Preprint typeset in JHEP style - HYPER VERSION
arXiv:1410.1616v2 [hep-th] 5 Nov 2014
Soft theorems in maximally supersymmetric theories
Zheng-Wen Liu Department of Physics, Renmin University of China, Beijing 100872, P. R. China
E-mail:
[email protected] Abstract: 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 super-Yang-Mills theory and N = 8 supergravity using super-BCFW recursion relations. We verify these theorems exactly by showing some examples. ArXiv ePrint: 1410.1616
Contents 1 Introduction
1
2 New soft graviton theorem and soft gluon theorem
2
2.1
Cachazo-Strominger’s new soft graviton theorem
3
2.2
Soft gluon theorem in Yang-Mills theory
4
2.3
Gravity soft operators as double copy of Yang-Mills soft operators
5
3 Soft theorem in N = 4 super-Yang-Mills theory
6
3.1
Super soft theorem in N = 4 SYM
7
3.2
MHV and NMHV Examples
9
4 Super soft theorem in N = 8 supergravity
11
4.1
Super soft theorem in N = 8 SUGRA
12
4.2
MHV Examples
14
4.3
SUGRA soft operators as double copy of SYM soft operators
18
5 Conclusion and discussions
18
A Alternative derivation of soft theorem in N = 4 SYM
20
B Sub-sub-leading soft operator in N = 8 SUGRA
23
1
Introduction
Over the past few decades, there was great progress on the understanding of the 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 an external leg tends to zero in both gravity and Yang-Mills [16–42]. The study on soft behavior of the amplitudes (in gravity) goes back to Steven Weinberg who proposed an universal leading soft-graviton behavior in the framework of S-matrix program more than 50 years ago [11]. The leading soft-graviton behavior of the amplitudes, also called “Weinberg’s theorem”, is uncorrected to all loop orders. At sub-leading order, the soft behavior of amplitudes for soft photon and for gravitons were also studied by using Feynman diagrams [12–14]. Recently, sub-leading and sub-sub-leading soft-graviton divergences of amplitudes were proposed [15] beyond Weinberg’s theorem. In [15], Cachazo and Strominger presented a proof for tree-level amplitudes of gravitons using BCFW recursion relation [8, 75–79] in spinor-helicity formulism [1, 6, 9]. Similarly, the sub-leading soft divergence for Yang-Mills amplitudes was obtained by same analysis in [16]. As pointed out in [19], the sub-leading divergence is vanishing in pure Yang-Mills amplitudes. However, we
–1–
will see that the sub-leading Yang-Mills soft operators are necessary for “KLT-construction” of gravity soft operators in this paper. On the other hand, a lot of remarkable progress has also been made on the understanding of the soft theorem from viewpoint of symmetry principles. The leading and sub-leading soft theorems are understood as Ward identities of BMS symmetry [40–42]. The leading [33] and subleading [32] soft photon theorems were interpreted as asymptotic symmetries of S-matrix in massless QED. Now aspects of the soft theorems have been investigated along many different directions [16–42]. The loop corrections of soft theorems were investigated in both gravity and Yang-Mills [17–20]. Various different methods were used to derive the soft theorems, including Feynman diagram approach [23] and conformal symmetry approach [31] for Yang-Mills, the Poincar´e symmetry and gauge symmetry approach [21, 22], ambitwistor string approach [30] for Yang-Mills and gravity. The new soft graviton theorem was also obtained from the soft gluon theorem via the Kawai-Lewellen-Tye relations [34]. The scattering equations [80–83], also called 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 [19, 29]. It is natural and interesting to investigate the soft theorems in supersymmetric theories. In this paper, we mainly focus on the soft theorems in the maximally supersymmetric theories, i.e., N = 4 super-YangMills (SYM) theory and N = 8 supergravity (SUGRA), in 4 spacetime dimensions. Great progress on analytical calculations of the scattering amplitudes, in particular tree-level, has been achieved in N = 4 SYM and N = 8 SUGRA. A lot of remarkable and interesting structures of scattering amplitudes were discovered [61–66]. Many novel methods for perturbative scattering amplitudes were proposed, and a large number of amplitudes were also analytically computed in various theories [3–5, 7, 53–60, 68]. For example, by solving super-BCFW recursion relation [7, 8, 75, 76], the compact analytical formulae for all tree amplitudes were presented in N = 4 SYM [53], also in massless QCD with up to four quark-anti-quark pairs [57]. 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 superBCFW 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 section 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 section 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 section 5, we conclude this paper with some brief discussions. The appendix A provides an alternative derivation of soft theorem in N = 4 SYM. The appendix B gives calculational details of the sub-sub-leading soft operator in N = 8 SUGRA.
2
New soft graviton theorem and soft gluon theorem
In this section we briefly review the new soft graviton theorem [15] and the soft gluon theorem [16, 69]. We also show that each order soft operator in gravity may be expressed as a double copy of the Yang-Mills
–2–
soft operators with gauge freedom. 2.1
Cachazo-Strominger’s new soft graviton theorem
An on-shell (n + 1)-point scattering amplitude including an external graviton with momentum ks may be denoted Mn+1 = Mn+1 (k1 , . . . , kn , ks ).
(2.1)
In the soft limit ks → 0, the amplitude Mn+1 behaves as Mn+1 (k1 , . . . , kn , ks ) = S (0) + S (1) + S (2) Mn (k1 , . . . , kn ) + O(ks2 ).
(2.2)
The soft operators are given by
S (0) =
n X Eµν kaµ kν a
a=1
S
(1)
S (2)
ks · ka
(2.3)
,
n X Eµν kaµ ksσ Jaσν = −i , ks · ka a=1 n ρµ ksσ Jaσν 1 X Eµν ksρ Ja , = − 2 ks · ka
(2.4) (2.5)
a=1
where Eµν is the polarization tensor of the soft graviton s and Jaµν is the total angular momentum of the ath external leg. It is easy to check that all the soft operators are gauge invariant [15]. The leading soft factor S (0) proposed by Steven Weinberg is uncorrected by all loop orders [11] while sub-leading and sub-sub-leading soft operators are not, as discussed in [17–21]. In spinor-helicity formulism, the momentum vector kµ of an on-shell massless particle may be represented as a bispinor, i.e., ˜ α˙ . kαα˙ = kµ σαµα˙ = λα λ
(2.6)
Introducing an infinitesimal soft parameter ǫ, one can write soft limit of the momentum ks of soft particle as ˜s ks → ǫ ks = ǫ λs λ
=⇒
λs → ǫδ λs ,
˜ s → ǫ1−δ λ ˜s. λ
(2.7)
Here different choices of δ in physical amplitudes can be linked to each other via the little group transformation, i.e., ˜ s , hs } = t−2hs M {λs , λ ˜ s , hs } , M {tλs , t−1 λ
(2.8)
λs → ǫ λs
(2.9)
where hs is helicity of the particle s. In this paper, one employs the holomorphic soft limit [7]:
˜ s remains in which only the holomorphic spinor λs tends to zero while the anti-holomorphic spinor λ unchangeable. The new soft graviton theorem (2.2) is then 1 (1) 1 (2) 1 (0) ˜ S + 2S + S Mn + O(ǫ0 ). Mn+1 (. . . , {ǫλs , λs }) = ǫ3 ǫ ǫ –3–
(2.10)
In the spinor-helicity formulism1 , the soft operators are given by n X [s, a] hx, ai hy, ai , hs, ai hx, si hy, si a=1 n ∂ 1 X [s, a] hx, ai hy, ai ˜ + λsα˙ , = ˜ aα˙ 2 a=1 hs, ai hx, si hy, si ∂λ
S (0) =
(2.11)
S (1)
(2.12)
n
S (2) =
∂2 1 X [s, a] ˜ ˜ . λsα˙ λsβ˙ ˜ aα˙ ∂ λ ˜ ˙ 2 hs, ai ∂λ aβ a=1
(2.13)
Here one has assigned soft graviton the helicity hs = +2, just a convention. The spinors λx , λy are two arbitrary choosen reference spinors and the freedom in this choice is equivalent to the gauge freedom. 2.2
Soft gluon theorem in Yang-Mills theory
The similar soft behavior of the scattering amplitudes appears also in Yang-Mills theory [16, 69]. In soft limit of the momentum of a gluon, ks → 0, an on-shell color-ordered Yang-Mills amplitude An+1 becomes (1) (0) (2.14) An+1 (k1 , . . . , kn , ks ) = SYM + SYM An (k1 , . . . , kn ) + O(ks ), where the leading soft (eikonal) factor [69] is
X Eµ kaµ , ks · ka
(0)
SYM ≡
(2.15)
a=1,n
while the sub-leading soft operator is given by (1)
SYM ≡ −i
X Eµ ksν Jaµν , k · k s a a=1,n
(2.16)
with Eµ the polarization vector of soft gluon. In spinor-helicity formulism, employing the holomorphic soft limit (2.9), the soft gluon theorem (2.14) may be expressed as ˜s }) = An+1 (. . . , {ǫλs , λ
1 (0) 1 (1) S + S An + O(ǫ0 ). ǫ2 ǫ
(2.17)
Taking the helicity of the soft gluon hs = +1 as a convention, the soft operators may be written as (0)
hx, ni hx, 1i + , hs, ni hx, si hs, 1i hx, si 1 ˜ ∂ ∂ 1 ˜ + , λsα˙ λsα˙ = ˜ nα˙ ˜ 1α˙ hn, si hs, 1i ∂λ ∂λ
SYM = (1)
SYM
(2.18) (2.19)
with λx , λy 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, the amplitudes An+1 and An in eq. (2.17) contain respective momentum conservation delta functions. With ˜ 1 and λ ˜ n in these two amplitudes this in mind, we can remove dependence of anti-holomorphic spinors λ by solving momentum conservation delta functions appropriately. This implies that the sub-leading soft divergence vanishes in color-ordered Yang-Mills amplitudes [19]. As we will see immediately, however, (1)
that the term SYM is necessary for constructing the gravity soft operators from Yang-Mills soft operators. In this paper, we mainly follow the notation of ref. [15]. The spinor products are defined as hi, ji = ǫαβ λiα λjβ = λiα λα j ˙ ˙ ˜α ˜ ˙ =λ ˜ iα˙ λ ˜ iα˙ λ and [i, j] = ǫα˙ β λ , and we use the convention s = hi, ji [i, j] whic is different from QCD convention. ij j jβ 1
–4–
2.3
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 [44]. In field theory limit, this relation expresses a gravity amplitude as a sum of the square of Yang-Mills amplitudes. The similar relation also exists between gravity soft operators and Yang-Mills soft operators. In [18], 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 choices 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 notations2 Eµ+ (λx )kaµ hx, ai = , ks · ka hx, si hs, ai Eµ+ (λx )ksν Jaµν 1 ˜ ∂ = λsα˙ S1 (s, a) ≡ −i , ˜ aα˙ ks · ka hs, ai ∂λ
S0 (x, s, a) ≡
(2.20) (2.21)
which are the fundamental building blocks for constructing gravity soft operators. Employing these notations, one can write the Yang-Mills soft operators as (0)
SYM = S0 (x, s, n) + S0 (x, s, 1), (1) SYM (1, s, n)
= S1 (s, 1) − S1 (s, n).
(2.22) (2.23)
Let us start with a simple relation that expresses a graviton polarization tensor as the product of gluon polarization vectors with same momentum, i.e., ± Eµν (k) = Eµ± (k) × Eν± (k) + Eν± (k) × Eµ± (k).
(2.24)
Here Eµν have been written in a symmetric form. By making use of this relation, the leading soft operator in gravity can be written as: S
(0)
n X 2 Eµ+ (λx )Eν+ (λy ) kaµ kaν = ks · ka a=1 + n X Eµ+ (λx )kaµ Eν (λy )kaν = 2ks · ka ks · ka ks · ka a=1
=
n X
ssaS0 (x, s, a)S0 (y, s, a)
(2.25)
a=1
where ssa = 2ks · ka = hs, ai [s, a] and the λx and λy are arbitrary reference spinors and the freedom in this choice is equivalent to the gauge freedom. This relation was presented in [49–51] and derived in [34, 48] by KLT realtion [44–46]. The sub-leading soft graviton operator can be expressed as S (1)
n X (ks · ka ) = −i a=1
=
n 1X
2
a=1
! Eµ+ (λx )kaµ Eν+ (λy )ksσ Jaσν + (x ↔ y) ks · ka ks · ka
ssa S0 (x, s, a) + S0 (y, s, a) S1 (s, a)
2
(2.26)
Here the first notation S0 is just famous eikonal factor in Yang-Mills amplitudes [1, 50, 51] and the ‘x’ denotes the ˜ s , λx ) of the polarization vector Eµ+ [6]. reference spinor λx in spinor representation Eα+α˙ (λs , λ
–5–
This relation with a special gauge choice was derived in [34] by KLT realtion [44–46]. Similarly, the sub-sub-leading soft operator may be written as n
S
(2)
1X ssa S1 (s, a)S1 (s, a) = 2
(2.27)
a=1
It is important to notice that operators product S1 (s, a)S1 (s, a) should be understood as: S1 (s, a)S1 (s, a) =
1 ˜ ˜ ∂2 . λ λ ˙ s α ˙ sβ ˜ ˜ ˙ ha, si2 ∂ λaα˙ ∂ λ aβ
(2.28)
In another words, the differentials only act on the amplitudes.
3
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 lowerpoint amplitudes examples to demonstrate the validity of super soft theorem, in particular the soft gluino divergence. Let us begin with a very brief introduction of N = 4 SYM and the superamplitudes in on-shell superspace. The N = 4 on-shell field consists of 8 bosons and 8 fermions and one can write it out as 1 gluon g +
h=1: 1 h= : 2 h=0: 1 h=− : 2 h = −1 :
4 gluinos ΓA 6 scalars SAB
(3.1)
¯A 4 gluinos Γ 1 gluon g −
Here A, B, . . . = 1, 2, 3, 4 are SU(4) R-symmetry indices and the scalar SAB is antisymmetric in indices A, B. The N = 4 on-shell superfield can be expanded as follows [10]: Φ(p, η) = g + (p) + η A ΓA (p) +
1 A B 1 ¯ D (p) + η 1 η 2 η 3 η 4 g − (p). η η SAB (p) + η A η B η C ǫABCD Γ 2! 3!
(3.2)
Here Grassmann odd variables η A transforms in a fundamental representation of the SU(4) R-symmetry. ˜ a (or momenIn super-momentum space, a color-ordered superamplitude is a function of spinors λa , λ tum pa ) and Grassmann variables ηa , i.e., ˜ 1 , η1 }, . . . , {λn , λ ˜n , ηn } . An ≡ An (Φ1 , . . . , Φn ) = An {λ1 , λ
(3.3)
The component field amplitudes are then obtained by projecting upon the relevant terms in the ηi expansion of the superamplitude. For detail, see [4, 5, 7].
–6–
3.1
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 [7, 75, 76] in spinor-helicity formulism [1, 2, 9]. Let us choose the soft particle and its adjacent particle to sfift: ˜ n (z) = λ ˜n − zλ ˜s, λ
λs (z) = λs + zλn ,
ηn (z) = ηn − zηs .
(3.4)
These shifts preserve the total momentum and super-momentum. Super-BCFW recursion gives: An+1 =
n−2 XZ a=1
˜s , ηs }, 1, . . . , a, {I(z ∗ ), ηI } d4 ηI AL {λs (z ∗ ), λ ×
1 ˜ n (z ∗ ), ηn (z ∗ )} . AR {−I(z ∗ ), ηI }, a + 1, . . . , n − 1, {λn , λ 2 PI
(3.5)
Here the integral over ηI denotes the sum over intermediate states in ordinary BCFW recursion [7]. The blackboard-bold style denotes the stripped superamplitude, An = An δ4 (p).
(3.6)
According to different Grassmann odd degrees, one can decompose the superamplitude into various Nk MHV sectors, i.e., k
N MHV An+1 =
Z +
d4 ηP AMHV (z ∗ ) 3 k−1 X n Z X
1 Nk MHV ∗ A (z ) P2 n
d4 ηPa AaN
m MHV
m=0 a=4
(za )
1 N(k−m−1) MHV A (za ). Pa2 n−a+3
(3.7)
As shown explicitly in [15], 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 Z ˜s , ηs }, {1, η1 }, {I, ηI } An+1 = d4 ηI AMHV {λs (z ∗ ), λ 3
1 ˜ n (z ∗ ), ηn (z ∗ )} . An {−I, ηI }, {2, η2 }, . . . , {λn , λ 2 PI
×
Graphically,
sˆ
1
n−1
1
An+1
n ˆ
=
Z
d4 ηI
AL
sˆ
n−1
2 ηI
(3.8)
ηI
AR
n ˆ
Here we have PI2 = ks + k1 z∗ = −
2
= hs, 1i [s, 1],
hs, 1i , hn, 1i
(3.9) (3.10)
λI = λ1 ,
(3.11)
˜I = λ ˜ 1 + hn, si λ ˜ s. λ hn, 1i
(3.12)
–7–
In on-shell resursion method, three-point amplitudes are seeds of the construction of higher-point amplitudes. In on-shell superspace, the three-point superamplitudes of N = 4 SYM are given by X3 δ4 (p) δ8 λαa ηaA , a=1 h12i h23i h31i 4 δ (p) = δ4 [12]η3A + [23]η1A + [31]η2A . [12][23][31]
A3MHV =
(3.13)
A3MHV
(3.14)
Then it is easy to get left 3-point superamplitude in eq. (3.8)
hn, 1i [s, 1] 4 hn, si ∗ ˜ {λ (z ), λ , η }, {1, η }, {I, η } = AMHV η − . δ η − η s s s 1 I I s 1 3 hn, si hn, 1i
(3.15)
Notice that the left superamplitude which contributes to BCFW construction 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 [59]. Inserting the 3-point superamplitude (3.15) into (3.8) and computing the integral over ηI give An+1 =
hn, 1i ˜1 + hn, si λ ˜ 2 , η2 }, . . . , {λn , λ ˜ n + hs, 1i λ ˜ s , η1 + hn, si ηs }, {λ2 , λ ˜ s , ηn + hs, 1i ηs } . An {λ1 , λ hn, si hs, 1i hn, 1i hn, 1i hn, 1i hn, 1i (3.16)
Dressing both sides of the above equation in respective appropriate momentum conservation delta functions, one obtains hn, 1i hn, si hs, 1i ˜ 2 , η2 }, . . . , {λn , λ ˜ n + hs, 1i λ ˜ s , η1 + hn, si ηs }, {λ2 , λ ˜ s , ηn + hs, 1i ηs } . (3.17) ˜1 + hn, si λ × An {λ1 , λ hn, 1i hn, 1i hn, 1i hn, 1i
A n+1 =
In the holomorphic soft limit (2.9), 1 hn, 1i 2 ǫ hn, si hs, 1i ˜ 1 + ǫ hn, si λ ˜ 2 , η2 }, . . . , {λn , λ ˜n + ǫ hs, 1i λ ˜ s , η1 + ǫ hn, si ηs }, {λ2 , λ ˜ s , ηn + ǫ hs, 1i ηs } . (3.18) ×An {λ1 , λ hn, 1i hn, 1i hn, 1i hn, 1i
A n+1 (ǫ) =
Performing Taylor expansion at ǫ = 0, we obtain the soft theorem 1 (1) 1 (0) + S S An + O(ǫ0 ) An+1 (ǫ) = ǫ2 SYM ǫ SYM
(3.19)
where (0)
SSYM = (1)
hn, 1i (0) = SYM , hn, si hs, 1i (1)
(0)
SSYM = SYM + ηsA FA ,
(3.20) (0)
FA
≡
1 ∂ ∂ 1 + . hs, 1i ∂η1A hn, si ∂ηnA
(3.21)
Let us expand the superamplitide An+1 in Grassmannian variables ηs An+1 (Φ1 , . . . , Φn , Φs ) = An+1 (Φ1 , . . . , Φn , gs+ ) + ηsA An+1 (Φ1 , . . . , Φn , ΓsA ) 1 + ηsA ηsB An+1 (Φ1 , . . . , Φn , SsAB ) + · · · . 2!
–8–
(3.22)
According to the degrees of the Grassmann odd ηs , we can express super soft theorem (3.19) as following: 1 (1) 1 (0) + S S An + O(ǫ0 ), (3.23) An+1 . . . , gs+ (ǫ) = ǫ2 YM ǫ YM 1 (0) (3.24) An+1 . . . , ΓsA (ǫ) = FA An + O(ǫ0 ), ǫ 0 An+1 . . . , SsAB (ǫ) = + O(ǫ0 ). (3.25) ǫ In the last equation, the term
0 ǫ
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 section 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 (0)
FA involves the first order derivatives with respect to the Grassmannian variables η1 and ηn . In fact, (0)
these two terms of FA 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. 3.2
MHV and NMHV 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 λs → ǫλs , the soft theorem of pure gluonic MHV amplitudes gives AMHV n+1 (ǫ) =
1 (0) MHV S A ǫ2 YM n
(3.26)
which is exact in ǫ. ¯ A , g+ , . . . , g + , g− , ΓB ) involving a gluino-anti-gluino pair. Using the Now we study the amplitude A(Γ soft gluino theorem (3.24), we get ¯ A , g+ , . . . , g + , g− , ΓB = δA 1 1 An g− , g+ , . . . , g + , g− + O(ǫ0 ). An+1 Γ B ǫ hs, 1i
(3.27)
in the holomorphic soft limit λs → ǫλs . We will show that there is no O(ǫ0 ) corrections in above relation. Noticing the SWI [1, 4]: ¯ A , g+ , . . . , g + , g− , ΓB = δA hn, si An+1 g− , g+ , . . . , g + , g− , g+ An+1 Γ n B hn, 1i
(3.28)
then using the soft gluon theorem (3.26), we have A δB
1 hn, 1i ǫ hn, si A ǫ hn, si An+1 g− , g+ , . . . , g + , g− , g+ = δB × 2 An g − , g + , . . . , g + , g − hn, 1i hn, 1i ǫ hn, si hs, 1i 1 1 A (3.29) An g − , g + , . . . , g + , g − = δB ǫ hs, 1i
in the holomorphic soft limit λs → ǫλs . This agrees with eq. (3.27). –9–
In the MHV sector, another a SWI involving two scalars is [1, 4]: h2, si2 − − + + + . An+1 S12 , g− , g+ , . . . , g + , S34 = 2 An+1 g , g , g , . . . , g , g h2, 1i
(3.30)
Using the soft gluon theorem (3.26) for the right hand side of above equation, one finds that λ →ǫλs h2, si2 (0) − − + + An+1 S12 , g − , g+ , . . . , g+ , S34 ==s==== 2 SYM An g , g , g , . . . , g h2, 1i = An+1 S12 , g− , g+ , . . . , g + , S34 ∼ O(ǫ0 ).
(3.31)
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 2-gluino amplitudes Next we turn to 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 6-point NMHV amplitudes involving a gluino-anti-gluino pair which were obtained by Feynman diagrams [73], also by solving supersymmetry Ward identity [72] and BCFW recursion [70]. + + ¯A The first example is A6 (g− , g − , Γ 3 , Γ4B , g , g ): 1
2
5
6
+ + ¯A =− A6 g1− , g2− , Γ 3 , Γ4B , g5 , g6 +
[4|2 + 3 |1i2 [3|2 + 4 |1i δA s234 [23][34] h56i h61i [2|3 + 4 |5i B [6|1 + 2 |3i2 [6|1 + 2 |4i δA . s612 [61][12] h34i h45i [2|6 + 1 |5i B
(3.32)
In the soft limit λ4 → ǫλ4 , [65]2 h53i2 [6|3 + 5 |4i ¯ A , Γ4B , g+ , g+ = δA 1 A6 g1− , g2− ,Γ + O(ǫ0 ) B 3 5 6 ǫ s35 [61][12] h34i h45i [23] h35i 1 1 − − − + + − − ¯C + A1 = δB A5 g1 , g2 , g3 , g5 , g6 + A5 g1 , g2 , Γ3 , Γ5C , g6 + O(ǫ0 ). ǫ h34i h45i
(3.33)
This agrees completely with the soft gluino theorem (3.24). Here we have used two 5-point amplitudes follows: [56]4 , [12][23][35][56][61] [65]2 [63] + ¯A = A5 g1− , g2− , Γ δA . 3 , Γ5B , g6 [12][23][35][61] B A5 g1− , g2− , g3− , g5+ , g6+ =
(3.34) (3.35)
The second example is the amplitude:
− + + ¯A =− A6 g1− , Γ 2 , g3 , Γ4B , g5 , g6 +
[4|2 + 3 |1i2 [2|3 + 4 |1i δA s234 [23][34] h56i h61i [2|3 + 4 |5i B [6|1 + 2 |3i2 [26] h34i δA . s612 [61][12] h34i h45i [2|6 + 1 |5i B
(3.36)
In the soft limit λ4 → ǫλ4 , we have [65]2 h53i2 [26] − + + A1 ¯A + O(ǫ0 ) A6 g1− , Γ , g , Γ , g , g = δ 4B 5 2 B 3 6 ǫ s35 [61][12] h45i [23] h35i − + 0 A1 1 ¯A A5 g1− , Γ = δB 2 , g3 , Γ5A , g6 + O(ǫ ) ǫ h45i – 10 –
(3.37)
where [56]4 [56] − + ¯A δA . , g , Γ , g = A5 g1− , Γ 5B 6 2 3 [26] [12][23][35][56][61] B
(3.38)
This also agrees with the soft gluino theorem (3.24). Similarly, after some calculation we have
→ǫλ4 1 1 ¯ A , g− , g− , Γ4B , g+ , g+ =λ=4= ¯ A , g− , g− , Γ5B , g+ + O(ǫ0 ), A6 Γ === A5 Γ 1 1 2 3 5 6 2 3 6 ǫ h45i λ →ǫλ5 1 1 − + − + + ¯A ¯A A5 g1− , Γ ==5==== A6 g1− , Γ 2 , g3 , Γ4B , g6 2 , g3 , g4 , Γ5B , g6 ǫ h45i 1 1 ¯ A , g− , g + , Γ6B + O(ǫ0 ), + A5 g1− , Γ 2 3 4 ǫ h56i λ5 →ǫλ5 1 1 − − + − − + + ¯A ¯A A5 Γ ====== A6 Γ 1 , g2 , g3 , Γ4B , g6 1 , g2 , g3 , g4 , Γ5B , g6 ǫ h45i 1 1 0 − − + ¯A A5 Γ + 1 , g2 , g3 , g4 , Γ6B + O(ǫ ), ǫ h56i →ǫλ3 0 ¯ A , g− , Γ3B , g − , g+ , g+ =λ=3= A6 Γ === + O(ǫ0 ), 1 2 4 5 6 ǫ λ3 →ǫλ3 1 1 A − ¯A − + + A6 g1 , Γ2 , Γ3B , g4 , g5 , g6 ====== δB A5 g1− , g2− , g4− , g5+ , g6+ + O(ǫ0 ), ǫ h23i →ǫλ3 1 1 A ¯ A , g+ , g+ =λ=3= A6 g1− , g2− , Γ3B , Γ === δB A5 g1− , g2− , g4− , g5+ , g6+ + O(ǫ0 ), 4 5 6 ǫ h34i 1 A 1 λ →ǫλ 2 − + − + ¯A δB A5 g1− , g3− , g4+ , g5− , g6+ + O(ǫ0 ), ==2==== A6 Γ 1 , Γ2B , g3 , g4 , g5 , g6 ǫ h12i 1 A 1 λ →ǫλ 2 ¯ A , g + , g− , g+ ==2==== δB A5 g1− , g3− , g4+ , g5− , g6+ + O(ǫ0 ), A6 g1− , Γ2B , Γ 3 4 5 6 ǫ h23i 0 →ǫλ2 ¯ A , g+ =λ=2= A6 g1− , Γ2B , g3− , g4+ , Γ === + O(ǫ0 ). 5 6 ǫ
(3.39)
(3.40)
(3.41) (3.42) (3.43) (3.44) (3.45) (3.46) (3.47)
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 [71].
4
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 at tree level. Then we verify the 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 the soft operators in N = 8 SUGRA and the 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 (N = 8 SUGRA) ∼ (N = 4 SYM) ⊗ (N = 4 SYM).
– 11 –
(4.1)
These on-shell fields form a CPT-self-conjugate supermultiplet and may be organized into a superfield Φ. With the help of the Grassmann odd variables η A , one can expand on-shell superfield Φ follows Φ(p, η) = h+ (p) + η A ψA (p) + +
1 1 A B η η vAB (p) + η A η B η C χABC (p) 2! 3!
1 A B C D η η η η SABCD (p) + · · · + η 1 η 2 η 3 η 4 η 5 η 6 η 7 η 8 h− (p). 4!
(4.2)
Here A, B, . . . = 1, 2, . . . , N are SU(8) R-symmetry indices and each state above is fully antisymmetric in these labels. In N = 8 on-shell superspace, there are also fundamental three-point superamplitudes: M3MHV = M3MHV =
δ4 (p) h12i h23i h31i δ4 (p)
2 δ
16
λα1 η1A + λα2 η2A + λα3 η3A ,
8 A A A 2 δ [12]η3 + [23]η1 + [31]η2 . [12][23][31]
(4.3) (4.4)
Here each is just the square of corresponding three-point superamplitude of N = 4 SYM. 4.1
Super soft theorem in N = 8 SUGRA
Now we start to derive the soft theorem. Consider an on-shell (n + 1)-point superamplitudes in N = 8 SUGRA with a soft particle3 ˜ 1 , η1 }, · · · , {λn , λ ˜ n , ηn }, {λs , λ ˜ s , ηs } . Mn+1 ≡ Mn+1 {λ1 , λ
(4.5)
˜ n (z) = λ ˜n − zλ ˜s, λ
(4.6)
Let us choose the following super-shift:
λs (z) = λs + zλn ,
ηn (z) = ηn − zηs .
Using the analysis similar to SYM, the super-BCFW recursion gives Mn+1 =
n−1 X
[s, a] hn, ai2 2 a=1 hs, ai hn, si ˜ n + hs, ai λ ˜ s , ηa + hn, si ηs }, . . . , {λn , λ ˜ s , ηn + hs, ai ηs } . ˜ a + hn, si λ × Mn . . . , {λa , λ hn, ai hn, ai hn, ai hn, ai
(4.7)
Here one has omitted the terms which stay finite in the holomorphic soft limits (2.9). Applying the deformation λs → ǫλs to above formula, one gets n−1 1 X [s, a] hn, ai2 2 ǫ3 a=1 hs, ai hn, si ˜ n + ǫ hs, ai λ ˜ s , ηa + ǫ hn, si ηs }, . . . , {λn , λ ˜ s , ηn + ǫ hs, ai ηs } . ˜ a + ǫ hn, si λ × Mn . . . , {λa , λ hn, ai hn, ai hn, ai hn, ai (4.8)
Mn+1 (ǫ) =
Performing Taylor expansion of M(ǫ) around ǫ = 0, one obtains the super soft theorem: 1 (0) 1 (1) 1 (2) S + 2S + S Mn+1 (ǫ) = Mn + O(ǫ0 ). ǫ3 ǫ ǫ 3
(4.9)
Here soft particle may be any one in N = 8 supermultiplet (4.1), including graviton (spin-2), gravitino (spin-3/2),
gravi-photon (spin-1), gravi-photino (spin-1/2) and scalar (spin-0).
– 12 –
Here leading soft factor is same with the one in non-supersymmetric gravity theory, S (0) =
n−1 X a=1
[s, a] hn, ai2 (0) 2 = S . hs, ai hn, si
(4.10)
The sub-leading soft operator consists of two parts: S
(1)
=
n−1 X a=1
[s, a] hn, ai hs, ai hn, si
˜ sα˙ ∂ + η A ∂ λ s ˜ aα˙ ∂ηaA ∂λ
(1)
= S (1) + ηsA SA ,
(4.11)
while the sub-sub-leading soft operator consists of three parts: 1 A B (2) η η S , 2 s s AB n 1 X [s, a] ˜ ˜ ∂2 , = λsα˙ λsβ˙ ˜ aα˙ ∂ λ ˜ ˙ 2 a=1 hs, ai ∂λ aβ (2)
S (2) = S (2) + ηsA SA +
(4.12)
S (2)
(4.13)
(2)
SA = (2)
SAB =
n X [s, a] ˜ ∂2 λsα˙ , ˜ aα˙ ∂η A hs, ai ∂ λ a a=1 n X ∂2 [s, a] . hs, ai ∂ηaB ∂ηaA
(4.14) (4.15)
a=1
See Appendix B for some calculational details. Expanding superamplitude Mn+1 in Grassman odd variables ηs , we have A Mn+1 Φ1 , . . . , Φn , Φs = Mn+1 Φ1 , . . . , Φn , h+ s + ηs Mn+1 Φ1 , . . . , Φn , ψsA 1 + ηsA ηsB Mn+1 Φ1 , . . . , Φn , vsAB + · · · . 2
Thus we can express the soft theorem (4.9) in N = 8 SUGRA as 1 (0) 1 (1) 1 (2) + S + 2S + S soft graviton: Mn+1 . . . , hs (ǫ) = Mn + O(ǫ0 ), ǫ3 ǫ ǫ 1 (1) 1 (2) soft gravitino: Mn+1 . . . , ψsA (ǫ) = S + SA Mn + O(ǫ0 ), ǫ2 A ǫ 1 (2) soft gravi-photon: Mn+1 . . . , vsAB (ǫ) = SAB Mn + O(ǫ0 ), ǫ 0 soft gravi-photino: Mn+1 . . . , χsABC (ǫ) = + O(ǫ0 ), ǫ 0 soft scalar: Mn+1 . . . , SsABCD (ǫ) = + O(ǫ0 ). ǫ
(4.16)
(4.17) (4.18) (4.19) (4.20) (4.21)
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 the soft theorems by some examples in the MHV sector of N = 8 SUGRA in detail. We will pay special attention to the soft gravitino divergences and the soft gravi-photon divergences. – 13 –
4.2
MHV Examples
For soft graviton theorem, in particular leading and sub-leading orders, there are a great deal of study and investigation on both theoretical derivations and special examples check so far [11, 15, 35]. In this subsection, we check soft theorem of N = 8 SUGRA amplitudes by some examples of the MHV sector. We mainly focus on leading and sub-leading soft gravitino divergences and 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 identities4 : − + + + A + + A h1, si Mn+1 h− Mn+1 h− 1 , h2 , h3 , . . . , hn , hs , 1 , ψ2 , h3 , . . . , hn , ψsB = δB h1, 2i 2 AB h1, si AB + + − − + + + = δ , v , h , . . . , h , v Mn+1 h− sCD CD 2 n 1 3 2 Mn+1 h1 , h2 , h3 , . . . , hn , hs , h1, 2i 3 − − + + + ABC h1, si ABC + + = δ , χ , h , . . . , h , χ Mn+1 h− sDEF DEF 2 n 1 3 3 Mn+1 h1 , h2 , h3 , . . . , hn , hs , h1, 2i 4 − − + ABCD h1, si ABCD + + + + = δ , S , h , . . . , h , S Mn+1 h− sEF GH EF GH 2 n 1 3 4 Mn+1 h1 , h2 , h3 , . . . , hn , hs h1, 2i
(4.22) (4.23) (4.24) (4.25)
1 ABCDEF GH ǫ SEF GH . With the help of these Ward identities, we can study the soft 4! divergences of amplitudes involving soft lower-helicity particle by using only soft graviton theorem.
where S ABCD =
Soft gravitino Here we study the soft gravitino divergences. For the right hand side of SWI (4.22), by using the soft graviton theorem we have A δB
h1, si h1, 2i
1 (0) 1 (1) − + + 0 S + S Mn h− 1 , h2 , h3 , . . . , hn + O(ǫ ). 2 ǫ ǫ
(4.26)
First we consider the leading order O(ǫ−2 ): A δB
n−1 X h1, ai [s, a] hn, ai 1 h1, si (0) − − + − + + + A 1 h , h , h , . . . , h = δ S M Mn h− n 1 n B 2 2 3 1 , h2 , h3 , . . . , hn . 2 ǫ h1, 2i ǫ h1, 2i hs, ai hn, si
(4.27)
a=2
In order to get sub-leading order, substituting soft operator S (1) into the right hand side of eq. (4.26), we have n
A δB
X [s, a] h1, ai 1 h1, si (1) − + + A 1 ˜ sα˙ ∂ Mn h− , h− , h+ , . . . , h+ . (4.28) S Mn h− λ n 1 , h2 , h3 , . . . , hn = δB 1 2 3 ˜ aα˙ ǫ h1, 2i ǫ hs, ai h1, 2i ∂λ a=2
Here the gauge freedom of S (0) is fixed by taking x = y = 1. 4
Here the generalized Kronecker delta-symbol is defined as A1 ...An δB ≡ 1 ...Bn
X
A
A
sgn(σ) δB1σ(1) · · · δBnσ(n) .
σ∈Sn
– 14 –
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: n−1 1 X [s, a] hn, ai − A + h , ψ , . . . , Q Φ , . . . , h M aB a n 2 n 1 ǫ2 hs, ai hn, si a=1
1 = 2 ǫ
A = δB
n−1
X [s, a] hn, ai [s, 2] hn, 2i − + A A + Mn h− Mn h− 1 , h2 , . . . , hn δB + 1 , ψ2 , . . . , ψaB , . . . , hn hs, 2i hn, si hs, ai hn, si a=3
1 ǫ2
n−1 X a=2
[s, a] hn, ai h1, ai − + + Mn h− 1 , h2 , h3 , . . . , hn . hs, ai hn, si h1, 2i
(4.29)
This gives the same result as eq. (4.27). Here one has used the SUSY Ward identity (4.22) and the operator QaA is defined by QaA h+ a ≡ ψaA ,
QaA ψaB ≡ vaAB ,
B − QaA ψaB ≡ δA ha ,
QaA h− a ≡ 0, · · · .
(4.30)
Roughly speaking, it make spin (or helicity) of the particle reduce by one half when an operator QaA act on this particle. Similarly, applying straightforwardly sub-leading soft gravitino theorem (4.18) to the left hand side of the identity (4.22) gives n
1 X [s, a] ˜ ∂ Mn h− , ψ2A , . . . , QaB Φa , . . . , h+ λsα˙ n 1 ˜ aα˙ ǫ a=1 hs, ai ∂λ n
A = δB
1 X [s, a] h1, ai ˜ ∂ , h− , h+ , . . . , h+ . Mn h− λsα˙ n 1 2 3 ˜ aα˙ ǫ hs, ai h1, 2i ∂λ
(4.31)
a=2
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 holomorphic soft limit λs → ǫλs , using leading soft graviton theorem5 the right hand side of the SWI (4.23) becomes 1 AB h1, si2 (0) − + + 0 δCD S Mn h− 1 , h2 , h3 , . . . , hn + O(ǫ ) 2 ǫ h1, 2i n 1 AB X h1, ai2 [s, a] − − + + h , h , h , . . . , h + O(ǫ0 ). M = δCD n n 1 2 3 ǫ h1, 2i2 hs, ai
(4.32)
a=2
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 n 1 X [s, a] AB AB− 0 = , v , . . . , v Mn+1 h− , . . . , QaD QaC Φa , . . . , h+ Mn h− s CD 2 n + O(ǫ ) 1 1 , v2 ǫ hs, ai a=1 ! n X 1 [s, 2] [s, a] − − − AB + AB + Mn h1 , h2 , . . . , hn δCD + Mn h1 , v2 , . . . , vaCD , . . . , hn = + O(ǫ0 ) ǫ hs, 2i hs, ai a=3 n
=
1 AB X [s, a] h1, ai2 − + + 0 Mn h− δCD 1 , h2 , h3 , . . . , hn + O(ǫ ). 2 ǫ hs, ai h1, 2i a=2
5
It is necessary to notice that gauge freedoms in soft factor S (0) are fixed by setting x = y = 1.
– 15 –
(4.33)
Here one has used the Ward identity (4.23). Next we analyse two 4-point amplitudes which have been computed by using both generating function method proposed in [54] and Feynman diagrams respectively in [58]. The first example is 4-gravi-photon amplitude: M4 v
AB
,v
CD
, vEF , vGH
where s, t, u are Mandelstam variables
2
2
= h12i [34]
1 AB CD 1 AB CD 1 ABCD δ δ + δ δ + δEF GH t EF GH u GH EF s
(4.34)
s = (k1 + k2 )2 = (k3 + k4 )2 = s12 = s34 ,
(4.35)
t = (k1 + k3 )2 = (k2 + k4 )2 = s13 = s24 ,
(4.36)
u = (k1 + k4 )2 = (k2 + k3 )2 = s14 = s23 .
(4.37)
In holomorphic soft limit λ4 → ǫλ4 , the amplitude becomes M4 v AB , v CD , vEF , ǫ vGH
1 = ǫ 1 ǫ
=
! h12i2 [34]2 AB CD h12i2 [34]2 AB CD h12i2 [34]2 ABCD + + δ δ δ δ δ h24i [24] EF GH h14i [14] GH EF h34i [34] EF GH ! [14] h12i4 AB CD [34] h12i2 ABCD [24] h12i4 AB CD δ δ + δ δ + δEF GH (4.38) h24i h13i2 EF GH h14i h23i2 GH EF h34i
which is exact in ǫ. In the last line, one used the momentum conservation conditions: h12i [24] = − h13i [34] and h21i [14] = − h23i [34]. Notice that following 3-point amplitudes of N = 8 SUGRA: CD M3 h− 1 , v2 , v3EF
M3 v1AB , h− 2 , v3EF M3 v1AB , v2CD , S3EF GH Then one finds that M4 v AB , v CD , vEF , ǫvGH
=
1 ǫ
=
h12i4 CD δEF , h23i2
(4.39)
=
h12i4 AB δEF , h31i2
(4.40)
ABCD = h12i2 δEF GH .
(4.41)
CD AB [24] [14] CD M3 v1AB , h− M3 h− 2 , v3EF δGH + 1 , v2 , v3EF δGH h24i h14i ! [34] M3 v1AB , v2CD , S3EF GH . + (4.42) h34i
Very nice! This is just result from soft gravi-photon theorem.
Another example is 2-scalar 2-gravi-photon amplitude: M4 v AB ,vCD , SEF GH , SIJKL 3! AB 3! AB 1 AB δCD ǫEF GHIJKL + δ[EF ǫGH]IJKLCD + δ[IJ ǫKL]EF GHCD . = h13i2 [23]2 s t u
(4.43)
We study the property of this amplitude when external particle vCD becomes soft. In holomorphic soft limit λ2 → ǫλ2 , this amplitude becomes [24] 1 [12] h13i2 h14i2 AB AB AB h14i2 δ[EF ǫGH]IJKLCD δCD ǫEF GHIJKL + 3! M4 v , ǫ vCD , SEF GH , SIJKL = 2 ǫ h12i h34i h24i [23] 2 AB + 3! h13i δ[IJ ǫKL]EF GHCD . (4.44) h23i – 16 –
Here used the momentum conservation conditions:
h14i [23] = and h13i [23] = − h14i [24]. [12] h34i
Notice the 3-pt MHV amplitudes: M3 h− 1 , S3EF GH , S4IJKL
Then we have6
h13i2 h14i2 = ǫEF GHIJKL h34i2
ABM N M3 v1AB , v3M N , S4IJKL = h13i2 δIJKL ABM N M3 v1AB , S3EF GH , v4M N = h14i2 δEF GH
1 [21] AB δCD M3 h− M4 v AB , ǫ vCD , SEF GH , SIJKL = 1 , S3EF GH , S4IJKL ǫ h21i 1 [23] ǫM N EF GHCD M3 v1AB , v3M N , S4IJKL + 2 h23i 1 [24] + ǫM N IJKLCD M3 v1AB , S3EF GH , v4M N . 2 h24i
(4.45) (4.46) (4.47)
(4.48)
By applying straightforwardly the soft gravi-photon theorem (4.19) to amplitude M4 v AB , vCD , SEF GH , SIJKL , one can also obtain the same result. Soft gravi-photino and soft scalar According to SUSY Ward identities (4.24) and (4.25), one finds + ABC , h+ Mn+1 h− 3 , . . . , hn , χsDEF 1 , χ2
∼ O(ǫ0 ),
ABCD + , h3 , . . . , h+ Mn+1 h− n , SsEF GH ∼ O(ǫ) 1 , S2
(4.49) (4.50)
in the holomorphic soft limit λs → ǫλs . These results accord with the soft theorems (4.20) and (4.21). Next we study the 2-scalar 2-gravi-photon amplitude (4.43) which have been discussed previously. In the holomorphic soft limit λ4 → ǫλ4 , since h13i2 [23]2 = h14i2 [24]2 ∼ ǫ2 , s ∼ t ∼ u ∼ ǫ, this amplitude becomes M4 (v − v + φφ) λ4 →ǫλ4 ∼ ǫ → 0.
(4.51)
In MHV sector of N = 8 SUGRA, there are also the amplitudes involving four scalars. Here we analyse a 4-scalar amplitude which was computed in [58]: M4 SABCD , SEF GH , SIJKL , SM N P Q tu su st = ǫABCDEF GH ǫIJKLM N P Q + ǫABCDIJKL ǫEF GHM N P Q + ǫABCDM N P Q ǫEF GHIJKL s t u 1 X sgn(σ) s ǫ11 12 13 14 31 32 43 44 ǫ21 22 23 24 41 42 33 34 + t ǫ11 12 13 14 21 22 43 44 ǫ31 32 33 34 41 42 23 24 + 2(4!)3 σ + u ǫ11 12 13 14 21 22 33 34 ǫ41 42 43 44 31 32 23 24 . (4.52) 6
Here one has used a identity 1 ABM N AB δ ǫM NIJ KLCD = 3!δ[EF ǫGH]IJ KLCD . 2 EF GH
– 17 –
Here {11 12 13 14 } denotes permutations of {A, B, C, D} and so forth [58]. In holomorphic soft limit λ4 → ǫλ4 , since s ∼ t ∼ u ∼ ǫ, the amplitude behaves as M4 (φφφφ) λ4 →ǫλ4 ∼ ǫ → 0.
(4.53)
A great deal of research shows that amplitudes vanish in soft scalar limit, which indicates a hidden global E7(7) symmetry of classical N = 8 SUGRA [7, 54, 87, 90]. This is consist with our soft theorem (4.21). 4.3
SUGRA soft operators as double copy of SYM soft operators
As discussed in subsection 2.3, the gravity soft operators 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 ηaA as follows: S1η (s, a) ≡
1 ∂ ηsA A . hs, ai ∂ηa
(4.54)
Then the soft operators in N = 4 SYM may be written as (0)
(0)
SSYM = SYM = S0 (x, s, n) + S0 (x, s, 1), (1) SSYM = S1 (s, 1) − S1 (s, n) + S1η (s, 1) − S1η (s, n) .
(4.55) (4.56)
The ‘KLT-like formula’ of the leading soft factor S (0) in gravity has been obtained in section 2.3. The sub-leading soft operator may be written as: S (1) =
n 1X 0 ssa S (x, s, a) + S0 (y, s, a) S1 (s, a) + S1η (s, a) 2 a=1
(4.57)
where λx and λy are arbitrary reference spinors. The sub-sub-leading soft operator may be expressed as S (2) =
n 1X ssa S1 (s, a) + S1η (s, a) S1 (s, a) + S1η (s, a) . 2
(4.58)
a=1
As mentioned in subsection 2.3, all derivatives in operators only act on amplitudes. These relations may be derived using the scheme proposed in [34] by super-KLT relation [44–48].
5
Conclusion and discussions
In this work, the super soft theorems were investigated systematically in 4D maximally N = 4 superYang-Mills theory and N = 8 supergravity. We have presented the super soft theorems with rigorous proofs at tree level. The main results are eq. (3.19) for SYM and eq. (4.9) for SUGRA. In N = 4 SYM, several simple examples were examined and the results were in agreement with the super soft theorem exactly. In N = 8 SUGRA, employing the SUSY Ward identities, we verified the soft gravitino and soft gravi-photon divergences by using leading and sub-leading soft graviton theorem in MHV sector. Several four-point amplitudes were also checked in details.
– 18 –
There are several further topics that are fascinating for us. First, properties of amplitudes with 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 soft gravitino divergences and soft graviphotino 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 [53], all tree-level superamplitudes in N = 4 SYM were expressed as compact analytical formulas. By taking 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 [56]. 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 [15, 40–43] and leading and sub-leading soft-photon theorems in massless QED [32] 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.
– 19 –
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 [53]. First of all, we have to summarize briefly main results of Drummond and Henn’s paper [53]. Threepoint amplitudes are fundamental in BCFW-construction of higher-point amplitudes. In section 3, threepoint 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 (n > 3) MHV superamplitudes ˜ a , ηa } = AnMHV {λa , λ
δ4 (p)δ8 (q) . h12i h23i · · · hn1i
(A.1)
The n-point MHV superamplitude is simple and compact, just as Parke-Taylor formula of the pure gluonic amplitude. The delta functions δ4 (p) and δ8 (q) are consequences of translation invariance and supersymmetry. Therefore all tree-level superamplitudes, not just MHV, contain delta function factor δ4 (p)δ8 (q) in N = 4 SYM. So it is very convenient to factor out the MHV superamplitude, An = AnMHV Pn .
(A.2)
˜ a and Grassmann variables η A and one can express this quantity as Here Pn is a function of spinors λa , λ a following form: Pn = PnMHV + PnNMHV + · · · + PnMHV . Of course PnMHV = 1, and the Nk MHV function PnN
k MHV
(A.3)
has Grassmann degree 4k.
Turning to the NMHV sector, the function PnNMHV is given by [53] X PnNMHV = Rn;ab .
(A.4)
2≤a
