Scattering in Conformal Gravity
Henrik Johansson Uppsala U. & Nordita July 13, 2017 Amplitudes 2017 Edinburgh
Based on work with: Josh Nohle [1707.02965]; Marco Chiodaroli, Murat Gunaydin, Radu Roiban [1408.0764, 1511.01740, 1512.09130, 1701.02519]; Gregor Kälin, Gustav Mogull [1706.09381]. Perturbative Einstein gravity (textbook)
de Donder = gauge
=
After symmetrization 100 terms ! higher order vertices…
103 terms complicated diagrams:
104 terms 107 terms 1021 terms On-shell simplifications
Graviton plane wave:
Yang-Mills polarization On-shell 3-graviton vertex:
=
Yang-Mills vertex
Gravity scattering amplitude: Yang-Mills amplitude st M GR(1, 2, 3, 4) = AYM(1, 2, 3, 4) AYM(1, 2, 3, 4) tree u tree ⌦ tree
Gravity processes = “squares” of gauge theory ones Generic gravities are double copies
Amplitudes in familiar theories are secretly related, for example:
(dilaton-axion)
(Maxwell-Einstein)
(Yang-Mills-Einstein) and many more…
(gauge sym) (gauge sym) = di↵eo sym ⌦ Generality of double copy Gravity processes = product of gauge theory ones - entire S-matrix Gauge theory Gravity Bern, Carrasco, HJ (’10) trees: → “squared” numerators loops: → Recent generalizations: à Theories that are not truncations of N=8 SG HJ, Ochirov; Chiodaroli, Gunaydin, Roiban à Theories with fundamental matter HJ, Ochirov; Chiodaroli, Gunaydin, Roiban à Spontaneously broken theories Chiodaroli, Gunaydin, HJ, Roiban à Form factors Boels, Kniehl, Tarasov, Yang à talks by Yang & Boels à Gravity off-shell symmetries from YM Anastasiou, Borsten, Duff, Hughes, Nagy à Classical (black hole) solutions Luna, Monteiro, O’Connell, White; Ridgway, Wise; Goldberger,... à Amplitudes in curved background Adamo, Casali, Mason, Nekovar à New double copies for string theory Mafra, Schlotterer, Stieberger, Taylor, Broedel, Carrasco... à CHY scattering eqs, twistor strings Cachazo, He, Yuan, Skinner, Mason, Geyer, Adamo, Monteiro,.. à Conformal gravity HJ, Nohle see talks by Goldberger; Mason; Schlotterer Motivation: (super)gravity UV behavior Old results on UV properties: 2 3 Ferrara, Zumino, Deser, Kay, Stelle, Howe, Lindström, SUSY forbids 1,2 loop div. R , R Green, Schwarz, Brink, Marcus, Sagnotti Pure gravity 1-loop finite, 2-loop divergent Goroff & Sagnotti, van de Ven With matter: 1-loop divergent ‘t Hooft & Veltman; (van Nieuwenhuizen; Fischler..) New results on UV properties: N=8 SG and N=4 SG 3-loop finite! Bern, Carrasco, Dixon, HJ, Kosower, Roiban; Bern, Davies, Dennen, Huang Beisert, Elvang, Freedman, Kiermaier, Morales, Stieberger; Björnsson, Green, Bossard, Howe, N=8 SG: no divergence before 7 loops Stelle, Vanhove, Kallosh, Ramond, Lindström, Berkovits, Grisaru, Siegel, Russo, and more…. First N=4 SG divergence at 4 loops Bern, Davies, Dennen, Smirnov, Smirnov (unclear interpretation, U(1) anomaly?) Evanescent effects: Einstein gravity Bern, Cheung, Chi, Davies, Dixon, Nohle =1, 2, 3SG N
need calculations: à talk by Roiban
5-loop finite ??? N=8 SG Outline
On-shell diffeomorphisms from gauge symmetry Color-kinematics duality & double copy examples: SQCD; magical SG; YM-Einstein. Generalization to conformal (super-)gravity New dimension-six gauge theory Deformations and extensions à more gravities Conclusion Gauge & diffeomorphism symmetry
kinematic numerators color factors cubic diagram form: propagators
Consider a gauge transformation
Invariance of requires that are linearly dependent
[Jacobi id. or Lie algebra] thus the combination vanishes. ‘‘Double copy always gravitates’’
Assume: gauge freedom can be exploited to find color-dual numerators
Then the double copy à Gravity describes a spin-2 theory invariant under (linear) diffeos
= 0 (gauge sym) (gauge sym) = di↵eo sym Chiodaroli, Gunaydin, ⌦ HJ, Roiban Color-kinematics duality
10 Color-kinematics duality Gauge theories are controlled by a hidden kinematic Lie algebra à Amplitude represented by cubic graphs: numerators color factors
propagators
Color & kinematic commutation numerators satisfy= = identity same− relations: − (a) T aT b T b(b)T a = f abcT c Figure 2: Pictorialcolor form of the basic color and kinematic Lie-algebraic relations: (a) the Jacobi relations for fields in the adjoint representation,ci and (b) the commutation relation for fields in a Jacobi generic complex representation. = = − identity − Due tokinematics the Lie algebra of then gauge symmetry, color factors obey simple linear relations i f dacf cbe f dbc(a)f cae = f abcf dce (b) arising from the Jacobi identities and commutation relations, Bern, Carrasco,f˜dˆaˆcˆf˜cˆˆbeˆ HJf˜d ˆˆbcˆf˜cˆaˆeˆ =Figuref˜aˆˆbcˆnf˜dˆcˆieˆ 2: Pictorialn formj of the= basicn colork and kinematic Lie-algebraic relations: (a) the Jacobi − c c = c , (2.3) (taˆ) kˆ (tˆb) ȷˆ (tˆb) kˆ (taˆ) ȷˆ = f˜aˆrelationsˆbcˆ (tcˆ) ȷˆ for ⇒ fieldsi in− thej adjointk representation, and (b) the commutation relation for fields in a ˆı kˆ − ˆı kˆ genericˆı complex! representation. and this is depicted diagrammatically in figure 2. The identity c c = c is understood to i − j k hold for triplets of diagrams (i, j, k)thatdiffer only by the subdiagrams drawn in figure 2, but otherwise have common graph structure. Due to the Lie algebra of the gauge symmetry, color factors obey simple linear relations
The linear relations among the color factorsarisingci imply from that the the Jacobi corresponding identities andkinematic commutation relations, coefficients ni/Di are in general not unique, as should be expected from the underlyinggauge ˆ ˆ ˆˆ ˆ ˆ dependence of individual (Feynman) graphs. f˜daˆcˆf˜cˆbeˆ f˜dbcˆf˜cˆaˆeˆ = f˜aˆbcˆf˜dcˆeˆ ˆ ˆ ȷˆ ˆ−ˆ ȷˆ ˆ ȷˆ ci cj = ck , (2.3) It was observed by Bern, Carrasco and one of the current autho(taˆrs) k (BCJ)(tb) [3],(t thatb) k ( withintaˆ) = f˜aˆbcˆ (tcˆ) ⇒ − ˆı kˆ − ˆı kˆ ˆı ! the (gauge) freedom of redefining the numerators there exist particularly nice choices, such and this is depicted diagrammatically in figure 2. The identity ci cj = ck is understood to that the resulting numerator factors ni obey the same general algebraic identities as the − hold for triplets of diagrams (i, j, k)thatdiffer only by the subdiagrams drawn in figure 2, color factors ci. That is, there is a numerator relation for every color Jacobi of commutation relation (2.45) and a numerator sign flip for everybut otherwise color factor have sign common flip (2.2): graph structure.
The linear relations among the color factors ci imply that the corresponding kinematic ni nj = nk ci cj = ck , (2.4a) − ⇔coefficients− ni/Di are in general not unique, as should be expected from the underlyinggauge n n c c . (2.4b) i →− i ⇔dependencei →− i of individual (Feynman) graphs. Amplitudes that satisfy these relations are saidIt to was exhibit observed color/kine by Bern,matics Carrasco duality. and An one of the current authors (BCJ) [3], that within important point is that a numerator factor ni theentering (gauge) different freedom kinematic of redefining relations the may numerators not there exist particularly nice choices, such take the same functional form in all such relations;that the rather, resulting momen numeratortum conservation factors ni andobey the same general algebraic identities as the changes of integration variables must be usedcolor to allign factors thec momentumi. That is,assignment there is a numerator between relation for every color Jacobi of commutation the three graphs participating in the relation.relation (2.45) and a numerator sign flip for every color factor sign flip (2.2): The relations in eq. (2.4) define a kinematic algebra of numerators, which is suggestive of an underlying kinematic Lie algebra. While not much is known about thnisi Lienj algebra,= nk ci cj = ck , (2.4a) − ⇔ − which should be infinite-dimensional due to the functional nature of the kinematic Jacobi ni ni ci ci . (2.4b) relations, in the special case of the self-dual sector of YM theory the kinematic algebra→− was ⇔ →− shown to be isomorphic to that of the area-preservingAmplitudes diffeomorph that satisfyisms [13]. these relations are said to exhibit color/kinematics duality. An
important point is that a numerator factor ni entering different kinematic relations may not take7 the same functional form in all such relations; rather, momentum conservation and changes of integration variables must be used to allign the momentum assignment between the three graphs participating in the relation. The relations in eq. (2.4) define a kinematic algebra of numerators, which is suggestive of an underlying kinematic Lie algebra. While not much is known about this Lie algebra, which should be infinite-dimensional due to the functional nature of the kinematic Jacobi relations, in the special case of the self-dual sector of YM theory the kinematic algebra was shown to be isomorphic to that of the area-preserving diffeomorphisms [13].
7 Gauge-invariant relations (pure adjoint theories)
A(1, 2,...,n 1,n)=A(n, 1, 2,...,n 1) cyclicity à (n-1)! basis n 1 A(1, 2,...,i,n,i+1,...,n 1) = 0 U(1) decoupling i=1 X (n-2)! basis Kleiss-Kuijf relations (‘89)
BCJ relations (‘08) (n-3)! basis
BCJ rels. proven via string theory by Bjerrum-Bohr, Damgaard, Vanhove; Stieberger (’09) and field theory proofs through BCFW: Feng, Huang, Jia; Chen, Du, Feng (’10 -’11) Relations used in string calcs: Mafra, Stieberger, Schlotterer, et al. (’11 -’17) Loop-level generalizations: Tourkine, Vanhove; Hohenegger, Stieberger; Chiodaroli et al. à talk by Tourkine; Schlotterer Gravity is a double copy of YM Gravity amplitudes obtained by replacing color with kinematics
double copy Bern, Carrasco, HJ
The two numerators can differ by a generalized gauge transformation à only one copy needs to satisfy the kinematic algebra
The two numerators can differ by the external/internal states à graviton, dilaton, axion (B-tensor), matter amplitudes
The two numerators can belong to different theories Equivalent to à give a host of different gravitational theories KLT at tree level for adj. rep. à talk by Schlotterer Which gauge theories obey C-K duality Bern, Carrasco, HJ (’08) Pure N=0,1,2,4 super-Yang-Mills (any dimension) Bjerrum-Bohr, Damgaard, Vanhove; Stieberger; Feng et al. Self-dual Yang-Mills theory O’Connell, Monteiro (’11) Mafra, Schlotterer, etc (’08-’11) Heterotic string theory Stieberger, Taylor (’14) Yang-Mills + F 3 theory Broedel, Dixon (’12) QCD, super-QCD, higher-dim QCD HJ, Ochirov (’15); Kälin, Mogull (‘17) Chiodaroli, Gunaydin, Generic matter coupled to N = 0,1,2,4 super-Yang-Mills Roiban; HJ, Ochirov (’14) Spontaneously broken N = 0,2,4 SYM Chiodaroli, Gunaydin, HJ, Roiban (’15) Yang-Mills + scalar ϕ3 theory Chiodaroli, Gunaydin, HJ, Roiban (’14) Bi-adjoint scalar 3 theory Bern, de Freitas, Wong (’99), Bern, Dennen, Huang; ϕ Du, Feng, Fu; Bjerrum-Bohr, Damgaard, Monteiro, O’Connell NLSM/Chiral Lagrangian Chen, Du (’13) Bargheer, He, McLoughlin; D=3 BLG theory (Chern-Simons-matter) Huang, HJ, Lee (’12 -’13) (Non-)Abelian Z-theory Carrasco, Mafra, Schlotterer (‘16) Dim-6 gauge theories: HJ, Nohle (‘17) Which gravity theories are double copies
Pure N=4,5,6,8 supergravity (2 < D < 11) KLT (‘86), Bern, Carrasco, HJ (’08 -’10) Einstein gravity and pure N=1,2,3 supergravity HJ, Ochirov (’14) D=6 pure N =(1,1) and N =(2,0) supergravity HJ, Kälin, Mogull (’17) Self-dual gravity O’Connell, Monteiro (’11) Closed string theories Mafra, Schlotterer, Stieberger (’11); Stieberger, Taylor (’14) Einstein + R 3 theory Broedel, Dixon (’12) Abelian matter coupled to supergravity Carrasco, Chiodaroli, Gunaydin, Roiban (‘12) HJ, Ochirov (’14 - ’15) Magical sugra, homogeneous sugra Chiodaroli, Gunaydin, HJ, Roiban (’15) (S)YM coupled to (super)gravity Chiodaroli, Gunaydin, HJ, Roiban (’14) Spontaneously broken YM-Einstein gravity Chiodaroli, Gunaydin, HJ, Roiban (’15) 2 Bargheer, He, McLoughlin; D=3 supergravity (BLG Chern-Simons-matter theory) Huang, HJ, Lee (’12 -’13) Born-Infeld, DBI, Galileon theories (CHY form) Cachazo, He, Yuan (’14) N =0,1,2,4 conformal supergravity HJ, Nohle (‘17) 4 ←ℓ2ℓ1→ 1 n3 n = ⌘ 3 2 ✓ ◆ 4s 5`s `s +10`2 2`2 +30µ 12+34 i✏(µ ,µ )( ) 1 2 1 2 13 30 1 2 12 34 s s u t t u 2 s s 2 2 15u `1`3 +(`1 `1)(`3 + `3 ) +2 u (5`1 + `2 10`1 +2`2) 13+24 + 2 s 8u2 30`u`u +60u(` ` + µ ) 60u " 1 3 1· 3 13 # + t u 14+23 $ 60t2 u 2i(13 24) ¯ ¯ ¯ ⇥`3 ✏(1, 2, 3, `1)+u ✏(1, 2, `1, `2) u2 ⇤ 2i(14 23) t u , ⇥ $ ⇤ t2 ℓ1→ 1 ℓ2↙ ⇥ 4 2 ⇤ n4 n = ⌘ 3 Super-QCD at✓ two loops◆ 3 (`t + `u)`s `s(`t + `u) 4s(p ` +2p ` + ` ` ) 12+34 1 1 2 1 2 2 4· 1 4· 2 1· 2 6s HJ, G. Kälin, G. Mogull s u 13+24 ⇥ ⇤ 6u $ 14+23 à talk by Mogull s t 6t (1) ⇥ (2) (3) (4)$ (5) ⇤ 12 34 13 24 14 23 +4i ✏(3, 4, `¯ , `¯ ) + ✏(2, 4, `¯ , `¯ ) ✏(1, 4, `¯ , `¯ ) - two-loop SQCD amplitude ⇥ 1 2 s 1 2 u 1 ⇤ 2 t - color-kinematics manifest +2i ✏(µ1,µ2)(12 34 + 13 24 14 + 23), (8) (9) (10) (6) (7)ℓ1→ 1 - planar + non-planar ℓ2↙ 1 4 2 2 n5 n = (4`1 `2 +3`2 2p4 `1 p4 `2) - Nf massless quarks ⌘ 3 15 · · · ✓ ◆ - integrand valid in D ≤ 6 (12 + 13 + 14 + 23 + 24 + 34), (11) (12) (13) (14) ⇥ (15) Identical formulas apply for theℓ1 MHV-sector→ 1 supergravity numerators, and there ℓ2↙ 1 4 2 e.g. simple SQCDare two numerator: more: n6 n = p4 (`1 `2) 2`1 `2 (12 + 13 + 14 + 23 + 24 + 34), ⌘ 3 15 · · ℓ1→ 1 ✓ ◆ Identical formulas apply for the MHV-sectorℓ2↙ supergravityℓ1→ numerators,1 and there [ =4 SG] 4 2 N (16) ℓ2(17)↙ (18) (19) N =(5.27)4 2 (20) are twoh more:alf-maximal supergravity numerator:3 n7 n = ✓ ◆ 3 2 ⌘ 2 2 2 ℓ1→ 1 ℓ1→ 1 ✓ ℓ1→ 1 ◆ ℓ1→ 1 ℓ1→ 1 ℓ2↙ ℓ2↙ ℓ2↙ ℓ2↙ ℓ2↙ [ =4 SG] 4 2 4 2 4t u 2 s s 4 t u2 4 2 12+34 N N =(5.27)n +(Ds +6) 3 n(`1 + `1 )`2 + `n1(`2 + `2 ) + 4ns p4 `1 +2p4 ,`2)+`1 `2 12s 3 3 0 3 3 3 1 · · · 13+24 ✓ ◆ ✓ ◆ +(21) ✓ (22)◆ ✓ (23) s◆ u (24)✓ ◆ 2 2 2 2 12u ℓ1→ 1 ℓ1→ℓ 11 ⇥ℓ ℓ 1→1 1 2 ℓ1→ 1 $ ⇤ 1↗ 2 @1↗ 2 A 14+23 ℓ2↙ [ =4 SG]ℓ2↙ ℓ2 Figure 5: A completeℓ2ℓ↙2 list of non-vanishing ℓ2↙ graphs and graphs corresponding to master 4 2 4 2 +4 2 4 2 s t n pure supergravities+(D 6)N Nn given↓ by3 += nn ↓ 3 + n cf., HJ, Ochirov (‘14)(5.28) 12t s numerators.4 The⇥ eight4 master graphs that we choose to work$ with are (1)–(5), (13), (19) ⇤ 3 0 ✓ 3 ◆ ✓ s 3◆ ¯ s 3 1 ¯ ¯ ¯ ¯ ¯ 2i(12 34) ✓ ◆ ✓ and◆ (22). While✓ tadpoles` ✏(1 and,◆2, external3, ` ✓) bubbles` ✏ are(1 dropped◆, 2, 3, from` )+ the finals ✏ amplitudes(1, 2, ` it, is` )+2✏(1, 3, ` , ` ) 2 ℓ 1⇥ 1 2 ℓ 2 1 2 ℓ 1 1 2 1 2 ⇤1 2 s 1↗ 2 1↗ 2 1↗ 2 useful to considerℓ2 them at intermediate ℓ2 steps of the calculation.ℓ2 ℓ 1 ℓ 1 2 (the D1↗ = 26 theory@1↗ is+( 2 Ds=(1,1)6) nsupergravity↓ 3t + n) ↓ 3 t + n ↓A 3 , 2i(14 23) ℓ2 ℓ2 N ¯ ¯ ¯ ¯ ¯ ¯ [ =4 SG] 4⇥`1✏(1, 2 , 3, `2)4 `2✏(1, 2 , 3, `4 1)+t ✏(2, 3, `1, `2)+2✏(1, 3, `1, `2) ⇤ t2 N N ↓ 3 = n ↓ 3 (5.28) ! 4 4 ✓ ◆ ✓ ◆ ✓ ◆ In the MHV sector we ¯ provide¯ 2i( two13 alternative24 ) solutions. The first of these ✓ ◆ ✓ ◆ ⇥✏(2 , 4, ` 1, `2) ⇤ ℓ As1 already2 explained,includesℓ 1 non-zero we2 drop numerators ℓ the 1 remaining corresponding2 u bubble-on-external-leg to bubble-on-external-leg and and tadpole tadpole 1↗ 2 1↗ 2 1↗ 2 ℓ2 ℓ2diagrams — diagramsℓ2 (17)–(24) in fig. 5. All of these diagrams have propagators of +(Ds 6) n ↓ diagrams3 + since n ↓ they3 integrate2+ in✏( toµ↓1 zero,µ2 in)(3 dimensional12 , 34 + regularization13 24 (see14 section+ 234.6),). 4 the form4 1/p 1/0 that are4 ill-defined in the on-shell limit (unless amputated away). ✓ ◆Pure half-maximal ✓ ◆ supergravity ⇠ ✓ numerators◆ ! (including =(1 , 1) but not = Their appearance in the solution follow from certain desirableN but auxiliary relationsN (2, 0)) are obtainedthat we by choose setting to imposeD on= theD . color-dual The numerators,=(2, 0) making supergravity them easier two-loop to find As already explained, we drop the remaining bubble-on-external-leg s andN tadpole numerators canthrough be obtained an ansatz. from As thewe shall above see, formulas contributions by from replacing these diagrams the modulus either vanish square diagrams since they integrate to zero inupon dimensional integration regularization or can be dropped because (see section the physical4.6). unitarity cuts are insensitive with an ordinaryto them. square However, and (D ass they6) potentially with (N canT give1). non-vanishing contributions to the Pure half-maximal supergravity numeratorsultraviolet (UV) (including divergences in=(1=2SQCD,wealsoprovideanalternativecolor-, 1) but not = N N N (2, 0)) are obtained by setting5.3 EnhancedDs = Ddual. ultraviolet solutionThe in=(2 which cancellations, such0) termssupergravity are manifestly in D two-loop=5 absent.2✏ Thedimensions two solutions may be found in separateN ancillary files. numerators can be obtained from the above formulas by replacing the modulus square In D =5 2✏ the two-loopTo aid the purediscussion half-maximal we generally supergravity represent numerators amplitude pictorially is known (as we have to be with an ordinary square and (D 6) with (N 1). UV-finites for all externalT helicity configurations. This was demonstrated in ref. [72]as an example of an enhanced cancellation. While a potentially valid counterterm seems 5.3 Enhanced ultraviolet cancellations in D =5 2✏ dimensions to exist, recent arguments confirm that half-maximal–18– D =5supergravityisfinite In D =5 2✏ the two-loopat pure two half-maximal loops [72, 99 supergravity–101]. Using amplitude the obtained is known pure supergravity to be amplitude (5.2) UV-finite for all external helicitywe confirm configurations. this enhanced This UV was cancelation. demonstrated In in the ref. all-chiral [72]as sector, the only UV- an example of an enhanceddivergent cancellation. integrals While are a the potentially planar and valid non-planar counterterm double seems boxes, divergences for which to exist, recent argumentsmay confirm be found that in half-maximal ref. [72]. It isD a=5supergravityisfinite simple exercise to show that these–43– contributions, when substituted into the two-loop supergravity amplitude (5.2), give an overall at two loops [72, 99–101]. Using the obtained pure supergravity amplitude (5.2) cancellation. we confirm this enhanced UV cancelation. In the all-chiral sector, the only UV- In the MHV sector, where there is a wider range of non-trivial integrals involved, divergent integrals are thethe planar UV and calculation non-planar provides double a useful boxes, check divergences on our for results. which Following the procedure may be found in ref. [72].outlined It is a simplein ref. [102 exercise], we considerto show the that limit these of contributions, small external momenta with respect when substituted into theto two-loop the loop momenta supergravityp amplitude` in the (5.2 integrand), give ( an5.2). overall This is formally achieved by | i| ⌧ | j| cancellation. taking p p for a small parameter , keeping only the leading term. The resulting i ! i In the MHV sector, whereintegrand there is is then a wider reduced range to of a non-trivial sum of vacuum-like integrals integrandsinvolved, of the form [103] the UV calculation provides a usefulD D check on our results. Following the procedure d `1d `2 1 D 2 D ⌫1 ⌫2 ⌫3 outlined in ref. [102], we consider the limit of small external momenta with= H respect(⌫1, ⌫2, ⌫3)( k ) , D 2 2 ⌫1 2 ⌫2 2 ⌫3 i⇡ 2 ( `1) ( `2) ( (`1 + `2 + k) ) to the loop momenta pi Z`j in the integrand (5.2). This is formally achieved by | | ⌧ | | (5.29) taking p p for a small parameter , keeping only the leading term. The resulting i ! i integrand is then reduced to a sum of vacuum-like integrands of the form [103]
D D d `1d `2 1 D 2 D ⌫1 ⌫2 ⌫3 = H (⌫1, ⌫2, ⌫3–33–)( k ) , D 2 2 ⌫1 2 ⌫2 2 ⌫3 2 ( ` ) ( ` ) ( (` + ` + k) ) Z i⇡ 1 2 1 2 (5.29)
–33– five-dimensional loop momenta, the two alternative double copies become identical
as mi = mi and ✏(µi,µj)=0. Since the external states of our numerators carry four-dimensional momenta, they cane be properly identified using the four-dimensional double copy (5.15). The
chiral and anti-chiral supergravity multiplets, H =4 and H =4, are associated with N N the chiral and anti-chiral =2multiplets,V =2 and V =2, respectively: N N N H =4 = V =2 V =2, H =4 = V =2 V =2. (5.21) N N ⌦ N N N ⌦ N five-dimensionalThe cross terms loop between momenta, external the states two alternativeV =2 and V double=2 give copies the extra become two identical=4 N N VN asmultipletsmi = mi and that✏ should(µi,µj)=0. be manually truncated away in the pure four-dimensional theory.Since the external states of our numerators carry four-dimensional momenta, they canIn termse be properly of the helicity identified sectors using of the SQCD four-dimensional the double copy double can be copy performed (5.15). in The each sector separately since the cross-terms between the sectors should integrate to chiral and anti-chiral supergravity multiplets, H =4 and H =4, are associated with zero. This vanishing is necessary in order for the gravitationalN N R-symmetry SU(4) to the chiral and anti-chiral =2multiplets,V =2 and V =2, respectively: emerge out of the R-symmetryN of the two gauge-theoryN factorsN SU(2) SU(2). Thus ⇥ the all-chiralH and=4 MHV= V sectors=2 V of=2, =4supergravitycanbeisolatedbyconsideringH =4 = V =2 V =2. (5.21) N N ⌦ N N N N ⌦ N color-dual numerators of = 2 SQCD belonging to the all-chiral and MHV sectors The cross terms betweenN external states V =2 and V =2 give the extra two =4 respectively. At two loops, these are the setsN of numeratorsN provided in the previousVN multiplets that should be manually truncated away in the pure four-dimensional section. theory. 5.2.1In terms One of loop the helicity sectors of SQCD the double copy can be performed in eachIn the sector all-chiral separately sector the since only the non-zero cross-terms one-loop between numerators the sectors are boxes. should The integrate relevant to zero.=(1 This, 1)-type vanishing double is necessary copy gives in the order following for the half-maximal gravitational supergravity R-symmetry numerator SU(4) to N N =(1,1) vs N =(2,0) supergravity emergein generic out dimensions: of the R-symmetry of the two gauge-theory factors SU(2) SU(2). Thus ⇥ In sixthe dimensions all-chiral and there MHV4 are sectors 1two half of -4maximal=4supergravitycanbeisolatedbyconsidering1 2 supergravities4 : 1 2 [ =4 SG] N N = n N +(Ds 6) n , (5.22) color-dual numerators3 of 2 = 2 SQCD3 2 belonging to the all-chiral3 2 and MHV sectors ✓ N◆ ✓ ◆ ✓ ◆ respectively. At two loops, these are the sets of numerators provided in the previous where we have replaced Nf using NV =2(1+ Nf )andN V = Ds 4. The modulus- section. square notation of the(graviton numerators + vector indicates multiplets multiplication) between the chiral and chiral-conjugate numerators: these are related by m m and ✏(µ ,µ ) ✏(µ ,µ ) 5.2.1 One loop i $ i i j ! i j (for real external states chiral conjugation is the same as complex conjugation). In the all-chiral sector the only non-zero one-loop numerators are boxes. The relevant Pure theories in D =4, 5, 6 dimensions can be obtainede by setting Ds = D. After plugging=(1, 1)-type in the double=2SQCDnumeratorsgivenearlier((graviton copy gives + tensor the following multiplets half-maximal) 4.15 supergravity)intoeq.(5.22 numerator), and N N inusing genericµ2 = dimensions:mm, one recovers the box numerator (5.10)ofthe( =0) ( =4) In terms of gravity numerators: N ⌦ N construction. 4 1 4 1 2 4 1 2 [ =4 SG] (1,1) à TheN N e=(2, 0)-type double= n copy gives+( theD followings 6) n =(2, 0), supergravity(5.22) N 3 2 3 2 N3 2 box numerator in✓ six dimensions:◆ ✓8 ◆ ✓ ◆ where we have replaced4 N1 f using N4 V =2(1+1 2 Nf )andN V =4 Ds 1 4.2 The modulus- [ =(2,0) SG] (2,0) à N N = n +(NT 1) n , (5.23) square notation of3 the numerators2 3 indicates2 multiplication 3 between2 the chiral and ✓ ◆ ✓ ✓ ◆◆ ✓ ✓ ◆◆ chiral-conjugate numerators: these are related by mi mi and ✏(µi,µj) ✏(µi,µj) 8The Grassmann-odd parameters ⌘I should not be literally$ squared in eq. (5.23) or! eq. (5.22); (for real external states chiral conjugationi is the same as complex conjugation). instead,same they simple should be relation shifted ⌘ iat two⌘i+4 inloops! one of the copiesHJ, G. Kälin, G. Mogull ! Pure theories in D =4, 5, 6 dimensions can be obtainede by setting Ds = D. After plugging in the =2SQCDnumeratorsgivenearlier(4.15)intoeq.(5.22), and N using µ2 = mm, one recovers the box numerator (5.10)ofthe( =0) ( =4) N ⌦ N construction. –31– The e=(2, 0)-type double copy gives the following =(2, 0) supergravity N N box numerator in six dimensions:8
4 1 4 1 2 4 1 2 [ =(2,0) SG] N N = n +(NT 1) n , (5.23) 3 2 3 2 3 2 ✓ ◆ ✓ ✓ ◆◆ ✓ ✓ ◆◆ 8 I The Grassmann-odd parameters ⌘i should not be literally squared in eq. (5.23) or eq. (5.22); instead, they should be shifted ⌘ ⌘ in one of the copies i ! i+4
–31– Magical and homogeneous SUGRAs Maxwell-Einstein 5d supergravity theories Gunaydin, Sierra, Townsend 1 1 R 1 I Jµ⌫ 1 x µ y e µ⌫⇢ I J K e = ˚a IJF F gxy@µ' @ ' + CIJK✏ F F A L 2 4 µ⌫ 2 6p6 µ⌫ ⇢ Lagrangian is entirely determined by constants Homogeneous supergravities
#f = #f = #f = #f = D D D D ··· D = — Luciani model (?) — - Homogenous scalar manifold D = — Generic non-Jordan Family — D = — Generic Jordan Family — de Wit, van Proeyen D = R D = C - Double copy: D = D = H
··· D = O - Magical theories ··· Theories identified from symmetry considerations and pt amplitudes I Chiodaroli, Gunaydin, I Two different realizations for generic Jordan family I Twin double-copy construction.HJ, Pairs Roiban of “twin supergravities"(‘15) obtained replacing adjoint with matter representation [Anastasiou, Borsten, Duff, Hughes, Marrani, Nagy, Zoccali, ]
Marco Chiodaroli (Uppsala Universitet) Maxwell-Einsteincf. HJ and YM-Einstein, Ochirov sugras as double copies(‘15) Dec , / Yang-Mills-Einstein theory (GR+YM) Chiodaroli, Gunaydin, HJ, Roiban (’14) GR + YM = YM (YM + 3) ⌦ GR+YM amplitudes are hµ⌫ Aµ A⌫ ⇠ ⌦ “heterotic” double copies Aµa Aµ a ⇠ ⌦
N = 0,1,2,4 YME N = 0,1,2,4 SYM YM + 3 supergravity - simplest type of gauged supergravity; R-symmetry gauged is more complicated - construction extends to SSB (Coulomb branch) Chiodaroli, Gunaydin, HJ, Roiban (’15) 19 Conformal Gravity
20 Conformal gravity Some properties of conformal gravity:
4-derivative action
(Weyl)2
invariant under local scale transformations:
propagator
States: 2 + 5 – 1 (Weyl invariance However: physical - graviton: 2 removes one state): S-matrix trivial - graviton ghost: 2 in flat space Maldacena; - vector ghost : 2 Adamo, Mason à negative-norm states à non-unitary à R2 gravities renormalizable K. Stelle (‘77) Conformal supergravity Some properties of conformal supergravity:
supersymmetric extensions of
invariant under local superconformal transformations
Total # on-shell states: 192 40 16 - graviton multipet: 32 8 4 - graviton ghost multipet : 32 8 4 - gravitino ghost multiplets: 4×32 2×8 4 - vector ghost multiplets: 0 8 4
”minimal” N=4 at one-loop: conformal anomaly present Fradkin, Tseytlin unless coupled to four N=4 vector multiplets. “non-minimal” N=4 (e.g. arise in twistor string) – may remove anomaly. Double copy for conformal supergravity? How to obtain conformal gravity and supersymmetric extensions?
Double copy ?
Problems: 1) simple poles à double poles 2) many strange states to account for 3) non-unitary theory (unitarity method?)
dimensional analysis: Ampl. has 4 derivatives:
at any multiplicity: Double copy for conformal gravity?
at any loop order:
The latter theory is marginal in D=6:
Guess for double copy:
(dimensional analysis)
marginal in D=6 marginal in D=4 HJ, Nohle Dimension-six gauge theory
Two dim-6 operators:
correct propagator but trivial S-matrix
3pt double copy is promising: (Broedel, Dixon)
amplitude violates U(1) R-symmetry ßà S-matrix is non-trivial
(3-graviton amplitude vanish ßà Gauss-Bonnet term) Dimension-six gauge theory
Candidate theory:
4pt ampl:
Check color-kinematics duality (BCJ relation):
Missing contribution: new dim-6 operator!
Add scalar, new operators: This modification is consistent with the Kleiss-Kuijf identity,
+ + + + + + A(1 This, 2 , modification3 , 4 )+A(2 is consistent, 1 , 3 , 4 with)+ theA(1 Kleiss-Kuijf, 3 , 2 , 4 identity,)=0, (3.11) and the left-hand-sides of eq. (3.8A(1) are, 2 modified, 3+, 4+)+ byA (2(t , 1u)=, 3+ , 4(+)+2uA(1s),, which3+, 2 , exactly4+)=0, (3.11) cancels the unwanted terms in the BCJ relations. and the left-hand-sides of eq. (3.8) are modified by (t u)= ( 2u s), which exactly Looking at the reside of the pole in eq. (3.9) it is clear that the term we added corre- cancels the unwanted terms in the BCJ relations. sponds to a propagating scalar with a factorization into two three-point amplitudes corre- Looking at the reside of the pole in eq. (3.9) it is clear that the term we added corre- sponding to the operator 'F 2.Indeed,inD = 6 this operator has engineering dimension sponds to a propagating scalar with a factorization into two three-point amplitudes corre- six, given that thesponding scalar has to a the regular operator two-derivative'F 2.Indeed,in kineticD term,= 6 this as the operator pole suggest. has engineering dimension Given than wesix, have given a scalar that the in the scalar spectrum, has a regular there is two-derivative one more operator kinetic of term, dimension as the pole suggest. 3 six that we can writeGiven down, than namely we have' . a Thus scalar we in expect the spectrum, a Lagrangian there is of one the more schematic operator of dimension form (DF)2 + F 3six+( thatD') we2 + can'F 2 write+ '3 down,. However, namely the'3. last Thus two we terms expect would a Lagrangian vanish if of the schematic we are dealing withform an ( adjointDF)2 + scalarF 3 +( thatD')2 only+ ' couplesF 2 + '3 through. However, the thef abc laststructure two terms con- would vanish if stants. The non-vanishingwe are dealing of these with terms an adjoint seems scalar to imply that a only coupling couples through through the the tensorf abc structure con- abc a b stants.c TheLagrangian non-vanishing using the of BCJ these relations terms for seems adjoint to fields, imply we a need coupling to look through at amplitudes the that tensor dF =Tr( T ,T T ); however, this cannot explain why an s-channel diagram shows up { }dabc =Tr(+ haveT a,T only+b T externalc); however, gluons. this From cannot inspecting explain the Feynman why an diagrams s-channel this diagram implies that shows the up in the color order AF (1 , 3 , 2 ↵ab, 4 ).a ↵ Furthermore,↵ the scalar cannot be a singlet under {C ,(T} ) and d+ tensors+ are first available in the four-, five- and six-point amplitudes, in the color order A(1 , 3 , 2 , 4 ). Furthermore, the scalar cannot be a singlet under the gauge group since then therespectively. term in1 eq.Indeed, (3.9 the) would rule of thumb not contribute is that each free to the↵, , single-trace index needs to be soaked up the gauge group since then the term in eq. (3.9) would not contribute to the single-trace tr(T a1 T a2 T a3 T a4 ) at four points.by a pair Nor of can gluons the before scalar we carry can study any the charge tensors. or From flavor this indices point of not view we can think a1 a2 a3 a4 tr(T T T T ↵ab) at four points. Nor can the scalar carry any charge or flavor indices not belonging to the gauge group,of sinceC 3.1as it a is Clebsch-Gordan The sourced Dimension-SixAnsatz by the coefor field cientdimension Lagrangian strength. for some auxiliary-six theory “bi-adjoint” representation. In belonging tothe the same gauge sense group, it might since be meaningful it is sourced to think by of the'↵ fieldas an strength. auxiliary field even though it We are forced to assume thatLet the us scalar summarize is in what some we achieved: unknown imposing representation color-kinematics of duality the (BCJ relations) up We areis forced propagating. to assume that the scalar is in some unknown representation of the to six points gives a unique Lagrangian for the dimension-six theory. It reads gauge group. Letgauge us denote group. the LetIn indices the us four-gluon denote of this amplitudethe representation indices we find of this contractions by representation↵, , of ,... the type, so byC that↵ab↵,C ↵ the,cd that,..., corresponds so that the ↵ scalar is ' . The newscalar ansatz is '↵to. for The an the internal new dimension-six ansatz1 scalar.aµ for Since⌫ 2 the gauge we1 dimension-six know3 theory1 that these is↵ 2 now gauge contributions1 ↵ theoryab ↵ a are isbµ now needed⌫ 1 to↵ cancel↵ some scalar= (inD someµF real) representationgF + (Dµ 'of )gauge+ gC group' (notFµ ⌫adjointF +) gd ' ' ' (3.30) pieces of theL pure-gluon2 diagrams, 3 we can2 assume that2 this contraction must3! equal to a tree- 1 1 1 1 1 1 1 1 1 1 aµ⌫ 2 3 levelaµ⌫ rank-four2 ↵ 2 adjoint3 3 tensor.↵ab ↵↵ Therea2 bµ are⌫ exactly↵ab ↵ two↵ a independentbµ↵ ⌫ such↵ tensors:↵ f ace f ebd ↵ = (DµF ) + =⇢gF(Dµ+F (D)µ'where+unknown) ⇢+FgFand Clebsh+gC the(D- covariantGordanµ''F)µ⌫+coeffF derivatives : gC + are⌧ 'gd, definedFµ⌫F' in'(symmetric) eqs.+' (3.1⌧(3.12)gd) and ('3.13').' The(3.12) scalar ' 2 3! 2 4 3! ↵cd L 2 L3! and2 (c transformsd). The4 relative in a real coe (auxilliary) cient is uniquely representation3! fixed by of the the symmetries gauge group, of thewithC generatortensor, (T a)↵ $ Assume: diagrams with internal scalars reduce↵ab to ↵ where ⇢, , ⌧ are unknownwhere ⇢, parameters,, ⌧andare the unknownand overall symmetric and coeC parameters, ↵cientab Clebsh-Gordanand cand be↵ and absorbedare coeC↵ group-theoryabcients intoand theCd↵ freeandare parameter tensorsd group-theory, which to⇢ arein be the only tensors Lagrangian. implicitly to defined be determined.This From fixesthrough their the relation appearance the two relations in the Lagrangian it is clear that we↵ab can choose C↵ab determined. From their appearance in the Lagrangian↵ab it is↵cd clearace thatedb weade canecb choose C 4pt BCJ relation à C C = f f + f f ↵ . (3.17) to be symmetric into its be last symmetric two indices in its and last similarly two indices chooseC↵ab andC↵cdd similarly↵ = f acetof edb be choose+ af totallyadef ecbd , symmetricto be a totally symmetric Using this relation all four-gluon amplitudes in the Lagrangian (3.12) can be computed on tensor. In the covariant6pt BCJ derivative relation à ofC↵ theabd↵ scalar=(T a) ↵(T b)↵ + C acC cb +(a b) , (3.31) tensor. In the covariant derivativea color-ordered of the scalar form. In particular, the ( ++) amplitude that we looked$ at before now ↵ ↵ a ↵ a takes thewhere form thes lastufficientD equationµ' to =compute@ isµ (non-trivially)' anyig tree(T amplitude) equivalentA ' with to external the tensor vectors! reduction in eq.(3.13) (3.25). ↵ ↵ a ↵ a 2 µ Dµ' = @µ' ig(T ) A+µ'+ i 2 12 2 2 (3.13) Furthermore, A all(1 the , 2 group-theory , 3 , 4 )= tensorsg h i transforms⇢ (u t covariantly) s under infinitesimal(3.18) group we have one more unknownWhich representation tensor (T fora) scalar↵ . We ?8 “Bi can34-adjoint2 think”, “auxiliary” of ( T a)↵ rep.as the generator of rotations,a ↵ implying that they satisfya the↵ three relations in eqs. (3.14)–(3.16). These five 2 h i we have one moresome unknown real representationThe tensor spinortensor (T prefactor) relations (.' Weis is real aresymmetric can su since thinkcientF under to ofis reduce real),( 1T )2 any thus interchangeas color it the structure is generatorantisymmetric so only appearing the of expression in in a its pure-gluon in last the two tree 2 $ some real representationindices. (' isparenthesis real sinceamplitude needF tois to be real), contractions plugged thus into itof thef isabc antisymmetric BCJtensors. relation, giving in its the last constraint two indices. Since C↵ab, d↵ ,(WeT havea)↵ computedare covariant the tree-level tensors pure-gluon of a Lie amplitudes algebra up they to multiplicity must transform eight, and 0=t ⇢2(u t) 2s (t u)=s(t u)(⇢2 2) (3.19) Since C↵ab, d↵ correctly,(T a)↵ underare infinitesimal covariantfound that tensors no group additional of rotations. a corrections Lie algebra This to$ impliesthe they Lagrangian must that transform they are needed satisfy for the it to relations satisfy color- kinematics duality. correctly under infinitesimal groupThus we rotations. conclude that This implies= ⇢. The that ambiguity they satisfy in the the sign relations reflect the fact that the a ↵ b ± b ↵ a abc c ↵ redefinition '↵(T ) '(↵Tin) the Lagrangian(T ) (T does) not= if change(T the) value, of any pure-gluon(3.14) 4 Conformal! (Weyl) 2 gravity (T a)↵ amplitude.(T b) Without(Tfbbae)↵ C( loss↵Teca)+ of f generalitycae= ifC↵abcbe ( we=T ci can)(↵ T a choose,)↵ C bc=, ⇢, which(3.14) is more convenient.(3.15) bae ↵ec Thecae ( ↵++)bea ↵ and (a +↵ +) abc four-gluon ↵ amplitudesa ↵ are now as follows f C + fThe C theory(T =) weid(T )+( CT ) , d +(T ) d =0. (3.15) (3.16) a ↵ a ↵ a ↵ 2 4.1 Conformal supergravity+ + i 2 2 12 We(T can) d now+( constrainT ) d the+( unknownTA(1) , 2d parameters, 3 ,=04 )=. ⇢ andg uh tensorsi , by(3.16) a careful analysis of 4 34 2 5 Explicit Amplitudes for (DF)2 and CG the tree amplitudes of the dimension-six Lagrangian. Givenh thati2 we want to constrain the We can now constrain the unknown parameters and+ tensors+ byi a2 careful2 13 analysis of A(1 , 2 , 3 , 4 )= ⇢ g uh i . (3.20) Gluon amplitudes in a gauge theory are color4 decomposed24 2 as the tree amplitudes of the dimension-six Lagrangian. Given that we want toh constraini the n 2 a a a Because we haven the= g F 3 operatorA( in (1) the, Lagrangian,(2),..., (n)) we Tr( alsoT (1) haveT (2) nonvanishingT (n) ) , am- (5.1) A ··· Sn/Zn plitudes in other helicity sectors.2X The–4– all-plus amplitude and one-minus amplitude are as 1Notewhere that the the (DFA)(2 and(1),F 3(2)operators,..., contains(n)) are quartic, color-ordered quintic and primitive sextic interactions amplitudes. that are needed for restoring gaugeAs a–4– invariance reference, of we the here linearized give operators. the MHV The gluon fact amplitudes that the C↵ab in,(T Yang-Millsa)↵ and d↵ theory,tensors show up at the same orders is suggestive of that they also cancel something unwanted in the (DF)2 and 3 4 F terms. YM + + + ij A (1 , 2 ,...,i ,...,j ,...,n )=i h i , (5.2) 12 23 n 1 h ih i ···h i and in = 4 super-Yang-Mills theory the MHV superamplitude is N (8)(Q) ASYM(1, 2,...,n)=i , (5.3) 12 23 n 1 –5– h ih i ···h i ↵A n ↵ A where the supermomenta are Q = i=1 i ⌘i . P
–8– Construction works!
Color-kinematics duality checked up to 8 pts ! (no new Feynman vertices beyond 6pt)
Double copy with YM agrees with conformal gravity: (Berkovits, Witten)
All-plus amplitude is non-zero à no susy extension of
Supersymmetry of conformal supergravity sits on the YM side: Generalizations and deformations
Curiously no interacting scalars are obtained from dimensional reduction Instead add regular scalars in adjoint…
color-kinematics fixes interactions Bi-adjoint Double copy: Maxwell-Weyl gravity: Double copy: Yang-Mills-Weyl
N=4 case: Witten’s twistor string! finally, deform with dim-4 operators: à Yang-Mills-Einstein-Weyl gravity
à Summary
Powerful framework for constructing scattering amplitudes in various gravitational theories – well suited for multi-loop UV calculations
Color-kinematics duality and gauge symmetry underlies consistency of construction. (Kinematic Lie algebra ubiquitous in gauge theory.)
Constructed new dim-6 theory using color-kinematics duality - theory has several unusual features. Checks: Explicitly up to 8pts tree level (loop level analysis remains…)
First construction of conformal gravity as a double copy: may simplify analysis of unresolved unitarity issues (if not yet resolved) may be an interesting UV regulator of Einstein (super)gravity (div. N=4 SG ßà U(1) anomaly cf. Bern, Edison, Kosower, Parra-Martinez)
An increasing number of gravitational theories exhibit double-copy structure (some in surprising ways) – more are likely to be found!