Jiro Kodaira and QCD Analysis of Structure Functions Dedicated to the memory of Jiro Kodaira T. Uematsu (Kyoto Univ.) at KEK workshop “Towards precision QCD physics” March 10, 2007

Jiro Kodaira’s Personal History Born　in　Nagano，Japan in 1951 Graduated from Kyoto Univ. in 1974 Finished Ph.D in 1979 1979 －1980　 RA at YITP, Kyoto Univ. 1980 －1982 JSPS Fellow at SLAC 1982 －1983 RA at Kobe University 1983 －2005 RA & AP at Hiroshima Univ. 2005 － 2006 Professor at KEK

Asymptotic freedom ’73 Kodaira’s paper EMC effects ’88 LHC starting ’07 1951 ’70 ’74 1979 2005 2006 ’80 ’82 ’83 under grad. SLAC Hiroshima KEK Born in Nagano Graduate course in Kyoto Univ. Kobe Yukawa Inst.

Kodaira’s contributions to perturbative QCD QCD higher order corrections to the polarized deep inelastic scattering 　Kyoto Group 1978-1979 　flavor-singlet nucleon polarized structure function Kodaira 1979 Soft gluon resummation Kodaira & Trentadue 1981-82 Spin correlation in top quark production 　　　　　　　 Kodaira & Parke, ….. 1998-99 Axial Anomaly plays an important role NLL order

QCD Higher Order Effects Physical quantity Anomalous dim. Next-Leading-log Leading-log Kyoto Group Kodaira et al. 1979 flavor non-singlet Phys.Rev.D20 (1979) 627; Nucl.Phys. B159 (1979) 99

Polarized DIS & Structure Functions nucleon spin incident lepton scattered quark scattered lepton or Structure tensor Unpolarized structure fns. polarized structure fns.

Flavor Singlet Case Axial anomaly Kodaira QCD correction Axial vector current Flavor Singlet　Case Axial anomaly Triangle anomaly 2-loop anomalous dim. does not vanish + other diagram Kodaira Nucl.Phys. B165 (1980)129 : # of flavors QCD correction

Parton distribution functions are scheme-dependent 1988 EMC at CERN “Spin Crisis” Altarelli-Ross Phys. Lett B212 (1988) 391 Gluon polarization contributes due to anomaly Or Parton distribution functions are scheme-dependent J. Kodaira and T. U. Nucl. Phys. B141 (1978) 497

Kodaira’s other contributions to pQCD Twist-3 effects & BC sum rule in Yasui, Tanaka, Nasuno, Tochimura; Sasaki, Matsuda, T.U. Twist-4 effects in Kawamura, Yasui, T.U. Spin correlation in top quark production Parke, Nasuno, Kiyo, Hori, Morii Polarized Drell-Yan processes Kawamura, Tanaka, Shimizu, Yokoya Small-x of in resummation Kiyo, Tochimura

Higher-twist effects in polarized structure fns. Twist-3 and twist-4 operators Tanaka's talk characteristics of higher-twist operators Represents quark and gluon correlations The number of relevant composite operators increases as n　increases Renormalization of higher-twist operators 　 Operator mixing through renormalization becomes quite complicated

Twist-3 effects in twist-2 twist-3 quark-gluon operators Mixing very complicated Burkhardt-Cottingham sum rule

Twist-4 effects on spin structure fn Flavor non-singlet Twist-4 effects Flavor singlet Off-shell Green’s functions in covariant gauge

Kodaira @ KEK An expert in the field of pQCD and expected to advise experimentalists about physics at LHC and ILC from deep understanding of the theory Leader’s role of “Minami-Tateya” (numerical analysis) group doing automatic computation of Feynman graphs and various simulations for the analyses of data from collider experiments International collaborations just started LAPTH Annecy, NIKHEF (GRACE-FORM project), …

Kodaira’s connections in the world SLAC S. Parke (Fermilab), L. Trentadue (Parma), 　　　　　　　　　L. Dixon, S. Drell, S. Brodsky,… RIKEN-BNL R.L.Jaffe (MIT), X.Ji (Maryland), W. Vogelsang, D. Boer, M. Stratmann,.. DESY Zeuthen J. Bluemlein, S. Moch, … NIKHEF J. Vermaseren, … LAPTH D. Perret-Gallix, F. Boudjema,.. CERN G. Altarelli, …. Firenze S. Catani, … Collaboration with KEK Minami-tateya Group

7th International Symposium on Radiative Corrections APPLICATION OF QUANTUM FIELD THEORY TO PHENOMENOLOGY Shonan Village, Japan October 2-7, 2005

Concluding remarks Very sad but his achievements will remain Take over Kodaira’s spirit for the research in high-energy physics and pass it to the younger generations Efforts should be taken toward precisions of QCD as well as Electro-weak physics Promote the international collaborations as Jiro Kodaira had tried to establish

Conclusion QCD is a Theory of Strong Interaction From Jiro’s Slide in Aug06 Conclusion QCD is a Theory of Strong Interaction Non-abelin Gauge Theory　 One of the “most” beautiful The. in 20th Century Forced to solve (at least) two problems now 1. do calculation for practical purpose (LHC, ILC, …) 2. reveal dynamics of gauge theory using the simplest but the most difficult one namely QCD