Gaussian limits for subcritical chaos

We present a simple criterion, only based on second moment assumptions, for the convergence of polynomial or Wiener chaos to Gaussian limit. exploit this criterion obtain new asymptotics partition functions two-dimensional directed polymers in sub-critical regime, including singular product between function and disorder. These results can also be applied KPZ Stochastic Heat Equation. As tool independent interest, we derive an explicit expansion which sharply approximates logarithm function.