Leptonic unitarity triangles: RGE running effects and $\mu$-$\tau$ reflection symmetry breaking

There are six leptonic unitarity triangles (LUTs) defined by six orthogonality conditions of the three-family lepton flavor mixing matrix in the complex plane. In the framework of the standard model or the minimal supersymmetric standard model, the evolutions of sides and inner angles of the six LUTs from a superhigh energy scale ${\mathrm{\Lambda }}_{\mathrm{H}}^{}$ to the electroweak scale ${\mathrm{\Lambda }}_{\mathrm{E}\mathrm{W}}^{}$ due to the renormalization-group equation (RGE) running are derived in the integral form for both Dirac and Majorana neutrinos. Furthermore, the LUTs as an intuitively geometrical language are applied to the description of the RGE-induced $\mu$-$\tau$ reflection symmetry breaking analytically and numerically.