The Higgs mode and \(v\)

Selection of the mode

The vacuum leaves the spatial cube unbroken, so the Higgs mode is invariant under \(\mathrm{Aut}(Q_3)\), acting on \(x,y,z\) with \(\tau\) fixed. \(\mathrm{Aut}(Q_3)\) is transitive on the vertices of each cube slice, so of the sixteen vertex modes an invariant one is constant on each slice, a function of \(\tau\) alone. On the two slices these are the constant \(\mathbf 1\) at \(\lambda=0\) and the alternating \((-1)^{\tau}\) at \(\lambda=2\). The constant lies in the kernel of the coboundary, \(d_0\mathbf 1=0\), with no edge content and no contour, and carries no vacuum expectation value; the alternating mode has \(d_0\psi\ne0\) on every temporal edge. The Higgs field is the temporal mode.

The Higgs as the temporal mode

The Higgs is the scalar mode of the vertex operator that alternates along the Euclidean time axis \(\tau\) and is constant on the three cube directions,

\[ \psi(x,y,z,\tau)=(-1)^{\tau},\qquad L_0\psi=2\,\psi,\qquad |\psi|^{2}=16. \]

It is a single eigenvector of \(L_0\) with \(\lambda_\psi=2\); flat across the disentangler plane, it is untouched by the deformation, \(L_\omega\psi=2\,\psi\).

Spectral weight and the master equation

With one surviving weight at \(\lambda_\psi=2\), the propagator weight and the spectral weight against the deformed vertex trace \(\Sigma_V=3.82333\) are

\[ G_\psi=\frac{1}{2+1}=\frac13,\qquad \Pi_\psi=\frac{G_\psi}{\Sigma_V}=\frac{1/3}{3.82333}=0.087184. \]

The mode is a colour singlet (\(\Theta_\psi=1\)) and a scalar (\(p=1\)). Through the master equation \(\text{scale}=m_P\,\Theta\,\Pi^{p}\exp(-1/\xi^{*})\), with \(\exp(-1/\xi^{*})=2.31952\times10^{-16}\),