Updates on ABJM Scattering Amplitudes

Sangmin Lee(Seoul National University)

17 June 2013

Exact Results in String/M Theory (pre Strings 2013 workshop)

Korea Institute for Advanced Study

Introduction

Scattering Amplitudes in Gauge Theories... beyond Feynman diagrams

Scattering amplitudes in 2003-2013

Twistor string theory

BCFW recursion relation

BCJ color/kinematics duality

Grassmannians, affine permutations

MHV vertices expansion

Amplitude/Wilson-loop duality

Polylogarithms, symbols, Hopf algebras

[Witten ’03]

[Cachazo,Svrcek,Witten ’04]

[Britto,Cachazo,Feng ’04][Britto,Cachazo,Feng,Witten ’05]

[Drummond,Henn,Korchemsky,Sokatchev ’06-’08][Alday,Maldacena ’07][Berkovits,Maldacena ’08]

[Beisert,Ricci,Tseytlin,Wolf ’08]

[Bern,Carrasco,Johansson ’08]

[Goncharov,Spradlin,Vergu,Volovich ’10][Duhr ’12]

[Arkani-Hamed,Cachazo,Cheung,Kaplan ’09] [Arkani-Hamed,Bourjaily, Cachazo,Gonchaorv,Postnikov,Trnka ’12]

N = 4 SYM N = 6 SCS

Scattering Amplitudes of ABJM theory

AdS4 x CP3AdS5 x S5

This talk is based on

S.L. [1007.4772]

Dongmin Gang, Yu-tin Huang, Eunkyung Koh, S.L., Arthur E. Lipstein [1012.5032]

Yu-tin Huang, S.L. [1207.4851]

Yu-tin Huang, Henrik Johansson, S.L. [1306.nnnn]

Joonho Kim, S.L. and friends, work in progress

Kinematics

d = 4, N = 4 SYM d = 3, N = 6 SCS

Lorentz:

Conformal:

Super-Conformal:

Super-charges:

SO(1, 2)SO(1, 3)

SO(2, 4) ' SU(2, 2)

32 24

⇢⇢

⇢⇢

Super-conformal symmetry

SO(2, 3) ' Sp(4, R)

PSU(2, 2|4) � SU(4)R OSp(4|6) � SO(6)R

Spinor helicity and Twistor in 4d

Twistors “linearize” conformal symmetry

Massless momentum in bi-spinor notation:

Twistor transformation

Paa = lala ! la∂

∂µa

Kaa =∂

∂la

∂

∂la! µa

∂

∂la

paa = lala

A(⇥, ⇥) ! A(⇥, µ) =Z

d2⇥ eiµ� ⇥� A(⇥, ⇥)

ZA ⌘ (⇥�, µ�)

MAB = ZA ∂

∂ZB � (trace) 2 SU(2, 2)

Lorentz invariants: hiji ⌘ eab(li)a(lj)b , [ij] ⌘ eab(li)a(lj)b

[Penrose 67, Witten 03]

[Berends, Kleiss, Troost, Wu, Xu... 80’s]

Spinor helicity and Twistor in 3d

Twistor ??

Massless momentum in bi-spinor notation

Twistor transformation ??

pab = lalb hiji ⌘ eab(li)a(lj)b

A(⇥) ! A(µ) =Z

d2⇥ eiµ�⇥� A(⇥)

ZA ⌘ (la, ∂/∂la)QM analogy !

Pab = lalb

Kab =∂2

∂la∂lb

Lab = la

∂

∂lb

Conformal generators act on amplitudes as if were “wave-functions”.An(l1, . . . , ln) An

JAB = Z(AZB) 2 Sp(4, R)

On-shell super-fields

SYM4

SCS6

(1+3+3+1)=(4+4)F = y4 + h I fI +

12 eI JKh Ih J yK + 1

2 eI JKh Ih JhKf4 .

F = f4 + h IyI +12 eI JKh Ih JfK + 1

6 eI JKh Ih JhKy4 ,

[U(N)�U(N)]gauge � SU(4)R

Z� : (N, N; 4) , �� : (N, N; 4) .

F = A+ + h IcI +12 h Ih JfI J +

16 eI JKLh Ih JhKcL + 1

6 eI JKLh Ih JhKhL A�

SU(N)gauge ⇥ SU(4)R

+1 �10+ 12 � 1

2

SU(4)R

helicity

1 14 46

⇢∂

∂h I , h J�

= dJI SO(6) Clifford algebra, only U(3) manifest.

Super-twistor

SYM4

SCS6

Configuration space: R2|3 � L = (l, h)

Phase space: R4|6 � Z = (L; ∂/∂L) = (l, h; ∂/∂l, ∂/∂h)

Super-conformal generators: JAB = Z[AZB)

Super-twistor: W = (⇤�, µ�, ⇥ I)

Super-conformal generators: J AB = WA ∂

∂WB

Super-QM analogy !

Recursion Relation[Gang,Huang,Koh,SL,Lipstein 1012]

BCFW recursion relation [Britto,Cachazo,Feng(,Witten) 04(05)]

Higher point amplitude from lower ones

An = Âr,s,h

AhL(z = zrs)

iP2

r···sA�h

R (z = zrs)

Idea: on-shell deformation of momenta

lj ! lj � zll , ll ! ll + zlj

(p2j = 0, p2

l = 0, p1 + · · ·+ pn = 0 unaffected)

An = An(z = 0) =Z dz

2piAn(z)

z

(deformed contour, fall-off at infinity, residue theorem)

BCFW recursion relation

Generalization to d > 4 [Arkani-Hamed,Kaplan 08]

pj ! pj + zq , pl ! pl � zq

pj · q = 0 , pl · q = 0 , q2 = 0

pj = (1, 1, 0, 0; 0, . . . , 0) , pl = (1, 1, 0, 0; 0, . . . , 0) , q = (0, 0, 1, i; 0, . . . , 0)

Naive generalization to 3d fails

lj ! lj � zll , ll ! ll + zlj (??)

Recursion relation in 3d [Gang,Huang,Koh,SL,Lipstein 1012]

Momentum conservation in 4d vs 3d

(4d) Âi

lili = 0

(3d) Âi

lili = 0

Use O(2,C) instead of GL(2,C) in 3d!

✓ljll

◆!

✓cos q � sin qsin q cos q

◆✓ljll

◆ ✓cos q =

z + 1/z2

, sin q =z � 1/z

2i

◆

GL(n, C) : li ! Mijlj , li ! (M�T)i

jlj

O(n, C) : li ! Rijlj

An = An(z = 1) =Z dz

2piAn(z)z � 1

Grassmannian Integral Formula

[Arkani-Hamed,Cachazo,Cheung,Kaplan 0907][SL 1007]

Grassmannian Integral formula in 4d [Arkani-Hamed,Cachzo,Cheung,Kaplan 09]

C = k

(i + k � 1)(i)

n

Mi GL(k)L ⇥ GL(n)R

(C · W)m = CmiWi

GL(k) “gauge” symmetry

Atree

n,k (W) =Z dk⇥nC

vol [GL(k)]d4k|4k(C · W)

M1

M2

· · · Mn�1

Mn

n: total number of external legs

k: number of (-) helicity gluons

Grassmannian Integral formula in 4d

only dependence on (super-)momenta

Cyclic + Superconformal symmetry manifest

Contour integral over Grassmannian manifold

= moduli space of k-planes in .Cn

= U(n)/[U(k)⇥ U(n � k)]

Proven to produce ALL tree amplitudes.

Useful in building up loop integrands.

Atree

n,k (W) =Z dk⇥nC

vol [GL(k)]d4k|4k(C · W)

M1

M2

· · · Mn�1

Mn

Grassmannian Integral formula in 3d [SL 1007]

C = k

2k

(i + k � 1)(i)

Mi = em1···mk Cm1(i)Cm2(i+1) · · ·Cmk(i+k�1)

(C · CT)mn = CmiCni

(C · L)m = CmiLi

GL(k)⇥ O(2k)

Atree

2k (L) =Z dk⇥2kC

vol [GL(k)]dk(k+1)/2(C · CT) d2k|3k(C · L)

M1

M2

· · · Mk�1

Mk

2k: total number of external legs

Grassmannian Integral formula in 3d

Contour integral over orthogonal-Grassmannian manifold

= moduli space of null k-planes in .C2k

= O(2k)/U(k)

Contours for ALL tree amplitudes under construction.

Atree

2k (L) =Z dk⇥2kC

vol [GL(k)]dk(k+1)/2(C · CT) d2k|3k(C · L)

M1

M2

· · · Mk�1

Mk

Grassmannian Integral formula in 3d

Superconformal symmetry

Atree

2k (L) =Z dk⇥2kC

vol [GL(k)]dk(k+1)/2(C · CT) d2k|3k(C · L)

M1

M2

· · · Mk�1

Mk

Cyclic symmetry

∂2

∂L∂L! killed by d(C · CT)L

∂

∂L! manifest

CT bC + bCTC = I2k⇥2k LL ! LT · (CT bC + bCTC) · L = 0

C · CT = 0 =) Mi Mi+1 = (�1)k�1Mi+k Mi+1+k

KK & BCJ Relations

[Huang,Johansson,SL 1306]

Color decomposition

Trace-based color decomposition

Full amplitude (permutation symm)

“Color-ordered” amplitude(cyclic symmetry only)

Atreen (pi, hi, ai) = Â

s2Sn/Zn

Tr(Tas(1) · · · Tas(n) )An(s(1h1), · · · , s(nhn))

Graph vs trace

f abc = Tr(Ta[Tb, Tc]) = Tr(TaTbTc)� Tr(TaTcTb)

f abc f cde = Tr([Ta, Tb][Td, Te])

= (abde)� (bade)� (abed) + (baed)

(n-1)!

Color decomposition

Jacobi identity

[[Ta1 , Ta2 ], Ta3 ] + [[Ta2 , Ta3 ], Ta1 ] + [[Ta3 , Ta1 ], Ta2 ] = 0

f a1a2b f ba3a4 + f a2a3b f ba1a4 + f a3a1b f ba2a4 = 0

1

2

4

3

+

1 4

2 3

1 4

2 3

+ = 0f a1a2b

[Ta, Tb] = f abcTc

Kleiss-Kuijf identities [Kleiss,Kuijf ’86]

Counting the number of independent graphsafter imposing Jacobi identities.

1

2

4

3

+

1 4

2 3

1 4

2 3

+ = 0

(n � 2)!1

· · ·

n

Color/Kinematics duality [Bern,Carrasco,Johansson ’08]

4-point amplitude

A4 =

=nscs

s+

ntctt

+nucu

u

1

2

4

3

+

1 4

2 3

1 4

2 3

+ +

1 4

2 3

s = (p1 + p2)2 , t = (p1 + p4)

2 , u = (p1 + p3)2

st u

1 =ss=

tt=

uu

Jacobi identity: cs + ct + cu = 0

Kinematic Jacobi identity ! ns + nt + nu = 0

Color/Kinematics duality [Bern,Carrasco,Johansson ’08]

Ai =(n�2)!

Âj=1

Qijnj

rank(Qij) = (n � 3)!

“Fundamental BCJ”

Graph combinatorics : Yang-Mills

Number of external legs

All possible graphs

Cyclic traces

After applying KK identities

After applying BCJ relations

(2n � 5)!!

n

(n � 1)!

(n � 2)!

(n � 3)!

Graph combinatorics : ABJM [Huang,Johansson,SL ’13]

Chern-Simons-matter theory in (1+2) dimensions

Quartic vertex with bi-fundamental maps

Jacobi-like identity

Graph combinatorics : ABJM

Legs 4 6 8 10 12

All graphs 1 9 216 9900 737100

Traces 1 6 72 1440 43200

KK 1 5 57 1144 ? ?

BCJ 1 4 57 1144 ? ?

n = 2k

(3k � 3)!k!(2k � 1)!2k�1

k!(k � 1)!2

Gravity = (gauge)^2 [Bern,Carrasco,Johansson ’08]

An(gauge) = Âi

ciniDi

(sum over all trivalent graphs)

=) An(gravity) = Âi

niniDi

color kinematics

product of scalar propagators

(sum over trivalent graphs only!)

Gravity = (gauge)^2 in 3d [Huang,Johansson ’12]

Same gravity from ... either Yang-Mills or Chern-Simons !!!

An(gauge) = Âi

ciniDi

(sum over all trivalent graphs)

=) An(gravity) = Âi

niniDi

color kinematics

product of scalar propagators

(sum over trivalent graphs only!)

Twistor-string-likeIntegral Formula

[Huang,SL 1207]

RSV-W formula [Witten ’03][Roiban,Spradlin,Volovich ’04]

An,k =1

vol(GL(2))

Z d2s1

d2s2

· · · d2sn(12)(23) · · · (n1)

� d4k|4k(C[s] · W)

d4k|4k(Cmi[s]W i) ⇥Z

d4k|4kzn

’i=1

d4|4(Zi � CVm [si]zm)

Connected prescription for twistor string theory

Integral over moduli space of degree (k - 1) curves in CP3|4

Contours in 4d

6- point NMHV amplitude

Parameter countingk

n - k

k - 2

n - k - 2k(n � k)� (2n � 4) = (k � 2)(n � k � 2)

dim[Gr(k, n)] d(C · W)/d(P) dn,k = dim[Gr(k � 2, n � 4)]

Recursive construction

An,3 =Z

~f=0dn�5t

hnf6 · f7 · · · fn�1 · fn 1

M1M2 · · · Mn�1Mn

NMHV amplitudes ( k = 3, all n )dn,k = n � 5

f j = (j � 2, j � 1, j)(j, 1, 2)(2, 3, j � 2)

Parity flip (k $ n � k)Soft limit

An ⇥ ⇤n � 1, 1⌅⇤n � 1, n⌅⇤n, 1⌅An�1

ln ! 0 ,

¯ln fixed

k

n - k

3 4 5 6 7

3456

Deformation of the integrand

hnf6 · f7 · · · fn�1 · fn

=⇥ Hn

F6 · F7 · · · Fn�1 · Fn

f j = (j � 2, j � 1, j)(j, 1, 2)(2, 3, j � 2)

=⇥ Hn

F6(t) · F7(t) · · · Fn�1(t) · Fn(t)

Fj(t) = (j � 2, j � 1, j)(j, 1, 2)(2, 3, j � 2)(1, 3, j � 1)

�t(1, 2, 3)(3, j � 2, j � 1)(j � 2, j � 1, j)(2, j � 2, j)

Fj = (j � 2, j � 1, j)(j, 1, 2)(2, 3, j � 2)(1, 3, j � 1)

rewrite

deform

Poles move, residues vary, but the integral remains unchanged!

Geometry of the deformed contour

Fj(1) = 0 for all j

=� C =

0

@A2

1 A22 · · · A2

nA1B1 A2B2 · · · AnBn

B21 B2

2 · · · B2n

1

A

Veronese map CP1 ! CPk

(A, B) ⇥ (Ak�1, Ak�2B, · · · , ABk�2, Bk�1)

Veronese map Gr(2, n) ! Gr(k, n)

Veronese Integral Formula

An,k =1

vol(GL(2))

Z d2s1

d2s2

· · · d2sn(12)(23) · · · (n1)

� d4k|4k(C[s] · W)

Contours in 3d

A2k ⇥⇤2k � 2, 1⌅

⇤2k � 2, e⌅⇤e, 1⌅A2k�2

l1 + il2 ⌘ e ! 0 (p1 + p2 ! 0)

Parameter counting

Soft limit

kk - 2k(k � 1)

2� (2k � 3) =

(k � 2)(k � 3)2

dim[OG(k, 2k)] d2k = dim[OG(k � 2, 2k � 4)]d(C · l)/d(P)

Twistor-string-like integral formula for ABJM [Huang,SL ’12]

Verified by explicit computation up to 8-point amplitude.

Consistency check with KK identities

“Twistor string theory” interpretation? [Cachazo,He,Yuan ’13]

J[s] = ’1ijk

(2i � 1, 2j � 1)

D2k[s] = det 2k�1

⇣A2k�1�j

i Bji

⌘ 2k�1

’j=1

d

2k

Âi=1

A2k�1�ji Bj

i

!

A2k =

1

vol(GL(2))

Z d2s1

d2s2

· · · d2s2k

(12)(23) · · · (2k, 1)� D

2k[s]�d2k|3k(CV [s] · L)

J[s]

cyclic symmetry

(2k-3)-dim surface in Gr(2,n)

KK identities vs Integral formula [Huang,SL ’12][Huang,Johansson,SL ’13]

Under (non-cyclic) permutations, the only non-trivial part is

Special KK identities

A2k =

1

vol(GL(2))

Z d2s1

d2s2

· · · d2s2k

(12)(23) · · · (2k, 1)� D

2k[s]�d2k|3k(CV [s] · L)

J[s]

1(12)(34) · · · (2k, 1)

KK identities vs Integral fromula [Huang,SL ’12][Huang,Johansson,SL ’13]

General KK identities via graphical representation

KK identities vs Integral fromula [Huang,SL ’12][Huang,Johansson,SL ’13]

On-shell graphs,Affine permutations,

Positive Grassmannian, etc.[Arkani-Hamed,Bourjaily, Cachazo,Gonchaorv,Postnikov,Trnka ’12]

[Joonho Kim, SL, others, work in progress]

On-shell graphs [Arkani-Hamed,Bourjaily,Cachazo,Gonchaorv,Postnikov,Trnka ’12]

On-shell graphs, permutations [Arkani-Hamed,Bourjaily, Cachazo,Gonchaorv,Postnikov,Trnka ’12]

Yang-Baxter in 4d and 3d [Arkani-Hamed,Bourjaily, Cachazo,Gonchaorv,Postnikov,Trnka ’12]

On-shell graphs in 3d [Joonho Kim, SL, others, work in progress]

d3(P)d6(Q)h14ih34i

Zd2ld3h

⇠ A6

Yang-Baxter via 3d Euler angles !

cf) BCJ duality for dimensional reduction to 2d

Summary

Summary

Twistor-string-like integral formula

KK/BCJ relations

BCFW-like recursion relation

Grassmannian integral formula- formal proof of Yangian invariance

On-shell graph, permutations, cluster algebra, ...

Amplitude/Wilson-loop duality?

Thank you