d¯−u¯Asymmetry in the Proton in Chiral Effective Theory

Abstract

We compute thed −ū asymmetry in the proton in chiral effective theory, including both nucleon and ∆ 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 ofd −ū as well as the integrated asymmetry can be well reproduced in terms of a transverse momentum cutoff parameter.The observation of thed −ū flavor asymmetry in the light quark sea of the proton [1][2][3][4] has been one of the seminal results in hadronic physics over the past two decades, leading to a major reevaluation of our understanding of the quark structure of the nucleon. In particular, the measurement revealed the importance of 5-quark Fock state components of the nucleon's wave function, and the crucial role played by chiral symmetry breaking. This asymmetry had been anticipated by Thomas [5] a decade earlier, and has subsequently been studied using various nonperturbative models [6][7][8][9][10]. However, despite some successes in reproducing the general features of the data, it has proved very difficult to obtain direct connection between the models and QCD.An important development in establishing a formal link between models ofd −ū and QCD came with the realization that the moments of parton distribution functions (PDFs) could be formally expanded in chiral effective field theory in terms of power series in the pion mass squared, m 2 π . The leading nonanalytic (LNA) contributions were found to depend on the (model-independent) long range structure of the pion cloud, with a characteristic m 2 π log m 2 π dependence [11,12]. This idea was later applied to the chiral extrapolation of lattice QCD moments of the u − d distribution to reconcile the lattice data at large m 2 π with experiment [13]. Initial calculations of pion loop effects were performed in the context of the "Sullivan" process [14], using pseudoscalar pion-nucleon coupling, which involves only the pion "rainbow" diagram. Analysis within the chiral effective theory for the pseudovector coupling reveals differences in the off-shell behavior of the loops [15], as well as the presence of additional pion bubble terms [16,17] at x = 0 [18]. The relationship between the pseudoscalar and pseudovector theories was recently discussed in Refs. [18,19].While the structure of the pion loops constrains the behavior of the PDF moments at small m 2 π , the total moments depend also on the short-distance contributions, parametrized by coefficients of analytic terms in the chiral expansion. In principle, these can be fitted to data, and the PDFs reconstructed from the moments assuming a functional form for the dependence on the parton momentum fraction x [20]. Recently, however, a method for computing the PDFs directly in x space has been developed, by matching nonlocal operators within heavy baryon (HB) chiral effective th...

Strange-quark asymmetry in the proton in chiral effective theory

“…In general we follow the notations introduced for the pion loop corrections in Refs. [40,41,52,53], with obvious extensions.…”

“…In the literature various prescriptions have been adopted, ranging from dimensional reg-ularization in traditional chiral perturbation theory [56] to sharp cutoffs in k ⊥ [40] or form factors in more phenomenological approaches [57]. Regularization with the help of a finiterange regulator has been advocated [58][59][60] as a practical method which reflects the finite size of the baryon to which the chiral field couples.…”

“…This methodology was applied by Salamu et al [40] to the case of pion loops and their effects on thed −ū asymmetry in the proton, using for illustration a simple sharp cutoff on the transverse momentum of the pion k ⊥ for the ultraviolet regulator. More recently Wang et al [41] generalized this approach to the SU(3) sector, using Pauli-Villars (PV) regularization to compute the various lowest order diagrams in the chiral SU(3) expansion, and obtain a range for S − consistent with available phenomenological constraints.…”

Pion structure function and SU(2) flavor asymmetry

“…1 (b) are known to play an important role in hadron structure. Within the same chiral effective theory framework, using an effective πN∆ interaction [14], the p → ∆ 0 π + splitting function can be written as a sum of three terms,…”

“…1 (a), the p → nπ + splitting function derived from chiral effective theory is expressed as a sum of on-shell and δ -function pieces [13,14],…”