Mixed Correlators and Global-Symmetry Sectors

A mixed-correlator bootstrap keeps several external operators and all OPE pairings related by permutation in one system. Global symmetry first decomposes each channel into irreducible sectors; reflection positivity then acts on vectors of OPE coefficients and produces positive-semidefinite (PSD) matrices, not independent nonnegative numbers. The construction below is Euclidean, assumes a reflection-positive CFT and Hermitian scalar external operators, and makes the representation basis, spin parity, degeneracy treatment, and correlator closure explicit.

Required background. Crossing Equations and Positivity supplies the scalar crossing equation and its Hilbert-space positivity hypotheses. Multiplets, Invariants, and Selection Rules supplies irreducible sectors and invariant tensors. Helpful background. Representations, Intertwiners, and Invariants supplies projector and basis-change methods.

Projectors resolve the exchanged representation

Let a real scalar primary $\phi_i$ transform in the fundamental of $O(N)$, with

$$\langle\phi_i(x)\phi_j(0)\rangle =\frac{\delta_{ij}}{(x^2)^{\Delta_\phi}}.$$

For $N\geq2$, the tensor product splits into singlet ($S$), symmetric-traceless ($T$), and antisymmetric ($A$) sectors. With the $12\to34$ pairing, orthogonal projectors are

$$\begin{aligned} P_S^{ij;kl}&=\frac1N\delta_{ij}\delta_{kl},\\ P_T^{ij;kl}&=\frac12(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}) -\frac1N\delta_{ij}\delta_{kl},\\ P_A^{ij;kl}&=\frac12(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk}). \end{aligned}$$

They obey $P_RP_{R'}=\delta_{RR'}P_R$ and $P_S+P_T+P_A=\delta_{ik}\delta_{jl}$. Thus

$$\langle\phi_i\phi_j\phi_k\phi_l\rangle =\frac{1}{(x_{12}^2x_{34}^2)^{\Delta_\phi}} \sum_{R=S,T,A}P_R^{ij;kl}\,\mathcal G_R(u,v).$$

This decomposition is meaningful only after fixing the order of the four group indices and the normalization of every projector. Those choices also fix the crossing matrix.

To see the signs rather than importing them, set $A=\delta_{ij}\delta_{kl}$, $B=\delta_{ik}\delta_{jl}$, and $C=\delta_{il}\delta_{jk}$. In the $s$-channel basis, $A=NP_S$, $B=P_S+P_T+P_A$, and $C=P_S+P_T-P_A$. The projectors in the $13\to24$ pairing therefore satisfy

$$\begin{pmatrix}P_S^{(t)}\\ P_T^{(t)}\\ P_A^{(t)}\end{pmatrix} = \begin{pmatrix} \frac1N&\frac1N&\frac1N\\ \frac{(N-1)(N+2)}{2N}&\frac{N-2}{2N}&-\frac{N+2}{2N}\\ \frac{N-1}{2}&-\frac12&\frac12 \end{pmatrix} \begin{pmatrix}P_S\\ P_T\\ P_A\end{pmatrix}.$$

Equating the two decompositions, including the prefactor ratio $(u/v)^{\Delta_\phi}$ and the interchange $u\leftrightarrow v$, gives three coupled crossing equations. A transpose may appear if coefficient vectors rather than projector vectors are used; the displayed derivation fixes which convention is meant. Repeating the exchange twice must return the original invariant tensor, which is a quick sign check. This $O(N)$ construction and its bootstrap implementation are developed explicitly in Kos et al. 2016, §§2–3.

Exchange of the two identical bosons adds a second selection rule. $P_S$ and $P_T$ are symmetric under $i\leftrightarrow j$, whereas $P_A$ is antisymmetric; the spin-$\ell$ three-point structure contributes $(-1)^\ell$. Consequently the $S$ and $T$ channels contain even spins and the $A$ channel odd spins. For fermionic external operators, parity-odd structures, or nonidentical fields, this conclusion must be recomputed rather than reused.

Two external scalars produce matrix positivity

Consider two normalized Hermitian scalar primaries $\sigma$ and $\epsilon$. A useful concrete case has a $\mathbb Z_2$ symmetry with $\sigma$ odd and $\epsilon$ even. The permutation closure of their four-point functions contains

$$\langle\sigma\sigma\sigma\sigma\rangle, \quad \langle\sigma\sigma\epsilon\epsilon\rangle, \quad \langle\sigma\epsilon\sigma\epsilon\rangle, \quad \langle\epsilon\epsilon\epsilon\epsilon\rangle,$$

together with orderings obtained from them. An even exchanged primary $\mathcal O$ can occur in both $\sigma\times\sigma$ and $\epsilon\times\epsilon$, so its OPE data form the real vector

$$\boldsymbol\lambda_{\mathcal O} =\begin{pmatrix} \lambda_{\sigma\sigma\mathcal O}\\ \lambda_{\epsilon\epsilon\mathcal O} \end{pmatrix}.$$

Its contribution to the coupled system is the rank-one PSD matrix

$$M_{\mathcal O} =\boldsymbol\lambda_{\mathcal O} \boldsymbol\lambda_{\mathcal O}^{\mathsf T} = \begin{pmatrix} \lambda_{\sigma\sigma\mathcal O}^2& \lambda_{\sigma\sigma\mathcal O}\lambda_{\epsilon\epsilon\mathcal O}\\ \lambda_{\sigma\sigma\mathcal O}\lambda_{\epsilon\epsilon\mathcal O}& \lambda_{\epsilon\epsilon\mathcal O}^2 \end{pmatrix}\succeq0.$$

The off-diagonal entry need not be positive: changing the sign of one OPE coefficient changes it while preserving positive semidefiniteness. This is why replacing matrix positivity by entrywise positivity is incorrect.

Odd operators occur in $\sigma\times\epsilon$ and carry the nonnegative weight $\lambda_{\sigma\epsilon\mathcal O}^2$ in a real orthonormal convention. After the position-dependent prefactors and crossing permutations are combined into vector-valued kernels, the complete system has the schematic but precise cone structure

$$\sum_{\mathcal O\in\mathrm{even}} \boldsymbol\lambda_{\mathcal O}^{\mathsf T} \vec V_{+,\Delta,\ell} \boldsymbol\lambda_{\mathcal O} +\sum_{\mathcal O\in\mathrm{odd}} \lambda_{\sigma\epsilon\mathcal O}^{2} \vec V_{-,\Delta,\ell} +\vec V_{\mathbf1}=0.$$

Each component of $\vec V_+$ is a real symmetric $2\times2$ matrix assembled from conformal blocks; $\vec V_-$ is scalar. The identity belongs to the even sector with coefficients fixed by the two-point normalization. This is the matrix crossing system used for the three-dimensional Ising model in Kos et al. 2016, §§2.1–2.2.

Degeneracy and basis covariance

Suppose $m$ orthonormal primaries $\mathcal O_\alpha$ share the same dimension, spin, and internal representation. Blocks cannot resolve the label $\alpha$, and the observable coefficient is

$$M_{ab}=\sum_{\alpha=1}^{m} \lambda_{a\alpha}\lambda_{b\alpha}, \qquad a,b\in\{\sigma\sigma,\epsilon\epsilon\}.$$

For every real vector $c$, $c^{\mathsf T}Mc=\sum_\alpha(c_a\lambda_{a\alpha})^2\geq0$. An orthogonal rotation among the degenerate primaries changes individual OPE coefficients but leaves $M$ unchanged. In a nonorthonormal exchanged basis with positive Gram matrix $G$, the invariant expression is $M=\Lambda G^{-1}\Lambda^{\mathsf T}$; writing $\Lambda\Lambda^{\mathsf T}$ without first orthonormalizing is generally wrong.

A real change of the external-pair basis acts by congruence, $M\mapsto U M U^{\mathsf T}$, provided correlator prefactors and crossing kernels are transformed with the same $U$. Congruence preserves positive semidefiniteness. It does not preserve individual matrix entries, so any claim formulated in terms of the sign of an off-diagonal entry is basis dependent.

Closing the correlator system

A proposed mixed system is closed only if every permutation and OPE pairing of every retained correlator can be expressed in the same set of tensor structures and correlators. The following checks separate a genuine closed system from a consistent but weaker subsystem.

Ingredient	Required check	Consequence of omission
Internal representations	Decompose every external tensor product and normalize all projectors	An exchange sector or multiplicity may be absent
Crossing matrices	Re-expand projectors after each generator of the permutation group and verify the group relations	Relative signs or dimensions can be wrong
Spin and parity	Combine exchange symmetry of the tensor structure with $(-1)^\ell$ and statistics	Forbidden spins can enter the sum
Degenerate primaries	Sum OPE outer products at fixed quantum numbers	Basis-dependent coefficients may be mistaken for observables
External pairs	Include every pair connected by the chosen permutations	Matrix blocks can be truncated inconsistently
Positivity hypotheses	Fix Hermiticity, reflection positivity, and a positive two-point metric	The PSD cone may not exist

Omitting a correlator is sometimes deliberate. The remaining crossing equations can still yield valid bounds because they impose fewer necessary conditions, but they cannot support a claim that relies on the omitted channel. Conversely, adding correlators without adding the corresponding tensor structures does not strengthen the system; it makes the equations incomplete.

The strongest defensible conclusion from a numerical or analytic mixed-system argument must therefore state the external operator set, global group and real form, representation multiplicities, parity assumptions, spacetime dimension, spectral gaps, and which crossing permutations were enforced. Positivity further requires a reflection-positive Euclidean theory, Hermitian external operators (or an explicitly conjugate basis), and positive two-point norms. General reviews of mixed and global-symmetry bootstrap systems emphasize these hypotheses in Poland, Rychkov, and Vichi 2019, §§III–IV.

Check: one operator versus a degenerate family

For one even primary, show that $M=\boldsymbol\lambda\boldsymbol\lambda^{\mathsf T}$ has determinant zero and is PSD. Then explain how a degenerate family can have positive determinant.

Solution

The eigenvalues of a rank-one outer product are $\lVert\boldsymbol\lambda\rVert^2$ and $0$, so $M\succeq0$ and $\det M=0$. For a family, $M=\sum_\alpha\boldsymbol\lambda_\alpha\boldsymbol\lambda_\alpha^{\mathsf T}$. It is still PSD, but it has positive determinant whenever at least two OPE vectors are linearly independent. Thus a full-rank numerical matrix at fixed $(\Delta,\ell,R)$ signals unresolved degeneracy, not a violation of positivity.

Handoff

The same matrix logic extends to spinning external operators after each three-point tensor-structure label is treated as an additional OPE-vector index; Spinning Correlators and Tensor Structures supplies those structures. Momentum-Space Correlators and Conformal Ward Identities next explains how distributional and anomaly terms modify Ward identities after Fourier transformation.

References

• Kos, F., Poland, D., Simmons-Duffin, D., and Vichi, A. (2016). “Precision Islands in the Ising and $O(N)$ Models.” Journal of High Energy Physics 08 (2016), 036. doi:10.1007/JHEP08(2016)036. Open version.
• Poland, D., Rychkov, S., and Vichi, A. (2019). “The Conformal Bootstrap: Theory, Numerical Techniques, and Applications.” Reviews of Modern Physics 91, 015002. doi:10.1103/RevModPhys.91.015002. Open version.