1. 1 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 R. Machleidt Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12 R. Machleidt University of Idaho

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

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

4. R. Machleidt 4 2N forces 3N forces4N forces Leading Order Next-to- Next-to Leading Order Next-to- Next-to- Next-to Leading Order Next-to Leading Order The Hierarchy of Nuclear Forces Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12

5. R. Machleidt Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12 5 2N forces 3N forces4N forces Leading Order Next-to- Next-to Leading Order Next-to- Next-to- Next-to Leading Order Next-to Leading Order The Hierarchy of Nuclear Forces

6. R. Machleidt 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).

7. R. Machleidt 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

8. R. Machleidt 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.

9. R. Machleidt 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”).

10. R. Machleidt 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.

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

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

13. R. Machleidt Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12 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): Application of the NN Pot. in the Schrodinger or Lippmann-Schwinger (LS) equation: non-perturbative summation of ladder diagrams (infinite sum): 1313 In diagrams: +++ …

14. R. Machleidt Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12 1414 The second kind: Application of the NN Pot. in the Schrodinger or Lippmann-Schwinger (LS) equation: non-perturbative summation of ladder diagrams (infinite sum): Application of the NN Pot. in the Schrodinger or Lippmann-Schwinger (LS) equation: non-perturbative summation of ladder diagrams (infinite sum): Divergent integral. Divergent integral. Regularize it: Regularize it: Cutoff dependent results. Cutoff dependent results. Renormalize to get rid of the cutoff dependence: Renormalize to get rid of the cutoff dependence: 14  Non-perturbative renormalization

15. R. Machleidt Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12 15 R. Machleidt 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): Application of the NN Pot. in the Schrodinger or Lippmann-Schwinger (LS) equation: non-perturbative summation of ladder diagrams (infinite sum): Divergent integral. Divergent integral. Regularize it: Regularize it: Cutoff dependent results. Cutoff dependent results. Renormalize to get rid of the cutoff dependence: Renormalize to get rid of the cutoff dependence: 15  Non-perturbative renormalization 15 With what to renormalize this time? Weinberg’s silent assumption: The same counter terms as before. (“Weinberg counting”)

16. There are several options for non-perturbative renormalization. I will discuss two of them: Infinite cutoff reno Infinite cutoff reno Finite cutoff reno Finite cutoff reno R. Machleidt Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/1216

17. In lower partial waves ( ≅ short distances), in general, no order by order convergence; data are not reproduced. 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. 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). 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! 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? Where are the systematic order by order improvements? R. Machleidt Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12 17 Option 1: Nonperturbative infinite-cutoff renormalization up to N3LO Observations and problems

18. R. Machleidt Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/1218

19. R. Machleidt Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12 19 In lower partial waves ( ≅ short distances), in general, no order by order convergence; data are not reproduced. 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. 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). 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! 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? Where are the systematic order by order improvements? R. Machleidt 19 Option 1: Nonperturbative infinite-cutoff renormalization up to N3LO Observations and problems No good! And Weinberg Counting fails.

20. Option 2: Rethink the problem from scratch EFFECTIVE field theory for Q ≤ Λχ ≈ 1 GeV. EFFECTIVE field theory for Q ≤ Λχ ≈ 1 GeV. So, you have to expect garbage above Λχ. 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). 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. So, stay away from territory that isn’t covered by the EFT. R. Machleidt Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12 20

21. Option 2: Nonperturbative using finite cutoffs ≤ Λχ ≈ 1 GeV. Goal: Find “cutoff independence” for a certain finite range below 1 GeV. R. Machleidt Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12 21 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).

22. R. Machleidt 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.

23. R. Machleidt Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12 23 Χ 2 /datum for the neutron-proton data as function of cutoff in energy intervals as denoted There are ranges of cutoff independence (“plateaus”)

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

25. R. Machleidt Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12 25 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! Renormalization Summary

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

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

28. Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/1228R. Machleidt 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.

29. R. Machleidt Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/1229 The 3NF at NNLO; used so far.

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

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

32. R. Machleidt Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/1232 The 3NF at NNLO; used so far. Small?

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

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

35. R. Machleidt Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/1235 The 3NF at NNLO; used so far. Small? Large?! 3NF contacts at N4LO Girlanda, Kievsky, Viviani, PRC 84, 014001 (2011) Spin-Orbit Force!

36. R. Machleidt36 A realistic, investigational approach: use Δ-less skip N3LO 3NF that may already solve some of your problems. include NNLO 3NF at N4LO start with contact 3NF, use one term at a time, e.g. spin-orbit

37. R. Machleidt Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/1237 … and then there Is also the Δ-full theory …

38. R. Machleidt Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/1238 … and then there Is also the Δ-full theory … … but we have no time left for that.

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

40. 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. 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. 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. Identify “Cutoff Independence” within a range ≤ Λχ ≈1 GeV. Most realistic approach (Lepage!). We have demonstrated this at NLO and NNLO. R. Machleidt Prog. Theory of Nuclear Forces FB20, Fukuoka, Japan, 08/21/12 40 Concerning the renormalization issue, we take the view:

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