Abstract
In this paper, we examine the leptonic flavor invariants in the minimal see-saw model (MSM), in which only two right-handed neutrino singlets are added into the Standard Model in order to accommodate tiny neutrino masses and explain cosmological matter-antimatter asymmetry via leptogenesis mechanism. For the first time, we calculate the Hilbert series (HS) for the leptonic flavor invariants in the MSM. With the HS we demonstrate that there are totally 38 basic flavor invariants, among which 18 invariants are CP-odd and the others are CP-even. Moreover, we explicitly construct these basic invariants, and any other flavor invariants in the MSM can be decomposed into the polynomials of them. Interestingly, we find that any flavor invariants in the effective theory at the low-energy scale can be expressed as rational functions of those in the full MSM at the high-energy scale. Practical applications to the phenomenological studies of the MSM, such as the sufficient and necessary conditions for CP conservation and CP asymmetries in leptogenesis, are also briefly discussed.
Highlights
Where the charged-le√pton mass matrix an√d the Dirac neutrino mass matrix are respectively given by Ml = Ylv/ 2 and MD = Yνv/ 2, and g is the coupling constant of the SU(2)L gauge group
In this paper, we examine the leptonic flavor invariants in the minimal seesaw model (MSM), in which only two right-handed neutrino singlets are added into the Standard Model in order to accommodate tiny neutrino masses and explain cosmological matter-antimatter asymmetry via leptogenesis mechanism
In ref. [18], implementing the mathematical tool of Hilbert series (HS) and plethystic logarithm (PL) from the invariant theory [19, 20], we have studied the algebraic structures of the invariant ring in the low-energy effective theory with three massive Majorana neutrinos and explicitly constructed all the 34 basic flavor invariants
Summary
In this subsection we sketch the indispensable mathematical ingredients of the invariant theory. For a reductive group, including all the finite groups and semi-simple Lie groups, the ring is finitely generated, in the sense that all the invariants in the ring could be expressed as the polynomials of a finite number of basic invariants. It is worth noting that not all the basic invariants are algebraically independent, and there may exist polynomial functions of the basic invariants that are identically equal to zero [32, 33]. A significant result is that the number of the primary invariants ( the Krull dimension of the ring) equals the number of the physical parameters in the theory. The total number of the factors r equals the Krull dimension of the ring, or the number of the primary invariants, while the power indices dk The MW formula reduces the computation of HS to several complex integrals, which can be performed by virtue of the residue theorem
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