1. Effective Field Theory Applied to Nuclei Evgeny Epelbaum, Jefferson Lab, USA PN12, 4 Nov 2004

2. Outline Introduction Few nucleons at very low energy Going to higher energies: chiral EFT 2 nucleons 3,4 and 6 nucleons Selected further topics Outlook

3. Nuclear A-body problem: QCDQCD p n π atomic nuclei How can effective (field) theory contribute? Provides dynamical input (systematic, consistent, QCD-based). Simplifies calculations in some cases (effective degrees of freedom). Major difficulties: Quantum mechanical many-body problem. - microscopic ab initio calculations: solved for any and ; bound state problem solved for any and. First results for the continuum available; spectra of nuclei using Green’s Function Monte Carlo method (restricted to local ) and the No-Core Shell Model including. - Shell Model; - Density Functional Theory. Underlying dynamics (i. e.: ).

4. Effective (Field) Theory and the nuclear many-body problem We cannot (yet) solve QCD at low E use chiral EFT to derive and to be applied in microscopic many- body calculations “Hybrid” approach: - from chiral EFT, - phenomenologically At very low E: pion-less EFT (i.e. nucleons inter- acting via ) In-medium chiral EFT Shell Model (SM) as an effective theory Use effective theory to get rid of the high-momentum components of no need for -matrix in SM calculations

5. models (Urbana-IX, Tuscon-Melbourne, …). Dynamical input: Works good in many cases but problems remain. AV 18, CD-Bonn, … (all with: χ 2 datum ~1) meson exchange currents via Siegert theorem or Riska prescription. Also conceptual problems: Relation to QCD?, inconsistent with each other! Structure of. Theoretical uncertainty? How to improve? Chiral EFT can help to solve these problems! Linked to QCD. Consistent and systematic framework. Theoretical uncertainly can be estimated. Straightforward to improve. Conventional approach to few-body systems Tensor analyzing powers for dd -> pt at E d =6.1 MeV A y versus θ CM for p 3 He reaction E CM =1.2 MeV E CM =1.69 MeV (from: www.unitn.it/convegni/download/FFLEEP.pdf)

6. Effective field theory identify the relevant degrees of freedom and symmetries, construct the most general Lagrangian consistent with, do standard quantum field theory with this Lagrangian. “ if one writes down the most general possible Lagrangian, including all terms consistent with the assumed symmetry principles, and then calculates S-matrix elements with this Lagrangian to any order in perturbation theory, the result will simply be the most general possible S-matrix consistent with analyticity, perturbative unitarity, cluster decomposition and the assumed symmetry princi- ples ” S.Weinberg, Physica A96 (79) 327

7. Few nucleons at very low energy (expansion in Q/M π ) Shallow (virtual) bounds states in S-waves: 1 S 0 channel: 3 S 1 channel: Nonperturbative problem, resummation is needed! Power counting (Kaplan, Savage, Wise ‘97): ; where: S-matrix in the 1 S 0 channel: equivalent to effective range expansion in the pure 2N case (using DR & Power Divergence Subtraction) 3 S 1 phase shift (Chen, Rupak, Savage 99) LO Nijmegen PSA NLO NNLO

8. Applications and extensions Chen, Rupak & Savage ’99; Chen & Savage ’99; Rupak ’00 (at N 4 LO accurate to 1% for ) M1 E1 M1+E1 (from: Chen & Savage ’99) Kong & Ravndal ’99, ’01; Butler & Chen ‘01 Chen, Rupak & Savage ‘99 Butler & Chen ’00; Butler, Chen & Kong ’01; Chen ‘01 Bedaque, Hammer, van Kolck ‘98; Gabbiani, Bedaque, Grieβhammer ‘00; Blankleider, Gegelia ‘01, … Platter, Hammer & Meißner ‘04 halo-nuclei: Bertulani, Hammer & van Kolck ’02; Bedaque, Hammer & van Kolck ‘03

9. Going to higher energies: chiral EFT If typical nucleon momenta, pions should be included as explicit degrees of freedom. chiral EFT (expect to work for ) mass gap Chiral symmetry of QCD Define: Chiral group SU(N f ) L X SU(N f ) R = group of independent rotations of in the flavor space. chiral invariant not chiral invariant strong interactions are approximately chiral invariant QCD vacuum is only invariant under spontaneous symmetry breaking Goldstone bosons (pions, due to ). (Leutwyler ’96)

10. Notice: chiral symmetry has to be realized nonlinearly. (worked out by: Weinberg ’68; Coleman, Callan, Wess & Zumino ’69 ) Degrees of freedom: Goldstone bosons (pions) and matter fields (N, Δ, …). Symmetries: Lorentz invariance, spontaneously broken chiral symmetry, … coefficients fixed by chiral symmetry ChPT = simultaneous expansion in energy and around the chiral limit (m q =M π =0) ππ,πN: perturbation theory (Goldstone bosons do not interact at E~0) LO, ~(Q/Λ) 2 NLO, ~(Q/Λ) 4 Soft scale: Q~p~M π ; Hard scale: Λ~Λ χ ~M ρ.

11. NN: perturbation theory does not work (deuteron, large a NN, …) Weinberg’s idea: Use chiral EFT to calculate. (Irreducible diagram = diagram that is not generated through iterations in the dynamical equation.) is not unique and can be derived in various ways, see e.g. Ordonez, Ray & van Kolck ‘94; Friar & Coon ‘94; Kaiser, Brockmann & Weise ‘97; Epelbaum, Glöckle & Meißner ‘98, ‘00; Higa & Robilotta 03, …. Generate observables by solving the dynamical equation: Notice: as a consequence of chiral symmetry; is bounded from below and for any there is a finite number of graphs to be calculated.

12. Two nucleons LO (Q 0 ): NLO (Q 2 ): N 2 LO (Q 3 ): N 3 LO (Q 4 ): 3π exchange (small), Kaiser ‘99, ‘00 2π exchange, Kaiser ‘01 Timeline 1990: Formulation by Weinberg. 1994: N 2 LO, energy-dependent, by Ordonez et al. 1998: N 2 LO, energy-independent, by Epelbaum et al. 2003-2004: N 3 LO by: - Entem, Machleidt; - Epelbaum et al. Important work by: Kaiser, van Kolck, Friar, Robilotta, …

13. Low-energy constants: known from the πN system fixed from NN data Valid at low momenta. Wrong behavior (grows) at large momenta needs to be regularized. We use the finite momentum cutoff Λ. (see P.Lepage, nucl-th/9706029 for more details) We use the novel regularization scheme for loop integrals introduced in E.Epelbaum et al., EPJA 19 (04) 125 (quicker convergence compared to DR).

14. Selected NN phase shifts at NLO, N 2 LO and N 3 LO 1S01S0 3S13S1 3P03P0 1D21D2 3P13P1 3D13D1 1F31F3 1G41G4 ε2ε2 N 2 LO NLO N 3 LO (from E.Epelbaum, W.Glöckle, Ulf-G.Meißner, nucl-th/0407037, to appear in Nucl. Phys. A) Λ=450…600 MeV

15. E lab =25 MeV E lab =50 MeV Differential cross section for np scattering NLON 2 LON 3 LOExp E d [MeV] -2.171…-2.186-2.189…-2.202-2.216…-2.223-2.225 A S [fm -1/2 ] 0.868…0.8730.874…0.8790.882…0.8830.8846(9) η 0.0256…0.02570.0255…0.02560.0254…0.02550.0256(4) Deuteron observables At large r :

16. 3,4,… nucleons Hierarchy of nuclear forces No 3NF parameter-free (Epelbaum et al. ‘01) First 3NF: D E LECs D, E fixed from 3 H BE and a Nd. (Epelbaum et al. ‘02) in progress… In collaboration with: A.Nogga, W.Glöckle, H.Kamada, Ulf-G.Meißner and H.Witala

17. Elastic Nd scattering at E N = 65 MeV Deuteron break up at E N = 65 MeV NLO NNLO

18. 3N and 4N binding energies Predictions for 6 Li ground and excited states (Calculation performed by A. Nogga, University of Washington, USA)

19. Selected further topics: chiral extrapolation in the NN system EFT data lattice gauge theory Today’s lattice calculations adopt large m q (or M π, since ), Chiral EFT might be used to extrapolate to physical values of M π. Beane & Savage ’03; Epelbaum, Meißner & Glöckle ‘03. see: Chiral extrapolation of the NN observables at NLO physical point uncertainty due to d 16 uncertainty due to D 1/a 1S0 [fm -1 ] 1/a 3S1 [fm -1 ] M.Fukugita et al., PRD 52 (95) (from E.Epelbaum, U.-G.Meißner, W.Glöckle NPA 714 (03) 535)

20. πN scattering length from πd scattering (in collaboration with: S.R.Beane, V.Bernard, Ulf-G.Meißner and D.R.Phillips ) In the limit of exact isospin symmetry at threshold: No πN data at very low energy. Extractions of a + and a - from the level shifts and lifetime of pionic hydrogen have large error bars. πd scattering length a πd measured with high accuracy. use chiral EFT to extract a + and a - from a πd (from Ulf-G.Meiβner et al., nucl-th/0301079) J.Gasser et al., EPJC 26 (02) 13 LO ChPT our calculation Novel power counting: where.

21. Isospin violation in nuclear reactions chiral invariant break chiral (and isospin) symm. includes in addition to isospin conserving terms: strong isospin breaking terms, electromagnetic isospin breaking terms (due to hard photons), coupling to (soft) photons. 2NF 3NF N 2 LØ van Kolck et al. ‘96 van Kolck et al. ‘98Friar et al. ‘99,‘03,‘04; Niskanen ‘02 N 2 LØ NLØ LØ em str em The 3NF depends on (δm) str, (δm) em, δM π and f 1. (Epelbaum et al. ‘04; J.L.Friar et al. ‘94) N 3 LØ f1f1

22. Summary Few-nucleon systems can be studied in chiral EFT approach in a systematic and model independent way. The 2N system has been analyzed at N 3 LO. Accurate results for deuteron and scattering observables at low energy. 3N, 4N and 6N systems have been studied at N 2 LO including the chiral 3NF. The results look promising. Many other applications have been performed. Outlook Few-nucleon systems at N 3 LO need V 3N, V 4N at N 3 LO. Electroweak probes in nuclear environment need currents! Reactions with pions. Going to higher energies: inclusion of the Δ-resonance.

24. Perspectives: Few-nucleon scattering Properties of light nuclei Electroweak reactions with nuclei Chiral V NN provides a basis for applications to other systems Reactions with pionic probes 3 He as neutron target Nuclear parity violation Astrophysical applications

25. Effective (field) theory and the nuclear many-body problem We cannot (yet) solve QCD at low energy Use chiral EFT to derive and to be applied in microscopic many-body calculations (see: S.Weinberg 90, 91; C.Ordóñez, L.Ray, U.van Kolck 96; U.van Kolck 94; E.E., W.Glöckle, U.-G.Meißner 98, 00,04; D.R.Entem, R.Machleidt 03; S.R.Beane et al 03; …). “Hybrid” approach: from chiral EFT, - phenomenologically. (see: S.Weinberg 92; T.-S.Park et al. 93,96,98,00,01,03; C.H.Hyun, T.-S.Park, D.-P.Min 01; S.R.Beane 98,99,04; V.Bernard, H.Krebs, U.-G.Meißner 00; L.E.Marcucci et al. 01; S.Ando et al. 02,03; …) At very low even π’s can be treated as heavy particles Use pion-less EFT [nucleons interacting via ] to describe few-nucleon systems, also in the presence of external sources (see: U. van Kolck 99; J.W.Chen, G.Rupak, M.J.Savage 99; X.Kong, F.Ravndal 99,00; G.Rupak 00; M.Butler et al. 00, 01; J.W.Chen 01; P.F.Bedaque, H.-W.Hammer, U. van Kolck 00; Gabbiani, Bedaque, Grieβhammer 00; Blankleider, Gegelia 01, … ). Use in-medium chiral EFT to describe nuclear structure properties (see: M.Lutz 00, M.Lutz, B.Friman, Ch.Appel 00; N.Kaiser, S.Fritsch, W.Weise 02, 03, 04). Shell Model (SM) as an effective theory (see: W.C.Haxton, C.-L.Song 00). Use effective theory to get rid of the high-momentum components of. The resulting has no hard core and can be used as input in SM calculations (no need for -matrix). (see: E.E. et al. 98,99; S.K.Bogner et al. 01,02,03; S.Fujii et al. 04; A.Nogga, S.K.Bogner, A.Schwenk 04)

26. Status of the few-body problem Both bound state and scattering problems can be accurately solved for any and. Coulomb problem in the continuum can be handled for 2 charged par- ticles (in configuration space only for local ). Properties of the ground and low-lying excited states are studied using the Green’s Function Monte Carlo method (restricted to local ) and the No-Core Shell Model including. Bound state problem can be accurately solved for any and. First re- sults for the continuum spectrum become available. Most advanced calculations are performed in configuration space only local. not yet included. 3N: 4N: 5…13N:

27. models (Urbana-IX, Tuscon-Melbourne, …). Dynamical input in most of the calculations: high-precision potentials (i.e.: χ 2 datum ~1) like AV 18, CD-Bonn, Nijm I,II, … Proton A y for elastic pd scattering Proton A y for pd -> γ 3 He at E p =150 MeV (from: J.Golak et al., PRC 62 (00) 054005) single nucleon Siegert theorem Riska prescription meson exchange currents via Siegert theorem or Riska prescription. Works good in many cases but problems remain. Also conceptual problems: Relation to QCD?, inconsistent with each other! Structure of. Theoretical uncertainty? How to improve? Chiral EFT can help to solve these problems! Linked to QCD. Consistent and systematic framework. Theoretical uncertainly can be estimated. Straightforward to improve.

28. A natural consequence of the chiral power counting: Hierarchy of nuclear forces