The Higgs mode and v

Selection of the mode

The vacuum leaves the spatial cube unbroken, so the Higgs mode is invariant under Aut(\(Q_3\)), acting on x,y,z with τ fixed. 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 τ alone. On the two slices these are the constant 1 at \(\lambda=0\) and the alternating (−1)^τ 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 d₀ψ≠0 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 τ and is constant on the three cube directions,

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

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

Spectral weight and the descent relation

Write \(M_{V,\omega}=L_{V,\omega}+I\) with eigenvalues \(\lambda_n+1\) and orthonormal eigenvectors \(|n\rangle\), so that \(M_{V,\omega}^{-1}=\sum_n|n\rangle\langle n|/(\lambda_n+1)\). The propagator weight of a normalised mode \(\phi\) is \(G_\phi=\langle\phi|M_{V,\omega}^{-1}|\phi\rangle\), which for a general mode spreads over the spectrum as \(\sum_\lambda\widehat c_\lambda^{\,2}/(\lambda+1)\). The Higgs mode is itself an eigenvector, so a single projection survives, \(\widehat c^{\,2}_{\lambda_\psi}=1\), and the sum leaves one term, \(G_\phi=1/(\lambda_\phi+1)\). With \(\lambda_\psi=2\), and against the deformed vertex trace \(\Sigma_V=3.82333\),

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

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