$Cos(2{\varphi }_h)$ asymmetry in $J/\psi$ production in unpolarized $ep$ collision

We present a calculation of the $\cos (2{\varphi }_h)$ asymmetry in $J/\psi$ production in electron-proton collisions at the future electron-ion collider (EIC), a useful channel to probe the transverse momentum dependent gluon distribution functions also known as gluon transverse momentum dependent parton distributions (TMDs). The dominant subprocess for the $J/\psi$ production is the virtual-photon-gluon fusion process ${\gamma }^{*}+g\to J/\psi +g$. The production of $J/\psi$ is calculated in the nonrelativistic quantum chromodynamics framework with the inclusion of both color singlet and color octet contributions. Numerical estimates of the $\cos (2{\varphi }_h)$ asymmetry are given in the kinematical region to be accessed by the future EIC. The asymmetry depends on the parametrization of the gluon TMDs, as well as on the long distance matrix elements. We use both Gaussian-type parametrization and the McLerran-Venugopalan model for the TMDs in the kinematical region of small $x$, where the gluons play a dominant role. The asymmetry may be useful to probe the ratio of the linearly-polarized and the unpolarized gluon distribution in the proton.

Figure 1

Feynman diagrams for the partonic subprocess ${\gamma }^{*}+g\to J/\psi +g$.

Figure 2

$\cos (2{\varphi }_h)$ asymmetry in $e+p\to e+J/\psi +X$ process. Left: As a function of $P_{h\perp }$ at $\sqrt{s}=100\mathrm{GeV}$ and $Q^2=15{\mathrm{GeV}}^2$. The integration ranges are $0.0015<x_B<0.1$, $0.1<z<0.9$ and $y$ is set by the values of $Q^2$, $s$, and $x_B$. Right: As a function of $y$ at $\sqrt{s}=120\mathrm{GeV}$ and at fixed $x_B$. The integration ranges are $0<P_{h\perp }<2$, $0.1<z<0.9$. For both plots CMSWZ set of LDMEs [48] is used.

Figure 3

$\cos (2{\varphi }_h)$ asymmetry in $e+p\to e+J/\psi +X$ process as function of $P_{h\perp }$ at $\sqrt{s}=100\mathrm{GeV}$ and $Q^2=15{\mathrm{GeV}}^2$. The integration ranges are $0.0015<x_B<0.1$, $0.1<z<0.9$ and y is set by the values of $Q^2$, $s$ and $x_B$. Left: contributions from color singlet and color octet states using the LDMEs set CMSWZ [48]. Right: comparing the asymmetry for different LDMEs sets: CMSWZ [48], SV [59], ZSSL [60], BK [61].

Figure 4

$\cos (2{\varphi }_h)$ asymmetry in $e+p\to e+J/\psi +X$ process as function of $P_{h\perp }$ at $\sqrt{s}=100\mathrm{GeV}$ and $Q^2=15{\mathrm{GeV}}^2$. The integration ranges are $0.0015<x_B<0.1$, $0.1<z<0.9$ and y is set by the values of $Q^2$, $s$ and $x_B$. Left: asymmetry for the set of LDMEs, CMSWZ [48]. Right: asymmetry for the set of LDMEs SV [59].

Figure 5

$\cos (2{\varphi }_h)$ asymmetry in $e+p\to e+J/\psi +X$ process as function of $P_{h\perp }$ at $\sqrt{s}=100\mathrm{GeV}$ and $Q^2=15{\mathrm{GeV}}^2$. The integration ranges are $0.0015<x_B<0.1$, $0.1<z<0.9$ and y is set by the values of $Q^2$, $s$ and $x_B$. Left: asymmetry for three different values of $r$. Right: asymmetry for two different values of Gaussian width parameter. For both plots CMSWZ set of LDMEs [48] is used.

Figure 6

$\cos (2{\varphi }_h)$ asymmetry in $e+p\to e+J/\psi +X$ process as function of $P_{h\perp }$ at $\sqrt{s}=150\mathrm{GeV}$, $x=0.01$ and $z=0.7$. Left: contribution to the $\cos (2\varphi )$ asymmetry coming from the individual states, as a function of $P_{h\perp }$ in the NRQCD framework using the color octet model. Right: comparison of the Gaussian and MV model (with two different values of $Q_{sg0}$). For both plots, CMSWZ set of LDMEs [48] is used.

Figure 7

$\cos (2{\varphi }_h)$ asymmetry in $e+p\to e+J/\psi +X$ process as function of $P_{h\perp }$ at $\sqrt{s}=150\mathrm{GeV}$, $x=0.01$ and $z=0.7$. Contribution to the $\cos (2{\varphi }_h)$ asymmetry coming from the color singlet and color octet models. CMSWZ set of LDMEs [48] is used.