{"id":"9ae5ba17-1f67-4624-a597-234344e60ddf","arxiv_id":"2610.37541","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The authors construct on-shell renormalization schemes and redundancy-free two-loop RGEs for six vector-like lepton models, then use the RG-improved effective potential to map vacuum instability scales across the parameter space.","lead":"This paper builds precision calculation tools for six vector-like lepton extensions of the Standard Model, renormalizing lepton masses on-shell and running the couplings at two loops. It then computes where the Higgs vacuum becomes unstable, finding rich behaviour and constraints that complement electroweak precision tests.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"Every numerical instability scale in Sec. 3 inherits the unprinted two-loop redundancy-free RGEs of Sec. 2.7, which rest entirely on the field-redefinition translation of Eqs. (27)-(35); no independent or diagrammatic check of those beta functions is shown.","rationale":"The reader's weakest_assumption and my stress-test converge on the same point: the two-loop redundancy-free beta functions are load-bearing but not shown. The paper itself acknowledges in Sec. 2.7 that generic literature beta functions do not apply because they assume hermitian field renormalization and keep off-diagonal redundant parameters, so the translation through V_L/R,0 is essential rather than cosmetic. Every high-scale quantity in Sec. 3—the effective potential, Lambda, and the fits of Table 3—depends on the running of lambda_Phi, y_t, and the VLL Yukawas; an error in the VLL-specific two-loop terms would shift all numerical bounds, while the OS scheme construction could remain correct. This is why the concern is load-bearing. It is not, however, an identified error: the one-loop structure of Secs. 2.3-2.6 is internally consistent, the zero-coupling limit gives the expected SM behaviour, and the triplet stabilization at small lambda_max has a plausible physical origin in SU(2)_L gauge contributions. The correct disposition is therefore to keep the reader's CONDITIONAL verdict rather than upgrade or reject: the condition is that the full RGE system be printed or independently checked. My concrete test is a coefficient-level comparison of the translated system against a direct RGBeta computation of the redundant system; that is a well-defined check that should settle whether the concern lands. I do not see a separate issue in the effective-potential or instability-scale definitions that would change the verdict; the acknowledged caveats about Goldstone contributions, OS-MS mu^2 conversion, and discontinuities are already internalised in the paper's uncertainty estimate and in the reader's conditional assessment.","tokens_in":30031,"tokens_out":8989,"duration_ms":67499,"concrete_test":"Publish or independently recompute the explicit two-loop redundancy-free RGE system for at least the L+E model: generate the full generic two-loop beta functions for the redundant set {y_ij, lambda_Li, lambda_Ei, lambda, bar{lambda}, M_iL, M_iE, M_L, M_E} with RGBeta, impose the parameterization conditions of Eqs. (27)-(35) order by order in hbar, and compare every coefficient of the resulting beta functions (including the Dirac-mass sector and the anti-hermitian shift Eq. (35)) with the system actually used to produce Figs. 4-5. Coefficient-by-coefficient agreement settles the translation; any mismatch localizes the affected terms and allows a rerun of the instability-scale scan.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central numerical outputs—Lambda(lambda_max) in Fig. 5, the fit parameters of Eq. (49)/Table 3, and the stability statements in the abstract—are obtained by solving the \"redundancy-free\" two-loop RGEs described in Sec. 2.7. The paper does not print the resulting RGE system; it states that generic beta functions from Refs. [64-68]/RGBeta are not directly applicable (Sec. 2.7) and that they are translated through the field redefinitions V_L/R,0, with the anti-hermitian field-RC shift of Eq. (35) asserted to be necessary. A sign or coefficient error at two loops in beta(lambda_Phi), beta(y_t), or beta(lambda_i) would propagate directly into every Lambda contour and into the approximate fit (49). The zero-VLL-coupling limit quoted in Sec. 3.3 reproduces the SM running, but that limit does not exercise the VLL-specific translation, so it cannot validate the new terms. This is a verification gap in the central numerical chain, not an identified algebraic mistake; the OS scheme construction itself is largely independent of it.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"The paper develops on-shell (OS) renormalisation schemes for six vector-like lepton (VLL) extensions of the Standard Model in which SM lepton masses receive seesaw-like contributions through up to 5×5 mass mixing. Three scheme variants (OS1, OS2, OS3) are defined, the finite parts of the fundamental counterterms are fixed by on-shell mass and field conditions, and the remaining parameters are renormalised in the MS scheme. The paper then constructs what it calls redundancy-free two-loop RGEs by translating generic literature RGEs through field redefinitions, solves these RGEs, computes the RG-improved effective potential at next-to-leading-log accuracy, and studies the scale Λ at which the electroweak vacuum becomes unstable. The main reported results are: singlet VLL models reduce to the SM-like instability picture in the zero-coupling limit; triplet VLL models are stable up to the Planck scale for λ_max ≲ 0.25; for larger VLL Yukawa couplings the instability scale can drop as low as 10^5 GeV; and an approximate fit formula Eq. (49) with parameters in Table 3 is provided for future phenomenological use.","tokens_in":30376,"tokens_out":6949,"duration_ms":54456,"significance":"If the two-loop RGE construction is correct, the paper makes a useful and fairly general contribution: it extends a previously known single-model OS scheme to six VLL models with up to 5×5 mixing, provides explicit counterterm matrices and coefficient matrices in Appendix A, defines three schemes with different singular limits, and gives a careful uncertainty treatment for the instability scale. The zero-coupling limits match the expected SM and SM-plus-triplet-gauge behaviour, which is a genuine consistency check of part of the setup. The scheme construction itself appears internally coherent, and the explicit statement that not all heavy masses can be on-shell renormalised is an honest limitation. However, the numerical stability conclusions rest on the unprinted two-loop RGE system of Section 2.7, and this is the main load-bearing gap.","major_comments":[{"comment":"The two-loop redundancy-free RGEs are the foundation of every numerical result in Section 3, but the RGE system itself is never printed. The text explains that generic RGEs from Refs. [64-68]/RGBeta are not directly applicable and are translated through the field redefinitions V_{L/R,0}, including the anti-hermitian field-renormalisation shift of Eq. (35), yet only an L⊕E illustration of the translation is given. Consequently the Λ(λ_max) curves in Fig. 5, the fit parameters in Eq. (49)/Table 3, and the abstract's instability-scale statements cannot be checked by a reader, and a sign or coefficient error in the translated β(λ_Φ), β(y_t), or β(λ_i) would propagate into all of them. The zero-VLL-coupling limits reported in Section 3.3 reproduce the expected SM and triplet-gauge running, but they do not exercise the VLL-specific Yukawa and mass-parameter translation. I am not requesting a reproduction of the published generic RGEs; I am requesting the original output of the translation, either as printed equations or as a machine-readable implementation, together with at least one non-trivial cross-check such as running in the redundant parametrisation with the generic RGEs or a direct two-loop diagrammatic calculation for one model. This is a verification gap rather than an identified algebraic mistake, and the OS scheme construction appears unaffected.","section":"Section 2.7, Eqs. (27)-(35)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The explicit translation formulas in Eqs. (27)-(35) are written for L⊕E only, with the statement that the structure is the same in the other models. The triplet models L⊕E_a, L⊕N_a, and (3/2)⊕E_a have different field content, hypercharges, and mass-matrix structures, and the central claim that triplet models remain stable up to the Planck scale for λ_max ≲ 0.25 depends on their VLL-specific two-loop terms. Readers therefore need either the corresponding translated beta functions for the triplet models or a direct verification that the field-redefinition mapping in those cases has been implemented and tested. Without this, the most distinctive phenomenological conclusion of the paper is not independently checkable.","section":"Section 2.7, model coverage","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"The neglect of Goldstone-boson contributions is acknowledged and stated to have been checked, but no quantitative result of that check is shown; a sentence or small table giving the maximum relative size of the neglected terms for the representative points of Fig. 2 would make the approximation easier to assess.","section":"Section 3.1","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The statement that other choices of nonzero Yukawa couplings do not change the analysis within the uncertainty bands would be more useful if the tested alternatives were listed or shown in an appendix, given that the fit in Eq. (49) depends on the number of active couplings.","section":"Section 3.3.2, footnote 8","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"There are a few typographical errors in the running headers, such as 'T able 1' and 'T able 2', and the paper would benefit from a final proofreading pass.","section":"Throughout","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and the OS scheme construction is the strongest part of the manuscript. The main risk is the unprinted two-loop RGE system of Section 2.7, which all Section 3 results inherit; if the authors provide the RGE system or a reproducible implementation and a non-trivial cross-check, I would be inclined to accept. I do not see a circularity problem with the instability-scale results themselves, since they are outputs rather than fitted targets, and the approximate fit of Eq. (49) is presented as a fit rather than as an independent prediction."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"Punchline first: the on-shell scheme construction is the real content of this paper and it is solid; the instability-scale maps are the advertisement, and every one of them rides on a set of two-loop beta functions the paper never prints.\n\nWhat is new: Ref [9] built one OS scheme for the L+E model; this extends the construction to all six models with up to 5x5 mixing, and the OS3 variant genuinely fixes the two singular limits (OS1 breaks when the Higgs-induced mixing vanishes, OS2 when the Dirac masses are degenerate). The redundancy between field redefinitions and Lagrangian parameters is treated carefully, and the anti-hermitian field-RC shift of Eq. (35) is a real subtlety that would be easy to miss. Appendix A prints the kappa matrices for all six models, so the OS1/OS2/OS3 definitions are actually checkable line by line. The authors also handle the mu^2 scheme-conversion artifact honestly by deriving mu^2 from the minimisation condition, and their uncertainty treatment makes sense given the discontinuous jumps in Lambda. The triplet stabilization below lambda_max around 0.25 and instability scales down to 10^5 GeV are the qualitative results people will actually use, and the Table 3 fit makes them easy to adopt.\n\nSoft spots, in proportion. The main one is Sec. 2.7, and the stress-test note survives contact with the paper: the two-loop RGEs come from translating generic literature beta functions through the field redefinitions of Eqs. (27)-(35), but the resulting system is never written down, and the zero-coupling limit quoted in Sec. 3.3 does not exercise the VLL-specific translation terms. A sign or coefficient error at two loops could shift every contour in Fig. 5 and every entry of Table 3. This is a verification gap, not an identified algebraic mistake, and the method itself is standard; but for a paper whose numerical headline depends on that step, printing the beta functions or providing an ancillary file should have been non-negotiable. Minor: the tree-level mass matrices live in a companion preprint by the same group [10], so the reader needs that paper in hand; scan data and code are absent; and the Eq. (49) fit is acknowledged to lose accuracy near the VLL mass scale, which the authors state outright.\n\nBottom line: this deserves a serious referee. I would send it to review and make the explicit beta functions, or an independent check of the translation, a condition of acceptance. The scheme construction is careful and reusable, and the qualitative stability picture will be cited. The reader's conditional verdict is about right, and the conditions are removable.","headline":"The OS scheme construction is the solid, genuinely new content; the headline instability-scale maps inherit unprinted two-loop beta functions, so send it to referees and make the RGEs a condition of acceptance.","tokens_in":30936,"tokens_out":5097,"would_cite":true,"duration_ms":34571,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":["12.15.Lk","12.60.-i"],"model":"deepseek/deepseek-v4-flash","headline":"For all six vector-like lepton models, the paper constructs on-shell renormalisation schemes that keep the observed SM lepton masses fixed despite up to 5×5 mixing, then uses two-loop running and the NLL effective potential to map the…","keywords":["vector-like leptons","on-shell renormalisation","renormalisation group equations","two-loop beta functions","effective potential","vacuum stability","Higgs instability scale","seesaw lepton masses"],"falsifier":"Compute the two-loop renormalisation-group equations for one model, say $L\\oplus E$, directly from the redundant parameter set that keeps off-diagonal Yukawas and Dirac masses, project onto the physical parameters, and compare with the translated redundancy-free $\\beta$-functions; any mismatch in the $1/\\epsilon$ poles would shift every quoted instability scale. A cheaper probe: evaluate a physical observable such as the muon-Higgs coupling or muon $g-2$ both at fixed one-loop order and through the NLL-resummed running, and check that the two agree within the paper's stated uncertainties.","tokens_in":31491,"feed_emoji":"⚛️","tokens_out":13468,"duration_ms":87008,"temperature":0.7,"pith_summary":"Vector-like lepton extensions of the Standard Model give the electron, muon, and tau their masses through seesaw-like mixing with heavy new fermions, which generates large radiative corrections and makes ordinary renormalisation impractical. The paper aims to show that all six of the simplest such models admit renormalisation schemes in which every SM-like lepton mass is fixed on-shell, at its observed pole value, even though the mixing matrices are up to $5\\times5$, while the remaining heavy masses are handled by a three-way scheme choice. To support this it derives redundancy-free two-loop $\\beta$-functions and an RG-improved effective potential at next-to-leading-log accuracy, and applies them to the vacuum stability question. The payoffs it claims are concrete: triplet-type VLL models keep the electroweak vacuum stable up to the Planck scale as long as the largest VLL Yukawa coupling stays below about 0.25, while in other parameter regions the instability scale can fall as low as $10^5$ GeV, and some regions are stabilised by the one-loop corrections to the input parameters themselves.","feed_headline":"Heavy lepton mixing can drop Higgs vacuum stability to 10^5 GeV","feed_subtitle":"On-shell schemes now cover all six vector-lepton models; triplets hold the vacuum to the Planck scale below λ ≈ 0.25.","key_machinery":"The construction rests on three linked objects. The first is the set of on-shell conditions on the renormalised lepton self-energies, pole conditions on chosen diagonal masses plus no-mixing and residue conditions on all fields, which fix the finite parts of the mass and field renormalisation constants. The second is the linear system $\\delta^{[0]}m = \\mathrm{Re}(\\kappa)\\cdot\\mathrm{Re}(\\delta^{[0]}M)+\\mathrm{Re}(\\delta^{[0]}u)$, in which the coefficient matrix $\\kappa$ is built from the tree-level diagonalisation matrices $U^f_{L,R}$ and, once inverted, expresses the fundamental counterterms through the on-shell mass counterterms; the three scheme variants are exactly different choices of rows and columns of $\\kappa$. The third is the field-redefinition translation that converts generic two-loop $\\beta$-functions from the literature into redundancy-free ones: unitary redefinitions $V_{L/R}$ eliminate the unphysical off-diagonal Yukawa and Dirac-mass parameters, and the accompanying anti-hermitian shift of the field renormalisation constants, fixed by unitarity and nonzero only with CP violation, keeps off-shell Green functions finite. The observable that carries the physics is the RG-improved effective potential $V_{\\mathrm{eff}}(h)$ with field-dependent lepton masses $m_{fa}(h)$, whose zero of the effective quartic coupling $\\lambda^{\\mathrm{eff}}_{\\Phi}$ defines the instability scale $\\Lambda$, with the MS value of $\\mu^2$ derived from the minimisation condition rather than from a one-loop scheme conversion that would leave a large $M^2_{L,E}$-proportional artifact.","core_discovery":"On the paper's own terms, the central result is that the six VLL models, the singlet pairs $L\\oplus E$, $L\\oplus N$, $L_{3/2}\\oplus E$ and the triplet pairs $L\\oplus E_a$, $L\\oplus N_a$, $L_{3/2}\\oplus E_a$, all admit mixed on-shell and $\\overline{\\text{MS}}$ renormalisation schemes. In each scheme the three SM lepton masses and a selected subset of heavy lepton masses satisfy on-shell pole conditions, all heavy states in the singlet models and four of the five correlated heavy states in the triplet models, while the remaining fundamental Lagrangian couplings are $\\overline{\\text{MS}}$-renormalised. Three scheme variants per model, OS1, OS2 and OS3, trade off numerical stability across the vanishing-mixing limit, the degenerate-mass limit, and general hierarchies. With these schemes the paper obtains the two-loop renormalisation-group running appropriate to the redundancy-free parameter set and computes the next-to-leading-log effective potential, finding that the instability scale is governed mainly by the largest VLL Yukawa coupling $\\lambda_{\\max}$, that triplet models stabilise the vacuum up to the Planck scale for $\\lambda_{\\max}\\lesssim 0.25$, and that larger couplings in all models can push the instability scale down to $10^5$ GeV with discontinuous jumps in $\\Lambda$ as the couplings vary.","pith_inferences":["The OS3 construction, which chooses which heavy mass stays on-shell according to whether $M_L>M_E$, reads as a general recipe for any extension with near-degenerate or dependent heavy masses; the paper does not claim this generality, but vector-like quark models seem the natural next test.","Since the anti-hermitian field-renormalisation shift vanishes without CP violation, internal scheme consistency offers a built-in cross-check specifically for CP-violating predictions such as electric dipole moments, a use the paper leaves implicit.","The discontinuous jumps in $\\Lambda$ suggest that near transition surfaces the instability scale is effectively multi-valued in the couplings, so future global fits should avoid smooth interpolation there; the paper reports the jumps but does not map where the transition surfaces sit relative to the EWPO-allowed regions."],"forward_implications":["If the schemes work as claimed, precision predictions in VLL models no longer require iterating tree-level Yukawa couplings to match the observed lepton masses; the on-shell conditions fix them directly, in all six models.","Triplet VLL models with $\\lambda_{\\max}\\lesssim 0.25$ predict a vacuum that stays stable up to and beyond the Planck scale, so vacuum stability alone cannot exclude that region of their parameter space.","Where the couplings are larger, instability scales of $10^5$ to $10^8$ GeV become typical inside the regions that electroweak precision data currently allow, making vacuum stability the sharper constraint once VLL masses exceed about 2 TeV.","The same setup transfers directly to further observables, as the paper notes: flavour- and CP-violating Higgs and $Z$ decays and lepton dipole moments can be computed with the same on-shell conditions and the same redundancy-free running.","The simple fits for $\\Lambda(\\lambda_{\\max})$ in Eq. (49) with the stated uncertainty bands give conservative estimates of the allowed coupling size for $\\lambda_{\\max}\\lesssim 1.5$ in continuous regions.","The minimisation-based derivation of the MS Higgs mass parameter avoids the residual two-loop artifact proportional to $M^2_{L,E}$, keeping the electroweak minimum at $h=\\sqrt{2}v$ in the RG-improved potential."],"supporting_citations":[{"why":"The prior on-shell renormalisation scheme for the $L\\oplus E$ model that this paper extends to all six models and replaces with three numerically improved scheme variants.","marker":"[9]"},{"why":"Introduced the six vector-like lepton models with seesaw-like mass contributions whose parameterisation the paper adopts.","marker":"[2]"},{"why":"Supplies the conventions for the Lagrangians, mass matrices, mixing matrices and Feynman rules, plus the electroweak precision constraints used for comparison.","marker":"[10]"},{"why":"The renormalisation procedure for the neutralino-type sector whose scheme structure the on-shell construction follows.","marker":"[56]"},{"why":"The one-loop implementation of that scheme logic for chargino and neutralino masses, adapted here to the VLL lepton sector with up to $5\\times5$ mixing.","marker":"[59]"},{"why":"The computer-algebra tool used to generate the generic two-loop beta functions that the paper then translates into redundancy-free form.","marker":"[69]"},{"why":"Supplies the standard one-loop radiative-correction formula for the effective potential used in the NLL computation.","marker":"[70]"},{"why":"The NNLO Standard Model vacuum-stability analysis whose SM instability scale around $10^{11}$ GeV serves as the baseline the VLL results are measured against.","marker":"[73]"},{"why":"Provides the $\\Lambda>10^5$ GeV stability criterion and the analogous $\\lambda_{\\max}$-based bound analysis that motivates the paper's exclusion procedure.","marker":"[43]"}],"fun_headline_variants":["Six vector-like lepton models get mixed on-shell MS renormalisation","Triplet lepton models keep vacuum stable to Planck scale below 0.25","Vector-like lepton mixing can push vacuum instability down to 10^5 GeV","Two-loop RG and improved potential computed for six vector-like lepton models"],"cache_read_input_tokens":0,"weakest_assumption_plain":"All running couplings, the effective potential, and every quoted instability scale inherit the two-loop $\\beta$-functions that the paper obtains by translating generic literature RGEs through field redefinitions in Eqs. (27-35), a translation whose full RGE system is not printed and is not compared against a direct diagrammatic computation.","fun_headline_variants_meta":{"raw":{"variants":["Six vector-like lepton models get mixed on-shell MS renormalisation","Triplet lepton models keep vacuum stable to Planck scale below 0.25","Vector-like lepton mixing can push vacuum instability down to 10^5 GeV","Two-loop RG and improved potential computed for six vector-like lepton models"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.000161,"raw_usage":{"cost":0.0001606332,"is_byok":false,"cost_details":{"upstream_inference_cost":0.0001606332,"upstream_inference_prompt_cost":3.15952e-05,"upstream_inference_completions_cost":0.000129038},"total_tokens":1793,"prompt_tokens":1084,"completion_tokens":709,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":921,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":636}},"tokens_in":1084,"tokens_out":709,"cost_source":"openrouter_usage","duration_ms":7916,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":636,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-28T03:36:08.865001+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"Compute the two-loop renormalisation-group equations for one model, say $L\\oplus E$, directly from the redundant parameter set that keeps off-diagonal Yukawas and Dirac masses, project onto the physical parameters, and compare with the translated redundancy-free $\\beta$-functions; any mismatch in the $1/\\epsilon$ poles would shift every quoted instability scale. A cheaper probe: evaluate a physical observable such as the muon-Higgs coupling or muon $g-2$ both at fixed one-loop order and through the NLL-resummed running, and check that the two agree within the paper's stated uncertainties.","supporting_citations":[{"cited_title":"On-shell Renormalization with Vector-like Leptons, One-loop Muon-Higgs Coupling and Muon g-2","cited_arxiv_id":"2407.09421","evidence_quote":"The prior on-shell renormalisation scheme for the $L\\oplus E$ model that this paper extends to all six models and replaces with three numerically improved scheme variants."},{"cited_title":"Anthropic solution to the magnetic muon anomaly: the charged see-saw","cited_arxiv_id":"1111.2551","evidence_quote":"Introduced the six vector-like lepton models with seesaw-like mass contributions whose parameterisation the paper adopts."},{"cited_title":"From Higgs physics to lepton flavour violation: current bounds and future prospects for vector-like lepton models","cited_arxiv_id":"2603.21414","evidence_quote":"Supplies the conventions for the Lagrangians, mass matrices, mixing matrices and Feynman rules, plus the electroweak precision constraints used for comparison."},{"cited_title":"Renormalization of the Minimal Supersymmetric Standard Model","cited_arxiv_id":"hep-ph/0204350","evidence_quote":"The renormalisation procedure for the neutralino-type sector whose scheme structure the on-shell construction follows."},{"cited_title":"Complete one-loop corrections to the mass spectrum of charginos and neutralinos in the MSSM","cited_arxiv_id":"hep-ph/0203159","evidence_quote":"The one-loop implementation of that scheme logic for chargino and neutralino masses, adapted here to the VLL lepton sector with up to $5\\times5$ mixing."},{"cited_title":"RGBeta: A Mathematica Package for the Evaluation of Renormalization Group $ \\beta $-Functions","cited_arxiv_id":"2101.08265","evidence_quote":"The computer-algebra tool used to generate the generic two-loop beta functions that the paper then translates into redundancy-free form."},{"cited_title":"Radiative Corrections as the Origin of Spon- taneous Symmetry Breaking","cited_arxiv_id":null,"evidence_quote":"Supplies the standard one-loop radiative-correction formula for the effective potential used in the NLL computation."},{"cited_title":"Higgs mass and vacuum stability in the Standard Model at NNLO","cited_arxiv_id":"1205.6497","evidence_quote":"The NNLO Standard Model vacuum-stability analysis whose SM instability scale around $10^{11}$ GeV serves as the baseline the VLL results are measured against."},{"cited_title":"Theoretical constraints on models with vector-like fermions","cited_arxiv_id":"2406.16050","evidence_quote":"Provides the $\\Lambda>10^5$ GeV stability criterion and the analogous $\\lambda_{\\max}$-based bound analysis that motivates the paper's exclusion procedure."}],"review_version":1}