Kaon and Pion Fragmentation Functions

TL;DR

This paper develops a unified framework to predict pion and kaon fragmentation functions from hadron-scale parton distributions using the DLY relation, connecting nonperturbative hadron structure to hadronisation via elementary fragmentation functions. By employing two DF inputs—SCI and realistic CSMs—it derives EFFs, feeds them into hadron jet cascade equations, and then evolves the complete FFs to experimental scales with an all-orders AO scheme that preserves momentum through a gluon-channel treatment. The study finds semiquantitative agreement between SCI and CSM FFs and demonstrates that the endpoint behaviour ($z\to 0,1$) is consistent with QCD expectations, while comparing predictions for hadron multiplicities and kaon/pion SU(3) breaking against data and phenomenological fits. The results offer a parameter-free, theoretically grounded benchmark for fragmentation that can guide future data analyses and extensions to other hadrons, including protons, and heavy-quark sectors, illuminating confinement-related phenomena via emergent hadron mass.

Abstract

The Drell-Levy-Yan relation is employed to obtain pion and kaon elementary fragmentation functions (EFFs) from the hadron-scale parton distribution functions (DFs) of these mesons. Two different DF sets are used: that calculated using a symmetry-preserving treatment of a vector $\times$ vector contact interaction (SCI) and the other expressing results obtained using continuum Schwinger function methods (CSMs). Thus determined, the EFFs serve as driving terms in a coupled set of hadron cascade equations, whose solution yields the complete array of hadron-scale fragmentation functions (FFs) for pion and kaon production in high energy reactions. After evolution to scales typical of experiments, the SCI and CSM FF predictions are seen to be in semiquantitative agreement. Importantly, they conform with a range of physical expectations for FF behaviour on the endpoint domains $z\simeq 0, 1$, e.g., nonsinglet FFs vanish at $z=0$ and singlet FFs diverge faster than $1/z$. Predictions for hadron multiplicities in jets are also delivered. They reveal SU$(3)$ symmetry breaking in the charged-kaon/neutral-kaon multiplicity ratio, whose size diminishes with increasing reaction energy, and show that, with increasing energy, the pion/kaon ratio in $e^+ e^- \to h X$ diminishes to a value that is independent of hadron masses.

Figures (10)

• Figure 1: Panel A. SCI valence quark parton distribution functions, obtained using Eqs. \ref{['pidfexp']}, \ref{['SCIuh']}, \ref{['sKdfexp']}, and the results listed in Table \ref{['Tab:DressedQuarks']}: ${\mathpzc s}_{K^-}(x ; \zeta_H)$ -- long-dashed red curve; ${\mathpzc u}_{K^+}(x ; \zeta_H)$ -- dot-dashed blue ; ${\mathpzc u}_{\pi^+}(x ; \zeta_H)$ -- solid purple; ${\mathpzc u}_{\pi^+}(x ; \zeta_H)$ in chiral limit ($h=0$) -- dashed green. Panel B. SCI elementary fragmentation functions, obtained from the results 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; $d_{\mathpzc u}^{\pi}(x ; \zeta_H)$ in chiral limit -- dashed green.
• Figure 2: SCI results for pion fragmentation functions, defined in Eqs. \ref{['Spi']}, \ref{['Npi']}. Solutions of cascade equations, Eq. \ref{['JetExplicit']} -- dashed purple curves. AO evolution of those curves to $\zeta=\zeta_2 := 2\,$GeV -- solid purple curves, with uncertainty bands obtained as described in Sect. \ref{['HSuncertainty']}. Comparison curves are inferences from: high-energy lepton-lepton, lepton-hadron and hadron-hadron scattering data Moffat:2021dji -- dotted brown curves, within like coloured bands; and electron-positron annihilation and lepton-nucleon semi-inclusive deep-inelastic scattering data AbdulKhalek:2022laj -- dot-dashed blue curves within like-coloured bands.
• Figure 3: SCI results for kaon fragmentation functions, defined in Eqs. \ref{['SK']} -- \ref{['NKs']}. Solutions of cascade equations, Eq. \ref{['JetExplicit']} -- dashed purple curves. AO evolution of those curves to $\zeta=\zeta_2 := 2\,$GeV -- solid purple curves, with uncertainty bands obtained as described in Sect. \ref{['HSuncertainty']}. Comparison curves are inferences from: high-energy lepton-lepton, lepton-hadron and hadron-hadron scattering data Moffat:2021dji -- dotted brown curves, within like coloured bands; and electron-positron annihilation and lepton-nucleon semi-inclusive deep-inelastic scattering data AbdulKhalek:2022laj -- dot-dashed blue curves within like-coloured bands.
• Figure 4: 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 in Eq. \ref{['ScaleFree']} -- dotted black. Panel B. Realistic elementary fragmentation functions, obtained from the $\pi, K$ curves in Panel A using Eqs. \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 5: CSM results for pion fragmentation functions, defined in Eqs. \ref{['Spi']}, \ref{['Npi']}. Solutions of cascade equations, Eq. \ref{['JetExplicit']} -- dashed purple curves. AO evolution of those curves to $\zeta=\zeta_2 := 2\,$GeV -- solid purple curves, with uncertainty bands obtained as described in Sect. \ref{['HSuncertainty']}. Comparison curves are inferences from: high-energy lepton-lepton, lepton-hadron and hadron-hadron scattering data Moffat:2021dji -- dotted brown curves, within like coloured bands; and electron-positron annihilation and lepton-nucleon semi-inclusive deep-inelastic scattering data AbdulKhalek:2022laj -- dot-dashed blue curves within like-coloured bands.