Measurements of $\mathrm{J} \mathord{\left/ {\vphantom {\mathrm{J} \mathrm{\psi }}} \right. \kern-\nulldelimiterspace} \mathrm{\psi }$ polarization in p$+$p collisions at STAR

Measurements of $\mathrm{J} \mathord{\left/ {\vphantom {\mathrm{J} \mathrm{\psi }}} \right. \kern-\nulldelimiterspace} \mathrm{\psi }$ production cross section and polarization can help understand $\mathrm{J} \mathord{\left/ {\vphantom {\mathrm{J} \mathrm{\psi }}} \right. \kern-\nulldelimiterspace} \mathrm{\psi }$ production mechanism in hadron collisions and distinguish among different models. $\mathrm{J} \mathord{\left/ {\vphantom {\mathrm{J} \mathrm{\psi }}} \right. \kern-\nulldelimiterspace} \mathrm{\psi }$ polarization could be characterized by the $\lambda_{\theta }$, $\lambda_{\varphi }$and $\lambda _{inv}$ polarization parameters, where $\lambda_{\theta }$ and $\lambda _{\varphi }$ are coefficients of positron polar and azimuthal angle distribution in the $\mathrm{J} \mathord{\left/ {\vphantom {\mathrm{J} \mathrm{\psi }}} \right. \kern-\nulldelimiterspace} \mathrm{\psi }$ rest frame with respect to a chosen polarization axis, while $\lambda_{inv}$ is a frame-independent variable calculable from $\lambda_{\theta }$ and $\lambda_{\varphi }$. $\mathrm{J} \mathord{\left/ {\vphantom {\mathrm{J} \mathrm{\psi }}} \right. \kern-\nulldelimiterspace} \mathrm{\psi }$ polarization parameters $\lambda_{\theta }$, $\lambda_{\varphi }$and $\lambda_{inv}$ in both helicity and Collins-Soper frames have been extracted from the STAR 2011 data in p$+$p collisions at $\sqrt s =500$ GeV, while only $\lambda_{\theta }$ in the helicity frame has been extracted from the STAR 2009 data in p$+$p collisions at $\sqrt s =200$ GeV. In this talk, we will present a new analysis to study $\mathrm{J} \mathord{\left/ {\vphantom {\mathrm{J} \mathrm{\psi }}} \right. \kern-\nulldelimiterspace} \mathrm{\psi }$ polarization using the STAR 2012 data to extract $\lambda _{\theta }$, $\lambda_{\varphi }$and $\lambda_{inv}$ in both the helicity and Collins-Soper frames in p$+$p collisions at $\sqrt s =200$ GeV.