Fusing Gauge Theory Tree Amplitudes Into Loop Amplitudes

Abstract

We identify a large class of one-loop amplitudes for massless particles that can be constructed via unitarity from tree amplitudes, without any ambiguities. One-loop amplitudes for massless supersymmetric gauge theories fall into this class; in addition, many non-supersymmetric amplitudes can be rearranged to take advantage of the result. As applications, we construct the one-loop amplitudes for $n$-gluon scattering in $N=1$ supersymmetric theories with the helicity configuration of the Parke-Taylor tree amplitudes, and for six-gluon scattering in $N=4$ super-Yang-Mills theory for all helicity configurations.

49 pages, SLAC-PUB-6563

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• Differential equations on unitarity cut surfaces
• Employing Helicity Amplitudes for Resummation
• Amplitude bases in generic EFTs
• Tree-Level Formalism
• Feynman Rules for the Rational Part of the Standard Model One-loop Amplitudes in the 't Hooft-Veltman ${\gamma }_5$ Scheme
• $n$-point amplitudes with a single negative-helicity graviton
• Colour Decompositions of Multi-quark One-loop QCD Amplitudes
• Proof of the Dual Conformal Anomaly of One-Loop Amplitudes in N=4 SYM
• Analytic Two-Loop Higgs Amplitudes in Effective Field Theory and the Maximal Transcendentality Principle
• The rational parts of one-loop QCD amplitudes III: The six-gluon case
• N=8 Supergravity 4-point Amplitudes
• One-loop Amplitudes in Six-Dimensional (1,1) Theories from Generalised Unitarity
• Prescriptive Unitarity for Non-Planar Six-Particle Amplitudes at Two Loops
• Angular Momentum Loss Due to Tidal Effects in the Post-Minkowskian Expansion
• Automating QCD amplitudes with on-shell methods
• N=8 Supergravity on the Light Cone
• Virtuous Trees at Five and Six Points for Yang-Mills and Gravity
• A Course in Amplitudes
• Next-to-Leading Order Gamma Gamma + 2-Jet Production at the LHC
• One-Loop Gluonic Amplitudes from Single Unitarity Cuts
• Amplitudes at Infinity
• Decomposition of one-loop QCD amplitudes into primitive amplitudes based on shuffle relations
• The two-loop n-point all-plus helicity amplitude
• Full Colour for Loop Amplitudes in Yang-Mills Theory
• Graviton Bending in Quantum Gravity from One-Loop Amplitudes
• Non-planar form factors of generic local operators via on-shell unitarity and color-kinematics duality
• Two-loop $\mathcal{N}=2$ SQCD amplitudes with external matter from iterated cuts
• On the associativity of 1-loop corrections to the celestial operator product in gravity
• One-loop monodromy relations on single cuts
• Six-Photon Amplitudes in Scalar QED
• The role of colour flows in matrix element computations and Monte Carlo simulations
• Manifestly Dual-Conformal Loop Integration
• The one-loop amplitudes for Higgs + 4 partons with full mass effects
• The Next-to-Simplest Quantum Field Theories
• The Black Hole Weak Gravity Conjecture with Multiple Charges
• Single Cut Integration
• Simplifying Algebra in Feynman Graphs, Part I:Spinors
• Color Dressed Unitarity and Recursion for Yang-Mills Two-Loop All-Plus Amplitudes
• Extracting analytical one-loop amplitudes from numerical evaluations
• All-Helicity Symbol Alphabets from Unwound Amplituhedra
• FDR, an easier way to NNLO calculations: a two-loop case study
• SUSY Ward identities, Superamplitudes, and Counterterms
• The six-point remainder function to all loop orders in the multi-Regge limit
• Relations of loop partial amplitudes in gauge theory by Unitarity cut method
• Maximal Unitarity for the Four-Mass Double Box
• Helicity Methods for High Multiplicity Subleading Soft and Collinear Limits
• Counting to one: reducibility of one- and two-loop amplitudes at the integrand level
• Symbol Alphabets from Tensor Diagrams
• Full-color three-loop three-point form factors in N=4 SYM
• Simplifying $D$-Dimensional Physical-State Sums in Gauge Theory and Gravity
• $\text{Tr}(F^3)$ supersymmetric form factors and maximal transcendentality Part II: $0<\mathcal{N}<4$ super Yang-Mills
• Unitarity Cuts of Integrals with Doubled Propagators
• S-Matrix Equivalence Theorem Evasion and Dimensional Regularisation with the Canonical MHV Lagrangian
• Three-loop octagons and n-gons in maximally supersymmetric Yang-Mills theory
• Manifesting enhanced cancellations in supergravity: integrands versus integrals
• One-Loop Maximal Helicity Violating Amplitudes in N=4 Super Yang-Mills Theories
• UV cancelations in gravity loop integrands
• Calculation of 1-loop Hexagon Amplitudes in the Yukawa Model
• Direct reduction of multiloop multiscale scattering amplitudes
• Spinor helicity structures in higher spin theories
• Primary Feynman rules to calculate the epsilon-dimensional integrand of any 1-loop amplitude
• Two-Loop Five-Parton Leading-Colour Finite Remainders in the Spinor-Helicity Formalism
• Correlation functions of local composite operators from generalized unitarity
• Building Bases of Loop Integrands
• String-motivated one-loop amplitudes in gauge theories with half-maximal supersymmetry
• MultivariateResidues: a Mathematica package for computing multivariate residues
• Solving for tadpole coefficients in one-loop amplitudes
• On General BCJ Relation at One-loop Level in Yang-Mills Theory
• Finite remainders of the Konishi at two loops in $\mathcal{N}=4$ SYM
• OPE for Super Loops
• Form factor recursion relations at loop level
• The Unitarity Method using a Canonical Basis Approach
• Augmented Recursion For One-loop Gravity Amplitudes
• On Cancellations of Ultraviolet Divergences in Supergravity Amplitudes
• Perfecting one-loop BCJ numerators in SYM and supergravity
• New ideas for handling of loop and angular integrals in D-dimensions in QCD
• Global Residues and Two-Loop Hepta-Cuts
• On the ABJM four-point amplitude at three loops and BDS exponentiation
• Constraints on anomalous dimensions from the positivity of the S-matrix
• The quintic interaction vertex in light-cone gravity
• BCJ Identities and $d$-Dimensional Generalized Unitarity
• R2SM: a package for the analytic computation of the R2 Rational terms in the Standard Model of the Electroweak interactions
• The SAGEX Review on Scattering Amplitudes, Chapter 7: Positive Geometry of Scattering Amplitudes
• Systematics of One-Loop Scattering Amplitudes in N=4 Super Yang-Mills Theories
• NLO QCD Predictions for $Wb\bar{b}$ Production in Association with up to Three Light Jets at the LHC
• Spinning Black Hole Scattering at $\mathcal{O}(G^3{S}^2)$: Casimir Terms, Radial Action and Hidden Symmetry
• Classical off-shell currents
• Gauge Theory Amplitudes, Scalar Graphs and Twistor Space
• On radiative QCD backgrounds to exclusive H -> b bbar production at the LHC and a photon collider
• One-Loop Calculations with BlackHat
• The rational parts of one-loop QCD amplitudes II: The five-gluon case
• Two-loop Sudakov Form Factor in ABJM
• On the Numerical Evaluation of Loop Integrals With Mellin-Barnes Representations
• On the low energy limit of one loop photon-graviton amplitudes
• Computation of multi-leg amplitudes with NJet
• One-Loop NMHV Amplitudes involving Gluinos and Scalars in N=4 Gauge Theory
• Perturbative expansion of N<8 Supergravity
• Prescriptive Unitarity with Elliptic Leading Singularities
• Binary Dynamics from Worldline QFT for Scalar-QED
• Non-planarity through unitarity in ABJM
• Bootstrapping a Two-Loop Four-Point Form Factor
• A minimal approach to the scattering of physical massless bosons
• Color ordering in QCD
• AdS super gluon scattering up to two loops: A position space approach
• Cutting the traintracks: Cauchy, Schubert and Calabi-Yau
• A super MHV vertex expansion for N=4 SYM theory
• Simplifying Algebra in Feynman Graphs, Part III: Massive Vectors
• Unitarity Method with Spurious Pole
• The Noncommutative S-Matrix
• The two-loop MHV amplitudes in N=4 supersymmetric Yang-Mills theory
• Two-Loop anomalous dimensions of QCD operators up to dimension-sixteen and Higgs EFT amplitudes
• A unitarity compatible approach to one-loop amplitudes with massive fermions
• Cross-Order Integral Relations from Maximal Cuts
• Worldsheet factorization for twistor-strings
• One-Loop QCD and Higgs to Partons Processes Using Six-Dimensional Helicity and Generalized Unitarity
• Gravity and Form Scattering and Renormalisation of Gravity in Six and Eight Dimensions
• An Elliptic Yangian-Invariant, `Leading Singularity'
• Review of AdS/CFT Integrability, Chapter V.1: Scattering Amplitudes - a Brief Introduction
• On a possibility of a UV finite N=8 supergravity
• Scattering Amplitudes and Toric Geometry
• A tale of tails through generalized unitarity
• Double-copy towards supergravity inflation with $\alpha$-attractor models
• The Q-cut Representation of One-loop Integrands and Unitarity Cut Method
• Hidden Analytic Relations for Two-Loop Higgs Amplitudes in QCD
• Scattering Amplitudes and N-Body Post-Minkowskian Hamiltonians in General Relativity and Beyond
• Determination of Boundary Contributions in Recursion Relation
• Gravitational Dyonic Amplitude at One-Loop and its Inconsistency with the Classical Impulse
• An $n$-point QCD two-loop amplitude
• Complex Factorisation and Recursion for One-Loop Amplitudes
• Frequency-domain approach to self-force in hyperbolic scattering
• Radiative Classical Gravitational Observables at $\mathcal{O}(G^3)$ from Scattering Amplitudes
• EFT matching from analyticity and unitarity
• Two-Loop QCD Corrections to the Higgs plus three-parton amplitudes with Top Mass Correction
• N=4 SYM NMHV Loop Amplitude in Superspace
• Integrands for QCD rational terms and N=4 SYM from massive CSW rules
• All-Multiplicity One-Loop Amplitudes in Born-Infeld Electrodynamics from Generalized Unitarity
• Quantum corrections to frame-dragging in scattering amplitudes
• Multi-spin soft bootstrap and scalar-vector Galileon
• On the two-loop hexagon Wilson loop remainder function in N=4 SYM
• Constructing Gravity Amplitudes from Real Soft and Collinear Factorisation
• Analytic Tadpole Coefficients of One-loop Integrals
• Global Poles of the Two-Loop Six-Point N=4 SYM integrand
• External leg corrections in the unitarity method
• R2 vertices for the effective ggH theory
• Generation Function For One-loop Tensor Reduction
• The Coulomb-branch S-matrix from massless amplitudes
• Non-factorisable contribution to $t$-channel single-top production
• On-shell diagrammatics and the perturbative structure of planar gauge theories
• Quantum corrections to tidal Love number for Schwarzschild black holes
• Structure functions at small x from world-lines. I: Unpolarized distributions
• Massive Quark-Gluon Scattering Amplitudes at Tree Level
• Does one need the O(epsilon)- and O(epsilon^2)-terms of one-loop amplitudes in an NNLO calculation ?
• Global Structure of Curves from Generalized Unitarity Cut of Three-loop Diagrams
• One-loop Feynman Integral Reduction by Differential Operators
• Cutting-Edge Tools for Cutting Edges
• Random polarisations of the dipoles
• All rational one-loop Einstein-Yang-Mills amplitudes at four points
• Dual-Conformal Regularization of Infrared Loop Divergences and the Chiral Box Expansion
• First-quantized N=4 Yang-Mills
• Automated Evaluation of One-Loop Six-Point Processes for the LHC
• Generalised Unitarity At One-Loop With Massive Fermions
• Form Factor and Boundary Contribution of Amplitude
• Consistency conditions from generalized-unitarity
• One-loop renormalization and the S-matrix
• PV-Reduction of Sunset Topology with Auxiliary Vector
• Imaginary parts and discontinuities of Wilson line correlators
• One-loop N=8 Supergravity Amplitudes from MHV Diagrams
• The BlackHat Library for One-Loop Amplitudes
• Reduction of General One-loop Integrals Using Auxiliary Vector
• Note on Bonus Relations for N=8 Supergravity Tree Amplitudes
• Feynman Rules for the Rational Part of One-loop QCD Corrections in the MSSM
• A compact representation of the 2 photon 3 gluon amplitude
• Gravity loop integrands from the ultraviolet
• A Forward Branching Phase-Space Generator
• Recent Progress in Perturbative Quantum Field Theory
• On Perturbative Gravity and Gauge Theory
• Effective Field Theory Islands from Perturbative and Nonperturbative Four-Graviton Amplitudes
• Integrability and unitarity
• On the Classification of Residues of the Grassmannian
• Complete Form Factors in Yang-Mills from Unitarity and Spinor Helicity in Six Dimensions
• Factorization of cubic vertices involving three different higher spin fields
• Tree-level amplitudes and dual superconformal symmetry
• Positivity Sectors and the Amplituhedron
• MHV amplitudes in N=2 SQCD and in N=4 SYM at one loop
• Computing real time correlation functions on a hybrid classical/quantum computer
• A Procedure g5anchor to Anchor ${\gamma }_5$ in Feynman Diagrams for the Standard Model
• Hidden Beauty in Multiloop Amplitudes
• Tree amplitudes of noncommutative U(N) Yang-Mills Theory
• Construction of Infrared Finite Observables in N=4 Super Yang-Mills Theory
• Manifest SO(N) invariance and S-matrices of three-dimensional N=2,4,8 SYM
• Massive Nonplanar Two-Loop Maximal Unitarity
• Reduction with Degenerate Gram matrix for One-loop Integrals
• Dispersion theoretic calculation of the $H\to Z+\gamma$ amplitude
• Recursion relations, Helicity Amplitudes and Dimensional Regularization
• Cyclic Mario Worlds -- Color-Decomposition for One-Loop QCD
• Anomalous Dimensions via on-shell Methods: Operator Mixing and Leading Mass Effects
• The n-point MHV one-loop Amplitude in N=4 Supergravity
• Module Intersection and Uniform Formula for Iterative Reduction of One-loop Integrals
• Higher point MHV amplitudes in N=4 Supersymmetric Yang-Mills Theory
• BCJ Relations for One-Loop QCD Integral Coefficients
• Scattering amplitudes at strong coupling for 4K gluons
• On-Shell Unitarity Bootstrap for QCD Amplitudes
• Collinear and Soft Limits of Multi-Loop Integrands in N=4 Yang-Mills
• Two-Loop Gravity amplitudes from four dimensional Unitarity
• The spinning self-force EFT: 1SF waveform recursion relation and Compton scattering
• A note on on-shell recursion relation of string amplitudes
• Positivity Bounds in Scalar Effective Field Theories at One-loop Level
• Natural Boundaries for Scattering Amplitudes
• Conservative Scattering of Reissner-Nordström Black Holes at Third Post-Minkowskian Order
• Non-planar BCFW Grassmannian Geometries
• Multi-loop Integrand Reduction with Computational Algebraic Geometry
• $W^{+}{W}^{-}$ + 3 Jet Production at the Large Hadron Collider in NLO QCD
• Color-Kinematics Duality for Sudakov Form Factor in Non-Supersymmetric Pure Yang-Mills Theory
• Conformally Regulated Direct Integration of the Two-Loop Heptagon Remainder
• Position-space cuts for Wilson line correlators
• Loop-level double-copy for massive fermions in the fundamental
• A Surprise in the Amplitude/Wilson Loop Duality
• One-loop N = 4 super Yang-Mills scattering amplitudes in d dimensions, relation to open strings and polygonal Wilson loops
• Testing and improving the numerical accuracy of the NLO predictions
• Two-loop Integral Reduction from Elliptic and Hyperelliptic Curves
• Anomalies without an action
• The SAGEX Review on Scattering Amplitudes, Chapter 12: Amplitudes and collider physics
• Poles At Infinity in On-shell Diagrams
• Recursion relations for the general tree-level amplitudes in QCD with massive dirac fields
• Six-jet production at lepton colliders
• On the Dimensional Regularization of QCD Helicity Amplitudes With Quarks
• D3-Brane Loop Amplitudes from M5-Brane Tree Amplitudes
• Universal Treatment of Reduction for One-Loop Integrals in Projective Space
• Emergent unitarity from the amplituhedron
• Higher-order tails and RG flows due to scattering of gravitational radiation from binary inspirals
• Obtaining One-loop Gravity Amplitudes Using Spurious Singularities
• FiniteFieldSolve: Exactly Solving Large Linear Systems in High-Energy Theory
• The dimension-shift conjecture for one-loop amplitudes
• Applications of Subleading-color Amplitudes in N=4 SYM Theory
• 6d Dual Conformal Symmetry and Minimal Volumes in AdS
• Supersymmetric Yang-Mills and Supergravity Amplitudes at One Loop
• The $\mathcal{N}=4$ Supergravity NMHV six-point one-loop amplitude
• Roots of Amplitudes
• Loop Amplitudes in an Extended Gravity Theory
• Nontrivial One-loop Recursive Reduction Relation
• Perturbation theory in (2,2) signature
• Gluonic evanescent operators: classification and one-loop renormalization
• Deciphering the Maximal Transcendentality Principle via Bootstrap
• Helicity conservation in Born-Infeld theory
• Recovering Infinities in Graviton Scattering Amplitudes using Cutkosky rules
• Diagrammar in an Extended Theory of Gravity
• Permutation Relations of Yangian Invariants, Unitarity Cuts, and Scattering Amplitudes
• One-Loop N = 4 Super Yang-Mills Scattering Amplitudes to All Orders in the Dimensional Regularization Parameter
• Dijet Invariant Mass Distribution in Top Quark Hadronic Decay with QCD Corrections
• Expansion of EYM Amplitudes in Gauge Invariant Vector Space
• ALPs, the on-shell way
• Next-to-Leading-Order Corrections to Weak Boson Production with a Massive Quark Jet Pair at Hadron Colliders
• Dual conformal constraints and infrared equations from global residue theorems in N=4 SYM theory
• Twistor and Polytope Interpretations for Subleading Color One-Loop Amplitudes
• Recursive prescription for logarithmic jet rate coefficients
• Reduction of one-loop integrals with higher poles by unitarity cut method
• Summary of Lepton Photon 2011
• Two-loop n-point anomalous amplitudes in N=4 supergravity
• Stable One-Dimensional Integral Representations of One-Loop N-Point Functions in the General Massive Case: I - Three Point Functions
• Spinor formalism for massive fields with half-integral spin
• Illustrations of Integrand-Basis Building at Two Loops
• Gluonic evanescent operators: two-loop anomalous dimensions
• Hard QCD at hadron colliders
• Perturbative Gravity and Twistor Space
• One-Loop Six Gluon Amplitude
• Nonlinear couplings and tree amplitudes in gauge theories
• Automated calculations for multi-leg processes
• Virtual QCD corrections to gluon-initiated diphoton plus jet production at hadron colliders
• A high-precision result for a full-color three-loop three-point form factor in $\mathcal{N}=4$ SYM
• Recursive Approach to One-loop QCD Matrix Elements
• Four photon production at the LHC: an application of 2->4 analytic unitarity
• Observables and Unconstrained Spin Tensor Dynamics in General Relativity from Scattering Amplitudes
• Quantum Effects for Black Holes with On-Shell Amplitudes
• Two-loop all-plus helicity amplitudes for self-dual Higgs boson with gluons via unitarity cut constraints
• Exploiting Twistor Techniques for One-loop QCD Amplitudes
• Scrambling and Entangling Spinning Particles
• Integral Reduction by Unitarity Method for Two-loop Amplitudes: A Case Study
• Generalized unitarity method for unstable particles
• All Two-Loop, Color-Dressed, Six-Point Amplitude Integrands in sYM
• Nonplanar On-shell Diagrams and Leading Singularities of Scattering Amplitudes
• All-loop-orders relation between Regge limits of $\mathcal{N}=4$ SYM and $\mathcal{N}=8$ supergravity four-point amplitudes
• Ultra-Violet Infinities and Counterterms in Higher Dimensional Yang-Mills
• Twistors, strings and supersymmetric gauge theories
• Classical gravitational scattering from a gauge-invariant double copy
• Rational Terms in Theories with Matter
• A Note on Loop Amplitudes in QED
• Minimally helicity violating, maximally simple scalar amplitudes in N=4 SYM
• Analytic structure of one-loop coefficients
• On the non-planar beta-deformed N=4 super-Yang-Mills theory
• Analytic two-Loop four-point form factor of the stress-tensor supermultiplet in $\mathcal{N}=4$ SYM
• Gauge theory one-loop amplitudes and the BCFW recursion relations
• One-loop inelastic amplitudes from tree-level elasticity in 2d
• Holonomies of gauge fields in twistor space 4: functional MHV rules and one-loop amplitudes
• All one-loop NMHV gluon amplitudes in N=1 SYM
• Infrared and transcendental structure of two-loop supersymmetric QCD amplitudes
• Massive particles and unitarity cuts
• Developments in perturbative QCD
• Note on solutions of scattering equations
• Locally Finite Observables in sYM
• Five-Point Two-Loop Amplitudes from Numerical Unitarity
• S-matrix equivalence restored
• On Higgs boson plus gluon amplitudes at one loop
• MHV Lagrangians for Yang--Mills and QCD
• Supersums for all supersymmetric amplitudes
• New N=4 SYM Path Integral
• Symmetric reduction of high-multiplicity one-loop integrals and maximal cuts
• Perturbative Methods for Superconformal Quantum Field Theories in String - Gauge Theory Dualities
• Scattering Amplitudes for Monopoles: Pairwise Little Group and Pairwise Helicity
• Integrated Unitarity for Scattering Amplitudes
• Scattering amplitudes in gauge theories with and without supersymmetry
• Loop Amplitudes in the Coulomb Branch of $\mathcal{N}=4$ Super-Yang-Mills Theory
• Precise evaluation of the Cut Constructible part of one loop amplitudes within D-dimensional unitarity
• Evaluation of Multi-Box Diagrams in Six Dimensions
• QCD corrections to e+ e- --> 4 jets
• On perturbative field theory and twistor string theory
• Infrared Finite Observables in N=8 Supergravity
• The double copy effective action: a quantum (chromodynamics) approach to space-time
• Accelerating Berends-Giele recursion for gluons in arbitrary dimensions over finite fields
• Application of BCFW-recursion relations and the Feynman-tree theorem to the four gluon amplitude with all plus helicities
• Recursive relations for a quiver gauge theory
• One-Loop Amplitudes in Supersymmetric QCD from MHV Vertices
• Unitarity Cuts: NLO Six-Gluon Amplitudes in QCD
• Two-loop Splitting Amplitudes
• Effective Field Theories for the LHC
• A Recursive Method to Calculate UV-divergent Parts at One-Loop Level in Dimensional Regularization
• Integrands of Less-Supersymmetric Yang-Mills at One Loop
• Computational Particle Physics for Event Generators and Data Analysis
• On-shell recursion relations for generic integrands
• Cross Section Evaluation by Spinor Integration I: The massless case in 4D
• Analytic Four-Point Lightlike Form Factors and OPE of Null-Wrapped Polygons
• Recursive structures in string field theory
• Quantum MHV Diagrams
• The Radial Action from Probe Amplitudes to All Orders
• Perturbative Unitarity Calls for An Action
• Quantum point particle approximation of spinning black holes and compact stars
• Analytic Scattering Amplitudes for QCD
• General One-loop Reduction in Generalized Feynman Parametrization Form
• Brane Tilings as On-shell Diagrams
• Feynman diagrams in terms of on-shell propagators
• Applications of higher order QCD
• Forward scattering amplitudes in the imaginary time formalism
• Equivalence of Coefficients Extraction of One-loop Master Integrals
• Constructing the general partial waves and renormalization in EFT
• Two-loop leading-color QCD corrections for Higgs plus two-jet production in the heavy-top limit
• Two-loop corrections to $gg\to \gamma \gamma$
• One-loop central-emission vertex for two gluons in $\mathcal{N}=4$ super Yang-Mills theory
• Next-to-leading order QCD corrections to ttbb production at the LHC
• On-shell Methods for Form Factors in N=4 SYM and Their Applications
• Cross Section Evaluation by Spinor Integration II: The massive case in 4D
• Leading singularities in higher-derivative Yang-Mills theory and quadratic gravity
• A new method for the numerical evaluation of one-loop amplitudes
• Extended DBI and its generalizations from graded soft theorems
• Co-homology of Differential Forms and Feynman diagrams
• A Tale Of Two Amplitudes In High Energy Physics
• Three-loop color-kinematics duality: A 24-dimensional solution space induced by new generalized gauge transformations
• On-shell Correlators and Color-Kinematics Duality in Curved Symmetric Spacetimes