First determination of $D^{*}$-meson fragmentation functions and their uncertainties at next-to-next-to-leading order

We present, for the first time, a set of next-to-next-to-leading order (NNLO) fragmentation functions (FFs) describing the production of charmed-meson $D^{*}$ from partons. Exploiting the universality and scaling violations of FFs, we extract the NLO and NNLO FFs through a global fit to all relevant data sets from single-inclusive $e^{+}{e}^{-}$ annihilation. The uncertainties for the resulting FFs as well as the corresponding observables are estimated using the Hessian approach. We evaluate the quality of the SKM18 FFs determined in this analysis by comparing with the recent results in literature and show how they describe the available data for single-inclusive $D^{*\pm }$-meson production in electron-positron annihilation. As a practical application, we apply the extracted FFs to make our theoretical predictions for the scaled-energy distributions of $D^{*\pm }$-mesons inclusively produced in top quark decays. We explore the implications of SKM18 for LHC phenomenology and show that our findings of this study can be introduced as a channel to indirect search for top-quark properties.

Figure 1

SKM18 fragmentation densities and their uncertainties (shaded bands) are shown for $z{D}_i^{D^{*\pm }}$ at the initial scale ${\mu }_0^2=18.5{\mathrm{GeV}}^2$ for $c$ and $b$ both at NLO (solid lines) and NNLO (dashed lines).

Figure 2

Fragmentation densities and their uncertainties (shaded bands) are shown for $z{D}_i^{D^{*\pm }}$ at ${\mu }^2=100{\mathrm{GeV}}^2$ for $c$, $b$ and gluon both at NLO (solid lines) and NNLO (dashed lines). Our results are also compared with the KKKS08 (dot-dashed lines) [32] and the AKSRV17 (short dashed lines) [11] results at NLO.

Figure 3

Fragmentation densities and their uncertainties (shaded bands) are shown for $z{D}_i^{D^{*\pm }}$ at ${\mu }^2=M_Z^2$ for $c$, $b$ and gluon at NLO (solid lines) and NNLO (dashed lines). Our results are also compared with the KKKS08 (dot-dashed lines) [32] and the AKSRV17 (short dashed lines) [11] results at NLO.

Figure 4

The NNLO/NLO ratios for SKM18 fragmentation densities and their uncertainties are shown for $z{D}_i^{D^{*\pm }}$ at ${\mu }^2=M_Z^2$ for $c$, $b$ and gluon at NLO (dashed lines) and NNLO (dot-dashed lines).

Figure 5

Our NLO (solid line) and NNLO (dashed line) results for the normalized total cross sections of $D^{*\pm }$-production compared with the KKKS08 ones [32] (dot-dashed line) at the scale $Q=M_Z$. Data from ALEPH [34] and OPAL [35] are also shown in this scale. The shaded bands refer to our uncertainty results at NLO (green band) and NNLO (yellow band).

Figure 6

Our NLO (solid line) and NNLO (dashed line) results for the normalized charm and bottom tagged cross sections of $D^{*\pm }$-production compared with the KKKS08 ones [32] (dot-dashed line) at the scale $Q=M_Z$. Data from ALEPH [34] and OPAL [35] are also shown in this scale. The shaded bands refer to our uncertainty results at NLO (green band) and NNLO (yellow band).

Figure 7

Fragmentation densities $z{D}_i^{D^{*\pm }}(z,{\mu }^2)$ are shown at ${\mu }^2=Q^2=25{\mathrm{GeV}}^2$ for the gluon and the charm and bottom quarks both at NLO (dashed lines) and NNLO (dot-dashed lines). As a common convention, here we set ${\mu }_F={\mu }_R=\mu$. The shaded bands indicate theoretical uncertainties when the scale is ranged as $Q/2\le \mu \le 2Q$.

Figure 8

$x_D$ distribution of $d{\mathrm{\Gamma }}^{\mathbf{NLO}}/d{x}_D$ for the polarized top quark decay at ${\mu }_F=m_t$. We also show the same distribution using the FFs presented by KKKS08 [32] at NLO.

Figure 9

$d\mathrm{\Gamma }(t\to D^{*\pm }+Jets)/d{x}_D$ as a function of $x_D$ at NLO for the polarized (dashed) and unpolarized (dots) top decays. We also show the same results at NNLO.