Phases with modular ground states for symmetry breaking by rank 3 and rank 2 antisymmetric tensor scalars

TL;DR

This paper explores how scalar fields in rank 3 and 2 antisymmetric tensor representations can lead to multiple modular ground states during symmetry breaking, revealing new phases with discrete Abelian symmetries relevant for grand unification.

Contribution

It introduces the concept of modular ground states with periodicity p in symmetry breaking, extending the Higgs mechanism to include discrete Abelian symmetry groups.

Findings

Ground states can be periodic with integer divisor p of N.

Fractional powers of scalar fields serve as order parameters.

New phases with discrete symmetry groups emerge for p > 1.

Abstract

Working with explicit examples given by the 56 representation in $SU(8)$, and the 10 representation in $SU(5)$, we show that symmetry breaking of a group ${\cal G}\supset {\cal G}_1 \times {\cal G}_2$ by a scalar in a rank three or two antisymmetric tensor representation leads to a number of distinct $modular$ ground states. For these broken symmetry phases, the ground state is periodic in an integer divisor $p$ of $N$, where $N>0$ is the absolute value of the nonzero $U(1)$ generator of the scalar component $\Phi$ that is a singlet under the simple subgroups ${\cal G}_1$ and ${\cal G}_2$. Ground state expectations of fractional powers $\Phi^{p/N}$ provide order parameters that distinguish the different phases. For the case of period $p=1$, this reduces to the usual Higgs mechanism, but for divisors $N\geq p>1$ of $N$ it leads to a modular ground state with periodicity $p$, implementing a discrete Abelian symmetry group $U(1)/Z_p$. This observation may allow new approaches to grand unification and family unification.