5.2 High-energy theories and varying constants

5.2.1 Kaluza–Klein

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 ([409Jump To The Next Citation Point], chapter 13)

∫ --- S = ---1---- ¯R √ − ¯gd5x. 12π2G5
Decomposing the 5-dimensional metric ¯g AB as
( g + A-μAνϕ2 Aμϕ2 ) ( μν AνM22 M 2 ) ¯gAB = M ϕ ϕ ,
where M is a mass scale, we obtain
1 ∫ ( ϕ2 ) √ --- S = ------- R − ----2F 2 ϕ − gd4x, (171 ) 16 πG ∗ 4M
where the 4-dimensional gravitational constant is G ∗ = 3πG5 ∕4∫ dy. The scalar field couples explicitly to the kinetic term of the vector field and cannot be eliminated by a redefinition of the metric: again, this is the well-known conformal invariance of electromagnetism in four dimensions. Such a term induces a variation of the fine-structure constant as well as a violation of the universality of free-fall. Such dependencies of the masses and couplings are generic for higher-dimensional theories and in particular string theory. It is actually one of the definitive predictions for string theory that there exists a dilaton, that couples directly to matter [484Jump To The Next Citation Point] and whose vacuum expectation value determines the string coupling constants [546Jump To The Next Citation Point].

In the models by Kaluza [269] and Klein [291] the 5-dimensional spacetime was compactified assuming that one spatial extra-dimension S1, of radius RKK. It follows that any field χ (xμ,y) can be Fourier transformed along the compact dimension (with coordinate y), so that, from a 4-dimensional point of view, it gives rise to a tower of of fields (n) μ χ (x ) of mas mn = nRKK. At energies small compared to −1 R KK only the y-independent part of the field remains and the physics looks 4-dimensional.

Assuming that the action (171View Equation) corresponds to the Jordan frame action, as the coupling ϕR may suggest, it follows that the gravitational constant and the Yang–Mills coupling associated with the vector field μ A must scale as

G ∝ ϕ−1, g−Y2M ∝ ϕ2∕G ∝ ϕ3. (172 )
Note that the scaling of G with ϕ (or time) is not the one of the gravitational constant that would be measured in a Cavendish experiment since Eq. (165View Equation) tells us that ( ) Gcav ∝ G∗ϕ −1 1 + 2ϕ1+3-.

This can be generalized to the case of D extra-dimensions [114] to

G ∝ ϕ −D, αi(mKK ) = Ki (D )G ϕ− 2 (173 )
where the constants Ki depends only on the dimension and topology of the compact space [525] so that the only fundamental constant of the theory is the mass scale M4+D entering the 4 + D-dimensional theory. A theory on ℳ4 × ℳD where ℳD is a D-dimensional compact space generates a low-energy quantum field theory of the Yang–Mills type related to the isometries of ℳ D [for instance [545] showed that for D = 7, it can accommodate the Yang–Mills group SU (3) × SU (2) × U (1)]. The two main problems of these theories are that one cannot construct chiral fermions in four dimensions by compactification on a smooth manifold with such a procedure and that gauge theories in five dimensions or more are not renormalizable.

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 D ≥ 4 dimensions, endowed with an invariant UV cut-off Λ, and containing a large number N of non-self-interacting matter species.

Ref. [295] used the variation (173View Equation) to constrain the time variation of the radius of the extra dimensions during primordial nucleosynthesis to conclude that|ΔRKK ∕RKK | < 1%. [28] took the effects of the variation of α ∝ R− 2 S KK and deduced from the helium-4 abundance that |ΔR ∕R | < 0.7% KK KK and |ΔRKK ∕RKK | < 1.1% respectively for D = 2 and D = 7 Kaluza–Klein theory and that −10 |ΔRKK ∕RKK | < 3.4 × 10 from the Oklo data. An analysis of most cosmological data (BBN, CMB, quasar etc..) assuming that the extra dimension scales as R0 (1 + Δt −3∕4) and R0 [1 + Δ ](1 − cosω(t − t0)) concluded that Δ 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. [336Jump To The Next Citation Point] evaluated the effect of such a couple variation of G 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.

5.2.2 String theory

There exist five anomaly-free, supersymmetric perturbative string theories respectively known as type I, type IIA, type IIB, SO(32) heterotic and E × E 8 8 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) gμν and a rank two antisymmetric tensor B μν. 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 N = 1 and the quantization of the left-moving modes imposes that the gauge group is either SO (32) or E8 × E8 depending on the fermionic boundary conditions. The effective tree-level action is (see, e.g., Ref. [237])

∫ [ 6 ] S = d10x √ − g-e− 2Φ M 8{R + 4□ Φ − 4(∇ Φ)2} − M-HF F AB + ... . (174 ) H 10 H 10 4 AB
When compactified on a 6-dimensional Calabi–Yau space, the effective 4-dimensional action takes the form
[ { } ] ∫ √ ---- ( ∇ϕ )2 1( ∇V )2 M 6 SH = d4x − g4ϕ M 8H R4 + ---- − -- ---6- − --H-F 2 + ... (175 ) ϕ 6 V6 4
where − 2Φ ϕ ≡ V6e couples identically to the Einstein and Yang–Mills terms. It follows that
M 24 = M H8ϕ, g −Y2M = M 6Hϕ (176 )
at tree-level. Note that to reach this conclusion, one has to assume that the matter fields (in the ‘dots’ of Equation (175View Equation) are minimally coupled to g4; see, e.g., [340Jump To The Next Citation Point]).

The strongly coupled SO (32) 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 N = 1, D = 10 supergravity, which takes the form, in the string frame,

∫ [ 2 ] S = d10x √ − g-M 6e−Φ e−ΦM 2R − F--+ ... (177 ) I 10 I I 10 4
where the dots contains terms describing the dynamics of the dilaton, fermions and other form fields. At variance with (174View Equation), the field Φ couples differently to the gravitational and Yang–Mills terms because the graviton and Yang–Mills fields are respectively excitation of close and open strings. It follows that MI can be lowered even to the weak scale by simply having expΦ small enough. Type I theories require D9-branes for consistency. When V6 is small, one can use T-duality (to render V6 large, which allows to use a quantum field theory approach) and turn the D9-brane into a D3-brane so that
∫ ∫ 10 √ -----− 2Φ 8 4 √ ---- −Φ 1 2 SI = d x − g10e M I R10 − d x − g4e -F + ... (178 ) 4
where the second term describes the Yang–Mills fields localized on the D3-brane. It follows that
M 2 = e− 2ΦV6M 8, g−2 = e−Φ (179 ) 4 I YM
at tree-level. If one compactifies the D9-brane on a 6-dimensional orbifold instead of a 6-torus, and if the brane is localized at an orbifold fixed point, then gauge fields couple to fields Mi living only at these orbifold fixed points with a (calculable) tree-level coupling c i so that
2 − 2Φ 8 −2 −Φ M 4 = e V6M I , gYM = e + ciMi. (180 )
The coupling to the field c i is a priori non universal. At strong coupling, the 10-dimensional E8 × E8 heterotic theory becomes M-theory on 10 1 R × S ∕Z2 [255]. The gravitational field propagates in the 11-dimensional space while the gauge fields are localized on two 10-dimensional branes.

At one-loop, one can derive the couplings by including Kaluza–Klein excitations to get [163]

b g−Y2M = M 6Hϕ − -a (RMH )2 + ... (181 ) 2
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, [340Jump To The Next Citation Point] considered the (N = 1,D = 10)-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 R (xμ) so that the effective 4-dimensional theory described by (175View Equation) is of the Brans–Dicke type with ω = − 1. Assuming that ϕ has a mass μ, and couples to the matter fluid in the universe as ∫ 10 √ ----- Smatter = d x − g10exp(− 2Φ )ℒmatter(g10), the reduced 4-dimensional matter action is

∫ S = d4x √ − g-ϕℒ (g). (182 ) matter matter
The cosmological evolution of ϕ and R can then be computed to deduce that 10 αE˙M ∕αEM ≃ 10 (μ∕1 eV )−2 yr−1. considered the same model but assumed that supersymmetry is broken by non-perturbative effects such as gaugino condensation. In this model, and contrary to [340], ϕ is stabilized and the variation of the constants arises mainly from the variation of R in a runaway potential.

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.

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