non-linear qcd dynamics and wilson lines...a saturation limit? froissart bound: σ tot ∝ ln2 s g....
TRANSCRIPT
![Page 1: Non-linear QCD Dynamics and Wilson Lines...a saturation limit? Froissart bound: σ tot ∝ ln2 s G. A. Chirilli (The Ohio State Uni.) Non-linear QCD dynamics QCD evolutionMay 28, 2015](https://reader036.vdocuments.net/reader036/viewer/2022071404/60f81344880d8f1f1d41d4dc/html5/thumbnails/1.jpg)
Non-linear QCD Dynamics and Wilson Lines
Giovanni Antonio Chirilli
The Ohio State University
QCD evolution 2015
JLAB - Newport News, VA May 26 - 30, 2015
G. A. Chirilli (The Ohio State Uni.) Non-linear QCD dynamics QCD evolution May 28, 2015 1 / 43
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Outline
High-energy/high-density QCD scattering processes.
BFKL equation and violation of unitarity.
High-energy Operator Product Expansion at NLO: high-energy
QCD factorization.
LO and NLO non-linear BK/JIMWLK evolution equation.
G. A. Chirilli (The Ohio State Uni.) Non-linear QCD dynamics QCD evolution May 28, 2015 2 / 43
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Problem: Structure of hadrons at high-energy
Understand the structure of hadrons at high energy.
Can parton density rise forever? Is there a saturation limit? The
cross section for scattering on a disc of radius R is bounded
σtot ≤ 2πR2.
What are the proper degrees of freedom at high-energy?
Can we calculate particles production cross-section in
high-energy nucleus-nucleus collisions?
G. A. Chirilli (The Ohio State Uni.) Non-linear QCD dynamics QCD evolution May 28, 2015 3 / 43
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Where do we probe high-energy/high-density QCD dynamics?
Nucleus-nucleus and proton-nucleus collisions
RHIC: Au-Au and p-Au collisions.
LHC: Pb-Pb and p-Pb collisions.
Future experiments:
Electron Ion Collider (EIC) at Brookhaven or JLAB.
Large Hadron electron Collider (LHeC) at CERN.
G. A. Chirilli (The Ohio State Uni.) Non-linear QCD dynamics QCD evolution May 28, 2015 4 / 43
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Incoherent-vs-Coherent
Do DGLAP equations describe high parton-density dynamics?
DGLAP is evolution equation towards dilute regime.
Incoherent Interactions
Bjorken Limit
q
P
*
e
−q2 = Q2 → ∞, (P + q)2 = s → ∞
xB =Q2
s + Q2fixed
resum αs lnQ2
Λ2QCD
G. A. Chirilli (The Ohio State Uni.) Non-linear QCD dynamics QCD evolution May 28, 2015 5 / 43
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Incoherent-vs-Coherent
Do DGLAP equations describe high parton-density dynamics?
DGLAP is evolution equation towards dilute regime.
Incoherent Interactions
Bjorken Limit
q
P
*
e
−q2 = Q2 → ∞, (P + q)2 = s → ∞
xB =Q2
s + Q2fixed
resum αs lnQ2
Λ2QCD
VS.
Regge Limit
Coherent Interactions
q
Q2 fixed, s → ∞
xB =Q2
s→ 0
resum αs ln1
xB
G. A. Chirilli (The Ohio State Uni.) Non-linear QCD dynamics QCD evolution May 28, 2015 5 / 43
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High Center of Mass Energy Limit
s Mandelstam variable for the C.M.E
ΛQCD: Scale at which perturbative expansion breaks down.
t Mandelstam variable for momentum transfer in scatt. processes.
Hig-energy (Regge limit) QCD
s ≫ ΛQCD, t αs ≪ 1
⇒ αs ln s ∼ 1 ⇒ ∀ n ∈ N (αs ln s)n ∼ 1 : Leading Log Approximation
Type of diagrams at Leading Log Approximation αs ln s ∼ 1
s ln s sln s)2
sln s)n} n
Need an evolution equation to resum αs ln s contributions: BFKL equation.
G. A. Chirilli (The Ohio State Uni.) Non-linear QCD dynamics QCD evolution May 28, 2015 6 / 43
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Leading Log Approximation in scatt. process at high energy
electron-proton/nucleus Deep Inelastic Scattering (DIS)
s = (q + P)2
〈P|T jµ(x)jν(y)|P〉
q
*
P
energy suppressed
} n
P
q
*
n
( )s lnqs
2
G. A. Chirilli (The Ohio State Uni.) Non-linear QCD dynamics QCD evolution May 28, 2015 7 / 43
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DIS cross section at Leading Log Approximation
BFKL: Leading Logarithmic Approximation αs << 1 (αs ln s)n ∼ 1
s
PRE−ASYMPTOTIC BEHAVIOR
BFKL PREDICTS ONLY
sectioncrossDIS
NON LINEAR EFFECTS
BFKL
ln s Froissart bound2
pQCD at LLA: σtot ∝ s∆
G. A. Chirilli (The Ohio State Uni.) Non-linear QCD dynamics QCD evolution May 28, 2015 8 / 43
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DIS cross section at Leading Log Approximation
BFKL: Leading Logarithmic Approximation αs << 1 (αs ln s)n ∼ 1
s
PRE−ASYMPTOTIC BEHAVIOR
BFKL PREDICTS ONLY
sectioncrossDIS
NON LINEAR EFFECTS
BFKL
ln s Froissart bound2
pQCD at LLA: σtot ∝ s∆
Can parton density rise forever? Is there
a saturation limit?
Froissart bound: σtot ∝ ln2 s
G. A. Chirilli (The Ohio State Uni.) Non-linear QCD dynamics QCD evolution May 28, 2015 8 / 43
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DIS cross section at Leading Log Approximation
BFKL: Leading Logarithmic Approximation αs << 1 (αs ln s)n ∼ 1
s
PRE−ASYMPTOTIC BEHAVIOR
BFKL PREDICTS ONLY
sectioncrossDIS
NON LINEAR EFFECTS
BFKL
ln s Froissart bound2
pQCD at LLA: σtot ∝ s∆
Can parton density rise forever? Is there
a saturation limit?
Froissart bound: σtot ∝ ln2 s
At very high energy recombination begins to compensate gluonproduction. Gluon density reaches a limit and does not grow anymore.
So does the total DIS cross sections. Unitarity is restored!
G. A. Chirilli (The Ohio State Uni.) Non-linear QCD dynamics QCD evolution May 28, 2015 8 / 43
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DIS cross section at Leading Log Approximation
BFKL: Leading Logarithmic Approximation αs << 1 (αs ln s)n ∼ 1
s
PRE−ASYMPTOTIC BEHAVIOR
BFKL PREDICTS ONLY
sectioncrossDIS
NON LINEAR EFFECTS
BFKL
ln s Froissart bound2
pQCD at LLA: σtot ∝ s∆
Can parton density rise forever? Is there
a saturation limit?
Froissart bound: σtot ∝ ln2 s
At very high energy recombination begins to compensate gluonproduction. Gluon density reaches a limit and does not grow anymore.
So does the total DIS cross sections. Unitarity is restored!
In order to take in to account rgluons recombination the evolution
equation for the structure function has to be non-linear.
G. A. Chirilli (The Ohio State Uni.) Non-linear QCD dynamics QCD evolution May 28, 2015 8 / 43
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DIS at high-energy: parton saturation Color Glass Condensate
*
xx
G. A. Chirilli (The Ohio State Uni.) Non-linear QCD dynamics QCD evolution May 28, 2015 9 / 43
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DIS at high-energy: parton saturation Color Glass Condensate
*
xx
coll.
G. A. Chirilli (The Ohio State Uni.) Non-linear QCD dynamics QCD evolution May 28, 2015 9 / 43
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DIS at high-energy: parton saturation Color Glass Condensate
*
xx
coll. coll.
G. A. Chirilli (The Ohio State Uni.) Non-linear QCD dynamics QCD evolution May 28, 2015 9 / 43
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DIS at high-energy: parton saturation Color Glass Condensate
*
xx
coll. coll.
coll.
G. A. Chirilli (The Ohio State Uni.) Non-linear QCD dynamics QCD evolution May 28, 2015 9 / 43
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DIS at high-energy: parton saturation Color Glass Condensate
*
xx
coll. coll.
coll.
G. A. Chirilli (The Ohio State Uni.) Non-linear QCD dynamics QCD evolution May 28, 2015 9 / 43
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DIS at high-energy: parton saturation Color Glass Condensate
*
xx
coll. coll.
coll.
higher energy
G. A. Chirilli (The Ohio State Uni.) Non-linear QCD dynamics QCD evolution May 28, 2015 9 / 43
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DIS at high-energy: parton saturation Color Glass Condensate
*
xx
coll. coll.
coll.
higher energyhigher energy
G. A. Chirilli (The Ohio State Uni.) Non-linear QCD dynamics QCD evolution May 28, 2015 9 / 43
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DIS at high-energy: parton saturation Color Glass Condensate
*
xx
coll. coll.
coll.
higher energyhigher energy higher energy
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DIS at high-energy: parton saturation Color Glass Condensate
*
xx
coll. coll.
coll.
higher energyhigher energy higher energy
Need a mechanism to stop the growth: parton-recombination
⇒ Non-linear evolution equation
G. A. Chirilli (The Ohio State Uni.) Non-linear QCD dynamics QCD evolution May 28, 2015 9 / 43
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Saturation Scale: euristic explanation
σonium−oniumtot ∝ α2
s x1⊥ x2⊥ e∆ Y
∆ =4αsNc
πln 2 > 0
Black − disk limit : σtot ≤ 2π R2
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Saturation Scale: euristic explanation
σonium−oniumtot ∝ α2
s x1⊥ x2⊥ e∆ Y
∆ =4αsNc
πln 2 > 0
Black − disk limit : σtot ≤ 2π R2
DIS case:
x1⊥ ∼ 1Q
size of dipole generated by γ∗
x2⊥ ∼ R ∼ 1ΛQCD
radius of hadron
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Saturation Scale: euristic explanation
σonium−oniumtot ∝ α2
s x1⊥ x2⊥ e∆ Y
∆ =4αsNc
πln 2 > 0
Black − disk limit : σtot ≤ 2π R2
DIS case:
x1⊥ ∼ 1Q
size of dipole generated by γ∗
x2⊥ ∼ R ∼ 1ΛQCD
radius of hadron
⇒ α2s
RQ
e∆Y ≤ 2π R2 With Y = log 1xB
G. A. Chirilli (The Ohio State Uni.) Non-linear QCD dynamics QCD evolution May 28, 2015 10 / 43
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Saturation Scale: euristic explanation
σonium−oniumtot ∝ α2
s x1⊥ x2⊥ e∆ Y
∆ =4αsNc
πln 2 > 0
Black − disk limit : σtot ≤ 2π R2
DIS case:
x1⊥ ∼ 1Q
size of dipole generated by γ∗
x2⊥ ∼ R ∼ 1ΛQCD
radius of hadron
⇒ α2s
RQ
e∆Y ≤ 2π R2 With Y = log 1xB
Qs ∼ α2s ΛQCD
(
1xB
)∆violation of unitarity for Q < Qs
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Saturation Scale: euristic explanation
σonium−oniumtot ≃ α2
s x1⊥ x2⊥ e∆ Y
∆ =4αsNc
πln 2 > 0
Black − disk limit : σtot ≤ 2π R2
DIS case on large nucleus
Qs ∼ A1/3(
1xb
)∆
for high enough energy (small-xB) or large nucleus
Qs ≫ ΛQCD ⇒ pQCD allowed.
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High-energy scattering in QCD
A(r,t)
phase factor for the high-energy scattering: Wilson-line operator
U(x⊥, v) = Pe−ig
c~
∫+∞
−∞dt xµAµ(x(t))
Pe∫+∞
−∞dtA(t) = 1 +
∫ +∞−∞ dt A(t) +
∫ +∞−∞ dt A(t)
∫ t
−∞ dt′ A(t′)
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Propagation in the shock wave: Wilson line (Spectator frame)
Boosted Field
Each path is weighted with the gauge factor Peig∫
dxµAµ
. Since the external
field exists only within the infinitely thin wall, quarks and gluons do not havetime to deviate in the transverse direction ⇒ we can replace the gauge factor
along the actual path with the one along the straight-line path.
x
z z’
yWilson Line x y
Uz = [∞p1 + z⊥,−∞p1 + z⊥]
[x, y] = Peig∫
1
0du(x−y)µAµ(ux+(1−u)y) pµ = αp
µ
1 + βpµ
2 + pµ
⊥
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Propagation in the shock wave: Wilson line (Spectator frame)
Boosted Field
Each path is weighted with the gauge factor Peig∫
dxµAµ
. Since the external
field exists only within the infinitely thin wall, quarks and gluons do not havetime to deviate in the transverse direction ⇒ we can replace the gauge factor
along the actual path with the one along the straight-line path.
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Propagation in the shock wave: Wilson line (Spectator frame)
Boosted Field
Each path is weighted with the gauge factor Peig∫
dxµAµ
. Since the external
field exists only within the infinitely thin wall, quarks and gluons do not havetime to deviate in the transverse direction ⇒ we can replace the gauge factor
along the actual path with the one along the straight-line path.
η>Y
η<Y
+ +...
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High-energy Operator Product Expansion
η>Y
η<Y
+ +...
〈B|T{jµ(x)jν(y)}|B〉 ≃∫
d2z1d2z2 ILOµν (z1, z2; x, y)〈B|tr{Uη
z1U† η
z2}|B〉
+αs
π
∫
d2z1d2z2d2z3 INLOµν (z1, z2, z3; x, y)〈B|tr{Uη
z1U†η
z3}tr{Uη
z3U†η
z2}|B〉
η = ln 1xB
|B〉 Target state
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Leading Order
[
〈T {jµ(x)jν(y)}〉A
]LO=
∫
d2z1d2z2
z412
ILOµν (x, y; z1, z2)tr{Uη
z1U†η
z2}
z1
z2
y x
〈B|T {jµ(x)jν(y)}|B〉 =∫
d2z1d2z2
z412
ILOµν (x, y; z1, z2)〈B|tr{Uη
z1U†η
z2}|B〉+ . . .
If we use a model to evaluate 〈B|tr{Uηz1
U†ηz2}|B〉 we can calculate the DIS
cross-section.
If we want to include energy dependence to the DIS cross section, we
need to find the evolution of 〈B|tr{Uηz1
U†ηz2}|B〉 with respect to the rapidity
parameter η.
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LO Impact Factor
Conformal invariance: (x+, x⊥)2 = −x2
⊥ ⇒ after the inversion x⊥ → x⊥/x2⊥ and
x+ → x+/x2⊥ x+ = x0+x3
√2
z1
y x
z2
Conformal vectors:
κ =1√sx+
(p1
s− x2p2 + x⊥)−
1√sy+
(p1
s− y2p2 + y⊥)
ζ1 =(p1
s+ z2
1⊥p2 + z1⊥)
, ζ2 =(p1
s+ z2
2⊥p2 + z2⊥)
Here x2 = −x2⊥; R = κ2(ζ1·ζ2)
2(κ·ζ1)(κ·ζ2)
ILOµν (z1, z2) =
R2
π6(κ · ζ1)(κ · ζ2)
∂2
∂xµ∂yν
[
(κ · ζ1)(κ · ζ2)−1
2κ2(ζ1 · ζ2)
]
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Regularization of the rapidity divergence
〈T {jµ(x)jν(y)}〉A ≃∫
d2z1d2z2 ILO(z1, z2)〈tr{Uη1z1
U† η1z2
}〉A
Matrix elements of Wilson lines: 〈tr{U(x)U†(y)}〉A are divergent
η1 η
1For light-like Wilson lines loop integrals
are divergent in the longitudinal
direction∫ ∞
0
dα
α=
∫ ∞
−∞dη = ∞
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Regularization of the rapidity divergence
Matrix elements of Wilson lines: 〈tr{U(x)U†(y)}〉A are divergent
η2 η
2For light-like Wilson lines loop integrals
are divergent in the longitudinal
direction∫ ∞
0
dα
α=
∫ ∞
−∞dη = ∞
Regularization by: slope
Uη(x⊥) = Pexp{
ig
∫ ∞
−∞du nµ Aµ(un + x⊥)
}
nµ = pµ1 + e−2ηp
µ2
At NLO the regularization by rigid cut-off is more convenient.
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Evolution Equation
η2
η1 αs(η1 − η2)Kevol ⊗
η2 η
2
Separate fields in quantum and classical according to low and large rapidity.
Formally we may write:
〈B|Oη1 |B〉 → 〈Oη1〉A → 〈O′η2 ⊗O′η1〉A
Integrate over the quantum fields and get one-loop rapidity evolution of the
operator O
〈Oη1〉A = αs(η1 − η2)Kevol ⊗ 〈O′η2〉A
Where in principle O and O′ are different operators.
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Non-linear evolution equation
Linear case Oη1 = αs∆η Kevol ⊗ Oη2
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Non-linear evolution equation
Linear case Oη1 = αs∆η Kevol ⊗ Oη2
Non-linear case Oη1 = αs∆η Kevol ⊗ {Oη2Oη2}
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Non-linear evolution equation
Linear case Oη1 = αs∆η Kevol ⊗ Oη2
Non-linear case Oη1 = αs∆η Kevol ⊗ {Oη2Oη2}
〈{Uη1x }ij〉A =
αs
2π2∆η
∫
d2z⊥(x − z)2
⊥
[
〈tr{Uη2x Uη2†
z }{Uη2z }ij〉A − 〈 1
Nc
{Uη2x }ij〉A
]
∆ = η1 − η2
{U†η1x }ij, {Uη1
x Uη1y }ij, {Uη1
x U†η1y }ij, {U†η1
x U†η1y }ij
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Non-linear evolution equation
Linear case Oη1 = αs∆η Kevol ⊗ Oη2
Non-linear case Oη1 = αs∆η Kevol ⊗ {Oη2Oη2}
〈{Uη1x }ij〉A =
αs
2π2∆η
∫
d2z⊥(x − z)2
⊥
[
〈tr{Uη2x Uη2†
z }{Uη2z }ij〉A − 〈 1
Nc
{Uη2x }ij〉A
]
∆ = η1 − η2
{U†η1x }ij, {Uη1
x Uη1y }ij, {Uη1
x U†η1y }ij, {U†η1
x U†η1y }ij
Obtain a set of rules that allow one to get the LO evolution of any trace
or product of traces of Wilson linesG. A. Chirilli (The Ohio State Uni.) Non-linear QCD dynamics QCD evolution May 28, 2015 21 / 43
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Leading order: BK equation
d
dηtr{UxU†
y} = KLOtr{UxU†y}+ ... ⇒
d
dη〈tr{UxU†
y}〉shockwave = 〈KLOtr{UxU†y}〉shockwave
x
a
b
b
a a
a
b
b
y(a) (b) (c) (d)
x xx* xx* x*x x*
x• =√
s2x− x∗ =
√
s2x+
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Non-linear evolution equation: BK equation
Uabz = 2tr{taUzt
bU†z} ⇒ (UxU†
y)η1 → (UxU†
y)η2 + αs(η1 − η2)(UxU†
z UzU†y)
η2
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Non-linear evolution equation: BK equation
Uabz = 2tr{taUzt
bU†z} ⇒ (UxU†
y)η1 → (UxU†
y)η2 + αs(η1 − η2)(UxU†
z UzU†y)
η2
U(x, y) ≡ 1 − 1
Nc
tr{U(x⊥)U†(y⊥)}
BK equation: Ian Balitsky (1996), Yu. Kovchegov (1999)
d
dηU(x, y) =
αsNc
2π2
∫
d2z (x − y)2
(x − z)2(y − z)2
{
U(x, z) + U(z, y) − U(x, y) − U(x, z)U (z, y)}
Alternative approach: JIMWLK (1997-2000)
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Non-linear evolution equation: BK equation
Uabz = 2tr{taUzt
bU†z} ⇒ (UxU†
y)η1 → (UxU†
y)η2 + αs(η1 − η2)(UxU†
z UzU†y)
η2
U(x, y) ≡ 1 − 1
Nc
tr{U(x⊥)U†(y⊥)}
BK equation: Ian Balitsky (1996), Yu. Kovchegov (1999)
d
dηU(x, y) =
αsNc
2π2
∫
d2z (x − y)2
(x − z)2(y − z)2
{
U(x, z) + U(z, y) − U(x, y) − U(x, z)U (z, y)}
Alternative approach: JIMWLK (1997-2000)
LLA for DIS in pQCD ⇒ BFKL (LLA: αs ≪ 1, αsη ∼ 1)
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Non linear evolution equation: BK equation
Uabz = 2tr{taUzt
bU†z} ⇒ (UxU†
y)η1 → (UxU†
y)η2 + αs(η1 − η2)(UxU†
z UzU†y)
η2
U(x, y) ≡ 1 − 1
Nc
tr{U(x⊥)U†(y⊥)}
BK equation: Ian Balitsky (1996), Yu. Kovchegov (1999)
d
dηU(x, y) =
αsNc
2π2
∫
d2z (x − y)2
(x − z)2(y − z)2
{
U(x, z) + U(z, y) − U(x, y) − U(x, z)U (z, y)}
Alternative approach: JIMWLK (1997-2000)
LLA for DIS in pQCD ⇒ BFKL (LLA: αs ≪ 1, αsη ∼ 1)
LLA for DIS in sQCD ⇒ BK eqn (LLA: αs ≪ 1, αsη ∼ 1, α2s A1/3 ∼ 1)
(s for semi-classical)
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Goal: Single-gluon cross-section in A-A collision
A1 and A2 are the atomic numbers of the two nuclei.
A2
A1
High−energy scatt.
Eventually one would like to obtain the classical gluon produced in
heavy-ion collisions: initial condition for Quark-Gluon-Plasma.
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Simplified problem: 1 ≪ A1 ≪ A2
Consider 1 ≪ A1 ≪ A2 ⇒ Qs1 ≪ Qs2
Nucleus A1 is considered as a dilute system: not densely packed as
nucleus A2.
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Simplified problem: 1 ≪ A1 ≪ A2
Consider 1 ≪ A1 ≪ A2 ⇒ Qs1 ≪ Qs2
Nucleus A1 is considered as a dilute system: not densely packed as
nucleus A2.
3-gluon vertex diagrams G.A.C., Yu. Kovchegov, D. Wertepny (2014)
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Simplified problem: 1 ≪ A1 ≪ A2
Box-type diagrams G.A.C., Yu. Kovchegov, D. Wertepny (2014)
More details and the result of the calculation in D. Wertepny’s talk on Friday
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Motivation: Why NLO correction of evolution equations?
How to take higher-order corrections into account (either for BFKL
or non-linear evolution equation).
Higher-order corrections are needed to improve phenomenology:
Determine the argument of the coupling constant.
Gives precision of LO.
Get the region of application of the leading order evolution
equation.
Check conformal invariance (in N=4 SYM)
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High-energy Operator Product Expansion
η>Y
η<Y
+ +...
〈B|jµ(x)jν(y)|B〉 ≃∫
d2z1d2z2 ILOµν (z1, z2; x, y)〈B|tr{Uη
z1U† η
z2}|B〉
+αs
π
∫
d2z1d2z2d2z3 INLOµν (z1, z2, z3; x, y)〈B|tr{Uη
z1U†η
z3}tr{Uη
z3U†η
z2}|B〉
η = ln 1xB
|B〉 Target state
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LO and NLO Impact Factor
T{jµ(x)jν(y)} =
∫
d2z1d2z2 ILOµν
(z1, z2, x, y)tr{Uη
z1U†η
z2}
+
∫
d2z1d2z2d2z3 INLOµν
(z1, z2, z3, x, y)[tr{Uη
z1U†η
z3}tr{Uη
z3U†η
z2} − Nctr{Uη
z1U†η
z2}]
LO Impact Factor diagram: ILO
z1
y x
z2
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LO and NLO Impact Factor
T{jµ(x)jν(y)} =
∫
d2z1d2z2 ILOµν
(z1, z2, x, y)tr{Uη
z1U†η
z2}
+
∫
d2z1d2z2d2z3 INLOµν
(z1, z2, z3, x, y)[tr{Uη
z1U†η
z3}tr{Uη
z3U†η
z2} − Nctr{Uη
z1U†η
z2}]
LO Impact Factor diagram: ILO
z1
y x
z2
NLO Impact Factor diagrams: INLO
z2
z’z1
y xz3
z2
zz’z1
y xz3
z
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NLO Photon Impact Factor
[
〈T {jµ(x)jν(y)}〉A
]LO=
∫
d2z1d2z2
z412
ILOµν (x, y; z1, z2)〈tr{Uη
z1U†η
z2}〉A
[
〈T {jµ(x)jν(y)}〉A
]NLO=
∫
d2z1d2z2
z412
d2z3
[
Iµν1 (z1, z2, z3) + I
µν2 (z1, z2, z3)
]
×[tr{Uz1U†
z3}tr{Uz3
U†z2} − Nctr{Uz1
U†z2}]
where Iµν2 (z1, z2, z3) is finite and conformal, while
Iµν1 (z1, z2, z3) =
αs
2π2ILOµν
z212
z213z2
23
∫ +∞
0
dα
αei sα
4Z3
is rapidity divergent.
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NLO Impact Factor
z2
z’z1
y xz3
z2
zz’z1
y xz3
zZ3 ≡ (x−z3)
2⊥
x+− (y−z3)
2⊥
y+
INLOµν (x, y; z1, z2, z3; η) = − ILO
µν × αs
2π
z213
z212z2
23
lnσs
4Z3 + conf.
The NLO impact factor is not conformal (Möbius) invariant ⇒ the color dipole
with the cutoff η = lnσ is not invariant.
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NLO Impact Factor
z2
z’z1
y xz3
z2
zz’z1
y xz3
zZ3 ≡ (x−z3)
2⊥
x+− (y−z3)
2⊥
y+
INLOµν (x, y; z1, z2, z3; η) = − ILO
µν × αs
2π
z213
z212z2
23
lnσs
4Z3 + conf.
The NLO impact factor is not conformal (Möbius) invariant ⇒ the color dipole
with the cutoff η = lnσ is not invariant.
However, if we define a composite operator (a - analog of µ−2 for usual OPE)
[tr{Uηz1
U†ηz2}]conf
= tr{Uηz1
U†ηz2}
+αs
4π
∫
d2z3
z212
z213z2
23
[1
Nc
tr{Uηz1
U†ηz3}tr{Uη
z3U†η
z2} − tr{Uη
z1U†η
z2}] ln
az212
z213z2
23
+ O(α2s )
the impact factor becomes conformal in the NLO.
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NLO Photon Impact Factor for BFKL pomeron in momentum space
G.A.C. and I. Balitsky (2013)
Iµν(q, k⊥)
=Nc
32
∫
dν
πν
sinh πν
(1 + ν2) cosh2 πν
(k2⊥
Q2
)12−iν{[(9
4+ ν2
)(
1 +αs
π+
αsNc
2πF1(ν)
)
Pµν1
+(11
4+ 3ν2
)(
1 +αs
π+
αsNc
2πF2(ν)
)
Pµν2
]
+14+ ν2
2k2⊥
(
1 +αs
π+
αsNc
2πF3(ν)
)
[
Pµν k2 + Pµν k2]
}
Pµν1 = gµν − qµqν
q2
Pµν2 =
1
q2
(
qµ − pµ2 q2
q · p2
)(
qν − pν2 q2
q · p2
)
Pµν =(
gµ1 − igµ2 − pµ2
q
q · p2
)(
gν1 − igν2 − pν2q
q · p2
)
Pµν =(
gµ1 + igµ2 − pµ2
q
q · p2
)(
gν1 + igν2 − pν2q
q · p2
)
The NLO Impact Factor in conventional pQCD took more then 10 years of
calculation by several groups and the result is not in a form useful for
phenomenology.
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Non-linear evolution equation at NLO G.A.C. and I. Balitsky
d
dηTr{UxU†
y} =
∫
d2z
2π2
(
αs
(x − y)2
(x − z)2(z − y)2+ α2
s KNLO(x, y, z)
)
[Tr{UxU†z}Tr{UzU
†y} − NcTr{UxU†
y}] +
α2s
∫
d2zd2z′
(
K4(x, y, z, z′){Ux,U†z′,Uz,U†
y}+ K6(x, y, z, z′){Ux,U†z′,Uz′ ,Uz,U†
z ,U†y})
KNLO is the next-to-leading order correction to the dipole kernel and K4 and K6 are the
coefficients in front of the (tree) four- and six-Wilson line operators with arbitrary white
arrangements of color indices.
We need to calculate some diagrams analytically (pen and paper).
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Definition of the NLO kernel
In general
d
dηtr{UxU†
y} = αsKLOtr{UxU†y}+ α2
s KNLOtr{UxU†y}+ O(α3
s )
α2s KNLOtr{UxU†
y} =d
dηtr{UxU†
y} − αsKLOtr{UxU†y}+ O(α3
s )
We calculate the “matrix element” of the r.h.s. in the shock-wave background
〈α2s KNLOtr{UxU†
y}〉 =d
dη〈tr{UxU†
y}〉 − 〈αsKLOtr{UxU†y}〉+ O(α3
s )
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Diagrams of the NLO gluon contribution
Diagrams with 2 gluons interaction
(II) (III) (IV) (V)
(VI) (VII) (VIII) (IX) (X)
(I)
(XIV)(XI) (XIII)(XII) (XV)
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Diagrams of the NLO gluon contribution
Diagrams with 2 gluons interaction
(XXVI) (XXVII)
(XVI) (XVII) (XVIII) (XIX) (XX)
(XXI) (XXIV) (XV)(XXII) (XXIII)
(XVIII) (XXIX) (XXX)
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Diagrams of the NLO gluon contribution
Diagrams with 2 gluons interaction
(XXXI) (XXXIII) (XXXIV)(XXXII)
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Diagrams of the NLO gluon contribution
"Running coupling" diagrams
(I) (II) (III) (IV) (V)
y
x
(VI) (VII) (VIII) (IX) (X)
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Diagrams of the NLO gluon contribution
1 → 2 dipole transition diagrams
q
k’
k
k’
k
q q
k’ kk’
q
k k
k’
q
k’
kk’
k
k’
k’
qk
q
k’k
(c)(b)(a) (d) (e)
(f) (g) (h) (i) (j)
x x*
x*x
x*
x x* x
*x
*x
*
x x x x*x
*x
*
x x x x
k
q
a
b
c
a
b
cd
a
b
cd
a
bc
d
a
a aaaa
b b
b bc
c c c
cd
d
ddd
cb
b
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Diagrams of the NLO gluon contribution
N = 4 SYM diagrams (scalar and gluino loops)
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Gluon contribution to the NLO kernel (I. Balitsky and G.A.C, 2007)
d
dηTr{UxU†
y} =αs
2π2
∫
d2z(
[Tr{UxU†z}Tr{UzU
†y} − NcTr{UxU†
y}]
×{(x − y)2
X2Y2
[
1 +αsNc
4π(11
3ln(x − y)2µ2 +
67
9− π2
3)]
−11
3
αsNc
4π
X2 − Y2
X2Y2ln
X2
Y2− αsNc
2π
(x − y)2
X2Y2ln
X2
(x − y)2ln
Y2
(x − y)2
}
+αs
4π2
∫
d2z′{
[Tr{UxU†z}Tr{UzU
†z′}{Uz′U
†y} − Tr{UxU†
z Uz′U†yUzU
†z′}
−(z′ → z)]1
(z − z′)4
[
− 2 +X′2Y2 + Y ′2X2 − 4(x − y)2(z − z′)2
2(X′2Y2 − Y ′2X2)ln
X′2Y2
Y ′2X2
]
+ [Tr{UxU†z}Tr{UzU
†z′}{Uz′U
†y} − Tr{UxU
†z′
UzU†yUz′U
†z} − (z′ → z)]
×[ (x − y)4
X2Y ′2(X2Y ′2 − X′2Y2)+
(x − y)2
(z − z′)2X2Y ′2
]
lnX2Y ′2
X′2Y2
})
Our result Agrees with NLO BFKL
It respects unitarity
G. A. Chirilli (The Ohio State Uni.) Non-linear QCD dynamics QCD evolution May 28, 2015 35 / 43
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Gluon contribution to the NLO kernel (I. Balitsky and G.A.C, 2007)
d
dηTr{UxU†
y} =αs
2π2
∫
d2z(
[Tr{UxU†z}Tr{UzU
†y} − NcTr{UxU†
y}]
×{(x − y)2
X2Y2
[
1 +αsNc
4π(11
3ln(x − y)2µ2 +
67
9− π2
3)]
−11
3
αsNc
4π
X2 − Y2
X2Y2ln
X2
Y2− αsNc
2π
(x − y)2
X2Y2ln
X2
(x − y)2ln
Y2
(x − y)2
}
+αs
4π2
∫
d2z′{
[Tr{UxU†z}Tr{UzU
†z′}{Uz′U
†y} − Tr{UxU†
z Uz′U†yUzU
†z′}
−(z′ → z)]1
(z − z′)4
[
− 2 +X′2Y2 + Y ′2X2 − 4(x − y)2(z − z′)2
2(X′2Y2 − Y ′2X2)ln
X′2Y2
Y ′2X2
]
+ [Tr{UxU†z}Tr{UzU
†z′}{Uz′U
†y} − Tr{UxU
†z′UzU
†yUz′U
†z} − (z′ → z)]
×[ (x − y)4
X2Y ′2(X2Y ′2 − X′2Y2)+
(x − y)2
(z − z′)2X2Y ′2
]
lnX2Y ′2
X′2Y2
})
NLO kernel = Running coupling terms + Non-conformal term + Conformal term
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Evolution equation for color dipoles in N = 4
( I. Balitsky and G.A.C. 2009)
d
dηTr{Uη
z1U†η
z2}
=αs
π2
∫
d2z3
z212
z213z2
23
{
1 − αsNc
4π
[π2
3+2 ln
z213
z212
lnz2
23
z212
]}
× [Tr{TaUηz1
U†ηz3
TaUηz3
U†ηz2} − NcTr{Uη
z1U†η
z2}]
− α2s
4π4
∫
d2z3d2z4
z434
z212z2
34
z213z2
24
[
1 +z2
12z234
z213z2
24 − z223z2
14
]
lnz2
13z224
z214z2
23
× Tr{[Ta,Tb]Uηz1
Ta′Tb′U†ηz2
+ TbTaUηz1[Tb′ ,Ta′ ]U†η
z2}(Uη
z3)aa′(Uη
z4− Uη
z3)bb′
NLO kernel = Non-conformal term + Conformal term.
Non-conformal term is due to the non-invariant cutoff α < σ = e2η in the rapidity
of Wilson lines.
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Evolution equation for color dipoles in N = 4
( I. Balitsky and G.A.C. 2009)
d
dηTr{Uη
z1U†η
z2}
=αs
π2
∫
d2z3
z212
z213z2
23
{
1 − αsNc
4π
[π2
3+2 ln
z213
z212
lnz2
23
z212
]}
× [Tr{TaUηz1
U†ηz3
TaUηz3
U†ηz2} − NcTr{Uη
z1U†η
z2}]
− α2s
4π4
∫
d2z3d2z4
z434
z212z2
34
z213z2
24
[
1 +z2
12z234
z213z2
24 − z223z2
14
]
lnz2
13z224
z214z2
23
× Tr{[Ta,Tb]Uηz1
Ta′Tb′U†ηz2
+ TbTaUηz1[Tb′ ,Ta′ ]U†η
z2}(Uη
z3)aa′(Uη
z4− Uη
z3)bb′
NLO kernel = Non-conformal term + Conformal term.
Non-conformal term is due to the non-invariant cutoff α < σ = e2η in the rapidity
of Wilson lines.
For the conformal composite dipole the result is Möbius invariant
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Evolution equation for composite conformal dipoles in N = 4 SYM
I. Balitsky and G.A.C (2009)
[Tr{Uηz1
U†ηz2}]conf
= Tr{Uηz1
U†ηz2}
+αs
4π
∫
d2z3
z212
z213z2
23
[1
Nc
tr{Uηz1
U†ηz3}tr{Uη
z3U†η
z2} − Tr{Uη
z1U†η
z2}] ln
az212
z213z2
23
+ O(α2s )
d
dη
[
Tr{Uηz1
U†ηz2}]conf
=αs
π2
∫
d2z3
z212
z213z2
23
[
1 − αsNc
4π
π2
3
]
[
Tr{TaUηz1
U†ηz3
TaUz3U†η
z2} − NcTr{Uη
z1U†η
z2}]conf
− α2s
4π4
∫
d2z3d2z4
z212
z213z2
24z234
{
2 lnz2
12z234
z214z2
23
+[
1 +z2
12z234
z213z2
24 − z214z2
23
]
lnz2
13z224
z214z2
23
}
× Tr{[Ta,Tb]Uηz1
Ta′Tb′U†ηz2
+ TbTaUηz1[Tb′ ,Ta′ ]U†η
z2}[(Uη
z3)aa′(Uη
z4)bb′ − (z4 → z3)]
Now Möbius invariant!
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NLO evolution of composite “conformal” dipoles in QCD
d
dη[tr{Uz1
U†z2}]conf =
αs
2π2
∫
d2z3
(
[tr{Uz1U†
z3}tr{Uz3
U†z2} − Nctr{Uz1
U†z2}]conf
× z212
z213z2
23
[
1 +αsNc
4π
(
b ln z212µ
2 + bz2
13 − z223
z213z2
23
lnz2
13
z223
+67
9− π2
3
)
]
+αs
4π2
∫
d2z4
z434
{[
− 2 +z14
2z223 + z24
2z213 − 4z2
12z234
2(z142z2
23 − z242z2
13)ln
z142z2
23
z242z2
13
]
× [tr{Uz1U†
z3}tr{Uz3
U†z4}{Uz4
U†z2} − tr{Uz1
U†z3
Uz4U†
z2Uz3
U†z4} − (z4 → z3)]
+z2
12z234
z213z24
2
[
2 lnz2
12z234
z214z2
23
+(
1 +z2
12z234
z213z2
24 − z214z2
23
)
lnz2
13z242
z142z2
23
]
× [tr{Uz1U†
z3}tr{Uz3
U†z4}tr{Uz4
U†z2} − tr{Uz1
U†z4
Uz3U†
z2Uz4
U†z3} − (z4 → z3)]
})
b = 113
Nc − 23nf I. Balitsky and G.A.C
KNLO BK = Running coupling part + Conformal "non-analytic" (in j) part
+ Conformal analytic (N = 4) part
Linearized KNLO BK reproduces the known result for the forward NLO BFKL
kernel Fadin and Lipatov (1998).
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NLO Balitsky-JIMWLK evolution equation
Scattering amplitudes for proton-Nucleus and Nucleus-Nucleus collisions
are described by matrix elements made of multiple Wilson lines.
Typical matrix elements in pA and AA collisions is 〈tr{UxU†yUwU
†z}〉
quadrupole operator.
To include energy dependence ⇒ need NLO Balitsky-JIMWLK evolution
equation.
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NLO Balitsky-JIMWLK evolution equation
a) b )
c) d)
e) f)
Sample of diagrams: a), b) are self-interactions; c), d) are pairwise interactions;
e), f) are triple interactions.
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Self interaction at NLO I. Balitsky and G.A.C. (2013)
a) b )
d
dη(U1)ij =
α2s
8π4
∫
d2z4d2z5
z245
{
Udd′
4 (Uee′
5 − Uee′
4 )
×([
2I1 −4
z245
]
f adef bd′e′(taU1tb)ij +(z14, z15)
z214z2
15
lnz2
14
z215
[
if ad′e′({td, te}U1ta)ij − if ade(taU1{td′ , te′})ij
]
)
+α2
s Nc
4π3
∫
d2z4 (Uab4 − Uab
1 )(taU1tb)ij ×1
z214
[11
3ln z2
14µ2 +
67
9− π2
3
]
I1 ≡ I(z1, z4, z5) =ln z2
14/z215
z214 − z2
15
[z214 + z2
15
z245
− (z14, z15)
z214
− (z14, z15)
z215
− 2]
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Pairwise Interaction at NLO I. Balitsky and G.A.C. (2013)
c) d)
d
dη(U1)ij(U
†2)kl =
α2s
8π4
∫
d2z4d2z5(A1 +A2 +A3) +α2
s Nc
8π3
∫
d2z4(B1 + NcB2)
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Pairwise Interaction at NLO I. Balitsky and G.A.C. (2013)
A1 =[
(taU1)ij(U2tb)kl + (U1tb)ij(taU2)kl
]
×[
f adef bd′e′Udd′
4 (Uee′
5 − Uee′
4 )(
− K − 4
z445
+I1
z245
+I2
z245
)]
K is the NLO BK kernel for N=4 SYM
A2 = 4(U4 − U1)dd′(U5 − U2)
ee′
{
i[
f ad′e′(tdU1ta)ij(teU2)kl − f ade(taU1td′)ij(U2te′)kl
]
J1245 lnz2
14
z215
+ i[
f ad′e′(tdU1)ij(teU2ta)kl − f ade(U1td′)ij(t
aU2te′)kl
]
J2154 lnz2
24
z225
}
J1245 ≡ J(z1, z2, z4, z5) =(z14, z25)
z214z2
25z245
− 2(z15, z45)(z15, z25)
z214z2
15z225z2
45
+ 2(z25, z45)
z214z2
25z245
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Pairwise Interaction at NLO I. Balitsky and G.A.C. (2013)
A3 = 2Udd′
4
{
i[
f ad′e′(U1ta)ij(tdteU2)kl − f ade(taU1)ij(U2te′ td′)kl
]
×[
J1245 lnz2
14
z215
+ (J2145 − J2154) lnz2
24
z225
]
(U5 − U2)ee′
+ i[
f ad′e′(tdteU1)ij(U2ta)kl − f ade(U1te′ td′)ij(taU2)kl
]
×[
J2145 lnz2
24
z225
+ (J1245 − J1254) lnz2
14
z215
]
(U5 − U1)ee′}
J1245 ≡ J (z1, z2, z4, z5)
=(z24, z25)
z224z2
25z245
− 2(z24, z45)(z15, z25)
z224z2
25z215z2
45
+2(z25, z45)(z14, z24)
z214z2
24z225z2
45
− 2(z14, z24)(z15, z25)
z214z2
15z224z2
25
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Pairwise Interaction at NLO I. Balitsky and G.A.C. (2013)
B1 = 2 lnz2
14
z212
lnz2
24
z212
×{
(U4 − U1)abi[
f bde(taU1td)ij(U2te)kl + f ade(teU1tb)ij(tdU2)kl
]
[(z14, z24)
z214z2
24
− 1
z214
]
+ (U4 − U2)abi[
f bde(U1te)ij(taU2td)kl + f ade(tdU1)ij(t
eU2tb)kl
]
[(z14, z24)
z214z2
24
− 1
z224
]}
B2 =[
2Uab4 − Uab
1 − Uab2
]
[(taU1)ij(U2tb)kl + (U1tb)ij(taU2)kl]
×{(z14, z24)
z214z2
24
[11
3ln z2
12µ2 +
67
9− π2
3
]
+11
3
( 1
2z214
lnz2
24
z212
+1
2z224
lnz2
14
z212
)
}
G. A. Chirilli (The Ohio State Uni.) Non-linear QCD dynamics QCD evolution May 28, 2015 41 / 43
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Triple interactions
I. Balitsky and G.A.C. (2013) (see also A. Grabovsky (2013))
e) f)
J12345 ≡ J (z1, z2, z3, z4, z5) = −2(z14, z34)(z25, z35)
z214z2
25z234z2
35
− 2(z14, z45)(z25, z35)
z214z2
25z235z2
45
+2(z25, z45)(z14, z34)
z214z2
25z234z2
45
+(z14, z25)
z214z2
25z245
G. A. Chirilli (The Ohio State Uni.) Non-linear QCD dynamics QCD evolution May 28, 2015 42 / 43
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Triple interactions
I. Balitsky and G.A.C. (2013)
d
dη(U1)ij(U2)kl(U3)mn
= iα2
s
2π4
∫
d2z4d2z5
{
J12345 lnz2
34
z235
× f cde[
(taU1)ij(tbU2)kl(U3tc)mn(U4 − U1)
ad(U5 − U2)be
− (U1ta)ij(U2tb)kl(tcU3)mn(U4 − U1)
da(U5 − U2)eb]
+ J32145 lnz2
14
z215
× f ade[
(U1ta)ij(tbU2)kl(t
cU3)mn(U4 − U3)cd(U5 − U2)
be
− (taU1)ij ⊗ (U2tb)kl(U3tc)mn(Udc4 − Udc
3 )(Ueb5 − Ueb
2 )]
+ J13245 lnz2
24
z225
× f bde[
(taU1)ij(U2tb)kl(tcU3)mn(U4 − U1)
ad(U5 − U3)ce
− (U1ta)ij(tbU2)kl(U3tc)mn(U4 − U1)
da(U5 − U3)ec]
}
G. A. Chirilli (The Ohio State Uni.) Non-linear QCD dynamics QCD evolution May 28, 2015 42 / 43
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Conclusions
Linear evolution equation cannot describe the dynamics of high-energy
QCD.
Dynamics of QCD at high-energy is non-linear.
Calculations of BK/JIMWLK evolution equations at NLO.
NLO photon impact factor.
NLO BK/JIMWLK evolution equation.
G. A. Chirilli (The Ohio State Uni.) Non-linear QCD dynamics QCD evolution May 28, 2015 43 / 43