Test of Lorentz invariance in $\beta$ decay of polarized ${}^{20}\mathrm{Na}$

Background: Lorentz invariance is key in our understanding of nature, yet relatively few experiments have tested Lorentz invariance in weak interactions.

Purpose: Our goal is to obtain limits on Lorentz-invariance violation in weak interactions, in particular rotational invariance in $\beta$ decay.

Method: We search for a dependence of the lifetime of ${}^{20}\mathrm{Na}$ nuclei on the nuclear spin direction. Such directional dependence would be evidence for Lorentz-invariance violation in weak interactions. A difference in lifetime between nuclei that are polarized in the east and west direction is searched for. This difference is maximally sensitive to the rotation of the Earth, while the sidereal dependence is free from most systematic errors.

Results: The experiment sets a limit of $2\times {10}^{-4}$ at 90% C.L. on the amplitude of the sidereal variation of the relative lifetime differences, an improvement by a factor 15 compared to an earlier result.

Conclusions: No significant violation of Lorentz invariance is found. The result sets limits on parameters of theories describing Lorentz-invariance violation.

Figure 1

Schematic view of the gas cell, detectors, and laser setup. Left: front view; right: top view.

Figure 2

The switching scheme of the ${}^{20}\mathrm{Na}$ beam and the polarization sequence. Gray areas denote the time window for changing the polarization state.

Figure 3

(a) The instantaneous $\beta$-decay rates averaged over all data of run III for one of the $\beta$-particle detectors. The blue (upper) and red (lower) data points are obtained with opposite polarization. The black data points (middle) are obtained without polarization. The bin width is 1 ms. (b) The experimental $\beta$ asymmetry [Eq. (4)]. The data points have been binned to 100 ms, reducing statistical scatter.

Figure 4

Experimental $\gamma$-rate data averaged over run III. (a) The red and blue data points are instantaneous $\gamma$ rates measured in 1 ms, obtained for opposite polarization directions. The blue points are mostly invisible, because they lie under the red points. The sudden drop in count rate at $T=2\text{s}$ is a result of a count rate background which is only present when the production beam is on. (b) The presence of the term $P\left(t\right)K$ in Eq. (6) as seen from the difference between the $\gamma$ rates with and without polarization. The data follow the $\beta$ asymmetry in Fig. 3. The jump at $T=2\text{s}$ is again due to the beam related background. The data were binned to 100 ms. (c) A small parity-odd dependence $P\left(t\right)L$ in Eq. (6) can be seen from the difference between the $\gamma$ rates obtained with opposite polarization. It has an instrumental origin. The data were binned to 100 ms.

Figure 5

A typical fit to a $\gamma$-rate spectrum of a single slice of 17 minute data. The fitted lines cover the data, measured as instantaneous rate in 10 ms. The overlapping blue and red (upper) data points are obtained with opposite polarization. The black (lower) data points are obtained without polarization. The time region (0–0.21) s where the polarization may not have lost memory of the previous 4.1 s polarization period is excluded from the fit. In addition, the region (1.98–2.02) s where the beam is switched from “on” to “off” is excluded. The bottom graph shows the residuals $\chi =(\text{data}-\text{fit})/{\sigma }_{\text{data}}$.

Figure 6

The dependence of ${\mathrm{\Delta }}_{\text{LIV}}$ on the asymmetry of the power of the laser for opposite polarizations (defined as fraction of the difference over the sum). This indicates that the polarizations were not completely balanced throughout the experiment. The laser asymmetry itself was found to be temperature dependent. Therefore, it is also a measure of the temperature of the experimental hall.

Figure 7

Correlation between pressure and temperature in the gas cell. The parameters shown already have the correlation with the laser asymmetry removed. The error on the data points are about $1\times {10}^{-2}{}^{\circ }\mathrm{C}$ and 1 mbar, indicating that other environmental parameters play a role in the data scatter.

Figure 8

The LIV observable $\frac{\mathrm{\Delta }\tau }{\tau }\frac{1}{P_{\text{eff}}}$ versus local sidereal time for data set III. For this figure data were binned to 8 bins of 3 sidereal hours width. Vertical error bars indicate the statistical uncertainty only. The curved line $y\left(t\right)=\xi N^{1}cos\mathrm{\Omega }t+\xi N^{2}sin\mathrm{\Omega }t$ was fitted to the data. The fit result has ${\chi }^2/\text{ndf}=3.2/6$ with $p=78\%$. Both $\xi N^1$ and $\xi N^2$ are consistent with zero (see Table 3).