Specifically, $${\mathbb{R}}^4=32(R_1{)}^{\mu \nu \rho \sigma }(R_2{{{{)}_{\mu }}^{\alpha }}_{\rho }}^{\beta }(R_3{{{{)}_{\alpha }}^{\gamma }}_{\nu }}^{\delta }(R_4{)}_{\beta \gamma \sigma \delta }$$$$-8(R_1{)}^{\mu \nu \rho \sigma }(R_2{{)}_{\mu \nu }}^{\alpha \beta }(R_3{{)}_{\rho \alpha }}^{\gamma \delta }(R_4{)}_{\sigma \beta \gamma \delta },$$plus all other permutations of the labels 1,2,3,4, where $(R_i{)}^{\mu \nu \rho \sigma }$ is the linearized Riemann tensor, $(p_i^{\nu }{p}_i^{\sigma }{\epsilon }_i^{\mu \rho }-p_i^{\nu }{p}_i^{\rho }{\epsilon }_i^{\mu \sigma }-p_i^{\mu }{p}_i^{\sigma }{\epsilon }_i^{\nu \rho }+p_i^{\mu }{p}_i^{\rho }{\epsilon }_i^{\nu \sigma })/2$. Alternatively, for scattering scalars minimally coupled to gravity, we simply replace ${\mathbb{R}}^4$ with $(s^4+t^4+u^4)/2$.

The string amplitude in Eq. (2) is not of the form in Eq. (12). Instead, $-\mathrm{\Gamma }(-s)/\mathrm{\Gamma }(1+s)$ can be written as ${\mathrm{\Sigma }}_{n=0}^{\infty }{f}_n(s)/(-s+n)$, where $f_n(s)=[{\mathrm{\Pi }}_{i=0}^n(-2s-1+i)]/[n!\mathrm{\Gamma }(2+2s)]$, i.e., the $g_n^2$ are replaced in ${\mathcal{M}}_{\mathrm{type}\text{−}\mathrm{II}}$ by analytic functions of $s$ of indefinite sign on each pole in $\mathcal{A}$.

The nongravitational amplitude in Eq. (5) has a very similar EFT expansion, but with the ${\mathbb{R}}^4$ stripped off and the indices $i$, $j$, $k$ starting at 1 rather than 0.