arxiv: 2512.24215 · v2 · submitted 2025-12-30 · ✦ hep-ph

Recognition: no theorem link

Neutrino Mass, Vacuum Stability and Higgs Inflation with Vector-Like Quarks and a Single Right-Handed Neutrino

Canan Karahan

Authors on Pith no claims yet

Pith reviewed 2026-05-16 18:55 UTC · model grok-4.3

classification ✦ hep-ph

keywords neutrino massvacuum stabilityHiggs inflationvector-like quarksright-handed neutrinorenormalization group evolutionType-I seesawelectroweak vacuum

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The pith

A Standard Model extension with n degenerate vector-like quarks and one right-handed neutrino stabilizes the electroweak vacuum to the Planck scale while generating viable neutrino masses and matching Higgs inflation observables to data.

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

The paper investigates an extension of the Standard Model that adds n degenerate down-type vector-like quarks together with a single right-handed neutrino. These additions modify the renormalization group running of the Higgs quartic coupling so that it stays positive up to the Planck scale. The right-handed neutrino simultaneously accounts for the observed light neutrino masses through the Type-I seesaw and improves the high-scale behavior of the potential. Using two-loop Standard Model running plus one-loop new particle effects with threshold matching, the authors locate parameter regions that achieve vacuum stability, produce acceptable neutrino masses, and yield inflationary predictions for the spectral index and tensor ratio that agree with the latest Planck and ACT data.

Core claim

The SM+(n)VLQ+RHN model produces RG trajectories that keep the Higgs potential stable to the Planck scale, generate phenomenologically viable neutrino masses via Type-I seesaw, and deliver n_s and r values consistent with Planck-LB-BK18 and ACT-LB-BK18 observations in the metric formulation of non-minimal Higgs inflation.

What carries the argument

Renormalization group evolution of the Higgs quartic coupling including one-loop contributions from degenerate vector-like quarks and the right-handed neutrino, with threshold matching, to obtain the RG-improved potential for non-minimal inflation.

If this is right

Specific regions of the parameter space (n, y_D, M_D) ensure the Higgs quartic remains positive to the Planck scale.
The Type-I seesaw mechanism from the single right-handed neutrino produces light neutrino masses consistent with experiment.
The computed inflationary observables n_s and r lie within the allowed contours of recent CMB data.
The VLQ contributions provide the primary stabilization effect, with the RHN playing a supporting role at high scales.

Where Pith is reading between the lines

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

Vector-like quarks in the stabilizing mass range could be searched for at high-energy colliders.
Removing the degeneracy assumption on the VLQs could enlarge the viable parameter space.
The same RG stabilization technique might address related issues such as coupling unification at high scales.
Future precision measurements of the Higgs self-coupling could test the required threshold corrections.

Load-bearing premise

The vector-like quarks must be degenerate with identical masses and Yukawa couplings, and the one-loop threshold matching plus the specific non-minimal coupling must suffice for accurate high-scale running.

What would settle it

Detection of a Higgs quartic coupling that turns negative at scales below the Planck scale in improved calculations, or inflationary observables outside the model's predicted range, or no evidence for vector-like quarks in the relevant mass window.

Figures

Figures reproduced from arXiv: 2512.24215 by Canan Karahan.

**Figure 2.** Figure 2: FIG. 2. Running of the Higgs quartic coupling [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Vacuum stability in the SM+( [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Running of the Higgs quartic coupling [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Predictions for the spectral index [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Predictions for the spectral index [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗

read the original abstract

We investigate a Standard Model extension containing $n$ degenerate down-type isosinglet vector-like quarks (VLQs) with masses $M_{\mathcal D}$ and Yukawa couplings $y_{\mathcal D}$, supplemented by a single right-handed neutrino (RHN), aiming to simultaneously address neutrino mass generation, electroweak vacuum stability, and Higgs inflation. The VLQs play the dominant role in stabilizing the Higgs potential through their impact on the renormalization-group evolution, while the RHN generates light neutrino masses via a Type-I seesaw mechanism and smooths the high-scale running of the Higgs quartic coupling in the inflationary regime. We perform a two-loop Standard Model renormalization-group equation analysis supplemented by the one-loop contributions of the VLQs and the RHN, with proper matching across their mass thresholds. Using these RG trajectories, we identify the regions in $(n,\, y_{\mathcal D},\, M_{\mathcal D})$ that stabilize the Higgs potential up to the Planck scale while satisfying experimental constraints. Employing the RG-improved Higgs potential in the metric formulation of non-minimal Higgs inflation, we compute the inflationary observables $n_s$ and $r$. The SM+$(n)$VLQ+RHN framework yields predictions consistent with the latest Planck-LB-BK18 and ACT-LB-BK18 data, while simultaneously ensuring electroweak vacuum stability and phenomenologically viable neutrino masses within well-defined regions of parameter space. For comparison, we also investigate the SM+$(n)$VLQ limit and present its vacuum stability and Higgs inflation predictions as a reference to quantify the stabilizing role of the VLQ sector.

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 combines n degenerate VLQs and one RHN to stabilize the vacuum, fit neutrino data, and match Higgs inflation observables, though one-loop RG additions leave some uncertainty in the high-scale behavior.

read the letter

This paper takes n degenerate vector-like down-type quarks plus one right-handed neutrino and shows they can stabilize the electroweak vacuum, generate viable neutrino masses, and produce Higgs inflation consistent with recent data. The core result is that certain ranges of n, the VLQ Yukawa, and mass allow the Higgs quartic to stay positive to the Planck scale while the seesaw works and the inflationary predictions land inside the Planck-LB-BK18 and ACT contours. The new part is the specific combination with exactly one RHN and degenerate VLQs treated together for all three issues. They do a two-loop SM running supplemented by one-loop new physics terms, match at thresholds, and then use the improved potential for the metric non-minimal inflation calculation. That joint treatment is a reasonable extension of earlier work on VLQs for stability or RHN for seesaw. It does the job of mapping out viable parameter space and comparing to the VLQ-only case to show the extra smoothing from the RHN. The claims stay within standard techniques, so the numerics look plausible on the surface. The main limitation is the one-loop approximation for the VLQ and RHN beta functions. Two-loop corrections from these particles could be comparable near the would-be instability and might move the stability regions or shift ns and r enough to change whether they fit the data. The degeneracy assumption also simplifies the coefficients; splitting the masses or Yukawas would alter the running. The abstract does not show error estimates or full tables, so the robustness is not fully clear from what's presented. This kind of model-building paper is useful for specialists who build extensions that hit multiple targets at once. A reader looking for concrete parameter regions that satisfy stability plus inflation data would find it worth reading. It deserves peer review because the framework is internally consistent and the results are falsifiable with better data or higher-order calculations. I would recommend sending it to referees, with a request to verify the size of the missing two-loop terms.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates a Standard Model extension with n degenerate down-type vector-like quarks (VLQs) of mass M_D and Yukawa y_D, plus a single right-handed neutrino (RHN). It performs a two-loop SM RGE analysis supplemented by one-loop VLQ and RHN contributions with threshold matching to identify parameter regions (n, y_D, M_D) that stabilize the Higgs potential to the Planck scale, generate viable neutrino masses via Type-I seesaw, and yield n_s and r from the RG-improved non-minimal Higgs potential (metric formulation) consistent with Planck-LB-BK18 and ACT-LB-BK18 data. A comparison to the SM+nVLQ case is included to isolate the VLQ stabilizing role.

Significance. If the results hold, this provides a minimal unified framework simultaneously addressing neutrino mass, electroweak vacuum stability, and Higgs inflation, with explicit viable parameter regions and a direct comparison quantifying each sector's contribution. The RG-improved potential approach for inflation and the use of latest cosmological data contours are standard strengths that make the predictions falsifiable.

major comments (2)

[RG evolution section (beta functions for λ)] The renormalization-group analysis (abstract and RG evolution section) employs two-loop SM beta functions for λ but only one-loop contributions from the VLQs and RHN, with one-loop threshold matching at M_D and the RHN mass. Near the would-be instability scale, two-loop corrections from the new fields can be comparable in size to the SM two-loop pieces and can alter the sign of β_λ or shift the location of the minimum; this directly impacts the central claim of stability up to the Planck scale and the subsequent n_s, r predictions.
[Parameter scan and degeneracy assumption] The VLQs are assumed degenerate with identical masses and Yukawas (abstract and parameter scan). This collapses the beta-function coefficients; any realistic mass or Yukawa splitting would modify the running of λ and the high-scale behavior, potentially moving the stability regions and the (n_s, r) points relative to the Planck-LB-BK18 and ACT-LB-BK18 contours.

minor comments (2)

[Abstract and results section] The abstract states that the RHN 'smooths the high-scale running' but the explicit numerical impact on β_λ from the RHN one-loop term versus the VLQ terms is not quantified in a dedicated plot or table.
[Inflation section] Notation for the non-minimal coupling ξ and the precise matching procedure across thresholds could be clarified with an explicit equation for the threshold correction to λ.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below, indicating planned revisions where appropriate.

read point-by-point responses

Referee: The renormalization-group analysis (abstract and RG evolution section) employs two-loop SM beta functions for λ but only one-loop contributions from the VLQs and RHN, with one-loop threshold matching at M_D and the RHN mass. Near the would-be instability scale, two-loop corrections from the new fields can be comparable in size to the SM two-loop pieces and can alter the sign of β_λ or shift the location of the minimum; this directly impacts the central claim of stability up to the Planck scale and the subsequent n_s, r predictions.

Authors: We agree that a complete two-loop treatment of the BSM contributions would be desirable for maximal precision near the would-be instability scale. Our analysis follows the common approach in the literature of retaining two-loop SM beta functions while adding one-loop BSM terms, which capture the dominant positive contribution to β_λ from the VLQ Yukawa y_D. For the parameter regions we identify, the stabilization occurs sufficiently below the Planck scale that the leading one-loop BSM effects control the sign of β_λ. In the revised manuscript we will add a dedicated paragraph in the RG section estimating the expected size of the omitted two-loop BSM corrections (based on the magnitude of the one-loop terms) and explicitly noting that a full two-loop BSM calculation is left for future work. revision: partial
Referee: The VLQs are assumed degenerate with identical masses and Yukawas (abstract and parameter scan). This collapses the beta-function coefficients; any realistic mass or Yukawa splitting would modify the running of λ and the high-scale behavior, potentially moving the stability regions and the (n_s, r) points relative to the Planck-LB-BK18 and ACT-LB-BK18 contours.

Authors: The degeneracy assumption is introduced to keep the parameter space minimal while isolating the stabilizing role of the VLQ sector. For small relative splittings the beta-function coefficients receive averaged contributions, so the qualitative location of the stability bands and the resulting (n_s, r) predictions are expected to remain similar. We will insert a short clarifying remark in the parameter-scan section stating that the degenerate case is adopted as a benchmark and that non-degenerate spectra would require a broader numerical scan, which we leave for future study. revision: partial

Circularity Check

1 steps flagged

Parameter scan over n, y_D, M_D selects regions that enforce stability and data consistency, rendering ns/r agreement fitted rather than predicted

specific steps

fitted input called prediction [Abstract (final paragraph) and sections on RG trajectories + inflationary observables]
"The SM+(n)VLQ+RHN framework yields predictions consistent with the latest Planck-LB-BK18 and ACT-LB-BK18 data, while simultaneously ensuring electroweak vacuum stability and phenomenologically viable neutrino masses within well-defined regions of parameter space."

The 'well-defined regions' are obtained by scanning n, y_D, M_D until both vacuum stability and the ns/r values fall inside the experimental contours; the quoted consistency is therefore the result of that selection, not an a-priori prediction of the RGEs.

full rationale

The paper scans the discrete parameter space (n, y_D, M_D) to locate intervals that keep the Higgs quartic positive to the Planck scale and produce ns, r inside the cited Planck/ACT contours after RG improvement. Once those intervals are chosen, the inflationary observables are read off from the same RG trajectories; the agreement is therefore a direct consequence of the selection criterion rather than an independent output of the equations. The underlying two-loop SM + one-loop new-physics RGEs themselves contain no circularity, but the load-bearing claim that the framework 'yields predictions consistent with data' reduces to the fitting step. No self-citation chains, self-definitional equations, or smuggled ansatze are present.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 2 invented entities

The central claim rests on standard QFT renormalization-group methods plus two new particle sectors whose effects are added at one-loop; no machine-checked proofs or external data anchors are supplied.

free parameters (3)

n
Integer number of degenerate VLQs chosen to achieve stability up to Planck scale.
y_D
Yukawa coupling of the VLQs scanned over ranges that keep the quartic positive.
M_D
Common mass of the VLQs used for threshold matching in the RG flow.

axioms (2)

standard math Two-loop Standard Model RGEs supplemented by one-loop VLQ and RHN contributions with proper threshold matching
Invoked in the abstract as the computational backbone.
domain assumption Type-I seesaw mechanism generates light neutrino masses from the single RHN
Standard assumption for neutrino mass generation in the model.

invented entities (2)

n degenerate down-type vector-like quarks no independent evidence
purpose: Dominant stabilization of the Higgs quartic via RG evolution
New particles introduced to alter the running of the Higgs self-coupling.
single right-handed neutrino no independent evidence
purpose: Generates neutrino masses and smooths high-scale Higgs running
Postulated to implement Type-I seesaw and assist inflation regime.

pith-pipeline@v0.9.0 · 5598 in / 1584 out tokens · 69405 ms · 2026-05-16T18:55:42.378813+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

64 extracted references · 64 canonical work pages · 23 internal anchors

[1]

computations of the Higgs effective potential. In literature, numerous extensions of the Standard Model have been proposed to address this issue, such as introducing additional scalar fields to raise the Higgs quartic coupling at high energies [17–20], or new fermions/gauge sectors that alter the renormalization-group (RG) running ofλ H [21–23]. In partic...

work page
[2]

For anSU(2) L singlet with hyperchargeY=− 1 3, the covariant derivative isD µ =∂ µ −ig 3T aGa µ −ig ′Y Bµ

Down-Type Isosinglet Vector-Like Quark Sector The kinetic and mass terms of down-type isosinglet VLQ sector are LD = Di /DD −M D DD,(3) whereM D is a gauge-invariant Dirac mass. For anSU(2) L singlet with hyperchargeY=− 1 3, the covariant derivative isD µ =∂ µ −ig 3T aGa µ −ig ′Y Bµ. Gauge invariance permits for a single renormalizable Yukawa operator: LY...

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[3]

Here, ˜H≡iσ 2H ∗ denotes theSU(2) L- conjugate Higgs doublet

Right-Handed Neutrino Sector The RHN kinetic and Majorana mass terms are LN = 1 2 N i /∂ N− 1 2 MN N cN.(6) The neutrino Yukawa interaction is LYuk N =−y N L eH N+ h.c.,(7) producing a Dirac massm D =y N v/ √ 2 after electroweak symmetry breaking. Here, ˜H≡iσ 2H ∗ denotes theSU(2) L- conjugate Higgs doublet. A light neutrino mass arises from the Type-I se...

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[4]

This relation follows from the standard diagonalization of the VLQ–SM mass matrix and is consistent with the formalism developed in vector-like quark studies [36, 37]

Experimental constraints on the Yukawa couplingy D and massM D In the SM extended by down-type isosinglet VLQs with the massM D and Yukawa interactiony D, the mass terms after electroweak symmetry breaking lead to a left-handed mixing angle defined by sinθ L ≃ yD v√ 2M D ,(11) wherev≃246 GeV is the Higgs vacuum expectation value. This relation follows fro...

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[5]

These constraints ensure that the theory remains perturbative, unitary, and free of Landau poles up to the Planck scale

Theoretical constraints on the Yukawa couplingy D In addition to the experimental limits discussed above, the down-type VLQ Yukawa couplingy D is subject to several theoretical consistency conditions. These constraints ensure that the theory remains perturbative, unitary, and free of Landau poles up to the Planck scale. a. Perturbativity and Landau–pole c...

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[6]

The point at which a given trajectory intersects the lineλ min = 0 signals the onset of vacuum instability for the corresponding (n, M D)

For fixed values of (n, M D), the minimum value of the Higgs quartic coupling,λ min, exhibits a monotonic decrease as the VLQ Yukawa couplingy D is increased. The point at which a given trajectory intersects the lineλ min = 0 signals the onset of vacuum instability for the corresponding (n, M D). Parameter regions with λmin >0 correspond to an absolutely ...

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Comparing the three panels, the range of Yukawa couplingsy D that keepsλ min >0 becomes more restrictive as VLQ massM D increase

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For sufficiently small Yukawa couplings, increasingnshiftsλ min upward and improves vacuum stability

At all three benchmark masses, the dependence on the number of VLQsnshows a characteristic two–regime behaviour. For sufficiently small Yukawa couplings, increasingnshiftsλ min upward and improves vacuum stability. However, beyond a certainy D value in each panel, the destabilizing∼ −n y 4 D contribution dominates the running, and the trend reverses: larg...

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Forn≥3, vacuum stability can be achieved within a finite interval of smally D, but this interval becomes progressively narrower asM D is increased

In particular, the casen= 1 andn= 2 never yields a stable electroweak vacuum for any of the benchmark 8 masses. Forn≥3, vacuum stability can be achieved within a finite interval of smally D, but this interval becomes progressively narrower asM D is increased. Taken together, these results show that, in the presence of a single RHN withy N = 0.42, restorin...

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Consequently, for eachn,λ min stays higher than in the full SM+(n)VLQ+RHN model

Removing the RHN Yukawa eliminates the negative contribution that previously pushedλ min downward at high scales. Consequently, for eachn,λ min stays higher than in the full SM+(n)VLQ+RHN model. This confirms that VLQs alone have a net stabilizing tendency at small Yukawa couplings

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However, without the RHN, this stabilizing effect is stronger and persists to larger values ofy D

As in the full model, increasing the number of VLQs shifts the running upward, delaying (or preventing) the 10 point at whichλ min turns negative. However, without the RHN, this stabilizing effect is stronger and persists to larger values ofy D

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In other words, the destabilizing regime sets in more softly because only the VLQ sector drives it, not the RHN

The characteristic transition seen previously — where large values ofneventually accelerate the fall ofλ min still appears, but with a reduced rate and at later scales. In other words, the destabilizing regime sets in more softly because only the VLQ sector drives it, not the RHN

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Here, for SM+(n)VLQ, even smallnvalues lead to a visibly less negativeλ min, and several trajectories remain close to or above zero

In the SM+(n)VLQ+RHN case,n= 1 andn= 2 never achieved stability in the allowed parameter space. Here, for SM+(n)VLQ, even smallnvalues lead to a visibly less negativeλ min, and several trajectories remain close to or above zero. Forn= 2, a vacuum–stable region exists only for the lightest VLQ massM D = 1.5 TeV, while forM D = 3.0 and 5.0 TeV then= 2 curve...

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In this case, vacuum stability is still governed primarily by the interplay betweennandy D

Overall, the SM+(n)VLQ scenario leads to improved electroweak vacuum stability compared to the SM+(n)VLQ +RHN model for all considered values ofnandM D, as reflected by larger values ofλ min and a broader region withλ min >0. In this case, vacuum stability is still governed primarily by the interplay betweennandy D. The absence of the RHN Yukawa contribut...

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Standard Model (1-loop) At one loop the SMβ-functions for the gauge couplings, top Yukawa and Higgs quartic coupling read [42] βSM,1L g1 = 41 10 g3 1,(A2) βSM,1L g2 =− 19 6 g3 2,(A3) βSM,1L g3 =−7g 3 3,(A4) βSM,1L yt =y t 9 2 y2 t −8g 2 3 − 9 4 g2 2 − 17 20 g2 1 ,(A5) βSM,1L λ = 24λ2 −6y 4 t + 9 8 g4 2 + 27 200 g4 1 + 9 20 g2 2g2 1 +λ 12y2 t −9g 2 2 − 9 5...

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Standard Model (2-loop top-sector terms) We also include the dominant two-loop terms involving the top Yukawa, matching exactly the structure used in the numerical implementation [43]: βSM,2L g1 =g 3 1 199 50 g2 1 + 27 10 g2 2 + 44 5 g2 3 − 17 10 y2 t ,(A7) βSM,2L g2 =g 3 2 9 10 g2 1 + 35 6 g2 2 + 12g2 3 − 3 2 y2 t ,(A8) βSM,2L g3 =g 3 3 11 10 g2 1 + 9 2 ...

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Down-type singlet VLQ contributions (1 loop) In the following, we present the RGE fory D together with the corresponding one-loop contributions relevant for the present analysis [24, 25]. βVLQ yD =y D 3 2 y2 t + 3 2 + 3n y2 D −8g 2 3 − 9 4 g2 2 − 1 4 g2 1 ,(A12) ∆βVLQ g1 = 4 15 ng3 1,(A13) ∆βVLQ g2 = 0,(A14) ∆βVLQ g3 = 2 3 ng3 3,(A15) ∆βVLQ yt = 3 2 n y2 ...

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[44, 45] βRHN yN =y N 5 2 y2 N + 3n y2 D + 3y2 t − 9 20 g2 1 − 9 4 g2 2 ,(A18) ∆βRHN λ =−2y 4 N + 4y2 N λ.(A19) 17

Majorana RHN contributions (1 loop) In the following, we present the RGE for the RHN Yukawa coupling and retain only its dominant one-loop con- tributions to the Higgs quartic coupling, which are sufficient to capture the impact of the RHN sector on vacuum stability. [44, 45] βRHN yN =y N 5 2 y2 N + 3n y2 D + 3y2 t − 9 20 g2 1 − 9 4 g2 2 ,(A18) ∆βRHN λ =−...

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Full RG system in theSM+ (n)V LQ+RHNFramework In this part, we summarize the RGEs for the couplings in theSM+ (n)V LQ+RHNframework. Collecting all contributions, the fullβ-functions used in the numerical analysis read βg1 = 1 16π2 βSM,1L g1 + 1 (16π2)2 βSM,2L g1 + 1 16π2 ∆βVLQ g1 ,(A20) βg2 = 1 16π2 βSM,1L g2 + 1 (16π2)2 βSM,2L g2 ,(A21) βg3 = 1 16π2 βSM,...

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