The Operator Product Expansion Beyond Perturbation Theory in QCD C.A. Dominguez Centre for Theoretical Physics & Astrophysics University of Cape Town Department of Physics, Stellenbosch University South Africa XII WORKSHOP ON PARTICLES & FIELDS MAZATLAN NOVEMBER 2009

QCD SUM RULES (Shifman, Vainshtein, Zakharov) 1979 – to date (a few kP) ANALYTICAL METHOD TO SOLVE QCD AT FERMI SCALES OPERATOR PRODUCT EXPANSION OF CURRENT CORRELATORS AT SHORT DISTANCES CAUCHY THEOREM IN THE COMPLEX ENERGY PLANE (QUARK-HADRON DUALITY) EXTENDS DOMAIN OF CHIRAL PT COMPLEMENTARY TOOL TO LATTICE QCD

SCOPE OF APPLICATIONS HADRONIC SPECTRUM: Masses & Couplings FORM FACTORS: Electromagnetic & hadronic & weak hadronic QCD: Quark masses & QCD strong coupling s A tool to confront QCD predictions with data e+e- → hadrons & τ → hadrons QCD at FINITE TEMPERATURE: Quark-gluon plasma. Chiral symmetry restoration. Quark-gluon deconfinement

QCDSR DEBUT MASS OF C

CONFRONTING QCD WITH DATA THIS TALK LIGHT QUARK MASSES s(M2) CONFRONTING QCD WITH DATA

QUARK MASSES CPT: Light quark mass ratios Lattice QCD QCD Sum Rules

Q C D SUM RULES Shifman-Vainshtein-Zakharov (1979)

Q C D

HADRONIC

CONFINEMENT STRONG MODIFICATION TO QUARK & GLUON PROPAGATORS NEAR THE MASS SHELL INCORPORATE CONFINEMENT THROUGH A PARAMETRIZATION OF PROPAGATOR CORRECTIONS IN TERMS OF QUARK & GLUON VACUUM CONDENSATES

QUARK CONDENSATE

GLUON CONDENSATE

FOUR-QUARK CONDENSATE

FOUR-QUARK CONDENSATE RESPONSIBLE FOR  - a1 MASS SPLTITING  (770) – a1 (1100) [V - A]| PQCD  0 (mq = 0) [V - A]|d=4  0 [V - A]|d=6  0

  hadrons

Q C D SUM RULES (SVZ)

QUARK-HADRON DUALITY

QUARK-HADRON DUALITY

PROBLEM WITH Im (S)|resonance e+ e-  hadrons  Im  (s)|V   hadrons  Im  (s)|V & Im  (s)|A PSEUDOSCALAR CHANNEL (beyond pole): Not measured & not measurable SYSTEMATIC UNCERTAINTY

SYSTEMATIC UNCERTAINTY CAD, Nasrallah, Schilcher (2007) CAD, Nasrallah, Röntsch, Schilcher (2008)

INTEGRATION KERNEL Δ5 (s) Analytic function  ds Im (s) 5(s) = 0

PURPOSE OF THE INTEGRATION KERNEL ENHANCE / SUPPRESS SPECIFIC CONTRIBUTIONS HADRONIC: resonance region: non-existing experimental data extend analysis beyond end-point of experimental data

Realistic Spectral Function Im Π s ≡ E2

HADRONIC SPECTRAL FUNCTION Pseudoscalar meson pole (pion, kaon) OK Resonances: (???)  → hadrons (JP = 0-) NOT FEASIBLE

PION (KAON) RADIAL EXCITATIONS π (1300): M = 1300 ± 100 MeV Γ = 200 – 600 MeV π (1800): M = 1812 ± 14 MeV Γ = 207 ± 13 MeV K (1460) & K (1830) Γ ≈ 250 MeV

SYSTEMATIC UNCERTAINTY MASS & WIDTH OF RESONANCES: NOT ENOUGH TO RECONSTRUCT HADRONIC SPECTRAL FUNCTION !!! HADRONIC BACKGROUND & CONSTRUCTIVE/DESTRUCTIVE INTERFERENCE COMPLETELY UNKNOWN

Δ5 (s) Δ5 (s) = 1 - a0 s – a1 s2 Δ5 (M12) = Δ5 (M22) = 0

FOPT αs(s0) & mq(s0) frozen FOPT αs(s0) & mq(s0) frozen. RG ⇨ after integration CIPT αs(s0) & mq(s0) running. RG ⇨ before integration

PHYSICAL QUANTITIES ARE INDEPENDENT OF S0 S0 DEPENDENCE PHYSICAL QUANTITIES ARE INDEPENDENT OF S0 IN PRACTICE : S0  1 – 3 GeV2

ERROR ANALYSIS ΛQCD = 365 – 397 MeV (ALEPH ’08) αs (M2) = 0.344 ± 0.005 ± 0.007 <αs G2> = 0.06 GeV4 within a factor 2 Ψ5(Q2)|6 LOOP ≈ Ψ5(Q2)|5 LOOP Ψ5(Q2)|RESONANCE : factor 5 smaller than PQCD

RESULTS ms (2 GeV) = 102 ± 8 MeV md (2 GeV) = 5.3 ± 0.4 MeV mu (2 GeV) = 2.9 ± 0.2 MeV (mu + md)/2 = 4.1 ± 0.2 MeV

SUMMARY A method to decrease substantially the systematic uncertainties from the hadronic resonance sector Future improvement from more precise ΛQCD & higher loop order in PQCD

DETERMINATION OF S(M2)  hadrons

  hadrons

STUDENTS: AN ACTIVE RESEARCH FIELD

IS THERE A CONFLICT BETWEEN QCD AND THE τ – HADRONIC DECAY DATA ? CHIRAL SUM RULES (V-A): SATURATION ? EXTRACTION OF CONDENSATES: COMPARE WITH RESULTS FROM e+ e- CONSISTENCY BETWEEN VECTOR AND AXIAL-VECTOR RESULTS?

HOW TO EXTEND THE ANALYSIS BEYOND THE KINEMATICAL END POINT OF THE DATA ??? s1 s0

QUARK-HADRON DUALITY

Calculable assuming e.g. (a) data follows a logarithmic fall-off (PQCD) (b) data has other functional relation, e.g. linear Assumptions testable in e+ e- (data exists) Δ(s0) is negligible

AXIAL-VECTOR CHANNEL τ- DECAY

CONCLUSIONS NO VIOLATIONS OF QUARK-HADRON DUALITY g-2 of the MUON A NEW TECHNIQUE TO EXTEND A QCD ANALYSIS BEYOND THE KINEMATICAL END POINT OF DATA IN τ – DECAY: PERFECT AGREEMENT BETWEEN QCD & DATA IN THE WIDE RANGE S0 = 3 – 10 GeV2 NO VIOLATIONS OF QUARK-HADRON DUALITY g-2 of the MUON

QCD SUM RULES QUARK MASS DETERMINATIONS CONFRONTING QCD WITH DATA s (q2) ,, g-2|muon , PQCD