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On-shell Mass Renormalisation and RG-improved Effective Potential in Vector-like Lepton Models

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Gregor Daberstiel, Kilian Möhling, Dominik Stöckinger, Hyejung Stöckinger-Kim

High Energy Physics - Phenomenology · Submitted 27 Oct 2026 · 77 references

Pith's verdict: 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.

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arXiv ID2610.37541 (version 1)
Pith ID3PNDCIHJ, Pith's permanent record of this paper and its review
CitationDominik St\"ockinger, Gregor Daberstiel, Hyejung St\"ockinger-Kim, Kilian M\"ohling. "On-shell Mass Renormalisation and RG-improved Effective Potential in Vector-like Lepton Models." arXiv:2610.37541 (2026). https://pith.science/paper/3PNDCIHJ
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The paper

The paper claims

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…

The reading

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.

What carries the argument

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.

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
Original abstract

We discuss the on-shell mass renormalisation in a class of six vector-like lepton (VLL) extensions of the Standard Model with seesaw-like contributions to the SM lepton masses. The schemes absorb large radiative corrections to the SM lepton masses arising from mixing between the SM and VLLs with up to $5\times5$ mixing matrices and allows for efficient computation of observables and the low-scale $\overline{\text{MS}}$ values of model parameters. We determine and solve the redundancy-free 2-loop renormalisation-group (RG) equations, compute the RG-improved effective potential and study the vacuum instability scales of each of the VLL models. We find rich phase behaviour and parameter regions that show instability scales as low as $10^5$ GeV as well as regions where the one-loop corrections to the input parameters can lead to a stable vacuum.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-28: 1 request, 3,790 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.23 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.96 · compares with prior results 0.11 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We discuss the on-shell mass renormalisation in a class of six vector-like lepton (VLL) extensions of the Standard Model with seesaw-like contributions to the SM lepton masses. The schemes absorb large radiative corrections to the SM lepton masses arising from mixing between the SM and VLLs with up to $5\times5$ mixing matrices and allows for efficient computation of observables and the low-scale $\overline{\text{MS}}$ values of model parameters. We determine and solve the redundancy-free 2-loop renormalisation-group (RG) equations, compute the RG-improved effective potential and study the vacuum instability scales of each of the VLL models. We find rich phase behaviour and parameter regions that show instability scales as low as $10^5$ GeV as well as regions where the one-loop corrections to the input parameters can lead to a stable vacuum.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We discuss the on-shell mass renormalisation in a class of six vector-like lepton (VLL) extensions of the Standard Model with seesaw-like contributions to the SM lepton masses. contribution 0.97 0.87 0.05 asserted 0.89
s1 The schemes absorb large radiative corrections to the SM lepton masses arising from mixing between the SM and VLLs with up to $5\times5$ mixing matrices and allows for efficient computation of observables and the low-scale $\overline{\text{MS}}$ values of model parameters. method 0.80 0.76 0.08 asserted 0.99
s2 We determine and solve the redundancy-free 2-loop renormalisation-group (RG) equations, compute the RG-improved effective potential and study the vacuum instability scales of each of the VLL models. contribution 0.54 0.96 0.13 asserted 1.00
s3 We find rich phase behaviour and parameter regions that show instability scales as low as $10^5$ GeV as well as regions where the one-loop corrections to the input parameters can lead to a stable vacuum. result 1.00 0.96 0.64 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.61 · evidence from simulation or computation 0.66.

What was examined, and what was not

sources read
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not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Illustration of the MS initial values λ MS(MZ ) and MMS L (MZ ) (including the uncertainty estimates from scale variation discussed in sec. 3.2) in L⊕E at the singular limits of OS1 and OS2. In both plots the couplings λ L,E i are set to 0. instead of Eq. (23). The corresponding choices of on-shell renormalised masses defining “OS2” renormalisation schemes are given in the respective column of Tab. 2. The corresponding coefficient matrix needed for solving for the RCs is obtained by dropping columns in κ corresponding to δλ (and δλ¯ in triplet models) as well as the rows correspondi
Figure 2. Plots of λ eff Φ (h) around the instability scale Λ for different parameter points (at ML = ME = 10 TeV) in the L ⊕ E and L ⊕ E a model. (a) shows the typical behaviour for moderate couplings, while (b) illustrates the formation of discontinuous jumps in the instability scale at large couplings. uncertainty is again driven by the QCD corrections. This uncertainty can be determined efficiently by linearising the system in the truncation error d dt∆Θ = ∂β(Θ) ∂Θ ∆Θ + ∆3β(Θ), for Θ = {λΦ, yt , gs}. (46) Here, ∆3β(Θ) denotes the 3-loop contribution to the β-functions purely governed by…
Figure 3. Comparison of the effective quartic Higgs coupling λ eff Φ (h) (solid) and the running MS coupling λ MS Φ (dashed) in L ⊕ E (with 4 active Yukawa couplings) and L ⊕ E a (with one active Yukawa coupling). The parameter points correspond to those already shown in Fig. 2a for the indicated VLL couplings set to 0.32. We also show the uncertainty estimate described in the text. are the same as the ones in Fig. 2a and correspond to the generic situation of a singlet model and the situation close to a discontinuous transition in a triplet model. The zeros of the curves correspond to the…
Figure 4. Contours of the vacuum instability scale Λ and the EWPO 2σ exclusion limits (blue hatched) for the L ⊕ E and L ⊕ E a models in the λ L,E 2 –ML,E plane (top) and λ L 2 –λ E 2 plane (bottom). The colours correspond to values of Λ > 1010 GeV (green), 108 – 1010 GeV (yellow), 106 – 108 GeV (orange) and Λ < 106 GeV (red). All VLL Yukawa couplings not shown are set to 0. and mass scale using simple parametric fits. The fit function is given by log10(Λ/1 GeV) = a0 + a1 − a0 1 + (a2λmax) 4 , (49) such that the fit function interpolates between a1 and a0 when λmax varies from 0 to ∞. The fit…
Figure 5. Plots of Λ(λmax) for the six VLL models. We show the numerical results (solid) with the uncertainty estimate (shaded) and the approximate fit (dashed) for 1 (red) and 4 (green) active Yukawa couplings (as indicated) and ML = ME ≡ M at 2 and 10 TeV. maximum of the BSM Yukawa couplings λmax, the number of Yukawa couplings close to this value and whether the model contains a singlet or triplet VLL. The resulting instability scale Λ was then studied in all models as a function of λmax for various patterns of couplings and masses. There is a complementarity to the parameter dependence of…

Figures reproduced from arXiv:2610.37541 by the authors.

References

Discussion

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