Scholarly article on topic 'On the soft limit of closed string amplitudes with massive states'

On the soft limit of closed string amplitudes with massive states Academic research paper on "Physical sciences"

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Abstract of research paper on Physical sciences, author of scientific article — M. Bianchi, A.L. Guerrieri

Abstract We extend our analysis of the soft behavior of string amplitudes with massive insertions to closed strings at tree level (sphere). Relying on our previous results for open strings on the disk and on KLT formulae we check universality of the soft behavior for gravitons to sub-leading order for superstring amplitudes and show how this gets modified for bosonic strings. At sub-sub-leading order we argue in favor of universality for superstrings on the basis of OPE of the vertex operators and gauge invariance for the soft graviton. The results are illustrated by explicit examples of 4-point amplitudes with one massive insertion in any dimension, including D = 4 , where use of the helicity spinor formalism drastically simplifies the expressions. As a by-product of our analysis we confirm that the ‘single valued projection’ holds for massive amplitudes, too. We briefly comment on the soft behavior of the anti-symmetric tensor and on loop corrections.

Academic research paper on topic "On the soft limit of closed string amplitudes with massive states"

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Nuclear Physics B 905 (2016) 188-216

www. elsevier. com/locate/nuclphysb

On the soft limit of closed string amplitudes with

massive states

M. Bianchi, A.L. Guerrieri *

Dipartimento di Fisica, Universita di Roma "Tor Vergata" and Sezione INFNdi Roma II "Tor Vergata", Via della Ricerca Scientifica, 00133 Rome, Italy

Received 24 December 2015; received in revised form 30 January 2016; accepted 3 February 2016

Available online 8 February 2016

Editor: Stephan Stieberger

Abstract

We extend our analysis of the soft behavior of string amplitudes with massive insertions to closed strings at tree level (sphere). Relying on our previous results for open strings on the disk and on KLT formulae we check universality of the soft behavior for gravitons to sub-leading order for superstring amplitudes and show how this gets modified for bosonic strings. At sub-sub-leading order we argue in favor of universality for superstrings on the basis of OPE of the vertex operators and gauge invariance for the soft graviton. The results are illustrated by explicit examples of 4-point amplitudes with one massive insertion in any dimension, including d = 4, where use of the helicity spinor formalism drastically simplifies the expressions. As a by-product of our analysis we confirm that the 'single valued projection' holds for massive amplitudes, too. We briefly comment on the soft behavior of the anti-symmetric tensor and on loop corrections. © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction and motivations

The connection between 'gravitational memory', 'soft behavior' of graviton scattering amplitudes and 'BvBMS symmetry' [1-6] seems to play a crucial in a recently proposed solution to the Information Paradox for Black Holes [7]. While waiting for a refined version of the argument, it is natural to ask the fate of the universal 'soft' behavior of graviton scattering amplitudes

* Corresponding author.

E-mail addresses: massimo.bianchi@roma2.infn.it (M. Bianchi), andrea.guerrieri@roma2.infn.it (A.L. Guerrieri). http://dx.doi.Org/10.1016/j.nuclphysb.2016.02.005

0550-3213/© 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

in a quantum theory of gravity such as closed string theory. The problem has been addressed for tree-level amplitudes with only mass-less gravitons in [8,9], relying on KLT formulae and OPE of the vertex operators, and in [10], relying on gauge invariance. Bosonic amplitudes with tachyons have been investigated to sub-leading order in [11,12].

In gravity theories, when one of the external graviton momenta goes soft i.e. k ^ 0 with k = Sk with k some fixed momentum, not only the leading S—1 and sub-leading behaviors 8° [13, 14], but also the next-to-subleading or sub-sub-leading behavior S+1 is universal [15]. Calling k^v the soft graviton polarization and k's its soft momentum, one has

Mn(1, 2,...,s,...,n)

ki 'ks'ki + ki 'ks Ji 'ks + ks Ji 'ks Ji 'ks

ks 'ki ks 'ki 2ks 'ki

Mn-1(1, 2,...s ...,n) + O(S2) (1)

where ki and Ji denote the 'hard' momenta and angular momentum operators. These results are valid at tree-level and are derived with the understanding that interactions be governed by minimal coupling.

In theories with closed strings, the conclusions, though quite independent of the number of (non-compact) space-time dimensions, depend on the nature of the higher derivative couplings [8]. R3 terms do not change the universal soft behavior of minimal coupling, while $R2 do modify even the leading term when $ is a massless scalar such as the dilaton. This happens in particular in the bosonic string and heterotic string at tree level1 and in the Type II compacti-fications preserving less than maximal super-symmetry.

The aim of the present investigation, that may be considered a follow up of [16], is to show that inclusion of massive external states does not spoil the universal 'soft' behavior (1) for Type II theories with maximal susy at tree level. In [16] open string amplitudes with massive external states as well as tachyons have been computed and shown to expose the expected behavior even when non-minimal interactions are considered. Neither F3 terms nor the coupling a'TF2, where T is the tachyon, change the universal soft behavior, based on minimal coupling. On the other hand $F2 terms do modify even the leading term when $ is a massless scalar. For color-ordered string amplitudes one gets the same universal behavior as in YM theories [17-29]

An(1, 2,...,s,...,n) ^

as-ks+1 as-ks-1

_ks'ks+1 ks ks — 1

fs :Js+1 fs :Js — 1

ks 'ks+1 ks ks— 1

An—1(1, 2,...s ...,n) + O(S) (2)

where as and ks denote the soft gluon polarization and momentum, so that f'sLV = k^aV — k^aV is its linearized field strength, while ks±i and Js±i denote the 'hard' momenta and angular momentum operators of the adjacent insertions. Relying on [16] and on KLT formulae, we presently analyze closed string amplitudes with massive external states. In the bosonic string case we will also consider tachyons as external states.

Amplitudes with massive external states have been considered earlier on [30-33], see also [34] for the case of 'light' string states and [35-41] as well as the review [42] for more phenomeno-logical applications. The plan of the paper is as follows.

1 M. B. would like to thank I. Antoniadis for stressing the tree level origin of this term in the heterotic string, which only

gets generated at one-loop in 4-dim Type II theories with 16 supercharges, such as after compactification on K3 x T2.

R3 term is forbidden due to supersymmetry.

In Section 2, we briefly review KLT formulae relating closed string to open string amplitudes and the 'single valued projection' suggested in [43,44]. Then we discuss how to relate the soft limit of closed string amplitudes with an arbitrary number of massive insertions to the soft limit of open string amplitudes with the same number of massive insertions in Section 3. In Section 4 and 5 we illustrate our point with explicit examples of 4-point amplitudes with one massive higher spin insertion (or tachyons in the bosonic case). We check the (non-)universal-ity of the soft behavior for bosonic string gravitons in Section 6 and discuss how to generalize the analysis to the case of anti-symmetric tensors. For the superstrings in D = 4 we rely on the spinor helicity formalism to simplify our expressions. Our conclusions are presented in Section 7.

2. From Veneziano to Shapiro-Virasoro according to KLT

Closed-string amplitudes, henceforth denoted by Mn to distinguish them from open-string amplitudes, denoted by An, can be efficiently computed relying on KLT formulae [45]. At the cost of being pedantic, in order to fix our notation and illustrate the KLT procedure, we start by briefly reviewing some 4-point string amplitudes involving tachyons or massless states.

In going from open to closed strings the mass shell condition becomes a'c(p/2)2 = (N - 1) that effectively amounts to the replacement a'o ^ a'c/4.2 As a result a closed string vertex operator can be expressed as the product of two open-string vertex operators, each carrying half of the total momentum. In formulae

Vci(H = H ® H,p) = Vpp(H, p/2)VRp(H, p/2), (3)

where p2 = m^ = 4m2H, and H = H ® H in general comprises several irreducible representations of the Lorentz group.

2.1. Four tachyons: M(T1, T2, T3, T4)

The simplest closed-string amplitude is the Shapiro-Virasoro amplitude M4(T1, T2, T3, T4) describing the scattering of four tachyons in the closed bosonic string. The tachyon vertex operator is

VT (z, z) = eipX(z~z) = eipp Xp(z)eip Xr(~z), (4)

with a'c p2 = +4 = —a'c M^. Up to an overall constant factor, one finds [46,47]

M4(Ti, T2, T3, T4) = n j d2z |z|acp3p411 - z|acp2p3

r(1 + ^p3p4W(1 + acp2p3)r(-1 - p3(p2 + p4))

r(-a2cp3p4)r(-a2cp2p3)r(2 + p3 (p2 + p4)) where use has been made of the integral

2 While the open string spectrum is given by a'0pM'N = N — 1 with N = SMax, the closed string spectrum is given by a' ,MN = 4(N - 1) = 2(NP + Nr - 2) due to level matching Np = NR = N = SMax/2.

I(a,n; b,m) = \ d2z |z|a |1 - z\bzn(1 - z)m

r(1 + n + f )T(1 + m + §)-(-1 - a++b) = r(-f)r(-|)r(2 + n + m + a+++b) ' Rewriting the amplitude as a function of the Mandelstam variables s, t, u yields

r(-1 - ofs)r(-1 - oFt)-(-1 - oFu)

MA(TU Tf, T3, T4) = n—-y--a ) ( a> , (7)

r(2 + O-S)-(2 + Oft)-(2 + ^u)

multiplying and dividing by -(-1 - a'ct/4), and using the relation -(z)-(1 - z) = n/sinnz produces the KLT relation [45]

M4(Tx, 72, T3, 74) = sin ^-^tj AL(T1,T2,T3,T4)AR(T1,T3,T2T), (8)

where A4(Ti, T2, T3, T4) denotes the Veneziano amplitude

A4(Tx,T2,Ts,T4) = I dxx-a0S-2(1 - x)-a'ot-2 = --—' 7 v-—, (9)

- (-1- ^s)- (-1 -«-ft)

- (-2 - af (s +1))

where we have used a'o ^ a'c/4. Henceforth we will set a'c = 2 for convenience. 2.2. Four massless superstring states: M4(£\, E2, E3, E4)

In Type II superstrings the tachyon is projected out. The lowest lying states in the NS-NS sector are massless. The massless vertex operator

VE = E,v(idX^ + XR + k^R^)eiki Xl e 2 Xrz (10)

with k2 = 0, = £^vkv = 0. Setting £/xv_ = Ev, = h,v, with n,vh,v = 0, describes gravitons, E,v = Ev, = v = n,v - k^kv - kvk, describes dilatons, while E,v = -Ev, = b,v describes anti-symmetric tensors (Kalb-Ramond fields). For later purposes, it is crucial to observe that gravitons and dilatons are even under L-R exchange, ^ = 1, while Kalb-Ramond fields are odd, ^ = -1. This implies that amplitudes with an odd number of Kalb-Ramond fields and an arbitrary number of gravitons and dilatons vanish.

The amplitude for 4 massless NS-NS states is well known. The expression is extremely lengthy and can be expressed more compactly in terms of the t8 tensor introduced by Brink, Green and Schwarz [48]. We refrain from doing so. Using KLT in the t-channel, one finds

M4(EX, E2, E3, E4) = sin (n ;Q AL(AX,A2,A3,A4)AR(A1,A3,A2,A4). (11)

Now writing [37]

L FL -(1 - s)-(1 -1)

AL(Ax,A2,A3,A4) = L ( ) (

st -(1 + u) with

(/1/2/3 f4) - 2(hf2)(f3f4) + cyclic 234

totally symmetric, and rewriting Fp ® FR « R4 + ... one can systematically derive the Type II 4-graviton amplitudes and the related ones for 0's and (an even number of) b's.

For instance, in D = 4, F4 is only non-vanishing for MHV (Maximally Helicity Violating) configurations i.e. (-, -, +, +) or permutations thereof. As a result, F4 = (12}2[34]2. Similarly, R4 = (12}4[34]4 for the MHV configurations, i.e. (-2, -2, +2, +2). Mixed amplitudes, with gravitons, dilatons and axions arise from combinations with Fp = FR, for instance, (-2, 0, +2, 0) = (-, -, +, +) ® (-, +, +, -) = (12}2(14}2[34]2[23]2 and (0, 0, 0, 0) = (-, -, +, +) ® (+, +, -, -) = (12}2(34}2[34]2[12]2, while (±2, ±2, ±2, 0) = 0, (±2, ±2, 0, 0) = 0, (±2, 0, 0, 0) = 0, irrespective of whether the h = 0 particle is a dilaton or an axion.

For bosonic strings the situation is richer. For open strings the tri-linear coupling is nonminimal. In addition to the standard Yang-Mills term, it contains an F 3-term, suppressed by a'. As mentioned in the introduction and discussed in [16], this does neither spoil universality of the soft behavior at leading order nor at subleading order, even in the case of massive insertions. For closed bosonic strings, in addition to minimal tri-linear terms (graviton, dilatons and Kalb-Ramond fields), there is a $R2 term (suppressed by a') and an R3-term (suppressed by (a')2). As shown in [8], the latter does not spoil the universality of the soft behavior while the former spoils it even at leading order. Barring the distinction between gravitons and dilatons, i.e. describing them in a unified fashion with £jXV = +EVM = htxv + $IJvV, one can regain a sort of universality of the soft behavior as advocated in [11,12]. Yet b^v behaves in a very different way due to its being odd under as we will see in Section 5.

2.3. Higher-point amplitudes

Closed-string amplitudes with massive insertions look extremely cumbersome and not very illuminating in D = 10, even at tree level (sphere). In D = 4, using the spinor helicity basis, formulae look more tractable. A possible strategy for systematic computations is to first use KLT relations in order to express closed-string amplitudes in terms of open-string amplitudes, and then compute open-string amplitudes for massive states by multiple factorizations of amplitudes with only massless insertions on massive poles in two-particle channels as in [16].

KLT relations incorporate the intrinsic non-planarity of closed-string amplitudes and rely on the monodromy properties of (color-ordered) open string amplitudes [45]. The basic idea is to parameterize the closed-string insertion points as zi = xi + iyi and notice that the integrand is an analytic function of the yi viewed as complex variables with branch points at ±i(xi - xj). One can then deform the integration contour from Imyi = 0 to Rey; = 0 so much so that zi and zi = x; - iyi become two independent real variables fi and ri that one can integrate over with Jacobian d(xi, yj)/d(fi, ,j) = (i/2)N. The correct monodromy around the branch points of the integrand (Koba-Nielsen factor, in units a'c = 2)

nte - zj)kikj +nij(~zl-z j)kkj +nj ^ ^(o^or,) - j)kikj +nij(n> - nj)kikj +nj i>j i>j

with nij and nij integer, is accounted for by the phase factor

$(o^,Or) = Y\exp[inkikj0[-(£• - fj)(ri - nj)]} i>j

that only depends on the orderings of and or but not on the variables f 's and n's themselves. The integrations decouples and can be performed explicitly. In particular, using SL(2) to fix 3

f's, there remain (n - 3)! orderings of the f's. For each of them, the independent choices of the contours in r that give a non-vanishing result give in fact all the same result. All in all there are (n - 3)![±(n - 3)!]2 terms for n odd or (n - 3)![2(n - 4)!][2(n - 2)!] for n even [45]. In particular, for n = 3, 4 there is only one term3

M3(123) = Ap(123)AR(123)

M4(1234) = sin(nkxk2) Ap (1 [2] 34) AR (2134). For n = 5 one has two terms

M5(12345) = sin(nkxk2) sin(nk3k4)^p(1[23]45)^R (21435)

+ sin(nk\ks) sin(nk2k4)Ap(1[32]45)AR (31425), (12)

while for n = 6 one has twelve terms

M6(123456) = sin(nkk) sin(nk4k5)Ap(1[234]56)

[ sin (nk3k5)Ap (215346) + sin(nk3k + k5))Ap (215436)} + Perm[234] = sin (nkk) sin(nk4k5)Ap(1 [234]56)

|sin(nkik3)AR (231546) + sin(nk3(ki + k2))ApR (321546)} + Perm[234].

In general, one has [49]

Mn(1, 2,...,n) = Ap(1, [2,...,n - 2],n - 1 ,n)

f(i1,...,i[n/2i-1)f(j1,...,j[n/2i-2)AR([i}, 1,n- [j},n)

[i},[j}

+ Perm[2,...,n - 2], (14)

where {¿} e Perm[2, ..., |_n/2J], [j} e Perm[Ln/2J + 1, ..., n - 2], with |_n/2J = (n - 1)/2 for n odd, and |_n/2J = n/2 - 1 for n even, while the relevant momentum kernels read [49]

m-1 / / m

f(i1,...im) = sin(ni1im) Y[ sin I n I suk + ^ $ikH k=1 \ \ l=k+1 m / / k-1

f (j1,...jm) = sin(nSj1n-1) siM n|sjkn-1 + ^ Sjijk , (15)

k=2 l=1

where sij = sij = ki kj, if i > j, and zero otherwise. Let us observe once again that KLT formulae are valid for all kinds of closed strings, Type II, Heterotic and Bosonic, at tree level and for any kind of insertions: tachyonic, mass-less or massive.

Similar formulae relating string amplitudes with only massless insertions to SYM amplitudes [50,51], see also [52], have been derived for open superstrings, whose validity we have given further support in [16]. MSST formulae read

3 Neglecting overall constants.

AST(1,p\2,...,n — 2],n — 1,n) = ^^ Fn\p\a]AYM(1,a\2,...,n — 2],n- 1,n)

a eSn—3

where the (n — 3)! x (n — 3)! dimensional matrices of generalized Euler integrals read

r n—2 n—2 k—1~ ,

■n—3r. n—4 / dziY[ z2a kik^

D(p) l=2 i<j k=2 m=1

Fn\p\a] = (—1)n—3(VaiT—4j ndziUz2ikikj HE (17)

with integration domain D(p) = {0 = zi < p(z2) < ... < p(zn—2) < zn—1 = 1 < zn =

Following the strategy outlined above, one can now combine the virtues of KLT and of MSST. For instance, at 5-points a closed (super)string amplitude with n massless and m = 5 — n massive states, according to KLT, reads

Mn,5— n(12345) = sin(^^12/2) sin(ns34/2)AL,5—n(1\23]45)AR,5—n(21435)

+ sin(ns13/2) sin(ns24/2)AL5—n(1\32]45)AR5—n (31425) (18)

In turn, the open string amplitude AL/—n(12345) can be computed factorizing AL/Rn 0(1...5) on 5 — n massive poles in two-particle channels. The massless amplitude AL/Rn 0(1...5) can be expressed in terms of AfYMn(1...5) thanks to MSST formula. The generalization, relating Mn,m

with arbitrary n and m to and the latter to AJ^^ 0 and finally to Asn™m is straightforward, but more and more cumbersome as the number of particles increases.

2.4. From open to closed via 'single-valuedprojection'

Although we will not fully exploit it in the following, an alternative and elegant expression of closed superstring amplitudes with massless insertions only in terms of SYM amplitudes at tree level has been found in [43,44] that exposes the cancellation of various MZV (Multiple Zeta Values) including rational multiples of Z2n in the a' expansion. The 'single-valued projection' formula reads4

Mn = J2 AYM (1,2p, 3p,...(n — 2)p,n,n — 1)

p,a,t€Sn—3

x ^0\2p, 3p,...(n — 2)p\2a, 3a,...(n— 2)a ]

Gn\a\T ]AYM (1, 2r, 3r,...(n— 2)r,n— 1,n) (19)

where ^0\p\a ] = Sklt\p\a ]\

S0 \p(2,...,n — 2)\a(2,...,n — 2)]

n-2 i-1

= n I —kikp(i) — 0a(p(i), p(j))kp(i)kp(j) I (20)

i=2\ j =2 J

with 0a(p(i), p(j)) = 1 if the ordering of (p(i), p(j)) is equal to the ordering of (a(i), a(j)) and zero otherwise. S0\p\a] is the 'super-gravity' limit of the KLT momentum kernel such that

4 Notice the exchange of n and n — 1 in A^M (1, 2p, 3p,. ..(n — 2)p, n, n — 1).

sin(na'kikj/2) ^ na'kikj/2 and the (n - 3)!x(n - 3)! matrix Gn[p\a] is given by the 'single-valued projection'

Gn[a \ t ] = 1 + Z3M3 + Z5M5 + ^ K3M3M3 + 2Z7M7 + ... = sv[FnW \ t ]}

= sv j 1 + Z2^2+Z3M3+Z22^4+Z5M5+Z2Z3^2M3 + Z23^6

+ 2 Z32M3M3+2Z7M7+Z2 Z5 P2M5+Z2Z3 P4M3+ ... J (21)

of the (n - 3)!x(n - 3)! matrix Fn[p\a] that appear in MSST formula. Not only all P2n matrices drop but also higher depth MZV's do as a result of properties of the M2k+1 matrices.

3. Soft limit from open to closed

When considering the soft behavior of string amplitudes one may expect corrections from standard field theory results due to the non-minimal higher-derivative terms in the coupling among mass-less states as well as with massive states. For open strings we have checked that this higher-derivative couplings coded in the OPE of the vertex operators do not spoil universality of the soft behavior at leading and sub-leading order. For completeness, let us now recall the argument [8,9,16]. The OPE of a massless vector boson vertex operator (in the q = 0 super-ghost picture) and a massive higher spin vertex operator (in the q = -1 super-ghost picture) reads

VA(as,ks)VM(Hs±1,Ps±1) -vm'(H [as,Hs±1 ,ks,ps±i],ks + ps±1) + ... (22)

2ksps±1

where M' denotes any state at the same mass level as the state M. For totally symmetric tensors of the first Regge trajectory at level N = I - 1 one has

A3(A1,H2,e,H3,e) = a1P23H^1^t H^..^ + pvuH3^2...^

+ p31,MH2W2...MVH3,vM2...w + O(a'p2). (23)

The leading term encodes minimal coupling. The sub-leading term is fixed by gauge invariance so that, barring some subtleties, to be dealt with momentarily, one gets

An+1,m(1,...s ...,n + m + 1)

^±J as ps+1 - £ ks-Hs+1 a d + as-ps+1 k d + £ as • 1 k • d 1

2ks •ps+1 2ks •ps+1 dH;+1 2ks-ps+1 dps+1 2ks-ps+1 dH;+1 J

ks ps-L. 1 d

T 0,s ps+1 as ^-An,m(1,...s ...,n + m + 1) + ..., (24)

2ks-ps+1 dps+1

for an amplitude with n massless and m massive states.

Before generalizing the above argument to the closed string case, let us deal with a couple of subtleties: the higher derivative terms in the tri-linear coupling A-H-H and the possible nondiagonal couplings A-H-H' that would spoil universality. First, higher derivative corrections to minimal coupling can only affect the sub-leading term that is fixed by gauge invariance wrt the soft gluon [10]. Second, for open superstrings already at the first massive level one finds two kinds of particles in the Neveu-Schwarz sector: C^vp and H^v. In addition to the 'diagonal' couplings V-C-C and V-H-H (and SUSY related) one should consider the mixed coupling V-H-C ^ a'Mp31-H2-C3:[a1p12] that exposes the singular soft factor 1/kp since Mc — Mh

but gets suppressed by an extra power of the soft momentum in the numerator. Lacking the leading 8—1 term that fixes also the sub-leading 80 term, thanks to gauge invariance, this kind of higher derivative non-diagonal couplings can at most affect the sub-sub-leading 8+1 (and higher) terms which are not expected to be universal.

Relying on KLT, similar arguments were advocated to warrant universality of closed super-string amplitudes to leading, sub-leading and sub-sub-leading order [8,9]. Indeed, the relevant OPE's of closed string vertex operators are simply the L+R combinations of the ones shown above for open strings. This implies that the leading behavior is completely fixed by the trilinear coupling. If this is minimal as for the superstrings one gets a universal behavior if it is not, as for the bosonic and heterotic strings one expects non-universality or some sort of generalization thereof [11]. The additional ingredients are two. First, KLT formulae produce amplitudes with non-planar duality, with the soft graviton that can attach to each of the 'hard' (massless or massive) legs. Second, not only the sub-leading but also the sub-sub-leading term is fixed by gauge invariance of the soft graviton [10]. We would like to stress that this is true also for amplitudes with massive insertions as we will now sketch and check with explicit examples later on. Given universality of the soft behavior of all open string amplitudes for granted [16] one schematically has

Mn+1 = £ nsin(nkk')/AL+1(...)AR+1(...) i i

« sin(nkk')i(Sf + SL) + -UL^KS^ + sR1 + ...)AR(...) ii

= (S(0) + S(r) + s(2) + ...)£n sin (nkk' )i AL(...)AR (...). (25)

One can easily check that sga = S^sR using momentum conservation, similarly Sga =

sLr)sR0) + sfsRl). Finally the sub-sub-leading sg^ = S^S^ + sfsf + S^S^ to be

checked on a case by case basis since SL/'r is not universal, but conspires with the permutation to give something universal. We will limit ourselves to check cancellation of n2 = 6£2 and similar terms that are forbidden by the single-valued projection [43,44]. At the cost of being pedantic we would like to reiterate that once the leading term is fixed and universal then sub-leading and sub-sub-leading terms follow thanks to gauge invariance of the soft graviton.

3.1. 4-point amplitudes with massive states

Let us consider first 4-point amplitudes. We already know that

M4(1234) = sin(n p1k4)AL(1234)AR(1324), (26)

allowing for a time-like p1, while we assume k4 to be light-like and 'soft' with 'polarization' E = aL ® aR. From KLT we also know that

A4(1234) = S43—1A3(123) and ^(1324) = 1A3(123), (27)

sj — = sj— + sj— + j — Z2 ksPj ksPisj—) , (28) with universal

ci — _ a'ki Si(0) —

atki k(ki

ci-l _ fiJj

Si(1) = kk-kikj

ci-l _ Si(2) —

f-Wik,- fi-Wiki

is not universal. In D — 4 there is only one gauge invariant non-vanishing derivative of f, i.e. uaua(upuy) or uaUa(u^uy) and W should reflect this structure (pretty much as J parallels f itself). The obvious guess is a mixed-symmetry tensor ('hook' Yang tableau) W[x(^]v) — pxd2/Bp^dpv ± .... Moreover, it is worth to notice that the factor Z2 — n2/6 in Eq. (28) comes from the expansion of the beta function appearing in the open string disk amplitudes with four external legs.

Combining the two amplitudes in Eq. (27), and using M3(123) — AL(123) AR(123) (up to an overall factor) as well as sin(np1k4) — np1k4 - n3(p1k4)3/6 + ..., we get

M4(1234) w [npk - n3(p1k4)3/6]S4-1S2-1M3(123).

p3 &4 p 1 p 1E4 p2

Expanding at leading order yields

M4(1234) w ( pCp + p3E4p2 p1 k4

p1 k4 p2k4 p3k4 p3k4 p2k4

and relying on momentum conservation, and on the standard trick p1 k4 1 1

M3(123),

p2k4 p3k4 p2k4 p3k4

we get

M4(1234) i

[ p1^4p1 , p2&4p2 , p3E4p3 1 p1k4

M3(123).

p2k4 p3k4

Only the symmetric (not necessarily trace-less) part contributes, thus exposing the violation of the principle of equivalence in presence of a massless dilaton. At sub-leading order one has

M4(1234)

p3k4 p\k4

p2k4 p\k4

jL fL jL fL J3 f4 J1 f4

p3k4 p1 k4

aR p2 aRp1

p2k4 pk

M3(123).

Expanding and combining the terms appearing in Eq. (35), one gets for the pole in pik4

p1aL fRjR + jLf4L aRp1 — p1E±(jL ± JR),

depending on the 'symmetry' of E4. Moreover, for the pole in p2k4 one gets

-(p1 + p3)aL fR jR - (JL + JL)fL aR p2 — p2£±(jR ± JR),

where in the last step we used the angular momentum conservation J + JR + J3L)A3(123) — 0. For the pole in p3k4 one gets the same result mutatis mutandis.

At sub-sub-leading order one has many terms ^4(1234)

: Plk4

P3k4 Plk4

uRwR uRwR

P2k4 Pik4

W3LuL4

P3k4 Plk4

aRP2 aRPi

P2k4 Plk4

jl fL jl fL J3 f 4 J1 f 4

P3k4 Plk4

f 4 J1

f 4 J2 P2k4 Plk4

- — Plk4 6

P3k4 Plk4

aRP2 aRPi

P2k4 Plk4

- Z2(k4P3 + k4P2)

P3aL PiaL

P3k4 Plk4

aR P2 aRPi

P2k4 Plk4

M3(123),

where uIL/R = k4k4a4s/R and WL/R = aL/R d2/dk4dk4 properly (anti-)symmetrized but not universal (for open strings).

After lengthy manipulations one reproduces

M4(1234);

I k4 J1<?4 Jxk4 k4 J2&4 J2k4 k4.J3E4.J3k4 I

M3(123),

P1k4 P2k4 P3k4 J '

where k4/1E4/1k4 = J1R4J1 involves the linearized Riemann tensor, and thus it is manifestly gauge-invariant. The n2 factor form the expansion of the KLT kernel at 4-point cancels exactly the Z2 appearing in the expansion of the open string amplitudes, thus implementing the single-valued projection discussed in Sec. 2.4.

3.2. 5-point amplitudes with massive states

Starting from the KLT expression for the 5-point closed string amplitude M5 (12 345) = sin(nk1 P2) sin(nP3P4) AS (1 \23]45)AR (21435)

+ sin(nk1P3) sin(nP2P4)AS (1 \32]45)AR (31425), (40)

where we assume that k2 = 0 (massless graviton) goes soft, k1 = 8tc1, with 8^0. In this limit, we know that

AS(12345) « S2—5AL(2345), AR(21435) « s4—2AR(2435), AS(13245) « S3—5AL(3245), and AR(31425) « s4—3AR(3425). (41)

Observing that

sin (^p3p4)AL(2345)AR (2435) = M4(2345) = sin(nP2P4)AL(3245)AR(3425), (42) one gets

M5(12 345) « [sin(nklP2)S2-5S4-2 + sin(пk1p3)S3-5Sl-3]M4(2345).

At leading order, Eq. (43) yields

k1P2 S2(05S1(02 + k1P3 S3(05S1(0) k P

i_1 k1Pl

3-5 S4-3 _ ^ Pi E1Pi _ sgrav

"1(0)'

At sub-leading order

k p [" <?2—5 o4—2 , <-.2-5 <-.4-2 k1p2 [¿1(0) ¿1(1) + ¿1(1) ¿1(0)

—V k1

+ k1P3

<-3-5 <-4-3 , <-3-5 <-4-3 ¿1(0) ¿1(1) + ¿1(1) ¿1(0)

Ek1 Ji E1Pi <-grav

kk = ¿1(1)" (45)

At sub-sub-leading order

2-5 4-2 2-5 4-2 2-5 4-2 ¿1(0) ¿1(2) + ¿1(2) ¿1(0) + ¿1(1) ¿1(1)

<-3-5 <-4-3 , <-3-5 <-4-3 , <-3-5 <-4-3 ¿1(0) ¿1(2) + ¿1(2) ¿1(0) + ¿1(1) ¿1(1)

Ek1 JiE1Jik1 JiR1Ji cgrav

JJJ- = = (46)

where the Z2 factors coming from the KLT kernel cancel exactly those produced by the expansion at the sub-sub-leading of the 5-point disk integral, as encoded by the single-valued projection.

3.3. 6-and higher-point amplitudes with massive states

Lastly, let us briefly focus on 6-point amplitudes. In this case one has twelve terms M6(123 456) = sin(nk1 p2) sin(np4p5)AL (1 [234]56)

{sin(nk1P3)As (231546) + sin(np3(k1 + P2)) Ar (321546)} + Perm[234], (47)

At leading order, we get

M6(123 456) « nk1 P2 ¿2-6Al (23 456)

[sin(nP4P5) sin(nP3 P2)S5-2AR (32 546)

+ sin(nP3P5) sin(nP4P2)AL (24 356)Sf(-2Ar(42 536)] + [2^3] + [2^4], (48)

that yields

M6(123456) « |n k1 p2¿2-6^5-2 + [2^3] + [2^4]} M5(23456),

exposing the expected universal terms at leading order, where non-planarity is restored by summing over permutations in KLT or 'single-valued map' formulae. Sub-leading and sub-sub-leading are more laborious but are fixed by gauge invariance, as repeatedly discussed above.

4. Closed superstring amplitudes with massive insertions

In this section we compute some amplitudes with insertions of massive string states. Later on we will examine their soft behavior.

Let us now consider closed superstrings and focus on the NS-NS sector. At the first massive level one finds a plethora of particles (all in all 214 = 128 x 128 = (44 + 84) x (44 + 84) d.o.f.) arising from the combinations [H^v © CilVf,]l ® [HMv © CmVp']r.

4.1. Three massless states one massive: M4(E1, E2, E3, K4 + L4 + U4)

Relying on KLT formulae one has M(E1, E2, E3, K4 + L4 + U4) = sin (n^^ As(A1,A2,A3,Ht + C4)Ar(A1,A3,A2,H4 + C4), (49)

with K + L + U = H ® H + C ® C + H ® C + C ® H. The highest spin state is the Konishi top state with s = 4 [53-56]. In D = 10 the explicit formula is extremely long and not very illuminating. We refrain for writing it down except for L = C ® C, whereby it reads

r(1 — i)r(1 — 2)r(1 — u)

M(E1, E2, E3, K4) = t 2 u 2 s 2

( 1, V( 2)r(1 + u )r(1 + 2)

n r 1 1 r 7 , r 7 Ia2k' , r 7

C4La1a2a3]+> C4\a1a2ki ]^-+> C4\a3a1ki ]^-+> C4\a2a3ki ] ——

k3ki k2k; k1ki i=3 i=2 i = 1

a2a3 a3a1 a1a2

+ C4\a1k2k3^—— + C4\a2k3k1^—— + C4\a3k1 k2] —— k2k3 k3k1 k1k2

We shall also study the soft behavior of the amplitude M(E1, E2, E3, K4). It is worth to notice that K4 = H ® H is a reducible tensor. The following decomposition holds

44 ® 44 = 450 © 910 © 495 © 44 © 36 © 1, (51)

(2, 0) ® (2, 0) = (4, 0) © (2, 1) © (0, 2) © (2, 0) © (0, 1) © (0, 0). (52)

In particular, this product contains the 10-dimensional analogue of a spin 4 state

SM1M2M3M4 = 1/2(HM1M2 HM3M4 + HM3M4 HM1M2 ) — 1/(9 X 4) ^^ H'flj8PkM .

i,k=1,2 j,l=3,4

In D = 4 the situation drastically simplifies. Focusing on the combinations of the SO(6) singlets H,v = Hjlv + H0(n,v + ol'pp) (with Hij = —H08ij/2) and C,vp = C0^fa'pkex,vp that couple to two gluons, one has 49 d.o.f. that assemble in five scalars, one vector, five spin-2 (5 states each), one spin 3 (7 states) and one spin 4 (9 states). Since the H"v couples to gluons with opposite helicity while H0/C0 couple to gluons with the same helicity, the open-string building blocks are

A(1 —, 2+, 3+ ,H++) , A(1+, 2+, 3+,H0/C0) , A(1 —, 2—, 3+H0/C0)

and the ones related to them by Lorentz transformations (acting on Hh), conjugation or permutations of the gluons.

For instance, the amplitude of 3 gravitons with the top component K+4 = u4«4 (recall p4 = k4 + k5 = u4U 4 + V5 D5) reads

M4(1—2, 2+2, 3+2,K4+4) = sin (—nk2k3) x As(1—2+3+H+2) ® Ar(1—3+2+H4+2) „ T(k3P4)T(1 + k2k3)T(1 + k1k3) \13]<14>4\45]2 \12]<14>4\45]2

= GNn--t--T—. (54)

N r (2 — k3P4)T(—k2k3)T(1 — k1k3) <12><23>mH <13><32>m3H

The amplitudes for the lower spin components of K follow performing SO(3) little group transformations on the above one. Similarly one can replace two gravitons with dilatons or axions

M4(10203+2K+4) = sin(—nk2k3)AL(1—2+3+H++) ® AR(1+3+2—H++)

„ rfe^ra + k2k3)T(1 + kk) [13]<14}4[45]2 [12]{24}4[45]2 = GNn------—. (55)

N r (2 - k3p4)r(-k2k3)r(1 — kxks) <12}<23}mH <13}<32}m3H ' '

Once again, amplitudes for the other helicity states of K obtain after SO(3) little group transformations. Note, for instance, that K+++0 = H++H+0 + H+0H++ while K+++ = H++H+0 — H+0H++. The former is even under the latter is odd.

One can also consider the 4 real (2 complex) s = 2 massive states corresponding to H0/C0 ® H2 ± H2 ® H0/C0 whose amplitudes with massless states obtain from combinations of A(1 —, 2+, 3+, H++) with A(1+, 2+, 3+, H0/C0) or A(1 —, 2—, 3+, H0/C0). For instance

M(1°2+23+2W+2) = sin (—nk2k3) Al(1—2+3+ H++ )Ar(1+3+2+H 0/C0)

„ r(k3P4)T(1 + k2k3)T(1 + k1k3) [13]<14}4[45]2 [12]m3H

= GNn--r--. (56)

N r (2 — k3P4) r (— k2k3) r (1 — kxk3) <12}<23}m3H <13}<32} ' ;

Finally amplitudes for the four scalars H0/C0 ® H°/C° obtain combining A(1+, 2+, 3+, H0/C0) or A(1—, 2—, 3+, H0/C0) with each other and with permutations thereof.

5. Bosonic string amplitudes with 'massive' insertions

5.1. Three-tachyons one-massless: M4(Ti, T2, E3, T4)

Consider now also the insertion of generic massless closed string states with k2 = 0

V£ (z,z) = £^vidX'xL(z)i~d XR (z)e'5 XL^eik Xr®, (57)

where ELV is transverse with respect to both indices kLELV = 0 = kvELV. Decomposing ELV = hLV + $LV + bLV into irreducible representations of the Lorentz group, hLV = hVL with nLVhVL= 0 describes the graviton, $LV = nLV — kLkV — kVkL with kk = 0 and kk = 1 describes the dilaton and bLV = —bVL the Kalb-Ramond field. Consider the amplitude:

M4(T1, T2, E3, T4)

= |cc eipiX(zx.zx) cC eip2X(z2,z2) J — idX E idX eik3X(z3, z3) cC eip4X (z4,Z4))

—- P3E3i^3\z\2p3p411 — z|2p2p3. (58)

P3 = PI + P2 + P4 «if» pi — _P1_, (59)

3 z31 z32 z34 z 1 — z

the amplitude reads as

f d-1 |Z|2P3P4 |1 — z\2p2pJ P4E3P4 + P2E3P2 — P2E3P4 — P4E3P2 \

J n \ |z|2 |1 — z|2 z(1 — z) z(1 —z)J

= P4E3P4I(2P3P4 — 2, 0; 2P2P4, 0) + P2E3P2I(2P3P4, 0; 2p2p3 — 2, 0) — (P2E3P4 + P4E3P2) I(2P3P4 — 2, 1; 2P2P3, —1)

1 ( —P4E3P4(k3P2)2 — P2E3P2(k3P4)2 + P2(E3 + E3)P4 k3P4 k3P2)

k3P1 k3P2 k3P4 r(1 + k3P4)r(1 + k3 P2)r(1 + k3P1) T(1 — k3P4)r(1 — k3 P2)r(1 — k3P1)

= --:-:-(P1 E3P4(k3P2)2 + P1E3P2(k3P4)2 + P2E3P4(k3P\)2)

k3P1 k3P2 k3P4

r(1 + k3P4)r(1 + k3 P2)r(1 + k3P1)

-. (60)

r(1 — k3P4)r(1 — k3 P2)r(1 — k3P1) y '

One concludes that only the symmetric part of ES = |(E3 + E'3) contributes due to symmetry under world-sheet parity under which h and $ are even while b is odd. The r functions in the above expression can be rearranged as

B(1,2, 3, 4)B(4,2, 3, 1) sin (nk3p2), (61)

r(1 + k3p2)V(1 + k3p4) B(1, 2, 3, 4) = ( + 3P2) ( + 3P4). (62)

r(1 — k3P1)

Moreover

— P4E3P4(k3P2)2 — P2E3P2(k3P4)2 + P2(E3 + E'3)p4 k3P4 k3P2

= —(p4a3 k3P2 — P2a3 k3p4)(p1^3 k3 p2 — P2^3 k3P1) (63)

so much so that

M4(71, T2, E3, T4) = si^n0 AL(T1,T2,A3,T1)AR(T4,T2,A3,T1), (64)

as expected.

5.2. Two-tachyons two-massless: M4(E1, E2, T3, T4)

Using KLT in the s-channel (1-2 or 3-4 exchange) one finds

M(E1, E2, T3, T4) = si^n2) AL(A1,A2,T3,T4)AR(A1 ,A2,T4,T>) (65)

so that the two-massless two-tachyon amplitude reads

. r _ r(1 + k1P3)T(1 + k1P4)r( —1 + k1k2)

M(E1, E2, T3, T4) =--(66)

( r(— k1P3)r(—k1 P4)r(2 — k1k2)

1 + k1P4 1 + k1P3 a1a2 — (a1P3 a2P3 + a1 P4 a2P4) + a1P3 a2P4—;--+ a1P4 a2P3

k1 P3 k1 p4

1 + k1P4 1 + k1 P3

a1a2 — (a 1P3 a2P3 + a1 P4 a2P4) + a1P3 CI2P4—--+a1P4 <22P3—;- . (67)

k1 P3 k1 P4

Replacing afaV = ELLV one gets

M(E-, E2, T3, T4) = I(s, t,u)ELVE^KlpKvo

T, t , r(1 + kip3)r(1 + kip4)r(—1 + kik2)

l(s,t,u) =--(68)

r(—kip3)r(—kip4)r(2 — kik2)

klv = nLV — (plpv + plpv) + plpv I+Ppi + PlP3v

kiP3 k-p4

that shows that only M(h/^1, h/$2, T3, T4) and M(b1, b2, T3, T4) are non-vanishing, as expected on the basis of world-sheet parity symmetry

5.3. Two-tachyons one-massless one-massive: M4(T1, T2, E3, K4) Using KLT in the s-channel (1-2 exchange) one finds

M(T-, T2, E3, K4) = sin (n2) Al(T-,T2,A3,h) ® Ar(T2,T-,A3,h) (69)

or more explicitly

KA(T T F ra r(1 + p-k3)r( —1 + k3P4)r(1 + P2k3) M(T-, T2, E3, K4) = ■

— 2a3Hp2 — 2a3Hk3

r(—pi k3)r(2 — k3p4)r(—p2k3)

1 + k3Pi

2 — k3 p4

(1 — k3P4 1 + k3p- \

P2HP2—--+ k3Hk3-—---+ 2p2Hk3

k3Pi 2 — k3P4 )

(k3p4p2HP2n , , , U1 1 + Pik3 0 „, 1 — k3P4\

— a3P2[ -:-:— (1 — k3P4) — k3Hk3----2p2Hk3---

V P2k3 Pik3 P2k3 P2k3 /

1 + k3 p 2

— 2a3Hp- — 2a3Hk3

2 — k3 p4

(1 — k3 P4 ~ 1 + k3P2 ~ \

P-Hp-—---+ k3 Hk3 -—---+ 2p-Hk3

k3 P2 2 — k3P4 /

~ (k3P4 PiHPin , , , 1 + P2k3 0 1 — k3P4\ a 3pH -:-:— (1 — k3P4) — k3Hk3----2p-Hk3---

\ p-k3 P2k3 p-k3 p-k3 /

where E3 = a3 ® a 3 and K4 = H ® H. Without much effort one can check that E± = a ® a ± a ® a with definite parity under ^ couple to K± = H ® H ± H ® H with the same parity.

6. Soft limit of closed string amplitudes with massive insertions

In this section, we study the soft limit of 4-point amplitudes with massive insertions. We start with the superstring and focus on the D = 4 case where the spinor helicity formalism largely simplifies the results. We then pass to consider the bosonic strings and study tachyon insertions, too. Finally we investigate the soft behavior for amplitudes with two Kalb-Ramond fields.

6.1. Soft limit of superstring amplitudes in the spinor helicity formalism

Restring the momenta and polarizations to D = 4 allows us to derive compact expressions for the universal soft operator in the spinor helicity formalism. For simplicity we focus on 4-point amplitudes with three massless and one massive external legs. In particular, we will consider the soft limit of the amplitudes in Eqs. (54), (55), and (55), computed using KLT. When the graviton with helicity h = +2 and momentum k3 goes to zero, we find

5o = ^ [13][23](12)2 V N <13><32>2k3P4

«s1 = Gn-

v N <3«>

<lq3>[31]„ d <2q3 >[32] „ d <4qs>[43] + <5qs>[53]

-U3 —+--U3 —+--

<13> dui (23> du2 2k3P4

d _ d [34]u^— + [35]U3-^

dU4 0U5

[13] fu 2 J23^ 2

2<31> \U3 dUUj + 2<32> \U3 du2 J

1 ( ~ d _ d \2' ( [34]ÏÏ3—- + [35fe —

4k3P4 \ dÏÏ4 du5 J

Applying the operators Si, i = 0, 1, 2 to the amplitudes

M3(*. 2+2, HT» = G <14>2'2f[12]2 , ,76)

we reproduce the soft expansions found respectively in Appendix A.1.1, A.1.2, and A.1.3

0 2 -U9 +4 [13][23]<12>2 <14>4[25]4

S M.3(1 2+ K+ ) = Gn l jl j< > < > \ ] (77)

3( 4 » N <13><32>2k3p4 m6

1 9 +9 +4 [13][23][35][25]3 < 12>< 14>4

S 1M3(1-22+2K4+4 » = (78)

s2M3(1-22+2K4+4) = 6Gn [13][23][35]2[625]2<13><14>4. (79)

4 » N <32>m62k3p4

Had we chosen the leg with momentum k1 to be soft in Eq. (54), we would have gotten a trivial result, since the interaction vertex vanishes M3(£+2, £+2, K+4) = 0. While our results are symmetric in the exchange of 2 ^ 3, when the external leg with momentum k2 is a graviton.

6.2. Soft limit of bosonic string amplitudes

6.2.1. M4O1, T2, £3, T4)

The simplest case to be considered is the amplitude with three tachyons and the one graviton M4(T1, T2, £3, T4)

M3(1-2, 2+2, K+4) = JON(74)

m * * r+4) G <14>2<24>2[15]2[25]2 (75) m3($1,$2, kj ) = Vgn-—--(75)

M4(Ti, T2, E3, T4)

= r(1 + k3P4)r(1 + k3P2)r(1+k3Pi) /P1E3P1 + P2E3P2 + P4E3PA (80)

r(1 — k3P4)r(1 — k3P2)r(1 — k3pi) \ k3p- k3P2 k3P4 )'

The dynamical factor in the above expression has a very special soft behavior r(1 + k3P4)r(1 + k3P2)r(1 + k3Pi) r(1 — k3P4)r(1 — k3P2)r(1 — k3Pi)

1 + Zi=3 k3Pir'(1) + I Y,i=3(k3Pi)2r"(1) + Y,i<j;ij=3 k3Pi k3Pjr'2(1) + O(S3) = - — Ei=3 k3Pir'(1) + - Ti=3(k3Pi)2 r"(1) + Ti<j;ij=3 k3Pik3Pjr'2(1) + O(S3)

= 1 + O(S3).

Eq. (81) does not spoil the soft behavior of the amplitude up to the sub-sub-leading order. This happens every time the dynamical factor depends on the soft momentum as in Eq. (81) and in all cases we are going to study we will always extract this factor. At this stage, the expansion of the amplitude yields

PiE3Pi , P2E3P2 , P4E3P4 ,

M4(Ti, T2, E3, T4) = —----+ —--+ O(S3). (82)

k3Pi k3P2 k3P4

Which agrees with the expected soft behavior since the three amplitude M3(T1, T2, T4) is just a number, so the action of the angular momentum operator gives zero.

6.2.2. M4(E-, E2, T3, T4)

When E1 = h1/$1 and E2 = h2/$2 the soft theorem would suggest the following expansion for the amplitude in Eq. (67)

0 ( P3ElP3 , P4E1P4 , k2E1k2 \ p— c P—

S M3 (E2, T3, T4) = —-+ —-+ , , -E2—; (83)

\ kiP3 kiP4 k-k2 J 2 2

. iC ^^ /k1 J2E1k2 , k1J3ElP3 , k1J4EiPA p— c P —

S M3(E2, T3, T4) = —--+ —-- -rE2 —

\ k-k2 kiP3 k-p4 J 2 2

k2E-p— k-E2P— — k-p— k2E-E2P—

2k- k2

P3E1E2P— k-p3 — P3E1P3 k-E2P— 2k-p3

P4E1P4 k-E2P— — P4E1E2P— k-p4

Sm (^ t3 T4) ^ f k- J2E1 J2ki + k-J3E-J3k- + P— ^ P—

\ 2k-k2 2k-p3 2k-p4 /2 2

k-p— p—E-E2k- — p—E-p— k-E2k- — E-E2(k-p—) 4k-k2

2p3E-E2k- k-p3 — k-E2k- P3E2P3 — (kiP3)2 E-E2

2p4E-E2k- k-p4 — k-E2k- P4E2P4 — (k-p4)2 E-E2 +--^-. (85)

Since Kalb-Ramond b-fields are odd under world-sheet parity we would expect zero because M3(b2, T3, T4) = 0. Following the steps reported in Appendix A.2 we find that at the sub-sub-leading order the soft behavior of the amplitude is not reproduced by the soft operator S2. In particular, there are additional terms that we expect coming from the M3(h1, h2, pj) vertex, Eq. (A.38).

For two Kalb-Ramond fields £]_,2 = bi,2 the amplitude at leading order O(S-1) is zero. The expansion starts at order O(S0)

\A0)ru h T T \ 1 U U , k1b2P- k2b1 P- , k1P-k2b1b2P-

M\ '(b1,b2, T1, T2) = -P-b1b2P-+------1------(86)

2 2k1k2 2k1k2

M41)(b1,b2, T1, T2) = -2hp-k2b1b2P- + 2hb2P- k2b1P- - 1 k2b1b2k1

k1p- p-b1b2k1 1 1

+--—--+ -k1k2 p-b1b2P- + -k1k2tr(b1b2)

2k1 k2 2 4

(k1P-)2tr(b1b2) (87)

It is worth to notice that there are only poles in k1k2, as expected since M3 (b, T, T) = 0 due to world-sheet parity. One can try to interpret the soft result as a factorization on the massless pole viz.

lim M4(bi,b2, T3, T4) = V M3(bi,b2,e(-ki — kz))^- Ms(e(ki + ki), T3, T4) ki^0 ¿-f 2k i ¿2

where e(k) collectively denotes the physical polarizations of the graviton and dilaton e,v = h^v + . Alternatively, since 2kik2 = —2(ki + k2)p3 = —2(ki + k2)p4, one can envisage a 'double soft limit', see e.g. [57-59],

lim Mn+2(bi,b2,H3,...,Hn+2)

ki ,k2^0

= V „ , ^ D(bi,b2; ki — k2)Mn(H3,...,Hn+2) (89)

i (ki + k2)pi

where our present computations suggest

V(bi,b2\ ki — k2) = (k2 — ki)biPi (ki — k2)b2Pi i 2

+ 4 [(ki — k2)2Pi — (ki — k2)Pi(ki — k2)]{bi ,b2}Pi. (90)

Clearly this issue deserves further investigation.5

6.2.3. M4(Ti, T2, £3, K4)

For simplicity we consider only the case in which K4[^, v, p, a] is the completely symmetric irreducible state. In this case due to ^-parity £3 = h3/<3 only. Applying the soft operators to the three level amplitude

5 We thank Paolo Di Vecchia and Raffaele Marotta for interesting and fruitful discussions on this and related issues.

M3(Tl, T2, K4) = K4 we expect the following behavior

P- P- P- PL 2 ' 2 ' 2 ' 2 J

5>M3 (Tu T2' K4) = (+ + K4

K3P4 /

V k3Pl

P- P- P- P-2 ' 2 ' 2 ' 2

51M3 (Tu T2' K4) —

/Pi£3Jik3 | P2E3Jk , P4E3J4^3^ V k3Pi

k3P2 ^3 P4 )

— 2 /P1£3P1 - P2£3P2 - P-£3 P4 j £

V k3Pi k3P2 k3P4

P- P- P- PL 2 ' 2 ' 2 ' 2 J

P- P- P-2 ' 2 ' 2

+ 2( -^£3)" + (P2£3)" + PPfk3P- )K4

52M3(Ti' T2 ' K4) = 2 ((PE)" + (P2£3)" + k3P-lPp-£3r ) K4

P- P- P-

3fk j.k . (k3P-)2

-7 k3Pi + k3P2 +----

4 k3 P4

- 3 /Pi£3Pi + P2£3P2 + P-£3P-

4 \ k3Pi k3P2 k3P4

k3' k3' P—' P—

. 3 ' 2 ' 2 J

J4 "v — P4[^—V + 4K4["'«'}'y] „r rv

9P4 9K4[ ' œ' f}' y]

Following the steps outlined in Appendix A.3, we reproduce the leading and sub-leading behavior as predicted by the soft theorem, but not the sub-sub-leading order. As for the amplitude M4(E1, E2, T3, T4) we are led to think that the mixing with the other degenerate string states spoil the soft theorem statement at this order.

7. Conclusions and outlook

We have extended our analysis of the soft behavior of string amplitudes with massive insertions to closed strings. Relying on our previous results for open strings and on KLT formulae we have checked universality of the soft behavior to sub-leading order for superstring amplitudes. At sub-sub-leading order we have argued in favor of universality on the basis of OPE of massless and massive vertex operators and gauge invariance with respect to the soft gravitons. We have also checked our statements against explicit 4-point amplitudes with one massive insertion in any dimension, including D = 4, where use of the helicity spinor formalism drastically simplifies all expressions. As a by-product of our analysis we have checked the cancellation of n2 arising from sin(na'ckikj) factors in KLT formula with those arising from open superstring amplitudes in the soft limit, at sub-sub-leading order. This is expected for the 'single valued projection' advocated in [43,44] to hold for massive amplitudes, too. This is comforting, being closed string theory of quantum gravity. Yet, our results are only valid at tree level and the proper extension to one- and higher-loops is still under debate in that IR divergences seem to produce non-universal log 8 terms [60] even in N = 4 SYM at one-loop, let alone supergravity or superstring theories. It would be very interesting to investigate this subject along the lines of [37,61] and establish

whether log 8 terms exponentiate, as usual for IR divergences, and in case which would be the relevant 'anomalous' dimension that governs this hopefully universal behavior. The approach proposed in [62,63] based on the second Nother theorem seems promising in this respect, though so far shown to be valid only at tree level.

Acknowledgements

We would like to thank Andrea Addazi, Marcus Berg, Marco Bochicchio, Dario Consoli, Giuseppe D'Appollonio, Paolo Di Vecchia, Yu-tin Huang, Raffaele Marotta, Francisco Morales, Tassos Petkou, Lorenzo Pieri, Oliver Schlotterer, Stephan Stieberger, Gabriele Veneziano, and Congkao Wen, for interesting discussions. This work is partially supported by the INFN network ST&FI "String Theory and Fundamental Interactions".

Appendix A. Expansion of the amplitudes

A.1. Soft limit of the amplitudes with the Konishi operator

In this section we give more details about the soft limit of the amplitudes in Eqs. (54), (55) and (56). As a preliminary step we consider the soft limit of the common dynamical factor when the momentum k3 becomes soft in any case.

r(ks^4)r(1 + k2k3)T(\ + kik3) k2k3 3

- + O(S3)

r (2 - k3P4) r (-^3) r (1 - k\kj) k3P4k\k2

<23)[32] + 0(83). (A.1)

<12>[21]k3^4

Combining this expression with the expansions of the different kinetic terms we will get the final result.

A.1.1. The amplitude M4(E-2, E+2, E+2, K+4)

The expansion of the kinematical term in Eq. (54) yields

[13]<14>8 [12][45]4 N <12><23>mH <13><32>

[13][12]<14>4 „„^4 A , ,*<13>[35] , ÄS2 <13>2[35]^ + O^

= Gn--' '-- <12>4[25]4 1 + 4S-———- + 6S ' „ + O(S3).

N <12><13><23><32>mH L V <12>[25] ^ <12>2[25]2 ( )

Combining the Eq. (A.1) with Eq. (A.2) we obtain up to order S terms

O(i-i) : Gn [13][23]<12)Z <14>4[25]4 (A.3)

( ) N <13><32>2k3P4 m6 ' '

0 [13][23][35][25]3 < 12>< 14>4

O(S°) : 4GN ]< 6 >< > (A.4)

<32>2k3P4m6

n(Si)- 6G [!3][23][35]2[25]2<13><14>4 (A5)

O(S) : 6Gn-<32>2k3P4m6-. (A.5)

A.1.2. The amplitude M4«1, <2' £+2, K+4) The kinematical term in Eq. (55) yields

[13][12](14}4(24}4[45]4

(12}(23}(13}(32}mH

— Gn [12][13](14}2(24}26 (12}4[25]2[15]2

(12}(23}(13}(32}mH

x (1 + 25 (+ ) + 52 ( (13}2[35]2 + (23}2[35]2)). (A.6)

V V (12}[25] ( 12}[15^ / V (12}2[25]2 (12}2[15]2//

Here we give the result of the expansion to be compared with the predictions dictated by the soft theorem.

, (12}2[13][23] (14}2(24}2[25]2[15]2

O(S-1) : GN X ' X ' X ' \ (A.7)

( ) N (13}(32}2k3P4 m6H '

0, 2G (12}[13][23] (14}2(24}2[25]2[15]2 ((13}[35] (23}[25]) (A§) ( ) : N (13}(32}2k3P4 m6H V [25] [15] / (.)

[13][23] (14}2(24}2[25]2[15]2 /(13}2[35]2 , (232}[25]2 O(5) : Gn ,,„. --6- —--+

(13}(32}2k3P4 m6H V [25]2 [15]

A.1.3. The amplitude M4(fa, E+2, E+2, H+2)

To expand the amplitude in Eq. (56) we need to expand only AL

[13][12](14}4[45]2

(12}(13}(23}(32}

[13][12](14}2[25]2(12W (13}[35] 2 (13}2[35]2 , — GN ^ ' 1 J \ ' 1 + j + S2X ' , L (A.10)

N (13}(23}(32} V (12}[25] (12}2[25]2

getting

1 [13][23](12}2 (14}2[25]2[12]2

O(S-1) : Gn M 7 V 7 , (A.11)

( ) N (13}(32}2k3P4 m4 ' '

0 [13][23](12} (14}2[25][35][12]2

O(S0) : 2Gn-—I———- W 4 (A.12)

(32}2k3P4 m4

[13][23] (13}(14}2[35]2[12]2 O(S) : Gn——I——V M ' 4 • (A.13)

(32}2k3P4 m4

A.2. The amplitude M4(£1' £2, T3, T4)

It is convenient to factor out the structure in Eq. (81), which has a trivial soft behavior, from the dynamical term in Eq. (68)

TV . Ï k1P3 k1P4 /1 ,

I (S' t'U) — ---——2 (1 + O(53))

(1 - k1k2)2

— -k1k2 k1P3 k1P^ + 2 + 3k1k^ + O(S3). (A.14)

The expansion of the kinematical structure £1K.£'^K can be organized as follows

£1&£2 K2 = £1^-1^22 K_ 1 + 2E1K0E2 1 + £1&o£2 Ko , (A.15)

K-1 = P^Pi + E^Ei (A.16)

k1P3 k1P4

P3 ® P4 P4 ® P3 Ko = 1 - P3 ® P3 - P4 ® P4 +--:-k1P4 +----k1P3. (A.17)

k1P3 k1P4

The expansion of the amplitude up to O(S) yields

M(£1, £2, T3, T4) =-U k1P,3 k11P4 £1K-EK-1 S \ kk 2 1

+ S0 (-2k1P3 ]k1P4 £Ko£21 - 2k1P3 k1 P4 £K-1£2K_ 1

V k1k2

+ S f- ^^ £1kO£2K0 - 4 k1P3 k1 P4 £1kO£2K-1

- 3k1k2k1P3 k1 p4 £1 K-1£2K-^ + O(S2). (A.18)

To make explicitly the expansion it is convenient to introduce the variables p+ = p3 + p4 and P- = p3 - p4. As far as £2 is concerned, all the bilinear expressions involving £2 are well organized

P-£2 p-= O(1); P+£2P- = -k1£2p- = O(S); P+£2P+= k^ = O(S2).

(A.19)

Starting with the tensorial structure £1K-1£^respectively for £1,2 = h/$ both symmetric (graviton and dilaton) and for £1 2 = b both anti-symmetric (Kalb-Ramond fields) we get up to O(1)

£1K-1£2K 1(S-2) = 1 (P^Pl + E^ - 2) p-£2p-

1 12 -1( ) 4 ( (k1P3)2 (k1P4)2 k1P3 k1pjP 2P fr Ftr-t (S-1) l( P3£1P3 , P4£1P^ f

£1K-1£2K-1(S ) = ^- —2 + a—?) P+£2P-

£1K-1£2K 1(S0) = 1 (E!^ + Ei£E± + 2_E£E^ p+£2p+. (A.20)

1 12 -1( ) 4 \ (kp)2 (k1P4)2 k1P3 k1pj P+ 2P+ ' '

b1K-1b2K-1 = ,P3b,1P4 p+b2P-. (A.21)

k1P3k1P4

The expansion of the structure 2£1K0£2 K_ 1 is up to O(1) 2£1K0£2 K-1 (S-1)

2 P- 1 k1 P4 k1P3 — I -P3£1£2—+-p-£2P- P3£1P-+P3£1P3---P3£1P4--I

-- -P3E1E2 — + 7P-£2P- P3£1P-+P3£1P3---P3£1P4--

k1P3 V 2 4 \ k1P3 k1P4 )

2 P- 1 k1 P4 k1 P3 + 7- P4£1£^— + 7P-£2P- -P4£1P- - P4£1 P3---+ P4£1P4"—

k1PA 2 4 \ k1P3 k1P4)/

M. Bianchi, A.L. Guerrieri /Nuclear Physics B 905 (2016) 188-216 211

2Î1^0£"2Kt-1(5°) — (p3£1£2P+ + 1P+E2P- (P3E1P4 - P3E1P3 k^)

+ 7-^ ( P4E1E2P+ + 1P+E-2P- ( P4E1P3+P4E1P4k1P3

k1P4 2 2 k1P4

(A.22)

ity-t P3b1b2P- P4b1b2P-

k1P3 k1P4

P3b1b2P+ P4b

k1P3 k1P4

Finally we consider the expansion of the structure £1K0£t2 K'0

1b2P+ P3b1P4 P4b1P3

-----+ —---:- P+b2P-. (A.23)

1P4 V k1P4 k1P3 /

EK0E2Kt^(SQ) — E1E2 - 1 P-{E1' £2)P- - kLP±P3E1f2P- + ^P4EE2P-

( 2 k1P3 k1P4

1 ^ /1 1 k1P3 k1P3

+ 7P-E2P- -P3E1P3 + -P4E1P4 - P3E1P4 + P3E1P3---+ P4E1P4-

2 \2 2 k1P4 k1P4

k1P3 ,, k1 P3 1 _ fk1P4\2 1 /k1P3V IT--P3C1 P4~--+ - P3C1 P3 -- + - P401 P4 --

- P3E1P4---P3E1P4---+ 7P3E1P3 -- + 7P4E1P4 7- • (A.24)

k1P4 k1 P4 2 \k1P3/ 2 \k1P4/

t t 1 k1P4 k1P3

b1^0b2^0 — -b1b2 + -P-{b1'b2}P- + --P3b1b2P- - --P4b1b2P-. (A.25)

2 k1P3 k1P4

Now we have all the ingredients to compute the full expansion of the amplitude. Consider first the symmetric case in which E1/2 — h/<. At leading order we have

k1P3 k1P4c ^ Ptrt

--—-t1K_1t2^ 1

k(1 k2 )

— -1fP3 E P k1P4 + E P4 k1 P3 - 2 P3g1PA e2 P

4 \ k1k2 k1P3 k1k2 k1 P4 k1 k2 /

— (P^PS + P^^ + №2 ) P- E2 P- (A.26)

V k1 P3 k1P4 k1 k2 / 2 2

which has the expected structure from the soft theorem.

The subleading order comes from three different contributions:

— -2- P3E1E2+ 1 P-E2P-( P3E1P- + P3E1P3 T1P4 - P3E1P4 ) k1 k2 ( 2 4 ( k1 p3 k1 p4 )

-2t1-^) P4E1E2P— + 1 P-£2P-( - P4E1P- - P4E1P3+ P4E1P4) k1 k2 2 4 k1 p3 k1 p4

(A.27)

12 x E1K-1E2 1 — -if p3 E1P3- 2p3£1p4 + P4E1P4 P-&2P-• (A.28)

2 k1 p 3 k1 p 4

The subleading contribution coming from

ctw 1 ( _ c- _ k1P4 , _ ^ _ k1P3

I1 x E1K-1E2KL1 — - P3E1 P3~j I :--+ P4E1P4——:- P+&2P-• (A.29)

2 \ k1k2 k1P3 k1k2 k1P4/

The sum of these three gives the answer expected from the soft graviton theorem

1 "P3£1P3 P+£2P- + 1 (P3£1P3 - P4£1P4) P+EliP--—--P4£\P4 P+£2P-

2k1P3 2 k1k2 2k1 P4

c c k1P4 , c c k1 P3 - P3 £1£2P^^— + P4£1£2P--r-r k1 k2 k1 k2

P3£1P3 c , 1 ,, c P4£1P4 c 1 C C

= "T7-P+£2P- + -P3£1£2P- - —-P+£2P- - -P4£1£2P-

2k1P3 2 2k1p4 2

k1P- k2£1£2P-- p+£2P-. (A.30)

2k1k2 2k1k2 It is straightforward to compare the last expression with the expected behavior k1 J2£1k2 P- c P- k2£1 P- * c k1P- , cc

—T-J--^T£2^T = , k1£2P--^- k2£1£2P- (A.31)

k1k2 2 2 2k1k2 2k1k2

k1J3£1P3 P-c P- c c k1P3 P3£1P3, c ,,,

—;--= P3£1£2P-—-----k1£2P- (A.32)

k1P3 2 2 2k1P3 2k 1P3

k1J4£1P4 P P- k1P4 , P4£1 P4, c ,,,

—-,--^r£2^T = P4£1 £2P- ^--+ —-k1£2P-. (A.33)

k1 P4 2 2 2k1P4 2k1 P4

The sub-sub-leading contribution comes from the sum of the following terms

M1 = -k1E3 k1P4£1Ko£2kO(So) - 4k1P3 k1P4£1Ko£2K_ 1(S-1) k1 k2

-3k1k2k1P3 k1P4£1K-1£2K_ 1(S-2) - 2k1p3k1P4 £1Ko£2K_ 1 (S0)

-2k1p3 k1p4 £1K-1£2K_ 1(S-1) - k1P3 k1P4£1K-1£2K_ 1(S0). (A.34)

In Eq. (A.34) we can recognize the structures predicted by the soft theorem

1 k1J3£1J3k1 1 1 1

----= -P3£1£2k1 - —-k1£2k1 P3£2P3 - -k1P3 ££ (A.35)

2 k1 p3 2 4k1p3 4

1 k1.J4£1.J4k1 11 1

--:-= -P4£1£2k1 - —-k1£2k1 P4£2P4 - Tk1P4 ££ (A.36)

2 k1P4 2 4k1P4 4

1 k1/4£1J4k^ p-£1£2k1 p-£1p- k1£2k1 £1£2(k1 p-)2 ^

--= k1 p------(A.37)

2 k1P4 2k1 k2 4k1 k2 4k1 k2

and additional terms

k1P-1 c c , k1k2 , k1p- k1£2P- k2£1P-

k2£1£2 P- +--r P-£1 £2P- + ■

2 2 2 2k1 k2 k1£2P- k2£1P-

(A.38)

For two Kalb-Ramond fields £1 2 = 2 the amplitude at order O(S-1) is zero. The expansion starts at order O(S0) with

P3b1P4 k1p- 1

mo = -r-:— P+b2P- P+b1b2P- - TP-{b1, b2ÎP-. (A.39)

k1 k2 2k1 k2 4

Note that p+b2 — -k1b2 is of order O(5), while p+b1 — -k2b1 is of order O(S°). At order O(5) the amplitude looks like

k1 P4 k1 P3 1

M1 — P3b1P4 P+b2P- - b1 b2-—---p+b1b2p- k1p-

k1 k2 2

1 1 k1 P-

-TP-{b1'b2}P-k1k2 + TP+{b1'b2}P+ + ^-¡—rP—b1b2P+• 4 4 2k1 k2

A.3. The amplitude M4(T1' T2' E3, K4)

(A.40)

The expansion is organized as follows. The dynamical term can be expanded as

T(1 + p1k3)T(-1 + k3P4)r(1 + P2k3) k3P1 k3P2 ,

r(-P1k3)r(2 - k3p4)r(-p2k3)

-(1 + 2k3P4 + 3(k3P4)2) + O(55).

(A.41)

The expansion of the open string amplitude AL can be expanded up to the O(S°) as

(-1) 1 / a3 P2 a3P1\ „

U ' — T I ----- P2HP2

13 P2 k3P1j

P2Hk3 k3P2

A0 — 50

«3 P2 «3 P1

k3P4 + --k3P4 - 2a3Hp1 + 2a3P4 P2HP2 - 2a3P2-

(A.42)

The expansion for Ar is obtained by exchanging the labels 1 ^ 2.

The expansion of the kinematical term of the amplitude M(T1' 72 E3, K4) can be easily yield multiplying the expansions of the open string amplitudes.

(-2) — (-1)

® aR_1)

K(-1) — A^ ® af + a(0 ® aR_1)

k(0) — a0 ® arr.

(A.43)

For simplicity we consider only the case in which cK4 is the completely symmetric irreducible state. To complete the expansion we need to disentangle the sub-leading contributions to each tC(i) term.

K(-2)(5-2) — (-P^P - P^Zi + 2 P1E3P^ K4

V k3P1 k3P2 k3P1k3P2j

P- P- P- PL 2 ' 2 ' 2 ' 2

K(-2)(5-1) — 0

K(-2)(50) — / p^Pi - + P^P^ K4

\2(k3P1)2 k3P1 k3P2 2(k3P2)2J

P—' P—' k3 ' k3

L 2 ' 2

(A.44)

(A.45)

( ) ' (k3P1)2 (k3P2)2 '

P- P-2 ' 2

+ 2( P^Pl - 2+ P^) k3P4 K4 \(k3 P1)2 k3P1 k3P2 (k3P2)2 /

P- P- P- PL 2 ' 2 ' 2 ' 2 J

, (P1£3)'^

2k3 p1

P- E- P' 2 ' 2

(p2E3)m 2k3P2

K(-1)(8°) = 2 (P^Pi + - 2 PlE3M — Hk3 — Hk3 (A.46)

\(k3Pi)2 (k3P2)2 k3P-k3P2/ 2 2

K(0)(80) = -£/v K4

P1 £> P2 k3Pi k3P2

P—, P— ,k3,k3 2 , 2 , 3, 3.

, (P1£3P11 , 0P1£3P4 P2£3P2, ,P2E3P4\ 2 7;-TTk3P4 + 2—----k3P4 - 2—-

V(k3P1 )2

P- P- P-

k3P1 (k3P2)2

k3P2 /

22 P1E3P1

(k3P4)2 + 2

(k3P1)2 P- P- P- P-2 , 2 , 2 , 2 P- P

P1E3P2 k3P1 k3P2

(k3P4)2 -

P2E3 P2 (k3P2)2

(k3P4)2

+ 4 P4E3P4 K4

/ (p^ + (P2£3)/\

V k3P1 k3P2 /

P- P-/ , —, —, k3 x , 2 , 2

/ + (P^V P4 K4

\ 2k3P1 2k3P2 /

(A.47)

Collecting the contributions to each term in the soft expansion, we reproduce the leading and sub-leading behavior as predicted by the soft theorem, but we find that the sub-sub-leading order is not.

^(-1) = k3P1 k3P2 k(-2)(8-2) k3P4

P1E3P1 , P2E3P2 , P4E3P4 \

k3 P4 /

P- P- P- PL 2 , 2 , 2 , 2

(A.48)

^(0) = k3P1 k3P2 k(-2)(8-1) + k3P1 k3P2 K(-1)(8-1) + 2k3P1 k3P2K-2>(8-2)

k3 P4 k3 P4

P1E3P1 P2 £3 P2 P-E3P4

V k3P1

P- P- P- k

2 , 2 , 2 , 3.

+2( -(P1f3)M+(P2 £3)^+(p:£3)\p-

(A.49)

For the sake of completeness we report the expression obtained at the sub-sub-leading order. M(1) = --

k3P1 k3P2 K4E3, P--, P2-] k3P1 k3P2 P4&3P4 P2-, P2-, P2-, P2-]

/P1&3P1 + P1&3P1 - 3 P1 £3P2 - P2£3P2 + P2£3P2\

V 8k3P1 8k3P4 4k3P4 8k3P2 8k3P4 /

X K4[—, —, k3 , k3]

+ / k3P2 P1E3P1 + k3P2 P1£3 P4 + k3P1 P2E3P2 - k3P1 P2£3 P4 \

V 4k3P1 2k3P4 4k3P2 2k3 P4 /

P- P- P-X K4[-,-,-,k3]

'k3P2 (piSsr + k3Pl (P2£3)^\ k [ P— P— ] k3P4 k3P4 J 2 ' 2 '

+ 1 (k3P2 (P&r - k3Pl (P2£3)^ K[ß ' P-' P-' P-].

(A.50)

References

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