1. R. Machleidt University of Idaho Nuclear forces from chiral EFT: The unfinished business

2. Outline Nuclear forces from chiral EFT: Nuclear forces from chiral EFT: Overview & achievements Overview & achievements Are we done? No! Are we done? No! Sub-leading many-body forces Sub-leading many-body forces Proper renormalization of chiral forces Proper renormalization of chiral forces Outlook: Will we ever be done? Outlook: Will we ever be done? R. Machleidt 2 The unfinished business Vietri, Italy, 22 May 2010

3. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 3 The chiral EFT approach to nuclear forces Clear connection to QCD: via symmetries Clear connection to QCD: via symmetries Degrees of freedom relevant to (low- energy) nuclear physics: pions and nucleons Degrees of freedom relevant to (low- energy) nuclear physics: pions and nucleons Systematic expansion: Q/Λ Systematic expansion: Q/Λ Controlled error, predictive power. Controlled error, predictive power. Explains the empirically known hierarchy of nuclear forces Explains the empirically known hierarchy of nuclear forces

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 The unfinished business Vietri, Italy, 22 May 2010

5. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 5 The Hierarchy of Nuclear Forces 2N forces Leading Order Next-to Leading Order Next-to- Next-to Leading Order Next-to- Next-to- Next-to Leading Order

6. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 6 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

7. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 7 N3LO Potential by Entem & Machleidt, PRC 68, 041001 (2003). NNLO and NLO Potentials by Epelbaum et al., Eur. Phys. J. A19, 401 (2004).

8. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 8 Applications of the chiral NN potential

9. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 9

10. R. Machleidt The unfinished business Vietri, Italy, 22 May 201010

11. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 11 Chiral NN potential at N 3 LO underbinds by ~1MeV/nucleon. (Size extensivity at its best.) Nucleus  E / A [MeV] 4 He1.08 (0.73 FY ) 16 O1.25 40 Ca0.84 48 Ca1.27 48 Ni1.21

12. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 12

13. R. Machleidt The unfinished business Vietri, Italy, 22 May 201013

14. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 14 The Hierarchy of Nuclear Forces 2N forces Leading Order Next-to Leading Order Next-to- Next-to Leading Order Next-to- Next-to- Next-to Leading Order 4N forces

15. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 15 R. Machleidt 15 4NF at N3LO (leading order) Essentially negligible, as to be expected.

16. Interim summary 2NF o.k. 2NF o.k. (except for the issue of non- perturbative renormalization) (except for the issue of non- perturbative renormalization) 4NF (negligibly) small … 4NF (negligibly) small … R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 16

17. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 17 3NF The Hierarchy of Nuclear Forces 2N forces 3N forces Leading Order Next-to Leading Order Next-to- Next-to Leading Order Next-to- Next-to- Next-to Leading Order The focus: The 3NF 4N forces

18. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 18 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 18 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 18 ✗ The Hierarchy of Nuclear Forces Next-to Leading Order (Q 2 ) Next-to- Next-to Leading Order (Q 3 ) Next-to- Next-to- Next-to Leading Order (Q 4 ) The chiral 3NF Leading Order (Q 0 ) Basic Diagrams N4LO (Q 5 )

19. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 19 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 19 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 19 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 19 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 19 ✗ The Hierarchy of Nuclear Forces Next-to Leading Order (Q 2 ) Next-to- Next-to Leading Order (Q 3 ) Next-to- Next-to- Next-to Leading Order (Q 4 ) The chiral 3NF Leading Order (Q 0 ) Basic Diagrams (1/M) n corrections N4LO (Q 5 ) Coon & Friar (1986)

20. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 20 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 20 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 20 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 20 ✗ The Hierarchy of Nuclear Forces Next-to Leading Order (Q 2 ) Next-to- Next-to Leading Order (Q 3 ) Next-to- Next-to- Next-to Leading Order (Q 4 ) The chiral 3NF Leading Order (Q 0 ) Basic Diagrams (1/M) n corrections N4LO (Q 5 ) The 3NF At N2LO : How well does it do?

21. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 21

22. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 22 Calculating the properties of light nuclei using chiral 2N and 3N forces “No-Core Shell Model “ Calculations by P. Navratil et al., LLNL

23. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 23 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

24. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 24 R. Machleidt 24 2N (N3LO) force only Calculating the properties of light nuclei using chiral 2N and 3N forces 2N (N3LO) +3N (N2LO) forces “No-Core Shell Model “ Calculations by P. Navratil et al., LLNL

25. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 25 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

26. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 26 Why do we need 3NFs beyond NNLO? The 2NF is N3LO; The 2NF is N3LO; consistency requires that all contributions are at the same order. consistency requires that all contributions are at the same order. There are unresolved problems in 3N, 4N scattering and nuclear structure. There are unresolved problems in 3N, 4N scattering and nuclear structure.

27. What’s there beyond NNLO? R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 27

28. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 28 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 28 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 28 ✗ The Hierarchy of Nuclear Forces Next-to Leading Order (Q 2 ) Next-to- Next-to Leading Order (Q 3 ) Next-to- Next-to- Next-to Leading Order (Q 4 ) The chiral 3NF Leading Order (Q 0 ) Basic Diagrams (1/M) n corrections N4LO (Q 5 )

29. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 29 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 29 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 29 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 29 ✗ The Hierarchy of Nuclear Forces Next-to Leading Order (Q 2 ) Next-to- Next-to Leading Order (Q 3 ) Next-to- Next-to- Next-to Leading Order (Q 4 ) The chiral 3NF Leading Order (Q 0 ) Basic Diagrams (1/M) n corrections N4LO (Q 5 )

30. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 30 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 30 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 30 ✗ The Hierarchy of Nuclear Forces Next-to Leading Order (Q 2 ) Next-to- Next-to Leading Order (Q 3 ) Next-to- Next-to- Next-to Leading Order (Q 4 ) The chiral 3NF Leading Order (Q 0 ) Basic Diagrams (1/M) n corrections N4LO (Q 5 )

31. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 31 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 31 ✗ The Hierarchy of Nuclear Forces Next-to Leading Order (Q 2 ) Next-to- Next-to Leading Order (Q 3 ) Next-to- Next-to- Next-to Leading Order (Q 4 ) The chiral 3NF Leading Order (Q 0 ) Basic Diagrams (1/M) n corrections N4LO (Q 5 ) ✗ ✗ ✗ ✗ Time derivative

32. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 32 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 32 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 32 ✗ The Hierarchy of Nuclear Forces Next-to Leading Order (Q 2 ) Next-to- Next-to Leading Order (Q 3 ) Next-to- Next-to- Next-to Leading Order (Q 4 ) The chiral 3NF Leading Order (Q 0 ) Basic Diagrams (1/M) n corrections N4LO (Q 5 ) ✗ ✗ ✗ ✗ 1-loop lead. vert.

33. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 33 Ishikawa & Robilotta, PRC 76, 014006 (2007) Bernard, Epelbaum, Krebs, Meissner, PRC 77, 064004 (2008) In progress The 3NF at N3LO explicitly One-loop, leading vertices

34. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 34 The ring diagrams From: Bernard, Epelbaum, Krebs, Meissner, PRC 77, 064004 (2008)

35. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 35 Proton-deuteron elastic scattering Black line: 2NF only Red dashed line: 2NF + 3NF(N2LO+2πN3LO) ≈ 2NF + 3NF(N2LO) From: Ishikawa & Robilotta, PRC 76, 014006 (2007) The 2πN3LO 3NF is negligible.

36. Interim balance The 3NF at N3LO may be generally weak. The 3NF at N3LO may be generally weak. Reason: only “leading” vertices are involved, which are known to be weak (e.g., from NN). Reason: only “leading” vertices are involved, which are known to be weak (e.g., from NN). Thus, the 3NF at N3LO may not solve any of the outstanding problems. Thus, the 3NF at N3LO may not solve any of the outstanding problems. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 36

37. What now? R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 37 Go to the next order!

38. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 38 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 38 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 38 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 38 ✗ The Hierarchy of Nuclear Forces Next-to Leading Order (Q 2 ) Next-to- Next-to Leading Order (Q 3 ) Next-to- Next-to- Next-to Leading Order (Q 4 ) The chiral 3NF Leading Order (Q 0 ) Basic Diagrams (1/M) n corrections N4LO (Q 5 ) ✗ ✗ ✗ ✗ 1-loop one c i vert. 1-loop lead. vert.

39. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 39 What to expect from those N4LO 1-loop diagrams? Compare to “similar” 2NF diagrams.

40. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 40 ✗ Small 1-loop lead. vert. 1-loop One c i vert. Corresponding 2NF contributions 2π

41. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 41 R. Machleidt 41 ✗ Small 1-loop lead. vert. 1-loop One c i vert. Corresponding 2NF contributions 2π 3π Kaiser (2000) Kaiser (2001) Small Large!!

42. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 42 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 42 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 42 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 42 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 42 ✗ The Hierarchy of Nuclear Forces Next-to Leading Order (Q 2 ) Next-to- Next-to Leading Order (Q 3 ) Next-to- Next-to- Next-to Leading Order (Q 4 ) The chiral 3NF Leading Order (Q 0 ) Basic Diagrams (1/M) n corrections N4LO (Q 5 ) ✗ ✗ ✗ ✗ 1-loop one c i vert. 1-loop lead. vert. Large!!

43. So, we will have to go to N4LO!? No, not necessarily. No, not necessarily. There is an alternative: There is an alternative: go to N3LO, but include Δ(1232) isobars explicitly. go to N3LO, but include Δ(1232) isobars explicitly. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 43

44. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 44 The 3NF in the “ Δ -full” theory

45. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 45 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 45 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 45 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 45 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 45 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 45 ✗ The Hierarchy of Nuclear Forces Next-to Leading Order (Q 2 ) Next-to- Next-to Leading Order (Q 3 ) Next-to- Next-to- Next-to Leading Order (Q 4 ) The chiral 3NF: Δ-less vs. Δ-full Leading Order (Q 0 ) N4LO (Q 5 ) Δ less Additional in Δ full 1-loops with Δ e.g., Illinois

46. However, there is a however … For reasons of consistency, the 2NF must then also include Δ’s, For reasons of consistency, the 2NF must then also include Δ’s, up to N3LO! up to N3LO! It will be a big mess! It will be a big mess! R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 46

47. Conclusions Substantial advances in chiral nuclear forces during the past decade. The major milestone of the decade: “high precision” NN pots. at N3LO, successful in nuclear structure. Substantial advances in chiral nuclear forces during the past decade. The major milestone of the decade: “high precision” NN pots. at N3LO, successful in nuclear structure. But there are still issues: But there are still issues: 3NFs beyond N2LO are needed! 3NFs beyond N2LO are needed! The 3NF at N3LO of the Δ-less theory may be weak and useless. The 3NF at N3LO of the Δ-less theory may be weak and useless. This calls for further 3NF contributions: This calls for further 3NF contributions: - Δ-less N4LO or - Δ-less N4LO or - Δ-full N3LO, - Δ-full N3LO, they will be large, but none will be easy. they will be large, but none will be easy. ChPT of nuclear forces has been fairly easy and straightforward, so far; but what’s left won’t be easy, no matter how you do it. ChPT of nuclear forces has been fairly easy and straightforward, so far; but what’s left won’t be easy, no matter how you do it. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 47

48. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 48 The End

49. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 49 Three-nucleon forces at N2LO TPE-3NF OPE-3NF Contact- 3NF No new parameters! One new parameter, D. One new parameter, E. Strategy: Adjust D and E to two few-nucleon observables, e.g., the triton and alpha-particle binding energies. Then predict properties of other light nuclei.

50. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 50 The Hierarchy of Nuclear Forces Leading Order Next-to Leading Order Next-to- Next-to Leading Order Next-to- Next-to- Next-to Leading Order Additional in Δ full + 1/M corrections, Kaiser ’98: large? 2pi & 3pi 2-loops with Δ The chiral 2NF: Δ -less vs. Δ -full N4LO (Q 5 ) The Hierarchy of Nuclear Forces Ordonez ’94, Kaiser ‘98 Krebs ‘07 Δ less

51. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 51 Either In summary:

52. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 52 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 52 The Hierarchy of Nuclear Forces Leading Order Next-to Leading Order Next-to- Next-to Leading Order Next-to- Next-to- Next-to Leading Order Additional in Δ full + 1/M corrections, Kaiser ’98: large? 2pi & 3pi 2-loops with Δ The chiral 2NF: Δ -less vs. Δ -full N4LO (Q 5 ) The Hierarchy of Nuclear Forces Ordonez ’94, Kaiser ‘98 Krebs ‘07 Δ less

53. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 53 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 53 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 53 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 53 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 53 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 53 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 53 ✗ The Hierarchy of Nuclear Forces Next-to Leading Order (Q 2 ) Next-to- Next-to Leading Order (Q 3 ) Next-to- Next-to- Next-to Leading Order (Q 4 ) The chiral 3NF: Δ-less vs. Δ-full Leading Order (Q 0 ) N4LO (Q 5 ) Δ less Additional in Δ full 1-loops with Δ e.g., Illinois

54. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 54 Or

55. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 55 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 55 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 55 The Hierarchy of Nuclear Forces Leading Order Next-to Leading Order Next-to- Next-to Leading Order Next-to- Next-to- Next-to Leading Order Additional in Δ full + 1/M corrections, Kaiser ’98: large? 2pi & 3pi 2-loops with Δ The chiral 2NF: Δ -less vs. Δ -full N4LO (Q 5 ) The Hierarchy of Nuclear Forces Ordonez ’94, Kaiser ‘98 Krebs ‘07 Δ less

56. R. Machleidt The unfinished business Vietri, Italy, 22 May 2010 56 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 56 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 56 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 56 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 56 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 56 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 56 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 56 ✗ The Hierarchy of Nuclear Forces Next-to Leading Order (Q 2 ) Next-to- Next-to Leading Order (Q 3 ) Next-to- Next-to- Next-to Leading Order (Q 4 ) The chiral 3NF: Δ-less vs. Δ-full Leading Order (Q 0 ) N4LO (Q 5 ) Δ less Additional in Δ full 1-loops with Δ e.g., Illinois