In the HEFT, we consider the Regge limit of the two-loop amplitudes for Higgs boson production in association with a jet, expanded to NNLL accuracy. We discuss the issue of the Regge cuts versus poles in this context, showing that the former cannot contribute through three loops, due to the simplicity of the colour structure of the amplitudes. Finally, we determine for the first time the Higgs impact factor at two-loop accuracy.

Italy

Vittorio Duca

Switzerland

Giulio Falcioni

The two-loop Higgs impact factor

High-energy factorization of 2 → 2 amplitudes in QCD has been recently pushed to the next-to-next-to-leading logarithmic order by determining the three-loop gluon Regge trajectory. This was based on computing multi-Reggeon exchanges using rapidity evolution in the shock-wave formalism, and disentangling between the Regge pole and Regge cut contributions. In the present paper we extend the relevant theoretical framework to 2 → 3 processes, and compute all multi-Reggeon exchanges necessary for extracting the two-loop Reggeon-gluon-Reggeon Lipatov vertex from 2 → 3 amplitudes. Then, specializing general amplitude methods to multi-Regge kinematics, we derive analytic expressions for non-planar two-loop $\textit{gg}$ → $\textit{ggg}$, $\textit{gq}$ → $\textit{ggq}$ and $\textit{qq}$ → $\textit{qgq}$ QCD amplitudes in that limit. Matching these to the multi-Reggeon computation, we determine the QCD Lipatov vertex in dimensional regularization at two loops through finite terms. We also determine the one-loop vertex through $$ \mathcal{O} $$ ($\textit{ϵ}$ ). All results are expressed in a compact form in terms of a basis of single-valued generalised polylogarithms, manifesting target-projectile symmetry and reality properties. Furthermore, our basis of functions is explicitly finite in the soft limit, featuring delicate cancellation of spurious rational poles by transcendental functions. Agreement between all three partonic channels, as well agreement of the maximal weight contributions with the super Yang-Mills Lipatov vertex provide robust checks of the result.

not found

Samuel Abreu

Giuseppe Laurentis

Einan Gardi

Calum Milloy

Leonardo Vernazza

The two-loop Lipatov vertex in QCD

We provide the all-loop structure of gauge-variant operators required for the renormalisation of Green’s functions with insertions of twist-two operators in Yang-Mills theory. Using this structure we work out an explicit basis valid up to 4-loop order for an arbitrary compact simple gauge group. To achieve this we employ a generalised gauge symmetry, originally proposed by Dixon and Taylor, which arises after adding to the Yang-Mills Lagrangian also operators proportional to its equation of motion. Promoting this symmetry to a generalised BRST symmetry allows to generate the ghost operator from a single exact operator in the BRST-generalised sense. We show that our construction complies with the theorems by Joglekar and Lee. We further establish the existence of a generalised anti-BRST symmetry which we employ to derive non-trivial relations among the anomalous dimension matrices of ghost and equation-of-motion operators. For the purpose of demonstration we employ the formalism to compute the N = 2, 4 Mellin moments of the gluonic splitting function up to 4 loops and its N = 6 Mellin moment up to 3 loops, where we also take advantage of additional simplifications of the background field formalism.

United Kingdom

Franz Herzog

Renormalization of gluonic leading-twist operators in covariant gauges

The high-energy limit of gauge-theory amplitudes features both a Regge pole and Regge cuts. We show how to disentangle these, and hence how to determine the Regge trajectory beyond two loops. While the nonplanar part of multiple Reggeon <math><mi>t</mi></math>-channel exchange forms a Regge cut, the planar part contributes to the pole along with the single Reggeon. With this, we find that the infrared singularities of the trajectory are given by the cusp anomalous dimension. By matching to recent QCD results, we determine the quark and gluon impact factors to two loops and the Regge trajectory to three loops.

Niamh Maher

CERN

Disentangling the Regge Cut and Regge Pole in Perturbative QCD

Using rapidity evolution equations we study two-to-two gauge-theory scattering amplitudes in the Regge limit. We carry out explicit computations at next-to-next-to-leading logarithmic accuracy through four loops and present new results for both infrared-singular and finite contributions to the amplitude. New techniques are devised in order to derive the colour structure stemming from three-Reggeon exchange diagrams in terms of commutators of channel operators, obtaining results that are valid for any gauge group, and apply to scattered particles in any colour representation. We also elucidate the separation between contributions to the Regge cut and Regge pole in the real part of the amplitude to all loop orders. We show that planar contributions due to multiple-Reggeon exchange diagrams can be factorised as a Regge pole along with the single-Reggeon exchange, and when this is done, the singular part of the gluon Regge trajectory is directly determined by the cusp anomalous dimension. We explicitly compute the Regge cut component of the amplitude through four loops and show that it is non-planar. From a different perspective, the new results provide important information on soft singularities in general kinematics beyond the planar limit: by comparing the computed corrections to the general form of the four-loop soft anomalous dimension we derive powerful constraints on its kinematic dependence, opening the way for a bootstrap-based determination.

Scattering amplitudes in the Regge limit and the soft anomalous dimension through four loops

We show that an iterative solution of rapidity evolution equations with a leading-order kernel renders the entire signature-odd tower of next-to-next-to-leading logarithmic (NNLL) contributions to partonic <math><mrow><mn>2</mn><mo>→</mo><mn>2</mn></mrow></math> amplitudes in the Regge limit computable in full color. The result, which universally holds in any gauge theory, corresponds to a set of three-Reggeon ladder diagrams. In this setting we compute the NNLL amplitude to four loops in dimensional regularization through finite corrections. Furthermore, by contrasting the result with the exponentiation properties of soft singularities we determine the four-loop correction to the soft anomalous dimension at this logarithmic accuracy. The latter features quartic Casimir contributions beyond those appearing in the cusp anomalous dimension. Finally, in the case of <math><mrow><mi>N</mi><mo>=</mo><mn>4</mn></mrow></math> super Yang-Mills, we also determine the finite hard function at four loops through NNLL in full color.

Climbing three-Reggeon ladders: Four-loop amplitudes in the high-energy limit in full color

We apply a fully automated extension of the <math><msup><mi>R</mi><mo>*</mo></msup></math> operation capable of calculating higher-loop anomalous dimensions of <math><mi>n</mi></math>-point Green’s functions of arbitrary, possibly nonrenormalizable, local quantum field theories. We focus on the case of the <math><mi>C</mi><mi>P</mi></math>-violating Weinberg operator of the Standard Model effective field theory whose anomalous dimension is so far known only at one loop. We calculate the two-loop anomalous dimension in full QCD and the three-loop anomalous dimensions in the limit of pure Yang-Mills theory. We find sizable two-loop and large three-loop corrections, due to the appearance of a new quartic group invariant. We discuss phenomenological implications for electric dipole moments and future applications of the method.

United States

Jordy de Vries

Netherlands

Ben Ruijl

Two- and three-loop anomalous dimensions of Weinberg’s dimension-six <math><mi>C</mi><mi>P</mi></math>-odd gluonic operator

We study long-distance singularities governing different physical quantities involving massless partons in perturbative QCD by using factorisation in terms of Wilson-line correlators. By isolating the process-independent hard-collinear singularities from quark and gluon form factors, and identifying these with the ones governing the elastic limit of the perturbative Parton Distribution Functions (PDFs) — δ(1 − x) in the large-x limit of DGLAP splitting functions — we extract the anomalous dimension controlling soft sin- gularities of the PDFs, verifying that it admits Casimir scaling. We then perform an independent diagrammatic computation of the latter using its definition in terms of Wil- son lines, confirming explicitly the above result through two loops. By comparing our eikonal PDF calculation to that of the eikonal form factor by Erdogan and Sterman and the classical computation of the closed parallelogram by Korchemsky and Korchemskaya, a consistent picture emerges whereby all singularities emerge in diagrammatic configurations localised at the cusps or along lightlike lines, but where distinct contributions to the anomalous dimensions are associated with finite (closed) lightlike segments as compared to infinite (open) ones. Both are relevant for resumming large logarithms in physical quantities, notably the anomalous dimension controlling Drell-Yan or Higgs production near threshold on the one hand, and the gluon Regge trajectory controlling the high-energy limit of partonic scattering on the other.

Relating amplitude and PDF factorisation through Wilson-line geometries

We calculate analytically the flavor non-singlet O(αs2) massive Wilson coefficients for the inclusive neutral current non-singlet structure functions F1,2,Lep(x,Q2) and g1,2ep(x,Q2) and charged current non-singlet structure functions F1,2,3ν(ν¯)p(x,Q2) , at general virtualities Q2 in the deep-inelastic region. Numerical results are presented. We illustrate the transition from low to large virtualities for these observables, which may be contrasted to basic assumptions made in the so-called variable flavor number scheme. We also derive the corresponding results for the Adler sum rule, the unpolarized and polarized Bjorken sum rules and the Gross–Llewellyn Smith sum rule. There are no logarithmic corrections at large scales Q2 and the effects of the power corrections due to the heavy quark mass are of the size of the known O(αs4) corrections in the case of the sum rules. The complete charm and bottom corrections are compared to the approach using asymptotic representations in the region Q2≫mc,b2 . We also study the target mass corrections to the above sum rules.

Germany

Johannes Blümlein

Abilio De Freitas

The complete O(αs2) non-singlet heavy flavor corrections to the structure functions g1,2ep(x,Q2) , F1,2,Lep(x,Q2) , F1,2,3ν(ν¯)(x,Q2) and the associated sum rules

We provide a complete and detailed study of the high-energy limit of four-parton scattering amplitudes in QCD, giving explicit results at two loops and higher orders, and going beyond next-to-leading logarithmic (NLL) accuracy. Building upon recent results, we use the techniques of infrared factorization to investigate the failure of the simplest form of Regge factorization, starting at next-to-next-to-leading logarithmic accuracy (NNLL) in ln( s/ | t |). We provide detailed accounts and explicit expressions for the terms responsible for this breaking in the case of two-loop and three-loop quark and gluon amplitudes in QCD; in particular, we recover and explain a known non-logarithmic double-pole contribution at two-loops, and we compute all non-factorizing single-logarithmic singular contributions at three loops. Conversely, we use high-energy factorization to show that the hard functions of infrared factorization vanish in d = 4 to all orders in the coupling, up to NLL accuracy in ln( s/ | t |). This provides clear evidence for the infrared origin of high-energy logarithms. Finally, we extend earlier studies to t -channel exchanges of color representations beyond the octet, which enables us to give predictions based on the dipole formula for single-pole NLL contributions at three and four loops.