⚛️ phenomenology

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

This paper investigates on-shell mass renormalisation and RG-improved effective potentials in six vector-like lepton extensions of the Standard Model, revealing diverse vacuum stability behaviors and instability scales as low as $10^5$ GeV while providing efficient computational schemes for handling large radiative corrections from SM-VLL mixing.

Original authors: Gregor Daberstiel, Kilian Möhling, Dominik Stöckinger, Hyejung Stöckinger-Kim

Published 2026-10-28

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Gregor Daberstiel, Kilian Möhling, Dominik Stöckinger, Hyejung Stöckinger-Kim

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The universe as we know it is built on a foundation of particles and forces, a delicate architecture that holds everything from the smallest atom to the largest galaxy together. At the heart of this structure lies the Higgs field, an invisible energy that permeates all of space and gives mass to elementary particles. Without it, the particles that make up matter would zip around at the speed of light, unable to clump together to form stars, planets, or people. For decades, physicists have studied this field with increasing precision, confirming that it behaves exactly as predicted by the Standard Model, our best theory of particle physics. Yet, the Standard Model is known to be incomplete; it cannot explain gravity, dark matter, or why the universe contains more matter than antimatter. This has led scientists to propose extensions to the theory, introducing new, heavier particles that interact with the known ones but remain hidden from direct detection. One such proposal involves "vector-like leptons," a class of heavy, mirror-image versions of the familiar electrons and neutrinos that could exist just beyond our current reach.

A team of researchers at the Technical University of Dresden has taken a deep dive into the implications of these hypothetical particles, specifically focusing on how they would alter the stability of the universe itself. In their study, they examined six different scenarios where these heavy leptons mix with the ordinary ones we observe. The central question they asked was whether the introduction of these new particles would cause the vacuum of space—the very ground state of our reality—to become unstable. If the vacuum were unstable, it could theoretically collapse into a different, lower-energy state, a catastrophic event that would rewrite the laws of physics. By developing new mathematical tools to calculate the behavior of these particles with extreme precision, the team mapped out the conditions under which our universe remains safe and the points where it might teeter on the edge of collapse.

The researchers began by acknowledging a significant challenge in theoretical physics: calculating the effects of new particles is notoriously difficult because the math often becomes tangled with redundant information. When particles mix, as they do in these models, the equations can produce multiple solutions that describe the same physical reality, making it hard to pin down exact values for mass and interaction strength. To solve this, the team constructed a rigorous framework for "renormalization," a process that allows physicists to separate the infinite, meaningless parts of a calculation from the finite, physical results. They developed a method that treats the masses of known particles, like the electron, as fixed, observable facts, while allowing the properties of the new, heavy particles to be calculated with high precision. This approach is akin to calibrating a scale: by ensuring the weight of a known object is correct, one can trust the measurement of a new, unknown object placed on the same scale. They applied this method to six distinct models, each featuring different combinations of heavy leptons, and solved complex equations that describe how these particles influence the Higgs field over time.

Their calculations revealed a landscape of possibilities that is far richer and more varied than previously understood. In models where the new particles form "triplets"—groups of three related heavy states—the vacuum of space tends to be remarkably stable. Even with the presence of these heavy particles, the universe remains secure up to the highest energy scales imaginable, far beyond the limits of our current particle accelerators. However, in models where the new particles are "singlets"—single, isolated heavy states—the situation is more precarious. Here, the interaction between the heavy particles and the Higgs field can drive the vacuum toward instability. The researchers found that if the interaction strength, or "coupling," between these new particles and the Higgs field is too strong, the energy of the vacuum can drop, creating a deeper, more stable state that our universe could fall into.

The study identified specific thresholds where this stability breaks down. For the singlet models, the team found regions where the universe could become unstable at energy scales as low as 100,000 gigaelectronvolts, a value significantly lower than the 100 billion gigaelectronvolts where the Standard Model alone predicts instability. This means that if these heavy particles exist and interact strongly with the Higgs field, the universe might be far less stable than we thought. Conversely, they also discovered that in certain parameter ranges, the quantum corrections from these particles could actually help stabilize the vacuum, pushing the point of instability to even higher energies. The researchers noted that the behavior of the vacuum is not always smooth; in some cases, small changes in the strength of the particle interactions can lead to sudden, discontinuous jumps in the stability scale, causing the universe to flip from a stable state to an unstable one almost instantly.

To make these findings useful for future research, the team provided a set of practical formulas that allow other scientists to estimate the stability of the vacuum based on the mass and interaction strength of these hypothetical particles. These formulas act as a guide, helping researchers determine which combinations of particle properties are physically viable and which would lead to a universe that could not exist. They emphasized that while their results are based on sophisticated simulations and mathematical derivations, they are robust enough to draw firm conclusions about the general behavior of these models. The work highlights a crucial complementarity between different types of constraints: while experiments measuring the precision of known particle interactions limit how heavy these new particles can be, the requirement of vacuum stability limits how strongly they can interact with the Higgs field. Together, these constraints carve out a narrow window of possibility where such particles could exist without destroying the universe.

The implications of this work extend beyond mere theoretical curiosity. By establishing a clear link between the properties of hypothetical heavy particles and the long-term fate of the universe, the study provides a powerful tool for testing the validity of these models. If future experiments at particle colliders discover evidence of vector-like leptons, physicists will be able to use these new formulas to immediately check whether the discovery is consistent with a stable universe. The researchers concluded that their method of calculation, which carefully handles the complexities of particle mixing and renormalization, sets a new standard for precision in this field. It allows for a detailed exploration of the "what if" scenarios of particle physics, ensuring that our understanding of the universe's stability is grounded in the most accurate calculations possible. The universe, it seems, is a delicate balance, and these new heavy particles could be the tipping point that either secures its future or seals its fate.

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

Problem Statement
Vector-like lepton (VLL) extensions of the Standard Model (SM) offer attractive alternatives to scalar sector extensions, particularly those incorporating seesaw-like mass contributions. In these models, observed SM lepton masses arise from mixing with heavy VLL states, leading to chirally enhanced radiative corrections. While these models can explain anomalies in lepton observables (e.g., the muon anomalous magnetic moment), they introduce significant theoretical challenges regarding renormalisation and vacuum stability.

A primary difficulty lies in constructing consistent renormalisation schemes. Unlike the SM, where physical masses can generally be on-shell renormalised, VLL models involve complex mixing matrices (up to $5 \times 5$ ) and parameter redundancies. Standard on-shell conditions may become numerically unstable or singular in specific limits (e.g., vanishing Higgs-induced mixing or degenerate Dirac masses). Furthermore, determining the stability of the electroweak (EW) vacuum in the presence of these new states requires precise higher-order calculations. Previous studies have shown that loop corrections can significantly alter constraints on VLL parameters and vacuum stability scales, yet a systematic treatment of on-shell mass renormalisation combined with two-loop Renormalisation Group (RG) improvement for all six minimal VLL models has been lacking.

Methodology
The authors develop a comprehensive framework for six specific VLL models ( $L \oplus E$ , $L \oplus N$ , $L^{3/2} \oplus E$ , $L \oplus E_a$ , $L \oplus N_a$ , and $L^{3/2} \oplus E_a$ ), where $L$ denotes a doublet and $E, N$ denote singlets or triplets with specific hypercharges.

Renormalisation Schemes:
- On-Shell (OS) Conditions: The authors define three distinct renormalisation schemes (OS1, OS2, OS3) for each model. These schemes impose on-shell conditions on all SM-like lepton masses and a selected subset of heavy VLL masses.
- Handling Redundancies: The construction addresses redundancies between field redefinitions and Lagrangian parameters. By eliminating unphysical off-diagonal Yukawa couplings and phases via field redefinitions, the authors derive "redundancy-free" sets of fundamental parameters.
- Scheme Variants:
  - OS1: Maximizes on-shell conditions but suffers from singularities when Higgs mixing vanishes.
  - OS2: Reduces on-shell conditions to avoid OS1 singularities but fails when Dirac masses are degenerate with large mixing.
  - OS3: A refined scheme that dynamically selects which masses are on-shell renormalised based on the hierarchy of Dirac mass terms ( $M_L$ vs $M_E$ ), ensuring numerical stability across the entire parameter space.
- Counterterms: The finite parts of the counterterms for fundamental parameters (Yukawa couplings, Dirac masses) are determined by inverting linear systems derived from the on-shell mass conditions. The remaining parameters are renormalised in the $\overline{\text{MS}}$ scheme.
Renormalisation Group Equations (RGEs):
- The authors derive two-loop $\beta$ -functions for the redundancy-free parameters. They explicitly account for the difference between generic literature results (which often assume hermitian field renormalisation matrices) and their specific treatment of field redefinitions. This leads to additional terms in the $\beta$ -functions to preserve the redundancy-free structure.
Effective Potential and Vacuum Stability:
- The RG-improved effective potential is computed at Next-to-Leading Log (NLL) accuracy. This combines one-loop fixed-order corrections (including contributions from VLLs and SM leptons via the Coleman-Weinberg formula) with two-loop RGE running.
- To resolve the fine-tuning problem associated with the Higgs mass parameter $\mu^2$ , the authors derive the $\overline{\text{MS}}$ value of $\mu^2$ directly from the minimisation condition of the effective potential ( $dV_{\text{eff}}/dh = 0$ ) rather than relying solely on tree-level relations, thereby avoiding large scheme-conversion artifacts.
- The instability scale $\Lambda$ is defined as the field value where a deeper minimum develops (i.e., where the effective quartic coupling $\lambda_{\text{eff}}$ vanishes).
Uncertainty Estimation:
- Uncertainties are estimated via parametric variations (Higgs mass, top mass, $\alpha_s$ ), scale variation (to estimate missing higher-order corrections), and RGE truncation errors. The authors propose taking the pointwise maximum of these uncertainties to handle discontinuous transitions in the instability scale.

Key Contributions and Results

Robust Renormalisation Schemes: The paper provides explicit, numerically stable renormalisation schemes for all six VLL models. The OS3 scheme is highlighted as the most robust, avoiding singularities found in simpler schemes by adapting the choice of on-shell masses to the parameter hierarchy.
Two-Loop RGEs: The authors provide the first redundancy-free two-loop $\beta$ -functions for these specific VLL models, correcting generic results to account for the specific field redefinitions required to eliminate unphysical parameters.
Vacuum Stability Analysis:
- Singlet Models: In models with VLL singlets, the vacuum tends to be less stable than in the SM. The instability scale $\Lambda$ can drop significantly (as low as $10^5$ GeV) depending on the VLL Yukawa couplings.
- Triplet Models: Models with VLL triplets exhibit a stabilising effect due to additional gauge contributions. For small Yukawa couplings ( $\lambda_{\text{max}} \lesssim 0.25$ ), the vacuum remains stable up to and beyond the Planck scale ( $\Lambda \sim 10^{19}$ GeV).
- Discontinuous Transitions: The study reveals rich phase behaviour, including discontinuous jumps in the instability scale as a function of coupling strengths. This occurs when the effective potential develops or loses secondary minima, a feature captured by the NLL-improved potential.
Complementarity with EWPOs: The analysis demonstrates that vacuum stability constraints are complementary to Electroweak Precision Observables (EWPO). While EWPOs constrain the ratio of Yukawa couplings to masses, vacuum stability provides direct constraints on the magnitude of Yukawa couplings, becoming the dominant constraint for VLL masses above $\sim 2$ TeV.
Analytic Fits: The authors provide simple parametric fits (Eq. 49) for the instability scale $\Lambda$ as a function of the maximum Yukawa coupling $\lambda_{\text{max}}$ , valid for continuous regions of the parameter space. These fits allow for quick estimation of stability bounds in phenomenological studies.

Significance and Claims
The paper claims that its setup allows for precise computations in VLL models while keeping physical lepton masses fixed to their observed values, a crucial requirement for phenomenological viability. The authors emphasize that quantum corrections are sizeable and essential for robust predictions, noting that loop corrections can significantly lower limits on VLL contributions to observables like the muon $g-2$ and alter vacuum stability bounds.

The work establishes that:

On-shell renormalisation of SM lepton masses in VLL models is feasible and necessary to avoid computationally expensive iterations to determine tree-level Yukawa couplings.
The vacuum stability of the EW vacuum is highly sensitive to the VLL sector, with triplet models offering potential stabilization up to the Planck scale, while singlet models generally lower the instability scale.
The derived redundancy-free two-loop RGEs and the specific treatment of the effective potential minimisation are critical for obtaining reliable instability scales, particularly in regions where discontinuous transitions occur.

The authors conclude that their framework can be extended to compute other observables, such as flavour- and CP-violating decays, to further constrain the VLL parameter space.

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