Dynamical completion of the $Z_{4R}$ R-parity construction from $Z_{12-I}$ orbifold compactification of the SO(32) heterotic string

A 2021 construction by J. E. Kim derives an $R$-parity-like discrete symmetry $Z_{4R}$ as a discrete subgroup of gauged $U(1)$ charges in a $Z_{12-I}$ orbifold compactification of the SO(32) heterotic string, whose visible sector is an anti-SU(5) grand unified theory with three families and whose hidden sector is a confining SU(9)$'$ gauge group. The construction, however, leaves its central dynamical steps unverified: the vacuum alignment $\langle H(10_R)\rangle=0$ with $\langle \bar H(\bar{10}_R)\rangle\neq0$ is imposed by hand; the SU(9)$'$ confinement scale, the condensate $C\sim\bar{36}'\cdot 9'\cdot 9'$, and the Higgs-pair masses it induces are not computed; and the $S_3$-based quark and lepton Yukawa matrices are not constructed. The open problem is to complete this dynamical and phenomenological verification — to determine, within the specific string-derived model, whether the assumed vacuum structure follows from the effective potential, where and how $Z_{4R}$ is broken, and whether the resulting low-energy spectrum and flavor structure are phenomenologically viable.

Background

In supersymmetric extensions of the Standard Model, $R$ parity — the discrete symmetry $R=(-1)^{3(B-L)+2s}$ acting on baryon number $B$, lepton number $L$, and spin $s$ — forbids the renormalizable baryon- and lepton-number violating operators that would otherwise mediate catastrophically fast proton decay, and it makes the lightest supersymmetric particle stable. A natural way to obtain it is as a remnant of a $U(1)$ $R$ symmetry, a symmetry that also rotates the superspace coordinate and supercharges; the discrete group $Z_{4R}$ (integers mod 4 with an $R$-symmetry action) contains ordinary matter parity $Z_2$ as a subgroup, forbids the supersymmetric Higgsino $\mu$ term at perturbative order, and is broken by nonperturbative effects down to an exact $Z_{2R}$ matter parity. Lee, Raby, Ratz, Ross, Schieren, Schmidt-Hoberg and Vaudrevange (2011) showed in the $E_8\times E_8'$ heterotic orbifold framework that $Z_{4R}$ is the unique such symmetry of the Minimal Supersymmetric Standard Model (MSSM, the minimal supersymmetric extension of the Standard Model) commuting with SO(10), once anomaly cancellation via the Green–Schwarz mechanism (cancellation of anomalous gauge-current divergences through the shift of a model-independent axion-like field) is imposed. J. E. Kim pursued a complementary, model-building route: constructing an explicit string compactification whose derived charge lattice contains $Z_{4R}$. Because rank-greater-than-8 unified groups cannot come from $E_8\times E_8'$, Kim compactified the SO(32) heterotic string on the $Z_{12-I}$ orbifold (a specific quotient of six compact internal dimensions by a twist of order 12, whose massless spectrum is organized in untwisted and twisted sectors, the latter distinguished by Wilson lines — constant background gauge fields along compact directions that split the twisted-sector fixed points), obtaining the rank-16 gauge group SU(5)$\times$SU(9)$'$ $\times U(1)^4$ with a single family-unified visible sector and a hidden-sector SU(9)$'$. The visible theory is an anti-SU(5) (flipped SU(5)) grand unified theory, the construction of Derendinger, Kim and Nanopoulos (1984) in which a family sits in $\bar{10}\oplus 5$ rather than $10\oplus\bar 5$, so that SU(5)$\times U(1)$ breaking is achieved by vacuum expectation values of two-index antisymmetric tensor Higgs fields $10$ rather than by adjoint-type fields. In Kim's 2021 model, the discrete $R$ charge $Q_R=\tfrac{9}{2}Q_1-Q_2-\tfrac{3}{2}Q_3+Q_X$ (a combination of the four gauged $U(1)$ charges) is $0$ mod 4 on all needed symmetry-breaking vacuum expectation values, so $U(1)_R$ breaks to $Z_{4R}$ at the grand-unification scale. The multiplicity-3 fixed points of one twisted sector supply a permutation symmetry $S_3$ (the symmetry group of three interchangeable objects, with one singlet, one sign-flip singlet $1'$, and one doublet) that singles out the third family and, together with the condensate of the confining hidden sector SU(9)$'$, removes all but one supersymmetric Higgs pair. Crucially, however, the construction assumes rather than derives its key dynamical ingredients: it demands that the $Q_R=+3$ Higgs field $H(10_R)$ acquire zero vacuum expectation value while its $Q_R=+4$ conjugate $\bar H(\bar{10}_R)$ acquires the symmetry-breaking expectation value (so that $Z_{4R}$ survives grand-unification breaking); it posits, without calculation, that SU(9)$'$ confines at a high scale and forms the condensate $C\sim\bar{36}'\cdot 9'\cdot 9'$ (in the source paper, the scalar component of the hidden-sector antisymmetric-tensor superfield $\bar{36}'$ condensed with the fermion components of the two fundamental $9'$ superfields), which carries $Q_R=2$ mod 4 and thereby breaks $Z_{4R}$ to $Z_{2R}$ while lifting the unwanted Higgs pairs; and it stops short of writing the quark and lepton Yukawa matrices in the $S_3$ basis. These gaps define the problem: verifying that the string-derived model actually realizes $R$ parity dynamically rather than by assignment.

Problem Statement

Within the specific $Z_{12-I}$ orbifold compactification of the SO(32) heterotic string constructed in Kim (2021) — whose four-dimensional gauge group is SU(5)$\times$SU(9)$'$ $\times U(1)^4$, whose visible sector is an anti-SU(5) family-unified theory with three families and $R$ charge $Q_R=\tfrac{9}{2}Q_1-Q_2-\tfrac{3}{2}Q_3+Q_X$, and whose hidden sector is SU(9)$'$ — determine whether the $Z_{4R}$ R-parity construction is dynamically realized, by establishing: (1) a concrete mechanism (or a no-go demonstration) that enforces the assumed vacuum alignment $\langle H(10_R)\rangle=0$ while $\langle\bar H(\bar{10}_R)\rangle\neq0$, so that $Z_{4R}$ is not broken below the grand-unification scale other than as intended; (2) the SU(9)$'$ confinement dynamics — the condensation scale of $C\sim\bar{36}'\cdot 9'\cdot 9'$ (including whether this chiral hidden sector in fact confines rather than lying in a conformal window, the range of matter content for which a supersymmetric gauge theory flows to an interacting scale-invariant fixed point instead of confining) and the quantitative mass spectrum induced for the vector-like Higgs pairs (pairs of fields with conjugate gauge quantum numbers, which can therefore acquire supersymmetric mass terms), fixing the scale of the residual breaking $Z_{4R}\to Z_{2R}$; and (3) the explicit quark and lepton Yukawa matrices in the $S_3$ singlet/doublet family basis selected by the fixed-point permutation symmetry, with their predicted mass hierarchies and mixings (CKM quark-mixing and PMNS lepton-mixing matrices). Items (1) and (2) may each be resolved by either branch: a no-go demonstration in (1), or a determination that the hidden sector lies in a conformal window in (2), is itself a resolution of the corresponding item, and any computation that presupposes the affirmative branch is then required only when that branch holds. These determinations bear on the single scientific objective — whether string compactification delivers an unbroken, phenomenologically viable $R$ parity in this model.

References

1. Jihn E. Kim, R parity from string compactification, Phys. Rev. D 104, 016012 (2021), DOI: 10.1103/PhysRevD.104.016012, arXiv:2103.14084, https://arxiv.org/abs/2103.14084
2. Jihn E. Kim, R-parity from string compactification, Phys. Rev. D 99, 093004 (2019), DOI: 10.1103/PhysRevD.99.093004, arXiv:1810.10796, https://arxiv.org/abs/1810.10796
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5. H. M. Lee, S. Raby, M. Ratz, G. G. Ross, R. Schieren, K. Schmidt-Hoberg, P. K. S. Vaudrevange, A unique $Z_4^R$ symmetry for the MSSM, Phys. Lett. B 694, 491 (2011), DOI: 10.1016/j.physletb.2010.10.038, arXiv:1009.0905, https://arxiv.org/abs/1009.0905
6. J. P. Derendinger, Jihn E. Kim, D. V. Nanopoulos, Anti-SU(5), Phys. Lett. B 139, 170 (1984), DOI: 10.1016/0370-2693(84)91238-3, https://doi.org/10.1016/0370-2693(84)91238-3