  A new diagnostics to break the degeneracy between the total neutrino mass
($M_{\nu}$) and the primordial power spectrum amplitude ($\sigma_{8}$) by using
the drifting coefficient of the field cluster mass function is presented.
Analyzing the data from the Cosmological Massive Neutrino Simulations, we first
determine the numerical mass functions of the field clusters at various
redshifts. Then, we compare the numerical results with the analytical model
characterized by a single parameter called the drifting coefficient which
measures the drifts of the collapse density threshold, $\delta_{c}$, from the
Einstein-de Sitter spherical value, $\delta_{sc}$, at a given mass scale. It is
found that the analytic model for the field cluster mass function is found to
work excellently even in the presence of massive neutrinos and that its
drifting coefficient evolves differently in the cosmologies with different
values of $M_{\nu}$. At low redshifts ($z\lesssim 0.3$) the more massive
neutrinos drift $\delta_{c}$ further from $\delta_{sc}$, while the opposite
trend is found at higher redshifts ($z\gtrsim0.3$). Speculating that this
distinct redshift-dependent effect of massive neutrinos on the drifting
coefficient of the field cluster mass function might help break the
$\sigma_{8}$-$M_{\nu}$ degeneracy, we also show that the sensitivity of this
new diagnostics to $M_{\nu}$ is high enough to discriminate the case of
$M_{\nu}=0.1\,{\rm eV}$ from that of massless neutrinos.
