Pomeron loop equations and phenomenological consequences

Cyrille Marquet

RIKEN BNL Research Center

ECT* workshop, January 2007

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Contents• The B-JIMWLK equations

- scattering off a dense target

• The dipole model equations- scattering off a dilute target

• The Pomeron loop equations- combining dense and dilute evolution- stochasticity in the QCD evolution

• Phenomenological consequences- diffusive scaling- implications for deep inelastic scattering- implications for particle production

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Introductionx : parton longitudinal momentum fraction

kT : parton transverse momentum

weak coupling regime

effective coupling

dense system of partons mainly gluons (small-x gluons)

transverse view of the hadron

high-energy scattering processes are sensitive to the small-x gluons

Regime of interest:

the dilute/dense separation is

caracterized by the saturation

scale Qs(x)

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The B-JIMWLK equations

scattering off a dense target

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Effective description of the hadron

the numerous small-x gluons are responsible for a large color field

which can be treated as a classical field

McLerran and Venugopalan (1994)

gggggqqqqqqgqqq .........hadron

α : large color fields created by the small-x gluons

effective wavefunction for the dressed hadron

][hadron YD

To describe a hadron dressed with many small-x gluons, we use an effective theory:

light-cone gauge : smallest value of longitudinal impulsion

is called the hadron rapidity

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The JIMWLK equationa functional equation for the rapidity evolution of

22][][ YY H

dY

d

Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov, Kovner2

][Y

the Wilson lines sum powers of α gS ~ 1

adjoint representation

study the high-energy scattering of simple projectiles (dipoles) off this dense hadron

the JIMWLK equation gives evolution of the hadron wavefunction for large enough Y

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Dipoles as test projectiles

))()((1

1][ uvuv FFc

WWTrN

T

][][][nn11nn11

2 vuvuvuvu TTDTT YY

u : quark space transverse coordinatev : antiquark space transverse coordinate

the dipole:qq

scattering amplitude off the dense target

JIMWLK equation → evolution equation for the dipole correlators

scattering of the quark:

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An hierarchy of equations

an hierarchy of equation involving correlators with more and more dipoles

Balitsky (1996)

YYYYYYTfTTTTT

zdT

dY

d)(

)()(

)(

2 22

22

uvzvuzuvzvuzuv vzzu

vu

2

1)()(

cYYY N

OTfTTTfTTdY

dzvuzzvuzzvuz

in the large Nc limit, the hierarchy is restricted to dipoles

NcS

for dipoles scattering off a dense target

BFKL saturation

general structure:

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Something is missingframe invariance requires that H is invariant under

the following transformation (dense-dilute duality)

22][][ YY H

dY

d

Kovner and Lublinsky (2005)color chargecolor field

the Wilson lines sum powers of gS δ/δρ ~ 1

is not invariant, it transform into

Balitsky (2005), Hatta, Iancu, McLerran, Stasto and Triantafyllopoulos (2006)

study the dilute regime ρ ~ gS also approach with effective action

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The dipole model equations

scattering off a dilute target

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The dipole model

N-1 gluons emitted at transverse coordinates 11,..., Nzz N dipoles ),(),...,,( 110 NN zzzz

in the large Nc limit, the emission cascade of soft gluons is a dipole cascade:

ansatz for the wavefuntion of a dilute hadron :

~ dipole creation operator

this transforms the functional equation for into a master equation for the

probabilities

splitting

no splitting

Iancu and Mueller (2004)Mueller, Shoshi and Wong (2005)

C.M., Mueller, Shoshi and Wong (2006)Hatta, Iancu, McLerran and Stasto (2006)

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Scattering of projectile dipoles

][][][nn11nn11

2 vuvuvuvu TTDTT YY

high-energy scattering of dipoles off this dilute hadron

obtained from T [α] after inverting

at lowest order with respect to αS :

dipole-dipole cross-section

from the master equation for the probabilities, one obtains the equation for the

dipole correlators

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A new hierarchy of equationsfor dipoles scattering off a dilute target

k = 1 the BFKL equation

I denoteY

nY TTT

nn11),,,,( nn11

)(vuvuvuvu

The equation for T(n) reads

k > 1 fluctuation terms uv

x

yz

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Structure of the fluctuation term

BFKL fluctuation, important when

general structure:

previous hierarchy of Iancu and Triantafyllopoulos:

analogous to recent toy models :

Iancu and Triantafyllopoulos (2005)

except for n = 1, there is more than BFKL

obtained requiring that the target dipoles scatter only once

Kovner and Lublinsky (2006)Blaizot, Iancu and Triantafyllopoulos (2006)Iancu, de Santana Amaral, Soyez and Triantafyllopoulos (2006)

work in progress

differences to understand

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The Pomeron-loop equations

combining dense and dilute evolution

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A stochastic evolutionby combining the evolution equations of the dense and dilute regimes,(counting the BFKL term only once), one gets

BFKLfluctuation, important when

saturation, important when

the QCD evolution is equivalent to a stochastic process

for instance the truncated hierarchy can bereformulated into a Langevin equation for a stochastic dipole amplitudeand the correlators are obtained by averaging the realizations:

Iancu and Triantafyllopoulos (2005)

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The sF-KPP equation

the reduction to one dimension introduces the noise strength parameter

high-energy QCD evolution = stochastic process in the universality classof reaction-diffusion processes, of the sF-KPP equation Iancu, Mueller and Munier (2005)

noise

r = dipole size

solutions of the deterministic partof the equation: traveling waves

the saturation scale:

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A stochastic saturation scale

v : average speed of the waves

(for ) DYSS 22 Q/Qln

DYDYP S

)Q/Q²(ln

exp1

)Q(ln22

S2S

the saturation scale is a stochasticvariable distributed according to

a Gaussian probability law:

corrections to the Gaussian law for improbable fluctuations also known

confirmed by exact results in the strong noise limitand numerical results for arbitrary values of the noise strength

C. M., Soyez and Xiao (2006)

C. M., Peschanski and Soyez (2006)

Soyez (2005)

The noise term introduce a stochastic saturation scale

D : dispersion coefficient

average saturation scale

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A new scaling law

: the diffusion is negligible and with )(Q)( YrTrT SY

we obtain geometric scaling

DYYrTrT SY)(Qln)( 22: the diffusion is important and

new regime: diffusive scaling

in the diffusive scaling regime

- the amplitudes are dominated by events that feature the hardest fluctuation of - in average the scattering is weak, yet saturation is the relevant physics

the average dipolescattering amplitude:

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Phenomenological consequences

diffusive scaling

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Geometric scaling and DIS data

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2

Q

Q

Wx

2photon virtuality Q2 = - (k-k’)2 >> QCD

*p collision energy W2 = (k-k’+p)2

this is seen in the data with

Stasto, Golec-Biernat and Kwiecinski (2001)

size resolution 1/Q

k

k’

p

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High-energy DIS

an intermediate energy regime:geometric scaling

HERA

it seems that HERA is probing

the geometric scaling regime

22 1~Q r

Y. Hatta, E. Iancu, C.M., G. Soyez and D. Triantafyllopoulos (2006)

)(Q)( YrTrT SY

In the diffusive scaling regime, saturation is the relevant physics

up to momenta much higher than the saturation scale

at higher energies, a newscaling law: diffusive scaling

no Pomeron (power-like) increase

DYYrTrT SY)(Qln)( 22

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Consequences for the observables

YDIS rTrdrbd

d )()Q,(22

222 22

222 )()Q,(

YDDIS rTrdrbd

d

geometric scaling regime:

DIS dominated by relatively hard sizes

DDIS dominated by semi-hard sizes

Sr Q1~

Sr Q1Q1

dipole size r

Y. Hatta, E. Iancu, C.M., G. Soyez and D. Triantafyllopoulos (2006)

Sr Q1~Q1~r

geometric scaling

diffusive scaling

both DIS and DDIS are dominatedby hard sizes

diffusive scaling regime:

Q1~ryet saturation is the relevant physics

the photon hits black spots

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Inclusive gluon productiongluon production is effectively described by a gluonic dipole (gg):

q : gluon transverse momentum

yq : gluon rapidity

))()((1

11][~

2 zzzz AAc

' W'WTrN

AT

scattering amplitude with

adjointWilson line

Y'Tzz

~

the other Wilson lines (coming from the

interaction of non-mesured partons) cancelhh

),()(~2.2

222 q

y

i yqrTerdbqdd

dqq

r

rq

result valid for any dilute projectile

the transverse momentum spectrum is obtained

from a Fourier transform of the dipole size r:

unintegrated

gluon distribution

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in forward particle production, the transverse momentum spectrum is obtained from the unintegrated gluon distribution of

the small-x hadron

Forward particle production

),( Yk

important in view of the LHC: large kT , small values of x

),(2

2 ykdykd

dk T

TT

kT , y

yT eksx 1

particle production at forward rapidities y(in hadron-hadron and heavy-ion collisions):

yT eksx 2

in the geometric scaling regime

is peaked around k ~ QS(Y)Y

),( Yk

),( Yk

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Consequences in particle productionE. Iancu, C.M. and G. Soyez (2006)

DY

k

DYYk S

)Q/²²(ln

exp1

),(2

In the diffusive scaling regime, flattens with increasing Y

Is diffusive scaling within the LHC energy range?

hard to tell: theoretically, we have a poor knowledge of the coefficient D

Y),( Yk

Consequences for RpA (~ ratio of gluon distribution) :

Kozlov, Shoshi and Xiao (2006)

kdd

dN

kdd

dN

NR hXpp

hXpA

collpA

2

21

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• Scattering off a dense targetB-JIMWLK equations

• Scattering off a dilute targetdipole model equations

• Pomeron loop equationscombining the dense and dilute regimes

high-energy QCD evolution stochastic process

this implies: geometric scaling at intermediate energiesdiffusive scaling at higher energies

• Phenomenological consequencesnew scaling laws in DIS and particle production for large momenta and small xof strong interest in view of the LHC

Conclusions