Heavy quarkonium wave functions at the origin and excited heavy quarkonium production via top quark decays at the LHC

The value of the quarkonium wave function at the origin is an important quantity for studying many physical problems concerning a heavy quarkonium. This is because it is widely used to evaluate the production and decay amplitudes of the heavy quarkonium within the effective field theory framework, e.g., the nonrelativistic QCD (NRQCD). In this paper, the values of the Schrödinger radial wave function or its first nonvanishing derivative at zero quark-antiquark separation, i.e., $|(|c\overline{c})[n]⟩$, $|(|b\overline{c})[n]⟩$, and $|(b\overline{b})[n]⟩$ quarkonium, have been tabulated under five potential models with new parameters for the heavy quarkonium. Moreover, the production of the lower-level Fock states $|(b\overline{Q})[1S]⟩$ and $|(b\overline{Q})[1P]⟩$, together with the higher excited Fock states $|(b\overline{Q})[nS]⟩$ and $|(b\overline{Q})[nP]⟩$ ($Q$ stands for the $c$ or $b$ quark; $n=2,\dots ,6$) through top quark decays has been studied with the new values of heavy quarkonium wave functions at the origin under the framework of NRQCD. At the LHC with the luminosity $\mathcal{L}\propto 10^{34}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$ and the center-of-mass energy $\sqrt{S}=14\mathrm{TeV}$, sizable heavy quarkonium events can be produced through top quark decays, i.e., $4\times 10^5{B}_c$ and $B_c^{*}$, and $2\times 10^4{\eta }_b$, and $\mathrm{\Upsilon }$ events per year can be obtained according to our calculation.

Figure 1

Feynman diagrams for the process $t(q_0)\to |(b\overline{Q})[n]⟩(q_3)+W^{+}(q_2)+Q(q_1)$, where $Q$ stands for the $c$ and $b$ quark in the left and right panels, respectively. $|(b\overline{Q})[n]⟩$ quarkonium stands for a heavy quarkonium Fock state.

Figure 2

Differential decay widths $d\mathrm{\Gamma }/d{s}_1$ and $d\mathrm{\Gamma }/d{s}_2$ for $t\to |(b\overline{c})[n]⟩+c{W}^{+}(n_f=3)$, where the diamond line, the dash-dotted line, the dotted line, the solid line, and the dashed line are for $|(b\overline{c})[1S]⟩$, $|(b\overline{c})[2S]⟩$, $|(b\overline{c})[3S]⟩$, $|(b\overline{c})[1P]⟩$, and $|(b\overline{c})[2P]⟩$, respectively.

Figure 3

Differential decay widths $d\mathrm{\Gamma }/d{s}_1$ and $d\mathrm{\Gamma }/d{s}_2$ for $t\to |(b\overline{b})[n]⟩+b{W}^{+}(n_f=4)$, where the diamond line, the dash-dotted line, the dotted line, the solid line, and the dashed line are for $|(b\overline{b})[1S]⟩$, $|(b\overline{b})[2S]⟩$, $|(b\overline{b})[3S]⟩$, $|(b\overline{b})[1P]⟩$, and $|(b\overline{b})[2P]⟩$, respectively.

Figure 4

Differential decay widths $d\mathrm{\Gamma }/d\cos {\theta }_{12}$ and $d\mathrm{\Gamma }/d\cos {\theta }_{13}$ for $t\to |(b\overline{c})[n]⟩+c{W}^{+}(n_f=3)$, where the diamond line, the dash-dotted line, the dotted line, the solid line, and the dashed line are for $|(b\overline{c})[1S]⟩$, $|(b\overline{c})[2S]⟩$, $|(b\overline{c})[3S]⟩$, $|(b\overline{c})[1P]⟩$, and $|(b\overline{c})[2P]⟩$, respectively.

Figure 5

Differential decay widths $d\mathrm{\Gamma }/d\cos {\theta }_{12}$ and $d\mathrm{\Gamma }/d\cos {\theta }_{13}$ for $t\to |(b\overline{b})[n]⟩+b{W}^{+}(n_f=4)$, where the diamond line, the dash-dotted line, the dotted line, the solid line, and the dashed line are for $|(b\overline{b})[1S]⟩$, $|(b\overline{b})[2S]⟩$, $|(b\overline{b})[3S]⟩$, $|(b\overline{b})[1P]⟩$, and $|(b\overline{b})[2P]⟩$, respectively.

Figure 6

The ratios $R_1[n]$ and $R_2[2]$ versus $s_1$ and $s_2$ for the channel $t\to |(b\overline{c})[n]⟩+c{W}^{+}(n_f=3)$. Here the dotted line, the dash-dotted line, the solid line, and the dashed line are for $|(b\overline{c})[2S]⟩$, $|(b\overline{c})[3S]⟩$, $|(b\overline{c})[1P]⟩$, and $|(b\overline{c})[2P]⟩$, respectively.