Eikonal phase matrix, deflection angle and time delay in effective field theories of gravity

The eikonal approximation is an ideal tool to extract classical observables in gauge theory and gravity directly from scattering amplitudes. Here we consider effective theories of gravity where in addition to the Einstein-Hilbert term we include non-minimal couplings of the type $R^3$, $R^4$ and $FFR$. In particular, we study the scattering of gravitons and photons of frequency $\omega$ off heavy scalars of mass $m$ in the limit $m\gg \omega \gg |\vec{q}\,|$, where $\vec{q}$ is the momentum transfer. The presence of non-minimal couplings induces helicity-flip processes which survive the eikonal limit, thereby promoting the eikonal phase to an eikonal phase matrix. We obtain the latter from the relevant two-to-two helicity amplitudes that we compute up to one-loop order, and confirm that the leading-order terms in $\omega$ exponentiate \`{a} la Amati, Ciafaloni and Veneziano. From the eigenvalues of the eikonal phase matrix we then extract two physical observables, to 2PM order: the classical deflection angle and Shapiro time delay/advance. Whenever the classical expectation of helicity conservation of the massless scattered particle is violated, i.e. the eigenvalues of the eikonal matrix are non-degenerate, causality violation due to time advance is a generic possibility for small impact parameter. We show that for graviton scattering in the $R^4$ and $FFR$ theories, time advance is circumvented if the couplings of these interactions satisfy certain positivity conditions, while it is unavoidable for graviton scattering in the $R^3$ theory and photon scattering in the $FFR$ theory. The scattering processes we consider mimic the deflection of photons and gravitons off spinless heavy objects such as black~holes.


Overview
One of the exciting applications of scattering amplitudes focuses on the computation of classical observables in gauge theory and gravity such as deflection angles and time delay/advance, or effective Hamiltonians describing the dynamics of binary systems. Early results in this direction date back to [1], where it was already noted that loop amplitudes contribute to classical processes, contradicting the erroneous belief of e.g. [2]. The intimate connection between loops and classical physics was sharpened in [3], and had already been applied in [4] to obtain the classical and quantum O(G 2 ) corrections to Newton's potential, where G is Newton's constant. In this approach, gravity is treated as an effective theory [5], where one can make predictions at low energy despite the non-renormalisability of the theory. More recently, a systematic approach employing scattering amplitudes in conjunction with unitarity [6,7] was developed to compute classical quantities in gauge theory and gravity. Classical [8] and quantum [8,9] corrections to Newton's potential can be obtained from a two-to-two scattering amplitude of two massive scalars, in particular narrowing down to terms that have discontinuities in the channel corresponding to the momentum transfer q of the process [5,3]. An additional simplification stems from the fact that in the unitarity-based calculation the cuts can be kept in four dimensions, as discrepancies with D-dimensional results only give rise to analytic terms, at least at one loop. Unitarity was also applied in [10][11][12][13] to compute the deflection angle for light or for gravitons passing by a heavy mass, a quantity that has the advantage of being gauge invariant. We also mention some of the efforts leading to the computation of the effective (Newton) potential at second [14,15], third [16][17][18][19], fourth [20][21][22][23][24][25][26][27], fifth [28,29] and sixth [30] post-Newtonian order, following the landmark computation at first post-Newtonian order [31]. In the post-Minkowskian framework, which is natural in the context of amplitudes, the current state of the art is at 3PM order [32,33], a result which was recently confirmed in [30,34]. Note also the effective one-body approach of [35], recently extended to incorporate the first and second post-Minkowskian corrections in [36,37], respectively. For other interesting approaches to extract classical observables in general relativity from amplitudes see [38][39][40][41][42][43][44][45][46][47][48].

Gravity with higher-derivative couplings
Much attention has been devoted to the study of effective theories of gravity obtained by adding higher-derivative interactions to the Einstein-Hilbert (EH) action. In particular, efforts have been made in [49,50] to confront such modifications with gravitational wave observations. It was also noted in [49] that for these effects to be measurable by experiments such as LIGO the cutoff of the effective theory must not be much larger than O(km −1 ). In [51] we initiated a study of the effects that these higher-derivative terms have on the Newtonian potential and deflection angle. In this paper we sharpen this study by rooting it in the eikonal approximation -specifically, applying it to three types of terms, denoted schematically as R 3 , R 4 and F F R, for which we compute the corresponding corrections to the deflection angle and time delay/advance. More in detail, the particular action we consider for the graviton, photon and a massive scalar has the form: where I 1 := R αβ µν R µν ρσ R ρσ αβ , G 3 := I 1 − 2R µνα β R βγ νσ R σ µγα , (1.2) while A few comments on the various couplings in (1.1) are in order here.
First, there are two types of R 3 terms, denoted as I 1 and G 3 above. Such terms arise naturally in the low-energy effective description of bosonic string theory. Their effects on gravitational scattering of different matter fields have been discussed recently in [51,52]; specifically for the scattering of two massive scalars, both independent structures I 1 and G 3 were found to contribute. On the other hand, for the helicity-preserving deflection of massless particles of spin 0, 1 and 2, it was shown in [51] that the G 3 interaction has no effect. Additional interesting features about the I 1 and G 3 couplings are that I 1 is the only coupling that contributes to pure graviton scattering up to four points [53,54] and is the two-loop counterterm in pure gravity [55], while G 3 is a topological term in six dimensions. In the following we will be concerned with (helicity-preserving and flipping) scattering of massless gravitons in the background produced by a massive scalar, in which case only the I 1 structure contributes, hence we will refer to it simply as the R 3 term, since no confusion can arise. Note that in the case of photons there is no R 3 contribution to the helicity-flipping process.
The second interaction we study is of the type R 4 . In principle there are 26 independent parity-even quartic contractions of the Riemann tensor [56], but only the seven which do not contain the Ricci scalar or tensor survive on shell in arbitrary dimensions, as can also be seen using field redefinitions [57][58][59]. In four dimensions these reduce to two independent parity-even structures [49,60], plus one parity-odd structure [50], as shown in (1.3). 1 In agreement with [60] we find that these interactions induce the following four-point graviton amplitudes: those with all-equal helicities, and the amplitude with two positive-and two negative-helicity gravitons (the MHV configuration). If β 2 in (1.3) is non-vanishing, then the all-plus and all-minus graviton amplitudes are independent. We also note that a particular contraction of four Riemann tensors appears in type-II superstring theories where it is the first higher-derivative curvature correction to the EH theory, and can be determined from four-graviton scattering [62].
The third interaction we consider is an F F R term, where F is the electromagnetic field strength. It is known to arise in string theory as well as from integrating out massive, charged electrons in the case of electrodynamics coupled to gravity, as discussed in [63,64], and considered more recently in [65,66].
We have also introduced in the action a minimally coupled massive scalar to represent a black hole 2 . Note that in (1.1) we have excluded terms quadratic in the curvatures since from an effective field theory/on-shell point of view they have no effect to any order in four dimensions, as shown recently in [59].

Physical observables from the eikonal phase matrix
We now come to the computation of the physical observables of interest -these are the classical deflection angle and the time delay/advance [67] experienced by massless gravitons and photons when they scatter off a (possibly charged) massive scalar. A method ideally suited for obtaining classical observables directly from amplitudes, without passing through intermediate, unphysical quantities, is the eikonal [68][69][70][71][72][73]. In this approach the relevant amplitudes are evaluated in an approximation where the momentum transfer | q | is taken to be much smaller than both the mass m of the heavy scalar and the energy ω of the massless particle, or more precisely taking m ω | q |. Crucial for this is a convenient parameterisation of spinor helicity variables for the massless particles in the eikonal limit. The amplitudes thus obtained are then transformed to impact parameter space via a two-dimensional Fourier transform. In this space the amplitudes are expected to exponentiate into an eikonal phase, from which one can extract directly the classical (and, if desired, quantum) deflection angle and time advance/delay. Recent applications of this method to this type of problem include [74] for the deflection angle of massless scalars up to 2PM, [11] for photons and fermions up to 2PM order, and up to 3PM order in [75][76][77]. We also note that [11] showed the equivalence of the eikonal method and the formalism based on the computation of an intermediate potential/Hamiltonian used for instance in [10-13, 51, 52].
An important point we wish to make is that in our case, because helicity-preserving as well as helicity-violating processes contribute, the eikonal phase is promoted to an eikonal phase matrix in the space of helicities of the external massless particles, with (+−) and (−+) being the diagonal entries associated to no-flip scattering (in a convention where all particles' momenta are outgoing), while (++) and (−−) are the off-diagonal entries, with helicity flip. The associated mixing problem has to be resolved in order to obtain the physical quantities of interest. Whenever the two eigenvalues of the eikonal phase matrix are distinct, a possible violation of causality at small impact parameter arises, as noticed already at tree level in [78]. See also [65,66,[79][80][81][82][83] for further discussions an resolutions of this issue in UV-complete theories, [71,78,84,85] for earlier appearances of the eikonal operator and [63,64] for related discussions involving helicity flip and no-flip amplitudes.

Summary of the paper
We now summarise our results. We have computed the graviton deflection angle and time delay/advance for the three interactions R 3 , R 4 and F F R, and in addition the photon deflection and time delay induced by the F F R interaction. The single most important qualitative difference with the EH theory is that the propagation and speed of the massless particle acquire a dependence on its polarisation in all cases except the graviton propagation in the presence of the F F R interaction. This generically leads to a time advance at small impact parameter b. Interestingly, in the case of graviton scattering due to R 4 and F F R, causality violation can be avoided if the coefficients of the interactions obey certain positivity constraints which, for R 4 , are in precise agreement with those of [86,87]. For the R 3 interaction our results are fully consistent with the tree-level findings of [78], extending them to one loop. Note that while we used a massive scalar, [78] used a coherent state to set up the background in which the graviton is deflected. Similarly, the F F R interaction induces super-luminal propagation of photons.
An important point is the dependence of the eikonal S-matrix S eik on the energy ω of the scattered massless particle. In the EH theory, S eik is expected to take the form S eik = e i(δ 0 +δ 1 +··· ) , where the subscript L denotes the loop order of δ L . For the leading eikonal δ 0 , this was proven for our kinematic set-up in [74], and it is generally expected that the δ L are linear in ω, although we are not aware of an all-order proof. Both for R 3 and F F R our results are perfectly aligned with this expectation up to 2PM order, resulting in an ω-independent deflection angle and time advance/delay. A novelty arises for R 4 where the corresponding eikonal phase (matrix) scales as ω 3 with the graviton frequency.
The rest of the paper is organised as follows. In Section 2 we discuss our kinematic set-up and provide explicit expressions for the spinor helicity variables associated to each massless particle in the eikonal limit. We then discuss some general aspects of the eikonal approximation, in particular the extraction of the phases δ L from the loop scattering amplitudes. We highlight consistency conditions relating amplitudes in impact parameter space at different loop orders and powers of ω which will then be explicitly checked in all cases considered. We also quote here the relevant formulae to derive the particle deflection angle and time advance/delay. Section 3 contains the computations of all tree-level and one-loop amplitudes relevant for our analysis. As a warm up we consider the EH theory, where we re-discuss the graviton deflection computation of [13]; while it is conventionally assumed that in the classical picture the helicity of the scattered massless projectile is unchanged, we show explicitly that this is the case in the eikonal limit: the flipped-helicity amplitudes are non-zero both at tree level and one loop, but are subleading once the eikonal limit is taken, resulting in a diagonal eikonal phase matrix. We then move on to present the relevant four-point two-scalar two-graviton amplitudes with and without helicity flip in the case of R 3 , R 4 and F F R, as well as the two-scalar two-photon amplitudes for the F F R case, all at tree and one-loop level. Some of these amplitudes have been calculated here for the first time. While at tree level we present exact expressions, at one loop we work in the eikonal approximation and we only consider cuts in the q 2 -channel which produce non-analytic terms arising from the long-range propagation of two massless particles, following the approach initiated in [4,8,9,88,89]. Although in the subsequent section we focus only on classical contributions, arising from triangle integral functions with one internal mass, we quote complete answers for the amplitudes up to one loop including box (needed to check the exponentiation of the tree-level phase matrix) and bubble integrals (generating O( ) corrections to the physical observables). Section 4 is dedicated to the computation of the leading and subleading eikonal matrices δ 0 and δ 1 , from which we will then obtain the O(G) and O(G 2 ) corrections to the deflection angle and time advance/delay for the four cases considered -scattering of gravitons in the presence of R 3 , R 4 and F F R terms, and scattering of photons induced by the F F R interaction. We also show the case of graviton scattering in EH to set the scene for the more complicated examples discussed later. In all cases we check the exponentiation of the tree-level eikonal phase matrix explicitly, providing important consistency checks of our calculations. Our main results are given in (4.40), (4.47), (4.55), and in (4.41), (4.48), (4.56), for the graviton deflection angle and time advance/delay in the R 3 , R 4 and F F R cases, while the photon deflection and time advance/delay in the F F R theory are given in (4.74) and (4.75), respectively.
A few appendices complete the paper, where we present relevant integrals, the Feynman rules used for some of our new computations, a list of tree amplitudes, and a derivation of the four-point graviton amplitudes in R 4 only based on little-group and dimensional analysis considerations.

From amplitudes to the deflection angle and time delay via the eikonal
In this section we first give a precise definition of the eikonal limit providing an explicit parametrisation for all the momenta and spinor-helicity variables we need. We then briefly review the connection between amplitudes in the eikonal limit (Fourier-transformed to impact parameter space) and the eikonal phase matrix, the deflection angle and the time delay.

Kinematics of the scattering
We begin by describing the kinematics of the scattering processes we consider. We denote by p 1 and p 2 the four-momenta of the incoming and outgoing scalars, respectively, with m being their common mass, while the momenta of the incoming and outgoing massless particles (gravitons or photons) are p 4 and p 3 . We will work in the centre of mass frame, with the following parameterisation: In our conventions we take all momenta to be outgoing, hence the minus signs in the expressions of p 1 and p 4 since particles 1 and 4 are incoming. We also have where p · q = 0 due to momentum conservation. Hence q lives in a two-dimensional space orthogonal to p. In this paper we define the Mandelstam variables as with s + t + u = 2m 2 . In this notation the spacelike momentum transfer squared is given by s, while t denotes the centre of mass energy squared, and ω is the energy of the scattered massless particle.
In the above parameterisation, the kinematic limit we are interested is which implies for the Mandelstam variables and for the energies of the massless particles For definiteness we choose p = | p |ẑ with | p | | q |, as implied by (2.4). In this approximation we can write the four-momentum p 3 of the massless particle in spinor notation as with q := q 1 + iq 2 andq := q 1 − iq 2 . One can then find an explicit parameterisation for the spinors associated to the null momenta p i = λ iλi , i = 3, 4, with the result Note the extra factors of i due to the negative energy-component of p 4 corresponding to an incoming particle.

Eikonal phase, deflection angle and time delay
In this section we briefly review relevant aspects of the eikonal approximation and the eikonal phase matrix which allows for an efficient extraction of the deflection angle and time delay/advance from scattering amplitudes. This topic was intensively studied in the context of gravity and string theory in the nineties [72,73]; for related recent work see also [90,91,46] and references therein.
First, we introduce the amplitude in impact parameter space A. This is defined as a Fourier transform of the amplitude A with respect to the momentum transfer q, where b is the impact parameter, and the number of dimensions will eventually be set to In the eikonal approximation the gravitational S-matrix can be written in the form [72,74] S eik = e i(δ 0 +δ 1 +··· ) , (2.10) where δ 0 is the leading eikonal phase, which is O(G), δ 1 the first subleading correction, of O(G 2 ), and the dots represent subsubleading contributions. Alternatively, one can write the S-matrix in impact parameter space as where the superscript indicates the loop order L and the subscript the power in the energy ω of the massless particle. That the maximal power of ω at a given loop order is L + 1 is a wellestablished fact in (super)gravity and we will see below that the R 3 corrections do not alter this expectation. However, we also find that the R 4 corrections lead to higher powers of ω starting at one loop, which is not surprising since higher-derivative corrections worsen the high-energy behaviour. In the effective field theory approach we adopt, we are not really interested in highenergy physics (or high-energy completions of the theory) -we use the eikonal approximation as an efficient and elegant tool to extract deflection angles and time delay/advances without passing through the computation of non gauge-invariant intermediate quantities such as effective potentials or Hamiltonians. Nevertheless it would interesting to check if in the R 4 case unitarity can be restored as well through exponentiation.
Equating (2.10) with (2.11) one gets as well as the condition which implies the consistency condition Thus, the contribution to the one-loop amplitude that is leading in ω, A ω 2 , does not provide any new information about the S-matrix. In general, it is only the term in ω , that provides new information entering δ L . We also note that (2.12)-(2.15) hold as matrix equations.
Note that a priori these statements are known to hold for EH gravity. The results in this paper show that (2.15) also holds for the higher-derivative interactions discussed here at least up to one loop. Of course the work of [72] on the eikonal limit of string amplitudes gives reason to believe that the exponentiation will work for higher-derivative interactions to all orders.
Finally, the particle deflection angle can be obtained from the eigenvalues δ (i) of the eikonal phase matrix δ. Using a saddle-point approximation [72,11,92] one finds, for small θ, where i runs over all eigenvalues of δ and b = | b |. For the time delay, we will use instead [93][94][95] (2.17)

The relevant scattering amplitudes
In this section we compute the relevant amplitudes needed to extract the deflection angle and time delay/advance induced by the various interactions in (1.1). At tree level we will present exact expressions; at one loop we only need to compute the part of the amplitude with a discontinuity in the s-channel 3 and we will write the relevant expressions after expanding them in the eikonal approximation (2.4) -this will be denoted in the following by the symbol. A direct extraction of the classical part of the deflection angle and time delay can be performed using triple cuts, and in an even more refined way using the holomorphic classical limit [96].
We chose instead to compute the one-loop amplitudes through two-particle cuts, which also determine the quantum part of the amplitude. The latter, despite not being used in the present paper, becomes essential when considering the exponentiation in the eikonal limit at higher orders [91].
We will begin our discussion with the simple case of EH gravity, quoting from [13] the relevant two-scalar two-graviton amplitude without helicity flip. We also compute the amplitude with helicity flip, and show that it does not contribute in the eikonal approximation, as correctly assumed in previous treatments. We will then move on to compute the relevant tree and oneloop amplitudes that are necessary in order to compute the corrections induced by the R 3 , R 4 and F F R terms in (1.1).
The two-particle cut diagrams relevant for the R 3 and R 4 cases are shown in Figure 1. The corrections induced by the F F R interaction need a separate analysis and we show the corresponding diagrams in Figures 2 and 3. For the case of the R n interaction both internal and external particles are gravitons, while in the case of F F R we either have external gravitons and internal photons, or viceversa. Figure 1: The two-particle cut diagrams for the R n interaction in the s = − q 2 -channel. In our conventions external momenta are all outgoing and internal loop momenta flow from left to right in the diagram.
A comment is in order here. Focusing on the cuts relevant for R n depicted in Figure 1, the case h 3 = h 4 corresponds to the massless particle flipping helicity upon interacting with the scalar, whereas h 3 = −h 4 corresponds to the helicity-preserving process, since in our conventions all external particles are outgoing. A simple way to take into account particle statistics is to sum over all values of the internal helicities h 1 and h 2 and divide the result by 2. 4

Four-point scalar/graviton scattering in EH gravity
The relevant tree-level amplitudes in the EH case are the two-scalar/two-graviton amplitudes in the two helicity configurations for the gravitons: 5 The computation of the four-point one-loop amplitude without helicity flip in the eikonal approximation (2.4) was performed in [13], with the result is a pure phase, with N h → 1 in the eikonal approximation. We have also computed the new amplitude with helicity flip in the same approximation, with the result

Four-point scalar/graviton scattering in EH + R 3
We now consider the amplitudes with addition of the R 3 interaction in (1.1): the helicitypreserving amplitude at tree-level is vanishing while the helicity-flip amplitude is [51] A (0) At one loop, the result of [51] for the no-flip amplitude gives:
where N h is defined in (3.3). The one-loop amplitude with helicity flip requires a new computation and the result in the eikonal approximation is (3.8)

Four-point scalar/graviton scattering in EH + R 4
In this section we consider the addition of an R 4 interaction to the EH action. Such interaction affects the two-scalar two-graviton amplitude at one loop and thus contributes to graviton deflection and time delay at order G 2 . In order to build this amplitude using the unitarity-based method we first need to find out the expression for the four-graviton tree-level amplitudes in the R 4 theory. We do so here starting from the Lagrangian in (1.3) in order to make contact with the notation of [49]; in Appendix D we present an alternative derivation only relying on little-group considerations and dimensional analysis, which does not require writing down any Lagrangian.
Deriving the four-graviton amplitudes from (1.3) is straightforward -we simply have to replace the four Riemann tensors in each term by their linearised form corresponding to the four on-shell gravitons. For particle i the well-known expression in momentum space is Since we are interested in helicity amplitudes, we choose the field strengths F (i) to be selfdual (negative helicity) or anti-selfdual (positive helicity), hence in spinor-helicity formalism their form is The building blocks in (1.4) are bilinear in Riemann tensors, and take the form and where · denotes Lorentz contractions and * denotes the usual Hodge dual which acts on the (anti-)selfdual field strengths as * F SD = F SD and * F ASD = −F ASD . Furthermore, given the form (3.11) these expressions are only non-vanishing if both particles i and j have the same helicity. In summary, if both gravitons have negative helicity (SD field strength) we have while if both gravitons have positive helicity (ASD field strength) we have With these results one easily arrives at with Note that if we do not allow the parity-odd coupling (β 2 = 0), then the coefficient of the all-plus and all-minus amplitudes are the same β + = β − := β.
The next step is to carry out one-loop amplitude calculations in the eikonal approximation, as done in previous sections. The result for the relevant amplitudes is: 20) where N h was introduced in (3.3).

Scattering with the F F R interaction
The last interaction we wish to consider is the F F R term in (1.1). From an on-shell point of view this is the simplest non-minimal modification of the coupling of photons to gravity. As we will show below this leads to new corrections to the bending and time delay/advance of light and graviton propagation in the background of a very massive scalar particle.
This new interaction modifies the three-point two-photon/one-graviton amplitude: which we will now use to construct the relevant amplitudes at tree level and one loop to compute deflection angles and time delay in the presence of this interaction. Note that this amplitude is determined by its helicity structure and dimensional analysis up to a normalisation which we fixed from the Feynman rule (B.3) following from our action (1.1).

Relevant amplitudes for graviton deflection
Using factorisation and Feynman diagrams we have computed the four-point amplitudes relevant for graviton deflection from a massive charged source (such as a charged black hole). The new F F R interaction involves two photons and one graviton, hence one cannot generate a tree-level correction to the amplitude with two scalars and two gravitons. The first corrections arise at one loop, from the cut diagrams in Figure 2.
For the cut diagram on the left-hand side of the figure, we need the tree-level scalar QED amplitude with two photons and two massive scalars [9] A (0) along with the modification to the two-graviton/two-photon amplitudes arising from the F F R coupling for both helicity configurations of the graviton: no flip, Note that the cut is non-vanishing only in the singlet configuration (internal photons with the same helicities). This is because the four-point amplitude with two photons and two gravitons induced by the F F R interaction is non-vanishing only for same-helicity photons.
We now move to the cut diagram on the right-hand side of Figure 2. The two-photon/twograviton EH amplitude only exists in the configuration where the gravitons and the photons have opposite helicity (see for instance [11]), (3.25) and thus it contributes only in the helicity-preserving process. Hence, in order to compute the cut we will only need the following two-scalar/two-photon amplitude involving an F F R interaction: Performing the calculation, it turns out that the right-hand side of Figure 2 does not produce any non-analytic term with an s-channel discontinuity when external kinematics are considered to be strictly four-dimensional.
Following the above considerations, the one-loop amplitudes in the eikonal limit can be computed entirely from the LHS of Figure 2, and are found to be 27) where again N h is the phase defined in (3.3), and Q denotes the charge of the classical source (the black hole).

Relevant amplitudes for photon deflection
It is interesting to study how this new F F R interaction affects the bending and time delay/advance of light. In order to do so, we now review the known two-scalar/two-photon amplitudes for minimally coupled photons [11], and present the new corresponding amplitudes induced by the F F R interaction, both at tree and one-loop level.
In the following we consider processes where the internal legs are gravitons. In the EH theory, for the two-photon two-scalar process, only the helicity-preserving amplitude is non vanishing 6 , both at tree level Figure 3: The two-particle cut diagrams in the s = −| q | 2 -channel contributing to photon deflection to first order in the F F R interaction. We only show the helicity-flip configuration since the helicitypreserving cuts vanish. The cut diagram on the RHS of the figure only contributes terms which are subleading in the eikonal limit.
and at one loop [11], where the phase factor N γ is We now discuss the corrections to the two-scalar two-photon amplitudes arising from one insertion of the F F R interaction. These come from a single graviton exchange between a minimally coupled scalar and the F F R three-point vertex. At tree level, only the helicity-flip amplitude contributes in the eikonal approximation, while the no-flip amplitude, already quoted in (3.26), is a contact term that is subleading in the eikonal limit (it does not have a pole in s = −| q | 2 ).
Moving to one loop, the relevant two-particle cuts for the (++) configuration are shown in Figure 3. We find that the amplitude with photons in the (++) helicity configuration in the eikonal approximation is

Eikonal phase matrix, deflection angle and time delay
In the previous section we have derived the relevant tree and one-loop amplitudes which we will now use to extract the deflection angle and time delay up to 2PM order (or O(G 2 )) generated by the addition of the various couplings in (1.1). The key quantity is the eikonal phase matrix δ, to be introduced below, of which we will compute the leading, δ 0 , and subleading contributions, δ 1 . As an important consistency check we will confirm that the leading-energy contribution of the one-loop amplitudes captures the required exponentiation of the leading-order eikonal phase matrix δ 0 .
In the following we focus on the classical contribution to δ. We stress that for the cases we consider, δ will be a 2 × 2 matrix: the diagonal entries correspond to the two amplitudes A (1 φ , 2 φ , 3 h 1 , 4 h 2 ) where the helicity of the massless particle is not flipped (which in our alloutgoing convention corresponds to h 1 = −h 2 ), while the off-diagonal ones correspond to the two helicity-flip processes (with h 1 = h 2 ).
As a final comment, we note that the combined effect of the interactions in (1.1) is simply the sum of the contributions of each interaction treated independently; hence we will study them separately, and begin our discussion by reviewing the computation in EH gravity.

Leading eikonal
The relevant tree-level amplitudes in EH gravity are given in (3.1). In the eikonal approximation (2.4) they become where the second amplitude is subleading compared to the first.
The amplitudes in impact parameter space are obtained from those in momentum space using (2.9). To compute them, we will use repeatedly the result therefore the leading eikonal phase matrix is where we omitted terms of O(D − 4) and finite terms which do not depend on b.
Next we consider the one-loop amplitudes (3.2) and (3.4). In order to check exponentiation (2.15) we only keep terms that are leading in energy in the eikonal approximation, i.e. O(ω 3 ) in momentum space (or O(ω 2 ) in impact parameter space). These are where the sum of the box integrals I 4 (s, t; m) + I 4 (s, u; m) was evaluated in D dimensions in [46] and is given in (A.4). Transforming to impact parameter space, we have (4.6) As expected from (2.15), we find that

Subleading eikonal
In momentum space, the subleading contribution to the eikonal phase matrix is extracted from the O(ω 2 ) contribution to the amplitude in (3.2): 7 (4.8) where I 3 (s; m) is given in (A.3), and as usual s = −| q | 2 . In the following we focus on the first term on the right-hand side of (A.3), since the log term only contributes quantum corrections. Using we obtain the subleading part of the amplitude in impact parameter space: (4.10) and finally, using (2.13), δ 1 : The eikonal phase matrix up to one loop in EH is then given by Note that this matrix is proportional to the identity, since the polarisation of the gravitons scattered by the classical source is unchanged. The deflection angle can now be extracted using (2.16). While the eigenvalues of δ are divergent in D = 4, the corresponding deflection angle is finite: This result agrees with the derivation of [13], and as expected matches the photon deflection angle [10,11], first computed by Einstein. 8 Another quantity of interest which can be extracted from the eigenvalues of the eikonal matrix is the time delay. Using (2.17) applied to the leading eikonal phase (4.4), we get (4.14) As is well known, in order to define the time delay in four dimensions we need to take the difference of two time delays as measured by an observer at b and one at a much larger distance b 0 b [78]. Doing so the pole in (4.14) drops out, and neglecting power-suppressed terms in b 0 one gets (4.15) in agreement with [101]. Including now also the contribution from δ 1 , we arrive at the result In the next sections we compute the corrections ∆θ X and ∆t X to the deflection angle (4.13) and time delay (4.15) in EH due to the inclusion of an interactions X in (1.1). The complete deflection angle and time delay will then be θ EH + ∆θ X and t EH + ∆t X .

Leading eikonal
The relevant new amplitudes are obtained by evaluating (3.5) and (3.6) in the eikonal limit (2.4), with the result where from (2.7) we have [34] 4 = q 4 . In order to transform to impact parameter space we rewrite b · q = bq +bq , (4.18) with b := (b 1 + ib 2 )/2, andb := (b 1 − ib 2 )/2 (and we recall our previous definitions q = q 1 + iq 2 , q = q 1 − iq 2 ), from which bb = b 2 /4. Then in b -space we have Hence the leading eikonal phase matrix δ 0 , including the first contribution from the R 3 interaction, has the form where δ 0,EH is given in (4.4), and where we have used (2.12).
Moving on to one loop, from (3.7) and (3.8) we obtain (4.23) Transforming to impact parameter space, and using (A.4), we arrive at The leading one-loop amplitude matrix in the eikonal approximation is then found to be One can then check the matrix relation 27) in agreement with (2.15). In writing (4.27) we have used that, (4.28) up to and including O (α /4) 2 .
Finally, it is interesting to compare the expression of the leading eikonal phase matrix (4.21) and of its eigenvalues to (3.20) and (3.22) of [78], respectively. In that paper, the process considered was that of a graviton moving in the background produced by a coherent state, while in our case it is produced by a heavy scalar. To perform this comparison, we note that Then the matrix δ 0 in (4.21) becomes 9 (4.30) Using G = κ 2 /(32π), the eigenvalues of (4.30) are (dropping the 1/ pole) From this result, a causality violation at small b was argued in [78] (see (3.22) of that paper). Indeed, proceeding as in (4.15), one obtains for the time delay at O(G) and for sufficiently small b the eigenvalue with the choice of negative sign may become negative, leading to a time advance. We also note that the form of the eigenvalues in (4.31) is in agreement with that of [78], with the replacement mω → ω 2 in (4.31), because the process studied in that paper only involves massless particles. We will come back to the time delay computation and add O(G 2 ) corrections in Section 4.2.3.

Subleading eikonal
We now go back to the one-loop amplitudes (3.7) and (3.8) and extract the triangle contributions which are the relevant terms contributing to the subleading eikonal matrix: (4.33) We can now transform to impact parameter space, using The amplitudes in impact parameter space then become (4.36) Using (2.12), we can extract the contribution of the R 3 interaction to the subleading eikonal matrix δ 1 : (4.37)

Deflection angle and time delay
We can proceed similarly to the EH case. In the previous sections we showed that the R 3 interaction introduced off-diagonal terms, i.e. the helicity of the scattered graviton can change.
The eigenvalues of the leading and subleading eikonal matrices (4.38) and (4.39) are Next we present the correction to the graviton deflection angle, both in terms of κ and G: (4.40) The deflection involving a graviton whose helicity is preserved in the scattering process has already been studied in [51], instead the flipped helicity case is presented here for the first time.

Graviton deflection angle and time delay in EH + R 4
In this section we consider the deflection of gravitons induced by eight-derivative couplings in the Lagrangian, which we collectively denote as R 4 . We will only consider the parity-even interactions in (1.3) in order to present more compact formulae, therefore we set β 2 = 0, and hence β + = β − = β in (3.16) and (3.20). Furthermore, since these interactions do not produce a three-graviton vertex, it is impossible to build any tree-level two-scalar two-graviton amplitude involving R 4 . Thus there is no tree-level (1PM) bending associated to the new term in the Lagrangian, and one has δ 0,R 4 = 0 , (4.42) and the leading contribution arises at 2PM order. Furthermore, since the R 4 term only produces a contact term four-graviton interaction, the resulting one-loop amplitudes does not contain any box integral. This is consistent with the absence of a tree-level contribution in (4.42) which, in the eikonal approximation, is expected to exponentiate, and would result at one loop in the appearance of box integrals. The same situation occurs for the graviton deflection due to F F R couplings discussed in Section 4.4.
The relevant one-loop amplitudes are given in (3.20), and from the massive triangle contributions we extract the following results in the eikonal approximation: which then translate in impact parameter space into (4.44) The subleading eikonal phase matrix resulting from the previous amplitudes is given by Similarly to the EH and the R 3 interaction we can extract the time delay arising from the R 4 interaction in (1.1), which in this case arises entirely from the subleading eikonal phase. Applying (2.17) to (4.46) we find ∆t (1,2) We can express (4.47) and (4.48) in terms of the couplings introduced in (1.3), using (3.17), (3.18) and (3.19). In the parity-even theory (β 2 = 0) we get β +β = β 1 , andβ − β = β 3 . In order to avoid a potential time-advance and associated causality violation, we need to require β 1 > 0 and β 3 > 0 . (4.49) Interestingly this positivity constraint is the same as derived from causality considerations in [86] and general S-matrix analyticity properties in [87].

Graviton deflection angle and time delay in EH + F F R
Next we focus our attention on graviton deflection in EH theory with the addition of an F F R coupling. As discussed in Section 3.4.1, at tree level there is no new two-scalar two-graviton amplitude generated by this interaction, hence In order to compute the subleading eikonal phase matrix, we look at the massive triangle contribution to the one-loop amplitudes in (3.27), (4.51) Using we obtain In this case the eikonal phase matrix is diagonal and the subleading contribution δ 1,FFR is immediately seen to be (4.54) The new contribution to the graviton deflection angle due to the F F R interaction is then obtained using (2.16): (4.55) Applying (2.17) to (4.54) we find the additional contribution to the time delay associated to the bending of a graviton in the F F R theory: (4.56) The bending in this case is due to the electric charge Q of the black hole, not to its mass, which does not appear in either (4.55) or (4.56). We conclude that in order to avoid possible causality violation due to time advance the coefficient of the F F R interaction must obey the positivity constraint α γ > 0 .

Photon deflection angle and time delay in EH + F F R
In this section we consider the photon deflection angle and the time delay/advance arising from the F F R interaction. Compared to the case of graviton bending considered in the previous section, there is a non-vanishing tree-level contribution to the deflection, thus we consider the leading and subleading eikonal cases separately.

Leading eikonal
The first contribution we consider arises from the EH tree-level amplitude (3.28), which in the eikonal approximation becomes 10 58) or, upon transforming to impact parameter, Note that (4.58) has the same form as the two-scalar two-graviton amplitude in the eikonal approximation, first equation in (4.1), as as consequence of the equivalence principle.
At tree-level the helicity-preserving F F R amplitude (3.26) is purely a contact term, while the helicity-flip amplitude is given in (3.31). The leading contribution in the eikonal limit is then where we used [34] 2 = −q 2 . Transforming the non-vanishing helicity-flip amplitude to impact parameter space we obtain Defining we can combine (4.59) and (4.61) into a single leading eikonal phase matrix 11 which, upon expanding around D = 4, reduces to (4.65) 10 We recall from Section 3.4.2 that A There is no need here to separate the EH and theF F Rcontributions, since we consider only photon bending coming from this source.
Next, in order to test the expected exponentiation property of the leading eikonal phase matrix, we consider the terms of O(ω 2 ) in the one-loop amplitudes. These are given in impact parameter space by which are obtained from (3.29) and (3.32). In matrix form, (4.67) Expanding around D = 4 we find thatÃ (1) ω 2 satisfies the matrix equation as expected.

Subleading eikonal
Next we consider the subleading eikonal phase. The only non-vanishing EH contribution comes from the one-loop massive triangles in the helicity-preserving amplitude (3.29), and reads Just as in the case of the leading eikonal phase, the bending angle of photons in pure EH comes is the same as the graviton bending (4.11) thanks to the equivalence principle.
The contributions coming from the F F R interaction are obtained from (3.33) and (3.32), and in impact parameter space are

C The tree-level amplitudes
In this appendix we collect for the reader's convenience all the tree-level amplitudes we have used in our derivations. All are consistent with the normalisations of (1.1), also we assume all momenta to be outgoing.

D The four-graviton amplitudes in R 4
In this appendix we show how the most generic four-graviton amplitude in an R 4 background can be constructed just from little-group considerations and dimensional analysis, without looking at any Lagrangian. We begin by noting that the coupling constant of the four-point amplitude has two powers of κ ([κ] = −1) and it is proportional to the coupling constant of the R 4 interaction β ([β] = −6). Furthermore, the nature of the new interaction implies that the four-point amplitude is just a contact term. Mass dimension and scaling under little-group transformations fix the form of the possible amplitudes completely: (D.5) However, using the Schouten identity, which in terms of these variables reads a + b + c = 0 , (D. 6) one can show that there is actually only one independent combination, which we will take to be the first of (D.5). We will then define the all-plus amplitude to be In the presence of a parity-invariant theory, the amplitude corresponding to (D.7) with all helicities flipped is simply obtained by replacing [ji] → ij , otherwise it should be considered to have an independent normalisation. (D.8) The derivation of these amplitudes from the Lagrangian (1.3) is presented in Section 3.3.