Dual color-ordered formula and DDM chain expression

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Dual color-ordered formula and DDM chain expression

NTU April 13

Chih-Hao Fu

NCTU

April 19, 2013

Based on work in collaboration with Yi-Jian Du and Bo Feng

arXiv[hep-th]:1212.6168, 1105.3503, 1110.4683, 1111.5691, 1304.2978

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Outline

• Color-kinematic duality – A brief introduction

• possible mechanism: gauge DOF? algebra? or...?

• cubic prescription for YM amplitudes

• Duality and formulations of YM and gravity amplitudes

• color-ordered formulation

• Del Duca-Dixon-Maltoni “half ladder” formulation

• KLT relation

• Remaining thoughts

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BCJ duality

• Definition of kinematic numerators through absorbing 4-pt contributions into cubic graphs

A(1234) = n _s

s − n _t

t A(1324) = − n u

u + n t

t

• Jacobi-like identity

[ Bern, Carrasco, Johansson(08)]

n _s + n _t + n _u = 0

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BCJ duality

• Definition of kinematic numerators through absorbing 4-pt contributions into cubic graphs

A(1234) = n s + s∆

s − n t + t∆

t A(1324) = − n _u + u∆

u + n _t + t∆

t

• Jacobi-like identity

[ Bern, Carrasco, Johansson(08)]

n _s +n _t +n _u +(s +t +u)∆ = 0

Generalized gauge invariance

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BCJ duality

An algebraic-like identity is satisfied between kinematic dependent numerators

f

^12e

f

^e34

+ f

^23e

f

^e14

+ f

^31e

f

^e24

= 0 l n

s

+ n

t

+ n

u

= 0

• BCJ numerators and Jacobi identities at 5-points (tree level):

A(12345) = n₁ s₁₂s₄₅+ n₂

s₂₃s₅₁+ n₃ s₃₄s₁₂+ n₄

s₄₅s₂₃+ n₅ s₅₁s₃₄, A(14325) = n₆

s₁₄s₂₅+ n₅ s₄₃s₅₁+ n₇

s₃₂s₁₄+ n₈ s₂₅s₄₃+ n₂

s₅₁s₃₂, A(13425) = n₉

s₁₃s₄₅+ n₅ s₃₄s₅₁+ n₁₀

s₄₂s₁₃

− n₈ s₂₅s₃₄+ n₁₁

s₅₁s₄₂, A(12435) = n₁₂

s12s35 + n₁₁

s24s51

− n₃ s43s12

+ n₁₃ s35s24

− n₅ s51s43

,

A(14235) = n₁₄ s₁₄s₃₅− n₁₁

s₄₂s₅₁− n₇ s₂₃s₁₄− n₁₃

s₃₅s₄₂ − n₂ s₅₁s₂₃, A(13245) = n₁₅

s₁₃s₄₅− n₂ s₃₂s₅₁− n₁₀

s₂₄s₁₃− n₄ s₄₅s₃₂ − n₁₁

s₅₁s₂₄,

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BCJ duality

Validity check at loop-level:

• N = 4 SYM

• At 4-pts, verified up to four loops

[Bern, Carrasco, Johansson(10)]

[Bern, Dixon,Dunbar,Perelstein, Rozowsky(98)]

[Bern, Carrasco, Dixon, Johansson, Roiban(12)]

• At 5-pts, up to three loops

[Carrasco, Johansson(12)]

[Yuan(12)]

• pure YM. Two-loops checked

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A new version of KLT relations

Double-copy expression

M(1

^α

, 2

^β

, 3

^γ

, 4

^δ

) = 1

s + 1

t

+ 1 u

= c

s

n

s

s + c

t

n

t

t + c

u

n

u

• A simpler tree level fromula

M = Y

cubic graphs i

c

i

n

i

D

i

• seems to generalize to loop levels!

M = Z d

^D

l

k

(2π)

^D

Y

cubic graphs i

c

i

n

i

(l

k

) D

i

(l

k

)

N = 8 supergravity, 4-pts up to four loops

[Bern, Dixon,Dunbar,Perelstein, Rozowsky(98)]

[Bern, Carrasco, Dixon, Johansson, Roiban(12)]

5-pts, checked up to two loops

[Carrasco, Johansson(12)]

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A new version of KLT relations

• Double-copy vs string low-energy limit KLT

Heterotic string theory → A “color” KLT relation for example

M full YM = X

α, β

A ˜ _scalar (n, α, 1)S[α|β]A _YM (1, β, n) s 123...n−1

[Kawai, Lewellen, Tye(86)]

[Bern, Freitas, Wong(00)]

[Bjerrum-Bohr, Feng, Damgaard, Sondargaard(10)]

[Bjerrum-Bohr, Damgaard, Sondargaard, Vanhove(10)]

M(1

^α

, 2

^β

, 3

^γ

, 4

^δ

) = A(4321)s ˜

21

(s

31

+ s

32

)A(1234) s

123

+ A(4321)s ˜

21

s

31

A(1234) s

123

+ A(4231)s ˜

21

s

31

A(1234) s

123

+ A(4321)(s ˜

21

+ s

23

)s

31

A(1324)

s

123

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Analytic construction of numerators

Algebra of generators of area-preserving diffeomorphism

• Self-dual YM

• Light-cone gauge YM

MHV

3−pt

→ L

k

= e

^−ik·x

(−k

⊥

∂

+

+ k

+

∂

⊥

), MHV

3−pt

→ ¯ L

k

= e

^−ik·x

(−¯ k

⊥

∂

+

+ k

+

∂ ¯

⊥

).

[Bjerrum-Bohr, Damgaard, Monteiro, O’Connell(11)(12)]

Algerba of generators of diffeomorphism (in Fourier basis) x

^a

→ g

^a

(x ) = x

^a

+

Z

d

^D

k

^a

(k)e

^ik·x

f (x ) → f (g (x )) = f (x ) +

Z

d

^D

k

^a

e

^ik·x

∂

a

f (x )

T

^k,a

= e

^ik·x

∂

a

,

T

^k¹^,a

, T

^k²^,b

= (−i )(δ

ac

k

1b

− δ

bc

k

2a

) e

^{i (k}¹^+k²^)·x

∂

c

= f

^(k¹^,a),(k²^,b)(k₁+k₂,c)

T

^k¹^+k²^,c

.

[Du, Feng, CF(12)]

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Analytic construction of numerators

• structure constants are NOT totallly anti-symmetric

η

ab

(k

1

− k

2

)

c

+ η

bc

(k

2

− k

3

)

a

+ η

ca

(k

3

− k

1

)

b

= f

^1,23

+ f

^2,31

+ f

^3,12

• Observed numerator relations are the collective work of four sets of Jacobi identities

n

^∗_s

+ n

_t^∗

+ n

^∗_u

= 0 l f

^3,4e

f

^2,e1

+ f

^2,3e

f

^4,e1

+ f

^4,2e

f

^3,e1

= 0 f

^3,4e

f

^e,12

+ f

^4,1e

f

^e,32

+ f

^1,3e

f

^e,42

= 0 f

^1,2e

f

^4,e3

+ f

^4,1e

f

^2,e3

+ f

^2,4e

f

^1,e3

= 0 f

^1,2e

f

^e,34

+ f

^2,3e

f

^e,14

+ f

^3,1e

f

^e,24

= 0

n

^∗_s

=

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Analytic construction of numerators

What about 4-pt vertex?

A(1234) = n

^∗s

s − n

^∗t

t + X

1^∗

A(1324) = − n

u^∗

u + n

^∗t

t + X

₂^∗

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Analytic construction of numerators

What about 4-pt vertex?

A(1234) = X

i

c

i

· n

^∗_s

s − n

^∗_t

t + X

1^∗

A(1324) = X

i

c

i

· − n

_u^∗

u + n

^∗_t

t + X

2^∗

Average over reference momenta, subject to the constraints

c

1

+ c

2

= 1, (c

1

+ c

2

) · X

1^∗

= 0, (c

1

+ c

2

) · X

2^∗

= 0.

X

i

c

i

·n

s^∗

= X

i

c

i

·

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Formulations of YM amplitudes: A sketchy review

• Color-ordered formulation

M(1

^α

, 2

^β

, 3

^γ

, . . . , n

^δ

) = X

σ∈S_n−1

tr (T

^α

T

^σ²

T

^σ³

. . . T

^σⁿ

)

×A(p

1

, p

σ₂

, . . . , p

σ_n

) properties of SU(N) color algebra

tr (T

^α

T

^β

) = δ

^αβ

f

^αβγ

= tr ([T

^α

, T

^β

]T

^γ

) (T

^α

)

_ij

(T

^α

)

_kl

= δ

il

δ

jk

− 1

N δ

ij

δ

kl

• Del Duca-Dixon-Maltoni “half ladder”/chain

M(1

^α

, 2

^β

, 3

^γ

, . . . , n

^δ

) = X

σ∈S_n−2

f

^ασ²^ρ²

f

^ρ²^σ³^ρ³

. . . f

^ρⁿ⁻²^σⁿ⁻¹^σⁿ

×A(p

1

, p

σ2

, . . . , p

σ_n−1

, p

n

)

[ Del Duca, Dixon, Maltoni(00)]

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Formulations of YM amplitudes

tr (T T . . . T )× A(1σ) color-ordered

formulation P 1

p2i

double-copy formulation

A(1σ)˜ × τ_1σ dual color- ordered formulation [Bern, Dennen(11)]

Feynman rules

OO

BCFW [Bjerrum-Bohr, Feng, Damgaard, Sondergaard,

Vanhove(10)]

gg

?

77

× A(1σn) Del Duca-Dixon- Maltoni chain(00) KK −relation

OO

(−)ⁿP α,β

˜A(n,α,1)S[α|β]A(1,β,n) s12...n−1

Jacobi identity KLT [Du, Feng, CF(11)]

oo

Jacobi identity

[Du, Feng, CF(12)]

//

˜ A(1σn)×

[Du, Feng, CF(13)]

OO

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Formulations of YM amplitudes

• Dual Del Duca-Dixon-Maltoni chain

(−)

ⁿ

P

α,β

A(n,α,1)S[α|β]A(1,β,n)˜ s12...n−1

KLT relation

vv ))

DDM chain Dual DDM chain

X σ

A(1, σ₁σ₂. . . σ_n−1, n) X i ,σ

c_iA(1, σ˜ ₁σ₂. . . σ_n−1, n)_i·

(Off-shell BCJ relation) [Du, Feng, CF(11)]

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Formulations of YM amplitudes

• Dual color-ordered formula

tr (T^αT^β) = δ^αβ

f^αβγ = tr ([T^α, T^β]T^γ)

↓

M = tr (T¹T². . . Tⁿ)A(12 . . . n) + . . .

X σ

tr (T¹T^σ2. . . T^σn)A(1, σ₁σ₂. . . σ_n) X σ

A(1, σ˜ ₁σ₂. . . σ_n)τ (1σ₂. . . σ_n)

color-ordered formula Dual color-ordered formula

DDM chain

OO

Dual DDM chain

OO

X σ

A(1, σ₁σ₂. . . σ_n−1, n) X i ,σ

c_iA(1, σ˜ ₁σ₂. . . σ_n−1, n)_i·

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Remaining Thoughts

• kinematic ansatz

n

s

= n(1234)

= [s

12

s

23

A(1234)]

det(k

i

· k

j

) 1

3 s

12

(s

13

− s

23

) n

t

= n(1423) ∼ s

14

(s

12

− s

42

) n

u

= n(1342) ∼ s

13

(s

14

− s

34

)

[Broedel, Carrasco(11)]

[Du, Feng, CF, work in progress]

• dual Del Duca-Dixon-Maltoni chain at loop level

• A proof of double-copy expression at loop-level

Improved large-z behavior [Boels, Isermann(12)]

(+ + + · · · +), LC construction at 1-loop [Boels, Isermann, Monteiro, O’Connell(13)]

• Algebra and double-copy from string perspective

[Bjerrum-Bohr, Damgaard, Johansson, Sondergaard(11)]

Dual color-ordered formula and DDM chain expression