Double-soft behavior for scalars and gluons from string theory

We compute the leading double-soft behavior for gluons and for the scalars obtained by dimensional reduction of a higher dimensional pure gauge theory, from the scattering amplitudes of gluons and scalars living in the world-volume of a Dp-brane of the bosonic string. In the case of gluons, we compute both the double-soft behavior when the two soft gluons are contiguous as well as when they are not contiguous. From our results, that are valid in string theory, one can easily get the double-soft limit in gauge field theory by sending the string tension to infinity.


Introduction and results
Soft theorems were in the 1950s and 1960s studied for photons [1,2,3,4] and gravitons [3,5,6], when they were recognized to be important consequences of local gauge invariance 1 . More recent discussions of the generic subleading behavior of soft gluons and gravitons were given in Refs. [8,9]. The very recent interest in the soft behavior of gravitons and gluons has arisen after work by Strominger and collaborators [10], showing that the soft-graviton behavior follows from Ward identities of extended Bondi, van der Burg, Metzner and Sachs (BMS) symmetry [11,12], and that tree-level graviton amplitudes in four spacetime dimensions have a universal behavior through second subleading order in the soft-graviton momentum [13]. This has stimulated the study of the subleading soft behavior in amplitudes with gluons and gravitons, which we will briefly summarize.
For gluons in arbitrary number of dimensions soft theorems were obtained in various ways at tree level [14,15,16,17,18], and discussed for quark-gluon amplitudes in QCD in Ref. [19]. Poincaré and gauge invariance as well as a condition arising from the distributional nature of scattering amplitudes have been used in Ref. [20] to strongly constrain the soft behavior for gluons and gravitons, while in Ref. [21] gauge invariance is shown to completely fix the first two leading terms (up to terms O(q 0 )) in the case of a gluon, and the first three leading terms (up to terms O(q 1 )) in the case of a graviton, for any number of space-time dimensions (q being the soft momentum).
Further study of the subleading soft-graviton theorems in arbitrary number of dimensions were performed in Refs. [22,23,24,25,26,27,28]. Soft theorems were also studied in the ambitwistor string [29,30], and their connection with effective field theories was discussed in Ref. [31]. As discussed in Refs. [32,33,21,34], one should note that soft gluon and graviton behaviors are in general modified by loop corrections.
Finally, soft gluon and graviton behavior have also been studied in the framework of string theory in Refs. [17,35,36,37]. In particular, soft theorems for the dilaton were discussed in Refs. [38,39,40,41,37] and for the anti-symmetric tensor in Ref. [37], while soft behavior in amplitudes with massive states was discussed in Ref. [42].
Double-soft theorems are now receiving increasing interest, and have been studied for scalars and gluons. The interest in the double-soft behavior arises originally from the analysis of the spontaneously symmetry breaking of a group G keeping a subgroup H unbroken. The presence of Nambu-Goldstone bosons, living in the coset space G/H, implies the vanishing of the scattering amplitude with a single soft Goldstone boson, giving rise to the famous Adler's zeros, while leading to a peculiar nonzero universal behaviour of the amplitude with two soft Goldstone particles [43,44]. The double-soft theorems for different kinds of scalars were computed in Ref. [45] using the CHY representation of the scattering amplitude [46]. The double-soft theorems for gluons were computed in Refs. [48,47] in four dimensions using the spinor helicity formalism and the BCFW recursion relations, and this result has since been confirmed in Ref. [47] by a calculation in an arbitrary number of spacetime dimensions, using the CHY representation of the scattering amplitude. Ref. [47] also uses superstring theory obtaining agreement with field theory results without extra string corrections. Finally, in Ref. [49] the so-called CSW method is used to compute the double-soft behavior in four dimensions.
In this work we compute the leading double-soft limit for gluons and scalars directly on the scattering amplitudes for gluons and scalars in the bosonic string. The corresponding result in the limiting field theory is then obtained by performing the field theory limit (α → 0). Our results are valid for any number of space-time dimensions and without fixing a particular gauge. In particular, we consider scattering amplitudes of massless open strings on a Dp-brane in the bosonic string and use them to derive double-soft theorems for both gauge fields and scalars. In fact, in the presence of a Dp-brane, the original Lorentz symmetry SO(1, d − 1) is broken into SO(1, p) ⊗ SO(d − 1 − p) and the original d-dimensional gauge field gives rise to a (p + 1)-dimensional gauge field that we in the following, for the sake of simplicity, call gluon, and d − p − 1 massless scalars 2 . They all live on the (p + 1)-dimensional world-volume of the Dp-brane.
We start by computing the color-ordered amplitude with (n + 2) gluons. We keep only the leading term when two contiguous gluons with momenta q 1 and q 2 become simultaneously soft, while the momenta k 1 . . . k n of the other gluons are kept finite. In this case, where the gluons are ordered as k 1 . . . k n−1 q 1 q 2 k n , we find that the leading term in the double-soft limit behaves as 1 q 2 1,2 and is given by: where g p+1 is the gauge coupling constant of the gauge theory living on the Dp-brane and is the scattering amplitude of n gluons in the bosonic string. As we explicitly show in the next sections the dependence on α in the soft factor drops out, while it is still present in the n-gluon amplitude. The field-theory limit, corresponding to α → 0, leaves the soft factor unchanged and acts only on M ng giving then just the n-gluon amplitude of Yang-Mills theory. Notice also that, unlike the case of a single soft behavior, the double-soft behavior cannot be written as the difference of two terms, one depending on k n−1 and the other on k n , because of the presence of the last term in Eq. (1.1) that contains both momenta.
We then consider the double-soft limit for two identical scalars in an amplitude with n gluons and we get the following double-soft behavior: where M ng is again given in Eq. (1.2). Finally, we get the same double-soft behavior for a scattering amplitude with n + 2 identical scalars when two contiguous scalars become soft: where M ns is the n-scalar amplitude in the bosonic string.
The three previous formulas are valid in an arbitrary (p + 1)-dimensional space-time and all follow from a d-dimensional gauge theory that, because of the presence of a Dpbrane, gives rise to a gauge theory in a (p+1)-dimensional space-time coupled to (d−p−1) scalar fields, which correspond to the components of the original d-dimensional gauge field along the directions outside the world-volume of the Dp-brane. Because of this, Eqs. (1.3) and (1.4) follow from Eq. (1.1) by neglecting the terms where the polarizations of the soft gluons are contracted with the momenta of all particles. This is a consequence of the fact that the scalars correspond to the components of the gluons in the directions orthogonal to the Dp-brane.
It is also interesting to note that the previous three soft amplitudes all involve the factor in the squared bracket of Eqs. (1.3) and (1.4) that is typical of the double-soft limit of Goldstone bosons [43]. Unlike the case of Goldstone bosons, however, in this case we get an additional singular factor 1 q 1 ·q 2 .
We have also computed the leading double-soft behavior of a color-ordered amplitude with n+2 gluons in the case where the two soft gluons are not next to each other as before, but have a hard gluon between them, which we take to be the one with momentum k n−1 . When the gluons are ordered as k 1 , k 2 . . . q 1 k n−1 q 2 k n , the leading double-soft behavior is given by Finally, for the color-ordered amplitude where the two soft gluons have more than one hard gluon between them, the double-soft behavior is just given by the product of two single-soft behaviors.
The paper is organized as follows. In Sect. 2 we discuss the double-soft limit for gluons in an amplitude with n gluons. Sect. 3 is devoted to the double-soft limit for two scalars in an amplitude with n gluons, while in Sect. 4 we discuss the leading double-soft limit for two scalars in an amplitude with n scalars. The last section is left for conclusions. In the Appendix we compute the double-soft limit of some double integrals used for computing the double-soft limits.

Double-soft behavior with n + gluons
In this section we consider the color-ordered scattering amplitude involving (n + 2) gauge fields living on the world-volume of a Dp-brane of the bosonic string and we compute the leading double-soft behavior when two contiguous gluons become simultaneously soft.
We denote with ( q 1 , q 1 ) and ( q 2 , q 2 ) the polarizations and momenta of the gluons that eventually will become soft and with ( i , k i ) the polarizations and momenta of the remaining gluons. We consider the color-ordered amplitude corresponding to the following permutation: k 1 , k 2 , . . . k n−1 , q 1 , q 2 , k n for which the Koba-Nielsen variables of the various gluons are ordered as z 1 ≥ z 2 ≥ z 3 · · · ≥ z n−1 ≥ w 1 ≥ w 2 ≥ z n .
It is convenient to write the amplitude with (n+2) gluons by exponentiating the derivative part of the vertex operators by introducing for each vertex operator a Grassmann variable, called θ i , i = 1 . . . n and φ a , a = 1, 2, and by writing the scattering amplitude as follows 3 : where g p+1 is the (p + 1)-dimensional gauge coupling constant 4 . Using the contraction we can contract the vertex operators of the states with momenta q 1 and q 2 , with those of the other states getting We can now perform the Grassmann integrals over φ 1 and φ 2 arriving at where by * a convolution of the integrals is understood, is the scattering amplitude of n gluons, and This expression contains two kinds of terms. The first one is without any dependence on the variables θ i and acts in the convolution at the end of Eq. (2.4) as a factor that multiplies the n-gluon amplitude M ng , while the second one contains the terms with θ i that, when acting in the convolution in Eq. (2.4), modify the structure of the n-gluon amplitude M ng . It can be shown that the first kind of terms is the leading one in the double-soft limit, while the second one is subleading. In this paper we compute only the leading one that is equal to It is convenient to introduce two new variables z n−1ŵa = w a for a = 1, 2. We get We want to extract from the previous sum of integrals the most singular term in the double-soft limit. In this limit, we can approximate z 2α [q 1 q 2 +(k n−1 +kn)(q 1 +q 2 )] n−1 with 1. The double-soft behavior of the various integrals is computed in Appendix A. Here we give only the final result: In conclusion, the leading double-soft behavior is given by: which behaves as 1 q 1 q 2 in the double-soft limit, i.e. when both q 1 and q 2 simultaneously go to zero. M ng is the n-gluon amplitude in the bosonic string. We see that, in the double-soft limit, the dependence on α of the soft factor in Eq. (2.9 ) cancels, while it is still kept in the n-gluon amplitude in Eq. (2.5).
One can check gauge invariance of the soft factor in Eq. (2.9 ) by substituting q 1 with q 1 . One gets: The original amplitude behaves as 1 . Saturating with q 1 , we find that the terms of order 1 q 1,2 cancel and we are left with a term of order q 0 1,2 that should be cancelled by the next to the leading term, which we have not yet computed.
One can also check gauge invariance with respect to the other soft particle by substituting q 2 with q 2 . One gets: which is equal to the expression in Eq. (2.11) by the substitutions k n−1 ↔ k n and q 1 ↔ q 2 .
In order to see the pole structure of the amplitude, it is convenient to write Eq. (2.10) as follows: This expression shows a non-trivial pole structure and the various pole singularities are all consistent with the structure of a color-ordered amplitude. In Fig. (1 [47] in the gauge chosen there, namely ( q 2 k n ) = ( q 1 k n−1 ) = 0.
In the final part of this section we discuss the case of a color-ordered amplitude where the two soft gluons are not next to each other. In particular, we consider the color-ordered amplitude where a hard gluon, say the one with momentum k n−1 , is between the two soft. The gluons are then ordered as k 1 k 2 . . . k n−2 q 1 k n−1 q 2 k n . The complete amplitude is equal to M 2g;ng = (2α ) (2.14) Since we are interested in the leading double-soft behavior, we can neglect the terms with θ i in the last four lines of the previous expression and keep in the curly bracket only the terms with momenta. It can be seen that the leading double-soft behavior is obtained by restricting ourselves to the following expression: In order to study the double-soft behavior of the four terms that appear in the last line of the previous equation, it is convenient to go to the variables x i with i = 1, 2, 3 that run from 0 to 1: The Jacobian of the transformation is equal to z 3 n−2 x 2 1 x 2 and the last three lines of Eq.
(2.15) become: Let us consider the first term of the square bracket in the last line of the previous equation. The leading double-soft limit is obtained for x 1 ∼ 1 and x 3 ∼ 1 and x 3 ∼ 0 for the two terms in the second square bracket, respectively. In this corner of the integration region z n−1 ≡ x 2 z n−2 and therefore we get where we have used that, for x 2 ∼ 1, we can write z n−1 = z n−2 x 1 . When multiplied with the first line of Eq. (2.15), one gets again the soft term in Eq. (2.19) times the amplitude with n gluons.
The leading double-soft behavior of the last term is a bit more complicated to extract. The relevant terms of Eq. (2.17) that one needs to consider, are: The dominant contribution comes from the region of the integrals around x 2 ∼ 1 and x 3 ∼ 1. It is convenient to first change variables to y i = 1 − x i for i = 2, 3 and then to y 2 = tu and y 3 = t(1 − u) with Jacobian equal to t. Then, Eq. (2.20) becomes where we have neglected irrelevant factors that go to 1 in the double-soft limit and, by introducing a cutoff , we have restricted the region of integration for t to a small interval around t = 0. Eq. (2.21) is equal to where, for x 2 ∼ 1, we have used that z n−1 = z n−2 x 1 . When we insert the result in Eq.
(2.22) in Eq. (2.17), we get again the double-soft factor times the amplitude with n gluons.
In conclusion, the double-soft behavior is given by the sum of the previous three terms: It is easy to check that the soft factor is gauge invariant up to terms of order q 0 1,2 as in the case of two contiguous soft gluons.
Finally, the double-soft behavior when the two soft gluons are separated by more than one leg is just the product of two single soft behaviors.

Double-soft behavior with 2 scalars and n gluons
In this section we compute the double-soft behavior of two scalars in an amplitude with n gluons. The amplitude with two scalars and n gluons can be obtained from that computed in the previous section by noticing that, in this case, we have to impose: We then get where M ng is given in Eq. (2.5) and

Double-soft behavior with n + 2 scalars
In this section we compute the double-soft behavior of two scalars in an amplitude with n additional scalar particles instead of gluons. The set-up and the procedure is as in the section for gluons. The scattering amplitude is given by: Contracting the vertex operators of the states with momenta q 1 and q 2 and remembering that, in this case, all momenta are orthogonal to all polarizations, we get We can now integrate over φ 1 and φ 2 M 2s;ns = g n−2 and (4.5) Similar to the case of gluons, we keep only the leading terms in the double-soft limit. It is also convenient to introduce the new variables w 1 = z n−1ŵ1 and w 2 = z n−1ŵ2 . Then, the previous expression becomes (4.6) Also in this case we have two kinds of terms. The first one is without any dependence on the variables θ i and the convolution multiplies the amplitude with n scalars, while the second term contains terms with θ i that modify the structure of M ns . Also in this case it can be shown that only the first term contributes to the leading double-soft limit. Therefore, here we concentrate on the first term taking the term z 2α [q 1 q 2 +(kn+k n−1 )(q 1 +q 2 )] n−1 = 1. We get the integral in Eq. (A.1) that has been computed in Eq. (A.25) and therefore for the first term in the squared bracket in Eq. (4.6), one gets: This means that the double-soft behavior in an amplitude with n + 2 scalars is given by 2q 1 q 2 k n (q 2 − q 1 ) + q 1 q 2 k n (q 1 + q 2 ) + q 1 q 2 + k n−1 (q 1 − q 2 ) + q 1 q 2 k n−1 (q 1 + q 2 ) + q 1 q 2 M ns . (4.8)

Conclusions
In this paper we have computed the leading double-soft behavior of the scattering amplitudes with gluons and scalars living in the world-volume of a Dp-brane of the bosonic string. The corresponding field-theory results can be obtained by just sending α to zero. Our results are valid in any number of space-time dimensions and for an arbitrary gauge choice. We have also for the first time considered and provided the results for the doublesoft limit of two gluons that are not contiguous, but separated by a hard gluon in the color-ordered amplitude.

A Computation of various integrals to the leading order in the double-soft limit
In this Appendix we compute the contribution of various integrals in the double-soft limit.
Let us start considering the first integral appearing in Eq. (2.8): and let us show that it is symmetric under the simultaneous exchange of k 2 ↔ k 3 and q 1 ↔ q 2 . It is convenient to introduce the variables: The integral becomes: It is possible to give an equivalent expression of the integral (A.1): Introducing again the variables in Eq. (A.2), we get Both representations can be collected in the formula: Let us now keep the variable x 2 and solve the delta function by eliminating either x 3 or x 4 . By eliminating x 3 = 1 − x 2 − x 4 one gets: Introducing the variables x 2 = ρω ; x 4 = ρ(1 − ω) one gets: By changing variable t = 1 − ω, one gets: Let us now eliminate instead x 4 = 1 − x 2 − x 3 . One gets: By defining again x 2 = ρω ; x 3 = ρ(1 − ω) one gets: This equation is obtained from Eq. (A.8) by changing q 1 ↔ q 2 and k 2 ↔ k 3 and we get I(k 2 , q 1 ; k 3 , q 2 ) = I(k 3 , q 2 ; k 2 , q 1 ) (A.12) Let us now compute it in the double-soft limit. Let us first change variables w 1 = z and w 2 = xw 1 in Eq. (A.1) getting Then, we use the following equation to rewrite Eq. (A.13) as follows Then we can use to get It is equal to Introducing, for the sake of simplicity, the momentum k 1 = −k 2 − k 3 − q 1 − q 2 , yields Keeping only the term with 1 in the exponent of the last term, we can easily compute the two integrals. One gets: Γ(2α (k 3 (q 1 + q 2 ) + q 1 q 2 ))Γ(2α (k 2 (q 1 + q 2 ) + q 1 q 2 )) Γ(2α ((k 3 + k 2 )(q 1 + q 2 ) + 2q 1 q 2 )) × Γ(−2α k 2 q 2 )Γ(−2α q 2 k 1 ) Γ(−2α q 2 (k 1 + k 2 )) − Γ(2α (k 3 (q 1 + q 2 ) + q 1 q 2 ) + 1)Γ(2α (k 2 (q 1 + q 2 ) + q 1 q 2 )) Γ(2α ((k 3 + k 2 )(q 1 + q 2 ) + 2q 1 q 2 ) + 1) It can be rewritten as follows that is equal to In the double-soft limit one gets: Because of the symmetry in Eq. (A.12), we must symmetrize the previous expression getting: In the second part of this section we compute the remaining nine integrals in Eq. (2.7) in the double-soft limit (q 1 and q 2 going simultaneously to zero).
The term with the factor (1 − w 1 )w 2 in the denominator is given by where we have kept only the first term (1) of the expansion of the last term in the previous equation. The last integral to compute is the one with the terms w 1 (1 − w 2 ) in the denominator: (1 − w 2 ) 2α q 2 k 2 w 2α q 2 k 3 2 = 1 0 dw 1 w 1 0 dw 2 (1 − w 1 ) 2α q 1 k 2 w 2α q 1 k 3 −1 1 (w 1 − w 2 ) 2α q 1 q 2 ×(1 − w 2 ) 2α q 2 k 2 w 2α q 2 k 3 2 q 2 · k 3 q 2 k 2 w −1 2 − q 1 q 2 q 2 k 2 (w 1 − w 2 ) −1 = 0 (A. 38) as one can see from Eqs. (A.34) and (A.26). Finally, the term involving I w 1 −w 2 ;w 1 −w 2 is directly seen to be subleading in the double-soft limit, due to the explicit factor of q 1 q 2 that multiplies this integral, and is thus not needed.