Tri-linear Couplings in an Heterotic Minimal Supersymmetric Standard Model

TL;DR

The authors compute, in the classical large-volume limit, the tri-linear superpotential couplings of a globally consistent heterotic MSSM constructed on a CY threefold with ${\mathbb Z}_2$ fundamental group and an $SU(5)$ bundle. They map MSSM fields to cohomology classes, and evaluate cubic couplings via triple pairings of these classes, finding non-zero up-type Yukawas and vanishing R-parity-violating terms at tree level, while down-type Yukawas are zero classically. A key result is that vector bundle moduli induce moduli-dependent $\mu$-terms and neutrino Dirac masses on loci with $n=1$ or $n=2$ massless Higgs pairs, with the full invariant moduli space dimension $h^1(\mathrm{ad}\tilde V)_+=51$ shaping these couplings. The work demonstrates a concrete mechanism for generating $\mu$-terms and neutrino masses in heterotic compactifications, while highlighting that down-sector Yukawas and potential R-parity violation are expected to arise from quantum effects; this constrains viable regions of moduli space and informs phenomenological viability of the model.

Abstract

We calculate, at the classical level, the superpotential tri-linear couplings of the only known globally consistent heterotic minimal supersymmetric Standard Model [ hep-th/0512149 ]. This recently constructed model is based on a compactification of the E_8 x E_8 heterotic string theory on a Calabi-Yau threefold with Z_2 fundamental group, coupled with a slope-stable holomorphic SU(5) vector bundle. In the observable sector the massless particle content is that of the three-family supersymmetric Standard Model with n=0,1,2 massless Higgs pairs, depending on the location in the vector bundle moduli space, and no exotic particles. We obtain non-zero Yukawa couplings for the three up-sector quarks, and vanishing R-parity violating terms. In particular, the proton is stable. Another interesting feature is the existence of tri-linear couplings, on the loci with massless Higgs pairs, generating μ-mass parameters for the Higgs pairs and neutrino mass terms, with specific vector bundle moduli playing the role of right-handed neutrinos.