The Quark Model — Question 10

Question 10

Current quark masses do not add up to the proton mass

For an illustrative specified renormalization convention take current rest energies $m_uc^2=2.2$ MeV and $m_dc^2=4.7$ MeV. Proton rest energy is 938.3 MeV. Its valence content is uud.

1. Compute the sum of valence current rest energies and its fraction of proton rest energy.

2. Compare with a constituent model assigning 313 MeV to each quark. Explain why the two mass definitions cannot be mixed.

Question 10 - Solution

Current quark masses do not add up to the proton mass

(a) Method and calculation

$$\boxed{(2m_u+m_d)c^2=2(2.2)+4.7=9.1\ {\mathrm{MeV}}}.$$$$\boxed{9.1/938.3=.00969839=0.969839\%}.$$

(b) Interpretation and further calculation

$$\boxed{3m_{\mathrm{constituent}}c^2=939\ {\mathrm{MeV}}}.$$$$M_pc^2=\text{full interacting QCD-state energy}\ne\sum m_{\mathrm{current}}c^2.$$

Physical check

Constituent masses include effective interaction and motion effects. The numerical remainder is not a scheme-independent decomposition of proton mass.