Valence and sea decomposition of proton-antiproton W production (${\mathcal{L}}_{VV},{\mathcal{L}}_{VS},{\mathcal{L}}_{SS}$)

For charged current production $u\bar{d}\to W^{+}$, charge conjugation of the antiproton distributions gives $u(x_{+})d(x_{-})+\bar{d}(x_{+})\bar{u}(x_{-})$. With a common sea quark distribution $S$, the valence quark terms are $u_v{d}_v$, the mixed terms are $u_{v}S+S{d}_v$, and the two sea assignments give $2S(x_{+})S(x_{-})$. Omitting the second beam assignment undercounts the pure sea contribution by two.