Massless positivity in graviton exchange

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Massless positivity in graviton exchange © Authors, 2021 Published version Herrero-Valea, Mario; Santos-Garcia, Raquel; Tokareva, Anna Herrero-Valea, M., Santos-Garcia, R., & Tokareva, A. (2021). Massless positivity in graviton exchange. Physical Review D, 104(8), Article 085022. https://doi.org/10.1103/physrevd.104.085022 2021 Massless positivity in graviton exchange Mario Herrero-Valea ,1,2,* Raquel Santos-Garcia,3,†and Anna Tokareva4,5,6,‡ 1SISSA, Via Bonomea 265, 34136 Trieste, Italy and INFN Sezione di Trieste, 34127 Trieste, Italy 2IFPU—Institute for Fundamental Physics of the Universe Via Beirut 2, 34014 Trieste, Italy 3Departamento de Física Teórica and Instituto de Física Teórica, IFT-UAM/CSIC, Universidad Autónoma de Madrid, Ciudad Universitaria de Cantoblanco, 28049 Madrid, Spain 4Department of Physics, University of Jyväskylä, P.O. Box 35 (YFL), FIN-40014 Jyväskylä, Finland 5Institute for Nuclear Research of Russian Academy of Sciences, 117312 Moscow, Russia 6Helsinki Institute of Physics (HIP), University of Helsinki, P.O. Box 64, 00014 Helsinki, Finland (Received 22 December 2020; accepted 29 September 2021; published 27 October 2021) We formulate positivity bounds for scattering amplitudes including exchange of massless particles. We generalize the standard construction through dispersion relations to include the presence of a branch cut along the real axis in the complex plane for the Maldestam variable s. In general, validity of these bounds requires the cancellation of divergences in the forward limit of the amplitude, proportional to t−1and logðtÞ. We show that this is possible in the case of gravitons if one assumes a Regge behavior of the amplitude at high energies below the Planck scale, as previously suggested in the literature, and that the concrete UV behavior of the amplitude is uniquely determined by the structure of IR divergences. We thus extend previous results by including a subleading logarithmic term, which we show to be universal. The bounds that we present here have the potential of constraining very general models of modified gravity and effective field theories of matter coupled to gravitation. DOI: 10.1103/PhysRevD.104.085022 I. INTRODUCTION Positivity bounds [1–5] have become standard tools in assessing the validity of low-energy effective field theories (EFT). By invoking the plausible existence of an ultraviolet (UV) completion satisfying reasonable properties such as Lorentz invariance, unitarity, and locality, positivity bounds exclude large regions of the parameter space of a given EFT by demanding the positivity of a certain combination of couplings. In particular, these bounds are obtained by combining the knowledge of the analytic structure of 2-to-2 scattering amplitudes with the optical theorem ImAðs; 0Þ¼sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1− 4m2 s rσðsÞ;ð1Þ which ensures positivity of the imaginary part of the scattering amplitude Aðs; tÞin the forward limit t→0. Applications of positivity bounds include the proof of the a-theorem [6,7], the study of chiral perturbation theory [8], effective Higgs models [9], quantum gravity [10,11], massive gravity and Galileons [12–16], higher spins [17], cosmology [18–21], string theory [22,23], and many more. Recently, a generalization of positivity bounds, named arcs, was proposed [24]. However, all these examples omit an important case of physical relevance, the exchange of massless particles. In that case, the scattering amplitude contains pathologies that impede one from taking the forward limit—a pole t−1 and a logarithmic divergence logðtÞ, due to exchange and production of massless particles. This is particularly relevant in the presence of gravity, since gravitons couple to all forms of matter. Although for energies below the Planck scale gravity could be ignored, its character as a long range force produces contributions to the scattering amplitude down to the deep infra-red (IR). Formally, the pathologies which come with the exchange of gravitons are never absent and cast a shadow on the validity of positivity bounds. Even if one trusts the decoupling limit and the validity of gravity-less positivity bounds, it would be desirable to find a way to extend them to include graviton exchange. There have been previous attempts to solve this issue by compactifying space-time down to three dimensions, where gravitons decompose in massive fields [25,26], but a general formalism applicable in more *[email protected] †[email protected] ‡tokare[email protected].ac.ru Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. PHYSICAL REVIEW D 104, 085022 (2021) 2470-0010=2021=104(8)=085022(9) 085022-1 Published by the American Physical Society varied situations, without requiring compactification, is still lacking. Recently, it was suggested that forward divergences in graviton exchange can be cancelled by assuming a Regge form for the high-energy limit of the scattering amplitude [27], which is expected to hold from string theory [28,29]. However, in [27] only the term t−1is cancelled and nothing is said about the logarithm. This is important, though, because due to crossing symmetry, equivalent logðsÞand logðuÞterms are expected to coexist in the scattering amplitude. These terms split the complex plane in sin two, with a branch cut along the real line for t→0. This obstructs the formulation of usual positivity bounds, which require one to deform an integration contour crossing the real axis. In this paper we construct new positivity bounds for theories with exchange of massless particles, provided that we cancel the divergences in the forward limit. We show that this is indeed possible when the massless states correspond to gravitons. Generalizing the results of [27], we prove that both the pole t−1and the logðtÞcan be eliminated, with the remaining pieces in the amplitude satisfying a positivity bound reminiscent of the standard case. Finally, we discuss the robustness of our result by showing agreement with previous works in the literature, formally deriving the bounds recently proposed by [25,26]. II. DISPERSION RELATIONS From now on we will consider ab →ab scattering amplitudes which include a massless particle coupled to thebosonicexternalstatesaand b. The presence of this massless state will produce poles in s,t,andufrom treelevel exchange, as well as logarithmic cuts logðsÞ,logðtÞ, and logðuÞindicating particle production, found at loop level in perturbation theory. Here s,t,anduare the Maldestam variables, with sthe energy in the centerof-mass frame squared. ucan always be eliminated by using sþtþu¼4m2, where we have assumed that both states aand bhavethesamemassm. From Cauchy’s integral theorem, one can write a family of dispersion relations for the amplitude Aðs; tÞ¼ðs−μÞn 2πiIγs dz Aðz; tÞ ðz−sÞðz−μÞn;ð2Þ with n≥1. The integration contour γsmust be taken as a small circle surrounding only the point z¼s, while the point z¼μis arbitrary provided that it lays outside the contour. We take μreal hereinafter. A key point in deriving positivity bounds lays on the behavior of the scattering amplitude at high energies. For massive particles, it can be proven that it satisfies the Froissart-Martin bound [30], which implies lim jsj→∞ Aðs; tÞ s2 ¼0;t<4m2:ð3Þ Alas, the formal proof of this bound cannot be applied to the exchange of massless particles. However, we will assume that this is still true for the cases considered here. We will justify this assumption later. Now we take the forward limit of (2). In the case of massless particles in the intermediate channel, this is divergent and cannot be taken exactly. We thus instead, in more generality,1expand the amplitude around the limit t→0− Aðs; 0−Þ≡Aðs; tÞjt→0− ¼fðsÞ tþgðsÞlogðtÞþA∘ðsÞþOðtÞ;ð4Þ where the limit is taken from the negative side of the real line. Here fðsÞand gðsÞare holomorphic functions. When the scattering amplitude is computed in perturbation theory, fðsÞcontains the residue on the pole of the massless propagator, while gðsÞis proportional to the βfunction of the ab →ab coupling. The analytic structure of Aðs; 0−Þis therefore controlled by A∘ðsÞ. This is analytic in the whole complex plane except for a branch cut running over the whole real line, due to production of massless particles and crossing symmetry [30]. The branch cut obstructs the standard derivation of positivity bounds, which uses a contour integral crossing the real line [2]. Here instead we note that for any real value of s, we can perform two different analytic continuations of the amplitude, by adding a small imaginary part siϵ which moves the point to the upper (down) part of the complex plane. Afterwards we can deform the integration contour to run above (below) the real axis plus a semicircumference at infinity, as shown in Fig. 1. This allows one to define two different realisations of (2) Aðsþiϵ;0−Þ¼ðs−μÞn 2πiZ∞ −∞ dz Aðzþiϵ;0−Þ ðz−sÞðz−μÞn;ð5Þ Aðs−iϵ;0−Þ¼ðs−μÞn 2πiZ−∞ ∞ dz Aðz−iϵ;0−Þ ðz−sÞðz−μÞn;ð6Þ where the circles at infinity vanish due to (3) and ϵmust be understood as infinitesimal. Here we keep it finite only on nonholomorphic tems. Subtracting both representations we get 1This form encodes all the cases of relevance to our knowledge. For exchange of scalars and vectors, both fðsÞand gðsÞare constant, while for gravitons, they behave as ∼s2. HERRERO-VALEA, SANTOS-GARCIA, and TOKAREVA PHYS. REV. D 104, 085022 (2021) 085022-2 Aðsþiϵ;0−Þ−Aðs−iϵ;0−Þ ¼ðs−μÞn 2πiZ∞ −∞ dz Aðzþiϵ;0−ÞþAðz−iϵ;0−Þ ðz−sÞðz−μÞn:ð7Þ In the physical region s∈Rand we have Aðs−iϵ;tÞ¼Aðsþiϵ;tÞ. Then ImAðsiϵ;0−Þ¼∓ðs−μÞn 2πZ∞ −∞ dz ReAðziϵ;0−Þ ðz−sÞðz−μÞn: ð8Þ We now take Aðsþiϵ;0−Þand use this result to rewrite it as Aðsþiϵ;0−Þ−iImAðsþiϵ;0−Þ ¼ReAðsþiϵ;0−Þ¼ðs−μÞn 2πZ∞ −∞ dz ImAðzþiϵ;0−Þ ðz−sÞðz−μÞn: ð9Þ This expression is reminiscent of the standard derivation of positivity bounds. However, in our case we have cancelled out the imaginary part of the amplitude, getting rid of the discontinuity explicitly. The integral in (9) runs over nonphysical values of z. This can be solved by splitting it in three integrals over f−∞;0g,f0;4m2gand f4m2;∞g. Performing a change of variables z→−zþ4m2in the first one and using crossing symmetry, (9) can be rewritten as Bðs; 0−Þ¼ðs−μÞn 2πZ∞ 4m2 dzImAðzþiϵ;0−Þ ðz−sÞðz−μÞnþð−1ÞnImA×ðzþiϵ;0−Þ ðz−4m2þsÞðz−4m2þμÞn;ð10Þ where we have defined Bðs; 0−Þ¼ReAðsþiϵ;0−Þ −ðs−μÞn 2πZ4m2 0 dz ImAðzþiϵ;0−Þ ðz−sÞðz−μÞn:ð11Þ Here A×ðs; 0−Þ¼Að−sþ4m2;0−ÞþOðtÞis the crossed amplitude in the uchannel. Now, by using the optical theorem (1) in the right-hand side of (10), we would be tempted to follow the standard derivation of positivity bounds, and conclude that 1 n! dn dsnBðs;0−Þjs¼0 ¼Z∞ 4m2 dz 2πzffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1− 4m2 z sσðzÞ znþ1þð−1Þnσ×ðzÞ ðz−4m2Þnþ1>0;ð12Þ for even n, after taking derivatives in both sides of (10). Nonetheless, this is not possible in the case at hand. Barring aside the issue of the forward limit divergences— which we will discuss later—we must note that the lefthand side of (10) can be IR divergent in perturbation theory. For finite masses, the integral piece in (11) takes care of these IR divergences, replacing them by m−2and logðm2Þ. However, this will not work in the massless case. To circumvent this issue, we take (10) and define instead the following function: ΣðjÞ¼1 2πiIγδ ds s3Bðs; 0−Þ ðs2þδ2Þ2jþ1ð13Þ where now n¼2jand δhas dimensions of energy squared. Using now (10) we find ΣðjÞ¼Z∞ 4m2 dzFðjÞðzÞ;ð14Þ FIG. 1. Integration contours in the complex plane for s.The zigzag line represents the branch cut. For points siϵ, the integration contour γsin the corresponding half of the complex plane is shown in red. The equivalent contours used in (5) are dotted in blue. The radius of the large semicircumferences γ ∞is jsj→∞. MASSLESS POSITIVITY IN GRAVITON EXCHANGE PHYS. REV. D 104, 085022 (2021) 085022-3 were we have performed the integral in sexplicitly, getting FðjÞðzÞ¼z3ImAðzþiϵ;0−Þ 2πðz2þδ2Þ2jþ1 þðz−4m2Þ3ImA×ðzþiϵ;0−Þ 2πððz−4m2Þ2þδ2Þ2jþ1:ð15Þ Notice that all dependence on μhas cancelled after integration, with δtaking its place in the denominators. The contour γδis the sum of two small circles enclosing the points s¼iδ, with δ>0. It acts as a sort of an IR regulator, but it is not constrained to be small. Note also that the right-hand side of (14) is positive definite for all j, since FjðzÞ>0within the integration regime from application of (1). In that case, we can conclude that ΣðjÞ¼1 2πiIγδ ds s3Bðs; 0−Þ ðs2þδ2Þ2jþ1>0;ð16Þ regardless of the shape of the scattering amplitude, which might be even unknown above a certain energy scale Λ. Indeed, let us assume that Aðs; tÞis known only within an EFT with validity up to2E∼Λ≫m,δ. In that case, we can split the integral on the right-hand side and rewrite (13) as ˆ ΣðjÞ¼Z∞ Λ2 dzFðjÞðzÞ;ð17Þ where ˆ ΣðjÞ¼ΣðjÞ−ZΛ2 4m2 dzFðjÞðzÞ:ð18Þ Again, the right-hand side of (17) is positive and we conclude ˆ ΣðjÞ>0:ð19Þ Expressions (16) and (19) are the massless version of positivity and beyond positivity bounds [3].Theystate that the contour integral in (13)—or the quantity ¯ ΣðjÞin (17)—which can be computed in an EFT provided that it is valid below Λ, has to be positive. They differ from standard positivity bounds in two manners, which encode the particularities of the massless exchange. First, we find that the imaginary part of the amplitude in (9) cancels out from the left-hand side, leaving only its real part. Second, the definition of Bðs; 0−Þalso includes an integral in the region 0≤s≤4m2, which regulates IR divergences for massive external fields, ensuring finiteness of the physical result. Notice that when only massive modes are exchanged, our bounds reduce trivially to the standard bounds in the absence of massless poles. From now on, all expressions can be equivalently used with either ΣðjÞor ˆ ΣðjÞ, the only difference being the lower limit of the integral in the right-hand side of the dispersion relation. However, its explicit positivity does not change. Provided that the amplitude Aðs; 0−Þis finite, these bounds are applicable and can lead to interesting constraints on the structure of EFT Lagrangians through the presence of the Wilson coefficients in ReAðs; 0−Þ. III. REGULARITY IN THE FORWARD LIMIT Although the bounds (16) and (19) are completely general and valid in the case of massless particles in the spectrum of the theory, they are meaningless in the presence of divergences in the forward limit, as is the case when gravitons are exchanged in the tchannel. In that case, the left-hand side of the bound is dominated by the tree-level contribution to the exchange, which is of the form Aðs; tÞ∝−R×s2 t;ð20Þ where Ris the residue in the pole of the graviton propagator. Since the limit t→0−is continuous—although divergent—in principle we can always use (16) to fix the sign of the divergence and conclude that R>0;ð21Þ which tells us that in order to agree with unitarity requirements, the graviton must not be a ghost. Although it is interesting to see this trivial condition for unitarity arising in this way, the information that it provides is scarce. If we want to extract more information from the positivity bounds (16) and (19) in the presence of a graviton in the spectrum, then we need to find a way to regularize the forward limit divergences. In [27] it is shown3that this is possible for the pole t−1if one takes a seemingly strong assumption about the scattering amplitude—that it takes the Regge form [31] ImAðs; tÞ¼rðtÞðα0sÞ2þlðtÞ1þζ logðα0sÞþO1 α0s; ð22Þ which we extend here with a subleading correction, above a certain energy scale E∼M. Here rðtÞencodes information 2Note that Λmight not be strictly the cutoff of the theory, but the energy at which the EFT is not a good approximation to the UV complete theory anymore. This might happen a few orders of magnitude below the cutoff. 3Note however that the bounds derived in [27] do not take into account the presence of the branch cut at all. HERRERO-VALEA, SANTOS-GARCIA, and TOKAREVA PHYS. REV. D 104, 085022 (2021) 085022-4 about the polarization of external states, while lðtÞis constrained to be negative lðtÞ<0and satisfies lð0Þ¼0, in order for the amplitude to unitarize at high energies. The scale α0is controlled by the value of the Regge scale Mas α0∼Oð1ÞM−2 . Note that in [27] the logarithmic correction that we include here is not considered. By assuming this behavior of the amplitude at large s,it can be easily shown that the right-hand side of (17) will also present divergences when t→0, which are thus controlled only by the large slimit of the integral. These can then be cancelled against those in the left-hand side, with the remaining finite piece satisfying its own version of positivity. Retaining only the leading term in the amplitude allows one to cancel the pole t−1but leaves the logarithmic divergence untouched. As we will see in a moment, the subleading correction that we have included accounts for the latter. Of course, at this point one could question the validity of the high-energy behavior (22). So far this is an assumption of our work, but one which is well justified in the case of gravitons for two different reasons. First, let us give a heuristic argument. If we believe that string theory provides a UV completion of gravitational interactions, then it can be shown that graviton mediated scattering amplitudes satisfy (22) at Oð1Þ,whereα0is the string scale [28,29,32].This happens due to the contribution of the tower of massive modes in the spectrum that are excited above these energies. Logarithmic corrections of similar form to the ones in (22) can also be found in certain cases [28] and we expect them to arise from string loops.4Second, if instead we assume an arbitrary subleading correction gðsÞ, it can be checked that the only choice that allows for cancelling the logarithmic divergence is precisely gðsÞ¼ζ=logðα0sÞ,asitisshownin Appendix. Therefore, from now on we assume (22).Note that by assuming (22), the bound (3) is automatically satisfied. We can then split the integral on the right-hand side of (17) in two Z∞ Λ2 dzFðjÞðzÞ¼ZM2  Λ2 dzFðjÞðzÞþZ∞ M2  dzFðjÞðzÞ:ð23Þ Calling Δ¼R∞ M2 dzFðjÞðzÞand using the Regge form of the scattering amplitude (22), we get Δ¼rðtÞα02þlðtÞ πZ∞ M2  dzz3þlðtÞ−4j1þζ logðα0zÞ;ð24Þ which can be computed explicitly in terms of the analytic continuation of the Gamma function. The forward limit can be taken in this expression after integration. Note that since lð0Þ¼0,t→0−, and lðtÞ<0, we have lðtÞ¼l0ð0Þtþl00ð0Þ 2t2þOðt3Þ;ð25Þ with l0ð0Þ>0. We thus get lim t→0− Δ¼rð0Þα02 π8 < : ððM2 Þ4−4j 4j−4þζα04j−4Γ½0;ð4j−4ÞlogðM2 α0ÞÞ;j>1 ð1 l0ð0Þt−l00ð0Þ 2l0ð0Þ2þlogðM2 α0ÞÞ −ζðγþlogðtÞþlog ½−l0ð0ÞlogðM2 α0ÞÞ;j¼1 ;ð26Þ up to terms which vanish when t¼0. Here γis the EulerMascheroni constant, Γðs; xÞ¼R∞ xdtts−1e−tis the incomplete Gamma function, and we have taken z≫m2;δ.We have also assumed that our external states satisfy ImA×ðs; tÞ¼ImAðs; tÞfrom crossing symmetry, which limits the application of our result to bosonic states. Fermions will introduce extra signs from crossing symmetry. We find that indeed the leading term in (22) produces a pole t−1, while the subleading correction gives a logðtÞ. However, note that they only exist when j¼1, while for j>1the result is completely regular. This is exactly the same kind of divergence that we find in the forward limit of ΣðjÞ, only present for j¼1as well.5Thus, we can expand both sides of expressions (13) and (17) in the limit t→0− and cancel divergences in the left-hand side against those in the right-hand side provided by Δ, with the rest of the terms remaining finite. It is particularly interesting to note that assuming Regge behavior, which is expected to arise in gravity, precisely allows for cancellation of those divergences produced in graviton scattering. As discussed in Appendix, this seems to be a unique result. Explicitly, using (26) we can now rewrite (17) for j¼1as 4Higher loop contributions like logðlog tÞare expected beyond one loop in the scattering amplitude. We expect them to cancel against higher loop corrections in the string theory. 5The divergent part of the amplitude for graviton exchange is proportional to s2. Thus, it vanishes from ΣðjÞwith j>1after evaluation of the residues in the pole. MASSLESS POSITIVITY IN GRAVITON EXCHANGE PHYS. REV. D 104, 085022 (2021) 085022-5 ˆ Σð1Þ R¼ZM2  Λ2 dzFð1ÞðzÞþrð0Þα02logðM2 α0Þ π −rð0Þα02 π l00ð0Þ 2l0ð0Þ2−rð0Þα02ζ πlog ½−l0ð0ÞlogðM2 α0Þ −rð0Þα02ζγ π;ð27Þ where we have introduced the regularized version of ˆ Σð1Þas ˆ Σð1Þ R¼ˆ Σð1Þþrð0Þα02 πζlogðtÞ− 1 l0ð0Þt;ð28Þ by taking all divergent terms to the left-hand side. By choosing the appropriate value of the combinations rð0Þα02=l0ð0Þand rð0Þα02ζ, the forward limit divergences can be cancelled, so that (27) remains regular. Note that, since α02>0and l0ð0Þ>0this also fixes the sign of ζ uniquely, although in a case by case way. Finally, we turn our attention to the explicit form of (27). Note that the integral along Λ2<s<M 2 must remain positive by application of (1). However, the rest of the terms do not have a definite sign. In particular, we cannot determine the overall sign of the right-hand side in (27) without knowing the value of l00ð0Þ, which we do not know. Nevertheless, all these terms come multiplied by the overall scale rð0Þα02. Thus, what we can do is to assess that the right-hand side is positive up to the order in which they become important. Meaning ˆ Σð1Þ R>−Oðrð0Þα02Þ;ð29Þ so that a small amount of positivity violation is allowed and controlled by the dynamics of the UV degrees of freedom. For j>1things are simpler. Since (26) is always convergent in this case, there is no need to expand nor to split the range of integration in (24). Thus, we simply recover our result (19), which remains valid ˆ Σðj>1Þ>0:ð30Þ Expressions (29) and (30) are the final results of our work. They represent positivity bounds whose left-hand sides can be computed in an EFT, as long as δ<Λ2, and whose value is constrained by features of the high-energy theory. IV. GRAVITATING SCALAR FIELD Now that we have derived useful positivity bounds in the presence of exchange of gravitons, let us test their validity with some well-known theories of scalar fields coupled to Einstein gravity. The first case that we examine is a free gravitating scalar field, with action S¼Zd4xffiffiffiffiffi jgj p−R 2κ2þ1 2∂μϕ∂μϕ;ð31Þ where κ2¼8πG¼M−2 P. In order to include the branch cut into the scattering amplitude ϕϕ →ϕϕ we must at least compute the first loop correction. Combining it with the tree-level amplitude we get, in the forward limit and after renormalization6 Aðs; 0−Þ¼−κ2s2 t− 33κ4s2 24π2ðlogðsÞþlogð−sÞÞ − 33κ4s2 24π2logðtÞ:ð32Þ Here we have used the de Donder gauge and set the renormalization scale μR¼1in the modified minimal subtraction scheme. This choice is harmless since its value always drops from the result. The integral in (11) vanishes for massless external fields. Thus Bðs; 0−Þ¼ReAðs; 0−Þ¼−κ2s2 t− 33κ4s2 24π2logðs2Þ − 33κ4s2 48π2logðt2Þ;ð33Þ and from this we can easily use (13) to compute Σð1Þ¼−κ2 t− 33κ4 24π23 2þlogðtÞþlogðδ2Þ;ð34Þ Σðj>1Þ¼yðjÞκ2 π2δ4j−4;ð35Þ where yðjÞ>0for all j. Here we have decided not to add the contribution from RΛ2 0dzFðjÞðzÞ, thus working with ΣðjÞ instead of ˆ ΣðjÞ. Cancelling the divergences using (28) determines rð0Þα02∼−l0ð0Þκ2and rð0Þα02ζ∼κ4. Thus the bounds read − 33κ4 24π23 2þlogðδ2Þ>−Oðrð0Þα02Þ;ð36Þ yðjÞκ2 π2δ4j−4>0:ð37Þ The first bound is however meaningless since the left-hand side is already comparable to the subleading terms in the right-hand side. This forbids us to conclude anything from Σð1Þ. On the other hand, the second bound is automatically 6The coefficient in front of the logarithms is gauge dependent. However, its sign is universal for the family of βgauges [33] explored here. HERRERO-VALEA, SANTOS-GARCIA, and TOKAREVA PHYS. REV. D 104, 085022 (2021) 085022-6 satisfied for all δ, confirming a trivial statement, that a free gravitating scalar field is a bona fide theory up to MP. V. SCALAR QED Even more interesting is to explore the case of scalar QED with a photon ϕ, an electron ψ, and an spectator field χ, as suggested in [26]. The action is S¼Zd4xffiffiffiffiffi jgj p−R 2κ2þ1 2∂μϕ∂μϕþ1 2∂μχ∂μχ þ1 2∂μψ∂μψ− 1 2 Λ2ψ2−λΛϕψ2:ð38Þ At energies below the mass of the electron Λ≪MP,ψ can be integrated out, leaving a generic EFT describing effective interactions between the rest of the fields S¼Zd4xffiffiffiffiffi jgj p−R 2κ2þ1 2∂μχ∂μχþ1 2∂μϕ∂μϕ −λ3Λ ð2πÞ2 ϕ3 3! þλ4 2π2 ϕ4 4! þDλ2κ2 Λ2ð∂ϕÞ4 þCλ2κ2 Λ2ð∂μϕ∂μχÞ2þ…;ð39Þ where the dots indicate further Λor κ2suppressed terms. In matching both actions, the Wilson coefficients Dand C must be determined by a direct comparison of a scattering amplitude. However here we are interested in exploring what positivity can say about them. Following [26] we focus on ϕχ →ϕχ, whose one-loop amplitude gives Aðs; 0−Þ¼−κ2s2 tþs2Cλ2κ2 Λ2− 11κ4s2 24π2logðtÞ − 11κ4s2 24π2ðlogðsÞþlogð−sÞÞ þ OðtÞ:ð40Þ Again, we have added the one-loop correction, with μR¼1, in order to make the branch cut explicit. From here we find Σð1Þ¼κ2 48 − 48 tþ48Cλ2 Λ2− 33κ2 π2 − 22κ2 π2logðtÞ− 22κ2 π2logðδ2Þ;ð41Þ Σðj>1Þ¼yðjÞκ4 π2δ4j−4:ð42Þ Cancelling the forward divergences we get again rð0Þα02∼−l0ð0Þκ2,rð0Þα02ζ∼κ4. From this the bound Σðj>1Þ>0is automatically satisfied. It also allows us to disregard the loop corrections in Σð1Þ, since they are subleading. We thus get Cκ2λ2 Λ2>−Oðrð0Þα02Þ:ð43Þ This result agrees with that of [25,26], where it is proposed from different arguments. This also proves the conjecture in their conclusions of new physics required at a scale ðrð0Þα02Þ−1=4<M Pin order to unitarize the theory. This can be seen from the fact that a direct matching between the EFT (39) and its partial UV completion (38) demands C<0with C∼Oð1Þ, which violates our bound. Thus, (38) needs to be completed at intermediate energies. VI. CONCLUSIONS In this paper we have derived new positivity bounds in the presence of exchange of massless particles between bosonic states. They generalize and formalize previous results in the literature. Provided that divergences in the forward limit can be ignored, our bounds can constrain the value of Wilson coefficients and other couplings in EFTs for which the existence of a plausible unitary, Lorentz invariant, and local UV completion is demanded. We have gone further and shown that in the case of exchange of gravitons, forward divergences can be cancelled by assuming a Regge behavior of the scattering amplitude, which is unique if one assumes analyticity of the function lðtÞ. Although peculiar, this form of the amplitude has been previously found in the literature on string theory. This leads to well-defined bounds which can now be used in the presence of gravity. We have shown how our bounds work in two simple examples. A free gravitating scalar field, where they are automatically satisfied, and scalar QED with a spectator field, for which they demand new physics below the Planck scale to unitarize the theory, as previously suggested by [25,26]. These new bounds open up a window to explore the theory space of phenomenological viable theories of (matter and) gravity. We believe that our results here have the potential to highly constrain different popular models currently used to investigate properties of black hole physics and cosmology. It would also be interesting to apply them to the exploration of unitarization mechanisms for graviton scattering [34–36]. ACKNOWLEDGMENTS We are grateful to Brando Bellazzini, Javi Serra, and Inar Timiryasov fordiscussions andcomments. Our work has been supported by the EuropeanUnion’s H2020ERC Consolidator Grant “Gravity from Astrophysical to Microscopic Scales” Grant Agreement No. GRAMS-815673 (M. H-V.), by the Spanish FPU Grant No. FPU16/01595 (R. S-G.) and by the Academy of Finland Grant No. 318319 (A. T.). The part of work of A. T. related to obtaining the bounds from imaginary poleswassupportedby theRussianScienceFoundationGrant MASSLESS POSITIVITY IN GRAVITON EXCHANGE PHYS. REV. D 104, 085022 (2021) 085022-7 No. 19-12-00393. We also wish to acknowledge networking support from COST action CA16104 “GWverse." APPENDIX: UNIVERSALITY OF THE SUBLEADING CORRECTION Let us address here the question on the uniqueness of the subleading correction to the amplitude in the Regge limit (22) required to cancel divergences in the forward limit. Let us start by noticing again that the leading term, which cancels the t−1contribution of the amplitude in the IR, was already proposed in earlier works [27] and can be obtained from a closed string amplitude after careful manipulation. In particular, the imaginary part of the string amplitude is not a regular function of sand t; it has instead an infinite set of Regge poles that require regularization. Hereinafter we will assume instead that the imaginary part of the Regge amplitude that we consider is regular in both arguments when s→∞,t→0. Going back to FðjÞðz; tÞ, defined in (15), let us examine the integral in (24) Δj¼Z∞ M2  dzFðjÞðz; tÞ:ðA1Þ Note that this integral can give a singularity at t→0only if it is divergent when t¼0but finite for some small finite t. In particular, for j¼1we obtain Δ1¼Z∞ M2  dz ImAðz; tÞ z3:ðA2Þ Since ImAðs; tÞis regular at t¼0from our assumption, we can Taylor expand it around this point ImAðs;tÞ¼ImAðs;0Þþ∂tImAðs;tÞjt¼0tþOðt2Þ;ðA3Þ in one to one correspondence to the series expansion of the Regge form (22), ImAðs; tÞ¼rðtÞðα0sÞ2þlðtÞð1þgðsÞÞ ∼ ¼ðα0sÞ2ð1þgðsÞÞ½rð0Þþtðr0ð0Þ−l0ð0ÞlogðsÞ; ðA4Þ where we have assumed the expansion (25). At this point we leave the form of the subleading correction gðsÞcompletely arbitrary. If we demand that the result of Δ1has the correct divergent structure we have Δ1¼Z∞ M2  dz zrðtÞz−lðtÞð1þgðzÞÞ¼a tþbðtÞþOð1Þ:ðA5Þ Here bðtÞstands for the remaining divergent terms at t→0, which include the one-loop log tterm among others. Changing the integration variable to log z¼σand plugging the small texpansion on the integrand, brings us to the condition rðtÞZ∞ log M2  dσe−ðl0ð0ÞtþOðt2ÞÞσð1þgðσÞÞ ¼a tþbðtÞþOð1Þ:ðA6Þ The leading term in the left-hand side can be computed explicitly and shown to cancel the at−1term, while for the rest we have rð0ÞZ∞ log M2  dσe−ðl0ð0ÞtÞσgðσÞ¼bðtÞþOð1Þ:ðA7Þ After multiplying by a step function under the integral sign, this becomes a Laplace transform. Although it requires regularization, its result is unique and therefore there exists a single function gðsÞwhich satisfies this identity. Since gðsÞ¼ζ=logðα0sÞdoes the work, we conclude that it is the only option. [1] A. Nicolis, R. Rattazzi, and E. Trincherini, J. High Energy Phys. 05 (2010) 095; 11 (2011) 128(E). [2] A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis, and R. Rattazzi, J. High Energy Phys. 10 (2006) 014. [3] B. Bellazzini, F. Riva, J. Serra, and F. Sgarlata, Phys. Rev. Lett. 120, 161101 (2018). [4] C. de Rham, S. 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