New geometric structures in scattering amplitudes The anatomy of - PowerPoint PPT Presentation

New geometric structures in scattering amplitudes ——————— The anatomy of scattering amplitudes in pure spinor superspace ——————— Oliver Schlotterer (AEI Potsdam) based on arXiv:1404.4986, arXiv:1408.3605: C. Mafra, OS and work in progress with M. Green and C. Mafra 22.09.2014

1 Goal of this talk • framework for amplitudes of gluon and graviton multiplet in 10 dim • both field theory and string theory, both type IIA and type IIB • manifest supersymmetry from pure spinor formalism

2 Goal of this talk • framework for amplitudes of gluon and graviton multiplet in 10 dim • both field theory and string theory, both type IIA and type IIB • manifest supersymmetry from pure spinor formalism Intuitive mapping between • cubic diagrams and kinematic factors • kinematic factors and worldsheet functions Make essential use of BRST symmetry in the pure spinor formalism [N. Berkovits hep-th/0001035]

3 Pure spinor superspace Bosonic pure spinor λ α defined by algebraic constraint λ α γ m αβ λ β = 0 ∀ m = 0 , 1 , . . . , 9 Pure spinor superspace (PSS) { x m , θ α , λ β } with component prescription � ( λγ m θ ) ( λγ n θ ) ( λγ p θ ) ( θγ mnp θ ) � = 1 BRST invariant & supersymmetric and automated in [C. Mafra 1007.4999]

4 Pure spinor superspace Bosonic pure spinor λ α defined by algebraic constraint λ α γ m αβ λ β = 0 ∀ m = 0 , 1 , . . . , 9 Pure spinor superspace (PSS) { x m , θ α , λ β } with component prescription � ( λγ m θ ) ( λγ n θ ) ( λγ p θ ) ( θγ mnp θ ) � = 1 BRST invariant & supersymmetric and automated in [C. Mafra 1007.4999] Driving force towards amplitudes in PSS: BRST charge ↔ eq’s of motion λ α � ∂ � ∂θ α + 1 λ α D α 2 k m ( γ m θ ) α ≡ = Q descends from gauge fixing worldsheet action [see Nathan’s talk and 1409.2510]

5 Outline scattering amplitudes gluon & gluino polariz. e m , χ α

6 Outline scattering amplitudes A α ( x, θ ) = e ik · x � 1 2 e m ( γ m θ ) α 10 dim N = 1 SYM superfields − 1 3 ( χγ m θ )( γ m θ ) α + θ 3 ke � gluon & gluino polariz. e m , χ α + θ 4 χk + θ 5 k 2 e + . . .

7 Outline scattering amplitudes part I based on 1404.4986 multiparticle superfields A α ( x, θ ) = e ik · x � 1 2 e m ( γ m θ ) α 10 dim N = 1 SYM superfields − 1 3 ( χγ m θ )( γ m θ ) α + θ 3 ke � gluon & gluino polariz. e m , χ α + θ 4 χk + θ 5 k 2 e + . . .

8 Outline scattering amplitudes Berends–Giele currents part II based on 1404.4986 part I based on 1404.4986 multiparticle superfields A α ( x, θ ) = e ik · x � 1 2 e m ( γ m θ ) α 10 dim N = 1 SYM superfields − 1 3 ( χγ m θ )( γ m θ ) α + θ 3 ke � gluon & gluino polariz. e m , χ α + θ 4 χk + θ 5 k 2 e + . . .

9 Outline scattering amplitudes part IV based on [to appear] BRST (pseudo-)invariants part III based on 1408.3605 Berends–Giele currents part II based on 1404.4986 part I based on 1404.4986 multiparticle superfields A α ( x, θ ) = e ik · x � 1 2 e m ( γ m θ ) α 10 dim N = 1 SYM superfields − 1 3 ( χγ m θ )( γ m θ ) α + θ 3 ke � gluon & gluino polariz. e m , χ α + θ 4 χk + θ 5 k 2 e + . . .

10 I. Multiparticle superfields Vertex operators for SYM states (unintegrated and integrated) V 1 ≡ λ α A 1 U 1 ≡ ∂θ α A 1 1 + 1 α + Π m A m 1 + d α W α 2 N mn F mn α , 1 superfields with known θ expansion ⊗ h = 1 fields as “bookkeeping var’s”

11 I. Multiparticle superfields Vertex operators for SYM states (unintegrated and integrated) 1 + 1 V 1 ≡ λ α A 1 U 1 ≡ ∂θ α A 1 α + Π m A m 1 + d α W α 2 N mn F mn α , 1 superfields with known θ expansion ⊗ h = 1 fields as “bookkeeping var’s” � � ∂ BRST invariance QV 1 = 0 and Q U 1 = ∂z V 1 = 0 equivalent to equations of motion (since Q = λ α D α on superfields) [E. Witten 1986] 2 D ( α A 1 β ) = γ m αβ A 1 m D α A 1 m = ( γ m W 1 ) α + k 1 m A 1 α D α W β 1 = 1 4 ( γ mn ) αβ F mn 1 = 2 k [ m 1 ( γ n ] W 1 ) α D α F mn 1

12 I. Multiparticle superfields Vertex operators for SYM states (unintegrated and integrated) 1 + 1 V 1 ≡ λ α A 1 U 1 ≡ ∂θ α A 1 α + Π m A m 1 + d α W α 2 N mn F mn α , 1 superfields with known θ expansion ⊗ h = 1 fields as “bookkeeping var’s” � � ∂ BRST invariance QV 1 = 0 and Q U 1 = ∂z V 1 = 0 equivalent to equations of motion (since Q = λ α D α on superfields) [E. Witten 1986] 2 D ( α A 1 β ) = γ m αβ A 1 components m D α A 1 m = ( γ m W 1 ) α + k 1 m A 1 ⇓ α k m e m = 0 D α W β 1 = 1 4 ( γ mn ) αβ F mn 1 = 2 k [ m � k αβ χ β = 0 1 ( γ n ] W 1 ) α D α F mn 1

13 Define multiparticle superfields via OPE (where z ij ≡ z i − z j ) � � U 1 ( z 1 ) U 2 ( z 2 ) ∼ z α ′ k 1 · k 2 − 1 ∂θ α A 12 α +Π m A m 12 + d α W α 12 + 1 2 N mn F mn + ∂ ∂z i ( . . . ) 12 12 1 + 1 Same structure as U 1 = ∂θ α A 1 α + Π m A m 1 + d α W α 2 N mn F mn with 1 � � α ( k 2 · A 1 ) + A 2 A 12 A 2 m ( γ m W 1 ) α − (1 ↔ 2) α = 1 2

14 Define multiparticle superfields via OPE (where z ij ≡ z i − z j ) � � U 1 ( z 1 ) U 2 ( z 2 ) ∼ z α ′ k 1 · k 2 − 1 ∂θ α A 12 α +Π m A m 12 + d α W α 12 + 1 2 N mn F mn + ∂ ∂z i ( . . . ) 12 12 1 + 1 Same structure as U 1 = ∂θ α A 1 α + Π m A m 1 + d α W α 2 N mn F mn with 1 � � α ( k 2 · A 1 ) + A 2 A 12 A 2 m ( γ m W 1 ) α − (1 ↔ 2) α = 1 2 � � m ( k 1 · A 2 ) + ( W 1 γ m W 2 ) − (1 ↔ 2) A 12 m = 1 A 1 p F 2 pm − A 1 2 2 ( k 2 · A 1 ) − (1 ↔ 2) W α 12 = 1 4 ( γ mn W 2 ) α F 1 mn + W α mn ( k 2 · A 1 ) + F 2 F 12 mn = F 2 p F 1 n ] p + 2 k 1 [ m ( W 1 γ n ] W 2 ) − (1 ↔ 2) [ m

15 Define multiparticle superfields via OPE (where z ij ≡ z i − z j ) � � U 1 ( z 1 ) U 2 ( z 2 ) ∼ z α ′ k 1 · k 2 − 1 ∂θ α A 12 α +Π m A m 12 + d α W α 12 + 1 2 N mn F mn + ∂ ∂z i ( . . . ) 12 12 1 + 1 Same structure as U 1 = ∂θ α A 1 α + Π m A m 1 + d α W α 2 N mn F mn with 1 � � α ( k 2 · A 1 ) + A 2 A 12 A 2 m ( γ m W 1 ) α − (1 ↔ 2) α = 1 2 � � m ( k 1 · A 2 ) + ( W 1 γ m W 2 ) − (1 ↔ 2) A 12 m = 1 A 1 p F 2 pm − A 1 2 2 ( k 2 · A 1 ) − (1 ↔ 2) W α 12 = 1 4 ( γ mn W 2 ) α F 1 mn + W α mn ( k 2 · A 1 ) + F 2 F 12 mn = F 2 p F 1 n ] p + 2 k 1 [ m ( W 1 γ n ] W 2 ) − (1 ↔ 2) [ m ⇓ 2 A 12 . . . α , A m 12 , W α 12 , F mn Four superfield rep’s ↔ 12 1 of the cubic vertex

16 12 + 1 V 12 ≡ λ α A 12 U 12 ≡ ∂θ α A 12 α + Π m A m 12 + d α W α 2 N mn F mn α , 12 Two-particle EOM ∼ = single-particle EOM ... 2 D ( α A 1 αβ A 1 β ) = γ m m D α A 1 m = ( γ m W 1 ) α + k 1 m A 1 α D α W β = 1 4 ( γ mn ) αβ F mn 1 1 = 2 k [ m 1 ( γ n ] W 1 ) α D α F mn 1

17 12 + 1 V 12 ≡ λ α A 12 U 12 ≡ ∂θ α A 12 α + Π m A m 12 + d α W α 2 N mn F mn α , 12 Two-particle EOM ∼ = single-particle EOM up to contact terms ∼ ( k 1 · k 2 ) 2 D ( α A 12 αβ A 12 m + ( k 1 · k 2 )( A 1 α A 2 β − A 2 α A 1 β ) = γ m β ) D α A 12 m = ( γ m W 12 ) α + k 12 m A 12 α + ( k 1 · k 2 )( A 1 α A 2 m − A 2 α A 1 m ) D α W β α W β α W β + ( k 1 · k 2 )( A 1 2 − A 2 12 = 1 4 ( γ mn ) αβ F mn 1 ) 12 = 2 k [ m 12 ( γ n ] W 12 ) α + ( k 1 · k 2 )( A 1 − A 2 D α F mn α F mn α F mn ) 12 2 1 + 2( k 1 · k 2 )( A [ n 1 ( γ m ] W 2 ) α − A [ n 2 ( γ m ] W 1 ) α ) where k m 12 ≡ k m 1 + k m 2

18 12 + 1 V 12 ≡ λ α A 12 U 12 ≡ ∂θ α A 12 α + Π m A m 12 + d α W α 2 N mn F mn α , 12 Two-particle EOM ∼ = single-particle EOM up to contact terms ∼ ( k 1 · k 2 ) 2 D ( α A 12 αβ A 12 m + ( k 1 · k 2 )( A 1 α A 2 β − A 2 α A 1 β ) = γ m β ) D α A 12 m = ( γ m W 12 ) α + k 12 m A 12 α + ( k 1 · k 2 )( A 1 α A 2 m − A 2 α A 1 m ) D α W β α W β α W β + ( k 1 · k 2 )( A 1 2 − A 2 12 = 1 4 ( γ mn ) αβ F mn 1 ) 12 = 2 k [ m 12 ( γ n ] W 12 ) α + ( k 1 · k 2 )( A 1 − A 2 D α F mn α F mn α F mn ) 12 2 1 + 2( k 1 · k 2 )( A [ n 1 ( γ m ] W 2 ) α − A [ n 2 ( γ m ] W 1 ) α ) � � ∂ BRST invariance QV 1 = 0 and Q U 1 = ∂z V 1 replaced by covariance QU 12 = ∂ QV 12 = ( k 1 · k 2 ) V 1 V 2 , ∂zV 12 + ( k 1 · k 2 )( V 1 U 2 − V 2 U 1 )

19 2 More particles by recursion . . . k 12 � � V 2 ( k 2 · A 1 ) + A 2 V 12 = 1 m ( λγ m W 1 ) − (1 ↔ 2) 1 2

20 2 3 More particles by recursion (replacing [1,2] by [12,3]) k 123 . . . k 12 � � V 3 ( k 3 · A 12 ) + A 3 V 123 = 1 m ( λγ m W 12 ) − (12 ↔ 3) � 1 2 BRST variation cancels propagators ∼ k 2 12 , k 2 123 of the cubic diagram V 123 = 1 2 ( k 2 123 − k 2 12 ) V 12 V 3 + 1 2 k 2 Q � 12 ( V 1 V 23 − V 2 V 13 )

21 2 3 More particles by recursion (replacing [1,2] by [12,3]) k 123 . . . k 12 � � V 3 ( k 3 · A 12 ) + A 3 V 123 = 1 m ( λγ m W 12 ) − (12 ↔ 3) � 1 2 BRST variation cancels propagators ∼ k 2 12 , k 2 123 of the cubic diagram 2 ( k 2 123 − k 2 2 k 2 V 123 = 1 12 ) V 12 V 3 + 1 Q � 12 ( V 1 V 23 − V 2 V 13 ) Moreover – totally antisymmetric component is BRST closed and exact V 123 = � V 123 + QH [123] = ⇒ V 123 + V 231 + V 312 = 0 Reproduce Jacobi identity among color tensors f 12 a f a 3 b +cyc(1 , 2 , 3) = 0 = ⇒ evidence for duality between color and kinematics [Bern, Carrasco, Johansson 0805.3993] [Mafra, OS, Stieberger 1104.5224]