The University of Tokyo Microscopic origin and universality classes of the three-body parameter The University of Tokyo Pascal Naidon Shimpei Endo Masahito Ueda

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

The Efimov 3-body parameter Zero-range condition with 𝑎→∞ − 1 𝑅 2 Efimov attraction R x 𝑅 2 = 2 3 ( 𝑥 2 + 𝑦 2 + 𝑧 2 ) Hyperradius The Efimov effect (1970) y z Parameters describing particles at low energy Scattering length a (2-body parameter) 1/𝑎 − - 𝜅 2 𝐸 1/𝑎 𝑎 Three-body parameter trimer Λ −1 3-body parameter dimer

The Efimov 3-body parameter Zero-range condition with 𝑎→∞ − 1 𝑅 2 Efimov attraction R x 𝑅 2 = 2 3 ( 𝑥 2 + 𝑦 2 + 𝑧 2 ) Hyperradius The Efimov effect (1970) y z Parameters describing particles at low energy Scattering length a (2-body parameter) 1/𝑎 − 𝐸 1/𝑎 - 𝜅 2 22.72 𝑎 Three-body parameter trimer dimer Λ −1 3-body parameter

Universality for atoms triplet scatt. length 𝑎 [ 𝑎 0 ] 𝑟 vdW [ 𝑎 0 ] 4He 7Li 6Li 39K 23Na 87Rb 85Rb 133Cs Microscopic determination? no universality of the scattering length r short-range details − 1 𝑟 6 van der Waals Two-body potential Effective three-body potential Hyperradius R short-range details − 1 𝑅 2 Efimov 𝑎 − [ 𝑎 0 ] 𝑎 − ≈−10 𝑟 𝑣𝑑𝑊 𝑟 vdW [ 𝑎 0 ] universality of the 3-body parameter

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 𝑎 − ≈−11 𝑟 𝑣𝑑𝑊 − 1 𝑅 2 Efimov Hyperradius R Three-body repulsion at 𝑅≈2 𝑟 𝑣𝑑𝑊

Interpretation: two-body correlation 𝜓 𝑘 =sin (𝑘𝑟 − 𝛿 𝑘 ) Asymptotic behaviour 𝜓 𝑘 𝑉(𝑟) Strong depletion Interatomic separation r Resonance 𝒂→∞ 𝜓 𝑘 𝑉(𝑟) 𝑟 0 ≈ 𝑟 𝑣𝑑𝑊 ∼ 1 2 𝑟 𝑒 = 0 ∞ 𝜓 0 𝑟 2 − 𝜓 0 𝑟 2 𝑑𝑟

Interpretation: two-body correlation Kinetic energy cost due to deformation Excluded configurations induced deformation squeezed equilateral Excluded configurations deformation 〈𝛼〉 Efimov elongated 𝑟 0

Confirmation 1: pair correlation model FModel = FEfimov x j(r12) j(r23) j(r31) (hyperangular wave function) (product of pair correlations) 3-body potential 𝑈 𝑅 = 𝜆 𝑅 2 + 𝑑 cos 𝜃 𝑑𝛼 𝜕Φ 𝜕𝑅 2 Hyperradius R [ 𝑟 0 ] Energy E [ 𝑟 0 −2 ] Pair model (for Lennard-Jones two-body interactions) Exact Efimov attraction

Confirmation 2: separable model Reproduces the low-energy 2-body physics Scattering length Effective range Last bound state …. 𝑉=𝜉 𝜒 〈𝜒| Parameterised to reproduce exactly the two-body correlation at zero energy. 𝜒 𝑞 =1−𝑞 0 ∞ 𝜓 0 𝑟 − 𝜓 0 (𝑟) sin 𝑞𝑟 𝑑𝑟 1/𝑎 − 𝜉=4𝜋 1 𝑎 − 2 𝜋 0 ∞ 𝜒 𝑞 2 𝑑𝑞 −1 - 𝜅 2

Confirmation 2: separable model 𝑎 − 𝑟 𝑣𝑑𝑊 𝑎 − =−10.86(1) 𝑟 𝑣𝑑𝑊 n 𝑉=𝜉 𝜒 〈𝜒| Parameterised to reproduce exactly the two-body correlation at zero energy. Hyperradius R Energy Exact Pair model Separable model Hyperradius R Integrated probability

Confirmation 2: separable model 𝑉=𝜉 𝜒 〈𝜒| Parameterised to reproduce exactly the two-body correlation at zero energy. Other potentials Potential 𝒂 − 𝜿 Yukawa -5.73 0.414 Exponential -10.7 0.216 Gaussian -4.27 0.486 Morse ( 𝑟 0 =1) -12.3 0.180 Morse ( 𝑟 0 =2) -16.4 0.131 Pöschl-Teller (𝛼=1) -6.02 0.367 at most 10% deviation -6.55 -11.0 -4.47 -12.6 -16.3 -6.23 0.204 -0.366 0.472 0.173 0.128 0.350 Separable model Exact calculations S. Moszkowski, S. Fleck, A. Krikeb, L. Theuÿl, J.-M.Richard, and K. Varga, Phys. Rev. A 62 , 032504 (2000).

Summary universal two-body correlation ∼ 𝑟 𝑒 effective range three-body deformation three-body repulsion three-body parameter ∼ 𝑟 𝑒 effective range

Two-body correlation universality classes ∝− 1 𝑟 𝑛 (𝑛>3) Power-law tails Faster than Power-law tails Step function correlation limit Universal correlation 𝜓 0 𝑟 =Γ 𝑛−1 𝑛−2 𝑟/ 𝑟 𝑛 1/2 𝐽 1/(𝑛−2) (2 𝑟/ 𝑟 𝑛 −(𝑛−2)/2 ) Probability density Van der Waals 0.0 1.0 2.0 3.0 4.0 Interparticle distance ( 𝑟 𝑣𝑑𝑊 )

Separable model 3-body parameter in units of the two-body effective range (= size of two-body correlation) Nuclear physics Atomic physics 𝜅=−0.364 1 𝑟 𝑒 2 −1 𝜅=−0.261 1 𝑟 𝑒 2 −1 Binding wave number at unitarity 𝜅=−0.2190 1 𝑟 𝑒 2 −1 ? Number of two-body bound states

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 classes of interaction P. Naidon, S. Endo, M. Ueda, PRA 90, 022106 (2014) P. Naidon, S. Endo, M. Ueda, PRL 112, 105301 (2014)

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Separable model 𝑉=𝜉 𝜒 〈𝜒| Parameterised to reproduce exactly the two-body correlation at zero energy. … 𝑎 − 𝑟 𝑣𝑑𝑊