• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

204 submissions , 162 unreviewed
5,030 questions , 2,184 unanswered
5,344 answers , 22,705 comments
1,470 users with positive rep
816 active unimported users
More ...

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

+ 3 like - 0 dislike

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
asked Oct 2, 2014 in Theoretical Physics by Trimok (955 points) [ no revision ]

1 Answer

+ 3 like - 0 dislike

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).

answered Oct 2, 2014 by 40227 (5,140 points) [ revision history ]

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).

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights