Production of doubly heavy-flavored hadrons at $e^{+}{e}^{-}$ colliders

Production of the doubly heavy-flavored hadrons ($B_c$ meson, doubly heavy baryons ${\mathrm{\Xi }}_{cc}$, ${\mathrm{\Xi }}_{bc}$, ${\mathrm{\Xi }}_{bb}$, their excited states, and antiparticles of them as well) at $e^{+}{e}^{-}$ colliders is investigated under two different approaches: LO (leading-order QCD complete calculation) and LL (leading-logarithm fragmentation calculation). The results for the production obtained by the LO and LL approaches, including the angle distributions of the produced hadrons with unpolarized and polarized incoming beams, the behaviors on the energy fraction of the produced doubly heavy-flavored hadron, and comparisons of results between the two approaches, are presented in tables and figures. Thus, characteristics of the production and uncertainties of the approaches are shown precisely, and it is concluded that only if the colliders run at the energies around the $Z$ pole (which may be called the $Z$ factories) and the luminosity of the colliders is as high as possible is the study of the doubly heavy-flavored hadrons completely accessible.

Figure 1

Feynman diagrams for the processes $e^{-}(q_1)+e^{+}(q_2)\to B_c(B_c^{*}\dots )(p_1)+b(p_2)+\overline{c}(p_3)$, while (a) and (b), being gauge invariant, are of fragmentation of $\overline{b}$-quark; (c) and (d), being gauge invariant, are of fragmentation of $c$-quark.

Figure 2

The differential cross sections $d\sigma /d\cos \theta$ for the production $e^{+}{e}^{-}\to B_c(B_c^{*},\dots )+b+\overline{c}$ under the complete QCD approach. In order to put all the results into the figure, the results of the $P$-wave meson production are multiplied by a factor of 10.

Figure 3

The differential cross sections $d\sigma /d\cos \theta$ for the production $e_R^{+}{e}_L^{-}\to B_c(B_c^{*},\dots )+b+\overline{c}$ (with polarized incoming beams) under complete QCD approach. In order to put all the results into the figure the results for $P$-wave states are multiplied by a factor ten.

Figure 4

The differential cross sections $d\sigma /d\cos \theta$ for the production $e_L^{+}{e}_R^{-}\to B_c(B_c^{*},\dots )+b+\overline{c}$ (with polarized incoming beams) under complete QCD approach. In order to put all the results into the figure, the results for $P$-wave states are multiplied by a factor of 10.

Figure 5

The differential cross sections $d\sigma /dz$ for the processes $e^{+}{e}^{-}\to B_c(B_c^{*},\dots )+b+\overline{c}$ by complete QCD approach, where $z$ is the energy fraction carried by the concerned meson. The results for the $P$-wave states are multiplied by a factor of 10.

Figure 6

The differential cross sections $d\sigma /dz$ for the processes $e^{+}{e}^{-}\to B_c(B_c^{*},\dots )+\cdots$ by $\overline{b}$-quark fragmentation under fragmentation approach, where $z$ is the energy fraction carried by the concerned meson. The results for the $P$-wave states are multiplied by a factor of 10.

Figure 7

The differential cross sections $d\sigma /dz$ for the processes $e^{+}{e}^{-}\to B_c(B_c^{*},\dots )+\cdots$ by $c$-quark fragmentation under fragmentation approach, where $z$ is the energy fraction carried by the concerned meson. The results for the $P$-wave states are multiplied by a factor of 10.

Figure 8

Differential cross sections $d\sigma /d\cos \theta$ for the production of the doubly heavy baryons under the complete QCD approach, where $\theta$ is the angle between the momenta of the final baryon and the initial electron. In order to show it in one figure, for ${\mathrm{\Xi }}_{bb}$, it was multiplied by a factor of 10.

Figure 9

Angle distributions of the polarized differential cross sections $d\sigma /d\cos \theta$ for the production of the doubly heavy baryons with $e_R^{+}{e}_L^{-}$ polarization states under the complete QCD approach, where $\theta$ is the angle between the momenta of the final baryon and the initial electron. In order to show it in one figure, for ${\mathrm{\Xi }}_{bb}$, it was multiplied by a factor of 10.

Figure 10

Angle distributions of the polarized differential cross sections $d\sigma /d\cos \theta$ for the production of the doubly heavy baryons with $e_L^{+}{e}_R^{-}$ polarization states under the complete QCD approach, where $\theta$ is the angle between the momenta of the final baryon and the initial electron. In order to show it in one figure, for ${\mathrm{\Xi }}_{bb}$, it was multiplied by a factor of 10.

Figure 11

The differential cross sections $d\sigma /dz$ in $pb$ for the production of the doubly heavy baryons ${\mathrm{\Xi }}_{cc}$, where $z$ is the energy fraction of the final baryon. The solid line represents the result obtained by complete QCD approach, dashed line represents that by fragmentation approach.

Figure 12

The differential cross sections $d\sigma /dz$ in $pb$ for the production of the doubly heavy baryons ${\mathrm{\Xi }}_{bc}$, where $z$ is the energy fraction of the final baryon. The solid and dashed lines represent the contributions from $|{(bc)}_{\overline{\mathbf{3}}},{{}^{1}S}_0⟩$ and $|{(bc)}_{\overline{\mathbf{3}}},{{}^{3}S}_1⟩$ obtained by complete QCD approach, dash-dot and dot lines represent the contributions from $|{(bc)}_{\overline{\mathbf{3}}},{{}^{1}S}_0⟩$ and $|{(bc)}_{\overline{\mathbf{3}}},{{}^{3}S}_1⟩$ obtained by fragmentation approach.

Figure 13

The differential cross sections $d\sigma /dz$ in $pb$ for the production of the doubly heavy baryons ${\mathrm{\Xi }}_{bb}$, where $z$ is the energy fraction of the final baryon. The solid line represents the result obtained by the complete QCD approach, the dashed line represents that by the fragmentation approach.