- 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

Estimating energy levels from lattice QCD correlation functions using a transfer matrix formalism

Debsubhra Chakraborty*, Dhruv Sood, Archana Radhakrishnan, and Nilmani Mathur†

  • *Contact author: debsubhra.chakraborty@tifr.res.in
  • †Contact author: nilmani@theory.tifr.res.in
PDF

Phys. Rev. D 112, 074506 – Published 17 October, 2025

DOI: https://doi.org/10.1103/x3kb-8zw8

Abstract

We present an efficient method for extracting energy levels from lattice QCD correlation functions by computing the eigenvalues of the transfer matrix associated with the lattice QCD Hamiltonian. While mathematically and numerically equivalent to the recently introduced Lanczos procedure [1], our approach introduces a novel prescription for removing spurious eigenvalues using a kernel density estimator and Gaussian-convoluted histogram method. This strategy yields a robust and stable estimate of the energy spectrum, outperforming the Cullum-Willoughby filtering technique in efficiency. In addition, we detail how this method can be applied to extract overlap factors from two-point correlation functions, as well as matrix elements from three-point functions with a current insertion. Furthermore, we extend the methodology to accommodate correlation matrices constructed from a variational basis of operators, with its block formulation. We demonstrate the efficacy of this framework by computing the two lowest energy levels for a broad range of hadrons, including several nuclei. Although the signal-to-noise ratio is not significantly improved, the extracted energy levels are found to be more reliable than those obtained with conventional techniques. Within a given statistical ensemble, the proposed method effectively captures both statistical uncertainties and systematic errors, including those arising from the choice of fitting window, making it a robust and practical tool for lattice QCD analysis.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (50)

  1. M. L. Wagman, Lanczos algorithm, the transfer matrix, and the signal-to-noise problem, Phys. Rev. Lett. 134, 241901 (2025).
  2. C. Michael, Adjoint sources in lattice gauge theory, Nucl. Phys. B259, 58 (1985).
  3. M. Luscher and U. Wolff, How to calculate the elastic scattering matrix in two-dimensional quantum field theories by numerical simulation, Nucl. Phys. B339, 222 (1990).
  4. S. Dürr, Z. Fodor, J. Frison, C. Hoelbling, R. Hoffmann, S. D. Katz, S. Krieg, T. Kurth, L. Lellouch, T. Lippert, K. K. Szabo, and G. Vulvert, Ab initio determination of light hadron masses, Science 322, 1224 (2008).
  5. C. T. H. Davies et al. (HPQCD Collaboration, UKQCD Collaboration, MILC Collaboration, and Fermilab Lattice Collaboration), High precision lattice QCD confronts experiment, Phys. Rev. Lett. 92, 022001 (2004).
  6. R. J. Dowdall, C. T. H. Davies, T. C. Hammant, and R. R. Horgan (HPQCD Collaboration), Precise heavy-light meson masses and hyperfine splittings from lattice QCD including charm quarks in the sea, Phys. Rev. D 86, 094510 (2012).
  7. Z. S. Brown, W. Detmold, S. Meinel, and K. Orginos, Charmed bottom baryon spectroscopy from lattice QCD, Phys. Rev. D 90, 094507 (2014).
  8. N. Mathur, M. Padmanath, and S. Mondal, Precise predictions of charmed-bottom hadrons from lattice QCD, Phys. Rev. Lett. 121, 202002 (2018).
  9. R. A. Briceño, J. J. Dudek, and R. D. Young, Scattering processes and resonances from lattice QCD, Rev. Mod. Phys. 90, 025001 (2018).
  10. G. Parisi, The strategy for computing the hadronic mass spectrum, Phys. Rep. 103, 203 (1984).
  11. G. P. Lepage, The analysis of algorithms for lattice field theory, in From Actions to Answers: Proceedings of the 1989 Theoretical Advanced Study Institute in Elementary Particle Physics (World Scientific, Singapore, 1989).
  12. W. Detmold and K. Orginos, Nuclear correlation functions in lattice QCD, Phys. Rev. D 87, 114512 (2013).
  13. W. Detmold and M. G. Endres, Signal/noise enhancement strategies for stochastically estimated correlation functions, Phys. Rev. D 90, 034503 (2014).
  14. M. Cè, L. Giusti, and S. Schaefer, Domain decomposition, multilevel integration, and exponential noise reduction in lattice QCD, Phys. Rev. D 93, 094507 (2016).
  15. D. Chakraborty, P. Srivastava, A. Kumar, and N. Mathur, Nuclear correlation functions using first-principle calculations of lattice quantum chromodynamics, Phys. Rev. D 110, 114505 (2024).
  16. D. C. Hackett and M. L. Wagman, Lanczos for lattice QCD matrix elements, arXiv:2407.21777.
  17. S. Kaniel, Estimates for some computational techniques in linear algebra, Math. Comput. 20, 369 (1966).
  18. Y. Saad, On the rates of convergence of the lanczos and the block-lanczos methods, SIAM J. Numer. Anal. 17, 687 (1980).
  19. B. G. R. de Prony, Essai experimental et analytique: Sur les lois de la dilatabilit e de fluides elastique et sur celles de la force expansive de la vapeur de l’alkool a diff erentes temp eratures, J. de l’ecole Polytech. 1, 24 (1975).
  20. G. T. Fleming, What can lattice QCD theorists learn from NMR spectroscopists?, in Proceedings of the 3rd International Workshop on Numerical Analysis and Lattice QCD (Springer-Verlag, 2005), pp. 143–152.
  21. S. Kunis, T. Peter, T. Roemer, and U. von der Ohe, A multivariate generalization of Prony’s method, Linear Algebra Appl. 490, 31 (2016).
  22. M. Fischer, B. Kostrzewa, J. Ostmeyer, K. Ottnad, M. Ueding, and C. Urbach, On the generalised eigenvalue method and its relation to Prony and generalised pencil of function methods, Eur. Phys. J. A 56, 206 (2020).
  23. S. R. Beane, W. Detmold, T. C. Luu, K. Orginos, A. Parreño, M. J. Savage, A. Torok, and A. Walker-Loud (NPLQCD Collaboration), High statistics analysis using anisotropic clover lattices: Single hadron correlation functions, Phys. Rev. D 79, 114502 (2009).
  24. S. R. Beane, W. Detmold, H.-W. Lin, T. C. Luu, K. Orginos, M. J. Savage, A. Torok, and A. Walker-Loud (NPLQCD Collaboration), High statistics analysis using anisotropic clover lattices: III. Baryon-baryon interactions, Phys. Rev. D 81, 054505 (2010).
  25. J. Ostmeyer, A. Sen, and C. Urbach, On the equivalence of Prony and Lanczos methods for Euclidean correlation functions, Eur. Phys. J. A 61, 26 (2025).
  26. R. Abbott, D. C. Hackett, G. T. Fleming, D. A. Pefkou, and M. L. Wagman, Filtered Rayleigh-Ritz is all you need, arXiv:2503.17357.
  27. A. Seleznev, Kernel density estimation in the study of star clusters, Open Astron. 25, 267 (2017).
  28. Peel, Austin, Lin, Chieh-An, Lanusse, François, Leonard, Adrienne, Starck, Jean-Luc, and Kilbinger, Martin, Cosmological constraints with weak-lensing peak counts and second-order statistics in a large-field survey, Astron. Astrophys. 599, A79 (2017).
  29. M. T. McCarthy and C. A. O’Callaghan, Peakdeck: A kernel density estimator-based peak calling program for dnasei-seq data, Bioinformatics 30, 1302 (2014).
  30. J. Sadiq, K. Dey, T. Dent, and E. Barausse, Reconstructing the LISA massive black hole binary population via iterative kernel density estimation, Phys. Rev. D 111, 063051 (2025).
  31. A. Poluektov, Kernel density estimation of a multidimensional efficiency profile, J. Instrum. 10, P02011 (2015).
  32. B. B. Brandt, F. Cuteri, G. Endrődi, and S. Schmalzbauer, The dirac spectrum and the BEC-BCS crossover in QCD at nonzero isospin asymmetry, Particles 3, 80 (2020).
  33. M. Bruno, L. Giusti, and M. Saccardi, On the prediction of spectral densities from Lattice QCD, Nuovo Cimento Soc. Ital. Fis. 47C, 197 (2024).
  34. W. Härdle, A. Werwatz, M. Müller, and S. Sperlich, Nonparametric and Semiparametric Models, Springer Series in Statistics (Springer-Verlag, Heidelberg, Germany2004), 1st ed., pp. XXVII, 300.
  35. J. Cullum and R. A. Willoughby, Computing eigenvalues of very large symmetric matrices—An implementation of a lanczos algorithm with no reorthogonalization, J. Comput. Phys. 44, 329 (1981).
  36. B. W. Silverman, Density Estimation for Statistics and Data Analysis (Chapman & Hall/CRC, London, 1998), repr. ed.
  37. G. R. Terrell and D. W. Scott, Variable kernel density estimation, Ann. Stat. 20, 1236 (1992).
  38. T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd ed. Springer Series in Statistics (Springer, New York, 2009).
  39. G. Wahba, Optimal convergence properties of variable knot, kernel, and orthogonal series methods for density estimation, Ann. Stat. 3, 15 (1975).
  40. P. Mills, Efficient statistical classification of satellite measurements, International J. Remote Sensing 32, 6109 (2011).
  41. scipy-community, scipy.linalg.eig, 1.14.1 (stable), https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.eig.html (2024).
  42. E. Polizzi, Density-matrix-based algorithm for solving eigenvalue problems, Phys. Rev. B 79, 115112 (2009).
  43. D. Hatton, C. T. H. Davies, B. Galloway, J. Koponen, G. P. Lepage, and A. T. Lytle (HPQCD Collaboration), Charmonium properties from lattice QCD+QED: Hyperfine splitting, j/ψ leptonic width, charm quark mass, and aμc, Phys. Rev. D 102, 054511 (2020).
  44. G. S. Bali, S. Collins, S. Heybrock, M. Löffler, R. Rödl, W. Söldner, and S. Weishäupl (RQCD Collaboration), Octet baryon isovector charges from Nf=2+1 lattice QCD, Phys. Rev. D 108, 034512 (2023).
  45. D. C. Hackett and M. L. Wagman, Block Lanczos for lattice QCD spectroscopy and matrix elements, Phys. Rev. D 112, 014514 (2025).
  46. N. S. Dhindsa, D. Chakraborty, A. Radhakrishnan, N. Mathur, and M. Padmanath, Precise study of triply charmed baryons (Ωccc), arXiv:2411.12729.
  47. A. Bazavov et al. (MILC Collaboration), Lattice QCD ensembles with four flavors of highly improved staggered quarks, Phys. Rev. D 87, 054505 (2013).
  48. A. Bazavov et al., B- and D-meson leptonic decay constants from four-flavor lattice QCD, Phys. Rev. D 98, 074512 (2018).
  49. N. Mathur and M. Padmanath, Lattice QCD study of doubly charmed strange baryons, Phys. Rev. D 99, 031501 (2019).
  50. C. DeTar and S.-H. Lee, Variational method with staggered fermions, Phys. Rev. D 91, 034504 (2015).

Outline

Information

Phys. Rev. D 112, 074506– Published 17 October, 2025
  • Received 7 December 2024
  • Accepted 5 September 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