Nevanlinna-Pick interpolation from uncertain data

TL;DR

The paper tackles the ill-posed problem of extracting real-time, inclusive QCD observables from Euclidean lattice data by extending Nevanlinna–Pick interpolation to propagate lattice uncertainties. By mapping $G(z)$ to a disk via the Cayley transform, enforcing Pick-consistency, and sampling Pick-consistent data within the lattice error volume, the authors obtain Wertevorrat bounds that encapsulate interpolation uncertainty. They define a contour-integral observable and propagate both statistical and systematic errors through the interpolation, illustrating the method on a Gaussian spectral-density model and showing how results improve with more input data and smaller input errors while highlighting numerical and methodological challenges. The approach yields bounded, first-principles error estimates for inclusive decay rates and serves as a foundation for applying this framework to concrete physical quantities, with future work addressing correlations, larger-energy data, and real-world lattice calculations. Overall, this work provides a principled pathway to connect Euclidean lattice outputs with Minkowski observables under controlled interpolation uncertainty, potentially expanding the reach of lattice QCD in inclusive processes.

Abstract

The calculation of inclusive processes that involve the production of many particles is a challenge for lattice QCD, a Euclidean-space method that is far removed from real-time, multiparticle production. A new approach to this problem based on Nevanlinna-Pick interpolation has been proposed by Bergamaschi et al. Here we extend their method by exploring the propagation of the statistical and systematic errors that accompany a lattice QCD calculation through this interpolation process. A simplified example of a multiparticle spectral function is studied with a focus on the possible applications of these methods to the calculation of inclusive heavy-particle decays.

Figures (10)

• Figure 1: Sketch of the contour $\mathcal{C'} = \mathcal{C}_1\cup \mathcal{C}_2\cup\mathcal{C_3}$ in the complex plane that might be used in Eq. \ref{['eq:contour2']} to calculate inclusive heavy particle decay. The crosses on the imaginary axis represent values of $z$ for which lattice data is available
• Figure 2: Cross-section plots from tests of Pick consistency for the case of interpolating from four lattice data points. The plots show seven independent tests in which pairs of trial Green's functions values were varied randomly within an error volume corresponding to an error scale $\xi=0.01$ while the other six results were held fixed at their exact values. Trial data which did not obey the Pick criterion are blue. Pick-consistent points were given three different colors depending on the average diameter of the Wertevorräte for the vector of complex Green's function value $\widetilde{G}^m$ corresponding to that point. Specifically, each of these $N=4$ data points has a corresponding Wertevorrat and we take the average of the diameters of these four Wertevorräte. Each plot shows 4000 random samples.
• Figure 3: Cross-section plots corresponding to the top row of plots shown in Fig. \ref{['fig:cross_section_plots']}. Here we have expanded the $x$ and $y$ scales by a factor of 65 to capture the complete Pick-consistent region explored in this study where only two of the eight possible lattice data values are randomly sampled in the error volume. This was accomplished by increasing the error scale from $\xi=0.01$ to $\xi = 0.65$ region. Each plot shows 8000 random samples.
• Figure 4: Cross-section plots indicating the location of 3725 Pick-consistent Green's functions values where the sample contains 50 random boundary points found through the constrained gradient ascent procedure, and the remaining points were gathered by following the lines connecting all pairs of the boundary points. The imaginary energies $z_n$ to which the lattice correspond were chosen as 10 equally spaced values along the interval $\{0.1i,2.0i\}$ and the error scale was $\xi=0.01$. The gradient ascent procedure was performed in the unit disk, and then all values were transformed back onto the complex plane to obtain the ten components $(1\le n \le 10)$$\widetilde{G}_n^m$ for each 3725 samples $1\le m \le 3725$.
• Figure 5: Histograms of the real (top) and imaginary (bottom) distributions of the $M$ differences $\mathcal{I}^{X,m}_\mathrm{max} - \left\langle\mathcal{I}^X_\mathrm{avg}\right\rangle$ (right) and similarly for the $M$ differences $\mathcal{I}^{X,m}_\mathrm{min} - \left\langle\mathcal{I}^X_\mathrm{avg}\right\rangle$ (left). The initial points $z_n$ were chosen as 10 equally spaced values along the interval $\{0.1i,2.0i\}$ and the error scale was $\xi = 0.01$.