String Memories ... Lost and Regained

We discuss stringy ${\alpha'}$ corrections to the gravitational wave signal generated in the merging of two black holes. We model the merging with two BPS compact massive objects in the heterotic string, described by standard vertex operators or coherent states. Despite the expected cubic suppression in ${\alpha'}$ w.r.t. the General Relativity result, at tree level the string corrections seem to leave a footprint or a memory on the gravitational wave signals within the sensitivity region of aLIGO/VIRGO and future interferometers. Including loop effects that broaden and destabilise the string resonances suggests a sort of lost stringy memory effect that can be regained through the analysis of the quasi normal modes in the ring-down phase.


Introduction
Direct detection of gravitational waves (GWs) [1,2,3,3,4,5] has opened a new chapter in the (theoretical) exploration of Black Hole physics [6,7]. Many modified gravity models -including higher-derivative gravity and massive gravity -as well as models that violate fundamental symmetries, such as Lorentz and CPT invariance, have been highly constrained by the GW data hitherto collected [8,9,10]. Thus, GW physics is running at the frontier of high precision tests of the gravitational sector of fundamental interactions. In this context, a natural question that one can ask is: "Can we probe quantum gravity with gravitational waves?" UV issues in quantum gravity have been debated for more than half a century, now. GWs may provide precious information on the quantum nature of the gravitational field in the IR region that can help discriminating among competing theories 1 . An exciting perspective is to 1 Another direction of the quantum gravity phenomenology that may be coded in GWs is the possible hints for the existence of Exotic Compact Objects (ECOs) [11] such as Fuzzballs [12] that may be exposed by searching for departures from the zero Love number bound, set by fundamental uncertainty principle limits on the BH radius [13], or from corrections to the ring-down modes [14,15]. search for possible hints of String Theory (ST) inside the GW data. String corrections typically produce higher-derivative terms in the low-energy effective Lagrangians that are suppressed by the Regge slope α and by the string coupling g s . Interesting constraints on the String Tension, the size of the internal Compactification Volume and limits on Large or Warped Extra Dimensions may be achieved in the near future. Moreover, string theory predicts the presence of stringy resonances (Regge recurrences) as well as Kaluza-Klein (KK) excitations, which can both leave imprints in the GW signal. Last but not least, the BH information may be stored in the form of quantum, possibly soft, hairs, as originally suggested in [16,17,18,19,20].
This may result into a multi-polar structure much richer than the one appearing in General Relativity 2 (GR) [25,26,27]. The correspondence between strings, branes, their bound-states and 'large' BHs has a long story that we cannot review here, see e.g. Refs. [28,29,30,31,32] for early and recent work with different perspectives.
In the attempt to expose α effects in GWs beyond GR, we use a toy model for BH merging within the framework of high-energy string scattering. Despite the complexity of the problem, we identify a simple case that may offer some insights in the dynamics of BH mergings in ST.
We will consider the tree-level merging of two very massive states, which we assume to retain a compactness comparable to BH's or neutron stars. We will further assume that the incoming states have the same quantum numbers as extremal 2-charge BHs and that the scattering produces a non-BPS charged BH state plus (soft) gravitons. The computation of the (treelevel) amplitude for this scattering process turns out to be remarkably simple in the heterotic string framework, where BPS micro-states just correspond to vertex operators or coherent states thereof 3 . Closed-string coherent states can be powerfully treated within the DDF approach [35,36,37,38,39], using similar techniques as in the case of open (super)strings [40,41].
This provides a simplified framework for our tree-level calculations, which enables us to obtain an exact result for the relevant scattering amplitudes. Our main purpose is to explore α deviations from the GR (or Supergravity) results. After showing the complete agreement of our final expressions with GR at the leading order, we confirm that the first stringy corrections arise at cubic order in α and as such would give too rapid a decay with the distance. Yet, the collective effect of all string resonances leads to a sizeable correction to the GW signal 4 .
Heuristic estimates show that forthcoming gravitational interferometers are actually close to access a non-negligible part of the parameter space of α corrections. Thus the logic behind our calculation is the following. As a first step, we will show that the process of two BPS BH-like 'compact' states merging into a non-BPS BH-like 'compact' state, with concomitant release of GW, can be computed in the context of heterotic string theory 5 . Then, by 'stripping off' the graviton polarization, we will use the 'truncated' scattering amplitude as a source for the gravitational wave equation. The GW profile is obtained from the convolution of the string source with the graviton propagator. This allows us to study how α corrections from the stringy-BH sources alter GW emission.
Not surprisingly, our results confirm gravitational memory, based on the celebrated Weinberg's soft graviton theorem [45], recently extended 6 in [48,49,50,51,52,53,54,55,56,57,58,59,60] and [61,62,63,64,65,66], and suggest an interpretation of the α effects as a sort of string memory. However taking into account their finite width beyond tree level, the GW profiles from string resonances are exponentially decaying in time and the effect seems to be lost. Nevertheless, the lost memory may be regained, if one looks at the quasi-normal modes (QNM's) [27,15] that govern the ring-down phase of the produced stringy BH. This effect may be enhanced for some string states that can be very long-lived [68]. This suggests that some string information may have milder decay in time, opening a (small) window for searches of the string (lost and regained) memory effect.
In order to further motivate our analysis, let us estimate the order-of-magnitude of the putative string corrections and argue that some portion of the parameter space of the highenergy phenomena can indeed be accessed and thus constrained by forthcoming gravitational interferometers 7 . At tree-level, scattering amplitudes acquire stringy corrections with respect to standard quantum field theory results in the form of powers of ν ≡ α k · p a , where k is the graviton four-momentum and p a the BH four-momenta. Considering BHs of mass around M BH 20 M ≈ 20· 10 30 Kg and a LIGO/VIRGO frequency of 10÷100 Hz, barring very special kinematics, one can reach values of ν O(1) even assuming a relatively high string tension (around the Grand Unification Scale or beyond). Such a surprising amplifier is provided by the BH mass entering the expression for ν, provided the massive (non)BPS states involved retain the required compactness.
The rest of the paper is organized as follows. In Section 2, we discuss our toy model for (BPS) BH mergers and GW production in heterotic string and compute the relevant amplitude at tree level. We briefly discuss loop corrections, exponentiation (eikonal approximation) and compare our analysis with the one in [69] for BH production in cosmic superstring collisions.
In Section 3 we analyze the role of the Regge recurrences that are stable at tree-level and produce a modulation of the GW signal w.r.t. GR. We will treat the corrections in three different schemes. First as higher-derivative corrections that individually seem to produce no effect at large distances. Second as a sum over (tree-level stable) resonances that has a sizeable effect on the GW profile in real space, coded in a power series in u/ a , with u = t − R the retarded time and a = α kp a /ω for a, b = 1, 2, 3. Third in the high-energy limit α kp a >> 1, both at fixed angle α kp a ≈ α kp b for all a, b, in Section 3.4, and small angle (Regge limit) α kp 1 << α kp 2 ≈ α kp 3 , in Section 3.5, relevant for Extreme Mass Ratio Inspirals (EMRI's). Then in Section 4 we discuss how string memories look irremediably lost once quantum effects producing a finite width are included and how they can be (partially) regained either by a collective effect or by their imprints on the QNM's governing the ring-down phase of the produced non-BPS BH. Section 6 contains our conclusions and final comments.

Gravitational Wave Production from BPS-BHs merging
A remarkably simple and interesting example of GW production in a 'macroscopic' high-energy inelastic string process is based on the heterotic string 8 scattering involving very massive, spinless and compact BPS states such as With some abuse of language we call BHs the large mass (non)BPS states with windings and Figure 1: Schematic representation of the selected scattering process 8 On a general six dimensional compact manifold with some non-trivial 1-cycle. momenta such that where P L and P R denote the internal L/R momenta and N L and N R the (mass) levels. States of this kind are also known as 'small' BH's (2-charge 'fuzz balls' [12]) in that they have no (or 'zero-area') horizon in the supergravity approximation but have a non vanishing entropy associated to the exponential growth of the degeneracy (à la Hardy-Ramanujan-Hagedorn), i.e.
These are very peculiar BHs, which can carry arbitrarily high spin [70,71]. More 'realistic' models for 'large' BH's in string theory have been proposed over the years [28,29,30,31,32,12] that do not allow a simple analysis of the merging as the one we perform here.
We consider one of the three BHs to be non-BPS since otherwise the kinematic conditions would trivialise the scattering amplitude in the soft limit, as shown 9 in appendix B. N L measure the level of L-excitation over the BPS ground-state with given P L . Last but not least, we assume the three massive states to be spin-less and compact, i.e. J a = 0 and R a ≈ G N M a . As we will see in Section 5, compact-ness can be achieved for coherent states, though it might be harder to achieve for mass eigenstates such as the ones we consider in this Section, for which R a ≈ α M a > G N M a , in the perturbative regime whereby g s < < 1.

Scattering Amplitude
The tree-level (sphere) scattering amplitude we consider is where C S 2 = 8π/g 2 s α is the normalisation constant of the sphere S 2 , V CKG is the volume of its conformal Killing group (CKG) PSL(2, C), k and h are respectively the momentum and polarization of the graviton, p a and ζ a , with a = 1, 2, 3, are respectively the momentum and the polarization of each one of the three BHs.
Setting α = 2 as usual and using the first non-canonical (0) superghost picture, the graviton vertex operator (see e.g. [67]) reads where X L(R) denotes the L-(R-) moving bosonic coordinates, ψ L the L-moving fermionic coordinates, k µ L = k µ R = (ω, k) is the null momentum of the graviton, with no internal components, i.e. P L = P R = 0. The split of the polarisation as h = h L ⊗ h R is often adopted.
In the same (0) superghost picture and with a splitting of the polarisation ζ = ζ L ⊗ ζ N R R , the massive (BPS) BH vertex operator reads The remaining two BH vertex operators can be expressed in the canonical (−1) picture.
The first (BPS) can be written as where K 1L = (p 1 , P 1L ), K 1R = (p 1 , P 1R ) and, again, For simplicity let us take the final non-BPS one to be 'excited' only with 'internal' bosonic oscillators viz., namely where K 3L = (p 3 , P 3L ), K 3R = (p 3 , P 3R ), N 3L = k M ax k=1 km k and, for later use, n 3 = k M ax k=1 k (for states in the first Regge trajectory N 3 = n 3 ).
BRST invariance requires the mass-shell conditions spelled out above as well as 'transversality' and 'traceless-ness' conditions ζ 3L P 3L = 0 = ζ 3L H L N 3 = P 3L H L N 3 = Tr(H L N 3 ), which we assume to be satisfied.
The issue of collinearity that afflicts the 3-BPS case, discussed in Appendix B, is solved since α K 1 K 2 = α (p 1 p 2 + P 1L P 2L ) = 2N 3 = 0. As a result, though the incoming BH's are BPS they are not mutually BPS since P 1L and P 2L are not collinear.
Choosing the 'polarisations' of all three massive states to be along the 'internal' directions, so much so that the resulting BHs have zero spin (J a = 0), contractions are easy to take, only a little bit more involved w.r.t. the 3-BPS case (equivalent to 3 massless in D = 10). In particular one can show that the terms k 1 ·ψh·ψ , in the graviton vertex, and K 2 ·ψζ 2 ·ψ, in the (BPS) BH (taken to be in the 0 picture), cannot contribute. The only difference w.r.t. the 3-BPS case is the presence of H 3L that can contract with exp ıP aL X L , as well as with ζ 2L ∂X.
The (vanishing) 3-vector boson YM vertex is then replaced by [70,71] (2.8) Obviously, at the expenses of more involved computations, one can also replace one or both the BPS BH's with non-BPS states with M a > |P aL |, or even consider coherent states, as we do later on along the lines of [37,38,39,40,41]. The crucial point is that in the soft limit k → 0 a non vanishing 3-point 'physical' amplitude be produced together with terms of higher-order in k which will be our main focus.
Assembling the various pieces, the amplitude takes the following form With the stringy identification of the 4-d Newton constant 10 G N = g 2 s α 4 64 πV (6) , (2.10) where V (6) is the volume of the compactification manifold and the un-normalized three-point amplitude coming from the factorisation the result can be written as where M 3−BH = (2πV (6) /g s α 3 )M 3−BH and the adimensional prefactor can be reabsorbed in the (not-directly measured) distance R traveled by the GW from the source (the merging) to the detector.
Finally, using the leading soft factor, introduced by Weinberg [45], namely we may recast Eq.(2.12) in the more compact and final form (2.14) The appearance of the soft factor S 0 should not be deceiving: the expression M 3+1 is exact, valid for any value of ω.
In principle, in order to have a reliable picture of the scattering process at least at the perturbative level, one should include loop corrections to the above tree-level amplitude. Although the one-loop (torus) contribution would not be hard to compute relying on 4-point massless amplitudes in D = 10 [72,73] and tackling the two-loop amplitude should be harder but doable, addressing higher-loops looks daunting and subtle to some extent [74].
In fact we would argue that this is not necessary for our purposes. As in GR, the process will take place in three phases: inspiralling, merger and ring-down. Denoting by b the 'impact' 10 The identification of the Newton constant comes from standard matching procedure in the heterotic supergravity.
parameter and by R 1 and R 2 the 'sizes' (gyration radii) of the 'compact' BPS incoming BH's, in the in-spiralling phase the two massive compact objects are well separated b > > R 1 , R 2 and exchange mostly massless quanta (gravitons), whose contribution can be resummed in the eikonal approximation 11 , leading to a computable phase shift. GW production in this phase has been studied and produces a spectrum in line with GR [46,47].
During the merger b < R 1 + R 2 , we can use the inelastic tree-level amplitude we computed and we get a correction to the GW signal in GR that we will study momentarily.
In the ring-down phase the non-BPS BH will relax to some (meta)stable configuration. We will briefly address the spectrum of quasi-normal modes (QNM) in the ring-down phase in Section 4 but we plan to investigate this issue more thoroughly in the future.
Before concluding this Section we would also like to briefly compare our present results with those on BH production in cosmic superstring collisions [69] and the ones on pair-production of miniBH [70,71,75]. In [69], the incoming states are far from being compact and the splitting and joining process is suppressed by the probability that two bits of the colliding cosmic superstrings come close enough. Then the process is further suppressed by the probability that the produced string be compact enough to behave as a BH. Most of the analysis is semi-classical and reliable in the regime of [69]. In [70,71,75] slightly different processes are considered, whereby very energetic massless initial states come so close that b < R S and a pair of mini-BH of small size (even T eV scale in principle in models with very low string tension) and opposite charge is produced. Here we have focussed on a sort of crossed channel whereby the BPS BH's (but non-mutually BPS) are in the initial state.

String Memories
Barring irrelevant constants, the 4-point amplitude BPS 1 + BPS 2 → non−BPS 3 + h µν can be re-written as where p a , ξ a collectively characterise momenta and polarisations of the massive (non) BPS states, with M 2 a = −p 2 a = |P L,a | 2 + N L . Note that, even though a p a = 0 = a P L,a = a P R,a , M 3 (p a , ξ a ) is 'physical' and non-zero since the three momenta p a are not necessarily collinear even in the soft limit k→0, contrary to what happens for mass-less quanta. Yet, the kinematics is rather scant in the soft limit, since 2p 1 p 2 = −M 2 3 +M 2 1 +M 2 2 and cyclic. The phase space deforms a bit due the emission of the massless graviton. See Appendix A for details.
Moreover, even for finite k

Gravitational Memory
In GR, the GW profile h µν , produced by a (transverse traceless) source S µν , obeys the following For the causal retarded wave propagation, the solution takes the form For the GW produced in a high-energy collision in GR, and observed at large distances R = with polarisation tensor dictated by the Weinberg's soft theorem [45] e µν (ω, where p a = +p a for out-going particles (η = +1 in Weinberg's notation), while p a = −p a for in-coming particles (η = −1 in Weinberg's notation), so that a p a = 0 = a∈out p a − a∈inp a .

String memories
Including stringy corrections, the GW profile e µν (t, x) is determined by the solution to the equation where (µν)| denote the symmetric, trace-less component. In the low-energy limit the Shapiro-Virasoro factor in the heterotic string amplitude, namely produces corrections to the GR results that can be written as a power series in α k·p a that starts at cubic order. Fourier-transforming in ω would produce corrections decaying faster than 1/R at large distances that would be totally negligible. On the other hand, including the contribution of the infinite tower of string resonances turns out to produce a sizeable effect.
Indeed, starting from where the last approximation is valid at large distances R = | x− y| > > L, so that 14) or, barring the overall and largely irrelevant non-zero factor M 3 (p a , ξ a ) =0 and anti-Fourier transforming, one finds where u = t − R is the retarded time. In order to compute the ω integral it is convenient to as an infinite sum of poles in ωà la Mittag-Leffler (ML), and obtain where we introduced the kinematical ratios The pole in ω = 0 reproduces the gravitational memory effect 13 . In addition to this, in (heterotic) string theory one finds genuine string corrections. Inserting the ML expansion in 13 Massless dilatons and axions do not contribute at this leading order.
the integral over ω (3.15), and adopting some reasonable prescription to deform the integration path, i.e. kp a → kp a − ı , one finds intriguing corrections ∆ s e µν (t, x) to the usual GR profile, suggesting some sort of string memory effect with a non-trivial (retarded) time dependence.
Before integration the correction reads Performing the integral, in the physical kinematic region where 1,2 > 0 and 3 < 0, for Notice that the typical time-scales of the signal are set by a , which we have estimated to be in the aLIGO/VIRGO range (ω ≈ 10 ÷ 100Hz) for 1/ √ α ≈ 10 15÷16 GeV and M a ≈ 10 ÷ 50M . Since the amplitude is of the same order as the leading GR contribution, we expect the correction to be detectable in the near future.
Although the resulting series cannot be resummed in general, for special values of the kinematical parameters a = np a they can we written in terms of known functions.
In this regard some examples are reported in Table 1.
For representative kinematical ratios, in Fig. 4 we plot the real and imaginary parts of the function δ a (λ a+1,a , λ a+2,a ; u/ a ), which according to Table 1 and its properties can be represented as a function of only one kinematical parameter, the other ones being fixed as in (3.36). Table 1: Some examples of 'rational' kinematical regimes. Note that contrary to gravitational memory, string memories have a non-trivial u dependence and that their origin lies in the possibility of an excited string state to emit gravitons through Regge resonance in highly inelastic processes like the one we have considered.
Moreover, recall that incoming BHs produce radiation that can be detected outside the future light-cone of the merging event, i.e. for u < 0, but also in the future u > 0, while the produced BH emits radiation only inside the future light-cone, i.e. u > 0, see Fig. 5.
In order to shed some light onto the physical implications that these α corrections might entail, and to expose a more transparent u dependence, we will focus on special/extreme kinematical regimes in the following.

Large ω a behaviour of the String Memories
Let us subtract the purely GR pole at ω = 0 and consider At large ω a , (keep in mind that 1 and 2 are 'in' and 3 is 'out') the dominant behaviour is Plugging this in the ω integral yields the expression that is ill-defined and requires regularisation. A reasonable choice 14 seems to be replacing e −βω with e −β|ω| that yields As visible from the plot in Fig. 6 the real part of the function E reg (u, β) behaves like 1/u, while the imaginary part tends to a constant. It is amusing to observe that a 1/u behaviour is 14 We thank A. Sen and B. Sahoo for pointing out a shortcoming in a preliminary version of this manuscript. similar to the one produced by the log ω terms appearing at one-loop in the case of D = 4, that have been identified in [66], following earlier work [61,62,63,64,65]. This seems to suggest that the inclusion of α -effects may emerge as a violation of the gravitational soft theorems. As previously suggested, log ω terms may be detected through GWs. One should keep in mind that the function βω = α (s log s + t log t + u log u) coincides with the leading log IR divergent terms appearing at one-loop in 4-graviton amplitudes in GR (or supergravity), with G N = α g 2 s /V 6 replacing α . Some kind of Regge behaviour is found in the case ε = 1 << 2 ≈ − 3 , whereby one of the merging BPS BH's is much lighter than the other two, i.e. M 1 << M 2 ∼ M 3 . This process is some times called the 'plunge' and leads to Extreme Mass-Ratio Inspirals (EMRIs) that will be one of the scientific goals of LISA mission. In this case one finds where w * satisfies the saddle-point equation where one can neglect 1/w. So one gets and finally, exponentiating the result, One should keep in mind that np 1 << np 2 , the result looks different at u > 0 from u < 0 whereby the imaginary part gets flipped. See plots in Fig. 7.
Both the (adimensional) mass-shift δN and the decay width γ N depend in a highly non-trivial fashion on the details of the state, i.e. the vibration modes n k and polarisation tensor, and on the string coupling g s . The study of this feature, that appears already at one-loop, remains beyond the purpose of our present analysis. Yet, we would like to mention that string states can be found with a very long lifetime, growing with the mass M as fast as T g −2 s M 5 [68]. In a semi-classical picture, these long-lived closed strings remain unbroken during their classical evolution. Emission of massless quanta provide the dominant decay channel of these and other massive string state [76]. Type II superstring one-loop amplitudes were evaluated in [77], for states in the Neveu-Schwarz (NS) sector, obtaining mass-shifts and decay rates as a function of the space-time dimension and the string scale.
In general these instabilities are experimentally relevant, encoding contributions that in nonlinear optics are termed 'evanescent' waves. The large number of resonances that are eventually produced in some phenomena, including super-oscillation, can actually trigger remarkable amplifier mechanisms, see e.g. [78]. On the other hand, barring the dispersion at a given level and relying on string analyses [76,68,77] suggests power-law expressions of the decay rate such as with α real and positive. For N → ∞, the high density of states allows to replace the infinite sum over the massive string resonances with an integration i.e.
This means that stringy resonances may survive in the GW signal as a cumulative effect. Yet, it seems quite unlikely that one could resolve individual peaks.

QNM's and ring-down phase
To make contact with the phenomenology of GWs, we may associate to γ N of Eq. (4.2) an exponential damping. This immediately calls for a consideration of quasi normal modes (QNM's) ω lmn [27] and echoes thereof [82,83]. QNM's represent unstable perturbations of a background metric. The real part Re(ω lmn ) is associated to the frequency of the unstable closed orbits of a (massless) probe, while the imaginary Im(ω lmn ) to the Lyapunov exponent γ ≈ τ −1 that governs the chaotic behaviour of geodesics around the 'photon-sphere' [15].
In GR, the uniqueness of the frequencies and damping times is customarily related to the "no hair" theorem. In this sense, the detection and identification of QNM's may provide a further possible test for GR in strong-field regimes such as BHs [79].
Indeed, while in-spiralling can be dealt with by means of a post-Newtonian analysis in GR 15 , the merging phase requires analytical tools: the perturbative approach ceases to be valid as none of the two BHs can be treated as a perturbation of the other (except possibly for EMRI's [81]). In our toy model this is accounted for by the highly inelastic amplitude we used as a source for the GW signal in the merging phase. In the ring-down phase, that represent the last part of the signal, perturbation methods can be still applied to the analysis, and with satisfactory results since the signal can reliably be decomposed in QNMs [27]. configuration (typically a rotating BH, described by Kerr metric [89]) will be predominant.
We expect the same to take place in a complete quantum theory of gravity such as string theory. The QNM's of a stringy (nonBPS) BH, such as the ones we have considered in our toy model, or a (nonBPS) fuzz-ball are characterised by their peculiar QNM's [15,27] and analysis of the ring-down signal may expose echoes [82,83,7] and novel multipolar structures [21,22,23,24] that could help discriminating between different models for the smooth horizonless compact object replacing the BH and its singular and paradoxical behaviour in GR.
Once again this interesting analysis, that is being performed for fuzz-balls, is beyond the scope of our investigation and we defer it to the future.

Coherent states of quasi BPS BH's
For the purpose of making our computations as simple as possible, so far, we have considered mass eigen-states of heterotic string on T 6 . We would like to generalise our analysis to quasi BPS coherent states using DDF operators [35,36,37,38,39,40,41] that can be made as compact as required for the validity of our analysis.

DDF operators for open bosonic strings
In order to fix the notation let us introduce the DDF operators that for the open bosonic string are defined as where i = 1, ..., D−2 (D = 26) and q 2 = 0. For convenience, we set q + = q i = 0 and q − = 0 from the start, so that q·X = −q + X − −q − X + +q i X i = −q − X + with strings one has to double the modes A i n → (A i n,L , A j n,R ) up to subtleties with the (generalised) momentum we will deal with momentarily.

Classical Profiles for Coherent States
In order to illustrate the dynamical profiles of (quasi)BPS coherent states we consider simple examples that correspond to different choices of the polarizations ζ µ n (ζ µ n ), or more precisely of the parameters λ µ n (λ µ n ). Closed-string coherent states satisfy A i n |C(λ,λ, p) = λ i n |C(λ, p) ,Ã i n |C(λ,λ, p) =λ i n |C(λ,λ, p) (5.4) Starting from the classical string profile n,L,(R) = λ n,L(R) ± λ * n,L(R) real polarizations, where the mass formula for a BPS state reads while for a (quasi)BPS state one has Using a simple ansatz for the coherent state polarizations of the form λ µ n = V µ e −αn n β where α and β are two free parameters, and V µ a (null) vector the BPS and (quasi)BPS states, provided with a coherent structure, display three-dimensional profiles as the ones displayed in the plots in Fig. 8, 9

Generalized momentum
In order to write down vertex operators for coherent states, we start by fixing the conserved charges P L and P R of each massive state, and choose the reference null momentum q of all states to have only the q − component i.e. q + = q I = q i = 0, where q ± = 1 √ 2 (q 0 ±q 3 ) and I = 1, 2 -space-time (x, y) -while i = 1, ...6 (internal).  The full 10-d momenta read so that K 2 L = 0, while K 2 R = P 2 R −P 2 L = 2−2N R , as desired. Notice that the two momenta differ only in the internal part.
For quasi (or non-)BPS states, K 2 L = −2N L , where N L is the excess with respect to the BPS ground-state N L = δ L . In this case, the momenta read with P 2 L + 2N L = P 2 R − 2 + 2N R such that K µ L = K µ R along the non-compact space-time directions. We should anyway keep in mind that K L = K L −(N L +1)q, and K R = K R −N R q with the tachyonic momenta K L = K R yet K 2 L = 2 = K 2 R . After this longish kinematic preamble, that should clarify issues on DLCQ raised in [37,38,39], for closed-string coherent states, we may proceed writing down vertex operators for the BPS or quasi-BPS BH-like coherent states using DDF operators [35,36].

Vertex operators
Recalling that K L = K L − q is null in D = 10, for BPS 'coherent' states we may choose where the level mathcing imposes N R = 1 2 (P 2 L −P 2 R )+1 and with polarizationsζ α n =λ A n (δ α A −q α K A ), α = 0, ...25 and A = 1, ...24 (bosonic string sector). The operatoratorial structures that appear explicitly read u n n ! n are the cycle index polynomials ... .
Integration over β implements level-matching and ζ µ n = λ i (δ µ i −q µ p i ), P n , S r,s , U as for the R-movers, with∂ X R replaced by ∂ X L .
Whether a coherent state is compact or not depends on the choice of the parameters λ µ n and λ µ n that determine the mass M a and gyration radius R a where theλ µ n are constrained by level-matching. We would like to have R a G N M a , where G N = α g 2 s / V (6) is the 4-d Newton constant and V (6) = V (6) 64π(α ) 3 is the adimensional compactification volume. This means that the parameters λ µ n andλ µ n should satisfy in order to fullfil the requirement on the compactness of the stringy BH's involved in the scattering process in the perturbative regime g s << 1. Among a variety of choices, a possible ansatz for the coherent state polarization is as above (see Figs. 8,9,10) λ µ n = V µ e −αn n β (5. 19) with V µ a (null) vector and α and β two tuneable parameters, the condition (5.18) leads to with Li n (x) the polylog function. In the extremely simple case in which β = 1 one has giving the possibility to tune α and the parameter (or in general the parameters), associated to the distribution of the coherent state harmonics. Notice that, even if this condition can be satisfied, the profound physical reason why the state is compact and behaves like a (small) BH is not completely obvious [28,29,30,31,32,12].

Interactions
In [40], amplitudes with coherent states for open bosonic string interacting with massless vector bosons were shown to expose the expected soft factor at tree level. Very much as for the amplitudes with heterotic string mass eigenstates studied in Section 3, we expect the amplitudes with heterotic string coherent interacting with graviton to expose Weinberg's soft factor and a more involved Shapiro-Virasoro dressing of the 3-point amplitude for the coherent states. To this purpose, we need the (non-vanishing on-shell) 3-point amplitude of coherent states.
For the R-movers (bosonic string) one can borrow the result from the open bosonic string [40], dropping the integrations over X 0 and β one has and The expression drastically simplifies whenζ For the L-movers (superstring), under the assumption that we consider only bosonic excitations over the BPS ground state and modulo some subtleties [41], we can also borrow from the open strings. Dropping the integrations over X 0 and β, the relevant amplitude is As above, there are major simplifications if ζ Combining the two expressions one gets the complete closed-string 3-point amplitude where A L a = A L a e −ıqaX 0 +ıβa (a = 1, 2), ζ n e −ınq 3 X 0 +ınβ 3 and ζ (a) n =ζ (a) n e −ınqaX 0 −ınβa . It is crucial to recall that q a q b = 0 since the q's are indeed all collinear, with only q − a = −1/p + a = 0. Moreover, the two integrations over β 1 and β 2 simply project the R-movers onto the level N R = 1 + 1 2 (P 2 L − P 2 R ) for a = 1, 2 (BPS states) for level-matching. Level-matching for the non-BPS state gives an infinite number of states with N R − (N L − δ L ) = 1 + 1 2 (P 2 L − P 2 R ). It is instructive to study more explicitly the amplitude (5.27) where the arguments of the polynomials are given bȳ One can plug this non-vanishing result into an inelastic heterotic string amplitude, such as the one computed with mass eigenstates in Sect. 3, and obtain the GW profile. We will not perform this laborious analysis here but we expect the result to be similar, since this is largely determined by the Shapiro-Virasoro dressing of the GR result 17 .

Conclusions and final comments
We investigated the α stringy corrections to the GW emitted during the merging of two BPS BHs. For this purpose, we used a toy model whereby small BH's are described by vertex operators in heterotic string or coherent state thereof. This allowed to compute the exact amplitude at tree level (sphere).
As expected, we found that the leading order corrections to the GW signal calculated in GR are of the order (α ) 3 . Although the suppression is cubic in α , which as such would produce a signal decaying too fast with the distance, using the full tree-level scattering amplitude, we find an imprinting in the GW signals due to the infinite tower of massive string resonances that we dub 'string memory'. This string footprint contributes to the falsifiability of this scenario, laying within the sensitivity region of aLIGO/VIRGO and future interferometers.
The effect of string resonances tends to be partly lost due to loop effects that broaden and shift the poles, providing a lost memory effect that can be partly regained in the GW signal both in the merging and in the ring-down phase, the latter governed by QNM [27,15] of the stringy version of the BH [21,22,23,24]. Indeed, we found that GWs can carry information on string resonances and that the signal, being polynomially rather than exponentially decaying in (retarded) time, does actually enable for the search of such string memory effect.
Our present work can be extended in several directions. In particular it would be very interesting to study the case of spinning BHs as well as to refine the analysis in the coherent state description.

Appendix
A Note on 2-body decay kinematics In the soft limit ω = 0, the resulting kinematics is the one of a 2-body decay / production.
In this frame np 3 = −M 3 = −| 3 |, while the other two scalar invariants read while the second is the three-point like coupling of higher spin states 18 with only internal momenta and polarisations [see BF]. Quite remarkably, This is a consequence of the BPS nature of the 3 BH states that requires collinearity of their full 10-dim left 'massless' momenta i.e. K 2 a = p 2 a + P 2 a = 0, since M 2 a = P 2 a = −p 2 a . As a consequence K 1 + K 2 = −K 3 implies K 1 K 2 = 0 and cyclic. But two light-like momenta in any dimension are 'orthogonal' only if they are 'parallel' i.e. collinear. Indeed (assuming both 'in' or both 'out') so that θ 12 = 0 as expected.
Whichever left 'polarisations' A a (dropping the L subscript) one chooses for the 3 BPS BHs, one gets since A a K b = A a K a ρ b,a = 0 due to the BRST condition A a K a = 0 and collinearity, namely K b = ρ b,a K a with ρ b,a some constants such that b =a ρ b,a = −1. Notice the somewhat confusing historical notation for which the first top index m is related to the second bottom index q by 0 ≤ m ≤ q, while the second top index n is related to the first bottom index p by 0 ≤ n ≤ p. It is understood that z = 0 and that a i −b j is not an integer for i = 1, ..n and j = 1, ...m (to avoid double poles in s). By judicious choice of the contour L one can make sense of the integral for any p, q and z with m, n in the allowed ranges 19 .

C Generalized Hypergeometric Functions and Meijer G-function
By analytic continuation, under the conditions of Theorem 3.1 in [86], the generalized hypergeometric function p F q admits a representation as a G-function of the form p F q (a 1 , ..., a p ; b 1 , ..., b q ; z) = after some manipulations the sum can be represented as x (D.10)