Understanding the biasing between the clustering properties of halos and the underlying dark matter distribution is important for extracting cosmological information from ongoing and upcoming galaxy surveys .
While on sufficiently larges scales the halo overdensity is a local function of the mass density fluctuations , on smaller scales the gravitational evolution generates non-local terms in the halo density field .
We characterize the magnitude of these contributions at third-order in perturbation theory by identifying the coefficients of the non-local invariant operators , and extend our calculation to include non-local ( Lagrangian ) terms induced by a peak constraint .
We apply our results to describe the scale-dependence of halo bias in cosmologies with massive neutrinos .
The inclusion of gravity-induced non-local terms and , especially , a Lagrangian k ^ { 2 } -contribution is essential to reproduce the numerical data accurately .
We use the peak-background split to derive the numerical values of the various bias coefficients from the excursion set peak mass function .
For neutrino masses in the range 0 \leq \sum _ { i } m _ { \nu _ { i } } \leq 0.6 eV , we are able to fit the data with a precision of a few percents up to k = 0.3 h { Mpc ^ { -1 } } without any free parameter .