The Mass Gap Approach to QCD. II. The non-perturbative renormalization program for the massive gluon fields

G.G. Barnafoldi; V. Gogokhia

arxiv: 2605.21547 · v1 · pith:JAVHOSV4new · submitted 2026-05-20 · ✦ hep-ph · hep-th

The Mass Gap Approach to QCD. II. The non-perturbative renormalization program for the massive gluon fields

V. Gogokhia , G.G. Barnafoldi This is my paper

Pith reviewed 2026-05-22 01:02 UTC · model grok-4.3

classification ✦ hep-ph hep-th

keywords QCDmass gapgluon propagatorconfinementnon-perturbative renormalizationdynamical massgauge structure

0 comments

The pith

A non-perturbative renormalization program for massive gluon fields confines them by denying mass-shell propagation in the mass gap approach to QCD.

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

The paper develops a non-perturbative multiplicative renormalization program for massive gluon fields inside the mass gap approach to QCD. It rests on fresh insights into the ground state dynamical and gauge structures that permit gluons to acquire mass dynamically. The full gluon propagator is worked out in detail, with its asymptotic forms, perturbation theory limit, and mass-shell peculiarities used to resolve the inconsistency of the canonical gauge. This structure ensures massive gluons cannot exist as mass-shell objects and therefore cannot enter the physical spectrum, while still allowing them inside the vacuum or hadrons. Euclidean expressions are supplied for lattice work, and the massive solution is shown to recover the free massless propagator when the gluon pole mass is set exactly to zero.

Core claim

Within the mass gap approach to QCD the non-perturbative renormalization of the massive gluon fields produces a full propagator whose mass-shell structure excludes on-shell propagation; massive gluons therefore remain confined to the vacuum and to the interior of hadrons, the canonical gauge inconsistency is removed, and the propagator reduces correctly to the free massless case when the pole mass vanishes.

What carries the argument

The full massive gluon propagator, whose mass-shell structure and asymptotic properties implement dynamical mass generation while enforcing off-shell confinement.

If this is right

Massive gluons cannot appear as physical particles in the spectrum.
Massive gluons are permitted only inside the vacuum or within hadrons.
The canonical gauge inconsistency of QCD is removed by the mass-shell analysis.
The massive solution limits exactly to the free massless propagator when the defined pole mass is set to zero.
Euclidean-metric expressions are available for immediate use in lattice simulations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same off-shell restriction could shape the effective forces inside hadrons by letting massive gluons mediate binding without free propagation.
Lattice runs using the supplied Euclidean forms could directly map the propagator and test the claimed absence of physical poles.
The approach may be compared with other non-perturbative methods such as Dyson-Schwinger equations to check consistency of the massive propagator.
If the ground-state structures generalize, similar renormalization programs could be written for other gauge theories exhibiting dynamical mass.

Load-bearing premise

The entire renormalization program takes previously formulated insights into the ground state dynamical and gauge structures of the mass gap approach to QCD as its fixed starting point.

What would settle it

A lattice computation of the gluon propagator in Euclidean space that directly searches for a mass-shell pole in the massive case would confirm or refute whether massive gluons remain strictly off-shell.

read the original abstract

We present a non-perturbative multiplicative renormalization program for the massive gluon fields. This has been done within the previously formulated the mass gap approach to QCD. It is based on a new insights into its ground state true dynamical and gauge structures. Our approach makes it possible for gluons to acquire mass dynamically. The corresponding full gluon propagator has been investigated in full details. Its asymptotic properties have been analysed, including the perturbation theory limit. The peculiarities of the mass-shell structure of the full massive gluon propagator has been discussed. The inconsistency of the canonical gauge in QCD is fixed. Our approach does not allow the massive gluons to be the mass-shell objects. This prevents them to appear in the physical spectrum (confinement of massive gluon states). The massive gluons may exist in the vacuum or inside hadrons only. Expressions in Euclidean metric for the lattice simulations are also present. We have also shown that the massive solution has a correct limit to the free massless gluon propagator, when the exactly defined gluon pole mass is to be formally put zero.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This extends the authors' mass-gap QCD series with a renormalization scheme for massive gluons and lattice expressions, but the no-mass-shell confinement claim rests entirely on un-rederived prior ground-state assumptions.

read the letter

The punchline is that this paper adds a multiplicative non-perturbative renormalization program for the massive gluon propagator inside their existing mass-gap framework, plus Euclidean expressions aimed at lattice work and a check that the massive solution limits correctly to the free massless propagator when the pole mass is set to zero. That limit and the lattice-ready forms are the concrete pieces a reader can actually use without buying the whole prior program. The propagator asymptotics and the discussion of mass-shell peculiarities are laid out in more detail than the abstract alone suggests. The claim that massive gluons cannot appear as mass-shell objects (hence confined) follows directly from the renormalization once the ground-state dynamical and gauge structures are accepted. Those structures are taken from the authors' earlier papers rather than rederived here, so the confinement statement inherits whatever gaps or choices exist in that foundation. The gluon pole mass appears as a free parameter that is formally set to zero for the massless limit, which keeps the construction from being fully parameter-free. This is narrow-gauge work for people already tracking the mass-gap approach or non-perturbative gluon propagators in QCD. A reader outside that circle would need the preceding papers to judge whether the renormalization actually closes the loop or just rearranges prior choices. The topic is central enough and the lattice expressions are specific enough that it should go to referees rather than a desk reject; the referees will have to decide whether the chain of assumptions from part I holds. I would not cite it in my own work unless I were already using their mass-gap setup, but I would send it out for review.

Referee Report

1 major / 2 minor

Summary. The manuscript presents a non-perturbative multiplicative renormalization program for massive gluon fields within the authors' mass-gap approach to QCD. It relies on new insights into the ground-state dynamical and gauge structures to derive the full gluon propagator, analyze its asymptotic properties (including the perturbative limit), discuss mass-shell peculiarities, and conclude that massive gluons cannot appear as mass-shell objects and are therefore confined to the vacuum or hadrons. The work also claims to resolve an inconsistency in the canonical gauge, supplies Euclidean expressions for lattice simulations, and demonstrates that the massive solution reduces to the free massless gluon propagator when the gluon pole mass is formally set to zero.

Significance. If the central construction holds, the result would supply a dynamical mechanism for gluon mass generation and confinement via the absence of physical poles in the propagator, while maintaining consistency with the massless limit of QCD. The provision of lattice-ready Euclidean expressions and the explicit massless reduction are concrete strengths that could facilitate numerical tests.

major comments (1)

[Abstract and introductory discussion of the renormalization program] The mass-shell analysis and the claim that massive gluons 'do not allow ... to be the mass-shell objects' (abstract) rest on the ground-state structures taken from prior mass-gap papers without rederivation or independent verification in the present manuscript. Because the renormalization program and confinement conclusion are presented as direct consequences of these structures, the absence of an explicit check here makes the central claim dependent on unexamined assumptions from the referenced earlier work.

minor comments (2)

[Abstract] The abstract contains minor grammatical issues ('the previously formulated the mass gap approach'; 'new insights' should be checked for number agreement; 'are also present' should be 'are also provided').
[Section introducing the propagator and renormalization constants] Notation for the gluon pole mass and the precise definition of the multiplicative renormalization constants should be introduced with explicit equations early in the text to avoid ambiguity when the massless limit is taken.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the single major comment below and describe the revisions we intend to make.

read point-by-point responses

Referee: The mass-shell analysis and the claim that massive gluons 'do not allow ... to be the mass-shell objects' (abstract) rest on the ground-state structures taken from prior mass-gap papers without rederivation or independent verification in the present manuscript. Because the renormalization program and confinement conclusion are presented as direct consequences of these structures, the absence of an explicit check here makes the central claim dependent on unexamined assumptions from the referenced earlier work.

Authors: This is Part II of a series on the mass-gap approach to QCD. The ground-state dynamical and gauge structures that determine the mass-shell properties of the gluon propagator were derived in full in Part I. The present work takes those structures as established input and develops the non-perturbative multiplicative renormalization program, constructs the complete propagator, examines its asymptotic and mass-shell behavior, supplies Euclidean expressions for lattice use, and demonstrates the correct massless limit. All steps that are new to this paper are carried out explicitly and are referenced to the prior derivations. To make the logical dependence clearer for readers who have not yet consulted Part I, we will insert a concise summary of the relevant ground-state features and their direct implications for the propagator in the revised introduction. revision: partial

Circularity Check

1 steps flagged

Central claim on gluon confinement via absent mass-shell poles rests on un-rederived prior ground-state structures

specific steps

self citation load bearing [Abstract]
"This has been done within the previously formulated the mass gap approach to QCD. It is based on a new insights into its ground state true dynamical and gauge structures. ... Our approach does not allow the massive gluons to be the mass-shell objects. This prevents them to appear in the physical spectrum (confinement of massive gluon states)."

The renormalization program and mass-shell analysis leading to confinement are based on insights from the authors' prior mass gap approach without independent derivation here. The key results on absent physical poles thus reduce to the validity of the self-cited prior ground-state structures.

full rationale

The paper's non-perturbative renormalization of the massive gluon propagator and the conclusion that massive gluons cannot be mass-shell objects (hence confined) are presented as direct consequences of 'new insights into its ground state true dynamical and gauge structures' taken from previous mass-gap work by the same authors. No independent derivation or explicit check of those structures appears in the current manuscript; the propagator analysis and mass-shell discussion therefore inherit whatever assumptions or gaps exist in the foundational papers. This is the single load-bearing point: if those structures do not hold, the claimed absence of physical poles and the confinement statement do not follow.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The program depends on the validity of the prior mass-gap approach and on the assumption that dynamical mass generation occurs through the ground-state structure; no independent evidence or machine-checked derivation is supplied in the available text.

free parameters (1)

gluon pole mass
The exactly defined gluon pole mass is introduced and can be formally set to zero to recover the massless propagator; its value is not derived from first principles in the abstract.

axioms (1)

domain assumption Ground state possesses true dynamical and gauge structures that permit dynamical mass acquisition for gluons
Invoked as the foundation for the entire non-perturbative renormalization program.

invented entities (1)

massive gluon fields no independent evidence
purpose: To implement dynamical mass and confinement within the renormalization scheme
Postulated as the central objects whose propagator is renormalized; no independent falsifiable signature outside the framework is given.

pith-pipeline@v0.9.0 · 5720 in / 1391 out tokens · 37626 ms · 2026-05-22T01:02:47.586126+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Our approach does not allow the massive gluons to be the mass-shell objects. This prevents them to appear in the physical spectrum (confinement of massive gluon states).
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the tadpole term … has been already replaced by M² … responsible for the first confinement phase transition

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages · 1 internal anchor

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The Mass Gap Approach to QCD.I. The true gauge and dynamical structures of its ground state

V. Gogokhia, G.G. Barnafoldi, The Mass Gap Approach to QCD. I. The true gauge and dynamical structures of its ground state, arXiv:2301.04561v5

work page internal anchor Pith review Pith/arXiv arXiv
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G.A. Korn, T.M. Korn, Mathematical Handbook (McGraw-Hill Book Company, 1968)

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Politzer, Reliable perturbative results for strong interactions?, Phys

H.D. Politzer, Reliable perturbative results for strong interactions?, Phys. Rev. Lett., 30 (1973) 134

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A. Cucchieri, D. Dudal, T. Mendes, N. Vandersickel, Modelling the Gluon Propagator in Landau Gauge: Lattice Estimates of pole masses and Damensional-Two Condensates, Phys. Rev. D, 85 (2012) 094513

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A. Jakovác, G.G. Barnaf´’oldi, P. Pósfay, Effect of quantum fluctuations in the high-energy cold nuclear equation of state and in compact star observations, Phys. Rev. C, 97 (2018) 025803. [25]Quark Matter 2022, Proc. of the XXIV Inter. Conf. on Ultra-Relativistic Nucleus Collisions, ed. by P. Braun-Munziger, B. Friman, J. Stachel, Nucl. Phys. A, 931 (2014) 1. 20

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V. Gogokhia, M. Vasuth, The non-perturbative analytical equation of state for the gluon matter, J. Phys. G: Nucl. Part. Phys., 37 (2010) 075015

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D. Zwanziger, Action from Gribov horizon, Nucl. Phys. B, 321 (1989) 591

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Huber, Nonperturbative properties of Yang-Mills theories, Phys

M.Q. Huber, Nonperturbative properties of Yang-Mills theories, Phys. Rep., 879 (2020) 1-92

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M. Baker, J.S. Ball, F. Zachariasen, Dual QCD: a review, Phys. Rep., 209 (1991) 73

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[1] [1]

The Mass Gap Approach to QCD.I. The true gauge and dynamical structures of its ground state

V. Gogokhia, G.G. Barnafoldi, The Mass Gap Approach to QCD. I. The true gauge and dynamical structures of its ground state, arXiv:2301.04561v5

work page internal anchor Pith review Pith/arXiv arXiv

[2] [2]

Marciano, H

W. Marciano, H. Pagels, Quantum Chromodynamics, Phys. Rep. C, 36 (1978) 137

work page 1978

[3] [3]

J. C. Taylor, Ward identities and charge of the Yang-Mills field, Nucl. Phys. B, 33 (1971) 436

work page 1971

[4] [4]

A. A. Slavnov, Ward identities in Gauge theories, Theor. Math. Phys., 10 (1972) 153

work page 1972

[5] [5]

Jaffe, E

A. Jaffe, E. Witten, Yang–Mills Existence and Mass Gap, http://www.claymath.org/prize−problems/, http: //www.arthurjaf f e.com

work page

[6] [6]

Itzykson, J.-B

C. Itzykson, J.-B. Zuber, Quantum Field Theory (McGraw-Hill Book Company, 1984)

work page 1984

[7] [7]

’t Hooft, Renormalizable Lagrangians for Massive Yang-Mills Fields, Nucl

G. ’t Hooft, Renormalizable Lagrangians for Massive Yang-Mills Fields, Nucl. Phys. B, 35 (1971) 167

work page 1971

[8] [8]

Lee, C.N

T.D. Lee, C.N. Yang, Theory of Charged Vector Mesons Interacting with the Electromagnetic Field, Phys. Rev., 128 (1962) 885

work page 1962

[9] [9]

Fujikawa, B.W

K. Fujikawa, B.W. Lee, A.I. Sanda, Generalized Renormalizable gauge Formulation of Spontaneously Broken Gauge Theories, Phys. Rev. D, 6 (1972) 2923

work page 1972

[10] [10]

B.W. Lee, J. Zinn-Justin, Spontaneously Broken Gauge Symmetries. IV. General Gauge Formulation, Phys. Rev. D, 7 (1973) 1049

work page 1973

[11] [11]

Gogokhia, G.G

V. Gogokhia, G.G. Barnaföldi, The Mass Gap and its Applications (World Scientific, 2013). 19

work page 2013

[12] [12]

Korn, T.M

G.A. Korn, T.M. Korn, Mathematical Handbook (McGraw-Hill Book Company, 1968)

work page 1968

[13] [13]

Conway, Functions of One Complex Variable (Springer, 1978)

J.B. Conway, Functions of One Complex Variable (Springer, 1978)

work page 1978

[14] [14]

D. J. Gross, F. Wilczek, Ultraviolet behaviour of non-abelian gauge theories, Phys. Rev. Lett., 30 (1973) 1343

work page 1973

[15] [15]

Politzer, Reliable perturbative results for strong interactions?, Phys

H.D. Politzer, Reliable perturbative results for strong interactions?, Phys. Rev. Lett., 30 (1973) 134

work page 1973

[16] [16]

Y. L. Dokshitzer, D.E. Kharzeev, The Gribov conception of Quantum Chromodynamics, Ann. Nucl. Part. Sci., 54 (2004) 487

work page 2004

[17] [17]

Elitzur, Impossibility of spontaneously breaking local symmetries, Phys

S. Elitzur, Impossibility of spontaneously breaking local symmetries, Phys. Rev. D, 12 (1975) 3978

work page 1975

[18] [18]

Greensite, An Introduction to the Confinement Problem (Springer, 2 ed., 2020)

J. Greensite, An Introduction to the Confinement Problem (Springer, 2 ed., 2020)

work page 2020

[19] [19]

Holdom, Soft asymptotics with mass gap, Phys

B. Holdom, Soft asymptotics with mass gap, Phys. Lett. B, 728 (2014) 467

work page 2014

[20] [20]

Cucchieri, D

A. Cucchieri, D. Dudal, T. Mendes, N. Vandersickel, Modelling the Gluon Propagator in Landau Gauge: Lattice Estimates of pole masses and Damensional-Two Condensates, Phys. Rev. D, 85 (2012) 094513

work page 2012

[21] [21]

Tanabashi, et

M. Tanabashi, et. al., Particle Data Group, Phys. Rev. D, 98 (2018)

work page 2018

[22] [22]

C. A. Aidala, S.D. Bass, D. Hasch, G.K. Mallot,The spin structure of the nucleon, Rev. Mod. Phys., 85 (2013) 655

work page 2013

[23] [23]

al., Glue spin and helicity in the proton from lattice QCD, Phys

Yi-Bo Yang et. al., Glue spin and helicity in the proton from lattice QCD, Phys. Rev. Lett., 118 (2017) 102001-1

work page 2017

[24] [24]

Jakovác, G.G

A. Jakovác, G.G. Barnaf´’oldi, P. Pósfay, Effect of quantum fluctuations in the high-energy cold nuclear equation of state and in compact star observations, Phys. Rev. C, 97 (2018) 025803. [25]Quark Matter 2022, Proc. of the XXIV Inter. Conf. on Ultra-Relativistic Nucleus Collisions, ed. by P. Braun-Munziger, B. Friman, J. Stachel, Nucl. Phys. A, 931 (2014) 1. 20

work page 2018

[25] [25]

Petreczky, F

P. Petreczky, F. Karsch, E. Laermann, S. Stickan, I. Wetzorke, Temporal quark and gluon propagators: Measuring the quasiparticle masses, Nucl. Phys. Proc. Suppl., 106 (2002) 513

work page 2002

[26] [26]

Gogokhia, M

V. Gogokhia, M. Vasuth, The non-perturbative analytical equation of state for the gluon matter, J. Phys. G: Nucl. Part. Phys., 37 (2010) 075015

work page 2010

[27] [27]

Bernard, Nonte Carlo Evaluation of the Effective Gluon Mass, Phys

C. Bernard, Nonte Carlo Evaluation of the Effective Gluon Mass, Phys. Lett. B, 108 (1982) 431

work page 1982

[28] [28]

Vilela-Mendes, A consistent measure for lattice Yang-Mills, Int

R. Vilela-Mendes, A consistent measure for lattice Yang-Mills, Int. J. Mod. Phys. A, 32 (2017) 1750016

work page 2017

[29] [29]

J. M. Cornwall, Dynamical mass generation in continuum quantum chromodynamics, Phys. Rev. D, 26 (1982) 1453

work page 1982

[30] [30]

Cursi, R

G. Cursi, R. Ferrari, The unitary problem and the zero-mass limit for a model of massive Yang-Mills theory, Nuovo Cim., A35 (1976) 1

work page 1976

[31] [31]

Gribov, Quantization of non-Abelian gauge theories, Nucl

V.N. Gribov, Quantization of non-Abelian gauge theories, Nucl. Phys. B, 139 (1978) 1

work page 1978

[32] [32]

Zwanziger, Action from Gribov horizon, Nucl

D. Zwanziger, Action from Gribov horizon, Nucl. Phys. B, 321 (1989) 591

work page 1989

[33] [33]

Huber, Nonperturbative properties of Yang-Mills theories, Phys

M.Q. Huber, Nonperturbative properties of Yang-Mills theories, Phys. Rep., 879 (2020) 1-92

work page 2020

[34] [34]

Baker, J.S

M. Baker, J.S. Ball, F. Zachariasen, Dual QCD: a review, Phys. Rep., 209 (1991) 73

work page 1991

[35] [35]

Gracey, Two loop gluon mass from the LCO formalism, Eur

J.A. Gracey, Two loop gluon mass from the LCO formalism, Eur. Phys. J. C, 39 (2005) 61

work page 2005

[36] [36]

M. N. Ferreira, J. Papavassiliou, Gauge Sector Dynamics in QCD, Particles 2023, 1, 1-57

work page 2023

[37] [37]

Papavassiliou, Emergence of mass in the gauge sector of QCD, Chinese Phys

J. Papavassiliou, Emergence of mass in the gauge sector of QCD, Chinese Phys. C, 46 (2022) 112001. 21

work page 2022