Amplitudes and the scattering equations,

Amplitudes and the Scattering Equations,Proofs and Polynomials Louise DolanUniversity of North Carolina at Chapel Hill Strings 2014, Princetonwork with Peter Goddard, IAS 1402.7374 [hep-

Amplitudes and the Scattering Equations,

Proofs and Polynomials

Louise DolanUniversity 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

•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

z a − zb = 0 The Scattering Equations

(Cachazo, He, Yuan 2013) (Fairlie, Roberts 1972)

k a2= 0, ∑

a ∈A

k a µ = 0, A = {1, 2, N.}

DG proved A(k1, k2, k n ) are ϕ3 and Yang-Mills gluon

field theory tree amplitudes , as conjectured by CHY

M¨obius Invariance z a → αza +β

a ∈A (z a − za+1)× Pffafian for Yang-Mills

Theintegrandand the Scattering Equationsare M¨obius invariant(CHY)

Massive Scattering Equations bf a (z, k) = 0, k a2= m2

The infinitesimal transformations δz a = ϵ1+ϵ2 za + ϵ3 z a2,

U(z + δz) ∼ U(z) + ∂U

∂za δz a , so the b f a satisfy the three relations

There are N − 3 independent Scattering Equations bf a= 0

Fixing z1 =∞, z2 = 1, z N = 0, there are N − 3 variables,

and generally (N − 3)! solutions z a (k). ˆf = f when m2 = 0.

A Single Scalar Field, Massless ϕ3

A single massless scalar field, ΨN = 1

Proof of the Formula of CHY for Massless ϕ3

A ϕ N (ζ) = A ϕ N (k1, k2+ ζℓ, k3, , k N −1 , k N − ζℓ),

For ℓ2 = ℓ · k2 = ℓ · k N = 0, these shifted, ordered field theory tree

amplitudes have simple poles in ζ, and A ϕ N (ζ) → 0 as ζ → ∞.

× A ϕ

N −m+2(¯π ζ

R m

which determines A ϕ (k1, k N ) for N > 3 from A ϕ (k1, k2, k3) = 1.

Our proof is to show A ϕ=A ϕsatisfies∗

N∏−1 b=5

b∏−2 a=3 (z a − z b)2

N∏−1 a=3

dz a

f a (z, ζ)

A pole at ζ m R comes from the integration region z a → 0,

m ≤ a ≤ N − 1 Let z a = x a z m , z m → 0,

m , k m , , k N −1 , k ζ

R m

Proof for Pure Gauge Theory

where the only difference from the scalar case is Ψo N, which is

related to the Pfaffian of the antisymmetric matrix MN with the2nd and Nth rows and columns removed,

All singularities in Ψo N are canceled by the numerator Ψo N

factorizes at the poles in the integrand ζ m L,R, since the Pfaffian

This demonstrates thatAYM

N (ζ = 0) satisfies the BCFW recurrence

relation, so thatAYM(k1, 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

d ρad κa κa

∏

s ∈N κ4s

×∏r <s;r ,s ∈N (ρ r − ρ s)4⟨0|J A1 (ρ1)J A2 (ρ2) J AN (ρ N)|0⟩/dg

r ̸=s;r∈N ρs ρj−ρr −ρ r = λs (ρ λj j−ρ s) are the link variables.

Fourier transforming to momentum space,

4-dimensional momenta k aα ˙ α = π aα π¯a ˙ α, 1≤ a ≤ N; α, ˙α = 1, 2.

n∏−1 b=2

dc iarb Cab (c(ζ)) ,

Mmn (ζ) =∑

zi

M ζi mn

Twistor String Equations imply the Scattering Equations

Polynomial Form for the Scattering Equations

Proof of the Polynomial Form of the Scattering Equations

k 1a12 am za1za2 zam , 1≤ m ≤ N − 3,

The N − 3 polynomial equations hm = 0, of degree m,

linear in each z a individually,

are equivalent to the Scattering Equations

By B´ezout’s theorem, they determine (N-3)! solutions for the

ratios of the z2 , , zN −1

Solutions to the Scattering Equations

The polynomial form of the Scattering Equations facilitates

computation of their solutions z a (k), due to the linearity of the equations in the individual variables z a

The Scattering Equations can be generalized to massive particles,

enabling the description of tree amplitudes for massive ϕ3 theory

In four dimensions, the Scattering Equations and the twistor stringequations are closely related

The proofs make it certain that both the twistor string and theScattering Equations approach are equivalent to gauge field theory

at tree level

This critical reasoning may provide insight into possible extensions

to loop level of these non-Lagrangian methods