Distinguishing Quarks and Gluons in Pion and Kaon Parton Distribution Functions

The leading-twist parton distribution functions of the pion and kaon are calculated for the first time using a rainbow-ladder truncation of QCD’s Dyson-Schwinger equations (DSEs) that self-consistently sums all planar diagrams. The nonperturbative gluon dressing of the quarks is thereby correctly accounted for, which in practice means solving the inhomogeneous Bethe-Salpeter equation (BSE) for the quark operator that defines the spin-independent quark distribution functions. An immediate consequence of using this dressed vertex is that gluons carry 35% of the pion’s and 30% of the kaon’s light-cone momentum, with the remaining momentum carried by the quarks. The scale associated with these DSE results is ${\mu }_0=0.78\mathrm{GeV}$. The gluon effects generated by the inhomogeneous BSE are inherently nonperturbative and cannot be mimicked by the perturbative QCD evolution equations. A key consequence of this gluon dressing is that the valence quarks have reduced support at low-to-intermediate $x$, where the gluons dominate, and increased support at large $x$. As a result, our DSE calculation of the pion’s valence quark distribution is in excellent agreement with the Conway et al. pion-induced Drell-Yan data, but nevertheless exhibits the $q_{\pi }(x)\simeq (1-x{)}^2$ behavior as $x\to 1$ predicted by perturbative QCD.

Figure 1

A representation of Eq. (2) for the quark distribution in a meson. The infinite sum of ladder gluon exchanges which dress the elementary vertex $\otimes =i{Z}_2\cancel{n}\delta (x-k·n/P·n)$ depicts the solution for the vertex ${\mathrm{\Gamma }}_q(x,p,n)$ defined in Eq. (3).

Figure 2

Results for the amplitudes $f_i^q(z,p^2)$ of the dressed quark vertex from Eq. (6). The active quark has momentum fraction $z$ relative to the parent dressed quark of virtuality $p^2$. The colored surfaces represent results for $u/d$ quarks, while the gray surface depicts $s$ quark results relevant to the kaon.

Figure 3

Top panel: Solid red band is our DSE result for the pion’s valence quark distribution [$q^{-}(x)=q(x)-\overline{q}(x)$]; blue band is the JAM analysis [31]; dashed-dotted curve is the DSE result from Ref. [26] which uses the WIA; the dashed curve is the NLO soft-gluon-resummation analysis from ASV [22]; and the data points are from Ref. [19]. All curves are at the scale $Q=5.2\mathrm{GeV}$ and, with the exception of JAM, have a $x\to 1$ exponent consistent with theory expectations. Bottom panel: Contrast between our DSE results for the pion and kaon PDFs, and $u_K(x)/u_{\pi }(x)$ which is compared to data from Ref. [46].