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5. Radiative Decays of Pions and Muons

Pion and muon decay processes are often associated with additional photons (see Table 2-3). The main source of radiative decays is `inner bremsstrahlung' of either the pion or muon in motion or of a daughter particle, as there is the muon or the electron, respectively. Most of those processes are indistiguishable from non-radiative decays, since the photon energy is low and the angle relative to the charged decay product small. More interesting regarding the physical relevance of their origin are processes that emit photons as the result of the finite size of the pions or due to QED corrections for the muons. These structure dependent radiative decays allow the determination of pion form factors or the strength of axial vector coupling in the case of the muon. (See [Bry82] and [Cri61] for a comprehensive introduction into the theory of radiative decays of pions and muons, respectively.) The measurement of the probability of radiative decays will also be a feature of the PiBeta detector. An additional test of the standard model can be achieved this way in addition to a reduction of the systematical error for the determination of the pion beta decay rate.

In general all measured or calculated decay rates do include radiative decays - regardless of the origin of the emitted photon. Their relevance for the upcoming measurement of the pion beta decay rate are twofold: i) Neglected radiative p +->e+ n µ decays would result in an additional uncertainty due to an underdetermined calibration process and ii) four-fold coincidences of pairs of µ+->e+ n eµ( g ) decays could fake an p b event. For both cases the relative strength and the emission angle between photon and charged particle have to be recorded. Hence, the calorimeter acceptances and a possible cluster recognition algorithm had to be studied.

In order to measure the exact p +->e+ n e decay rate also the radiative mode has to be included. This requires an identification of the actual decay process and a determination of radiative decays. The most likely process is the p ->µ->e chain with a 14% probability for the radiative decay µ+->e+ n eµ g (at a g threshold energy of 10 MeV [Cri61]). This is identified by the presence of a charged particle and a total energy of at most half the muon mass[20]. Since the signature of the p b event is determined by the presence of two photons with nearly identical energies, the p +->e+ n e peak clearly can be identified by an energy exceeding 55 MeV [Law98, Ass95]. Furthermore the difference in lifetime will give different signatures in the target.

There are two major concerns regarding the decay p +->e+ n e g . In the unlikely case of a low energy neutrino it contributes to the background of the p b event. Due to the high chamber efficiency, the probability to have two clumps in opposite clusters without a charged particle identification calculates[21] -> to be 1.91*10-13. From kinematics it can be seen that only large emission angles, and therefore high photon energies, can be problematic. For lower angles a cluster summing algorithm, which will be explained in the subsequent section, will take care of radiative decays. The relative emission angle between positron and photon in the following is denoted as f .

Figure 5-1 The kinematics of the decay p +->e+ n e g . The calculation follows 3 body decay kinematics. The allowed region here is bound by the dotted lines (x, y or z=1); where x corresponds to the photon energy defined by (in the rest frame of the pion); y and z are equally defined for the electron and the neutrino, respectively.

Regarding a simulation of the entire PiBeta calorimeter the relative acceptance for a p +->e+ n e g decay, for instance, with f >0° is at 86.2% while it reduces to 14.63% for f >30°. Since the µ+->e+ n e µ g process is generated with a much higher probability by bremsstrahlung processes the relative acceptance for f > 30° reduces to 0.032%. A similar reduction can be observed for cuts on the photon energy. The acceptances for radiative decays, relative to the correspondent non-radiative decays, are plotted below in Figure 5-2.


Figure 5-2 Photon energy plotted against the relative emission angle. The acceptance of the calorimeter then is plotted logarithmically in the projection of the z-axis. The plot for µ+->e+ n eµ g (lower plot) looks similar; but is more pronounced at lower energies and angles. The acceptances are listed in Table 5-1 down below.

From this figure one can extract the relative calorimeter acceptances after applying kinematic cuts. It shall be summarized in Table 5-1.

Kinematic Cut
Acceptance in per cent
relative to p +->e+ n e
Acceptance in per cent
relative to µ+->e+ n eµ
f greater than
p +->e+ n e g
µ+->e+ n eµ g


[Epsilon] g greater than

0 MeV
0.5 MeV
1 MeV
2 MeV
3.5 MeV
5 MeV
7 MeV
10 MeV
Table 5-1 Acceptances for radiative pion and muon decay modes for two kinematic cuts at different thresholds.

[20] The so-called Michel edge at 52.83 MeV

[21] Here a cut on 50 MeV positrons and muons at a relative angle of >170° is assumed (this was chosen due the trigger thresholds and the kinematics of the p b -decay assuming 3 deg angular resolution). This leads to a BR of 0.47*10-8 for p +e+ n e g . For the parametrization of this calculation see [Bry82]. The charged particle tracking inefficiency comprises MWPCs and PV hodoscope and, hence, is about 4.0*10-5.

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