Amplitudes and the scattering equations,

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 ϕ3 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 ∮ A(k1 , k2 , , kN ) = O ΨN (z, k, ϵ) ∏′ a∈A ∏ / dza dω, fa (z, k) (za − za+1 ) a∈A O encircles the zeros of fa (z, k), fa (z, k) ≡ ∑ ka · kb =0 za − zb b∈A The Scattering Equations b̸=a (Cachazo, He, Yuan 2013) (Fairlie, Roberts 1972) ∑ ka2 = 0, kaµ = 0, A = {1, 2, N.} a∈A DG proved A(k1 , k2 , kn ) are ϕ3 and Yang-Mills gluon field theory tree amplitudes , as conjectured by CHY za → M¨ obius Invariance ∮ A(k1 , k2 , , kN ) = ∏′ a∈A O ΨN (z, k, ϵ) ∏′ a∈A αza +β γza +δ , ∏ / dza dω fa (z, k) (za − za+1 ) a∈A ∏ 1 ≡ (z1 − z2 )(z2 − zN )(zN − z1 ) fa (z, k) f (z, k) a a∈A ∏ (αδ − βδ) ∏ ′ → , (γza + δ)2 fa (z, k) a∈A a̸=1,2,N a∈A ΨN (z, k, ϵ) is M¨obius invariant, ∏ ΨN = for ϕ3 , ΨN = a∈A (za − za+1 ) × Pffafian for Yang-Mills The integrand and the Scattering Equations are M¨obius invariant (CHY) Massive Scattering Equations fa (z, k) = 0, ∏ ∏ m2 U(z, k) ≡ (za − zb )−ka ·kb (za − za+1 )− a from Aϕ (k1 , k2 , k3 ) = Our proof is to show Aϕ = Aϕ satisfies ∗ ∮ ∏N−2 AϕN (ζ) ∼ ∏ N−1 N−1 ∏ b−2 ∏ ∏ dza za N−1 a=4 (1 − za ) (za − zb ) (1 − z3 )zN−1 fa (z, ζ) a=3 b=5 a=3 a=3 R comes from the integration region z → 0, A pole at ζm a m ≤ a ≤ N − Let za = xa zm , zm → 0, ∏N−1 a=3 dza = ∏m−1 a=3 ζR dza dzm ∏N−1 a=m+1 dxa , ζR ResζmR AϕN = Aϕm (k1 , k2 m , k3 , , km−1 , −¯ πmm ) × 2¯ πm · R R ζm ζm ϕ AN−m+2 (¯ πm , km , , kN−1 , kN ), Similarly for ResζmL AϕN So proving the formula for Aϕ (k1 , , kN ) by induction ℓ Proof for Pure Gauge Theory ∏N−2 ∮ AN (ζ) ∼ YM ΨoN ∏ N−1 N−1 ∏ b−2 ∏ ∏ dza za N−1 a=4 (1 − za ) (za − zb ) (1 − z3 )zN−1 fa (z, ζ) a=3 b=5 a=3 a=3 where the only difference from the scalar case is ΨoN , which is related to the Pfaffian of the antisymmetric matrix MN with the 2nd and Nth rows and columns removed, ΨoN = (−1)N Pf MN (z; k ζ ; ϵζ )(2,N) N ∏ (za − za+1 ), a=1 det M ≡ (Pf M)2 , ζ− ζ± ¯ ϵζ+ = ℓ − 2(ζ/k2 · kN )kN , ϵ2 = ℓ; ϵ4 , ℓ¯2 = ℓ¯ · k2 = ℓ¯ · kN = 0, ℓ · ℓ¯ = All singularities in ΨoN are canceled by the numerator ΨoN L,R factorizes at the poles in the integrand ζm , since the Pfaffian does As zm → 0, Pf MN (k1 , , kN ; ϵ1 , , ϵN ; z3 , , zN−1 )(2,N) ∑ ∼ Pf, Mm (k1 , , km−1 , −¯ πm ; ϵ1 , , ϵm−1 , ϵs ; z3 , , zm−1 )(2,m) s × Pf MN−m+2 (¯ πm , km , , kN ; ϵs , ϵm , , ϵN ; xm+1 , , xN−1 )(1,N−m+2) , and N−1 ∏ m−2 ∏ a=2 a=2 N−m (za − za+1 ) → zm−1 zm (za − za+1 ) N−1 ∏ (xa − xa+1 ) a=m This demonstrates that AYM N (ζ = 0) satisfies the BCFW recurrence relation, so that AYM (k1 , kN ), computed from the scattering equations, are equal to the Yang Mills field theory tree amplitudes Twistor String Theory (4d) k µ σµ αα˙ ≡ kαα˙ = πα π ¯α˙ , ( Z= ) α π , ω α˙ Conjugate twistor variables ( ) ω ¯α W = , π ¯α˙ W ·Z =ω ¯α πα + π ¯α˙ ω α˙ , ( and twistor string worldsheet fields, Z (ρ) = ) λα (ρ) µα˙ (ρ) Fourier transform gluon vertex operators according to helicity: ∫ iκW ·Z (ρ) J A , V+A (W , ρ) = dκ κ e ∫ V−A (Z , ρ) = κ3 dκ δ (κZ (ρ) − Z ) J A ψ ψ ∫ ∏ Tree M ϵ1 ϵN{= ⟨0|e (n−1)q0 s∈N}δ (κs Z (ρs ) − Zs ) ∏ ∑ dρa dκa ∏ × exp i j∈P κj Wj · Z (ρj ) |0⟩ N a=1 s∈N κs κa / ∏ × r