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One knows the formula for the open string $4$-tachyon amplitude for the disk topology in the bosonic string theory : it is proportional to the $s \leftrightarrow t \leftrightarrow u$ symmetrisation of the Veneziano amplitude $B(- \alpha's-1, - \alpha't-1)$.

However, what is the explicit value of the open string $4$-tachyon (one-loop) amplitude for the cylinder/annulus topology in the bosonic string theory ?

If possible, what is the expression of this amplitude in a series of poles of $s$ or $t$, analogous to the known series for the Veneziano amplitude ?

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    $\begingroup$ Since there are two boundaries at one-loop, there will be more than one possibility to insert the tachyon vertex operators. See if there is a discussion in Green-Schwarz-Witten vol II. $\endgroup$ – suresh Oct 2 '14 at 12:23
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    $\begingroup$ Open string one-loop amplitudes are for example explained in Section 8.1 of Green, Schwarz, Witten's book, see some details here. $\endgroup$ – Dilaton Oct 2 '14 at 22:06

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