Space–time variation of the electron-to-proton mass ratio in a Weyl model

Abstract

Seeking a possible explanation for recent data indicating a space–time variation of the electron-to-proton mass ratio within the Milky Way, we consider a phenomenological model where the effective fermion masses depend on the local value of the Weyl tensor. We contrast the required values of the model’s free parameters with bounds obtained from modern tests on the violation of the weak equivalence principle and we find that these quantities are incompatible. This result indicates that the variation of nucleon and electron masses through a coupling with the Weyl tensor is not a viable model.

Introduction

The search for space–time dependence of fundamental constants plays a fundamental role in the continuous efforts to put in firmer empirical grounds our current physical theories and, at the same time, explore the possibilities of exotic physics that might become manifest trough small deviations. The experimental research can be grouped into astronomical and local methods. The latter ones include geophysical methods such as the natural nuclear reactor that operated about 1.8 × 10⁹ years ago in Oklo, Gabon [18], [45], [26], the analysis of natural long-lived β decays in geological minerals and meteorites [41] and laboratory measurements such as comparisons of rates between clocks with different atomic numbers [46], [2], [35], [8], [24], [44]. The astronomical methods are based mainly on the analysis of high-redshift quasar absorption systems. Most of the reported data are, as expected, consistent with null variation of fundamental constants. Nevertheless, there are reports of intriguing results. For instance Webb et al. [55] and Murphy et al. [38] have reported observations made with the Keck telescope which suggest a smaller value of the fine structure constant (α) at high redshift as compared with its local value. However, an independent analysis performed with VLT/UVES data gave null results [50]. Furthermore, a recent analysis using VLT/UVES data suggests also a variation in α but in the opposite sense, that is, α appears to be larger in the past [54]. The discrepancy between Keck/HIRES and VLT/UVES is yet to be resolved. In particular, the two studies rely on data from different telescopes observing different hemispheres and it was pointed out that the Keck/Hires and VLT/UVES observations can be made consistent in the case where the fine structure constant is spatially varying [54].

Focusing on a different quantity, observations of molecular hydrogen in quasar absorption systems can be used to set constraints on the electron-to-proton mass ratio μ ≡ m_e/m_p at high redshift [29], [53], [34], while the present value of μ can be constrained using comparisons of different transitions in atomic clocks [8], [24], [44]. Furthermore, the observed temperature isotropy of microwave background radiation can be used to set bounds on the spatial variation of μ at extragalactic scales [4]. Surprisingly, a recent analysis of ammonia spectra in the Milky Way suggests a spatial variation of μ [36], [31], [32]. The study, comparing the spectral lines of the ammonia inversion transition and rotational transitions of other molecules with different sensitivities to the parameter μ, finds a statistically significant velocity offset that when interpreted in terms of a variation in μ gives Δμ/μ = (2.2 ± 0.7) × 10⁻⁸. This will be the focus of the present paper. If we assume that the latter is not the result of some fluke and systematic experimental error, and thus take the result quite seriously, we are naturally led to the following question: What would be the simplest modification of our present physical theories that might account for such phenomena? One of the simplest possibilities one can think of is that the effective value of the coupling constants changes with space–time location. In this sense we note that, within the context of theories that are at the fundamental level background independent, the study of possible space–time dependence of fundamental constants is often considered as equivalent to the search for the existence of dynamical fields which couple to the gauge fields and/or to ordinary matter in ways that mimic the ordinary coupling constants.

There have been several proposals along those lines with various different motivations. Some of them arise from proposals for basic theories that arise in the search for unification of the four fundamental laws of physics such as string-derived field theories [58], [33], [3], [19], [20], [21], related brane-world theories [59], [60], [43], [12] and Kaluza–Klein theories [27], [30], [56], [25], [42]. There are also phenomenological models where a scalar field ϕ couples to the Maxwell tensor F_μν and are characterized by Lagrangian density terms such as −B_F(ϕ)F_μνF^μν/4 [7], [6], [39], [57], [22] and/or to the matter fields Ψ_i(i labeling the field flavor) as $B_i(\varphi ){\overline{\Psi }}_i{\Psi }_i$[39], [28], [13], [5], [40], [57], [22]. If these terms were added to the standard Lagrangian densities, it is quite clear that, in the first case, it would result in something like an effective fine structure constant $1/e_{\mathit{effective}}^2=1/e^2+B_F(\varphi )$ while in the second case the effective masses of the elementary particles would be given by $m_i^{\mathit{eff}}=m_i+B_i(\varphi )$. Then, if the field took space–time dependent values, a feature that often requires the new field to be quite light so its value is not too rigidly tied to the minima of any self-interaction potential, then the effective fine structure constant and/or effective masses might look space–time dependent. This part of the story is quite clear, however, one can not focus on just this aspect of the theory when considering it. In fact, it is often the case that the most important bounds on the theory do not arise from the direct search for this dependence but from the effects of the direct exchange of quanta of this putative field would have on the behavior of ordinary matter. The fact that the scalar field must be light, as we have just described, indicates that this quanta exchange would not be drastically suppressed by a large mass in its propagator [52]. This generically leads to modification of the free fall and very often to signals that would mimic violations of the weak equivalence principle (WEP).

In fact, the connection between the theories involving spacetime dependence of coupling constants and the WEP was recognized already in the 1960s by Dicke [23] (see page 163). More recently, Damour [17] noted the extreme difficulty in mimicking the behavior of the mass of an object with any coupling that is not the gravitational one. This means that if the modified theory is required to be covariant, then the only possibility to ensure an exact compliance with the WEP is essentially to restrict the new fields to couple to matter in the same way as gravity, and this is impossible if we want the effective coupling constants to be scalar functions of some new fields. Consequently, one finds that the two issues are generically considered simultaneously in attempts to deal with possible variations of fundamental constants [19], [20], [21], [7], [3], [42], [48], [5], [39], [57], [22]. Thus, it is clear that we must face this connection in any attempt to deal with the problem at hand, and that in so doing we must consider the most modern and stringent bounds that are currently available on the possible violations of the WEP.

The basic idea of this manuscript is that rather than considering new fields which play the role of modifying the effective value of the fundamental constants, we can introduce non-standard aspects of well known fields in order to play that role. The long range fields in nature are the electromagnetic and gravitational ones. The use of the former in the desired context does not seem as a promising possibility because, for one it is very well understood and tested over very wide class of regimes, even at the quantum level (i.e., QED), and its enormous strength implies that any small modification would have very noticeable effects. The latter, on the other hand, is still far from being well understood (particularly its quantum aspects), and secondly, it seems conceivable that an exotic type of its coupling to matter might exist without having been detected so far. Considerations along these lines have led to proposals where the curvature of space–time might affect the propagation of matter fields in rather unusual ways [16], which might be viewed as violating of the strict letter of the equivalence principle without destroying overall general covariance.² In this manuscript we will explore this issue and show that, despite this early optimistic assessment, the value of the model’s parameters needed to explain the observed variation of μ in the Milky Way can be ruled out by the bounds on those parameters emerging from experimental tests of the WEP.

The paper is organized as follows. In Section 2 we discuss the astronomical data that suggest a variation of μ in the Milky Way. In Section 3 we describe a general theoretical model with an “exotic” coupling between matter and gravitation; in Section 3.1 we present a subclass of models where gauge invariance is preserved. Section 4 is devoted to determine the value of a combination of the free parameters for the models presented in Section 3 as implied by the astronomical data discussed in Section 2. In Section 5 we obtain bounds on another combination of the free parameters of the models using the latest tests of the WEP. We end with a brief discussion and some conclusive remarks in Section 6.