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On the addition of an $SU(2)$ quadruplet of scalars to the Standard Model

Darius Jurčiukonis, Luís Lavoura

TL;DR

This work analyzes the bounded-from-below (BFB) conditions for the Standard Model extended by a single SU(2) quadruplet of scalars, considered with hypercharges $Y=3/2$ or $Y=1/2$ (the latter with a reflection symmetry). It derives exact analytical expressions for the boundaries of the phase space of the scalar potential’s quartic sector, introducing gauge-invariant combinations $F_1$, $F_2$, $F_4$, and $F_5$ and the dimensionless parameters $r$, $\boldsymbol{\delta}$, and $\boldsymbol{\gamma}_5$ (and analogously $\boldsymbol{\texteta}$ for the $Y=1/2$ case). The authors show that, in practice, testing BFB reduces to scanning along a few special lines on the boundary (rather than across surfaces), and they provide explicit line choices and the associated analytic conditions; this yields a dramatic speed-up (up to ~1700x) compared with brute-force minimization. They also discuss the convexity/concavity of boundary sheets, justify the line-based approach, and confirm the results numerically across large random parameter samples. The findings enable efficient and exact characterization of the viable parameter space for these extended scalar sectors, with potential implications for neutrino mass mechanisms and Higgs-gauge couplings.

Abstract

We consider the extension of the Standard Model through an $SU(2)$ quadruplet of scalars with hypercharge either $3/2$ or $1/2$ (with an additional reflection symmetry in the latter case). We establish, through exact analytical equations, the boundaries of the phase spaces of the gauge-invariant terms that appear in the (renormalizable) scalar potentials. We devise procedures for the determination of necessary and sufficient bounded-from-below conditions on those potentials; we emphasize that one mostly needs to scan the scalar potential over a few \emph{lines}, instead of \emph{surfaces}, in order to establish the boundedness-from-below; this fact allows one to reduce by three orders of magnitude the computational time devoted to that establishment.

On the addition of an $SU(2)$ quadruplet of scalars to the Standard Model

TL;DR

This work analyzes the bounded-from-below (BFB) conditions for the Standard Model extended by a single SU(2) quadruplet of scalars, considered with hypercharges or (the latter with a reflection symmetry). It derives exact analytical expressions for the boundaries of the phase space of the scalar potential’s quartic sector, introducing gauge-invariant combinations , , , and and the dimensionless parameters , , and (and analogously for the case). The authors show that, in practice, testing BFB reduces to scanning along a few special lines on the boundary (rather than across surfaces), and they provide explicit line choices and the associated analytic conditions; this yields a dramatic speed-up (up to ~1700x) compared with brute-force minimization. They also discuss the convexity/concavity of boundary sheets, justify the line-based approach, and confirm the results numerically across large random parameter samples. The findings enable efficient and exact characterization of the viable parameter space for these extended scalar sectors, with potential implications for neutrino mass mechanisms and Higgs-gauge couplings.

Abstract

We consider the extension of the Standard Model through an quadruplet of scalars with hypercharge either or (with an additional reflection symmetry in the latter case). We establish, through exact analytical equations, the boundaries of the phase spaces of the gauge-invariant terms that appear in the (renormalizable) scalar potentials. We devise procedures for the determination of necessary and sufficient bounded-from-below conditions on those potentials; we emphasize that one mostly needs to scan the scalar potential over a few \emph{lines}, instead of \emph{surfaces}, in order to establish the boundedness-from-below; this fact allows one to reduce by three orders of magnitude the computational time devoted to that establishment.

Paper Structure

This paper contains 58 sections, 107 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Multiplets:
  4. Invariants:
  5. Dimensionless parameters:
  6. Gauge $a=0$:
  7. Potential:
  8. Quadruplet with hypercharge $3/2$
  9. Extra term:
  10. Bounds on $\epsilon^2$:
  11. Parameters $x$, $y$, and $z$:
  12. Boundary:
  13. Points:
  14. Lines:
  15. Bounding surface:
  16. ...and 43 more sections

Figures (6)

  • Figure 1: Two perspectives of the phase space for case $Y=3/2$. The lines and the points---except $\widetilde{P}_5$---defined in the text are explicitly displayed. In the left panel the phase space is displayed transparent, so that the lines may be seen behind each other; in the right panel it is opaque.
  • Figure 2: The projections of the phase space for case $Y=3/2$ on the planes $\gamma_5$vs.$\delta$ (left panel) and $\epsilon$vs.$\delta$ (right panel). See further comments on this figure in the text.
  • Figure 3: Two perspectives of the boundary of phase space for case $Y=1/2$ with reflection symmetry. The points and lines defined in the text are explicitly displayed.
  • Figure 4: Left panel: the projection of phase space for case $Y=1/2$ with reflection symmetry onto the $\gamma_5$vs.$\delta$ plane. The blue line, which is not displayed, coincides with the magenta and green--brown lines, just as the whole sheet 1 of the boundary of phase space. Sheet 2 is between the magenta and purple lines. Sheet 3 is among the purple, yellow, and green--brown lines. Sheet 4 is the whole area under the parabola. Dashed lines of constant $\eta$ are marked in various colours. Right panel: the projection of the same phase space onto the $\eta$vs.$\delta$ plane. The blue line, which is not displayed, coincides with the yellow line, just as the whole sheet 4 of the boundary of phase space. The purple line, which is not displayed, coincides with the magenta line, just as the whole sheet 2 of the boundary of phase space. Sheet 3 is the whole area under the magenta line, black dashed line, and part of the green segment; sheet 1 is the area under the magenta and green--brown lines. Dashed lines of constant $\gamma_5$ are marked in various colours.
  • Figure 5: Two perspectives of the boundary of phase space. The points and lines displayed are defined in the text. The blue points in left panel indicate the convex part of the surface $Q_5=0$; green points correspond to the part of that surface that is neither concave nor convex.
  • ...and 1 more figures