Tree-level scattering amplitudes in N = 4 SYM from integrability Tomek Łukowski Mathematical Institute, University of Oxford New Geometric Structures in Scattering Amplitudes University of Oxford 23.09.2014 Based on : L. Ferro, TŁ, C. Meneghelli, J. Plefka, M. Staudacher – 1212.0850 L. Ferro, TŁ, C. Meneghelli, J. Plefka, M. Staudacher – 1308.3494 N. Kanning, TŁ, M. Staudacher – 1403.3382 L. Ferro, TŁ, M. Staudacher – 1407.6736 Tomek Łukowski (University of Oxford) 23.09.2014 1 / 20
Introduction Main focus : Understand and use integrable structures present in four-dimensional quantum field theories. Quantum integrability – concept originating from 1+1 dimensional quantum systems. → Existence of an infinite dimensional symmetry. Integrability in 1+3 dimensions: integrable structures come from some dual two-dimensional description. Focus on the planar limit of maximally supersymmetric Yang-Mills theory ( N = 4 SYM) in four dimensions: scaling dimensions ↔ energies of worldsheet excitations [many authors, 2003-] polygonal Wilson loops ↔ GKP string excitations [Benjamin’s and Pedro’s talks] scattering amplitudes at strong coupling ↔ minimal surfaces [Alday, Maldacena, Sever, Vieira] scattering amplitudes at weak coupling ↔ inhomogeneous spin chains [this talk] Tomek Łukowski (University of Oxford) 23.09.2014 2 / 20
Our motivation Integrability proved its usefulness in finding all-loop and finite coupling results for scaling dimensions of gauge invariant operators. We hope the history will repeat itself for scattering amplitudes. We aim in constraining or constructing scattering amplitudes using powerful tools of integrable models, e. g. quantum inverse scattering method (QISM). Amplitudes suffer from infrared divergencies. Most popular method to regulate – dimensional regularization. Away from four dimensions large part of the nice structure disappears. Spectral parameters promise a new way of regulating divergencies while staying in four dimensions! Tomek Łukowski (University of Oxford) 23.09.2014 3 / 20
Amplitudes in N = 4 SYM We consider color-ordered scattering amplitudes of superfields Φ = G + + ˜ η C ǫ ABCD Γ D + 1 η D ǫ ABCD G − η A Γ A + 1 η A ˜ η B S AB + 1 η A ˜ η B ˜ η A ˜ η B ˜ η C ˜ 2 ! ˜ 3 ! ˜ 4 ! ˜ The amplitudes A n , k are labeled by two numbers: number of particles – n η 4 k , MHV level – ˜ k = 2 , . . . n − 2 , η 4 A n , 3 + . . . + ˜ η 4 k − 8 A n , k − 2 A n = A n , 2 + ˜ All particles are massless: p 2 = 0 ⇒ p α ˙ α = λ α ˜ λ ˙ α . On-shell superspace – Λ A = ( λ α , ˜ λ ˙ α , ˜ η A ) Parke-Taylor formula for MHV amplitudes : [Parke, Taylor] δ 4 ( P ) δ 8 ( Q ) � ij � = ǫ αβ λ α i λ β A n , 2 = � 12 �� 23 � . . . � n 1 � , j Tomek Łukowski (University of Oxford) 23.09.2014 4 / 20
Twistors - natural coordinates to describe scattering amplitudes W A = (˜ µ α , ˜ λ ˙ α , ˜ η A ) Twistor variables: [Penrose] where ˜ µ is the Fourier transform of λ . Conformal symmetry [Witten] ∂ � W A A n , k = 0 i ∂ W B i i Z A = ( λ α , µ ˙ α , η A ) Momentum twistors: [Hodges] Dual conformal symmetry [Drummond, Henn, Korchemsky, Sokatchev], [Drummond,Ferro] A n , k ∂ � Z A A n , 2 = 0 i ∂ Z B i i Yangian algebra generators in twistor space [Drummond, Henn, Plefka] � � ∂ ∂ ∂ ∂ J AB = J AB = � W A ˆ � W A W C � v i W A , − ( i ↔ j ) + i i j i ∂ W B ∂ W C ∂ W B ∂ W B i i j i i i < j i Analogous expressions for momentum twistors. v i – evaluation representation parameters. Tomek Łukowski (University of Oxford) 23.09.2014 5 / 20
BCFW recursion relation for scattering amplitudes in N = 4 SYM BCFW recursion relation (based on the residue theorem): [Arkani-Hamed, Bourjaily,Cachazo, Caron-Huot, Trnka] + O ( g 2 ) Example solution to the tree-level BCFW recursion relation A 6 , 3 = One can associate a permutation to each on-shell diagram. One can associate an integral over an auxiliary real/complex Graßmannian to each such diagram. All such integrals are Yangian invariant for suitable integration contours. Real Graßmannians – on-shell diagrams correspond to cells of positive Graßmannian. [Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, Trnka] Complex Graßmannians – on-shell diagrams related to residues of Graßmannian integrals Tomek Łukowski (University of Oxford) 23.09.2014 6 / 20
From amplitudes to spin chains We consider planar theory → color ordered amplitudes scattering amplitude in N = 4 SYM ↔ ( p ) su ( 2 , 2 | 4 ) spin chain ↔ particle spin chain site number of particles n ↔ length of spin chain ↔ MHV degree k ? i , ¯ a α b ˙ α c A spin chain state is a polynomial/function in oscillators ¯ i , ¯ i acting on the Fock vacuum and constraint by ( c i – central charge of su ( 2 , 2 | 4 ) ) 2 + n a i − n b i − n c i = c i i , ˜ λ ˙ amplitude is a function/distribution of λ α α η A i , ˜ i with the constrained ( h i – superhelicity) � � 2 + λ i ∂ λ i ∂ η i ∂ ∂λ i − ˜ − ˜ A = 2 ( 1 − h i ) A ∂ ˜ ∂ ˜ η i λ i Task : use QISM to construct Yangian invariants of the inhomogeneous gl ( N | M ) spin chain Tomek Łukowski (University of Oxford) 23.09.2014 7 / 20
Yangian invariance = monodromy eigenproblem Alternative way of defining Yangian invariance for inhomogeneous spin chains M A B ( u ) | Ψ � = δ A B | Ψ � . ( ⋆ ) The monodromy matrix is defined as . . . . . . M ( u ) = L 1 ( u , v 1 ) . . . L n ( u , v n ) = � , u . . . . . . s n , v n s 1 , v 1 s k , v k s k + 1 , v k + 1 with the Lax operators � � � e A B J A B L i ( u , v i ) = N ( u , v i ) ( u − v i ) + = � , u i A , B s , v i Expanding the monodromy matrix around u → ∞ we find M A B ( u ) = δ A B + 1 uJ A B + 1 J A B + . . . u 2 ˆ Monodromy eigenproblem is equivalent to demanding Yangian invariance: | Ψ � is annihilated by all Yangian generators! Tomek Łukowski (University of Oxford) 23.09.2014 8 / 20
Quantum Inverse Scattering Method [Frassek, Kanning, Ko, Staudacher] Solution to ( ⋆ ) can be found using the Algebraic Bethe Ansatz. Focus on highest weight representations of su ( 2 ) and define A ( u ) B ( u ) M ( u ) = C ( u ) D ( u ) The monodromy eigenproblem is equivalent to the conditions: A ( u ) | Ψ � = D ( u ) | Ψ � = | Ψ � B ( u ) | Ψ � = C ( u ) | Ψ � = 0 Two oscillator realizations of the algebra ( symmetric and dual realizations) J A B = ¯ J A B = − ¯ a A a B , ¯ b B b A Consider a particular (inhomogeneous) quantum space V s 1 ⊗ . . . ¯ ¯ V s k ⊗ V s k + 1 ⊗ V s n Tomek Łukowski (University of Oxford) 23.09.2014 9 / 20
Quantum Inverse Scattering Method [Frassek, Kanning, Ko, Staudacher] Construct a reference state, which is highest weight, that is C ( u ) | Ω � = 0 (¯ i ) s i | ¯ b 2 0 � for i = 1 , . . . , k | Ω � = ω 1 ⊗ . . . ⊗ ω n , ω i = a 1 i ) s i | 0 � (¯ for i = k + 1 , . . . , n and make a Bethe ansatz for the Yangian invariant in the form | Ψ � = B ( u 1 ) . . . B ( u F ) | Ω � It is Yangian invariant if and only if the Bethe equations are satisfied k Q ( u ) u − v i − s i − 1 � Q ( u + 1 ) = , u − v i − 1 i = 1 k n u − v i − s i − 2 u − v i + s i � � = 1 u − v i − 2 u − v i i = 1 i = k + 1 with the Baxter polynomial Q ( u ) = � F i = 1 ( u − u i ) . Tomek Łukowski (University of Oxford) 23.09.2014 10 / 20
Solving Bethe equations From Bethe equations to permutations ( v + i = v i ± s i 2 + 2, v − i = v i ∓ s i 2 ): n n � ( u − v + � ( u − v − i ) = i ) i = 1 i = 1 All solutions are of the form v + σ ( i ) = v − i for some permutation σ ! Sample invariants: | Ψ � 2 , 1 = (¯ a 2 ) s 2 | 0 � b 1 · ¯ σ 2 , 1 = 1 2 2 1 | Ψ � 3 , 1 = (¯ a 2 ) s 2 (¯ b 1 · ¯ b 1 · ¯ a 3 ) s 3 | 0 � σ 3 , 1 = 1 2 3 2 3 1 | Ψ � 3 , 2 = (¯ a 3 ) s 1 (¯ a 3 ) s 2 | 0 � b 1 · ¯ b 2 · ¯ σ 3 , 2 = 1 2 3 3 1 2 with the Fock vacuum | 0 � = | ¯ 0 � ⊗ . . . ⊗ | ¯ 0 � ⊗ | 0 � ⊗ . . . ⊗ | 0 � Tomek Łukowski (University of Oxford) 23.09.2014 11 / 20
From Yangian invariants to Graßmannian integrals We represent harmonic oscillators as ∂ i , ¯ a A i , b A i ↔ W A a A b A ¯ i = i ∂ W A i Building blocks for invariants � u � ∂ B ij ( u ) = W i · ∂ W j and the Fock vacuum k � δ 4 | 4 ( W i ) | 0 � = i = 1 Using the integral representation of B -operators � d α ( W i · ∂ W j ) u = α W i · ∂ W j α 1 + u e one obtains, after change of variables, the integral over the Graßmannian space G ( 2 , 4 ) : d 2 × 2 C � δ 4 | 4 ( C · W ) | Ψ � 4 , 2 = ( 12 ) 1 + v − 4 − v − 1 ( 23 ) 1 + v − 1 − v − 2 ( 34 ) 1 + v − 2 − v − 3 ( 41 ) 1 + v − 3 − v − 4 1 0 c 13 c 14 where C = and ( ij ) = c 1 i c 2 j − c 2 i c 1 j 0 1 c 23 c 24 Tomek Łukowski (University of Oxford) 23.09.2014 12 / 20
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