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D. Kharzeev Nuclear Theory Group @ BNL

Alice Club, CERN TH, May 14, 2007

Non-linear evolution in QCDand hadron multiplicity predictions for the LHC

Based on work with E. Levin, M. Nardi, K. Tuchin

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Two questions:

1. What is the mechanism of multi-particle production at high energies in QCD?

2. What are the implications for high-energy evolution and for the energy dependence?

A possible answer:strong semi-classical color fields

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Strings vs partons in high energy QCD

string picture:

color string = longitudinal color fields

parton picture:

“Weizsacker-Williams” gluons = transverse color fields

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€

Aμ =e

4π R − v→

R→ ⎛

⎝ ⎜

⎞ ⎠ ⎟(1,v

→

)

• Lienard-Weichert potential of a moving charge

€

R2 = (z − z(t '))2 + x⊥2 (1− v 2)

• Electro-magnetic fields:

€

E→

=e

4π

1− v 2

(R − R→

v→

)3R→

− v→

R ⎛ ⎝ ⎜

⎞ ⎠ ⎟+

e

4π (R − R→

v→

)3R→

× R→

− v→

R ⎛ ⎝ ⎜

⎞ ⎠ ⎟× a

→ ⎛

⎝ ⎜

⎞

⎠ ⎟

€

H→

=R→

× E→

R

a

vR

E

≈Ez

What is the structure of the classical fields?warm-up: electrodynamics

transverselongitudinal

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The space-time picture of high-energy interactions in QCD

1. Fast (large y) partons live for a long time;2. Parton splitting probability is ~ s y - not small!

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The origin of classical background field

Gluons with large rapidity and large occupation numberact as a background field for the production of slower gluons

static field sources

“Color Glass Condensate”

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What is the dynamics of non-linear evolution in QCD?

Parton splitting in the background of the color field? (generalization of the linear QCD evolution equations - BFKL, DGLAP)

GLR, MQ,JIMWLK,BK equations

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Renormalization group

Emitted partons become a part of the classical fieldfor slower partons; “slow” and “fast” are relative

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Parton production in the background field

Parton propagator in the background field

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Mean field approach: BK equation

Let us compute an imaginary part of the gluon propagator in the background field:

where the S-matrix is related to the imaginary scattering amplitude

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The equation

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Equivalent form:

where

is the BFKL splitting kernel;initial conditions are provided e.g. by MV model:

Is this evolution equation unique?

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What are the properties of the color field at high energies?

The field created by faster moving partons is seen by the slower produced partons as:

• Static

• Constant in space

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Why static?

The lifetime of a field configuration is

(y is the rapidity distance from the beam);The ratio is

for BFKL,>>1

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Why constant in space?

at rapidity y, the field is constant at distances up to

gluon production occurs at y-y, at distances

The ratio is “large”: BFKL yields R ~ 10

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What is the mechanism of gluon production in strong, constant, static

color field at weak coupling? Schwinger-like gluon pair production:

whereSU(3): G.Nayak,P.Nieuwenhuizen,hep-ph/0504070

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Integrated spectrum:

Average transverse momentum (saturation scale):

Saturation momentum is a measure of the field strength

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Towards the evolution equation

the energy density of the field

grows with rapidity:

this is just the energy conservation!

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The evolution equation

Sudakov-type factor neededto avoid double counting(no gluons produced between Y and Y’)

DK, E. Levin,to appear

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Equation for the saturation momentum(differential form)

where

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The solution

Initial condition

RHIC phenomenology (KLN):

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Properties of the solution

for moderate energies, power growth with the intercept (for s ~ 0.3) ~ 0.25;

at very high energies, a universal limit!

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Phenomenology

central Au-Au collisions:

GeV2

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Energy dependence of multiplicity

powergrowth

new evolutionequation

logarithmicfit

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Predictions for the LHC

KLN, hep-ph/0408050

pp: little change; Pb-Pb: decrease by ~ 30%

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Summary

The hadron multiplicity measurements at the LHC will enable us to understand the nature of multiparticle production and the origin of parton evolution at high energies