Brane World versus Kaluza-Klein

It is important to realise the difference between the brane world and the better known Kaluza-Klein scenario. In Kaluza-Klein all fields feel the extra dimensions whereas in the brane world, only a subset of the fields (gravity and moduli fields in string theory) feel all the extra dimensions. This simple fact has very important physical implications regarding the possible values of the fundamental scale. An explicit way to see the difference is comparing the low-energy effective actions for perturbative heterotic strings and type I strings. In the heterotic case, both gravity and the gauge fields live on the full 10-dimensional spacetime corresponding to a standard Kaluza-Klein scenario. The low-energy effective action in 10 dimensions takes the form:

$$S\ = \ M^8\ \int d^{10}x\sqrt{-G}\ e^{-\phi}\left( {\cal R}\ + \ M^{-2}\ F_{MN}^2 \ +\ \cdots \right),$$ (47)

where $M=1/\sqrt{\alpha'}$ is the string scale and $\phi$ is the dilaton field. Upon compactification to 4-dimensions each of the two terms in the action above will get a volume factor coming from the integration of the 6 extra dimensions. This gives us an expression for the gravitational and gauge couplings (the numerical coefficients of each of the two terms above) of the form:

$$M_{Planck}^2\ \sim \ e^{-\phi}\ M^8 r^6\qquad \alpha_{GUT}^{-1}\ \sim\ e^{-\phi}M^6 r^6$$ (48)

where $r$ is the overall size of the extra dimensions. Taking the ratio of those expressions the volume factors cancel and we get $M_{Planck}^2\sim \alpha_{GUT}^{-1} M^2$. Therefore for $\alpha_{GUT}$ not much different from 1 (as expected) we have to have the fundamental scale $M$ to be of the same order of magnitude as the gravitational scale $M_{Planck}\sim 10^{19}$ GeV. This was the old belief that the string scale was the Planck scale. Things are very different in the brane world scenario as we can see for the case of the type I string. For a configuration with the standard model spectrum belonging to a Dp-brane, the low energy action in 4-dimensions takes the form:

$$S\ =\ -\frac{1}{2\pi} \int d^4x\sqrt{-g}\left({r^6\ M^8}{\... ...4}{(rM)^{p-3}}{\ e^{-\phi}}\ F_{\mu\nu}^2\ +\ \cdots \right)$$ (49)

Comparing the coefficient of the Einstein term with the physical Planck mass $M_{Planck}^2$ and the coefficient of the gauge kinetic term with the physical gauge coupling constant $\alpha_p$($\sim 1/24$ at the string scale), we find the relation:

$$M^{7-p}\ =\ \frac{\alpha_p}{\sqrt 2}\ M_{Planck}\ r^{p-6}$$ (50)

from which we can easily see that if the Standard Model fits inside a D3-brane, for instance, we may have $M$ substantially smaller than $M_{Planck}$ as long as the sizes of the extra dimensions are large enough. Given the fact that we do not have a way to fix the size of the extra dimensions we can take advantage of our ignorance and follow a bottom-up approach considering different possibilities motivated by phenomenological inputs. Several scenarios have been proposed depending on the value of the fundamental scale. The four main scenarios at present correspond to

1. $M\sim M_{Planck}$. This is just the old perturbative heterotic string case corresponding to compactification scale close to the Planck scale. There is nothing wrong with this possibility. Research over the years has shown difficult to obtain gauge coupling unification in this case.
2. $M\sim M_{GUT}\sim 10^{16}$ GeV. Obtained for $r\sim 10^{-30} cm$ in the expression above. This proposal [111] was made precisely to `solve' the gauge coupling unification problem in string theories. This requires a compactification scale of order $10^{14}$ GeV. Recent progress has been made [42] in looking for three generation models realising this scenario from the Horava-Witten construction but, so far, not from type I models.
3. $M\sim M_I\sim 10^{10-12}$ GeV. If the world is a D3-brane we can see that this scale is obtained from the equations above for $r\sim 10^{-23}cm$. This proposal [112], was based on the special role played by the intermediate scale $M_I$ in different ideas beyond he standard model. Particularly the scale of supersymmetry breaking in gravity mediated supersymmetry breaking scenario. This then allows to identify the string scale with the supersymmetry breaking scale and opens up the room for non supersymmetric string models to be relevant at low-energies. Explicit models realising this scenario will be discussed in the next section.
4. $M\sim M_{EW}\sim 10^3$ GeV. This is obtained for overall radius $r\sim 10^{-12}cm$ above and if only two of the six dimensions were large this would have given us the famous $r\sim 1 mm$ quoted as the extreme case of the brane world scenario since lengths bigger than this would have been observed by deviations of gravity. This is the most popular scenario [113] due to its proximity with experiment. It has opened up a completely new approach towards looking for physics beyond the standard model at present and future experiments, especially after the work of Arkani-Hamed, Dimopoulos and Dvali [114] where a detailed analysis was done about the possible experimental, astrophysical and cosmological constraints of this scenario which range from comparisons with Van der Waals forces in molecules to overcooling of supernovae, especially supernova 1987a. Concrete string models realising this scenario do not exist. We will discuss some attempts in the next section.

Notice that only the first scenario was possible following the standard Kaluza-Klein approach in the perturbative heterotic string models. The brane world opened up the possibility of the next three scenarios as well as any other scale in the range $M_{EW}<M<M_{Planck}$.