1. R. Machleidt Collaborators: E. Marji, Ch. Zeoli University of Idaho The nuclear force problem: Have we finally reached the end of the tunnel? 474-th International Heraeus Seminar Bad Honnef, Germany, February 12 – 16, 2011

2. Outline Historical perspective Historical perspective 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 The end of the tunnel? The end of the tunnel? R. Machleidt 2 The Nuclear Force Problem Bad Honnef, 14 February 2011

3. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 20113 The circle of history is closing!

4. 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. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 4

5. 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. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 5

6. R. Machleidt 6 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 Nuclear Force Problem Bad Honnef, 14 February 2011

7. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 7 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

8. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 8 N3LO Potential by Entem & Machleidt, PRC 68, 041001 (2003). NNLO and NLO Potentials by Epelbaum et al., Eur. Phys. J. A19, 401 (2004).

9. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 9 Applications of the chiral NN potential at N3LO

10. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 10 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

11. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 11 … including the chiral 3NF at N2LO

12. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 12 Calculating the properties of light nuclei using chiral 2N and 3N forces “No-Core Shell Model “ Calculations by P. Navratil et al., LLNL

13. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 13 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

14. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 14 R. Machleidt 14 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

15. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 15 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

16. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 16 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.

17. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 17 The 3NF at NNLO; used so far.

18. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 18 R. Machleidt Nuclear forces from chiral EFT EFB21, Salamanca, 08-31-2010 18 R. Machleidt Nuclear forces from chiral EFT EFB21, Salamanca, 08-31-2010 18 The 3NF at NNLO; used so far. Small? Large!! See contribution to This Seminar By H. Krebs.

19. So, we are obviously not done! Subleading few-nucleon forces: N4LO in Δ-less or N3LO in Δ-full. Subleading few-nucleon forces: N4LO in Δ-less or N3LO in Δ-full. Renormalization of chiral nuclear forces Renormalization of chiral nuclear forces R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 19 I will focus now on this one. Some of the more crucial open issues:

20. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 20 “I about got this one renormalized”

21. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 21 The issue has produced lots and lots of papers; this is just a small sub-selection.

22. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 22 So, what’s the problem with this renormalization?

23. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 23 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”).

24. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 24 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.

25. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 25 The first kind: “NN Potential”: “NN Potential”: irreducible diagrams calculated perturbatively. irreducible diagrams calculated perturbatively. Example: Example: Counter terms  perturbative renormalization (order by order)

26. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 26 R. Machleidt 26 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.

27. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 27 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): 2727 In diagrams: +++ …

28. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 2828 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: 28  Non-perturbative renormalization

29. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 29 R. Machleidt 29 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: 29  Non-perturbative renormalization 29 With what to renormalize this time? Weinberg’s silent assumption: The same counter terms as before. (“Weinberg counting”)

30. Weinberg counting fails already in Leading Order (for Λ  ∞ renormalization) 3S1 and 1S0 (with a caveat) renormalizable with LO counter terms. 3S1 and 1S0 (with a caveat) renormalizable with LO counter terms. However, where OPE tensor force attractive: However, where OPE tensor force attractive: 3P0, 3P2, 3D2, … 3P0, 3P2, 3D2, … a counter term a counter term must be added. must be added. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 30 “Modified Weinberg counting” for LO Nogga, Timmermans, v. Kolck PRC72, 054006 (2005):

31. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 31 Quantitative chiral NN potentials are at N3LO. So, we need to go substantially beyond LO.

32. Nonperturbative or perturbative? Nonperturbative or perturbative? Infinite cutoff or finite cutoff? Infinite cutoff or finite cutoff? R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 32 Renormalization beyond leading order – Issues

33. Renormalization beyond leading order – Options 1 Continue with the nonperturbative infinite-cutoff renormalization. 2 Perturbative using DWBA. 3 Nonperturbative using finite cutoffs ≤ Λχ ≈ 1 GeV. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 33

34. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 201134 Option 1: Nonperturbative infinite-cutoff renormalization up to N3LO

35. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 35 S=1 T=1 NLO NNL O LO N3LO Different partial waves are windows on different ranges of the force.

36. In lower partial waves ( ≅ short distances), in some cases convergence, in some not; data are not reproduced. In lower partial waves ( ≅ short distances), in some cases convergence, in some not; 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? 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? Where are the systematic order by order improvements? Where are the systematic order by order improvements? R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 36 Option 1: Nonperturbative infinite-cutoff renormalization up to N3LO Observations and problems

37. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 37 In lower partial waves ( ≅ short distances), in some cases convergence, in some not; data are not reproduced. In lower partial waves ( ≅ short distances), in some cases convergence, in some not; 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? 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? Where are the systematic order by order improvements? Where are the systematic order by order improvements? R. Machleidt 37 Option 1: Nonperturbative infinite-cutoff renormalization up to N3LO Observations and problems No good!

38. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 38 Option 2: Perturbative, using DWBA (Valderrama ‘09) Renormalize LO non-perturbatively with infinite cutoff using modified Weinberg counting. Renormalize LO non-perturbatively with infinite cutoff using modified Weinberg counting. Use the distorted LO wave to calculate higher orders in perturbation theory. Use the distorted LO wave to calculate higher orders in perturbation theory. At NLO, 3 counterterms for 1S0 and 6 for 3S1: a power-counting scheme that allows for systematic improvements order by order emerges. At NLO, 3 counterterms for 1S0 and 6 for 3S1: a power-counting scheme that allows for systematic improvements order by order emerges. Results for NN scattering o.k., so, in principal, the scheme works. Results for NN scattering o.k., so, in principal, the scheme works. But how practical is this scheme in nuclear structure? But how practical is this scheme in nuclear structure? LO interaction has huge tensor force, huge wound integral; bad convergence of the many-body problem. Impractical! LO interaction has huge tensor force, huge wound integral; bad convergence of the many-body problem. Impractical!

39. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 39 R. Machleidt 39 Option 2: Perturbative, using DWBA (Valderrama ‘09) Renormalize LO non-perturbatively with infinite cutoff using modified Weinberg counting. Renormalize LO non-perturbatively with infinite cutoff using modified Weinberg counting. Use the distorted LO wave to calculate higher orders in perturbation theory. Use the distorted LO wave to calculate higher orders in perturbation theory. At NLO, 3 counterterms for 1S0 and 6 for 3S1: a power-counting scheme that allows for systematic improvements order by order emerges. At NLO, 3 counterterms for 1S0 and 6 for 3S1: a power-counting scheme that allows for systematic improvements order by order emerges. Results for NN scattering o.k., so, in principal, the scheme works. Results for NN scattering o.k., so, in principal, the scheme works. But how practical is this scheme in nuclear structure? But how practical is this scheme in nuclear structure? LO interaction has huge tensor force, huge wound integral; bad convergence of the many-body problem. Impractical! LO interaction has huge tensor force, huge wound integral; bad convergence of the many-body problem. Impractical! For considerations of the NN amplitude o.k. But impractical for nuclear structure applications.

40. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 40 What now?

41. Option 3: 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 convert the garbage into the good stuff (Epelbaum & Gegelia ‘09). The garbage may even converge, but that doesn’t convert the garbage into the good stuff (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 The Nuclear Force Problem Bad Honnef, 14 February 2011 41

42. Option 3: Nonperturbative using finite cutoffs ≤ Λχ ≈ 1 GeV. Goal: Find “cutoff indepence” for a certain finite range below 1 GeV. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 201142 Very recently, a systematic investigation of this kind has been conducted by us at NLO 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).

43. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 201143 Cutoff dependence of NN Phase shifts at NLO Where is the range of cutoff independence??? 400 1000 400

44. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 201144 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.

45. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 201145 Χ 2 /datum for the neutron-proton data as function of cutoff in energy intervals as denoted There is a range of cutoff independence!

46. Conclusions R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 46 Chiral effective field theory is a useful tool to deal with the nuclear force problem. Chiral effective field theory is a useful tool to deal with the nuclear force problem. 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 issues: But there are still issues: Subleading 3NFs: additional and stronger 3NFs are needed (see next talk by H. Krebs). Subleading 3NFs: additional and stronger 3NFs are needed (see next talk by H. Krebs). Renormalization: more subtle, more controversial. Renormalization: more subtle, more controversial.

47. 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: may be o.k. for the NN amplitude (cf. work of Valderrama); but impractical in nuclear structure applications, tensor force (wound integral) too large. Perturbative beyond LO: may be o.k. for the NN amplitude (cf. work of 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!). I have demonstrated this at NLO (NNLO and N3LO to come, stay tuned). Identify “Cutoff Independence” within a range ≤ Λχ ≈1 GeV. Most realistic approach (Lepage!). I have demonstrated this at NLO (NNLO and N3LO to come, stay tuned). R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 47 Our views on reno

48. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 48 Have we finally finally reached the end of the tunnel? Not quite, But certainly we see the light at the end of the tunnel! And so,