Such coupling terms naturally appear when compactifying a higher-dimensional theory. As an
example, let us recall the compactification of a 5-dimensional Einstein–Hilbert action ([409],
chapter 13)
In the models by Kaluza [269] and Klein [291] the 5-dimensional spacetime was compactified assuming
that one spatial extra-dimension , of radius
. It follows that any field
) can be Fourier
transformed along the compact dimension (with coordinate
), so that, from a 4-dimensional point of
view, it gives rise to a tower of of fields
of mas
. At energies small
compared to
only the
-independent part of the field remains and the physics looks
4-dimensional.
Assuming that the action (171) corresponds to the Jordan frame action, as the coupling
may
suggest, it follows that the gravitational constant and the Yang–Mills coupling associated with the vector
field
must scale as
This can be generalized to the case of extra-dimensions [114] to
In such a framework the variation of the gauge couplings and of the gravitational constant arises from
the variation of the size of the extra dimensions so that one can derives stronger constraints that by
assuming independent variation, but at the expense of being more model-dependent. Let us mention the
works by Marciano [345] and Wu and Wang [550] in which the structure constants at lower energy are
obtained by the renormalization group, and the work by Veneziano [515] for a toy model in
dimensions, endowed with an invariant UV cut-off
, and containing a large number
of
non-self-interacting matter species.
Ref. [295] used the variation (173) to constrain the time variation of the radius of the extra dimensions
during primordial nucleosynthesis to conclude that
. [28] took the effects of the
variation of
and deduced from the helium-4 abundance that
and
respectively for
and
Kaluza–Klein theory and
that
from the Oklo data. An analysis of most cosmological data
(BBN, CMB, quasar etc..) assuming that the extra dimension scales as
and
concluded that
has to be smaller than 10–16 and 10–8 respectively [311],
while [330] assumes that gauge fields and matter fields can propagate in the bulk, that is in the extra
dimensions. Ref. [336
] evaluated the effect of such a couple variation of
and the structures constants on
distant supernova data, concluding that a variation similar to the one reported in [524] would make the
distant supernovae brighter.
There exist five anomaly-free, supersymmetric perturbative string theories respectively known as type I,
type IIA, type IIB, SO(32) heterotic and heterotic theories (see, e.g., [420]). One of
the definitive predictions of these theories is the existence of a scalar field, the dilaton, that
couples directly to matter [484] and whose vacuum expectation value determines the string
coupling constant [546]. There are two other excitations that are common to all perturbative
string theories, a rank two symmetric tensor (the graviton)
and a rank two antisymmetric
tensor
. The field content then differs from one theory to another. It follows that the
4-dimensional couplings are determined in terms of a string scale and various dynamical fields (dilaton,
volume of compact space, …). When the dilaton is massless, we expect three effects: (i) a scalar
admixture of a scalar component inducing deviations from general relativity in gravitational
effects, (ii) a variation of the couplings and (iii) a violation of the weak equivalence principle.
Our purpose is to show how the 4-dimensional couplings are related to the string mass scale,
to the dilaton and the structure of the extra dimensions mainly on the example of heterotic
theories.
To be more specific, let us consider an example. The two heterotic theories originate from the fact that
left- and right-moving modes of a closed string are independent. This reduces the number of supersymmetry
to and the quantization of the left-moving modes imposes that the gauge group is either
or
depending on the fermionic boundary conditions. The effective tree-level action is (see,
e.g., Ref. [237])
The strongly coupled heterotic string theory is equivalent to the weakly coupled type I string
theory. Type I superstring admits open strings, the boundary conditions of which divide the number of
supersymmetries by two. It follows that the tree-level effective bosonic action is
,
supergravity, which takes the form, in the string frame,
At one-loop, one can derive the couplings by including Kaluza–Klein excitations to get [163]
when the volume is large compared to the mass scale and in that case the coupling is no more universal. Otherwise, one would get a more complicated function. Obviously, the 4-dimensional effective gravitational and Yang–Mills couplings depend on the considered superstring theory, on the compactification scheme but in any case they depend on the dilaton. As an example, [340] considered the (
)-supergravity model derived from the heterotic
superstring theory in the low energy limit and assumed that the 10-dimensional spacetime is compactified
on a 6-torus of radius
so that the effective 4-dimensional theory described by (175
) is of the
Brans–Dicke type with
. Assuming that
has a mass
, and couples to the matter fluid in
the universe as
, the reduced 4-dimensional matter action is
Ref. [290] considers a probe D3-brane probe in the context of AdS/CFT correspondence at finite temperature and provides the predictions for the running electric and magnetic effective couplings, beyond perturbation theory. It allows to construct a varying speed of light model.
To conclude, superstring theories offer a natural theoretical framework to discuss the value of the fundamental constants since they become expectation values of some fields. This is a first step towards their understanding but yet, no complete and satisfactory mechanism for the stabilization of the extra dimensions and dilaton is known.
It has paved the way for various models that we detail in Section 5.4.
http://www.livingreviews.org/lrr-2011-2 |
Living Rev. Relativity 14, (2011), 2
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