1. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-20141 The Theory of Nuclear Forces: Eight Decades of Struggle Supported by the US Department of Energy Kavli Institute for Theoretical Physics China Present Status of Nuclear Interaction Theory August 25 to September 19, 2014, Bejing, China R. Machleidt, University of Idaho

2. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-20142 Outline Historical overview Historical overview The divers current status The divers current status The EFT approach The EFT approach Some open issues in chiral EFT: Some open issues in chiral EFT: Proper renormalization of chiral forces Proper renormalization of chiral forces Uncertainty quantification Uncertainty quantification Sub-leading many-body forces Sub-leading many-body forces Conclusions Conclusions

3. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-20143

4. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-20144 The circle of history is closing.

5. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-20145 This was just the history in a nutshell. The history was, of course, much richer. Let me show this for the last two decades. It is best to distinguish between phenomenological and “first principal” approaches.

6. Current phenomenological approaches/models/potentials The Moscow potential, Kukulin et al.; hybrid model, short- range 6-quark bag, long-range meson-exchange. The Moscow potential, Kukulin et al.; hybrid model, short- range 6-quark bag, long-range meson-exchange. The high-precision NN potentials The high-precision NN potentials - Argonne V18 (1995) - Argonne V18 (1995) - Nijmegen (1994) - Nijmegen (1994) - CD-Bonn (1996 & 2001) - CD-Bonn (1996 & 2001) Meson-theory based, very accurate. Meson-theory based, very accurate. Phenomenological three-nucleon forces (3NFs): Phenomenological three-nucleon forces (3NFs): - Urbana (1995) - Urbana (1995) - Tucson-Melbourne (1975-1999) - Tucson-Melbourne (1975-1999) - Illinois (2001-2010) - Illinois (2001-2010) - CD-Bonn + Δ (Deltuva, Sauer, 2003) - CD-Bonn + Δ (Deltuva, Sauer, 2003) R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-20146

7. Phenomenology, cont’d Non-local INOY potentials (“inside non-local, outside Yukawa”), Doleschall (2000-2004). The 2NF reproduces triton and alpha energy. Non-local INOY potentials (“inside non-local, outside Yukawa”), Doleschall (2000-2004). The 2NF reproduces triton and alpha energy. JISP potentials (J-matrix Inverse Scattering Potential), non-local interaction in the form of a matrix in oscillator space in each partial wave; reproduces not only the NN data also light nuclei up to A=16 (“JISP16”) due to variations of the off-shell behavior. No 3NF needed. JISP potentials (J-matrix Inverse Scattering Potential), non-local interaction in the form of a matrix in oscillator space in each partial wave; reproduces not only the NN data also light nuclei up to A=16 (“JISP16”) due to variations of the off-shell behavior. No 3NF needed. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-20147

8. First principal approaches to nuclear forces Lattice QCD Lattice QCD - NPLQCD Collaboration, S. R. Beane et al. - NPLQCD Collaboration, S. R. Beane et al. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-20148

9. Lattice QCD, cont’d Lattice QCD, cont’d - HAL QCD Collaboration, T. Hatsuda, S. Aoki et al. - HAL QCD Collaboration, T. Hatsuda, S. Aoki et al. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-20149

10. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201410 Lattice QCD, cont’d Lattice QCD, cont’d - HAL QCD Collaboration, T. Hatsuda et al. - HAL QCD Collaboration, T. Hatsuda et al.

11. First Principal approaches, cont’d Effective Field Theory (EFT) approaches Effective Field Theory (EFT) approaches (two- and many-body forces) (two- and many-body forces) - pion-less - pion-less - pion-full - pion-full * Δ-less * Δ-less * Δ-full * Δ-full R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201411

12. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201412 The chiral EFT approach QCD at low energy is strong. QCD at low energy is strong. Quarks and gluons are confined into colorless hadrons. Quarks and gluons are confined into colorless hadrons. Nuclear forces are residual forces (similar to van der Waals forces) Nuclear forces are residual forces (similar to van der Waals forces) Separation of scales Separation of scales

13. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201413 Calls for an EFT: Calls for an EFT: soft scale: Q ≈ m π, hard scale: Λ χ ≈ m ρ ; pions and nucleons are relevant d.o.f. soft scale: Q ≈ m π, hard scale: Λ χ ≈ m ρ ; pions and nucleons are relevant d.o.f. Low-momentum expansion: (Q/Λ χ ) ν Low-momentum expansion: (Q/Λ χ ) ν with ν bounded from below. with ν bounded from below. Most general Lagrangian consistent with all symmetries of low-energy QCD, particularly, chiral symmetry which is spontaneously broken. Most general Lagrangian consistent with all symmetries of low-energy QCD, particularly, chiral symmetry which is spontaneously broken. Weakly interacting Goldstone bosons = pions. Weakly interacting Goldstone bosons = pions. π-π and π-N perturbatively π-π and π-N perturbatively NN has bound states: NN has bound states: (i) NN potential perturbatively (i) NN potential perturbatively (ii) apply nonpert. in LS equation. (ii) apply nonpert. in LS equation. (Weinberg) (Weinberg)

14. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201414 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

15. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201415 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

16. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201416 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 NNL O N3LO

17. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201417 N3LO Potential by Entem & Machleidt, PRC 68, 041001 (2003). NNLO and NLO Potentials by Epelbaum et al., Eur. Phys. J. A19, 401 (2004).

18. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201418 N3LO Potential by Entem & Machleidt, PRC 68, 041001 (2003). NNLO and NLO Potentials by Epelbaum et al., Eur. Phys. J. A19, 401 (2004). NNLO 1.00 1.87 6.09 2.95

19. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201419 How does this compare to conventional meson theory?

20. Main differences Chiral perturbation theory (ChPT) is an expansion in terms of small momenta. Chiral perturbation theory (ChPT) is an expansion in terms of small momenta. Meson theory is an expansion in terms of ranges (masses). Meson theory is an expansion in terms of ranges (masses). R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201420

21. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201421 The nuclear force in the meson picture Short Inter- mediate Long range Taketani, Nakamura, Sasaki (1951): 3 ranges

22. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201422 2N 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

23. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201423 ChPT Conventional meson theory OPE TPE Short range

24. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201424 What is the physics of contact terms? Contact terms take care of the short range structures without resolving them. Consider the contribution from the exchange of a heavy meson ++ + …

25. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201425 ChPT Conventional meson theory OPE TPE Short range Resonance Saturation Epelbaum et al., PRC 65, 044001 (2002)

26. Chiral EFT claims to be a theory, while “meson theory” is a model. Chiral EFT claims to be a theory, while “meson theory” is a model. Chiral EFT has a clear connection to QCD, while the QCD-connection of the meson model is more hand- woven. Chiral EFT has a clear connection to QCD, while the QCD-connection of the meson model is more hand- woven. In ChPT, there is an organizational scheme (“power counting”) that allows to estimate the size of the various contributions and the uncertainty at a given order (i.e., the size of the contributions we left out). In ChPT, there is an organizational scheme (“power counting”) that allows to estimate the size of the various contributions and the uncertainty at a given order (i.e., the size of the contributions we left out). Two- and many-body force contributions are generated on an equal footing in ChPT. Two- and many-body force contributions are generated on an equal footing in ChPT. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201426 Question: When everything is so equivalent to conventional meson theory, why not continue to use conventional meson theory?

27. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201427 But: Meson-models are superior at higher Energies. Green and red curves: Chiral NN pots. Black line: CD-Bonn

28. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201428 So, chiral EFT wants to be a theory. How true is that?

29. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201429 If EFT wants to be a theory, it better be renormalizable. 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”).

30. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201430 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.

31. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201431 The first kind: “NN Potential”: “NN Potential”: irreducible diagrams calculated perturbatively up to a fixed order. Example: irreducible diagrams calculated perturbatively up to a fixed order. Example: Counter terms  perturbative renormalization (order by order)

32. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201432 R. Machleidt 32 The first kind: “NN Potential”: “NN Potential”: irreducible diagrams calculated perturbatively up to a fixed order. Example: irreducible diagrams calculated perturbatively up to a fixed order. Example: Counter terms  perturbative renormalization (order by order) This is fine. No problems.

33. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-20143333 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): 33 In diagrams: +++ …

34. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-2014343434 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:  Non-perturbative renormalization

35. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201435 Some Results from non-perturbative renormalization Infinite cutoff: no reasonable power counting scheme, no order-by-order improvement (Idaho group). Infinite cutoff only at LO, higher orders perturbatively (Valderrama; Gegelia): How to implement in nuclear structure calculations? Also: huge tensor force. Finite cutoff (below the hard scale): cutoff independence for the range 450-800 MeV, substantial improvements from NLO to NNLO to N3LO (Idaho group).

36. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201436 R. Machleidt Chiral NFs NTSE2014, June 23-27, 2014 36 NLO Cutoff = 450-800 MeV

37. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201437 R. Machleidt Chiral NFs NTSE2014, June 23-27, 2014 37 R. Machleidt Chiral NFs NTSE2014, June 23-27, 2014 37 NLO NNLO Cutoff = 450-800 MeV

38. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201438 R. Machleidt Chiral NFs NTSE2014, June 23-27, 2014 38 R. Machleidt Chiral NFs NTSE2014, June 23-27, 2014 38 R. Machleidt Chiral NFs NTSE2014, June 23-27, 2014 38 NLO NNLO Cutoff = 450-800 MeV NLONNLO Cutoff = 450-600 MeV N3LO

39. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201439 Note, however, 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.

40. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201440 The plateaus improve with increasing order.

41. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201441 R. Machleidt 41 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

42. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201442 The plateaus improve with increasing order.

43. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201443 Non-perturbative reno using finite cutoffs ≤ Λχ ≈ 1 GeV. For this, we have shown: Cutoff independence for a certain finite range below 1 GeV (shown for NLO, NNLO, and N3LO). Order-by-order improvement of the predictions. This is what you want to see in an EFT! Renormalization Summary

44. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201444 Chiral three-nucleon forces (3NF) On another topic:

45. 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 The 3NF at NNLO; used so far. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201445

46. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201446 The 3NF at NNLO; used so far. Now, showing only 3NF diagrams.

47. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201447 Calculating the properties of light nuclei using chiral 2N and 3N forces “No-Core Shell Model “ Calculations by P. Navratil et al., LLNL

48. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201448 2N (N3LO) force only Calculating the properties of light nuclei using chiral 2N and 3N forces “No-Core Shell Model “ Calculations by P. Navratil et al., LLNL

49. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201449 R. Machleidt 49 2N (N3LO) force only Calculating the properties of light nuclei using chiral 2N and 3N forces “No-Core Shell Model “ Calculations by P. Navratil et al., LLNL 2N (N3LO) +3N (N2LO) forces

50. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201450 R. Machleidt 50 Oxygen Calcium

51. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201451 Cutoff = 450-600 MeV NLO NNLO N3LO How about some predictive power?

52. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201452 R. Machleidt 52 The Ay puzzle is NOT solved by the 3NF at NNLO. Analyzing Power Ay p-d p- 3 He 2NF only 2NF+3NF Calculations by the Pisa Group 2NF only 2NF+3NF

53. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201453 The 2NF is N3LO; The 2NF is N3LO; consistency requires that all contributions are included up to the same order. consistency requires that all contributions are included up to the same order. There are unresolved problems in 3N and 4N scattering, and nuclear structure. There are unresolved problems in 3N and 4N scattering, and nuclear structure. And so, we need 3NFs beyond NNLO, because …

54. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201454 The 3NF at NNLO; used so far. Back to the drawing board.

55. The 3NF at NNLO; used so far. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201455

56. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201456 The 3NF at NNLO; used so far. Small? Apps of N3LO 3NF: Triton: Skibinski et al., PRC 84, 054005 (2011). Not conclusive. Neutron matter: Hebeler, Schwenk and co-workers, PRL 110, 032504 (2013). Not small!(?) N-d scattering (Ay): Witala et al. Small!

57. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201457 N-d A y calculations by Witala et al.

58. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201458 The 3NF at NNLO; used so far. Small? Apps of N3LO 3NF: Triton: Skibinski et al., PRC 84, 054005 (2011). Not conclusive. Neutron matter: Hebeler, Schwenk and co-workers, PRL 110, 032504 (2013). Not small!(?) N-d scattering (Ay): Witala et al. Small!

59. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201459 The 3NF at NNLO; used so far. Small?

60. The 3NF at NNLO; used so far. Small? R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201460 1-loop graphs: 5 topologies 2PE 2PE-1PE Ring Contact-1PE Contact-2PE

61. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201461 The 3NF at NNLO; used so far. Small? R. Machleidt Chiral EFT TRIUMF, Feb. 21, 2013 61 1-loop graphs: 5 topologies 2PE 2PE-1PE Ring Contact-1PE Contact-2PE

62. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201462 The 3NF at NNLO; used so far. Small? Large?! Many new isospin/spin/momentum structures.

63. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201463 The 3NF at NNLO; used so far. Small?

64. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201464 The 3NF at NNLO; used so far. Small? 3NF contacts at N4LO Girlanda, Kievsky, Viviani, PRC 84, 014001 (2011) Spin-Orbit Force!

65. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201465 … and then there Is also the Δ-full theory …

66. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201466 … and then there Is also the Δ-full theory … … but we have no time left for that.

67. R. Machleidt Theory of Nucl. Forces KITPC Beijing 08-25-201467 Conclusions The research on nuclear interaction theory continues to be exciting and divers such that, after eight decades, a workshop on this topic is still very timely. The research on nuclear interaction theory continues to be exciting and divers such that, after eight decades, a workshop on this topic is still very timely. Presently, lattice QCD and the chiral EFT approach appear to be the most promising ones. Presently, lattice QCD and the chiral EFT approach appear to be the most promising ones. But there are still some not so subtle “subtleties” to be taken care of: But there are still some not so subtle “subtleties” to be taken care of: The renormalization of the chiral 2NF, The renormalization of the chiral 2NF, Sub-leading 3NFs, Sub-leading 3NFs, The predictive power. The predictive power.