String Memories ... openly retold

We identify string corrections to the EM memory effect. Though largely negligible in the low-energy limit, the effect become relevant in high-energy collisions and in extreme events. We illustrate our findings in a simple unoriented bosonic string model. Thanks to the coherent effect of the infinite tower of open string resonances, the corrections are non-perturbative in $\alpha'$, modulated in retarded time and slowly decaying even at large distances from the source. Remarkably compact expressions obtain for special choices of the kinematics in tree-level 4-point amplitudes. We discuss further corrections occurring at higher-points and the exponential damping resulting from broadening and shifting of the massive poles due to loops. Finally we estimate the range of the parameters and masses for detectability in semi-realistic (Type I) contexts and propose a rationale for this string memory effect.


Introduction
Thanks to the universal behaviour of soft photons and gravitons [1], a memory effect is expected to take place both in electro-dynamics [2] and in geometro-dynamics [3].
In a series of papers [4] a triangle of equivalences Soft Behaviour [1,5] -BMS Symmetry [6][7][8] -Memory Effect [9] has been put forward and proposals for experimental tests have been suggested [10]. While gravitational memory implies a distortion of the detector after a GW has passed through [2], EM memory corresponds to a residual velocity (called a kick) for the charged particles of the detector [2].
Since String Theory is a consistent quantum theory of gravity and electro-magnetism, one expects a similar story to be told. In fact, the leading soft behaviour of string scattering amplitudes is the same as in field theory [11]. Since the soft behavior determines the profile of electro-magnetic (EM) and gravitational waves (GW) at large distances from the source, one would naively expect no string corrections to the memory effect 1 . Indeed standard lowenergy expansions of string scattering amplitudes would produce corrections in α that are highly suppressed at large distances and would be totally negligible. However, taking into account the infinite tower of string resonances changes the story completely.
The coherent effect of string resonances, that are crucial for the finiteness and consistency of the theory, play a key role in the non-trivial corrections to the GW profile produced in a 'stringy' BH merger in the heterotic string [14]. Here we argue that a similar phenomenon takes place for EM waves in the Veneziano model a.k.a. open bosonic strings. Actually we consider a variant that requires internal dimensions and Wilson lines [15][16][17] or in modern language D-branes and Ω-planes [18,20,21].
We focus on the simplest non-trivial scattering amplitudes on the disk with insertion of a single photon 2 , which expose a pole in the frequency at ω = 0, reproducing the EM

EM memory and string corrections
In classical electro-dynamics, the EM field produced by a source J µ is given by retarded potential A ret µ . In Fourier space, at large distances from the source In QED, the leading behavior (as k → 0) of an amplitude with a soft photon and n hard particles with charge Q a is dictated by [1] A QED n+1 (a, k; where g is the charge quantum. Stripping off the photon polarisation a µ (k), the amplitude becomes a source for the 'classical' EM potential that at large distances assumes the form with n µ =k µ /ω = (1, x/R). Integrating over ω, the pole at ω = 0 produces a constant shift of A > µ (t, x) at late retarded time u = t − R w.r.t. A < µ (t, x), which is known as 'electromagnetic memory'.
In String Theory (ST), whenever a massless abelian vector boson is present in the spectrum as in Veneziano model, the (transverse) 'current' that sources the EM potential A µ is given by In the low-energy limit, Veneziano amplitude or its generalisations can be expanded in powers of α k·p j . Fourier-transforming in ω back to t would produce string corrections to the retarded potential decaying faster than 1/R, that would be totally negligible at large distances.
On the other hand, for α k·p j ≈ 1, including the contribution of the infinite tower of string resonances turns out to produce a coherent effect that is non-perturbative in α and corrects even the leading 1/R terms. Indeed, starting from and integrating over ω yields contributions from the tower of poles lying on the real axis (at tree level). These form various series in the variables ζ j = exp(−iu/2α n·p j ), that may give rise to detectable signals Contrary to the 'standard' EM memory, which is a DC effect relating the behavior at u > 0 to the one at u < 0, the 'EM string memory' is modulated i.e. depends on u. Quite remarkably ∆ s A µ = θ(u)∆ s A > µ + θ(−u)∆ s A < µ inherits peculiar duality properties from the parent string amplitudes i.e. ∆ s A > µ + ∆ s A < µ = 0. In theories with open unoriented bosonic strings, massless vector bosons are ubiquitous. At tree level (disk) 'color-ordered' amplitudes 5 are cyclic and expose simple poles in channels corresponding to sums of consecutive momenta. In particular soft poles in kp arise as usual when a photon is inserted on a charged leg. This is accompanied by 'massive' poles 1/2α kp+n that are responsible for the string corrections to the EM memory. Complete amplitudes require summing over non-cyclic orderings and expose all the expected soft poles as well as massive ones. Multi-particle channels involving a photon, that give rise to sub-leading corrections to the soft behavior in field theory, produce new towers of massive poles and further string corrections to the EM memory.
Actually, massive string resonances are unstable and acquire finite width and massshifts due to loop effects. As a consequence an exponential damping of the string memory will result [14]. Still, assuming g s <<1, this may be a small effect that can be taken into account in a detailed analysis of the signal, very much as QNM's with Imω = 0 in the Ring-down phase of BH mergers.

Setup
Let us identify a convenient setup with a massless photon and two charged (massive) scalars that allow to illustrate the EM string memory. To this end we compute a 4-point amplitude that admits a closed-form expression for special kinematics. We then extend our analysis to higher points, where new structures appear and briefly address higher loops that mark the onset log corrections.
In the presence of an Ω25-plane, dilaton tadpole cancellation requires the introduction of 2 13 D25-branes, resulting in the gauge group SO(8192) [25,26]. Neglecting closed strings and the coupling to gravity, one can work in a local setting with D3-branes and Ω3-planes well separated from the remaining branes. By T-duality this is equivalent to a configuration of D25-branes with Wilson lines [15][16][17].
More specifically we consider a 4-dimensional configuration ( Fig. 1) with one D3-brane on top of an Ω3-plane, giving rise to an O(1) gauge group (no vector bosons), and one D3brane (together with its image) parallel to the Ω3-plane and separated from it by a distance d in one of the 22 'internal' directions giving rise to a U (1) gauge group with a massless photon like in Maxwell theory 6 . In addition to the massless photon, the low-mass spectrum contains neutral and charged tachyons. Discarding the neutral ones (a 'symmetric' singlet of O(1) and the neutral singlet of U (1)), we have a singly charged scalar φ + and its conjugate φ − , stretching from the U (1) D3-brane to the O(1) D3-brane and vice versa, with mass α M 2 ±1 = −1 + δ 2 (where δ 2 = d 2 /α ) and a doubly charged scalar χ +2 and its conjugate χ −2 , stretching from the U (1) D3-brane to its image and vice versa, with mass α M 2 ±2 = −1 + 4δ 2 . For δ > 1 the 'tachyons' are massive. We safely assume this to be the case and largely neglect the extra dimensions henceforth. The vertex operators for φ + and χ −2 take the standard 'tachyonic' form At tree level (disk) non-vanishing 3-point amplitudes are: √ g s is the open string coupling, and the 'minimal' couplings of φ ± and χ ∓2 to

4-pt amplitude
As a 4-point amplitude with a single photon insertion, one can consider Including the two contributions A Aφφχ and A φφAχ in Fig. 2 yields Setting k = ωn = ω(1, n) and defining the 'scattering lengths' [14] a = 2α np a : 3 a=1 a = 0 , (1.9) thanks to 3 a=1 k·p a = −k 2 = 0, the amplitude can be written as where the function (a generalisation of Veneziano amplitude) has simple poles at ω = 0 as well as at ω 1 = −n 1 −1 and ω 3 = −n 3 −1 and admits an expansionà la Mittag-Leffler (ML) of the form where the ratios (1.13) satisfy λ 3,1 +λ 2,1 =−1 and cyclic. The same applies to H 23 (ω) after 1 ↔ 2 exchange. For the EM wave profile at large distances one has where 2α is an overall, non-zero factor, that can be absorbed into the largely unknown distance R from the source. Anti-Fourier transforming, one finds The pole at ω=0 reproduces the EM memory DC effect. In addition to this, one finds genuine (open) string corrections ∆ s A µ (t, x) to the retarded potential. Adopting some reasonable prescription to deform the integration path, i.e. kp a →kp a −i , one can perform the integral and get series in ζ j = e iu/ j . Note that the effect is non-perturbative in α as j = 2α np j . Assuming φ + (p 1 ) and φ + (p 2 ) to be incoming (p µ =−p µ phys ) and χ −2 (p 3 ) and a(k) to be outgoing (p µ =+p µ phys ), in the physical kinematic region one has 1,2 > 0 and Recall that the dependence on the position is coded in n·p j =−E j (1− n v j ) with n= x/R. The series have finite radii of convergence that may exclude the physical domain |ζ j |=1. Yet they can be summed explicitly for special choices of the kinematics.
In the very-high-energy limit u/| j | << 1, there are two possible regimes: fixed angle for | 1 | ≈ | 2 | ≈ | 3 | and Regge | 1 | << | 2 | ≈ | 3 |. In the former case, as visible in the plots in Fig. 1.2, the real part of the signal gets a constant shift, while the imaginary part  Table 1. Some examples of 'rational' kinematical regimes. has a linear behaviour in u. In the latter case, the real part has a long plateau in u and the imaginary part has a sudden jump and then flattens down. These peculiar features of the stringy signal should allow to discriminate it from the standard EM memory or other field-theory effects.

Generalizations
One can generalise the analysis to higher-points and higher-loops or to more realistic models with open and unoriented strings.
We are interested in the case n A = 1. The photon can be inserted on the U (1) end of any of the n 'tachyon' legs. For a given color-ordering at tree level, one has with z j < z 0 < z j+1 for some j. Integrating over z 0 near z j (z j+1 ) produces a soft pole 1/2α kp j (1/2α kp j+1 ) and a series of 'massive' ones. Additional massive poles come from multi-particle channels involving the photon. These are present in QED and arise from photons inserted on internal lines. No soft poles are exposed for generic choices of the hard momenta since the full amplitude includes insertions of A(k) in between φ + (3) and φ − (4) as well as the exchanges φ + (1) ↔ φ + (4) and φ − (2) ↔ φ − (3). Let us focus on the indicated 'color ordering' [φ + (1), A(k), φ − (2), φ − (3), φ + (4)]. Setting z 4 = ∞, z 3 = 0 and z 1 = 1 with 0 < z A = z < 1 and 0 < z 2 = yz < z (i.e. 0 < y < 1) one has Expanding (1−yz) 2α p 1 p 2 = ∞ N =0 (−2α p 1 p 2 ) N y N z N /N !, the integrals in z and y decouple. Setting 2α = 1 and using momentum conservation one can factor out the function . (2.8) Thanks to F N (k, p a ) ω=0 = 1, the leading behavior at the soft pole ω = 0 is as expected.
In addition there are four infinite sets of simple poles at 9) The first two correspond to photon insertion on an external 'tachyon' leg, as in the case of 4-point amplitudes, the last two to insertion on an 'internal' leg (multi-particle channel) 7 , a novel feature of 5-and higher-point amplitudes. Using the ML expansion, the string corrections to A µ (t, x) at large distance read N (ζ 23 ) + S N (ζ 23 ) and S (14) N (ζ 14 ). In addition one has to include the contributions arising from different 'color orderings'. The double series S (i) N (ζ i ) look hard to write in closed form. Assuming particle 1 and 4 to be incoming and 2 and 3 as well as the photon to be outgoing, the simplest choice of kinematics is taking 1 = implies photon emission perpendicular to the incident beams ( n· p = 0). Then 2 = 3 implies E 2 = E 3 = E − ω 2 and p 2,3 =± q− ω 2 n, with n· q = 0. In this special kinematic regime, setting =2α E one finds ζ 1 =ζ −1 2 =ζ −2 23 =ζ 2 14 = exp(iu/ ). Moreover p 1 p 2 = p 3 p 4 =t, p 1 p 3 = p 2 p 4 =û and p 1 p 4 + ω =ŝ = p 2 p 3 − ω with ω = −n i andŝ+t+û = 0. Even in this simple case, unless one further specialises the remaining 4-point kinematics, the final expressions are not very illuminating and we refrain from displaying them here 8 . Moreover the ratios of the functions S(ζ) to the residual 4-point 'tachyon' amplitude depend onŝ,t andû.

Loops
One-loop annulus and Möbius-strip amplitudes can be written down and analysed in general and in particular in the U (1)×O(1) model. The combinatorics of boundaries is more involved than at the disk level. So let us focus on the planar case, whereby all the insertions are on the same boundary of an annulus. For a given color-ordering, one has where η(τ ) is Dedekind function, G(z) = − log[ϑ 1 (z)/ϑ 1 (0)] − (πz 2 /τ ) is the bosonic propagator on the annulus with τ =iτ 2 . The moduli space includes loci where the annulus degenerates. The UV region τ →0 produces the closed-string tachyon pole (off-shell) and the dilaton tadpole that cancels for SO(8192) or its Wilson-line breakings. When z 0 collides with the adjacent points one gets log-'deformed' poles giving rise to the one-loop corrections to the EM memory and to the string memories. When all but one of the points collide one gets the one-loop corrections to the mass and width of the particle. Other degenerations that start to appear at higher loops produce quantum corrections to lowerpoint amplitudes that require a case-by-case analysis but do not affect our main result in so fa as g s << 1.

Type I models
In more realistic models with open and unoriented superstrings, that allow an embedding of the Standard Model or (supersymmetric) extensions thereof, U (1) gauge bosons are typically anomalous and massive 9 [28]. Yet, massless non-anomalous combinations exist that can play the role of the SM hyper-charge [30]. Mutatis mutandis and barring issues such as moduli stabilisation and supersymmetry breaking, open superstring amplitudes with photons and charged scalars or fermions present the same structure as in our O(1)×U (1) bosonic model and the analysis of the EM String Memory proceeds along the same steps. Different (holographic) scenari are possible [31], whereby QCD is described by some strongly coupled sector ('color branes') and the electro-weak sector is coded in a weakly coupled sector ('flavour branes'). These configurations involve large fluxes and strong warping that lower the effective string tension. Although we expect genuine string corrections to EM memory also in these contexts, at present we cannot support our statement with explicit computations of the relevant amplitudes.
Discussion. While the coherent effect of the infinite tower of string resonances may well give rise to a detectable modulated EM 'memory' we have not given any formal argument why it should happen altogether. Our results do not point towards a DC effect as in the standard memory in gravity and in EM [2,3,9]. Neither they seem to be related to the proposed 'string memory effect' that involve large gauge transformations of the Kalb-Ramond field [13]. Instead the effect we find is oscillatory and it is tempting to conjecture that it be related to the 'global' part of the infinite (but broken) higher spin symmetries of string theory. In particular, redundancies in the definition of the physical states, Ψ ∼ Ψ + Q BRST Λ involved in the scattering process may effect the 'gauge' chosen for the incoming string states w.r.t. the gauge chosen for the outgoing string states. Note that not only states in the first Regge trajectory but also states with mixed symmetry and lower spin admit complicated 'gauge symmetries' that may be exposed in certain regimes [32][33][34].
Moreover, in a given model, only for a special range of the parameters and the masses of the particles involved in the collision the effect may give rise to a measurable signal. A reasonable time-scale, compatible with present detector resolutions, would be ∆t ≈ α E ≈ 10 −15 s = 1f s that in turn would require E ≈ 10 15 GeV , well beyond the present and near future accelerators even for T eV -scale (super)strings with α = E −2 s ≈ 10 −6 GeV −2 [35]. An alternative more promising scenario is a collision of macroscopic objects such as (open) string coherent states [36][37][38] with large mass (and spin) E >> 10 15 GeV . In this case, even for ∆t >> 1f s one could hope to recognise the string imprints in the EM radiation.