Production and polarization of prompt $\mathrm{\Upsilon }(n\mathrm{S})$ in the improved color evaporation model using the $k_T$-factorization approach

We calculate the polarization of prompt $\mathrm{\Upsilon }(n\mathrm{S})$ production in the improved color evaporation model at leading order employing the $k_T$-factorization approach. We present the polarization parameter ${\lambda }_{\vartheta }$ of prompt $\mathrm{\Upsilon }(n\mathrm{S})$ as a function of transverse momentum in $p+p$ and $p+\overline{p}$ collisions to compare with data in the helicity, Collins-Soper and Gottfried-Jackson frames. We also present calculations of the bottomonium production cross sections as a function of transverse momentum and rapidity. This is the first $p_T$-dependent calculation of bottomonium production and polarization in the improved color evaporation model. We find agreement with both bottomonium cross sections and polarization measurements.

Figure 1

The $p_T$ dependence of prompt $\mathrm{\Upsilon }(1\mathrm{S})$ production at $\sqrt{s}=7\mathrm{TeV}$ in the ICEM obtained by varying the bottom quark mass (blue), the factorization scale in the range $0.5<{\mu }_F/m_T<2$ (magenta), and the renormalization scale in the range $0.5<{\mu }_R/m_T<2$ (green) is compared with the CMS midrapidity data [22].

Figure 2

The $p_T$ dependence of prompt $\mathrm{\Upsilon }(1\mathrm{S})$ production at $\sqrt{s}=7\mathrm{TeV}$ in the ICEM with combined mass and renormalization scale uncertainties (blue) and that in the CEM using collinear factorization approach (magenta). The CMS midrapidity data [22] from Fig. 1 are also shown.

Figure 3

The $p_T$ dependence of prompt $\mathrm{\Upsilon }(1\mathrm{S})$ production at $\sqrt{s}=7\mathrm{TeV}$ and $2<y<4.5$ in the ICEM with combined mass and renormalization scale uncertainties is compared with the LHCb data [27].

Figure 4

The $p_T$ dependence of prompt $\mathrm{\Upsilon }(1\mathrm{S})$ production at $\sqrt{s}=7\mathrm{TeV}$ and $|y|<0.7$ in the ICEM with combined mass and renormalization scale uncertainties is compared with the D0 data [28].

Figure 5

The $p_T$ dependence of prompt $\mathrm{\Upsilon }(2\mathrm{S})$ production at $\sqrt{s}=7\mathrm{TeV}$ and $2<y<4.5$ in the ICEM with combined mass and renormalization scale uncertainties is compared with the CMS midrapidity data [22].

Figure 6

The $p_T$ dependence of prompt $\mathrm{\Upsilon }(2\mathrm{S})$ production at $\sqrt{s}=7\mathrm{TeV}$ and $2<y<4.5$ in the ICEM with combined mass and renormalization scale uncertainties is compared with the LHCb data [27].

Figure 7

The $p_T$ dependence of prompt $\mathrm{\Upsilon }(3\mathrm{S})$ production at $\sqrt{s}=7\mathrm{TeV}$ in the ICEM with combined mass and renormalization scale uncertainties is compared with the CMS midrapidity data [22].

Figure 8

The $p_T$ dependence of prompt $\mathrm{\Upsilon }(2\mathrm{S})$ production at $\sqrt{s}=7\mathrm{TeV}$ and $2<y<4.5$ in the ICEM with combined mass and renormalization scale uncertainties is compared with the LHCb data [27].

Figure 9

The ratio of ${\chi }_{b2}(1\mathrm{P})$ to ${\chi }_{b1}(1\mathrm{P})$ in the ICEM with combined mass and renormalization scale uncertainties at $\sqrt{s}=8\mathrm{TeV}$ at central rapidity $|y|<1.5$ (a) and at forward rapidity $2<y<4.5$ (b) assuming $F_{{\chi }_{b1(1\mathrm{P})}}=F_{{\chi }_{b2(1\mathrm{P})}}$. The CMS data [29] and the LHCb data [30] are also shown in (a) and (b) respectively.

Figure 10

The rapidity dependence of prompt $\mathrm{\Upsilon }(1\mathrm{S})$ (blue solid), $\mathrm{\Upsilon }(2\mathrm{S})$ (magenta dashed), and $\mathrm{\Upsilon }(3\mathrm{S})$ (green dot-dashed) production at $\sqrt{s}=7\mathrm{TeV}$ integrated over $p_T<30\mathrm{GeV}$ in the ICEM with combined mass and renormalization scale uncertainties are compared with the LHCb data [27].

Figure 11

The $p_T$ dependence of the polarization parameter ${\lambda }_{\vartheta }$ for prompt $\mathrm{\Upsilon }(1\mathrm{S})$ (a), $\mathrm{\Upsilon }(2\mathrm{S})$ (b), and $\mathrm{\Upsilon }(3\mathrm{S})$ (c) production in the helicity frame at $\sqrt{s}=7\mathrm{TeV}$ in the ICEM using the “low $p_T$” $c_{\mathcal{Q}}$’s with mass uncertainties are compared to the LHCb data in the range $2.2<y<3$ [31].

Figure 12

The $p_T$ dependence of the polarization parameter ${\lambda }_{\vartheta }$ for prompt $\mathrm{\Upsilon }(1\mathrm{S})$ production in the helicity frame at $\sqrt{s}=1.8\mathrm{TeV}$ with $|y|<0.4$ in the ICEM using the “low $p_T$” $c_{\mathcal{Q}}$’s [13] with mass uncertainties are compared to the CDF data [32].

Figure 13

The $p_T$ dependence of the polarization parameter ${\lambda }_{\vartheta }$ for prompt $\mathrm{\Upsilon }(1\mathrm{S})$ (a), $\mathrm{\Upsilon }(2\mathrm{S})$ (b), and $\mathrm{\Upsilon }(3\mathrm{S})$ (c) production in the helicity frame at $\sqrt{s}=7\mathrm{TeV}$ in the ICEM using the “high $p_T$” $c_{\mathcal{Q}}$’s [13] with mass uncertainties are compared to the CMS data at midrapidity in the range $|y|<0.6$ [34].

Figure 14

The $p_T$ dependence of the polarization parameter ${\lambda }_{\vartheta }$ for prompt $\mathrm{\Upsilon }(1\mathrm{S})$ production in the Collins-Soper frame at $\sqrt{s}=7\mathrm{TeV}$ and $2.2<y<3$ in the ICEM using the “low $p_T$” $c_{\mathcal{Q}}$’s [13] with mass uncertainties are compared to the LHCb data [31].

Figure 15

The $p_T$ dependence of the polarization parameter ${\lambda }_{\vartheta }$ for prompt $\mathrm{\Upsilon }(1\mathrm{S})$ production in the Gottfried-Jackson frame at $\sqrt{s}=7\mathrm{TeV}$ and $2.2<y<3$ in the ICEM using the “low $p_T$” $c_{\mathcal{Q}}$’s [13] with mass uncertainties are compared to the LHCb data [31].