Spectroscopy as a test of Coulomb’s law: A probe of the hidden sector

High precision spectroscopy can provide a sensitive tool to test Coulomb’s law on atomic length scales. This can then be used to constrain particles such as extra “hidden” photons or minicharged particles that are predicted in many extensions of the standard model, and which cause small deviations from Coulomb’s law. In this paper we use a variety of transitions in atomic hydrogen, hydrogenic ions, and exotic atoms to probe Coulomb’s law. This extends the region of pure Coulomb’s law tests to larger masses. For hidden photons and minicharged particles this region is already tested by other astrophysical and laboratory probes. However, future tests of true muonium and muonic atoms are likely to probe new parameter space and therefore have good discovery potential for new physics. Finally, we investigate whether the discrepancy between the theoretical calculation of the $2s_{1/2}^{F=1}-2p_{3/2}^{F=2}$ transition in muonic hydrogen and its recent experimental measurement at PSI can be explained by the existence of a hidden photon. This explanation is ruled out by measurements of the Lamb shift in ordinary hydrogen.

Figure 1

Summary of the current bounds on hidden photons (compilation from 14, 17 updated with 2, 3). We note that in addition to the bound labeled “Coulomb,” the bounds labeled “Earth,” “Jupiter,” and “Rydberg” also originate from tests of Coulomb’s law. The best bounds that we derive from atomic spectra are represented by the black lines. The dashed black line is from the Lamb shift in atomic hydrogen. The dotted black line is a combined bound from the $1s_{1/2}-2s_{1/2}$ and $2s_{1/2}-8s_{1/2}$ transitions in atomic hydrogen. The solid black line is the bound obtained from the Lamb shift in hydrogenlike helium ions. The light orange areas correspond to regions suggested by astrophysical and cosmological puzzles [18, 19, 20, 21, 22]. The brown region is derived from measurements of the Rydberg constant, and represents bounds already obtained from atomic spectra [1, 2, 3].

Figure 2

Feynman diagrams for the potential between two charged particles, Eq. (2.3). In particular the second diagram gives the hidden photon contribution to the interaction between two charged particles, leading to a modification of Coulomb’s law.

Figure 3

The blue (top at low masses) curve denotes the bound on hidden photons obtained from the $2s_{1/2}-2p_{1/2}$ transition in atomic hydrogen. We use a conservative error of $2\Delta M^{*}=2\times {10}^{-10}\mathrm{eV}$, where $\Delta M^{*}$ is defined in (3.5). Note that for same $n$ transitions $\chi$ dies off correctly for both small and large $m_{{\gamma }^{\prime }}$. The red (bottom at low masses) curve shows the naive bound from the $1s_{1/2}-2s_{1/2}$ transition in atomic hydrogen. It has an incorrect behavior at small $m_{{\gamma }^{\prime }}$. The green (middle at low masses) curve gives the correct bound obtained by combining the $1s_{1/2}-2s_{1/2}$ with the $2s_{1/2}-8s_{1/2}$ transition, according to the procedure described in Sec. 3a2. The green and blue bounds will turn out to be the best ones that we can derive from atomic spectra, and are combined together in Fig. 1. For comparison we depict earlier bounds on hidden photons as the color filled regions. Those corresponding to pure tests of Coulomb’s law (also obtained from atomic spectra) are highlighted in brown (upper left corner) [1, 2, 3]. The remaining white region corresponds to unexplored parameter space.

Figure 4

$\chi$ bounds for the $2s_{1/2}-2p_{1/2}$ transition in hydrogenic ions with $Z=2$ (black, bottom curve), $Z=15$ (green, middle curve), and $Z=110$ (yellow dashed curve, speculative). At the $2\Delta M$ level the uncertainties are $3\times {10}^{-9}\mathrm{eV}$ [24], $6\times {10}^{-4}\mathrm{eV}$ [32, 33], and 8 eV [33], respectively. We can see that as $Z$ increases our bounds move away from the white region and therefore become less useful.

Figure 5

The black dashed curve (bottom dashed curve) shows a speculative bound on $\chi$ for the $2s_{1/2}-2p_{1/2}$ transition in true muonium. This is formed using only an estimate of the theoretical uncertainty of $\sim 0.1\mathrm{GHz}$ (see Appendix ). If the experimental result is measured to a similar precision and agrees with theory, we will be able to form a bound similar to this and penetrate new parameter space. The green solid (top at low masses) curve shows an actual bound formed from the $2s_{1/2}^{F=1}-2p_{3/2}^{F=2}$ transition in muonic hydrogen. There is a large theoretical error $2\Delta M^{*}=1.2\times {10}^{-3}\mathrm{eV}$. This is caused by the wide range of possible values of $r_p$. The red dashed curve (top dashed curve) uses the muonic hydrogen transition to form a speculative bound. The error here is taken to be just the $2\sigma$ experimental uncertainty of $6\times {10}^{-6}\mathrm{eV}$ [25]. Finally, the blue (solid, bottom at low masses) curve shows a bound obtained by combining the measurement of the Lamb shift in ordinary hydrogen and the $2s_{1/2}^{F=1}-2p_{3/2}^{F=2}$ transition in muonic hydrogen. Using these two measurements we do not need an additional determination of the proton radius.

Figure 6

Existing bounds for MCPs (see, e.g. 14). Note that the Lamb shift bound is produced using the method from 6. We modify this method to consider other atomic spectra.

Figure 7

Vacuum polarization diagram leading to a deviation from Coulomb’s law in the presence of minicharged particles.

Figure 8

The $1s_{1/2}-2s_{1/2}$ (black, top curve at high masses) and $2s_{1/2}-2p_{1/2}$ [6] (green, bottom curve at high masses) bounds, both at the $2\Delta M^{*}$ level. The $1s_{1/2}-2s_{1/2}$ transition is renormalized using $2s_{1/2}-8s_{1/2}$ as in the hidden photon case. For comparison we show previous bounds from purely laboratory experiments in orange (gray). Our bounds do not penetrate new parameter space.