Custodial Naturalness Mechanism

• Custodial Naturalness is a symmetry-based framework that combines classical scale invariance with an extended custodial (SO(6)) symmetry to protect the Higgs mass and generate the electroweak scale.
• The minimal realization augments the Standard Model with a new complex scalar, right-handed neutrinos, and a U(1)_X gauge boson, resulting in a Higgs that emerges as an elementary pseudo-Nambu-Goldstone boson.
• Radiative corrections via the Coleman-Weinberg mechanism trigger dimensional transmutation and a naturally suppressed Higgs vacuum expectation value, addressing the little hierarchy problem.

Searching arXiv for papers on "Custodial Naturalness" and related custodial-symmetry naturalness frameworks. Custodial Naturalness is a symmetry-based mechanism for explaining the separation between the electroweak scale and ultraviolet completions of the Standard Model. It combines classical scale invariance with an enhanced scalar-sector custodial symmetry, and both are spontaneously broken by dimensional transmutation at a new intermediate scale. In its minimal realization, the scalar sector is enlarged by a single new complex scalar field charged under a new $\mathrm{U}(1)_X$ gauge symmetry which partially overlaps with $B-L$, and the Standard Model-like Higgs boson arises as an elementary pseudo-Nambu-Goldstone boson of the spontaneously broken $\mathrm{SO}(6)$ custodial symmetry (Trautner, 13 May 2025). Related formulations also develop the mechanism at the level of general principles and simplest extensions, including hidden-sector realizations, neutrino portals, and dark-matter candidates (Boer et al., 13 Feb 2025, Boer et al., 30 Jul 2025).

1. Definition and symmetry assumptions

Custodial Naturalness is based on two guiding principles: classical scale invariance and an enlarged custodial symmetry in the scalar sector (Trautner, 13 May 2025). Classical scale invariance means that no dimensionful parameters appear at tree level. In particular, the usual Higgs mass term $\mu_H^2 |H|^2$ is absent above some high scale $\Lambda_{\rm high}$, such as the Planck scale (Trautner, 13 May 2025). The enlarged custodial symmetry is an exact $\mathrm{SO}(6)$ symmetry imposed on the scalar potential at the high scale, under which the four real components of the Standard Model Higgs doublet $H$ and the two real components of a new complex scalar $\Phi$ are assembled into a real 6-dimensional vector (Boer et al., 2024).

At $\Lambda_{\rm high}$ the most general renormalizable, $\mathrm{SO}(6)\times \mathrm{U}(1)_X$-invariant scalar potential is

$B-L$0

with no $B-L$1 or $B-L$2 terms present (Trautner, 13 May 2025). Because gauge and Yukawa couplings do not respect the full $B-L$3, this custodial symmetry is described as an accidental custodial symmetry of the scalar potential at the cutoff (Boer et al., 2024).

The general mechanism is therefore not merely the absence of a Higgs mass term. It is the simultaneous imposition of scale invariance and extended custodial symmetry, followed by their radiative breaking. This symmetry structure is the basis for the claim that the Higgs mass is protected while the electroweak scale is still generated dynamically (Boer et al., 13 Feb 2025).

2. Minimal realization: field content, gauge structure, and scalar sector

The minimal realization extends the Standard Model by a complex scalar $B-L$4, three right-handed neutrinos $B-L$5, and a new $B-L$6 gauge boson $B-L$7 (Trautner, 13 May 2025). The scalar $B-L$8 is a singlet under $B-L$9 but is charged under $\mathrm{SO}(6)$0, and both $\mathrm{SO}(6)$1 and $\mathrm{SO}(6)$2 carry the same $\mathrm{SO}(6)$3 charge (Boer et al., 2024). The right-handed neutrinos are included to cancel $\mathrm{SO}(6)$4 anomalies (Trautner, 13 May 2025).

The $\mathrm{SO}(6)$5 charge assignment is chosen so that $\mathrm{SO}(6)$6 is a linear combination of hypercharge $\mathrm{SO}(6)$7 and $\mathrm{SO}(6)$8,

$\mathrm{SO}(6)$9

with $\mu_H^2 |H|^2$0 in a benchmark (Trautner, 13 May 2025). This choice preserves the $\mu_H^2 |H|^2$1 boundary condition and allows $\mu_H^2 |H|^2$2 to feel strong gauge-induced radiative effects (Trautner, 13 May 2025).

Below $\mu_H^2 |H|^2$3, the running couplings split and the scalar potential becomes

$\mu_H^2 |H|^2$4

with no mass terms ever present in the Lagrangian (Trautner, 13 May 2025). In the exact $\mu_H^2 |H|^2$5 limit at the high scale one has $\mu_H^2 |H|^2$6 (Boer et al., 2024).

This minimal setting has the same number of parameters as the Standard Model (Boer et al., 2024). In one formulation, the free input parameters are $\mu_H^2 |H|^2$7 at $\mu_H^2 |H|^2$8 together with $\mu_H^2 |H|^2$9 for maximal custodial symmetry (Boer et al., 2024). In another equivalent summary, all new parameters $\Lambda_{\rm high}$0 at $\Lambda_{\rm high}$1 can be traded against the measured Fermi scale, $\Lambda_{\rm high}$2, and $\Lambda_{\rm high}$3, so that the minimal model has precisely the same number of free parameters as the Standard Model (Trautner, 13 May 2025).

3. Radiative breaking, dimensional transmutation, and hierarchy generation

Quantum effects break both scale invariance and the accidental $\Lambda_{\rm high}$4 symmetry (Boer et al., 2024). The dominant radiative effects come from the $\Lambda_{\rm high}$5 gauge coupling $\Lambda_{\rm high}$6, its kinetic mixing $\Lambda_{\rm high}$7 with hypercharge, the top Yukawa $\Lambda_{\rm high}$8, and $\Lambda_{\rm high}$9, which drive the running of $\mathrm{SO}(6)$0 to negative values at some scale $\mathrm{SO}(6)$1 (Trautner, 13 May 2025). At that scale a Coleman-Weinberg-type vacuum develops,

$\mathrm{SO}(6)$2

and scale invariance is spontaneously broken (Trautner, 13 May 2025).

In the one-loop Coleman-Weinberg treatment, the effective potential in $\mathrm{SO}(6)$3 is

$\mathrm{SO}(6)$4

where the sum runs over all Standard Model and new fields, including $\mathrm{SO}(6)$5, $\mathrm{SO}(6)$6, $\mathrm{SO}(6)$7, the top quark, and scalars (Boer et al., 2024). A flat direction arises when

$\mathrm{SO}(6)$8

and the generated vacuum expectation value is exponentially small compared to the cutoff (Boer et al., 2024).

A second flat direction exists provided

$\mathrm{SO}(6)$9

which yields

$H$0

The true minimum is aligned predominantly along the $H$1 axis because the $H$2-violating renormalization-group running from top-Yukawa and gauge interactions splits $H$3, $H$4, and $H$5 just enough to leave a suppressed Higgs vacuum expectation value (Boer et al., 2024).

In the minimal realization, the induced Higgs vacuum expectation value satisfies

$H$6

so that $H$7 arises naturally because $H$8 and $H$9 remain close as a remnant of the custodial symmetry (Trautner, 13 May 2025). This is the core hierarchy-generating mechanism: the intermediate scale $\Phi$0 arises via dimensional transmutation in the $\Phi$1–$\Phi$2 sector, and the electroweak scale emerges via a custodially suppressed portal to $\Phi$3 (Trautner, 13 May 2025).

4. Higgs as a pseudo-Nambu-Goldstone boson and the naturalness argument

The spontaneous breaking $\Phi$4 yields five Goldstone bosons (Trautner, 13 May 2025). Four of them are eaten by $\Phi$5, $\Phi$6, and $\Phi$7, while the remaining scalar $\Phi$8 is a pseudo-Nambu-Goldstone boson (Trautner, 13 May 2025). In this construction, the Standard Model Higgs is therefore an elementary, not composite, pseudo-Nambu-Goldstone boson of $\Phi$9 (Boer et al., 2024).

Its tree-level mass squared is

$\Lambda_{\rm high}$0

which is suppressed by the small custodial-symmetry breaking, namely the difference $\Lambda_{\rm high}$1 (Trautner, 13 May 2025). A related expression emphasizes that the Higgs mass receives its leading suppression from the small splitting among $\Lambda_{\rm high}$2 and $\Lambda_{\rm high}$3 induced by radiative effects (Boer et al., 2024). In the general treatment, the Higgs mass is written as

$\Lambda_{\rm high}$4

with the splitting controlled by

$\Lambda_{\rm high}$5

This formulation makes explicit that kinetic mixing and additional Yukawa couplings provide leading explicit custodial-symmetry violation (Boer et al., 13 Feb 2025).

The protection mechanism is stated in symmetry terms. Because the tree-level potential respects scale invariance and a large $\Lambda_{\rm high}$6 global symmetry, no hard $\Lambda_{\rm high}$7 term can ever appear (Boer et al., 2024). Radiative corrections generate masses only through the small splittings of quartics induced by explicit $\Lambda_{\rm high}$8 breakings, and these enter the Higgs mass linearly in the one-loop beta functions (Boer et al., 2024). The Higgs is thus an elementary pseudo-Nambu-Goldstone boson whose would-be shift symmetry protects $\Lambda_{\rm high}$9 from large additive quantum corrections (Boer et al., 2024). This is the sense in which Custodial Naturalness is presented as solving the little hierarchy problem while keeping the Higgs elementary (Trautner, 13 May 2025).

5. Spectrum and experimental signatures

After symmetry breaking, the physical spectrum contains the Standard Model Higgs with $\mathrm{SO}(6)\times \mathrm{U}(1)_X$0, a heavy $\mathrm{SO}(6)\times \mathrm{U}(1)_X$1 gauge boson, and a light dilaton-like scalar $\mathrm{SO}(6)\times \mathrm{U}(1)_X$2 (Trautner, 13 May 2025).

The heavy $\mathrm{SO}(6)\times \mathrm{U}(1)_X$3 arises from $\mathrm{SO}(6)\times \mathrm{U}(1)_X$4 breaking. Its mass is approximately

$\mathrm{SO}(6)\times \mathrm{U}(1)_X$5

in the simplified description (Trautner, 13 May 2025), while a more detailed expression is

$\mathrm{SO}(6)\times \mathrm{U}(1)_X$6

with small $\mathrm{SO}(6)\times \mathrm{U}(1)_X$7–$\mathrm{SO}(6)\times \mathrm{U}(1)_X$8 mixing of order $\mathrm{SO}(6)\times \mathrm{U}(1)_X$9 (Boer et al., 2024). Its couplings to Standard Model fermions are fixed by their $B-L$00 charges, and in particular the Drell-Yan coupling to $B-L$01 is $B-L$02 (Boer et al., 2024). Direct LHC dilepton resonance searches already exclude $B-L$03 (Boer et al., 2024). The predicted viable range is $B-L$04 (Boer et al., 2024).

The dilaton $B-L$05 is the radial mode associated with the breaking of scale invariance. Its mass is parametrically controlled by the Coleman-Weinberg beta function,

$B-L$06

and it is typically light (Trautner, 13 May 2025). The minimal papers describe it as having mass in the few-tens of GeV range (Trautner, 13 May 2025), with a representative value $B-L$07 in the abstract-level summary (Boer et al., 2024). More general realizations broaden the range to $B-L$08 with very small mixing $B-L$09 (Boer et al., 13 Feb 2025).

Its mixing with the Higgs is tiny. The published minimal realization quotes $B-L$10 (Trautner, 13 May 2025), while the preprint formulation gives $B-L$11 (Boer et al., 2024). This leads to suppressed visible decays, and the scalar is described as almost invisible, yet potentially discoverable at Higgs factories, sometimes even via displaced vertices for long lifetimes (Trautner, 13 May 2025).

No extra top partners or compositeness is required in the minimal realization (Trautner, 13 May 2025). The Higgs couplings remain Standard Model-like (Trautner, 13 May 2025). The phenomenological profile of the mechanism is therefore unusually concentrated in a fixed-coupling multi-TeV $B-L$12 and a light dilaton-like scalar.

6. Generalizations, robustness, and related custodial frameworks

A broader treatment presents Custodial Naturalness as a general mechanism together with minimal realization and simplest extensions which populate Higgs-, gauge-, and neutrino-portals and introduce candidates for particle dark matter (Boer et al., 13 Feb 2025). One extension adds a vector-like fermion pair with Yukawa interaction

$B-L$13

which generates one heavy Dirac sterile neutrino $B-L$14 plus see-saw-like effects in the active sector (Boer et al., 13 Feb 2025). Another adds two vector-like fermion pairs stable by accidental $B-L$15 symmetries, yielding two-component WIMP dark matter with correct relic density only near $B-L$16 (Boer et al., 13 Feb 2025).

The same general treatment states that the mechanism remains stable under inclusion of new sources of explicit custodial symmetry violation, as well as under variations of boundary conditions at the high scale (Boer et al., 13 Feb 2025). In particular, the leading custodial-breaking parameters $B-L$17 and $B-L$18 can be kept small, $B-L$19, so that the little hierarchy $B-L$20 emerges without fine tuning (Boer et al., 13 Feb 2025).

A distinct hidden-sector realization removes the extension of the Standard Model gauge group and reduces the custodial symmetry to $B-L$21 (Boer et al., 30 Jul 2025). In that model, the four real components of the Higgs doublet plus a new real singlet field $B-L$22 form a 5-plet, and an additional $B-L$23-odd real scalar singlet $B-L$24 is automatically stable and a dark-matter candidate produced via freeze-in with moderate couplings (Boer et al., 30 Jul 2025). The most minimal scenario requires a UV completion at around $B-L$25, while including right-handed neutrinos can push this scale to $B-L$26 (Boer et al., 30 Jul 2025). This suggests that the core mechanism is not tied uniquely to the $B-L$27 implementation, although the minimal published realization remains the $B-L$28 model with a heavy $B-L$29 and light dilaton.

A separate line of work studies canonical $B-L$30 custodial symmetry in multi-Higgs-doublet potentials through basis-covariant and representation-theoretical methods (Plantey et al., 2024). There, a basis-covariant test of $B-L$31 is presented as ensuring “custodial naturalness” in the specific sense that large quartic couplings or misaligned CP phases that would otherwise generate $B-L$32 are forbidden or aligned (Plantey et al., 2024). That usage concerns protection of the electroweak $B-L$33 parameter and oblique parameters in NHDMs, rather than the specific hierarchy-generating mechanism based on classical scale invariance and $B-L$34 or $B-L$35 scalar symmetry.

7. Phenomenological tests, mass correlations, and cosmological implications

Custodial Naturalness is presented as experimentally testable at colliders, Higgs factories, and gravitational-wave observatories (Boer et al., 13 Feb 2025). In collider terms, the principal targets are a $B-L$36 in the $B-L$37 range and a light dilaton-like scalar (Boer et al., 13 Feb 2025). HL-LHC and HE-LHC can probe the $B-L$38 up to $B-L$39, and future $B-L$40 $B-L$41 colliders up to $B-L$42 (Boer et al., 2024). Future Higgs factories or the HL-LHC can probe Higgs-dilaton mixing down to $B-L$43 in the hidden-sector realization (Boer et al., 30 Jul 2025).

The general mechanism paper also identifies a specific correlation between the Higgs and top quark masses (Boer et al., 13 Feb 2025). Because the top Yukawa strongly drives the running of $B-L$44, the condition $B-L$45 selects a narrow band in the $B-L$46 plane, approximately

$B-L$47

with $B-L$48 and $B-L$49 (Boer et al., 13 Feb 2025). In that summary, $B-L$50 must lie near the lower end of its PDG range $B-L$51 to obtain $B-L$52 (Boer et al., 13 Feb 2025).

The cosmological evolution is described as featuring a strongly supercooled first-order phase transition (Boer et al., 13 Feb 2025). In the classically scale-invariant sector, bubbles of the broken $B-L$53 vacuum nucleate at a temperature $B-L$54, and the resulting gravitational-wave signal can lie in the millihertz-hertz band accessible to LISA, DECIGO, or Einstein Telescope, depending on $B-L$55 (Boer et al., 13 Feb 2025). The hidden-sector realization similarly states that a strongly supercooled first-order phase transition at $B-L$56 can produce a gravitational-wave signal potentially visible at future experiments (Boer et al., 30 Jul 2025).

A common misconception is that custodial symmetry in these models refers only to the Standard Model relation $B-L$57 at tree level. In the Custodial Naturalness mechanism proper, the relevant custodial structure is an enlarged scalar-sector symmetry, $B-L$58 in the minimal realization or $B-L$59 in the hidden-sector variant, and its spontaneous breaking is directly responsible for the pseudo-Nambu-Goldstone nature of the Higgs (Trautner, 13 May 2025, Boer et al., 30 Jul 2025). A plausible implication is that the term “custodial” is being used in two related but distinct senses across the literature: one associated with precision electroweak protection, and the other with hierarchy protection through an enlarged scalar-sector symmetry.