Conformal Higgs Potential

• Conformal Higgs potential is defined by the absence of explicit mass terms, employing a quartic scalar sector and classical scale or Weyl invariance.
• It appears in various models including conformal Standard Model extensions, Weyl-invariant Higgs–dilaton systems, and other gravitational formulations.
• Key mechanisms such as the Coleman–Weinberg effect, threshold matching, and condensate dynamics generate the effective tachyonic mass needed for electroweak symmetry breaking.

Searching arXiv for the core paper and adjacent conformal-Higgs literature. Searching for (Oda, 2018) and related conformal Higgs potential works. A conformal Higgs potential is a Higgs-sector potential organized by classical scale invariance or local Weyl invariance, so that explicit dimension-two mass terms are absent at the fundamental level and the scalar sector is initially quartic. In this class of constructions, the electroweak mass parameter is not inserted by hand; it is generated only after conformal symmetry is broken radiatively, gravitationally, or by gauge/frame fixing. The subject therefore spans several related frameworks: classically conformal Standard Model extensions, Weyl-invariant Higgs–dilaton systems, conformal gravity models, and Weyl-quadratic theories in which the Higgs potential emerges from the gravitational sector itself (Oda, 2018, Bounakis et al., 2017, Oda, 2020).

1. Symmetry principle and basic definitions

The defining structural feature is the absence of a fundamental Higgs mass term. In classically conformal formulations one writes, for the scalar sector, a quartic potential such as

$V(H)=\lambda(\mu)\,(H^\dagger H)^2,$

or its multi-scalar generalizations, and imposes scale invariance at the ultraviolet scale, often taken to be $M_{Pl}$ (Gorsky et al., 2014). In locally Weyl-invariant formulations the fields transform under

$$g_{\mu\nu}\to \Omega^2(x)\,g_{\mu\nu},\qquad \phi\to \Omega^{-1}(x)\,\phi,\qquad H\to \Omega^{-1}(x)\,H,$$

so the scalar potential must be homogeneous of degree four and the scalar-curvature coupling is fixed to the conformal value, conventionally written as $1/12$ in the Lagrangian or $\xi=1/6$ in the non-minimal coupling (Oda, 2018, Bars et al., 2013, Cléry et al., 2023).

A standard conformal starting point couples the Higgs doublet $H$ to an additional scalar $\phi$, often interpreted as a dilaton or conformal factor. In one representative conformal BSM construction the Lagrangian contains

$$\mathcal L_c= -\frac{1}{2\xi^2} C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma} +\frac{1}{12}\phi^2 R +\frac12 g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi -g^{\mu\nu}(D_\mu H)^\dagger(D_\nu H) -V(\phi,H)+\mathcal L_m,$$

with the unique renormalizable conformally invariant scalar potential

$$V(\phi,H)=\lambda_\phi \phi^4+\lambda_{\phi H}\phi^2(H^\dagger H)+\lambda_H(H^\dagger H)^2,$$

and $\lambda_i>0$ for positivity at large field values (Oda, 2018).

A closely related Weyl-invariant Standard Model plus dilaton uses

$M_{Pl}$0

while the coefficient $M_{Pl}$1 of $M_{Pl}$2 and $M_{Pl}$3 is fixed as the unique Weyl-invariant scalar-curvature coupling (Bars et al., 2013). This establishes the general rule: conformal symmetry fixes the allowed operators severely, but it does not determine a unique phenomenology.

2. Canonical tree-level forms

At tree level, conformal Higgs potentials are quartic polynomials in the available scalar invariants. Their precise form depends on whether the framework uses a dilaton, extra singlets, extra gauge sectors, or Weyl geometry.

Framework	Scalar content	Tree-level conformal potential
Conformal BSM	$M_{Pl}$4	$M_{Pl}$5
Weyl-invariant SM + dilaton	$M_{Pl}$6	$M_{Pl}$7
Classically conformal $M_{Pl}$8 model	$M_{Pl}$9	$$g_{\mu\nu}\to \Omega^2(x)\,g_{\mu\nu},\qquad \phi\to \Omega^{-1}(x)\,\phi,\qquad H\to \Omega^{-1}(x)\,H,$$0
Two-singlet minimal conformal model	$$g_{\mu\nu}\to \Omega^2(x)\,g_{\mu\nu},\qquad \phi\to \Omega^{-1}(x)\,\phi,\qquad H\to \Omega^{-1}(x)\,H,$$1	$$g_{\mu\nu}\to \Omega^2(x)\,g_{\mu\nu},\qquad \phi\to \Omega^{-1}(x)\,\phi,\qquad H\to \Omega^{-1}(x)\,H,$$2

These quartic forms recur across otherwise distinct programs. In the classically conformal $$g_{\mu\nu}\to \Omega^2(x)\,g_{\mu\nu},\qquad \phi\to \Omega^{-1}(x)\,\phi,\qquad H\to \Omega^{-1}(x)\,H,$$3 model the Higgs direction is taken to be exactly flat at $$g_{\mu\nu}\to \Omega^2(x)\,g_{\mu\nu},\qquad \phi\to \Omega^{-1}(x)\,\phi,\qquad H\to \Omega^{-1}(x)\,H,$$4, with

$$g_{\mu\nu}\to \Omega^2(x)\,g_{\mu\nu},\qquad \phi\to \Omega^{-1}(x)\,\phi,\qquad H\to \Omega^{-1}(x)\,H,$$5

so electroweak breaking is entirely deferred to the infrared (Iso et al., 2012). In the conformal neutrino option the high-scale scalar potential is likewise massless,

$$g_{\mu\nu}\to \Omega^2(x)\,g_{\mu\nu},\qquad \phi\to \Omega^{-1}(x)\,\phi,\qquad H\to \Omega^{-1}(x)\,H,$$6

with heavy-neutrino masses generated only after spontaneous breaking of scale invariance (Brdar et al., 2018). In the conformal Higgs-triplet model, the most general tree-level quartic potential contains the doublet $$g_{\mu\nu}\to \Omega^2(x)\,g_{\mu\nu},\qquad \phi\to \Omega^{-1}(x)\,\phi,\qquad H\to \Omega^{-1}(x)\,H,$$7, triplet $$g_{\mu\nu}\to \Omega^2(x)\,g_{\mu\nu},\qquad \phi\to \Omega^{-1}(x)\,\phi,\qquad H\to \Omega^{-1}(x)\,H,$$8, and singlet $$g_{\mu\nu}\to \Omega^2(x)\,g_{\mu\nu},\qquad \phi\to \Omega^{-1}(x)\,\phi,\qquad H\to \Omega^{-1}(x)\,H,$$9, with no dimensionful parameters because classical conformal invariance forbids all mass terms (Okada et al., 2015).

This multiplicity of quartic realizations shows that “conformal Higgs potential” is not a single algebraic ansatz. It is a symmetry class of potentials defined by the absence of relevant scalar operators.

3. Dynamical origin of the Higgs mass parameter

The central dynamical problem is to recover a negative effective Higgs mass-squared from a theory that forbids it at tree level. Several mechanisms appear in the literature.

In the conformal BSM model of Oda, the key step is the Wick rotation of the conformal factor in the Euclidean functional integral. Because the scalar $1/12$0 plays the role of the gravitational conformal factor, one analytically continues

$1/12$1

which flips the sign of every term even in $1/12$2. The scalar potential therefore changes from

$1/12$3

to

$1/12$4

Once conformal symmetry is broken radiatively and $1/12$5 acquires a Planck-scale vacuum expectation value $1/12$6, the low-energy Higgs potential becomes

$1/12$7

so the induced Higgs mass term is tachyonic. In unitary gauge this gives

$1/12$8

with conformal symmetry itself broken by a Coleman–Weinberg minimum near $1/12$9 (Oda, 2018).

A second route is radiative portal generation. In the classically conformal $\xi=1/6$0 model, $\xi=1/6$1 breaking occurs through the Coleman–Weinberg mechanism, and a small negative $\xi=1/6$2 is radiatively generated around $\xi=1/6$3 TeV. After $\xi=1/6$4, the mixed quartic becomes

$\xi=1/6$5

and electroweak breaking follows from

$\xi=1/6$6

The same logic reappears in the conformal Higgs-triplet model, where $\xi=1/6$7 is generated by Coleman–Weinberg dynamics and the portal couplings $\xi=1/6$8 and $\xi=1/6$9 induce the low-energy mass terms of the doublet and triplet (Iso et al., 2012, Okada et al., 2015).

A third route is threshold generation by heavy states. In the conformal neutrino option, a singlet $H$0 obtains a one-loop vacuum expectation value, producing heavy Majorana masses $H$1. Integrating out $H$2 and $H$3 generates a negative Higgs-mass-squared term at one loop,

$H$4

which then triggers electroweak breaking (Brdar et al., 2018).

A fourth route is condensate-driven symmetry breaking. In the condensate mechanism of conformal symmetry breaking, the tachyonic mass term is replaced by a linear term generated by the top condensate,

$H$5

so the stationarity condition gives

$H$6

and

$H$7

The paper quotes $H$8 GeV from this mechanism (Pervushin et al., 2012).

4. Gravity, conformal frames, and Weyl geometry

Conformal Higgs potentials are frequently formulated in Jordan and Einstein frames, with the non-minimal coupling to curvature playing a decisive role. A Jordan-frame action may be written as

$H$9

with

$\phi$0

After the Weyl map $\phi$1, the potential becomes

$\phi$2

Frame-covariant quantisation using the Vilkovisky–DeWitt effective action was used to show that the Higgs potential can be derived in a way that is independent of the choice of conformal frame, and the resulting vacuum-stability analysis gives frame-independent bounds on $\phi$3 (Bounakis et al., 2017).

In Weyl geometry, the gravitational sector can itself generate the Higgs potential. In one Weyl-conformal model, a $\phi$4 term in the Jordan frame leads after gauge fixing to the Einstein-frame potential

$\phi$5

which is a perfect square, has vanishing vacuum energy at the minimum, and is accompanied by a Proca mass

$\phi$6

for the Weyl gauge field after the would-be dilaton is absorbed (Oda, 2020). A related Weyl-quadratic construction introduces the combination

$\phi$7

uses local Weyl symmetry to set $\phi$8, and obtains at low energies an Einstein-frame Higgs potential whose small-field expansion contains a negative quadratic term,

$\phi$9

so electroweak symmetry breaking follows for $$\mathcal L_c= -\frac{1}{2\xi^2} C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma} +\frac{1}{12}\phi^2 R +\frac12 g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi -g^{\mu\nu}(D_\mu H)^\dagger(D_\nu H) -V(\phi,H)+\mathcal L_m,$$0 (Ghilencea, 2018).

The same non-minimal structure is also central in inflationary realizations. In Higgs pole inflation the Jordan-frame coupling

$$\mathcal L_c= -\frac{1}{2\xi^2} C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma} +\frac{1}{12}\phi^2 R +\frac12 g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi -g^{\mu\nu}(D_\mu H)^\dagger(D_\nu H) -V(\phi,H)+\mathcal L_m,$$1

makes the effective Planck mass vanish at

$$\mathcal L_c= -\frac{1}{2\xi^2} C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma} +\frac{1}{12}\phi^2 R +\frac12 g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi -g^{\mu\nu}(D_\mu H)^\dagger(D_\nu H) -V(\phi,H)+\mathcal L_m,$$2

After Weyl rescaling, the Einstein-frame kinetic term has a pole and the potential

$$\mathcal L_c= -\frac{1}{2\xi^2} C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma} +\frac{1}{12}\phi^2 R +\frac12 g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi -g^{\mu\nu}(D_\mu H)^\dagger(D_\nu H) -V(\phi,H)+\mathcal L_m,$$3

remains finite only if

$$\mathcal L_c= -\frac{1}{2\xi^2} C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma} +\frac{1}{12}\phi^2 R +\frac12 g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi -g^{\mu\nu}(D_\mu H)^\dagger(D_\nu H) -V(\phi,H)+\mathcal L_m,$$4

This yields a flat plateau for inflation once the canonical field

$$\mathcal L_c= -\frac{1}{2\xi^2} C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma} +\frac{1}{12}\phi^2 R +\frac12 g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi -g^{\mu\nu}(D_\mu H)^\dagger(D_\nu H) -V(\phi,H)+\mathcal L_m,$$5

is introduced (Cléry et al., 2023).

5. Renormalization-group criticality, Planck-scale structure, and vacuum stability

A major branch of the subject studies whether the Higgs potential approaches a critical, nearly flat form near the Planck scale. One proposal imposes exact conformal symmetry at $$\mathcal L_c= -\frac{1}{2\xi^2} C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma} +\frac{1}{12}\phi^2 R +\frac12 g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi -g^{\mu\nu}(D_\mu H)^\dagger(D_\nu H) -V(\phi,H)+\mathcal L_m,$$6 through the boundary conditions

$$\mathcal L_c= -\frac{1}{2\xi^2} C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma} +\frac{1}{12}\phi^2 R +\frac12 g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi -g^{\mu\nu}(D_\mu H)^\dagger(D_\nu H) -V(\phi,H)+\mathcal L_m,$$7

With these conditions, one-loop renormalization-group evolution is argued to reproduce $$\mathcal L_c= -\frac{1}{2\xi^2} C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma} +\frac{1}{12}\phi^2 R +\frac12 g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi -g^{\mu\nu}(D_\mu H)^\dagger(D_\nu H) -V(\phi,H)+\mathcal L_m,$$8, while the RG-improved potential

$$\mathcal L_c= -\frac{1}{2\xi^2} C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma} +\frac{1}{12}\phi^2 R +\frac12 g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi -g^{\mu\nu}(D_\mu H)^\dagger(D_\nu H) -V(\phi,H)+\mathcal L_m,$$9

develops a second minimum at $$V(\phi,H)=\lambda_\phi \phi^4+\lambda_{\phi H}\phi^2(H^\dagger H)+\lambda_H(H^\dagger H)^2,$$0 almost degenerate with the electroweak minimum at $$V(\phi,H)=\lambda_\phi \phi^4+\lambda_{\phi H}\phi^2(H^\dagger H)+\lambda_H(H^\dagger H)^2,$$1 (Gorsky et al., 2014).

This Planck-scale criticality is closely connected to the vacuum-stability problem of the Standard Model. In the pure SM, the Higgs self-coupling can run negative at scales $$V(\phi,H)=\lambda_\phi \phi^4+\lambda_{\phi H}\phi^2(H^\dagger H)+\lambda_H(H^\dagger H)^2,$$2–$$V(\phi,H)=\lambda_\phi \phi^4+\lambda_{\phi H}\phi^2(H^\dagger H)+\lambda_H(H^\dagger H)^2,$$3 GeV for the central top mass, producing metastability. In the conformal BSM framework with an extra scalar $$V(\phi,H)=\lambda_\phi \phi^4+\lambda_{\phi H}\phi^2(H^\dagger H)+\lambda_H(H^\dagger H)^2,$$4, the $$V(\phi,H)=\lambda_\phi \phi^4+\lambda_{\phi H}\phi^2(H^\dagger H)+\lambda_H(H^\dagger H)^2,$$5–$$V(\phi,H)=\lambda_\phi \phi^4+\lambda_{\phi H}\phi^2(H^\dagger H)+\lambda_H(H^\dagger H)^2,$$6 mixing modifies the $$V(\phi,H)=\lambda_\phi \phi^4+\lambda_{\phi H}\phi^2(H^\dagger H)+\lambda_H(H^\dagger H)^2,$$7-function of $$V(\phi,H)=\lambda_\phi \phi^4+\lambda_{\phi H}\phi^2(H^\dagger H)+\lambda_H(H^\dagger H)^2,$$8 and the threshold at $$V(\phi,H)=\lambda_\phi \phi^4+\lambda_{\phi H}\phi^2(H^\dagger H)+\lambda_H(H^\dagger H)^2,$$9 changes matching conditions, which stabilizes the running of $\lambda_i>0$0 up to $\lambda_i>0$1 (Oda, 2018).

The same large-field question appears in curved space. In inflationary backgrounds one has $\lambda_i>0$2 and an effective mass

$\lambda_i>0$3

A frame-covariant computation gives, for the present central top mass $\lambda_i>0$4 GeV, that for $\lambda_i>0$5 GeV a stable vacuum requires $\lambda_i>0$6–$\lambda_i>0$7, while a new local maximum and minimum appear for $\lambda_i>0$8 and additional instabilities are avoided only if $\lambda_i>0$9 (Bounakis et al., 2017).

Other conformal constructions emphasize the existence of a second high-scale minimum rather than absolute stability. In one Higgs–gravity setup, the RG-improved quartic can cross zero around $M_{Pl}$00–$M_{Pl}$01 GeV and produce a second minimum near $M_{Pl}$02, which is $M_{Pl}$03 orders of magnitude above the observed $M_{Pl}$04 GeV. The same framework then invokes a singular Euclidean instanton with action $M_{Pl}$05 to generate

$M_{Pl}$06

and requiring $M_{Pl}$07 reproduces $M_{Pl}$08 (Shaposhnikov et al., 2018).

6. Variants, phenomenology, and controversies

Conformal Higgs potentials appear in a wide range of model-building programs. The minimal conformal extension of the Higgs sector that remains stable under RG evolution up to $M_{Pl}$09 GeV adds two real singlets, one of which acquires a vacuum expectation value and mixes with the physical Higgs, while the other is stabilized by a $M_{Pl}$10 symmetry and can act as a Higgs-portal dark matter candidate. In that construction the pseudo-Goldstone boson of scale invariance acquires

$M_{Pl}$11

through the Coleman–Weinberg potential, and the Higgs–singlet mixing angle is generically $M_{Pl}$12 (Helmboldt et al., 2016). In the classically conformal Higgs-triplet model, one representative set of Planck-scale boundary conditions yields $M_{Pl}$13 TeV and a $M_{Pl}$14 GeV SM-like Higgs after RG running and Coleman–Weinberg breaking of $M_{Pl}$15 (Okada et al., 2015).

At lower compositeness scales, conformal symmetry has also been combined with little-Higgs ideas. In conformal little Higgs models the tree-level scalar sector contains only collective quartics, while the Higgs mass parameter arises from Coleman–Weinberg loops. A representative low-energy potential is

$M_{Pl}$16

with

$M_{Pl}$17

and $M_{Pl}$18 generated from collective quartics (Ahmed et al., 2023).

The subject also contains genuine disagreements. One line of work argued that the Higgs modulus can be interpreted as the conformal degree of freedom in a conformal gravity background without a tree-level Higgs potential, but a later one-loop analysis found that genuine radiative effects cancel the local functional measure that supported this interpretation. The same calculation nevertheless generated a Coleman–Weinberg-type effective potential and a nonzero vacuum expectation value for the Higgs modulus (Bhattacharjee et al., 2010). Another divergence concerns the physical status of the $M_{Pl}$19 GeV state. Several conformal models identify it with an ordinary Higgs, a mixed Higgs–singlet state, or a pseudo-Goldstone boson of broken scale invariance (Oda, 2018, Helmboldt et al., 2016). By contrast, some versions of the conformal Higgs model claim that no elementary massive Higgs particle survives and interpret the observed resonance as a composite gauge-field excitation $M_{Pl}$20 (Nesbet, 2010, Nesbet, 2021).

Taken together, these developments define the conformal Higgs potential as a research area rather than a single model. Its common thesis is that the Higgs mass term should be induced, not postulated: by Coleman–Weinberg dimensional transmutation, by portal thresholds, by condensates, by Weyl-frame dynamics, by $M_{Pl}$21 terms, or, in Oda’s conformal BSM construction, by Wick rotation of the conformal factor itself (Oda, 2018).