# Probing unification scenarios with atomic clocks

###### Abstract

We discuss the usage of measurements of the stability of nature’s fundamental constants coming from comparisons between atomic clocks as a means to constrain coupled variations of these constants in a broad class of unification scenarios. After introducing the phenomenology of these models we provide updated constraints, based on a global analysis of the latest experimental results. We obtain null results for the proton-to-electron mass ratio and for the gyromagnetic factor (both of these being at the confidence level). These results are compatible with theoretical expectations on unification scenarios, but much freedom exists due to the presence of a degeneracy direction in the relevant parameter space.

###### pacs:

12.10.Dm,06.20.Jr,06.30.Ft## I Introduction

The observational evidence for the acceleration of the universe demonstrates that canonical theories of cosmology and particle physics are incomplete, if not incorrect. Several few-sigma hints of new physics exist, but so far these are smoke without a smoking gun; it’s time to actively search for the gun.

The LHC evidence for the Higgs particle strongly suggests that fundamental scalar fields are among nature’s building blocks. An obvious follow-up question is whether such fields also play a role in cosmology. They have been invoked to explain paradigms such as inflation, cosmological phase transitions or dynamical dark energy, but the most direct way to infer their presence is (arguably) to search for spacetime variations of nature’s fundamental constants Martins (2002); Garcia-Berro et al. (2007); Uzan (2011). It is known that fundamental couplings run with energy, and many particle physics and cosmology models suggest that they should also roll with time. One example are cosmological models with dynamical scalar fields, including string theory.

Astrophysical measurements have led to claims for Murphy et al. (2004); Reinhold et al. (2006); Webb et al. (2011) and against Srianand et al. (2007); King et al. (2008); Thompson et al. (2009) variations of the fine-structure constant and the proton-to-electron mass ratio at redshifts . An ongoing Large Programme at European Southern Observatory’s Very Large Telescope should soon clarify matters, but a resolution may have to wait for a forthcoming generation of high-resolution ultra-stable spectrographs which include these measurements among their key science drivers. Answering this question can also shed light on the enigma of dark energy Parkinson et al. (2004); Nunes and Lidsey (2004); Doran (2005); Amendola et al. (2012).

Any Grand-Unified Theory predicts a specific relation between variations of and , and therefore simultaneous measurements of both provide key consistency tests. Our work revisits the ideas of Coc et al. (2007); Luo et al. (2011) and applies them in the same spirit as Vieira et al. (2012); Perez-Garcia and Martins (2012), by using the most recent tests of the stability of fundamental constants using atomic clock comparisons to obtain direct constraints on the phenomenological parameters characterizing these unification scenarios.

## Ii Phenomenology of unification

We wish to describe phenomenologically a class of models with simultaneous variations of several fundamental couplings, such as the fine-structure constant , the proton-to-electron mass ratio and the proton gyromagnetic ratio . The simplest way to do this is to relate the various changes to those of a particular dimensionless coupling, typically . Then if and

(1) |

we have and so forth.

The relations between the couplings will be model-dependent. In this section we follow Coc et al. (2007); Luo et al. (2011), considering a class of grand unification models in which the weak scale is determined by dimensional transmutation and further assuming that relative variation of all the Yukawa couplings is the same. Finally we assume that the variation of the couplings is driven by a dilaton-type scalar field (as in Campbell and Olive (1995)). With these assumptions one finds that the variations of and are related through

(2) |

where and can be taken as free phenomenological (model-dependent) parameters. Their absolute value can be anything from order unity to several hundreds, although physically one usually expects them to be positive. (Nevertheless, for our present purposes they can be taken as free parameters to be constrained by data.)

For our purposes it’s natural to assume that particle masses and the QCD scale vary, while the Planck mass is fixed. We then have

(3) |

(since the mass is simply the product of the Higgs VEV and the corresponding Yukawa coupling) and

(4) |

The latter equation is the more model-dependent one, as it requires modeling of the proton. At a phenomenological level, the choice , can also describe the limiting case where varies but the masses don’t. Further useful relations can be obtained Flambaum (2003); Flambaum and Tedesco (2006); Flambaum et al. (2004) for the g-factors for the proton and neutron

(5) |

(6) |

These allow us to transform any measurement of a combination of constants (for example , and ) into a constraint on the parameter space. For atomic clocks, the relevant g-factors are those for Rubidium and Caesium, so these need to be related to those of the nucleons. The way to do this stems from Luo et al. (2011); Flambaum (2003); Flambaum and Tedesco (2006); Flambaum et al. (2004). Using a simple shell model one has

(7) |

(8) |

for our purposes in the following section, the following derived relation is also useful

(9) |

A more accurate phenomenological description (motivated from experimental results and including a dependence on and the spin-spin interaction) leads to

(10) |

(11) |

Notice that these coefficients are very small, particularly in the last parametrization.

Clock | (yr) | Ref. | |
---|---|---|---|

Hg-Al | Rosenband et al. (2008) | ||

Cs-SF | Shelkovnikov et al. (2008) | ||

Cs-H | Fischer et al. (2004) | ||

Cs-Sr | Blatt et al. (2008) | ||

Cs-Hg | Fortier et al. (2007) | ||

Cs-Yb | Peik et al. (2004) | ||

Cs-Rb | ( | Bize et al. (2005) | |

Cs-Yb | Peik (2010) | ||

Cs-Rb | ( | Guena et al. (2012) |

## Iii Constraints from atomic clocks

By measuring the rate of two different atomic clocks one obtains a constraint on the relative shift of the corresponding characteristic frequencies. These are proportional to certain products of fundamental couplings, and thus the measurement can be translated into a constraint of the drift of that combination. Different clock comparisons are sensitive to different products of these couplings, and therefore a combined analysis can in principle lead to constraints on each of them.

Table 1 shows the existing constraints for several pairs of clocks. Since the Hg-Al comparison yields a direct constraint on Rosenband et al. (2008), one can use the combined dataset to obtain constraints in the - plane. In Luo et al. (2011) this was done for the first 7 entries on the table, and we reproduce this (for comparison purposes) in the top panel of Fig. 1. The bounds coming from Rubidium and Ytterbium clocks (lines 6 and 7) have since improved to those in lines 8 and 9, and a reanalysis leads to the improved constraints in the bottom panel of Fig. 1.

Notice that the new measurements must replace the old ones in the analysis, as they are not independent—in both cases, the improved results are primarily due to a longer comparison time for the same set of clocks. With the more recent measurements the degeneracy direction is significantly changed. This is due to the fact that the Rubidium measurement (which is now the most sensitive one, apart from the -only one) is not sensitive to .

From this combined analysis we can also calculate the 95% confidence intervals for both parameters, finding

(12) |

(13) |

the latter can be equivalently expressed in terms of

(14) |

These should be compared to the result of Rosenband et al. (2008) for the fine-structure constant (also at the 95% confidence level)

(15) |

There is no evidence of variations. The bound for is almost as strong the one for , whereas the one for is significantly weaker. This highlights the importance of improved experimental bounds using pairs of clocks with different sensitivities to , and .

The formalism in the previous section can be used to obtain constraints on the R-S parameter space, shown in Fig. 2. As we pointed out, the relations between the gyromagnetic rations and are given by Eqs. (7-8) for a simple shell model, while a better phenomenological description yields Eqs. (10-11). Fig. 2 presents the results for both assumptions, quantifying the importance of this theoretical uncertainty: with current experimental data this is not critical, but as measurements improve better theoretical calculations will become necessary.

Here there is a degeneracy between the two parameters, so that only a combination of them can be reasonably well constrained. The degeneracy direction can be characterized by , and the allowed region has a relatively large uncertainty due to the fact that is less sensitive than to the values of and . The naively expected values (for typical GUT models) of , Coc et al. (2007); Luo et al. (2011) are allowed by the experimental results. By separately fixing each of them we find the following conservative bounds for the other

(16) |

(17) |

These values are in agreement, at the 95% confidence, with both methods of calculation depicted in Fig. 2.

## Iv Conclusions

We have considered the latest tests of the stability of nature’s fundamental constants using atomic clocks and discussed their usage as a tool to constrain unification scenarios. A global analysis of existing measurements, assuming the tight bound of Rosenband et al. Rosenband et al. (2008), allows us to obtain separate updated constraints on and .

It’s worth comparing our results with the ones recently found by Guena et al. (2012). Although they use a different parametrization that does not explicitly include (and also a slightly different dataset), the results of both works agree in the case of : at two sigma Guena et al. (2012) find , while we find the marginally tighter . On the other hand they find a relatively weak bound for the ratio of the quark mass to the QCD mass scale, while we find a comparatively stronger bound for ; the difference is due to the fact that atomic clock experiments are more sensitive to than to the ratio .

We carried out a first analysis of the impact of atomic clock measurements on the phenomenological parameters describing the class of varying fundamental coupling scenarios under consideration: , related to QCD physics, and , related to electroweak/Higgs physics. These measurements are only sensitive to a particular combination of these parameters. The experimental results are in agreement with theoretical expectations on unification scenarios.

This R-S degeneracy can be broken by measurements in astrophysical systems that have different sensitivities to these parameters. Two examples of such systems are main sequence stars and neutron stars, for which parts of the R-S parameter space have been previously explored in Vieira et al. (2012); Perez-Garcia and Martins (2012). We will discuss these issues in a future publication.

###### Acknowledgements.

This work was done in the context of the project PTDC/FIS/111725/2009 from FCT (Portugal), with additional support from grant PP-IJUP2011-212 (funded by U. Porto and Santander-Totta). The work of CJM is supported by a Ciência2007 Research Contract, funded by FCT/MCTES (Portugal) and POPH/FSE (EC). We are grateful to Nelson Nunes and Jean-Philippe Uzan for their comments and suggestions.## References

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