$N$-jettiness subtractions for $gg\to H$ at subleading power

$N$-jettiness subtractions provide a general approach for performing fully-differential next-to-next-to-leading order (NNLO) calculations. Since they are based on the physical resolution variable $N$-jettiness, ${\mathcal{T}}_N$, subleading power corrections in $\tau ={\mathcal{T}}_N/Q$, with $Q$ a hard interaction scale, can also be systematically computed. We study the structure of power corrections for 0-jettiness, ${\mathcal{T}}_0$, for the $gg\to H$ process. Using the soft-collinear effective theory we analytically compute the leading power corrections ${\alpha }_s\tau \ln \tau$ and ${\alpha }_s^2\tau {\ln }^3\tau$ (finding partial agreement with a previous result in the literature), and perform a detailed numerical study of the power corrections in the $gg$, $gq$, and $q\overline{q}$ channels. This includes a numerical extraction of the ${\alpha }_s\tau$ and ${\alpha }_s^2\tau {\ln }^2\tau$ corrections, and a study of the dependence on the ${\mathcal{T}}_0$ definition. Including such power suppressed logarithms significantly reduces the size of missing power corrections, and hence improves the numerical efficiency of the subtraction method. Having a more detailed understanding of the power corrections for both $q\overline{q}$ and $gg$ initiated processes also provides insight into their universality, and hence their behavior in more complicated processes where they have not yet been analytically calculated.

Figure 1

An estimate of the missing power corrections $\mathrm{\Delta }\sigma ({\tau }_{\mathrm{cut}})$ based on their functional form at NLO (green), NNLO (blue), and ${\mathrm{N}}^3\mathrm{LO}$ (orange). The solid (dashed) lines correspond to the (N)LL power corrections, showing the possible improvement by removing the LL corrections. On the left, the estimate is normalized to the full ${\mathrm{N}}^n\mathrm{LO}$ contribution, while on the right it is normalized to the LO cross section (assuming a 30% correction of each perturbative order relative to the previous order). In both cases, the bands around the LL estimate illustrates a factor of 3 variation.

Figure 2

Representative NLO diagrams for Category 1. In (a) a gluon becomes soft, and in (b) two gluons become collinear. Collinear particles are shown in light blue, soft particles in orange. The power counting of the hard-scattering operators and Lagrangian insertions is explicitly indicated.

Figure 3

Representative NLO diagrams for Category 2. In (a) a quark becomes soft, and in (b) a quark and a gluon become collinear. The power counting of the hard-scattering operators and Lagrangian insertions is explicitly indicated.

Figure 4

Two-loop hard-collinear contributions, which are used to compute the LL divergence at subleading power. The grey circle represents a one-loop hard virtual correction. (a) shows the contributions to category 1, when two gluons become collinear, and (b) shows the contributions to category 2 (b), when a quark and a gluon become collinear.

Figure 5

Representative diagrams contributing to ${\sigma }_{gq}$, where either a soft quark crosses the cut (a), or a collinear quark crosses the cut (b). One-loop corrections to these diagrams give rise to the $C_F(C_F+C_A){\ln }^3\tau$ structure at NNLO.

Figure 6

Diagram contributing to ${\sigma }_{q\overline{q}}$, arising from a subleading operator where a collinear gluon crosses the cut. There is no corresponding soft diagram. Unlike for the other channels, this channel is IR finite at lowest order, so there is no constraint from the cancellation of soft and collinear IR poles.

Figure 7

Fit to the $\mathcal{O}({\alpha }_s)$ nonsingular corrections for beam thrust in the $gg$ channel (top row) and the $gq$ channel (bottom row). The fit functions are defined in Eq. (35) and yield the solid red line, while the dashed blue line shows the result when only the leading coefficient from the red solid fit is retained. (The light dashed orange line shows an extrapolation of the fit result beyond the fit region.) The plots in the right column are identical to those in the left column, but show the absolute value on a logarithmic scale.

Figure 8

Fit to the $\mathcal{O}({\alpha }_s^2)$ nonsingular corrections for beam thrust in the $gg$ channel (top row) and the $gq$ channel (bottom row). The fit functions are defined in Eq. (35) and yield the solid red line (whose continuation by a light orange dashed line shows the extrapolation outside the fit region). The dashed blue line shows the result when only the leading coefficient from the red solid fit is retained, while the dotted red line shows the result when only the full set of logarithms at this power are retained. The plots in the right column are identical to those in the left column, but show the absolute value on a logarithmic scale.

Figure 9

Fit to the nonsingular corrections for beam thrust in the $q{q}^{\prime }$ channel. Results are shown at NLO (top row) and NNLO (bottom row). The fit functions are defined in Eq. (35) and yield the solid red line (whose continuation by a light orange dashed line shows the extrapolation outside the fit region). The dashed blue line shows the result when only the leading coefficient from the red solid fit is retained, while the dotted red line shows the result when only the full set of logarithms at this power are retained. The plots in the right column are identical to those in the left column, but show the absolute value on a logarithmic scale.

Figure 10

Power corrections $\mathrm{\Delta }\sigma ({\tau }_{\mathrm{cut}})$ for the $\mathcal{O}({\alpha }_s)$ contributions in the $gg$ channel (top row), the $gq$ channel (middle row), and the sum of both channels (bottom row). The plots in the right column are identical to those in the left column, but show the absolute value on a logarithmic scale.

Figure 11

Power corrections $\mathrm{\Delta }\sigma ({\tau }_{\mathrm{cut}})$ for the $\mathcal{O}({\alpha }_s^2)$ contributions in the $gg$ channel (top row), the $gq$ channel (middle row), and the sum of both channels (bottom row). The plots in the right column are identical to those in the left column, but show the absolute value on a logarithmic scale.

Figure 12

Leading-logarithmic power correction in the rapidity spectrum at $\mathcal{O}({\alpha }_s)$ (top row) and $\mathcal{O}({\alpha }_s^2)$ (bottom row), for the $gg$ (dotted), $gq$ (dashed), and $gg+gq$ (solid) channels using ${\tau }_{\mathrm{cut}}={10}^{-3}$. The colored curves show the standard definition of ${\mathcal{T}}_0$ in the leptonic frame, while the gray curves show the definition of ${\mathcal{T}}_0^{\mathrm{had}\mathrm{cm}}$ in the hadronic frame. The exponential growth of the power corrections for ${\mathcal{T}}_0^{\mathrm{had}\mathrm{cm}}$ are clearly visible. The plots in the right column are identical to those in the left column, but show the absolute value on a logarithmic scale.