Allowed interactions in bosonic string theory

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?

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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. –  Danu Apr 7 '14 at 10:07
So the Standard Model is to be viewed as an effective field theory? –  JamalS Apr 7 '14 at 10:15
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. –  Danu Apr 7 '14 at 10:18

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.

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