# Pion decay: why is the positron electron neutrino decay suppressed?

I'm reading through M. Thomson's Modern Particle Physics to better understand why the positron electron neutrino decay channel is suppressed relative to the muon and muon neutrino decay route.

I know it has to do with helicity supression, but I can't quite formulate a description of what helicity supression is concisely by reading Thomson's explanation. If anyone could concisely explain this to me it would be greatly appreciated!

• This is discussed near the very beginning in Srednicki web.physics.ucsb.edu/~mark/qft.html, see page 14 – innisfree May 2 '16 at 11:44
• PS if you understand it after reading that, you can write an answer to your own question – innisfree May 2 '16 at 11:45
• Unfortunately I only have the M. Thomson book. – DarthPlagueis May 2 '16 at 15:03
• The Srednicki book is available for download legitimately on his webpage, linked above – innisfree May 2 '16 at 15:04

## 1 Answer

In an imaginary world where the leptons and the neutrinos are all massless, they would exist only in the left-handed helicity version.

In this imaginary world the decay of the pion in lepton plus neutrino would be prohibited: you could not conserve both the total spin (zero) and the total momentum, in that this would imply emitting two particles both with the spin in the direction of motion (i.e. right handed).

In our real world the leptons are not massless, so that they can be emitted with a component along the direction of motion. Heuristically, the probability of this kind of emission grows bigger with the mass of the lepton (being zero for zero mass), so that the decay in heavier particles (ie muons) is favorable wrt lighter particles (electrons) one.

• Why not a virtual Tau? – Viktor May 2 '16 at 13:25
• The pion mass is around 140 MeV, the tau lepton mass is around 1700 MeV so you have a clear energetic barrier. BTW we are speaking of real, not virtual particles! – Arnaldo Maccarone May 2 '16 at 13:31
• See this is three paragraphs, I asked for concise..so in a sentence? – DarthPlagueis May 2 '16 at 15:02
• In a sentence: if the leptons were massless the decay would be prohibited by spin and momentum conservation law, therefore the decay amplitude is heuristically growing with growing lepton mass – Arnaldo Maccarone May 3 '16 at 8:28