Finite-size scaling, Intermittency and the QCD critical point

Based on an effective action for the 3d Ising critical point we construct a Ginzburg-Landau free energy for the proton density to describe the QCD thermodynamics in the critical region. We argue that the relation between finite-size scaling in configuration space and intermittency in momentum space is a valuable tool for the detection of the QCD critical point in ion collisions. We find that the critical region is very narrow along both chemical potential ($\mu$) and temperature ($T$) directions supporting that wide ranged beam energy scans with fixed size nuclei are unlikely to reach the critical region. Furthermore, we present a systematic procedure leading to the detection of the critical point through combined measurements of intermittency indices and freeze-out thermodynamic parameters $(\mu,T)$ for protons. Exploiting previous NA49 measurement of the intermittency index $\phi_2$ and the freeze-out parameters $(\mu,T)$ for protons produced in central Si+"Si" collision at $158$A GeV we predict the approach to the critical point in peripheral Ar+Sc collisions of the NA61/SHINE experiment at maximum SPS (CERN) energy.