Aharonov-Bohm effect and nucleon-nucleon phase shifts on the lattice

TL;DR

The paper addresses the challenge of determining nucleon-nucleon interactions from QCD in a non-perturbative way, circumventing the Euclidean-to-Minkowski analytic continuation problem by using finite-volume energy shifts. It introduces a background gauge field with zero field strength (Aharonov-Bohm configuration) that induces controllable energy shifts in the two-nucleon sector, equivalently realized as twisted boundary conditions, allowing extraction of phase shifts at arbitrary small momenta from ground-state energies on relatively small lattices. The analysis combines lattice QCD with an effective field theory of nucleons and derives a generalized Luscher relation linking the finite-volume spectrum to the scattering phase shifts, applicable up to momentum scales below the pion mass and providing practical guidance on lattice sizes and twist values. The approach reduces computational demands (lattice sizes around 5–7 fm) while enabling phase-shift determinations across a range of momenta, and it suggests extensions to other hadronic interactions such as pion-pion scattering under similar background-field twists.

Abstract

We propose a method for the lattice QCD computation of nucleon-nucleon low-energy interactions. It consists in simulating QCD in the background of a ''electromagnetic" field whose potential is non-vanishing, but whose field strength is zero. By tuning the background field, phase-shifts at any (but small) momenta can be determined by measuring the shift of the ground state energy. Lattice sizes as small as 5 Fermi can be sufficient for the calculation of phase shifts up to momenta of order of $m_π/2$.

Figures (4)

• Figure 1: The lattice with periodic boundary conditions (and two dimensions suppressed) is represented by the surface of the outer torus. The fictitious solenoid (inner ring) generates a magnetic vector potential $\vec{A}$ along direction $z$ (wrapped around the torus). The magnetic field is confined inside the ring and vanishes at the surface of the torus, where the lattice is.
• Figure 2: Sum of graphs determining the tw0-nucleon scattering amplitude in the effective theory. The vertices include interactions with an arbitrary number of derivatives.
• Figure 3: $S(\frac{MEL^2}{4\pi^2},\phi)$ as a function of $ME$ for three values of $\phi=0,\pi/2$ and $\pi$.
• Figure 4: Ground state energy (in MeV) of two nucleons in the spin singlet (left) and triplet (right) channel as a function of the background field. The three curves correspond, from bottom to top, $L=5, 7$ and $10$$fm$.