Dark matter glueball candidate from a $G(2)$--$E_6$--$E_7$ exceptional grand unified theory: $G$-parity, spin, mass and stability

RESUMO

We determine the gauge-invariant identity, mass scale and ultraviolet stability of the dark matter candidate arising from \(G(2)\to SU(3)_C\) in the exceptional \(G(2)\!-\!E_6\!-\!E_7\) construction. The broken \(G(2)\) sector contains odd \(1^{+-}\) and \(0^{--}\) channels, whereas the scalar \(0^{++}\sim X\bar X\) state is even and unprotected. For \(m_X\simeq5.7\times10^{13}\,\mathrm{GeV}\), weak-binding reference masses are \(M_{2X}\simeq1.14\times10^{14}\,\mathrm{GeV}\) and \(M_{3X}\simeq1.71\times10^{14}\,\mathrm{GeV}\), while the exact pole masses remain nonperturbative. Gauge-invariant Fröhlich--Morchio--Strocchi (FMS) operators, Hall--Post bounds, \(Y\)-junction arguments and the pure-\(SU(3)\) glue spectrum favor \(1^{+-}\) in their controlled regimes without excluding a deeply bound \(0^{--}\) state. We then test whether this dark grading survives the full chiral exceptional embedding. An on-shell analysis finds no independent purely dark odd operator through dimension seven: the first nonvanishing basis appears at dimension nine. So the minimal one-copy exceptional embedding does not provide an exact ultraviolet dark \(G\)-parity. Moving the dark Higgs from the common \(\mathbf{1463}_H\) parent to a separated \(\mathbf{1539}_H\) removes the scalar-parent obstruction and the relevant dimension-nine exceptional parents vanish on the selected pure-dark component in the undressed limit. A genuinely gauged or geometric \(\mathbb Z_{2,D}\) would therefore leave a bosonic parity after dark Higgsing. Its extension to the mirror-free theory nevertheless fails because the required \(G(2)\) conjugation also conjugates color, \(\mathbf3_C\leftrightarrow\bar{\mathbf3}_C\). The remaining obstruction to exact dark matter stability is therefore ultraviolet and chiral, rather than low-energy or purely scalar.