Low-energy universality and scaling of van der Waals forces

Abstract

At long distances interactions between neutral ground state atoms can be described by the Van der Waals potential V (r) = − P ∞ n=6 Cn/r n . In the ultra-cold regime atom-atom scattering is dominated by s-waves phase shifts given by an effective range expansion p cot δ0(p) = −1/α0 + r0p 2 /2 + . . . in terms of the scattering length α0 and the effective range r0. We show that while for these potentials the scattering length cannot be predicted, the effective range is given by the universal low energy theorem r0 = A + B/α0 + C/α 2 0 where A,B and C depend on the dispersion coefficients Cn and the reduced di-atom mass. We confront this formula to about a hundred determinations of r0 and α0 and show why the result is dominated by the leading dispersion coefficient C6. Universality and scaling extends much beyond naive dimensional analysis estimates. Van der Waals (VdW) forces appear ubiquitously in many contexts of atomic, molecular, nuclear and particle physics. They account for long range dipole fluctuations between charge neutral atomic and molecular systems [1] with implications on the production of Bose-Einstein condensates of ultra-cold atoms and molecules [2]. The intermediate range nucleon-nucleon interaction due to two pion exchange also exhibits this VdW behaviour based on chiral symmetry [3] providing a justification for the liquid drop model of nuclei [4]. The short distance gluon exchange interaction between (colour neutral) hadrons also display this kind of interaction [5,6]. Van der Waals forces, however, diverge when naively extrapolated to short distance scales [7,8]. The study of such problems in a variety of situations will certainly shed light on the usefulness of renormalization ideas within the specific context of quantum mechanics (see e.g. Ref.[9]).Fundamental work for neutral atoms was initiated in Refs. [10,11,12] (see also [13]),within a quantum-defect theoretical viewpoint. In this letter we systematically show that these simplified approaches work and analyze why they succeed. VdW forces are extremely simple in this case and are described by the potentialwhere C n are the VdW coefficients which are computed ab initio from intensive electronic orbital atomic structure calculations (see e.g. Ref.[14] for a compilation). Usually, only the terms with n = 6, 8, 10 are retained although the series is expected to diverge asymptotically, C n ∼ n! [15]. The impressive calculation in Hydrogen up to C 32 [16] exhibits the behaviour C n ∼ (1/2) n n! at relatively low n-values. The potential (1) holds for distances much larger than the ionization length l I = / √ 2m e I (I is the ionization potential) which usually is a few a.u. In the Born-Oppenheimer approximation the quantum mechanical problem consists of solving the Schrödinger equation for the two atoms apart a distance r,where U (r) = 2µV (r)/ 2 is the reduced potential, µ = m 1 m 2 /(m 1 + m 2 ) the reduced di-atom mass, k = p/ = 2π/λ the wavenumber, and u k (r) the reduced wave function. For our purposes, it is convenient to write the reduce...

Renormalization versus strong form factors for one-boson-exchange potentials

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“…We use the same numbers and notation as in Ref. [7]. Here AVMD means taking m a 1 = √ 2m ρ ≃ 1107MeV and PDG taking m a 1 = 1230MeV.…”

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