Hilbert Series for Flavor Invariants of the Standard Model

TL;DR

The paper computes the Hilbert series $H(t)$ counting flavor invariants in the lepton sector of the three-generation see-saw Standard Model and, as a bonus, in the four-generation quark sector. It employs the Molien-Weyl integral together with the plethystic exponential to construct $H(t)$ and uses the plethystic logarithm to extract generators and relations, including a multi-graded variant $H(t_u,t_d)$. For three-generation leptons, the Hilbert series has $d_N=114$, $d_D=162$, with $ ext{dim }V=48$ and $p=21$, and $PL[H(t)]=3 t^2 + 5 t^4 + 9 t^6 + 10 t^8 + 19 t^{10} + 40 t^{12} + 66 t^{14} + 92 t^{16} + 70 t^{18} - O(t^{20})$; for four-generation quarks, $d_N=46$, $d_D=110$, $ ext{dim }V=64$, $p=17$, and $PL[H(t)]=2 t^2 + 3 t^4 + 4 t^6 + 6 t^8 + 2 t^{10} + 4 t^{12} + 2 t^{14} + 4 t^{16} + 4 t^{18} + t^{20} - O(t^{24})$, with a lengthy multi-graded series also provided. Significantly, the results offer exact, finitely generated descriptions of flavor invariants, connect neutrino mass generation and quark flavor structure to invariant-theory counting, and mirror gauge-invariant operator counting in related $ ext{N}=1$ supersymmetric theories.

Abstract

The Hilbert series is computed for the lepton flavor invariants of the Standard Model with three generations including the right-handed neutrino sector needed to generate light neutrino masses via the see-saw mechanism. We also compute the Hilbert series of the quark flavor invariants for the case of four generations.