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Lattice-QCD Validation of Hadron Mass and Trace-Anomaly Decomposition Sum Rules

Dennis Bollweg1, Heng-Tong Ding2, Xiang Gao3,*, Ran Luo2,†, and Swagato Mukherjee3

  • *Contact author: xgao@bnl.gov
  • †Contact author: ranluo@mails.ccnu.edu.cn
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Phys. Rev. Lett. 137, 061901 – Published 7 August, 2026

DOI: https://doi.org/10.1103/5n46-717z

Abstract

We present the first lattice-QCD validation of multiple sum rules associated with quark-gluon decomposition of hadron mass by computing all relevant tensor components of the quark and gluon energy-momentum tensor matrix elements from first principles. We achieve this through nonperturbative renormalization of the QCD energy-momentum tensor, including its trace, in a gradient-flow scheme, followed by continuum extrapolations, two-loop matching to the MS¯ scheme, and zero-flow-time extrapolations. These ingredients enable a direct and simultaneous verification, in a common renormalization scheme and scale, of multiple energy-density-based and trace-based mass decomposition sum rules proposed in the literature. We demonstrate the framework for the ηc and J/ψ charmonia using three fine lattice spacings with a physical strange-quark and near-physical up- and down-quark masses. We present the first lattice-QCD results for the gravitational form factor C¯. We find sizable gluonic contributions to charmonia masses at the hadronic scale, ∼15% in the Lorcé and Metz-Pasquini-Rodini decompositions. The trace-anomaly contribution in the Ji sum rule is ∼6%, while the gluonic component of the trace anomaly in the Hatta-Rajan-Tanaka sum rule is ∼35%. The method is general and can be straightforwardly adopted for lattice-QCD calculations of mass and spin decompositions as well as gravitational form factors of other hadrons and nuclei.

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References (96)

  1. H. Fritzsch and M. Gell-Mann, Current algebra: Quarks and what else?, eConf C720906V2, 135 (1972).
  2. D. J. Gross and F. Wilczek, Ultraviolet behavior of non-Abelian gauge theories, Phys. Rev. Lett. 30, 1343 (1973).
  3. H. D. Politzer, Reliable perturbative results for strong interactions?, Phys. Rev. Lett. 30, 1346 (1973).
  4. J. C. Collins, A. Duncan, and S. D. Joglekar, Trace and dilatation anomalies in gauge theories, Phys. Rev. D 16, 438 (1977).
  5. N. K. Nielsen, The energy momentum tensor in a non-Abelian quark gluon theory, Nucl. Phys. B120, 212 (1977).
  6. M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Remarks on Higgs boson interactions with nucleons, Phys. Lett. 78B, 443 (1978).
  7. Y. Hatta, A. Rajan, and K. Tanaka, Quark and gluon contributions to the QCD trace anomaly, J. High Energy Phys. 12 (2018) 008.
  8. X. Ji and Y. Liu, Quantum anomalous energy effects on the nucleon mass, Sci. China Phys. Mech. Astron. 64, 281012 (2021).
  9. X.-D. Ji, A QCD analysis of the mass structure of the nucleon, Phys. Rev. Lett. 74, 1071 (1995).
  10. X.-D. Ji, Breakup of hadron masses and energy—momentum tensor of QCD, Phys. Rev. D 52, 271 (1995).
  11. C. Lorcé, On the hadron mass decomposition, Eur. Phys. J. C 78, 120 (2018).
  12. C. Lorcé, A. Metz, B. Pasquini, and S. Rodini, Energy-momentum tensor in QCD: Nucleon mass decomposition and mechanical equilibrium, J. High Energy Phys. 11 (2021) 121.
  13. M. V. Polyakov, Generalized parton distributions and strong forces inside nucleons and nuclei, Phys. Lett. B 555, 57 (2003).
  14. C. Lorcé, H. Moutarde, and A. P. Trawiński, Revisiting the mechanical properties of the nucleon, Eur. Phys. J. C 79, 89 (2019).
  15. M. V. Polyakov and P. Schweitzer, Forces inside hadrons: Pressure, surface tension, mechanical radius, and all that, Int. J. Mod. Phys. A 33, 1830025 (2018).
  16. V. D. Burkert, L. Elouadrhiri, and F. X. Girod, The pressure distribution inside the proton, Nature (London) 557, 396 (2018).
  17. V. D. Burkert, L. Elouadrhiri, F. X. Girod, C. Lorcé, P. Schweitzer, and P. E. Shanahan, Colloquium: Gravitational form factors of the proton, Rev. Mod. Phys. 95, 041002 (2023).
  18. A. Accardi et al., Electron Ion Collider: The next QCD frontier: Understanding the glue that binds us all, Eur. Phys. J. A 52, 268 (2016).
  19. R. Abdul Khalek et al., Snowmass 2021 white paper: Electron Ion Collider for high energy physics, arXiv:2203.13199.
  20. D. Boer et al., Gluons and the quark sea at high energies: Distributions, polarization, tomography, arXiv:1108.1713.
  21. D. Müller, D. Robaschik, B. Geyer, F. M. Dittes, and J. Hořejši, Wave functions, evolution equations and evolution kernels from light ray operators of QCD, Fortschr. Phys. 42, 101 (1994).
  22. D. Müller, D. Robaschik, B. Geyer, F. M. Dittes, and J. Hořejši, Wave functions, evolution equations and evolution kernels from light ray operators of QCD, Fortschr. Phys. 42, 101 (1994).
  23. X.-D. Ji, Gauge-invariant decomposition of nucleon spin, Phys. Rev. Lett. 78, 610 (1997).
  24. A. V. Radyushkin, Scaling limit of deeply virtual Compton scattering, Phys. Lett. B 380, 417 (1996).
  25. X.-D. Ji, Deeply virtual Compton scattering, Phys. Rev. D 55, 7114 (1997).
  26. J. E. Mandula, G. Zweig, and J. Govaerts, Representations of the rotation reflection symmetry group of the four-dimensional cubic lattice, Nucl. Phys. B228, 91 (1983).
  27. M. Gockeler, R. Horsley, E.-M. Ilgenfritz, H. Perlt, P. E. L. Rakow, G. Schierholz, and A. Schiller, Lattice operators for moments of the structure functions and their transformation under the hypercubic group, Phys. Rev. D 54, 5705 (1996).
  28. D. C. Hackett, P. R. Oare, D. A. Pefkou, and P. E. Shanahan, Gravitational form factors of the pion from lattice QCD, Phys. Rev. D 108, 114504 (2023).
  29. D. C. Hackett, D. A. Pefkou, and P. E. Shanahan, Gravitational form factors of the proton from lattice QCD, Phys. Rev. Lett. 132, 251904 (2024).
  30. D. Brommel, M. Diehl, M. Gockeler, P. Hagler, R. Horsley, D. Pleiter, P. E. L. Rakow, A. Schafer, G. Schierholz, and J. M. Zanotti, Structure of the pion from full lattice QCD, Proc. Sci. LAT2005 (2006) 360 [arXiv:hep-lat/0509133].
  31. P. E. Shanahan and W. Detmold, Gluon gravitational form factors of the nucleon and the pion from lattice QCD, Phys. Rev. D 99, 014511 (2019).
  32. D. A. Pefkou, D. C. Hackett, and P. E. Shanahan, Gluon gravitational structure of hadrons of different spin, Phys. Rev. D 105, 054509 (2022).
  33. M. Löffler, P. Wein, T. Wurm, S. Weishäupl, D. Jenkins, R. Rödl, A. Schäfer, and L. Walter (RQCD Collaboration), Mellin moments of spin dependent and independent PDFs of the pion and rho meson, Phys. Rev. D 105, 014505 (2022).
  34. C. Alexandrou et al., Moments of nucleon generalized parton distributions from lattice QCD simulations at physical pion mass, Phys. Rev. D 101, 034519 (2020).
  35. G. S. Bali, S. Collins, M. Göckeler, R. Rödl, A. Schäfer, and A. Sternbeck, Nucleon generalized form factors from two-flavor lattice QCD, Phys. Rev. D 100, 014507 (2019).
  36. S. Bhattacharya, K. Cichy, M. Constantinou, X. Gao, A. Metz, J. Miller, S. Mukherjee, P. Petreczky, F. Steffens, and Y. Zhao, Moments of axial-vector GPD from lattice QCD: Quark helicity, orbital angular momentum, and spin-orbit correlation, J. High Energy Phys. 01 (2025) 146.
  37. S. Bhattacharya, K. Cichy, M. Constantinou, X. Gao, A. Metz, J. Miller, S. Mukherjee, P. Petreczky, F. Steffens, and Y. Zhao, Moments of proton GPDs from the OPE of nonlocal quark bilinears up to NNLO, Phys. Rev. D 108, 014507 (2023).
  38. H. Dutrieux, R. G. Edwards, C. Egerer, J. Karpie, C. Monahan, K. Orginos, A. Radyushkin, D. Richards, E. Romero, and S. Zafeiropoulos (HadStruc Collaboration), Towards unpolarized GPDs from pseudo-distributions, J. High Energy Phys. 08 (2024) 162.
  39. B. Wang, F. He, G. Wang, T. Draper, J. Liang, K.-F. Liu, and Y.-B. Yang (χQCD Collaboration), Trace anomaly form factors from lattice QCD, Phys. Rev. D 109, 094504 (2024).
  40. W. Sun, Y. Chen, P. Sun, and Y.-B. Yang (χQCD Collaboration), Gluons in charmoniumlike states, Phys. Rev. D 103, 094503 (2021).
  41. Y.-B. Yang, Y. Chen, T. Draper, M. Gong, K.-F. Liu, Z. Liu, and J.-P. Ma, Meson mass decomposition from lattice QCD, Phys. Rev. D 91, 074516 (2015).
  42. Y.-B. Yang, J. Liang, Y.-J. Bi, Y. Chen, T. Draper, K.-F. Liu, and Z. Liu, Proton mass decomposition from the QCD energy momentum tensor, Phys. Rev. Lett. 121, 212001 (2018).
  43. F. He, P. Sun, and Y.-B. Yang (χQCD Collaboration), Demonstration of the hadron mass origin from the QCD trace anomaly, Phys. Rev. D 104, 074507 (2021).
  44. B. Hu, H. Du, X. Jiang, K.-F. Liu, P. Sun, and Y.-B. Yang, Unveiling the strong interaction origin of baryon masses with lattice QCD, arXiv:2411.18402.
  45. A. Metz, B. Pasquini, and S. Rodini, Revisiting the proton mass decomposition, Phys. Rev. D 102, 114042 (2020).
  46. R. Narayanan and H. Neuberger, Infinite N phase transitions in continuum Wilson loop operators, J. High Energy Phys. 03 (2006) 064.
  47. M. Luscher, Trivializing maps, the Wilson flow and the HMC algorithm, Commun. Math. Phys. 293, 899 (2010).
  48. M. Lüscher, Properties and uses of the Wilson flow in lattice QCD, J. High Energy Phys. 08 (2010) 071; 03 (2014) 092(E).
  49. M. Luscher and P. Weisz, Perturbative analysis of the gradient flow in non-Abelian gauge theories, J. High Energy Phys. 02 (2011) 051.
  50. H. Suzuki, Energy–momentum tensor from the Yang–Mills gradient flow, Prog. Theor. Exp. Phys. 2013, 083B03 (2013); 2015, 079201(E) (2015).
  51. H. Makino and H. Suzuki, Lattice energy–momentum tensor from the Yang–Mills gradient flow—inclusion of fermion fields, Prog. Theor. Exp. Phys. 2014, 063B02 (2014); 2015, 079202(E) (2015).
  52. Y. Taniguchi, S. Ejiri, R. Iwami, K. Kanaya, M. Kitazawa, H. Suzuki, T. Umeda, and N. Wakabayashi (WHOT-QCD Collaboration), Exploring Nf=2+1 QCD thermodynamics from the gradient flow, Phys. Rev. D 96, 014509 (2017); 99, 059904(E) (2019).
  53. Y. Taniguchi, S. Ejiri, K. Kanaya, M. Kitazawa, A. Suzuki, H. Suzuki, and T. Umeda (WHOT-QCD Collaboration), Energy-momentum tensor correlation function in Nf=2+1 full QCD at finite temperature, EPJ Web Conf. 175, 07013 (2018).
  54. A. Shindler, Moments of parton distribution functions of any order from lattice QCD, Phys. Rev. D 110, L051503 (2024).
  55. A. Francis, P. Fritzsch, R. V. Harlander, R. Karur, J. Kim, J. T. Kohnen, G. Pederiva, D. A. Pefkou, A. Rago, A. Shindler, A. Walker-Loud, and S. Zafeiropoulos, Gradient flow for parton distribution functions: First application to the pion, Phys. Rev. Lett. 136, 171903 (2026).
  56. A. Francis, P. Fritzsch, R. Karur, J. Kim, G. Pederiva, D. A. Pefkou, A. Rago, A. Shindler, A. Walker-Loud, and S. Zafeiropoulos, Moments of parton distributions functions of the pion from lattice QCD using gradient flow, Phys. Rev. D 113, 074520 (2026).
  57. F. J. Belinfante, Consequences of the postulate of a complete commuting set of observables in quantum electrodynamics, Phys. Rev. 128, 2832 (1962).
  58. A. V. Belitsky and A. V. Radyushkin, Unraveling hadron structure with generalized parton distributions, Phys. Rep. 418, 1 (2005).
  59. A. Freese, Noether’s theorems and the energy-momentum tensor in quantum gauge theories, Phys. Rev. D 106, 125012 (2022).
  60. X. Ji, Y. Liu, and A. Schäfer, Scale symmetry breaking, quantum anomalous energy and proton mass decomposition, Nucl. Phys. B971, 115537 (2021).
  61. R. V. Harlander, Y. Kluth, and F. Lange, The two-loop energy–momentum tensor within the gradient-flow formalism, Eur. Phys. J. C 78, 944 (2018); 79, 858(E) (2019).
  62. E. Follana, Q. Mason, C. Davies, K. Hornbostel, G. P. Lepage, J. Shigemitsu, H. Trottier, and K. Wong (HPQCD and UKQCD Collaborations), Highly improved staggered quarks on the lattice, with applications to charm physics, Phys. Rev. D 75, 054502 (2007).
  63. A. Bazavov et al. (HotQCD Collaboration), Equation of state in (2+1)-flavor QCD, Phys. Rev. D 90, 094503 (2014).
  64. A. Bazavov et al., Meson screening masses in (2+1)-flavor QCD, Phys. Rev. D 100, 094510 (2019).
  65. T. Izubuchi, L. Jin, C. Kallidonis, N. Karthik, S. Mukherjee, P. Petreczky, C. Shugert, and S. Syritsyn, Valence parton distribution function of pion from fine lattice, Phys. Rev. D 100, 034516 (2019).
  66. X. Gao, L. Jin, C. Kallidonis, N. Karthik, S. Mukherjee, P. Petreczky, C. Shugert, S. Syritsyn, and Y. Zhao, Valence parton distribution of the pion from lattice QCD: Approaching the continuum limit, Phys. Rev. D 102, 094513 (2020).
  67. X. Gao, N. Karthik, S. Mukherjee, P. Petreczky, S. Syritsyn, and Y. Zhao, Towards studying the structural differences between the pion and its radial excitation, Phys. Rev. D 103, 094510 (2021).
  68. X. Gao, N. Karthik, S. Mukherjee, P. Petreczky, S. Syritsyn, and Y. Zhao, Pion form factor and charge radius from lattice QCD at the physical point, Phys. Rev. D 104, 114515 (2021).
  69. X. Gao, A. D. Hanlon, S. Mukherjee, P. Petreczky, P. Scior, S. Syritsyn, and Y. Zhao, Lattice QCD determination of the Bjorken-x dependence of parton distribution functions at next-to-next-to-leading order, Phys. Rev. Lett. 128, 142003 (2022).
  70. X. Gao, A. D. Hanlon, N. Karthik, S. Mukherjee, P. Petreczky, P. Scior, S. Syritsyn, and Y. Zhao, Pion distribution amplitude at the physical point using the leading-twist expansion of the quasi-distribution-amplitude matrix element, Phys. Rev. D 106, 074505 (2022).
  71. X. Gao, A. D. Hanlon, N. Karthik, S. Mukherjee, P. Petreczky, P. Scior, S. Shi, S. Syritsyn, Y. Zhao, and K. Zhou, Continuum-extrapolated NNLO valence PDF of the pion at the physical point, Phys. Rev. D 106, 114510 (2022).
  72. X. Gao, A. D. Hanlon, J. Holligan, N. Karthik, S. Mukherjee, P. Petreczky, S. Syritsyn, and Y. Zhao, Unpolarized proton PDF at NNLO from lattice QCD with physical quark masses, Phys. Rev. D 107, 074509 (2023).
  73. X. Gao, W.-Y. Liu, and Y. Zhao, Parton distributions from boosted fields in the Coulomb gauge, Phys. Rev. D 109, 094506 (2024).
  74. X. Gao, A. D. Hanlon, S. Mukherjee, P. Petreczky, Q. Shi, S. Syritsyn, and Y. Zhao, Transversity PDFs of the proton from lattice QCD with physical quark masses, Phys. Rev. D 109, 054506 (2024).
  75. H.-T. Ding, X. Gao, A. D. Hanlon, S. Mukherjee, P. Petreczky, Q. Shi, S. Syritsyn, R. Zhang, and Y. Zhao, QCD predictions for meson electromagnetic form factors at high momenta: Testing factorization in exclusive processes, Phys. Rev. Lett. 133, 181902 (2024).
  76. I. Cloet, X. Gao, S. Mukherjee, S. Syritsyn, N. Karthik, P. Petreczky, R. Zhang, and Y. Zhao, Lattice QCD calculation of x-dependent meson distribution amplitudes at physical pion mass with threshold logarithm resummation, Phys. Rev. D 110, 114502 (2024).
  77. H.-T. Ding, X. Gao, S. Mukherjee, P. Petreczky, Q. Shi, S. Syritsyn, and Y. Zhao, Three-dimensional imaging of pion using lattice QCD: Generalized parton distributions, J. High Energy Phys. 02 (2025) 056.
  78. X. Gao, J. He, R. Zhang, and Y. Zhao, Systematic uncertainties from Gribov copies in lattice calculation of parton distributions in the Coulomb gauge, Chin. Phys. Lett. 41, 121201 (2024).
  79. D. Bollweg, X. Gao, J. He, S. Mukherjee, and Y. Zhao, Transverse-momentum-dependent pion structures from lattice QCD: Collins-Soper kernel, soft factor, TMDWF, and TMDPDF, Phys. Rev. D 112, 034501 (2025).
  80. X. Gao, S. Mukherjee, Q. Shi, F. Yao, and Y. Zhao, Skewness-dependent moments of pion GPD from nonlocal quark-bilinear correlators, Phys. Rev. D 113, 014505 (2026).
  81. A. Hasenfratz and F. Knechtli, Flavor symmetry and the static potential with hypercubic blocking, Phys. Rev. D 64, 034504 (2001).
  82. S. Navas et al. (Particle Data Group), Review of particle physics, Phys. Rev. D 110, 030001 (2024).
  83. A. Ramos and S. Sint, Symanzik improvement of the gradient flow in lattice gauge theories, Eur. Phys. J. C 76, 15 (2016).
  84. M. Luscher, Chiral symmetry and the Yang–Mills gradient flow, J. High Energy Phys. 04 (2013) 123.
  85. See Supplemental Material at http://link.aps.org/supplemental/10.1103/5n46-717z for details on the gradient-flow renormalization of the EMT, lattice setup, correlation-function analysis, continuum and zero-flow-time extrapolations, MS¯ matching, and checks of charm-quark disconnected contributions.
  86. E. Shintani, R. Arthur, T. Blum, T. Izubuchi, C. Jung, and C. Lehner, Covariant approximation averaging, Phys. Rev. D 91, 114511 (2015).
  87. M. V. Polyakov and B.-D. Sun, Gravitational form factors of a spin one particle, Phys. Rev. D 100, 036003 (2019).

  88. The sign difference of C¯(0) between ηc and J/ψ stems from different Minkowski and polarization-normalization conventions used in the standard Lorentz-covariant parametrization of EMT matrix elements for pseudoscalar [32] and vector [87] mesons.

  89. K. Tanaka, Three-loop formula for quark and gluon contributions to the QCD trace anomaly, J. High Energy Phys. 01 (2019) 120.
  90. K. Tanaka, Twist-four gravitational form factor at NNLO QCD from trace anomaly constraints, J. High Energy Phys. 03 (2023) 013.
  91. K. Tanaka, Proton mass decompositions in the NNLO QCD, arXiv:2511.18310.
  92. S. Rodini, A. Metz, and B. Pasquini, Mass sum rules of the electron in quantum electrodynamics, J. High Energy Phys. 09 (2020) 067.
  93. X. Ji, Proton mass decomposition: Naturalness and interpretations, Front. Phys. (Beijing) 16, 64601 (2021).
  94. P. Boyle, A. Yamaguchi, G. Cossu, and A. Portelli, grid: A next generation data parallel c++ QCD library, Proc. Sci. LATTICE2015 (2016) 023 [arXiv:1512.03487].
  95. C. Lehner et al., grid python toolkit (gpt), https://github.com/lehner/gpt.
  96. K. G. Chetyrkin, J. H. Kuhn, and M. Steinhauser, rundec: A Mathematica package for running and decoupling of the strong coupling and quark masses, Comput. Phys. Commun. 133, 43 (2000).

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Phys. Rev. Lett. 137, 061901– Published 7 August, 2026
  • Received 28 January 2026
  • Accepted 24 June 2026
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