Energy-momentum tensor in QCD: nucleon mass decomposition and mechanical equilibrium. (English) Zbl 1521.81451
Summary: We review and examine in detail recent developments regarding the question of the nucleon mass decomposition. We discuss in particular the virial theorem in quantum field theory and its implications for the nucleon mass decomposition and mechanical equilibrium. We reconsider the renormalization of the QCD energy-momentum tensor in minimal-subtraction-type schemes and the physical interpretation of its components, as well as the role played by the trace anomaly and Poincaré symmetry. We also study the concept of “quantum anomalous energy” proposed in some works as a new contribution to the nucleon mass. Examining the various arguments, we conclude that the quantum anomalous energy is not a genuine contribution to the mass sum rule, as a consequence of translation symmetry.
MSC:
| 81V05 | Strong interaction, including quantum chromodynamics |
| 81T15 | Perturbative methods of renormalization applied to problems in quantum field theory |
| 81T50 | Anomalies in quantum field theory |
| 81T25 | Quantum field theory on lattices |
| 81V35 | Nuclear physics |
| 81T17 | Renormalization group methods applied to problems in quantum field theory |
Keywords:
perturbative QCD; spacetime symmetries; anomalies in field and string theories; renormalization regularization and renormalonsReferences:
| [1] | M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Remarks on Higgs Boson Interactions with Nucleons, Phys. Lett. B78 (1978) 443 [INSPIRE]. |
| [2] | J.F. Donoghue, On the contents of the nucleon, in Physics with Light Mesons, (1987) [INSPIRE]. |
| [3] | X.-D. Ji, A QCD analysis of the mass structure of the nucleon, Phys. Rev. Lett.74 (1995) 1071 [hep-ph/9410274] [INSPIRE]. |
| [4] | X.-D. Ji, Breakup of hadron masses and energy-momentum tensor of QCD, Phys. Rev. D52 (1995) 271 [hep-ph/9502213] [INSPIRE]. |
| [5] | Lorcé, C., On the hadron mass decomposition, Eur. Phys. J. C, 78, 120 (2018) |
| [6] | Rodini, S.; Metz, A.; Pasquini, B., Mass sum rules of the electron in quantum electrodynamics, JHEP, 09, 067 (2020) |
| [7] | A. Metz, B. Pasquini and S. Rodini, Revisiting the proton mass decomposition, Phys. Rev. D102 (2020) 114042 [arXiv:2006.11171] [INSPIRE]. · Zbl 07414500 |
| [8] | Ji, X., Proton mass decomposition: naturalness and interpretations, Front. Phys. (Beijing), 16, 64601 (2021) |
| [9] | X. Ji and Y. Liu, Quantum anomalous energy effects on the nucleon mass, Sci. China Phys. Mech. Astron.64 (2021) 281012 [arXiv:2101.04483] [INSPIRE]. |
| [10] | X. Ji, Y. Liu and A. Schäfer, Scale symmetry breaking, quantum anomalous energy and proton mass decomposition, Nucl. Phys. B971 (2021) 115537 [arXiv:2105.03974] [INSPIRE]. · Zbl 07408774 |
| [11] | E.A. Dudas and D. Pirjol, A Virial theorem in quantum field theory, Phys. Lett. B260 (1991) 186 [INSPIRE]. |
| [12] | Gaite, J., The relativistic virial theorem and scale invariance, Phys. Usp., 56, 919 (2013) |
| [13] | D. Davies, Cardy’s Conjecture and the Spectrum of Infrared Strongly-Coupled Quantum Field Theories, arXiv:2005.07209 [INSPIRE]. |
| [14] | M.V. Polyakov, Generalized parton distributions and strong forces inside nucleons and nuclei, Phys. Lett. B555 (2003) 57 [hep-ph/0210165] [INSPIRE]. · Zbl 1008.81514 |
| [15] | Polyakov, MV; Schweitzer, P., Forces inside hadrons: pressure, surface tension, mechanical radius, and all that, Int. J. Mod. Phys. A, 33, 1830025 (2018) |
| [16] | Lorcé, C.; Moutarde, H.; Trawiński, AP, Revisiting the mechanical properties of the nucleon, Eur. Phys. J. C, 79, 89 (2019) |
| [17] | Hatta, Y.; Rajan, A.; Tanaka, K., Quark and gluon contributions to the QCD trace anomaly, JHEP, 12, 008 (2018) · Zbl 1405.81144 |
| [18] | Tanaka, K., Three-loop formula for quark and gluon contributions to the QCD trace anomaly, JHEP, 01, 120 (2019) · Zbl 1409.81167 |
| [19] | Callan, CG Jr; Coleman, SR; Jackiw, R., A New improved energy-momentum tensor, Annals Phys., 59, 42 (1970) · Zbl 1092.83502 |
| [20] | S.R. Coleman and R. Jackiw, Why dilatation generators do not generate dilatations?, Annals Phys.67 (1971) 552 [INSPIRE]. |
| [21] | F. Villars, On the Energy-Momentum Tensor of the Electron, Phys. Rev.79 (1950) 122 [INSPIRE]. · Zbl 0038.40705 |
| [22] | Rohrlich, F., The Self-Stress of the Electron, Phys. Rev., 77, 357 (1950) · Zbl 0035.27401 |
| [23] | Jukawa, J.; Umezawa, H., On the Problem of Covariance in Quantum Electrodynamics, I, Prog. Theor. Phys., 6, 112 (1951) · Zbl 0045.13405 |
| [24] | Takahashi, Y.; Umezawa, H., On the Problem of Covariance in Quantum Electrodynamics, I, Prog. Theor. Phys., 8, 193 (1952) · Zbl 0047.44902 |
| [25] | E.B. Manoukian, Selfstress and Renormalization Group, Phys. Rev. D12 (1975) 1199 [INSPIRE]. |
| [26] | T. Kashiwa, Trace identity and selfstress in quantum electrodynamics, Prog. Theor. Phys.62 (1979) 250 [INSPIRE]. |
| [27] | Shafranov, VA, Plasma Equilibrium in a Magnetic Field, Rev. Plasma Phys., 2, 103 (1966) |
| [28] | R.M. Kulsrud, MHD description of plasma, Basic Plasma Physics I, North Holland (1983). |
| [29] | L.D. Faddeev, L. Freyhult, A.J. Niemi and P. Rajan, Shafranov’s virial theorem and magnetic plasma confinement, J. Phys. A35 (2002) L133 [physics/0009061] [INSPIRE]. · Zbl 1031.76061 |
| [30] | M. Laue, Zur Dynamik der Relativitätstheorie, Annalen Phys.340 (1911) 524 [INSPIRE]. · JFM 42.0725.02 |
| [31] | A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn and V.F. Weisskopf, A New Extended Model of Hadrons, Phys. Rev. D9 (1974) 3471 [INSPIRE]. |
| [32] | A. Chodos, R.L. Jaffe, K. Johnson and C.B. Thorn, Baryon Structure in the Bag Theory, Phys. Rev. D10 (1974) 2599 [INSPIRE]. |
| [33] | I. Zahed and G.E. Brown, The Skyrme Model, Phys. Rept.142 (1986) 1 [INSPIRE]. |
| [34] | C. Cebulla, K. Goeke, J. Ossmann and P. Schweitzer, The Nucleon form-factors of the energy momentum tensor in the Skyrme model, Nucl. Phys. A794 (2007) 87 [hep-ph/0703025] [INSPIRE]. |
| [35] | L. Landau and E. Lifshitz, The Classical Theory of Fields, Addison-Wesley Press (1951). · Zbl 0043.19803 |
| [36] | S. Chandrasekhar and E. Fermi, Problems of Gravitational Stability in the Presence of a Magnetic Field, Astrophys. J.118 (1953) 116 [Erratum ibid.122 (1955) 208] [INSPIRE]. |
| [37] | Parker, EN, Tensor Virial Equations, Phys. Rev., 96, 1686 (1954) · Zbl 0057.23208 |
| [38] | I.Y. Kobzarev and L.B. Okun, Gravitational interaction of fermions, Zh. Eksp. Teor. Fiz.43 (1962) 1904 [INSPIRE]. · Zbl 0124.45204 |
| [39] | H. Pagels, Energy-Momentum Structure Form Factors of Particles, Phys. Rev.144 (1966) 1250 [INSPIRE]. |
| [40] | K. Nishijima and A.H. Singh, Normalization of Bethe-Salpeter Amplitudes, Phys. Rev.162 (1967) 1740 [INSPIRE]. |
| [41] | S. Cotogno, C. Lorcé, P. Lowdon and M. Morales, Covariant multipole expansion of local currents for massive states of any spin, Phys. Rev. D101 (2020) 056016 [arXiv:1912.08749] [INSPIRE]. |
| [42] | X.-D. Ji, Gauge-Invariant Decomposition of Nucleon Spin, Phys. Rev. Lett.78 (1997) 610 [hep-ph/9603249] [INSPIRE]. |
| [43] | Lorcé, C., The relativistic center of mass in field theory with spin, Eur. Phys. J. C, 78, 785 (2018) |
| [44] | Lorcé, C., Relativistic spin sum rules and the role of the pivot, Eur. Phys. J. C, 81, 413 (2021) |
| [45] | Lorcé, C.; Mantovani, L.; Pasquini, B., Spatial distribution of angular momentum inside the nucleon, Phys. Lett. B, 776, 38 (2018) · Zbl 1380.81486 |
| [46] | A. Freese and G.A. Miller, Forces within hadrons on the light front, Phys. Rev. D103 (2021) 094023 [arXiv:2102.01683] [INSPIRE]. |
| [47] | C. Lorcé, Charge Distributions of Moving Nucleons, Phys. Rev. Lett.125 (2020) 232002 [arXiv:2007.05318] [INSPIRE]. · Zbl 1437.81171 |
| [48] | X.-D. Ji, Lorentz symmetry and the internal structure of the nucleon, Phys. Rev. D58 (1998) 056003 [hep-ph/9710290] [INSPIRE]. |
| [49] | C. Eckart, The Thermodynamics of irreversible processes. 3. Relativistic theory of the simple fluid, Phys. Rev.58 (1940) 919 [INSPIRE]. · Zbl 0027.03107 |
| [50] | P.G. Bergmann ed., Introduction to the Theory of Relativity, Dover, New York (1976). |
| [51] | L. Smarr and J.W. York Jr., Kinematical conditions in the construction of space-time, Phys. Rev. D17 (1978) 2529 [INSPIRE]. |
| [52] | A. Pais and S.T. Epstein, Note on Relativistic Properties of Self-Energies, Rev. Mod. Phys.21 (1949) 445 [INSPIRE]. · Zbl 0035.13301 |
| [53] | K.A. Milton, S.A. Fulling, P. Parashar, P. Kalauni and T. Murphy, Stress tensor for a scalar field in a spatially varying background potential: Divergences, “renormalization”, anomalies, and Casimir forces, Phys. Rev. D93 (2016) 085017 [arXiv:1602.00916] [INSPIRE]. |
| [54] | V.D. Burkert, L. Elouadrhiri and F.X. Girod, The pressure distribution inside the proton, Nature557 (2018) 396 [INSPIRE]. |
| [55] | A. Shayit, S.A. Fulling, T.E. Settlemyre and J. Merritt, Vacuum energy density and pressure inside a soft wall, arXiv:2107.10439 [INSPIRE]. |
| [56] | Meyer, HB, Finite temperature sum rules in lattice gauge theory, Nucl. Phys. B, 795, 230 (2008) · Zbl 1219.81195 |
| [57] | M. Cheng et al., The QCD equation of state with almost physical quark masses, Phys. Rev. D77 (2008) 014511 [arXiv:0710.0354] [INSPIRE]. |
| [58] | A. Bazavov et al., Equation of state and QCD transition at finite temperature, Phys. Rev. D80 (2009) 014504 [arXiv:0903.4379] [INSPIRE]. |
| [59] | P.E. Shanahan and W. Detmold, Pressure Distribution and Shear Forces inside the Proton, Phys. Rev. Lett.122 (2019) 072003 [arXiv:1810.07589] [INSPIRE]. |
| [60] | Yanagihara, R.; Iritani, T.; Kitazawa, M.; Asakawa, M.; Hatsuda, T., Distribution of Stress Tensor around Static Quark-Anti-Quark from Yang-Mills Gradient Flow, Phys. Lett. B, 789, 210 (2019) · Zbl 1406.81068 |
| [61] | R. Yanagihara and M. Kitazawa, A study of stress-tensor distribution around the flux tube in the Abelian-Higgs model, PTEP2019 (2019) 093B02 [Erratum ibid.2020 (2020) 079201] [arXiv:1905.10056] [INSPIRE]. · Zbl 1528.81188 |
| [62] | R. Yanagihara, M. Kitazawa, M. Asakawa and T. Hatsuda, Distribution of Energy-Momentum Tensor around a Static Quark in the Deconfined Phase of SU(3) Yang-Mills Theory, Phys. Rev. D102 (2020) 114522 [arXiv:2010.13465] [INSPIRE]. · Zbl 1406.81068 |
| [63] | Synge, JL, Relativity: The General theory (1960), Amsterdam: North-Holland, Amsterdam · Zbl 0090.18504 |
| [64] | RQCD collaboration, Direct determinations of the nucleon and pion σ terms at nearly physical quark masses, Phys. Rev. D93 (2016) 094504 [arXiv:1603.00827] [INSPIRE]. |
| [65] | Y.-B. Yang et al., Proton Mass Decomposition from the QCD Energy Momentum Tensor, Phys. Rev. Lett.121 (2018) 212001 [arXiv:1808.08677] [INSPIRE]. |
| [66] | K.-F. Liu, Proton mass decomposition and hadron cosmological constant, Phys. Rev. D104 (2021) 076010 [arXiv:2103.15768] [INSPIRE]. |
| [67] | P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Graduate Texts in Contemporary Physics, Springer-Verlag, New York (1997) [DOI] [INSPIRE]. · Zbl 0869.53052 |
| [68] | D. Kharzeev, Quarkonium interactions in QCD, Proc. Int. Sch. Phys. Fermi130 (1996) 105 [nucl-th/9601029] [INSPIRE]. |
| [69] | Y. Hatta, A. Rajan and D.-L. Yang, Near threshold J/ψ and Υ photoproduction at JLab and RHIC, Phys. Rev. D100 (2019) 014032 [arXiv:1906.00894] [INSPIRE]. |
| [70] | C.D. Roberts, D.G. Richards, T. Horn and L. Chang, Insights into the emergence of mass from studies of pion and kaon structure, Prog. Part. Nucl. Phys.120 (2021) 103883 [arXiv:2102.01765] [INSPIRE]. |
| [71] | Pauli, W., Relativitätstheorie (1921), Berlin: Teubner, Berlin · JFM 48.0977.01 |
| [72] | Staruszkiewicz, A., Relativistic transformation laws for thermodynamical variables with application to the classical electron theory, Nuovo Cim. A, 45, 684 (1956) |
| [73] | C. Möller, Thermodynamics in the Special and the General Theory of Relativity, Bernardini Festschrift, G. Puppi, Academic Press, Berlin (1968). |
| [74] | R.C. Tolman, On the Use of the Energy-Momentum Principle in General Relativity, Phys. Rev.35 (1930) 875 [INSPIRE]. · JFM 56.0743.05 |
| [75] | Tolman, RC, Relativity, thermodynamics, and cosmology (1934), Toronto: Oxford University Press, Toronto · Zbl 0009.41304 |
| [76] | M. Burkardt, Impact parameter space interpretation for generalized parton distributions, Int. J. Mod. Phys. A18 (2003) 173 [hep-ph/0207047] [INSPIRE]. · Zbl 1035.81597 |
| [77] | Z. Abidin and C.E. Carlson, Hadronic Momentum Densities in the Transverse Plane, Phys. Rev. D78 (2008) 071502 [arXiv:0808.3097] [INSPIRE]. |
| [78] | X. Ji, Fundamental Properties of the Proton in Light-Front Zero Modes, Nucl. Phys.B (2020) 115181 [arXiv:2003.04478] [INSPIRE]. · Zbl 1472.81312 |
| [79] | Gnendiger, C., To d, or not to d: recent developments and comparisons of regularization schemes, Eur. Phys. J. C, 77, 471 (2017) |
| [80] | K.G. Wilson, Quantum field theory models in less than four-dimensions, Phys. Rev. D7 (1973) 2911 [INSPIRE]. |
| [81] | J.C. Collins, Renormalization: An Introduction to Renormalization, The Renormalization Group, and the Operator Product Expansion, vol. 26 of Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge (1986) [DOI] [INSPIRE]. · Zbl 1094.53505 |
| [82] | Breitenlohner, P.; Maison, D., Dimensional Renormalization and the Action Principle, Commun. Math. Phys., 52, 11 (1977) |
| [83] | N.K. Nielsen, The Energy Momentum Tensor in a Nonabelian Quark Gluon Theory, Nucl. Phys. B120 (1977) 212 [INSPIRE]. |
| [84] | H. Suzuki, Energy-momentum tensor from the Yang-Mills gradient flow, PTEP2013 (2013) 083B03 [Erratum ibid.2015 (2015) 079201] [arXiv:1304.0533] [INSPIRE]. · Zbl 07406705 |
| [85] | H. Makino and H. Suzuki, Lattice energy-momentum tensor from the Yang-Mills gradient flow — inclusion of fermion fields, PTEP2014 (2014) 063B02 [Erratum ibid.2015 (2015) 079202] [arXiv:1403.4772] [INSPIRE]. · Zbl 1331.81199 |
| [86] | J.C. Collins, A. Duncan and S.D. Joglekar, Trace and Dilatation Anomalies in Gauge Theories, Phys. Rev. D16 (1977) 438 [INSPIRE]. |
| [87] | R. Tarrach, The renormalization of FF, Nucl. Phys. B196 (1982) 45 [INSPIRE]. |
| [88] | G. Bonneau, Zimmermann identities and renormalization group equation in dimensional renormalization, Nucl. Phys. B167 (1980) 261 [INSPIRE]. |
| [89] | G. Bonneau, Trace and Axial Anomalies in Dimensional Renormalization Through Zimmermann Like Identities, Nucl. Phys. B171 (1980) 477 [INSPIRE]. |
| [90] | S. Caracciolo, A. Montanari and A. Pelissetto, Operator product expansion on the lattice: A Numerical test in the two-dimensional nonlinear sigma model, JHEP09 (2000) 045 [hep-lat/0007044] [INSPIRE]. · Zbl 0990.81086 |
| [91] | J. Collins, Foundations of perturbative QCD, vol. 32, Cambridge University Press (2013) [INSPIRE]. |
| [92] | W.A. Bardeen, A.J. Buras, D.W. Duke and T. Muta, Deep Inelastic Scattering Beyond the Leading Order in Asymptotically Free Gauge Theories, Phys. Rev. D18 (1978) 3998 [INSPIRE]. |
| [93] | J.C. Collins, Normal Products in Dimensional Regularization, Nucl. Phys. B92 (1975) 477 [INSPIRE]. |
| [94] | K.G. Wilson, Anomalous dimensions and the breakdown of scale invariance in perturbation theory, Phys. Rev. D2 (1970) 1478 [INSPIRE]. |
| [95] | Peskin, ME; Schroeder, DV, An Introduction to quantum field theory (1995), Reading, U.S.A.: Addison-Wesley, Reading, U.S.A. |
| [96] | M.E. Luke, A.V. Manohar and M.J. Savage, A QCD Calculation of the interaction of quarkonium with nuclei, Phys. Lett. B288 (1992) 355 [hep-ph/9204219] [INSPIRE]. |
| [97] | M.B. Voloshin and V.I. Zakharov, Measuring QCD Anomalies in Hadronic Transitions Between Onium States, Phys. Rev. Lett.45 (1980) 688 [INSPIRE]. |
| [98] | Novikov, VA; Shifman, MA, Comment on the ψ′ → J/ψππ Decay, Z. Phys. C, 8, 43 (1981) |
| [99] | V.A. Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Are All Hadrons Alike? , Nucl. Phys. B191 (1981) 301 [INSPIRE]. |
| [100] | M.A. Shifman, Anomalies and Low-Energy Theorems of Quantum Chromodynamics, Sov. Phys. Usp.32 (1989) 289 [INSPIRE]. |
| [101] | D.E. Kharzeev, Mass radius of the proton, Phys. Rev. D104 (2021) 054015 [arXiv:2102.00110] [INSPIRE]. |
| [102] | H.J. Rothe, A Novel look at the Michael lattice sum rules, Phys. Lett. B355 (1995) 260 [hep-lat/9504012] [INSPIRE]. |
| [103] | H.J. Rothe, Lattice energy sum rule and the trace anomaly, Phys. Lett. B364 (1995) 227 [hep-lat/9508005] [INSPIRE]. |
| [104] | K.G. Wilson, Confinement of Quarks, Phys. Rev. D10 (1974) 2445 [INSPIRE]. |
| [105] | C. Michael, Lattice action sum rules, Nucl. Phys. B280 (1987) 13 [INSPIRE]. |
| [106] | C. Michael, Lattice sum rules for the color fields, Phys. Rev. D53 (1996) 4102 [hep-lat/9504016] [INSPIRE]. |
| [107] | F. Karsch, SU(N) Gauge Theory Couplings on Asymmetric Lattices, Nucl. Phys. B205 (1982) 285 [INSPIRE]. |
| [108] | J. Engels, F. Karsch and K. Redlich, Scaling properties of the energy density in SU(2) lattice gauge theory, Nucl. Phys. B435 (1995) 295 [hep-lat/9408009] [INSPIRE]. |
| [109] | J. Engels, F. Karsch and T. Scheideler, Determination of anisotropy coefficients for SU(3) gauge actions from the integral and matching methods, Nucl. Phys. B564 (2000) 303 [hep-lat/9905002] [INSPIRE]. |
| [110] | H.B. Meyer, A Calculation of the shear viscosity in SU(3) gluodynamics, Phys. Rev. D76 (2007) 101701 [arXiv:0704.1801] [INSPIRE]. |
| [111] | S. Caracciolo, P. Menotti and A. Pelissetto, One loop analytic computation of the energy momentum tensor for lattice gauge theories, Nucl. Phys. B375 (1992) 195 [INSPIRE]. · Zbl 0875.53033 |
| [112] | S. Capitani and G. Rossi, Deep inelastic scattering in improved lattice QCD. 1. The First moment of structure functions, Nucl. Phys. B433 (1995) 351 [hep-lat/9401014] [INSPIRE]. |
| [113] | M. Gockeler et al., Lattice operators for moments of the structure functions and their transformation under the hypercubic group, Phys. Rev. D54 (1996) 5705 [hep-lat/9602029] [INSPIRE]. |
| [114] | G.S. Bali, K. Schilling and C. Schlichter, Observing long color flux tubes in SU(2) lattice gauge theory, Phys. Rev. D51 (1995) 5165 [hep-lat/9409005] [INSPIRE]. |
| [115] | M. Gockeler et al., Perturbative renormalization of bilinear quark and gluon operators, Nucl. Phys. B Proc. Suppl.53 (1997) 896 [hep-lat/9608033] [INSPIRE]. |
| [116] | S. Caracciolo, G. Curci, P. Menotti and A. Pelissetto, The Energy Momentum Tensor on the Lattice: The Scalar Case, Nucl. Phys. B309 (1988) 612 [INSPIRE]. · Zbl 0959.81576 |
| [117] | S. Caracciolo, G. Curci, P. Menotti and A. Pelissetto, Renormalization of the Energy Momentum Tensor and the Trace Anomaly in Lattice QED, Phys. Lett. B228 (1989) 375 [INSPIRE]. · Zbl 0959.81576 |
| [118] | S. Caracciolo, G. Curci, P. Menotti and A. Pelissetto, The Energy Momentum Tensor for Lattice Gauge Theories, Annals Phys.197 (1990) 119 [INSPIRE]. · Zbl 0875.53033 |
| [119] | S. Caracciolo, P. Menotti and A. Pelissetto, Analytic determination at one loop of the energy momentum tensor for lattice QCD, Phys. Lett. B260 (1991) 401 [INSPIRE]. |
| [120] | A. Buonanno, G. Cella and G. Curci, Lattice energy-momentum tensor with Symanzik improved actions, Phys. Rev. D51 (1995) 4494 [hep-lat/9506008] [INSPIRE]. |
| [121] | Del Debbio, L.; Patella, A.; Rago, A., Space-time symmetries and the Yang-Mills gradient flow, JHEP, 11, 212 (2013) · Zbl 1342.81271 |
| [122] | L. Giusti and M. Pepe, Energy-momentum tensor on the lattice: Nonperturbative renormalization in Yang-Mills theory, Phys. Rev. D91 (2015) 114504 [arXiv:1503.07042] [INSPIRE]. |
| [123] | R.V. Harlander, Y. Kluth and F. Lange, The two-loop energy-momentum tensor within the gradient-flow formalism, Eur. Phys. J. C78 (2018) 944 [Erratum ibid.79 (2019) 858] [arXiv:1808.09837] [INSPIRE]. |
| [124] | M. Dalla Brida, L. Giusti and M. Pepe, Non-perturbative definition of the QCD energy-momentum tensor on the lattice, JHEP04 (2020) 043 [arXiv:2002.06897] [INSPIRE]. · Zbl 1436.81100 |
| [125] | G. Panagopoulos, H. Panagopoulos and G. Spanoudes, Two-loop renormalization and mixing of gluon and quark energy-momentum tensor operators, Phys. Rev. D103 (2021) 014515 [arXiv:2010.02062] [INSPIRE]. · Zbl 07408698 |
| [126] | H. Suzuki and H. Takaura, t → 0 extrapolation function in the small flow time expansion method for the energy-momentum tensor, PTEP2021 (2021) 073B02 [arXiv:2102.02174] [INSPIRE]. · Zbl 1499.81079 |
| [127] | Y.-B. Yang et al., Meson Mass Decomposition from Lattice QCD, Phys. Rev. D91 (2015) 074516 [arXiv:1405.4440] [INSPIRE]. |
| [128] | χQCD collaboration, Gluons in charmoniumlike states, Phys. Rev. D103 (2021) 094503 [arXiv:2012.06228] [INSPIRE]. |
| [129] | χQCD collaboration, Demonstration of the hadron mass origin from the QCD trace anomaly, Phys. Rev. D104 (2021) 074507 [arXiv:2101.04942] [INSPIRE]. |
| [130] | H. Hata, On the ‘Trace Anomaly From Path Integral Measure’, Prog. Theor. Phys.65 (1981) 1052 [INSPIRE]. |
| [131] | N.N. Bogolyubov and D.V. Shirkov, Introduction to the theory of quantized fields, vol. 3, John Wiley & Sons (1959). · Zbl 0088.21701 |
| [132] | G. ’t Hooft, Renormalization of Massless Yang-Mills Fields, Nucl. Phys. B33 (1971) 173 [INSPIRE]. |
| [133] | Bélusca-Maïto, H.; Ilakovac, A.; Mađor-Božinović, M.; Stöckinger, D., Dimensional regularization and Breitenlohner-Maison/’t Hooft-Veltman scheme for γ_5applied to chiral YM theories: full one-loop counterterm and RGE structure, JHEP, 08, 024 (2020) · Zbl 1454.83123 |
| [134] | D.B. Zacrep and B.L. Young, Trace and Ward-Takahashi Identity Anomalies in an SU(3) Current Model with Energy Momentum Tensor, Phys. Rev. D12 (1975) 513 [INSPIRE]. |
| [135] | A.Y. Morozov, Anomalies in Gauge Theories, Sov. Phys. Usp.29 (1986) 993 [INSPIRE]. |
| [136] | Z.-E. Meziani and S. Joosten, Origin of the Proton Mass? Heavy Quarkonium Production at Threshold from Jefferson Lab to an Electron Ion Collider, in Probing Nucleons and Nuclei in High Energy Collisions: Dedicated to the Physics of the Electron Ion Collider, pp. 234-237 (2020) [DOI] [INSPIRE]. |
| [137] | Y. Hatta and D.-L. Yang, Holographic J/ψ production near threshold and the proton mass problem, Phys. Rev. D98 (2018) 074003 [arXiv:1808.02163] [INSPIRE]. |
| [138] | GlueX collaboration, First Measurement of Near-Threshold J/ψ Exclusive Photoproduction off the Proton, Phys. Rev. Lett.123 (2019) 072001 [arXiv:1905.10811] [INSPIRE]. |
| [139] | K.A. Mamo and I. Zahed, Diffractive photoproduction of J/ψ and Υ using holographic QCD: gravitational form factors and GPD of gluons in the proton, Phys. Rev. D101 (2020) 086003 [arXiv:1910.04707] [INSPIRE]. |
| [140] | Wang, R.; Evslin, J.; Chen, X., The origin of proton mass from J/Ψ photo-production data, Eur. Phys. J. C, 80, 507 (2020) |
| [141] | R. Boussarie and Y. Hatta, QCD analysis of near-threshold quarkonium leptoproduction at large photon virtualities, Phys. Rev. D101 (2020) 114004 [arXiv:2004.12715] [INSPIRE]. |
| [142] | O. Gryniuk, S. Joosten, Z.-E. Meziani and M. Vanderhaeghen, Υ photoproduction on the proton at the Electron-Ion Collider, Phys. Rev. D102 (2020) 014016 [arXiv:2005.09293] [INSPIRE]. |
| [143] | R. Wang, W. Kou, C. Han, J. Evslin and X. Chen, Proton and deuteron mass radii from near-threshold ϕ-meson photoproduction, Phys. Rev. D104 (2021) 074033 [arXiv:2108.03550] [INSPIRE]. |
| [144] | Zeng, F.; Wang, X-Y; Zhang, L.; Xie, Y-P; Wang, R.; Chen, X., Near-threshold photoproduction of J/ψ in two-gluon exchange model, Eur. Phys. J. C, 80, 1027 (2020) |
| [145] | Wang, R.; Kou, W.; Xie, Y-P; Chen, X., Extraction of the proton mass radius from the vector meson photoproductions near thresholds, Phys. Rev. D, 103, L091501 (2021) |
| [146] | P. Sun, X.-B. Tong and F. Yuan, Perturbative QCD analysis of near threshold heavy quarkonium photoproduction at large momentum transfer, Phys. Lett. B822 (2021) 136655 [arXiv:2103.12047] [INSPIRE]. |
| [147] | Y.-P. Xie and V.P. Goncalves, Near threshold heavy vector meson photoproduction at LHC and EicC, arXiv:2103.12568 [INSPIRE]. |
| [148] | Clausius, R., On a mechanical theorem applicable to heat, Phil. Mag., 40, 122 (1870) |
| [149] | W. Lucha and F.F. Schoberl, The Relativistic Virial Theorem, Phys. Rev. Lett.64 (1990) 2733 [INSPIRE]. · Zbl 1050.81545 |
| [150] | W. Lucha and F.F. Schoberl, Relativistic virial theorems, Mod. Phys. Lett. A5 (1990) 2473 [INSPIRE]. · Zbl 1050.81545 |
| [151] | Schectman, RM; Good, RH, Generalizations of the Virial Theorem, Am. J. Phys., 25, 219 (1957) · Zbl 0077.44708 |
| [152] | M. Born, W. Heisenberg and P. Jordan, Zur Quantenmechanik. II, Z. Phys.35 (1926) 557 [INSPIRE]. · JFM 52.0963.01 |
| [153] | Finkelstein, BN, Über den Virialsatz in der Wellenmechanik, Z. Phys., 50, 293 (1928) · JFM 54.0978.07 |
| [154] | Fock, V., Bemerkung zum Virialsatz, Z. Phys., 63, 855 (1930) · JFM 56.1327.05 |
| [155] | Killingbeck, J., Methods of Proof and Applications of the Virial Theorem in Classical and Quantum Mechanics, Am. J. Phys., 38, 590 (1970) |
| [156] | Crawford, F., Footnote to the quantum mechanical virial theorem, Am. J. Phys., 57, 555 (1989) |
| [157] | Puff, RD; Gillis, NS, Fluctuations and transport properties of many-particle systems, Annals Phys., 46, 364 (1968) |
| [158] | Gersch, HA, Another derivation of the quantum virial theorem, Am. J. Phys, 47, 555 (1979) |
| [159] | Kleban, P., Virial theorem and scale transformations, Am. J. Phys, 47, 883 (1979) |
| [160] | J. Rafelski, Virial Theorem and Stability of Localized Solutions of Relativistic Classical Interacting Fields, Phys. Rev. D16 (1977) 1890 [INSPIRE]. |
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