Running Coupling in Small-x Evolution Yuri Kovchegov The Ohio State University Based on work done in collaboration with Heribert Weigert, hep-ph/ and hep-ph/ and with Javier Albacete, arXiv: [hep-ph]

Preview  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:

Introduction

DIS in the Classical Approximation The DIS process in the rest frame of the target is shown below. It factorizes into with rapidity Y=ln(1/x Bj )

DIS in the Classical Approximation The dipole-nucleus amplitude in the classical approximation is A.H. Mueller, ‘90 1/Q S Color transparency Black disk limit, But: no energy dependence in this approximation!

Quantum 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 that when  S >1 this parameter is  S ln s ~ 1.

Resumming Gluonic Cascade In the large-N C limit of QCD the gluon corrections become color dipoles. Gluon cascade becomes a dipole cascade. A. H. Mueller, ’93-’94 We need to resum dipole cascade, with each final state dipole interacting with the target. Yu. K. ‘99

Nonlinear Evolution Equation Defining rapidity Y=ln s we can resum the dipole cascade I. Balitsky, ’96, HE effective lagrangian Yu. K., ’99, large N C QCD  Linear part is BFKL, quadratic term brings in damping initial condition

Nonlinear 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 Disk Limit Limit

Running Coupling Corrections

What Sets the Scale for the Running Coupling? In order to perform consistent calculations it is important to know the scale of the running coupling constant in the evolution equation. There are three possible scales – the sizes of the “parent” Dipole and “daughter” dipoles. Which one is it?

What Sets the Scale for the Running Coupling? transverse plane

Main 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.

Leading Order Corrections A B C UV divergent ~ ln  UV divergent ~ ln  ? The lowest order corrections to one step of evolution are target dipole

Diagram 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 transverse size 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 Orders Let’s insert fermion bubbles to all orders:

Virtual Diagram: Graph C Concentrating on UV divergences only we write All running coupling corrections assemble into the physical coupling.

Real Diagram: Graph B Again, concentrating on UV divergences only we write Running coupling corrections do not assemble into anything one could express in terms of the physical coupling !!!

Real Diagram: Graph A Looks like resummation without diagram A does not make sense after all. Keeping the UV divergent parts we write:

Real Diagrams: A+B Adding the two diagrams together we get Two graphs together give results depending on physical couplings only! They come in as “triumvirate”!

Extracting the UV Divergence from Graph A We can add and subtract the UV-divergent part of graph A: + UV-finite UV-divergent

Extracting the UV Divergence from Graph A In principle there appears to be no unique way to extract the UV divergence from graph A. Which coordinate should we keep fixed as we integrate over the size of the quark-antiquark pair? gluon Need to integrate over One can keep either or fixed (Balitsky, hep-ph/ ).

Extracting the UV Divergence from Graph A gluon We decided to fix the transverse coordinate of the gluon:

Results: Transverse Momentum Space The resulting JIMWLK kernel with running coupling corrections is where 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 Space To Fourier-transform the kernel into transverse coordinate space one has to integrate over Landau pole(s). Since no one knows how to do this, one is left with 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/ Renormalon corrections may be large…

Running 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: where

Numerical Solution

Running Coupling: Numerical Solution J. Albacete, Y.K. arXiv: [hep-ph]

A 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!

Relative Contribution of the Subtraction Term Subtraction term decreases with rapidity Y!

Solution of the Full Equation Subtraction term introduces a significant correction, lowering the solution.

Geometric 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.

Geometric Scaling in DIS Geometric scaling has been observed in DIS data by Stasto, Golec-Biernat, Kwiecinski in ’00. Here they plot the total DIS cross section, which is a function of 2 variables - Q 2 and x, as a function of just one variable:

Geometric 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!

Running Coupling vs Fixed Coupling J. Albacete, Y.K. arXiv: [hep-ph] is the “scaling variable” Slopes are different! RC is steeper!

NLO BFKL +

NLO BFKL Since we know corrections to all orders, we know them at the lowest order and can find their contribution to the NLO BFKL intercept. However, in order to compare that to the results 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. To compare to CCFL we need to write down an equation for the unintegrated gluon distribution.

NLO BFKL At the leading twist level we define the gluon distribution by

( is the LO BFKL eigenvalue.) NLO BFKL Defining the intercept by acting with the NLO kernel on the LO eigenfunctions we get with in agreement with the results of Camici, Ciafaloni, Fadin and Lipatov!

BFKL with Running Coupling We can also write down an expression for the BFKL equation with running coupling corrections (H. Weigert, Yu.K. ‘06): Note: the above equation includes all corrections exactly!

BFKL with Running Coupling We can also write down an expression for the BFKL equation with running coupling corrections: If one rescales the unintegrated gluon distribution: then one gets in agreement with what was conjectured by Braun (hep- ph/ ) and by Levin (hep-ph/ ) based on postulating bootstrap to work even for running coupling (though for a differently normalized gluon distribution).

More Recent Developments

Running Coupling In Gluon Production We want to include running coupling corrections into gluon production 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 Production 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 Production When the dust settles we get where  coll is the IR cutoff (resolution). It appears that saturation effects can not prevent non-perturbative effects from coming in at the fragmentation level: there are always going to be factors of non-perturbatively large  S in the inclusive cross section! Yu.K., H. Weigert, arXiv: [hep-ph]

Conclusions  We have derived the BK/JIMWLK evolution equations with the running coupling corrections. Amazingly enough they 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.

Conclusions  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-N f 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 N C : JIMWLK To do calculations beyond the large-N C limit on has to use a functional integro-differential equation written by Iancu, Jalilian-Marian, Kovner, Leonidov, McLerran and Weigert (JIMWLK): where the functional Z  can then be used for obtaining wave function-averaged observables (like Wilson loops for DIS):

Going Beyond Large N C : JIMWLK The JIMWLK equation has been solved on the lattice by K. Rummukainen and H. Weigert For the dipole amplitude N(x 0,x 1, Y), the relative corrections to the large-N C limit BK equation are < ! Not the naïve 1/N C 2 ~ 0.1 ! The reason for that is dynamical, and is largely due to saturation effects suppressing the bulk of the potential 1/N C 2 corrections.