Connecting the Hadron Mass Scale to the Fundamental Mass Scale of Quantum Chromodynamics

Establishing an explicit connection between the long distance physics of confinement and the dynamical interactions of quarks and gluons at short distances has been a long-sought goal of quantum chromodynamics. Using holographic QCD, we derive a direct analytic relation between the scale $\kappa$ which determines the masses of hadrons and the scale $\Lambda_{s}$ which controls the predictions of perturbative QCD at very short distances. The resulting prediction $\Lambda_{s}=0.341\pm0.032$ GeV in the $\overline{MS}$ scheme agrees well with the experimental average $0.339\pm0.016$ GeV. We also derive a relation between $\Lambda_{s}$ and the QCD string tension $\sigma$. This connection between the fundamental hadronic scale underlying the physics of quark confinement and the perturbative QCD scale controlling hard collisions can be carried out in any renormalization scheme.