On the soft limit of open string disk amplitudes with massive states

We discuss the soft behaviour of open string amplitudes with gluons and massive states in any dimension. Notwithstanding non-minimal couplings of massive higher spin states to gluons, relying on OPE and factorization, we argue that the leading and sub-leading terms are universal and identical to the ones in Yang-Mills theories. In order to illustrate this, we compute some 4-point amplitudes on the disk involving gluons, massive states and, for the bosonic string, tachyons. For the superstring, we revisit the structure of the massive super-multiplets at the first massive level and rewrite the amplitudes in D = 4 in the spinor helicity formalism, that we adapt to accommodate massive higher spin states. We also check the validity of a recently obtained formula relating open superstring amplitudes for mass-less states at tree-level to SYM amplitudes, by factorisation on two-particle massive poles. Finally we analyse the holomorphic soft limit of superstring amplitudes with one massive insertion.


Introduction and motivations
Recently the soft behaviour of scattering amplitudes has received renewed attention in connection with the extended BvBMS symmetry [1][2][3][4][5][6]. It has been long known that gauge theory and gravity amplitudes expose universal behaviours when one of the mass-less external momenta is 'soft' i.e. k → 0 [7][8][9]. In both cases the leading behaviour is singular, i.e. goes as δ −1 if k = δk withk some fixed momentum [7]. The sub-leading terms can be derived from the leading ones and are largely fixed by gauge invariance. In particular, in gauge theories the sub-leading behaviour δ 0 is universal, too. In gravity not only the sub-leading behaviour δ 0 but also the next-to-subleading or sub-sub-leading behaviour δ +1 is universal [8][9][10].
The problem of what happens when loops or non-minimal higher derivative couplings are included was addressed in [11][12][13][14][15]. At the loop level IR divergences tend to spoil the analysis. Yet, in supersymmetric theories such as N = 4 SYM, one can define loop integrands recursively and check that they expose the expected soft behaviour at all loops and for any choice of (massless) external legs. This may be viewed as a further constraint on (loop) amplitudes derived without resorting to standard perturbative methods (see e.g. [16] for a recent pedagogical review).
When non-minimal interactions are considered, the result depends on the specific choice [14]. In gauge theories, F 3 terms do not change the universal soft behaviour of minimal coupling, while φF 2 do modify even the leading term when φ is a massless scalar. Similarly, in gravity theories R 3 terms do not change the universal soft behaviour of minimal coupling, while φR 2 do modify even the leading term when φ is a massless scalar such as the dilaton.
These results are largely independent of the number of space-time dimensions and in particular apply to string theory in critical dimension and in lower dimensions [17][18][19][20][21][22]. One has to distinguish between open and closed strings and between bosonic, super and heterotic strings. In [14], the soft behaviour has been shown to be governed by the OPE of the vertex operators. As a result both open and closed superstring amplitudes with external massless states expose the expected soft behaviour, while closed bosonic string amplitudes don't, due to the tree-level non minimal coupling φR 2 with the dilaton. Open bosonic string amplitudes behave universally despite the presence already at tree level of the non-minimal F 3 correction to the standard Yang-Mills coupling and the coupling T F 2 to the tachyon 1 . For the heterotic string at tree level, the soft behaviour of massless vector bosons is universal, since the trilinear coupling is purely of Yang-Mills type, while the soft behaviour of the graviton is non-universal due to φR 2 coupling 2 . No R 3 term however appears due to supersymmetry.
Aim of the present paper is to extend the analysis to open 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 not received much attention in the literature. See however [23][24][25][26][27][28] for recent work on the subject and [29][30][31][32][33] and the review [34] for mass-less amplitudes with emphasys on the exchange of massive higher spin states.
The analysis is interesting in two respects. On the one hand couplings of string states are generically non-minimal, although probably unique. On the other hand, gravity and gauge interactions emerge quite naturally in string theory and one would expect the soft behaviour of scattering amplitudes to expose some universality thanks to gauge invariance.
Plan of the paper is as follows. We start with open bosonic strings. After reviewing tri-linear couplings on the disk of tachyons, vector bosons and higher spin massive states, we compute some explicit 4-point amplitudes involving tachyons and massive states. We then consider open superstrings on the disk and perform a similar analysis in an arbitrary number of dimension D ≤ 10. For convenience and for comparison with the existing literature we rewrite superstring amplitudes in D = 4 in the spinor helicity formalism, that we adapt to accommodate massive higher spin states, after revisiting the structure of the massive super-multiplets. We also check the validity of a recently obtained formula relating open superstring amplitudes for mass-less states to SYM amplitudes at tree-level [35,36], by factorizing 5-point amplitudes on the first massive pole and recovering our previous formulae. We explain how to generalise this procedure to an arbitrary number of massive external states. We then discuss the soft behaviour of open string amplitudes with gluons and massive states in any dimension and argue that the leading and sub-leading terms are universal and identical to the SYM case, relying on OPE and factorization. We then check this explicitly for the amplitudes, we previously computed. We also analyse the holomorphic soft limit of superstring amplitudes with one massive state and check consistency with our expectations. Finally we will draw our conclusions and identify interesting directions for future investigation. Various appendices contain technical details that are included for completeness.

Open bosonic string amplitudes
In order to check the soft behavior of four-point amplitudes on the disk in the open bosonic string, we summarize in Appendix A all the possible tri-linear couplings involving the vertex operators up to the first massive level.
2 M. B. would like I. Antoniadis to stressing the tree level origin of this term in the heterotic string, which only is generated at one-loop in 4-dim Type II theories with 16 supercharges, such as after compactification on K3 × T 2

Vertex operators
Our first goal is to compute scattering amplitudes with the insertion of vertex operators up to the first massive level. Up to normalization factors, the tachyon vertex operator is the gluon vertex operator is and the first massive level vertex operator is While the choice of the tachyon vertex operator is essentially unique, the choice of the vertex operators for the massless gluon A and for the massive state H is not unique. It is always possible to add the null operator c ik∂Xe ikX = δ BRST e ikX to c V A . For H one can choose a linear combination of the operator V B = B µ i∂ 2 X µ e ipX (z) and a generic H(z) = H µν i∂X µ i∂X ν e ipX (z), with H an arbitrary two-index symmetric tensor. Nonetheless, due to BRST invariance, one has the freedom to fix the gauge in which B µ = 0 and H µν is symmetric, traceless and traverse as in Eq. (2.3).

Chan-Paton factors and twist symmetry
Although we will mostly consider 'color-ordered' amplitudes on the disk, we would like to review some relevant aspect of the group theory structure. Disk amplitudes are cyclically invariant and can be dressed with Chan-Paton factors [37] where t a with a = 1, . . . N 2 are the generators of U (N ) 3 . In modern terms this corresponds to the fact that open strings carry multiplicities associated to the D-branes they end on. A(1, 2, . . . n) are called 'color-stripped' or 'color-ordered' amplitudes or simply sub-amplitudes. They enjoy such remarkable properties as [38] • Cyclic symmetry: A(k, k+1, . . . , n, 1, 2, . . . k−1) = A(1, 2, . . . n) • Twist symmetry: A(n, n−1, . . . , 2, 1) = n i=1 ω i A(1, 2, . . . , n−1, n) • Dual identity: A(1, 2, . . . n) + A(2, 1, 3, . . . n) + . . . + A(2, 3, . . . , n−1, 1, n) = 0 where ω S = ±1 is the eigenvalue in the state S of the 'twist' operator Ω, that exchanges the two ends of the strings. In particular ω A = −1 while ω T,H = +1. In general ω S = (−1) N S where N S is the level of S. Pretty much as in gauge theory, complete amplitudes are obtained by summing over non-cyclical permutations of color-dressed amplitudes. At 3points one simply has For the amplitude with three tachyons A(1, 3, 2) = A(3, 2, 1) = +A (1,2,3), so that In general one gets d abc ('anomaly coefficients' or cubic Casimir) when 3 i=1 ω i = +1 and f abc (structure constants) when 3 i=1 ω i = −1. In particular all couplings A(S, S, A) ∼ f abc whichever the state S. Moreover, at least for totally symmetric tensors in the first Regge trajectory, the dominant term at low energy is i.e. string theory tries to be as 'minimal' as it can! Yet there are higher derivative corrections to this, as we will see momentarily.

Four-point bosonic string amplitude
In this section we collect some open bosonic string amplitudes involving massless, massive and tachyonic states. Details of the computations can be found in Appendix B4. For simplicity we consider color-ordered amplitudes. Complete amplitudes arise after multiplying by the relevant Chan-Paton factors tr(t 1 t 2 t 3 t 4 ) and summing over non-cyclic permutations. In fact, exploiting 'twist symmetry' i.e. Ω invariance, one can further reduce the sum to three terms (instead of six). For notational simplicity we will drop all adimensional constants (including powers of g s ) that are irrelevant for our analysis and the δ-function of momentum conservation (2π) D δ D ( i p i ) will be understood. To help recognising the light-like momenta we will denote them by k's, while tachyonic and massive momenta will be denoted by p's. Starting with Veneziano amplitude (four tachyons) the two-tachyons two-vectors amplitude satisfy a i p 4 = 0, and finally the two-tachyons-vector-tensor amplitude Later on we will check that they enjoy the expected behavior in the soft limit. It is worth to notice that in D = 4, the tensor H corresponds to a massive spin 2 particle, while the C corresponds to a massive pseudoscalar. Up to normalization factors, In the canonical q = −1 super-ghost picture their vertex operators are For our purposes it is necessary to consider also vertex operators in the q = 0 superghost picture (3.4) Higher spin massive states in the first Regge trajectory are described by vertex operators of the form with α p 2 = (1−s) and H totally symmetric, transverse and trace-less. Their tri-linear couplings to the vector bosons schematically read In D = 4 the above expressions drastically simplify if one resorts to the spinor helicity formalism and adapt it so as to encompass massive states.

Supersymmetry
Although we will only consider bosonic states in the NS sector of the open superstring, it is worth discussing the structure of the super-multiplet at the first massive level [39].
In addition to the NS states H and C we have two spin 3/2 fermions of opposite chirality (in D = 10) that combine to give a massive spin 3/2 fermion. In the canonical q = −1/2 super-ghost picture their vertex operators read and where S α is a spin field of conformal dimension 5/8 in the 16 irrep of SO (1,9) and W α µ = :C α ψ µ : is an excited spin field of conformal dimension 13/8 in the 144 irrep of SO (1,9).
The N = (1, 0) supersymmetry charge in D = 10 is the gaugino vertex at zero momentum In the q = +1/2 super-ghost picture one has Acting with Q (−1/2) α on V Ψ and V Ψ one gets combinations of the NS vertex operators V C and V H as well as the 'auxiliary' vertices V B and V E in the canonical q = −1 picture, yielding very schematically 4 Similarly acting with Q (+1/2) α on the NS vertex operators V C and V H as well as on the 'auxiliary' vertices V B and V E yields very schematically 5 and

Dimensional reduction to D = 4
For obvious reasons we are particularly interested in the dimensional reduction to D = 4. The massive N = (1, 0) super-multiplet in D = 10 at the first level yields a long multiplet of the N = 4 super-algebra comprising 128 bosonic and as many fermionic states. In order not to burden the notation µ, ν, . . . are now 4-dim indices, while i, j, . . . denote the internal 6 dimensions. The origin of the bosonic fields is as follows since a massive vector in D = 4 is equivalent to a massive anti-symmetric tensor, and It is perhaps not surprising that these be in one-to-one correspondence with the (bosonic) fields in the N = 4 super-current multiplet, upon dualizing the six massive H µ,i into as many massive anti-symmetric tensors H µν,i = ε µνλρ p λ H ρ i /M . It is amusing to decompose this massive multiplet into massive multiplets of the N = 1 super-algebra In the case of a Z 3 orbifold, whereby x i → z I , z * I the multiplicities can be expressed in terms of dimensions of irreps of SU (3) i.e. 6 → 3+3 * , 14 → 8+3+3 * and 14 → 1+1+6+6 * . Once again, it is not surprising that the multiplet content {H µν , 2ψ µ , Z µ } be in one-to-one correspondence with the currents {T µν , Σ µ ,Σ µ , J µ } in the N = 1 super-current multiplet of Ferrara and Zumino [40].
For later purposes, note that H µν , with η µν H µν = 0 = p µ H µν belongs in a spin-2 supermultiplets with 8 bosonic and as many fermionic d.o.f. whose vector boson is

Four-point amplitudes (superstring) and spinor helicity basis
For simplicity, we will only consider amplitudes with a single massive external state: Depending on the choice of incoming particles these correspond to production, annihilation or 3-body decay of the massive state. In view of this, it is useful to restrict to 4-dimensional momenta and polarisations and work in the helicity basis whereby null momenta are expressed in terms of on-shell Weyl spinors of opposite chirality k αα = k µ σ µ αα = u αūα (3.22) and resort to the standard notation u α (k) → |k ūα(k) → [k|, so much so that Schoutens's identity entails 12 34 + 13 42 + 14 23 =0 and a similarly for |k]'s.
Positive and negative helicity polarisations can be expressed as where q αα = v αvα is an arbitrary light-like momentum that encodes the gauge freedom. Also for massive particles it proves convenient to express their momenta and polarisations in terms of null momenta and Weyl spinors. Setting p αα =k αα +q αα =u αūα +v αvα one has p 2 =2kq=−m 2 =uvūv. Helicity of a massive particle is not Lorentz invariant. For later purposes it proves convenient to explicitly identify the precise Lorentz transformations that map massive helicity states into one another. Let us choose the basis {u α , v α } for Left-handed spinors with uv= uv =0 and {ūα,vα} for Right-handed spinors withūv=[uv] =0. Dropping indices for simplicity, the Lorentz group SL(2, C) × SL(2, C) act as with a, b, c, d ∈ C such that ad − bc = 1 and Rū =ū =āū +bv Rv =v =cū +dv (3.27) It is easy to check that any symplectic product is invariant i.e. i j = ij and The Lorentz transformations that leave the time-like momentum p invariant form an SO(3) subgroup with and with R x/y =L * x/y will prove particularly useful in the following.
For a massive vector boson, with p=uū+vv the three helicity states are 6 The complex circular polarisations w ± can be combined into real ones It is easy to check that L x maps w x into w 0 (up to a sign L x w x =−w 0 ) and vice versa For a massive tensor boson (s=2), the five helicity states can be taken to be

. With a judicious choice of super-ghost pictures and c-ghost insertions one has
Following the steps detailed in Appendix D.1, one finally gets where the contractions are performed in a self-explanatory fashion and

A AAAC in D=4 helicity basis
In D=4 C µνρ is equivalent to a (pseudo) scalar C 0 =ε λµνρ p λ C µνρ /6M . In the helicity basis one has two independent color-ordered amplitudes The latter reads . (3.40)

Amplitude A AAAH in D≤10
Let us now consider A AAAH . With a judicious choice of super-ghost pictures and c-ghost insertions one has Following the steps detailed in Appendix D.2, one finally finds After laborious manipulations, this amplitude can be written in the compact symmetric form

A AAAH in D=4 helicity basis
Let us first consider the amplitudes involving the scalar component of H and start with The amplitude can be written in the very compact form which is identical up to a phase to A(1 + 2 + 3 + C 0 ), for normalised states. Consider a different choice for the helicity of the vectors in the amplitude with H 0 : Consider now the the amplitude for the spin-2 tensor H 2 and three vector bosons Setting p=k 4 +k 5 , the simplest amplitude to compute is the one for the state with polarisation H ++ = 4| 4||5]|5] that reads The other amplitudes obtain in a straightforward way, after repeatedly applying L x and L y as outlined above. The final result can be compactly written as H c 00 for properly normalized polarization tensors.

Higher-point open superstring amplitudes from SYM
In [35,36] Mafra, Schlotterer and Stieberger (MSS) have obtained a beautiful formula that allows one to express open superstring amplitudes for massless external states on the disk to SYM amplitudes at tree level. The formula is reminiscent of the KLT (Kawai, Lewellen, Tye) relations [41] and reads The formula (3.49) follows from a tree-level CFT computation using the pure spinor formalism [35] and its soft limits and other properties were checked in [36]. A pure RNS derivation of Eq. (3.49) has been given in [42], the proof is based on a revisited S-matrix approach [43] .We will here check that it is consistent with factorization on massive string states in two-particle channels i.e.
where A 3 (V V H) is physical (decay rate, width) and can be computed for arbitrary states following the strategy outlined in appendix C. This is nothing but for s 12 =−α M 2 H . For simplicity will only consider mass-less 5-point amplitudes producing 4-point amplitudes with 3-massless and 1-massive state in D=4 and briefly mention how to generalize the procedure to an arbitrary number of mass-less and massive external states. In particular we give the relevant formula for mass-less 6-point amplitudes and sketch, at least in the MHV case, how to get the 4-massless and 1-massive at 5-points or the 2-massless and 2-massive at 4-points.

5-points in D=4 helicity basis
The 5-point color-ordered amplitude for open superstring massless gluons reads [35,36] where F are multiple hyper-geometric functions Since in D=4 any 5-pt amplitude is either MHV or antiMHV with A M HV MSS have checked the correct factorization on the massless poles [35,36]. Here we will check consistency in the massive two-particle channel. To this end one has to take the residue at the pole s 12 → −1 respectively of F (12345) and F (13245). Starting from the expression and making use of One can check the factorization case by case, fixing the helicity of the external gluons. Before embarking in the computations, notice that only SO(6) singlet bosons can appear in the two-gluon channel. Following the dimensional reduction we previously revisited in some detail, one only has H µν , C µνρ =C 0 ε λµνρ p λ /m C and δ ij H ij =−η µν H µν =−3H 0 , after decomposing H µν =H tt µν +H 0 (η µν +α p µ p ν ). Let us start with A(1 − 2 − 3 + 4 + 5 + ). In this case we expect that only C and 3H 0 =η µν H where B(1, 1)=B(1+2α k 1 k 2 , 1+2α k 1 k 3 ). The result coincides with the one we previously derived using standard world-sheet techniques.
Consider now the amplitude A(1 − 2 + 3 − 4 + 5 + ). As shown in Tab. 1 in appendix D.4, if we take k 1αα =u αūα and k 2ββ =v βvβ , with k 1 , k 2 such that 2k 1 · k 2 =p 2 =−1/α , we find that only the spin-2 polarization v α v βūαūβ contributes at the massive pole. With this choice, we have Res . The result coincides with the one we previously derived using standard world-sheet techniques.
The last case is the amplitude A(1 + 2 + 3 − 4 − 5 + ) in which, as for the first case, only H 0 and C 0 get exchanged in the s 12 channel.

Res
The result coincides with the one previously derived using standard world-sheet techniques.

6-points and higher point amplitudes
Open-string amplitudes with more than one massive insertion look somewhat cumbersome and not very illuminating in D=10. In D=4, in the spinor helicity basis, formulae look more compact. A possible strategy for systematic computations is to derive amplitudes for massive states by multiple factorization of amplitudes for massless states on massive poles in two-particle channels. For open superstrings in turn one can rely on the MSS formula [35,36], relating string amplitudes to SYM amplitudes, whose validity we have given further support earlier on.  For an arbitrary number of external massless legs n a priori one has N k MHV amplitudes with k = 0, . . . [n/2]−2. These, and susy related ones, are needed to compute amplitudes for generic massive states by factorization. Summarizing one can start with A SY M,k n+2m , then derive A ST,k n+2m and finally obtain A ST,k n,m by factorization on the assigned two-particle massive poles. Notice that the initial helicity configuration should be chosen compatibly with the choice of massive states, i.e. at the first level H 0 /C 0 couple to gluons with the same helicity while H 2 couples to gluons with opposite helicity. Reverting the argument, the allowed helicity configurations in SYM constrain the allowed amplitudes in superstring theory.

General arguments
In [14] the soft limits of massless string amplitudes was studied both explicitly (up to 6point amplitudes) and abstractly by making use of OPE analysis. The conclusion was that disk amplitudes of gluons behave exactly as in Yang-Mills theory at tree level both for the open superstring and for the open bosonic string. Indeed one expects universal behaviour at leading (δ −1 ) and sub-leading (δ 0 ) order, in formulae A n (1, 2, . . . , s, . . . , n) where J i denotes the angular momentum operator acting on particle i and f µν s = k µ s a ν s − k ν s a µ s , as by now usual. We would like to extend the analysis of [14] based on the OPE to open string amplitudes with massive states.
The leading and subleading soft behaviours are captured by the OPE of the soft gluon integrated vertex with the adjacent (integrated) vertices. Using and similarly for z s−1 one gets where ... includes massive string states which do not contribute to the leading singularity since (H[a s , a s±1 , k s , k s±1 ], k s + k s±1 ) + ...

(4.4)
where V M denotes the vertex operator of a massive state, with momentum p = k s + k s±1 and polarisation H that can be expressed in terms of a s , a s±1 , k s , k s±1 .
Expanding the denominator as one immediately sees that at most the sub-leading (regular δ 0 behaviour) might be affected. However the tri-linear coupling A−A−H contains at least one soft momentum k s and this produces a further suppression by δ +1 . This holds true also for the tachyon since the . ., while for C µνρ at the first massive level of the superstring one has A C−A−A = C µνρ a µ 1 a ν 2 (k 1 − k 2 ) ρ . Let us now consider the case where the soft gluon is attached to a massive (or tachyonic) leg The first term is the string analogue of minimal coupling that is leading in the soft limit k 1 → 0. Gauge invariance fixes the sub-leading term to be as expected. Indeed, for colorordered amplitudes with n + 1 gluons and no massive states one finds [14,22] A n+1 (1, . . . s . . . , n+1) = ±1 2k s ·k s±1 A n (1, . . .ŝ, . . . , n+1)| a s+1 =a s+1 as·k s±1 −asks·a s±1 +ksas·a s±1 k s+1 =k s+1 +ks +. . .
Including m massive states, if the soft gluon is adjacent to two hard gluons the above analysis continues to apply. When at least one of the adjacent legs is massive, let's say the one in position s + 1, with spin one has (4.11) where . . . denotes the additional terms in the tri-linear V -H-H coupling. Barring a couple of subtleties, we will deal with later on, expanding in k s one gets (schematically) that completes the action of f s :J s+1 on A n,m (1, . . .ŝ . . . , n + m + 1) at sub-leading order. Now let us deal with two subtleties: the higher derivative terms in the tri-linear coupling A-H-H and the possible non-diagonal couplings A-H-H that would spoil universality. The former is easy to dispose of, higher derivative corrections to minimal coupling can only affect the sub-leading term that is fixed by gauge invariance wrt the soft gluon starting from the low-derivative terms coded in the OPE. The latter requires more attention. For open superstrings, as we have seen, already at the first massive level one finds two kinds of particles in the Neveu-Schwarz sector: C µνρ (3-index anti-symmetric tensor, 84 d.o.f.) and H µν (2-index symmetric traceless tensor, 44 d.o.f.). 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 ≈ α M p 31 ·H 2 ·C 3 :[a 1 p 12 ] that exposes the singular soft factor 1/kp since M C = M H but gets suppressed by an extra power of the soft momentum in the numerator. Lacking the leading δ −1 term that fixes also the sub-leading δ 0 term, thanks to gauge invariance, this kind of higher derivative non-diagonal couplings can at most affect the sub-sub-leading δ +1 (and higher) terms which are not expected to be universal. Although the situation gets exponentially more intricate the higher the mass level and spin, we conclude that no correction are to be expected wrt the standard YM case in the soft behaviour for open string amplitudes involving massive states.
For illustrative purposes, we will explicitly check the above statements in the soft limit of some 4-point amplitudes with massive string states at the first level. Differently to the case of amplitudes with only mass-less external states that factorise on 3-point amplitudes, that would vanish for real momenta due to collinearity, when some of the external states are massive, the soft limit can produce physical 3-point amplitudes e.g. widths or decay rates of massive states into lower mass ones.

Soft limit of
In this case the limit k 1 → 0 is straightforward. Consider first the expansion of the factor 14) The expansion of the full amplitude reads showing the expected singular behavior in both the s and the t channels, whereas the term of order O(δ 0 ) vanishes because being the 3-tachyon amplitude a constant independent from the momenta.

Soft limit of
Consider the amplitude in Eq. (B.4) once we have expressed the Euler beta function in terms of Euler gamma functions As already verified, the combination of Euler gamma functions in the above expression contributes in the limit δ → 0 as 1 + O(δ 2 ), thus can be neglected. We have two terms of order δ −1 : Using the identity Eq. (4.18) can be written as where The leading soft contribution is, as expected, The order O(δ 0 ) contribution to the amplitude reads − a 1 a 2 k 1 p 3 k 1 k 2 + 1 2 In the above expression we recognize the expected behavior (4.27)

Soft limit of A(T 1 , T 2 , A 3 , H 4 )
Consider the amplitude in Eq. (B.6). Let us first discuss the soft limit of the kinematical factor It is then convenient to multiply the tensorial part of the amplitude for the above expansion in order to identify more easily the contributions up to O(δ 0 ): The leading order O(δ −1 ) behaves as expected being A(T 1 , T 2 , H 4 ) = p 12 2 H p 12 2 . Look at the subleading contribution: (4.34) It is easy to verify that the above expressions coincide with: We recall that the angular momentum for a spin-2 particle takes the form (4.36)

Soft limit of
The amplitude is given in Eq (3.36). Let us study the soft behaviour when k 3 → 0. Recall that s + t + u = 1/α The Veneziano factor B(1, 1) yields (4.37) To leading order the polarisation dependent factor yields combining the two one gets, as expected (4.39) To sub-leading order δ 0 , one gets as expected, where Actually ∂/∂p 4 acts trivially in this case].
With little more effort one would get the same result for k 1 → 0, while for k 2 → 0 the only contributions come from 'standard' soft behaviour of gluons hitting adjacent gluons.
This gives support to our general conclusion that superstring amplitudes with n massless and m massive external legs on the boundary of the disk behave universally in the soft limit.

Soft limit of
The amplitude is given in Eq (3.43). Let us study the soft behaviour when k 3 → 0. Recall that s + t + u = 1/α as for A(A 1 , A 2 , A 3 , C 4 ). Following the same steps one finds to leading order where use of m 2 H = −p 2 4 = −2k 1 k 2 + O(δ) has been made. At subleading order one finds several terms i.e.
Summing the terms within squared parenthesis everything can be written in terms of f 1 , f 2 , f 3 (as dictated by gauge invariance at this order) finding with J 2 µ ν given above and As above ∂/∂p 4 acts trivially in this case. This gives further support to our general arguments on the soft limit.

Holomorphic soft limit
In this section we verify that open string amplitudes with massive external states enjoy the same universal behaviour as YM amplitudes in the holomorphic soft limit [12]. In this limit the holomorphic spinor u s of a positive helicity gluon (inserted between leg 1 and leg n) is scaled to zero u = δû s . In SYM the leading behaviour as δ −2 is governed by the operator the sub-leading behaviour as δ −1 is governd by the operator For MHV amplitude the sub-leading term vanishes and the procedure exponentiates [12]. In general it is convenient to use momentum conservation to express twoū's in terms of the remaining ones and the u's. In our case, an obvious choice is to expressū 4 andū 5 that appear in the definition of the massive momentum p = k 4 + k 5 = u 4ū4 + u 5ū5 . When taking derivatives one has to take into account the mass constraint m 2 = −(k 4 + k 5 ) 2 as we will see momentarily.
The momentum of the massive particle is the sum of two massless momenta p = k 4 + k 5 with the constraint 45 [54] = m 2 C . This constraint implies 13 [13] .
The leading contribution to the holomorphic soft limit is easily identified to be
The sub-leading contribution is expected to be In the presence of the mass constraint, the derivatives wrtū 1,3 are replaced by Writing the three-point function A(1 + , 3 + , C 0 ) in a slightly different way we need to evaluate only the derivative of the tri-linear coupling respect to m C . Finally we find [13]m C 12 23 23 [23] 13 [13] + 12 [12] 13 [13] . (5.11) that exposes the expected behaviour, too.
In order to complete our analysis, we consider the case in which the soft momentum is k 3 . Let's first expand the amplitude in Eq. (5.1) up to the order δ −1 At leading order, the soft operator is simply with the help of Schouten's identity. The expected leading order behavior looks like

A(A − , A + , A − , C)
Consider now the amplitude Taking the limit in which u 2 → 0, we have 12 [12] 13 [13] + 23 [23] 13 [13] . (5.21) For the leading term one finds for the sub-leading term that behaves as expected in the holomorphic soft limit. In this case we will not consider the limit in which k 3 → 0 since the three-point amplitude A(1 − , 2 + , C 0 ) = 0 vanishes.  Following the same algebraic manipulations as in Sec. 5.1 it can be shown that these two terms reproduce the sub-leading soft term of the expansion in Eq. (5.24).

Conclusions
We have computed several open bosonic and super-string scattering amplitudes on the disk with massive and tachyonic external states in critical dimension as well as in D = 4 (for the superstring, using the spinor helicity basis).
We have then checked their universal behaviour when massless gluons go soft, despite the presence of higher derivative couplings, and offered a general argument to this effect based on world-sheet OPE. We have also checked consistency of the factorisation on the first massive pole of the MSS formula obtained in [35,36] relating open superstring amplitudes on the disk to tree-level SYM amplitudes.
We have only briefly considered closed strings. For gravitons, even in the presence of massive external legs, one would expect a universal soft behaviour up to sub-sub-leading order (δ +1 ) [8][9][10][47][48][49] M n (1, 2, . . . , s, . . . , n) = (6.1) i =s This should hold true at tree-level and with the understanding that interactions be governed by minimal couplings. While in closed Type II superstrings on the sphere the soft limit of amplitudes with massless states is the same as in gravity at tree level, for bosonic strings -and in fact for the heterotic string, too -the presence of a φR 2 vertex with the dilaton spoils the universal behave even at leading order, in that a soft graviton attached to a hard graviton can produce a hard dilaton thus producing a mixed amplitude 8 Using KLT relations [41] one can efficiently compute closed amplitudes with massive external states as 'squares' of open string amplitudes with massive external states, like the ones we have considered in the present investigation. We plan to carry out this analysis in simple cases and study the soft behaviour at tree level confirming universality, respectively lack of it, in the case of the closed superstring (both Type IIA and Type IIB), respectively in the case of the bosonic or heterotic string due to the presence of the φR 2 terms [52]. We hope to shed further light on the soft behaviour of the Kalb-Ramond field, the dilaton [50] and the other moduli fields [51]. It would also be interesting to investigate the soft behaviour of loop amplitudes and to test the validity of the new proposal [53,54] of getting the graviton from the collinear limit of two gluons beyond tree level and in the presence of massive external states.
following the discussion in section 2.2, we will explicitly include the relevant Chan-Paton factors f abc or d abc that make the full 'amplitude' Bose symmetric.

• T T T vertex
where we used the identity (p i +p j ) 2 =−2m 2 T +2p i p j =−m 2 T , so that 2α p i p j =m 2 T =− 1 α for all i, j. The symbol z ij stands for z i −z j . In order to simplify the notation, from now on we will introduce the notation In general, P i is contracted always with the i-th polarization vector/tensor. Exploiting 'transversality' i.e. p µ i t i µ... =0, we will always replace the sum in Eq. (A.2) with: The amplitude can be rewritten in a manifestly gauge invariant form:

B Open bosonic string four-point amplitudes
In this appendix we sketch the computation of the open bosonic string amplitudes involving massive and tachyonic states. For simplicity we consider color-ordered amplitudes. Complete amplitudes arise after multiplying by the relevant Chan-Paton factors tr(t 1 t 2 t 3 t 4 ) and summing over non-cyclic permutations. In fact, exploting 'twist symmetry' i.e. Ω invariance, one can reduce the sum to three terms (instead of six). Exploiting conformal invariance we choose to fix z 1 → ∞, z 2 =1, z= z 12 z 34 z 13 z 24 and z 4 =0. A factor g 2 s,ap (α ) (D/2−4)/2 (2π) D δ D ( i p i ) is always understood.

(D.4)
The second kind of contractions yields in a self-explanatory index-free notation. Including the momentum factor and the (super-)ghost correlators yields (barring δ(Σp) etc)