1. 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

2. 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

3. 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

4. QCDSR DEBUT
MASS OF C

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

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

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

8. Q C D

9. HADRONIC

10. 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

13. QUARK CONDENSATE

15. GLUON CONDENSATE

16. FOUR-QUARK CONDENSATE

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

19.   hadrons

21. Q C D SUM RULES (SVZ)

22. QUARK-HADRON DUALITY

23. QUARK-HADRON DUALITY

27. 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

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

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

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

32. Realistic Spectral Function
Im Π
s ≡ E2

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

35. 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

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

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

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

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

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

47. 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

48. 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

49. DETERMINATION OF S(M2)  hadrons

51.   hadrons

54. STUDENTS: AN ACTIVE RESEARCH FIELD

56. 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?

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

61. QUARK-HADRON DUALITY

62. 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

63. AXIAL-VECTOR CHANNEL τ- DECAY

66. 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

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