Testing R-parity with geometry

Abstract

We present a complete classification of the vacuum geometries of all renormalizable superpotentials built from the fields of the electroweak sector of the MSSM. In addition to the Severi and affine Calabi-Yau varieties previously found, new vacuum manifolds are identified; we thereby investigate the geometrical implication of theories which display a manifest matter parity (or R-parity) via the distinction between leptonic and Higgs doublets, and of the lepton number assignment of the right-handed neutrino fields.We find that the traditional R-parity assignments of the MSSM more readily accommodate the neutrino see-saw mechanism with non-trivial geometry than those superpotentials that violate R-parity. However there appears to be no geometrical preference for a fundamental Higgs bilinear in the superpotential, with operators that violate lepton number, such as νHH, generating vacuum moduli spaces equivalent to those with a fundamental bilinear.

Chiral rings, Futaki invariants, plethystics, and Gröbner bases

“…[12,17,[26][27][28][29]). Moreover, its properties have also been exploited to study the phenomenology of the standard model, ranging from question of vacuum structure to operator selection [14,15,[30][31][32][33][34][35].…”

“…The electroweak sectors of minimal supersymmetric standard model (MSSM) with renormalizable superpotentials are classified in [32]. The simplest case is generated by LH and HH where L stands for the lepton doublets and H,H stand for the up and down types of Higgs doublets.…”

“…The next simplest case is generated by LLe and LHe where e stands for the lepton singlet. From [32], the HS is…”

Standard model plethystics

“…The flat directions of the MSSM have been identified in [5]. Combining these directions of thought, a long programme was launched to study the vacuum geometry of the MSSM and its relatives [6][7][8][9][10][11]: under the guiding principle that "interesting geometry is coextensive with interesting physics", the ultimate goal is to use geometric and topological properties of VMS as a selection rule for operators in the Standard Model Lagrangian. Specifically, if the VMS were to be found to have some special form in the mathematical sense, which (1) cannot be explained in terms of symmetries relating the relevant degree of freedom in the low energy effective field theory; and (2) is very unlikely to have occurred by chance, then this special form should be regarded as a consequence of some unknown physics.…”

“…In such sense, we are placing very restrictive constraints on allowed physical processes that are mediated by certain operators. Already, many interesting features have been found, such as the VMS of the electro-weak sector being an affine cone over the classical Veronese surface, a structure ruined by addition of R-parity-violating operators, or the sensitive dependence of the geometry on the number of generations, or the appearance of Calabi-Yau varieties, etc [9][10][11]. Supersymmetry and the VMS thus provide us with a low energy window of how geometry can guide certain phenomenological questions.…”