$\bar{d}-\bar{u}$ asymmetry in the proton in chiral effective theory

We compute the $\bar{d}-\bar{u}$ asymmetry in the proton in chiral effective theory, including both nucleon and $\mathrm{\Delta }$ degrees of freedom, within both the relativistic and heavy baryon frameworks. In addition to the distribution at $x>0$, we compute the corrections to the asymmetry arising from zero momentum contributions from pion rainbow and bubble diagrams at $x=0$, which have not been accounted for in previous analyses. We find that the empirical $x$ dependence of $\bar{d}-\bar{u}$ as well as the integrated asymmetry can be well reproduced in terms of a transverse momentum cutoff parameter.