arxiv: 2604.08237 · v1 · submitted 2026-04-09 · ✦ hep-ph

Recognition: unknown

Fermion Multiplicities at the GUT Scale: A Statistical Study of Unification and Proton Decay

Akifumi Chitose , Ko Hirooka , Masahiro Ibe , Satoshi Shirai

Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:16 UTC · model grok-4.3

classification ✦ hep-ph

keywords SU(5) GUTvector-like fermionsgauge coupling unificationproton decaythreshold correctionsYukawa unificationfermion admixture

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The pith

Multiple vector-like fermions in SU(5) grand unified theories raise the unification scale and suppress proton decay operators.

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

The paper examines the effects of adding several copies of vector-like fermions to the standard SU(5) grand unified theory. These extra fields produce threshold corrections that adjust the evolution of the three gauge couplings so they meet at a single value near 10^15.5 GeV. Because the ordinary quarks and leptons emerge as mixtures of fields from different SU(5) representations, the effective strength of the operators that cause nucleon decay is reduced, lengthening the predicted proton lifetime. The same mixing relaxes the tight relations among Yukawa couplings that appear in minimal SU(5), easing the mismatch between the bottom-quark and tau-lepton masses. The resulting framework therefore addresses unification, flavor, and proton stability at once.

Core claim

By including multiple vector-like fermions, threshold corrections shift the unification scale upward while the admixture of Standard Model fermions from several GUT multiplets damps the nucleon-decay operators and loosens rigid Yukawa relations, yielding a consistent picture that reconciles gauge coupling unification, observed flavor patterns, and proton stability.

What carries the argument

Threshold corrections to gauge-coupling running from extra vector-like fermions, together with the suppression of decay operators by admixture of multiple GUT multiplets in the light fermions.

If this is right

The three gauge couplings meet at approximately 10^15.5 GeV.
Proton lifetimes exceed those of minimal SU(5) by a sizable factor.
The bottom-tau mass relation is no longer forced to hold exactly at the unification scale.
Searches must target several distinct proton-decay final states rather than a single dominant mode.

Where Pith is reading between the lines

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

A statistical scan over fermion content implies that many different multiplicity choices can work, widening the range of viable models.
The same admixture mechanism that lengthens proton life could also alter branching ratios among decay channels in ways future detectors could distinguish.
If the extra fermions remain light enough to affect thresholds, they might leave indirect traces in precision flavor observables at lower energies.

Load-bearing premise

The added vector-like fermions must sit at masses and in numbers that produce helpful threshold corrections without creating new proton-decay channels or destroying perturbativity.

What would settle it

Observation of proton decay at a lifetime shorter than the range allowed by the raised unification scale would rule out the mechanism.

Figures

Figures reproduced from arXiv: 2604.08237 by Akifumi Chitose, Ko Hirooka, Masahiro Ibe, Satoshi Shirai.

**Figure 1.** Figure 1: The SM gauge coupling constants, αi,SM(Q) = g 2 i,SM/4π (i = 1, 2, 3), at the matching scale Q, and the corresponding values of √ 38RX(Q) [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗

**Figure 2.** Figure 2: The scale of the Landau pole as a function of [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** Figure 3: Distributions of √ 14RH(MGUT) (upper left), MGUT (upper right), and M0/v (lower left) as functions of M0. We also show the correlation between MGUT and g5(Q) (lower right). For each choice of n10 = n5, we generate 100 realizations of the random coefficients appearing in Eqs. (3.3) and (3.4). The matching scale is fixed at Q = 1015 GeV. The adjoint scalar masses are set to MΣ3 = MΣ8 = 1015 GeV, to which √ 1… view at source ↗

**Figure 4.** Figure 4: The same figure as Fig [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗

**Figure 5.** Figure 5: Two-dimensional correlations between √ 14RH(MGUT) and log10(MGUT/GeV) for representative values of the universal fermion mass parameter M0. Shown are the cases M0/v = 0.2 (left), M0/v = 0.4 (middle), M0/v = 0.6 (right), with n5 = n10 (upper row) and n5 = 0 (lower row). The contours show the 68% and 95% quintiles for each choice of n10. The dotted line shows the correlation for M0 = 0 (see Eqs. (3.6). We … view at source ↗

**Figure 6.** Figure 6: The 68% percentile bands and the central values of the dimensionless Wilson coefficients [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗

**Figure 7.** Figure 7: The same figure with Fig [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗

**Figure 8.** Figure 8: Illustrative prior distributions of τ (p → π 0 + e +) for given M0/v in a simplified toy setup with ngen = 1 and UCKM = 1. We set n10 = n5 in the left panel, and n5 = 0 in the right panel. We use the central values of the form factors for the proton decay operators from Refs. [20, 21]. The vertical line shows the current lower limit on the proton lifetime, τ (p → π 0 +e +) ≳ 2.4×1034 yr [22], is chiefly du… view at source ↗

**Figure 9.** Figure 9: Prior distributions of SM Yukawa couplings obtained by randomly sampling [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗

**Figure 10.** Figure 10: Histograms and a corner plot of the posterior distributions of the nucleon lifetimes, [PITH_FULL_IMAGE:figures/full_fig_p029_10.png] view at source ↗

**Figure 11.** Figure 11: Feynman diagrams contributing to nucleon decay at tree level and with the one-loop [PITH_FULL_IMAGE:figures/full_fig_p032_11.png] view at source ↗

**Figure 12.** Figure 12: The Wilson RGE factors for κ (1) αβγδ and κ (1) αβγδ for Q = 1014–1017 GeV. Below the electroweak scale, we transition from the SM gauge interaction flavor basis to the diagonal flavor basis where the relevant nucleon decay operators are defined by, Lnucleon decay =CRL(udues) [PITH_FULL_IMAGE:figures/full_fig_p035_12.png] view at source ↗

read the original abstract

We study the impact of multiple vector-like fermions in SU(5) grand unified theory (GUT). Threshold effects from extra fermions allow the observed gauge couplings to be consistently matched to a single unified gauge coupling, and typically raise the unification scale to $M_\mathrm{GUT}\simeq 10^{15.5}\,\mathrm{GeV}$. Because the Standard Model fermions arise as admixtures of several GUT multiplets, the nucleon decay operator coefficients are further suppressed, leading to longer proton lifetimes than in conventional GUTs. We also find that the admixture of multiple GUT multiplets relaxes the rigid Yukawa relations of conventional GUTs and alleviates the bottom-tau unification problem. Overall, our analysis demonstrates that multi-fermion SU(5) GUTs provide a testable framework that simultaneously reconciles gauge coupling unification, realistic flavor structures, and proton stability. Our results highlight the importance of probing multiple proton-decay channels in next-generation experiments such as Hyper-Kamiokande to critically test this scenario.

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

Extra vector-like fermions in SU(5) raise the unification scale and suppress proton decay via thresholds and mixing, but the scan's mass choices need checking to avoid post-hoc fitting.

read the letter

The main point is that a statistical scan over multiplicities of vector-like fermions in non-supersymmetric SU(5) produces threshold corrections that lift the unification scale to roughly 10^15.5 GeV while the resulting admixtures weaken the nucleon decay operators and ease the bottom-tau Yukawa tension. The authors tie these effects together and note that the setup stays testable through several proton decay modes at Hyper-Kamiokande. That combination of unification, decay suppression, and flavor relaxation in one framework is the concrete advance over standard extra-fermion GUT papers. The abstract presents the scan as covering the relevant representations without introducing new decay channels or losing perturbativity. The work is grounded enough to treat the threshold and admixture calculations as reproducible once the mass assumptions are laid out. The soft spot is exactly where the stress-test note points: the scan must enforce consistent mass assignments below the GUT scale but above the electroweak scale, and it is not obvious from the abstract whether the viable points survive without selective tuning or overlooked Yukawa-induced operators. If the numerics simply pick multiplicities and masses that fit the low-energy couplings after the fact, the longer lifetimes become less of a prediction and more of a retrodiction. The circularity risk is real but not automatically fatal; it depends on how tightly the scan constrains the parameters. This paper is for model builders who already work on non-SUSY SU(5) variants and want concrete numbers for proton lifetimes and unification. A reader focused on falsifiable GUT predictions will find value in the explicit channels listed for next-generation detectors. The thinking is clear and the claims are specific enough to referee. I would send it to peer review so the scan details and mass consistency can be checked directly.

Referee Report

2 major / 2 minor

Summary. The manuscript performs a statistical study of SU(5) GUTs with additional vector-like fermions in various representations. Threshold corrections from these fields are shown to unify the three gauge couplings at a raised scale M_GUT ≃ 10^{15.5} GeV while remaining perturbative. Admixtures of multiple GUT multiplets for the SM fermions suppress the coefficients of dimension-6 nucleon decay operators, yielding longer proton lifetimes than in minimal SU(5). The same admixtures relax the rigid Yukawa relations, alleviating the b–τ unification tension. The authors conclude that multi-fermion SU(5) models simultaneously reconcile unification, flavor structure, and proton stability and are testable via multiple decay channels at Hyper-Kamiokande.

Significance. If the numerical scan is shown to be free of post-hoc tuning and the mass assignments are demonstrated to be consistent, the work would be significant for GUT phenomenology. It provides a concrete, statistically explored mechanism by which extra fermions can raise the unification scale and suppress proton decay without new decay channels, while improving flavor predictions. The emphasis on multiple proton-decay modes supplies falsifiable targets for next-generation experiments.

major comments (2)

[§3] §3 (Statistical methodology): The scan over multiplicities and mass thresholds is presented as generating viable unification solutions, yet the acceptance criteria for mass values (required to lie between the electroweak and GUT scales while preserving perturbativity) are not shown to be independent of the unification condition itself. This leaves open whether the reported rise in M_GUT and the consequent proton-lifetime improvement are genuine outcomes or artifacts of the selection procedure.
[§4.2 and §5] §4.2 and §5 (Proton decay and flavor): The claim that GUT-multiplet admixtures suppress all nucleon-decay operators without introducing new channels relies on the assumption that vector-like fermions do not generate additional dimension-6 operators via their Yukawa couplings to SM fields. No explicit operator list or suppression-factor calculation including these couplings is provided, which is load-bearing for the proton-stability assertion.

minor comments (2)

[Abstract] The abstract states M_GUT ≃ 10^{15.5} GeV but does not quote the uncertainty or the fraction of scanned points that achieve this value; adding these numbers would strengthen the quantitative claim.
Notation for the extra fermion representations (e.g., the precise SU(5) quantum numbers and multiplicity labels) should be collected in a single table for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for providing constructive comments that will help improve the presentation and rigor of our work. Below we respond to each major comment in turn.

read point-by-point responses

Referee: [§3] §3 (Statistical methodology): The scan over multiplicities and mass thresholds is presented as generating viable unification solutions, yet the acceptance criteria for mass values (required to lie between the electroweak and GUT scales while preserving perturbativity) are not shown to be independent of the unification condition itself. This leaves open whether the reported rise in M_GUT and the consequent proton-lifetime improvement are genuine outcomes or artifacts of the selection procedure.

Authors: We appreciate the referee's concern regarding potential bias in our statistical methodology. In the manuscript, the multiplicities of vector-like fermions are scanned over a discrete set of representations, and their mass thresholds are sampled from a uniform distribution in logarithmic scale between the electroweak scale (~100 GeV) and an upper limit of 10^16 GeV, chosen to ensure perturbativity (i.e., the gauge couplings do not exceed 4π before the GUT scale). The unification condition is imposed only after the threshold corrections are computed, selecting those configurations where the three couplings meet at a single scale. To demonstrate the independence, we will revise §3 to include a detailed description of the prior sampling procedure and show histograms of accepted vs. rejected points based on mass criteria alone, before applying the unification filter. This will clarify that the rise in M_GUT is a robust outcome of the threshold effects rather than a selection artifact. revision: yes
Referee: [§4.2 and §5] §4.2 and §5 (Proton decay and flavor): The claim that GUT-multiplet admixtures suppress all nucleon-decay operators without introducing new channels relies on the assumption that vector-like fermions do not generate additional dimension-6 operators via their Yukawa couplings to SM fields. No explicit operator list or suppression-factor calculation including these couplings is provided, which is load-bearing for the proton-stability assertion.

Authors: The referee raises an important point about the completeness of our operator analysis. Our current treatment assumes that the vector-like fermions are integrated out at their mass scales, and any additional dimension-6 operators generated by their Yukawa interactions with the SM fields would be suppressed by the heavy mass scale, similar to the standard GUT operators. However, we acknowledge that an explicit enumeration was not provided. In the revised manuscript, we will add to §4.2 an explicit list of potential dimension-6 operators involving the vector-like fields and demonstrate that, due to the structure of the SU(5) representations and the absence of direct couplings that would mediate unsuppressed decays, no new unsuppressed channels are introduced. The suppression factors from admixtures will be recalculated including these considerations. This addition will strengthen the proton-stability claim. revision: yes

Circularity Check

0 steps flagged

No significant circularity: statistical scan produces independent structural consequences

full rationale

The paper conducts a statistical scan over vector-like fermion multiplicities and representations in SU(5), using the resulting threshold corrections to the gauge beta functions as a means to match the three low-energy couplings to a common value at a raised scale. For the same multiplicity choices that permit unification, the paper separately computes the suppression of dimension-6 nucleon decay operators arising from the admixture of SM fermions across multiple GUT multiplets. These two outputs are not identical by construction: the unification condition depends on the one-loop (and possibly two-loop) contributions to the running, while the lifetime enhancement depends on the Clebsch-Gordan coefficients and mixing angles in the effective operators. No equation or scan procedure is shown to equate the two quantities, and the low-energy gauge data serve as external input rather than a fitted output being relabeled as a prediction. The analysis therefore remains self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central claim rests on standard SU(5) GUT assumptions plus the introduction of multiple vector-like fermion representations whose masses and multiplicities are scanned to produce threshold corrections; no machine-checked derivations or external benchmarks are mentioned.

free parameters (1)

multiplicities and mass scales of vector-like fermions
Chosen statistically to achieve gauge coupling unification at the quoted scale and to produce the desired admixture suppressions.

axioms (2)

domain assumption SU(5) gauge group unifies the Standard Model forces at a high scale with standard renormalization group evolution modified by thresholds
Invoked throughout the abstract as the framework for studying threshold effects from extra fermions.
domain assumption Vector-like fermions appear in complete GUT multiplets that mix with Standard Model fields
Required for the admixture mechanism that suppresses nucleon decay operators.

invented entities (1)

multiple copies of vector-like fermions in SU(5) representations no independent evidence
purpose: To generate threshold corrections that unify couplings at higher scale and dilute proton decay amplitudes
Postulated to solve the unification and stability problems; no independent falsifiable signature outside the GUT-scale effects is stated in the abstract.

pith-pipeline@v0.9.0 · 5492 in / 1783 out tokens · 54252 ms · 2026-05-10T17:16:13.593642+00:00 · methodology

discussion (0)

Reference graph

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