The MHV Lagrangian and hidden Wilson lines Piotr Kotko IFJ PAN - PowerPoint PPT Presentation

INSTITUTE OF NUCLEAR PHYSICS THE HENRYK NIEWODNICZAŃSKI POLISH ACADEMY OF SCIENCES The MHV Lagrangian and hidden Wilson lines Piotr Kotko IFJ PAN based on: P .K., A. Stasto, JHEP 1709 (2017) supported by: DEC-2011/01/B/ST2/03643 DE-FG02-93ER40771 Light Cone 2018, May 14-18, 2018

Outline Anna’s talk: Recursion relations for off-shell MHV currents contain an object ˜ J which has a structure exactly like on-shell MHV amplitude but with spinor products continued off-shell. • ˜ J can be constructed from a straight-infinite Wilson line along a polarization vector • On the other hand ˜ J corresponds to the so-called MHV vertices in the Cachazo-Svrcek-Witten (CSW) construction. What is the connection between the two? 1

Outline Anna’s talk: Recursion relations for off-shell MHV currents contain an object ˜ J which has a structure exactly like on-shell MHV amplitude but with spinor products continued off-shell. • ˜ J can be constructed from a straight-infinite Wilson line along a polarization vector • On the other hand ˜ J corresponds to the so-called MHV vertices in the Cachazo-Svrcek-Witten (CSW) construction. What is the connection between the two? This talk: • Lagrangian for the CSW method (the light-front Yang-Mills Lagrangian after certain canonical field transformation) [P . Mansfield (2006)] • Exact solution to the field transformation – constructed from non-light-like Wilson lines, similar to those in ˜ J • Consequences 1

MHV amplitudes Spinor algebra Spinor products: ˙ i λ β β ˜ i ˜ β � ij � = u − ( k i ) u + ( k j ) ≡ ǫ αβ λ α λ ˙ α [ ij ] = u + ( k i ) u − ( k j ) ≡ ǫ ˙ j , λ α ˙ j i ≡ u + ( k i ) , ˜ where u ± ( k i ) = 1 2 ( 1 ± γ 5 ) u ( k i ) and λ α λ ˙ α i ≡ u − ( k i ) . Momenta k i are light-like. Parke-Taylor amplitudes 1 � 12 � 4 � 1 − , 2 − , 3 + , . . . , n + � = M � 12 � � 23 � . . . � n 1 � 1 S.J. Parke, T.R. Taylor, Phys.Rev.Lett. 56, 2459 (1986) 2

Cachazo-Svrcek-Witten (CSW) Method (1) General idea Glue any amplitude from the MHV amplitudes continued off-shell. 3

Cachazo-Svrcek-Witten (CSW) Method (1) General idea Glue any amplitude from the MHV amplitudes continued off-shell. Off-shell continuation of spinors If k is light-like, we have α = λ k α ˜ α ˜ λ ˙ α k α ˙ λ k ˙ = ⇒ λ k α = k α ˙ q / [ kq ] α where q is auxiliary light-like momentum. If k is off-shell we define the off-shell continuation of spinor in the same way: λ ( ∗ ) α ˜ λ ˙ α k α = k α ˙ q 3

Cachazo-Svrcek-Witten (CSW) Method (1) General idea Glue any amplitude from the MHV amplitudes continued off-shell. Off-shell continuation of spinors If k is light-like, we have α = λ k α ˜ α ˜ λ ˙ α k α ˙ λ k ˙ = ⇒ λ k α = k α ˙ q / [ kq ] α where q is auxiliary light-like momentum. If k is off-shell we define the off-shell continuation of spinor in the same way: λ ( ∗ ) α ˜ λ ˙ α k α = k α ˙ q MHV vertices + i − . . . + � ij � 4 . . . . . � 12 � � 23 � . . . � n 1 � ≡ . + . . . j − + The spinor products are made from off-shell spinors � ij � = ǫ αβ λ ( ∗ ) α λ ( ∗ ) β . i j 3

Cachazo-Svrcek-Witten (CSW) Method (2) Example: NMHV amplitude M ( 1 − , 2 − , 3 − , 4 + , 5 + ) 2 − 2 − 2 − 1 − 1 − 3 − 3 − + + = 3 − − 1 − − 5 + 4 + 4 + 5 + 4 + 5 + 2 − 2 − 1 − 3 − 1 − 3 − + − + − 4 + 4 + 5 + 5 + 4

Cachazo-Svrcek-Witten (CSW) Method (2) Example: NMHV amplitude M ( 1 − , 2 − , 3 − , 4 + , 5 + ) 2 − 2 − 2 − 1 − 1 − 3 − 3 − + + = 3 − − 1 − − 5 + 4 + 4 + 5 + 4 + 5 + 2 − 2 − 1 − 3 − 1 − 3 − + − + − 4 + 4 + 5 + 5 + The result: [ 45 ] 4 � 1 − , 2 − , 3 − , 4 + , 5 + � M = [ 12 ] [ 23 ] [ 34 ] [ 45 ] [ 51 ] 4

Yang-Mills action on the light-front (1) Yang-Mills action where: � A µ = A µ S Y − M = − 1 F µν = ˆ g ′ [ D µ , D ν ] i a t a d 4 x Tr F µν F µν √ � t a , t b � 4 D µ = ∂ µ − ig ′ ˆ 2 f abc t c = i A µ 5

Yang-Mills action on the light-front (1) Yang-Mills action where: � A µ = A µ S Y − M = − 1 F µν = ˆ g ′ [ D µ , D ν ] i a t a d 4 x Tr F µν F µν √ � t a , t b � 4 D µ = ∂ µ − ig ′ ˆ 2 f abc t c = i A µ Light-cone coordinates Basis vectors: 1 1 1 ε ± η = ( 1 , 0 , 0 , − 1 ) , ˜ η = ( 1 , 0 , 0 , 1 ) , ⊥ = ( 0 , 1 , ± i , 0 ) √ √ √ 2 2 2 Contravariant coordinates: v + = v · η , v − = v · ˜ η , v • = v · ε + v ⋆ = v · ε − ⊥ , ⊥ u · v = u + w − + u − w + − u • w ⋆ − u ⋆ w • Scalar product: � p + , p • , p ⋆ � x ≡ ( x − , x • , x ⋆ ) , p ≡ Three-vectors: 5

Yang-Mills action on the light-front (2) Yang-Mills action in transverse fields only A · η = A + = 0 • Light cone gauge: • Integration of A − fields out of the action 6

Yang-Mills action on the light-front (2) Yang-Mills action in transverse fields only A · η = A + = 0 • Light cone gauge: • Integration of A − fields out of the action � dx + � � S ( LC ) L ( LC ) + − + L ( LC ) ++ − + L ( LC ) + −− + L ( LC ) Y − M [ A • , A ⋆ ] = ++ −− 6

Yang-Mills action on the light-front (2) Yang-Mills action in transverse fields only A · η = A + = 0 • Light cone gauge: • Integration of A − fields out of the action � dx + � � S ( LC ) L ( LC ) + − + L ( LC ) ++ − + L ( LC ) + −− + L ( LC ) Y − M [ A • , A ⋆ ] = ++ −− � L ( LC ) d 3 x Tr ˆ A • � ˆ + − [ A • , A ⋆ ] = − A ⋆ � A • � A • � L ( LC ) ++ − [ A • , A ⋆ ] = − 2 ig ′ d 3 x Tr γ x ˆ ∂ − ˆ A ⋆ , ˆ � A ⋆ � A ⋆ � L ( LC ) d 3 x Tr γ x ˆ ∂ − ˆ A • , ˆ −− + [ A • , A ⋆ ] = − 2 ig ′ � � A ⋆ � � A • � L ( LC ) ∂ − ˆ A • , ˆ ∂ − ˆ A ⋆ , ˆ ++ −− [ A • , A ⋆ ] = − g 2 d 3 x Tr ∂ − 2 − where γ x = ∂ − 1 γ x = ∂ − 1 − ∂ • , − ∂ ⋆ . 6

The MHV action (1) Transformation of fields 1 ( A • , A ⋆ ) → ( B • , B ⋆ ) 1 P . Mansfield, JHEP 03 (2006) 037 7

The MHV action (1) Transformation of fields 1 ( A • , A ⋆ ) → ( B • , B ⋆ ) 1 Transformation is canonical such that B • = B • [ A • ] � d 3 y δ B • c ( y ) ∂ − A ⋆ a ( x ) ∂ − B ⋆ a ( x ) = c ( y ) δ A • 1 P . Mansfield, JHEP 03 (2006) 037 7

The MHV action (1) Transformation of fields 1 ( A • , A ⋆ ) → ( B • , B ⋆ ) 1 Transformation is canonical such that B • = B • [ A • ] � d 3 y δ B • c ( y ) ∂ − A ⋆ a ( x ) ∂ − B ⋆ a ( x ) = c ( y ) δ A • 2 The vertex (+ + − ) is removed L ( LC ) + − [ A • , A ⋆ ] + L ( LC ) ++ − [ A • , A ⋆ ] = L ( LC ) + − [ B • , B ⋆ ]   � δ B • a ( x )   � �  A • ( y )  D ⋆ , γ y ˆ d 3 y Tr  t c  c ( y ) = ω x B • a ( x )   δ A •     where ω x = ∂ • ∂ ⋆ ∂ − 1 − . 1 P . Mansfield, JHEP 03 (2006) 037 7

The MHV action (2) Solution to the transformations in momentum space ∞ � ˜ a = ˜ ˜ ⊗ ˜ b 1 . . . ˜ A • B • Ψ a { b 1 ... b n } B • B • a + n b n n = 2 ∞ � ˜ a = ˜ Ω ab 1 { b 2 ... b n } ˜ ⊗ ˜ b 1 ˜ b 2 . . . ˜ A ⋆ B ⋆ B ⋆ B • B • a + n b n n = 2 8

The MHV action (2) Solution to the transformations in momentum space ∞ � ˜ a = ˜ ˜ ⊗ ˜ b 1 . . . ˜ A • B • Ψ a { b 1 ... b n } B • B • a + n b n n = 2 ∞ � ˜ a = ˜ Ω ab 1 { b 2 ... b n } ˜ ⊗ ˜ b 1 ˜ b 2 . . . ˜ A ⋆ B ⋆ B ⋆ B • B • a + n b n n = 2 The MHV action � � ˜ B ⋆ � dx + � � S ( LC ) B • , ˜ L ( LC ) + − + L ( LC ) −− + + · · · + L ( LC ) = −− + ··· + + . . . Y − M 8

The MHV action (2) Solution to the transformations in momentum space ∞ � ˜ a = ˜ ˜ ⊗ ˜ b 1 . . . ˜ A • B • Ψ a { b 1 ... b n } B • B • a + n b n n = 2 ∞ � ˜ a = ˜ Ω ab 1 { b 2 ... b n } ˜ ⊗ ˜ b 1 ˜ b 2 . . . ˜ A ⋆ B ⋆ B ⋆ B • B • a + n b n n = 2 The MHV action � � ˜ B ⋆ � dx + � � S ( LC ) B • , ˜ L ( LC ) + − + L ( LC ) −− + + · · · + L ( LC ) = −− + ··· + + . . . Y − M where the MHV vertex is: L ( LC ) −− + ··· + = ˜ V b 1 ... b n −− + ··· + ⊗ ˜ B ⋆ b 1 ˜ B ⋆ b 2 ˜ B • b 3 . . . ˜ B • b n � 2 � p + v ∗ 4 ˜ V −− + ··· + ( p 1 , . . . , p n ) = 1 ˜ n ! ( g ′ ) n − 1 1 21 p + ˜ v ∗ 1 n ˜ v ∗ n ( n − 1 ) ˜ v ∗ ( n − 1 )( n − 2 ) . . . ˜ v ∗ 21 2 � � � � i + p + j / p + v ∗ ( i )( j ) = − p • i + p + p • j / p + with ˜ v ( i )( j ) = − p ⋆ p ⋆ ∼ [ ij ] , ˜ ∼ � ij � . i j i j 8

The diagrammatic content of transformations (1) Solution B • [ A • ] B • = + 1 1 1 + + + 1 + . . . s 2 s 2 s 3 s 2 s 3 s 2 s 3 • Vertical dashed lines – energy denominators:    , E p = p ⋆ p •  �      D 1 ... i = 2  E initial − E j       p + j ∈ intermediate • Triple gluon vertices – helicity ( − + +) . 9