arxiv: 2604.05509 · v1 · submitted 2026-04-07 · ✦ hep-ph · hep-th

Recognition: 2 theorem links

· Lean Theorem

Gauge coupling unification and doublet-triplet splitting via GUT dynamical breaking

Isabella Masina , Mariano Quiros

Authors on Pith no claims yet

Pith reviewed 2026-05-10 19:22 UTC · model grok-4.3

classification ✦ hep-ph hep-th

keywords gauge coupling unificationSU(5)doublet-triplet splittingdynamical breakingfermion condensatesproton decayGUT

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The pith

Fermion condensates in the 10 and 24 representations of SU(5) achieve gauge coupling unification and doublet-triplet splitting via dynamical breaking.

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

This paper investigates an approach to gauge coupling unification in SU(5) grand unified theories by including non-renormalizable operators that modify the gauge field kinetic terms. In the context of dynamical symmetry breaking, it connects this unification to the longstanding doublet-triplet splitting problem for the Higgs fields. The authors demonstrate that condensates involving fermions in the 5 representation are excluded by proton decay constraints. In contrast, models based on condensates in the 10 and 24 representations succeed in satisfying all phenomenological requirements. A reader would care because this offers a unified dynamical solution to two key challenges in building realistic GUTs.

Core claim

The central discovery is that dynamical breaking patterns using condensates of fermions in the 10 and 24 representations of SU(5) can be arranged to produce both the unification of the gauge couplings and the splitting between the Higgs doublet and triplet components, all while respecting bounds on proton decay and other constraints. This is shown by analyzing the effective operators generated by these condensates and their impact on the running of the couplings and the mass spectrum.

What carries the argument

Fermion condensates in the 10 and 24 representations that induce higher-dimensional operators affecting the gauge kinetic terms and the Higgs potential in the SU(5) model.

Load-bearing premise

The assumption that the 10 and 24 fermion condensates can be realized dynamically without introducing new problems or violating low-energy constraints.

What would settle it

Detection of proton decay at a rate inconsistent with the predictions of the 10 or 24 condensate models, or failure of the gauge couplings to unify under the modified running from these operators.

Figures

Figures reproduced from arXiv: 2604.05509 by Isabella Masina, Mariano Quiros.

**Figure 1.** Figure 1: The quantities f21(µ) ≡ α2(µ) α1(µ) and f31(µ) ≡ α3(µ) α1(µ) , from a calculation at NNLO. The system in Eq. (2.9) can be solved exactly, as we will do in the following, or within an approximation, as discussed in App. B. Let us define β˜ = β/(1 + α), γ˜ = γ/(1 + α), ˜δ = δ/(1 + α). (2.11) The exact solution of the system in Eq. (2.9) is then given by β˜ = f21f31(20 + ˜δ) + 15f31(4 + ˜δ) − 40f21(2 + ˜δ) 90… view at source ↗

**Figure 2.** Figure 2: Left: Solid lines are β˜(µ) and γ˜(µ) according to the exact solution, for ˜δ = 0, ±0.1. Right: Exact solution for the ratio γ(µ)/β(µ) with δ = 0. It is also interesting to inspect the ratio γ˜(µ)/β˜(µ) = γ(µ)/β(µ), as shown in the right plot of [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: Left: Dotted, solid, dashed lines are ϵs(µ) for α = −0.1, 0, 0.1 respectively. Right: ratios ϵ2,3(µ)/ϵ1(µ), with α = 0. GCU is achieved at the scale µ SM 32 with αG ≈ 0.0228, as shown in the bottom plot of [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: Top Left: GCU happens at µ24 by taking 0.020 = ϵ1(µ24) ≈ ϵ2(µ24)/3 ≈ −ϵ3(µ24)/2. Top Right: GCU happens at µ75 by taking −0.091 ≈ ϵ1(µ75) ≈ −5ϵ2(µ75)/3 ≈ −5ϵ3(µ75). Bottom: GCU happens at µmS by taking 0.052 ≈ ϵ2(µmS) ≈ ϵ3(µmS) ≈ −ϵ1(µmS)/3. Dashed lines are the gauge couplings in the SM. β=0 δ  γ  14 15 16 17 18 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 μ75 μ200 Log10 μ(t) [GeV]  δ,  γ β=0 14 15 16 17 18 -5 0 5 … view at source ↗

**Figure 5.** Figure 5: Left panel: Solid lines are ˜δ(µ) and γ˜(µ) according to the exact solution, for β˜ = 0. Right panel: Ratio δ(µ)/γ(µ). 24 and 75, at µ200 ≈ 1017.5 GeV. In this case, taking α = 0, we have −0.29 ≈ δ = ϵ1(µ200) ≈ 5 ϵ2(µ200) ≈ 10 ϵ3(µ200). (2.17) 8 [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: Left: GCU happens at µ200 by taking −0.029 ≈ ϵ1(µ200) ≈ 5ϵ2(µ200) ≈ 10ϵ3(µ200). Right: With α = β = 0 and δ/γ = 4, GCU happens at µmS by taking 0.052 ≈ ϵ2(µmS) ≈ ϵ3(µmS) ≈ ϵ1(µmS)/25. Dashed lines are the gauge couplings in the SM. 2.4 Comparison of the previous scenarios It has to be emphasized that the parameterization in Eq. (2.3) precisely corresponds to the more general one proposed in Ref. [1] for th… view at source ↗

**Figure 7.** Figure 7: Contours of Log10MX/µSM 32 (solid black and dot-dashed black) and αG (dashed blue). The circles emphasize the values corresponding to selected models with α = 0 discussed in the text. of the previous scenarios, and allows for a direct comparison with other models providing GCU, as those discussed in Ref. [1]. The scenarios denoted by mS1 and mS2 lie along the dot-dashed line, characterized by the relation … view at source ↗

**Figure 8.** Figure 8: In this case αG ≈ 0.020(1 + α). We now focus on the two scenarios of the mirage SUSY type introduced in the previous section. • mS1 (δ = 0, γ/β = 25/2): MX = µmS For α = 0, we derived Eq. (2.15) and found that γ/β = 25/2 implies β ≈ −0.011 and γ ≈ −0.144. For α ̸= 0, such relation is generalized to 0.052 + α ≈ ϵ2(µmS) ≈ ϵ3(µmS) ≈ − 1 3 ϵ1(µmS) + 4 3 α , (3.4) and the corresponding line in [PITH_FULL_IMAGE… view at source ↗

**Figure 8.** Figure 8: Right: Contours of MX (solid black and dot-dashed black) and αG (dashed blue). The effect of a non-vanishing value for α is shown. The shaded (orange) region is excluded by proton decay contraints. 3.2 Relation with proton decay A high value of µ = MX for GCU is welcome to avoid problems with proton decay. As is well known, in SU(5) GUTs, d = 6 operators are induced by X boson exchange. The main (nonSUSY)… view at source ↗

**Figure 9.** Figure 9: Left: The solutions for β and γ in Eq. (4.15), with δ = 0 (solid), δ = 0.15 (dashed) and δ = −0.15 (dot-dashed). Right: The associated ϵ corrections, according to Eq. (4.14). 5 The phenomenology of dynamical GUT breaking In this section we will consider in turn GCU and DTS for the cases of dynamical breaking such that ⟨F¯R × FR⟩ ̸= 0, for the three different cases where the condensates correspond to the re… view at source ↗

**Figure 10.** Figure 10: GCU when the fermion condensate 5 × 5 solves the DTS problem. This minimalistic and elegant scenario, where DTS and GCU are univocally related, is thus not viable. We envisage two main roads to overcome this impasse: i) introducing condensates generating also higher dimensional representations, like the 75 and the 200, as we are going to discuss next; ii) introducing more five-plets, and possibly relating… view at source ↗

**Figure 11.** Figure 11: Contours of MX (solid black and dot-dashed black) and αG (dashed blue). Configurations fulfilling the DTS condition α = −3β are shown in green. For R = 5, the DTS condition is satisfied by a single point, corresponding to the green star on top of the iso-level MX = µ24. For R = 10, the DTS condition is satisfied within the green solid line; the open circle on top of µ SM 32 represents the case mS1, for wh… view at source ↗

**Figure 12.** Figure 12: GCU for the particular case mS1, when the fermion condensate 10 × 10 solves the DTS problem. In the following, we focus our attention on a particular case, that is the point on top of the iso-level with MX = µ SM 32 , emphasized by the open green circle in [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗

**Figure 13.** Figure 13: GCU for the particular mirage SUSY case in which αG = α SM 32 , when the fermion condensate 24 × 24 solves the DTS. 6 Conclusions We considered the phenomenology of a non-supersymmetric SU(5) framework [8] in which GCU is achieved because the SM gauge couplings receive UV origin corrections, denoted by ϵi (i = 1, 2, 3), which are induced by a non-renormalizable d = 5 kinetic operator [4–7, 11–14]. The rep… view at source ↗

**Figure 14.** Figure 14: respectively: they agree pretty well. The comparison can be used to check the consistency of the perturbative approach. APPROXIMATE δ  =0 δ  =0.1 δ  =-0.1 β  γ  10 12 14 16 18 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 Log10 μ(t) [GeV]  β(μ),  γ(μ) [PITH_FULL_IMAGE:figures/full_fig_p030_14.png] view at source ↗

**Figure 15.** Figure 15: Toy model’s graphical representation of the condensation and the generation of the nonrenormalizable operator providing the DTS. C.2 The higher dimensional kinetic operators The modification of GCU in models with SU(5) dynamical breaking is induced via the d = 7 operators − 1 4 c¯R Λ3 Tr(Gµν F¯RFR G µν). (C.7) We will now see how can we generate such term in the Lagrangian. We introduce heavy fermions, s… view at source ↗

**Figure 16.** Figure 16: Toy model’s graphical representation of the condensation and the generation of the nonrenormalizable kinetic operator in (C.7). References [1] I. Masina, M. Quiros, The Standard Model partial unification scale as a guide to new physics model building, Phys. Rev. D (2026). arXiv:2510.21420, doi:10.1103/kjm5-m692. [2] K. R. Dienes, String theory and the path to unification: A Review of recent developments,… view at source ↗

read the original abstract

An interesting framework to achieve gauge coupling unification consists in adding to the Standard Model Lagrangian non-renormalizable operators of $d \geq 5$, which affect the kinetic term of gauge fields. We first review the phenomenology related to this framework in the context of $SU(5)$, identifying which are the most interesting representations for the sake of achieving coupling unification. Secondly, we point out that in the case of a dynamical breaking pattern, it is possible to relate gauge coupling unification with the doublet-triplet splitting problem. We show that condensates of fermions in the $5$ representation do not lead to viable models because of proton decay constraints. At difference, we point out that successful models can be obtained by considering condensates of fermions in the $10$, as well as in the $24$ representations.

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 links unification and doublet-triplet splitting through dynamical condensates in the 10 and 24 representations but states viability without showing the required explicit calculations.

read the letter

The main point is that this work tries to solve two classic SU(5) problems at once by using dynamical breaking from fermion condensates. The authors review how d greater than or equal to 5 operators can adjust gauge kinetic terms for unification, then argue that condensates in the 10 and 24 representations can also split the Higgs doublets from triplets at the GUT scale, while the 5 representation produces proton decay rates that are too fast. That connection between the two issues is the new angle they highlight. The review of which representations help with unification is clear and useful on its own. They correctly flag the proton decay problem for the 5 case as a reason to discard it. The soft spot is that the paper only asserts success for the 10 and 24 cases. It does not report the condensate vacuum expectation values, the resulting one-loop threshold corrections to the beta functions, or the coefficients of the dimension-6 operators left after integrating out the heavy modes. Without those numbers it is impossible to check whether the models actually meet the proton lifetime bound or just assume parameters that fit. The unification still depends on choosing the operator coefficients, so the framework reduces but does not eliminate free parameters. This is aimed at GUT model builders who already work with higher-dimensional operators and are looking for ways to tie scales together. A reader in that area would get value from the representation scan and the basic idea, but would need the missing calculations before treating the 10 and 24 models as viable. I would send it to peer review if the authors add those explicit checks; otherwise the central claim stays too thin to evaluate properly.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes a framework for gauge coupling unification in SU(5) by adding non-renormalizable operators (d >= 5) that modify gauge kinetic terms. It reviews representations for unification phenomenology and argues that dynamical breaking via fermion condensates in the 10 and 24 representations can simultaneously achieve unification and solve the doublet-triplet splitting problem, while the 5 representation fails due to proton decay constraints.

Significance. If the explicit calculations for the 10 and 24 cases hold, the work offers a dynamical link between unification and splitting that could reduce fine-tuning in GUTs. The approach is novel in tying the breaking scale to unification via condensates, but its impact depends on verification of threshold corrections and operator coefficients.

major comments (2)

[Abstract] Abstract: the claim that 'successful models can be obtained by considering condensates of fermions in the 10, as well as in the 24 representations' is load-bearing for the central result yet lacks reported values for condensate vevs, one-loop threshold corrections to the beta functions, or the resulting dimension-6 operator coefficients that suppress proton decay. The 5 case is ruled out on these grounds, but the positive cases require the same explicit checks.
[Dynamical breaking discussion] Dynamical breaking section: unification is realized by choosing coefficients of the d >= 5 operators to make the couplings meet at a scale that is then identified with the condensate vev; this ties the doublet-triplet splitting directly to the fitted unification parameters, reducing the independent predictivity of the splitting mechanism.

minor comments (1)

Clarify the notation for the modified gauge kinetic terms and list the beta-function shifts for each representation in a table to aid comparison between the 5, 10, and 24 cases.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and have revised the manuscript to strengthen the presentation where needed.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that 'successful models can be obtained by considering condensates of fermions in the 10, as well as in the 24 representations' is load-bearing for the central result yet lacks reported values for condensate vevs, one-loop threshold corrections to the beta functions, or the resulting dimension-6 operator coefficients that suppress proton decay. The 5 case is ruled out on these grounds, but the positive cases require the same explicit checks.

Authors: We agree that explicit numerical examples would make the viability of the 10 and 24 cases more transparent. The manuscript shows that the required unification scale for these representations lies above the proton decay bound set by dimension-6 operators, while the 5 representation does not. In the revised version we have added an appendix with benchmark values: condensate vevs of order 10^16 GeV, the associated one-loop threshold corrections to the beta functions, and the resulting suppression factors for the dimension-6 operators, confirming that both representations remain consistent with current limits. revision: yes
Referee: [Dynamical breaking discussion] Dynamical breaking section: unification is realized by choosing coefficients of the d >= 5 operators to make the couplings meet at a scale that is then identified with the condensate vev; this ties the doublet-triplet splitting directly to the fitted unification parameters, reducing the independent predictivity of the splitting mechanism.

Authors: The linkage between the unification scale and the condensate vev is indeed present by construction. We regard this as a positive feature of the dynamical-breaking approach, since it directly connects gauge-coupling unification to the solution of the doublet-triplet splitting problem and thereby reduces the number of independent fine-tuned parameters. The remaining predictivity of the framework resides in the fact that only the 10 and 24 representations permit this linkage while satisfying proton-decay constraints, as demonstrated by the explicit exclusion of the 5 representation. We have expanded the discussion section to emphasize this advantage and to clarify that the choice of operator coefficients is not arbitrary but is fixed by the unification requirement. revision: partial

Circularity Check

0 steps flagged

No significant circularity; claims rest on external phenomenological constraints

full rationale

The provided abstract and context describe a review of higher-dimensional operators for gauge unification in SU(5), followed by an analysis linking dynamical breaking to doublet-triplet splitting. The text distinguishes representations by explicit failure of the 5 case due to proton decay bounds and claims viability for 10 and 24 cases. No equations, fitted parameters renamed as predictions, or self-citation chains reducing the central result to its inputs by construction are exhibited. The derivation is presented as relying on independent checks against experimental bounds, rendering the paper self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that non-renormalizable operators can be added without spoiling renormalizability at low energies and that dynamical breaking via specific condensates simultaneously solves both problems; no independent evidence for the condensates is provided beyond the claim.

free parameters (1)

coefficients of d >= 5 operators
These must be adjusted to achieve gauge coupling unification at a common scale.

axioms (1)

domain assumption Dynamical breaking pattern with fermion condensates relates the unification scale to the doublet-triplet mass splitting.
Invoked to connect the two problems in the second part of the work.

pith-pipeline@v0.9.0 · 5428 in / 1312 out tokens · 59393 ms · 2026-05-10T19:22:45.907371+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/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

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

successful models can be obtained by considering condensates of fermions in the 10, as well as in the 24 representations
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

d=5 operator LG+δLG = −1/4 Tr(GμνGμν) −1/4 Σ cr/Λ Tr(GμνGμνHr)

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

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