Relativistic corrections to double $B_c$ meson production in $e^+e^-$ annihilation

TL;DR

The paper analyzes relativistic corrections to double $B_c$ production in $e^+e^-$ annihilation within NRQCD factorization, covering energies from threshold to $2m_Z$. By computing both $ ext{γ}^*/Z^0$-propagated processes and including the $ ext{NLO}(v^2)$ expansion, it shows that relativistic corrections reduce the color-singlet cross sections by about $40\%$ (K $ oughly$ 0.6), a magnitude comparable to NLO QCD effects, while color-octet contributions are negligible. It provides cross sections, differential distributions in $ heta$ and $p_T$, and azimuthal asymmetries, and discusses theoretical uncertainties from LDMEs and quark masses. The findings imply that observing double $B_c$ production at future $e^+e^-$ colliders will be challenging, motivating exploration of alternative production modes and further refinement of relativistic corrections in the NRQCD framework.

Abstract

Within the framework of nonrelativistic QCD (NRQCD) factorization, we investigate the relativistic corrections to the production of double $B_c$ mesons in $e^+e^-$ annihilation. The study covers center-of-mass energies from the production threshold up to $2m_Z$, considering both the photon and $Z^0$-boson propagated processes. We find that the relativistic corrections are significant, with the corresponding $K$ factors of approximately 0.6. The azimuthal asymmetry, angular distribution, and transverse momentum distribution are also presented.

Figures (7)

• Figure 1: Feynman diagrams for $e^-(k_1)+e^+(k_2)\rightarrow \gamma^*/Z^0\rightarrow H_1(p_1)+H_2(p_2)$ at the tree level. $H_1, H_2$ are $B_c^+(c\bar{b})$ and $B_c^-(b\bar{c})$ (or their respective excited states, $B_c^{*\pm}, h_{bc}^{\pm}, \chi_{bcJ}^{\pm} (J=0,1,2)$), respectively. The permutation diagrams can be obtained by reversing the quark lines and exchanging $b,c$ quarks. (a) and (b) diagrams belong to QCD channels, (c) and (d) diagrams belong to EW channels.
• Figure 2: (Color online) Cross section $\sigma$ versus c.m. energy $\sqrt{s}~(E_{cm}=\sqrt{s})$. The red, green, black color lines correspond to $\gamma$-propagated processes, $Z$-propagated processes, and the sum of both, respectively. The solid line represents LO results, while the dashed line represents NLO($v^2$) results.
• Figure 3: (Color online) Cross section $\sigma$ versus c.m. energy $\sqrt{s}~(E_{cm}=\sqrt{s})$ for the s-channel $\gamma$-$g$ type process and t-channel process. The red and blue lines represent s-channel and t-channel processes, respectively. The solid and dashed lines represent LO and NLO($v^2$) results, respectively. The t-channel results are scaled up by a factor of 100.
• Figure 4: (Color online) $\sigma_{NLO(v^2)}/\sigma_{LO}$ versus c.m. energy. The left, middle, and right panels correspond to the production of double S-wave $B_c$ meson, $B_c^{*+}+X$, and $B_c^++X$, respectively, where $X$ denotes $h_{bc}^-$ or $\chi_{bcJ}^-(J=0,1,2)$.
• Figure 5: (Color online) Azimuthal asymmetry versus c.m. energy. The red solid line represents LO and blue dashed line represents NLO($v^2$) result. The vertical dashed line marks $\sqrt{s}=m_Z$. The asymmetry is exactly zero for $B_c^++B_c^-$ production.