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

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

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.


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?


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.


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


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


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.


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


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∆




s


BFKL PREDICTS ONLY

sectioncrossDIS

NON LINEAR EFFECTS

BFKL



Can parton density rise forever? Is there

a saturation limit?

Froissart bound: σtot ∝ ln2 s




s


BFKL PREDICTS ONLY

sectioncrossDIS

NON LINEAR EFFECTS

BFKL




a saturation limit?


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!




s


BFKL PREDICTS ONLY

sectioncrossDIS

NON LINEAR EFFECTS

BFKL




a saturation limit?


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.


DIS at high-energy: parton saturation Color Glass Condensate

*

xx



*

xx

coll.



*

xx

coll. coll.



*

xx

coll. coll.

coll.



*

xx

coll. coll.

coll.



*

xx

coll. coll.

coll.

higher energy



*

xx

coll. coll.

coll.

higher energyhigher energy



*

xx

coll. coll.

coll.

higher energyhigher energy higher energy



*

xx

coll. coll.

coll.

higher energyhigher energy higher energy

Need a mechanism to stop the growth: parton-recombination

⇒ Non-linear evolution equation


Saturation Scale: euristic explanation

σonium−oniumtot ∝ α2

s x1⊥ x2⊥ e∆ Y

∆ =4αsNc

πln 2 > 0

Black − disk limit : σtot ≤ 2π R2




s x1⊥ x2⊥ e∆ Y

∆ =4αsNc

πln 2 > 0


DIS case:

x1⊥ ∼ 1Q

size of dipole generated by γ∗

x2⊥ ∼ R ∼ 1ΛQCD

radius of hadron




s x1⊥ x2⊥ e∆ Y

∆ =4αsNc

πln 2 > 0


DIS case:

x1⊥ ∼ 1Q



radius of hadron

⇒ α2s

RQ

e∆Y ≤ 2π R2 With Y = log 1xB




s x1⊥ x2⊥ e∆ Y

∆ =4αsNc

πln 2 > 0


DIS case:

x1⊥ ∼ 1Q



radius of hadron

⇒ α2s

RQ

e∆Y ≤ 2π R2 With Y = log 1xB

Qs ∼ α2s ΛQCD

(

1xB

)∆violation of unitarity for Q < Qs



σonium−oniumtot ≃ α2

s x1⊥ x2⊥ e∆ Y

∆ =4αsNc

πln 2 > 0


DIS case on large nucleus

Qs ∼ A1/3(

1xb

)∆

for high enough energy (small-xB) or large nucleus

Qs ≫ ΛQCD ⇒ pQCD allowed.


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′)


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

⊥



Boosted Field


dxµAµ






Boosted Field


dxµAµ




η>Y

η<Y

+ +...


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


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.


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)

]


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 = ∞


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.


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.


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





〈{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

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+


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



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)



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



d

dηU(x, y) =

αsNc

2π2

∫

d2z (x − y)2

(x − z)2(y − z)2

{



LLA for DIS in pQCD ⇒ BFKL (LLA: αs ≪ 1, αsη ∼ 1)


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



d

dηU(x, y) =

αsNc

2π2

∫

d2z (x − y)2

(x − z)2(y − z)2

{



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)


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.


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.



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)



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


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)


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


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


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


NLO Photon Impact Factor

[


]LO=

∫

d2z1d2z2

z412

ILOµν (x, y; z1, z2)〈tr{Uη

z1U†η

z2}〉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.


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.


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.


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.


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).


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 )


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)




(XXVI) (XXVII)

(XVI) (XVII) (XVIII) (XIX) (XX)

(XXI) (XXIV) (XV)(XXII) (XXIII)

(XVIII) (XXIX) (XXX)




(XXXI) (XXXIII) (XXXIV)(XXXII)



"Running coupling" diagrams

(I) (II) (III) (IV) (V)

y

x

(VI) (VII) (VIII) (IX) (X)



1 → 2 dipole transition diagrams

q

k’

k

k’

k

q q

k’ kk’

q

k k

k’

q

k’

kk’

qq

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



N = 4 SYM diagrams (scalar and gluino loops)


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}]

×{(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


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}]

×{(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


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.


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


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


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


z2}[(Uη

z3)aa′(Uη

z4)bb′ − (z4 → z3)]

Now Möbius invariant!


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).


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.


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.


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]


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)



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



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



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

)

}


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


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]

}


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.


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