On the soft limit of closed string amplitudes with massive states

We extend our analysis of the soft behaviour of string amplitudes with massive insertions to closed strings at tree level (sphere). Relying on our previous results for open strings on the disk and on KLT formulae we check universality of the soft behaviour for gravitons to sub-leading order for superstring amplitudes and show how this gets modified for bosonic strings. At sub-sub-leading order we argue in favour of universality for superstrings on the basis of OPE of the vertex operators and gauge invariance for the soft graviton. The results are illustrated by explicit examples of 4-point amplitudes with one massive insertion in any dimension, including D=4, where use of the helicity spinor formalism drastically simplifies the expressions. As a by-product of our analysis we confirm that the `single valued projection' holds for massive amplitudes, too. We briefly comment on the soft behaviour of the anti-symmetric tensor and on loop corrections.


Introduction and motivations
The connection among 'gravitational memory', 'soft behaviour' of graviton scattering amplitudes and 'BvBMS symmetry' [1][2][3][4][5][6] seems to play a crucial in a recently proposed solution to the Information Paradox for Black Holes [7]. While waiting for a refined version of the argument , it is natural to ask the fate of the universal 'soft' behaviour of graviton scattering amplitudes in a quantum theory of gravity such as closed string theory. The problem has been addressed for tree-level amplitudes with only mass-less gravitons in [8; 9], relying on KLT formulae and OPE of the vertex operators, and in [10], relying on gauge invariance. Bosonic amplitudes with tachyons have been investigated to sub-leading order in [11; 12].
In gravity theories, when one of the external graviton momenta goes soft i.e. k → 0 with k = δk withk some fixed momentum, not only the leading δ −1 and sub-leading behaviours δ 0 [13; 14], but also the nextto-subleading or sub-sub-leading behaviour δ +1 is universal [15]. Calling h µν s the soft graviton polarisation and k µ s its soft momentum, one has M n (1, 2, . . . , s, . . . , n) ≈ where k i and J i denote the 'hard' momenta and angular momentum operators. These results are valid at tree-level and are derived with the understanding that interactions be governed by minimal coupling.
In theories with closed strings, the conclusions, though quite independent of the number of (non-compact) space-time dimensions, depend on the nature of the higher derivative couplings [8]. R 3 terms do not change the universal soft behaviour of minimal coupling, while φR 2 do modify even the leading term when φ is a massless scalar such as the dilaton. This happens in particular in the bosonic string and heterotic string at tree level 1 and in the Type II compactifications preserving less than maximal super-symmetry.
Aim of the present investigation, that may be considered a follow up of [16], is to show that inclusion of massive external states does not spoil the universal 'soft' behaviour (1) for Type II theories with maximal susy at tree level. In [16] open string amplitudes with massive external states as well as tachyons have been computed and shown to expose the expected behavior even when non-minimal interactions are considered. Neither F 3 terms nor the coupling α ′ T F 2 , where T is the tachyon, change the universal soft behaviour, based on minimal coupling. On the other hand φF 2 terms do modify even the leading term when φ is a massless scalar. For color-ordered string amplitudes one gets the same universal behaviour as in YM theories [17][18][19][20][21][22][23][24][25][26][27][28][29] A n (1, 2, . . . , s, . . . , n) ≈ where a s and k s denote the soft gluon polarisation and momentum, so that f µν s = k µ s a ν s −k µ s a ν s is its linearised field strength, while k s±1 and J s±1 denote the 'hard' momenta and angular momentum operators of the adjacent insertions. Relying on [16] and on KLT formulae, we presently analyse closed string amplitudes with massive external states. In the bosonic string case we will also consider tachyons as external states.
In Section 2, we briefly review KLT formulae relating closed string to open string amplitudes and the 'single valued projection' suggested in [43; 44]. Then we discuss how to relate the soft limit of closed string amplitudes with an arbitrary number of massive insertions to the soft limit of open string amplitudes with the same number of massive insertions in Section 3. In Section 4 and 5 we illustrate our point with explicit examples of 4-point amplitudes with one massive higher spin insertion (or tachyons in the bosonic case). We check the (non) universality of the soft behaviour for bosonic string gravitons in Section 6 and discuss how to generalise the analysis to the case of anti-symmetric tensors. For the superstrings in D = 4 we rely on the spinor helicity formalism to simplify our expressions. Our conclusions are presented in Section 7.

From Veneziano to Shapiro-Virasoro according to KLT
Closed-string amplitudes, henceforth denoted by M n to distinguish them from open-string amplitudes, denoted by A n , can be efficiently computed relying on KLT formulae [45]. At the cost of being pedantic, in order to fix our notation and illustrate the KLT procedure, we start by briefly reviewing some 4-point string amplitudes involving tachyons or massless states.
In going from open to closed strings the mass shell condition becomes α ′ c (p/2) 2 = (N − 1) that effectively amounts to the replacement α ′ o → α ′ c /4 2 . As a result a closed string vertex operator can be expressed as the product of two open-string vertex operators, each carrying half of the total momentum. In formulae where p 2 = m 2 H = 4m 2 H , and H = H ⊗H in general comprises several irreducible representations of the Lorentz group.
1 M. B. would like to thank I. Antoniadis for stressing the tree level origin of this term in the heterotic string, which only gets generated at one-loop in 4-dim Type II theories with 16 supercharges, such as after compactification on K3 × T 2 . R 3 term is forbidden due to supersymmetry. 2 While the open string spectrum is given by α ′ op M 2 N = N − 1 with N = S M ax , the closed string spectrum is given by α ′ cl M 2 N = 4(N − 1) = 2(N L + N R − 2) due to level matching N L = N R = N = S M ax /2.

Four tachyons:
The simplest closed-string amplitude is the Shapiro-Virasoro amplitude M 4 (T 1 , T 2 , T 3 , T 4 ) describing the scattering of four tachyons in the closed bosonic string. The tachyon vertex operator is Up to an overall constant factor, one finds [46; 47] M 4 (T 1 , T 2 , T 3 , T 4 ) = π d 2 z |z| α ′ c p3p4 |1 − z| α ′ In type II superstrings the tachyon is projected out. The lowest lying states in the NS-NS sector are massless. The massless vertex operator with . For later purposes, it is crucial to observe that gravitons and dilatons are even under L-R exchange, Ω = 1, while Kalb-Ramond fields are odd, Ω = −1. This implies that amplitudes with an odd number of Kalb-Ramond fields and an arbitrary number of gravitons and dilatons vanish. The amplitude for 4 massless NS-NS states is well known. The expression is extremely lengthy and can be expressed more compactly in terms of the t 8 tensor introduced by Brink, Green and Schwarz [48]. We refrain from doing so. Using KLT in the t-channel, one finds 3 Now writing [37] A totally symmetric, and rewriting F 4 L ⊗ F 4 R ≈ R 4 + . . . one can systematically derive the Type II 4-graviton amplitudes and the related ones for φ's and (an even number of) b's. For bosonic strings the situation is richer. For open strings the tri-linear coupling is non-minimal. In addition to the standard Yang-Mills term, it contains an F 3 -term, suppressed by α ′ . As mentioned in the introduction and discussed in [16], this does neither spoil universality of the soft behaviour at leading order nor at subleading order, even in the case of massive insertions. For closed bosonic strings, in addition to minimal tri-linear terms (graviton, dilatons and Kalb-Ramond fields), there is a φR 2 term (suppressed by α ′ ) and an R 3 -term (suppressed by (α ′ ) 2 ). As shown in [8], the latter does not spoil the universality of the soft behaviour while the former spoils it even at leading order. Barring the distinction between gravitons and dilatons, i.e. describing them in a unified fashion with E µν = +E νµ = h µν + φ µν , one can regain a sort of universality of the soft behaviour as advocated in [11; 12]. Yet b µν behaves in a very different way due to its being odd under Ω, as we will see in Section 5.

Higher-point amplitudes
Closed-string amplitudes with massive insertions look extremely cumbersome and not very illuminating in D = 10, even at tree level (sphere). In D = 4, using the spinor helicity basis, formulae look more tractable. A possible strategy for systematic computations is to first use KLT relations in order to express closed-string amplitudes in terms of open-string amplitudes, and then compute open-string amplitudes for massive states by multiple factorizations of amplitudes with only massless insertions on massive poles in two-particle channels as in [16].
KLT relations incorporate the intrinsic non-planarity of closed-string amplitudes and rely on the monodromy properties of (colour-ordered) open string amplitudes [45]. The basic idea is to parameterize the closed-string insertion points as z i = x i + iy i and notice that the integrand is an analytic function of the y i viewed as complex variables with branch points at ±i(x i − x j ). One can then deform the integration contour from Imy i = 0 to Rey i = 0 so much so that z i andz i = x i − iy i become two independent real variables ξ i and η i that one can integrate over with Jacobian ∂(x i , y j )/∂(ξ i , η j ) = (i/2) N . The correct monodromy around the branch points of the integrand (Koba-Nielsen factor, in units with n ij andn ij integer, is accounted for by the phase factor that only depends on the orderings σ ξ and σ η but not on the variables ξ's and η's themselves. The integrations decouples and can be performed explicitly. In particular, using SL(2) to fix 3 ξ's, there remain (n − 3)! orderings of the ξ's. For each of them, the independent choices of the contours in η that give a nonvanishing result give in fact all the same result. All in all there are (n − 3)![ 1 2 (n − 3)!] 2 terms for n odd or (n − 3)![ 1 2 (n − 4)!][ 1 2 (n − 2)!] for n even [45]. In particular, for n = 3, 4 there is only one term 3 and . For n = 5 one has two terms while for n = 6 one has twelve terms In general, one has [49] where {i} ∈ Perm[2, . . . , ⌊n/2⌋], {j} ∈ Perm[⌊n/2⌋+1, . . . , n−2], with ⌊n/2⌋ = (n−1)/2 for n odd, and ⌊n/2⌋ = n/2−1 for n even, while the relevant momentum kernels read [49] f whereŝ ij = s ij = k i k j , if i > j, and zero otherwise. Let us observe once again that KLT formulae are valid for all kinds of closed strings, Type II, Heterotic and Bosonic, at tree level and for any kind of insertions: tachyonic, mass-less or massive. Similar formulae relating string amplitudes with only massless insertions to SYM amplitudes [50; 51], see also [52], have been derived for open superstrings, whose validity we have given further support in [16]. MSST formulae read where the (n−3)! × (n−3)! dimensional matrices of generalised Euler integrals read Following the strategy outlined above, one can now combine the virtues of KLT and of MSST. For instance, at 5-points a closed (super)string amplitude with n massless and m = 5−n massive states, according to KLT, reads n+2m,0 and finally to A SY M n+2m is straightforward, but more and more cumbersome as the number of particles increases.

From open to closed via 'single-valued projection'
Although we will not fully exploit it in the following, an alternative and elegant expression of closed superstring amplitudes with massless insertions only in terms of SYM amplitudes at tree level has been found in [43; 44] that exposes the cancellation of various MZV (Multiple Zeta Values) including rational multiples of ζ 2n in the α ′ expansion.
The 'single-valued projection' formula reads 4 with θ σ (ρ(i), ρ(j)) = 1 if the ordering of (ρ(i), ρ(j)) is equal to the ordering of (σ(i), σ(j)) and zero otherwise. S 0 [ρ|σ] is the 'super-gravity' limit of the KLT momentum kernel such that sin(πα that appear in MSST formula. Not only all P 2n matrices drop but also higher depth MZV's do as a result of properties of the M 2k+1 matrices.

Soft limit from open to closed
When considering the soft behaviour of string amplitudes one may expect corrections from standard field theory results due to the non-minimal higher-derivative terms in the coupling among mass-less states as well as with massive states. For open strings we have checked that this higher-derivative couplings coded in the OPE of the vertex operators do not spoil universality of the soft behaviour at leading and sub-leading order. For completeness, let us now recall the argument [8; 9; 16]. The OPE of a massless vector boson vertex operator (in the q = 0 super-ghost picture) and a massive higher spin vertex opeartor (in the q = −1 super-ghost picture) reads where M ′ denotes any state at the same mass level as the state M . For totally symmetric tensors of the first Regge trajectory at level N = ℓ − 1 one has The leading term encodes minimal coupling. The sub-leading term is fixed by gauge invariance so that, barring some subtleties, to be dealt with momentarily, one gets for an amplitude with n massless and m massive states. Before generalising the above argument to the closed string case, let us deal with a couple of subtleties: the higher derivative terms in the tri-linear coupling A-H-H and the possible non-diagonal couplings A-H-H ′ that would spoil universality. First, higher derivative corrections to minimal coupling can only affect the sub-leading term that is fixed by gauge invariance wrt the soft gluon [10]. Second, for open superstrings already at the first massive level one finds two kinds of particles in the Neveu-Schwarz sector: C µνρ and H µν . In addition to the 'diagonal' couplings V -C-C and V -H-H (and SUSY related) one should consider the mixed coupling V -H-C ≈ α ′ M p 31 ·H 2 ·C 3 :[a 1 p 12 ] that exposes the singular soft factor 1/kp since M C = M H but gets suppressed by an extra power of the soft momentum in the numerator. Lacking the leading δ −1 term that fixes also the sub-leading δ 0 term, thanks to gauge invariance, this kind of higher derivative nondiagonal couplings can at most affect the sub-sub-leading δ +1 (and higher) terms which are not expected to be universal.
Relying on KLT, similar arguments were advocated to warrant universality of closed super-string amplitudes to leading, sub-leading and sub-sub-leading order [8; 9]. Indeed, the relevant OPE's of closed string vertex operators are simply the L+R combinations of the ones shown above for open strings. This implies that the leading behavior is completely fixed by the trilinear coupling. If this is minimal as for the superstrings one gets a universal behavior if it is not, as for the bosonic and heterotic strings one expects non universality or some sort of generalization thereof [11]. The additional ingredients are two. First, KLT formulae produce amplitudes with non-planar duality, with the soft graviton that can attach to each of the 'hard' (massless or massive) legs. Second, not only the sub-leading but also the sub-sub-leading term is fixed by gauge invariance of the soft graviton [10]. We would like to stress that this is true also for amplitudes with massive insertions as we will now sketch and check with explicit examples later on. Given universality 7 of the soft behavior of all open string amplitudes for granted [16] one schematically has One can easily check that S L/R is not universal, but conspires with the permutation to give something universal. We will limit ourselves to check cancellation of π 2 = 6ζ 2 and similar terms that are forbidden by the single-valued projection [43; 44]. At the cost of being pedantic we would like to reiterate that once the leading term is fixed and universal then sub-leading and sub-sub-leading terms follow thanks to gauge invariance of the soft graviton.

4-point amplitudes with massive states
Let us consider first 4-point amplitudes. We already know that allowing for a time-like p 1 , while we assume k 4 to be light-like and 'soft' with 'polarisation' E = a L ⊗ a R . From KLT we also know that where with universal is not universal. In D = 4 there is only one gauge invariant non vanishing derivative of f , i.e. u αūα (u β u γ ) or u αūα (ūβūγ) and W should reflect this structure (pretty much as J parallels f itself). The obvious guess is a mixed-symmetry tensor ('hook' Yang tableau) W [λ(µ]ν) = p λ ∂ 2 /∂p µ ∂p ν ± . . .. Moreover, it is worth to notice that the factor ζ 2 = π 2 /6 in Eq. (28) comes from the expansion of the beta function appearing in the open string disk amplitudes with four external legs. Combining the two amplitudes in Eq. (27), and using M 3 (123) = A L 3 (123)A R 3 (123) (up to an overall factor) as well as sin(πp 1 k 4 ) = πp 1 k 4 − π 3 (p 1 k 4 ) 3 /6 + . . ., we get Expanding at leading order yields and relying on momentum conservation, and on the standard trick we get Only the symmetric (not necessarily trace-less) part contributes, thus exposing the violation of the principle of equivalence in presence of a massless dilaton. At sub-leading order one has M 4 (1234) Expanding and combining the terms appearing in Eq. (35), one gets for the pole in p 1 k 4 depending on the 'symmetry' of E 4 . Moreover, for the pole in p 2 k 4 one gets where in the last step we used the angular momentum conservation (J L 1 + J L 2 + J L 3 )A 3 (123) = 0. For the pole in p 3 k 4 one gets the same result mutatis mutandis.
At sub-sub-leading order one has many terms After lengthy manipulations one reproduces where k 4 J 1 E 4 J 1 k 4 = J 1 R 4 J 1 involves the linearised Riemann tensor, and thus it is manifestly gaugeinvariant. The π 2 factor form the expansion of the KLT kernel at 4-point cancels exactly the ζ 2 appearing in the expansion of the open string amplitudes, thus implementing the single-valued projection discussed in Sec. 2.4.

5-point amplitudes with massive states
where we assume that k 2 1 = 0 (massless graviton) goes soft, k 1 = δk 1 , with δ→0. In this limit, we know that Observing that At leading order, Eq. (43) yields At sub-leading order At sub-sub-leading order where the ζ 2 factors coming from the KLT kernel cancel exactly those produced by the expansion at the sub-sub-leading of the 5-point disk integral, as encoded by the single-valued projection.

6-and higher-point amplitudes with massive states
Lastly, let us briefly focus on 6-point amplitudes. In this case one has twelve terms At leading order, we get exposing the expected universal terms at leading order, where non-planarity is restored by summing over permutations in KLT or 'single-valued map' formulae. Sub-leading and sub-sub-leading are more laborious but are fixed by gauge invariance, as repeatedly discussed above.

Closed superstring amplitudes with massive insertions
In this section we compute some amplitudes with insertions of massive string states. Later on we will examine their soft behavior.
Let us now consider closed superstrings and focus on the NS-NS sector. At the first massive level one finds a plethora of particles (all in all 2 14 = 128 × 128 = (44 + 84) × (44

Three massless states one massive:
Relying on KLT formulae one has with K + L + U = H ⊗H + C ⊗C + H ⊗C + C ⊗H. The highest spin state is the Konishi top state with s = 4 [53][54][55][56]. In D = 10 the explicit formula is extremely long and not very illuminating. We refrain for writing it down except for L = C ⊗C, whereby it reads We shall also study the soft behavior of the amplitude M(E 1 , E 2 , E 3 , K 4 ). It is worth to notice that K 4 = H ⊗ H is a reducible tensor. The following decomposition holds In particular, this product contains the 10-dimensional analogue of a spin 4 state In D = 4 the situation drastically simplifies. Focussing on the combinations of the SO(6) singlets H µν = H tt µν + H 0 (η µν + α ′ p µ p ν ) (with H ij = −H 0 δ ij /2) and C µνρ = C 0 √ α ′ p λ ε λµνρ that couple to two gluons, one has 49 d.o.f. that assemble in five scalars, one vector, five spin-2 (5 states each), one spin 3 (7 states) and one spin 4 (9 states). Since the H tt µν couples to gluons with opposite helicity while H 0 /C 0 couple to gluons with the same helicity, the open-string building blocks are

11
The amplitudes for the lower spin components of K follow performing SO(3) little group transformations on the above one. Similarly one can replace two gravitons with dilatons or axions Once again, amplitudes for the other helicity states of K obtain after SO(3) little group transformations. Note, for instance, that K +++0 The former is even under Ω, the latter is odd.
One can also consider the 4 real (2 complex) s = 2 massive states corresponding to H 0 /C 0 ⊗H 2 ± H 2 ⊗H 0 /C 0 whose amplitudes with massless states obtain from combinations of Finally amplitudes for the four scalars H 0 with each other and with permutations thereof.

Three-tachyons one-massless:
Consider now also the insertion of generic massless closed string states with k 2 = 0 where E µν is transverse with respect to both indices k µ E µν = 0 = k ν E µν . Decomposing E µν = h µν +φ µν +b µν into irreducible representations of the Lorentz group, h µν = h νµ with η µν h νµ = 0 describes the graviton, φ µν = η µν − k µkν − k νkµ withkk = 0 and kk = 1 describes the dilaton and b µν = −b νµ the Kalb-Ramond field. Consider the amplitude: Since the amplitude reads as One concludes that only the symmetric part of E S = 1 2 (E 3 + E t 3 ) contributes due to symmetry under world-sheet parity Ω, under which h and φ are even while b is odd.
The Γ functions in the above expression can be rearranged as where so much so that as expected.

Two-tachyons one-massless one-massive:
Using KLT in the s-channel (1-2 exchange) one finds or more explicitly where E 3 = a 3 ⊗ã 3 and K 4 = H ⊗H. Without much effort one can check that E ± = a ⊗ã ±ã ⊗ a with definite parity under Ω couple to K ± = H ⊗H ±H ⊗ H with the same parity.

Soft limit of closed string amplitudes with massive insertions
In this section, we study the soft limit of 4-point amplitudes with massive insertions. We start with the superstring and focus on the D = 4 case where the spinor helicity formalism largely simplifies the results. We then pass to consider the bosonic strings and study tachyon insertions, too. Finally we investigate the soft behaviour for amplitudes with two Kalb-Ramond fields.

Soft limit of superstring amplitudes in the spinor helicity formalism
Restring the momenta and polarisations to D = 4 allows us to derive compact expressions for the universal soft operator in the spinor helicity formalism. For simplicity we focus on 4-point amplitudes with three massless and one massive external legs. In particular, we will consider the soft limit of the amplitudes in Eqs. (54), (55), and (55), computed using KLT. When the graviton with helicity h = +2 and momentum k 3 goes to zero, we find 14 Applying the operators S i , i = 0, 1, 2 to the amplitudes we reproduce the soft expansions found respectively in Appendix A.
Had we chosen the leg with momentum k 1 to be soft in Eq. (54), we would have gotten a trivial result, since the interaction vertex vanishes M 3 (E +2 2 , E +2 3 , K +4 4 ) = 0. While our results are symmetric in the exchange of 2 ↔ 3, when the external leg with momentum k 2 is a graviton.

Soft limit of bosonic string amplitudes
The simplest case to be considered is the amplitude with three tachyons and the one graviton M 4 (T 1 , T 2 , E 3 , T 4 ) The dynamical factor in the above expression has a very special soft behavior Eq. (81) does not spoil the soft behavior of the amplitude up to the sub-sub-leading order. This happens every time the dynamical factor depends on the soft momentum as in Eq. (81) and in all cases we are going to study we will always extract this factor. At this stage, the expansion of the amplitude yields Which agrees with the expected soft behavior since the three amplitude M 3 (T 1 , T 2 , T 4 ) is just a number, so the action of the angular momentum operator gives zero.
When E 1 = h 1 /φ 1 and E 2 = h 2 /φ 2 the soft theorem would suggest the following expansion for the amplitude in Eq. (67) (83) Since Kalb-Ramond b-fields are odd under world-sheet parity we would expect zero because M 3 (b 2 , T 3 , T 4 ) = 0. Following the steps reported in Appendix A.2 we find that at the sub-sub-leading order the soft behavior of the amplitude is not reproduced by the soft operator S 2 . In particular, there are additional terms that we expect coming from the M 3 (h 1 , h 2 , φ I ) vertex, Eq. (A.38). For two Kalb-Ramond fields E 1,2 = b 1,2 the amplitude at leading order O(δ −1 ) is zero. The expansion starts at order O(δ 0 ) It is worth to notice that there are only poles in k 1 k 2 , as expected since M 3 (b, T , T ) = 0 due to worldsheet parity. One can try to interpret the soft result as a factorization on the massless pole viz.
where e(k) collectively denotes the physical polarisations of the graviton and dilaton e µν = h µν + φ µν . Alternatively, since 2k 1 k 2 = −2(k 1 + k 2 )p 3 = −2(k 1 + k 2 )p 4 , one can envisage a 'double soft limit', see e.g. [57][58][59], where our present computations suggest Following the steps outlined in Appendix A.3, we reproduce the leading and sub-leading behavior as predicted by the soft theorem, but not the sub-sub-leading order. As for the amplitude M 4 (E 1 , E 2 , T 3 , T 4 ) we are led to think that the mixing with the other degenerate string states spoil the soft theorem statement at this order.

Conclusions and outlook
We have extended our analysis of the soft behaviour of string amplitudes with massive insertions to closed strings. Relying on our previous results for open strings and on KLT formulae we have checked universality of the soft behaviour to sub-leading order for superstring amplitudes. At sub-sub-leading order we have argued in favour of universality on the basis of OPE of massless and massive vertex operators and gauge invariance with respect to the soft gravitons. We have also checked our statements against explicit 4-point amplitudes with one massive insertion in any dimension, including D = 4, where use of the helicity spinor formalism drastically simplifies all expressions. As a by-product of our analysis we have checked the cancellation of π 2 arising from sin(πα ′ c k i k j ) factors in KLT formula with those arising from open superstring amplitudes in the soft limit, at sub-sub-leading order. This is expected for the 'single valued projection' advocated in [43; 44] to hold for massive amplitudes, too. This is comforting, being closed string theory of quantum gravity. Yet, our results are only valid at tree level and the proper extension to one-and higher-loops is still under debate in that IR divergences seem to produce non-universal log δ terms [60] even in N = 4 SYM at one-loop, let alone supergravity or superstring theories. It would be very interesting to investigate this subject along the lines of [37; 61] and establish whether log δ terms exponentiate, as usual for IR divergences, and in case which would be the relevant 'anomalous' dimension that governs this hopefully universal behaviour. The approach proposed in [62; 63] based on the second Nöther theorem seems promising in this respect, though so far shown to be valid only at tree level.
Here we give the result of the expansion to be compared with the predictions dictated by the soft theorem. It is convenient to factor out the structure in Eq. (81), which has a trivial soft behavior, from the dynamical term in Eq. (68) I(s, t, u) = − k 1 p 3 k 1 p 4 (1 − k 1 k 2 ) 2 (1+O(δ 3 )) = −k 1 k 2 k 1 p 3 k 1 p 4 1 k 1 k 2 +2+3k 1 k 2 +O(δ 3 ). (A.14) The expansion of the kinematical structure E 1 KE t 2 K t can be organized as follows The expansion of the amplitude up to O(δ) yields