C.A. Dominguez

Centre for Theoretical Physics & AstrophysicsUniversity of Cape Town

Department of Physics, Stellenbosch UniversitySouth Africa

Work done with J. Bordes, P. Moodley, J. Peñarrocha, K. Schilcher

GELL-MANN-OAKES-RENNER RELATION IN QCD:

CHIRAL CORRECTIONS FROM SUM RULES

QCD- 2010 Montpellier

GMOR RELATION:A QCD LOW ENERGY THEOREM (NLO)

)0(

,,|)()(

0|))0()((|0)(

5

425

dujixAxJ

JxJTexdiq

ij

iqx

GMOR RELATION:A QCD LOW ENERGY THEOREM (NLO)

225 20||0)()0( Mfdduumm du

)1(20||0)()0( 225 Mfdduumm du

IMPORTANCE OF δπ

• 1) INTRINSIC

• 2) χPT: δπ = (4 Mπ2 / fπ

2) ( 2 Lr8 – Hr

2)

• 3) Lattice QCD: Lr8

SUBTLETIES22

5 20||0)()0( Mfdduumm du

• 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)

IS δπ = 0 ???

• QCD CORRECTIONS:

• HADRONIC CORRECTIONS:

∂µ Aµ (x) = 2 fπ2 Mπ

2 φπ(x) + Σn 2 fn2 Mn

2 φn(x)

Q C D SUM RULESShifman-Vainshtein-Zakharov

HADQCD

iqx

qq

JxJTexdiq

)()(

0|))0()((|0)(

22

42

QUARK-HADRON DUALITY

)(|)(Im1

)()(2

1)0()0( 5555|)(|55

0

0

sss

dsss

s

ds

i HAD

s

s

QCD

sC th

QCD FINITE ENERGY SUM RULE

Δ5(s): ANALYTIC KERNEL

PQCD

)4,...,0(ln),(

)]([),(8

3])()([)(

2

1

0

)(5

4

0

)(52

25

is

cs

ssmms

ji

jij

i

is

i

idu

)(|)(Im)(|)(Im 05242

5 sssMsMfs RESPPPHAD

HADRONIC ψ5(s)

Realistic Spectral Function

Im G

E2

PION RADIAL EXCITATIONS

• π (1300): M = 1300 ± 100 MeV• Γ = 200 – 600 MeV

• π (1800): M = 1812 ± 14 MeV• Γ = 207 ± 13 MeV

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

)(|)(Im1

)()(2

1)0()0( 5555|)(|55

0

0

sss

dsss

s

ds

i HAD

s

s

QCD

sC th

QCD FINITE ENERGY SUM RULE

Δ5(s): ANALYTIC KERNEL

Δ5 (s)

• Δ5 (s) = 1 - a0 s – a1 s2

• Δ5 (M12) = Δ5 (M2

2) = 0

Realistic Spectral Function

IMPACT OF KERNEL Δ5(s)

Im G

E2

Δ5 (s)

• Δ5 (s) ≡ Pn(s) ⇨ Legendre

Polynomials

• (with global constraints)

• TUNED TO STRONGLY QUENCH THE HADRONIC RESONANCE

CONTRIBUTION TO THE FESR

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

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))

δπ (%)

FOPT CIPT FIXED µµ2 = 2 – 50 GeV2

6.5 0.9 7.0 0.8 5.6 1.1

δπ = 6.2 1.6 %

32 )5267(|0||0 MeVqq GeV

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>

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]