Flat space amplitudes in a highest-weight basis · 2017. 2. 22. · Asymptotically Flat Spacetimes...
Transcript of Flat space amplitudes in a highest-weight basis · 2017. 2. 22. · Asymptotically Flat Spacetimes...
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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SGP@MIT
Toseewherethisnon-trivialactionon
theboundarydatacomesfrom,letus
considerscatteringfromaspacetime
perspectivewheretheinandoutstates
comefromandexitthepastandfuture
boundariesofourspacetime
ScatteringfromaSpacetime Perspective
02/22/17 6
conformal
compactification
./
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massiveparticlesexithere
masslessparticlesexithere
masslessparticlesenterhere
massiveparticlesenterhere
thepointat∞
SGP@MIT
02/22/17 7
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massiveparticlesexithere
masslessparticlesexithere
masslessparticlesenterhere
massiveparticlesenterhere
thepointat∞
45 ateachpoint
constanttimeslices
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ScatteringfromaSpacetime Perspective
SGP@MIT
U(1)Example
02/22/17 8
constanttimeslices
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|Ψ⟩
• Withthisinmindwecanunderstandthe
U(1)iterationofthetriangle
• Gauss’slawisaconstraintequationrelating
electricfluxtocharges
9
:
SGP@MIT
U(1)Example
02/22/17 9
constanttimeslices
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21
|Ψ⟩
Ashtekar
1980’s ./
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Ψ ;<
Ψ =>?
Pushslicetonullinfinity&constraint
equationscanbeusedtorelateradiationto
changeinchargevelocities
SGP@MIT
U(1)Example
02/22/17 10
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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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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)
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Flatspaceamplitudesinahighest-weightbasis:Anaturalchoiceforstudyingimplicationsofsuperrotations?
SABRINAGONZALEZPASTERSKI