Tackling inverse problems for PDFs from lattice QCD

Abstract

In this kick-off presentation for the "Recent developments in QCD" session at Baryons 2025 I will tie together the recent progress made on the extraction of parton distribution functions (PDFs) in lattice QCD and the long standing efforts in solving the inverse problem in the form of spectral function reconstruction.

Figures (6)

• Figure 1: (a) The eigenvalues of the kernel matrix $\mathfrak{K}$ for different maximum values of available Ioffe time. Full access to the Brillouin zone corresponds to eigenvalues of order unity (orange open squares), while reduced access to limited Ioffe time values is accompanied by an exponential decrease in the eigenvalues, rendering inversion ill-conditioned. (b) Reconstruction (colored symbols) of a mock PDF (denoted $q(x)$, dashed line) from direct inversion in the presence of constant relative errors $\Delta D/D$. Note that as access to the Brillouin zone is reduced (left) full (center) $4/5\nu_{\rm max}$ and (right) $0.2 \nu_{\rm max}$ the reconstruction becomes unreliable even in the presence of only minute statistical errors (plots reprinted from Karpie:2019eiq).
• Figure 2: A possible application of neural networks in the reconstruction of PDFs: starting from a single parameter input layer representing the Bjorken-x value at which we wish to evaluate the PDF, a set of hidden layers is spanned which eventually are projected into a single output layer representing the PDF value (figure reprinted from Karpie:2019eiq).
• Figure 3: (left) Mock PDFs used in our benchmark study. Featuring similar convex behavior at values $x>0.4$, model PDF B (red) exhibits a downward trend leading to a vanishing intercept, while model PDF A (gray) exhibits a monotonous increase towards smaller Bjorken x. (right) The values of the matrix elements according to the two mock PDFs, evaluated over a large range of Ioffe time (plots reprinted from Karpie:2019eiq).
• Figure 4: (top row) Backus-Gilbert reconstruction (red) of the mock PDFs (gray dashed) based on raw input data. Note the deviations of the reconstruction for $x<0.5$ from the true solution not captured by uncertainty bands. (bottom row) Improved BG reconstruction based on preconditioned data using fit data (plots reprinted from Karpie:2019eiq).
• Figure 5: Best phenomenological fit (red) obtained from using the function $f_{\rm fit}(x) = c x^a (1-x)^b$ to reproduce the Ioffe-time data of mock PDF A (left) and mock PDF B (right). For the data used in the fit see the right panel of \ref{['fig:mockPDF']}. In addition to the best fit result we also plot several deformations used as variations of the default model $m(x)$ in the Bayesian approaches to PDF reconstruction (plots reprinted from Karpie:2019eiq).