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Re: Question Radiative muon decay (Tom Roberts)
Date: 25 Nov, 2013
From: Gumplinger Peter <Gumplinger Peter>

Hi Tom,

> The PDG lists the radiative muon decay channel with a branching ratio of
> (1.4+-.4)%, with the note: for gamma energies > 10MeV (presumably in the
> muon rest frame). This decay is mu+ => e+ anti_nu_mu nu_e gamma.

The radiative decay mode cannot be clearly separated from the Michel decay with a soft-photon. If the photon is (cannot) be detected in an experiment, the G4MuonDecayChannelWithSpin calculates the Michel electron spectrum correctly including O(1) radiative corrections.

If the experiment is sensitive to gammas, the radiative muon decay branching ratio (to be used for the simulation) depends on the gamma threshold of the experiment. The class G4MuonRadiativeDecayChannelWithSpin produces photons as can be seen in the attachement where y = 2E_g/m_mu.

A useful figure of how the SM differential branching ratio fraction changes with y can be found in:

(or it can be extracted from the attachment by normalizing the integral to 1.4% for E>10MeV.) References are Fronsdal and Ueberall, 1959 and Eckstein and Pratt, 1959.

> This channel is not implemented in the normal Decay process, but is
> included in DecayWithSpin. Unfortunately, its branching ratio is much
> too small -- it is 1.4% for ALL gammas, which makes it about 0.08% for
> energy > 10MeV. So it looks to me like the note was not noticed.

The 1.4% appears only in examples, but thanks, for pointing this out! The correct implementation would be to resample the decay branch until a photon is produced that is larger than the threshold with the corresponding branching ratio supplied by the user.

> If I assume the energy distribution is correct, then the overall
> branching ratio is about 24%. That's quite large and cannot reasonably
> be ignored. Why is the conventional wisdom "muons decay into an electron
> plus two neutrinos", when a quarter of the time they don't?

See, above.

This leaves us with another problem, however, and that is how to sample now the Michel energy distribution for the "normal" muon decay in G4MuonDecayChannelWithSpin where currently the O(1) radiative corrections have been obtained by integrating over all gamma energies and angles.

> This can be a serious error for experiments involving muon decay
> backgrounds, because for high-energy muons, those gammas from ~24% of
> the decays get boosted to significant energies. Even for stopped muons,
> some 5-10% of the decays have gammas above the pair-production
> threshold. Surely many experiments ought to see this....

Yes, those experiments need very carefully drafted muon decay routines that account for their finite photon sensitivity (in energy and, possibly, angle) for the radiative decay, balanced by a modified Michel decay routine that takes the explicit generation of some part of the external bremsstrahlung spectrum into account as well.

> Does anybody know the real story here? What should we do about it? IMHO
> the Decay process is in error by omitting a channel with a branching
> ratio of ~24%, as is DecayWithSpin.

At this point, Geant4 provides a routine that allows the careful sampling to O(1) rad. corrections of the Michel electrons and it provides a routine that samples the e and gamma in rad. muon decay. An experiment would typically simulate the rad. decay as a background to their observable by adjusting the strength (BR) in their final fit.


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