Perturbative vacuum stability and dilaton stabilization of the three-generation standard-like model from the tachyonic ten-dimensional heterotic-string vacuum

A tachyon-free, three-generation, standard-like heterotic-string model was constructed as a compactification of the tachyonic ten-dimensional heterotic-string vacuum, with all geometrical moduli projected out so that only the dilaton remains unfixed (Faraggi, Matyas and Percival, 2020). The model is explicitly non-supersymmetric, and its perturbative viability is unsettled: at one loop it develops tadpoles from its anomalous $U(1)$ (a would-be Fayet–Iliopoulos D-term) and from the non-vanishing vacuum energy of the boson–fermion mismatch, the dilaton is stabilized only by an uncomputed racetrack suggestion, and stability beyond one loop is uncontrolled. The problem is to establish or rule out a perturbatively stable vacuum for this model: compute the one-loop scalar potential with both tadpoles and the effect of vacuum shifts, demonstrate or exclude dilaton stabilization at finite coupling with small positive vacuum energy, and assess higher-order corrections. Follow-up work through 2025 has computed one-loop cosmological constants for the model class and D-term uplifts in related vacua, but explicitly leaves the full tadpole analysis, the racetrack dilaton potential, and higher-order stability open.

Background

The heterotic string combines a right-moving superstring with a left-moving bosonic string, so that four-dimensional vacua can contain chiral fermions and gauge groups containing the Standard Model. In the free fermionic formulation the internal degrees of freedom of a $Z_2\times Z_2$ toroidal orbifold compactification are represented by worldsheet fermions, and a choice of boundary-condition basis vectors together with a generalized GSO (GGSO) phase matrix — the set of signs and phases that determines which sectors of the worldsheet fermions survive in the string spectrum — fixes the four-dimensional gauge group and matter content. Supersymmetry is not automatic: besides the supersymmetric ten-dimensional heterotic vacua and the non-supersymmetric but tachyon-free $SO(16)\times SO(16)$ vacuum, ten-dimensional heterotic string theory admits tachyonic non-supersymmetric vacua. Replacing the supersymmetry-generating basis vector $S$ by $\tilde S$ produces four-dimensional 'tilde-S' models that can be regarded as compactifications of such a tachyonic ten-dimensional vacuum: spacetime supersymmetry is broken explicitly at the string scale, while the tachyons are projected out by other GGSO projectors. Faraggi, Matyas and Percival used this route to construct a tachyon-free, three-generation standard-like model whose asymmetric boundary conditions project all geometrical moduli out of the massless spectrum — the Kähler and complex-structure fields of the compact space are absent, so the internal geometry is frozen rather than merely stabilized — leaving the dilaton, the field whose expectation value sets the string coupling, as the only flat direction.

The model nevertheless carries three admitted instabilities. First, one combination $U(1)_A$ of its six $U(1)$ gauge symmetries is anomalous; as in supersymmetric vacua the anomaly is removed by the Green–Schwarz mechanism, but the anomalous symmetry generates at one loop a tadpole — the analog of a Fayet–Iliopoulos D-term, a one-loop contribution to the potential of an anomalous gauge field whose appearance signals that the vacuum sits at an unstable point. Second, since the numbers of bosonic and fermionic states differ at the massive levels, the one-loop vacuum energy is non-zero and itself sources a tadpole; the authors speculated that the two tadpoles might be made to conspire and cancel, noting that the anomalous-$U(1)$ contribution is proportional to a trace over massless fermions whose sign can be changed by GGSO phases, but stated that a comprehensive analysis of the scalar potential and possible vacuum shifts was beyond their scope and 'possibly out of reach' of contemporary tools given the large number of scalar fields in a quasi-realistic spectrum. Third, the fixing of the dilaton was only conjectured via a racetrack mechanism — hidden-sector gaugino condensation in two or more hidden gauge groups at different scales whose contributions to the dilaton potential nearly cancel and can produce a minimum — with no potential computed, no condensation scales determined, and no demonstration of a phenomenologically acceptable vacuum energy; the paper also stresses that even a favourable one-loop result would leave viability at higher orders of string perturbation theory open. The argument that all non-geometrical flat directions are lifted was moreover borrowed from a supersymmetric model with the same internal space rather than derived in the non-supersymmetric model itself. Later work by the same collaboration has computed one-loop cosmological constants for wide samples of these vacua at the free fermionic point — the highly symmetric point in field space at which the free-fermionic description applies exactly — with tadpoles excluded, and studied D-term uplifts in related non-supersymmetric orbifolds, but the stability of this specific quasi-realistic vacuum remains undetermined.

Problem Statement

Establish, or rule out, the perturbative viability of the non-supersymmetric, tachyon-free, three-generation standard-like heterotic-string model constructed as a compactification of the tachyonic ten-dimensional heterotic-string vacuum by Faraggi, Matyas and Percival (arXiv:1912.00061), in which all geometrical moduli are projected out and only the dilaton remains. Concretely, with the model fixed by the published boundary-condition basis vectors and GGSO phase matrix: (i) compute the one-loop scalar potential of the model, including the tadpoles generated by the anomalous $U(1)_A$ (the would-be Fayet–Iliopoulos D-term) and by the non-vanishing one-loop vacuum energy arising from the boson–fermion mismatch at massive levels, and determine whether the instabilities they signal can be cancelled, suppressed, or balanced against each other without reintroducing tachyonic states or destroying the three-generation standard-like spectrum; (ii) demonstrate or exclude the stabilization of the dilaton at finite coupling with small positive (or otherwise phenomenologically acceptable) vacuum energy; and (iii) assess whether stability, if achieved at one loop, persists under higher-order perturbative and non-perturbative corrections. The existence or absence of a perturbatively stable vacuum for this model is decided jointly by these three components.

References

1. Alon E. Faraggi, Viktor G. Matyas, Benjamin Percival, Stable Three Generation Standard-like Model From a Tachyonic Ten Dimensional Heterotic-String Vacuum, Eur. Phys. J. C 80 (2020) 337, DOI: 10.1140/epjc/s10052-020-7894-x, arXiv:1912.00061, https://arxiv.org/abs/1912.00061
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6. Eman Basaad, Luke A. Detraux, Alonzo R. Diaz Avalos, Alon E. Faraggi, Benjamin Percival, Vacuum Energy in Non-Supersymmetric Quasi-Realistic Heterotic-String Vacua with Fixed Moduli, Eur. Phys. J. C 85 (2025) 209, DOI: 10.1140/epjc/s10052-024-13733-2, arXiv:2408.03928, https://arxiv.org/abs/2408.03928
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