We study consequences of the Kawai-Lewellen-Tye (KLT) relations applied to
tree amplitudes in toroidal compactifications of string theory to four
dimensions. The closed string tree amplitudes with massless external states
respect a global SU(4)xSU(4) symmetry, which is enhanced to the SU(8)
R-symmetry of N=8 supergravity in the field theory limit. Our analysis focuses
on two aspects: (i) We provide a detailed account of the simplest
SU(8)-violating amplitudes. We classify these processes and derive explicit
superamplitudes for all local 5- and 6-point operators with SU(4)xSU(4)
symmetry at order alpha'^3. Their origin is the dilatonic operator exp(-6 phi)
R^4 in the closed-string effective action. (ii) We expand the 6-point closed
string tree amplitudes to order alpha'^3 and use two different methods to
isolate the SU(8)-singlet contribution from exp(-6 phi) R^4. This allows us to
extract the matrix elements of the unique SU(8)-invariant supersymmetrization
of R^4. Their single-soft scalar limits are non-vanishing. This demonstrates
that the N=8 supergravity candidate counterterm R^4 is incompatible with
continuous E_7(7) symmetry. From the soft scalar limits, we reconstruct to
quadratic order the SU(8)-invariant function of scalars that multiplies R^4,
and show that it satisfies the Laplace eigenvalue equation derived recently
from supersymmetry and duality constraints.Comment: 23 pages, published versio