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Recent Progress in the Theory of Nuclear Forces

20 th International IUPAP Conference on Few-Body Problems in Physics August 20-25, 2012, Fukuoka, Japan. Recent Progress in the Theory of Nuclear Forces. R. Machleidt University of Idaho. Outline. The history of the progress Nuclear forces from chiral EFT:

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Recent Progress in the Theory of Nuclear Forces

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  1. 20th International IUPAP Conference on Few-Body Problems in Physics August 20-25, 2012, Fukuoka, Japan Recent Progress in the Theory of Nuclear Forces R. Machleidt University of Idaho Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  2. Outline • The history of the progress • Nuclear forces from chiral EFT: Basic ideas and current status • The open issues • Proper renormalization of chiral forces • Sub-leading many-body forces • Outlook Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  3. Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  4. 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 Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  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 Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  6. N3LO Potential by Entem & Machleidt, PRC 68, 041001 (2003). NNLO and NLO Potentials by Epelbaum et al., Eur. Phys. J. A19, 401 (2004). Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  7. This is, of course, all very nice; However there is a “hidden” issue here that needs our attention: Renormalization Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  8. So, what’s this Renormalization about? See also contributions by Gegelia, Ando, Harada, Kukulin. Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  9. 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”). Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  10. 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. Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  11. The first kind: • “NN Potential”: • irreducible diagrams calculated perturbatively. • Example: Counter terms • perturbative renormalization • (order by order) Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  12. The first kind: • “NN Potential”: • irreducible diagrams calculated perturbatively. • Example: This is fine. No problems. Counter terms • perturbative renormalization • (order by order) R. Machleidt Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12 12

  13. 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: + + + … Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12 13 13

  14. 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 Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12 14 14

  15. 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 Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12 15 15 15

  16. There are several options for non-perturbative renormalization.I will discuss two of them: • Infinite cutoff reno • Finite cutoff reno Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  17. 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? Not Weinberg Counting! • Where are the systematic order by order improvements? Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  18. Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  19. Option 1: Nonperturbative infinite-cutoff renormalization up to N3LO Observations and problems No good! And Weinberg Counting fails. • 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? Not Weinberg Counting! • Where are the systematic order by order improvements? R. Machleidt Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12 19

  20. Option 2: 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 mean it’s proper renormalization (it may be “peratization”, Epelbaum & Gegelia ‘09). • So, stay away from territory that isn’t covered by the EFT. Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  21. Option 2: 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 and NNLO 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). Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  22. 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. Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  23. Χ2/datum for the neutron-proton data as function of cutoff in energy intervals as denoted There are ranges of cutoff independence (“plateaus”) Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  24. The plateaus improve with increasing order. Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  25. Renormalization Summary Best and most realistic Option: Nonperturbative using finite cutoffs ≤ Λχ ≈ 1 GeV. For this, we have shown:Cutoff independence for a certain finite range below 1 GeV (shown for NLO and NNLO). Order-by-order improvement of the predictions. This is what you want to see in an EFT! Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  26. On another topic: Chiral three-nucleon forces (3NF) Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  27. The 3NF at NNLO; used so far. Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  28. See talks at this conference by Bacca, Fonseca, Otsuka, Quaglioni, Sekiguchi, Viviani, Witala; and contributions in Parallel IIIc and IVc this afternoon. Medium-mass nuclei: Nuclear and neutron matter: Bogner, Furnstahl, Hebeler, Nogga, Schwenk. Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  29. The 3NF at NNLO; used so far. Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  30. The 3NF at NNLO; used so far. Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  31. The 3NF at NNLO; used so far. Apps of N3LO 3NF: Triton: Skibinski et al., PRC 84, 054005 (2011). Neutron matter: Hebeler, Schwenk, and co-workers, arXiv:1206.0025. Not small!(?) Small? Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  32. The 3NF at NNLO; used so far. Small? Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  33. The 3NF at NNLO; used so far. Small? 1-loop graphs: 5 topologies 2PE 2PE-1PE Ring Contact-1PE Contact-2PE Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  34. The 3NF at NNLO; used so far. Small? Large?! Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  35. 3NF contacts at N4LO Girlanda, Kievsky, Viviani, PRC 84, 014001 (2011) The 3NF at NNLO; used so far. Small? Large?! Spin-Orbit Force! Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  36. A realistic, investigational approach: • use Δ-less • include NNLO 3NF • skip N3LO 3NF • at N4LO start with • contact 3NF, use • one term at a time, • e.g. spin-orbit • that may already • solve some of your • problems.

  37. … and then there Is also the Δ-full theory … Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  38. … and then there Is also the Δ-full theory … … but we have no time left for that. Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  39. Conclusions • The circle of history that was opened by Yukawa in 1935 is closing. • One major milestone of the past decade: “high precision” NN pots. at N3LO. • But there are still some “subtleties” to be taken care of: • The renormalization of the chiral 2NF • Sub-leading 3NFs Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  40. Concerning the renormalization issue,we take the view: • Forget about non-perturbative infinite-cutoff reno: not convergent (in low partial waves ≅ short distances), should not be a surprise; no clear power counting scheme, no systematic improvements order by order. • Perturbative beyond LO: academically interesting (cf. good work by Valderrama); but impractical in nuclear structure applications, tensor force (wound integral) too large. • Identify “Cutoff Independence” within a range ≤ Λχ ≈1 GeV. Most realistic approach (Lepage!). We have demonstrated this at NLO and NNLO. Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

  41. The 3NF issue • The 3NF at NNLO is insufficient. • The 3NF at N3LO (in the Δ-less theory) may be weak. • However, large 3NFs with many new structures to be expected at N4LO (of Δ-less). Construction is under way. • Order by order convergence of the chiral 3NF may be questionable. • There will be many new 3NFs in the near future. Too many? • Don’t get overwhelmed. For a while just do what you can do---on an investigational basis. Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

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