Spontaneous breaking of gauge groups to discrete symmetries

Journal of High Energy Physics, Aug 2017

Many models of beyond Standard Model physics connect flavor 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.

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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)


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Bradley L. Rachlin, Thomas W. Kephart. Spontaneous breaking of gauge groups to discrete symmetries, Journal of High Energy Physics, 2017, pp. 110, Volume 2017, Issue 8, DOI: 10.1007/JHEP08(2017)110