Virasoro–Shapiro amplitude pole residue ($s_n=4(n-1)/{\alpha }^{\prime }$)

For $a=-1-{\alpha }^{\prime }s/4$, $b=-1-{\alpha }^{\prime }t/4$ and $a+b+c=1$, the amplitude has generic simple poles at $a=-n$. The gamma function residue and recurrence giveThe residue polynomial has degree $2n$, consistent with closed-string exchange up to spin $2n$. When $a=-n+{\delta }_a$ and $b=-m+{\delta }_b$, the reciprocal factor $1/\Gamma (a+b)$ removes a putative double pole: the leading ratio is ${\left(\genfrac{}{}{0ex}{}{n+m}{n}\right)}^2(1/{\delta }_a+1/{\delta }_b)$. Thus channel singularities must be counted using the kinematic constraint, not by multiplying independent numerator poles.