Parton distributions in the virtual photon up to NNLO in QCD and - PowerPoint PPT Presentation

Parton distributions in the virtual photon up to NNLO in QCD and factorization scheme dependence Ken Sasaki (Yokohama National U.) In collaboration with T. Uematsu (Kyoto U.) T. Ueda (KEK) RADCOR 2007 at GGI, Florence

Introduction & Motivation Deep-inelastic electron-nucleon scattering Nucleon structure functions :Bjorken variable :mass squared of the probe photon Parton distribution functions (PDFs) inside a nucleon probe  necessary for the analysis of semi-inclusive reactions • factorization-scheme dependent target Some assumptions made to extract PDFs from data DIS Rather difficult to see the features of factorization schemes 1/26

Introduction & Motivation Future linear collider experiment (e.g. ILC) Two-photon process Viewed as a deep-inelastic electron-photon scattering We can study the structures of photon probe large Highly virtual photon target Deep inelastic scattering target pQCD gives definite predictions for • pdf in the virtual photon :mass squared of the target photon • A good playground to see the scheme-dependence of pdf 2/26

in Perturbative QCD For highly virtual photon target ( ) : QCD scale parameter (Uematsu’ talk) (LO) (NLO) (NNLO) Hadronic piece can also be dealt with perturbatively Definite prediction of , its shape and magnitude, is possible LO, NLO Uematsu-Walsh (1981,1982) (OPE+RG method) NNLO Ueda-Uematsu-Sasaki Physical Review D75, 114009 (2007) hep-ph/0703296 Motivated by the calculation of 3-loop anomalous dimensions Vogt-Moch-Vermaseren (2004,2006) 3/26

QCD-improved Parton Model Approach :Quark DF with helicity of the virtual photon ( ) : Gluon DF with helicity : Photon DF with helicity Unpolarized PDFs In LO in QED coupling : Photon DF does not evolve Singlet quark DF: Non-singlet quark DF: 4/26

QCD-improved Parton Model Approach Parton distribution functions in the virtual photon Factorization: DGLAP evolution equation :Splitting fn. of photon into quark and gluon :Splitting fn. of quark and gluon 5/26

QCD-improved Parton Model Approach Taking moments: Expanding in powers of Introducing 6/26

QCD-improved Parton Model Approach Solution to DGLAP evolution eq. Initial conditions (factorization scheme dependent) finite photon matrix element Splitting functions anomalous dimensions When parameters calculated in scheme are used, we obtain PDFs in the virtual photon up to NNLO in scheme 7/26

QCD-improved Parton Model Approach Solution 8/26

Moment Sum Rule for Since LO ( ) for even NLO ( ) NNLO ( ) 9/26

Moment Sum Rule for LO : NLO : 10/26

Moment Sum Rule for NNLO : 11/26

Moment Sum Rule for NNLO : 12/26

Factorization-scheme independent combinations Since is the moments of are physically measurable quantities, each term, , is factorization-scheme (FS) independent In the NS sector, the following combinations are FS independent Bardeen-Buras (1979) Nucleon structure func 13/26

PDFs in scheme ---- ----- ----- In the expression of , put and 14/26

PDFs in scheme Non-singlet quark 15/26

PDFs in scheme Singlet quark 16/26

PDFs in scheme Gluon 17/26

Factorization scheme dependence Scheme-dependent PDFs scheme scheme independent DIS scheme Altarelli-Ellis-Martinelli(1978) is given by the naïve parton model expression to all orders (hadronic coefficient functions are the same as the tree level) scheme Gluck-Reya-Vogt (1992) Photonic coefficient function in NLO becomes negative and divergent for . It is included into quark PDFs. 18/26

PDFs in DIS scheme In DIS scheme, the coefficient fns are those at the tree level In the expression of , put and 19/26

PDFs in DIS scheme Singlet quark (NLO) (NNLO) (LO) 20/26

PDFs in scheme In scheme, photon coefficient fn is absorbed into quark distribution fns, so that Expressing and as Then we get 21/26

PDFs in scheme Singlet quark (NLO) (NNLO) (LO) 22/26

PDFs in scheme Singlet quark (NLO) (NNLO) (LO) for 23/26

PDFs in scheme Large behavior Behavior at 24/26

Summary PDFs (quark and gluon) in the virtual photon were investigated up to NNLO ( ) in the kinematical region PDFs were studied in , DIS and schemes Scheme dependences appear at large Singlet quark 25/26

Summary Gluon distribution 26/26