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Nuclear forces from chiral effective field theory: Achievements and challenges R. Machleidt University of Idaho R. Machleidt 1 Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

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Outline Nuclear forces from chiral EFT: Nuclear forces from chiral EFT: Basic ideas and results Basic ideas and results Are we done? No! Are we done? No! Proper renormalization of chiral forces Proper renormalization of chiral forces Sub-leading many-body forces Sub-leading many-body forces Is THE END near? Is THE END near? R. Machleidt 2 Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

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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 Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 3

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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 Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 4

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R. Machleidt 5 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 Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ NN phase shifts up to 300 MeV Red Line: N3LO Potential by Entem & Machleidt, PRC 68, (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

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ N3LO Potential by Entem & Machleidt, PRC 68, (2003). NNLO and NLO Potentials by Epelbaum et al., Eur. Phys. J. A19, 401 (2004).

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/20118 This is, of course, all very nice; However there is a “hidden” issue here that needs our attention: Renormalization

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ “I about got this one renormalized”

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ The issue has produced lots and lots of papers; this is just a small sub-selection.

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ Now, what’s so important with this renormalization?

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ 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”).

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ 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.

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ The first kind: “NN Potential”: “NN Potential”: irreducible diagrams calculated perturbatively. irreducible diagrams calculated perturbatively. Example: Example: Counter terms perturbative renormalization (order by order)

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ R. Machleidt 15 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.

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ 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): 1616 In diagrams: +++ …

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ 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: 17 Non-perturbative renormalization

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ R. Machleidt 18 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: 18 Non-perturbative renormalization 18 With what to renormalize this time? Weinberg’s silent assumption: The same counter terms as before. (“Weinberg counting”)

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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 Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ “Modified Weinberg counting” for LO Nogga, Timmermans, v. Kolck PRC72, (2005):

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ Quantitative chiral NN potentials are at N3LO. So, we need to go substantially beyond LO.

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Nonperturbative or perturbative? Nonperturbative or perturbative? Infinite cutoff or finite cutoff? Infinite cutoff or finite cutoff? R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ Renormalization beyond leading order – Issues

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ Nonperturbative or perturbative? Nonperturbative or perturbative? Infinite cutoff or finite cutoff? Infinite cutoff or finite cutoff? Renormalization beyond leading order – Issues

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In lower partial waves ( ≅ short distances), in general, no order by order convergence; data are not reproduced. In lower partial waves ( ≅ short distances), in general, no order by order convergence; 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 Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ Option 1: Nonperturbative infinite-cutoff renormalization up to N3LO Observations and problems

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ In lower partial waves ( ≅ short distances), in general, no order by order convergence; data are not reproduced. In lower partial waves ( ≅ short distances), in general, no order by order convergence; 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 24 Option 1: Nonperturbative infinite-cutoff renormalization up to N3LO Observations and problems No good!

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ Option 2: Perturbative, using DWBA and finite cutoff (Valderrama ‘11) Renormalize LO non-perturbatively with finite cutoff using modified Weinberg counting. Renormalize LO non-perturbatively with finite 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!

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ R. Machleidt 26 Renormalize LO non-perturbatively with finite cutoff using modified Weinberg counting. Renormalize LO non-perturbatively with finite 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. Option 2: Perturbative, using DWBA and finite cutoff (Valderrama ‘11)

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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 Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ Nonperturbative or perturbative? Nonperturbative or perturbative? Infinite cutoff or finite cutoff? Infinite cutoff or finite cutoff? Renormalization beyond leading order – Issues

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Option 3: Nonperturbative using finite cutoffs ≤ Λχ ≈ 1 GeV. Goal: Find “cutoff independence” for a certain finite range below 1 GeV. R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ 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).

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ 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.

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ Χ 2 /datum for the neutron-proton data as function of cutoff in energy intervals as denoted There is a range of cutoff independence! Investigations At NNLO and N3LO Are under way.

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ Chiral three-nucleon forces (3NF)

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ N 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

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ N 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

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ N 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.

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ Calculating the properties of light nuclei using chiral 2N and 3N forces “No-Core Shell Model “ Calculations by P. Navratil et al., LLNL

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ N (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

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ R. Machleidt 39 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

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ Analyzing Power Ay p-d p- 3 He 2NF only 2NF+3NF Calculations by the Pisa Group 2NF only 2NF+3NF The Ay puzzle is NOT solved by the 3NF at NNLO.

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ And so, we need 3NFs beyond NNLO, because … 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.

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ The 3NF at NNLO; used so far.

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ Ishikawa & Robilotta, PRC 76, (2007) Bernard, Epelbaum, Krebs, Meissner, PRC 77, (2008) Bernard et al., arXiv: The 3NF at N3LO explicitly One-loop, leading vertices

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 The 3NF at NNLO; used so far. Small?!

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ ✗ Small 1-loop lead. vert. 1-loop One c i vert. Corresponding 2NF contributions 2π

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ ✗ Small 1-loop lead. vert. 1-loop One c i vert. Corresponding 2NF contributions 2π 3π Kaiser (2000) Kaiser (2001) Small Large!!

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ The 3NF at NNLO; used so far. Small?! Large?!

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R. Machleidt 48 The N4LO 3NF 2PE-1PE diagrams Many new isospin/spin/momentum structures, some similar to N3LO 2PE-1PE 3NF. Still a lot to come in the 3NF business. Stay tuned!

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ The 3NF at NNLO; used so far. Small?! Large?! And there is also the Δ-full version

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ Small?! The 3NF at NNLO; used so far. Large?! And there is also the Δ-full version

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ What’s the better approach is a matter of taste Bochum (E. Epelbaum & H. Krebs) pursues Δ-full Idaho (R. M. & D. R. Entem) Δ-less

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Conclusions R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ The chiral EFT approach has substantially advanced our understanding of nuclear forces. The chiral EFT approach has substantially advanced our understanding of nuclear forces. One major milestone of the past decade: “high precision” NN pots. at N3LO. One major milestone of the past decade: “high precision” NN pots. at N3LO. But there are still some open issues: But there are still some open issues: The renormalization of the chiral 2NF The renormalization of the chiral 2NF Sub-leading 3NFs Sub-leading 3NFs

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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: academically interesting (cf. good work by Valderrama); but impractical in nuclear structure applications, tensor force (wound integral) too large. Perturbative beyond LO: academically interesting (cf. good work by 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 Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ Concerning the renormalization issue, we take the view:

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3NF issue The 3NF at NNLO is insufficient. The 3NF at NNLO is insufficient. The 3NF at N3LO (in the Δ-less theory) is probably weak. The 3NF at N3LO (in the Δ-less theory) is probably weak. However, large 3NFs with many new structures to be expected at N3LO of Δ-full or N4LO at Δ-less. Construction is under way. However, 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. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/201154

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R. Machleidt Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/ And so, we are not yet completely done with the nuclear force problem, but …

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