ar X iv : h ep - p h / 03 02 20 8 v 3 1 9 M ay 2 00 3 Time varying α in N = 8 extended Supergravity

There has been some evidence that the fine structure “constant” α may vary with time. We point out that this variation can be described by a scalar field in some supergravity theory in our toy model, for instance, the N = 8 extended supergravity in four dimensions which can be accommodated in M-theory. Email address: [email protected],[email protected] There exists a lot of analysis about the observational constraints on possible variation of the fine structure constant α [1] in time. A number of absorption systems in the spectra of distant quasars suggests a smaller value of α in the past, with a favored value of the change δα α ∼= (−0.72 ± 0.18) × 10 over the red-shift range 0.5 ≤ z ≤ 3.5 [2]. And the analysis of the isotope abundances in the Oklo natural reactor operated 1.8 Gyr ago gives a new constraint, |δα| α ∼ 10 [3]. Some theoretical issues of a varying α are discussed in [4, 5, 6]. There are two scenarios to explain the variation of α. One possibility is that there was a first order phase transition between the time at which the quasar light was emitted and the present. But if the Oklo natural reactor confirms the above data which indicates that α at the red-shift z ∼ 0.13 is different from its present value, it is most possible that α varied with time continuously. On the other hand, in string or M-theory, the couplings in the effective field theory depend on the expectation values of some dynamical scalar fields such as the dilation and other string moduli, then the coupling “constants” in general vary with time, if they are not trapped in the minimum of a potential in an early time. Naturally the fine structure constant varies continuously in this scenario. In a four dimensional effective field theory, the change of the fine structure constant is controlled by a dynamical scalar field φ, which may be a combination of several canonically normalized moduli scalars [4, 5, 6]. The change in α during the last Hubble time requires that the scalar field φ should be extraordinarily light, with a mass comparable to the present Hubble scale H0 ∼ 10eV [4]. It is usually difficult to find a candidate for this scalar field in a fundamental theory. However it was found that one can describe the present state of quasi-exponential expansion of the universe in a broad class of models based on four dimensional N = 8 extended supergravity [7, 8, 9]. And it is important that there are scalars whose masses in N ≥ 2 supergravity are quantized in units of the Hubble constant H0 corresponding to DS solutions: m H 0 = −n, where n are some positive integers of the order 1. The minus sign says that the scalar fields are tachyonic, consistent with the fact that α becomes larger and larger. In particular, there is a scalar whose mass square of the scalar field m = −6H in N = 8 supergravity [7]. And we know N=8 supergravity with de Sitter maximim and one scalar field has only non-Abelian gauge fields SO(3)× SO(5) or SO(4)× SO(4). But in our real world the supersymmetry must be broken and then the Abelian gauge field will appear in our toy model. In an effective field theory or M-theory, the photon kinetic term reads f(φ)FμνF μν (1) where f(φ) is a function of φ. The most general expansion of the function α(φ) about its