Measurement of proton, deuteron, triton, and $\alpha$ particle emission after nuclear muon capture on Al, Si, and Ti with the AlCap experiment

Background: Heavy charged particles after nuclear muon capture are an important nuclear physics background to the muon-to-electron conversion experiments Mu2e and COMET, which will search for charged lepton flavor violation at an unprecedented level of sensitivity.

Purpose: The AlCap experiment aimed to measure the yield and energy spectra of protons, deuterons, tritons, and $\alpha$ particles emitted after the nuclear capture of muons stopped in Al, Si, and Ti in the low-energy range relevant for the muon-to-electron conversion experiments.

Methods: Individual charged particle types were identified in layered silicon detector packages and their initial energy distributions were unfolded from the observed energy spectra.

Results: The proton yields per muon capture were determined as $Y_p\left(\text{Al}\right)=26.64\left(28\text{stat.}\right)\left(77\text{syst.}\right)\times {10}^{-3}$ and $Y_p\left(\text{Ti}\right)=26.48\left(35\right)\left(80\right)\times {10}^{-3}$ in the energy range 3.5–20.0 MeV, and as $Y_p\left(\text{Si}\right)=52.5\left(6\right)\left(18\right)\times {10}^{-3}$ in the energy range 4.0–20.0 MeV. Detailed information on yields and energy spectra for all observed nuclei are presented in the paper.

Conclusions: The yields in the candidate muon stopping targets, Al and Ti, are approximately half of that in Si, which was used in the past to estimate this background. The reduced background allows for less shielding and a better energy resolution in these experiments. It is anticipated that the comprehensive information presented in this paper will stimulate modern theoretical calculations of the rare process of muon capture with charged particle emission and inform the design of future muon-to-electron conversion experiments.

Figure 1

Sobottka-Wills data of charged particle emission after nuclear muon capture on silicon (digitized from Ref. [14]). The data below the valley at 1.4 MeV is due to nuclear recoil, mainly from neutron emission, and the data above is due to charged particle emission. The total number of charged particles emitted per muon capture between 1.4 and 26 MeV is $0.15\left(2\right)$. Sobottka and Wills note that the valley has a large statistical uncertainty and that the exponential decay parameter is 4.6 MeV.

Figure 2

Demonstration of particle identification method using the AlCap simulation. By plotting the energy deposited in the thin detector, $E_1$, against the total energy deposited in both layers, $E_1+E_2$, different bands corresponding to different particle types appear. The band of punch-through protons is vetoed by a third layer situated behind the second layer.

Figure 3

Diagram (left) and photo (right) of the AlCap apparatus. In the diagram, the gray circle shows the outer wall of the vacuum chamber (30 cm in diameter and 60 cm in height), the black lines indicate lead shielding, the red lines indicate silicon detectors, and the blue line is the muon stopping target. The veto scintillator was not used in the analysis.

Figure 4

Measured energy in a thin silicon target for different beam momenta from data (black) and simulation (red). The data points are from beam tuning scans with the muon momentum set to $p_{\text{beam}}$, and the simulation points are from muons generated with kinetic energy $E_{\text{MC}}$ upstream of the entrance window. As the beam momentum increases, more muons punch through the target. Since stopped muons deposit more energy than punch-through muons, the stopping peak is at higher energies than the punch-through peak and appears as a bulge at the highest energy shown here. Energies above 1200 keV are from capture products not included in the first-stage simulation.

Figure 5

Diagrams of the two pulse analysis techniques used in this paper. The standard pulse analysis (left) was used in the main analyses, and the template fitter pulse analysis (right) was used in the thick active-target silicon dataset in Sec. 8b. In the standard pulse analysis, the amplitude and time of a digitized pulse (black) are defined as the peak minus pedestal, and the time the pulse crosses 20% of its amplitude, respectively. In the template fitter pulse analysis, individual templates (blue) are combined and fit (green) to the digitized pulse (black). The energy and time of the subpulses are defined in the same way as the standard pulse analysis.

Figure 6

X- and $\gamma$-ray spectra used to determine the number of muon stops in the targets. The fine splitting of the $2p$ level was included. Nearby peaks in the Al and Ti datasets were due to muon stops in lead.

Figure 7

Time between hits in the first two layers for different charged particles as selected by the data-driven particle-identification method.

Figure 8

$E_1$ vs $E_1+E_2$ and $E_1+E_2$ vs $E_1+E_2+E_3$ plots for the Al dataset demonstrating the third-layer veto.

Figure 9

Comparison between the two particle identification methods on the Al dataset: (top) data-driven method, and (bottom) power-law function method where $\mathrm{\Delta }E=E_1$ and $E=E_1+E_2$.

Figure 10

Measured charged particle energy spectra for aluminum (top), silicon (middle) and titanium (bottom) from the right detector package.

Figure 11

Arrival times of protons in the Al dataset. An exponential fit plus a constant (red line) obtains a muonic atom lifetime consistent with the literature.

Figure 12

Simulated energy loss $E_i^{MC}-E_m^{MC}$ as a function of $E_i$ for protons (top) and tritons (bottom) emitted from the Al target that stop in the second layers of the left detector package (left) and right detector package (right). Note the difference in the upper $E_i^{MC}$ cutoff between protons and tritons is due to protons that punch through the second layer, and the difference in $E_i^{MC}-E_m^{MC}$ between the left and the right side is because the muon beam stops slightly closer to the back of the stopping target and so particles detected in the left detector package have passed through less of the target to reach the detector. This same simulation data are used to construct the response matrices for unfolding.

Figure 13

The average initial energy spectra combining left and right detector packages for different simulated muon stopping positions. As can be seen, the average is resistant to the assumed stopping depth.

Figure 14

Initial energy spectra of charged particles emitted after nuclear muon capture on aluminum (top), silicon (middle), and titanium (bottom). The statistical uncertainties are represented by the error bars whereas the systematic uncertainties are the color-shaded areas for each energy bin. Overall systematic uncertainties due to detector misalignment and detector placement (see Table 4) are not included here.

Figure 15

Initial energy spectra for protons measured with data from a thin (blue) and thick (red) aluminum stopping target.

Figure 16

Proton response matrix for escape correction in active target analysis. For fully contained protons, the initial and measured energies are equal. Above 9 MeV protons can escape and the measured energies become smaller than the initial energies.

Figure 17

(left) Figure showing fitting procedure used in the active-target analysis. (right) Comparison between the active-target and main silicon analyses.

Figure 18

Energy spectra of protons, deuterons and tritons from the AlCap experiment (points) and TWIST experiment (shaded regions). AlCap overall systematic uncertainties due to detector misalignment and detector placement, typically 2.6% for protons and deuterons (see Table 4) are not included.

Figure 19

Comparison between AlCap's active-target analysis and the result of Sobottka and Wills [14] with statistical error bars added as inferred from their paper.