Integrable Deformations for N = 4 SYM and ABJM Amplitudes Till - PowerPoint PPT Presentation

Integrable Deformations for N = 4 SYM and ABJM Amplitudes Till Bargheer IAS Princeton Dec 12, 2014 Based on 1407.4449 with Yu-tin Huang, Florian Loebbert, Masahito Yamazaki Graßmannian Geometry of Scattering Amplitudes Walter Burke Institute Workshop, California Institute of Technology

Motivation On-shell integrand for planar N = 4 and ABJM well understood How to integrate? Regularization breaks conformal symmetry But even for finite ratio function: No practical way to integrate ⇒ Try to deform integrand, preserving as much symmetry as possible Till Bargheer — Integrable Deformations for N = 4 SYM and ABJM Amplitudes — Caltech — 12 Dec 2014 1 / 18

Deformed On-Shell Diagrams in N = 4 SYM (I) � Ferro, Łukowski, Meneghelli � Deformed three-point amplitudes: Plefka, Staudacher, 2012 1 3 � δ 4 ( P ) δ 4 ( ˜ d α 2 d α 3 Q ) δ 4 | 4 ( C ◦ ·W ) ≃ = α 1+ a 2 α 1+ a 3 [12] 1+ a 3 [23] 1 − a 1 [31] 1+ a 2 2 3 c 1 = a 1 ≡ a 2 + a 3 , c 2 = − a 2 , c 3 = − a 3 2 1 3 � δ 4 ( P ) δ 8 ( Q ) d α 1 d α 2 δ 8 | 8 ( C • ·W ) ≃ = α 1+ a 1 α 1+ a 2 � 12 � 1 − a 3 � 23 � 1+ a 1 � 31 � 1+ a 2 1 2 c 1 = a 1 , c 2 = a 2 , c 3 = − a 3 ≡ − a 1 − a 2 2 n � � J a ∈ psu (2 , 2 | 4) , J a = f abc � J b i J c u i J a Yangian: j + i 1 ≤ i<j ≤ n i =1 � Beisert, Broedel � u + j = u − u ± Invariance conditions: σ ( j ) , j = u j ± c j Rosso, 2014 Deformed helicities: h j = 1 − C j , h j = 1 − c j Till Bargheer — Integrable Deformations for N = 4 SYM and ABJM Amplitudes — Caltech — 12 Dec 2014 2 / 18

Deformed On-Shell Diagrams in N = 4 SYM (II) Bigger on-shell diagrams can be obained by gluing: Products & Fusion 1 n i − → Y 1 Y 2 Y ′ Y j m m +1 Reduced diagram ≃ permutation σ u + j = u − u ± j = u j ± c j Invariance conditions from gluing: σ ( j ) � Beisert, Broedel � General deformed on-shell diagram: Rosso, 2014 � n F − 1 � d α j δ 4 k | 4 k ( C · W ) , Y ( W i , a i ) = α 1+ a j j =1 j Edge deformation parameters α i equal central charges on internal lines, fully determined by external u i , c i via left-right paths Till Bargheer — Integrable Deformations for N = 4 SYM and ABJM Amplitudes — Caltech — 12 Dec 2014 3 / 18

Deformations in ABJM: Four-Vertex δ 3 ( P ) δ 6 ( Q ) The Basic diagram in A 4 = = ABJM is the four-vertex: � 12 � � 23 � Till Bargheer — Integrable Deformations for N = 4 SYM and ABJM Amplitudes — Caltech — 12 Dec 2014 4 / 18

Deformations in ABJM: Four-Vertex δ 3 ( P ) δ 6 ( Q ) The Basic diagram in � A 4 ( z ) = = z � 12 � 1 − z � 23 � 1+ z ABJM is the four-vertex: � TB, Huang, Loebbert � Admits a deformation, parameter z Yamazaki, 2014 Can show invariance under level-zero osp (6 | 4) Till Bargheer — Integrable Deformations for N = 4 SYM and ABJM Amplitudes — Caltech — 12 Dec 2014 4 / 18

Deformations in ABJM: Four-Vertex δ 3 ( P ) δ 6 ( Q ) The Basic diagram in � A 4 ( z ) = = z � 12 � 1 − z � 23 � 1+ z ABJM is the four-vertex: � TB, Huang, Loebbert � Admits a deformation, parameter z Yamazaki, 2014 Can show invariance under level-zero osp (6 | 4) n � � J a = f abc � J b i J c u i J a Level-one generators: j + i 1 ≤ i<j ≤ n i =1 Till Bargheer — Integrable Deformations for N = 4 SYM and ABJM Amplitudes — Caltech — 12 Dec 2014 4 / 18

Deformations in ABJM: Four-Vertex δ 3 ( P ) δ 6 ( Q ) The Basic diagram in � A 4 ( z ) = = z � 12 � 1 − z � 23 � 1+ z ABJM is the four-vertex: � TB, Huang, Loebbert � Admits a deformation, parameter z Yamazaki, 2014 Can show invariance under level-zero osp (6 | 4) n � � J a = f abc � J b i J c u i J a Level-one generators: j + i 1 ≤ i<j ≤ n i =1 �� � � � � P αβ = � L ( α P γβ ) − Q ( αA Q β ) u k P αβ 1 jγ + δ ( α k A − ( j ↔ k ) γ D j + 2 k j k 1 ≤ j<k ≤ n k Invariance under � P implies full Y( osp (6 | 4)) invariance. u j Constraints: u k z = z = u 1 − u 2 . u 1 = u 3 , u 2 = u 4 , u j − u k Till Bargheer — Integrable Deformations for N = 4 SYM and ABJM Amplitudes — Caltech — 12 Dec 2014 4 / 18

Deformations in ABJM: Gluing Construct bigger deformed diagrams from four-vertex Products: Invariance trivial, no constraints 1 n Y (1 . . . n ) = Y 1 (1 . . . m ) Y 2 ( m + 1 . . . n ) n � � Y 1 Y 2 J a = f abc � J b i J c u i J a j + i 1 ≤ i<j ≤ n i =1 m m +1 Till Bargheer — Integrable Deformations for N = 4 SYM and ABJM Amplitudes — Caltech — 12 Dec 2014 5 / 18

Deformations in ABJM: Gluing Construct bigger deformed diagrams from four-vertex Products: Invariance trivial, no constraints 1 n Y (1 . . . n ) = Y 1 (1 . . . m ) Y 2 ( m + 1 . . . n ) n � � Y 1 Y 2 J a = f abc � J b i J c u i J a j + i 1 ≤ i<j ≤ n i =1 m m +1 � Y ′ ( . . . ) = d 2 | 3 Λ i d 2 | 3 Λ j δ 2 | 3 ( Λ i − iΛ j ) Y ( . . . , i, j, . . . ) Fusing: ☛ ✟ i − → Constraint: u i = u j Y ′ Y j ✡ ✠ Till Bargheer — Integrable Deformations for N = 4 SYM and ABJM Amplitudes — Caltech — 12 Dec 2014 5 / 18

Deformations in ABJM: Gluing Products: Invariance trivial, no constraints 1 n Y (1 . . . n ) = Y 1 (1 . . . m ) Y 2 ( m + 1 . . . n ) n � � Y 1 Y 2 J a = f abc � J b i J c u i J a j + i 1 ≤ i<j ≤ n i =1 m m +1 � Y ′ ( . . . ) = d 2 | 3 Λ i d 2 | 3 Λ j δ 2 | 3 ( Λ i − iΛ j ) Y ( . . . , i, j, . . . ) Fusing: ☛ ✟ i − → Constraint: u i = u j Y ′ Y j ✡ ✠ Can construct all deformed diagrams by iterated product and fusion Till Bargheer — Integrable Deformations for N = 4 SYM and ABJM Amplitudes — Caltech — 12 Dec 2014 5 / 18

Deformations in ABJM: General Diagrams Fundamental vertex is four-valent ⇒ General 2 k -point diagram can be drawn with k straight lines z 6 u 1 z 1 z 4 z 5 u 2 z 2 z 3 u 3 u 4 u j Characterized by order-two permutation σ , σ 2 = 1 One evaluation parameter on each line, u j = u σ ( j ) u k z = Vertex parameters z i completely fixed by constraint u j − u k Till Bargheer — Integrable Deformations for N = 4 SYM and ABJM Amplitudes — Caltech — 12 Dec 2014 6 / 18

Deformations in ABJM: R-Matrix Formalism (I) Reformulate invariants in terms of R-matrix construction. j Λ ′′ j R jk ( z ) ◦ f = f z k k Λ ′ � d Λ ′ d Λ ′′ A 4 ( z )( Λ j , Λ k , iΛ ′ , iΛ ′′ ) f ( Λ ′′ , Λ ′ ) ( R jk ( z ) ◦ f )( Λ j , Λ k ) ≡ R-matrix kernel is four-point amplitude Just a reformulation of gluing ⇒ R jk ( z ) trivially preserves invariance R-operator intertwines two single-particle representations Permutes evaluation parameters: J a ( . . . , u j , u k , . . . ) R jk ( z ) = R jk ( z ) � � J a ( . . . , u k , u j , . . . ) (on space of invariants) Till Bargheer — Integrable Deformations for N = 4 SYM and ABJM Amplitudes — Caltech — 12 Dec 2014 7 / 18

Deformations in ABJM: R-Matrix Formalism (II) R-operator permutes evaluation parameters: J a ( . . . , u j , u k , . . . ) R jk ( z ) = R jk ( z ) � � J a ( . . . , u k , u j , . . . ) Invariants are chains of R-matrices acting on vacuum Ω 2 k : k � δ 2 | 3 ( Λ 2 j − 1 + iΛ 2 j ) Y 2 k = R i ℓ ,j ℓ ( z ℓ ) . . . R i 1 ,j 1 ( z 1 ) Ω 2 k , Ω 2 k = j =1 Vacuum permutation: σ = [1 , 2][3 , 4] . . . [2 k − 1 , 2 k ] Action of R-matrices conjugate the permutation: Y 2 k → R ij Y 2 k ⇒ σ → [ i, j ] · σ · [ i, j ] = Till Bargheer — Integrable Deformations for N = 4 SYM and ABJM Amplitudes — Caltech — 12 Dec 2014 8 / 18

Deformations in ABJM: R-Matrix Formalism (II) R-operator permutes evaluation parameters: J a ( . . . , u j , u k , . . . ) R jk ( z ) = R jk ( z ) � � J a ( . . . , u k , u j , . . . ) Invariants are chains of R-matrices acting on vacuum Ω 2 k : k � δ 2 | 3 ( Λ 2 j − 1 + iΛ 2 j ) Y 2 k = R i ℓ ,j ℓ ( z ℓ ) . . . R i 1 ,j 1 ( z 1 ) Ω 2 k , Ω 2 k = j =1 Vacuum permutation: σ = [1 , 2][3 , 4] . . . [2 k − 1 , 2 k ] Action of R-matrices conjugate the permutation: Y 2 k → R ij Y 2 k = ⇒ σ → [ i, j ] · σ · [ i, j ] 1 2 3 4 5 6 Till Bargheer — Integrable Deformations for N = 4 SYM and ABJM Amplitudes — Caltech — 12 Dec 2014 8 / 18

Deformations in ABJM: R-Matrix Formalism (II) R-operator permutes evaluation parameters: J a ( . . . , u j , u k , . . . ) R jk ( z ) = R jk ( z ) � � J a ( . . . , u k , u j , . . . ) Invariants are chains of R-matrices acting on vacuum Ω 2 k : k � δ 2 | 3 ( Λ 2 j − 1 + iΛ 2 j ) Y 2 k = R i ℓ ,j ℓ ( z ℓ ) . . . R i 1 ,j 1 ( z 1 ) Ω 2 k , Ω 2 k = j =1 Vacuum permutation: σ = [1 , 2][3 , 4] . . . [2 k − 1 , 2 k ] Action of R-matrices conjugate the permutation: Y 2 k → R ij Y 2 k = ⇒ σ → [ i, j ] · σ · [ i, j ] 2 3 1 4 5 6 Till Bargheer — Integrable Deformations for N = 4 SYM and ABJM Amplitudes — Caltech — 12 Dec 2014 8 / 18