A Three-Family Standard-like Orientifold Model: Yukawa Couplings and Hierarchy

TL;DR

The paper investigates Yukawa coupling hierarchies in a supersymmetric three-family Standard-like string model realized via intersecting D6-branes on a Z2xZ2 orientifold, with an M-theory lift to a G2 manifold. Yukawa couplings originate from worldsheet instantons tied to triangles formed by brane intersections, leading to couplings that are suppressed by the corresponding triangle areas. By analyzing configurations with branes near symmetric positions and by introducing controlled brane displacements, the authors show how lepton-quark and Up-down splittings arise, yielding hierarchical Yukawas that depend on the internal geometry and moduli. These results illuminate how, given SUSY-breaking and Higgs sector details, the model can accommodate two or one massive fermion generations and offer a geometric mechanism for fermion mass hierarchies in string-derived theories.

Abstract

We discuss the hierarchy of Yukawa couplings in a supersymmetric three family Standard-like string Model. The model is constructed by compactifying Type IIA string theory on a Z_2 x Z_2 orientifold in which the Standard Model matter fields arise from intersecting D6-branes. When lifted to M theory, the model amounts to compactification of M-theory on a G_2 manifold. While the actual fermion masses depend on the vacuum expectation values of the multiple Higgs fields in the model, we calculate the leading worldsheet instanton contributions to the Yukawa couplings and examine the implications of the Yukawa hierarchy.

Figures (6)

• Figure 1: The initial symmetric configuration of the $A, B, C'$ sectors of branes, associated with the $U(1)_{8,8'}$ , $U(2)_L$ and $\{U(3)_C, U(1)_1 \}$ sectors, are denoted by dashed, dotted, and solid lines, respectively. The intersections denoted by $\alpha=(1,2,3)$ and $\gamma=(i,ii)$ correspond to the appearance of Higgs and left-handed families, respectively.
• Figure 2: Intersection areas of the branes in the third torus. The thin solid lines denote the lattice, the thick solid lines-the $U(3)_C$ branes, the dotted lines-$U(2)_L$ branes and the dashed ones- $U(1)_{8,8'}$ branes. Again $\alpha=(1,2,3)$ and $\gamma=(i,ii)$ denote the location of the three Higgs fields and the two left-handed quark families, respectively.
• Figure 3: The brane configurations in the first torus, depicting the breaking on $U(4)$ Pati-Salam symmetry down to $U(3)_C\times U(1)_1$. The $U(1)_1$ branes (denoted by dash-dotted line) are positioned in a $Z_2$ symmetric way relative to $U(3)_C$ branes (denoted by a solid line). The separation between them is $\eta R_{1,2}^{(1)}$ in the respective $x$- and $y$-directions. The relevant interesection area in the first torus, contributing to the lepton Yukawa coupling is denoted by a shaded area. [$U(2)_L$ and $U(1)_{8,8'}$ branes are denoted by a dotted and a dashed line, respectively.]
• Figure 4: The splitting of A-type branes (associated with $U(1)_{8,8'}$ and denoted by the dashed lines) from the orientifold planes. For simplicity the figure shows only the fundamental domain of each of the three two-tori. The solid and dotted lines denote the $U(3)_C$ and $U(2)_L$ branes, respectively.
• Figure 5: Relevant intersection areas in the third toroidal lattice for the Up-sector.