Quark mass dependence of light resonances and phase shifts in elastic - - PowerPoint PPT Presentation

Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary

Quark mass dependence of light resonances and phase shifts in elastic ππ and πK scattering

Jenifer Nebreda, J. R. Peláez and G. Ríos

Universidad Complutense de Madrid

Hadron 2011, Munich June 13-17, 2011

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

Motivation Phase shifts Mπ dependence in Standard ChPT Phase shifts Mπ dependence in Unitarized ChPT Comparison of ChPT and lattice results Light resonances dependence on ˆ m 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

Motivation

Lattice: rigorous QCD results with quarks and gluons. Growing interest in scattering and scalar sector. Caveat: small, realistic quark masses are difficult to implement. ChPT: QCD dependence on quark masses as an expansion. We can compare: Lattice multi-hadron states calculations → phase shifts and scattering lengths vs. standard ChPT (model independent) or UChPT (to go higher in √s) Lattice spectrum calculations → masses vs. UChPT

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 Chiral Perturbation Theory

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

Chiral Perturbation Theory Weinberg, Gasser & Leutwyler

Low energy effective theory of QCD with: DOF: Pseudo-Goldstone Bosons of the spontaneous chiral symmetry breaking SU(Nf)L × SU(Nf)R → SU(Nf)V Nf=2 → π’s Nf=3 → π’s, K’s and η expansion in masses and momenta Leff = L2 + L4 + L6 + · · · parameters: Low Energy Constants (LECs) Nf=2 → 4 l’s (one loop) and 7 r’s (two loops) Nf=3 → 8 L’s (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 ππ scattering in SU(2) standard ChPT

ππ scattering in SU(2) standard ChPT: Already calculated to 1 and 2 loops*, we study the phases dependence on ˆ m = mu+md

2

. Advantages:

• SISTEMATIC EXPANSION, MODEL INDEPENDENT
• some lattice groups already giving results

for I=2 phases and scattering lenghts** Limitations:

• only low energy region
• no resonances.

*J. Bijnens, G. Colangelo, G. Ecker, J. Gasser and M. E. Sainio, Phys. Lett. B 374, 210 (1996) ** 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 ππ scattering in SU(2) standard ChPT

Standard SU(2) ChPT amplitudes with LECs from

• G. Colangelo, J. Gasser and H. Leutwyler, Nucl. Phys. B 603, 125 (2001)

O(p4) LECs (×10−3) O(p6) LECs(×10−4) lr

1

• 3.98 ± 0.62

rr

1

• 0.60

lr

2

1.89 ± 0.23 rr

2

1.28 l3 0.82 ± -3.80 rr

3

• 1.68

lr

4

6.17 ± 1.39 rr

4

• 1.00

rr

5

1.52 ± 0.42 rr

6

0.40 ± 0.04

Statistical error, not systematic

Change ˆ m ⇒ change on M2

π = 2 ˆ

mB0 ⇒ change on fπ

(one more O(p6) parameter: rr

f ≈ 0 ± 1.2 × 10−4)

Uncertainties in phase shifts: Montecarlo Gaussian Sampling.

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 ChPT δ dependence on Mπ

Phase shifts vs. Momentum, increasing Mπ Phases vs. energy → ˆ m dependence from the threshold’s shift. Better to plot phases vs. momentum.

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 ChPT δ dependence on Mπ 10 20 30 40 50 60 70 50 100 150 200 250 300

One loop

δ00 O(p4)

Mπ = 139.57 MeV

• 14
• 12
• 10
• 8
• 6
• 4
• 2

50 100 150 200 250 300

δ20 O(p4)

5 10 15 20 50 100 150 200 250 300

δ11 O(p4)

• 0.5

0.5 1 1.5 2 2.5 50 100 150 200 250 300 p (MeV)

δ02 O(p4)

• 2
• 1

1 2 3 50 100 150 200 250 300 p (MeV)

δ22 O(p4)

Standard 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 Standard ChPT δ dependence on Mπ 10 20 30 40 50 60 70 50 100 150 200 250 300

One loop

δ00 O(p4)

Mπ = 139.57 MeV Mπ = 230 MeV

• 14
• 12
• 10
• 8
• 6
• 4
• 2

50 100 150 200 250 300

δ20 O(p4)

5 10 15 20 50 100 150 200 250 300

δ11 O(p4)

• 0.5

0.5 1 1.5 2 2.5 50 100 150 200 250 300 p (MeV)

δ02 O(p4)

• 2
• 1

1 2 3 50 100 150 200 250 300 p (MeV)

δ22 O(p4)

Standard 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 Standard ChPT δ dependence on Mπ 10 20 30 40 50 60 70 50 100 150 200 250 300

One loop

δ00 O(p4)

Mπ = 139.57 MeV Mπ = 230 MeV Mπ = 300 MeV

• 14
• 12
• 10
• 8
• 6
• 4
• 2

50 100 150 200 250 300

δ20 O(p4)

5 10 15 20 50 100 150 200 250 300

δ11 O(p4)

• 0.5

0.5 1 1.5 2 2.5 50 100 150 200 250 300 p (MeV)

δ02 O(p4)

• 2
• 1

1 2 3 50 100 150 200 250 300 p (MeV)

δ22 O(p4)

Standard 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 Standard ChPT δ dependence on Mπ 10 20 30 40 50 60 70 50 100 150 200 250 300

One loop

δ00 O(p4)

Mπ = 139.57 MeV Mπ = 230 MeV Mπ = 300 MeV Mπ = 420 MeV

• 14
• 12
• 10
• 8
• 6
• 4
• 2

50 100 150 200 250 300

δ20 O(p4)

5 10 15 20 50 100 150 200 250 300

δ11 O(p4)

• 0.5

0.5 1 1.5 2 2.5 50 100 150 200 250 300 p (MeV)

δ02 O(p4)

• 2
• 1

1 2 3 50 100 150 200 250 300 p (MeV)

δ22 O(p4)

Standard 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 Standard ChPT δ dependence on Mπ 10 20 30 40 50 60 70 50 100 150 200 250 300

One loop

δ00 O(p4)

Mπ = 139.57 MeV Mπ = 230 MeV Mπ = 300 MeV Mπ = 420 MeV

• 14
• 12
• 10
• 8
• 6
• 4
• 2

50 100 150 200 250 300

δ20 O(p4)

5 10 15 20 50 100 150 200 250 300

δ11 O(p4)

• 0.5

0.5 1 1.5 2 2.5 50 100 150 200 250 300 p (MeV)

δ02 O(p4)

• 2
• 1

1 2 3 50 100 150 200 250 300 p (MeV)

δ22 O(p4)

If threshold is "subtracted": VERY SOFT Mπ 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 Standard ChPT δ dependence on Mπ 10 20 30 40 50 60 70 50 100 150 200 250 300

One loop

δ00 O(p4)

Mπ = 139.57 MeV Mπ = 230 MeV Mπ = 300 MeV Mπ = 420 MeV

• 14
• 12
• 10
• 8
• 6
• 4
• 2

50 100 150 200 250 300

δ20 O(p4)

5 10 15 20 50 100 150 200 250 300

δ11 O(p4)

• 0.5

0.5 1 1.5 2 2.5 50 100 150 200 250 300 p (MeV)

δ02 O(p4)

• 2
• 1

1 2 3 50 100 150 200 250 300 p (MeV)

δ22 O(p4)

If threshold is "subtracted": VERY SOFT Mπ dependence! ρ channel phase shift

decreases?? ❳❳❳

③

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 ChPT δ dependence on Mπ 10 20 30 40 50 60 70 50 100 150 200 250 300

Two loops

δ00 O(p6)

Mπ = 139.57 MeV

• 14
• 12
• 10
• 8
• 6
• 4
• 2

50 100 150 200 250 300

δ20 O(p6)

5 10 15 20 50 100 150 200 250 300

δ11 O(p6)

• 0.5

0.5 1 1.5 2 2.5 50 100 150 200 250 300 p (MeV)

δ02 O(p6)

• 2
• 1

1 2 3 50 100 150 200 250 300 p (MeV)

δ22 O(p6)

Standard 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 Standard ChPT δ dependence on Mπ 10 20 30 40 50 60 70 50 100 150 200 250 300

Two loops

δ00 O(p6)

Mπ = 139.57 MeV

• 14
• 12
• 10
• 8
• 6
• 4
• 2

50 100 150 200 250 300

δ20 O(p6)

5 10 15 20 50 100 150 200 250 300

δ11 O(p6)

• 0.5

0.5 1 1.5 2 2.5 50 100 150 200 250 300 p (MeV)

δ02 O(p6)

• 2
• 1

1 2 3 50 100 150 200 250 300 p (MeV)

δ22 O(p6)

Attention: 2 loops is just NLO for D waves

❩❩ ❩ ⑦

Standard 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 Standard ChPT δ dependence on Mπ 10 20 30 40 50 60 70 50 100 150 200 250 300

Two loops

δ00 O(p6)

Mπ = 139.57 MeV Mπ = 230 MeV

• 14
• 12
• 10
• 8
• 6
• 4
• 2

50 100 150 200 250 300

δ20 O(p6)

5 10 15 20 50 100 150 200 250 300

δ11 O(p6)

• 0.5

0.5 1 1.5 2 2.5 50 100 150 200 250 300 p (MeV)

δ02 O(p6)

• 2
• 1

1 2 3 50 100 150 200 250 300 p (MeV)

δ22 O(p6)

Standard 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 Standard ChPT δ dependence on Mπ 10 20 30 40 50 60 70 50 100 150 200 250 300

Two loops

δ00 O(p6)

Mπ = 139.57 MeV Mπ = 230 MeV Mπ = 300 MeV

• 14
• 12
• 10
• 8
• 6
• 4
• 2

50 100 150 200 250 300

δ20 O(p6)

5 10 15 20 50 100 150 200 250 300

δ11 O(p6)

• 0.5

0.5 1 1.5 2 2.5 50 100 150 200 250 300 p (MeV)

δ02 O(p6)

• 2
• 1

1 2 3 50 100 150 200 250 300 p (MeV)

δ22 O(p6)

Standard 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 Standard ChPT δ dependence on Mπ 10 20 30 40 50 60 70 50 100 150 200 250 300

Two loops

δ00 O(p6)

Mπ = 139.57 MeV Mπ = 230 MeV Mπ = 300 MeV Mπ = 420 MeV

• 14
• 12
• 10
• 8
• 6
• 4
• 2

50 100 150 200 250 300

δ20 O(p6)

5 10 15 20 50 100 150 200 250 300

δ11 O(p6)

• 0.5

0.5 1 1.5 2 2.5 50 100 150 200 250 300 p (MeV)

δ02 O(p6)

• 2
• 1

1 2 3 50 100 150 200 250 300 p (MeV)

δ22 O(p6)

Standard 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 Standard ChPT δ dependence on Mπ 10 20 30 40 50 60 70 50 100 150 200 250 300

Two loops

δ00 O(p6)

Mπ = 139.57 MeV Mπ = 230 MeV Mπ = 300 MeV Mπ = 420 MeV

• 14
• 12
• 10
• 8
• 6
• 4
• 2

50 100 150 200 250 300

δ20 O(p6)

5 10 15 20 50 100 150 200 250 300

δ11 O(p6)

• 0.5

0.5 1 1.5 2 2.5 50 100 150 200 250 300 p (MeV)

δ02 O(p6)

• 2
• 1

1 2 3 50 100 150 200 250 300 p (MeV)

δ22 O(p6)

Standard ChPT

Still soft Mπ

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 Standard ChPT δ dependence on Mπ 10 20 30 40 50 60 70 50 100 150 200 250 300

Two loops

δ00 O(p6)

Mπ = 139.57 MeV Mπ = 230 MeV Mπ = 300 MeV Mπ = 420 MeV

• 14
• 12
• 10
• 8
• 6
• 4
• 2

50 100 150 200 250 300

δ20 O(p6)

5 10 15 20 50 100 150 200 250 300

δ11 O(p6)

• 0.5

0.5 1 1.5 2 2.5 50 100 150 200 250 300 p (MeV)

δ02 O(p6)

• 2
• 1

1 2 3 50 100 150 200 250 300 p (MeV)

δ22 O(p6)

Still soft Mπ

dependence

ρ channel phase shift

decreases?? ❳❳❳

③

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(p4) IAM partial waves: t(s) ≃ t2

2(s)

t2(s) − t4(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(p6) IAM partial waves: t(s) ≃ t2

2(s)

t2(s) − t4(s) + t42

t2 − t6

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 Unitarized ChPT δ dependence on Mπ

SU(2) Unitarized ChPT phase shifts vs. Momentum

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 δ dependence on Mπ

Unitarized SU(2) ChPT amplitudes with LECs: One loop O(p4) LECs (×10−3) lr

1(µ)

−3.7 ± 0.2 lr

2(µ)

5.0 ± 0.4 lr

3(µ)

0.8 ± 3.8 lr

4(µ)

6.2 ± 5.7 Two loops Set A Set D O(p4)(x10−3) lr

1(µ)

• 5.0
• 4.0

lr

2(µ)

1.7 1.2 lr

3(µ)

0.8 0.8 lr

4(µ)

6.5 6.5 O(p6)(x10−4) rr

1(µ)

• 0.6
• 0.6

rr

2(µ)

1.3 1.5 rr

3(µ)

• 1.7
• 3.3

rr

4(µ)

2.0 0.9 rr

5(µ)

2.0 1.7 rr

6(µ)

• 0.6
• 0.7

rr

f (µ)

• 1.4
• 1.8

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 δ dependence on Mπ

20 40 60 80 100 120 100 200 300 400 500 p (MeV)

One loop

δ00 O(p4)

Mπ = 139.57 MeV

• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 p (MeV)

δ20 O(p4)

20 40 60 80 100 120 140 160 180 100 200 300 400 500 p (MeV)

δ11 O(p4)

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 Unitarized ChPT δ dependence on Mπ

20 40 60 80 100 120 100 200 300 400 500 p (MeV)

One loop

δ00 O(p4)

Mπ = 139.57 MeV Mπ = 230 MeV

• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 p (MeV)

δ20 O(p4)

20 40 60 80 100 120 140 160 180 100 200 300 400 500 p (MeV)

δ11 O(p4)

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 Unitarized ChPT δ dependence on Mπ

20 40 60 80 100 120 100 200 300 400 500 p (MeV)

One loop

δ00 O(p4)

Mπ = 139.57 MeV Mπ = 230 MeV Mπ = 300 MeV

• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 p (MeV)

δ20 O(p4)

20 40 60 80 100 120 140 160 180 100 200 300 400 500 p (MeV)

δ11 O(p4)

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 Unitarized ChPT δ dependence on Mπ

20 40 60 80 100 120 100 200 300 400 500 p (MeV)

One loop

δ00 O(p4)

Mπ = 139.57 MeV Mπ = 230 MeV Mπ = 300 MeV Mπ = 420 MeV

• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 p (MeV)

δ20 O(p4)

20 40 60 80 100 120 140 160 180 100 200 300 400 500 p (MeV)

δ11 O(p4)

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 Unitarized ChPT δ dependence on Mπ

20 40 60 80 100 120 100 200 300 400 500 p (MeV)

One loop

δ00 O(p4)

Mπ = 139.57 MeV Mπ = 230 MeV Mπ = 300 MeV Mπ = 420 MeV

• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 p (MeV)

δ20 O(p4)

20 40 60 80 100 120 140 160 180 100 200 300 400 500 p (MeV)

δ11 O(p4)

ρ channel phase shift

increases!!

contradiction with ChPT??

❳❳ ❳ ③

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 Unitarized ChPT δ dependence on Mπ

20 40 60 80 100 120 140 160 180 100 200 300 400 500 p (MeV)

Two loops

δ00 O(p6)

Mπ = 139.57 MeV

• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 p (MeV)

δ20 O(p6)

20 40 60 80 100 120 140 160 180 100 200 300 400 500 p (MeV)

δ11 O(p6)

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 Unitarized ChPT δ dependence on Mπ

20 40 60 80 100 120 140 160 180 100 200 300 400 500 p (MeV)

Two loops

δ00 O(p6)

Mπ = 139.57 MeV Mπ = 230 MeV

• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 p (MeV)

δ20 O(p6)

20 40 60 80 100 120 140 160 180 100 200 300 400 500 p (MeV)

δ11 O(p6)

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 Unitarized ChPT δ dependence on Mπ

20 40 60 80 100 120 140 160 180 100 200 300 400 500 p (MeV)

Two loops

δ00 O(p6)

Mπ = 139.57 MeV Mπ = 230 MeV Mπ = 300 MeV

• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 p (MeV)

δ20 O(p6)

20 40 60 80 100 120 140 160 180 100 200 300 400 500 p (MeV)

δ11 O(p6)

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 Unitarized ChPT δ dependence on Mπ

20 40 60 80 100 120 140 160 180 100 200 300 400 500 p (MeV)

Two loops

δ00 O(p6)

Mπ = 139.57 MeV Mπ = 230 MeV Mπ = 300 MeV Mπ = 420 MeV

• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 p (MeV)

δ20 O(p6)

20 40 60 80 100 120 140 160 180 100 200 300 400 500 p (MeV)

δ11 O(p6)

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 Unitarized ChPT δ dependence on Mπ

20 40 60 80 100 120 140 160 180 100 200 300 400 500 p (MeV)

Two loops

δ00 O(p6)

Mπ = 139.57 MeV Mπ = 230 MeV Mπ = 300 MeV Mπ = 420 MeV

• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 p (MeV)

δ20 O(p6)

20 40 60 80 100 120 140 160 180 100 200 300 400 500 p (MeV)

δ11 O(p6)

ρ channel phase shift

increases!!

contradiction with ChPT??

❳❳ ❳ ③

Bound state: phase jumps 2π

(Levinson’s theorem) ✓ ✓ ✓ ✓ ✴

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 I=1 J=1 channel behavior

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

Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary I=1 J=1 channel behavior

For a simple Breit-Wigner parametrization: t(s) = −√sM Γ(p)/2p s − M2 + iMΓ(p) with Γ(p) = ΓR p pR 3 we get a positive phase shift derivative: ∂δ(p) ∂(M2

π) = − ∂δ(p)

∂(p2

R) =

4MΓ(p)

• 4p2 − 4p2

R

2 + M2Γ(p)2 > 0. 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

Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary I=1 J=1 channel behavior

For a simple Breit-Wigner parametrization: t(s) = −√sM Γ(p)/2p s − M2 + iMΓ(p) with Γ(p) = ΓR p pR 3 we get a positive phase shift derivative: ∂δ(p) ∂(M2

π) = − ∂δ(p)

∂(p2

R) =

4MΓ(p)

• 4p2 − 4p2

R

2 + M2Γ(p)2 > 0. 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

Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary I=1 J=1 channel behavior

For a simple Breit-Wigner parametrization: t(s) = −√sM Γ(p)/2p s − M2 + iMΓ(p) with Γ(p) = ΓR p pR 3 we get a positive phase shift derivative: ∂δ(p) ∂(M2

π) = − ∂δ(p)

∂(p2

R) =

4MΓ(p)

• 4p2 − 4p2

R

2 + M2Γ(p)2 > 0. 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

Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary I=1 J=1 channel behavior

Introducing Blatt-Weisskopf modification: Γ(p) = ΓR p pR 2l+1 Dl(pRr) Dl(pr) ≡ ˜ Γ(p)Dl(pRr) Dl(pr) the phase shift derivative is given by: ∂δ(p) ∂(M2

π) ≃ 1 + p4 R(r2)′

4p4

R

M˜ Γ(p) Estimation of r2 matching LO ChPT at low p: r2 = 1 g2f 2

π

M Mπ + O(M0

π) ⇒ 1 + p4 R(r2)′ = 1 −

M p4

R

2g2f 2

πM3 π

< 0

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 I=1 J=1 channel behavior

Introducing Blatt-Weisskopf modification: Γ(p) = ΓR p pR 2l+1 Dl(pRr) Dl(pr) ≡ ˜ Γ(p)Dl(pRr) Dl(pr) = ˜ Γ(p)1 + (pRr)2 1 + (pr)2 the phase shift derivative is given by: ∂δ(p) ∂(M2

π) ≃ 1 + p4 R(r2)′

4p4

R

M˜ Γ(p) Estimation of r2 matching LO ChPT at low p: r2 = 1 g2f 2

π

M Mπ + O(M0

π) ⇒ 1 + p4 R(r2)′ = 1 −

M p4

R

2g2f 2

πM3 π

< 0

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 I=1 J=1 channel behavior

Introducing Blatt-Weisskopf modification: Γ(p) = ΓR p pR 2l+1 Dl(pRr) Dl(pr) ≡ ˜ Γ(p)Dl(pRr) Dl(pr) = ˜ Γ(p)1 + (pRr)2 1 + (pr)2 the phase shift derivative is given by: ∂δ(p) ∂(M2

π) ≃ 1 + p4 R(r2)′

4p4

R

M˜ Γ(p) Estimation of r2 matching LO ChPT at low p: r2 = 1 g2f 2

π

M Mπ + O(M0

π) ⇒ 1 + p4 R(r2)′ = 1 −

M p4

R

2g2f 2

πM3 π

< 0

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 I=1 J=1 channel behavior

Introducing Blatt-Weisskopf modification: Γ(p) = ΓR p pR 2l+1 Dl(pRr) Dl(pr) ≡ ˜ Γ(p)Dl(pRr) Dl(pr) = ˜ Γ(p)1 + (pRr)2 1 + (pr)2 the phase shift derivative is given by: ∂δ(p) ∂(M2

π) ≃ 1 + p4 R(r2)′

4p4

R

M˜ Γ(p) Estimation of r2 matching LO ChPT at low p: r2 = 1 g2f 2

π

M Mπ + O(M0

π) ⇒ 1 + p4 R(r2)′ = 1 −

M p4

R

2g2f 2

πM3 π

< 0

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 I=1 J=1 channel behavior

Introducing Blatt-Weisskopf modification: Γ(p) = ΓR p pR 2l+1 Dl(pRr) Dl(pr) ≡ ˜ Γ(p)Dl(pRr) Dl(pr) = ˜ Γ(p)1 + (pRr)2 1 + (pr)2 the phase shift derivative is given by: ∂δ(p) ∂(M2

π) ≃ 1 + p4 R(r2)′

4p4

R

M˜ Γ(p) < 0 Estimation of r2 matching LO ChPT at low p: r2 = 1 g2f 2

π

M Mπ + O(M0

π) ⇒ 1 + p4 R(r2)′ = 1 −

M p4

R

2g2f 2

πM3 π

< 0

✖✕ ✗✔ ❙ ❙ ❙ ❙ ❙ ❙ ♦ ✖✕ ✗✔

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 I=1 J=1 channel behavior

Introducing Blatt-Weisskopf modification: Γ(p) = ΓR p pR 2l+1 Dl(pRr) Dl(pr) ≡ ˜ Γ(p)Dl(pRr) Dl(pr) . the phase shift derivative is given by: ∂δ(p) ∂(M2

π) ≃ 1 + p4 R(r2)′

4p4

R

M˜ Γ(p) < 0 The phase shift goes down for low p and near Mπ = Mphys

π

Agreement with standard and 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 I=1 J=1 channel behavior

0.1 0.2 0.3 0.4 0.5 0.6 0.7 20 40 60 80 100 p (MeV)

δ11 (deg.) O(p4) mπ phys mπ = 350 MeV

Unitarized ChPT At low p the phase shift decreases as in standard 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 I=1 J=1 channel behavior

0.1 0.2 0.3 0.4 0.5 0.6 0.7 20 40 60 80 100 p (MeV)

δ11 (deg.) O(p4) mπ phys mπ = 350 MeV

Unitarized ChPT However at higher p the phase shift grows At low p the phase shift decreases as in standard 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

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

I=2 J=0 phase shift at one loop Mπ=139.57 MeV

• 30
• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 600 p (MeV)

δ20 (deg.) mπ = mπ phys

• 30
• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 600 p (MeV)

δ20 (deg.) mπ = mπ phys

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 Scalar I=2 wave

I=2 J=0 phase shift at one loop Mπ=396 MeV

• 30
• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 600 p (MeV)

δ20 (deg.) mπ = mπ phys mπ = 396 MeV

• 30
• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 600 p (MeV)

δ20 (deg.) mπ = mπ phys mπ = 396 MeV

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 Scalar I=2 wave

I=2 J=0 phase shift at one loop Mπ=420 MeV

• 30
• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 600 p (MeV)

δ20 (deg.) mπ = mπ phys mπ = 396 MeV mπ = 420 MeV

• 30
• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 600 p (MeV)

δ20 (deg.) mπ = mπ phys mπ = 396 MeV mπ = 420 MeV

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 Scalar I=2 wave

I=2 J=0 phase shift at one loop Mπ=444 MeV

• 30
• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 600 p (MeV)

δ20 (deg.) mπ = mπ phys mπ = 396 MeV mπ = 420 MeV mπ = 444 MeV

• 30
• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 600 p (MeV)

δ20 (deg.) mπ = mπ phys mπ = 396 MeV mπ = 420 MeV mπ = 444 MeV

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 Scalar I=2 wave

I=2 J=0 phase shift at one loop Mπ=524 MeV

• 30
• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 600 p (MeV)

δ20 (deg.) mπ = mπ phys mπ = 396 MeV mπ = 420 MeV mπ = 444 MeV mπ = 524 MeV

• 30
• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 600 p (MeV)

δ20 (deg.) mπ = mπ phys mπ = 396 MeV mπ = 420 MeV mπ = 444 MeV mπ = 524 MeV

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 Scalar I=2 wave

I=2 J=0 phase shift at one loop Mπ=524 MeV

• 30
• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 600 p (MeV)

δ20 (deg.) mπ = mπ phys mπ = 396 MeV mπ = 420 MeV mπ = 444 MeV mπ = 524 MeV

• 30
• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 600 p (MeV)

δ20 (deg.) mπ = mπ phys mπ = 396 MeV mπ = 420 MeV mπ = 444 MeV mπ = 524 MeV

Standard ChPT Unitarized ChPT Limited to very low momenta Improves behavior at higher 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

I=2 J=0 phase shift at two loops Mπ=139.57 MeV

• 30
• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 600 p (MeV)

δ20 (deg.) mπ = mπ phys

• 30
• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 600 p (MeV)

δ20 (deg.) mπ = mπ phys

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 Scalar I=2 wave

I=2 J=0 phase shift at two loops Mπ=396 MeV

• 30
• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 600 p (MeV)

δ20 (deg.) mπ = mπ phys mπ = 396 MeV

• 30
• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 600 p (MeV)

δ20 (deg.) mπ = mπ phys mπ = 396 MeV

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 Scalar I=2 wave

I=2 J=0 phase shift at two loops Mπ=420 MeV

• 30
• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 600 p (MeV)

δ20 (deg.) mπ = mπ phys mπ = 396 MeV mπ = 420 MeV

• 30
• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 600 p (MeV)

δ20 (deg.) mπ = mπ phys mπ = 396 MeV mπ = 420 MeV

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 Scalar I=2 wave

I=2 J=0 phase shift at two loops Mπ=444 MeV

• 30
• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 600 p (MeV)

δ20 (deg.) mπ = mπ phys mπ = 396 MeV mπ = 420 MeV mπ = 444 MeV

• 30
• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 600 p (MeV)

δ20 (deg.) mπ = mπ phys mπ = 396 MeV mπ = 420 MeV mπ = 444 MeV

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 Scalar I=2 wave

I=2 J=0 phase shift at two loops Mπ=524 MeV

• 30
• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 600 p (MeV)

δ20 (deg.) mπ = mπ phys mπ = 396 MeV mπ = 420 MeV mπ = 444 MeV mπ = 524 MeV

• 30
• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 600 p (MeV)

δ20 (deg.) mπ = mπ phys mπ = 396 MeV mπ = 420 MeV mπ = 444 MeV mπ = 524 MeV

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 Scalar I=2 wave

I=2 J=0 phase shift at two loops Mπ=524 MeV

• 30
• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 600 p (MeV)

δ20 (deg.) mπ = mπ phys mπ = 396 MeV mπ = 420 MeV mπ = 444 MeV mπ = 524 MeV

• 30
• 25
• 20
• 15
• 10
• 5

100 200 300 400 500 600 p (MeV)

δ20 (deg.) mπ = mπ phys mπ = 396 MeV mπ = 420 MeV mπ = 444 MeV mπ = 524 MeV

Standard ChPT Unitarized ChPT Bends down faster than 1 loop No improvement Works better than Standard ChPT at high p 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

• ne 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 Mπ=139.57 MeV

• 4
• 2

2 4 100 200 300 400 500 600 p (MeV)

δ22 (deg.) mπ = mπ phys

• 4
• 2

2 4 100 200 300 400 500 600 p (MeV)

δ22 (deg.) mπ = mπ phys

One loop 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 Tensor I=2 wave

I=2 J=2 phase shift in standard ChPT Mπ=396 MeV

• 4
• 2

2 4 100 200 300 400 500 600 p (MeV)

δ22 (deg.) mπ = mπ phys mπ = 396 MeV

• 4
• 2

2 4 100 200 300 400 500 600 p (MeV)

δ22 (deg.) mπ = mπ phys mπ = 396 MeV

One loop 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 Tensor I=2 wave

I=2 J=2 phase shift in standard ChPT Mπ=444 MeV

• 4
• 2

2 4 100 200 300 400 500 600 p (MeV)

δ22 (deg.) mπ = mπ phys mπ = 396 MeV mπ = 444 MeV

• 4
• 2

2 4 100 200 300 400 500 600 p (MeV)

δ22 (deg.) mπ = mπ phys mπ = 396 MeV mπ = 444 MeV

One loop 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 Tensor I=2 wave

I=2 J=2 phase shift in standard ChPT Mπ=524 MeV

• 4
• 2

2 4 100 200 300 400 500 600 p (MeV)

δ22 (deg.) mπ = mπ phys mπ = 396 MeV mπ = 444 MeV mπ = 524 MeV

• 4
• 2

2 4 100 200 300 400 500 600 p (MeV)

δ22 (deg.) mπ = mπ phys mπ = 396 MeV mπ = 444 MeV mπ = 524 MeV

One loop 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 Tensor I=2 wave

I=2 J=2 phase shift in standard ChPT Mπ=524 MeV

• 4
• 2

2 4 100 200 300 400 500 600 p (MeV)

δ22 (deg.) mπ = mπ phys mπ = 396 MeV mπ = 444 MeV mπ = 524 MeV

• 4
• 2

2 4 100 200 300 400 500 600 p (MeV)

δ22 (deg.) mπ = mπ phys mπ = 396 MeV mπ = 444 MeV mπ = 524 MeV

One loop Two loops 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 Mπ

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) πK → πK: resonances K ∗(892) and κ. Change of ˆ m = mu+md

2

and ms ⇒ change of M2

π, M2 K, M2 η, fπ, fK , fη.

Applicability in SU(3): 0 < Mπ 400 MeV ⇒ MK 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

ˆ m dependence - Light vector mesons - Mass

0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 0.5 1 1.5 2 2.5 3 0.5 1 2 3 4 5 6 7 8 9

Mπ/Mπ phys m ^ / m ^

phys

Mρ/Mρ phys

0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 0.5 1 1.5 2 2.5 3 0.5 1 2 3 4 5 6 7 8 9

Mπ/Mπ phys m ^ / m ^

phys

Mρ/Mρ phys MK*/MK* phys

Both masses increase slower than Mπ 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

ˆ m dependence - Light vector mesons - Width

0.2 0.4 0.6 0.8 1 1.2 1.4 0.5 1 1.5 2 2.5 3 0.5 1 2 3 4 5 6 7 8 9

Mπ/Mπ phys m ^ / m ^

phys

Γρ/Γρ phys

0.2 0.4 0.6 0.8 1 1.2 1.4 0.5 1 1.5 2 2.5 3 0.5 1 2 3 4 5 6 7 8 9

Mπ/Mπ phys m ^ / m ^

phys

Γρ/Γρ phys ΓK*/ΓK* 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

ˆ m dependence - Light vector mesons - Width

0.2 0.4 0.6 0.8 1 1.2 1.4 0.5 1 1.5 2 2.5 3 0.5 1 2 3 4 5 6 7 8 9

Mπ/Mπ phys m ^ / m ^

phys

Γρ/Γρ phys

0.2 0.4 0.6 0.8 1 1.2 1.4 0.5 1 1.5 2 2.5 3 0.5 1 2 3 4 5 6 7 8 9

Mπ/Mπ phys m ^ / m ^

phys

Γρ/Γρ phys ΓK*/ΓK* phys

Width decrease in accordance with phase space reduction: ΓV = g2 1 8π |p|3 M2

V

(black lines)

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

ˆ m dependence - Light vector mesons - Coupling

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 0.5 1 1.5 2 2.5 3 0.5 1 2 3 4 5 6 7 8 9

Mπ/Mπ phys m ^ / m ^

phys

gρππ/gρππ phys

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 0.5 1 1.5 2 2.5 3 0.5 1 2 3 4 5 6 7 8 9

Mπ/Mπ phys m ^ / m ^

phys

gρππ/gρππ phys gK*πK/gK*πK phys

Coupling to two mesons independent of ˆ m (assumption in 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

ˆ m dependence - Light vector mesons - KSFR

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 0.5 1 1.5 2 2.5 3 0.5 1 2 3 4 5 6 7 8 9

Mπ/Mπ phys m ^ / m ^

phys

(Mρ/fπ)/(Mρ phys/fπ phys)

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 0.5 1 1.5 2 2.5 3 0.5 1 2 3 4 5 6 7 8 9

Mπ/Mπ phys m ^ / m ^

phys

(Mρ/fπ)/(Mρ phys/fπ phys) (MK*/fπ)/(MK* phys/fπ phys)

Fulfill the KSFR relation for different ˆ m: g ≃ MV/2 √ 2fπ

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

ˆ m dependence - Light scalar mesons - Mass

0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 0.5 1 1.5 2 2.5 3 0.5 1 2 3 4 5 6 7 8 9

Mπ/Mπ phys m ^ / m ^

phys

Mσ/Mσ phys

0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 0.5 1 1.5 2 2.5 3 0.5 1 2 3 4 5 6 7 8 9

Mπ/Mπ phys m ^ / m ^

phys

Mσ/Mσ phys Mκ/Mκ 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

ˆ m dependence - Light scalar mesons - Width

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0.5 1 1.5 2 2.5 3 0.5 1 2 3 4 5 6 7 8 9

Mπ/Mπ phys m ^ / m ^

phys

Γσ/Γσ phys

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0.5 1 1.5 2 2.5 3 0.5 1 2 3 4 5 6 7 8 9

Mπ/Mπ phys m ^ / m ^

phys

Γσ/Γσ phys Γκ/Γκ 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

ˆ m dependence - Light scalar mesons - Width

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0.5 1 1.5 2 2.5 3 0.5 1 2 3 4 5 6 7 8 9

Mπ/Mπ phys m ^ / m ^

phys

Γσ/Γσ phys

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0.5 1 1.5 2 2.5 3 0.5 1 2 3 4 5 6 7 8 9

Mπ/Mπ phys m ^ / m ^

phys

Γσ/Γσ phys Γκ/Γκ phys

Width decrease not explained by phase space reduction: ΓS = g2 1 8π |p| M2

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

ˆ m dependence - Light scalar mesons - Coupling

0.5 1 1.5 2 2.5 3 3.5 0.5 1 1.5 2 2.5 3 0.5 1 2 3 4 5 6 7 8 9

Mπ/Mπ phys m ^ / m ^

phys

gσππ/gσππ phys

0.5 1 1.5 2 2.5 3 3.5 0.5 1 1.5 2 2.5 3 0.5 1 2 3 4 5 6 7 8 9

Mπ/Mπ phys m ^ / m ^

phys

gσππ/gσππ phys gκπK/gκπK phys

Strong ˆ m dependence of coupling to two 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

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 (f0(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 (f0(600)), κ(800), ρ(770) and K ∗(892) resonances increasing ˆ m. Vector mesons vector resonances mass grows slower than Mπ, 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 (f0(600)), κ(800), ρ(770) and K ∗(892) resonances increasing ˆ m. Vector mesons vector resonances mass grows slower than Mπ, coupling to two mesons almost independent of Mπ, 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 (f0(600)), κ(800), ρ(770) and K ∗(892) resonances increasing ˆ m. Vector mesons vector resonances mass grows slower than Mπ, coupling to two mesons almost independent of Mπ, 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 (f0(600)), κ(800), ρ(770) and K ∗(892) resonances increasing ˆ m. Vector mesons vector resonances mass grows slower than Mπ, coupling to two mesons almost independent of Mπ, 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 (f0(600)), κ(800), ρ(770) and K ∗(892) resonances increasing ˆ m. Vector mesons vector resonances mass grows slower than Mπ, coupling to two mesons almost independent of Mπ, 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 (f0(600)), κ(800), ρ(770) and K ∗(892) resonances increasing ˆ m. Vector mesons vector resonances mass grows slower than Mπ, coupling to two mesons almost independent of Mπ, 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 Mπ 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 Mπ 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 Mπ dependence : Standard ChPT very soft Mπ 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 Mπ dependence : Standard ChPT very soft Mπ 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 Mπ dependence : Standard ChPT very soft Mπ 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 Mπ dependence : Standard ChPT very soft Mπ 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 Mπ dependence : Standard ChPT very soft Mπ 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 Mπ dependence : Standard ChPT very soft Mπ 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 Mπ dependence : Standard ChPT very soft Mπ 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 Mπ dependence : Standard ChPT very soft Mπ 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 2π 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 Mπ dependence *: Standard ChPT very soft Mπ 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 2π 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

Motivation Standard ChPT Unitarized ChPT ChPT vs. lattice Resonances Summary

ms dependence of σ, ρ, κ 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

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

ms dependence - Light vector mesons - Mass & Width

0.9 0.95 1 1.05 1.1 1.15 1.2 0.9 1 1.1 1.2 0.8 0.9 1.0 1.1 1.2 1.3 1.4

ms / ms phys

Mρ/Mρ phys

0.9 0.95 1 1.05 1.1 1.15 1.2 0.9 1 1.1 1.2 0.8 0.9 1.0 1.1 1.2 1.3 1.4

ms / ms phys

Mρ/Mρ phys MK*/MK* phys

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0.9 1 1.1 1.2 0.8 0.9 1.0 1.1 1.2 1.3 1.4

MK/MK phys

Mρ/Mρ phys MK*/MK* phys Γρ/Γρ phys

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0.9 1 1.1 1.2 0.8 0.9 1.0 1.1 1.2 1.3 1.4

MK/MK phys

Mρ/Mρ phys MK*/MK* phys Γρ/Γρ phys ΓK*/ΓK* phys

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

ms dependence - Light vector mesons - Mass & Width

0.9 0.95 1 1.05 1.1 1.15 1.2 0.9 1 1.1 1.2 0.8 0.9 1.0 1.1 1.2 1.3 1.4

ms / ms phys

Mρ/Mρ phys

0.9 0.95 1 1.05 1.1 1.15 1.2 0.9 1 1.1 1.2 0.8 0.9 1.0 1.1 1.2 1.3 1.4

ms / ms phys

Mρ/Mρ phys MK*/MK* phys

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0.9 1 1.1 1.2 0.8 0.9 1.0 1.1 1.2 1.3 1.4

MK/MK phys

Mρ/Mρ phys MK*/MK* phys Γρ/Γρ phys

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0.9 1 1.1 1.2 0.8 0.9 1.0 1.1 1.2 1.3 1.4

MK/MK phys

Mρ/Mρ phys MK*/MK* phys Γρ/Γρ phys ΓK*/ΓK* phys

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

ms dependence - Light vector mesons - Coupling

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 0.9 1 1.1 1.2 0.8 0.9 1.0 1.1 1.2 1.3 1.4

MK/MK phys ms / ms phys

gρππ/gρππ phys

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 0.9 1 1.1 1.2 0.8 0.9 1.0 1.1 1.2 1.3 1.4

MK/MK phys ms / ms phys

gρππ/gρππ phys gK*πK/gK*πK phys

Coupling to two mesons constant

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

ms dependence - Light vector mesons - KSFR

0.2 0.4 0.6 0.8 1 1.2 0.9 1 1.1 1.2 0.8 0.9 1.0 1.1 1.2 1.3 1.4

MΚ/MΚ phys ms / ms phys

(Mρ/fπ)/(Mρ phys/fπ phys)

0.2 0.4 0.6 0.8 1 1.2 0.9 1 1.1 1.2 0.8 0.9 1.0 1.1 1.2 1.3 1.4

MΚ/MΚ phys ms / ms phys

(Mρ/fπ)/(Mρ phys/fπ phys) (MK*/fπ)/(MK* phys/fπ phys)

KSFR relation well satisfied for different ms

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

ms dependence - Light scalar mesons - Mass & Width

0.9 0.95 1 1.05 1.1 1.15 1.2 0.9 1 1.1 1.2 0.8 0.9 1.0 1.1 1.2 1.3 1.4

ms / ms phys

Mσ/Mσ phys

0.9 0.95 1 1.05 1.1 1.15 1.2 0.9 1 1.1 1.2 0.8 0.9 1.0 1.1 1.2 1.3 1.4

ms / ms phys

Mσ/Mσ phys Mκ/Mκ phys

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0.9 1 1.1 1.2 0.8 0.9 1.0 1.1 1.2 1.3 1.4

MK/MK phys

Mσ/Mσ phys Mκ/Mκ phys Γσ/Γσ phys

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0.9 1 1.1 1.2 0.8 0.9 1.0 1.1 1.2 1.3 1.4

MK/MK phys

Mσ/Mσ phys Mκ/Mκ phys Γσ/Γσ phys Γκ/Γκ phys

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

ms dependence - Light scalar mesons - Mass & Width

0.9 0.95 1 1.05 1.1 1.15 1.2 0.9 1 1.1 1.2 0.8 0.9 1.0 1.1 1.2 1.3 1.4

ms / ms phys

Mσ/Mσ phys

0.9 0.95 1 1.05 1.1 1.15 1.2 0.9 1 1.1 1.2 0.8 0.9 1.0 1.1 1.2 1.3 1.4

ms / ms phys

Mσ/Mσ phys Mκ/Mκ phys

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0.9 1 1.1 1.2 0.8 0.9 1.0 1.1 1.2 1.3 1.4

MK/MK phys

Mσ/Mσ phys Mκ/Mκ phys Γσ/Γσ phys

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0.9 1 1.1 1.2 0.8 0.9 1.0 1.1 1.2 1.3 1.4

MK/MK phys

Mσ/Mσ phys Mκ/Mκ phys Γσ/Γσ phys Γκ/Γκ phys

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

ms dependence - Light scalar mesons - Coupling

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 0.9 1 1.1 1.2 0.8 0.9 1.0 1.1 1.2 1.3 1.4

MK/MK phys ms / ms phys

gσππ/gσππ phys

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 0.9 1 1.1 1.2 0.8 0.9 1.0 1.1 1.2 1.3 1.4

MK/MK phys ms / ms phys

gσππ/gσππ phys gκπK/gκπK phys

Jenifer Nebreda, U. Complutense de Madrid Quark mass dependence of light resonances and phase shifts