  A gauge theory with gauge group G defined in D>4 space-time dimensions can be
broken to a subgroup H on four dimensional fixed point branes, when
compactified on an orbifold. Mass terms for extra dimensional components of
gauge fields A_i (brane scalars) might acquire (when allowed by the brane
symmetries) quadratically divergent radiative masses and thus jeopardize the
stability of the four-dimensional theory. We have analyzed Z_2
compactifications and identified the brane symmetries remnants of the higher
dimensional gauge invariance. No mass term is allowed for D=5 while for D>5 a
tadpole \epsilon^{ij}F_{ij}^\alpha can appear when there are U_\alpha(1)
factors in H. A detailed calculation is done for the D=6 case and it is
established that the tadpole is related, although does not coincide, with the
U_\alpha(1) anomaly induced on the brane by the bulk fermions. In particular,
no tadpole is generated from gauge bosons or fermions in real representations
