Symmetry breaking by bi-fundamentals

TL;DR

This paper classifies all symmetry breaking patterns in intersecting brane models involving bi-fundamental and rank-2 tensor Higgs fields, extending previous work to symplectic groups and analyzing the scalar potential's minima.

Contribution

It generalizes symmetry breaking analysis to symplectic groups and mixed bi-fundamentals, providing a comprehensive classification of minima in the scalar potential.

Findings

Extended symmetry breaking patterns to symplectic groups.

Derived block-diagonal matrix solutions for scalar potential minima.

Identified conditions for maximum block size and true minima.

Abstract

We derive all possible symmetry breaking patterns for all possible Higgs fields that can occur in intersecting brane models: bi-fundamentals and rank-2 tensors. This is a field-theoretic problem that was already partially solved in 1973 by Ling-Fong Li. In that paper the solution was given for rank-2 tensors of orthogonal and unitary group, and U(N x U(M) and O(N) x O(M) bi-fundamentals. We extend this first of al to symplectic groups. When formulated correctly, this turns out to be straightforward generalization of the previous results from real and complex numbers to quaternions. The extension to mixed bi-fundamentals is more challenging and interesting. The scalar potential has up to six real parameters. Its minima or saddle points are described by block-diagonal matrices built out of K blocks of size p x q. Here p=q=1 for the solutions of Ling-Fong Li, and the number of possibilities for p x q is equal to the number of real parameters in the potential, minus 1. The maximum block size is p x q=2 x 4. Different blocks cannot be combined, and the true minimum occurs for one choice of basic block, and for either K=1 or K maximal, depending on the parameter values.