Polarization of prompt $J/\psi$ and $\mathrm{\Upsilon }(1\mathrm{S})$ production in the color evaporation model

We calculate the polarization of prompt $J/\psi$ and $\mathrm{\Upsilon }(1\mathrm{S})$ production using the color evaporation model at leading order. We present the polarization parameter ${\lambda }_{\vartheta }$ as a function of center-of-mass energy and rapidity in $p+p$ collisions. We also compare the $x_F$ dependence to experimental results in $p+\mathrm{Cu}$ and $\pi +\mathrm{W}$ collisions and predict the $x_F$ dependence in $p+\mathrm{Pb}$ collisions at fixed-target energies. At energies far above the $Q\overline{Q}$ production threshold, we find the prompt $J/\psi$ and $\mathrm{\Upsilon }(1\mathrm{S})$ production to be longitudinally polarized with ${\lambda }_{\vartheta }^{J/\psi }=-0.51_{-0.16}^{+0.05}$ and ${\lambda }_{\vartheta }^{\mathrm{\Upsilon }(1\mathrm{S})}=-0.69_{-0.02}^{+0.03}$. Both prompt $J/\psi$ and prompt $\mathrm{\Upsilon }(1\mathrm{S})$ are also longitudinally polarized at central rapidity, becoming transversely polarized at the most forward rapidities.

Figure 1

The orientation of the $z$ axis is indicated by the dashed arrowed line. Two proton arrows indicate the incoming beam directions. If the quarks in the final state heavy quark-antiquark pair have the same helicity, then the total angular momentum along the $z$ axis, $J_z$, is zero, while if they have opposite helicity, then $J_z=\pm 1$.

Figure 2

The energy dependence of the polarization parameter ${\lambda }_{\vartheta }$ for production of prompt $J/\psi$ (solid), direct $J/\psi$ (dashed), direct ${\chi }_{c2}(1\mathrm{P})$ (dot-dashed), and direct $\psi (2\mathrm{S})$ (dot-dot-dashed).

Figure 3

The energy dependence of the polarization parameter ${\lambda }_{\vartheta }$ for production of prompt $\mathrm{\Upsilon }(1\mathrm{S})$ (solid), direct $\mathrm{\Upsilon }(1\mathrm{S})$ (dashed), direct ${\chi }_{b2}(2\mathrm{P})$ (dot-dashed), direct $\mathrm{\Upsilon }(2\mathrm{S})$ (dot-dot-dashed), direct ${\chi }_{b2}(2\mathrm{P})$ (dot-dot-dot-dashed), and direct $\mathrm{\Upsilon }(3\mathrm{S})$ (dotted). The result is shown for $\sqrt{s}>20\mathrm{GeV}$ to be above the $B\overline{B}$ threshold.

Figure 4

The rapidity dependence of the polarization parameter ${\lambda }_{\vartheta }$ for the production of prompt $J/\psi$ at $\sqrt{s}=20\mathrm{GeV}$ (solid), 38.8 GeV (dashed), 200 GeV (dot-dashed), and 7000 GeV (dotted). The distributions are symmetric around $y=0$.

Figure 5

The rapidity dependence of the polarization parameter ${\lambda }_{\vartheta }$ for the production of prompt $\mathrm{\Upsilon }(1\mathrm{S})$ at $\sqrt{s}=20\mathrm{GeV}$ (solid), $\sqrt{s}=38.8\mathrm{GeV}$ (dashed), 200 GeV (dot-dashed), and 7000 GeV (dotted). The distributions are symmetric around $y=0$.

Figure 6

The $x_F$ dependence of the polarization parameter ${\lambda }_{\vartheta }$ for prompt production of $J/\psi$ in $p+\mathrm{Cu}$ collisions at $\sqrt{s_{NN}}=38.8\mathrm{GeV}$ is compared to the E866/NuSea data [27, 28]. The horizontal uncertainties are the experimental bin widths.

Figure 7

The $x_F$ dependence of the polarization parameter ${\lambda }_{\vartheta }$ for prompt production of $J/\psi$ in $\pi +\mathrm{W}$ collisions at $\sqrt{s}=22\mathrm{GeV}$ are compared to the CIP data [30].

Figure 8

The $x_F$ dependence of the polarization parameter ${\lambda }_{\vartheta }$ for prompt production of $\mathrm{\Upsilon }(1\mathrm{S})$ in $p+\mathrm{Cu}$ collisions at $\sqrt{s}=38.8\mathrm{GeV}$ using CTEQ6L1 and varying $m_b$ (blue solid), GRV98 LO and varying $m_b$ (red dashed), CTEQ6L1 and varying $Q$ (magenta solid), and the data (box). The horizontal uncertainties on the E866/NuSea data [29] are the bin widths.

Figure 9

The $x_F$ dependence of the polarization parameter ${\lambda }_{\vartheta }$ for production of $J/\psi$ in $p+\mathrm{Pb}$ at $\sqrt{s_{NN}}=72\mathrm{GeV}$ (blue dashed) and 115 GeV (red solid).

Figure 10

The $x_F$ dependence of the polarization parameter ${\lambda }_{\vartheta }$ for production of $\mathrm{\Upsilon }(1\mathrm{S})$ in $p+\mathrm{Pb}$ at $\sqrt{s_{NN}}=72\mathrm{GeV}$ (blue dashed) and 115 GeV (red solid).