Because the <gamma function> has simple poles at the nonpositive integers and no zeros, the $s$-dependent numerator $\Gamma(-\alpha's/4)$ has poles at
$$
-\frac{\alpha's}{4}=-n,
\qquad n=0,1,2,\ldots,
$$
or
$$
\boxed{s=M_n^2=\frac{4n}{\alpha'}}.
$$
Factorization 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 $n\geq1$. There is no negative-$M^2$ pole, consistently with the absence of a tachyon in <Type II superstring theory>.

Solution

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

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{wiki=Type_II_string_theory}

Type II superstring theory is a ten-dimensional closed oriented superstring theory with spacetime supersymmetry and no tachyon. Its massive closed-string levels have squared masses separated by $4/\alpha'$.


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 Incoming links: Type II superstring theory