Crossing symmetry, transcendentality and the Regge behaviour of 1d CFTs

TL;DR

This work develops a complete Polyakov-Mellin bootstrap framework for one-dimensional CFTs, enabling analytic access to CFT data at tree and one loop for scalar EFTs on AdS$_2$. It fixes the PM basis in 1d by introducing necessary contact terms, including a new gradient term for O($N$) theories, and cross-checks results against independent transcendentality-based methods. The authors derive closed-form CFT data in terms of harmonic sums, explore the Regge limit and its relation to large-twist data, and obtain universal constraints on large-twist behavior in Regge-bounded theories with finitely many exchanges. They further extend the analysis to loop level, confirm derivative relations, and map out the qualitative structure of EFT expansions and their Regge/regge-like consequences. Overall, the paper provides a robust, cross-validated toolkit for 1d PM bootstrap with clear pathways to higher dimensions and defect theories, highlighting deep links between Regge limits, transcendentality, and CFT data.

Abstract

We develop the technology for Polyakov-Mellin (PM) bootstrap in one-dimensional conformal field theories (CFT$_1$). By adding appropriate contact terms, we bootstrap various effective field theories in AdS$_2$ and analytically compute the CFT data to one loop. The computation can be extended to higher orders in perturbation theory, if we ignore mixing, for any external dimension. We develop PM bootstrap for $O(N)$ theories and derive the necessary contact terms for such theories (which also involves a new higher gradient contact term absent for $N=1$). We perform cross-checks which include considering the diagonal limit of the $2d$ Ising model in terms of the $1d$ PM blocks. As an independent check of the validity of the results obtained with PM bootstrap, we propose a suitable basis of transcendental functions, which allows to fix the four-point correlators of identical scalar primaries completely, up to a finite number of ambiguities related to the number of contact terms in the PM basis. We perform this analysis both at tree level (with and without exchanges) and at one loop. We also derive expressions for the corresponding CFT data in terms of harmonic sums. Finally, we consider the Regge limit of one-dimensional correlators and derive a precise connection between the latter and the large-twist limit of CFT data. Exploiting this result, we study the crossing equation in the three OPE limits and derive some universal constraints for the large-twist limit of CFT data in Regge-bounded theories with a finite number of exchanges.

Figures (6)

• Figure 1: $f_{\Delta}(\Delta_{\phi}+1)~vs ~ \Delta$
• Figure 2: $\left|\gamma_n\right|~vs~n$
• Figure 3: Plots of $\, {\cal F}_{\Delta}(1/2+it)\,$ vs $\Delta$ for ${\Delta_{\phi}}=1$ and two values of $t$. One can see that the plot is peaked for a value of $\Delta$ that increases roughly with $\sqrt{t}$.
• Figure 4: As $m$ increases $F(m,n=1)$ goes to $0$, also $G(m,n=0)$ goes to $0$ which is expected. Note that $q_{dis}'(\Delta_\phi)=2$ and $\Delta_{\phi}=\frac{1}{8}.$
• Figure 5: Mellin-Barnes type integration contour where the path of integration is parallel to imaginary vertical axis. The contour separates chains of poles which lie entirely on the right and the chains of poles that lie entirely on the left.