Dirac neutrinos and family symmetries

Neutrino physics has made substantial progress in the last 15 years on the experimental side, in particular, through the measurement of neutrino oscillations. However, from a theoretical perspective the understanding and interpretation of the data with respect to the lepton mass and mixing problem is unsatisfatory. Moreover, one of the main unresolved problems of lepton physics is whether neutrinos are Dirac or Majorana particles. Though Majorana neutrinos are favoured by most theorists, at present there is no incontrovertible experimental evidence which would hint at the Majorana nature. Two other major puzzles related to neutrinos are the smallness of neutrino masses compared to the mass of the electron and the specific form of the lepton mixing matrix. The most popular mechanism that can explain the smallness of neutrino masses is the so-called seesaw mechanism. While the type I seesaw mechanism is mainly applied to Majorana neutrinos, the type II seesaw mechanism has been invented for generating small vacuum expectation values of scalar fields. It can thus be used to obtain small vacuum expectation values of Higgs doublets in order to obtain small Dirac neutrino masses. In this case, the large seesaw scale occurs in the Higgs potential in one or several mass terms. The fact that the fermion flavour eigenstates do not coincide with the mass eigenstates, gives rise to the quark and lepton mixing matrices. While the quark mixing matrix deviates only slightly from the unit matrix, the lepton mixing matrix is far from it and experiments have revealed that its form is rather specific, namely close to tri- bimaximal mixing. One possible way to understand this feature could be the invariance of the action of the theory under family symmetries (horizontal symmetries), i.e. symmetries acting in family space for each type of fermion field. All experimental evidence suggests that there are three fermion families. Therefore, we choose three- dimensional representations of such a family symmetry group for the different types of fermionic fields. The choice of the symmetry group and the involved representations partly relate the Yukawa couplings and, therefore, also the mass and the mixing matrices to Clebsch-Gordan coefficients, which are specific to the representations of the family symmetry group. The proposed project aims at the investigation of family symmetries in the context of Dirac neutrinos, within the framework of renormalizable extensions of the standard model with three right-handed neutrino gauge singlets and an arbitrary number of Higgs doublets. The implementation of a family symmetry group is supposed to determine, at least partly, the structure of the lepton mixing matrix, while a type II seesaw mechanism is assumed to be responsible for small Dirac neutrino masses. Since up to now the family symmetry group is completely unknown, within the above framework we will use group representation theory and the classification of Clebsch-Gordan coefficients in order to arrive at statements and conclusions as general as possible, valid for large classes of groups.

The origin of the fermion mass spectra and the quark and lepton mixing matrices is one of the biggest mysteries in particle physics. Based on the assumption that specific features of the mass spectra and the mixing matrices originate from symmetries, we have derived general results in the case of Abelian symmetry groups.The mentioned mystery consists of the following issues:the strong hierarchy in all three mass spectra of charged fermions (up-type quarks, down- type quarks and charged leptons),the smallness of the mixing angles in the quark (CKM) mixing matrix, but the existence of two large mixing angles in the lepton (PMNS) mixing matrix,and the smallness of the neutrino masses (six orders of magnitude smaller than the electron mass).While the last issue is believed to find an explanation beyond the Standard Model in the so- called seesaw-mechanism, where a mass scale much higher than the electroweak scale takes care of the suppression of the neutrino masses, for the other two issues one can attempt to find a symmetry reason. During the project we provided a complete classification of mass matrices with texture zeros, which for all practical purposes can be thought of as being enforced by Abelian symmetries, and performed an exhaustive discussion of the predictions of all viable cases obtained for the lepton sector with both Dirac and Majorana neutrinos. Since we had already developed the necessary methods, we could also apply them in the quark sector with general and symmetric quark mass matrices. It turned out that texture zeros are relatively weak in their predictions. For instance in the lepton sector with Dirac neutrinos, they can at most predict the smallest neutrino mass. We concluded therefore that texture zeros can only be responsible for some features of the mass spectra, but not for the global mass and mixing picture.Concerning non-Abelian groups, we were able to complete the mathematical classification of all finite subgroups of SU(3)the most popular flavour groups for model buildingby identifying the structure of two classes of SU(3)-subgroups which had hitherto been unexplored.