Resolving the structure of bound states using lattice quantum field theories

Abstract

This work presents the first lattice calculation of a two-to-two particle matrix element of a local current. This exploratory calculation is performed using a leading-order pionless effective field theory of two nucleons in a finite 3D spatial volume, where the Hamiltonian can be diagonalized exactly for moderate volumes. By considering a range of couplings where the theory supports a deuteron-like bound state, we determine the finite-volume spectra and matrix elements of the conserved local vector current. Using the Lüscher formalism, we constrain the infinite-volume, purely hadronic amplitude for this theory. Using previously derived formalism, we then map the finite-volume matrix elements to scattering amplitudes describing a reaction coupling two-particle states via a current insertion, $\2+\Jc \to \2$. We then use a recently derived relation between this class of amplitudes and the bound-state elastic form factor to directly constrain the infinite-volume form factor. By varying over a range of values of the coupling of the theory, we explore the effects of this analysis for deep-bound states and shallow-bound states. We reproduce the expected result that for deep bound states, the finite-volume formalism is largely unnecessary, while for shallow bound states, it is absolutely critical to obtain a sensible result. We present a detailed outline of the analysis of this class of matrix elements, including the determination of the charge radius of the bound state. In the shallow bound state limit, we find good agreement with the prediction stemming from the anomalous threshold.

Figures (10)

• Figure 1: Shown is a qualitative depiction of the $\mathbf{2}+\mathcal{J}^\mu\to\mathbf{2}$ amplitude, $\mathcal{W}^\mu$, and its relation to the bound state form factor, $f_B$. Note $Q^2=-(P_f-P_i)^2$.
• Figure 2: Shown is the finite-volume spectra for the $A_1^{(+)}$ cubic irreps for a range of values of $c$ and fixed values of $L = 10$ and $M = 50$. The errors are introduced as discussed in the main body of text. The solid lines show the non-interacting spectrum, and the dashed line shows the two-particle threshold.
• Figure 3: Shown is the procedure for constraining the scattering amplitude from the finite-volume spectrum using the Lüscher quantization condition in Eq.\ref{['eq:QC']} for $c=0.5$. Using the finite-volume energy levels in Fig. \ref{['fig:spectra1']}, the ERE parametrization in Eq. \ref{['eq:ERE']} is numerically constrained. In (a), we show the result of inverting this procedure, allowing us to predict the finite-volume spectrum as a function of the lattice size in any frame, where we present the case of the rest frame. The red line is the predicted energy levels with error, while the points are the obtained finite-volume spectrum, and the dashed grey lines are the non-interacting energy levels. In (b), we show the resulting $1 \sigma$ band for $q^\star \cot \delta / M$ alongside the finite-volume spectrum and the bound state condition, $-\sqrt{-q^{\star 2}}$.
• Figure 4: In the top panel, we show the resulting parametrized $q^\star \cot \delta / M$ for the different values of the coupling $c$ considered in Fig. \ref{['fig:spectra1']} as a function of the COM momentum compared to the bound state condition, as given in Eq. \ref{['eq:BS_pole']}. In the bottom panel, we show the expected finite-volume corrections on the matrix elements as a function of the coupling of the theory.
• Figure 5: Shown are the values of $2E_n\,\widetilde{\mathcal{R}}= 2E_n\,\mathcal{M}^2\, \mathcal{R}$ for the ground state of all volumes and values of the coupling. The errors are due to the propagation of errors of the fitted effective range parameters. The straight lines are the infinite-volume prediction of $2E_n\,\widetilde{\mathcal{R}}$, which is equal to $g^2$ .