Volume Dependence of N-Body Bound States

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.

11 pages, 3 figures; Haskell code provided as ancillary file. Added discussion of single-volume extrapolations. Matches published version

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• Pion-less effective field theory for atomic nuclei and lattice nuclei
• Extrapolation of nuclear structure observables with artificial neural networks
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• Large-scale exact diagonalizations reveal low-momentum scales of nuclei
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• Singlet channel scattering in a Composite Higgs model on the lattice
• Impurity Lattice Monte Carlo for Hypernuclei
• Extrapolating Lattice QCD Results using Effective Field Theory
• Lattice QCD constraints on the parton distribution functions of ${}^3\text{He}$
• The isospin-3 three-particle $K$-matrix at NLO in ChPT
• Few-body bound states and resonances in finite volume
• The Tjon Band in Nuclear Lattice Effective Field Theory
• Volume extrapolation via eigenvector continuation
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• Quantum Monte Carlo formalism for dynamical pions and nucleons
• Three-particle formalism for multiple channels: the $\eta \pi \pi +K\overline{K}\pi$ system in isosymmetric QCD
• Charged-particle bound states in periodic boxes
• P-Wave Two-Particle Bound and Scattering States in a Finite Volume including QED
• The three-pion $K$-matrix at NLO in ChPT
• Complex scaling in finite volume
• Efficient few-body calculations in finite volume
• Spurious poles in a finite volume
• Spectrum of light nuclei in a finite volume
• A Vision for the Science of Rare Isotopes
• Exactness of the normal-ordered two-body truncation of three-nucleon forces
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• Toward scalable quantum computations of atomic nuclei
• Perturbative application of next-to-leading order pionless EFT for $A\le 3$ nuclei in a finite volume
• An Efficient Learning Method to Connect Observables
• Chiral Effective Field Theory after Thirty Years: Nuclear Lattice Simulations