1. R. Machleidt, University of Idaho D. R. Entem, University of Salamanca Sub-Leading Three-Nucleon Forces in Chiral Perturbation Theory

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 three-nucleon forces Sub-leading three-nucleon forces Outlook Outlook R. Machleidt2 Sub-Leading 3NFs ECT* Trento 07/11/2011

3. From QCD to nuclear physics via chiral EFT (in a nutshell) 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 R. Machleidt3 Sub-Leading 3NFs ECT* Trento 07/11/2011

4. Calls for an EFT Calls for an EFT soft scale: Q ≈ m π, hard scale: Λ χ ≈ m ρ ; pions and nucleon relevant d.o.f. soft scale: Q ≈ m π, hard scale: Λ χ ≈ m ρ ; pions and nucleon relevant d.o.f. Low-energy expansion: (Q/Λ χ ) ν Low-energy 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. Most general Lagrangian consistent with all symmetries of low-energy QCD. 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) R. Machleidt4 Sub-Leading 3NFs ECT* Trento 07/11/2011

5. R. Machleidt5 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 Sub-Leading 3NFs ECT* Trento 07/11/2011

6. R. Machleidt6 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 Sub-Leading 3NFs ECT* Trento 07/11/2011

7. R. Machleidt7 N3LO Potential by Entem & Machleidt, PRC 68, 041001 (2003). NNLO and NLO Potentials by Epelbaum et al., Eur. Phys. J. A19, 401 (2004). Sub-Leading 3NFs ECT* Trento 07/11/2011

8. R. Machleidt8 Applications of the chiral NN potential at N3LO Sub-Leading 3NFs ECT* Trento 07/11/2011

9. R. Machleidt9 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 Sub-Leading 3NFs ECT* Trento 07/11/2011

10. R. Machleidt10 Sub-Leading 3NFs ECT* Trento 07/11/2011

11. R. Machleidt11 … including the chiral 3NF at N2LO Sub-Leading 3NFs ECT* Trento 07/11/2011

12. R. Machleidt12 Sub-Leading 3NFs ECT* Trento 07/11/2011

13. R. Machleidt13 Calculating the properties of light nuclei using chiral 2N and 3N forces “No-Core Shell Model “ Calculations by P. Navratil et al., LLNL Sub-Leading 3NFs ECT* Trento 07/11/2011

14. R. Machleidt14 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 Sub-Leading 3NFs ECT* Trento 07/11/2011

15. R. Machleidt15 R. Machleidt 15 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 Sub-Leading 3NFs ECT* Trento 07/11/2011

16. R. Machleidt16 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 Sub-Leading 3NFs ECT* Trento 07/11/2011

17. R. Machleidt17 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. Sub-Leading 3NFs ECT* Trento 07/11/2011

18. So, let’s take a close look at the development of chiral 3NFs. R. Machleidt18 Sub-Leading 3NFs ECT* Trento 07/11/2011

19. R. Machleidt19 The 3NF at NNLO; used so far. Sub-Leading 3NFs ECT* Trento 07/11/2011

20. R. Machleidt20 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 Sub-Leading 3NFs ECT* Trento 07/11/2011

21. R. Machleidt 21 The ring diagrams From: Bernard, Epelbaum, Krebs, Meissner, PRC 77, 064004 (2008) R. Machleidt21 Sub-Leading 3NFs ECT* Trento 07/11/2011

22. R. Machleidt 22 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. R. Machleidt22 Sub-Leading 3NFs ECT* Trento 07/11/2011

23. R. Machleidt23 R. Machleidt Nuclear forces from chiral EFT EFB21, Salamanca, 08-31-2010 23 Small? The 3NF at NNLO; used so far. Sub-Leading 3NFs ECT* Trento 07/11/2011

24. R. Machleidt24 R. Machleidt Nuclear forces from chiral EFT EFB21, Salamanca, 08-31-2010 24 R. Machleidt Nuclear forces from chiral EFT EFB21, Salamanca, 08-31-2010 24 Small? The 3NF at NNLO; used so far. Sub-Leading 3NFs ECT* Trento 07/11/2011

25. R. Machleidt 25 What to expect from those N4LO 1-loop diagrams? Compare to “similar” 2NF diagrams. R. Machleidt25 Sub-Leading 3NFs ECT* Trento 07/11/2011

26. R. Machleidt 26 ✗ Small 1-loop lead. vert. 1-loop One c i vert. Corresponding 2NF contributions 2π R. Machleidt26 Sub-Leading 3NFs ECT* Trento 07/11/2011

27. R. Machleidt 27 27 ✗ Small 1-loop lead. vert. 1-loop One c i vert. Corresponding 2NF contributions 2π 3π Kaiser (2000) Kaiser (2001) Small Large!! R. Machleidt27 Sub-Leading 3NFs ECT* Trento 07/11/2011

28. R. Machleidt The Nuclear Force Problem Mumbai, 22 November 2010 28 R. Machleidt Nuclear forces from chiral EFT EFB21, Salamanca, 08-31-2010 28 R. Machleidt Nuclear forces from chiral EFT EFB21, Salamanca, 08-31-2010 28 Small? The 3NF at NNLO; used so far. Large!! R. Machleidt28 Sub-Leading 3NFs ECT* Trento 07/11/2011

29. By the way, the full story is more complicated, See next slide. R. Machleidt29 Sub-Leading 3NFs ECT* Trento 07/11/2011

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 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 ) ✗ ✗ ✗ ✗ R. Machleidt30 Sub-Leading 3NFs ECT* Trento 07/11/2011

31. But to keep a long story short, Let’s not bother with those 1/M corrections. R. Machleidt31 Sub-Leading 3NFs ECT* Trento 07/11/2011

32. R. Machleidt The Nuclear Force Problem Mumbai, 22 November 2010 32 R. Machleidt Nuclear forces from chiral EFT EFB21, Salamanca, 08-31-2010 32 R. Machleidt Nuclear forces from chiral EFT EFB21, Salamanca, 08-31-2010 32 Small? The 3NF at NNLO; used so far. Large!! R. Machleidt32 Sub-Leading 3NFs ECT* Trento 07/11/2011

33. A variation of the theory is possible: Introduce Δ degrees of freedom R. Machleidt33 Sub-Leading 3NFs ECT* Trento 07/11/2011

34. R. Machleidt The Nuclear Force Problem Mumbai, 22 November 2010 34 R. Machleidt Nuclear forces from chiral EFT EFB21, Salamanca, 08-31-2010 34 R. Machleidt Nuclear forces from chiral EFT EFB21, Salamanca, 08-31-2010 34 Small? The 3NF at NNLO; used so far. R. Machleidt34 Sub-Leading 3NFs ECT* Trento 07/11/2011

35. R. Machleidt 35 pi-N Lagrangian with two derivatives (“next-to-leading” order) Bernard et al. ‘97 R. Machleidt35 Sub-Leading 3NFs ECT* Trento 07/11/2011

36. R. Machleidt The Nuclear Force Problem Mumbai, 22 November 2010 36 R. Machleidt Nuclear forces from chiral EFT EFB21, Salamanca, 08-31-2010 36 R. Machleidt Nuclear forces from chiral EFT EFB21, Salamanca, 08-31-2010 36 Small? The 3NF at NNLO; used so far. R. Machleidt36 Sub-Leading 3NFs ECT* Trento 07/11/2011

37. R. Machleidt The Nuclear Force Problem Mumbai, 22 November 2010 37 R. Machleidt Nuclear forces from chiral EFT EFB21, Salamanca, 08-31-2010 37 R. Machleidt Nuclear forces from chiral EFT EFB21, Salamanca, 08-31-2010 37 Small? Large!! The 3NF at NNLO; used so far. R. Machleidt37 Sub-Leading 3NFs ECT* Trento 07/11/2011

38. So, with Δ’s you have to go “only” to N3LO (to catch the large 3NFs), But notice that there are many more diagrams At N3LO with Δ’s as compared to Δ-less. So, it’s not necessarily simpler. Notice further that in the Δ-full theory the 2NF Gets also more complicated. R. Machleidt38 Sub-Leading 3NFs ECT* Trento 07/11/2011

39. R. Machleidt39 Sub-Leading 3NFs ECT* Trento 07/11/2011

40. Summarize: Δ-full vs. Δ-less (2NF and 3NF). R. Machleidt40 Sub-Leading 3NFs ECT* Trento 07/11/2011

41. R. Machleidt The Nuclear Force Problem Mumbai, 22 November 2010 41 R. Machleidt Nuclear forces from chiral EFT EFB21, Salamanca, 08-31-2010 41 R. Machleidt Nuclear forces from chiral EFT EFB21, Salamanca, 08-31-2010 41 R. Machleidt41 Sub-Leading 3NFs ECT* Trento 07/11/2011

42. R. Machleidt The Nuclear Force Problem Mumbai, 22 November 2010 42 R. Machleidt Nuclear forces from chiral EFT EFB21, Salamanca, 08-31-2010 42 R. Machleidt Nuclear forces from chiral EFT EFB21, Salamanca, 08-31-2010 42 R. Machleidt42 Sub-Leading 3NFs ECT* Trento 07/11/2011

43. What’s the better approach is a matter of taste Bochum pursues Δ-full Idaho Δ-less R. Machleidt43 Sub-Leading 3NFs ECT* Trento 07/11/2011

44. R. Machleidt The Nuclear Force Problem Mumbai, 22 November 2010 44 R. Machleidt Nuclear forces from chiral EFT EFB21, Salamanca, 08-31-2010 44 R. Machleidt Nuclear forces from chiral EFT EFB21, Salamanca, 08-31-2010 44 Small? The 3NF at NNLO; used so far. Large!! R. Machleidt44 Sub-Leading 3NFs ECT* Trento 07/11/2011

45. The Δ-less N4LO (ν=5) 3NF in more detail: Tree diagrams and 1-loop graphs 1-loop graphs: 5 topologies 2PE 2PE-1PE Ring Contact-1PE Contact-2PE R. Machleidt45 Sub-Leading 3NFs ECT* Trento 07/11/2011 Pisa e_i

46. The N4LO 3NF 2PE diagrams R. Machleidt46 Sub-Leading 3NFs ECT* Trento 07/11/2011

47. The N4LO 3NF 2PE-1PE diagrams R. Machleidt47 Sub-Leading 3NFs ECT* Trento 07/11/2011 Many new isospin/spin/momentum structures, some similar to N3LO 2PE-1PE 3NF.

48. Conclusions R. Machleidt 48 Substantial advances in chiral nuclear forces during the past decade. The major milestone of the decade: “high precision” NN pots. at N3LO; good for nuclear structure. Substantial advances in chiral nuclear forces during the past decade. The major milestone of the decade: “high precision” NN pots. at N3LO; good for nuclear structure. But there are still substantial open issues concerning three-nucleon forces (3NF), which is why we are here. But there are still substantial open issues concerning three-nucleon forces (3NF), which is why we are here. Large 3NFs with many new structures to be expected at N3LO of Δ-full or N4LO at Δ-less. Construction is under way. Large 3NFs with many new structures to be expected at N3LO of Δ-full or N4LO at Δ-less. Construction is under way. There will be many new 3NFs to check out in the near future. Stay tuned. There will be many new 3NFs to check out in the near future. Stay tuned. R. Machleidt48 Sub-Leading 3NFs ECT* Trento 07/11/2011

49. And a specific request to 3NF appliers In the 2PE 3NF at NNLO, consider the c_i LECs as free parameters: Is there a parameter set that substantially improves the Nd Ay? In the 2PE 3NF at NNLO, consider the c_i LECs as free parameters: Is there a parameter set that substantially improves the Nd Ay? Is there any combination of isospin, spin, and momentum operators of 3 nucleons that particularly improves Ay? Is there any combination of isospin, spin, and momentum operators of 3 nucleons that particularly improves Ay? R. Machleidt Sub-Leading 3NFs ECT* Trento 07/11/201149