Pion and Kaon Fragmentation Functions from Continuum Schwinger Function Methods

Abstract

Using the Drell-Levy-Yan relation, the pion and kaon elementary fragmentation functions (EFFs) are obtained from their hadron-scale parton distribution functions (DFs). These EFFs serve as driving terms in the hadron cascade equations, whose solution yields the complete array of hadron-scale fragmentation functions (FFs) for pion and kaon production in high energy reactions. Evolved to experimental scales, the continuum Schwinger function methods (CSMs) predictions satisfy QCD endpoint behavior: nonsinglet FFs vanish at $z=0$, singlet FFs diverge faster than $1/z$. Jet multiplicity predictions reveal SU(3) symmetry breaking in the charged/neutral kaon ratio, decreasing with energy, and show the pion/kaon ratio in $e^+e^-$ collisions asymptotes to a mass-independent value.

Figures (4)

• Figure 1: Panel A. Dressed valence quark parton distribution functions evaluated using CSMs in Ref. Cui:2020tdf: ${\mathpzc s}_{K^-}(x ; \zeta_H)$ -- long-dashed red curve; ${\mathpzc u}_{K^+}(x ; \zeta_H)$ -- dot-dashed blue; ${\mathpzc u}_{\pi^+}(x ; \zeta_H)$ -- solid purple; scale-free DF -- dotted black. Panel B. Realistic elementary fragmentation functions, obtained from the $\pi, K$ curves in Panel A using Eq. \ref{['DLYR']}. $d_{\mathpzc s}^{K^-}(x ; \zeta_H)$ -- long-dashed red curve; $d_{\mathpzc u}^{K^+}(x ; \zeta_H)$ -- dot-dashed blue; $d_{\mathpzc u}^{\pi^++\pi^0}(x ; \zeta_H)$ -- solid purple.
• Figure 2: CSM results for kaon fragmentation functions, defined in Eqs. \ref{['SK']} -- \ref{['NKs']}. Solutions of cascade equations, Eq. \ref{['JetEq']} -- dashed purple curves. All-orders evolution of those curves to $\zeta=\zeta_2 := 2\,$GeV -- solid purple curves, with uncertainty bands obtained as described in Sect. (5.2) in Ref. Xing:2025eip. Comparison curves are inferences from: high-energy lepton-lepton, lepton-hadron and hadron-hadron scattering data Moffat:2021dji -- dotted brown curves, within like colored bands; and electron-positron annihilation and lepton-nucleon semi-inclusive deep-inelastic scattering data AbdulKhalek:2022laj -- dot-dashed blue curves within like-colored bands.
• Figure 3: $\zeta$-dependence of the charged/neutral kaon multiplicity ratio in Eq. \ref{['RKzeta0']}. CSM prediction -- solid purple curve; SCI prediction in Ref. Xing:2025eip -- dashed purple curve. Data are empirical estimates from Refs. BESIII:2025mbcTPCTwoGamma:1983lrvTPCTwoGamma:1984eojTASSO:1988jmaDELPHI:2000ahn.
• Figure 4: Hadron multiplicity structure function in Eq. \ref{['FhNorm']} at $\zeta = 10.5\,$GeV -- solid purple curves with associated uncertainty bands -- compared with data drawn from Ref. BaBar:2013yrg. Panel A. $h = \pi^\pm$. Panel B. $h = K^\pm$.