Next-to-next-to-leading order nonresonant corrections to threshold top-pair production from $e^{\mathbf{+}}{e}^{\mathbf{-}}$ collisions: Endpoint-singular terms

We analyze the subleading nonresonant contributions to the $e^{+}{e}^{-}\to W^{+}{W}^{-}b\overline{b}$ cross section at energies near the top-antitop threshold. These correspond to next-to-next-to-leading-order (NNLO) corrections with respect to the leading-order resonant result. We show that these corrections produce $1/ϵ$ endpoint singularities which precisely cancel the finite-width divergences arising in the resonant production of the $W^{+}{W}^{-}b\overline{b}$ final state from on-shell decays of the top and antitop quarks at the same order. We also provide analytic results for the $(m_t/\Lambda {)}^2$, ($m_t/\Lambda$) and $(m_t/\Lambda {)}^0\log \Lambda$ terms that dominate the expansion in powers of ($\Lambda /m_t$) of the complete set of NNLO nonresonant corrections, where $\Lambda$ is a cut imposed on the invariant masses of the $bW$ pairs that is neither too tight nor too loose ($m_t{\Gamma }_t\ll {\Lambda }^2\ll m_t^2$).

Figure 1

Two-loop resonant NRQCD graph with a Coulomb potential.

Figure 2

Two-loop forward-scattering amplitude diagrams with $b{W}^{+}\overline{t}$ cuts contributing to the NLO nonresonant cross section. $\overline{b}{W}^{-}t$ cuts and symmetric diagrams are not shown. This figure is reproduced from Ref. 28.

Figure 3

$\mathcal{O}({\alpha }_s)$ virtual corrections to the two-loop forward-scattering diagrams with $b{W}^{+}\overline{t}$ cuts which are endpoint-singular. $\overline{b}{W}^{-}t$ cuts and symmetric diagrams are not shown. The $e^{+}{e}^{-}$ external legs have not been drawn, except for the $h_{4a}$ diagram.

Figure 4

$\mathcal{O}({\alpha }_s)$ real-gluon corrections to the two-loop forward-scattering diagrams with $b{W}^{+}\overline{t}g$ cuts which are endpoint-singular. $\overline{b}{W}^{-}tg$ cuts and symmetric diagrams are not shown. The $e^{+}{e}^{-}$ external states have been omitted.

Figure 5

Endpoint-singular NNLO nonresonant contribution to the $t\overline{t}$ cross section computed at $s=4m_t^2$ as a function of the invariant-mass cut $\Delta M_t=m_t(1-\sqrt{1-{\Lambda }^2/m_t^2})$. The solid (blue) line corresponds to ${\sigma }_{\mathrm{nonres}}^{(2),\mathrm{ep}}$ in (52) dropping the $1/{ϵ}_{\mathrm{ep}}$ term and using ${\alpha }_s\equiv {\alpha }_s({\mu }_{\mathrm{soft}})$ with ${\mu }_{\mathrm{soft}}=30\mathrm{GeV}$. The shaded (blue) band shows the same result varying ${\mu }_{\mathrm{soft}}$ in the interval 15–60 GeV. For comparison, the NLO nonresonant contribution (dashed red line), ${\sigma }_{\mathrm{nonres}}^{(1)}$ from 28, is also shown.

Figure 6

Relative sizes of the nonresonant corrections with respect to the $t\overline{t}$ LO cross section in percent: ${\sigma }_{\mathrm{nonres}}^{(2),\mathrm{ep}}/{\sigma }_{t\overline{t}}^{(0)}$ (upper blue lines) and ${\sigma }_{\mathrm{nonres}}^{(1)}/{\sigma }_{t\overline{t}}^{(0)}$ (lower black lines, from 28). Solid (dashed) lines correspond to an invariant-mass cut $\Delta M_t=35\mathrm{GeV}$ ($\Delta M_t=15\mathrm{GeV}$). The renormalization scale in the NNLO nonresonant contribution has been set to ${\mu }_{\mathrm{soft}}=30\mathrm{GeV}$, and we have chosen ${\alpha }_s(30\mathrm{GeV})=0.142$ for the value of the QCD coupling that enters also in the LO result.