Superstring amplitudes with the pure spinor formalism Carlos R. - PowerPoint PPT Presentation

Superstring amplitudes with the pure spinor formalism Carlos R. Mafra STAG Research Centre and Mathematical Sciences, University of Southampton, UK C.R. Mafra (Southampton) String amplitudes 11.06.2019 1 / 17

Motivation Calculate the L -loop N -point superstring amplitude Find hidden structures in superstring and field-theory amplitudes Simplify the calculations and the results C.R. Mafra (Southampton) String amplitudes 11.06.2019 2 / 17

Motivation Calculate the L -loop N -point superstring amplitude Find hidden structures in superstring and field-theory amplitudes Simplify the calculations and the results C.R. Mafra (Southampton) String amplitudes 11.06.2019 2 / 17

Motivation Calculate the L -loop N -point superstring amplitude Find hidden structures in superstring and field-theory amplitudes Simplify the calculations and the results C.R. Mafra (Southampton) String amplitudes 11.06.2019 2 / 17

D=10 SYM theory F 2 + χ α γ m � αβ D m χ β Describes gluon and gluino with action Covariant description using 10D superfields (Witten’86) A α ( x , θ ) , A m ( x , θ ) , W α ( x , θ ) , F mn ( x , θ ) Linearized equations of motion D α A β + D β A α = γ m αβ A m , D α A m = ( γ m W ) α + ∂ m A α D α W β = 1 4( γ mn ) β α F mn , D α F mn = 2 ∂ [ m ( γ n ] W ) α where D α is covariant derivative, { D α , D β } = γ m αβ ∂ m . Have standard θ expansions, e.g A m ( x , θ ) = a m − ( ξγ m θ ) − 1 8( θγ m γ pq θ ) F pq + . . . C.R. Mafra (Southampton) String amplitudes 11.06.2019 3 / 17

Pure Spinor Formalism (Berkovits’ 00) 1 � ∂ x m ∂ x m + α ′ p α ∂θ α − α ′ w α ∂λ α � d 2 z � S = 2 πα ′ Σ g Pure spinor λ α and its conjugate momenta w α : αβ λ β = 0 λ α γ m Manifest spacetime SUSY, pure spinor superspace (PSS): ( λ α , θ α ) Covariant BRST Quantization: Q = λ α D α Free CFT up to PS constraint with “simple” OPEs, e.g. p α ( z i ) θ β ( z j ) → δ β α , z ij ≡ z i − z j z ij C.R. Mafra (Southampton) String amplitudes 11.06.2019 4 / 17

Pure Spinor Formalism Massless N -point tree-level prescription (Berkovits ‘00): � � A tree = � V 1 V 2 V 3 U 4 . . . U n � Vertex operators V = λ α A α ( x , θ ) , QV = 0 U = ∂θ α A α + A m Π m + d α W α + 1 2 N mn F mn , QU = ∂ V Use OPEs to integrate non-zero modes Surviving zero modes give rise to pure spinor superspace expressions λ α λ β λ γ f αβγ ( x , θ ) � . . . � non-vanishing only for � ( λγ m θ )( λγ n θ )( λγ p θ )( θγ mnp θ ) � = 1 C.R. Mafra (Southampton) String amplitudes 11.06.2019 5 / 17

Pure Spinor Superspace Cohomology BRST charge Q = λ α D α and SYM equations of motion are closely related D α A β + D β A α = γ m αβ A m , D α A m = ( γ m W ) α + ∂ m A α D α W β = 1 4( γ mn ) β α F mn , D α F mn = 2 ∂ [ m ( γ n ] W ) α Amplitudes are expressions in the BRST cohomology of pure spinor superspace Study PSS cohomology and use it as tool to simplify and antecipate amplitude calculations (CM ‘10) C.R. Mafra (Southampton) String amplitudes 11.06.2019 6 / 17

Multiparticle SYM superfields (CM, Schlotterer ‘14, ‘15) K B ∈ { A B α , A m B , W α B , F mn B } Recursive definition of multiparticle superfields inspired by OPE computations and satisfy generalized EOMs 12 = 1 2 ( k 2 · A 1 ) − (1 ↔ 2) W α 4( γ mn W 2 ) α F 1 mn + W α 12 = 1 mn + ( k 1 · k 2 )( A 1 D α W β 4( γ mn ) αβ F 12 α W β 2 − A 2 α W β 1 ) Can be defined to satisfy generalized Jacobi identities (Elliot Bridges) 0 = K A ℓ ( B ) + K B ℓ ( A ) ℓ ( A ) is the left-to-right Dynkin bracket, e.g ℓ (123) = [[1 , 2] , 3] = 123 − 213 − 312 + 321 C.R. Mafra (Southampton) String amplitudes 11.06.2019 7 / 17

Towards FT tree amplitudes String tree-level amplitudes written in terms of multiparticle vertices V B = λ α A B α having BRST variations QV 1 = 0 QV 12 = s 12 V 1 V 2 QV 123 = ( k 1 · k 2 )( V 1 V 23 + V 13 V 2 ) + ( k 12 · k 3 ) V 12 V 3 String amplitudes reduce to FT amplitudes in the α ′ → 0 limit Guess FT amplitudes from BRST invariance Write down terms V A V B V C in the BRST cohomology with the correct poles QA SYM (1 , 2 , . . . , n ) = 0 , A SYM (1 , 2 , . . . , n ) � = Q ( something ) C.R. Mafra (Southampton) String amplitudes 11.06.2019 8 / 17

Tree-level FT amplitudes 2 3 3 4 A (1 , 2 , 3 , 4) = + s 12 s 23 1 4 2 1 = � V 12 V 3 V 4 � + � V 23 V 4 V 1 � s 12 s 23 In the BRST cohomology with correct kinematic poles Rewrite using Berends-Giele current M 12 ≡ V 12 / s 12 , M 1 ≡ V 1 A (1 , 2 , 3 , 4) = � ( M 12 M 3 + M 1 M 23 ) M 4 � Generalizes to N -pts � A SYM (1 , 2 , . . . , n ) = � M X M Y M n � XY =12 ... n − 1 C.R. Mafra (Southampton) String amplitudes 11.06.2019 9 / 17

String tree-level amplitude String tree amplitude can be rewritten in terms of ( N − 3)! SYM amplitudes (CM, Schlotterer, Stieberger) � N − 2 k − 1 � � � s mk � � | z ij | − s ij A = A SYM (1 , 2 , . . . , N )+ P (2 , . . . , N − 2) z mk m =1 i < j k =2 Recursive techniques to compute α ′ expansion of associated integrals: Drinfeld method (Drummond, Ragoucy 2013; Broedel et al 2013) 1 Berends-Giele method (CM, Schlotterer 2016) 2 C.R. Mafra (Southampton) String amplitudes 11.06.2019 10 / 17

1-loop string amplitudes String 1-loop amplitudes from OPEs (CM, Schlotterer ‘18) Building blocks V A , T A , B , C , T m A , B , C , D , T mn etc with BRST algebra ... QT 1 , 2 , 3 = 0 QT 12 , 3 , 4 = s 12 ( V 1 T 2 , 3 , 4 − V 2 T 1 , 3 , 4 ) QT 12 , 34 , 5 = s 12 ( V 1 T 2 , 34 , 5 − V 2 T 1 , 34 , 5 ) + s 34 ( V 3 T 12 , 4 , 5 − V 4 T 12 , 3 , 5 ) QT m 1 , 2 , 3 , 4 = k m 1 V 1 T 2 , 3 , 4 + (1 ↔ 2 , 3 , 4) QT m 12 , 3 , 4 , 5 = s 12 ( V 1 T m 2 , 3 , 4 , 5 − V 2 T m 1 , 3 , 4 , 5 ) + k m 12 V 12 T 3 , 4 , 5 k m � � + 3 V 3 T 12 , 4 , 5 + (3 ↔ 4 , 5) Guess FT integrands from BRST invariance (CM, Schlotterer 2014) d D ℓ � A (1 , 2 , 3 , . . . , n ) = (2 π ) D � A (1 , 2 , 3 , . . . , n | ℓ ) � QA (1 , 2 , 3 , . . . , n | ℓ ) = 0 C.R. Mafra (Southampton) String amplitudes 11.06.2019 11 / 17

FT 1-loop integrand Amplitude with cubic graphs (Bern, Carrasco, Johansson ‘10) Propagators with Mandelstams and loop momentum ℓ m The 4-point 1-loop FT integrand is in the BRST cohomology V 1 T 2 , 3 , 4 A (1 , 2 , 3 , 4 | ℓ ) = ℓ 2 ( ℓ − k 1 ) 2 ( ℓ − k 12 ) 2 ( ℓ − k 123 ) 2 . C.R. Mafra (Southampton) String amplitudes 11.06.2019 12 / 17

5-point 1-loop integrand 3 2 4 1 5 ℓ (2 ℓ m − k 1 m ) V 1 T m � � 2 , 3 , 4 , 5 + V 1 T 23 , 4 , 5 + (2 , 3 | 2 , 3 , 4 , 5) Overall 5pt integrand is BRST closed! C.R. Mafra (Southampton) String amplitudes 11.06.2019 13 / 17

5-point 2-loop integrand 5pt closed-string amplitude computed with the (non-minimal) pure spinor formalism led to kinematic building blocks: T A , B | C , D and T m A , B , C | D , E (Gomez, CM, Schlotterer ‘15) Minimal pure spinor representation with BRST algebra (CM, Schlotterer ‘15) QT 1 , 2 | 3 , 4 = 0 QT 12 , 3 | 4 , 5 = s 12 ( V 1 T 2 , 3 | 4 , 5 − V 2 T 1 , 3 | 4 , 5 ) QT m 1 , 2 , 3 | 4 , 5 = k m 1 V 1 T 2 , 3 | 4 , 5 + k m 2 V 2 T 1 , 3 | 4 , 5 + k m 3 V 3 T 1 , 2 | 4 , 5 Essentially the same algebraic structure as before! C.R. Mafra (Southampton) String amplitudes 11.06.2019 14 / 17

2-loop 5pt topologies BCJ identities: master diagram (a) (Carrasco, Johansson ‘11) PSS representation 2 N ( a ) 1 , 2 , 3 | 4 , 5 ( ℓ ) ≡ ( ℓ m + ℓ m − k 123 m ) T m 1 , 2 , 3 | 4 , 5 +( T 12 , 3 | 4 , 5 + T 13 , 2 | 4 , 5 + T 23 , 1 | 4 , 5 ) C.R. Mafra (Southampton) String amplitudes 11.06.2019 15 / 17

2-loop 5-pt integrand BRST-invariant 2-loop 5-pt integrand (CM, Schlotterer ‘15) BCJ identities satisfied by construction, gravity amplitudes for free The solution for numerators look intuitive, hope for N -pt solution 4-point 3-loop FT integrand under construction C.R. Mafra (Southampton) String amplitudes 11.06.2019 16 / 17

Conclusions Many simplying features not discussed today for lack of space and time! The multiloop amplitudes have a nice underlying BRST algebra structure Compact expressions in 10D PS superspace; the benefits of 10D SYM Patterns are building up, closed formula solution? Can we find generating series of FT loop amplitudes? How much of the recent simplicity in 4D scattering amplitudes can be explained from a 10D perspective? C.R. Mafra (Southampton) String amplitudes 11.06.2019 17 / 17