Mass Hierarchy of Z(2) Monopoles

Abstract

In this work we establish every spherically symmetric non-Abelian Z(2) monopole generated by su(2) embeddings in the SU(4) Yang-Mills-Higgs model minimally broken to SO(4) by a symmetric second-rank tensor Higgs field. We find new monopole solutions associated with index 4 and index 10 embeddings. These solutions belong to su(2) multiplets that are higher dimensional than triplets. Properties of these monopoles such as their mass and radius are calculated in the vanishing potential limit. A parallel between this result and the Standard Model hierarchy of fermion masses is considered.

Figures (3)

• Figure 1: A diagram of the weight space for the symmetric second-rank tensor representation of $SU(4)$. The highest weight is $\Lambda = \ket{2\ 0\ 0}$ and each arrow represents a lowering operation in the direction of some root $\alpha_i$. The resulting weight is determined using \ref{['lower once']} and \ref{['lower twice']}. Each state $\ket{m_1\ m_2\ m_3}$ can be lowered, at most $m_i>0$ times in the $i$-th direction. For example the highest weight can be lowered twice by $\alpha_1$, the second one once by $\alpha_2$, and so forth until every path has been taken into account.
• Figure 2: Profile functions of every $\mathbb{Z}_2$ monopole as functions of the dimensionless radial variable $\xi=ver$ in the vanishing potential limit $\lambda\rightarrow0$. Decaying gauge potential $K(\xi)$ is drawn in solid lines. Types of rising scalar profiles differ depending on the branching rules of the $su(2)$ embedding. (a) Index 1, $H(\xi)/\xi$ of the triplet dashed; (b) Index 2, two triplets $H_1(\xi)/\xi$ and $H_2(\xi)/\xi$ dashed are identical; (c) Index 4, quintuplet $H(\xi)/\xi$ dashed; (d) Index 10, quintuplet $H_1(\xi)/\xi$ dashed and triplet $H_2(\xi)/\xi$ dash-dotted. Singlets are always omitted.
• Figure 3: Hamiltonian radial densities of $\mathbb{Z}_2$ monopoles of indices 1, 2, 4 and 10, functions $\mathscr{H}(\xi)$ of the dimensionless variable $\xi=ver$ in the vanishing potential limit. Their respective integrals yield the particles mass while the critical points define their radii. These values are gathered in Table \ref{['results']}.