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Abstract We present a factorization theorem of the partonic Drell-Yan off-diagonal processes g q ¯ qg → γ ∗ + X $$ g\overline{q}(qg)\to {\gamma}^{\ast }+X $$ in the kinematic threshold regime z = Q 2 / s ̂ → 1 $$ z={Q}^2/\hat{s}\to 1 $$ at general subleading powers in the (1 − z) expansion. Focusing on the first order of the expansion (next-to-leading power accuracy with respect to the leading power q q ¯ $$ q\overline{q} $$ channel), we validate the bare factorization formula up to O α s 2 $$ \mathcal{O}\left({\alpha}_s^2\right) $$ . This is achieved by carrying out an explicit calculation of the generalized soft function in d-dimensions using the reduction to master integrals and the differential equations method. The collinear function is a universal object which we compute from an operator matching equation at one-loop level. Next, we integrate the soft and collinear functions over the convolution variables and remove the remaining initial state collinear singularities through PDF renormalization. The resulting expression agrees with the known cross section in the literature.

Abstract We calculate the generalized soft functions at O $$ \mathcal{O} $$ ( α s 2 $$ {\alpha}_s^2 $$ ) at next-to-leading power accuracy for the Drell-Yan process at threshold. The operator definitions of these objects contain explicit insertions of soft gauge and matter fields, giving rise to a dependence on additional convolution variables with respect to the leading power result. These soft functions constitute the last missing ingredient for the validation of the bare factorization theorem to NNLO accuracy. We carry out the calculations by reducing the soft squared amplitudes into a set of canonical master integrals and we employ the method of differential equations to evaluate them. We retain the exact d-dimensional dependence of the convolution variables at the integration boundaries in order to regulate the fixed-order convolution integrals. After combining the soft functions with the relevant collinear functions, we perform checks of the results at the cross-section level against the literature and expansion-by-regions calculations, at NNLO and partly at N3LO, finding agreement.

Abstract The off-diagonal parton-scattering channels g + γ * and q + ϕ * in deep-inelastic scattering are power-suppressed near threshold x → 1. We address the next-to-leading power (NLP) resummation of large double logarithms of 1 − x to all orders in the strong coupling, which are present even in the off-diagonal DGLAP splitting kernels. The appearance of divergent convolutions prevents the application of factorization methods known from leading power resummation. Employing d-dimensional consistency relations from requiring 1/ϵ pole cancellations in dimensional regularization between momentum regions, we show that the resummation of the off-diagonal parton-scattering channels at the leading logarithmic order can be bootstrapped from the recently conjectured exponentiation of NLP soft-quark Sudakov logarithms. In particular, we derive a result for the DGLAP kernel in terms of the series of Bernoulli numbers found previously by Vogt directly from algebraic all-order expressions. We identify the off-diagonal DGLAP splitting functions and soft-quark Sudakov logarithms as inherent two-scale quantities in the large-x limit. We use a refactorization of these scales and renormalization group methods inspired by soft-collinear effective theory to derive the conjectured soft-quark Sudakov exponentiation formula.

Abstract We present a factorization theorem valid near the kinematic threshold z = Q 2 / s ̂ → 1 $$ z={Q}^2/\hat{s}\to 1 $$ of the partonic Drell-Yan process q q ¯ → γ ∗ + X $$ q\overline{q}\to {\gamma}^{\ast }+X $$ for general subleading powers in the (1 − z) expansion. We then consider the specific case of next-to-leading power. We discuss the emergence of collinear functions, which are a key ingredient to factorization starting at next-to-leading power. We calculate the relevant collinear functions at O α s $$ \mathcal{O}\left({\alpha}_s\right) $$ by employing an operator matching equation and we compare our results to the expansion-by- regions computation up to the next-to-next-to-leading order, finding agreement. Factorization holds only before the dimensional regulator is removed, due to a divergent convolution when the collinear and soft functions are first expanded around d = 4 before the convolution is performed. This demonstrates an issue for threshold resummation beyond the leading-logarithmic accuracy at next-to-leading power.

Abstract We sum the leading logarithms α s n ln 2 n − 1 1 − z $$ {\alpha}_s^n{\ln}^{2n-1}\left(1-z\right) $$ , n = 1, 2, . . . , near the kinematic threshold z = m H 2 / s ̂ → 1 $$ z={m}_H^2/\hat{s}\to 1 $$ at next-to-leading power in the expansion in (1 − z) for Higgs production in gluon fusion. We highlight the new contributions compared to Drell-Yan production in quark-antiquark annihilation and show that the final result can be obtained to all orders by the substitution of the colour factor C F → C A , confirming previous fixed-order results. We also provide a numerical analysis of the next-to-leading power leading logarithms, which indicates that they are numerically relevant.

Abstract We resum the leading logarithms α s n ln2n − 1(1 − z), n = 1, 2, . . . near the kine-matic threshold z = Q 2/ŝ → 1 of the Drell-Yan process at next-to-leading power in the expansion in (1 − z). The derivation of this result employs soft-collinear effective theory in position space and the anomalous dimensions of subleading-power soft functions, which are computed. Expansion of the resummed result leads to the leading logarithms at fixed loop order, in agreement with exact results at NLO and NNLO and predictions from the physical evolution kernel at N3LO and N4LO, and to new results at the five-loop order and beyond.

We present a method of merging the exclusive LO-matched high energy resummation of High Energy Jets (HEJ) with the parton shower of PYTHIA which preserves the accuracy of the LO cross sections and the logarithmic accuracy of both resummation schemes across all of phase space. Predictions produced with this merging prescription are presented with comparisons to data from experimental studies and suggestions are made for further observables and experimental cuts which highlight the importance of both high energy and soft-collinear effects.