Annihilation-in-flight of polarised positrons with polarised electrons as an analyser of the positron polarisation from muon decay

The use of annihilation-in-flight of arbitrarily polarised positrons with arbitrarily polarised electrons as an analyser for the positron polarisation from muon decay is discussed. Analysing powers for the longitudinal and the two transverse positron polarisation components are derived and algorithms for the simulation of polarised muon decay and of annihilation-in-flight are given.


Introduction
The chiral nature of the charged weak interaction manifests itself through the appearance of pseudoscalar terms like decay asymmetries and lepton longitudinal polarisations both of which involve measurements of particle spins. The corresponding measurements in nuclear β decay [1], pion and muon decay [2][3][4] proved that the charged weak interactions prefers leptons of negative helicity and antileptons of positive helicity. This experimental evidence has led to the concept of lefthanded fermions (V -A hypothesis) and the standard model [5][6][7], which has been extremely successful in describing experimental data. Still, the question remains, if extensions to the standard model are needed. Effects of additional, weaker, interactions could be found with the help of precision measurements.
These investigations can be pursued especially well in the case of muon decay. It can be described by the most general, local four-fermion point interaction Hamiltonian [8]. It contains ten complex coupling constants to be determined by experiment. The observables can be expressed in terms of a chiral Hamiltonian in chargechanging form characterized by fields of definite handedness [9]. The matrix element is given by [10,11]: The index γ = S, V, T labels the type of interaction (4scalar, 4-vector, 4-tensor), while the indices ε, µ = R, L indicate the chirality (R, L: right-, lefthanded) of the particle spinors. The standard model predicts g V LL = 1, with all a e-mail: fetscher@phys.ethz.ch other couplings being zero. In fact, with a selected set of µ decay experiments it has been possible to derive a lower limit for g V LL and upper limits for the absolute values of all other 9 coupling constants [10].
The chiral nature of the weak interaction implies the special importance of spin-dependent measurements as decay asymmetries and polarisation measurements, while measurements of scalar quantities as the Michel parameter ρ which parametrizes the energy spectrum of the decay positrons do not constrain the possible interactions sufficiently. In fact, a precise measurement of ρ in agreement with the standard model prediction ρ = 3/4 does not imply the V -A interaction. Indeed, it has been shown that any linear combination of the six complex coupling constants g S RR , g S RL , g S LR , g S LL , g V RR , and g V LL results in ρ = 3/4 [12]. A measurement of the longitudinal polarisation (decay parameter ξ ), on the other hand, allows to derive stringent limits to five of the ten complex coupling constants, if it turns out to be in agreement with the standard model value of 1, within errors [12,13]: Note that because the left-hand side of (2) contains the sum of positive squares, the standard model value of exactly ξ = 1 excludes any righthanded contributions for the electron (first lower index of the coupling constant).
In striking contrast to that, the measurement of possible transverse polarisation components does not allow to derive such stringent limits, because any sizeable transverse component is due to interference between different interactions, so that the resulting amplitude may well be negative. The energy dependence of the transverse component P T 1 , which lies in the plane of the muon polarisation and positron momentum, is described by the decay parameters η and η [11,14] or, in the older publications, by α/A and β/A [15,16]. The energy dependence of the transverse component P T 2 , which lies perpendicular to the mentioned plane, is described by the decay parameters α /A and β /A. In terms of the coupling constants of (1) these four parameters are given by: In contrast to our conclusions for the precise measurement of P L , i.e., the ξ parameter, a precise measurement of the four parameters η, η , α /A and β /A does not permit to exclude any coupling constants, since in the standard model all four parameters are equal to zero, and as interference terms, they can cancel each other mutually. However, with our knowledge that g V LL > 0.96 (90% c.l.) [10,11], we can neglect terms in second order of the nine complex couplings consistent with zero. With the approximation g V LL ≈ 1 we obtain: The specific advantage of the measurement of the transverse components here becomes obvious: One is able to measure the scalar coupling g S RR linearly and, since P T 2 violates time reversal invariance, one gets separate limits for the real and for the imaginary part of this coupling constant with a corresponding gain in sensitivity as compared to the measurements of the longitudinal polarisation P L ± ∆P L . This would yield ∆ξ /ξ = ∆P L /P L and give the following limit to g S RR according to (2): since P L ≈ 1. This obviously is not as sensitive as the direct measurement of the transverse polarisation due to the proportionality to √ ∆P L . The argument above is equivalent to the relation which follows from the definition of the polarisation, |P| ≤ 1. Moreover, a measurement of P L , as an absolute measurement, is quite difficult due to the small available electron polarisation of 7%. The measurement of P T 1 and P T 2 is simplified by the fact that both components are small or zero, so that the experimental errors are of the same order of magnitude as the values themselves and therefore much less critical than in the case of P L .
For intermediate energies (a few MeV up to ≈ 53 MeV) the annihilation-in-flight of positrons with polarised electrons in a magnetised foil has been shown to efficiently analyse both the transverse [17,18] and the longitudinal [19][20][21] polarisation. The cross section in the c.m. system for this process had been calculated already in the late 1950's and early 1960's [22,23]. From this cross section the analysing power for the longitudinal polarization has been derived [19,24].
The experimental method for the measurement of the transverse polarisation is identical for both measurements performed up to date. The first experiment at PSI (PSIPT I) used four big NaI detectors as a calorimeter [17]. With a slightly modified setup the longitudinal polarisation was also measured [20,21]. The second experiment (PSIPT II) used parts of the same setup, but, among other improvements, replaced the four NaI detectors by 127 BGO crystals [18,25]. Since the measurement of P T makes use of a precessing muon polarisation vector (see Sect. 5.1), the determination of the respective analysing powers turns out to be quite demanding. They depend not only on the energies of the positron and of the two annihilation quanta, but also on the azimuthal angle of orientation ψ of the gamma rays (for the definition of ψ see Figs. 2 and 4). For PSIPT I these analysing powers were derived for positrons emitted in the direction of the symmetry axis of the setup and for the angles ψ = 0 • , 45 • , 90 • , and 135 • , corresponding to the four possible orientations of NaI detector pairs [26].
For PSIPT II, with its finegrained BGO detector wall, ψ was measured for each event separately. In addition to that, the data analysis was performed by comparing the measured data with Monte Carlo generated polarisation distributions, one for each of the four decay parameters η, η , α /A and β /A [27]. For this simulation the emission of positrons of arbitrary polarisation and emitted into arbitrary directions from the rotating coordinate system of the muon has been calculated. The dependence of the annihilation rate on time as well as on the azimuthal angle α and the polar angle χ with respect to the fixed axes of the experiment is derived for the first time, thereby including all three polarisation components (106). As a by-product of this result, a new possible method to measure the longitudinal polarisation is presented.
This comprehensive overview of the theory underlying this kind of experiment generalises previous results, as used for PSIPT I, by deriving distributions and analysing powers for arbitrary polarisation values and directions of emission of the positrons. It establishes the relations between coupling constants, decay parameters and polarisation components. It also clarifies the hitherto neglected difference between P T 1 (x, ϑ), as predicted by theory, and P 1 (x), as obtained in the experiment. Finally it derives algorithms to simulate polarised muon decay and the polarisation dependent annihilation.

Polarisation vector of the decay positron
The polarisation vector of the e + is described most easily in the coordinate system defined by the muon polarisation vector P µ and the positron momentum k e (see Fig. 1): Here, ϑ is the angle between positron momentum k e and muon polarisation P µ , and x is the reduced positron energy: with W µe = (m 2 µ + m 2 e )/(2m µ ) as the maximum energy of the positron. The orthogonal unit vectorsx 0 ,ŷ 0 ,ẑ 0 are defined as follows:ẑ x 0 =ŷ 0 ×ẑ 0 . The components of P e then are given by [11,14]: The functions F i (x) are given by: In the standard model, ξ = ξ = 1. Radiative corrections to the polarisation observables have been calculated recently including terms with x 0 [28]. Their influence, however, is only substantial towards the soft end of the e + spectrum. Neglecting terms proportional to x 0 = 9.67 × 10 −3 and using the canonical (V -A) values for the experimentally well-determined decay parameters ρ = 3/4 [29], δ = 3/4 [30] and ξ = 1 [31][32][33], I obtain According to (18) and (19) the transverse polarisation depends on the polar emission angle ϑ both in the numerator and in the denominator. In the experimental setup described below ϑ is distributed symmetrically around ϑ = 90 • . There the polarisation is determined by summing over events with different ϑ which eliminates the (already small) ϑ dependence in the denominator. As a result of this procedure both components can be factorized as follows: The energy distributions P 1 (x) and P 2 (x) contain the full dependence on the decay parameters (η, η ) and (α /A, β /A), respectively, and thus form the basis for the determination of these four decay parameters.

Kinematics of annihilation-in-flight
We regard the reaction It is convenient to use reduced four-momenta π i for this reaction: where the particles' energy and three-momentum are given by E i and k i , respectively, and m designates the positron mass. Table 1 lists the reduced four-momenta of the particles of the annihilation reaction (36) in the laboratory and in the c.m. system. The center of momentum (c.m.) of the positronelectron system moves with the reduced velocity with the associated γ-factor The absolute value of the reduced momentum of each photon in the c.m. system is given by We designate the photon with the higher energy in the Lab system as particle # 3. It is emitted in the c.m. system under the polar angle ϑ * and the azimuthal angle ϕ * . The Table 1. Labelling of the reduced fourmomenta π i = p i /m of the particles of the annihilation reaction (36) in the laboratory and in the c.m. system. The positron mass is given by m resulting momenta π * 3 and π * 4 of the two photons in the c.m. system are then: With the definition of particle # 3 as given above, the angular region is restricted to The momenta of the two photons in the lab system as functions of the polar c.m. angle ϑ * and the azimuthal c.m. (as well as lab) angle ϕ * ≡ ϕ are obtained by a Lorentz boost: The relations between the polar lab angles ϑ 3 and ϑ 4 and the c.m. polar angle ϑ * are: The relations between the polar lab angles ϑ 3 and ϑ 4 and the lab energies ε 3 and ε 4 are: The opening angle ϑ 34 between the lab momenta of the two annihilation quanta is: 4 Annihilation cross section for arbitrarely polarised electrons and positrons The matrix elements for the annihilation of arbitrarely polarised electrons and positrons have been calculated by W.H. McMaster [23]. There the polarisation vectors of the e − and the e + , in this paper labelled as Q and P, repectively, are extended to the Stokes vectors where the first entry stands for the intensity of the beam, while the other three are the usual components of the polarisation, with P c , P a , and P b corresponding to the z, x, and y component, respectively. The differential cross section in the c.m. system, finally, is given by where T is a 4 × 4 matrix as given by (69) of [23]. After correcting for an obvious error 1 and summing over the unobserved polarisation states of the two annihilation quanta, we obtain: with the abbreviations 5 Annihilation as an analyser of the positron polarisation

Introduction
A realistic experimental setup for the measurement of the positron polarisation consists of a polarised positron source, a magnetised foil as polarisation analyser and a highly granular electromagnetic calorimeter as detector for the annihilation quanta. This is shown in Fig. 2 which displays the experimental setup of the recently published measurement of the transverse positron polarisation from muon decay [18,25]. The measurement of the two transverse positron polarisation components P T 1 and P T 2 relies on the fact that the annihilation cross section for transversely polarised e + with transversely polarised e − leads to an anisotropic azimuthal distribution of the two annihilation quanta, as will be shown below. By precessing the muon polarisation in a homogeneous magnetic field the transverse polarisation will precess with the same angular frequency ω. This, in turn, leads to a corresponding rotation of the annihilation quanta distribution. It finally results in a harmonic variation of the annihilation rate of a given detector pair with azimuthal orientation ψ. Neglecting effects of multiple scattering I will deduce the amplitude and phase of this annihilation signal. The effects of multiple scattering will only effect the amplitude, but not the general form thus obtained. This change can be readily obtained by a Monte Carlo simulation specific to a given experiment.
The following calculation, although performed specifically for the derivation of the two transverse components, includes also the longitudinal polarisation. In fact, it will be shown that it can be applied just as well to a measurement of P L , for which a rotation of P µ is not required. Indeed, measurements of P L prefer to use an unpolarised sample of muons [20,21,24], unless one wishes to measure the angular dependence of P L as the experiment of the Louvain-la-Neuve-PSI-ETH collaboration does [34].

Global and local coordinate systems
For the description of the experiment I use a series of righthanded coordinate systems, each of them adapted to a special purpose:  1. The global coordinate system K with base vectorsx,ŷ, andẑ as shown in Fig. 2. Vectors given in component form relate to this system unless noted otherwise. 2. The coordinate system K µ 0 of the muon. This system is used with the muon polarisation P µ as polar axis to describe the decay most naturally. Since P µ is precessing in the (x, y) plane with angular frequency ω, its base vectors are time-dependent and are given bŷ 3. The coordinate system K e 0 of the positron. Here the polar axis is given by the momentum of the positron which has been emitted at polar angle ϑ 0 and azimuthal angle ϕ 0 with respect to K µ 0 . Its base vectorsx 0 ,ŷ 0 , andẑ 0 , expressed in the muon's coordinate system K µ 0 , are: In the global coordinate system K its base vectors are functions of the time t of the decay and are given by: symmetry axis one obtains: The polarisation vector of the e + is described most easily in the coordinate system K e 0 (see Sect. 2, (13)): where P L designates the longitudinal positron polarisation. According to (34)-(35) they can be factorised as The three components P 1 , P 2 and P 3 are functions of the reduced positron energy x and also depend on several muon decay parameters [11,16]. 4. The direction of the positron in the global system K is given by a polar angle χ and an azimuthal angle α. This defines the coordinate system K ν with unit vectorsx ν , y ν andẑ ν :x with the abbreviations For each detector pair (i, j) with an azimuthal orientation angle ψ the expected annihilation rate will be calculated as a function of the precessing muon polarisation and thereby as a function of time. For the calculation of the annihilation cross section the polarisation vectors P and Q of the e + and the e − , respectively, are needed in the individual coordinate system K ν of the νth positron as well as in the global system K.

The annihilation cross section of McMaster (see Sect. 3)
has been calculated for the c.m. system of the e + e − pair in a plane given by the momentum of the positron and the momenta of the annihilation quanta (see Fig. 3). There the momentum of the positron is opposite to the polar axisẑ used. In the following section I transform the annihilation cross section (55) into the global system.

Annihilation cross section in the global coordinate system
In this section the annihilation cross section in the global coordinate system is derived for an ensemble of positrons emitted in different directions. Figure 4 shows the front view as seen in the direction ofẑ. A µ + decays at r 0 . The e + is emitted under the azimuthal angle α and the polar angle χ (not shown). The e + -trajectory intersects with the BGO front plane at r 1 . The two annihilation quanta hit the BGO front plane at r 3 and r 4 (E 3 ≥ E 4 ). The difference vector r 34 = r 3 − r 4 forms the angle ψ with thexaxis (0 ≤ ψ < 2π). In the global system the positron polarisation amounts to while the electron polarisation is fixed in the positive or negative x-direction: I calculate the annihilation cross section by transforming P and Q into the system K of McMaster: The polarisation terms in the annihilation cross section (55) are then given by The elements of the transformation matrix M, given as column vectors, are: with the following abbreviations: At this point I substitute the polarisation components P x , P y , and P z in the global system with the individual polarisation components P 1 , P 2 and P 3 , thereby making use of (70)-(72) and (76)-(79). The latter three components contain only the information from the matrix element of the decay, while the angular information is separated. I also make use of the fact that χ 1 and expand both M and P up to first order in χ: The polarisation terms are then: The cross section, finally, in this approximation is given by: The behaviour of this result is illustrated in Fig. 5 for completely transversely polarised positrons (P T 1 = 1) and electrons (Q = 1) (which implies P µ = 1) with E 3 = E 4 = 50 m. There the normalized annihilation rate is displayed in function of the azimuthal angle ψ. At time t = 0, P T Q, and the intensity distribution of the annihilation quanta has the shape of a horizontally lying "figure of eight" in the polar diagram. At t = π/(2ω), P T ⊥ Q, and the "figure of eight" has moved by π/4. The maximum of the intensity lies on the bisector of the angle ωt between the two spins. Thus the "figure of eight" moves with angular frequency ω/2. For a fixed detector pair at azimuthal angle ψ the time dependence is still given by the angular frequency ω due to the two symmetric lobes of the "figure of eight".  This is due to the fact that the cross section depends on time by a harmonic with angular frequency ω, but on the azimuthal orientation angle ψ of the two annihilation quanta by a harmonic of 2ψ. By transforming the electron and the positron polarisation into the system of McMaster which is oriented at the azimuthal angle ψ terms with cos ψ and sin ψ appear in second order . In contrast to that the time dependence of the positron polarisation is due to the precession of the muon polarisation and thus does not effect the electron polarisation. Therefore terms with cos ωt and sin ωt enter (105) only linearly.

Time dependent annihilation rate
The normalized time distribution is then given by: with By summing over annihilation events for a given angle ψ, but distributed symmetrically with respect to the azimuthal angle α of the positron, the spatial dependence of the distribution (106) on the longitudinal positron polarisation component P 3 vanishes. The remaining time distri-bution is: The quantities G and H are the effective analysing powers. In the experiment of Burkard et al. [17,26] they were calculated for the four angles ψ = 0 • , 45 • , 90 • and 135 • , corresponding to the four different possible orientations of the detector pairs for the four NaI detectors used there. With the 127 BGO detectors of this experiment representing a continuous surface for the detection of the annihila-  tion quanta, the full angular dependence of the analysing powers as given by (107) and (108) has been derived to get the best possible sensitivity for the transverse polarisation. They are explicitly functions of ψ and implicitly of the c.m. energy of the e + e − system and of the polar emission angle of the photons in the c.m. system. For the analysis of the data it is more convenient to make use of the measured photon energies E 3 and E 4 in the following way: with Note that due to energy loss of the positron E 3 + E 4 = E 1 < m + E 1 , where E 1 is the positron initial energy. The analysing powers A x and their corresponding Figures of Merit in the lab system G x , A y and G y as well as A z and G z are displayed as contour lines in Figs. 6-11, respectively. Their absolute values |A j | are remarkably large in much of the kinematically allowed region. In Appendix A these three analysing powers and their corresponding Figures of Merit are given.

Measurement of the longitudinal polarisation
The longitudinal polarization can be determined according to (55). One needs, however, to turn the magnetised analyser foil by an angle (usually 45 • ) in order to get a component of the e − polarisation parallel to P L [21,24]. To restore the symmetry of the experiment one measures alternatingly with the foil once turned to the right and then to the left. This procedure has to be controlled carefully in order not to introduce systematic errors. A possible alternative is a measurement with the setup used for the measurement of the transverse polarization, where the analysing foil is perpendicular to the average positron direction of flight. In this case one should use unpolarised muons or average over the time-dependent parts of (105). One then gets a small signal proportional to χ. A measurement of the rate asymmetry, obtained from measurements with positive and with negative electron polarisation, in the range 80 • ≤ χ ≤ 100 • , for example, will show a linear dependence on χ with a change of sign at χ = 90 • . The disadvantage of the smaller signal could be outweighed by the gain of symmetry of this setup.

Appendix A: Analysing powers and figures of Merit
In the c.m.system, the figure of Merit (FoM) G j (j = x, y, z) is defined as: With the laboratory energy u given, the free parameter is the energy difference v: The cross section in the lab system is therefore: .
The FoM G j in the lab system is then: Appendix B: Simulation of muon decay

B.1 Probability density function
Given the two-dimensional normalised decay probability (see the denominator of (18) to (20)) g(x, y) = d 2 Γ dx dy = F IS (x) + P µ cos ϑF AS (x) (B.1) with (see (14)) In (B.2) standard V -A values have been assumed and terms proportional to the electron mass have been neglected.

B.2 Density functions of x and y
The marginal probability density function (p.d.f.) of x is The conditional density function of y is obtained for a given value x = x µ : The two density functions are thus: with the uniformly distributed random number This equation can be solved numerically with Newton's method. There is a problem at x = 0, however, where the derivatives of the polynomial are equal to zero. With a threshold energy x t > 0 we can neglect this problem. This reduces the range for R x µ to

B.4 Cumulative distribution function of y
One obtains accordingly the conditional cumulative distribution function of y for the value x µ resp. α µ , as obtained from the random variable R x µ , by generating the uniformly distributed random variable R y µν with the range The solutions of this quadratic equation are whereby only the upper sign constitutes a physical solution. This expression is not determined for α µ = 0. This problem can be repaired by transforming it into the following expression: (B.22) Appendix C: Simulation of the polarisation dependent annihilation

C.1 Differential cross section
The simulation of the differential annihilation cross section for arbitary electron and positron polarisation is determined according to (55) in the system of McMaster. Figure 12 shows the geometry. The annihilation takes place in thexẑ plane.
The system K ν hasẑ ν = −ẑ, with the azimuthal angle ϕ betweenx ν andx (see Fig. 13). The polarisation vectors P = (P c , P a , P b ) of the e + and Q = (Q c , Q a , Q b ) of the e − ((64) and (65)) used there are replaced by the components used in the system K ν of the positron: ⎛ The polarisation Q of the electron is transformed accordingly. We finally obtain: (C.8)  The differential cross section thus becomes: or, in in a more compact form: dσ dΩ = f 1 (cos ϑ * ) + f 2 (cos ϑ * ) cos 2ϕ + f 3 (cos ϑ * ) sin 2ϕ , (C.10) with f 1 (cos ϑ * ) = σ 0 + X L σ z + X T (σ x + σ y ) 2 (C.11) The functions f i , expressed through the terms A, B, C und D, are: (C.14)

C.2 Integral cross section
As mentioned in Sect. 3, (43)-(44), the angular region is restricted to The integration of the differential annihilation cross section (C.10) over the azimutal angle ϕ yields: