Chiral Extrapolations of light resonances Departamento de Física Teórica II. Universidad Complutense de Madrid Chiral Extrapolations of light resonances from dispersion relations and Chiral Perturbation Theory Guillermo Ríos In colaboration with C. Hanhart and J. R. Peláez PRL100:152001(2008) A. Gómez-Nicola and J. R. Peláez PRD77:056006(2008) C. Hanhart and J. R. Peláez in progress

Introduction Why changing the pion mass? The LATTICE provides rigorous and systematic QCD results in terms of quarks and gluons. Caveat: small, realistic, quark masses are hard to implement. ChPT provides the correct QCD dependence of quark masses as an expansion... Unitarized ChPT (IAM) generates poles in ππ scattering amplitudes associated to resonances not present in the lagrangian without a priori assumptions We can study the resonances in Unitarized ChPT for larger pion masses and provide a reference for lattice studies

Inverse Amplitude Method (Dispersive derivation) Outline Inverse Amplitude Method (Dispersive derivation) Results Pole movements Resonance mass dependence with pion mass Resonance width dependence. Comparision with phase space Comparision with lacttice Summary

Chiral Perturbation Theory Weinberg, Gasser & Leutwyler ’s Goldstone Bosons of the spontaneous chiral symmetry breaking SU(2)L SU(2)R  SU(2)V QCD degrees of freedom at low energies << 4f~1 GeV ChPT is the most general expansion in energies of a lagrangian made only of pions compatible with the QCD symmetry breaking Leading order parameters: breaking scale fπ and mπ At 1-loop, QCD dynamics encoded in Low Energy Constants ππ scattering: l1,…,l4 Determined from experiment ChPT is the QCD Effective Theory but is limited to low energies

Elastic Inverse Amplitude Method (Dispersive derivation) The analytic structure of 1/t (right cut, left cut and possible poles) allows us to write a dispersion relation for 1/ t substracted at the Adler zero, sA On the right cut we use elastic unitarity and approximate the Adler zero by its LO approximation Right cut imaginary part known exactly LO very good approximation for s’ far from sA

Elastic Inverse Amplitude Method (Dispersive derivation) The analytic structure of 1/t (right cut, left cut and possible poles) allows us to write a dispersion relation for 1/ t substracted at the Adler zero, sA On the left cut we use ChPT Left cut weighted at low energies where ChPT valid even more suppresed when s is in the physical and resonace region (near right cut)

Elastic Inverse Amplitude Method (Dispersive derivation) The analytic structure of 1/t (right cut, left cut and possible poles) allows us to write a dispersion relation for 1/ t substracted at the Adler zero, sA This is exactly the dispersion relation for –t4 / t22 except for the pole contribution

Elastic Inverse Amplitude Method (Dispersive derivation) The pole contributions read We use ChPT to evaluate the derivatives of t at the Adler zero This is a low energy point → ChPT perfectly justified

physical and resonance Elastic Inverse Amplitude Method (Dispersive derivation) In the end we arrive at If we set A(s)=0 we get the standard IAM. This is the case of the p wave Note: A(s) counts O(p6) and is numerically small except near s=sA - A(s) The differences with the standard IAM are less than 1% in the physical and resonance region

Elastic Inverse Amplitude Method (Dispersive derivation) The IAM satisfies exact unitarity and the chiral series is recovered when reexpanding at low energies Describes data up to ~ 1 GeV Generates poles on the second Riemann sheet associated to resonances. In ππ scattering we find the ρ and the σ Derived from analyticity, unitarity and ChPT. No model dependencies, just aproximations. Use of ChPT perfectly justified, always used at low energies Left cut and Pole Contribution correct up to NLO. Quark mass dependence in LC un PC correct up to NLO Has the Adler zero at its NLO approximation and has no spurious pole

Elastic Inverse Amplitude Method fit to data Fit pion scattering data (μ=770 MeV) Only fit l1 and l2 lr1 = -3.7 ± 0.2 lr2 = 5.0 ± 0.4 For l3 and l4 we take the values of Gasser and Leutwyler lr3 = 0.8 ± 3.8 lr4 = 6.2 ± 5.7 - lri does not depend on mπ

Elastic Inverse Amplitude Method for narrow resonances Mρ = 753 MeV Γρ = 150 Mev Im t11 Im t00 we take as a definition for the wide sigma Mσ = 443 MeV Γσ = 432 MeV

Changing the pion mass (Applicability) How high can we make the pion mass? We do not want to spoil the chiral expansion SU(3) ChPT works well with kaon masses ~ 494 MeV with mπ < 500 MeV we are OK But we are working with SU(2) and there are not kaons. We do not want to reach the kaon threshold For mπ = 500 MeV we have mK ~ 600 MeV Being generous mπ = 500 MeV is our limit of applicability

Comparison of (770) and f0(600) or  movements with increasing pion mass Becomes narrower, reaches real axis at threshold (also increasing) and jumps into the 1st sheet (bound state) The rho Becomes narrower,reaches real axis below threshold on 2nd sheet and split in two poles The sigma

Pole Movements (normalized to mπ) sigma pole rho pole

Pole Movements - ρ pole - σ pole The rho poles reach the real axis just at threshold and one of them jumps into the 1st sheet becoming a bound state. the other one stays on the 2nd sheet in almost the same position The sigma goes below threshold with a siezable width. Meets its partner on the real axis and they split in two poles on the real axis. One of them moves toward threshold and jumps into the 1st sheet becoming a bound state two asymmetric poles in different sheets for scalar channels indicates siezable molecular component Morgan, Pennington. PRD48 (1993) 1185 Baru et al. PLB586(2004)53 Similar pole movements are found with quarks models E. van Beveren et al. PRD74,037501(2006) and with Roy equations G. Colangelo, private communication

Simple interpretation of pole movements These movements are generic for s-wave and p-wave resonances Simple S-matrix parametrization (in k-space) with resonance pole at k=k0 – iγ : Partial wave with angular momentum p-wave poles s-wave poles

Simple interpretation of pole movements In s-plane this reads: For p-wave: -Reach the real axis at threshold For s-wave: -Can reach threshold with a sizeable width -Reach the real axis below threshold

Simple interpretation of pole movements ρ pole movement k-plane

Simple interpretation of pole movements σ pole movement k-plane This behavior is generic for an s-wave resonance. The point where it occurs is especific of QCD and is what we estimate mπ = (329 ± 4) MeV

Mass dependence on mπ The rho mass grows slower than sigma

Width dependence on mπ

Width behavior comparison with phase space For a narrow vector particle (like the rho) the decay width is given by Phase space Coupling to pions We can calculate the width variation due to phase space reduction and compare with our results. The difference give the dependence of the coupling constant on the pion mass

Width behavior comparison with phase space Γρ / Γρphys phase space Γρ / Γρphys IAM Width behavior explained by phase space Coupling constant almost independent of mπ

Γσ / Γσphys phase space Γσ / Γσphys IAM Very bad approximation for In contrast, the sigma “width” does not follow the decrease in phase space of a Breit-Wigner resonance: Γσ / Γσphys phase space Γσ / Γσphys IAM Very bad approximation for a wide resonance as the sigma Differences with only phase space reduction g dependence on mπ The dynamics of the sigma decay depends strongly on the pion-quark mass Recall that some pion-pion vertices in ChPT depend on the pion mass.

Rho mass comparison with lattice We find a cualitative agreement with lattice results

Outlook. In progress... Calculation at NNLO Main diffuculty: More LECs at NNLO LECs and poorly known We use lattice data to constrain their values a20 dependence on mπ fπ dependence on mπ ETMC hep-lat/0701012 JLQCD hep-lat/0810.1360 S. Beane et al. PRD77,014505(2008) S. Beane et al. PRD77,014505(2008)

We have calculated the ρ and σ pole mass and width Summary We have calculated the ρ and σ pole mass and width dependence on mπ from the IAM (model independent up to NLO) Both resonances become bound states for large mπ but they approach this limit in a very different way The rho mass grows slower than the sigma mass The ρππ coupling is almost mπ independent while the σππ coupling is strongly mπ dependent We find a qualitative agreement with lattice calculations for the rho mass To do so we have presented a modified IAM to account properly for the Adler zero region

Thank you!