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

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 G superset of G(1) x 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 G(1) and 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 >= p > 1 of Nit 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. (C) 2015 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license.