Spinning Mellin Bootstrap: Conformal Partial Waves, Crossing Kernels and Applications

TL;DR

This work develops a comprehensive Mellin-space framework for conformal partial waves with arbitrary spinning external operators, exploiting a new basis that factorises external spins from the exchanged spin. It derives explicit crossing kernels in terms of ${}_4F_3$ hypergeometric functions and constructs inversion formulae using continuous Hahn polynomials to extract OPE data, including ${ m O}(1/N)$ anomalous dimensions for general double-trace flows. The authors validate the approach with a consistency check in the free $O(N)$ model and extend the analysis to leading and subleading twists, partially-conserved currents, and large-spin limits, with applications to scalar and spinning double-trace deformations in diverse dimensions and epsilon-expansions. The results provide precise, analytic access to OPE coefficients and anomalous dimensions across scalar and spinning sectors, offering a powerful tool for analytic bootstrap in theories with higher-spin symmetry and large-$N$ dynamics.

Abstract

We study conformal partial waves (CPWs) in Mellin space with totally symmetric external operators of arbitrary integer spin. The exchanged spin is arbitrary, and includes mixed symmetry and (partially)-conserved representations. In a basis of CPWs recently introduced in arXiv:1702.08619, we find a remarkable factorisation of the external spin dependence in their Mellin representation. This property allows a relatively straightforward study of inversion formulae to extract OPE data from the Mellin representation of spinning 4pt correlators and in particular, to extract closed-form expressions for crossing kernels of spinning CPWs in terms of the hypergeometric function ${}_4F_3$. We consider numerous examples involving both arbitrary internal and external spins, and for both leading and sub-leading twist operators. As an application, working in general $d$ we extract new results for ${\cal O}\left(1/N\right)$ anomalous dimensions of double-trace operators induced by double-trace deformations constructed from single-trace operators of generic twist and integer spin. In particular, we extract the anomalous dimensions of double-trace operators $[\mathcal{O}_JΦ]_{n,l}$ with ${\cal O}_J$ a single-trace operator of integer spin $J$.

Figures (12)

• Figure 1: An illustration of the defining feature of the basis \ref{['IntegralBasisAdS']} dual to the bulk cubic couplings \ref{['Ical']}. The non-vanishing coefficients of the bulk couplings are exactly the same in number as the CFT structures that are set to zero. This condition has a unique solution which defines the basis proposed in Sleight:2017fpc and which is employed in this work.
• Figure 2: Pictorial representation of the action of higher-spin transformations on CFT operators/bulk fields via the bulk-to-boundary map. $\bf n$ lables the OPE structures. The basis used in this work should be thought of as an analytic continuation in dimensions of the higher-spin generators which one recovers for $\tau_1=d-2-k$ with $0\leq k< J_1$. In the bulk such analytic continuation is natural when going off-shell and considering the action of the deformed bulk gauge symmetries on off-shell couplings. The bulk to boundary map then gives the corresponding families of cubic structures on the boundary.
• Figure 3: Plot of $\delta\gamma_{1,0}$ (vertical axis) in various dimensions in the interval $\tau\in[\tfrac{d-2}{2},\tfrac{d}{2}]$ (horizontal axis) for increasing values of $\Delta>\tfrac{d-2}{2}$ (color-bar in each graph).
• Figure 4: Plot of $\delta\gamma_{2,0}$ (vertical axis) in various dimensions in the interval $\tau\in[\tfrac{d-2}{2},\tfrac{d}{2}]$ (horizontal axis) for increasing values of $\Delta>\tfrac{d-2}{2}$ (color bar).
• Figure 5: Plot of $\delta\gamma^{[\Phi\Phi]}_{l^\prime|0,0}$ in $d=2$ on the left and $d=3$ on the right, in the interval $\tau\in[\tfrac{d-2}{2},\tfrac{d}{2}]$ for even values of $l^\prime$.