First Monte Carlo analysis of fragmentation functions from single-inclusive $e^{+}{e}^{-}$ annihilation

We perform the first iterative Monte Carlo (IMC) analysis of fragmentation functions constrained by all available data from single-inclusive $e^{+}{e}^{-}$ annihilation into pions and kaons. The IMC method eliminates potential bias in traditional analyses based on single fits introduced by fixing parameters not well constrained by the data and provides a statistically rigorous determination of uncertainties. Our analysis reveals specific features of fragmentation functions using the new IMC methodology and those obtained from previous analyses, especially for light quarks and for strange quark fragmentation to kaons.

Figure 1

Workflow of the iterative Monte Carlo fitting strategy. In the upper diagram (red lines) an iteration begins at the prior sampler and a given number of fits are performed, generating an ensemble of posteriors. After the initial iteration, with a flat sampler, the generated posteriors are used to construct a multivariate Gaussian sampler for the next iteration. The lower diagram (with blue lines) summarizes the workflow that transforms a given prior into a final posterior.

Figure 2

Normalized IMC volume versus number of iterations for pions (red lines) and kaons (blue lines). The approximate convergence of the volumes is indicated by the colored regions.

Figure 3

Fragmentation functions computed from 100 (pink), 200 (black), 500 (green), $10^3$ (yellow) and $10^4$ (red for pions, blue for kaons) fits, normalized to the latter. The uncertainties for the 200 (black shaded) and $10^4$ results are indicated by the bands.

Figure 4

Normalized yield of IMC fits versus ${\chi }^2/N_{\mathrm{dat}}$ for the training (blue forward hashed), validation (green backward hashed), and combined (red dotted) samples for $\pi$ (left panel) and $K$ production (right panel).

Figure 5

Ratio of experimental single-inclusive $e^{+}{e}^{-}$ cross sections to the fitted values versus $z$ (or $z_{\min }$ for OPAL data [32, 33]) for pion production. The experimental uncertainties are indicated by the black points, with the fitted uncertainties denoted by the red bands. For the BABAR data [40] the prompt data set is used.

Figure 6

As in Fig. 5, but for kaon production.

Figure 7

Fragmentation functions for $u^{+}$, $d^{+}$, $s^{+}$, $c^{+}$, $b^{+}$ and $g$ into ${\pi }^{+}$ (red bands) and $K^{+}$ (blue bands) mesons as a function of $z$ at the input scale ($Q^2=1{\mathrm{GeV}}^2$ for light-quark flavors and gluon, $Q^2=m_q^2$ for the heavy quarks $q=c$ and $b$). A random sample of 100 posteriors (yellow curves for ${\pi }^{+}$, green for $K^{+}$) is shown together with the mean and variance (red and blue bands).

Figure 8

Iterative convergence of the ${\pi }^{+}$ (red bands) and $K^{+}$ (blue bands) fragmentation functions for the $u^{+}$, $d^{+}$, $s^{+}$, $c^{+}$, $b^{+}$ and $g$ flavors (in individual columns) at the input scale. The first row shows the initial flat priors (single yellow curves for ${\pi }^{+}$ and green curves for $K^{+}$) and their corresponding posteriors (error bands). The second and third rows show selected intermediate snapshots of the IMC chain, and the last row shows the priors and posteriors of the final IMC iteration.

Figure 9

Evolution of the $u^{+}$, $d^{+}$, $s^{+}$, $c^{+}$, $b^{+}$ and $g$ fragmentation functions to ${\pi }^{+}$ (red curves) and $K^{+}$ (blue curves) with the scale, from the input scale $Q^2=1{\mathrm{GeV}}^2$ (solid) to $Q^2=10{\mathrm{GeV}}^2$ (dot-dashed), $Q^2=100{\mathrm{GeV}}^2$ (dashed) and $Q^2=M_Z^2$ (dotted).

Figure 10

Comparison of the JAM fragmentation functions (solid curves) for ${\pi }^{+}$ (red curves) and $K^{+}$ (blue curves) with the HKNS [12] (dashed curves) and DSS [10] (dotted curves) parametrizations at the input scale $Q^2=1{\mathrm{GeV}}^2$ for the light-quark and gluon distributions, and $Q^2=10$ and $20{\mathrm{GeV}}^2$ for the $c^{+}$ and $b^{+}$ flavors, respectively.

Figure 11

Comparison of the JAM fragmentation functions (solid curves) for ${\pi }^{+}$ (red curves) and $K^{+}$ (blue curves) with the HKNS [12] (dashed curves), DSS [10] (dotted curves) and AKK [16] (dot-dashed curves) evolved to a common scale $Q^2=M_Z^2$. Note that the fragmentation functions $D(z)$ are shown rather than $zD(z)$.