Hilbert series and plethystics: paving the path towards 2HDM- and MLRSM-EFT

Abstract

Effective Field Theory (EFT) technique is one of the most elegant ways to capture the impact of high scale theory, if any, at some low energy by incorporating higher mass dimensional (≥ 5) effective operators (O i ). The low energy EFT is described in terms of only light degrees of freedom (DOF) which can appear on-shell. An essential task while developing the EFT framework is to compute these O i 's. Hilbert Series (HS) is a novel and mathematically robust method to construct the complete set of gauge invariant independent, effective operators. The HS requires the knowledge of the transformation properties of the light DOF and the covariant derivatives under the internal gauge symmetries and conformal groups. The Hilbert Series method, by its virtue, automatically takes care of the redundancies in the operator set due to the Equations of Motion (EOMs) of fields and Integration by Parts (IBPs) with impeccable accuracy.In this paper, we have adopted this methodology to construct the complete set of independent operators up to dimension-6 in the "Warsaw"-like basis for two different Beyond Standard Model scenarios -Two Higgs Doublet Model (2HDM) and Minimal Left-Right Symmetric Model (MLRSM). For both these cases, we have calculated the corrections to the scalar, gauge boson and fermion mass spectra due to the dimension-6 operators. The additional contributions to all the Feynman vertices are computed and their impact on different observables, namely Weak mixing angle, Fermi constant, ρ and oblique (S, T, U ) parameters. We have further discussed how the magnetic moments of charged leptons and production and decay of the massive BSM particles, e.g., charged scalar and different rare processes are affected in the presence of effective operators. We have also constructed the effective scalar four-point interactions and commented on the possible reinvestigation of the theoretical constraints, e.g., unitarity and vacuum stability within these frameworks.

Characters and group invariant polynomials of (super)fields: road to “Lagrangian”

“…We have also catalogued higher dimensional effective operators up to dimension-6 in Table 5 and categorized the operators based on their composition in terms of scalars (φ), fermions (ψ) and the field strength tensor of gauge bosons (X ). Note, the covariant form of these operators (for N f = 1) were discussed in great detail in [45]. So, we do not repeat the same here.…”

“…Again, unlike the quantum fields we can not simply treat it as another degree of freedom because such an inclusion can introduce redundancies within the operator sets by virtue of integration by parts (IBP) and equation of motion of fields (EOM). It has been discussed in [6,9,45] how some of the effective operator structures can be removed in favor of others by paying close attention to IBP and EOM. Thus the incorporation of D μ is a highly involved task.…”

“…The categorization of the non-covariant HS output of mass dimension-6 has been shown in Table 48. See [45] for their covariant forms. All the operators are provided for general N f .…”

One-loop matching in the SMEFT extended with a sterile neutrino

“…refs. [7][8][9]. A common feature of all these EFTs is that they predict new processes that are completely absent in the renormalisable SM.…”

“…Corrections to the gauge boson propagators. There are four diagrams contributing to the self-energy of B µ , see figure 9. The diagram (a) gives…”