Controlled realization of maximal-vacuum-energy selection of the weak scale in a UV-complete landscape

Cheung and Saraswat (arXiv:1811.12390) proposed that the electroweak hierarchy is explained by a 'principle of maximum vacuum energy': if a light scanning field phi shifts Higgs-sector parameters, the classical vacuum energy is generically maximized at critical boundaries where the weak scale is hierarchically small, and some early-universe mechanism could localize the Universe there. The authors themselves state what remains unresolved: the localization mechanism (inflationary volume weighting, anthropic selection, or ghost/non-local operators) is model-dependent and experimentally unverifiable; the cosmology it demands is restrictive (low-scale inflation with reheating below the weak scale); the scanning field's excursion may be trans-Planckian and in tension with quantum-gravity conjectures; the statistical distribution of the cosmological-constant landscape needed for the anthropic argument is not derived from any fundamental theory; and the principle has only been worked out quantitatively in a phi-scanned MSSM that cannot accommodate the 125 GeV Higgs mass. The open problem is the construction (or exclusion) of a complete, controlled realization of this selection principle in which the landscape statistics are derived rather than assumed, the selection dynamics is cosmologically and quantum-gravitationally consistent, and the resulting electroweak model is phenomenologically viable with falsifiable predictions.

Background

The electroweak hierarchy problem is the question of why the Higgs mass parameter is some 32 orders of magnitude below the Planck mass when quantum corrections from any heavy states coupled to the Higgs naturally drag it up to the cutoff. 'Naturalness' is the traditional demand that no such fine-tuning occur; conventional solutions (supersymmetry, compositeness) have not been confirmed experimentally. A different route is cosmological selection of the weak scale, pioneered in the relaxion mechanism of Graham, Kaplan and Rajendran (arXiv:1504.07551), in which a light axion-like field dynamically scans the Higgs mass during inflation and gets trapped at a small value.

Cheung and Saraswat (arXiv:1811.12390) observed a universal geometric fact underlying such scans: if one plots the classical vacuum energy $V_{\mathrm{vac}}$ of the electroweak theory as a function of a parameter such as the Higgs mass-squared (which a background scanning field $\phi$ effectively shifts), then the boundaries separating phases with different vacua — for example the transition between unbroken and broken electroweak symmetry (electroweak symmetry breaking, EWSB) — are exactly where $V_{\mathrm{vac}}$ changes from flat to concave down. Near-criticality, i.e. fine-tuning of the weak scale, corresponds to these boundaries. After radiative corrections from heavy states coupled to the Higgs (two-loop effects through heavy superpartners such as the top squark (the top quark's supersymmetric partner), or one-loop effects of Higgs 'partners' such as the Higgsino), these boundary regions are generically perturbed into local maxima of $V_{\mathrm{vac}}(\phi)$. They then proposed a 'principle of maximum vacuum energy': some mechanism localizes the Universe at such a maximum, which automatically selects a hierarchically small weak scale generated radiatively, and predicts the new states to be roughly a loop factor ($\sim 4\pi$) heavier than in conventional naturalness, pushing collider targets toward multi-TeV to 100 TeV scales.

The paper proposed several localization mechanisms, each with stated weaknesses: (i) inflationary volume weighting — during eternal inflation, Hubble patches with larger vacuum energy expand faster, so vacua near the maximum dominate the Universe's volume; this is described by a volume-weighted Fokker-Planck equation (a diffusion-drift equation for the probability distribution of $\phi$ across inflating patches), but depends on the measure ambiguities of eternal inflation and, in their implementation, requires a long period of low-scale inflation with a Standard Model reheating temperature below the weak scale to avoid thermally destabilizing the selected vacuum; (ii) anthropic selection — combining Weinberg's anthropic bound on the cosmological constant (CC, the vacuum energy density; Weinberg, Phys. Rev. Lett. 59, 2607 (1987)) with the $\phi$ landscape, which however requires a specific statistical distribution $n(\rho)$ of CC values across vacua that is not derived from any fundamental theory; and (iii) exotic ghost fields (fields of wrong-sign kinetic term) or non-local operators, which are of dubious consistency. To give the scanning potential the needed periodic structure over a super-Planckian range while keeping control, the paper invoked axion monodromy (a periodic potential superposed on a non-periodic one) or clockwork models (chains of fields generating an exponentially large effective decay constant), whose exact ultraviolet (UV, short-distance) completion is unspecified. The only quantitative model studied was the $\phi$MSSM, in which $\phi$ scans the parameters of the Minimal Supersymmetric Standard Model; footnote 4 of that paper concedes it does not reproduce the observed 125 GeV Higgs mass without extending the model. The authors flag a further concern: the field excursion of $\phi$ may need to exceed the Planck scale, which conjectured quantum-gravity constraints such as the swampland distance conjecture (the claim, still conjectural, that large field excursions in a consistent quantum gravity necessarily bring towers of new states into play; Ooguri, Palti, Shiu, arXiv:1810.05506) would forbid. Their conclusions explicitly leave open whether the approach can be realized in models beyond the MSSM, such as composite Higgs theories.

Problem Statement

Construct, or establish a no-go result for, a controlled realization of the 'principle of maximum vacuum energy' selection of the electroweak scale in which all load-bearing ingredients are derived within one consistent theory rather than assumed. Realizing vacuum-energy-maximization selection requires, jointly:

1. Landscape statistics from the theory: derive, within a UV-complete framework (for example a controlled string-theory landscape), the joint statistical distribution of vacua over (i) Higgs-sector parameters scanned by the field $\phi$ or an equivalent landscape coordinate and (ii) the remaining vacuum-energy contribution $\rho_0$, such that vacua with a hierarchically small weak scale and an anthropically acceptable cosmological constant (in the sense of Weinberg's bound) are selected — replacing the ansatz distributions $n(\rho_0)$ (exponential or Gaussian) that the source paper shows to require a super-exponentially large number of vacua. A distribution that is postulated rather than derived does not satisfy this facet, even if it is embedded in an internally consistent effective field theory.

2. Controlled selection dynamics: specify the mechanism that localizes the Universe at or near a local maximum of $V_{\mathrm{vac}}$ with respect to the scanned parameters — inflationary volume weighting (solving or controlling the measure problem of eternal inflation), anthropic selection, or an alternative — with a field excursion and cosmological history that are compatible with any quantum-gravity constraints established within the chosen framework (that is, derived rather than conjectured), that explicitly address any tension with conjectural constraints such as the swampland distance conjecture instead of silently assuming them away or elevating them to theorems, and that do not demand an observationally problematic history (e.g. a reheating temperature below the weak scale, unless justified).

3. Viable phenomenology: show the resulting electroweak model reproduces the observed 125 GeV Higgs mass and all collider and cosmological constraints, and derive explicit, falsifiable relations among measurable parameters (mass spectra, couplings, or precision observables) that distinguish vacuum-energy-maximization selection from conventional naturalness.

A no-go answer must state its assumptions precisely (on landscape statistics, measure, or field range) and demonstrate that no theory satisfying them can select a small weak scale via vacuum-energy maximization. The assumptions must be independently motivated rather than question-begging: in particular, a no-go may not be obtained by postulating as a premise the truth of constraints that this statement identifies as conjectural (such as the swampland distance conjecture), the unsolvability of the eternal-inflation measure problem, or the non-derivability of the required joint distribution of vacua, since each of these would restate part of the target as an assumption. The three facets are jointly the completion boundary: a construction resolving only one facet (for example an assumed-landscape effective model, or an abstract statistics argument with no electroweak realization) leaves the source question open.

References

1. Clifford Cheung, Prashant Saraswat, 'Mass Hierarchy and Vacuum Energy' (2018), arXiv:1811.12390, DOI:10.48550/arXiv.1811.12390, https://arxiv.org/abs/1811.12390
2. Peter W. Graham, David E. Kaplan, Surjeet Rajendran, 'Cosmological Relaxation of the Electroweak Scale' (2015), arXiv:1504.07551, DOI:10.48550/arXiv.1504.07551, https://arxiv.org/abs/1504.07551
3. Steven Weinberg, 'Anthropic Bound on the Cosmological Constant', Physical Review Letters 59, 2607 (1987), DOI:10.1103/PhysRevLett.59.2607, https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.59.2607
4. Hirosi Ooguri, Eran Palti, Gary Shiu, Cumrun Vafa, 'Distance and de Sitter Conjectures on the Swampland', Physics Letters B 788, 180 (2019), arXiv:1810.05506, DOI:10.1016/j.physletb.2018.11.018, https://arxiv.org/abs/1810.05506
5. Alessandro Strumia, Daniele Teresi, 'Cosmological constant: relaxation vs multiverse' (2019), arXiv:1904.07876, DOI:10.48550/arXiv.1904.07876, https://arxiv.org/abs/1904.07876
6. Gian F. Giudice, Alexandros Kehagias, Antonio Riotto, 'The Selfish Higgs' (2019), arXiv:1907.05370, DOI:10.48550/arXiv.1907.05370, https://arxiv.org/abs/1907.05370
7. Alessandro Strumia, Daniele Teresi, 'Relaxing the Higgs mass and its vacuum energy by living at the top of the potential' (2020), arXiv:2002.02463, DOI:10.48550/arXiv.2002.02463, https://arxiv.org/abs/2002.02463
8. Csaba Csaki, Raffaele Tito D'Agnolo, Michael Geller, Ameen Ismail, 'Crunching Dilaton, Hidden Naturalness' (2021), arXiv:2007.14396, DOI:10.48550/arXiv.2007.14396, https://arxiv.org/abs/2007.14396
9. Gian F. Giudice, Matthew McCullough, Tevong You, 'Self-organised localisation' (2021), arXiv:2105.08617, DOI:10.48550/arXiv.2105.08617, https://arxiv.org/abs/2105.08617
10. Raffaele Tito D'Agnolo, Daniele Teresi, 'Sliding Naturalness: Cosmological Selection of the Weak Scale' (2021), arXiv:2109.13249, DOI:10.48550/arXiv.2109.13249, https://arxiv.org/abs/2109.13249
11. Arthur Hebecker, 'Lectures on Naturalness, String Landscape and Multiverse' (2020), arXiv:2008.10625, DOI:10.48550/arXiv.2008.10625, https://arxiv.org/abs/2008.10625
12. Thomas Steingasser, David I. Kaiser, 'Higgs Criticality beyond the Standard Model' (2023), arXiv:2307.10361, DOI:10.48550/arXiv.2307.10361, https://arxiv.org/abs/2307.10361
13. Oleksii Matsedonskyi, 'Hierarchies from Landscape Probability Gradients and Critical Boundaries' (2023), arXiv:2311.10139, DOI:10.48550/arXiv.2311.10139, https://arxiv.org/abs/2311.10139
14. Susobhan Chattopadhyay, Dibya S. Chattopadhyay, Rick S. Gupta, 'Cosmological selection of a small weak scale from large vacuum energy: a minimal approach' (2024), arXiv:2407.15935, DOI:10.48550/arXiv.2407.15935, https://arxiv.org/abs/2407.15935
15. Raffaele Tito D'Agnolo, Manuel Ettengruber, Lian-Tao Wang, 'Landscapes at Colliders' (2025), arXiv:2512.18001, DOI:10.48550/arXiv.2512.18001, https://arxiv.org/abs/2512.18001