Production and polarization of direct $J/\psi$ to $\mathcal{O}({\alpha }_s^3)$ in the improved color evaporation model in collinear factorization

We calculate the production and polarization of direct $J/\psi$ in the improved color evaporation model at $\mathcal{O}({\alpha }_s^3)$ in the collinear factorization approach. We present the first calculation of polarization parameters ${\lambda }_{\vartheta }$, ${\lambda }_{\phi }$, and ${\lambda }_{\vartheta \phi }$ in the helicity and the Collins-Soper frames as well as the frame-invariant polarization parameter $\tilde{\lambda }$ as a function of transverse momentum. We find agreement with both $J/\psi$ cross sections and the invariant polarization parameters at small and moderate $p_T$.

Figure 1

The $p_T$ dependence of inclusive $J/\psi$ production at $\sqrt{s}=7\mathrm{TeV}$ with $|y|<0.9$ in the ICEM. The combined mass, renormalization scale, factorization scale, and $F_{J/\psi }$ uncertainties are shown in the band and compared to the ALICE data [22]. The ALICE data are measured while assuming $J/\psi$ production is unpolarized.

Figure 2

The $p_T$ dependence of inclusive $J/\psi$ production at $\sqrt{s}=7\mathrm{TeV}$ with $2.5<y<4$ (blue region) and $|y|<2$ in the ICEM (red region). The combined mass, renormalization scale, factorization scale, and $F_{J/\psi }$ uncertainties are shown in the band. They are compared to the ALICE data [22] (blue squares) and the ATLAS data [23] (red squares), respectively.

Figure 3

The $p_T$ dependence of inclusive $J/\psi$ production at $\sqrt{s}=7\mathrm{TeV}$ in the polarized collinear ICEM (this calculation) (blue region), in the polarized ICEM using the $k_T$-factorization [13] (magenta region), in the unpolarized collinear ICEM [12] (green region). They are compared to the LHCb data [28] assuming that the $J/\psi$ polarization is totally transverse, ${\lambda }_{\vartheta }=+1$ (red squares), and totally longitudinal, ${\lambda }_{\vartheta }=-1$ (blue squares). The LHCb data assuming ${\lambda }_{\vartheta }=0$ lie between the red and blue points and are not shown.

Figure 4

(a) The polar anisotropy parameter (${\lambda }_{\vartheta }$), (b) the azimuthal anisotropy parameter (${\lambda }_{\phi }$), and (c) the polar-azimuthal correlation parameter (${\lambda }_{\vartheta \phi }$) in the helicity frame at $\sqrt{s}=7\mathrm{TeV}$ in the ICEM. The combined mass, renormalization scale, and factorization scale uncertainties are shown in the band and compared to the LHCb data [24] (blue) and the ALICE data [25] (red).

Figure 5

(a) The polar anisotropy parameter (${\lambda }_{\vartheta }$), (b) the azimuthal anisotropy parameter (${\lambda }_{\phi }$), and (c) the polar-azimuthal correlation parameter (${\lambda }_{\vartheta \phi }$) in the Collins-Soper frame at $\sqrt{s}=7\mathrm{TeV}$ in the ICEM. The combined mass, renormalization scale, and factorization scale uncertainties are shown in the band and compared to the LHCb data [24] (blue) and the ALICE data [25] (red).

Figure 6

(a) The polar anisotropy parameter (${\lambda }_{\vartheta }$), (b) the azimuthal anisotropy parameter (${\lambda }_{\phi }$), and (c) the polar-azimuthal correlation parameter (${\lambda }_{\vartheta \phi }$) in the helicity frame at $\sqrt{s}=7\mathrm{TeV}$ in the ICEM. The combined mass, renormalization scale, and factorization scale uncertainties are shown in the band and compared to the CMS data [26] in $|y|<0.6$ (blue) and $0.6<|y|<1.2$ (red).

Figure 7

(a) The polar anisotropy parameter (${\lambda }_{\vartheta }$), (b) the azimuthal anisotropy parameter (${\lambda }_{\phi }$), and (c) the polar-azimuthal correlation parameter (${\lambda }_{\vartheta \phi }$) in the Collins-Soper frame at $\sqrt{s}=7\mathrm{TeV}$ in the ICEM. The combined mass, renormalization scale, and factorization scale uncertainties are shown in the band and compared to the CMS data [26] in $|y|<0.6$ (blue) and $0.6<|y|<1.2$ (red).

Figure 8

(a) The polar anisotropy parameter (${\lambda }_{\vartheta }$), (b) the azimuthal anisotropy parameter (${\lambda }_{\phi }$), and (c) the polar-azimuthal correlation parameter (${\lambda }_{\vartheta \phi }$) (c) at $\sqrt{s}=7\mathrm{TeV}$ in the ICEM. The ICEM results in the Collins-Soper frame in $|y|<0.6$ (solid) and in $2<y<4.5$ (dot-dashed), and the results in the helicity frame in $|y|<0.6$ (dashed) and in $2<y<4.5$ (dotted) are shown. Only the baseline results are shown here.

Figure 9

(a) The angular distribution of the ICEM direct $J/\psi$ production in the Collins-Soper frame and (b) in the helicity frame at $p_T=12\mathrm{GeV}$. They represent the same angular distribution separated by one rotation. We constructed the angular distribution in the Collins-Soper frame based on the data collected in the $10<p_T<15\mathrm{GeV}$ bin by the LHCb Collaboration [24], and it is shown in (c) for comparison.

Figure 10

The polar anisotropy parameter (${\lambda }_{\vartheta }$) of direct $J/\psi$ production at $\sqrt{s}=7\mathrm{TeV}$ with $2<y<4.5$ in the Collins-Soper frame calculated in the collinear ICEM in this work (blue) and in the $k_T$-factorized ICEM in Ref. [13]. Both calculations include only the $gg$ contributions and are compared to the LHCb data [24].

Figure 11

The $p_T$ dependence of the frame-invariant polarization parameter, $\tilde{\lambda }$ in the ICEM (a) compared to the LHCb data [24] (blue) and the ALICE data [25] (red) at low $p_T$; (b) compared to the CMS data [26] in the most central rapidity bin (blue) and the next-most-central rapidity bin (red) at high $p_T$; and (c) the baseline ICEM results in $|y|<0.6$ (solid) and $2<y<4.5$ (dashed) for all $p_T$. Data measured in the Collins-Soper frame are presented as solid points, and those in the helicity frame are presented as open points. The data in the helicity frame are displaced by 0.05 GeV for visualization purposes.