The three Standard Model gauge couplings, when renormalization-group evolved in the minimal supersymmetric Standard Model, meet far more precisely at a high scale than the threshold corrections that generic grand unified theories (GUTs) predict. Bhattiprolu and Wells (2024) turned this observation into the 'precision unification conjecture' — supersymmetry is correct and high-scale GUT threshold corrections are highly suppressed — and showed it favors superpartner masses from a few TeV up to the PeV scale in minimal supergravity and minimal anomaly-mediated scenarios, with Higgsino or wino thermal dark matter. They explicitly left open what ultraviolet dynamics could suppress the thresholds to the required smallness and did not embed the viable parameter points into any fully specified unified theory. The open problem is to identify and explicitly construct such a mechanism — a GUT, orbifold GUT, or string compactification whose calculable threshold spectrum yields the required unification quality for the identified spectra.

Background

Gauge coupling unification is the empirical fact that the running strengths α1,α2,α3\alpha_1,\alpha_2,\alpha_3 of the three Standard Model (SM) gauge interactions, evolved by the renormalization group (RG) from measured low-energy values toward high energies, approach a common value. In the minimal supersymmetric Standard Model (MSSM), the supersymmetric particle content changes the RG equations so that the couplings meet far more closely than in the non-supersymmetric SM, which has long been viewed as evidence for both supersymmetry (SUSY) and a grand unified theory (GUT) — a single simple gauge group such as SU(5)SU(5) or SO(10)SO(10) broken to the SM at a high scale.

In any GUT the heavy remnants of unification (the additional gauge bosons, Higgs-sector fields, and other multiplets with GUT-scale masses) split the three measured couplings at the unification scale by so-called threshold corrections: one-loop shifts 1/gi2−1/gj21/g_i^2-1/g_j^2 given, since Hall (1981), by logarithmic mass sums over the split multiplets weighted by their Dynkin indices (measures of the size of each representation's gauge charge). In typical four-dimensional supersymmetric GUTs the remnants of high-dimensional representations make these corrections large; as quantified by Ellis and Wells (2015, 2017), the natural misfit is 'of order hundreds' in the standard measure ρλmin⁡\rho_\lambda^{\min} — the minimum over the sliding scale μ\mu of ρλ\rho_\lambda, where ρλ/48π2≡∑i≠j(1/gi2−1/gj2)2\rho_\lambda/48\pi^2 \equiv \sqrt{\sum_{i\neq j}(1/g_i^2-1/g_j^2)^2} with GUT-normalized couplings (Bhattiprolu and Wells 2024, eq. (1)). The MSSM couplings, by contrast, meet with ρλmin⁡\rho_\lambda^{\min} far smaller.

Bhattiprolu and Wells (2024, arXiv:2309.12954) took this discrepancy seriously as the 'precision unification conjecture': SUSY is a correct principle of nature and the high-scale threshold corrections are much smaller than naive GUT expectations. Adopting the concrete criterion ρλmin⁡<20\rho_\lambda^{\min}<20 (roughly a 3σ3\sigma deviation from exact unification), they scanned the MSSM with a common superpartner threshold and two high-scale scenarios — minimal supergravity (mSUGRA, gravity-mediated SUSY breaking with comparable superpartner masses) and minimal anomaly-mediated supersymmetry breaking (mAMSB, where the gaugino masses — the fermionic superpartners of the gauge bosons — arise from the superconformal anomaly in a distinctive hierarchy) — requiring the lightest Higgs boson mass mh=125.25±3m_h=125.25\pm3 GeV. Precision unification plus the Higgs constraint selects superpartner masses from a few TeV up to the PeV scale, with the Higgsino mass parameter ∣μ∣|\mu| of order one to a few hundred TeV, and leaves regions where a Higgsino or wino lightest supersymmetric particle (the fermionic superpartners of the Higgs bosons and of the electroweak gauge bosons, respectively) reproduces the observed thermal dark-matter abundance. The authors were explicitly agnostic about why the thresholds are suppressed, mentioning only speculative options (an orbifold GUT — extra-dimensional unification in which boundary conditions project out the harmful heavy representations — possibly arising from string compactification), and they provided no embedding of the viable points into a fully specified unified theory.

Known routes toward small thresholds exist but do not answer the question. Highly anisotropic string compactifications (Hebecker and Trapletti 2005) can preserve gauge unification with a high string scale. In F-theory GUTs with high-scale SUSY, Hebecker and Unwin (2014) showed the in-principle-calculable classical F-theory corrections can compensate imprecise unification, but at the price of a lowered GUT scale in tension with proton decay, which they found difficult to evade because the higher Landau-level modes of the heavy X,YX,Y gauge bosons are hard to localize. Precision unification can also be engineered at the electroweak scale through gaugino-mass patterns such as mirage mediation (a scheme in which anomaly- and gravity-mediated contributions combine so that the three gaugino masses appear to unify at a 'mirage' scale below the GUT scale; Raby, Ratz and Schmidt-Hoberg 2010; Krippendorf et al. 2013) or widely separated gauginos with TeV Higgsinos (Suryanarayana Mummidi and Patel 2020) — mechanisms within the MSSM spectrum that do not address the high-scale GUT thresholds at all. Secondary qualifications on the identified spectra remain active: Higgsino thermal dark matter requires an extremely pure Higgsino under direct-detection limits (Martin 2025; Bhattiprolu, Martin and Wells 2026), wino dark matter is pressured by indirect detection, and avoiding gravitino overproduction can require a reheat temperature Treh≲m3/2/20T_{\mathrm{reh}}\lesssim m_{3/2}/20 (Moroi, Murayama and Yamaguchi 1993).

Problem Statement

Identify the ultraviolet dynamics underlying the precision-unification conjecture of Bhattiprolu and Wells (2024), and demonstrate it in an explicit construction. Concretely: construct a fully specified unified theory — a four-dimensional GUT, an orbifold GUT, or a string compactification — in which (i) the high-scale threshold corrections to the mismatch among the three gauge couplings are calculable and collectively suppressed enough that, across the mSUGRA and mAMSB parameter regions the source paper identifies as precision-unification viable (superpartner masses from a few TeV to the PeV scale, ∣μ∣|\mu| of order one to a few hundred TeV, mh≈125m_h\approx125 GeV, and a Higgsino or wino thermal dark-matter candidate), the precision-unification criterion ρλmin⁡<20\rho_\lambda^{\min}<20 (with ρλmin⁡\rho_\lambda^{\min} the minimum-over-scale mismatch measure defined in the background) continues to hold after the theory's own GUT-scale spectrum is included — supported both by a suppression argument and by evaluation at the source's viable spectra (or a stated representative set), not at only a single selected parameter point; and (ii) the construction specifies the heavy-particle spectrum and symmetry-breaking sector concretely enough that the threshold shifts can be computed and shown compatible with the dark-matter/cosmological qualifications noted by the source (Higgsino purity under direct detection, wino pressure from indirect detection, and the gravitino reheat bound). Both (i) and (ii) are required.

References

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