Pion Creation and Neutron Decay

a) Calculate the threshold energy in MeV for the creation of charged pions by photons incident on stationary protons,

b) Neutrons β-decay via

If the neutron is at rest, what is the maximum possible velocity for the electron in units of c? (Note that m_p = 938.27 Mev/c², m_n = 939.57 Mev/c², m_π+ = 139.57 Mev/c², and m_e = 0.51 Mev/c²).

SOLUTION

a) The threshold energy for the creation of a neutron n and a pion π⁺ in the center of mass frame is simply the sum of their masses (in units where c = 1. The 4-momentum of the neutron/pion system in the center of mass frame is

(1)

We must also calculate the energy of the incident particles in the center of mass frame. However, we may use the relativistic invariance of a product of 4-vectors (which can be written for a complex system as well as for a single particle). The 4-momentum in the lab (stationary proton) frame is

(2)

Taking the squares of (1) and (2) and equating them, we obtain

(3)

Note that in these units |p_γ| = ε_γ. Therefore, we have for the threshold energy

(4)

Substituting the values given in the problem

b) Consider the neutron decay following the scheme

where we consider the proton and neutrino as one complex particle. From energy and momentum conservation, we find

(5)

(6)

(7)

(8)

Substituting (6) into (8), we have

(9)

We are looking for the maximum possible momentum p_e- of the electron. It depends only on one variable,  where  so the maximum p_e- corresponds to a minimum  Using (5) and (7), we obtain

(10)

Substituting p²_e-  from (9) into (10) yields

(11)

From this equation, the minimum of  corresponds to a minimum of  Now,

(12)

The minimum of (12) corresponds to Therefore

(13)

where we have substituted the masses in energy units. On the other hand

(14)

From (14), we have

Therefore in regular units

.