Question

A pion decays into a muon and a neutrino. Let’s model this decay as A → B+C, where A has a mass of M, B is a massless particle of energy 1/4 Mc^2 in the rest frame of A, and C is a massive particle of unknown mass and energy. Physicist Olga observes particle A moving in the positive x direction, and measures A to have a mass of M=100 MeV/c^2, and an Energy of 1500 MeV before decay.

a) If particle B is traveling in the same direction that particle A was traveling before the decay, what is the energy of B in Olga’s frame of reference?

b) How does it compare to B’s energy in the rest frame of A?

Answer 1

Mass of A= M

In the rest frame of A (Centre of momentum Frame):

mass of B,MB=0

energy of B,K.EB=(1/4)MeV/C^2

mass of C,MC=? (unknown)

energy of C,K.EC=? (unknown)

In Olga's Frame (Laboratory Frame of refrence)

mass of A,M=100 MeV/C^2

Energy of A,K.EA=1500 MeV/C^2

From relativistic K.E equation:

with above data we have for rest frame of A:

Therefore lorentz factor for rest frame of A is 16.

Total energy avalable in rest frame of A is rest mass of A,ETC=100MeV/C^2

From Energy conservation we must have:

Energy before decay = enegy after decay

Total energy of A= Total energy of B+ Total energy of C (TEC)=1/4+K.EC+MC=100

->> ETC = K.EC+MC=100-0.25=99.75 MeV/C^2

From momentum coservation we must have:

Total Momentum before decay = Total Momentum after decay=momentum of B +momentum of C

But A is at rest in its frame of refrenace so Momentum before decay is zero.Thus

momentum of B +momentum of C =0 =>magnitude of momentum of B = magnitude of momentum of C

From relativistic energy equation:

we have momentum of B

agan from relation

with EC=99.75 MeV/C^2  and PC=0.5 MeV/c

we have

Thus energy of C in rest frame of A is:

Ecm=99.75-99.7487= 0.0513 MeV/C^2 ...(A)

a)Answer

The energy in Lab frame as measure by Olga from inverse(C has same direction as A) lorentz energy momentum equation:

b)Answer

In rest frame of A energy is =0.0513 MeV/C^2 while as in Olga's refrence frame it is 8.8 MeV/c^2 . Thus the energy in lab frame is about two hundred times more.