Muon polarization in the MEG experiment: predictions and measurements
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Abstract
The MEG experiment makes use of one of the world’s most intense low energy muon beams, in order to search for the lepton flavour violating process \(\mu ^{+} \rightarrow \mathrm{e}^{+} \gamma \). We determined the residual beam polarization at the thin stopping target, by measuring the asymmetry of the angular distribution of Michel decay positrons as a function of energy. The initial muon beam polarization at the production is predicted to be \(P_{\mu } = 1\) by the Standard Model (SM) with massless neutrinos. We estimated our residual muon polarization to be \(P_{\mu } = 0.86 \pm 0.02 ~ \mathrm{(stat)} ~ { }^{+ 0.05}_{0.06} ~ \mathrm{(syst)}\) at the stopping target, which is consistent with the SM predictions when the depolarizing effects occurring during the muon production, propagation and moderation in the target are taken into account. The knowledge of beam polarization is of fundamental importance in order to model the background of our \({\mu ^+ \rightarrow e^+ \gamma }\) search induced by the muon radiative decay: \(\mu ^{+} \rightarrow \mathrm{e}^{+} \bar{\nu }_{\mu } \nu _\mathrm{e} \gamma \).
Keywords
Monte Carlo Systematic Uncertainty Standard Model Prediction Monte Carlo Event Depolarize Effect1 Introduction
Low energy muon physics experiments frequently use copious beams of “ surface muons”, i.e. muons generated by pions decaying at rest close to the surface of the pion production target, such as those produced at meson factories (PSI and TRIUMF). In the Standard Model (SM) with massless neutrinos, positive (negative) muons are fully polarized, with the spin opposite (parallel) to the muon momentum vector, that is \(P_{\mu } = 1\) for positive muons, at the production point; the muon polarization can be partially reduced by the muon interaction with the electric and magnetic fields of the muon beam line as well as with the muon stopping target. The degree of polarization at the muon decay point affects both the energy and angular distribution of the muon decay products i.e. Michel positrons and \(\gamma 's\) from the normal \({\mu ^+ \rightarrow e^+ \nu \bar{\nu }}\) and radiative muon decay \(\mu ^{+} \rightarrow \mathrm{e}^{+} \bar{\nu }_{\mu } \nu _\mathrm{e} \gamma \). The muon decay products are an important background when searching for rare decays such as \(\mu ^{+} \rightarrow \mathrm{e}^{+} \gamma \); a precise knowledge of their distribution is therefore mandatory. We report on the determination of the residual muon polarization in the PSI \(\pi \)E5 [1] channel and MEG beam line [2] from the data collected by the MEG experiment between 2009 and 2011. Clear signs of the muon polarization are visible in the Michel positron angular distribution; the measured polarization is in good agreement with a theoretical calculation (see Sect. 2) based on the SM predictions and on the beam line characteristics.
The MEG experiment at the Paul Scherrer Institute (PSI) [3] has been searching for the lepton flavour violating decay \(\mu ^{+} \rightarrow \mathrm{e}^{+} \gamma \) since 2008. Preliminary results were published in [4, 5] and [6]. The analysis of the MEG full data sample is under way and will soon be published. A detailed description of the experiment can be found in [2]. A high intensity surface muon beam (\(\sim \)3\(\times 10^{7} \upmu ^+/\mathrm{s}\)), from the \(\pi \)E5 channel and MEG beam line, is brought to rest in a \(205~\mathrm{\upmu m}\) slanted plastic target, placed at the centre of the experimental setup. The muon decay products are detected by a spectrometer with a gradient magnetic field and by an electromagnetic calorimeter. The magnetic field is generated by a multicoil superconducting magnet (COBRA) [7, 8], with conventional compensation coils; the maximum intensity of the field is \(1.26~\mathrm{T}\) at the target position. The positron momenta are measured by sixteen drift chambers (DCH) [9], radially aligned, and their arrival times by means of a Timing Counter (TC)
[10, 11, 12], consisting of two scintillator arrays, placed at opposite sides relative to the muon target. The momentum vector and the arrival time of photons are measured in a 900 liter Cshaped liquid xenon photon detector (LXe) [13, 14], equipped with a dense array of 846 UVsensitive PMTs. A dedicated trigger system [15, 16] allows an efficient preselection of possible \(\mu ^{+} \rightarrow \mathrm{e}^{+} \gamma \) candidates, with an almost zero deadtime. The signals coming from the DCH, TC and LXe detectors are processed by a custommade waveform digitizer system (DRS4) [17, 18] operating at a maximum sampling speed close to \(2~\mathrm{GHz}\). Several calibration tools are in operation, allowing a continuous monitoring of the experiment [19, 21]. Dedicated prescaled trigger schemes collect calibration events for a limited amount of time (few hours/week). A complete list of the experimental resolutions (\(\sigma \)’s) for energies close to the kinematic limit \(m_{\mu }/2\) can be found in [6]; the most relevant being: \(\sim \)340 \(\mathrm{keV}/c\) for the positron momentum, \(\sim \)10 \(\mathrm{mrad}\) for the positron zenith angle and \(\sim \) 1 and \(\sim \) 3 mm for the positron vertex along the two axes orthogonal to the beam direction.
The beam axis defines the zaxis of the MEG reference frame. The part of the detector preceeding the muon target is called the “ UpStream” (US) side and that following the muon target is called the “ DownStream” (DS) side. The zenith angle \(\theta \) of the apparatus ranges from \({\approx }60^{\circ }\) to \({\approx }120^{\circ }\), with \(\left( 60^{\circ }  90^{\circ }\right) \) defining the DSside and \(\left( 90^{\circ }  120^{\circ }\right) \) defining the USside. The SM prediction is \(P^{z}_{\mu } = 1\) for muons travelling along the positive zaxis.
2 Theoretical issues
 1.
effects at the production stage, close to and within the production target;
 2.
effects along the beam line up to the stopping target;
 3.
effects during the muon moderation and stopping process in the target.
2.1 Depolarization at the production stage
Since the angular divergence of the beam is not zero, the average muon polarization \(P_{\mu }\) along the muon flight direction does not coincide with \(P^{z}_{\mu }\) where z is the direction of the muon beam (the beam acceptance at the source is \(150~\mathrm{msr}\) and the angular divergence is \(450~\mathrm{mrad}\) in the horizontal and \(120~\mathrm{mrad}\) in the vertical direction).
A more important effect is due to “ cloud muons”, i.e. muons originating from pion decays in flight, in or close to the production target, and accepted by the beam transport system. These muons have only a small net polarization due to their differing acceptance kinematics which leads to an overall reduction of the beam polarization, based on studies performed at LAMPF [22] and measurements we made at the \(\pi \)E5 channel at PSI. The latter involved the fitting of a constant cloud muon content to the limited region of the measured muon momentum spectrum, around the kinematic edge at \({\approx }29.79~\mathrm{MeV}/c\). This was crosschecked by direct measurements of negative cloud muons at the MEG central beam momentum of \(28~\mathrm{MeV}/c\), where there is no surface muon contribution on account of the charge sign (muonic atom formation of stopped negative muons). The cloud muon content was found to be consistent from both measurements when taking the kinematics and crosssections of positive and negative pions into account. This leads to an estimated depolarization of \(\left( 4.5 \pm 1.5~\% \right) \), which is the singlemost important effect at the production stage.
2.2 Depolarization along the beam line
2.3 Depolarization during the muon moderation and stopping processes.
The largest muon depolarization effect is expected to take place in the MEG muon stopping target. The behaviour of positive muons in matter is extensively discussed in the literature (for a review [24]). After a rapid moderation and thermalization of muons in matter, muonium (\(\mu ^{+} \mathrm{e^{}}\)) is formed and further thermalized by collisions. The muon polarization is unaffected during the muonium formation and thermalization and subsequent decay.
So, we can assume that even in our case the strong magnetic field quenches any depolarizing effect.
The last point to be addressed is that muons reach the target centre under different angles within a \({\approx }1 \times 1~\mathrm{cm}^{2}\) beam spot. This angular spread corresponds to an apparent depolarization, since \(P_{\mu }\) does not coincide with \(P^{z}_{\mu }\). Using the full MEG Monte Carlo (MC) simulation we evaluated that the angular divergence at the target corresponds to a cone of \({<}20^{\circ }\) opening angle, corresponding to \({\approx }3~\%\) apparent depolarization.
2.4 Total depolarization
Summary of main depolarizing effects (%)
Source  (%) 

Multiple scattering in the production target  0.3 
Cloud muons  4.5 
Muon transport along the beam line  0.8 
Muon interactions with the MEG target  Negligible 
Muon angular spread at the target  3.0 
Total  8.6 
3 Expected Michel positron spectrum from polarized muons
The differential decay width for \(x = 1\) at \(\theta _\mathrm{e} = 70^{\circ }\) is about twice that at \(\theta _\mathrm{e} = 110^{\circ }\). Inspection of Fig. 2 shows that detectable effects are expected in the MEG data sample, even if the MEG apparatus is not the best suited for polarization measurements due to the relatively small angular range, centred around \(\theta _\mathrm{e} = 90^{\circ }\).
4 Results of the measurement
4.1 Generalities
In the previous section we showed that polarization effects can be observed in the angular distributions of highenergy positrons from Michel decays. In addition to that, the distribution of highenergy photons from Radiative Muon Decay (RMD) is expected to be affected by the polarization; however its associated error is very large, because of the intrinsic uncertainties in the analysis method, mainly related to the determination of the photon emission angle, and because of the presence in this data sample of a large background of photons from other sources (e.g. bremsstrahlung, annihilation in flight, pileup of lower energy gamma’s ...). We will therefore disregard this item.
It is important to note that in Eq. 6 the quantization axis is the muon spin direction; however, surface muons are expected to be fully polarized in the backward direction, i.e. along the negative zaxis. Therefore, the polar angle \(\theta \) in the MEG reference frame is related to \(\theta _\mathrm{e}\) in Eq. 6 by \(\theta = 180^{\circ }  \theta _\mathrm{e}\). Hence, the excess in the theoretical angular distribution Eq. 6 for \(\theta _\mathrm{e} < 90^{\circ }\) corresponds to an excess for \(\theta > 90^{\circ }\) in the experimental angular distribution, i.e. on the USside.
4.2 Analysis of Michel positrons

in the first one, we compared the energy integrated experimental angular distribution of Michel positrons with that obtained by a detailed Geant3based MC simulation of those events, as seen in the MEG detector, with the muon polarization as a free input parameter;

in the second one, we measured the USDS asymmetry \(A\left( E_\mathrm{e^{+}} \right) \) and the ratio \(R\left( E_\mathrm{e^{+}} \right) \) as a function of positron energy and fit them with the expected phenomenological forms, after unfolding the detector acceptance and response.
4.2.1 MC simulation
The MEG MC simulation is described in details in [4, 31]. Michel positrons were generated in the stopping target (the full simulation of the muon beam up to the stopping target described above was not used since it is much slower and does not bring significant advantages in this case) with a minimum energy of \(40~\mathrm{MeV}\) and a muon polarization \(P_{\mu }\) varying between 0 and \(1\) in steps of 0.1. A smaller step size of 0.05 was used between \(0.8\) and \(1\), close to the expected value (Sect. 2). Separate samples of MC events were produced for each polarization value and the positron energy and direction were generated according to the theoretical energyangle distribution corresponding to this polarization. Positrons were individually followed within the fiducial volume and their hits in the tracking system and on the timing counters were recorded; a simulation of the electronic chain converted these hits into anodic and cathodic signals which were processed by the same analysis algorithms used for real data. Modifications of the apparatus configuration during the whole period of data taking were simulated in detail, following the information recorded for each run in the experiment database. The position and spatial orientation of the target varied slightly each year, as well as trigger and acquisition thresholds, beam spot centre and size and the drift chamber alignment calibration constants. Some of the drift chambers suffered from instabilities, with a time scale from days to weeks, with their supply voltages finally set to a value smaller than nominal. The supply voltage variations, chamber by chamber, were also followed in the simulation on a run by run basis. However, voltage instabilities do not significally affect the polarization measurement. Since drift chamber wires run along the zaxis, a non operating chamber produces the same effect on US and DS if the beam is perfectly centred on the target, while it gives a second order contribution to the USDS asymmetry when the beam is not perfectly centred. The number of MC events generated using the global configuration (target position, alignment ...) corresponding to a given year is proportional to the actual amount of data collected in that year.
4.2.2 Data sample
The data sample contains the events collected between 2009 and 2011 by a prescaled trigger requiring only a timing counter hit above the threshold (so called “ trigger 22”). The analysis procedure requires an accurate preselection of good quality tracks: strict selection cuts are applied in order to single out tracks with good angular and momentum resolutions, well matched with at least one timing counter hit and with the decay vertex reconstructed within the target volume. A fiducial volume cut is included to avoid efficiency distorsions at the borders of the acceptance. The sample and the selection criteria are essentially those used to identify Michel events for the absolute normalization of the MEG data (see [5, 6]). About 37k (2009), 65k (2010) and 115k (2011) positron tracks passed all selection cuts, for a total of about \(2.1 \times 10^{5}\) events. The same criteria were applied to the MC tracks; about \(1.3 \times 10^{5}\) events passed all selections for each polarization value.
4.2.3 Comparison between MC and data
We also show in Fig. 5 the comparison between data (blue points) and MC (red line) positron energy spectra on the US (left) and DS (right) sides.
The red vertical lines in the bottom plots define the energy region where the polarization fit is performed (\(46~\mathrm{MeV} < E < 53~\mathrm{MeV}\)). The agreement between data and MC is generally quite good for the spatial coordinates, while some (\({<}10\,\%\)) discrepancies can be observed in the energy spectra and expecially in their ratios, even in the fit region. Data/MC ratios are consistent with unity for \(48~\mathrm{MeV} < E < 52~\mathrm{MeV}\), but exhibit some systematic differences close to the threshold (\(E \approx 4548~\mathrm{MeV}\)) and in the upper edge (\(E > 52~\mathrm{MeV}\)). Such discrepancies are due to the fact that the MC simulation is not able to perfectly reproduce the experimental energy resolution: for instance \(\sigma _{E} \approx 340~\mathrm{keV}\) for data and \({\approx }260~\mathrm{keV}\) for MC at \(E = 52.83~\mathrm{MeV}\). However, if one looks at both bottom plots together, one sees that the differences are clearly correlated; then, they tend to cancel out when one uses \(A\left( E_\mathrm{e^{+}} \right) \) or \(R\left( E_\mathrm{e^{+}} \right) \) as analysis tools. We also note that the differences are particularly relevant in the year 2010 sample, when the beam centre was displaced with respect to the target centre by some \(\mathrm{mm}\). (See Sect. 4.2.7 dedicated to the analysis of systematic uncertainties.)
The general agreement between data and MC for all reconstructed variables demonstrates our ability to correctly simulate the behaviour of the apparatus.
4.2.4 Efficiency correction for MC and data
The efficiency for the full reconstruction of a positron event is composed of two parts: the absolute efficiency \(\epsilon (Track)\) for producing a track satisfying all trigger and software requirements and the relative efficiency \(\epsilon \left( TC \left Track \right. \right) \) of having a TC hit, given a track. Both efficiencies are functions of the positron energy and emission angles and can be different on the US and DS sides because of intrinsic asymmetries of the experimental apparatus.
The \(\epsilon \left( TC \left Track \right. \right) \) efficiency was separately computed for MC and real events. In the case of MC this calculation is straightforward. In the more complicate case of real data, we selected positrons collected by a different prescaled trigger (so called “ trigger 18”) requiring only loose conditions on the number and the topological sequence of fired drift chambers, and selected the fraction of tracks with an associated good TC hit within this sample. The MC and data \(\epsilon \left( TC \left Track \right. \right) \) efficiency matrices were then used to correct the \(\theta \) angular distributions, \(A\left( E_\mathrm{e^{+}} \right) \) and \(R\left( E_\mathrm{e^{+}} \right) \).
We then applied the same correction functions to all MC samples and we checked that the polarization values extracted by fitting \(A\left( E_\mathrm{e^{+}} \right) \) and \(R\left( E_\mathrm{e^{+}} \right) \) were consistent with those generated. The correction functions were also applied to the data since the good agreement between MC and data shown in Figs. 4 and 5 gives us confidence of the correct apparatus response to positron events.
4.2.5 Results of first strategy: angular distribution
According to Eq. 6 and to the definition of \(\theta \), we expect to observe an asymmetric distribution for large values of the polarization, with an excess on the USside (\(\theta > 90^{\circ }\)) and a symmetric distribution for null polarization. Figure 7 shows a clear disagreement between data and MC for \(P_{\mu } = 0\) and a good agreement for \(P_{\mu } = 0.85\). The simulation well reproduces the USDS asymmetry observed in the data, as well as the dip for \(\theta \approx 90^{\circ }\). Angular distributions for \(P_{\mu } = 0.8\) and for \(P_{\mu } = 0.9\) do not significantly differ from that shown for \(P_{\mu } = 0.85\): the comparison between data and MC gives strong indications for a large polarization, \( \left( 0.8  0.9 \right) \), but it is not precise enough to single out a value of \(P_{\mu }\), with its uncertainty.
4.2.6 Results of second strategy: USDS asymmetry and ratio
The average \(\chi ^{2}/d.o.f.\) of the fits is 0.74, mainly determined by the points close to the threshold. In both plots the yellow line represents the best fit, while the two green lines show the \(\pm 1~\sigma \) band, obtained by adding or subtracting the sum of statistic and systematic uncertainties (see next section for the discussion of systematic uncertainties).
Results of the polarization fit year by year
2009  2010  2011  Global  

\(\langle P \rangle \pm \Delta P\)  \( 0.85 \pm 0.05\)  \( 0.71 \pm 0.05\)  \( 0.92 \pm 0.04\)  \( 0.856 \pm 0.021 \) 
\(\chi ^{2}\)/d.o.f.  0.90  1.37  0.34  0.74 
The polarization value measured in 2010 sample is significantly lower than in the other two years; however, the larger values of the \(\chi ^{2}\)/d.o.f. suggests that this result is of lower quality and less reliable. The observed deviation in 2010 data is discussed in the next section and is reflected in the associated systematic error.
Note that the energy region above \(53~\mathrm{MeV}\), where the data and MC errors are quite large, does not affect the result, since the fit was limited to 46–53 MeV. We checked that these results do not depend on the fitting interval by eliminating one bin at the lower bound and/or one bin at the upper bound: in all cases the fit results agreed with (9) within the statistical error.
4.2.7 Systematic uncertainties
 1.
Energy scale. The energy scale and resolution are determined in MEG, as discussed in [5], by fitting the Michel positron energy spectrum with the convolution of the theoretical spectrum (including radiative corrections) of the detector acceptance and of a resolution curve, in the form of a partially constrained triple Gaussian shape. The position of the Michel edge, used as a reference calibration point, is determined with a precision of \(\delta E_\mathrm{e^{+}} \sim 30~~\mathrm{keV}\). The effect of this uncertainty was evaluated by varying the reconstructed energy of our events by a factor \(\left( 1 \pm \delta E_\mathrm{e^{+}}/\bar{E}_\mathrm{e^{+}} \right) \), where \(\bar{E}_\mathrm{e^{+}} = 52.83~\mathrm{MeV}\) is the position of the Michel edge, and repeating the analysis. The polarization value determined by the average of \(A\left( E_\mathrm{e^{+}} \right) \) and \(R\left( E_\mathrm{e^{+}} \right) \) fits increases by 0.0029 when \(\delta E_\mathrm{e^{+}}/\bar{E}_\mathrm{e^{+}}\) is added and decreases by \(0.0052\) when \(\delta E_\mathrm{e^{+}}/\bar{E}_\mathrm{e^{+}}\) is subtracted.
 2.
Angular bias. The angular resolution is determined by looking at tracks crossing the chamber system twice (double turn method), as discussed in [4, 5]. The uncertainty on the \(\theta \) and \(\phi \) scales varies between 1 and \(3~\mathrm{mrad}\). The effect of this uncertainty on the angular scale was (conservatively) evaluated by modifying both the reconstructed polar angles by \(\pm 3~\mathrm{mrad}\) and repeating the analysis. The measured polarization decreases (increases) by \(0.013\) (\(+0.025\)).
 3.
Target position. The target position with respect to the centre of the COBRA magnet is measured by means of an optical survey and checked by looking at the distribution of the positron vertex of reconstructed tracks. The discrepancies between the two methods are at the level of a fraction of a mm. Since in our analysis we require that the reconstructed positron vertex lies within the target ellipse, an error on the target position can alter the positron selection. We assumed a conservative estimate of a target position uncertainty of \(\pm 1~\mathrm{mm}\) on all coordinates and, as previously, added or subtracted it and repeated the analysis. The effect was to decrease (increase) the polarization by \(0.022\) (0.016).
 4.
MCbased corrections. The MC corrections, inserted to take into account the absolute tracking efficiency, are based on the position of the target as measured by the optical survey and on the nominal location of the beam centre. A variation of these parameters produces a variation on the correction functions, applied year by year to MC and data. We estimated the size of this effect by generating MC samples with a displaced beam and target (\(\pm 1~\mathrm{mm}\) shift as previously) and null polarization and determined new tracking efficiency correction functions. Such functions were then applied to the data and MC: the measured polarization decreased (increased) by \(0.035\) (0.036).
 5.
Threshold effects. The response of the MEG tracking system close to the momentum threshold (\(E_\mathrm{e^{+}} \approx 45~\mathrm{MeV}\)) depends in general on the polar angles and can be significantly distorted when the beam and target are not centred, causing fictitious differences between the US and DS sides. In 2010 the beam centre to target centre displacement was maximal, corresponding to more than \(3~\mathrm{mm}\) in the horizontal plane and just over \(3~\mathrm{mm}\) in the vertical plane, producing an asymmetric USDS energy threshold, with the DS spectrum systematically higher than the US one for \(E_\mathrm{e^{+}} < 47~\mathrm{MeV}\). The beam and target displacement were introduced in the MC, but the simulation for 2010 did not result in a good agreement with the data in the region close to the energy threshold. We then estimated the systematic effect due to the angular dependence of the energy threshold by removing the 2010 sample from the fit: the polarization decreases by \(0.047\), a difference twice larger than the statistical error. The \(\chi ^{2}/d.o.f.\) of the fit improved a bit from 0.74 to 0.71. A better fit quality was observed also on MC events by removing the simulated data corresponding to the year 2010 configuration.
 6.
Higher order corrections to the theoretical formula in Eq. 6. The effect of second and higher order contributions and of taking into account the finite electron mass to the muon decay rate is discussed in some detail in [32, 33, 34, 35]. The conclusion is that they are smaller than the first order correction and therefore we can deduce that the effect of including them in Eq. 6 for extracting the polarization from Fig. 8 is not larger than the effect of the first order correction that is 0.3 %. Hence this value can be assumed as a conservative estimation of the systematic error due to higher order corrections.
 7.
Offtarget muon decays. A conservative estimation of offtarget muon decays as discussed in Sect. 4.2 is 0.5 %, that is 0.004 on the polarization value.
Main systematic uncertainties and their effect on polarization
Source  (\(\Delta P\)) 

Energy scale  \(\left( +0.0029, 0.0052 \right) \) 
Angular scale  \(\left( +0.025, 0.013 \right) \) 
Target position  \(\left( +0.016, 0.022 \right) \) 
Tracking efficiency  \(\left( +0.036, 0.035 \right) \) 
Energy threshold  \(0.047\) 
Higher order corrections  \(\pm 0.003\) 
Offtarget decsy  \(\pm 0.004\) 
Total (in quadrature)  \(\left( +0.047, 0.064 \right) \) 
5 Summary and conclusions
Notes
Acknowledgments
We are grateful for the support and cooperation provided by PSI as the host laboratory and to the technical and engineering staff of our institutes. This work is supported by SNF Grant 200021_137738 (CH), DOE DEFG0291ER40679 (USA), INFN (Italy) and MEXT KAKENHI 22000004 and 26000004 (Japan). Partial support of the Italian Ministry of University and Research (MIUR) Grant RBFR08XWGN, Ministry of University and Education of the Russian Federation and Russian Fund for Basic Research Grants RFBR 142203071 are acknowledged.
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