Scattering amplitudes in Einstein-Yang-Mills and other

supergravity theories

Radu Roiban Pennsylvania State U.

Based on work with M. Chiodaroli, M. Gunaydin and H. Johansson

and Z. Bern, JJ Carrasco, W-M. Chen, Johansson

S-matrix–oneofthetoolstoexploringtheUVbehaviorofgravity

-WhatistheactualUVbehaviorofN=8supergravity?

NotphilosophicalquesAons,butrathertechnicalones

-IsthereaQFTofgravitythatisUV-finite?

BytryingtoanswerthemwearelikelytolearnalotaboutthestructureandproperAesofgravityandsupergravitytheories

Color/kinemaAcsduality&

Doublecopy

ScaPeringequaAons

Stringtheory

Twistorandambitwistorstring(s)

AdS/CFT&

integrability

Duality

Generalprinciples

Smatrix

Newphysics

NewmathemaAcs

Coffee

QFT

Color/kinemaAcsduality&

Doublecopy

ScaPeringequaAons

Stringtheory

Twistorandambitwistorstring(s)

AdS/CFT&

integrability

Duality

Generalprinciples

Smatrix

Newphysics

NewmathemaAcs

Coffee

QFT

PerturbaAvegravityandsupergravityhavealonghistory

-1974:‘tHoo^&Veltman:1-loopfinitenessofpuregravityusingFeynmanrules

-1985:Goroff&Sagno`:2-loopdivergenceofpuregravityusingFeynmanrules

-1991:vandeVen:confirmaAonof2-loopdivergenceinbackgroundfieldmethod

-1981:Howe&Lindstrom;Kallosh:symmetriesvscounterterms1.0

-1986:Kawai,Lewellen,Tye:(KLT)relaAonbetweentreeamplitudesofopenandclosedstringtheoriesmaximalsusyQFTrelaAons

-1993:Bern,Dunbar,Shimada:stringmethodsinperturbaAve(super)gravityresurrecAonoftheKLTrelaAons

-1995-2010:(generalized)unitarity+KLTrelaAons:gaugetheorycutsmaximalsupergravitycutsloops

-2010:Bern,Carrasco,Johansson:color/kinemaAcsdualityandadouble-copyrelaAonbetweenN=4sYMandN=8supergravity;clean&directatlooplevel

-sincethen:Manysupergravitytheories,withvariousamountsofsupersymmetryhavebeenshowntoberelatedtopairsofgaugetheories

QuesAon(s):Areall(super)gravitytheoriesrelatedtoapairofgaugetheories?Ifnotall,whyandwhichonesare(not)?

Plan

-  Color/kinemaAcsdualityandthedouble-copyconstrucAon-  Gravitysymmetriesfromsymmetriesof(Yang-Mills)^2:-  Infinitefamilies:examples

-allN=2SGswithabelianvectorfieldsandhomogeneousscalarmanifolds-someN=2SGswithnon-abelianvectorfields;someall-mulAplicityamp’s

-WhattodowhentheGodslooktheotherway -Abiasedoutlook

Textbookapproach:scaPeringamplitudesfromFeynmanrules

ScaPeringamplitudesandcolor/kinemaAcsduality

F aµ⌫ = @µA

a⌫ � @⌫A

aµ + g fabcAb

µAc⌫L = �1

4F aµ⌫F

µ⌫a +matter

AL�loop

m = iL gm�2+2LX

i2G3

Z LY

l=1

dDpl(2⇡)D

1

Si

niCiQ↵i

p2↵i

ni = ni(p↵ · p� , ✏ · p↵, . . . )

A(0)4 (1, 2, 3, 4) = g2

✓csns(p, ✏)

s+

ctnt(p, ✏)

t+

cunu(p, ✏)

u

◆cs = fA1A2BfBA3A4

�gfabc(⌘µ⌫(k1 � k2)

⇢+ 2 more)

i⌘µ⌫�ab

p2 + i✏

•  GeneralformofanL-loopamplitude

Example:4-pttreeamp:1

2 3

4

ns(p, ✏) + nt(p, ✏) + nu(p, ✏) = 0cs + ct + cu = 0

etc.

•  Color/kinemaAcsduality Bern,Carrasco,Johansson

AL�loop

m = iL gm�2+2LX

i2G3

Z LY

l=1

dDpl(2⇡)D

1

Si

niCiQ↵i

p2↵i

Thegeneralpicture/conjecture:adualitybetweencolorandkinemaAcs

− −d c

a b

1

2

3

m

d

a b

c

1

2

3

m

1

2

3

m

d

a b

c

= 0

suchthat,whenrequiredbygaugeinv.,

•  For(s)YMtheoriesinanydimensionwithcertainaddiAonalmaPer

Ci + Cj + Ck = 0

ni = ni(p↵ · p� , ✏ · p↵, . . . )

adjointrep:Bern,Carrasco,Johanssonnon-adjointrep:Chiodaroli,Jin,RR;Johansson,Ochirov

Chiodaroli,Gunaydin,Johansson,RR

ni + nj + nk = 0

•  Presentinmanytheories:YM+maPer,QCD,Coulombbranch,,Z-theory,BLG,ABJM,…aswellascertainformfactorsandcorrelaAonfcts.

�3

•  ImpliesnontrivialrelaAonsbtwamplitudes(L=0)andintegrands(L>0)

Gravityfromgaugetheory:Giventwogaugetheorieswithduality-saAsfyingm-pointamplitudes,thescaPeringamplitudesofasupergravityis(thedouble-copyconstrucAon)

ML�loop

m = iL+1

⇣2

⌘m�2+2L X

i2G3

Z LY

l=1

dDpl(2⇡)D

1

Si

niniQ↵i

p2↵i

Expectedtoholdtoalllooporders;parAalargumentsavailable-  Explicitlytestedinvarioussusyandnon-susytheoriesw/&w/omaPer-  Atvariouslooporders(1through4loopsinN=4andN=8SG)-  Capturessubtlefieldtheoryeffects,suchasanomalies-  ExtendedtoclassicalsoluAonsofeqsofmoAon

Bern,Carrasco,Johansson

Spectrumofthe(super)gravitytheory:Tensorproductofspectraofthetwogaugetheoriessuchthatthefieldsofgaugetheoriesformasingletunderthegaugegroup

N<4:Chiodaroli,Jin,RR;Johansson,OchirovChiodaroli,Gunaydin,Johansson,RR

Luna,Monteiro,Nicholson,O'Connell,White

Whenduality-saAsfyingrepsexistbutarenotavailable,doublecopyissAllsurprisinglysimpleandstructured Bern,Carrasco,Chen,Johansson,RR

Gravitysymmetriesfromsymmetriesof(Yang-Mills)^2:

•  NonabeliangaugesymmetryofYMtheoriesisgone•  Gaugetheoryglobalsymmetriessurviveandcaneitherremainglobal(e.g.R-symmetry)orbegaugedEnhancement::

generaldiscussion:Anastasiou,Borsten,DuffetalR1 ⇥R2 ! R {Ti

j , T i0j0 , Qi

↵Qj0↵, Qi0

↵Qj0↵}

Gravitysymmetriesfromsymmetriesof(Yang-Mills)^2:

•  NonabeliangaugesymmetryofYMtheoriesisgone•  Gaugetheoryglobalsymmetriessurviveandcaneitherremainglobal(e.g.R-symmetry)orbegaugedEnhancement::•  Emergentglobalsymmetries–e.g.dualitysymmetries

R1 ⇥R2 ! R {Tij , T i0

j0 , Qi↵Qj0

↵, Qi0↵Qj0

↵}

U(1) : q = hL � hR

Gravitysymmetriesfromsymmetriesof(Yang-Mills)^2:

•  NonabeliangaugesymmetryofYMtheoriesisgone•  Gaugetheoryglobalsymmetriessurviveandcaneitherremainglobal(e.g.R-symmetry)orbegaugedEnhancement:•  Emergentglobalsymmetries–e.g.dualitysymmetries•  But….Whatmakesatheoryofspin-2parAclesatheorygravityis…diffeomorphisminvariance

linearizeddiffeomorphism

gravitondouble-copypolarizaAonvector

LinearizeddiffeomorphismsrelatedtoYMlinearizedgaugetransf’s

Alldouble-copytheoriesarediffeomorphism-invariant:

M =X

�

n�(✏1(p1), ✏2, . . . )n�(✏1(p1), ✏02, . . . )

D�

R1 ⇥R2 ! R {Tij , T i0

j0 , Qi↵Qj0

↵, Qi0↵Qj0

↵}

U(1) : q = hL � hR

✏µ⌫(p) 7! ✏(µ(p)✏0⌫)(p)

7! p(µ✏0⌫)(p) + p(⌫✏µ)(p)

Gravitysymmetriesfromsymmetriesof(Yang-Mills)^2:

•  But….Whatmakesatheoryofspin-2parAclesatheorygravityis…diffeomorphisminvariance

Alldouble-copytheoriesarediffeomorphism-invariant:

LinearizeddiffeomorphismsrelatedtoYMlinearizedgaugetransf’s

0 =X

�

n�(p1, ✏2, . . . )c�D�

�!

-structureof-JacobiidenAAesforc�

n�

n�, n�&c� havethesameproperAes =) �M = 0

M =X

�

n�(✏1(p1), ✏2, . . . )n�(✏1(p1), ✏02, . . . )

D�

A =X

�

n�(✏1(p1), ✏2, . . . )c�D�

�M =X

�

n�(p1, ✏2, . . . )n�(✏1(p1), ✏02, . . . )

D�+ (n $ n)

linearizeddiffeomorphism

gravitondouble-copypolarizaAonvector✏µ⌫(p) 7! ✏(µ(p)✏0⌫)(p)

7! p(µ✏0⌫)(p) + p(⌫✏µ)(p)

ThereverseimplicaAonisfarfromobvious;consider…

ThereverseimplicaAonisfarfromobvious;consider“proofbyexhausAon”1)  PickaclassofsupergraviAes2)  ConstructtheirscaPeringamplitudesandcompare3)  Repeat

Asuitableclass:N=2supergraviAes

-Nontrivial:maPercontentdoesnotuniquelydeterminethetheory

-Their5DoriginuniquelyidenAfiesthembytheirthree-pointamplitudes-Lagrangiansareknownexplicitcomparisoncanbecarriedout�!

-Someofthemhaveglobalsymmetries;canbegauged-Maxwell-EinsteinvsYang-Mills-Einstein-DifferentconstrucAonsareusefulfordifferentpurposes

-Thedouble-copyalwaysgivesSG,butwhichone?

GeneralN=25DMaxwell-EinsteinLagrangian(suppressfermions):

e�1L = �R

2� 1

4aIJ

F I

µ⌫

F Jµ⌫ � 1

2gxy

@µ

'x@µ'y +e�1

6p6C

IJK

✏µ⌫⇢��F I

µ⌫

F J

⇢�

AK

�

'funcAonsof

graviphoton+maPer Gunaydin,Sierra,Townsend

V(⇠) =⇣23

⌘3/2CIJK⇠I⇠J⇠K = 1•  Scalarmanifold:

Canonical basis:

C000 = 1 , C00i = 0

C0ij =12�ij

Cijk = arbitrary

ConstrucAonsofN=2SGs:•  (N=2)=(N=2)x(N=0)or(N=2)=(N=1)x(N=1)

•  EachoffersseveralopAons;focus(fornow)onaparAcularcaseoftheformer

-Gaugetheory1:N=2sYMwithasingle½-hypermulApletinpseudo-realrepR-Obeyscolor-kinemaAcsduality

-Gaugetheory2:YMw/(q+2)adj.scalarsandrfermionsinpseudo-realrepR

-Color-kinemaAcsdualityrequires: {�a,�b} = 2�ab

-Whythese?-SimplegaugetheorymaPercontent-ContainsthetwopossiblerealizaAonofvectorfields-PrePylargemanifestglobalsymmetry–

-MorereasonsrelatedtothespectrumSO(q + 2)

L = � 14F

aµ⌫F

aµ⌫ + 12 (Dµ�a)a(Dµ�a)a + i

2�↵Dµ�µ�↵

+ g2�

aa�a �↵ �

↵�5T a�� � g2

4 f abef cde�aa�bb�ac�bd

Chiodaroli,Gunaydin,Johansson,RR

ConstrucAonsofN=2SGs:

•  EachoffersseveralopAons;focus(fornow)onaparAcularcaseoftheformer

-Gaugetheory1:N=2sYMwithasingle½-hypermulApletinpseudo-realrepR

-Obeyscolor-kinemaAcsduality-Gaugetheory2:YMw/(q+2)adj.scalarsandrfermionsinpseudoreal-realrepR

-Color-kinemaAcsdualityrequires: {�a,�b} = 2�ab

-The4Dbosonicspectrum: A�1� = �⌦A� , h� = A� ⌦A� ,

A0� = �⌦A� , iz0 = A+ ⌦A� ,

Aa� = A� ⌦ �a , iza = �⌦ �a ,

A↵� = �� ⌦ (U��)↵ , iz↵ = �+ ⌦ (U��)↵ ,

-RepresentaAonsofthegaugegroupforbidotherproducts

�Aa

+, a+,�

a�G�

�Aa

�, a�, �

a�G�

��+,'1,'2,��

�R

L = � 14F

aµ⌫F

aµ⌫ + 12 (Dµ�a)a(Dµ�a)a + i

2�↵Dµ�µ�↵

+ g2�

aa�a �↵ �

↵�5T a�� � g2

4 f abef cde�aa�bb�ac�bd

•  (N=2)=(N=2)x(N=0)or(N=2)=(N=1)x(N=1)

Chiodaroli,Gunaydin,Johansson,RR

E.g.of3-pointamplitudesandproperAesofhigher-pointamplitudes:

•  N=2factor:

•  N=0factor:

•  Thedouble-copy:

A(0)3

�1Aa

�, 2��, 3�+

�= ig

h12i2

h23i TaV �1

A(0)3

�1�aa, 2��, 3��

�=

igp2h23i(�aC�1)(T aV �1)

GeneralproperAesofamplitudesofthisdouble-copy:

M(0)3

�1Aa

�, 2A↵�, 3z�

�=�

⇣

2p2

⌘h12i2(U t�aCU)↵�

limpn!0

M(0)n

�. . . , nz�

�= 0 = lim

pn!0M(0)

n

�. . . , nz�

�

WhichsupergraviAesarethese?

Scalarstakevaluesina(locally-)homogeneousspace!

GivenbytheprepotenAal(notincanonicalform)

V(⇠) =p2�⇠0(⇠1)2 � ⇠0(⇠i)2

�+ ⇠1(⇠↵)2 + �i

↵�⇠i⇠↵⇠�

-3-pointamplitudesarereproducescorrectly(nontrivialchoiceofU)

-⇣(U tC�aU) =�1 , �i�i

�

deWit,vanProeyen

Throughdouble-copycancomputeloopamplitudes;no1-loopfinitetheories

Chiodaroli,Gunaydin,Johansson,RR

Canbypassuseofmanifestc/k-saAsfyingreps.Bern,Carrasco,Chen,,Johansson,RR

ThecurrentperspecAveonhowtogaugeaglobal(non-R)symmetryofSG

•  Minimalcouplingswithspin-0andspin-1/2fields

std3-pointS-matrixelements(sameasinagaugetheory)

-Requirethatthiscanbefactorized…

…andthatareLorentz-invariant

fromstandarddim.-4operator(4dcounAng)

fromdim.-3operator(4dcounAng)

UniquelocalopAon:trilinearscalaroperator

2.Minimalcouplingsmusthaveadouble-copystructure

fABCfabc�Aa�Bb�Cc

1.Thesymm.tobegauged--visibleinonegaugetheories:AAµ ⇠ Aµ ⌦ �A

Homogeneoustheories: A�1� =�⌦A� , A0

�=�⌦A�

Aa� = A� ⌦ �a , A↵� = �� ⌦ (U��)↵

-Adjointrep.ofthegaugegroupmustfitinsidethevectorrep.ofso(q+2)

vector of so(q + 2)

-Many“spectator”vectorfields+moreeconomical:truncateawayrepRfermionsgenericJordanfamily!

V(⇠) =p2�⇠0(⇠1)2 � ⇠0(⇠i)2

�

-Gaugetheory1:N=2super-Yang-Millstheory-Obeyscolor-kinemaAcsduality

-Gaugetheory2:Yang-MillstheorywithadjointscalarsnV

-Color-kinemaAcsdualitydemandsthatobeystheJacobiidenAtyFabc

L = � 14F

aµ⌫F

aµ⌫ + 12 (Dµ�a)a(Dµ�a)a

+ g�3 Fabcfabc�

aa�bb�cc � g2

4 fabefcde�aa�bb�ac�bd

Lagrangian:

GaugedgenericJordanfamilySGs: V(⇠) =p2�⇠0(⇠1)2 � ⇠0(⇠i)2

�

-Gaugetheory1:N=2super-Yang-Millstheory-Obeyscolor-kinemaAcsduality

-Gaugetheory2:Yang-MillstheorywithadjointscalarsnV

-Color-kinemaAcsdualitydemandsthatobeystheJacobiidenAtyFabc

+CovariantderivaAves+extratermsM4 =SO(nv, 2)

SO(nv)⇥ SO(2)⇥ SU(1, 1)

U(1)

•  ComparisonwithLagrangian:-precisefieldmap-preciseparametermap-3-and4-pointamplitudesexplicitlychecked

�Fabc = 2igsgfabc 6= 0 i↵ a, b, c = 2, . . . , nV

L = � 14F

aµ⌫F

aµ⌫ + 12 (Dµ�a)a(Dµ�a)a

+ g�3 Fabcfabc�

aa�bb�cc � g2

4 fabefcde�aa�bb�ac�bd

GaugedgenericJordanfamilySGs: V(⇠) =p2�⇠0(⇠1)2 � ⇠0(⇠i)2

�

-Gaugetheory1:N=2super-Yang-Millstheory-Obeyscolor-kinemaAcsduality

-Gaugetheory2:Yang-MillstheorywithadjointscalarsnV

-Color-kinemaAcsdualitydemandsthatobeystheJacobiidenAtyFabc

+CovariantderivaAves+extraterms

With3-scalarcoupling,globalsym.oftheN=0factorbecomeslocalSGsymmetry

M4 =SO(nv, 2)

SO(nv)⇥ SO(2)⇥ SU(1, 1)

U(1)

Similartree-levelconstrucAonfromscaPeringequaAonPOV Cachazo,He,Yuan

•  ComparisonwithLagrangian:-precisefieldmap-preciseparametermap-3-and4-pointamplitudesexplicitlychecked

�Fabc = 2igsgfabc 6= 0 i↵ a, b, c = 2, . . . , nV

Throughdouble-copycancomputeloopamplitudes;sameUVprop’sasMESGTs

L = � 14F

aµ⌫F

aµ⌫ + 12 (Dµ�a)a(Dµ�a)a

+ g�3 Fabcfabc�

aa�bb�cc � g2

4 fabefcde�aa�bb�ac�bd

Explicittree-levelEYMamplitudesfromdouble-copy

•  Onegraviton,(m-1)gluons,single-trace

•  Twogravitons,(m-2)gluons,single-trace

•  Threegravitons,(m-3)gluons,single-trace

•  Four&fivegravitons–nottoobadeither•  Simplerexpressionsthanfromothertechniques Nandan,Ple|a,SchloPerer,Wen

ApanoramicviewofN=2xN=0theories Chiodaroli,Gunaydin,Johansson,RR

Straigh}orwardextensionto N=4SGandcertain N=1and N=0(S)Gs

or--Double-copyw/oexplicitcolor/kinemaAcs

1.  Startwithapairofgaugetheoryamplitudes(thisisthedata)2.  Constructnaïvedouble-copy;proceedtocorrectit3.  Findcutsofnaïvedouble-copy4.  FindcutsofthedesiredSGamplitudeusingKLT5.  Comparewith3.andshakethedifferencetoagraph-likepresentaAon6.  IdenAfymissingterms(contactterms);stepthroughallcuts

InprogressBern,Chen,Carrasco,Johansson,RR

Amiracleoccurs:contacttermsaresimpleandarebuiltfromtheviolaAonofthekinemaAcJacobirelaAonsbytheiniAalamplitudes:skipsteps2/3-5

BCJdiscrepancyfuncAon: − −d c

a b

1

2

3

m

d

a b

c

1

2

3

m

1

2

3

m

d

a b

c

= 0= J

n1 + n2 + n3 = Jc1 + c2 + c3 = 0

WhattodowhentheGodslooktheotherway

Useonlygaugetheorydata

Example:onelooporanygeneralizedcutmadeupoftwo4-pointamplitudefactors

C4⇥4YM =

X

i1,i2

ni1i2ci1i2

d(1)i1d(2)i2

�i1i2 ⌘ ni1i2 � nBCJi1,i2 = d(1)i1

k(2)(i2) + d(2)i2k(1)(i1)

X

i1,i2

�i1i2ci1i2

d(1)i1d(2)i2

= 0 =X

i1,i2

�i1i2nBCJi1i2

d(1)i1d(2)i2

J•,i2 ⌘X

i1

ni1i2 = di2X

i1

k(1)(i1) Ji1,• ⌘X

i2

ni1i2 = di1X

i2

k(2)(i2)

l1

l21

2 3

4

ProperAesofgaugeparameters:

Gaugetheorycut: TransformaAonrelaAngittoc/k-saAsfyingone:

C4⇥4SG =

X

i1,i2

nBCJi1i2 n

0BCJi1i2

d(1)i1d(2)i2

=ni1i2n

0i1i2

d(1)i1d(2)i2

� 1

d(1)1 d(2)1

�J•,1J

01,• + J1,•J

0•,1

�

BCJdiscrepancyfuncAons:

Supergravitycut(thereareseveralequivalentvariants):

•  Seriesofformulaeforalltypesofgeneralizedcuts(derivaAon/proofinprogress)

•  Worksforasymmetricdouble-copies

•  StarAngpointcanbeanyrepresentaAonofamplitudes,includingFeynmandiagrams

•  Novelwaytofindgravitytree-levelamplitudesadaptedtocubicgraphs

•  StaytunedforapplicaAonstoN=8SGat5loops

Abiasedoutlook

-R-symmetrygaugingandapplicaAonstogauge/stringdualityDoN=4sYMcorrelaAonfctshaveadouble-copystructure?PerturbaAontheoryincurvedspace?

-IdenAfydouble-copyconstrucAonsofother,moregeneral,familiesofSGs

-QuesAonremains:DoallN=2SGshaveadouble-copyconstrucAon?Ifnot,whynot?

-SystemaAcstudyofamplitudesinN=2theoriesatoneloopandbeyondArethereanyfinitetheories?(hypermulApletsnecessary)

-Discusseddouble-copyconstrucAonsoflargeclassesofME&YMESGTsneedsmassivemaPerandmaPerinnon-adjointrep