New method of precise measurement of positronium hyperfine splitting
A. Ishida, G. Akimoto, Y. Sasaki, T. Suehara, T. Namba, S. Asai, T. Kobayashi, H. Saito, M. Yoshida, K. Tanaka, A. Yamamoto
aa r X i v : . [ h e p - e x ] A p r New method of precise measurement of positronium hyperfine splitting
A. Ishida a, ∗ , G. Akimoto a , Y. Sasaki a , T. Suehara b , T. Namba b , S. Asai a , T. Kobayashi b , H. Saito c , M. Yoshida d , K. Tanaka d , andA. Yamamoto d a Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan b International Center for Elementary Particle Physics (ICEPP), The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan c Department of General Systems Studies, Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan d High Energy Accelerator Research Organization (KEK), 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan
Abstract
The ground state hyperfine splitting of positronium, ∆ HFS , is sensitive to high order corrections of QED. A new calculation upto O( α ln α ) has revealed a 3 . σ discrepancy between the QED prediction and the experimental results. This discrepancy mighteither be due to systematic problems in the previous experiments or to contributions beyond the Standard Model. We proposean experiment to measure ∆ HFS employing new methods designed to remedy the systematic errors which may have a ff ected theprevious experiments. Our experiment will provide an independent check of the discrepancy. The prototype run has been finishedand a result of ∆ HFS = .
380 4 ± .
008 4 GHz(41 ppm) has been obtained. A measurement with a precision of O(ppm) is expectedwithin a few years.
Keywords: quantum electrodynamics (QED), positronium, hyperfine splitting (HFS)
1. Introduction
Positronium (Ps), a bound state of an electron and a positron,is a purely leptonic system which allows for very sensitivetests of Quantum ElectroDynamics (QED). The precise mea-surement of the hyperfine splitting between orthopositronium(o-Ps, 1 S ) and parapositronium (p-Ps, 1 S ) (Ps-HFS) pro-vides a good test of bound state QED. Ps-HFS is expected tobe relatively large (for example compared to hydrogen HFS)due to a relatively large spin-spin interaction, and also due tothe contribution from vacuum oscillation (o-Ps → γ ∗ → o-Ps).The contribution from vacuum oscillation is sensitive to newphysics beyond the Standard Model.Figure 1 shows the measured and theoretical values of Ps-HFS. The combined value from the results of the previous 2experiments is ∆ expHFS = .
388 65(67) GHz(3 . α ln α ) corrections to the theoretical prediction whichnow stands at ∆ thHFS = .
391 69(41) GHz(2 . σ ) between ∆ expHFS and ∆ thHFS might either be due to the common systematic uncertain-ties in the previous experiments or to new physics beyond theStandard Model.There are two possible common systematic uncertaintiesin the previous experiments. One is the unthermalized o-Pscontribution which results in an underestimation of the mate-rial e ff ect. This e ff ect has already been shown to be signifi-cant [4, 5, 6] in the o-Ps lifetime puzzle. The other is the uncer- ∗ Corresponding author (TEL: + / FAX: + Email address: [email protected] (A. Ishida)
HFS [ GHz ] Experimentalaverage Theory(Kniehl et al., 2000)Mills et al., 1983Ritter et al., 1984
Figure 1: Measured and theoretical values of Ps-HFS. tainty in the magnetic field uniformity which was cited as themost significant systematic error by previous experimenters.
2. Theory of Experiment ff ect The energy levels of the ground state of Ps are shown as afunction of static magnetic field in Figure 2. Due to techni-cal di ffi culties in directly stimulating ∆ HFS , we make an indi-rect measurement by stimulating the transition ∆ mix . This is thesame approach as previous experiments. The relationship be-tween ∆ HFS and ∆ mix is approximately given by the Breit-Rabiequation ∆ mix ≃ ∆ HFS (cid:16) √ + x − (cid:17) , (1) October 1, 2018
HFS D mix -> H m z = L +> H m z = L Ý> , ¯ß> H m z = ± L - - - - E p - Ps - E o - Ps B @ T D E @ GH z D Figure 2: Zeeman energy levels of Ps in its ground state. The arrows ↑ , ↓ meansthe spin up and down of electron, and the arrows ⇑ , ⇓ means the spin up anddown of positron. in which x = g ′ µ B B / h ∆ HFS , g ′ = g (cid:16) − α (cid:17) is the g factor fora positron (electron) in Ps [7], µ B is the Bohr magneton, B isthe static magnetic field, and h is the Plank constant.In a static magnetic field, the | S , m z i = | , i , where S isthe total spin of Ps and m z is the magnetic quantum number ofPs along with z-axis (direction of static magnetic field), statemixes with the | , i state hence the | + i state annihilates into2 γ -rays with a lifetime of about 8 ns (with our experimentalconditions). The | , ± i states annihilate into 3 γ -rays with alifetime of about 140 ns. When a microwave field with a fre-quency of ∆ mix is applied, transitions between the | + i state andthe | , ± i states are induced so that the 2 γ -ray annihilationrate increases and the 3 γ -ray annihilation rate decreases. Thischange of annihilation rates is our experimental signal.Our experimental resonance line shape is obtained using den-sity matrix. We use the basis for four spin eigenstates of Ps as( ψ , ψ , ψ , ψ ) ≡ ( | , i , | , i , | , i , | , − i ). We apply a mag-netic field B ( t ) = B e z + B e x cos ( ω t ) , (2)where e z , e x are the unit vectors for z, x direction respectively, B is magnetic field strength of microwaves, and ω is the fre-quency of microwaves. Then the Hamiltonian H becomes H = − − i2 γ s x − y yx − i2 γ t − y − i2 γ t y − i2 γ t h ∆ HFS , (3)where y = C y g ′ µ B B h ∆ HFS cos ( ω t ), C y is a constant, γ s = Γ p-Ps π ∆ HFS , γ t = Γ o-Ps π ∆ HFS , Γ p-Ps is the decay rate of p-Ps, and Γ o-Ps is that of o-Ps.The most recent and precise experimental values are Γ o-Ps = .
040 1(7) µ s − [4] and Γ p-Ps = .
990 9(17) ns − [8]. From the time-dependent Schr¨odinger equation, the 4 × ρ ( t ) is given byi ~ ˙ ρ = H ρ − ρ H † , (4)where the i,j-element of ρ ( t ) is defined as ρ ij ( t ) ≡h ψ i | ψ ( t ) ih ψ ( t ) | ψ j i . If we take the initial state to be unpolarized, ρ (0) = diag (cid:16) , , , (cid:17) .The 2 γ -ray annihilation probability S γ and 3 γ -ray annihila-tion probability S γ , between t = t and t = t , are obtainedby S γ = Γ p-Ps Z t t ρ ( t ) dt , (5) S γ = Γ o-Ps Z t t ( ρ ( t ) + ρ ( t ) + ρ ( t )) dt . (6) ff ect Forming Ps needs material to provide electrons, but materialaround Ps makes electric field and changes ∆ HFS . This e ff ectis called the Stark e ff ect. The material e ff ect must be properlyconsidered to evaluate ∆ HFS in vacuum. In the previous exper-iments, the material e ff ect on ∆ HFS was considered to be pro-portional to the material (gas) density. ∆ HFS was measured atvarious gas density, and they were extrapolated linearly to zerodensity [1, 2]. But this extrapolation method can make largesystematic error enough to account for the discrepancy between ∆ expHFS and ∆ thHFS .Formed Ps has the initial energy of O(eV). Positronium losesits energy when it collides with material, and finally its energybecomes to room temperature ( ∼ /
30 eV). This process iscalled thermalization. The material e ff ect between t = t and t = t is proportional to Z t t f ( t ) ( ρ ( t )( B , − ρ ( t )( B = dt , (7)where f ( t ) ∼ n σ v ( t ) is the collision rate of Ps with material, n is the number density of material, σ is the typical cross sectionof collision, and v ( t ) is the mean velocity of Ps. In the previousexperiments, timing information was not measured so that t = t = ∞ . The e ff ect is proportional to the material densityif the thermalization occurs much faster than the o-Ps lifetime,but the thermalization time scale becomes large especially atlow material density, which makes nonlinear e ff ect on ∆ HFS .The nonlinear e ff ect can be estimated using the thermaliza-tion model [9], dE av ( t ) dt = − p m Ps E av ( t ) E av ( t ) − kT ! r π σ m nM , (8)where E av ( t ) is the average Ps energy, m Ps is the Ps mass, M isthe mass of the gas molecule, T is the temperature of the gas, k is the Boltzmann constant, and q π σ m nM means the collisione ff ect. The solution of this equation is [10] E av ( t ) = + Ae − bt − Ae − bt ! kT , (9)2 ENSITY (amagat)0 0.5 1 1.5 2 2.5 3 ( G H z ) H F S ∆ Constant 203.4Slope -0.006804
DATA FITTING WITH NONLINEAR EFFECTPhys. Rev. A 30, 1331 (1984) ) QED α ln α O( Figure 3: (Color online) Thermalization e ff ect on ∆ HFS . The circles and errorbars are the data of Ref. [1], the solid line is the linear fit, the dashed line isthe best fit including Ps thermalization e ff ect, and the red band is the O( α ln α )QED prediction in vacuum. where b = q π σ m nM √ m Ps kT , A = √ E − √ kT √ E + √ kT , and E ≡ E av (0) is the initial energy of Ps.Figure 3 shows the best fit result of the previous experimen-tal data from Ref. [1] considering nonlinear material e ff ect us-ing Equations (7) and (9) with parameters E = .
07 eV and σ m = . [11], compared with linear fitting. The nonlin-ear e ff ect can be clearly seen at low density, and it can explainthe discrepancy between ∆ expHFS and ∆ thHFS (the best fit value in-cluding nonlinear e ff ect is ∆ HFS = .
392 80(95) GHz). But σ m = ±
10 Å from Ref. [9], which is not consistent with thevalue of Ref. [11], a ff ects only about 5 ppm and cannot explainthe discrepancy. Therefore, new independent measurement ofPs thermalization is needed. Our new methods will significantly reduce the systematic er-rors present in previous experiments; the Ps thermalization ef-fect and the non-uniformity of the magnetic field. The mainimprovements in our experiment are the large bore supercon-ducting magnet, β -tagging system, and high performance γ -raydetectors. Details are discussed in the following sections.
3. Prototype Run
The prototype run of the measurement with new methods hasbeen performed.
A schematic diagram of the experimental setup of the proto-type run is shown in Figure 4.
A large bore superconducting magnet is used to produce themagnetic field B ∼ .
866 T which induces the Zeeman split-ting. The bore diameter of the magnet is 800 mm, and its length
Figure 4: Schematic diagram of the experimental setup of the prototype run(top view in magnet). is 2 m. The large bore diameter means that there is good uni-formity in the magnetic field in the region where Ps is formed.Furthermore, the magnet is operated in persistent current mode,making the stability of the magnetic field better than ± β -tagging system and timing information The positron source is 19 µ Ci (700 kBq) of Na (Eckert &Ziegler POSN-22). A plastic scintillator (NE102A) 10 mm indiameter and 0.2 mm thick is used to tag positrons emitted fromthe Na. The scintillation light is detected by fine mesh pho-tomultiplier tubes (PMT: HAMAMATSU H6614-70MOD) andprovides a start signal which corresponds to the time of Ps for-mation. The timing resolution is 1.0 ns (1 σ ). The positron thenenters the microwave cavity, forming Ps in the gas containedtherein.Ps decays into photons that are detected with LaBr (Ce)scintillators (Saint-Gobain BrilLanCe TM γ -detectionresults in decay curves of Ps as shown in Figure 5. The timinginformation is used to improve the accuracy of the measurementof ∆ HFS as follows:1. Imposing a time cut means that we can select well ther-malized Ps, reducing the unthermalized o-Ps contribution.It should also be possible to precisely measure the contri-butions of unthermalized o-Ps, and of material e ff ects (weplan to make such measurements in future runs).2. A time cut also allows us to avoid the prompt peak (con-tributions of simple annihilation and of fast p-Ps decay),which greatly increases the S / N of the measurement (byabout a factor of 20).3 .1.3. High performance γ -ray detectors Six γ -ray detectors are located around the microwave cavityto detect the 511 keV annihilation γ -rays. LaBr scintillators,1.5 inches in diameter and 2 inches long are used. The scintilla-tion light is detected by fine mesh PMT through the UVT lightguide. Without light guide, LaBr scintillators have good en-ergy resolution (4% FWHM at 511 keV) and timing resolution(0.2 ns FWHM at 511 keV), and have a short decay constant(16 ns). The good energy resolution and the high counting rateof LaBr results in very good overall performance for measur-ing Zeeman transitions. In particular the good energy resolu-tion allows us to e ffi ciently separate 2 γ events from 3 γ events,negating the need to use a back-to-back geometry to select 2 γ events, thus greatly increasing the acceptance of our setup.This γ -ray detector system greatly reduces the statistical er-ror in the measurement. Microwaves are produced by a local oscillator signal gener-ator (ROHDE & SCHWARZ SMV 03) and amplified to 500 Wwith a GaN amplifier (R&K A2856BW200-5057-R).The microwave cavity is made with oxygen-free copper;the inside of the cavity is a cylinder 128 mm in diameter and100 mm long. The side wall of the cavity is only 2 mm thickin order to allow the γ -rays to e ffi ciently escape. The cav-ity is operated in the TM mode. The resonant frequency is2.856 6 GHz and Q L = , ±
50. The cavity is filled withgas (90% N and 10% iso-C H ) with a gas-handling system.Iso-C H is used as the quenching gas to remove background2 γ -ray annihilation. The Monte Calro simulation to use in analysis is performedusing Geant4 [12]. The low energy physics package PENE-LOPE [13] is used and the geometry of the experimental setupis carefully input. The simulation is produced at each magneticfield strength and gas density.
The prototype run was performed from 2 July 2009 to 24September 2009 using the large bore magnet with no compensa-tion (compensation magnets to reduce the uniformity to O(ppm)are planned but are not yet installed). In the overall period, thetrigger rate was about 3.6 kHz and the data acquisition rate wasabout 650 Hz. The data acquisition was performed using NIMand CAMAC system.The trigger signal is the coincidence signal of β -tagging sys-tem and γ -ray detectors. Timing information of all PMT are ob-tained with a 2 GHz direct clock counting Time-to-Digital Con-verter (TDC: GNC-060) [4, 5]. A charge ADC (CAEN C1205)is used to measure the energy information of the LaBr crystalswhile another charge ADC is used to measure the energy in-formation of the plastic scintillators. A crate controller (TOYOCC / NET) is used and obtained data are stored in a HDD of aLinux PC via Ethernet. The Zeeman transition has been mea-sured at various magnetic field strengths with a fixed RF fre-
TIME (ns)0 50 100 150 200 C O UN T S ( / ke V / n s / s ) -4 -3 -2 -1 m_h_time_ind_ew7_la0_n150_x1400 Entries 19415Mean 17.67RMS 41.48 -1) TIMING SPECTRA at 0.8658169 T (LaBr
TIMING WINDOW
Figure 5: (Color online) Decay curves of Ps. The red line is on-resonance(0.865 816 9 T) RF ON, and the black line is RF OFF. The timing window of35–155 ns is also indicated. The decay rate of Ps increases with RF because ofthe Zeeman transition. quency and power. The transition resonance lines are obtainedat two gas densities (1.350 1 amagat and 0.891 6 amagat).
Figure 5 shows examples of measured timing spectra. Theprompt peak coming from annihilation and |−i decay is fol-lowed by the decay curve of | + i and m z = ± t = S OFF can be expressed as S OFF = A OFF (cid:16) S MC3 γ + β S MC2 γ (cid:17) , (10)where A OFF is a normalizing constant, β is a 2 γ/ γ decay ratioof RF-OFF, S MC2 γ is a Monte Calro simulated energy spectrumof 2 γ decay, and S MC3 γ is that of 3 γ decay. On the other hand,the energy specrum of RF-ON S ON can be expressed as S ON = A ON (cid:16) S MC3 γ, in + Γ S MC2 γ, in (cid:17) + A OFF (cid:16) S MC3 γ, out + β S MC2 γ, out (cid:17) , (11)where A ON is a normalizing constant and Γ is a 2 γ/ γ decayratio of RF-ON. The subscript “in” and “out” means the spec-trum of γ -rays from Ps decays in the volume where RF poweris applied or not applied, respectively.Typical fitted results are also shown in Figure 6. Figure 6(a)is RF-OFF, and 6(b) is RF-ON. The energy spectrum is mea-sured at di ff erent magnetic field strengths. The fitting range is370–545 keV. All the spectra are fitted successfully using MI-NUIT [14].We take Γ − β as an amount of Zeeman transition. An exampleof resonance line obtained is shown in Figure 7. Resonancelines can be fitted by S γ / S γ (RF-ON) − S γ / S γ (RF-OFF) .The theoretical function is calculated from Equations (5) and(6) numerically using RKF45 formula, with contributions frompick-o ff and slow positron. The typical B is 14.2 G. The RF-OFF function is obtained by substitute 0 for B . Then the free4 _h_data_la0Entries 9002Mean 293.5RMS 107.4 ENERGY (keV)150 200 250 300 350 400 450 500 550 600 C O UN T S ( / V / s ) m_h_data_la0Entries 9002Mean 293.5RMS 107.4 TYPICAL FIT WITH MC (RF-OFF)
FITTING RANGE (a) Typical fitting of energy spectrum (RF-OFF). The black pointsare the data, the red line is S MC2 γ , the blue line is S MC3 γ , and the pinkline is sum of the Monte Calro spectra. m_h_data_la0Entries 9002Mean 309.4RMS 120 ENERGY (keV)150 200 250 300 350 400 450 500 550 600 C O UN T S ( / V / s ) m_h_data_la0Entries 9002Mean 309.4RMS 120 TYPICAL FIT WITH MC (RF-ON)
FITTING RANGE (b) Typical fitting of energy spectrum (RF-ON). The black pointsare the data, the red line is S MC2 γ, in , the blue line is S MC3 γ, in , the greenline is S MC2 γ, out , the aqua line is S MC3 γ, out , and the pink line is sum ofthe Monte Calro spectra. A OFF and β are fixed with the fitted valuesof RF-OFF, so that the free parameters are A ON and Γ .Figure 6: (Color online) Typical energy spectra and its fitting with Monte Calrosimulation. The data at 1.350 1 amagat, 0.865 816 9 T are shown and the spectraare normalized by the live time. MAGNETIC FIELD (T) ) β - Γ T RAN S I T I O N ( / ndf χ χ RESONANCE LINE at 1.3501 amagat
Figure 7: Resonance line at 1.350 1 amagat gas density. The circles and errorbars are the data, and the solid line is the best fit result. The error bars includeerrors from statistics of data, statistics of Monte Calro simulation, uncertaintyof RF power, and uncertainty of Q L value of the RF cavity. Table 1: Fitting result of the resonance lines. These uncertainties include errorsfrom statistics of data, statistics of Monte Calro simulation, uncertainty of RFpower, and uncertainty of Q L value of the RF cavity. Gas density ∆ HFS
Relative error χ / ndf(amagat) (GHz) (ppm)1.350 1(71) 203.368 3(55) 27 0.9100.891 6(23) 203.379 3(70) 34 0.483parameters of fitting are ∆ HFS and C y . The fitting results aresummarized in Table 1. We use −
33 ppm / amagat [1] as thematerial e ff ect and obtain ∆ HFS in vacuum.
Systematic errors of the prototype run are summarized in Ta-ble 2.1.
Magnetic Field . The largest uncertainty in the prototyperun is non-uniformity of the magnetic field. The weightednon-uniformity is 10.4 ppm. The o ff set and reproducibilityof the magnetic field is measured to be 2 ppm. The calibra-tion uncertainty of NMR magnetometer is 1 ppm. Theseuncertainties are doubled because ∆ HFS is approximatelyproportional to square of the magnetic field strength.2.
Monte Calro Simulation . The magnetic field dependenceof β is not exactly reproduced by the energy spectra fit-ting with Monte Calro simulation. From the di ff erence be-tween theoretical dependence and fitted result, this e ff ectis estimated to be 18 ppm. The statistical uncertainty ofMonte Calro simulation is 17 ppm.3. RF System . The uncertainty of Q L value of RF cavitycomes from reproducibility of the RF environment ( ∼ Q L measurement method (0.6%).Its e ff ect on ∆ HFS is estimated to be 6 ppm. RF power un-certainty comes from reproducibility of the RF environ-ment ( ∼ / K). It contributes to the error on ∆ HFS by 5 ppm. The uncertainty of RF frequency is 5 ppm. Itdirectly a ff ects ∆ HFS because ∆ HFS is approximately pro-portional to inverse of RF frequency.4.
Material E ff ect . Thermalization of Ps can a ff ect ∆ HFS byup to 20 ppm, but it has been not yet measured. Gasdensity dependence has not been measured in the proto-type run, so we have used the value from previous ex-periment. The uncertainty of the density dependence is4 ppm / amagat [1], and the uncertainty from iso-C H isestimated to be less than 7.7 ppm / amagat. These result in7 ppm uncertainty of ∆ HFS .Other uncertainties are considered to be negligible. The sys-tematic errors discussed above are summed in quadrature.
The value of ∆ HFS obtained from the prototype run is ∆ HFS = .
380 4 ± .
002 2(stat .,
11 ppm) ± .
008 1(sys .,
40 ppm) GHz , (12)5 able 2: Summary of Systematic errors of the prototype run. Source Errors in ∆ HFS (ppm)
Magnetic Field:
Non-uniformity 21O ff set and reproducibility 4NMR measurement 2 Monte Calro Simulation:
Magnetic field dependence 18Statistics 17
RF System:Q L value of RF cavity 6RF power 5RF frequency 5 Material E ff ect: Thermalization of Ps < The following improvements are planned for future measure-ments:1. Compensation magnets will be installed and O(ppm) mag-netic field uniformity is expected to be achieved.2. The Monte Calro simulation will be studied to reproducethe magnetic field dependence of energy spectra of Ps de-cays. Statistics of simulation will be reduced to O(ppm).3. The errors from RF system will be reduced to O(ppm) bycarefully controlling the environment (especially the tem-perature) of the experiment.4. Measurements at various pressures of gas will be per-formed to estimate the material e ff ect (the Stark e ff ect).The accumulation of these measurements will result in anO(ppm) statistical error within a few years.5. The timing information allows for a measurement of Psthermalization as a function of time [4, 5, 6]. We can thusprecisely measure the material e ff ect including the ther-malization e ff ect.
4. Conclusion
A new experiment to measure the Ps-HFS which re-duces possible common uncertainties in previous experimentshas been constructed and the prototype run has been fin-ished. A value of ∆ HFS = .
380 4 ± .
002 2(stat . ) ± .
008 1(sys . ) GHz (41 ppm) has been obtained, which is consis-tent with both of the previous experimental values and with thetheoretical calculation. Development of compensation magnetsis underway with a view to obtaining O(ppm) magnetic fieldhomogeneity for the final run. The final run will start soon. Anew result with an accuracy of O(ppm) will be obtained within a few years which will be an independent check of the discrep-ancy between the present experimental values and the QED pre-diction. References [1] M. W. Ritter, P. O. Egan, V. W. Hughes, and K. A. Woodle, Phys. Rev. A30 (1984) 1331.[2] A. P. Mills, Jr. and G. H. Bearman, Phys. Rev. Lett. 34 (1975) 246; A. P.Mills, Jr., Phys. Rev. A 27 (1983) 262.[3] B. A. Kniehl and A. A. Penin, Phys. Rev. Lett. 85 (2000) 5094; K. Mel-nikov and A. Yelkhovsky, Phys. Rev. Lett. 86 (2001) 1498; R. J. Hill,Phys. Rev. Lett. 86 (2001) 3280.[4] Y. Kataoka, S. Asai, and T. Kobayashi, Phys. Lett. B 671 (2009) 219.[5] Y. Kataoka,
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MINUIT Function Minimization and Error Analysis, ver-sion 94.1