- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Nevanlinna-Pick interpolation from uncertain data

Sarah Fields and Norman Christ

PDF

Phys. Rev. D 113, 054502 – Published 4 March, 2026

DOI: https://doi.org/10.1103/brwx-k3xf

Abstract

The calculation of inclusive processes that involve the production of many particles is a challenge for lattice quantum chromodynamics (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.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (31)

  1. A. Rothkopf, Inverse problems, real-time dynamics and lattice simulations, EPJ Web Conf. 274, 01004 (2022).
  2. A. Rothkopf, Bayesian inference of real-time dynamics from lattice QCD, Front. Phys. 10, 1028995 (2022).
  3. G. Aarts, K. Fukushima, T. Hatsuda, A. Ipp, S. Shi, L. Wang, and K. Zhou, Physics-driven learning for inverse problems in quantum chromodynamics, Nat. Rev. Phys. 7, 154 (2025).
  4. L. Del Debbio, A. Lupo, M. Panero, and N. Tantalo, Approaches to the inverse problem, Proc. Sci., EuroPLEx2023 (2024) 019.
  5. M. T. Hansen, H. B. Meyer, and D. Robaina, From deep inelastic scattering to heavy-flavor semileptonic decays: Total rates into multihadron final states from lattice QCD, Phys. Rev. D 96, 094513 (2017).
  6. J. Bulava and M. T. Hansen, Scattering amplitudes from finite-volume spectral functions, Phys. Rev. D 100, 034521 (2019).
  7. M. Hansen, A. Lupo, and N. Tantalo, Extraction of spectral densities from lattice correlators, Phys. Rev. D 99, 094508 (2019).
  8. G. Bailas, S. Hashimoto, and T. Ishikawa, Reconstruction of smeared spectral function from Euclidean correlation functions, Prog. Theor. Exp. Phys. 2020, 043B07 (2020).
  9. P. Gambino, S. Hashimoto, S. Mächler, M. Panero, F. Sanfilippo, S. Simula, A. Smecca, and N. Tantalo, Lattice QCD study of inclusive semileptonic decays of heavy mesons, J. High Energy Phys. 07 (2022) 083.
  10. A. Barone, S. Hashimoto, A. Jüttner, T. Kaneko, and R. Kellermann, Approaches to inclusive semileptonic B(s)-meson decays from Lattice QCD, J. High Energy Phys. 07 (2023) 145.
  11. M. Bruno, L. Giusti, and M. Saccardi, Spectral densities from Euclidean lattice correlators via the Mellin transform, Phys. Rev. D 111, 094515 (2025).
  12. A. Patella and N. Tantalo, Scattering amplitudes from Euclidean correlators: Haag-Ruelle theory and approximation formulae, J. High Energy Phys. 01 (2025) 091.
  13. L. Del Debbio, A. Lupo, M. Panero, and N. Tantalo, Bayesian solution to the inverse problem and its relation to Backus–Gilbert methods, Eur. Phys. J. C 85, 185 (2025).
  14. T. Bergamaschi, W. I. Jay, and P. R. Oare, Hadronic structure, conformal maps, and analytic continuation, Phys. Rev. D 108, 074516 (2023).
  15. L. Zhang, Y. Yu, and E. Gull, Minimal pole representation and analytic continuation of matrix-valued correlation functions, Phys. Rev. B 110, 235131 (2024).
  16. J. Horak, J. M. Pawlowski, J. Rodríguez-Quintero, J. Turnwald, J. M. Urban, N. Wink, and S. Zafeiropoulos, Reconstructing QCD spectral functions with Gaussian processes, Phys. Rev. D 105, 036014 (2022).
  17. J. Karpie, K. Orginos, A. Rothkopf, and S. Zafeiropoulos, Reconstructing parton distribution functions from Ioffe time data: From Bayesian methods to neural networks, J. High Energy Phys. 04 (2019) 057.
  18. J. Fei, C.-N. Yeh, and E. Gull, Nevanlinna analytical continuation, Phys. Rev. Lett. 126, 056402 (2021).
  19. M. Salg, F. Romero-López, and W. I. Jay, Bayesian Analysis and analytic continuation of scattering amplitudes from Lattice QCD, Phys. Rev. D 112, 114502 (2025).
  20. S. Iskakov, A. Hampel, N. Wentzell, and E. Gull, TRIQS/Nevanlinna: Implementation of the Nevanlinna analytic continuation method for noise-free data, Comput. Phys. Commun. 304, 109299 (2024).
  21. L. Huang and S. Liang, Reconstructing lattice QCD spectral functions with stochastic pole expansion and Nevanlinna analytic continuation, Phys. Rev. D 109, 054508 (2024).
  22. R. Abbott, W. I. Jay, and P. R. Oare, Moment problems and bounds for matrix-valued smeared spectral functions, arXiv:2508.01377.
  23. M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, QCD and resonance physics. Theoretical foundations, Nucl. Phys. B147, 385 (1979).
  24. M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, QCD and resonance physics: Applications, Nucl. Phys. B147, 448 (1979).
  25. M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, QCD and resonance physics. The rho-omega mixing, Nucl. Phys. B147, 519 (1979).
  26. E. Braaten, S. Narison, and A. Pich, QCD analysis of the tau hadronic width, Nucl. Phys. B373, 581 (1992).
  27. F. Le Diberder and A. Pich, Testing QCD with tau decays, Phys. Lett. B 289, 165 (1992).
  28. P. Boyle, R. J. Hudspith, T. Izubuchi, A. Jüttner, C. Lehner, R. Lewis, K. Maltman, H. Ohki, A. Portelli, and M. Spraggs (RBC Collaboration and UKQCD Collaboration), Novel—Vus—Determination using inclusive strange τ decay and lattice hadronic vacuum polarization functions, Phys. Rev. Lett. 121, 202003 (2018).
  29. C. Alexandrou et al. (Extended Twisted Mass Collaboration), Inclusive hadronic decay rate of the τ lepton from lattice QCD: The u−s flavor channel and the Cabibbo angle, Phys. Rev. Lett. 132, 261901 (2024).
  30. A. Nicolau, The Nevanlinna-pick interpolation problem, in Proceedings of the Summer School in Complex and Harmonic analysis, and related topics (2016), https://erepo.uef.fi/handle/123456789/15782.
  31. T. Mpmath Development Team, mpmath: A python Library for Arbitrary-Precision Floating-Point Arithmetic (Version 1.2.1) (2020), http://mpmath.org/.

Outline

Information

Phys. Rev. D 113, 054502– Published 4 March, 2026
  • Received 3 November 2025
  • Accepted 12 January 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation