Collinear Anomalous Dimensions 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.

Fermionic corrections to quark and gluon form factors in four-loop QCD

We analytically compute all four-loop QCD corrections to the photon-quark and Higgs-gluon form factors involving a closed massless fermion loop. Our calculation of non-planar vertex integrals confirms a previous conjecture for the analytical form of the non-fermionic contributions to the collinear anomalous dimensions of quarks and gluons.

Four-loop collinear anomalous dimensions in QCD and [formula omitted] super Yang-Mills

We calculate the collinear anomalous dimensions in massless four-loop QCD and N=4 supersymmetric Yang-Mills theory from the infrared poles of vertex form factors. We give very precise numerical approximations and a conjecture for the complete analytic results in both models we consider.

Infrared structure of SU(N) \xd7 U(1) gauge theory to three loops

We study the infrared (IR) structure of SU(N ) × U(1) (QCD × QED) gauge theory with nf quarks and nl leptons within the framework of perturbation theory. In particular, we unravel the IR structure of the form factors and inclusive real emission cross sections that contribute to inclusive production of color neutral states, such as a pair of leptons or single W/Z in Drell-Yan processes and a Higgs boson in bottom quark annihilation, in Large Hadron Collider (LHC) in the threshold limit. Explicit computation of the relevant form factors to third order and the use of Sudakov’s K + G equation in SU(N ) × U(1) gauge theory demonstrate the universality of the cusp anomalous dimensions (AI, I = q, b). The abelianization rules that relate AI of SU(N ) with those from U(1) and SU(N) × U(1) can be used to predict the soft distribution that results from the soft gluon emission subprocesses in the threshold limit. Using the latter and the third order form factors, we can obtain the collinear anomalous dimensions (BI) and the renormalisation constant Zb to third order in perturbation theory. The form factors, the process independent soft distribution functions can be used to predict fixed and resummed inclusive cross sections to third order in couplings and in leading logarithmic approximation respectively.

Cusp and Collinear Anomalous Dimensions in Four-Loop QCD from Form Factors.

We calculate the complete quark and gluon cusp anomalous dimensions in four-loop massless QCD analytically from first principles. In addition, we determine the complete matter dependence of the quark and gluon collinear anomalous dimensions. Our approach is to Laurent expand four-loop quark and gluon form factors in the parameter of dimensional regularization. We employ finite field and syzygy techniques to reduce the relevant Feynman integrals to a basis of finite integrals, and subsequently evaluate the basis integrals directly from their standard parametric representations.

Four-loop quark form factor with quartic fundamental colour factor

We analytically compute the four-loop QCD corrections for the colour structure (dFabcd)2 to the massless non-singlet quark form factor. The computation involves non-trivial non-planar integral families which have master integrals in the top sector. We compute the master integrals by introducing a second mass scale and solving differential equations with respect to the ratio of the two scales. We present details of our calculational procedure. Analytical results for the cusp and collinear anomalous dimensions, and the finite part of the form factor are presented. We also provide analytic results for all master integrals expanded up to weight eight.

Anomalous dimension of subleading-power N-jet operators. Part II

We continue the investigation of the anomalous dimension of subleading-power N-jet operators. In this paper, we focus on the operators with fermion number one in each collinear direction, corresponding to quark (antiquark) initiated jets in QCD. We investigate the renormalization effects induced by the soft loop and compute the one-loop mixing of time-ordered products involving power-suppressed SCET Lagrangian insertions into N-jet currents through soft loops. We discuss fermion number conservation in collinear directions and provide explicit results for the collinear anomalous dimension matrix of the currents. The Feynman rules for the power-suppressed SCET interactions in the position-space formalism are collected in an appendix.

The Sudakov form factor at four loops in maximal super Yang-Mills theory

The four-loop Sudakov form factor in maximal super Yang-Mills theory is analysed in detail. It is shown explicitly how to construct a basis of integrals that have a uniformly transcendental expansion in the dimensional regularisation parameter, further elucidating the number-theoretic properties of Feynman integrals. The physical form factor is expressed in this basis for arbitrary colour factor. In the nonplanar sector the required integrals are integrated numerically using a mix of sector-decomposition and Mellin-Barnes representation methods. Both the cusp as well as the collinear anomalous dimension are computed. The results show explicitly the violation of quadratic Casimir scaling at the four-loop order. A thorough analysis concerning the reliability of reported numerical uncertainties is carried out.

The principle of maximal transcendentality and the four-loop collinear anomalous dimension

We use the principle of maximal transcendentality and the universal nature of subleading infrared poles to extract the analytic value of the four-loop collinear anomalous dimension in planar mathcal{N} = 4 super-Yang-Mills theory from recent QCD results, obtaining {widehat{mathcal{G}}}_0^{(4)}=-300{zeta}_7-256{zeta}_2{zeta}_5-384{zeta}_3{zeta}_4 . This value agrees with a previous numerical result to within 0.2%. It also provides the Regge trajectory, threshold soft anomalous dimension and rapidity anomalous dimension through four loops.

Konishi form factor at three loops in N=4 supersymmetric Yang-Mills theory

We present the first results on the third order corrections to on-shell form factor (FF) of the Konishi operator in ${\cal N}=4$ supersymmetric Yang-Mills theory using Feynman diagrammatic approach in modified dimensional reduction ($\overline {DR}$) scheme. We show that it satisfies KG equation in $\overline {DR}$ scheme while the result obtained in four dimensional helicity (FDH) scheme needs to be suitably modified not only to satisfy the KG equation but also to get the correct ultraviolet (UV) anomalous dimensions. We find that the cusp, soft and collinear anomalous dimensions obtained to third order are same as those of the FF of the half-BPS operator confirming the universality of the infrared (IR) structures of on-shell form factors. In addition, the highest transcendental terms of the FF of Konishi operator are identical to those of half-BPS operator indicating the probable existence of deeper structure of the on-shell FF. We also confirm the UV anomalous dimensions of Konishi operator up to third order providing a consistency check on the both UV and universal IR structures in ${\cal N}=4$.

Four-loop photon quark form factor and cusp anomalous dimension in the large-Nc limit of QCD

We compute the four-loop QCD corrections to the massless quark-anti-quark-photon form factor Fq in the large-Nc limit. From the pole part we extract analytic expressions for the corresponding cusp and collinear anomalous dimensions.

A planar four-loop form factor and cusp anomalous dimension in QCD

We compute the fermionic contribution to the photon-quark form factor to four-loop order in QCD in the planar limit in analytic form. From the divergent part of the latter the cusp and collinear anomalous dimensions are extracted. Results are also presented for the finite contribution. We briefly describe our method to compute all planar master integrals at four-loop order.

The last of the simple remainders

We compute the n-point two-loop form factors of the half-BPS operators Tr(phi_{AB}^n) in N=4 super Yang-Mills for arbitrary n >2 using generalised unitarity and symbols. These form factors are minimal in the sense that the n^{th} power of the scalar field in the operator requires the presence of at least n on-shell legs. Infrared divergences are shown to exponentiate as for amplitudes, reproducing the known cusp and collinear anomalous dimensions at two loops. We define appropriate infrared-finite remainder functions and compute them analytically for all n. The results obtained by using the known expressions of the integral functions involve complicated combinations of Goncharov multiple polylogarithms, but we show that much simpler expressions can in fact be derived using the symbol of transcendental functions. For n=3 we find a very compact remainder function expressed in terms of classical polylogarithms only. For arbitrary n>3 we are able to write all the remainder functions in terms of a single compact building block, expressed as a sum of classical polylogarithms augmented by two multiple polylogarithms. The decomposition of the symbol into specific components is crucial in order to single out a natural combination of multiple polylogarithms. Finally, we analyse in detail the behaviour of these minimal form factors in collinear and soft limits, which deviates from the usual behaviour of amplitudes and non-minimal form factors.

The three-loop form factor in $ \mathcal{N} = {4} $ super Yang-Mills

In this paper we study the Sudakov form factor in N=4 super Yang-Mills theory to the three-loop order. The latter is expressed in terms of planar and non-planar loop integrals. We show that it is possible to choose a representation in which each loop integral has uniform transcendentality. We verify analytically the expected exponentiation of the infrared divergences with the correct values of the three-loop cusp and collinear anomalous dimensions in dimensional regularisation. We find that the form factor in N=4 super Yang-Mills can be related to the leading transcendentality part of the quark and gluon form factors in QCD. We also study the ultraviolet properties of the form factor in D>4 dimensions, and find unexpected cancellations, resulting in an improved ultraviolet behaviour.

BFKL equation for the adjoint representation of the gauge group in the next-to-leading approximation at [formula omitted] SUSY

We calculate the eigenvalues of the next-to-leading kernel for the BFKL equation in the adjoint representation of the gauge group SU(Nc) in the N=4 supersymmetric Yang–Mills model. These eigenvalues are used to obtain the high energy behavior of the remainder function for the 6-point scattering amplitude with the maximal helicity violation in the kinematical regions containing the Mandelstam cut contribution. The leading and next-to-leading singularities of the corresponding collinear anomalous dimension are calculated in all orders of perturbation theory. We compare our result with the known collinear limit and with the recently suggested ansatz for the remainder function in three loops and obtain the full agreement providing that the numerical parameters in this ansatz are chosen in an appropriate way.

On form factors in $ \mathcal{N} = 4 $ SYM

In this paper we study the form factors for the half-BPS operators $\mathcal{O}^{(n)}_I$ and the $\mathcal{N}=4$ stress tensor supermultiplet current $W^{AB}$ up to the second order of perturbation theory and for the Konishi operator $\mathcal{K}$ at first order of perturbation theory in $\mathcal{N}=4$ SYM theory at weak coupling. For all the objects we observe the exponentiation of the IR divergences with two anomalous dimensions: the cusp anomalous dimension and the collinear anomalous dimension. For the IR finite parts we obtain a similar situation as for the gluon scattering amplitudes, namely, apart from the case of $W^{AB}$ and $\mathcal{K}$ the finite part has some remainder function which we calculate up to the second order. It involves the generalized Goncharov polylogarithms of several variables. All the answers are expressed through the integrals related to the dual conformal invariant ones which might be a signal of integrable structure standing behind the form factors.

The quark and gluon form factors to three loops in QCD through to $ \mathcal{O}\left( {{\epsilon^2}} \right) $

We give explicit formulae for the O(eps) and O(eps^2) contributions to the unrenormalised three loop QCD corrections to quark and gluon form factors. These contributions have at most transcendentality weight eight. The O(eps) terms of the three-loop form factors are required for the extraction of the four-loop quark and gluon collinear anomalous dimensions. The O(eps^2) terms represent an irreducible contribution to the finite part of the form factors at four-loops. For the sake of completeness, we also give the contributions to the one and two loop form factors to the same transcendentality weight eight.

Calculation of the quark and gluon form factors to three loops in QCD

We describe the calculation of the three-loop QCD corrections to quark and gluon form factors. The relevant three-loop Feynman diagrams are evaluated and the resulting three-loop Feynman integrals are reduced to a small set of known master integrals by using integration-by-parts relations. Our calculation confirms the recent results by Baikov et al.\ for the three-loop form factors. In addition, we derive the subleading ${\cal O}(\e)$ terms for the fermion-loop type contributions to the three-loop form factors which are required for the extraction of the fermionic contributions to the four-loop quark and gluon collinear anomalous dimensions. The finite parts of the form factors are used to determine the hard matching coefficients for the Drell-Yan process and inclusive Higgs-production in soft-collinear effective theory.

Beyond cusp anomalous dimension from integrability

We study the first sub-leading correction O ( ( ln s ) 0 ) to the cusp (minimal) anomalous dimension in the high spin expansion of finite twist operators in N = 4 SYM theory. Since this approximation is still governed by a linear integral equation (derived already from the Bethe ansatz equations in a previous paper), we finalise it better in order to study the weak and strong coupling regimes. In fact, we emphasise how easily the weak coupling expansion can be obtained, confirms the known four loop results and predicts the higher orders. Eventually, we pay particular attention to the strong coupling regime showing agreement and predictions in comparison with the string expansion; speculations on the ‘universal’ part (upon subtracting the collinear anomalous dimension) are brought forward.

Four-loop collinear anomalous dimension inN=4Yang-Mills theory

We report a calculation in N = 4 Yang-Mills of the four-loop term g (4) in the collinear anomalous dimension g(A) which governs the universal subleading infrared structure of gluon scattering amplitudes. Using the method of obstructions to extract this quantity from the 1/e singularity in the four-gluon iterative relation at four loops, we find g (4) = -1240.9 with an estimated numerical uncertainty of 0.02%. We also analyze the implication of our result for the strong coupling behavior of g(A), finding support for the string theory prediction computed recently by Alday and Maldacena using AdS/CFT.