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
ˇ
c
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
1
× G
2
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
G
1
and G
2
. 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
p
. This observation may allow new approaches to grand unification
and family unification.
© 2015 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
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
U
(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
p
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.
We
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
p
, 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/
© 2015 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.
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