Quark number sum rule (${\int }_0^1[f_f(x)-{\bar{f}}_f(x)]dx=N_f$)

The parton distribution function is a number density. Integrating the quark distribution minus the antiquark distribution counts the net flavour charge of the hadron, since every sea quark-antiquark pair contributes zero. The corresponding flavour-vector conserved current counts particles minus antiparticles. For a proton, the net up/down numbers are two and one; for a neutron they are one and two. The rule fixes the difference of distributions, not their sum, and does not imply absence of a sea.

With only up/down valence quarks and no antiquarks, the quark number sum rule and leading parton model expression give the charge-squared number moment $\int F_2(x)dx/x={\sum }_f{Q}_f^2{N}_f$. The squared charges are $4/9$ for up and $1/9$ for down, so the proton has $8/9+1/9=1$ and the neutron has $4/9+2/9=2/3$. Restricting the flavour list to up/down does not itself remove antiquarks; the no-sea assumption is essential for these individual moments.