  Positivity bounds are powerful tools to constrain effective field theories.
Utilizing the partial wave expansion in the dispersion relation and the full
crossing symmetry of the scattering amplitude, we derive several sets of
generically nonlinear positivity bounds for a generic scalar effective field
theory: We refer to these as the $PQ$, $D^{\rm su}$, $D^{\rm stu}$ and
$\bar{D}^{\rm stu}$ bounds. While the $PQ$ bounds and $D^{\rm su}$ bounds only
make use of the $s\leftrightarrow u$ dispersion relation, the $D^{\rm stu}$ and
$\bar{D}^{\rm stu}$ bounds are obtained by further imposing the
$s\leftrightarrow t$ crossing symmetry. In contradistinction to the linear
positivity for scalars, these inequalities can be applied to put upper and
lower bounds on Wilson coefficients, and are much more constraining as shown in
the lowest orders. In particular we are able to exclude theories with soft
amplitude behaviour such as weakly broken Galileon theories from admitting a
standard UV completion. We also apply these bounds to chiral perturbation
theory and we find these bounds are stronger than the previous bounds in
constraining its Wilson coefficients.
