Modern Theory of Nuclear Forces

Transcription

1 Evgeny Epelbaum PAX Meeting, Stockholm, Modern Theory of Nuclear Forces Evgeny Epelbaum, Ruhr-Universität Bochum Outline Chiral EFT for nuclear forces Some hot topics (work in progress) Deuteron breakup at COSY Summary Chiral EFT for Nuclear Forces E.E., Hammer, Meißner, Rev. Mod. Phys., in press Kalantar-Nayestanaki, E.E., Nucl. Phys. News. 17 (07) 22 E.E., Prog. Part. Nucl. Phys. 57 (06) 654 Intro & motivation Two nucleons at N 3 LO Three nucleons at N 2 LO Light nuclei

2 Conventional approach: NN force One-pion exchange as the longest-range contribution (Yukawa 35) Phenomenological parametrization of the shorter-range terms. Available vectors and isovectors. Hermiticity + isospin conservation + invariance under rotations, space reflections and time reversal only 12 structures left for : spin-momentum The corresponding scalar functions of are to be fit to data. isospin np scattering at 96 MeV Modern phenomenological potentials: AV18, CD Bonn, Nijm I/II, Reid93, nearly perfect description of pp and np data below ~ 350 MeV with χ 2 /datum ~ 1 typically about parameters Conventional approach: 3N force replacement of field interactions by two-body action-at-a-distance potentials is a poor approximation in nuclear physics. Indeed, 3N calculations based on NN potentials show evidence for missing 3N forces (e.g. the underbinding of 3 H by about 1 MeV). Phenomenological parametrization of the 3NF seems not feasible: too many possible structures (> 100) too scarce data base available need guidance from a theory too involved calculations Three-nucleon force models Fujita-Miyazawa, Brazil, Tucson-Melbourne, Urbana IX, Illinois,

3 Successes and failures Elastic scattering observables 65 MeV 135 MeV Deuteron breakup Space STAR, 13 MeV V 2N +V 3N V 2N +V 3N V 2N, V 2N +V 3N V 2N V 2N Spectra of light nuclei Inclusion of the 3NF sometimes leads to improvements, sometimes not. Situation, in part, chaotic. Need a theoretical approach which would: be based on QCD, yield consistent many-body forces, be systematically improvable, allow for error estimation chiral effective field theory Chiral Perturbation Theory Weinberg, Gasser, Leutwyler, Bijnens,, Bernard,, Kaiser, Meißner ner, Eckard, Pich, asymptotically observed states as effective DOF EFT spontaneously broken approximate χ-symmetry of QCD plays a crucial role soft (M π ) and hard (M ρ ) scales well separated At very low energy, the only relevant DOF are Goldstone bosons. Derivative expansion: low-energy constants Gasser, Leutwyler M [MeV] mass gap ω (782) ρ (770) π (140) Scattering amplitude can be calculated via a perturbative expansion in and

4 Few nucleons: from ChPT to ChEFT Meson & single-baryon sectors: : chiral perturbation theory Weinberg, Gasser, Leutwyler, Bijnens, Bernard, Kaiser, Meißner, Ecker, Pich, Few/many baryon systems: non-perturbative physics ( 2 H, 3 H, 3 He, ). Hierarchy of scales for non-relativistic ( ) nucleons: π-less EFT with local few-n interactions chiral EFT (cf. pnrqcd), instantaneous (nonlocal) potentials due to exchange of multiple Goldstone bosons rigorously derivable in ChPT internucleon potential [MeV] zero-range operators chiral expansion of multi-pion exchange Weinberg s approach Weinberg 91, 92 irreducible contributions to be derived in ChPT enhanced reducible contributions must be summed up to infinite order separation between the nucleons [fm] Hierarchy of nuclear forces Two-nucleon force Three-nucleon force Four-nucleon force Q 0 Q 2 Q 3 Q 4 Explains the observed hierarchy of nuclear forces: (values from: Pudliner et al., Phys. Rev. Lett. 74 (95) 4396)

5 Two-nucleon force Ordonez et al. 94; Friar & Coon 94; Kaiser et al. 97; E.E. et al. 98, 03; Kaiser 99-01; Higa, Robilotta 03; Chiral expansion of the 2N force: (0) (2) (3) (4) V 2N = V 2N +V 2N + V 2N + V 2N + LO: 2 LECs 24 LECs fit to np data NLO: renormalization of 1π-exchange 7 LECs renormalization of contact terms 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) + isospin-breaking corrections van Kolck et al. 93, 96; Friar et al. 99, 03, 04; Niskanen 02; Kaiser 06; E.E. et al. 04, 05, 07; Results based on EFT with explicit Δ(1232) degrees of freedom available up to N 2 LO Ordonez, Ray, van Kolck 96; Kaiser, Gerstendorfer, Weise 98; Krebs, E.E., Meißner 07, 08 Two nucleons up to N 3 LO Entem, Machleidt 04; E.E., Glöckle, Meißner 05 Neutron-proton phase shifts at N 3 LO Neutron-proton scattering at 50 MeV dσ/dω [mb/sr] N 2 LO N 3 LO PWA A y EE, Glöckle, Meißner Entem, Machleidt Deuteron observables Evidence of the 2π-exchange from the partial wave analysis Rentmeester et al. 99, 03 Energy-dependent boundary condition b EM + [Nijm78; 1π; 1π+2π]

6 Three-nucleon force at N 2 LO van Kolck 94; E.E. et al. 02 to be determined from >2N data such as e.g. 3 H BE and Nd scattering length Three nucleons up to N 2 LO E.E. et al. 02; Kistryn et al. 05; Witala et al. 06; Ley et al. 06; Stephan et al. 07; Differential cross section in elastic Nd scattering N 2 LO NLO Polarization observables in elastic Nd scatering E N = 22:7 MeV EN = 90 MeV N 2 LO

7 Deuteron breakup at N 2 LO d + p p + p + 19 MeV The so-called Symmetric-Constant- Energy geometry Ley et al. 06; data taken at the Cologne FN Tandem accelerator p p p n α d d + p p + p MeV N 2 LO Kistryn et al. 05, KVI data More nucleons No-Core-Shell-Model results for 10 B, 11 B, 12 C and 13 N 2 LO Navratil et al., PRL 99 (2007) He and 6 NLO and N 2 LO Nogga et al., NPA 737 (2004) 236

8 Hot topics Three-nucleon force at N 3 LO Three-nucleon force and the D-term Chiral EFT with explicit Δ(1232) isobar Heavier systems: nuclear lattice simulations Three-nucleon force at N 3 LO Ishikawa, Robilotta 07; Bernard, E.E., Krebs, Meißner 07; to appear no unknown LECs contribute N 3 LO corrections to the 3NF are parameter-free! analytic expressions available both in coordinate and momentum space

9 Three-nucleon force at N 3 LO Ishikawa, Robilotta 07; Bernard, E.E., Krebs, Meißner 07; to appear For example, for ring diagrams one obtains: Bernard, E.E., Krebs, Meißner, PRC 77 (08) Complicated expressions in momentum space (3-point function), much more compact expressions in r-space 1 3 known constants isospin operators spin-space operators Yukawa-like functions 2 Next step: partial-wave decomposition Analytical calculations tedious (more than 100 operators!) Work in progress: numerical partial wave decomposition Golak et al. 09 not feasible matrix, ~ 10 5 x 10 5 can be reduced to depends on spin & isospin 5 dim. integral a feasible task for modern supercomputers, work in progress Three-nucleon force and the D-termD First nonvanishing 3N-force contribution appears at next-to to-next next-to to-leading order van Kolck 94, E.E. et al. 02 D E D-term figures prominently in various reactions NN NNπ Nd Nd Hanhart et al. 00, Baru et al. 09, Filin et al. 09, E.E. et al. 02, Nogga et al. 05, Navratil et al. 07 NN dlν l dπ γnn Park et al. 03, Ando et al. 02, 03, Nakamura et al. 07 Lensky et el. 05, 07 Gardestig et al. 06

10 p-wave π-production and the D-termD Hanhart, van Kolck, Miller 00; Baru, EE, Haidenbauer, Hanhart, Kudryavtsev, Lensky, Meißner 09 Loops start to contribute at N 3 LO Up to N 2 LO, D is the only unknown LEC Simultaneous description of pn ppπ -, pp pnπ + and pp dπ + nontrivial consistency check of chiral EFT N 2 LO D 1 S 0 for pp pnπ +, pp dπ + ; 3 S 1 for pn ppπ - 3 S 1 for pp pnπ +, pp dπ + ; 1 S 0 for pn ppπ - pp dπ + : angular asymmetry pn ppπ - : diff. cross section pp pnπ + : angular asymmetry overall best results for d ~ 3 Chiral EFT with explicit Δ(1232) Δ couples strongly to the πn-system and has low excitation energy scale of certain LECs in expect much better convergence in the formulation based on Preis to pay: calculations much (!) more involved, more LECs which sets the Two-nucleon force in EFT with and without Δ Δ-less theory Δ-full theory: additional graphs LO NLO Ordonez, Ray & van Kolck. 96, Kaiser, Gerstendorfer & Weise 98 N 2 LO Krebs, E.E., Meißner EPJA 32 (2007) 127

11 Δ-isobar & the two-nucleon force Krebs, E.E., Meißner EPJA 32 (2007) 127 2π-exchange up to N 2 LO NLO with Δ NNLO without Δ a much better convergence for the potential when Δ is included explicitly clearly visible in NN peripheral waves 3 F 3 partial wave up to N 2 LO NNLO-Δ NNLO NLO-Δ NLO LO (OPE) Δ-isobar & the three-nucleon force E.E., Krebs, Meißner NPA 806 (2008) 65 Δ-less theory Δ-full theory: additional graphs NLO N 2 LO N 3 LO Δ contributions at N 3 LO are large! Long-range part is parameter free Much richer spin/isospin structure compared to the Illinois model Complete analysis still to be done isoscalar central potential Illinois model Ring (Δ), chiral EFT

12 Nuclear Lattice Simulations Borasoy, E.E., Krebs, Lee, Meißner, Eur. Phys. J. A31 (07) 105, Eur. Phys. J. A34 (07) 185, Eur. Phys. J. A35 (08) 343, Eur. Phys. J. A35 (08) 357, E.E., Krebs, Lee, Meißner, Eur. Phys. J A40 (09) 199, Eur. Phys. J A41 (09) 125, Phys. Rev. Lett 104 (10) , arxiv: [nucl-th] Pions and nucleons as point-like particles on the lattice (typical lattice size ~ 20 fm) Use Monte Carlo to evaluate path integral h Z A (t) = hª 0 A j exp( th )jª 0 A i E 0 A = lim t! 1 d i dt ln ZA (t) systematic ab initio approach to few- & many-nucleon systems Lattice simulations of light nuclei E.E., Krebs, Lee, Meißner 10 Simulations for 6 Li, L=9.9 fm Simulations for 12 C, L=13.8 fm

13 Deuteron breakup at COSY Deuteron breakup at COSY Nd breakup as a laboratory for 3N forces For the first time, we now have the theory & machinery to derive 3NF in a systematic way, calculate corrections and test their effects in 3N continuum. PWD, Faddeev equations Energy range MeV is ideal to test chiral 3NF, no systematic breakup measurements exist! New, independent way to fix the LECs D and E implications for spectra of (light) nuclei, pion production and electroweak reactions; interesting tests of the theory (e.g. MuSun PSI)

14 Deuteron breakup at COSY 3NF effects in pd breakup at E N =135 MeV Find phase space regions sensitive to particular isospin-spin-space operators guidence for the measurement tests for the theory More details in the talk by Pia Thörngren Engblom Combined with measurements at higher energies, will allow to test theoretical predictions for energy dependence of polarization observables from: Kuros-Zolnierczuk et al., Few-Body Syst. 34 (2004) 259 Summary Chiral EFT allows for quantitative understanding of the 3NF and predicts rich isospin-spin-space structure The proposed Nd breakup COSY will allow to test novel chiral 3NF and provide a new way to determine the LECs D and E Implications for spectra of light nuclei, pion production, electroweak reactions, etc. Important step towards precision nuclear physics