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REVIEW 4 major objections 5 minor 76 references

Misconceptions About the Physics of the QCD Trace Anomaly from Renormalization in a Reducible Basis

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The QCD trace anomaly is scheme-independent and must be renormalized separately from the traceless part of the energy-momentum tensor; reducible-basis renormalization that mixes them generates an unphysical, scheme-dependent quark–gluon spl

desk verdict Useful unification of reducible-basis EMT schemes, but the central x=0 claim is underderived and the paper states one side of the debate rather than settling it. read the letter →

arxiv 2608.01002 v1 pith:PMC5ISUQ submitted 2026-08-02 hep-ph hep-latnucl-th

Chen Yang This is my paper
classification hep-phhep-latnucl-th
keywords QCDtraceanomalyenergy-momentumtensorrenormalizationreducibleoperatorbasisnucleonmasssumruleconfinementforceLorentzsymmetrydimensionalregularizationschemedependence
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that the QCD trace anomaly is a regulator-independent, scheme-independent, scale-invariant object that must be renormalized in isolation from the traceless part of the energy-momentum tensor, because the two belong to distinct irreducible representations of the Lorentz group. It claims that recently proposed reducible-basis renormalization schemes, which mix these parts, generate a spurious scheme-dependent parameter $x$ that redistributes the anomaly between “quark” and “gluon” contributions to nucleon mass. The only symmetry-allowed choice is $c_2=4/d$ with $x=0$, which assigns the whole anomaly to the gluon operator; even then the paper insists this assignment is a physical infrared statement, not a renormalization choice. Accepting this restores a clean decomposition of nucleon mass into quark kinetic energy, gluon classical energy, and a purely quantum anomaly energy that obeys the virial 3:1 ratio, and makes the anomaly a confining force. The debate matters because it decides whether separate quark and gluon contributions to nucleon mass are physically meaningful or artifacts of a renormalization scheme.

What carries the argument

The central object is the block-diagonal renormalization matrix of Eq. (3), where the traceless quark and gluon operators mix only with each other and the trace-anomaly operator has renormalization factor 1. The paper contrasts this with the reducible-basis matrix of Eq. (4), whose off-diagonal entries mix the trace and traceless parts; the scheme-dependent parameter $x$ in Eqs. (5)–(7) is the signature of that mixing. The identity $c_2=1+c_0(1-d/4)$ with the requirement $c_2=4/d$ is the mechanism that fixes $x=0$.

What would settle it

Compute the one-loop trace of the renormalized quark energy-momentum tensor in the chiral limit within the reducible basis and check whether the non-zero trace predicted by Eq. (5) changes any physical matrix element of the total energy-momentum tensor; alternatively, extract the separate quark and gluon contributions to the nucleon mass from lattice QCD and compare with the $x=0$ prediction. A non-zero trace that leaves all physical observables unchanged confirms the split is a scheme artifact; a measured split requiring $x\neq 0$ falsifies the paper’s conclusion.

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Extended reading notes

Core claim

The paper’s central claim is that the renormalization matrix of the QCD energy-momentum tensor has a block-diagonal structure: the traceless quark and gluon operators mix with each other, while the trace (anomaly) operator renormalizes independently. Reducible-basis renormalization violates this structure by mixing a $c_2 g_{\mu\nu} F^2$ term with the traceless gluon and quark operators, producing renormalized quark and gluon energy-momentum tensors that each contain a piece of the anomaly, with a scheme-dependent coefficient $x=(\alpha_s/4\pi)(N_f/3)(1-c_0)$. The paper shows that imposing Lorentz symmetry on the renormalized operators forces $c_2=4/d$ and $x=0$, so the apparent “quark vs gl

Load-bearing premise

The argument stands on the premise that the traceless and trace parts of the QCD energy-momentum tensor are distinct irreducible representations of the Lorentz group and therefore cannot mix under renormalization; if such mixing is allowed, the scheme-dependent split becomes a legitimate choice rather than an artifact.

Editorial extensions

If this is right

  • In the chiral limit the nucleon mass is the sum of three renormalized Hamiltonian components: quark kinetic energy, gluon classical energy, and trace-anomaly energy; the anomaly contributes $M/4$ and each traceless component contributes $3\langle x\rangle M/4$, with the 3:1 ratio fixed by Lorentz symmetry.
  • The trace anomaly appears in the Hamiltonian as an emergent infrared scale, contradicting claims that it resides only in the spatial part of the energy-momentum tensor; this reshapes how nucleon mass structure is analyzed.
  • Only the traceless parts of the quark and gluon energy-momentum tensors are connected to experimentally measurable PDF moments; the scheme-dependent anomaly split is not observable.
  • The color-Lorentz force on quarks gains a confining contribution from the trace anomaly, which can be extracted from off-forward matrix elements of the energy-momentum tensor.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the paper is right, lattice calculations that use a reducible basis and quote separate quark and gluon anomaly contributions should be re-matched to $x=0$ before comparison with phenomenology; otherwise the quoted split is not a physical observable.
  • The block-diagonal argument should extend to higher-rank tensor operators; if so, recent decompositions of spin-3 and other higher-twist energy-momentum components may face similar scheme artifacts and should be checked for Lorentz-block mixing.
  • The paper’s claim that the anomaly is scale-independent suggests a testable prediction: any scheme that introduces scale dependence into the anomaly part of a quark or gluon energy-momentum tensor should contaminate the nucleon mass sum rule, so a precision lattice extraction of the separate contributions at different scales could expose the artifact.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper argues that the recently introduced 'reducible-basis' renormalization of the QCD energy-momentum tensor (EMT) improperly mixes the trace anomaly with regular traceless operators, producing a scheme-dependent split of the anomaly into quark and gluon pieces. Section II introduces a unified reducible basis (Eq. (4)) and states that the resulting renormalized quark and gluon EMTs acquire trace terms proportional to a parameter x (Eqs. (5)–(6)), with the one-loop result x = (alpha_s/4pi) N_f/3 (1-c_0) (Eq. (7)). The paper concludes that Lorentz symmetry forces c_2 = 4/d and hence x = 0, so that the entire anomaly must be assigned to the gluon operator in its IR form. Section III applies this to the nucleon mass sum rule and to the color-Lorentz force, arguing that the traceless and trace parts contribute separately and that attempts to redistribute the anomaly are unphysical. The central technical step, Eq. (7), is quoted rather than derived, and the symmetry argument relies on the block-diagonal structure of Eq. (3), which is itself the point at issue.

Significance. If the paper's central claim were established, it would overturn a substantial body of recent work (MPR, HRT, and related papers) and would impose a unique convention for the quark/gluon decomposition of the trace anomaly. The paper usefully organizes the existing schemes through the parameter c_0, and it correctly emphasizes that the trace anomaly is an IR physical operator independent of UV regulator details. However, the manuscript does not provide a derivation of its key one-loop result, and its main symmetry argument is not sufficient to rule out finite renormalizations. The paper would be valuable as a sharp statement of one side of an active controversy, but in its present form the central conclusion is not demonstrated.

major comments (4)
  1. [Section II, Eq. (7)] The one-loop expression for x is the quantitative core of the paper, but it is introduced as 'By perturbative calculations or the same procedure as Ref. [34]' with no derivation. No Z-factors, no Feynman diagrams, and no details of the dimensional-regularization limit are given. Since Eq. (7) is used to identify the scheme dependence that the paper calls unphysical, this missing derivation is load-bearing. The author should either provide the computation or state explicitly where in the cited literature it appears and what assumptions are involved.
  2. [Section II, Eqs. (3) and (5)] The argument that 'Lorentz symmetry' forces c_2 = 4/d and x = 0 conflates Lorentz covariance with scheme choice. Even if the UV counterterm matrix is block diagonal in an irreducible basis, one can add a finite Lorentz-invariant counterterm -x/4 g^{mu nu} F_R^2 to the quark operator and +x/4 g^{mu nu} F_R^2 to the gluon operator. This operation does not break Lorentz symmetry; it merely redistributes a scalar operator between the two components. The paper never demonstrates that x != 0 violates a Ward identity or changes any measurable quantity. Without such a demonstration, the statement 'the only choice is c_2 = 4/d with associated x = 0' is an assertion of a convention, not a consequence of Lorentz symmetry.
  3. [Section II, paragraph after Eq. (9)] The paper claims the ambiguity in x is 'unphysical and unnecessary' and that the reducible-basis operators 'involve additional ill-defined ambiguities not accessible through experiments.' But it never identifies an observable — a nucleon matrix element, a form factor, or a Ward identity — that depends on x. The reducible-basis literature already concedes that the split is scheme dependent; the disagreement is only about whether this is a legitimate bookkeeping choice. If all physical quantities are x-independent, then the paper's conclusion that these schemes are 'misconceptions' is not supported. The author needs to exhibit a concrete physical consequence that is incorrectly predicted by the x != 0 choices.
  4. [Section II, Eq. (3)] The block-diagonal structure of Eq. (3) is assumed, not derived. The text states that the traceless and trace parts belong to different Lorentz irreps and therefore 'there is no physics reason that the renormalization of two parts shall be mixed.' But an operator basis that is reducible contains evanescent combinations such as (c_2 - 4/d) g^{mu nu} F^2, which vanish in four dimensions but mix with the trace anomaly through 1/epsilon poles in dimensional regularization. The paper's conclusion that mixing is forbidden is thus equivalent to the assumption that the split must be done in an irreducible basis. This is a circularity: the conclusion x = 0 is built into the premise of Eq. (3).
minor comments (5)
  1. [Abstract and Introduction] The abstract states that the anomaly is 'independent of the particular UV regulator used,' and the text correctly nuances this by saying the bare forms are evanescent and the physical IR result is regulator independent. The wording in the abstract could mislead; suggest rephrasing to distinguish the bare-operator form from the physical anomaly.
  2. [Section II, Eq. (4)] The notation for the quark operator, \bar\psi i\overleftrightarrow{D}_{(\mu}\gamma_{\nu)}\psi, is not consistently defined. The symmetrization and the index placement should be spelled out, especially because the trace part in d dimensions depends on the gamma-matrix convention.
  3. [Section II, 'To restore' paragraph] The sentence 'This apparently is a common pitfall in dimensional regularization in which the number of physical gluon polarization is d-2' is unclear. The relation between the factor 1/d in the definition of c_2 and the number of gluon polarizations needs elaboration, or the remark should be removed.
  4. [Section III A, Eq. (17)] Writing 1/4 g^{00} in the Hamiltonian is unconventional; since g^{00}=1 in the rest frame, the expression could be simplified. Also, the decomposition H = H_c + H_a relies on the earlier assumption that the trace part contributes only through \hat T^{00}; this should be stated explicitly.
  5. [Throughout] There are several typographical and grammatical errors, e.g., 'how thereducible operator basisrenormalization methods that fail' in the Introduction, 'these re-normalized quark and gluon operators' with a hyphen, and 'the trace anomaly is also free from scale evolutions' (should be 'evolution'). A careful editing pass is needed.

Circularity Check

1 steps flagged · score 6.0 of 10

The central 'only x=0' conclusion restates the no-mixing premise rather than deriving it from Lorentz symmetry.

  1. self definitional [Section II, after Eqs. (5)-(7)]
    "To restore the physical Lorentz symmetry, the only choice is c2=4/d with associated x=0. In other words, the only symmetry-allowed reducible basis choice is to associate gμν with 1/d factor in dimensional regularization and minimal subtraction because the trace produces a factor of d."

    In Eq. (5), x is defined as the coefficient of the g^{μν}(F^2)_R trace term in the renormalized quark EMT. Setting x=0 is exactly requiring the renormalized quark EMT to contain no trace part. The paper justifies this by asserting that the traceless and trace parts belong to different irreducible representations of the Lorentz group and therefore cannot mix (Eq. (3) and Introduction). But that is the very point at issue: a finite counterterm proportional to g^{μν}F^2 can be added to the quark operator and subtracted from the gluon operator without breaking Lorentz covariance; it only shifts the bookkeeping of a scalar operator. The argument that x≠0 'breaks Lorentz symmetry' is not derived from Lorentz covariance alone but from the prior assumption that mixing between the trace and tracele

full rationale

The paper's main critical claim—that reducible-basis renormalization introduces unphysical scheme dependence and that only x=0 with c2=4/d is symmetry-allowed—is built on the assumption that the traceless and trace parts of the QCD EMT are separate Lorentz irreps and hence cannot mix. The block-diagonal structure of Eq. (3) is asserted rather than derived; the 'Lorentz symmetry' argument in Section II then concludes x=0, which is equivalent to that asserted block-diagonal structure. A reader who allows finite mixing, as the cited reducible-basis papers do, will not find a derivation here that x must vanish; the paper only reiterates its premise. This is a genuine circular step in the central argument. However, the paper also makes several independent points that do not reduce to this premise: the four-loop perturbative mixing results, the lattice power-divergence concern, and the measurability argument for traceless operators. These give the paper independent content and keep the overall circularity from being total. The nucleon mass sum rule and Virial theorem are cited from earlier work and are external to this paper's own derivation. Overall, the central conclusion is partially circular but not wholly so, hence a score of 6.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The paper contributes no data fits. Its argument rests on the standard trace-anomaly formalism plus two contested premises (non-mixing of Lorentz irreps, and the IR-fixed physical decomposition). These premises are not derived; they are the substance of the dispute with the reducible-basis literature. The scheme label c0 is a hand-chosen parameter that organizes the criticized schemes.

free parameters (1)
  • c0 = arbitrary (scheme label)
    The paper introduces c0 to parametrize the reducible operator basis (c2 = 1 + c0(1-d/4)). The scheme dependence of x, x = alpha_s/(4 pi) N_f/3 (1-c0), is the paper's central demonstration, but c0 is chosen by hand to label schemes, not fitted to data.
assumptions (5)
  • domain assumption The trace anomaly operator (beta/2g)(F^2)_R is UV finite, regulator independent, scheme independent, and scale independent.
    Invoked in Section I and the abstract as the standard textbook result (Refs. [1-3,11-14]); it is the foundation of the critique.
  • ad hoc to paper The traceless and trace parts of the QCD EMT are separate irreducible representations of the Lorentz group and must not mix under renormalization.
    Used in Section II (Eq. 3 block-diagonal form) to force c2=4/d and x=0. This is exactly the premise the reducible-basis papers (Refs. [32-42]) dispute.
  • ad hoc to paper The physical quark/gluon split of the trace anomaly is fixed by the IR operators, gamma_m m bar-psi psi and (beta/2g)F^2, not by UV renormalization choices.
    Assumed in Section I and used to declare the decompositions in Eqs. (5)-(6) unphysical; it is the interpretive core of the paper.
  • domain assumption In the chiral limit the bare quark EMT is traceless, so a renormalized trace term is an artifact.
    Used in Section II to argue that the x-dependent trace in Eq. (5) violates Lorentz symmetry. Standard in the chiral limit, but the conclusion only holds if mixing is forbidden (axiom 2).
  • standard math The nucleon mass is the rest-frame Hamiltonian expectation value, and the Virial theorem relates traceless contributions to 3 times the anomaly.
    Used in Section III.A to derive Eqs. (14)-(16); standard results from Refs. [15,16,45-48].

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Pith. "Pith review of Misconceptions About the Physics of the QCD Trace Anomaly from Renormalization in a Reducible Basis." pith.science (2026). https://pith.science/paper/PMC5ISUQ

@misc{pith2026260801002,
  author       = {Pith},
  title        = {Pith review of: Misconceptions About the Physics of the QCD Trace Anomaly from Renormalization in a Reducible Basis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PMC5ISUQ}},
  note         = {Machine review of arXiv:2608.01002}
}
abstract

The QCD trace anomaly is a well-established textbook result in quantum field theory with several prominent features: (1) it arises from the quantum breaking of scale symmetry at ultraviolet (UV) scales, yet is independent of the particular UV regulator used, whether lattice or dimensional regularization; (2) although it is nominally proportional to (${\cal O}(\alpha_s)$), it is free of renormalization-scheme ambiguity; and (3) it is free of UV divergences and is therefore scale independent. Unfortunately, these important features have been undermined in the recently introduced reducible-basis renormalization, leading to misunderstandings of anomaly-related nucleon physics, including the origins of nucleon mass and internal forces.

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Works this paper leans on

76 extracted references · 30 canonical work pages

  1. [34]

    Hatta, A

    Y . Hatta, A. Rajan, and K. Tanaka, JHEP12, 008 (2018), arXiv:1810.05116 [hep-ph]

    arXiv 2018

  2. [1]

    M. E. Peskin and D. V . Schroeder,An Introduction to quantum field theory(Addison-Wesley, Reading, USA, 1995)

    1995

  3. [2]

    M. D. Schwartz,Quantum Field Theory and the Standard Model(Cambridge University Press, 2014)

    2014

  4. [3]

    J. C. Collins,Renormalization : An Introduction to Renormal- ization, the Renormalization Group and the Operator-Product Expansion, Cambridge Monographs on Mathematical Physics, V ol. 26 (Cambridge University Press, Cambridge, 1984)

    work page 1984

  5. [4]

    C. G. Callan, Jr., S. R. Coleman, and R. Jackiw, Annals Phys. 59, 42 (1970)

    work page 1970

  6. [5]

    S. R. Coleman and R. Jackiw, Annals Phys.67, 552 (1971)

    work page 1971

  7. [6]

    ’t Hooft and M

    G. ’t Hooft and M. J. G. Veltman, Nucl. Phys. B44, 189 (1972)

    1972

  8. [7]

    C. G. Bollini and J. J. Giambiagi, Nuovo Cim. B12, 20 (1972)

    1972

Show all 76 references
  1. [8]

    H. J. Rothe,Lattice Gauge Theories : An Introduction (Fourth Edition), V ol. 43 (World Scientific Publishing Company, 2012)

    2012

  2. [9]

    Noether, Nachrichten von der Gesellschaft der Wis- senschaften zu G ¨ottingen, Mathematisch-Physikalische Klasse 1918, 235 (1918)

    E. Noether, Nachrichten von der Gesellschaft der Wis- senschaften zu G ¨ottingen, Mathematisch-Physikalische Klasse 1918, 235 (1918)

    1918

  3. [10]

    D. N. Blaschke, F. Gieres, M. Reboud, and M. Schweda, Nucl. Phys. B912, 192 (2016), arXiv:1605.01121 [hep-th]

    2016 arXiv

  4. [11]

    S. D. Joglekar and B. W. Lee, Annals Phys.97, 160 (1976)

    1976

  5. [12]

    S. L. Adler, J. C. Collins, and A. Duncan, Phys. Rev. D15, 1712 (1977)

    1977

  6. [13]

    J. C. Collins, A. Duncan, and S. D. Joglekar, Phys. Rev. D16, 438 (1977)

    1977

  7. [14]

    N. K. Nielsen, Nucl. Phys. B120, 212 (1977)

    1977

  8. [15]

    Ji, Phys

    X.-D. Ji, Phys. Rev. Lett.74, 1071 (1995), arXiv:hep- ph/9410274

    1995

  9. [16]

    Ji, Phys

    X.-D. Ji, Phys. Rev. D52, 271 (1995), arXiv:hep-ph/9502213

    1995 arXiv

  10. [17]

    Caracciolo, G

    S. Caracciolo, G. Curci, P. Menotti, and A. Pelissetto, Annals Phys.197, 119 (1990)

    1990

  11. [18]

    Caracciolo, P

    S. Caracciolo, P. Menotti, and A. Pelissetto, Nucl. Phys. B375, 195 (1992)

    1992

  12. [19]

    Jackiw and K

    R. Jackiw and K. Johnson, Phys. Rev.182, 1459 (1969)

    1969

  13. [20]

    S. L. Adler and W. A. Bardeen, Phys. Rev.182, 1517 (1969)

    1969

  14. [21]

    J. S. Schwinger, Phys. Rev.128, 2425 (1962)

    1962

  15. [22]

    Fujikawa, Phys

    K. Fujikawa, Phys. Rev. Lett.42, 1195 (1979)

    1979

  16. [23]

    Fujikawa, Phys

    K. Fujikawa, Phys. Rev. D29, 285 (1984)

    1984

  17. [24]

    K. G. Wilson, Phys. Rev. D10, 2445 (1974)

    1974

  18. [25]

    L. H. Karsten and J. Smit, Nucl. Phys. B183, 103 (1981)

    1981

  19. [26]

    Kawai, R

    H. Kawai, R. Nakayama, and K. Seo, Nucl. Phys. B189, 40 (1981)

    1981

  20. [27]

    H. S. Sharatchandra, H. J. Thun, and P. Weisz, Nucl. Phys. B 192, 205 (1981)

    1981

  21. [28]

    P. H. Ginsparg and K. G. Wilson, Phys. Rev. D25, 2649 (1982)

    1982

  22. [29]

    Kerler, Phys

    W. Kerler, Phys. Rev. D23, 2384 (1981)

    1981

  23. [30]

    Seiler and I

    E. Seiler and I. O. Stamatescu, Phys. Rev. D25, 2177 (1982), [Erratum: Phys.Rev.D 26, 534 (1982)]

    1982

  24. [31]

    Fujikawa, Z

    K. Fujikawa, Z. Phys. C25, 179 (1984)

    1984

  25. [32]

    Makino and H

    H. Makino and H. Suzuki, PTEP2014, 063B02 (2014), [Erra- tum: PTEP 2015, 079202 (2015)], arXiv:1403.4772 [hep-lat]

    2014 arXiv

  26. [33]

    R. V . Harlander, Y . Kluth, and F. Lange, Eur. Phys. J. C78, 944 (2018), [Erratum: Eur.Phys.J.C 79, 858 (2019)], arXiv:1808.09837 [hep-lat]

    2018 arXiv

  27. [35]

    Tanaka, JHEP01, 120 (2019), arXiv:1811.07879 [hep-ph]

    K. Tanaka, JHEP01, 120 (2019), arXiv:1811.07879 [hep-ph]

    2019 arXiv

  28. [36]

    Lorc ´e, Eur

    C. Lorc ´e, Eur. Phys. J. C78, 120 (2018), arXiv:1706.05853 [hep-ph]

    2018 arXiv

  29. [37]

    Lorc ´e, Springer Proc

    C. Lorc ´e, Springer Proc. Phys.238, 635 (2020), arXiv:1811.02803 [hep-ph]

    2020 arXiv

  30. [38]

    Ahmed, L

    T. Ahmed, L. Chen, and M. Czakon, JHEP01, 077 (2023), arXiv:2208.01441 [hep-ph]

    2023 arXiv

  31. [39]

    Rodini, A

    S. Rodini, A. Metz, and B. Pasquini, JHEP09, 067 (2020), arXiv:2004.03704 [hep-ph]

    2020

  32. [40]

    A. Metz, B. Pasquini, and S. Rodini, Phys. Rev. D102, 114042 (2020), arXiv:2006.11171 [hep-ph]

    2020 arXiv

  33. [41]

    Lorc ´e, A

    C. Lorc ´e, A. Metz, B. Pasquini, and S. Rodini, JHEP11, 121 (2021), arXiv:2109.11785 [hep-ph]

    2021 arXiv

  34. [42]

    Tanaka, (2025), arXiv:2511.18310 [hep-ph]

    K. Tanaka, (2025), arXiv:2511.18310 [hep-ph]

    2025

  35. [43]

    Bollweg, H.-T

    D. Bollweg, H.-T. Ding, X. Gao, R. Luo, and S. Mukherjee, (2026), arXiv:2601.13070 [hep-lat]. 5

    2026

  36. [44]

    S. Moch, B. Ruijl, T. Ueda, J. A. M. Vermaseren, and A. V ogt, Phys. Lett. B825, 136853 (2022), arXiv:2111.15561 [hep-ph]

    2022 arXiv

  37. [45]

    Ji and Y

    X. Ji and Y . Liu, Sci. China Phys. Mech. Astron.64, 281012 (2021), arXiv:2101.04483 [hep-ph]

    2021 arXiv

  38. [46]

    Ji, Front

    X. Ji, Front. Phys. (Beijing)16, 64601 (2021), arXiv:2102.07830 [hep-ph]

    2021 arXiv

  39. [47]

    X. Ji, Y . Liu, and A. Sch ¨afer, Nucl. Phys. B971, 115537 (2021), arXiv:2105.03974 [hep-ph]

    2021 arXiv

  40. [48]

    H. J. Rothe, Phys. Lett. B364, 227 (1995)

    1995

  41. [49]

    Ji and C

    X. Ji and C. Yang, Research9, 1155 (2026), arXiv:2503.01991 [hep-ph]

    2026

  42. [50]

    Ji and C

    X. Ji and C. Yang, Nucl. Phys. B1024, 117342 (2026), arXiv:2508.16727 [hep-ph]

    2026

  43. [51]

    X. Ji, G. A. Miller, and C. Yang, (2026), arXiv:2605.00339 [hep-ph]

    2026 arXiv

  44. [52]

    Einstein, Annalen der physik18, 639 (1905)

    A. Einstein, Annalen der physik18, 639 (1905)

    1905

  45. [53]

    Houet al., Phys

    T.-J. Houet al., Phys. Rev. D103, 014013 (2021), arXiv:1912.10053 [hep-ph]

    2021 arXiv

  46. [54]

    Abdel-Rehim, C

    A. Abdel-Rehim, C. Alexandrou, M. Constantinou, K. Had- jiyiannakou, K. Jansen, C. Kallidonis, G. Koutsou, and A. Va- quero Aviles-Casco (ETM), Phys. Rev. Lett.116, 252001 (2016), arXiv:1601.01624 [hep-lat]

    2016 arXiv

  47. [55]

    Alexandrou, M

    C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, K. Jansen, C. Kallidonis, G. Koutsou, A. Vaquero Avil ´es- Casco, and C. Wiese, Phys. Rev. Lett.119, 142002 (2017), arXiv:1706.02973 [hep-lat]

    2017 arXiv

  48. [56]

    Y .-B. Yang, J. Liang, Y .-J. Bi, Y . Chen, T. Draper, K.- F. Liu, and Z. Liu, Phys. Rev. Lett.121, 212001 (2018), arXiv:1808.08677 [hep-lat]

    2018 arXiv

  49. [57]

    Alexandrou, S

    C. Alexandrou, S. Bacchio, M. Constantinou, J. Finkenrath, K. Hadjiyiannakou, K. Jansen, G. Koutsou, H. Panagopou- los, and G. Spanoudes, Phys. Rev. D101, 094513 (2020), arXiv:2003.08486 [hep-lat]

    2020 arXiv

  50. [58]

    D. C. Hackett, D. A. Pefkou, and P. E. Shanahan, Phys. Rev. Lett.132, 251904 (2024), arXiv:2310.08484 [hep-lat]

    2024 arXiv

  51. [59]

    Liu, Phys

    K.-F. Liu, Phys. Rev. D104, 076010 (2021), arXiv:2103.15768 [hep-ph]

    2021 arXiv

  52. [60]

    M. V . Polyakov and H.-D. Son, JHEP09, 156 (2018), arXiv:1808.00155 [hep-ph]

    2018 arXiv

  53. [61]

    Won, H.-C

    H.-Y . Won, H.-C. Kim, and J.-Y . Kim, JHEP05, 173 (2024), arXiv:2310.04670 [hep-ph]

    2024 arXiv

  54. [62]

    Freese, Phys

    A. Freese, Phys. Rev. D111, 034047 (2025), arXiv:2412.09664 [hep-ph]

    2025 arXiv

  55. [63]

    R. G. Sachs, Phys. Rev.126, 2256 (1962)

    1962

  56. [64]

    M. V . Polyakov and P. Schweitzer, Int. J. Mod. Phys. A33, 1830025 (2018), arXiv:1805.06596 [hep-ph]

    2018 arXiv

  57. [65]

    R. L. Jaffe, Phys. Rev. D103, 016017 (2021), arXiv:2010.15887 [hep-ph]

    2021 arXiv

  58. [66]

    P. A. M. Dirac, Rev. Mod. Phys.21, 392 (1949)

    1949

  59. [67]

    Weinberg, Phys

    S. Weinberg, Phys. Rev.150, 1313 (1966)

    1966

  60. [68]

    J. B. Kogut and D. E. Soper, Phys. Rev. D1, 2901 (1970)

    1970

  61. [69]

    Burkardt, Int

    M. Burkardt, Int. J. Mod. Phys. A18, 173 (2003), arXiv:hep- ph/0207047

    2003

  62. [70]

    G. A. Miller, Phys. Rev. Lett.99, 112001 (2007), arXiv:0705.2409 [nucl-th]

    2007 arXiv

  63. [71]

    G. A. Miller, Phys. Rev. C112, 045204 (2025), arXiv:2507.14388 [hep-ph]

    2025 arXiv

  64. [72]

    Liu, Phys

    K.-F. Liu, Phys. Lett. B849, 138418 (2024), arXiv:2302.11600 [hep-ph]

    2024 arXiv

  65. [73]

    Liu, (2026), arXiv:2605.04163 [hep-ph]

    K.-F. Liu, (2026), arXiv:2605.04163 [hep-ph]

    2026 arXiv

  66. [74]

    Tanaka, D

    M. Tanaka, D. Fujii, and M. Kawaguchi, Phys. Rev. D112, 054048 (2025), arXiv:2507.21220 [hep-ph]

    2025 arXiv

  67. [75]

    Fujii and M

    D. Fujii and M. Tanaka, Phys. Lett. B870, 139872 (2025), arXiv:2507.23786 [hep-ph]

    2025

  68. [76]

    Fujii, A

    D. Fujii, A. Iwanaka, and M. Tanaka, Phys. Rev. D112, 094051 (2025), arXiv:2507.18690 [hep-ph]

    2025

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Reviewed August 6, 2026 · model on record in the stance chip above.