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

2. 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)

3. Motivation

4. DIS Structure Functions

5. 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.

6. 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)

7. 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

8. 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.

9. 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

10. Nonlinear evolution
dashed line =
all interactions
with the target

11. 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

12. 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.

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

14. 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?

15. 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.

16. 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) .

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

18. 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!!!

19. Running Coupling Corrections to All Orders
Let’s insert fermion bubbles to all orders:

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

21. 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 !!!

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

23. 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”!

24. 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

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

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

27. Inclusive gluon production

28. 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

29. 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

30. 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

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

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

33. 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)

34. 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.)

35. Gluon dipole
The gluon dipole amplitude is

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

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

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

39. 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?

40. Calculation

41. 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

42. 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) .

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

44. 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.

45. Redefined produced gluon

46. Redefined produced gluon

47. 3-Gluon vertex graphs Need to include corrections to graphs
like this too.

48. 3-Gluon vertex graphs Quark bubbles on propagators give
while the vertex bubbles give two new scales QA and QB :

49. Results

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

51. The monster scale
note that it is complex!

52. 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:

53. 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 )

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

55. 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).

56. 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.