Why is π0 created in the high-energy collision p+p→p+p+π0?
Answer
Just summing together all the comments and providing some more explicit calculations, we have conservation of four-momentum (which is the amalgamation of the conservation of energy and conservation of momentum), we have:
pμ1+pμ2=p′μ1+p′μ2+pμπ
Taking the inner product of each side with itself, we get:
⟨pμ1|pμ1⟩+2⟨pμ1|pμ2⟩+⟨pμ2|pμ2⟩=⟨p′μ1|p′μ1⟩+2⟨p′μ1|p′μ2⟩+2⟨p′μ1|pμπ⟩+⟨p′μ2|p′μ2⟩+2⟨p′μ2|pμπ⟩+⟨pμπ|pμπ⟩
We note that pμp′μ=⟨pμ|p′μ⟩ is invaraiant in all frames of reference and that pμpμ=m2c2, we can therefore simplify:
2m2pc2+2⟨pμ1|pμ2⟩=4m2pc2+4mpmπc2+m2πc2
If we consider that the second proton is initially at rest we have: pμ1=(Ec,→p) and therefore:
2mpE=2m2pc2+4mpmπc2+m2πc2
Rearranging we get:
E=mpc2+2mπc2+m2πc22mp
Using the constants mp=938 MeV/c2 and mπ=139.6 MeV/c2, we get:
E=1.228 GeV⟹T=289 MeV
So if a proton with 289 MeV of kinetic energy collided with a stationary proton, there is a chance that a pion will be produced.
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