Joe's Big Book of String

1 = \int [d\zeta] \Delta_{FP}(g^\zeta) \delta(g^\zeta - \hat{g})

Since \Delta multiplies the delta-function it is defined only when its argument is on the gauge slice, but that is all that we will need. Now insert this into the functional integral 3.3.1. Use the gauge invariance of measure times e^{-S} to write

[dX dg] e^{-S(X,g)} = [dX^\zeta dg^\zeta] e^{-S(X^\zeta,g^\zeta)}

Now only X^\zeta and g^\zeta appear, so rename dummy variables

(X^\zeta,g^\zeta) \to (X,g)

and integrate over g to obtain 3.3.14. In particular, the obscure footnote on page 87 is avoided.

So how low can these scales be? The simplest bounds are that the compactification scale be at least $10^{-4}$ eV (the inverse of 1 mm) and the string scale be at least 1 TeV. The latter comes from the non-observation of string physics at particle accelerators. The former comes from the Cavendish experiment: if the extra dimension were larger the inverse square law would be seen to turn into an inverse cube. Remarkably, it is not easy to improve on these bounds. Proton stability, cosmology, and many other constraints must be taken into account, and unless the model is very contrived these may raise the scales somewhat, but this still leaves open the possibility of a very new picture of physics above the weak scale.

By the way, the discussion of figure 18.2 is for gauge fields in 4 dimensions and gravity in 5, but more generally the gauge fields may live in 4 and gravity in 4+k, with the rest at the string scale (that is, the gauge fields live on a 3-brane embedded in some higher space). The dimensionless gravitational coupling is $\kappa^2 E^{2+k} R^k$. Setting this to O(1) determines the approximate string scale. The reader can check that for k=2, the largest possible size (1 mm) gives also the lowest possible string scale (1 TeV). Incidentally, A. Hashimoto points out that the kink in figure 18.2 has been exaggerated, and corresponds to k = 3 rather than k = 1 as intended.

There have been various discussions over the years of large dimensions in which both gauge fields and gravity propagate. As noted in figure 18.1, taken by itself this is inconsistent with perturbative unification of gauge interactions and gravity, because the gauge coupling becomes strong while the gravitational coupling is still very weak. Again, strong interactions allow new possibilities. One that has been recently discussed is a hybrid, a kink in the gravitational coupling at a very low scale plus a kink in the gauge couplings at a higher scale such that they still meet. It has been argued that this can produce the accelerated gauge unification that is needed when the string scale is lowered.

One review of this subject is N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, "Phenomenology, Astrophysics and Cosmology of Theories with Sub-Millimeter Dimensions and TeV Scale Quantum Gravity," hep-ph/9807344.