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We develop the technology for Polyakov-Mellin (PM) bootstrap inone-dimensional conformal field theories (CFT$_1$). By adding appropriatecontact terms, we bootstrap various effective field theories in AdS$_2$ andanalytically compute the CFT data to one loop. The computation can be extendedto higher orders in perturbation theory, if we ignore mixing, for any externaldimension. We develop PM bootstrap for $O(N)$ theories and derive the necessarycontact terms for such theories (which also involves a new higher gradientcontact term absent for $N=1$). We perform cross-checks which includeconsidering the diagonal limit of the $2d$ Ising model in terms of the $1d$ PMblocks. As an independent check of the validity of the results obtained with PMbootstrap, we propose a suitable basis of transcendental functions, whichallows to fix the four-point correlators of identical scalar primariescompletely, up to a finite number of ambiguities related to the number ofcontact 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 forthe corresponding CFT data in terms of harmonic sums. Finally, we consider theRegge limit of one-dimensional correlators and derive a precise connectionbetween the latter and the large-twist limit of CFT data. Exploiting thisresult, we study the crossing equation in the three OPE limits and derive someuniversal constraints for the large-twist limit of CFT data in Regge-boundedtheories with a finite number of exchanges.

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