Personal View QCD old boys view of development
摂動論的Ｑ Ｃ Ｄ の発展 －Personal View－ QCD old boy’s view of development of p. QCD 植松恒夫 (京大理 ) 「核子構造研究の新展開2011」 @KEK January 7 -8, 2011
Introduction QCD以前 黎明期 • 核子の構造の研究 (60年代）deep inelastic scattering • Current Algebra Bjorken scaling 1968 • SLAC-MIT experiment Parton分布関数 L. W. Mo SLAC-PUB-660 (1969) Parton Model Feynman，Bjorken-Paschos 核子構造関数
Scalingの破れ • Broken scaling inv. くりこみ群 ε-展開 • Critical phenomena (物理量） Kogut-Susskind Scale-inv. Parton Model • スケーリングの破れ • 場の理論でのくりこみ群的解析 fixed point theory power breaking anomalous dim. asymptotically free theory logarithmic breaking fixed point theory asymptotically free theory
Various Ideas leading to QCD Large PT Bjorken Scaling Large Angle Scattering Current Algebra Adler-Dashen Bjorken Light-cone Fritzsch-Gell-Mann Parton Picture Feynman Brodsky-Farrar, Matveev et al. OPE K. Wilson Renormalization Group Asymptotic freedom Gross-Wilczek, Politzer Perturbative QCD Gell-Mann-Low
第１期 1973－1970年代 勃興期 Color gauge theory (Color octet gluon) Nambu 1966 Fritzsch-Gell-Mann-Leutwyler 1973 QCD Weinberg 1973 Asymtotic freedom Gross-Wilczek, Politzer, ‘t Hooft 1973 ・ short distance ・・・漸近的自由 ・ long distance ・・・閉じ込め QCD Lagrangian : coupling const. No free parameter except for
Effective coupling constant Fundamental constant of QCD “dimensional transmutation’’ S. Bethke
Leading Order (LO) analysis • OPE+RG analysis of structure function Gross-Wilczek, Politzer, … • DGLAP evolution equation for PDFs Dokshitzer-Diakonov-Troyan, Gribov -Lipatov, Altarelli-Parisi, … • Q 2 dep. of fragmentation fn. Georgi-Politzer, Owens, T. U. , … • Polalrized structure functions Ahmed-Ross, Sasaki
Mass singularityの因子化（Factorization） • quark-gluonの系を摂動論で扱うと質量 0の粒子の 4元運 動量が0になることから生じる赤外発散（infrared divergence）や質量 0のgluonがquarkとcollinearに放出され ることによる質量特異性(mass singularity）が現れる． mass (collinear) singularity • parton描像でhadronのcross section singularな因子は hadron分布関数へ吸収 mass singularity
Factorization （因子化） 物理量 Collinear factorization short distance long distance 非摂動論 convolution 摂動論 scheme- dep Structure fn. OPE Q 2 dependence DGLAP equation PDF (Q 2 dep) DGLAP eq. or OPE+RGE
Splitting functions “Plus distribution” infrared cancel Kinoshita-Lee-Nauenberg -Nakanishi Theorem real emission virtual correction
e+e-hadrons jet production Sterman-Weinberg jet infrared/collinear-safe quantity Event shapes: Thrust etc. 3 -jet event @DESY 1979 半角δのconeに全エネルギーの (1 -ε)倍が放出される
Factorization in hadron collisions DIS, e+ e－→h. X, Drell-Yan, hadron-hadron semi-inclusive・・
Higher-twist effects ＋ Twist-3 or 4 twist-2 twist-4 twist-6 twist=dim-spin • Twist-4 4 -quark 2 -quark&1 gluon operators • Anomalous dimensions of 4 -quark operators Gottlieb(78) , Okawa(80) • Polarized structure fn. twist-3 contributes to g 2 Kodaira-Tanaka-TU-Yasui (96) • twist-4 effects to g 1 Kawamura et al (97)
QCD higher-order (NLO) effects Moment of structure fn. LO 1 -loop NLO NNLO 2 -loop Floratos et al (78) 1 -loop anomalous dim. coefficient fn. RG eq. LO 1 -loop Bardeen et al (78) Scheme independent anomalous dim. coefficient fn. 1 -loop tree NLO 2 -loop 1 -loop NNLO 3 -loop 2 -loop Moch et al (04)
第２期 1980年代 中間期 • QCDの摂動論および非摂動論での様々な 理論的整備/実験の進歩 • Unpolarized & polarized DIS data SLAC DESY • EMC effects Spin Crisis CERN • Polarized structure functions • Twist-2, twists-3 effects mixing complicated renormalization of higher-twist ops. • Small-x behavior BFKLeq. BK eq.
Polarized DIS & Structure Functions spin incident lepton spin nucleon scattered lepton Structure tensor or scattered quark Unpolarized structure fns.
1988 EMC at CERN “Spin Crisis” “quarks carries only a small fraction of nucleon spin” in contradiction with naïve quark model But this is relativistic quantities ! Altarelli-Ross Phys. Lett B 212 (1988) 391 Gluon polarization contributes due to anomaly Parton distribution functions are scheme -dependent J. Kodaira and T. U. Nucl. Phys. B 141 (1978) 497
1 st moment sum rule Kyoto Group Kodaira et al. 1979 flavor non-singlet Phys. Rev. D 20 (1979) 627; Nucl. Phys. B 159 (1979) 99 Now Larin-Vermaseren (1991) Axial anomaly flavor singlet Kodaira Nucl. Phys. B 165 (1980)129 QCD correction
• Coefficient functions: van Neerven-Zijlstra (1994) • Transversity Chiral-odd structure fn. twist-2 op. 2 -loop anomalous dim. Hayashigaki, Kanazawa, Koike; Kumano, Miyama • Higher-twist-3 contributes in leading order to • Many structure functions : Collins function Asymmetry in Semi-inclusive DIS Transversity & Collins and Sivers asymmetries : Sivers function
Small-x region physics • Large logs の足し上げ BFKL方程式 • Non-linear evolution equation BK方程式 gluon分布のsaturation color glass condensation (CGC) recombination
Soft-gluon resummation • 2 mass scales: e. g. Q 2&m 2 or Q 2&QT 2 large logs appear at kinematical boundaries spoil perturbative expansions e. g. (real and virtual contributions highly unbalanced) Sudakov resummation=resumming the doublelogarithmic perturbative contributions
Threshold Resummation と書くと，以下が示される Sterman(’ 87), Catani&Trendadue(’ 89, ’ 91) LL NLL (Trentadue and Kodaira Catani, D’Emilio and Trentadue) NNLL
第３期 1990年代－現在 Precision of QCD: LO→NNLO 検証から精密化へ • QCDの量子補正をNNLOの精度に高める Splitting functions 1 -loop 2 -loop 3 -loop # of diagrams LO 18 NLO 350 NNLO 9607 Gross-Wilczek, Politzer(1974) Coefficient functions. Floratos-Ross-Sachrajda(1978) Moch-Vermaseren-Vogt (2004)
Recent topics in QCD Higgs production String-inspired QCD Multi-parton amplitudes Color Glass Condensation Global PDF analysis Perturbative QCD Ad. S/CFT Heavy quark mass effects Precision QCD
Global PDF Analysis Collaborations • MSTW (MRST) LO NNLO • CTEQ LO NLO Tevatron jet analysis • NNPDF NLO Neural networks • ABKM NNLO heavy quark effects • GRV, AAC NLO polarized PDFs
QCD and HERA data
QCD @ LHC era RHICからLHCへ • CMS@7 Te. V pp collision ～ RHIC heavy ion collision Ridge structure pseud-rapidty (η） correlation • ALICE heavy ion collision high multiplicities • ATLAS & CMS ジェット抑制（Jet quenching) Higgs production ｇluon fusion Diffractive production of Higgs
Heavy quark mass effects • heavy quark coefficient functions deep inelastic lepton-hadron scattering for mass factorization: Buza et al. (‘ 96) heavy quark massive operator light quark coeff. fn matrix element coeff. fn • low では heavy quarkはradiativeに作られる heavy quark high では light quarkと同等 matching condition：
Fixed and variable flavor number schemes • Fixed Flavor Number Scheme (FFNS) u, d, s: massless heavy quarks (H): c, b, s are radiatively generated for large log spoils perturbation • Zero Mass Variable Flavor Number Scheme (ZMVFNS) ZMVFNS shows discontinuity at transition → • General mass variable flavor number (GMVFN)scheme To achieve smooth transition
QCD and String Theory • Ad. S/CFT対応 String on Ad. S 5×S 5～N=4 SCFT Maldacena • 5次元Ad. S時空での弦のsemi-classicalな振る舞いと boundaryの４次元Yang-Mills理論の対応関係 Polyakov et al • Ad. S/CFT対応（弦／ゲージ双対性）とform factorお よび大角度散乱と強結合領域でのDIS構造関数 Polchinski et al • String理論でエネルギーについてベキ的振舞い 5次元方向の寄与によるwarp factor • Ad. S/QCD (Holographic QCD) spectrum, decay width, Pomeron, BFKL anom. dim.
multi-parton amplitudes + + MHV － + + • MHV振幅に対するParke-Taylor公式の拡張（spinor-helicity） • Witten Commun. Math. Phys. 252(2004)189 N=4超対称理論で運動量空間からFourier変換で得たtwistor spaceにおける散乱振幅を弦理論のインスタントンの寄与に 結びつけた • Cachazo-Svrcek-Witten (CSW) JHEP 09(2004)006 最大にヘリシティーを破る(MHV)振幅をvertexに拡張し，一般 のMHVでないhelicity振幅を計算するルールを与えた • 任意の 1 -loop multi-leg amplitudes A～(Box)+(triangle)+(bubble)+(tadpole)
Nucleon’s total angular momentum proton spin sum rule gluon orbital angular mom. quark orbital angular mom. at Sizable contributions are from gluon pol. Or from orbital angular mom. Asymptotic values from evolution eq. and
Semi-inclusive Deep Inelastic Scattering (SIDIS) Transverse-spin asymmetry at leading-twist in SIDIS fragmentation function : Collins function Transverse distribution in the initial state Sivers function Earlier claim was that Sivers asymmetry vanish time reversal inv. Final-state interaction important for Sivers fn. not to vanish Wilson lines in the op. def required by gauge inv.
Generalized Parton distribution (GPD) See Wakamatsu’s talk non-forward proton matrix elements can be measured at Deeply virtual Compton scattering (DVCS) null vector skewdness parameter Bjorken variable
: Generalized Parton distribution (GPD) Experimentally known Lattice calculation Not in accordance with exp.
Spin-dependent fragmentation function See Seidl’s talk @BELLE at KEK Chiral-odd fragmentation function e. g. Collins function e+e- collider KEK-B at Tsukuba unpolarized Collins function
Buza-Matiounine-Smith-van Neerven (BMVN) prescription
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