Universal three-body spectrum of charged particles with resonant short-range interactions beyond the Born-Oppenheimer approximation: finite-mass-ratio corrections and the microscopic origin of the three-body parameter

Mochizuki and Nishida (2024) developed a zero-range Bethe-Peierls theory for charged particles interacting via Coulomb plus short-range potentials when the Bohr radius and the scattering length both greatly exceed the potential range, and applied it to three equally charged particles at infinite scattering length within a variational Born-Oppenheimer approximation, finding an infinite ladder of bound states that successively become resonances under increasing Coulomb repulsion, with the spectrum fixed by an input three-body parameter. Two coupled gaps between this prediction and a predictive theory remain open. First, the three-body spectrum is known only in the adiabatic limit of infinite heavy-to-light mass ratio with a specific variational ansatz, so nonadiabatic and finite-mass-ratio corrections, and hence the domain of validity of the Born-Oppenheimer spectrum, are undetermined. Second, the three-body parameter has no microscopic determination or established universality class for charged systems, in contrast to the van der Waals universality of the neutral case. No existing work determines the finite-mass-ratio spectrum or the microscopic origin of the three-body parameter for charged systems. The problem asks for the beyond-Born-Oppenheimer three-body spectrum at finite mass ratios together with the microscopic determination or universality class of the three-body parameter.

Background

Low-energy scattering of two particles through a short-range potential of range $R$ is governed, at leading order, by the $s$ -wave scattering length $\tilde a$ ; when $|\tilde a|\gg R$ the details of the potential drop out and observables become universal functions of $\tilde a$ alone (Braaten and Hammer 2006). At $\tilde a=\infty$ (unitarity) the two-body system is scale invariant and maximally strongly interacting. If the particles also carry charge, the Coulomb interaction adds a long-range $1/r$ tail whose strength defines the Bohr radius $a_0$ ; low-energy scattering is then parameterized by the Coulomb-modified effective-range expansion in $\tilde a$ , the effective range $\tilde r$ , and $a_0$ . Mochizuki and Nishida (2024) constructed the zero-range theory of charged particles in the regime $a_0,\tilde a,k^{-1}\gg R$ by generalizing the Bethe-Peierls boundary condition, the contact condition imposed on the two-body wave function as the interparticle separation $r\to 0$ (at unitarity, $\psi\propto 1/r$ ), derived rigorously as a self-adjoint extension of the Hamiltonian, so that the Coulomb logarithm enters the contact condition alongside $1/\tilde a$ . For two charged particles this yields infinite resonances for a repulsive Coulomb potential, one of which becomes a bound state with increasing inverse scattering length, and infinite bound states with no resonances for an attractive Coulomb potential. The paper's own $\alpha$ - $\alpha$ example shows the limits of the zero-range description: applied naively where $a_0$ is not much larger than $R$ , it mispredicts the measured 8Be ground-state resonance energy by a factor of about 20, reduced to a factor of 2 once the effective range $\tilde r$ is retained. For three equally charged particles at unitarity the authors employ the Born-Oppenheimer (BO) approximation, the molecular-physics separation in which two heavy particles of mass $M$ are held fixed while the light particle of mass $m\ll M$ adjusts adiabatically, together with a specific variational two-centered trial wave function for the light particle. The light-particle ground state then acts as an effective potential between the heavy particles, a function of the heavy-particle separation $d$ : repulsive, proportional to $1/d$ , at separations large compared with the Bohr radius, and attractive inverse-square, proportional to $-c^2/d^2$ with the dimensionless coefficient $c=0.567\ldots$ defined by $c=e^{-c}$ , below it. An inverse-square attraction stronger than a critical coupling produces the Efimov effect, an infinite geometric ladder of three-body bound states obeying discrete scale invariance; the phase of the ladder is fixed by a three-body parameter, here denoted $\kappa_*$ , which in any zero-range theory cannot be predicted but must be supplied as an input encoding short-distance three-body physics. Solving the heavy-particle motion under this induced attraction plus the mutual Coulomb repulsion, the authors find infinite three-body bound states that successively turn into resonances as the Coulomb repulsion increases, recovering the neutral Efimov limit as the Coulomb interaction vanishes. The result is established only in the BO limit $M\gg m$ and with the specific variational ansatz. Earlier related work brackets but does not settle the problem: Hammer and Higa (2008) studied the breaking of discrete scale invariance by a long-range interaction in a one-dimensional inverse-square-plus-Coulomb toy model; Schmickler, Hammer and Volosniev (2019) derived universal scaling laws for shallow bound states of charged particles, finding that an accurate description requires both the Coulomb-modified scattering length and the effective range unless the Coulomb interaction is very weak; and Schmickler, Hammer and Hiyama (2019) performed exact three- and four-body calculations for equal-mass charged bosons with resonant short-range attraction, finding that universal features survive only for weak Coulomb interaction and that the $\alpha$ -cluster sector is strongly non-universal. For neutral particles, by contrast, the microscopic origin of the three-body parameter is understood: it falls into universality classes fixed by the range scale, such as van der Waals universality (Naidon, Endo and Ueda 2014). No analogous result exists when the Coulomb interaction is present.

Problem Statement

For three equally charged particles in three dimensions interacting via pairwise repulsive Coulomb potentials $1/r$ and a short-range potential of range $R$ tuned to infinite two-body scattering length $\tilde a$ (unitarity), in the universal regime $a_0,\tilde a\gg R$ where $a_0$ is the Bohr radius: Mochizuki and Nishida's prediction of an infinite ladder of three-body bound states and resonances was derived in the adiabatic limit $M/m\to\infty$ (two heavy particles of mass $M$ , one light particle of mass $m$ ) using a specific variational two-centered trial wave function for the light particle, and the spectrum is fixed only up to the three-body parameter $\kappa_*$ taken as input. Two coupled questions complete the microscopic foundation of this universal three-body Coulomb spectrum. (1) Beyond the Born-Oppenheimer approximation: determine the zero-range three-body spectrum at finite mass ratio $M/m$ , and establish the mass ratios and Coulomb strengths for which the Born-Oppenheimer spectrum is quantitatively reliable, including the size and sign of the nonadiabatic shifts of the bound-state and resonance energies. (2) Microscopic origin of the three-body parameter: determine $\kappa_*$ from microscopic finite-range two-body (and, if required, three-body) interactions, or establish its universality class, in particular whether $\kappa_*$ is fixed, up to the discrete scale factor, by the Bohr radius $a_0$ , by the range $R$ , or by another combination of the physical scales, analogous to the van der Waals universality of the neutral three-body parameter. The two questions jointly convert the 2024 existence result into a predictive theory: without (1) the error bars of the predicted spectrum are unknown, and without (2) its overall scale is unfixed.

References

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