Fragmentation functions of the pion, kaon, and proton in the NLO approximation: Laplace transform approach

Using repeated Laplace transform, We find an analytical solution for DGLAP evolution equations for extracting the pion, kaon and proton Fragmentation Functions (FFs) at NLO approximation. We also study the symmetry breaking of the sea quarks Fragmentation Functions, $D_{\bar q}^h (z,Q^2)$ and simply separated them according to their mass ratio. Finally, we calculate the total Fragmentation Functions of these hadrons and compare them with experimental data and those from global fits. Our results show a good agreement with the FFs obtained from global parameterizations as well as with the experimental data.

💡 Research Summary

The paper presents a novel analytical approach to solving the Dokshitzer‑Gribov‑Lipatov‑Altarelli‑Parisi (DGLAP) evolution equations for fragmentation functions (FFs) at next‑to‑leading order (NLO) using repeated Laplace transforms. Starting from the standard DGLAP equations for non‑singlet, singlet, and gluon sectors, the authors introduce the variables ν = ln(1/z) and τ = ∫α_s(Q′²)dlnQ′²/(4π) to recast the integro‑differential equations into ordinary convolution forms. A first Laplace transform from ν‑space to s‑space decouples the convolutions, yielding simple exponential solutions for the non‑singlet FFs: f_ns(s,τ) = e^{τΦ_ns(s)} f_ns⁰(s), where Φ_ns(s) contains the Laplace‑transformed LO and NLO splitting functions. For the coupled singlet‑gluon system, a second Laplace transform from τ‑space to U‑space converts the equations into linear algebraic relations. The strong coupling α_s(τ) is approximated by a simple exponential series α_s(τ)/(4π) ≈ a₀ + a₁ e^{‑b₁τ}, allowing the authors to express the transformed terms as shifted functions in U. The resulting algebraic system (Eqs. 30‑31) is solved iteratively, producing the singlet and gluon moments f(s,τ) and g(s,τ). Inverse Laplace transforms then return the solutions to the physical (z, Q²) space.

Initial conditions are taken from the HKNS global fit at Q₀² = 4.5 GeV², ensuring that the analytical method can be benchmarked against established parametrizations. The authors compute the full set of FFs for the pion, kaon, and proton, including non‑singlet, singlet, and gluon components, and compare the results with the AKK, DSS, and HKNS global analyses. Across the full z range, the analytical FFs agree closely with these fits, confirming the correctness of the method.

A distinctive contribution of the work is a simple model for sea‑quark symmetry breaking. The total sea‑quark FF D̄_q(z,Q²) is first obtained as the difference between the singlet and non‑singlet FFs. The authors then assume that each sea‑quark flavor contributes proportionally to its mass ratio relative to a reference quark (e.g., D̄_u/D_c ≈ m_u/m_c). Introducing a universal mass‑ratio factor B and a normalization constant A (extracted from HKNS), they write D_q(z,Q²) = B·A·D̄_q(z,Q²) for each flavor. Table 1 lists the fitted A and B values, and the resulting flavor‑separated sea‑quark FFs are incorporated into the total hadron FFs.

Finally, the total fragmentation functions F_H(z,Q²) = ∑_q e_q²