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3 Efimov’s Universal Three-Body Effect
3.2 Relation Between Total Cross Section and Scattering
Length for Two Particles at Low Energies
Consider spin-less particles with no internal degrees of freedom interacting through
short-range interaction. This means there exists a range r 0 beyond which the relative
motion of the particles is almost free. Efimov effect arises when the two-particle
interaction is nearly resonant. From scattering theory, we know that at low energies
when k r
−1
0 , only the s-wave is scattered and has a nonzero phase shift δ 0 , which
is given by
tan δ 0 ≈ −ka
(3.1)
Here, k is the relative wave number between two particles and a is the scattering
length. Also, the s-wave scattering amplitude is expressed as:
f =
1
k cot δ 0 − ik
(3.2)
From Eq. (3.1), it is clear that for two-body interaction to be resonant at low
energies, the scattering length has to be much larger than the range r 0 . Also, according
to optical theorem which essentially arises from the unitarity relation:
Im( f ) =
4πσ tot
k
(3.3)
we get from Eq. (3.2),
σ tot =
4π sin
2
δ 0
k 2
→ k→0 4πa
2
(3.4)
It is thus clear that in the limit when a → ±∞, the factor sin
2
δ 0 in Eq. (3.4)
approaches its maximum value (called the unitarity limit). Near unitarity, scattering
length a (positive or negative) is a single parameter which determines low-energy
scattering (i.e., positive energies) as well as the binding energy of a weakly bound
state below the two-body breakup threshold (negative energy). This bound state exists
only for positive scattering length and its binding energy is close to
2
/(ma
2
), where
m is the mass of the particles.
3.3 Efimov Effect in Three-Boson System
For three bosons with position coordinates,
r 1 ,
r 2 ,
r 3 , let us define Jacobi coordinates
3 Efimov’s Universal Three-Body Effect
3.2 Relation Between Total Cross Section and Scattering
Length for Two Particles at Low Energies
Consider spin-less particles with no internal degrees of freedom interacting through
short-range interaction. This means there exists a range r 0 beyond which the relative
motion of the particles is almost free. Efimov effect arises when the two-particle
interaction is nearly resonant. From scattering theory, we know that at low energies
when k r
−1
0 , only the s-wave is scattered and has a nonzero phase shift δ 0 , which
is given by
tan δ 0 ≈ −ka
(3.1)
Here, k is the relative wave number between two particles and a is the scattering
length. Also, the s-wave scattering amplitude is expressed as:
f =
1
k cot δ 0 − ik
(3.2)
From Eq. (3.1), it is clear that for two-body interaction to be resonant at low
energies, the scattering length has to be much larger than the range r 0 . Also, according
to optical theorem which essentially arises from the unitarity relation:
Im( f ) =
4πσ tot
k
(3.3)
we get from Eq. (3.2),
σ tot =
4π sin
2
δ 0
k 2
→ k→0 4πa
2
(3.4)
It is thus clear that in the limit when a → ±∞, the factor sin
2
δ 0 in Eq. (3.4)
approaches its maximum value (called the unitarity limit). Near unitarity, scattering
length a (positive or negative) is a single parameter which determines low-energy
scattering (i.e., positive energies) as well as the binding energy of a weakly bound
state below the two-body breakup threshold (negative energy). This bound state exists
only for positive scattering length and its binding energy is close to
2
/(ma
2
), where
m is the mass of the particles.
3.3 Efimov Effect in Three-Boson System
For three bosons with position coordinates,
r 1 ,
r 2 ,
r 3 , let us define Jacobi coordinates