We present a simple systematic method to study candidate counterterms in N=8
supergravity. Complicated details of the counterterm operators are avoided
because we work with the on-shell matrix elements they produce. All n-point
matrix elements of an independent SUSY invariant operator of the form D^{2k}
R^n +... must be local and satisfy SUSY Ward identities. These are strong
constraints, and we test directly whether or not matrix elements with these
properties can be constructed. If not, then the operator does not have a
supersymmetrization, and it is excluded as a potential counterterm. For n>4, we
find that R^n, D^2 R^n, D^4 R^n, and D^6 R^n are excluded as counterterms of
MHV amplitudes, while only R^n and D^2 R^n are excluded at the NMHV level. As a
consequence, for loop order L<7, there are no independent D^{2k}R^n
counterterms with n>4. If an operator is not ruled out, our method constructs
an explicit superamplitude for its matrix elements. This is done for the 7-loop
D^4 R^6 operator at the NMHV level and in other cases. We also initiate the
study of counterterms without leading pure-graviton matrix elements, which can
occur beyond the MHV level. The landscape of excluded/allowed candidate
counterterms is summarized in a colorful chart.Comment: 25 pages, 1 figure, published versio
