  The four-body equations of Alt, Grassberger and Sandhas are solved for $\nH$
scattering at energies below three-body breakup threshold using various
realistic interactions including one derived from chiral perturbation theory.
  After partial wave decomposition the equations are three-variable integral
equations that are solved numerically without any approximations beyond the
usual discretization of continuum variables on a finite momentum mesh. Large
number of two-, three- and four-nucleon partial waves are considered until the
convergence of the observables is obtained. The total $\nH$ cross section data
in the resonance region is not described by the calculations which confirms
previous findings by other groups. Nevertheless the numbers we get are slightly
higher and closer to the data than previously found and depend on the choice of
the two-nucleon potential. Correlations between the $A_y$ deficiency in $\nd$
elastic scattering and the total $\nH$ cross section are studied.
