Massless Partons Research Articles

Radiation from a gluon-gluino colour-singlet dipole at N3LO

We compute the quantum chromodynamics (QCD) corrections to the decay of a neutralino to gluinos and partons at next-to-next-to-next-to-leading order (N3LO) in the strong coupling constant αs, integrated separately over the phase-space of two, three, four or five particles in the final state. The resulting matrix elements are related to the quark-gluon antenna functions, completing the set of integrated antenna functions in final-final kinematics required for the extension of the antenna subtraction scheme to N3LO. For a model with massless partons (quarks and gluons) and an arbitrary number of adjoint fermions, we obtained the following new results to Oαs3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O}\\left({\\alpha}_s^3\\right) $$\\end{document}: the inclusive cross section for the decay of a neutralino, the renormalization of the effective coupling of a neutralino to a gluino and gluon(s), the gluino collinear anomalous dimension, the gluino contribution to the parton collinear anomalous dimensions and the neutralino-gluino-gluon(s) vertex form factors. A special case of this model is the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 super Yang-Mills theory, where we can relate some of our findings to known results.

Top tree amplitudes for higher order calculations

We present compact analytic results for tree-level amplitudes containing a tt¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ t\\overline{t} $$\\end{document} pair accompanied by up to four massless partons, tt¯gg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ t\\overline{t} gg $$\\end{document}, tt¯ggg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ t\\overline{t} ggg $$\\end{document}, tt¯gggg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ t\\overline{t} gggg $$\\end{document}, tt¯qq¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ t\\overline{t}q\\overline{q} $$\\end{document}, tt¯qq¯g\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ t\\overline{t}q\\overline{q}g $$\\end{document}, tt¯qq¯gg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ t\\overline{t}q\\overline{q} gg $$\\end{document} and tt¯qq¯q′q¯′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ t\\overline{t}q\\overline{q}{q}^{\\prime }{\\overline{q}}^{\\prime } $$\\end{document}. The results, obtained using BCFW on-shell recursion, are based both on previous published results and on the new calculations performed in this paper. These amplitudes are sufficient to calculate the production of a tt¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ t\\overline{t} $$\\end{document} pair and zero, one, or two light parton jets, with the option to include the tree-level decays t → bνe+ and t¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{t} $$\\end{document} → b¯e\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{b}e $$\\end{document}−ν¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{\ u} $$\\end{document} efficiently. Our results are part of the NNLO corrections to tt¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ t\\overline{t} $$\\end{document} production including the decay correlations for on-shell top quarks.

Infrared Singularities of Multileg QCD Amplitudes with a Massive Parton at Three Loops

We derive the structure of three-loop anomalous dimensions governing infrared singularities of QCD amplitudes with one massive and an arbitrary number of massless external partons. The contributions of tripole and quadrupole correlations involving a massive parton are studied in detail. The analytical expression of tripole correlations between one massive and two massless partons is obtained at three loops for the first time. We regularize the infrared divergences in the soft matrix element in a novel approach, where no extra scale dependence is involved, and the calculation can be performed in momentum space. Our results are essential to improve the theoretical predictions of single top and top quark pair productions at hadron colliders.

A massive variable flavour number scheme for the Drell-Yan process

The prediction of differential cross-sections in hadron-hadron scattering processes is typically performed in a scheme where the heavy-flavour quarks (c, b, tc,b,t) are treated either as massless or massive partons. In this work, a method to describe the production of colour-singlet processes which combines these two approaches is presented. The core idea is that the contribution from power corrections involving the heavy-quark mass can be numerically isolated from the rest of the massive computation. These power corrections can then be combined with a massless computation (where they are absent), enabling the construction of differential cross-section predictions in a massive variable flavour number scheme. As an example, the procedure is applied to the low-mass Drell-Yan process within the LHCb fiducial region, where predictions for the rapidity and transverse-momentum distributions of the lepton pair are provided. To validate the procedure, it is shown how the n_fnf-dependent coefficient of a massless computation can be recovered from the massless limit of the massive one. This feature is also used to differentially extract the massless N^3LON3LO coefficient of the Drell-Yan process in the gluon-fusion channel.

Higgs production in bottom-quark fusion: Matching beyond leading order

We compute the total cross-section for Higgs boson production in bottom-quark fusion using the so-called FONLL method for the matching of a scheme in which the b-quark is treated as a massless parton to that in which it is treated as a massive final-state particle, and extend our previous results to the case in which the next-to-next-to-leading-log five-flavor scheme result is combined with the next-to-leading-order O(αs3) four-flavor scheme computation.

Analytic integration of soft and collinear radiation in factorised QCD cross sections at NNLO

Within the framework of local analytic sector subtraction, we present the full analytic integration of double-real and real-virtual local infrared counterterms that enter NNLO QCD computations with any number of massless final-state partons. We show that a careful choice of phase-space mappings leads to simple analytic results, including non-singular terms, that can be obtained with conventional integration techniques.

Three-loop soft function for energetic electroweak boson production at hadron colliders

We calculate the three-loop soft function for the production of an electroweak boson (Higgs, γ, W±, Z) with large transverse momentum at a hadron collider. It is the first time a soft function for a three-parton process is computed at next-to-next-to-next-to-leading order (N3LO). As a technical novelty, we perform the calculation in terms of forward-scattering-type loop diagrams rather than evaluating phase space integrals. Our three-loop result contains color-tripole contributions and explicitly confirms predictions on the universal infrared structure of QCD scattering amplitudes with three massless parton legs. The soft function is a central ingredient in the factorized cross section for electroweak boson production near the kinematic endpoint (threshold), where the invariant mass of the recoiling hadronic radiation is small compared to its transverse momentum. Our result is required for predictions of the near-threshold cross sections at N3LO and for the resummation of threshold logarithms at primed next-to-next-to-next-to-leading logarithmic (N3LL′) accuracy.

Higgs Boson Production in Bottom-Quark Fusion to Third Order in the Strong Coupling.

We present the inclusive cross section at next-to-next-to-next-to-leading order (N^{3}LO) in perturbative QCD for the production of a Higgs boson via bottom-quark fusion. We employ the five-flavor scheme, treating the bottom quark as a massless parton while retaining a nonvanishing Yukawa coupling to the Higgs boson. We find that the dependence of the hadronic cross section on the renormalization and factorization scales is substantially reduced. For judicious choices of the scales the perturbative expansion of the cross section shows a convergent behavior. We present results for the N^{3}LO cross section at various collider energies. In comparison to the cross section obtained from the Santander matching of the four- and five-flavor schemes, we predict a slightly higher cross section, though the two predictions are consistent within theoretical uncertainties.

A forest formula to subtract infrared singularities in amplitudes for wide-angle scattering

For any hard QCD amplitude with massless partons, infrared (IR) singularities arise from pinches in the complex planes of loop momenta, called pinch surfaces. To organize and study their leading behaviors in the neighborhoods of these surfaces, we can construct approximation operators for collinear and soft singularities. A BPHZ-like forest formula can be developed to subtract them systematically.In this paper, we utilize the position-space analysis of Erdoğan and Sterman for Green functions, and develop the formalism for momentum space. A related analysis has been carried out by Collins for the Sudakov form factors, and is generalized here to any wide-angle kinematics with an arbitrary number of external momenta. We will first see that the approximations yield much richer IR structures than those of an original amplitude, then construct the forest formula and prove that all the singularities appearing in its subtraction terms cancel pairwise. With the help of the forest formula, the full amplitude can also be reorganized into a factorized expression, which helps to generalize the Sudakov form factor result to arbitrary numbers of external momenta. All our analysis will be on the amplitude level.

Tree-level splitting amplitudes for a quark into four collinear partons

We compute in conventional dimensional regularisation the tree-level splitting amplitudes for a quark parent in the limit where four partons become collinear to each other. This is part of the universal infrared behaviour of the QCD scattering amplitudes at next-to-next-to-next-to-leading order (N3LO) in the strong coupling constant. Further, we consider the iterated limit when m′ massless partons become collinear to each other within a bigger set of m collinear partons, as well as the limits when one gluon or a q overline{q} pair or two gluons become soft within a set of m collinear partons.

Triple (and quadruple) soft-gluon radiation in QCD hard scattering

We consider the radiation of three soft gluons in a generic process for multi- parton hard scattering in QCD. In the soft limit the corresponding scattering amplitude has a singular behaviour that is factorized and controlled by a colorful soft current. We compute the tree-level current for triple soft-gluon emission from both massless and massive hard partons. The three-gluon current is expressed in terms of maximally non-abelian irreducible correlations. We compute the soft behaviour of squared amplitudes and the colour correlations produced by the squared current. The radiation of one and two soft gluons leads to colour dipole correlations. Triple soft-gluon radiation produces in addition colour quadrupole correlations between the hard partons. We examine the soft and collinear singularities of the squared current in various energy ordered and angular ordered regions. We discuss some features of soft radiation to all-loop orders for processes with two and three hard partons. Considering triple soft-gluon radiation from three hard partons, colour quadrupole interactions break the Casimir scaling symmetry between quarks and gluons. We also present some results on the radiation of four soft gluons from two hard partons, and we discuss the colour monster contribution and its relation with the violation (and generalization) of Casimir scaling. We also compute the first correction of \U0001d4aa(1/ {N}_c^2 ) to the eikonal formula for multiple soft-gluon radiation with strong energy ordering from two hard gluons.

Relating amplitude and PDF factorisation through Wilson-line geometries

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.

Fitting the b-quark PDF as a massive-b scheme: Higgs production in bottom fusion

We show that a simple and accurate approach to the computation of hadron collider processes involving initial-state b quarks can be obtained by introducing an independently parametrized b PDF. We use the so-called FONLL method for the matching of a scheme in which the b quark is treated as a massless parton to that in which it is treated as a massive state, and extend it to the case in which the b quark PDF is not necessarily determined by perturbative matching conditions. This generalizes to hadronic collisions analogous results previously obtained for deep-inelastic scattering. The results corresponds to a “massive b” scheme, in which b mass effects are retained, yet the b quark is endowed with a PDF. We specifically study Higgs production in bottom fusion, and show that our approach overcomes difficulties related to the fact that in a standard massive four-flavor scheme b-quark induced processes only start at high perturbative orders.

Anisotropic flows and the shear viscosity of the QGP within an event-by-event massive parton transport approach

We have developed an event-by-event relativistic kinetic transport approach to study the build up of the anisotropic flows v_{n}(p_T) for a system at fixed eta /s(T). The partonic approach describe the evolution of massless partons which imply epsilon =3p as Equation of State (EoS). We extend previous studies to finite partonic masses tuned to simulate a system that expand with an EoS close to the recent lQCD results. We study the role of EoS and the effect of eta /s(T) ratio on the build up of v_n(p_T) up to n=5 for two beam energies: RHIC energies at sqrt{s}=200 GeV and LHC energies at sqrt{s}=2.76 TeV. We find that for the two beam energies considered the suppression of the v_n(p_T) due to the viscosity of the medium have different contributions coming from the cross over or QGP phase. We shows that in ultra-central collisions (0–0.2%) the v_n(p_T) have a stronger sensitivity to the T dependence of eta /s that increases with the order of the harmonic n. Finally, we discuss the results for the integrated flow harmonics langle v_{n} rangle in ultra-central collisions pointing-out how the relative strength of langle v_{n} rangle depend on the colliding energies as well as on the freeze-out dynamics.

Z boson production in bottom-quark fusion: a study of b-mass effects beyond leading order

We compute the total cross-section for Z boson production in bottom-quark fusion, applying to this case the method we previously used for Higgs production in bottom fusion. Namely, we match, through the FONLL procedure, the next-to-next-to-leading-log five-flavor scheme result, in which the b quark is treated as a massless parton, with the next-to-leading-order mathcal{O}(alpha _s^3) four-flavor scheme computation in which bottom is treated as a massive final-state particle. Also, we add to our formalism the possibility of varying the heavy quark matching scale. The results obtained with the FONLL formalism can thus be compared directly to recent results obtained in various approximations, and used as a proxy to assess and discuss the issues of scale dependence and treatment of heavy quarks. Finally, We use our results in order to improve the prediction for the total Z production cross-section.

Heavy quark jet fragmentation

In this paper we study the fragmentation of a parton into a jet containing a heavy quark. When heavy quarks are involved in a jet, the quark mass can lead to a numerically significant correction to the jet cross section and its substructure. With this motivation, we calculated the heavy quark mass effects to next-to-leading order in αs on the fragmentation functions to a jet (FFJs) and the jet fragmentation functions (JFFs), where the former describes fragmentation of parton into a jet and the latter describes fragmenting processes inside a jet. The finite size of the heavy quark mass does not change the ultraviolet behaviors, but it can give significant corrections to the finite contributions. When we take the zero mass limit, we find that the FFJs and the JFFs reproduce established results for massless partons. If we define the heavy quark jet as one that include at least one heavy (anti-)quark, the tagged heavy quark jet production is sensitive to the heavy quark mass and produces large logarithms of the mass. Taking advantage of the FFJs and JFFs, we formulate a factorization theorem for heavy quark jet production in order to resum these large logarithms systematically. As an application, we study inclusive b-jet production and show phenomenological implications due to keeping a non-zero quark mass.

The double-soft integral for an arbitrary angle between hard radiators

We consider the double-soft limit of a generic QCD process involving massless partons and integrate analytically the double-soft eikonal functions over the phase-space of soft partons (gluons or quarks) allowing for an arbitrary relative angle between the three-momenta of two hard massless radiators. This result provides one of the missing ingredients for a fully analytic formulation of the nested soft-collinear subtraction scheme described in Caola et al. (Eur Phys J C 77(4):248, 2017).

Initial State fluctuations from midperipheral to ultracentral collisions in a transport approach

We study the build up of the anisotropic flows vn for a fluid at fixed η/s(T) by means of an event-by-event transport approach. Usually in the partonic approach are used massless partons implying a ε = 3p Equation of State (EoS). In this paper we extend previous studies to finite partonic masses tuned to simulate a fluid with an EoS close to the recent lQCD results. In particular we study the role of the equation of state and the effect of the η/s ratio and its T dependence on the build up of the vn (pT) for Au+Au collisions at and for Pb + Pb collisions at . We find that the sensitivity to the EoS of the vn (pT) increase with the order of the harmonics n. In particular we find a mass ordering for the elliptic flow at low pT. We find that for the two different beam energies considered the suppression of the vn (pT) due to the viscosity of the medium have different contributions coming from the cross over or QGP phase depending on the collision energies. The study reveals that in ultra-central collisions (0 − 0.2%) the vn (pT) have a stronger sensitivity to the T dependence of η/s in the QGP phase and this sensitivity increases with the order of the harmonic n.

An automated subtraction of NLO EW infrared divergences

In this paper a generalisation of the Catani–Seymour dipole subtraction method to next-to-leading order electroweak calculations is presented. All singularities due to photon and gluon radiation off both massless and massive partons in the presence of both massless and massive spectators are accounted for. Particular attention is paid to the simultaneous subtraction of singularities of both QCD and electroweak origin which are present in the next-to-leading order corrections to processes with more than one perturbative order contributing at Born level. Similarly, embedding non-dipole-like photon splittings in the dipole subtraction scheme discussed. The implementation of the formulated subtraction scheme in the framework of the Sherpa Monte-Carlo event generator, including the restriction of the dipole phase space through the alpha -parameters and expanding its existing subtraction for NLO QCD calculations, is detailed and numerous internal consistency checks validating the obtained results are presented.

$b$ Quark Mass Effects in Associated Production

In this work we study an extension of the commonly used 5F scheme, where $b$ quarks are treated as massless partons, in which full mass effects are retained in both the initial and in the final state. We name this scheme 5F massive scheme (5FMS). We implement this scheme in the \Sherpa Monte Carlo event generator at MEPS@NLO accuracy, and we compare it for two relevant cases for the LHC: $b\bar{b} \rightarrow H$ and $pp\rightarrow Zb$.