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Boson+jet production in hadron collisions. Here $p_V$ and $p_J$ are the momenta of the color singlet boson and the jet, and $R$ is the jet radius. By definition $\vec{q}_T=\vec{p}^J_{T}+\vec{p}^{\,V}_{T}$. The modes relevant for the observable $q_T$ include the soft modes with momentum $p_s$, and the collinear modes along the two beam directions ($n_1$ and $n_2$) and the jet direction ($n_J$). Small-angle soft modes are taken as an independent degree of freedom from those emitted from the jet at wide angle, and its momentum is denoted as $p_t$. The $n_1$-collinear and $n_2$-collinear modes and soft modes all have a transverse momentum $\sim q_T$, while the $n_J$-collinear modes carry most of the jet momentum.

Illustration of the $\log R$ dependence in $dn(q_T)/dq_T$ for the $qg$ channel at LL. Here we take $\alpha_s(q_T)\approx 0.18$ with $q_T\approx 10~$GeV around the peak region.

Illustration of the $\log R$ dependence in $dn(q_T)/dq_T$ for the $qg$ channel at LL. Here we take $\alpha_s(q_T)\approx 0.18$ with $q_T\approx 10~$GeV around the peak region.

Effects of $\log R$ resummation illustrated in the high $p_T^J$ (left plot) and low $p_T^J$ (right plot) cases. Here, NLL$_{\rm p}$ stands for the NLL resummed results excluding NGLs. The label ``partial log $R$" corresponds to the results from setting $\mu_j = \mu_h$ and $\mu_t = \mu_b$.

Effects of $\log R$ resummation illustrated in the high $p_T^J$ (left plot) and low $p_T^J$ (right plot) cases. Here, NLL$_{\rm p}$ stands for the NLL resummed results excluding NGLs. The label ``partial log $R$" corresponds to the results from setting $\mu_j = \mu_h$ and $\mu_t = \mu_b$.

The effect of NGL resummation illustrated in the high $p_T^J$ (left plot) and low $p_T^J$ (right plot) cases. Here, NLL stands for the full results calculated from (\ref{eq:dsigmaResumFinal}), and NLL$_\text{p}$ corresponds to the results without NGL resummation.

The effect of NGL resummation illustrated in the high $p_T^J$ (left plot) and low $p_T^J$ (right plot) cases. Here, NLL stands for the full results calculated from (\ref{eq:dsigmaResumFinal}), and NLL$_\text{p}$ corresponds to the results without NGL resummation.

Comparison between the NLL cross section calculations with \textsc{Pythia} simulations, in the high $p_T^J$ case (top row) and the low $p_T^J$ case (bottom row). In all the plots, the red curves are the theoretical predictions with the scale choice in (\ref{eq:scaleChoice}), and the error bands are shown as the shaded regions. The histograms are the \textsc{Pythia} results at parton (dashed lines) and hadron (solid lines) levels.

Comparison between the NLL cross section calculations with \textsc{Pythia} simulations, in the high $p_T^J$ case (top row) and the low $p_T^J$ case (bottom row). In all the plots, the red curves are the theoretical predictions with the scale choice in (\ref{eq:scaleChoice}), and the error bands are shown as the shaded regions. The histograms are the \textsc{Pythia} results at parton (dashed lines) and hadron (solid lines) levels.

Comparison between theoretical calculations with experimental data for the process: $pp\to Z^0/\gamma^*\to e^+ e^-$ and $\mu^+\mu^-$. The left plot shows the comparison between our NLL result of $d\sigma/dq_T$ with the measurement in \cite{Sirunyan:2018cpw}, where the solid red curve is the result with the scale choice in (\ref{eq:scaleChoice}) and the shaded region indicates the error band from scale variation. The solid black and orange curves are respectively the LO result and the LO result including only logarithmic terms (LO singular). The right plot shows the comparison between our prediction of azimuthal angle decorrelation with the measurement in \cite{Chatrchyan:2013tna}, where $\Delta\phi(Z,j_1)$ is defined as the azimuthal angle between the $Z$ boson and the leading jet.

Comparison between theoretical calculations with experimental data for the process: $pp\to Z^0/\gamma^*\to e^+ e^-$ and $\mu^+\mu^-$. The left plot shows the comparison between our NLL result of $d\sigma/dq_T$ with the measurement in \cite{Sirunyan:2018cpw}, where the solid red curve is the result with the scale choice in (\ref{eq:scaleChoice}) and the shaded region indicates the error band from scale variation. The solid black and orange curves are respectively the LO result and the LO result including only logarithmic terms (LO singular). The right plot shows the comparison between our prediction of azimuthal angle decorrelation with the measurement in \cite{Chatrchyan:2013tna}, where $\Delta\phi(Z,j_1)$ is defined as the azimuthal angle between the $Z$ boson and the leading jet.