Patrick Otto Ludl

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). Although 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.

Is S4 the horizontal symmetry of tri-bimaximal lepton mixing?

We determine the symmetry groups under which the charged-lepton and the Majorana-neutrino mass terms are invariant. We note that those two groups always exist trivially, i.e. independently of the presence of any symmetries in the Lagrangian, and that they always have the same form. Using this insight, we re-evaluate the recent claim that, whenever lepton mixing is tri-bimaximal, S4 is the minimal unique horizontal-symmetry group of the Lagrangian of the lepton sector, with S4 being determined by the symmetries of the lepton mass matrices. We discuss two models for tri-bimaximal mixing which serve as counterexamples to this claim. With these two models and some group-theoretical arguments we illustrate that there is no compelling reason for the uniqueness of S4.

Maximal CP Violation in Lepton Mixing from a Model with Delta(27) flavour Symmetry

A bstractWe propose a simple mechanism which enforces

On the finite subgroups of U(3) of order smaller than 512

\left| {{U_{\mu j}}} \right| = \left| {{U_{\tau j}}} \right|\forall j = 1,2,3

A complete survey of texture zeros in the lepton mass matrices

in the lepton mixing matrix U. This implies maximal atmospheric neutrino mixing and a maximal CP-violating phase but does not constrain the reactor mixing angle θ13. We implement the proposed mechanism in two renormalizable seesaw models which have features strongly resembling those of models based on a flavour symmetry group Δ(27). Among the predictions of the models, there is a determination, although ambiguous, of the absolute neutrino mass scale, and a stringent correlation between the absolute neutrino mass scale and the effective Majorana mass in neutrinoless double-beta decay.

Comments on the classification of the finite subgroups of SU(3)

We use the SmallGroups Library to find the finite subgroups of U(3) of order smaller than 512 which possess a faithful three-dimensional irreducible representation. From the resulting list of groups we extract those groups that cannot be written as direct products with cyclic groups. These groups are the basic building blocks for models based on finite subgroups of U(3). All resulting finite subgroups of SU(3) can be identified using the well-known list of finite subgroups of SU(3) derived by Miller, Blichfeldt and Dickson at the beginning of the 20th century. Furthermore, we prove a theorem which allows us to construct infinite series of finite subgroups of U(3) from a special type of finite subgroups of U(3). This theorem is used to construct some new series of finite subgroups of U(3). The first members of these series can be found in the derived list of finite subgroups of U(3) of order smaller than 512. In the last part of this work we analyze some interesting finite subgroups of U(3), especially the group , which is closely related to the important SU(3)-subgroup S4.

Residual Z2×Z2 symmetries and lepton mixing

A bstractWe perform a systematic and complete analysis of texture zeros in the lepton mass matrices and identify all viable and maximally restrictive cases of pairs (Mℓ, MD ) and (Mℓ, ML), where Mℓ, MD and ML are the charged-lepton, Dirac neutrino and Majorana neutrino mass matrices, respectively. To this end, we perform a thorough analysis of textures which are equivalent through weak-basis permutations. Furthermore, we introduce numerical measures for the predictivity of textures and apply them to the viable and maximally restrictive texture zero models. It turns out that for Dirac neutrinos these models can at most predict the smallest neutrino mass and the CKM-type phase of the mixing matrix. For Majorana neutrinos most models can, in addition, predict the effective neutrino mass for neutrinoless double beta decay.

Maximal atmospheric neutrino mixing from texture zeros and quasi-degenerate neutrino masses

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 Blichfeldts work the two series (C) and (D) of finite subgroups of SU(3) are defined. While the group series Δ(3n2) and Δ(6n2) (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 show that (C) and (D) have the structures and , respectively. Furthermore, we show that, while the (C)-groups can be interpreted as irreducible representations of Δ(3n2), the (D)-groups can in general not be interpreted as irreducible representations of Δ(6n2).

Correlations of the elements of the neutrino mass matrix

Abstract We consider two novel scenarios of residual symmetries of the lepton mass matrices. Firstly we assume a Z 2 × Z 2 symmetry G l for the charged-lepton mass matrix and a Z 2 symmetry G ν for the light neutrino mass matrix. With this setting, the moduli of the elements of one column of the lepton mixing matrix are fixed up to a reordering. One may interchange the roles of G l and G ν in this scenario, thereby constraining a row, instead of a column, of the mixing matrix. Secondly we assume a residual symmetry group G l ≅ Z m ( m > 2 ) which is generated by a matrix with a doubly-degenerate eigenvalue. Then, with G ν ≅ Z 2 × Z 2 the moduli of the elements of a row of the lepton mixing matrix get fixed. Using the library of small groups we have performed a search for groups which may embed G l and G ν in each of these two scenarios. We have found only two phenomenologically viable possibilities, one of them constraining a column and the other one a row of the mixing matrix.

A complete survey of texture zeros in general and symmetric quark mass matrices

Abstract It is well known that, in the basis where the charged-lepton mass matrix is diagonal, there are seven cases of two texture zeros in Majorana neutrino mass matrices that are compatible with all experimental data. We show that two of these cases, namely B 3 and B 4 in the classification of Frampton, Glashow and Marfatia, are special in the sense that they automatically lead to near-maximal atmospheric neutrino mixing in the limit of a quasi-degenerate neutrino mass spectrum. This property holds true irrespective of the values of the solar and reactor mixing angles because, for these two cases, in the limit of a quasi-degenerate spectrum, the second and third row of the lepton mixing matrix are, up to signs, approximately complex-conjugate to each other. Moreover, in the same limit the aforementioned cases also develop a maximal CP-violating CKM-type phase, provided the reactor mixing angle is not too small.