Scalar-tensor mechanisms in which a gravitational-sector scalar non-minimally coupled to curvature suppresses Standard Model mass parameters by an exponential Weyl factor, notably the equivalence-principle mechanism of Calmet (Phys. Rev. D 77, 047502 (2008)) and the induced-gravity variant of Lin (Commun. Theor. Phys. 68, 223 (2017)), reproduce the observed electroweak–Planck hierarchy without extra dimensions or TeV-scale new physics, but as implemented they relocate the fine-tuning into the gravitational sector: a tuned dilaton mass and self-coupling (of order 10−5210^{-52}) and a negative cosmological constant in one variant, a hand-chosen vacuum ratio and roughly sixty residual orders of cosmological-constant smallness in the other. Two questions decide whether this class of models genuinely solves the gauge hierarchy problem: whether electroweak-scale quantities, once suppressed by the relevant Weyl factor (Ω∼10−17\Omega\sim10^{-17} in the Planck-scale-fundamental variant, e−υh/η∼10−15e^{-\upsilon_h/\eta}\sim10^{-15} in the weak-scale-fundamental variant), are stable against quantum corrections computed in the Einstein frame — an analysis that has never been performed — and whether the required gravitational-sector parameters can be fixed dynamically rather than tuned.

Background

The gauge hierarchy problem asks why the Higgs vacuum expectation value (VEV) — about 246 GeV, setting the electroweak scale — is some seventeen orders of magnitude below the Planck scale that governs gravity. In quantum field theory the Higgs mass parameter receives radiative corrections that scale with the ultraviolet cutoff, so if the Standard Model (SM) remains valid up to the Planck scale the corrections are enormous unless the bare parameter cancels them to extreme precision. Proposed cures include weak-scale supersymmetry (a symmetry pairing each particle with a superpartner of opposite spin, whose cancellations protect the Higgs mass), technicolor (a new strong interaction that breaks electroweak symmetry dynamically), and the Randall–Sundrum (RS) scenario, in which a warped extra dimension generates an exponential hierarchy; there the separation is stabilized by the Goldberger–Wise mechanism, which shows that the warp-setting modulus can be fixed by dynamics rather than by hand.

A less explored route uses the gravitational sector itself. In scalar-tensor (Jordan–Brans–Dicke-type) gravity one adds a scalar field non-minimally coupled to the curvature, e.g. through a term M2Ω2(ϕ)RM^2\Omega^2(\phi)R with Ω(ϕ)\Omega(\phi) the coupling function. Passing to the Einstein frame — where the metric is Weyl-rescaled (conformally transformed) by Ω\Omega so that the gravitational action takes the standard Einstein form — multiplies every SM mass parameter by powers of Ω\Omega. Calmet (Phys. Rev. D 77, 047502 (2008); arXiv:0708.2767) showed that if the scalar develops a vacuum expectation value with Ω∼10−17\Omega\sim10^{-17}, then a Planck-scale Jordan-frame Higgs VEV vv becomes a physical VEV v0=Ωvv_0=\Omega v of electroweak size, mimicking RS without extra dimensions; the new scalar couples only gravitationally to matter, and the hierarchy is attributed to a violation of the Einstein equivalence principle (the statement that local nongravitational physics is independent of spacetime position, so particle masses are not fundamental constants but frame-dependent combinations). As implemented, however, the mechanism requires the dilaton mass — the dilaton being the non-minimally coupled gravitational scalar just introduced — mϕ^∼1010.5m_{\hat\phi}\sim10^{10.5} GeV to be tuned to obtain Ω∼10−17\Omega\sim10^{-17}, a negative bare cosmological constant, and a dilaton self-coupling λϕ^∼−5×10−52\lambda_{\hat\phi}\sim-5\times10^{-52} to keep the effective vacuum energy small; the paper conjectures that a supergravity extension (a supersymmetric theory of gravity, with a singlet fermion superpartner canceling the radiative corrections of the dilaton self-interactions and of graviton loops) could protect these parameters, but no such model is constructed, and no loop or renormalization-group analysis establishes that electroweak-scale quantities remain small under quantum corrections in the Einstein frame.

Lin (Commun. Theor. Phys. 68, 223 (2017); arXiv:1405.4821) proposed the complementary variant in which the fundamental scale is at the weak scale and the Planck mass is dynamically generated by spontaneous breaking of a dilaton-like field's symmetry: with a non-minimal coupling M2e2ϕ/ηR/2M^2e^{2\phi/\eta}R/2, all SM masses are suppressed as m→m e−υh/ηm\to m\,e^{-\upsilon_h/\eta}, and choosing υh/η≃35\upsilon_h/\eta\simeq35 gives the observed hierarchy. The bare cosmological constant is likewise suppressed by e−4υh/η∼10−60e^{-4\upsilon_h/\eta}\sim10^{-60}, yet roughly sixty further orders of smallness are still needed; the ratio υh/η\upsilon_h/\eta is input by hand; and above the weak scale thermal restoration of the symmetry lowers the effective Planck mass so the theory becomes strongly coupled and the effective description breaks down.

The status of such frame-based arguments at the quantum level is itself unsettled: Kamenshchik and Steinwachs (Phys. Rev. D 91, 084033 (2015)) showed that the Jordan and Einstein frames, equivalent classically on shell, are inequivalent off shell — different renormalization schemes and effective potentials — so the quantum meaning of a suppression asserted in one frame requires explicit calculation. Related classically scale-invariant theories (theories containing no dimensionful parameters at the classical level) with a dynamically generated Planck mass are known to destabilize that mass in the presence of explicit symmetry breaking such as matter and cosmological-constant terms (Kannike, Raidal, Spethmann, Veermäe, JHEP 04 (2017) 026), illustrating the residual-sensitivity pattern shared by this family. What remains open, at the sources' own generality, is whether this purely gravitational mechanism class can be completed into a genuine solution.

Problem Statement

For the class of scalar-tensor (equivalence-principle-based) mechanisms in which a gravitational-sector scalar non-minimally coupled to the curvature generates, through the Weyl factor Ω\Omega of the transformation to the Einstein frame, an exponential suppression of all Standard Model mass parameters — the Planck-scale-fundamental construction of Calmet and the weak-scale-fundamental construction of Lin being the two defining instances — determine whether this class can be completed into a genuine solution of the gauge hierarchy problem. The overarching objective decomposes into two jointly required, scientifically linked results:

(1) Radiative stability. Establish, by explicit one-loop effective-potential or renormalization-group analysis in a fixed frame and scheme, whether quantum corrections to the Higgs mass parameter in the Einstein frame are suppressed by the same Weyl factor as the tree-level vacuum expectation value in each model (Ω∼10−17\Omega\sim10^{-17} in the Planck-scale-fundamental variant, e−υh/η∼10−15e^{-\upsilon_h/\eta}\sim10^{-15} in the weak-scale-fundamental variant), or whether Standard Model and gravitational-sector loops (including the unsuppressed Higgs quartic self-coupling visible in the Einstein-frame Lagrangian) reintroduce a hierarchy between the physical electroweak scale and the Planck scale.

(2) Dynamical stabilization. Either construct a dynamics within this mechanism class — for example a Goldberger–Wise-type stabilization mechanism or the supergravity extension of the dilaton sector conjectured by Calmet — that fixes the required gravitational-sector parameters without fine-tuning, in the standard sensitivity-measure sense (e.g. the Barbieri–Giudice criterion), so that a mere reparametrization of the same tuned values does not qualify: for at least one of the two defining variants, fixing that variant's parameters (the dilaton mass setting Ω\Omega, the self-coupling of order 10−5210^{-52} and the sign of the cosmological constant in the Planck-fundamental variant; the ratio υh/η≃35\upsilon_h/\eta\simeq35 and the residual roughly sixty orders of bare cosmological-constant smallness in the weak-scale-fundamental variant, whose effective theory is moreover strongly coupled above the weak scale and needs a controlled ultraviolet completion); or establish, for the mechanism class as a whole, that no such dynamics exists.

At the sources' own generality: whether any purely gravitational, equivalence-principle-based stabilization of the electroweak scale can explain the observed hierarchy rather than merely relocate its fine-tuning into the gravitational sector. This question is answered affirmatively by an explicit dynamical construction for either one of the two defining variants, and negatively by a demonstration that no such construction exists for the class as a whole; establishing either branch (together with deliverable (1)) settles the task.

References

  1. Xavier Calmet, Equivalence principle and the gauge hierarchy problem, Phys. Rev. D 77, 047502 (2008). DOI: 10.1103/PhysRevD.77.047502 (arXiv:0708.2767). https://doi.org/10.1103/PhysRevD.77.047502
  2. Chun-Shan Lin, Large Hierarchy from Non-Minimal Coupling, Commun. Theor. Phys. 68, 223 (2017). DOI: 10.1088/0253-6102/68/2/223 (arXiv:1405.4821). https://doi.org/10.1088/0253-6102/68/2/223
  3. Hiroki Matsui, Yoshio Matsumoto, Gravitational Relaxation of Electroweak Hierarchy Problem, Int. J. Theor. Phys. 61, 182 (2022). DOI: 10.1007/s10773-022-05168-w (arXiv:1608.08838). https://doi.org/10.1007/s10773-022-05168-w
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