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Nuclear forces from chiral effective field theory: Achievements and challenges

Nuclear forces from chiral effective field theory: Achievements and challenges. R. Machleidt University of Idaho. Outline. Nuclear forces from chiral EFT: Basic ideas and results Are we done? No! Proper renormalization of chiral forces Sub-leading many-body forces Is THE END near?.

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Nuclear forces from chiral effective field theory: Achievements and challenges

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  1. Nuclear forces from chiral effective field theory: Achievements and challenges R. Machleidt University of Idaho Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  2. Outline • Nuclear forces from chiral EFT: Basic ideas and results • Are we done? No! • Proper renormalization of chiral forces • Sub-leading many-body forces • Is THE END near? Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  3. From QCD to nuclear physics via chiral EFT (in a nutshell) • QCD at low energy is strong. • Quarks and gluons are confined into colorless hadrons. • Nuclear forces are residual forces (similar to van der Waals forces) • Separation of scales Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  4. Calls for an EFT soft scale: Q ≈ mπ ,hard scale: Λχ ≈ mρ ; pions and nucleon relevant d.o.f. • Low-energy expansion: (Q/Λχ)ν with ν bounded from below. • Most general Lagrangian consistent with all symmetries of low-energy QCD. • π-π and π-N perturbatively • NN has bound states: (i) NN potential perturbatively (ii) apply nonpert. in LS equation. (Weinberg) Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  5. 3N forces 4N forces 2N forces Leading Order The Hierarchy of Nuclear Forces Next-to Leading Order Next-to-Next-to Leading Order Next-to-Next-to-Next-to Leading Order Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  6. NN phase shifts up to 300 MeV Red Line: N3LO Potential by Entem & Machleidt, PRC 68, 041001 (2003). Green dash-dotted line: NNLO Potential, and blue dashed line: NLO Potential by Epelbaum et al., Eur. Phys. J. A19, 401 (2004). LO NLO NNLO N3LO Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  7. N3LO Potential by Entem & Machleidt, PRC 68, 041001 (2003). NNLO and NLO Potentials by Epelbaum et al., Eur. Phys. J. A19, 401 (2004). Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  8. This is, of course, all very nice; However there is a “hidden” issue here that needs our attention: Renormalization Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  9. “I about got this one renormalized” Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  10. The issue has produced lots and lots of papers; this is just a small sub-selection. Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  11. Now, what’s so important with this renormalization? Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  12. The EFT approach is not just another phenomenology. It’s field theory. The problem in all field theories are divergent loop integrals. The method to deal with them in field theories: 1. Regularize the integral (e.g. apply a “cutoff”) to make it finite. 2. Remove the cutoff dependence by Renormalization (“counter terms”). Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  13. For calculating pi-pi and pi-N reactions no problem. However, the NN case is tougher, because it involves two kinds of (divergent) loop integrals. Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  14. The first kind: • “NN Potential”: • irreducible diagrams calculated perturbatively. • Example: Counter terms • perturbative renormalization • (order by order) Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  15. The first kind: • “NN Potential”: • irreducible diagrams calculated perturbatively. • Example: This is fine. No problems. Counter terms • perturbative renormalization • (order by order) R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 15

  16. The second kind: • Application of the NN Pot. in the Schrodinger or Lippmann-Schwinger (LS) equation: non-perturbative summation of ladder diagrams (infinite sum): In diagrams: + + + … Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 16 16

  17. The second kind: • Application of the NN Pot. in the Schrodinger or Lippmann-Schwinger (LS) equation: non-perturbative summation of ladder diagrams (infinite sum): • Divergent integral. • Regularize it: • Cutoff dependent results. • Renormalize to get rid of the cutoff dependence: • Non-perturbative renormalization Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 17 17

  18. The second kind: • Application of the NN Pot. in the Schrodinger or Lippmann-Schwinger (LS) equation: non-perturbative summation of ladder diagrams (infinite sum): • Divergent integral. • Regularize it: • Cutoff dependent results. • Renormalize to get rid of the cutoff dependence: With what to renormalize this time? Weinberg’s silent assumption: The same counter terms as before. (“Weinberg counting”) • Non-perturbative renormalization R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 18 18 18

  19. Weinberg counting fails already in Leading Order(for Λ  ∞ renormalization) • 3S1 and 1S0 (with a caveat) renormalizable with LO counter terms. • However, where OPE tensor force attractive: 3P0, 3P2, 3D2, … a counter term must be added. Nogga, Timmermans, v. Kolck PRC72, 054006 (2005): “Modified Weinberg counting” for LO Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  20. Quantitative chiral NN potentials are at N3LO. So, we need to go substantially beyond LO. Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  21. Renormalization beyond leading order –Issues • Nonperturbative or perturbative? • Infinite cutoff or finite cutoff? Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  22. Renormalization beyond leading order –Issues • Nonperturbative or perturbative? • Infinite cutoff or finite cutoff? Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  23. Option 1: Nonperturbative infinite-cutoff renormalization up to N3LO Observations and problems • In lower partial waves (≅ short distances), in general, no order by order convergence; data are not reproduced. • In peripheral partial waves (≅ long distances), always good convergence and reproduction of the data. • Thus, long-range interaction o.k., short-range not (should not be a surprise: the EFT is designed for Q < Λχ). • At all orders, either one (if pot. attractive) or no (if pot. repulsive) counterterm, per partial wave: What kind of power counting scheme is this? • Where are the systematic order by order improvements? Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  24. Option 1: Nonperturbative infinite-cutoff renormalization up to N3LO Observations and problems No good! • In lower partial waves (≅ short distances), in general, no order by order convergence; data are not reproduced. • In peripheral partial waves (≅ long distances), always good convergence and reproduction of the data. • Thus, long-range interaction o.k., short-range not (should not be a surprise: the EFT is designed for Q < Λχ). • At all orders, either one (if pot. attractive) or no (if pot. repulsive) counterterm, per partial wave: What kind of power counting scheme is this? • Where are the systematic order by order improvements? R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 24

  25. Option 2: Perturbative, using DWBA and finite cutoff (Valderrama ‘11) • Renormalize LO non-perturbatively with finite cutoff using modified Weinberg counting. • Use the distorted LO wave to calculate higher orders in perturbation theory. • At NLO, 3 counterterms for 1S0 and 6 for 3S1: a power-counting scheme that allows for systematic improvements order by order emerges. • Results for NN scattering o.k., so, in principal, the scheme works. • But how practical is this scheme in nuclear structure? • LO interaction has huge tensor force, huge wound integral; bad convergence of the many-body problem. Impractical! Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  26. Option 2: Perturbative, using DWBA and finite cutoff (Valderrama ‘11) • Renormalize LO non-perturbatively with finite cutoff using modified Weinberg counting. • Use the distorted LO wave to calculate higher orders in perturbation theory. • At NLO, 3 counterterms for 1S0 and 6 for 3S1: a power-counting scheme that allows for systematic improvements order by order emerges. • Results for NN scattering o.k., so, in principal, the scheme works. • But how practical is this scheme in nuclear structure? • LO interaction has huge tensor force, huge wound integral; bad convergence of the many-body problem. Impractical! For considerations of the NN amplitude o.k. But impractical for nuclear structure applications. R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 26

  27. Option 3: Rethink the problem from scratch • EFFECTIVE field theory for Q ≤ Λχ ≈ 1 GeV. • So, you have to expect garbage above Λχ. • The garbage may even converge, but that doesn’t convert the garbage into the good stuff (Epelbaum & Gegelia ‘09). • So, stay away from territory that isn’t covered by the EFT. Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  28. Renormalization beyond leading order –Issues • Nonperturbative or perturbative? • Infinite cutoff or finite cutoff? Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  29. Option 3: Nonperturbative using finite cutoffs ≤ Λχ ≈ 1 GeV.Goal: Find “cutoff independence” for a certain finite range below 1 GeV. Very recently, a systematic investigation of this kind has been conducted by us at NLO using Weinberg Counting, i.e. 2 contacts in each S-wave (used to adjust scatt. length and eff. range), 1 contact in each P-wave (used to adjust phase shift at low energy). Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  30. Note that the real thing are DATA (not phase shifts), e.g., NN cross sections, etc. Therefore better: Look for cutoff independence in the description of the data. Notice, however, that there are many data (about 6000 NN Data below 350 MeV). Therefore, it makes no sense to look at single data sets (observables). Instead, one should calculate with N the number of NN data in a certain energy range. Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  31. Χ2/datum for the neutron-proton data as function of cutoff in energy intervals as denoted Investigations At NNLO and N3LO Are under way. There is a range of cutoff independence! Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  32. Chiral three-nucleon forces (3NF) Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  33. 3N forces 4N forces 2N forces Leading Order The Hierarchy of Nuclear Forces Next-to Leading Order Next-to-Next-to Leading Order Next-to-Next-to-Next-to Leading Order R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 33

  34. 3N forces 4N forces 2N forces Leading Order The Hierarchy of Nuclear Forces Next-to Leading Order Next-to-Next-to Leading Order Next-to-Next-to-Next-to Leading Order R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 34

  35. 3N forces 4N forces 2N forces Leading Order The Hierarchy of Nuclear Forces Next-to Leading Order Next-to-Next-to Leading Order The 3NF at NNLO; used so far. Next-to-Next-to-Next-to Leading Order R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 35

  36. Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  37. Calculating the properties of light nuclei usingchiral 2N and 3N forces “No-Core Shell Model “ Calculations by P. Navratil et al., LLNL Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  38. Calculating the properties of light nuclei usingchiral 2N and 3N forces “No-Core Shell Model “ Calculations by P. Navratil et al., LLNL 2N (N3LO) force only Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  39. Calculating the properties of light nuclei usingchiral 2N and 3N forces “No-Core Shell Model “ Calculations by P. Navratil et al., LLNL 2N (N3LO) +3N (N2LO) forces 2N (N3LO) force only R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 39

  40. p-d Analyzing Power Ay 2NF+3NF Calculations by the Pisa Group 2NF only The Ay puzzle is NOT solved by the 3NF at NNLO. p-3He 2NF+3NF 2NF only Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  41. And so, we need 3NFs beyond NNLO, because … • The 2NF is N3LO; • consistency requires that all contributions are included up to the same order. • There are unresolved problems in 3N and 4N scattering, and nuclear structure. Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  42. The 3NF at NNLO; used so far. Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  43. The 3NF at N3LO explicitly One-loop, leading vertices Ishikawa & Robilotta, PRC 76, 014006 (2007) Bernard, Epelbaum, Krebs, Meissner, PRC 77, 064004 (2008) Bernard et al., arXiv:1108.3816 Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  44. The 3NF at NNLO; used so far. Small?! Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  45. ✗Small Corresponding 2NF contributions 2π 1-loop lead. vert. 1-loop One ci vert. Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  46. ✗Small Corresponding 2NF contributions 2π 3π Small 1-loop lead. vert. Kaiser (2000) Large!! 1-loop One ci vert. Kaiser (2001) Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

  47. The 3NF at NNLO; used so far. Small?! Large?! R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 47

  48. The N4LO 3NF 2PE-1PE diagrams Still a lot to come in the 3NF business. Stay tuned! Many new isospin/spin/momentum structures, some similar to N3LO 2PE-1PE 3NF.

  49. And there is also the Δ-full version The 3NF at NNLO; used so far. Small?! Large?! R. Machleidt R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 49 49

  50. And there is also the Δ-full version The 3NF at NNLO; used so far. Small?! Large?! R. Machleidt R. Machleidt R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 50 50 50

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