Microscopic origin and universality classes of the three-body parameter

1. Microscopic origin and universality classes of the three-body parameter PascalNaidon The University of Tokyo Masahito Ueda Shimpei Endo

2. 3 particles (bosons or distinguishable) with resonant two-body interactions • single-channel two-body interaction • no three-body interaction

3. The Efimov 3-body parameter Parameters describing particles at low energy The Efimov effect (1970) y x (2-body parameter) Scattering length a z Hyperradius Zero-range condition with Three-body parameter - R dimer Efimov attraction trimer 3-body parameter

4. The Efimov 3-body parameter Parameters describing particles at low energy The Efimov effect (1970) y x (2-body parameter) Scattering length a z Hyperradius Zero-range condition with 22.72 Three-body parameter - trimer R dimer Efimov attraction 3-body parameter

5. Universality for atoms Microscopic determination? no universality of the scattering length 133Cs 4He 85Rb triplet scatt. length r 7Li 23Na 39K 87Rb short-range details van der Waals 6Li Two-body potential universality of the 3-body parameter HyperradiusR short-range details Efimov Effective three-body potential

6. Three-body with van der Waals interactions Phys. Rev. Lett. 108 263001 (2012) J. Wang, J. D’Incao, B. Esry, C. Greene Lennard-Jones potentials supporting n = 1, 2, 3, ...10 s-wave bound states Efimov HyperradiusR Three-body repulsion at

7. Interpretation: two-body correlation Two-body correlation Asymptoticbehaviour Strong depletion Interatomic separation r Resonance

8. Interpretation: two-body correlation Kinetic energy cost due to deformation Excluded configurations squeezed equilateral Excluded configurations induced deformation deformation Efimov elongated

9. Confirmation 1: pair correlation model FModel = FEfimovx j(r12) j(r23) j(r31) (product of pair correlations) (hyperangular wave function) 3-body potential Pair model (for Lennard-Jones two-body interactions) Exact Energy E [] Efimov attraction Hyperradius R []

10. Confirmation 2: separable model Parameterised to reproduce exactly the two-body correlation at zero energy. • Reproduces the low-energy 2-body physics • Scattering length • Effective range • Last bound state • …. -

11. Confirmation 2: separable model Parameterised to reproduce exactly the two-body correlation at zero energy. n Pair model Separable model Exact Energy Integrated probability HyperradiusR HyperradiusR

12. Confirmation 2: separable model Parameterised to reproduce exactly the two-body correlation at zero energy. Other potentials at most 10% deviation -0.366 -6.55 Separable model 0.204 -11.0 -4.47 0.472 Exact calculations -12.6 0.173 -16.3 0.128 -6.23 0.350 S. Moszkowski, S. Fleck, A. Krikeb, L. Theuÿl, J.-M.Richard, and K. Varga, Phys. Rev. A 62 , 032504 (2000).

13. Summary two-body correlation universal effective range three-body deformation three-body repulsion universal three-body parameter effective range

14. Two-body correlation universality classes Faster than Power-law tails Power-law tails Probability density Van der Waals 4.0 3.0 1.0 2.0 0.0 Interparticle distance () Universal correlation Step function correlation limit

15. Separable model 3-body parameter in units of the two-body effective range (= size of two-body correlation) Nuclear physics Atomic physics Binding wave number at unitarity ? Number of two-body bound states

16. Summary The 3-body parameter is (mostly) determined by the low-energy 2-body correlation. Reason: 2-body correlation induces a deformation of the 3-body system. • Consequences: the 3-body parameter • is on the order of the effective range. • has different universal values for distinct classesof interaction P. Naidon, S. Endo, M. Ueda, PRA 90, 022106 (2014) P. Naidon, S. Endo, M. Ueda, PRL 112, 105301 (2014)

17. Obrigado pela suaattenção!