Particule production and saturation

Cyrille MarquetSPhT, Saclay

ISMD 2005, Kromeriz, Czech Republic

• IntroductionBjorken limit and Regge limit of perturbative QCD

• High-energy QCD (the Regge limit) and saturationscattering matrix for high-energy partonsqq dipoles, gg dipoles, multipoles, … observables at small-x

• HERA Phenomenologyforward jetsvector mesons, DVCSdiffractive jets

• Conclusion and outlook

Contents

Introduction

The Bjorken limit of pQCDConsider a collision of hadronic particules with a center-

of-mass energy W and a hard scale Q >> QCD

• The Bjorken limit: Q² , W² with Q²/W² fixed ( xBj in DIS)

• Operator product expansion• At leading twist:

collinear factorizationgluon distributionDGLAP evolution

• Higher twists suppressed by powers of Q²

• Scattering amplitudes decrease with increasing Q²

Transverse view of

the proton in DIS

The Regge limit of pQCD

• The Regge limit:W² with Q² fixed (xBj 0 in DIS)

• One has to introduce a new scale:the saturation scale Qsat(W²)

Consider a collision of hadronic particules with a center-of-mass energy W and a hard scale Q >> QCD

• If W is such that Qsat(W²) < Q,no higher-twist effectskT-factorization, unintegrated gluon distribution, BFKL evolutionscattering amplitudes increase with increasing W

• If W is such that Qsat(W²) > Q, density effects are important (higher-twist)need to go beyond the OPE,strong gluon fields, CGC, saturation …scattering amplitudes approach unitarity limit

Qsat(W²)

High-energy QCD(the Regge limit)

• For an incoming quark of color i, at transverse position x:

The action of the S matrix is

Scattering matrix for high-energy partons

target...,, iin x

)/1(target...,,)( 2WOjWS ijF

jinout xx

• For a gluon: the same with the adjoint Wilson line WA

• Wilson lines WF and WA: the degrees of freedom of high-energy QCD

),(exp)( xATdxigPW aaSF xx Y = log(W²) : total rapidity

Tqq(x, x’,Y): the scattering amplitude of a qq dipole off the target:

Tqq(x, x’; y, y’,Y): the scattering amplitude of two qq dipoles:

Tgg(x, x’,Y): the scattering amplitude of a gg dipole:

and more generally any multipole

Dipoles and multipoles

t

abA

bF

aF zWTWTWTr )....(....))()'(( xx

tFF

cqq WWTr

NYT ))()'((11),',( xxxx

tAA

cgg WWTr

NYT ))()'((

111),',( 2 xxxx

tAFFF

cqq WWTrWWTr

NYT ))()'(())()'((11),',;',( 2

)2( yyxxyyxx

(2)

• Instead of directly the Wilson lines, colorless combinations arise as the degrees of freedom:

• We have denoted target.target. t

Simplest illustration : DIS

r: transverse size of the dipole

b: impact parameter

z: longitudinal momentum fraction of the quark

qqzrrdzbdd )Q,,( 222*

2* pSfd f

does not depend on z in the high-energy limit

the qq dipole amplitude Tqq(r, b, Y) appears

2

22 )Q,,()Q,( zrdzr

Y: total rapidity

);,()Q,( 222 YbrTbdrrd qqDIS

Observables at small-x

• Particule production phenomenology: jet cross-sections, heavy-quark production, diffractive vector mesons production, di-lepton production, multiplicities …have been studied in this high-energy QCD framework

The same dipole amplitudes enter in the formulation of

inclusive, diffractive, exclusive cross-sections

Y[A], and therefore Tqq, Tgg, Tqqg … are mainly non-perturbative, however the Y evolution is computable (in the leading logarithmic approximation)

for more on these equations, see Larry McLerran’s talk tomorrow

and Robi Peschanski’s talk sunday

)()( YTKYTdYdH

dYd

qqqqYY

• More generally, any cross-section is a function of Tqq, Tgg, Tqqg …

• The more exclusive the final state is, the more complicated the corresponding multipoles are

• How does one compute Tqq, Tgg, Tqqg …? With ][][][ AfADAAf Yt

HERA phenomenologyfor particule production

* -proton collisions

Forward-jet production• proton + * forward-jet + X

photon virtuality: Qjet transverse momentum: kwith Q k » QCD and xBj <<1, small-x effets expected

• photon qq dipole and jet emission gg dipole

C.M., R. Peschanski and C. Royon, Phys. Lett. B 599 (2004) 236

C.M. and C. Royon, in preparation

• the different observables are well described by BFKL and saturation models

• NLOQCD is a factor 2 below the data at small-x

data: see Leif Joensson’s talk later today

Diffractive vector-meson productionS. Munier, A. Stasto and A. Mueller, Nucl. Phys. B 603 (2001) 427

),()Q,,()Q,( 22 zrzrdzr V

22.22 )Q,();,(

161 reYbrTbdrd

dtd biq

qq

)Q,,( 2zr ),( zrV

t = -q²

• the S-matrix is extracted from the data for • S(1/r 1Gev, b 0, x 5.10-4) 0.6

HERA is entering the saturation regime

biqqq eYbrTbd .2 );,();,( YbrT qq or

need a parametrization for

Diffractive J-Psi production (1)H. Kowalski and D. Teaney, Phys. Rev. D 68 (2003) 114005

dipole amplitude: ansatz for the b dependence

),()Q,,()Q,( /22 zrzrdzr PsiJ

22.22 )Q,();,(

161 reYbrTbdrd

dtd biq

qq

))()/1,(exp(1);,( 22 bTrxxgarYbrT qq 2

)( bebT Y = log(1/x)

Diffractive J-Psi production (2)E. Gotsman, E. Levin, M. Lublinsky, U. Maor and E. Naftali, Acta Phys. Polon. B34 (2003) 3255

• dipole amplitude obtained from a numerical solution of the BK equation

• ansatz for the b dependence in the initial condition

2222 );,()Q,( YbrTrrdbd qq

),()Q,,()Q,( /22 zrzrdzr PsiJ

Deeply Virtual Compton Scattering

• they compute

• they assume

L. Favart and M. Machado, Eur. Phys. J C29 (2003) 365Eur. Phys. J C34 (2004) 429

Bt

te

dtd

dtd

0

2222

0)Q,();,(

161 rYbrTbdrd

dtd

qqt

0

1

t

dtd

B

Bartels Golec-biernatKowalski model

• to do better and compute , one needs a model for

• need an analysis of the BK equation at non zero momentum transfer:

biqqq eYbrTbd .2 );,(

dtd

with t = -q²

C.M. and G. Soyez, Nucl. Phys. A, in pressC.M., R. Peschanski and G. Soyez, Nucl. Phys. A 756 (2005) 399

))/1,(exp(1 22 rxxgar

Y = log(1/x)

• Diffractive photon dissociation is the dominant contribution to the diffractive cross-section diff at large MX in DIS:

elas: involves the qq dipole fluctuation, dominant for small-mass final states dissoc: involves higher Fock state fluctuations: qqg, …dominant for large-mass final states

Diffractive jet production (1)

= Q²/MX² <<1

dissocelasdiff

rapidity gap

= log(1/xpom)xpom<<1target

proton

k: transverse momentum of the final-state gluon

C. M., Nucl. Phys. B 705 (2005) 319

K. Golec-Biernat and C. M., Phys. Rev. D 71 (2005) 114005

• 1/k0: typical size at which the S-matrices are cut off

observable strongly sensitive to unitarity effects

• measuring could select between saturation and Regge-based models

kddMd

X 2

0 k

modeldependent

kddMdk

X

dissoc2

2

k²1/k²

modelindependent

modelindependent

k0

Tqq and Tqq

(2)

Diffractive jet production (2)kmax/QS = independent of Q², QS

1.5

saturation predictions for HERA:

RHIC phenomenologysee Larry McLerran’s talk tomorrow

quark-antiquark pair productionsee Hiro Fujii’s talk sunday

recent review on particule production and saturation at RHIC: J. Jalilian-Marian and Y. Kovchegov, hep-ph/0505052

• Particule-production cross-sections are sensitive to the small-x regime of QCD they contain important complementary information w.r.t. DIS for Tqq but also for Tgg, Tqqg, … on impact parameter/momentum transfer dependence

• Diffractive vector meson production at HERA: saturation models with ansatz for the impact parameter profile work quite well but that is not evidence for saturation need to start working with the momentum transfer

• Jet production in diffraction at HERA: great place to look for saturation effect can distinghuish between soft models and saturation

Conclusions

• Universality of Tqq:there are several parametrizations for Tqq

but could we describe everything that Tqq should describe with only one? new global analysis

• Has RHIC really provided evidence for saturation? waiting for the LHCor listen to Larry McLerran tomorrow

Outlook

RHIC phenomenologysee also Larry McLerran’s talk tomorrow

see recent review: J. Jalilian-Marian and Y. Kovchegov, hep-ph/0505052

R. Baier, A. Kovner and U. Wiedemann, Phys. Rev. D 68 (2003) 054009D. Kharzeev, Y. Kovchegov and K. Tuchin, Phys. Rev. D 68 (2003) 094013E. Iancu, K. Itakura and D. Triantafyllopoulos, Nucl. Phys. A 742 (2004) 182J.P. Blaizot, F. Gélis and R. Venugopalan, Nucl. Phys. A 743 (2004) 13J.Albacete, N. Armesto, A. Kovner, C. Salgado and U. Wiedemann, Phys. Rev. Lett 92 (2004) 082001

Nuclear modification factor in deuteron-gold collisions (1)

);,()/1log()(1 20

/1

022

beamgg

gXdA

brTbdr

rr

rkrJdrkkdd

dN

kdddN

kdddN

NR hXpp

hXdA

colldA

2

21

with the parton-level cross-section

predictions with a toy-model for Tgg and with a numerical solution of the BK equation

Nuclear modification factor in deuteron-gold collisions (2)

first comparisons to the data:

D. Kharzeev, Y. Kovchegov and K. Tuchin,Phys. Lett. B 599 (2004) 23D. Kharzeev, E. Levin and M. Nardi, Nucl.Phys. A 747 (2005) 609

A. Dumitru, A. Hayashigaki and J. Jalilian-Marian, hep-ph/0506308

recent work:

shows the importance

of both x and DGLAP

evolutions

shows the importance

of the quark component

Azimutal correlationsD. Kharzeev, E. Levin and L. McLerran, Nucl. Phys. A 748 (2005) 627

J. Jalilian-Marian and Y. Kovchegov, Phys. Rev. D 70 (2004) 114017N. Nikolaev, W. Schäfer, B. Zakharov and V. Zoller, hep-ph/0504057R. Baier, A. Kovner, M. Nardi and U. Wiedemann, hep-ph/0506126

but: correlators with product of up to four Wilson lines enter in the formulation

of the cross-section

preliminary data:predictions using kT-factorization assumption

Other Observables• Dilepton production

electromagnetic probe very clear signal, no fragmentation functionbut need data

• Heavy quark productionsee Hiro Fujii’s talk sunday

N. Armesto and M. Braun, Eur. Phys. J C22 (2001) 351B. Kopeliovich and A. Tarasov, Nucl. Phys. A 710 (2002) 180K. Tuchin, Phys. Lett. B 593 (2004) 66N. Nikolaev and W. Schäfer, Phys. Rev. D 71 (2005) 014023J.P. Blaizot, F. Gélis and R. Venugopalan, Nucl. Phys. A 743 (2004) 57

B. Kopeliovich, J. Raufeisen and A. Tarasov, Phys. Lett. B 503 (2001) 91F. Gélis and J. Jalilian-Marian, Phys. Rev. D 66 (2002) 094014M. Betemps, M. Gay Ducati, M. Machado and J. Raufeisen, Phys. Rev. D 67 (2003) 114008R. Baier, A. Mueller and D. Schiff, Nucl. Phys. A 741 (2004) 358