Scattering amplitudes are expected to admit a factorised structure in special kinematic limits, such as the Regge, soft and collinear limits. However, less is known about the precise mechanisms through which factorisation of $\textit{n}$-particle scattering amplitudes is realised at high perturbative orders, where more complex structures arise. Starting with the soft anomalous dimension, in this work we investigate the multi-particle collinear limits of massless amplitudes at three- and four-loop orders. Using colour conservation and rescaling symmetry, we show how strict collinear factorisation of multiple massless final-state coloured particles is realised, and provide results for the corresponding splitting amplitude soft anomalous dimensions. In particular, we demonstrate through four loops that the conditions on the structure of the massless soft anomalous dimension that are required by strict collinear factorisation in all two-particle collinear limits, are sufficient to guarantee such factorisation also in any multiple collinear limit. Then, assuming that strict collinear factorisation of massless partons holds also for amplitudes containing massive coloured particles, we derive new constraints on the soft anomalous dimension from multi-collinear limits.

Germany

Claude Duhr

United Kingdom

Einan Gardi

Switzerland

Sebastian Jaskiewicz

Jonas Lübken

Italy

Leonardo Vernazza

Infrared singularities and the collinear limits of multi-leg scattering amplitudes

The consistency of collinear factorization violation with PDF factorization has been an outstanding challenge and subject of considerable debate. In this work we demonstrate their compatibility using a factorization theorem for non-global jet observables. Our analysis relies on consistency relations derived from renormalization conditions in effective field theory. We verify these relations through an explicit computation at three-loop order and show that the double-logarithmic evolution sourcing the super-leading logarithms reduces to single-logarithmic DGLAP running below the lowest perturbative scale. The crucial ingredient reconciling the two evolutions is a perturbative Glauber contribution to the low-energy matrix elements which breaks soft-collinear factorization at the cross section level but restores PDF factorization.

Thomas Becher

Patrick Hager

United States

Matthias Neubert

Dominik Schwienbacher

Low-energy theory of jet processes and PDF factorization

The leading and next-to-leading order QCD predictions for Higgs boson pair production at hadron colliders suffer from a significant mass renormalisation scheme uncertainty related to the choice of the top-quark mass. The functional dependence of the result on the value of the intermediate quark mass can be understood in the high-energy limit using the Method of Regions and the tools of Soft-Collinear Effective Theory. In this work, we study the origin of the sizeable logarithmic mass corrections in the $\textit{gg}$ → $\textit{HH}$ amplitudes at leading and next-to-leading power in the limit $\textit{s}$, |$\textit{t}$|, |$\textit{u}$| ≫ $$ {m}_t^2 $$ ≫ $$ {m}_H^2 $$ . We argue that the mass corrections follow a predictable factorised pattern that can be exploited to simplify their computation. We present results for the leading power leading logarithmic corrections, our analysis leads to a significant reduction in the theoretical uncertainty of the double Higgs production amplitudes at scattering energies ≳ 1 TeV due to the top-quark mass scheme.

Stephen Jones

Robert Szafron

Yannick Ulrich

The structure of quark mass corrections in the $\textit{gg}$ → $\textit{HH}$ amplitude at high-energy

<p>We analyze the low-energy dynamics of gap-between-jets cross sections at hadron colliders, for which phase factors in the hard amplitudes spoil collinear cancellations and lead to double (“super-leading”) logarithmic behavior. Based on a method-of-regions analysis, we identify three-loop contributions from perturbative active-active Glauber-gluon exchanges with the right structure to render the cross section consistent with PDF factorization below the gap veto scale. The Glauber contributions we identify are unambiguously defined without regulators beyond dimensional regularization.</p>

Factorization Restoration through Glauber Gluons

We present a factorization theorem of the partonic Drell-Yan off-diagonal processes   <math> <mi>g</mi> <mover> <mi>q</mi> <mo>¯</mo> </mover> <mfenced> <mi>qg</mi> </mfenced> <mo>→</mo> <msup> <mi>γ</mi> <mo>∗</mo> </msup> <mo>+</mo> <mi>X</mi> </math>  $$ g\overline{q}(qg)\to {\gamma}^{\ast }+X $$  in the kinematic threshold regime   <math> <mi>z</mi> <mo>=</mo> <msup> <mi>Q</mi> <mn>2</mn> </msup> <mo>/</mo> <mover> <mi>s</mi> <mo>̂</mo> </mover> <mo>→</mo> <mn>1</mn> </math>  $$ z={Q}^2/\hat{s}\to 1 $$  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   <math> <mi>q</mi> <mover> <mi>q</mi> <mo>¯</mo> </mover> </math>  $$ q\overline{q} $$  channel), we validate the bare factorization formula up to   <math> <mi>O</mi> <mfenced> <msubsup> <mi>α</mi> <mi>s</mi> <mn>2</mn> </msubsup> </mfenced> </math>  $$ \mathcal{O}\left({\alpha}_s^2\right) $$ . 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.

Austria

Alessandro Broggio

Threshold factorization of the Drell-Yan quark-gluon channel and two-loop soft function at next-to-leading power

We calculate the generalized soft functions at   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (  <math> <msubsup> <mi>α</mi> <mi>s</mi> <mn>2</mn> </msubsup> </math>  $$ {\alpha}_s^2 $$ ) 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.

CERN

Next-to-leading power two-loop soft functions for the Drell-Yan process at threshold

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.

Martin Beneke

Mathias Garny

China

Jian Wang

Large-x resummation of off-diagonal deep-inelastic parton scattering from d-dimensional refactorization

We present a factorization theorem valid near the kinematic threshold   <math> <mi>z</mi> <mo>=</mo> <msup> <mi>Q</mi> <mn>2</mn> </msup> <mo>/</mo> <mover> <mi>s</mi> <mo>̂</mo> </mover> <mo>→</mo> <mn>1</mn> </math>  $$ z={Q}^2/\hat{s}\to 1 $$  of the partonic Drell-Yan process   <math> <mi>q</mi> <mover> <mi>q</mi> <mo>¯</mo> </mover> <mo>→</mo> <msup> <mi>γ</mi> <mo>∗</mo> </msup> <mo>+</mo> <mi>X</mi> </math>  $$ q\overline{q}\to {\gamma}^{\ast }+X $$  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   <math> <mi>O</mi> <mfenced> <msub> <mi>α</mi> <mi>s</mi> </msub> </mfenced> </math>  $$ \mathcal{O}\left({\alpha}_s\right) $$  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.

Threshold factorization of the Drell-Yan process at next-to-leading power

We sum the leading logarithms $$ {\alpha}_s^n{\ln}^{2n-1}\left(1-z\right) $$ , = 1 2 near the kinematic threshold $$ z={m}_H^2/\hat{s}\to 1 $$ at next-to-leading power in the expansion in (1 ) 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 → , 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.

Netherlands

Leading-logarithmic threshold resummation of Higgs production in gluon fusion at next-to-leading power

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.