  Chiral primary operators annihilated by a quarter of the supercharges are
constructed in the four dimensional N=4 Super-Yang-Mills theory with gauge
group SU(N). These quarter-BPS operators share many non-renormalization
properties with the previously studied half-BPS operators. However, they are
much more involved, which renders their construction nontrivial in the fully
interacting theory. In this paper we calculate order g^2 two-point functions of
local, polynomial, scalar composite operators within a given representation of
the SU(4) R-symmetry group. By studying these two-point functions, we identify
the eigenstates of the dilatation operator, which turn out to be complicated
mixtures of single and multiple trace operators.
  Given the elaborate combinatorics of this problem, we concentrate on two
special cases. First, we present explicit computations for quarter-BPS
operators with scaling dimension Delta < 8. In this case, the discussion
applies to arbitrary N of the gauge group. Second, we carry out a leading plus
subleading large N analysis for the particular class of operators built out of
double and single trace operators only. The large N construction addresses
quarter-BPS operators of general dimension.
