TFPT scientific source excerpt
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000240 | \]
000241 | so Newton's constant is read as a boundary-spectral ratio rather than as an isolated SI datum.
000242 | \end{remark}
000243 | 
000244 | \begin{lemma}[Two-sheet seam-even zero-mode normalization]
000245 | \label{lem:two-sheet-zero-mode-normalization}
000246 | If the seam-even Higgs zero mode is normalized across both sheets of the oriented double cover, then its branch weight is
000247 | \[
000248 | w_H
000249 | =
000250 | 2\,\gcar\,\beta_{\mathrm{rad}}^2
000251 | \exp\!\left[-\frac{\alpha^{-1}(0)+\delta_{\mathrm{ph}}}{5}\right].
000252 | \]
000253 | With
000254 | \[
000255 | \mu_\Phi^2(\chigeozero)=\frac{\chigeozero^2}{8\pi^2}w_H^2,
000256 | \qquad
000257 | \lambda_\Phi=\frac{1}{16\pi^2},
000258 | \]
000259 | the canonical Higgs vacuum expectation value is
000260 | \begin{equation}
000261 | \label{eq:carrier-dressed-v}
000262 | \vgeo
000263 | =
000264 | \Mpl\,\gcar\,\beta_{\mathrm{rad}}^2\,
000265 | \exp\!\left[-\frac{\alpha^{-1}(0)+\delta_{\mathrm{ph}}}{5}\right].
000266 | \end{equation}
000267 | \end{lemma}
000268 | 
000269 | \begin{proof}[Proof sketch]
000270 | The two-sheet seam-even normalization doubles the unnormalized zero-mode weight and therefore produces the stated $w_H$. Since the canonical Higgs field is $\Phi_{\mathrm{can}}=\sqrt{Z_H}\,\Phi$, its vacuum relation is
000271 | \[
000272 | \vgeo^2=\frac{Z_H\mu_\Phi^2(\chigeozero)}{\lambda_\Phi}
000273 | =
000274 | \frac{\chigeozero^2}{8\pi^2}w_H^2.
000275 | \]
000276 | Using $\Mpl^2=\chigeozero^2/(2\pi^2)$ from \TFPTxref{prop:canonical-bosonic-normalization} gives $\vgeo=\Mpl w_H/2$, hence \Cref{eq:carrier-dressed-v}.