Soft theorems in maximally supersymmetric theories

In this paper we study the supersymmetric generalization of the new soft theorem which was proposed by Cachazo and Strominger recently. At tree level, we prove the validity of the super soft theorems in both N=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{N}=4$$\end{document} super-Yang-Mills theory and N=8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{N}=8$$\end{document} supergravity using super-BCFW recursion relations. We verify these theorems exactly by showing some examples.


Introduction
Over the past few decades, there was great progress on the understanding of physical and mathematical structures of the perturbative scattering amplitudes in gauge and gravity theories. Among these remarkable developments, a recent advance is the soft behavior of scattering amplitudes when one or more external legs tend to zero in both gravity and Yang-Mills .
The study on soft behavior of the amplitudes (in gravity) goes back to Steven Weinberg who proposed the universal leading soft-graviton behavior using the Feynman diagrams technique more than 50 years ago [28]. The leading softgraviton behavior of the amplitudes, also called "Weinberg's theorem", is uncorrected to all loop orders. At sub-leading order, the soft photon behavior and the soft graviton behavior were also studied by using Feynman diagrams [29][30][31].
On the other hand, a lot of remarkable progress has also been made on the understanding of the soft theorem from the viewpoint of symmetry principles.
The leading and sub-leading soft graviton theorems are understood as Ward identities of BMS symmetry [25][26][27]. The leading [18] and sub-leading [17] soft photon theorems were interpreted as asymptotic symmetries of S-matrix in massless QED. Now aspects of soft theorems have been investigated along many different directions . Loop corrections of soft theorems were investigated in both gravity and Yang-Mills [2][3][4][5]. Various different methods were used to derive the soft theorems, including Feynman diagram approach [8] and conformal symmetry approach [16] for Yang-Mills, the Poincaré symmetry and gauge symmetry approach [6,7], ambitwistor string approach [15] for Yang-Mills and gravity. The new soft graviton theorem was also obtained from the soft gluon theorem via the Kawai-Lewellen-Tye (KLT) relations [19]. The scattering equations [42][43][44][45], or Cachazo-He-Yuan (CHY) formulae, were used to study the soft theorems in arbitrary number of dimensions. The soft divergence in string theory was also investigated in [4,14].
These previous works provide a solid foundation for our study.
We will systematically study the soft theorems in N = 4 SYM and N = 8 SUGRA using super-BCFW recursion relations in this work. We will pay special attention to soft gravitino divergence, soft gravi-photon divergence for SUGRA and soft gluino divergence for SYM. This paper is organized as follows. In the next section, we briefly review soft theorems in both gravity and Yang-Mills. In Sect. 3, we present the super soft theorem in N = 4 super-Yang-Mills with a rigorous proof in detail. There we also provide some lower-point amplitudes to test the validity of the super soft theorem, especially for soft gluino divergence.
In Sect. 4, we study the super soft theorem in N = 8 supergravity in detail. As examples, we show that the super soft theorem is consistent with SUSY Ward identity in the MHV sector. Several four-boson amplitudes are also examined exactly.
In Sect. 5, we conclude this paper with some brief discussions. Appendix A provides an alternative derivation of soft theorem in N = 4 SYM. Appendix B gives calcula-tional details of the sub-sub-leading soft operator in N = 8 SUGRA.

New soft graviton theorem and soft gluon theorem
In this section we briefly review the new soft graviton theorem [32] and the soft gluon theorem [1,71]. We also show that each order soft operator in gravity may be expressed as a double copy of Yang-Mills soft operators with gauge freedom.

Cachazo-Strominger's new soft graviton theorem
An on-shell (n + 1)-point scattering amplitude including an external graviton with momentum k s may be denoted (2.1) In the soft limit k s → 0, the amplitude M n+1 behaves as The soft operators are given by 3) where E μν is the polarization tensor of the soft graviton s and J μν a is the total angular momentum of the ath external leg. It is easy to check that all the soft operators are gauge invariant [32]. The leading soft factor S (0) proposed by Steven Weinberg is uncorrected by all loop orders [28] while sub-leading and sub-sub-leading soft operators are not, as discussed in [2][3][4][5][6].
In the spinor-helicity formulism, the momentum vector k μ of an on-shell massless particle may be represented as a bispinor, i.e., (2.6) Introducing an infinitesimal soft parameter , one can write soft limit of the momentum k s of soft particle as Here different choices of δ in physical amplitudes can be linked to each other via the little group transformation, i.e., where h s is helicity of the particle s. In this paper, one employs the holomorphic soft limit [56]: in which only the holomorphic spinor λ s tends to zero while the anti-holomorphic spinorλ s remains unchangeable. The new soft graviton theorem (2.2) is then In the spinor-helicity formulism, 1 the soft operators are given by Here one has assigned soft graviton the helicity h s = +2, just a convention. Spinors λ x , λ y are two arbitrary choosen reference spinors and the freedom in this choice is equivalent to the gauge freedom.

Soft gluon theorem in Yang-Mills theory
The similar soft behavior of the scattering amplitudes appears also in Yang-Mills theory [1,71]. In the soft limit of the momentum of a gluon, k s → 0, an on-shell color-ordered Yang-Mills amplitude A n+1 becomes where the leading soft (eikonal) factor [71] is 1 In this paper, we mainly follow the notation of ref. [32]. The spinor products are defined as i, j = αβ λ iα λ jβ = λ iα λ α j and [i, j] = αβλ iαλ jβ =λ iαλα j , and we use the convention (2.15) while the sub-leading soft operator is given by (2.16) with E μ the polarization vector of soft gluon.
In the spinor-helicity formulism, employing the holomorphic soft limit (2.9), the soft gluon theorem (2.14) may be expressed as Taking the helicity of the soft gluon h s = +1 as a convention, the soft operators may be written as with λ x , λ y arbitrary choosen reference spinors and the freedom in this choice is equivalent to the gauge freedom.
It is important to note that two amplitudes in the soft theorem are both unstripped. In other words, amplitudes A n+1 and A n in Eq. (2.17) contain respective momentum conservation delta functions. With this in mind, we can remove dependence of anti-holomorphic spinorsλ 1 andλ n in these two amplitudes by imposing momentum conservation delta functions appropriately. This implies that the sub-leading soft divergence vanishes in color-ordered Yang-Mills amplitudes [4].
As we will see immediately, however, that the operator S (1) YM is necessary for constructing gravity soft operators from Yang-Mills soft operators.

Gravity soft operators as double copy of Yang-Mills soft operators
There exists a remarkable relation between gravity amplitudes and Yang-Mills amplitudes. At tree level, Kawai, Lewellen and Tye (KLT) found that one can express a closed string amplitude as a sum of the square of open string amplitudes [72]. In field theory limit, this relation expresses a gravity amplitude as a sum of the square of color-ordered Yang-Mills amplitudes. The similar relation also exists between gravity soft operators and Yang-Mills soft operators. In [3], the gravity soft operators were expressed as a double copy of Yang-Mills soft operators with a special gauge choice which associated with the special choice of shifted external legs in BCFW recursion. In this subsection, we rewrite this relation with gauge freedom. First of all, for the sake of convenience, introduce two notations 2 which are the fundamental building blocks for constructing gravity soft operators. Employing these notations, one can write the Yang-Mills soft operators as Let us note a simple relation that expresses a graviton polarization tensor as the product of gluon polarization vectors with same momentum, i.e., Here E μν have been written in a symmetric form. By making use of this relation, the leading soft operator in gravity can be written as: where s sa = 2k s · k a = s, a [s, a] and the λ x and λ y are arbitrary reference spinors and the freedom in this choice is equivalent to the gauge freedom. This relation was presented in [73][74][75] and derived in [19,76] by KLT realtion [72,77,78]. The sub-leading soft graviton operator can be expressed as This relation with a special gauge choice was derived in [19] by KLT realtion [72,77,78]. 2 Here the first notation S 0 is just famous eikonal factor in Yang-Mills amplitudes [39,73,74] and the 'x' denotes the reference spinor λ x in spinor representation E + αα (λ s ,λ s , λ x ) of the polarization vector E + μ [40].
Similarly, the sub-sub-leading soft operator may be written as It is important to notice that here the operator product S 1 (s, a)S 1 (s, a) should be understood as: In another words, the differential only acts on the amplitudes.

Soft theorem in N = 4 super-Yang-Mills theory
We turn to study the soft theorems in supersymmetric theories. In this section, we present the soft theorem in N = 4 super-Yang-Mills theory with a rigorous proof at tree level. We also give some lower- The N = 4 on-shell superfield can be expanded as follows [79]: Here Grassmann odd variables η A transforms in a fundamental representation of the SU(4) R-symmetry.
In super-momentum space, a color-ordered superamplitude is a function of spinors λ a ,λ a (or momentum p a ) and Grassmann variables η a , i.e., The component field amplitudes are then obtained by projecting upon the relevant terms in the η i expansion of the superamplitude. For detail, see [54][55][56].

Super soft theorem in N = 4 SYM
Here we derive the soft theorem in N = 4 SYM with the help of super-BCFW recursion relation [34,35,56] in the spinorhelicity formulism [39,41,80]. Let us choose the soft particle and its adjacent particle to shift: These shifts preserve the total momentum and super-momentum. Super-BCFW recursion gives: Here the integral over η I denotes the sum over intermediate states in ordinary BCFW recursion [56]. The blackboardbold style denotes the stripped superamplitude, According to different Grassmann odd degrees, one can decompose the superamplitude into various N k MHV sectors, i.e., As shown explicitly in [32], the singular terms only come from the term with a = 1 in Eq. (3.5), or the first term of the right hand side in Eq. (3.7) under the holomorphic soft limit (2.9). So we drop the terms from contributions with a > 1, and write Graphically, Here we have In on-shell resursions of tree-level amplitudes, three-point amplitudes are seeds for generating higher-point amplitudes.
In on-shell superspace, the three-point superamplitudes of N = 4 SYM are given by (3.14) Then it is easy to get the left three-point superamplitude in Eq. (3.8) Notice that the left superamplitude in Eq. (3.8) is 3-point anti-MHV one. In fact, this corresponds to the super-shift (3.4) and in this case the helicity of soft gluon takes positive one [63]. Inserting the 3-point superamplitude (3.15) into Eq. (3.8) and computing the integral over η I give Dressing both sides of the above equation in respective appropriate momentum conservation delta functions, one obtains (3.17) In the holomorphic soft limit (2.9), Performing Taylor expansion at = 0, we obtain the soft theorem (3.21) Let us expand the superamplitude A n+1 in Grassmannian variables η s According to the degrees of the Grassmann odd η s , we can express super soft theorem (3.19) as following: In the last equation, the term 0 implies that there is no singular term. The soft gluon operators in N = 4 SYM are identical to the ones in pure Yang-Mills. As mentioned in Sect. 2, the sub-leading soft gluon divergence is also vanishing in N = 4 SYM. As we expected, the amplitudes involve more types of particle, including gluon, gluino and scalar in N = 4 SYM. More interestingly, we find the soft divergence of amplitudes involving a soft fermionic gluino. Notice that the leading soft gluino operator F (0) A involves the first order derivative with respect to the Grassmannian variables η 1 and η n . In fact, these two terms of F (0) A change helicity of corresponding external leg in respectively. And this preserves the total helicity as well as SU(4) R-symmetry before and after soft gluino emission. We also provide an alternative derivation of the soft theorem in N = 4 SYM in Appendix A. In the next subsection, we will check soft theorem by some examples in detail. We will pay special attention to soft gluino theorem.

MHV examples
In the remainder of this section, we verify the soft theorem presented above by some examples in detail. We take special care of the property of amplitudes when a gluino leg becomes soft.
The simplest example is MHV sector. In this sector, one can study the amplitudes involving an arbitrary number of external legs. In the holomorphic soft limit λ s → λ s , the soft theorem of pure gluonic MHV amplitudes gives ) involving a gluino-anti-gluino pair. Using the soft gluino theorem (3.24), we get (3.27) in the holomorphic soft limit λ s → λ s . We will show that there is no O( 0 ) corrections in above relation. Noticing the supersymmetry Ward identity (SWI) [39,54]: then using the soft gluon theorem (3.26), we have in the holomorphic soft limit λ s → λ s . This agrees with Eq. (3.27).
In the MHV sector, another a SWI involving two scalars is [39,54]: Using the soft gluon theorem (3.26) for the right hand side of above equation, one finds that It agrees with the soft theorem (3.25). This also shows that the MHV amplitudes involving two scalars remain invariant under rescaling of momentum of one of scalars.

Six-point NMHV examples
Next we turn to the Next-to-MHV (NMHV) sector. In this sector, it is difficult to check the amplitudes which consist of an arbitrary number of external legs. Here we mainly check 6point NMHV amplitudes involving a gluino-anti-gluino pair which were obtained by Feynman diagrams [81], also by solving supersymmetry Ward identity [82] and BCFW recursion [83].
The first example is : In the soft limit λ 4 → λ 4 , This agrees completely with the soft gluino theorem (3.24).
Here we have used two 5-point amplitudes follows: , (3.34) The second example is the amplitude: In the soft limit λ 4 → λ 4 , we have  This also agrees with the soft gluino theorem (3.24). Similarly, after some calculation we have

46)
Here we omit some details. These examples strongly support the soft gluino theorem (3.24). In addition, the same checks can be done for other six-point NMHV amplitudes in N = 4 SYM, for example tree-level four-gluino two-gluon amplitudes and six-gluino amplitudes [84].

Super soft theorem in N = 8 supergravity
The N = 8 Supergravity is the most symmetric quantum field theory in 4 dimensions. In this section, we study super soft theorem in N = 8 SUGRA. First we derive the soft theorem by using super-BCFW recursion relations at tree level. Then we verify soft divergences of scattering amplitudes of N = 8 SUGRA, in particular soft gravitino and soft gravi-photon divergences, by some MHV tree-level amplitudes exactly. We also give "KLT-like relations" between soft operators in N = 8 SUGRA and ones in N = 4 SYM at the end of this section.
The N = 8 SUGRA consists of 256 massless on-shell fields which can be characterized as (4.1) These on-shell fields form a CPT-self-conjugate supermultiplet and may be organized into a single on-shell superfield . With the help of the Grassmann odd variables η A , one can expand on-shell superfield follows Here A, B, . . . = 1, 2, . . . , N are SU(8) R-symmetry indices and each state above is fully antisymmetric in these labels.

(4.7)
Here one has omitted terms which stay finite in the holomorphic soft limits (2.9). Applying the deformation λ s → λ s to above formula, one gets s, a n, s 2 × M n . . . , λ a ,λ a + n, s n, a λ s , η a + n, s n, a η s , . . . , λ n ,λ n + s, a n, a λ s , η n + s, a n, a η s .
Here the leading soft factor is same with the one in nonsupersymmetric gravity theory, [s, a] n, a 2 s, a n, s 2 = S (0) . (4.10) The sub-leading soft operator consists of two parts: [s, a] n, a s, a n, s while the sub-sub-leading soft operator consists of three parts: soft gravitino: M n+1 . . . , ψ s A ( )  There are more contents in N = 8 SUGRA. Scattering amplitudes of N = 8 SUGRA involve more types of particle. That is, every (hard or soft) external leg in amplitudes may be any particle of 4D N = 8 SUGRA. Thus the soft graviton theorem (4.17) in SUGRA incorporates the soft graviton theorem for pure graviton amplitudes. Besides the soft graviton theorem, one obtains leading and sub-leading soft gravitino divergences and leading soft gravi-photon divergence. Also one finds that there are no soft gravi-photino divergence and soft scalar divergence.
In the next subsection, we will check soft theorems by some examples in the MHV sector of N = 8 SUGRA in detail. We will pay special attention to the soft gravitino divergence and the soft gravi-photon divergence.

MHV examples
For the soft graviton theorem, in particular leading and subleading orders, there are a great deal of study and investigation on both theoretical derivations and special examples check so far [20,28,32]. In this subsection, we check the soft theorem of N = 8 SUGRA amplitudes by some examples of the MHV sector. We mainly focus on the leading and the sub-leading soft gravitino divergences and the leading soft gravi-photon divergence, as well as the property of amplitudes with soft scalar.
First of all, we analyse a special class of amplitudes which are proportional to MHV amplitudes of gravitons. For such amplitudes, we can check soft gravitino divergences or soft gravi-photon divergences by using only soft graviton theorem, Eq. (2.2) or Eq. (4.17). In MHV sector of N = 8 SUGRA, there exists the following supersymmetry Ward identities: 4

Soft gravitino
Here we study the soft gravitino divergence. For the right hand side of SWI (4.22), by using the soft graviton theorem we have Bn .
In order to get the sub-leading order, substituting soft operator S (1) into the right hand side of Eq. (4.26), we have (4.28) Here the gauge freedom of S (0) is fixed by taking x = y = 1.
Next we study the soft-gravitino divergence by using directly the soft gravitino theorem (4.18). From the left hand side of the identity (4.22), using the leading soft gravitino theorem (4.18), one can obtain:  This give the same result as the one from the soft graviton theorem, Eq. (4.28).

Soft gravi-photon
Since there is only leading soft gravi-photon divergence, just Weinberg's leading soft graviton theorem is need. In the holo-morphic soft limit λ s → λ s , using leading soft graviton theorem 5 the right hand side of the SWI (4.23) becomes On the other hand, by applying straightforwardly the soft gravi-photon theorem (4.19) to the left hand side of the SWI (4.23), one gets
Next we analyse two 4-point amplitudes which have been computed by using both generating function method proposed in [58] and Feynman diagrams respectively in [62]. The first example is 4-gravi-photon amplitude: where s, t, u are Mandelstam variables (4.37) In holomorphic soft limit λ 4 → λ 4 , the amplitude becomes  [34]. Notice that following 3-point amplitudes of N = 8 SUGRA: Then one finds that Very nice! This is just the result from soft gravi-photon theorem.
Another example is 2-scalar 2-gravi-photon amplitude: We study the property of this amplitude when external particle v C D becomes soft. In holomorphic soft limit λ 2 → λ 2 , this amplitude becomes  [24].
Note the following three-point MHV amplitudes: Then we have 6 (4.48) By applying straightforwardly the soft gravi-photon theorem (4.19) to amplitude M 4 v AB , v C D , S E FG H , S I J K L , one can also obtain the same result.

SUGRA soft operators as double copy of SYM soft operators
As discussed in Sect. 2.3, the gravity soft operator can be expressed as double copy of gauge theory soft operators. This also occurs in supersymmetric theories. In the end of this section, we write the soft operators in N = 8 SUGRA in terms of a sum of some products of soft operators in N = 4 SYM. First introducing a new operator involving the derivative with respect to Grassmann odd variable η A a as follows: (4.54) Then the soft operators in N = 4 SYM may be written as where λ x and λ y are arbitrary reference spinors. The subsub-leading soft operator may be expressed as (s, a) . (4.58) As mentioned in Sect. 2.3, all derivatives in operators only act on amplitudes. These relations may be derived using the scheme proposed in [19] by super-KLT relation [72,76-78, 91,92].

Conclusion and discussions
In this work, the super soft theorems were investigated systematically in 4D maximally N = 4 super-Yang-Mills theory and N = 8 supergravity. We have presented the super soft theorems with rigorous proofs at tree level. The main results are Eq. There are several further topics that are fascinating for us. First, properties of amplitudes involving soft fermion should be investigated more systematically. In this paper, we discussed the soft gluino divergence for color-ordered amplitudes of N = 4 SYM and the soft gravitino divergence and the soft gravi-photino divergence for N = 8 SUGRA amplitudes. It will be interesting to study the properties of amplitudes involving soft fermion in other theories.
Second, it will be interesting to find other methods to derive the soft theorems. Let us take an example. In [57], all tree-level superamplitudes in N = 4 SYM were expressed as compact analytical formulas. By taking the soft limit directly, it gives the soft theorem as shown in Appendix A. The similar formulas for all tree-level superamplitudes in N = 8 SUGRA were also obtained in [60]. We will also study soft theorem through these formulas in future work.
Finally, more on the relations between the soft theorems and symmetry principle should be understood.
Although the leading and sub-leading soft graviton theorems in gravity [25][26][27]32,93] and leading and sub-leading soft-photon theorems in massless QED [17] were interpreted as symmetries of S-matrixes in recent works, very limited information was known for other soft divergences. Our particular interest is to explore the remarkable relations between super soft theorems and local supersymmetry.
Acknowledgments The author would like to thank Professor Jun-Bao Wu for suggesting this project as well as numerous discussions and valuable comments through out all the stages of the work. He would also especially like to thank Professor Chuan-Jie Zhu for his support, guidance, discussions and careful reading of the manuscript. He is also grateful to Da-Ping Liu, Wen-Jian Pan, Yu Tian, Gang Yang and Hongbao Zhang for various helpful discussions. This work was supported by the National Natural Science Foundation of China under Grants No. 11135006.
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited. Funded by SCOAP 3 / License Version CC BY 4.0.

Appendix A: Alternative derivation of soft theorem in N = 4 SYM
In this appendix, we rederive the soft theorem of N = 4 SYM by using formulaes for all tree-level superamplitudes which were given in [57].
First of all, we have to summarize briefly main results of Drummond and Henn's paper [57]. Three-point amplitudes are fundamental in BCFW-construction of higher-point amplitudes. In Sect. 3, three-point MHV and Googly (or anti-MHV) superamplitudes of N = 4 SYM have been presented. By solving super-BCFW resursion, it is easy to obtain n-point The n-point MHV superamplitude is simple and compact, just as Parke-Taylor formula of the pure gluonic amplitude. The delta functions δ 4 ( p) and δ 8 (q) are consequences of translation invariance and supersymmetry. Therefore all treelevel superamplitudes, not just MHV, contain delta function factor δ 4 ( p)δ 8 (q) in N = 4 SYM. So it is very convenient to factor out the MHV superamplitude, Here P n is a function of spinors λ a ,λ a and Grassmann variables η A a and one can express this quantity as following form: where and the Grassmann odd quantity n;ab is defined by n;ab ≡ n|x na x ab |θ bn + n|x nb x ba |θ an Obviously, this quantity is independent of η 1 and η n . In fact, it is relevant to a special gauge choice.
Similarly, NNMHV function also may be constructed as follows Here has involved a quantity which is a generalization of the R-invariant, For details, see [57]. More generally, all the P N k MHV n functions can be written in terms of the quantities R n;b 1 a 1 ;b 2 a 2 ;...;b r a r ;ab . It is somewhat surprising that all quantities (A.10) are independent of η 1 and η n and so are all P N k MHV n , i.e., In fact, this reflects the special gauge choice of shifted legs in BCFW recursion. Now we turn to study soft theorem. The n-point MHV superamplitude is not only the simplest in all n-point amplitudes, but a common factor for all amplitudes. So we begin with MHV sectors. (A.14) In the holomorphic soft limit λ s → λ s , it gives O( −1 ) order soft divergence. When the soft particle is scalar, it is also easy to see that MHV superamplitudes are invariant under the holomorphic soft rescaling λ s → λ s . Another task is to see the function P n+1 in the holomorphic soft limit. First we consider soft behavior of NMHV function: 7 7 Here we fix gauge such that the function P NMHV n+1 is independent of η n and η s .  Using the same method, we can show that the similar conclusion holds for all N k MHV sectors of N = 4 SYM, i.e., (A. 28) in the holomorphic soft limit λ s → λ s . Both A MHV n+1 and P n+1 have no singular term in the holomorphic soft limit λ s → λ s of a scalar. This implies that there exists no singularity when a external scalar becomes soft in an on-shell amplitude.

Appendix B: Sub-sub-leading soft operator in N = 8 SUGRA
In this appendix, we compute the sub-sub-leading soft operator in N = 8 SUGRA by a Taylor expansion in detail.