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R. Machleidt, University of Idaho D. R. Entem , University of Salamanca

Sub-Leading Three-Nucleon Forces in Chiral Perturbation Theory. R. Machleidt, University of Idaho D. R. Entem , University of Salamanca. Outline. Nuclear forces from chiral EFT: Overview & achievements Are we done? No! Sub-leading three-nucleon forces Outlook.

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R. Machleidt, University of Idaho D. R. Entem , University of Salamanca

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

  2. Outline • Nuclear forces from chiral EFT: Overview & achievements • Are we done? No! • Sub-leading three-nucleon forces • Outlook 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. • Quarks and gluons are confined into colorless hadrons. • Nuclear forces are residual forces (similar to van der Waals forces) • Separation of scales Sub-Leading 3NFs ECT* Trento 07/11/2011

  4. Calls for an EFT soft scale: Q ≈ mπ ,hard scale: Λχ ≈ mρ ; pions and nucleon relevant d.o.f. • Low-energy expansion: (Q/Λχ)ν with ν bounded from below. • Most general Lagrangian consistent with all symmetries of low-energy QCD. • π-π and π-N perturbatively • NN has bound states: (i) NN potential perturbatively (ii) apply nonpert. in LS equation. (Weinberg) Sub-Leading 3NFs ECT* Trento 07/11/2011

  5. 3N forces 4N forces 2N forces Leading Order The Hierarchy of Nuclear Forces Next-to Leading Order Next-to-Next-to Leading Order Next-to-Next-to-Next-to Leading Order Sub-Leading 3NFs ECT* Trento 07/11/2011

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

  9. Chiral NN potential at N3LO underbinds by ~1MeV/nucleon. (Size extensivity at its best.) Sub-Leading 3NFs ECT* Trento 07/11/2011

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

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

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

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

  14. Calculating the properties of light nuclei usingchiral 2N and 3N forces “No-Core Shell Model “ Calculations by P. Navratil et al., LLNL 2N (N3LO) force only Sub-Leading 3NFs ECT* Trento 07/11/2011

  15. Calculating the properties of light nuclei usingchiral 2N and 3N forces “No-Core Shell Model “ Calculations by P. Navratil et al., LLNL 2N (N3LO) +3N (N2LO) forces 2N (N3LO) force only R. Machleidt Sub-Leading 3NFs ECT* Trento 07/11/2011 15

  16. p-d Analyzing Power Ay 2NF+3NF Calculations by the Pisa Group 2NF only The Ay puzzle is NOT solved by the 3NF at NNLO. p-3He Sub-Leading 3NFs ECT* Trento 07/11/2011

  17. Why do we need 3NFs beyond NNLO? • The 2NF is N3LO; • consistency requires that all contributions are at the same order. • 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. Sub-Leading 3NFs ECT* Trento 07/11/2011

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

  20. The 3NF at N3LO explicitly One-loop, leading vertices Ishikawa & Robilotta, PRC 76, 014006 (2007) Bernard, Epelbaum, Krebs, Meissner, PRC 77, 064004 (2008) In progress Sub-Leading 3NFs ECT* Trento 07/11/2011

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

  22. Proton-deuteron elastic scattering Black line: 2NF only Red dashed line: 2NF + 3NF(N2LO+2πN3LO) ≈ 2NF + 3NF(N2LO) The 2πN3LO 3NF is negligible. From: Ishikawa & Robilotta, PRC 76, 014006 (2007) R. Machleidt Sub-Leading 3NFs ECT* Trento 07/11/2011 22

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

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

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

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

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

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

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

  30. The chiral 3NF ✗ Leading Order (Q0) Basic Diagrams (1/M)n corrections Next-to Leading Order (Q2) The Hierarchy of Nuclear Forces The Hierarchy of Nuclear Forces Next-to-Next-to Leading Order (Q3) ✗ ✗ ✗ ✗ Next-to-Next-to-Next-to Leading Order (Q4) N4LO (Q5) R. Machleidt R. Machleidt R. Machleidt R. Machleidt R. Machleidt Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 The unfinished business Vietri, Italy, 22 May 2010 Sub-Leading 3NFs ECT* Trento 07/11/2011 30 30 30 30 30

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  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 e_i Pisa Sub-Leading 3NFs ECT* Trento 07/11/2011

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

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

  48. Conclusions R. Machleidt • 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. • 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. Sub-Leading 3NFs ECT* Trento 07/11/2011 48

  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? • Is there any combination of isospin, spin, and momentum operators of 3 nucleons that particularly improves Ay? Sub-Leading 3NFs ECT* Trento 07/11/2011

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