Blois workshopK. Itakura (CEA/Saclay)1 Perturbative Odderon in the Color Glass Condensate in...

Blois workshop K. Itakura (CEA/Saclay) 1

Perturbative Odderon Perturbative Odderon in the in the ColorColor GlassGlass CondensateCondensate

in collaboration with

E. Iancu (Saclay),

L. McLerran & Y. Hatta (BNL)

Kazunori Itakura

(SPhT, CEA/Saclay

KEK in two weeks)

based on hep-ph/0501171


OutlineOutlineIntroduction Odderon in Regge theory and in perturbative QCD

Why Odderon in CGC??

C-odd operators in CGC Relevant operators for dipole-CGC & 3quark-CGC scatterings

Odderon evolutions dipole-CGC scattering decomposition of the Balitsky equation,

BFKL equation in weak-field regime

3quark-CGC scattering new equation, reduces to the BKP eq.

in weak-field regime

Summary


OdderonOdderon = Leading “C-odd” exchange in hadron scatt. at high energies.

“C-odd” counterpart of the Pomeron (see Ewerz’s talk)

Odderon /Odderon / Introduction (I)Introduction (I)

Regge theory “soft” Odderon [Lukaszuk-Nicolescu ’73]

Elastic amplitude odd under “crossing” (a+ba+b vs “crossed” a+b a+b)

A-- : “particle-particle scatt” – “particle-antiparticle scatt”

ODD under charge conjugation pp

}{ ),(),(2

1),( tsAtsAtsA baab

--

-

*

Perturbative QCD “hard” Odderon

three reggeized gluon exchange in C-odd

state (exists only for )

C-odd three gluon operator

3Nccbaabc AAAd

*

* Experimental status not conclusive so far…


The BKP equation for 3 gluonsThe BKP equation for 3 gluons [Bartels, Kwiecinski-Praszalowicz ‘80]

F: amplitude for exchange of three reggeized gluons in a color singlet C-odd state Pair-wise interaction between two gluons among three

BFKL evolution HBFKL

The physical amplitude is obtained after convoluting

the impact factor of the projectile

Two solutions for BKP eq. with 3 gluons:

Janik-Wosiek (‘99) , Bartels, Lipatov & Vacca (‘00)

Perturbative Odderon /Perturbative Odderon / Introduction (II)Introduction (II)

FHFHFHkkkFY BFKLBFKLBFKL 312312

321 ),,(

1odd 1odd

**

*

*

Y


Why Odderon in CGC? / Why Odderon in CGC? / Introduction (III)Introduction (III)

Perturbative Pomeron in the Color Glass Condensate

dipole-CGC scattering ( dipole operator + JIMWLK equation)

The relevant operator for the Pomeron (see talks by Venugopalan, Iancu)

Two reggeized gluon exchange in linear regime

two Wilson lines in nonlinear regime BFKL equation

But n-reggeon dynamics (BKP) is also important at high energy Need to investigate n-reggeon dynamics in the CGC which is in princip

le applicable for n-reggeons.

The first step: 3 gluon exchange in linear regime Odderon !

What is the relevant operator for the Odderon exchange???What is the relevant operator for the Odderon exchange???

Can we reproduce the BKP equation in the CGC???Can we reproduce the BKP equation in the CGC???

22

)(4

1)( ay

axyx Nc

gVVtr }{ ),(exp

xxdxigPVx


Determine the relevant operators for scatt. btw a projectile and the CGC

A projectile traverses a strong “random” gauge field created by the CGC.

- the eikonal approximation

- The operator is evaluated with

averaging over the color field

W[]: weight function randomness

General strategies in CGCGeneral strategies in CGC

ex)Dipole-CGC scattering: the relevant operator leads to the Balitsky eq.

0|)()( yqxq inin 0|)()( yqxq outout

Compute the evolution equations from the JIMWLK equation

JIMWLK eq. = evolution equation for the weight function in the target. easily converted into the equations for operators.can be made simple for gauge invariant operators

IR finiteness manifest


Transition from C-odd to C-even dipole states

　　　　　　　　　　　　　　　　　　　　　

Relevant operator

- anti-symmetric under the exchange of x and y: O(x,y) = - O(y,x)

- imaginary part of the dipole operator.

Weak field expansion leading order is 3 gluons

gauge invariant combination! ( + c)

C-odd operatorC-odd operator in in dipole-CGCdipole-CGC scatt.scatt.

0|)()()()( ][ xqyqyqxq 0|)()()()( ][ xqyqyqxq


C-odd operator inC-odd operator in 3-quark--CGC3-quark--CGC scatt. scatt.

Consider the scattering of a color singlet “3-quark state” and transition from C-even to C-odd 3 quark states

Relevant operator

“baryonic Wilson lines”

Weak field expansion

3 gluons with d-symbol, gauge invariant

________ ________ all the possible ways of attaching

0|kzjy

ix

ijk qqq


Evolution of the Evolution of the dipole Odderondipole OdderonEvolution eq. for the dipole Odderon “imaginary part” of the Balitsky eq.

couple to the Pomeron N(x,y) = 1- 1/Nc Re tr(V+xVy)

becomes equivalent to Kovchegov-Szymanowski-Wallon (‘04)

if one assumes factorization <NO> <N><O>.

initial condition computable with a classical gauge field + color averaging

or in an extended McLerran-Venugopalan model (Jeon-Venugopalan ‘05)

linear part = the BFKL eq. (but with C-odd initial condition)

reproduces the BKP solution with the largest intercept

found by Bartels, Lipatov & Vacca (KSW,04)

intercept reduces due to saturation: <O(x,y)> decreasing as <N(x,y)> 1

Evolution of N(x,y) is also modified due to Odderon: 2 Odderons 1 Pomeron

BFKL

**

*

*

*

1odd

)/1ln( x

*


Evolution of the Evolution of the dipole Odderondipole Odderon (I (II)I)

The presence of imaginary part (odderon) affects the evolution equation for the scattering amplitude N(x,y).

Balitsky equation new contribution!

- Two Odderons can merge into one Pomeron!

N=1, O=0 is the stable fixed point.


3-quark--Odderon operator3-quark--Odderon operatorBaryonic Wilson line operatorBaryonic Wilson line operator

multiplying the identity

One can rewrite 3quark-Odderon operator as manifestly gauge invariant

reduces to dipole-Odderon operator when two coordinates are the same

Oproton(x,z,z) = O(x,z) diquark ~ antiquark

can compute nonlinear evolution equation for Oproton(x,y,z) complicated


: weak field limit of

The BKP equation appears as the equation for 3 point Green function

with infra-red singularities removed

Evolution of Evolution of 3quark-Odderon3quark-Odderon operator operatorin the weak-field limitin the weak-field limit

),,( zyxOproton

cz

by

ax

abcdzyxf ),,(

xyzB


Relation to the traditional approachRelation to the traditional approachTraditional description CGC formalism

Our operator partly contains the

information of the impact factor

Gauge invariant

impact factorgauge invariance

BKP equation

}{}{

...),,(2

...),,(),,(3),,(12

),,(

xxxf

zxxfyxxfzyxf

zyxOproton

LC wavefunction


SummarySummary• Identified the relevant operator for C-odd Odderon exchange in dipole-CGC scattering imaginary part of the dipole operator (2pt fnc),

O(x,y) = [ tr(Vx+ Vy) – tr(Vy

+ Vx) ] / 2iNc. in 3-quark--CGC scattering a 3 point fnc constructed from baryonic Wilson line operator Both reduce to 3 gluons with d-symbol in the weak-field limit

• Evolution equations for these operators JIMWLK eq. dipole--CGC scattering Imaginary part of the Balitsky eq. Nonlinear terms represent coupling to the Pomeron. 3-quark--CGC scattering Complicated in the nonlinear (strong field) regime Reproduce the BKP equation in the weak-field limit

Blois workshopK. Itakura (CEA/Saclay)1 Perturbative Odderon in the Color Glass Condensate in...

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