Grand unified theories that remain supersymmetric down to energies of 10310^{3}-10510^{5} GeV were shown by Dimopoulos, Raby, and Wilczek in 1981 to push the gauge-unification scale up to the vicinity of the Planck mass, preserving the successful weak-mixing-angle prediction while enormously suppressing gauge-boson-mediated proton decay — at the cost of four acknowledged unresolved consequences: unspecified early-Universe initial conditions for superheavy states that never reach thermal equilibrium; the unconstrained possibility of unexpectedly light scalar bosons mediating faster nucleon decay; the difficulty of realizing supersymmetry breaking ('heavy color') at 10310^{3}-10510^{5} GeV without so many multiplets that gauge couplings become nonperturbative below unification, together with an oversimplified Higgs sector; and the ambiguous particle interpretation of monopoles and superheavy gauge bosons heavier than the Planck mass (Compton radius inside Schwarzschild radius) absent a unification that includes gravity. Later literature resolved parts of this agenda — inflation answers the relic and initial-conditions problem; dimension-five color-triplet-Higgs proton decay (p→K+νˉp\to K^{+}\bar{\nu}) was established, computed at two loops, and bounded by Super-Kamiokande (excluding minimal supersymmetric SU(5)); explicit dynamical supersymmetry-breaking and doublet-triplet-splitting mechanisms were constructed; string theory is a candidate gravity-inclusive framework, with the monopole/black-hole issue sharpened into the still-unproven Weak Gravity Conjecture. The conjunction, however, remains open: no complete, internally consistent, experimentally viable supersymmetric grand unified theory satisfying all four conditions exists, and no no-go theorem excludes the program. The problem is to construct such a theory.

Background

By the late 1970s the Standard Model of particle physics was successful enough that attention turned to unifying its forces — the strong interaction (gauge group SU(3), the source of 'color' charge) and the electroweak interaction (SU(2)xU(1)) — into a single grand unified theory (GUT) based on one larger simple group, the minimal example being SU(5). In such a theory the three coupling constants, which drift apart at low energies according to renormalization-group equations, meet at a common value at a huge unification scale MM. Georgi, Quinn, and Weinberg computed this scale in ordinary (non-supersymmetric) SU(5) and found M∼1015M\sim 10^{15} GeV, together with a successful prediction for the weak mixing angle sin⁡2θW\sin^{2}\theta_{W} that fixes the relative strength of the weak and electromagnetic interactions. The same theories predict proton decay mediated by superheavy X and Y gauge bosons with a lifetime of order 103110^{31} years, then being sought in underground water detectors. Two structural problems afflicted them. First, the gauge hierarchy problem: quantum corrections involving GUT-scale particles would drag the electroweak scale (∼102\sim 10^{2} GeV) up toward MM unless parameters are tuned to many decimal places. Second, the early-Universe phase transition that breaks the unified group would unavoidably produce superheavy magnetic monopoles (stable point defects carrying isolated magnetic charge) at unacceptably high relic abundance, as Preskill quantified in 1979.

Supersymmetry (SUSY) — the postulated symmetry pairing each fermion with a bosonic superpartner and vice versa — offered a way out, since boson and fermion loops cancel and stabilize the electroweak scale. Dimopoulos and Georgi constructed the explicit softly broken supersymmetric SU(5), whose low-energy content is the Minimal Supersymmetric Standard Model (MSSM): every Standard Model particle plus a superpartner below roughly 10310^{3}-10510^{5} GeV. Dimopoulos, Raby, and Wilczek — the source of the present problem — then computed the effect of this enlarged low-energy spectrum on coupling unification. The sin⁡2θW\sin^{2}\theta_{W} prediction survives almost unchanged and remains successful, but the unification scale jumps to ∼1019\sim 10^{19} GeV in their one-loop estimate (modern two-loop analyses with threshold effects place it near 101610^{16} GeV), i.e., near the Planck mass MPl≈1.2×1019M_{Pl}\approx 1.2\times 10^{19} GeV where gravity becomes as strong as the gauge forces. Gauge-boson-mediated proton decay is thereby enormously suppressed. The authors closed by acknowledging four unresolved implications. First, gauge bosons this massive would never reach thermal equilibrium in the rapidly expanding early Universe, so the initial state must be determined 'from some as yet unknown dynamics, presumably involving particle production in strong gravitational fields'. Second, 'one cannot ... necessarily preclude the existence of unexpectedly light scalar bosons mediating the nucleon decay at a faster rate': color-triplet scalar partners of the Higgs field can induce baryon-number-violating operators of dimension five (suppressed by only one power of a heavy mass, hence far more dangerous than the usual dimension-six gauge-boson exchange). Third, if supersymmetry is broken by new strong dynamics — 'heavy color' or 'supercolor' — too many light multiplets would make the SU(2) coupling nonperturbative below unification; they bounded the allowed extra content (no more than ten supersymmetric chiral SU(2) doublets beyond the three known families, suggesting at most five families) and noted it is 'difficult to understand why it becomes strong at 10310^{3}-10510^{5} GeV', while admitting their Higgs-sector treatment was oversimplified. Fourth, monopoles and some superheavy gauge bosons with masses above MPlM_{Pl} would have Compton radii inside their Schwarzschild radii — 'they are black holes' — and the paper could only hint that a unification scale near the Planck mass might presage 'a larger unification including gravity', without a theory of it.

Much of the subsequent history of particle physics engages these points. Inflationary cosmology (Guth; Linde) — a postulated epoch of exponential spatial expansion in the very early Universe — was invented partly to dilute the monopole overproduction and now sets the standard answer to the early-Universe initial-conditions problem, with gravitational particle production developed into the theory of reheating after inflation (Kofman, Linde, and Starobinsky). Dimension-five proton decay was established within a year (Dimopoulos, Raby, Wilczek; Sakai and Yanagida; Weinberg) — with the kaon mode p→K+νˉp\to K^{+}\bar{\nu} dominant — computed definitively at two loops by Hisano, Murayama, and Yanagida, and bounded by Super-Kamiokande (2014: no signal, τ/B>5.9×1033\tau/B>5.9\times 10^{33} yr), which excludes minimal supersymmetric SU(5) (Murayama and Pierce). Dynamical supersymmetry breaking (initiated by Witten) matured into many explicit models (reviewed by Intriligator and Seiberg; gauge mediation reviewed by Giudice and Rattazzi), and the doublet-triplet splitting problem — why the electroweak Higgs doublets remain light while their color-triplet partners inside the unified multiplet are superheavy — was posed by Sakai, with proposed solutions such as the missing-partner mechanism of Grinstein (a superpotential structure in which the color-triplet partners naturally sit at the unification scale, so only the doublets remain light). String theory, which describes particles as extended one-dimensional objects rather than points, provides a candidate framework unifying gauge forces with gravity — most notably the heterotic string of Gross, Harvey, Martinec, and Rohm — and the monopole/black-hole consistency question reappears there as the Weak Gravity Conjecture (Arkani-Hamed, Motl, Nicolis, and Vafa), still unproven. The same authors reviewed the mature quantitative success of supersymmetric coupling unification in 1991. Yet no construction satisfying all four conditions simultaneously is known, and model-building and experimentation around them remain active (dimension-five suppression: Haba et al.; experimental frontier: Baum et al.).

Problem Statement

Complete the program opened by Dimopoulos, Raby, and Wilczek (1981): construct a single, fully specified, internally consistent supersymmetric grand unified theory that simultaneously addresses the four implications the source paper itself acknowledged as unresolved consequences of the near-Planck unification scale.

(i) Perturbative unification: with the model's complete field content, the three Standard Model gauge couplings must unify at a single scale while preserving the successful weak-mixing-angle prediction, and no coupling may become nonperturbative below that scale (the source's multiplicity bound on light multiplets must be respected: no more than ten supersymmetric chiral SU(2) doublets beyond the three known families). Unification must be demonstrated at two-loop accuracy including threshold effects (corrections from particles at their mass thresholds).

(ii) Supersymmetry breaking: the theory must specify and realize a mechanism of supersymmetry breaking at a naturalness-compatible scale — the source window is 10310^{3}-10510^{5} GeV — communicated to the visible sector consistently with flavor-changing, collider, and precision constraints, together with a realistic Higgs sector going beyond the oversimplification the source acknowledges, in which the doublet-triplet splitting is achieved without a cancellation among independent parameters tuned to many significant figures.

(iii) Nucleon stability: all baryon-number-violating operators of the theory — in particular the dimension-five operators mediated by color-triplet scalars, whose possible lightness the source flagged as not excludable — must predict nucleon lifetimes consistent with current experimental limits (at present the Super-Kamiokande 2014 bound, partial lifetime τ/B(p→K+νˉ)>5.9×1033\tau/B(p\to K^{+}\bar{\nu})>5.9\times 10^{33} yr at 90% confidence level), with the dominant decay modes and their rates computed.

(iv) Cosmology and gravity: the theory must provide a consistent cosmological history for its superheavy states (magnetic monopoles, superheavy gauge bosons, the gravitino — the graviton's superpartner — or hidden-sector relics, that is, fields coupled to the visible sector only indirectly; initial conditions, production, and relic abundances), and it must be consistent with gravity at the near-Planck unification scale: either embedded in an explicitly named consistent quantum-gravity construction, or shown to be a controlled effective field theory up to its own unification scale, together with an account of the status of any monopoles and superheavy gauge bosons at or above the model's own unification scale whose Compton radius would fall inside their Schwarzschild radius (the source's one-loop estimate places unification near MPlM_{Pl}, where such states arise; this clause applies when the model's own unification scale is at or above MPl/2M_{Pl}/\sqrt{2}, whereas the modern two-loop scale of a minimal spectrum near 101610^{16} GeV lies below that threshold).

These four conditions are jointly necessary consistency requirements of the single overarching objective — a complete supersymmetric unification — rather than independent deliverables: each constrains the same model from a different side, and a candidate theory fails the objective if it fails any one of them. The target theory specifies the full gauge group, field content, superpotential, and breaking sector; its observables entering conditions (i)-(iv) follow from it; and it is consistent with the experimental constraints relevant to those conditions.

References

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