$\overline{d}-\overline{u}$ Asymmetry in the Proton in Chiral Effective Theory

We compute the $\overline{d}-\overline{u}$ asymmetry in the proton in chiral effective theory, including both nucleon and $\mathrm{\Delta }$ degrees of freedom, within the relativistic and heavy baryon frameworks. In addition to the distribution at $x>0$, we compute the corrections to the asymmetry 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 $\overline{d}-\overline{u}$ as well as the integrated asymmetry can be well reproduced in terms of a transverse momentum cutoff parameter.

Figure 1

Contributions to the pion light-cone momentum distributions in the proton, from the pion rainbow diagram with (a) a nucleon or (b) $\mathrm{\Delta }$ intermediate state, and (c) from the pion bubble diagram.

Figure 2

Contributions to $\overline{D}-\overline{U}$ from the diagrams in Fig. 1 as a function of the cutoff parameters $\mu$ and $\mathrm{\Lambda }$. The nucleon on-shell and $\delta$-function, and $\mathrm{\Delta }$ on-shell, end-point, and $\delta$-function contributions are shown individually, together with the pion bubble contribution and the total. The calculations are compared with data from the NMC [1], HERMES [2], and E866 [4] experiments, and from a recent lattice QCD calculation [29].

Figure 3

Flavor asymmetry $\overline{d}-\overline{u}$ from the $N$ and $\mathrm{\Delta }$ intermediate states, and the total, for cutoffs $\mu =0.3$ and $\mathrm{\Lambda }=0.2\mathrm{GeV}$, compared with the asymmetry extracted at leading order from the E866 Drell-Yan data [4] at $Q^2=54{\mathrm{GeV}}^2$. The band indicates the uncertainty on the total distribution from the cutoff parameters (for $\mu$ between 0.1 and 1 GeV) and from the empirical $\overline{D}-\overline{U}$ normalization.