PS5 - 253a/Schwartz Due Problem Set 5 1 Radioactive decay...

253a/Schwartz
Due October 12, 2010
Problem Set 5
1. Radioactive decay. The muon decays to an electron and two neutrinos through an intermediate
massive particle called the W − boson. The muon, electron and W − all have charge - 1 .
a) Write down a Lagrangian that would allow for μ
−
→ e
−
ν
e
ν
μ
. Assume the W and other
particles are all scalars, and the e − , ν
e
and ν
μ
are massless. Call the coupling g .
b) Calculate |M|
2
for this decay in the limit that the W mass, m
W
, is large.
c) The decay rate ( =
1
lifetime
) is proportional to |M|
2
. The coupling g should be dimension-
less (like the coupling e for the photon), but appears dimensionful because we ignore
spin. If the W spin were included, you would get extra factors of p μ , which would turn
into a factor of s
√
= m
μ
in |M| 2 . Use dimensional analysis to figure out what power of
m
μ
should be there. Also, throw in a
1
(2 π ) 3
for the 3-body phase space.
d) Look up the muon mass and lifetime on pdglive.lbl.gov. Pick some reasonable perturba-
tive value for g and use the muon mass and lifetime to esimate the W mass.
e) The tauon, τ