Precision measurement of the β-asymmetry in spin-polarized ^{37}\mathrm{K} decay
B. Fenker, A. Gorelov, D. Melconian, J.A. Behr, M. Anholm, D. Ashery, R.S. Behling, I. Cohen, I. Craiciu, G. Gwinner, J. McNeil, M. Mehlman, K. Olchanski, P.D. Shidling, S. Smale, C.L. Warner
aa r X i v : . [ nu c l - e x ] J a n Precision Measurement of the β Asymmetry in Spin-Polarized K Decay
B. Fenker,
1, 2
A. Gorelov, D. Melconian,
1, 2, ∗ J.A. Behr, M. Anholm,
3, 4
D. Ashery, R.S. Behling,
1, 6
I. Cohen, I. Craiciu, G. Gwinner, J. McNeil,
7, 3
M. Mehlman,
1, 2
K. Olchanski, P.D. Shidling, S. Smale, and C.L. Warner Cyclotron Institute, Texas A&M University, 3366 TAMU, College Station, Texas 77843-3366, USA Department of Physics and Astronomy, Texas A&M University,4242 TAMU, College Station, Texas 77843-4242, USA TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia V6T 2A3, Canada Department of Physics and Astronomy, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel Department of Chemistry, Texas A&M University,3012 TAMU, College Station, Texas 77843-3012, USA Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada (Dated: September 27, 2018)Using
Triumf ’s neutral atom trap,
Trinat , for nuclear β decay, we have measured the β asym-metry with respect to the initial nuclear spin in K to be A β = − . syst (13) stat (5) pol , a0.3% measurement. This is the best relative accuracy of any β -asymmetry measurement in a nucleusor the neutron, and is in agreement with the standard model prediction − . β -decay measurements, and improvethe value of V ud measured in this mirror nucleus by a factor of 4. PACS numbers: 23.40.Bw, 32.80.Pj, 12.15.-y, 12.60.-i, 13.30.Ce, 14.60.StKeywords: β decay, atom trap, optical pumping, β asymmetry Nuclear β -decay correlation experiments were instru-mental in establishing the standard model (SM) chargedweak interaction as a theory with spin-1 W ± bosons, cou-pling only to left-handed neutrinos through a vector mi-nus axial-vector ( V − A ) current. Precision measurementscontinue to probe this structure [1]. Extensions to the SMpropose that parity symmetry, which is maximally vio-lated in the weak interaction, is restored at some higherenergy scale by extending the SU (2) L ⊗ U (1) Y elec-troweak gauge group to include a right-handed SU (2) R sector. Manifest left-right symmetric models have anangle ζ which mixes the weak ( W L,R ) eigenstates toform mass eigenstates with masses M , , characterizedby δ = ( M /M ) [2].Atom and ion trapping techniques [3–6], and progressin neutron decay measurements [7, 8], have allowed cor-relation parameters in β decay to be measured with im-proved precision recently, increasing their sensitivity asprobes of non-SM physics. We present here an exper-iment combining a magneto-optical trap (MOT) withoptical pumping (OP) to produce a set of nearly idealconditions: an isomerically selected source of highly po-larized [9] β -decaying atoms that are cold and localizedwithin an exceptionally open geometry. We measure thecorrelation between the spin of a parent K nucleus andthe momentum of the outgoing β + , given by the decayrate [10]: d Γ angular dE β d Ω β ∝ b m e E β + P nucl · (cid:16) A β p β E β (cid:17) , (1)where we have neglected terms that cancel in the asym-metry measurement of our geometry. In this expression, m e , E β , and p β are the mass, total energy, and momen- tum of the positron, P nucl is the polarization of the parentnucleus, and b and A β are correlation parameters whosevalues depend on the symmetries inherent in the weakinteraction. We take the SM value b = 0 for this Let-ter, consistent with the E β dependence of our observedasymmetry as shown below. We will consider non-SMphysics that depend on E β in a future publication [11].The β asymmetry has been measured previously in theneutron and ten different nuclei. The focus of this workis the mixed I π = 3 / + → / + Fermi/Gamow-Teller β + decay of K, which has a half-life of 1 .
236 51(94) s [12]and Q EC = 6 .
147 47(23) MeV [13]. The transition to theground state of Ar dominates with a branching ratioof 97.99(14)% [14]. The next most significant branch isto an excited 5 / + state at 2 . A GT β = − .
6. All other branchesto excited states are below 0 .
03% [15].The corrected comparative half-life for K is F t =4605 . ± . Q EC values given above. The F t values for transi-tions between T = 1 / F t value for 0 + → + decays via: F t mirror = 2 F t + → + f A f V ρ , (2)where f A /f V = 1 . ρ = C A M GT C V M F is the ratio of Gamow-Tellerand Fermi coupling constants ( C A /C V ) and matrix el-ements ( M GT /M F ). Equation (2) with F t + → + =3072 . ρ = 0 . β asymmetry including thepossibility of right-handed currents is given by [10, 17]: A β = ρ (1 − y ) I +1 − ρ q II +1 (1 − xy )(1+ x ) + ρ (1+ y ) , (3)where x ≈ ( δ − ζ ) / (1 − ζ ) and y ≈ ( δ + ζ ) / (1 + ζ ) arenonzero in left-right symmetric models. The SM predic-tion for K is found by setting x = y = 0 . With the abovevalue of ρ derived from the measured F t value, the resultis A SM β = − . ρ is suchthat the sensitivity of A β to its uncertainty is reducedcompared to other observables; e.g., for the ν asymme-try it is nearly 2 × bigger, B SM ν = − . ρ varies considerably among K and the otherwell-studied mirror nuclei ( Ne, Na and Ar) makingeach nucleus complementary to the others as each willhave different dependencies on beyond the SM physics.Recoil-order and radiative corrections to A β [18] areincluded in our analysis. For isobaric analog decays, theinduced 1st-order tensor form factor is very small (onlypresent because of isospin symmetry breaking), and allbut the very small induced pseudoscalar and q expan-sion of the Fermi and Gamow-Teller form factors [19] aregiven by the conserved vector current (CVC) hypothe-sis using measured electromagnetic moments [18]. Thesecorrections combine to add ≈ − . E β /E to the ex-pression for A β .The experiment described here was performed withthe Triumf
Neutral Atom Trap (
Trinat ) [20, 21].
Triumf ’s radioactive ion beam facility,
Isac , delivered8 × K ions/s, 0.1% of which were neutralizedand trapped. Background from the decay of untrappedatoms in the collection MOT was avoided by pushing thetrapped atoms every second by a pulsed laser beam to asecond MOT [22] where the precision measurement tookplace, depicted in Fig. 1.Once the atoms are collected in the second MOT, weapply a sub-Doppler cooling scheme unique to potas-sium [23]. Since the atoms can only be polarized whilethe MOT is off, we alternate between periods of trappingand polarizing the atoms. To optimize the shutoff timeof the MOT’s magnetic field, we employ an alternating-current MOT (ac MOT) [24]. Once atoms are pushedfrom the first trap and cooled, a series of 100 cycles be-gins, where each cycle consists of 1 . K nuclei and collecting polarized decay data, fol-lowed by 3 . σ ± ) flipped every 16 s.While the MOT light and magnetic fields are off, weoptically pump the atoms on the D (4 s / → p / ) tran-sition with circularly polarized light. This technique di-rectly polarizes the nucleus via the hyperfine coupling ofthe atomic and nuclear spins. It also lets us measure P nucl nondestructively by probing the atoms with a pulsed355 nm UV laser and detecting the resulting photoions Electrostatichoops E l ec t r o n M C P R ec o il M C P Laserlight (Anti-)HelmholtzcoilsMirror with275 µ m-thick SiCsubstrate229 µ m-thick Be foil BC408scintillator90 mm z x re-entrant flangeand collimator40 × × Si-strip detector
FIG. 1. The
Trinat detection chamber. To polarize theatoms along the β -detection (ˆ z -) axis, optical pumping lightis brought in at a 19 ◦ angle with respect to the ˆ z axis andreflected off thin mirrors mounted within a β collimator on thefront face of the reentrant flanges. Thin Be foils behind themirrors separate the Si strip and scintillator β detectors fromthe 1 × − Torr vacuum of the chamber. Magnetic fieldcoils provide the Helmholtz (optical pumping, 2 Gauss) andanti-Helmholtz (MOT) fields. Glassy carbon and titaniumelectrostatic hoops produce a uniform electric field of 150 to535 V / cm in the ˆ x direction to guide shakeoff electrons andions towards microchannel plate detectors. with the recoil MCP detector. The UV photons can onlyionize atoms from the 4 p excited state which fully polar-ized atoms cannot populate, so the rate of photoions is asensitive probe of P nucl . Since 1 − P nucl is small, its deter-mination to 10% precision is sufficient to achieve [9, 25]: P σ + nucl = 99 . P σ − nucl = − . x , while adelay-line anode readout of the MCP provides positionsensitivity to image the other axes. Since the MOT’scycling transition produces a relatively large fraction ofatoms in the 4 p state, the position of the atoms is wellknown while the MOT is on. When the MOT light isoff, very few atoms are available to be photoionized, andthe trap position must be inferred from observations im-mediately before and after the polarized phase. Fromthese measurements, we observed that the atom cloudmoved 0 . . to 16 . . The entire cloud was illumi-nated by the OP light of 20 mm diameter (1 /e ) through-out the optical-pumping cycle.To identify decays that occurred within the region ofoptical pumping, we detect the low-energy shakeoff elec-trons (SOE) by sweeping them with an electric field to-wards an MCP and observing them in coincidence withthe β + . At least one SOE is present for every β + de-cay [28, 29] because the Ar − ion is unstable.To detect the nuclear decay products, we employ a FIG. 2. Scintillator spectrum in coincidence with its DSSSDand the electron MCP, showing a very clean selection of β -decay events originating from the trapping region. The Geant4 comparison shows residuals consistent with statis-tics. The vertical dashed blue line shows the energy thresholdused to exclude Compton-scattered annihilation radiation. pair of β telescopes along the vertical polarization axis(Fig. 1). Each consists of a thin double-sided Si-stripdetector (DSSSD) backed by a 35-mm thick BC408 scin-tillator. The 300- µ m thick DSSSDs are segmented into1-mm strips on both sides, providing position and ∆ E information. Because of its low efficiency for detecting γ rays, it also suppresses the background from 511-keVannihilation radiation.The plastic scintillators and DSSSDs were calibratedby comparing the observed spectra to a Geant4 simu-lation. For the plastic scintillators, we assumed a lin-ear calibration and a detector resolution with a 1 / √ E dependence. The calibration was performed using thescintillator spectrum in coincidence with a SOE withoutadding the energy deposited in the DSSSD. The calibra-tion spectrum included both β + events and the Comptonedge of the 511-keV annihilation radiation. The result-ing spectra including the DSSSD coincidence, shown inFig. 2 for one detector, agree well with the simulationover the entire observed E β range.The asymmetry is calculated by comparing the ob-served rate of β particles in the two detectors. Since theexperiment uses two symmetric detectors and reversesthe sign of the polarization, we use the superratio tech-nique which reduces many systematic uncertainties (seeRefs. [30, 31] for details).The data analysis was performed blind by temporar-ily culling an unknown fraction, up to 1%, of β -decayevents from the analysis. All analysis cuts, corrections,and uncertainties were finalized on the biased data. Thecomplete data set was then reanalyzed in this predefinedway to obtain the final results presented here.A detailed representation of the geometry of Fig. 1was included in the Geant4 simulation [32, 33]. Theposition of each decay was randomly sampled from the
FIG. 3. Top: The physics superratio of a subset of the data(points) fit to a
Geant4 simulation (filled band, with thewidth indicating its statistical uncertainty) where the onlyfree parameter was the value of ρ . Bottom: Difference be-tween the data and Geant4 , and the small size of the recoil-order+radiative corrections (ROC). observed distribution, modeled as a Gaussian ellipsoidand included the effects of the cloud’s expansion anddrift. We used the emstandard opt3 variation of thestandard physics lists as well as nondefault values of 1 µ mfor the cut-for-secondaries parameter and a range factorof f R = 0 .
002 in order to simulate the low- E β scatter-ing of β + more accurately [34]. The multiple scatter-ing (MSC) of e ± was simulated with the Urban MSCmodel of Ref. [35] to avoid the nonphysical behavior ofthe Goudsmit-Saunderson MSC model [36] observed inRef. [34].The simulation was tested by directly comparing thefraction of β + that backscattered out of the plastic scin-tillator. A large fraction of these events have the distinctsignature of depositing energy in two different pixels ofthe DSSSD. The number of these backscattered events,normalized by the number of events leaving energy onlyin one pixel, was found to differ by only (2 . ± . β ’s finite helicity[ p β /E β of Eq. (1)]. The observed asymmetry is comparedto the Geant4 simulation in order to obtain the best-fitresults for the input asymmetry.Although our geometry is very open, β scattering offof volumes such as the opposite β telescope, electro-static hoops, etc. (see Fig. 1), must be accounted for by Geant4 . Simulations indicate that 1.60% of accepted
FIG. 4. Shakeoff electron TOF spectrum with respect to the β + , showing all data at an electric field of 150 V / cm. Thisspectrum constrains the production of metastable Ar − with τ = 260(25) ns [37] to be less than 4%, while the TOF cuteliminates any possible contribution. Overlaid is a simulation(dotted line) of the TOF from atoms that escaped the trapbefore decaying from an electrostatic hoop, where the onlyfree parameter is the normalization fixed to times ≥
43 ns.While this simulation reproduces the longer TOF very well,it does not explain all of the background (red hatched area)under the main peak of good events within our TOF cuts(dashed vertical lines). events scattered by ≥ ◦ before being detected, leadingto an effective h cos θ i = 0 . Geant4 simulations therefore apply a 2.30% correction due to β scattering. Using a combination of our data and somefrom the literature, we assign a systematic uncertaintywhich is 5.6% of the correction (see Table I), as explainedin the Supplemental Material [25].Accounting for our measured h P i = 99 . A obs = − . χ /
123 = 0 . β + (Fig. 4) has the expected large, narrow peak near t =10 ns, the good events we use in our analysis. The peaksat 24 ,
39, and 53 ns come from electrons that do not firethe MCP, but produce a secondary e − that is re-collectedby the electric field which is registered by the MCP. Wecan simulate most of the broad TOF structure to be back-ground from decays of atoms stuck to the SiC mirrorsand electrostatic hoops. The same simulation suggests anunresolved peak at 12 ns from the electrode nearest thetrapping region, but this does not account for the major-ity of the total background under the good peak: 0.28%.We conservatively assume that this unknown backgroundis either fully polarized or unpolarized atoms and makea correction A β = A obs × . β de-tector differences, and β scattering), this cancellation isnot exact. Independently, we adjusted the trap position,size, temperature, drift velocity, and other parameterswithin the Geant4 simulation, obtaining the systematicuncertainties shown in Table I.
TABLE I. Uncertainty budget for A β . Each entry is given asthe absolute uncertainty, and correction factors and the rangevaried are listed where applicable. Polarization uncertainties,detailed in Ref. [9], are statistically independent.Source Correction UncertaintySystematicsBackground 1.0014 0.0008 β scattering a . ± µ m) 0.0004Trap ( σ + vs σ – ) sail velocity (typ . ± µ m / ms) 0.0005temperature (typ . ± . a (15 . +3 . − . mm) 0.0004Si-strip energy agreement ( ± σ → ± σ ) 0.0002threshold (60 →
40 keV) 0.0001Shakeoff electron TOF region ( ± . → ± . a ( ± µ m) 0.0001Thicknesses Be window a ( ± µ m) 0.000 09Si-strip a ( ± µ m) 0.000 01Scintillator only vs. E + ∆ E a → ± . / keV) 0.000 01Total systematics 0.0013Statistics 0.0013Polarization 1.0088 0.0005Total 1.0338 0.0019 a Denotes sources that are related to β + scattering. The final result is A β = − . syst (13) stat (5) pol , (4)where the third uncertainty combines the systematicand statistical uncertainties on the polarization measure-ment [9]. This result has the lowest relative uncertaintyof any measurement of the β asymmetry in a nuclearsystem to date. Since the simulation includes the recoil-order and radiative corrections, this result may be di-rectly compared to A SM β given earlier.Figure 5 shows the allowed parameter space in themanifest left-right model. We vary ρ at each ( ζ, δ ) co-ordinate to minimize the χ over all observables ( F t , A β and B ν ). The K limit includes our previous B ν mea-surement [38], but is dominated by the present A β result.Assuming ζ = 0 from other experiments (particu-larly Ref. [16]), our result implies δ = 0 . +45 − and amass for a W R coupling to right-handed ν R greater than340 GeV / c at 90% confidence, a slight improvement overthe P β /A β
310 GeV / c limit [2, 40]. Much of the pa-rameter space in left-right symmetric models has beenexcluded by other measurements. Constraints from po-larized muon decay [48] are relaxed if the ν Rµ is heavy(as e.g. in Ref. [49]). LHC searches directly exclude W R with mass < . /c if the right-handed gauge cou-pling g R = g L [39], while our K results imply g R < / c W R . Manifest models with M W ′ < M W and FIG. 5. Constraints on manifest L-R symmetric models fromnuclear and neutron [39] β decay: CKM unitarity [16]; theratio of β + polarization to A β of N and
In [2, 40]; A β of mixed GT/F Ne [14, 41–43]; the β + polarization of Ccompared to O [44]; and the weighted average of A β fromthree recent pure-GT cases [45–47]. V R ud considerably less than unity are also constrained by β decay correlations [2].If we make the assumption that the SM completelydescribes the β decay of K, we can use the result totest the CVC hypothesis. Combining the present re-sult for A β with the previous measurement of B ν [38],we find ρ = 0 . F t value of Ref. [12], leads to V ud = 0 . K, a greater than 4 × improvement over the previousvalue [12]. Isospin-mixing calculations [14] contribute0 . . V ud to other nuclear β -decay measurements in Fig. 6. Our K result has the same accuracy as Ne [42] and im-proves a CVC test at
I > / T = 1 / h V ud i mirror = 0 . . × less precisethan the 0 + → + result [16] and slightly better than theneutron.We have used a highly polarized, laser-cooled sourceof K to measure the β asymmetry in its decay to be A β = − . ± . W R coupling to right-handed ν ’s as wellas improving the value of V ud from mirror transitions.The high precision of our nuclear polarization measure-ment on the atom cloud is enabling a further program ofimproved A β , B ν , and recoil asymmetry measurements.We acknowledge Triumf / Isac staff, in particular forTiC target preparation, and the remaining authors ofRef. [9] for previous polarization development. Sup-ported by the Natural Sciences and Engineering Re-search Council of Canada, the Israel Science Founda-
FIG. 6. Measurements of V ud comparing the values fromthe neutron [39], Al [51], and the T = 1 / Ne [42], Na [52], Ar [43], the previous value for K [12],and the present work. The averages (uncertainties) in V ud de-termined from 0 + → + [16] and mirror transitions are shownas the solid (dashed) lines. tion, and the U.S. Department of Energy, Office of Sci-ence, Office of Nuclear Physics under Award No. DE-FG03-93ER40773 and No. DE-FG02-11ER41747. Tri-umf receives federal funding via a contribution agree-ment through the National Research Council of Canada. ∗ [email protected][1] Barry R Holstein, “Precision frontier in semileptonic weakinteractions: introduction and overview,” J. Phys. G ,110301 (2014).[2] E. Thomas, R. Prieels, M. Allet, K. Bodek, J. Camps,J. Deutsch, F. Gimeno-Nogues, J. Govaerts, J. Lang,O. Naviliat-Cuncic, I. Pepe, P. Quin, N. Severijns, andJ. Sromicki, “Positron polarization in the decay of polar-ized N: a precision test of the Standard Model,” Nucl.Phys.
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PrecisionMeasurement of the β Asymmetry inSpin-Polarized K Decay
B. Fenker, A. Gorelov, D. Melconian, J.A. Behr, M.Anholm, D. Ashery, R.S. Behling, I. Cohen, I. Craiciu,G. Gwinner, J. McNeil, M. Mehlman, K. Olchanski,P.D. Shidling, S. Smale, and C.L. Warner
ATOMIC PHYSICS
Given the precision measurement of our Letter, weinclude here some additional details about the atomicphysics methods for the interested reader. Certainatomic effects produce negligible uncertainties on thedetermination of the nuclear polarization P needed todeduce the value of A β for K, and we explore thesethrough more detailed measurements made possible bylarger quantities of stable K atoms. We take the oppor-tunity to provide some qualitative guides to our detailedpublication on our polarization methods in Ref. [1].There are several features of our optical pumping andprobing method that we want to emphasize. We probethe small unpolarized fraction, so not much precision isrequired. Our probe is parasitic — unlike most moredirect methods, it does not alter the polarization duringprobing. We also measure P throughout the duty cycleof polarization, so we can choose the best times of theduty cycle to determine A β . Optical pumping tests on stable K Qualitative description
For optical pumping of small densities of atoms, thereare two depolarization mechanisms. We measure opti-cally the degree of imperfectly circularly polarized light(see Section 2.3 of Ref. [1]). We then fit the excited statepopulation mentioned in our Letter for a parameter B x ,an average magnetic field perpendicular to the opticalpumping ˆ z axis. The result, after a full detailed charac-terization of optical pumping using the well-establishedoptical Bloch equations (OBE, described in Ref. [1]), de-termines the population of unpolarized atomic states.Most important is the population of two almost-pumpedground states ( F = 2 M F = 1 and F = 1 M F = 1) with nu-clear polarization 1 / /
6: determining their popula-tion supplies the precision needed for this Letter. Larmorprecession governed by B x does not change F , while themeasured imperfect circular polarized light can change F by optical pumping, so once we quantify the two depo-larization mechanisms the OBE’s give us the populationswe need.The tail/peak ratio of the excited state population de-termines 1 − P . Given that 1 − P is less than 0 .
01, and
FIG. S1. Fluorescence of K in 4 S / to 4 P / transitionduring optical pumping. Note the log-log scale showing thepeak fluorescence, the region dominated by falling B x [ t ], andthe tail due to imperfect polarization. that the nuclear polarization is ≥ . − P = 0 . ± . K at long times in Fig. S1(or photoions in K in Ref. [1]), the polarization wouldbe 100% with no uncertainty.Thus effectively, a single parameter, B x , is fit to the K excited state population data. All other parametersare fixed by independent measurements on K, and high-statistics independent data on K in the same geometry.The K data we describe in this Supplemental Materialis helpful in lending confidence to our model, but in theend not essential to the K polarization result.
Time dependence of B x We mention in Section 2.3 of Ref. [1] that we measurethe time-dependence of B x with Hall probes as the MOTquadrupole B field falls. This provides reasonable accu-racy, albeit with vacuum system open without detectorsinstalled, and suggests the depolarization from this com-ponent is unimportant in the part of the duty cycle usedfor A β data.To test this with all detectors in place, we opticallypump stable K atoms. K has almost the same hyper-fine structure as K, so after adjusting laser frequenciesexperimentally the OBE predict almost the same results.We show in Fig. S1 the dependence of the fluorescenceof the 4 P / state as a function of time, along with opti-cal pumping calculations including the time dependenceof B x . The region from 60 to 200 µ s after the opti-cal pumping starts is better modelled if we include thistime-changing B x , with the fall time fixed to the Hall FIG. S2. Optical pumping tail/peak ratio for K, optimizedby changing one set of uniform-field Helmholtz coils. probe measurement of τ = 130 µ s. In Fig. S1 the MOTquadrupole field was turned off at − µ s. We waitedlonger for the MOT field to decay away before we startedoptical pumping K (see Fig. 10 of Ref. [1]), and thensimply waited for this field to decay away to take the A β data. The tail/peak ratio, and hence the deduced po-larization of K at OP times used for β decay, do notdepend on whether or not the decaying MOT quadrupolefield is included in the theory. Nor is the goodness of fitwith and without this effect changed in the K photoiondata (Fig. 8 and Fig. 10 of Ref. [1]).With data of this sort, we can tune parameters to op-timize the polarization. Figure S2 shows optimization ofone perpendicular uniform magnetic B field by trimmingthe current through a Helmholtz coil along one perpen-dicular axis, after similar optimization of the other axis.This effectively aligns the total B field with the OP laserlight axis. The OP laser light axis is in turn aligned me-chanically with the β detector axis, to optical alignmentaccuracy of 1 mm in 1 m, or 0 .
001 rad, producing negli-gible misalignment accuracy on cos θ β ˆ I of less than 10 − .We also learn that the polarization difference fromunity depends quadratically on applied B ⊥ , in agreementwith our OP calculation. The result in Fig. S2 is con-sistent with a small average horizontal field that is notzeroed out with our uniform applied field. In our Kdata we are also able to fit for this effective B x , and findit is consistent with the values found for K, as wouldbe expected since the atom clouds are located at almostthe same position in the apparatus.
Spatial gradients of the B field
Given the dying rem-nants of the time-changing MOT quadrupole field, it isnatural to consider whether spatial gradients of the mag-netic field can make gradients of the polarization acrossthe atom cloud. In particular, a finite dP/dz could inprinciple perturb A β significantly. However, the possi-ble residual dB/dz of < .
01 G / cm detunes the optical pumping laser by negligible Zeeman shifts, so negligible dP/dz is produced. For measurement of future β -decayobservables, polarization gradients along the other axesare being studied (by fast CMOS camera) and minimized(by the standard trick of unbalancing Helmholtz coils). Metastable Ar − atoms If nothing else happens, β + decay of a potassium atompopulates a negative Ar ion. The ground state of this ionis, of course, unstable, and dissociates in negligible time.There is a known metastable state of the Ar − ion withlifetime τ = 260 ns [2]. We mention here that this statemakes negligible contribution to A β systematic uncer-tainties.The angular distribution of the β + is quite differentin singles versus in coincidence with the recoil (i.e. the ν ). So it is important in measuring A β that the shakeoffelectrons be detected with no bias from the recoil direc-tion. A metastable Ar − could in principle move in z firstbefore releasing the electron, thus biasing the critical co-incidence.We can fit for a tail with the known lifetime in the β –shakeoff electron TOF spectra like Fig. 4 of our Letter(but to longer times than shown). The population of theAr − is less than 4%, which could produce a less than0.08% correction to A β using a 40-mm diameter shakeoffelectron detector. However, such a tail is excluded by thetime width of the β -shakeoff coincidence in Fig. 4 usedfor A β , so the possible distortion to A β vanishes. β (BACK)SCATTERING A primary concern of any β asymmetry measurementis the effect of β scattering before entering the detector.These events will have an apparent initial direction thatis incorrect and will therefore bias the results – especiallyin the case of large-angle backscatters. A separate publi-cation is in preparation where we will describe in greaterdetail our estimates of these effects [3], but in order forthe reader of our Letter to understand how we arrived atthe correction and uncertainty for β scattering in TableI of the Letter, we provide the plots comparing our datato our Geant4 simulation which led to these results.One comparison of the data to
Geant4 could be madeby looking at events where the β backscatters off of onedouble-sided Si-strip detector (DSSSD) into the both thescintillator and DSSSD of the opposite β telescope. How-ever, given the small ( ∼ . β togo from one telescope to the other, these events are ex-tremely rare, . − of non-scattering events. The effectof β backscattering on the A β measurement in our geom-etry is highly suppressed: the 20 candidate events in ourdata set are too few to serve as a meaningful benchmark0 FIG. S3. Comparison of
Geant4 with the observed fractionof events backscattering out of the scintillator through a 2ndpixel in the DSSSD ∆ E detector. The bottom plot shows thepercent difference between Geant4 and observations. for
Geant4 . Although negligible, these events were ve-toed in the analysis of A β .A much more frequent type of backscatter we are ableto measure experimentally are events in one β telescopewhere the scintillator and two pixels in the correspondingDSSSD are all above threshold [4]. These “scintillator-backscatter” events correspond to a β entering a pixelin the DSSSD of one of the β telescopes, leaving energyin the plastic scintillator, and then backscattering outthrough a different pixel of the same DSSSD detector.Note that this is a very clean measurement: the triple-coincidence between the shake-off electron MCP, theDSSSD and the scintillator greatly suppresses γ eventsand other backgrounds, and in particular the shake-offelectron coincidence ensures the decay occurred from thetrap.Figure S3 shows the fraction of scintillator-backscatterevents normalized to the number of good events as ob-served by each β telescope. These are compared tothe fraction predicted by the Geant4 simulation, whichcan be seen to be quite favourable when using the non-standard
Geant4 options listed in the Letter: the av-erage difference is only (+2 . ± . Geant4 simulations.The same
Geant4 simulation is used to predict theeffect of β scattering on the A β measurement. Giventhe position of an event in the DSSSD and assuming thedecay occurred from the trap center, we are able to calcu-late the angle between the polarization direction and themomentum of the β . If the β scattered before enteringthe detector, this calculated angle will be wrong, mostdramatically for events which backscattered off of a vol-ume opposite the telescope in which it was detected. Toestimate the effect, we performed a simulation looking atevents which fired the β telescope in the same direction FIG. S4.
Geant4 simulation showing the effect on the A β measurement due to β scattering. The dominant peak at∆ cos θ ≈ β telescope directlyfrom the trap; the events below this peak are ones where the β scattered before entering the detector and which lead to anincorrect angle reconstruction. We have divided these eventsinto two regions: 0.72(8)% of events we labelled “backscatter”events, and 0.88(10)% “scatter” (see text). Instead of thetrue cos θ , β -scattering effects lead to an effective cos θ thatis attenuated by 2.3%. as the initial nuclear polarization (so cos θ calc ≈
1) andcompared this to the actual cos θ of the generated event.Figure S4 shows the distribution of simulated events asa function of the true cos θ minus that which we calculatebased on the position in the DSSSD. The main peakat ∆ cos θ ≈
0, containing 98 . β telescope with minimalscattering; most of the width of this peak is due to thefinite position resolution of the DSSSD (1 mm strips)and finite size of the cloud of atoms. The events belowthis main peak correspond to events which scattered offthe opposite β telescope, one of the electrostatic hoopsand/or one of the other volumes shown in Fig. 1 of theLetter.All together, Geant4 predicts that scattered eventsreduce the observed asymmetry by 1 / h cos θ eff i = 1 . Geant4 simulations, in-cludes this 2.30% correction for β scattering. To assigna systematic uncertainty, we consider three regions inFig. S4: “not scattered” events are those where ∆ cos θ ≥− . θ ≤− .
5; and the rest, − . < ∆ cos θ < − .
085 are “scat-tered”. We varied the fraction of events in the “scat-tered” and “backscattered” regions to estimate a system-atic uncertainty on h cos θ eff i . For the “backscattered”region, we use our result from Fig. S3 to assign an un-certainty of 5 . σ upper-limit of the differenceshown. We have no data of our own to constrain the frac-tion of “scattered” events, so for these we assign a 10%uncertainty, consistent with the accuracy of a Geant4 simulation we ran compared to literature data on fewMeV electron transmission through thin materials intoangles of 10 −
75 degs [5, 6]. The result is a ± . h cos θ eff i and an absolute systematic un-certainty of ± . A β , which is 5 .
6% of the totalcorrection. This is the systematic uncertainty assigneddirectly to β scattering in Table I of the Letter. Note thatthere are five other entries in this table of uncertainties(labelled with a superscript “a”) which also contributeto β scattering, albeit to a lesser extent and less directly. ∗ [email protected][1] B. Fenker, J.A. Behr, D. Melconian, R.M.A. Anderson,M. Anholm, D. Ashery, R.S. Behling, I. Cohen, I. Craiciu,J.M. Donohue, C. Farfan, D. Friesen, A. Gorelov, J. Mc-Neil, M. Mehlman, H. Norton, K. Olchanski, S. Smale, O. Th´eriault, A.N. Vantyghem, and C.L. Warner, “Pre-cision measurement of the nuclear polarization in laser-cooled, optically pumped K,” New J. Phys. , 073028(2016).[2] I. Ben-Itzhak, O. Heber, I. Gertner, and B. Rosner, “Pro-duction and mean-lifetime measurement of metastableAr − ions,” Phys. Rev. A , 4870 (1988).[3] B. Fenker, A. Gorelov, D. Melconian, J.A. Behr, M. An-holm, D. Ashery, R.S. Behling, I. Cohen, I. Craiciu,G. Gwinner, J. McNeil, M. Mehlman, K. Olchanski,P.D. Shidling, S. Smale, and C.L. Warner, (unpublished).[4] B. Fenker, Ph.D. Thesis, Texas A & M University (2016).[5] J. A. Lonergan, C. P. Jupiter, and G. Merkel, J. App.Phys. , 678 (1970).[6] D. H. Rester and J. H. Derrickson, J. App. Phys.42