- 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

Scalar size of the pion from lattice QCD

Konstantin Ottnad* and Georg von Hippel

  • *Contact author: kottnad@uni-mainz.de
PDF

Phys. Rev. D 112, 034504 – Published 13 August, 2025

DOI: https://doi.org/10.1103/qh7q-9nyy

Abstract

We present a lattice QCD calculation of the pion scalar form factor and associated radii with fully controlled systematics. Lattice results are computed on a large set of 17 gauge ensembles with Nf=2+1 Wilson Clover-improved sea quarks. These ensembles cover four values of the lattice spacing between a=0.049  fm and a=0.086  fm, a pion mass range of 130–350 MeV, and various physical volumes. A precise determination of the notorious quark-disconnected contributions facilitates an unprecedented momentum resolution for the form factor, particularly on large and fine ensembles in the vicinity of physical quark mass. A large range of source-sink separations 1.0  fm≲tsep≲3.25  fm is used to reliably extract the relevant ground state matrix elements at vanishing and nonvanishing momentum transfer. This allows us for the first time to obtain the scalar radii from a z-expansion parametrization of the Q2 dependence of the resulting form factors rather than a simple, linear approximation at small momentum transfer. The physical extrapolation for the radii is carried out using three-flavor NLO chiral perturbation theory to parametrize the quark mass dependence in terms of three low-energy constants, including the first lattice determination of L4r. Systematic uncertainties on the final results related to the ground state extraction, form factor parametrization as well as the physical extrapolation are accounted for via model averages based on the Akaike information criterion. Our physical results for the pion singlet and octet scalar radii are ⟨rS2⟩π0=0.550(32)stat(39)sys[50]total  fm2 and ⟨rS2⟩π8=0.443(16)stat(45)sys[47]total  fm2, respectively. The values of the relevant low energy constants are given by f0=116.5(5.5)stat(13.7)sys[14.8]total  MeV, L4r(μ=770  MeV)=+0.38(09)stat(15)sys[18]total×10−3 and L5r(μ=770  MeV)=+0.58(0.38)stat(1.08)sys[1.14]total×10−3. For the light quark contribution to the pion scalar radius, we obtain ⟨rS2⟩πl=0.515(28)stat(32)sys[43]total  fm2, resulting in a value of ℓ¯4=3.99(15)stat(17)sys[23]total for the pertinent low-energy constant in two-flavor chiral perturbation theory.

View figure in article

Physics Subject Headings (PhySH)

See Also

Low-Energy Constants of Chiral Perturbation Theory from Pion Scalar Form Factors in Nf=2+1-Flavor Lattice QCD with Controlled Errors

Georg von Hippel and Konstantin Ottnad
Phys. Rev. Lett. 135, 071904 (2025)

Article Text

References (65)

  1. J. Heitger, F. Joswig, P. L. J. Petrak, and A. Vladikas, Ratio of flavour non-singlet and singlet scalar density renormalisation parameters in Nf=3 QCD with Wilson quarks, Eur. Phys. J. C 81, 606 (2021); 82, 104(E) (2022).
  2. J. Gasser and H. Leutwyler, Low-energy expansion of meson form-factors, Nucl. Phys. B250, 517 (1985).
  3. G. von Hippel and K. Ottnad, companion Letter, Low-energy constants of chiral perturbation theory from pion scalar form factors in Nf=2+1-flavor lattice QCD with controlled errors, Phys. Rev. Lett. 135, 071904 (2025).
  4. 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.
  5. B. Sheikholeslami and R. Wohlert, Improved continuum limit lattice action for QCD with Wilson fermions, Nucl. Phys. B259, 572 (1985).
  6. M. Lüscher and P. Weisz, On-shell improved lattice gauge theories, Commun. Math. Phys. 97, 59 (1985); 98, 433(E) (1985).
  7. M. Lüscher and S. Schaefer, Lattice QCD without topology barriers, J. High Energy Phys. 07 (2011) 036.
  8. M. Lüscher and S. Schaefer, Lattice QCD with open boundary conditions and twisted-mass reweighting, Comput. Phys. Commun. 184, 519 (2013).
  9. M. A. Clark and A. D. Kennedy, Accelerating dynamical fermion computations using the rational hybrid Monte Carlo (RHMC) algorithm with multiple pseudofermion fields, Phys. Rev. Lett. 98, 051601 (2007).
  10. S. Kuberski, Low-mode deflation for twisted-mass and RHMC reweighting in lattice QCD, Comput. Phys. Commun. 300, 109173 (2024).
  11. D. Mohler and S. Schaefer, Remarks on strange-quark simulations with Wilson fermions, Phys. Rev. D 102, 074506 (2020).
  12. M. Lüscher, Properties and uses of the Wilson flow in lattice QCD, J. High Energy Phys. 08 (2010) 071; 03 (2014) 092(E).
  13. M. Bruno, T. Korzec, and S. Schaefer, Setting the scale for the CLS 2+1 flavor ensembles, Phys. Rev. D 95, 074504 (2017).
  14. Y. Aoki et al. (Flavour Lattice Averaging Group (FLAG), FLAG review 2021, Eur. Phys. J. C 82, 869 (2022).
  15. D. Dolgov et al. (LHPC, TXL Collaborations), Moments of nucleon light cone quark distributions calculated in full lattice QCD, Phys. Rev. D 66, 034506 (2002).
  16. Gunnar S. Bali, S. Collins, and A. Schäfer, Effective noise reduction techniques for disconnected loops in Lattice QCD, Comput. Phys. Commun. 181, 1570 (2010).
  17. T. Blum, T. Izubuchi, and E. Shintani, New class of variance-reduction techniques using lattice symmetries, Phys. Rev. D 88, 094503 (2013).
  18. E. Shintani, R. Arthur, T. Blum, T. Izubuchi, C. Jung, and C. Lehner, Covariant approximation averaging, Phys. Rev. D 91, 114511 (2015).
  19. A. Agadjanov, D. Djukanovic, G. von Hippel, H. B. Meyer, K. Ottnad, and H. Wittig, Nucleon sigma terms with Nf=2+1 flavors of O(a)-improved Wilson fermions, Phys. Rev. Lett. 131, 261902 (2023).
  20. D. Djukanovic, G. von Hippel, H. B. Meyer, K. Ottnad, M. Salg, and H. Wittig, Electromagnetic form factors of the nucleon from Nf=2+1 lattice QCD, Phys. Rev. D 109, 094510 (2024).
  21. D. Djukanovic, G. von Hippel, H. B. Meyer, K. Ottnad, M. Salg, and H. Wittig, Precision calculation of the electromagnetic radii of the proton and neutron from lattice QCD, Phys. Rev. Lett. 132, 211901 (2024).
  22. 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.
  23. L. Giusti, T. Harris, A. Nada, and S. Schaefer, Frequency-splitting estimators of single-propagator traces, Eur. Phys. J. C 79, 586 (2019).
  24. C. McNeile and C. Michael (UKQCD Collaboration), Decay width of light quark hybrid meson from the lattice, Phys. Rev. D 73, 074506 (2006).
  25. V. Gülpers, G. von Hippel, and H. Wittig, Scalar pion form factor in two-flavor lattice QCD, Phys. Rev. D 89, 094503 (2014).
  26. A. Stathopoulos, J. Laeuchli, and K. Orginos, Hierarchical probing for estimating the trace of the matrix inverse on toroidal lattices, SIAM J. Sci. Comput. 35, S299 (2013).
  27. E.-H. Chao, R. J. Hudspith, A. Gérardin, J. R. Green, H. B. Meyer, and K. Ottnad, Hadronic light-by-light contribution to (g−2)μ from lattice QCD: A complete calculation, Eur. Phys. J. C 81, 651 (2021).
  28. M. Cè et al., Window observable for the hadronic vacuum polarization contribution to the muon g−2 from lattice QCD, Phys. Rev. D 106, 114502 (2022).
  29. D. Djukanovic, G. von Hippel, S. Kuberski, H. B. Meyer, N. Miller, K. Ottnad, J. Parrino, A. Risch, and H. Wittig, The hadronic vacuum polarization contribution to the muon g−2 at long distances, J. High Energy Phys. 04 (2025) 098.
  30. L. Maiani, G. Martinelli, M. L. Paciello, and B. Taglienti, Scalar densities and baryon mass differences in lattice QCD with Wilson fermions, Nucl. Phys. B293, 420 (1987).
  31. S. Güsken, U. Löw, K. H. Mütter, R. Sommer, A. Patel, and K. Schilling, Nonsinglet axial vector couplings of the baryon octet in lattice QCD, Phys. Lett. B 227, 266 (1989).
  32. J. Bulava, M. Donnellan, and R. Sommer, On the computation of hadron-to-hadron transition matrix elements in lattice QCD, J. High Energy Phys. 01 (2012) 140.
  33. S. Capitani, M. Della Morte, G. von Hippel, B. Jäger, A. Jüttner, B. Knippschild, H. B. Meyer, and H. Wittig, The nucleon axial charge from lattice QCD with controlled errors, Phys. Rev. D 86, 074502 (2012).
  34. V. Gülpers, G. von Hippel, and H. Wittig, The scalar radius of the pion from lattice QCD in the continuum limit, Eur. Phys. J. A 51, 158 (2015).
  35. J. Koponen, F. Bursa, C. T. H. Davies, R. J. Dowdall, and G. P. Lepage, Size of the pion from full lattice QCD with physical u, d, s and c quarks, Phys. Rev. D 93, 054503 (2016).
  36. R. J. Hill and G. Paz, Model independent extraction of the proton charge radius from electron scattering, Phys. Rev. D 82, 113005 (2010).
  37. G. Lee, J. R. Arrington, and R. J. Hill, Extraction of the proton radius from electron-proton scattering data, Phys. Rev. D 92, 013013 (2015).
  38. S. Aoki et al., Review of lattice results concerning low-energy particle physics, Eur. Phys. J. C 77, 112 (2017).
  39. G. Colangelo, S. Dürr, and C. Haefeli, Finite volume effects for meson masses and decay constants, Nucl. Phys. B721, 136 (2005).
  40. K. P. Burnham and D. R. Anderson, Multimodel inference: Understanding AIC and BIC in model selection, Sociol. Methods Res. 33, 261 (2004).
  41. S. Borsanyi et al. (BMW Collaboration), Ab initio calculation of the neutron-proton mass difference, Science 347, 1452 (2015).
  42. E. T. Neil and J. W. Sitison, Improved information criteria for Bayesian model averaging in lattice field theory, Phys. Rev. D 109, 014510 (2024).
  43. H. Akaike, A new look at the statistical model identification, IEEE Trans. Autom. Control 19, 716 (1974).
  44. H. Akaike, Information theory and an extension of the maximum likelihood principle, in Selected Papers of Hirotugu Akaike, edited by E. Parzen, K. Tanabe, and G. Kitagawa (Springer, New York, 1998), pp. 199–213.
  45. S. Borsanyi et al., Leading hadronic contribution to the muon magnetic moment from lattice QCD, Nature (London) 593, 51 (2021).
  46. P. Zyla et al. (Particle Data Group), Review of particle physics, Prog. Theor. Exp. Phys. 2020, 083C01 (2020). and 2021 update.
  47. A. Bazavov et al. (MILC Collaboration), Results for light pseudoscalar mesons, Proc. Sci. LATTICE2010 (2010) 074 [arXiv:1012.0868].
  48. S. R. Beane, W. Detmold, P. M. Junnarkar, T. C. Luu, K. Orginos, A. Parreño, M. J. Savage, A. Torok, and A. Walker-Loud, SU(2) low-energy constants from mixed-action lattice QCD, Phys. Rev. D 86, 094509 (2012).
  49. S. Borsányi, S. Dürr, Z. Fodor, S. Krieg, A. Schäfer, E. E. Scholz, and K. K. Szabó, SU(2) chiral perturbation theory low-energy constants from 2+1 flavor staggered lattice simulations, Phys. Rev. D 88, 014513 (2013).
  50. S. Dürr et al. (BMW Collaboration), Lattice QCD at the physical point meets SU(2) chiral perturbation theory, Phys. Rev. D 90, 114504 (2014).
  51. P. A. Boyle et al., Low energy constants of SU(2) partially quenched chiral perturbation theory from Nf=2+1 domain wall QCD, Phys. Rev. D 93, 054502 (2016).
  52. J. Gasser and H. Leutwyler, Chiral perturbation theory to one loop, Ann. Phys. (N.Y.) 158, 142 (1984).
  53. J. Bijnens, G. Colangelo, and P. Talavera, The vector and scalar form-factors of the pion to two loops, J. High Energy Phys. 05 (1998) 014.
  54. G. Amoros, J. Bijnens, and P. Talavera, Kℓ4 form-factors and π−π scattering, Nucl. Phys. B585, 293 (2000); B598, 665(E) (2001).
  55. G. Colangelo, J. Gasser, and H. Leutwyler, ππ scattering, Nucl. Phys. B603, 125 (2001).
  56. J. Liang, A. Alexandru, Y.-J. Bi, T. Draper, K.-F. Liu, and Y.-B. Yang (χQCD Collaboration), Detecting the flavor content of the vacuum using the Dirac operator spectrum, Phys. Rev. D 110, 094513 (2024).
  57. G. S. Bali, S. Collins, W. Söldner, and S. Weishäupl (RQCD Collaboration), Leading order mesonic and baryonic SU(3) low energy constants from Nf=3 lattice QCD, Phys. Rev. D 105, 054516 (2022).
  58. M. F. M. Lutz, Y. Heo, and R. J. Hudspith, QCD in the chiral SU(3) limit from baryon masses on lattice QCD ensembles, Phys. Rev. D 110, 094046 (2024).
  59. J. Gasser and H. Leutwyler, Chiral perturbation theory: Expansions in the mass of the strange quark, Nucl. Phys. B250, 465 (1985).
  60. R. J. Dowdall, C. T. H. Davies, G. P. Lepage, and C. McNeile, Vus from π and K decay constants in full lattice QCD with physical u, d, s and c quarks, Phys. Rev. D 88, 074504 (2013).
  61. A. Jüttner, Quark disconnected diagrams in chiral perturbation theory—the scalar form factor, Proc. Sci. LATTICE2012 (2012) 196 [arXiv:1212.2559].
  62. Jülich Supercomputing Centre, JUQUEEN: IBM Blue Gene/Q Supercomputer System at the Jülich Supercomputing Centre, J. Large-Scale Res. Facil. 1, A1 (2015).
  63. Jülich Supercomputing Centre, JUWELS Cluster and Booster: Exascale Pathfinder with Modular Supercomputing Architecture at Juelich Supercomputing Centre, J. Large-Scale Res. Facil. 7, A138 (2021).
  64. R. G. Edwards and B. Joó (SciDACLHPC, UKQCD Collaborations), The chroma software system for lattice QCD, Nucl. Phys. B, Proc. Suppl. 140, 832 (2005).
  65. M. Lüscher, S. Schaefer et al., openqcd, http://luscher.web.cern.ch/luscher/openQCD/.

Outline

Information

Phys. Rev. D 112, 034504– Published 13 August, 2025
  • Received 2 April 2025
  • Accepted 2 July 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