Precise Determination of the Proton's Gluon Cloud Geometry from HERA data
The transverse shape of the proton's small-$x$ gluon distribution is determined from exclusive $J/\psi$ photoproduction at HERA. We derive analytic expressions for the coherent and incoherent diffractive cross sections in the hotspot model at leading twist, enabling a global fit to all 104 available H1 and ZEUS data points, spanning three decades in $t$, with $\chi^2/{\rm ndf}=0.77$. The gluonic hotspots are resolved into a perturbative Gaussian core of size 0.105(2) fm surrounded by a nonperturbative exponential halo of range 0.220(15) fm, in agreement with the gluon-field correlation length of the QCD vacuum and with the core-halo structure of flux tubes recently determined on the lattice. This shape is independent of $x_{I\!\!P}$ and of the assumed number of hotspots or hotspot repulsion. The gluonic geometry evolve only through a slow transverse diffusion of the hotspot centres with $\alpha'_{\rm eff.}=0.046(28)~{\rm GeV}^{-2}$, while the incoherent cross section at small $|t|$ is dominated by shot-noise fluctuations at the order of one gluon per hotspot.