Precision Measurement of Orthopositronium Decay Rate Using SiO_2 Powder

The intrinsic decay rate of orthopositronium formed in ${\rm SiO_2}$ powder is measured using the direct $2\gamma$ correction method such that the time dependence of the pick-off annihilation rate is precisely determined using high energy-resolution germanium detectors. As a systematic test, two different types of ${\rm SiO_2}$ powder are used with consistent findings. The intrinsic decay rate of orthopositronium is found to be $7.0396\pm0.0012 (stat.)\pm0.0011 (sys.)\mu s^{-1}$, which is consistent with previous measurements using ${\rm SiO_2}$ powder with about twice the accuracy. Results agree well with a recent $O(\alpha^2)$ QED prediction, varying $3.8-5.6$ experimental standard deviations from other measurements.


Introduction
Positronium (Ps), the bound state of an electron and positron, is a pure quantum electro-dynamical system providing a highly sensitive field for testing bound state quantum electrodynamics (QED). The triplet (1 3 S 1 ) state of Ps, orthopositronium (o-Ps), will in most cases decay into three photons due to odd-parity under the C-transformation. Because o-Ps has a lifetime about 1140 times longer compared to the singlet state, parapositronium (p-Ps), this enables direct and precise measurement of the intrinsic decay rate of orthopositronium, λ o-Ps [1,2], although obtained values are much larger, i.e., 5.2 and 8.2 experimental standard deviations, than a recent non-relativistic QED calculation (7.039 979(11) µs −1 ) corrected up to O(α 2 ) [3]. To elucidate discrepancies, a variety of experiments have since been carried out to search for the exotic decay mode of o-Ps, resulting in no evidence so far [4,5,6,7,8,9,10,11,12,13].
As some fraction of o-Ps inevitably results in 'pick-off' annihilations due to collisions with atomic electrons of the target material, the observed o-Ps decay rate λ obs is a sum of the intrinsic o-Ps decay rate λ o-Ps and the pick-off annihilation rate into 2γ's, λ pick , i.e., λ obs (t) = λ 3γ + λ pick (t). (1) Contributions from exotic decays are assumed to be zero because they are confirmed to be less than 200 ppm [4,5,6,7,8,9,10,11,12,13]. λ pick (t) is proportional to the rate of o-Ps collisions with the target materials, i.e.; λ pick = nσ a v(t), where n is the density of the target, σ a is the annihilation cross section, and v(t) the time dependent velocity of o-Ps. Due to the thermalization process of o-Ps, this necessitates expressing λ pick as a function of time whose properties are dependent on the surrounding materials.
In previous measurements [1,2], λ obs 's were measured by varying the densities of the target materials (n) or size of the cavities. For the uniform distribution of o-Ps in the cavity, λ pick is depending on the size and shape of the cavity due to the relation, v(t) = (v (t)/4)S/V , wherev (t) is the average o-Ps velocity, S is the cavity surface area, and V is the volume of the cavity.
The extrapolation to zero gas density or infinite cavity size was expected to yield the decay rate in a vacuum, λ 3γ , under the assumption of quick thermalization with constant o-Ps velocity. However, this assumption contains a serious systematic error as pointed out in reference [14] in that it would make the obtained decay rate larger than the true value; i.e., the time-spectrum fitting was carried out using a simple exponential function, dN(t)/dt = N 0 exp(−λ 3γ t) + C, where C represents the flat spectrum of accidental events. In order to cleanly eliminate the contribution from the prompt annihilations and fast p-Ps decay component, the decay spectrum needs to be fitted for times after these components decay away. A somewhat fast, beginning time of the fitting range was adopted to obtain the high statistics: the cavity experiment typically used t start = 170 ns for all runs [2], while the gas experiment [1] used t start = 180 ns for all gases except the lowest pressure Ne and neopentane runs. Because it takes several hundred nsec for o-Ps to thermalize via elastic collisions with surrounding molecules to an ultimate energy of about 0.03 eV depending on the type of target material [15,16], such fast t start tends to result in larger values of λ 3γ . Moreover, in the cavity experiment, o-Ps is not well thermalized due to the small escape and collision rates.
The new technique called the direct 2γ correction method was introduced to overcome the problem regarding the extrapolation. The following section briefly explains its concept and procedure. Once a precise thermalization function is obtained, λ pick (t) will contain all information about the process. The population of o-Ps at time t, N(t) can be expressed as The energy distribution of photons from the 3-body decay is continuous below the steep edge at 511 keV, whereas the pick-off annihilation is 2-body which produces a 511 keV monochromatic peak. Energy and timing information are simultaneously measured with high-energy resolution germanium detectors such that λ pick (t)/λ 3γ can be determined from the energy spectrum of the emitted photon. Providing the ratio is determined as a function of time, the intrinsic decay rate of o-Ps, λ 3γ , can be directly obtained by fitting the observed time spectrum. The direct 2γ correction method makes any extrapolation procedure unnecessary and precluding use of the hypothesis of linear dependence at vacuum limit or o-Ps behavior at the material surface.

Experiment
The direct 2γ correction method first applied in 1995 [17] provided consistent results with QED predictions, yet several problems remained: (i) accuracy was 400 ppm, being worse than those of the other experiments [1,2], (ii) decay rates systematically increased before t start = 200 ns due to an unknown reason, and therefore to remove this uncertainty, final results were obtained using data after 220 ns, and (ii) systematic error regarding the Stark effect was not estimated. Improving the method by considering these problems will be described in the later sections. We employed a 22 Na positron source (dia., 2 mm) with approximate strength of 0.03µCi, being sandwiched between two sheets of plastic scintillators (NE104) and held by a cone made of aluminized mylar (Fig. 1). The scintillators and mylar were 100 µm-and 25 µm-thick, respectively, corresponding to a total area density of about 17 mg/cm 2 . The cone is situated at the center of a cylindrical 50 mm-diameter vacuum container made of 1 mm-thick glass, being filled with SiO 2 powder and evacuated down to 5 × 10 −2 Torr. Two different types of SiO 2 powder (Nippon Aerosil Ltd.) were used (Table 1), with the biggest difference being in the mean distance between grains (340 vs. 200 nm) such that different pick-off ratio would be obtained. Using these powders, two 6-month runs were performed (RUN1 and 2). To remove water molecules absorbed on the grain surface, the powder was heated just before evacuation.
While traveling through the scintillators, the source-emitted positrons deposit  Table 1 Characteristics of SiO 2 powders used in the measurements. energy from 50 − 100 keV which produces scintillation photons transmitted to the trigger PMT (Hamamatsu H3165-04). The magnetic field around the assembly is measured to be 0.5±0.1 Gauss, equivalent to terrestrial magnetism which barely contributes to the mixing between o-Ps and p-Ps estimated to be 3 × 10 −11 . The observed o-Ps is therefore regarded as a pure o-Ps sample.
Two high-purity coaxial germanium detectors (Ortec GEM38195), referred to as Ge0 and Ge1, precisely measured the thermalization process, respectively having crystal sizes of 58.3 and 60.3 mm in dia. and 73.8 and 67.4 mm in length, and a solid angle of 5.7 and 6.0% of 4π. Energy resolutions were measured using several line γ sources, with typical resultant values of 0.53 and 0.64 keV in sigma for Ge0 and Ge1, respectively, at 514 keV. Lead sheets 2.0-mmthick were placed in front of each detector for suppressing contributions from simultaneous low-energy γ hits from the 3γ decay of o-Ps.
Four large cylindrical NaI(Tl) scintillators (Scionix 127A127/5; 127 mm (φ) × 127 mm (t)) simultaneously measured the time and energy information from each decay. Due to their higher efficiency and faster time response, the time spectrum can be fit to the o-Ps decay rate determination. At 662 keV, their A new time-to-digital converter (TDC) was employed, jointly developed with the High Energy Accelerator Research Organization (KEK) and customized for the present measurement. This direct clock-counting type TDC, referred to as the 2-GHz TDC, has a time-resolution of 0.5 nsec with known accuracy of 1-ppm, equipped with 8 channels being able to accommodate multiple signals. The time range for each channel is 32 µs, and the integral non-linearity (INL) is expected to be extremely small at < 15 ppm. To provide a systematic check, we used a 200-MHz internal clock-based TDC (Hoshin C006) with 5 ns time resolution.
The electronic system consists of three sections: the trigger, NaI(Tl), and Ge sections. Each has a latch veto structure to prevent overlapping events. The main latch of the trigger section is set by the signals themselves and released with system reset. To ensure precise measurement, the latches of the NaI(Tl) and Ge sections are also set by themselves and released by reset signals from the trigger section. The trigger PMT signal is fed into a fast leading discriminator whose output provides common start signals for the TDCs as well as their reset signals. Each output signal from the photo-multipliers to the NaI(Tl)s is fed into three ADCs and a fast leading edge discriminator that provides stop signals for the TDCs and gate timing for the ADCs. One ADC with a 3-µs gate width, called the NAI-WD (Lecroy 2249W), measures the whole charge for the duration of the signal, with slewing correction being carried out using energy information. The other two ADCs eliminate pile-up events at the tail of the signal and base-line fluctuations, respectively. The NAI-NW (mainly Lecroy 2249W) ADC measures the signal charge with a narrower gate width (250 ns for RUN1 and 400 ns for RUN2) comparable to the intrinsic decay time constant of the NaI(Tl) scintillator, while the NAI-BS (Lecroy 2249A) ADC measures the base-line condition of the signal (180 ns width) just prior to the event. All gates for these ADCs are individual and gate timings are optimized for signal timings.
Signal outputs from the Ge detectors are used for timing measurements and obtaining precise energy information. One signal is fed into a fast-filter amplifier (FFA)(Ortec 579) whose output is used as the stop signal for the TDCs, two auxiliary ADC signals, and three discriminators whose thresholds are set at −50, −100, and −150 mV. Each FFA output is fed into another highresolution TDC having fine resolution of 250 ps. Signal timings are determined utilizing signal shapes calculated from different threshold discriminating times and extrapolating to the intrinsic timing [18]. Good time resolution of 4 ns is obtained, as is efficient rejection power for slow rise signal components known to disturb Ge timing spectroscopy. Similar to the NaI(Tl) detectors, the two auxiliary ADCs effectively reject pile-up events by measuring the signal charge with a narrower gate (2.0(1.3) µs width for RUN2(1)) and earlier timing gate (200 ns width just prior to the signal). Precise time and energy resolutions can be obtained with this rejection scheme. The other output from each Ge detector is independently fed into a SHAPER spectroscopy amplifier (Ortec 673). Amplified with a 6-µs time constant, the uni-polar output is provided to a Wilkinson type peak-holding ADC called GE-WD that provides a precise energy spectrum. For RUN1 and RUN2, 2.7 × 10 9 β + and 2.0 × 10 9 β + events were recorded, respectively. Each run had a total data acquisition period of about six months. Area temperature was maintained within ±0.5 • C to ensure stability of the amplifiers, ADCs, and TDCs.

Analysis and discussion
The ratio λ pick (t)/λ 3γ is determined using the energy spectrum measured by the Ge detectors. The energy spectrum of the o-Ps decay sample, referred to as the o-Ps spectrum, is obtained by subtracting accidental contributions from the measured spectrum. The 3γ-decay continuum spectrum is calculated using Monte Carlo simulation in which the geometry and various material distributions are reproduced in detail. Three photons are generated according to the O(α) corrected energy spectrum [19], followed by successive photo-electric, incoherent, and coherent interactions with materials until all photon energy is either deposited or escapes from the detectors. The response function of the detectors is determined based on the measured spectrum of monochromatic γ-rays emitted from 152 Eu, 85 Sr, and 137 Cs, with this function being used in the simulation. We refer to the obtained spectrum as the 3γ-spectrum which is normalized to the o-Ps spectrum with the ratio of event numbers within the energy domain 480-505 keV. The region is chosen to minimize, the Compton scattering effect from pick-off events, dependence of the detector efficiencies on energy, and the O(α) correction to the 3γ-spectrum. Figure 3(a) shows good agreement between the o-Ps spectrum and 3γ spectrum below 508 keV, where the pick-off annihilation peak is evident at the edge of the 3γ-decay continuum. It should be noted that the lead sheets in front of the Ge detectors effectively suppress the contribution from events in which two low-energy γ's emitted from a 3γ-decay simultaneously hit one detector and that the 3γ-spectrum well reproduces such simultaneous events. Figure 3(b) shows an enlarged view of the observed o-Ps spectrum after subtracting the 3γ-spectrum, where good agreement with a detector response function [20,21] is present. The centroids of the peak spectrum are obtained as 510.997 +0.003 −0.024 (keV) for RUN1 and 510.995 +0.008 −0.059 (keV) for RUN2. Good consistency with 511.0 (keV) indicates successful subtraction of the 3γ contribution such that the resultant peak can be regarded as pure pick-off annihilation samples. Obtained ratios of λ pick /λ 3γ at a typical time window of 150 − 700 ns are 0.01049(8) and 0.01582(9) for RUN1 and RUN2, respectively. The weighted mean time within the range for each run is 287.5 ns (RUN1) and 284.3 ns (RUN2). The large difference between the two runs originates from a different mean distance of powder grains (Table 1) since the lambda ratio is a function of time. Figure 4 shows results when the procedures described above for a typical time window were applied to multiple, narrower time windows. Since the fractional energy loss of o-Ps per collision with SiO 2 powder and the collision rate are both dependent on its energy, the time dependence of the average kinetic energy of o-Ps at time t, E(t) can be derived from the Boltzmann equation where m P s is the mass of o-Ps, T room temperature, and k B the Boltzmann constant. The last term, the momentum transfer cross section of SiO 2 is expanded in terms of E(t), i.e., the coefficients a j represent the effect of effective mass at the surface of the SiO 2 grain and mean distance between the grains. Since the pick-off rate is proportional to the average velocity of the o-Ps, the ratio θ(t) ≡ λ pick (t)/λ 3γ can be expressed by a differential equation [22,23], i.e., where C is a constant, θ ∞ ≡ θ(t → ∞), and the last summation term in Eq. 3 is replaced with an arbitrary real number β. The measured λ pick (t)/λ 3γ 's are fit with Eq. 4, and the first-order differential equation is numerically solved using the Runge-Kutta method. The data point at a time window of 40−45 ns is used for a fixed initial value. Fitting is carried out using four parameters: C, θ ∞ , β, and the deviation from fixed point δ found to be −0.27 × 10 −4 and −0.32 × 10 −4 for RUN1 and RUN2, respectively. These small values indicate that the fitted values rarely depend on the initial condition of the Runge-Kutta method. Figure 4 shows best fit results using the MINUIT library [24], where the pickoff rate cannot be assumed as constant, even in SiO 2 powder where the collision rate is expected to be higher.  pile-up events, a base-line cut condition was applied, and to further reject pile-up events, events with small differences between two ADC values (wide and narrow gates) were selected.
After selecting events in the energy window of 370 − 440 keV, we fit resultant time spectrum using the least square method, i.e., where ǫ pick and ǫ 3γ are the detection efficiencies for pick-off annihilations and 3γ decays. R stop in the function is the measured stop rate of the NaI(Tl) scintillators, representing the fact that time interval measurement accepts only the first γ as a stop signal. λ pick /λ 3γ is about 1% due to the low-density of the SiO 2 powder, i.e., the ratio of error propagation to decay rate is suppressed by a factor of 100. The value of ǫ pick /ǫ 3γ is estimated using Monte Carlo simulation, yielding 0.0651(7) and 0.0607(6) for RUN 1 and RUN2, respectively. The time spectrum is corrected in terms of the R stop (order of 1 kHz) before fitting. Fitting stop time is fixed at 5000 ns. Figure 6 shows obtained fitting as a function of fitting start time for both runs, where values are stable with respect to fitting start time except those before 80 ns. Since the fitting χ 2 's for both runs rapidly increase before 90 ns due to the tail effect of prompt events, values at t = 100 ns are taken as the final results. The reduced χ 2 's at t = 100 ns are 1.006 and 1.051 for RUN 1 and RUN 2, respectively. Note the good correspondence between runs. Since no large difference in thermalization properties are apparent in Fig. 6, this indicates that our method correctly takes into account thermalization. The obtained decay rates are λ 3γ = 7.03991 ± 0.0017(stat.) µs −1 for RUN 1 and λ 3γ = 7.03935 ± 0.0017(stat.) µs −1 for RUN 2. −120 and +153 −151 and +148 Table 3 Summary of systematic errors. See text for details of each item.
The sources of systematic errors are summarized in Table 3 and each item is depicted in the following discussions.
TDC module related errors: Calibration accuracy (< 1 ppm) and stability of the calibration oscillator (2 ∼ 3 ppm) are known as they are product specifications, while integral (< 15 ppm) and differential (negligible) non linearities are evaluated using basic performance of the 2-GHz TDC. Cut condition errors: Systematic errors for base-line selections and pile-up rejections were assessed by changing the cut criteria. The base-line cut was intentionally changed from ±3σ to ±5σ where the σ represents standard deviation of the base-line distribution. The decay rate results are compared with the nominal result with ±4σ, and systematic error is estimated as −17 ppm, +89 ppm for RUN1 and −11 ppm, +23 ppm for RUN2. Regarding the pile-up events rejection, the WD-NW cut condition was changed from ±25keV to ±40keV to estimate deviation from the nominal condition which is ±30 keV, and is estimated as −6 ppm, +45 ppm for RUN1 and −20 ppm for RUN2. The dependence on the energy cut condition and trigger PMT threshold are considered to have a negligible effect.
Monte Carlo simulation related errors: The predominant contribution to total systematic error is produced by uncertain normalization. That is, the number of pick-off events are determined by subtracting the normalized 3-γ spectrum of Monte Carlo simulation from the o-Ps spectrum, where changing the normalization factor affects the λ pick (t)/λ 3γ values and eventually propagates to the final result. Since the sharp fall-off of the 3γ-spectrum at 511 keV is solely produced by the good Ge energy resolution of σ = 0.5 keV, this subtraction only affects the lower side of the pick-off spectrum such that improper subtraction results in asymmetry of the pick-off spectrum shape. Comparison of the asymmetries of the pick-off peak shape and the prompt peak annihilation spectrum is a good parameter for estimating this systematic error. The 1 σ error is assessed as ±99 ppm for RUN1 and ±113 ppm for RUN2.
Other MC simulation related errors: The relative efficiency of the NaI(Tl) scintillators, ε pick /ε 3γ , has an uncertainty of +5% evaluated using a comparison of the energy spectrum of data and that of the simulation which reproduces data within an uncertainty of 2% except for the Compton free region where the deviation is 5% (+7 ppm). SiO 2 powder density in the MC simulation is conservatively changed by ±10% although the uniformity is known to be within a few %; a change resulting in an error of < ±55 ppm.
Stark shift errors: The Stark shift stretches the lifetime of Ps atoms, i.e., a perturbative calculation shows that the shift is proportional to a square of the effective electric field E such that △λ 3γ /λ 3γ = 248 · (E/E 0 ) 2 [25], where E 0 = m 2 e e 5 / 4 ≈ 5.14 × 10 9 V /cm. E is defined as the root-mean-square electric field sensed by o-Ps during its lifetime. Calculations have estimated two contributions exist based on measurements of the electrical charge-up on the primary grains of silica powders and electrical dipole moment on the surface of grains [18]. The charge-up is partly intrinsic depending on powder specifications and partly due to positron depositions from the β + source. The effect, however, is negligible in both cases, i.e., on the level of 10 −2 ppm at most. Silanol functional groups on the surface of the powder grain behave as an electrical dipole moment creating an effective field around the grains. Average densities are known to be as 2.5/nm 2 and 0.44/nm 2 for RUN 1 and 2, respectively. Accordingly, the effective field can be analytically calculated such that the contribution to the o-Ps decay rate is determined to be −21 ppm for RUN1 and −6 ppm for RUN2. These estimations were confirmed using results  Vertical line shows the O(α 2 )-corrected NRQED prediction [3]. Small vertical lines on the error bars indicate the size of errors solely due to statistics while error bars represent total ambiguities including systematic errors.
from precise hyper-fine-structure (HFS) interval measurements of ground state Ps in silica powder [26], where the interval is proportional to the size of Stark effect. Considering the difference in powder densities used, the HFS results are consistent with our estimation.
Other sources of systematic errors: Error contribution due to the Zeeman effect is estimated using the measured absolute magnetic field around the positronium assembly (−5 ppm). Since the 3-γ pick-off process can only occur at a certain ratio, the calculated relative frequency σ 3γ /σ 2γ ∼ 1/378 is consistent with previous measurements [27], being −26 ppm for RUN1 and −42 ppm for RUN2. The probability of the excited state (n=2) of Ps is about 3 × 10 −4 [28], which could make the intrinsic decay rate 19 ppm smaller due to the low decay rate ( 1 8 λ 3γ ) The above discussed systematic errors are regarded as independent contributions such that the total systematic error can be calculated as their quadratic sum, resulting in −120 ppm, +153 ppm for RUN1 and −151 ppm, +148 ppm for RUN2.

Conclusions
The decay rate of o-Ps formed in SiO 2 was measured using a direct 2γ correction method in which the thermalization effect of o-Ps is accounted for and integrated into the time spectrum fitting procedure. Results were obtained using two runs with different types of SiO 2 powders, i.e., λ o−Ps (RUN1) = 7.03991 ±