20 research outputs found
Comments on the classification of the finite subgroups of SU(3)
Many finite subgroups of SU(3) are commonly used in particle physics. The
classification of the finite subgroups of SU(3) began with the work of H.F.
Blichfeldt at the beginning of the 20th century. In Blichfeldt's work the two
series (C) and (D) of finite subgroups of SU(3) are defined. While the group
series Delta(3n^2) and Delta(6n^2) (which are subseries of (C) and (D),
respectively) have been intensively studied, there is not much knowledge about
the group series (C) and (D). In this work we will show that (C) and (D) have
the structures (C) \cong (Z_m x Z_m') \rtimes Z_3 and (D) \cong (Z_n x Z_n')
\rtimes S_3, respectively. Furthermore we will show that, while the (C)-groups
can be interpreted as irreducible representations of Delta(3n^2), the
(D)-groups can in general not be interpreted as irreducible representations of
Delta(6n^2).Comment: 15 pages, no figures, typos corrected, clarifications and references
added, proofs revise
On a possible relationship between lepton mixing and the stability of dark matter
I comment on the proposal that the stability of dark matter may be due to an
unbroken Z_2 symmetry contained in the partially broken lepton flavour symmetry
group. I remark that (1) there is no Z_2 symmetry apparent in the lepton mass
spectrum and in lepton mixing, (2) predictive models of this type may be
constructed by using a lepton flavour symmetry group with three inequivalent
singlets, to which the three left-handed-lepton gauge-SU(2) doublets are
assigned, and (3) some predictions for the lepton masses and mixings are likely
to be altered by radiative contributions to the neutrino mass matrix. I
construct two models of this type in which the conserved Z_2 originates in a
lepton flavour symmetry group D_4.Comment: 13 page
Finite flavour groups of fermions
We present an overview of the theory of finite groups, with regard to their
application as flavour symmetries in particle physics. In a general part, we
discuss useful theorems concerning group structure, conjugacy classes,
representations and character tables. In a specialized part, we attempt to give
a fairly comprehensive review of finite subgroups of SO(3) and SU(3), in which
we apply and illustrate the general theory. Moreover, we also provide a concise
description of the symmetric and alternating groups and comment on the
relationship between finite subgroups of U(3) and finite subgroups of SU(3).
Though in this review we give a detailed description of a wide range of finite
groups, the main focus is on the methods which allow the exploration of their
different aspects.Comment: 89 pages, 6 figures, some references added, rearrangement of part of
the material, section on SU(3) subgroups substantially extended, some minor
revisions. Version for publication in J. Phys. A. Table 12 corrected to match
eq.(256), table 14 and eq.(314) corrected to match the 2-dimensional irreps
defined on p.6
Principal series of finite subgroups of SU(3)
We attempt to give a complete description of the "exceptional" finite
subgroups Sigma(36x3), Sigma(72x3) and Sigma(216x3) of SU(3), with the aim to
make them amenable to model building for fermion masses and mixing. The
information on these groups which we derive contains conjugacy classes, proper
normal subgroups, irreducible representations, character tables and tensor
products of their three-dimensional irreducible representations. We show that,
for these three exceptional groups, usage of their principal series, i.e.
ascending chains of normal subgroups, greatly facilitates the computations and
illuminates the relationship between the groups. As a preparation and testing
ground for the usage of principal series, we study first the dihedral-like
groups Delta(27) and Delta(54) because both are members of the principal series
of the three groups discussed in the paper.Comment: 43 pages, no figures; typos corrected, clarifications and references
added, version matches publication in J. Phys.
Theory of Neutrino Masses and Mixing
We motivate the usage of finite groups as symmetries of the Lagrangian. After
a presentation of basic group-theoretical concepts, we introduce the notion of
characters and character tables in the context of irreducible representations
and discuss their applications. We exemplify these theoretical concepts with
the groups S_4 and A_4. Finally, we discuss the relation between tensor
products of irreducible representations and Yukawa couplings and describe a
model for tri-bimaximal lepton mixing based on A_4.Comment: 23 pages, lecture presented at IV International Pontecorvo Neutrino
Physics School, September 26 - October 6, 2010, Alushta, Crimea, Ukrain
Two-parameter neutrino mass matrices with two texture zeros
We reanalyse Majorana-neutrino mass matrices M_nu with two texture zeros, by
searching for viable hybrid textures in which the non-zero matrix elements of
M_nu have simple ratios. Referring to the classification scheme of Frampton,
Glashow and Marfatia, we find that the mass matrix denoted by A1 allows the
ratios (M_nu)_{mu mu} : (Mnu)_{tau tau} = 1:1 and (M_nu)_{e tau} : (Mnu)_{mu
tau} = 1:2. There are analogous ratios for texture A2. With these two hybrid
textures, one obtains, for instance, good agreement with the data if one
computes the three mixing angles in terms of the experimentally determined
mass-squared differences Delta m^2_21 and Delta m^2_31. We could not find
viable hybrid textures based on mass matrices different from those of cases A1
and A2.Comment: 10 pages, no figures, minor changes, some references adde
Correlations of the elements of the neutrino mass matrix
Assuming Majorana nature of neutrinos, we re-investigate, in the light of the
recent measurement of the reactor mixing angle, the allowed ranges for the
absolute values of the elements of the neutrino mass matrix in the basis where
the charged-lepton mass matrix is diagonal. Apart from the derivation of upper
and lower bounds on the values of the matrix elements, we also study their
correlations. Moreover, we analyse the sensitivity of bounds and correlations
to the global fit results of the neutrino oscillation parameters which are
available in the literature.Comment: 37 pages, 146 figures, minor corrections, 17 additional figures,
version for publication in JHE
Abelian symmetries in multi-Higgs-doublet models
N-Higgs doublet models (NHDM) are a popular framework to construct
electroweak symmetry breaking mechanisms beyond the Standard model. Usually,
one builds an NHDM scalar sector which is invariant under a certain symmetry
group. Although several such groups have been used, no general analysis of
symmetries possible in the NHDM scalar sector exists. Here, we make the first
step towards this goal by classifying the elementary building blocks, namely
the abelian symmetry groups, with a special emphasis on finite groups. We
describe a strategy that identifies all abelian groups which are realizable as
symmetry groups of the NHDM Higgs potential. We consider both the groups of
Higgs-family transformations only and the groups which also contain generalized
CP transformations. We illustrate this strategy with the examples of 3HDM and
4HDM and prove several statements for arbitrary N.Comment: 33 pages, 2 figures; v2: conjecture 3 is proved and becomes theorem
3, more explanations of the main strategy are added, matches the published
versio