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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) ×  SU (2) ×  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.

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