Ill-Posedness in Limited Discrete Fourier Inversion and Regularization for Quasi Distributions in LaMET

TL;DR

This paper analyzes the ill-posed nature of limited inverse discrete Fourier transforms in LaMET, demonstrating how Tikhonov regularization can stabilize solutions and enable uncertainty quantification in lattice QCD quasi distributions.

Contribution

It provides a systematic study of the inverse Fourier problem in LaMET, showing how Tikhonov regularization addresses instability and offers a rigorous framework for uncertainty quantification.

Findings

Inverse Fourier transform in LaMET is moderately ill-posed with exponential sensitivity.

Tikhonov regularization effectively stabilizes solutions in toy models and real lattice QCD data.

Regularized solutions align with physics-driven extrapolation methods.

Abstract

We systematically investigated the limited inverse discrete Fourier transform of the quasi distributions from the perspective of inverse problem theory. This transformation satisfies two of Hadamard's well-posedness criteria, existence and uniqueness of solutions, but critically violates the stability requirement, exhibiting exponential sensitivity to input perturbations. To address this instability, we implemented Tikhonov regularization with L-curve optimized parameters, demonstrating its validity for controlled toy model studies and real lattice QCD results of quasi distribution amplitudes. The reconstructed solutions is consistent with the physics-driven $\lambda$-extrapolation method. Our analysis demonstrates that the inverse Fourier problem within the large-momentum effective theory (LaMET) framework belongs to a class of moderately tractable ill-posed problems, characterized by distinct spectral properties that differ from those of more severely unstable inverse problems encountered in other lattice QCD applications. Tikhonov regularization establishes a rigorous mathematical framework for addressing the underlying instability, enabling first-principles uncertainty quantification without relying on ansatz-based assumptions.