Next to subleading soft-graviton theorem in arbitrary dimensions

ICTP-SAIFR/2014-005

arXiv:1407.5982v1 [hep-th] 22 Jul 2014

Next to subleading soft-graviton theorem in arbitrary dimensions

Chrysostomos Kalousios,a Francisco Rojas,b

1

a

ICTP South American Institute for Fundamental Research Instituto de F´ısica Te´orica, UNESP-Universidade Estadual Paulista R. Dr. Bento T. Ferraz 271 - Bl. II, 01140-070, S˜ao Paulo, SP, Brasil b

Instituto de F´ısica Te´orica, UNESP-Universidade Estadual Paulista R. Dr. Bento T. Ferraz 271 - Bl. II, 01140-070, S˜ao Paulo, SP, Brasil

Abstract We study the soft graviton theorem recently proposed by Cachazo and Strominger. We employ the Cachazo, He and Yuan formalism to show that the next to subleading order soft factor for gravity is universal at tree level in arbitrary dimensions.

1

[email protected], [email protected]

1

Introduction

The study of the soft graviton amplitudes dates back to Weinberg [1, 2] where the leading soft behavior was obtained. In [3, 4, 5] a new soft graviton theorem, conjectured to be the Ward identities of a new symmetry of the quantum gravity S-matrix1 , has been proposed. Cachazo and Strominger [8] have recently shown that the new conjectured soft behavior, through subleading and next-to-subleading orders in the soft expansion, has a universal form in four spacetime dimensions at tree level2 . An extension to gluons for the first subleading soft behavior at tree level was reported in [13]. Using Feynman diagram techniques, the first subleading theorem has also been confirmed in [14]. In [15] it has been demonstrated that the conformal invariance of tree level gauge theory amplitudes in four spacetime dimensions determines the form of the first subleading theorem. Very recently it has been shown that the form of the first subleading term in the soft expansion in D dimensions is highly constrained by the requirements from Poincar´e symmetry and gauge invariance [16]. Subsequently, the authors of [17] have shown that on-shell gauge invariance determines the complete form of the first two subleading soft graviton theorems. Using the Cachazo, He, Yuan (CHY) formula [18], the universality of the soft behavior to first subleading order has been shown to hold in D dimensions [19, 20]. The purpose of the present note is to use the CHY formula to prove the universal nature of the next-to-subleading soft graviton theorem in arbitrary dimensions. Studies on loop corrections to subleading soft theorems have been presented in [21, 22, 23]. Progress in the context of string theory has been reported in [24, 25] and also in [26, 27] relevant for recent twistor constructions. More recent progress on soft theorems has appeared in [28, 29]. The conjecture of [8] states, for an on-shell tree level n-graviton amplitude Mn , that   1 (0) (1) (2) 2 Mn = S + S + λS + O(λ ) Mn−1 , (1) λ where n is taken to be the soft particle with momentum kn and we scale the momentum kn → λkn and take the limit when λ approaches to zero. In the above S

(0)

=

n−1 X ǫµν k µ k ν a a

a=1

kn·ka

(2)

is Weinberg’s soft theorem with ǫµν denoting the polarization tensor of the soft graviton and the gravitational constant has been set to 1. The conjectured forms of the subleading and next-to-subleading theorems are S

(1)

=

n−1 X ǫµν k µ knλJ λν a

a=1

kn·ka

a

n−1

,

S

(2)

1 X ǫµν knρ Jaρµ knλJaλν = . 2 a=1 kn·ka

(3)

In order to treat gluon and graviton polarizations on an equal footing one chooses to write the graviton polarization for the ath particle as ǫaµν → ǫaµ ǫaν 1

(4)

This new proposed symmetry is an extension of the Bondi, van der Burg, Metzner and Sachs (BMS) symmetry [6, 7] 2 Early results for soft photons at subleading order were obtained in [9, 10, 11, 12].

1

with ǫa ·ka = ǫa ·ǫa = 0. The subleading contributions to the soft theorem depend on the total angular momentum operator, which is 3 Jaµν = ka[µ

∂ ∂ + ǫ[µ a ∂kaν] ∂ǫaν]

(5)

for the ath particle. Note that in using this formula one should consider the polarization vectors ǫµa to be independent of the momenta kaµ . This paper is organized as follows. In Section 2 we review the CHY formalism [18] for tree level graviton amplitudes which is valid in arbitrary dimensions and, in this language, we set up the computation for the expansion of the amplitude up to nextto-subleading order in the soft parameter. We finish this section by stating the new soft theorem of Cachazo and Strominger [8] extended to D dimensions. In Section 3 we explicitly evaluate the tree level n-graviton amplitude at next-to-subleading order in the soft expansion. In Section 4 we compute the action of the conjectured S (2) operator (3) onto the (n − 1)-graviton amplitude, as stated in (1), and show that it perfectly matches with the next-to-subleading amplitude M (2) of Section 3, thus proving the theorem.

2

Review and setup of the problem

In this section we briefly review the CHY construction [18] for tree level graviton amplitudes. A key object is the scattering equations n X ka·kb b6=a

σab

= 0,

a, b = 1, . . . , n.

(6)

with σab ≡ σa − σb , where the σa are in general complex valued quantities. Due to the SL(2, C) symmetry of (6), these constitute a system of n − 3 independent equations for the set {σa } and one can arbitrarily fix three of the σa variables. We will call σi , σj , σk the three fixed σs. The gauge fixed amplitude is Z n Y Mn = [dσ]n−4 dσn δ(fan ) En , (7) a6=i,j,k

where we have employed the useful short notation fan

≡

n X ka·kb b6=a

σab

,

[dσ]n−4 ≡ (σpq σqr σrp )(σij σjk σki )

n Y

dσc .

(8)

c6=p,q,r

In the above, En is defined to be 2 En = 4 det(Ψxy xy )/σxy ,

(9)

where Ψxyz... xyz ′ ... is obtained from the 2n × 2n antisymmetric matrix Ψ after removing rows x, y, z, . . . and columns x, y, z ′ , . . . with 1 ≤ x < y ≤ n. The explicit expression of Ψ is given by   A −C T Ψ= (10) C B 3

We follow the convention A(µ Bν) = Aµ Bν + Aν Bµ and A[µ Bν] = Aµ Bν − Aν Bµ .

2

with the n × n matrices A, B, C given by Aab

ka·kb = δa6=b , σab

Bab

ǫa·ǫb = δa6=b , σab

n

X ǫa·kc ǫa·kb Cab = δa6=b − δab , σab σac c6=a

(11)

where we use δa6=b ≡ 1 − δab in order to avoid cluttering our equations. In [18] it was shown that the quantity En is independent of the choice of x and y. In order to expand the delta function appearing in (7) in powers of λ we separate it into two parts ! n−1 ! n−1 n−1 n X Y X Y k ·k k ·k k ·k 1 a b n b a n δ +λ δ(fan ) = δ λ σ σ σan nb ab b6=a a6=i,j,k b=1 a6=i,j,k (12)   1 = δ(fnn−1 ) δ (0) + δ (1) + λδ (2) + O(λ2 ), λ where we define δ (0) =

n−1 Y

δ(fan−1),

δ (1)

a6=i,j,k

δ (2)

n−1 X kl·kn ′ n−1 δ (fl ) = σln l6=i,j,k

"

n−1 Y

#

δ(fan−1 ) ,

a6=i,j,k,l

" # n−1 n−1 n−1 km·kn ′ n−1 Y 1 X kl·kn ′ n−1 X δ (fl ) δ (fm ) δ(fbn−1 ) = 2 l6=i,j,k σln σ mn m6=i,j,k,l b6=i,j,k,l,m   n−1 n−1 2 Y 1 X kl·kn ′′ n−1 + δ (fl ) δ(fbn−1 ). 2 l6=i,j,k σln b6=i,j,k,l

(13)

(14)

We also need to expand En in (7) to second order in λ En = En(0) + λEn(1) + λ2 En(2) + O(λ2 ).

(15)

Plugging (12) and (15) into (7) we get Mn =

1 (0) M + Mn(1) + λMn(2) + O(λ2 ), λ n

(16)

where Mn(0) Mn(2)

= =

Z

Z

[dσ]n−4 dσn δ(fnn−1 )δ (0) En(0) ,

Mn(1)

=

Z

[dσ]n−4 dσn δ(fnn−1 )(δ (1) En(0) + δ (0) En(1) ),

[dσ]n−4 dσn δ(fnn−1 )(δ (2) En(0) + δ (1) En(1) + δ (0) En(2) ). (17)

The soft theorem conjectures that the following equality should hold Mn(i) = S (i) Mn−1 ,

i = 0, 1, 2.

(18)

Weinberg’s soft theorem, i.e., Mn(0) = S (0) Mn−1 , can be derived as follows. To evaluate Mn(0) in (17) we also need En(0) , the leading contribution to the determinant (9), which is 3

2 En(0) = Cnn En−1 . In order to see that, we can set λ = 0 in En . Then all the elements of th the (n−2) row vanish apart from the last one which equals −Cnn . Similarly all elements of the (n − 2)th column are zero apart from the last one which is Cnn . Expansion of the determinant along the aforementioned row and column will yield another extra sign which completes the proof. Separating all the dependence on σn in Mn(0) , i.e., Z Z (0) (0) 2 Mn = [dσ]n−4 δ En−1 dσn δ(fnn−1 )Cnn , (19)

we can explicitly evaluate the integral over σn . We may treat the δ-distributions as poles and since the integrands are regular when σn → ∞ we evaluate them by deforming the contour and using the residue theorem. Doing this one obtains Z n−1 X (ǫn·ka )2 n−1 2 (20) dσn δ(fn )Cnn = . kn·ka a=1

Putting everything together into (19) yields Z n−1 n−1 Y X (ǫn·ka )2 (0) [dσ]n−4 δ(fln−1)En−1 Mn = k ·k n a a=1 l6=i,j,k =

n−1 X (ǫn·ka )2 a=1

kn·ka

(21)

Mn−1 .

From (2) one can easily see that S (0) Mn−1 is precisely the last line of (21), thus proving Weinberg’s leading soft-graviton theorem. The computation of (18) for i = 1 in arbitrary dimensions was performed in [19, 20]. In the next section we start the computation of the next to subleading soft contribution (i = 2) by evaluating Mn(2) in (17). Then, in Section 4, we will evaluate the action of S (2) on Mn−1 . We will compare both sides of (18) by matching terms that contain the same support from the δ-distributions and we will find perfect matching, thus, proving the theorem.

3

Evaluation of Mn(2)

We split the evaluation of Mn(2) into three parts Z Z (2) Mn = [dσ]n−4 (m1 + m2 + m3 ) , mi = dσn δ(fnn−1)δ (3−i) En(i−1) . 3.1

(22)

Evaluation of m1

Using (14), the first contribution, m1 , to Mn(2) is n−1 n−1 X X 1 n−1 ′ n−1 δ ′ (fm ) δ (fl ) m1 = En−1 2 m6=i,j,k,l

l6=i,j,k

1 + En−1 2

n−1 X

δ ′′ (fln−1 )

n−1 Y

b6=i,j,k,l

l6=i,j,k

4

n−1 Y

b6=i,j,k,l,m

δ(fbn−1 ) I2 ,

δ(fbn−1 ) I1 (23)

where we have isolated the integration over σn to the following integral Z C2 I = kl·kn km·kn dσn δ(fnn−1) nn . σnl σnm Therefore, in (23), we have I1 = I|m6=l and I2 = I|m=l . We now move on to compute the integral (24). We find ! # (" n−1 n−1 X (ǫn·kl )2 km·kn ǫn·kc km·kn ǫn·kl ǫn·kl X kn·kc − −2 + (l ↔ m) I= 2 σml kn·kl c6=l σlc σ σ lc ml c6=l ) n−1 X (ǫn·kc )2 + kl·kn km·kn δm6=l σlc σmc kn·kc c6=l,m ( n−1 n−1 X X (ǫn·kc )2 ǫn·kl kn·kc − 2 ǫn·kc kn·kl + (kl·kn )2 + (ǫ ·k ) n l 2 σlc kn·kc σlc2 c6=l c6=l !2 ) n−1 n−1 X ǫn·kl X kn·kc ǫn·kc δml − + kl·kn σlc kn·kl σlc

(24)

(25)

c6=l

c6=l

The first line in (25) is the contribution of a double pole at σn = σl and a double pole at σn = σm , whereas the second line in (25) comes from the contribution of a single pole of the integrand at σn = σc , for all c 6= l, m, n. The first term in the third line comes from a single pole at σn = σc for all c 6= l, n and the remaining of (25) comes from a third order pole at σn = σl . 3.2

Evaluation of m2

For the evaluation of m2 we need to expand (9) to order λ. The derivative of the determinant of a n × n matrix with entries Tab can be obtained from the formula n

n

XX d dTab a det(T ) = (−1)a+b Mb , dλ dλ a=1 b=1

(26)

where Mba denotes the determinant of the matrix obtained by removing the ath row and the bth column of T . Applying it onto En in equation (9) yields  n n  dEn X X a+b dAab ˜a a+b+n dCab ˜n+a a+b dBab ˜n+a (−1) = ψb + 2(−1) ψb + (−1) ψn+b . dλ dλ dλ dλ a=1 b=1

(27)

Here we have used the short notation 4 det(Ψ12a 12b ) ψ˜ba ≡ δa6={1,2} δb6={1,2} . 2 σ12

(28)

For convenience and without loss of generality we have chosen to remove the first two rows and the first two columns in (9). In (27) we have also used the identity ψ˜ba = −ψ˜ab . 5

The derivatives of the different matrix elements are 1 dBab dAab = (δan kn·kb + δbn ka·kn ) δa6=b , = 0, dλ σab dλ dCab ǫa·kn ǫa·kn = δbn δa6=b − δab δa6=n . dλ σab σan

(29)

Putting this into (27) yields n−1  X 1  dEn a+n n a n n−1 a ˜ ˜ ˜ (−1) ka·kn ψa + (−1) ǫa·kn ψn+a + (−1) ǫa·kn ψn+a , =2 dλ σ a=1 na

(30)

a where we have also used the identity ψ˜an+a = −ψ˜n+a . Note that all the dependence in λ ˜ is now contained in the ψ determinants only, which also need to be evaluated at λ = 0 at the end. We further need to isolate any encounter of σn in (30), since we eventually want to integrate over that variable. We find

 n−1  X n+b ǫn·kb a b ǫn·ǫb a (−1) ψb − (−1) ψn+b−1 , σ σ nb nb b=1  n−1  X n+b ǫn ·kb n+a−1 b ǫn ·ǫb n+a−1 (−1) = −Cnn ψb − (−1) ψn+b−1 , σ σ nb nb b=1

ψ˜an = Cnn n ψ˜n+a

(31)

a 2 a ψ˜n+a = −Cnn ψn+a−1 ,

where we have dropped the tilde sign to denote the further removal of the rows and columns that contain the variable σn , that is ψba denotes the determinant En−1 after the removal of the ath row and the bth column. Then En(1)

n−1 X n−1 X (−1)a+b  n+a−1 kn·ka ǫn·kb ψba − kn·ǫb ǫn·ǫa ψn+b−1 = 2Cnn σna σnb a=1 b=1  a + (−1)n (ǫn·ka kn·ǫb − kn·ka ǫn·ǫb ) ψn+b−1

+

2 2Cnn

n−1 X (−1)n a=1

σna

(32)

a ǫa·kn ψn+a−1 .

We recall that m2 takes the form m2 =

Z

dσn δ(fnn−1 )

n−1 n−1 X kn·kl ′ n−1 Y n−1 δ (fl ) δ(fm )En(1) σ ln l6=i,j,k m6=i,j,k,l

thus, we will need the following integrals Z I3 ≡ −kn·kl dσn δ(fnn−1) I4 ≡ −kn·kl

Z

Cnn , σnl σna σnb

(34)

2 Cnn . σnl σna

(35)

dσn δ(fnn−1 ) 6

(33)

The integral I4 is directly obtained from (25) since I4 = −(kn·ka )−1 I|m=a . For I3 we find   ǫn·ka 1 I3 = kn·kl + (a ↔ l) + (a ↔ b) δl6=a δl6=b δa6=b kn·ka σal σab " !   n−1 ǫn·kl 1 ǫn·kc kn·kc ǫn·ka 1 ǫn·ka 1 X 1 + − + kn·kl − δl6=a δab 2 2 kn·ka σal c6=a σac ǫn·ka kn·ka kn·kl σal kn·ka σal # + (l ↔ a) + ({a, l, b} → {l, b, a})

+ ǫn·kl

"

(36)

n−1 n−1 X n−1 X kn·kc ǫn·kd 1 1 X ǫn·kc − − 2 ǫn·kl c6=l σlc kn·kl ǫn·kl c6=l d6=l σlc σld !2  n−1 n−1 X X kn·kc kn·kc  1 1 δab δbl . + + 2 kn·kl c6=l σlc (kn·kl )2 c6=l σlc

As a check, note that from this expression the quantity I3 /(kn·kl ) is symmetric under the exchange of any two pairs of (l, a, b) which is evident from the original definition in (34). We now write m2 making explicit the linear combination of the different types of minors we have, i.e., m2 = 2

n−1 X

δ

′

(fln−1)

l6=i,j,k

n−1 Y

n−1 δ(fm )Dl ,

(37)

m6=i,j,k,l

where Dl ≡

n−1 X n−1 X

a6=l b6=l,a

+

n−1 X


n+a−1 a c1 ψba + c2 ψn+b−1 + c3 ψn+b−1 I3{l6=a, l6=b, a6=b}

a c4 ψn+a−1 +

a6=l

+

l c9 ψl+n−1 .

n−1 X a6=l


n+a−1 a l c5 ψla + c6 ψn+l−1 + c7 ψn+l−1 + c8 ψn+a−1 I3{l=b, l6=a}

(38)

The coefficients ci are c1 = (−1)a+b kn·ka ǫn·kb ; c2 = −(−1)a+b ǫb·kn ǫn·ǫa ; c3 = (−1)n (ǫn·ka ǫb·kn − kn·ka ǫn·ǫb ) ; c4 = (−1)n (ǫn·ka kn·ǫa − kn·ka ǫn·ǫa ) I3{a=b, l6=a} + (−1)n ǫa·kn I4{l6=a} ; c5 = (−1)a+l (kn·ka ǫn·kl − kn·kl ǫn·ka ) ; c6 = (−1)a+l (ǫa·kn ǫn·ǫl − ǫl·kn ǫn·ǫa ) ; c7 = (−1)a+l+n (ǫn·ka ǫl·kn − kn·ka ǫn·ǫl ) ; c8 = (−1)a+l+n (ǫn·kl ǫa·kn − kn·kl ǫn·ǫa ) ; c9 = (−1)n (ǫn·kl kn·ǫl − kn·kl ǫn·ǫl ) I3{l=a=b} + (−1)n ǫl·kn I4{l=a} . In the above we have used 3.3

ψba

=

−ψab ,

a ψn+b

=

−ψan+b ,

n+a ψn+b

=

n+b −ψn+a ,

ψaa

=

n+a ψn+a

(39) = 0.

Evaluation of m3

ab ab We define ψ˜cd and ψcd to be respectively the determinants En and En−1 after the removal of the rows a, b and the columns c, d.

7

For the evaluation of m3 we need to take the second derivative of (9) with respect to λ. From (30) we have ! n−1 ˜n ˜a X ˜n d ψ d ψ d ψ d2 En 1 (−1)a+n ka·kn a + (−1)a ǫa·kn n+a + (−1)n−1 ǫa·kn n+a . =2 dλ2 σ dλ dλ dλ a=1 na (40) With the definition θij to be 0 when i > j and −1 when i < j we find n−1  dψ˜an X 1  cn cn n,n+c (−1)n+c−1 kc·kn ψ˜an + (−1)n−1 ǫc·kn ψ˜a,n+c + (−1)c ǫc·kn ψ˜an = dλ σ c=1 cn

+

n−1 X

(−1)n+θac

c6=a

n dψ˜n+a = dλ

n−1 X c=1

ǫc·kn ˜n,n+c ψ , σcn ac

 1  n,n+c n,n+c cn (−1)n+c kc·kn ψ˜n,n+a + (−1)c−1 ǫc·kn ψ˜n,a+n + (−1)n ǫc·kn ψ˜c,n+a σcn +

n−1 X

(−1)n+θac

c6=a

ǫc·kn ˜nc ψ , σcn n+a,n+c

n−1  a X dψ˜n+a 1  a,n+c an an (−1)n+c kc·kn ψ˜c,n+a + (−1)n ǫc·kn ψ˜c,n+a + (−1)c−1 ǫc·kn ψ˜n+a,n+c = dλ σ c=1 cn

n−1  X (−1)θac  n+c ac n+θac ac c an ˜ ˜ ˜ + (−1) kc·kn ψn,n+a + (−1) ǫc·kn ψn+a,n+c + (−1) ǫc·kn ψn+a,n+c . σcn c6=a

(41) We can further expand the n and 2n rows and columns of the minors appearing in (41). ab cd With the help of the identity ψcd = ψab we arrive at the following result 2 En(2) = Cnn A1 + Cnn A2 + A3 ,

where A1 =

n−1 X n−1 X ǫa·kn ǫc·kn a=1 c=1

σna σnc

a,n+c−1 ψc,n+a−1

+

n−1 X n−1 X ǫa·kn ǫc·kn a=1 c6=a

σna σnc

(42)

ac ψn+a−1,n+c−1 ,

(43)

n−1 X n−1 X n−1 X (−1)b+c a,n+b−1 ǫa·kn (ǫb·kn ǫn·kc − kc·kn ǫb ǫn )ψc,n+a−1 A2 =2 σ σ σ a=1 c=1 b=1 na nb nc

+2

n−1 X n−1 X n−1 X (−1)b+c+n+θab a=1 c=1 b6=a

+2

σna σnb σnc

ǫa·kn

(44)

  a,n+c−1 ab (kb·kn ǫn·kc − kc kn ǫn·kb )ψc,n+a−1 + (ǫb·kn ǫc·ǫn − ǫb·ǫn ǫa·kn )ψn+a−1,n+b−1

n−1 X n−1 X n−1 X (−1)b+c+θab+θac a=1 c6=a b6=a

σna σnb σnc

ab ǫa·kn (ǫc·kn ǫn·kb − kb·kn ǫc·ǫn )ψn+a−1,n+c−1 ,

8

A3 =

n−1 X n−1 X n−1 X n−1 X (−1)a+b+c+d+θad +θcb a=1 c=1 b6=a d6=a

+2

n−1 X n−1 X n−1 X n−1 X (−1)a+b+c+d+n+θad a=1 c=1 b=a d6=a

+2

n−1 X n−1 X n−1 X n−1 X a=1 c=1 b6=c d6=a

+

σna σnb σnc σnd

σna σnb σnc σnd

bc ka·kn kc·kn ǫn·kb ǫn·kd ψad n+b−1,n+c−1 +ǫb·ǫn ǫd·ǫn ǫa·kn ǫc·kn ψn+a−1,n+d−1



ad ka·kn ǫn·kd (kb·kn ǫc·ǫn − ǫn·kb ǫc·kn )ψb,n+c−1

(−1)a+b+c+d+θad +θcb ad ka·kn ǫb·ǫn ǫc·kn ǫn·kn ψn+b−1,n+c−1 σna σnb σnc σnd

n−1 X n−1 X n−1 X n−1 X (−1)a+b+c+d (ǫn·ka ǫn·kb ǫc·kn ǫd·kn + ka·kn kb·kn ǫc·ǫn ǫd·ǫn σ σ σ σ na nb nc nd a=1 c=1 b=1 d=1 b,n+c−1 −ka·kn ǫn·kb ǫc·kn ǫd·ǫn − ǫn·ka kb·kn ǫc·ǫn ǫd·kn ) ψa,n+d−1

+2

n−1 X n−1 X n−1 X n−1 X (−1)a+b+c+d+n+θad a=1 c=1 b=1 d6=a

σna σnb σnc σnd

b,n+c−1 ǫa·kn ǫd·kn (kb·kn ǫc ǫn − ǫn·kb ǫc·kn )ψn+a−1,n+d−1 .

(45) In order to finish the calculation of m3 the only new integral we need to evaluate is Z 1 , (46) I5 = dσn δ(fnn−1 ) σna σnb σnc σnd

for which we obtain

 !2  n−1 n−1 X X kn·kl  kn·kl 1 1  δab δbc δcd + I5 = 2 2 (kn·ka ) σal kn·ka σal l6=a l6=a " # ) ( n−1 1 X kn·kl 1 1 1 δa6=b δbc δcd + cyclic {a, b, c, d} + − kn·kb σab kn·kb σbl σab l6=b ( ) 1 1 + δab δc6=d + cyclic{b, c, d} σac σad kn·ka ) ( 1 1 δa6=b δcd + cyclic{b, c, d} . + σbd σba kn·kb

4

(47)

Action of S (2) on the amplitude

From (3), the complete expression for S (2) including the spin contribution can be written as (2) (2) (2) S (2) = Sorb + Sso + Sspin , (48) where the orbital, spin-orbit and spin parts are respectively given by n−1

(2) Sorb

1 X orb ∂2 = , K 2 a=1 aµν ∂kaµ ∂kaν

(2) Sso

=

n−1 X

so Kaµν

a=1

9

∂2 , ∂kaµ ∂ǫaν

n−1

(2) Sspin

1 X spin ∂2 = , K 2 a=1 aµν ∂ǫaµ ∂ǫaν (49)

with (ǫn·ka )2 knµ knν , kn·ka ǫn·ka ǫn·ǫa ǫn·ka ǫa·kn ǫnν knµ + knµ knν , ≡ǫa·kn ǫnµ ǫnν − ǫn·ǫa ǫnµ knν − kn·ka kn·ka (ǫa·kn )2 ǫa·kn ǫn·ǫa (ǫn·ǫa )2 ≡ ǫnµ ǫnν − ǫn(µ knν) + knµ knν . kn·ka kn·ka kn·ka

orb Kaµν ≡kn·ka ǫnµ ǫnν − ǫn·ka ǫn(µ knν) + so Kaµν spin Kaµν

Then the action of S (2) on the amplitude is Z n−1 Y (2) (2) δ(fln−1)En−1 [dσ]n−4 S Mn−1 = S l6=i,j,k

=

Z

(50)

(51)

[dσ]n−4 (s1 + s2 + s3 + s4 ) ,

where we have separated the calculation into the following four parts s1 =

(2) En−1 Sorb n−1 X

n−1 Y

δ(fln−1 ), l6=i,j,k

s2 =

n−1 X

orb ∂En−1 Kaµν ∂kaµ a=1

n−1 Y so ∂En−1 ∂ s3 = Kaµν δ(fln−1), ∂ǫ ∂k aν aµ a=1 l6=i,j,k

s4 =

n−1 Y

n−1 ∂ Y δ(f n−1 ), ∂kaν l6=i,j,k l

(52)

δ(fln−1) S (2) En−1 .

l6=i,j,k

In the subsequent computations we will make use of the identities n

X kµ ∂fln kµ d = l δl6=a + δla , ∂kaµ σla σ ld d6=l

∂ 2 fln = 0, ∂kaµ ∂kaν

(53)

and also n−1

X 1
∂En−1 a b (−1)a+b kb µ ψba + (−1)a+b+n+1 ǫb µ ψn+b−1 + (−1)n ǫb µ ψn+b−1 =2 , ∂kaµ σab b6=a

∂En−1 =2 ∂ǫaµ

n−1 X b6=a


1 b a n+a−1 . (−1)a+b+n kb µ ψn+a−1 + (−1)n+1 kb µ ψn+a−1 + (−1)a+b ǫb µ ψn+b−1 σab

(54) In the following we omit the upper index of the scattering equations fln−1 and we simply write them as fl . 4.1

Evaluation of s1

We find n−1 n−1 n−1 n−1 X Y X X 1 orb ∂fm ∂fl ′ ′ Kaµν δ(fb ) δ (fm ) δ (fl ) s1 = En−1 2 ∂kaµ ∂kaν a=1 l6=i,j,k

b6=i,j,k,l,m

m6=i,j,k,l

1 + En−1 2

n−1 X

δ ′′ (fl )

l6=i,j,k

10

n−1 Y

n−1 X

orb ∂fl ∂fl Kaµν δ(fb ) . ∂k ∂k aµ aν a=1

b6=i,j,k,l

(55)

After some straightforward algebra and using (50) and (53) we obtain n−1 X

orb Kaµν

a=1

∂fm ∂fl = δml I2 + δm6=l I1 . ∂kaµ ∂kaν

(56)

thus, comparing with (23), we obtain the desired result s1 = m1 . 4.2

Evaluation of s2 and s3

The combination s2 + s3 has the same delta function support as m2 , thus, we will compare these two expressions. For s2 we obtain n−1 X

s2 =

′

δ (fl )

l6=i,j,k

n−1 Y

δ(fm )

m6=i,k,j,l

n−1 X

orb Kaµν

∂En−1 ∂fl ∂kaµ ∂kaν

(57)

so Kaµν

∂En−1 ∂fl ∂ǫaν ∂kaµ

(58)

a=1

and for s3 we get s3 =

n−1 X

δ ′ (fl )

l6=i,j,k

n−1 Y

δ(fm )

m6=i,k,j,l

n−1 X a=1

After some tedious but straightforward algebra and using (50), (53) and (54) we can expand s2 + s3 in the same form of m2 as shown in (37) and (38). We have explicitly computed each of the coefficients of the corresponding expansion for s2 + s3 and see that they all precisely match those of (39), thus, arriving at s2 + s3 = m2 as expected. 4.3

Evaluation of s4

Having matched all the previous terms on both sides, our last task is to show that s4 = m3 . From (52), (48) and (49) we have s4 =

n−1 Y

δ(fln−1)

l6=i,j,k

n−1  X 1 a=1

2

orb Kaµν

 2 1 spin ∂ 2 En−1 ∂ 2 En−1 so ∂ En−1 + Kaµν + K . ∂kaν ∂kaµ ∂ǫaν ∂kaµ 2 aµν ∂ǫaν ∂ǫaµ

(59)

Appropriately differentiating (54) we find n−1 n−1

XX 1 ∂ 2 En−1 ac (−1)b+c+θba +θac kb µ kc ν ψab =2 ∂kaν ∂kaµ σ σ ab ca b6=a c6=a

ac ab +(−1)a+b+n+θac (kb µ ǫc ν + ǫc µ kb ν )ψb,n+c−1 + (−1)b+c+n+θba (kb µ ǫc ν + ǫc µ kb ν )ψa,n+c−1  a,n+c−1 b,n+c−1 a,n+c−1 + (−1)b+c ψa,n+b−1 ) +(−1)a+b (ǫb µ ǫc ν + ǫc µ ǫb ν )ψc,n+b−1 − ǫb µ ǫc ν (ψc,n+b−1

+2

n−1 X n−1 X

b6=a c6=a,b

+(−1)

a+b+θbc +θac

1 bc (−1)a+b+n+θcb (kb µ ǫc ν + ǫc µ kb ν )ψa,n+c−1 σab σca


ac bc (ǫb µ ǫc ν + ǫc µ ǫb ν )ψn+b−1,n+c−1 − ǫb µ ǫc ν ψn+b−1,n+c−1 , 11

(60)

n−1 X n−1 X 1  ∂ 2 En−1 ac ab (−1)b+n kb µ kc ν ((−1)c+θca ψb,n+a−1 + (−1)a+θab ψa,n+a−1 ) =2 ∂ǫaν ∂kaµ σab σac b6=a c6=a a,n+a−1 a,n+c−1 +(−1)b+c (kb µ ǫc ν − ǫc µ kb ν )ψb,n+c−1 − (−1)b+c kb µ ǫc ν ψb,n+a−1 ac +ǫb µ kc ν ((−1)b+c+θab+θac − 1)ψn+a−1,n+b−1 a,n+a−1 b,n+a−1 b,n+a−1 +ǫb µ kc ν ((−1)a+b ψa,n+b−1 + (−1)a+c ψc,n+b−1 − ψa,n+b−1 ) a,n+c−1 b,n+c−1 +ǫb µ ǫc ν (−1)c+n+θab ((−1)b ψn+a−1,n+b−1 − (−1)a ψn+a−1,n+b−1 )

+2

2

n−1 X n−1 X

1  a,n+a−1 (−1)b+c+n+θcb kb µ kc ν ψbc σ σ ab ca b6=a c6=a,b

i

b,n+a−1 a,n+a−1 +(−1)a+c+n+θcb ǫb µ ǫc ν ψn+b−1,n+c−1 + (−1)b+c+n+θcb ǫb µ ǫc ν ψn+b−1,n+c−1  bc +(−1)a+c+θba +θbc ǫb µ kc ν ψn+a−1,n+b−1 , n−1 X n−1 X

∂ En−1 =2 ∂ǫaν ∂ǫaµ b6=a

c6=a

1 σab σac

  h a,n+a−1 b,n+a−1 + ψa,n+a−1 kb µ kc ν (−1)b+c ψc,n+a−1

n+a−1,n+c−1 a,n+a−1 +(−1)b+c+θba ǫb µ ǫc ν ψn+a−1,n+b−1 − (−1)a+b (kb µ kc ν + kc µ kb ν )ψb,n+a−1 i  b,n+a−1 a,n+a−1 . +(−1)c+n+θca (kb µ ǫc ν + ǫc µ kb ν ) (−1)b ψn+a−1,n+c−1 − (−1)a ψn+a−1,n+c−1

(61)

(62)

We now have all the ingredients to perform the comparison of s4 with m3 . The algebra is tedious but straightforward. We have performed the analysis and found agreement of the two expressions which completes the proof of the soft-graviton theorem. Acknowledgments We thank Wei He for useful discussions. We also thank Anastasia Volovich and Michael Zlotnikov for correspondence. In the final stages of the writing of this paper we learned of the work of Michael Zlotnikov [30] who was also embarked on our same computations. C.K. would like to thank the organizers of the Simons workshop during which this project was completed. The work of C.K. is supported by the S˜ao Paulo Research Foundation (FAPESP) under grants 2011/11973-4 and 2012/00756-5. The work of F.R is supported by FAPESP grant 2012/05451-8.

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