We revisit models of gauge mediated supersymmetry breaking where messenger
parity is violated. Such a symmetry is usually invoked in order to set to zero
potentially dangerous hypercharge D-term tadpoles. A milder hypothesis is that
the D-tadpole vanishes only at the first order in the gauge coupling constant.
Then the next order leads to a contribution to the sfermion masses which is of
the same magnitude as the usual radiative one. This enlarges the parameter
space of gauge mediated models. We first give a completely general
characterization of this contribution, in terms of particular three-point
functions of hidden sector current multiplet operators. We then explore the
parameter space by means of two simple weakly coupled models, where the
D-tadpole arising at two-loops has actually a mild logarithmic divergence.Comment: 13 pages + 9 pages of appendix, 1 figure; v2: some clarifying
comments added, version to appear in JHE
