The high-energy limit of DIS and DDIS cross-sections in QCD

Cyrille Marquet

Service de Physique Théorique CEA/Saclay

based onY. Hatta, E. Iancu, C.M., G. Soyez and D. Triantafyllopoulos, hep-ph/0601150, NPA in press

Contents• Introduction

kinematics and notations

• The Good and Walker picture realized in pQCD- eigenstates of the QCD S-matrix at high energy: color dipoles- the virtual photon expressed in the dipole basis- inclusive and diffractive cross-sections in terms of dipole amplitudes

• High-energy QCD predictions for DIS and DDIShigh-energy QCD: see Kharzeev, Venugopalan, McLerran, Soyez, Motyka, Peschanski and Munier

- the geometric scaling regime - the diffusive scaling regime- consequences for the inclusive and diffractive cross-sections

• Conclusionstowards the LHC

p p’

diffractive massof the final state:

MX2 = (p-p’+q)2

some events

are diffractive

see the lectures by Christophe Royon

Kinematics and notations

photon virtuality Q2 = - (k-k’)2 > 0

*p collision energy W2 = (k-k’+p)2

22

2

Q

Q

Wx

the *p total cross-section:DIS(x, Q2)

high-energy limit: x << 1

Bjorken x:22

2

Q

Q

XM xpom = x/

rapidity gap: = ln(1/xpom)

high-energy limit: xpom<<1

the diffractive cross-section: DDIS(, xpom, Q2)

k

k’

size resolution 1/Q Q >> QCD

The Good and Walker picturerealized in perturbative QCD

Description of diffractive events

hadronic projectile before the collisionafter the collision:

hadronic particules

target which

remains intact

picture in the target rest frame:

gggggqqqqqqgqqqP .........

n

n

n ecP n

n

nn ecSPSF ˆ: eigenstates of the interaction

which can only scatter elastically

ne

nnn eSeS ˆ

the final state

is then some multi-particule state

Ptot T2 2

Pel TPdiff T 2 22

PP

ineldiff TT

elastic scattering amplitude: Pn

nnel TiTciPSPiA 2ˆ1nS1

Good and Walker (1960)

Eigenstates of the QCD S-matrix

Eigenstates?The interaction should conserve their spin, polarisation, color, momentum …

Degrees of freedom of perturbative QCD: quarks and gluons

In general, we don’t know , andne nc nS

In the high-energy limit, eigenstates are simple, provided a Fourier transform of the

transverse momenta: color singlet combinations of quarks and gluons. For instance:

),();,( jqiqij yx),();,();,( agjqiqT a

ji zyx

colorless quark-antiquark pair:

colorless quark-antiquark-gluon triplet:

x, y, z : transverse coordinates

In the large-Nc limit, further simplification: the eigenstates are only made of dipoles

),( yxne other quantum numbersaren’t explicitely written

),(),(ˆ yxyx xySS

),();...;,( 11 NN yxyx,

the eigenvalues depend only on the transverse coordinates

The virtual photon inthe dipole basis (I)

For the virtual photon in DIS and DDIS, we do know them: the wavefunction ofthe virtual photon is computable from perturbation theory.

...* gqqqq

For an arbitrary hadronic projectile, we don’t know nc

wavefunction computed from QED at lowest order in em

)();()Q,( 22* kkk qqkd k : quark transverse momentum

k

-k

x

y

For instance, the component:qq

)();()Q,(~

222* yxyx qqyxdd

with transverse coordinates:

x : quark transverse coordinatey : antiquark transverse coordinate

To describe higher-mass final states, we need to include states with more dipoles

y

z

x

The virtual photon inthe dipole basis (II)

With the component:gqq

)();(1)Q,(

~ 2/12

22222* yxyx qqzdNCgyxdd cFS

),();();(2 agqqTzdg a

S zyx22 )()( zyzy

zxzx

from QCD at order gS

x

y

z1

zN-1

… In principle, all are computable from perturbation theorync

Large-Nc limit: we include an arbitrary number of gluons in the photon wave function

),(),...,,();,...,( 1100 NNNN Yc zzzzzz

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

)()(...)Q,(~

02

02

1

222* yzxzyx NNN

zdzdyxdd

Formula for DIS(x, Q2) Via the optical theorem: DIS(x, Q2) = 2 Im[Ael(x, Q2)] :

12

12

1

2222 ...)Q,(

~2²)Q,( N

NDIS zdzdyxddx yx

0

1211...1;,,...,, 011

YYNN

NSSSYP

yzzzxzyzzx

Y0 specifies the frame in whichthe cross-section has been calculated,

therefore DIS is independent of Y0

Y = ln(1/x): the total rapidity

average over the target wave function

Two consequences:

- One can use the simplest expression (obtained for Y0 = 0)

YDIS Tyxddx xyyx 2

222 )Q,(~

2²)Q,( xyxy ST 1with

- One can derive an hierarchy of evolution equations for the dipole amplitudes

YTxy

YTT ...

4321 zzzz see the lectures by Gregory Soyez

Y

Y0

arbitrary

2nn cP

Formula for DDIS(, xpom, Q2)

12

12

1

2222pom ...)Q,(

~²)Q,,( N

NDDIS zdzdyxddx yx

For the diffractive cross-section one obtains:

2

111211

...1)/1ln(;,,...,,

yzzzxz

yzzxN

SSSP NN

ln(1/ )

YTT ...

4321 zzzz

now we needall the

= ln(1/xpom)

new factorizationformula

the are easily computable with a Monte CarloNPbut we need to solve the hierarchy (or the equivalent Langevin equation)

Salam (1995)

see the lectures by Gregory Soyez

in the BFKL approximation, we recover the formulae of Bialas and Peschanski (1996)

High-energy QCD predictions forthe DIS and DDIS cross-sections

The dipole scattering amplitudes

• The dipole amplitudes , , … should be obtained from the recentlyderived Pomeron-loop equation

• This is a Langevin equation, for an event-by-event dipole amplitude ,the physical amplitudes , … are them obtained from after properaveraging

• Although one cannot solve this equation yet, some properties of the solutions havebeen obtained, exploiting the similarities between the Pomeron-loop equation andthe s-FKPP equation well-known in statistical physics

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

As explained in the lectures by G. Soyez and S. Munier:

),( yxYT),( yxYT

YTxy

YTT ...

4321 zzzz

YTxy

Iancu, Mueller and Munier (2005)

C.M., R. Peschanski and G. Soyez (2006)

Brunet, Derrida, Mueller and Munier (2006)

The different high-energy regimes

)(Q)( 22 YfT SYyxxy

in an intermediate energy regime,it predicts geometric scaling:

HERA

it seems that HERA is probing

the geometric scaling regime

at higher energies, a new scalingtakes over : diffusive scaling due to

the fluctuations which have beenamplified as the energy increased

In the diffusive scaling regime, saturation is the relevant physics

up to momenta much higher than the saturation scale

22 )(1~Q yx

Stasto, Golec-Biernat and Kwiecinski (2001)

The geometric scaling regime )(Q)( 22 YfT SY

yxxy

this is seen in the data with 0.3

saturation models with the features

of this regime fit well F2 data

and they give predictions which describe accurately a number of observables at

HERA (F2D, FL, DVCS, vector mesons)

and RHIC (nuclear modification factor in d-Au)

Golec-Biernat and Wüsthoff (1999)Bartels, Golec-Biernat and Kowalski (2002)

Iancu, Itakura and Munier (2003)

The diffusive scaling regime

D : diffusion coefficient

Blue curves: different realizations of

Red curve: the physical amplitude

),( yxYT

YTxy

: average speed

YTxy

Y1

Y2 >Y1

1DY 2DY

12 YY 2

02 Q)(ln yx

DYYgT SY)(Q)(ln 22yxxy

The dipole amplitude is a function of one variable:

We even know the functional form for : DYS 22 Q)(ln yx

the smallest size controls the correlators

DYYErfcT SY)(Q)(ln

21 22yxxy

DYYErfcTT SY

)(Q)(ln21 22

2211 yx

yxyx

Consequences on the observables

YDIS rTrdrbd

d )()Q,(~

22

222

yxr

2222

2 )()Q,(~

YDDIS rTrdrbd

d

geometric scaling regime:

DIS dominated by relatively hard sizes

DDIS dominated by semi-hard sizes

Sr Q1~

Sr Q1Q1

totaldiff

inte

gran

d

totaldiffgeometric scaling

diffusive scaling

Sr Q1~Q1~rdipole size r

dipole size

both DIS and DIS are dominatedby hard sizes

diffusive scaling regime:

Q1~ryet saturation is the relevant physics

Some analytic estimatesAnalytical estimates for DIS(x, Q2) in the diffusive scaling regime:

valid for

2

2

2

)exp(1ln

ZZ

xDFbd

d DIS

f

femc eNF 2212

xD

Z S

1lnQ²Qln 2

3

2

2

)2exp(1ln

4 ZZ

xDFbd

d DDIS

And for the diffractive cross-section (integrated over at fixed x)

xDxD S 1lnQ²Qln1ln 2

with and

Physical consequences: up to momenta Q² much bigger than the saturation scale

• the cross-sections are dominated by small dipole sizes • there is no Pomeron (power-like) increase• the diffractive cross-section is dominated by the scattering of the quark-antiquark

component

Q1~r

• QCD in the high-energy limit provides a realization of the Good and Walker picture, and allows to make accurate QCD predictions for diffractive observables

• From the recently-derived Pomeron-loop equation, a new picture emerges for DIS.

• In an intermediate energy regime: geometric scaling- inclusive and diffractive experimental data indicate that HERA probes this regime

• At higher energies: diffusive scaling - up to values of Q² much higher than the saturation scale , saturation is the relevant physics- cross-sections are dominated by rare events, in which the photon hits a black spot, that he sees dense (at saturation) at the scale Q²- the features expected when are extended up to much higher Q²

• Towards the LHC- the energy there may be high enough to see the diffusive scaling regime- we want to say more before LHC starts: determination of and D ? work in progress

Conclusions and Outlook

Recovering known results

22222 )Q,(

~²)Q,1,(

YDDIS Tyxddx xyyx

- The diffractive cross-section for close to 1:)Q,(

~2r )Q,(

~2r

Y

22

22

22

)()()(

2)/1ln(

xyzyxzzyxzyzzxyx

TTTTTzd

2

xy

2222pom )Q,(

~²)Q,,(

TyxddxDDIS yx

- The diffractive cross-section for smaller 1, but not too small(with only the and components):qq gqq

- When making the assumption , we also recover the evolution equation for ²)Q,,( pomx

dd

DDIS

YYYTTTT zyxzzyxz

Kovchegov and Levin (2000)

same for total and diffractive cross-section(and also exclusif processes, jet production…)

YTxy