TFPT scientific source excerpt
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000143 | =
000144 | -\int d^4x\sqrt g
000145 | \left(
000146 | \frac{1}{24\pi^2}R
000147 | +
000148 | \frac{1}{48\pi^2}\Phi^\dagger\Phi
000149 | \right)
000150 | +
000151 | \mathcal E_\Sigma,
000152 | \]
000153 | and
000154 | \[
000155 | \begin{aligned}
000156 | \Gamma_D^{(4)}
000157 | ={}&
000158 | \int d^4x\sqrt g
000159 | \Big(
000160 | \frac{1}{16\pi^2}|D\Phi|^2
000161 | +\frac{1}{16\pi^2}(\Phi^\dagger\Phi)^2
000162 | +\frac{1}{96\pi^2}R\,\Phi^\dagger\Phi
000163 | \\
000164 | &\qquad
000165 | +\frac{1}{192\pi^2}\tr\mathbb F^2
000166 | +\frac{1}{6M^2}R^2
000167 | +\cdots
000168 | \Big)
000169 | +\mathcal A_\Sigma.
000170 | \end{aligned}
000171 | \]
000172 | The seam-even scale channel $\chigeo$ is the unique scalar geometric mode on the retained branch;
000173 | together with the retained $R^2$ term it forms a positive scalaron block, so no independent
000174 | conformal negative mode survives as a separate physical degree of freedom.
000175 | Therefore
000176 | \[
000177 | \Gamma_{\mathrm{grav}}
000178 | :=
000179 | -6\chigeo^2\mathcal E_D+\Gamma_D^{(4)}
000180 | \]
000181 | has low-curvature form
000182 | \[
000183 | \Gamma_{\mathrm{grav}}
000184 | =
000185 | \int d^4x\sqrt g
000186 | \left(
000187 | \frac{\Mpl^2}{2}R
000188 | +
000189 | \frac{\Mpl^2}{12M^2}R^2
000190 | +
000191 | |D\Phi|^2
000192 | -V(\Phi,\chigeo)
000193 | -\sum_a \frac{1}{4g_a^2}\tr F_a^2
000194 | +\cdots
000195 | \right),
000196 | \]
000197 | with
000198 | \[
000199 | \Mpl^2=\frac{\chigeozero^2}{2\pi^2},
000200 | \qquad
000201 | Z_H=\frac{1}{16\pi^2},
000202 | \qquad
000203 | \Phi_{\mathrm{can}}:=\sqrt{Z_H}\,\Phi=\frac{\Phi}{4\pi},
000204 | \]
000205 | and therefore
000206 | \begin{equation}
000207 | \label{eq:induced-einstein}
000208 | \Mpl^2=\frac{\chigeozero^2}{2\pi^2},
000209 | \qquad
000210 | \vgeo^2=Z_H\,\frac{\mu_\Phi^2(\chigeozero)}{\lambda_\Phi}.
000211 | \end{equation}
000212 | After \TFPTxref{thm:absolute-spectral-planck-closure}, the first identity is equivalently
000213 | \[
000214 | \frac{\Mpl^2}{\lambda_\Sigma^2}=\frac{\rho_\star}{2\pi^2}.
000215 | \]
000216 | \end{theorem}
000217 | 
000218 | \begin{proof}[Proof sketch]
000219 | The operator \Cref{eq:relative-bosonic-endomorphism} controls the seam-even residue expansion of
000220 | \Cref{def:canonical-residues}. Mellin expansion of the relative zeta pair fixes the order-one and
000221 | order-zero residues without any profile choice, so the Einstein term appears before the Higgs
000222 | kinetic, quartic, mixed-curvature, gauge-curvature, and $R^2$ terms. The coefficient of
000223 | $|D\Phi|^2$ is therefore the wave-function factor $Z_H=1/(16\pi^2)$. After the canonical rescaling
000224 | $\Phi_{\mathrm{can}}=\sqrt{Z_H}\,\Phi$, the Einstein and Higgs branches are read from one common