Exclusive vs. DiffractiveVM production in DIS at small-x or off nuclei

Cyrille Marquet

Columbia University

based onF. Dominguez, C.M. and B. Wu, Nucl. Phys. A823 (2009) 99, arXiv:0812.3878

+ work in progress

Motivations

• proton vs. nucleus target

in e+p collisions at HERA, both can be measuredalready at rather low |t| (~0.5 GeV2), the diffractive process is considered a background

in e+A collisions at a future EIC/LHeC, at accessible values of |t|, the nucleus is broken upit is crucial to understand and quantify the transition from exclusive to diffractive scattering

predictions of what happens with nuclei work in progress

• low vs. high momentum transfer

the target is intact (low |t|)saturation models work well

)M,,()Q,,()M,Q,( 2V

22V

2 zrzrdzr V upper part described with the overlap function:

exclusive process diffractive processthe target has broken-up (high |t|)

BFKL Pomeron exchange works well

interaction at small :

)M,,( 2VzrV

)Q,,( 2zr

description of both within the same framework ? possible at low-xDominguez, C.M. and Wu, (2009)

22

22

Q

Q

W

Mx V

Outline

• Saturation and the Color Glass Condensatescattering off a high-energy hadron/nucleusVM production off the CGCthe McLerran-Venugopalan model

• The process ep → eVYunified formula (low and high t)comparison with HERA data

• The process eA → eVYthe Woods-Saxon averagingthree distinct momentum-transfer regimes

The saturation momentum

gluon kinematics

recombination cross-section

gluon density per unit areait grows with decreasing x

the saturation regime: for with

recombinations important when

for a given value of k², the saturation regime in a nuclear wave functionextends to a higher value of x compared to a hadronic wave function

McLerran and Venugopalan (1994)

• the CGC: an effective theory to describe the saturation regime

lifetime of the fluctuations

in the wave function ~

high-x partons ≡ static sources

low-x partons ≡ dynamical fields

the idea in the CGC is to take into account saturation via strong classical fields

• gluon recombination in the hadronic wave function

The Color Glass Condensate

gggggqqqqqqgqqq .........hadron CGC][hadron xD

small-x gluons as radiation field

),(,

z zFD cc

valence partonsas static random

color source separation between

the long-lived high-x partons

and the short-lived low-x gluons

CGC wave function

classical Yang-Mills equations

• the CGC wave function

the solution gives 3.03/12 ~),(Q xAAxs

the evolution of with x is a renormalization-group equation2

][x

Jalilian-Marian, Iancu, McLerran,Weigert, Leonidov, Kovner (1997-2002)

from , one can obtainthe unintegrated gluon distribution,

as well as any n-parton distributions

2][x

• the small-x evolution

in the A+=0 gauge

Scattering off the CGC

scattering of a quark:

• this is described by Wilson lines

dependence kept implicit in the following

in the CGC framework, any cross-section is determined by colorless combinations ofWilson lines , averaged over the CGC wave function ][][

2 SDS xx ][S

))()((1

1][ xyxy FFc

WWTrN

T

x : quark space transverse coordinatey : antiquark space transverse coordinate

the dipole scattering amplitude:qq

this is the most common averagefor instance it determines deep inelastic scattering

• the 2-point function or dipole amplitude

xTxy

VM production off the CGC• the diffractive cross section

overlap functions

amplitude

)Q,,( 2zr)M,,( 2

VzrV

][)2/)(,( xyyxyx TT qq

xqqqqiq

VYp

TTebdbdrdrddt

d)','(),(')M,Q,'()M,Q,('

4

1 )'.(222V

2*2V

222*

brbrrr bb

target average at the cross-section level:

contains both broken-up and intact events

conjugate amplitude

r : dipole size in the amplituder’ : dipole size in the conjugate amplitude

• the exclusive part

xqqxqqxqqqq TTTT )','(),()','(),( brbrbrbr

2.22

V22

*

),( )M,Q,( 4

1xqq

iqVpp

Tebdrddt

dbrr b

one needs to compute a 4-point function, possible in the MV model for

obtained by averaging at the level of the amplitude:

2][x

ep → eVY

The MV model

µ2 characterizes the density of color charges along the projectile’s path

with this model for the CGC wavefunction squared, it is possible to compute n-point functions

• a Gaussian distribution of color sources

2][

is the two-dimensional massless propagator

)()( )()'(),'(),( 22 zyzxyx GGzdzzzzz cddc

• applying Wick’s theorem

when expanding in powers of α and averaging,

all the field correlators can be expressed in terms of

)','(),( brbr qqqq TT),'(),( yx zz dc

the difficulty is to deal with the color structure

Fujii, Gelis and Venugopalan (2006)

Analytical results

qqqq TS 1

• the 4-point function

(using transverse positions and not sizes here)

the x dependence can also be consistently included, andshould be obtained from the BK equation (now available at NLO)

for now, we are just using models

linearizing, we recover the BFKL formula → high-t OKthe exclusive part is also contained → low-t OK

• recovering known limits

)(rFx

Exclusive vs. diffractive

• as a function of t

exclusive production:the proton undergoes elastic scatteringdominates at small |t|

diffractive production :the proton undergoes inelastic scatteringdominates at large |t|

exclusive→ exp. fall at -t < 0.7 GeV2

diffractive→ power-law tail at large |t|

• two distinct regimes

Dominguez, C.M. and Wu, (2009)

the transition point is where thedata on exclusive production stop

The eA → eVY case

exclusive production is called coherent diffractionthe nucleus undergoes elastic scattering, dominates at small |t|

intermediate regime (absent with protons)the nucleus breaks up into its constituents nucleons, intermediate |t|

then there is fully incoherent diffractionthe nucleons undergo inelastic scattering, dominates at large |t|

From protons to nuclei• qualitatively, one expects three contributions

averaged with the Woods-Saxon distribution

position of the nucleons

• the dipole-nucleus cross-section Kowalski and Teaney (2003)

Kowalski, Lappi and Venugopalan (2007)application for inclusive DIS off nuclei (F2):

how to bring nucleons in the picture ?

Hard diffraction off nuclei

in diffraction, averaging at the level of the amplitudecorresponds to a final state where the nucleus is intact

averaging at the cross-section levelallows the breakup of the nucleus into nucleons

• the Woods-Saxon averaging

Kowalski, Lappi, C.M. and Venugopalan (2008)

coherent diffraction→ steep exp. fall at small |t|breakup into nucleons→ slower exp. fall at 0.05 < -t < 0.7 GeV2

incoherent diffraction→ power-law tail at large |t|

• three regimes as a function of t:

next step: computation for vector mesons

results for t-integrated structure functions

Conclusions

• Vector meson production is an important part of the physics program at an eA collider

it allows to understand coherent vs. incoherent diffraction

• The CGC provides a framework for QCD calculations in the small-x regime

explicit calculations possible in the MV model for the CGC wave function

• VM production off the proton understood, preliminary results for the nucleus case

coherent diffraction→ steep exp. fall at small |t|breakup into nucleons→ slower exp. fall at 0.05 < -t < 0.7 GeV2

incoherent diffraction→ power-law tail at large |t|