Superconnections and the Higgs Field

TL;DR

This paper interprets the Higgs field as a superconnection in a mathematical framework, deriving relations for gauge couplings, the Weinberg angle, and particle mass ratios within a geometric model.

Contribution

It introduces a geometric model of the Higgs field using superconnections on a superbundle, deriving physical parameters from mathematical structures.

Findings

Reformulates the U(1) Higgs model as Ginzburg-Landau theory.

Relates the U(2) model to electroweak theory with specific coupling constants.

Predicts mass ratios of W, Z, and Higgs bosons at tree level.

Abstract

Within the mathematical framework of Quillen, one interprets the Higgs field as part of the superconnection on a superbundle. We propose to take as superbundle the exterior algebra obtained from a Hermitian bundle with structure group U(n). Spontaneous symmetry breaking appears as a consequence of a non-vanishing scalar curvature. The U(1) Higgs model reformulates the Ginzburg-Landau theory, while the U(2) model relates to the electroweak theory with the relation $g^2=3g4^2$ for the gauge coupling constants, the formula $\sin^2\theta=1/4$ for the Weinberg angle, and the ratio $ m_W^2 : m_Z^2 : m_H^2 = 3 : 4 : 12 $ for the masses (squared) of the W, Z, and Higgs boson (at tree level).