Kaon and pion parton distribution amplitudes to twist three

We compute all kaon and pion parton distribution amplitudes (PDAs) to twist three and find that only the pseudotensor PDA can reasonably be approximated by its conformal limit expression. At terrestrially accessible energy scales, the twist-two and pseudoscalar twist-three PDAs differ significantly from those functions commonly associated with their forms in QCD’s conformal limit. In all amplitudes studied, $SU(3)$ flavor-symmetry breaking is typically a 13% effect. This scale is determined by nonperturbative dynamics; namely, the current-quark-mass dependence of dynamical chiral symmetry breaking. The heavier quark is favored by this distortion; for example, support is shifted to the $s$-quark in the negative kaon. It appears, therefore, that at energy scales accessible with existing and foreseeable facilities, one may obtain reliable expectations for experimental outcomes by using these “strongly dressed” PDAs in formulas for hard exclusive processes. Following this procedure, any discrepancies between experiment and theory will be significantly smaller than those produced by using the conformal-limit PDAs. Moreover, the magnitude of any disagreement will either be a better estimate of higher-order, higher-twist effects or provide more realistic constraints on the Standard Model.

Figure 1

Functions characterizing the dressed quark propagator in the DB truncation. Upper panel: $u/d$-quark functions, ${\sigma }_{S,V}^{u/d}(p^2)$, and their solution (open circles and squares, respectively) and interpolation functions (solid and long-dashed curves, respectively). Lower panel: $s$-quark functions, ${\sigma }_{S,V}^s(p^2)$, with the same legend.

Figure 2

Two-particle twist-two PDAs, computed at ${\zeta }_2$: pion, Eq. (30) (dark blue dot-dashed curve); kaon, Eqs. (31), (32) (black solid curve); ${\phi }_{\pi }^{\text{model}}(u)$, Eq. (33) (dark green dashed curve); and asymptotic distribution, Eq. (5) (indigo dotted curve).

Figure 3

Pseudoscalar two-particle, twist-three PDAs, computed at ${\zeta }_2$. Upper panel: Pion, prediction in Eq. (35) (black solid curve); QCD sum rules estimate (dark green dot-dashed curve) [29]; and ${\omega }_P^{\text{asy}}$ in Eq. (26) (indigo dotted curve). Lower panel: Kaon, DB result (black solid curve); RL result (green dot-dashed curve); QCD sum rules estimate (red dashed curve) [29]; and pion prediction in Eq. (35) (blue dotted curve).

Figure 4

Pseudotensor two-particle, twist-three PDAs, computed at ${\zeta }_2$: pion, ${\upsilon }_{\pi }(u)$ in Eq. (30) (dark blue dot-dashed curve); kaon in DB truncation, ${\upsilon }_K^{\mathrm{DB}}(u)$ in Eqs. (31), (32) (black solid curve); kaon in RL truncation ${\upsilon }_K^{\mathrm{RL}}(u)$ in Eqs. (31), (32) (dark green dashed curve); and QCD sum rules result for the kaon from Ref. [29] (red long-dashed curve). We do not plot the asymptotic form in Eq. (25) because it is effectively indistinguishable from our prediction for the pion.