A Conformal Basis for Flat Space Amplitudes · Motivation from Asymptotic Symmetries vRecent studies of low energy limits of scattering in gauge theories led to understanding connections

AConformalBasisforFlatSpaceAmplitudesSABRINAGONZALEZPASTERSKI

Ascatteringbasismotivatedbyasymptoticsymmetries?

v Planewave⇒ Highestweightscattering[arXiv:1701.00049, arXiv:1705.01027,… S.Pasterski,S.H.Shao,A.Strominger]

v raisond’être: Preferredw.r.t.Superrotations[arXiv:1404.4091, arXiv:1406.3312, arXiv:1502.06120 ...]

5/9/17 SGP@RUTGERS 2

MotivationfromAsymptoticSymmetriesv Recentstudiesoflowenergylimitsofscatteringingaugetheoriesledtounderstandingconnectionsbetweensofttheorems,asymptoticsymmetries,&memoryeffects


SoftTheorems

MemoriesSymmetries

e𝑃"𝐽"$

v Totheextentthatonceonevertexwasknown/hypothesizedtobepresent,theotherscouldbe‘filledin’therebyreaffirmingtheconjecture.

MotivationfromAsymptoticSymmetriesv RecastsofttheoremasWardidentityfor‘largegaugetransformations’thatactnon-triviallyonboundary data.


hout|a�(q)S|ini =⇣S

(0)� + S

(1)�⌘hout|S|ini+O(!)

S(0)� =X

k

(pk · ✏�)2

pk · q S(1)� = �iX

k

pkµ✏�µ⌫q�Jk�⌫pk · q




(0)� + S


S(0)� =X

k

(pk · ✏�)2

pk · q S(1)� = �iX

k





(0)� + S


S(0)� =X

k

(pk · ✏�)2

pk · q S(1)� = �iX

k


asFouriermodeoffieldoperator

𝑞&'() ⇔ 𝑞&+(+ & lim0→2 ⇔∫ 𝑑𝑢eiq·x = e�i!u�i!r(1�q·x)

MotivationfromAsymptoticSymmetries



𝑖2

𝑖7

𝑖8

𝒥7

𝒥8

v Keypointofthecorrespondencesofttheorem⟺ Wardidentityisrelatingmomentumspace(softlimit)topositionspace(boundary@NullInfinity)whereASGisdefined




𝑆; ateachpointconstanttimeslices

𝑖2

𝑖7

𝑖8

𝒥7

𝒥8

massiveparticlesexithere

masslessparticlesenterhere

massiveparticlesenterhere

thepointat∞ v Keypointofthecorrespondencesofttheorem⟺ Wardidentityisrelatingmomentumspace(softlimit)topositionspace(boundary@NullInfinity)whereASGisdefined

masslessparticlesexithere



𝑖2

𝑖7

𝑖8

𝒥7

𝒥8

BMS1960’s

v Interestedinsetofdiffeomorphismsthatpreserveclassofasymptoticallyflatmetrics,characterizedbyradialfall-offnearnullinfinity



𝑖2

𝑖7

𝑖8

𝒥7

𝒥8

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

•RadialExpansion:

•ASGthatpreservesthisexpansion:

CoordinateConventions:

z = ei� tan✓

2�zz =

2

(1 + zz)2

f+ = f+(z, z) @zY+z = 0



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

•RadialExpansion:

•ASGthatpreservesthisexpansion:

CoordinateConventions:

z = ei� tan✓

2�zz =

2

(1 + zz)2

f+ = f+(z, z) @zY+z = 0

RadiativeData

Superrotations

MotivationfromAsymptoticSymmetriesv Fromthesofttheorem⟺ Wardidentity perspectivethesuperrotationactioncorrespondstothesubleading softgravitontheorem

v Letustakeacloserlookatthesuperrotation vectorfieldnearnullinfinity:

• NoticewehavetwocopiesoftheWittalgebrasince𝑌 isany2DCKV• Also,𝑢𝜕? prefersRindler energyeigenstates


⇠+|J+ = Y +z@z +u

2DzY

+z@u + c.c.

MotivationfromAsymptoticSymmetriesv Fromthesofttheorem⟺ Wardidentity perspectivethesuperrotationactioncorrespondstothesubleading softgravitontheorem

v Ratherthanusingthesubleading softfactortoestablisha4DWardidentityforthisasymptoticsymmetry[arXiv:1406.3312]one can massage it tolook like a2Dstresstensorinsertion [arXiv:1609.00282]


S(1)� = �iX

k


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:

Motivation:RecapvWe’vehighlightedcertainaspectsofthesofttheorem⟺ Wardidentityprogramrelevanttowhatwewilldonext:

• HowtoconnectbetweensoftlimitsandpositionspaceASG’s

• Thesuperrotation asymptotickillingvectorfields

• Thesubleading softtheoremasa2Dstresstensor


Motivation:RecapvWe’vehighlightedcertainaspectsofthesofttheorem⟺ Wardidentityprogramrelevanttowhatwewilldonext:

• HowtoconnectbetweensoftlimitsandpositionspaceASG’s

• Thesuperrotation asymptotickillingvectorfields

• Thesubleading softtheoremasa2Dstresstensor


EnhancementfromLorentztoVirasoro

Preferredhighest-weightbasis

ConstructingHighest-WeightSolutionsv Startinarbitrarydimensions𝑹A,C7A [arXiv:1705.01027] thenconsiderexampleswhered=2[arXiv:1701.00049 +...]

vWouldliketoseeifpossible(andifso,how)togobackandforthbetweenS-matrixelementsinstandardplanewaveversushighest-weight bases


𝑚 = 0

𝑚 ≠ 0𝑤 ∈ 𝑹C

Δ ∈𝑑2 + 𝑖𝑹M2

Δ ∈𝑑2 + 𝑖𝑹

×

×

𝑒±Q'⋅S ⇔ 𝜙U±(𝑋";𝑤)

ConstructingHighest-WeightSolutions


𝑚 = 0

𝑚 ≠ 0

𝑒±Q'⋅S ⇔ 𝜙U±(𝑋";𝑤) 𝑤 ∈ 𝑹C

Δ ∈𝑑2 + 𝑖𝑹M2

Δ ∈𝑑2 + 𝑖𝑹

×

×

PrincipalContinuousSeriesofSO(1,d+1)canonicalreference

directionwhenm=0



𝑚 ≠ 0𝑤 ∈ 𝑹C𝑝 = 𝑚��

��

𝑞

ds2Hd+1=

dy2 + d~z · d~zy2

p(y, ~z) =

✓1 + y2 + |~z|2

2y,~z

y,1� y2 � |~z|2

2y

◆

pµ(y0, ~z 0) = ⇤µ⌫ p

⌫

qµ(~w) =�1 + |~w|2 , 2~w , 1� |~w|2

�

qµ(~w 0) =

��@ ~w 0

@ ~w

��1/d

⇤µ⌫q

⌫(~w)

ConstructingHighest-WeightSolutionsvUsingthattheLorentzgroupSO(1,d+1)in𝑹A,C7A actsastheconformalgroupon𝑹C definethemassivescalarconformalprimarywavefunction to:

• satisfythe(d+2)-dimensionalmassiveKlein-Gordonequationofmassm:

• transformcovariantly asascalarconformalprimaryoperatorinddimensionsunderanSO(1,d+1)transformation:


✓@

@X⌫

@

@X⌫�m2

◆��(X

µ; ~w) = 0

�� (⇤µ⌫X

⌫ ; ~w 0(~w)) =

��@ ~w 0

@ ~w

��/d

��(Xµ; ~w)



𝑚 ≠ 0𝑤 ∈ 𝑹C𝑝 = 𝑚��

��

𝑞

ds2Hd+1=

dy2 + d~z · d~zy2

p(y, ~z) =

✓1 + y2 + |~z|2

2y,~z

y,1� y2 � |~z|2

2y

◆

pµ(y0, ~z 0) = ⇤µ⌫ p

⌫

qµ(~w) =�1 + |~w|2 , 2~w , 1� |~w|2

�

qµ(~w 0) =

��@ ~w 0

@ ~w

��1/d

⇤µ⌫q

⌫(~w)

G�(p; q) =1

(�p · q)� =

✓y

y2 + |~z � ~w|2

◆�

G�(p0; q0) =

��@ ~w 0

@ ~w

��/d

G�(p; q)

ConstructingHighest-WeightSolutionsv Thedesiredpropertiesaremetbytheconvolution:


�±�(X

µ; ~w) =

Z

Hd+1

[dp]G�(p; ~w) exp [±imp ·X ]

𝑃 vInterpretationasbulk-to-boundarypropagationinmomentumspace

vHaveplanewave⇒ highest-weight,whataboutreverse?

ConstructingHighest-WeightSolutionsv Ifwedefinetheshadowforascalaras

vTheactiononourscalarwavefunctionsshowslineardependencebetweenweightsΔ andd − Δ

SGP@RUTGERS

eO�(~w) ⌘ �(�)

⇡d2�(�� d

2 )

Zdd ~w 0 1

|~w � ~w 0|2(d��)O�(~w

0)

f�±�(X; ~w) = �±

d��(X; ~w)

𝑤 ∈ 𝑹CΔ ∈𝑑2 + 𝑖𝑹M2

×

5/9/17 22

ConstructingHighest-WeightSolutionsv Theorthogonalityconditions

v Implywecangointheoppositedirectionhighest-weight⇒ planewave


Z

Hd+1

[dp]G d2+i⌫(p; ~w1)G d

2+i⌫(p; ~w2) =

2⇡d+1 �(i⌫)�(�i⌫)

�(d2 + i⌫)�(d2 � i⌫)�(⌫ + ⌫)�(d)(~w1 � ~w2) + 2⇡

d2+1 �(i⌫)

�(d2 + i⌫)�(⌫ � ⌫)

1

|~w1 � ~w2|2(d2+i⌫)

Z 1

�1d⌫ µ(⌫)

Zdd ~wG d

2+i⌫(p1; ~w)G d2�i⌫(p2; ~w) = �(d+1)(p1, p2)

µ(⌫) =�(d2 + i⌫)�(d2 � i⌫)

4⇡d+1�(i⌫)�(�i⌫)

e±imp·X = 2

Z 1

0d⌫ µ(⌫)

Zdd ~w G d

2�i⌫(p; ~w) �±d2+i⌫

(Xµ; ~w)


×

[arXiv:1404.5625]

ConstructingHighest-WeightSolutionsv AndweseethattheKlein-Gordoninnerproduct

evaluatedbetweensolutionsinourbasisisofadistributionalnature

SGP@RUTGERS


×

5/9/17 24

(�1,�2) = �i

Zdd+1Xi [�1(X) @X0�⇤

2(X)� @X0�1(X)�⇤2(X)]

⇣�±

d2+i⌫1

(Xµ; ~w1),�±d2+i⌫2

(Xµ; ~w2)⌘

= ±2d+3⇡2d+2

md

�(i⌫1)�(�i⌫1)

�(d2 + i⌫1)�(d2 � i⌫1)

�(⌫1 � ⌫2) �(d)(~w1 � ~w2)

± 2d+3⇡3d2 +2

md

�(i⌫1)

�(d2 + i⌫1)�(⌫1 + ⌫2)

1

|~w1 � ~w2|2(d2+i⌫1)

MasslessHighest-WeightSolutionsvWehavebeenusingpropertiesofthebulk-to-boundarypropagatoronthemomentumspacehyperboloid(conformalcovariance,orthogonality,completeness)toconvertbetweenplanewavesandhighestweightsolutions.

v Byformingthecombination𝜔 = +;_

wecanfurtherusetheboundarybehaviorof𝐺U toexplorethemasslessanalog:

SGP@RUTGERS5/9/17 25

G�(y, ~z; ~w) �!m!0

⇡d2�(�� d

2 )

�(�)yd��(d)(~z � ~w) +

y�

|~z � ~w|2� + · · ·

�±d2+i⌫

(X; ~w) �!m!0

⇣m2

⌘� d2�i⌫ ⇡

d2�(i⌫)

�(d2 + i⌫)

Z 1

0d! !

d2+i⌫�1e±i!q(~w)·X

+⇣m2

⌘� d2+i⌫

Zdd~z

1

|~z � ~w|2( d2+i⌫)

Z 1

0d!!

d2�i⌫�1 e±i!q(~z)·X

MasslessHighest-WeightSolutionsv Thefirsttermsatisfiesthedesiredpropertiesofamasslesshighestweightsolutiononitsown.

v ItcanbeidentifiedasaMellin transformoftheenergy,inwhichthereferencedirectionisthesameasthemomentum.

SGP@RUTGERS

𝑤 ∈ 𝑹CΔ ∈𝑑2 + 𝑖𝑹

×

5/9/17 26

'±�(X

µ; ~w) ⌘Z 1

0d! !��1 e±i!q·X�✏! =

(⌥i)��(�)

(�q(~w) ·X ⌥ i✏)�

e±i!q·X�✏! =

Z 1

�1

d⌫

2⇡!� d

2�i⌫ (⌥i)d2+i⌫�(d2 + i⌫)

(�q ·X ⌥ i✏)d2+i⌫

, ! > 0

MasslessHighest-WeightSolutionsv Photon

v Graviton

SGP@RUTGERS


×

5/9/17 27

✓@

@X�

@

@X��µ⌫ � @

@X⌫

@

@Xµ

◆A�±

µa (X⇢; ~w) = 0 A�±µa (⇤⇢

⌫X⌫ ; ~w 0(~w)) =

@wb

@w0a

��@ ~w0

@ ~w

��(��1)/d

⇤ �µ A�±

�b (X⇢; ~w)

@�@⌫h�µ;a1a2

+ @�@µh�⌫;a1a2

� @µ@⌫h��;a1a2

� @⇢@⇢hµ⌫;a1a2 = 0

h�,±µ1µ2;a1a2

(⇤⇢⌫X

⌫ ; ~w 0(~w)) =@wb1

@w0a1

@wb2

@w0a2

��@ ~w0

@ ~w

��(��2)/d

⇤ �1µ1

⇤ �2µ2

h�,±�1�2;b1b2

(X⇢; ~w)

h�,±µ1µ2;a1a2

= h�,±µ2µ1;a1a2

,

h�,±µ1µ2;a1a2

= h�,±µ1µ2;a2a1

, �a1a2h�,±µ1µ2;a1a2

= 0

MasslessHighest-WeightSolutionsv Theshadowislinearlyindependent.

v DemandingconformalprofilefixesresidualgaugetransformationsbutwithingaugeequivalenceclasscanreturntoMellin representative.


A�,±µa (Xµ; ~w) =

@aqµ(�q ·X ⌥ i✏)�

+@aq ·X

(�q ·X ⌥ i✏)�+1qµ

@

@Xµ

✓@aq ·X

(�q ·X ⌥ i✏)�

◆−const.


×

AmplitudeTransformsv ItisusefultopointoutthattheabovetransformscanbeapplieddirectlytotheS-matrixelements.


eA(�i, ~wi) ⌘nY

k=1

Z

Hd+1

[dpk]G�k(pk; ~wk) A(±mipµi )

eA(�i, ~w0i(~wi)) =

nY

k=1

��@ ~w0

k

@ ~wk

��k/d

eA(�i, ~wi)Massivescalar

𝑚 = 0 eA(�i, ~wi) ⌘nY

k=1

Z 1

0d!k!

��1k A(±!kq

µk )

AmplitudeTransformsv Notethattransformingmomentumspaceamplitudesdirectly,isanalternativetopreviousapproaches[hep-th/0303006,arXiv:1609.00732]towardsflatspaceholography,whichhavelookedatafoliationofMinkowski spacetoreproduceAdS/CFT,dS/CFToneachslice.


𝑖2

𝒥7

𝒥8

𝑖7

𝑖8

AmplitudeTransformsv From[arXiv:1705.01027],summarizedhere,weknowthatwecanequivalentlyconsiderplanewaveorhighest-weightscatteringstatesontheprincipalcontinuousseries.

Ø Sothebasismotivatedbythesubleading soft-theoremisokay butisituseful?

v Lookatd=2examples:

ØMassivescalar3ptnear-extremal decay[arXiv:1701.00049]ØMHVMellinamplitudes



MassiveScalar3ptv Ford=2,weusetheprojectivecoordinate𝑤, forthecelestialsphere𝐶𝑆; attheboundaryofthelightcone fromtheorigin.𝑤 undergoesmobius transformationswhenthespacetime undergoesLorentztransformations

w =X1 + iX2

X0 +X3w ! aw + b

cw + d

v Thehighestweightstatesnowlooklikequasi-primariesunderSL(2,C)

��,m

✓⇤µ

⌫X⌫ ;

aw + b

cw + d,aw + b

cw + d

◆= |cw + d|2��,m (Xµ;w, w)

𝑋


MassiveScalar3ptv Lorentzcovarianceisbuiltintothedefinitionofthebasis.Ifnon-zero/finite4DLorentzcovariancedictates2D-correlatorform.

v Thebehavioroflow-point“correlationfunctions”isstronglydictatedbymomentumconservationinthebulk.SpecialscatteringconfigurationscanbeusedtogetWittendiagram-likeresults.

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(✏)

[arXiv:1701.00049]

MHVMellinvMomentumconservationstronglydictatestheformoflowpointMellin amplitudes.IfwethinkofcorrelationfunctionsofMellin operators,weseethecontactnatureofthetwopointfunctionalreadyfromthescalarMellin modes:

v ForMHVamplitudes(andanytheorywithscaleinvariance)onefindsthattheMellintransformedamplitudeshaveaconservation-of-weight


a�(q) ⌘Z 1

0d! !i�a(!, q) h0|a�0(q0)a†�(q)|0i = (2⇡)4�(�� 0)�(2)(w1 � w2)

A�1,··· ,�n(wi, wi) ⌘nY

k=1

Z 1

0d!k!

i�kk A(!kq

µk ) An =

nY

k=1

Z 1

0d!k!

i�kk

hiji4

h12ih23i...hn1i�4(X

k

pk)

hiji = 2p!i!j(wi � wj) A /

Z 1

0dssi

P�k�1 = 2⇡�(

X�k)

MHVMellinv Onceyoutellmethedirectionsofscattering,thefrequenciesinthemellin integralgetfixed,ie themomentumconservingdeltafunctionslocalizethefrequencyintegrals(andthensome).Fora2 → 2 processwithhelicities(− −++)


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

h� =i

2�

h+ = 1 +i

2�

h� = 1 +i

2�

h+ =i

2�

MHVMellinv On-shell+momentumconservingkinematicsrestrict2 → 2 referencedirectionstolieonacirclewithinthecelestialsphere

vMHV3pthasnosupportin(1,3)signaturebutcananalyticallycontinueto(2,2)signaturewithindependentrealcoordinates


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

MHVMellinv OnecanthenuseaslightlymodifiedBCFW,combinedwithMellin andinverseMellin transformstocheckconsistencyofthe4pt result.


A��++(�i, zi, zi) = |1� z|✓z24z14

◆2+i�1✓z13z14

◆2+i�4

⇥Z 1

�1

dU

U

Z 1

�1

d�P

2⇡

ZdzP dzP A��+(�1,�2,�P ; ezj , ezj)A++�(�3,�4,��P ; ezj , ezj)

MHVMellinv Stillmuchmoresingularthanonemighthopetohaveifthesuperrotation-inspiredputativeCFT2 dualcouldactuallybemanifested…Options?

Ø Shadow

v HaveMellin&Mellin+ShadowasequallygoodbasesforscatteringØ Give‘standard’non-contact2ptterms,4ptalsopromising


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)

MHVMellinvMorecuriously,issueofwhatlinearcombinationofbasestouseconnectsbacktosofttheoremsinitiatingthisinvestigation

v Themodecombinationthatdecouplesinthesoftlimit(iezerosoftfactor)ispreciselyalinearcombinationofMellinandMellin+shadowinthelimitwhereImΔ = 0.v Alsosinglehelicitybasisbecomesmorenatural.


a� ⌘ a�(!x)�1

2⇡

Zd

2w

1

z � w

@wa+(!y)

Ascatteringbasismotivatedbyasymptoticsymmetries?


vAsymptoticsymmetry/softphysicsinvestigationmotivatedbydesiretoconstrainS-matrixviapromotingmoresymmetriesas‘physical’

vLedtoasuperrotationiterationthathintedatLorentz→Virasoro+putativestresstensorviasubleadingsoftfactor

vFindthatthestatespreferredbythisactionindeedformabasisforsingleparticlescatterers.

vSecrethopeforOPE⟷Amplituderecursionrelationstatement?

vIntermediateobstaclestofleshingouttheputativedualseemtoatleastofferresolutionstosomeissuesthataroseinthestudyofthesoftsectoralone.

AConformalBasisforFlatSpaceAmplitudesSABRINAGONZALEZPASTERSKI

Documents

A Conformal Basis for Flat Space Amplitudes · Motivation from Asymptotic Symmetries vRecent studies of low energy limits of scattering in gauge theories led to understanding connections