We discuss renormalization in a toy model with one fermion field and one real scalar field phi, featuring a spontaneously broken discrete symmetry which forbids a fermion mass term and a phi^3 term in the Lagrangian. We employ a renormalization schem
e which uses the MSbar scheme for the Yukawa and quartic scalar couplings and renormalizes the vacuum expectation value of phi by requiring that the one-point function of the shifted field is zero. In this scheme, the tadpole contributions to the fermion and scalar selfenergies are canceled by choice of the renormalization parameter delta_v of the vacuum expectation value. However, delta_v and, therefore, the tadpole contributions reenter the scheme via the mass renormalization of the scalar, in which place they are indispensable for obtaining finiteness. We emphasize that the above renormalization scheme provides a clear formulation of the hierarchy problem and allows a straightforward generalization to an arbitrary number of fermion and scalar fields.
It has been suggested that residual symmetries in the charged-lepton and neutrino mass matrices can possibly reveal the flavour symmetry group of the lepton sector. We review the basic ideas of this purely group-theoretical approach and discuss some
of its results. Finally, we also list its shortcomings.
We present a general framework for models in which the lepton mixing matrix is the product of the maximal mixing matrix U_omega times a matrix constrained by a well-defined Z_2 symmetry. Our framework relies on neither supersymmetry nor non-renormali
zable Lagrangians nor higher dimensions; it relies instead on the double seesaw mechanism and on the soft breaking of symmetries. The framework may be used to construct models for virtually all the lepton mixing matrices of the type mentioned above which have been proposed in the literature.
We investigate the possibility that the first column of the lepton mixing matrix U is given by u_1 = (2,-1,-1)^T/sqrt{6}. In a purely group-theoretical approach, based on residual symmetries in the charged-lepton and neutrino sectors and on a theorem
on vanishing sums of roots of unity, we discuss the finite groups which can enforce this. Assuming that there is only one residual symmetry in the Majorana neutrino mass matrix, we find the almost unique solution Z_q x S_4 where the cyclic factor Z_q with q = 1,2,3,... is irrelevant for obtaining u_1 in U. Our discussion also provides a natural mechanism for achieving this goal. Finally, barring vacuum alignment, we realize this mechanism in a class of renormalizable models.
We construct a model for tri-bimaximal lepton mixing which employs only family symmetries and their soft breaking; neither vacuum alignment nor supersymmetry, extra dimensions, or non-renormalizable terms are used in our model. It is an extension of
the Standard Model making use of the seesaw mechanism with five right-handed neutrino singlets. The scalar sector comprises four Higgs doublets and one complex gauge singlet. The horizontal symmetry of our model is based on the permutation group S_3 of the lepton families together with the three family lepton numbers--united this constitutes a symmetry group Delta(6infty^2). The model makes no predictions for the neutrino masses.
Using the seesaw mechanism, we construct a model for the light-neutrino Majorana mass matrix which yields trimaximal lepton mixing together with maximal CP violation and maximal atmospheric-neutrino mixing. We demonstrate that, in our model, the ligh
t-neutrino mass matrix retains its form under the one-loop renormalization-group evolution. With our neutrino mass matrix, the absolute neutrino mass scale is a function of |U_e3| and of the atmospheric mass-squared difference. We study the effective mass in neutrinoless double beta-decay as a function of |U_e3|, showing that it contains a fourfold ambiguity.
We consider trimaximal lepton mixing, defined by |U_{alpha 2}|^2 = 1/3 for all alpha = e, mu, tau. This corresponds to a two-parameter lepton mixing matrix U. We present a model for the lepton sector in which trimaximal mixing is enforced by softly b
roken discrete symmetries; one version of the model is based on the group Delta(27). A salient feature of our model is that no vacuum alignment is required.
We discuss a realization of the non-abelian group O(2) as a family symmetry for the lepton sector. The reflection contained in O(2) acts as a mu-tau interchange symmetry, enforcing--at tree level--maximal atmospheric neutrino mixing and a vanishing m
ixing angle theta_13. The small ratio m_mu/m_tau (muon over tau mass) gives rise to a suppression factor in the mass of one of the pseudoscalars of the model. We argue that such a light pseudoscalar does not violate any experimental constraint.
We derive a general expression for Delta rho (or, equivalently, for the oblique parameter T) in the SU(2) x U(1) electroweak model with an arbitrary number of scalar SU(2) doublets, with hypercharge +-1/2, and an arbitrary number of scalar SU(2) sing
lets. The experimental bound on Delta rho constitutes a strong constraint on the masses and mixings of the scalar particles in that model.
We present general expressions for the oblique parameters S, T, U, V, W, and X in the SU(2)xU(1) electroweak model with an arbitrary number of scalar SU(2) doublets, with hypercharge 1/2, and an arbitrary number of scalar SU(2) singlets.
