Muon Decay and Special Relativity
John R. Newport Ph.D., Spring 2007
Muon decay is the reaction: + e+ + + e.
(muon initially at rest)
calculated lifetime from quantum mechanics: = k/m5, k = 1923h7/(G2c4)
measure lifetime from PDG: = 2.197034x10-5 s
* mechanism of muon production:
very high energy protons from sun =>
protons collide with air molecules in upper atmosphere =>
collision produces pions, electrons,....
pions decay to muons
* measured kinetic energy of muons at surface of earth (average), 4GeV,
so =E/mc2 = 37.7 (extremely relativistic!)
=> v/c = 0.99965
* decay length (no special relativity).
L = c = 660 m;
the collision typically occurs at altitudes around 15km
(verified by detectors placed at various altitudes)
prediction: should be no muons reaching the surface
actual: about 1 muon per minute, per cm2 of surface area
* decay length (with special relativity)
=> , L => L = 38*660m = 25 km.
prediction: lots of muons reaching the surface! consistent with actual
event A: muon born
event B: muon dies (decays)
the reference frame of the muon is proper;
the proper time interval is t0 =
in the reference from of the earth, t = t0 = .
* muons actually created 660 m above surface:
easy to verify that this does not happen, move detectors to mountain top,
result is flux very close to surface value
* increase in kinetic energy
K = mgh = mc2 (gh/c2) = E0(gh/c2) = E0(1.6 x 10-12) for h = 15km
(allowing g to vary with height makes it even smaller)
=> particles do not significantly increase speed due to fall