N = 4 Scattering Amplitudes and the Regularized Gramannian Matthias - PowerPoint PPT Presentation

N = 4 Scattering Amplitudes and the Regularized Graßmannian Matthias Staudacher Institut f¨ ur Mathematik und Institut f¨ ur Physik Humboldt-Universit¨ at zu Berlin & AEI Potsdam & CERN Geneva Strings 2014, Princeton 25 June 2014 1 1

Based on L. Ferro, T. � Lukowski, C. Meneghelli, J. Plefka and M. Staudacher, 1212.0850 & 1308.3494 R. Frassek, N. Kanning, Y. Ko and M. Staudacher, 1312.1693 N. Kanning, T. � Lukowski and M. Staudacher, 1403.3382 And to appear. 2 2

Related Work D. Chicherin, S. Derkachov and R. Kirschner, 1306.0711 & 1309.5748 N. Beisert, J. Br¨ odel and M. Rosso, 1401.7274 J. Br¨ odel, M. de Leeuw and M. Rosso, 1403.3670 & 1406.4024 3 3

A Case for 3+1 Dimensions Nature prefers Yang-Mills theory in exactly 1+3 dimensions: Coordinates x µ , momenta p µ . So let us stay there! α = ˙ 1 , ˙ Split index µ = 0 , 1 , 2 , 3 into spinorial indices α = 1 , 2 and ˙ 2 . α . Interesting bijection R 1 , 3 → Hermitian(2 × 2) , p µ �→ p α ˙ Explicitly: � p 0 + p 3 � p 1 − i p 2 α = p α ˙ p 1 + i p 2 p 0 − p 3 Gluons are labeled by momenta p µ with p 2 = p µ p µ = det p α ˙ α = 0 and α = λ α ˜ helicity ± 1 . Momentum factors: p α ˙ λ ˙ α . 4 4

Super-Spinor-Helicity and Amplitudes There is a beautiful extension to maximally supersymmetric N = 4 theory: One introduces for each leg j a Graßmann spinor η A j where A = 1 , 2 , 3 , 4 . α = � j and Q α A = � j ˜ With P α ˙ j λ α α j λ α j η A λ ˙ j the (color stripped) tree amplitudes for n particles are the known [ Drummond, Henn ‘08 ] distributions n 1 n − 1 δ 4 ( P α ˙ α ) δ 8 ( Q α A ) 2 � 12 �� 23 � . . . � n − 1 , n �� n 1 � P n ( { λ j , ˜ · λ j , η j } ) , = · · · · · · · · β ˜ ℓ ˜ λ ˙ where � ℓ m � = � αβ λ α ℓ λ β α β λ ˙ m and [ ℓ m ] = � ˙ m . α ˙ All external helicity configurations are generated by expansion in the η A j . Super-helicity k corresponds to the terms of order η 4 k . 5 5

Graßmannian Integrals and Amplitudes, I A Graßmannian space Gr( k, n ) is the set of k -planes intersecting the origin of an n -dimensional space. k = 1 is ordinary projective space. “Homogeneous” coordinates are packaged into a k × n matrix C = ( c aj ) . C and A · C with A ∈ GL( k ) correspond to the same “point” in Gr( k, n ) . j , ˜ Build super-twistors W A λ ˙ j , η A j = (˜ j ) w. Fourier conjugates λ α j → ˜ j . µ α µ α α Graßmannian integral formulation of tree-level N k − 2 MHV n amplitudes: d k · n C δ 4 k | 4 k ( C · W ) � A n,k = vol(GL(k)) (1 . . . k )(2 . . . k + 1) . . . ( n . . . n + k − 1) The ( i i + 1 ...i + k − 1) are the n cyclic k × k minors. Integration is along “suitable contours”. [ Arkani-Hamed, Cachazo, Cheung, Kaplan ‘09 ] 6 6

Graßmannian Integrals and Amplitudes, II For “most” points on G( k, n ) we may use the GL( k ) symmetry to write   · · · c 1 ,k +1 c 1 ,k +2 c 1 ,n . . . ... . . . C = . . . I k × k     c k,k +1 c k,k +2 · · · c k,n The Graßmannian integral A n,k simplifies to � k � n k n i = k +1 dc ai � � � a =1 δ 4 | 4 � W A c ai W A � a + i (1 . . . k )(2 . . . k + 1) . . . ( n . . . n + k − 1) a =1 i = k +1 Fourier-transforming back to spinor-helicity space, all tree-level N k − 2 MHV n amplitudes may be obtained. 7 7

Symmetries The amplitudes enjoy N = 4 superconformal symmetry ( A , B = 1 . . . 8 ): J AB · A n,k = 0 , J AB ∈ psu (2 , 2 | 4) with However, there is also a “non-local” dual superconformal symmetry: J AB · A n,k = 0 , ˜ ˜ J AB ∈ psu (2 , 2 | 4) dual with Commuting J and ˜ J , one obtains Yangian symmetry. [ Drummond, Henn, Plefka ‘09 ] ∂ With “local” generators J AB = W A j − supertrace , where W A are j j j ∂ W B super-twistors, we can succinctly express it as n J AB = J AB = ˆ � � J AB J AC J CB − ( i ↔ j ) , j i j j =1 i<j This is how integrability first appeared in the planar scattering problem. 8 8

Dual Graßmannian Integrals and Amplitudes In the dual description one can employ 4 | 4 super momentum-twistors Z A j . With ˆ k = k − 2 , there is an equivalent “dual” description on Gr(ˆ k, n ) : [ Mason, Skinner ‘09; Arkani-Hamed et.al. ‘09 ] k · n ˆ d ˆ δ 4ˆ k | 4ˆ k ( ˆ α ) δ 8 ( Q α A ) A n,k = δ 4 ( P α ˙ C · Z ) C � vol(GL(ˆ (1 . . . ˆ k ) . . . ( n . . . ˆ � 12 �� 23 � . . . � n 1 � k)) k − 1) Note that the k = 2 MHV part factors out. The fact that the two formulations are related by a simple change of variables is due to dual conformal invariance, and thus Yangian invariance. 9 9

Deformed Symmetries [ Ferro, � Lukowski, Meneghelli, Plefka, MS ‘12 ] Of particular interest is the central charge generator of gl (4 | 4) : n ∂ ∂ ∂ − ˜ � − η A λ ˙ C = with c j = λ α + 2 c j α j j j ∂ ˜ ∂η A ∂λ α λ ˙ α j j j j =1 For overall psu (2 , 2 | 4) we have C = 0 . So we can relax the “local” condition c j = 0 . This deforms the super helicities h j = 1 − 1 2 c j . This yields something well-known: The Yangian in evaluation representa- tion. Deforming the c j switches on the parameters v j . More below. n n J AB = J AB = ˆ � � � J AB J AC J CB v j J AB − ( i ↔ j ) + , j i j j j =1 i<j j =1 10 10

Deformed Graßmannian Integrals [ Ferro, � Lukowski, MS, in preparation ] One could then ask how the Graßmannian contour formulas are deformed. The final answer is exceedingly simple, and very natural. Define c j v ± j = v j ± 2 Requiring Yangian invariance, we find, with v + j + k = v − j for j = 1 , . . . , n d k · n C δ 4 k | 4 k ( C · W ) � 1 . . . ( n, ... , k − 1) 1+ v + (1 , ... , k ) 1+ v + vol(GL(k)) k − 1 − v − k − v − n Note that it is not really the Graßmannian space Gr( k, n ) as such that is deformed, but the integration measure on this space. GL( k ) preserved! 11 11

Deformed Dual Graßmannian Integrals [ Ferro, � Lukowski, MS, in preparation ] It is equally natural to ask how the dual Graßmannian integrals deform. Using the parameters v ± j , we found δ 4 ( P α ˙ α ) δ 8 ( Q α A ) n × � 12 � 1+ v + 1 . . . � n 1 � 1+ v + 2 − v − 1 − v − k · n ˆ d ˆ δ 4ˆ k | 4ˆ k ( ˆ C · Z ) C � × vol(GL(ˆ 1+ v + 1+ v + n . . . ( n, ... , ˆ k +1 − v − k)) k − v − (1 , ... , ˆ k ) k − 1) ˆ n − 1 ˆ The number of deformation parameters equals n − 1 since v + j + k = v − for j = 1 , . . . , n . j Note that both the MHV-prefactor and the contour integral are deformed. 12 12

Why? Why should we consider this deformation? Here are some of the reasons: • We shall see that it is very natural from the point of view of integrability. • In fact, constructing amplitudes by integrability (arguably) requires it. • Amplitudes are related to the spectral problem, where it is indispensable. • Most importantly: It promises to provide a natural infrared regulator! The last point was our original motivation. Interestingly, we recently learned that this deformation had been already studied as an infrared regulator in twistor theory in the early seventies by Penrose and Hodges. 13 13

Meromorphicity Lost and Gained Let us take another look at the deformed Graßmannian contour integral: d k · n C δ 4 k | 4 k ( C · W ) � 1 . . . ( n, ... , k − 1) 1+ v + (1 , ... , k ) 1+ v + vol(GL(k)) k − 1 − v − k − v − n Choosing the parameters v ± j to be non-integer, we see that the poles in the variables c aj generically turn into branch points. Important point: We can no longer use the BCFW recursion relations, as they are based on the residue theorem, which does not apply anymore. Sounds bad? What we can hope to gain is complete meromorphicity in suitable combi- nations of the deformation parameters v ± j . This should fix the contours. 14 14

A Toy Meromorphicity Experiment Consider Euler’s first integral, the beta function B ( v 1 , v 2 ) . � 1 1 dc c 1 − v 1 (1 − c ) 1 − v 2 0 For v 1 , v 2 ∈ N Euler derived ( v 1 − 1)!( v 2 − 1)! ( v 1 + v 2 − 1)! . The analytic continuation for arbitrary v 1 , v 2 ∈ C is Γ ( v 1 ) Γ ( v 2 ) Γ ( v 1 + v 2 ) . Meromorphic in both v 1 and v 2 . This is not obvious from the integral. This problem was fixed by [ Pochhammer ‘90 ] : 1 1 � dc (1 − e 2 π iv 1 )(1 − e 2 π iv 2 ) c 1 − v 1 (1 − c ) 1 − v 2 C where the contour C goes at least two times through the cut: [ Wikipedia, the free encyclopedia ] 15 15

Yangian Invariants as Spin Chain States, I [ Frassek, Kanning, Ko, MS ‘13; Chicherin, Derkachov, Kirschner ‘13 ] How to construct, generally and systematically, Yangian invariants? It was recently proposed to identify them as special spin-chain states | Ψ � . How does the Yangian appear for spin chains with gl ( m | n ) symmetry? Package the “local” generators J AB into a Lax operator L j ( u, v ′ j ) : j 1 e AB J AB L j ( u, v ′ j ) = 1 + = j u − v ′ j s v Then build up a monodromy matrix M AB ( u, { v ′ j } ) : . . . . . . M ( u ) = L 1 ( u, v ′ 1 ) . . . L n ( u, v ′ n ) = . . . . . . s v Here multiplication is both a tensor product and a matrix product. 16 16