Transverse momentum resummation for $s$-channel single top quark production at the LHC

We study the soft gluon radiation effects for the $s$-channel single top quark production at the LHC. By applying the transverse momentum dependent factorization formalism, the large logarithms about the small total transverse momentum ($q_{\perp }$) of the single-top plus one-jet final state system are resummed to all orders in the expansion of the strong interaction coupling at the accuracy of next-to-leading Logarithm (NLL). We compare our numerical results with pythia and find that both the $q_{\perp }$ and ${\varphi }^{*}$ observables from pythia are consistent with our prediction. Furthermore, we point out that the soft gluon radiation effects from the final state become significant in this process, especially for the boosted kinematical region.

Figure 1

Feynman diagrams contributing to the soft functions at the NLO.

Figure 2

The $q_{\perp }$ distribution from the asymptotic result (blue dashed line), NLO calculation (red dotted line), resummation prediction (black solid line), parton shower Monte Carlo prediction by pythia8 (green solid line), and $Y$-term (orange dot-dashed line) for the $s$-channel single top quark (a) and anti-top quark (b) production at the $\sqrt{S}=13\mathrm{TeV}$ LHC with $|y_J|<3$, $|y_t|<3$ and $P_{J\perp }>30\mathrm{GeV}$. The resummation and renormalization scales are choose as $\mu ={\mu }_{\text{Res}}={\mu }_{\text{ren}}=m_{tJ}=(p_t+p_J{)}^2$.

Figure 3

The scale uncertainties for the $s$-channel single top quark (a) and anti-top quark (b) production at the $\sqrt{S}=13\mathrm{TeV}$ LHC with $|y_J|<3$, $|y_t|<3$ and $P_{J\perp }>30\mathrm{GeV}$. The resummation and renormalization scales are varied from $m_{tJ}/2$ to $2m_{tJ}$.

Figure 4

The ratio of the resummation and pythia8 prediction for the $s$-channel single top quark (a) and anti-top quark (b) production at the $\sqrt{S}=13\mathrm{TeV}$ LHC with $|y_J|<3$, $|y_t|<3$ and $P_{J\perp }>30\mathrm{GeV}$. The resummation and renormalization scales are choose as $\mu ={\mu }_{\text{Res}}={\mu }_{\text{ren}}=m_{tJ}$.

Figure 5

The normalized $W$-piece prediction for the $s$-channel single top quark (a) and anti-top quark (b) production at the $\sqrt{S}=13\mathrm{TeV}$ LHC with $|y_J|<3$, $|y_t|<3$ and $P_{J\perp }>30\mathrm{GeV}$. The red-dashed line denotes the $W$-piece prediction with only Drell-Yan–like Sudakov factor, while the dotted black and solid blue lines label the results from the $W$-piece with and without $m_{tJ}>1\mathrm{TeV}$ cut. The resummation and renormalization scales are choose as $\mu ={\mu }_{\text{Res}}={\mu }_{\text{ren}}=m_{tJ}$.

Figure 6

The normalized distribution of ${\varphi }^{*}$ for top quark (a) and anti-top quark (b) production at the 13 TeV LHC with $|y_{t,J}|<3$ and $P_{J\perp }>30\mathrm{GeV}$. The resummation and renormalization scales are choose as $\mu ={\mu }_{\text{Res}}={\mu }_{\text{ren}}=m_{tJ}$. The blue solid and red dashed line represent the resummation prediction with full Sudakov factor and only Drell-Yan–like Sudakov factor, respectively. The black dotted line labels the prediction from pythia8.

Figure 7

The prediction for $pp\to W^{\prime +}\to t+\mathrm{jet}$ at the $\sqrt{S}=13\mathrm{TeV}$ LHC with $|y_J|<3$, $|y_t|<3$ and $P_{J\perp }>30\mathrm{GeV}$. The red-dashed line denotes the $W$-piece prediction with only Drell-Yan–like Sudakov factor, while the solid blue line labels the result from the $W$-piece with full Sudakov factor. The resummation and renormalization scales are choose as $\mu ={\mu }_{\text{Res}}={\mu }_{\text{ren}}=m_{W^{\prime }}$.