2 Dec2010 Wei-MingChen N= 6Chern-SimonsMatterTheory DualitiesforLoopAmplitudesof

Dualities for Loop Amplitudes of N = 6 Chern-Simons Matter Theory

Wei-Ming Chen

NTU

2 ^nd Dec 2010

Based on JHEP 1111 (2011) 057 (1107.2710)

with Yu-tin Huang(UCLA)

Motivation

▸ Construct an simple example of odd-loop amplitudes in

ABJM theory

Motivation

▸ Construct an simple example of odd-loop amplitudes in ABJM theory

▸ Get insight for Amplitude/Wilson Loop duality in ABJM theory

N = 4 SYM one-loop 4-point Amplitude:

A

[− ^(−s/µ _ǫ

⁾

− ^(−t/µ _ǫ

⁾

+

log

( ^−s _−t ) + const + O(ǫ)]

N = 4 SYM one-loop 4-point Wilson loop:

g²N

8π²

[− ^{(−˜ µ}

_ǫ

⁾

− ^{(−˜ µ}

_ǫ

⁾

+

log

(

24

) + const + O(ǫ)]

N = 6 ABJM two-loop 4-point Wilson loop:

−(

)

[− ^(−µ _(2ǫ)

⁾

− ^(−µ _(2ǫ)

⁾

+

log

(

24

) + const + O(ǫ)]

Outline

Introduction

One-Loop and Two-Loop Amplitudes

Conclusion and Discussion

Outline

Introduction

One-Loop and Two-Loop Amplitudes

Conclusion and Discussion

Field Content

N = 6 Superconformal Cherm-Simon Matter theory (ABJM)

Guage Fields: A ^a _b A ^{ˆ a} _ˆ

b U ( N )

² _∋a × U ( N )

² ∋ˆ a

Matter Fields: (φ ^I , ψ ^I ) ^a _{ˆ a} (¯φ I , ¯ ψ _I ) ^a _{ˆ a} I ∈ SU ( 4 )

On-shell Variables and Superfields

Three-Dimensional Kinematics,

p ^ab = (σ ^µ ) ^ab p _µ = λ ^a λ ^b , if p ² = 0 N = 6 Superfields, I ∈ SU ( 3 )

Φ(λ,η) = φ ⁴ + η ^I ψ _I + 1

2 ǫ _IJK η ^I η ^J φ ^K + 1

3 ! ǫ _IJK η ^I η ^J η ^k ψ ₄ Φ(λ,η) = ¯ψ ¯ ⁴ + η ^I φ ¯ I + 1

2 ǫ IJK η ^I η ^J ψ ¯ ^K + 1

3 ! ǫ IJK η ^I η ^J η ^K φ ¯ 4

General n-point Tree Amplitude

▸ Tree level n-point amplitude, only n = even exists.

▸ Color ordered amplitude A ⁿ (λ i , η _i ) ,

T. Bargheer, F. Loebbert, C. Meneghelli, 1003.6120

A ˆ ⁿ = ˆ A ⁿ (Φ( 1 ) ^A _{A ¯}

1

Φ( ¯ 2 ) ^B B ^¯

. . . ¯ Φ( n ) ^B B ^¯

)

= ∑

σ∈(S

×S

)/C

A ⁿ (σ 1 , . . . σ n )δ ^A _B

δ ^B _¯ ^¯

A

. . . δ ^B _¯ ^¯

A

General n-point Tree Amplitude

T. Bargheer, F. Loebbert, C. Meneghelli, 1003.6120

▸ R-symmetry invariance Ð→ η

▸ momentum and momentum supercharge conservation Ð→ A n = δ ³ ( P )δ ³ ( Q ^αI )δ ³ ( Q ^I _α ) f (λ) , Q ^αI ≡ ∑ i λ ^α _i η ^I _i

▸ Lorentz invariance and Dilatation invariance Ð→ f with λ weight − 4

▸ Consistent with field theory computation, A 4 = i δ ³ ( P )δ ³ ( Q ^αI )δ ³ ( Q _α ^I )

⟨ 41 ⟩⟨ 12 ⟩

Dual Conformal Symmetry

J.M. Drummond, J. Henn, V.A. Smirnov, E. Sokatchev, hep-th/0607160

First observed from one-loop to three loop four-point gluon scattering amplitudes

p _i = x _i − x _i+1 I [ x _i ^µ ] = x _i ^µ

x _i ² , K = IPI

I [A 4 ] = x ₁ ² x ₂ ² x ₃ ² x ₄ ² A 4 , integrand level

A Example of Dual Conformal Symmetry

x

x

x

x

x

A ^1−Loop ₄ = A ^Tree 4 L, L = ∫ d ⁴ l

( 2π ) ⁴

st

l ₁ ² ( l ₁ + p ₁ ) ² ( l ₁ + p ₁ + p ₂ ) ² ( l ₁ − p ₄ ) ² Ð→ ∫ d ⁴ x 5

( 2π ) ⁴

x ₁₃ ² x ₂₄ ²

x ₅₁ ² x ₅₂ ² x ₅₃ ² x ₅₄ ²

A Example of Dual Conformal Symmetry

x ⁵ x ²

x 3

x 1

x ⁴

▸ Dual conformal covariant: I [A ^Tree 4 ] = x ₁ ² x ₂ ² x ₃ ² x ₄ ² A ^Tree 4

▸ Dual conformal invariant: I [ L ] = L

Dual Superconformal Symmetry

Dongmin Gang, Yu-tin Huang, Eunkyung Koh, Sangmin Lee, Arthur E. Lipstein 1012.5032

Dual superspace is parametrized by x , θ, y :

x _i,i+1 ^αβ ≡ x _i ^αβ − x _i+1 ^αβ = p ^αβ _i = λ ^α i λ ^β _i θ _i,i+1 ^Iα ≡ θ ^Iα i − θ _i+1 ^Iα = q _i ^Iα = λ ^α i η _i ^I , y _i,i+1 ^IJ ≡ y _i ^IJ − y _i+1 ^IJ = r _i ^IJ = η i ^I η ^J _i

Amplitude transforms covariantly under dual superconformal symmetry:

I [A ⁿ ] = ∏ ⁿ

i=1

√ x _i ² A ⁿ

= ∏ ⁿ

i=1

√

x _i ² A ^Tree n L (only valid in integrand level)

I [ L ] = L

Generalized Unitarity Cut

▸ Assume tree-amplitudes are known

▸ Unitarity cut, only L + 1 propagators at most can be cut.

S = 1 + iT , S ^† S = 1 ⇒ i ( T ^† − T ) = T ^† T

A∣ ^cut = A ¹ A ²

▸ Generalized unitarity Cut, cut number larger than L + 1 is possible.

A∣ ^cut = A ¹ A ² ⋯A ⁿ

Procedure to Construct Amplitudes

▸ Guess all possible dual superconformal invariant integrands

▸ Cut all possible dual superconformal invariant integrands

▸ Match cut-integrands with product of tree amplitudes

Outline

Introduction

One-Loop and Two-Loop Amplitudes

Conclusion and Discussion

One-Loop Amplitude

▸ Embed 3D into 5D, − T ² − U ² + V ² + W ² + Y ² = 0

▸ Degree of freedom of 3D, 5 − 1(light cone

condition) − 1(identification of rescaling T → ρ T ) = 3 I ₄ ^1−loop = ∫ D ³ X ₅ 4ǫ ( 5, 1,2,3,4 )

X ₅₁ ² X ₅₂ ² X ₅₃ ² X ₅₄ ²

= ∫ d ³ x 5

( 2π ) ³

2x ₅₁ ² ǫ _µνρ x ₂₁ ^µ x ₃₁ ^ν x ₄₁ ^ρ + 2x ₃₁ ² ǫ _µνρ x ₅₁ ^µ x ₂₁ ^ν x ₄₁ ^ρ

x ₁₅ ² x ₂₅ ² x ₃₅ ² x ₄₅ ²

One-Loop Amplitude

i A ^Tree 4 ( 1, 2, 3, 4 ) I ₄ ^1−loop ∣ ^cut = A ^Tree 4 ( 1, 2, l ₂ , − l ₁ )A ^Tree 4 (− l ₂ , 3, 4, l ₁ ) I ^1−loop = 0

l ₂ l ₁

p ₄ p ₁

p ₂

p ₃

Two-Loop Amplitude

Possible scalar integrals:

x₃ x1

x₄ x₅ x₆ x₂

I

x₂

x₃ x1

x₅ x₆ x₄

I

x6

x₅ x1

x₃ x₂ x₄

I

x₄ x₂

x₃ x1

x₆ x₅

I

One more possible integrals:

I _0s = ∫ D ³ X ₅ D ³ X ₆ 16ǫ ( 5,1,2,3, 4 )ǫ( 6, 1, 2, 3, 4 ) X ₅₁ ² X ₅₃ ² X ₅₄ ² X ₅₆ ² X ₆₁ ² X ₆₃ ² X ₆₂ ² X ₄₂ ² These integrals are not linearly independent:

2I _0s = I _1s − I _2s + I _3s + I _3t + I _4s .

Two-Loop Amplitude

Match cut

l4 l₁ l₂

l3 p₄

p₃

p₁

p₂

p₄

p₃

p₁

p₂ l₅ l₆

l₇

A ^Tree 4 ∑

i

( c _i I _is + c _i ^′ I _it )∣ ^cut

= { A ^{Tree 4} ( 1, 2, − l ₃ , l ₂ )A ^Tree 4 (− l ₂ , − l ₃ , l ₄ , − l ₁ )A ^Tree 4 (− l ₁ , l ₄ , 3, 4 ) 0

⇒ A ^2−Loop = ( N

K ) ² A ^Tree ₄ [− I _0s + I _1s + ( s ↔ t )]

Outline

Introduction

One-Loop and Two-Loop Amplitudes

Conclusion and Discussion

Amplitude/Wilson Loop Duality

▸ N = 6 ABJM two-loop 4-point Amplitude:

−(

)

A

[− ^{(−s /µ”} _(2ǫ)

⁾

− ^{(−t /µ”} _(2ǫ)

⁾

+

log

( ^−s _−t ) + const + O(ǫ)]

Amplitude/Wilson Loop Duality

▸ N = 6 ABJM two-loop 4-point Amplitude:

−(

)

A

[− ^{(−s /µ”} _(2ǫ)

⁾

− ^{(−t /µ”} _(2ǫ)

⁾

+

log

( ^−s _−t ) + const + O(ǫ)]

▸ N = 6 ABJM two-loop 4-point Wilson loop:

−(

)

[− ^(−µ _(2ǫ)

⁾

− ^(−µ _(2ǫ)

⁾

+

log

(

24

) + const + O(ǫ)]

N = 4 SYM one-loop 4-point Amplitude:

A

8π²

[− ^(−s/µ _ǫ

⁾

− ^(−t/µ _ǫ

⁾

+

log

( ^−s _−t ) + const + O(ǫ)]

N = 4 SYM one-loop 4-point Wilson loop:

g²N

8π²

[− ^{(−˜ µ}

_ǫ

⁾

− ^{(−˜ µ}

_ǫ

⁾

+

log

(

24

) + const + O(ǫ)]

Amplitude/Wilson Loop Duality in 3D

+ = +

+ = + +

...

String Picture for Amplitude/Wilson Loop Duality

T-duality

p4

p3 p2

p1

p6

p5

p4

p3 p2

p1

p6

p5

D(−1) D3

AdS5 AdSg5

N = 4SYM

String Picture for Amplitude/Wilson Loop Duality

T-duality?

p4

p3 p2

p1

p6

p5

p4

p3 p2

p1

p6

p5

D0 D2

AdS4 AdSg4

N = 6ABJM

Conclusion

▸ One loop four-point integrand exists and can be integrated to zero.

▸ Two loop four-point amplitude/Wilson loop duality:

N = 4 SYM

N = 6 ABJM

A

hW

i

A

hW

i

Thank You