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

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
Zero-range condition with 𝑎→∞
− 1 𝑅 2 Efimov attraction R x
𝑅 2 = ( 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

4. The Efimov 3-body parameter
Zero-range condition with 𝑎→∞
− 1 𝑅 2 Efimov attraction R x
𝑅 2 = ( 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

5. 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

6. Three-body with van der Waals interactions
Phys. Rev. Lett (2012)
J. Wang, J. D’Incao, B. Esry, C. Greene
Lennard-Jones potentials supporting n = 1, 2, 3, s-wave bound states
𝑎 − ≈−11 𝑟 𝑣𝑑𝑊
− 1 𝑅 2 Efimov
Hyperradius R
Three-body repulsion at 𝑅≈2 𝑟 𝑣𝑑𝑊

7. 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
∼ 𝑟 𝑒 = 0 ∞ 𝜓 𝑟 2 − 𝜓 0 𝑟 2 𝑑𝑟

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

9. 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 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

10. 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

11. 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

12. 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 , (2000).

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

14. 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 )

15. Separable model
3-body parameter in units of the two-body effective range
(= size of two-body correlation)
Nuclear physics
Atomic physics
𝜅=− 𝑟 𝑒 2 −1
𝜅=− 𝑟 𝑒 2 −1
Binding wave number at unitarity
𝜅=− 𝑟 𝑒 2 −1 ?
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 classes of interaction
P. Naidon, S. Endo, M. Ueda, PRA 90, (2014)
P. Naidon, S. Endo, M. Ueda, PRL 112, (2014)

17. Obrigado pela sua attenção!

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