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Physics Letters B 742 (2015) 231–235

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Phases with modular ground states for symmetry breaking by rank 3

and rank 2 antisymmetric tensor scalars

Stephen L. Adler

Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA

a r t i c l e i n f o a b s t r a c t

Article history:

Received

26 November 2014

Accepted

14 January 2015

Available

online 28 January 2015

Editor: M.

Cveti

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 ⊃ G

× G

by a scalar in a rank three or two an-

tisymmetric

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 > 0is the absolute

value of the nonzero U (1) generator of the scalar component Φ that is a singlet under the simple sub-

groups

and G

. Ground state expectations of fractional powers Φ

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 > 1of N it leads to a modular ground state with periodicity p, implementing a dis-

crete

Abelian symmetry group U(1)/ Z

. This observation may allow new approaches to grand uniﬁcation

and family uniﬁcation.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

The possibilities for constructing grand uniﬁed models depend

crucially on the pathways available for symmetry breaking, and the

corresponding structures of the vacuum or ground state. For the

familiar case of SU(5) grand uniﬁcation, breaking to the standard

model is accomplished by assuming a scalar in the 24 representa-

tion,

which has a singlet component under the SU(2) × SU(3) sub-

group,

with zero U (1) generator. Hence the group after symmetry

breaking, when the singlet scalar component acquires a nonzero

vacuum expectation in a vacuum in a U (1) eigenstate with eigen-

value

0, is SU(2) × SU(3) × U (1). Discussions of symmetry break-

ing

by second rank [1] and third rank [2] antisymmetric tensors

have assumed a ground state that completely breaks the analo-

gous

U (1), since the singlet component of the scalar under the

simple subgroups of the initial gauge group typically has a nonzero

(1) generator, with absolute value that we denote by N. We shall

show in this paper that the phase with completely broken U(1) is

only the simplest of a set of symmetry breaking phases, the rest

of which have discrete U (1)/Z

residual symmetry, with p an in-

teger

divisor of N, corresponding to a modular ground state that is

periodic in the U (1) generator with period p.

begin in Section 2 by considering a model that we recently

proposed [3] for SU(8) family uniﬁcation, with SU(8) broken by a

E-mail address: adler@ias.edu.

scalar in the rank three antisymmetric tensor 56 representation.

We review the arguments that consistency of symmetry breaking

by a third rank antisymmetric tensor scalar ﬁeld in the 56 rep-

resentation,

together with the requirement of clustering, requires

a ground state structure of modularity 15 in the U(1) genera-

tor.

This can be achieved by a ground state that has periodicity

p in the U (1) eigenvalue, with p any integer divisor of 15, that

is p = 1, 3, 5, 15. We show that the case p = 1 corresponds to the

calculation of [2]. Transforming from U (1) eigenvalue space to ω

space, where −15ω is the phase angle of the component of the 56

that attains a vacuum expectation value, we see that this case cor-

responds

to the usual Higgs mechanism where the vacuum picks

a value of ω in the range 0 ≤ ω < 2π . The cases of p > 1 corre-

spond

in ω space to the vacuum picking a value of ω in the range

0 ≤ ω < 2π/p, since the period p modularity of the ground state

leads to a discrete Abelian symmetry U (1)/Z

, which maps the

wedge sectors 2πn/p ≤ ω < 2π (n + 1)/p, for n = 0, 1, ..., p − 1,

into one another. This scenario has been analyzed in the context

of Abelian models, for the case when p is equal to the U (1) gener-

ator

of a condensing scalar ﬁeld, in the presence of a second scalar

ﬁeld with U (1) generator 1, by Krauss and Wilczek [4], by Banks

[5], and by Preskill and Krauss [6]. We review their conclusions

showing that although the U (1) gauge ﬁeld gets a mass, the mod-

ularity

of the ground state leads to conservation of U (1) generator

charges modulo p, and to other residual long range effects. Thus,

the p > 1 symmetry breaking phases contain new physics not evi-

dent

from the p = 1case.

http://dx.doi.org/10.1016/j.physletb.2015.01.037

0370-2693/

SCOAP

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