Measurement of Muon Capture on the Proton to 1% Precision and
Determination of the Pseudoscalar Coupling
The MuCap experiment at the Paul Scherrer Institute has measured the rate of muon capture from the singlet state of the muonic hydrogen atom to a precision of 1 %. A muon beam was stopped in a time projection chamber filled with 10-bar, ultrapure hydrogen gas. Cylindrical wire chambers and a segmented scintillator barrel detected electrons from muon decay. is determined from the difference between the disappearance rate in hydrogen and the free muon decay rate. The result is based on the analysis of decays, from which we extract the capture rate sand derive the proton’s pseudoscalar coupling .
pacs:23.40.-s, 24.80.+y, 13.60.-r, 14.20.Dh, 11.40.Ha, 29.40.Gx
We report a measurement of the rate of ordinary muon capture (OMC),
from the singlet state of the muonic hydrogen atom. The analysis uses the complete data set of the MuCap experiment, with significantly smaller systematic and statistical uncertainties compared to our earlier publication Andreev et al. (2007).
For the low momentum transfer in process (1), the standard model electroweak interaction reduces to an effective Fermi interaction between the leptonic and hadronic weak currents. While the leptonic current has a simple structure, the hadronic current between nucleon states is modified by QCD, as expressed in a model-independent way by the introduction of form factors. Since second-class currents are suppressed, muon capture on the proton involves and , the vector and magnetic form factors in the vector current, as well as and , the axial and pseudoscalar form factors in the axial current Bernard et al. (2002); Gorringe and Fearing (2004); Kammel and Kubodera (2010). The first three are well known and contribute only around 0.4% uncertainty to the determination of Beringer et al. (2012). Our measurement of determines , the least well known of these form factors.
The pseudoscalar term in the axial nucleon current has played a significant role in the understanding of weak and strong interactions. Initial estimates were based on the concept of a partially conserved axial current, followed by the recognition of its deeper significance as a consequence of chiral symmetry and its spontaneous and explicit breaking Nambu (1960). These ideas were foundations for explaining the generation of hadronic masses and the development of chiral perturbation theory (ChPT), the effective field theory of low-energy QCD. Based on well-known low-energy constants,
was derived within ChPT Bernard et al. (2002, 1994), with good convergence to two-loop order Kaiser (2003). Though lattice QCD has advanced to unquenched calculations of and Yamazaki et al. (2009); Alexandrou et al. (2011), the precision of the ChPT prediction in Eqn. 2 remains unmatched and stands to be tested experimentally.
Muon capture on hydrogen is the most direct means to determine . Such experiments are complicated by the fact that negative muons stopped in hydrogen form not only atoms, but subsequently molecules where the capture rate differs significantly. Prior to MuCap, the most precise capture rate was measured in liquid hydrogen (LH) Bardin et al. (1981a), where forms rapidly. The value of extracted under these conditions depends critically on the poorly known ortho-to-para transition rate, Bardin et al. (1981b); Clark et al. (2006). The related, rare radiative muon capture (RMC) process is less sensitive to , but the first experimental result on Wright et al. (1998) disagreed with theory. For a discussion of this puzzling situation and muonic processes in hydrogen, see Bernard et al. (2002); Gorringe and Fearing (2004); Kammel and Kubodera (2010).
The MuCap experiment was designed to significantly reduce the density-dependent formation of molecules by employing a gas density of (relative to LH). In these conditions about 97% of the muon captures occur in the singlet state. The experimental concept is sketched in Figure 1(a). A 34 MeV/ muon beam was stopped in a time projection chamber (TPC) filled with 10-bar, ultrapure hydrogen gas Egger et al. (2011). The TPC was used to discriminate between muons that stop in the gas and those that reach wall materials, where capture proceeds much faster than in hydrogen. Arriving muons were detected by an entrance scintillator (SC) and a proportional chamber (PC), and tracked in the TPC. Outgoing decay electrons were detected by concentric multiwire proportional chambers (ePC1 and ePC2) and a segmented scintillator barrel (eSC). The decay times were histogrammed and fit to an exponential. The difference between the observed disappearance rate and the free muon decay rate Webber et al. (2011) is attributed to muon capture, .
The new data reported here were collected during the 2006 (R06) and 2007 (R07) running periods in the E3 muon channel at the Paul Scherrer Institute. Properties of the new data sets are compared to our published result (R04) Andreev et al. (2007) in Table 1. Besides the muon-electron pairs from stops in hydrogen, additional systematic data included decays and data collected when the target gas was doped with elemental impurities (nitrogen, water and argon).
Although the experimental methodology closely followed that of our first result, several hardware upgrades implemented between R04 and R06 led to significantly enhanced performance. In R06 the TPC was operated with about 2.5 times higher gas gain than in R07. As this affects critical chamber parameters, the comparison of the two runs provides an invaluable consistency check.
Events with multiple muons in the TPC (pileup) need to be rejected as they distort the extracted disappearance rate. The maximum rate of the dc muon beam employed in R04 was throttled to minimize pileup. In R06 and R07, the loss of events to pileup was largely eliminated by the introduction of a 25-kV, fast-switching electrostatic kicker Barnes and Wait (2004). The detection of a muon traversing the SC triggered the kicker, which deflected the beam for a period of 25.6 s. The beam extinction factor was around 100 and the rate of pileup-free data was three times larger in R06/7 than in R04.
Another essential change was the greatly improved isotopic and chemical purity of the TPC gas. When deuterium is present (concentration ), muons can form atoms, which, due to a Ramsauer-Townsend minimum in the scattering cross section Kammel and Kubodera (2010), can diffuse out of the fiducial volume and distort the disappearance rate. To separate the hydrogen into its isotopic components, a new cryogenic distillation column was installed. Periodic gas samples were analyzed externally using atomic mass spectrometry Döbeli et al. (2008). For the limits on listed in Table 1, transfer to leads to distortions of less than 0.74 sand 0.12 sfor the R06 and R07 run periods, respectively. A higher sensitivity of the atomic mass spectrometer was responsible for the improved limit in R07.
In the presence of impurities, muons preferentially transfer from to atoms, distorting the disappearance rate. Extended baking of the TPC and increased flux and filtering in the continuous gas purification system Ganzha et al. (2007) led to a fourfold reduced impurity level compared to R04. Moreover, the installation of a humidity sensor before R06 allowed monitoring of the dominant chemical impurity.
Our novel hydrogen TPC was key to the experiment. In the sensitive volume of cm with an applied field of 2 kV/cm, ionization electrons from stopping muons drifted down towards the readout plane with a velocity s. They were amplified in a multiwire proportional chamber region with 72 anodes perpendicular and 36 strip wires parallel to the beam. Anode and cathode signals were discriminated with three energy thresholds and read out by TDCs in 200 ns time intervals. The event display in the center of Figure 1(a) shows the - projection of a typical muon stop in the TPC. The threshold keV is indicated by green pixels. Blue pixels denote the threshold keV, which was set just below the muon’s Bragg peak. The threshold keV [red pixels in Figure 1(c)] was set to record nuclear recoils from muon captures on impurities. In addition to the primary TDC-based readout of the TPC, new 12-bit, 25 MHz flash analog-to-digital converters recorded selectively triggered events.
About 30 TB of raw data were processed at the NCSA supercomputing facility in a multistage procedure. Muon stop candidates were constructed from the SC time and TPC track information. Muon pileup events, flagged by the entrance counters, were rejected. Because the combined inefficiency of the entrance counters was less than , the residual pileup distorted the observed muon disappearance rate by less than 0.5 s. Contiguous pixel regions in the TPC were then fit to a straight line, as indicated by the black line along the muon trajectory in Fig. 1(a). In the case of large angle scattering [Fig. 1(b)] a two-line fit was applied. The muon stop location (red circle) was identified as the most downstream pixel. The muon track requirements were optimized so as to minimize possible distortions to while suppressing events where the muon could have left the hydrogen gas. Muons that stopped within a fiducial volume were accepted. The minimum track length was 3.2 cm, and the maximum fit was 2. was stable against variation of the track length and cuts. However, variations in the fiducial volume boundaries produced statistically disallowed deviations, for which a systematic uncertainty of s was assigned.
Electron tracks were constructed from coincidences between an eSC segment (comprising four photomultiplier tubes) and hits in the two ePCs (each requiring an anode and at least one cathode plane). In the R06 and R07 run periods, the time and gain stabilities of the eSC were verified by recording their signals in 8-bit, 450 MHz waveform digitizers. While the TPC gain was insufficient to produce electron tracks with contiguous pixels, a virtual track in the TPC was reconstructed from hits in the eSC and ePCs, as indicated by the red line in Figure 1(a). A cut of mm was placed on the impact parameter between the muon stop and electron vector. This loose cut significantly reduces backgrounds (c.f. Fig. 3 in Andreev et al. (2007)) while minimizing distortions of introduced by a time-dependent acceptance due to diffusion. Although atoms diffuse only at the mm scale, changes in vs. were observed. This dependence was used to fixed the single parameter of a diffusion model in good agreement with theory Adamczak (2006). For the applied cut, the model was used to determine small corrections sand sfor R06 and R07, respectively. To check that was insensitive to the electron track definition, we also constructed coincidences requiring different combinations of anode and cathode planes within the ePCs. This revealed slightly nonstatistical variations in , which were fully covered by a 1.8 ssystematic uncertainty.
In extreme cases of example Fig. 1(b), muons scatter through large angles, leave the TPC volume, and stop on surrounding materials. Because of the lower TPC gain during R07, there were often gaps in the tracks of scattered muons, making it difficult to reliably identify these events. Moreover, the recoil proton could deposit enough energy in the TPC to trigger the threshold, mimicking an acceptable muon stop. However, these events were unlikely to deposit enough energy at the scattering vertex to exceed the -threshold on neighboring anodes. In the analysis of the R06 and R07 data sets, we required at least two consecutive anodes at the end of the muon track.
This cut introduced a subtle systematic effect. Electrons that traversed the muon’s drifting ionization charge occasionally deposited enough additional energy to elevate a muon’s signal above the threshold. In rare instances a muon stop with a single anode was promoted to a stop with two neighboring anodes. Such events would pass the - scatter cut described above with a decay-time-dependent acceptance and therefore distort the extracted disappearance rate. Because positive muons are sensitive to the charge interference effect but do not capture on nuclei, we were able to measure the induced distortion Kiburg (2011). The method was supplemented by neutron data collected in 8 large liquid scintillator detectors: muons scattered into materials were found to yield copious neutrons from nuclear capture. The resulting corrections were sand sfor R06 and R07, respectively. The correction was sensitive to the threshold, which we decreased in R07, suppressing the interference effect.
As illustrated in Fig. 1(c), nuclear capture on impurities was identified by the presence of an threshold signal in the TPC. This allowed continuous in situ monitoring of the yield of these events. The average values for over the three data sets are given in Table 1 and track well with the humidity sensor readings. To calibrate the necessary correction, special runs were conducted in which the hydrogen gas was doped with known amounts of nitrogen or water vapor. The changes in the disappearance rate and were measured relative to the pure, undoped hydrogen data. Scaling by the observed then determined the corrections for residual impurities: sand sin R06 and R07, respectively.
To obtain the final muon disappearance rate, the muon stops and electron tracks were first sorted into muon-electron pairs. The decay time, , was histogrammed and fit with the function over the range 160 ns 19000 ns. The bin width was fixed at 80 ns, while , and were free parameters. To avoid analysis bias, the exact clock frequency was hidden from the analyzers. After it was revealed, we obtained
Because the fit was slightly larger than expected for both R06 and R07, these values reflect inflated statistical uncertainties following the -factor prescription Beringer et al. (2012). The three-parameter fitting procedure was complemented by applying a full kinetics fit which included all atomic- and molecular-state effects as well as water and nitrogen impurities; the result was consistent within 0.2 s.
In order to check the consistency of our result, we examined changes in with respect to variations in data selection. The fit start and stop times were varied over a range of several microseconds and the parameters remained stable. Only statistical variations were observed when the data were sorted chronologically by run number. Since many of the subtle couplings between the muon and electron definitions are geometrical, the observed stability of the result with respect to azimuth was a critical cross-check.
|Effect||Corrections and uncertainties [s]|
|- scatter removal|
|Fiducial volume cut|
|Entrance counter ineff.|
|Electron track def.|
Table 2 summarizes the aforementioned corrections to our result as well as the systematic uncertainties. Two additional corrections are required to correctly express as:
Here is a calculable bound-state effect Überall (1960); Von Baeyer and Leiter (1979), while accounts for the around 3% of muons that capture from molecular states. The latter depends on and and is derived from fits to simulated data generated with the precise experimental conditions (gas density , background level, impurity concentrations, and fit ranges). We used Kammel and Kubodera (2010) and a newly determined value of s Knaack (2012), which was measured by admixing ppm of argon to the TPC’s hydrogen. It agrees with the previous world average Kammel and Kubodera (2010) but is three times more precise. Evaluating Eq. 5 and including the updated positive muon decay rate of s Beringer et al. (2012); Webber et al. (2011), we determine the singlet capture rates:
We also update slightly our previous publication Andreev et al. (2007) using the latest values for , , and , to obtain Accounting for correlated systematics among these three data sets, we report a final, combined result
where all form factors are evaluated at . Equation (9) quantifies the dependence of the theoretical capture rate on the choice of , relative to value used in Ref. Czarnecki et al. (2007), and on , relative to the latest Beringer et al. (2012). The two uncertainties in the equation stem from limited knowledge of and radiative corrections. Setting to gives
where the two uncertainties arise from the error propagation of and , respectively. If we would have updated to 1.275, as advocated in Blucher and Marciano () and supported by recent measurements of the neutron -decay asymmetry Mund et al. (2012); Mendenhall et al. (2012), the extracted from MuCap would have increased to 8.34.
Figure 2 illustrates the excellent agreement with the theoretical prediction, Eq. (2), and highlights MuCap’s reduced sensitivity to the molecular parameter . This answers the long-standing challenge of an unambiguous measurement of , generated by the mutual inconsistency of earlier experiments (OMC, RMC) and their strong sensitivity to . Corroborating values for are obtained in recent analyses Marcucci et al. (2012); Gazit (2009) of an earlier 0.3% measurement of muon capture on He Ackerbauer et al. (1998), with uncertainties limited by theory. MuCap provides the most precise determination of in the theoretically clean atom and verifies a fundamental prediction of low-energy QCD.
We are grateful to the technical staff of the collaborating institutions, in particular of the host laboratory PSI. We thank M. Barnes, G. Wait, and A. Gafarov for the design and development of the kicker, the Demon collaboration for providing neutron detectors, the AMS team at the ETH Zürich for the deuterium measurements, and A. Adamczak, N. Bondar, D.B. Chitwood, P.T. Debevec, T. Ferguson, J. Govaerts, S. Kizilgul, M. Levchenko, and C.S. Özben for their contributions. This work was supported in part by the U.S. NSF, the U.S. DOE and CRDF, PSI, the Russian Academy of Sciences and the Grants of the President of the Russian Federation. NCSA provided essential computing resources.
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