Standard-like heterotic-string vacuum with a low-scale anomaly-free Z′: explicit x-sector construction and low-energy validation

Free-fermionic heterotic-string compactifications generically predict extra U(1) gauge symmetries, but constructing a quasi-realistic vacuum in which one remains unbroken down to the TeV scale is severely constrained: the family-universal U(1)_ζ descending from E6 is anomalously broken in the standard-like model class unless the GGSO projections preserve complete E6 multiplets. Athanasopoulos, Faraggi and Mehta (Phys. Rev. D 89, 105023 (2014)) supplied a recipe — retaining the x-sector spacetime vector bosons so that the chiral spectrum fills complete E6 27s under a standard-like gauge group — and posed as open problems the explicit three-generation string vacuum realizing it, the renormalization-group identification of the light Higgs doublets and the fermion-mass hierarchy, and the string-realized discrete symmetries required for proton longevity, and a self-dual spinor–vector-dual model with broken SO(10). Later literature resolved that last open problem (Faraggi–Rizos 2015), which now anchors the string-derived Z′ phenomenology program, but the standard-like x-sector construction itself, its Yukawa/RG analysis, and its proton-stability discrete symmetries remain unpublished.

Background

The heterotic string unifies the gauge and gravitational degrees of freedom of perturbative string theory, and its four-dimensional vacua are the main testing ground for string-derived particle phenomenology. In the free fermionic formulation the internal worldsheet degrees of freedom are represented by free fermions, and a vacuum is specified by boundary-condition basis vectors — the phases the fermions pick up around the non-contractible loops of the worldsheet torus — together with the phases of the one-loop Generalized Gliozzi–Scherk–Olive (GSO) projections; these data must obey modular-invariance constraints, and the massless spectrum is obtained sector by sector by applying the GSO projections. Quasi-realistic vacua are built on the NAHE set of five basis vectors, whose observable gauge symmetry is SO(10) times three U(1) factors; further basis vectors break SO(10) to one of its maximal subgroups — SU(5)×U(1) ('flipped SU(5)'), SU(3)×SU(2)×U(1)^2 ('standard-like', i.e. the Standard Model gauge group plus extra Abelian factors), SO(6)×SO(4) (Pati–Salam) or left-right subgroups — and reduce the chiral generations to three.

Extra U(1) gauge factors are abundant in these vacua, but a Z′ boson — the massive gauge boson of an extra U(1) factor — surviving to the TeV scale must satisfy several simultaneous conditions: anomaly freedom (an anomalous U(1) is broken at a high scale by its gauge anomaly); family universality; charges admitting an E6 embedding, so that the additional doublet states required for anomaly cancellation preserve the agreement of the gauge-coupling running with the measured values of $\sin^2\theta_W$ and $\alpha_3$; and compatibility with neutrino-mass suppression via a high-scale seesaw (heavy Majorana masses for right-handed neutrinos). These conditions are in known mutual tension. In the standard-like and flipped-SU(5) classes the breaking pattern $E_6\to SO(10)\times U(1)_\zeta$ removes states from the $\mathbf{27}$ of $E_6$ and renders the family-universal combination $U(1)_\zeta=U(1)_1+U(1)_2+U(1)_3$ anomalous, whereas the left-right classes keep $U(1)_\zeta$ anomaly free but produce extra doublets whose charges lack the E6 embedding required by the gauge-coupling data. The SO(10)-embedded $U(1)_{B-L}$-type combination must be broken at a high scale to suppress the tau-neutrino mass, and high-scale breaking reintroduces dangerous proton-decay operators: proton stability and light-neutrino masses pull the Z′ breaking scale in opposite directions, so remnant discrete symmetries are needed to forbid the dimension-four and dimension-five proton-decay mediating operators.

Athanasopoulos, Faraggi and Mehta (2014) identified a way to satisfy the anomaly and E6-embedding conditions in the standard-like class: retain the spacetime vector bosons of the x-sector — the basis vector $x=\{\bar\psi^{1,\dots,5},\bar\eta^{1,2,3}\}$ with all eight observable-sector complex fermions periodic, whose vector bosons combine with the untwisted gauge generators to enhance the observable $SO(16)$ to $E_8$. Each twisted sector $\alpha$ is then complemented by its $\alpha+x$ partner, so the chiral spectrum fills complete $E_6$ $\mathbf{27}$ representations decomposed under the standard-like group, and $U(1)_\zeta$ can be anomaly free with E6-embedded charges, as in the $SU(6)\times SU(2)$ string models. They displayed a string-inspired high-scale spectrum and charge assignment realizing this structure, but not a fully derived string vacuum, and posed the completion of the program as their open question. An alternative route is offered by the spinor–vector duality: a redefinition of the GGSO phases that exchanges the number of spinorial $\mathbf{16}\oplus\overline{\mathbf{16}}$ and vectorial $\mathbf{10}$ representations of SO(10) in $Z_2\times Z_2$ fermionic orbifolds; self-dual models, with equal numbers of spinorial and vectorial states, also produce complete E6 multiplets without the gauge symmetry being enhanced to $E_6$.

Problem Statement

The source (Athanasopoulos, Faraggi and Mehta, Phys. Rev. D 89, 105023 (2014)) posed four linked open tasks serving one objective: a complete, explicit standard-like heterotic-string vacuum with a light Z′ and validated low-energy phenomenology. One component — the construction of a self-dual three-generation model under the spinor–vector duality with SO(10) broken — has since been carried out by Faraggi and Rizos (Nucl. Phys. B 895 (2015) 233), who thereby also realized an explicit three-generation string-derived vacuum with an anomaly-free low-scale U(1)_ζ by the alternative route. The surviving open problem is:

(1) Construct an explicit, fully consistent three-generation standard-like free-fermionic heterotic-string vacuum implementing the x-sector recipe of the source: boundary-condition basis vectors and GGSO projections that retain the x-sector spacetime vector bosons, break SO(10) to the standard-like subgroup, produce the three-generation high-scale spectrum with the charge assignments displayed in the source, and leave the family-universal $U(1)_\zeta$ anomaly free with an $E_6$ embedding of the light-state charges.

(2) In that model, carry out the detailed renormalization-group and Yukawa-coupling analysis that (2a) identifies which electroweak doublets play the role of the light Higgs doublets of the Minimal Supersymmetric Standard Model (MSSM), (2b) determines the fermion-mass hierarchies, and (2c) determines the low-energy scalar-sector properties, namely the electroweak-symmetry-breaking conditions and the resulting Higgs mass spectrum (the light MSSM Higgs mass and the CP-even, CP-odd and charged Higgs masses). The light MSSM Higgs doublets, the fermion-mass hierarchies, and the low-energy scalar sector follow from the string-level Yukawa couplings of the constructed vacuum and their renormalization-group evolution.

(3) Specify the additional discrete symmetries — concretely realized as remnants of the string U(1) symmetries in the complete model — that suppress the proton-decay mediating superpotential couplings (the $QQD$, $ud\bar D$, $dND$, $ueD$, $QL\bar D$ terms and their string-scale induced descendants), while remaining compatible with neutrino-mass generation.

These three components jointly support the single stated objective: without (1) the model remains string-inspired rather than string-derived; without (2) the low-energy identification of the Z′ charge assignments and Higgs sector is untested; without (3) the vacuum does not respect proton-stability limits.

References

1. P. Athanasopoulos, A. E. Faraggi, V. M. Mehta, 'Light Z′ in heterotic string standardlike models', Physical Review D 89, 105023 (2014). DOI: 10.1103/PhysRevD.89.105023; arXiv:1401.7153. https://doi.org/10.1103/PhysRevD.89.105023
2. A. E. Faraggi, J. Rizos, 'A light Z′ heterotic-string derived model', Nuclear Physics B 895, 233–247 (2015). DOI: 10.1016/j.nuclphysb.2015.03.031; arXiv:1412.6432. https://doi.org/10.1016/j.nuclphysb.2015.03.031
3. V. M. Mehta, 'Light U(1)s in heterotic string models', Journal of Physics: Conference Series 631, 012078 (2015). DOI: 10.1088/1742-6596/631/1/012078; arXiv:1503.04671. https://doi.org/10.1088/1742-6596/631/1/012078
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7. A. McEntaggart, A. E. Faraggi, M. Guzzi, 'Precision studies for string derived Z′ dynamics at the LHC', European Physical Journal C 83, 54 (2023). DOI: 10.1140/epjc/s10052-023-11223-5; arXiv:2211.15905. https://doi.org/10.1140/epjc/s10052-023-11223-5
8. A. E. Faraggi, M. D. Goodsell, 'M_W in string derived Z′ models', European Physical Journal C 84, 589 (2024). DOI: 10.1140/epjc/s10052-024-12900-9; arXiv:2312.13411. https://doi.org/10.1140/epjc/s10052-024-12900-9
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