Volume Dependence of N-Body Bound States

TL;DR

This paper derives formulas for how the binding energy of N-body quantum states changes in finite cubic volumes, aiding lattice calculations in nuclear, hadronic, and atomic physics.

Contribution

It provides a general finite-volume correction formula for N-body bound states applicable to various compositions and angular momenta, including methods to extract asymptotic normalization coefficients from finite-volume data.

Findings

Derived leading exponential volume dependence for two-cluster breakups.

Presented two methods to determine asymptotic normalization coefficients from finite-volume data.

Validated formulas with exactly solvable systems and numerical examples.

Abstract

We derive the finite-volume correction to the binding energy of an N-particle quantum bound state in a cubic periodic volume. Our results are applicable to bound states with arbitrary composition and total angular momentum, and in any number of spatial dimensions. The only assumptions are that the interactions have finite range. The finite-volume correction is a sum of contributions from all possible breakup channels. In the case where the separation is into two bound clusters, our result gives the leading volume dependence up to exponentially small corrections. If the separation is into three or more clusters, there is a power-law factor that is beyond the scope of this work, however our result again determines the leading exponential dependence. We also present two independent methods that use finite-volume data to determine asymptotic normalization coefficients. The coefficients are useful to determine low-energy capture reactions into weakly bound states relevant for nuclear astrophysics. Using the techniques introduced here, one can even extract the infinite-volume energy limit using data from a single-volume calculation. The derived relations are tested using several exactly solvable systems and numerical examples. We anticipate immediate applications to lattice calculations of hadronic, nuclear, and cold atomic systems.