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Sorry if this is a duplicate. I have read that string theory has a good explanation for the "hierarchy" problem of the force strength disparity between the fundamental forces because as gravity weakens proportional to the inverse square law the effect is intensified by "leakage" into "extra dimensions". Then I read about the weakness of he cosmological constant begin the worse predicted value of physics in some order of magnitude $100^{120}$ powers weaker than what the vacuum energy of quantum mechanics might predict. I was wondering if the extra dimensions that "leak" gravity might also "leak" the energy of the cosmological constant and does string theory give some prediction as to how much? It may be the case that it's only gravity that is affected by extra dimensions for some very specific reason so my question may be very naive at best.

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Gravity does not "leak" in string theory, at least not in any meaningful sense from the viewpoint of the effective four-dimensional theory we are interested in. The prediction for the strength of four-dimensional gravity is simply the size of the effective four-dimensional gravitational constant $G^{(4)}$. Likewise, the cosmological constant does not leak. It is simply the vacuum expectation value of energy in the effective four-dimensional theory.

The actual value of these things depends of course on the original ten-dimensional gravitational constant $G^{(10)}$ which is generally believed to be on the order of its corresponding Planck scale since there's nothing else setting a scale a priori. But even more crucially it depends on how you get to your four-dimensional effective theory. There is no generic prediction of string theory for this, this is the landscape of "string vacua": Possible compactifications of the six extra dimension that lead to a consistent low-energy effective theory.

For instance, in a simple model in which the six dimensions are simply rolled up as circles of radius $r_i$, we get something like $G^{(10)} = \left(\prod_{i=4}^9 r_i\right) G^{(4)}$, but the rest of the resultant theory is nothing like the Standard Model. The general idea holds, though - the size of the compactified extra dimensions determines the ratio between the four- and ten-dimensional gravitational strengths.

The cosmological constant - the vacuum energy - is different: Many stringy model preserve supersymmetry, but unbroken supersymmetry imposes zero energy of the ground state, meaning supersymmetric models cannot straightforwardly generate a non-zero cosmological constant. Non-supersymmetric models can, but the quantum corrections to this vacuum energy are typically both large and hard to compute. It is not the generic size of the extra dimensions, but the actual details of the quantum theory generated that determine this.

Finally, compactification is not the only thing that can generate scales. Braneworld models do not arrive at the effective four-dimensional theory solely through the dimensional reduction on compactified dimensions, but conceive of our universe as a "brane" inside a (not necessarily compactified) higher-dimensional space, and the restriction of physics to this brane then yields the effective theory.

For instance, in the Randall-Sundrum model, the "full" universe is five-dimensional and there's a gradient over this universe where the weak and the gravitational force are "equal" on one end, but the hierarchy between them emerges as one moves to the other end, where "our universe" sits. This resolves the hierarchy problem not through the specific geometry of the extra dimensions, but through the specific physics assumed to be realized in them.

In the end, asking what "string theory" predicts for the value of the gravitational or cosmological constant is as non-sensical as asking what "quantum field theory" predicts as the mass of some particle - it wholly depends on what specific theory you're looking at. In QFT, we have settled upon the Standard Model as best describing our world, but there is no "Standard Model of string theory" as of yet. Phenomenological investigations of all sorts of models and their specific predictions are still on-going.

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