First Millimeter-wave Spectroscopy of the Ground-state Positronium

We report on the first measurement of the Breit-Wigner resonance of the transition from {\it ortho-}positronium to {\it para-}positronium. We have developed an optical system to accumulate a power of over 20 kW using a frequency-tunable gyrotron and a Fabry-P\'{e}rot cavity. This system opens a new era of millimeter-wave spectroscopy, and enables us to directly determine both the hyperfine interval and the decay width of {\it p-}Ps.


Introduction
Positronium (Ps) [1], a bound state of an electron and a positron, is a good system with which to precisely study quantum electrodynamics (QED) in the bound state [2]. Ground-state positronium has two spin eigenstates: orthopositronium (o-Ps, Spin = 1, 3γ-decay, lifetime = 142 ns [3]) and para-positronium (p-Ps, Spin = 0, 2γ-decay, lifetime = 125 ps [4]). The energy level of o-Ps is higher than that of p-Ps by the hyperfine structure (∆ Ps HFS ), whose value is about 203 GHz (wavelength = 1.5 mm). The first measurement of ∆ Ps HFS was performed in 1952 [5], and precise measurements were performed in the 1970s and 1980s [6]. All these experiments were indirect measurements using the Zeeman splitting of about 3 GHz caused by a static magnetic field of about 1 T. There is a discrepancy of 3.9 standard deviations (15 ppm) between the indirectly measured values [6] and theoretical predictions [7]. Non-thermalized Ps and non-uniformity of the static magnetic field might be underestimated systematic errors [8]. It is of great importance to re-measure ∆ Ps HFS using a method totally different from the previous experiments. Some independent experiments (using quantum interference [9], optical lasers [10], and a precise magnetic field and considering time evolution [11]) have been performed, but have not yet reached a sufficient level of precision to address the observed discrepancy.
Determination of ∆ Ps HFS by directly measuring the transition from o-Ps to p-Ps is a completely new method, which is free from systematic uncertainties due to the static magnetic field. This experiment has not yet been performed due to the many Email address: miyazaki@icepp.s.u-tokyo.ac.jp (A. Miyazaki) technological difficulties regarding the use of millimeter waves. Since the transition from o-Ps to p-Ps is strongly suppressed due to their short lifetime, high-power millimeter-wave radiation, of over 10 kW, is required. In this paper we present the first results of the measurement of the Breit-Wigner resonance of the transition, achieved thanks to the development of high-power millimeter-wave techniques. We directly determine both ∆ Ps HFS and the decay width of p-Ps (Γ p-Ps ) through the Breit-Wigner resonance. Figure 1 shows our experimental setup, composed of a gyrotron and a Fabry-Pérot resonant cavity. Millimeter-wave radiation is produced by a gyrotron, a cyclotron-resonance-maser fast wave device. An electron beam gyrating in a strong magnetic field (∼ 7 T) excites a resonant mode (millimeter waves) of a cavity in the gyrotron, whose output power is highest (> 100 W) in the millimeter-wave range. After the first observation of the transition from o-Ps to p-Ps at 202.89 GHz [12], we have developed a new gyrotron (FU CW GI) operating in the TE 52 mode with an internal mode converter [13]. This gyrotron works in pulsed operation (duty ratio 30%, repetition rate 5 Hz). The output millimeter wave beam has a Gaussian profile. As shown in Table 1, the frequency of the gyrotron can be tuned between 201 GHz and 205 GHz by using gyrotron cavities of different radius, and high power is obtained for all cavities. A far off-resonance point (180.56 GHz) is obtained by using a different operating mode (TE 42 mode). The frequency is precisely measured (±1 kHz) using a heterodyne detector (Virginia Diodes Inc., WR5.1 Even Harmonic Mixer). The electron beam current is monitored and fed back to control the voltage of the heater of the gyrotron's electron gun. The power of the output beam was thus stabilized to within ∼10 % during each measurement (lasting a few days). The beam from the gyrotron is guided into the Fabry-Pérot cavity, which consists of a gold mesh plane mirror (diameter = 50 mm, line width = 200 µm, separation = 140 µm, thickness = 1 µm) and a copper concave mirror (diameter = 80 mm, curvature = 300 mm). The cavity length is precisely controlled (∼100 nm) by a piezoelectric stage under the copper mirror ( Fig. 1). The accumulated power in the cavity is measured using the radiation transmitted through a hole (diameter = 0.6 mm) at the center of the copper mirror. This transmitted radiation is monitored by a pyroelectric detector. The gold mesh mirror is fabricated on a water-cooled high-resistivity silicon plate (thickness = 1.96 mm). This substrate is also used as the window of the gas chamber. CST Microwave Studio [14] is used to simulate the interference of millimeter waves between the mesh mirror and the silicon plate (refractive index = 3.45). A high reflectivity (∼99.1 %) and low loss (∼ 0.3 %) are predicted at frequencies around 203 GHz. The power accumulated in the Fabry-Pérot cavity is designed to be over 20 kW when the power of input radiation is over 100 W.

Experimental setup
The absolute accumulated power is measured as shown in Fig. 2. The ratio between the accumulated power and the radiation transmitted through the hole on the copper mirror is calibrated using the beam produced by the gyrotron. A chopper splits the beam in order to simultaneously measure the transmitted signal and the beam power.This reduces systematic uncertainties due to time-dependent (a few minutes) instability of the gyrotron output. The chopper is synchronized to the gyrotron pulses, and switches the propagation direction from one pulse to the next. Half of the pulses are totally absorbed in a Teflon box filled with water (46 ml) whose temperature increase is used to estimate the power P, while the other half are passed to the copper mirror, where the power transmitted through its hole is measured by the pyroelectric detector (output voltage = V tr ). The calibration factor C is defined as C ≡ 2P/V tr [kW/V]. The small difference in shape between the accumulated beam in the Fabry-Pérot cavity and the Gaussian beam from the gyrotron is corrected by measuring the spatial distribution of the beam using an IR camera to image the temperature increase of a polyvinyl chloride sheet placed in the beam.
The beam reflected from the copper mirror (Fig. 2) creates a standing wave between the mirror and the gyrotron. To obtain and correct the shape of this standing wave, the piezoelectric stage under the copper mirror is moved by a few wavelengths during the measurement. The reflected beam also changes the oscillation efficiency of the gyrotron, as reported in Ref. [15]. We split the Gaussian beam using a half mirror placed before the chopper to monitor the beam power using another pyroelectric detector, and correct for this change of the efficiency. The ratio C is measured at three different positions (distance of 10 cm) of the concave mirror. Differences between these three measurements are later assigned as a systematic uncertainty. Using this method, the accumulated power is assured to be over 20 kW, consistent with a rough estimation considering the finesse and coupling of the cavity [16].
Positronium is formed in the gas chamber in which the Fabry-Pérot cavity is placed (Fig. 1). A positron emitted from a 22 Na source (1 MBq) is tagged by a thin plastic scintillator (thickness = 0.1 mm, NE-102), and the γ rays produced in its annihilation are detected by four LaBr 3 (Ce) crystals. Photomultipliers (HAMAMATSU R5924-70) are used to detect optical photons from the scintillators, and charge-sensitive ADCs are used to measure the energy. One essential improvement from Ref. [12] is the use of neopentane (C-(CH 3 ) 4 ) gas (25 • , 1 atm) as an electron source for Ps production. Neopentane, which provides high stopping power and efficient Ps production, does not absorb millimeter waves because it is a symmetric alkane gas molecule. The use of neopentane also prevents an increase of Ps production in an electromagnetic field [17], as was reported in the first indirect measurement using RF (3 GHz) [5] and studied in detail with a static electric field [18].
The measurement was performed at the eight frequencies listed in Table 1. The data acquisition system is triggered (∼600 Hz) by the coincidence between signals from the plastic scintillator and a pair of back-to-back LaBr 3 (Ce) crystals. During the data acquisition, energy calibrations are performed every 30 minutes. The gas temperature is maintained at less than 30 • C using water-cooling. We measure C before and after the measurement at each frequency; since these two measurements are consistent, their mean is used in the analysis.

Analysis
To enhance o-Ps events, we require that the time difference between the plastic scintillator signal and the coincidence signal of the LaBr 3 (Ce) detectors is between 50 ns and 250 ns. Accidental events from pileup in the plastic scintillator are reduced by requiring that the charge measured by long (1000 ns) and short (60 ns) gate ADCs are consistent. The remaining number of accidental events is estimated by considering events falling in the time window between 850 ns and 900 ns, and is subtracted from the signal sample. We also apply an energy selection, between 494 keV and 536 keV, for the two back-to-back γ rays. Figure 3(a) shows the measured time spectra at a frequency of 203.51 GHz and accumulated power of 67.4 kW. The data are shown separately for events collected during (beam-ON) or outside (beam-OFF) gyrotron pulses, after the rejection of accidental events and the energy selection. The lifetime of beam-ON events (τ ON = 108.2 ± 3.1 ns) is significantly shorter than that of beam-OFF events (τ OFF = 131.3 ± 2.7 ns) because of the transition from o-Ps to p-Ps. This decrease in lifetime is consistent with the theoretical prediction of the transition, and results in an enhancement of the event rate during the beam-ON period. Figure 3(b) shows energy spectra measured by the LaBr 3 (Ce) crystals for all events within the time window, when the opposing LaBr 3 (Ce) scintillator has detected a 511 keV γ-ray. The beam-OFF spectrum consists of pick-off annihilation (quenching by an electron in a gas molecule) [19] and 3γ-decays of o-Ps. After all selections, the event rates in beam-ON and beam-OFF periods are R ON = 548 mHz and R OFF = 455 mHz, respectively.
We obtain the reaction cross-section σ of the transition from o-Ps to p-Ps by comparing the measured S /N ≡ (R ON − R OFF )/R OFF with the value simulated using photon flux information. The photon flux (C × V tr ) in the Fabry-Pérot cavity is continuously monitored by measuring the V tr waveform using a sampling ADC (sampling rate of 0.5 kHz). We estimate the position of Ps formation and relative detection efficiencies of 2γ-and 3γ-decays using GEANT4 simulation [20]. The transition probability is calculated using the simulated Ps positions and the theoretical distribution of the beam within the cavity. We then obtain the simulated dependence of S /N on σ, and numerically solve the equation S /N(σ) = (R ON −R OFF )/R OFF . The advantage of using S /N is that the least well constrained parameters used in the simulation (absolute source intensity, detector misalignment, and stopping position of positrons) are canceled out. We also measure S /N when the Fabry-Pérot cavity does not accumulate millimeter waves, in which case S /N is consistent with zero. Figure 4 shows the obtained cross-sections versus frequency. The data far off-resonance (180.56 GHz) demonstrate the absence of fake signals. These data are fitted by a Breit-Wigner function of the angular frequency ω where ω 0 is 2π∆ Ps HFS , A is the Einstein A coefficient of this transition, and Γ is the natural width of the transition. Using the decay width of o-Ps (Γ o-Ps ) and Γ p-Ps , Γ is expressed by Since A and Γ o-Ps are much smaller than Γ p-Ps , Γ p-Ps can be approximated by Γ. We therefore treat ∆ Ps HFS , Γ p-Ps and A as the three parameters to be determined in the fit.
The systematic errors are summarized in Table 2. We estimate the systematic error on C from the measurement of the water temperature (10%) and correction of the spatial distribution (10%). This was combined with the variations of C observed under different reflection conditions. The standard deviation of this fluctuation is between 9% and 20% for the different gyrotron cavities. At each frequency, C was scanned within its error to estimate the effect on the three fitting parameters.
The Stark effect due to the electric field of gas molecules induces a shift in ∆ Ps HFS . This effect is estimated from the measurements in nitrogen gas used in Ref. [6], assuming that it depends linearly on the number of density and the scattering crosssection obtained in Doppler-broadening measurements [21]. The shift is corrected (+460 ppm) and the amount of this correction is conservatively assigned as a systematic error. A linear extrapolation is sufficient at the current experimental precision, however, as has recently been pointed out [11], the effect of non-thermalized Ps distorts the linearity by around 10-20 ppm, and may be problematic for more precise measurements. We also estimate an uncertainty due to detection efficiencies using GEANT4 simulation. Since S /N is used to obtain the cross-sections, only the relative efficiency between 2γ-and 3γdecays are considered. Energy spectra of beam-OFF events are fitted with the simulated spectra of 2γ-and 3γ-decays, in which their ratio is taken as a free parameter. This ratio is nothing but the pick-off annihilation probability of beam-OFF events, given by fitting the time spectra. The lifetime of o-Ps decreases from 142 ns to approximately 131 ns due to this effect (the pickoff annihilation probability is about 8%). Relative differences of the two pick-off annihilation probabilities determined using these different methods are between 1% and 17% at the different frequencies, and are assigned as a systematic uncertainty to the simulated S /N. These errors are propagated to obtained cross-sections and then to the three fitting parameters.

Result and discussion
The systematic errors discussed above are independent, and are therefore summed quadratically to calculate the total systematic error. The results are listed in Table 3. This is the first direct measurement of both ∆ Ps HFS and Γ p-Ps . These all are consistent with the theoretical predictions [7][22] [23]. This is an essential step for the precise millimeter-wave spectroscopy of energy levels in Ps, but the current direct measurement of ∆ Ps HFS is far from the required level of precision to test the observed discrepancy (15 ppm) in previous indirect measurement. However, this method has a great potential to solve the anomaly because this is the only measurement which does not use the static magnetic field for the Zeeman splitting. In addition to the experimental uncertainties due to the magnetic field, the direct measurement is free from the higher order corrections to the Zeeman effect [24]. We now discuss three improvements to achieve accuracy of 10 ppm level for ∆ Ps HFS : 1. Using a high-intensity positron beam (intensity of 7×10 7 e + /s is available in KEK [26]) would increase the statistics by four orders of magnitude because only a few kHz Ps is formed inside the Fabry-Pérot cavity using the 22 Na source. The statistical error becomes smaller than 10 ppm. 2. Positronium will be formed in vacuum using an efficient Ps converter (conversion efficiency is around 20-50% [25]). The Stark effect (460 ppm at 1 atm) and nonthermalization effect of Ps (about 10-20 ppm) can be eliminated. Since there is no pick-off annihilation in vacuum, S /N will be also improved significantly by a factor of two. 3. Using a megawatt (MW) class gyrotron [15][27] would enable us to precisely (better than 0.3 %) monitor the real power with a calorimeter. The present accuracy (20 %) of the power estimation is mainly limited by uncertainty of the effective power in the Fabry-Pérot cavity. The systematic error due to the power can be better than 10 ppm.