Completing the heterotic $\mathbb{Z}_6$-II orbifold MSSM: all-orders supersymmetric vacua and full phenomenological viability of the Buchmüller–Hamaguchi–Lebedev–Ratz model

The $\mathbb{Z}_6$-II heterotic orbifold compactification of Buchmüller, Hamaguchi, Lebedev and Ratz (2007) was the first construction in which the exact supersymmetric Standard Model spectrum emerges from the $E_8\times E_8$ heterotic string with local SO(10) grand unification, a supersymmetric vacuum that lifts all exotic states, a top Yukawa of order the gauge coupling, and singlet-suppressed lighter-generation Yukawas. The authors themselves framed the work as 'first steps towards a fully realistic theory' and listed what remains to be done: an all-orders proof that the $B-L$-preserving singlet vacuum solves all F-term equations (including non-perturbative contributions), the dynamical $B-L$-breaking mechanism generating the $\mu$-term, seesaw neutrino masses and $R$-parity, suppression of proton-decay operators, moduli and dilaton stabilization, string threshold corrections and precision gauge unification, quantitative Yukawa couplings, the geometric (blow-up) interpretation of exotic decoupling, and systematic exploration of the surrounding model space. Later literature by the same network (the 'heterotic mini-landscape') resolved substantial parts of the model-space, $R$-parity, proton-stability and blow-up questions in successor models (not in the source model itself), but the central core — exact all-orders F-flatness of a chosen vacuum with stabilized moduli, numerical Yukawa constants, and model-specific threshold corrections — remains open. These items form one linked program with a single explicit objective: establishing (or refuting) a fully realistic, all-orders-controlled string-derived MSSM vacuum in this framework.

Background

The heterotic $E_8\times E_8$ superstring is a candidate unified theory of gravity and particle physics in which the gauge group of the ten-dimensional theory is broken by compactification on a six-dimensional internal space. In orbifold compactifications the internal space is a torus $T^6$ divided by a discrete twist symmetry; for the '$\mathbb{Z}_6$-II' orbifold the twist acts as a $60^\circ$, $120^\circ$ and $180^\circ$ rotation in the three complex planes (equivalently the twist vector $v_6=\frac{1}{6}(1,2,-3)$). Twisted strings localized at the fixed points of the twist, together with continuous Wilson lines (background gauge fields winding around torus cycles that further break the gauge group), generate chiral families, while the gauge symmetry at a fixed point can be larger than the four-dimensional one — 'local grand unification': quarks and leptons sit in complete SO(10) multiplets $\mathbf{16}$ at fixed points even though the four-dimensional gauge group is only the Standard Model one. Buchmüller, Hamaguchi, Lebedev and Ratz (2007) constructed such a model in which three families appear as three $\mathbf{16}$-plets of SO(10) — two localized at fixed points with local SO(10) symmetry (in the first twisted sector) and the third assembled from untwisted and higher twisted sectors, with its $\mathbf{16}$-plet structure following from standard-model anomaly cancellation rather than from local grand unification — the four-dimensional gauge group and chiral spectrum are exactly those of the Minimal Supersymmetric Standard Model (MSSM), up to vector-like 'exotic' states, the top-quark Yukawa coupling is of order the gauge coupling, and the remaining Yukawa couplings are suppressed by powers of Standard-Model singlet vacuum expectation values (VEVs), in the manner of the Froggatt–Nielsen mechanism (flavour hierarchies from suppression powers of fields carrying flavour charge). At the perturbative level, a globally supersymmetric vacuum requires all F-terms to vanish, $F_i=\partial W/\partial\phi_i=0$ for the effective superpotential $W$, and all D-terms to vanish, $D^a=0$, the latter enforced along 'D-flat directions' generated by holomorphic gauge-invariant monomials. Vacua that preserve $B-L$ (the gauged anomaly-free combination of baryon minus lepton number) are singled out: one Higgs pair is automatically massless with a $\mu$-term (the supersymmetric Higgs mass parameter) generated only after $B-L$ breaking, matter parity ($R$-parity, the $\mathbb{Z}_2$ remnant when $B-L$ is broken by two units) is exact in the $B-L$-preserving vacuum and forbids fast proton decay from dimension-four operators, while $B-L$ breakdown at an intermediate scale induces only small $R$-parity-violating effects, and singlet VEVs of $B-L$ charge $-2$ give right-handed neutrinos Majorana masses, feeding the seesaw mechanism for light-neutrino masses. Supersymmetry itself is to be broken by gaugino condensation — the formation of a strong-coupling condensate of the hidden sector's gauginos (the spin-$1/2$ superpartners of its gauge bosons), inherited from the second $E_8$ — which by dimensional transmutation produces a hierarchically small gravitino mass, but this requires stabilizing the moduli and dilaton (the string compactification's radius and coupling fields), and the authors note that the common practice of imposing $\langle W\rangle=0$ constrains only the total superpotential including non-perturbative moduli potentials, not the charged-matter VEVs. Two further standard hurdles of string model building complete the picture: string threshold corrections (the one-loop, moduli-dependent corrections to gauge couplings) must be computed before the model's gauge-coupling unification can be compared with the measured values, and orbifolds are singular points in the moduli space of smooth Calabi–Yau manifolds (the Ricci-flat six-dimensional internal spaces underlying supersymmetric string compactifications) endowed with vector bundles (the 'non-standard embedding' gauge backgrounds), so the singlet VEVs that lift the exotic states should have a geometric interpretation as resolutions ('blow-ups') of the orbifold singularities.

Problem Statement

For the Buchmüller–Hamaguchi–Lebedev–Ratz $\mathbb{Z}_6$-II heterotic orbifold model — and, where an answer requires it, the surrounding 'fertile patch' of $\mathbb{Z}_6$-II local-GUT compactifications it seeded (the class of $\mathbb{Z}_6$-II orbifold models with localized SO(10) $\mathbf{16}$-plets and exact MSSM spectra, of which the source construction is the prototype) — complete the program the source paper explicitly left open, namely to establish (or definitively refute) a fully realistic string-derived supersymmetric Standard Model from this construction. The components, which jointly serve this single objective, are: (i) prove that a specific singlet VEV configuration — in particular the $B-L$-preserving one — satisfies all $F_i=0$ conditions to all orders in the effective superpotential, including non-perturbative contributions and higher-order singlet couplings beyond the orders analyzed, so that the vacuum is exactly supersymmetric (not merely at finite order); (ii) compute the quantitative Yukawa coupling constants in such a chosen supersymmetric vacuum — the order-one coefficients and higher-order operators — to the numerical values of the charged-fermion masses and of the CKM and PMNS mixing parameters; (iii) construct the dynamical mechanism and scale of spontaneous $B-L$ breaking in a supersymmetry-preserving way, generating the $\mu$-term, Majorana neutrino masses via the seesaw, and $R$-parity-violating effects small enough to satisfy the present proton-decay and flavour constraints; (iv) demonstrate suppression of the higher-dimensional baryon- and lepton-number-violating (proton-decay) operators — directly generated or induced by integrating out heavy vector-like exotics — to levels below the present experimental bounds on proton decay, with whatever extra discrete symmetries or vacuum alignments this requires; (v) provide an explicit moduli and dilaton stabilization mechanism, consistent with the hidden-sector gaugino condensation assumed for supersymmetry breaking and with a cosmological constant below the present observational bound; (vi) compute the one-loop string threshold corrections and localized-field contributions for this model and show whether precision gauge-coupling unification survives; (vii) explain the decoupling of exotic states geometrically, i.e. relate the singlet VEVs that lift exotics to resolutions (blow-ups) of the orbifold singularities in smooth Calabi–Yau compactifications with non-standard embedding, clarifying which exotics are absent on smooth backgrounds; and (viii) determine systematically whether the $E_8\times E_8$ and SO(32) heterotic orbifold or smooth compactifications with localized SO(10) $\mathbf{16}$-plets contain inequivalent models whose phenomenological properties are comparable to, or improved on, those of the source construction (for instance fewer singlets, a simpler $\mu$-term, or better proton stability). Components (i)–(vi) concern the singlet vacuum of the Buchmüller–Hamaguchi–Lebedev–Ratz construction, while (vii) and (viii) are class-level questions — (vii) over the resolutions of the surrounding $\mathbb{Z}_6$-II local-GUT fertile patch and (viii) over the $E_8\times E_8$ and SO(32) orbifold or smooth compactifications it names. The program asks whether the all-orders supersymmetric $B-L$-preserving vacuum of component (i) exists — that is, whether it sits at an exactly supersymmetric, moduli-stabilized point whose effective theory below the string scale is the MSSM, satisfying the conditions of components (i)–(vi) together with the geometric (vii) and landscape (viii) determinations — or whether an obstruction to its existence can be demonstrated.

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