qcd at high-energy - lpthe.jussieu.frsoyez/data/talks/2006-05-zakopane.pdf · qcd at high-energy...
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QCD at high-energySaturation and fluctuation effects
Gregory Soyez
Cracow School of Theoretical Physics, XLVI Course, Zakopane, May-June 2006
G. Soyez Zakopane, May 27-June 05 2006 QCD at high-energy – p. 1/64
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Outline
Lecture 1: Evolution towards high-energy
Motivation
Leading log approximation: BFKL equation
Saturation effects: BK equation
fluctuations
Lecture 2: Properties of the scattering amplitudes
BK, statistical physics and geometric scaling
fluctuations, reaction-diffusion and diffusive scaling
Applications
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Evolution towards high-energy
G. Soyez Zakopane, May 27-June 05 2006 QCD at high-energy – p. 3/64
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Motivation
x
Q2N
onpe
rtur
bativ
e
log(Q2)
log(1/x
)
Energy W 2 = Q2/x
Rapidity Y = log(1/x)
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Motivation
x
Q2N
onpe
rtur
bativ
e
log(Q2)
log(1/x
)
DGLAP
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Motivation
x
Q2N
onpe
rtur
bativ
e
log(Q2)
log(1/x
)
DGLAP
BFKL
G. Soyez Zakopane, May 27-June 05 2006 QCD at high-energy – p. 4/64
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Motivation
x
Q2N
onpe
rtur
bativ
e
log(Q2)
log(1/x
)
DGLAP
BFKL
Q = Qs(Y )
G. Soyez Zakopane, May 27-June 05 2006 QCD at high-energy – p. 4/64
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Motivation
x
Q2N
onpe
rtur
bativ
e
log(Q2)
log(1/x
)
DGLAP
BFKL
Q = Qs(Y )
How to describe
this in QCD ?
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Motivation
Rough estimates:
Saturation scale: ↔ partons are covering the proton
xg(x,Q2)︸ ︷︷ ︸
# gluons
Q−2
︸︷︷︸
parton size
= πR2p
︸︷︷︸
proton size
The gluon distribution behaves like: xg(x,Q2) ∝ x−λ
⇒ Q2s(Y ) ∝ R−2
p x−λ
Rp ≈ 1 fm ≈ 5 GeV−1
Q2s ≈ 1GeV2 at x = 10−4
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Soft gluon emission
Need for some careful treatment because of Bremsstrahlung:
p
kz =xp
x≪ 1
p
kz1 =x1p
kz =xp
x≪ x1 ≪ 1
Probability of emission
dP ∼ αs
dk2
k2
dx
x
In the small-x limit∫ 1
x
dx1
x1∼ αs log(1/x)
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Soft gluon emission
Need for some careful treatment because of Bremsstrahlung:
p
kz =xp
x≪ 1
p
kz1 =x1p
kz =xp
x≪ x1 ≪ 1
px1 ≪ 1
x2 ≪ x1
x ≪ xn
xn ≪ xn−1
n-gluon emission:
∫ 1
x
dxn
xn
∫ 1
xn
dxn−1
xn−1. . .
∫ 1
x2
dx1
x1∼ 1
n!αn
s logn(1/x)
−→ At small x: need to be resummed:
∞∑
n=0
1
n!αn
s logn(1/x) ≈ eωY
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Dipole picture
[Mueller, 93]
Consider a fast-moving qq dipole Rapidity: Y = log(s)
Y0
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Dipole picture
[Mueller, 93]
Consider a fast-moving qq dipole Rapidity: Y = log(s)
Y0 ≪ Y1
Probability αK of emission
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Dipole picture
[Mueller, 93]
Consider a fast-moving qq dipole Rapidity: Y = log(s)
Y0 ≪ Y1 ≪ Y2
Probability αK of emission
Independent emissions
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Dipole picture
[Mueller, 93]
Consider a fast-moving qq dipole Rapidity: Y = log(s)
Y0 ≪ Y1 ≪ Y2 ≪ Y3
Probability αK of emission
Independent emissions
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Dipole picture
[Mueller, 93]
Consider a fast-moving qq dipole Rapidity: Y = log(s)
Y0 ≪ Y1 ≪ Y2 ≪ Y3
n(r, Y ) dipoles
of size r
Probability αK of emission
Independent emissions
Large-Nc approximation
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Dipole splitting
≡+++
x
y
z+
2
=
[x − z
(x− z)2− y − z
(y − z)2
]2
=α
2π
(x − y)2
(x − z)2(z− y)2
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BFKL evolution (1/4)
How to observe this system ?
Scattering amplitude
T (r, Y ) ≈ α2sn(r, Y )
Count the number of dipoles of a given size
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BFKL evolution (2/4)
Consider a small increase in rapidity
x
y
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BFKL evolution (2/4)
Consider a small increase in rapidity ⇒ splitting
x
y
x
yz
∂Y T (x,y;Y )
T (x, z;Y )
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BFKL evolution (2/4)
Consider a small increase in rapidity ⇒ splitting
x
y
x
yz
x
yz
∂Y T (x,y;Y )
T (x, z;Y ) + T (z,y;Y )
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BFKL evolution (2/4)
Consider a small increase in rapidity ⇒ splitting
x
y
x
yz
x
yz
x
yz
∂Y T (x,y;Y )
T (x, z;Y ) + T (z,y;Y ) − T (x,y;Y )
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BFKL evolution (2/4)
Consider a small increase in rapidity ⇒ splitting
x
y
x
yz
x
yz
x
yz
∂Y T (x,y;Y )
= α
∫
d2z(x− y)2
(x − z)2(z− y)2[T (x, z;Y ) + T (z,y;Y ) − T (x,y;Y )]
[Balitsky, Fadin, Kuraev, Lipatov, 78]
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BFKL evolution (3/4)
Solution ? Use Mellin space
T (x,y) = T (|x − y|) = T (r) = r2γ ⇒ ∂Y T (r) = αχ(γ)T (r)
χ(γ) is the BFKL eigenvalues:
χ(γ) = 2ψ(1) − ψ(γ) − ψ(1 − γ)
γ
χ(γ
)
10.90.80.70.60.50.40.30.20.10
14
12
10
8
6
4
2
0
b-independent solution:
T (r) =
∫ c+i∞
c−i∞
dγ
2iπT0(γ) exp
[
αχ(γ)Y − γ log
(r20r2
)]
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BFKL evolution (4/4)
Solution in the saddle point approximation
T (r, Y ) ≈ 1√Y
r
r0eωY exp
[
− log2(r2/r20)
2αχ′′(1/2)Y
]
with ω = 4α log(2) ≈ 0.5
Fast growth of the amplitude
Intercept value too large
Same with r0/r → k/k0
problem of diffusion in the infrared
Violation of the froissart bound:T (Y ) ≤ C log2(s) T (r, b) ≤ 1
Y = 2Y = 1.5Y = 1
Y = 0.5Y = 0
log(k2/k20)
T
1050-5-10
6
5
4
3
2
1
0
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BFKL and BK evolution
Let us reconsider one step of the evolution
Rapidity increase ⇒ Splitting into 2 dipoles
x
y
Y + δY
z
∂Y 〈Txy〉 = α
∫
z
Mxyz[〈Txz〉 + 〈Tzy〉 − 〈Txy〉︸ ︷︷ ︸
Linear BFKL
]
[Balitsky, Fadin, Kuraev, Lipatov, 78]
Solution: eωY but violates unitarity
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BFKL and BK evolution
Let us reconsider one step of the evolution
Rapidity increase ⇒ Splitting into 2 dipoles
x
y
Y + δYx
yz
∂Y 〈Txy〉 = α
∫
z
Mxyz[〈Txz〉 + 〈Tzy〉 − 〈Txy〉︸ ︷︷ ︸
Linear BFKL
−〈TxzTzy〉︸ ︷︷ ︸
Unitarity
]
[Balitsky 96]
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Saturation effects: BK evolution
x
yz
Proportional to T 2
important when T ≈ 1
〈T 〉,⟨T 2
⟩, ...: JIMWLK/Balitsky equations (at large Nc)
Mean-field approximation:⟨T 2
⟩= 〈T 〉2 (BK equation)
∂Y 〈Txy〉 = α
∫
z
Mxyz[〈Txz〉 + 〈Tzy〉 − 〈Txy〉︸ ︷︷ ︸
Linear BFKL
−〈Txz〉 〈Tzy〉︸ ︷︷ ︸
Unitarity
]
[Kovchegov 99]
Most simple perturbative evolution equation including BFKL + saturation
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Saturation effects: pros
Improvements due to this new term:
0 ≤ T (x,y) ≤ 1 ⇒ unitarity preserved
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Saturation effects: pros
Improvements due to this new term:
0 ≤ T (x,y) ≤ 1 ⇒ unitarity preserved
Cut the diffusion to the infrared
unintegratedgluon distribution
BFKLBK
Y = 0
Y = 0, 4, 8, 12, 16, 20
log(k2/k20)
n(k
)
403020100-10
10000
1000
100
10
1
0.1
0.01
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Saturation effects: fan diagrams
Equivalence with “usual” Feynman graphs:
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Saturation effects: fan diagrams
Equivalence with “usual” Feynman graphs:
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Saturation effects: fan diagrams
Equivalence with “usual” Feynman graphs:
BFKL ladder
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Saturation effects: fan diagrams
Equivalence with “usual” Feynman graphs:
BFKL ladder
fan diagram
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Fluctuations
[E. Iancu, D. Triantafyllopoulos, 05]Also A. Mueller, S. Munier, A. Shoshi, S. Wong
Consider evolution of⟨T (2)
⟩
Usual BFKL ladder ∂Y
⟨T (2)
⟩∝
⟨T (2)
⟩
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Fluctuations
[E. Iancu, D. Triantafyllopoulos, 05]Also A. Mueller, S. Munier, A. Shoshi, S. Wong
Consider evolution of⟨T (2)
⟩
Usual BFKL ladder ∂Y
⟨T (2)
⟩∝
⟨T (2)
⟩
fan diagram −→ saturation effects ∂Y
⟨T (2)
⟩∝
⟨T (3)
⟩
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Fluctuations
[E. Iancu, D. Triantafyllopoulos, 05]Also A. Mueller, S. Munier, A. Shoshi, S. Wong
Consider evolution of⟨T (2)
⟩
Usual BFKL ladder ∂Y
⟨T (2)
⟩∝
⟨T (2)
⟩
fan diagram −→ saturation effects ∂Y
⟨T (2)
⟩∝
⟨T (3)
⟩
splitting −→ fluctuations, pomeron loops ∂Y
⟨T (2)
⟩∝ 〈T 〉
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Why fuctuations
Why do we need fluctuations ?
The fluctuation term acts as a seed for⟨T 2
⟩
Then, it grows like 2 pomerons !
∼ eωY ∼ α2se
2ωY
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Why fuctuations
Why do we need fluctuations ?
The fluctuation term acts as a seed for⟨T 2
⟩
Then, it grows like 2 pomerons !
∼ eωY ∼ α2se
2ωY
Comparable for rapiditiesY ∼ 1
ωPlog(1/α2
s) ∼ 1αS
log(1/α2s)
⇒ need to include everything.
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Fluctuation term
Computation of the fluctuation term
Expected to be important for dilute target⇒ Target made of dipoles
x1
y1x2
y2
uz
v
∫
uvz
α
2π
(u− v)2
(u− z)2(z− v)2︸ ︷︷ ︸
BFKL splitting
α2sA0(x1y1|uz)α2
sA0(x2y2|zv)︸ ︷︷ ︸
interaction (2GE)
1
α2s
∇2u∇2
v 〈Tuv〉︸ ︷︷ ︸
dipole density
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Evolution hierarchy
⇒ complicated hierarchy
∂Y
⟨
T (2)(x1,y1;x2,y2)⟩
=α
2π
∫
d2z(x2 − y2)
2
(x2 − z)2(z − y2)2
[⟨
T (2)(x1,y1;x2, z)⟩
+⟨
T (2)(x1,y1; z,y2)⟩
−⟨
T (2)(x1,y1;x2,y2)⟩
−⟨
T (3)(x1,y1;x2, z; z,y2)⟩
+ (1 ↔ 2)]
+α
2π
(αs
2π
)2∫
uvz
MuvzA0(x1y1|uz)A0(x2y2|zv)∇2u∇2
v
⟨
T (1)(u,v)⟩
Saturation: important when T (2) ∼ T (1) ∼ 1 i.e. near unitarity
Fluctuations: important when T (2) ∼ α2sT
(1) or T ∼ α2s i.e. dilute regime
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Langevin equation
Infinite hierarchy ≡ Langevin equation
∂Y Txy =α
2π
∫
z
Mxyz [Txz + Tzy − Txy − TxzTzy]
+1
2
α
2π
αs
2π
∫
uvz
A0(xy|uz)|u− v|(u− z)2
√
∇2u∇2
vTuv ν(u,v, z;Y )
where ν is a Gaussian white noise
〈ν(u,v, z;Y )〉 = 0
〈ν(u,v, z;Y )ν(u′,v′, z′;Y ′)〉 = δ(αY − αY ′)δ(2)(u− v′)δ(2)(z− z′)δ(2)(v − u′)
Hierarchy obtained by averaging events with different realization of thenoise
problem: non-local & off-diagonal noise !
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Langevin equation
Simple example of noise term: zero space dimension
∂tu(t) =√
2κu ν(t) with 〈ν(t)ν(t′)〉 = δ(t− t′)
discrete t Itô⇒ u(tj + δt) = u(tj) + δt√
2κu νj 〈νiνj〉 =1
δtδij
⇒ F (uj+1) = F (uj) + δt√
2κujνjF′(uj) + δt2κujν
2jF
′′(uj)
⇒ ∂t 〈F (u)〉 = κ 〈uF ′′(u)〉
F (u) = un ⇒ ∂t 〈un〉 = n(n− 1)κ⟨un−1
⟩
corresponding to the fluctuation term.
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Target vs. projectile
targ.
proj.BFKL Saturation Fluctuations
duality: target-proectile symmetry ≡ boost invariance
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Target vs. projectile
targ.
proj.BFKL Saturation Fluctuations
duality: target-proectile symmetry ≡ boost invariance
Projectile wavefunction evolution
BFKL & saturation from dipole splitting
fluctuations ≡ gluon merging (or recombination, saturation)
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Target vs. projectile
targ.
proj.BFKL Saturation Fluctuations
duality: target-proectile symmetry ≡ boost invariance
Projectile wavefunction evolution
BFKL & saturation from dipole splitting
fluctuations ≡ gluon merging (or recombination, saturation)
Target wavefunction evolution
BFKL & fluctuations from dipole splitting
saturation ≡ multiple scatterings
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Reaction-diffusion
Reaction-diffusion process Aγσ
A + A
Master equation: Pn ≡ proba to have n particles
∂tPn = γ (n− 1)Pn−1︸ ︷︷ ︸
gain
− γ nPn︸ ︷︷ ︸
loss
+σ n(n+ 1)Pn+1︸ ︷︷ ︸
gain
−σ n(n− 1)Pn︸ ︷︷ ︸
loss
Particle densities: we observe a subset of k particles
⟨nk
⟩≡
∞∑
N=k
N !
(N − k)!PN
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Reaction-diffusion
Reaction-diffusion process Aγσ
A + A
Master equation: Pn ≡ proba to have n particles
∂tPn = γ (n− 1)Pn−1︸ ︷︷ ︸
gain
− γ nPn︸ ︷︷ ︸
loss
+σ n(n+ 1)Pn+1︸ ︷︷ ︸
gain
−σ n(n− 1)Pn︸ ︷︷ ︸
loss
Evolution equation:⟨nk
⟩≡ particle density/correlators
∂t
⟨nk
⟩= γ k
⟨nk
⟩+ γ k(k − 1)
⟨nk−1
⟩− σ k(k + 1)
⟨nk+1
⟩
Scattering amplitude for this system off a target
A(t) =∞∑
k=0
(−)k⟨nk
⟩
t0
⟨T k
⟩
t−t0
t0-independent ⇒ ∂t
⟨T k
⟩= γ
⟨T k
⟩
︸ ︷︷ ︸
BFKL
− γ⟨T k+1
⟩
︸ ︷︷ ︸
sat.
+σ⟨T k−1
⟩
︸ ︷︷ ︸
fluct.
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Reaction-diffusion & QCD
For QCD particle = (effective) dipoles
Dipole plitting ≡ BFKL kernel γ ∼ α(x − y)2
(x − z)2(z− y)2
Effective dipole merging
σ(x1y1,x2y2 → uv)
∼ αα2s∇2
u∇2v
Muvz log2
[(x1 − u)2(y1 − z)2
(x1 − z)2(y1 − u)2
]
log2
[(x2 − v)2(y2 − z)2
(x2 − z)2(y2 − v)2
]
Remarks:
merging not always positive
fluctuations = gluon-number fluctuations
Can be obtained from projectile or target point of view
Known at large Nc.
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Color Glass Condensate (1/4)
ρ
small-x gluon
fast partons
k
x- x+
z
t
Α[ρ]+ = xP+
Effective theory for High-Energy QCD:
Theory for the gluonic field : Color
Small-x ≡ classical field radiated by frozen fast gluonsLarge-x ≡ random distribution of color sources: Glass
Large occupation number: Condensate
Equation for the probability distribution of the color charge WY [ρ]
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Color Glass Condensate (2/4)
ρ
small-x gluon
fast partons
k
x- x+
z
t
Α[ρ]+ = xP+
fast gluons: frozen, source for slow partons
(DµFµν)a = δν+ρa(x−,x⊥)
Random source: correlators computed using the probability distributionWY [ρ]
⟨Ai
aAia
⟩=
∫
DρWY [ρ]AiaA
ia
Strong field A ∼ 1/g (equivalent to n ∼ 1/αs or T ∼ 1)G. Soyez Zakopane, May 27-June 05 2006 QCD at high-energy – p. 27/64
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Color Glass Condensate (3/4)
WY [ρ]
δY
WY [ρ]
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Color Glass Condensate (3/4)
WY [ρ]
δY
WY [ρ]
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Color Glass Condensate (3/4)
WY [ρ]
δY
WY [ρ]
≡
WY +δY [ρ]
Evolution
∂YWY [ρ] =1
2
∫
xy
δ
δρax
χabxy[ρ]
δ
δρay
WY [ρ]
[Jalilan-Marian,Iancu,McLerran,Weigert,Leonidov,Kovner,97-00]
Wilson Line
Vx = P exp
[
ig
∫
dx−A+(x−,x⊥)
]
In the weak field limit −→ BFKL.
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Color Glass Condensate (4/4)
2r ~ Q 21/
q
rγ ∗
P
Dipole
Proton
S-matrix: γ∗ → qq → V †xVy
SY =
∫
DA+WY [A]1
Nc
tr(V †xVy)
Wilson Line
Vx = P exp
[
ig
∫
dx−A+(x−,x⊥)
]
G. Soyez Zakopane, May 27-June 05 2006 QCD at high-energy – p. 29/64
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Summary
Ladder-type diagrams ⇒ BFKL equation
∂Y 〈Txy〉 =α
2π
∫
z
(x − y)2
(x− z)2(z− y)2[〈Txz〉 + 〈Tzy〉 − 〈Txy〉]
unitarity violations
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Summary
Unitarity corrections: Add fan diagrams ⇒ Balitsky equation
∂Y 〈Txy〉 =α
2π
∫
z
(x − y)2
(x − z)2(z − y)2[〈Txz〉 + 〈Tzy〉 − 〈Txy〉 − 〈TxzTzy〉]
infinite hierarchy: Balitsky/JIMWLK
for⟨
T(2)xz,zy
⟩
= 〈Txz〉 〈Tzy〉: BK equation
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Summary
gluon-number fluctuations: add fluctuations in the target
∂Y
⟨
T (2)x1y1,x2y2
⟩∣∣∣flucu
=1
2
α
2π
(αs
2π
)2∫
uvz
MuvzA0(1|uz)A0(2|zv)∇2u∇2
v 〈Tuv〉
equivalent to a reaction-diffusion problem
projectile-target duality
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Asymptotic solutions for scattering amplitudes
impact-parameter-independent BK
BK at nonzero momentum transfer
including fluctuations
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b-independent BK
[Munier, Peschanski,05]
b
ryx
T (x,y)
T (r;b)
T (r)
Momentum space:
T (k) =1
2π
∫d2r
r2eir.kT (r) =
∫dr2
r2J0(kr)T (r)
BK equation
∂Y T (k) =α
π
∫dp2
p2
[
p2T (p) − k2T (k)
|k2 − p2| +k2T (k)
√
4p4 + k4
]
− αT 2(k)
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Link with statistical physics (1/2)
[S. Munier, R. Peschanski]
b-independent situation: momentum space (L = log(k2/k20))
∂αY T (k) = χBFKL(−∂L)T (k) − T 2(k)
Diffusive approximation:
χBFKL(−∂L) = χ( 12 ) + 1
2 (∂L + 12 )2
Time t = αY , Space x ≈ log(k2), u ∝ T
∂tu(x, t) = ∂2xu(x, t) + u(x, t) − u2(x, t)
Fisher-Kolmogorov-Petrovsky-Piscounov (F-KPP)
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Link with statistical physics (2/2)
t = 0 t1
X(t1)
2t1
X(t2)
3t1
X(t3)
Asymptotic solution:
travelling wave
u(x, t) = u(x − vct)
Position: X(t) = X0 + vct
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Travelling waves
x
u
linear
∂tu = ∂2xu+ u = ω(−∂x)u
with ω(γ) = γ2 + 1
⇒ u(x, t) =∫
dγ2iπ
u0(γ) exp[ω(γ)t− γx]
x
γc
γ0
The minimal speed is
selected during evoution
vc
γc
γ
spee
d
32.521.510.50
5
4
3
2
1
0
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Asymptotic behaviour (1/2)
More generaly, it is true under the conditions:
0 is an unstable solution, 1 is a stable solution
The initial condition is steep enough
The linear equation admits solution of the form
Tlin =
∫ c+∞
c−i∞
dγ
2iπa0(γ) exp [ω(γ)Y − γL]
⇒ Travelling waves with critical speed
vc = minω(γ)γ
or vc = ω(γc)γc
= ω′(γc)
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Asymptotic behaviour (1/2)
More generaly, it is true under the conditions:
0 is an unstable solution, 1 is a stable solutionBFKL growth, BK damping
The initial condition is steep enoughColour transparency
The linear equation admits solution of the form
Tlin =
∫ c+∞
c−i∞
dγ
2iπa0(γ) exp [ω(γ)Y − γL]
BFKL: ω(γ) = αχ(γ) = α [2ψ(1) − ψ(γ) − ψ(1 − γ)]
⇒ Travelling waves with critical speed
vc = minω(γ)γ
or vc = ω(γc)γc
= ω′(γc)
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Asymptotic behaviour (2/2)
γc
γ
χ(γ
)
10.90.80.70.60.50.40.30.20.10
14
12
10
8
6
4
2
0
BK equation: linear part ≡ BFKL kernel
γc = 0.6275
vc = 4.8834α
Tail of the front:
T (k, Y ) = T
(k2
Q2s(Y )
)
≈ log
(k2
Q2s(Y )
) ∣∣∣∣
k2
Q2s(Y )
∣∣∣∣
−γc
exp
(
− log2(k2/Q2s(Y ))
αχ′′(γc)Y
)
Geometric scalingSaturation Scale:
log(Q2s(Y )) = vcY − 3
2γc
log(Y ) − 3
γ2c
√
2π
αχ′′(γc)
1√Y
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Geometric scaling
Numerical simulations: α = 0.2
Y = 25Y = 20Y = 15Y = 10Y = 5Y = 0
log(k2)
T
4035302520151050-5
10
1
0.1
0.01
0.001
1e-04
1e-05
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Geometric scaling
Numerical simulations: α = 0.2
Y = 25Y = 20Y = 15Y = 10Y = 5Y = 0
log(k2)
T
4035302520151050-5
10
1
0.1
0.01
0.001
1e-04
1e-05
N = 0.5N = 0.1N = 0.05N = 0.01
Y
1 Ylo
g(
Qs2(Y
)Q
s2(0
)
)
504540353025201510
1.4
1.2
1
0.8
0.6
0.4
0.2
0
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Geometric scaling
[Kwiecinski, Stasto; 01]
Observed in the HERA data for F p2
Y = 25Y = 20Y = 15Y = 10Y = 5Y = 0
log(k2)
T
4035302520151050-5
10
1
0.1
0.01
0.001
1e-04
1e-0510
-1
1
10
10 2
10 3
10-3
10-2
10-1
1 10 102
103
E665
ZEUS+H1 high Q2 94-95H1 low Q2 95ZEUS BPC 95ZEUS BPT 97
x<0.01
all Q2
τ
σ totγ*
p [µ
b]
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Another approach
log(Q2)
Y
DGLAP
BFKL
BK
Qs(Y )
[Mueller, Triantafyllopoulos, 02]
Idea:saturation cuts Q2 < Q2
s
⇒ BK ≡ BFKL + boundary
BFKL solution:
T =
∫dγ
2iπT0(γ) exp
[αχ(γ)Y − γ log(Q2/µ2)
]
Conditions:
Saddle point:αχ′(γc)Y − log(Q2s(Y )/µ2) = 0
Barrier: αχ(γc)Y − γc log(Q2s(Y )/µ2) = 0
⇒ γcχ′(γc) = χ(γ) and log(Q2
s/µ2) = αχ′(γc)Y .
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Asymptotic solutions
The full BK equation
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Coordinate vs. momentum space
Question: do we have the same properties for the full BK equation ?
∂Y 〈Txy〉 =α
2π
∫
d2zMxyz [〈Txz〉 + 〈Tzy〉 − 〈Txy〉 − 〈Txz〉 〈Tzy〉]
Problem:
x
z
y
bxz
bzy
bxy
Diffusion/non-locality in b
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New form of the BK equation
[C. Marquet, R. Peschanski, G.S., 05]
Solution: go to momentum space
T (k,q) =
∫
d2x d2y eik.xei(q−k).y T (x,y)
(x− y)2
new form of the BK equation
∂Y T (k,q) =α
π
∫d2k′
(k − k′)2
T (k′,q) − 1
4
[k2
k′2+
(q − k)2
(q − k′)2
]
T (k,q)
− α
2π
∫
d2k′ T (k,k′)T (k − k′,q− k′)
locality in q for the BFKL term
Asymptotic solutions: study the linear kernelWe need a superposition of waves
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More BFKL dynamics
Solutions of the full BFKL kernel:
Tlin(k,q) =
∫ 12+i∞
12−i∞
dγ
2iπeαχ(γ)Y fγ(k,q)φ0(γ,q)
with
fγ(k,q) =Γ2(γ)
Γ2(
12 + γ
)2
|k|∣∣∣q
4k
∣∣∣
2γ−1
2F1
(
γ, γ; 2γ;q
k
)
2F1
(
γ, γ; 2γ;q
k
)
︸ ︷︷ ︸
−→1 when k≫q
−(γ → 1−γ)
⇒ Tlin(k,q) =
∫ 12+i∞
12−i∞
dγ
2iπφ0(γ,q) exp
[
αχ(γ)Y − γ log
(k2
q2
)]
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Geometric scaling
⇒ geometric scaling for the full BK equationSaturation Scale: same Y dependence as previously
Q2s(Y ) ∼ q2 exp
[
vcY − 3
2γc
log(Y ) − 3
γ2c
√
2π
αχ′′(γc)
1√Y
]
∼ q2Ω2s(Y )
Tail of the front: same slope γc
T (k, Y ) = T
(k2
q2Ω2s(Y )
)
≈ log
(k2
q2Ω2s(Y )
) ∣∣∣∣
k2
q2Ω2s(Y )
∣∣∣∣
−γc
Note: more careful treatment gives Q2s = Ω2(Y ) max(q2, Q2
T )
Predicts geometric scaling for t-dependent processes
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Numerical simulations (1/2)
Dependence on momentum transfer k: travelling waves
Rp
T(p
;q)
1e+081e+061000010010.01
1000
100
10
1
0.1
0.01
0.001
1e-04
1e-05
RpT
(p;q
)1e+081e+061000010010.01
0.1
0.01
0.001
1e-04
1e-05
1e-06
1e-07
1e-08
formation of a travelling wave at large p (or k) ⇒ Geometric scaling
cut-off effect in the infrared region
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Numerical simulations (2/2)
Saturation scale
log(Rq) = 3log(Rq) = 2log(Rq) = 1log(Rq) = 0
log(Rq) = −1log(Rq) = −2log(Rq) = −5
Y
1 Ylo
g(
Q2 s(Y
)Q
2 s(Y
0)
)
2520151050
2
1.5
1
0.5
0
log(Rq)lo
g(RQ
s)
3210-1-2-3-4-5
12
11
10
9
8
7
Y dependence: converges to vc
q dependence: scales like a constant or linearly (Y = 25)
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Solutions
Fluctuation effects
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Event evolution
[E. Iancu, A. Mueller, S. Munier, 04]
no b-dependence + coarse-graining −→ Langevin equation
∂Y T (k, Y ) = αKBFKL ⊗ T (k, Y ) − αT 2(k, Y ) + α√
κα2sT (k, Y ) ν(k, Y )
with 〈ν(k, Y )〉 = 0
〈ν(k, Y )ν(k′, Y ′)〉 = 1αδ(Y − Y ′) kδ(k − k′)
Diffusive approximation
∂tu(x, t) = ∂2xu+ u− u2 +
√2κu ν(x, t)
stochastic F-KPP
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Numerical solution
Numerical solution of
∂Y T (k, Y ) = αKBFKL ⊗ T (k, Y ) − αT 2(k, Y ) + α√
κα2sT (k, Y ) ν(k, Y )
Dealing with the noise term: du =√
2κuν(t) ⇒ ∂t 〈F (u)〉 = κ 〈uF ′′(u)〉Associated probability
〈F (u)〉 =
∫
duF (u)P (u, t)∂t⇒ ∂tP (u, t) = κ∂2
u [uP (u, t)]
Including the initial contition u(t = 0) = u0, we get
Pt(u0 → u) ≡ probability to go from u0 to u in a time t.
Define the cumulative probability Fu0,t(u) =∫ u
0−dvPt(u0 → v).
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Numerical solution
Numerical solution of
∂Y T (k, Y ) = αKBFKL ⊗ T (k, Y ) − αT 2(k, Y ) + α√
κα2sT (k, Y ) ν(k, Y )
Rapidity step δY :
Step 1: Use probability: 0 < y < 1 uniform random variable
Tnoise(k, Y ) = F−1T (k,Y ),δY
(y)
Step 2: Apply the remaining equation
T (k, Y + δY ) = Tnoise(k, Y ) + δY[αKBFKL ⊗ Tnoise(k, Y ) − αT 2
noise(k, Y )]
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Event properties (1/2)
[G.S., 05]
0 (BK)0.0010.010.11
κ
Y
〈X(Y
)〉−〈X
(1000)〉
Y−
1000
2500020000150001000050000
1
0.98
0.96
0.94
0.92
0.9
0.88
0.86
Decrease of the velocity/exponent of the saturation scaleFor αs ≪ 1 (not true here)[A. Mueller, S. Munier, E. Brunet, B. Derrida], see S. Munier’s talk
v∗ →α2
sκ→0vBK − απ2γcχ
′′(γc)
2 log2(α2sκ)
+ . . .
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Event properties (2/2)
[G.S., 05]
k
T
1e+201e+151e+101000001
1
0.8
0.6
0.4
0.2
0
linear fit
κ = 0.1
κ = 1
κ = 5
Y
(dis
per
sion
)2
50403020100
14
12
10
8
6
4
2
0
Dispersion of the events
∆log[Q2s(Y )] ≈
√
DdiffαY with Ddiff ∼α2
sκ→0
1
| log3(α2sκ)|
.
No important dispersion in early stages of the evolution !
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Averaged amplitude
[G.S., 05]
BK
κ = 1
κ = 0.1
k
〈T〉
1025
1020
1015
1010
105
1
10
1
0.1
0.01
0.001
10−4
10−5
BK
sFKPP
numeric
k
〈T〉
1e+251e+201e+151e+101000001
100
1
0.01
1e-04
1e-06
1e-08
1e-10
1e-12
1e-14
Clear effect of fluctuations
Violations of geometric scaling (not in early stages )
Agrees with predictions
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Correlators
[G.S., 05]
〈T 〉2〈T 〉〈T 2〉
k
am
plitu
de
10251020101510101051
1000
100
10
1
0.1
0.01
0.001
10−4
10−5
Dense regime:⟨T 2
⟩≈ 〈T 〉2
Dilute regime:⟨T 2
⟩≈ 〈T 〉 (pre-asymptotics!)
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Short parenthesis
log(Q2)
YQs(Y )
[Mueller, Triantafyllopoulos, 02]
vetoed
BFKL
log(Q2)
YQs(Y )
[Mueller, Shoshi, 04]
vetoed
vetoed
BFKL
Saturation Saturation
+ fluctuations
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High-energy behviour
[Y. Hatta, E. Iancu, C. Marquet, G.S., D. Triantafyllopoulos, 06]
Evolution with saturation & fluctuations ≡
superposition of unitary front(with geometric scaling)
with a dispersion(yielding geometric scaling violations)
〈T (r, Y )〉 =
∫
dρs Tevent(ρ− ρs)1√πσ
exp
(
− (ρs − ρs)2
σ2
)
with ρ = log(1/r2), ρs = log(Q2s)
Tevent(ρ− ρs) =
1 r > Qs
(r2Q2s)
γ r < Qs
k
T
1e+201e+151e+101000001
1
0.8
0.6
0.4
0.2
0
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High-energy behviour
log(r20/r
2)
T
2520151050-5
1
0.8
0.6
0.4
0.2
0
dispersion ∼ DY
Y not too large ⇒ small dispersion ⇒ 〈T 〉 ≈ Tevent ⇒ geometric scaling
Y very high ⇒ dominated by disperion i.e. 〈T 〉 ≈ Tsat
log(r20/r
2)
T
2520151050-5
1
0.8
0.6
0.4
0.2
0
NB.:⟨T 2
⟩= 〈T 〉
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Strong noise limit
Intermediate energies High energies
Mean field (BK) Fluctuations
Geometric scaling Diffusive scaling
〈T 〉 = f[log(k2/Q2
s)]
〈T 〉 = f[
log(k2/Q2s)/
√DY
]
⟨T (k)
⟩= 〈T 〉k
⟨T (k)
⟩= 〈T 〉
At high-energy, amplitudes are dominated by black spots i.e.rare fluctuations at saturation
true for strong fluctuations
asymptotically true in general
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Phase space & scaling
saturation:
log(Q2)
Y
saturation
geom.scaling
DLA
√Y
Qs2(Y)
log(Q2)
Y
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Phase space & scaling
saturation: saturation+fluctuations:
log(Q2)
Y
saturation
geom.scaling
DLA
√Y
Qs2(Y)
log(Q2)
Y
log(Q2)
Y
√Y
Y
sat.
diffusivescaling
DLA
<Qs2(Y)>
log(Q2)
Y
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Description of γ∗p data
Application: description of F p2 data.
Factorisation formula: (see C. Marquet’s talk for more details)
σγ∗pL,T
d2b=
∫
d2r
∫ 1
0
dz∣∣ΨL,T (z, r;Q2)
∣∣2T (r,b;Y )
z
1−z
r
b
Ψ ≡ photon wavefunction γ∗ → qq: QED process
T ≡ scattering amplitude from high-energy QCD.
∫
d2b T (r, b, Y ) = 2πR2p T (r;Y ) F2 =
Q2
4παe
[
σγ∗pL + σγ∗p
T
]
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Describing F2
Saturation fit: [Iancu, Itakura, Munier]
〈T (r, Y )〉 =
8
>
>
<
>
>
:
(r2Q2s)
γce−2 log2(rQs)
CY r < Qs
1 − e−a−b log2(rQs) r > Qs
Q2s(Y ) = λY , ρs = log(Q2
s)
Saturation+fluctuations fit: [in preparation]
〈T (r, Y )〉 =∫dρs T (r, ρs)
1√πσe−
(ρs−ρs)2
σ2
T (r, ρs) =
8
>
>
<
>
>
:
r2Q2s r < Qs colour transparency
1 r > Qs
log(1/r2)
T(r
,Y)
420-2-4-6-8
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
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Describing F2
Saturation fit: [Iancu, Itakura, Munier]
〈T (r, Y )〉 = (r2Q2s)
γce−2 log2(rQs)
CY → r2Q2s
Saturation+fluctuations fit: [in preparation]
〈T (r, Y )〉 =∫dρs T (r, ρs)
1√πσe−
(ρs−ρs)2
σ2 → 12erfc
(− log(r2Q2
s)√2DY
)
limit
log(1/r2Q2s)
T(r
,Y)
6420-2-4
1
0.8
0.6
0.4
0.2
0
limit
log(1/r2Q2s)/
√Y
T(r
,Y)
3210-1-2
1
0.8
0.6
0.4
0.2
0
Y →∞−→ Geometric scaling Y →∞−→ Diffusive scaling
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Describing F2
10-1 1 101 102 103
0
1
2
0.01
10-1 1 101 102 103
0.0061
10-1 1 101 102 103
0.0037
10-1 1 101 102 103
0.0022
10-1 1 101 102 103
0.0
1.0
2.0
0.00135
0.0
1.0
2.0
8.2 10-4 5 10-4 3 10-4 1.8 10-4
0.0
1.0
2.0
1.1 10-4
0.0
1.0
6.7 10-5 4 10-5 2.5 10-5 1.5 10-5
0.0
1.0
9 10-6
10-1 1 101 102 1030.0
0.5
1.0
5.5 10-6
10-1 1 101 102 103
3.3 10-6
10-1 1 101 102 103
2 10-6
10-1 1 101 102 103
1.2 10-6
10-1 1 101 102 1030.0
0.5
1.0
7.5 10-7
Both fitscan describe
the datafor x ≤ 0.01
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Conclusion
Effects of saturation
Evolution equations for high-energy QCDBalitsky, Balitsky-Kovchegov (dipole), JIMWLK (CGC)
Good knowledge of the asymptotic solutionsTravelling waves → geometric scaling, saturation scale ∝ exp(αvcY )
Effects of fluctuations
Known at large-Nc
analytical solutions: αs ≪ 1
numerical solutions: coherent with statistical-physics analog
Consequences on saturation (e.g. geometric scaling violations)black spots ⇒ Diffusive scaling
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Perspectives
phenomenological tests:
do we observe geometric scaling at nonzero momentum transfer ?
predictions for LHC ? diffusive scaling at high-energy ?
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Perspectives
phenomenological tests:
do we observe geometric scaling at nonzero momentum transfer ?
predictions for LHC ? diffusive scaling at high-energy ?
theoretical questions:
importance of geometric scaling violations
analytical predictionsspeed, dispersion coefficientspomeron loops, triple pomeron vertex
numerical simumations:go beyond local-noise approximationinclude impact parameter
beyond large-Nc
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