  We apply Connes-Chamseddine spectral action to the $U(1)_{\rm B-L}$-
extension of the Standard Model. We show that in order for the scalar potential
to reach its minimum for a non-zero value of the new complex scalar field, thus
triggering the breaking of B-L symmetry, a constraint on the quartic coupling
constants must be satisfied at unification scale. We then explore the
renormalization flow of this model in two opposite scenarios for the neutrino
sector, and show that this constraint is not compatible with the pole masses of
the top quark and SM Higgs boson. We also show that the model suffers from a
mass-splitting problem similar to the doublet-triplet splitting problem of
Grand Unified Theories. We discuss potential implications for the
Noncommutative Geometry program.
