$KX(3872)$ interaction and correlation function

We investigate the $K X(3872)$ interaction and the corresponding correlation function, assuming the $X(3872)$ to be a molecular state of $D \bar D^*$ and $D^* \bar D$ with isospin $I=0$ and positive $C$-parity. The interaction is treated within the fixed-center approximation (FCA) to the Faddeev equations, in which the $X(3872)$ is taken as a cluster of its constituents and the kaon interacts with the $D^*$ and $D$ components. The three-body scattering amplitude is evaluated using the Fixed Center Approximation (FCA) to the Faddeev equations, improved by taking the FCA amplitude as an optical potential which is later unitarized by means of the Lippmann-Schwinger equation. We find a narrow resonant structure about 50~MeV below the $K^+ X(3872)$ threshold with a width of approximately 1~MeV, and determine the $K X(3872)$ scattering length $a = (0.39 - i\,0.00)$~fm and effective range $r_0 = (1.16 - i\,1.66)$~fm. The corresponding correlation function is evaluated and shows a clear deviation from unity at low momenta, characteristic of a strongly attractive interaction leading to a bound state. These predictions are tied to the molecular nature of the $X(3872)$ and can be measured experimentally via measurements of the $K X(3872)$ correlation function and three-body invariant mass distributions.