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Indirect probes of heavy neutral leptons through the dipole portal at FCC-ee

This paper investigates how virtual Majorana heavy neutral leptons interacting via a gauge-invariant dipole portal modify $Z$ -pole observables at FCC-ee, demonstrating that with improved Standard Model predictions, the facility can probe dipole coefficients as small as $0.14\,\mathrm{TeV}^{-2}$ for 1 TeV HNLs, achieving sensitivity competitive with HL-LHC searches while quantifying the impact of additional ultraviolet contributions.

Original author: Natascia Vignaroli

Published 2026-11-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original author: Natascia Vignaroli

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 author. For technical accuracy, refer to the original paper. Read full disclaimer

The universe is filled with particles that we can see and touch, like the electrons that power our lights and the protons that form the core of every atom. But there are also invisible players, such as neutrinos, which zip through everything without leaving a trace. For decades, physicists have wondered if there are even heavier, stranger cousins to these neutrinos, hidden in the shadows of known physics. These hypothetical particles, called heavy neutral leptons, could hold the key to understanding why the universe has more matter than antimatter and why neutrinos have mass at all. The challenge is that if these particles are very heavy, they are too massive to be created directly in current particle accelerators. They would be like a ghost that is too heavy to be caught in a net, yet its presence might still be felt by the way it disturbs the water around it.

To find these ghosts, scientists at the Future Circular Collider (FCC-ee), a proposed machine that will smash electrons and positrons together with unprecedented precision, are looking for subtle ripples in the behavior of known particles. Specifically, they are studying how the Z boson, a carrier of the weak nuclear force, behaves when it interacts with other particles. If heavy neutral leptons exist, they would not just sit idle; they would interact with the Z boson and other forces through a specific mechanism known as a "dipole portal." This interaction is like a temporary, invisible handshake between the heavy particle and the forces of nature, occurring only for a fleeting moment before the heavy particle vanishes again. Even though the heavy particle never appears on the screen, its brief existence would slightly alter the speed and strength of the Z boson's interactions, leaving a fingerprint that precise instruments could detect.

A recent study by Natascia Vignaroli maps out exactly how to read these fingerprints. The research focuses on a scenario where these heavy particles are Majorana fermions, meaning they are their own antiparticles, and interact through the dipole portal. The author calculated the complex loops of virtual particles that would form if these heavy leptons existed, determining how they would shift the properties of the Z boson and the way it decays into other particles. The calculation was rigorous, accounting for the fact that the heavy particles are not just simple points but are connected to the Higgs field and the electromagnetic field in a way that respects the fundamental symmetries of the universe. The study found that if these heavy particles exist with a mass of one trillion electron volts, the FCC-ee could detect their influence even if the strength of their interaction is as small as 0.14 inverse teraelectron volts squared. This level of sensitivity is a significant leap forward, allowing the machine to probe a range of interactions that are currently out of reach for direct searches.

However, the study also highlights a crucial caveat: the signal from these heavy particles is not unique. Other unknown physics could produce the exact same ripples in the data. The research shows that the dipole interaction changes the behavior of the Z boson in four specific ways. If other, unrelated forces were also at play, they could mimic these changes, making it impossible to say for certain that the heavy leptons are the cause. The paper demonstrates that unless scientists can assume that no other new forces are interfering, the ability to pinpoint the heavy leptons disappears. In a world where only the dipole interaction is active, the FCC-ee would be a powerful microscope, capable of seeing these heavy particles indirectly. But if the universe is more complex, with multiple new forces acting at once, the signal becomes ambiguous.

The study also looked at whether the Higgs boson, the particle responsible for giving mass to others, could offer a second clue. The heavy leptons would also affect how the Higgs decays into light particles like photons. The researchers found that while this provides a small amount of extra information, it is not enough to solve the ambiguity on its own. The primary power remains in the precision measurements of the Z boson. When compared to the High-Luminosity Large Hadron Collider, which looks for these particles by trying to create them directly, the FCC-ee approach offers a different kind of advantage. While the collider searches for the heavy particle itself, the FCC-ee looks for the shadow it casts. For particles with a mass of one trillion electron volts, the precision of the FCC-ee is competitive with, and in some cases superior to, the direct search capabilities of the collider.

Ultimately, this work provides a clear roadmap for what to expect if these heavy particles exist and how to distinguish their signal from background noise. It confirms that the FCC-ee will be able to test the existence of these heavy neutral leptons through their virtual effects, provided that the universe does not contain a confusing mix of other new forces that hide the signal. The findings suggest that if the dipole interaction is the dominant new physics, the FCC-ee will be able to probe coefficients as small as 0.14 inverse teraelectron volts squared for a one-trillion-electron-volt particle. This represents a profound deepening of our ability to see the invisible, turning the collider into a detector for the ghosts of the subatomic world. The study does not claim to have found these particles, but it defines the exact conditions under which they could be found, setting a precise target for the next generation of physics experiments.

Technical Summary: Indirect probes of heavy neutral leptons through the dipole portal at FCC-ee

Problem Statement
Heavy Neutral Leptons (HNLs) are a natural extension of the Standard Model (SM) neutrino sector. While direct searches typically rely on active-sterile mixing, HNLs can also interact via higher-dimensional operators, specifically the gauge-invariant dipole portal. This portal couples neutral fermions directly to electroweak field strengths. The challenge addressed in this work is to quantify the indirect sensitivity of future precision measurements at the FCC-ee (Future Circular Collider in $e^+e^-$ mode) to these dipole interactions. Unlike direct searches, indirect probes rely on virtual HNL effects modifying electroweak gauge-boson propagation and lepton couplings. A critical theoretical difficulty is distinguishing the specific signature of the dipole portal from other potential ultraviolet (UV) completions that could generate similar low-energy effective operators.

Methodology
The author employs a gauge-invariant Effective Field Theory (EFT) approach to calculate the virtual effects of Majorana HNLs interacting via dimension-six dipole operators ( $O_B$ and $O_W$ ) involving the hypercharge and weak isospin field strengths.

Model Implementation: The analysis utilizes a gauge-invariant Lagrangian implemented in FeynRules and exported to FeynArts and UFO formats. The model includes the necessary contact interactions (e.g., $N\ell W\gamma$ , $N\nu WW$ ) required by gauge symmetry, which are fixed by the same Wilson coefficients as the three-point dipoles.
Loop Calculation: The study computes one-loop electroweak vacuum polarizations (self-energies) and lepton vertex corrections induced by two dipole insertions. The calculation is performed in dimensional regularization, separating finite threshold corrections from UV divergences.
- Bosonic Sector: The calculation derives the transverse self-energies for neutral ( $\gamma, Z$ ) and charged ( $W$ ) gauge bosons. The neutral sector self-energy matrix is shown to factorize into a rank-one structure dependent on the dipole couplings.
- Fermionic Sector: Direct lepton-current corrections are calculated, including vertex diagrams and external-leg wave-function renormalization. The analysis distinguishes between singlet and triplet current operators in the Warsaw basis.
Matching and EFT Interpretation: The results are matched onto the Standard Model Effective Field Theory (SMEFT) at the HNL mass scale ( $M_N$ ). This involves calculating finite dimension-six thresholds (proportional to $M_N^2 C^2$ ) and identifying dimension-eight counterterms required to absorb divergences associated with momentum-dependent curvature terms.
Phenomenological Forecast: The derived matching corrections are translated into shifts in Z-pole observables (widths, asymmetries, and ratios) using a specific input scheme ( $\alpha, M_W, M_Z$ ). A likelihood analysis is performed using projected FCC-ee experimental precision and varying assumptions for future SM theoretical uncertainties (conservative vs. aggressive scenarios).
UV Sensitivity Analysis: To assess the robustness of the indirect probe, the study profiles the sensitivity against independent UV contributions. It tests scenarios where additional dimension-six operators (lepton currents and mixed bosonic operators) are allowed to vary, potentially canceling the dipole signal.
Radiative Higgs Decays: The paper also evaluates the contribution of the dipole portal to radiative Higgs decays ( $h \to \gamma\gamma$ and $h \to Z\gamma$ ) to determine if these channels provide complementary information beyond the Z-pole.

Key Contributions

Gauge-Invariant Calculation: The work provides a complete one-loop calculation of dipole-induced corrections, explicitly including the non-Abelian contact interactions and Goldstone boson partners mandated by gauge invariance. This ensures the cancellation of gauge-dependent terms and the correct relation between neutral and charged sectors.
Finite Matching Corrections: The author derives explicit finite matching corrections for dimension-six bosonic operators ( $Q_{HB}, Q_{HW}, Q_{HWB}$ ) and lepton-current operators ( $Q_{Hl}^{(1)}, Q_{Hl}^{(3)}$ ) arising from two dipole insertions.
Rank-One Correlation: The analysis demonstrates that the dipole-induced self-energy matrix possesses a rank-one structure, leading to specific correlations between oblique parameters ( $X^2 = WY$ ) and the $S$ -parameter.
UV Degeneracy Quantification: The paper rigorously quantifies the "blind directions" in the parameter space. It shows that while the dipole portal generates three specific combinations of Wilson coefficients, four independent local UV directions exist at the Z-pole. Consequently, without assumptions about the UV completion, pole data alone cannot uniquely constrain the dipole coefficients.
FCC-ee Reach Projections: The study provides quantitative sensitivity projections for FCC-ee, comparing them with LEP precision and HL-LHC direct search projections.

Results

Sensitivity under Dipole Dominance: Assuming dipole dominance (i.e., independent UV coefficients vanish at a scale $\Lambda_B = r_B M_N$ ), FCC-ee can probe equal, same-sign hypercharge and weak dipole coefficients ( $|C_B| = |C_W|$ ) as small as 0.14 TeV $^{-2}$ (aggressive SM theory scenario) and 0.22 TeV $^{-2}$ (conservative scenario) for an HNL mass of 1 TeV. This corresponds to a reach of approximately $0.14 \text{ TeV}^{-2}$ for $r_B=2$ .
Mass Dependence: The sensitivity improves approximately as $1/M_N$ for fixed dipole coefficients due to the $M_N^2$ enhancement in the threshold corrections.
Impact of UV Contributions: If independent lepton-current and mixed-bosonic operators are allowed to vary freely, the sensitivity to the dipole coefficients is significantly weakened. Specifically, allowing the two independent lepton currents to vary degrades the reach from 0.1431 TeV $^{-2}$ to 0.2208 TeV $^{-2}$ . If all four independent local directions are free, the finite local-pole dipole bound disappears entirely, as the UV contributions can perfectly compensate the dipole signal.
Radiative Higgs Decays: The inclusion of radiative Higgs decay constraints ( $h \to Z\gamma$ vs $h \to \gamma\gamma$ ) offers only a modest improvement, reducing the coefficient-region radius by at most ~7% (conservative) and ~1% (aggressive) at 1 TeV.
Complementarity with HL-LHC: At $M_N = 1$ TeV, the indirect FCC-ee reach is competitive with or stronger than the HL-LHC same-sign dilepton projections in specific benchmark directions (A, B, and C). The indirect probe retains sensitivity even for Dirac HNLs (which do not produce same-sign dileptons), provided the coupling bilinears are matched.

Significance and Claims
The paper claims that electroweak precision measurements at FCC-ee constitute a powerful, complementary probe to direct searches for HNLs, capable of testing dipole interactions for masses well beyond the direct production threshold. The primary significance lies in the rigorous establishment of the conditions required for interpretation: the paper emphasizes that a precision constraint on the dipole portal is only valid if additional UV contributions are suppressed or correlated by specific symmetries in the UV model.

The author explicitly states that the indirect reach is not a model-independent discovery potential; rather, it defines a target for UV models where additional electroweak effects are controlled. The work highlights that improved SM theoretical predictions are essential to realizing the full potential of the FCC-ee experimental precision. Furthermore, the study clarifies that while the dipole portal predicts correlated effects across multiple observables, the presence of independent local operators in a general UV completion can obscure these correlations, necessitating a combined approach of direct and indirect searches to unambiguously identify the nature of any new physics signal.

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