Standard ChPT δ dependence on M π Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary 0 20 70 δ 20 O(p 6 ) δ 11 O(p 6 ) -2 60 15 -4 50 -6 40 10 -8 30 -10 δ 00 O(p 6 ) 20 5 -12 10 -14 0 0 0 50 100 150 200 250 300 0 50 100 150 200 250 300 0 50 100 150 200 250 300 2.5 3 M π = 139.57 MeV M π = 230 MeV δ 02 O(p 6 ) δ 22 O(p 6 ) 2 2 1.5 1 Two loops 1 0 0.5 Standard ChPT -1 0 -0.5 -2 0 50 100 150 200 250 300 0 50 100 150 200 250 300 p (MeV) p (MeV) Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Standard ChPT δ dependence on M π Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary 0 20 70 δ 20 O(p 6 ) δ 11 O(p 6 ) -2 60 15 -4 50 -6 40 10 -8 30 -10 δ 00 O(p 6 ) 20 5 -12 10 -14 0 0 0 50 100 150 200 250 300 0 50 100 150 200 250 300 0 50 100 150 200 250 300 2.5 3 M π = 139.57 MeV M π = 230 MeV δ 02 O(p 6 ) δ 22 O(p 6 ) M π = 300 MeV 2 2 1.5 1 Two loops 1 0 0.5 Standard ChPT -1 0 -0.5 -2 0 50 100 150 200 250 300 0 50 100 150 200 250 300 p (MeV) p (MeV) Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Standard ChPT δ dependence on M π Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary 0 20 70 δ 20 O(p 6 ) δ 11 O(p 6 ) -2 60 15 -4 50 -6 40 10 -8 30 -10 δ 00 O(p 6 ) 20 5 -12 10 -14 0 0 0 50 100 150 200 250 300 0 50 100 150 200 250 300 0 50 100 150 200 250 300 2.5 3 M π = 139.57 MeV M π = 230 MeV δ 02 O(p 6 ) δ 22 O(p 6 ) M π = 300 MeV 2 2 M π = 420 MeV 1.5 1 Two loops 1 0 0.5 Standard ChPT -1 0 -0.5 -2 0 50 100 150 200 250 300 0 50 100 150 200 250 300 p (MeV) p (MeV) Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Standard ChPT δ dependence on M π Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary 0 20 70 δ 20 O(p 6 ) δ 11 O(p 6 ) -2 60 15 -4 50 -6 40 10 -8 30 -10 δ 00 O(p 6 ) 20 5 -12 10 -14 0 0 0 50 100 150 200 250 300 0 50 100 150 200 250 300 0 50 100 150 200 250 300 2.5 3 M π = 139.57 MeV M π = 230 MeV δ 02 O(p 6 ) δ 22 O(p 6 ) M π = 300 MeV 2 2 M π = 420 MeV 1.5 1 Two loops 1 0 0.5 Standard ChPT -1 0 Still soft M π -0.5 -2 dependence 0 50 100 150 200 250 300 0 50 100 150 200 250 300 p (MeV) p (MeV) Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Standard ChPT δ dependence on M π Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary 0 20 70 δ 20 O(p 6 ) δ 11 O(p 6 ) -2 60 15 -4 50 ρ channel -6 40 10 phase shift -8 30 decreases ?? ❳❳❳ -10 δ 00 O(p 6 ) 20 ③ 5 -12 10 -14 0 0 0 50 100 150 200 250 300 0 50 100 150 200 250 300 0 50 100 150 200 250 300 2.5 3 M π = 139.57 MeV M π = 230 MeV δ 02 O(p 6 ) δ 22 O(p 6 ) M π = 300 MeV 2 2 M π = 420 MeV 1.5 1 Two loops 1 0 Still soft M π 0.5 -1 dependence 0 -0.5 -2 0 50 100 150 200 250 300 0 50 100 150 200 250 300 p (MeV) p (MeV) Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Unitarized ChPT Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Inverse Amplitude Method Inverse Amplitude Method Truong, Dobado, Herrero, Peláez Elastic IAM partial waves satisfy exact unitarity SS † = 1 ⇒ Im t − 1 = − σ O ( p 4 ) IAM partial waves: t 2 2 ( s ) t ( s ) ≃ t 2 ( s ) − t 4 ( s ) It is derived from a dispersion relation: exact on the elastic right cut, left cut and substraction constants approximated within NLO ChPT, fully renormalized, no spurious parameters. Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Inverse Amplitude Method Inverse Amplitude Method Truong, Dobado, Herrero, Peláez Elastic IAM partial waves satisfy exact unitarity SS † = 1 ⇒ Im t − 1 = − σ O ( p 6 ) IAM partial waves: t 2 2 ( s ) t ( s ) ≃ t 2 ( s ) − t 4 ( s ) + t 42 t 2 − t 6 It is derived from a dispersion relation: exact on the elastic right cut, left cut and substraction constants approximated within NLO ChPT, fully renormalized, no spurious parameters. Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Unitarized ChPT δ dependence on M π Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary SU(2) Unitarized ChPT phase shifts vs. Momentum Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Unitarized ChPT δ dependence on M π Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Unitarized SU(2) ChPT amplitudes with LECs: Two loops Set A Set D O ( p 4 )( x10 − 3 ) l r 1 ( µ ) -5.0 -4.0 l r 2 ( µ ) 1.7 1.2 One loop l r 3 ( µ ) 0.8 0.8 O ( p 4 ) LECs ( × 10 − 3 ) l r 4 ( µ ) 6.5 6.5 l r 1 ( µ ) − 3 . 7 ± 0 . 2 O ( p 6 )( x10 − 4 ) l r 2 ( µ ) 5 . 0 ± 0 . 4 r r 1 ( µ ) -0.6 -0.6 l r 3 ( µ ) 0 . 8 ± 3 . 8 r r 2 ( µ ) 1.3 1.5 l r 4 ( µ ) 6 . 2 ± 5 . 7 r r 3 ( µ ) -1.7 -3.3 r r 4 ( µ ) 2.0 0.9 r r 5 ( µ ) 2.0 1.7 r r 6 ( µ ) -0.6 -0.7 r r f ( µ ) -1.4 -1.8 Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Unitarized ChPT δ dependence on M π Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Unitarized ChPT 0 180 120 δ 00 O(p 4 ) δ 20 O(p 4 ) 160 δ 11 O(p 4 ) -5 140 100 120 80 -10 100 60 80 -15 60 40 40 -20 20 20 0 -25 0 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 p (MeV) p (MeV) p (MeV) M π = 139.57 MeV One loop Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Unitarized ChPT δ dependence on M π Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Unitarized ChPT 0 180 120 δ 00 O(p 4 ) δ 20 O(p 4 ) 160 δ 11 O(p 4 ) -5 140 100 120 80 -10 100 60 80 -15 60 40 40 -20 20 20 0 -25 0 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 p (MeV) p (MeV) p (MeV) M π = 139.57 MeV One loop M π = 230 MeV Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Unitarized ChPT δ dependence on M π Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Unitarized ChPT 0 180 120 δ 00 O(p 4 ) δ 20 O(p 4 ) 160 δ 11 O(p 4 ) -5 140 100 120 80 -10 100 60 80 -15 60 40 40 -20 20 20 0 -25 0 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 p (MeV) p (MeV) p (MeV) M π = 139.57 MeV One loop M π = 230 MeV M π = 300 MeV Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Unitarized ChPT δ dependence on M π Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Unitarized ChPT 0 180 120 δ 20 O(p 4 ) 160 -5 140 100 120 80 -10 100 60 80 -15 60 40 40 -20 20 δ 00 O(p 4 ) δ 11 O(p 4 ) 20 0 -25 0 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 p (MeV) p (MeV) p (MeV) M π = 139.57 MeV One loop M π = 230 MeV M π = 300 MeV M π = 420 MeV Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Unitarized ChPT δ dependence on M π Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Unitarized ChPT ρ channel phase shift 0 180 120 δ 20 O(p 4 ) 160 increases !! ❳❳ -5 140 ③ ❳ 100 contradiction 120 80 -10 with ChPT?? 100 60 80 -15 60 40 40 -20 20 δ 00 O(p 4 ) δ 11 O(p 4 ) 20 0 -25 0 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 p (MeV) p (MeV) p (MeV) M π = 139.57 MeV One loop M π = 230 MeV M π = 300 MeV M π = 420 MeV Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Unitarized ChPT δ dependence on M π Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Unitarized ChPT 180 0 180 160 δ 20 O(p 6 ) 160 δ 11 O(p 6 ) -5 140 140 120 120 -10 100 100 80 80 -15 60 60 δ 00 O(p 6 ) 40 40 -20 20 20 0 -25 0 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 p (MeV) p (MeV) p (MeV) M π = 139.57 MeV Two loops Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Unitarized ChPT δ dependence on M π Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Unitarized ChPT 180 0 180 160 δ 20 O(p 6 ) 160 δ 11 O(p 6 ) -5 140 140 120 120 -10 100 100 80 80 -15 60 60 δ 00 O(p 6 ) 40 40 -20 20 20 0 -25 0 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 p (MeV) p (MeV) p (MeV) M π = 139.57 MeV Two loops M π = 230 MeV Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Unitarized ChPT δ dependence on M π Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Unitarized ChPT 180 0 180 160 δ 20 O(p 6 ) 160 δ 11 O(p 6 ) -5 140 140 120 120 -10 100 100 80 80 -15 60 60 δ 00 O(p 6 ) 40 40 -20 20 20 0 -25 0 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 p (MeV) p (MeV) p (MeV) M π = 139.57 MeV Two loops M π = 230 MeV M π = 300 MeV Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Unitarized ChPT δ dependence on M π Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Unitarized ChPT 180 0 180 160 δ 20 O(p 6 ) 160 δ 11 O(p 6 ) -5 140 140 120 120 -10 100 100 80 80 -15 60 60 δ 00 O(p 6 ) 40 40 -20 20 20 0 -25 0 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 p (MeV) p (MeV) p (MeV) M π = 139.57 MeV Two loops M π = 230 MeV M π = 300 MeV M π = 420 MeV Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Unitarized ChPT δ dependence on M π Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Bound state: phase jumps 2 π (Levinson’s theorem) ✓ ρ channel ✓ ✓ phase shift 180 0 180 ✴ ✓ 160 δ 20 O(p 6 ) 160 δ 11 O(p 6 ) increases !! ❳❳ -5 140 140 ❳ ③ contradiction 120 120 -10 with ChPT?? 100 100 80 80 -15 60 60 δ 00 O(p 6 ) 40 40 -20 20 20 0 -25 0 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 p (MeV) p (MeV) p (MeV) M π = 139.57 MeV Two loops M π = 230 MeV M π = 300 MeV M π = 420 MeV Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
I =1 J =1 channel behavior Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Crude, intuitive model of I =1 J =1 channel behavior Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
I =1 J =1 channel behavior Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary For a simple Breit-Wigner parametrization: � p t ( s ) = −√ sM Γ( p ) / 2 p � 3 Γ( p ) = Γ R with s − M 2 + iM Γ( p ) p R we get a positive phase shift derivative: ∂δ ( p ) π ) = − ∂δ ( p ) 4 M Γ( p ) R ) = � 2 + M 2 Γ( p ) 2 > 0 . ∂ ( M 2 ∂ ( p 2 4 p 2 − 4 p 2 � R The phase shift grows as the ρ approaches threshold. Intuitive behavior but opposed to ChPT at low momentum. Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
I =1 J =1 channel behavior Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary For a simple Breit-Wigner parametrization: � p t ( s ) = −√ sM Γ( p ) / 2 p � 3 Γ( p ) = Γ R with s − M 2 + iM Γ( p ) p R we get a positive phase shift derivative: ∂δ ( p ) π ) = − ∂δ ( p ) 4 M Γ( p ) R ) = � 2 + M 2 Γ( p ) 2 > 0 . ∂ ( M 2 ∂ ( p 2 4 p 2 − 4 p 2 � R The phase shift grows as the ρ approaches threshold. Intuitive behavior but opposed to ChPT at low momentum. Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
I =1 J =1 channel behavior Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary For a simple Breit-Wigner parametrization: � p t ( s ) = −√ sM Γ( p ) / 2 p � 3 Γ( p ) = Γ R with s − M 2 + iM Γ( p ) p R we get a positive phase shift derivative: ∂δ ( p ) π ) = − ∂δ ( p ) 4 M Γ( p ) R ) = � 2 + M 2 Γ( p ) 2 > 0 . ∂ ( M 2 ∂ ( p 2 4 p 2 − 4 p 2 � R The phase shift grows as the ρ approaches threshold. Intuitive behavior but opposed to ChPT at low momentum. Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
I =1 J =1 channel behavior Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Introducing Blatt-Weisskopf modification: � p � 2 l + 1 D l ( p R r ) Γ( p ) D l ( p R r ) Γ( p ) = Γ R D l ( pr ) ≡ ˜ p R D l ( pr ) the phase shift derivative is given by: π ) ≃ 1 + p 4 R ( r 2 ) ′ ∂δ ( p ) M ˜ Γ( p ) ∂ ( M 2 4 p 4 R Estimation of r 2 matching LO ChPT at low p : M p 4 M 1 r 2 = R ( r 2 ) ′ = 1 − R + O ( M 0 π ) ⇒ 1 + p 4 < 0 g 2 f 2 M π 2 g 2 f 2 π M 3 π π Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
I =1 J =1 channel behavior Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Introducing Blatt-Weisskopf modification: � p � 2 l + 1 D l ( p R r ) Γ( p ) D l ( p R r ) Γ( p ) 1 + ( p R r ) 2 Γ( p ) = Γ R D l ( pr ) ≡ ˜ D l ( pr ) = ˜ p R 1 + ( pr ) 2 the phase shift derivative is given by: π ) ≃ 1 + p 4 R ( r 2 ) ′ ∂δ ( p ) M ˜ Γ( p ) ∂ ( M 2 4 p 4 R Estimation of r 2 matching LO ChPT at low p : M p 4 M 1 r 2 = R ( r 2 ) ′ = 1 − R + O ( M 0 π ) ⇒ 1 + p 4 < 0 g 2 f 2 M π 2 g 2 f 2 π M 3 π π Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
I =1 J =1 channel behavior Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Introducing Blatt-Weisskopf modification: � p � 2 l + 1 D l ( p R r ) Γ( p ) D l ( p R r ) Γ( p ) 1 + ( p R r ) 2 Γ( p ) = Γ R D l ( pr ) ≡ ˜ D l ( pr ) = ˜ p R 1 + ( pr ) 2 the phase shift derivative is given by: π ) ≃ 1 + p 4 R ( r 2 ) ′ ∂δ ( p ) M ˜ Γ( p ) ∂ ( M 2 4 p 4 R Estimation of r 2 matching LO ChPT at low p : M p 4 M 1 r 2 = R ( r 2 ) ′ = 1 − R + O ( M 0 π ) ⇒ 1 + p 4 < 0 g 2 f 2 M π 2 g 2 f 2 π M 3 π π Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
I =1 J =1 channel behavior Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Introducing Blatt-Weisskopf modification: � p � 2 l + 1 D l ( p R r ) Γ( p ) D l ( p R r ) Γ( p ) 1 + ( p R r ) 2 Γ( p ) = Γ R D l ( pr ) ≡ ˜ D l ( pr ) = ˜ p R 1 + ( pr ) 2 the phase shift derivative is given by: π ) ≃ 1 + p 4 R ( r 2 ) ′ ∂δ ( p ) M ˜ Γ( p ) ∂ ( M 2 4 p 4 R Estimation of r 2 matching LO ChPT at low p : M p 4 M 1 r 2 = R ( r 2 ) ′ = 1 − R + O ( M 0 π ) ⇒ 1 + p 4 < 0 g 2 f 2 M π 2 g 2 f 2 π M 3 π π Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
I =1 J =1 channel behavior Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Introducing Blatt-Weisskopf modification: � p � 2 l + 1 D l ( p R r ) Γ( p ) D l ( p R r ) Γ( p ) 1 + ( p R r ) 2 Γ( p ) = Γ R D l ( pr ) ≡ ˜ D l ( pr ) = ˜ p R 1 + ( pr ) 2 the phase shift derivative is given by: ✗✔ π ) ≃ 1 + p 4 R ( r 2 ) ′ ∂δ ( p ) M ˜ Γ( p ) < 0 ∂ ( M 2 4 p 4 ✖✕ ♦ ❙ R ❙ ❙ Estimation of r 2 matching LO ChPT at low p : ❙ ❙ ✗✔ ❙ M p 4 M 1 r 2 = R ( r 2 ) ′ = 1 − R + O ( M 0 π ) ⇒ 1 + p 4 < 0 g 2 f 2 M π 2 g 2 f 2 π M 3 ✖✕ π π Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
I =1 J =1 channel behavior Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Introducing Blatt-Weisskopf modification: � p � 2 l + 1 D l ( p R r ) Γ( p ) D l ( p R r ) Γ( p ) = Γ R D l ( pr ) ≡ ˜ D l ( pr ) . p R the phase shift derivative is given by: π ) ≃ 1 + p 4 R ( r 2 ) ′ ∂δ ( p ) M ˜ Γ( p ) < 0 ∂ ( M 2 4 p 4 R The phase shift goes down for low p and near M π = M phys π Agreement with standard and unitarized ChPT. Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
I =1 J =1 channel behavior Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary 0.7 δ 11 (deg.) O(p 4 ) 0.6 m π phys m π = 350 MeV 0.5 Unitarized ChPT 0.4 0.3 At low p the phase shift 0.2 ✛ decreases 0.1 as in standard ChPT 0 0 20 40 60 80 100 p (MeV) Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
I =1 J =1 channel behavior Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary 0.7 δ 11 (deg.) O(p 4 ) However at higher p ✛ 0.6 the phase shift grows m π phys m π = 350 MeV 0.5 Unitarized ChPT 0.4 0.3 At low p the phase shift 0.2 ✛ decreases 0.1 as in standard ChPT 0 0 20 40 60 80 100 p (MeV) Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Standard and unitarized ChPT phase shifts vs. lattice results ChPT J. Nebreda, J.R. Peláez and G. Ríos, Phys. Rev. D 83: 094011(2011) Lattice J. Dudek et al., Phys.Rev. D 83: 071504 (2011) K. Sasaki and N. Ishizuka, Phys. Rev. D 78, 014511 (2008) Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Scalar I=2 wave Scalar I=2 wave - one loop Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Scalar I=2 wave M π =139.57 MeV I=2 J=0 phase shift at one loop Standard ChPT Unitarized ChPT 0 0 m π = m π phys m π = m π phys -5 -5 -10 -10 -15 -15 δ 20 (deg.) δ 20 (deg.) -20 -20 -25 -25 -30 -30 0 100 200 300 400 500 600 0 100 200 300 400 500 600 p (MeV) p (MeV) Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Scalar I=2 wave M π =396 MeV I=2 J=0 phase shift at one loop Standard ChPT Unitarized ChPT 0 0 m π = m π phys m π = m π phys m π = 396 MeV m π = 396 MeV -5 -5 -10 -10 -15 -15 δ 20 (deg.) δ 20 (deg.) -20 -20 -25 -25 -30 -30 0 100 200 300 400 500 600 0 100 200 300 400 500 600 p (MeV) p (MeV) Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Scalar I=2 wave M π =420 MeV I=2 J=0 phase shift at one loop Standard ChPT Unitarized ChPT 0 0 m π = m π phys m π = m π phys m π = 396 MeV m π = 396 MeV -5 m π = 420 MeV -5 m π = 420 MeV -10 -10 -15 -15 δ 20 (deg.) δ 20 (deg.) -20 -20 -25 -25 -30 -30 0 100 200 300 400 500 600 0 100 200 300 400 500 600 p (MeV) p (MeV) Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Scalar I=2 wave M π =444 MeV I=2 J=0 phase shift at one loop Standard ChPT Unitarized ChPT 0 0 m π = m π phys m π = m π phys m π = 396 MeV m π = 396 MeV -5 m π = 420 MeV -5 m π = 420 MeV m π = 444 MeV m π = 444 MeV -10 -10 -15 -15 δ 20 (deg.) δ 20 (deg.) -20 -20 -25 -25 -30 -30 0 100 200 300 400 500 600 0 100 200 300 400 500 600 p (MeV) p (MeV) Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Scalar I=2 wave M π =524 MeV I=2 J=0 phase shift at one loop Standard ChPT Unitarized ChPT 0 0 m π = m π phys m π = m π phys m π = 396 MeV m π = 396 MeV -5 m π = 420 MeV -5 m π = 420 MeV m π = 444 MeV m π = 444 MeV m π = 524 MeV m π = 524 MeV -10 -10 -15 -15 δ 20 (deg.) δ 20 (deg.) -20 -20 -25 -25 -30 -30 0 100 200 300 400 500 600 0 100 200 300 400 500 600 p (MeV) p (MeV) Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Scalar I=2 wave M π =524 MeV I=2 J=0 phase shift at one loop Standard ChPT Unitarized ChPT 0 0 m π = m π phys m π = m π phys m π = 396 MeV m π = 396 MeV -5 m π = 420 MeV -5 m π = 420 MeV m π = 444 MeV m π = 444 MeV m π = 524 MeV m π = 524 MeV -10 -10 -15 -15 δ 20 (deg.) δ 20 (deg.) -20 -20 -25 -25 -30 -30 0 100 200 300 400 500 600 0 100 200 300 400 500 600 p (MeV) p (MeV) Improves behavior Limited to very low at higher momenta momenta Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Scalar I=2 wave Scalar I=2 wave - two loops Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Scalar I=2 wave M π =139.57 MeV I=2 J=0 phase shift at two loops Standard ChPT Unitarized ChPT 0 0 m π = m π phys m π = m π phys -5 -5 -10 -10 -15 -15 δ 20 (deg.) δ 20 (deg.) -20 -20 -25 -25 -30 -30 0 100 200 300 400 500 600 0 100 200 300 400 500 600 p (MeV) p (MeV) Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Scalar I=2 wave M π =396 MeV I=2 J=0 phase shift at two loops Standard ChPT Unitarized ChPT 0 0 m π = m π phys m π = m π phys m π = 396 MeV m π = 396 MeV -5 -5 -10 -10 -15 -15 δ 20 (deg.) δ 20 (deg.) -20 -20 -25 -25 -30 -30 0 100 200 300 400 500 600 0 100 200 300 400 500 600 p (MeV) p (MeV) Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Scalar I=2 wave M π =420 MeV I=2 J=0 phase shift at two loops Standard ChPT Unitarized ChPT 0 0 m π = m π phys m π = m π phys m π = 396 MeV m π = 396 MeV -5 m π = 420 MeV -5 m π = 420 MeV -10 -10 -15 -15 δ 20 (deg.) δ 20 (deg.) -20 -20 -25 -25 -30 -30 0 100 200 300 400 500 600 0 100 200 300 400 500 600 p (MeV) p (MeV) Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Scalar I=2 wave M π =444 MeV I=2 J=0 phase shift at two loops Standard ChPT Unitarized ChPT 0 0 m π = m π phys m π = m π phys m π = 396 MeV m π = 396 MeV -5 m π = 420 MeV -5 m π = 420 MeV m π = 444 MeV m π = 444 MeV -10 -10 -15 -15 δ 20 (deg.) δ 20 (deg.) -20 -20 -25 -25 -30 -30 0 100 200 300 400 500 600 0 100 200 300 400 500 600 p (MeV) p (MeV) Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Scalar I=2 wave M π =524 MeV I=2 J=0 phase shift at two loops Standard ChPT Unitarized ChPT 0 0 m π = m π phys m π = m π phys m π = 396 MeV m π = 396 MeV -5 m π = 420 MeV -5 m π = 420 MeV m π = 444 MeV m π = 444 MeV m π = 524 MeV m π = 524 MeV -10 -10 -15 -15 δ 20 (deg.) δ 20 (deg.) -20 -20 -25 -25 -30 -30 0 100 200 300 400 500 600 0 100 200 300 400 500 600 p (MeV) p (MeV) Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Scalar I=2 wave M π =524 MeV I=2 J=0 phase shift at two loops Standard ChPT Unitarized ChPT 0 0 m π = m π phys m π = m π phys m π = 396 MeV m π = 396 MeV -5 m π = 420 MeV -5 m π = 420 MeV m π = 444 MeV m π = 444 MeV m π = 524 MeV m π = 524 MeV -10 -10 -15 -15 δ 20 (deg.) δ 20 (deg.) -20 -20 -25 -25 -30 -30 0 100 200 300 400 500 600 0 100 200 300 400 500 600 p (MeV) p (MeV) Works better than Standard ChPT at high p Bends down faster than 1 loop No improvement No clear improvement either Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Tensor I=2 wave Tensor I=2 wave D waves are zero at tree level: IAM cannot be applied at one or two loops one and two-loops amplitudes are only LO and NLO Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Tensor I=2 wave I=2 J=2 phase shift in standard ChPT M π =139.57 MeV One loop Two loops m π = m π phys m π = m π phys 4 4 δ 22 (deg.) δ 22 (deg.) 2 2 0 0 -2 -2 -4 -4 0 100 200 300 400 500 600 0 100 200 300 400 500 600 p (MeV) p (MeV) Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Tensor I=2 wave M π =396 MeV I=2 J=2 phase shift in standard ChPT One loop Two loops m π = m π phys m π = m π phys 4 4 m π = 396 MeV m π = 396 MeV δ 22 (deg.) δ 22 (deg.) 2 2 0 0 -2 -2 -4 -4 0 100 200 300 400 500 600 0 100 200 300 400 500 600 p (MeV) p (MeV) Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Tensor I=2 wave M π =444 MeV I=2 J=2 phase shift in standard ChPT One loop Two loops m π = m π phys m π = m π phys 4 4 m π = 396 MeV m π = 396 MeV δ 22 (deg.) δ 22 (deg.) m π = 444 MeV m π = 444 MeV 2 2 0 0 -2 -2 -4 -4 0 100 200 300 400 500 600 0 100 200 300 400 500 600 p (MeV) p (MeV) Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Tensor I=2 wave M π =524 MeV I=2 J=2 phase shift in standard ChPT One loop Two loops m π = m π phys m π = m π phys 4 4 m π = 396 MeV m π = 396 MeV δ 22 (deg.) δ 22 (deg.) m π = 444 MeV m π = 444 MeV m π = 524 MeV m π = 524 MeV 2 2 0 0 -2 -2 -4 -4 0 100 200 300 400 500 600 0 100 200 300 400 500 600 p (MeV) p (MeV) Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Tensor I=2 wave M π =524 MeV I=2 J=2 phase shift in standard ChPT One loop Two loops m π = m π phys m π = m π phys 4 4 m π = 396 MeV m π = 396 MeV δ 22 (deg.) δ 22 (deg.) m π = 444 MeV m π = 444 MeV m π = 524 MeV m π = 524 MeV 2 2 0 0 -2 -2 -4 -4 0 100 200 300 400 500 600 0 100 200 300 400 500 600 p (MeV) p (MeV) Works up to higher p No improvement Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Scalar and vector mesons dependence on M π Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Quark mass dependence Generalization to SU ( 3 ) of previous work on SU ( 2 ) *. Elastic channels: ππ → ππ : resonances ρ and σ (comparison to SU ( 2 ) results) π K → π K : resonances K ∗ ( 892 ) and κ . m = m u + m d and m s ⇒ Change of ˆ 2 change of M 2 π , M 2 K , M 2 η , f π , f K , f η . Applicability in SU ( 3 ) : 0 < M π � 400 MeV ⇒ M K � 600 MeV (Being optimistic!) * C. Hanhart, J.R. Pelaez and G. Rios, Phys. Rev. Lett. 100 , 152001 (2008) Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Light vector mesons: ρ and K ∗ ( 892 ) Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary m dependence - Light vector mesons - Mass ˆ ^ / m ^ ^ / m ^ m m phys phys 0.5 1 2 3 4 5 6 7 8 9 0.5 1 2 3 4 5 6 7 8 9 1.3 1.3 1.25 1.25 M ρ /M ρ phys M ρ /M ρ phys M K * /M K * phys 1.2 1.2 1.15 1.15 1.1 1.1 1.05 1.05 1 1 0.95 0.95 0.9 0.9 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 M π /M π phys M π /M π phys Both masses increase slower than M π Agreement with SU(2) analysis (blue line)* * C. Hanhart, J.R. Pelaez and G. Rios, Phys. Rev. Lett. 100 , 152001 (2008) Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary m dependence - Light vector mesons - Width ˆ ^ / m ^ ^ / m ^ m m phys phys 0.5 1 2 3 4 5 6 7 8 9 0.5 1 2 3 4 5 6 7 8 9 1.4 1.4 1.2 1.2 1 1 0.8 0.8 Γ ρ / Γ ρ phys Γ ρ / Γ ρ phys Γ K * / Γ K * phys 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 M π /M π phys M π /M π phys C. Hanhart, J.R. Pelaez and G. Rios, Phys. Rev. Lett. 100 , 152001 (2008) Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary m dependence - Light vector mesons - Width ˆ ^ / m ^ ^ / m ^ m m phys phys 0.5 1 2 3 4 5 6 7 8 9 0.5 1 2 3 4 5 6 7 8 9 1.4 1.4 1.2 1.2 1 1 0.8 0.8 Γ ρ / Γ ρ phys Γ ρ / Γ ρ phys Γ K * / Γ K * phys 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 M π /M π phys M π /M π phys Width decrease in accordance with phase space reduction: | p | 3 Γ V = g 2 1 (black lines) M 2 8 π V Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary m dependence - Light vector mesons - Coupling ˆ ^ / m ^ ^ / m ^ m m phys phys 0.5 1 2 3 4 5 6 7 8 9 0.5 1 2 3 4 5 6 7 8 9 1.2 1.2 1.15 1.15 g ρππ /g ρππ phys g ρππ /g ρππ phys g K* π K /g K* π K phys 1.1 1.1 1.05 1.05 1 1 0.95 0.95 0.9 0.9 0.85 0.85 0.8 0.8 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 M π /M π phys M π /M π phys m (assumption in Coupling to two mesons independent of ˆ some lattice works) Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary m dependence - Light vector mesons - KSFR ˆ ^ / m ^ ^ / m ^ m m phys phys 0.5 1 2 3 4 5 6 7 8 9 0.5 1 2 3 4 5 6 7 8 9 1.2 1.2 1.15 1.15 (M ρ /f π ) / (M ρ phys /f π phys ) (M ρ /f π ) / (M ρ phys /f π phys ) (M K* /f π ) / (M K* phys /f π phys ) 1.1 1.1 1.05 1.05 1 1 0.95 0.95 0.9 0.9 0.85 0.85 0.8 0.8 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 M π /M π phys M π /M π phys m : Fulfill the KSFR relation for different ˆ √ g ≃ M V / 2 2 f π Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Light scalar mesons: σ and κ Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary m dependence - Light scalar mesons - Mass ˆ ^ / m ^ ^ / m ^ m m phys phys 0.5 1 2 3 4 5 6 7 8 9 0.5 1 2 3 4 5 6 7 8 9 1.8 1.8 1.7 1.7 M σ /M σ phys M σ /M σ phys M κ /M κ phys 1.6 1.6 1.5 1.5 1.4 1.4 1.3 1.3 1.2 1.2 1.1 1.1 1 1 0.9 0.9 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 M π /M π phys M π /M π phys Mass split into two branches Agreement with SU(2) analysis * C. Hanhart, J.R. Pelaez and G. Rios, Phys. Rev. Lett. 100 , 152001 (2008) Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary m dependence - Light scalar mesons - Width ˆ ^ / m ^ ^ / m ^ m m phys phys 0.5 1 2 3 4 5 6 7 8 9 0.5 1 2 3 4 5 6 7 8 9 1.8 1.8 1.6 1.6 Γ σ / Γ σ phys Γ σ / Γ σ phys Γ κ / Γ κ phys 1.4 1.4 1.2 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 M π /M π phys M π /M π phys C. Hanhart, J.R. Pelaez and G. Rios, Phys. Rev. Lett. 100 , 152001 (2008) Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary m dependence - Light scalar mesons - Width ˆ ^ / m ^ ^ / m ^ m m phys phys 0.5 1 2 3 4 5 6 7 8 9 0.5 1 2 3 4 5 6 7 8 9 1.8 1.8 1.6 1.6 Γ σ / Γ σ phys Γ σ / Γ σ phys Γ κ / Γ κ phys 1.4 1.4 1.2 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 M π /M π phys M π /M π phys Width decrease not explained by phase space reduction: | p | Γ S = g 2 1 M 2 8 π S Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary m dependence - Light scalar mesons - Coupling ˆ ^ / m ^ ^ / m ^ m m phys phys 0.5 1 2 3 4 5 6 7 8 9 0.5 1 2 3 4 5 6 7 8 9 3.5 3.5 3 3 2.5 2.5 g σππ /g σππ phys g σππ /g σππ phys g κπ K /g κπ K phys 2 2 1.5 1.5 1 1 0.5 0.5 0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 M π /M π phys M π /M π phys m dependence of coupling to two mesons Strong ˆ Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Summary Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Summary Chiral extrapolation of the parameters of the σ ( f 0 ( 600 ) ), κ ( 800 ) , ρ ( 770 ) and K ∗ ( 892 ) resonances increasing ˆ m . Vector mesons Scalar mesons Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Summary Chiral extrapolation of the parameters of the σ ( f 0 ( 600 ) ), κ ( 800 ) , ρ ( 770 ) and K ∗ ( 892 ) resonances increasing ˆ m . Vector mesons vector resonances mass grows slower than M π , Scalar mesons Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Summary Chiral extrapolation of the parameters of the σ ( f 0 ( 600 ) ), κ ( 800 ) , ρ ( 770 ) and K ∗ ( 892 ) resonances increasing ˆ m . Vector mesons vector resonances mass grows slower than M π , coupling to two mesons almost independent of M π , Scalar mesons Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Summary Chiral extrapolation of the parameters of the σ ( f 0 ( 600 ) ), κ ( 800 ) , ρ ( 770 ) and K ∗ ( 892 ) resonances increasing ˆ m . Vector mesons vector resonances mass grows slower than M π , coupling to two mesons almost independent of M π , KSFR is well satisfied for different quark masses. Scalar mesons Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Summary Chiral extrapolation of the parameters of the σ ( f 0 ( 600 ) ), κ ( 800 ) , ρ ( 770 ) and K ∗ ( 892 ) resonances increasing ˆ m . Vector mesons vector resonances mass grows slower than M π , coupling to two mesons almost independent of M π , KSFR is well satisfied for different quark masses. Scalar mesons very different behavior from vector mesons: two branches, Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Summary Chiral extrapolation of the parameters of the σ ( f 0 ( 600 ) ), κ ( 800 ) , ρ ( 770 ) and K ∗ ( 892 ) resonances increasing ˆ m . Vector mesons vector resonances mass grows slower than M π , coupling to two mesons almost independent of M π , KSFR is well satisfied for different quark masses. Scalar mesons very different behavior from vector mesons: two branches, σ and κ show different quantitative but similar qualitative behavior, Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Summary Chiral extrapolation of the parameters of the σ ( f 0 ( 600 ) ), κ ( 800 ) , ρ ( 770 ) and K ∗ ( 892 ) resonances increasing ˆ m . Vector mesons vector resonances mass grows slower than M π , coupling to two mesons almost independent of M π , KSFR is well satisfied for different quark masses. Scalar mesons very different behavior from vector mesons: two branches, σ and κ show different quantitative but similar qualitative behavior, coupling to two mesons shows stronger M π dependence. Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Summary We have presented recent results for the phase shifts M π dependence : Standard ChPT Unitarized ChPT Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Summary We have presented recent results for the phase shifts M π dependence : Standard ChPT very soft M π dependence once threshold is "subtracted", Unitarized ChPT Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Summary We have presented recent results for the phase shifts M π dependence : Standard ChPT very soft M π dependence once threshold is "subtracted", surprising decrease of phase in vector channel, Unitarized ChPT Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Summary We have presented recent results for the phase shifts M π dependence : Standard ChPT very soft M π dependence once threshold is "subtracted", surprising decrease of phase in vector channel, S2 wave: agreement with lattice only at very low p , Unitarized ChPT Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Summary We have presented recent results for the phase shifts M π dependence : Standard ChPT very soft M π dependence once threshold is "subtracted", surprising decrease of phase in vector channel, S2 wave: agreement with lattice only at very low p , D2 wave: fair agreement with lattice at 1 loop, spoilt at 2 loops. Unitarized ChPT Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Summary We have presented recent results for the phase shifts M π dependence : Standard ChPT very soft M π dependence once threshold is "subtracted", surprising decrease of phase in vector channel, S2 wave: agreement with lattice only at very low p , D2 wave: fair agreement with lattice at 1 loop, spoilt at 2 loops. Unitarized ChPT S2 wave: better agreement with lattice at high p , Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Summary We have presented recent results for the phase shifts M π dependence : Standard ChPT very soft M π dependence once threshold is "subtracted", surprising decrease of phase in vector channel, S2 wave: agreement with lattice only at very low p , D2 wave: fair agreement with lattice at 1 loop, spoilt at 2 loops. Unitarized ChPT S2 wave: better agreement with lattice at high p , similar results at one and two loops, Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Summary We have presented recent results for the phase shifts M π dependence : Standard ChPT very soft M π dependence once threshold is "subtracted", surprising decrease of phase in vector channel, S2 wave: agreement with lattice only at very low p , D2 wave: fair agreement with lattice at 1 loop, spoilt at 2 loops. Unitarized ChPT S2 wave: better agreement with lattice at high p , similar results at one and two loops, reconciles ρ ChPT behavior with naive expectation, Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Summary We have presented recent results for the phase shifts M π dependence : Standard ChPT very soft M π dependence once threshold is "subtracted", surprising decrease of phase in vector channel, S2 wave: agreement with lattice only at very low p , D2 wave: fair agreement with lattice at 1 loop, spoilt at 2 loops. Unitarized ChPT S2 wave: better agreement with lattice at high p , similar results at one and two loops, reconciles ρ ChPT behavior with naive expectation, bound states seen as 2 π jump in phase shift (Levinson’s). Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary Summary We have presented recent results for the phase shifts M π dependence *: Standard ChPT very soft M π dependence once threshold is "subtracted", surprising decrease of phase in vector channel, S2 wave: agreement with lattice only at very low p , D2 wave: fair agreement with lattice at 1 loop, spoilt at 2 loops. Unitarized ChPT S2 wave: better agreement with lattice at high p , similar results at one and two loops, reconciles ρ ChPT behavior with naive expectation, bound states seen as 2 π jump in phase shift (Levinson’s). *C. Hanhart, J.R. Pelaez and G. Rios, Phys. Rev. Lett. 100 , 152001 (2008) Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts
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