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

∏
/
1
dza
dω,
2
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 ,

∏
/
1
dza
dω
2
fa (z, k)
(za − za+1 )
a∈A

∏
1
1
≡ (z1 − z2 )(z2 − zN )(zN − z1 )
fa (z, k)

f
(z,
k)
a
a∈A
∏ (αδ − βδ) ∏ ′ 1
→
,
(γza + δ)2
fa (z, k)
a∈A

a̸=1,2,N

a∈A

ΨN (z, k, ϵ) is M¨obius invariant,
∏
ΨN = 1 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 )− 2
a
∂U
= −fa U,
∂za

ka2 = m2
is M¨obius invariant,

a∈A

∑ ka · kb
m2
m2
fa (z, k) =
+
+
,
za − zb
2(za − za+1 ) 2(za − za−1 )
b∈A
b̸=a

implying fa (z) → fa (z)

(γza + δ)2
.
(αδ − βγ)

The infinitesimal transformations δza = ϵ1 +ϵ2 za + ϵ3 za2 ,
∂U
U(z + δz)∼ U(z) +
δza , so the fa satisfy the three relations
∂za
∑
∑
∑
fa = 0,
za fa = 0,
za2 fa = 0.
a∈A

a∈A

a∈A

There are N − 3 independent Scattering Equations fa = 0.
Fixing z1 = ∞, z2 = 1, zN = 0, there are N − 3 variables,
and generally (N − 3)! solutions za (k). fˆ = f when m2 = 0.


Total Amplitudes
For example, N = 4,
(
ns
nt
nu )
Aabcd (k1 , k2 , k3 , k4 ) = g 2 fabe fecd
+ fbce fead

+ fcae febd
s
t
u
((
)
2
=g
tr (Ta Tb Tc Td ) + tr (Td Tc Tb Ta ) A(1234)
(
)
+ tr (Ta Tc Td Tb ) + tr (Tb Td Tc Ta ) A(1342)
)
(
)
+ tr (Ta Td Tb Tc ) + tr (Tc Tb Td Ta ) A(1423) ,
(
)
ns = ϵ1 · ϵ2 (k1 −k2 )α + 2ϵ1 · k2 ϵ2α − 2ϵ2 · k1 ϵ1α
(
)
× ϵ3 · ϵ4 (k3 − k4 )α + 2ϵ3 · k4 ϵα4 − 2ϵ4 · k3 ϵα3
(
)
+ ϵ1 · ϵ3 ϵ2 ·ϵ4 − ϵ1 · ϵ4 ϵ2 · ϵ3 s,
ns
nt
A(1234) =
+ .
s = (k1 + k2 )2 , t = (k2 + k3 )2 , u = (k1 + k3 )2

s
t
A(k1 , k2 , k3 , k4 ) =A(1234).


A Single Scalar Field, Massless ϕ3
A single massless scalar field, ΨN = 1.
∮ ∏
∏
/
′
dza
1
ϕ
A (k1 , k2 , . . . , kN ) =
dω
fa (z, k)
(za − za+1 )2
O
a∈A

a∈A

1 1
Aϕ (k1 , k2 , k3 , k4 ) = + ,
s t
total
ϕ
A = A (k1 , k2 , k3 , k4 )+Aϕ (k1 , k3 , k2 , k4 ) + Aϕ (k1 , k4 , k2 , k3 )
(

)
1 1 1
=2
+ +
s
t
u


Proof of the Formula of CHY for Massless ϕ3
AϕN (ζ) = AϕN (k1 , k2 + ζℓ, k3 , . . . , kN−1 , kN − ζℓ),
For ℓ2 = ℓ · k2 = ℓ · kN = 0, these shifted, ordered field theory tree
amplitudes have simple poles in ζ, and AϕN (ζ) → 0 as ζ → ∞.
AϕN (ζ) = −

∑ Resζ Aϕ
i

i

The poles ζi occur where

ζ 2
(πm
)

N

ζi − ζ

ζ 2
= 0 or (¯
πm
) = 0, i.e. at

L
R
ζ = sm /2πm · ℓ ≡ ζm
, and ζ = −¯sm /2¯
πm · ℓ ≡ ζm
,

3 ≤ m ≤ N − 1,

with residues given by
ζR

ζR

ResζmR AϕN = Aϕm (k1 , k2 m , k3 , . . . , km−1 , −¯
πmm )
×

1

2¯
πm ·
R
ζm
ζR

ϕ
AN−m+2 (¯
πm , km , . . . , kN−1 , kNm ),

πm ≡ −k2 − k3 − . . . − km ,

ℓ

2
2
π
¯m ≡ −km − k3 − . . . − kN ; sm = πm
, ¯sm = π
¯m
.


Aϕ (k1 , k2 , . . . , kN ) = AϕN (ζ = 0)
]
N−1
∑ [ 2πm · ℓ
2¯
πm · ℓ
ϕ
ϕ
=−2
ResζmL AN −
ResζmR AN
sm
¯sm

∗

m=3

which determines Aϕ (k1 , . . . kN ) for N > 3 from Aϕ (k1 , k2 , k3 ) = 1.
Our proof is to show Aϕ = Aϕ satisfies ∗.
∮ ∏N−2
AϕN (ζ)

∼

∏
N−1
N−1
∏ b−2
∏
∏ dza
za N−1
2
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 − 1.
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 )
×

1

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
2
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+
2 = ℓ − 2(ζ/k2 · kN )kN , ϵ2 = ℓ; ϵ4 ,
ℓ¯2 = ℓ¯ · k2 = ℓ¯ · kN = 0, ℓ · ℓ¯ = 2.


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 4 (κZ (ρ) − Z ) J A ψ 1 . . . ψ 4 .
∫
∏
Tree M ϵ1 ...ϵN{= ⟨0|e (n−1)q0 s∈N}δ 4 (κs Z (ρs ) − Zs )
∏
∑
dρa dκa ∏
4
× exp i j∈P κj Wj · Z (ρj ) |0⟩ N
a=1
s∈N κs
κa
/
∏
× r

δ 4 (κs Z (ρs ) − Zs )
Z (ρ) = Z0 + Z−1 ρ + · · · + Z−n+1 ρn−1 , polynomial of order n − 1,
so
∑
∏
r
Z (ρ) = s∈N κ1s Zs r ̸=s;r ∈N ρρ−ρ
,
where κs Z (ρs ) = Zs .
s −ρr
The positive helicity vertices become
ei

∑

j∈P

κj Wj ·Z (ρj )

where cjs =

κj
κs

= ei
∏

∑

j∈P

∑

s∈N

ρj −ρr
r ̸=s;r ∈N ρs −ρr

cjs Wj ·Zs

=

λj
λs (ρj −ρs )

are the link variables.

Fourier transforming to momentum space,
(
)
∫∏
∑
Mϵ1 ...ϵN = ⟨r1 , rn ⟩2 (ρr1 − ρrn )2
δ 2 πj − r ∈N cjr πr
j∈P
(
)
∑
∏
¯s + i∈P π
¯i cis
× s∈N ′ δ 2 π
∏
∏N
dκa dρa
1
× N
a=1
a=1 (ρa −ρa+1 )
κa .
a̸=r1 ,rn

4-dimensional momenta kaαα˙ = πaα π
¯aα˙ , 1 ≤ a ≤ N; α, α˙ = 1, 2.
{a ∈ A : a = i ∈ P, r ∈ N , m + n = N}, ρa ≡ za .
Link variables cir ≡
πiα =

λi
λr (zi −zr )

∑

r ∈N

satisfy:

cir πrα ,

−¯
πr α˙ =

∑
i∈P

π
¯i α˙ cir .

BCFW in Twistor String Theory link variables
∮
Mmn (ζ) = Kmn

Mmn (ζ) =

i
∑ Mζmn
,
ζ − ζi
z
i

O

Fmn (c(ζ))

m−1
∏
∏ n−1
a=2 b=2

dcia rb
,
Cab (c(ζ))

Mmn = Mmn (0) = −

∑1
i
Mζmn
.
ζ

i
z
i

Analysis of the poles and residues proves BCFW, demonstrating
equivalence between the twistor string amplitudes and Yang Mills.


Twistor String Equations imply the Scattering Equations
2

∑ ki · kb
∑ ⟨πi , πr ⟩[¯
πi , π
¯j ]
πi , π
¯r ] ∑ ⟨πi , πj ⟩[¯
=
+
zi − zb
zi − zr
zi − zj
r
j

b

∑ ⟨πi , πr ⟩[¯
πi , π
¯r ]

zi − zr

r

=−

∑ cis ⟨πs , πr ⟩[¯
πi , π
¯j ]cjr

=

1
2

∑
rsj

∑ ⟨πi , πj ⟩[¯
πi , π
¯j ]
zi − zj

j

zi − zr

rsj

=

∑

λi λj ⟨πr , πs ⟩[¯
πi , π
¯j ](zr − zs )
λr λs (zi − zr )(zi − zs )(zj − zr )(zj − zs )

cir cjs

rsj

= − 12

∑
rsj

So

∑

ki ·kb
b zi −zb

⟨πr , πs ⟩[¯
πi , π
¯j ]
zi − zj

λi λj ⟨πr , πs ⟩[¯

πi , π
¯j ](zr − zs )
.
λr λs (zi − zr )(zj − zs )(zi − zs )(zj − zr )

= 0, and similarly

2ka · kb = ⟨πa , πb ⟩[¯
πa , π
¯b ];

∑

kr ·kb
b zr −zb

= 0.

⟨πa , πb ⟩ ≡ πaα πbα , [¯
πa , π
¯b ] ≡ π
¯aα˙ πbα˙


Polynomial Form for the Scattering Equations
For a subset U ⊂ A,
kU ≡

∑

ka ,

zU ≡

a⊂U

∏

zb ,

b⊂U

then the Scattering Equations
∑ ka · kb
=0
za − zb
b∈A
b̸=a

are equivalent to the homogeneous polynomial equations
∑
kU2 zU = 0, 2 ≤ m ≤ N − 2,
U⊂A
|U|=m

where the sum is over all

N!
m!(N−m)!

subsets U ⊂ A with m elements.


Proof of the Polynomial Form of the Scattering Equations
p µ (z) ≡

∑
a∈A

p 2 (z) =

∑
a,b

2p 2 (z)

kaµ
,
z − za

∑

kaµ = 0,

a

ka · kb
1 ∑ 1 ∑ ka · kb
=
=0

(z − za )(z − zb )
2 a z − za
(za − zb )
b̸=a

∏

(z − zc ) =

c∈A

=

∑

2ka · kb

c̸=A
c̸=a,b

N−2
∑

∑

z N−m−2

where U = {b ∈ A : b ∈
/ U}. Using
∑

U∈A
|U|=m

kU2 zU = 0.

∏

a,b∈A

m=0

hm ≡

ka2 = 0,

U⊂A
|U|=m

∑
S⊂U
|S|=2

(z − zc )

zU

∑

kS2 = 0

S⊂U
|S|=2

kS2 = kU2 = kU2 , then


z1 → ∞, z2 fixed, zN → 0,

Amplitudes in terms of Polynomial Constraints
∮
AN =

O

ΨN (z, k)

z2
zN−1

N−3
∏

1
h
(z,
k)
m
m=1

hm+1

1
=
z1 →∞ z1
m!

hm = lim

∏
2≤a
∑

(za − zb )

N−2
∏
a=2

2
k1a
z z . . . zam ,
1 ...am a1 a2

za dza+1
.
(za − za+1 )2

1 ≤ m ≤ N − 3,

a1 ,a2 ,...,am ̸=1,N

ai uneq.

The N − 3 polynomial equations hm = 0, of degree m,
linear in each za 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
N=4
2 z + k 2 z = 0,
h1 = k12
2
13 3

2 /k 2 = −k · k /k · k .
z3 /z2 = −k12
1
2 1
3
13

N=5
2 z + k 2 z + k 2 z = 0,
h1 = k12
2
13 3
14 4
2 z z + k 2 z z + k 2 z z = 0,

h2 = k123
2 3
124 2 4
134 3 4

eliminating z4 yields a quadratic equation for z3 /z2 .


Comments
The polynomial form of the Scattering Equations facilitates
computation of their solutions za (k), due to the linearity of the
equations in the individual variables za .
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 string
equations are closely related.
The proofs make it certain that both the twistor string and the
Scattering 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.