Collinear Limits Research Articles

Moyal deformations, W1+∞ and celestial holography

We consider the Moyal deformation of self-dual gravity. In the conformal primary basis, holomorphic collinear limits of the amplitudes of this theory show that it enjoys a perturbatively exact symmetry algebra LW∧ that generalises Lw∧, the loop algebra of the wedge algebra of w1+∞, which appears in self-dual gravity.

Transverse momentum structure of proton within the basis light-front quantization framework

We obtain the leading-twist valence quark transverse-momentum-dependent parton distribution functions (TMD PDFs) for the proton within the basis light-front quantization (BLFQ) framework. Our results are consistent with lattice QCD calculations and our previous results for the collinear limit. We also obtain consistency with the Soffer-type bounds. Within our approach, we find that six T-even TMDs in the leading twist are all independent of each other, and previously found model-dependent relations do not hold. This is a promising sign that our results are representative of future, more extensive treatments of QCD. Furthermore, we obtain a non-trivial x-dependence of the 〈(p⊥)2〉 and some consistency with the Gaussian ansatz but only in the small (p⊥)2 region. Those features suggest our results may be a useful alternative in future experimental extractions.

Three-Point Energy Correlator in N=4 Supersymmetric Yang-Mills Theory.

An analytic formula is given for the three-point energy correlator at leading order in maximally supersymmetric Yang-Mills theory (N=4 sYM). This is the first analytic calculation of a three-parameter event shape observable, which provides valuable data for various studies ranging from conformal field theories to jet substructure. The associated class of functions define a new type of single-valued polylogarithms characterized by 16 alphabet letters, which manifest a D_{6}×Z_{2} dihedral symmetry of the event shape. With the unexplored simplicity in the perturbative structure of the three-point energy correlator, all kinematic regions including collinear, squeezed and coplanar limits are now available.

Pentagon Wilson loop with Lagrangian insertion at two loops in mathcal{N} = 4 super Yang-Mills theory

We compute the two-loop result for the null pentagonal Wilson loop with a Lagrangian insertion (normalized by the Wilson loop without insertion) in planar, maximally supersymmetric Yang-Mills theory. This finite observable is closely related to the Amplituhedron, and it is reminiscent of finite parts of planar two-loop five-particle scattering amplitudes. We verify that, up to this loop order, the leading singularities are given by the same conformally invariant expressions that appear in all-plus pure Yang-Mills amplitudes. The accompanying weight-four transcendental functions are expressed in terms of the pentagon functions space known from planar two-loop five-particle amplitudes, but interestingly only a subset of the functions appears. Being a function of four dimensionless variables, the observable has interesting asymptotic limits. We verify that our analytic result is consistent with soft and collinear limits, and find an intriguingly simple pattern in the multi-Regge limit. Thanks to the new result we can also conjecturally predict, for general kinematics, the maximal weight piece of the planar three-loop five-particle all-plus amplitude in pure Yang-Mills theory. Motivated by the Amplituhedron geometry, we investigate positivity properties of the integrated answer. Generalizing previous results at four particles, we find numerical evidence that the two-loop five-particle result has uniform sign in a kinematic region suggested by the loop Amplituhedron.

Next-to-leading power endpoint factorization and resummation for off-diagonal “gluon” thrust

The lack of convergence of the convolution integrals appearing in next-to-leading-power (NLP) factorization theorems prevents the applications of existing methods to resum power-suppressed large logarithmic corrections in collider physics. We consider thrust distribution in the two-jet region for the flavour-nonsinglet off-diagonal contribution, where a gluon-initiated jet recoils against a quark-antiquark pair, which is power-suppressed. With the help of operatorial endpoint factorization conditions, we obtain a factorization formula, where the individual terms are free from endpoint divergences in convolutions and can be expressed in terms of renormalized hard, soft and collinear functions in four dimensions. This allows us to perform the first resummation of the endpoint-divergent SCETI observables at the leading logarithmic accuracy using exclusively renormalization-group methods. The presented approach relies on universal properties of the soft and collinear limits and may serve as a paradigm for the systematic NLP resummation for other 1 → 2 and 2 → 1 collider physics processes.

NNLO photon fragmentation within antenna subtraction

We report on our recent progress towards including the photon fragmentation contribution in next-to-next-to-leading order (NNLO) QCD predictions for photon production cross sections. This extension to previous NNLO calculations requires the identification of the photon in singular parton-photon collinear limits. We discuss how these limits can be subtracted within antenna subtraction using fragmentation antenna functions and we outline their integration.

The Effective Vector Boson Approximation in high-energy muon collisions

Due to the inclination for forward gauge radiation, lepton colliders beyond a few TeV are effectively electroweak (EW) boson colliders, suggesting the treatment of EW bosons as constituents of high-energy leptons. In the context of a muon collider, we revisit the validity of W and Z parton distribution functions (PDFs) at leading order in 2 → n process. We systematically investigate universal and quasi-universal power-law and logarithmic corrections that arise when deriving (polarized) weak boson PDFs in the collinear limit. We go on to survey a multitude of 2 → n processes at sqrt{s} = 2–30 TeV via polarized and unpolarized EW boson fusion/scattering. To conduct this study, we report a public implementation of the Effective W/Z and Weizsäcker-Williams Approximations, which we collectively call the Effective Vector Boson Approximation, into the Monte Carlo event generator MadGraph5_aMC@NLO. This implementation lays the groundwork for developing matrix-element matching prescriptions involving EW parton showers and renormalized EW PDFs. To further with this agenda, we give recommendations on using W/Z PDFs.

Universal structure of radiative QED amplitudes at one loop

We present two novel results about the universal structure of radiative QED amplitudes in the soft and in the collinear limit. On the one hand, we extend the well-known Low-Burnett-Kroll theorem to the one-loop level and give the explicit relation between the radiative and non-radiative amplitude at subleading power in the soft limit. On the other hand, we consider a factorisation formula at leading power in the limit where the emitted photon becomes collinear to a light fermion and provide the corresponding one-loop splitting function. In addition to being interesting in their own right these findings are particularly relevant in the context of fully-differential higher-order QED calculations. One of the main challenges in this regard is the numerical stability of radiative contributions in the soft and collinear regions. The results presented here allow for a stabilisation of real­virtual amplitudes in these delicate phase-space regions by switching to the corresponding approximation without the need of explicit computations.

Soft spin correlations in final-state parton showers

We introduce a simple procedure that resolves the long-standing question of how to account for single-logarithmic spin-correlation effects in parton showers not just in the collinear limit, but also in the soft wide-angle limit, at leading colour. We discuss its implementation in the context of the PanScales family of parton showers, where it complements our earlier treatment of the purely collinear spin correlations. Comparisons to fixed-order matrix elements help validate our approach up to third order in the strong coupling, and an appendix demonstrates the small size of residual subleading-colour effects. To help probe wide-angle soft spin correlation effects, we introduce a new declustering-based non-global spin-sensitive observable, the first of its kind. Our showers provide a reference for its single-logarithmic resummation. The work in this paper represents the last step required for final-state massless showers to satisfy the broad PanScales next-to-leading logarithmic accuracy goals.

One-loop same helicity YM amplitudes from BG currents

We propose and prove a new formula for the one-loop all same helicity Yang-Mills amplitudes. These amplitudes are seen to arise as a sum of products of two tree-level Berends-Giele currents connected by an effective propagator. To make sense of the propagators one needs to introduce the so-called region, or dual momenta. The formula is proven by observing that it readily implies the correct collinear limit properties. The only non-trivial part of the proof is establishing that our formula for the amplitude is invariant under shifts of the region momenta.

Celestial OPEs and w1+\u221e algebra from worldsheet in string theory

Celestial operator product expansions (OPEs) arise from the collinear limit of scattering amplitudes and play a vital role in celestial holography. In this paper, we derive the celestial OPEs of massless fields in string theory from the worldsheet. By studying the worldsheet OPEs of vertex operators in worldsheet CFT and further examining their behaviors in the collinear limit, we find that new vertex operators for the massless fields in string theory are generated and become dominant in the collinear limit. Mellin transforming to the conformal basis yields exactly the celestial OPEs in celestial CFT. We also derive the celestial OPEs from the collinear factorization of string amplitudes and the results derived in these two different methods are in perfect agreement with each other. Our final formulae of celestial OPEs are applicable to general dimensions, corresponding to Einstein-Yang-Mills theory supplemented by some possible higher derivative interactions. Specializing to 4D, we reproduce all the celestial OPEs for gluon and graviton in the literature. We consider various string theories, including the open and closed bosonic string, as well as the closed superstring theory with mathcal{N} = 1 and mathcal{N} = 2 worldsheet supersymmetry. In the case of mathcal{N} = 2 string, we also derive all the overline{mathrm{SL}left(2,mathbb{R}right)} descendant contributions in the celestial OPE; the soft sector of such OPE just yields the w1+∞ algebra after rewriting in terms of chiral modes. Our stringy derivation of celestial OPEs thus initiates the first step towards the microscopic realization of celestial CFT dual to string theory in flat spacetime.

Celestial operator product expansions and w1+\u221e symmetry for all spins

The operator product expansion of massless celestial primary operators of arbitrary spin is investigated. Poincaré symmetry is found to imply a set of recursion relations on the operator product expansion coefficients of the leading singular terms at tree-level in a holomorphic limit. The symmetry constraints are solved by an Euler beta function with arguments that depend simply on the right-moving conformal weights of the operators in the product. These symmetry-derived coefficients are shown not only to match precisely those arising from momentum-space tree-level collinear limits, but also to obey an infinite number of additional symmetry transformations that respect the algebra of w1+∞. In tree-level minimally-coupled gravitational theories, celestial currents are constructed from light transforms of conformally soft gravitons and found to generate the action of w1+∞ on arbitrary massless celestial primaries. Results include operator product expansion coefficients for fermions as well as those arising from higher-derivative non-minimal couplings of gluons and gravitons.

Dissecting the collinear structure of quark splitting at NNLL

We explore the collinear limit of final-state quark splittings at order {alpha}_s^2 . While at general NLL level, this limit is described simply by a product of leading-order 1 → 2 DGLAP splitting functions, at the NNLL level we need to consider 1 → 3 splitting functions. Here, by performing suitable integrals of the triple-collinear splitting functions, we demonstrate how one may extract {mathrm{mathcal{B}}}_2^q(z) , a differential version of the coefficient {mathrm{mathcal{B}}}_2^q that enters the quark form factor at NNLL and governs the intensity of collinear radiation from a quark. The variable z corresponds to the quark energy fraction after an initial 1 → 2 splitting, and our results yield effective higher-order splitting functions, which may be considered as a step towards the construction of NNLL parton showers. Further, while in the limit z → 1 we recover the standard soft limit results involving the CMW coupling with scale kt, the z dependence we obtain also motivates the extension of the notion of a physical coupling beyond the soft limit.

Bootstrapping a Two-Loop Four-Point Form Factor.

We compute the two-loop four-point form factor of a length-3 half-BPS operator in planar N=4 SYM theory, which belongs to the class of two-loop five-point scattering observables with one off-shell color-singlet leg. A new bootstrapping strategy is developed to obtain this result by starting with an ansatz expanded in terms of master integrals and then solving the master coefficients via various physical constraints. We find that consistency conditions of infrared divergences and collinear limits, together with the cancellation of spurious poles, can fix a significant part of the ansatz. The remaining degrees of freedom can be fixed by one simple type of two-double unitarity cut. Full analytic results in terms of both symbol and Goncharov polylogarithms are provided.

Celestial superamplitude in mathcal{N} = 4 SYM theory

Celestial amplitude is a new reformulation of momentum space scattering amplitudes and offers a promising way for flat holography. In this paper, we study the celestial amplitudes in mathcal{N} = 4 Super-Yang-Mills (SYM) theory aiming at understanding the role of superconformal symmetry in celestial holography. We first construct the superconformal generators acting on the celestial superfield which assembles all the on-shell fields in the multiplet together in terms of celestial variables and Grassmann parameters. These generators satisfy the superconformal algebra of mathcal{N} = 4 SYM theory. We also compute the three-point and four-point celestial super-amplitudes explicitly. They can be identified as the conformal correlation functions of the celestial superfields living at the celestial sphere. We further study the soft and collinear limits which give rise to the super-Ward identity and super-OPE on the celestial sphere, respectively. Our results initiate a new perspective of understanding the well-studied mathcal{N} = 4 SYM amplitudes via 2D celestial conformal field theory.

The two-loop remainder function for eight and nine particles

Two-loop MHV amplitudes in planar mathcal{N} = 4 supersymmetric Yang Mills theory are known to exhibit many intriguing forms of cluster-algebraic structure. We leverage this structure to upgrade the symbols of the eight- and nine-particle amplitudes to complete analytic functions. This is done by systematically projecting onto the components of these amplitudes that take different functional forms, and matching each component to an ansatz of multiple polylogarithms with negative cluster-coordinate arguments. The remaining additive constant can be determined analytically by comparing the collinear limit of each amplitude to known lower-multiplicity results. We also observe that the nonclassical part of each of these amplitudes admits a unique decomposition in terms of a specific A3 cluster polylogarithm, and explore the numerical behavior of the remainder function along lines in the positive region.

Spin-orbit coupling from a two-component self-consistent approach. II. Non-collinear density functional theories.

We revise formal and numerical aspects of collinear and non-collinear density functional theories in the context of a two-component self-consistent treatment of spin-orbit coupling. Theoretical and numerical analyses of the non-collinear approaches confirm their ability to yield the proper collinear limit and provide rotational invariance of the total energy for functionals in the local-density or generalized-gradient approximations (GGAs). Calculations on simple molecules corroborate the formal considerations and highlight the importance of an effective screening algorithm to provide the sufficient level of numerical stability required for a rotationally invariant implementation of non-collinear GGA functionals. The illustrative calculations provide a first numerical comparison of both previously proposed non-collinear formulations for GGA functionals. The proposed screening procedure allows us to effectively deal with points of small magnetization, which would otherwise be problematic for the evaluation of the exchange-correlation energy and/or potential for non-collinear GGA functionals. Both previously suggested formulations for the non-collinear GGA are confirmed to be adequate for total energy calculations, provided that the screening is achieved on a sufficiently fine grid. All methods are implemented in the Crystal program.

Non-abelian infrared divergences on the celestial sphere

We consider the infrared factorisation of non-abelian multi-particle scattering amplitudes, and we study the form of the universal colour operator responsible for infrared divergences, when expressed in terms of coordinates on the ‘celestial sphere’ intersecting the future light-cone at asymptotic distances. We find that colour-dipole contributions to the infrared operator, to all orders in perturbation theory, have a remarkably simple expression in these coordinates, with scale and coupling dependence factorised from kinematics and colour. Generalising earlier suggestions in the abelian theory, we then show that the infrared operator can be computed as a correlator of vertex operators in a conformal field theory of Lie-algebra-valued free bosons on the celestial sphere. We verify by means of the OPE that the theory correctly predicts the all-order structure of collinear limits, and the tree-level factorisation of soft real radiation.

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

We compute in conventional dimensional regularisation the tree-level splitting amplitudes for a gluon parent which splits into four collinear partons. 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. Combined with our earlier results for a quark parent, this completes the set of tree-level splitting amplitudes required at this order. We also study iterated collinear limits where a subset of the four collinear partons become themselves collinear.

Transverse momentum dependent PDFs at N3LO

We compute the quark and gluon transverse momentum dependent parton distribution functions at next-to-next-to-next-to-leading order (N3LO) in perturbative QCD. Our calculation is based on an expansion of the differential Drell-Yan and gluon fusion Higgs production cross sections about their collinear limit. This method allows us to employ cutting edge multiloop techniques for the computation of cross sections to extract these universal building blocks of the collinear limit of QCD. The corresponding perturbative matching kernels for all channels are expressed in terms of simple harmonic polylogarithms up to weight five. As a byproduct, we confirm a previous computation of the soft function for transverse momentum factorization at N3LO. Our results are the last missing ingredient to extend the qT subtraction methods to N3LO and to obtain resummed qT spectra at N3LL′ accuracy both for gluon as well as for quark initiated processes.