{"id":"c55d89a5-4118-4374-be77-ca4daa5fd0d5","arxiv_id":"2009.11448","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"The minimal SU(5) x U(2)_H product group unification is excluded by dimension-6 proton decay, while an extension with Peccei-Quinn-scale incomplete SU(5) multiplets predicts proton lifetimes within DUNE and Hyper-K reach.","lead":"This preprint shows that the simplest product group unification model, based on SU(5) x U(2)_H, is already ruled out by proton decay experiments. It then presents an extension whose proton lifetime falls within the reach of upcoming experiments like DUNE and Hyper-K, tied to axion dark matter.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Extended-model proton lifetime predictions rely on the light color-triplet exchange remaining subdominant, but the triplet-decay mixing admitted in footnote 12 can generate unsuppressed dimension-5 operators; this contribution is neither computed nor bounded.","rationale":"The reader identified the same load-bearing assumption: the proton lifetime predictions assume dimension-6 dominance, while footnote 12 admits a mixing that can introduce dimension-5 contributions. My review confirms this is the weakest point of the central argument. The minimal-model exclusion is robust under the stated perturbativity assumption: even with the uncertainty in the exact value of the g_1H^2 upper bound, the required g_1H^2 ≳ 6 for viability is far above the quoted 3.68 bound, and the conclusion is not sensitive to the discrepancy I noted in the one-loop estimate. The extended-model prediction, by contrast, depends on an unquantified operator contribution. The paper's own footnote 12 is an explicit limitation statement: it acknowledges that nonzero ¯Φ′–¯Φ mixing is needed for triplet decay and that sizable mixing changes both the proton lifetime and branching fractions. No calculation or bound is supplied for this effect. The external citation [55] indicates prior work on exactly this issue, but the present manuscript does not carry that calculation into its numerical results. Because the concern is already reflected in the reader's CONDITIONAL verdict, my pass does not change the verdict; it reinforces it. The concrete test I propose would settle the issue by computing the dimension-5 amplitude with the actual light-triplet spectrum and mixing required by the model, and comparing its rate with the quoted dimension-6 rate across the allowed parameter space.","tokens_in":18028,"tokens_out":6583,"duration_ms":69508,"concrete_test":"Add to the superpotential an explicit ¯Φ′–¯Φ mixing term with coefficient ε M_G (or the lowest-dimension operator realizing the mixing described in footnote 12), compute the effective dimension-5 proton decay amplitude induced by integrating out the light Φ′, ¯Φ′ triplets at μ5, and include the wino/higgsino dressing contribution. Then scan ε over the range needed for the triplets to decay before Big Bang Nucleosynthesis, and compare the resulting τ(p→K^+ν̄) with the quoted dimension-6 p→π^0 e^+ lifetime from the SSARD scans. If the dimension-5 contribution is within one order of magnitude of, or larger than, the dimension-6 contribution for any allowed ε, the extended-model claim that dimension-6 decay dominates and the quoted DUNE/Hyper-K signature are not established; the branching-fraction prediction should also be rechecked.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central positive claim is that, in the extended SU(5)×U(2)_H model, the proton lifetime is dominated by dimension-6 X-boson exchange and lies in the DUNE/Hyper-K reach, roughly τ(p→π^0 e^+) ~ 10^34–10^35 yr for μ5 ~ 10^10–10^12 GeV. This claim requires that the light color triplets introduced in Eqs. (31) and (38) do not mediate competing proton decay channels.\n\nFootnote 12 explicitly concedes that the PQ-charge assignment allows mixing between ¯Φ′ and ¯Φ, and that this mixing is what lets the Φ′, ¯Φ′ triplets decay into MSSM fields. The footnote also cites Ref. [55] for the statement that sizable mixing affects the proton lifetime and branching fractions. In a supersymmetric theory, once the light triplets couple to 10 and ¯5 matter through such mixing, integrating them out at μ5 generates dimension-5 proton decay operators of the type Q Q Q L, with the usual wino/higgsino dressing. With μ5 ~ 10^10–10^12 GeV, these operators can easily dominate over the dimension-6 contribution, unless the relevant couplings are very small.\n\nThe paper does not compute the dimension-5 Wilson coefficients, does not bound the mixing parameter, and does not quantify the resulting p→K^+ν̄ width or the shift in the p→π^0 e^+ branching fraction. Therefore the statement that 'dimension-6 proton decay dominates' is an assumption embedded in the numerical scan, not a demonstrated property of the model. Since the predicted lifetime and final state both depend on which operator dominates, this is the most load-bearing weakness of the extended-model conclusion.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper analyzes proton decay in the product-group unification model SU(5)×U(2)_H, where an R-symmetry forbids the usual 5-5bar Higgs bilinear and the dimension-5 proton decay operators, while matter is still embedded in 10 and 5bar multiplets. Using gauge-coupling matching, the authors find that the minimal model has a low X-boson mass, and that the proton lifetime falls below the Super-Kamiokande limits unless g_1H^2 ≳ 6, far above the perturbativity bound g_1H^2 ≲ 3.68 derived from requiring a separation between the GUT scale and the cutoff. They therefore conclude that the minimal setup is excluded by dimension-6 proton decay. The paper then extends the model with an incomplete 5+5bar pair and a 2+2bar pair, so that the color-triplet components obtain a mass μ5 associated with Peccei-Quinn symmetry breaking while the doublets remain at the GUT scale. These light triplets modify the gauge-coupling running, raising M_X as M_X ≃ (M_G0/μ5)^{2/15} M_X0, and for μ5 ~ 10^10–10^12 GeV the predicted p→π^0 e^+ lifetime is ~10^34–10^35 years, in reach of DUNE and Hyper-K. The paper also discusses axion/Wino dark matter and predicts an axion-photon coupling three times larger than in the complete-multiplet KSVZ case.","tokens_in":18536,"tokens_out":7541,"duration_ms":72651,"significance":"If the central claims hold, the paper is significant: it shows that the minimal SU(5)×U(2)_H product-group unification is already excluded by dimension-6 proton decay, and it identifies a natural extension with a falsifiable proton-decay signature that connects the GUT threshold corrections to the Peccei-Quinn scale. The use of the R-symmetry to solve the doublet-triplet splitting and dimension-5 problems without fine-tuning is attractive, and the paper makes concrete, testable predictions for DUNE and Hyper-K, including a distinctive branching-ratio pattern if dimension-6 exchange dominates. The minimal-model exclusion appears robust because the required g_1H^2 > 6 is well separated from the perturbativity bound near 3.7. However, the positive claim for the extended model is conditional on an unquantified assumption about the mixing of the new light color triplets, which can regenerate dimension-5 proton decay; this is the main reason the manuscript needs revision.","major_comments":[{"comment":"The central positive claim that dimension-6 X-boson exchange dominates in the extended model is not established. Footnote 12 concedes that the PQ-charge assignment allows a mixing between ¯Φ′ and ¯Φ through which the new color triplets can decay into MSSM fields, and it cites Ref. [55] for the statement that sizable mixing affects the proton lifetime and branching fractions. In a supersymmetric theory, integrating out those light triplets at μ5 ~ 10^10–10^12 GeV can generate dimension-5 Q Q Q L operators; unless the mixing and Yukawa couplings are sufficiently small, these operators can dominate the dimension-6 amplitude and shift the dominant channel toward p→K^+ν̄. The paper does not compute the dimension-5 Wilson coefficients, does not bound the mixing parameter, and does not propagate the resulting uncertainty into the branching fractions used in the figures. Since some nonzero mixing is needed for the triplets to decay without long-lived colored relics, the assumption of negligible mixing is not automatic. The authors should compute or tightly constrain the dimension-5 contribution (or impose and justify a symmetry/parameter regime that suppresses it) before claiming that the quoted lifetimes are predictions of the model.","section":"Sec. 4.2, Footnote 12; Sec. 5"},{"comment":"The quantitative lifetime predictions rest on the one-loop relation M_X ≃ (M_G0/μ5)^{2/15} M_X0, obtained with λv = M_G to maximize the effect. Because the exponent is small, M_X and hence τ_p are sensitive to the actual doublet threshold mass and to higher-order threshold corrections; the paper itself calls the perturbativity bound 'naive' but still uses g_1H^2 = 3.2 as a reference value. Please quantify the uncertainty in M_X and τ_p from varying λv around M_G, from two-loop RGEs, and from the choice of the matching scale. This is needed to support the statement that the lifetime is 'in reach' for μ5 ~ 10^10–10^12 GeV, rather than merely within an optimistic upper range.","section":"Sec. 4.1, Eq. (32); Sec. 5"}],"minor_comments":[{"comment":"The caption attributes the bound g_1H^2 ≤ 3.68 to Eq. (34), but Eq. (34) is the extended-model bound with value 3.29; the minimal-model bound is Eq. (14).","section":"Fig. 1 caption"},{"comment":"The equalities g_1H^2(⟨Q⟩) = g_1H^2(Λ/4π) = ... should be written as inequalities '≲' or '≤', since the text uses them as upper bounds.","section":"Eqs. (14) and (34)"},{"comment":"The mass notation M_{Q_α^β+¯Q_α^β} is undefined and difficult to parse; please define the mass eigenstates and use consistent notation for the VEV scale ⟨Q⟩.","section":"Eq. (3)"},{"comment":"The relation between the cutoff Λ and the confinement scale is stated only verbally; clarifying this identification would make the derivation of Eq. (14) easier to check.","section":"Sec. 2.1"},{"comment":"There are several notation inconsistencies and typos: 'g^2_H1' appears instead of 'g^2_1H', 'production group unification' appears in the Introduction, and 'Expmental Limit' appears in Fig. 3.","section":"Sec. 5 and throughout"}],"recommendation":"major_revision","confidential_remarks":"The minimal-model exclusion part is solid and could almost stand alone. The main risk is the extended-model claim: the authors acknowledge in a footnote exactly the channel that can spoil dimension-6 dominance, but they do not compute it. Ref. [55] appears to contain relevant results and should be used quantitatively. I would ask for a concrete estimate or an explicit, justified suppression mechanism for the dimension-5 contribution; without that, the paper should be reframed as presenting an optimistic scenario rather than a prediction."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This one is worth your time if you care about SUSY GUT options. The genuinely new result is the exclusion of minimal SU(5) × U(2)_H: with the perturbativity requirement g1H^2 ≤ 3.68, Eq. (8) forces MX low enough that dimension-6 X-boson exchange puts the proton lifetime below Super-K limits. That argument is clean, robust to the rough one-loop treatment, and I don't see a way around it. The authors deserve credit for stating it plainly. The second new item is the extension with incomplete 5 + bar5 pairs whose light triplets lift MX through gauge-coupling running; tying mu5 to the PQ scale makes a concrete target: tau(p -> pi0 e+) around 10^34–10^35 yr for mu5 ~ 10^10–10^12 GeV, with a wino+axion dark matter admixture. The factor-three enhancement of the axion-photon coupling (caγγ = 2/3 rather than 8/3) is a nice, checkable extra prediction.\n\nThe soft spots. The extended-model headline is conditional on dimension-6 dominance, and that is assumed, not demonstrated. Footnote 12 concedes that the PQ charge assignment allows PhiBar'-PhiBar mixing, which is precisely what lets the light triplets decay into MSSM fields, and it cites Ref. [55] for the statement that sizable mixing changes the proton lifetime and branching fractions. In a SUSY theory, once those triplets couple to 10 and bar5 matter, integrating them out at mu5 generates dimension-5 Q Q Q L operators. The paper neither computes the Wilson coefficients nor bounds the mixing. If the mixing is not tiny, p->K+ nu can easily dominate, and the quoted reach numbers for DUNE/Hyper-K lose their meaning. That is the load-bearing weakness. Secondary issues are minor: Eq. (32) is an approximation, the perturbativity bound is one-loop, the SSARD scans are not publicly reproducible, and the lifetime plots have no uncertainty bands. All of those would be fixable in a revision or with a companion code depository.\n\nWho should read it: SUSY GUT model builders and experimentalists who want a concrete benchmark for DUNE/Hyper-K proton decay channels. The minimal-model exclusion should survive as a known constraint. The extended-model prediction is a reasonable target, but it needs a dimension-5 analysis before I'd bet on it. Recommendation: send it to peer review; the minimal-model part is solid and the extended-model part is important enough to referee, with the clear mandate that the referee request a quantitative treatment of triplet mixing or a stated viable regime where the mixing is suppressed.","headline":"The minimal-model exclusion is solid and worth knowing; the extended-model lifetime claim is a target, not a proof, because dimension-5 decay from the admitted triplet mixing is never computed.","tokens_in":19043,"tokens_out":2902,"would_cite":true,"duration_ms":27775,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Minimal SU(5)×U(2)_H product-group unification is already excluded by dimension-6 proton decay; an axion-linked extension reaches observable lifetimes.","keywords":["proton decay","product group unification","SU(5) x U(2)_H","doublet-triplet splitting","dimension-6 proton decay","Peccei-Quinn symmetry","axion dark matter","pure gravity mediation"],"falsifier":"A proton decay experiment that measures $p\\to\\pi^0 e^+$ with a lifetime below about $10^{34}$ years would rule out the extended model's parameter range, which centers on $10^{34}$–$10^{35}$ years for $\\mu_5$ in $10^{10}$–$10^{12}$ GeV; a null result above about $10^{35}$ years would push the Peccei-Quinn-linked scenario beyond next-generation reach.","tokens_in":17818,"feed_emoji":"⚛️","tokens_out":8074,"duration_ms":69382,"temperature":0.7,"pith_summary":"This paper asks whether the simplest product-group unification model, SU(5)×U(2)_H, can survive proton decay bounds. It argues that the minimal version is already excluded: the gauge-coupling matching conditions force the SU(5) X bosons to be light, and dimension-6 proton decay then shortens the proton lifetime below current experimental limits unless the hidden U(1)_H coupling is non-perturbatively large. The paper then shows that adding a 5 and 5-bar pair whose color triplets remain light, with their mass set by Peccei-Quinn breaking at $10^{10}$–$10^{12}$ GeV, shifts the matching conditions so the X bosons become heavy enough. In that extended model the proton lifetime reaches roughly $10^{34}$–$10^{35}$ years, which would be visible to the next generation of proton decay searches, and the dark matter can be a mixture of a Wino-like neutralino and the axion.","feed_headline":"Proton decay rules out the minimal product-group model","feed_subtitle":"Adding light color triplets tied to the axion restores lifetimes near 10^34 to 10^35 years, visible to coming experiments.","key_machinery":"The load-bearing object is the gauge-coupling matching condition at the unification scale, Eq. (8), which ties the heavy SU(5) X-boson mass to $g_{1H}^2(M_G)$, the hidden U(1)_H gauge coupling. Because the standard-model hypercharge is a diagonal subgroup of SU(5) and U(1)_H, a small hidden coupling drags $M_X$ down exponentially through $M_X = M_G \\exp(\\cdots)$, shortening the proton lifetime. The extension works through Eq. (32), a second matching relation: a light $5+\\bar 5$ pair with triplet mass $\\mu_5$ deflects the running and raises $M_X$ by the factor $(M_{G0}/\\mu_5)^{2/15}$, with the doublet partners made heavy by pairing with $2+\\bar 2$ fields. The mechanism that makes $\\mu_5$ naturally small is the Peccei-Quinn symmetry, whose breaking scale $10^{10}$–$10^{12}$ GeV produces both the axion and the triplet mass $\\lambda_P \\langle P\\rangle$.","core_discovery":"The paper's central claim is that the minimal SU(5)×U(2)_H product-group unification model is excluded by dimension-6 proton decay, and that a modest extension—one extra $5+\\bar 5$ pair with a companion $2+\\bar 2$ pair—restores viability. The decisive relation is Eq. (8), which expresses the X-boson mass $M_X$ in terms of the hidden gauge coupling $g_{1H}^2$ at the GUT scale; requiring perturbativity up to the cutoff caps $g_{1H}^2$ at about 3.68, and at that value $M_X$ is too small. Fig. 1 shows the proton lifetime then falls below current bounds unless $g_{1H}^2 \\gtrsim 6$. In the extended model, Eq. (32) gives $M_X \\simeq (M_{G0}/\\mu_5)^{2/15} M_{X0}$, so color-triplet masses $\\mu_5 \\lesssim 10^{12}$ GeV raise $M_X$ enough to bring the lifetime to $10^{34}$–$10^{35}$ years. The paper identifies $\\mu_5$ with the Peccei-Quinn breaking scale, making the axion a natural dark-matter component alongside the Wino LSP.","pith_inferences":["The same light-incomplete-multiplet rescue could be attempted in SU(5)×U(3)_H product unification, but the paper notes there is no equally motivated intermediate mass scale; the Peccei-Quinn connection is specific to the U(2)_H case.","If triplet–Higgs mixing is not negligible, dimension-5 operators could shorten the lifetime below the quoted range, so the $10^{34}$–$10^{35}$ year window is conditional on that mixing being small.","The correlation between proton lifetime and axion abundance means combined null results from proton decay, Wino dark matter searches, and axion searches can jointly constrain the model's parameter space."],"forward_implications":["The minimal SU(5)×U(2)_H model, without extra incomplete multiplets, cannot satisfy both perturbativity ($g_{1H}^2 \\le 3.68$) and proton decay bounds (need $g_{1H}^2 \\gtrsim 6$).","Adding one $5+\\bar 5$ pair with light color triplets raises the X-boson mass and lifts the proton lifetime into the $10^{34}$–$10^{35}$ year range.","If $\\mu_5$ is tied to Peccei-Quinn breaking, the axion naturally accompanies the Wino LSP as dark matter, and the axion-photon coupling is enhanced by a factor of three relative to a GUT-complete $5+\\bar 5$ axion multiplet.","Because dimension-6 decay dominates, the model predicts clean two-body channels $p\\to\\pi^0 e^+$ and $p\\to\\pi^0 \\mu^+$ as the primary signatures."],"supporting_citations":[{"why":"Defines the SU(5)×U(2)_H product-group unification model whose gauge structure and matter content this paper analyzes.","marker":"[9]"},{"why":"Provides the earlier product-group unification construction with the R-symmetry that forbids the 5-5bar Higgs bilinear and dimension-5 operators.","marker":"[10]"},{"why":"Supplies the current experimental lower limits on the proton lifetime that the minimal model fails to satisfy.","marker":"[15]"},{"why":"Gives the lattice hadron matrix elements used in the dimension-6 decay amplitude.","marker":"[21]"},{"why":"Axion framework used to motivate identifying the triplet mass with the Peccei-Quinn breaking scale.","marker":"[16, 17]"},{"why":"Provides the simplest pure gravity mediation spectrum used as the reference supersymmetry-breaking framework for the lifetime calculation.","marker":"[33]"},{"why":"Numerical engine that runs the gauge couplings, computes the supersymmetric spectrum, and evaluates the proton lifetime.","marker":"[37]"},{"why":"Shows that sizable mixing between the new triplets and the MSSM fields would affect the proton lifetime and branching fractions, the main caveat to dimension-6 dominance.","marker":"[55]"}],"fun_headline_variants":["Proton decay kills minimal product-group unification","Minimal product-group model excluded; axion triplets restore viability","Axion-linked triplets bring proton lifetime into DUNE and Hyper-K reach","Dimension-six proton decay rules out minimal product-group model"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The numerical lifetimes assume the new light color-triplet fields mix only weakly with the ordinary matter and Higgs fields of the supersymmetric standard model, so X-boson exchange remains the dominant proton decay channel; the paper notes that sizable mixing would alter the lifetime and branching fractions but does not compute that effect.","fun_headline_variants_meta":{"raw":{"variants":["Proton decay kills minimal product-group unification","Minimal product-group model excluded; axion triplets restore viability","Axion-linked triplets bring proton lifetime into DUNE and Hyper-K reach","Dimension-six proton decay rules out minimal product-group model"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001644,"raw_usage":{"total_tokens":6553,"prompt_tokens":985,"completion_tokens":5568,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":601,"completion_tokens_details":{"reasoning_tokens":5499}},"tokens_in":601,"tokens_out":5568,"duration_ms":35425,"temperature":1.0,"reasoning_tokens":5499,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-27T22:55:15.297859+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A proton decay experiment that measures $p\\to\\pi^0 e^+$ with a lifetime below about $10^{34}$ years would rule out the extended model's parameter range, which centers on $10^{34}$–$10^{35}$ years for $\\mu_5$ in $10^{10}$–$10^{12}$ GeV; a null result above about $10^{35}$ years would push the Peccei-Quinn-linked scenario beyond next-generation reach.","supporting_citations":[{"cited_title":"Upper Bound of Proton Lifetime in Product-Group Unification","cited_arxiv_id":"hep-ph/0303123","evidence_quote":"Defines the SU(5)×U(2)_H product-group unification model whose gauge structure and matter content this paper analyzes."},{"cited_title":"Universality in Pure Gravity Mediation","cited_arxiv_id":"1302.5346","evidence_quote":"Provides the simplest pure gravity mediation spectrum used as the reference supersymmetry-breaking framework for the lifetime calculation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Numerical engine that runs the gauge couplings, computes the supersymmetric spectrum, and evaluates the proton lifetime."}],"review_version":1}