Weakly Coupled Bulk Fields from Connected Correlators

Suppressed connected correlators support a perturbative bulk-field description only when the relevant single-trace operators are normalized, spectrally isolated, and uniformly few enough that weighted sums over species remain small. In a matrix large-$N$ sector, a normalized three-point coefficient fixes the scaling of a cubic interaction at leading order. The product of two such coefficients must then reproduce the corresponding exchange order in the connected four-point function. This is a conditional perturbative inference, not a theorem that factorization alone produces a finite local bulk theory.

Required background. Large-N Factorization and Classical Bulk Scaling gives the connected hierarchy. Single-Trace, Multi-Trace, and Collective-Field Organization for Bulk Criteria identifies the candidate one- and multiparticle sectors.

Helpful background. Scalar Two- and Three-Point Functions fixes CFT normalization. Power Counting and Predictive Order explains how interaction order becomes a prediction.

From a normalized three-point function to a cubic vertex

Begin with fixed-length scalar single-trace primaries in Euclidean signature, where “fixed length” means that the number of constituent matrix fields does not grow with $N$. Normalize them by

$$\langle \mathcal O_i(x)\mathcal O_j(0)\rangle =\frac{\delta_{ij}}{\lvert x\rvert^{2\Delta_i}},$$

and define $C_{ijk}$ through

$$\langle \mathcal O_i(x_1)\mathcal O_j(x_2)\mathcal O_k(x_3)\rangle = \frac{C_{ijk}} {\lvert x_{12}\rvert^{\Delta_i+\Delta_j-\Delta_k} \lvert x_{23}\rvert^{\Delta_j+\Delta_k-\Delta_i} \lvert x_{31}\rvert^{\Delta_k+\Delta_i-\Delta_j}}.$$

For fixed-length, unit-normalized single-trace primaries in an orientable closed adjoint or matrix expansion, at fixed couplings and kinematics,

$$C_{ijk}=\frac{c_{ijk}}{N}+O(N^{-3}), \qquad c_{ijk}=O(1).$$

The $N^{-3}$ correction assumes the closed orientable genus sequence. Normalized multi-traces can instead have order-one coefficients, while vector, tensor, fundamental-boundary, nonorientable, and correlated limits can change the leading or subleading powers. The expansion family and operator class must therefore be stated rather than inferred from the symbol $N$.

Now choose canonically normalized scalar fields in Euclidean AdS$_{d+1}$,

$$\begin{aligned} S_E={}&\frac12\sum_i\int d^{d+1}X\,\sqrt g\, \bigl[(\nabla\varphi_i)^2+m_i^2\varphi_i^2\bigr] \\ &+\frac1{3!}\sum_{ijk}g_{ijk} \int d^{d+1}X\,\sqrt g\, \varphi_i\varphi_j\varphi_k+\cdots . \end{aligned}$$

For the chosen scalar quantization—standard or, when allowed, alternate—the mass and selected boundary dimension obey

$$m_i^2L^2=\Delta_i(\Delta_i-d).$$

A canonical bulk scalar has mass dimension $(d-1)/2$, so

$$[g_{ijk}] =(d+1)-\frac32(d-1) =\frac{5-d}{2}.$$

The dimensionless cubic coupling is therefore

$$\lambda_{ijk}:=g_{ijk}L^{(5-d)/2}.$$

Let $\lambda^{\mathrm{eff}}_{ijk}$ denote this $\lambda_{ijk}$ when the displayed nonderivative interaction is the sole EOM-reduced scalar cubic. If derivative scalar cubics are retained, $\lambda^{\mathrm{eff}}_{ijk}$ is the on-shell linear combination that survives integrations by parts, equations of motion, and the declared boundary prescription. A single scalar three-point structure determines this effective combination, not every off-shell cubic separately.

For generic nonextremal dimensions in the direct-integral convergence domain,

$$\Delta_i+\Delta_j-\Delta_k>0 \quad\text{cyclically}, \qquad \Delta_i+\Delta_j+\Delta_k>d,$$

fix the normalized bulk-to-boundary propagators by the unit CFT two-point functions and evaluate the tree-level integral of their product. It gives a finite dimensionless factor $\mathcal A_{ijk}$ such that

$$C_{ijk} =\mathcal A_{ijk}(d,\Delta_i,\Delta_j,\Delta_k)\, \lambda^{\mathrm{eff}}_{ijk}, \qquad \lambda^{\mathrm{eff}}_{ijk}=O(N^{-1}).$$

The action-sign convention is included in $\mathcal A_{ijk}$. Outside the direct convergence domain, $\mathcal A_{ijk}$ means the renormalized analytic continuation of the triple bulk-to-boundary integral in the chosen scheme. This is a reproducible numerical matching relation once the operator basis, kinetic terms, quantization, and boundary prescription are fixed; it is not a convention-free equality. The generic scalar integral is evaluated in Freedman et al. 1999, § 2, eqs. (18)–(25), printed pp. 6–8, PDF. Derivative scalar cubics change the dimension-dependent matching and powers of $L$ while retaining the one separated-point scalar structure; spinning fields can introduce additional tensor structures. The normalization-first workflow applies in each case.

Extremal configurations are exceptional. If, for example, $\Delta_i=\Delta_j+\Delta_k$, a local cubic coefficient can vanish while the AdS integral diverges, leaving a finite correlator after analytic continuation; boundary terms and single-/multi-trace mixing can enter at the same order. A nonzero extremal $C_{ijk}$ therefore need not imply a finite nonzero local $g_{ijk}$ by the generic formula D’Hoker et al. 1999, § 1, printed pp. 2–4, PDF.

The four-point consistency check

Let $\mathcal G_{1234}(u,v)$ be the reduced four-point correlator after its kinematic prefactor is removed. One nondegenerate single-trace primary $\mathcal O_a$ contributes in a chosen OPE channel as

$$\left.\mathcal G_{1234}^{\mathrm{conn}}(u,v)\right|_{a} =C_{12a}C_{34a}\, G_{\Delta_a,J_a}(u,v) =O(N^{-2})$$

at fixed dimensions and cross-ratios. In the bulk, the corresponding tree Witten diagram contains two cubic vertices, so its single-trace residue scales as

$$\lambda^{\mathrm{eff}}_{12a}\lambda^{\mathrm{eff}}_{34a} =O(N^{-2}).$$

A Witten exchange diagram is not one conformal block: its block decomposition contains the exchanged single-trace block together with double-trace towers required by AdS propagation and crossing. The single-trace pole and residue provide the clean order check; the full Witten diagram is the bulk object. Fitzpatrick and Kaplan 2013, § 2.4, printed pp. 14–15, PDF formulate the normalized single-trace input and perturbative reconstruction.

For a nonderivative canonical scalar interaction $g_4\varphi^4/4!$, $[g_4]=3-d$ and $\lambda_4:=g_4L^{3-d}$ is dimensionless. A four-scalar vertex with $r$ derivatives instead has $[g_{4,r}]=3-d-r$ and dimensionless coefficient $g_{4,r}L^{3-d-r}$. A local quartic vertex can contribute at the same order, $\lambda_4=O(N^{-2})$, or at a more suppressed order. It changes contact and double-trace data without changing the conclusion that two $1/N$ cubics give an $N^{-2}$ exchange residue. Crossing symmetry constrains the sum over exchange and contact contributions in all channels.

The off-shell split is partly basis dependent. A local redefinition

$$\varphi_i\longrightarrow \varphi_i+a_i{}^{jk}\varphi_j\varphi_k+\cdots$$

moves interactions proportional to equations of motion between cubic, derivative, and contact terms. When dimensions and quantum numbers overlap, the boundary description has the parallel freedom to rotate a nominal single trace into normalized multi-traces. For an isolated nondegenerate primary, its pole location and residue are invariant correlator data. Inside an exactly degenerate subspace, individual OPE coefficients and assigned residues rotate with the orthonormal basis; the invariant objects are the full correlator and the projected residue or OPE matrix at the shared pole.

Uniformity over species and kinematics

With $M(N)$ possible intermediate single-trace primaries, a channel contribution is

$$\left.\mathcal G^{\mathrm{conn}}_{1234}(u,v)\right|_{\mathcal S} = \sum_{a=1}^{M(N)} C_{12a}C_{34a}\, G_{\Delta_a,J_a}(u,v).$$

Termwise $O(N^{-2})$ counting is not a uniform bound on this sum. On a compact kinematic set $\mathcal K$, a sufficient bound on the full coefficients is

$$\sum_{a=1}^{M(N)} N^2\lvert C_{12a}(N)C_{34a}(N)\rvert \sup_{(u,v)\in\mathcal K} \lvert G_{\Delta_a,J_a}(u,v)\rvert \le K,$$

with $K$ independent of $N$. A fixed number of uniformly bounded terms satisfies this automatically. Replacing the full coefficients by their leading $c_{ija}/N$ terms additionally requires a genus expansion whose remainders are uniform over the entire $N$-dependent family. A growing spectrum requires an actual weighted-sum estimate.

For a Hermitian identical external scalar in a unitary CFT, the squared OPE coefficients are nonnegative. Define their exact mean rescaled weight by

$$\overline w_N :=\frac1{M(N)}\sum_{a=1}^{M(N)} N^2 C_{\mathcal O\mathcal Oa}^{2}.$$

Then the unweighted cumulative OPE weight is the identity

$$W_M :=\sum_{a=1}^{M(N)} C_{\mathcal O\mathcal Oa}^{2} =\frac{M(N)\,\overline w_N}{N^2}.$$

If $\overline w_N=\Theta(1)$ and $M(N)\sim N^\alpha$, then $W_M\sim N^{\alpha-2}$. At $M\sim N^2$, $W_M$ is order one. The block-weighted correlator can likewise be order one only if its weighted block average stays nonzero and $O(1)$; an upper bound on blocks supplies only an upper ceiling. A particularly transparent lower-bound test uses a reflection-positive radial configuration such as $0<z=\bar z<1$, where the contributing block data have controlled sign.

A bulk loop has the same potential species hazard, but cancellations and decoupling must be retained. Let $\lambda_a=\ell_a/N$ denote the relevant dimensionless vertex involving species $a$, let $m_a$ be its mass, and let $E$ be the external bulk energy. For a loop insertion with one independently summed species label and two additional $1/N$ vertices, relative to the corresponding order-one tree or kinetic term,

$$\epsilon_{\mathrm{loop}} =\frac1{N^2}\sum_{a=1}^{M(N)} \ell_a^2F_a(m_aL,EL;\Lambda L,\mu L), \qquad \lvert\epsilon_{\mathrm{loop}}\rvert \le \frac{M(N)}{N^2} \sup_a\lvert\ell_a^2F_a\rvert .$$

Here $\Lambda$ is the EFT cutoff, $\mu$ is the renormalization scale, and $F_a$ is the regulated and renormalized dimensionless loop function in the declared scheme. The correction is $\Theta(M/N^2)$ only if the species-averaged weighted loop function approaches a nonzero order-one limit. Different flavor tensors can remove the enhancement or create more independent sums. In every case, “each vertex is small” does not by itself imply a uniformly weak expansion.

Near-degenerate mixing can erase particle labels

Growing multiplicity and near-degeneracy are different failures. To isolate the second, take one normalized single trace $\mathcal O_s$ and one normalized double trace or other primary $\mathcal O_m$ with the same quantum numbers. In their leading orthonormal basis, model the dilatation operator by

$$\mathsf D= \begin{pmatrix} \Delta_s & \mu/N \\ \mu/N & \Delta_m \end{pmatrix}, \qquad \tan(2\theta) =\frac{2\mu/N}{\Delta_s-\Delta_m}.$$

The exact eigenvalues are

$$\Delta_\pm =\frac{\Delta_s+\Delta_m}{2} \pm \sqrt{ \frac{(\Delta_s-\Delta_m)^2}{4} +\frac{\mu^2}{N^2} }.$$

Write the unmixed level spacing as $\lvert\Delta_s-\Delta_m\rvert\sim N^{-\beta}$ with $\mu=O(1)$. Then the mixing-to-spacing ratio scales as $N^{\beta-1}$:

• for $\beta<1$, the mixing angle vanishes and the original particle label is perturbatively stable;
• for $\beta=1$, mixing is generically order one;
• for $\beta>1$, the off-diagonal term dominates and the eigenstates approach order-one mixtures.

The two-state model diagnoses a collision, but a finite pair can still be diagonalized into a two-field EFT. To test a finite-field truncation itself, take

$$M(N)=N^\alpha$$

primaries with the same quantum numbers in a shrinking dimension band of width $B_N\sim N^{-\beta}$. Give each a uniformly bounded normalized coefficient $C_{\mathcal O\mathcal Oa}=c_a/N$ and a nonzero nearest-neighbor mixing matrix element $\mu_a/N$, with $c_a$ and $\mu_a$ of order one. The typical level spacing and local mixing-to-spacing ratio are then

$$\delta\Delta_{\mathrm{nn}} \sim\frac{B_N}{M(N)} \sim N^{-(\alpha+\beta)}, \qquad \frac{\mu_a/N}{\delta\Delta_{\mathrm{nn}}} \sim N^{\alpha+\beta-1}.$$

If $\alpha+\beta\ge1$, neighboring states reorganize by order-one angles. Because $M(N)\to\infty$, no $N$-independent finite list of the original or diagonalized fields contains the whole mixed band. The especially sharp choice $\alpha=\beta=1/2$ has

$$M=N^{1/2}, \qquad \delta\Delta_{\mathrm{nn}}\sim N^{-1}, \qquad W_M=O(N^{-3/2}).$$

Thus every three-point coefficient is $O(N^{-1})$ and even the unweighted cumulative OPE weight vanishes, yet adjacent mixing is order one and the number of relevant eigenstates diverges. This is the requested dense near-degenerate adversarial fixture: small individual correlators no longer justify a simple finite-field bulk EFT because both uniform spectral isolation and an $N$-independent truncation fail. The strongest surviving statement is that the fully diagonalized band may admit a perturbative collective, continuum, or string-like description after a uniform resummation; no particular such completion follows from the data given here.

What the correlators do and do not establish

A perturbative AdS EFT needs more than small connected correlators. Below a declared gap, the CFT should admit an approximate Fock space generated by finitely many low-dimension single traces, and its Mellin amplitudes should have the polynomial behavior expected of an effective derivative expansion Fitzpatrick and Kaplan 2013, §§ 1 and 6, printed pp. 1–4 and 31–32, PDF. Factorization supplies an interaction parameter, spectral isolation supplies candidate field content, and polynomial boundedness supplies locality information. These are independent tests. Heemskerk et al. 2009, §§ 2.4, 4, and 7–8 establish a leading nontrivial $O(N^{-2})$ scalar four-point locality result under their stated large-$N$ and large-gap assumptions, not a universal order-by-order or nonperturbative reconstruction theorem.

For a fixed normalized matrix-sector operator set, let $[\mathcal G_4]_{G_{\Delta_a,J_a}}$ denote the coefficient of the indicated conformal block. The controlled inference is

$$\begin{aligned} C_{12a},C_{34a}=O(N^{-1}) &\Longrightarrow \lambda^{\mathrm{eff}}_{12a}, \lambda^{\mathrm{eff}}_{34a}=O(N^{-1}), \\ \lambda^{\mathrm{eff}}_{12a}, \lambda^{\mathrm{eff}}_{34a}=O(N^{-1}) &\Longrightarrow \bigl[\mathcal G_4^{\mathrm{conn}}\bigr]_{G_{\Delta_a,J_a}} =C_{12a}C_{34a}=O(N^{-2}). \end{aligned}$$

provided the generic Witten matching factor is finite and nonzero. The last order is a prediction to test against the four-point data, not a second assumption to hide inside the first step. Agreement supports weak interactions in that sector. It does not by itself establish a large gap, a local derivative expansion, a unique field basis, or a nonperturbative bulk.

Reproducible inference and evidence ceiling

Conventions. State the CFT two-point normalization, the single-trace matrix-sector limit, the bulk kinetic normalization, the AdS radius, the boundary quantization, and the field basis. Without them, a numerical “bulk coupling” is not reproducible.

Observable and check. Match a normalized $C_{ijk}$ to $\lambda^{\mathrm{eff}}_{ijk}$, then test the associated single-trace conformal-block coefficient—or, after introducing Mellin variables, its pole residue—at order $N^{-2}$. Keep contact terms and double-trace data in the complete crossing-symmetric correlator.

Domain and controls. Take $N\to\infty$ at fixed couplings, operator set, dimensions, and cross-ratios in a declared compact kinematic region. Bound weighted species sums, level spacings, and loop functions. Regge, bulk-point, threshold, extremal, and growing-species limits require separate uniform estimates.

Logical status and uncertainty. The result is a leading-order conditional inference. Its uncertainties include subleading $1/N$ corrections, the exact Witten normalization factor, contact and scheme choices, higher-derivative terms, loops, operator mixing, and truncation above the gap. The evidence licenses weak interactions only for the isolated tested sector; it does not prove a finite local field content or a nonperturbative bulk.

Scalar Two- and Three-Point Functions develops the CFT coefficients. Bulk Fields and Boundary Operators and Spinning Fields, Forms, and Mixed-Symmetry Operators own the mass and spin dictionaries. Contact Witten Diagrams and Exchange Witten Diagrams and Conformal Blocks own the detailed bulk extraction. HKLL Reconstruction for Free Bulk Fields owns reconstructed operators.

Continue to Sparse Spectra, Large Gaps, and Semiclassical Bulk Criteria to combine this interaction test with central-charge and spectral-gap data.

Common pitfalls

Checking one coefficient but not the sum. Perturbation theory is controlled by cumulative channel and loop sums. State the number of species and bound the relevant weighted sums uniformly in the chosen kinematic window.

Reading a bulk number from an unnormalized CFT operator. The power of $N$ and the numerical coefficient both change under operator rescaling. Fix the CFT two-point function and the bulk kinetic term before matching.

Treating exchange versus contact as fully observable. The complete correlator and the properly projected pole data are invariant. The off-shell division among vertices can move under field redefinitions, and individual residues rotate within a degenerate operator subspace.

Exercises

Cubic exchange order

Suppose normalized single-trace scalar primaries have $C_{12a}=c_{12a}/N$ and $C_{34a}=c_{34a}/N$, with one isolated exchanged operator $\mathcal O_a$. What is the $N$-scaling of its tree exchange? What scaling of the dimensionless quartic coupling can contribute at the same order?

Solution — cubic exchange order

The single-trace exchange residue is proportional to

$$C_{12a}C_{34a} =\frac{c_{12a}c_{34a}}{N^2},$$

so it is $O(N^{-2})$ at fixed dimensions and cross-ratios. In canonical bulk normalization, this is $\lambda^{\mathrm{eff}}_{12a}\lambda^{\mathrm{eff}}_{34a}$ with each effective dimensionless cubic coupling of order $N^{-1}$. A dimensionless nonderivative quartic coupling $\lambda_{1234}=O(N^{-2})$ contributes to the full connected correlator at the same order. A smaller quartic is allowed; a larger one would violate the assumed connected hierarchy unless a stated symmetry or cancellation removes its contribution.

Growing-multiplicity stress test

Let $M(N)=N^\alpha$ exchanged primaries have comparable squared OPE coefficients $C_{\mathcal O\mathcal Oa}^2=c^2/N^2$. Determine the unweighted cumulative scaling for $\alpha<2$, $\alpha=2$, and $\alpha>2$. What extra information decides the correlator itself?

Solution — growing multiplicity and block control

The total squared OPE weight is

$$\sum_{a=1}^{M(N)}C_{\mathcal O\mathcal Oa}^{2} =c^2N^{\alpha-2}.$$

It vanishes for $\alpha<2$, is order one for $\alpha=2$, and grows for $\alpha>2$. At $\alpha=2$, every individual coefficient is small but the unweighted total OPE weight is order one. Deciding the correlator requires upper and lower control of the block-weighted average, including masses, spins, degeneracies, signs in the selected configuration, and kinematic growth. Counting coefficients alone does not determine it.

Near-degenerate mixing and a dense band

In the two-state dilatation matrix above, take $\mu=2$, $\Delta_s-\Delta_m=N^{-\beta}$, and $N\gg1$. Determine the scaling of $\tan(2\theta)$ for $\beta=1/2$, $1$, and $3/2$. Which cases preserve the original single-trace particle label perturbatively?

Next take $M=N^{1/2}$ primaries in a band of width $B_N=N^{-1/2}$, with nearest-neighbor mixing $2/N$ and $C_{\mathcal O\mathcal Oa}=c/N$. Estimate the typical level spacing, mixing-to-spacing ratio, and unweighted total OPE weight. Does an $N$-independent finite-field truncation survive?

Solution — near-degenerate mixing

The mixing angle obeys

$$\tan(2\theta)=4N^{\beta-1}.$$

For $\beta=1/2$ it vanishes as $4N^{-1/2}$, so the original label is stable. For $\beta=1$ it is $4$, an order-one rotation. For $\beta=3/2$ it grows as $4N^{1/2}$, and the eigenstates approach order-one mixtures. Thus only $\beta<1$ preserves the naive particle label perturbatively.

For the band,

$$\delta\Delta_{\mathrm{nn}} \sim\frac{N^{-1/2}}{N^{1/2}} =N^{-1}, \qquad \frac{2/N}{\delta\Delta_{\mathrm{nn}}}=O(1), \qquad W_M=c^2N^{-3/2}.$$

The total OPE weight vanishes, but adjacent states mix by order-one angles and their number grows as $N^{1/2}$. No fixed finite list is closed under the mixing or captures the whole band. A collective description may survive after diagonalizing and uniformly resumming all $M(N)$ states, but a simple finite-field EFT does not follow.

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

• D’Hoker, Eric, Daniel Z. Freedman, Samir D. Mathur, Alec Matusis, and Leonardo Rastelli. 2000. “Extremal Correlators in the AdS/CFT Correspondence.” In The Many Faces of the Superworld: Yuri Golfand Memorial Volume, edited by Mikhail A. Shifman, 332–360. Singapore: World Scientific. DOI. Open PDF.
• Fitzpatrick, A. Liam, and Jared Kaplan. 2013. “AdS Field Theory from Conformal Field Theory.” Journal of High Energy Physics 02, 054. DOI. Open PDF.
• Freedman, Daniel Z., Samir D. Mathur, Alec Matusis, and Leonardo Rastelli. 1999. “Correlation Functions in the CFT$_d$/AdS$_{d+1}$ Correspondence.” Nuclear Physics B 546: 96–118. DOI. Open PDF.
• Heemskerk, Idse, João Penedones, Joseph Polchinski, and James Sully. 2009. “Holography from Conformal Field Theory.” Journal of High Energy Physics 10, 079. DOI. Open PDF.