Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

==== 1. Compute total spectral coefficient ====

A2tot=∑isiA2(i)(bulk, boundary; σ,η,R).A_2^{\rm tot}=\sum_i s_i A_2^{(i)}(\text{bulk, boundary};\,\sigma,\eta,R).A2tot=i∑siA2(i)(bulk, boundary;σ,η,R).
Include ghosts (subtract scalar ghost contribution) and vector gauge-fixing corrections (physical polarizations only).
# Match to induced 1/G1/G1/G: identify coefficient multiplying ∫d4xgR\int d^4x\sqrt{g}R∫d4xgR in WWW. With a sharp cutoff Λ\LambdaΛ,

1Gind=c3ℏ  κind  Λ2withκind∝ℏc3⋅A2tot×(numerical factor from t-integral).\frac{1}{G_{\rm ind}} = \frac{c^3}{\hbar}\; \kappa_{\rm ind}\; \Lambda^2 \quad\text{with}\quad
\kappa_{\rm ind} \propto \frac{\hbar}{c^3}\cdot A_2^{\rm tot}\times (\text{numerical factor from }t\text{-integral}).Gind1=ℏc3κindΛ2withκind∝c3ℏ⋅A2tot×(numerical factor from t-integral).
(We keep ℏ\hbarℏ explicit; the algebraic prefactors will come out of the integration.)
# Compute κind\kappa_{\rm ind}κind symbolically (function of Nvec,Nferm,σ,η,RN_{\rm vec},N_{\rm ferm},\sigma,\eta,RNvec,Nferm,σ,η,R) and then numerically for chosen Λ\LambdaΛ.