A Toolbox for $q_T$ and $0$-Jettiness Subtractions at N$^3$LO

TL;DR

The article develops the three-loop infrared structure for two color-singlet resolution variables, 0-jettiness $\mathcal{T}_0$ and transverse momentum $q_T$, deriving the complete differential subtraction terms needed for N$^3$LO calculations and constructing the full three-loop beam and soft functions. It leverages RGEs and consistency relations between factorization limits to obtain the $z\to1$ eikonal limits of the beam-function coefficients and provides numerical studies illustrating the impact of the three-loop terms on resummed predictions. The work underpins N$^3$LO$+$PS matching and high-order resummations ($\mathrm{N^3LL'}$ and $\mathrm{N^4LL}$), and it offers practical subtraction formulae and a path to estimate unknown beyond-threshold contributions. Subsequent developments confirmed the predicted eikonal terms and delivered complete three-loop integrated subtraction terms for both $\mathcal{T}_0$ and $q_T$.

Abstract

We derive the leading-power singular terms at three loops for both $q_T$ and 0-jettiness, $\cal{T}_0$, for generic color-singlet processes. Our results provide the complete set of differential subtraction terms for $q_T$ and $\cal{T}_0$ subtractions at N$^3$LO, which are an important ingredient for matching N$^3$LO calculations with parton showers. We obtain the full three-loop structure of the relevant beam and soft functions, which are necessary ingredients for the resummation of $q_T$ and $\cal{T}_0$ at N$^3$LL$'$ and N$^4$LL order, and which constitute important building blocks in other contexts as well. The nonlogarithmic boundary coefficients of the beam functions, which contribute to the integrated subtraction terms, are not yet fully known at three loops. By exploiting consistency relations between different factorization limits, we derive results for the $q_T$ and $\cal{T}_0$ beam function coefficients at N$^3$LO in the $z\to 1$ threshold limit, and we also estimate the size of the unknown terms beyond threshold.

Figures (8)

• Figure 1: Residual scale dependence of the integrated resummed $\mathcal{T}_0$ soft function for $i = q$ (left) and $i = g$ (right). Shown are the relative deviations from the NNLL$'$ result $S_{i,\mathrm{cut}}^\mathrm{central}$ at the central scale $\mu_S = \mathcal{T}_\mathrm{cut}$.
• Figure 2: Residual scale dependence of the resummed integrated $\mathcal{T}_N$ beam function for $i = d$ (top left), $u$ (top right), $\bar{d}$ (bottom left), and $g$ (bottom right). Shown are the relative deviations from the NNLL$'$ result $B_{i,\mathrm{cut}}^\mathrm{central}$ at the central scale $\mu_B = \sqrt{t_\mathrm{cut}}$.
• Figure 3: Comparison of the full beam function coefficients to their leading eikonal (LP) and next-to-eikonal (NLP) expansion at NLO (top) and NNLO (bottom). The $u$-quark channel is shown on the left and the gluon channel on the right. In each case we also show the sum of all nondiagonal partonic channels for comparison.
• Figure 4: Approximate ansatzes for the NNLO (top) and N$^3$LO (bottom) kernels, in the $u$-quark (left) and gluon (right) channels.
• Figure 5: Residual scale dependence of the resummed $q_T$ soft function in Fourier space for $i = q$ (left) and $i = g$ (right). Shown are the relative deviations from the NNLL$'$ result $\tilde{S}_i^\mathrm{central}$ at the central scales $(\mu_S, \nu_S) = (\mu, \nu) = (b_0/b_T, b_0/b_T)$.