# Proton to neutron conversion at relativistic energies

## Main Question or Discussion Point

The reaction p→n+e+e is common inside nucleus.
But it not considered when we talk about free particle and reason is simply given mass of products being larger than reactants.
Now my question is if there is a high energy proton having total energy in order of 2 GeV or let it to be 100 GeV (To even account mass of intermediate W boson) then mass-energy issue should not be in picture. Will then this reaction be possible for proton as free particle.
What I think is technically it should be possible as energy and momentum be conserved and all quantum numbers conserved to the extent as inside nucleus.
Am I missing something??

If conversion being possible at 100GeV and not at 2 GeV then also suggest how it happens inside nucleus as inside nucleus 100GeV is not possible.

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The reaction p→n+e+e is common inside nucleus.
But it not considered when we talk about free particle and reason is simply given mass of products being larger than reactants.
Now my question is if there is a high energy proton having total energy in order of 2 GeV or let it to be 100 GeV (To even account mass of intermediate W boson) then mass-energy issue should not be in picture. Will then this reaction be possible for proton as free particle.
What I think is technically it should be possible as energy and momentum be conserved and all quantum numbers conserved to the extent as inside nucleus.
Am I missing something??
No, you can't do that while conserving energy and momentum. Since part of kinetic energy would need to turn into additional rest mass, momentum will have to decrease (i.e. it won't be conserved).

A simpler way to see it is to change to a coordinate system where your free proton is stationary. A stationary proton can't decay purely by conservation of energy; thus any moving free proton can't do that either.

ChrisVer
Gold Member
The reaction p→n+e+e is common inside nucleus.
But it not considered when we talk about free particle and reason is simply given mass of products being larger than reactants.
Now my question is if there is a high energy proton having total energy in order of 2 GeV or let it to be 100 GeV (To even account mass of intermediate W boson) then mass-energy issue should not be in picture. Will then this reaction be possible for proton as free particle.
What I think is technically it should be possible as energy and momentum be conserved and all quantum numbers conserved to the extent as inside nucleus.
Am I missing something??

If conversion being possible at 100GeV and not at 2 GeV then also suggest how it happens inside nucleus as inside nucleus 100GeV is not possible.
That is not true for a single isolated proton. An isolated proton will always have at its rest frame energy equal to its rest mass, so at its rest frame it won't be able to decay. Since decaying or not is not frame-dependent, the proton beta decay cannot occur for an isolated proton (seeing proton as a single particle and not a composite one).