Spectroscopic searches for a fifth force and Dark Matter

A very light, new boson \(\phi\) can induce a fifth force which leads to a modification of the Coulomb potential in atoms for the force between two particles \(i,j\),
$$
Z\frac{\alpha}{r}\longrightarrow Z\frac{\alpha}{r}+\frac{y_iy_j}{4\pi}\frac{e^{-m_\phi r}}{r}\,.
$$
Precision measurements of atomic transition frequencies can in principle be used to set stringent bounds on whether such a deviation exists. The main difficulty with this approach is the many-body nature of the electronic wavefunction for most atoms, which means that an exact standard model prediction for the transition frequency is not possible. A number of approaches have been proposed, such as using isotope shifts [Berengut et al 2018] [Frugiuele et al 2018] , to reduce the sensitivity to electronic structure. An exciting alternative is to exploit Rydberg states with principal quantum numbers of \(n \require{ams} \gtrapprox 100\). Advantages of Rydberg states include the ability to tune the size of the atomic wavefunction \(\propto n^2 a_0\) relative to the range of the fifth force, and the availability of a great number of transitions for each isotope and element, enabling more complex searches for systematic deviations. Furthermore, Rydberg electrons only weakly interact with the nucleus and the remaining core electrons, raising the possibility of precise {ab initio} calculations with developments in existing theory.

While fifth forces are more general, very light Dark Matter with masses \(m_\text{DM} \ll \) eV can be probed by these searches. This type of Dark Matter is particularly well motivated by fits to the small scale spectrum [Hui et al 2018]

For example a light scalar dark matter field \(\phi\) with a coupling to SM fermions \(f\)
$$ \mathcal{L} = -\frac{\phi}{\Lambda^2}m_f f\bar f $$
leads to a time-dependent variation of the fermion masses [Stadnik, Flambaum 2015]
$$ m_f \to m_f\left( 1+\frac{\phi}{\Lambda} \right)$$
The relic abundance of this light Dark Matter is set by the misalignment mechanism, leading to oscillations in the field which induce an oscillating mass
$$ \frac{\delta m_f}{m_f}= \frac{\phi_0}{\Lambda}\cos (m_\phi t)$$
as well as a fifth force between the fermions \(f\).

We propose to set improved constraints using optical spectrocopy of Rydberg states in heavier atoms such as Rb, Cs and Sr that are easily laser cooled and trapped. For example the Durham group is already able to perform such measurements with \(\sim$kHz\) absolute precision in Sr. In parallel we recommend funding the theoretical development of the required atomic physics “phenomenology”.

Precision tests using few-body systems

The problem of direct comparison with theoretical calculations can be solved by using light atoms such as the isotopes of hydrogen and helium, where highly accurate standard-model calculations of the atomic wavefunction are possible. An example is provided by the well-known spectroscopy of the \(1S\rightarrow2S\) transition in hydrogen that is at the heart of the proton radius puzzle. During the next years we propose to build dedicated experiments for precision spectroscopy in these light atomic species, using both laser cooling and alternative cooling methods based on Stark and Zeeman deceleration [Liu et al 2018] to produce cold samples suitable for precision spectroscopy. As well as contributing to improved measurements of the proton radius, precision measurements of Rydberg states in these atoms would improve the constraints on fifth forces considerably. For example, comparing the Rydberg constant measured from low Hydrogen states with \(r=a_0\) with the Rydberg constant measured in Rydberg states with \(n \approx 30\), which corresponds to \(r\approx 10^3 a_0\), allows the derivation of limits on new bosons with masses between eV and keV.
Existing searches achieve a precision of \(\delta R_\infty =\mathcal{O}(10^{-6})\), which translates in limits on the product of couplings to electrons and protons of \(y_Py_e\lesssim10^{-12}\) for these masses [Liu et al 2017] [Karshenboim 2010] . We propose that an improvement of between 3-6 orders of magnitude is possible, by creating for example an atomic fountain of hydrogen atoms. Such measurements would also contribute to searches for CPT violation by providing reference measurements to compare with antihydrogen.

We note that the few and many-body approaches are complementary; in H and He theory predictions are very precise but high-precision measurements are hard, whereas in atoms with higher \(Z\) such as \(^{88}\)Sr or \(^{174}\)Yb, very high experimental precision can be achieved, but theoretical calculations are challenging.
As well as increasing the discovery potential for new physics, comparing measurements in heavy and light Rydberg atoms will help to advance both the experimental state-of-the-art for light atoms and atomic structure calculations in high-\(Z\) Rydberg atoms.

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