Fragmentation Functions of neutral mesons $π^0$ and $k ^0$ with Laplace transform approach

With an analytical solutions of DGLAP evolution equations based on the Laplace transform method , we find the fragmentation functions (FFs) of neutral mesons, $π^0$ and $k ^0$ at NLO approximation. We also calculated the total fragmentation functions of these mesons and compared them with experimental data and those from global fits. The results show a good agreements between our solutions and other models and also are compatible with experimental data.

💡 Research Summary

This paper presents a novel analytical approach to calculating the fragmentation functions (FFs) of neutral mesons, specifically the π⁰ and K⁰, within the framework of Quantum Chromodynamics (QCD). The core achievement is the application of the Laplace transform method to solve the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations at the Next-to-Leading Order (NLO) approximation, providing closed-form solutions for parton-to-meson fragmentation probabilities.

The methodology hinges on a technique introduced by Block et al. It involves a clever change of variables from the standard momentum fraction z and energy scale Q² to ν = ln(1/z) and an integrated coupling constant τ. Subsequently, successive Laplace transforms convert the intricate integro-differential DGLAP equations into more tractable algebraic equations in complex ’s’ and ‘U’ spaces. After solving these algebraic equations, the desired FFs in the physical (z, Q²) space are retrieved via inverse Laplace transforms. The paper meticulously outlines this procedure for the non-singlet, singlet, and gluon sectors, with explicit formulas for the Laplace-transformed NLO splitting functions provided in an appendix.

To validate their analytical framework rather than perform a new global fit, the authors adopt the well-established HKNS parameterization for the initial FFs at a starting scale Q₀ = 4.5 GeV. They first compute the FFs for the charged mesons π⁺ and K⁺ using their Laplace method. Leveraging charge conjugation symmetry (which relates particle and antiparticle fragmentation, e.g., D_{π⁺}^{ū} = D_{π⁻}^{u}), they then derive the FFs for π⁻ and K⁻. The FFs for the neutral mesons are finally constructed as the average of their charged counterparts: D_{π⁰} = (D_{π⁺} + D_{π⁻})/2 and similarly for K⁰. Contributions from heavier sea quarks (strange, charm, bottom) are estimated using a simple parameterization based on mass ratios.

The results demonstrate excellent agreement. Figures 1 and 2 show that the extracted parton-level FFs for π⁰ and K⁰ (for up quarks, gluons, etc.) align remarkably well with those from major global fit collaborations like AKK, DSS, and HKNS. Furthermore, the authors compute the “total” fragmentation function, which is directly comparable to experimental data from e⁺e⁻ annihilation experiments. As shown in Figures 3 and 4, their predictions for the π⁰ and K⁰ total FFs agree satisfactorily with measured data from the TASSO, ALEPH, TOPAZ, and OPAL collaborations across a range of energy scales.

In conclusion, this work successfully implements an analytical Laplace transform solution for the QCD evolution of fragmentation functions. The strong consistency between their results, established global fits, and experimental data robustly validates the proposed method. This technique offers a rigorous, algebraically controlled alternative to purely numerical evolution codes and provides a solid foundation for incorporating neutral meson data into future global analyses of fragmentation functions.