Measurement of the neutron decay electron-antineutrino angular correlation by the aCORN experiment
M. T. Hassan, W. A. Byron, G. Darius, C. DeAngelis, F. E. Wietfeldt, B. Collett, G. L. Jones, A. Komives, G. Noid, E. J. Stephenson, F. Bateman, M. S. Dewey, T. R. Gentile, M. P. Mendenhall, J. S. Nico
MMeasurement of the neutron decay electron-antineutrino angular correlation by theaCORN experiment
M. T. Hassan, ∗ W. A. Byron, † G. Darius, C. DeAngelis, and F. E. Wietfeldt
Department of Physics and Engineering Physics, Tulane University, New Orleans, LA 70118
B. Collett and G. L. Jones
Physics Department, Hamilton College, Clinton, NY 13323
A. Komives
Department of Physics and Astronomy, DePauw University, Greencastle, IN 46135
G. Noid and E. J. Stephenson
CEEM, Indiana University, Bloomington, IN 47408
F. Bateman, M. S. Dewey, T. R. Gentile, M. P. Mendenhall, ‡ and J. S. Nico National Institute of Standards and Technology, Gaithersburg, MD 20899, USA (Dated: December 29, 2020)The aCORN experiment measures the neutron decay electron-antineutrino correlation ( a -coefficient) using a novel method based on an asymmetry in proton time-of-flight for events where thebeta electron and recoil proton are detected in delayed coincidence. We report the data analysis andresult from the second run at the NIST Center for Neutron Research, using the high-flux cold neutronbeam on the new NG-C neutron guide end position: a = − . ± . ± . a = − . ± . ± . λ = G A /G V is λ = − . ± . ∗ current address: Los Alamos National Laboratory, Los Alamos, NM 87545, USA. † current address: Dept. of Physics, University of Washington, Seattle, WA 98195, USA ‡ current address: Lawrence Livermore National Laboratory, Livermore, CA 94550, USA a r X i v : . [ nu c l - e x ] D ec I. INTRODUCTION
The free neutron decays into a proton, electron, and antineutrino via the charged-current weak interaction. Thisis the simplest example of nuclear beta decay. In contrast to beta decay of most nuclei, the dynamics of neutrondecay are undisturbed by nuclear structure effects. Experimental observables can be directly related to fundamentalparameters in the theory. As a result, neutron decay is an excellent laboratory for studying details of the weak nuclearforce and searching for hints of physics beyond the Standard Model (SM). The important experimental features ofneutron decay are described by the formula of Jackson, Treiman, and Wyld [1], which gives the decay probability N of a spin-1/2 beta decay system in terms of the neutron spin polarization P , the beta electron total energy andmomentum E e , p e , and the antineutrino total energy and momentum E ν , p ν N ∝ τ n E e | p e | ( Q − E e ) (cid:20) a p e · p ν E e E ν + b m e E e + P · (cid:18) A p e E e + B p ν E ν + D ( p e × p ν ) E e E ν (cid:19)(cid:21) . (1) Q = 1293 keV is the neutron-proton mass difference, m e is the electron mass, and τ n is the neutron lifetime. Hereand throughout velocity is in units with c = 1. The parameters a , A , B , and D are correlation coefficients whichare measured by experiment. We note that a , b are parity conserving, A , B are parity violating, and D violatestime-reversal symmetry. The Fierz interference parameter b is zero in the SM; it would be generated by the presenceof scalar or tensor weak currents. Neglecting recoil order effects, the values of the other coefficients are related to twobasic parameters in the theory: the nucleon weak vector and axial vector coupling constants G V and G A . Writingtheir ratio as λ = G A /G V we have [1] τ n = 2 π (cid:126) ( G V + 3 G A ) m e f R a = 1 − λ λ A = − { λ } + λ λ B = − { λ } − λ λ D = 2 Im { λ } λ (2)where f R is the value of the integral over the Fermi energy spectrum. There are two main motivations for precisionmeasurements of neutron decay observables.The first is to accurately determine the values of G V and G A . These constants appear not only in neutron decaybut in many other weak interaction processes involving free neutrons and protons that are important in astrophysics,cosmology, solar physics, and neutrino detection [2, 3]. The value of G V gives the first element V ud of the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix: G V = G F V ud , where G F is the universal weak coupling constantobtained from the muon lifetime. A very important low energy test of the Standard Model is the unitarity of the firstrow of the CKM matrix | V ud | + | V us | + | V ub | = 1 . (3)The term | V ub | is small enough to be neglected so in practice this is a precise comparison of V ud and V us . A realviolation of this unitarity condition would be a clear sign of new physics Beyond the Standard Model (BSM) at thelow energy, precision frontier. For example supersymmetry loop corrections could cause a departure from equation 3at the few 10 − level and reveal new physics that lies beyond present constraints from the Large Hadron Collider [4].The second motivation is to search for small discrepancies in the values of these observables that could resultfrom BSM physics. We see from equation 2 that a measurement of τ n and any one of a , A , or B determine thereal values of G V and G A , but new physics could introduce dependencies on additional new parameters. A usefulmodel-independent self-consistency test is obtained from the Mostovoy parameters [5] F = 1 + A − B − a = 0 F = aB − A − A = 0 . (4)which follow algebraically from the relations in equation 2. Inserting the Particle Data Group 2020 (PDG 2020) [6]recommended values we have F = 0 . ± . F = 0 . ± . a -coefficient is the largest contributor to the uncertainties in F and F .We note that recoil order corrections will cause F and F to differ from zero at the 10 − level, but those correctionsare calculable. Important model-dependent tests for new physics can be made with neutron decay observables. Therelative values of a , A , and B can be related to the strength of hypothetical right-handed weak forces and scalar andtensor forces [7, 8]. Gardner and Zhang have shown that a comparison of a and A at the 10 − level can place sharplimits on possible conserved-vector-current (CVC) violation and second-class currents [9]. Possible extensions to theStandard Model, such as supersymmetry or left-right symmetric models, could lead to observable departures from thepredictions in equations 2.Figure 1 summarizes the current experimental results for G A and G V . The PDG 2020 recommended value λ = − . ± . A and a from 1986 to 2019, and theuncertainty is expanded by a factor of 2.6 due to poor agreement. The most recent and precise results for the betaasymmetry A from the PERKEO II,III [10, 11] and UCNA [12] experiments are in good agreement and give a morenegative value of λ . The neutron lifetime averages from beam method and ultracold storage experiments significantlydisagree, see for example [13]. The value of G V = 1 . × − GeV − from an evaluation of 222 measurementsof 20 superallowed beta decay systems [14] agrees moderately with the CKM unitarity requirement (using the PDG2020 V us [6]), differing by 1.2 σ . But a 2018 calculation of electroweak box diagram contributions to the “universal”radiative correction ∆ R by Seng, et al. [15] shifts the superallowed result down to G V = 1 . × − GeV − which would violate CKM unitarity by 3.4 σ . In a recent paper Czarnecki, Marciano, and Sirlin [16] recommendan intermediate value for ∆ R and hence G V . Clearly the present experimental situation with G A and G V is notsatisfactory and additional measurements, with careful attention to evaluating systematic effects and uncertainties,are needed. II. EXPERIMENTAL METHOD
The traditional method for measuring the a -coefficient is from the shape of the recoil ion energy spectrum. If thebeta electron and antineutrino momenta are anticorrelated, the average recoil momentum is reduced, which shiftsweight to the low-energy part of the spectrum. Until recently all measurements of the neutron a -coefficient usedsome variation of this method and achieved results that were systematically limited at the 5 % level [17–19]. Themethod used by aCORN, first proposed by Yerozolimsky and Mostovoy [20, 21], relies on a novel time-of-flight (TOF)asymmetry that does not require precise proton spectroscopy. The aCORN method is illustrated in figure 2. Assumea pointlike cold neutron source on the axis between a set of opposing electron and proton detectors with a uniformaxial magnetic field applied throughout. Electron and proton collimators, shown schematically as cylindrical tubes,lie on the axis. When a cold neutron, which is effectively at rest, decays, a beta electron, antineutrino, and proton areemitted. Due to their helical motion in the magnetic field B , the collimators impose a maximum transverse momentumof p ⊥ (max) = eBr/
2, where r is the collimator radius, for detected electrons and protons. An electrostatic mirrorcontaining a uniform axial electric field, produced by a pair of grids at ground and +3 kV as shown, causes all neutrondecay protons to be accelerated and directed toward the proton detector. Electrons in the energy range of interestmust be emitted into the right hemisphere to be detected. The momentum acceptances for the electron and protonin this scheme are shown in figure 2 (middle). These are cylinders in momentum space and the proton acceptanceextends to both sides of the origin. Now consider the antineutrino momentum acceptance for coincidence-detectionevents where the electron momentum is (cid:126)p e as shown. Conservation of momentum requires (cid:126)p ν = − ( (cid:126)p e + (cid:126)p p ) so theantineutrino momentum acceptance is a cylinder equivalent to the proton acceptance cylinder but displaced from theorigin by − (cid:126)p e . If we neglect the kinetic energy of the proton (751 eV maximum) the electron and antineutrino mustshare the total decay energy Q = 1293 keV and conservation of energy requires | (cid:126)p ν | = Q − (cid:112) p e + m e . So for the given (cid:126)p e the antineutrino momentum must lie on the intersection of the cylinder and sphere shown in figure 2 (bottom),which is indicated by the gray regions marked I and II. Region I (II) antineutrinos are correlated (anticorrelated)with (cid:126)p e and have equal solid angles from the origin. If the a -coefficient is zero, the number of coincidence eventsassociated with regions I and II will be equal. If not there will be an asymmetry. The same is true when we sum overall values of (cid:126)p e for detectable electrons. In reality the neutron source is not a point but a cylindrical beam passingthrough the electrostatic mirror perpendicular to (cid:126)E and (cid:126)B , so most decay vertices are off axis. For off-axis decaysthe proton and electron momentum acceptances are elliptical cylinders and the geometric construction is somewhatmore complicated, but the result is essentially the same and solid angles of regions I and II remain equal.In the experiment we measure the beta electron energy and proton TOF, the time between electron and protondetection, for neutron decay events where both were detected. The data form the characteristic wishbone shapeshown in figure 3. Region I antineutrinos are correlated with the electron momentum direction, so the associatedprotons have larger momentum and axial velocity and the events lie on the lower wishbone branch (group I). Region IIantineutrinos are anticorrelated with electron momentum, so the protons have smaller momentum and axial velocityand the events lie on the upper wishbone branch (group II). The gap between the wishbone branches corresponds tothe kinematically forbidden gap between regions I and II in figure 2 (bottom). At beta energy above about 400 keVthe regions overlap and the wishbone branches merge. A vertical slice at beta energy E , depicted in figure 3, contains N I events in the lower branch and N II events in the upper branch. Using equation 1 we have N I ( II ) ( E ) = F ( E ) (cid:90) (cid:90) (1 + av cos θ eν ) d Ω e d Ω I ( II ) ν (5)where F ( E ) is the beta energy spectrum, v is the beta velocity (in units of c ), θ eν is the angle between the electronand antineutrino momenta, and d Ω e , d Ω I ( II ) ν are elements of solid angle of the electron and antineutrino (group I,II) momenta. The integrals are taken over the momentum acceptances shown in figure 2. Since by construction thetotal solid angle products are equal for the two groups: Ω e Ω Iν = Ω e Ω IIν , we find that the a -coefficient is related tothe wishbone asymmetry X ( E ) by X ( E ) = N I ( E ) − N II ( E ) N I ( E ) + N II ( E ) = av (cid:0) φ I ( E ) − φ II ( E ) (cid:1) av ( φ I ( E ) + φ II ( E )) (6)The functions φ I ( E ) and φ II ( E ) are defined as φ I ( E ) = (cid:82) d Ω e (cid:82) I d Ω ν cos θ eν Ω e Ω Iν and φ II ( E ) = (cid:82) d Ω e (cid:82) II d Ω ν cos θ eν Ω e Ω IIν , (7)where again the integrals are taken over the momentum acceptances. Equations 7 can be understood as the averagevalue of cos θ eν for detection regions I and II. These are geometrical functions that depend only on the transversemomentum acceptances of the proton and electron so they can be calculated precisely from the known axial magneticfield and collimator geometries.The second term in the denominator of equation 6 has a numerical value less than 0.005 in the energy range ofinterest (100 keV–380 keV), so we can treat it as a small correction and write X ( E ) = af a ( E ) [1 + δ ( E )] + δ ( E ) (8)with f a ( E ) = 12 v (cid:0) φ I ( E ) − φ II ( E ) (cid:1) (9)and δ ( E ) = − av (cid:0) φ I ( E ) + φ II ( E ) (cid:1) . (10)The other small correction δ ( E ) in equation 8 comes from our neglect of the proton’s kinetic energy in the momentumspace discussion of figure 2. If we account for this energy, the antineutrino momentum sphere is slightly oblong andthe solid angles of groups I and II differ by approximately 0.1 %. This causes a small (about 1 % relative) intrinsicwishbone asymmetry that is independent of the a -coefficient; it is straightforward to compute by Monte Carlo to theneeded precision.Omitting the small corrections we see that X ( E ) = af a ( E ); the experimental wishbone asymmetry is proportionalto the a -coefficient and the dimensionless geometric function f a ( E ). In analyzing the data we take the approach ofassuming a perfectly uniform axial magnetic field and exact collimator configuration, and use the precisely computed f a ( E ) shown in figure 4. We then treat nonuniformities and uncertainties in the measured magnetic field magnitudeand shape and the collimator geometry as systematic effects applied to the result.aCORN runs on a nominally unpolarized neutron beam. If the beam were slightly polarized, there would be anadditional contribution to the wishbone asymmetry from the antineutrino asymmetry correlation B term in equation1, giving X ( E ) = af a ( E ) + P Bf B ( E ) (11)where P and B are the neutron polarization and B -coefficient, and f B ( E ) is a similarly calculated geometric functionfor P B . Because neutron polarization is more axially peaked than p e , f B ( E ) is on average 40 % larger than f a ( E ).Also | B/a | ≈
10. So even with P (cid:28) X ( E ) with the magnetic field in the up and down directions was attributed to a neutron polarization P ≈ III. THE aCORN APPARATUS
We describe here briefly the main components of the aCORN apparatus. More details can be found in previouspublications [23–25]. Figure 5 shows a cross section view of the aCORN tower.The 36.3 mT axial main magnetic field is produced by a vertical array of 24 individual flat coils supplied in series.Each coil contains 121 turns of 2 cm × µ m layer of copper divided into 63 parallel thin bands by photolithography, produced by Polyflon [26, 27]. These 63bands were held at potentials established by a chain of equal precision resistors to approximate the linear boundarycondition. The neutron beam was allowed to pass through the wall on both sides, each side scattering about 1 % ofthe beam by the PTFE and scattering/absorbing about 0.1 % of the beam by the copper. The end potentials wereset by grids of 100 µ m wires. The grid on the bottom (electron) side was at +3 kV and the grid on the top (proton)side was at ground.The proton detector was a 600 mm , 1000 µ m thick surface barrier detector held at -28 kV to accelerate protons toa detectable energy. Figure 6 shows an overhead view, looking down from the top of the tower. Detector componentswere located off-axis to prevent neutron decay electrons emitted in the upward direction from backscattering on theproton detector and returning to the beta spectrometer, where they would be detected with the wrong energy andwrong sign of cos θ eν . A focusing fork and ring act as a lens to focus all protons exiting the proton collimator ontothe active area of the detector. The proton detector is cooled by a copper panel attached to a liquid nitrogen coolingsystem.aCORN employed a novel backscatter-suppressed beta spectrometer, illustrated in cross-section in figure 7. Thebeta energy detector was a 5 mm thick, 280 mm diameter circular slab of Bicron BC-408 plastic scintillator, viewedby 19 Photonis XP3372 8 stage 7.6 cm (3 inch) hexagonal photomultiplier tubes (PMT’s). Surrounding the energydetector was an array of eight veto detectors, each composed of a 10 mm thick BC-408 plastic scintillator and adiabaticacrylic light guide viewed by a Burle 8850 12 stage 5.1 cm (2 inch) PMT. The spectrometer was mounted on the towerbelow the bottom flux return end plate. The axial magnetic field was high at the entrance to the spectrometer butdropped quickly below it to about 1 mT at the energy detector. All beta electrons with kinetic energy >
100 keV thatwere accepted by the beta collimator passed through the opening at the top of the veto array and struck the activearea of the energy detector, as verified by Monte Carlo simulation. Approximately 5 % were expected to backscatterfrom the plastic scintillator without depositing their full energy. This may lead to a large systematic effect, discussedin section VI E 3. To mitigate this a backscatter veto array was used; the majority of backscattered electrons strucka veto paddle and were vetoed. The overall veto efficiency for backscattered electrons was measured to be (92 ± Sn and
Bi). During production runs, in situ calibration measurements were madeat approximately 48 hour intervals to monitor slow gain drifts in the beta spectrometer and enable correction in thedata analysis. Details of the design, construction, and characterization of the aCORN beta spectrometer can be foundin reference [25].The main vacuum chamber of aCORN was a vertical aluminum tube 3 m tall and 28 cm inner diameter. It wasjoined at the top and bottom to the iron endplates by o-ring seals. A 250 l/s turbomolecular pump was mountedon the beta spectrometer chamber and a 370 l/s helium cryopump was attached to the beam dump. A set of threeliquid nitrogen cooled copper cryopanels extended from the top of the main chamber to the bottom of the protoncollimator to provide high conductance pumping of water and volatiles released by the plastic scintillator in the betaspectrometer. During normal operation the pressure at the top of the electrostatic mirror was about 8 × − Pa(6 × − torr). IV. MODIFICATIONS FOR THE NG-C RUN
A previous publication [24] describes the aCORN apparatus as it was used for the first measurement on the NG-6beamline at the NIST Center for Neutron Research (NCNR) [28] in 2013–2014. The experiment was moved, withsome modifications, to the new high-flux beamline NG-C in 2015 for a second run. This section describes thosemodifications.
A. Neutron beam and collimation
In 2013 a second guide hall was commissioned at the NCNR with four new supermirror guides. The end position onnew guide NG-C was designated for fundamental neutron physics experiments and aCORN was the first experimentto run there. NG-C is a ballistic curved supermirror guide 11 cm ×
11 cm at the exit with a measured capture fluxof 8 . × cm − s − . Details of the design of the NG-C guide and other guides in the new guide hall can be foundin [29]. Because NG-C is curved, a bismuth filter is not needed to remove fast neutrons and gammas, which improvesneutron transmission to the experiment. A 180-cm long secondary focusing supermirror guide was installed to reducethe beam cross section to a 6 cm × LiF apertures. Its interior was lined with Li glass to absorb scattered neutrons. The collimator reduced thebeam divergence and delivered a 3.1-cm diameter circular beam to the experiment. The capture flux in the neutrondecay region of aCORN was measured to be 6 . × cm − s − , about a factor of ten higher than the equivalentmeasurement with the experiment installed on NG-6, but with a beam area that was a factor of two smaller, resultingin an overall factor of five increase in the wishbone event rate from neutron decay.At the end of NG-C is a 2.4-m deep pit available to experiments that need part of the apparatus below floor level.For aCORN we constructed a false floor inside the pit at 40 cm below the main floor level for better access to thebeta spectrometer and the field mapping apparatus when installed. B. Electrostatic mirror
The departure from a perfectly axial electric field in the vicinity of the upper grounded end grid, where the protonspass through, resulted in the largest systematic correction (5.2 %) and uncertainty (1.1 %) in the result from theNG-6 run [22]. Guided by a 3D COMSOL [30] model along with a Monte Carlo proton transport simulation, wemade some improvements to the upper grid geometry to reduce this effect. We replaced the linear wire upper gridwith an electroformed square mesh copper grid containing 100 µ m threads spaced by 2 mm, purchased from PrecisionEforming [31]. We also redesigned the upper aluminum support ring to locate it entirely outside the thick PTFEtube, thereby increasing the open inner diameter at the top to 10.9 cm. The new upper grid can be seen in the photoin figure 8. These adjustments reduced the size of the electrostatic mirror correction by more than a factor of three(see the discussion in section VI E 1). The lower +3 kV grid was left unchanged; protons do not pass close to thelower grid and the electrostatic effect on electrons passing through it is negligible. C. Data Acquisition
Electronic pulses from the 19 beta energy channels, 8 backscatter veto channels, and the proton detector weresent to two PIXIE-16 modules [32] which are 12 bit, 100 MSPS multiplexing analog to digital converters. For theNG-C run we made two changes to the PIXIE-16 firmware: i) certain calculations that were not needed, such asconstant fraction discrimination ratios, were removed in order to increase throughput; and ii) the energy calculationfor all channels was switched from a trapezoidal filter to the charge to digital conversion (QDC) mode. In the QDCmode three timings are specified: 1) time before the event trigger to begin saving data (0.6 µ s for electrons, 1 µ s forprotons), 2) the pre-pulse time window (0.5 µ s for electrons, 0.5 µ s for protons), and 3) the pulse time window (0.3 µ s for electrons, 2 µ s for protons). The energy is then calculated as the average number of counts per channel in thepulse window minus the average number of counts per channel in the pre-pulse window. We found that this switchdid not noticeably affect energy resolution or linearity, but it significantly lowered the effective energy threshold whichwas useful for all channels but was particularly helpful for the proton channel. In the NG-6 data analysis [22] therewas a 3 % systematic correction to the a -coefficient due to loss of protons below threshold. Such a correction was notneeded in the NG-C data analysis. V. THE NG-C RUN aCORN ran on the NG-C end position at the NCNR from August 2015 to September 2016 and collected a total of3758 beam hours of neutron decay data. The raw coincidence event rate was 171 s − , The neutron decay wishboneevent rate, after background subtraction, was 0.9 s − , about a factor of three higher than in the previous run onNG-6.We collected data in both axial magnetic field directions in order to monitor and correct for a possible effect dueto residual polarization of the neutron beam. The first data run was magnetic field up (B up ), for 1097 beam hours.The second run was magnetic field down (B down ), for 2178 hours. The magnetic field was returned to B up for thefinal 482 hours. The following protocol was followed whenever the magnetic field was reversed: 1) the detectors andcollimation insert were removed and the field mapper installed; 2) the existing axial and transverse magnetic fieldswere mapped and compared to the previous maps; 3) the leads to the main magnet supply were reversed and alltrim coils were de-energized; 4) the magnetic field was mapped and trimmed to specification in the new direction; 5)the field mapper was removed and the detectors and collimation insert reinstalled and aligned. The entire processof reversing the magnetic field took about two weeks, completed mostly during NCNR refueling shutdown periods.Figure 9 shows results of on-axis axial and transverse field maps made in June 2015, prior to the first production run,and December 2015, just before the first field reversal and with the trim settings unchanged. Drifts in the field shapeover the six month span are evident in the plots. We attribute the increase in axial field near the top and bottomof the tower to relaxation of the flux return endplates. Our target uncertainty for the axial field is ± . i.e. the electrostatic mirror and proton collimator, as they can cause a false wishbone asymmetry. As can be seen infigure 9 (bottom) the newly trimmed field in June met our target of < µ T, but in December the transverse field ina region near the bottom of the proton collimator exceeded the target. However the associated systematic effect wassmall (see section VI E 2).Figure 10 shows a transverse field map taken 5.1 cm off-axis. At each z position the field was measured in stepsof 30 ◦ as the mapper carriage rotated. The data were fit to a Fourier series function: B trans ( θ ) = b + b cos( θ − θ ) + b cos 2( θ − θ ). The constant term b is dominated by the small misalignment angle between the Hall probeand the field axis and is not interesting. The cos θ coefficient b gives the uniform transverse field off axis. The cos 2 θ coefficient b results from a transverse gradient. The parameters θ and θ are constant phase offsets.Figure 11 shows results of alignment checks of the collimation insert made at various times during the run. Mea-surements were made using an optical system consisting of a theodolite, a pentaprism that rotates the line of sight by90 ◦ , and a series of precision reticules installed in the insert, all in a very well measured geometry. Usually indepen-dent measurements were made by two people as a double-check. The electrostatic mirror alignment was consistentlywithin our target of 1 mrad. The proton collimator had a much stricter target of 0.1 mrad which was generally metor slightly exceeded. VI. DATA REDUCTION AND ANALYSIS
For each aCORN event the PIXIE system recorded the energy and time of 31 signals: 19 beta energy PMTs, 8beta veto PMTs, 2 copies of the proton preamp output, and 2 copies of a level-discriminated proton pulse. An eventwas defined in firmware as any two of the above signals above threshold within a 100 ns time window. Two copies ofthe proton detector signal were used so that a single proton would produce an event. Most noise and dark currentfrom individual PMTs did not produce events. These raw data were written to disk along with header informationcontaining run parameters at a rate of about 5 TB per day. An online data distiller preprocessed raw data andremoved much of the background. The distiller included all events that were within a time window 10 µ s before to 1 µ s after each proton event. Events outside this window could not be a neutron decay coincidence and were discarded.Data bottlenecks within the PIXIE could cause events to enter the data stream out of time order, but each eventcontained an accurate time stamp used by the distiller to correct the time order. The distiller produced distilled datafiles at a rate of about 8 GB per day (a factor of >
600 reduction) that became the archival data. Raw data were notsaved, except for a small sample kept each day for diagnostic purposes.A data reducer was then used to convert the distilled data into reduced data files, individual text files each containing160 s of coincidence event data, for analysis. The reducer combined individual beta PMT events into complete betaenergy and time, or discarded them as noise, assigned a veto state to each, and calculated the beta-proton time offlight (TOF) for each proton event within the 11 µ s time window. The reduced data were organized into series of upto 1000 files, about two days of data, collected under essentially the same experimental conditions. Each series had anassociated beta energy calibration obtained from in situ calibration source measurements completed every two days.Data were divided into groups, each containing several equivalent series totaling approximately 100 beam hours,for analysis. Data were then sorted into a raw wishbone plot, a plot of proton TOF vs. beta energy, applying thecalibration data for each series separately, with a proton energy cut applied as shown in figure 12. A typical rawwishbone plot is shown in figure 13. Neutron decays are contained in the “wishbone” structure of delayed coincidenceevents. A. Data Blinding Strategy
The nature of aCORN does not allow an easy way to add an arbitrary blinding constant to the wishbone data. Butthe possibility of residual neutron polarization offers a useful data blinding strategy. An unknown neutron polarizationwould add an offset to the wishbone asymmetry, as shown in equation 11, that is undetectable in the analysis of datafrom a single magnetic field direction. In the NG-6 run a presumed neutron polarization of only 0.6 % produced an8.4 % shift in the value of the a -coefficient for each field direction [22]. Our blinding strategy was as follows:1. A small subset of the aCORN collaboration, the polarimetry group, measured the aCORN neutron beampolarization in situ using polarized He NMR in an auxiliary experiment and analyzed the result, which wasnot revealed to other collaboration members.2. The magnetic field up (B up ) data only were fully analyzed, including all systematic corrections and uncertainties,and a result for the a -coefficient was obtained and locked. The polarimetry group did not participate in thisanalysis.3. The magnetic field down (B down ) data were analyzed using the same procedures and corrections, withoutadjustment.4. The polarimetry “box” was opened and the result compared to the a -coefficients from the B up and B down analyses. B. Background Subtraction and Dead Time Correction
The PIXIE system is complicated and exhibits dead time effects at several time scales. First there is a dead timefor each channel that depends on the time structure of its signal pulses. This was 300 ns for beta PMT channels and3000 ns for proton detector channels. Because the analog to digital conversion is multiplexed, an additional deadtimeof several µ s can occur for a group of channels on a single module when data rates are high. Finally, when the modulememory is full, the entire module of 16 channels is dead for about 3 µ s while data is transferred to the host computer.During the NG-6 run the data rate was sufficiently low that the longer dead times were not apparent in the data, butduring the NG-C run, with about five times higher data rate, such affects did appear. We found that a 4 µ s deadtime for all events, applied in the analysis, was sufficient to remove all nonphysical time correlation effects betweenchannels in the data.In the NG-6 data analysis described in [22, 24] we were able to treat each electron event within the coincidencetime window of a proton (10 µ s before the proton to 1 µ s after) as a separate coincidence event. The same protoncould be associated with several different coincidence events, but at most one would be a neutron decay because theneutron decay rate was quite low. Any others were background coincidences where the electron and proton eventswere uncorrelated in time. As a result, the background in the raw wishbone was completely flat and structureless,lacking the usual exponential shape of a random time spectrum, and background subtraction was relatively simple.Due to the longer time scale dead time effects observed in the NG-C data, we were unable to use the same method.Instead we kept only the earliest electron in the 11 µ s wide coincidence time window of each proton and discardedany others. This change produced three important effects:1. A random background coincidence could preempt a neutron decay event if the background electron event oc-curred earlier in time. This removed an estimated 20 % of usable neutron decays from the data with a resultingloss of statistics.2. Neutron decay protons appear in a coincidence region of (3–4.5) µ s after the beta electron as can be seen in figure13. If an event appeared in this region, there could not have been an earlier electron event during the previous5 µ s, otherwise the coincidence region event would have been preempted. Note that earlier electrons correspondto longer proton TOF in the wishbone plot. This enforced the > µ s dead time requirement described above.3. The random background coincidences now have the usual exponential time structure. A more intricate methodis needed to subtract background and correct for dead time.We begin with the assumption that the raw wishbone plot contains only neutron decay coincidence events andrandom background coincidences. This is reasonable because we do not expect physical correlations in backgroundevents in the time range (1–10) µ s. The vast majority of background comes from gamma rays produced by neutroncapture in the electrostatic mirror, collimator, and other nearby materials. The remainder is radioactive decay, guidehall background, and cosmic rays. Weak decays from neutron capture may produce correlations, but at much longertimes. The others produce only prompt coincidences well within 1 µ s.Consider a vertical slice of the raw wishbone at a particular beta energy. Let the neutron decay wishbone functionbe bounded by proton TOF values t and t . For t < t the background has an exponential shape B ( t < t ) = c e Rt (12)and for t > t a similar exponential shape B ( t > t ) = c e Rt . (13)Note that these are positive exponentials because larger t (larger proton TOF) corresponds to earlier electron eventtime. The rate parameter R is the same in both regions; it is the random background electron event rate at the energyof this wishbone slice. The constants c and c are different; their ratio c /c < t < t < t , for a given proton event.The values of R , c , and c are found by fitting the data simultaneously in the two regions outside the neutron decaywindow.Inside the neutron decay window the background shape is more complicated; at each point in t it depends on theprobability that a background electron was not preempted by a neutron decay electron prior to that point, i.e. B ( t < t < t ) = c ( t ) e Rt (14)with c ( t ) = c − ( c − c ) (cid:82) t t N ( t (cid:48) ) dt (cid:48) (cid:82) t t N ( t (cid:48) ) dt (cid:48) . (15)Here N ( t ) is the neutron decay wishbone function that can be obtained by subtracting the background B ( t ) fromthe measured spectrum and applying the dead time correction factor e − R ( t − t ) . We start with an estimate for N ( t )and find the background B ( t ) using equations 12–15. Subtracting B ( t ) from the measured wishbone slice yields animproved, measured result for N ( t ) and we repeat the process iteratively until the resulting background subtractedwishbone function N ( t ) is stable (typically three iterations). We note that c /c ≈ .
99 in the beta energy range ofinterest (100 keV–400 keV) so the function c ( t ) in equation 15 affects the background subtraction at the 1 % level.This background subtraction algorithm was extensively tested using pseudodata and it worked very effectively.Figure 14 shows a 20 keV wide vertical slice (blue) of the raw wishbone (figure 13), centered at 100 keV, the lowestbeta energy that was used in the final analysis. Also shown is the same slice after background subtraction (green), i.e. the measured neutron decay wishbone function N ( t ). The background outside the neutron decay window is flatand without apparent structure. The bottom plot in the figure is a fit of the same background-subtracted slice to azero-slope line with the neutron decay window (3–4.6) µ s excluded. The variation in counts is consistent with Poissonstatistical fluctations. Figure 15 shows similar plots for a 20 keV wide vertical slice of the raw wishbone (figure 13)centered at 380 keV, the highest beta energy that was used in the final analysis. As can be seen here, the signal tobackground ratio (S/B) was strongly dependent on beta energy. In the energy range used in the analysis, E e = 100keV–380 keV, the average S/B was 0.2.During the experimental run, as a systematic check, we collected 19 hours of beam data with the polarity ofthe electrostatic mirror reversed. This prevented all neutron decay protons from reaching the proton detector with0minimal effect on background coincidences. Data from this run are shown in figure 16, again 20 keV wide slicescentered at beta energies 100 keV and 380 keV. Other than the expected exponential there is no apparent structurein the background inside or outside the neutron decay window. The green points are after background subtractionusing the same algorithm as for the neutron decay data described above, and fitting to a zero-slope line. A fullbackground-subtracted and deadtime-corrected wishbone plot is shown in figure 17, obtained from the data shown infigure 13. Blue points are positive and red points are negative (due to background subtraction). C. Energy Calibration Fit
The absolute beta energy calibration was monitored during the run by collecting data from in situ conversionelectron sources (
Sn and
Bi) 3–4 times per week, interleaved with the neutron decay data series. A more robustand precise energy calibration was obtained later for each data group using the neutron decay beta spectrum. Figure18 (top) shows the wishbone energy spectrum, which is the background subtracted wishbone data (figure 17) summedover proton TOF. The corresponding theoretical spectrum is the Fermi beta energy spectrum F ( E e ) = E e | p e | ( Q − E e ) found in equation 1, modified by the constraints on beta and proton momenta for coincidence events imposed by theaCORN collimation. The solid curve in figure 18 (top) is this theoretical spectrum computed numerically using thecollimator diameters, axial magnetic field strength, and neutron beam geometry. The theoretical function was fit tothe data to minimize chi-squared, with four variable free parameters: • An overall multiplicative scale factor • A linear energy calibration slope • A linear energy calibration offset • The theoretical function was convoluted with a normalized Gaussian energy response function G ( E, E (cid:48) ) = √ πCE (cid:48) exp( − ( E − E (cid:48) ) /CE (cid:48) ), based on the expected √ E resolution-width dependence of the scintillator detector.The constant C was a free parameter in the fit.Acceptable fits were obtained as illustrated in figure 18. With this method the wishbone data were self-calibrating forbeta energy. This result also supports the success of the background subtraction, the absence of extraneous structurein the data, and the effectiveness of the backscatter suppression which obviated the need for a low energy tail inthe beta response function. We note that the wishbone energy spectrum in figure 18 is insensitive to the wishboneasymmetry and the value of the a -coefficient so these fits had no bearing on the asymmetry analysis (section VI D),other than to provide the absolute beta energy scale. D. Wishbone Asymmetry Analysis, Magnetic Field Up
To calculate the wishbone asymmetry X ( E ) for data taken with the magnetic field up direction (B up ), we startwith 20-keV wide vertical slices of the background-subtracted wishbone plot (figure 17) for each data group. Thebackground-subtracted histograms (green) in figures 14, 15 are examples of these. From equation 6 we have X ( E ) = N I ( E ) − N II ( E ) N I ( E ) + N II ( E ) (16)where N I ( E ) and N II ( E ) are the counts in the fast (left) and slow (right) peaks, respectively, for each energy slice,summed over all B up data groups. Because the fast and slow peaks tend to overlap a bit, we are faced with thequestions of which TOF bin to use to separate them, and how to apportion the counts within that bin. For this weuse Monte Carlo data as a guide. We take a high-statistics Monte Carlo wishbone slice for each beta energy, andfind the TOF bin and its apportionment that reproduces the exactly correct wishbone asymmetry based on the inputvalue of the a -coefficient. This can always be done in spite of the slight overlap of the fast and slow peaks. We expecta systematic uncertainty in this procedure that will be small for low beta energy where the overlap is negligible,and large for beta energy above ≈
400 keV where the overlap becomes significant. To estimate this uncertainty, weassume that the correct apportionment of the TOF separation bin lies somewhere between 100 % of its counts tothe slow peak and 100 % to the fast peak, and assign this full range a 95 % C.L. ( ± σ ). It then follows that the1 σ systematic uncertainty equals one-half the counts in the separation bin divided by the total counts in the fastand slow wishbone peaks. Figure 19 shows the average systematic uncertainty in the wishbone asymmetry using this1prescription, compared to the Poisson statistical uncertainty in X ( E ) for all B up data. We restrict the a -coefficientanalysis to the energy range where this systematic uncertainty is less than the statistical, i.e. up to 380 keV.The wishbone data for beta energy ≤
80 keV has a number of issues: • Beta electrons may have zero axial momentum and still satisfy the transverse momentum acceptance, adding atail to the wishbone TOF. • Some beta electrons will miss the active region of the beta spectrometer, as shown by Monte Carlo simulation,complicating the geometric function f a ( E ). • The wishbone signal/background is very poor in this energy region and the background subtraction is imperfect.From the above considerations we choose the energy range 100 keV–380 keV for the a -coefficient analysis. Theuncorrected wishbone asymmetry X ( E ) for all B up data is shown in figure 20. E. Systematic Effects and Corrections
1. Electrostatic Mirror
The electrostatic mirror was designed to provide an approximately uniform axial electric field in the proton transportregion. Protons associated with group I and II wishbone events tend to have different trajectories inside the mirror sothe presence of transverse electric fields will cause a bias in their transmission within the proton collimator. ThroughMonte Carlo studies we found that a 0.1 % uniform transverse electric field, relative to the axial, produces a 0.5% false wishbone asymmetry. Due to the precision of its construction and alignment (see figure 11) the uniformtransverse field was much smaller than this. However it is unfortunately not possible to avoid significant transverseelectric fields in the vicinity of the upper (grounded) wire grid. In the NG-6 run this effect gave the largest correctionto the result: (5 . ± .
1) % [22]. For the NG-C run the grid support structure was modified and the upper linear gridwas replaced with the crossed wire grid shown in figure 8. A detailed 3D COMSOL [30] model, depicted in figure21, was built to calculate the resulting electric field shape. This field map was input to the aCORN proton transportMonte Carlo to calculate the beta-energy dependent correction shown in figure 22, an overall relative correction to thewishbone asymmetry of (1 . ± .
27) %. The uncertainty was calculated using a standard 20 % relative uncertaintythat we chose and assigned to all Monte Carlo corrections in this experiment. We regard this uncertainty to be anoverestimate as the electric field and proton transport calculations are expected to be much more accurate than 20%.
2. Magnetic Field
The proton collimation is affected by both the shape and absolute value of the magnetic field in the proton transportregion. In particular, a transverse magnetic field will cause a bias in proton collimation and thence a false wishboneasymmetry. For a radially symmetric transverse field the effect averages out. The absolute value of the magnetic fieldis used to calculate the geometric function f a ( E ); an error in the absolute field will result in a proportional error inthe a -coefficient result.Through Monte Carlo analysis we have found that the false asymmetry is proportional to the average magnitudeof the transverse magnetic field in the proton transport region. An average of 4 µ T produces a wishbone asymmetryof ∆ X = − . × − which is about 0.5 % of the a -coefficient asymmetry. Based on the field maps, the averagetransverse field in the B up configuration was 1 µ T giving a systematic error in the asymmetry of ∆ X = − . × − and we assign an uncertainty equal to the size of the correction.The absolute axial magnetic field was determined from NMR measurements on a glass cell filled with spin-polarized He that was lowered into the proton collimator from above. For B up the result was B axial = 36.39(11) mT, whichleads to an uncertainty of 0.3 % in the calculated f a ( E ).
3. Electron Backscatter
Approximately 5 % of electrons that strike the active energy detector will backscatter from it and the energydeposited is incomplete, producing a low energy tail in the electron response function. Such backscattered events havetwo undesirable effects: 1) they tend to fill in the gap between the wishbone branches (see figure 3) and confound2our ability to cleanly separate group I and group II events; and 2) they systematically shift events from group II intogroup I, causing a false positive wishbone asymmetry. The backscatter veto system in the beta spectrometer was usedto mitigate this problem. Electrons may also scatter from the beta collimator with similar effect. These cannot bevetoed, but the collimator was designed to limit the probability of a scattered electron to reach the beta spectrometerto 0.3 %, as verified in a PENELOPE simulation. Electron scatter from other materials or residual gas, and electronBremsstrahlung, were investigated during the NG-6 run and found to be negligible [24].Our best test for electron backscatter effects was in the wishbone data. We looked for an excess of events in the gapbetween the wishbone branches and compared to a Monte Carlo wishbone that included a low energy scattering tail.Figure 23 shows a combined background-subtracted wishbone plot with all B up data. The choice of the gap region,indicated in green, required some optimization. We want to use a large region while avoiding the tails of the wishbonebranches and avoiding low energies where the background subtraction uncertainty is large. A nonzero total of countsin this region can be attributed to non-vetoed backscattered electrons but would also include contributions fromelectron collimator scattering, electron scattering from the wire grid (section VI E 4), and proton collimator scattering(section VI E 9). The number of counts in the chosen gap is 62 ± σ upper limit due to non-vetoedbackscattered electrons, and zero to be the lower limit. We generated Monte Carlo wishbone data, including a flattail in the electron energy response function, and varied the tail area to achieve a count rate in the gap region thatequals the 1 σ upper limit. The resulting tail area was 0.59 % of the peak, which produces an average false asymmetryof +1.5 %. Therefore our systematic error due to electron backscatter is (+0 . ± .
75) %.
4. Electron Energy Loss in Grid
Beta electrons pass through the positive grid at the bottom of the electrostatic mirror. The grid is composed ofparallel wires, diameter 100 µ m, made of 2 % beryllium copper with approximately 1 µ m coatings of nickel and gold.The wire spacing is 2 mm, so the geometric probability of striking a grid wire is approximately 5 %. When an electronstrikes a wire the main systematic effect comes from energy loss. A beta electron will generally pass through thewire and lose typically about 100 keV. Electrons may also be scattered into a different direction, but to first orderthe probability of scattering into the collimator acceptance is the same as the probability of scattering out of it, andbecause the wishbone asymmetry is insensitive to beta collimation this does not create a systematic error. Energyloss in a grid wire is similar to backscatter from the beta spectrometer in its effect, but instead of producing a broadlow energy tail it produces a small low energy shoulder on the energy response function, which is less of a problem.Energy loss in the grid and its effect on the electron energy response was calculated using the NIST ESTAR database [33]. The associated error in the wishbone asymmetry is (+1.0 ±
5. Beta Energy Calibration
As discussed in section VI C, the most precise beta energy calibration comes from a fit to the wishbone data. Thecombined calibration from all B up data gives an overall energy uncertainty of σ ( E ) = ± σ ( X ) = a ∂f a ( E ) ∂E σ ( E ) (17)which has an average value of 0.27 % in the energy range 100 keV–380 keV.
6. Proton Energy Threshold
Protons associated with group I and II coincidence events differ in kinetic energy by an average of 380 eV. Bothgroups of protons are preaccelerated by the electrostatic mirror and then, after passing through the proton collimator,accelerated to a final energy of about 30 keV by the proton focusing electrodes and detector. While this difference inenergy is a small fraction of the detected energy, protons near threshold nevertheless contain a slightly higher fractionof group II protons. If these are not completely counted, a false negative wishbone asymmetry results. In the NG-6aCORN run about 1.2 % of protons were excluded by the PIXIE threshold which lead to a 3.0 % false asymmetry[22]. For the NG-C run we significantly lowered the PIXIE energy threshold (see section IV C). Figure 24 shows a3fit of a typical proton energy spectrum fit to a Gaussian plus a 4 th order polynomial background function to extractthe Gaussian component. The fraction of events excluded by the threshold is less than 0.02 % and the resulting falseasymmetry is negligible.
7. Collimator Insert Alignment
A small angular misalignment φ coll (radians) of the proton collimator is equivalent to a uniform transverse magneticfield B trans = φ coll B axial . Figure 11 shows a summary of the collimator alignment measurements. The variation inresults obtained by two independent observers for the same misalignment strongly suggests that the overall variationis due mostly to measurement error rather than differences in the actual misalignment. Therefore we take the meanmisalignment and the standard deviation (square root of variance) from all nine measurements: φ coll = (0 . ± . axial = 0.0364 T we have B trans = (3 . ± . µ T. Using the Monte Carlo result described in sectionVI E 2, this results in ∆ X = ( − . ± . × − , where the standard 20 % Monte Carlo uncertainty has been includedin quadrature. Note that this effective transverse magnetic field is independent of that measured by the field mapperas the collimator was not present when the maps were made. Therefore we treat the collimator misalignment as anindependent source of error.Similarly, a misalignment of the electrostatic mirror would introduce an approximately uniform transverse electricfield. From Monte Carlo analysis we found that a 1 mrad misalignment will produce a false wishbone asymmetry of∆ X = − × − . The mean and standard deviation of the measured values shown in figure 11 is φ mirror = (0 . ± . X = − . ± . × − .
8. Residual Gas Interactions
Protons travel about 2 m from the decay region to the detector. If a proton interacts with residual gas during thistrip it may be neutralized or scattered. Neutralized protons cause neutron decay events to be eliminated and theymay introduce a false wishbone asymmetry due to the slight velocity-dependence of the neutralization probability.Scattered protons result in a larger TOF in the wishbone plot which may also result in a false wishbone asymmetry.Monte Carlo analyses showed that proton scattering and neutralization have opposite-sign effects on the asymmetry,and that their relative probability depends on the gas species. We accounted for this effect by collecting data for 134hours with a deliberately higher pressure in the chamber, effected by partially closing a gate valve to the turbopump.The average pressure in the proton collimator during the high pressure run was 1 . × − Pa (1 . × − torr),compared to the normal pressure of 8 . × − Pa (6 . × − torr), a factor of 22 higher. Residual gas analyzer(RGA) measurements indicated that the gas was dominated by hydrogen and water (due to outgassing from the betaspectrometer plastic scintillator) at both pressures.Comparing the wishbone asymmetry from the high pressure run, from beta energy 100 keV–380 keV, to that of theproduction B up data, we found an average difference ∆ X = − . ± . σ X = 0 . /
22 = 3 . × − .
9. Proton Scattering from the Collimator
A large number of neutron decay protons strike the aluminum knife edge elements of the proton collimator. ASRIM Monte Carlo study showed that for protons with energy in the range 2–3 keV, about 90 % of those will beabsorbed in the aluminum, 9.5 % will emerge as neutral hydrogen atoms, and the remaining 0.5 % emerge as bareprotons, having lost an average of 2/3 of their kinetic energy. Many of those will subsequently strike the collimatoragain and be removed but some fraction will be detected with TOF that is systematically too large. Absorbedand neutralized protons are not detected and cause no systematic effect. Because protons are accelerated by theelectrostatic mirror they have a minimum possible axial momentum while in the collimator. This sets an upper limiton the TOF for unscattered protons in the wishbone plot. Scattered neutron decay protons would appear beyondthis maximum as a broad tail several µ s in width, and we can study this effect in the wishbone plot. This effect isinsensitive to beta energy, so it is useful to look at relative high beta energy where the statistical uncertainty dueto background subtraction is smaller. We use the beta energy range 400 keV–600 keV. Figure 25 (top) shows thetotal B up wishbone proton TOF spectrum summed from 400 keV–600 keV compared to the equivalent Monte Carloproton TOF spectrum. Figure 25 (bottom) is the same with an expanded vertical scale. We choose 1- µ s wide regionsjust before and after the wishbone TOF peak where the Monte Carlo counts are zero and take the difference of theirsums, post-wishbone minus pre-wishbone, which is 2296 ± . ± . X = − . ± .
10. Proton Focusing
The proton detector focusing system was designed to focus all neutron decay protons that were accepted by theproton collimator onto the active region of the surface barrier detector. The focusing efficiency, while very good, wasnot perfect. A small fraction of protons may strike the focusing electrodes or an inactive region of the detector, ormiss the detector entirely. Because the average kinetic energies of the fast (group I) and slow (group II) protons differslightly at the exit of the collimator, and the focusing efficiency is expected to depend on kinetic energy, imperfectproton focusing will lead to a systematic error in the wishbone asymmetry. This effect was studied computationallyand experimentally.A simulation of the focusing assembly and related apparatus was developed using the software suite AMaze byField Precision [34]. The relative positions of the surface barrier proton detector and ring and fork electrodes wereaccurately measured using a FARO coordinate measuring device [35]. An auxiliary simulation produced neutrondecay protons in the decay region and transported them to the exit of the proton collimator. These proton momentawere then fed into the AMaze simulation to track them to the detector.We fabricated a set of thin aluminum detector masks that blocked different regions of the detector face. One ofthese (the “R4” mask) blocked a central circle 24.8 mm in diameter, leaving a ring of width 3 mm at the outer edgeof the active region exposed to detect protons. Neutron decay data were collected with the various masks installedin 1–2 day runs. The resulting background-subtracted wishbone event rates were compared to the rates found in thesimulation using the same mask geometries which enabled us to fix the absolute position of the detector system inspace relative to the neutron beam and collimator. The simulation then computed the focusing efficiency. Figure26 shows a simulation of 10 neutron decay protons, out of which 146 struck the focusing ring (green circles) and154 struck the inactive region of the detector (red circles). No protons missed the detector assembly entirely. Theresulting focusing efficiency was 99.97 %. A 45-hour run with the R4 mask in place produced a wishbone event rateof (3 . ± . × − s − , or (0 . ± .
17) % of the normal unmasked rate, consistent with the AMaze simulation.From the simulation of the B up proton assembly the systematic error in the wishbone asymmetry was determinedto be ∆ X/X = − . ± . F. Wishbone Asymmetry Result, Magnetic Field Up
To produce the corrected wishbone asymmetry, we started with [ X ( E ) − δ ( E )] / [1 + δ ( E )] (see equation 8) andadded the systematic corrections described above. This can be seen in figure 20. The corrected wishbone asymmetrywas then divided by the geometric function f a ( E ) to give the measured value of the a -coefficient for each energy slice,shown in figure 27. These were then fit to a constant to obtain the overall result a = − . ± . ± . up ) . (18) G. Wishbone Asymmetry Analysis, Magnetic Field Down
After finalizing the B up result, we analyzed the B down data in the same way, except that four systematic effectswere analyzed independently for B down :1. Magnetic field shape:
In the B down field maps the average transverse magnetic field magnitude was 2 µ T,a factor of two larger than in the B up field maps, so the systematic correction was correspondingly larger, andas before we assign an uncertainty equal to the correction, giving ∆ X = ( − . ± . × − .2. Absolute magnetic field:
Independent He NMR measurements were made in the B down configuration withthe result B axial = 0.03624(11) T. Because the geometric function f a ( E ) was calculated using B axial = 0.0364T, a correction of (0 . ± .
3) % to the wishbone asymmetry was needed.53.
Proton scattering:
While the effect of proton scattering from the collimator should be the same for B up andB down , it was analyzed independently using the method described in section VI E 9. The count rate differencein 1- µ s wide regions just before and after the wishbone TOF peak was smaller, − ± X = 0 ± . Proton focusing:
In order to accomodate the change in sign of the E × B force, a separate proton focusingassembly with slightly different geometry was used for the B down run. The systematic error in the wishboneasymmetry was estimated from the B up analysis to be ∆ X/X = 0 ± . X ( E ) for the combined B down data, both uncorrected (blue dots) with statistical error bars, and with all corrections(red squares), are shown in the top plot of figure 28. The bottom plot shows the corrected wishbone asymmetry,divided by the geometric function f a ( E ), giving the measured a -coefficient for each beta energy slice. These were fitto a constant to produce the overall a -coefficient result for B down a = − . ± . ± . down ) . (19) VII. RESULT AND DISCUSSION
The difference in the results from the B up and B down runs is a (B down ) − a (B up ) = 0 . ± . . (20)Attributing this difference to a residual neutron polarization gives P = (5 . ± . × − , consistent with zero, usingequation 11. At this point in the analysis we unblinded by revealing the directly measured neutron polarization, P < . × − (90 % C.L.), an upper limit that confirmed the null polarization. The direct neutron polarizationmeasurement on NG-C is described in detail in another publication [36]. We combine the B up and B down results forthe aCORN NG-C run a = − . ± . ± . . (21)The error budget for the combined result is shown in Table I. In producing this table we used the standard deviationof the mean for the independent systematic uncertainties, i.e. the enumerated list in section VI G.This result is in good agreement with the result of the aCORN NG-6 run: a = − . ± . ± . a -coefficient, using statistical uncertainties only. The onlysystematic correction and uncertainty that was applied equally to both measurements was the effect of electron energyloss in the positive grid of the electrostatic mirror; the others were all evaluated independently. Therefore we removethe grid uncertainty from both, compute the standard deviation of the mean of the two systematics uncertainties,and then add the grid uncertainty back in quadrature. The result is a = − . ± . ± . , (22)or with the statistical and systematic uncertainties combined in quadrature: a = − . ± . λ = G A /G V , λ = − . ± . . (23)Figure 29 shows a summary of four neutron a -coefficient measurements from the past 50 years. The 2020 resultfrom the aSPECT experiment [37], which used an electromagnetic retardation spectrometer to measure the protonenergy spectrum, is the most precise. The overall agreement of these is good in spite of the slight tension (1.7 σ )between the aSPECT and aCORN results. The weighted average of these is a = − . ± . . (24)The effects of the new a -coefficient results on the world average for λ are less satisfactory. Figure 30 shows anideogram, in the style of the Particle Data Group ([6], p. 16), of precise determinations of λ = G A /G V from theneutron decay beta asymmetry ( A − coefficient) [10–12, 38–40] and the electron-antineutrino correlation ( a -coefficient)[37] and this work. Also included is a determination from the ratio of the A -coefficient to B -coefficient in a combined6 TABLE I. A summary of systematic corrections and uncertainties for the value of the a -coefficient in the combined NG-Cresult. The third column lists the absolute uncertaintes and the fourth column is relative to our final result for | a | . Thecombined uncertainty is the quadrature sum of statistical and systematic.systematic correction σ uncertainty relative uncertainty e scattering − . . . . . . . . . . . . . . . . . − . . . . . . . . . . . . − . . . . . . . . . . . . . experiment [41]. The distribution is unfortunately bimodal with poor overall agreement ( χ ν = 43 . / . λ = − . ± . √ .
37 = 2 .
32. The aSPECT result adds weight to the more positivenumber favored by older beta asymmetry experiments. The aCORN result is in better accord with recent betaasymmetry experiments. In particular it is troubling that the most precise results for the A - and a -coefficients[11, 37], both published within the past two years, disagree by 3 standard deviations. New precision experiments, inparticular additional measurements of the neutron a -coefficient at the < a -coefficient (equation 24) F = 1 + A − B − a = 0 . ± . F = aB − A − A = 0 . ± . . (26)The value of F now exceeds zero by 3 σ indicating a strong deviation, for the first time using this test, from theStandard Model prediction. This follows mainly from the disagreement in the value of λ between aSPECT [37] andPERKKEO III [11]. VIII. ACKNOWLEDGEMENTS
This work was supported by the National Science Foundation, U.S. Department of Energy Office of Science, andNIST (US Department of Commerce). We thank the NCNR for providing the neutron facilities used in this work, andfor technical support, especially Eli Baltic, Daniel Ogg, Dan Adler, George Baltic, and the NCNR Research FacilitiesOperations Group. [1] J. D. Jackson, S. B. Treiman, and H. W. Wyld, Nuclear Physics , 206 (1957).[2] D. Dubbers, Nucl. Phys. A527 , 239 (1991).[3] J. Barranco, G. Miranda, and T. I. Rashba, JHEP , 021 (2005).[4] S. Bauman, J. Erler, and M. J. Ramsey-Musolf, Phys. Rev. D , 035012 (2013).[5] Y. Mostovoy and A. Frank, JETP Lett. , 38 (1976).[6] P. A. Zyla, et al. (Particle Data Group), Prog. Theor. Exp. Phys. , 083C01 (2020). [7] A. N. Ivanov, M. Pitschmann, and N. I. Troitskaya, Phys. Rev. D , 073002 (2013).[8] T. Bhattacharya, et al. , Phys. Rev. D , 054508 (2016).[9] S. Gardner and C. Zhang, Phys. Rev. Lett. , 5666 (2001).[10] D. Mund, et al. , Phys. Rev. Lett. , 172502 (2013).[11] B. M¨arkisch, et al. , Phys. Rev. Lett. , 242501 (2019).[12] M. A. Brown, et al. , Phys. Rev. C , 035505 (2018).[13] F. E. Wietfeldt and G. L. Greene, Rev. Mod. Phys. , 1173 (2011).[14] J. C. Hardy and I. S. Towner, Phys. Rev. C , 025501 (2015).[15] C-Y. Seng, M. Gorchtein, H. H. Patel, and M. J. Ramsey-Musolf, Phys. Rev. Lett. , 241804 (2018).[16] A. Czarnecki, W. J. Marciano, and A. Sirlin, arXiv:1907.06737 (2019).[17] V. K. Grigor’ev, A. P. Grishen, V. V. Vladimirskii, E. S. Nikolaevskii, and D. P. Zharkov, Sov. J. Nucl. Phys. , 239(1968).[18] C. Stratowa, R. Dobrozemsky, and P. Weinzierl, Phys. Rev. D , 3970 (1978).[19] J. Byrne et al. , J. Phys. G , 1325 (2002).[20] S. Balashov and Yu. Mostovoy, Russian Research Center Kurchatov Institute Preprint IAE-5718 /2, Moscow (1994).[21] B. G. Yerozolimsky, et al. , arXiv:nucl-ex/0401014 (2004).[22] G. Darius, et al. , Phys. Rev. Lett. , 042502 (2017).[23] F. E. Wietfeldt, et al. , Nucl. Instr. Meth. A611 , 207 (2009).[24] B. Collett, et al. , Rev. Sci. Instr. , 083503 (2017).[25] T. Hassan, et al. , Nucl. Instr. Meth. A , 023101 (2009).[30] COMSOL, Inc., Burlington, MA, USA.[31] Precision Eforming, LLC, Cortland, NY, USA.[32] XIA LLC, Newark, CA.[33] https://physics.nist.gov/PhysRefData/Star/Text/ESTAR.html[34] Field Precision, LLC, Albuquerque, NM, USA.[35] FARO Technologies, Lake Mary, FL, USA.[36] B. C. Schafer, et al. , arXiv:2009.05149 [physics.ins-det], submitted to Nucl. Instr. Meth. (2020).[37] M. Beck, et al. , Phys. Rev. C , 055506 (2020).[38] P. Bopp, et al. , Phys. Rev. Lett. , 919 (1986).[39] B. G. Yerozolimsky, et al. , Phys. Lett. B , 240 (1997).[40] P. Liaud, et al. , Nucl. Phys. A , 53 (1997).[41] Yu. Mostovoi, et al. , Phys. Atom. Nucl. , 1955 (2001).[42] J. Fry, et al. , EPJ Web Conf. , 04002 (2019). -14.6 x10 -6 -14.5-14.4-14.3 G A ( G e V - ) -6 G V (GeV -2 ) λ (PDG 2020) τ n (beam ave.) τ n (UCN ave.) CKMunitaritysuperallowed β decay FIG. 1. A summary of experimental constraints on the nucleon weak coupling constants G A and G V . The purple band isthe PDG 2020 [6] recommended value for λ from neutron decay parameters A and a , including a scale factor of (cid:112) χ ν = 2 . (cid:112) χ ν = 1 .
5) bands arederived from the neutron lifetime averages for the beam and UCN storage experiments. The brown vertical band shows G V from superallowed beta decay [14] and the dashed lines indicate the shift due to the calculation of ∆ R by Seng, et al. [15]. Thered vertical band shows the CKM matrix unitarity condition using the PDG recommended value of V us [6]. proton acceptance electron acceptance ! p e − ! p e ! p υ II I proton detector electron detectorneutron source electron collimatorproton collimator +3 kV ! B ! E electrostatic mirror FIG. 2. An illustration of the aCORN experimental method. Top: A neutron source, shown as a point source here, lies on axisbetween a set of proton and electron detectors. A uniform axial magnetic field (cid:126)B is present throughout. Electron and protoncollimators act to limit the transverse momenta of detected electrons and protons from neutron decay. An electrostatic mirrorproduces an approximately uniform electric field (cid:126)E in the decay region that accelerates and directs all protons toward the protondetector, but beta electrons in the energy range of interest must be emitted into the right hemisphere to be detected. Middle: Amomentum space plot showing the cylindrical momentum acceptances of electrons and protons. Bottom: A momentum spaceconstruction of the acceptance for antineutrinos from neutron decay, when the detected electron momentum was (cid:126)p e as shownand the proton was also detected. Conservation of energy and momentum restricts the antineutrino momentum to the shadedregions I and II which have equal solid angle from the source. Region I is correlated with (cid:126)p e and region II is anticorrelated, sothe asymmetry in events associated with each region measures the a -coefficient. p r o t on T O F ( µ s ) group Igroup II N II N I FIG. 3. A Monte Carlo simulation of aCORN data, proton TOF vs. beta energy for coincidence events. The fast protonbranch (group I) is associated with neutron decays where the antineutrino momentum was in region I in figure 2. The slowproton branch (group II) is associated with decays where the antineutrino momentum was in region II. The sums N I and N II are used to compute the wishbone asymmetry for each beta energy slice. f a (E) FIG. 4. The dimensionless geometric function f a ( E ), computed numerically from the aCORN geometry and a 36.4 mTuniform magnetic field (see equations 8, 9). backscatter-suppressedbeta spectrometerneutron beamiron flux returnbeta collimatorproton collimatorelectrostatic mirrormagnetic field coilsproton detectorhigh voltage cablesand connections liquid nitrogenfill system FIG. 5. A cross section view of the aCORN tower showing the arrangement of major components. The neutron beam passesthrough from right to left. FIG. 6. An overhead view of the proton detector assembly showing the positions of the surface barrier detector and focusingelectrodes.
FLOOR
XP3372 XP3372 XP3372 vacuumchamber
FIG. 7. An interior cross section view of the backscatter-suppressed beta spectrometer. Dimensions are in mm. FIG. 8. The redesigned upper end of the electrostatic mirror used in the NG-C run, showing the new square mesh grid andlarger open diameter. a x i a l m a gn e ti c f i e l d on a x i s ( m T ) β collimator proton collimatorelectrostaticmirror June 2015 December 20151086420 t r a n s v e r s e m a gn e ti c f i e l d on a x i s ( µ T ) FIG. 9. The axial (top) and transverse (bottom) magnetic fields measured by the robotic field mapper on axis. The June2015 maps were made after reversing and trimming the field. The December 2015 maps were made just prior to the next fieldreversal. The difference shows typical drift over six months with unchanged trim coil settings. Gray shaded regions in thebottom plot indicate the < µ T target for the transverse field in the electrostatic mirror and proton collimator. t r a n s v e r s e m a gn e ti c f i e l d o ff a x i s ( µ T ) θ component cos 2 θ component FIG. 10. The transverse field map measured 5.1 cm off axis using the robotic field mapper. At each z position the fieldis measured at 30 ◦ intervals as the mapper rotates. The result is Fourier decomposed into a cos θ component that gives theuniform transverse field and a cos 2 θ that corresponds to a transverse gradient. m i s a li gn m e n t ( m r a d ) proton collimator targetelectrostatic mirror target proton collimator electrostatic mirror8 Jun 2015 16 Dec 2015 19 Jan 2016 2 Aug 2016 1 Aug 2016 FIG. 11. A summary of optical insert alignment checks made over the course of the run. Multiple points are independentmeasurements made by two people. c oun t s FIG. 12. A typical proton energy singles spectrum. The peak on the right is protons. The noise/background forms a peak onthe left due to the soft energy threshold of the PIXIE. The shaded region is the applied proton energy window. p r o t on T O F ( µ s ) FIG. 13. A typical raw wishbone obtained from approximately 100 hours of reduced data, using the proton energy cut shownin figure 12. -400-2000200400 b ac kg r ound s ub t r ac t e d c oun t s χ ν = 88.1/107 = 0.8225 x10 c oun t s β = 100 keV FIG. 14. Top: A 20-keV wide wishbone slice centered at beta energy 100 keV (blue), and the same wishbone slice aftersubtracting background (green). Bottom: The same background subtracted slice fit to a horizontal line, excluding the neutrondecay region (3–4.6 µ s). Error bars are statistical. -200-1000100200 b ac kg r ound s ub t r ac t e d c oun t s χ ν = 124.3/119 = 1.0440003000200010000 c oun t s β = 380 keV FIG. 15. Top: A 20-keV wide wishbone slice centered at beta energy 380 keV (blue), and the same wishbone slice aftersubtracting background (green). Bottom: The same background subtracted slice fit to a horizontal line, excluding the neutrondecay region (3.1–4.1 µ s). Error bars are statistical. c oun t s β = 100 keV χ ν = 143.2/140 = 1.02200150100500-50 c oun t s β = 380 keV χ ν = 146.3/140 = 1.05 FIG. 16. 20-keV wide wishbone slices centered at 100 and 380 keV for a data series where the polarity of the electrostaticmirror was reversed, so neutron decay protons could not be detected. The upper curve (blue) is the raw wishbone and thelower curve (green) is after subtracting background and fitting to a horizontal line. Error bars are statistical. p r o t on T O F ( µ s ) FIG. 17. A background-subtracted wishbone plot (data from figure 13). Blue points are positive and red are negative. -2000-1000010002000 r e s i du a l ( d a t a - f it ) c oun t s χ ν = 41.7 / 35 = 1.19 wishbone energy spectrum best fit theoretical spectrum FIG. 18. Top: The wishbone energy spectrum, i.e. the background subtracted wishbone (figure 17) summed over proton TOFand the best fit theoretical spectrum. Bottom: Fit residuals (data minus fit). Error bars are statistical.
10 x10 -3 un ce r t a i n t y i n X ( E ) FIG. 19. The estimated systematic uncertainty in computing the wishbone asymmetry X ( E ) from the data, compared to thePoisson statistical uncertainty. -0.10-0.08-0.06-0.04-0.020.00 w i s hbon e a s y mm e t r y X ( E ) FIG. 20. The wishbone asymmetry X ( E ) for the combined B up data, uncorrected with statistical error bars, and with allcorrections. z y E y (V/m) +500+200-200-5000 FIG. 21. A COMSOL finite element map of the transverse electric field inside the NG-C electrostatic mirror, in the regionnear the upper wire grid through which the protons pass. a dd iti v e w i s hbon e a s y mm e t r y c o rr ec ti on FIG. 22. The electrostatic mirror correction calculated by proton transport Monte Carlo using the 3D COMSOL model ofthe electric field. The red curve is a smoothed average obtained by fitting the Monte Carlo data to a second order polynomial.Error bars are statistical. p r o t on T O F ( µ s ) up total combined wishbone FIG. 23. A combined background-subtracted wishbone plot with all B up data. The event total in the region outlined in greenwas used to test for the presence of a low energy tail in the detected electron response function due to electron scattering.The inset shows the same green-outlined region with an expanded color scale. Blue points are positive counts, red points arenegative due to the background subtraction. p r o t on c oun t s FIG. 24. A typical proton energy spectrum (blue) fit to a 4 th order polynomial background function plus a Gaussian (red).The resulting Gaussian alone is shown in green. The soft energy threshold of the PIXIE-16 takes effect below channel 27. Theslight loss of protons below threshold has a negligible effect on the wishbone asymmetry.
300 x10 c oun t s up ) Monte Carlo 6000400020000-2000 c oun t s up ) pre-and post-wishbone regions Monte Carlo FIG. 25. Top: The total B up wishbone proton TOF spectrum summed from 400–600 keV (blue) compared to the equivalentMonte Carlo proton TOF spectrum (red). Bottom: The same plot with an expanded vertical scale and statistical error bars.The 1 µ s wide regions pre- and post-wishbone used to estimate the proton scattering tail are shown in green. ● ●●●●●● ●●● ●●●●●● ●●●● ●●● ●●● ●●●● ●● ●●● ●●● ●●●● ●●●●● ●●●●● ●●● ●● ●● ●●●●● ●●●●● ● ●● ●● ●●● ●●●●●●● ●●●●●● ●●● ● ●● ●●●● ●●●● ●●● ● ● ●●● ●● ●●●● ● ●●●●● ●● ●● ● ●●●● ●●●● ●●● ●● ●●● ●● ●●●●● ●● ● ● ●● ● ●● ●● ●● ●●● ● ● ●●● ●●● ●●●●●● ●●● ●● ● ● ●● ● ●● ●● ●●●● ●● ●● ● ●●●● ●●● ● ●● ●●● ●● ● ●●●● ● ●● ●●● ●● ●●●●● ●●●● ● ●● ● ● ●● ● ●●● ●●● ●● ●● ● ●● ●●● ● ● ●●● ● ● ●● ● ●● ●● ● ●●● ●● ●●● ●● ● ● ●●● ● ●● ● − − − Horizontal Coordinate (cm) V e r t i c a l C oo r d i na t e ( c m ) count detector x (cm) d e t ec t o r y ( c m ) FIG. 26. Results from a proton focusing simulation tracking 1 million neutron decay protons from the proton collimator tothe detector. Green and red circles are protons striking the focusing ring and detector inactive region, respectively. The thinblack circle indicates the active region of the surface barrier detector. -0.20-0.15-0.10-0.050.00 a - c o e ff i c i e n t B up result fit average: a = -0.1083 ± 0.0020 (stat) χ ν = 13.8/14 = 0.99 FIG. 27. The corrected B up wishbone asymmetry (see figure 20), divided by the geometric function f a ( E ) (see figure 4),giving the measured a -coefficient for each beta energy slice. These were fit to a constant to produce the a -coefficient result forthe B up data. Error bars are statistical. -0.20-0.15-0.10-0.050.00 a - c o e ff i c i e n t B down result fit average: a = -0.1069 ± 0.0019 (stat) χ ν = 17.2/14 = 1.23-0.10-0.08-0.06-0.04-0.020.00 w i s hbon e a s y mm e t r y X ( E ) FIG. 28. Top: The wishbone asymmetry X ( E ) for the combined B down data, uncorrected with statistical error bars, andwith all corrections. Bottom: The corrected B down wishbone asymmetry, divided by the geometric function f a ( E ), giving themeasured a -coefficient for each beta energy slice. These were fit to a constant to produce the a -coefficient result for the B down data. Error bars are statistical. -0.120-0.115-0.110-0.105-0.100-0.095-0.090 a - c o e ff i c i e n t Stratowa, et al. (1978) [18] Byrne, et al. (2002) [19] Beck, et al. (2020) [35] aCORN combined(this work)best fit: a = -0.10486 ± 0.00075 χ ν = 3.51/3 = 1.17 FIG. 29. A summary of neutron a -coefficient measurements from the past 50 years. -1.29 -1.28 -1.27 -1.26 -1.25 -1.24 -1.23 λ = G A / G V Bopp, et al. (1986) [38]Yerozolimsky, et al. (1997) [39]Liaud, et al. (1997) [40]Mostovoi, et al. (2001) [41]Mund, et al. (2013) [10]Brown, et al. (2018) [12]Maerkisch, et al. (2019) [11]Beck, et al. (2020) [37]aCORN combined (this work) -1.2754 ± 0.0011 (expanded error)