Constraints on $s-\bar s$ asymmetry of the proton in chiral effective theory

We compute the $s-\bar s$ asymmetry in the proton in chiral effective theory, using phenomenological constraints based upon existing data. Unlike previous meson cloud model calculations, which accounted for kaon loop contributions with on-shell intermediate states alone, this work includes off-shell terms and contact interactions, which impact the shape of the $s-\bar s$ difference. We identify a valence-like component of $s(x)$ which is balanced by a $\delta$-function contribution to $\bar s(x)$ at $x=0$, so that the integrals of $s$ and $\bar s$ over the experimentally accessible region $x > 0$ are not equal. Using a regularization procedure that preserves chiral symmetry and Lorentz invariance, we find that existing data limit the integrated value of the second moment of the asymmetry to the range $-0.07 \times 10^{-3} \leq \langle x(s-\bar s) \rangle \leq 1.12 \times 10^{-3}$ at a scale of $Q^2=1 $GeV$^2$. This is too small to account for the NuTeV anomaly and of the wrong sign to enhance it.