Running coupling corrections to inclusive gluon production Yuri Kovchegov The Ohio State University based on arXiv:1009.0545 [hep-ph] in collaboration with William Horowitz

Outline Motivation: phenomenological success of rc small-x evolution Structure functions in DIS Inclusive production Calculation: dispersion relations, etc Results: “septumvirate of couplings”, complex (& conjugate) momentum scale(s)

Motivation

DIS Structure Functions

How to Calculate Observables Start by finding the classical field of the McLerran-Venugopalan model. Continue by including the quantum corrections of the nonlinear BK/JIMWLK evolution equations. Works for structure functions of hadrons and nuclei, as well as for gluon production in various hadronic collisions.

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)

DIS in the Classical Approximation The dipole-nucleus amplitude in the classical approximation is A.H. Mueller, ‘90 Black disk limit, Color transparency 1/QS

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 aS ln s, such that when aS << 1 and ln s >>1 this parameter is aS ln s ~ 1.

Resumming Gluonic Cascade In the large-NC 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 dashed line = all interactions with the target

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

Non-linear evolution Theoretically nothing is wrong with it: preserves unitarity (black disk limit), no BFKL IR diffusion problem Phenomenologically there is a problem though: LO BFKL intercept is way too large Full NLO calculation (order- kernel): tough (see Balitsky and Chirilli ’08). Instead let’s try to “simply” fix the scale of the coupling.

What Sets the Scale for the Running Coupling? 1 transverse plane 2

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?

Preview The answer is that the running coupling corrections come in as a “triumvirate” of couplings (H. Weigert, Yu. K. ’06; I. Balitsky, ’06; Gardi et al ‘06): cf. Braun ’94, Levin ‘94 The scales of three couplings are somewhat involved.

Main Principle To set the scale of the coupling constant we will first calculate the (quark loops) corrections to LO gluon production cross section to all orders. We then will complete to the full QCD beta-function by replacing (Brodsky, Lepage, Mackenzie ’83 – BLM prescription) .

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

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 m 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 quark loop corrections assemble into the running coupling .

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

Running Coupling BK Here’s the BK equation with the running coupling corrections (H. Weigert, Yu. K. ’06; I. Balitsky, ’06; Gardi et al ‘06): where

What does the running coupling do? Slows down the evolution with energy / rapidity. down from about at fixed coupling Albacete ‘07

Comparison of rcBK with HERA F2 Data from Albacete, Armesto, Milhano, Salgado ‘09

Inclusive gluon production

Gluon Production in Proton-Nucleus Collisions (pA): Classical Field To find the gluon production cross section in pA one has to solve the same classical Yang-Mills equations for two sources – proton and nucleus. Yu. K., A.H. Mueller ‘98

Gluon Production in pA: McLerran-Venugopalan model The diagrams one has to resum are shown here: they resum powers of Yu. K., A.H. Mueller, ‘98

Gluon Production in pA: McLerran-Venugopalan model Classical gluon production: we need to resum only the multiple rescatterings of the gluon on nucleons. Here’s one of the graphs considered. Yu. K., A.H. Mueller, ‘98 Resulting inclusive gluon production cross section is given by With the gluon-gluon dipole-nucleus forward scattering amplitude

Factorial series Experiments often measure the nuclear modification factor In this classical case it is known exactly Kharzeev Yu. K. Tuchin ‘03

Factorial series Expanding in powers of yields Kharzeev Yu. K. Tuchin ‘03

Including Quantum Evolution To understand the energy dependence of particle production in pA one needs to include quantum evolution resumming graphs like this one. This resums powers of a ln 1/x = a Y. Yu. K., K. Tuchin, ‘01 The rules accomplishing the inclusion of quantum corrections are Proton’s LO wave function Proton’s BFKL wave function  and where the dipole-nucleus amplitude N is to be found from (Yu. K., Balitsky)

Gluon Production in pA Amazingly the gluon production cross section reduces to a kT-factorization expression (Yu. K., Tuchin, ’01; cf. Gribov, Levin, Ryskin ‘83): with the proton and nucleus unintegrated gluon distributions defined by with NGp,A the scattering amplitude of a GG dipole on p or A. (Includes multiple rescatterings and small-x evolution.)

Gluon dipole The gluon dipole amplitude is

Gluon Production in pA A simplified diagrammatic way of thinking about this result is

CGC Gluon Production in AA This is an unsolved problem. Albacete and Dumitru ‘10, following KLN ‘01, approximate the full unknown solution by

CGC Multiplicity Prediction CGC prediction by Albacete and Dumitru ‘10 for LHC multiplicity and its centrality dependence was quite successful.

Problem We have running-coupling NG , which we inserted into fixed coupling formulas (note that A&D run their couplings ad hoc with kT): Does this formalism survive if we include running-coupling corrections? If yes, what are the scales of the couplings?

Calculation

LO Gluon Production The answer is which is the same kT-factorization First consider LO gluon production (A+=0 light-cone gauge): The answer is which is the same kT-factorization formula if we notice that at LO

Main Principle To set the scale of the coupling constant we will first calculate the (quark loops) corrections to LO gluon production cross section to all orders. We then will complete to the full QCD beta-function by replacing (Brodsky, Lepage, Mackenzie ’83 – BLM prescription) .

Bremsstrahlung graphs Start by putting quark bubbles on this graph: problem: no scale here!?

Gluon Resolution Scale Resolution: redefine “gluon” production to include qqbar (GG) pairs with the invariant mass Then Yu. K., H. Weigert ‘07 The scale is still largely arbitrary, but now it has a physical meaning.

Redefined produced gluon

Redefined produced gluon

3-Gluon vertex graphs Need to include corrections to graphs like this too.

3-Gluon vertex graphs Quark bubbles on propagators give while the vertex bubbles give two new scales QA and QB :

Results

The answer for LO+rc The final result can be written rather compactly as: with the “septumvirate” of running couplings, except for…

The monster scale note that it is complex!

Simplified form In an AA collision, with saturation effects included (as a cutoff), the formula can be simplified to: Note that the monster scale becomes simple in the collinear limit:

Ansatz for the full expression Inspired by the LO result we make the following ansatz for the gluon production in pA including running coupling corrections with the new unintegrated gluon distributions (note that )

Integrated multiplicity dN/dy depends on how one defines Qs : If then If then

Multiplicity vs centrality We conclude that rc is not a defining factor in the centrality dependence of mutiplicity per participant, contrary to KLN interpretation of their results. Perhaps more importantly one needs two different saturation scales Qs1 and Qs2 to get the right centrality dependence. (cf. Drescher et al ’06, Kuhlman el at ’06).

Conclusions rc corrections to non-liner small-x evolution make the agreement with the DIS and AA data easier to achieve kT-factorization ansatz survives inclusion of running coupling corrections, at least at LO, after a slight redefinition of gluon distributions We have obtained a formula for LO gluon production + rc corrections. We have constructed an ansatz for the rc corrections to the pA gluon production including multiple rescatterings and small-x evolution.