Quark changes in beta decay

In beta decay, a weak interaction changes the flavour, or type, of one quark inside a nucleon. This changes a neutron into a proton, or a proton into a neutron. An electron or positron and a corresponding antineutrino or neutrino are also emitted.

A neutron contains one up quark and two down quarks, written $$udd$$. A proton contains two up quarks and one down quark, written $$uud$$.

Beta-minus decay: neutron to proton

In beta-minus decay, one down quark changes into an up quark. The nucleon’s quark composition therefore changes from $$udd$$ to $$uud$$: a neutron becomes a proton. An electron and an electron antineutrino are emitted:

Beta-plus decay: proton to neutron

In beta-plus decay, one up quark changes into a down quark. The nucleon’s composition changes from $$uud$$ to $$udd$$: a proton in a suitable nucleus becomes a neutron. A positron and an electron neutrino are emitted:

Charge and lepton number in beta decay

Charge is conserved in each decay. In beta-minus decay, the proton has charge $$+1$$ and the electron has charge $$-1$$, giving the same total charge as the original neutral neutron. In beta-plus decay, the neutron has charge $$0$$ and the positron has charge $$+1$$, matching the original proton’s charge. The neutrino and antineutrino have no electric charge.

Lepton number is also conserved. The electron has lepton number $$+1$$ and the electron antineutrino has lepton number $$-1$$. In beta-plus decay, the positron has lepton number $$-1$$ and the electron neutrino has lepton number $$+1$$. In each case, the emitted particles’ lepton numbers add to zero, matching the initial nucleon’s lepton number.

Conservation laws in particle interactions

A proposed particle interaction can be tested by adding up certain quantities for all the initial particles and comparing them with the totals for all the final particles. A conservation law requires a quantity to have the same total before and after the interaction. Individual particles may be created or transformed, but the required totals must remain unchanged.

Electric charge, baryon number and lepton number must each be conserved. A mismatch in any one of these totals rules out the proposed interaction.

Baryon and lepton numbers

Baryon number distinguishes baryons from antibaryons. A baryon has baryon number $$+1$$ and an antibaryon has baryon number $$-1$$. At the quark level, a quark has baryon number $$+\frac{1}{3}$$ and an antiquark has baryon number $$-\frac{1}{3}$$. Add the values for all the relevant particles on each side of an interaction.

Leptons have lepton number $$+1$$ and antileptons have lepton number $$-1$$. Any particle that is neither a lepton nor an antilepton has lepton number zero.

For example, consider the proposed interaction $$n \rightarrow p + e^- + \bar{\nu}_e$$. The neutron $$n$$ and proton $$p$$ are baryons, the electron $$e^-$$ is a lepton, and the antineutrino $$\bar{\nu}_e$$ is an antilepton.

• Charge: the initial total is $$0$$; the final total is $$+1-1+0=0$$, in units of the elementary charge.
• Baryon number: the initial total is $$+1$$; the final total is also $$+1$$.
• Lepton number: the initial total is $$0$$; the final total is $$+1-1=0$$.

These three totals match, so none of these conservation laws rules out the interaction.

Strangeness conservation

Strangeness is another quantum number to compare between the initial and final particles. A strange quark has strangeness $$-1$$, while an antistrange quark has strangeness $$+1$$. Use the particle’s quark content or other supplied data to find its strangeness, then add the values on each side.

Strangeness must be conserved in strong and electromagnetic interactions, but it can change in a weak interaction. A change in total strangeness therefore rules out a proposed strong or electromagnetic interaction, though it does not by itself rule out a weak interaction.

Energy and momentum conservation

Total energy and total momentum must also be conserved. Energy accounting includes the rest energy of the particles as well as other forms, such as kinetic energy. Rest energy may be converted into other forms or vice versa, but the total must remain the same.

Momentum is a vector, so direction matters: the vector sum of the initial momenta must equal the vector sum of the final momenta. Matching momentum magnitudes alone is not sufficient.

If a proposed interaction fails a required conservation law, it cannot occur as written. Passing the conservation-law tests does not, by itself, establish that the interaction occurs.