Experimental Constraints on Left-Right Symmetric Models from Muon Decay

R. Bayes,* J. F. Bueno, A. Hillairet,* Yu. I. Davydov, P. Depommier, W. Faszer, C. A. Gagliardi, A. Gaponenko, D. R. Gill, A. Grossheim, P. Gumplinger, M.D. Hasinoff, R. S. Henderson, J. Hu,7,x D.D. Koetke, R. P. MacDonald, G.M. Marshall, E. L. Mathie, R. E. Mischke,7,k K. Olchanski, A. Olin,* R. Openshaw, J.-M. Poutissou, R. Poutissou, V. Selivanov, G. Sheffer, B. Shin,7,{ T.D. S. Stanislaus, R. Tacik, and R. E. Tribble

The maximal parity violation of the standard model (SM) charged weak interaction is empirically based.Many natural SM extensions restore parity conservation at a higher mass scale with additional weak couplings.Muon decay is an excellent laboratory to search for these couplings because the purely leptonic process can be calculated very precisely within the SM.This Letter presents a high-precision measurement of the energy-angle spectrum of the positrons emitted in polarized muon decay, which provides new limits for the mass and mixing angle of the heavy W in a class of left-right symmetric models [1].
The most general Lorentz-invariant, derivative-free muon decay matrix element [2] is described by 10 complex, model-independent couplings (g ) involving leftand right-handed leptons ( and label the electron and muon, respectively) in scalar, vector, and tensor interactions ().In the SM, g V LL ¼ 1, and the other nine constants are zero.When only the positron energy and direction are measured, the muon decay spectrum can be described by four parameters [3], which are bilinear combinations of the g : the Michel parameter , as well as , P , and .The differential decay rate is then with x 0 ¼ m e E max ; P ¼ j P j; cos ¼ P Á pe j P jj pe j : The neutrino masses are neglected; E max ¼ 52:828 MeV.
Radiative corrections [4] are not explicitly shown but are significant and must be evaluated within the SM to a precision comparable to the experiment.The polarization of the muon from pion decay begins as P and may evolve over the 2:2 s mean lifetime of the muon to become P at the time of decay.The SM predictions are ¼ ¼ 3=4, P ¼ ¼ 1, and ¼ 0. Precision measurements of these parameters test the SM predictions and are sensitive to extensions to the SM.
Prior to the TRIUMF weak interaction symmetry test (TWIST) experiment, , , and P were known with uncertainties in the range of ð3:5-8:5Þ Â 10 À3 [5].Intermediate TWIST results have already reduced those uncertainties to ð0:7-3:8Þ Â 10 À3 [6,7].TWIST has now realized its goal of making about an order of magnitude improvement in each of the parameters.These final results of the experiment supersede those in our previous publications.
TWIST used highly polarized positive muons from pions decaying on the surface of a graphite production target irradiated with 500 MeV protons.The muons were transported by the TRIUMF M13 surface muon channel to the entrance of a 2 T superconducting solenoidal magnet and were guided by the field along its symmetry axis into the detector array [8].A thin (239 m) trigger scintillator identified muons entering the detector.The muons were ranged to stop predominately in a thin metal foil at the center of a symmetric stack of high-precision, low-mass planar multiwire chambers.There were 6 proportional and 22 drift chambers, surrounded by helium gas, on each side of the stopping target.Ionization of tracks from decay positrons was sampled by the chambers, and drift times were recorded by time-to-digital converters.
With a muon rate of ð2-5Þ Â 10 3 s À1 , data sets of 10 9 events could be obtained within a few days.Data sets were taken with two foil stopping targets: Ag (30:9 m thick) and Al (71:6 m thick), each >99:999% pure.Sets were also taken with deliberately altered to assist in studies of possible systematic errors.A pair of time expansion chambers [9] was inserted upstream of the solenoid to determine the incoming muon beam characteristics.Because they caused multiple scattering and hence muon depolarization, the time expansion chambers were removed during most data taking.This phase of the experiment was completed in 2007.
Analyzed events were collected into two-dimensional (2D) distributions of positron angle and momentum (or energy) whose shape depends on the decay parameters.These distributions from data were compared to similar ones derived from a GEANT3 simulation [10].Both were subjected to essentially the same analysis, allowing biases and inefficiencies to be included in an equivalent way to reduce the dependence of the result on the specific analysis procedure.This places great importance on the accuracy and detail of the simulation, which includes not only standard physics processes but also a detailed description of the beam, magnetic field, geometry, and detector response.Decay parameters were extracted by fitting the 2D data distribution to that of a base distribution of simulated events, plus simulated distributions corresponding to the first derivatives of the spectrum with respect to decay parameters (or combinations thereof), yielding fit coefficients Á, Á, and Á.The decay parameters used in the generation of the simulation were hidden, so the analysis was ''blind'' [6].
Fourteen data sets were used to extract and , seven with each of the Ag and Al targets.Only nine sets were used for P ; the other five were acquired to test consistency and systematic effects with altered beam, magnetic field, or muon multiple scattering, where the depolarization was not optimally controlled.The residuals for the fit of one nominal data set in units of standard deviations () are shown in Fig. 1.A histogram of the residuals in the fiducial region, summed over all sets, has a mean of À0:003 AE 0:005 and ¼ 1:002 AE 0:004.Also shown is the range of ðp; cosÞ used to determine the decay parameters.The fiducial cuts are symmetric for upstream and downstream decays and were selected to maximize sensitivity to the decay parameters while reducing systematic uncertainties.For all 14 data sets, there were 11 Â 10 9 events, of which 0:55 Â 10 9 passed all event selection criteria and were within the fiducial region.Simulation sets were typically 2.7 times larger than the corresponding data set.The consistency of the data sets (statistical uncertainties only) was assessed by fits to constant means for the values of Á, Á, and ÁP , which gave reduced 2 values of 14:0=13, 17:7=13, and 9:7=8, respectively.
The procedure of fitting the difference of two spectra in terms of derivatives also provides a natural tool for the evaluation of systematic uncertainties.The simulation was validated through comparison with observables in the data that do not depend on muon decay parameter values, and the resulting uncertainties were factored into estimates of the systematic uncertainties.The sensitivities were obtained from the effect on the decay parameters when an identified source of systematic uncertainty was changed (often by an exaggerated amount) in one of the spectra.This was typically achieved with two simulated spectra.The systematic uncertainties are listed, along with the statistical errors, in Table I.
Notable improvements in the systematic uncertainties for and compared to our intermediate results [6] were achieved for positron interactions, chamber response, and momentum calibration.The positron interactions systematic addresses the possible inaccuracy in the simulation of reproducing positron energy loss in the stopping target and detector elements, primarily due to bremsstrahlung, deltaray production, and ionization.It was better constrained by comparisons of identified interactions observed in the data and in the simulation.Chamber response refers to the conversion of drift times to spatial information used in track fitting to evaluate the momentum and angle of each decay positron.This was improved by more precise monitoring and control of atmospheric influences that could change the chamber cell geometry.In addition, a method was devised [11] to calibrate the chambers' spacetime relations, for each plane, in both data and simulation, thereby reducing reconstruction biases.The maximum positron energy provides a calibration feature that was used to reduce the energy scale uncertainty.Since energy loss varies with the track angle linearly in 1=ðcosÞ due to the planar geometry of the detector, the region near the kinematic end point of 52:8 MeV=c was matched for data and simulation for small bins of cos.The data-simulation relative energy calibration procedure has undergone improvements to become more robust to fitting conditions.
The asymmetry parameter is also subject to uncertainties from these sources, but they are overshadowed by uncertainties unique to depolarization, as shown in Table I.Depolarization in the fringe field and in the muon stopping target result in P < P and constitute the largest contributions to systematic uncertainties for P .These uncertainties were improved considerably for this analysis compared to the intermediate result [7].Improvements in the beam steering reduced the fringe field depolarization for a nominal data set to only 2:5 Â 10 À3 .The uncertainty depends on the accuracy of simulating the muon spin evolution as the beam passes through significant radial field components at the solenoid entrance.The essential ingredients are an accurate field map and precise knowledge of the position and direction of the muons, as provided by the time expansion chambers.Depolarization in the stopping target from muon spin relaxation is assessed from the measured time dependence of the asymmetry.
After revealing the hidden parameters, the results for the three decay parameters are consistent with the SM predictions.While the generalized matrix element treatment of Ref. [2] does not constrain the sign of deviations from the SM values for , , and , the product P = is constrained to be 1:0 and is 1.0 in the SM.This quantity defines the asymmetry between cos ¼ AE1 at the maximum decay positron energy.Our decay parameter values combined to give P = ¼ 1:001 92 þ0:001 67 À0:000 66 (the errors account for significant correlations), and the initial evaluation of P = showed that the value for the Ag data was higher than that for Al by 3:8.This apparent contradiction initiated an exhaustive reconsideration of effects that might have been overlooked in the blind analysis.The review showed that effects such as þ ! e þ X 0 decays (where X 0 is a long-lived unobserved particle), an incorrect value of the parameter, or plausible errors in the radiative correction implementation were not responsible for the unexpected P = value.While no obvious mistake was uncovered in the estimates of the systematic uncertainties previously considered, we found that two corrections had been missed.A small correction was added for muon radiative decay (< 1 Â 10 À4 for the Ag data and negligible for Al).Another correction was made for each set to account for a difference between the mean muon stopping position for data and simulation.We also concluded that the uncertainties for the two targets were sufficiently different to merit dividing the systematic uncertainties into common and target-dependent categories.The target-independent systematics are unchanged from the blind analysis.Separate uncertainties for bremsstrahlung were computed, and an additional sensitivity to the muon stopping position in the target was added.
With these changes the central values of and decreased from the blind results by 0.000 14 and 0.000 23, respectively.P is unchanged and its error reduced after including information from the measurement of in the five sets not used for P .All uncertainties changed by <0:000 06.The difference between targets for P = is reduced to $1, and P = ¼ 1:001 79 þ0:001 56 À0:000 71 .The revised results are compared to prior results in Fig. 2. The values, including the uncertainties from Table I, are ¼ 0:749 77 AE 0:000 12ðstatÞ AE 0:000 23ðsystÞ; ¼ 0:750 49 AE 0:000 21ðstatÞ AE 0:000 27ðsystÞ; P ¼ 1:000 84 AE 0:000 29ðstatÞ þ0:001 65 À0:000 63 ðsystÞ: The decay parameters measured by TWIST contribute to a larger set derived from other muon decay observables that can be analyzed in terms of the weak couplings g .A global analysis [6,12] imposes P = 1:0 and yields P = > 0:999 09 (90% C.L.), compared to the pre-TWIST lower limit P = > 0:996 82 (90% C.L.) 041804-3 [13].The global analysis confirms consistency with the SM, where g V LL is the only nonzero term.The TWIST results restrict the upper limits of other terms.For example, the limit on the total right-handed muon coupling is reduced by a factor of 6 from the pre-TWIST value to <8:2 Â 10 À4 (90% C.L.).Left-right symmetric models extend the SM with a righthanded W [1].In the generalized (or nonmanifest) model no assumptions are made about the ratio of right-to lefthanded couplings (g R =g L ) or the form of the right-handed Cabibbo-Kobayashi-Maskawa matrix.In this case, the TWIST result for provides the best limit on the mixing angle between the light and heavy mass eigenstates W 1 and W 2 .Our limit is jðg R =g L Þj < 0:020 (90% C.L.), compared to the pre-TWIST limit of jðg R =g L Þj < 0:066.The lower limit on the mass of W 2 [ðg L =g R Þm 2 ] has been increased from 400 to 578 GeV=c 2 .Coupled constraints on the mass for ðg L =g R Þm 2 and the mixing angle are shown in Fig. 3, where our limits are derived from a correlated 2D probability distribution from our measurements.These improved constraints will significantly impact predictions from the class of left-right symmetric models where the neutrinos are light compared to the muon mass.

FIG. 1 (
FIG.1 (color).Residuals for the fit of one nominal data set to simulation in units of standard deviations.The fiducial region is outlined.

FIG. 2 .
FIG. 2. Summary of published central values and total uncertainties for , , and P [5-7,14,15], along with the results of this analysis.Vertical lines represent the SM values.

TABLE I .
Systematic uncertainties and statistical errors for , , and P .