Bulk Causality from Lorentzian Inversion and Dispersive Sum Rules

The Lorentzian inversion formula reconstructs analytic-in-spin OPE data from a correlator’s double discontinuity. Dispersive sum rules similarly relate low-energy coefficients to Lorentzian spectral data, but only after specifying Regge bounds and subtractions. Their bulk interpretation constrains exchange and contact couplings; an omitted arc term can invalidate a claimed positivity or locality bound.

Required background. Bulk interaction scaling supplies the EFT order. The Lorentzian inversion formula and subtracted dispersion relations supply the imported results.

Helpful background. Flat-space subtracted dispersion relations supplies the distinct S-matrix analogue.

Double discontinuity and subtractions

Schematically, the coefficient function is

$$c(\Delta,J)=\frac{\kappa_{\Delta+J}}4 \int_0^1dz\,d\bar z\,\mu(z,\bar z) G_{J+d-1,\Delta-d+1}(z,\bar z) \operatorname{dDisc}\mathcal G(z,\bar z).$$

The double discontinuity removes terms analytic around the crossed-channel cut and is positive for appropriate identical-scalar configurations. The formula converges for spins above a threshold set by Regge growth; lower spins require subtractions or separate data Caron-Huot 2017.

Contact terms can have vanishing double discontinuity at a perturbative order, so inversion from cuts determines nonlocal exchange data only up to a polynomial ambiguity. A conformal dispersive representation makes this freedom explicit as subtraction data Carmi and Caron-Huot 2020.

First application: one crossed-channel exchange

Insert the double discontinuity of a primary of twist $\tau_\chi$. Expanding the inversion kernel at large $J$ gives

$$\gamma_{n,J}^{(\chi)} =-A_{n,\chi}J^{-\tau_\chi}+O(J^{-\tau_\chi-2}),$$

reproducing the lightcone result without summing direct-channel blocks individually. The residue fixes a long-range bulk exchange coefficient after operator normalizations are supplied. A local contact coefficient remains a subtraction datum unless extra Regge falloff, crossing equations, or additional correlators fix it.

Positive dispersive functionals can constrain combinations of higher-derivative bulk couplings, but their sign follows from the positive spectral input and the absence or controlled evaluation of boundary arcs—not from the word “causality” alone.

Adversarial control: violate the Regge bound

Apply an unsubtracted formula to a correlator growing as $\sigma^{1-J_0}$ faster than the assumed bound. Closing the contour leaves a nonzero arc at infinity. Dropping it changes the contact coefficients and can manufacture a false positivity inequality. Adding the required subtractions restores the formula but introduces constants that the discontinuity does not determine.

The evidence ceiling is analytic-in-spin OPE or bulk-EFT information under explicit unitarity, analyticity, Regge, and subtraction hypotheses. It is not a nonperturbative bulk locality theorem. Regge and eikonal scattering examines the high-energy data that control those hypotheses.

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

• Caron-Huot, S. (2017), “Analyticity in Spin in Conformal Theories,” Journal of High Energy Physics 2017(09), 078. arXiv:1703.00278.
• Carmi, D., and Caron-Huot, S. (2020), “A Conformal Dispersion Relation: Correlations from Absorption,” Journal of High Energy Physics 2020(09), 009. arXiv:1910.12123.