- 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

Bayesian analysis and analytic continuation of scattering amplitudes from lattice QCD

Miguel Salg1,*, Fernando Romero-López1,†, and William I. Jay2,‡

  • *Contact author: miguel.salg@unibe.ch
  • †Contact author: fernando.romero-lopez@unibe.ch
  • ‡Contact author: william.jay@colostate.edu
PDF

Phys. Rev. D 112, 114502 – Published 2 December, 2025

DOI: https://doi.org/10.1103/ty19-xvvw

Abstract

We present a novel procedure for analyzing the lattice QCD spectrum via the finite-volume formalism to obtain constraints on multihadron scattering amplitudes at both real and complex energies. This approach combines a Bayesian reconstruction of the scattering amplitude on the real axis with Nevanlinna interpolation for analytic continuation to complex-valued energies. The method is nonparametric, inherently accounting for parametrization dependence within the uncertainty. We demonstrate the applicability of this approach using both toy data and real lattice QCD data in resonant systems from the Hadron Spectrum and the Baryon Scattering collaborations.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (102)

  1. P. Koppenburg (LHCb Collaboration), List of hadrons observed at the LHC, Report No. LHCb-FIGURE-2021-001, 2021, see 2022 update online.
  2. Z. Liu and R. E. Mitchell, New hadrons discovered at BESIII, Sci. Bull. 68, 2148 (2023).
  3. J. Bulava et al., Hadron spectroscopy with lattice QCD, in Snowmass 2021 (2022), .
  4. M. Lüscher, Volume dependence of the energy spectrum in massive quantum field theories. 2. Scattering states, Commun. Math. Phys. 105, 153 (1986).
  5. M. Lüscher, Two particle states on a torus and their relation to the scattering matrix, Nucl. Phys. B354, 531 (1991).
  6. K. Rummukainen and S. A. Gottlieb, Resonance scattering phase shifts on a nonrest frame lattice, Nucl. Phys. B450, 397 (1995).
  7. C. h. Kim, C. T. Sachrajda, and S. R. Sharpe, Finite-volume effects for two-hadron states in moving frames, Nucl. Phys. B727, 218 (2005).
  8. V. Bernard, M. Lage, U.-G. Meißner, and A. Rusetsky, Resonance properties from the finite-volume energy spectrum, J. High Energy Phys. 08 (2008) 024.
  9. L. Leskovec and S. Prelovsek, Scattering phase shifts for two particles of different mass and non-zero total momentum in lattice QCD, Phys. Rev. D 85, 114507 (2012).
  10. M. Göckeler, R. Horsley, M. Lage, U.-G. Meißner, P. E. L. Rakow, A. Rusetsky, G. Schierholz, and J. M. Zanotti, Scattering phases for meson and baryon resonances on general moving-frame lattices, Phys. Rev. D 86, 094513 (2012).
  11. R. A. Briceño, Two-particle multichannel systems in a finite volume with arbitrary spin, Phys. Rev. D 89, 074507 (2014).
  12. R. A. Briceño, J. J. Dudek, and R. D. Young, Scattering processes and resonances from lattice QCD, Rev. Mod. Phys. 90, 025001 (2018).
  13. J. Bulava et al. (Baryon Scattering (BaSc) Collaboration), Two-pole nature of the Λ(1405) resonance from lattice QCD, Phys. Rev. Lett. 132, 051901 (2024).
  14. J. Bulava et al. (Baryon Scattering (BaSc) Collaboration), Lattice QCD study of πΣ−K− N scattering and the Λ(1405) resonance, Phys. Rev. D 109, 014511 (2024).
  15. A. Rodas, J. J. Dudek, and R. G. Edwards (Hadron Spectrum Collaboration), Quark mass dependence of ππ scattering in isospin 0, 1, and 2 from lattice QCD, Phys. Rev. D 108, 034513 (2023).
  16. R. A. Briceño, J. J. Dudek, R. G. Edwards, and D. J. Wilson, Isoscalar ππ,KK¯,ηη scattering and the σ,f0,f2 mesons from QCD, Phys. Rev. D 97, 054513 (2018).
  17. S. Prelovsek, S. Collins, D. Mohler, M. Padmanath, and S. Piemonte, Charmonium-like resonances with JPC=0++, 2++ in coupled DD¯, DsD¯s scattering on the lattice, J. High Energy Phys. 06 (2021) 035.
  18. R. A. Briceño, J. J. Dudek, R. G. Edwards, and D. J. Wilson, Isoscalar ππ scattering and the σ meson resonance from QCD, Phys. Rev. Lett. 118, 022002 (2017).
  19. J. J. Dudek, R. G. Edwards, C. E. Thomas, and D. J. Wilson (Hadron Spectrum Collaboration), Resonances in coupled πK−ηK scattering from quantum chromodynamics, Phys. Rev. Lett. 113, 182001 (2014).
  20. A. J. Woss, C. E. Thomas, J. J. Dudek, R. G. Edwards, and D. J. Wilson, b1 resonance in coupled πω, πϕ scattering from lattice QCD, Phys. Rev. D 100, 054506 (2019).
  21. D. J. Wilson, R. A. Briceño, J. J. Dudek, R. G. Edwards, and C. E. Thomas, Coupled ππ,KK¯ scattering in P-wave and the ρ resonance from lattice QCD, Phys. Rev. D 92, 094502 (2015).
  22. A. Rothkopf, Inverse problems, real-time dynamics and lattice simulations, EPJ Web Conf. 274, 01004 (2022).
  23. 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).
  24. L. Del Debbio, T. Giani, and M. Wilson, Bayesian approach to inverse problems: An application to NNPDF closure testing, Eur. Phys. J. C 82, 330 (2022).
  25. J. M. Pawlowski, C. S. Schneider, J. Turnwald, J. M. Urban, and N. Wink, Yang-Mills glueball masses from spectral reconstruction, Phys. Rev. D 108, 076018 (2023).
  26. J. Horak, J. M. Pawlowski, J. Turnwald, J. M. Urban, N. Wink, and S. Zafeiropoulos, Nonperturbative strong coupling at timelike momenta, Phys. Rev. D 107, 076019 (2023).
  27. 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).
  28. A. Candido, L. Del Debbio, T. Giani, and G. Petrillo, Bayesian inference with Gaussian processes for the determination of parton distribution functions, Eur. Phys. J. C 84, 716 (2024).
  29. H. Dutrieux, J. Karpie, K. Orginos, and S. Zafeiropoulos, Simple nonparametric reconstruction of parton distributions from limited Fourier information, Phys. Rev. D 111, 034515 (2025).
  30. H. Dutrieux, J. Karpie, C. J. Monahan, K. Orginos, A. Radyushkin, D. Richards, and S. Zafeiropoulos, Inverse problem in the LaMET framework, arXiv:2504.17706.
  31. J. Fei, C.-N. Yeh, and E. Gull, Nevanlinna analytical continuation, Phys. Rev. Lett. 126, 056402 (2021).
  32. T. Bergamaschi, W. I. Jay, and P. R. Oare, Hadronic structure, conformal maps, and analytic continuation, Phys. Rev. D 108, 074516 (2023).
  33. V. Gribov, Strong Interactions of Hadrons at High Energies: Gribov Lectures on Theoretical Physics, Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology (Cambridge University Press, Cambridge, England, 2023).
  34. I. Matuschek, V. Baru, F.-K. Guo, and C. Hanhart, On the nature of near-threshold bound and virtual states, Eur. Phys. J. A 57, 101 (2021).
  35. S. M. Dawid, Z. T. Draper, A. D. Hanlon, B. Hörz, C. Morningstar, F. Romero-López, S. R. Sharpe, and S. Skinner, Two- and three-meson scattering amplitudes with physical quark masses from lattice QCD, Phys. Rev. D 112, 014505 (2025).
  36. P. Boyle, F. Erben, V. Gülpers, M. T. Hansen, F. Joswig, M. Marshall, N. P. Lachini, and A. Portelli, Physical-mass calculation of ρ(770) and K*(892) resonance parameters via ππ and Kπ scattering amplitudes from lattice QCD, Phys. Rev. D 111, 054510 (2025).
  37. P. Boyle, F. Erben, V. Gülpers, M. T. Hansen, F. Joswig, M. Marshall, N. P. Lachini, and A. Portelli, Light and strange vector resonances from lattice QCD at physical quark masses, Phys. Rev. Lett. 134, 111901 (2025).
  38. W. I. Jay and E. T. Neil, Bayesian model averaging for analysis of lattice field theory results, Phys. Rev. D 103, 114502 (2021).
  39. A. Rodas, J. J. Dudek, and R. G. Edwards (Hadron Spectrum Collaboration), Determination of crossing-symmetric ππ scattering amplitudes and the quark mass evolution of the σ constrained by lattice QCD, Phys. Rev. D 109, 034513 (2024).
  40. I. Steinwart, On the influence of the kernel on the consistency of support vector machines, J. Mach. Learn. Res. 2, 67 (2002), https://www.jmlr.org/papers/v2/steinwart01a.html.
  41. H. Jeffreys, Some tests of significance, treated by the theory of probability, Math. Proc. Cambridge Philos. Soc. 31, 203 (1935).
  42. R. E. Kass and A. E. Raftery, Bayes factors, J. Am. Stat. Assoc. 90, 773 (1995).
  43. H. Jeffreys, The Theory of Probability (Oxford University Press, Oxford, 1998).
  44. R. D. Cousins, The Jeffreys–Lindley paradox and discovery criteria in high energy physics, Synthese 194, 395 (2017).
  45. D. V. Lindley, A statistical paradox, Biometrika 44, 187 (1957).
  46. J. R. Peláez, A. Rodas, and J. R. de Elvira, f0(1370) controversy from dispersive meson-meson scattering data analyses, Phys. Rev. Lett. 130, 051902 (2023); J. R. Pelaez, A. Rodas, and J. R. de Elvira 132, 239901(E) (2024).
  47. G. Pick, Über die Beschränkungen analytischer Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden, Math. Ann. 77, 7 (1915).
  48. R. Nevanlinna, Über beschränkte Funktionen die in gegebenen Punkten vorgeschriebene Werte annehmen, Ann. Acad. Sci. Fenn. Ser. A 13 (1919).
  49. R. Nevanlinna, Über beschränkte analytische Funktionen, Ann. Acad. Sci. Fenn. Ser. A 32 (1929).
  50. A. Nicolau, The Nevanlinna-Pick Interpolation Problem, in Proceedings of the Summer School in Complex and Harmonic Analysis, and Related Topics, edited by J. Gröhn, J. Heittokangas, R. Korhonen, and J. Rättyä (Publications of the University of eastern Finland, 2016), https://mat.uab.cat/~artur/data/nevanlinna-pick.pdf.
  51. J. Schur, Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind., J. Reine Angew. Math. 148, 122 (1918).
  52. W. J. Blaschke, Eine Erweiterung des Satzes von Vitali über Folgen analytischer Funktionen, in Berichte über die Verhandlungen der Königlich-Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Vol. 67 (Königlich-Sächsische Gesellschaft der Wissenschaften, Leipzig, 1915), pp. 194–200.
  53. S. R. Garcia, J. Mashreghi, and W. T. Ross, Finite Blaschke Products and Their Connections (Springer, Cham, 2018).
  54. J. Fei, C.-N. Yeh, D. Zgid, and E. Gull, Analytical continuation of matrix-valued functions: Carathéodory formalism, Phys. Rev. B 104, 165111 (2021).
  55. K. Nogaki and H. Shinaoka, Bosonic Nevanlinna analytic continuation, J. Phys. Soc. Jpn. 92, 035001 (2023).
  56. L. Castillejo, R. H. Dalitz, and F. J. Dyson, Low’s scattering equation for the charged and neutral scalar theories, Phys. Rev. 101, 453 (1956).
  57. B. Palka, An Introduction to Complex Function Theory (Springer-Verlag, New York, 1991).
  58. X.-Y. Liu, C. S. Chen, and A. Karageorghis, Conformal mapping for the efficient solution of poisson problems with the Kansa-RBF method, J. Sci. Comput. 71, 1035 (2017).
  59. P. Kythe, Computational Conformal Mapping (Birkhäuser, Boston, MA, 1998).
  60. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 10th ed., Applied Mathematics Series Vol. 55 (United States Department of Commerce, National Bureau of Standards, Washington, DC, 1972).
  61. J. J. Dudek, R. G. Edwards, and C. E. Thomas (Hadron Spectrum Collaboration), Energy dependence of the ρ resonance in ππ elastic scattering from lattice QCD, Phys. Rev. D 87, 034505 (2013). 90, 099902(E) (2014).
  62. C. Morningstar, J. Bulava, B. Singha, R. Brett, J. Fallica, A. Hanlon, and B. Hörz, Estimating the two-particle K-matrix for multiple partial waves and decay channels from finite-volume energies, Nucl. Phys. B924, 477 (2017).
  63. M. A. Branch, T. F. Coleman, and Y. Li, A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems, SIAM J. Sci. Comput. 21, 1 (1999).
  64. M. Bruno et al., Simulation of QCD with Nf=2+1 flavors of non-perturbatively improved Wilson fermions, J. High Energy Phys. 02 (2015) 043.
  65. T. D. Blanton, A. D. Hanlon, B. Hörz, C. Morningstar, F. Romero-López, and S. R. Sharpe, Interactions of two and three mesons including higher partial waves from lattice QCD, J. High Energy Phys. 10 (2021) 023.
  66. R. G. Edwards, B. Joó, and H.-W. Lin, Tuning for Three-flavors of anisotropic clover fermions with stout-link smearing, Phys. Rev. D 78, 054501 (2008).
  67. H.-W. Lin et al. (Hadron Spectrum Collaboration), First results from 2+1 dynamical quark flavors on an anisotropic lattice: Light-hadron spectroscopy and setting the strange-quark mass, Phys. Rev. D 79, 034502 (2009).
  68. T. D. Blanton, F. Romero-López, and S. R. Sharpe, I=3 three-pion scattering amplitude from lattice QCD, Phys. Rev. Lett. 124, 032001 (2020).
  69. Z. T. Draper, A. D. Hanlon, B. Hörz, C. Morningstar, F. Romero-López, and S. R. Sharpe, Interactions of πK, ππK and KKπ systems at maximal isospin from lattice QCD, J. High Energy Phys. 05 (2023) 137.
  70. S. M. Dawid, Z. T. Draper, A. D. Hanlon, B. Hörz, C. Morningstar, F. Romero-López, S. R. Sharpe, and S. Skinner, QCD predictions for physical multimeson scattering amplitudes, Phys. Rev. Lett. 135, 021903 (2025).
  71. S. Weinberg, Pion scattering lengths, Phys. Rev. Lett. 17, 616 (1966).
  72. J. Gasser and H. Leutwyler, Chiral perturbation theory to one loop, Ann. Phys. (N.Y.) 158, 142 (1984).
  73. M. Cè, A. Gérardin, G. von Hippel, H. B. Meyer, K. Miura, K. Ottnad, A. Risch, T. San José, J. Wilhelm, and H. Wittig, The hadronic running of the electromagnetic coupling and the electroweak mixing angle from lattice QCD, J. High Energy Phys. 08 (2022) 220.
  74. S. L. Adler, Consistency conditions on the strong interactions implied by a partially conserved axial vector current, Phys. Rev. 137, B1022 (1965); 139, AB2(E) (1965).
  75. M. Fischer, B. Kostrzewa, L. Liu, F. Romero-López, M. Ueding, and C. Urbach, Scattering of two and three physical pions at maximal isospin from lattice QCD, Eur. Phys. J. C 81, 436 (2021).
  76. I. Caprini, G. Colangelo, and H. Leutwyler, Mass and width of the lowest resonance in QCD, Phys. Rev. Lett. 96, 132001 (2006).
  77. C. Hanhart, J. R. Peláez, and G. Ríos, Quark mass dependence of the rho and sigma from dispersion relations and Chiral Perturbation Theory, Phys. Rev. Lett. 100, 152001 (2008).
  78. R. H. Dalitz and S. F. Tuan, The energy dependence of low energy K−-proton processes, Ann. Phys. (N.Y.) 8, 100 (1959).
  79. R. H. Dalitz and S. F. Tuan, A possible resonant state in pion-hyperon scattering, Phys. Rev. Lett. 2, 425 (1959).
  80. M. H. Alston, L. W. Alvarez, P. Eberhard, M. L. Good, W. Graziano, H. K. Ticho, and S. G. Wojcicki, Study of resonances of the Σ−π system, Phys. Rev. Lett. 6, 698 (1961).
  81. P. L. Bastien, M. Ferro-Luzzi, and A. H. Rosenfeld, Sigma decay modes of pion-hyperon resonances, Phys. Rev. Lett. 6, 702 (1961).
  82. S. Navas et al. (Particle Data Group), Review of particle physics, Phys. Rev. D 110, 030001 (2024).
  83. J. A. Oller and U.-G. Meißner, Chiral dynamics in the presence of bound states: Kaon-nucleon interactions revisited, Phys. Lett. B 500, 263 (2001).
  84. A. V. Anisovich, A. V. Sarantsev, V. A. Nikonov, V. Burkert, R. A. Schumacher, U. Thoma, and E. Klempt, Hyperon III: K−p−πΣ coupled-channel dynamics in the Λ(1405) mass region, Eur. Phys. J. A 56, 139 (2020).
  85. T. Hyodo and M. Niiyama, QCD and the strange baryon spectrum, Prog. Part. Nucl. Phys. 120, 103868 (2021).
  86. M. Mai, Review of the Λ(1405) A curious case of a strangeness resonance, Eur. Phys. J. ST 230, 1593 (2021).
  87. E. Oset, On ambiguities of sign determination of the S-matrix from energy levels in a finite box, Eur. Phys. J. A 49, 32 (2013).
  88. J. J. Dudek, R. G. Edwards, and D. J. Wilson (Hadron Spectrum Collaboration), An a0 resonance in strongly coupled πη, KK¯ scattering from lattice QCD, Phys. Rev. D 93, 094506 (2016).
  89. W. Jay, Approaching the inverse problem: Toward lattice QCD calculations of inclusive hadronic quantities, in 41st International Symposium on Lattice Field Theory (2025), arXiv:2501.12259.
  90. M. T. Hansen, R. A. Briceño, R. G. Edwards, C. E. Thomas, and D. J. Wilson (Hadron Spectrum Collaboration), Energy-dependent π+π+π+ scattering amplitude from QCD, Phys. Rev. Lett. 126, 012001 (2021).
  91. M. Garofalo, M. Mai, F. Romero-López, A. Rusetsky, and C. Urbach, Three-body resonances in the φ4 theory, J. High Energy Phys. 02 (2023) 252.
  92. R. Brett, C. Culver, M. Mai, A. Alexandru, M. Döring, and F. X. Lee, Three-body interactions from the finite-volume QCD spectrum, Phys. Rev. D 104, 014501 (2021).
  93. M. Mai, A. Alexandru, R. Brett, C. Culver, M. Döring, F. X. Lee, and D. Sadasivan (GWQCD Collaboration), Three-body dynamics of the a1(1260) resonance from lattice QCD, Phys. Rev. Lett. 127, 222001 (2021).
  94. H. Yan, M. Mai, M. Garofalo, U.-G. Meißner, C. Liu, L. Liu, and C. Urbach, ω meson from lattice QCD, Phys. Rev. Lett. 133, 211906 (2024).
  95. https://www.id.unibe.ch/hpc
  96. S. Duane, A. D. Kennedy, B. J. Pendleton, and D. Roweth, Hybrid Monte Carlo, Phys. Lett. B 195, 216 (1987).
  97. C. Gattringer and C. B. Lang, Quantum Chromodynamics on the Lattice (Springer, Berlin, 2010), Vol. 788.
  98. I. P. Omelyan, I. M. Mryglod, and R. Folk, Symplectic analytically integrable decomposition algorithms: Classification, derivation, and application to molecular dynamics, quantum and celestial mechanics simulations, Comput. Phys. Commun. 151, 272 (2003).
  99. T. Takaishi, Choice of integrator in the hybrid Monte Carlo algorithm, Comput. Phys. Commun. 133, 6 (2000).
  100. M. Hasenbusch, Speeding up the hybrid Monte Carlo algorithm for dynamical fermions, Phys. Lett. B 519, 177 (2001).
  101. M. Hasenbusch and K. Jansen, Speeding up lattice QCD simulations with clover improved Wilson fermions, Nucl. Phys. B659, 299 (2003).
  102. T. Takaishi and P. de Forcrand, Testing and tuning new symplectic integrators for hybrid Monte Carlo algorithm in lattice QCD, Phys. Rev. E 73, 036706 (2006).

Outline

Information

Phys. Rev. D 112, 114502– Published 2 December, 2025
  • Received 30 June 2025
  • Accepted 14 October 2025
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