Spinning-off stringy electro-magnetic memories
SSpinning-off stringy electro-magnetic memories
A. Aldi, M. Bianchi, M. Firrotta a Dipartimento di Fisica, Universit`a di Roma Tor VergataVia della Ricerca Scientifica 1, 00133, Roma, Italy b INFN sezione di Roma Tor VergataVia della Ricerca Scientifica 1, 00133 Roma, Italy
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We extend and generalise the string corrections to the EM memory to the TypeI superstring including spin effects. Very much as in the simpler bosonic string context, therelevant corrections are non-perturbative in α (cid:48) , slowly decaying (as 1 /R ) at large distancesand modulated in retarded time u = t − R . For spin N states in the first Regge trajectorythey entail a sequence of N derivatives wrt u on the ‘parent’ N = 0 amplitude. We alsobriefly discuss how to include loop effects, that broaden and shift the string resonances, andhow to modify our analysis for macroscopic semi-classical coherent states, whose collisionsmay lead to detectable string memory signals in viable Type I models. a r X i v : . [ h e p - t h ] J a n ontents Introduction
Recently [1] we have shown that even at large distances from a string collision the pro-duced EM wave profile receives corrections wrt the one predicted by QED that can codeinformations on string resonances.The present investigation is a generalisation and extension of the analysis performed in[1] and, for gravitational waves (GWs), in [2]. In particular we will consider spin effects andfermions in the case of open unoriented Type I superstrings. Not too surprisingly we findstring corrections to the ‘spin memory’ effects that are expected to take place for photonsand gravitons [3] as a result of new soft theorems [4], that generalise the well known softtheorems [5].As a consequence of the leading soft theorems the passing of a wave through a detectorleaves a ‘memory’. In the EM case, the memory consists in a kick (remnant velocity) to thecharged particles in the detector[6]. In the gravity case, the memory consists in a permanentdisplacement of the particles in the detector [7]. The new ‘spin memory’ effect [8] impliesthat (massless) particles that counter-rotate with respect to one another acquire a time-delay induced by the flux of angular momentum carried by GWs through the orbits. Forcharged particles a similar effect takes place. A deep connection between BMSvB symmetry[9–11], old and new soft theorems [4, 5] and old and new Memory Effects [6, 8] has beenestablished in [3]. Experimental tests have been proposed, too [12].String Theory predicts a similar soft behaviour [13]. In the standard low-energy ex-pansions in α (cid:48) , string corrections are suppressed and become almost negligible at large– 1 –istances, where the EM and gravitational waves (GWs) seem to retain their ‘classical’profiles, leaving no obvious room for any string memory . Yet taking into account theinfinite tower of string resonances drastically changes the situation.In the heterotic string, for instance, a toy model for ‘BH merger’ was studied in [2]whereby the large masses of the colliding (BPS) objects leads to a regime with non negligiblecorrections to the GWs produced in the process. An analogous phenomenon for EM waveswas shown to take place in the (unoriented) open bosonic string, a variant of Venezianomodel with internal dimensions and Wilson lines [14–16], that can be more intuitivelydescribed in terms of D-branes and Ω-planes [17–20].For single photon insertions the relevant amplitudes display the familiar soft pole at ω = 0, that leads to the known EM memory effect, plus an infinite tower of simple poleson the real axis, associated to the string resonances. The latter lead to significant stringcorrections to the EM wave profile even at large distance from the collision R >> L thatgive rise to power series in the variable(s) ζ = exp( − iu/ α (cid:48) n · p ), where u = t − R is theretarded time, n = (1 , (cid:126)x/R ) and p is the 4-momentum of one of the charged particlesinvolved in the process. This reveals the non-perturbative nature of the effect in α (cid:48) , thatis completely hidden in the usual low-energy expansion.At low energy i.e. for u >> α (cid:48) n · p , due to destructive interference the effect is washedout. Yet, collisions of cosmic strings or BH mergers may produce a detectable signal for u ≈ α (cid:48) n · p , i.e. at high energy. The same happens in chiral Type I models [21], where morereliable order of magnitude estimates can be made for phenomenological purposes.Very much as in General Relativity and Electro-dynamics, sub-leading terms give riseto ‘spin memory’ effects that we address in the present investigation starting from low butnon-zero spin and passing later on to consider higher-spins which are the hallmark of stringtheory.Plan of the paper is as follows.In Section 1 we briefly review the EM memory effect both at leading and sub-leadingorder and describe string corrections thereof.In Section 2 we generalize the results of [1] for spin-zero colliding objects in the openunoriented bosonic string to colliding fermions and bosons in the Type I superstring.In Section 3 we present new results for the bosonic string and Type I superstring inthe case with higher spins and analyse the modification due to spin effects.In Section 4 we discuss higher-point amplitudes and briefly mention higher-loop con-tributions.In Section 5, after estimating the order of magnitude of the colliding objects for theeffect to be detectable in phenomenologically viable models such as Type I superstringswith large or warped extra dimensions and TeV scale tension, we discuss how macroscopicsemi-classical coherent states can meet the required scales of mass and spin.In Section 6 we draw our conclusions and discuss some open issues. A different version of the ‘string memory effect’ was studied by [22] that is related to large gaugetransformations of the Kalb-Ramond field B MN . Multi-photon emission would encode non-linear aspects of DBI action, as suggested by Cobi Sonnen-schein. – 2 –
EM memory and string corrections
At large distances from the source R = | (cid:126)x | >> | (cid:126)x (cid:48) | ≈ L , the retarded potential A ret µ producedby an EM current J µ in Fourier space is given by (cid:101) A µ ( ω, (cid:126)x ) = (cid:90) d x (cid:48) e iω | (cid:126)x − (cid:126)x (cid:48) | π | (cid:126)x − (cid:126)x (cid:48) | (cid:101) J µ ( ω, (cid:126)x (cid:48) ) ≈ e iωR πR (cid:98) J µ ( ω, (cid:126)k = ω(cid:126)n ) . (1.1)where (cid:126)n = (cid:126)x/R is the unit vector in the direction of the observer. In QED the ‘source’ canbe any collision, described by a ‘stripped’ amplitude (cid:98) J µ ( k ; p j ) = δ A n +1 ( a, k ; p j ) δa µ ( k ) . (1.2)where k = ω (1 , (cid:126)n ) and a µ ( k ) is the photon polarisation. In the soft limit k → A QEDn +1 ( a, k ; p j ) = g n (cid:88) j =1 Q j a · p j + a · J j · kk · p j A QEDn ( p j ) + ... (1.3)with g the charge quantum and Q j the charges of the ‘hard’ particles, with momenta p j and angular momenta J µνj = p µj ∂ νp j − p νj ∂ µp j + S µν that act on the remaining hard amplitude.Absorbing the ω independent (but non-zero) factor A QEDn ( p j ) into a redefinition of R andwith the understanding that J must be replaced by some average spin one has (cid:101) A µ ( ω, (cid:126)x ) = g e iωR ωR (cid:88) j Q j p µj + J µνj k ν np j + ... , (1.4)where n µ = k µ /ω = (1 , (cid:126)n ). After Fourier Tranform (FT) in ω , the soft pole at ω = 0produces a constant shift of A >µ ( t, (cid:126)x ) at u = t − R > A <µ ( t, (cid:126)x ) at u = t − R < A µ receives corrections in powers of α (cid:48) k · p j . The FT in ω produces terms that vanish faster than1 /R at large R and that are thus totally negligible at macroscopic distances. Yet as arguedin the introduction, when the masses of the colliding objects are such that α (cid:48) k · p j ≈
1, theinfinite tower of string resonances produces sizeable corrections even to the leading 1 /R terms (cid:101) A µ ( ω, (cid:126)x ) = (cid:90) d x (cid:48) e iω | (cid:126)x − (cid:126)x (cid:48) | π | (cid:126)x − (cid:126)x (cid:48) | (cid:101) J µ ( ω, (cid:126)x (cid:48) ; p j ) ≈ e iωR πR δ A STn +1 ( a, k ; p j ) δa µ ( k ) (cid:12)(cid:12)(cid:12) k µ = ω (1 ,(cid:126)n ) . (1.5) The notation (cid:101) G ( ω, (cid:126)x ) is for Fourier transform w.r.t. t , while (cid:98) G ( ω, (cid:126)k ) is for full transform, also w.r.t. (cid:126)x . – 3 –he corrections are non-perturbative in α (cid:48) , since they form series in ζ j = exp( − iu/ α (cid:48) n · p j ) (1.6)that modulates in u the ‘EM string memory’ given by A STµ ( t, (cid:126)x ) = A QEDµ ( t, (cid:126)x ) + ∆ s A µ ( t, (cid:126)x ) . (1.7)As shown in [1], the unoriented open bosonic string corrections ∆ s A µ = θ ( u )∆ s A >µ + θ ( − u )∆ s A <µ satisfy peculiar duality properties ∆ s A >µ + ∆ s A <µ = 0, that reflect planarduality of disk amplitudes. We will confirm the universal nature of this result for Type Isuperstrings and spinning states.For simplicity, we only consider open-string insertions on the boundary. At treelevel (disk) ‘color-ordered’ amplitudes display soft poles in kp for charged ‘hard’ legs withmomentum p adjacent to the soft photon leg with momentum k . String ‘massive’ poles onthe real axis at 2 α (cid:48) kp = − n are responsible for the string corrections to the EM memory.Due to quantum loop effects the string resonances acquire a finite width and a mass-shift that are perturbative in g s and lead to an exponential damping in u of the stringmemory [2]. To a large extent the unstable string resonances play the role of quasi-normalmodes (QNM’s) with Im ω (cid:54) = 0 of the produced ‘remnant’, an unstable (non-BPS) massive,possibly spinning, state. As shown by Green and Schwarz [25], open and unoriented superstrings in D = 10 areconsistent only for the choice SO (32) of the Chan-Paton group . In lower dimensions,as originally shown in [14–16] and later on refined in [26, 27], the gauge group can bebroken and the rank can be reduced. Recently new constraints from co-bordism havebeen identified [28]. Wilson lines in the original D9-brane description are T-dual to branedisplacements. Here we work in a local setting with D3-branes and Ω3-planes, equivalentto D9-branes with Wilson lines [14–16] and ignore global consistency conditions that canbe fulfilled by adding well-separated D-branes and/or (discrete) fluxes. More specificallywe consider a 4-dimensional Type I-like configuration, shown in Fig. 1, with one D3-braneon top of an (cid:102) Ω3 − -plane, giving rise to an O (1) gauge group, and one D3-brane parallel tobut separated from the (cid:102) Ω3 − -plane, giving rise to a U (1) gauge group.In order to expose the string corrections to the EM memory effect, we consider a4-point amplitude with a single massless U (1) photon a µ , two charged (massive) gaugini λ (+1) , with ends on the U (1) D3-brane and on the O (1) D3-brane, mass α (cid:48) M ± = δ (where Closed-string insertions on the bulk of the disk or the projective plane have been carefully reconsideredin [29]. Non-perturbative corrections due to stringy instantons, see for instance [30] can be considered that areexpected to be largely suppressed for g s << For unoriented open and closed bosonic strings with Ω25-plane, dilaton tadpole cancellation requires2 D25-branes, leading to the gauge group SO (8192) [31, 32]. – 4 – igure 1 . Representative picture of the Ω3 /D δ = d /α (cid:48) ) and one doubly-charged scalar, from the U (1) D3-brane to its image, with mass α (cid:48) M ± = 4 δ .The vertex operators for the U (1) gauge boson A µ in the 0-picture reads as usual V (0) A = a · ( i∂X + 2 α (cid:48) Ψ k · Ψ) e ik · X (2.1)with k = 0 and k · a ( k ) = 0. For the singly-charged gaugini in the canonical − / V λ = √ α (cid:48) λ Aα S α Σ A e − ϕ/ e iK · X (2.2)with K = ( p µ ; ± (cid:126)d/α (cid:48) ), S α ( α = 1 ,
2) a chiral SO (1 ,
3) spin field and Σ A ( A = 1 , ...
4) achiral SO (6) spin field. For the doubly-charged BPS scalar χ − in the canonical − V χ = √ α (cid:48) χ i Ψ i e − ϕ e iK · X (2.3)with K = ( p µ ; ± (cid:126)d/α (cid:48) ) and i = 1 , ... (cid:126)d · (cid:126)χ = 0.In addition to the ‘minimal’ couplings of λ ± and χ ∓ to the photon, one has thegauge-invariant Yukawa couplings A λ ± λ ± χ ∓ = g op √ α (cid:48) ε αβ χ AB (3) λ Aα (1) λ Bβ (2) (2.4)where χ AB = Γ iAB χ i and g op = √ g s is the open string coupling.This Yukawa coupling is the dimensional reduction of the minimal coupling in D = 10 A ΛΛ A = g op f rst A M Γ M ( ab ) Λ a (1)Λ b (2) (2.5)which is zero for collinear momenta K ∼ K ∼ K since 16-components commuting chiralspinors in D = 10 with collinear momenta u a ( K ) ∼ u a ( K ) can be chosen to satisfy u a ( K )Γ Mab u b ( K ) = K M and A M K M = 0.A non-vanishing 3-point amplitude requires one non-BPS state at least. However,replacing χ ± with H ± , at the first massive level, whose emission vertex reads V H = H ij Ψ i ∂X j e − ϕ e iK · X (2.6)– 5 –ith α (cid:48) K = − i.e. α (cid:48) m H = 1 + 4 | (cid:126)d | α (cid:48) and (cid:126)d i H ij = 0 = δ ij H ij would not work! Therelevant 3-point amplitude A λ ± λ ± H ∓ = g op √ α (cid:48) ε αβ λ Aα (1)Γ iAB λ Bβ (2) H ij (3)( d j − d j ) (2.7)vanishes since (cid:126)d , (cid:126)d and (cid:126)d are collinear ( (cid:126)d = (cid:126)d = (cid:126)d and (cid:126)d = − (cid:126)d − (cid:126)d = − (cid:126)d ), so that (cid:126)d − (cid:126)d = 0. The simplest way around, at the same mass level, is to consider V C = √ α (cid:48) C ijk Ψ i Ψ j Ψ k e − ϕ e iK · X (2.8)with α (cid:48) m C = 1 + 4 | (cid:126)d | α (cid:48) and d i C ijk = 0. The relevant 3-point amplitude is A λ ± λ ± C ∓ = g op √ α (cid:48) ε αβ λ Aα (1)Γ ijkAB λ Bβ (2) C ijk (3) (2.9)which is non-zero even if (cid:126)d = (cid:126)d = (cid:126)d and (cid:126)d = − (cid:126)d . Let us then consider the 4-point amplitude with a single photon insertion, two massivesingly charged BPS gaugini λ + and a massive non BPS doubly-charged scalar C − viz. A = g op C D (cid:89) j =0 (cid:90) dz j V CKG (cid:104)V a ( k, z ) V λ + ( K , z ) V λ + ( K , z ) V C − ( K , z ) (cid:105) (2.10)where C D = ( g op α (cid:48) ) − . The two contributions A Aλ + λ + C − and A λ + λ + AC − are shown inFig. 2. Figure 2 . Amplitude with one U (1) photon, two massive BPS gaugini λ and a massive non BPSscalar C . The final result reads A = g op √ α (cid:48) λ Aα (1)Γ ijkAB λ Bβ (2) C ijk (3) (cid:26)(cid:18) a · p ε αβ + a · σ αβ · kk · p − a · p ε αβ k · p (cid:19) × (2.11)Γ(2 α (cid:48) k · p + 1)Γ(2 α (cid:48) k · p + 1)Γ(1 − α (cid:48) k · p ) + (1 ↔ (cid:27) – 6 –here σ αβµν is the Lorentz generator for chiral spinors. It is convenient to set k = ωn = ω (1 , (cid:126)n ) and define the ‘scattering lengths’ [2] (cid:96) a = 2 α (cid:48) np a : (cid:88) a =1 (cid:96) a = 0 , (2.12)since (cid:80) a =1 k · p a = − k = 0, by momentum conservation. Then one can write A = g op √ α (cid:48) λ Aα (1)Γ ijkAB λ Bβ (2) C ijk (3) (cid:26)(cid:18) a · p ε αβ + ωa · σ αβ · n(cid:96) − a · p ε αβ (cid:96) (cid:19) H ( ω )+(1 ↔ (cid:27) (2.13)where the analytic function H ( ω ) = 1 ω Γ(1+ ω(cid:96) )Γ(1+ ω(cid:96) )Γ(1 − ω(cid:96) ) (2.14)has simple poles at ω = 0 as well as at ω(cid:96) = − n − ω(cid:96) = − n −
1. The Mittag-Leffler (ML) expansion reads H = 1 ω + ∞ (cid:88) n =1 ( − n (cid:96) n !( ω(cid:96) + n ) Γ(1 − n λ , )Γ(1 + n λ , ) + ∞ (cid:88) n =1 ( − n (cid:96) n !( ω(cid:96) + n ) Γ(1 − n λ , )Γ(1 + n λ , ) (2.15)where λ b,a = (cid:96) b (cid:96) a = n · p b n · p a = k · p b k · p a . (2.16)such that λ , + λ , = −
1. One gets H ( ω ) from H ( ω ) after 1 ↔ ω H = H spin13 =1 − ∞ (cid:88) n =1 ( − n ( n − ω(cid:96) + n ) Γ(1 − n λ , )Γ(1+ n λ , ) − ∞ (cid:88) n =1 ( − n ( n − ω(cid:96) + n ) Γ(1 − n λ , )Γ(1+ n λ , )(2.17)that has no soft pole anyway.At large distances R >> L one has (cid:101) A µ ( ω, (cid:126)x ) = g op e iωR πR (cid:88) j Q j n · p j F µj ( ω, (cid:126)n ; p j ) (cid:98) A ( p j ) (2.18)where F µ = H p µ ε αβ + H spin ( n · σ ) µαβ , F µ = H p µ ε αβ + H spin ( n · σ ) µαβ , F µ = 12 p µ ε αβ ( H + H )(2.19)Note that the ‘spin’ terms J µνa = L µνa + S µνa act on the (non-zero) 3-point amplitude (cid:98) A αβ ofEq. (2.9) and some spin-average is understood in the unpolarised case. With this provisowe are ready to perform the FT in ω to the time coordinate viz. A µ ( t, (cid:126)x ) = g op πR (cid:88) j Q j n · p j (cid:90) + ∞−∞ dω π e − iωu F µj ( ω, (cid:96) j ) . (2.20)– 7 –he soft pole at ω =0 EM is responsible for the EM memory (DC effect). The constantterm in ω produces the spin correction to the EM memory effect [8, 23], already discussedabove, whereby the Lorentz generators act on the ‘hard’ 3-point (non-zero unpolarised)amplitude. The infinite tower of poles on the real axis generate the superstring corrections∆ s A µ ( t, (cid:126)x ).Deforming the integration path from the real line, i.e. kp a → kp a − i(cid:15) , as required bycausality, and integrating in ω yield infinite series in the variables ζ j = e iu/(cid:96) j , with (cid:96) j =2 α (cid:48) np j , that expose the non-perturbative nature of the effect in α (cid:48) . As already mentionedbefore, in the usual low-energy expansion in powers of α (cid:48) this effect would be completelyhidden.Assuming the two BPS gaugini to be incoming λ + ( p ) and λ + ( p ) ( p µ = − p µ phys ) andthe photon a ( k ) as well as the non-BPS scalar C − ( p ) to be outgoing ( p µ =+ p µ phys ), onehas (cid:96) , > (cid:96) < u = t − R > ( > ) s A µ ( t, (cid:126)x ) = − g πR (cid:40) n · p ∞ (cid:88) n =1 ( − ) n n ! Γ(1 − n λ , )Γ(1+ n λ , ) ( p µ − n (cid:96) nJ µ ) e in u/(cid:96) + 1 n · p ∞ (cid:88) n =1 ( − ) n n ! Γ(1 − n λ , )Γ(1+ n λ , ) ( p µ − n (cid:96) nJ µ ) e in u/(cid:96) − p µ n · p ∞ (cid:88) n =1 ( − ) n n ! (cid:18) Γ(1 − n λ , )Γ(1+ n λ , ) + Γ(1 − n λ , )Γ(1+ n λ , ) (cid:19) e in u/(cid:96) (cid:41) (2.21)where nJ µj = n ν J νµj , while at u = t − R < ( < ) s A µ ( t, (cid:126)x ) = g πR (cid:40) n · p ∞ (cid:88) n =1 ( − ) n n ! Γ(1 − n λ , )Γ(1+ n λ , ) ( p µ − n (cid:96) nJ µ ) e in u/(cid:96) + 1 n · p ∞ (cid:88) n =1 ( − ) n n ! Γ(1 − n λ , )Γ(1+ n λ , ) ( p µ − n (cid:96) nJ µ ) e in u/(cid:96) (2.22) − p µ n · p (cid:32) ∞ (cid:88) n =1 ( − ) n n ! Γ(1 − n λ , )Γ(1+ n λ , ) e in u/(cid:96) + ∞ (cid:88) n =1 ( − ) n n ! Γ(1 − n λ , )Γ(1+ n λ , ) e in u/(cid:96) (cid:33) (cid:41) The dependence on the position of the observer is coded in (cid:126)n = (cid:126)x/R , that appears in n · p j = − E j (1 − (cid:126)n(cid:126)v j ) both in the denominators and in the exponents. Before trying andsumming the above expressions, let us stress that in general they give contributions thatare of the same order of magnitude as the ‘standard’ QED memories, although they aremodulated in u . Anyway they decay as 1 /R and their ‘intensity’ I ∼ | A | ≈ O (1), in thatthere is no further α (cid:48) suppression if u ≈ (cid:96) i . The infinite series we found have finite radii of convergence and the physical values | ζ j | =1may be outside the domain (circle) of analyticity. However, the sum is possible in closed– 8 –orm for special kinematical regimes and analytic continuation can be explicitly carriedout. Working in the CoM frame for clarity, one has (cid:126)p = (cid:126)p = − (cid:126)p , (cid:126)p = − (cid:126)k = − ω(cid:126)n , so that E = E phys3 = (cid:113) M + ω , E , = − E phys1 , = (cid:113) M , + | (cid:126)p | (2.23)with E phys1 = (cid:102) M + M − M (cid:102) M , E phys2 = (cid:102) M + M − M (cid:102) M , | (cid:126)p | = (cid:113) F ( M , M , (cid:102) M )2 (cid:102) M , (2.24)where (cid:102) M = E + ω and F ( x, y, z ) = x + y + z − xy − yz − zx is the ubiquitous ‘fakesquare’. Setting µ = M (cid:102) M , µ = M (cid:102) M and cos θ = (cid:126)x · (cid:126)pR | (cid:126)p | (2.25) F ( M , M , (cid:102) M ) turns out to be positive when 0 <µ , µ < µ − µ ) − µ + µ )+1 > µ = µ = µ = M / (cid:102) M and λ , = −
12 + 12 cos θ (cid:112) − µ , λ , = − −
12 cos θ (cid:112) − µ , (2.26)For instance, θ = π/ i.e. cos θ = 0 corresponds to (cid:96) = (cid:96) = − (cid:96) so that λ , ≡ λ , = − , λ , ≡ λ , = − , λ , ≡ λ , = 1 . (2.27)Using this particular choice of λ a,b in (2.21) and (2.22), for u > ( > ) s A µ ( t, (cid:126)x ) = − g πR (cid:18) p µ + i nJ µ ∂ u n · p + p µ + i nJ µ ∂ u n · p − p µ n · p (cid:19) ∞ (cid:88) n =1 ( − ) n n ! Γ(1+ n )Γ(1 − n ) e in u/(cid:96) = g πR (cid:18) p µ + i nJ µ ∂ u n · p + p µ + i nJ µ ∂ u n · p − p µ n · p (cid:19) e iu/(cid:96) (cid:112) e iu/(cid:96) (2.28)with n · p j = − E j (cid:16) − (cid:126)x(cid:126)v j πR (cid:17) , while for u < ( < ) s A µ ( t, (cid:126)x ) = g πR (cid:18) p µ + i nJ µ ∂ u n · p + p µ + i nJ µ ∂ u n · p − p µ n · p (cid:19) ∞ (cid:88) n =1 ( − ) n n ! Γ(1+2 n )Γ(1+ n ) e in u/(cid:96) = g πR (cid:18) p µ + i nJ µ ∂ u n · p + p µ + i nJ µ ∂ u n · p − p µ n · p (cid:19) (cid:32) (cid:112) e iu/(cid:96) − (cid:33) (2.29)Notice that although the series have finite radii of convergence ( | ζ | < | ζ | < / ζ = ± i and ζ , = ± i/
4. In fact planar duality entails ∆ ( > ) s A µ + ∆ ( < ) s A µ = 0 very much– 9 –s in [1]. The main difference with respect to the latter is the presence of extra terms dueto the spin of the two gaugini. This affects both the ‘standard’ EM memory, includingthe spin memory, and the string corrections thereof in the form of single derivatives wrt to u ∼ log ζ . We will see later on that higher-spin states produce higher derivatives wrt to u .The main difference wrt to the EM memory is the modulation in u with periods set by (cid:96) j .It is worth recalling that in the case of closed (heterotic) strings [2], analytic continuationin the same kinematical regime as above produces log terms that are not periodic in u . The series encoding the string corrections to the EM memory can be summed for other‘rational’ values of the kinematical variables λ a,b .In Table 1, we list a set of ‘rational’ kinematical values used to plot the real andimaginary parts of the string corrections to the EM wave profile at fixed large R as afunction of u/(cid:96) in Fig. 3. Clearly in order for the string effect to be measurable one needsa time resolution ∆ t ≈ (cid:96) ∼ α (cid:48) E (1 − (cid:126)n · (cid:126)v ). λ λ λ λ λ λ -1/2 -1/2 -2 1 1 -2-1/3 -2/3 -3 2 1/2 -3/2-1/4 -3/4 -4 3 1/3 -4/3-1/5 -4/5 -5 4 1/4 -5/4-2/3 -1/3 -3/2 1/2 2 -3-3/4 -1/4 -4/3 1/3 3 -4 Table 1 . Some examples of ‘rational’ kinematical regimes.
For instance taking the value λ , = − / λ , = − /
3, Eq.(2.21) yields∆ ( > ) s A µ ( t, (cid:126)x ) = g πR (cid:40) (cid:20) ( p µ + inJ µ ∂ u ) n · p − p µ n · p (cid:21) (cid:34) (cid:18)
1+ 427 e iu(cid:96) (cid:19) − / (cid:32) / (cid:32) (cid:114)
1+ 427 e iu(cid:96) (cid:33) / + 2 / e iu(cid:96) (cid:32) (cid:114)
1+ 427 e iu(cid:96) (cid:33) − / (cid:33)(cid:35) + (cid:20) ( p µ + inJ µ ∂ u ) n · p − p µ n · p (cid:21)(cid:34) (cid:18) − e iu(cid:96) (cid:19) − / (cid:32) / e iu(cid:96) (cid:32) (cid:114) − e iu(cid:96) (cid:33) − / − / (cid:32) (cid:114) − e iu(cid:96) (cid:33) / (cid:33)(cid:35)(cid:41) (2.30)– 10 – igure 3 . Real and imaginary part of the Type I superstring correction to the EM memory forsome rational kinematical values, with spin in two opposite directions ( nJ > nJ < while Eq. (2.22) yields∆ ( < ) s A µ ( t, (cid:126)x ) = g πR (cid:40) (cid:20) ( p µ + inJ µ ∂ u ) n · p − p µ n · p (cid:21) (cid:20) (cid:32)(cid:113) e − iu(cid:96) (cid:33) − cosh (cid:32)
13 sinh − (cid:32) √ e − iu (cid:96) (cid:33)(cid:33) − (cid:21) + (cid:20) ( p µ + inJ µ ∂ u ) n · p − p µ n · p (cid:21) (cid:20) (cid:32)(cid:113) − e − iu(cid:96) (cid:33) − (cid:20) cos (cid:18)
16 cos − (cid:18) − e − iu(cid:96) (cid:19)(cid:19) − √ (cid:32)
13 sin − (cid:32) √ e − iu (cid:96) (cid:33)(cid:33) (cid:21) − (cid:21)(cid:41) . (2.31)Otherwise starting for instance with the values λ , = − / λ , = − /
4, the resultfor Eq.(2.21) is the following∆ ( > ) s A µ ( t, (cid:126)x ) = g πR e iu/(cid:96) (cid:40) (cid:20) p µ + inJ µ ∂ u ) n · p + ( p µ + inJ µ ∂ u ) n · p − p µ n · p (cid:21)(cid:34) F (cid:18) , , ,
34 ; − e iu(cid:96) (cid:19) + 52 e iu(cid:96) F (cid:18) , , ,
32 ; − e iu(cid:96) (cid:19) (cid:35) + − e iu(cid:96) (cid:20) p µ + inJ µ ∂ u ) n · p + ( p µ + inJ µ ∂ u ) n · p − p µ n · p (cid:21) (cid:34) F (cid:18) , ,
76 ; 34 ,
54 ; − e iu(cid:96) (cid:19) (cid:35)(cid:41) (2.32)– 11 –hile for Eq. (2.22) the result reads∆ ( < ) s A µ ( t, (cid:126)x ) = g πR (cid:26) (cid:20) ( p µ + inJ µ ∂ u ) n · p − p µ n · p (cid:21) (cid:20) F (cid:18) , ,
34 ; 13 ,
23 ; − e − iu(cid:96) (cid:19) − (cid:21) + 43 (cid:20) ( p µ + inJ µ ∂ u ) n · p − p µ n · p (cid:21) (cid:20) e − iu (cid:96) F (cid:18) , , ,
53 ; − e − iu(cid:96) (cid:19) − e iu (cid:96) F (cid:18) , , ,
43 ; − e − iu(cid:96) (cid:19) − (cid:21)(cid:27) . (2.33)In addition to the special ‘rational’ kinematical regimes, it is interesting to consider theultra-high-energy limit | u | << | (cid:96) j | . Two possible regimes can take place: fixed angle | (cid:96) | ≈| (cid:96) | ≈ | (cid:96) | and Regge | (cid:96) | << | (cid:96) | ≈ | (cid:96) | . We have already discussed these regimes in [1, 2],to which we refer the interested reader for the detailed analysis. Nothing new and specialhappens in the Type I superstring case wrt the unoriented open bosonic string since themain character in the play is the scalar amplitude of the Veneziano model.One can generalise the analysis in various other directions. We will consider in turn theinclusion of spinning states, higher-points and higher-loops. More realistic (chiral) modelswith open and unoriented strings will be addressed later on, together with macroscopicsemi-classical coherent states that may have the desired mass, spin and compactness forthe effect to be detectable. In order to investigate the effect of higher spins in the simplest possible context, we moveback to where we started from: unoriented open bosonic string. We focus on the U (1) × O (1)model, built in [1], that mutatis mutandis coincides with the Type I model in the previoussection. We can consider an amplitude with one U (1) photon, two singly-charged ‘tachyons’ φ + and a doubly-charged higher-spin state H − N in the first Regge trajectory. Later on wewill sketch how the analysis needs to be modified for the superstring.The relevant open bosonic string vertex operators are V a ( k ) = a · i∂Xe ik · X , V φ ( K ) = √ α (cid:48) e iK · X V H = √ α (cid:48) − N i N H µ µ ...µ N ∂X µ ∂X µ ...∂X µ N e iK · X (3.1)Henceforth we set H N = ζ ⊗ N for convenience. The disk amplitude A = g op C D (cid:89) j =0 (cid:90) dz j V CKG (cid:10) V a ( k, z ) V φ + ( K , z ) V φ + ( K , z ) V H − ( K , z ) (cid:11) (3.2)involves two possible colour-ordered contributions: A φφAH and A AφφH . Focussing on theformer, putting 2 α (cid:48) = 1, and performing the Wick contractions yield A = g op C D (cid:90) z z dz (cid:89) i 1) + he x (cid:0) √ e x − e x (cid:1)(cid:3)(cid:0) e x + √ e x (cid:1) − h (cid:33) (3.6)Note that the presence of (cid:96) = √ α (cid:48) n · p in the denominator should not be worrisome.String corrections to the memory effects are visible in the high-energy limit √ α (cid:48) n · p ≈ ω(cid:96) + 1 = − n as well as for the pole at 1 + ω(cid:96) = − n .The structure of the other contributions to the amplitude is similar. The lowest termswith h = 1 reproduce the straight-forward generalisation of the spin memory, upon anti-symmetrization of a and n = k/ω . Higher derivative terms in u up to N th order are peculiarto string theory and to the specific higher-spin state considered (first Regge trajectory).The overall effect is quite dramatic as visible in Fig.4, where the EM wave profiles areplotted for different values of N and n · ζ .One simple extension to the above analysis is to consider higher-spin states in the firstRegge trajectory for Type I superstring compactified on tori [33]. In the N-S sector therelevant vertex operator is V NSH S = √ α (cid:48) − N H µ µ ...µ N e − ϕ ψ µ ∂X µ ...∂X µ N e iP X (3.7)with P µ H µ µ ...µ N = 0 and η µν H µ µ ...µ N = 0. In the R sector the relevant vertex operatoris V R Λ S = √ α (cid:48) − N Λ αAµ µ ...µ N e − ϕ/ S α Σ A ∂X µ ...∂X µ N e iP X (3.8)with ( P µ σ µα ˙ α + ... )Λ αAµ µ ...µ N = 0 and η µν Λ µ µ ...µ N = 0.– 13 – ul - - - ( δ ) λ =- / 2, n · ζ > = = = = ul - ( δ ) λ =- / 2, n · ζ > ul - - - ( δ ) λ =- / 2, n · ζ < = = = = ul - ( δ ) λ =- / 2, n · ζ < ul - - ( δ ) λ =- / 4, n · ζ > = = = = ul - - ( δ ) λ =- / 4, n · ζ < ul - - ( δ ) λ =- / 4, n · ζ < = = = = ul - - ( δ ) λ =- / 4, n · ζ > Figure 4 . Real and imaginary part of the string corrections to the EM spin memory for λ = − / , − / n · ζ (cid:39) − . For each choiceof λ there are four corresponding profiles that differ for the spin values N = 2 , , , 5, respectively. The computation of the amplitude with a single photon insertion, two massive BPSgaugini and a bosonic higher-spin state (with an internal ψ i replacing ψ µ , for simplicity)is straightforward, though tedious. The result is largely fixed by Lorentz invariance and– 14 –-symmetry. As in the bosonic string case, it involves two possible colour-ordered con-tributions: A λλAH and A AλλH . Focussing on the former, setting H (cid:48) N = χ i ⊗ ζ ⊗ N , putting2 α (cid:48) = 1 and performing the Wick contractions yields A λλAH = g op C D χ AB (3) λ αA (1) λ βB (2) (cid:90) z z dz (cid:89) i 1) one finds A c . o . [1 + , a ( k ) , − , + , − ] = g op (cid:90) dz (cid:90) dy (cid:18) δ CA δ DB − yz + δ DA δ CB yz (cid:19) (4.8) (cid:34) ε αβ ¯ ε ˙ α ˙ β (cid:18) a · p − z − a · p z (1 − y ) − a · p z (cid:19) + (cid:32) σ µναβ ¯ ε ˙ α ˙ β z (1 − y ) − ε αβ ¯ σ µν ˙ α ˙ β z (1 − z ) (cid:33)(cid:35) × (1 − z ) kp (1 − yz ) P P (1 − y ) kp z ( kp + kp + P P )+1 y P P . Notice the appearance of the spin operators acting on the spin 1/2 fermions. Using(1 − yz ) P P − r = ∞ (cid:88) N =0 ( r − P P ) N N ! y N z N (4.9) Since N = 4 multiplets (both massless and massive) are non-chiral, gaugini with same charge appearwith both chiralities. With the chosen charge assignment K λ + = ( p µ , (cid:126)d ) while K ¯ λ − = ( p µ , − (cid:126)d ). – 16 –ith r = 0 , z and y independently and get combinations of theform F N ( k, p j ) = Γ( P P + k ( p + p )+ N +1)Γ( kp +1)Γ( P P − kp + N +1) Γ( P P + k ( p + p )+ N +1)Γ( kp +1)Γ( P P − kp + N +1) (4.10)where we set 2 α (cid:48) = 1 for simplicity. Note that P , P , = p , p , − | (cid:126)d | while P P = p p + | (cid:126)d | P P = p p + | (cid:126)d | (4.11)Plugging into the (color-ordered) amplitude one finds A c . o . [1 + , a ( k ) , − , + , − ] = g op ∞ (cid:88) N =0 ( r − P P − k ( p + p )) N N ! F N ( k, p a ) × (cid:20) ε αβ a · ( p ¯ ε ˙ α ˙ β + k · J )( P P + k ( p + p )+ N +1) kp ( P P − kp + N +1)( P P − kp + N +1) (cid:0) δ r, δ CA δ DB + δ r, δ DA δ CB U N − ,N − (cid:1) − ¯ ε ˙ α ˙ β a · ( p ε αβ + k · J ) kp ( P P − kp + N +1) (cid:0) δ r, δ CA δ DB + δ r, δ DA δ CB V N − ,N − (cid:1) − ε αβ a · ( p ¯ ε ˙ α ˙ β + k · J )( P P − kp + N +1)( P P − kp + N +1) (cid:0) δ r, δ CA δ DB + δ r, δ DA δ CB Z N − ,N − (cid:1) (cid:21) . (4.12)where U N − ,N − , V N − ,N − , Z N − ,N − are given by U N − ,N − = ( P P − kp + N +1)( P P − kp + N +1)( P P + k ( p + p )+ N +1)( P P + N ) V N − ,N − = ( P P − kp + N +1)( P P − kp + N )( P P + k ( p + p )+ N )( P P + N ) Z N − ,N − = ( P P − kp + N +1)( P P − kp + N +1)( P P + k ( p + p )+ N )( P P + N ) (4.13)are the required ‘correction factors’ to F N due to powers of y and z , see Eq.s (4.6) and(4.9). While J a are the spin generators in the spin 1/2 (L/R) representation pertaining tothe (massive) gaugini of the two chiralities( J ) µν ˙ α ˙ β = − ¯ σ µν ˙ α ˙ β , ( J ) µναβ = − σ µναβ , ( J ) µν ˙ α ˙ β = ¯ σ µν ˙ α ˙ β . (4.14)Since F N ( k, p a ) (cid:12)(cid:12) ω =0 = 1, the soft pole ω = 0 produces the expected result that includesboth the leading current terms as well as the subleading spin terms. As mentioned before,the structure of the poles is very similar to the bosonic case except for the spin terms andthe absence of tachyon poles. As already mentioned, earlier on, one can include the effect of string loops. As a resultstring resonances would shift in mass and acquire a finite width. In the perturbativeregime g s << u (not in R ) of the string memory [2]. Very much as in the mergerof BHs or other very compact gravitating objects that leave some long-lived ‘remnant’, thecorrections due to the unstable open (super)string resonances will behave as quasi-normalmodes (QNM’s) with Im ω (cid:54) = 0 of the resulting unstable (non-BPS) massive open-stringstate. The narrower long-lived resonances would characterise and dominate the late timeEM-wave signal.While open and unoriented string amplitudes at higher loop can in principle be com-puted, their analysis and the systematic extraction of the necessary informations on mass-shifts and partial widths would be a formidable task that goes beyond the scope of thepresent investingation.Our only concern here is the following. In order for the effect that we found to give adetectable EM-wave signal, special range of the parameters and masses should be consid-ered, whereby the usual low-energy expansion would not apply. At present, time resolutionsthat can be achieved are ∆ t ≈ α (cid:48) E ≈ − s = 1 f s . In turn this would require an irre-alistic value E ≈ GeV for collisions of ‘microscopic’ open-string objects, even in thefavourable T eV -scale (super)string scenari with α (cid:48) = E − s ≈ − GeV − [34].For this reason we need to consider macroscopic objects such as open-string coherentstates, possibly in phenomenologically viable Type I chiral models. Let us consider very massive, possibly highly charged macroscopic objects such as opencosmic strings [35]. A very promising approach to describe semi-classical macroscopicstrings relies on the use of DDF operators [36] to build (open) string coherent states[35, 37, 38]. In fact, even in less favourable scenari with Planck scale string tension √ α (cid:48) = M − GUT ≈ − GeV − , one can construct states with large (average) mass, charge and spin.For collisions of such macroscopic objects, even for ∆ t >> f s , one can find a reasonablerange of masses.In the following we will turn our attention on semi-classical coherent states. Firstly wework in the bosonic string context and then extend the analysis to the Type I superstringin the NS sector.Let’s consider coherent states in the open bosonic string. Following [35, 37, 38] thevertex operator for a coherent state can be written as V C ( ζ n , p, q ; z ) = √ α (cid:48) exp ∞ (cid:88) m,n =1 ζ n · ζ m mn S n,m e − i ( n + m ) q · X + ∞ (cid:88) n =1 ζ n ·P n n e − inq · X e ip · X ( z )(5.1)where q µ is a null momentum i.e q = 0, p µ is a tachyonic momentum i.e p = 1 /α (cid:48) ,constrained by 2 α (cid:48) p · q = 1. BRST invariance is guaranteed by the fact that ζ n · p = ζ n · q = 0with polarizations of the form ζ µn = λ i ( δ iµ − α (cid:48) p i q µ ) .The S n,m operators are symmetric in n, m and read S n,m ( z ) = n (cid:88) h =1 h Z n − h ( U ( n ) (cid:96) ) Z m + h ( U ( m ) (cid:96) ) (5.2)– 18 –hile the ‘generalised’ momentum operators P µn are given by P µn = 1 √ α (cid:48) n (cid:88) (cid:96) =1 i∂ (cid:96) X µ ( (cid:96) − Z n − (cid:96) ( U ( n ) s ) (5.3)The cycle index polynomial Z n ( u s ) encode the functional dependence of both S n,m and P µn on the operators U ( k ) s U ( k ) s = − ik q · ∂ s X ( s − V ph C = √ α (cid:48) exp (cid:18) ζ · i∂X √ α (cid:48) e − iq · X + ip · X (cid:19) (5.5)In order to simplify the computations, the complex polarization ζ µ are chosen to satisfy ζ · ζ = 0. Coherent states are characterized by macroscopic parameters such as the gyrationradius R C , the momentum P µ C and the total angular momentum J µν C . In particular theaverage size of the coherent state is determined by R C = (cid:104) V C | ( X − X cl ) | V C (cid:105) = 2 α (cid:48) ∞ (cid:88) n =1 | ζ n | n . (5.6)The average momentum is proportional to the average level number (cid:104) N (cid:105) = (cid:80) ∞ n =1 | ζ n | andreads P µ C = (cid:104) V C | P µ | V C (cid:105) = p µ − (cid:104) N (cid:105) q µ (5.7)As a consequence the average mass takes the following form M C = −(cid:104) V C | P | V C (cid:105) = 1 α (cid:48) (cid:0) (cid:104) N (cid:105) − (cid:1) (5.8)Finally the total angular momentum is J µν = L µν + S µν . Barring the orbital part L µν ,determined by the zero modes, the average spin is given by (cid:104) S ij (cid:105) = (cid:104) V C | S ij | V C (cid:105) = 1 α (cid:48) ∞ (cid:88) n =1 n Im ( λ ∗ in λ jn ) (5.9) (cid:104) S i − (cid:105) = (cid:104) V C | S i − | V C (cid:105) = 12 α (cid:48) p + (cid:88) n> ∞ (cid:88) m = −∞ n Im ( λ ∗ in λ m · λ n − m ) (5.10)In order to study the EM memory effect generated by a string source where semi-classical objects are present, one can consider the scattering amplitude of two scalars φ ,one photon a µ and a coherent state of photons A = g op C D (cid:89) j =0 (cid:90) dz j V CKG (cid:68) V φ ( K , z ) V φ ( K , z ) V a ( k, z ) V ph C ( p , q , z ) (cid:69) (5.11)– 19 –ithout much loss of generality, one can choose the null momentum of the coherent state q µ to be collinear to the photon momentum k µ i.e q · k = 0. Then performing the rele-vant contractions, detailed in [37], and putting 2 α (cid:48) = 1 one gets the following scatteringamplitude A φφa C = g op e − (cid:98) ζ · p (cid:32) a · p k · p − a · p k · p − a · (cid:98) ζ k · p ( k · p − k · p (cid:33) Γ(1 + k · p )Γ(1 + k · p )Γ(1 − k · p ) . (5.12)where (cid:98) ζ = ζe − iqx and integration over x with a factor e i ( p + p + p + k ) x is understoodthat implements momentum conservation level by level. Including the other contribution,generated by the exchange (1 ↔ α (cid:48) (cid:104) M (cid:105) = (cid:104) N (cid:105) >> t >> f s .The generalization of this computation to Type I superstringa can be obtained usinga general NS coherent vertex operator [38], in the canonical superghost picture, restrictedto the form V NS C ( − = √ α (cid:48) (cid:90) dθ e − ϕ ζ θ · ψ exp (cid:18) ζ · i∂X √ α (cid:48) e − iq · X + ik · X (cid:19) (5.13)where ζ µθ = θζ µ with θ the N = 1 world-sheet Grassmann variable. The Type I superstringcorrections to the EM wave profile assume pretty much the same form as in the bosonicstring case. The role of (minimal chiral) supersymmetry, possibly broken at a lower energyscale, is to make the Type I model phenomenologically viable. So far we have considered ‘unrealistic’ non-chiral Type I models corresponding to toroidalcompactifications with D-brane separation, T-dual to Wilson line breaking [39].Starting from the first chiral Type I model [40], a plethora of chiral semi-realistic modelshave been found and investigated. One of the general features is that open-string U (1) vec-tor bosons are anomalous and become massive via a St¨uckelberg mechanism that involvesclosed-string axions [41]. In Type I string model building, one of the basic phenomeno-logical constraints is that the combination associated to weak hyper-charge Y = (cid:80) a c a Q a remain massless, before the ‘standard’ Higgs-Englert-Brout mechanism take place [42].Our analysis can be easily generalised to this case. For each U (1) a factor that con-tributes to Y ( i.e. with c a (cid:54) = 0) one can compute simple amplitudes with singly chargedchiral fermions (replacing the non-chiral N = 4 gaugini) and doubly-charged Higgs-likescalars with one or more U (1) a photon insertions. Barring the internal parts of the vertexoperators, where Σ A and Ψ i would be replaced by Ramond ground-states and NS chiralprimary operators, respectively, the rest of the computation remains essentially unaltered.What matters is the non-vanishing of the Yukawa coupling, replacing Γ ABi , found in the N = 4 setting, between chiral fermions and scalars . Yukawa couplings of both massless and ‘light’ massive string states have been investigated recently in[43]. – 20 –n many cases the ‘trivial’ O (1) group would be replaced by some non-abelian factor,that would anyway play a marginal ‘spectator’ role in the computations. The main dif-ference would be the position of the poles, resulting from a mass spectrum of the form α (cid:48) M = n + β , where β corresponds to some intersection angle or some internal mag-netic field. The series would look slightly different but they can be summed for specialkinematics. In any case, the ‘universal’ nature of the string corrections to the EM (andgravitational) memory effect would not be spoiled by the ‘real’ (mass)shift of the poles.What alters the effect significantly is an ‘imaginary’ shift (width) due to the instabilityof the string resonances beyond tree-level. Assuming that g s is stabilised by fluxes or(non)perturbative effects at a small value, the leading corrections to the mass and widthshould scale like √ g s N Q/ √ α (cid:48) where N is the level and Q is the charge (typically ± ± u >> α (cid:48) n · p .One should however keep in mind that closed-string fluxes, necessary for moduli sta-bilisation, and the consequent warping may change the situation quite a bit and lead tocontexts similar to the ones pursued in the quest of Holographic QCD [45] or in emergentscenari for gravity, axions and ‘dark’ photons . We have confirmed in the Type I superstring and with higher-spin insertions that thecoherent effect of the infinite tower of string resonances gives rise to a new modulatedEM ‘memory’. At tree-level this string memory is oscillatory and different from the DCmemory effect in gravity and in EM [6–8]. It is also different from the recently proposed‘string memory effect’, related to large gauge transformations of the Kalb-Ramond field[22]. Including loops, the poles in ω would shift and broaden, producing a time-decayingsignal of the same kind as for quasi-normal modes (QNMs). One should keep in mindthat though fully established the effect we find can be measured with presently availabledetectors only if the states involved in the process are macroscopic like coherent open stringstates, describing for instance cosmic superstrings.We have already argued in [1] that the present string memory may be related to the‘global’ part of the infinite (but broken) higher spin symmetries of string theory that isimplemented via BRST transformations: Ψ ∼ Ψ + Q BRST Λ. The corresponding hierarchyof ‘gauge symmetries’ may be exposed in certain extreme regimes [47–49]. 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