All-loop S-matrix of planar N = 4 Super Yang-Mills from Yangian symmetry Song He Simon Caron-Huot & SH, arXiv: 1112.1060. Crete Center for Theoretical Physics, University of Crete April 24th, 2012
Plan of the talk • Motivations • Review of the S-matrix in planar N = 4 SYM • The S-matrix from symmetries • New differential equations • Solving the equations • Jumpstarting amplitudes • Two-loop MHV • Two-loop NMHV • Three-loop MHV • Two-dimensional kinematics • Outline of a derivation • Summary and outlook 24. Apr, 2012, Song He: All-loop S-matrix of planar N = 4 Super Yang-Mills from Yangian symmetry 2 / 20
Motivations • Integrability in AdS/CFT duality Planar N = 4 SYM and IIB superstring theory on AdS 5 × S 5 are integrable. • • From dimensions of local operators to other important observables, such as correlation functions, Wilson loops and scattering amplitudes. • Integrability as a hidden, infinite-dimensional symmetry: the psu (2 , 2 | 4) Yangian. • S-matrix program: N = 4 SYM as the simplest QFT • Remarkable structures of amplitudes in gauge theories and gravity, which are completely obscured in textbook formulation of QFT. • Planar N = 4 SYM has the nicest S-matrix, and can be viewed as our new har- monic oscillator. It serves as a toy model for general gauge theories and gravity. • Towards a dual formulation of QFT from S-matrix program, which manifests the structures of amplitudes: symmetries constrains everything? 24. Apr, 2012, Song He: All-loop S-matrix of planar N = 4 Super Yang-Mills from Yangian symmetry 3 / 20
Review of the S-matrix in planar N = 4 SYM • All the on-shell states in N = 4 SYM can be combined into an on-shell superfield, Φ := G + + η A Γ A + 1 2! η A η B S AB + 1 Γ D + 1 3! ε ABCD η A η B η C ¯ 4! ε ABCD η A η B η C η D G − , α = λ α ¯ which depends on the Grassmann variable η A , and a null momenta p α ˙ λ ˙ α . • All color-ordered amplitudes are then packaged into a superamplitude A ( { λ i , ¯ λ i , η i } ) , which has an expansion in terms of Grassmann degrees 4 k + 8 , n − 3 i λ i ¯ A n = A n, MHV + A n, NMHV + · · · + A n, MHV = δ 4 ( � λ i ) δ 0 | 8 ( � i λ i η i ) � A n,k , � 12 �� 23 � · · · � n 1 � k =0 where A n,k denotes the N k MHV amplitude, with MHV tree, A tree n, MHV , stripped off. N = 4 SYM is a superconformal field theory, which should be reflected in the • structure of scattering amplitudes. The tree-level S-matrix is invariant under this psu (2 , 2 | 4) symmetry: { q α q A β , s A s ˙ α α , d , r A A , ¯ α , m αβ , ¯ α , ¯ α , p α ˙ A , k α ˙ B } . At loop level, the m ˙ α ˙ ˙ superconformal symmetry of the S-matrix is broken by infrared divergences. 24. Apr, 2012, Song He: All-loop S-matrix of planar N = 4 Super Yang-Mills from Yangian symmetry 4 / 20
Review of the S-matrix in planar N = 4 SYM: dual symmetries A dual conformal symmetry has been observed at both weak [ Drummond Henn • Smirnov Sokatchev 2006 ] and Alday strong couplings [ Maldacena 2007 ]. The symmetry has been generalized to a dual super- Drummond Henn conformal symmetry [ Korchemsky Sokatchev 2008 ] of the dual chiral superspace, i ¯ x α ˙ − x α ˙ λ ˙ α i − 1 = λ α α α θ αA − θ αA i − 1 = λ α i η A i , i . i i The tree-level S-matrix is invariant under the dual psu (2 , 2 | 4) symmetry. • An all-loop, exponentiated ansatz for MHV amplitude in 4 − 2 ǫ dimensions has been Dixon Kosower 2003 ] [ Bern Dixon Anastasiou Bern proposed, which encodes infrared and collinear behavior [ Smirnov 2005 ], � ∞ ∞ �� n, 0 ( ℓǫ ) + C ( ℓ ) + E ( ℓ ) A BDS g 2 ℓ � cusp ( ǫ ) A (1) � � g 2 ℓ A ( ℓ ) Γ ( ℓ ) = 1 + n ( ǫ ) := exp n ( ǫ ) . n ℓ =1 ℓ =1 • MHV loop amplitudes satisfy an anomalous Ward identity for the dual conformal Drummond Henn symmetry [ Korchemsky Sokatchev 2007 ]. For n = 4 , 5 , the only solution is given by the BDS ansatz, since there is no cross-ratios. A finite remainder function of 3( n − 5) cross- ratios is allowed for n -point MHV amplitude, e.g. u 1 = x 2 13 x 2 36 etc. for n = 6 . 46 x 2 14 x 2 24. Apr, 2012, Song He: All-loop S-matrix of planar N = 4 Super Yang-Mills from Yangian symmetry 5 / 20
Review of the S-matrix in planar N = 4 SYM: Wilson loops • There is strong evidence for a duality between MHV amplitude and a null polygonal ] [ Brandhuber Heslop Wilson loop in dual spacetime [ Drummond Korchemsky ] [ Bern Dixon Kosower Roiban Spradlin Vergu Volovich 2008 ]. On Travaglini 2007 Sokatchev 2007 the string side, (fermionic) T-duality maps the original superconformal symmetry of Beisert Ricci Berkovits the amplitude to the dual symmetry of the Wilson loop [ Maldacena 2008 ] [ Tseytlin Wolf 2008 ], and their closure is the Yangian symmetry, y [ psu (2 , 2 | 4)] [ Drummond Henn ]. Plefka 2009 • A generalized duality between the superamplitude and a supersymmetric Wilson Skinner 2010 ][ Caron-Huot Mason loop has been derived at the integrand level [ ], although a rigorous 2010 UV regularization for the super-loop has not been carried out [ Belitzky Korchemsky ], Sokatchev 2011 W n = 1 � Tr P e − A n ( λ i , ¯ � A ( x i ,θ i ) � . λ i , η i ) = W n ( x i , θ i )(1 + O ( ǫ )) , N c • The super Wilson loop in chiral formalism obscures one chiral half of supercon- formal symmetries. As a natural generalization, Wilson loops in non-chiral N = 4 Beisert Beisert SH superspace generally manifest the full symmetry [ Caron-Huot ] [ Vergu 2012 ] [ Schwab Vergu 2012 ]. 2011 One can obtain amplitudes by setting ¯ • θ = 0 , but there is no obvious way to de- fine non-chiral amplitudes dual to non-chiral Wilson loops. They contain additional terms, which can play a role for compensating symmetry anomalies of amplitudes. 24. Apr, 2012, Song He: All-loop S-matrix of planar N = 4 Super Yang-Mills from Yangian symmetry 6 / 20
Review of the S-matrix in planar N = 4 SYM: momentum twistors It is convenient to introduce unconstrained momentum-twistor variables [ Hodges • 2009 ], i , x α ˙ Z i = ( Z a i , χ A i ) := ( λ α α λ iα , θ αA λ iα ) , i i which are twistors of the dual (super)space. Then one can construct invariants, u 1 = � 1234 �� 4561 � four-bracket : e.g. � ijkl � := ε abcd Z a i Z b j Z c k Z d l , � 1245 �� 3461 � . i � jklm � + cyclic ) δ 0 | 4 ( χ A R-invariant : [ i j k l m ] := � ijkl �� jklm �� klmi �� lmij �� mijk � . • Using momentum twistors, which form fundamental representation of the dual psu (2 , 2 | 4) , all the generators become first-order differential operators, n n ∂ ∂ A , ¯ � ¯ α , ¯ � S ˙ Q a A = ( Q α α Z a Q A a = ( S A Q A s A χ A A ) := , α = ¯ α ) := , ˙ ˙ i i ∂χ A ∂Z a i i i =1 i =1 n n ∂ ∂ α , M αβ , ¯ � � K a Z a R A B = R A χ A b = ( P α ˙ α , K α ˙ β , D ) := , B := . M ˙ α ˙ i i ∂Z b ∂χ B i i i =1 i =1 24. Apr, 2012, Song He: All-loop S-matrix of planar N = 4 Super Yang-Mills from Yangian symmetry 7 / 20
Review of the S-matrix in planar N = 4 SYM: current status All tree amplitudes are known by BCFW recursions [ Britto Cachazo • ], e.g. NMHV tree, Feng 2004 A tree n, 1 = � 1 <i<j<n [1 i i +1 j j +1] , which are built from leading singularities, or (generally-shifted) R-invariants. All leading singularities are Yangian invariant, and correspond to contour integrals on the Grassmannian G ( k, n ) [ Arkani-Hamed Cachazo Cheung Kaplan 2009 ]. • The Yangian-invariant planar integrand of all-loop amplitudes/Wilson loops is Arkani-Hamed Bourjaily known recursively [ Cachazo Caron-Huot Trnka 2010 ], but it is difficult to perform integrals. The n -point, N k MHV, ℓ -loop amplitude is of the form “ G ( k, n ) leading-singularities” • × “pure, transcendental degree 2 ℓ functions of 3( n − 5) cross-ratios”. It is convenient to use “symbol” for transcendental functions as iterated integrals [ Goncharov Spradlin Vergu Volovich 2010 ], � d log X 1 ( x 1 ) . . . d log X m ( x m ) ⇒ S [ F ] = X 1 ⊗ . . . ⊗ X m . F ( x ) = x 1 < ··· <x m <x • All one-loop amplitudes are known using leading singularity method (or generalized-unitarity), but higher-loop integral basis are lacking. Recent advances Smirnov 2010 ][ Goncharov Spradlin have reached two-loop MHV [ Del Duca Duhr Vergu Volovich 2010 ], NMHV and three-loop MHV [ Kosower Roiban 2011 ][ Dixon Drummond 2011 ][ Caron-Huot SH 2011 ]. It is promising to go to all loops in the near future! Vergu Henn 24. Apr, 2012, Song He: All-loop S-matrix of planar N = 4 Super Yang-Mills from Yangian symmetry 8 / 20
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