A Conformal Basis for Flat Space Amplitudes · Motivation from Asymptotic Symmetries vRecent...
Transcript of A Conformal Basis for Flat Space Amplitudes · Motivation from Asymptotic Symmetries vRecent...
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 ...]
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MotivationfromAsymptoticSymmetriesv Recentstudiesoflowenergylimitsofscatteringingaugetheoriesledtounderstandingconnectionsbetweensofttheorems,asymptoticsymmetries,&memoryeffects
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SoftTheorems
MemoriesSymmetries
e𝑃"𝐽"$
v Totheextentthatonceonevertexwasknown/hypothesizedtobepresent,theotherscouldbe‘filledin’therebyreaffirmingtheconjecture.
MotivationfromAsymptoticSymmetriesv RecastsofttheoremasWardidentityfor‘largegaugetransformations’thatactnon-triviallyonboundary data.
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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
MotivationfromAsymptoticSymmetriesv RecastsofttheoremasWardidentityfor‘largegaugetransformations’thatactnon-triviallyonboundary data.
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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
MotivationfromAsymptoticSymmetriesv RecastsofttheoremasWardidentityfor‘largegaugetransformations’thatactnon-triviallyonboundary data.
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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
asFouriermodeoffieldoperator
𝑞&'() ⇔ 𝑞&+(+ & lim0→2 ⇔∫ 𝑑𝑢eiq·x = e�i!u�i!r(1�q·x)
MotivationfromAsymptoticSymmetries
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𝑞&'() ⇔ 𝑞&+(+ & lim0→2 ⇔∫ 𝑑𝑢eiq·x = e�i!u�i!r(1�q·x)
𝑖2
𝑖7
𝑖8
𝒥7
𝒥8
v Keypointofthecorrespondencesofttheorem⟺ Wardidentityisrelatingmomentumspace(softlimit)topositionspace(boundary@NullInfinity)whereASGisdefined
MotivationfromAsymptoticSymmetries
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𝑞&'() ⇔ 𝑞&+(+ & lim0→2 ⇔∫ 𝑑𝑢eiq·x = e�i!u�i!r(1�q·x)
𝑆; ateachpointconstanttimeslices
𝑖2
𝑖7
𝑖8
𝒥7
𝒥8
massiveparticlesexithere
masslessparticlesenterhere
massiveparticlesenterhere
thepointat∞ v Keypointofthecorrespondencesofttheorem⟺ Wardidentityisrelatingmomentumspace(softlimit)topositionspace(boundary@NullInfinity)whereASGisdefined
masslessparticlesexithere
MotivationfromAsymptoticSymmetries
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𝑖2
𝑖7
𝑖8
𝒥7
𝒥8
BMS1960’s
v Interestedinsetofdiffeomorphismsthatpreserveclassofasymptoticallyflatmetrics,characterizedbyradialfall-offnearnullinfinity
MotivationfromAsymptoticSymmetries
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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
MotivationfromAsymptoticSymmetries
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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
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⇠+|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]
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S(1)� = �iX
k
pkµ✏�µ⌫q�Jk�⌫pk · q
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
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Motivation:RecapvWe’vehighlightedcertainaspectsofthesofttheorem⟺ Wardidentityprogramrelevanttowhatwewilldonext:
• HowtoconnectbetweensoftlimitsandpositionspaceASG’s
• Thesuperrotation asymptotickillingvectorfields
• Thesubleading softtheoremasa2Dstresstensor
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EnhancementfromLorentztoVirasoro
Preferredhighest-weightbasis
ConstructingHighest-WeightSolutionsv Startinarbitrarydimensions𝑹A,C7A [arXiv:1705.01027] thenconsiderexampleswhered=2[arXiv:1701.00049 +...]
vWouldliketoseeifpossible(andifso,how)togobackandforthbetweenS-matrixelementsinstandardplanewaveversushighest-weight bases
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𝑚 = 0
𝑚 ≠ 0𝑤 ∈ 𝑹C
Δ ∈𝑑2 + 𝑖𝑹M2
Δ ∈𝑑2 + 𝑖𝑹
×
×
𝑒±Q'⋅S ⇔ 𝜙U±(𝑋";𝑤)
ConstructingHighest-WeightSolutions
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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
ConstructingHighest-WeightSolutions
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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:
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✓@
@X⌫
@
@X⌫�m2
◆��(X
µ; ~w) = 0
�� (⇤µ⌫X
⌫ ; ~w 0(~w)) =
����@ ~w 0
@ ~w
������/d
��(Xµ; ~w)
ConstructingHighest-WeightSolutions
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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:
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�±�(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
×
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ConstructingHighest-WeightSolutionsv Theorthogonalityconditions
v Implywecangointheoppositedirectionhighest-weight⇒ planewave
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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)
𝑤 ∈ 𝑹CΔ ∈𝑑2 + 𝑖𝑹M2
×
[arXiv:1404.5625]
ConstructingHighest-WeightSolutionsv AndweseethattheKlein-Gordoninnerproduct
evaluatedbetweensolutionsinourbasisisofadistributionalnature
SGP@RUTGERS
𝑤 ∈ 𝑹CΔ ∈𝑑2 + 𝑖𝑹M2
×
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(�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:
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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
×
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'±�(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
𝑤 ∈ 𝑹CΔ ∈𝑑2 + 𝑖𝑹
×
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✓@
@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.
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A�,±µa (Xµ; ~w) =
@aqµ(�q ·X ⌥ i✏)�
+@aq ·X
(�q ·X ⌥ i✏)�+1qµ
@
@Xµ
✓@aq ·X
(�q ·X ⌥ i✏)�
◆−const.
𝑤 ∈ 𝑹CΔ ∈𝑑2 + 𝑖𝑹
×
AmplitudeTransformsv ItisusefultopointoutthattheabovetransformscanbeapplieddirectlytotheS-matrixelements.
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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.
SGP@RUTGERS5/9/17 30
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
SGP@RUTGERS5/9/17 31
SGP@RUTGERS5/9/17 32
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
SGP@RUTGERS5/9/17 33
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
SGP@RUTGERS5/9/17 34
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(− −++)
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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
SGP@RUTGERS5/9/17 36
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.
SGP@RUTGERS5/9/17 37
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
SGP@RUTGERS5/9/17 38
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.
SGP@RUTGERS5/9/17 39
a� ⌘ a�(!x)�1
2⇡
Zd
2w
1
z � w
@wa+(!y)
Ascatteringbasismotivatedbyasymptoticsymmetries?
5/9/17 SGP@RUTGERS 40
vAsymptoticsymmetry/softphysicsinvestigationmotivatedbydesiretoconstrainS-matrixviapromotingmoresymmetriesas‘physical’
vLedtoasuperrotationiterationthathintedatLorentz→Virasoro+putativestresstensorviasubleadingsoftfactor
vFindthatthestatespreferredbythisactionindeedformabasisforsingleparticlescatterers.
vSecrethopeforOPE⟷Amplituderecursionrelationstatement?
vIntermediateobstaclestofleshingouttheputativedualseemtoatleastofferresolutionstosomeissuesthataroseinthestudyofthesoftsectoralone.
AConformalBasisforFlatSpaceAmplitudesSABRINAGONZALEZPASTERSKI