The electroweak hierarchy (naturalness) problem: why does the Higgs mass sit near the electroweak scale rather than the Planck scale?

In the Standard Model the Higgs mass parameter is a free constant unprotected by any symmetry, and quantum corrections from generic heavy physics shift the Higgs mass-squared by amounts of order the heavy scale squared, making the observed electroweak scale ($v\approx 246$ GeV, $m_H\approx 125$ GeV) appear fine-tuned against the Planck scale ($M_{Pl}\approx 1.2\times 10^{19}$ GeV) by roughly 30 orders of magnitude in $m_H^2$. Since the canonical formulations (Weinberg 1976; 't Hooft 1979; Susskind 1979), no proposed resolution — supersymmetry, composite Higgs, extra dimensions, relaxion dynamics, gravitational scale invariance, asymptotic safety, or anthropic selection — has been accepted: the mechanism stabilizing the electroweak scale, or a rigorous demonstration that none is needed, remains unknown.

Background

The Standard Model (SM) of particle physics describes matter as quantum fields with gauge interactions. Masses of the W and Z bosons and of fermions arise through electroweak symmetry breaking (EWSB): a scalar Higgs field acquires a vacuum expectation value $v\approx 246$ GeV, and the observed Higgs boson has mass $m_H\approx 125$ GeV (discovered by the ATLAS and CMS collaborations in 2012). The only other dimensional constant in fundamental physics is gravitational: the reduced Planck scale $M_{Pl}\approx 1.2\times 10^{19}$ GeV, the energy at which gravity becomes strong. The ratio $v/M_{Pl}\sim 10^{-16}$ — about 32 orders of magnitude in energy-squared — is the electroweak–Planck hierarchy.

The problem is that scalar masses are uniquely sensitive to physics at higher energies. Computing the one-loop correction to the Higgs mass-squared with an ultraviolet (UV) cutoff $\Lambda$ (the maximum energy up to which the theory is applied) gives $\delta m_H^2\sim \Lambda^2/(16\pi^2)$ times a combination of couplings, dominated by the top-quark Yukawa contribution; if $\Lambda\sim M_{Pl}$ this exceeds the observed $m_H^2$ by over 30 orders of magnitude, so the underlying mass parameter must cancel the correction to that accuracy. In the modern Wilsonian (effective-field-theory) view the point survives the removal of cutoff artifacts: any heavy states of mass $M$ coupled to the Higgs shift $m_H^2$ by $\sim g^2 M^2/(16\pi^2)$, so the smallness of $v$ relative to any heavier threshold requires an explanation. 't Hooft's naturalness criterion (1979) makes this precise: a theory is natural if every small dimensionless ratio is protected by a symmetry that is restored when the ratio vanishes. Small fermion masses are protected by chiral symmetry (the symmetry relating left- and right-handed fermion components), and the small photon mass by gauge symmetry, but no known symmetry protects a fundamental scalar's mass. Weinberg (1976) and Susskind (1979) formulated this as the hierarchy problem; Susskind's technicolor (QCD-like new strong dynamics replacing the elementary Higgs) was the first proposed response.

Many resolutions have been proposed and none is accepted. Supersymmetry (a symmetry relating bosons and fermions whose loop contributions cancel the quadratic corrections) predicts partner particles near the electroweak scale, but the LHC has found none. Large extra dimensions (Arkani-Hamed, Dimopoulos, Dvali 1998) lower the fundamental gravitational scale; composite-Higgs theories make the Higgs a pseudo-Nambu–Goldstone boson (a light collective mode protected by an approximate symmetry); the relaxion (Graham, Kaplan, Rajendran 2015) uses cosmological evolution with axion-like fields to scan the Higgs mass dynamically; exactly scale-invariant gravity and asymptotic-safety approaches (Shaposhnikov–Wetterich 2010, which predicted $m_H\approx 126$ GeV but assumes no heavier particle thresholds) attempt to derive the scale from gravitational dynamics; anthropic arguments invoke selection within a multiverse. Giudice's reviews (2008, 2017) document the criterion's history and the post-LHC 'post-naturalness' reassessment of whether the criterion is well posed. An empirical hint without an accepted explanation is the near-criticality of the SM vacuum: the measured 125 GeV mass places the vacuum close to the boundary between absolute stability and metastability (Buttazzo et al. 2013).

Problem Statement

Explain, within a fundamental theory containing the Standard Model and gravity, why the Higgs mass parameter is of order the electroweak scale ($v\approx 246$ GeV, $m_H\approx 125$ GeV) rather than of order the highest relevant physical scale such as the Planck scale $M_{Pl}\approx 1.2\times 10^{19}$ GeV, given that a heavy state of mass $M$ coupled to the Higgs with coupling $g$ shifts the Higgs mass-squared by $\delta m_H^2\sim g^2M^2/(16\pi^2)$, so the observed value appears to require cancellations accurate to roughly one part in $10^{30}$. The question has three possible resolutions: (a) identify and establish the mechanism — a protecting symmetry, new dynamics, cosmological selection, or another principle — by which the electroweak scale is stabilized against ultraviolet corrections, and derive the observed scale from it while showing that the required cancellation in $m_H^2$ is no finer than a small explicit factor of order ten, rather than the one part in $10^{30}$ that the Standard Model alone demands; (b) establish rigorously that within quantum field theory coupled to gravity no symmetry, dynamics, or selection principle can protect the electroweak scale against corrections of order $g^2M^2/(16\pi^2)$, so that the observed value is left unexplained; or (c) establish rigorously that no such mechanism is needed — that is, state explicitly which naturalness criterion the answer adopts, defend that criterion against 't Hooft's symmetry-protection criterion, and show that under it the smallness of $m_H^2$ relative to the highest physical scale requires no dynamical explanation. The problem is posed at its natural generality: no particular mechanism, model class, or observable is prescribed, and any proposed answer must be consistent with all established experimental constraints, including the measured Higgs properties at 125 GeV, LHC search limits, precision electroweak and flavor data, and cosmology.

References

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