1. GELL-MANN-OAKES-RENNER RELATION CHIRAL CORRECTIONS FROM SUM RULES
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 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. ∂µ Aµ (x) = 2 fπ2 Mπ2 φπ(x) + Σn 2 fn2 Mn2 φn(x)
IS δπ = 0 ???
QCD CORRECTIONS:
HADRONIC CORRECTIONS:
∂µ Aµ (x) = 2 fπ2 Mπ2 φπ(x) + Σn 2 fn2 Mn2 φn(x)

7. Q C D SUM RULES Shifman-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 SYSTEMATIC UNCERTAINTY
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 Function IMPACT OF KERNEL Δ5(s)
Im G E2

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

19. FOPT αs(s0) & mq(s0) frozen
FOPT αs(s0) & mq(s0) frozen. RG ⇨ after integration CIPT αs(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)
=  (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))

24. δπ (%) δπ = 6.2  1.6 % 6.5  0.9 7.0  0.8 5.6  1.1 FOPT CIPT
δπ (%)
FOPT
CIPT
FIXED µ
µ2 = 2 – 50 GeV2
6.5  0.9
7.0  0.8
5.6  1.1
δπ = 6.2  1.6 %

25. Lr8 (νχ=Mρ) = (0.88  0.24)  10-3 Jamin (CHPT) 2002
δπ (%)
Hr2 (νχ=Mρ)
(in 10-3)
6.2  1.6
this work
- (5.1  1.8)
4  2
Dominguez, Nasrallah, Schilcher *
4.7  1.7
Jamin**
(4.3  1.3)
(3.4  1.5)
From a determination of <s-bar s> / <u-bar u> using same Δ5(s)
** From a determination of <s-bar s> / <u-bar u>

26. 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]