论文标题
中微子混合空间的几何形状
Geometry of the neutrino mixing space
论文作者
论文摘要
我们研究中微子混合矩阵的物理区域的几何结构，这是光谱标准的单位球的一部分。几何区域的每个矩阵都是统一PMN矩阵的凸组合。与不同数量的附加中微子相对应的分离子集被描述为单位球面的相对内部。我们确定了Carathéodory的数量，表明最多有四个尺寸的统一矩阵对于表示中微子几何区域的任何基质都是必需的。对于与一个和两个其他中微子状态相对应的矩阵，Carathéodory的数字分别为两个和三个。此外，我们讨论了与不同的数学结构相关的体积，尤其是与单一和正交组以及光谱规范的单位球相关的体积。我们将获得的体积与在当前情况下具有三个中微子家族的CP持续和CP侵入案例的物理可接受的混合矩阵区域的体积进行比较，而三个中微子家族和场景与中微子混合矩阵的维度高于三个。
We study a geometric structure of a physical region of neutrino mixing matrices as part of the unit ball of the spectral norm. Each matrix from the geometric region is a convex combination of unitary PMNS matrices. The disjoint subsets corresponding to a different minimal number of additional neutrinos are described as relative interiors of faces of the unit ball. We determined the Carathéodory's number showing that, at most, four unitary matrices of dimension three are necessary to represent any matrix from the neutrino geometric region. For matrices which correspond to scenarios with one and two additional neutrino states, the Carathéodory's number is two and three, respectively. Further, we discuss the volume associated with different mathematical structures, particularly with unitary and orthogonal groups, and the unit ball of the spectral norm. We compare the obtained volumes to the volume of the region of physically admissible mixing matrices for both the CP-conserving and CP-violating cases in the present scenario with three neutrino families and scenarios with the neutrino mixing matrix of dimension higher than three.
