Naturally Light Composite Higgs as a Protected Collective Eigenmode

Abstract

Standard routes to a light composite Higgs either rely on tuning a single channel near criticality or protect a pseudo-Nambu--Goldstone coordinate of a coset. We introduce a third mechanism class in which the protected object is an \emph{eigenvalue} of the renormalized multi-operator scalar kernel of the strong sector. Two basis-invariant diagnostics, a sector participation number $\Psec$ and a sector gap ratio $\Rcoll$, identify collective lightness, but they cannot distinguish an accidental small determinant from a protected zero mode; the missing discriminator is the microscopic sensitivity $\hat{I}{”}_g=\left|\partial \ln |m_H^2|/\partial \ln g\right|$. A protected collective Higgs is defined by $\Rcoll\gg1$, $\Psec>1$, and $\hat{I}{”}_g=\mathcal{O}(1)$. We prove that this class is nonempty. A rank-one TC--DTC locking invariant forbids tree-level aligned curvature, while universal vectorlike DQCD bridge fermions, massless in the microscopic Lagrangian but acquiring a common DQCD constituent mass, obey $\partial_h^2\sum_A\Tr\mathcal{M}_A^2\big|_0=0$. The aligned scalar is therefore lifted only at joint two-spurion order, $m_{\mathrm{br}}^2=-(d_X/2\ddot{I}{€}^2)g_T^2{g}_D^2(f_B^4/f_H^2)L_X$, giving $\hat{I}{”}_{g_T}=\hat{I}{”}_{g_D}=2$ at leading logarithmic order. The same DTC topology admits a collective top completion and a vector-decoupling route to reducing the positive technicolor contribution to $S$. The mechanism is falsifiable by sector-restricted lattice spectroscopy and coupling-response scans.

6 pages, 1 figure