Running Coupling in Running Coupling in Small-x EvolutionSmall-x Evolution

Yuri KovchegovYuri Kovchegov

The Ohio State UniversityThe Ohio State University

Based on work done in collaboration with Heribert Weigert, Based on work done in collaboration with Heribert Weigert, hep-ph/0609090 and hep-ph/0612071 and with Javier hep-ph/0609090 and hep-ph/0612071 and with Javier Albacete, arXiv:0704.0612 [hep-ph] Albacete, arXiv:0704.0612 [hep-ph]

PreviewPreview

Our goal here is to include running coupling corrections into BFKL/BK/JIMWLK small-x evolution equations.

The result is that the running coupling corrections come in as a “triumvirate” of couplings:

(...)

(...)(...)

S

SS

IntroductionIntroduction

DIS in the Classical ApproximationDIS in the Classical Approximation

The DIS process in the rest frame of the target is shown below.It factorizes into

),(),( *2* YxNQx qqBj

Atot

with rapidity Y=ln(1/xBj)

DIS in the Classical ApproximationDIS in the Classical Approximation

The dipole-nucleus amplitude inthe classical approximation is

x

QxYxN S 1

ln4

exp1),(22

A.H. Mueller, ‘90

1/QS

Colortransparency

Black disklimit,

22tot R

But: no energy dependence in this approximation!

Quantum EvolutionQuantum Evolution

As energy increases

the higher Fock states

including gluons on top

of the quark-antiquark

pair become important.

They generate a

cascade of gluons.

These extra gluons bring in powers of S ln s, such thatwhen S << 1 and ln s >>1 this parameter is S ln s ~ 1.

Resumming Gluonic CascadeResumming Gluonic Cascade

In the large-NIn the large-NC C limit oflimit of

QCD the gluon correctionsQCD the gluon corrections

become color dipoles. become color dipoles.

Gluon cascade becomes Gluon cascade becomes

a dipole cascade.a dipole cascade.

A. H. Mueller, ’93-’94A. H. Mueller, ’93-’94

We need to resumdipole cascade, with each finalstate dipoleinteracting withthe target. Yu. K. ‘99

NonlinearNonlinear EvolutionEvolution EquationEquation

)],,(),,(),,(),,(),,([

2

),,(

1220101220

212

202

201

22

210

YxxNYxxNYxxNYxxNYxxN

xx

xxd

N

Y

YxxN CS

Defining rapidity Y=ln s we can resum the dipole cascade

I. Balitsky, ’96, HE effective lagrangianYu. K., ’99, large NC QCD

Linear part is BFKL, quadratic term brings in damping

x

QxYxxN S 1

ln4

exp1)0,,(22

0110 initial condition

Nonlinear Equation: SaturationNonlinear Equation: Saturation

Gluon recombination tries to reduce the number of gluons in the wave function. At very high energy recombination begins to compensate gluon splitting. Gluon density reaches a limit and does not grow anymore. So do total DIS cross sections. Unitarity is restored!

Black DiskBlack Disk

LimitLimit

s3ln~

Running Coupling CorrectionsRunning Coupling Corrections

What Sets the Scale for the Running What Sets the Scale for the Running Coupling?Coupling?

(???)SIn order to perform consistent calculationsit is important to know the scale of the runningcoupling constant in the evolution equation.

There are three possible scales – the sizes of the “parent” Dipole and “daughter” dipoles . Which one is it? 202101 ,, xxx

)],,(),,(),,(),,(),,([

2

),,(

1220101220

212

202

201

22

210

YxxNYxxNYxxNYxxNYxxN

xx

xxd

N

Y

YxxN CS

What Sets the Scale for the Running What Sets the Scale for the Running Coupling?Coupling?

)],,(),,(),,(),,(),,([

2

),,(

1220101220

212

202

201

22

210

YxxNYxxNYxxNYxxNYxxN

xx

xxd

N

Y

YxxN CS

01x

0

1

202x

12xtransverseplane

Main PrincipleMain Principle

To set the scale of the coupling constant we will first calculate the corrections to BK/JIMWLK evolution kernel to all orders.

We then would complete to the QCD beta-function

by replacing .

fS N

fN

12

2112

fC NN

26 fN

Leading Order CorrectionsLeading Order Corrections

A B C

UV divergent ~ ln

UV divergent ~ ln ?

The lowest order corrections to one step of evolution arefS N

targetdipole

Diagram ADiagram A

If we keep the transverse coordinates of the quark and the antiquark fixed, then the diagram would be finite.

If we integrate over the transversesize of the quark-antiquark pair, then it would be UV divergent. ~ ln

Why do we care about this diagram at all? It does not even have the structure of the LO dipole kernel!!!

Running Coupling Corrections to All Running Coupling Corrections to All OrdersOrders

Let’s insert fermion bubbles to all orders:

Virtual Diagram: Graph CVirtual Diagram: Graph CConcentrating on UV divergences only we write

S

2

2 /1ln1

All running coupling correctionsassemble into the physical coupling .S

Real Diagram: Graph BReal Diagram: Graph B

Again, concentrating on UV divergences only we write

?]/1ln1[ 22

2

Running coupling correctionsdo not assemble into anything one could express in terms ofthe physical coupling !!!S

Real Diagram: Graph AReal Diagram: Graph A

Looks like resummation without diagram A does not make sense after all.

222

22

2

]/1ln1[

/1ln

Keeping the UV divergent parts we write:

Real Diagrams: A+BReal Diagrams: A+BAdding the two diagrams together we get

S

SS

22

2

22

]/1ln1[

]/1ln1[

Two graphs together give results depending on physical couplings only! They come in as “triumvirate”!

Extracting the UV Divergence from Extracting the UV Divergence from Graph AGraph A

We can add and subtract the UV-divergent part of graph A:

+UV-finite

UV-divergent

Extracting the UV Divergence from Extracting the UV Divergence from Graph AGraph A

In principle there appears to be no unique way to extract the UVdivergence from graph A. Which coordinate should we keep fixed as we integrate over the size of the quark-antiquark pair?

gluon

,1z

1,2z

Need to integrate over

21 zz

One can keep either or fixed (Balitsky, hep-ph/0609105).1z 2z

Extracting the UV Divergence from Extracting the UV Divergence from Graph AGraph A

gluon

,1z

1,2z

We decided to fix the transverse coordinate of the gluon:

21 )1( zzz

z

Results: Transverse Momentum Results: Transverse Momentum SpaceSpace

)(

)()(

)2(

'4);,(

2)()(

4

22

1010

Qe

qdqdK

S

SSii

22

22xzq'xzq q'q

q'qq'q

zxx

The resulting JIMWLK kernel with running coupling correctionsis

where22

2222

22

222222

2

2 )/(ln)/(ln)/(lnln

q'qq'q

q'qq'q

q'qq'q'qq

Q

The BK kernel is obtained from the above by summing over all possible emissions of the gluon off the quark and anti-quark lines.

q

q’

Results: Transverse Coordinate SpaceResults: Transverse Coordinate Space

)(

)()(

)2(

'4);,(

2)()(

4

22

1010

Qe

qdqdK

S

SSii

22

22xzq'xzq q'q

q'qq'q

zxx

To Fourier-transform the kernel

into transverse coordinate space one has to integrate overLandau pole(s). Since no one knows how to do this, one is leftwith the ambiguity/power corrections.

The standard way is to use a randomly chosen (usually PV) contour in Borel plane and then estimate power corrections to it by picking the renormalon pole. This is done by Gardi, Kuokkanen, Rummukainen and Weigert in hep-ph/0609087. Renormalon corrections may be large…

Running Coupling BKRunning Coupling BK

Let us ignore the Landau pole for now. Then after the Fourier transform we get the BK equation with the running coupling corrections:

]),,(),,(),,(),,(),,([

)/1(

)/1()/1(2

)/1()/1(

2

),,(

1220101220

212

202

21202

212

202

212

212

202

202

22

210

YxxNYxxNYxxNYxxNYxxN

xxR

xx

x

x

x

x

xdN

Y

YxxN

S

SSSS

C

xx

where

221

220

221

220

2120

221

220

221

220

2220

221

2221

22022 )/(ln)(ln)(ln

lnxx

xxxx

xx

xxxxR

xx

Numerical SolutionNumerical Solution

Running Coupling: Numerical SolutionRunning Coupling: Numerical Solution

J. Albacete, Y.K. arXiv:0704.0612 [hep-ph]

A Word of CautionA Word of Caution

When we performed a UV subtraction we left out a part of the kernel. Hence the evolution equation is incomplete unless we put that UV-finite term back in. Adding the term back in removes the dependence of the procedure on the choice of the subtraction point!

termfinite UV

)],,(),,(),,(),,(),,([

)/1(

)/1()/1(2

)/1()/1(

2

),,(

1220101220

212

202

21202

212

202

212

212

202

202

22

210

YxxNYxxNYxxNYxxNYxxN

xxR

xx

x

x

x

x

xdN

Y

YxxN

S

SSSS

C

xx

Relative Contribution of the Relative Contribution of the Subtraction TermSubtraction Term

)(YQr S Subtraction term decreases with rapidity Y!

Solution of the Full EquationSolution of the Full Equation

Subtraction term introduces a significant correction,lowering the solution.

Geometric ScalingGeometric Scaling

Geometric scaling is the property of the solution of nonlinear evolution equation. The solution leads to dipole amplitude(and structure functions) being a function of just one variable

inside the saturation region (Levin, Tuchin ‘99) and beyond

(Iancu, Itakura, McLerran ’02). The latter extension is called extended geometric scaling.

geomSS krQyQrNyrN /1/1,)(),(

SS QryQrNyrN /1,)(),(

Geometric Scaling in DISGeometric Scaling in DISGeometric scaling has been observed in DIS data by Stasto, Golec-Biernat, Kwiecinski in ’00.

Here they plot the totalDIS cross section, whichis a function of 2 variables- Q2 and x, as a function of just one variable:

2

2( )S

Q

Q x

Geometric ScalingGeometric Scaling

At high enough rapidity we recover geometric scaling, all solutions fall on the same curve. This has been known for fixed coupling: running coupling did not “spoil” that! However, the shape of the scaling function is different in the running coupling case!

)(YQr S

Running Coupling vs Fixed CouplingRunning Coupling vs Fixed Coupling

J. Albacete, Y.K. arXiv:0704.0612 [hep-ph]

)(YQr S

is the “scaling variable”

Slopes are different! RC is steeper!

NLO BFKL +NLO BFKL +

NLO BFKLNLO BFKL

Since we know corrections to all orders, we know them at the lowest order and can find their contribution to theNLO BFKL intercept. However, in order to compare that to theresults of Fadin and Lipatov ’98 and of Camici and Ciafaloni ’98(CCFL) we need to find the NLO BFKL kernel for the same observable.

Here we have been dealing with the dipole amplitude N. Tocompare to CCFL we need to write down an equation for theunintegrated gluon distribution.

fS N

...),(

),(),()/1ln(

22

22

QxNK

QxNKQxNx

BFKLNLOS

BFKLS

NLO BFKLNLO BFKL

At the leading twist level we define the gluon distribution by

2

22

01

),()()1(

2),( 01

k

Ykkekd

SNYxN Si

C

xk

( is the LO BFKL eigenvalue.)

NLO BFKLNLO BFKL

Defining the intercept by acting with the NLO kernel on the LO eigenfunctions we get

)(

12ln1)(),(

2

2

2

22

f

MS

CNLOLO NkN

k

qqkKqd

with)(

3

10)1(')(')()( 2

in agreement with the results of Camici, Ciafaloni, Fadin and Lipatov!

)1()()1(2)(

BFKL with Running CouplingBFKL with Running CouplingWe can also write down an expression for the BFKL equationwith running coupling corrections (H. Weigert, Yu.K. ‘06):

),()(

))(()(

)(),())((

)(

2

2

),(2

22

22

22

22

2Yk

k

q

q

kYqqd

N

Y

Yk

S

SSS

c

qk

qkqk

qk

Note: the above equation includes all corrections exactly!

fS N

BFKL with Running CouplingBFKL with Running CouplingWe can also write down an expression for the BFKL equationwith running coupling corrections:

),()(

))(()(

)(),())((

)(

2

2

),(2

22

22

22

22

2Yk

k

q

q

kYqqd

N

Y

Yk

S

SSS

c

qk

qkqk

qk

If one rescales the unintegrated gluon distribution:)(

),(),(

~2k

YkYk

S

then one gets

),(

~

)(),(

~

)(

2

)(

))(()(

2

),(~

22

2

22

222

2Yk

q

kYq

k

qqd

N

Y

Yk

S

SSc

qkqkqk

in agreement with what was conjectured by Braun (hep-ph/9408261) and by Levin (hep-ph/9412345) based on postulating bootstrap to work even for running coupling (though for a differently normalized gluon distribution).

More Recent DevelopmentsMore Recent Developments

Running Coupling In Gluon Running Coupling In Gluon ProductionProduction

We want to include running coupling corrections into gluonproduction cross section. Let’s start inserting bubbles:

These two chains give

Adding bubbles in the vertices and the coupling due to gluon emission we get

Running Coupling In Gluon Running Coupling In Gluon ProductionProduction

We have a problem: the result still depends on bare coupling!

In the end the problem is with the definition of gluon production.To properly define gluon production cross section we need to introduce resolution for gluons, such that diagrams like this would also contribute to gluon production:

Running Coupling In Gluon Running Coupling In Gluon ProductionProduction

When the dust settles we get )( 2collS

where coll is the IR cutoff (resolution).

It appears that saturation effects can not prevent non-perturbative effects from coming in at the fragmentationlevel: there are always going to be factors of non-perturbativelylarge S in the inclusive cross section!

Yu.K., H. Weigert, arXiv:0712.3732 [hep-ph]

ConclusionsConclusions

We have derived the BK/JIMWLK evolution equationswith the running coupling corrections. Amazingly enoughthey come in as a “triumvirate” of running couplings.

We solved the full BK equation with running coupling corrections numerically. We showed that running coupling corrections tend to slow down the evolution. At high rapidity we obtained geometric scaling behavior.

)/1(

)/1()/1(2

212

202

R

xx

S

SS

ConclusionsConclusions

We have derived the BFKL equation with the running coupling corrections. The answer confirms the conjecture of Braun and Levin, based on postulating bootstrap to all orders, though for the unintegrated gluon distribution with a non-traditional normalization.

We have independently confirmed the results of Camici, Ciafaloni, Fadin and Lipatov for the leading-Nf NLO BFKL intercept.

We showed that the inclusive cross section always has a factor of a non-perturbatively large coupling.

Backup Slides

Going Beyond Large NGoing Beyond Large NCC: JIMWLK: JIMWLK

To do calculations beyond the large-NTo do calculations beyond the large-NCC limit on has to use a functional limit on has to use a functional

integro-differential equation written by integro-differential equation written by Iancu, Jalilian-Marian, Kovner, Iancu, Jalilian-Marian, Kovner,

Leonidov, McLerran and Weigert (JIMWLK):Leonidov, McLerran and Weigert (JIMWLK):

21[ ( , )] [ ( )]

2 ( ) ( ) ( )S

ZZ u v Z u

Y u v u

where the functional Z can then be used for obtaining wave function-averaged observables (like Wilson loops for DIS):

[ ] [ ]

[ ]

D Z OO

D Z

Going Beyond Large NGoing Beyond Large NCC: JIMWLK: JIMWLK

The JIMWLK equation has been solved on the lattice by K. Rummukainen and H. Weigert

For the dipole amplitude N(x0,x1, Y), the relative corrections to the large-NC limit BK equation are < 0.001 ! Not the naïve 1/NC

2

~ 0.1 !

The reason for that is dynamical, and is largely due to saturation effects suppressing the bulk of the potential 1/NC

2 corrections.