Muon polarization in the MEG experiment: predictions and measurements

# Muon polarization in the MEG experiment: predictions and measurements

The MEG Collaboration
###### 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 . 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 by the Standard Model (SM) with massless neutrinos. We estimated our residual muon polarization to be 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 search induced by the muon radiative decay: .

\thankstext

[*]e1Corresponding author: fabrizio.cei@pi.infn.it \thankstext[]daggerDeceased. \thankstext[]Presently at INFN, Laboratori Nazionali di Frascati, via E.Fermi, 40, 00044 Frascati (Roma) Italy

## 1 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 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 from the normal and radiative muon decay . The muon decay products are an important background when searching for rare decays such as ; a precise knowledge of their distribution is therefore mandatory. We report on the determination of the residual muon polarization in the PSI E5 PE5 () channel and MEG beam line MEGdet () from the data collected by the MEG experiment between and . 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 Section 2) based on the SM predictions and on the beam line characteristics.

The MEG experiment at the Paul Scherrer Institute (PSI) PSI () has been searching for the lepton flavour violating decay since 2008. Preliminary results were published in meg2009 (); meg2010 () and meg2013 (). 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 MEGdet (). A high intensity surface muon beam (), from the E5 channel and MEG beam line, is brought to rest in a slanted plastic target, placed at the centre of the experimental set-up. 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 multi-coil superconducting magnet (COBRA) thin-cable (); ootani_2004 (), with conventional compensation coils; the maximum intensity of the field is at the target position. The positron momenta are measured by sixteen drift chambers (DCH) Hildebrandt2010111 (), radially aligned, and their arrival times by means of a Timing Counter (TC) Dussoni2010387 (); DeGerone:2011te (); DeGerone:2011zz (), 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 liter C-shaped liquid xenon photon detector (LXe) Sawada2010258 (); Mihara:2011zza (), equipped with a dense array of 846 UV-sensitive PMTs. A dedicated trigger system trigger2013 (); Galli:2014uga () allows an efficient preselection of possible candidates, with an almost zero dead-time. The signals coming from the DCH, TC and LXe detectors are processed by a custom-made waveform digitizer system (DRS4) ritt_2004_nim (); Ritt2010486 () operating at a maximum sampling speed close to . Several calibration tools are in operation, allowing a continuous monitoring of the experiment Baldini:2006 (); calibration_cw (); papa_2010 (). Dedicated prescaled trigger schemes collect calibration events for a limited amount of time (few hours/week). A complete list of the experimental resolutions (’s) for energies close to the kinematic limit can be found in meg2013 (); the most relevant being: for the positron momentum, for the positron zenith angle and and for the positron vertex along the two axes orthogonal to the beam direction.

The beam axis defines the -axis 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 of the apparatus ranges from to , with defining the DS-side and defining the US-side. The SM prediction is for muons travelling along the positive -axis.

## 2 Theoretical issues

The E5 channel is a high-intensity low-energy pion and muon beam line in the momentum range. Surface muons have a kinetic energy of and a muon momentum of and are produced fully polarized along the direction opposite to their momentum vector. Several depolarizing effects can reduce the effective polarization along the beam line. They are classified into three groups:

• effects at the production stage, close to and within the production target;

• effects along the beam line up to the stopping target;

• 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 along the muon flight direction does not coincide with where is the direction of the muon beam (the beam acceptance at the source is and the angular divergence is in the horizontal and in the vertical direction).

One such depolarizing effect is due to the multiple scattering in the target, which modifies the muon direction leaving the spin unaffected. Surface muons have a maximum range in the carbon production target of . The average broadening angle due to multiple scattering is then given by (see for instance Pifer ()):

 ⟨cosα⟩=1−21lX0[(30P)0.5−1], (1)

where is the muon momentum in and is the muon path in the target in units of carbon radiation lengths (). We obtain , a contribution of less than .

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 VanDyck:1979xr () and measurements we made at the 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 . This was cross-checked by direct measurements of negative cloud muons at the MEG central beam momentum of , 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 cross-sections of positive and negative pions into account. This leads to an estimated depolarization of , which is the single-most important effect at the production stage.

### 2.2 Depolarization along the beam line

The MEG beam line comprises of several different elements: quadrupole and bending magnets, fringing fields, an electrostatic separator, a beam transport solenoid and the COBRA spectrometer. The equation of motion of the muon spin is described, even in a spatially varying magnetic field such as the COBRA spectrometer, by the Thomas equation Jackson ():

 d→sdt=emc→s×[(g2−1+1γ)→B−(g2−1)γγ+1(→β⋅→B)→β (2) −(g2−γγ+1)→β×→E],

where , and are the muon velocity, electric charge and mass, is the speed of light, , is the muon gyromagnetic factor and and are the electric and magnetic field vectors. In principle this equation is valid only for uniform fields, but it gives correct results even in our case since any effect due to the non-uniformity of the magnetic field is many orders of magnitude smaller than the Lorentz force in the weak gradient field of COBRA. From Eq. 2 we can obtain the time evolution of the longitudinal polarization, defined as the projection of the spin vector along the momentum vector, which is given by:

 d(→s⋅→β)dt=−emc→s⊥⋅[(g2−1)→β×→B+(gβ2−1β)→E], (3)

where is the projection of the spin vector in the plane orthogonal to the muon momentum. In this equation, the first contribution is due to the muon magnetic moment anomaly () and the second to the presence of an electric field. In the MEG beam line the first term is associated with the guiding elements (quadrupole and bending magnets), while the second term is associated with the electrostatic separator. The geometrical parameters of the beam elements and their field intensities are PE5 (): for the deflecting magnets the length is and the vertical field is ; for the electrostatic separator the length is , the gap between the plates and the applied voltage . The COBRA spectrometer has a weak spatially varying magnetic field, which muons are subjected to while travelling on the US-side of the magnet, after being focused by the beam transport solenoid; the average vertical component of the COBRA magnetic field around the muon trajectory is of order of and its contribution to the spin rotation is about one order of magnitude smaller than the one of the bending magnets. With these parameters we evaluated a spin rotation of due to the magnetic component and of due to the electrostatic component. Note that the longitudinal polarization is, by definition, referred to the muon velocity, while the polarization we are interested in is the one in the beam direction, our natural quantization axis. Therefore the spin rotation results in a depolarizing effect of ; this is confirmed by a numerical integration of the Thomas and Lorentz equations along the MEG 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 Brewer ()). After a rapid moderation and thermalization of muons in matter, muonium () is formed and further thermalized by collisions. The muon polarization is unaffected during the muonium formation and thermalization and subsequent decay. Muonium interaction with the magnetic field in vacuum is described by a hyperfine Hamiltonian, which includes the muon-electron spin-spin interaction and the Larmor interaction of both spins with the external field. On the basis defined by the total spin and by its projection along the quantization axis , the muonium wavefunction is a superposition of a triplet state () and of a singlet state (). If one assumes muons to be fully polarized in the longitudinal direction when they enter the target and electrons in the target to be unpolarized, the initial state of the muonium formation is a mixture of the state and the combination of and . The coefficients of this combination and their time evolution can be calculated as functions of the ratio , where is the external magnetic field and . While the component is a pure state and is constant, the other oscillates with time; one can calculate its time average, which translates into an average longitudinal polarization given by:

 ⟨P∥(x)⟩=12(1+x21+x2). (4)

Since at the position of the MEG target is , we obtain an average residual polarization of : any depolarizing effect is quenched by the strong magnetic field. However, muons are propagating in a dense medium and not in vacuum; therefore the muonium interaction with the material medium should be taken into account, making a detailed calculation impossible. We therefore used available experimental data, i.e. direct measurements of the muon residual polarization after crossing different targets immersed in external magnetic fields. The MEG target is a layered structure of polyethylene and polyethylene terephthalate (PET), for which no direct measurement is available; we assume this material to behave like polyethylene Swanson (); Buhler1 (); Buhler2 ().

With zero magnetic field, the residual muon polarization is and reaches for increasing magnetic fields. Figure 1 shows the value of muon residual polarization as a function of the magnetic field intensity (adapted from Buhler2 ()): the polarization saturates at for a magnetic field intensity of , while the central value of the COBRA magnetic field is .

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 beam spot. This angular spread corresponds to an apparent depolarization, since does not coincide with . Using the full MEG Monte Carlo (MC) simulation we evaluated that the angular divergence at the target corresponds to a cone of opening angle, corresponding to apparent depolarization.

### 2.4 Total depolarization

In conclusion, the main depolarizing effects are due to cloud muons and beam divergence. The average final polarization along the beam axis () is:

 ⟨Pzμ⟩=(−0.91±0.03), (5)

where the systematic uncertainty takes into account the uncertainties in this computation. The various contributions are listed in Tab. 1.

## 3 Expected Michel positron spectrum from polarized muons

The angular distribution of Michel positrons was calculated in detail by several authors including the effect of the electron mass and the first order radiative corrections kuno_2001 (); kinoshita_1959 (); Arbuzov:2001ui (). The bidimensional energy-angular distribution for polarized decaying at rest, neglecting the electron mass, takes the following form:

 d2Γ(μ+→e+ν¯ν)dxdcosθe=mμ5GF2192π3 x2[F(x)+PμcosθeG(x)] F(x) =f0(x)+α2πf1(x)+O(α2) G(x) =g0(x)+α2πg1(x)+O(α2) f0(x) = (3−2x)g0(x)=(2x−1) (6)

where is the polarization along a selected axis, () and is the angle formed by the positron momentum vector and the polarization axis. Expressions for and neglecting the electron mass are available in kinoshita_1959 (); the MC simulations in the following are based on Eq. 6 including first order radiative corrections and neglecting the electron mass. Formulae incorporating the dependence on electron mass for all terms in Eq. 6 are presented in Arbuzov:2001ui ().
We show in Fig. 2 the angular distribution from Eq. 6 in the range for different values of .

The differential decay width for at is about twice that at . 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 .

## 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 high-energy positrons from Michel decays. In addition to that, the distribution of high-energy 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, pile-up 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 -axis. Therefore, the polar angle in the MEG reference frame is related to in Eq. 6 by . Hence, the excess in the theoretical angular distribution Eq. 6 for corresponds to an excess for in the experimental angular distribution, i.e. on the US-side.

A very powerful way to study the muon polarization is to compare the energy spectra, integrated over the angular acceptance, on the US () and on the DS () sides. In Fig. 3 we show the expected asymmetry between and as a function of positron energy :

 A(Ee+)=((dN/dEe+)US−(dN/dEe+)DS)((dN/dEe+)US+(dN/dEe+)DS) (7)

in the upper part and the ratio:

 R(Ee+)=(dN/dEe+)US(dN/dEe+)DS (8)

in the lower part for three representative polarization values: (red dotted line), (black dashed line) and (blue continuous line). First order R.C. are taken into account and have a effect on both asymmetry and ratio.

### 4.2 Analysis of Michel positrons

Experimentally measured angular distributions are a result of the convolution of the expected theoretical distributions with the detector response, acceptance and thresholds, whose non-uniformities can mimic angular asymmetries or create fictitious ones. Topological requirements and quality cuts needed to define and fit charged particle tracks also introduce angle-dependent non-uniformities. In particular, the tracking algorithm has a lower efficiency for positrons emitted with small longitudinal momenta, resulting in a dip in the angular distribution of Michel positrons for (see later Fig. 7). MEG positrons are mainly produced by muon decays in the stopping target, with a significant fraction () decaying off-target, in beam elements or in the surrounding helium gas. However, this contribution can be minimized by requiring the reconstructed positron decay vertex to lie within the target volume. The fraction of the positrons decaying off-target and reconstructed on the target was evaluated by a complete MC simulation of the muon trajectory along the PSI/MEG beam line up to the stopping target and of the subsequent muon decay. This fraction was found to be smaller than 0.5% and can be considered as a source of systematic uncertainty assuming, very conservatively, the same effect on the polarization measurement. In summary, an analytical prediction of the experimental distribution is rather complicated; hence, we decided to measure the muon polarization by means of two different analysis strategies:

• in the first one, we compared the energy integrated experimental angular distribution of Michel positrons with that obtained by a detailed Geant3-based 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 US-DS asymmetry and the ratio 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 meg2009 () and softwaretns (). 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 and a muon polarization varying between and in steps of . A smaller step size of was used between and , close to the expected value (section 2). Separate samples of MC events were produced for each polarization value and the positron energy and direction were generated according to the theoretical energy-angle 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 significantly affect the polarization measurement. Since drift chamber wires run along the -axis, 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 US-DS 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 and by a pre-scaled trigger requiring only a timing counter hit above the threshold (so called “trigger 22”). The analysis procedure requires an accurate pre-selection 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 meg2010 (); meg2013 ()). About (), () and () positron tracks passed all selection cuts, for a total of about events. The same criteria were applied to the MC tracks; about events passed all selections for each polarization value.

#### 4.2.3 Comparison between MC and data

The comparisons between the reconstructed positron vertex coordinates , and for data (blue points) and MC (red line, normalized to the data) are shown in Fig. 4, top and bottom left; at the bottom right the same comparison for the reconstructed azimuthal angle at the positron emission point is shown.

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. In the upper part of the figure we report the superimposed data and MC distributions, while in the lower part we show the ratios data/MC as a function of the positron energy (in ). All spectra are corrected for the left-right correction factors which will be discussed in the next section.

The red vertical lines in the bottom plots define the energy region where the polarization fit is performed (). The agreement between data and MC is generally quite good for the spatial coordinates, while some () 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 , but exhibit some systematic differences close to the threshold () and in the upper edge (). Such discrepancies are due to the fact that the MC simulation is not able to perfectly reproduce the experimental energy resolution: for instance for data and for MC at . 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 or as analysis tools. We also note that the differences are particularly relevant in the year sample, when the beam centre was displaced with respect to the target centre by some . (See section 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 for producing a track satisfying all trigger and software requirements and the relative efficiency 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 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 pre-scaled 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 efficiency matrices were then used to correct the angular distributions,