JHEP07(2018)016 Springer July 3, 2018 May 12, 2018 : June 20, 2018 : : , Published Received Accepted 1 Published for SISSA by https://doi.org/10.1007/JHEP07(2018)016 [email protected] , . 4 . 3 1805.00394 The Authors. c Scattering Amplitudes, Higher Spin Symmetry

We develop a formalism for describing the most general notion of tree-level , [email protected] Also at Lebedev Institute, Moscow. 1 [email protected] Theoretical Physics Group, Blackett Laboratory, Imperial College London, SW7 2AZ,E-mail: U.K. Open Access Article funded by SCOAP conformal higher spins in AdS Keywords: ArXiv ePrint: allows us to encodeleads these to scattering compact momentum statesof space efficiently conformal expressions in higher for spin terms all theory.all of While finite those ‘twistor-spinors’. some tree-level with of 3-point these This only amplitudes others 3-point standard which amplitudes two-derivative vanish are (including higher non-vanishing. spin We external also states), comment there on are the many generalization to scattering of ing to ill-defined amplitudes.produce We characterize finite the tree set amplitudes, ofmassless noting admissible higher scattering that spins obeying states there two-derivative which equations are ofas more motion. a such We prime use states example, conformal than gravity whereand just the a standard set ‘ghost’ of massless scattering spin states 1 includes particle. the usual An Einstein extension graviton of the usual spinor helicity formalism Abstract: scattering amplitudes in 4dobey conformal higher-derivative higher equations spin of motion, theory. therewhich As are may conformal contribute many to distinct higher their on-shell spin scattering, external some fields states of which grow polynomially with time, lead- Tim Adamo, Simon Nakach and Arkady A. Tseytlin Scattering of conformal higher spin fields JHEP07(2018)016 11 8 22 34 31 4 3 11 ]). ]) is a formally consistent higher spin 19 14 5 – 19 – derivatives in the kinetic term and thus 1 22 s 15 , Locality for a symmetric traceless higher-spin 3 32 1 25 – 1 – 10 17 27 7 6 14 ) implies the presence of 2 s 1 ...a 1 a φ 2.2.1 Conformal2.2.2 gauge Transverse2.2.3 gauge Two-derivative formulation of conformal gravity = ( s φ In this paper we only consider the case of 4 dimensions and bosonic integer spin fields. 5.1 Free fields5.2 and momentum eigenstates 3-point amplitudes of CHS theory 3.1 Twistor-spinors and3.2 linearised Bach equations Momentum eigenstates 3.3 Polarizations and double copy 2.1 Tree-level scattering2.2 in higher-derivative theories Linearized spectrum of conformal gravity 2.3 Scattering states in CHS theory 1 field non-unitarity. Despite this, there arefor many study, reasons including why its CHS good theory UVand is behaviour, higher-spins an relationship in interesting with topic anti-de other Sitter conformal space field theories (cf. [ 1 Introduction Conformal higher spin (CHS) theorymodel (see, that e.g., has [ atheory and local spin action 2 Weyl with gravity a to all flat-space spins. vacuum that generalizes spin 1 Maxwell C Deriving 3-point amplitudes from twistor space 6 Scattering in AdS background A Helicity structure of conformal gravitonB modes Counting CHS degrees of freedom 4 3-point amplitudes in conformal5 gravity Free fields and 3-point amplitudes in CHS theory 3 Twistor-spinor representation of states of conformal gravity Contents 1 Introduction 2 Free higher-derivative fields and the S-matrix JHEP07(2018)016 particle. s 2 ] and partially 20 + 1) degrees of freedom s ( satisfying the 2-derivative s (0) s φ ]. This is equivalent to defining 12 , , we give the generalised definition of action this separation becomes obvious upon 11 2 1 CHS equation in transverse traceless 2 C ]. ]. s > 23 26 , – – 2 – 12 24 , 11 state decomposes into the s term or by switching on a constant curvature resulting in the ghost R modes on external lines [ + 1) degrees of freedom of a CHS field are ghost-like. s s degrees of freedom may be interpreted as ‘physical’ (‘unitary’) ones = 0. The latter has further on-shell gauge invariance which reduces ( s s (0) s φ  = 0, one can always choose a special solution s ] in the second). ]), it is natural to define the CHS scattering amplitudes by keeping only those φ s 21 22 equation  s The plan of the paper is as follows: in section We observe that this issue can be resolved by augmenting the standard spinor helicity With this extended definition for the tree-level S-matrix, one still faces the obstacle One may wonder if, at least at tree-level, a more general definition of the CHS S-matrix Ignoring the fact that the ‘physical’ and ‘ghost’ degrees of freedom do not actually In such a higher-derivative theory, the definition of asymptotic states and scattering For example, in the case of the conformal gravity with 2 scattering. We focus inthe detail massless spin on 2 the Einstein example graviton of there is conformal also gravity, an whereadding oscillating in to ‘ghost’ addition spin the to 1 actionmode mode an decoupling which Einstein from themassless spin [ 2 Einstein one (and becoming massive in the first case [ capture all finite tree-level 3-pointin amplitudes which of massive the states theory.include can This spinor be is indices described analogous of by to the augmenting the little the way group spinor [ helicitythe formalism tree-level to S-matrix in CHS theory and clarify which external states are admissible for of CHS theory. formalism to include states carrying aalso spinor known index of as the twistor conformal indices.all group. on-shell Such The indices states resulting are of twistor-spinor formalism CHS allows theory, us and to can encode be used to provide compact expressions which of finding an efficientderivative formalism theory for encoding in the fourits on-shell dimensions, momentum, scattering spin, an states. and on-shell Unlike positive/negativerequired CHS helicity a to label. state two- distinguish how is Additional a ‘polarization’ not spin data uniquely is specified by address in this paper. Asset we of shall ‘ghost’ see, modes in intolatter a oscillating momentum lead modes representation to and one modes formally canamplitudes whose separate infinite and curvature the contributions thus grows appear to in to the time. becan on-shell The unsuitable be for action included, scattering. along or with However, tree-level the the scattering massless oscillating modes, modes into the set of admissible scattering states. infinite-dimensional symmetry algebra of thesponding CHS tree-level theory S-matrix appears is to trivial imply [ that the corre- is possible. That is,the can ‘ghost’ the modes space leading of to scattering non-vanishing states scattering be amplitudes? extended This to is also the include question some we of while the rest of the separate on a flatstates background [ (in particular, instandard the massless sense spin ofthe Hilbert scattering space amplitudes of with asymptotic the usual, two-derivative LSZ reduction. The underlying amplitudes is non-trivial. Givengauge the free spin spin the number of independentThese solutions massless to spin the two of the standard massless spin JHEP07(2018)016 ]. 3 12 (2.1) , 10 background. 4 totally symmetric s . By evaluating this contains a derivation 4 C ). The free field equation extends the twistor-spinor ]. 2) , 1 5 − ) s s ( a a ··· α 3 a α 2 a 1 is a transverse and traceless differential a ] (and thus including effectively all modes as ( ) η 6 s , ( b 5 + ) s ) ( s a a explains the counting of on-shell CHS degrees – 3 – P ··· 2 B a  1 a ( ∂ We then describe the scattering states for a generic spin = 3 ) s = 0, where ( ) ]. a . Appendix CHS fields, and we obtain an expression for all 3-point tree s ( b 28 2 δφ ]. s ) defined up to linearised gauge transformations φ ) x 27 s ( ( ) b s ( ) s a which is totally symmetric in its indices [ generalisation of infinitesimal local diffeomorphisms (parametrized by ( φ a s s CHS fields satisfy higher-derivative equations of motion, they contain P , we consider the generalisation to CHS scattering in a background with a s ) = x 6 ( describes the helicity structure of linearised conformal gravity in the two s 4 a ··· A 1 a φ ) and conformal transformations (parametrized by 1) Since spin In section Using the twistor-spinor formalism, we give an expression for all 3-point tree ampli- In general, the separation of the field potential into ‘oscillating’ and ‘growing’ modes is The oscillating mode corresponding to theRelated massless work spin describing 1 possible state is 3-point ghost-like vertices as for it higher originates derivative spin from 0 a and 1 conformal fields − 3 4 s ( CHS field. a many more on-shell degrees of freedom (d.o.f.) than ordinary two-derivative fields.time-like Indeed, component of the metric fluctuation and thusin contributes with general negative dimension sign usingexternal to states) the a appeared energy. 2-derivative in formulation [ [ which are the spin  can be written as operator of order 2 of motion. A freetensor bosonic CHS gauge field in 4 dimensions is a rank of the 3-point CHS amplitudes from the formulation of CHS theory in twistor space2 [ Free higher-derivative fields andExternal the states S-matrix in any scattering process are given by free field solutions of the equations cosmological constant. We commentexpressions on for the 3-point amplitudes structure forAppendix of massless on-shell higher states spin and statesgauges evaluate in discussed our an in AdS section of freedom and scattering states in terms of curvatures. Appendix tudes in conformal gravityessentially involve the two same spin as in 1 theformalism states Einstein-Maxwell theory. to and Section one generic Einstein spin amplitudes graviton; of these CHS are theory. shell states of conformaltensors, gravity. curvatures, and We the also double-copy4-derivative comment representation vector on of theory the conformal [ gravity relation in between terms polarization of a tudes (with complexified kinematics) inexpression conformal on gravity specific in section on-shell states, we find that the only non-vanishing 3-point ampli- s gauge-dependent, even though the characterisationwe of give scattering a states useful is gauge-invariant descriptionon not. of the In scattering corresponding section states in curvatures. conformal gravity The based resulting twistor-spinor formalism encodes all on- can be used as a scattering state. JHEP07(2018)016 int L (2.6) (2.2) (2.3) (2.4) . 3 . ]. For instance, i x φ from the classical φ 4 i 30 3 , d ε ε 2 29 =1 Z 4 i ε λ , 1 P 1 6 ε + 1) d.o.f. correspond to Φ= s =

= 0 ( ) s 2 y int ( int Φ , L δ in the action evaluated on ) δ 4 x ,S ( ε ) i 2 ) φ , k i ··· φ x, y ε x 1 ·  ε ( k = 0 (2.5) ∆( 3 i =1 x i X y φ 4 ) e d – 4 – 2 d x d  · ) = Z Z x n ( 1 2 [3] ) + = Φ x 0 ( i without solving some other equation which is of order 2 φ to use as the external states. As a toy example, consider a i s ε the only on-shell states in CHS theory which are suitable for B) family of solutions in terms of plane waves: } i , ) = (A + B ,S 4 φ x =1 i { X ( not + 1) on-shell d.o.f.. Many of these int φ S s ( ) = s + x ( 0 S [4] are free field solutions which can be expanded in a basis of plane waves, Φ = } i S φ has mass dimension 4. The free equation of motion { there are λ s ) is the propagator (i.e., the inverse of the kinetic operator in the action), and obey the higher-derivative equations of motion in a strict sense, satisfying the lin- For a higher-derivative theory, the same procedure can be used; the only subtlety is Two of the d.o.f. correspond to the standard two-derivative massless higher spin fields, is conformally invariant with the scalar assigned the conformal weight zero and the x, y 0 admits a two-parameter (A S coupling ∆( is the interaction part of the Lagrangian. what free field solutions four-derivative scalar theory on a flat background while a 4-point amplitude is the coefficient of Here, the a 3-point amplitude isaction obtained evaluated by on extracting the coefficient of The notion of scattering amplitudesissues, in but higher at derivative theoriesassociated tree-level is action an fraught with functional. elementary potential can definition In be can defined two-derivative be theories, by simplyon applied tree-level computing a scattering as the particular amplitudes multi-linear long partthe solution as of free which the there equations classical is is action of built an evaluated motion (recursively) with from specified superpositions asymptotic of behaviour solutions [ to and ear CHS equation ofin derivatives. order 2 2.1 Tree-level scattering in higher-derivative theories growing modes with polynomial (rather than pure oscillatory) asymptoticwhich sit behaviour. trivially inside thederivative space solutions of solutions are to thescattering linearised CHS in equations. Minkowski These space. two- There are other on-shell states which are pure oscillatory at spin JHEP07(2018)016 s (2.7) (2.8) scattering ) , mode expected a 0)). The mode ) , 2.4 3 0 k , ]). 0 + , 31 2 ), the set of scattering k growing = (1 + 2.4 1 a k n , x are solutions of the equations · ) := ( gives the expected finite 3-point 3 } i k a φ . + amplitudes. The modes which this int 2 { k S ! i + 6= 0, e.g., k 1 k k i ( 3 =1 · e i X n x

· ,K (4) – 5 – ) are clearly suitable for scattering — they are ! x n i λ δ 4 k d 2.6 ∼ 3 =1 3 i Z X λ

M ∼ (4) momentum-conserving δ 3 , M ∂ ∂K · finite λ n i ∼ − 3 ]. This, in turn, would indicate that the definition for the tree-level S-matrix M 32 [ is an arbitrary time-like vector (i.e. s a  scattering states in CHS theory is composed of only two-derivative massless spin n of the theory. In the case of the conformal scalar theory ( s If this were the case, the tree-level S-matrix of CHS theory would be rather trivial: This motivates a definition of the S-matrix in a higher-derivative theory: tree-level The pure oscillatory A-modes in ( there is strong evidence that all amplitudes of such two-derivative external states in CHS free fields; this would meanhelicity that there are onlyis two actually such equivalent modes to forreduction that each singles of spin, out two-derivative one all theories: each of the of the standard admissible two-derivative scattering LSZ states on the external legs. 2.2 Linearized spectrum ofBased on conformal the gravity example ofthe the four-derivative scattering conformal scalar, states it in iszero-rest-mass tempting a fields to assume generic of that higher-derivative appropriatespin theory spin. are Indeed, simply one the might two-derivative conclude that the space of of motion which leaddefinition to singles out as admissible externalstates states will be referredstates to as is the precisely set the of ordinary plane waves. But if one wishes toconservation, obtain it finite is tree-level clear amplitudes, thatexternal supported the states. on growing overall modes 4-momentum must be excluded from thescattering set amplitudes of are allowed givenusual, by extracting with the the same added multi-linear piece constraint of that the the action free as fields and similarly for the 3-point interactions involving two or three of the growing modes. However, if one of the external states is a growing mode, one finds from ( which is undefined. One can interpret this ‘amplitude’ in a purely distributional sense as level. Evaluated on threeamplitude oscillatory of external a states, cubic scalar theory (we ignore the overall numerical factors) parametrized by B then growsfor linearly in a time; generic this theory is with precisely higher-derivative the equations of motionjust (cf. the [ usualB-modes, plane on wave solutions the other to hand, the lead two-derivative to wave equation. un-defined (or The divergent) amplitudes growing even at tree- where JHEP07(2018)016 (2.9) (2.16) (2.12) (2.13) (2.14) (2.15) (2.10) (2.11) ]. . 35 , 5 , = 0 ): a a , = 0 ) on a flat background is , 2 . 2.12 . 2.1 = 0 c abcd  c c = 0 are symmetric traceless constant V C α , ∂ c c is the Weyl curvature tensor of the k ∂ ab ab , a four-derivative theory of gravity ab η abcd η ab , h ab η abcd C 1 2 A + . , k 1 2 | C b a g x = 0 − | k  · and B − b k a a ]: i = 0 ac ∂ a p  k b ab h V 36 i x ) e 4 ab b ∂ + a 4 – 6 – x ∂ h b ∂ d · 2 − +   a + b b  n Z ∂  b k − a 2 V ab encoding local diffeomorphisms and conformal trans- Fortunately, there are other scattering states in CHS ∂ cb = ε a 1 conformal gravity ab ∂ 5 2 α h = ab )B ]. + B c + k ∂ ab δh · b ab ] = 34 ab and , ): ∂ g n δh h [ a a ( 2  S ∂ 33 2.5  , = (A 1 3 ab 23 := , h c V 12 , 11 = 2 version of the linearised gauge freedom ( s again is a time-like vector and A = 0, with the remaining gauge freedom is a dimensionless coupling constant and a a a n ε h Conformal invariance can always be used to fix a traceless gauge for linear perturba- The truncation of CHS theory to the spin 2 sector serves to capture all essential For discussions of relations between the Einstein and Weyl actions see also [ 5 where tensors related by four algebraic constraints following from ( It is easy to see that plane wave solutions to this equation are given by It has the benefitMinkowski of background, reducing which the are free equations of motion of conformal gravity around a to the spin-2 analogue of ( 2.2.1 Conformal gauge One such gauge-fixing is ‘conformal’ gauge [ To determine the modes of the theory,of one must the fix linearised this gauge equations freedom of and find motion. the solutions with the gauge parameters formations, respectively. tions, where metric. The theory; we now explain how to define them infeatures a of gauge-invariant the manner. problemcorresponding of classifying non-linear higher-derivative, theory higher-spin is scatteringgoverned by states. the The action theory are zero [ JHEP07(2018)016 ]. 36 and (2.20) (2.18) (2.19) (2.17) ab 2 [ is again  ) that the ab . This im- a h A v , 1 and a , n ,  u · ) a v n b ab k ), so that η 1 2 + . b ; there are residual gauge − 2.14 4 = 6. In this case A n ab a a = 0 − n 4 = 14 independent parame- k b 4 ( have helicity ,B ab v ), now the symmetric traceless . − k × ) the transverse gauge k B ab 1 modes are encoded by a linear ab · + 9 · 2 a  b n 2.16 n u − = 0 × n i 2.12 5 9 a a a + v × − ab + ,  are pure oscillatory and therefore suitable A k   a · ab k k k · · i u u v ab – 7 – η , h ab ab 2 5 η η 1 2 1 2 − = 0 2, but the helicity a − −  k ab , . a a 2, while the four in A b are growing and will not lead to well-defined amplitudes. h contains two. It can be shown (see appendix b k k u  a b b k v a ∂ u ab = 0 ab + are precisely the Einstein gravitons, which form a consistent k b + + ab k k k ab b b · a · B k k u a a : u n a k a v u  2 5 u   matrices, i.e. the spin-1 modes appear to pick up a growing part (see = i = = i = i have helicity must obey ab ab B ab ab A , and 8 residual gauge transformations parametrized by ab ab both δ δ B A ). However, instead of the constraints ( ]. 2 modes in A ab are constant vectors. Thus there are 2 δ δ ,B  a for details). 38 ,B ab v , 2.15 contains four while B A , ab 37 a u ab At this point, one might say na¨ıvely that we have characterised the scattering states The residual gauge freedom can be used to distribute these six d.o.f. in such a way each encode modes of helicity ab Once again, this leaves usB with six on-shell d.o.f.: 2 combination of appendix It is straightforward tosingle constant show vector that there is a residual gauge freedom parametrized by a given by ( matrices A 2.2.2 Transverse gauge To see this, consider instead of the conformal gauge ( In this gauge the free equations of motion still take the form ( of conformal gravity: the modesfor encoded scattering, by while A those inBut B this isoscillatory statement metric is perturbations premature: is not the gauge invariant! decomposition of the modes into growing and that A two d.o.f. inThe B helicity two-derivative sub-sector of the theory. where ters in A plies that there aregravity six [ overall on-shell d.o.f., matching the known counting for conformal transformations of the form These conditions do not completely fix the freedom in A JHEP07(2018)016 , ) = ab φ ( ab V (2.24) (2.22) (2.23) (2.21) F . Thus + 2 modes contains ϕ 2 A priori )  φ . φ the helicity a 4) conformal  k ( , 1 2 , and ). The equations and  = ), where the spin- . c φ a ( b L  form. Alternatively, c n V b 2.12 ab 2 ) is the same (up to a 1 4 represents the growing + F ϕ 2 2.9 ab ϕ , − − ϕ F c 1 2 ab b 1 4 in ( ˜ S c ϕR − ) 2 ∇ x φ of dilatations and , C ) + ·  b b  a  n ab ϕ ϕ 2 g a a R ϕ = − 1 3 0 is an oscillating mode while b − L ϕ b − a ab 2 Einstein graviton modes b 2 ab ϕ + (1 + 2i  1 2 R ab ab = 0 theory the field ϕ  + | S ( – 8 – ) ) to rewrite it in the g φ 1 4 b | x 2 b · ab a p +  ( = 0 so that ϕ n ab ∇ ) could have a linearly growing piece, but an explicit ϕ 2i = 2 ˆ G +  − ab W R 2.21 ϕ L (1  ab and | g ] g ∼ | 1 2 ) ϕ 5 ab p = A − h − φ ab  = R , where the decomposition into modes is independent of the gauge choice. 0 W ≡ are L contains linearly growing terms. In the transverse gauge, the Einstein ab 0 ab h are constant matrices whose entries are determined by ˆ L G ab ab is the field strength of the gauge potential b ˜ S a , is related to the gauge potential corresponding to the conformal boosts, ] ending up with [ b b ab ∂ 39 ab S ϕ − Observing that in 4 dimensions the Weyl Lagrangian It is clear that if a metric perturbation is purely oscillatory, then its associated (lin- This seems to contradict what we found in the conformal gauge ( b b 1 modes are purely oscillatory while the curvature associated with the helicity a Here ∂ one can introduce an auxiliaryone tensor may start with a formulation ofgroup Weyl gravity [ as a gauge theory of the SO(2 from the point ofmode view and of should the not original be included in the settotal of derivative) as asymptotic states. consider its 2-derivative reformulation by introducingThis extra is fields a in generalization addition of toby replacing the an the metric. equivalent 4-derivative one scalar with Lagrangian following two from independent fields: both oscillating and growing modes, with the scale of the latter being related to calculation shows that these growingcourse terms matches cancel. what The was remaining found oscillatory in curvature conformal of 2.2.3 gauge. Two-derivative formulation ofTo get conformal another gravity perspective on the spectrum of states in conformal gravity, it is useful to of the form (cf. appendix where the curvature associated with ( earised) curvature tensor willthat also the curvatures be associated with purely the oscillatory. helicity In conformalencoded gauge, this by means B modes have the samehave the pure same oscillatory growing curvatures. curvatures, The and spin-1 the modes are spin-2 encoded modes by a encoded metric by perturbation B 1 modes were purelythat oscillatory. we are The trying resolution towhich of characterise are the this not scattering apparent gauge states paradox invariant. byassociated lies with Instead, looking in we at should metric the perturbations, look fact at the linearised curvature tensors JHEP07(2018)016 ) ) a b ab b ) to a , ϕ ( term 2.15 ) one ∂ (2.30) (2.26) (2.27) (2.28) (2.29) (2.25) ab as the bc h 2.30 h 2 in ( . Fixing + 2 2.23 ac that follow ab ab F ··· , , under a we may then h ϕ in ( ab ab and fixing the + F ab . ab = 0 ..., λg ϕ  ab a ϕ a h ζ and b − b + = should be the same ) −  ab b -part of = ab a b ϕ λ ) on ab a a 2 , , δg ( ∂ ) is invariant, in addition c c ∂ ab 2 ϕ η = and we defined ∆ 2.18 b ), ∆ ∂ − − a 2.23 2 cancels out and one recovers h 1 2 b h = 0 ; a ab  ( a g ∂ a = R ) in terms of the 3 fields O 2 1 b = a ab ab + g ∂ − . Integrating out ϕ ab 2.23 1 6 a ab ab h ab g ∂ + 2 h − h of the latter being related as in ( 2 b a a , δ ab ab ∂ ∆ c ab -gauge invariant combination 1 2 ϕ R ζ ζ c =  b . = 2 , a ab , ϕ – 9 – − g = 0, but instead one may impose a gauge condi- R = = 0 = 0 , χ is that the action for ( a ab − = 0 ; ) = 0 via the g ) contains the Einstein graviton part plus a growing a 2 a a  b 1 2 ) that the 3-point scattering of one Einstein graviton a c c h ...... ζ ab ( ˆ a ab G ϕ ( − + + h O 2.23 ab  , h c c describes a massless purely-oscillating graviton mode, b ab g ab + ϕ and b 3 1 ) (and their traces and derivatives) that b ϕ R . + 2b ) ab 2 ∂ b = 0 a ) ab ϕ − b χ ∆ ; − ϕ ζ ). a ab 2.27 ( a ) = 0. ab h ( b ∂ ab ˆ a ),( a G b ∇ ϕ b ∂ 2.22 a +  a ( a 2 ∂ ∂ ∂ 2.26 ab = 2 − ) by the harmonic gauge on + h + 2 (A-mode) and two massless vectors described by b ab =  ) with the polarization tensor B ab 1 2 ab F ab δϕ 2.25 ab a ϕ − h ϕ ∂ 2.15 . = is covariant derivative corresponding to = ) are ) satisfied automatically. Using the on-shell gauge invariance of ab )). This way we recover the 2+2+2=6 count of on-shell degrees of freedom, with a ab is the parameter of gauge transformations corresponding to local conformal boosts. ϕ is that it formally extends off-shell and also to the non-linear level. In particular, ab h ∇ = 0 and thus 2.23 a 2.28 a R  ζ a a 2.25 An advantage of the 2-derivative representation ( We then conclude that The linearised equations of motion for the fields The reason for the decoupling of b ϕ -symmetry ( a and b it is implied by thecontained structure in of ( as in Einstein-Maxwell theory, since such a vertex may only come from the describes a massless vector,mode as and in ( the polarization tensors of (cf. ( the scattering states beingand represented by by the the massless Einstein vector graviton b A with ( set ζ it follows from eqs.( Here linearized Einstein operator: the reparametrizations and Weyl rescalings by the TT gauge ( from ( Here One can fix thistion symmetry on by setting b finds (ignoring total derivatives)the that Weyl all Lagrangian dependence ( on b to the usual reparametrizations and Weyl rescalings of the metric where JHEP07(2018)016 s = 1, is the (2.33) (2.32) (2.31) 1 , 2 s,h ν CHS field ν 1) in time. 3 there are s + 1) on-shell − ≥ 2, the number s s ( s s , where ˆ s > ◦ s,h ν , . The . This decomposition is + 1) s s ) to compute the 4-point , it can be shown that (see − − s,h h = 2, we saw that ν s ( − 2.23 s 2 states (Einstein graviton) as , so that for s = ˆ s  ,..., 1 2 . This indicates that the spin 1 s s,h and with the structure of the free . − = ν 6 . For ] s,h ˆ 6 ν , + 1) 5 s in derivatives, many of these states will s =1 ( ]. , . . . , s h X h , s s 27 1 2 = := = 1 s = – 10 – ˆ s,h ν h ν CHS field of helicity s,h s ν ] can be divided equally into positive and negative ⇒ 1 s =1 . h X a , where 1 h − − h ]. = 8 CHS field [ s,h purely oscillatory modes, and it follows that these are distributed s from the structure of linearised field strengths. It is also consistent ˆ 1 states (spin-1 mode). ν 2 spins in any dimension in [ s B  s < be the number of on-shell d.o.f. in a negative helicity spin- = ◦ s s,h ν ν ) is the total number of growing d.o.f. at spin B s ). We shall reach the same conclusion via a different route in section 4 below. ν ]. The existence of such a 2-derivative local action at an interacting level is an open question for 6 From now on, we refer to a mode as ‘growing’ (‘oscillatory’) if the mode’s curvature = 1 + 1 = 2. In general, one can show that 2.23 Let us note that a 2-derivative formulation is known for all CHS fields but so far only at the quadratic 2. One may still attempt to construct interacting 3-point vertices using an indirect light-cone approach 6 2 , 2 with the 2-derivative formulationCHS of partition CHS function fields [ [ level [ s > as developed for low where ˆ field contains such that there isderived a in single appendix one at each integer helicity The number of suchmore growing modes modes increases than quadraticallynumber oscillatory of with modes. growing modes Writing forappendix the spin Since the equationscorrespond of to motion modes are which of grow order at 2 least linearly and at most of order ( ν which is consistent withsector: the overall degree of freedom counting in the negative helicity is growing (oscillatory) inof a scattering states plane (or waved.o.f. purely basis. oscillatory for modes) For a CHShelicity. increases free fields with Let spin- with which correspond to helicity is in terms ofoscillatory their linearised curvatures. curvature Those tensors modeseven are which, when, the in suitable in a states a planestates for wave in particular basis, scattering. conformal gauge, lead gravity This therefore their towell contains is potentials purely as two true two helicity are helicity growing. The space of scattering It would be interestingamplitudes also involving to the vector use b the 2-derivative action2.3 ( Scattering statesTo in summarize the CHS above theory discussion, the gauge-invariant characterisation of scattering states in ( JHEP07(2018)016 ab h (3.1) (3.2) (3.3) different as s ab h of a negative helicity states are precisely can be decomposed , respectively. At the abcd s ab C  αβγδ h , ), so Einstein gravitons αβγδ 3.2 = 0 , and Ψ = 0 ˙ δ αβγδ ˙ γ ˙ Ψ β ˙ ˙ α β αβγδ β e Ψ . The helicity , ∂ Ψ s ˙ ˙ α α  α α = 0 ,..., abcd 2 C – 11 – , ∂ , ∂  d , ∂ 1 a = 0 = 0  ∂ ˙ ˙ δ δ scattering states in CHS theory contains 2 ˙ ˙ γ γ ˙ ˙ β β s ˙ ˙ α α e e Ψ Ψ ˙ ˙ β α After demonstrating how the free field equations of conformal β α ∂ 7 ∂ ˙ α α ∂ indices, as well as the usual spinor indices familiar from the spinor ]. 42 – twistor 40 corresponding to a positive helicity perturbation and Ψ ˙ δ ˙ γ ˙ β , etc. ) can also be written in terms of the linearised Weyl tensor ˙ ˙ α β ˙ α e Ψ  Solutions to the standard zero-rest-mass equations of linearised Einstein gravity, In four dimensions, the Weyl tensor decomposes into the self-dual (SD) and anti-self- The basic idea is that CHS fields are most naturally represented in terms of objects In summary, the set of spin- 2.13 This connection is alternatively known as the local twistor connection or Cartan conformal connection , 7 αβ are trivially solutions toform the a four-derivative subsector Bach ofequations equations which the are ( solutions. not strictly two-derivative But in of nature. courseon there space-time [ are other solutions to the Bach with perturbation. Spinor indices are raised and lowered using the 2d Levi-Civita symbols linear level, this is equivalentinto to positive the statement and that negative the helicityare fluctuation parts. then The free field equations for these helicity sectors in ( which are often known as the linearised Bach equation. dual (ASD) parts, given by totally symmetric spinors gravity are recovered in thissition ‘twistor-spinor’ is formalism, achieved for we show momentum eigenstates. how the3.1 d.o.f. decompo- Twistor-spinors andThe linearised free Bach field equations equations of conformal gravity written in terms of the metric fluctuation Indeed, the formalism extends naturally to CHS fields ofwhich all carry integer spins. helicity formalism. These objectsconnection acts are covariantly. simply tensors upon which a conformally invariant Having established that the standardsible two-derivative LSZ-type asymptotic reduction scattering misses states out incan admis- a the higher-derivative various theory, different a modesthe natural of free question a field? is: higher-derivative In how fieldmodes this be of section, distinguished conformal we at gravity, develop which the a serves level formalism as of that a allows toy example us of to conformal isolate higher the spin on-shell theory. kinds of modes: onethe each two-derivative of massless helicity higherthe spin higher-derivative fields; CHS equations the of others motion. arise as oscillatory3 solutions of Twistor-spinor representation of states of conformal gravity JHEP07(2018)016 ]). 44 (3.7) (3.8) (3.9) (3.4) (3.5) (3.6) (3.10) , , is C   δ S A ) spinor indices, C C αβγ B , ) 0 Ψ ˙ α ˙ α α ˙ β . A  ) , which can be carried by ! + ( ) ) x x . B ( ( αβγρ takes values in the (complexi- , ˙ S α α ˙ ˜ α s s , Ψ ρ α ! ˙ = 0 A γ ˙ δ

˙ ˙ ˙ α β α ∇ and scalar curvature Λ of the Levi- α  δ ∇ α , ˙ 0 ˙ ˙ γ γ α γ = α ˙ ˙ ˙ δ ˙ α α ) = β αβ ˙ δ β twistor index  α  x B ˙ α ˙ αβ ( γ S A Λ + e Ψ A ˙ α ˙ αβ ˙ ρ α + − ˙ 0 α δ ˙ αβ ˙ ˙ β D α γ P  ˙ ˙ α α α , corresponding to a field valued in a rank four vector − , t e Ψ ∇ αβ ˙ – 12 – ρ ,S

γ ! = , ! β = ∇ ˙ ˙ α C ) ) αβ α := Φ α ˜ t t C x x T βα ˙ γ ( ( ; the four values of the index are decomposed into 2-  B D ˙

( C ) α α ˙ ˜ ˙ t t αβ α B Twistor indices are equivalent to SL(4 α =   ) α

˙ α P 8 ). In terms of geometric data, the potential A = α ( Aβ local twistor index C ]). A T , D ) = x C (4 ( ) 44 + ( A, B, C, . . . , ˙ β sl A B T 43 ˙ αβ T α ˙ α F α ∇ ] = ( = ˙ is the Schouten tensor, β β B ˙ is the Levi-Civita connection and the 1-form β D T ˙ α ˙ ˙ αγ α , α ˙ α α α α P ∇ D [D The reasons for considering twistor-valued objects on space-time are two-fold. First of Let us introduce a new kind of index, called a More precisely, this will be a 8 bundle over space-time whose fibresthe are ‘local’ copies of prefix the for (flat) much twistor of space this of paper, Minkowski space-time. as We we drop are not concerned with comparison to global twistors. and the action on higher valence twistor indices can be deduced by theall, Leibniz objects rule (cf. with [ twistor indicestion are is spinors itself of conformally the invariant. conformal This group and is the evident twistor from connec- the curvature of the connection: written in terms of theCivita trace-free Ricci connection. curvature The Φ actionthe of rule: this connection on twistor-indexed quantities is given by where where fied) conformal algebra The key property of twistorinvariant indices connection is on that they space-time,conformal are connection. known acted On as on any by the 4d a space-time, (local) particular this conformally- twistor twistor connection connection is or given locally Cartan by Twistor indices can beponents paired of with the ordinary resulting spinor field indices, are with trace-free, the for constraint instance, that com- space-time fields (cf. [ and will be denotedspinors by of opposite chirality.fields are For decomposed instance, into rank spinor one fields as: covariant and contravariant twistor JHEP07(2018)016 : Aβγδ (3.14) (3.15) (3.16) (3.11) (3.12) (3.13) (3.17) ) imply ]. 3.17 46 , 45 ). conformal gravity in 3.2 on-shell, which can be . , , . = 0 αβγδ ) = 0 γδ = 0 ! non-linear ˙ ˙ ˙ δ δ βα αβγδ ˙ γ ˙ ( γ γ ˙ ˙ α α , ˙ α , β − ˙ δ ˜ γ e Ψ ! = Ψ . ˙ γ ˙ β ]. obeys − = 0 βγδ α ˙ δ βγδ α ˜ γ 47 = 0 ˙ γ ˙ α ˙ αβγδ αβγδ ˙ β in terms of Ψ Ψ β , βγδ γ ): β ˙ ˙ Ψ Ψ α β α αβγδ ∂ β ! e Ψ

Ψ Aβγδ 3.6 ˙ ˙ δ ˙ β δ ˙ ˙ ˙ – 13 – β γ Γ αβγδ ˙ β γ ˙ = β ˙ β ˙ γ ⇒ β β ∂ ˙ ∂ α α β ˙ ˜ γ e α

Ψ , ∂ D α Aβγδ =

∂ Γ ˙ = 0 δ = 0 ˙ = γ ˙ β ˙ δ ˙ γ A αγδ ˙ β ˙ e α Γ αβγδ A ˙ γ β Ψ e Γ ˙ β β β D and the component Ψ ∂ ) as a linear field on Minkowski space-time, and impose a free equation ) this is equivalent to a system of equations for the components of Γ βγδ ( A 3.9 Aβγδ = Γ obeys the positive helicity free field equation of conformal gravity. ˙ δ ) and ( ˙ γ ˙ Aβγδ β ˙ α 3.7 e Ψ It should be noted that there is also a description of A similar story holds for the positive helicity free field equation. In this case, one has Consider a twistor-spinor field of the form that terms of the twistor connection.level, Indeed, to conformal a gravity gauge isYang-Mills theory equations equivalent, at of of the the the non-linear twistor twistor connection connection, [ with the Bach equations given by the obeying an equation of motion in Minkowski space. Just like the negative helicity case, the coupled equations ( a twistor-spinor field substituted back into the first equation to give i.e. the negative helicity free-field equation of conformal gravity in ( From ( The second of these equations defines Let us treat Γ of motion using the twistor connection in ( where Γ conformally invariant theory suchorder as equations conformal of gravity. motion withhigher-derivative The respect equations second to on the the reason local components is twistor of that connection a often first- twistor-valued correspond object to [ which is conformally covariant. Hence, this formalism is ideally suited to describing a JHEP07(2018)016 ) α ,B (3.23) (3.24) (3.19) (3.21) (3.22) (3.18) (3.20) ˙ α ) spinor ˜ C B 2 fields). , − = ( A B Rarita-Schwinger 2 3 − ]. Dotted SL(2 ]). Define the following 48 34 , 22 The operator , , matching the on-shell counting 9 , , } , = 0 x 1 2 α . · α ˙ k . α − i B  ˜ e B α α . ˙ δ α B,B λ = 0 ] = λ ˜ λ { ˙ α , λ α ˙ γ α β ˜ ˙ λ = 0 α β λ B B [ ∂ i ˜ λ ∂ 2 β α ∂ ∂x , etc. λ i λ − Aβγδ αβ A − – 14 –  ˙ α Γ , B  + i α ˜ λ A λ 1 2 ∂B ) spinor indices are negative chirality and we use the notation: , ˙ ∂ obey ˙ α = β α C := C ˜ β ˜ λ , . These choices are, of course, constrained by the fact B A = } A ∂ ∂ ] = = = 0 B ˙ ˙ α α α C is β Aβγδ ˙ ). Consider a negative helicity field constructed as: ˜ α ˜ β B λ B Γ [ ˜ λ β β ∂ ∂B , since its role is to convert the helicity B,B 3.17 Aβγδ { i ≡ to give: ˙ α βλ h ˜ B ) and ( , ˙ β ) becomes a simple PDE in on-shell momentum space: is an on-shell (massless) 4-momentum. ˙ α  3.13 ˙ ˙ α α ˜ ˜ λ β 3.21 ˙ α β ˜ λ λ = = ˙ β ˙ ˜ α β α ˙ β k ˜ λ Since there are three total d.o.f., we should be able to construct three distinct states by The components of this operator are constrained by a single condition, which descends ≡ helicity lowering operator Our conventions for the 4-dimensional spinor helicity formalism follow [ ] 9 ˜ β ˜ λ making different choices for that the resulting fields must satisfy the equations of motion. Toindices begin, are positive consider chirality, un-dotted a SL(2 negative [ which reduces the d.o.f. tofor three, parametrized conformal by gravity discussed above. the constraint ( This can be solved for Assuming that the components of The condition imposed on Γ as dictated by conformal invariance. from the twistor geometrytwistor-indexed underlying differential the operator, construction which (cf. acts [ on on-shell momenta: is a field into a negative helicityIts conformal gravity components field have (which mass must dimension include helicity To see the counting offind on-shell solutions states, of it is ( useful to go to a momentum eigenstate basis to where 3.2 Momentum eigenstates JHEP07(2018)016 , ) 1 2 1 2 ; it − − α 3.15 λ (3.26) (3.29) (3.30) (3.27) (3.28) (3.25) . This α ∝ λ α . Since the , B ˙ α ˜ β ! ˙ α . ˜ λ , ] ! ˜ β i | of conformal gravity ! ˙ , it is easy to see that β spinor x ˜ α α | β ε trivially satisfies ( 0 a λ ˙ a λ 1 2 β λ h has mass dimension h α α ˙ i α . The equations of motion 2

a x α ˜ λ α κ αβγδ i − B 2 , since these backgrounds are ˙ = κ λ α 4 = 0. , − ˜ β x . 1. Furthermore, conformal grav- A · =

k

x − x i · α · αβγδ e k = k i = i B ˙ α Ψ e ,B e A ˙ ˜ β α A ) : the coupling δ = 0. In summary, the Einstein mode ˙ δ } α α N λ ) λ α ∂ δ γ γ B x ,B λ λ πG ,B β γ β 8 } : , κ λ λ λ λ α o x √ 0 α ˙ β α β { ( ˜ λ – 15 – β , a a = ˙ α β κ λ 0 = ( α { = λ κ } = α = , x = ] has mass dimension } ˜ β 0 αβγδ α κ αβγδ ˜ λ [ Ψ B,B g αβγδ Ψ { n = 0 and carries the opposite little group weight to Ψ B,B i 6 = { 1 mode. Note that although this is a solution of the Bach equa- } a λ α h − B,B { Einstein: Spin-1: which satisfies α a to a constant of mass dimension +1, but no such object can be constructed from ) is dimensionless, whereas Growing: The third linearly independent solution is the growing state: With this in mind, a negative helicity Einstein graviton corresponds to By singling out this two-derivative solution inside the space of solutions to the four- B 2.9 such growing states must be excluded from the set external states in defining an S-matrix. which is specifiedlinearised by field a strength choice grows linearly of with a constant, mass dimension + tion (without solving anysince lower-derivative it equations), is it purely is oscillatory. a suitable state for scattering Taking into account the mass dimensionthe and field little strength group weight of corresponds to a helicity There are two other linearlyof independent motion. solutions, which The solve firstspinor the of 4-derivative these equations is specifiedstate is by referred the to choice as of a a ‘spin-1’ constant, state: mass dimension leading to the usual negative helicity Einstein graviton: the dimensionful scales of thisset solution allow us to fix the spinors and scalescorresponds to: at hand. Thus, we set ity cannot distinguish between Minkowskirelated space by and a (A)dS duces conformal a transformation. cosmological constant Selectingbackground, Λ, this the of corresponds Einstein mass to dimension Λ solution = +2. therefore 0. intro- Since weremains consider to a determine Minkowski the constant of proportionality. Since by virtue of obeying the zero-rest-mass equation derivative Bach equations, masscoupling scales constant are of introduced Einsteinin by ( gravity necessity. One of these is the helicity Einstein graviton: this is a field whose Weyl spinor Ψ JHEP07(2018)016 (3.39) (3.40) (3.31) (3.33) (3.41) (3.36) (3.37) (3.38) (3.32) (3.34) (3.35) . ! α λ ] . , ˜ λ | x ! γ · x ] . | k β i ˜ λ ˙ β ) α e , h Rarita-Schwinger field γ a α ) [˜ i ˙ 2 ˙ x δ α 3 2 ! α ˜ λ ˜ a , ,A ˙ ˙ λ − α γ ˙ 0 α i 2 ˜ ˜ λ λ α ˜ ˙ A β − β

˜ λ ] = 1 . The various objects appearing α κ = ( ˜ ( ˙ } A , ˙ a [ . α = = = A . 1 2 ˙ ˜ α A = ˜ , A A A ˜ − A  ˙ δ ˜ ˙ ˙ α γ , A, α , ˙ β { ˜ λ = 0 ∂ ˙ ] = 0 α 1 2 ∂λ ˙ α α α i α e 2 = 0 Ψ a λ [˜ ,A ˙ − δ . i ] = ˙ ∂λ ∂A γ , ˙ x

α β ,A ˙ · − α γ k A + x A ˜ λ i , · β – 16 – [ α ˜ e ˙ e Γ ˙ ,A k , α α A x  i · γ ˜ α ∂ A 1 2 e ,A k A ∂λ o i β ˙ e δ ˙ ˙ C α := α ) e

, x ˙ ˜ δ λ ˜ ˙ ˙ = ˜ λ , i λ δ α ˙ γ A α ] = i ˜ ˜ a λ 1 2 α e ˜ λ α x C ˙ γ , κ ˙ , λ β β ˙ β A γ − 0 ˜ λ [ 0 h A ˜ λ ˙ ˜ λ β n   ˙ A β ˜ A λ ˙ ˜ ] = λ α = = = A ˙ α α : ˜ λ ( ˙ } } } ˜ α ˙ ˙ ˙ A = ˜ κ λ α α α [ ˙ A ˜ ˜ ˜ δ A A A ˙ γ = = ˙ β ˙ ˙ δ δ ˜ ˜ ˜ ˙ ˙ A A, A, A, γ γ ˙ ˙ { { { β β e Γ ˙ ˙ α g α e e Ψ Ψ A A helicity raising operator Spin-1: The decomposition of these three d.o.f. into distinct states is accomplished in the same The construction for the positive helicity sector proceeds in a similar fashion. In this Einstein: Growing: and the corresponding field strengths are: The constant spinors appearing in this decomposition have mass dimensions as dictated by conformal invariance. way as in the negativemode: helicity case. The result is one Einstein, one spin-1 and one growing As expected, there are three d.o.f.in parametrized the by definition of the helicity raising operator have mass dimensions With some elementarynents of assumptions, this translates into a conditionwhich can on be solved the for compo- in terms of the momentum space differential operator case, the conformal gravityvia field a is constructed from a helicity + This field obeys a constraint JHEP07(2018)016 ˙ α a , ˜ α (3.44) (3.43) (3.45) (3.46) (3.42) a spinors , 1 2 x · − k i e ˙ δ , ˜ λ , ˙ γ β ˜ λ ˙ a 6= 0 β α ˜ ] λ in terms of a dimensionless ˙ a ˜ α λ , ˙ ˜ β λ ] ˜ correspond to additional d.o.f., i β [ ˜ λ ˜ λ α ˙ α ˙ γδ α ω ω λ ˜ ˜ β  β h ˜ [ λ , , ˙ α α αβ drops out at the level of the gauge = = a  . ˙ , β ˙ ] β , 6= 0 α α ˙ = ˜ α = ˜ λ ] a x ˙ a · αβ ˜ a ˙ spinors can then be fixed as ˜ λ α a k , ˜ , (+2) α ˜ i β ˙ a α α α e [˜ ˜ (+2) abcd a a β , ab α ε , β , – 17 – , ε = 1 = [˜ = i ˙ β i = 0 ˜ a ,R ab ˙ i 6 α β λ h x a λ · h ˜ a h k i α β β λ e h a λ , and assume that the mass dimension . Since the growing states are excluded from this class, δ α ab ab = λ ] setting the overall scale. λ ε , h γ ˜ λ α ) are normalized to obey λ = ˜ β β β ˙ β = 0 λ ,[ 2) α 3.38 i 6 i ˙ αβ − λ ( α ˙ δ ε a λ β λ ˙ γ h h  ˙ β ˙ ) and ( α  = 3.27 2) − ( abcd . Then the polarization tensors for negative and positive helicity Einstein modes R α ω In summary, the twistor-spinor formalism provides an easy way to capture all of the One could worry that the constant spinors confirming that they correspond tothat negative all and positive dependence helicity oninvariant Einstein field gravitons. the strengths, Note constant as required spinors in the Einstein sector. which are the usualformalism. expressions for It Einstein isassociated graviton a with polarizations these straightforward in polarizations, exercise the to spinor calculate helicity the linearised curvature tensors This normalization can bespinor viewed as expressing are given respectively by: for some constant polarization appearing in ( To enable comparison with theuseful results to obtained have in expressions thestandard twistor-spinor for metric formalism, the perturbation it scattering willthis be states entails of finding polarization conformalconformal tensors gravity gravity for in field. the terms To Einstein do of graviton this, the and we spin-1 write modes of the d.o.f. of conformal gravityraising/lowering in operators a serve way as that ‘polarizations’:operator, is we manifestly by can conformally selecting single invariant. which out d.o.f. The the helicity appear individual in states3.3 the of the on-shell Polarizations theory. and double copy already fix each spinor up to a scale. The with scalar products gates of their negative helicity counterparts. but this is not the case. Indeed, the conditions It is easy to see that in each case, the positive helicity fields are simply the helicity conju- JHEP07(2018)016 , ]: 22 ] and is the (3.47) (3.49) (3.51) (3.50) (3.48) 51 F . x · k double copy i e  ) ˙ β ˜ λ α ( ˙ . ˜ a ) (as well as two scalar β δ a λ α γ a 3.45 λ ˙ δ ) . ˙ ˙ γ β  α ˜ a a α . ( ˙ αβ ˙ α  ˜ λ x · ˜ λ 1 2 k i = e = + ˙ ) β ) ˙ , β ˙ δ ˙ ˙ αβ α ˜ λ ˜ λ (+1) α (+1) α α γ ( ˙ ( ˙ = 0 ˜ a ˜ a but at the linearised level there is no distinc- β ab β λ λ 10 F is the gauge covariant derivative and α α – 18 – , e , ε a λ ˙ ˙ λ β α ∂ ]. D ˜ a 1 2 ˜ a γδ  ˙ α  55 α ˙ – ˜ a β = λ ˙ ) α ˙ β  β 53 ˙ = , α αβ a 1 2 α Φ 1) ( 22 , where ˙ + , α − 2 λ 1 ( α ) ) gauge theory has an ambitwistor string description [ δ e = λ 2 ˙ γ ) β DF ]. λ 1) ˙ αβ β − 52 ( α λ DF ε α ( a ˙ δ ˙ γ  ˙ β ˙ α  ]. The two gauge theories which form the basis for the conformal supergravity  ], it was shown that certain ‘non-minimal’ conformal gravities obey = 50 ), while the other two terms are contributions from a linearised Ricci tensor 28 , 1) ]? The linearised equations of motion for this four-derivative theory are 49 − There are important differences between non-minimal conformal gravity and the stan- In [ As for the spin-1 sector, we can take the same polarization tensors as those used in [ ( abcd 3.28 In particular, non-minimal conformal gravities have additional scalars which couple to the graviton 28 R 10 of [ through an arbitrary function, cf. [ It is easy to seepositive that and taking negative symmetric helicity squares Einsteinpolarizations, of graviton corresponding these polarizations to polarization ( aBut vectors generates dilaton what the and about axion, the as polarization expected data from corresponding the to double the copy). four-derivative gauge theory Indeed, in double copy the polarizationssymmetric of products the gravitational of theory the shouldtheory, be polarizations one expressible in as has the only appropriateare gauge the given theories. negative in and spinor In helicity positive Yang-Mills form helicity by: gluon polarization vectors, which also plays an interesting rolebosonic in string the theory construction [ of scattering amplitudes in heteroticdard and ‘minimal’ conformal gravity wetion study, and we expect some remnant of the double copy to be visible in the polarization data. constructed by tensoring togetherries [ kinematic numerators fromdouble two copy different are gauge Yang-Millsmotion theory theo- and and a a couplingtheory gauge constant theory is of with schematically mass four-derivative ( field equations dimension strength. of +1. This The ( kinetic term of this latter These Ricci tensor contributions canthus decouple be from removed any by scattering a amplitude conformal calculations. transformation, and will in the sense that kinematic numerators of the theory’s scattering amplitudes can be The first term isof a ( linearised ASD Weyl spinor, corresponding to the desired behaviour For the negative helicity mode, this leads to a linearised curvature JHEP07(2018)016 ) ), 3.18 3.52 (3.52) (3.53) in ( true of a A B theory. The ; such modes not a 2 ) n DF theory gives precisely 2 ) , x · DF k ]. The same is i e 56  ) ˙ β ˜ a and time-like vector α ( ˙ a ˜ λ . ˙ α αβ ˜ a  α + a ), so the amplitude is a function only of the ) 3 β = λ – 19 – MHV) or two negative and one positive helicity a ˙ α α ∝ (0) α ( e 2 λ ˙ λ β ˙ α  ∝  1 λ ) for the scattering states in conformal gravity (plus some = i , for some polarization B (0) ab 3.47 x · F k i e ), ( x · 3.45 n a B ). ∼ is the field strength of a gauge potential. Clearly, gluons are a consistent g a ). There are also two spin-1 growing modes (one each of negative and positive ) and the polarizations of the scattering states in ( 3.31 A ab MHV configuration, momentum conservation dictates that all un-dotted momen- F in ( 3.50 3.50 A The only potentially non-vanishing 3-point amplitudes involve two positive helicity Therefore, we expect any formula for 3-point amplitudes to depend explicitly on these From this, we conclude that there are five on-shell d.o.f. in the ( Finally, the theory also includes purely oscillatory solutions with a polarization A and one negative helicityexternal external fields fields ( (MHV). ThisIn the follows from thetum spinors integrability are of proportional the ( self-duality equations. or helicity raising/lowering operators for eachtors particle. as Indeed, higher-derivative ‘polarization’ one data canAmplitudes think for for of the the these external opera- states specificexplicit in choices helicity a for states scattering these process. of polarizations. conformal gravity are obtained by making fixed by specifying thehigher-derivative helicities theory of such the as externalstate conformal of particles gravity. positive/negative [ helicity The isstate notion not being unique: of scattered one an is must an external additionallyformalism, Einstein scattering graviton specify this or is whether a captured the spin-1 by field. the choice In of terms helicity of lowering/raising the operator, twistor-spinor 4 3-point amplitudes inThe conformal twistor-spinor gravity representation ofgive scattering a states compact for expression conformaltwo-derivative gravity for theories, enables all Poincar´ecovariance us means of to that the a tree-level 3-point 3-point amplitude amplitudes is in uniquely that theory. In of motion in ations strict ( sense. Taking symmetricthe products polarizations between ( the Yang-Millsexpected polariza- scalars), as dictated by the double copy. which is helicity zero, having both SD and ASDspectrum parts consists (of of: equal positive magnitude). ative and helicity negative spin-1 helicity growing (Yang-Mills) modes; gluons; and positive a and scalar neg- which obeys the four-derivative equations The little group weight ofof this freedom; polarization this indicates is that confirmed it by corresponds computing to the a linear scalar field degree strength associated with ( subsector of solutions totions these ( equations, sohelicity): we get anotherare copy excluded of from the the set gluon of polariza- acceptable scattering states. where JHEP07(2018)016 ) ) are 4.3 (4.2) (4.3) (4.4) (4.1) i 3.37 a , ! i ˜ λ and ˜ i i λ a 3 =1 i X ,

) passes some basic (4) ! have mass dimension i δ 4.1 ˜ λ , j i λ A !  ). The first of these ( ) results in: i · ˜ λ 3 i =1 i i X 4 λ B [3 1] 3.38 3.25

2 3 . Alternatively, we can simply =1 [2 3] i (4) X δ and ) for the lowering operators. The C [1 2]

. j 2] 2 ) and ( ) and ( e (4) C 2 A δ 3.27 · a · [˜ i i 1 3.27 i ) = 0 1 3.37 2 A 1 B 1 + 1 3 a  ) or ( h , a 2 ) that is actually non-vanishing. Note that [1 2] h 5 + e C 4 2 · ), and acts on everything to its right, including 4.4 – 20 – , 3.25 [1 2] [3 1] 3 − [2 3] ) comes by evaluating it when the three external A [2 3] 2 (1 3.32 + 3 4.1 ε κ ε κ  M ), which means that the amplitude is, in fact, zero by ]. The opposite is true of the MHV configuration, and 2 + ) = ) = 4 2 57 , + + , + 3 3 , , 3 56 [2 3] 2 , + 1 ˜ a [1 2] [3 1] 2 is given by ( , a , 3 ( 1 1 3 ]. a A a i A ( ( · e 3 C 3 1 59 , −M B M 2] = 1 thanks to the initial normalization of the external wave-functions, M ], which is summarized in appendix 33  2 3 a e 59 C ) = , · + 2 3 , A 58 ) and then check that it is correct. It is easy to see that ( ,  + MHV 3-point amplitude of conformal gravity is given by: is the dimensionless coupling constant, and particle 1 has been chosen to have ) for the raising operators, and ( ε 2 34 ε , 4.1 = 1 and [˜ 1 = a i Therefore, it is only the amplitude ( But what about more general scattering configurations? In particular, we are free to The first substantive check on ( This formula can be derived directly from a formulation of conformal gravity in twistor The 3 3.38 ( 1 3 1 M a is anti-symmetric under theM interchange of thecrossing-symmetry. two positive helicity Einstein gravitons: h with all othersconstant vanishing spinors as parametrizing spin-1 a modes result in of ( momentum conservation. Here, scatter any combination ofshow Einstein that gravitons the and only spin-1 non-vanishing modes. results are It in is the configurations: straightforward to This is theensures expected that result: the tree-level theflat S-matrix background embedding of [ Einstein of states Einstein in gravity conformal gravity into vanishes conformal on gravity a operators, as expected.zero, Further, so combinations the coefficienthas of mass the dimension coupling +1, constant as and required momentum for conserving a delta conformallyparticles function invariant are theory. all Einstein gravitons. Plugging in ( the momentum conserving delta function. space [ posit ( consistency tests. The formula is linear in each of the external helicity raising/lowering where negative helicity whilst 2ators and 3 for have each positive particleor helicity. can The ( take helicity any raising/lowering ofdifferential oper- operator the allowed scattering state forms; namely ( the two should benegative helicity). related by complex conjugation (i.e., exchanging positive helicity for dotted momentum spinors [ JHEP07(2018)016 (4.8) (4.7) (4.5) (4.6) MHV MHV , ]. ]. . When i ! 61 61 i ˜ λ i λ ) is correct. 3 =1 i X 4.1 ,

! (4) i δ ˜ λ i . λ  i ! 3 =1 i i X ˜ λ 4 3 1 i i h

λ 2 . (4) i 2 3 3 δ h =1 i ) on this configuration, we re-write X   1 2 2 ˙ h β

4.1 ˜ ∂ λ 3 ). This proves that ( 6 ∂ ˙ (4) β [3 1] B δ α ˙ · α ∂ 2 1 4 2 ˜ λ ∂k [2 3] 3.47 ˙ A ˙ β β 2 ˜ ˜ β β  [1 2] ]. This is simply the vector-vector-graviton ), ( [2 3] [1 2] i 2 ] + 1 56 C − ˜ · – 21 – β ˙ α ε κ 3 3.45 ˜ β [1 B 2 ) acting on the on-shell momenta of particle   ) = + + ε κ  → 3 2 3.20 , i g A 2 ). 4 +2) fields [ ) = ˜ a ). B i 3 1 , , + 1 3 ) can also be evaluated — formally, at least — on growing i h 2.23 2 3 a 4.5 , +1 h ( , + 3 1 2 4.6 1 2 h , − M 3 1 ) ˜ ), ( β B ( · 4.1 3 1 4.1 A M ]. However, at 3-points there exists a special degenerate configuration  3 60 C · 2 MHV sector, this configuration is given by a negative helicity growing mode and B  the differential operator ( ε A i = C For the Observe that ( The helicity-conjugate MHV amplitudes are captured in the obvious way by the fol- This result (as well as the vanishing of all other external configurations for the 3 M these contributions decouple leaving: which matches the ‘strange’ 3-point amplitude of conformal gravity observed in [ The un-dotted entries ofthe full this external polarization, momentum,conservation proportional in are to amplitudes what involving a the generically derivative growing lead with mode. to respect But the in to breaking this special of configuration, 4-momentum involving a single growing mode which results in finite,two well-defined positive helicity amplitudes Einstein [ gravitons.the To evaluate growing ( mode’s twistor polarization as a momentum space differential operator: expected sense: theypolynomial are growth not of the supported mode curvature, onmomentum which manifests overall conserving itself 4-momentum as delta derivatives conservation of function. due‘amplitudes’ the overall in to Such conformal the highly gravity have distributional previouslystring been expressions theory computed for in [ 3-point the context of twistor- with evaluated on specific configurationsis for the the helicity external conjugate states, of the ( only non-vanishingmodes. result In this case, the resulting 3-point amplitudes are generically undefined in the lowing analogue of ( amplitudes of helicity ( amplitude of Einstein-Maxwell theory,conformal gravity as given expected by ( from the two-derivative formulation amplitude) has been confirmed byspace-time, direct using calculation on-shell from polarizations the ( conformal gravity action on ing/lowering operators drops out of the gauge-invariant scattering amplitude: Not surprisingly, the kinematic part of this expression matches the expected result for meaning that explicit dependence on the constant spinors appearing in the helicity rais- JHEP07(2018)016 (5.2) (5.3) (5.1) ]. ] (see 2 12 ) by the 11 , 10 3.9 -tuples [ , s ) s ( ˙ β ) , s ( ˙ α 1 twistor indices. Since e = 0 Ψ − ) with all negative chirality s s s β ( s ˙ β α )  s +1) ( s ˙ ( α β ··· e Ψ , 1 1) ) β s − CHS field is encoded in its linearised 1 totally symmetric covariant twistor 1 ( s ˙ α ( β s = 0 − )  A s s ( in derivatives of the potential, traceless -derivative equations of motion for a free + β s +1) s ) s s ( ( β β ). ) 1) s ( − 3.2 – 22 – , ∂ α s ( ) defined on a Minkowski background. The action Ψ A s are totally symmetric. In terms of these ASD and ˙ = 0 Γ β 3.6 ˙ s ) ) β ˙ s s α β ( (  ˙ β β D ) ) s s ( ( ··· ˙ α α 1 ˙ β e Ψ Ψ 1 , which is of order ˙ ) ) α , s s )  ( ( s b ( α ) ) ˙ = β s s ) ( ( ) s a be a twistor-spinor field with s ( α -tuple of indices, and anti-symmetric between the ( α C s b ∂ can be obtained from twistor-spinors with ) s +1) = 2 Bach equations in ( ( s s a ( s β C + 1 totally symmetric negative chirality spinor indices, which obeys the free 1) s − be considered as evidence that growing modes should be included in the set s ]). The symmetries of this Weyl tensor mean that it can be decomposed into is the twistor connection ( ( ˙ A β β 63 , not Weyl tensor, 62 s Let Γ Note that the existence of this well-defined amplitude involving a single growing mode Note also that the quantum-mechanical definition of a growing mode as an asymptotic state will still 11 of the twistor connectionLeibniz on multiple rule. symmetric twistor Assuming indices that follows the from component ( ofbe Γ problematic even if thesolution multi-linear returns piece some of finite the result. classical action evaluated on the corresponding classical indices, and field equation where D Just as the 4-derivativetwistor-spinors equations with of a single motion twistor forCHS index, field conformal the of gravity 2 spin canthis be construction obtained builds from much naturally briefer on than that in the for previous conformal section; further gravity, details our can exposition also be will found be in [ generalizing the 5.1 Free fields and momentum eigenstates where the spinors Ψ SD parts of the (linearised) higher spin curvature, the free field equations become the CHS gauge fields. Thespin- dynamical part of aon spin- each symmetric also [ anti-self-dual and self-dual (or negative and positive helicity) parts: 5 Free fields and 3-pointThe twistor-spinor amplitudes formalism in also CHS encodesthe free example theory of CHS conformal fields gravity, this of allowsfor any us the integer to scattering spin. states write of down Building CHS a theory,all on momentum which eigenstate 3-point in basis tree turn translates amplitudes into of compact the expressions theory. for To do this we work directly with the curvatures of should of scattering states: inRather, it general, demonstrates growing that modes certaining lead degenerate a configurations to fixed may un-defined number exist of scattering in growing which amplitudes. modes scatter- may ‘accidentally’ lead to finite amplitudes. JHEP07(2018)016 s s (5.9) (5.6) (5.7) (5.8) (5.4) )) 3.14 , a spin- 2 )–( ), as desired. 5.2 3.11 (cf. ( have been eliminated, +1) s ( +1) , β s ( x 1) β · k . − , 1) i s e ( − s A +1) ( +1) s . α . , s ( +1) ˙ ( , , , . β s β ( 1) β 1) = 0 = 0 = 0 − λ − s = 0 = 0 = 0 = 0 ) ) s . . . (5.5) ( s s ( 1) ) ) ) ) ( ( A +1) s s s s α ˙ coupled equations for the components of − β β s oscillatory modes. This decomposition is ( ( ( ( e Γ ) ) ( s γ γ Ψ ˙ ( s s s β ˙ βγ ( ( s αγ 2) 1) A ˙ α α 1) ≡ 1) 2) ) in terms of the local twistor connection gives − − B e s − s except for Ψ Ψ Ψ − – 23 – − obeying the obvious symmetry properties, and ( ( s ) ) s s ˙ ( ( α s s ( = 5.3 +1) ( ( ˙ ˙ A α βα ˙ ˙ s β β ˙ ˙ ( βα βα e +1) Γ ) ) Γ Γ +1) s ˙ β s s = 0, this component can be identified with a spin- s ˙ β Γ Γ ( +1) CHS field, this means finding a twistor-spinor repre- ( ( β ( s β ˙ − β β β β 1) ) β ( s s − − ∂ ) ∂ ∂ − β 1) ( D 1) s . s ) ) ( β − ( s s − 1) 1 s ( ( α s ( B − α βγ ( Γ s A βγ A ( 1) 2) ˙ αβγ 2) e A Γ − − 1) growing and 2 s 2) s − Γ ( ( s − ( α α − s ˙ α ( 1 Ψ s α α α ˙ β ( Γ Γ β s ˙ ˙ β β ∂ a totally symmetric negative chirality spinor. β β + 1) on-shell d.o.f., which are split evenly between negative and positive ∂ ∂ s ( +1) s s coupled equations for the components of Γ ( , which imply β s 1) there are +1) − s s ( s ( ˙ β α 1) − Following the lesson of conformal gravity, we look for momentum eigenstate solutions The twistor-spinor representation of free CHS fields can now be used to provide explicit For the positive helicity sector, the conjugate construction holds. That is, we begin Unpacking the equation of motion ( s ( A e which are obtained in aspin. helicity raised/lowered For manner a from negativesentative zero-rest-mass of helicity fields the spin- of form: lower momentum eigenstate expressions for scatteringCHS states. field has As discussed inhelicities. section These d.o.f. canat then be spin decomposed into growingdiscussed and in purely detail oscillatory in modes; appendix This equation isΓ equivalent to a system of for the totally positive chirality spinor part of with a twistor-spinor field impose an equation of motion it, until allleaving components the of single Γ free field equation: This is precisely the negative helicity free field equation of CHS theory in ( Starting from the upper-most equation, each of these relations can be fed into the one below a system of linearised ASD Weyl spinor: with Ψ spinor indices obeys Γ JHEP07(2018)016 is = ) , it h ) h 6= 0. h − (5.14) (5.15) (5.11) (5.12) (5.13) (5.10) ) s − h ( s ) obeys encodes ( α − are fixed α s a ( 5.9 1) λ 1) α , − − λ s x in terms of a · ( ) s ). ( k h α i ) of conformal A 1) − e B s , B − ( 5.11 s )) α ( s 3.18 1) a ( A = 1), and − β k 1 B . λ − κ ) ) s I h o ( ( − α 1) α k I , ˙ α − β λ , s linearly independent choices ˜ ) λ ( B h x | ∂ α · s = 0 I − k s i ( −| e 2) , k α ) ( − which satisfies s ∂ s a ( ( I β s h = 0 A α ,...,B (by definition, 2 λ − 2 twistor indices. 1 κ ( ˙ 2 ). This translates — with some mild h ) α ˜ λ α h s − 1 = +1) I B α s α s β ) − ( λ 1 3.20 s β λ α ( 1) | β λ ) ,B 2) ) I − h | s α − s ( − s (  – 24 – ( α ( + i α i . The remaining components of 2 ( A Ψ s 1 2) B − 2 B,B − A − 1 s n  . Γ ( = h 1 ). Labelling this family of solutions by an integer A ) , =0 s A 1 k | − ( ˙ I α scattering states are then obtained by making choices of X B β | C 1 totally symmetric twistor indices. The field ( ) ↔ 1)) scales with the opposite little group weight to B s s d.o.f. in this set of symmetric spinors, matching the count − ( = − 1 ) h ˙ α α h ( s through the relations ∂ 1) ˜ λ α Ψ − 1) ). One can show that there are s +1) − } 2 ∂ λ − ( ) — into a set of constraints on the helicity lowering operator: s k s ) ( 1) α ( s − , but since these are not admissible scattering states we will not h 5.6 a − s A − ( s s ( has helicity 3.22 B α B ( α ) ) α k s ( ( ( ˙ a 1. The resulting components can be shown to satisfy ( α β ) ) h h s B is a generalisation of the helicity lowering operator ( − ( κ − ( α having mass dimension 1) = B,...,B − is the momentum space operator ( 1) s { 1) ( is a coupling constant of dimension 1 − − , . . . , s s A A which are constant (in position space), leading to purely oscillatory fields. Of s ( h ( , the helicity raising operator components and spinor fields are: α B C κ α h 1) = 0 B B − The helicity lowering components associated with growing modes are also easily de- The negative helicity spin- These relations allow us to determine all of the components of s k ( α , . . . , s a totally symmetric constantWith spinor the of proviso dimension that follows that Ψ duced from appendix of freedom counting in1 appendix where B course, these choices are nothelicity arbitrary: field the equation resulting ( which spinor are field purely must satisfy oscillatory the and negative solve the field equation (this also follows from the degree by the set for for the negative helicitythe sector CHS of curvature CHS via theory. The highest-rank spinor, with set of negative chirality symmetric spinors: Sure enough, there are assumptions akin to ( which must hold for all values of the remaining gravity, now an object with a constraint, derived from the twistor geometry: where where JHEP07(2018)016 , 1) of − s (  all (5.23) (5.20) (5.22) (5.17) (5.19) (5.21) (5.16) (5.18) B +1 A obeying are then 3 s − s , 1 +2 s 2 − x 1 · h s particles have k i , . . . , s , , e 3 [3 1] . s 1 3 [2 3] )) s = 1  s 1) = 0 ( − 2 ˙ − h β s 1 k ˜ λ s [1 2] , MHV and negative for − +1) ) ) s ) s I h ( − o ( [3 1] ( ˙ K − β ˙ ˙ α 3 α +2 k 1) I + and spin . 1 ˙ s β J ˜ λ α 1) +1 − s CHS field via: s 2 ˜ ) A 2 − 2 A ( s + h s | ˙ ∂λ s s β A ( I − 2 . = 0 − ˜ 1 I [2 3] B s A s 3 s ( −| A e Γ 2) s k α ˙ 1 ( ˙ = α − ∂ s ˜ a α + | B ( [1 2] I h 2 D β B  K ) ,..., ( s κ | 1 2 J ˙ λ β K β MHV amplitude reads: − ≤ 1 = I I B ˙ A ˙ β α 2 A 3 ) 1 1 ˜ , s ˜ ˜ λ A β ( C A , ˙ x ∂ | , β · I 3 ) ˙ I K k β | ∂λ s ) s A i − ( ˜ h A e  1 ˙ J – 25 – α ( + − i , s − 2 B determining a spin- e I Ψ ˜ 1 3 A, 2) +1) −  A s ( s 1) − n , s (  s K 2 K ˙ − = ( α s s B B 3 ( | ˜ λ B =0 , 3 k | 1 B A J ˙ ˜ ≥ I X β | 1) C | ↔ A 1)) 1 − )! A J s s 2 − = ( 1 A 1)! h ˙ s ( β ( B ˙ 1) 1) ˜ K β λ − , − A i − − ˜ 2 B λ 3 1 s 1 k ) ( s A s = − ( h B ( − s − ( )! 1 ˙ 1 A s β s 3 ( s 1)! ) +1) ˙ s − β s k particle has negative helicity, the spin ( 3 ( ( − = s ˙ ˜ a − β α 1 | + 2 h 1 B s A 2 a totally symmetric constant spinor of dimension I s s | s κ e ( Γ 6= 0. ( ) 1)  ) = h h ) − s − − 1) ( s s ( − ( ˙ +( ˙ β s β ( ˜ λ ˙ a β h MHV or MHV. There are only mild constraints on the allowed spins of the external independent scattering states, labelled by their integer helicity ) ∼ N ˜ h A s − 3 s ( ˙ β M ˜ where the spin positive helicity, multi-index labels obey For spins satisfying these constraints, the 3-point either states: if particles 2MHV), and then 3 the have spins the of same the helicity sign three (positive external for states must obey: a 5.2 3-point amplitudesThe of twistor-spinor CHS representation theory forthe CHS tree-level fields 3-point allows amplitudes us of to the write theory. expressions Once for again, these 3-point amplitudes are The given by: with ˜ via the relations As in the negative helicityin case, terms this of allows us a to set solve of for symmetric all spinors, of the components of with a helicity raising operator subject to the constraint treat them explicitly here. The positive helicity states are derived in a similar manner, JHEP07(2018)016 (5.29) (5.30) (5.24) (5.27) (5.28) (5.25) (5.26) ) has the 5.22 . ,  ! MHV amplitude, as i )! ˜ λ 2 ), and contractions 1 mode of the spin 1)! i s λ − , − − 1 3.33 3 3 =1 1 i s X s ( (

2 As expected, ( s , (4) , . . . , s δ 1) 1) 12 − − = 2 3 +1 s 3 ( MHV amplitudes with the helic- 1 s defined by ( , + ( 1)! s B 3 . − h 2 )! − i A s = 1: 3 +1 ∀ ˜ = 1 ), we have stripped off an overall factor 1 1)! C 3 s 1 s s s ,A 1) ) = 0 − [3 1] + − defining the helicity h 1 3 − 2 − 2 1) 5.22 1 s 1 3 s as well as an overall momentum conserving s − +1 , s s 1) ( ( s 2 2 ( 2 ε − s s α ( 1 3 + 1 [2 3] − , s s s B 2 ( 3 1 – 26 – λ 2 , − s α 1 ]. s s 1 a 1) ( s − − ( 23 1) 1 [1 2] ,A , 3 s := ) = 0 ( − 1) 3 ) K ( α 1 M 12 s −  a 3 , ( , s 1 s 1 ), it can be seen that s 2 ( κ − 11 N 1 A 2 , s ), where the negative helicity can take any of the values s s 1 1 3 5.22 s  B h , s ε κ 1 2 2 i − ( , s  constant spinor 3 1 ) = ) s 1 s 3 ( s h M 2 . From ( , s − − N 1 1 2 s , s ) it is easy to see that the configuration in which each external state has − := 1 s 1 K leads to vanishing amplitudes: 5.20 − ( ,..., i 3 s 1 ), (  − M = 5.15 However, it is easy to see that there are other helicity configurations which lead The amplitudes for specific helicity configurations of the external states are read off The only ingredients in this formula are the twistor ‘polarizations’ of each external 1 Recall that only ‘anti-holomorphic’ square bracket contractions can appear in the s field. 12 h 1 for the dimension s all angle bracket contractions vanish. and we have used the normalisation while a non-vanishing amplitude is obtained for Here, there is an overall spin-dependent numerical factor to non-vanishing amplitudes.ity Consider configuration the ( − family of This is in linestates with in CHS the theory claim vanishes that [ the S-matrix of standard two-derivative massless HS and is linear in each particle’s twistor polarization. from this formulaing by ( inserting thehelicity appropriate twistor polarizations. For instance, us- particle powers of the momentum space differential operator of the on-shell momentum spinorsappropriate of mass each dimension external for particle. a scattering amplitude in a conformally invariant theory, is a spin-dependent normalisation constant.of In ( the dimensionless CHS couplingdelta constant function. and JHEP07(2018)016 . ˙ α α x (6.1) in terms 6= 0. In 2 up to the − R boundary is 1 4 4 Λ of the standard ) which leads to 4 → − instead of 2 − 0, where the analogue of R < = 3 | Λ | 0. Instead, one expects momen- < , manifest full momentum conservation 2 ˙ α ) α 2 x 13 x d ˙ α never α x d (1 + Λ – 27 – background with Λ 4 = 2 = 0 in the affine Minkowski space charted by metric is just the analytic continuation Λ 0) background the notion of ‘scattering’ is uniquely s 4 2 d x < with the metric: 4 ]. In any theory with (classical) conformal invariance, calculating 66 0) background, the analogue of a scattering amplitude is not uniquely ]). There is a mathematically consistent S-matrix propagating data from ]). In other words, the calculation of an AdS scattering amplitude in a ]). This means that our definition of a scattering state in AdS should be 66 67 > . 69 – – of the sphere. 64 0 is the cosmological constant. In these coordinates, the AdS 66 R < As above, we first discuss the case of the conformal gravity before generalising to the The second consequence is that there is a new dimensionful parameter Λ in play which For a higher-derivative theory like CHS, the altered boundary conditions of AdS have For clarity, let us focus on an AdS This slightly non-standard looking AdS metric, up to a rescaling of coordinates to take into account that 13 4 where Λ the hypersurface (3-sphere) 1 + Λ S of the radius full CHS theory. Consider AdS amended to be any solutionfinite to multi-linear the pieces free of equationsAdS the (linearised isometries action, around consistent AdS with momentum conservation was not available in Minkowski space.new As solutions we to will the see, linearised this equations parameter allows of us motion to which construct vanish in the flat space limit. conservation. But AdS scatteringas amplitudes there is notum global conservation space-like only Killing in vector thedirection when directions transverse parallel Λ to to the thementa AdS (cf. boundary boundary; [ is conservation in replaced the by a singularity in the transverse mo- half-space (cf. [ conformally invariant theory is the same as in flatimportant space, consequences. up First to of boundarybackground all, conditions. used our the classification criteria of that scattering such states states on lead a to Minkowski amplitudes supported on momentum classical action, evaluated on solutionsboundary to behaviour. the linearised It equationsthis of should fashion motion be are with easily noted specified related that,analytic to continuation at those [ obtained tree-level, from AdS thesuch in-in amplitudes formalism AdS obtained on ‘amplitudes’ in dS space at by an tree-level is equivalent to calculating the amplitude in a flat can define ‘scattering’ incontrast, terms in of an the anti-de in-indefined Sitter in formalism terms on (Λ of the boundary observable correlation patch functions. of dS.a By tree-level scattering amplitude is again defined in terms of a multi-linear piece of the The underlying conformal invariance ofstudy CHS ‘scattering’ theory of suggests CHS thata fields it de on should a Sitter be background possible (Λ defined with to (cf. a [ cosmological constantpast Λ to future infinity, but its elements are physically unobservable. Alternatively, one 6 Scattering in AdS background JHEP07(2018)016 ˙ δ ˙ γ ˙ β = 0 α (6.6) (6.7) (6.8) (6.9) (6.2) (6.3) (6.4) (6.5) γ 2 (6.10) ]). For x 70 and ˜ . . , )) ˙ αβγδ 0, although the γ = 0 , = 0 = 0 3.25 < 2 ˙ , δ ! x ˙ γδ γ ˙ δ ˙ ˙ γδ δ β ˙ γ ! ˙ γ ˙ ˙ α βα ˙ β ˙ ˙ ˙ β βα β α ˜ ˙ γ γ α α ˙ δ γ , β 0 . ˜ γ e Ψ 1 + Λ β γ α − δ . ! ∇

= 0 = ˙ ˙ δ γ ˙ β β ˙ βγδ ˙ γ α  ˙ β λ α β  , our previous basis of momen- ˙ ˙ α 4 βγδ α Ψ 0 α , β α ˙ e αβ Ψ β λ ˙ ˙ αβγδ x  = D x β β Ψ · γ ˙ β k ˙ δ Λ β ∇ ˙ i ˙ γ β 2Λ β ∇ ˙ e reads β β ˙

x α δ x A

4 α λ 2 + e Γ γ ). The linearised Bach equations for neg- x ˙ − κ ∇ β ˙ α λ β α β = 6.1 D 2 Λ βγδ λ ∇ ˙ A β ) in affine Minkowski coordinates (cf. [ A 1 + Λ γ B 0. In this limit, the hypersurface 1 + Λ = – 28 – = 0 = B ˙ α 6.5 − C of Minkowski space. , β → = B x )  I αβγδ ˙ )–( βγδ α , , = 0 2 α α Ψ ˙ x β Aβγδ 6.4 Ψ A ! β ˙ ) is advantageous as there is a manifest and smooth flat Γ β = 0 = 0 β ∇ ˙ 2 Λ δ + ( ˙ ∂ βγδ ˙ 6.1 α γ ˙ ˙ ˙ α α α αβγδ α 1 + Λ ˙ α αβγδ γ β Ψ ∇ γ ˜ γ + ∇ ˙ β

Einstein: β − ˙ = β ˙ δ β ∇ ˙ ˙ ˙ α γ αβγδ ˙ α β γ ˙ ˙ α β D = D β e Ψ ˙ ∂ β β ∇ Aβγδ is the Levi-Civita connection of ( Γ is a helicity lowering operator. ˙ ˙ β α β α A to rewrite the equations ( D 4 ∇ B Of course, the 2-derivative Einstein graviton solution remains for Λ In this metric, the twistor connection for AdS where helicity lowering operator picks up a non-trivial dotted component (cf. ( where all spinor indices arein now raised flat and space, lowered we with want theof to usual plane classify Levi-Civita wave linearly symbols. form, independent encoded As solutions by to a these twistor-spinor coupled equations Although the twistor-spinor formalism carriestum over eigenstate to solutions AdS does not.of AdS To see this, itthe is negative useful to helicity exploit sector theconjugation) one conformal finds flatness (the positive helicity sector is given by the obvious As in Minkowski space, theseleaving relations only can the be usual used linearised to Bach eliminate equations the in fields terms of the Weyl spinors In terms of the components of the twistor spinors, these equations are equivalent to where ative and positive helicitytwistor-spinors as free before: fields are expressed via the action of this connection on space limit given byapproaches simply the conformal taking boundary Λ Working in the coordinates ( JHEP07(2018)016 ] ) 72 only , 6.10 1 = 4 (6.13) (6.11) (6.12) be de- 33 − ). ) not 14 3.27 as well as the 3.27 ) disappears in x , up to potential ) can 4 , ) which exists 6.12 6= 0. Instead, we find 3.29 , -dependence in ( ! . Indeed, on the AdS x 4 ˙ 3.29 α ! ˜ ∂ λ β ∂ . ) for an Einstein mode is a α ˙ α λ ! β , one can now consider their 2i Λ α β ). ) when Λ x 6.10 4 λ λ − Λ 2 ˙ α 4.6 6.8

x β α i a 2 x κ i − )–( ), (

= λ a 6.7 4.1 ) remain valid in AdS h κ A ˙ α ) modes taken together represent the well-known 4.6 ˜ λ i 2 = Λ 6.12 ), ( A

constant spinor that appeared in ( – 29 – ,B B 4.1 = 1 2 ! A − α B ∂ ∂λ ˙ α ˜ λ ) and spin-2 ( ] that has only scalar gauge invariance and thus carries 5 2i Λ 21 − 6.11

Spin-2: κ = ). Spin-1: A the curvature associated with this mode is perfectly finite. A 6.12 1 Λ − = 2 ]. The truncation to scattering states in Minkowski space removes precisely x is the same mass-dimension 71 α a At first, it may seem that the twistor polarization ( Having obtained a basis of linearised states on AdS At this point, it is clear that Minkowski space is a singular background from the However, a straightforward calculation reveals that the mode ( Note that the above spin-1 ( 14 effective degrees of freedom. which are suitable for evaluation in the formulae ( partially-massless graviton mode [ momenta. Fortunately, this can be— rectified interpreted by as remembering that helicity the lowering/raisingmomentum twistor operators eigenstates. polarizations — This should meanswith that be we a thought can of momentum replace space as thepolarizations acting derivative. linear are on The then resulting positive and negative helicity Einstein indicates that this shouldscattering produce amplitudes a of result Einstein which gravity, where is the proportional constant to ofunsuitable the proportionality Minkowski is for space Λ. ourbut 3-point the amplitude twistor formulae. polarization of These the are Einstein momentum mode space is now formulae, a function of ‘scattering.’ We leave a more generalthat analysis of the this 3-point problem to amplitude futureboundary expressions work, but contributions ( conjecture to thethe scattering AdS of amplitude. three Einsteinwith modes; To Neumann the support boundary on-shell conditions this relationship and between conjecture, the conformal (renormalized) we Einstein-Hilbert gravity action consider [ when Λ = 0.gravity This [ is a reflectionthese of problematic the modes well-known from ‘linearizationmay the instability’ be external of the states case conformal of thatthe well-defined the spin amplitudes. space 2 of However, states scattering it ( states is enlarged away from Λ = 0 to include it does not makeboundary sense to refer to it asperspective a ‘growing’ of mode perturbative inthe conformal AdS flat gravity: space the limit, spin where two it mode is ( replaced by the growing mode ( an additional spin 2 mode: Although this mode has quadratic polynomial dependence on the space-time coordinates, where formed into a solution of the equations of motion ( Similarly, the spin-1 mode is given by a deformation of its flat space form ( JHEP07(2018)016 , ! ), one (6.17) (6.18) (6.14) (6.15) (6.16) 0 limit α 3 , ∂ 4.1 ∂λ → ! α 1 i . ˜ λ λ ˙ α i . This reduces α λ . ˙ + α ) , MHV amplitude α 3 ˙ α 1 ! ∂ 3 =1 ˜ λ i ˜ i λ ∂ ! X ∂K ˜ λ 3 i ∂ i ˜ λ λ ˙

˙ α α 3 λ i α 2 ˜ λ λ (4) ∂ + 3 δ =1 2

MHV 3-point amplitude ∂K i 5 X 3 =1 ˜ λ 2  [3 1] i X 2 2 κ

= 2 λ

[2 3] (4) K 5 δ (4) + [3 1] [1 2]  ) δ 2 i Λ 1 2 ) K − ˜ λ [2 3] K 1   = λ [1 2]  2 3 the tree-level S-matrix of the Einstein ∂ A ∂λ := ( ) explicitly breaks 4-momentum conser- ·  3 to (1 + Λ 3 ˙ 1 K α ∂ (1 + Λ 4 2 ∂ α  ∂λ times ∂λ 2 6  2 [3 1] 6 ∂ – 30 – ]. [3 1] 2 ∂λ ,B 2Λ [2 3] 2 75  ,K [2 3] − ), note that [1 2] ˙ α  [1 2] α ε 2Λ 2 ]. This ‘bulk contribution’ does not include boundary 4.1 ∂ 3 ∂ [2 3] = − κ ∂λ ∂K 74 ) contractions do not contribute. The remaining differential ,  3 ˙ α + ε ∂ α A 3 73 4i Λ [3 2] [ , ∂ ∂λ 3 · −  κ + 4  Λ ∂K B 2 + 2 , ) = − 2Λ 2i Λ 2 MHV amplitude on AdS ε κ + [3 2] (1 − 4 3 − , Λ 3 = + 2 ) = M  , [2 3] i + 2 − 3  ∂ , κ (1 ∂λ + Λ 3 MHV configuration of ( ), confirming that the Einstein gravitons within the conformal gravity are good 2 3 = , ∂ 3 M − 6.1 ∂λ ). In other words, the tree-level S-matrix of conformal gravity evaluated on Einstein e -derivatives in the C (1 ·  λ Λ 3 2 This gives an additional perspective on the vanishing of the amplitudes for the Einstein In the 6.17 A M agrees with a formula obtainedof for Einstein the gravity ‘bulk in contribution’contributions AdS to to the the 3-point fullsuch AdS boundary amplitude, contributions decouple but [ manifests a smooth flat space limit, where of ( states is zero in antheory. interesting More way: generally, the it normalised is amplitude zero vation, as expected formetric AdS ( amplitudes. It isAdS easy scattering to states. see that this is compatible withsector the of conformal gravity in Minkowski space, which arises by taking the Λ Note that the leadingof coefficient Einstein of gravity, as Λ requiredgravity. is by precisely Also, the the embedding the of flat differential Einstein operator space gravity (1 inside + the Λ conformal The wave operator in the totalthe momenta can expression be for denoted the by since operators act only onmake the the replacement overall 4-momentum delta function, which means that we can and similarly for contractionsobtains with different particle labels. Feeding these into ( JHEP07(2018)016 4 . (6.20) (6.21) (6.22) (6.19) 1)! states of , − s 3 ! s  1) − , )! 1 − 2 3 3 s − ( field. Evaluating , s ! s s s ˙ i α ∂ − s ˜ λ ˜ λ ! 2 + − i ∂ i 1) 3 s 1 1 s ˜ λ λ − + 1 i s s s + 1 ( − 2 s λ 3 s s α =1 i )! ( X λ + 2 3 1)! ( =1 1 i s X

s [3 1] − 2i Λ) 3 (4) −

s MHV amplitude, and is pro- 3 δ − 1 + s [2 3] ( (4) ) that the helicity decomposition 2 s δ s ( ) + K

s 1 − ( 2 1  2.2 s − s n ). s 1 κ s 3-point − − = 3 [1 2] 1 4 6.13 s s 3 1) + (1 + Λ ( s 2 − ) s 2 κ s ) i ( 2 s 1)! A K 6 κ 3 1 i 6.17 1 −  h 1 s – 31 – 1 2 − 2 3 i 1 s ,B h s ε κ − 1 1 2 ! ] for the AdS h − 2 i) 3 (1 + Λ 1 s ε 1) suggests that there may be a way to isolate the tree- s 12 − 1 − ( 3 × s + 1 − ( κ s 2 − α ) on these polarizations results in: ∂ coupling associated with a massless spin- 1 s s ( ∂λ 1) ) = Λ s 4i Λ 3 − 1
s − ) on the polarizations ( 5.22 1 − ( − , s s ˙ s α 2 − 4.6 ˜ λ 3 , s s ) = 1 + − s 2i Λ) 2 3 s − − , ( ( 1) − Λ 3 2

− , s M + κ (1 1 + ( = Λ 3 is the dimension 1 1) M s − := κ s ) ( s A A similar phenomenon occurs for the ‘scattering’ of general CHS fields on an AdS For completeness, the MHV 3-point AdS amplitude evaluated on Einstein states is ( MHV 3-point formula ( n A A Helicity structure ofHere conformal we give graviton some details modes concerning theof claim on-shell in conformal section graviton states isric. gauge-dependent at the level of the (linearised) met- many helpful comments on a draft.the beginning TA thanks of N. this Arkani-Hamed project. forquestions. useful AAT conversations is at grateful This to work R. Metsaev wassupported for by partially useful an discussions supported Imperial of by related Collegethe Junior the Russian Research STFC Science Fellowship. grant AAT Foundation was ST/P000762/1. grant also 14-42-00047 supported TA at by is Lebedev Institute. analogous to the embedding of the Einstein gravity insideAcknowledgments of the conformal gravity. We thank Thales Azevedo, Henrik Johansson and Tristan McLoughlin for discussions and This matches the resultportional found to in the flat [ spaceThe 3-point overall amplitude constant for factor masslesslevel two-derivative Λ higher S-matrix spin of fields. a massless higher spin theory within the AdS amplitudes of CHS theory, with the spin-dependent normalisation where the background. Ignoring theCHS rest theory of are represented the on spectrum, AdS by the the two-derivative twistor helicity polarizations: given by the obvious helicity conjugate of ( obtained by evaluating ( JHEP07(2018)016 (B.1) (A.3) (A.4) (A.1) (A.2) ) and , ω )) 0 2 tensors. . , . 5.2  0 CHS field by      ab ω, i , s  , 0 )   − ˜ i = ( ab T , ˜ T   iθ a ) k ab  + −
e ˜ are helicity T 0 0 00 0 0 00 0 1 0 0 1 − − = T T      − (  T − 1 states appear only in the − − = T A R  T A   ( + ˜ ˜ − T T + . ) + ) leads to A R T + : , ˜      + 3 T 0 0 ) + 2.20 x , A + + θ = 0 θ 1 states appear only in the oscillatory , ˜      T 0). Helicity is then determined by the ) ), ( + ,  + s 0  ), specializing to the above momentum , ( T sin cos 0 − i T β ( ab ab , ) 1 tensors, while
θ
+ iθ s 2.19 +  0 ( θ 2.16 ,   T A α 0 0 0 e −− ( −− sin  ), ( Ψ i – 32 – + T ), ( T ) 0 1 1 0 0 0 0 0 0 0 = ), one finds that −  s iω A = (1 ( T 2 −− 10 0 cos 0 00 0 0 0 1  2.18 −− are 6 free independent polarization constants. α      ) in a particular frame, where a ) ˙ 2.12 R B 2.17 s A      − + n  ( =  ) the helicity α + + A ab = ab T  ∂ 2.15

R A.3 R ++ ++ ) they are also present in the growing B-mode. Note that it −− −− and T T T T from ( A.4 represent helicity ++ ++ ,T −− , −− −− A B  B ab B A      ˜ , T  0 + + = = T i 1 0 ++ iθ  ab ab 2 and − B ++ and B ++  B A i ,  T T e ab  T −− = ++ ++ 0 00 0 1 0 0 0 0 0 0 A T A B      , R = = = ++  A ab ab  T B A Doing the same in the TT gauge ( Choosing the conformal gauge ( Consider A T R In this appendix we demonstrateworking the directly counting of with on-shell (gauge-invariant) states fieldhelicity for strengths. sector. the Let spin- The us linearised focus, CHS e.g., equations on of motion the in negative this sector are (cf. ( A-mode, in the TT gaugeis ( also possible to makegrowing a part gauge of choice the for potential. which the helicity B Counting CHS degrees of freedom While in the conformal gauge ( where indicating that frame and fixing the residual gauge ( These satisfy: Introduce the helicity basis tensors: choose the unit time-likebehaviour under vector rotations as in the plane transverse to JHEP07(2018)016 . ) ) 2 1 h λ − − − ( B.4 6= 0. s (B.6) (B.7) (B.8) (B.2) (B.3) (B.4) (B.5) s 1) − . In each s ( ˙ α which obeys ˜ λ s 1) h < s 2 − − 1 s ( . A further ≤ ˙ α s ˜ with insertions of has mass dimension β : , − s , ) x x · , . . . , s , h − x -tuple of spinor indices k · − for 2 i s k . s e i ( x = 1 ) e h · α negative helicity scattering l k a − , − i 1) . s s e x ( − ) indicates that this solution · x ˙ ˙ s , · α α k ( k ˜ ˜ i ˙ x β β α i · e ˙ ˜ B.4 α e k β ) 2) ) i s )) s e − 1) , h ( s α s ) ( β ( − x s x β · ˙ s ( α λ ( k λ ) β ˙ 2) i α s insertions. The construction terminates )) s λ l e 1, it is clear that there are in total − α α ˙ ) s − α λ 1) ) ( s )) s λ ˜ − ( β ( s α − 1) ˙ ( α growing states which must make up the re- α s α s a 1) ( β − λ α x 1 − purely oscillatory solutions is built by taking: s α λ ( s ≤ s x 1) ) α l ( x ) – 33 – s α κ ( l − and is constrained so that 2 λ ( 1 distinct forms for the growing mode of helicity α h 1 s α ( ( ) ( α B s = α 1 a s λ ≤ coupling constant associated with a massless, two- − ( , or λ 2 ) λ ) − β α and the constant spinor ) s s = h s s ( = ) ( s ( a . These are precisely the h ( ) λ β ) β h ) s 1. ( − h , β ) s − ( s ) growth is given by: ( ( s increases, the growth of the field weakens. For example, s α λ l = ( β − − ( β l − − λ ) . As 1 ) λ ( α − ) α s λ 1) 6= 0. Each such solution is purely oscillatory, and Ψ κ l s s ( s ( ( ( = Ψ − 2 − a x α ) ( α ) presented above. . β s = ( ) 2) h ) h s . s Ψ Ψ s ) − ( − O s κ ( s ) − s β ( g( α s ( carrying mass dimension +1. Our goal is to count linearly inde- ) ) 2.33 β s s = α − ) Ψ ) ( s λ − s ) g( α ( ( ) s g( α ( h β Ψ ) β ) − s ) Ψ h s ( s ( ( α − α ( α a mode with Ψ has mass dimension s 2, with a single growing field of the form is a constant totally symmetric spinor of mass dimension 1) − growing fields can be constructed by replacing powers of − − ) 6= 0. Counting the little group weight in ( s has mass dimension 1 s is the mass dimension 1 s of mass dimension ( ( ) ˙ α l α h s − s ˜ ( β a − κ κ α ˙ αs and obeys λ A similar method works for growing modes of helicities This leaves us to account for the It is now easy to see that a family of ) ˜ β ) can be partitioned among , confirming formula ( s s s ( h ( 2 − α h α at helicity At each stage, we see− that there are for growing modes of helicity case, we simply have to count the number of ways in which the Counting little group weightshelicity tells us thatAs this the is number a of field insertionsthe of helicity of helicity mainder of the negative helicitywith sector. the To highest do possible this, polynomial we growth: first construct the solution of ( where where corresponds to a field ofstates helicity of CHS field of spin where a corresponds to a field of helicity where derivative higher spin field. Another solution is provided by pendent solutions to thesegauge equations; where we solutions work take a in ‘helicity a lowered’ momentum form: eigenstate basis and inOne a simple solution is the standard zero-rest-mass field of helicity with the spinor Ψ JHEP07(2018)016 (C.7) (C.3) (C.4) (C.5) (C.6) (C.1) (C.2) , which , # PT ) K + , K , charted with J . 3 B A ) 3 , . 1)) J f 3 | , 3] − CP ∂Z , I K − K µ 1 | [ A B s 3 ∂ 1] ( t − f = 2 | i µ ]. Here, we review the 1 [ ∧ O ∂Z K s | 1 , ) e t K ∂ i , λ 12 − − , , i PT I 3 J ] ∧ 3 ) e ( t A − 10 s 1 λ I J µ i [ − N 1 A A ) is evaluated on the momentum H i + t ( ( t 3 i ∧ 2 K K λ ∈ − s ( I B B C.4 3 ) e 2 1 ) A 1) f f ¯ δ = λ J g λ − i | | ( − s )! t ! I K | 2 ( , and K =0 | ¯ 3 K δ | A B A ∞ 3 | CHS field. t − K I X −| f | K i d | PT 3 | s +2 | s 3 ∂Z λ ∧ )! 1 K t , | | ( J s 1 + | | − | 2 t 3 ∂ J Z | ¯ s δ J 1 Z 3 , f J t i ∧ – 34 – i | D t − J d s i J t | − | d − ). The variational data for this twistor action are 1

3)) | − | I A I PT α s Z ( | I A 3 Z ( | Z − =0 K | , λ = A 1) ∞ B !( s ˙ 2 X α | | K 1) − | K f ] = 1 − µ J − s B | ( K i ( | | )! s =0 3 | I | ( ! O A | ˜ | = ( g, f + X J C , 1 B i [ | , K J | | A B A 1 = ! + SD PT | Z + ( ) I S − = = | 1 J K A 1 | J − 1) 1) s | H I K ( ¯ − − ∂f A B 3 i 1 ∈ = " s s f ( | ( | = ∧ 1) ∂Z B I A i K I | | MHV 3-point amplitude. This proceeds from the formulation of the self- − I 1 f A s ∂ g A ( 1 A N ], it was shown that the SD sector of interacting CHS theory is described in g g is the holomorphic volume form on ∧ 12 , Z Z 3 3 10 D MHV 3-point amplitude is given by extracting the cubic part of this action: To arrive at a purely momentum space expression, ( In [ PT Z It is easy to see that the action of twistor derivatives on these momentum eigenstates is eigenstates: with the multi-indices taking values in terms of the spins of the external states. The where D twistor space by the action functional: homogeneous coordinates cohomology classes which encode the ASD and SD d.o.f. of a spin The 3-point amplitude formulaeway presented from in the this formulationderivation of paper of the CHS can theory bedual in derived sector twistor in of space a CHS [ is systematic theory an in open terms subset of of an the action three-dimensional functional complex on projective twistor space space C Deriving 3-point amplitudes from twistor space JHEP07(2018)016 , 2 ) t 02 ] 1 λ J. Nucl. (C.8) (2012) 2 , , t 01 JHEP + , 2 ]. λ (2002) 1376 ( (2) JHEP δ , 133 SPIRE (1985) 233 hep-th/0207212  IN | [ MHV amplitude in [ K | 2 119 t ) K ) to fix 3 + ]. J ˜ λ A 2 (2003) 59 3 (1989) 97 t ) for the A I + SPIRE | ]. A Phys. Rept. . 2 ]. IN 1 K 5.22 = 1, and the integrals over twistor , ˜ | 3 λ B 664 t ][ B Theor. Math. Phys. B 231 1 2 ]. ) t t , J K [1 2] [2 3] 5 SPIRE − B + SPIRE , S I ). This enables us to express the ampli- IN 2 ) = 1 IN A 3 ˜ 3 × ˜ C ][ λ 3 SPIRE ˜ ][ A λ ( 5 K IN I 3 3.33 t − Partition function of free conformal higher spin (2) A ][ J δ Phys. Lett. 1 Nucl. Phys. AdS − + , , I B – 35 – , t 2 ]. A ˜ ( λ K Effective action in a higher-spin background 2 K B arXiv:0709.4392 t 3 B 3 [3 1] [2 3] [ ˜ Cubic interaction in conformal theory of integer higher C + A SPIRE ]. Conformal supergravity = J 1 IN )! Conformal higher spin theory and twistor space actions ˜ A λ 2 2 ( t ( ][ 1)! ), which permits any use, distribution and reproduction in arXiv:1406.3542 s K [ arXiv:1604.08209 ]. − (2) B 2 − SPIRE [ δ 3 arXiv:1309.0785 A 1 (2012) 062 IN ) s [ s 1 ( ( )! ][ λ 3 1 06 SPIRE 1)! s 3 s t IN − − 3 − (2014) 113 CC-BY 4.0 s ][ have been performed. At this point, the remaining integrals over d − 2 1 + s 3 s 2 On limits of superstring in Ordinary-derivative formulation of conformal totally symmetric arbitrary spin Ordinary-derivative formulation of conformal low spin fields λ On partition function and Weyl anomaly of conformal higher spin fields ( This article is distributed under the terms of the Creative Commons 08 JHEP ( λ s 2 (2013) 598 ( ,  d (2017) 485401 Conformal higher spin theory 1) arXiv:1012.2103 3 (2) µ [ 3 t − 3 s 2 δ ]. ]. d t is the momentum space operator ( JHEP 2 2 +( × 2 s t B 877 A 50 , t K d arXiv:0707.4437 B [ ˜ SPIRE SPIRE C Z hep-th/0201112 IN IN ) Phys. theory Phys. bosonic fields (2011) 048 [ [ 064 [ spin fields in four-dimensional space-time s can be done against the delta functions ( P. H¨ahneland T. McLoughlin, M. Beccaria, X. Bekaert and A.A. Tseytlin, R.R. Metsaev, X. Bekaert, E. Joung and J. Mourad, A.A. Tseytlin, A.A. Tseytlin, A.Y. Segal, R.R. Metsaev, E.S. Fradkin and A.A. Tseytlin, E.S. 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