Flatspaceamplitudesinahighest-weightbasis:Anaturalchoiceforstudyingimplicationsofsuperrotations?

SABRINAGONZALEZPASTERSKI

ATriangleofRelations• OverthepastfewyearsStrominger andcollaboratorshavebeenstudyinghowasymptotic

symmetrygroupsmanifestthemselvesinscatteringmatrixelementsviasoftfactors

• softfactors⇒Wardidentitiesimplyingconstraintsonthe" -matrix

whichcorrespondtoalargerclass ofsymmetrytransformations

• Whatresultedisapatternofconnectionsbetweentraitsoflowenergyradiationthat

appearedwithmultipleiterations:thisturnsintoafillintheblankexerciseonceonevertexof

anewiterationismotivated

02/22/17 2

SoftTheorems

MemoriesSymmetriesSGP@MIT

ATriangleofRelations• OverthepastfewyearsStrominger andcollaboratorshavebeenstudyinghowasymptotic

symmetrygroupsmanifestthemselvesinscatteringmatrixelementsviasoftfactors

• softfactors⇒Wardidentitiesimplyingconstraintsonthe" -matrix

whichcorrespondtoalargerclass ofsymmetrytransformations

• Whatresultedisapatternofconnectionsbetweentraitsoflowenergyradiationthat

appearedwithmultipleiterations:thisturnsintoafillintheblankexerciseonceonevertexof

anewiterationismotivated

02/22/17 2

SoftTheorems

MemoriesSymmetries

e

SGP@MIT

ATriangleofRelations• OverthepastfewyearsStrominger andcollaboratorshavebeenstudyinghowasymptotic

symmetrygroupsmanifestthemselvesinscatteringmatrixelementsviasoftfactors

• softfactors⇒Wardidentitiesimplyingconstraintsonthe" -matrix

whichcorrespondtoalargerclass ofsymmetrytransformations

• Whatresultedisapatternofconnectionsbetweentraitsoflowenergyradiationthat

appearedwithmultipleiterations:thisturnsintoafillintheblankexerciseonceonevertexof

anewiterationismotivated

02/22/17 2

SoftTheorems

MemoriesSymmetries

e#$

SGP@MIT

ATriangleofRelations• OverthepastfewyearsStrominger andcollaboratorshavebeenstudyinghowasymptotic

symmetrygroupsmanifestthemselvesinscatteringmatrixelementsviasoftfactors

• softfactors⇒Wardidentitiesimplyingconstraintsonthe" -matrix

whichcorrespondtoalargerclass ofsymmetrytransformations

• Whatresultedisapatternofconnectionsbetweentraitsoflowenergyradiationthat

appearedwithmultipleiterations:thisturnsintoafillintheblankexerciseonceonevertexof

anewiterationismotivated

02/22/17 2

SoftTheorems

MemoriesSymmetries

e#$%$&

SGP@MIT

ATriangleofRelations• OverthepastfewyearsStrominger andcollaboratorshavebeenstudyinghowasymptotic

symmetrygroupsmanifestthemselvesinscatteringmatrixelementsviasoftfactors

• softfactors⇒Wardidentitiesimplyingconstraintsonthe" -matrix

whichcorrespondtoalargerclass ofsymmetrytransformations

• Whatresultedisapatternofconnectionsbetweentraitsoflowenergyradiationthat

appearedwithmultipleiterations:thisturnsintoafillintheblankexerciseonceonevertexof

anewiterationismotivated

02/22/17 2

SoftTheorems

MemoriesSymmetries

e#$%$&

Inthismannerabrandnewiterationwas

completed(ASG~ pastdecadevs50’s-60’s).

Thisiterationisrelatedtoageneralizationof

Lorentztransformationsandhasmotivated

lookingat"-matrixelementsinanewbasis

withdefiniteSL(2,C)weights

SGP@MIT

ATriangleofRelations

02/22/17 3

SoftTheorems

MemoriesSymmetries

i)Weinberg– photonO(() )

ii)Weinberg– gravitonO(() )

iii)Cachazo &Strominger – gravitonO(1)

i)Liénard-Wiechert /Bieri &Garfinkle

ii)Zeldovich &Polnarev /Christodoulou

iii)Pasterski,Strominger,&Zhiboedov

(global) (asymptotic)

i)e-charge largeU(1)

ii)#$ supertranslations

iii)*+, superrotations

Goalforthistalk:1)explaintheverticesofthissofttriangletodemonstratethemost

recentspin-relatediteration

2) howthismotivatesa changeof

basis(currentworkwithS.H.Shao

andA.Strominger)

SGP@MIT

"-matrixConstraintsfromSymmetries• Noether’s Theorem:ContinuousSymmetries⇒ ConservationLaws•MoreSymmetries⇒MoreConstraintson"-matrix

•ModusOperandi:

Ø Lookforlargersetof“physical”symmetries

ØMotivateviapropertiesoflowenergyscattering

02/22/17 4

SoftTheorems

MemoriesSymmetriesSGP@MIT

"-matrixConstraintsfromSymmetriesWheredothese“extra”symmetriescomefrom?

• Boundaryconditionscan’tbetoorestrictivesoastodisallowtypicalscattering

processes,largerclassofboundaryconditionsgivesalargergroupofsymmetries

thatpreservethisclassofboundaryconditions

Ø Theseareourextrasymmetries

02/22/17 5

SoftTheorems

MemoriesSymmetriesSGP@MIT

"-matrixConstraintsfromSymmetriesWheredothese“extra”symmetriescomefrom?

• Boundaryconditionscan’tbetoorestrictivesoastodisallowtypicalscattering

processes,largerclassofboundaryconditionsgivesalargergroupofsymmetries

thatpreservethisclassofboundaryconditions

Ø Theseareourextrasymmetries

02/22/17 5

SoftTheorems

MemoriesSymmetries

Classofgaugetransformations

thatactnon-triviallyonthe

boundarydatawhichislarger

thanthestandardglobalU(1)of

E&Mor#$ and%$&ofMinkowski

space

SGP@MIT

Toseewherethisnon-trivialactionon

theboundarydatacomesfrom,letus

considerscatteringfromaspacetime

perspectivewheretheinandoutstates

comefromandexitthepastandfuture

boundariesofourspacetime

ScatteringfromaSpacetime Perspective

02/22/17 6

conformal

compactification

./

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.1

20

21

SGP@MIT

Toseewherethisnon-trivialactionon

theboundarydatacomesfrom,letus

considerscatteringfromaspacetime

perspectivewheretheinandoutstates

comefromandexitthepastandfuture

boundariesofourspacetime

ScatteringfromaSpacetime Perspective

02/22/17 6

conformal

compactification

./

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20

21

massiveparticlesexithere

masslessparticlesexithere

masslessparticlesenterhere

massiveparticlesenterhere

thepointat∞

SGP@MIT

02/22/17 7

./

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20

21

massiveparticlesexithere

masslessparticlesexithere

masslessparticlesenterhere

massiveparticlesenterhere

thepointat∞

45 ateachpoint

constanttimeslices

./

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21

ScatteringfromaSpacetime Perspective

SGP@MIT

U(1)Example

02/22/17 8

constanttimeslices

./

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20

21

|Ψ⟩

• Withthisinmindwecanunderstandthe

U(1)iterationofthetriangle

• Gauss’slawisaconstraintequationrelating

electricfluxtocharges

9

:

SGP@MIT

U(1)Example

02/22/17 9

constanttimeslices

./

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.1

20

21

|Ψ⟩

Ashtekar

1980’s ./

.0

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20

21

Ψ ;<

Ψ =>?

Pushslicetonullinfinity&constraint

equationscanbeusedtorelateradiationto

changeinchargevelocities

SGP@MIT

U(1)Example

02/22/17 10

./

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20

21

Ψ ;<

Ψ =>?

@

A

9B=>C ∝1A5

9FGH ∝1A

• Lookatradialfalloffsandisolatethefree

datacorrespondingtoradiativemodes

• FindtheASGthatpreservethesefall-offs

SGP@MIT

U(1)ExampleSomemoredetails:

•RadialExpansion:

•ASGthatpreservesthisexpansion:

•ModeExpansion:

•ConstraintEquation:

02/22/17 11

Az(r, u, z, z) = Az(u, z, z) +1P

n=1

A(n)z (u,z,z)

rn

Au(r, u, z, z) =1rAu(u, z, z) +

1Pn=1

A(n)u (u,z,z)rn+1

Fur = Au

Fzz = @zAz � @zAz

Fuz = @uAz

ds2 = �du2 � 2dudr + 2r2�zzdzdz

z = ei� tan✓

2�zz =

2

(1 + zz)2

CoordinateConventions:

�✏Az(u, z, z) = @z✏(z, z)

Aµ

(x) = e

X

↵=±

Zd

3q

(2⇡)31

2!q

h✏

↵

⇤

µ

(~q)a↵

(~q)eiq·x + ✏

↵

µ

(~q)a↵

(~q)†e�iq·xi

@uAu = @u(DzAz +DzAz) + e2ju[arXiv:1407.3789]

SGP@MIT

U(1)ExampleTwokeypoints:

•SaddlepointatlargeApicksoutagaugebosonmomentumpointinginthesamedirectionas

whereanobservernearnullinfinitywoulddetectit.Asaresult,oneendsupwithamode

expansionwheretheangularintegrallocalizes,and(J, L) remainasFourierconjugates.

• ∫ OJpicksoutL → 0.Assuchwecanrelatethesoftfactorstotheconstraintequations:Fouriertransformofapole

() isastepfunction

02/22/17 12

eiq·x = e�i!u�i!r(1�q·x)Az(u, z, z) = � i

8⇡2

p2e

1 + zz

Z 1

0d!

⇥a+(!x)e

�i!u � a�(!x)†e

i!u⇤⇒

hzn+1, zn+2, ...|a�(q)S|z1, z2, ...i = S(0)�hzn+1, zn+2, ...|S|z1, z2, ...i+O(1)

S(0)� =X

k

eQkpk · ✏�

pk · q

[arXiv:1407.3789,arXiv:1505.00716 ]

SGP@MIT

SomeConventions:

U(1)Example

02/22/17 13

IntegratetheconstraintequationalongJ

Thesoftfactorindicatesthattypicalscattering

processeswillproduceanonzeroJ integrated

electricfield.

�Au = 2Dz�Az + e2Z

duju

S(0)±p = eQ

p · ✏±

p · qpµ = m�(1, ~�)

Er =Q

4⇡r21

�2(1� ~� · n)2

�Az = � e

4⇡✏⇤+z !S(0)+

� e4⇡ lim

!!0![Dz ✏⇤+z S(0)+

p +Dz ✏⇤�z S(0)�p ] = �e2 Q

4⇡1

�2(1�~�·n)2

[arXiv:1505.00716 ]

SGP@MIT

U(1)Example

02/22/17 14

•Upshot:TheresidueoftheWeinbergpoleindicatesanonzerovalueforcertain

low-energyradiationobservablesaka“memoryeffects”

•Sincesettingthesemodestozerowouldtrivializetheallowedscatteringevents,we

getwiththisclassofboundaryconditionsalargerclassofgaugetransformations

thatpreservetheradialorderofthefalloffswhileshiftingtheboundaryvaluesaka

“largegaugetransformations”(seeStrominger [arXiv:1308.0589 arXiv:1312.2229] for how softtheoremscanbe

usedtoconstructWardidentitiesfortheseasymptoticsymmetries)

J

symmetry

memory

RS

SGP@MIT

MidpointRecap• FromtheU(1)examplewe’veseenhowfromsofttheoremswecanextractlowenergy

radiationobservables(memoryeffects),whicharenon-zerointypicalscatteringprocessesand

thusobtainalargerasymptoticsymmetrygroupforlessstringentboundaryconditions

• Forthegravitycase,wecanwritedownametricexpansion,holdingontothenotionofnull

infinityforspacetimes thatareonlyasymptoticallyflat,lookatthevectorfieldsfor

diffeomorphismswhichpreservethesefalloffs

02/22/17 15

SoftTheorems

MemoriesSymmetries

e#$%$&

SGP@MIT

MidpointRecap• FromtheU(1)examplewe’veseenhowfromsofttheoremswecanextractlowenergy

radiationobservables(memoryeffects),whicharenon-zerointypicalscatteringprocessesand

thusobtainalargerasymptoticsymmetrygroupforlessstringentboundaryconditions

• Forthegravitycase,wecanwritedownametricexpansion,holdingontothenotionofnull

infinityforspacetimes thatareonlyasymptoticallyflat,lookatthevectorfieldsfor

diffeomorphismswhichpreservethesefalloffs

02/22/17 15

SoftTheorems

MemoriesSymmetries

e#$%$&

Boththesubleading softfactorandtheASGit

correspondstowillmotivatelookingat" -

matrixelementsinanewbasiswithdefinite

SL(2,C)weights,whilethememoryeffect

helpsmotivatethisextended-BMSsymmetry

as“physical.”

SGP@MIT

AsymptoticallyFlatSpacetimes

02/22/17 16

OT5 = −O@5 + OX5 + OY5 + OZ5

BMS1960’s

Wanttoconsidernon-trivial

gravitationalbackgroundsthat

are“close”tobeingflat

Ø Approachflatspacetime

farawayfromsources

SGP@MIT

AsymptoticallyFlatSpacetimesSomemoredetails:•RadialExpansion:

•ASGthatpreservesthisexpansion:

•ModeExpansion:[SS radiativedata

•ConstraintEquations:

02/22/17 17

CoordinateConventions:

ds2 = �du2 � 2dudr + 2r2�zzdzdz + 2mBr du2

+�rCzzdz2 +DzCzzdudz +

1r (

43Nz � 1

4@z(CzzCzz))dudz + c.c.�+ ...

⇠+ =(1 +u

2r)Y +z@z �

u

2rDzDzY

+z@z �1

2(u+ r)DzY

+z@r +u

2DzY

+z@u + c.c

+ f+@u � 1

r(Dzf+@z +Dzf+@z) +DzDzf

+@r

z = ei� tan✓

2�zz =

2

(1 + zz)2

f+ = f+(z, z) @zY+z = 0Guu = 8⇡GTM

uu Guz = 8⇡GTMuz

SGP@MIT

AsymptoticallyFlatSpacetimesThephysicalobservablemodesconjugatetothe

ASGtransformationsaregravitational

displacementandspinmemoryeffects([arXiv:1502.06120],potentiallyobservable[arXiv:1702.03300])

02/22/17 18SGP@MIT

AsymptoticallyFlatSpacetimesTheirvaluescanbeextractedfromtheleadingas

wellasamorerecentsubleading ([arXiv:1404.4091])softgravitontheorems.

02/22/17 19

hzn+1, zn+2, ...|a�(q)S|z1, z2, ...i =⇣S(0)� + S(1)�

⌘hzn+1, zn+2, ...|S|z1, z2, ...i+O(!)

S(0)� =X

k

(pk · ✏�)2

pk · q

S(1)� = �iX

k

pkµ✏�µ⌫q�Jk�⌫pk · q

SGP@MIT

AsymptoticallyFlatSpacetimes• Letustakeacloserlookatthesuperrotation vectorfieldnearnullinfinity:

• NoticewithtwocopiesofWittalgebrasince\ isany2DCKV

• Also,J]> prefersRindler energyeigenstates

• Ratherthanusingthesubleading softfactortoestablishaWardidentityforthisasymptotic

symmetry([arXiv:1406.3312])one can massage it tolook like 2Dstresstensor

([arXiv:1609.00282])

02/22/17 20

⇠+|J+ = Y +z@z +u

2DzY

+z@u + c.c.

Tzz ⌘ i8⇡G

Rd2w 1

z�wD2wD

wRduu@uCww

hTzzO1 · · · Oni =nX

k=1

hk

(z � zk)2+

�zkzkzk

z � zkhk +

1

z � zk(@zk � |sk|⌦zk)

�hO1 · · · Oni

h =1

2(s+ 1 + iER) h =

1

2(�s+ 1 + iER)

� = h+ h s = h� h

WeightConventions:

SGP@MIT

FinalRecap• Weseethatatriangleofconnectionsbetweensofttheorems,memoryeffects,andasymptotic

symmetrygroupshasledtoaniterationinvolvinganextensionoftheSL(2,C)Lorentz

transformationstoarbitrary2DCKVs.

• Moreoverthepreferredbasisfortheactionofthissymmetryon"-matrixelementsisonewith

definiteSL(2,C)weightsratherthanthestandardplanewavebasis.

02/22/17 21

SoftTheorems

MemoriesSymmetries

e#$%$&

SGP@MIT

FinalRecap• Weseethatatriangleofconnectionsbetweensofttheorems,memoryeffects,andasymptotic

symmetrygroupshasledtoaniterationinvolvinganextensionoftheSL(2,C)Lorentz

transformationstoarbitrary2DCKVs.

• Moreoverthepreferredbasisfortheactionofthissymmetryon"-matrixelementsisonewith

definiteSL(2,C)weightsratherthanthestandardplanewavebasis.

02/22/17 21

SoftTheorems

MemoriesSymmetries

e#$%$&

Isahighest-weightbasisanaturalchoiceforstudyingimplicationsofsuperrotations?

Ø Recastflatspaceamplitudesinthisformandsee

whatfeaturesarise…

SGP@MIT

MassiveScalars• Considerthecelestialsphere[45 astheboundaryofthelightcone fromtheorigininMinkowski

spacetime.Theprojectivecoordinate^ undergoesmobius transformationswhenthe

spacetime undergoesLorentztransformations

02/22/17 22

w =X1 + iX2

X0 +X3w ! aw + b

cw + d•WelookforsolutionstothemassiveKlein-Gordonequation

parameterizedbyaweightΔ andareferencedirection^ such

thatittransformsasaquasi-primaryunderSL(2,C)

✓@

@X⌫

@

@X⌫�m2

◆��,m(Xµ;w, w) = 0

��,m

✓⇤µ

⌫X⌫ ;

aw + b

cw + d,aw + b

cw + d

◆= |cw + d|2���,m (Xµ;w, w)

`

SGP@MIT

[arXiv:1701.00049]

MassiveScalars• Ifwelookedatsinglehyperbolicsliceofconstantunitdistancefromtheoriginina(,b describedbycoordinates:

02/22/17 23

ds2H3=

dy2 + dzdz

y2

pµ(y, z, z) =

✓1 + y2 + |z|2

2y,Re(z)

y,Im(z)

y,1� y2 � |z|2

2y

◆

•Wecanconstructabulktoboundarypropagatorthat

transformscovariantly underSL(2,C)

G�(y, z, z;w, w) =

✓y

y2 + |z � w|2

◆�

G�(y0, z0, z0;w0, w0) = |cw + d|2�G�(y, z, z;w, w)

SGP@MIT

[arXiv:1701.00049]

MassiveScalars• Ifwethinkof(Y, Z) ascoordinatesinmomentumspace,wecanusethisbulk-to-boundary

propagatortoconstructthedesiredsolutions:

02/22/17 24

•Moreover,wecanapplythistransformdirectlytothe

amplitudetoconvert"-matrixelementstoahighestweight

basis:

#

SGP@MIT

[arXiv:1701.00049]

�±�,m(Xµ

;w, w) =

Z 1

0

dy

y3

Zdzdz G�(y, z, z;w, w) exp

h±im pµ(y, z, z)Xµ

i

A�1,··· ,�n(wi, wi) ⌘nY

i=1

Z 1

0

dyiy3i

Zdzidzi G�i(yi, zi, zi;wi, wi)A(mj p

µj )

A�1,··· ,�n

✓awi + b

cwi + d,awi + b

cwi + d

◆=

nY

i=1

|cwi + d|2�i

!A�1,··· ,�n(wi, wi)

MassiveScalars•IfwelookattheKlein-Gordoninnerproductfortwosuchstateswithdistinctreferencevectors

wegetanintegralthatwecanmakesenseofinadistributionalmannerifwetakeΔc = d + .ef

02/22/17 25

(�+1 ,�

+2 ) = �i

Zd3 ~X

h�+�1,m

(Xµ;w1, w1)@X0�+⇤�2,m

(Xµ;w2, w2)� @X0�+�1,m

(Xµ;w1, w1)�+⇤�2,m

(Xµ;w2, w2)i

= 2(2⇡)3m�2

Z 1

0

dy

y3

ZdzdzG�1(y, z, z;w1, w1)G

⇤�2

(y, z, z;w2, w2)

(�+1 ,�

+2 ) = 64⇡5m�2 1

(�1 +�⇤2 � 2) |w1 � w2|�1+�⇤

2�(�1 + �2)

SGP@MIT

[arXiv:1701.00049]

MassiveScalars•Thewayinwhich2Dconformalsymmetrydictates

theformofcorrelationfunctionsisseenhereas

Lorentzinvariancewherethephysicalspacetime is

whatistypicallytheabstractembeddingspace

•Othereffortstowardsaflatspaceholographicdescription[hep-th/0303006,arXiv:1609.00732]havelookedatafoliationofMinkowski spaceto

reproduceAdS/CFT,dS/CFToneachsliceor

convolvingm=0highestweightstatesinex.onlythe

forwardlightcone regiontoreproduceCFT-like

correlators.

02/22/17 26

./

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21

.0

SGP@MIT

[arXiv:1701.00049]

MassiveScalars•Here,thebulktoboundarypropagatoractsontheamplitudeinmomentumspace.Assuchit

probestheamplitudeatallenergyscales(vsonlyneedingtoknowmatrixelementsbetween

lowenergystatesinaneffectivetheory)

•Thebehavioroflow-point“correlationfunctions”isstronglydictatedbymomentum

conservationinthebulk.SpecialscatteringconfigurationscanbeusedtogetWittendiagram-

likeresults.

02/22/17 27

2(1 + ✏)m ! m+m

A(wi, wi) =i2

92⇡6��(�1+�2+�3�2

2 )�(�1+�2��32 )�(�1��2+�3

2 )�(��1+�2+�32 )

p✏

m4�(�1)�(�2)�(�3)|w1 � w2|�1+�2��3 |w2 � w3|�2+�3��1 |w3 � w1|�3+�1��2+O(✏)

SGP@MIT

[arXiv:1701.00049]

m=0,MHV•ByholdingL ≡ h

5i fixedaswetakej tozero,wecanseehowthemasslesslimitofour

transformcanbededucedfromtheboundarybehaviorofourmomentumspacebulk-to-

boundarypropagator:

•Thismotivatesdefiningmasslesshighestweightstatesintermsofafrequencymellin transform,

whichobeyanorthogonalityconditionforΔ = 1 + .e (unitaryprincipalseriesrepresentation)

02/22/17 SGP@MIT 28

G�(y, z, z;w, w) ⇠ Cy2���2(z � w) +y�

|z � w|2� + ...

a�(q) ⌘Z 1

0d! !i�a(!, q)

h0|a�0(q0)a†�(q)|0i = 2(2⇡)4�(�� �0)�(2)(q � q0)

m=0,MHV•ItiscurioushowthisparticularchoiceofweightsproducesintegralsoverallenergiesthatarewelldefinedforthetreelevelMHVamplitudes(atleastinadistributionalsense):

A�1,··· ,�n(wi, wi) ⌘nY

k=1

Z 1

0d!k!

i�kk A(!kq

µk )

hiji = 2p!i!j(wi � wj)

A /Z 1

0dssi

P�k�1 = 2⇡�(

X�k)

02/22/17 SGP@MIT 29

An =nY

k=1

Z 1

0d!k!

i�kk

hiji4

h12ih23i...hn1i�4(X

k

pk)

m=0,MHV

02/22/17 SGP@MIT 30

•Thelowpointamplitudesareessentiallyfixedbythe4Dkinematics.TheMHV4ptisthefirst

wherenon-contactconfigurationsareallowed,althoughacrossratioconstraintremainsfrom

themannerinwhichfournullmomentasummingtozeroarenotlinearlyindependent.Fora

2 → 2 processwithhelicities(− −++)

h1 = � i

2�1 h1 = 1� i

2�1

h2 = � i

2�2 h2 = 1� i

2�2

h3 = 1 +i

2�3 h3 =

i

2�3

h4 = 1 +i

2�4 h4 =

i

2�4

A4 =(�1)1�i�2+i�3⇡

2

⌘5

1� ⌘

�1/3�(Im[⌘])

⇥ �(�1 + �2 � �3 � �4)4Y

i<j

zh/3�hi�hj

ij zh/3�hi�hj

ij

m=0,MHV

02/22/17 SGP@MIT 31

•Although3pt masslessscatteringrestrictsthemomentatobecollinear,onecangoto(2,2)

signaturewheretheZ arerealvariables

•AndthenuseBCFW,combinedwithmellin andinversemellin transformstocheckconsistencyof

the4pt result.

A3(�i; zi, zi) = ⇡(�1)i�1sgn(z23)sgn(z13) �(X

i

�i)�(z13)�(z12)

z�1�i�312 z1�i�1

23 z1�i�213

m=0,MHV– Howfarcanwetakethis?

02/22/17 SGP@MIT 32

•Curiousthoughts:Istheresenseinwhichwecouldrephraserecursionrelationsfor4Dmassless

scatteringamplitudes,likeBCFW,intermsof2DOPEstatements?

•Notthereyet,ex.singularbehaviorforlowpointfunctionswouldseemprohibitivetosuchan

interpretation,buthaveoptioncombiningashadow,smearedsolution:

stilluseMHVmellin amplitudesasbuildingblocks

O+i�(w, w) = �+

i�(w, w) + C+,�

Zd2z

1

(z � w)2+i�(z � w)i����i�(z, z)

O�i�(w, w) = ��

i�(w, w) + C�,�

Zd2z

1

(z � w)i�(z � w)2+i��+�i�(z, z)

Flatspaceamplitudesinahighest-weightbasis:Anaturalchoiceforstudyingimplicationsofsuperrotations?

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