Spontaneous breaking of gauge groups to discrete symmetries
HJE
Spontaneous breaking of gauge groups to discrete symmetries
Bradley L. Rachlin 0 1
Thomas W. Kephart 0 1
0 Nashville , TN 37235 , U.S.A
1 Department of Physics and Astronomy, Vanderbilt University
Many models of beyond Standard Model physics connect avor symmetry with a discrete group. Having this symmetry arise spontaneously from a gauge theory maintains compatibility with quantum gravity and can be used to systematically prevent anomalies. We minimize a number of Higgs potentials that break gauge groups to discrete symmetries of interest, and examine their scalar mass spectra.
Beyond Standard Model; Discrete Symmetries; Gauge Symmetry; Sponta-
1 Introduction 2
Lie group invariant potentials
2.1
Gauge group irreps containing discrete gauge singlets SO(3) potentials
Vaccuum alignments for spontaneous symmetry breaking
Vacuum expectation values and mass spectra
{ i {
to describe the quark sector, as well as Ma and collaborators [3, 4] who used
= A4 to
describe the lepton sector. Many other choices for
have subsequently been used in model
building, several of which will be discussed below. For an early brief review of possible
discrete groups that can be used for SM extensions see [5]. Recent extensive reviews with
more complete and up to date bibliographies are also available. See for instance [6{9].
Extending the SM by a discrete group is not without its perils. Global discrete
symmetries are violated by gravity [10]. (As an example, consider the case when a star collapses
to a black hole. The no hair theorem tells us initial baryon number is lost and hence gravity
causes a global discrete symmetry to be violated. Similarly, global continuous symmetries
are violated by gravity. See e.g., [11{13], where it is argued that gravity also spoils the
Peccei-Quinn solution to the strong CP problem.) In addition, the discrete group can be
anomalous [14], it can lead to unwanted cosmic defects [15], etc. To avoid as many of these
problems as possible the most expedient approach is to gauge the discrete symmetry, i.e.,
{ 1 {
extend the SM by a continuous gauge group G in such a way that no chiral anomalies are
produced. Then one breaks this gauge group to the desired discrete group, G !
, where
now
is e ectively anomaly free and avoids problems with gravity.
Various examples of Lie groups breaking to discrete groups have been discussed in the
literature, but only in a few cases have the details of the minimization of the scalar potential
and the extraction of the scalar spectrum been investigated. Here we plan to include these
important details for many of the discrete groups of interest via the following procedure:
(i) First we provide irreps of G that contain trivial
singlets. These results are
summarized in the appendix.
(ii) Next we set up scalar potentials V with scalars in one of these irreps.
(iii) Then we nd a vacuum expectation value (VEV) via the Reynolds operator [16, 17]
(related to the perhaps more familiar Molien series [18]) that can break G to .
(iv) Next we minimize V to show that the VEV indeed does properly break the symmetry.
(v) Finally, we provide the spectrum of scalar masses at the
level after the breaking.
Our calculations are carried out with Mathematica and checked by hand where practical.
Many of the methods we employ were developed in work by Luhn [19] and by Merle
and Zwicky [20], where some of the results summarized here can be found. We believe
our results will be of interest to many model builders, since it will allow them to include
the minimal set of scalars necessary to break a gauge symmetry to a discrete symmetry of
interest. A few examples that go beyond the minimal set of scalars are also included, where
the symmetry breaking is carried out from a nonminimal G irrep or from a non-minimal G.
2
Lie group invariant potentials
Our task in this section is to construct Higgs potentials invariant under Lie groups G for
speci c irreps. But rst we must see which irreps are suitable for spontaneous symmetry
breaking (SSB), i.e., irreps whose decompositions include a trivial singlet of the desired
subgroup
G to which we hope to break. Using the Mathematica package
decomposeLGreps [21] along with GAP to generate the groups [22], one can easily produce tables of
branching rules from Lie group irreps to subgroup irreps and
nd such singlets. We have
done this for a number of cases and have included them in a short appendix for convenience
and to make the paper self contained.
2.1
Gauge group irreps containing discrete gauge singlets
The discrete groups
we will discuss and the gauge groups where they can be minimally
embedded are A4; S4; A5
SO(3); Q6; T 0; O0; I0
SU(2); and T7; (27); PSL(2; 7)
SU(3).
Some of these discrete groups can also be embedded non-minimally. For example,
we include the case A4
SU(3). Minimal and non-minimal embedding of other discrete groups can be handled in a way similar to what is discussed here, and we hope that the examples we discuss (...truncated)