Amplitudes and the Scattering Equations, Proofs and Polynomials - PowerPoint PPT Presentation

Amplitudes and the Scattering Equations, Proofs and Polynomials Louise Dolan University of North Carolina at Chapel Hill Strings 2014, Princeton (work with Peter Goddard, IAS) 1402.7374 [hep-th], The Polynomial Form of the Scattering Equations 1311.5200 [hep-th], Proof of the Formula of Cachazo, He and Yuan for Yang-Mills Tree Amplitudes in Arbitrary Dimension 1111.0950 [hep-th], Complete Equivalence Between Gluon Tree Amplitudes in Twistor String Theory and in Gauge Theory

See also Freddy Cachazo, Song He, and Ellis Yuan (CHY) 1309.0885 [hep-th], Scattering of Massless Particles: Scalars, Gluons and Gravitons 1307.2199 [hep-th], Scattering of Massless Particles in Arbitrary Dimensions 1306.6575 [hep-th], Scattering Equations and KLT Orthogonality Edward Witten, hep-th/0312171, Perturbative Gauge Theory as a String theory in Twistor Space Nathan Berkovits, hep-th/0402045, An Alternative String Theory in Twistor Space for N=4 SuperYang-Mills

Outline • Tree amplitudes from the Scattering Equations in any dimension • M¨ obius invariance and massive Scattering Equations • Proof of the equivalence with ϕ 3 and Yang-Mills field theories • In 4d: link variables, twistor string ↔ the Scattering Equations • Direct proof of equivalence between twistor string and field theory gluon tree amplitudes • Polynomial form of the Scattering Equations

Tree Amplitudes ∮ ∏ ∏ / 1 dz a ′ A ( k 1 , k 2 , . . . , k N ) = Ψ N ( z , k , ϵ ) d ω, ( z a − z a +1 ) 2 f a ( z , k ) O a ∈ A a ∈ A O encircles the zeros of f a ( z , k ), ∑ k a · k b f a ( z , k ) ≡ = 0 The Scattering Equations z a − z b b ∈ A b ̸ = a ( Cachazo , He , Yuan 2013) . . . ( Fairlie , Roberts 1972) ∑ k 2 k µ A = { 1 , 2 , . . . N . } a = 0 , a = 0 , a ∈ A DG proved A ( k 1 , k 2 , . . . k n ) are ϕ 3 and Yang-Mills gluon field theory tree amplitudes , as conjectured by CHY.

z a → α z a + β M¨ obius Invariance γ z a + δ , ∮ ∏ ∏ / 1 dz a ′ A ( k 1 , k 2 , . . . , k N ) = Ψ N ( z , k , ϵ ) d ω ( z a − z a +1 ) 2 f a ( z , k ) O a ∈ A a ∈ A ∏ ∏ 1 1 ′ f a ( z , k ) ≡ ( z 1 − z 2 )( z 2 − z N )( z N − z 1 ) f a ( z , k ) a ∈ A a ∈ A a ̸ =1 , 2 , N ∏ ∏ ( αδ − βδ ) 1 ′ → f a ( z , k ) , ( γ z a + δ ) 2 a ∈ A a ∈ A Ψ N ( z , k , ϵ ) is M¨ obius invariant, Ψ N = ∏ Ψ N = 1 for ϕ 3 , a ∈ A ( z a − z a +1 ) × Pffafian for Yang-Mills The integrand and the Scattering Equations are M¨ obius invariant (CHY).

� k 2 a = m 2 Massive Scattering Equations f a ( z , k ) = 0, ∏ ( z a − z b ) − k a · k b ∏ ( z a − z a +1 ) − m 2 U ( z , k ) ≡ is M¨ obius invariant, 2 a < b a ∈ A ∑ m 2 m 2 k a · k b ∂ U = − � � f a U , f a ( z , k ) = + 2( z a − z a +1 ) + 2( z a − z a − 1 ) , ∂ z a z a − z b b ∈ A b ̸ = a f a ( z )( γ z a + δ ) 2 implying � f a ( z ) → � ( αδ − βγ ) . The infinitesimal transformations δ z a = ϵ 1 + ϵ 2 z a + ϵ 3 z 2 a , U ( z + δ z ) ∼ U ( z ) + ∂ U δ z a , so the � f a satisfy the three relations ∂ z a ∑ ∑ ∑ � z a � a � z 2 f a = 0 , f a = 0 , f a = 0 . a ∈ A a ∈ A a ∈ A There are N − 3 independent Scattering Equations � f a = 0. Fixing z 1 = ∞ , z 2 = 1 , z N = 0, there are N − 3 variables, f = f when m 2 = 0 . ˆ and generally ( N − 3)! solutions z a ( k ).

Total Amplitudes For example, N = 4, A abcd ( k 1 , k 2 , k 3 , k 4 ) = g 2 ( ) n s n t n u f abe f ecd s + f bce f ead t + f cae f ebd u = g 2 (( ) tr ( T a T b T c T d ) + tr ( T d T c T b T a ) A (1234) ( ) + tr ( T a T c T d T b ) + tr ( T b T d T c T a ) A (1342) ) ( ) + tr ( T a T d T b T c ) + tr ( T c T b T d T a ) A (1423) , ( ) ϵ 1 · ϵ 2 ( k 1 − k 2 ) α + 2 ϵ 1 · k 2 ϵ 2 α − 2 ϵ 2 · k 1 ϵ 1 α n s = ( ) ϵ 3 · ϵ 4 ( k 3 − k 4 ) α + 2 ϵ 3 · k 4 ϵ α 4 − 2 ϵ 4 · k 3 ϵ α × 3 ( ) ϵ 1 · ϵ 3 ϵ 2 · ϵ 4 − ϵ 1 · ϵ 4 ϵ 2 · ϵ 3 + s , A (1234) = n s s + n t t . s = ( k 1 + k 2 ) 2 , t = ( k 2 + k 3 ) 2 , u = ( k 1 + k 3 ) 2 A ( k 1 , k 2 , k 3 , k 4 ) = A (1234) .

A Single Scalar Field, Massless ϕ 3 A single massless scalar field, Ψ N = 1. ∮ ∏ ∏ / 1 dz a ′ A ϕ ( k 1 , k 2 , . . . , k N ) = d ω ( z a − z a +1 ) 2 f a ( z , k ) O a ∈ A a ∈ A A ϕ ( k 1 , k 2 , k 3 , k 4 ) = 1 s +1 t , A total = A ϕ ( k 1 , k 2 , k 3 , k 4 )+ A ϕ ( k 1 , k 3 , k 2 , k 4 ) + A ϕ ( k 1 , k 4 , k 2 , k 3 ) ( 1 ) s + 1 t + 1 = 2 u

Proof of the Formula of CHY for Massless ϕ 3 A ϕ N ( ζ ) = A ϕ N ( k 1 , k 2 + ζℓ, k 3 , . . . , k N − 1 , k N − ζℓ ) , For ℓ 2 = ℓ · k 2 = ℓ · k N = 0, these shifted, ordered field theory tree amplitudes have simple poles in ζ , and A ϕ N ( ζ ) → 0 as ζ → ∞ . Res ζ i A ϕ ∑ A ϕ N N ( ζ ) = − ζ i − ζ i m ) 2 = 0 or (¯ m ) 2 = 0, i.e. at The poles ζ i occur where ( π ζ π ζ ζ = s m / 2 π m · ℓ ≡ ζ L π m · ℓ ≡ ζ R m , and ζ = − ¯ s m / 2¯ m , 3 ≤ m ≤ N − 1 , with residues given by 1 m ( k 1 , k ζ R π ζ R m A ϕ N = A ϕ 2 , k 3 , . . . , k m − 1 , − ¯ Res ζ R m m ) m 2¯ π m · ℓ π ζ R m , k m , . . . , k N − 1 , k ζ R × A ϕ N − m +2 (¯ m N ) , m π m ≡ − k m − k 3 − . . . − k N ; s m = π 2 π 2 π m ≡ − k 2 − k 3 − . . . − k m , ¯ m , ¯ s m = ¯ m .

A ϕ ( k 1 , k 2 , . . . , k N ) = A ϕ N ( ζ = 0) [ 2 π m · ℓ ] N − 1 ∑ N − 2¯ π m · ℓ m A ϕ m A ϕ = − 2 Res ζ L Res ζ R ∗ N s m ¯ s m m =3 which determines A ϕ ( k 1 , . . . k N ) for N > 3 from A ϕ ( k 1 , k 2 , k 3 ) = 1. Our proof is to show A ϕ = A ϕ satisfies ∗ . ∮ ∏ N − 2 ∏ N − 1 N − 1 b − 2 N − 1 ∏ ∏ ∏ a =3 z a a =4 (1 − z a ) dz a A ϕ ( z a − z b ) 2 N ( ζ ) ∼ (1 − z 3 ) z N − 1 f a ( z , ζ ) b =5 a =3 a =3 A pole at ζ R m comes from the integration region z a → 0, m ≤ a ≤ N − 1. Let z a = x a z m , z m → 0,

∏ N − 1 a =3 dz a = ∏ m − 1 ∏ N − 1 a =3 dz a dz m a = m +1 dx a , 1 m ( k 1 , k ζ R π ζ R m A ϕ N = A ϕ 2 , k 3 , . . . , k m − 1 , − ¯ Res ζ R m m ) m 2¯ π m · ℓ π ζ R m , k m , . . . , k N − 1 , k ζ R × A ϕ N − m +2 (¯ N ) , m m m A ϕ Similarly for Res ζ L N . So proving the formula for A ϕ ( k 1 , . . . , k N ) by induction.

Proof for Pure Gauge Theory ∮ ∏ N − 2 ∏ N − 1 N − 1 b − 2 N − 1 ∏ ∏ ∏ a =3 z a a =4 (1 − z a ) dz a Ψ o ( z a − z b ) 2 A YM N ( ζ ) ∼ N (1 − z 3 ) z N − 1 f a ( z , ζ ) b =5 a =3 a =3 where the only difference from the scalar case is Ψ o N , which is related to the Pfaffian of the antisymmetric matrix M N with the 2nd and Nth rows and columns removed, N ∏ Ψ o N = ( − 1) N Pf M N ( z ; k ζ ; ϵ ζ ) (2 , N ) ( z a − z a +1 ) , a =1 det M ≡ (Pf M) 2 , ℓ − 2( ζ/ k 2 · k N ) k N , ϵ ζ − = ℓ ; ϵ ζ ± ϵ ζ + = ¯ 4 , 2 2 ℓ 2 = ¯ ¯ ℓ · k 2 = ¯ ℓ · k N = 0 , ℓ · ¯ ℓ = 2 .

All singularities in Ψ o N are canceled by the numerator. Ψ o N factorizes at the poles in the integrand ζ L , R m , since the Pfaffian does. As z m → 0, Pf M N ( k 1 , . . . , k N ; ϵ 1 , . . . , ϵ N ; z 3 , . . . , z N − 1 ) (2 , N ) ∑ π m ; ϵ 1 , . . . , ϵ m − 1 , ϵ s ; z 3 , . . . , z m − 1 ) (2 , m ) ∼ Pf, M m ( k 1 , . . . , k m − 1 , − ¯ s π m , k m , . . . , k N ; ϵ s , ϵ m , . . . , ϵ N ; x m +1 , . . . , x N − 1 ) (1 , N − m +2) , × Pf M N − m +2 (¯ and N − 1 m − 2 N − 1 ∏ ∏ ∏ ( z a − z a +1 ) → z m − 1 z N − m ( z a − z a +1 ) ( x a − x a +1 ) m a = m a =2 a =2 This demonstrates that A YM N ( ζ = 0) satisfies the BCFW recurrence relation, so that A YM ( k 1 , . . . k N ), computed from the scattering equations, are equal to the Yang Mills field theory tree amplitudes.

Twistor String Theory (4d) k µ σ µ α ˙ α ≡ k α ˙ α = π α ¯ π ˙ α , Conjugate twistor variables ( π α ) ( ¯ ) ω α ω α π α + ¯ α ω ˙ α , Z = , W = , W · Z = ¯ π ˙ ω ˙ α π ˙ ¯ α ( λ α ( ρ ) ) and twistor string worldsheet fields, Z ( ρ ) = . µ ˙ α ( ρ ) Fourier transform gluon vertex operators according to helicity: ∫ d κ V A κ e i κ W · Z ( ρ ) J A , + ( W , ρ ) = ∫ κ 3 d κ δ 4 ( κ Z ( ρ ) − Z ) J A ψ 1 . . . ψ 4 . V A − ( Z , ρ ) = ∫ ⟨ 0 | e ( n − 1) q 0 ∏ Tree M ϵ 1 ...ϵ N = s ∈N δ 4 ( κ s Z ( ρ s ) − Z s ) { } i ∑ | 0 ⟩ ∏ N ∏ d ρ a d κ a s ∈N κ 4 × exp j ∈P κ j W j · Z ( ρ j ) a =1 s κ a / × ∏ r < s ; r , s ∈N ( ρ r − ρ s ) 4 ⟨ 0 | J A 1 ( ρ 1 ) J A 2 ( ρ 2 ) . . . J A N ( ρ N ) | 0 ⟩ dg

δ 4 ( κ s Z ( ρ s ) − Z s ) Z ( ρ ) = Z 0 + Z − 1 ρ + · · · + Z − n +1 ρ n − 1 , polynomial of order n − 1, so Z ( ρ ) = ∑ ∏ ρ − ρ r 1 κ s Z s ρ s − ρ r , where κ s Z ( ρ s ) = Z s . s ∈N r ̸ = s ; r ∈N The positive helicity vertices become j ∈P κ j W j · Z ( ρ j ) = e i ∑ e i ∑ ∑ s ∈N c js W j · Z s j ∈P ∏ ρ j − ρ r where c js = κ j λ j ρ s − ρ r = λ s ( ρ j − ρ s ) are the link variables. r ̸ = s ; r ∈N κ s Fourier transforming to momentum space, M ϵ 1 ...ϵ N = ⟨ r 1 , r n ⟩ 2 ( ρ r 1 − ρ r n ) 2 ∫ ∏ j ∈P δ 2 ( ) π j − ∑ r ∈N c jr π r s ∈N ′ δ 2 ( ) × ∏ π s + ∑ ¯ i ∈P ¯ π i c is × ∏ N ∏ N 1 d κ a d ρ a . a =1 a =1 ( ρ a − ρ a +1 ) κ a a ̸ = r 1 , rn