In positron emission, a proton decays into a neutron, electron, and neutrino. Since the mass of a proton is less than that of a neutron, does that mean that energy is converted into mass in the reaction? Beta decay#β+ decay says that some of the binding energy goes into converting a proton into a neutron, but that's not really what I'm looking for. So, does the extra mass result from the conversion of energy into mass?
[Physics] ny Violation of conservation of mass in positron emission
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A neutron is not a proton and an electron lumped together (as your question seems to suggest you think)
A hydrogen atom is a bound state of an electron and a proton (bound by the electromagnetic force) whereas a neutron is a bound state of three quarks (bound by the strong force).
You might be tempted to think that a neutron is also a bound state of an electron and a proton because a neutron can decay into an electron and a proton and the neutron is also slightly more massive than a proton. But you'd be wrong. Here's why:
Both a neutron and a proton are bound states of three quarks. Beta decay can convert a neutron to a proton like this $$udd \rightarrow uud + e^- + \bar{\nu_e}$$
Here, one of the down quarks in the neutron gets converted into an up quark, an electron and an antineutrino through the mediation of the weak force. Since the up quark has a charge of $+2/3$ and the down quark $-1/3$, that explains the difference in charge of a neutron and a proton.
That's what turns a neutron into a proton, not some kind of ejection of an electron out of the neutron, but a genuine transmutation of a fundamental particle (a down quark) through the weak force.
Protons and neutrons are very similar particles. Although they have different charges, as far as the strong force is concerned they are almost identical. So changing a proton into a neutron and back isn't considered decay because you aren't changing the number of nucleons. More precisely the baryon number remains constant.
If we describe the proton and neutron as a bound state of three quarks$^1$ then a proton is two up and one down quarks while a neutron is two down and one up quarks. Interchanging them requires changing an up quark to a down quark and vice versa, which happens by emission/absorption of an electron or positron. The number of quarks doesn't change and the process is reversible.
The process we normally describe as proton decay is altogether more radical. There are actually several different possible mechanisms for proton decay, but they all involve the proton disappearing completely leaving behind just a positron and two photons. This has to involve the creation of a hypothetical and so far unobserved particle called an X boson. The Standard Model does not include the X boson so as far as the Standard model is concerned the proton cannot decay.
$^1$ Caution! Not literally true!
Best Answer
The distinction between energy and mass is really a matter of semantics here and is not really a fundamental distinction. We tend to call it "mass" when energy is in a bound state, i.e. an object such as a proton or an atomic nucleus that more or less holds together and stays in one place.
You can call positron emission "energy to mass conversion" if you like, but that's really an arbitrary distinction between the supposed "mass" of a proton/neutron and the "energy" of a nuclei. They're both energies of various kinds when you look close enough anyway!
The mass of a proton is mostly not the mass of the three valence quarks, but the energy in the quark-gluon field that surrounds them. Similarly, the mass of an atomic nucleus is not just the total mass of all the protons and neutrons in it; there is a significant correction to the mass from nuclear binding energy (which contributes negatively, reducing the total mass).
Even the masses of quarks, electrons, neutrinos etc. is really an energy of interaction with the Higgs field.
So most-to-all of what you ordinarily call "mass" is actually a sum of various energies of constituents and binding energies for different levels of structure.