The electroweak hierarchy problem, formulated in the regulator-independent 'threshold naturalness' language of de Gouvêa, Hernández and Tait (2014): a theory is natural when every finite correction δμ2\delta\mu^{2} to the Higgs mass-squared parameter from integrating out heavy states is no larger than the electroweak scale, δμ2≲v2\delta\mu^{2}\lesssim v^{2}. The known protective structures — softly broken supersymmetry and super-renormalizable coupling shielding — are incomplete or experimentally squeezed, and motivated high scales (Planck, grand unification, Peccei–Quinn/axion, seesaw neutrinos with thermal leptogenesis) all generate threshold corrections far above v2v^{2}. The open problem is to construct, or rule out, a broadly applicable, calculable ultraviolet structure in which all finite threshold corrections to μ2\mu^{2}, including mixed gravitational ones, satisfy the criterion while retaining the high-scale mechanisms. The 2014–2026 literature shows continued partial constructions (relaxion, sliding naturalness, softened gravity, twin Higgs, discrete-symmetry seesaw) and an unsettled fine-tuning criterion, but no accepted resolution: the problem is open.

Background

In the Standard Model (SM) of particle physics, electroweak symmetry breaking is driven by the Higgs field HH through the potential V=−μ2∣H∣2+λ∣H∣4V=-\mu^{2}|H|^{2}+\lambda|H|^{4}, whose dimensionful mass-squared parameter μ2\mu^{2} sets the Higgs vacuum expectation value v=μ2/λ=246v=\sqrt{\mu^{2}/\lambda}=246 GeV and, with it, the WW and ZZ boson and fermion masses. The hierarchy (or naturalness) problem is the observation that quantum corrections shift μ2\mu^{2} by amounts that naively scale with the heaviest mass in the theory. Textbook arguments invoke quadratic divergences from loops of fields coupled to the Higgs (famously the top quark), but such divergent pieces are regulator-dependent artifacts absorbed into renormalization; assigning them physical meaning tacitly assumes, for example, that spacetime resembles a lattice or that stringy top resonances exist at the cutoff. de Gouvêa, Hernández and Tait instead proposed a concrete, regulator-independent measure: the finite threshold corrections δμ2\delta\mu^{2} that arise when heavy particles of mass MM are integrated out, which are physical, calculable within quantum field theory, and insensitive to the ultraviolet (UV) completion (the short-distance theory above the heavy masses) except through the couplings of the heavy states to the Higgs. Their working criterion for naturalness is δμ2≲v2\delta\mu^{2}\lesssim v^{2}, with δμ2≲(100 GeV)2\delta\mu^{2}\lesssim(100~\mathrm{GeV})^{2} used in numerical estimates — a convention, not a derived principle.

Applied to concrete new-physics scenarios, the criterion yields sharp statements. A heavy fermion with an order-one Yukawa coupling to the Higgs contributes δμ2∼λ2M2/16π2\delta\mu^{2}\sim\lambda^{2}M^{2}/16\pi^{2} at one loop, so it must be lighter than roughly 1 TeV or weakly coupled. A heavy fermion coupled to the SM only through gravity contributes at three loops through mixed gravity–SM diagrams, δμ2∼λt2(16π2)3MΨ4MPl4MΨ2\delta\mu^{2}\sim\frac{\lambda_{t}^{2}}{(16\pi^{2})^{3}}\frac{M_{\Psi}^{4}}{M_{\mathrm{Pl}}^{4}}M_{\Psi}^{2}, which caps such states at MΨ≲1014M_{\Psi}\lesssim10^{14} GeV — well below the Planck mass, so even gravitationally-closeted states eventually destabilize the weak scale. In the type-I seesaw mechanism for neutrino masses (where heavy right-handed neutrinos of mass MM generate light neutrino masses via mν∼λν2v2/Mm_{\nu}\sim\lambda_{\nu}^{2}v^{2}/M), demanding δμ2≲v2\delta\mu^{2}\lesssim v^{2} bounds M≲8M\lesssim8–14×10314\times10^{3} TeV, a result also obtained by Vissani and by Casas, Espinosa and Hidalgo; but successful conventional thermal leptogenesis (the generation of the cosmic baryon asymmetry from heavy-neutrino decay) typically requires M≳109M\gtrsim10^{9} GeV, so most versions of leptogenesis are 'ruled out by naturalness'. In axion solutions of the strong CP problem (the unexplained smallness of the CP-violating angle of the strong interaction), a global Peccei–Quinn (PQ) symmetry broken at a scale fa≳109f_{a}\gtrsim10^{9} GeV (required by astrophysics) introduces a heavy radial scalar, and the mixed quartic portal coupling λnew∣H∣2∣Φ∣2\lambda_{\mathrm{new}}|H|^{2}|\Phi|^{2} — forbidden by no symmetry — generates δμ2∼λnewλρ16π2fa2\delta\mu^{2}\sim\frac{\lambda_{\mathrm{new}}\lambda_{\rho}}{16\pi^{2}}f_{a}^{2}; Farina, Pappadopulo and Strumia showed that λnew\lambda_{\mathrm{new}} acquires irreducible radiative lower bounds at two to three loops, so it cannot simply be set to zero. Grand unified theories (GUTs), with unification scale MGUT∼1016M_{\mathrm{GUT}}\sim10^{16} GeV, 'by themselves certainly signal extreme fine-tuning'. Two protective structures are known: softly broken supersymmetry (SUSY), where superpartners cancel the corrections but broken SUSY creates its own hierarchy problem requiring stops (the scalar superpartners of the top quark) near the TeV scale — now under strong experimental pressure — and the super-renormalizable portal κ∣H∣2Φ\kappa|H|^{2}\Phi to a scalar singlet, which shields μ2\mu^{2} from a heavy scale MΦM_{\Phi} with only a logarithmic correction δμ2∼−κ216π2log⁡(MΦ2/Mh2)\delta\mu^{2}\sim-\frac{\kappa^{2}}{16\pi^{2}}\log(M_{\Phi}^{2}/M_{h}^{2}).

The situation after the Higgs discovery, with no superpartners or top partners found at the LHC, is therefore: the SM alone is natural (no heavy thresholds), but every well-motivated extension involving high mass scales — dark matter, neutrino masses, baryogenesis, strong CP, unification, and ultimately gravity — generates finite threshold corrections to μ2\mu^{2} that exceed v2v^{2} unless additional protective dynamics intervenes. Whether broadly applicable UV structures beyond softly broken SUSY and singlet shielding exist, whether the Planckian/gravitational contributions can ever be controlled without a complete quantum theory of gravity, and whether the naturalness criterion itself can be sharpened from a convention into a derived principle, are all open.

Problem Statement

Under which circumstances can the Standard Model, augmented by heavy fields, accommodate — without large fortuitous cancellations — multiple widely separated mass scales? In the concrete threshold formulation of de Gouvêa, Hernández and Tait, this becomes: does there exist a plausible, broadly applicable, calculable ultraviolet structure — beyond the two protective mechanisms analyzed in the source paper, softly broken supersymmetry and shielding by a super-renormalizable coupling κ∣H∣2Φ\kappa|H|^{2}\Phi to a scalar singlet — in which every finite, regulator-independent threshold correction δμ2\delta\mu^{2} to the Higgs mass-squared parameter from integrating out heavy states satisfies δμ2≲v2\delta\mu^{2}\lesssim v^{2}, including the corrections induced by Planck-scale gravitational dynamics, while the theory still retains the well-motivated high scales: Peccei–Quinn/axion physics, grand unification, and seesaw neutrino masses with a viable baryogenesis mechanism such as conventional thermal leptogenesis? Alternatively, establish a controlled no-go result showing that any such completion forces some δμ2\delta\mu^{2} far above v2v^{2}, so that either a degree of fine-tuning or the abandonment of one of the high-scale mechanisms is unavoidable. These facets — the gravitational corrections, the radiative lower bounds on portal couplings such as λnew\lambda_{\mathrm{new}} (the mixed quartic λnew∣H∣2∣Φ∣2\lambda_{\mathrm{new}}|H|^{2}|\Phi|^{2} between the Higgs and a heavy scalar, forbidden by no symmetry), the tension with leptogenesis and GUT scales, and the sharpening of the fine-tuning criterion itself — are jointly one objective: to determine whether, and how, the electroweak scale can coexist naturally with the high scales that independent physics arguments demand.

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