Taming higher-derivative interactions and bootstrapping gravity with soft theorems
aa r X i v : . [ h e p - t h ] O c t Taming higher-derivative interactions and bootstrapping gravitywith soft theorems
Ra´ul Carballo-Rubio,
1, 2, ∗ Francesco Di Filippo,
1, 2, † and Nathan Moynihan ‡ SISSA, International School for Advanced Studies,Via Bonomea 265, 34136 Trieste, Italy INFN Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy The Laboratory for Quantum Gravity & Strings,Department of Mathematics & Applied Mathematics,University Of Cape Town, Private Bag, 7701 Rondebosch, South Africa
Abstract
On-shell constructibility is redefining our understanding of perturbative quantum field theory.The tree-level S-matrix of constructible theories is completely determined by a set of recurrencerelations and a reduced number of scattering amplitudes. In this paper, we revisit the on-shellconstructibility of gravitational theories making use of new results on soft theorems and recurrencerelations. We show that using a double complex shift and an all-line soft deformation allows usto relax the technical conditions for constructibility, in order to include more general propagatorsand higher-derivative interactions that prevent using conventional Britto-Cachazo-Feng-Witten(BCFW) shifts. From this result we extract a set of criteria that guarantee that a given gravita-tional action has the same tree-level S-matrix in Minkowski spacetime as general relativity, whichimplies the equivalence at all orders in perturbation theory between these classical field theories onasymptotically flat spacetimes. As a corollary we deduce that the scattering amplitudes of generalrelativity and unimodular gravity are the same for an arbitrary number of external particles (aslong as the S-matrix of the latter is unitary), thus extending previous works that were able to dealonly with n = 4 and n = 5 amplitudes. . INTRODUCTION A quantum field theory is said to be “constructible” [1–5] if its amplitudes A n (1 h h ... n h n )for n > n ⋆ ∈ Z can be determined recursively by means of recurrence relations, the initialconditions (or seeds) of which are given by the set of amplitudes with n ≤ n ⋆ . In con-structible theories, scattering amplitudes are therefore determined by the principles that fixthe form of the recurrence relations, and a reduced number of amplitudes. This representsa huge simplification with respect to the traditional calculation using Feynman diagrams,in which amplitudes have to be evaluated independently for each value of n , with the cal-culation quickly becoming cumbersome with increasing values of the latter integer (see [6]for instance).This higher calculation efficiency has found a large number of applications. Together withthe use of spinor-helicity variables, these techniques allow arriving at particularly elegantand simple expressions for A n (1 h h ... n h n ) (e.g., [7–9]). In this paper, we exploit thesemethods in order to extract information regarding the physical equivalence of different theo-ries through the complete calculation of their S-matrices. The equivalence theorem(s) statesthat the S-matrix is blind to (nonlinear) local field redefinitions in quantum field theory[10–12], and so computing the full (tree-level) S-matrix of two field theories and showingtheir equality is a useful way to determine physical equivalence around certain classical back-grounds. Computing the full S-matrix for any number n of external legs is a daunting (ifnot impossible) task in the traditional approach using Feynman diagrams even if restrictingto tree-level processes, but constructibility makes it possible.Moreover, we also highlight here that on-shell methods permits us to make precise state-ments regarding the possibility of deriving general relativity from Lorentz invariance, aclassical problem that goes back to Kraichnan [13, 14], Gupta [15, 16] and Feynman [17].The result that general relativity is the only nonlinear theory that can be obtained frommassless particles of spin 2 (gravitons, in the following) is frequently quoted in the literaturebut is not often scrutinized. In particular, the technical assumptions that are necessaryto prove such a result (that must certainly exist) remain obscure. This is partially due tointrinsic limitations of previous analyses, that were typically off-shell and hence focused onthe derivation of the Einstein-Hilbert action (in the following, we will refer to this as “off-shell constructibility” to distinguish it from the standard notion of on-shell constructibility).2ere, we stress that modern on-shell methods provide a convenient mathematical frameworkfor the analysis of this problem, in which these assumptions can be fleshed out. In fact, wethink that this is one of the major achievements of our discussion.The outline of this paper is as follows. We start in section II with a brief review of theconstructibility of general relativity, mentioning some of the classic results regarding the off-shell constructibility of the Einstein-Hilbert action as well as more recent on-shell results,and highlighting (to the best of our knowledge, for the first time) their interplay. SectionIII contains our main result, that determines the most general theories that have, accordingto their soft behavior, the same S-matrix (around flat spacetime) as general relativity. Theproof of this result is given in section IV. In section V we revisit some theories of modifiedgravity on the light of the previous discussion. We close the paper with a brief conclusionssection. II. GENERAL RELATIVITY AS A CONSTRUCTIBLE THEORY
Let us start with a brief summary of known results regarding the derivation of theEinstein-Hilbert action from the information encoded in the massless spin-2 representationof the Poincar´e group [18]. First of all, it is necessary to keep in mind that one of the maingoals of this approach is obtaining the features associated with diffeomorphism invarianceas a consequence of a non-geometric set of principles that can be formulated entirely withinthe framework of quantum field theory in flat spacetime. Aside from the first works [13–17],Deser’s derivation [19] seems to be the best-known approach to this problem, but we wouldalso like to point out the thorough analysis contained in Huggins’ PhD thesis [20] as wellas Wald’s analysis [21–23] (see also [24, 25]). More recent discussions include [26–30]. Thestarting point is the observation that the Einstein-Hilbert action S EH can be written, up toa boundary term and performing an expansion g ab = η ab + κh ab , as S EH = 12 κ Z d x √− gR ( g )= Z d x √− η ∞ X k =2 κ k − Γ a b c a b c i j ...i k − j k − ( k ) h i j × ... × h i k − j k − ¯ ∇ a h b c ¯ ∇ a h b c , (1)where κ = 8 πGc − and ¯ ∇ is the covariant derivative associated with η ab (we are beinggeneral enough to include the possibility that coordinates other than Cartesian are used).3he Einstein-Hilbert action displays specific values for the coefficients { Γ ( k ) } ∞ k =2 . Let us notethat, when going from the first line in Eq. (1) to the second line, there is a boundary termthat is discarded and that therefore cannot be recovered in this approach (this was stressedin [26]).The claim that general relativity is off-shell constructible would be precisely that the set { Γ ( k ) } ∞ k ≥ is uniquely determined from the knowledge of Γ (2) . Indeed, Γ (2) can be used todetermine a Noether current associated with translation invariance, the so-called canonicalstress-energy tensor. This Noether current can be used in order to couple the field h ab toitself, which introduces nonzero coefficients Γ (3) and might allow the unique determinationof the latter. This procedure can be then applied recursively. We can identify a number ofissues with this procedure:a) Field redefinitions: these redefinitions change the form of the action without modifyingthe actual physics of the theory. In other words, there are different sets of coefficients { Γ ( k ) } ∞ k ≥ that nevertheless lead to the same on-shell behavior (it is also worth keepingin mind that the values of these coefficients depend on the choice of gauge-fixing).It is not clear how and why this procedure would be able to pick a specific off-shellrealization (in particular, the one that corresponds precisely to the Einstein-Hilbertaction).b) Non-uniqueness: a specific set of values for Γ (2) , and a particular Noether currentderived from them, leads to general relativity [19]. However, the coefficients Γ (3) cannotbe uniquely determined from Γ (2) , as Noether currents are not uniquely defined (it isalways possible to add identically conserved pieces to these currents). This featureshows up at every step of the iterative procedure, thus leading to a cumulative non-uniqueness [26, 29] (see also the explicit discussion of this issue in App. A). Additionalarguments would be needed in order to discard these other solutions, but it is not clearwhether these arguments exist.c)
Higher-derivative interactions: there is no reason to consider from the beginning anansatz such as the one on the second line of Eq. (1), containing interactions that areonly quadratic on the derivatives of the field h ab (let us recall that this approach doesnot assume by construction diffeomorphism invariance from the beginning, so that thissymmetry cannot be used in order to reduce the number of possible interaction terms4n the initial ansatz). This feature has been always put by hand [26, 28] without fur-ther justification. This issue is entangled with the nature of the identically conservedpieces that can be added to the stress-energy tensor. Disregarding higher-derivativeinteractions becomes more questionable due to the existence of higher-derivative theo-ries with just two degrees of freedom that reduce to gravitons at the linear level [31, 32](see also [33, 34]).Aside from these issues, there are two additional points that, while minor, serve neverthelessto illustrate the difference with respect to the on-shell approach discussed below:d) Prior knowledge of general relativity: attempts at deriving the Einstein-Hilbert actionhave been typically contaminated with the knowledge of the desired outcome. Thiswas thoroughly discussed in [26], including using the Hilbert prescription to obtainthe stress-energy tensor which, however, is not necessary as one could equally use thecanonical stress-energy tensor derived using only flat-spacetime notions, as emphasizedin [29]. Nevertheless, it would be desirable to find approaches that make even moreclear that there are no traces of geometric notions associated with curved spacetimes.e)
The root of constructibility: off-shell approaches fail to justify what makes generalrelativity special so that it is off-shell constructible. The work [19] strongly suggeststhat off-shell constructibility is associated with gauge invariance. However, the on-shell techniques described below present a different take on this issue, as on-shellconstructibility is a much more general feature of interacting quantum field theories.Let us now turn our attention to the on-shell description in terms of scattering ampli-tudes. A given set { Γ ( k ) } ∞ k =2 can be used in order to determine the set of amplitudes { A n (1 h h ... n h n ) } ∞ n =3 . In fact, the subset of coefficients in the action with k ≤ n uniquelydetermines the n -point amplitudes: { Γ ( k ) } nk =2 → A n (1 h h ... n h n ) . (2)More explicitly, { Γ ( k ) } nk =2 permits us to write down the relevant Feynman rules, that can bethen used in order to calculate the amplitudes A n (1 h h ... n h n ). The inverse statement isnot true, in particular due to point (a) above regarding field redefinitions.Working with the set { A n (1 h h ... n h n ) } ∞ n =3 instead permits us to avoid the issues asso-ciated with the off-shell approach, that is focused on { Γ ( k ) } ∞ k =2 . It is straightforward to see5hat point (a) is avoided, as the S-matrix is invariant under field redefinitions [10–12]. Point(d) is trivially dealt with, as the discussion is now framed in terms of the language of stan-dard quantum field theory in flat spacetime. Understanding how the situation with respectto points (b), (c) and (e) may change requires that we recall some results obtained usingmodern techniques for the calculation of amplitudes, including the notion of constructibilitydefined in the introduction.General relativity has been shown to be constructible via the BCFW relations [35–38].For the purposes of this paper, we just need to recall that n ⋆ = 3 in this case, meaningthat 3-point amplitudes are enough to determine all the remaining n -point amplitudes.This procedure is unique once the 3-point amplitudes are fixed. This strongly alleviatespoint (b) regarding the non-uniqueness of the off-shell approach that was present at everystep, making this issue more manageable, as non-uniqueness is clearly confined to the seed A (1 h h h ). Regarding point (c), the recursive derivation based on the BCFW relationsis also limited by construction to interactions that are at most quadratic in the derivativesof the field h ab (see [32] for an explicit discussion). At first sight this may suggest that theon-shell approach would suffer from the same drawback of the off-shell approach. However,in this paper we see that, using soft theorems, it is possible to deal with higher-derivativeinteractions. Regarding the last point (e), the on-shell approach shows that constructibilityis a more general feature of (effective) quantum field theories [3, 5, 39, 40].One may say that a shortcoming of the on-shell approach is that it does not allowus to obtain the Einstein-Hilbert action uniquely. That is, once the set of amplitudes { A n (1 h h ... n h n ) } ∞ n =3 is obtained, we know that the Einstein-Hilbert action is one of thepossible off-shell realizations leading to these amplitudes. However, as stressed above, it isnot possible to carry out the inverse procedure and evaluate the action in a unique way.But it is important to recall that the off-shell approach suffers from the very same issue,as discussed above. Hence, we can conclude that the on-shell approach is more convenientin the sense that it offers a number of improvements with respect to the off-shell approach,without any actual drawbacks. 6 II. OUR MAIN STATEMENT
In this section, we formulate a general set of criteria that must be met in order todetermine completely the tree-level S-matrix of a given gravitational action with two localdegrees of freedom, showing its equivalence with general relativity on asymptotically flatspacetimes at all orders in perturbation theory. This set of criteria is general enough toinclude higher-derivative interactions and is formulated without relying on particular off-shell symmetries (such as diffeomorphism invariance). Aside from the conditions below, weassume the standard requisites of locality and unitarity [41, 42], and we work in D = 4dimensions.Let us assume that there exists a gravitational action such that:A) Describes two degrees of freedom that, at the linear level, correspond to masslessgravitons;B) Has the same 3-point amplitudes as general relativity;then, it follows that1) The soft graviton theorem with the standard leading, subleading andsub-subleading contributions is verified.Furthermore, if we also assume that:C) The propagator behaves for large (off-shell) momentum as T µνρσ ( p ) / ( p ) m/ , where T µνρσ ( p ) represents an arbitrary tensorial structure (perhaps Lorentz violating) con-taining 0 ≤ m ≤ p ;D) k -point interaction vertices, with k ≥
4, have at most I ( k ) ≤ I ⋆ ( k ) = 2( k −
1) powersof momenta, while for 3-point interaction vertices we demand that I (3) < − m/ V. DERIVATIONA. Soft theorem from a double complex shift
We will follow [43] for the derivation of the soft graviton theorem (see also [44]). Thatthe soft standard soft theorem applies with no modification to the theories satisfying (A-B)above is, in fact, a direct consequence of the discussion in [43]. Let us start with a briefreview of the steps in this derivation, up to the point in which it is possible to formulate theresult we want to highlight. Let us introduce the double complex deformation | ˆ s i = ǫ | s i − z | X i , | ˆ i ] = | i ] − ǫ h js ih ji i | s ] + z h jX ih ji i | s ] , | ˆ j ] = | j ] − ǫ h is ih ij i | s ] + z h iX ih ij i | s ] . (3)Under this deformation, a given amplitude A n +1 becomes a function of two complex variables z and ǫ , ˆ A n +1 ( z, ǫ ). The arguments below are formulated in the region of C defined by z ≪ ǫ . The limit ǫ → z → z ≪ ǫ always) corresponds thento the (holomorphic) soft limit in which the momentum of the particle s vanishes. Thearguments below do not depend on the particular choice of particles i and j inside the set k ∈ [1 , n ] and the arbitrary spinor | X i [43].The possible poles of ˆ A n +1 ( z, ǫ ) can come only from internal momenta becoming on-shell,which means that this function is meromorphic, given that locality implies that these polesshould be always associated with propagators of the form1( χ ˆ p s + ˆ P J ) , (4)where ˆ P J = P l ∈ J ˆ p l , J ⊂ [1 , n ] and χ ∈ { , } . There are two kinds of poles. The first classof poles, { ǫ k } , are linear in z , and their location approaches the origin as z →
0. Poles inthe second class, { ¯ ǫ m } , satisfy lim z → ¯ ǫ m = ¯ ǫ m = O ( ǫ ). We can see from Eq. (4) that thereare k ∈ [1 , n ] poles in the first class that are, moreover, simple poles, for which χ = 1 and J has just a single element:(ˆ p s + ˆ p k ) = h ˆ sk i [ s ˆ k ] = ( ǫ − ǫ k ) h sk i [ sk ] , ǫ k = z h Xk ih sk i . (5)8t is not difficult to show that, if χ = 0 or J has more than one element, the correspondingpoles must be in the second class.Given that ˆ A n +1 ( z, ǫ ) is meromorphic, we can always write it asˆ A n +1 ( z, ǫ ) = n X k =1 Res ǫ = ǫ k ˆ A n +1 ǫ − ǫ k + D ( z, ǫ ) + O ( ǫ ) , (6)where D ( z, ǫ ) is a meromorphic function containing all the poles in the second class. It isimportant to stress that the amplitude ˆ A n +1 ( z, ǫ ) may have a residue at ǫ = ∞ , but thiscontribution would be contained in the O ( ǫ ) part of the previous equation. Therefore, thesoft limit ǫ → z ≪ ǫ , we can writeˆ A n +1 ( z ≪ ǫ, ǫ ) = n X k =1 Res ǫ = ǫ k ˆ A n +1 ǫ (cid:16) ǫ k ǫ + ... (cid:17) + D ( z ≪ ǫ, ǫ ) + O ( ǫ ) . (7)It is then clear that the divergent behaviour in the limit ǫ → O ( ǫ ) in the soft limit, and write simplyˆ A n +1 ( z ≪ ǫ, ǫ ≪
1) = n X k =1 Res ǫ = ǫ k ˆ A n +1 ǫ (cid:16) ǫ k ǫ + ... (cid:17) + O ( ǫ ) . (8)This equation can be now used in order to obtain soft theorems, as discussed in [43]. Inparticular, it can be used in order to show that the certain theories of gravity satisfy thestandard soft graviton theorem.In order to do so, we need the form of the residues Res ǫ = ǫ k ˆ A n +1 , which is fixed by thecondition (A) and unitarity. Unitarity implies the factorization of amplitudes around simplepoles ǫ = ǫ k that come only from 2-particle channels (e.g., [41, 45]), so that the amplitudefactorizes into a product of a 3-point amplitude and an n -point amplitude,Res ǫ = ǫ k ˆ A n +1 = X h k ˆ A ( z, ǫ k ) ˆ A n ( z, ǫ k ) h sk i [ sk ] , (9)where ˆ A ( z, ǫ k ) is a 3-point on-shell amplitude and the sum is performed over the helicity h k of the internal particle that goes on-shell. Soft theorems can be calculated directly fromthe multiplicative factor in front of ˆ A n ( z, ǫ k ) and the Laurent expansion around z = 0 ofthe latter [43]. 9ondition (B) implies that 3-point on-shell amplitudes are identical to those of generalrelativity. From Eqs. (8) and (9) above, it follows that the soft graviton theorem could bemodified only through modifications of the 3-point on-shell amplitudes. Hence, condition(B) fixes the form of the soft graviton theorem to be the standard one.For completeness, let us write explicitly the form of the (positive helicity) soft gravitontheorem that can be directly obtained from Eqs. (8) and (9) by taking the following steps(we refer the reader to [43] for a more detailed discussion). The Laurent expansion of Eq.(8) around z = 0 gives a series of poles in z , the order of which is dependent on the theoryunder consideration. For gravitational theories with the same 3-point amplitudes as generalrelativity, the coefficients of both z − and z − terms must vanish on-shell (which leads toWeinberg’s formulation of the equivalence principle [46]), since the final amplitude shouldnot have any poles in z due to the requirement of locality. The remaining finite z piece ofthe expansion is exactly the (positive-helicity) soft graviton theorem, A n +1 = ( ..., ǫ | s i , | s ]) = (cid:18) ǫ S (0) + 1 ǫ S (1) + 1 ǫ S (2) (cid:19) A n + O ( ǫ ) . (10)The O ( ǫ ) terms are not universal, or their form is not known. On the other hand, thequantities S ( k ) are operators that act on the amplitude A n in the previous equation, and aregiven by [43, 47] S (0) = n X a =1 [ sa ] h sa i h xa ih ya ih xs ih ys i , S (1) = 12 n X a =1 [ sa ] h sa i (cid:18) h xa ih xs i + h ya ih ys i (cid:19) D sa , S (2) = 12 n X a =1 [ sa ] h sa i D sa , (11)where D sa = | s ] b ∂ | a ] b . This derivative operator comes directly from the expansion of ˆ A n ( z ) in z in Eq. (9). The S (2) term was first identified in [47], and it will be of essential importancefor our discussion in Sec. IV B.Before ending this section, it is interesting to recall that the information encoded inEq. (10) is enough to fix the n = 4 and n = 5 maximal helicity violating (MHV) gravi-ton amplitudes [47]. Regarding the other amplitudes with n = 4 and n = 5 externalgravitons, it is straightforward to show that condition (D) above guarantees they can befixed either by means of a BCFW shift such as the one used in order to derive the auxil-iary recursion relations in [37] affecting all particles in the amplitudes A (+ , + , + , − ) and10 (+ , + , + , + , − ), or a CSW shift [48] affecting all particles in the amplitudes A (+ , + , + , +)and A (+ , + , + , + , +). These shifts fail however in order to deal with higher values of n ,unless the numbers of powers of momenta are more restricted than our condition (D); seefor instance [32] for an explicit discussion. Hence, we can conclude that the n = 4 and n = 5amplitudes in the theories satisfying (A-B) are the same as in general relativity, but alsothat we will need to consider a different strategy to deal with n ≥ ǫ | s ]. B. All-line shift and on-shell recurrence relations from the soft theorem
Having established that the the standard soft graviton theorem is satisfied, let us nowexploit this information in order to construct the scattering amplitudes for n ≥
6. Let usconsider the effect of the following all-line shift on A n for n ≥ p i = p i (1 − a i z ) , i ∈ [1 , n ] . (12)The momentum conservation constraint, n X i =1 ˆ p i = z n X i =1 a i p i = 0 , (13)has a nontrivial solution (the trivial solution would correspond to all the coefficients a i beingequal) for z = 0 only for n > D + 1, where D is the dimension of spacetime. For D = 4,there is a nontrivial solution to Eq. (13) only for n ≥
6, which are nevertheless the onlyremaining cases we have to deal with. Particular solutions to Eq. (13) were given in [39],taking into account that only D of the momenta can be linearly independent, as explainedin the following. Let us consider a subset K ⊂ { i } ni =1 of D + 1 indices chosen from theindices labelling the external particles, that inherits the order of { i } ni =1 . Given j ∈ K , wecan then define a j = ( − j D ! ε r r ...r D ε b b ...b D ( p r ) b ( p r ) b ... ( p r D ) b D , (14)where r s takes all the values in K \{ j } for every s ∈ [1 , D ], and ε r r ...r D is the D -dimensionalLevi-Civita symbol. This provides a solution of Eq. (13) such that the coefficients { a j } j ∈ K are nonzero while the remaining coefficients, with indices on the set { i } ni =1 \ K , are identically11anishing. Given that Eq. (13) is linear, we can consider linear combinations of thesesolutions for different choices of K ⊂ { i } ni =1 in order to generate solutions with all coefficients { a i } ni =1 nonzero (for instance, for n = 6 we need to consider at least two of these subsetsof indices that must be, of course, different). For generic configurations of the externalmomenta, these coefficients will moreover be distinct. In summary, in the following we willalways limit our discussion to the case in which all coefficients a i are distinct and nonzero.One could allow for some of these coefficients to be the same, which would correspond tomulti-soft limits, although we do not need to consider these cases for the present discussion.The shift in Eq. (12) is such that the momentum of the particle s becomes soft for z = z s = 1 /a s . We can characterize the way in which the soft limit is approached defining z = (1 − ǫ ) /a s , where ǫ ≪ z s . We canthen deform the spinor-helicity variables of the soft momentum holomorphically, such thatthe soft limit ǫ → | ˆ s i = ǫ | s i , (15)or antiholomorphically as | ˆ s ] = ǫ | s ] . (16)The helicity of each external particle will determine which of these is chosen in each case[5], in order to obtain the best possible bound on the behavior with large z in Sec. IV C.The shift in Eq. (12) defines a complexification ˆ A n ( z ) of n -point amplitudes, such thatthe physical amplitudes are given by A n = ˆ A n (0). Let us define the function of complexvariable f n ( z ) = ˆ A n ( z ) z . (17)This function exhibits three different kinds of singularities. Aside from the trivial z = 0pole, the remaining singularities are inherited from the singularity structure of the physicalamplitude A n , resulting in factorization poles corresponding to internal momenta going on-shell and around which the amplitude factorizes into the product of two sub-amplitudes,and soft poles at z = 1 /a i due to ˆ p i becoming soft. Factorization poles arise whenever Let us remark that our definition of f n ( z ) does not contain additional multiplicative factors of the form1 / (1 − a i z ) σ for σ >
0, that are typically considered when exploiting soft theorems in order to deriveon-shell recursion relations [39, 49]. The reason is that, in theories satisfying the soft theorem (10), thiswould originate additional poles coming from the O ( ǫ ) pieces, thus preventing the recursive evaluationof scattering amplitudes. z [49], so that collinear limits yieldno factorization poles. Indeed, the singular structure of the corresponding propagator in thecollinear case is proportional to1ˆ p i · ˆ p j = 1 p i · p j (1 − a i z )(1 − a j z ) . (18)Hence, for this particular shift, collinear limits yield only soft poles in z . This will be ofimportance later.The possible existence of soft poles in theories with massless particles implies that theshift (12) would be generally useless (from the perspective of constructing the tree-levelS-matrix) unless the soft behavior, namely whether or not there are soft poles and theircorresponding degree and residues, is completely determined.The existence of poles at z = 1 /a i is guaranteed from the soft theorem in Eq. (10). Otherfactorization channels lead to poles arising from a quadratic equation, the two roots of whichwill be denoted as { z ± I } [39, 49]. There are no additional poles, as all poles of ˆ A n ( z ) must beassociated either with one of the internal momenta becoming on-shell or one of the externalmomenta becoming soft following our assumptions [41].Let us assume that, for | z | → ∞ , the complex amplitude ˆ A n ( z ) satisfiesˆ A n ( z ) ∝ z δ ( n ) , (19)for some integer (that may depend on n ) δ ( n ) < . (20)Integrating f n ( z ) on a contour γ that encloses all its poles, taking the contour to z → ∞ ,13nd applying Cauchy’s residue theorem, permits us to write A n = − X I Res z = z ± I ˆ A n ( z ) z − n X i =1 Res z =1 /a i ˆ A n ( z ) z . (21)The left-hand side of the previous equation contains the physical n -point amplitude of realmomenta A n . The right-hand side contains the contributions from the z = 0 poles. Let usstress that there is no contribution from the residue at infinity as we are assuming that Eq.(20) holds. We will determine in Sec. IV C the situations in which this is indeed verified,which will lead to condition (D).We have separated the two kinds of contributions from the poles of ˆ A ( z ) in Eq. (21). Thepoles in the first term of the right-hand side correspond to internal momenta going on-shell.Due to the factorization properties of scattering amplitudes [50], it follows that this termcan be written as a sum of products of lower-point amplitudes A k We have shown in Sec. IV A that the standard soft graviton theorem is satisfied while,in Sec. IV B, we have discussed that an all-line shift would allow us to construct scatteringamplitudes in a recursive manner from the information encoded in the soft theorem. Thisconstructibility condition relies on the validity of the shift, namely Eq. (20), being satisfied.In this section we show that this condition is satisfied under the conditions (C-D) in Sec. III.In order to do so, we need to obtain suitable bounds on the behavior of ˆ A n ( z ) for | z | → ∞ .The simplest bound on the behavior of the complex amplitudes with z one can obtain followsfrom the analysis of individual Feynman diagrams (see, e.g., [50] for an extended discussion).Given a particular helicity arrangement, we make the optimal choice regarding polariza-tion vectors. This means that for gravitons with positive helicity we will use holomorphicshift (15), while for gravitons with negative helicity we use the antiholomorphic shift (16).Hence, the leading contribution from the polarization vectors of an n -point amplitude is( z − ) n .Each propagator contributes with a z − factor, which follows directly from condition(C). On the other hand, let us assume that k -point interaction vertices display a leadingbehavior z I ( k ) . Let us start considering individual Feynman diagrams with vertices of thesame valence, which display the asymptotic behaviorˆ A n ( z ) ∝ ( z − ) n ( z I ) ( n − / ( k − ( z − ) ( n − k )( k − . (24)The validity of the shift (12) implies then that δ ( n ) ≤ − n + I n − k − − n − kk − < . (25)15his equation must be satisfied for all values of n ≥ n ≥ k ≥ 3, which leads tocondition (D): I ( k ) ≤ I ⋆ ( k ) = 2( k − . (26)This equation represents at this stage of the discussion a necessary condition, given that wehave only considered a particular class of diagrams in order to derive it. However, under thisnecessary condition, changing the valence of the vertices inside specific subdiagrams with m ≤ n legs does not modify the restriction of the bound (24) to these subdiagrams, thatis always proportional to z − . Hence, Eq. (26) is in fact necessary and sufficient. One cancheck that this is consistent with the discussion of the same all-line shift in [51]. Besides ofthe all-line shift being valid, theories in which I ( k ) does not saturate the inequality in Eq.(26) can display bonus relations in the sense of [52, 53].On the other hand, there may be vertices that vanish when one of their legs is on-shell(an example of this kind of theory is provided in Sec. V C). These vertices can display aleading behavior z J ( k ) , where J is determined taking into account the number of external(with at least one external leg attached to them) and internal vertices in a given diagram.Let us recall that the number of internal vertices is constrained by i ( k ) ≤ n ( k − k − − k − . (27)Assuming that J > I , the worst-case scenario is the one in which the inequality above issaturated. One obtains then J ( k ) ≤ J ⋆ ( k ) = 2( k − − ( k − I ( k ) . (28)For I = I ⋆ , we see that J ⋆ = I ⋆ . Hence, vertices that vanish when at least one leg is on-shellcannot go beyond J ⋆ = 2( k − − I ( k − . APPLICATIONSA. Constructibility of general relativity from soft theorems A straightforward application of the general result above is the case of general relativityitself. Of course, general relativity is known to be constructible using the Britto-Cachazo-Feng-Witten (BCFW) shift [37, 38]. However, it is still interesting to understand whetherthe amplitudes of general relativity could be evaluated from the information encoded insoft theorems (in fact, a different proof of this statement has been recently presented in[54]). General relativity certainly satisfies the criteria (A-D) above, as its interactions arequadratic in the momenta ( I = 2), and is therefore constructible following the proceduredescribed in this paper. B. Scattering amplitudes of unimodular gravity Another interesting modification of general relativity is unimodular gravity, which is de-fined in terms of a different action that is invariant under transverse diffeomorphisms (and,in some formulations, Weyl transformations [55, 56]). Even though the classical field equa-tions in vacuum are Einstein manifolds as in general relativity, a proof of its constructibilitywas lacking. Previous proofs of the constructibility of general relativity do not immediatelyapply to unimodular gravity due to the different off-shell behavior of the propagator which,as discussed in [57], must contain triple poles in p in order to reproduce the standardNewtonian potential (this seems to have been missed in [58]). In other words, the off-shellpropagator contains a piece proportional to p µ p ν p ρ p σ p . (29)Under a BCFW shift this propagator diverges as z , hence the original proof of constructibilityfor general relativity (that rests on the more standard 1 /z scaling of propagators) cannot beapplied to unimodular gravity. Nevertheless, the knowledge of the classical field equationsof the theory strongly suggests that the scattering amplitudes of unimodular gravity mustenjoy the same properties as the ones in general relativity, and therefore that the failureof showing the constructibility of unimodular gravity stems only from technical limitationsassociated with this particular shift. 17e can apply our general result discussed above in order to close the issue of the con-structibility of the scattering amplitudes of unimodular gravity. Unitarity, enforced as thecondition S † S = on the S-matrix, implies that scattering amplitudes factorize around sim-ple poles when internal momenta become on-shell [41, 45]. Hence, the contributions from theoff-shell pieces in the propagator containing higher-order poles in p must necessarily cancelon-shell in order to guarantee unitarity. If this cancellation does not take place, it wouldfollow that the tree-level S-matrix of unimodular gravity is not be unitary. This cancellationcan be seen explicitly for some of the n = 4 and n = 5 amplitudes, that have been evaluatedusing spinor-helicity variables and shown to be equivalent to the corresponding amplitudesin general relativity [57].The analysis in our paper permits to conclude that this equivalence extends to the com-plete S-matrix. Unimodular gravity satisfies the condition (A) as it is equivalent to theFierz-Pauli theory at the linear level [55]. (B) and (D) are also satisfied, as unimodulargravity shares the 3-point amplitudes with general relativity and its interaction vertices arequadratic in the momenta [57, 58]. On the other hand, its propagator behaves as 1 /p forlarge (off-shell) momenta, thus satisfying (C), with m = 4. We can then conclude that thetree-level S-matrix of unimodular gravity is either the same as in general relativity, or it isnon-unitary in the sense that S + S = does not hold, and that this follows necessarily fromthe basic principles and requirements in our general discussion. C. Minimally modified theories of gravity Let us consider the theories of modified gravity introduced in [31] (see also [32–34]). Allthe known examples of these theories that are radiatively stable have a Lagrangian densityof the form L = √− g G ( K , R ) , (30)where R is the 3-dimensional Ricci scalar and K = K ij K ij − K is quadratic in the extrinsiccurvature K ij . On the other hand, G ( K , R ) is a function that satisfies the constraintspresented in [31] such that the theory propagates two local degrees of freedom. However,on general grounds, we can consider a perturbative expansion of this Lagrangian around18inkowski spacetime, which leads to L √− g = c R R + c K K + c RK RK + c R R + · · · , c X m Y n = 1 n ! ∂ m + n L ∂ X n ∂ Y n (cid:12)(cid:12)(cid:12)(cid:12) R = K =0 . (31)In this perturbative expansion, the quantities K and R are expanded as well as K = K (2) + K (3) + ... ∼ p h + p h + ..., R = R (1) + R (2) + ... ∼ p h + p h + ..., (32)where the superindex indicates the number of powers of h ab (the perturbation with respectto the flat metric η ab ), and the right-hand side in each of these equations indicates schemat-ically the behavior of the different terms in momentum space. This family of theories isdetermined by the couplings c X m Y n (which must satisfy some constraints) in their pertur-bative expansion, with general relativity being included as a particular choice. In fact, wecan see that the first two terms in the expansion are proportional (up to rescalings and aboundary term) to the 4-dimensional Ricci scalar. This means that the first part of theLagrangian is nothing but the Fierz-Pauli Lagrangian, so that this theory satisfies condition(A).Evaluating the R term gives enough information to derive the propagator [32], D µνρσ ( p ) = F µνρσ ( p ) − c cp / δ µ δ ν δ ρ δ σ , (33)where F µνρσ ( p ) is the usual propagator derived from the Fierz-Pauli action in the de Dondergauge. Hence, the propagator satisfies condition (C). On the other hand, the on-shell 3-pointamplitudes were derived in [32] and were found to be the same as in general relativity, sothat (B) is satisfied.The last condition (D) has to do with the behavior of the interaction vertices with themomenta. There are two kinds of interaction vertices in the theory with Lagrangian (30).The first class encompasses k -point vertices that are nonzero off-shell. These are of the form K (2) [ R (1) ] k − or R (2) [ R (1) ] k − and therefore have precisely I = I ⋆ = 2( k − k -point vertices that vanish identically when one of theirlegs is on-shell. These have J = 2 k ≥ J ⋆ . We can conclude that (D) is not satisfied. Hence,even if the use of soft theorems improves the situation, it is still not possible to prove thatthe S-matrix in these theories is the same as in general relativity with the arguments in thispaper. 19 . Most general 3-point graviton amplitudes Condition (B) in Sec. III assumes that the 3-point amplitudes of gravitons take the formthat is obtained in general relativity. However, it is straightforward to check that the proofof constructibility still holds if we relax condition (B) but keep (A), (C) and (D) unchanged.Hence, we devote this section to discuss in more detail the possible freedom that may beallowed when relaxing this condition.It is well-known that on-shell 3-point amplitudes (of complex momenta) can be fixed com-pletely from kinematical considerations and little group scaling (hence, ultimately, Lorentzinvariance). There are two independent on-shell 3-point amplitudes, that we can choose tobe A (1 +2 +2 +2 ) and A (1 +2 +2 − ) without loss of generality. If (A) holds, little-groupscaling fixes these amplitudes to be A (1 +2 +2 − ) = κ [12] [13] [23] , A (1 +2 +2 +2 ) = ζ κ [12] [13] [23] , (34)where ζ ∈ R is a dimensionless constant and κ = 8 πG . If ζ = 0 we recover the 3-pointamplitudes of general relativity.One may be tempted to argue that all these deformations of general relativity, that forma one-parameter family, are constructible. However, on dimensional grounds we can seethat a nonzero A (1 +2 +2 +2 ) must be associated with higher-derivative interactions (seealso [59]). Hence, any theory with ζ = 0 would fail to satisfy (D) and, therefore, is notconstructible using the arguments provided in this paper. If a more powerful treatmentallows to relax condition (D) (perhaps, removing it completely), then one would be ableto find the constructible theory that would result from the 3-points above with ζ = 0. Itis worth mentioning that there is a natural candidate to be associated with the outcomeof this hypothetical procedure, namely the theory known as Einsteinian cubic gravity [60–62], which contains a cubic term in the curvature with the right dimension to generate thenonzero amplitude A (1 +2 +2 +2 ) in Eq. (34).The expectation of the existence of a more powerful treatment can be further motivatedby considering field redefinitions in general relativity. As we have discussed in Sec. V A, itis possible to construct the tree-level S-matrix of general relativity exploiting the gravitonsoft theorem. However, we also know that the S-matrix is invariant under (nonlinear) localfield redefinitions, and so we can use this freedom to do a general redefinition of the form20 −→ h + κ p + q − ¯ ∇ p h q . Under such a field redefinition, we find that the cubic interaction inthe Einstein-Hilbert action (1) changes as h ¯ ∇ h −→ h ¯ ∇ h + κ p + q − h q ¯ ∇ p +2 h q + ..., (35)where the ellipsis indicates terms that are better behaved for large momenta and henceignored. Provided that q > 1, this redefinition does not affect the propagator, but doesintroduce (2 q )-point vertices containing p + 2 derivatives. For a generic value of p , theseadditional vertices spoil the constructibility, since condition (D) in our criteria is, in general,no longer met after this field redefinition is performed. That general relativity is indeedconstructible illustrates that the direct counting of derivatives in interaction vertices cannotencapsulate all the physics.We think that it is interesting to keep studying whether higher-derivative gravitationaltheories with I ≥ k can be shown to be constructible using other arguments. A hypotheticalproof of constructibility of scattering amplitudes from n = 3 that succeeds at relaxing (D)would imply the existence of only two independent gravitational theories satisfying (A) and(C): general relativity and Einsteinian cubic gravity. Aside from this extension, it wouldalso be interesting to study the interplay between our results and recent related works suchas [63, 64]. Let us also mention for completeness that, for higher dimensional Yang-Millsoperators (that appear as the first term in an α ′ expansion in bosonic string theory and inother low energy effective string actions [65]) such as F = f abc F aνµ F bρν F cµρ , the soft theoremsremain unchanged [43, 66]. This might lead us to expect that gravitational amplitudesformed via the KLT relations also will not spoil the soft theorems, which is not true: thepossible KLT products of two higher-order Yang-Mills amplitudes contains contributionsfrom both R terms at order α ′ and φR terms at order α ′ . On the other hand, Einsteiniancubic gravity does not contain terms that spoil the soft theorems, but does modify the 3-point amplitudes as we have seen in Eq. (34). Higher-dimensional operators than the onesconsidered in Einsteinian cubic gravity cannot alter the 3-point amplitudes and, providedthey only contain helicity-2 modes, also cannot spoil the soft theorems.21 I. CONCLUSIONS In this paper we have formulated a set of criteria to determine the on-shell equivalence ofgravitational theories, also discussing the relation with previous work regarding the deriva-tion of general relativity from the principles of special relativity. In particular, we have shownthat using soft theorems leads to an improved treatment of higher-derivative interactions.While our main result can be applied to both general relativity and unimodular gravity,showing the equivalence of their S-matrices, a crucial point of failure for higher-derivativetheories is condition (D). It may be possible to relax condition (D), in the same way thatthe information encoded in soft theorems has allowed us to go up to I = I ⋆ = 2( k − k -point interaction vertices) instead of simply I = 2. However,this would require further improvement in the understanding of the large z behaviour ofthe amplitudes. Our discussion illustrates that this is not merely a technical point but thatit has important practical implications for the understanding of the possible equivalence ofgravitational theories. This provides, in our opinion, a strong motivation for further researchin this direction. Appendix A: On the non-uniqueness in Deser’s derivation For completeness, let us analyze in more detail the source of non-uniqueness in the deriva-tion of the Einstein-Hilbert action following Deser’s procedure [19], in order to justify theimportance of point b) in our discussion in Sec. II. After Padmanabhan stressed this issuein [26], Deser argued in [28] that it should be possible to deal with this non-uniquenessperforming suitable field redefinitions. However, as pointed out in [29], it is only possibleto do this at the lowest order in the iterative off-shell procedure, which means that fieldredefinitions are not enough in order to remove the ambiguities in this procedure. Here, wewant to provide a thoroughly explicit illustration of this point.Let us use the same schematic notation as in [26, 28], in which the Fierz-Pauli equations[67] are written as D abcd h cd = 0 , (A1)where D abcd is a certain second-order differential operator satisfying¯ ∇ a D abcd = 0 . (A2)22he first iteration yields equation of the form equations [67] are written as D abcd h cd = λT ab (1) ( h ) + λ ∆ ab (1) ( h ) , (A3)where λ is a coupling constant, T ab (1) ( h ) the stress-energy tensor that yields the correct cou-pling in general relativity, and ∆ ab (1) ( h ) a superpotential (i.e., an identically conserved tensor).The claim in [28] is that ∆ ab (1) ( h ) can be removed by a field redefinition h ab = ˜ h ab + λ Θ(˜ h ) , (A4)where Θ ab satisfies the equation D abcd Θ ab (˜ h ) = ∆ ab (1) (˜ h + λ Θ) . (A5)This equation is well-posed, as both sides of it are identically conserved. However, this fieldredefinition implies that Eq. (A3) reads D abcd ˜ h cd = λT ab (1) (˜ h + λ Θ) = λT ab (1) (˜ h ) + λ W ab (1) (˜ h ) . (A6)The second identity can be alternatively seen as the definition of the tensor W ab (1) . Thecontribution proportional to the latter is O ( λ ), and comes from the intrinsically nonlinearnature of the iterative procedure. This piece should be taken into account in the seconditeration, in which the field equations read D abcd ˜ h cd = λT ab (1) (˜ h ) + λ T ab (2) (˜ h ) + λ W ab (1) (˜ h ) + λ ∆ ab (2) (˜ h ) . (A7)∆ ab (2) (˜ h ) is again a superpotential and, therefore, can be dealt with at this order by anothershift similar to the one in Eq. (A4) but adding O ( λ ) terms. Let us focus our attention on W ab (1) (˜ h ), which is not identically conserved. In fact, it is straighftorward to show that λ ¯ ∇ a W ab (1) (˜ h ) = ¯ ∇ a h T ab (1) ( h ) − T ab (1) (˜ h ) i = O ( λ ) , (A8)where the O ( λ ) terms in the last identity arise from the λ ∆ ab (1) in Eq. (A3). This impliesthat W ab (1) is not conserved and, moreover, that this lack of conservation appears at O ( λ )in Eq. (A7) and therefore cannot be pushed forward to the next interation that includes O ( λ ) terms. Hence, it is impossible to remove completely the term ∆ ab (1) ( h ) doing a fieldredefinition, as the fact that the stress-energy tensor itself depends on h ab leads to an addi-tional residue W ab (1) that is not conserved. Similarly, trying to absorb the ∆ ab ( n ) terms resultsinto non-removable terms W ab ( n ) , which illustrates how this non-uniqueness manifests at everystep in the off-shell procedure. 23 CKNOWLEDGMENTS The authors would like to thank Freddy Cachazo, Alfredo Guevara, Stefano Liberatiand Dimitar Ivanov for useful discussions. RCR and NM are grateful for the hospitalityof Perimeter Institute, where this work was conceived. This research was supported inpart by Perimeter Institute for Theoretical Physics. 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