Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 306 / 1 / v / Solution

Because the gamma function has simple poles at the nonpositive integers and no zeros, the $s$-dependent numerator $\Gamma (-{\alpha }^{\prime }s/4)$ has poles atorFactorization of a scattering amplitude identifies each pole with an intermediate on-shell state. The Type II superstring mass spectrum therefore contains a massless level at $n=0$ and an infinite equally spaced tower in squared mass for $n\ge 1$. There is no negative-$M^2$ pole, consistently with the absence of a tachyon in Type II superstring theory.