Loop Integral Bases and Multi-Trace Mixing

Loop correlators are most useful when expanded in a spectral or integral basis that exposes intermediate conformal families. Degenerate multi-trace operators then require a mixing matrix, not a single anomalous dimension read from one correlator. Scheme-independent eigenvalues and properly normalized OPE projections emerge only after the two-point Gram matrix and all available external probes are included.

Required background. Loop Witten diagrams and bulk EFT renormalization supplies the finite loop correlator.

Helpful background. Operator mixing matrices supplies the algebra. Crossing kernels and conformal 6j symbols supplies changes of channel.

Spectral bases and degenerate subspaces

The split representation replaces each internal bulk propagator by boundary harmonic functions and a spectral integral. Gluing vertices then yields conformal partial waves, so poles in the spectral parameter identify intermediate dimensions. Integral bases built from $D$-functions are efficient in position space, while spectral bases make cuts and mixing transparent; they are complementary rather than unique.

Let $\{\mathcal O_a\}$ be degenerate double-trace primaries at leading order. Their two-point functions and dilatation matrix are

$$\langle\mathcal O_a\mathcal O_b\rangle=G_{ab}, \qquad D_{ab}=\Delta_0G_{ab}+N^{-2}\Gamma_{ab}+\cdots.$$

Physical first-order anomalous dimensions solve the generalized eigenproblem $\Gamma v=\gamma Gv$. A coefficient extracted from one four-point function is only a weighted average over the vectors to which that external pair couples.

First application: a two-operator mixing sector

Suppose two double-trace operators share $(\Delta_0,\ell)$ and $G=\mathbf1$, with

$$\Gamma=\begin{pmatrix}a&b\\ b&c\end{pmatrix}.$$

The eigenvalues are $(a+c\pm\sqrt{(a-c)^2+4b^2})/2$. A correlator whose external OPE vector is $(1,0)$ measures moments such as $a$ and $(\Gamma^2)_{11}$, not either eigenvalue unless $b=0$. A second independent correlator supplies another projection and permits diagonalization. This logic is essential in supergravity correlators with many degenerate double traces Aprile et al. 2018.

The spectral residue must be divided by the norm $v^TGv$ before it is called an OPE coefficient. Contact counterterms can shift analytic pieces but cannot change a correctly isolated nondegenerate cut residue.

Adversarial control: one correlator, one claimed eigenvalue

Rotate the degenerate basis by an orthogonal matrix. The coefficient of a named basis operator changes, while the eigenvalues of $G^{-1}\Gamma$ and the full correlator do not. If a reported “anomalous dimension” changes under this rotation, it was an unresolved average. Numerical diagonalization without the Gram metric produces the same error when $G\ne\mathbf1$.

The evidence ceiling is the spectrum and OPE projections within the probed degenerate sector and perturbative order. Unmeasured external operators can leave directions in $\Gamma$ undetermined. Bulk field redefinitions identify a distinct off-shell ambiguity that persists even after spectral mixing is solved.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Aharony, O., Alday, L. F., Bissi, A., and Perlmutter, E. (2017), “Loops in AdS from Conformal Field Theory,” Journal of High Energy Physics 2017(07), 036. arXiv:1612.03891.
• Aprile, F., Drummond, J. M., Heslop, P., and Paul, H. (2018), “Quantum Gravity from Conformal Field Theory,” Journal of High Energy Physics 2018(01), 035. arXiv:1706.02822.