See-saw Enhancement of Neutrino Mixing due to the Right-handed Phases

We study the see-saw enhancement mechanism in presence of the right-handed phases of the Dirac neutrino mass matrix and the Majorana mass matrix. The enhancement condition given by S mirnov is modified. We point out that the see-saw enhancement could be obt ained due to the right-handed phases even if the Majorana matrix is proportional to the unit matrix. We show a realistic Dirac mass matrix which ca uses the see-saw enhancement. Permanent address:Science Education Laboratory, Ehime University, 79 0 Matsuyama, JAPAN The see-saw mechanism of the neutrino mass generation gives a very natural and elegant understanding for the smallness of neutrino masses[1]. The recent obs erved solar neutrino deficit[2] and muon neutrino deficit in the atomospher ic neutrino flux[3] s timulate the systematic study of the neutrino mixings[4]. In the standpoint of the quark-lepton unification in most GUT groups, the Dirac mass matrix of neutrinos is similar to the one of quarks, therefore, the neut rino mixings turn out to be typically of the same order of magnitude as the q uark mixings. Howeve r, some authors pointed out that the large neutrino mixing between the different generations could be obtained in the see-saw mechanism as a consequence of c ertain structure of the right-handed Majorana mass matrix[5,6,7]. That is the so called see-saw en hancement[7] of the neutrino mixing due to the cooperation between the Dirac and Majorana mass matrices, which needs a higher representation of the Higgs fields such as 126 in SO(10). In this paper, we modify the enhancement conditions[7], which were given by S mirnov in presence of the right-handed phases of the Dirac mass matrix and th e Majorana mass marix. So, we point out that the see-saw enhancement could be obtained even if the Majora na matrix is proportional to the unit matrix, i.e., there is no hierarchy in the Majorana mas s matrix. This enhancement is caused by the right-handed phases, which never appear in the case of the quark mixing. The large enhancement is obtained for s ome textures of the Dirac mass matrix. In order to see the role of the right-handed phases of the Dirac mass matrix cl early, we work in the nearest neighbour interaction(NNI) basis[8], which is one specific weak basis and does not imply any loss of generality for the Dir ac mass matrix. In this basis, the Dirac mass matrix with three generations is written as follows: