Multi-particle correlators with higher KK modes I: a bootstrap approach

We bootstrap tree-level supergravity four-point correlators on AdS$_5\times$S$^5$ with one external half-BPS double-particle operator and three half-BPS single-particle operators. Our only input is the consistency of the operator product expansion of $SU(N)$ ${\cal N} = 4$ super Yang-Mills theory at large $N$ and large 't Hooft coupling. Even though the leading order OPE does not close on double-particle operators, but involves triple-particle operators, the CFT data of the double-particle operators, both long and protected, is sufficient to uniquely fix the correlators. We then verify that our results for the four-point correlators with one double-particle and three single-particle operators are reproduced by the appropriate double-particle limit of the five-point tree-level correlators of single-particle operators, with arbitrary Kaluza-Klein levels, recently conjectured in arXiv:2507.14124. Our study thus provides further evidence for the latter result.