Approximate Bulk Locality from Spectral and Mellin Data

Bulk locality is not read from one OPE coefficient or one low-energy pole. It is a controlled inference from a package of data: a sparse light spectrum, a parametrically separated heavy scale, large-$N$ factorization, Mellin amplitudes with the expected exchange poles, and bounded behavior in the kinematic domains used to construct an AdS derivative expansion. Under those hypotheses, a low-degree Mellin polynomial represents finitely many local contact interactions and the heavy gap estimates the first omitted term. The conclusion is approximate locality below a cutoff, not an exact local bulk at all energies.

Required background. Sparse Spectra, Large Gaps, and Semiclassical Bulk Criteria defines the single-trace gap used here; Bulk Interaction Scaling and Effective Cutoffs supplies the bulk power counting; and Mellin-Space CFT Correlators supplies the Mellin representation and crossing conventions.

Helpful background. Large-Gap Constraints and CFT-Side Locality Tests develops the CFT constraints in detail, while The CFT Regge Limit and Boundedness explains why boundedness is independent information.

Mellin data and the local derivative expansion

For four identical scalar primaries of dimension $\Delta_\phi$, write the connected reduced correlator schematically as

$$\mathcal G_{\mathrm{conn}}(u,v) = \int\frac{ds\,dt}{(4\pi i)^2}\, u^{s/2}v^{(t-2\Delta_\phi)/2} \mu_{\Delta_\phi}(s,t)\,\mathcal M(s,t),$$

where $\mu_{\Delta_\phi}$ is the standard product of Gamma functions and the third Mellin variable obeys

$$s+t+\widetilde u=4\Delta_\phi.$$

The Gamma-function measure supplies the double-trace pole structure. Additional poles of $\mathcal M$, together with their residues, encode exchanged single-trace operators. Polynomial pieces have no such exchange poles and are the Mellin images of AdS contact vertices. For a scalar bulk field, a contact term with $2k$ derivatives gives a crossing-symmetric polynomial of degree $k$ in $s,t,\widetilde u$. Thus

$$\mathcal M_{\mathrm{contact}} =a_0 +a_1(s^2+t^2+\widetilde u^2) +a_2\,st\widetilde u+\cdots$$

is not merely a convenient fit: after choosing an operator basis modulo integrations by parts and equations of motion, each independent polynomial structure corresponds to a finite family of local AdS interactions. The correspondence was made precise for perturbative AdS effective field theory by Heemskerk, Penedones, Polchinski, and Sully 2009, §§3–5 and by the Mellin-space construction of Penedones 2011.

Two qualifications are essential. First, contact terms are basis-dependent: field redefinitions can move coefficients between vertices without changing correlators. The invariant datum is the full correlator, not a preferred bulk Lagrangian coefficient. Second, a polynomial description is useful only over a declared Mellin domain. A finite polynomial evaluated far above the heavy scale is an extrapolation beyond the derivative expansion.

A low-degree polynomial and its first omitted term

Suppose the light single-trace exchanges have been separated explicitly and the remaining Mellin amplitude is well approximated in the domain

$$\lvert s\rvert,\ \lvert t\rvert,\ \lvert\widetilde u\rvert \lesssim E^2L^2 \ll \Delta_{\mathrm{gap}}^2$$

by a polynomial $P_K$ of degree $K$. Here $L$ is the AdS radius and $\Delta_{\mathrm{gap}}\simeq M_{\mathrm{gap}}L$ is the dimension of the first omitted heavy single-trace field. Integrating out one heavy field illustrates the estimate. In a local flat-space patch its propagator expands as

$$\frac{g_{\phi\phi H}^{\,2}}{M_{\mathrm{gap}}^2-\widehat s} = \frac{g_{\phi\phi H}^{\,2}}{M_{\mathrm{gap}}^2} \left[ 1+\frac{\widehat s}{M_{\mathrm{gap}}^2} +\cdots+ \left(\frac{\widehat s}{M_{\mathrm{gap}}^2}\right)^K \right],$$

with analogous crossed-channel terms. In AdS, the exact exchange is a sequence of Mellin poles rather than this single propagator, but below the gap its analytic part has the same derivative expansion. If the highest retained interaction contains $2K$ derivatives, a natural estimate for the next term is

$$\frac{\delta\mathcal M_{K+1}}{\mathcal M_{\mathrm{lead}}} \sim c_{K+1} \left(\frac{EL}{\Delta_{\mathrm{gap}}}\right)^{2K+2},$$

where $c_{K+1}$ is not fixed by the gap alone. A useful error estimate therefore states both the expansion parameter and the assumption that the dimensionless Wilson coefficients are not anomalously large.

As the first application, consider a crossing-symmetric residual

$$P_2(s,t,\widetilde u) =a_0+a_1(s^2+t^2+\widetilde u^2).$$

It licenses a two-vertex description: a zero-derivative $\phi^4$ interaction plus one independent four-derivative contact structure, modulo redundant operators. If data are restricted to $EL\leq \eta\Delta_{\mathrm{gap}}$ with $\eta<1$, then the first generic six-derivative contribution is suppressed by order $\eta^6$ relative to the leading natural coefficient. This estimate does not determine its sign and does not include $1/N$ loops; those are separate errors.

Poles, boundedness, and what each hypothesis contributes

A locality inference combines logically distinct observations:

• Sparse light data limit the number of bulk fields that must remain explicit below the cutoff.
• A heavy gap makes the expansion in $E/M_{\mathrm{gap}}$ possible.
• Large-$N$ factorization suppresses connected correlators and organizes bulk loops.
• Exchange-pole residues test whether the same OPE coefficients consistently determine different channels.
• Polynomial boundedness prevents high-energy behavior from undoing the derivative expansion.
• Crossing and unitarity constrain, but do not by themselves prove, an approximately local bulk description.

In particular, pole locations alone do not constrain an arbitrary entire addition to $\mathcal M$. Local contact ambiguities are polynomial at each fixed order; a general entire function can share all desired poles while growing too rapidly to arise from a local EFT with a finite cutoff hierarchy. Regge causality supplies a further filter. In theories with a large higher-spin gap, excessive higher-derivative graviton couplings lead to causality problems unless new higher-spin states enter at the corresponding scale, as shown by Camanho, Edelstein, Maldacena, and Zhiboedov 2016.

Adversarial amplitude with acceptable low-energy data

Let $P_K$ and $\mathcal M_{\mathrm{poles}}$ reproduce all low-energy information being tested. Define the crossing-symmetric entire deformation

$$\delta\mathcal M =\varepsilon\sum_{x\in\{s,t,\widetilde u\}} \left[ \exp\left(\frac{x^2}{\Lambda^4}\right) -\sum_{n=0}^{K} \frac{x^{2n}}{n!\Lambda^{4n}} \right].$$

It introduces no poles, is crossing symmetric, and begins beyond every polynomial coefficient retained through the chosen order. Consequently, a finite collection of low-energy OPE data cannot distinguish

$$\mathcal M_{\mathrm{good}} =\mathcal M_{\mathrm{poles}}+P_K \quad\text{from}\quad \mathcal M_{\mathrm{bad}} =\mathcal M_{\mathrm{good}}+\delta\mathcal M$$

to that accuracy. Yet $\mathcal M_{\mathrm{bad}}$ grows exponentially along suitable Regge directions and is not polynomially bounded. It therefore fails the stated local-EFT criterion. This counterexample isolates the missing hypothesis: matching finitely many poles and Taylor coefficients establishes neither the required high-energy bound nor a controlled infinite derivative expansion.

The strongest surviving statement is narrower: the tested correlator admits a finite-order low-energy contact parametrization in the sampled domain. Calling that parametrization a local bulk EFT additionally requires a gap, a uniform error bound, acceptable Regge behavior, and consistency across correlators.

Orders of limits and evidence ceiling

The useful hierarchy is

$$N\to\infty,\qquad \Delta_{\mathrm{gap}}\to\infty,\qquad \frac{EL}{\Delta_{\mathrm{gap}}}\to0,$$

but these operations answer different questions. Taking $N\to\infty$ first removes bulk loops; it does not suppress string-scale or other heavy-field corrections. Taking the gap large at fixed $E$ suppresses higher derivatives; allowing $E$ to scale with the gap removes that suppression. Regge limits probe a kinematic region in which a low-energy truncation need not be uniform. Finite $N$, finite gap, and finite Mellin-domain errors must therefore be quoted separately.

Evidence cutoff: 25 July 2026. The structural Mellin/EFT results cited here support a conditional, perturbative reconstruction in specified large-$N$, large-gap regimes. They do not prove that an arbitrary CFT satisfying a few spectral tests has an exact local bulk, determine its nonperturbative completion, or convert polynomial fits into evidence of quantum gravity in nature. Detailed Mellin poles, bulk-point singularities, Regge dispersion, and operator-locality questions are handled in their dedicated later chapters; CFT data extraction remains with the conformal-bootstrap volume.

Exercises

1. A residual Mellin amplitude is $P=a+b(s^2+t^2+\widetilde u^2)$. Explain why two measurements cannot identify a unique off-shell bulk Lagrangian even if they determine $a$ and $b$.

   Solution

   Correlators determine on-shell contact structures. Integrations by parts, use of the leading equations of motion, and local field redefinitions move coefficients among off-shell vertices while leaving the correlator invariant. The data identify two independent on-shell polynomial structures, not a unique Lagrangian basis.

2. If $\Delta_{\mathrm{gap}}=20$, $EL=4$, and the retained polynomial includes four derivatives, estimate the natural relative size of the first six-derivative term.

   Solution

   The expansion ratio is $EL/\Delta_{\mathrm{gap}}=0.2$. A six-derivative term is naturally of order $0.2^6=6.4\times10^{-5}$ relative to a leading coefficient of comparable natural size. This is not a rigorous bound unless the next Wilson coefficient is bounded.

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

• Camanho, Xian O.; Edelstein, José D.; Maldacena, Juan; and Zhiboedov, Alexander. “Causality Constraints on Corrections to the Graviton Three-Point Coupling.” Journal of High Energy Physics 2016, 020 (2016). doi:10.1007/JHEP02(2016)020.
• Heemskerk, Idse; Penedones, João; Polchinski, Joseph; and Sully, James. “Holography from Conformal Field Theory.” Journal of High Energy Physics 2009, 079 (2009). doi:10.1088/1126-6708/2009/10/079.
• Penedones, João. “Writing CFT Correlation Functions as AdS Scattering Amplitudes.” Journal of High Energy Physics 2011, 025 (2011). doi:10.1007/JHEP03(2011)025.