Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory Till Bargheer Max–Planck–Institut für Gravitationsphysik Albert–Einstein–Institut Potsdam–Golm Germany Uppsala University November 3, 2009 arXiv:0905.3738 (with N. Beisert, W. Galleas, F. Loebbert, T. McLoughlin) November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory -1 / 36
Outline 1 Overview and Motivation 2 Intriguing Results about Amplitudes in N = 4 SYM 3 A Closer Look at Tree-Level Symmetries 4 Outlook: Loops November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 0 / 36
Contents 1 Overview and Motivation 2 Intriguing Results about Amplitudes in N = 4 SYM 3 A Closer Look at Tree-Level Symmetries 4 Outlook: Loops November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 1 / 36
Why N = 4 Super Yang-Mills Theory ◮ N = 4 super Yang-Mills theory (SYM) is a supersymmetric cousin of QCD. ◮ Many things computable due to large amount of symmetry. ◮ Relation to gravity (strings) via the AdS/CFT correspondence. N = 4 SYM as a mathematical toy model (of more realistic theories) whose structure we can eventually understand. November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 1 / 36
Features of N = 4 SYM Theory Symmetry ◮ Superconformal theory, symmetry group psu (2 , 2 | 4) . ◮ Conformal to all orders, β ( g ) = 0 . Duality ◮ Dual to IIB string theory on AdS 5 × S 5 . ◮ Weak ↔ strong duality. ◮ Very successful comparison of string and gauge spectrum. Integrability in the Planar N c → ∞ Limit ◮ Appearance of integrability has led to tremendous progress in the study of anomalous dimensions of local gauge invariant operators. ◮ Will integrability also help us in the case of scattering amplitudes? ◮ Relation between anomalous dimensions and scattering amplitudes: Cusp anomalous dimension! Realistic hopes that the planar theory can be solved to all orders! November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 2 / 36
Scattering Amplitudes in Conformal Theories Conformal: Scattering amplitudes are not well-defined in conformal theories: There are ∞ -ranged interactions (massless gauge bosons) ◮ ⇒ No notion of non-interacting asymptotic states. ◮ ⇒ IR divergences. Regulate: In order to make sense of scattering processes, need to introduce IR-regulator ( D = 4 − 2 ε ) ⇒ Breaks conformal symmetry. ◮ Regulator drops out of tree-level amplitudes. ◮ Loop amplitudes diverge as ε → 0 . 2 1 3 n A n . . . 4 . 5 Scattering amplitudes only make sense in the regulated theory. November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 3 / 36
Contents 1 Overview and Motivation 2 Intriguing Results about Amplitudes in N = 4 SYM 3 A Closer Look at Tree-Level Symmetries 4 Outlook: Loops November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 4 / 36
Color Ordering Color Structure ◮ Device for decomposing amplitudes into smaller gauge-invariant pieces. ◮ Feynman rules factorize into a color part and a kinematical part. ◮ Color factor in amplitudes is a product of traces of color matrices T a . Decomposition In U( N c ) , SU( N c ) gauge theory, product of k traces comes with factor 1 /N k c . ⇒ In the planar (large- N c ) limit, only single-trace terms contribute. ⇒ Amplitudes can be color-decomposed: � Tr( T a σ (1) · · · T a σ ( n ) ) ˆ A n ( { p i , a i } ) ∼ A n ( p σ ( i ) ) . σ ∈ S n / Z n Color-stripped partial amplitudes ˆ A n ◮ Depend only on the kinematics (momenta, polarization, helicity). ◮ Particles have a definite ordering. ◮ Amplitudes are cyclic in their arguments (particles). From now on, consider only the color-stripped amplitudes ˆ A n ! November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 4 / 36
Spinor-Helicity Variables Particles One-particle states specified by: Momentum p µ , Polarization vector ε µ . Fermions ψ A / ¯ ψ A Gluons G ± Scalars φ AB Helicity h ± 1 ± 1 / 2 0 Witten Momentum Spinors [ hep-th/0312171 ] ◮ Four-momenta can be written as 2 × 2 hermitean matrices p a ˙ b = ( σ µ ) a ˙ ⇒ p µ p µ = − det( p a ˙ b p µ , b ) . ◮ Massless on-shell condition: det( p a ˙ b ) = 0 . b as product of two spinors λ a (chiral), ˜ b (antichiral). ⇒ Can write p a ˙ ˙ λ b is hermitean, can choose ˜ Since p a ˙ λ = ± ¯ λ : p a ˙ ˙ k = ± λ a b k ¯ b λ k . Polarization ¯ λ → e − iφ ¯ λ → e iφ λ , ◮ Gauge freedom: λ . ◮ For given helicity h , choice of λ uniquely determines polarization vector ε . Trade ( p µ , ε µ ) for ( λ a , ¯ λ ˙ a , h ) . Efficient variables for amplitudes and symmetries! November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 5 / 36
The Simplest Tree Amplitudes Gluon Amplitudes: h k = ± 1 ◮ Can classify amplitudes by helicities of external particles. ◮ Treat all particles as incoming (changes sign of h for outgoing ones). ˆ A ++++ ... + = 0 (supersymmetry) [ Grisaru, Pendleton, v. Nieuwenhuizen ][ Grisaru Pendleton ] n A + ... + − + ... + ˆ = 0 (supersymmetry) n ˆ A + ... + − + ... + − + ... + MHV ( M aximal H elicity V iolating) n ˆ A + ... + − + ... + − + ... + − + ... + Next-to-MHV n · · · MHV Gluon Amplitudes [ Parke Extremely simple form at tree level! Taylor ][ Berends Giele ] δ 4 ( P ) � jk � 4 ˆ MHV , tree A ∼ � 12 �� 23 �� 34 � · · · � n 1 � , n n � λ k ¯ Lorentz invariants: � jk � := λ a j ε ab λ b Total momentum: P = λ k , k . k =1 Cachazo, Feng ][ Britto, Cachazo Other Tree-Level Amplitudes [ Britto Feng, Witten ] Tree-level amplitudes can be constructed recursively: BCFW relations. November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 6 / 36
Loop Amplitudes Loop Technology ◮ Compute loop amplitudes efficiently: Generalized unitarity. [ Dunbar, Kosower ] Bern, Dixon Planar four-leg amplitude to four loops, [ Bern, Dixon Smirnov ][ Bern, Czakon, Dixon Kosower, Smirnov ] Six-leg amplitude at two loops. [ Bern, Dixon, Kosower, Roiban Spradlin, Vergu, Volovich ][ 0903.3526 ] Vergu ◮ Massless scattering amplitudes problematic: Single massless particle can decay into collinear particles. ⇒ IR divergences when integrating over collinear momenta. ⇒ Amplitudes divergent as ε → 0 , typically 1 /ε 2 at each loop order. Exponentiation ◮ IR divergences exponentiate into divergent prefactor. [ Tejeda-Yeomans ] Sterman A All − loop = A divergent ( ε ) · A finite Symmetries (conformal) of remainder might be broken by regulator. ◮ Simplification in the planar limit N c → ∞ ? Symmetry enhancement? November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 7 / 36
Loop Amplitudes: BDS Conjecture Observation in the planar limit N c → ∞ : [ Anastasiou, Bern Dixon, Kosower ][ Bern, Dixon Smirnov ] Higher loops of 4-gluon amplitude have iterative structure: ◮ Each loop order only depends on previous loop orders. ◮ Surprising! Usually new functional dependence at each loop level. BDS Conjecture : Finite part of n -gluon amplitude vanishes to all loops: � � A n ( p ) ∼ A tree 2 D cusp ( λ ) M (1) ( p ) exp n ( p ) + F ( p ) . n , one-loop M (1) Depends only on tree-level A tree and cusp dimension D cusp ( λ ) . n n Cusp Dimension ◮ D cusp ( λ ) is independent of particle number and kinematics! ◮ Integrability: All-order integral equation for D cusp ( λ ) . [ Staudacher ] Eden ◮ Expansions for weak and strong coupling. [ Beisert, Eden Staudacher ][ Basso, Korchemsky ] Kotanski Correctness BDS conjecture falsified for n ≥ 6 . Bern, Dixon, Kosower Drummond, Henn . . . [ Roiban, Spradlin, Vergu, Volovich ][ Korchemsky, Sokatchev ] November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 8 / 36
Amplitudes in the AdS/CFT Dual AdS/CFT: Equates N = 4 SYM and type IIB strings on AdS 5 × S 5 . [ Maldacena 9711200 ] Gluon Scattering [ Maldacena ] Alday ◮ Gluons ↔ Lightlike open strings ending on D3-branes at z = ∞ . ◮ IR regularization: One D3-brane x µ away from the boundary, z = z IR . ◮ z IR → ∞ : Amplitude dominated by minimal surface over boundary. [ Gross Mende ] ◮ “T-Dualize” x µ → y µ : Boundary of surface becomes polygonal: − → ◮ Polygon composed of lightlike gluon momenta: y j − y j − 1 = p j . AdS/CFT: Polygon is Wilson loop in N = 4 SYM! String amplitude ≡ VEV of Wilson loop. Computation of the minimal surface agrees with the BDS conjecture, taking the cusp anomalous dimension (computed from integrability) at strong coupling. November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 9 / 36
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