Gabriele TravagliniQueen Mary University of London

(+ Humboldt University, Berlin + Durham University + University of Rome “Tor Vergata”)

with

Andi Brandhuber, Paul Heslop, Brenda Penante, Donovan Young

Brandhuber, Penante, GT, Young 1412.1019 [hep-th] & 1502.06626 [hep-th] + in progressBrandhuber, Heslop, GT, Young, to appear tomorrow

Amplitudes 2015, ETH Zürich, 6th July 2015

The dilatation operator of N=4 SYM, amplitudes and Yangian symmetry

• Calculation of one-loop dilatation operator from two-point correlators with on-shell methods (Brandhuber, Penante, GT, Young)

‣ from MHV diagrams

‣ from generalised unitarity applied to two-point functions

- single-scale problem, unitarity particularly simple

• Derive the action of the Yangian on the dilatation operator from the Yangian symmetry of amplitudes (Brandhuber, Heslop, GT, Young)

‣ substantiate the idea that there exists a Yangian symmetry in N=4 SYM with different manifestations

Two themes

• General form of two-point functions of primary operators in a conformal theory:

‣ = classical dimension, = anomalous dimension (assume momentarily no mixing)

‣ expanding in g:

‣ anomalous dimension extracted from log divergence

‣ pole in 1/𝜖 in dimensional regularisation

• Definition of the dilatation operator

‣ ZAB = renormalisation constants for the operators {OA}

h0| O(x1)¯O(x2) |0i ⇠ 1

((x12)2)

�0(1� � log(x

212⇤

2))

�0 �

h0|O(x1)O(x2)|i ⇠ 1

((x12)2)�0+�x12 := x1 � x2

Dilatation operator

HAB = µ@

@µlogZAB

• Scalar operators:

- A1, …, BL = 1,…, 4 fundamental R-symmetry indices

‣ This sector is closed at one loop

‣ calculation of anomalous dimension mapped to that of the eigenvalues of an integrable Hamiltonian (Minahan & Zarembo)

• Perturbative calculation

‣ at one loop in the planar limit only nearest neighbours interact. Effectively the calculation is equivalent to that of the two-point correlator

The SO(6) sector

OA1B1,A2B2,...,ALBL(x) = Tr(�A1B1�A2B2 · · ·�ALBL)(x)

⌦(�AB�CD)(x1)(�A0B0

�C0D0)(x2)↵

• General structure of

‣ from Minahan & Zarembo: AUV = 1/2, BUV = -1, CUV = 1 (more accurately: )

⌦(�AB�CD)(x1)(�A0B0

�C0D0)(x2)↵

Trace Permutation Identity

φAB

φCD

φC'D'

φA'B'

φAB

φCD

φC'D'

φA'B'

φAB

φCD

φC'D'

φA'B'

x1

x2

H =LX

i=1

(I� P+1

2T)ii+1 Hamiltonian of an integrable spin chain

= A ✏ABCD✏A0B0C0D0 + B ✏ABA0B0✏C0D0CD + C ✏ABC0D0✏A0B0CD

A = AUV ⇥⇥�/(8⇡2)

⇤⇥�1/(4⇡2

x

212)

�2 ⇥ (1/✏) + finite

• In x space, prototypical UV divergence from

‣ for later applications, go to momentum space:

I(x12) :=

Z 1/⇤

d

4x

1

[(x� x1)2]

2[(x� x2)

2]

2⇠ 2⇡

2

(x

212)

2log(x

212⇤

2)

I(x12) :=

Zd

D

L

(2⇡)De

iLx12

Zd

D

L1

(2⇡)D1

L

21(L1 � L)2

Zd

D

L3

(2⇡)D1

L

23(L3 + L)2

=

p1

p2

p3

p4

p5

p345

x x

p6

p1

p2

p3

p3

p4 p

5

p6

x1

x2

x1

x2

x

x2

x1

Double bubble

where x12 := x1 � x2 and

�(x) := �⇡2�D2

4⇡2�⇣D2� 1

⌘ 1

(�x2 + i")D2 �1

, (2.6)

is the scalar propagator in D dimensions. Note that I(x12) has UV divergences arisingfrom the regions z ! x1 and z ! x2.

Because the MHV diagram method is formulated in momentum space, it is useful torecast I(x12) as an integral in momentum space. Doing so one finds that

I(x12) =

Z 4Y

i=1

dDLi

(2⇡)Dei(L1+L2)·x12

L21 L

22 L

23 L

24

(2⇡)D �(D)⇣ 4X

i=1

Li

⌘

=

ZdDL

(2⇡)DeiL·x12

ZdDL1

(2⇡)DdDL3

(2⇡)D1

L21 (L� L1)2 L2

3 (L+ L3)2,

(2.7)

where L := L1 + L2. The integral over L1 and L3 is the product of two bubble integralswith momenta as in Figure 2, which are separately UV divergent.

Figure 2: The double-bubble integral relevant for the computation of I(x12).

These divergences arise from the region L1, L3 ! 1. The leading UV divergence of (2.7)is equal to

I(x12)|UV =1

✏· 1

8⇡2· 1

(4⇡2x212)

2. (2.8)

3 The one-loop dilatation operator from MHV rules

In this section we compute the UV-divergent part of the coe�cients A, B, C defined in(2.2), representing the trace, permutation and identity flavour structures, respectively. Inorder to compute these three coe�cients, it is su�cient to consider one representativeconfiguration for each one. We will choose the following helicity (or SU(4)) assignments:

ABCD A0B0C 0D0

Tr 1234 2413P 1213 34241l 1213 2434

(3.1)

There is a single MHV diagram to compute, represented in Figure 3. It consists of onesupersymmetric four-point MHV vertex,

VMHV(1, 2, 3, 4) =�(4)(

P4i=1 Li

)�(8)(P4

i=1 `i⌘i)

h12ih23ih34ih41i , (3.2)

3

=1

✏

1

8⇡2

1

(4⇡2x

2)2+ finiteFT of

• Back to 2004!

‣ MHV diagrams: new perturbative expansion of Yang-Mills theory (Cachazo, Svrcek, Witten 2004)

- vertices are off-shell continuation of the MHV amplitudes, to each internal leg with momentum L one associates the spinor

- connect vertices using scalar propagators

- derivation from lightcone quantisation of YM + change of variables in the path integral with Jacobian = 1 (Mansfield; Gorsky, Rosly)

‣ also works for loops, and without supersymmetry (cut-constructible parts) (Brandhuber, Spence, GT 2004)

Calculation from MHV diagrams

�↵ ! L↵↵ ⇠↵ = reference spinor ⇠↵

(Brandhuber, Penante, GT, Young)

• Only one MHV diagram!

‣ choose three R-symmetry assignments contributing to one structure at a time

MHV

x1

x2

L1

L2

L3

L4

where x12 := x1 � x2 and

�(x) := �⇡2�D2

4⇡2�⇣D2� 1

⌘ 1

(�x2 + i")D2 �1

, (2.6)

is the scalar propagator in D dimensions. Note that I(x12) has UV divergences arisingfrom the regions z ! x1 and z ! x2.

Because the MHV diagram method is formulated in momentum space, it is useful torecast I(x12) as an integral in momentum space. Doing so one finds that

I(x12) =

Z 4Y

i=1

dDLi

(2⇡)Dei(L1+L2)·x12

L21 L

22 L

23 L

24

(2⇡)D �(D)⇣ 4X

i=1

Li

⌘

=

ZdDL

(2⇡)DeiL·x12

ZdDL1

(2⇡)DdDL3

(2⇡)D1

L21 (L� L1)2 L2

3 (L+ L3)2,

(2.7)

where L := L1 + L2. The integral over L1 and L3 is the product of two bubble integralswith momenta as in Figure 2, which are separately UV divergent.

Figure 2: The double-bubble integral relevant for the computation of I(x12).

These divergences arise from the region L1, L3 ! 1. The leading UV divergence of (2.7)is equal to

I(x12)|UV =1

✏· 1

8⇡2· 1

(4⇡2x212)

2. (2.8)

3 The one-loop dilatation operator from MHV rules

In this section we compute the UV-divergent part of the coe�cients A, B, C defined in(2.2), representing the trace, permutation and identity flavour structures, respectively. Inorder to compute these three coe�cients, it is su�cient to consider one representativeconfiguration for each one. We will choose the following helicity (or SU(4)) assignments:

ABCD A0B0C 0D0

Tr 1234 2413P 1213 34241l 1213 2434

(3.1)

There is a single MHV diagram to compute, represented in Figure 3. It consists of onesupersymmetric four-point MHV vertex,

VMHV(1, 2, 3, 4) =�(4)(

P4i=1 Li

)�(8)(P4

i=1 `i⌘i)

h12ih23ih34ih41i , (3.2)

3

h13ih24ih12ih34i

h13ih24ih23ih14i

�1

T :

P :

I :{

A ✏ABCD✏A0B0C0D0 + B ✏ABA0B0✏C0D0CD + C ✏ABC0D0✏A0B0CD

AMHV

�1�AB , 2�CD , 3�A0B0 , 4�C0D0

�=

T P I

• New integral to compute:

‣ Double bubble x one MHV vertex

• Work out the integrands:

‣ for P: nothing to be done! (-1 x double bubble, or -1)

‣ for I:

‣ next using CSW prescription and momentum conservation:

UV divergent manifestly finite

1 +[⇠|L1L2|⇠][⇠|L3L4|⇠][⇠|L2L3|⇠][⇠|L1L4|⇠]

= 1 +[⇠|L1L|⇠][⇠|L3L|⇠]

[⇠|(L� L1)L3|⇠][⇠|L1(L+ L3)|⇠]

DROP!

h13ih24ih23ih14i = 1 +

h12ih34ih23ih14i

Summarising: BUV = -1, CUV = 1

I(x12) :=

Zd

D

L

(2⇡)De

iLx12

Zd

D

L1

(2⇡)D1

L

21(L1 � L)2

Zd

D

L3

(2⇡)D1

L

23(L3 + L)2

AMHV(1�AB , 2�CD , 3�A0B0 , 4�C0D0 )

• Summary: AUV = 1/2, BUV = -1, CUV = 1 (after similar manipulations on Tr)

• Comments:

‣ ξ-dependence: drops out at the end of the calculation

- similar to lightcone gauge…

- explicitly: by Lorentz invariance, final result can depend only on the combination

- note that the result cannot depend on as we have NOT introduced a spinor ξα !

‣ self-energies: vanish with MHV diagrams (Brandhuber, Spence, GT 04)

- in lightcone gauge self-energies are finite and do not contribute to anomalous dimension (Belitsky, Derkachov, Korchemsky, Manashov)

‣ nice derivation from MHV rules in twistor space (Koster, Mitev, Staudacher)

[⇠|L2|⇠] = 0

L · ⇠

• Compute correlator from cuts

‣ Earlier applications (Engelund & Roiban)

‣ Ingredients:

- on-shell amplitudes (no off-shell continuations) & cut propagators

‣ Quickly reconstruct the nontrivial cases:

- for I:

- second term:

- Kite integral, UV finite in four dimensions

- Keep just the 1, drop the rest, or CUV = 1.

- Similarly for the Tr structure

Once again, with generalised unitarity

Figure 2: The single cut diagram contributing to the dilatation operator at one loop.

1l : A(1�12 , 4�34 , 3�24 , 2�13) =h13ih24ih23ih14i . (2.10)

Three observations are in order here. First, we note that the same integrands as in theapproach of [1] have appeared, with the important di↵erence that, in that paper, the spinorsassociated with the on-shell momenta are given by the appropriate o↵-shell continuationfor MHV diagrams. Here the spinors for the cut loop momenta do not need any o↵-shellcontinuation. Furthermore, for the case of the P integrand there is obviously no di↵erencebetween the two approaches, and the resulting integral is given by a double bubble whereall the four propagators are cut. In the other two cases, this integral is dressed by theappropriate amplitude. Finally, we note that the colour factor associated with all diagramsis obtained from the contraction

· · · (T bT a)ij

· · ·Tr(T aT bT cT d) · · · (T dT c)lm

· · · = · · ·N2�im

�lj

· · · , (2.11)

where the trace arises from the amplitude and the factors · · · (T bT a)ij

· · · and · · · (T dT c)lm

· · ·from the operators (and we indicate only generators corresponding to the fields being con-tracted). We now proceed to construct the relevant integrands.

The trace integrand

In this case the relevant amplitude (which multiplies four cut propagators) can be rewrittenas2

h13i h24ih12i h34i =

Tr+(`1 `3 `4 `2)

(`1 + `2)2(`3 + `4)2= �2(`1 · `3)

L2, (2.12)

where we have used `1 + `2 = �(`3 + `4) := L. Having rewritten the amplitude in termsof products of momenta, we lift the four cut momenta o↵ shell. The resulting integral hasthe structure of a product of two linear bubbles,

� 2

L2

Z

dDL1

(2⇡)DLµ

1

L21 (L� L1)2

Z

dDL3

(2⇡)DL3µ

L23 (L+ L3)2

. (2.13)

Using the fact thatZ

dDK

(2⇡)DKµ

K2(K ± L)2= ⌥Lµ

2Bub(L2) , (2.14)

2We define Tr+(abcd) := habi [bc ] hcdi [da].

4

h13ih24ih23ih14i = 1 +

h12ih34ih23ih14i = 1 +

h12ih34ih23ih14i

[34]

[34]

where

Bub(L2) :=

Z

dDK

(2⇡)D1

K2(K + L)2, (2.15)

we find that (2.13) is equal to 1/2 times a double bubble. Using (1.4) we finally arriveat AUV = 1/2. Note that in the definitions of AUV, BUV, and CUV, a factor of �/(8⇡2) ⇥�

1/(4⇡2x212)

�2 ⇥ (1/✏) will always be understood, with � := g2YMN .

The P integrand

No calculation is needed in this case, and the result is simply given by minus a cut double-bubble integral. Lifting the cut integral to a full loop integral we get BUV = �1.

The 1l integrand

The relevant amplitude in this case is

h13i h24ih23i h14i = 1 +

h12i h34ih23i h14i . (2.16)

Thus the first term in (2.16) gives the cut double-bubble integral, whereas we can useon-shell identities to rewrite the second term as

h12i h34ih23i h14i =

h12i h34i [34]h23i h14i [34] = � L2

2(`1 · `4) . (2.17)

Lifting the cut propagators of the second integral to full propagators, it is immediate tosee that this term produces the integral represented in Figure 3. This integral is finite infour dimensions and thus does not contribute to CUV. We then conclude that CUV = 1.

Figure 3: The finite integral corresponding to the term in (2.17). This integral is irrelevant forthe calculation of the dilatation operator.

For later convenience, we explicitly write down the form of the UV-divergent part of thecorrelator (2.4),

⌦�

�AB

�CD

�

i

j

(x1)�

�A

0B

0�C

0D

0�

l

m

(x2)↵

�

�

�

UV

=1

✏· �

8⇡2

⇣

�2(x12)�i

m

�lj

⌘⇣1

2✏ABCD

✏A

0B

0C

0D

0 � ✏ABA

0B

0✏CDC

0D

0 + ✏ABC

0D

0✏A

0B

0CD

⌘

.(2.18)

In terms of a spin-chain Hamiltonian, this can be represented as [9]

H2 =�

8⇡2

⇣1

2Tr + 1l � P

⌘

. (2.19)

5

� L2

2(`1 · `4)�!

(Brandhuber, Penante, GT, Young)

• Several generalisation/extensions possible

‣ SU(2|3) sector, higher loops….

• Yangian symmetry is thought to be a fundamental property of N=4 super Yang-Mills

• Two slightly different manifestations on

‣ amplitudes

‣ dilatation operator

• Goal: derive the action of the Yangian on the dilatation operator from the Yangian of amplitudes

Amplitude Yangian &Dilatation Operator Yangian

(Brandhuber, Heslop, GT, Young, to appear)

• Fact 1: Tree-level amplitudes in N=4 SYM are Yangian invariant (covariant) (Drummond, Henn, Plefka)

‣ level-zero generators JA ⟶ ordinary superconformal group

‣ level-one generators

- are non-local densities acting on particles i and j

‣ level-one generators ⟶ dual superconformal group

- level-one generator p(1) associated to momentum p is (related to) dual conformal K

- level-one generator q(1) associated to q-supersymmetry is (related to) dual superconformal generator S

‣ amplitudes are covariant under dual superconformal transformations (Drummond, Henn, Korchemsky, Sokatchev)

QAij := fA

CBJBi JC

j

QA =X

i<j

QAij

Amplitude Yangian

• Fact 2: The complete one-loop dilatation operator is Yangian invariant up to boundary terms (Dolan, Nappi, Witten)

‣ Equivalent to showing

- H = Hii+1 , where H12 acts on sites 1 and 2

‣ proof is based on acting on irreps of PSU(2,2|4)

‣ LHS ~ one loop, RHS ~ tree level. Proof relies on identity

• Next: derive Fact 2 from Fact 1

Dilatation operator Yangian

LX

i=1

⇥h(j)� h(j � 1)

⇤/j = 1 h (j) = jth harmonic number

[QA , H] ⇠ JA1 � JA

L

[QA12 , H12] ⇠ JA

1 � JA2

• Main tool: an intriguing formula relating the dilatation operator to four-point superamplitudes found by Zwiebel (+ unpublished work of Beisert)

• Building blocks of this formula:

‣ tree-level four-point superamplitude

- recall: at one-loop, only two fields interact, 2 →2 structure

‣ tree-level minimal form factors

- represent the states on which the dilatation operator acts

‣ effectively computes two-particle cuts of one-loop minimal form factors of (non-protected) operators (Wilhelm)

• Idea: use known action of Yangian generators on amplitudes to derive action on the dilatation operator

⌦0|��1 · · ·�L)(0) |�1 · · ·�L

↵

• States & single-trace operators

‣ A state corresponds to a single-trace operator

‣ The letters : (and symmetrised covariant derivatives D acting on them)

‣ oscillator representation of the states (Gunaydin & Marcus)

- a, b bosonic oscillators, d fermionic oscillators

‣ map to the states:

• Spinor-helicity translation:

Tr��1 · · ·�L

�(x)

�i F↵� , ↵ABC , �[AB], ↵A, F ↵�

F $ b†b† , $ b†d† , �$ d†d† , $ a†d†d†d† , F $ a†a†d†d†d†d† , D $ a†b†

a†↵ $ �↵ , b†↵ $ �↵, d†A $ ⌘A

a↵ $ @↵ , b↵ $ @↵, dA $ @A

[a↵, a†� ] = ��↵ , [b↵, b

†� ] = ��↵ , {dA, d†B} = �BA , ↵,� = 1, 2, ↵, � = 1, 2, A = 1, . . . , 4

• States in spinor-helicity language:

‣ combine

‣ a state is a polynomial in the ’s satisfying the physical state condition (vanishing central charge)

• Examples:

‣ half-BPS

‣ Konishi

• = tree-level minimal form factor of the corresponding operator (Wilhelm)

‣ E.g. half-BPS

· · ·�12�12 · · · $ (⌘11⌘21)(⌘

12⌘

22)

· · · ✏ABCD�AB�CD · · · $ ✏ABCD(⌘A1 ⌘B1 )(⌘C2 ⌘

D2 )

· · · (⌘11⌘21)(⌘12⌘22) · · · =⌦0��� · · ·�12�12 · · ·

�(0)

��· · ·�12�12 · · ·↵

⇤a :=��↵, �↵, ⌘A

�

P (⇤1, . . . ,⇤L)

P (⇤1, . . . ,⇤L) ⇤

←R-symmetry

← position

• Zwiebel’s formula: (second term slightly rewritten)

‣ phase-space measure (mod little group)

‣ superamplitude

‣ P(1,2) represents the operator/state

• Connection to dilatation operator

‣ integrating out the momentum delta function one gets: (Zwiebel)

‣ (similarly for )

‣ Neatly reproduces Beisert’s harmonic action form of the complete dilatation operator at one loop

A(1, 2, 3, 4) =�(4)

�Pi �i�i

��(8)

�Pi �i⌘i

�

h12ih23ih34ih41i

H12|1, 2i =

Zd⇤3d⇤4 A(1, 2, 3, 4)

hP (�4,�3) �

✓h12ih34i

◆2

P (1, 2)i

d⇤i := d2�id2�id

4⌘i

�01 = �1 cos ✓ � ei��2 sin ✓ , �0

2 = �1 sin ✓ + ei��2 cos ✓ �0, ⌘0

“integrated form”

“un-integrated form”

| · · · 1, 2, · · ·↵

H12|1, 2i = � 1

⇡

Z 2⇡

0d�

Z ⇡2

0d✓

he2i�P (10, 20)� P (1, 2)

i

• Connection to form factors (Wilhelm)

‣ first term is the two-particle cut of the one-loop form factor of the operator represented by P

- contains IR divergences (triangle) and also UV divergences (bubbles)

‣ second term subtracts the IR divergence of the same cut

‣ Leftover = discontinuity of a (UV-divergent) bubble, whose coefficient is ~ the dilatation operator

- note: the discontinuity of a bubble if finite

H12|1, 2i =

Zd⇤3d⇤4 A(1, 2, 3, 4)

hP (�4,�3) �

✓h12ih34i

◆2

P (1, 2)i

=

q

l2

l1

p1

p2

(a) (b)

p1

p2

q

Figure 1: In Figure (a) we show the diagram calculating the cut in the q2-channel ofthe Sudakov form factor (2.3). The cross denotes a form factor insertion. A seconddiagram with legs 1 and 2 swapped has to be added and doubles up the result of the firstdiagram. The result of this cut is given by (twice) a cut one-mass triangle function,depicted in Figure (b).

The q2-cut of the form factor (i.e. its discontinuity in the q2-channel) is obtainedfrom the diagram on the left-hand side of Figure 1, whose expression is3

F (1)(q2)!

!

q2−cut= 2

"

dLIPS(l1, l2; q) F(0)(l1, l2; q)A

(0)#

φ12(p1),φ12(p2),φ34(l1),φ34(l2)$

,

(2.5)where the Lorentz invariant phase space measure is

dLIPS(l1, l2; q) := dDl1 dDl2 δ

+(l21)δ+(l22)δ

D(l1 + l2 + q) , (2.6)

and q is given in (2.4). The tree-level component amplitude appearing in (2.5),A(0)

#

φ12(p1),φ12(p2),φ34(l1),φ34(l2)$

, can be extracted from Nair’s superamplitude[18]

AMHV := gn−2 (2π)4δ(4)#

n%

i=1

λiλi

$

δ(8)#

n%

i=1

λiηi$

n&

i=1

1

⟨ii+ 1⟩ , (2.7)

where λn+1 ≡ λ1. The result is

A(0)#

φ12(p1),φ12(p2),φ34(l1),φ34(l2)$

=⟨l1l2⟩⟨12⟩⟨l21⟩⟨2l1⟩

. (2.8)

The other quantity appearing in (2.5), F (0) is the tree-level expression for the formfactor (2.3), which is trivially equal to 1. Thus, we get

F (1)(q2)!

!

q2−cut= 2

"

dLIPS(l1, l2; q)⟨12⟩⟨l1l2⟩⟨2l1⟩⟨l21⟩

= −2 q2"

dLIPS(l1, l2; q)1

(l2 + p1)2.

(2.9)

3In this and the following formulae we omit a power of the ’t Hooft coupling, defined as a :=(g2N)/(16π2)(4πe−γ)ϵ. Note that this is 1/2 the ’t Hooft coupling defined in [12].

4

1

23

4A

P -=

x1

x2

x

x1

x2

L L1

2

P (1, 2)

Matthias Wilhelm’s seminar

• Summarising:

‣ Unintegrated form:

‣ integrated form:

• It is not at all obvious to see how the relation

H12|1, 2i =

Zd⇤3d⇤4 A(1, 2, 3, 4)

hP (�4,�3) �

✓h12ih34i

◆2

P (1, 2)i

H12|1, 2i = � 1

⇡

Z 2⇡

0d�

Z ⇡2

0d✓

he2i�P (10, 20)� P (1, 2)

i

[QA12 , H12] ⇠ JA

1 � JA2

is realised when acting on the integrated form

Amplitudes to the rescue!

• Act with level-one generator p(1) on un-integrated form:

‣ from DHP

• Preliminary check: half-BPS operators, e.g. Tr (𝜙12 𝜙12)

‣ First line vanishes since (Brandhuber, Spence, GT, Yang)

‣ Second line: only constant part of d survives

• Result of explicit integration of RHS:

P�12�12

(�4,�3) = r P�12�12

(1, 2)

r :=

✓h12ih34i

◆2

Qij =⇣m �

j ↵��↵ + m �

j ↵��↵ � dj�

�↵�

�↵

⌘pi �� + qj↵Cq

Ci↵ � (i $ j)

[Q12, H12]|1, 2i = Q12

Zd⇤3d⇤4 A(1, 2, 3, 4)

⇥P (�4,�3)� r P (1, 2)

⇤

�Z

d⇤3d⇤4 A(1, 2, 3, 4)⇥Q�4,�3P (�4,�3)� r Q12P (1, 2)

⇤

[Q12, H12]|�12�12i = P�12�12

(1, 2)

Zd⇤3d⇤4 A(1, 2, 3, 4) · r

⇥p3 � p4 � (p1 � p2)

⇤

[Q12, H12]|1, 2i = 2(p1 � p2)|1, 2i

• Ingredients of the general proof (for arbitrary states):

‣ after IBP, combination of generators acting on amplitude is

‣ related to dual conformal K, which annihilates amplitude

‣ (Q13 + Q14 + Q23 + Q24) A = 0 since

- JC is a symmetry of the amplitude

- proportional to the (vanishing) dual Coxeter number of PSU(2,2|4)

- alternative proof: use that Yangian on amplitudes is compatible with cyclicity!

‣ thus

- remaining term already computed in half-BPS case…

X

i<j

Qij

Q12 +Q34 =X

i<j

Qij � (Q13 +Q14 +Q23 +Q24)

(Q13 +Q14 +Q23 +Q24)A = fA

CB(J1 + J2)BJC � 1

2fACBf

BCD (J1 + J2)

D

fACBf

BCD

[Q12, H12]|1, 2i = P (1, 2)

Zd⇤3d⇤4 A(1, 2, 3, 4) · r

⇥p3 � p4 � (p1 � p2)

⇤

• Result:

• Comments:

1. very simple extension to show that a similar formula holds if Q is the level-one generator associated to supersymmetry q:

2. not obvious to see this result on the “integrated form” of Zwiebel’s formula (without amplitudes)!

3. RHS looks like a tree-level quantity!

4. can check other commutators

[Q12, H12]|1, 2i = 2(p1 � p2)|1, 2i

[Q12, H12]|1, 2i = 2(q1 � q2)|1, 2i