Euclidean radial OPE convergence in a unitary CFT

Given: The selected radial data, a positive Hilbert inner product linear in its first argument, a complete orthonormal energy eigenfamily for the selected channel, finite-norm states, the radial cut/composition identity, coefficient/state compatibility, and separated insertions. For the separate four-point clause, also assume the normalized real scalar, positive discrete primary spectrum, identity gap, nonnegative reflection-positive spectral coefficients, crossed-channel identity dominance, and the stated source normalization.

Radial quantization of a plane CFT: $\tau =\log \mathrm{\mid }z\mathrm{\mid }$ is Euclidean radial time; larger radii are to the left in a radially ordered product.

Radial state/operator data in a selected CFT model: $\mathcal{H}$ is a complex Hilbert space with inner product linear in the first argument; radial insertion states, their norms, and the declared reflection/adjoint are model data.

Radially ordered operator product expansion: the Hilbert or weak-correlator OPE interpretation, channel family, branch data, and global-conformal block grouping are declared separately.

Parseval equivalences for an orthonormal family: under Countable Choice, completeness of an orthonormal family in a real or complex Hilbert space implies finite-subset-net convergence and Parseval's identity.

Cauchy–Schwarz: $\mathrm{\mid }⟨x,y⟩\mathrm{\mid }\le \mathrm{\parallel }x\mathrm{\parallel }\mathrm{\parallel }y\mathrm{\parallel }$, with equality exactly for dependent pairs: $\mathrm{\mid }⟨u,v⟩\mathrm{\mid }\le \mathrm{\parallel }u\mathrm{\parallel }\mathrm{\parallel }v\mathrm{\parallel }$ in a real or complex inner-product space.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$): ${\mathrm{A}\mathrm{C}}_{\omega }$ selects from a countable family of nonempty sets.

A positive discrete Laplace asymptotic determines cumulative growth: if $L(s)={\sum }_n{a}_n{e}^{-s{E}_n}\sim s^{-\alpha }$ with positive discrete weights, then ${\sum }_{E_n\le E}{a}_n\sim E^{\alpha }\mathrm{/}\mathrm{\Gamma }(\alpha +1)$.

Euler's Gamma function on the right half-plane: for $q>0$, $\mathrm{\Gamma }(q)={\int }_0^{\mathrm{\infty }}{t}^{q-1}{e}^{-t}dt$, and this integral is finite.

The channel sector ${\mathcal{H}}_{\ast }$ is a closed linear subspace of $\mathcal{H}$ with its inherited Hilbert structure. It has a complete orthonormal family $(e_{\alpha }{)}_{\alpha \in I}$ of eigenvectors of the model's radial energy operator $K$: for each $\alpha$, $e_{\alpha }\in D(K)$ and $K{e}_{\alpha }=E_{\alpha }{e}_{\alpha }$ for some $E_{\alpha }\ge 0$. No continuous-spectrum expansion is included.

For the chosen separation circle, the fused-pair state $\mathrm{\Phi }$ and reflected complementary state $\mathrm{\Psi }$ belong to ${\mathcal{H}}_{\ast }$ and have finite Hilbert norm.

The radial cut/composition rule for this selected model identifies the correlation functional with the Hilbert overlap $⟨\mathrm{\Psi },\mathrm{\Phi }⟩$ on the separating circle. This is an adopted gluing axiom for the chosen correlation functional.

The selected OPE's complete label set ${\mathrm{\Lambda }}_{\mathrm{O}\mathrm{P}\mathrm{E}}$ includes every descendant, degeneracy, and multiplicity label and is bijective with the complete eigenfamily $(e_{\alpha }{)}_{\alpha \in I}$ by a map $b_{\mathrm{O}\mathrm{P}\mathrm{E}}:{\mathrm{\Lambda }}_{\mathrm{O}\mathrm{P}\mathrm{E}}\to I$. The declared finite-subset partial-sum net is indexed by all finite subsets of ${\mathrm{\Lambda }}_{\mathrm{O}\mathrm{P}\mathrm{E}}$ and maps under $b_{\mathrm{O}\mathrm{P}\mathrm{E}}$ to the full finite-subset net over $I$, with no omitted or additional terms. If $c_{\alpha }:=⟨\mathrm{\Phi },e_{\alpha }⟩$ and $d_{\alpha }:=⟨\mathrm{\Psi },e_{\alpha }⟩$, the term for $\lambda \in {\mathrm{\Lambda }}_{\mathrm{O}\mathrm{P}\mathrm{E}}$ is $d_{b_{\mathrm{O}\mathrm{P}\mathrm{E}}(\lambda )}\stackrel{‾}{c_{b_{\mathrm{O}\mathrm{P}\mathrm{E}}(\lambda )}}$ under the selected radial reflection/adjoint convention. This exhaustive correspondence and compatibility are model data or separately proved; completeness alone does not imply them.

In the separated-insertion clause, $x,y,z_1,\dots ,z_n$ are pairwise distinct, $n\ge 1$, and satisfy the displayed strict separation inequality, so there is a circular radial slice separating the fused pair from the complementary insertions.

In the four-point clause, a fixed countable enumeration of all descendant labels $j$, including degeneracies and multiplicities, has real nonnegative energy $E_j={\mathrm{\Delta }}_j+n_j$, with primary dimension ${\mathrm{\Delta }}_j\ge 0$ and integer descendant level $n_j\ge 0$. The selected model supplies a closed Hilbert sector ${\mathcal{H}}_4\subseteq \mathcal{H}$ and a family of linear isometries $R_{\theta }:{\mathcal{H}}_4\to {\mathcal{H}}_4$ indexed by $\theta \in \mathbb{R}\mathrm{/}(2\pi \mathbb{Z})$, with $R_0=I$ and $R_{\theta +\phi }=R_{\theta }{R}_{\phi }$. For each $j$, it supplies a finite-norm vector $u_{j,0}\in {\mathcal{H}}_4$ and a nonnegative weight ${\lambda }_j$. Set $u_{j,\theta }:=R_{\theta }{u}_{j,0}$, $b_j(\theta ):={\lambda }_j⟨u_{j,0},u_{j,\theta }⟩$, and $a_j:=b_j(0)={\lambda }_j\mathrm{\parallel }{u}_{j,0}{\mathrm{\parallel }}^2\ge 0$. Cauchy–Schwarz and $\mathrm{\parallel }{u}_{j,\theta }\mathrm{\parallel }=\mathrm{\parallel }{u}_{j,0}\mathrm{\parallel }$ give $\mathrm{\mid }{b}_j(\theta )\mathrm{\mid }\le a_j$. The term at $\rho =r{e}^{i\theta }$ is $b_j(\theta )r^{E_j}$, with $r=\mathrm{\mid }\rho \mathrm{\mid }$ and all angular dependence in the rotation overlap $b_j(\theta )$, so its modulus is at most $a_j{r}^{E_j}$. For real $0<r<1$, the angle-zero overlap is $L(\beta )={\sum }_j{a}_j{e}^{-\beta E_j}$, where $r=e^{-\beta }$, and this series is finite for each $\beta >0$. These are explicit spectral, rotation, and reflection-positivity data of the selected model.

In the two-pair geometry with $r\in (0,1)$, the crossed-channel OPE has a unit-normalized identity term and its remaining contribution is $o((1-r{)}^{-4{\mathrm{\Delta }}_{\varphi }})$ as $r\uparrow 1$. Thus the unnormalized overlap satisfies $L(-\log r)\sim (1-r{)}^{-4{\mathrm{\Delta }}_{\varphi }}$. This identity-dominance asymptotic is an explicit selected-model input, not a consequence of the gap alone or a theorem about every unitary CFT.

For the normalized conformal-block sum, use the source convention $⟨\varphi (x_1)\varphi (x_2)\varphi (x_3)\varphi (x_4)⟩=\frac{g(u,v)}{\mathrm{\mid }{x}_{12}{\mathrm{\mid }}^{2{\mathrm{\Delta }}_{\varphi }}\mathrm{\mid }{x}_{34}{\mathrm{\mid }}^{2{\mathrm{\Delta }}_{\varphi }}},$ where $u=z\bar{z}$ and $v=(1-z)(1-\bar{z})$, and the two-pair positions are $x_1=\rho ,x_2=-\rho ,x_3=-1,x_4=1$. The global-conformal block tail is this same descendant tail multiplied by $\mathrm{\mid }{x}_{12}{\mathrm{\mid }}^{2{\mathrm{\Delta }}_{\varphi }}\mathrm{\mid }{x}_{34}{\mathrm{\mid }}^{2{\mathrm{\Delta }}_{\varphi }}$, grouped by intermediate primary dimension.