Evgeny Epelbaum KHuK Jahrestagung, GSI, 25.10.2007 Evgeny Epelbaum Forschungszentrum Jülich & Universität Bonn Outline Motivation & Introduction Few nucleons in chiral EFT: where do we stand Work in progress: 3NF and 4NF at N 3 LO Isospin dependence of the nuclear forces Nuclear forces and the role of Δ excitation Few-baryon systems with strangeness Lattice QCD & lattice EFT Summary and outlook
Motivation nuclear forces currents ab initio methods for few (<13?) nucleons many-body methods for more nucleons Nuclear Standard Model high-precision 2N potentials (AV18, CD Bonn, ) models for 3N force (Brazilian, TM, Illinois, ) meson-exchange currents (Siegert/Riska prescr.) works well in most cases but: relation to QCD?? inconsistent nuclear forces & currents theoretical uncertainty?? how to improve?? why does the 2N force dominate?? chiral EFT: a systematic and model-independent framework
Chiral Perturbation Theory Weinberg, Gasser, Leutwyler, Bernard, Kaiser, Meißner, Goldstone bosons + matter fields weakly interacting Goldstone bosons instead of strongly interacting quarks and gluons unknown low-energy constants (LECs) in the effective Lagrangian CHPT = simultaneous expansion in energy and about the χ limit keeping write down the most general consistent with the χ-symmetry of QCD compute S-matrix elements in perturbation theory (power counting) fix the low-energy constants & make predictions... Model-independent & systematically improvable approach!
Example: ππ scattering 1-loop, all vertices from tree, 1 insertion from Predictive power? 2-loops, all vertices from 1-loop, 1 insertion from tree, 1 insertion from 2 insertions from # of LECs: 2 7 53 S-wave ππ scattering length LO: (Weinberg 66) NLO: (GL 83) NNLO: (Bijnens et al. 95) NNLO + dispersion relations: (Colangelo et al. 01) (from Caprini et al. hep-ph/0509266)
Generalization to few nucleons: non-perturbative problem! shallow bound states 2 H, 4 He, NN interaction does not vanish for E; M ¼! 0 Few nucleons at very low energies: π-less EFT 3 S 1 2N: power counting designed to reproduce the effective range expansion, Kaplan, Savage & Wise 97, separable forces simple, analytic calculations 3N, 4N, reactions with external probes highly non-trivial, important astrophysical applications, universal properties of few-body systems Braaten & Hammer, Phys. Rept. 428 (2006) 259, LO NLO NNLO Nijmegen PSA Chen, Rupak & Savage NPA653 (1999) Including pions: Weinberg s approach (chiral EFT) Weinberg 90, 91 irreducible contributions can be calculated using ChPT enhanced reducible contributions must be summed up to infinite order χ-symm. constrained parametrized
Few-nucleon forces in chiral EFT Two-nucleon force Three-nucleon force Four-nucleon force Q 0 Q 2 Q 3 Q 4 work in progress 2 nucleon force 3 nucleon force 4 nucleon force
Nucleon-nucleon force up to N 3 LO Ordonez et al. 94; Friar & Coon 94; Kaiser et al. 97; E.E. et al. 98, 03; Kaiser 99-01; Higa et al. 03; Chiral expansion for the 2N force: (0) (2) (3) (4) V 2N = V 2N +V 2N + V 2N + V 2N + LO: 2 LECs NLO: renormalization of 1π-exchange renormalization of contact terms 7 LECs leading 2π-exchange N 2 LO: renormalization of 1π-exchange subleading 2π-exchange N 3 LO: renormalization of 1π-exchange 15 LECs renormalization of contact terms sub-subleading 2π-exchange 3π-exchange (small) + 1/m and isospin-breaking corrections
Neutron-proton phase shifts up to N 3 LO N 3 LO np scattering at 50 MeV dσ/dω [mb/sr] N 2 LO NLO A y N2 LO N 3 LO PWA Deuteron binding energy & asymptotic normalizations A s and η d Entem & Machleidt; E.E. & Meißner; see also Pavon-Valderrama & Ruiz Arriola (infinite Λ framework)
Three-nucleon force up to N 2 LO Weinberg 91; Coon & Friar 94; van Kolck 94; E.E. et al., 98, 02 Chiral expansion for the 3N force: (2) (3) (4) V 3N = V 3N + V 3N + V 3N + NLO: no contributions to the 3NF suppression due to 0 no irreducible contributions N 2 LO: first nonvanishing contributions Determination of the LECs c i known from πn scattering c 1,3,4 D E D and E fixed from BEs of light nuclei ( 3 H, 4 He, 10 B) and/or nd doublet scatt length E.E. et al. 02; Nogga et al. 05; Navratil et al. 07 D contributes to many processes which can be used to extract its value: pp! pn¼ + Hanhart et al. 00 º e d! e pp; ºd! ºpn; ¹ d! º ¹ nn Ando et al. 02, 03 ¼ d! nn Gardestig & Phillips 06 pp! de + º e Park et al. 03; Nakamura et al. 07 Where to test: 3N (and 4N??) continuum, spectra of light nuclei
Nd elastic scattering Deuteron break-up Cross section & vector analyzing power SCRE configuration at E d =19 MeV E.E., PPNP 57 (2006) 654 Ley et al., PRC 73 (2006) 064001 E N =10 MeV E N =65 MeV Polarization transfer coefficients Witała et al., PRC 73 (2006) 044004 Cross section for E p =65 MeV, θ 1,2 = 15 Kistryn et al., PRC 68 (2003) 054004; 73 (2005) 044006 E N =22.7 MeV
-29 More nucleons (discrete spectrum) α-particle 6 Li ground and excited states -28-27 -26-25 -24 Nogga et al., NPA 737 (2004) 236 10 B, 11 B, 12 C and 13 C states dominated by p-shell configurations Navratil et al., PRL 99 (2007) 042501
Work in progress: N 3 LO corrections to the 3NF N 2 LO results: significant theor. uncertainty, some problems remain need to go to N 3 LO rich isospin-spin-space structure of the 3NF at N 3 LO not yet explored in few-body studies N 3 LO corrections to the 3NF contain no additional parameters Long-range contributions Bernard, EE, Krebs, Meißner, underway Ishikawa & Robilotta 07
4N force: the dominant contributions (N 3 LO) E.E., PLB 639 (2006) 456; EPJA, to appear, arxiv:0710.4250 [nucl-th] McManus, Riska 80; Robilotta 85 4NF is parameter-free Typical strength of individual terms is of the order of few 100s kev First estimation of 4NF effects in 4 He available, Rozpedzik et al. 06, dramatic cancellations! typically of the order ~ 10 a more careful study necessary, applications called for
Isospin dependence of the nuclear forces strong IV terms em (hard) terms chiral invariant break chiral (and isospin) symm. virtual photons Isospin-violating two-nucleon forces van Kolck et al. 93, 96, 98; Friar et al. 99, 03, 04; Niskanen 02; E.E. et al. 05; Kaiser 06 LØ NLØ N 2 LØ N 3 LØ some N 4,5 LØ long-range contributions up to N 3 LØ depend on ±m str N ; ±m em N ; ±M¼; 2 ±g ¼N IV contact terms could, in principle, be extracted from the upcoming Nijmegen PWA Isospin-violating three-nucleon forces E.E. et al. 04; Friar et al. 04 N 2 LØ N 3 LØ up to N 3 LØ depend on ±m str N ; ±m em N ; ±M¼; 2 f 1 big effects predicted (charge-symmetry conserving)
Nuclear forces and the role of Δ excitation Δ-isobar is known to be important due to: low excitation energy: strong coupling to the πn system ( large) expect: better convergence & applicability at higher energy in EFT with Δ s NLO Δ- less EFT Δ- full EFT: additional graphs Ordonez et al. 96, Kaiser et al. 98 N 2 LO Krebs et al. 07 Δ contributions to the 2NF worked out up to N 2 LO; The LECs and determined from πn S/P-wave threshold parameters; Observe improved convergence for the 2NF and for peripheral NN phase shifts; Contributions to the 3NF and isospin-violating terms underway, EE et al., to appear. calculated in the first Born approximation NNLO-Δ NNLO-Δ NLO-Δ NLO-Δ LO (OPE)
Few-baryon systems with strangeness Chiral EFT at LO Haidenbauer, Polinder & Meißner 06, 07 2 LECs fixed from g A + SU(6) Total cross section as a function of p lab 5 LECs fixed from 35 YN data points Polinder, Haidenbauer & Meißner, NPA 779 (2006) 244 LO results for S=-2 available; extension to NLO in progress Major problem: only very few data available (compared to the S=0 sector) To be checked: convergence, role of relativitye
Lattice QCD & few-baryon systems Quark mass dependence of the nuclear force Bulgac et al. 97; Beane et al. 02, 06; E.E. & Meißner 02 interesting question by itself chiral extrapolations for lattice QCD calculations constrains on time variation of fundamental coupling constants B d [MeV] 1=a1 S 0 [fm 1 ] 1=a3 S 1 [fm 1 ] Beane et al. 06 Fukugita et al. 95 First results for YN scattering available (no χ-extrapolations yet ) Beane et al. 06 Strong nucleon mass shift: Beane et al. 06 ±m str N = 2:26 0:57 0:42 0:10 MeV Hadronic potentials from lattice QCD Pennanen et al. 00; Cook & Fiebig 04; Ishii et al. 07; Nemura et al. 07; Detmold et al. 07
Few nucleons on the lattice Borasoy, E.E., Krebs, Lee, Meißner, EPJA 31 (2007) 105 We used the effective Lagrangian for nucleons and instantaneous pions which reproduces the chiral LO NN interaction. Findings good agreement with the data in the 2N sector promising results for 3N, 4N systems: Deuteron properties encouraging results for CPU time scaling with the # of particles Density correlations for the deuteron with spin in the +z direction CPU time scaling Phase shifts can also been extracted, Borasoy et al. 07; extension to NLO in progress An earlier work along these lines: Muller et al. 99; Lee et al. 04; Borasoy et al. 05; Seki et al. 05
Summary Chiral EFT results in the 2N sector at N 3 LO are in very good agreement with the data. N 2 LO calculations in >2N systems look promising. Leading 4NF worked out, N 3 LO corrections to the 3NF underway. IV contributions to the nuclear force worked out, LECs need to be determined. Δ-contributions worked out upto N 2 LO, improved convergence demonstrated. Encouraging results for in the S=-1 sector at LO in chiral EFT First results from Lattice QCD in the NN and YN sectors available. First results for nuclear lattice simulations at LO in chiral EFT Outlook Few nucleons at N 3 LO. Further investigation on the role of Δ-isobar. Further work on nuclear lattice lattice simulations. Few nucleons with electroweak & pionic probes.