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Sphere-level Ramond-Ramond couplings in Ramond-Neveu-Schwarz formalism
Hamid R. Bakhtiarizadeh, Mohammad R. Garousi
Department of Physics, Ferdowsi University of Mashhad, P.O. Box 1436, Mashhad, Iran Received 15 March 2014; received in revised form 4 May 2014; accepted 5 May 2014
Editor: Stephan Stieberger
Abstract
We calculate in detail the sphere-level scattering amplitude of two Ramond-Ramond (RR) and two Neveu-Schwarz-Neveu-Schwarz (NSNS) vertex operators in type II superstring theories in Ramond-Neveu-Schwarz (RNS) formalism. We then compare the expansion of this amplitude at order a'3 with the eight-derivative couplings of the gravity and B-field that have been recently found based on S-dual and T-dual Ward identities. We find exact agreement. Moreover, using the above S-matrix element, we have found various couplings involving the dilaton field, and shown that they are also fully consistent with these Ward identities.
© 2014 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3.
1. Introduction
Many aspects of superstring theory can be captured by studying its low energy supergravity effective action. The stringy effects, however, appear in higher-derivative and genus corrections to the supergravity. These corrections may be extracted from the most fundamental observables in the superstring theory which are the S-matrix elements [1,2]. These objects have various hidden structures such as the Kawai-Lewellen-Tye (KLT) relations [3] which connect sphere-level S-matrix elements of closed strings to disk-level S-matrix elements of open string states. There
E-mail addresses: hamidreza.bakhtiarizadeh@stu-mail.um.ac.ir (H.R. Bakhtiarizadeh), garousi@um.ac.ir (M.R. Garousi).
http://dx.doi.Org/10.1016/j.nuclphysb.2014.05.002
0550-3213/© 2014 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.Org/licenses/by/3.0/). Funded by SCOAP3.
1 are similar relations between disk-level S-matrix elements of open and closed strings and disk- 1
2 level S-matrix elements of only open string states [4-6]. On the other hand, the S-matrix elements 2
3 should expose the dualities of superstring theory [7,8] through the corresponding Ward identi- 3
4 ties [9]. These identities can be used as generating function for the S-matrix elements, i.e., they 4
5 establish connections between different elements of the scattering amplitude of n supergravitons. 5
6 Calculating one element explicitly, then all other elements of the S-matrix may be found by the 6
7 Ward identities [10-12]. 7
8 Alternatively, the effective actions may be calculated directly by implementing the string du- 8
9 alities. The consistency of the effective action of type IIB superstring theory with S-duality has 9
10 been used in [13-30] to find genus and instanton corrections to the four Riemann curvature 10
11 corrections to the type IIB supergravity. The consistency of non-abelian D-brane action with T- 11
12 duality has been used in [31] to find various commutators of the transverse scalar fields in the 12
13 D-brane effective action. They have been verified by the corresponding S-matrix elements in 13
14 [32]. Using the consistency of the effective actions with the string dualities, some of the eight- 14
15 derivative corrections to the supergravity and four-derivative corrections to the D-brane/O-plane 15
16 world-volume effective action have been found in [33-39]. The complete list of the couplings at 16
17 these orders which are fully consistent with the string dualities, however, are still lacking. 17
18 The effective actions of type II superstring theories at the leading order of a' are given by the 18
19 type II supergravities which are invariant under the string dualities. The first higher-derivative 19
20 correction to these actions is at eight-derivative order. The Riemann curvature corrections to the 20
21 supergravities, t8t8R4, have been found in [1] from the a'-expansion of the sphere-level S-matrix 21
22 element of four graviton vertex operators. These couplings have been extended in [2] to include 22
23 all other couplings of four NSNS states by extending the Riemann curvature to the generalized 23
24 Riemann curvature, i.e., 24
26 RabCd = RabCd [C$;b]" ] + 2e-*o/2 Hab[c;d] (1) 26
27 2 27
28 where R is the generalized Riemann curvature. The resulting couplings are fully consistent with 28
29 the corresponding S-matrix elements. The S-matrix elements of four massless states in the type II 29
30 superstring theories have only contact terms at order a'3. The S-dual and T-dual Ward identities 30
31 of the S-matrix elements then dictate that the NSNS couplings must be combined with the appro- 31
32 priate RR couplings to be consistent with these Ward identities. This guiding principle has been 32
33 used in [36] to find various on-shell couplings between two RR and two gravity/B-field states 33
34 and between four RR states at order a'3. 34
35 The dilaton term in (1) is canceled when transforming it to the string frame [37,38]. As a result, 35
36 there is no on-shell dilaton coupling between four NSNS fields in the string frame. It has been 36
37 speculated in [38] that there may be no dilaton couplings between all higher NSNS fields at 37
38 eight-derivative level. As we will see in this paper, however, the consistency of the couplings 38
39 t8t8R4 with the S-dual and T-dual Ward identities produces various non-zero couplings between 39
40 the dilaton and the RR fields in the string frame. In this paper, we are going to examine these 40
41 couplings as well as the couplings found in [36] with the explicit calculation of the sphere-level 41
42 S-matrix element of two RR and two NSNS vertex operators in the RNS formalism. 42
43 The outline of the paper is as follows: We begin with Section 2 which is the detail calculations 43
44 of the sphere-level S-matrix element of two RR and two NSNS vertex operators in the RNS 44
45 formalism. In Section 3, we compare the contact terms of these amplitudes at order a'3 for the 45
46 gravity/B-field with the corresponding couplings that have been found in [36]. In Section 4, 46
47 we study the dilaton couplings. Using the T-dual and S-dual Ward identities on the S-matrix 47
1 element of RR five form field strength, we find various couplings at order a'3 in both the string 1
2 and Einstein frames. We show that the dilaton couplings in the Einstein frame are fully consistent 2
3 with the corresponding S-matrix element at order a'3. In Section 5, we briefly discuss our results. 3
5 2. Scattering amplitude 5
7 The scattering amplitude of four RR states or two RR and two NSNS states in the pure spinor 7
8 formalism have been calculated in [40]. In this section, we are going to calculate the scattering 8
9 amplitude of two RR and two NSNS states in the RNS formalism [46,47]. In this formalism, 9
10 the tree level scattering amplitude of two RR and two NSNS states is given by the correlation 10
11 function of their corresponding vertex operators on the sphere world-sheet. Since the background 11
12 superghost charge of the sphere is = 2, one has to choose the vertex operators in the appro- 12
13 priate pictures to produce the compensating charge = -2. One may choose the RR vertex 13
14 operators in (-1/2, -1/2) picture, one of the NSNS vertex operators in (-1, -1) and the other 14
15 one in (0,0) picture. The final result, should be independent of the choice of the ghost picture. 15
16 Using the above picture for the vertex operators, the scattering amplitude is given by the 16
17 following correlation function [46,47]: 17
18 18 4 / 2 \
19 2 /fT v(-1/2,-1/2), - ,..(-1,-1), ,v(0,0), \ 19
^~/ nd2z- n yR-R1/2,-1/2)(zJ,-ZJ)VN^1i1)(Z3,-Z3)VN^Ns(Z4,-Z4)
20 ^ /11 M\1I RR 'NSNS v^3^3/'NSNS^4''-I'1 20
21 i = 1 j = 1 ' 21
22 where the vertex operators are1 22
24 VR-1/2,-1/2)(Zjj 24
26 = (P- rj(n))AB : e-*(zj)/2SA(Zj)e'kj: e-^j)/2SB(~Zj)e'kj: 26
27 vN-NS-1)(z3, *3) = ^v : e-*(z3)f\z3)eik3-X(z3) : v(hW*3*™ : 27
28 vNSNS(z4,-4) = S4ap : {BXa(z4) + i*4 • ffa (z4))eik4X(z4) : 29
30 x (dXP(Z4) + ik4 • f f P(Z4))eik4• X(z4) : (3) 30
32 where the indices A, B,. • • are the Dirac spinor indices and P- = |(1 - yu) is the chiral pro- 32
33 jection operator which makes the calculation of the gamma matrices to be with the full 32 x 32 33
34 Dirac matrices of the ten dimensions. The RR polarization tensors e(n-1), e2n-1) appear in r1(n), 34
35 r2(n) which are defined as 35
36 a 36
37 ri(n) = an(Fi)^v. •n (4) 37
39 where the n-form (Fi)^1...fln = 1 (dCi)^1...fln is the linearized RR field strength, and the factor 39
40 an = -1 in the type IIA theory and an = i in the type IIB theory [4]. The polarization tensors 40
41 of the NSNS fields are given by e3, e4. The polarization tensor is symmetric and traceless for 41
42 graviton, antisymmetric for B-field and for dilaton it is 42
uv _ _ uUov _ fukv
44 3 = ^ n - - liki) (5) 44
45 V8 45
47 1 Our conventions in the string theory side set a' = 2. 47
10 11 12
20 21 22
where li is an auxiliary vector which satisfies ki ■ li = 1 and is the dilaton polarization which is one. The on-shell relations for the vertex operators are kf = 0, ki ■ ei = 0, and ei ■ ki = 0. The normalization of the amplitude (2) will be fixed after fixing the conformal symmetry of the integrand.
Substituting the vertex operators (3) into (2), and using the fact that there is no correlation between holomorphic and anti-holomorphic for the world-sheets which have no boundary, one can separate the amplitude to the holomorphic and the anti-holomorphic parts as
A - (P-r1M)AB(P-r2(M)fD£3l,v£4ap f !"[
where the holomorphic part is
d hiAc ®
e-<fi(Z2)/2 : e-
Hz3) :)[(: Sa(Zi) : Sc(z2) : f^(z3) :)
X (: eik1'X(zi) : eik2-X(Z2) : eik3-X(z3) : dXa(z4)eik4'X(Z4) ^
+ (: Sa(ZI) : Sc(Z2) : f^(zs) : ik4 • ffa(z4) :)
ik4-X(z4) :)j
0ikvX(zD
eikrX(z2) . eiksX(zs)
and the anti-holomorphic part I]fD is given by similar expression.
In calculating the correlators (6), one needs the world-sheet propagators for the holomorphic and anti-holomorphic fields [46,47]. Using the standard sphere propagators, one can easily calculate the correlators of the bosonic fields as
ki 'kj
= ¡: eikvX(zi) : eik2X(z2)
eik3X(z3) : eik4X(z4) :\ _
. eik•X(Z1) : eik2X(z2) : eik3X(z3) : dXa(z4)eik4X(Z4) _ J^ikfz^P
(. e-$(zi)/2 . e-t(Z2)/2 . e-t(Z3) Z-1/4Z-1/2Z-1/2 \: e : e : e :/_ Z12 Z13 Z23
where zij = zi - Zj. Using the conservation of momentum and the on-shell condition k4 ■ e4 = 0, one can write J23=i ^z^1 = J22=i i'kazi:41z-41z3i. This relation will be useful later on to check that the integrand is invariant under SL(2,R) x SL(2,R) transformations.
To calculate the correlators involving the fermion and the spin operators, one may use the Wick-like rule for the correlation function involving an arbitrary number of fermion fields and two spin operators [41,42].2 Using this rule, one finds the following results for the fermion correlators which appear in (7):
Sa(Z1) : Sc(Z2) : f'x(z3 ) :) _ ^ {Y^C-1) ACzu SaZ) : Sc(Z2) : f^(z3) : ik4 • ffa(z4) :)
3/4 -1/2 -1/2
k4,z1/4Z3-11/2Z3-21/2Z4-11 Z-2^ {yakl^C-1)
+ Z-21Z431(Z41Z32 + Z31Z42^(/"C-1) Ac - ^(/C-1) AC]\
2 See [43-45], for the correlation function of fermion fields with four spin operators.
10 11 12
20 21 22
1 The fractional power of zij will be converted to the integer power when the ghost correlator in (8) 1
2 multiplied the above correlators. 2
3 Replacing the correlators (9) and (8) into the scattering amplitude (6), and using the on-shell 3
4 conditions along with the conservation of momentum, one can easily check that the integrand 4
5 of the scattering amplitude is invariant under SL(2,R) x SL(2,R) transformations which is the 5
6 conformal symmetry of the z-plane. Fixing this symmetry by setting z1 = 0, z2 = z, z3 = 1 and 6
7 z4 = to, one finds the following result: 7
9 A =-iK ^ r(-s/8)r(-t/8)r(-u/8) k (10) 8
10 8 r(1 + s/8)r(1 + t/8)r(1 + u/8) J 10
11 where Gamma functions are the standard Gamma functions that appear in four closed string 11
12 amplitude [1], and the closed string kinematic factor is 12
14 K = (P-rm)AB(P-r2(m))CDe3ive4apK*C ® KVBD (11) ^
15 In the kinematic factor, there is an implicit factor of delta function 510(k1 + k2 + k3 + k4) imposing 15
16 conservation of momentum. The Mandelstam variables s = -8k1 ■ k2, u = -8k 1 ■ k3 and t = 16
17 -8k2 ■ k3 satisfy s +1 + u = 0, and the kinematic factor in the holomorphic part is 17
19 Kla _ 1
Kac = 8
Фа+k*)(y*c-1) AC+k4Vc-1) AC - wvc-1) 19
22 ■ V IV /AC w )AC ^4 V w ) AC 22
21 + (kmc-1)ac -\[k4^C-1)ac - K(YaC-1)a 21
23 \ -
24 + k4na(ylC-1)AC])
26 The kinematic factor in the anti-holomorphic part is similar to the above expression. We have 26
27 normalized the amplitude (10) to be consistent with the field theory couplings found in [36]. The 27
28 background flat metric in the Mandelstam variables and in the kinematic factor is in the string 28
29 frame. That is why we have normalized the amplitude by the dilaton factor е-2фо. On the other 29
30 hand, the graviton and the dilaton have the standard kinetic term or standard propagator only 30
31 in the Einstein frame. The massless poles of the amplitude (10) then indicates that the external 31
32 gravitons in the amplitude (10) are in the Einstein frame. 32
33 As a double check of the amplitude (10), one should be able to relate this amplitude to the 33
34 product of open string amplitudes of two spinors and two gauge bosons using the KLT prescrip- 34
35 tion [3]. According to the KLT prescription, the sphere-level amplitude of four closed string 35
36 states is given by 36
37 I 37
A = — sin(nk2 ■ k3)Aopen('/8,i/8) ® Aopen(t/8,u/8) (13)
38 A = sin(nk2 ■ k3)Aopen(s/8,t/8) ® Aopen(t/8,u/8)
39 where Aopen(s/8,t/8) is the disk-level scattering amplitude of four open string states in the s -1 39
40 channel which has been calculated in [48], 40
42 Aopen(s/8,t/8) = -iKe-f0 r(-;/)f(-{/8)K (14) 42
43 r(1 + u/8) 43
44 where the Mandelstam variables are the same as in the closed string amplitude. The open string 44
45 kinematic factor K depends on the momentum and the polarization of the external states [48]. 45
46 We have normalized the amplitudes (14) and (13) to be consistent with the normalization of the 46
47 amplitude (10). 47
1 To find the sphere-level scattering amplitude of two RR and two NSNS states, one has to 1
2 consider the open string amplitude of two spinors and two gauge bosons. The kinematic factor 2
3 for this case is [48] 3
5 K(U1,U2,Z3,Z4) =--7=
- t(u2Y • Z4U1 k4 • Z3 - U2Y • Z3U\k,3 • Z4 - «2K • k4U\Z3 • Z4)
® £ ^ z' , i = 3, 4
uAA ^ (P-rm)
2su2Y • Z3Y • k + k4)Y • Z4U1 5
9 where u1, u2 are the spinor polarizations and f3, Z4 are the gauge boson polarizations. They 9
10 satisfy the following on-shell relations 10
12 kj = 0, ki^i = 0, (Y^kiC-1) ABuB = 0 (16) 12
13 Using these relations, and the identity 13
15 UC(nl^YaC-1 - naiykC-1 + naVC-1 + Y^YkYaC-1)ca^A = {YaklC-1)ac^A^C 15
16 one can write the open string kinematic factor (15) in terms of the holomorphic kinematic fac- 16
17 tor (12) as 17
K(U1,U2,Z3,Z4) = -4iV2uAuCz3|Z4aKAC (!7) 19
21 Similarly for the antiholomorphic part, i.e., 21
22 K (u 1,u 2,h,Z4) = -4iV2u Bu DhvhpKBD 22
24 Using the above relations and r(x)r(l - x) = n/ sin(nx), and substituting the following rela- 24
25 tions in (13) 25
26 i v iv 26
27 " \AB 27
28 1 1 - 1(n) 28
29 uC ®uD ^ (P-r2(m))CD (18) 29
one recovers the amplitude (10), as expected. While the open string kinematic factor (15) is the final result for the S-matrix element of two gauge bosons and two open string spinors, the closed
32 string kinematic factor (11) is not yet the final result. The external closed string states are bosons, 32
hence, the Dirac matrices in the kinematic factor must appear in the trace operator which should
34 then be evaluated explicitly to find the final kinematic factor of the closed string amplitude. 34
35 The kinematic factor (12) has one term which contains three antisymmetric gamma matrices 35 and all other terms contain only one gamma matrix. As a result, the closed string kinematic
37 factor (11) has four different terms, each one has one of the following factors: 37
^B/D n \CD(, ,o r—1\
41 Tfpv = {P-r1(n))AB(P-r2(m))CD (Y°C-1) c (Y ?pvC-1)bd 40
42 T™kl = (P- r1(n))AB(P-r2(m))CD{YaklC-1)AC (ytC-1)bd 42
43 C1^ = (P-rKn))AB(P-r2im))CD(YaXlC-A)Ac (Y^C-1)BD (19) 43
45 which can be written in terms of the RR field strengths and the trace of the gamma matrices. 45
46 Using the above factors, one may then separate the closed string kinematic factor to the following 46
47 parts: 47
39 txx = (P-rl(n))AB(P-r2(m))CD{Yac-1) ac(YtC-1) b- 39
10 11 12
20 21 22
K _ K1 + K2 + K3 + K4
K1 _ — ( - u)2k4ak4pe3k^4fl + (t - u)2k4^(k'^S3^iS4ap
4 k4p + £3^x£4,i.a)) + 2tk\((t - u)k% (£3^84^ + £3p^£4Xa)
- (t - u)k4pS3a^S4l^ + (2tk%S3ap - (t - u)k4aB3^
- 2uk1;((t - u)kl^(S3^p84ak + 83^841^) - (t - u)k4p£3a^£4x^
- (2uklE30lp + (t - u)k4a83lp)84ik + 2tkl83ap(84Xi + 84iX^]T[ K2 _ 256 [(u - t)k4ak4l83pv84^I + k4i((t - u)k^83pv84ai
+ 2(tk^ - uk^83av84^|]T2a^|V
K3 _ 256 [(u - t)k4ak4X83vp84i? + k4x{(t - u)klp83vp84ia
+ 2(tkp - ukPp)83va84i^i\T"klIv s2
K4 _ 256[k4ak4p83pi!,84vk]T4
Note that the tensor T2 (T3) is totally antisymmetric with respect to its last three indices, hence, the indices of the NSNS momenta and polarization tensors in K2 (K3) which contract with this tensor, must be antisymmetrized. Similarly the tensor T4 is totally antisymmetric with respect to its first and its second three indices, so the momenta and the polarization tensors in K4 should be antisymmetrized accordingly.
One may try to write the polarization tensors and the momenta of the NSNS states in the form of e[l[akv]kp] which is the generalized Riemann curvature in the momentum space. Such manipulation has been done in [40] for finding the couplings of two RR and two NSNS states in the pure spinor formalism. However, we are interested in this paper in the form of couplings which are manifestly invariant under the linear T-duality and S-duality. This form of couplings may not be in terms of the generalized Riemann curvature.
To proceed further and write the kinematic factors (21) in terms of the momenta and the polarization tensors of the external states, one has to find the explicit form of the tensors T1; ■■■ ,T4 in terms of the metric nlv and the RR fields strengths F1, F2. Using the properties of the charge conjugation matrix and the Dirac matrices (see e.g., Appendix B in [4]), one can write the tensors T1; ■■■ ,T4 as
(_ 1)2m(m+1)r a
rr _ -( 1) ■■ .a"amF1iM^nF2vv-vm Tr^y^ny^y^m )
2 n\m\
rpoppv r2 _ -
(-1) 2
2 m(m+1)a
2 n\m\
(— 1)\n(n+1)a a tali_ ( 1 ^ anan
rr, Tall
r3 _ -
2 n\m\
-F1i1-inF2vv-vm Tr( yryl1-lnyppvyv1-vm) ■F1i1-inF2vyvm Tr(yTyI1-Inyallyv1-vm )
(_ 1)2m(m+1)a a
allppv — ( 1)_anam F F Tr(. all. lv• InvlpvvvV• • vm)
4 _ 2 t t • InF2v1• • • vm Y Y y y )
10 11 12
20 21 22
(22) 47
1 In the chiral projection operator P- = 2 (1 - y11 ), 1 corresponds to the RR field strength Fn and 1
2 y11 corresponds to F10-n which is the magnetic dual of Fn at the linear order. One may ignore 2
3 y11 and assume that 1 < n < 9. The corresponding couplings then produce corrections to the 3
4 democratic form of the supergravity [49]. 4
5 The above traces indicate that when the difference between n and m is an odd number, these 5
6 tensors are zero. This is what one expects because there is no couplings between the RR fields 6
7 in the type IIA theory in which the RR field strengths have even rank, and the type IIB theory 7
8 in which the RR field strengths have odd rank. When the difference between n and m is an even 8
9 number, the traces are not zero. One can easily verify that the traces are zero for n = m + 8. For 9
10 n = m + 6 case, the traces in T1, T2, T3 are zero and T4 becomes totally antisymmetric. However, 10
11 the corresponding kinematic factor K4 has k4ak4p, so the kinematic factor K is zero in this case 11
12 too. Therefore, there are three cases to consider, i.e., n = m, n = m + 2 and n = m + 4. 12
13 For the case that n = m + 4, one can easily find T1 = 0 and T2 = T3. A prescription for 13
14 calculating the traces is given in Appendix A. Using it, one finds the tensors T2, T4 to be 14
15 15 t 1)n(n+1)a2
16 T°ppv _ _16( )_anFvpp° 16
17 2 = (n - 4)! 12 17
18 (_ 1)n(n+1)a2 18 ^akifipv _ 1)_a± r^ («afipvipk kfi jjvpia _ a^vpka\
19 T4 = 16 (n - 4)! Pv' F12 + n F12 n F12 ) 19
20 , A\(rr vipkfia . j^vpfiiak -^vpfikai j^pfi^kav 20
21 - (n - 4AF12 + F12 - F12 - F12 21
22 + Fvfiikap - Fvpikafi)J (23) 22
24 Here we have used the fact that an-4 = an, and have used the following notation for F12s: 24
25 pvpfia _ F vpfia fI1 —ln-4 25
26 F12 — F 1l1 '"ln-4 F 2 26
27 Fvpilfika = F vpifik fll1 '"lln-5a (24) 27
27 F12 — F 1|1 '"ln-5 F 2 (24) 27
28 Note that F1 is n-form and F2 is (n - 4)-form. Replacing the tensors T1, ■■■ ,T4 into the kine- 28 matic factor (20), one finds the amplitude (10) for one RR n-form, one RR (n - 4)-form
and two NSNS states. We have checked that the amplitude satisfies the Ward identity corre-
31 sponding to the NSNS gauge transformations. The kinematic factor can be further simplified 31
32 after specifying the NSNS states. As we will see in the next section, the amplitude is non-zero 32
33 only when the two NSNS states are antisymmetric. For the cases that n = m and n = m + 2, the 33
traces have been calculated in Appendix A.
To find the couplings which are produced by the amplitude (10), one has to expand the Gamma
36 functions in (10) at low energy, i.e., 36
38 r(-s/8)r(-t/8)r(-u/8) _ 29 s2 + su + u2 38
39 r(1 + s/8)r(1 + t/8)r(1 + u/8) stu ) 32 ) ( ) 39
41 where dots refer to higher order contact terms. The first term corresponds to the massless poles in 41
42 the Feynman amplitude of two RR and two NSNS fields which are reproduced by the supergrav- 42
43 ity couplings. We have done this calculation in Appendix B. All other terms correspond to the 43
44 on-shell higher-derivative couplings of two RR and two NSNS fields in the momentum space, 44
45 i.e., 45
46 „ k2e-2*0 ( s2 + su + u2 \ 46
47 Ac = -ii-2Z(3)--32-Z(5) + K (26) 47
1 Since the above amplitude contains only the contact terms, one has to be able to rewrite it in 1
2 terms of the RR and the NSNS field strengths. Moreover, the contact terms (26) should satisfy 2
3 the T-dual and S-dual Ward identities as well [9-11]. The couplings of two RR field strengths 3
4 and two Riemann curvatures/B-field strengths at eight-derivative level have been found in [36] 4
5 by imposing the above Ward identities on the four generalized Riemann curvature couplings [2]. 5
6 In the next section we will compare those couplings with the corresponding contact terms in (26). 6
8 3. Gravity and B-field couplings 8
10 In this section we are going to simplify the kinematic factor in (26) for the specific NSNS 10
11 states which are either graviton or B-field, and compare them with the couplings that have been 11
12 found in [36]. These couplings have structure F(n)F(n)RR, F(n) F(n) HH, F(n) F(n-2) RH and 12
13 F(n)F(n-4)HH where R stands for the Riemann curvature and H stands for the derivative of 13
14 B-field strength. The coupling with structure F(n) F(n-4) RR has been found to be zero. Using 14
15 the explicit form of the Ti tensors in (23), we have found that K = 0 when the two NSNS states 15
16 are symmetric and traceless. Therefore, there is no on-shell higher-derivative coupling between 16
17 one RR field strength F(n), one F(n-4) and two gravitons, as expected. 17
19 3.1. F(n)F(n)RR 19
21 To find the contact terms with structure F(n) F(n) RR, one should first simplify the kinematic 21
22 22 factors in (21) when the two NSNS polarization tensors are symmetric and traceless. One should
23 then use the explicit form of the tensors T1, ■ ■■ ,T4 calculated in (79). Using the totally antisym- 23
metric property of the RR field strengths and taking the on-shell relations into account, one can
write the kinematic factor (20) as
27 K = n[(n - 1)^(n - 2)sF^ kfxvph3kph4pvk4ak4^ + F^'xv[-h3pv(tkf - ukp) 27
28 2 28
29 X (h4/ipk4a - h4apk4\i) - uh3vPk4a(h4fipk4x - h4f>xk4p) + th3^f'k4¡x 29
30 X (h4avk4p - h4Vpk4a)]) + F1Q2x(h3axh4f [u2kxvk1p + tk2v(tk2p - 2ukXp)\ 30
32 + uh3/xV(tkp - ukf)(h4Vpk4a - h4apk4v) 32
33 +1 [-h3aV(tkf - ukf)(h4Vpk4x - h4xpk4v) 33
35 - uh3f {h4vpk4ak4x - k4v(h4xpk4a + h4apk4x - h4axk4p))])] (27) 35
36 where h3, h4 are the graviton polarizations and 1 < n < 9. In order to compare the above 36
37 kinematic factor with the eight-derivative couplings found, we have to write the couplings in 37
38 both cases in terms of independent variables. To this end, we have to first write the RR field 38
39 strengths in the above kinematic factor in terms of the RR potential. Then using the conserva- 39
40 tion of momentum and the on-shell relations, one may write it in terms of the momentum k1 , 40
41 k2, k3 and in terms of the independent Mandelstam variables s, u. Moreover, one should write 41
42 k3 ■ h4 = -ki ■ h4 - k2 ■ h4 and h4 ■ k3 = -h4 ■ ki - h4 ■ k2 to rewrite the kinematic factor in terms 42
43 of the independent variables. The Ward identity corresponding to the gauge transformations and 43
44 the symmetry of the amplitude under the interchange of 1 ^ 2 and 3 ^ 4 can easily be verified 44
45 in this form. Transforming the F(n)F(n)RR couplings found in [36] to the momentum space and 45
46 doing the same steps as above to write them in terms of the independent variables, we have found 46
47 exact agreement between (27) and the couplings F(n)F(n)RR for n = 1,2, 3,4, 5. 47
1 One can easily extend the couplings with structure F(5)F(5)RR to F(n)F(n)RR with 6 < 1
2 n < 9. Using F(5)F(5)RR couplings, one can use the dimensional reduction on a circle, y — 2
3 y + 2n, and find the 9-dimensional couplings with structure F(5)F(5)RR which have no Killing 3
4 index. Then under the linear T-duality, these couplings transforms to the couplings with structure 4
5 Fy6F(6)RR. Following [36], one can easily complete the y-index and find the 10-dimensional 5
6 couplings with structure F(6)F(6)RR. They produce the correct couplings since it is impossible 6
7 to have F(6)F(6)RR couplings in which the RR field strengths have no contraction. Repeating 7
8 the above steps, one finds F(n)F(n)RR with 6 < n < 9. 8
10 3.2. F(n)F(n)HH 10
12 To find the contact terms with structure F(n)F(n)HH, one should first simplify the kinematic 12 factors in (21) when the two NSNS polarization tensors are antisymmetric. Then, one should use the explicit form of the tensors T1, ■ ■ ■ ,T4 in (79). Using the totally antisymmetric property of
15 the RR field strengths and taking the on-shell relations into account, one can write the kinematic 15
16 factor (20) in this case as 16
18 K = 28b3ap [2n(t2 + u2)bfF%kAxkAp + 4b4px{(u - t)Fu(tk% - ukll)ka4 19 20 + n[tFia(tk$ - ukj") + ^F^]k4| + n[t^F^tf - (n - 1)sF1;2|apk4|) 20
22 - u{Fa2 (tk$ - uk\) + (n - 1)sFa2lXpk4i)]k4P) + nb4^(2(tk2- - uk\) 22
23 x [-2tFl2Pka + (n - 1)sFl2vapk4v] + 2uk'a [2F1P2l (tk% - uk^) + (n - 1)s 23 pvXlk4v] + (n - 1)s[(2tF12lPpka + (n - 2)sFgvaPpk4v) - 2FlPlxp
x FP2v^k4v] + (n - 1)s[{2tF^Ppka + (n - 2)sF^vaPpk4v) - 2FaxPlp 25
26 X (tk$ - ukj") + (n - 2)sFapvXlpk4v]k4p)] (28) 27
28 where b3, b4 are the B-field polarization tensors. Writing the above kinematic factor and the 28
29 couplings F(n)F(n)HH found 4in [36] in terms of the independent variables, we have found 29
30 that they are exactly identical for n = 1, 2, 3,4, 5. Using the consistency of the couplings with 30
31 the linear T-duality, one can easily extend the F(5)F(5)HH couplings to F(n)F(n)HH with 31
32 6 < n < 9. 32
34 3.3. F(n)F(n-2)HR 34
36 To find the contact terms with structure F(n)F(n-2)HR, one should first simplify the kine- 36
37 matic factors in (21) when one of the NSNS polarization tensors is symmetric and traceless, and 37
38 the other one is antisymmetric. Then, one should use the explicit form of the tensors T1, ■ ■ ■ ,T4 38
39 in (80). In this case, one finds 39
42 K = - ^7b3ap [F1phv4P [u2k1vk1p + tk2v(tk2p - 2ukp] + (2u2Fjf h4Pp^ 42
43 + (n - 2)s[(n - 3)sF1aIpvpXlh4^pk4x + 2uFP2vpk(h4Xpka - h4apk4x)])k4v 43
44 ( )( ) ( ) 44
45 + 2uFp2(tkp - ukp}(h4vpk4 - h4a pk4v) - (n - 2)sF1pvk (tkp - ukp) 45
46 x (h4vpk4X - h4Xpk4v)] (29) 46
1 Writing the above kinematic factor and the couplings F(n)F(n-2)HR found in [36] in terms of 1
2 the independent variables, we have found that they are exactly identical for n = 3,4, 5. 2
3 Here also one can use the dimensional reduction on F(5)F(3)HR couplings and consider the 3
4 9-dimensional couplings F(5)F(3)HR which have no y-index. Then under the linear T-duality 4
5 one finds Fy(6)Fy(4)HR. Since it is impossible to have coupling F(6)F(4)HR in which the RR 5
6 field strengths have no contraction with each other, one can find all F(6)F(4)HR couplings by 6
7 completing the y-index in the above F(6) f!^4) HR couplings. So the couplings corresponding the 7
8 kinematic factor (29) for 6 < n < 9 can easily be read from the F(5)F(3)HR couplings. 8
10 3.4. F(n)F(n-4)HH 10
12 To find the contact terms with structure F(n)F(n-4)HH, one should first simplify the kine- 12
matic factors in (21) when the NSNS polarization tensors are antisymmetric. Then, one should
14 use the explicit form of the Ti tensors in (23). In this case, one finds 14
16 K = 28 b3apb4,xk4p [2F0flxp (tk* - uk\4 + 2uF^>xpka + (n - 4^ "XfVk4V ] (30) 16
17 2 17
18 Writing it in terms of the independent variables, one finds that it is invariant under the interchange 18
19 of 3 ^ 4, and it satisfies the Ward identity corresponding to the B-field gauge transformation. 19
20 The minimum value of n is 5, so we consider n = 5 and compare it with the couplings with 20
21 structure F(1)F(5)HH. 21
22 The F(1)F(5)HH couplings have been found in [36] by using the dimensional reduction on 22
23 the couplings with structure F(2)F(4) HR which have been verified by explicit calculation in the 23
24 previous section. Then under the T-duality the couplings with structure F(2)F(4)HRy where the 24 index y is the Killing index, transform to the couplings with structure F(1)Fy(5) HHy. One can
26 complete the y-index to find the couplings with structure F(1)F(5)HH in the string frame,3 i.e., 26
28 Y / 10 / 28
S ^ o I d G\8Fh,kFmnpqr,s Hhpq,mHkrs,n
29 K 2 J 29
30 + 4Fh,kFkmnpq,rHmns,hHpqr,s '2Fh,kFkmnpq,hHmns,rHpqr,s] (31) 30
31 3 5 31
32 where y = a'3f(3)/25. There is one extra term in the couplings that have been found in [36] 32 which is zero on-shell. Moreover, there is an extra factor of -1/2 in the above action which is
34 resulted from completing the Killing y-index and considering the fact that three are two B-field 34
35 strengths in the couplings with structure F(1)Fy(5)HHy. Transforming the above action to the 35
36 momentum space, and writing the couplings in terms of the independent variables, we have 36
37 found exact agreement with the kinematic factor in (30). 37
38 One can use the dimensional reduction on above F(1)F(5)HH couplings and consider the 38
39 9-dimensional couplings F(1)F(5)HH which have no y-index. Then under the linear T-duality 39
40 one finds F(2)F(6)HH. Completing the y-index, one finds all couplings in which the RR field 40
41 strengths have contraction with each other. In this case, however, it is possible to have coupling 41
42 F(2)F(6)HH in which the RR field strengths have no contraction with each other. We add all 42
43 such couplings with unknown coefficients, and constrain them to be consistent with the kinematic 43
44 factor (30). We find the following result for F(2)F(6)HH couplings: 44
46 3 Note that in writing the field theory couplings we have used only the lowercase indices and the repeated indices are 46
47 contracted with the metric . 47
1 S D d 10xV-G[8Fht, kFmnpqrt ,sHhpq,mHkrs,n + 8Fhm,nFnkpqrs,tHhkp,qHmrs,t
2 K J 2
3 + 4Fht,kFkmnpqt,rHmns,hHpqr,s 2Fht,kFkmnpqt,hHmns,rHpqr,s] (32) 3
4 The first term and the couplings in the second line are the couplings that can be read from the 4
5 T-duality of the couplings (31). The second term is the coupling in which the RR field strengths 5
6 have no contraction with each other, hence, it could not be read from the T-duality of (31). 6
7 The couplings with structure F(n-4)F(n)HH for n > 6 can easily be read from the T-duality 7
8 of the couplings (32) because it is impossible to have such couplings in which the RR field 8
9 strengths have no contraction with each other. The result is 9
11 1 Y f d 10v rZr[8F F n n 11
S D / , 2 1 d A * G\Qгhal' ■ 'an-5,kгmnpqral' ■ an-5 ,snhpq,mnkrs,n
12 (n - 5)! K2 J 12
13 2Fhal' ■ ■ an-5,kFkmnpqay ■ ■ an-5,h Hmns,r Hpqr,s 13
+ 4Fha1 ■ ■ ■ an-5,kFkmnpqa1 ■ ■ ■ an-5,r Hmns,h Hpqr,s
15 ] 15
16 + 8(n 5)Fhmal' ■ ■ an-6,nFnkpqrsal' ■ ■ an-6,t Hhkp,q Hmrs,t \ (33) 16
17 The number of indices a1 ■ ■ ■ an-m in the RR field strengths is such that the total number of the 17
18 indices of F(n) must be n. For example, Fnkpqrsav■ ■ an-6,t is Fnkpqrs,t for n = 6 and is zero for 18
19 n < 6. We have checked that the above couplings are consistent with the kinematic factor (30) 19
20 for n > 5. 20
22 4. Dilaton couplings 22
24 In this section we are going to simplify the kinematic factor in (26) for the cases that one 24
25 or both of the NSNS states are dilatons. One has to use (5) for the dilaton polarization. There 25
26 are three cases to consider. The first case is when one of the polarizations is the dilaton and 26
27 the other one is antisymmetric. The kinematic factor in this case is non-zero for n = m + 2. 27
28 So the non-zero couplings should have structure F(n)F(n-1)H^ where $ stands for the second 28
29 derivatives of the dilaton. The second case is when one of the polarization tensors is the dilaton 29
30 and the other one is symmetric and traceless. The kinematic factor in this case is non-zero for 30
31 n = m. So the non-zero couplings in this case should have structure F(n)F(n)R$. The third case 31
32 is when both of the NSNS polarizations are the dilatons. The kinematic factor in this case is 32
33 also non-zero for n = m. The non-zero couplings should have the structure F(n)In all 33
34 cases, we have found that the auxiliary vector l of the dilaton polarization (5) is canceled in the 34
35 kinematic factors, as expected. Let us begin with the first case. 35
37 4.1. F(n)F(n-2)H$ 37
39 Replacing the tensors Ti for n = m + 2 case which are calculated in (80), into (21), one 39
40 finds the following result for the kinematic factor (20) when one of the polarization tensors is 40
41 antisymmetric and the other is the dilaton polarization (5): 41
43 K =-(n - 7)s~ 2t $3b4apk4v [2Ff2v(tk2a - uka) + (n - 2)sF1a2Pv^k4^] (34) 43
44 28 2 44
45 As has been discussed already, the external states in the contact terms (26) are in the Einstein 45
46 frame whereas the background metric niv is in the string frame. Hence, to find the appropriate 46
47 couplings in a specific frame, one has to either transform the external graviton states to the string 47
1 frame or transform the background metric nIv to the Einstein frame, i.e., e$0/2nIv. We choose 1
2 the latter transformation to rewrite the contact terms (26) in the Einstein frame. 2
3 Since the couplings are in the Einstein frame, the natural question is whether the string frame 3
4 couplings F(n) F(n-2) HR, produce all the couplings F(n)F(n-2)H$ in the Einstein frame? In 4
5 fact the transformation of the Riemann curvature from the string frame to the Einstein frame 5
6 is [37] 6
8 Rabcd Rabcd lC*;b]d] + -~ (35) 7
9 V2 9
10 where $ is the perturbation of dilaton, i.e., & = + In above equation dots refer to the 10
11 terms with two dilatons in which we are not interested. We have transformed the string frame 11
12 couplings F(n) F(n-2) HR to the Einstein frame and found that the resulting F(n) F(n-2) cou- 12
13 plings are not consistent with the kinematic factor in (34). This indicates that there must be some 13
14 new couplings with structure F(n)F(n-2)H$ in the string frame. The combination of these cou- 14
15 plings and the couplings with structure F(n) F(n-2) HR, then must be consistent with (34) when 15
16 transforming them to the Einstein frame. This constraint can be used to find the dilaton couplings. 16
17 We will find the couplings in both string and Einstein frames for n < 5. In Section 5, we extend 17
18 the string frame couplings to 1 < n < 9. 18
19 The new couplings in the string frame must be consistent with the linear T-duality. So we will 19
20 first find the new couplings for the case of n = 5 by using the consistency with (34) and then find 20
21 the couplings for other values of n < 4 by using the consistency with the linear T-duality. To find 21
22 the string frame couplings with structure F(5)F(3)H$, we consider all possible on-shell con- 22
23 tractions of terms with structure F(5)F(3)H$ with unknown coefficients. This can be performed 23
24 using the new field theory motivated package for the Mathematica "xTras" [50]. Transforming 24
25 the combination of these couplings and the couplings with structure F(5)F(3)H$ found in [36], 25
26 to the Einstein frame and constraining them to be consistent with the kinematic factor (34), one 26
27 finds some relations between the unknown coefficients. Replacing these relations into the general 27
28 couplings with structure F(5)F(3)H$, one finds the following couplings in the string frame: 28
29 29 4 y r _
30 S ^ T 2 I d G[6Hhrs,m Fknqrs ,p Fmnp ,q + 3Hhrs,mFknprs,qFmnp,q 30
31 3 K 2 J 31
32 Hhrs ,kFmnprs ,qFmnp ,q 6Fknprs ,qFmnp ,qHhmr ,s 32
33 + 6Fmnp,qFknpqr,sHhmr,s + 3Fhmn,pFmnpqr,sHkqr,s1&,hk (36) 33
35 Plus some other terms which contains some of the unknown coefficients. However, these terms 35
36 vanish when we write the field strengths in terms of the corresponding potentials and use the 36
37 on-shell relations to write the result in terms of the independent variables. That means these 37
38 terms are canceled using the Bianchi identities for the RR and for the B-field strengths and 38
39 the on-shell relations. As a result, these terms can safely be set to zero. We refer the reader to 39
40 Section 4.3 in which similar calculation has been done in more details. Since we have considered 40
41 all contractions without fixing completely the Bianchi identities, the above couplings are unique 41
42 up to using the Bianchi identities. That is, one may find another action which is related to the 42
43 above action by using the Bianchi identities and the on-shell relations. 43
44 Having found the string frame couplings F(n)F(n-2)for n = 5 in (36), we now apply the 44
45 T-duality transformations on them to find the corresponding couplings for other n. We use the 45
46 dimensional reduction on the couplings (36) and find the couplings with structure Fy5)Fy(3)H$. 46
47 Under the linear T-duality transformations, the RR field strength F^ transforms to F(n-1) with 47
1 no Killing index, the B-field with no y-index is invariant and the perturbation of dilaton trans- 1
2 forms as (see e.g., [36]) 2
$ ^ $--= hyy (37)
5 V2 5 66
7 where hiv is the metric perturbation, i.e., giv = niv + 2Khiv. The couplings in F(5)F(3)H$ 7
8 corresponding to the second term above should be canceled with the couplings with structure 8
9 F(4)F(2)HRyy. The terms corresponding to the first term in (37) have structure F(4)F(2)H$. 9
10 The result for all terms in (36) is the following couplings in the string frame: 10
12 S D 4 2 I d X ^ G[2Hhrs,mFkqrs,p Fmp,q + 2Hhrs ,mFkprs ,qFmp ,q 12
13 K J 13
14 Hhrs,kFmprs ,qFmp ,q 4Fkprs ,qFmp ,qHhmr ,s 14
16 + 4Fmp ,qFkpqr ,s Hhmr ,s 2Fhm,pFmpqr ,sHkqr,s\^,hk (38) 16
17 The transformation of the combination of the above couplings and the couplings with structure 17
18 F(4)F(2)HR which have been found in [36], to the Einstein frame is the following: 18
20 v C 10 ,- j, r 20
S D Y ' ^10 -- - " ' -- - -- -
^f d 10x V-Ge
2Fhm,nFmnpq,r Hkpq,r^,hk 2Fhm,nFnkpq,rHmpq,r&,hk
22 22 + 6Fhm,nFnpqr,kHmpq,r&,hk 8Fhm,nFmkpq,rHnpq,r^,hk
24 2 24 + 12Fhm,nFmkpr,q Hnpq,r&,hk + ~Z Fhm,nFmkpq,rHkpq,r&,hn
25 3 25
26 + 12Fhm,nFmnqr,p Hhkq,r&,kp 6Fhm,nFmnqr,p Hhqr,k&,kp 26
28 + 6Fhm,nFmkqr,pHhqr,n&,kp
30 We have checked that it is consistent with (34) for n = 4. In writing the above result we have used 30
31 the Bianchi identities and the on-shell relations to simplify the couplings. However, one may still 31
32 use these identities to rewrite the above couplings in a simpler form. It would be interesting to 32
33 find the minimum number of terms in which the Bianchi identities have been used completely. 33
34 Having found the string frame couplings F(n)F(n-2)Нф for n = 4 in (38), we now apply 34
35 the T-duality transformations on them to find the corresponding couplings for n = 3. To find 35
36 the couplings with structure F(3) F(1) Нф, one has to use the dimensional reduction on the cou- 36
37 plings (38) and find the couplings with structure Fy4)F^2)Нф. Then under the linear T-duality 37
38 transformations, they transform to the couplings with structure F(3)F(1)Hф. In the string frame 38
39 they are given by 39
41 у I 10 ,--41
42 S —^ 8 2 I d G^,hk[Hhrs ,mFkrs ,qFm,q Hhrs,kFmrs,qFm,q 42
2Fkrs,qFm,qHhmr,s 2 Fm, q Fkqr, s Hhm r ,s + Fh ,pFpqr,s Hkqr,s
] (40) 43
45 We have checked that the transformation of the combination of the above couplings and the cou- 45
46 plings with structure F(3)F(1)HR, to the Einstein frame produces exactly the couplings which 46
47 are consistent with (34) for n = 3. 47
1 4.1.1. Consistency with the S-duality 1
2 The dilaton couplings in (38) are in the type IIA theory whereas the couplings in (36) and (40) 2
3 are in the type IIB theory. The effective action in the type IIB theory should be invariant under 3
4 the S-duality, as a result, the couplings in (36) and (40) should be consistent with the S-duality. 4
5 The standard S-duality transformations are in the Einstein frame, so we have to transform the 5
6 couplings with structure F(n)F(n-2) HR and F(n)F(n-2)H$ to the Einstein frame and then study 6
7 their compatibility with the S-duality for n = 5, 3. 7
8 The extension of the couplings in (31) to the S-duality invariant form has been found in [36]. 8
9 Apart from the overall dilaton factor in the Einstein frame which is extended to the SL(2,Z) 9
10 invariant Eisenstein series E3/2, each coupling should be extended to the SL(2,R) invariant form, 10
11 e.g., the first term in (31) is extended to 11
12 T - I 2$0 \ 12 Hhqr,m^^,nkN ^^0Hmps,k = 2Fn,k\Hhqr,mHmps,k e 0 Fhqr,mFmps,k)
+ ^/2K^,nk(Hhqr,mFmps,k + Fhqr,mHmps,k) + ''' (41)
13 ' Lhqr,mJ ,nkN MM0Hmps,k — n,^^±hqr,^i±mps,k ° L hqr,m* mps,kj 13
16 where dots refer to the terms with non-zero axion background in which we are not interested. 16
17 We refer the interested reader to [36] for the definitions of H, M and N. The terms in (31) cor- 17
18 respond to the first term in the above SL(2,R) invariant set. The terms in the S-duality invariant 18
19 action which correspond to the second line above have structure F(5) F(3) H$. We have checked 19
20 explicitly that these terms are reproduced exactly by transforming the couplings (36) and the cou- 20
21 plings with structure F(5) F(3) HR (see Eq. (35) in [36]) to the Einstein frame. In other words, 21
22 these couplings are fully consistent with the kinematic factor (34) for n = 5. The couplings corre- 22
23 sponding to the last term in (41) are the S-duality prediction for four RR couplings with structure 23
F(5)F(1)F(3)F(3).
25 The transformation of the couplings (40) and the couplings with structure F(3)F(1)HR 25
26 (see Eq. (37) in [36]) to the Einstein frame, produces the following couplings with structure 26
27 F(3)F(1)H$ in the Einstein frame: 27
Y f d 10x
3 Fh,mFnkp,qHnkp,q&,hm + 8Fh,mFkpq,nHmkp,q^,hn 29
31 + 4Fh,mFnkp,qHmkp,q&,hn + 4Fh,mFmkp,qHnkp,q&,hn 8Fh,mFnpq,kHhpq,m^,nk
16Fh,mFhpq,nHmkp,q^,nk + 8Fh,mFhpq,nHmpq,k^,nk
3 which are consistent with the kinematic factor (34) for n = 3. Using the on-shell relations, one 3
35 finds that the above amplitude is invariant under the transformation 35
37 F(3) H; H -F(3) (43) 37
38 It is also invariant under the following transformation: 38
40 F(1) d& -F(1) (44) 40
1 1 Using these properties, one should extended the amplitude (42) to the S-duality invariant form.
The dilaton factor in (42) can be rewritten as e-3$0/2 x e$0. The first factor is extended to
the SL(2,Z) invariant function E3/2 after including the one-loop result and the nonperturba-
tive effects [13]. The second factor combines with the dilaton and the RR scalar to produce the
following SL(2,R) invariant term:
,mn Fm,n &,hk) (45) 47
1 Using the standard SL(2,R) transformation of the dilaton and the RR scalar, i.e., t —> jt+q 1
2 where t = C + ie, one finds the above term is invariant under the SL(2,R) transformation.4 2
3 The RR two-form and the B-field should appear in the following SL(2,R) invariant term: 3
10 11 12
20 21 22
Hmnq,pNHmnp,q — Hmnq,pFmnp,q Fmnq,pHmnp,q
Therefore, the SL(2,Z) invariant extension of the action (42) has no coupling other than
F(3)F(1)H$.
4.2. F(n)F(n)R$
Replacing the tensors Ti for n = m case which are calculated in (78), into (21), one finds the following result for the kinematic factor (20) when one of the polarization tensors is symmetric and traceless and the other one is the dilaton polarization (5):
(n - 5)[F12h^v{u2k1,Âk1V + tk2,Atk2v - 2uk^))
+ ns((n - DsF^h
k4ak4^ - F^fa^ - uk1)(h4IÀvk4a - h4avk4^))]
The kinematic factor is zero for n _ 5. So there is no higher-derivative coupling between two F(5), one graviton and one dilaton in the Einstein frame. This result is consistent with the S-duality because F(5) and the graviton in the Einstein frame are invariant under the S-duality whereas the dilaton is not invariant under the S-duality.
Now we are going to find the couplings with structure F(n)F(n)R$ for n _ 1, 2, 3,4, 5 in the string frame. To this end, we have to first transform the string frame couplings with structure F(n)F(n)RR which have been found in [36], to the Einstein frame. If they do not produce the kinematic factor (47), then one has to consider new couplings with structure F(n)F(n)R$. So let us begin with the case of n _ 5. Transforming the couplings with structure F(5)F(5)RR (see Eq. (27) in [36]) to the Einstein frame, we have found that they produce the couplings with structure F(5)F(5)R$ which are not zero, i.e., they are not consistent with (47). As a result, one has to consider new couplings in the string frame with structure F(5)F(5)R$ to cancel them. We consider all such on-shell couplings with unknown coefficients, and constrain them to cancel the above F(5)F(5)R$ couplings. This constraint produces some relations between the coefficients. Replacing them into the general couplings with structure F(5)F(5)R$, one finds the following couplings:
'RhmnpFkqrst ,n Fmpqrs,t RhmnpFmqrst,pFknqrs,t
3 к2 J
+ 3RhmnpFmpqrt,sFknqrs,t 3RhmnpFmpqrs,t Fknqrs,t + Rh mkn Fnpqrt ,sFmpqrs ,t] ф,hk
Plus some other terms which contains some of the unknown coefficients. However, these terms vanish when we write the field strengths in terms of the corresponding potentials and use the on-shell relations to write the result in terms of the independent variables. As a result, these terms can safely be set to zero. We refer the reader to Section 4.3 in which similar calculation has been done in more details.
4 It has been observed in [51] that e® F(1) A d@ is invariant under the Z2 subgroup of the SL(2, R) group.
10 11 12
20 21 22
1 Using the above couplings in the string frame, one can perform the dimensional reduction 1
2 on a circle and finds the couplings with structure F(5)F!(5) R$. Under the linear T-duality, they 2
3 produce the following couplings with structure F(4)F(4) R$ in the type IIA theory: 3
S D 4 2 I d G[RhmnpFkrst,nFmprs,t + RhmnpFmrst,pFknrs,t
6 K2 J 6
7 + 2RhmnpFmprt,sFknrs,t 3Rhmnp Fmprs,tFknrs,t 7 + Rh mknFnprt ,s Fmprs,t 1&, hk (49)
10 The transformation of the above couplings and the couplings with structure F(4)F(4)RR (see 10
11 Eq. (26) in [36]), to the Einstein frame produces the following couplings: 11
12 . 12
*Y i 10 i 3$ /2
13 S D 2 I d Ge Y3Fmkqr,sFnpqr,sRhnkp&,hm Fmqrs,kFnpqr,s Rhnkp&,hm 13
14 K * 14
15 2Fmkqr,s Fnpqs,r Rhnkp&,hm Fmkqr,sFnqrs,pRhnkp&,hm 15
16 Fkpqs,rFnpqr,s Rhnmk&,hm] (50) 16
18 which are exactly consistent with (47) for n = 4. 18
19 The couplings with structure F(3)F(3)R$ in the string frame can be found from the 19
20 couplings (49) by applying the dimensional reduction and finding the terms with structure 20
21 Fy3) Fy3) R$. Then under T-duality they produce the following couplings in the type IIB theory: 21
23 S D -8_2 I d 10W-G[-Rhmnp Fkst ,nFmps,t - RhmnpFmst ,p Fkns,t 23
24 K ' 24
25 + RhmnpFmpt ,s Fkns ,t - 3 RhmnpFmps ,t Fkns ,t + Rh mknFnpt ,sFmps ,t ]@,hk (51) 25
26 We have checked that the transformation of the above couplings and the couplings with structure 26
27 F(3)F(3)RR (see Eq. (20) in [36]), to the Einstein frame are exactly consistent with (47) for 27
28 n = 3. The S-duality transformations of these couplings are discussed in the next section. 28
29 Having found the couplings with structure F(3)F(3)R$ in (51), we now construct the cou- 29
30 plings with structure F(2)F(2)R$ in the type IIA theory. Under the dimensional reduction on 30
31 the above couplings, the couplings with structure F^3) F(3) R$ produce the following couplings 31
32 under the T-duality: 32
34 S ^ 8 2 I d ^ G[RhmnpFkt ,nFmp ,t + RhmnpFmt ,pFkn ,t
35 K I 35
3Rhmnp Fmp,t Fkn,t + RhmknFnt ,s Fms ,t ]&,hk (52) 36
38 Note that in the first term in the second line of (51) there is no contraction between the two 38
39 RR field strengths. Hence, this term does not produce coupling with structure F-(3)Fy(3)R$. That 39
40 is why this term does not appear in (52). The transformation of the above couplings and the 40
41 couplings with structure F(2)F(2)RR (see Eq. (21) in [36]), to the Einstein frame is given by the 41
42 following couplings: 42
43 y f __43
44 S D 12 — I d 10W-Ge $0/2[Fhm,nFkp,qRmkpq&,hn - Fhm,nFkp,qRmnpq&,hk 44
46 Fhm,nFkp,nRmkpq&,hq Fhm,nFnk,pRmkpq&,hq] (53) 46
47 which are exactly consistent with (47) for n = 2. 47
1 There is no contraction between the RR field strengths in (52), hence, the dimensional re- 1
2 duction on a circle does not produce couplings with structure Fy(2)Fy(2)R0. As a result, the 2
3 linear T-duality indicates that there is no new coupling with structure F(1)F(1)R< in the string 3
4 frame. We have checked that the transformation of the couplings with structure F(1)F(1)RR (see 4
5 Eq. (22) in [36]), to the Einstein frame are consistent with (47) for n _ 1. In fact both are zero in 5
6 this case. 6
8 4.2.1. Consistency with the S-duality 8
The Einstein frame couplings F(n)F(n)R< for n _ 1, 3, 5 are in the type IIB theory, so they
should be consistent with the S-duality. We have seen that for n _ 1, 5 the couplings are zero
which are consistent with the S-duality because it is impossible to construct the SL(2,R) invari-12 12
ant term from one dilaton or from one dilaton and two RR scalars. Note that the dilaton and
the axion in E3/2 are constant, so we cannot consider the derivative of E3/2 which produces
d<e 3</2 at weak coupling. In fact the contact terms in (26) represent the couplings of four quantum states in the presence of constant dilaton background. It is totally nontrivial to extend the amplitude (10) to non-constant dilaton background, i.e., it is not trivial to take into account 17 the derivatives of the dilaton background. That amplitude would produce higher-point functions. 17 The couplings with structure F(3) F(3) R<, however, are not zero. That means it is possible to construct the SL(2,R) invariant couplings which contains one dilaton and two RR 2-forms. 20 In fact the S-duality invariant couplings which include such couplings have been constructed 20 in [36], i.e.,
24 S D — d 10xE3/2V-G[mThq,nM ,rmHknp,hRmpkq 4Hrpr h
M ,mk Hhnq,rRkqmp 24
33 Hhqr n^^,rmHknp,hRmpkq
25 T T 25
26 - 4Hnpq,mM,qhHmpr,kRkrhn + 4Hnpq,hM,qmHknr,hRmrkp 26
27 - 2Hnpq,hM,mhHnpr,kRmrkq + 2Hmnq,hM,rkHnpq,kRprhm 27
28 T ] 28
29 - 2Hmnp,k^^,rmHnpq,hRqrhk\ (54) 29
30 Each term is invariant under the SL(2,R) transformations. For zero axion background, each term 30
31 has the following couplings: 31
34 $ 34
35 = 2e Fr,m(Fhqr,nHknp,h + Hhqr,nFknp,h)Rmpkq 35
36 + *f2K$,rm(e$° Fhqr,nFknp,h e $0 Hhqr,nHknp,h}Rmpkq (55) 36
38 The couplings corresponding to the terms in the first and second lines of (55) have been found in 38
39 [36]. The couplings corresponding to the last term above have been found in [52]. The couplings 39
40 corresponding to the first term in the third line of (55) are the couplings F(3)F(3)R$ in the 40
41 Einstein frame. We have checked it explicitly that they are consistent with (47) for n = 3. 41
43 4.3. F(n)F(n)$$ 43
45 Replacing the tensors Ti for n = m case which are calculated in (78), into (21), one finds the 45
46 following result for the kinematic factor (20) when both the NSNS polarization tensors are the 46
47 dilaton polarization (5): 47
10 11 12
20 21 22
K = Ф3Ф4 \(n — 5)stuFi2 + AnF^ {(n — 5)suki^k4a
+ (n — 5)stk2ak4^ + [(n — 5)2s2 — 8tUk4«k4M)]
2«k4^ + |(n — j) s — otujk4œk4^j (56)
We are going to find the couplings with structure F(n)F(n)фф for n = 1, 2, 3,4, 5 which are consistent with the above kinematic factor. For n = 5, only the last term survives. The coupling in the Einstein frame is
S D 14 f d 10x^-Ge—3ф0/2[FhpqrSmFkpqrs,nФмф,mn] (57)
6 к2 J
which can easily be extended to the S-duality invariant form. However, the couplings in the string frame which can be studied under the linear T-duality, are not so easy to read from the kinematic factor (56). So we consider all possible on-shell couplings with structure F(5)F(5)фф in the string frame with unknown coefficients, i.e.,
S D~ d1 °xV — G[C1 Fnkpqr,s Fnkpqr,s &,hm&,hm 2 к2 J
+ C2Fnkpqr,sFnkpqs,r^,hm^,hm + C3Fmkpqr,sFnkpqr,s^,hm^,hn + C4Fmkpqr,sFnkpqs,r^,hm^,hn + C5Fkpqrs,nFmkpqr,s^,hm^,hn + C6Fkpqrs,mFkpqrs,n^,hm^,hn + C1 Fhnpqr,s Fmkpqr,s^,hm^,nk + C8Fhnpqr,s Fmkpqs,r^,hm^,nk + C9 Fhnpqr,s Fmpqrs ,k^,hm^,nk + C10Fhpqrs,nFmpqrs,k^,hm^,nk + C11Fhpqrs,mFnpqrs,k^,hm^,nk
+ C12Fhpqrs,nFkpqrs,m^,hm^,nk 1 (58)
and find the coefficients by imposing the constraint that the couplings in the Einstein frame are given by the above equation.
To find the coefficients, one has to consider the string frame couplings with structure f(5)f(5)rr which have been found in [36], and the string frame couplings with structure F(5)F(5)Rф which have been found in (48). Both of them produce couplings with structure F(5)F(5)фф when transforming them to the Einstein frame (35). Transforming all couplings to the Einstein frame and constraining them to be identical with the coupling (51), one finds the following couplings in the string frame:
Lf d 10x J=S
TT Fmnpqr ,s(Fmnpqr ,s^,hk 10Fknpqr,s^,hm
Plus the following terms which contains some of the unknown coefficients:
S э 2Y
I d 10x V=G
C5Fkpqrs,nFmkpqr,s Ф,hm Ф,hn
5 C2,Fnkpqr,sFnkpqr,s^,hm^,hm + "J0 C11Fnkpqr,s Fnkpqr,s^,hm^,hm
+ C2Fnkpqr,sFnkpqs,r^,hm^,hm + C6Fkpqrs,mFkpqrs,n^,hm^,hn
C5 Fmkpqr,s Fnkpqr,s^,hm^,hn 5C6Fmkpqr,s Fnkpqr,s^,hm^,hn
2C11Fmkpqr,sFnkpqr,s^,hm^,hn + 4C5Fmkpqr,sFnkpqs,r^,hm^,hn
+ 20C6Fmkpqr,s Fnkpqs,r^,hm^,hn + C1 Fmkpqr,s Fnkpqs,r^,hm^,hn 1
2 C9Fmkpqr,sFnkpqs,r^,hm^,hn + 6C11Fmkpqr,sFnkpqs,r^,hm^,hn
10 11 12
20 21 22
10 11 12
20 21 22
2C12 Fmkpqr,sFnkpqs,r^,hm^,hn + Cl2Fhpqrs,nFkpqrs,m&,hm&,nk + ClFhnpqr,sFmkpqr,s^,hm^,nk + C8Fhnpqr,sFmkpqs,r^,hm^,nk + C9Fhnpqr,s Fmpqrs,k&,hm&,nk C11Fhpqrs,nFmpqrs,k^,hm^,nk
C12Fhpqrs,nFmpqrs,k^,hm^,nk + C11Fhpqrs,mFnpqrs,k^,hm^,nk
However, using the Bianchi identity and the on-shell relations, one finds that they are zero. To see this explicitly, consider for example the terms with coefficient C2, i.e.,
5 C2Fnkpqr,s Fnkpqr,s^,hm^,hm + C2Fnkpqr,sFnkpqs,r^,hm^,hm
To apply the Bianchi identity we write the RR field strength in terms of the RR potential. To impose the on-shell relations, we first transform the couplings to the momentum space and then impose the on-shell relations. One finds
-48C2(k1 .k2) C1hmnpC2npqrk1qk1rk2hk2m
which can easily be observed that it is zero using the totally antisymmetric property of the RR potential. Simile calculation shows that all other terms in (60) vanishes.
We now apply the T-duality transformations on the couplings (59) to find the string frame couplings with structure F(4)F(4)фф in the type IIA theory. To this end, we use the dimensional reduction on the couplings (59) and find the couplings with structure Fy5)Fy5)фф. Under the linear T-duality transformations, they transforms to the couplings with structure F(4)F(4)фф and some other terms involving Ryy in which we are not interested. The couplings with structure F(4)F(4)фф are
rf d », V=G
6 Fmnpq,r (Fmnpq8Fknpq,r^,hm)^,hk
The transformation of the above couplings, the couplings in (49) and the couplings with structure F(4)F(4)RR (seeEq. (26) in [36]), to the Einstein frame produces the following couplings:
S D 1Y f d 10xV-G, 2 к2 J
. 0 Fnkpq,rFnkpq,r^,hmL-
гФ.кшФ
+ , Fmkpq,rFnkpq,rФ,hmФ,hn
+ л Fhpqr,nFmpqr,kФ,hmФ
which are fully consistent with the kinematic factor (56) for n = 4. In writing the above result, we have used the Bianchi identity and the on-shell relations to simplify the result.
To find the string frame couplings with structure F(3)F(3)$$ in the type IIB theory, one has to use the dimensional reduction on the couplings (63) and find the couplings with structure F(4)F(4)$$. Then under the linear T-duality transformations, they transform to the following couplings with structure F(3)F(3)$$:
3 Fmnp,q (Fmnp6Fknp,qФ,hm)Ф,hk
10 11 12
20 21 22
46 We have checked that the transformation of the above couplings, the couplings in (51) and the 46
47 couplings with structure F(3)F(3)RR (see Eq. (20) in [36]), to the Einstein frame produces the 47
1 couplings with structure F(3)F(3)$$ which are consistent with the kinematic factor (56) for 1
2 n = 3. We will study the S-duality of these couplings in the next section. 2
3 Applying the T-duality transformations on the couplings (65), one finds the following cou- 3
4 plings in the type IIA theory in the string frame: 4
6 S D Y2 i d 10xV-G[-2Fmn,q(Fmn,q&,hk - 4Fkn,q&,hm)&,hk] (66) 6
7 K J 7
8 The transformation of the above couplings, the couplings in (52) and the couplings with structure 8
9 f(2)F(2)RR (see Eq. (21) in [36]), to the Einstein frame produces the following couplings: 9
10 1 f 10
11 S D 1Y2 d 10xV-G[9Fhk,pFhm,n®,kp®,mn + 10Fhm,nFnk,p&,hk&,mp 11
12 2 K J 12
13 - 9Fhm,nFhn,k&,kp&,mp - Fhk,pFhm,n&,mp&,nk] (67) 13
14 which are consistent with the kinematic factor (56) for n = 2. 14
15 Finally, applying the T-duality transformations on the couplings (66), one finds the following 15
16 couplings in the type IIB theory in the string frame: 16
18 S D Y2 f d 10xV-G[-4Fm,q(Fm,q&M - 2Fk,q&,hm)&,hk] (68) 18
19 K J 19
20 We have checked that the transformation of the above couplings and the couplings with struc- 20
21 ture F(1)F(1)RR (see Eq. (22) in [36]), to the Einstein frame produces couplings with structure 21
22 F(1)F(1)$$ which are consistent with the kinematic factor (56) for n = 1. 22
24 4.3.1. Consistency with the S-duality 24
25 The Einstein frame couplings F(n)F(n)$$ for n = 1, 3, 5 are in the type IIB theory, so they 25
26 should be consistent with the S-duality. For n = 5, the coupling is given in (57). The S-duality 26
27 invariant extension of this coupling is 27
29 S D--1 Y2 f d 10xV-GE3/2(Fhpqrs,mFkpqrs,nTr[MhhkM-1n]) (69) 29
30 12 K J 30
31 where the SL(2,R) invariant combination of the dilaton and the RR scalar is 31
32 1 [ ] 32
33 --Tr[M,hk M-Jn] = 2e2$0 Fh,kFm,n + k 2 $,hk$,mn (70) 33
34 4 34
35 The second term corresponds to the coupling (57). The first term is the S-duality prediction for 35
36 the couplings of two RR 4-forms and two RR scalars. 36
37 To study the S-duality of couplings for n = 3 case, consider the following S-duality invariant 37
38 action that has been found in [36]: 38
42 1Tr[M,nhM-m]HTmpnkM0Hnpr,h - M- M-1
S D 4 1 -10 K2
I d 10xV-GE3/2 1Tr[M,nhM"k]HTmp,hMoHmpr,k
- 2Tr[M,nhM-m]HTmp,kMoHnpr,h - ^rMhmM-nWmprkMoUnprh
44 where in the presence of zero axion background, the SL(2,R) invariant HTM0H has the fol- 44
lowing terms:
47 HT M0H = e-$0 HH + e$0 F(3)F(3) (72) 47
1 Using (70) and the above expression, one finds the S-duality invariant action (71) has four 1
2 different terms. Terms with structure ННфф which have been verified by the corresponding 2
3 S-matrix element in [52], terms with structure HHF(1)F(1) which have been verified by the 3
4 corresponding S-matrix element in Section 3.2, terms with structure F(3)F(3)фф and terms with 4
5 structure f(3)F(3)F(1)F(1). We have checked explicitly that the couplings in (71) with struc- 5
6 ture F(3)F(3)фф are consistent with the kinematic factor (56) for n = 3. The couplings in (71) 6
7 with structure F(3)F(3)F(1)F(1) are the prediction of the S-duality for the couplings of two RR 7
8 2-forms and two RR scalars. 8
9 To study the S-duality of couplings for n = 1 case, consider the following S-duality invariant 9
10 action that has been found in [52]: 10
¡2 SD^id 10x
S D K^f d 10xV-GE3/2 ^[M^mM-^
been verified by the S-matrix element of four dilatons in [52] and the couplings with structures
S D Y I d 10x
J d 10x V-G
T Fa1---an ,s Fa1---an ,s^,hk
42 The couplings with structure F(n)F(n)R< are the following: 42
. d 10-
(n - 2)! к2
«Л4 12
4 — I - -,rnaJ 1 nm I I
14 b [ ] [ ] 14
15 + ^Tr [M,nmM-hl] Tr[M,hk M-m] (73) 15
16 where the constants a, b satisfy the relation a + b = 1. Using the expression (70), one finds 16
17 the above action has three different couplings. The couplings with structure $$$$ which have 17
19 f(1)F(1)$$ and F(1)F(1)F(1)F(1). We have found that the couplings with structure F(1)F(1)$$
20 are reproduced by the kinematic factor (56) for n = 1. This fixes the constants to be a = -1 and 20
21 b = 2. The couplings in (73) with structure F(1)F(1)F(1)F(1) are the prediction of the S-duality 21
22 for the couplings of four RR scalars. These couplings have been confirmed in [52] to be consistent 22
23 with the linear T-duality of the couplings with structure F(3) F(3) F(3) F(3). 23
25 5. Discussion 25
27 In this paper, we have examined in details the calculation of the S-matrix element of two RR 27
28 and two NSNS states in the RNS formalism to find the corresponding couplings at order a'3. 28
29 For the gravity and B-field couplings, we have found perfect agreement with the eight-derivative 29
30 couplings that have been found in [36]. For the dilaton couplings in the Einstein frame, we have 30
31 found that the couplings are fully consistent with the S-dual multiplets that have been found 31
32 in [36]. We have also found the couplings with structure F(3)F(1)H$ which are singlet under 32
33 the SL(2,R) transformation. 33
34 Unlike the four NSNS couplings which have no dilaton in the string frame, we have found 34
35 that there are non-zero couplings between the dilaton and the RR fields in the string frame. The 35
36 couplings with structure F(n)F(n)$$ are the following: 36
( 1)! ,sFmal ■■■an-l ,s®,hm 38
44 S D ---^r— ¡d 10xV-GRhmnpFkta ■■■an-2 ,nFmpayan-2 ,t 44
av ■ ■ an-2 ,pFkna1 ■ ■ ■ an-2 ,t
+ (n - 2)Rhmn
47 3RhmnpFmpa 1■ ■ -an-2 ,t Fkna1■ ■ ■ an-2 ,t + RhmknFnta1■ ■ ■an-2 ,sFmsa1■ ■ ■ an-2 ,t~\Ф,hk (74) 47
45 - ! 45
46 + RhmnpFmta1-■ ■ an-2 ,pFkna1-■ ■ an-2 ,t + (n - 2)Rhmnp Fmpta1-■ -an-3 ,s Fknsay ■ -an-3 ,t 46
And the couplings with structure F(n)F(n 2)H0 are the following:
10 11 12
20 21 22
S э —
J d 10x V—G
(n 3)Hhrs,m Fkqrsa1 ■■■an—4 ,pFmpa1 ■■■an—4 ,q 1
(n — 3)! к2
+ Hhrs ,mFkrsa1---an—3 ,qFma1---an—3 ,q ( — 2Hh mr ,s Fkrsa1 ■■■an—3 ,qFma1---an—3 ,q + 2Hh mr,sFkqra1 ■■■an—3 ,sFma1 ■■■an—3 ,q
Hh rs .kFj
hrs,kFrsa1 ■■■an—2 ,qFa1 ■■■an—2 ,q
+ Hkqr,sFpqra1---an—3 ,s Fha1 ■■■an—3,p
The number of indices a1 ••• an-m in the RR field strengths is such that the total number of the
indices of F(n) must be n. For example, Fkqrsa1...an—4,p is Fkqrs,p for n = 4 and is zero for n < 4. We have shown that the transformation of the above couplings and the couplings with structure F(n) F(n—2) HR and F(n) F(n) RR to the Einstein frame are fully consistent with the S-duality and with the corresponding S-matrix elements. The above couplings have been found in this paper for 1 < n < 5. However, using the fact that for 6 < n < 9, it is impossible to have couplings in which the RR field strengths have no contraction with each other, one finds that the consistency of the couplings with the linear T-duality requires the above couplings to be extended to 1 < n < 9.
Using the pure spinor formalism, the S-matrix element of two NSNS and two RR states has been also calculated in [40]. The kinematic factor in this amplitude is given by
¿jmnpqm'n'p'q'a1---aMb1---bN r r f
R mnm'n' Rpqp'q ' Fa1---aM
,iFb1---bN,j
where R is the generalized Riemann curvature (1), and the tensor u is given in terms of the trace of the gamma matrices as
ijmnpqm' n' p' q' a1---aMb1"-bN
= —32
28Ngmpgm'q ' g'pgj(n' gp')k Tr(ynY
M !N !
— (SM + eN)gm'q'glqgjin'gp')k Tr (YmnpY
+ 1 SNgi[q\gjp'Tr(y \mnp]ya1 -aMym'n'p'yb1-bN^
nYa1—aM ykyb1'"bN\
av-aM ykyb1 -bN\
where cp = (-1)р+1/16л/2 and sN = (-1) 2N(N-1\ Writing the above kinematic factor in terms of the independent variables, we have checked that it is exactly identical to the kinematic factor (20) for the cases that N = M and N = M - 4. However, for the case that N = M - 2 the above result is different from (20). In fact the factor (eM + eN) in the third line above is zero for N = M — 2 whereas the corresponding kinematic term in (20) which is K2 + is nonzero. We think there must be a typo in the above amplitude, i.e., the factor (eM + eN) should be (iN-MsM + eN). With this modification, we find agreement with the kinematic factor (20) even for N = M - 2.
The S-duality invariant couplings in Sections 4.1.1 and 4.3.1 predict various couplings for four RR fields. These couplings may be confirmed by the details study of the S-matrix element of four RR vertex operators. This S-matrix element has been calculated in [40] in the pure spinor formalism. This amplitude can also be calculated in the RNS formalism using the KLT prescription (13) and using the S-matrix element of four massless open string spinors which has been calculated in [48]. In both formalisms the amplitude involves various traces of the gamma matrices which
10 11 12
20 21 22
have to be performed explicitly, and then one can compare the eight-derivative couplings with the four RR couplings predicted by the S-duality. We leave the details of this calculation to the further works.
The consistency of the NSNS couplings with the on-shell linear T-duality and S-duality has been used in [36] and in the present paper to find various four-field couplings involving the RR fields. Since the four-point function (26) has only contact terms, the Ward identities corresponding to the T-duality and the S-duality of the scattering amplitude appear as the on-shell linear dualities is the four-field couplings. On the other hand, one may require the higher derivative couplings to be consistent with the nonlinear T-duality and S-duality without using the on-shell relations. This may be used to find the eight-derivative couplings involving more than four fields. A step in this direction has been taken in [54] to find the gravity and the dilaton couplings which are consistent with the off-shell S-duality. It would be interesting to extended the four-field on-shell couplings found in [36] and in the present paper to the couplings which are invariant under the off-shell T-duality and S-duality.
Acknowledgements
This work is supported by Ferdowsi University of Mashhad under grant 3/27102-1392/02/25. Appendix A. Evaluation of traces for n = m and n = m + 2
In this appendix, we calculate the traces (22) for the cases that n = m and n = m + 2.
For the case that n = m, one can easily observes that T2 = T3. We use the following algorithm for performing the trace T1: There are three possibilities for contracting the first gamma ya with the other gammas. It contracts with one of the gammas in y11'''1n, with yT, and with one of the gammas in yvv''Vn. The symmetry factor and the sign of each contraction can easily be evaluated using the gamma algebra [yI,yV} = -2nIV. After performing these contractions, there is only one contraction for the leftover gammas. The leftover gammas for the first contraction is Tr(yI2'• • 1nYTYV1 • •Vn). This trace can easily be evaluated because yI2'''Inyt must be contracted with yV1' ' ' Vn .The reason is that the latter gammas are totally antisymmetric which cannot contract among themselves. The leftover gammas for the second contraction is Tr(yw'' InyV1 • •Vn) which has again one contraction. The leftover gammas for the third contraction is Tr(yIr' ' InyryV2-• •Vn). There is only one possibility for the gammas yTyV2 ' ' 'Vn to contract with the first bunch of gammas yIv' 'In. Taking the symmetry factors and the appropriate signs, one finds the result for T1.
For the other traces, one should note that after performing the first contractions, the leftover gammas have more than one contraction. However, the above algorithm can be used iteratively to reach to the trace which has only one contraction. After performing all the above contractions, one lefts with the trace of the identity matrix which is 32. The results are the following:
(_ 1 )n(n+1)^2
T- = 16-n[n(F?2T + F- natFi2\
T"P'JV = 16
(-1)n(n+1)a2
2 " n!
3n^(Ff2V - F??)
n! /_ FVpPo _ fppvo _ fpovp fpovp FVppo FVopp\
+ (n — 2)P r!2 r!2 r12 + F12 + F12 + F12 !
T^XiPpv _ 16 ( 1) ( + ) an
4 ^ n!
-6naVpn|VFx2 + 6^nPÀnIpF1V2a - napnIpF$
10 11 12
+ naPVXp F I1 + npXnIPFn - nap,nIpF'$ + naPVXp FI2 ) +
(n - 2)!
x -qPiF^" + -qP^F?1 - F?1 + q^iF^" - r)^kFpiva
_ nPl FPavX . naP FpavX , p FPaVI nap FpXvi fi^ FXavp
n F1 2 + n F1 2 + n F1 2 n F1 2 'I F1 2 + ^P^F^avP - n"PFiXvP - qPiFVP + qP^FViP® + ^P^F?^
„aPj7VIp „PXj7VaPI I „a^vXPI)
'I F12 n F12 + n F12 ! +
(n - 3)!
vpiPXa 12
vpXPia 12
vpaPiX
vpPiXa
piXvPa 12
piavPX 12
pXavPi 12
pPlvXa 12
- F 12 + F 12 + F 12 + F 12 - F 12 + F 12
j-,vikpPa j-,viapPX j-,vXapPi j-,vPipXa j-,vPXpia j-,vPapiX) + F12 - F12 + F12 - F12 + F12 - F12 !
10 11 12
20 21 22
where we have used the following notation in the above equations:
F12 = Fip F^ = FU
■■'In
1I1---Inr2
1l1-"In-1 f2
Fvpp° = F vp F F12 _ F1|1-"In-2 r2
FvPIpXa _ f vpi F
F12 _ F1|1-"In-3 r2
I1—In-2Po
I1~In-2PXa
Replacing the above tensors T1, ■■■ ,T4 into the kinematic factor (20), one finds the amplitude (10) for two RR «-forms and two NSNS states. We have checked that the amplitude satisfies the Ward identity corresponding to the NSNS gauge transformations. We have also checked that the kinematic factor vanishes when one of the NSNS states is symmetric and the other one is antisymmetric.
For the case that n = m + 2, one finds T2 = -T3. Using the above algorithm, one finds the following results for the tensors T1, T2, T4:
T°T = 16
(n - 2)! (-1)n2
n Tpta F12
(n - 2)!
.[3npaFV1pp + (n - 2)(Fvpap - F^ - FV1ppP° - FPpOV)]
T^kiPpv _ 16 ( 1)
(n - 2)!
6{nlpnPIF™ - npanPIF$ + /VF1) + 3(n - 2)
vXap + ^xp FvIaP
x (-nIPF-
~,vpXa pa -rpVpXi
+ naPFu
- npaF
- npaF1
xp J7VPia
- n F12
aP FvpaX nxp Fvpai
F 1 O + n F 1
+ n F12
' + nx
FVPIX + n^PF-
iP FvpXa _ npa FvpXi nxp Fvpia
F12 - n F12 - n F1
+ nPa. + n F12
FVPIX + n^PF-
pXav 12
pXav 12
— nXP FPiav npa Fpi'Xv) (n - 2)!
/77. PiXavp
- n f12 + n F12 ) + ,_A \ t
vPiXpa
(n - 4)!
+ F12 - F12
viXapP
vpPaiX 12
vppXia
I 77 _ J7wp
+ FvpiapX F12
vpiXPa
pPXavi 12
pPiavX 12
+ FvpPiXa FvpXapa
pPlXva 12
piXavP ) 12
20 21 22
1 where we have used among other things the fact that an-2 = an. In above tensors, we have used 1
2 the following notation for F12s which are different from the ones in (79): 2
J?GT _ -p GT J?ii1"iIn-2
4 F12 — F 1|1 '"In-2 F 2 4
5 pvpfio = F vpP
F12 — FlMl--Mn-3 f2
34 Fn = (-1)2(10-p)(9-p) * FW-n (84)
36 and are related to the RR potentials as
6 fvpip^® — F vpiP Fl1"'In-4k® (81) 6
7 F12 — F 1|1 '"In-4 F 2 (81) 7
8 Note that F1 is n-form and F2 is (n - 2)-form. Replacing the above tensors T1, ••• ,T4 into the 8
9 kinematic factor (20), one finds the amplitude (10) for one RR n-form, one RR (n - 2)-form and 9
10 two NSNS states. We have checked that the amplitude satisfies the Ward identity corresponding 10
11 to the NSNS gauge transformations. We have also checked that the kinematic factor vanishes 11
12 when one of the NSNS states is symmetric and the other one is antisymmetric. 12
14 Appendix B. Massless poles 14
16 In this appendix we calculate the S-matrix element of two RR and two gravitons in the type II 16
17 supergravities and compare the result with the leading order terms of the string theory S-matrix 17
18 element (10) which is given by the following amplitude: 18
19 19 25K 2e-2$0
20 Aow = i-K (82) 20
21 stu 21
22 where the explicit form of the kinematic factor K for various external states has been given in 22
23 Sections 3 and 4. 23
24 The S-matrix element in field theory is independent of the field redefinitions which present 24
25 different forms of the same theory. We use the democratic formulation of the supergravity which 25
26 has been found in [49]. The RR part of this action is reproduced in the double field theory [53], 26
27 so this part of the action is manifestly invariant under the T-duality. In the Einstein frame, the 27
28 action is given as 28
29 / 9 v 29
1 di^d1® - 1 e-®\H|2 - 1
31 2K2 j \ 2 2 4 " I 31
33 The nonlinear RR field strengths satisfy the following duality relation: 33
30 5 = ¿/d10*^ R - 2 - ±e-*\H |2 - i¿ e 1 (5-n)* |Fn|2) (83) 30
38 = e-BJ2dCn-i (85) 38
39 n=1 n=1 39
where the product is the wedge product. The linear field strengths, e.g., H = dB, are defined
41 such that 41
43 H^ = ± cyclic permutations 43
44 , 44
45 If A isa p-form, i.e., A = p. A^1...^pdxIH A dxtlp, and B isa q-form then 45
46 (p + q). 46
47 (A A B)^v..^pvv..vq = --^—A[^v..^pBvv..Vq ] (86) 47
10 11 12
20 21 22
where the bracket notation means antisymmetrization in the usual way, e.g., A^BV] = 1 (A^BV — AVB,Â). The square of the RR field strength in (83) is \Fn\2 = 1 F^v.. MnF ■ ■ In.
Given the supergravity (83), one can calculate different propagators, vertices, and subsequently the scattering amplitudes for massless NSNS and RR states. To have the normalizations of fields in (83) to be consistent with the normalization of the string theory amplitude (10), we use the following perturbations around the flat background, i.e.,
B(2) = 2Kb(2) ;
§I^V — + 2Kh„V;
0 = 00 + V2k$; C(n) = 2V2K€(n)
Note that the coefficient of the RR kinetic term in the conventional form of the supergravity is twice the RR kinetic term of the action (83). On the other hand we have normalized the string theory amplitude (10) to be consistent with the normalization of fields in [36]. The normalization of fields in [36] are consistent with the conventional form of the supergravity. Therefore, we have added an extra factor of V2 in the normalization of the RR potentials in compare with the normalization of the RR fields in [36].
The scattering amplitude of two NSNS and two RR fields in the supergravity is given by the following Feynman amplitude:
A = As + Au + At + Ac
where the massless poles and contact term depends on the polarization of the external states. When the two NSNS states are dilaton, the massless poles in s- and u-channels, and the contact terms are given as
As = [V,n)F(n)hTv [Gh]iv,Xp [Vh$3$4}kp
Au = ^F^foCd-1) ]
Ac = GJ7 (n)J7(n)^^.
I1 ■ ■ ■ In— 1
[Gc(n—1) ■ ■In.
V1-■ ■ Vn—1\G , ^
-1 [VC(n — 1)04^(n) W■ ■ Vn— 1
The massless pole in the t-channel is the same as Au in which the particle labels of the external dilatons are interchanged, i.e., At = Au(3 ^ 4). Using the supergravity (83), one finds the propagators and the vertices in the above amplitudes to be
[Gh]iv,kp = — TT2 ( Vl'kVvp VipVvi . VivVip
[Gc(n)]
I1 ■ ■ ■ In
in! V\ V2 = nl2 '
■ ■ nin]
[%304h]kp = —^K^k? — 2k3 ■ k4nkf^j
^F^F^h
]kp = ik1 (2nF(kP) — n'kpFn) 1
F((,) 03C(n—1)-W ■ ■ Vn—1
GF<n)F,(n)0304 = —2lK
\[2(n — 1)!
(5 — n)Fikvv ■ ■ vn—1 k
1/5 — n
where we have used the notation (79). Replacing them in (89), one finds the following results:
10 11 12
20 21 22
10 11 12
20 21 22
4iK2 / 1
A =__iAv,{T7.-hXhP
4n(F12hpkXkp - -(5n - 9)F^k3 -U
2iic2n -, ,
Au = 7-T^(5 - n)2(F12)Xp(k1 + k3)X(k1 + k3)P
(n - 1)!u
2iK2 n -, ,
Ai = --—(5 - n)2(F12)Xp(k1 + k4)X(k1 + k4)P
(n - 1)!i
Ac = -2iK
1/5 - n
Replacing these result into (88), one finds the field theory scattering amplitude of two RR and two dilatons which should be compared with the corresponding amplitude in (82). When we use the on-shell relations to write both amplitudes in terms of the independent variables, we find exact agreement between the two amplitudes for n = 1, 2, 3,4, 5.
When the two NSNS states are B-fields, the massless poles in s- and w-channels, and the contact terms are given as
As = [VF(n) F(«n) hTv [Gh\^v,xp [Vhb3b4 \Xp + [vF(n)F(n)0 ][GV ][Vtpb3b4 ]
Au = [V,
F(n)b3C(n-3)J
-||1 • • • In-3
[Gc(n-3) ]|r. . in_
V1- • • Vn-3
?(n) ]
Vr • • Vn-3
+ [VVF(n) b3C(n+1) ]I1 In+1 [GC(n+X) W • • 4n+1 V1 Vn+1 [VVC(n+1)b4Fn V • • Vn+1
Ac. = V
F,(n)F2(n)b3b4
The vertices in the first line of Aw are non-zero for n > 3. The graviton and the RR propagators,
as well as the vertex [V(n)Anu have been given in (90). The dilaton propagator and all other
F1 F2 h
vertices can be calculated from (83). They are i
G* = -2
(Vb3b4h)Xp = -2iK
2 (k3 • k4 nXp - kX kp - kpkX) Tr(b3 • b4)
- k3 • b4 • b3 •U nXp + 2 k(Xb4
k(Xh,P)
b3 ^4 + 2k4Xb3P)-b4- k3
+ 2k3 • b4(Xb3p) • k4 - k3 • k4 b •bpp + bX • bp)
Vb3b4<p = -iV2x[2k3 • b4 • b3 • k4 -k3 • k4Tr(b3 • b4)]
VF<n)F2(n)^ = -iK
(V/F(n)^C(n-3))
(5 - n)F1 • F2
F(n)b3C(n-3))V1- • • Vn-3
■PIV1- • • Vn-3^3
(V (n)
F\n)b3C(n+1))V1- • • Vn+1
= -2iK
(n - 3)!
k b3X[v1 F1V2 • • • Vn+1]
2iK (n -
(n + 1)! ■ 2)(n + 1)
(n) = '
F,(n)F2(n)b3b4
b3[|1|2 F1|3-• • 4n+2]b4
1112 f243 • • 4n+2 + (3 ^ 4) (93)
Replacing them in (92), one finds the massless poles and the contact terms of the scattering amplitude of two B-fields and two RR fields. We have compared the resulting amplitude (88)
10 11 12
20 21 22
with the corresponding string theory amplitude (82) and find exact agreement for n = 1, 2, 3,4, 5 1
when we write both amplitudes in terms of the independent variables. We have done similar 2
calculations for all other external states and find agreement with the string theory result. 3
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