1. 7/11/20161 Harmonic Effective Field Theory U. van Kolck University of Arizona with B. Barrett (Arizona) J. Rotureau (Arizona) I. Stetcu (Washington) Supported in part by US DOE Background by S. Hossenfelder

2. 7/11/20162 Outline  Why  (Pionless) Effective Field Theory  Life in the Harmonic Box  Trapped Fermions  Trapped Nucleons Fighting for Freedom  Conclusion & Outlook

3. 7/11/20163 Game Plan QCD Pionful EFT lattice NCSM, … ~1 GeV ~100 MeV ~30 MeV Pionless EFT Cluster EFT (Chi PT) Faddeev* eqs ?

4. 7/11/20164 Facts of Life  there is always* an underlying theory * except, maybe, at the Planck scale all interactions among low-energy d.o.f.s allowed by symmetries  there is always* a “model space” renormalization-group invariance to tame arbitrary UV cutoff “power counting” non-analytic, from loops normalization parameters truncate …want… there are always* these errors

5. 7/11/20165 Harmonic-Oscillator Box As A grows, given computational power limits number of accessible one-nucleon states IR cutoff “No-Core Shell Model” Lattice Box nuclear matter Stetcu et al ’06 … Mueller et al ’99 finite nuclei few nucleons Lee et al ’05 … few atoms Stetcu et al ’07 … in addition to UV cutoff

6. 7/11/20166 HO Basis : internal (Jacobi) coordinates reduced dimensions, but difficult antisymmetrization : Slater-determinant Navratil, Kamuntavicius + Barrett ‘00 Ormand ‘05 single particle code `a la code: REDSTICK generalized Laguerre polynomial maximum number of excitations

7. 7/11/20167 QCD: gauge theory of quarks deuteron QED: gauge theory of electrons and nuclei He4 dimer cf. Any EFT will do; for definiteness, pionless.

8. 7/11/20168 Fukugita et al. ‘95 quenched lattice (incomplete) NLO unitarity limit (incomplete) NLO cf. atoms as magnetic field varies Scale emerges QCD near a Feshbach resonance in pion mass Feshbach resonance unitarity limit MIT webpage Beane, Bedaque, Savage + v.K. ’02 Beane + Savage ’03 Epelbaum et al. ’03 … cf. Beane et al. ‘06

9. 7/11/20169 contact EFT degrees of freedom: nucleons symmetries: Lorentz, P, T expansion in: multipole non-relativistic omitting spin, isospin Universality: first orders apply also to atoms where two-body sector ~ effective-range expansion v.K. ’97 ’99 Kaplan, Savage + Wise ’98 Gegelia ’98 …

10. 7/11/201610 v.K. ’97 Kaplan, Savage + Wise ’98 Gegelia ‘98 Bedaque, Hammer + v.K. ’99 Hammer + Mehen ’00 … Platter, Hammer + Meissner ’04 NNLO not LO LO NLOLO NNLO

11. 7/11/201611 Untrapped nucleons pairs triplets EFT PC effectively justifies (modified) cluster approximation LO pairs parameters fitted to d, t, a ground-state energies works within ~30% predicted 4He excited, 6Li ground energies Stetcu, Barrett +v.K., ‘07 but parameters proliferate: e.g., at NLO two more 2-body parameters can we fit them to scattering data? Yes, trap them!

12. 7/11/201612 Trapped fermions I. Bloch optical trapping standing waves lasers low-tunneling regime test our method (band insulator) universal behavior only low-energy scale given by b untrapped limit significant trap effects some semi-analytical results known

13. 7/11/201613 Life in the Box LO NLO NNLO etc. two-body reduced mass Rotureau, Stetcu, Barrett + v.K., ‘09 S waves only in LO

14. 7/11/201614 input onedetermine e.g. lowest level LO NLO NNLO calculate other levels input second leveldetermine input third level Stetcu, Barrett, Vary + v.K. ’08 Rotureau, Stetcu, Barrett + v.K. ‘10 etc. calculate other levels

15. 7/11/201615 Busch et al. ’98 Blume + Greene ’02 Block + Holthaus ’02 Bolda, Tiesinga + Julienne ’02... LO NLO, NNLO Where do levels come from? untrapped bound state scattering states

16. 7/11/201616 Harmonic-Oscillator BoxLattice Box Parameters fitted to E from Busch et al. ‘98 Luescher ‘91 cf. Fukuda + Newton ‘54 Lattice EFTHarmonic EFT

17. 7/11/201617 RG runningLO

18. 7/11/201618 NLO RG running Etc.

19. 7/11/201619 Stetcu, Barrett, Vary + v.K. ’08 Rotureau, Stetcu, Barrett + v.K. ‘10 NNLO Hamiltonian fully diagonalized: worse than NLO! at unitarity cf. Luu et al. ‘10

20. 7/11/201620 lowest states: free-space bound states other states: scattering states phase-shift info, for example: S-wave phase shift for particle/lighter b.s. scattering binding energy info include few-body forces Rotureau, Stetcu, Barrett, Birse + v.K. ‘10 1) 2)

21. 7/11/201621 pairs Trapped two-component fermions: no two-body force three-body force in LOs up to NNLO + HO is physics S wave only in LOs fit to data Stoeferle et al (ETH) ‘06 atoms e.g. (independent of w since b only scale) no fit

22. 7/11/201622 at unitarity Werner+ Castin ‘06 Stetcu, Barrett, Vary + v.K., ’07 Rotureau, Stetcu, Barrett, Birse + v.K. ‘10

23. 7/11/201623 LO NNLO Errors at unitarity

24. 7/11/201624 at unitarity Chang + Bertsch ‘07 von Stecher et al. ‘07 Alhassid, Bertsch + Fang ‘08 cf. Stetcu, Barrett, Vary + v.K., ’07 Rotureau, Stetcu, Barrett, Birse + v.K. ‘10

25. 7/11/201625 Stetcu, Barrett, Vary + v.K., ’07 Kerstner + Duan ’07 Rotureau, Stetcu, Barrett, Birse + v.K. ‘10 inversion of g.s. parity! NNLO

26. 7/11/201626 3-body energy above dimer g.s. use two levels, eliminate : better precision at smaller cutoffs better dimer inside trap cf. von Stecher, Blume + Greene ‘08Rotureau, Stetcu, Barrett, Birse + v.K. ‘10

27. 7/11/201627 Trapped nucleons up to NLO S wave only add two-body force three-body force in LOs + HO is not physics triplets one parameter Rotureau, Stetcu, Barrett + v.K. in preparation

28. 7/11/201628 NLO

29. 7/11/201629 errors due to finite UV increase expt Dilg et al. ‘71 NLO

30. 7/11/201630 collapse Thomas ‘35 LO withOUT 3BF fit 3BF to triton BE Toelle, Hammer + Metsch ‘10 similar for bosons

31. 7/11/201631 Conclusion & Outlook EFT can be solved in HO basis with scattering input Pionless EFT in NCSM similar to trapped atoms near a Feshbach resonance Convergence improves with increasing order Few-body binding energies and scattering parameters can be calculated More extensive calculations with nucleons needed