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Effective Field Theory Applied to Nuclei

PN12, 4 Nov 2004. Effective Field Theory Applied to Nuclei. Evgeny Epelbaum, Jefferson Lab, USA. 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. Q C D. π. p. n.

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Effective Field Theory Applied to Nuclei

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

  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. QCD π p n Nuclear A-body problem: atomic nuclei 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.: ). How can effective (field) theory contribute? • Provides dynamical input (systematic, consistent, QCD-based). • Simplifies calculations in some cases (effective degrees of freedom).

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

  5. Ay versus θCM for p 3He reaction AV 18, CD-Bonn, … (all with: χ2datum~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? Conventional approach to few-body systems Dynamical input: Tensor analyzing powers for dd -> pt at Ed=6.1 MeV models (Urbana-IX, Tuscon-Melbourne, …). Works good in many cases but problems remain. Chiral EFT can help to solve these problems! • Linked to QCD. • Consistent and systematic framework. • Theoretical uncertainly can be estimated. • Straightforward to improve. ECM=1.2 MeV ECM=1.69 MeV (from: www.unitn.it/convegni/download/FFLEEP.pdf)

  6. “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 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.

  7. 1S0 channel: 3S1 channel: Few nucleons at very low energy (expansion in Q/Mπ) Shallow (virtual) bounds states in S-waves: Nonperturbative problem, resummation is needed! Power counting(Kaplan, Savage, Wise ‘97): S-matrix in the 1S0 channel: 3S1 phase shift (Chen, Rupak, Savage 99) ; where: LO NLO NNLO Nijmegen PSA (using DR & Power Divergence Subtraction) equivalent to effective range expansion in the pure 2N case

  8. M1+E1 E1 M1 (from: Chen & Savage ’99) Applications and extensions Chen, Rupak & Savage ’99; Chen & Savage ’99; Rupak ’00 (at N4LO accurate to 1% for ) 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 Bertulani, Hammer & van Kolck ’02; Bedaque, Hammer & van Kolck ‘03 halo-nuclei:

  9. (expect to work for ) notchiral invariant chiral invariant Define: Chiral group SU(Nf)LXSU(Nf)R= group of independent rotations of in the flavor space. strong interactions are approximately chiral invariant QCD vacuum is only invariant under spontaneous symmetry breakingGoldstone bosons (pions, due to ). Going to higher energies: chiral EFT If typical nucleon momenta , pions should be included as explicit degrees of freedom. mass gap chiral EFT Chiral symmetry of QCD (Leutwyler ’96)

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

  11. Generate observables by solving 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, … . • NN: perturbation theory does not work (deuteron, large aNN, …) Weinberg’s idea: • Use chiral EFT to calculate . (Irreducible diagram = diagram that is not generated through iterations in the dynamical equation.) as a consequence of chiral symmetry; is bounded from below and for any there is a finite number of graphs to be calculated. Notice:

  12. 2π exchange,Kaiser ‘01 3π exchange (small),Kaiser ‘99, ‘00 Two nucleons Timeline 1990: Formulation by Weinberg. N2LO, energy-dependent, by Ordonez et al. 1994: LO (Q0): 1998: N2LO, energy-independent, by Epelbaum et al. 2003-2004:N3LO by: NLO (Q2): • Entem, Machleidt; • Epelbaum et al. Important work by: Kaiser, van Kolck, Friar, Robilotta, … N2LO (Q3): N3LO (Q4):

  13. Low-energy constants: fixed from NN data known from the πN system • 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, N2LO and N3LO 3S1 3P0 1S0 N3LO N2LO NLO 1D2 3P1 3D1 ε2 1G4 1F3 Λ=450…600 MeV (from E.Epelbaum, W.Glöckle, Ulf-G.Meißner, nucl-th/0407037, to appear in Nucl. Phys. A)

  15. Differential cross section for np scattering Elab=25 MeV Elab=50 MeV Deuteron observables At large r :

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

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

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

  19. lattice gauge theory EFT data Beane & Savage ’03; Epelbaum, Meißner & Glöckle ‘03. see: Chiral extrapolation of the NN observables at NLO uncertainty due to d16 1/a1S0 [fm-1] 1/a3S1 [fm-1] uncertainty due to D physical point M.Fukugita et al., PRD 52 (95) Selected further topics: chiral extrapolation in the NN system • Today’s lattice calculations adopt large mq(or Mπ, since ), • Chiral EFT might be used to extrapolate to physical values of Mπ. (from E.Epelbaum, U.-G.Meißner, W.Glöckle NPA 714 (03) 535)

  20. S.R.Beane, V.Bernard, Ulf-G.Meißner and D.R.Phillips) (in collaboration with: use chiral EFT to extract a+ and a- from aπd Novel power counting: where . πN scattering length from πd scattering 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 lifetimeof pionic hydrogen have large error bars. • πd scattering length aπd measured with high accuracy. J.Gasser et al., EPJC 26 (02) 13 LO ChPT our calculation (from Ulf-G.Meiβner et al., nucl-th/0301079)

  21. break chiral (and isospin) symm. chiral invariant includes in addition to isospin conserving terms: 3NF 2NF Isospin violation in nuclear reactions • strong isospin breaking terms , • electromagnetic isospin breaking terms (due to hard photons) , • coupling to (soft) photons . NLØ LØ N2LØ em str em van Kolck et al. ‘98 Friar et al. ‘99,‘03,‘04; Niskanen ‘02 van Kolck et al. ‘96 N2LØ N3LØ f1 The 3NF depends on (δm)str, (δm)em, δMπ and f1. (Epelbaum et al. ‘04; J.L.Friar et al. ‘94)

  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 N3LO. Accurate results for deuteron and scattering observables at low energy. • 3N, 4N and 6N systems have been studied at N2LO including the chiral 3NF.The results look promising. • Many other applications have been performed. Outlook • Few-nucleon systemsat N3LO need V3N, V4N at N3LO. • Electroweak probes in nuclear environment need currents! • Reactions with pions. • Going to higher energies: inclusion of the Δ-resonance.

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

  24. 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)

  25. 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 . Status of the few-body problem 3N: 4N: 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. 5…13N:

  26. Relation to QCD? • , inconsistent with each other! • Structure of . • Theoretical uncertainty? • How to improve? Proton Ay for elastic pd scattering Proton Ay for pd -> γ3He at Ep=150 MeV single nucleon Siegert theorem Riska prescription (from: J.Golak et al., PRC 62 (00) 054005) Dynamical input in most of the calculations: high-precision potentials (i.e.: χ2datum~1) like AV 18, CD-Bonn, Nijm I,II, … models (Urbana-IX, Tuscon-Melbourne, …). meson exchange currents via Siegert theorem or Riska prescription. Works good in many cases but problems remain. Also conceptual problems: Chiral EFT can help to solve these problems! • Linked to QCD. • Consistent and systematic framework. • Theoretical uncertainly can be estimated. • Straightforward to improve.

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

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