Abstract
We develop the technology for Polyakov-Mellin (PM) bootstrap in one- dimensional conformal field theories (CFT1). By adding appropriate contact terms, we bootstrap various effective field theories in AdS2 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.
Highlights
Conformal symmetry puts stringent constraints on the structure of the correlators
We develop the technology for Polyakov-Mellin (PM) bootstrap in onedimensional conformal field theories (CFT1)
We perform two non-trivial consistency checks — a) we show that the diagonal limit of the 2d Ising model can be expanded in terms of the 1d PM blocks and b) we show that the fermionic generalised free field (GFF) correlator in one dimension can be expanded in terms of the bosonic 1d PM blocks
Summary
Conformal symmetry puts stringent constraints on the structure of the correlators. One interesting fact about all unitary Conformal Field Theories (CFTs) is that the local operators in the theory, which are labeled by their scaling dimension (∆) and spin ( ), satisfy an algebra, called the Operator Product Expansion (OPE). We use crossing symmetry, combined with properties of the one-dimensional OPE, to fix the rational functions, and this allows to find correlators up to one loop with only a finite number of ambiguities, which correspond precisely to the contact terms that one needs to add to the sum over Witten exchange diagrams in order to build a complete basis of Polyakov blocks. We shall consider the crossing equation in all the three OPE limits (namely s-, t- and u-channel), and observe that in two out of three the OPE is controlled by operators with large dimension (∆), while the third is dominated by the identity This allows, exploiting the Regge-limit expansions previously derived, to put some constraints on the CFT data order by order in 1/∆. The appendices supplement many computational details we used in the main text
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