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NNLO QCD Analysis of the Virtual Photon Structure
Functions
NNLO QCD Analysis of the Virtual Photon Structure
Functions
& Tsuneo Uematsu (Kyoto Univ. )
Talk at Radcor07 at GGIOctober 1-5, 2007
Takahiro Ueda (KEK)
Ken Sasaki (Yokohama National Univ.)
Yoshio Kitadono (Hiroshima/Kyoto Univ.)
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1. Introduction and Motivation
Two-photon process in e+e - collision
Photon structure functions
in the future linear collider ILC
Two-photon process
for the case of virtual photon ( ) target
as in the case of nucleon structure
functionsand
and
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e+e - collision
& QCD interactions
: Bjorken variable
: Mass squared of probe photon
: Mass squared of target photon
In the kinematic region:
structure fns. and perturbatively calculable !
Virtual-Photon KinematicsVirtual-Photon Kinematics
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in Perturbative QCD in Perturbative QCD Simple parton model
Point-like contributiondominates
Walsh-Zerwas (1973)(LO) (NLO)
Witten (1977)
Hadronic piece
Bardeen-Buras (1979)
(LO) (NLO) Walsh-TU (1981,1982)
: QCD scale parameter
OPE+RGE
lowest order in
Hadronic piece can also be dealt with perturbatively
Definite prediction of , its shape and magnitude, is possible
•For real photon target ( )
•For highly virtual photon target ( )
QCD improved parton modelGluck-Reya (1983),…
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Motivated by the calculation of 3-loop anomalous dimensions Vogt-Moch-Vermaseren (2004,2006)
Present MotivationPresent Motivation
We extend the analysis of virtual photon structure functions to NNLO ( )
Moch-Vermaseren-Vogt (2002, 2006)NNLO extension
K. Sasaki, T. Ueda and T.U., Phys. Rev.D75 (2007) 114009
For parton distributions inside virtual photon in the NNLO, see Ken Sasaki’s next talk
Here we also consider target mass effects to NNLO
Y. Kitadono, K. Sasaki, T. Ueda and T.U., in preparation
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Forward photon-photon scattering
cf. Structure tensor no spin averaged
Anti-symmetric part and
polarized photon structure functions
unpolarized photon structure functionsSymmetric part
Structure FunctionsStructure Functions
Spin-averaged Structure Tensor
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•
• The OPE near the light-cone
• Spin- twist-2 operators
light-cone
: over relevant ops.
(hadronic ops. + photon op.)
( -th moment) (dominant)
quark : (flavor singlet)
(flavor non-singlet)
gluon :photon :
QCD with massless quarks with flavors
2. Theoretical framework : OPE and RGE
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• Moment sum rule of
• Inverse Mellin transformation: -space -space
we take
for even
RG improved coefficients
Photon matrix element
Perturbatively calculable when
Solution of RG eq. for coefficient functions : QCD beta function: anomalous dimensionsExpand up to NNLO ( )
numerically inverted
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Anomalous dimension matrixAnomalous dimension matrix
Anomalous dimensions of hadronic operators
Mixing anomalous dimensions of photon-hadron ops.
4 x 4 matrix
1-loop anom. dim.
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Moment Sum Rule for
Moment Sum Rule for
• Summarized as
NNLO ( )
LO ( )
NLO ( )
for even
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• QCD -function
• Anomalous dimensions of hadronic operators
• Hadronic coefficient functions van Neerven-Zijlstra (1991,1992)
Tarasov-Vladimirov-Zarkov (1980)Larin-Vermaseren (1993)
1-loop 2-loop 3-loop
Moch-Vermaseren-Vogt (2004)
1-loop 2-loop 3-loop
tree 1-loop 2-loop
Renomalization Group parametersRenomalization Group parameters
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LO :
NLO :
Coefficients of the moment sum rule
Coefficients of the moment sum rule
2-loop
1-loop1-looptree tree
NNLO :
1-loop 2-loop 2-loop 1-loop
3-loop 3-loop 2-loop
Photon coefficient function
Mixing anomalous dimensions
Photon matrix elements
New evaluation needed
known
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• Mixing anomalous dimensions
No exact expressions for 3-loop yet But the compact parametrization of 3-loop photon-parton
splitting functions (and exact ) available (the deviation < 0.1 %)
1-loop
2-loop
3-loop
A.Vogt, S.Moch and J.Vermaseren Acta Phys.Polon. B37,683(2006)
We adopt this parametrization
published
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• Photon coefficient function
, are obtained by the replacement of group factors
[example]
2-loop1-loop
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– Unrenormalized gluon matrix elements of hadronic operators were calculated in -space
– We obtain , after taking moments,
renormalization and the replacement of group factors
Perturbatively calculable when
1-loop 2-loop
Matiounine-Smith-van Neerven (1998)
Photon matrix elements
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x
0 0.2 0.4 0.6 0.8 1
)2
/P2
ln
(Q 4 e
f 3
n
2F
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7 LO
NLO
NNLO
LO
NLO
NNLO
LO
NLO
NNLO
LO
NLO
NNLO
=200MeV 2=1GeV2 P2=30GeV2=4 Qfn
LO NLO NNLO
Numerical Plot3. Numerical Analysis
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Numerical Plot
x0 0.2 0.4 0.6 0.8 1
4 e
f 3
n
LF
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7 LO
NLO
LO
NLO
LO
NLO
=200MeV 2=1GeV2 P2=30GeV2=4 Qfn
LO NLO
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Sum rule : first momentSum rule : first moment
By taking the limit we get a sum rule
Note: and develop singularities at
which are cancelled by a factor
vanishing of anom. dim of energy-mom tensor
e.g.Corrections 8-10
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• For the virtual photon target, the maximal value of the Bjorken variable
due to the constraint In the nucleon case • We study the Target Mass Effects (TME) based on Operator Pr
oduct Expansion (OPE) taking into account the trace terms of the matrix elements of the operators.
• This amounts to consider Nachtmann moments. By using the orthogonality in the Gegenbauer polynomial we project out the definite spin contributions.
4. Target mass effects for virtual photon structures Y.Kitadono, K.Sasaki, T.Ueda and T.U.
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Nachtmann moments for
Inverting the moments
where
and
where
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where the four functions F, G, H and FL are defined as Mellin-inverted
moments :
with and defined as ordinary n-th moment given as
obtained to NNLO
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(1) F, G, H, FL Numerical Results
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(2)
at larger
x
No TME
TME
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(3) at larger
x
Note with TME
No TME
TME
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Comparison with existing experimental data Comparison with existing experimental data
: Effective photon structure fn.
PLB483 (2000) 373
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• The virtual photon structure function investigated in the kinematical region
• Definite predictions made up to NNLO ( )• NNLO corrections appear sizable at large x• We also analyzed the virtual photon structure
function to NLO ( )• The target mass effects (TME) investigated• TME becomes sizable at larger x. TME reduces
and enlarges at larger x.
5. Summary
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• Power corrections of the form due to higher-twist effects should also be studied
• Here we have treated active flavors as massless quarks Off course, surely we have to take into account mass effe
cts of heavy quark flavors• So far we have studied the NNLO QCD correction to the 1
st moment of We should also interested in full NNLO analysis of
and• Experimental confrontation in the future is anticipated
Future subjectsFuture
subjects
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Extra slides
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Moment Sum Rule for
LO ( )
NLO ( )
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LO :
NLO :
Coefficients of the moment sum rule
Coefficients of the moment sum rule
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