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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
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