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C.A. Dominguez Centre for Theoretical Physics & Astrophysics University of Cape Town

GELL-MANN-OAKES-RENNER RELATION IN QCD: CHIRAL CORRECTIONS FROM SUM RULES. C.A. Dominguez Centre for Theoretical Physics & Astrophysics University of Cape Town Department of Physics, Stellenbosch University South Africa Work done with J. Bordes, P. Moodley, J. Pe ñarrocha, K. Schilcher.

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C.A. Dominguez Centre for Theoretical Physics & Astrophysics University of Cape Town

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  1. GELL-MANN-OAKES-RENNER RELATION IN QCD: CHIRAL CORRECTIONS FROM SUM RULES C.A. Dominguez Centre for Theoretical Physics & Astrophysics University of Cape Town Department of Physics, Stellenbosch University South Africa Work done with J. Bordes, P. Moodley, J. Peñarrocha, K. Schilcher QCD- 2010 Montpellier

  2. GMOR RELATION:A QCD LOW ENERGY THEOREM (NLO)

  3. GMOR RELATION:A QCD LOW ENERGY THEOREM (NLO)

  4. IMPORTANCE OF δπ • 1) INTRINSIC • 2) χPT: δπ= (4 Mπ2 / fπ2) ( 2 Lr8 – Hr2) • 3) Lattice QCD: Lr8

  5. SUBTLETIES • RENORMALIZATION • LOG [QUARK MASS] SINGULARITIES • NORMAL ORDERING • HIGHER ORDER QUARK MASS CORRECTIONS Broadhurst & Generalis (81-82) Jamin & Münz (95) Chetyrkin, Dominguez, Pirjol, Schilcher (95) Dominguez, Nasrallah, Schilcher (08)

  6. IS δπ = 0 ??? • QCD CORRECTIONS: • HADRONIC CORRECTIONS: ∂µ Aµ (x) = 2 fπ2 Mπ2φπ(x) + Σn 2 fn2 Mn2φn(x)

  7. QCD SUM RULESShifman-Vainshtein-Zakharov

  8. QUARK-HADRON DUALITY

  9. QCD FINITE ENERGY SUM RULE Δ5(s): ANALYTIC KERNEL

  10. PQCD

  11. HADRONIC ψ5(s)

  12. Realistic Spectral Function Im G E2

  13. PION RADIAL EXCITATIONS • π (1300): M = 1300 ± 100 MeV • Γ = 200 – 600 MeV • π (1800): M = 1812 ±14 MeV • Γ = 207 ± 13 MeV

  14. PROBLEM • Hadronic pseudoscalar spectral function: • NOT DIRECTLY MEASURABLE • Knowledge of mass & width of resonances: • NOT ENOUGH TO RECONSTRUCT SPECTRAL FUNCTION INELASTICITY, NON-RESONANT BACKGROUND, INTERFERENCE: ??? SYSTEMATIC UNCERTAINTY

  15. QCD FINITE ENERGY SUM RULE Δ5(s): ANALYTIC KERNEL

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

  17. Realistic Spectral FunctionIMPACT OF KERNEL Δ5(s) Im G E2

  18. Δ5 (s) • Δ5 (s) ≡ Pn(s) ⇨ Legendre Polynomials • (with global constraints) • TUNED TO STRONGLY QUENCH THE HADRONIC RESONANCE CONTRIBUTION TO THE FESR

  19. FOPTαs(s0) & mq(s0) frozen. RG ⇨ after integrationCIPTαs(s0) & mq(s0) running. RG ⇨ before integration(µ2 = s)FIXED µ2 = 2 – 50 GeV2

  20. INPUT • αS (M2) = 0.344  0.0009(Davier et al., 2008) = 0.342  0.0012(Pich, 2010) • (mu + md) / 2 = 4.1  0.2 MeV (Dominguez et al., 2009) = 3.9  0.5 MeV (Lattice ETMC, 2008) . O (m4) ; <αs G2> : NEGLIGIBLE . IMPACT OF π1(1300) & π2(1800): < 6 % • (On account of Δ5(s))

  21. δπ(%) δπ= 6.2  1.6 %

  22. Lr8 (νχ=Mρ) = (0.88  0.24)  10-3Jamin (CHPT) 2002 • From a determination of <s-bar s> / <u-bar u>using same Δ5(s) ** From a determination of <s-bar s> / <u-bar u>

  23. EARLIER DETERMINATIONS OF δπ • PQCD: only to two or three loop order • PQCD: different values of αs • HADRONIC: strongly model dependent spectral functions [Δ5(s) = 1]

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