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Photoproduction of the heavy quarkonium at the ILC

We study the photoproduction of the heavy quarkonium at the future International Linear Collider (ILC) within the nonrelativistic QCD theory. We focus on the production channel via the subprocess $γγ\to |[Q\bar{Q'}]_{\bf 1}(n)>+Q'+\bar{Q}$, where $Q$ and $Q'$ stand for heavy $c$- or $b$-quark, respectively. $|[Q\bar{Q'}]_{\bf 1}(n)>$ stands for color-singlet $S$-wave quarkonium, i.e., $η_{c}(|[c\bar{c}]_{\bf 1}(^1S_0)>)$, $J/ψ(|[c\bar{c}]_{\bf 1}(^3S_1)>)$, $B_{c}(|[c\bar{b}]_{\bf 1}(^1S_0)>)$, $B^*_{c}(|[c\bar{b}]_{\bf 1}(^3S_1)>)$, $η_{b}(|[b\bar{b}]_{\bf 1}(^1S_0)>)$, and $Υ(|[b\bar{b}]_{\bf 1}(^3S_1)>)$, respectively. To improve the calculation efficiency, we adopt the improved helicity amplitude approach to deal with the difficulty of calculating the expressions for the yields when the quark masses cannot be neglected. Total and differential photoproduction cross sections, together with their uncertainties, have been presented. It is noted that sizable amount of $|c\bar{c}>$-charmonium and $|c\bar{b}>$-quarkonium events can be generated at the ILC. More specifically, we predict $(2.8^{+1.0}_{-0.7})\times 10^{6}$ $η_c$, $(5.4^{+1.9}_{-1.3})\times 10^{6}$ $J/ψ$, $(8.3^{+2.2}_{-1.8})\times 10^{4}$ $B_c$, $(4.3^{+1.1}_{-0.9})\times 10^{5}$ $B_c^*$, $(9.0^{+1.7}_{-1.4})\times 10^{3}$ $η_b$, and $(1.6\pm0.3)\times 10^{4}$ $Υ$ events to be generated in one operation year at the ILC under the condition of $\sqrt{S}=500$ GeV and ${\cal L}\simeq 10^{36}$cm$^{-2}$s$^{-1}$.