Proton mass decomposition and hadron cosmological constant

Keh Fei Liu

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11 Scopus citations


Lattice results on sigma terms and global analysis of parton momentum fractions are used to give the quark and glue fractions of the proton mass and rest energy. The mass decomposition in terms of the trace of the energy-momentum tensor is renormalization group invariant. The decomposition of the rest energy from the Hamiltonian and the gravitational form factors are scheme and scale dependent. The separation of the energy-momentum tensor into the traceless part, which is composed of the quark and glue parton momentum fractions, and the trace part has the minimum scheme dependence. We identify the glue part of the trace anomaly as the vacuum energy from the glue condensate in the vacuum. From the metric term of the gravitational form factors, which is the stress part of the stress-energy-momentum tensor, we find that the trace part of the rest energy, dominated by , gives a constant restoring pressure that balances that from the traceless part of the Hamiltonian to confine the hadron, much like the cosmological constant Einstein introduced for a static universe. From a lattice calculation of in the charmonium, we deduce the associated string tension, which turns out to be in good agreement with that from a Cornell potential, which fits the charmonium spectrum.

Original languageEnglish
Article number076010
JournalPhysical Review D
Issue number7
StatePublished - Oct 1 2021

Bibliographical note

Funding Information:
U.S. Department of Energy Oak Ridge National Laboratory U.S. Department of Energy National Science Foundation National Energy Research Scientific Computing Center

Funding Information:
The author is indebted to X. Ji, C. Lorcé, Y. Hatta, D. Kharzeev, A. Metz, B. Pasquini, S. Rodini, M. Constantinou, M. Polyakov, P. Schweitzer, and Y. B. Yang for fruitful discussions. He also thanks T. J. Hou for providing the CT18 data and G. Wang for help with the figures. This work is partially supported by the U.S. DOE Grants No. DE-SC0013065 and No. DE-AC05-06OR23177, which is within the framework of the TMD Topical Collaboration. This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725. This work used Stampede time under the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation Grant No. ACI-1053575. We also thank the National Energy Research Scientific Computing Center (NERSC) for providing HPC resources that have contributed to the research results reported within this paper. We acknowledge the facilities of the USQCD Collaboration used for this research in part, which are funded by the Office of Science of the U.S. Department of Energy.

Publisher Copyright:
© 2021 Published by the American Physical Society

ASJC Scopus subject areas

  • Nuclear and High Energy Physics


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