Extensions of the noncommutative Standard Model and the weak order one condition

In the derivation of the Standard Model from the axioms of noncommutative geometry the scalar sector is given by a finite Dirac operator which has to satisfy the first-order condition. However, the general solution to this constraint still has unphysical terms which must be fine-tuned to zero. Some of them can be removed by the so-called second-order condition. However, the first-order condition generally does not survive in extensions to models with gauge groups larger that U(1) x SU(2) x SU(3). In this paper we show that in the U(1)(B-L)-extension one can implement a weaker form of the first-order condition which, we argue, is necessary in order for noncommutative Gauge theory to make sense at all, and that this condition reduces the amount of fine-tuning to the off-diagonal terms in the Yukawa mass matrices for the leptons and quarks. It follows that the weak order one condition imposed on a the B-L-extended model yields exactly the same constraint as the much more restrictive, and, we believe, less well motivated, second-order condition imposed on the Standard Model alone.