• 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.


PO is now at the Physics Department of Bielefeld University!

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

205 submissions , 163 unreviewed
5,075 questions , 2,226 unanswered
5,348 answers , 22,757 comments
1,470 users with positive rep
818 active unimported users
More ...

  Allowed interactions in bosonic string theory

+ 5 like - 0 dislike

In a quantum field theory, only a finite set of interactions are allowed, determined by the Lagrangian of the theory which specifies the interaction vertex Feynman rules. In string theory, an $m$-point scattering amplitude is schematically given by,

$$\mathcal{A}^{(m)}(\Lambda_i,p_i) = \sum_{\mathrm{topologies}} g_s^{-\chi} \frac{1}{\mathrm{Vol}} \int \mathcal{D}X \mathcal{D}g \, \, e^{-S} \, \prod_{i=1}^m V_{\Lambda_i}(p_i)$$

where $S$ is the Polyakov action (Wick rotated), and $V_{\Lambda_i}(p_i)$ is a vertex operator corresponding to a state $\Lambda_i$ with momentum $p_i$. For example, for the tachyon,

$$V_{\mathrm{tachyon}} = g_s \int \mathrm{d}^2 z \, e^{ip_i \cdot X}$$

What I find troubling is that it seems the bosonic string theory does not impose any restrictions on which interactions may occur. For example, I could insert the vertex operators for a photon to compute photon scattering in the theory. But in the Standard Model direct interaction between photons is not permitted, but it could scatter via a fermion loop. So, how come any interaction is permitted in this theory? If this is an issue, how does the situation change in the case of the superstring?

This post imported from StackExchange Physics at 2014-04-13 14:41 (UCT), posted by SE-user JamalS

asked Apr 7, 2014 in Theoretical Physics by JamalS (895 points) [ revision history ]
edited Apr 19, 2014 by dimension10
I think that a general field theory also has infinitely many interactions, namely all those that are allowed by the symmetries of the theory (so not just anything). It is just in an appropriate low energy limit that you can ignore all but a few terms in your calculations. I guess this is intimately connected to issues of renormalization.

This post imported from StackExchange Physics at 2014-04-13 14:41 (UCT), posted by SE-user Danu
So the Standard Model is to be viewed as an effective field theory?

This post imported from StackExchange Physics at 2014-04-13 14:41 (UCT), posted by SE-user JamalS
It sure is! I'm pretty sure nobody thinks that this is the final theory... Furthermore, I believe it to be a key (philosophical) realization when studying these topics.

This post imported from StackExchange Physics at 2014-04-13 14:41 (UCT), posted by SE-user Danu

1 Answer

+ 2 like - 0 dislike

String theory reduces to ordinary field theory in the infinite string tension limit. In this limit, the massive modes are decoupled and we are left with solely, the massless modes.

In fact, Bern and Kosower (Please, see a modern review by Christian Schubert. ) proved that the computation of the field theory amplitudes from the string amplitudes at the infinite string tension limit has many advantages:

The group theory and symmetry factors of the loops are all embedded in the string amplitude and thus are trivial. A string loop diagram includes a sum of many field theory loop diagrams. Gamma matrix manipulations are not needed in general. In addition, the integration over the string moduli space for multi-loop string amplitudes reduces to a sum over corners of the moduli space orbifold in the infinite tension limit. Also, the compactification of extra dimensions (in the appropriate string theory) adds only trivial factors to the field theory amplitude.

The main difficulty is that the string theory amplitudes are on-shell, but in certain cases it can be extended to an off-shell magnitude.

Referring to your specific question, the bosonic string is not the appropriate string theory for QED photon scattering amplitudes because it does not contain massless fermions in the spectrum. For (spinor) QED, one needs to consider an appropriate heterotic string as Bern and Kosower did in their article. When the appropriate theory is taken, the number of loops of the string amplitude is the same of the field theory number of loops in the infinite string tension limit.

However, from an open bosonic string theory, one can obtain pure gluon interactions and in fact, the gluonic beta function was obtained from this formalism even prior to Bern and Kosower work. (Of course, the QED beta function can also be obtained from the Heterotic theory).

It is important to mention that the string magnitudes are consistent only when the conformal symmetry is maintained. Thus, in this formalism, masses are obtained from infrared regularization.

This post imported from StackExchange Physics at 2014-04-13 14:41 (UCT), posted by SE-user David Bar Moshe
answered Apr 7, 2014 by David Bar Moshe (4,355 points) [ no revision ]

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