JHEP05(2018)208 Springer May 21, 2018 May 31, 2018 : : , March 12, 2018 : Accepted Published Received a Published for SISSA by https://doi.org/10.1007/JHEP05(2018)208 [email protected] , and Jan Plefka b . 3 Matin Mojaza a 1802.05999 The Authors. c Conformal and W Symmetry, Scattering Amplitudes

We argue that the scattering of gravitons in ordinary Einstein gravity possesses , [email protected] Institut f¨urPhysik and IRIS Adlershof,Zum Humboldt-Universit¨atzu Großen Berlin, Windkanal 6, 12489Max-Planck-Institut Berlin, f¨urGravitationsphysik, Albert-Einstein-Institut, Germany Am 1, M¨uhlenberg 14476 Potsdam, Germany E-mail: [email protected] b a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: To motivate the underlying prescription,symmetry of we gluon demonstrate amplitudes that inifest terms formulating of reversal the momenta and conformal and cyclic polarizationof symmetry. vectors requires graviton Similarly, man- our amplitudes formulation reliestude of on function. the a conformal manifestly symmetry permutation symmetric form of the ampli- Abstract: a hidden conformal symmetrythis at conformal tree symmetry level is in indicated anyreminiscent by number of the of dilaton the dimensions. soft conformal theorem invariance The in of presence string gluon of theory, tree-level and it amplitudes is in four dimensions. Florian Loebbert, Hidden conformal symmetry inscattering tree-level graviton JHEP05(2018)208 ] 1 25 31 10 3 29 6 24 18 – 1 – ], which provides a concrete, yet still mysterious, 3 , 2 21 0 α 3 f M 1 Clearly symmetries play a decisive role in constraining the dynamics of quantum field theories and the Poincar´einvariance ofpractical formalism scattering to amplitudes compute is these:momentum a translational conserving symmetry built-in-feature delta-function is of of guaranteedvariant the any by modulo the amplitude gauge overall which transformations. inin turn Importantly, four must however, dimensions tree-level be are invariant gluon under Lorentz the amplitudes in- larger group of conformal transformations. While trees, also at the fieldMills theory level. theory This was representation liftedBern, of Carrasco gravity to and as an Johansson the (BCJ) “square” entirelyprescription [ of for new Yang- how level Yang-Mills amplitudes throughgravitational (at the amplitudes tree upon color-kinematics and replacing loop-level) duality color may of degrees be of combined freedom into by kinematical ones. properties. In their perturbativefluctuations quantization about through the gluons vacuum andin or gravitons their Minkowski as scattering spacetime, weak amplitudes respectively, field within striking have string been similarities theory, discovered. theallow Through for KLT a a relations unifying representation UV-completion between of open graviton tree-amplitudes and as closed products string of color-stripped amplitudes gluon [ 1 Introduction Although Yang-Mills (YM) theorylocal and symmetries, Einstein’s their theory actions of look gravity both rather are different built and upon lead to very distinct quantum C On gauge invariance of theD dilaton soft Conformal theorem symmetry of two-dilaton-two-graviton amplitude 6 Summary and outlook A Conformal generator on stripped amplitudes B Computation of 3 Conformal symmetry of Yang-Mills amplitudes 4 Soft dilatons, conformal generators5 and graviton scattering Conformal symmetry of graviton amplitudes Contents 1 Introduction 2 Poincar´eand conformal symmetry in momentum space JHEP05(2018)208 ]. 1 ]). 12 = 0, , 20 2 11 k massless all ], who used an 4 ] for 10 ]. The above findings , 9 19 2. However, the existence d > = 2. ) — and not in terms of helicity ]. Extensive works of Strominger d ] (up to collinear configurations [ 8 4 [ ¯ λ 2 ∂ ∂λ∂ = , tree-level Yang-Mills amplitudes in four dimensions µ ∆=1 ¯ λ K – 2 – and momentum space λ ], and as especially pointed out in [ 18 – ]. However, a similar relation does not immediately carry over to 13 7 – 5 ] — the non-preservation of the on-shell constraints is demonstrated. -dimensional treatment and a scaling dimension different from unity. 4 d symmetries in quantum field theories, i.e. the appearance of symmetries at the We shall begin our discussion with an analysis of some general features of massless In fact, it has been demonstrated that Einstein gravity in AdS space can be obtained Given the close connection between the conformally invariant gluon tree-amplitudes Remember that using momentum spinors hidden 1 spinors as done inThis [ is very subtle,fulfilled on as the core hypersurface properties of of the amplitudes on-shell such constraints. asare gauge We annihilated expose invariance by the are the special impact only conformal generator of these since we require a scattering amplitudes and conformal symmetry.tion Employing and a special representation conformal of generatorsmomentum the in vectors dilata- terms (here of referred differential to operators as in polarization and momenta and polarizations.which Since generically these do variables not obeyan commute on-shell immediate with constraints, puzzle the concerning e.g. actionwould the of be interpretation natural derivatives, of to we those translate are results.shell the confronted constraints above In statements are with four into unambiguously spinor-helicity dimensions, resolved. space it However, where this the route on- seems not practical here theorems do indicate that conformalthe symmetry plays present a work role was for largelydilatations gravity and motivated amplitudes. by special In the conformal fact, transformations appearancedilaton in field, of the as the soft derived conformal in theorem [ generatorsabout for the of the soft string dilaton theory are formulated in terms of differential operators in massless particle et al. speculate aboutparticle the scattering processes existence in ofmaps four a gluon tree-amplitudes dimensions. hidden in BMS Along MinkowskiThe symmetry space Lorentz these symmetry to [ lines, of the four-dimensional a celestialgroup Minkowski recent sphere space on at then prescription acts infinity the as [ celestial the 2d sphere. conformal While the status of this program is still indefinite, soft conformal symmetry of graviton tree-amplitudes beyond from conformal gravity [ flat space. Further clues towardsthe a dicovery hidden of symmetry novel of subleading graviton soft-graviton amplitudes theorems emerged [ from an inspection of theruled Einstein-Hilbert out action, due a to naiveof the conformal dimensionful gravitational symmetry coupling is in level immediately of observables which arebeen non-manifest a or recurrent non-existent themeality at in discussed the recent above. level years of — So the just it action, as might has is not the be case entirely for misguided the to color-kinematics explore du- the question of a formal symmetry of gluon tree-amplitudeselegant was representation first of fomulated the by conformal Witten generators [ in spinor-helicity and twistorand variables. graviton trees, themetry natural for question the arises former whether leaves the any imprint existence on of the conformal structure sym- of the latter. Of course, from this is a consequence of the classical conformal symmetry of the Yang-Mills action, the con- JHEP05(2018)208 = i (2.3) (2.1) (2.2) -stripped δ -function with δ i . ∀ , s i µ is the so-called ··· 1 . n i ) ) -dimensional A n i d ( n,µ has the property of being linear A , since its presence seems unex- s i n µ i  A , . . . , k 1 , k ··· ( hidden n 1 i = 0 , and A µ i i s )   µ ) denotes the -function: P  · . The function = ( δ i P µ i δ s ( k i ··· ) – 3 – µ d 1 ( th polarization tensor was exposed. The momenta µ ··· =1 ) = i δ n i 1  i n massless particles are described by a function on the ) ≡ P = i ( n,µ ) n s = A µ P s ( ··· i µ , . . . , k δ 1 µ 1 P µ k ··· ε ( 1 i n µ i ε A , k = = 0 n i A k amplitude. We will consider massless states carrying (symmetric) po- · i k understood as P stripped for each state, i.e. have to obey the on-mass shell and transversality conditions s µ ··· 1 In the final part, we put our conjecture of the conformal symmetry of graviton ampli- We move on to consider graviton scattering and explain how the string theory soft- µ ε , . . . , n where only the linear dependence onand the polarization vectors1 describing the scattering of massless particles with labels or simply larization tensors of the form in support of an overall momentum conserving Here and throughout thisits work argument dimension entering the conformal generators. 2 Poincar´eand conformal symmetry inIn momentum momentum space, space amplitudes of of the full permutation operator.these When observations combined imply with dilatation theconformal and invariance algebra. Poincar´esymmetry, of We tree-levelpected refer graviton to with amplitudes this regard under symmetry tothe the as the symmetry full is dimensionful emphasized gravitational by coupling. the fact The that hidden we character require of a multiplicity dependent scaling amplitudes are annihilated bymanifest the cyclic generators and of reversal the symmetrytion of symmetry. conformal Yang-Mills That algebra. amplitudes is, is in Here,is our taken the found momentum by only role space full formulation, if of permuta- specialform. we conformal act invariance Said on differently, the the graviton special amplitude conformal in generator a maps manifestly the permutation amplitude symmetric to the kernel gravity. A careful analysismomentum space suggests — that in thetation analogy formulation of to of the the graviton this Yang-Mills amplitude.stage conformal case of While symmetry — the the in requires paper performed a our analysis particular statements is about very represen- instructive, the at conformaltudes this symmetry to are the still test. conjectural. We verify that, up to and including multiplicity six, tree-level graviton We then demonstrate that annevertheless analogue be of found, the if an conformal explicit invariancestripped cyclic in amplitude and four reversal is dimensions symmetrization performed. of can the delta-function dilaton limit indicates the conformal invariance of graviton amplitudes in ordinary Einstein issues by performing a re-analysis of the conformal symmetry of Yang-Mills amplitudes. JHEP05(2018)208 is ]). a ) we 25 (2.6) (2.7) (2.4) (2.5) – . as well 2.2 22
n µ i in ( A ¯ k . ν i represents in k -function, the δ = 0 − even i n,µν ν i s k ¯ k A a µ i n 6= k which is conveniently i X · µ i − a k  µν can be decomposed into a + − = 1 and we have chosen the η ν i µ i i ), where   i = ¯ k 2 = 0. Notice that by imposing µ i · k  a − i ( 1 → d describes a multiplet containing k k conditions. Amplitudes of polar- = 2, the massless tensor product -function and on-shell conditions. = · √ µ i δ ν i s  a  µν i µ i µν dilaton µν graviton ε  ) = ε ε i ] and the recent discussions in [ . 2 k = i µν , . Of course the choice of the state ( ] for a detailed discussion). That is, by  = 0 and  − on-shell · η 21 n . d  ∂ 2 ,  i µν A · √ An elementary physical state should corre-  ¯ k 27 = 0 separately, i.e. to strip off the , i k n the graviton polarization tensor or the dilaton n = 0 µν dilaton 23 ) + – 4 – A i j A = µν graviton i k k ε i ( = 1 and , ε · W i W = ’s are additionally constrainted by gauge invariance; ) i k either µ i k  a µν i ( 6= ε n j dilaton µν graviton the constraints are resolved by setting ) X ε 6= i a µν = To consider ε µν dilaton = ε 2 a µν µν i 2 ε ε  − · d  ) is obeyed on the support of the √ 2.4 − , k ν i µ i  k µ i is not a Lorentz vector (see e.g. [ ]. Specifically, the symmetric tensor  a n µ i 6= i  X 26 , ) = − i 13 k ( = is an auxiliary momentum which obeys µ a i k = 0, we effectively constrain ¯ k i  µν graviton · ε i In any expressionamplitude, involving it not will the beas implicitly full assumed in amplitude that the these distribution, endarbitrary constraints of leading but are any to enforced only an manipulation on the ambiguity applied in stripped to the form of a stripped amplitude. invariance in coordinate space),on-shell as conditions of well that aschoosing state by some (see state imposing with also the label [ constraints induced via the  The stripped amplitude. constraints from must Poincar´esymmetry be resolved.the This can momentum be of achieved one by eliminating of the external states using momentum conservation (translational traceless and a trace part Here normalizations such that ( state characterized by theboth polarization the tensor graviton and athis scalar case component, a the reducible dilaton.sentations, stripped The once amplitude respective it that is becomes contractedprojector an with [ amplitude for irreducible repre- Polarization tensors of physicalspond states. to an irreducible representation ofmay the also Lorentz group. work with However, for reduciblefor tensor describing representations, the as physical for modes instance of done closed strings. in string For theory where we have introduced the generator of gauge transformations The above equation ( We note that ized states described in termsthey of must the be invariant underwritten the gauge as transformation We will refer to both of these sets of conditions as JHEP05(2018)208 (2.8) (2.9) = ∆. (2.10) (2.16) (2.12) (2.13) (2.14) (2.15) (2.11) i , ,ν i k ∂ µν i iS − , i.e. whose polarization i i µ k s W ∂ i ), can be written as , on the whole amplitude ∆ . ,ν  } i 2.2 , i  . − i , ] ∂ µν i i , thus from hereon ∆  µ k = 0: ,S i W µν J i ∂ µν i ∂ µ i i ∆ ) ·   i, iS + i as in ( iS i · k The momentum space generators  i ∂ s − . µν − . ,K + 2 k µ · i
) η i  µ k i ∆ i µ µ i )  µ  = P i, ∂ k k ∂ . ∆ ∂ ( ( ν i ∆) + 2 i ··· ν i . ∆ k D k  ,D 1 not commute with ∆) − i, d δ i µ − of the amplitude read µ 2∆) on the conformal dimension ∆ is stressed i − −  − G − 2 ∂ 1 k − i ,J i i − i d ∂ ν µ k ∆ ∂X ν  = ( − n µ i P =1 ∂ 2 ∂  i K X s k µ i d µ i ) )] = µ – 5 – := 1 2 + (1  − k ν [( = P ∈ { i ··· P µ i d ( µ 1 ( X i = =  and µ ∆ W ∆ ∂ ∂P in general i i ε , δ ∂δ i, = µ G ] = 0. k µν ∆ ∆ i i ∆ ∂ G ] = ( ] = µ i,  i D 2 i µν i D ·  − [ does to be the same for all legs S i · )] = , k i i µ ,  P ∆ µ ∆ ] = The above representation of the conformal generators does ( µ i ∆ , k K K we employ the shorthand notation [ µ , δ ∆ K µ ∆ ,W ,K K X i [ µ ∆ K [ K [ takes us off the constraint on-shell surface in momentum space. For + ∆ i µ ∆ k ∂ K ,J · µ i i k k = = i µ is the ‘spin’ operator which, for states with integer spin i ∆ P i, µν i D S The dilatation operator only suffers from not commuting with momentum conservation, leave the kinematic constraint hypersurface of on-shell scattering amplitudes invariant. Hence, in general completeness we note that [ i.e. we have Similarly, overall momentum conservation is generally not preserved: nor does it commute with the on-shell conditions since it will be of particular importanceKinematic for hypersurface. us. not To be explicit, the generator is realized via the standard tensor product: The dependence of the generators In the following, weThe consider action ∆ of the above generators Here tensor is symmetric and described by of the conformal algebra acting on a single leg where for any four-vector Conformal transformations in momentum space. JHEP05(2018)208 j  · i (3.1) (3.2) k color- + 1 in , can be j 4 k n Indepen- n − · d i , we study A 2 k µ ∆ ) are explic- K , 2.3 . We note that ) n is a homogeneous A n ) ] to describe relations n A , . . . , n 28 − (1 . n 2 , 2 − A d ] for explicit expressions. We = (4 n a =4 A = 0 d n ∆=1 T = 4 and for ∆ = 1, i.e. A n d ··· A 1 -dimensional treatment of scattering  ∆=0 a d In order to better understand the im- 1) T D − Tr[ denoting from here on a basis of ) (∆ n n A ,...,n – 6 – X + (2 -point stripped amplitude d ) of the special conformal generator . Using this, it is easy to show that the dilatation P n n 2 − − amplitudes. The above sum runs over all non-cyclic . − n 4 2.15 , which can equivalently be expressed as a sum over d  g ) = P ( n δ partial A , . . . , n ∆ ) = D ) were used, together with -stripped amplitudes, it is useful to rewrite the conformal generators δ = 4 in the context of a general , . . . , n or simply , by use of the chain rule, see appendix 3.1 j d (1 n 6= A i ) and ( ] by using the spinor-helicity formalism with conformal generators represented the color-group generators and , for 4 j  2.16 dimensions the canonical scaling dimension of the gluon is ∆ = a · d T i  In four dimensions, the classical scaling dimension of the YM field is ∆ = 1. Here we wish to understand how the conformal invariance of YM scattering amplitudes When acting on In 2 = 4 YM amplitudes. where ( dilatation invariance is also obtainedan for arbitrary the number multiplicity of dependent dimensions choice ∆ = dently of the spacetimefunction dimension, of the the momenta ofoperator degree annihilates 4 the YM amplitude at tree level in with decomposed partial permutations of the labelspermutations with 1 one label kept fix. ance features when restricting thedecomposed analysis into to colorless four partial dimensions. amplitudes YM amplitudes manifests itself in amplitudes written in momentum space and howImplications it of relates to representation the deficiencies. plications on-shell of constraints. the on-shell deficiencythe ( conformal transformations of tree-level YM amplitudes, where we expect to see invari- scattering amplitudes, this was firstwork demonstrated in in [ a manifestlyin four-dimensional frame- spinor space.itly Importantly, resolved. in this formalism the on-shell conditions ( notice that such differential operatorsamong were amplitudes considered of recently in different [ theories. 3 Conformal symmetry ofYang-Mills Yang-Mills theory amplitudes in four dimensions is classically conformally invariant. At the level of of) the terms givend above may vanish or cancel, as we will seein in terms a of differential moment operators in ofand the the case Lorentz invariant of kinematical variables, These commutators should, however, be considered on specific amplitudes, where (some JHEP05(2018)208 i F on µ = 4 ∆=1 = 4, µ (3.4) (3.5) (3.6) (3.3) ∆ d K d K , n A i , n W i A µ k µ i = 1. However, we ∂ , we find: . ,  n takes the amplitude − 2 A 2 − , keeping ∆ = 1 fixed. ). Nevertheless, in the d d =4 =4 =4 = = 0 = 3 = 0 d d d ∆=1 ∆=1 ∆=1 µ ∆=1 ∆=1 , as well as on-shell gauge 3.3 µ i K  , i under a Lorentz transformation. . G µ = 4. This leads to the question of µ i p ) vanish in four dimensions. Using . ) k n d p 6= 0 ( A n i f n =1 i 2.15 X from the amplitude. The coefficients F A = 0 + i µ i ∆=1 µ  + µ µ i ∆))  i  ∂ i k − G F n =1 i µ µ i i X ∆)  ) and ( (1 k n and – 7 – − = n -dimensional properties of A n =1 n µ i =1 d µ i n i X i  X 2.12 k A ) 4 + n = + (1 P A Poincar´einvariance implies that the action of − n ( n 2∆) µ i δ µ ∆=1 d  A A ( i − K µ ∆ ) 2 ∆) W = 4 is realized in momentum space. µ i K P µ k ( d − − ), presumably as a consequence of ( ∂ ∂P d d ∂δ − 2.7 = = ( = ( = n n n n A A A A ] ] ] i i )] 2 i 3  are some Poincar´einvariant functions of the kinematic variables, which = 0, which are generic P · , k ( i ,W i n µ ∆ µ G ∆ , δ A , k i K µ ∆ K [ µ ∆ [ is not a Lorentz vector, it transforms into W K K µ [ and [  i have the unwanted feature of being dependent on the ambiguity in resolving the F does not in general give zero, i.e. i G n On the other hand, the explicit examples also reveal that the naive application of Invariance under special conformal transformations is, however, delicate. For YM takes the form While A 3 n Does this mean that fullin conformal momentum invariance space? of tree-level YM amplitudes cannot be seen where the ordinary scalingalso dimension notice of that this the relation gluon is becomes valid ∆ in any = to number of dimensions this ambiguity We interpret this as a feature of the underlying conformal symmetry of YM theory in where inherit definite homogeneity degrees in and Poincar´econstraints, cf. ( case of explicit lower point examples (to be discussed below) we observe that regardless of Symmetrization prescription. A None of these commutatorswith vanishes ∆ generically = on 1 theoff only stripped the two amplitude, on-shell out and surface of in how in four conformal momentum commutators invariance space, in vanish. even for Hence theory, not all offirst the Lorentz commutators invariance and in linearity in ( invariance the polarization vectors JHEP05(2018)208 ), 2.7 does (3.7) (3.8) (3.9) (3.11) (3.12) (3.10) n in ( A a , i on k 1 inherit cyclic 6) this claim µ ∆ . − =2 n i K X n ) is not satisfied · A 1)] ] still bares some n > n , does not commute  n 2 3.12 µ ∆ A − [ n K = C 1 µ ∆ n, . . . , k . ( · K n 1) , n , A by hand, and we denote this n 2 n A , being permutation symmetric, , n 1) ,  µ ∆ µ ∆ A 1) j − K k K . · n, . . . , ) is that ( i 6= . ) as follows: ) + ( n k ] 1 n n ). However, since A 3.11 , . . . , n, − ] = 0 n 3.1 A A j>i X n n µ 2.7 ∆ . Remarkably, our explicit checks show that (2 1) 1 A n n [ − , . . . , n K =2 ] = − [ i n X n A 2 A n – 8 – n , C C A − [ (1 j n ) = ) = ( n k C µ ∆=1 ] = · A 6 that [ n , K 1 ) 5 A k , [ 1 4 n , . . . , n , . . . , n ,...,n − , =3 2 2 C , i.e. 2 j X n , , , µ n ∆ X C − (1 (1 = 3 K n n n = Cyc(1 A A 2 n are discussed below. At higher points (i.e. for 1 k 2 · n 1 is a fully permutation symmetric differential operator, it seems natural ] = µ , while the inequality shows that the naive application of ∆ n 5, by summing over all possible ways of resolving the constraints for a fixed Understanding this feature is a nonlinear problem that requires further n , k A K C [ ≤ i 4 n k n C 1 − =1 i X n − = The reason why we point out the choice ( At this point, it is useful to note that the stripped partial amplitudes n annihilates the expression for any value of ∆. At least for k 4 µ ∆ sensitivity to the ambiguity in resolvingfor the all and Poincar´econstraints, ( choices. investigation beyond the scope of this paper, which liesK on graviton scattering. The point The details for each remains conjectural. for the specific choice of resolving the Poincar´econstraints we systematically find for where the first equalitycommutes is with the trivialnot statement preserve that the permutation properties of the support of momentum conservation and on-shell conditions, i.e. In the explicit examples, to be discussed below, we find that manipulation by the symbol Of course, the stripped amplitude and stripped symmetrized amplitude are identical on and the on-shell conditions, aswith prescribed the on-shell by ( conditions, weby it. cannot expect But since these symmetrythat properties it to should be preserve preserved theby permutation manifesting symmetries the of the cyclic amplitude. and We reversal can ensure symmetries this of and reversal invariance from the color trace in ( In order to verify these symmetries, one generically has to resolve momentum conservation JHEP05(2018)208 ) − 1 2.7 , we here k n (3.16) (3.17) (3.18) (3.13) (3.14) (3.15) − µ i A k = 4 k , 6= 1, also terms 12 e ) 2 k ). It is straightforward , − 2 ), specifically 1 k 12 k · e ( ) 2 · 2 ,  3.11 3 k ( µ 2  ·  − 2 1 3 , 31  e ( + ) = + 6= 0 , − 31 1 ). We then find e i . µ 1 3 k 31  ) F k · will not be provided here, but can be e 1 µ i 23 ) 3 k  4 + ] = 0 1 e  ] = 0 4 k 2 A − 3 4 − · =1 k A and resolved the constraints from momentum i 3 [ X ( A 2 µ 3 [ 4 j k ·   3 ( – 9 –  C and ( = ( 4 · C · 12  i 4 2 e 2 −  µ ∆ k −  A µ ∆=1 2 1 K = 23 = − = e K 3 ) + is obtained by imposing ( 1 ij 1 µ ∆=1 2 A e k k 4 k 23 · K − · e A 4 1 ) µ  ∆=1  = 3 k K 3 The three-point stripped YM amplitude takes the form k = ( − The expression for and 3 2 k 23 A ( s · − 1 contribute. By considering instead the cyclic and reversal symmetrized  1 2 13 µ i s k − ] = 3 = A [ are some nonzero functions of the kinematic variables. For ∆ 12 3 s i C , F 3 k , such as the photon decoupling property, the Kleiss-Kuijf relations, and perhaps even − n 2 which holds only forvanish, not ∆ only = for 1. ∆a = special 1, It property but moreover at for turns four any points value out of that that ∆. we the do This not coefficients is, find as of at in the higher the points. three-point example, where proportional to form we find as conjectured Four-point example. straightforwardly computed from just four Feynman diagramsThe using partial textbook stripped prescriptions. amplitude k The three-point amplitude isHowever, as of we will course see next, very thethrough four-, special the five-, and same (e.g. six-point procedure, amplitudes it expose but this vanishes where symmetry for ∆ = real 1 momenta). becomes a crucial choice. form, which is readily obtained from the above expression, it is easily checked that for any value of ∆ we have conservation by imposing to compute which clearly does not vanish. Considering instead the cyclic and reversal symmetrized all symmetries of the amplitude can beThree-point easily example. implemented as full permutation symmetry. where we introduced the notation find what seems tospace be amplitudes. the It conformal seems symmetryA plausible that of if YM one theory couldthe at manifest BCJ the all relations, symmetry level the properties of sensitivitydisappears. of of momentum In these fact, statements this to is the what we prescription observe of in resolving the graviton ( case to be discussed later, where we want to make here is that by manifesting the cyclic and reversal symmetries of JHEP05(2018)208 ]. ) is 31 , (3.20) (3.21) (3.19) 30 3.5 ] and reads , 29 4 ν 4  ). In addition, the numerator ··· 3 in a manifestly gauge invariant 1 k ν 1 -tensor [ 4  . 8 + 4 t A in each pair, making it manifestly µ 4 2 k i k ), we again find that while ( ) is again generically not annihilated . 6= 0 ν + i ··· 1 F ↔ 3.11 3.11 1 k MHV. To ensure that the manifestation i µ µ 1 ( i  ] = 0 k 5 µ − 4 -point espressions have been fully exposed in [ 5 ν A =1 n 4 i [ X = 5 4 C = – 10 – k ,...,µ ) to hold iff ∆ = 1. The checks here have all been 5 1 ν Imposing ( A 1 µ ∆=1 ,µ 3.12 K 8 5 t , i.e. without using on-shell conditions. The absence of µ ∆=1 8 t 23 K s 4 12 s defined by imposing ( 6 = iff ∆ = 1. An additional, possibly related curiosity is that the Depending on parametrization, the five-point YM amplitude con- A 4 Up to five points, the Yang-Mills tree amplitude in four dimensions µ being a tensor with symmetry under exchange of any pairs of indices ∆ A 8 K t as given above is gauge invariant iff ∆ = 1. This can be shown using 4 A and j k on · i k µ ∆ K . After manifesting the cyclic and reversal symmetries as in the previous examples, ) to be satisfied, = 2 and antisymmetry under the exchange ij 3.5 µ } s ∆=1 i Curiously, this invariance is also seen when writing is annihilated by We thank C. Mafra and O. Schlotterer, as well as J. Bourjaily for providing us with explicit momentum ν K 5 i 4 µ 4 Soft dilatons, conformalA generators hint and on graviton special scattering observed conformal when invariance considering of the tree-level soft graviton behavior of amplitudes massless in closed field stringsspace theory in expressions for any is YM string tree-amplitudes. theory The lower find ( by we nevertheless find our conjectureperformed ( numerically. Six-point example. is either MHV (maximallyof helicity conformal violating) symmetry, or as foundthis in helicity configuration, the we previous have also examples,parametrization explicitly is it checked the not consists six-point of just case. some a Depending 6-7000 special on individual feature terms. of Also in this case, while we do find as conjectured This latter check has been performed numerically only and is satisfied iff ∆ = 1. satisfied, we still have By performing the cyclic and reversal symmetrization as in the previous examples, we then only the symmetry propertiesa of similarly compact expressionreflect for a higher special point symmetry amplitudes at suggests four these points. Five-point observations example. to sists of some 400 individual terms. gauge invariant. Here we emphasizehas that manifest cyclic and reversalthe symmetry denominator (in does fact, not it manifest has theseA full properties. permutation Nevertheless, symmetry), we but findaction that of in this form form. This form is obtained by employing the so-called with { JHEP05(2018)208 ] 17 and – (4.3) (4.4) (4.5) (4.6) (4.1) (4.2) a, b, c i 15 k at order S , ,  ) ,     q j j 0  k k j j ) α ¯ z z 2 · · ( q ) ) ( O i i − − . ¯   i i i i O ¯ z z ¯
θ θ : + i ( ( µ ¯  j j ∂ ) + and which is not of interest νσ . i ν i n 6= 6= 0 i i with the polarization tensor ¯  X X j J 2 in the limit where the corre- α k | 0 0 i q d i − µρ 2 2 i k α α z 0 z i J ν ¯  α | : r r , . . . , k ∂ − j 1 µ i c z q ¯  k + + c σ z · ( | , z q 0 ) ) − 0 i 2 2 i ρ j j | α α i are Graßmann variables introduced to n k n d 2 2 q ¯ c   z 2 b ) ) j j + z i | f f j j M z ¯ M ¯ θ θ θ
¯ z z ) 2 ( ( − i − S , µ d  · · ) symmetry which is fixed by igravitational constant, and integration is over i i − , . . . , n n ν i q d ¯  , modulo SL(2 θ n 1 abc · k i i d dθ {  µ k z µ i i dV d ∂ k π k n n =1 =1 ν i κ 2 i i Y Y  k , . . . , k is the  ]. The results of those works, important to us, are the following. 1 n Z Z 0 =1 d − k π i X 18 ( α can be fixed to any point in the complex plane and the indices i κ 8 × × ν – k , c can be written as i ∂ 0 +1 = 15 k α n µ i q,µν i , z 0 ¯  ε k b , α n d  i =1 M n i , z  κ i f a M . Normal ordering : : means that all derivatives act to the right. In bosonic z P ) = = ν q ¯  = S ]), which is proportional to the inverse string tension µ µν i ( q  J P -point bosonic string amplitude of massless closed states carrying momenta 15 n By explicitly calculating the integral over one of the = (cf. [ µν q The soft, symmetrically polarizedε state carries momentum string theory, but not in superstringq theory, an additional operator contributes to where sponding momentum is softthat compared the to soft the behaviordescribed other by of momenta, the a it expression symmetrically has been polarized, shown massless [ closed string state can be The points are any three fromexponentiate the the set integrand (the where the Koba-Nielsen variables with been studied in [ Consider for simplicity scatteringformalism amplitudes (the in same bosonic string analysisThe theory and using result the appliespolarizations operator in the RNS formalism of superstrings). (bosonic, heterotic, or in the NS-NS sector of superstrings). These limits have recently JHEP05(2018)208 - δ (4.7) before ) ], which 2 1 q such an 0 limit of ]. This is ( + 1)-point B O 32 → n ), but 0 + 1)-point α ) + n 4.2 n ) can be removed 4.5 , . . . , k . for further discussion). 1 ) -stripped scattering am- in ( 2 k δ ( B q 0 ( S α n -function cancel so that one O with the overall ( δ f M 0 α n S can, for the first three orders in ) + i n f ) consistently reproduces the tree-
M ). 0 ) +1 -point amplitude: ) involve the same ( α , and after pulling the soft operator n P 4.4 n 4.7 q ( M δ 4.4 becomes the -function through the soft operators to , . . . , k an unknown, but in principle calculable, ) was also derived in field theory from P δ 0 1 0 ∂ α n k α · n ( 4.4 ]. Consistency of the two different methods 0 f . This can be easily seen, by first noting that M q f α M n d 19 M ) – 12 – ) + , yielding immediately the known form for the ], including the recently discovered subsubleading P in powers of P ( ( 33 δ being kept outside. We would like to understand to which satisfies ( ), all derivatives of the , +1 e h S δ n , which clarifies the problems and shows how the above e P , under the assumption that no terms proportional to S 0 21 µν graviton 0 ( +1 = f M α ε α 3 δ n +1 , which is the integral representation in ( ) = 0 n f M α M n , q n M f M The field theory limit of ( ), where instead of 2 ), and its field theory limit. We provide in appendix q -function ( , . . . , k δ 1 4.4 O k ( ), but neither approach immediately yields relations among amplitudes. 0 0. ) is therefore not a soft theorem in the usual sense, where amplitudes map +1 α n ) thus indicates that there is a well defined way of taking the ) can be interpreted in terms of amplitudes, in particular in the field theory with one leg off shell enters [ up to . Another way is to commute the 4.4 → 4.4 0 M ] for any number of dimensions 6 0 α 4.4 4.4 ) 3 -point momentum conservation, in the last line. Note that the nontriviality in the above expression lies in the q α n 0 34 M , α n + current ) as well as the prefactor M 32 P appear , ( q 8 δ entering ( -expansion, be written as a differential equation on µν q + , as it enters in ( It is useful to note that the theorem in ( q 0 0 This assumption holds at tree-level to all orders, in fact, as a corollary of the KLT relations [ ε 6 α α n n P -point µν ( f f for the graviton, thedue normal to ordering the symbolgraviton in tensor soft the properties operator, operator and of secondly by noting that this operatorstate has that been the shown amplitude to factorizes into have two copies of Yang-Mills amplitudes. existence of a differential operator Soft graviton theorem. level soft theorem of theterm graviton [ [ Because of plitude the path we will takeδ here. Crucial for thethrough success the of this resulting approach isgenerically that ends after up expanding with an expression including the M analysis in the simplest caseequations of can be turnedamplitude into a statement forobtain a a representation statement of on the the three-point level stripped of amplitudes in analogy to the discussion of [ η n of obtaining ( M For obtaining these, there are at least two ways to proceed: one is to analyze carefully the momentum conservation from what extent ( limit where on-shell gauge invariance of imposing momentum conservation, hence theThis tilde (see remark appendix is important,function. because Eq. ( both sidesto of amplitudes and ( this willthe be integration important over to the us. moduli of This the ‘soft soft theorem’ is state instead in stating that here. The operators act on JHEP05(2018)208 ), , P ) ( 2 term δ . The (4.8) (4.9) q 1 ¯ ), the q (4.12) (4.13) (4.14) (4.10) (4.11) ). This ( · µν − q P η q O P ( + − δ )) + P . P ( 0 ( δ ) = ) α δ n . P P ,σ ( f i M ∂ = 2  gauge invariant δ · = 0 and Lorentz i , ¯ q ∂ ν q P ) = 1 i · ρ i ∂  2( q ¯ P q µ ) · ( · − d  ∆=0 q P -function by making i δ ] and the discussion ¯ q ) q δ k · + D + V P P 26 = ) ]: − S ( , . δ P 2 . i ) i,µσ ( manifestly 17 + . 0 13 2 k δ S – α n q S 2 = 2 = 2 ( W . ρµ 0 ¯ i − q 15 = 0 and ) δ S = · M O S 2 ) = 0, = 2 ( q P e S q -stripped amplitude, i.e. ¯ V ) ( q 2 + ( q is directly multiplied by · δ q ) + q S σ · 0 O 0 ). To obtain this result, momentum P q δ + i P ) for ¯ α ρ n + ( S q k P q
δ 2.5 ( 2 = 0, as will be further discussed below. f ) + + M ) δ W , 2 q P P S P ) · ( n ( =1 ( − µ i δ i X δ δ ¯ 2 q k ) holds for = ( ν P W which annihilates − graviton = 0 q ∂ S i S α · n 4.7 V ) where ) takes the form [ − – 13 – ,S W q = ) in terms of amplitudes is immediately solved in = f M i ν ) k ¯ 4.4 q ) are given by the local operators W ∂ 4.10 was moved to the left of the 4.4 µ P + µ ∆=0 S ( P q -function δ ,S ) ( δ δ i K 0 W q ) S − δ µ ) S graviton W V arises by use of momentum conservation from the leading ) becomes the proper q + S P ) S δ µν ) i ( + S η k 4.2 P q δ + ( ( ∂ h δ 2 ) and · + 2 W The soft behavior of the dilaton is obtained by replacing the − d ∆=0 q P − S P d κ ( 1 ( ( D ) on the leading term gives δ d √ δ = 0, while for the second term we can use the identity as well as the on-shell conditions − √ 4.9 (1 + 7 P i ) = q =  · = 2 · , q . The operator i 0 n δ q k contains the operator ] α , the ‘soft theorem’ in ( n S i , due to the propagator-pole at  f 32 n µν dilaton M =1 W 1 i ε X S )) → , . . . , k − i 1  is the generator of gauge transformations ( 2.6 k = ( i φ W ); i.e. the projector ( W 0 + S α n The factor 2 in the soft operator 4.4 7 M type in figure in ( can also be obtained by firstfirst expanding term the is zero because leading to the same result. expressions. For the contributions tointegral expression ( defined in ( From an analysis of Feynmann diagrams, these terms must correspond to diagrams of the invariance of use of the following, easily checked relation: Notice that Here conservation was used where the coefficients of as well as the non-local operators Assuming for simplicity that allcan hard take ¯ states are either gravitons or dilatons, such that we polarization tensor ofaround the ( soft state with the projector (cf. [ Hence, the problem of understandingthe ( case of the graviton and, inSoft this dilaton case, ( theorem. the property [ JHEP05(2018)208 ] ) q 0 19 α 4.16 gauge (4.16) (4.17) (4.15) , ) 2 ]. The full q ( . ) 14 O 2 , q ( + 13 manifestly 0 O α n ) for ∆ = 0, as first ) after the projection M i 2.8  ∂ takes a 4.11 · was immediately dropped, ) = 0 + 0 i 2 α n k q W i ( k S from a hard external leg. M ∂ O · q q e i ) is universal; i.e. it describes the q ) + .  · ) n · 2 i is identically zero (i.e. without using q q k 4.10 i (  appearing in ( ∂ O n =1 i · X i q, . . . , k − k µ ∆=0 + K = i ] for different string theory setups and in [ = 0 + – 14 – ) = i 0 2 can be immediately dropped. However, such a 17 α n q – W ( and , as well as the expansion of the momentum shift 0 M W 15 O , . . . , k α n S 1 W + k S M ∆=0 ). Evaluating the shift operator, we can thus write ( ( 0 0 ], the expression ( 2 α n is done, because as we will argue in a moment, we can q D α n q = ( 16 V M 0 M O S i α n  W + ∂ S f i M · k ) ). Notice that we can write ) ∂ and If one assumes that the amplitude P · q ) = ( q 2 δ W + . 4.10 q ( i S ) was derived in [ k W ]. This gives a first glimpse at the role played by the conformal algebra Emission of a soft particle with momentum O ( S i q 19 = 1+  + 4.10 · i · 0 i k α n ∂ q k · q f M e n ) =1 i X P Figure 1. ( ) vanish for the dilaton. There, however, the operator − δ The leading part of the above soft theorem was already understood in works dating Remarkably, the operators W 4.5 S operator: in the form should drop out in ( where we used that The operator invariant form, on which theon-shell action constraints), of therepresentation operator of the amplitudeparticularly useful is for generically us. not Let us known, therefore and give an the alternative argument argument seems for why not this term in field theory.soft As behavior shown of in the [ in dilaton ( in any stringbut theory, we meaning wish that here the to operators scrutinize of the order arguments for this. pointed out in [ in the context of graviton scattering, on which weback will to elaborate the below. 1970ssoft in behavior relation ( to scale renormalization in string theory [ effectively set onto the dilaton, are exactly the conformal generators defined in ( The splitting into JHEP05(2018)208 ) 4.17 (4.18) . ) can, as , the first µν 0 . 4.10 , which does η q α n,i,αβ 0 α n ∼ which complete M 2 ν i M q )  i  µ i can be exploited to  · th state is a graviton i -expansion is spoiled. i q  q ( αβ on (b) η 0 q α · n i ) + M k i  · ). If the hard external states are ). Dilatons are represented by dashed i   ( 2b q β 4.18 q 0 α α n,i,αβ q M = 0 and for a dilaton β i n =1  i X g α i  )  ·  P – 15 – g comprises only derivatives acting on the polar- ( )  δ . In consequence, we will assume that the operator P V ( = S δ C
i σ  ∂ ρ i d vanishes without employing any resummation. The same applies to the ), and this turns out to be consistent with our observations in + W 8 S . i,µσ 4.10 The operator (a) S 5 ρµ i . S V ). The second term on the other hand corresponds to the emission of , and hence the quadratic dependence of hard states are gravitons. In that case the soft theorem in ( S i q σ  · 2a n q i ρ k q q 2 Diagrammatic interpretation of the terms in ( . V = 0, the operator n =1 i S X ]). Since for a graviton we have q · 16 i  can be dropped in ( Note that if all reference vectors used to define the polarization vectors are chosen to be equal and such 8 W or dilaton, only onecase of where these all terms contributes. For clarity we considerthat from here onoperator the also [ term corresponds to the decaysoft of an (see internal figure graviton toa two external hard dilatons, graviton one of fromtaken them the to soft be dilaton ingoing,hard leg instead graviton (see into of figure a outgoing, hard dilaton. the diagram Thus, depending shows on that whether a the soft dilaton turns a In field theory, we interpret the above terms as on-shell emissions from external states (see The operator ization vectors rewrite these terms more transparently as the original expression to thea full gauge resummation invariance is condition. that TheThis the potential is danger order-by-order further of gauge discussed such in invarianceS appendix in the the subsequent section The final zero followsnot from necessitate on-shell manifest gauge gaugerelies invariance invariance. on of the the Notice, addition amplitude however, of that an the infinite rewriting number in of terms ( of higher order in Figure 2. lines while gravitons correspond to wavy lines. The soft particle carries momentum JHEP05(2018)208 0 n → , 0 (4.21) (4.22) (4.19) (4.20) ) 2 α q ( , ) in terms number of ) is not mul- O 2 . n q  ( 4.21 f , this equation 2 ) + M O 0
n even ] q + νρ 0 , B α n µ [ , . . . , k f M ∂ = 0 ) th state being a dilaton i 0 i n k φ → -field which in four dimensions ( 0 2 lim M B 0 α 8 − ] d α n i ) -fields at tree level. )). q f M ∆=0 B − i , . . . , φ 0 e 1 4.7 δ ∆=0 D k 1 S ( 12 D ): at leading order −
0 ) α n are taken to be symmetric, traceless − − P ν i ) [2 ( M 4.21 φ (2  ) δ ν P vanishes, and we end up with the consis-
µ i P ( ) P  ( δ ∂ φ∂ V P δ · µ The obstacles for intepreting ( ( S = 2 = ∂ δ q ) q 0 i P µν ·  α – 16 – µν i n ∂ g · i ε + · f k 1 2 M gravitons. This line of arguments applies, however, q is not necessary for an even subset of the set of δ q -point amplitudes with the ( 0 S n n − ) → µν i 0 n =1 ε P α R i X ( ) + ] lim  δ ) was tacitly, but importantly, used. This corollary states 2 h g − d 2 d κ µ − ∆=0 ∆=0 − d (cf. the discussion around ( hard states comprise both gravitons and an √ 2.16 d κ √ K D 1 √ n -function, and moreover that the equation includes a derivative x µ q + -point graviton amplitude in field theory. To obtain the stripped d δ q − n d Furthermore, at tree level pure graviton scattering is completely symmetry to generic tree-level scattering processes involving dila- + ) = 2 ) [2 Z 9 Z , q 2 d P n ( ∆=0 1 κ δ 2 D -field couples only quadratically to the graviton, scattering processes in- = − B , . . . , k 1 )(2 k is the stripped ( P φ ( n -function at order δ 0 + ) the contribution from the operator δ h α n M low-energy S There is, however, an immediate corollary of ( Translating this M 4.19 In four dimensions, these statements hold even at loop level as a consequence of U(1) charge conserva- 0 = 9 where amplitude, the commutator ( tion, because the dilaton formsis a dual complex U(1) to multiplet a together pseudo-scalar with field. the of this reduces to dilatons, and thus tracelessnesspolarization of tensors. Consequences for graviton scattering. of field theory scattering amplitudes aretiplied the by facts the that momentum the formal expression where it should bepolarization tensors understood corresponding that to all also to the case where the tons and gravitons, wedilatons conclude are that non-zero. onlydescribed amplitudes by with the an Einstein-Hilbertof even term ( number in the of action. external tency condition Therefore, taking the limit Since also the volving only gravitons and dilatons do not involve virtual rather than a graviton.tree-level This scattering expression amplitudes of pointsreason gravitons at with for a an this peculiarity: odd can in(in number be the the of understood Einstein field dilatons frame) from theory vanish. that the limit, The the low-energy dilaton action couples of only the quadratically to string, gravitons: which shows where in the second line we sum over just argued, be rewritten in the form JHEP05(2018)208 - ) 2 ). ). δ n − d 2.15 4.27 4.27 (4.26) (4.27) (4.28) (4.30) (4.24) (4.25) (4.29) (4.23) , n , 0 M α . n 0 µ ∆ which obeys ( f α also annihilates the . n M K 0 0 0 ) f 2 α M n α n → n − P 0 0 0 d lim f ( M f α α n M → δ 0 0 i . lim f k µ α M ∆= 2 → = . ∂ 0 0 i -stripped field theory ampli- lim K · α ) 2 n δ − n → , 0 i q lim d n ) − α M d
∆=0 n d # =1 = 0 i X ∆
D − 0 ) 2 α D n P n − − (2 indeed takes a form that obeys ( ( d f
− M δ 0 ) ) ) reduces to (2 . 0 i d α 3 k
P 2 P → ) ( ∂ ( for ∆ = 0
f lim exists for which M δ · δ ) n α − P 4.21 for ∆ = 0 ( i P q P d − k α δ n ∂ ( 0 ∂ P δ · -stripped amplitude by noting that for ∆ = f α · , – 17 – M n , field theory tree-level graviton amplitudes are µ ∂ n q =1 δ i P · q n X , f M ∆ = ∂ q 2 0 = 0 n =1 n − i X → d ) + n 0 = 0 + + α P : n n M − ( 0 ) lim )∆ δ ) α n µ ∆ M P P M µ ∆=0 ( ( P µ ∆ f K M ( δ δ µ ∆ ) K -graviton field theory scattering amplitude. We may translate µ ∆=0 δ 2 K µ n ) n K − P q K d ( ) ) was used. Hence, we find that P µ δ µ ( ∆=0 q P δ µ ∆= h ( K into the stripped amplitude, we pull the momentum conserving δ 4.22 µ is the K = h 0 q µ n α n n q " 0 = f M M 0 = = M ) provided we are working on a representation of 0 = µ P ∆ our explicitly derived result for ( n δ K B M = n , such a representation can be obtained by making the permutation symmetry of n ] this consistency condition was also discussed from a string theory perspective. M M 19 At subleading order, the above equation ( -stripped graviton amplitude: amplitude In appendix In the following sectiontude we will argue that,graviton at scattering least manifest. for the where the corollary ( δ It follows that all generators of the conformal algebra annihilate the tree-level graviton Here, this into an invariance statementwe for have the we arrive at the statement where in the second line we introduce a nontrivial scaling parameter Hence, assuming that a representation of We may rewrite this in the following suggestive way: In order to turn function through the specialmodulo conformal Lorentz invariance generator of using the commutation relation ( homogeneous functions of degreeref. 2 [ in the momenta. This is indeed a well-known fact. In that for any number of external states JHEP05(2018)208 (5.1) (5.2) (5.3) (5.4) (5.5) is obtained n 6) ) are satisfied. , . M Tree-level gravi- 4 , 4) 3 , 4.30 , , 3 ]). In this work, the 5 , , 5) 1 (2 , 37 , , 5) 2 P , , (2 ) and ( 3 36 4 6 + , , , ¯ i A 4 1 , , 3) 4.27 35 6) 1 , s , , (3 4 , h 3 5 , , (2 ¯ n 2 A 6) 5 4 , , , ¯ M , A 5) 5 5 (1 , 2 , , 4 5) 4 3) 4 − 1 ¯ , , , A 4) indicates a sum over all permutations , n , 4 2 , 2 3  , , 4) , 3 , (2 d 3 , ) as well as the implication of special confor- 3 , 2 6 2 , , (1 3 κ , ¯ A 3 2 , (2  ) , (1 ¯ 2 (1 = 0. The stripped amplitude A P ) 5 4.27 , 6 35 – 18 – (1  P A 3) s A 5 (1 ( , · 4 24 δ A 45 2 +  ], which relate closed string amplitudes to a double- s , A s 34 1 = 34 13 s 12 12 (1 s are the color-decomposed partial amplitudes as defined n 3 is 12 is is n with − is − iA M ¯ + + ( A . The term -function become irrelevant when acting on the manifestly δ n µν g indicating that the polarization vectors can be distinguished, ε A 4) = 5) = 6) = 3) = and n , , , , . Being interested in graviton amplitudes, however, we need to ≡ ]. ¯ i → A n 3 4 5 2 ¯  , , , , ν n A  -dimensional gravitational constant. As remarked, the polarization 35 2 3 4 ¯ , (1 , , , µ A d → 3  2 3 27 (1 ) are satisfied — at least for three, four, five, and six external gravitons. i , ,  4 M 2 (1 , and , j 5 M 3 and 4. The graviton amplitudes are then given by 4.30 k (1 , 6 · M i M k , or simply we take i  = 2 denoting the n ¯ A d ), with the bar on ij → s κ i Interestingly, it turns out that by manifesting full permutation symmetry of the 3.1  with vectors in the twolarization copies tensor of by Yang-Mills amplitudes are identified with the graviton po- Here in ( i.e. in set ¯ of the indices 2 ton amplitudes can be constructed fromthrough Yang-Mills the amplitudes so-called in various KLT ways. relationscopy One [ of way open is string amplitudes throughlimit a (for momentum kernel a with brief arelations well-defined review field at on theory three, the four, field five and theory six relations, points see will e.g. be [ used, to be precise: mal symmetry ( This is reminiscent ofarising the from Yang-Mills case. stripping off Additionally,permutation the we symmetric observe amplitude. that We the will ambiguities detail theseConstructing results graviton amplitudes in from the Yang-Mills following. amplitudes. amplitudes in four dimensions.or full The permutation analogous property symmetry.recently noticed for in The graviton [ importance amplitudes is of Bose manifesting thisstripped graviton symmetry amplitudes, was the identity also ( In this sectiontheory we amplitudes. study the WeOur relations would study derived like of in toexposed conformal the understand the transformations previous when necessity of to ( sectioncyclic Yang-Mills manifest permutation on amplitudes and physical explicit reversal in properties symmetry, field in momentum of order the space to stripped observe the amplitude, invariance namely properties of 5 Conformal symmetry of graviton amplitudes JHEP05(2018)208 ), 11 = 2, 3.4 ) (as (5.9) (5.6) (5.7) (5.8) (5.10) (5.11) d in the 2.7 n C As for . ) n 10 ,  n ] will be used to indicate n , )) M , . . . , k [ 1 n . will be trivial before symmetrization. i P ,  1 . n G to the form k k µ i , . . . , n ( . . n We would like to establish explicitly 2 k f n 6= 0 , on ) k in this context. M ∆ , n i =1 (1 i f X 8 n F ∝ P µ i ,...,n on + ( i M  X (1 i n µ ∆ G P n F =1 µ i i X ! K µ i ] = M k 1  n n = – 19 – n n =1 =1 M n i i [ ) = X ) vanishes, reminiscent of what happens for Yang- X See Footnote n ) )] = M . = P 5.9 n P ( n ,  = 0 ). The amplitude is fully permutation symmetric, i.e. δ ). While in both cases we lack a rigorous understanding µ the dependence of ∆=0 n , . . . , n M n momenta and polarization vectors. i 2 K µ 2.7 k ∆ 3.5 M , 1 n K − =1 (1 n i ∆=0 n , . . . , k P 1 K − · M = ,  variables, however, reintroduces q n 1 k n k | ( n in f ), we have of course that [ M n n P = 3.9 P f = 0 holds, we find denotes a function of q · f i  Note that for Note that if all reference vectors used to define the polarization vectors are chosen to be equal and such 10 11 The symmetrization that Hence, for ∆ = 0Mills the amplitudes second for sum ∆ in =of ( 1, this cf. feature, ( we notice thatwhere ∆ gravity = is 0 known is to the classical be scaling conformal. dimension of Rephrasing a the graviton above, in we thus have generically In all cases to be discussed below, we curiously find that described in the previous section.Poincare invariance Let constrains us the first action point of out some general features. As in ( where by equality we meanin on resolving all the previous Poincar´econstraints as discussions). prescribed by ( Conformal invariance of graviton amplitudes. whether tree-level graviton amplitudes can be considered conformally invariant in the sense Here Yang-Mills case ( however, it is useful toby make this explicitly symmetry permutation manifest symmetrizing atthis it. the manipulation: level The of notation the stripped amplitude we have as a consequence of Bose symmetry.but We stress holds that modulo this identity is momentum generically conservation not manifest, and on-shell conditions. For our purposes, by implementing the relations ( JHEP05(2018)208 , ) ) 2 3 3) A 5.9 , 5.14 2 (5.13) (5.14) (5.15) (5.16) (5.17) (5.12) , in ( (1 ) for i 3 G A . 3.13 and i ] = 0 F n M [ n , , and we again resolve P 2 i ]. These results are insen- µ k = 0 3) ∂ , 38 n 2 , , n =1 M
i X ) (1 3 3 P . A ( A 2 )) suggests that the above nontrivial δ µ ∆ 2 3) , n ). K − = 3.6 , d 2 , A 3 3 , 2 ∆= A (1 ] = 0 M n 3 − n D 12 A d M ) [ . As discussed in that section, choosing the and ] = n ). Now, using the expression in ( 1, but not otherwise. If instead of P – 20 – n ( 4 )) P -dependence of the scaling dimension ∆ in ( − δ ∆ = 2.7 M n µ ∆ [ ∼ n 4.22 K = 4(1 + ∆) 3 P 3 ) ] = 0 (cf. appendix M n P M ( ] = 0, can be alternatively stated in terms of the two conditions ). The µ ∆ M (notice that the latter is also permutation symmetric), we δ [ n , 2 n K 2 n − M 2.7 ] P [ d 3 i At three points the KLT momentum kernel is unity, and the n ) by numerical evaluation of the coefficients k · A P j [ ] = 0 ∆=  i 3 n µ k ∂ = 4 Yang-Mills case (cf. e.g. ( C up to multiplicity four. For mulitplicity five and six we have checked D ∂ i 5.13 M d d [ P =1 n n i ] or P ) holds for any ∆. P 2 3 A ) in the previous section [ ] furnishes a representation of the field theory amplitude which obeys the µ 3 5.13 ∆=0 n arise from a deficiency of preserving permutation symmetry. In fact, it turns P ] = 0 and K i 4.27 M n [ F n M [ P n P ij s On the amplitude which only depends on Lorentz invariants, the vanishing of the summed momen- ∂ i 12 we consider observe that ( tum derivative,P i.e. Poincar´esymmetry as prescribed byone ( observes that which coincidentally vanishes for ∆ = three-point graviton amplitude isMills simply amplitude: given by the square of the color-ordered Yang- The stripped graviton amplitude thus reads sitive to the ambiguitiesconstraints as in described obtaining in thesuggests ( stripped that these amplitude results bystraightforward do manner. resolving not the follow Poincar´e from theThree-point Lagrangian description example. of gravity in a establish our conjecture ofordinary full Einstein conformal gravity. invariance Wespacetime have of dimension verified tree-level these graviton statementsthe amplitudes analytically above in and conjecture for ( generic using four-dimensional kinematical data generated by S@M [ peculiar value the above results together with (cf. ( Hence, condition ( functions out that in the studied examples up to (and including) six points we have for any value of ∆. In particular, this implies This similarity to the JHEP05(2018)208 ). ∆ 4 D k = 4 (6.1) + (5.18) d 2 k , we find ) are ful- n + 1 M ) in the case k ( 5.13 ) are fulfilled. − contains, in ex- 5.12 . 6 = 5.13 2 . 4 2 M n ) − k ) are also fulfilled in ) and ( 4 d ν 4  5.13 5.12 ) and ( obtained by using the ex- ··· n 1 ν 1 5.12  M 4 µ 4 ) and ( k terms. Permutation symmetrizing ··· 5.12 Gravity : ∆ = 1 8 µ 1 k = 6, the amplitude 4 ν 4 n 3 upon replacement of , terms, that in the end sum to zero. 2 -function) under dilatations generated by ,...,µ , 1 δ 12 ν – 21 – 1 ): at 10 ,µ 8 t ∼ ( 5.13 boosts the number of terms at least by two orders of , 13 µ ∆ s At five- and six-points we find again by the same proce- K + 1 23 16 s 4 ) and ( 12 n − s ) d i )). 5.12 At four points starting from the stripped amplitude − 4.21 in the KLT relations, which in terms of the stripped amplitude reads = ( 4 4 we provide in addition an explicit demonstration of ( A M D YM : ∆ = ) for 3.19 = 2, respecively, for which the conformal symmetry of the respective theories is well d Let us summarize our observations in some more detail. Clearly, gluon and graviton In appendix requires to have These values for ∆and become multiplicity independent in the spacetime dimensions prescription, we observe similaritiesmetrized) Yang-Mills to amplitude the in four special dimensions, conformal but no symmetry fullamplitudes of conformal are the symmetry. both (unsym- tude invariant (including under the Poincar´esymmetry. momentum conserving Invariance of the full ampli- can indeed be turned intothe full an conformal invariance algebra, statement though of atMoreover, tree-level the the cost graviton of formulation amplitudes a of under multiplicity dependent theand scaling symmetry dimension. polarization in vectors, terms whichhinges of on differential does the operators full not in symmetrization leave of momenta the the graviton on-shell amplitude. constraint Dropping the surface symmetrization invariant, magnitude, ending naively with some 6 Summary and outlook The aim of the present paperton was limit to for identify the graviton consequences scattering. of We the found string that theory soft the dila- indications for a conformal symmetry The details should bycomplexity now in be observing clear, ( panded but form let us and makeincreases before a this any number remark reduction, by about athree roughly the factor labels), 10 computational of while 5! the action (since of the KLT expression is already symmetric in of two-dilaton+two-graviton scattering, wherethe this discussion equation below is ( also expectedFive- to and hold six-point (cf. checks. dure as detailed in the three-point case that the relations ( This expression has the virtuefestly of permutation being symmetric manifestly in gauge the invariant. labelsIt It 1 can is be additionally readily mani- this checked special using case. this expression that ( by the exact samefilled. We procedure have additionally, as like instarting in the from the case the of three-point manifestly Yang-Mills theory, gaugepression case also ( invariant studied that expression the for ( relations Four-point example. JHEP05(2018)208 ], ), ). P ( 40 6.1 (6.2) , δ 39 = 4 and -function d δ with µ ∆ . Since in the K d , takes the form n of special conformal trans- A µ ∆ K . i G µ i k ]), or by considering the twistor action n =1 i X ) appears to be related to the incompati- 42 = 0 for ∆ = 1. In the case of gravity we , + i 6.2 41 i G = 0 for any value of ∆ and F i µ i in (  F i – 22 – n F =1 i X = = 0, if the constraints from momentum conservation n ). We note that, somewhat unsatisfactorily, different i A F µ ∆ 3.11 may also be transferred to the Yang-Mills case, once we K ) of ∆, which imply dilatation invariance of the full amplitude. 6.1 with the on-shell constraints — at least for the conformal Yang-Mills theory in in any dimension µ ∆ K = 0 for ∆ = 0. i G A natural question is whether the above conformal symmetry of graviton amplitudes Another point is whether the finding of a conformal symmetry of tree-level graviton The observations of this paper lead to a tower of follow-up questions. The most For the case of fully permutation symmetrized graviton amplitudes, however, this am- The special conformal generator commutes with the momentum conserving The non-vanishing of the coefficients of our results. can be deduced fromconvenient starting the point Einstein-Hilbert might be action.which the employs simplification only In of cubic interactions this order (see action also to as [ expressed approach in this [ problem, a a form that manifestly preservesof the these on-shell statements. constraints should distill the physical content amplitudes manifest all symmetries of themension. amplitude and If this allow would for indeed a be multiplicity the dependent case, scaling it di- would emphasize the need for an interpretation pressing point is thecurious meaning coincidences of or the is conformal therebe a properties important deeper to found overcome significance the here. behinddue specific them? symmetrization Do to prescription In they the employed particular, represent use here, it which of is would momentum and polarization variables. Formulating our observations in erator independently of thewhich value guarantees of also ∆, dilatation invariance weHence, and may for a choose this vanishing the commutator multiplicity of scaling dependentformal choice dimension symmetry of as of the in scaling tree-level ( permutation dimension graviton symmetric we amplitudes, form. observe if full con- the amplitudes are in a manifestly observation to the factcyclization that and not reversal. all symmetries of the YMbiguity seems amplitude to are play manifestgravity no case after role manifest and symmetrization we leads to find invariance under the special conformal gen- 4d. We thus make theor physical full symmetries permutation of symmetry, the respectively.for amplitudes In ∆ manifest, the = i.e. case cyclic/reversal 1, ofare we Yang-Mills resolved theory then as in indeed prescribedways find of by resolving ( these constraints do not always give this result. We attribute the latter In the case offind Yang-Mills theory we find that only for the values ( bility of formations. Poincar´einvariance implies thatYang-Mills the or action gravity of amplitude, here this collectively generator denoted on by the stripped known. The most involved analysis concerns the generator JHEP05(2018)208 ]. 53 ] for ]. It 56 18 – 15 ]. The latter 48 – 46 ]). Exploring these di- 54 ]. 35 ] concerning a similar question ], valid for tree-level graviton 55 ]. In relation to our observations 50 , 45 49 = 4 super Yang-Mills model, this conformal 6= 4. Alternatively, it would be interesting to – 23 – d N ]), as well as by Hodges’ determinant expression for MHV ] and references therein). Intriguingly, these approaches were 7 44 , 43 ] (see also [ ], could provide another path towards a proof of the conformal properties 6 52 , ]. Here, the two-dimensional conformal symmetry of these correlators orig- 51 12 , 11 ]. In fact, it should be very enlightening to make connection to twistor methods for On the other hand, these modern formulations of gravity amplitudes can be taken as a Another interesting direction would be to turn the logic of this paper around: assuming Our motivation to look for conformal properties of graviton amplitudes was largely Recently, tree-level gluon amplitudes were mapped to correlators on the 2d celestial We speculate that BCFW recursion relations [ 43 Yang-Mills and gravity amplitudes? motivation to search for the imprintsgravitational of the double conformal symmetry copy. of gluonextension Intriguingly, scattering in of we their note 4d that Yang-Mills theory, in the the maximally supersymmetric conformal symmetry of gravitonmultiple tree-amplitudes, soft graviton what limits? are Consideringin the the analysis the implications of Yang-Mills for [ case, singlewhich implications strong or the constraints observed conformal may invariance haskinematics be on duality. the expected. KLT Does relations or it SimilarlyAnd the constrain color- how one the can may KLT our wonder kernel results or be the seen BCJ in kinematic the numerators? new formalism of Cachazo, He and Yuan [ define a finite remainder functionsuch by a factoring quantity out the hasrections divergent conformal pieces should properties and reveal to (e.g. whether seegeneric in whether the hidden the symmetry observed sense of conformal of graviton invariance scattering [ at or tree not. level reflects a gravity and try tograviton apply amplitude, which the for conformal four-dimensionalFor generic symmetry helicities helicity has generators configurations, been however, to the deriveddimensions. respective the already amplitudes Nevertheless, one-loop in are it divergent [ four-point in wouldpersist four be for interesting to the see finite whether contributions the or conformal for properties based on a detailedwould analysis be of important thefor string to field theory see theory soft whether graviton dilatonalso amplitudes one limit play at can derived a loop in role deduce order, there. [ similar and A whether consequences first the approach of conformal could soft generators be limits to consider perturbative Einstein-Hilbert sphere [ inates from the Lorentzto understand symmetry which of role the theextend 4d this 4d analysis conformal gluon to symmetry amplitudes. the plays graviton on It case. the would celestial sphere be and interesting to symmetry of amplitudes in conformalanalysis (super)gravity is might realized, help see in e.g. translating [ our statements into spinor-helicityamplitudes or [ twistor variables. presented in this paper.in In the fact, recursive constructibility permutation of symmetry graviton seems tree-amplitudes also [ to play a pivotal role gravity amplitudes (see [ motivated by Maldacena’s embedding ofin tree-level curved Einstein space gravity [ intograviton conformal amplitudes gravity making Bose symmetryin manifest pure [ Einstein gravity, it would also be interesting to understand how the conformal of [ JHEP05(2018)208 ]. 63 , (A.1) (A.2) (A.3) (A.4) 20 -point stripped n . j  · , , . . . , n . i }  , -stripped amplitudes in terms of = = 1 ji ]. The latter is in turn related to δ e ij ∂ 57 s, w, e i, j { = ∀ ), on , e ij = e j ) ∂ p 2.8 X · = 0 is a property of the graviton, but does i isupergravity double copies. mysterious features seen in recent years in gravity andAcknowledgments Yang-Mills theory. dimensional symmetry algebra known asa the dual Yangian superconformal [ symmetryin [ in the sense thatOne they may determine thus the wonder form whether of these the extended tree-level scattering conformal amplitudes symmetries [ leave an imprint in symmetry is in fact not only lifted to a JHEP05(2018)208 ) C , = 3 = 0. (B.1) (B.2) (A.5) (A.6) i n k must be indicate =1 n i R n P A i =
and = 0.

il . il P li 2 w s L q w ∂ ∂  
∂ j ji problem of not being il ij j = ji s z s w 2 w ∂ i K ∂ ∂ · ) is fixed (the SL(2 , thus − ∂ k i il ij = 0. To see the problem jl ) says that there µ s i s li Θ q q z w ∂ 4.2 , where w 0 . + 2 4.4 jl j α R + n + 2 s . Since there is no integration n + | li 6= = 3. I i X technical 1 il in ( w P q L n il + ∞ z n -point momentum conservation, w ∂ s I , + ∂ il n ∂ = ij 1 − s abc s = 0, with 2 ij , ji µ = ∂ ) 3 s ∂ j w q j 3 z ∂ ji dV li Q | ∂ z z n Θ w 2 jl = 0 2 I w + il · | s ∂ d s − 3 i , 3 z 2 jl + z i n k and + Θ z . ), where overall momentum conservation + z w il 2 ( A − n µ s i ji + li d ) by i k ∂ 2 4.4 w 1 w A 2 0 ji z n + 2 z ∂ ij i 2 ∂ k | iSupersymmetries  2 , 4 , k  Commun. Math. · ↔ 4 ,  3 1 

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D.3 3 2 · hep-th/0312171 ][ t , k k [ 2 manifestly k − 32 4) + 2 , CC-BY 4.0 4 permutation symmetry in (2008) 085011 − k 3 , Stueckelberg approach to 6d conformal gravity Ordinary-derivative formulation of conformal low spin fields Einstein Gravity from Conformal Gravity = = 32 This article is distributed under the terms of the Creative Commons ↔ − Nucl. Phys. 2 -function has been stripped off according to ( , , 3 4 δ 1 k D 78 Perturbative gauge theory as a string theory in twistor space F F (2004) 189 (1 g − 2 φ = 2 252 does commute with permutation symmetrization, we could also have started with 4 arXiv:0707.4437 M k µ [ 0  µ 0 K and Quantum Symmetries 064 Phys. Rev. Copy of Gauge Theory Phys. Open Strings K This expression is R.R. Metsaev, J. Maldacena, R.R. Metsaev, Z. Bern, J.J.M. Carrasco and H. Johansson, E. Witten, H. Kawai, D.C. Lewellen and S.H.H. Tye, Z. Bern, J.J.M. Carrasco and H. Johansson, [5] [6] [7] [3] [4] [1] [2] References manifesting the 3 Open Access. 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