Valence-quark distribution functions in the kaon and pion

We describe expressions for pion and kaon dressed-quark distribution functions that incorporate contributions from gluons which bind quarks into these mesons and hence overcome a flaw of the commonly used handbag approximation. The distributions therewith obtained are purely valence in character, ensuring that dressed quarks carry all the meson’s momentum at a characteristic hadronic scale and vanish as $(1-x{)}^2$ when Bjorken-$x\to 1$. Comparing such distributions within the pion and kaon, it is apparent that the size of $SU(3)$-flavor symmetry breaking in meson parton distribution functions is modulated by the flavor dependence of dynamical chiral symmetry breaking. Corrections to these leading-order formulas may be divided into two classes, responsible for shifting dressed-quark momentum into glue and sea quarks. Working with available empirical information, we build an algebraic framework that is capable of expressing the principal impact of both classes of corrections. This enables a realistic comparison with experiment which allows us to identify and highlight basic features of measurable pion and kaon valence-quark distributions. We find that whereas roughly two thirds of the pion’s light-front momentum is carried by valence dressed quarks at a characteristic hadronic scale; this fraction rises to 95% in the kaon; evolving distributions with these features to a scale typical of available Drell-Yan data produces a kaon-to-pion ratio of $u$-quark distributions that is in agreement with the single existing data set, and predicts a $u$-quark distribution within the pion that agrees with a modern reappraisal of $\pi N$ Drell-Yan data. Precise new data are essential in order to validate this reappraisal and because a single modest-quality measurement of the kaon-to-pion ratio cannot be considered definitive.

Figure 1

Valence-quark PDFs at the hadronic scale, ${\zeta }_H$, defined by Eqs. (14) and (17): $u_V^K(x)$, solid (black) curve; ${\overline{s}}_V^K(x)$, dashed (blue) curve; $u_V^{\pi }(x)$, dot-dashed (green) curve; and $[u_V^K(x)+{\overline{s}}_V^K(x)]/2$ dotted (red) curve.

Figure 2

$u^K(x)/u^{\pi }(x)$ at $\zeta =5.2\mathrm{GeV}$: solid (black) curve, obtained via LO evolution from ${\zeta }_H=0.51\mathrm{GeV}$ assuming 5% of the kaon’s momentum is carried by glue at this hadronic scale; dashed (green) curve, zero momentum carried by gluons; and dot-dashed (blue) curve, 10% of the kaon’s momentum carried by glue. For comparison, an analysis of $\pi N$ Drell-Yan data suggests that 29% of the pion’s momentum is carried by glue at ${\zeta }_H$, as explained in connection with Eq. (28). The long-dashed (purple) curve is the DSE prediction in Ref. [24], obtained using numerical solutions of realistic gap and Bethe-Salpeter equations. (Data in this figure are from Ref. [5]. The dotted (red) line marks a value of unity for the ratio. It is drawn to highlight the domain upon which one might be confident empirically that $u^K(x)/u^{\pi }(x)\ne 1$, viz. $x\gtrsim 0.8$.)

Figure 3

$x{u}^{\pi }(x;{\zeta }_{5.2})$. Solid (black) curve, our prediction, expressed in Eqs. (32)–(33); dot-dot-dashed (purple) curve, result obtained when sea-quark and gluon contributions are neglected at ${\zeta }_H$, i.e. using $u_V^{\pi }(x)$ from Eqs. (14) and (17); dashed (blue) curve, first DSE prediction [38]; and data, Ref. [4], are rescaled according to the reanalysis described in Ref. [40], from which the dot-dashed (green) curve is drawn. The dotted (red) curve is the result obtained using a Poincaré-covariant regularization of a contact interaction, Eq. (36).