the high-energy limit of dis and ddis cross-sections in qcd cyrille marquet service de physique...
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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