Abstract
The computation of the parton distribution functions (PDF) or distribution amplitudes (DA) of hadrons from first principles lattice QCD constitutes a central open problem in high energy nuclear physics. In this study, we present and evaluate the efficiency of several numerical methods, well established in the study of inverse problems, to reconstruct the full x-dependence of PDFs. Our starting point are the so called Ioffe time PDFs, which are accessible from Euclidean time simulations in conjunction with a matching procedure. Using realistic mock data tests, we find that the ill-posed incomplete Fourier transform underlying the reconstruction requires careful regularization, for which both the Bayesian approach as well as neural networks are efficient and flexible choices.
Highlights
Direct inversionIn order to invert the relations (1.2) and (1.3) direct approaches have been proposed and implemented in the literature [37]
Where p is an arbitrary hadron momentum, z is a space-like separation, τ 3 denotes a flavor Pauli matrix, γα refers to a gamma matrix acting in spin space, W (z; 0) to the 0 → z Wilson line, and ν = p · z represents a Lorentz invariant quantity known as the Ioffe time
Using realistic mock data tests, we find that the ill-posed incomplete Fourier transform underlying the reconstruction requires careful regularization, for which both the Bayesian approach as well as neural networks are efficient and flexible choices
Summary
In order to invert the relations (1.2) and (1.3) direct approaches have been proposed and implemented in the literature [37]. On the other hand the trapezoid rule has a relative error that always increases with ν for any number of interpolating points This improved integration scheme achieves its design goal of having a constant integration error for all values of ν. The proposed integration scheme can prove valuable because it can significantly reduce the number of points required to discretize the integral resulting a smaller maximum value of ν for which the problem is no longer ill-defined It has been proposed in [37], that the unphysical oscillations in the related quasi-PDF inverse problem can be controlled by considering the derivative of the integral equation with respect to ν or z3. In the main part of this paper we will explore several modern methods for treating inverse problems and compare their efficiency in dealing with the uncertainties introduced by the truncation of the integration regime
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