A tale of two exponentiations in N=8 supergravity
Paolo Di Vecchia, Andrés Luna, Stephen G. Naculich, Rodolfo Russo, Gabriele Veneziano, Chris D. White
BBOW-PH-167NORDITA 2019-076QMUL-PH-19-20CERN-TH-2019-133
A tale of two exponentiations in N = 8 supergravity Paolo Di Vecchia a,b , Andr´es Luna c , Stephen G. Naculich d , Rodolfo Russo e ,Gabriele Veneziano f,g and Chris D. White e a NORDITA, KTH Royal Institute of Technology and Stockholm University,Roslagstullsbacken 23, SE-10691 Stockholm, Sweden b The Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17,DK-2100 Copenhagen, Denmark c Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and Astronomy,University of California at Los Angeles, California 90095 d Department of Physics, Bowdoin College, Brunswick, ME 04011 USA e Centre for Research in String Theory, School of Physics and Astronomy,Queen Mary University of London, 327 Mile End Road, London E1 4NS, UK f Theory Department, CERN, CH-1211 Geneva 23, Switzerland g Coll´ege de France, 11 place M. Berthelot, 75005 Paris, France
Abstract
The structure of scattering amplitudes in supergravity theories continues to be of interest.Recently, the amplitude for 2 → N = 8 supergravity was presented at three-loop order for the first time. The result can be written in terms of an exponentiated one-loopcontribution, modulo a remainder function which is free of infrared singularities, but containsleading terms in the high energy Regge limit. We explain the origin of these terms from a well-known, unitarity-restoring exponentiation of the high-energy gravitational S -matrix in impact-parameter space. Furthermore, we predict the existence of similar terms in the remainderfunction at all higher loop orders. Our results provide a non-trivial cross-check of the recentthree-loop calculation, and a necessary consistency constraint for any future calculation at higherloops. [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] a r X i v : . [ h e p - t h ] A ug Introduction
Scattering amplitudes in gauge and gravity theories continue to be intensively studied, due to awide variety of both formal and phenomenological applications. Our focus in this paper is N = 8supergravity in four spacetime dimensions, which is of interest for a number of reasons. Firstly,it may prove to be an ultraviolet finite theory of perturbative quantum gravity [1–5], and in anycase has a special status as its amplitudes arise in the low energy limit from type II superstringtheory [6]. Secondly, calculations in maximally supersymmetric theories can be simpler than inless symmetric scenarios, making such theories the ideal frontier for developing new calculationaltechniques. Thirdly, there are a number of conjectures regarding the structure of amplitudes inmaximally supersymmetric theories, which higher-order computations are able to shed light on.One of the simplest amplitudes in terms of external multiplicity is that of four-graviton scatter-ing, results for which have been previously calculated at one-loop [6–9] and two-loop [10–13] order.In the maximally supersymmetric theory, the tree-level result factors out, such that the amplitudemay be written in the form i M = i M (0)4 (cid:32) ∞ (cid:88) L =1 M ( L )4 (cid:33) , (1.1)where M (0) is the tree-level contribution, and M ( L ) an implicitly defined correction factor at L -loop order. The latter is infrared divergent, such that M ( L ) has a leading 1 /(cid:15) L pole in d = 4 − (cid:15) spacetime dimensions. Additional structure arises, however, from the fact that infrared divergencesin gravity theories are known to exponentiate [11, 14–19], where the logarithm of the soft (IR-divergent) part of the amplitude terminates at one-loop order, in marked contrast to (non-Abelian)gauge theories [20–29]. This motivates the following ansatz for the all-order amplitude: i M = i M (0)4 exp[ M (1)4 ] F , (1.2)where M (1)4 is the full one-loop correction factor, including also its infrared singular part, and F isan infrared finite remainder function , commencing at two-loop order. Indeed, results for the latterhave been presented at two-loop order for a variety of supergravity theories in ref. [11–13], andtheir implications discussed further in refs. [19, 30].Recently, the four-graviton scattering amplitude in N = 8 supergravity has been obtained atan impressive three-loop order [31]. The authors compared their results with the form of eq. (1.2),confirming that the three-loop remainder function is infrared finite. This itself provided a highlynon-trivial cross-check of their results. However, as in previous studies [13, 19, 30], they also exam-ined the behaviour of the remainder function in the high energy Regge limit . This corresponds tohighly forward high energy scattering, such that the centre of mass energy is much greater thanthe momentum transfer. The authors of ref. [31] noted in particular the curious property thatthe remainder function, although infrared finite, contains leading contributions in the high energylimit, suggesting that their structure can be explained using known results regarding high energyand / or soft limits. Indeed this is the case, as we will show in this paper.High energy scattering in gauge and gravity theories has been studied for many decades. Forexample, generic scattering behaviour in the Regge limit formed a crucial ingredient in the S-matrix programme of strong interactions, which predated the discovery of QCD (see e.g. [32] for areview). Obtaining similar behaviour in perturbative quantum field theory has been pursued overmany decades, with relevant work in (super-)gravity including [33–42]. Recently, methods from2auge theory have been used to analyse gravitational physics, including clarifying the relationshipsbetween both theories in certain kinematic limits [15, 17, 19, 43–49]. Of particular relevance hereis the outcome of the studies, started in the late 1980’s [50–64], of high-energy (transplanckian)gravitational scattering in the Regge-asymptotics regime in both string and field theories (see [67]for a recent review). Indeed, in order to explain the three-loop findings of ref. [31], we will use avery well-established property of gravitational scattering in the leading Regge limit, namely thatthe S-matrix has a certain exponential structure in transverse position (i.e. impact parameter)space, in terms of the so-called eikonal phase . This may be expanded in the gravitational couplingconstant, before being Fourier transformed to momentum-transfer space order-by-order in pertur-bation theory. Given that a product in position space is not a product (but rather a convolution)in momentum space, the exponentiated eikonal phase in the former does not directly lead to an ex-ponential form in momentum space. The upshot of this is that by making the ansatz of eq. (1.2) inmomentum space, a mismatch occurs, giving leading Regge contributions in the remainder function.We will explicitly verify the form of the two- and three-loop remainder functions in the (leading)Regge limit. Furthermore, we will use our findings to predict additional terms at higher loops, beforeforming a conjecture for the leading Regge behaviour of the remainder function at arbitrary orderin perturbation theory. Our results provide a cross-check of the three-loop calculation in ref. [31],whilst also setting consistency constraints on any future higher-loop calculations.The structure of our paper is as follows. In section 2, we review previous results about fixed orderresults for supergravity amplitudes, and also the exponentiation of the position space amplitudein terms of the eikonal phase. In section 3, we verify the form of the remainder function up tothree-loop order in the leading Regge limit. In section 4, we extend our analysis to arbitrary ordersin perturbation theory. Finally, in section 5 we discuss our results and conclude. As discussed above, the remainder function F of eq. (1.2) is defined after subtracting the one-loopcontribution from the logarithm of the 4-graviton scattering amplitude. It thus begins at two-looporder, and we may then consider the perturbative expansion F = 1 + ∞ (cid:88) L =2 F ( L )4 , (2.1)where F ( L )4 is the L -loop contribution, including coupling factors other than those associated withthe tree-level amplitude. Explicit results for the two-loop contribution (in a variety of supergravitytheories) have been presented in ref. [11–13]. To present results, we label 4-momenta as shown infigure 1, from which we may define the Mandelstam invariants s = ( p + p ) , t = ( p − p ) , u = ( p − p ) . (2.2) A different high-energy regime, at fixed scattering angle, was also considered at about the same time withinstring theory [65, 66]. In the following we will often refer to transverse space (momentum) as, simply, space (momentum), but it isimportant to stress that longitudinal momentum (energy) are never converted into the corresponding space (time)variables. This distinction is also important [50, 51] to recover classical General Relativity expectations from theeikonal approximation when the eikonal phase is parametrically large. p p p Figure 1: Labelling of 4-momenta for the four-graviton scattering process.Note that all 4-momenta in figure 1 are physical (e.g. rather than all outgoing), so that we aredealing with the physical scattering region s ≥ , t, u ≤ . (2.3)Furthermore, momentum conservation implies s + t + u = 0, so that only two Mandelstam invariantsare independent. The Regge limit may then be formally defined as s (cid:29) − t . Alternatively, definingthe dimensionless ratio x = − ts , (2.4)the Regge limit corresponds simply to x →
0. Until recently, only the O ( (cid:15) ) contribution of thetwo-loop remainder function was known, whose leading behaviour in the Regge limit may be writtenas [19] F (2)4 = x (cid:16) α G s (cid:17) (cid:26) − π log x − π log x + π + 4 π (2.5)+ iπ (cid:20)
43 log x + 4 log x + 8 (cid:18) π (cid:19) (1 − log x ) + 16 ζ (cid:21) (cid:27) + O ( x ) + O ( (cid:15) ) , where we introduced the parameter α G ≡ G N π (4 π ) (cid:15) Γ (1 − (cid:15) )Γ(1 + (cid:15) )Γ(1 − (cid:15) ) = G N π + O ( (cid:15) ) . (2.6)Given that F (2)4 is O ( x ), it vanishes in the strict Regge limit. However, the results of ref. [31] havenow demonstrated that this is not true at higher orders in the dimensional regularisation parameter (cid:15) , nor at higher-loop level. In fact, the result in eq. (6.5) of ref. [31] is F (2)4 = α G s π (cid:20) ζ (cid:15) + (cid:18) π − ζ log( − t ) (cid:19) (cid:15) + O ( x ) + O ( (cid:15) ) (cid:21) , F (3)4 = − iα G s π ζ + O ( x ) + O ( (cid:15) ) . (2.7)These contributions are non-vanishing as x →
0; we will explain their origin in the followingsections. The (cid:15) contribution to F (2)4 is not explicitly written in (6.5) of [31], but can be deduced from the ancillary filesattached to the arxiv version of [31]. The apparent sign discrepancy between F (3) and eq. (6.5) results from ourchoosing s > s < .. Figure 2: A representative (crossed) ladder graph, where all particles are gravitons. The sum of allsuch diagrams in the leading Regge limit builds up the exponentiated amplitude of eq. (2.8).
The Regge limit of forward scattering consists of highly energetic particles that barely glance offeach other. As such, any exchanged radiation must be soft (i.e. have an asymptotically small 4-momentum), and the emitting particles are then said to be in the eikonal approximation . One maythen show [50, 51] that the dominant behaviour at arbitrary loop orders is given by the (crossed)horizontal ladder graphs of figure 2, in which all particles are gravitons. Furthermore, this situationdoes not depend on the amount of supersymmetry: in the leading Regge limit, the amplitude isdominated by the exchanged particle of highest spin, namely the graviton. It is then possible tosum such graphs to all perturbative orders by working at fixed impact parameter x ⊥ , a ( d − eikonal amplitude as (see e.g. [68]) i M eik . = 2 s (cid:90) d d − x ⊥ e − i q ⊥ · x ⊥ (cid:16) e iχ ( x ⊥ ) − (cid:17) , (2.8)where the quantity iχ ( x ⊥ ) is known as the eikonal phase , and is given in d = 4 − (cid:15) dimensionsby [50, 51] iχ ( x ⊥ ) = − iG N s Γ(1 − (cid:15) ) ( π x ⊥ ) (cid:15) (cid:15) . (2.9)In eq. (2.8), q ⊥ is the ( d − x ⊥ .In terms of the above Mandelstam invariants, one has t (cid:39) −| q ⊥ | in the leading Regge limit. Theexponentiation of the amplitude in terms of a large eikonal phase has the important consequenceof restoring partial-wave unitarity, which is violated as s → ∞ at each loop order due to gravitonexchange [50,51]. Equation (2.8) has a well-defined physical interpretation [50–52], in which iχ rep-resents the phase shift experienced by one of the incoming particles in the field set up by the other,thus forming a link between old-fashioned quantum mechanical scattering theory and perturbativeQFT approaches (see e.g. ref. [69] for an excellent review). Importantly, the exponentiation occursin position space. To obtain the momentum-space amplitude at a given order in perturbation the-ory, one must Taylor expand the exponential in the Newton constant G N , before carrying out theFourier transform: i M eik . = 2 s ∞ (cid:88) n =1 n ! (cid:90) d d − x ⊥ e − i q ⊥ · x ⊥ [ iχ ( x ⊥ )] n . (2.10) This result holds at finite (cid:15) and its validity is unrelated to the problem of infrared singularities originating in the (cid:15) →
5n each term, the product of phase factors [ iχ ( x ⊥ )] n becomes a convolution in momentum space,which may itself be given a direct physical interpretation. First, one may express the position-spaceeikonal phase as an inverse Fourier transform: iχ ( x ⊥ ) = − πiG N s (cid:90) d d − k ⊥ (2 π ) d − e i k ⊥ · x ⊥ ( − k ⊥ + iε ) , (2.11)where iε denotes the usual Feynman prescription. This allows us to rewrite eq. (2.10) as i M eik . = 2 s ∞ (cid:88) n =1 ( − πiG N s ) n n ! (cid:90) d d − x ⊥ e − i q ⊥ · x ⊥ (cid:32) n (cid:89) i =1 (cid:90) d d − k i ⊥ (2 π ) d − e i k i · x ⊥ ( − k i ⊥ + iε ) (cid:33) = 2 s (2 π ) d − ∞ (cid:88) n =1 ( − πiG N s ) n n ! n (cid:89) i =1 (cid:18)(cid:90) d d − k i ⊥ (2 π ) d − − k i ⊥ + iε ) (cid:19) δ ( d − (cid:32) q ⊥ − n (cid:88) i =1 k i ⊥ (cid:33) . (2.12)Each term in the second line consists of a momentum space Feynman integral, with n particlesbeing exchanged, each described by a standard propagator in ( d − q ⊥ , namely thetotal momentum transfer that is conjugate to the impact parameter. As we will see in the followingsection, it is precisely the lack of a simple product in momentum space that leads to the presenceof the non-trivial remainder function of eq. (2.7). Having seen how to describe the leading Regge limit of the four-graviton amplitude in supergravityto all orders via the eikonal phase, we now have everything we need to explain the results of ref. [31],presented here in eq. (2.7). To obtain the L -loop remainder function, we may start with eq. (2.10),and identify n = L + 1. Substituting eq. (2.9) then yields i M eik . = 2 s ∞ (cid:88) L =0 L + 1)! (cid:18) − iG N s Γ(1 − (cid:15) ) π (cid:15) (cid:15) (cid:19) L +1 (cid:90) d − (cid:15) x ⊥ e − i q ⊥ · x ⊥ (cid:0) x ⊥ (cid:1) ( L +1) (cid:15) = 2 s πiG N s q ⊥ ∞ (cid:88) L =0 L ! (cid:20) − iG N s Γ(1 − (cid:15) ) (cid:15) (cid:18) π q ⊥ (cid:19) (cid:15) (cid:21) L Γ(1 − (cid:15) )Γ(1 + L(cid:15) )Γ(1 − ( L + 1) (cid:15) ) . (3.1)The second line allows us to identify the Regge limit of the tree-level amplitude from the L = 0term: i M (0)4 = 8 πiG N s − t + O ( x ) . (3.2)Examining the one-loop term then allows us to construct the correction factor entering eq. (1.2) M (1)4 = i M (1)4 i M (0)4 = − iG N s(cid:15) Γ (1 − (cid:15) )Γ(1 + (cid:15) )Γ(1 − (cid:15) ) (cid:18) π − t (cid:19) (cid:15) . (3.3)6et us now rewrite eq. (3.1) as i M eik . = i M (0)4 ∞ (cid:88) L =0 L ! (cid:20) − iG N s(cid:15) Γ (1 − (cid:15) )Γ(1 + (cid:15) )Γ(1 − (cid:15) ) (cid:18) π − t (cid:19) (cid:15) (cid:21) L × (cid:26) Γ L (1 − (cid:15) )Γ(1 + L(cid:15) )Γ L − (1 − (cid:15) )Γ L (1 + (cid:15) )Γ(1 − ( L + 1) (cid:15) ) (cid:27) . (3.4)Were it not for the term in curly brackets, we would find that the full momentum-space amplitude issimply the tree-level amplitude multiplied by the exponential of the one-loop correction of eq. (3.3).By comparing eqs. (1.2) and (3.4), we thus find that the remainder function is given by F = exp (cid:2) − M (1)4 (cid:3) ∞ (cid:88) L =0 (cid:2) M (1)4 (cid:3) L L ! (cid:26) Γ L (1 − (cid:15) )Γ(1 + L(cid:15) )Γ L − (1 − (cid:15) )Γ L (1 + (cid:15) )Γ(1 − ( L + 1) (cid:15) ) (cid:27) + O ( x ) . (3.5)This is a complete all-orders expression in the leading Regge limit x →
0, which may be system-atically expanded in G N to obtain the result at a given loop order. Performing such an expansion(also in the dimensional regularisation parameter (cid:15) ), one finds F = 1 + α G s π (cid:20) ζ (cid:15) + (cid:18) π − ζ log( − t ) (cid:19) (cid:15) + O ( (cid:15) ) (cid:21) + α G s π (cid:20) − iζ + O ( (cid:15) ) (cid:21) + O ( α G ) + O ( x ) . (3.6)in agreement with eq. (2.7) and thus precisely confirming the results of ref. [31]. We can nowgo further than this, however, and predict the structure of the remainder function in the leadingRegge limit at higher orders in perturbation theory. In the previous section, we obtained a general expression, eq. (3.5), for the remainder function F in the leading Regge limit, and confirmed the results of a recent three-loop calculation (whichalso necessarily included O ( (cid:15) ) at O ( G N )). However, the all-order nature of eq. (3.5), in both G N and (cid:15) , means that we can expand this further. In doing so, we predict the existence of non-zeroterms in the remainder function at four-loop order and beyond. This potentially provides a highlynon-trivial cross-check of any future calculations in perturbative gravity.We have expanded eq. (3.5) to 16 orders in G N , finding that all poles in (cid:15) vanish. This is to beexpected, given the aforementioned fact that all infrared singularities in gravity are generated bythe exponentiation of the one-loop amplitude [11,14–19]. Turning to the O ( (cid:15) ) terms of the leadingenergy remainder F = F , + O ( (cid:15) ) + O ( x ), we may write the L -loop contribution as F ( L )4 , = ( iG N s ) L f ( L ) , (4.1) See footnote 9. f (2) = 0 f (7) = ¯ ζ f (12) = 14! ¯ ζ + ¯ ζ ¯ ζ + ¯ ζ ¯ ζ f (3) = ¯ ζ f (8) = ¯ ζ ¯ ζ f (13) = 12 ¯ ζ ¯ ζ + 12 ¯ ζ ¯ ζ + ¯ ζ f (4) = 0 f (9) = 13! ¯ ζ + ¯ ζ f (14) = 13! ¯ ζ ¯ ζ + ¯ ζ ¯ ζ + 12 ¯ ζ + ¯ ζ ¯ ζ (4.2) f (5) = ¯ ζ f (10) = 12 ¯ ζ + ¯ ζ ¯ ζ f (15) = 15! ¯ ζ + 12 ¯ ζ ¯ ζ + ¯ ζ ¯ ζ ¯ ζ + 13! ¯ ζ + ¯ ζ f (6) = 12 ¯ ζ f (11) = 12 ¯ ζ ¯ ζ + ¯ ζ f (16) = 13! ¯ ζ ¯ ζ + 14 ¯ ζ ¯ ζ + ¯ ζ ¯ ζ + ¯ ζ ¯ ζ + ¯ ζ ¯ ζ with ¯ ζ n = 2 ζ n /n . Despite the rather formidable nature of eq. (3.5), we see that the results for the O ( (cid:15) ) contributions have a simple form. It is apparent that the arguments of the zeta functions ineach term in the sums are such that they form a partition of L into a sum of odd integers greaterthan one. The generating function for the number of such partitions is ∞ (cid:89) j =1 − z j +1 = 1+ z + z + z + z + z +2 z +2 z +2 z +3 z +3 z +4 z +5 z +5 z + O (cid:0) z (cid:1) (4.3)so the coefficient of z L on the right-hand side of eq. (4.3) tells us the number of individual terms ineach f ( L ) of eq. (4.2). We then find that we can summarise all of eq. (4.2) as the compact formula F ( L )4 , = ( iG N s ) L (cid:88) p r ( L ) (cid:89) j n j ! (cid:18) ζ L j L j (cid:19) n j , (4.4)where the sum is over all restricted partitions of L , as mentioned above, the L j ’s are the distinctodd integers entering in the partition and n j is the number of times each L j appears, so we have L = (cid:88) j L j n j . (4.5)In fact, one may observe that eqs. (4.1), (4.2) and (4.4) may be compactly summarized by F , = 1 + ∞ (cid:88) L =2 F ( L )4 , = exp ∞ (cid:88) j =1 ( iG N s ) j +1 ¯ ζ j +1 = e − iG N sγ Γ (1 + iG N s ) exp (cid:20) log (cid:18) πiG N s sin( πiG N s ) (cid:19)(cid:21) = e − iG N sγ Γ(1 − iG N s )Γ(1 + iG N s ) . (4.6)The same result can be obtained from the (cid:15) → χ ( x ⊥ ) (see ref. [19] for a similar observation). Denoting the (cid:15) terms of eqs. (2.9) and (3.3) by χ and M (1)4 , respectively, we may write χ = − G N s (cid:0) log( π x ⊥ ) + γ (cid:1) , e M (1)4 , = e iG N sγ (cid:18) π q ⊥ (cid:19) − iG N s . (4.7) We would like to thank Henrik Johansson for this observation.
8e can then perform the Fourier transform of eq. (2.8) in d = 4 (i.e. restricting to the (cid:15) -independentpart) (cid:90) d x ⊥ e − i q ⊥ · x ⊥ e iχ ( x ⊥ ) = 4 πiG N s q ⊥ e − iG N sγ (cid:18) π q ⊥ (cid:19) − iG N s Γ(1 − iG N s )Γ(1 + iG N s ) = 4 πiG N s q ⊥ e M (1)4 , F , , (4.8)and check explicitly that the last step is consistent with the result of eq. (4.6) . This derivationcan be seen as a proof of the result (4.4) for the (cid:15) contribution, but we stress in any case that acomplete all-order expression for the remainder function (which is more powerful than a finite-order (cid:15) expansion) has already been given in eq. (3.5).The next unknown order in the four-graviton amplitude is four loops. It is easily checked fromeq. (3.5) that, as at two loops, the O ( (cid:15) ) contribution to the remainder function (in the leadingRegge limit) vanishes. However, there is a nonzero contribution beyond this, given by F (4)4 = − G N s ) ζ (cid:15) + O ( (cid:15) ) + O ( x ) . (4.9)We do not expect this result to be explicitly confirmed in the near future: calculating the O ( (cid:15) ) partof the four-loop amplitude would presumably be first carried out as part of a five-loop calculation!An interesting observation is that the above results respect the conjectured uniform transcen-dentality property of amplitudes in theories with maximal supersymmetry. That is, we can associatea transcendental weight n with the zeta value ζ n , where all rational coefficients are taken to haveweight zero. The sum of weights at O ( (cid:15) m ) and L -loop order is then w = L + m. (4.10)Beyond the leading order, the Regge limit breaks this uniform transcendentality property, as, forinstance, one approximates ln( − u/s ) ∼ x losing the transcendental contribution of the logarithm.Since the leading eikonal does not depend on the number of supersymmetries, the uniform weightproperty for N = 8 supergravity manifest in (4.10) is inherited by the lower supersymmetric cases.We stress that this property of the leading term is exact to all orders, not just the (cid:15) order consideredabove. For the amplitude itself at a given loop order, there is a dominant pole ∼ (cid:15) L coming from the exponentiated IR singularity in the one-loop contribution. All subleading terms in (cid:15) (in the leading Regge limit) come from expanding Euler gamma functions, and the coefficients ofall such expansions have increasing uniform weight as the power of (cid:15) increases. Thus, this accountsprecisely for the dependence of eq. (4.10). In this paper, we examined the form of the four-graviton scattering amplitude in N = 8 supergrav-ity, which was recently calculated at three-loop order [31]. It is conventional to define a remainderfunction for this amplitude, constituting what is left upon factoring out the tree-level amplitude, Note that, for (cid:15) →
0, the whole effect of F , boils down to a renormalization of an unobservable (and notexplicitly written) infinite Coulomb phase originating from the leading eikonal resummation. O ( (cid:15) ) part of the two-loop result, revealed the existence of leading terms in the remainderfunction in Regge’s high energy limit, at non-negative powers of the dimensional regularisationparameter (cid:15) .In this paper, we have shown that these contributions follow precisely from the known expo-nentiation of the four-graviton amplitude in position space, in terms of the so-called eikonal phase .At a given order in perturbation theory, a product of one-loop amplitudes occurs, which becomesa convolution in momentum space, whose physical interpretation is that the transverse momentumtransfer (conjugate to the impact parameter) must be democratically shared between the exchangedgravitons at that order. This in turn means that the amplitude does not straightforwardly exponen-tiate in momentum space, and we have derived an all-orders expression – in both the gravitationalcoupling G N and dimensional regularisation parameter (cid:15) – for the remainder function in the Reggelimit. As well as confirming the results of ref. [31], we also predict explicit contributions at higherloop orders. We obtained a particularly convenient combinatorial form for the O ( (cid:15) ) contributions,which we showed can be directly obtained from the leading eikonal expression in d = 4. The higherloop remainder function respects maximal transcendentality to all orders.There are a number of possible extensions of our analysis. Firstly, one could look at predictingthe structure of subleading terms in the Regge limit (see e.g. refs. [30, 44, 54, 57, 61–64, 70] forprevious work in this area). Secondly, it would be interesting to extend the analysis discussedin this paper to higher loops by starting from the integral expressions for the four- and five-loopamplitudes of refs. [71, 72]. Finally one can study the remainder function in theories with lessthan maximal supersymmetry. This is not independent of the exploration of subleading eikonalcontributions. Indeed, the leading Regge behaviour would be expected to be the same for lesssupersymmetric gravity theories, given that this kinematic regime is dominated by the exchangeonly of leading soft particles of highest spin (i.e. the graviton). Three-loop calculations in non-maximal supergravity theories do not yet exist, thus our results already provide a highly usefulconstraint. Acknowledgments
We thank Zvi Bern, Lance Dixon, Henrik Johansson, Julio Parra-Martinez, Lorenzo Magnea, andCristian Vergu for useful conversations. AL is supported in part by the Department of Energyunder Award Number DESC000993. The research of SGN is supported in part by the NationalScience Foundation under Grant No. PHY17-20202. CDW and RR are supported by the UK Sci-ence and Technology Facilities Council (STFC) Consolidated Grant ST/P000754/1 “String theory,gauge theory and duality, and/or by the European Union Horizon 2020 research and innovationprogramme under the Marie Ck(cid:32)lodowska-Curie grant agreement No. 764850 “SAGEX”. AL andSGN wish to thank the Centre for Research in String Theory at Queen Mary University of Londonfor hospitality. PDV, RR and GV would like to thank the Galileo Galilei Institute for hospitalityduring the workshop “String theory from the worldsheet perspective” where they started discussingthis topic. PDV was supported as a Simons GGI scientist from the Simons Foundation grant 4036341344 AL. The research of PDV is partially supported by the Knut and Alice Wallenberg Foun-dation under grant KAW 2018.0116. 10 eferences [1] Z. Bern, L. J. Dixon, and R. Roiban, “Is N = 8 supergravity ultraviolet finite?,”
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