Measurement of the formation rate of muonic hydrogen molecules

Background: The rate ${\lambda }_{pp\mu }^{}$ characterizes the formation of $pp\mu$ molecules in collisions of muonic $p\mu$ atoms with hydrogen. In measurements of the basic weak muon capture reaction on the proton to determine the pseudoscalar coupling $g_P^{}$, capture occurs from both atomic and molecular states. Thus knowledge of ${\lambda }_{pp\mu }^{}$ is required for a correct interpretation of these experiments.

Purpose: Recently the MuCap experiment has measured the capture rate ${\mathrm{\Lambda }}_{\mathrm{S}}^{}$ from the singlet $p\mu$ atom, employing a low-density active target to suppress $pp\mu$ formation [V. Andreev et al. (MuCap Collaboration), Phys. Rev. Lett. 110, 012504 (2013)]. Nevertheless, given the unprecedented precision of this experiment, the existing experimental knowledge in ${\lambda }_{pp\mu }^{}$ had to be improved.

Method: The MuCap experiment derived the weak capture rate from the muon disappearance rate in ultrapure hydrogen. By doping the hydrogen with 20 ppm of argon, a competing process to $pp\mu$ formation was introduced, which allowed the extraction of ${\lambda }_{pp\mu }^{}$ from the observed time distribution of decay electrons.

Results: The $pp\mu$ formation rate was measured as ${\lambda }_{pp\mu }^{}=(2.01\pm 0.{06}_{\mathrm{stat}}\pm 0.{03}_{\mathrm{sys}})\times {10}^6{\text{s}}^{-1}$. This result updates the ${\lambda }_{pp\mu }^{}$ value used in the abovementioned MuCap publication.

Conclusions: The $2.5\times$ higher precision compared to earlier experiments, and the fact that the measurement was performed under nearly identical conditions as the main data taking, reduces the uncertainty induced by ${\lambda }_{pp\mu }^{}$ to a minor contribution to the overall uncertainty of ${\mathrm{\Lambda }}_{\mathrm{S}}^{}$ and $g_P^{}$, as determined in the MuCap experiment. Our final value for ${\lambda }_{pp\mu }^{}$ shifts ${\mathrm{\Lambda }}_{\mathrm{S}}^{}$ and $g_P^{}$ by less than one-tenth of their respective uncertainties compared to our results published earlier.

Figure 1

Kinetics of negative muons in hydrogen containing a single chemical impurity $Z$. The large circles represent the different muonic states that can form. The black arrows denote transitions between muonic states, while the colored arrows with a small circle at the beginning of the line indicate muon disappearance due to the weak-interaction processes of decay or nuclear capture. The arrow labeled ${\lambda }_{\mathrm{pf}}^{}$ is dashed to indicate that its rate is about 240 times smaller than that of ${\lambda }_{\mathrm{of}}^{}$.

Figure 2

Comparison of a theoretical calculation [9] and experimental measurements of the molecular formation rate ${\lambda }_{pp\mu }^{}$. Red squares denote liquid-hydrogen targets (Bleser et al. [33], Conforto et al. [34]), the green cross denotes a solid-target measurement (Mulhauser et al. [36]), and the blue circles denote two measurements in a gaseous hydrogen environment (Bystritsky et al. [35] and this paper). The shaded region corresponds to an updated world average of the experimental results, excluding the outlying solid-target data point.

Figure 3

Calculated muonic-state populations for (a) the hydrogen density in the MuCap experiment, $\varphi =0.01$, and (b) ${\mathrm{LH}}_2,\varphi =1$. In the MuCap experiment, 97% of all captures proceeded from the $p\mu$ singlet state, while in ${\mathrm{LH}}_2$ capture takes place predominantly from $pp\mu$ molecules.

Figure 4

Simplified cross-sectional view of the MuCap detector setup. Neutron detectors not shown. The main components are described in the text. (The figure is reproduced from Ref. [10].)

Figure 5

(a) Fit to the decay electron time spectrum using Eq. (13). The data are shown as the black line. The colored curves depict the time-dependent contributions from the kinetic states; the black dashed line is the fitted sum. (b) The normalized residuals between the data and the fit function, $(N_i-N\left(t_i\right))/{\sigma }_i$, indicate good agreement in the fitted range 0.12–20 $\mu \text{s}$.

Figure 6

Fit results (blue points) versus fit start time for (a) ${\mathrm{\Lambda }}_{pp\mu }^{}$, (b) ${\mathrm{\Lambda }}_{p\mathrm{Ar}}^{}$, and (c) ${\mathrm{\Lambda }}_{\mathrm{Ar}}^{}$. The variation of each rate is consistent with the expectations from the $1\sigma$ statistically allowed set-subset deviation (solid red line).