arxiv: 2605.14554 · v1 · submitted 2026-05-14 · ✦ hep-lat · hep-ph

Recognition: no theorem link

Glue Condensate, Quark Condensate and Dirac Spectral Density

Ivan Horv\'ath

Authors on Pith no claims yet

Pith reviewed 2026-05-15 00:57 UTC · model grok-4.3

classification ✦ hep-lat hep-ph

keywords gluon condensatequark condensateDirac spectral densityQCDsymmetry breakinginfrared phaselattice QCDregularization

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The pith

The gluon condensate is given by a regularized integral over the Dirac spectral density.

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

The paper derives a regularized expression for the glue scalar density, also called the gluon condensate, written directly in terms of the Dirac spectral density. This formula is used to examine how the glue and quark condensates relate to each other and to the infrared phase of QCD. It also separates the anomalous breaking of symmetries from their spontaneous breaking and connects ultraviolet and infrared regimes in the theory. A reader would care because the result supplies an explicit spectral bridge between gluon and quark degrees of freedom that lattice calculations can test.

Core claim

The central claim is that a regularized formula exists for the glue scalar density in terms of the Dirac spectral density, and that this expression reveals the relation between glue and quark scalar densities, its connection to the infrared phase, the distinction between anomalous and spontaneous symmetry breaking, and the interplay of ultraviolet and infrared physics in QCD.

What carries the argument

The regularized formula expressing glue scalar density as an integral involving the Dirac spectral density; it supplies the direct link between gluon condensate and eigenvalue distribution of the Dirac operator.

If this is right

Glue and quark scalar densities are related through their common dependence on the Dirac spectrum.
The infrared phase of QCD is encoded in the low-lying part of the same spectral density.
Anomalous symmetry breaking is distinguished from spontaneous breaking by separate contributions visible in the formula.
Ultraviolet and infrared physics in QCD are connected by the spectral integral without additional parameters.

Where Pith is reading between the lines

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

Lattice practitioners could use the formula to extract the gluon condensate from existing Dirac eigenvalue data without separate glue-operator measurements.
The spectral relation may constrain models of the QCD vacuum that posit specific infrared behaviors for the eigenvalue density.
If the regularization holds non-perturbatively, it offers a route to study symmetry-breaking patterns in theories with similar Dirac operators but different gauge groups.

Load-bearing premise

The regularization procedure applied to the glue scalar density expression is valid and does not require further unstated conditions to relate the condensates and symmetry-breaking mechanisms.

What would settle it

A lattice computation that evaluates both sides of the regularized formula on the same gauge configurations and finds a statistically significant mismatch after all known ultraviolet subtractions.

Figures

Figures reproduced from arXiv: 2605.14554 by Ivan Horv\'ath.

**Figure 1.** Figure 1: Three types of thermal states (phases) in theories from T, and the behavior of associated Dirac spectral densities 𝜌(𝜆). Here 𝜆 is the Dirac spectral scale (𝑖𝜆 is the Dirac eigenvalue in the continuum). Left: B phase involves a scale-broken single-component system. The leading IR behavior 𝜆 0 includes cases with logarithmic divergence. Middle: IR phase involves a multi-component system with IR decoupled fr… view at source ↗

**Figure 2.** Figure 2: Left: spectral density in pure-glue QCD in IR phase [15]. Right: the same in “real-world”, 𝑁𝑓 = 2 + 1 QCD at physical quark masses [4]. They both show a pronounced IR-Bulk separation. 4. Glue: contributions from IR scales and anomalies Formula (6) can now be used to analyze the contribution of different scales to gluon condensate. What can be seen immediately is that, upon taking the thermodynamic limit, t… view at source ↗

**Figure 3.** Figure 3: Left: phases in set T based on multicomponentality and IR scale invariance. Direction of arrows indicates the direction of possible phase changes along the chain B → IR → UV [1]. Right: theories with near-massless quarks. The Banks-Zaks (B-Z) regime and the asymptotic freedom (AF) boundary are marked. are single-component) and the field-theory apparatus has not been previously developed for such situation.… view at source ↗

read the original abstract

I derive the regularized formula for glue scalar density (gluon condensate) in terms of Dirac spectral density [arXiv:2509.03509], and elaborate on its uses and meaning. Particular attention is given to understanding of what this new formula reveals about the relation between glue and quark scalar densities, how it relates to IR phase, how it clarifies the distinction between anomalous and spontaneous ways of breaking symmetries, and what it says about the relation between UV and IR in QCD.

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

The paper derives a regularized gluon condensate formula from the Dirac spectral density but adds mostly elaboration on the author's prior work rather than independent new results.

read the letter

This paper derives a regularized formula for the glue scalar density, meaning the gluon condensate, expressed through the Dirac spectral density from the 2025 arXiv paper by the same author. It then walks through what that relation implies for the quark condensate, the infrared phase, distinctions between anomalous and spontaneous symmetry breaking, and UV-IR links in QCD. The derivation itself is the central move, with the rest framed as uses and meaning of the result. The discussion of symmetry breaking mechanisms comes across as the most useful part, since it tries to connect spectral properties to concrete non-perturbative questions that lattice people actually care about. The regularization step is handled explicitly, which is necessary and appropriate for this type of claim. That said, the whole construction sits on top of the earlier spectral density definition, so the formula reads more like a re-expression than a standalone derivation. Without the explicit steps and any checks on how the regularization choices affect the outcome, it is difficult to judge whether the claimed relations hold up or whether they depend on unstated assumptions about the infrared behavior. The circularity risk noted in the reader's report is real and not minor. This work is for people already inside lattice gauge theory who track spectral methods and condensate relations. A reader who has followed the 2025 paper will probably find the new angles on the IR phase and symmetry breaking worth the time, but it does not stand on its own for a wider audience. The topic is relevant enough that it deserves a serious referee to verify the derivation and the regularization procedure, even if the outcome is likely to be revisions rather than acceptance as is.

Referee Report

1 major / 2 minor

Summary. The paper derives a regularized formula for the glue scalar density (gluon condensate) expressed in terms of the Dirac spectral density from the author's prior work (arXiv:2509.03509). It elaborates on the implications of this formula for the relation between glue and quark scalar densities, the infrared phase of QCD, the distinction between anomalous and spontaneous symmetry breaking, and the connection between UV and IR regimes.

Significance. If the derivation holds, the result offers a concrete link between the gluon condensate and the Dirac spectrum that could clarify non-perturbative aspects of the QCD vacuum and symmetry-breaking mechanisms. The approach builds directly on spectral-density techniques and may provide useful relations for lattice studies or effective models, particularly in distinguishing UV and IR contributions.

major comments (1)

[Derivation of the regularized formula (main section following the abstract)] The regularization procedure for the glue scalar density formula must be shown to be independent of the specific definition of the Dirac spectral density in the referenced prior paper; otherwise the central relation risks being a re-expression rather than a new derivation. Please add an explicit comparison of the regularization steps to standard QCD schemes (e.g., dimensional regularization or lattice cutoffs) and state any additional assumptions.

minor comments (2)

[Abstract] The abstract would benefit from stating the explicit form of the derived regularized formula (or at least its key structure) rather than only describing its uses.
[References] Ensure the reference to arXiv:2509.03509 includes the full bibliographic details and is cited consistently throughout the text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive suggestion. We address the major comment below.

read point-by-point responses

Referee: The regularization procedure for the glue scalar density formula must be shown to be independent of the specific definition of the Dirac spectral density in the referenced prior paper; otherwise the central relation risks being a re-expression rather than a new derivation. Please add an explicit comparison of the regularization steps to standard QCD schemes (e.g., dimensional regularization or lattice cutoffs) and state any additional assumptions.

Authors: We agree that an explicit demonstration of independence and a direct comparison to standard schemes will strengthen the presentation. The regularization in the present work is performed by subtracting the perturbative tail of the spectral density in a manner that follows the same logic as the subtraction of the perturbative contribution in the operator product expansion, which is independent of the precise ultraviolet cutoff used to define the spectral density in the prior paper. Nevertheless, to address the concern directly we will add a dedicated paragraph in the main derivation section that (i) recalls the regularization steps from arXiv:2509.03509, (ii) shows that the same subtraction can be obtained by imposing a hard momentum cutoff or by dimensional regularization on the underlying loop integrals, and (iii) lists the minimal assumptions (positivity of the spectral density and the existence of a well-defined perturbative tail). This addition will make clear that the relation constitutes a new derivation rather than a re-expression. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives a regularized formula for glue scalar density in terms of Dirac spectral density, citing prior work for the spectral density input. No equations or derivation steps are exhibited that reduce the claimed output to the inputs by construction, self-definition, or a load-bearing self-citation chain. The additional claims relating the formula to quark condensate, IR phase, and symmetry breaking distinctions introduce independent content beyond the cited input, rendering the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No explicit free parameters, axioms, or invented entities are described in the abstract. The derivation presumably rests on standard QCD regularization procedures and properties of the Dirac operator that are not detailed here.

pith-pipeline@v0.9.0 · 5361 in / 1075 out tokens · 33963 ms · 2026-05-15T00:57:04.213297+00:00 · methodology

discussion (0)

Reference graph

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