  If neutrinos are Dirac particles the existence of light right-handed
neutrinos $\nu_{R}$ is implied. Those would contribute to the effective number
of relativistic neutrino species $N_{{\rm eff}}$ in the early Universe. With
pure standard model interactions, the contribution is negligibly small. In the
presence of new interactions, however, the contribution could be significantly
enhanced. We consider the most general effective four-fermion interactions for
neutrinos (scalar, pseudo-scalar, vector, axial-vector and tensor), and compute
the contribution of right-handed neutrinos to $N_{{\rm eff}}$. Taking the
Planck 2018 measurement of $N_{{\rm eff}}$, strong constraints on the effective
four-fermion coupling are obtained, corresponding to interaction strengths of
$10^{-5}\sim10^{-3}$ in units of the Fermi constant. This translates in new
physics scales of up to 43 TeV and higher. Future experiments such as CMB-S4
can probe or exclude the existence of effective 4-neutrino operators for Dirac
neutrinos. Ways to avoid this conclusion are discussed.
