Optimized binning for response function reconstruction via Chebyshev expansions

TL;DR

This work tackles the challenge of reconstructing nuclear response functions from integral transforms without performing ill-posed inversions. It introduces a stochastic regularized density-of-states approach to adaptively bin spectra in equal-area bins, guided by Chebyshev moments to rebuild $R(q,\omega)$ with transparent uncertainty bounds. Demonstrated on the deuteron using a chiral NN interaction in a harmonic-oscillator basis, the method achieves excellent agreement with exact theory and experimental data for both electric dipole and longitudinal responses, and provides a general framework for ab initio lepton-nucleus cross sections in larger nuclei. The approach promises robust, uncertainty-quantified predictions for neutrino-nucleus and electron-nucleus scattering across a range of many-body systems.

Abstract

We propose an optimized histogram binning strategy to reconstruct nuclear response functions via the Chebyshev expansion bound-state method. Our approach employs a stochastic regularization of the density of states to define adaptive, equal-area bins. Using the deuteron solved in a harmonic-oscillator basis with a chiral interaction, we benchmark on dipole and longitudinal responses, obtaining excellent agreement with exact theory and experiment. This general framework readily extends to other many-body systems and opens the door to new ab initio calculations of lepton-nucleus cross sections in medium-mass nuclei.

Figures (5)

• Figure 1: Binning determined by exact diagonalization of the full Hamiltonian and the regularized DOS for the response ($N_{\textrm{max}} = 200, \hbar \Omega = 8 \ \text{MeV}$).
• Figure 2: Comparison of the reconstructed response ($\lambda = 0.025\,$MeV, $N_{\textrm{mom}}=6000$), experimental data from Ref. Arenhovel:1990yg and the theoretical result of Ref. Bampa:2011fq for different model space parameters.
• Figure 3: Calculation of the longitudinal response $R_{\rm L}(q,\omega)$ at different $q$ values and comparison with calculations from Refs. Emmons:2020aovEfros:1993xy with multipoles 0 through 4 and 6, respectively ($N_{\textrm{max}} = 200$, $\hbar \Omega = 8 \,$MeV, $\lambda = 0.025/0.05\,$MeV, $N_{\textrm{mom}}=6000$).
• Figure 4: Comparison of the reconstructed longitudinal response ($N_{\textrm{max}}=200$, $\hbar \Omega=8\,$MeV) to experimental data and a calculation, both found in Ref. deuteronlongitudinalbates.
• Figure : Estimate DOS and determine bins