Moduli Dependent mu-Terms in a Heterotic Standard Model

TL;DR

The paper develops a formalism to compute moduli-dependent Higgs $\mu$-terms in a heterotic standard model by forming a cubic product of cohomology groups involving vector-bundle moduli $\phi$, Higgs $H$, and Higgs conjugate $\bar{H}$. It reveals two restrictive Leray spectral sequence–based selection rules, the $(p,q)$ and the $[s,t]$ rules, which drastically restrict which vector-bundle moduli can couple to $H\bar{H}$, reducing the allowed moduli from $19$ to $4$ in a concrete vacuum. Applying the framework to a specific heterotic vacuum with ${\mathbb{Z}_3 \times \mathbb{Z}_3}$ symmetry and dual elliptic fibrations, the authors explicitly compute the non-vanishing cubic terms $\phi H \bar{H}$ and show that only four moduli contribute to the moduli-dependent mu-terms, yielding mu-terms of the form $\lambda_{iab}\,\langle\phi_i\rangle H_a \bar{H}_b$. They emphasize that the coupling coefficients $\lambda_{iab}$ are moduli-dependent (not fixed by intersection theory) and would be renormalized by the Kähler potential, highlighting a refined structure for mu-term generation in realistic heterotic vacua.

Abstract

In this paper, we present a formalism for computing the non-vanishing Higgs mu-terms in a heterotic standard model. This is accomplished by calculating the cubic product of the cohomology groups associated with the vector bundle moduli (phi), Higgs (H) and Higgs conjugate (Hbar) superfields. This leads to terms proportional to phi H Hbar in the low energy superpotential which, for non-zero moduli expectation values, generate moduli dependent mu-terms of the form <phi> H Hbar. It is found that these interactions are subject to two very restrictive selection rules, each arising from a Leray spectral sequence, which greatly reduce the number of moduli that can couple to Higgs-Higgs conjugate fields. We apply our formalism to a specific heterotic standard model vacuum. The non-vanishing cubic interactions phi H Hbar are explicitly computed in this context and shown to contain only four of the nineteen vector bundle moduli.