Open string 4-tachyon amplitude for cylinder/annulus topology in bosonic string theory

Question

One knows the formula for the open string $4$-tachyon amplitude for the disk topology in the bosonic string theory : it is proportortionnal 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 serie of poles of $s$ or $t$, analogeous to the known series for the Veneziano amplitude ?

This post imported from StackExchange Physics at 2014-10-02 12:02 (UTC), posted by SE-user Trimok

Answer 1

Open string one-loop amplitudes are the subject of the Section 8.1 of Green, Schwarz, Witten's book on string theory (it is at the very beginning of volume 2). In particular, one can find the N tachyons amplitude at one-loop for the annulus, with the N tachyons inserted on the same side of the annulus, as formula (8.1.35) and the more general case, with K tachyons on one side and N-K tachyons on the other side, as formula (8.1.77). These formulas are integral expressions, similar to the integral expression of the Veneziano amplitude. The various poles expansions for various limits in the space of external momenta are discussed after the statement of the formulas (even if these series expansions are not written explicitely, they should be easy to recover from the discussion).

Comments

Before OP goes on a goose chase, you should say there is no simple sum-of-poles expansion for loop diagrams, you get cuts from loops, since the intermediate momenta are indeterminate and integrated over. The cut discontinuity can be found from the Veneziano amplitudes by Cutkowsky rules or their analogs, the pole structure is only from the tree level stuff, I believe the full string loop amplitude is in principle uniquely reconstructed from the physical cut-discontinuity (which is determined from the tree level amplitude) by Mandelstam-style dispersion relations, although once you have the intuitive string world-sheet methods, you don't bother with that stuff, you just do the integral moduli space (but then you are taking for granted that this is a consistent unitary expansion).