Efimov Physics and the Three-Body Parameter

For three identical bosons with resonant short-range interactions, the scattering length does not renormalize the three-body problem. A new datum fixes the phase of a logarithmic short-distance oscillation. This three-body parameter produces discrete rather than continuous scale invariance and controls Efimov spectra, recombination features, and any many-body prediction sensitive to three-body correlations.

Required background. Effective Range, Shallow Poles, and Universality Windows defines the two-body zero-range window.

Helpful background. Three-Body Renormalization and Universality develops the full integral-equation and renormalization-group treatment.

Hyperradial origin of the Efimov effect

At $|a|\gg R$, consider hyperradii $R\ll\rho\ll|a|$. For three identical bosons in the resonant $s$-wave channel, the lowest hyperradial equation takes the scale-invariant form

$$\left[-\frac{\mathrm d^2}{\mathrm d\rho^2} -\frac{s_0^2+1/4}{\rho^2}\right]f(\rho) =-\kappa^2 f(\rho), \qquad s_0\simeq1.00624.$$

In the intermediate region the zero-energy solutions are

$$f(\rho)\propto\sqrt\rho\, \sin\!\left[s_0\ln(\rho/\rho_*)\right].$$

Both oscillatory solutions are equally singular, so the two-body boundary condition does not select their phase. The length $\rho_*$, equivalently a binding momentum $\kappa_*$ defined by a stated spectral convention, is the required three-body parameter.

Discrete scale invariance

At unitarity the asymptotic trimer spectrum is

$$\kappa_n=\kappa_*e^{-n\pi/s_0}, \qquad E_n=-\frac{\kappa_n^2}{m},$$

up to the convention used to label $n$. Consecutive length scales differ by

$$\lambda_0=e^{\pi/s_0}\simeq22.7,$$

and binding energies by $\lambda_0^2\simeq515$. Scaling $\rho$ by $\lambda_0$ returns the same short-distance phase. This limit-cycle behavior and its effective-field-theory renormalization are derived in Bedaque, Hammer, and van Kolck 1999.

At finite $a$, Efimov states meet atom–dimer or three-atom thresholds at log-periodically related values of $a$. Their exact ratios receive range corrections and depend on how $\kappa_*$ is defined.

Inelasticity and loss

Deep dimers provide decay channels. A common zero-range parameter $\eta_*$ makes the short-distance phase complex and broadens Efimov features. The pair $(\kappa_*,\eta_*)$, not $a$ alone, is then the minimal leading three-body input. Three-body recombination coefficients have the dimensional form

$$L_3(a)=\frac{|a|^4}{m} \,F_\pm\!\left(s_0\ln(|a|\kappa_*),\eta_*\right)$$

within the zero-range window. The log-periodic dimensionless functions $F_+$ and $F_-$ describe the $a>0$ and $a<0$ branches, respectively; their threshold structure need not be the same.

Which systems exhibit the effect

The numerical $s_0$ and factor $22.7$ apply to three identical bosons with resonant pair interactions. Fermionic statistics can forbid the relevant $s$ wave; mass-imbalanced mixtures have threshold mass ratios and different scaling factors. Spin, dimensionality, confinement, and which pairwise channels are resonant must be specified before transferring the formula.

In a many-body gas, density introduces $k_n\sim n^{1/3}$. Dependence on $k_n/\kappa_*$ is allowed even at two-body unitarity. A statement that a resonant Bose gas depends only on density therefore omits a leading symmetry-allowed input unless a regime or transient observable suppresses three-body sensitivity.

Common pitfalls

Using $a$ as the only bosonic input. Three-body renormalization requires $\kappa_*$, and loss requires inelasticity information.

Applying the factor $22.7$ to every three-body system. It is specific to the identical-boson value of $s_0$.

Counting cutoff stability without a three-body counterterm. Apparent stability over a narrow range can conceal log-periodic cutoff dependence.

Exercises

Derive the energy ratio

Use the Efimov momentum spectrum to find $E_{n+1}/E_n$.

Solution

$\kappa_{n+1}/\kappa_n=e^{-\pi/s_0}$ and $E\propto-\kappa^2$, so $E_{n+1}/E_n=e^{-2\pi/s_0}\simeq1/515$. The shallower state has the smaller magnitude.

Show why one datum is needed

What transformation of $\rho_*$ leaves the zero-energy hyperradial wave function unchanged up to an overall sign?

Solution

Replacing $\rho_*$ by $e^{\pi/s_0}\rho_*$ shifts the sine argument by $-\pi$, changing only its sign. Thus the boundary condition is periodic in $\ln\rho_*$ and one three-body datum fixes its phase modulo the discrete scaling factor.

Continue

Universal Relations and Tan Contact separates two-body contact information from three-body effects. Few-Body Data in the Virial Expansion shows where trimers enter thermodynamics. Resonant Bose Matter and Metastable Branches compares the Efimov and loss scales with the observation time.

References

• Bedaque, Paulo F., H.-W. Hammer, and U. van Kolck. “Renormalization of the Three-Body System with Short-Range Interactions.” Physical Review Letters 82 (1999): 463–467. DOI.

Further reading

• Braaten, Eric, and Hans-Werner Hammer. “Universality in Few-Body Systems with Large Scattering Length.” Physics Reports 428 (2006): 259–390. DOI.
• Efimov, Vitaly. “Energy Levels Arising from Resonant Two-Body Forces in a Three-Body System.” Physics Letters B 33 (1970): 563–564. DOI.
• Naidon, Pascal, and Shimpei Endo. “Efimov Physics: A Review.” Reports on Progress in Physics 80 (2017): 056001. DOI.