One-loop string amplitudes in AdS${}_5\times$S${}^5$: Mellin space and sphere splitting

Abstract

We study string corrections to one-loop amplitudes of single-particle operators ${\mathcal{O}}_p$ in $Ad{S}_5\times S^5$. The tree-level correlators in supergravity enjoy an accidental 10d conformal symmetry. Consequently, one observes a partial degeneracy in the spectrum of anomalous dimensions of double-trace operators and at the same time equality of many different correlators for different external charges $p_{i=1,2,3,4}$. The one-loop contribution is expected to lift such bonus properties, and its precise form can be predicted from tree-level data and consistency with the operator product expansion. Here we present a closed-form Mellin space formula for $⟨{\mathcal{O}}_{p_1}{\mathcal{O}}_{p_2}{\mathcal{O}}_{p_3}{\mathcal{O}}_{p_4}⟩$ at order $({\alpha }^{\prime }{)}^3$, valid for arbitrary external charges $p_i$. Our formula makes explicit the lifting of the bonus degeneracy among different correlators through a feature we refer to as `sphere splitting'. While tree-level Mellin amplitudes come with a single crossing symmetric kernel, which defines the pole structure of the $Ad{S}_5\times S^5$ amplitude, our one-loop amplitude naturally splits the $S^5$ part into two separate contributions. The amplitude also exhibits a remarkable consistency with the corresponding flat space IIB amplitude through the large $p$ limit.

32 pages, extended introduction containing summary of results, 2 appendices

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