An examination of the hierarchy problem beyond the Standard Model

TL;DR

This paper explores the hierarchy problem beyond the Standard Model by linking the Higgs mass to the automorphism group of a hypothetical field theory, suggesting a connection to the Monster group and implications for quantum gravity and gauge theories.

Contribution

It proposes a novel conjecture connecting the Higgs hierarchy problem to the Monster group and explores implications for gauge theories and quantum gravity at specific dimensions.

Findings

Calculated the automorphism group size close to the Monster group.

Predicted a 125.4 GeV scalar boson mass from the conjecture.

Identified a specific dimension (d=32) where quantum gravity mass matches Planck scale.

Abstract

As the Higgs field is a weak isospin doublet of the SU(2) symmetry, the Standard Model requires any symmetry solution to the Higgs hierarchy problem to be SU(2) invariant, a constraint on the type of the symmetry. However, the hierarchy problem is about the size. The size of SU(2) for the Higgs boson can be calculated by $|$SU$_2(\ell)|=\ell^3-\ell$, having the Higgs mass $M_H=1/\ell_{H}\approx125$ GeV. To find the origin of the relative smallness of the Higgs mass in Planck units, alternatively, we search for the origin of such a large order assuming that it stems from an unknown field theory X beyond the Standard Model. Accordingly, this order, which corresponds to the quantum of the Higgs field, should determine the order of quantum/core of X symmetry, its automorphism group. We calculate $|$Aut(X)$|\approx8.2\times 10^{53}$, close to the order of the Monster sporadic group, $|\mathbb M|\approx 8.1\times 10^{53}$, the automorphism group of the Monster CFT, which we therefore conjecture to be X. To examine this conjecture, we calculate the mass of a scalar boson whose SU(2) order is determined by $|\mathbb M|$, observing a 125.4 GeV boson mass and a 245.7 GeV VEV. The Monster CFT does not have any spin-1 operators and Kac-Moody symmetry. Therefore, based on the CFT/(A)dS correspondences, it only describes pure gravity without the gauge fields. In search of a gauge theory candidate, we promote SU(2) (double cover of SO(3)), to SO($d$), and show that the same $\mathbb M$-symmetric vacuum configuration reaches the Planck mass of quantum gravity precisely at $d=32$ (with 99\% accuracy). Then, the spin-1 boson mass of the eligible gauge candidates, SO(32) and $E_8\times E_8$, is calculated to be 80.9 GeV. Further, several pieces of evidence are provided supporting the conjecture.