Analytical results for $O({\alpha }_s)$ radiative corrections to $e^{+}{e}^{-}\to t{\overline{t}}^{\uparrow }$ up to a given gluon energy cut

We determine the $O({\alpha }_s)$ radiative corrections to polarized top quark pair production in $e^{+}{e}^{-}$ annihilations with a specified gluon energy cut. We write down fully analytical results for the unpolarized and polarized $O({\alpha }_s)$ cross sections $e^{+}{e}^{-}\to \overline{t}t(G)$ and $e^{+}{e}^{-}\to \overline{t}{t}^{\uparrow }(G)$ including their polar orientation dependence relative to the beam direction. In the soft-gluon limit we recover the usual factorizing form known from the soft-gluon approximation. In the limit when the gluon energy cut takes its maximum value we recover the totally inclusive unpolarized and polarized cross sections calculated previously. We provide some numerical results on the cutoff dependence of the various polarized and unpolarized cross sections and discuss how the exact results numerically differ from the approximate soft-gluon results.

Figure 1

Orthonormal spin basis ${\overrightarrow{e}}_T$, ${\overrightarrow{e}}_N$, and ${\overrightarrow{e}}_{\ell }$ for the top quark. Also shown are the beam plane (light gray, respectively, yellow) and the event plane (dark gray, respectively, green).

Figure 2

Phase-space diagram with gluon cut.

Figure 3

Dependence of the relative fraction $(h_{\mathrm{eik}}^{\prime }-h_{\mathrm{SGA}}^{\prime })/h_{\mathrm{SGA}}^{\prime }$ on the scaled gluon energy cutoff parameter $\lambda /{\lambda }_{\max }$ where ${\lambda }_{\max }=(1-\xi )/2$. Curves are shown for the three center-of-mass energies $\sqrt{s}=400$ (dotted line), 500 (dashed line), and 1000 GeV (full line).

Figure 4

Dependence of the total rate (solid line: exact NLO; dash-dotted line: eikonal; dashed line: SGA) on the scaled gluon energy cutoff parameter $\lambda /{\lambda }_{\max }$ where ${\lambda }_{\max }=(1-\xi )/2$. Also shown are the respective cutoff independent LO Born term contributions (horizontal dotted lines). Curves are shown for the three center-of-mass energies $\sqrt{s}=400$, 500, and 1000 GeV.

Figure 5

Dependence of the relative difference of the exact cross section and the eikonal approximation on $\lambda /{\lambda }_{\max }$ where ${\lambda }_{\max }=(1-\xi )/2$ for center-of-mass energies $\sqrt{s}=400$ (dotted line), 500 (dashed line), and 1000 GeV (full line).

Figure 6

Dependence of the ratio $\sigma (\mathrm{\text{hard}})/\sigma (\mathrm{\text{full}})$ on $\lambda /{\lambda }_{\max }$ in the hard region where $\lambda$ denotes a lower cutoff. Curves are shown for the three center-of-mass energies $\sqrt{s}=400$ (dotted line), 500 (dashed line), and 1000 GeV (full line).

Figure 7

Dependence of the differential rate $d\sigma /d\cos \theta$ on $\cos \theta$ in the soft region. Curves are shown for the three center-of-mass energies $\sqrt{s}=400$ (dotted line), 500 (dashed line), and 1000 GeV (full line) and three upper cutoff values $\lambda /{\lambda }_{\max }=0.2$, 0.4, and 0.8 (from bottom to top).

Figure 8

Dependence of the forward-backward asymmetry $A_{\mathrm{FB}}$ on the (upper) cutoff $\lambda /{\lambda }_{\max }$ in the soft region (full line). Curves are shown for the three center-of-mass energies $\sqrt{s}=400$, 500, and 1000 GeV. Also shown are the respective cutoff independent LO Born term contributions (horizontal dashed lines).

Figure 9

Dependence of the longitudinal polarization $P^{\ell }$ on the (upper) cutoff $\lambda /{\lambda }_{\max }$ in the soft region (full line). Curves are shown for the three center-of-mass energies $\sqrt{s}=400$, 500, and 1000 GeV. Also shown are the respective cutoff independent LO Born term contributions (horizontal dashed lines).

Figure 10

Dependence of the fractional deviation of the longitudinal polarization $\delta (P^{\ell })$ on the (upper) cutoff $\lambda /{\lambda }_{\max }$ in the soft region. Curves are shown for the three center-of-mass energies $\sqrt{s}=400$ (dotted line), 500 (dashed line), and 1000 GeV (full line).

Figure 11

Dependence of the forward-backward asymmetry $A_{\mathrm{FB}}$ on $\lambda /{\lambda }_{\max }$ in the hard region where $\lambda$ denotes a lower cutoff. Curves are shown for the three center-of-mass energies $\sqrt{s}=400$ (dotted line), 500 (dashed line), and 1000 GeV (full line). The straight lines indicate the Born term level results.

Figure 12

Dependence of the longitudinal polarization $P^{\ell }$ on $\lambda /{\lambda }_{\max }$ in the hard region where $\lambda$ denotes a lower cutoff. Curves are shown for the three center-of-mass energies $\sqrt{s}=400$ (dotted line), 500 (dashed line), and 1000 GeV (full line). The straight lines indicate the Born term level results.

Figure 13

Dependence of the longitudinal polarization from a longitudinal polarized gauge boson $P_L^{\ell }$ on $\lambda /{\lambda }_{\max }$ in the hard gluon region where $\lambda$ denotes a lower cutoff. Curves are shown for the three center-of-mass energies $\sqrt{s}=400$ (dotted line), 500 (dashed line), and 1000 GeV (full line).

Figure 14

Goodness of the class 1 relations against radiative corrections using an upper gluon energy cut. Dependence of the ratios $(H_U^1-H_F^{1\ell })/H_U^1(\mathrm{\text{Born}})$ (solid line), and $(H_F^4-H_U^{4\ell })/H_U^1(\mathrm{\text{Born}})$ (dashed line) on $\lambda /{\lambda }_{\max }$ where $\lambda$ denotes an upper cutoff. Curves are shown for the center-of-mass energy $\sqrt{s}=500\mathrm{GeV}$ in the soft region.

Figure 15

The same as in Fig. 14 for the hard region.

Figure 16

Goodness of the class 2 relations against radiative corrections using an upper gluon energy cut. Dependence of the ratios $(H_L^1-H_L^2)/H_U^1(\mathrm{\text{Born}})$ (solid line), $H_L^{4\ell }/H_U^1(\mathrm{\text{Born}})$ (dashed line), $(H_A^{1T}-H_A^{2T})/H_U^1(\mathrm{\text{Born}})$ (dash-dotted line), and $(H_I^{4T}-H_A^{3N})/H_U^1(\mathrm{\text{Born}})$ (dotted line) on $\lambda /{\lambda }_{\max }$ where $\lambda$ denotes an upper cutoff. Curves are shown for the center-of-mass energy $\sqrt{s}=500\mathrm{GeV}$.

Figure 17

The same as in Fig. 16 for the hard region.