The high-energy limit of DIS and
DDIS cross-sections in QCD
based on
Y. Hatta, E. Iancu, C.M., G. Soyez and D. Triantafyllopoulos, hep-ph/0601150, NPA in press
Cyrille Marquet
Service de Physique Théorique
CEA/Saclay
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 DDIS
high-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
• Conclusions
towards the LHC
Kinematics and notations
k’
see the lectures by Christophe Royon
some events
k
are diffractive
Q >> QCD
size resolution 1/Q
photon virtuality Q2 = - (k-k’)2 > 0
*p collision energy W2 = (k-k’+p)2
Bjorken x:
x
diffractive mass
of the final state:
MX2 = (p-p’+q)2
Q2
W  Q2
2
the *p total cross-section:DIS(x, Q2)
high-energy limit: x << 1

p’
p

Q2
M X  Q2
2
xpom = x/
rapidity gap:  = ln(1/xpom)
the diffractive cross-section: DDIS(, xpom, Q2)
high-energy limit: xpom<<1
The Good and Walker picture
realized in perturbative QCD
Description of diffractive events
Good and Walker (1960)
picture in the target rest frame:
after the collision:
hadronic particules
hadronic projectile before the collision
target which
remains intact
P  qqq  qqqg  ...  qqq......ggggg

the final state F  Sˆ P 
en
: eigenstates of the interaction
which can only scatter elastically
P  cn en
n
elastic scattering amplitude: Ael  i P 1Sˆ P  i
 tot  2 T
P
 el  T
2
P
 diff  T
n
is then some multi-particule state
Sˆ en  Sn en
n
S
2
P
c T
2
n
n
inel
diff

n
1  Sn
i T
 T
2
P
P
 T
2
P
cn en
Eigenstates of the QCD S-matrix
In general, we don’t know
en , cn and Sn
Degrees of freedom of perturbative QCD: quarks and gluons
Eigenstates?
The interaction should conserve their spin, polarisation, color, momentum …
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:
colorless quark-antiquark pair:
ij q(x, i); q (y, j)
colorless quark-antiquark-gluon triplet:
x, y, z : transverse coordinates
Tjia q(x, i); q (y, j); g(z, a)
In the large-Nc limit, further simplification: the eigenstates are only made of dipoles
en  (x, y)
,
(x1, y1);...; (xN, y N )
Sˆ (x, y)  Sxy (x, y)
other quantum numbers
aren’t explicitely written
the eigenvalues depend only on the transverse coordinates
The virtual photon in
the dipole basis (I)
For an arbitrary hadronic projectile, we don’t know
cn
For the virtual photon in DIS and DDIS, we do know them: the wavefunction of
the virtual photon  *  qq  qq g  ... is computable from perturbation theory.
For instance, the qq component:
 *  d 2k  (k, Q2) q(k); q (k)
k : quark transverse momentum
wavefunction computed from QED at lowest order in em
with transverse coordinates:
~
 *  d xd y  (xy, Q2) q(x); q (y)
2
2
k
-k
x
y
x : quark transverse coordinate
y : antiquark transverse coordinate
To describe higher-mass final states, we need to include states with more dipoles
The virtual photon in
the dipole basis (II)
With the qq g component:
x

~
2
2
  d xd y  (xy, Q ) 1  gS2 CF Nc d 2z   q(x); q (y)


xz  yz
2 z  T a q(x); q (y); g(z, a) 

g
d
S
2
2

(xz) (yz)

from QCD at order g
z
1/2
*
2
2
y
S
Large-Nc limit: we include an arbitrary number of gluons in the photon wave function
~
 *  d 2xd2 y  (xy, Q2) d 2z0...d 2zN (z0x)(zN y)
…
x
z1
zN-1
y
N 1
cN (z0,..., zN; Y ) (z0, z1),..., (zN 1, zN )
In principle, all
N-1 gluons emitted at transverse coordinates
cn are computable from perturbation theory
z1,..., zN 1
 N dipoles
(z0, z1),..., (zN 1, zN )
Formula for DIS(x, Q2)
Via the optical theorem: DIS(x, Q2) = 2 Im[Ael(x, Q2)] :
2
~
2
2
2
 DIS (x, Q²)  2 d xd y  (xy, Q )  d 2z1...d 2zN 1
average over the
target wave function
N 1
Pn  cn
2
PN x, z1,..., zN 1, y; Y0  1Sxz1 Sz1z2...SzN 1y
Y Y0
Y = ln(1/x): the total rapidity
Y0 specifies the frame in which
Y0
the cross-section has been calculated,
therefore DIS is independent of Y0
arbitrary
Y
Two consequences:
- One can use the simplest expression (obtained for Y0 = 0)
2
~
2
 DIS (x, Q²)  2 d xd y  (xy, Q ) Txy
2
2
Y
with Txy  1  Sxy
- One can derive an hierarchy of evolution equations for the dipole amplitudes
Tz1z2Tz3z4...
see the lectures by Gregory Soyez
Txy Y
Y
Formula for DDIS(, xpom, Q2)
For the diffractive cross-section one obtains:
2
~
2
2
2
 DDIS (, xpom, Q²)  d xd y  (xy, Q )  d 2z1...d 2zN 1
new factorization
formula
N 1
PN x, z1,..., zN 1, y; ln( 1/ ) 1Sxz1 Sz1z2...SzN 1y
ln(1/ )
now we need
all the Tz z Tz z ...
1 2
 = ln(1/xpom)
in the BFKL approximation, we recover the formulae of Bialas and Peschanski (1996)
the
PN are easily computable with a Monte Carlo
Salam (1995)
but we need to solve the hierarchy (or the equivalent Langevin equation)
see the lectures by Gregory Soyez
3 4
2

Y
High-energy QCD predictions for
the DIS and DDIS cross-sections
The dipole scattering amplitudes
As explained in the lectures by G. Soyez and S. Munier:
•
The dipole amplitudes Txy Y , Tz1z2Tz3z4... , … should be obtained from the recently
Y
derived Pomeron-loop equation
Mueller and Shoshi (2004); Iancu and Triantafyllopoulos (2005); Mueller, Shoshi and Wong (2005)
•
This is a Langevin equation, for an event-by-event dipole amplitude TY (x, y) ,
the physical amplitudes Txy Y, … are them obtained from TY (x, y) after proper
averaging
•
Although one cannot solve this equation yet, some properties of the solutions have
been obtained, exploiting the similarities between the Pomeron-loop equation and
the s-FKPP equation well-known in statistical physics
Iancu, Mueller and Munier (2005)
Brunet, Derrida, Mueller and Munier (2006)
C.M., R. Peschanski and G. Soyez (2006)
The different high-energy regimes
at higher energies, a new scaling
takes over : diffusive scaling due to
the fluctuations which have been
amplified as the energy increased
in an intermediate energy regime,
it predicts geometric scaling:
HERA
Txy Y  f (xy)2 QS2 (Y )
it seems that HERA is probing
the geometric scaling regime
Q2 ~ 1 (xy)2
In the diffusive scaling regime, saturation is the relevant physics
up to momenta much higher than the saturation scale
The geometric scaling regime
Txy Y  f (xy)2 QS2 (Y )
this is seen in the data with   0.3
saturation models with the features
of this regime fit well F2 data
Golec-Biernat and Wüsthoff (1999)
Bartels, Golec-Biernat and Kowalski (2002)
Iancu, Itakura and Munier (2003)
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)
Stasto, Golec-Biernat and Kwiecinski (2001)
Txy
The diffusive scaling regime
Y
Blue curves: different realizations of TY (x, y)
Y2 >Y1
Red curve: the physical amplitude Txy
DY 2
DY 1
Y
 : average speed
Y1
D : diffusion coefficient
ln(xy)2 Q02 
 Y 2Y 1
The dipole amplitude is a function of one variable:

Txy Y  g ln(xy)2 QS2 (Y ) DY

We even know the functional form for ln(xy)2 QS2   DY :

Txy Y  1 Erfc ln(xy)2 QS2 (Y ) DY
2


Tx1y1Tx2y2  1 Erfc ln(xy)2 QS2 (Y ) DY
2
Y

the smallest size controls the correlators
Consequences on the observables
d DIS  2 dr2~(r, Q2) 2 T(r)

Y
d 2b
d DDIS   dr2~(r, Q2) 2 T(r) 2

Y
d 2b
dipole size r  xy
diffusive scaling total
diff
geometric scaling regime:
geometric scaling total
diff
1 Q  r  1 QS
DDIS dominated by semi-hard sizes
integrand
DIS dominated by relatively hard sizes
r ~ 1 QS
diffusive scaling regime:
both DIS and DIS are dominated
by hard sizes r ~ 1 Q
yet saturation is the relevant physics
r ~1 Q
dipole size r
r ~ 1 QS
Some analytic estimates
Analytical estimates for DIS(x, Q2) in the diffusive scaling regime:
valid for
Dln1 x  lnQ² QS2  Dln1 x
d DIS  F Dln 1 x exp( Z 2)
d 2b
Z2
with F  Ncem
e2
2  f
12
and
f
ln Q² Q S2
Z
Dln 1 x
And for the diffractive cross-section (integrated over  at fixed x)
d DDIS  F Dln 1 x exp( 2Z 2)
d 2b
4
Z3
Physical consequences: up to momenta Q² much bigger than the saturation scale
•
•
•
the cross-sections are dominated by small dipole sizes r ~ 1 Q
there is no Pomeron (power-like) increase
the diffractive cross-section is dominated by the scattering of the quark-antiquark
component
Conclusions and Outlook
•
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
Recovering known results
- The diffractive cross-section for  close to 1:
~
2
 DDIS ( 1, x, Q²)  d 2 xd 2 y  (xy, Q2) Txy
~
~
 (r, Q2)
 (r, Q2)
2
Y
Y
same Txy Y for total and diffractive cross-section
(and also exclusif processes, jet production…)
- The diffractive cross-section for  smaller 1, but not too small
(with only the qq and qq g components):
2
~
2
2
2
 DDIS (, xpom, Q²)  d xd y  (xy, Q )  Txy

  ln( 1/ ) 

d 2z
(xy)2
 T
xz
2 (xz)2(zy)2 

 Tzy

2

 TxzTzy

 T
2

xy


 

2
- When making the assumption TxzTzy Y  Txz Y Tzy Y , we also
recover the evolution equation for d  DDIS (, xpom, Q²)
Kovchegov and Levin (2000)
d