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R. Machleidt, University of Idaho University of Idaho The missing three-nucleon forces: Where are they? 2009 Mardi Gras “Special Symmetries and Ab Initio.

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1 R. Machleidt, University of Idaho University of Idaho The missing three-nucleon forces: Where are they? 2009 Mardi Gras “Special Symmetries and Ab Initio Methods for Light Nuclei” Workshop Louisiana State University, 18-20 February, 2009

2 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-092 Outline Nuclear forces from Chiral Perturbation Theory (ChPT): Current status Nuclear forces from Chiral Perturbation Theory (ChPT): Current status The 3NF at N2LO The 3NF at N2LO The 3NF at N3LO: Weak expectations The 3NF at N3LO: Weak expectations N4LO or Δ-isobars? N4LO or Δ-isobars? Summary Summary

3 The ChPT 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 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-093

4 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-094 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

5 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-095 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 Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-096 Phase shifts up to 300 MeV Red Line: N3LO Potential by Entem & Machleidt, PRC 68, 041001 (2003). Green dashed line: NNLO Potential, and blue dotted line: NLO Potential by Epelbaum et al., Eur. Phys. J. A19, 401 (2004).

7 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-097 R. Machleidt 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 Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-098

9 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-099 R. Machleidt 9 Deuteron wave functions of N3LO potentials red = Idaho, green = Bochum/Juelich Black lines = high-precision pots.

10 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0910 R. Machleidt 10 Results from applications of chiral two- nucleon forces Latest Coupled Cluster results for A≤48 using N3LO Latest Coupled Cluster results for A≤48 using N3LO

11 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0911 R. Machleidt 11 Nucleus  E / A [MeV] 4 He1.08 (0.73 FY ) 16 O1.25 40 Ca0.84 48 Ca1.27 48 Ni1.21 Chiral NN potential at N 3 LO underbinds by ~1MeV/nucleon. (Size extensivity at its best.) ? V_low-k

12 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0912 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

13 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0913 R. Machleidt 13 4NF at N3LO (leading order) Essentially negligible, as to be expected.

14 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 Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0914

15 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0915 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

16 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0916 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 16 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 16 ✗ 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 )

17 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0917 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 17 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 17 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 17 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 17 ✗ 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)

18 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0918 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 (1/M) n corrections N4LO (Q 5 )

19 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0919 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.

20 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0920 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 Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0921 R. Machleidt 21 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 2N (N3LO) force only

22 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0922

23 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0923 Courtesy of M. Viviani; latest data from TUNL.

24 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0924 R. Machleidt 24 Recent Adv. in Th. of Nucl. Forces EXOCT07, Catania, 14-June-07 n-d at 3 MeV Ay puzzle at NLO and NNLO (2NF+3NF) NLO NNLO The Ay puzzle is NOT solved by the 3NF at NNLO.

25 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0925 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.

26 What 3NFs are generated by ChPT beyond NNLO? R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0926

27 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0927 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 27 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 27 ✗ 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 )

28 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0928 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 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 Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0929 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 Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0930 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 ) ✗ ✗ ✗ ✗ Time derivative

31 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0931 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 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 ) ✗ ✗ ✗ ✗ 1-loop lead. vert.

32 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0932 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

33 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0933 The ring diagrams From: Bernard, Epelbaum, Krebs, Meissner, PRC 77, 064004 (2008)

34 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0934 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.

35 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 Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0935

36 What now? R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0936 Go to the next order!

37 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0937 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 37 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 37 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 37 ✗ 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.

38 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0938 What to expect from those N4LO 1-loop diagrams? Compare to “similar” 2NF diagrams.

39 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0939 ✗ Small 1-loop lead. vert. 1-loop One c i vert. Corresponding 2NF contributions 2π

40 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0940

41 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0941 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 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 Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0942 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 Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0943

44 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0944 pi-N Lagrangian with two derivatives (“next-to-leading” order) Bernard et al. ‘97

45 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0945 R. Machleidt 45 The Hierarchy of Nuclear Forces Δ less Leading Order Next-to Leading Order Next-to- Next-to Leading Order Next-to- Next-to- Next-to Leading Order Additional in Δ full Ordonez ’94, Kaiser ‘98 Krebs ‘07 Chiral 2NF

46 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0946 R. Machleidt 46 Δ-full theory Raises NLO, Better convergence

47 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0947 The 3NF in the Δ -full theory

48 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0948 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 48 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 48 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 48 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 48 R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 48 ✗ 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

49 However, there is a however … For reasons of consistency, the 2NF must then also include Δ’s---up to N3LO! For reasons of consistency, the 2NF must then also include Δ’s---up to N3LO! So, what does that look like? So, what does that look like? R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0949

50 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0950 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 Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0951 Either In summary:

52 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0952 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 Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0953 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 Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0954 Or

55 R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0955 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 Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0956 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

57 Whatever we decide (Δ-less or Δ-full), it will be impossible to do a complete calculation anytime soon; so what’s a good strategy? Assume “resonance saturation” for the 2NF, i.e., stay with the present Δ-less N3LO 2NF (this has flaws!) Assume “resonance saturation” for the 2NF, i.e., stay with the present Δ-less N3LO 2NF (this has flaws!) Add the Δ-full 3NF at N3LO. Add the Δ-full 3NF at N3LO. R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0957

58 Conclusions 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 Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0958

59 at N4LO in Δ-less or at N4LO in Δ-less or at N3LO in Δ-full. at N3LO in Δ-full. R. Machleidt Missing 3NFs: Where are they? Mardi Gras, LSU, 02-18-0959 The missing three-nucleon forces: Where are they? They are

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