Soft Theorems from Effective Field Theory Andrew Larkoski MIT - PowerPoint PPT Presentation

Soft Theorems from Effective Field Theory Andrew Larkoski MIT AJL, D. Neill, I. Stewart 1412.3108 SCET, March 25, 2015

A (1 , . . . , N, s ) → S (0) ( s ) A (1 , . . . , N ) N gT i � ǫ s · p i (Prehistory) � Weinberg 1964, 1965 S (0) � gauge ( s ) = p s · p i i =1 Gauge invariance: Charge conservation N κQ i � ǫ µν s p iµ p iν � S (0) � grav ( s ) = p s · p i i =1 Gauge invariance: Momentum conservation & universal coupling of gravity 2

� � S (0) ( s ) + S (2) ( s ) A (1 , . . . , N, s ) → A (1 , . . . , N ) N gT i � ǫ µ s p ν s J i Low 1958 � µν Burnett, Kroll 1967 S (2) � gauge ( s ) = p s · p i i =1 Gauge invariance: Anti-symmetry of angular momentum tensor ∂ J i + Σ i µν = p i [ µ µν ∂p ν ] i Proofs of Low-Burnett-Kroll at tree-level: BCFW recursion relations Casali, 2014 Conformal symmetry of tree-level 4D gauge theory amplitudes AJL, 2014 Bern, Davies, Di Vecchia, Gauge and Lorentz invariance Nohles, 2014 3

Tree-level: p s · p i p s ( p j + p k ) 2 ≪ 1 H ( p H + p s ) 2 = 1 1 − 2 p H · p s + · · · p 2 p 4 H H Loop-level: p s · p i p s ( p i + ℓ ) 2 ∼ 1 p i H 1 1 ( ℓ + p i + p s ) 2 = ℓ ( ℓ + p i ) 2 + 2( ℓ + p i ) · p s 4

λ order loop-order A [0] (1 , . . . , N, s ) → A [0](0) (1 , . . . , N, s ) ( ∼ λ − 2 ) + A [0](1) (1 , . . . , N, s ) ( ∼ λ − 1 ) + A [0](2) (1 , . . . , N, s ) ( ∼ λ 0 ) + O ( λ 1 ) p s ∼ Q λ 2 “soft” p c ∼ Q (1 , λ 2 , λ ) external collinear 5

A [0](0) (1 , . . . , N, s ) : Tree-level A (1 , . . . , N, s ) → S (0) ( s ) A (1 , . . . , N ) In SCET: p s � 0 |O (0) N | p 1 , . . . , p N � = A [0] O (0) N + · · · N � O (0) O (0) L (0) � � � � �� � � � � � 0 � p 1 , . . . , p N , p s = 0 � T N , � p 1 , . . . , p N , p s int + . . . N n i , soft i = S (0) ( s ) A [0] N + . . . , p s ( p − � � i n i ) · ǫ s S (0) ( s ) = ⊗ = ¯ u ( p i ) · − gT i p i ( p − i n i ) · p s for fermions 6

A [0](1) (1 , . . . , N, s ) , A [0](2) (1 , . . . , N, s ) : Tree-level Power-count Low-Burnett-Kroll Operator N gT i � ǫ µ s p ν s J i � µν S (2) � gauge ( s ) = p s · p i i =1 Propagator factor: p i · p s = (¯ n · p i )( n · p s ) + ( n · p i )(¯ n · p s ) + p i ⊥ · p s ⊥ 2 2 � �� � � �� � � �� � ∼ λ 3 ∼ λ 2 ∼ λ 4 1 2 4 p i ⊥ · p s ⊥ n · p i ) 2 ( n · p s ) 2 + O ( λ 0 ) = n · p i )( n · p s ) − p i · p s (¯ (¯ 7

A [0](1) (1 , . . . , N, s ) , A [0](2) (1 , . . . , N, s ) : Tree-level Power-count Low-Burnett-Kroll Operator N gT i � ǫ µ s p ν s J i � µν S (2) � gauge ( s ) = p s · p i i =1 Angular momentum factor: ∂ J iµν = p i [ µ + Σ iµν ∂ p ν ] i � n · p i ¯ � ∂ ∂ J iµν = ∂ ( n · p i ) + n [ µ p i ⊥ [ µ n ν ] ∂p ν ] 2 i ⊥ � � ∂ n · p i ∂ n · p i ¯ ∂ + + ¯ ∂ ( n · p i ) + n [ µ ¯ n · p i ) + Σ iµν p i ⊥ [ µ n [ µ n ν ] n ν ] ∂p ν ] 2 2 ∂ (¯ i ⊥ + O ( λ 1 ) , 8

A [0](1) (1 , . . . , N, s ) , A [0](2) (1 , . . . , N, s ) : Tree-level Power-count Low-Burnett-Kroll Operator N gT i � ǫ µ s p ν s J i � µν S (2) � gauge ( s ) = p s · p i i =1 Putting it together: p i ⊥ = 0 RPI choice: n · p i = 0 On-shell: N 2 ǫ µ s p ν � n · p i ¯ � ∂ RPI � S (2) s gT i � n · p i ) + Σ i � gauge ( s ) ≃ n [ µ ¯ n ν ] µν ( n · p s )(¯ n · p i ) 2 ∂ (¯ i =1 ( ∼ λ 0 ) 9

A [0](1) (1 , . . . , N, s ) = � 0 | T {O (1) L (0) � Tree-level n i , soft } N , i + T {O (0) L (1) � n i , soft }| p 1 , ..., p N , p s � N , i Lagrangians: Operators: ∂ i ⊥ = i ¯ n / 2 p n ⊥ · p s ⊥ (1) ≃ 0 O (1) 2 n · p n ¯ p n , p s N (RPI) µ 2 p µ = ig ¯ n / O (1) ≃ 0 } n i n ⊥ N p n n 2 n · p n ¯ (1) (kinematics) No non-trivial contribution at λ -1 for fermions 10 2)

A [0](2) (1 , . . . , N, s ) ≃ � 0 | T {O (2) L (0) � n i , soft } Tree-level N , i + T {O (0) L (2) � n i , soft }| p 1 , ..., p N , p s � N , i Operators: Generated by RPI expansion of label δ -functions in O (0) p s N N N ∂ O (2 δ ) ) = − � � C (0) � � n · i ∂ n ) X κ i � � { Q i } ⊗ δ (¯ n i · Q i − ¯ n i (0) N ∂ ¯ n k · Q k N k =1 i =1 N � � � n k · gB ( n k ) A T κ k A Y κ i ⊗ T ¯ n i (0) s i =1 Generated by RPI expansion of collinear fields in O (0) p s N O (2 r ) N N ) = C (0) � � � � n · i ∂ n ) X κ i � � { Q i } ⊗ δ (¯ n i · Q i − ¯ n i (0) N N k =1 i =1 ,i � = k N t µ � � � � � X κ k gB ( n k ) A T κ k A Y κ i k δ (¯ n k · Q k − ¯ n · i ∂ n ) n k (0) ⊗ T n i (0) × sµ n k · Q k ¯ i =1 11

A [0](2) (1 , . . . , N, s ) ≃ � 0 | T {O (2) L (0) � n i , soft } Tree-level N , i + T {O (0) L (2) � n i , soft }| p 1 , ..., p N , p s � N , i Operators: Generated by RPI expansion of label δ -functions in O (0) p s N N N ∂ O (2 δ ) ) = − � � C (0) � � n · i ∂ n ) X κ i � � { Q i } ⊗ δ (¯ n i · Q i − ¯ n i (0) N ∂ ¯ n k · Q k N k =1 i =1 orbital angular momentum N � � � n k · gB ( n k ) A T κ k A Y κ i ⊗ T ¯ n i (0) s i =1 Generated by RPI expansion of collinear fields in O (0) p s N O (2 r ) N N ) = C (0) � � � � n · i ∂ n ) X κ i � � { Q i } ⊗ δ (¯ n i · Q i − ¯ n i (0) N N k =1 i =1 ,i � = k N t µ � � � � � X κ k gB ( n k ) A T κ k A Y κ i k δ (¯ n k · Q k − ¯ n · i ∂ n ) n k (0) ⊗ T n i (0) × sµ n k · Q k ¯ i =1 only non-zero for fermions 12

A [0](2) (1 , . . . , N, s ) ≃ � 0 | T {O (2) L (0) � n i , soft } Tree-level N , i + T {O (0) L (2) � n i , soft }| p 1 , ..., p N , p s � N , i Lagrangians: p µ s p µ p 2 = ig ¯ n / + ig ¯ n / p s ⊥ ν 1 = i ¯ n / (2) ⊥ , γ µ s ⊥ 2[ γ ν s ⊥ ⊥ ] p n n 2 n · p n ¯ 2 n · p n ¯ 2 n · p n ¯ p n , p s (2) p s p s (2) + ⊗ ⊗ p i p i (2) n µ n ν ǫ sµ p s ν � + 1 � p µ ⊥ , γ µ − p ν 2[ γ ν i i = ¯ u ( p i ) · ( − g ) ⊥ ] s ⊥ s ⊥ p − n i · p s n i · p s i ( n i · p s ) for fermions 13

A [0](2) (1 , . . . , N, s ) ≃ � 0 | T {O (2) L (0) � n i , soft } Tree-level N , i + T {O (0) L (2) � n i , soft }| p 1 , ..., p N , p s � N , i Putting it all together: A [0](2) (1 , . . . , N, s ) = S (2) ( s ) A (1 , · · · , N ) u ( p i ) ˜ A N = ¯ A N for fermions � 2 ǫ sµ p sν n i · p i ¯ ∂ S (2) n [ µ n ν ] iψ A N = g n i · p i )( n i · p s ) ¯ u ( p i ) T i i ¯ i (¯ 2 ∂ (¯ n i · p i ) n ν ] / � n i ¯ 2( n i · p s ) + 1 + γ [ µ ⊥ n ν ] 4 + p [ µ 4[ γ µ ˜ i ⊥ , γ ν ⊥ ] A N i s ⊥ 14

A [0](2) (1 , . . . , N, s ) ≃ � 0 | T {O (2) L (0) � n i , soft } Tree-level N , i + T {O (0) L (2) � n i , soft }| p 1 , ..., p N , p s � N , i Putting it all together: A [0](2) (1 , . . . , N, s ) = S (2) ( s ) A (1 , · · · , N ) u ( p i ) ˜ A N = ¯ A N O (2 δ ) orbital angular momentum for fermions N � 2 ǫ sµ p sν n i · p i ¯ ∂ S (2) n [ µ n ν ] iψ A N = g n i · p i )( n i · p s ) ¯ u ( p i ) T i i ¯ i (¯ 2 ∂ (¯ n i · p i ) n ν ] / � n i ¯ 2( n i · p s ) + 1 + γ [ µ ⊥ n ν ] 4 + p [ µ 4[ γ µ ˜ i ⊥ , γ ν ⊥ ] A N i s ⊥ spin angular momentum O (2 r ) L (2) N RPI was necessary for universal factorized form! 15

One-loop soft theorem 16

A [1](0) (1 , . . . , N, s ) : Loop-level A [1](0) N +1 s = S [0](0) ( s ) A [1](0) + S [1](0) ( s ) A [0](0) N N Universality of leading soft factor persists to one-loop A [1](1) (1 , . . . , N, s ) : A [1](1) N +1 s ≃ 0 All possible operator and Lagrangian contributions can be set to zero by RPI 17

A [1](2) N +1 s = S [0](2) ( s ) A [1 , hard](0) Loop-level N + A [0](0) I [0](2 L ) S [1](0) ( s ) N N N � � � + A [0](0) I [0](2 L ) k ( x ) E [1] µ � s ( n k ) ( x ) + I [0](2 L ) k κ ( x ) E [1] µ ν � κ � d d x s ( n k )( n k ) ( x, x ) N N µ N µ ν k =1 N − ∂ A [0](0) � � Split [0](0) E [1] µ � n kµ + A [0](0) I [0](0 r ) k (0) E [1] µ � κ κ � N + s ( n k ) (0) ¯ s ( n k ) (0) N N µ ∂ ¯ n k · Q k k =1 N [1](2) ( P k → k, s ) A [0](0) � + Split (1 , . . . , P k , . . . , N ) N k =1 N [0](2) ( P k → k, s ) Split [1](0) ( l → l ) A [0](0) � + Split (1 , . . . , l, . . . , P k , . . . , N ) N k =1 l � = k N − ∂ A [0](0) � � Split [1](0) E [0] µ � κ n kµ + A [0](0) I [1](0 r ) k E [0] µ � κ � N + s ( n k ) (0) ¯ s ( n k ) (0) N N µ ∂ ¯ n k · Q k k =1 N �� � [1](2 X 2 � k ) J [1](2 X k L ) + J [1](2 X k ∂ ) E [0] � E [0] � κ κ � + s [ n k ]2 + J N N N s [ n k ]3 k =1 N � d d x J [1](2 X k L k ′ ) µ ( x ) E [0] � κ � + s ( n k ′ )[ n k ] µ ( x ) N k,k ′ =1 18