ( )
vz = 0 Ls ∼ f(a, T)
No when
adiss T → 0
It is straight forward to check that
Another interesting limit to consider: a/T 1
So far we have studied , but clearly we could also think of
Ls(a, T, v)
and i.e. characterizes the dissociation of a qq-pair of fixed size in a plasma with a given degree of anisotropy . Analogously ¡for ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡.
Using our results for the screening length, we can study the behavior and adiss(T, `, v)
Tdiss(a, `, v) adiss(T, `, v) Tdiss(a, `, v) Tdiss(a, `, v) `
2 4 6 8 10 12 0.00 0.05 0.10 0.15 0.20 0.25
0.00 0.05 0.10 0.15 0.20 0.25 2 4 6 8 10 12
lTdiss l a
diss
al Tl x x x x z z z z Numerical results: As explained before, even at zero temperature a meson of size will dissociate if the anisotropy is increased above and the proportionality constant is a decreasing function of the velocity
at ¡rest ¡ moving ¡along ¡the ¡z-‑direc>on ¡( ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡) ¡ v = 0.45
adiss(T = 0, `) ∝ 1/`
`
Numerical results for motion within the transverse plane:
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Tdiss(v)/Tdiss(0) Tdiss(v)/Tdiss(0) v v extending ¡along ¡z ¡ extending ¡along ¡x ¡
The behavior is qualitatively analogous to that of the isotropic case.
Tdiss(v) ' Tdiss(0)(1 v2)1/4
[Liu ¡et ¡al.] ¡
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Tdiss(v)/Tdiss(0) Tdiss(v)/Tdiss(0) v v extending ¡along ¡z ¡ extending ¡along ¡x ¡ a = 0 a = 0 a = 25
There is a limiting velocity even at zero temperature!
vlim < 1
Numerical results for motion outside the transverse plane:
5 10 15 20 0.0 0.2 0.4 0.6 0.8 1.0
Limiting velocity for a fixed anisotropy and , meson oriented along the x-direction and moving along the z-direction vlim al
T = 0
5 10 15 20 0.0 0.2 0.4 0.6 0.8 1.0
Limiting velocity for a fixed anisotropy and , meson oriented along the x-direction and moving along the z-direction vlim al
T = 0
In the case of ultra-relativistic motion and :
a/T 1
adiss ⇠ 1 ` (1 v2
lim)1/2 if ✓v 6= ⇡/2
Tdiss ∼ 1 ` (1 − v2
lim)1/4 if ✓v = ⇡/2
mesons moving with arbitrary velocities and orientations.
at zero temperature.
at zero temperature .
background metric, so any anisotropy that gives rise to a qualitatively similar metric will yield qualitatively similar results.
F = 1 + 11 24a2u2 + ✓ F4 + 7 12a4 log u ◆ u4 + O(u6) , B = 1 − 11 24a2u2 + ✓ B4 − 7 12a4 log u ◆ u4 + O(u6) , H = 1 + 1 4a2u2 − ✓2 7B4 − 5 4032a4 − 1 6a4 log u ◆ u4 + O(u6) The near-boundary behavior of metric functions: B4(a, T) F4(a, T) and with