= Odd-level pole cancellation in the symmetric Veneziano amplitude
{title2=$\operatorname*{Res}A_{\rm sym}\propto[1+(-1)^n]\prod_{j=1}^n(\alpha't+j+1)$}
For identical open-string tachyons, $s+t+u=-4/\alpha'$. At $\alpha's=n-1$, the ordered <Veneziano amplitude> residues in the $s,t$ and $s,u$ terms differ by $(-1)^n$, so their symmetric sum cancels odd-level $s$-channel poles at generic $t$. In particular the massless vector does not couple to this identical-scalar singlet amplitude. It remains a state of the open-string spectrum and is visible in suitable ordered or color-dependent amplitudes. Pole absence due to an external-state selection rule must not be interpreted as absence of the exchanged state from the theory.
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