Jacob L. Bourjaily - Jagiellonian Universityth- · The Shocking Simplicity of Scattering Amplitudes...

The Vernacular of the S-MatrixThe All-Orders S-Matrix for Three Massless Particles

Consequences of Quantum Mechanical Consistency Conditions

Jacob L. BourjailyCracow School of Theoretical Physics

LVI Course, 2016A Panorama of Holography

Wednesday, 25th May Cracow School of Theoretical Physics, Zakopane Part I: The Vernacular of the S-Matrix

Jacob L. BourjailyCracow School of Theoretical Physics

LVI Course, 2016A Panorama of Holography

Organization and Outline

1 Spiritus Movens: a moral parableA Simple, Practical Problem in Quantum ChromodynamicsThe Shocking Simplicity of Scattering Amplitudes

2 The Vernacular of the S-MatrixPhysically Observable Data Describing Asymptotic StatesMassless Momenta and Spinor-Helicity Variables(Grassmannian) Geometry of Momentum Conservation

3 The All-Orders S-Matrix for Three Massless ParticlesThree Particle Kinematics and Helicity AmplitudesNon-Dynamical Dependence: Coupling Constants & Spin/Statistics

4 Consequences of Quantum Mechanical Consistency ConditionsFactorization and Long-Range Physics: Weinberg’s TheoremUniqueness of Yang-Mills Theory and the Equivalence PrincipleThe Simplest Quantum Field Theory: N=4 super Yang-Mills

A Simple, Practical Problem in Quantum ChromodynamicsThe Shocking Simplicity of Scattering Amplitudes (a parable)

Supercomputer Computations in Quantum ChromodynamicsConsider the amplitude for two gluons to collide and produce four: gg→gggg.

Before modern computers, this would have been computationally intractable220 Feynman diagrams

, thousands of terms

In 1985, Parke and Taylor took up the challengeusing every theoretical tool available

and the world’s best supercomputers

final formula fit into 8 pages

Supercomputer Computations in Quantum ChromodynamicsConsider the amplitude for two gluons to collide and produce four: gg→gggg.Before modern computers, this would have been computationally intractable

220 Feynman diagrams

220 Feynman diagrams, thousands of terms

In 1985, Parke and Taylor took up the challenge

using every theoretical tool available

The Discovery of Incredible, Unanticipated SimplicityThey soon guessed a simplified form of the amplitude

(checked numerically):

—which naturally suggested the amplitude for all multiplicity!

=〈a b〉4

〈1 2〉〈2 3〉〈3 4〉〈4 5〉〈5 6〉〈6 1〉δ2×2(λ·λ)

(≡δ2×2( n∑

λαa λαa))

Here, we have used spinor variables to describe the external momenta:

7→ pααa ≡ pµaσααµ

a + p3a p1

a−ip2a

p1a+ip2

a p0a− p3

≡ λαa λαa

⇔ “ a〉[a ”

Notice that pµpµ=det(pαα)

=0 for massless particles.

which is made manifest

The (local) Lorentz group, SL(2)L×SL(2)R, acts on λa and λa, respectively.The Grassmannian G(k, n): the linear span of k vectors in Cn.Momentum conservation becomes thegeometric statement:

λ⊂ λ⊥ and λ⊂λ⊥.

Thus, Lorentz invariants must be constructed out of determinants:

〈a b〉≡ det(λa, λb), [a b]≡ det(λa, λb)

λβbThe action of the little group corresponds to:

(λa, λa

)7→ (ta λa, t–1

a λa)

: Ψhaa 7→ t–2ha

a Ψhaa

The Discovery of Incredible, Unanticipated SimplicityThey soon guessed a simplified form of the amplitude (checked numerically):

=〈a b〉4

〈1 2〉〈2 3〉〈3 4〉〈4 5〉〈5 6〉〈6 1〉δ2×2(λ·λ)

(≡δ2×2( n∑

λαa λαa))

a + p3a p1

a−ip2a

p1a+ip2

a p0a− p3

≡ λαa λαa

⇔ “ a〉[a ”

(λa, λa

)7→ (ta λa, t–1

a λa)

a Ψhaa

=〈a b〉4

〈1 2〉〈2 3〉〈3 4〉〈4 5〉〈5 6〉〈6 1〉δ2×2(λ·λ)

(≡δ2×2( n∑

λαa λαa))

a + p3a p1

a−ip2a

p1a+ip2

a p0a− p3

≡ λαa λαa

⇔ “ a〉[a ”

(λa, λa

)7→ (ta λa, t–1

a λa)

a Ψhaa

=〈a b〉4

〈1 2〉〈2 3〉〈3 4〉〈4 5〉〈5 6〉〈6 1〉δ2×2(λ·λ)

(≡δ2×2( n∑

λαa λαa))

a + p3a p1

a−ip2a

p1a+ip2

a p0a− p3

≡ λαa λαa

⇔ “ a〉[a ”

(λa, λa

)7→ (ta λa, t–1

a λa)

a Ψhaa

=〈a b〉4

〈1 2〉〈2 3〉〈3 4〉〈4 5〉〈5 6〉〈6 1〉δ2×2(λ·λ)

(≡δ2×2( n∑

λαa λαa))

a + p3a p1

a−ip2a

p1a+ip2

a p0a− p3

≡ λαa λαa

⇔ “ a〉[a ”

(λa, λa

)7→ (ta λa, t–1

a λa)

a Ψhaa

=〈a b〉4

〈1 2〉〈2 3〉〈3 4〉〈4 5〉〈5 6〉〈6 1〉δ2×2(λ·λ)

(≡δ2×2( n∑

λαa λαa))

a + p3a p1

a−ip2a

p1a+ip2

a p0a− p3

≡ λαa λαa

⇔ “ a〉[a ”

(λa, λa

)7→ (ta λa, t–1

a λa)

a Ψhaa

=〈a b〉4

〈1 2〉〈2 3〉〈3 4〉〈4 5〉〈5 6〉〈6 1〉δ2×2(λ·λ)

(≡δ2×2( n∑

λαa λαa))

a + p3a p1

a−ip2a

p1a+ip2

a p0a− p3

≡ λαa λαa

⇔ “ a〉[a ”

(λa, λa

)7→ (ta λa, t–1

a λa)

a Ψhaa

=〈a b〉4

〈1 2〉〈2 3〉〈3 4〉〈4 5〉 · · · 〈n 1〉δ2×2(λ·λ)

(≡δ2×2( n∑

λαa λαa))

a + p3a p1

a−ip2a

p1a+ip2

a p0a− p3

≡ λαa λαa

⇔ “ a〉[a ”

(λa, λa

)7→ (ta λa, t–1

a λa)

a Ψhaa

=〈a b〉4

〈1 2〉〈2 3〉〈3 4〉〈4 5〉 · · · 〈n 1〉δ2×2(λ·λ)

(≡δ2×2( n∑

λαa λαa))

a + p3a p1

a−ip2a

p1a+ip2

a p0a− p3

≡ λαa λαa

⇔ “ a〉[a ”

(λa, λa

)7→ (ta λa, t–1

a λa)

a Ψhaa

=〈a b〉4

〈1 2〉〈2 3〉〈3 4〉〈4 5〉 · · · 〈n 1〉δ2×2(λ·λ)

(≡δ2×2( n∑

λαa λαa))

a + p3a p1

a−ip2a

p1a+ip2

a p0a− p3

≡ λαa λαa

⇔ “ a〉[a ”

(λa, λa

)7→ (ta λa, t–1

a λa)

a Ψhaa

Physically Observable Data Describing Asymptotic StatesMassless Momenta and Spinor-Helicity Variables(Grassmannian) Geometry of Momentum Conservation

On What Data Does a Scattering Amplitude Depend?A scattering amplitude, An, can be a generally complicated(?) function of all

the physically observable data describing each of the particles involved.

Physical data for the ath particle: |a〉

pµa momentum

, on-shell: p2a m2

•• qa all the non-kinematical quantum

numbers of a (color, flavor, . . . )

Although a Lagrangian formalism requires that we use polarization tensors,it is impossible to continuously define polarizations for each helicity statewithout introducing unobservable (gauge) redundancy

—e.g. for σa =1:

εµa ∼ εµa + α(pa)pµaSuch unphysical baggage is almost certainly responsible for the incredible

obfuscation of simplicity in the traditional approach to quantum field theory.

pµa momentum

, on-shell: p2a m2

—e.g. for σa =1:

pµa momentum

, on-shell: p2a m2

—e.g. for σa =1:

pµa momentum

, on-shell: p2a m2

—e.g. for σa =1:

• pµa momentum

, on-shell: p2a m2

—e.g. for σa =1:

Physical data for the ath particle: |a〉• pµa momentum

, on-shell: p2a m2

a =0•• qa all the non-kinematical quantum

—e.g. for σa =1:

Physical data for the ath particle: |a〉• pµa momentum

, on-shell: p2a m2

• ma mass

• qa all the non-kinematical quantumnumbers of a (color, flavor, . . . )

—e.g. for σa =1:

Physical data for the ath particle: |a〉• pµa momentum, on-shell: p2

a m2a =0

• ma mass

—e.g. for σa =1:

a m2a =0

—e.g. for σa =1:

a m2a =0

• σa spin

, helicity ha∈{σa, . . . , σa}• qa all the non-kinematical quantum

—e.g. for σa =1:

a m2a =0

• σa spin, helicity ha∈{σa, . . . , σa}

—e.g. for σa =1:

a m2a =0

• σa spin, helicity ha∈{σa, . . . , σa}

—e.g. for σa =1:

a m2a =0

• σa spin, helicity ha =±σa (ma =0)

—e.g. for σa =1:

a m2a =0

• σa spin, helicity ha =±σa (ma =0)

—e.g. for σa =1:

a m2a =0

• σa spin, helicity ha =±σa (ma =0)• qa all the non-kinematical quantum

—e.g. for σa =1:

a m2a =0

—e.g. for σa =1:

a m2a =0

Although a Lagrangian formalism requires that we use polarization tensors,

it is impossible to continuously define polarizations for each helicity statewithout introducing unobservable (gauge) redundancy

—e.g. for σa =1:

a m2a =0

—e.g. for σa =1:

a m2a =0

Although a Lagrangian formalism requires that we use polarization tensors,it is impossible to continuously define polarizations for each helicity statewithout introducing unobservable (gauge) redundancy—e.g. for σa =1:

a m2a =0

εµa ∼ εµa + α(pa)pµa

Such unphysical baggage is almost certainly responsible for the incredibleobfuscation of simplicity in the traditional approach to quantum field theory.

a m2a =0

εµa ∼ εµa + α(pa)pµa

Such unphysical baggage is almost certainly responsible for the incredibleobfuscation of simplicity in the traditional approach to quantum field theory.

a m2a =0

Making Masslessness Manifest: Spinor-Helicity VariablesTo avoid constraining each particle’s momentum to be null, van der Waerden

introduced (in 1929!) spinor-helicity variables to make this always trivial.

7→ pααa

≡ pµaσααµ

a+ p3a p1

a−ip2a

p1a+ip2

a p0a− p3

≡ λαa λαa

⇔ “ a〉[a ”

• Notice that det(pααa)

a)2 (p1

a)2 (p2

a)2 (p3

, for massless particles.

This can be made manifest by writing pααa as an outer product of 2-vectors.

|a〉ha 7→ t-2haa |a〉ha

•When pa is real(pa∈R3,1

), pααa =(pααa )†

, which implies that(λαa )∗=±λαa .

(but allowing for complex momenta, λa and λa become independent.)

det(λ

≡〈a b〉

det(λ

≡ [a b]

7→ pααa

≡ pµaσααµ

a+ p3a p1

a−ip2a

p1a+ip2

a p0a− p3

≡ λαa λαa

⇔ “ a〉[a ”

a)2 (p1

a)2 (p2

a)2 (p3

det(λ

≡〈a b〉

det(λ

≡ [a b]

pµa 7→ pααa

≡ pµaσααµ

a+ p3a p1

a−ip2a

p1a+ip2

a p0a− p3

≡ λαa λαa

⇔ “ a〉[a ”

a)2 (p1

a)2 (p2

a)2 (p3

det(λ

≡〈a b〉

det(λ

≡ [a b]

pµa 7→ pααa ≡ pµaσααµ

a+ p3a p1

a−ip2a

p1a+ip2

a p0a− p3

≡ λαa λαa

⇔ “ a〉[a ”

a)2 (p1

a)2 (p2

a)2 (p3

det(λ

≡〈a b〉

det(λ

≡ [a b]

pµa 7→ pααa ≡ pµaσααµ =

a+ p3a p1

a−ip2a

p1a+ip2

a p0a− p3

≡ λαa λαa

⇔ “ a〉[a ”

a)2 (p1

a)2 (p2

a)2 (p3

det(λ

≡〈a b〉

det(λ

≡ [a b]

a+ p3a p1

a−ip2a

p1a+ip2

a p0a− p3

≡ λαa λαa

⇔ “ a〉[a ”

a)2 (p1

a)2 (p2

a)2 (p3

, for massless particles.This can be made manifest by writing pααa as an outer product of 2-vectors.

det(λ

≡〈a b〉

det(λ

≡ [a b]

a+ p3a p1

a−ip2a

p1a+ip2

a p0a− p3

≡ λαa λαa

⇔ “ a〉[a ”

a)2 (p1

a)2 (p2

a)2 (p3

= 0, for massless particles.

det(λ

≡〈a b〉

det(λ

≡ [a b]

a+ p3a p1

a−ip2a

p1a+ip2

a p0a− p3

≡ λαa λαa

⇔ “ a〉[a ”

a)2 (p1

a)2 (p2

a)2 (p3

= 0, for massless particles.This can be made manifest by writing pααa as an outer product of 2-vectors.

det(λ

≡〈a b〉

det(λ

≡ [a b]

a+ p3a p1

a−ip2a

p1a+ip2

a p0a− p3

)≡ λαa λαa

⇔ “ a〉[a ”

a)2 (p1

a)2 (p2

a)2 (p3

det(λ

≡〈a b〉

det(λ

≡ [a b]

a+ p3a p1

a−ip2a

p1a+ip2

a p0a− p3

)≡ λαa λαa

⇔ “ a〉[a ”

a)2 (p1

a)2 (p2

a)2 (p3

det(λ

≡〈a b〉

det(λ

≡ [a b]

a+ p3a p1

a−ip2a

p1a+ip2

a p0a− p3

)≡ λαa λαa

⇔ “ a〉[a ”

a)2 (p1

a)2 (p2

a)2 (p3

), pααa =(pααa )†, which implies that

(λαa )∗=±λαa .

det(λ

≡〈a b〉

det(λ

≡ [a b]

a+ p3a p1

a−ip2a

p1a+ip2

a p0a− p3

)≡ λαa λαa

⇔ “ a〉[a ”

a)2 (p1

a)2 (p2

a)2 (p3

det(λ

≡〈a b〉

det(λ

≡ [a b]

a+ p3a p1

a−ip2a

p1a+ip2

a p0a− p3

)≡ λαa λαa

⇔ “ a〉[a ”

a)2 (p1

a)2 (p2

a)2 (p3

det(λ

≡〈a b〉

det(λ

≡ [a b]

a+ p3a p1

a−ip2a

p1a+ip2

a p0a− p3

)≡ λαa λαa

⇔ “ a〉[a ”

• pααa is unchanged by (λa, λa) 7→ (taλa, t-1a λa)

—the action of the little group.This can be made manifest by writing pααa as an outer product of 2-vectors.

det(λ

≡〈a b〉

det(λ

≡ [a b]

a+ p3a p1

a−ip2a

p1a+ip2

a p0a− p3

)≡ λαa λαa

⇔ “ a〉[a ”

• pααa is unchanged by (λa, λa) 7→ (taλa, t-1a λa)—the action of the little group.

det(λ

≡〈a b〉

det(λ

≡ [a b]

a+ p3a p1

a−ip2a

p1a+ip2

a p0a− p3

)≡ λαa λαa

⇔ “ a〉[a ”

Under little group transformations, wave functions transform according to:

det(λ

≡〈a b〉

det(λ

≡ [a b]

a+ p3a p1

a−ip2a

p1a+ip2

a p0a− p3

)≡ λαa λαa

⇔ “ a〉[a ”

det(λ

≡〈a b〉

det(λ

≡ [a b]

a+ p3a p1

a−ip2a

p1a+ip2

a p0a− p3

)≡ λαa λαa

⇔ “ a〉[a ”

• The (local) Lorentz group, SL(2)L×SL(2)R, acts on λa and λa, respectively.Therefore, Lorentz-invariants must be constructed using the determinants:

det(λ

≡〈a b〉

det(λ

≡ [a b]

a+ p3a p1

a−ip2a

p1a+ip2

a p0a− p3

)≡ λαa λαa

⇔ “ a〉[a ”

• The (local) Lorentz group, SL(2)L×SL(2)R, acts on λa and λa, respectively.

Therefore, Lorentz-invariants must be constructed using the determinants:

det(λ

≡〈a b〉

det(λ

≡ [a b]

a+ p3a p1

a−ip2a

p1a+ip2

a p0a− p3

)≡ λαa λαa

⇔ “ a〉[a ”

det(λ

≡〈a b〉

det(λ

≡ [a b]

a+ p3a p1

a−ip2a

p1a+ip2

a p0a− p3

)≡ λαa λαa

⇔ “ a〉[a ”

det(λ

≡〈a b〉

det(λ

≡ [a b]

a+ p3a p1

a−ip2a

p1a+ip2

a p0a− p3

)≡ λαa λαa ⇔ “ a〉[a ”

det(λ

≡〈a b〉

det(λ

≡ [a b]

a+ p3a p1

a−ip2a

p1a+ip2

a p0a− p3

)≡ λαa λαa ⇔ “ a〉[a ”

det(λ

)≡〈a b〉 det

)≡ [a b]

a+ p3a p1

a−ip2a

p1a+ip2

a p0a− p3

)≡ λαa λαa ⇔ “ a〉[a ”

εαβλαa

≡〈a b〉 εαβ λαa

≡ [a b]

a+ p3a p1

a−ip2a

p1a+ip2

a p0a− p3

)≡ λαa λαa ⇔ “ a〉[a ”

det(λ

)≡〈a b〉 det

)≡ [a b]

a+ p3a p1

a−ip2a

p1a+ip2

a p0a− p3

)≡ λαa λαa ⇔ “ a〉[a ”

det(λ

)≡〈a b〉 det

)≡ [a b]

a+ p3a p1

a−ip2a

p1a+ip2

a p0a− p3

)≡ λαa λαa ⇔ “ a〉[a ”

det(λ

)≡〈a b〉 det

)≡ [a b]

The Grassmannian Geometry of Kinematical ConstraintsThus, all the kinematical data can be described by a pair of (2×n) matrices:

λ ≡

1 λ12 λ

13 · · · λ1

λ21 λ

23 · · · λ2

≡(λ1

)λ ≡

1 λ12 λ

13 · · · λ1

λ21 λ

23 · · · λ2

≡(λ1

writing λa∈C2 for a column, λα∈Cn for a row.• Because Lorentz transformations mix the rows of each matrix, λα, λα,

the little group allows for rescaling

, the invariant content of the data is:

The “two–plane” λ:

the span of 2 vectors λα∈Cn

•Momentum conservation:

λ⊂λ⊥ and λ⊂ λ⊥

(taking all the momenta to be incoming)

δ4(∑

a pµa)

= δ2×2(∑

≡ δ2×2(λ·λ)

λ ≡

1 λ12 λ

13 · · · λ1

λ21 λ

23 · · · λ2

≡(λ1

)λ ≡

1 λ12 λ

13 · · · λ1

λ21 λ

23 · · · λ2

≡(λ1

)writing λa∈C2 for a column, λα∈Cn for a row.

• Because Lorentz transformations mix the rows of each matrix, λα, λα,

δ4(∑

a pµa)

= δ2×2(∑

≡ δ2×2(λ·λ)

λ ≡(λ1

1 λ12 λ

13 · · · λ1

λ21 λ

23 · · · λ2

≡(λ1

λ ≡(λ1

1 λ12 λ

13 · · · λ1

λ21 λ

23 · · · λ2

≡(λ1

δ4(∑

a pµa)

= δ2×2(∑

≡ δ2×2(λ·λ)

λ ≡(λ1

1 λ12 λ

13 · · · λ1

λ21 λ

23 · · · λ2

≡(λ1

λ ≡(λ1

1 λ12 λ

13 · · · λ1

λ21 λ

23 · · · λ2

≡(λ1

δ4(∑

a pµa)

= δ2×2(∑

≡ δ2×2(λ·λ)

λ ≡(λ1

1 λ12 λ

13 · · · λ1

λ21 λ

23 · · · λ2

)≡(λ1

)λ ≡(λ1

1 λ12 λ

13 · · · λ1

λ21 λ

23 · · · λ2

)≡(λ1

δ4(∑

a pµa)

= δ2×2(∑

≡ δ2×2(λ·λ)

λ ≡(λ1 λ2 λ3 · · · λn

)≡(λ1

)λ ≡

(λ1 λ2 λ3 · · · λn

)≡(λ1

δ4(∑

a pµa)

= δ2×2(∑

≡ δ2×2(λ·λ)

λ ≡(λ1 λ2 λ3 · · · λn

)≡(λ1

)λ ≡

(λ1 λ2 λ3 · · · λn

)≡(λ1

δ4(∑

a pµa)

= δ2×2(∑

≡ δ2×2(λ·λ)

λ ≡(λ1 λ2 λ3 · · · λn

)≡(λ1

)λ ≡

(λ1 λ2 λ3 · · · λn

)≡(λ1

andthe little group allows for rescaling

δ4(∑

a pµa)

= δ2×2(∑

≡ δ2×2(λ·λ)

λ ≡(λ1 λ2 λ3 · · · λn

)≡(λ1

)λ ≡

(λ1 λ2 λ3 · · · λn

)≡(λ1

• Because Lorentz transformations mix the rows of each matrix, λα, λα, andthe little group allows for rescaling

δ4(∑

a pµa)

= δ2×2(∑

≡ δ2×2(λ·λ)

λ ≡(λ1 λ2 λ3 · · · λn

)≡(λ1

)λ ≡

(λ1 λ2 λ3 · · · λn

)≡(λ1

• Because Lorentz transformations mix the rows of each matrix, λα, λα, andthe little group allows for rescaling, the invariant content of the data is:

δ4(∑

a pµa)

= δ2×2(∑

≡ δ2×2(λ·λ)

λ ≡(λ1 λ2 λ3 · · · λn

)≡(λ1

)λ ≡

(λ1 λ2 λ3 · · · λn

)≡(λ1

δ4(∑

a pµa)

= δ2×2(∑

≡ δ2×2(λ·λ)

λ ≡(λ1 λ2 λ3 · · · λn

)≡(λ1

)λ ≡

(λ1 λ2 λ3 · · · λn

)≡(λ1

δ4(∑

a pµa)

= δ2×2(∑

≡ δ2×2(λ·λ)

λ ≡(λ1 λ2 λ3 · · · λn

)≡(λ1

)λ ≡

(λ1 λ2 λ3 · · · λn

)≡(λ1

δ4(∑

a pµa)

= δ2×2(∑

≡ δ2×2(λ·λ)

λ ≡(λ1 λ2 λ3 · · · λn

)≡(λ1

)λ ≡

(λ1 λ2 λ3 · · · λn

)≡(λ1

δ4(∑

a pµa)

= δ2×2(∑

≡ δ2×2(λ·λ)

λ ≡(λ1 λ2 λ3 · · · λn

)≡(λ1

)λ ≡

(λ1 λ2 λ3 · · · λn

)≡(λ1

δ4(∑

a pµa)

= δ2×2(∑

≡ δ2×2(λ·λ)

λ ≡(λ1 λ2 λ3 · · · λn

)≡(λ1

)λ ≡

(λ1 λ2 λ3 · · · λn

)≡(λ1

δ4(∑

a pµa)

= δ2×2(∑

≡ δ2×2(λ·λ)

λ ≡(λ1 λ2 λ3 · · · λn

)≡(λ1

)λ ≡

(λ1 λ2 λ3 · · · λn

)≡(λ1

The Grassmanian G(k, n):

the span of k vectors in Cn

δ4(∑

a pµa)

= δ2×2(∑

≡ δ2×2(λ·λ)

λ ≡(λ1 λ2 λ3 · · · λn

)≡(λ1

)λ ≡

(λ1 λ2 λ3 · · · λn

)≡(λ1

δ4(∑

a pµa)

= δ2×2(∑

≡ δ2×2(λ·λ)

λ ≡(λ1 λ2 λ3 · · · λn

)≡(λ1

)λ ≡

(λ1 λ2 λ3 · · · λn

)≡(λ1

λ⊂λ⊥ and λ⊂ λ⊥(taking all the momenta to be incoming)

δ4(∑

a pµa)

= δ2×2(∑

≡ δ2×2(λ·λ)

λ ≡(λ1 λ2 λ3 · · · λn

)≡(λ1

)λ ≡

(λ1 λ2 λ3 · · · λn

)≡(λ1

δ4(∑

a pµa)

= δ2×2(∑

≡ δ2×2(λ·λ)

λ ≡(λ1 λ2 λ3 · · · λn

)≡(λ1

)λ ≡

(λ1 λ2 λ3 · · · λn

)≡(λ1

δ4(∑

a pµa)

= δ2×2(∑

≡ δ2×2(λ·λ)

λ ≡(λ1 λ2 λ3 · · · λn

)≡(λ1

)λ ≡

(λ1 λ2 λ3 · · · λn

)≡(λ1

δ4(∑

a pµa)

= δ2×2(∑

a pααa

≡ δ2×2(λ·λ)

λ ≡(λ1 λ2 λ3 · · · λn

)≡(λ1

)λ ≡

(λ1 λ2 λ3 · · · λn

)≡(λ1

δ4(∑

a pµa)

= δ2×2(∑

aλαaλαa

≡ δ2×2(λ·λ)

λ ≡(λ1 λ2 λ3 · · · λn

)≡(λ1

)λ ≡

(λ1 λ2 λ3 · · · λn

)≡(λ1

δ4(∑

a pµa)

= δ2×2(∑

aλαaλαa

)≡ δ2×2

(λ·λ)

λ ≡(λ1 λ2 λ3 · · · λn

)≡(λ1

)λ ≡

(λ1 λ2 λ3 · · · λn

)≡(λ1

•Momentum conservation: λ⊂λ⊥ and λ⊂ λ⊥(taking all the momenta to be incoming)

δ4(∑

a pµa)

= δ2×2(∑

aλαaλαa

)≡ δ2×2

(λ·λ)

λ ≡(λ1 λ2 λ3 · · · λn

)≡(λ1

)λ ≡

(λ1 λ2 λ3 · · · λn

)≡(λ1

δ4(∑

a pµa)

= δ2×2(∑

aλαaλαa

)≡ δ2×2

(λ·λ)

λ ≡(λ1 λ2 λ3 · · · λn

)≡(λ1

)λ ≡

(λ1 λ2 λ3 · · · λn

)≡(λ1

δ4(∑

a pµa)

= δ2×2(∑

aλαaλαa

)≡ δ2×2

(λ·λ)

λ ≡(λ1 λ2 λ3 · · · λn

)≡(λ1

)λ ≡

(λ1 λ2 λ3 · · · λn

)≡(λ1

δ4(∑

a pµa)

= δ2×2(∑

aλαaλαa

)≡ δ2×2

(λ·λ)

λ ≡(λ1 λ2 λ3 · · · λn

)≡(λ1

)λ ≡

(λ1 λ2 λ3 · · · λn

)≡(λ1

δ4(∑

a pµa)

= δ2×2(∑

aλαaλαa

)≡ δ2×2

(λ·λ)

λ ≡(λ1 λ2 λ3 · · · λn

)≡(λ1

)λ ≡

(λ1 λ2 λ3 · · · λn

)≡(λ1

δ4(∑

a pµa)

= δ2×2(∑

aλαaλαa

)≡ δ2×2

(λ·λ)

Three Particle Kinematics and Helicity AmplitudesNon-Dynamical Dependence: Coupling Constants & Spin/Statistics

Building Blocks: the S-Matrix for Three Massless ParticlesMomentum conservation and Poincare-invariance uniquely fix the kinematical

dependence of the amplitude for three massless particles (to all loop orders!).

f (λ1, λ2, λ3)

f q1,q2,q3〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉δ2×2(λ·λ)

〈12〉h3–h1–h2〈23〉h1–h2–h3〈31〉h2–h3–h1

λ⊥≡(〈23〉〈31〉〈12〉

)⊃λ

λ ≡(λ1

1 λ12 λ1

1 λ22 λ2

)h1 + h2 + h3 ≤ 0

h1 + h2 + h3 ≥ 0

−−−−−−−→〈a b〉→O(ε)

O(ε−(h1+h2+h3)

−−−−−−→[a b]→O(ε)

O(ε(h1+h2+h3)

)f (λ1λ1, λ2λ2, λ3λ3)δ2×2

(λ·λ)

f (λ1, λ2, λ3)

f q1,q2,q3[2 3]4

[1 2] [2 3] [3 1]δ2×2(λ·λ)

[12]h1+h2–h3[23]h2+h3–h1[31]h3+h1–h2 λ ≡(λ1

1 λ12 λ1

λ21 λ2

2 λ23

)λ⊥≡

([23] [31] [12]

)⊃λ

f (λ1, λ2, λ3)

f q1,q2,q3〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉δ2×2(λ·λ)

〈12〉h3–h1–h2〈23〉h1–h2–h3〈31〉h2–h3–h1

λ⊥≡(〈23〉〈31〉〈12〉

)⊃λ

λ ≡(λ1

1 λ12 λ1

1 λ22 λ2

)h1 + h2 + h3 ≤ 0

h1 + h2 + h3 ≥ 0

−−−−−−−→〈a b〉→O(ε)

O(ε−(h1+h2+h3)

−−−−−−→[a b]→O(ε)

O(ε(h1+h2+h3)

)f (λ1λ1, λ2λ2, λ3λ3)δ2×2

(λ·λ)

f (λ1, λ2, λ3)

f q1,q2,q3[2 3]4

[1 2] [2 3] [3 1]δ2×2(λ·λ)

[12]h1+h2–h3[23]h2+h3–h1[31]h3+h1–h2 λ ≡(λ1

1 λ12 λ1

λ21 λ2

2 λ23

)λ⊥≡

([23] [31] [12]

)⊃λ

f (λ1, λ2, λ3)

f q1,q2,q3〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉δ2×2(λ·λ)

〈12〉h3–h1–h2〈23〉h1–h2–h3〈31〉h2–h3–h1

λ⊥≡(〈23〉〈31〉〈12〉

)⊃λ

λ ≡(λ1

1 λ12 λ1

1 λ22 λ2

)h1 + h2 + h3 ≤ 0

h1 + h2 + h3 ≥ 0

−−−−−−−→〈a b〉→O(ε)

O(ε−(h1+h2+h3)

−−−−−−→[a b]→O(ε)

O(ε(h1+h2+h3)

f (λ1λ1, λ2λ2, λ3λ3)δ2×2(λ·λ)

f (λ1, λ2, λ3)

f q1,q2,q3[2 3]4

[1 2] [2 3] [3 1]δ2×2(λ·λ)

[12]h1+h2–h3[23]h2+h3–h1[31]h3+h1–h2 λ ≡(λ1

1 λ12 λ1

λ21 λ2

2 λ23

)λ⊥≡

([23] [31] [12]

)⊃λ

f (λ1, λ2, λ3)

f q1,q2,q3〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉δ2×2(λ·λ)

〈12〉h3–h1–h2〈23〉h1–h2–h3〈31〉h2–h3–h1

λ⊥≡(〈23〉〈31〉〈12〉

)⊃λ

λ ≡(λ1

1 λ12 λ1

1 λ22 λ2

h1 + h2 + h3 ≤ 0

h1 + h2 + h3 ≥ 0

−−−−−−−→〈a b〉→O(ε)

O(ε−(h1+h2+h3)

−−−−−−→[a b]→O(ε)

O(ε(h1+h2+h3)

f (λ1λ1, λ2λ2, λ3λ3)δ2×2(λ·λ)

f (λ1, λ2, λ3)

f q1,q2,q3[2 3]4

[1 2] [2 3] [3 1]δ2×2(λ·λ)

[12]h1+h2–h3[23]h2+h3–h1[31]h3+h1–h2

λ ≡(λ1

1 λ12 λ1

λ21 λ2

2 λ23

λ⊥≡([23] [31] [12]

)⊃λ

f (λ1, λ2, λ3)

f q1,q2,q3〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉δ2×2(λ·λ)

〈12〉h3–h1–h2〈23〉h1–h2–h3〈31〉h2–h3–h1

λ⊥≡(〈23〉〈31〉〈12〉

)⊃λ

λ ≡(λ1

1 λ12 λ1

1 λ22 λ2

h1 + h2 + h3 ≤ 0

h1 + h2 + h3 ≥ 0

−−−−−−−→〈a b〉→O(ε)

O(ε−(h1+h2+h3)

−−−−−−→[a b]→O(ε)

O(ε(h1+h2+h3)

f (λ1λ1, λ2λ2, λ3λ3)δ2×2(λ·λ)

f (λ1, λ2, λ3)

f q1,q2,q3[2 3]4

[1 2] [2 3] [3 1]δ2×2(λ·λ)

[12]h1+h2–h3[23]h2+h3–h1[31]h3+h1–h2

λ ≡(λ1

1 λ12 λ1

λ21 λ2

2 λ23

λ⊥≡([23] [31] [12]

)⊃λ

f (λ1, λ2, λ3)

f q1,q2,q3〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉δ2×2(λ·λ)

〈12〉h3–h1–h2〈23〉h1–h2–h3〈31〉h2–h3–h1

λ⊥≡(〈23〉〈31〉〈12〉

)⊃λ

λ ≡(λ1

1 λ12 λ1

1 λ22 λ2

h1 + h2 + h3 ≤ 0

h1 + h2 + h3 ≥ 0

−−−−−−−→〈a b〉→O(ε)

O(ε−(h1+h2+h3)

−−−−−−→[a b]→O(ε)

O(ε(h1+h2+h3)

f (λ1λ1, λ2λ2, λ3λ3)δ2×2(λ·λ)

f (λ1, λ2, λ3)

f q1,q2,q3[2 3]4

[1 2] [2 3] [3 1]δ2×2(λ·λ)

[12]h1+h2–h3[23]h2+h3–h1[31]h3+h1–h2

λ ≡(λ1

1 λ12 λ1

λ21 λ2

2 λ23

λ⊥≡([23] [31] [12]

)⊃λ

f (λ1, λ2, λ3)

f q1,q2,q3〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉δ2×2(λ·λ)

〈12〉h3–h1–h2〈23〉h1–h2–h3〈31〉h2–h3–h1

λ⊥≡(〈23〉〈31〉〈12〉

λ ≡(λ1

1 λ12 λ1

1 λ22 λ2

h1 + h2 + h3 ≤ 0

h1 + h2 + h3 ≥ 0

−−−−−−−→〈a b〉→O(ε)

O(ε−(h1+h2+h3)

−−−−−−→[a b]→O(ε)

O(ε(h1+h2+h3)

f (λ1λ1, λ2λ2, λ3λ3)δ2×2(λ·λ)

f (λ1, λ2, λ3)

f q1,q2,q3[2 3]4

[1 2] [2 3] [3 1]δ2×2(λ·λ)

[12]h1+h2–h3[23]h2+h3–h1[31]h3+h1–h2

λ ≡(λ1

1 λ12 λ1

λ21 λ2

2 λ23

)λ⊥≡

([23] [31] [12]

f (λ1, λ2, λ3)

f q1,q2,q3〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉δ2×2(λ·λ)

〈12〉h3–h1–h2〈23〉h1–h2–h3〈31〉h2–h3–h1

λ⊥≡(〈23〉〈31〉〈12〉

)⊃λ

λ ≡(λ1

1 λ12 λ1

1 λ22 λ2

h1 + h2 + h3 ≤ 0

h1 + h2 + h3 ≥ 0

−−−−−−−→〈a b〉→O(ε)

O(ε−(h1+h2+h3)

−−−−−−→[a b]→O(ε)

O(ε(h1+h2+h3)

f (λ1λ1, λ2λ2, λ3λ3)δ2×2(λ·λ)

f (λ1, λ2, λ3)

f q1,q2,q3[2 3]4

[1 2] [2 3] [3 1]δ2×2(λ·λ)

[12]h1+h2–h3[23]h2+h3–h1[31]h3+h1–h2

λ ≡(λ1

1 λ12 λ1

λ21 λ2

2 λ23

)λ⊥≡

([23] [31] [12]

)⊃λ

f (λ1, λ2, λ3)

f q1,q2,q3〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉δ2×2(λ·λ)

〈12〉h3–h1–h2〈23〉h1–h2–h3〈31〉h2–h3–h1

λ⊥≡(〈23〉〈31〉〈12〉

)⊃λ

λ ≡(λ1

1 λ12 λ1

1 λ22 λ2

h1 + h2 + h3 ≤ 0

h1 + h2 + h3 ≥ 0

−−−−−−−→〈a b〉→O(ε)

O(ε−(h1+h2+h3)

−−−−−−→[a b]→O(ε)

O(ε(h1+h2+h3)

f (λ1λ1, λ2λ2, λ3λ3)δ2×2(λ·λ)

f (λ1, λ2, λ3)

f q1,q2,q3[2 3]4

[1 2] [2 3] [3 1]δ2×2(λ·λ)

[12]h1+h2–h3[23]h2+h3–h1[31]h3+h1–h2

λ ≡(λ1

1 λ12 λ1

λ21 λ2

2 λ23

)λ⊥≡

([23] [31] [12]

)⊃λ

f (λ1, λ2, λ3)

f q1,q2,q3〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉δ2×2(λ·λ)

〈12〉h3–h1–h2〈23〉h1–h2–h3〈31〉h2–h3–h1

λ⊥≡(〈23〉〈31〉〈12〉

)⊃λ

λ ≡(λ1

1 λ12 λ1

1 λ22 λ2

h1 + h2 + h3 ≤ 0

h1 + h2 + h3 ≥ 0

−−−−−−−→〈a b〉→O(ε)

O(ε−(h1+h2+h3)

−−−−−−→[a b]→O(ε)

O(ε(h1+h2+h3)

)f (λ1λ1, λ2λ2, λ3λ3)δ2×2

(λ·λ)

f (λ1, λ2, λ3)

f q1,q2,q3[2 3]4

[1 2] [2 3] [3 1]δ2×2(λ·λ)

[12]h1+h2–h3[23]h2+h3–h1[31]h3+h1–h2

λ ≡(λ1

1 λ12 λ1

λ21 λ2

2 λ23

)λ⊥≡

([23] [31] [12]

)⊃λ

f (λ1, λ2, λ3)

f q1,q2,q3〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉δ2×2(λ·λ)

〈12〉h3–h1–h2〈23〉h1–h2–h3〈31〉h2–h3–h1

λ⊥≡(〈23〉〈31〉〈12〉

)⊃λ

λ ≡(λ1

1 λ12 λ1

1 λ22 λ2

h1 + h2 + h3 ≤ 0

h1 + h2 + h3 ≥ 0

−−−−−−−→〈a b〉→O(ε)

O(ε−(h1+h2+h3)

−−−−−−→[a b]→O(ε)

O(ε(h1+h2+h3)

)f (λ1λ1, λ2λ2, λ3λ3)δ2×2

(λ·λ)

f (λ1, λ2, λ3)

f q1,q2,q3[2 3]4

[1 2] [2 3] [3 1]δ2×2(λ·λ)

[12]h1+h2–h3[23]h2+h3–h1[31]h3+h1–h2

λ ≡(λ1

1 λ12 λ1

λ21 λ2

2 λ23

)λ⊥≡

([23] [31] [12]

)⊃λ

f (λ1, λ2, λ3)

f q1,q2,q3〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉δ2×2(λ·λ)

〈12〉h3–h1–h2〈23〉h1–h2–h3〈31〉h2–h3–h1

λ⊥≡(〈23〉〈31〉〈12〉

)⊃λ

λ ≡(λ1

1 λ12 λ1

1 λ22 λ2

h1 + h2 + h3 ≤ 0

h1 + h2 + h3 ≥ 0

−−−−−−−→〈a b〉→O(ε)

O(ε−(h1+h2+h3)

−−−−−−→[a b]→O(ε)

O(ε(h1+h2+h3)

)f (λ1λ1, λ2λ2, λ3λ3)δ2×2

(λ·λ)

f (λ1, λ2, λ3)

f q1,q2,q3[2 3]4

[1 2] [2 3] [3 1]δ2×2(λ·λ)

[12]h1+h2–h3[23]h2+h3–h1[31]h3+h1–h2

λ ≡(λ1

1 λ12 λ1

λ21 λ2

2 λ23

)λ⊥≡

([23] [31] [12]

)⊃λ

f (λ1, λ2, λ3)

f q1,q2,q3〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉δ2×2(λ·λ)∝

〈12〉h3–h1–h2〈23〉h1–h2–h3〈31〉h2–h3–h1

λ⊥≡(〈23〉〈31〉〈12〉

)⊃λ

λ ≡(λ1

1 λ12 λ1

1 λ22 λ2

h1 + h2 + h3 ≤ 0

h1 + h2 + h3 ≥ 0

−−−−−−−→〈a b〉→O(ε)

O(ε−(h1+h2+h3)

−−−−−−→[a b]→O(ε)

O(ε(h1+h2+h3)

)f (λ1λ1, λ2λ2, λ3λ3)δ2×2

(λ·λ)

f (λ1, λ2, λ3)

f q1,q2,q3[2 3]4

[1 2] [2 3] [3 1]δ2×2(λ·λ)∝

[12]h1+h2–h3[23]h2+h3–h1[31]h3+h1–h2 λ ≡(λ1

1 λ12 λ1

λ21 λ2

2 λ23

)λ⊥≡

([23] [31] [12]

)⊃λ

f (λ1, λ2, λ3)

f q1,q2,q3〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉δ2×2(λ·λ)∝

〈12〉h3–h1–h2〈23〉h1–h2–h3〈31〉h2–h3–h1

λ⊥≡(〈23〉〈31〉〈12〉

)⊃λ

λ ≡(λ1

1 λ12 λ1

1 λ22 λ2

h1 + h2 + h3 ≤ 0

h1 + h2 + h3 ≥ 0

−−−−−−−→〈a b〉→O(ε)

O(ε−(h1+h2+h3)

−−−−−−→[a b]→O(ε)

O(ε(h1+h2+h3)

)f (λ1λ1, λ2λ2, λ3λ3)δ2×2

(λ·λ)

f (λ1, λ2, λ3)

f q1,q2,q3[2 3]4

[1 2] [2 3] [3 1]δ2×2(λ·λ)∝

[12]h1+h2–h3[23]h2+h3–h1[31]h3+h1–h2 λ ≡(λ1

1 λ12 λ1

λ21 λ2

2 λ23

)λ⊥≡

([23] [31] [12]

)⊃λ

f (λ1, λ2, λ3)

f q1,q2,q3〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉δ2×2(λ·λ)∝

〈12〉h3–h1–h2〈23〉h1–h2–h3〈31〉h2–h3–h1

λ⊥≡(〈23〉〈31〉〈12〉

)⊃λ

λ ≡(λ1

1 λ12 λ1

1 λ22 λ2

h1 + h2 + h3 ≤ 0

h1 + h2 + h3 ≥ 0

−−−−−−−→〈a b〉→O(ε)

O(ε−(h1+h2+h3)

−−−−−−→[a b]→O(ε)

O(ε(h1+h2+h3)

f (λ1λ1, λ2λ2, λ3λ3)δ2×2(λ·λ)

f (λ1, λ2, λ3)

f q1,q2,q3[2 3]4

[1 2] [2 3] [3 1]δ2×2(λ·λ)∝

[12]h1+h2–h3[23]h2+h3–h1[31]h3+h1–h2 λ ≡(λ1

1 λ12 λ1

λ21 λ2

2 λ23

)λ⊥≡

([23] [31] [12]

)⊃λ

f (λ1, λ2, λ3)

f q1,q2,q3〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉δ2×2(λ·λ)∝

〈12〉h3–h1–h2〈23〉h1–h2–h3〈31〉h2–h3–h1

λ⊥≡(〈23〉〈31〉〈12〉

)⊃λ

λ ≡(λ1

1 λ12 λ1

1 λ22 λ2

h1 + h2 + h3 ≤ 0

h1 + h2 + h3 ≥ 0

−−−−−−−→〈a b〉→O(ε)

O(ε−(h1+h2+h3)

−−−−−−→[a b]→O(ε)

O(ε(h1+h2+h3)

f (λ1λ1, λ2λ2, λ3λ3)δ2×2(λ·λ)

f (λ1, λ2, λ3)

f q1,q2,q3[2 3]4

[1 2] [2 3] [3 1]δ2×2(λ·λ)∝

[12]h1+h2–h3[23]h2+h3–h1[31]h3+h1–h2 λ ≡(λ1

1 λ12 λ1

λ21 λ2

2 λ23

)λ⊥≡

([23] [31] [12]

)⊃λ

f (λ1, λ2, λ3)= f q1,q2,q3〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉δ2×2(λ·λ)

∝ 〈12〉h3–h1–h2〈23〉h1–h2–h3〈31〉h2–h3–h1

λ⊥≡(〈23〉〈31〉〈12〉

)⊃λ

λ ≡(λ1

1 λ12 λ1

1 λ22 λ2

)h1 + h2 + h3 ≤ 0

h1 + h2 + h3 ≥ 0

−−−−−−−→〈a b〉→O(ε)

O(ε−(h1+h2+h3)

−−−−−−→[a b]→O(ε)

O(ε(h1+h2+h3)

)f (λ1λ1, λ2λ2, λ3λ3)δ2×2

(λ·λ)

f (λ1, λ2, λ3)= f q1,q2,q3[2 3]4

[1 2] [2 3] [3 1]δ2×2(λ·λ)

∝ [12]h1+h2–h3[23]h2+h3–h1[31]h3+h1–h2 λ ≡(λ1

1 λ12 λ1

λ21 λ2

2 λ23

)λ⊥≡

([23] [31] [12]

)⊃λ

f (λ1, λ2, λ3)= f q1,q2,q3〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉δ2×2(λ·λ)

∝ 〈12〉h3–h1–h2〈23〉h1–h2–h3〈31〉h2–h3–h1

λ⊥≡(〈23〉〈31〉〈12〉

)⊃λ

λ ≡(λ1

1 λ12 λ1

1 λ22 λ2

)h1 + h2 + h3 ≤ 0

h1 + h2 + h3 ≥ 0

−−−−−−−→〈a b〉→O(ε)

O(ε−(h1+h2+h3)

−−−−−−→[a b]→O(ε)

O(ε(h1+h2+h3)

)f (λ1λ1, λ2λ2, λ3λ3)δ2×2

(λ·λ)

f (λ1, λ2, λ3)= f q1,q2,q3[2 3]4

[1 2] [2 3] [3 1]δ2×2(λ·λ)

∝ [12]h1+h2–h3[23]h2+h3–h1[31]h3+h1–h2 λ ≡(λ1

1 λ12 λ1

λ21 λ2

2 λ23

)λ⊥≡

([23] [31] [12]

)⊃λ

f (λ1, λ2, λ3)= f q1,q2,q3〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉δ2×2(λ·λ)

∝ 〈12〉h3–h1–h2〈23〉h1–h2–h3〈31〉h2–h3–h1

λ⊥≡(〈23〉〈31〉〈12〉

)⊃λ

λ ≡(λ1

1 λ12 λ1

1 λ22 λ2

)h1 + h2 + h3 ≤ 0

h1 + h2 + h3 ≥ 0

−−−−−−−→〈a b〉→O(ε)

O(ε−(h1+h2+h3)

−−−−−−→[a b]→O(ε)

O(ε(h1+h2+h3)

)f (λ1λ1, λ2λ2, λ3λ3)δ2×2

(λ·λ)

f (λ1, λ2, λ3)= f q1,q2,q3[2 3]4

[1 2] [2 3] [3 1]δ2×2(λ·λ)

∝ [12]h1+h2–h3[23]h2+h3–h1[31]h3+h1–h2 λ ≡(λ1

1 λ12 λ1

λ21 λ2

2 λ23

)λ⊥≡

([23] [31] [12]

)⊃λ

f (λ1, λ2, λ3)= f q1,q2,q3〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉δ2×2(λ·λ)

∝ 〈12〉h3–h1–h2〈23〉h1–h2–h3〈31〉h2–h3–h1

λ⊥≡(〈23〉〈31〉〈12〉

)⊃λ

λ ≡(λ1

1 λ12 λ1

1 λ22 λ2

h1 + h2 + h3 ≤ 0

h1 + h2 + h3 ≥ 0

−−−−−−−→〈a b〉→O(ε)

O(ε−(h1+h2+h3)

−−−−−−→[a b]→O(ε)

O(ε(h1+h2+h3)

)f (λ1λ1, λ2λ2, λ3λ3)δ2×2

(λ·λ)

f (λ1, λ2, λ3)= f q1,q2,q3[2 3]4

[1 2] [2 3] [3 1]δ2×2(λ·λ)

∝ [12]h1+h2–h3[23]h2+h3–h1[31]h3+h1–h2

λ ≡(λ1

1 λ12 λ1

λ21 λ2

2 λ23

)λ⊥≡

([23] [31] [12]

)⊃λ

f (λ1, λ2, λ3)

= f q1,q2,q3〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉δ2×2(λ·λ) ≡ A3

(+,−,−

∝ 〈12〉h3–h1–h2〈23〉h1–h2–h3〈31〉h2–h3–h1

λ⊥≡(〈23〉〈31〉〈12〉

)⊃λ

λ ≡(λ1

1 λ12 λ1

1 λ22 λ2

)h1 + h2 + h3 ≤ 0

h1 + h2 + h3 ≥ 0

−−−−−−−→〈a b〉→O(ε)

O(ε−(h1+h2+h3)

−−−−−−→[a b]→O(ε)

O(ε(h1+h2+h3)

)f (λ1λ1, λ2λ2, λ3λ3)δ2×2

(λ·λ)

f (λ1, λ2, λ3)

= f q1,q2,q3[2 3]4

[1 2] [2 3] [3 1]δ2×2(λ·λ) ≡ A3

(−,+,+

∝ [12]h1+h2–h3[23]h2+h3–h1[31]h3+h1–h2 λ ≡(λ1

1 λ12 λ1

λ21 λ2

2 λ23

)λ⊥≡

([23] [31] [12]

)⊃λ

f (λ1, λ2, λ3)

= f q1,q2,q3〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉δ2×2(λ·λ) ≡ A3

(+,−,−

∝ 〈12〉h3–h1–h2〈23〉h1–h2–h3〈31〉h2–h3–h1

λ⊥≡(〈23〉〈31〉〈12〉

)⊃λ

λ ≡(λ1

1 λ12 λ1

1 λ22 λ2

)h1 + h2 + h3 ≤ 0

h1 + h2 + h3 ≥ 0

−−−−−−−→〈a b〉→O(ε)

O(ε−(h1+h2+h3)

−−−−−−→[a b]→O(ε)

O(ε(h1+h2+h3)

)f (λ1λ1, λ2λ2, λ3λ3)δ2×2

(λ·λ)

f (λ1, λ2, λ3)

= f q1,q2,q3[2 3]4

[1 2] [2 3] [3 1]δ2×2(λ·λ) ≡ A3

(−,+,+

∝ [12]h1+h2–h3[23]h2+h3–h1[31]h3+h1–h2 λ ≡(λ1

1 λ12 λ1

λ21 λ2

2 λ23

)λ⊥≡

([23] [31] [12]

)⊃λ

f (λ1, λ2, λ3)

= f q1,q2,q3〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉δ2×2(λ·λ) ≡ A3

(+,−,−

∝ 〈12〉h3–h1–h2〈23〉h1–h2–h3〈31〉h2–h3–h1

λ⊥≡(〈23〉〈31〉〈12〉

)⊃λ

λ ≡(λ1

1 λ12 λ1

1 λ22 λ2

)h1 + h2 + h3 ≤ 0

h1 + h2 + h3 ≥ 0

−−−−−−−→〈a b〉→O(ε)

O(ε−(h1+h2+h3)

−−−−−−→[a b]→O(ε)

O(ε(h1+h2+h3)

)f (λ1λ1, λ2λ2, λ3λ3)δ2×2

(λ·λ)

f (λ1, λ2, λ3)

= f q1,q2,q3[2 3]4

[1 2] [2 3] [3 1]δ2×2(λ·λ) ≡ A3

(−,+,+

∝ [12]h1+h2–h3[23]h2+h3–h1[31]h3+h1–h2 λ ≡(λ1

1 λ12 λ1

λ21 λ2

2 λ23

)λ⊥≡

([23] [31] [12]

)⊃λ

f (λ1, λ2, λ3)

= f q1,q2,q3〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉δ2×2(λ·λ) ≡ A3

(+,−,−

∝ 〈12〉h3–h1–h2〈23〉h1–h2–h3〈31〉h2–h3–h1

λ⊥≡(〈23〉〈31〉〈12〉

)⊃λ

λ ≡(λ1

1 λ12 λ1

1 λ22 λ2

)h1 + h2 + h3 ≤ 0

h1 + h2 + h3 ≥ 0

−−−−−−−→〈a b〉→O(ε)

O(ε−(h1+h2+h3)

−−−−−−→[a b]→O(ε)

O(ε(h1+h2+h3)

)f (λ1λ1, λ2λ2, λ3λ3)δ2×2

(λ·λ)

f (λ1, λ2, λ3)

= f q1,q2,q3[2 3]4

[1 2] [2 3] [3 1]δ2×2(λ·λ) ≡ A3

(−,+,+

∝ [12]h1+h2–h3[23]h2+h3–h1[31]h3+h1–h2 λ ≡(λ1

1 λ12 λ1

λ21 λ2

2 λ23

)λ⊥≡

([23] [31] [12]

)⊃λ

f (λ1, λ2, λ3)

= f q1,q2,q3〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉δ2×2(λ·λ) ≡ A3

(+,−,−

∝ 〈12〉h3–h1–h2〈23〉h1–h2–h3〈31〉h2–h3–h1

λ⊥≡(〈23〉〈31〉〈12〉

)⊃λ

λ ≡(λ1

1 λ12 λ1

1 λ22 λ2

)h1 + h2 + h3 ≤ 0

h1 + h2 + h3 ≥ 0

−−−−−−−→〈a b〉→O(ε)

O(ε−(h1+h2+h3)

−−−−−−→[a b]→O(ε)

O(ε(h1+h2+h3)

)f (λ1λ1, λ2λ2, λ3λ3)δ2×2

(λ·λ)

f (λ1, λ2, λ3)

= f q1,q2,q3[2 3]4

[1 2] [2 3] [3 1]δ2×2(λ·λ) ≡ A3

(−,+,+

∝ [12]h1+h2–h3[23]h2+h3–h1[31]h3+h1–h2 λ ≡(λ1

1 λ12 λ1

λ21 λ2

2 λ23

)λ⊥≡

([23] [31] [12]

)⊃λ

f (λ1, λ2, λ3)

= f q1,q2,q3〈3 1〉〈2 3〉3

〈1 2〉〈2 3〉〈3 1〉δ2×2(λ·λ) ≡ A3

2 ,12 ,−

∝ 〈12〉h3–h1–h2〈23〉h1–h2–h3〈31〉h2–h3–h1

λ⊥≡(〈23〉〈31〉〈12〉

)⊃λ

λ ≡(λ1

1 λ12 λ1

1 λ22 λ2

)h1 + h2 + h3 ≤ 0

h1 + h2 + h3 ≥ 0

−−−−−−−→〈a b〉→O(ε)

O(ε−(h1+h2+h3)

−−−−−−→[a b]→O(ε)

O(ε(h1+h2+h3)

)f (λ1λ1, λ2λ2, λ3λ3)δ2×2

(λ·λ)

f (λ1, λ2, λ3)

= f q1,q2,q3[3 1][2 3]3

[1 2] [2 3] [3 1]δ2×2(λ·λ) ≡ A3

( 12 ,+

∝ [12]h1+h2–h3[23]h2+h3–h1[31]h3+h1–h2 λ ≡(λ1

1 λ12 λ1

λ21 λ2

2 λ23

)λ⊥≡

([23] [31] [12]

)⊃λ

f (λ1, λ2, λ3)

= f q1,q2,q3〈3 1〉〈2 3〉3

〈1 2〉〈2 3〉〈3 1〉δ2×2(λ·λ) ≡ A3

2 ,12 ,−

∝ 〈12〉h3–h1–h2〈23〉h1–h2–h3〈31〉h2–h3–h1

λ⊥≡(〈23〉〈31〉〈12〉

)⊃λ

λ ≡(λ1

1 λ12 λ1

1 λ22 λ2

)h1 + h2 + h3 ≤ 0

h1 + h2 + h3 ≥ 0

−−−−−−−→〈a b〉→O(ε)

O(ε−(h1+h2+h3)

−−−−−−→[a b]→O(ε)

O(ε(h1+h2+h3)

)f (λ1λ1, λ2λ2, λ3λ3)δ2×2

(λ·λ)

f (λ1, λ2, λ3)

= f q1,q2,q3[3 1][2 3]3

[1 2] [2 3] [3 1]δ2×2(λ·λ) ≡ A3

( 12 ,+

∝ [12]h1+h2–h3[23]h2+h3–h1[31]h3+h1–h2 λ ≡(λ1

1 λ12 λ1

λ21 λ2

2 λ23

)λ⊥≡

([23] [31] [12]

)⊃λ

f (λ1, λ2, λ3)

= f q1,q2,q3δ2×4

(λ·η)

〈1 2〉〈2 3〉〈3 1〉δ2×2(λ·λ) ≡ A(2)

∝ 〈12〉h3–h1–h2〈23〉h1–h2–h3〈31〉h2–h3–h1

λ⊥≡(〈23〉〈31〉〈12〉

)⊃λ

λ ≡(λ1

1 λ12 λ1

1 λ22 λ2

)h1 + h2 + h3 ≤ 0

h1 + h2 + h3 ≥ 0

−−−−−−−→〈a b〉→O(ε)

O(ε−(h1+h2+h3)

−−−−−−→[a b]→O(ε)

O(ε(h1+h2+h3)

)f (λ1λ1, λ2λ2, λ3λ3)δ2×2

(λ·λ)

f (λ1, λ2, λ3)

= f q1,q2,q3δ1×4

(λ⊥·η

)[1 2] [2 3] [3 1]

δ2×2(λ·λ) ≡ A(1)3

∝ [12]h1+h2–h3[23]h2+h3–h1[31]h3+h1–h2 λ ≡(λ1

1 λ12 λ1

λ21 λ2

2 λ23

)λ⊥≡

([23] [31] [12]

)⊃λ

Coupling Constant Constraints: Scaling and Spin/StatisticsThe coupling constants f q1,q2,q3 are quantum-number-dependent constants

which define the theory.

Because all the kinematical dependence is fixed,these couplings cannot ‘run’.

• Dimensional analysis shows that the mass-dimension of the coupling is:

[ f q1,q2,q3 ]=[mass]1−|h1+h2+h3|

• Consider a theory involving only particles with integer spin σ∈Z:

A(1+σ

q1, 2-σ

q2, 3-σ

= f q1,q2,q3

(〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉

)σδ2×2(λ·λ)

Bose statistics requires that A be symmetric under the exchange 2↔3;

even-spin: f q1,q2,q3 must be totally symmetricodd spin: f q1,q2,q3 must be totally antisymmetric

which define the theory. Because all the kinematical dependence is fixed,these couplings cannot ‘run’.

[ f q1,q2,q3 ]=[mass]1−|h1+h2+h3|

A(1+σ

q1, 2-σ

q2, 3-σ

= f q1,q2,q3

(〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉

)σδ2×2(λ·λ)

[ f q1,q2,q3 ]=[mass]1−|h1+h2+h3|

A(1+σ

q1, 2-σ

q2, 3-σ

= f q1,q2,q3

(〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉

)σδ2×2(λ·λ)

[ f q1,q2,q3 ]=[mass]1−|h1+h2+h3|

A(1+σ

q1, 2-σ

q2, 3-σ

= f q1,q2,q3

(〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉

)σδ2×2(λ·λ)

[ f q1,q2,q3 ]=[mass]1−|h1+h2+h3|

A(1+σ

q1, 2-σ

q2, 3-σ

= f q1,q2,q3

(〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉

)σδ2×2(λ·λ)

[ f q1,q2,q3 ]=[mass]1−|h1+h2+h3|

A(1+σ

q1, 2-σ

q2, 3-σ

= f q1,q2,q3

(〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉

)σδ2×2(λ·λ)

[ f q1,q2,q3 ]=[mass]1−|h1+h2+h3|

A(1+σ

q1, 2-σ

q2, 3-σ

)= f q1,q2,q3

(〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉

)σδ2×2(λ·λ)

[ f q1,q2,q3 ]=[mass]1−|h1+h2+h3|

A(1+σ

q1, 2-σ

q2, 3-σ

)= f q1,q2,q3

(〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉

)σδ2×2(λ·λ)

[ f q1,q2,q3 ]=[mass]1−|h1+h2+h3|

A(1+σ

q1, 2-σ

q2, 3-σ

)= f q1,q2,q3

(〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉

)σδ2×2(λ·λ)

even-spin: f q1,q2,q3 must be totally symmetric

odd spin: f q1,q2,q3 must be totally antisymmetric

[ f q1,q2,q3 ]=[mass]1−|h1+h2+h3|

A(1+σ

q1, 2-σ

q2, 3-σ

)= f q1,q2,q3

(〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉

)σδ2×2(λ·λ)

[ f q1,q2,q3 ]=[mass]1−|h1+h2+h3|

A(1+σ

q1, 2-σ

q2, 3-σ

)= f q1,q2,q3

(〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉

)σδ2×2(λ·λ)

[ f q1,q2,q3 ]=[mass]1−|h1+h2+h3|

A(1+σ

q1, 2-σ

q2, 3-σ

)= f q1,q2,q3

(〈2 3〉4

〈1 2〉〈2 3〉〈3 1〉

)σδ2×2(λ·λ)

Factorization and Long-Range Physics: Weinberg’s TheoremUniqueness of Yang-Mills Theory and the Equivalence PrincipleThe Simplest Quantum Field Theory: N=4 super Yang-Mills

Channeling Some Consequences of FactorizationIn [arXiv:0705.4305], Benincasa and Cachazo described how elementary

considerations of locality and unitarity strongly restricts the choice ofcoupling constants, and hence possible quantum field theories.

Consider the behavior of any local, unitarity theory in a factorization limit:

(〈12〉3

〈2 I〉〈I 1〉[34]3

[I 3][4 I]

with u≡(p2+p4)2

•Homework: use the result, together with the analogous u- and t-channels todetermine the form of A4

and show that if σ>2 all factorizations vanish.

This is Wienberg’s theorem

—proving that long-range physics requires σ≤2.

(〈12〉3

〈2 I〉〈I 1〉[34]3

[I 3][4 I]

with u≡(p2+p4)2

(〈12〉3

〈2 I〉〈I 1〉[34]3

[I 3][4 I]

with u≡(p2+p4)2

(〈12〉3

〈2 I〉〈I 1〉[34]3

[I 3][4 I]

with u≡(p2+p4)2

⇒ ∼

(〈12〉3

〈2 I〉〈I 1〉[34]3

[I 3][4 I]

with u≡(p2+p4)2

⇒ ∼

(〈12〉3

〈2 I〉〈I 1〉[34]3

[I 3][4 I]

with u≡(p2+p4)2

⇒ ∼(〈12〉[34]

with u≡(p2+p4)2

⇒ ∼(〈12〉[34]

with u≡(p2+p4)2

⇒ ∼(〈12〉[34]

with u≡(p2+p4)2

⇒ ∼(〈12〉[34]

with u≡(p2+p4)2

•Homework:

use the result, together with the analogous u- and t-channels todetermine the form of A4

⇒ ∼(〈12〉[34]

with u≡(p2+p4)2

and show that if σ>2 all factorizations vanish.This is Wienberg’s theorem

⇒ ∼(〈12〉[34]

with u≡(p2+p4)2

•Homework: use the result, together with the analogous u- and t-channels todetermine the form of A4 and show that if σ>2 all factorizations vanish.

⇒ ∼(〈12〉[34]

with u≡(p2+p4)2

•Homework: use the result, together with the analogous u- and t-channels todetermine the form of A4 and show that if σ>2 all factorizations vanish.This is Wienberg’s theorem

⇒ ∼(〈12〉[34]

with u≡(p2+p4)2

•Homework: use the result, together with the analogous u- and t-channels todetermine the form of A4 and show that if σ>2 all factorizations vanish.This is Wienberg’s theorem—proving that long-range physics requires σ≤2.

Quantum Consistency Conditions from Cauchy’s TheoremUsing Cauchy’s theorem to relate the three factorization channels to each

other, Benincasa and Cachazo prove in [arXiv:0705.4305] following:

σ=1: the coupling constants satisfy a Jacobi identity!

whatever quantum numbers distinguish mutually interacting spin-1particles, they form the adjoint representation of a Lie algebra!

σ=2: multiple spin-2 particles can always be decomposed intomutually non-interacting sectors

—there is at most one graviton!

the coupling strength of any spin-2 particle to itself must be thesame as its coupling to any other field

—the equivalence principle!

σ=2: multiple spin-2 particles can always be decomposed intomutually non-interacting sectors—there is at most one graviton!

the coupling strength of any spin-2 particle to itself must be thesame as its coupling to any other field—the equivalence principle!

Jacob L. Bourjaily - Jagiellonian Universityth- · The Shocking Simplicity of Scattering Amplitudes...

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