Correlation functions of stress-tensor multiplets in N=4 SYM Paul - PowerPoint PPT Presentation

Correlation functions of stress-tensor multiplets in N=4 SYM Paul Heslop New Geometric Structures in Scattering Amplitudes September 25th, 2014 Oxford based on work with: Eden, Korchemsky, Sokatchev (arxiv:1108.3557,1201.5329) Ambrosio, Eden, Goddard, Taylor (arXiv:1312.1163) Chicherin, Doobary, Eden, Korchemsky, Mason, Sokatchev, to appear soon

Outline Four-point correlation functions in planar N = 4 SYM Summarise progress over last three years Amplitudes in planar N = 4 SYM four- and five- point amplitude integrand from 4 point correlator Higher point correlators: Twistor approach

Correlators (Correlation functions of gauge invariant operators) Gauge invariant operators: gauge invariant products (ie traces) of the fundamental fields Simplest operator tr φ 2 ( φ one of the scalars) The simplest non-trivial correlation function is G 4 := �O ( x 1 ) ¯ O ( x 2 ) O ( x 3 ) ¯ O ( x 4 ) � O = Tr ( φ 12 φ 12 ) O ∈ stress energy supermultiplet. We consider correlators of all operators in this multiplet. Later discuss higher points too

Why are they interesting? AdS/CFT Supergravity/String theory on AdS 5 × S 5 = N =4 super Yang-Mills Correlation functions of gauge invariant operators in SYM ↔ string scattering in AdS Contain data about anomalous dimensions of operators and 3 point functions via OPE → integrability / bootstrap Finite Big Bonus of last 3 years Correlators contain all scattering amplitudes (more later)

Analytic superspace Stress-tensor multiplet best described using analytic superspace [ GIKOS , Hartwell Howe ] Analytic superspace = Grassmannian of ( 2 | 2 ) planes in C 4 | 4 � z i � a i � � z i � � θ i β i θ i ∼ 0 u i 0 c i 0 u i z : 2 × 4 matrix, Grassmannian of 2 planes in C 4 = Minkowski space = lines in twistor space u : 2 × 4 matrix, Grassmannian of 2 planes in C 4 = internal space ρ i := θ i ¯ Solve odd part of local Sl ( 2 | 2 ) : θ i ∼ θ i + β i u i ⇒ u i ∼ ρ i 6d coordinates for Minkowski and internal space β = 1 X i ) AB = (¯ α ˙ (¯ ( X i ) AB = ( z i ) α A ( z i ) β z i ) A ˙ z i ) B ˙ β ǫ ˙ 2 ǫ ABCD ( X i ) AB α (¯ B ǫ αβ b ǫ ab = 1 Y i ) IJ = (¯ ( ¯ ( Y i ) IJ = ( u i ) a I ( u i ) b u i ) Ia (¯ u i ) J 2 ǫ IJKL ( Y i ) KL . J ǫ ab

In the standard chart corresponding to the usual space-time x coordinates and corresponding internal coordinates y we put: Standard chart � − x i � 1 2 � � ˆ ˆ ¯ ¯ z i = ( 1 2 x i ) z i = ( 0 2 1 2 ) ⇒ z i = z i = 0 2 1 2 � 1 2 � − y i � � ˆ ˆ ¯ ¯ u i = ( 1 2 y i ) u i = ( 0 2 1 2 ) ⇒ u i = u i = 0 2 1 2 θ i = ( 0 2 ρ i ) .

Superspace: correlation functions stress-energy supermultiplet (half BPS) ρ = 0 , y ) = O ( x , y ) + . . . + ρ 4 L ( x ) , T ( x , ρ, ¯ O ( x , y ) = Tr ( φ IJ φ KL ) Y IJ Y KL correlation function of T s: ρ -expansion organised in powers of ρ 4 k Superspace expansion (cf superamplitude) G n | ¯ ρ = 0 := �T ( 1 ) T ( 2 ) . . . T ( n ) � � ρ 4 � � ρ 8 � � ρ 4 ( n − 4 ) � G n ; 2 + · · · + = G n ; 0 + G n ; 1 + G n ; n − 4

Integrands = correlators with Lagrangian insertions Loop corrections ⇒ Lagrangian insertions. 1 loop correlator � �T ( 1 ) . . . T ( n ) � ( 1 ) = d 4 x 0 �L ( x 0 ) T ( 1 ) . . . T ( n ) � ( 0 ) � d 4 x 0 d 4 ρ 0 �T ( 0 ) T ( 1 ) . . . T ( n ) � ( 0 ) = so the Born-level ( n + 1 ) -point correlator defines the 1 loop integrand: G ( 1 ) n ; k = G ( 0 ) n + 1 ; k + 1 ℓ -loops ⇒ ℓ Lagrangian insertions ⇒ n + ℓ -point tree correlator G ( ℓ ) n ; k = G ( ℓ − 1 ) n + 1 ; k + 1 = · · · = G ( 0 ) n + ℓ ; k + ℓ NB parameter space n , k , ℓ → n , k Amplitudes need n , k , ℓ : Many different amplitudes contained in each correlator.

Hidden permutation symmetry for the four-point correlator G ( 0 ) n ; n − 4 is “tree-level MHV” correlator G n ; n − 4 = ( S n symmetric ρ 4 ( n − 4 ) invariant) × f ( x i ) Crossing symmetry of super correlator under simultaneous ( x i , ρ i , y i ) → ( x j , ρ j , y j ) ⇒ permutation symmetry of f ( x i ) But G ( 0 ) n ; n − 4 is the four-point ( n − 4 ) -loop integrand Four-point correlator is given in terms of ∞ � a ℓ � d 4 x 5 . . . d 4 x 4 + ℓ f ( ℓ ) ( x 1 , . . . , x 4 + ℓ ) f ( x i ; a ) = ℓ ! ℓ = 1 Hidden symmetry: f ( ℓ ) ( x 1 , . . . x 4 + ℓ ) = f ( ℓ ) ( x σ 1 , . . . x σ 4 + ℓ ) ∀ σ ∈ S 4 + ℓ The symmetry mixes external variables x 1 , . . . x 4 with integration variables x 5 . . . x 4 + ℓ

1-, 2- and 3-loop integrands Entire 4-pnt correlator defined (perturbatively) via f ( ℓ ) ◮ conformal weight 4 at each point ◮ permutation invariant ◮ No double poles (from OPE) Naively equivalent to: degree (valency) 4 graphs on 4 + ℓ points graph edge = 1 x 2 ij ( But: we are also allowed numerator lines ⇒ degree ≥ 4 graphs). Don’t need to label graph since we sum over permutations ⇒ sum over all possible ways of labelling

1 f ( 1 ) = � 1 ≤ i < j ≤ 5 x 2 ij f ( 2 ) = x 2 12 x 2 34 x 2 56 + S 6 perms � 1 ≤ i < j ≤ 6 x 2 ij f ( 3 ) = ( x 4 12 )( x 2 34 x 2 45 x 2 56 x 2 67 x 2 73 ) + S 7 perms � 1 ≤ i < j ≤ 7 x 2 ij Unique (planar) possibilities

Four- and five-loops f ( 4 ) = � � f ( 5 ) = � � � � � � Very compact writing All come with coefficients 1,-1 (determined using technique of [ Bourjaily, DiRe, Shaikh, Spradlin, Volovich ]) From 6 loops we start to see integrands with the coefficient 2 (and also 0), the first being:

Relation to amplitudes triality between three objects in N = 4 SYM [ Alday Maldacena , Drummond Henn Korchemsky Sokatchev , Brandhuber Travaglini PH , Mason Skinner , Caron-Huot , Alday Eden Korchemsky Maldacena Sokatchev , Eden Korchemsky Sokatchev PH , Adamo Bullimore Mason Skinner , ... ]

Triality Full planar superamplitude super Wilson loop (vev) MHV tree �T ( x 1 , ρ 1 , y 1 ) T ( x 2 , ρ 2 , y 2 ) . . . T ( x n , ρ n , y n ) � lim �T ( x 1 , ρ 1 , y 1 ) T ( x 2 , ρ 2 , y 2 ) . . . T ( x n , ρ n , y n ) � tree x 2 i i + 1 → 0 Full super-correlation function ( y s completely factorise)

Superspaces: superamplitudes Use Nair’s N =4 on-shell superspace, all particles → superparticle super-particle Φ( p , η ) = G + ( p ) + ηψ + η 2 φ ( p ) + η 3 ¯ ψ ( p ) + η 4 G − ( p ) All amplitudes → superamplitudes A ( x i ) → A ( x i , η i ) super-amplitude structure: A ( x i , η i ) = � η 8 � � η 12 � � η 16 � � η 4 ( n − 2 ) � A MHV + A NMHV + A NNMHV + ... + A MHV � � � η 4 � ˆ � η 8 � ˆ � η 4 ( n − 4 ) � ˆ ˆ = A tree A MHV + A NMHV + A NNMHV + ... + A MHV MHV

Superamplitude/ supercorrelation function duality Superduality �T ( 1 ) . . . T ( n ) � � A n � 2 lim ( x , ρ, ¯ ρ = 0 , y ) = ( x , η ) �T ( 1 ) . . . T ( n ) � tree A tree x 2 i i + 1 → 0 n ; 0 n ; MHV duality works at the level of the integrand... Amplitude written in terms of dual/region momenta p i = x i − x i + 1 OR better momentum twistors (Hodges). Points where consecutive supertwistor lines intersect in the lightlike limit.

Correlator amplitude duality at 4,5 points But G ( ℓ ) 5 ; 1 = G ( ℓ + 1 ) at the integrand level 4 ; 0 Both four-point and five-point amplitudes are given in terms of the same objects: f -graphs � f ( ℓ ) ℓ ! := F ( ℓ ) d 4 x 5 . . . d 4 x 4 + ℓ = ( M 4 ) 2 external factor × lim 4 x 2 i i + 1 → 0 ( mod 4 ) � f ( ℓ + 1 ) := F ( ℓ ) d 4 x 6 . . . d 4 x 5 + ℓ external factor × lim = 2 M 5 M 5 5 ℓ ! x 2 i i + 1 → 0 ( mod 5 ) We can determine four points and five-point amplitude completely from the four-point correlator Beyond 5 points eg 6-point limit = M 6 M 6 + NMHV 2 .

Amplitude information from f ( 2 ) (octahedron) Having expanded in fermionic variables, we now expand in the coupling a : M n := 1 + a M ( 1 ) + a 2 M ( 2 ) + a 3 M ( 3 ) + . . . n n n Octahedron f ( 2 ) → F ( 2 ) 4 , F ( 1 ) 5 � � 2 ( 1 ) F ( 2 ) = 2 M ( 2 ) M ( 1 ) F ( 1 ) = M ( 1 ) + + M 5 4 4 4 5 5 Graphically at four points: 2 2 5 1 → → 3 6 5 6 1 3 4 4

Graphically at five -points (one loop): 23 October 2013 12:00 2 1 5 3 → 6 4 Summing all permutations gives the sum over 1 mass box functions = parity even 1-loop 5-point amplitude Also a well known parity odd part O ( ǫ ) but important eg in BDS To see this let’s consider the next order...

Four-points, 3 loop (from f ( 3 ) ) We have F ( 3 ) = 2 M ( 3 ) + M ( 1 ) 4 M ( 2 ) 4 4 4 7 4 5 3 7 4 5 → → 6 1 6 2 2 3 1 7 5 4 → 2 3 6 1 Graphically: the four external points we pick need to be connected consecutively: four-cycle. Four-cycle splits the planar graph into two pieces. Correspond to product terms.

Five-points, 2 loop and parity odd 1 loop (from f ( 3 ) ) We have F ( 2 ) = M ( 2 ) ( 2 ) + M ( 1 ) ( 1 ) + M 5 M 5 5 5 5 Distinguish contributions by topology: If the 5-cycle “splits” the ( 1 ) ( 2 ) f -graph it contributes to M ( 1 ) otherwise to M ( 2 ) 5 M + M 5 5 5 6 2 5 7 1 1 4 2 3 7 6 4 3 5 2 6 3 5 1 7 4 pentagon 2 2 loop ladder box × box � ( 1 ) ( 1 ) Two equations ( M ( 1 ) boxes; M ( 1 ) + M 5 M = = products). 5 5 5 Solve eqns gives the full (parity even and odd) amplitude M ( 1 ) 5 .