Pion-nucleon scattering around the delta-isobar resonance

Transcription

1 Pion-nucleon scattering around the delta-isobar resonance Bingwei Long (ECT*) In collaboration with U. van Kolck (U. Arizona)

2 What do we really do Fettes & Meissner Standard ChPT Isospin 3/2 What we want to do

3 Outline Why delta? why EFT? Standard ChPT power counting Power counting of resumming delta Some technicalities P-wave phase shifts Summary

4 DOFs of nuclear physics N, Δ excitations Baryons 940MeV 1232MeV 1440MeV (Roper) Mesons (mediating long-range forces) 140MeV (5~6)00MeV 770MeV 782MeV break-down scale most conventioal chiral EFT

5 Delta in nuclear forces Delta as an explicit DOF better convergence of EFT expansions, at the price of more DOFs Delta-ful Δ short range physics Delta-less Delta becomes shortrange physics EFT expansion in powers of / ~ 0.5 m

6 Why effective field theory? Several non-eft or quasi-eft approaches describe pion-nucleon data well K-matrix based models Unitarized meson-exchange Unitarized chiral perturbation theory (ChPT)... But, with a proper EFT Model-independent, controlled low-energy approximation to QCD Provides inputs to other nuclear reactions through low-energy constants (LECs) Natural framework to take data from lattice QCD LQCD data EFT in cont. Energy levels by EFT LECs nuclear reactions

7 Standard ChPT counting : generic momenta --- πn scattering chiral symmetry derivative couplings nucleon prop.... one more derivative M : scale of underlying theory... pion loop suppressed by But, perturbative series does not generate resonances naturalness

8 Power counting ingredients Pascalutsa & Phillips (2003) nucleon Compton scattering Bedaque, Hammer, UvK(2002) n-alpha scattering (1) resum Δ pole diags. within the resonance window not necessary to resum (2) dressed Δ propagator enhanced by relative to bare one (3) u-channel not enhanced LO pi-n scattering : Breit-Wigner Higher orders systematic improve over Breit-Wigner

9 Road map NLO selfenergy NNLO selfenergy NNLO πnn vertex func. Standard ChPT starts here

10 Fettes & Meissner (2001) Delta-ful, w/o resummation can t get the pole PSA points Solid line : lab momentum Δ 300 Tang & Ellis (1998) Delta-ful, with resummation Curves : different fits Unitarity not preserved around Delta peak standard counting with delta prop. dressed to only LO imbalance of accuracies in vertex & delta prop CM pion kin. eng.

11 Heavy-baryon ChPT Time-ordered diagrams pair Non-Goldstone boson with above threshold Integrating out anti-n constraint of micro. causality indistinguishable from other short-distance physics Why explicity delta field? Resumming p-wave contact int. seems to break RG inv. mass dim. too high to faciliate renormalization

12 Time to work... Only the imaginary part needed

13 deal with two-loop graphs Define δ so that with a narrow resum window smaller Δ field ren. const., divergent optical theorem can be ignored pion momenta

14 NNLO Renormalization : loop divergences... smaller All absorbed into

15 Effective Lagrangian f : pion decay constant g : nucleon axial coupling A g : A coupling : 2 4 matrices in isospin (spin) space iso-vector contraction constrained by Lorentz inv. N couplings spatial-vector contraction - Remove redundancy due to integrations by part, EOM - Next-to-leading N - No new low-energy constant (LEC) coupling : pure relativistic correction Cohen, Friar, Miller, & van Kolck (1996)...

16 Effective Lagrangian - Next-to-next-to-leading N couplings : 1 LEC 1/m 2 relativistic corrections

17 Results LO : recovering the Breit-Wigner formula P33 partial-wave amplitude (P-wave, I = 3/2, & J=3/2) 2 free parameters : with leading coupling NLO : vanishes in CMS

18 Results NNLO : beyond the Breit-Wigner formula Δ axial coupling NNLO coupling non-resonant background 1 free parameters at NNLO :

19 Results NNLO : non-delta P-waves P11, P13, & P31 contribution of the delta Large N c limit:

20 phase shifts Fitted to the phase-shift analysis of SAID (George Washington group) LO EFT NNLO EFT Blue band: est. theo. error Green dots: PSA (SAID)

21 other P-waves Still at O(Q) Roper N*(1440)

22 Including Roper (N*) A preliminary study before looking into the Roper region consider intermediate kinematic domain : or P11 All S, P-waves To answer : How much the Roper helps?

23 Outlook & Summary Model-independent framework to study pion-nucleon system Power counting that unifies pole and background contributions Extracting delta-related LECs other nuclear reactions, eg. nucl-force Pushing for higher-order calculations To the Roper : N*(1440)