TFPT scientific source excerpt
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000115 | \begin{theorem}[Admissible 1PI graph expansion]
000116 | \label{thm:admissible-1pi-graph-expansion}
000117 | Around any admissible background field $\phi$, the effective action admits the loop expansion
000118 | \[
000119 | \Gamma_k[\phi]
000120 | =
000121 | S_{\mathrm{adm}}[\phi]
000122 | +\frac{\hbar}{2}\operatorname{STr}\log\!\bigl(S_{\mathrm{adm}}^{(2)}[\phi]+R_k\bigr)
000123 | +\sum_{\ell\ge 2}\hbar^\ell\Gamma_{\ell,k}^{\mathrm{1PI}}[\phi].
000124 | \]
000125 | Each coefficient $\Gamma_{\ell,k}^{\mathrm{1PI}}$ is the sum of admissible one-particle-irreducible
000126 | Feynman graphs with propagator
000127 | \[
000128 | G_{k,\phi}:=\bigl(S_{\mathrm{adm}}^{(2)}[\phi]+R_k\bigr)^{-1}
000129 | \]
000130 | and vertices given by the higher functional derivatives $S_{\mathrm{adm}}^{(n)}[\phi]$.
000131 | \end{theorem}
000132 | 
000133 | \begin{proof}
000134 | Set $\varphi=\phi+\sqrt{\hbar}\eta$ and expand
000135 | \[
000136 | S_{\mathrm{adm}}[\phi+\sqrt{\hbar}\eta]
000137 | =
000138 | S_{\mathrm{adm}}[\phi]
000139 | +\frac{\hbar}{2}\langle \eta,S_{\mathrm{adm}}^{(2)}[\phi]\eta\rangle
000140 | +\sum_{n\ge 3}\frac{\hbar^{n/2}}{n!}S_{\mathrm{adm}}^{(n)}[\phi](\eta^n).
000141 | \]
000142 | The regulated functional integral therefore becomes a Gaussian measure with covariance
000143 | $G_{k,\phi}$ dressed by higher interaction vertices. Taking the logarithm produces the connected
000144 | diagram expansion, and the Legendre transform removes one-particle-reducible pieces. The resulting
000145 | coefficients are exactly the admissible 1PI graph sums recorded above.
000146 | \end{proof}
000147 | 
000148 | \begin{definition}[Canonical renormalized 1PI package]
000149 | \label{def:renormalized-low-energy-package}
000150 | Let $\GrenTFPT$ denote the category whose objects are renormalized low-energy packages
000151 | \[
000152 | \mathcal G=
000153 | \bigl(
000154 | \{\Sigma_f^{\mathrm{TFPT}}(p)\}_f,\,
000155 | G_F^{\mathrm{TFPT}},\,
000156 | \mathcal R_{\det},\,
000157 | \mathcal B_{\mathrm{cos}},\,
000158 | \mathcal A_{\mathrm{pole}}
000159 | \bigr)
000160 | \]
000161 | built canonically from the rigid object $\Tstar$. They are extracted from the infrared limit of the
000162 | exact admissible flow,
000163 | \[
000164 | \Gamma_{\mathrm{TFPT}}^{\mathrm{ren}}[\Phi_c]
000165 | :=
000166 | \lim_{k\to0}\Gamma_k[\Phi_c],
000167 | \]
000168 | equivalently the Legendre transform of the retained relative generating functional after removing
000169 | the infrared regulator. Here $\mathcal A_{\mathrm{pole}}$ collects the dressed inverse propagators,
000170 | $\mathcal R_{\det}$ the determinant-line response kernels, and $\mathcal B_{\mathrm{cos}}$ the
000171 | FRW / reheating / Boltzmann interface data on the closed branch. Morphisms are RG-compatible field
000172 | reparameterizations preserving the branch-fixed pole and response data.