Rescattering corrections and self-consistent metric in Planckian scattering

Starting from the ACV approach to transplanckian scattering, we present a development of the reduced-action model in which the (improved) eikonal representation is able to describe particles' motion at large scattering angle and, furthermore, UV-safe (regular) rescattering solutions are found and incorporated in the metric. The resulting particles' shock-waves undergo calculable trajectory shifts and time delays during the scattering process --- which turns out to be consistently described by both action and metric, up to relative order $R^2/b^2$ in the gravitational radius over impact parameter expansion. Some suggestions about the role and the (re)scattering properties of irregular solutions --- not fully investigated here --- are also presented.


Introduction
Interest in the gravitational S-matrix at transplanckian energies [1][2][3][4] has revived in the past few-years [5][6][7], when explicit solutions of the so-called reduced-action model [4] have been found [5]. The model is a much simplified version of the ACV eikonal approach [1,3] to transplanckian scattering in string-gravity, and is valid in the regime in which the gravitational radius R ≡ 2G √ s is much larger than the string length λ s ≡ √ α ′ , so that string-effects are supposed to be small.
The reduced-action model (sec. 4) was derived by justifying the eikonal form of the S-matrix at impact parameter b on the basis of string dynamics and by then calculating the eikonal itself (of order ∼ Gs ≫ 1) in the form of a 2-dimensional action, whose power series in R 2 /b 2 corresponds to an infinite sum of proper irreducible diagrams (the "multi-H" diagrams [3,4]), evaluated in the high-energy limit. The model admits a quantum generalization [6] of the S-matrix in the form of a path-integral -with definite boundary conditions -of the reduced-action exponential itself.
The main feature of the model and of its boundary conditions is the existence of a critical impact parameter b c ∼ R such that, for b > b c the S-matrix matches the perturbative series and is unitary, while for b < b c the field solutions are complex-valued and the elastic S-matrix is suppressed exponentially. The suppression exponent is of order Gs ∼ R 2 λ 2 P (λ P being the Planck length) or, if we wish, of the same order as the entropy of a black-hole of radius R. From various arguments we believe that in the region in which b < b c (that is, b is smaller than the gravitational radius), a classical gravitational collapse is taking place.
A key issue related to the collapse region is the possible existence of information loss at quantum level, which in the ACV model shows up as lack of S-matrix unitarity. According to [6] such unitarity loss is mostly related to a restrictive boundary condition, which is required for the solutions of the model be UV-safe. If such condition is relaxed, further solutions show up, which could contribute to unitarity, but are irregular -i.e., dominated by planckian distances, region in which the model itself is inadequate.
The main motivation of the present paper is to improve and complete some unsatisfactory aspects of the reduced-action model which might be crucial at planckian distances, but show up already at distances of order R ≡ 4GE, the gravitational radius. The new model that we shall present here features two improvements. Firstly, the eikonal representation itself is embedded in three dimensions, so as to be able to describe the motion of the Breit-frame during the scattering process. By comparison, the usual two-dimensional representation is inadequate outside the regime of very small angle scattering, and in particular in the collapse region. In fact, the energy-momentum of the impinging particles is taken to be an external light-like source -without any deflection -which generates gravitational fields characterized by Aichelburg-Sexl-like shock waves [8]. However, the resulting action predicts the Einstein deflection angle [1] and corrections to it [3], at variance with the original assumption of frozen sources. Instead, we know from the beginning [9] that in the classical limit the gravitational equations should predict both particle motion and fields in a self-consistent way. For that to be possible, the improved eikonal representation is needed.
The additional important improvement of the present model is the treatment of rescattering corrections of the produced gravitons. Such effects were argued in [10] to be of particular interest for the irregular solutions of the model, perhaps explaining their relationship to collapse and their contribution to unitarity. But, independently of such a feature, we shall argue here that even for regular (UV-safe) solutions, rescattering corrections are needed in order to achieve self-consistency of motion and metric at higher orders in the eikonal expansion.
To start with, we deal with the self-consistency problem at leading level, in which the eikonal refers to frozen undeflected sources, the metric contains two Aichelburg-Sexl (AS) shock waves, while the action predicts the Einstein deflection angle θ E = R/b, as function of the impact parameter b and of the gravitational radius R(E) = 4GE (2E = √ s being the invariant mass of the system). The problem in introducing the particle motion is that the kinematical corrections implementing it are of the same relative order (θ 2 E or higher) as the irreducible dynamical corrections to the Einstein deflection. How to disentangle the former terms from the latter ones?
A hint about solving the question above was provided in ref. [3], where it was shown that -given the particular Coulombic form of the leading eikonal -it is possible to take into account the motion of the Breit-frame of the particles without affecting the S-matrix eigenvalue which is still provided by the naive two-dimensional Fourier transform. Such observation is the basis for our treatment of motion in sec. 2, which suggests writing a modified metric, which is self-consistent at leading level. The main difference with the previous one [5] lies in the introduction of two-body "shifts" of 't Hooft type [11], summarizing the action of the leading S-matrix at two-body level. That is not enough however. The reduced-action model contains the longitudinal fields h ++ (h −− ) characterized by AS shock waves centered on the particles with profile functions 2πRa(x) (2πRā(x)); furthermore, it generates a gravitational wave also with a field h = ∇ 2 φ defined by the H-diagram. The longitudinal field is here calculated in the improved eikonal representation in sec. 3 and found to be consistent with AS waves [8] which are delayed in time and rotated in space, as suggested in sec.. 2. The transverse field, on the other hand, provides corrections to the leading eikonal profiles in a and a, and causes modifications of the metric inside the light-cone by providing important rescattering corrections.
We then set up a perturbative procedure, in order to deal with the self-consistency problem at higher orders in the eikonal expansion. At next order we find the H-diagram deflection [3] which again appears as a feature of the action, but is not incorporated in the metric, which would require further modifications of the energy-momentum tensor. Surprisingly, we find that the class of multi-H diagrams considered in [1] is not sufficient in order to provide a satisfactory metric at this level but we need to go one further step, and calculate rescattering corrections, which carry the information due to two-body shifts for the produced gravitons. Such shifts occur in an approximate solution to the rescattering equations under consideration since a long time [12], which is here worked out to completion in sec. 4. The results so obtained complete the picture of the improved eikonal representation in sec. 3 for corrections of relative order R 2 /b 2 .
Finally, in sec. 5 we somewhat change subject, and we address the gross features of the ultraviolet-sensitive solution of the model, whose importance lies in the fact that they could play a role for the recovery of unitarity and/or the related information loss. We then summarize the essential results of the paper and their consequences for action and metric in sec. 6, by discussing also some suggestions which arise from our preliminary analysis of singular solutions.
2 Improved leading eikonal description and particle motion 2.1 Eikonal representation of the scattering amplitude ACV [3] have shown that the leading contributions to the high-energy elastic scattering amplitude come from the s-channel iteration of soft-graviton exchanges, which can be represented by effective ladder diagrams as in fig. 1. The purpose of the present section is to recall the method of resumming the effective ladder contributions to all orders so as to provide the so-called eikonal representation for the elastic S matrix. This representation is here "improved" in the sense that we do not make a separation of longitudinal and transverse variables by neglecting the leading scattering angle. Instead, by following [3], Figure 1: One-and two-rung effective ladder diagrams determining the elastic S-matrix in the eikonal approximation. Solid lines: on-shell external particles; dashed lines: eikonal gravitational exchanges.
we use the exact 3-dimensional phase-space of the on-shell particles at each gravitational eikonal exchange and we prove that -due to the Coulombic form of the exchange -the S-matrix is nevertheless provided by a 2-dimensional transform.
It is important to note that the exchanged gravitons (dashed lines) are associated to a propagator −i/Q 2 and are coupled to the colliding particles/strings (solid lines) with an interaction strength equal to α ≡ Gs/ for each pair of vertices. Furthermore, the particles' lines are on-shell along the effective ladder, as proved in ref [1] by the sum over inelastic excitations also. The generic ladder is thus built by iteration of the basic rung In the center-of-mass (CM) frame where p 1 = (E, 0, 0, E), p 2 = (E, 0, 0, −E), Q = (Q 0 , Q), s = 4E 2 , the two mass-shell delta functions in eq. (3) become and, for the same reason, the two external mass-shell deltas in eq. (2) constrain Q 0 = 0, cos θ Q ≡p 1 ·Q = | Q|/2E, Q 2 = − Q 2 . In this way, the 4D integration d 4 q = dq 0 | q| 2 d| q| d cos θ dφ reduces to a 2D integral 1 By taking into account the expression of cos θ and cos θ Q in terms of | q|, | Q| and E and using the identity it is easy to evaluate the azimuthal integral in eq. (5) and, more interestingly, to show that it can be rewritten as the azimuthal integral of a pure 2D propagator: where Q and q are fictitious 2D vectors with the same modulus as Q and q respectively, and φ ′ is the angle between them. With this trick, M 2 can be rewritten as a 2D convolution: It is now straightforward to generalize that procedure to the n-rung ladder, by deriving the recursion formula It should be noted that such simplification is due to the peculiar form of the graviton interaction from which eq. (7) follows. Finally, the eikonal S matrix (with overall momentum conservation factored out) is given by the sum of the ladder diagrams with the usual combinatorial factor 1/n! and with the final-state mass-shell deltas removed: By applying a 2D Fourier transform to eq. (11a), convolutions become standard products and the r.h.s. of eq. (11a) reduces to an exponential series: where we interpret the 2D Fourier transform of the one-graviton amplitudeM 1 as eikonal action A eik . Actually, the definition of A eik (and of all convolutions involved in this calculation) requires the introduction of an infrared cutoff Q 0 ∼ 1/L in order to regularize the "Coulomb" divergence typical of long-range interactions ∼ 1/Q 2 : where b ≡ |b| and L ≡ 2e −γ E /Q 0 . A couple of remarks are in order: • The "position" variable b, being conjugate to Q, is usually interpreted as impact parameter of the collision, and δ 0 (b, s) as the associated phase shift. However, Q is a fictitious 2D vector whose polar angle is actually undefined, therefore b cannot be thought of as a purely transverse (to p 1 ) vector.
• The dependence ofS on L amounts to a b-independent phase, and therefore it is unimportant in the determination of the scattering angle. However such a phase depends on the total energy √ s. This may cause a cutoff-dependent time evolution of the scattered particles, hopefully without physical consequences.
Both issues will be discussed in detail in the next section.

Wave packet motion induced by the S-matrix
The purpose of this section is to derive the motion of two quantum particles subject to a scattering amplitude given by the leading eikonal result of [3] and recalled in sec. 2.1. Suppose we prepare a state of two well separated light-like free particles in the past. In a first-quantization description 2 the wave function is given by In the far future the wave function is expressed in terms of the S-matrix as follows: It is convenient to perform the calculation in the CM frame. As usual, we define the CM coordinate X and relative coordinate x X ≡ x 1 + x 2 2 conjugated to total momentum P ≡ p 1 + p 2 (15a) x ≡ x 1 − x 2 conjugated to relative momentum p ≡ p 1 − p 2 2 (15b) For simplicity, we limit ourselves to states which have zero total momentum, by setting With this restriction the CM coordinate is completely delocalized, but we can build a state representing a localized wave packet in the relative coordinate: where p = p 1 = − p 2 is the relative momentum, p 0 = (0, 0, E) is its mean value and σ a measure of its broadening,Ñ is a normalization factor, 2E p = 2| p| is the total CM energy, while x 0 is the relative position at time t 0 . In fact, by inserting eqs. (16) into eq. (13) we are left with an integral in p whose phase since the shape ofψ forces p ≃ p 0 . Going back to particles' coordinates, we confirm that in the past each particle moved at the speed of light in the direction of p 0 : On the other hand, by substituting eqs. (16) in the outgoing wave (14) together with the energy-momentum conserving eikonal S-matrix (11) we find where Q and b are auxiliary 2D vectors, the former being determined by the condition Q 2 = −Q 2 . In this case, the stationarity of Φ out with respect to independent variations of p, k and b yields the asymptotic motion in the future, as follows. In the plane of scattering p, k we introduce the polar angles θ i of momenta and coordinates, and write (23) where we used the notation θ ij ≡ θ i − θ j , b ≡ |b|, r ≡ | x| and the relations E k = E p , | Q| = 2E p | sin(θ kp /2)|.
We can now determine the five unknowns r, θ k , θ p and b by finding the stationarity point of the phase with respect to variations of the five integration variables E p , θ k , θ p and b, while t 0 and x 0 are known parameters. We have Figure 2: Kinematics of the wave packets' relative coordinates.
where ǫ(x) denotes the sign of x. The first equation causes b to be aligned with Q (φ ′ = 0) and to have modulus b = 2Gs The second equation yields a further relation involving the physical impact parameter b 0 : allowing us to determine the (attractive) scattering angle θ ≡ −θ kp > 0 The third equation identifies the outgoing trajectory to be the straight line with the same impact parameter of the incoming one, thus implying angular momentum conservation. Note that this trajectory intersects the incoming one at the point b = b cos θ/2, 0, sin θ/2) , as is apparent from fig. 2. The fourth equation provides the motion of the outgoing relative coordinate meaning that each particle keeps on moving at the speed of light. A very peculiar behaviour of our result is that the motion of the outgoing particles suffers a time delay given by the energy-derivative of the action with respect to a motion with the same deflection but at constant speed. This is easily seen by projecting the incoming trajectory (17) on p and by choosing the origin of time t 0 = 0 when x in is (or would be) at the intersection point b of the trajectories (31,34), i.e. r 0 cos θ x 0 p = b sin(θ/2), so that eqs. (34,32) become showing that x out = b at t = t delay . Let us stress that this time delay is peculiar of gravity, since the energy dependence of the amplitude stems from the gravitational coupling α = Gs. In other theories, like electrodynamics, the coupling is independent 3 of the particles' energies, and one obtains analogous equations with t delay = 0.

Scattering description by trajectory shifts
We have seen that the leading eikonal S-matrix provides a definite scattering angle and wave-packet motion. It also provides a gravitational metric, associated to the gravitaional fields h µν = g µν − η µν , whose longitudinal terms are expressed in terms of the eikonal amplitude a(x) as follows [5] 1 where and we actually deal with the situation before collision (t < 0). The collision process, according to the reduced action model to be described in sec. 4, introduces further (transverse) components of the metric field, and modifies the expressions (37) by corrections of relative order R 2 /b 2 and higher. Nevertheless, at leading level, starting from the (improved) eikonal representation only, it should be possible to describe the scattering on the basis of an associated metric. Here we suggest a simple way to do it, which is based on 't Hooft's understanding [11] of the S-matrix as a coordinate shift.
The starting point is to write the leading order metric before collision, provided by the expressions (37) as follows: We see that it consists of two Aichelburg-Sexl (AS) shock waves [8] travelling against each other. If only one is present -the one at x + = 0, say -, a test particle impinging on it will acquire a shift in the x − direction: As a consequence, after the shift, the test particle will be deflected. In particular, if the particle was moving at fixed x − and x, the scattering angle θ(x) is given by tan(θ/2) = πR|∇ā(x)|, see fig. 2. Figure 3: Space-time diagram of the double-shift picture of a test particle (solid blue) by the shock-wave of particle 2 (not shown) and then by the shifted shock wave of particle 1 (thick pink line). In the upper part, the motion in the longitudinal plane is shown. In the lower part, the coordinate ξ denote the transverse direction starting from the center of mass (CM) of the two colliding particles. At the point A (B) the test particle leaves the first (second) wave front.
The scattering angle formula is derived by either calculating the action 2πGsā(x) of eq. (12) from the shift (40) as in ref. [5] and then using eq. (30), or by computing the geodesics by standard methods.
The next question is: what happens when both AS waves are present? In order to understand that, consider a test particle at x − = −ε < 0, travelling along with particle 1 (x − = 0) shortly before it. Then, at x + = 0 (t = −z = −ε/2) the test particle will collide with particle 2 and will acquire the shift (40), while the location of particle 1 will be forced to do the same with ∆x − 1 = 2πRā(0), evaluated at the "on shell" point x ≡ b 0 /2 + ξ = 0 (ξ = −b 0 /2), see fig. 3. Next, the test particle trajectory is deflected on a trajectory z ≃ t cos θ(x), x ≃ −t sin θ(x), therefore, due to the cos θ factor, it will be shortly hit by particle 1, travelling at t − z = const, and thus will acquire the shift To sum up, the test particle acquires two shifts separated by a short time interval O (ε), which vanishes in the ε → 0 limit. As a consequence, it is natural to assume that particle 1 acquires two shifts also: whereã ≡ a(0) is actually a possibly singular parameter describing the scattering of particle 1 on itself, which is here taken to be a b 0 -independent constant,ã >ā(0). Taking into account both shifts, particle 1 goes from z = 0 − to ∆z = πR(ã −ā) at time ∆t = πR(ã +ā) = 2πRā + ∆z ≡ t D + ∆z and is then deflected by an angle θ(x = 0) on the trajectory 4 where we recall that x = b 0 /2 + ξ. The total collision time -from z = 0 − to z R (t C ) = 0 -is therefore (45) From the stationarity condition of the collision time with respect to ξ, we obtain the scattering angle (46) is consistent with the action determination, at leading level: Besides providing the scattering angle, the double-shift picture has a strict analogy with the wave packet motion in sec. 2.2. The former can be summarized in 3 steps: a) The "come-back" motion from z = 0 − back to z = 0 + , taking the time t D = 2πRā(0) = 2πR a(b) which is just the retardation time with respect to travel at the speed of light; b) the motion along the residual shift ∆z, taking time where we have defined the angular shiftθ as in fig. 3; c) The motion along the deflected trajectory, taking time By comparison, in the wave packet motion the delay time is just the same, while steps (b) (travel to point x 0 = b joining incoming and outgoing trajectories) and (c) (travel to z R = 0) both take time ξ tan(θ/2). Although different, the expressions (48) and (49) correspond to the total time whose difference with the former is of relative order R 2 /b 2 , because both θ andθ are of order R/b. Therefore the two pictures coincide at leading order. Similarly, the outgoing impact parameter is which differs from the incoming one by relative order R 2 /b 2 also. Furthermore both differences vanish in the limit θ =θ, which is obtained by setting This model, that we could call the "linear model", could be described by the scattering of extended sources, in which the energy of each source inside a disk of radius r increases like r.
We conclude that the double-shift picture provides us with some understanding of the evolution of the metric at collision and after, which is satisfactory at leading level, even if a full account of the collision process is lacking. The basic features argued so far are: • The existence of a delay time t D = 2πR a(b) which causes a shift of either shock wave around z = 0 in the form around particle 1, with a similar one for particle 2; • The deflection of the particle trajectories, causing a rotation by the scattering angle θ for t > t B and reaching z R = 0 at the collision time Figure 4: Simplest eikonal diagrams with one insertion of the h −− field (red cross). The parametrization of momenta is choosen so as to simplify the the calculations.
where the rotated coordinates are Therefore, the particle motion occurs explicitly in the expression of the metric, that we could call at this point, a self-consistent leading metric.
3 Shifted shock-wave fields from the improved eikonal model We have seen before how the improved eikonal model is able to describe particle motion at leading level (sec. 2.2), and how this feature can be incorporated by trajectory shifts in the corresponding shock-waves (sec. 2.3). We want now to show how the shock-wave fields can be explicitly derived from the sum of diagrams describing eikonal scattering, as defined in sec. 2.1. We confirm in this way the delay shifts introduced previously, leading eventually to the scattered energy-momentum. In order to calculate the fields h ++ and h −− we should consider the eikonal diagrams ( fig. 1) in presence of a linearly coupled external source (T ++ , T −− ) which generates the fields by a functional derivative of the corresponding semiclassical S-matrix or, in other words, of the corresponding action. We can write and compute the result by inserting the T −− source on the eikonal diagrams, in all possible ways.
Since the external current diagrams generate insertions of momentum q and thus generally do not conserve energy-momentum, we should restore time-ordered integrations in order to evaluate them properly. Since h −− is coupled to p + only, the p − conservation is not affected, and it is sufficient to consider x + -ordered diagrams as depicted in fig. 4. Let us first consider the small scattering angle kinematics θ ≪ 1, in which the longitudinal dynamics is separated from the transverse one. Σ n,m In order to understand the issue, let us start from the simplest diagrams in fig. 4. We notice that, while the Q-exchange in the eikonal line has negligible is still negligible, but the q + leakage should be considered in the p + evolution.
In the example of fig. 4 we choose to couple the T −− current to initial and final particles of type 1 symmetrically, 5 so that a q + -independent factor of κ 2 p + /(2q 2 ) is factored out. In this way, the lowest-order diagram in fig. 4.a yields the Born-level field where a 0 (x) = 1 2π log L 2 x 2 is the leading profile function of the shock-wave. On the other hand, the Q-exchange yields the leading eikonal ia 0 (b) coupled to the charge 2πGp − (p + + 1 2 q + ) = 2πG(s + √ s 1 2 q + ), where 1 2 q + = i∂/∂x − according to eq. (57). Therefore, diagram 4.b predicts a (subleading) contribution to because of the opposite sign in q + and x + . We shall interpret the results (58) as the first order expansion in which is retarded (advanced) for x + > 0 (x + < 0). Therefore, the result (59) confirms the delay-shift of sec. 2.3, in a form which is appropriate for the calculation of the real part of the action ℜA, or to the principal-value prescription of the x − , x + propagator. 5 Furthermore, we shall weight the initial and final states in (56) with a wave function having energy centered around p + = √ s = p − 2 , which amounts to setting p The higher-order contributions to the shifted field are computed from their definition in fig. 5. In order to do the explicit calculation, we have to explain in more detail how the x + -ordering just mentioned is generated from the usual t-ordering in perturbation theory. We shall work off energy-shell, but on mass-shell for the intermediate particles so that, in the example of fig. 6 Therefore, to each eikonal exchange, we assign the energy differences in the center of mass frame ∆ i = ∆(k i , k i−1 ), e.g., and so on. 6 The ordered time integrations, up to time t of the field, yield therefore the factor where the last expression is obtained after 1 ↔ 2 symmetrization, because of the integrations on ∆ 1 , ∆ 2 induced by d 3 q 1 , d 3 q 2 , with a factor which is symmetrical under q i permutations.
Finally the factorized ∆ i integrations induced by the angular ones on the q i 's are done separately, by using the identity in which the integration contour is closed in the upper (lower) half-plane for t > 0 (t < 0) by noting that corrections due to the large ∆ contour are exponentially small in  This means that eikonal echanges occurring both before and after the insertion are suppressed, 7 because of Θ(t)Θ(−t) = 0, so that the whole sum reduces to the diagrams in fig. 7. In order to compute them we first diagonalize the scattering amplitude in impact parameter b space and we then compute the inserted field by Fourier transform in the q µ variables, in order to yield its x − , x + , x dependence. The non-trivial point about impact parameter transform is that it is 2-dimensional, while the integration variables q 1 and Q − q 1 (in, say fig. 6) are 3-dimensional in the finite angle kinematics. Fortunately, the conditions ∆ 1 = 0 = ∆ 2 fix the 3-components of both q 1 and Q, by determining the directions of the respective Breit frames. We thus have the Bessel transform 1 2 which, by the clever identity (7) yields the same result as the 2-dimensional azimuthal average which is a 2-dimensional convolution diagonalized in b-space.
We thus obtain that the amplitude in front of the field insertion resums to the exponentiated result and cancels out with the factor S −1 in eq. (56).
We are left with the calculation of the h −− field itself. Notice that in fig. 7.b the insertion is on the incoming leg p, while in fig. 7.a it occurs on the final leg k = p − Q. Since | p − Q| = | p| because of the mass-shell plus energy-conservation constraints, the difference is just a rotation k = p R by the angle sin θ 2 = Q/ √ s. When inserted in the 7 Of course, a more refined calculation may introduce a spread ∆t ∼ 1/ √ s in the Θ-functions so that overlap of subsequent contributions is allowed for a limited time. The uncertainty so introduced in the time of shift is expected to be larger than 1/ √ s because of the number of coherent reinteractions, but in any case smaller than R, which would correspond to the maximal number Gs/ of leading eikonal scatterings. field propagator it leads to a rotated variable 8 x R with a mass-shell condition 9 where we have neglected the exponent dependence on is a very small variable if we make the natural assumption | q| ≪ √ s, insuring that the field insertion does not modify the scattering angle. Finally, by taking the value p + = √ s + 1 2 q + of the initial energy as we did in the beginning, we get for the field a factorized value proportional to δ(x − ) which translates Θ(t) into Θ(x + ). The complete resummation in fig. 5 is thus summarized in the exponentiated form An identical calculation with the exchange (+ ↔ −) yields an analogous result for the field h ++ (x). The overall situation is depicted in fig. 8, where we show the shifted (and slightly rotated) shock waves, on which the source particles lie before and after the collision (solid lines). The evolution of the particles during the collision actually depends on the reference frame used to describe their motion. For instance, a time-like observer that crosses first shock 2 (the one on the bottom-right) describes the scattering in two steps: firstly the shock 2 drags shock 1 and particle 1 by an amount 2πRā(x) in the x − direction; then shock 1 drags shock 2 and particle 2 in the x + direction. In this frame particle 1 travels along the dashed blue line in fig. 8. A symmetric description is given by an observer crossing first shock 1, which "sees" particle 2 moving along the dashed red line.
4 Rescattering corrections to the reduced-action model 4

.1 Semiclassical field equations
The transplanckian field equations in the ACV proposal [3] were based on two main groups of results. Firstly, the ACV investigation of string-gravity showed that at transplanckian energies Gs ≫ 1 (R ≡ 2G √ s ≫ 1/ √ s) the eikonal representation -with an eikonal operator which is calculable in principle by expanding in λ s /b and R/b -yields a good representation of the scattering amplitude, and incorporates both string-and stronggravity effects. Secondly, in the regime b R ≫ λ s in which string effects are supposed to be small, the irreducible eikonal diagrams are much in correspondence with the effective action of Lipatov and co-workers [13,14], who calculated a Regge-graviton emission vertex [15] which is the building block of the effective lagrangian used in [4]. Finally, the equations of motion of the latter -by neglecting rescattering terms -were shown to yield a shockwave solution for the fields which is the basis for the reduced-action model investigated in detail in later years [5].
Here, we have considered so far only the leading graviton-exchange kernel in the eikonal, and we have improved the amplitude representation based on it, so as to include a motion of the Breit-frame on which the exchange is defined. Next, we want to reconsider the field equations in 4-dimensions -including rescattering -and we shall provide a solution for the fields which fits very well in our understanding of scattering developed so far, by adding corrections which are of relative order R 2 /b 2 and higher.
In the effective action framework, the elastic S-matrix of the tree diagrams in fig. 9 is given in terms of the classical solutions of the lagrangian equation of motion as whereh ++ ≡ (2/κ)h −− andh −− ≡ (2/κ)h ++ are just rescaled versions of the longitudinal fields considered so far, Φ is related to the transverse field which is proportional to h = ∇ 2 Φ, and represents (up to an unconventional but convenient factor of κ) the energy-momentum tensor of the colliding particles. The usual metric components are given by [5] h Figure 9: Diagrams providing subleading contributions to the eikonal approximation: (a) the "H-diagram" representing the first correction of (relative) order (R/b) 2 ; (b) a "multi-H" diagram of order (R/b) 4 involving only Lipatov's vertices (red disks); (c) the first diagram with a rescattering vertex (green triangle) is of order (R/b) 3 , but actually vanishes on-shell; (d) the first nonvanishing rescattering diagram is of order (R/b) 4 .
where ∇ 2 denotes the transverse laplacian and the ǫ's denote graviton polarizations to be specified below.
The lagrangian is conveniently written in terms of real light-cone variables x ± , ∂ ± ≡ ∂/∂x ± and a complex transverse variable z ≡ where the longitudinal fields have a mostly transverse propagator and the (complex) Φ field a mostly longitudinal one, of a graviton emission term incorporating Lipatov's vertex [15], and finally, of a rescattering term which is supposed to take into account the rescattering diagrams of fig. 9.c,d. This term is quadratic in Φ, and is likely to play a role when the latter is large. Notice in particular in eq. (75) the current J , which describes transverse graviton emission [13] in the so-called H-diagram [3] and is non-local because of the inverse laplacian needed to find it. For this reason it has been convenient to introduce the complex field Φ, which corresponds to two graviton polarizations. The detailed computation shows that and, more generally, using a notation with real-valued indices, that where so that we have the polarizations The resulting equations of motion are given by where the rescattering terms are set on the r.h.s., with similar equations forh −− and Φ * . Notice that in eqs. (81) the impinging particles are represented by the frozen external sources T ++ and T −− written in eq. (72), which pertain to the collinear kinematics used before. According to this small-angle approach, scattering comes out because of a nontrivial action, but is not taken into account in the energy-momentum of the particles. Shortly, we shall take a somewhat different attitude.

The reduced-action without rescattering
We start recalling [4] that, if the rescattering terms are dropped, eqs. (81) admit a shockwave solution of the form where the metric component h −− = (κ/2)h ++ = 2πRa(x)δ(x − ) are of AS type [8] while the transverse field Φ has support inside the whole light-wedges x + x − ≥ 0. The latter propagation, of retarded plus advanced type, corresponds to the principal value part of the Feynman propagators, as is appropriate for the real part of the amplitude. The field equations (81) induce a set of differential equations on a,ā and φ, which can be solved on the basis of proper boundary conditions which make them regular, i.e., ultraviolet safe. Furthermore, by replacing the expressions (82) into (81) and omitting the rescattering term, we obtain the ACV equations of motion and the corresponding reduced action where we have freely performed some integration by parts, assuming a smooth enough ultraviolet behaviour of the solutions, and we have replaced the current J by its reduced counterpart H: We state below some of the properties of the RAM solutions [5] in the particularly simple case of axisymmetric sources like particle-ring scattering -which is relevant for the azimuthal averaged particle-particle case. By assuming the real fields a(r 2 ),ā(r 2 ) and the complex field φ(r 2 ) to be functions of r 2 ≡ |x| 2 only, and by calling s(r 2 ),s(r 2 ) the axisymmetric sources, we soon realize that the action A R can be recast in one-dimensional form. This is because the current H defined by eq. (84) is simply obtained in this case by settingḢ = −2ȧȧ -where the dot denotes the r 2 -derivative -and is therefore real and axisymmetric also. As a consequence, the interaction term involves ℜφ only and is proportional to the aā kinetic term as follows Here we have replaced φ by the auxiliary field ρ(r 2 ) -a sort of renormalized squared distance -defined by which incorporates the φ-a-ā interaction. Furthermore, the field φ is now taken to be realvalued, describing the T T polarization only. The external axisymmetric sources s ands are able to approximately describe the particle-particle case by setting s(r 2 ) = δ(r 2 )/π, s(r 2 ) = δ(r 2 − b 2 )/π, where the azimuthal ACV averaging procedure is assumed. The equations of motion, derived from the reduced-action (85) and specialized to the case of particles at impact parameter b (with the axisymmetric sources just quoted), provide the profile functionṡ from the analogue of eq. (81a), and the ρ-field (or φ-field) from the analogue of eq. (81b). Eq. (88) shows a "Coulomb" potential in ρ-space, which is repulsive for ρ > 0, acts for r > b and plays an important role in the tunneling phenomenon [10]. By replacing the equation of motion (87) into eq. (85), the reduced action can be expressed in terms of the ρ field only, and takes the simple form which is the one we shall consider at quantum level in the following. The effective metric generated by the axisymmetric fields ρ, a andā was calculated [5] on the basis of the complete form of the shock-wave (82) and is given by This metric is dynamically generated and may be regular or singular at short distances, depending on the behaviour of the field solutions themselves. It is not fully consistent, however, since it does not take into account the longitudinal shifts that the fields cause to each other. In the following sections we shall compute such shifts at subleading level and we shall present the improved expression of the self-consistent metric.

The H-diagram: scattering angle and shifts
The first nontrivial use of the reduced action (83) is the calculation of the first order correction to the Einstein deflection [3] which is due to the H-diagram contribution to the action ( fig. 9.a). The latter is obtained by expanding the action (83) and the corresponding equations of motion (quoted in eqs. (87,88) for the axisymmetric case) in the parameter R 2 /b 2 where R ≡ 2G √ s is the gravitational radius and b is the impact parameter conjugated to the transverse momentum Q (and related to the true impact parameter b 0 by eq. (29)). By expanding the equations of motion we obtain first order corrections to the profile function a(x) = a 0 (x) + a 1 (x) + · · · (91) and to the transverse field as consequence of the Lipatov's vertex J in eq. (75). It is then straighforward to obtain for the action [5] where and the H-diagram contribution is In the following, we specialize the expression (96) to the axisymmetric case 10 in which, according to eq. (86), and -by the equations of motion (88) -we obtain the result The complete result, including the LT polarization would be [3] A R = 2πGs(a 0 + a H + · · · ) = Gs log and provides -by stationarity of the eikonal phase -the scattering angle The recollection above suggests to generalize the calculation of the shifted field h −− in sec. 3 to the next order in the R/b expansion, by including the H-diagram in the eikonal. In so doing we find 3 kinds of insertions of the h −− source, illustrated in fig. 10: (a) insertions on the on-shell propagators, which can be done in a similar way as we did the leading one; (b) insertions on the particle propagators of the H-dagram itself, and (c) insertions on the emitted transverse field h = ∇ 2 φ. The latter are new and involve the rescattering vertex, which is omitted in the RAM calculation and will be included in the next subsection.
Insertions of type (a) above add up to the leading shift a 0 (b) and are done by replacing p + with p + + q + in the corresponding energy charge. 11 However, Gs a H ∼ G 3 s 2 contains two p + factors and thus acquires a factor of 2, as follows The outcome is that the overall shift 2∆ is proportional toā(0) = a(b), as given by the perturbative formula (59): and thus it extends to the H-diagram the relationship between shift and energy-derivative of the action suggested in sec. 2.

Rescattering solutions with shifted fields
Consider now the full field equations, starting from (81a). While, by replacing in the l.h.s. the shock-wave (82), we generate terms proportional to κ √ sδ(x − ), the rescattering term in the r.h.s. yields instead the structure which suggests an x − translation, i.e., a shift similar to those just found at leading and H-diagram level with the eikonal insertions. Therefore, we make the Ansatz where ∆,∆, ∆ φ ,∆ φ (x, b) are shift variables to be determined. We take a similar Ansatz for energy-momentum too, because we know from last section the latter is really shifted at leading level. By then formally expanding the δ-function (105a) in R/x − , we get two equations from the zeros of the δ and δ ′ contributions. The first equation is simply the reduced-action model (RAM) equation for a, which reads |∂| 2 a + 2πR 2 ∂ 2 (φ * ∂ * 2 a) − |∂| 2 (φ * |∂| 2 a) + c.c. + 1 2 δ 2 (x) = 0 (106) and reduces, in the axisymmetric case and for real-valued φ, to The latter is equivalent to eq. (88) and yields the solution a(r 2 ) = 1 2π where we have introduced the infrared cutoff parameter L by setting a(L 2 ) = 0. Since by eqs. (99) above [5] ρ(r 2 ) ≃ a(r 2 ) approaches the leading value for b ≫ R a(r 2 ) ≃ a 0 (r 2 ) = 1 2π log L 2 r 2 + · · · (110) and diverges logarithmically for r 2 → 0. The second equation (from the vanishing of the δ ′ -coefficient) involves the rescattering terms and is supposed to determine the shift parameter ∆, which occurs in the combination D(x) ≡ ∆(x)a(x), as follows where we have set the rescattering terms on the r.h.s. and we notice in particular the energy-momentum shift which, according to sec. 3, has the contribution emerging from the eikonal insertions up to R 2 /b 2 accuracy. In the axisymmetric limit, eq. (111) takes the simplified form where we note that the rescattering source J r (r 2 ) vanishes for r 2 ≤ b 2 , as a consequence of the equations of motion (109). It is then straightforward to find a solution for ∆(r 2 ) which is regular everywhere, in the form In other words, the shift parameter of the longitudinal field takes over the (constant) leading value a(b) from the energy-momentum tensor and adds an r 2 -dependent term of relative order R 2 /b 2 which, for r 2 ≫ b 2 , is related to the kinetic term of the RAM action This means that, while the RAM action is sufficient in order to describe the scattering parameters at relative order R 2 /b 2 (H diagram), the rescattering terms are needed in order to describe the form of the fields, like shock-wave shifts, and thus the metric properties at a comparable level of accuracy. It is amusing to check the result (115) -which is based on the rescattering equations and on the energy momentum shift in eq. (111) -by using the direct insertions (b) and (c) on the H-diagram mentioned before, and depicted in fig. 10. By keeping track of the flow of energy charges we calculate the above insertions on the imaginary parts (app. A) and we find that diagram (b) is already counted by Lipatov's vertices, while diagram (c) contributes the absorptive part The q + -dependent part, by dividing out the rapidity factor (2/π) dk + /k + = 2Y /π (as requested in order to obtain the real part of the diagram from the dispersion relations) and the leading field a 0 (r 2 ) fits with the displacement form i(∆ R + i∆ I )i∂ − = −(∆ R + i∆ I )∂ − , by thus providing the shift 2πR∆ H from past to future, where We recognize here the 2-dimensional Laplacian Green function G 0 (x − x ′ ) applied to the H-diagram density. By translating it to the azimuthal-averaged formalism we can use the one-dimensional form and apply it to the rescattering current J r (r 2 ). The result is just identical to the r.h.s. of eq. (115) with the ∆ 0 contribution subtracted out. In particular, for r ≫ b eq. (119) factorizes and yields ∆ H (∞) = −a H , thus recovering the form (117) of the total shift. Our overall interpretation of the present findings is that a large-distance observer will see the particle shock-waves to suffer a total time delay 2πR A(b), directly related to the action and thus to the scattering angle because of the contribution of the rescattering shift. On the other hand, a test particle parallel to particle 2, traveling at transverse distance r ≃ b from particle 1 12 will see a profile function 2πR a(r 2 ) related to its own scattering angle which turns out to be the same in the limit r = b. 13 Furthermore, the test-particle will suffer a time delay that, due to the double-shift picture of sec. 2.3, is ∆t−∆z = 2πR a(x), close to the short-distance value 2πR a(b) = ∂A R /∂ √ s coming from the eikonal insertions of sec. 3 Finally, while the rescattering contribution is needed for the large distance shifts, its contribution to the action starts at order (R/b) 4 because the diagram in fig. 9.c (formally of order (R/b) 3 ) vanishes by the property ǫ(x + )δ(x + ) = 0. Therefore it does not affect the consistency of the metric and action descriptions at order (R/b) 2 . 14 Of course, a symmetrical calculation yields the shift πRǫ(x − )∆(x) for the field h ++ , provided one exchanges light-cone indices + ↔ − and the profile functions and shifts a ↔ā, ∆ ↔∆.

Shift modification for the transverse field
We now go over to the second basic equation (81b) which essentially describes rescattering properties ( fig. 9.c,d) of the transverse field h = ∇ 2 φ, once emitted by the longitudinal fields. We note that, by replacing the RAM fields (105) in the r.h.s. of eq. (81b), we obtain the structure Let us recall that, according to eqs. (112,115), the longitudinal rescattering predicts a shift ∆ = a(b). We then decide to expand the shift ∆ φ of the transverse field around the constant value ∆ = a(b) = 0, which gives the leading order of the longitudinal shift. By this method, it is easy to evaluate a (formally) first order modification ∆ (1) ≡ ∆ − ∆ φ of the transverse-field shift, which however is also of leading order in the R/b expansion. By expanding the first term in the l.h.s. of eq. (81b) we obtain so that we can easily match the left and the r.h.s. to get the equation, in the axisymmetric limit, 2|∂| 4 (φ∆ (1) ) = ∂ * 2 (ā∂ 2 φ) + c.c. , where we work in the regime r ≫ b withā(r 2 ) = a(r 2 ) ≃ 1 2π log L 2 r 2 . The expression of φ is found by the RAM equation which was provided before (88). By inserting proper boundary conditions, in our regime r ≫ b we can set φ = 1 4 [a(r 2 )−a(b 2 )] 2 and tentatively look for a solution ∆ (1) = λa(r 2 )+µ, where λ and µ are constants. A simple calculation shows that so that ∆ φ is r 2 -dependent and of leading order, while the correction ∆ (1) is sizeable. Furthermore, ∆ φ becomes cutoff independent, and pretty small at r = b. The above conclusion may be unpalatable from a calculational standpoint, but is natural on physical grounds because rescattering occurs for the φ field at relatively leading level and at all distances, so that the parameter a(b) has no particular role and a sizeable difference ∆ − ∆ φ is expected. One might ask, at this point, whether the RAM hierarchy for the action is really satisfied or not. Fortunately it remains, because the integral over x + , x − of the rescattering vertex with the longitudinal field just vanishes (ǫ(x − )δ(x − ) = 0) or, in other words, there is no R 3 /b 3 contribution to the action from diagram 9.c. Furthermore, the shift does not change the x + , x − integration in the H-diagram and therefore rescattering contributions.
On the other hand, the location of the shock-wave and the evolution of particle trajectories and geodesics does change from the point of view of the metric, so that rescattering is needed in order to have self-consistent calculations even at relative order R 2 /b 2 .

Irregular solutions and their (re)scattering properties
All preceding arguments hold for the UV-safe solutions of the RAM model in eqs. (87,88) which, by definition satisfy the condition ρ(0) = 0. Due to the form of ρ(r 2 ) in eq. (86), ρ(0) = 0 would imply thatφ ∼ −ρ(0)/r 2 has a short-distance singularity, ∇φ has an outgoing flux −ρ(0) and therefore h = ∇ 2 φ ∼ −ρ(0)δ(r 2 )/(πR) 2 has a singular δ-function contribution. The latter behaviour would call for large short-distance effects -possibly regularized by the string -which are expected, but not considered, in the RAM model. Restricting the solutions by ρ(0) = 0 is possible only if the impact parameter is larger than some critical value b c ∼ R which signals a possible classical collapse. In fact, the regular solution (99) of eq. (88) has continuous derivative at τ ≡ r 2 = b 2 provided a condition which has real-valued solutions only for On the other hand, the complex solutions for b < b c lead to an exponential damping [6] of the S-matrix of type ∼ exp(−Gs) -with exponent of the order of a black-hole entropy ER -which leads, eventually, to a violation of unitarity of the RAM model [7,10].
From the above discussion, some questions arise: What happens to the rescattering solutions for b < b c , when they are irregular? Do we see any sign of a possible collapse in the latter, perhaps in relation with the unitarity problem and to a "fall in the center" mechanism [16]?
Let us then consider a class of real-valued RAM solutions with ρ(0) = 0 for b < b c , as follows. We start from the general solution for ρ(0) = 0 where t b is now a free parameter which ρ(0) > 0 depends on. Then we choose t b ≡ t m so as to minimize ρ(0), i.e., dρ(0)/dt b | tm = 0. The parameters of this "minimal" solution ρ m (τ ) are given by The corresponding rescattering current is The above current is now non-vanishing for τ < b 2 , and, in addition, has a non-integrable behaviour for τ → 0 + . This means that large rescattering amplitudes are built in the short-distance region λ s < r, b < R and, as a consequence, that exchange and emission of (massive) string states can no longer be neglected. Nevertheless, we find it instructive to provide here a preliminary analysis of irregular solutions in our effective-theory framework in which only graviton intermediate states are considered. Basic string effects -like graviton reggeization and ensuing string production at λ s [1], as well as diffractive and central string emission induced by tidal forces [1,17] 15 should certainly be estimated in the near future in order to see how they affect the picture, but we feel that our effective approach may still provide suggestions and questions to be answered, and is anyway needed as a ground for the estimates just mentioned.
In order to get a better insight on the evolution of irregular solutions, we cut-off the current (131) below r = λ s ≪ R, and we compute the total charge, related in sec. 4 to the H-diagram action 2πGs a H and to the shift ∆ H as follows: where the singular integration has been performed by a careful integration by parts and in the last equality we have introduced the δ-function singularity in the ρ-derivative: This means that we think of ρ(τ ) = 0 for τ ≤ 0 (a region which is felt by the twodimensional model through the outgoing flux) and that the δ-function occurs because of the τ -discontinuity. Thus, strictly speaking, we discover that the singular solutions are not bona fide solutions of the equations of motion up to τ = 0 − unless we introduce a singular "external force"ρ = ρ(0)δ ′ (τ ) at τ = 0. We can do that by starting from the action [10] A 2πGs = d 2 x as +ās − and by supplementing the Lipatov's current H of eq. (84) with a singular external current By then replacing such expression into eq. (134), i.e., we find that the action enforces eq. (133) and becomes, on the equation of motion, thus determining the singular term in the action. 16 We are now able to look at the scattering and shift properties of the irregular solutions, by keeping in mind that we need in this case the external current (135), that we think generated by the short-distance string dynamics and/or by possibly collapsed matter. Compared to the complex solutions -which have a quantum-tunneling interpretation and cause a probability suppression -the irregular ones may provide a probability source or alternatively may carry away the information loss.
By thus using the eikonal representation of the S-matrix with the action (137) and by carefully computing b-and √ s-derivatives, we get the scattering angle and the time delay where, by eqs. (129,135) Such results can be further specified by using and the outcomes are plotted in fig. 11. We note the strong increase of the action in the region b ≃ λ s due to the positive singular contribution -implying in particular a motion with many turns in the region λ s b R. Actually, large scattering angles are reached pretty soon around b = b c as solutions of the equation but, due to the strong increase of the action derivative ( fig. 11), the saddle-points are confined to either the critical region b 2 c − b 2 = O (λ 2 s ) or to the small-b region (b/R) 3 = 16 Note that we interpret the distribution δ(τ ) as a step-function located around τ = 0 with width λ 2 s and height λ −2 s , hence the identification δ(0) = λ −2 s . O (λ 2 s /R 2 ), for all real values of θ s . 17 Finite b-values, in the region λ 2 s ≪ b 2 < b 2 c − λ 2 s are strongly suppressed like exp(−GsR 2 /λ 2 s ). The above observation suggests that the singular solutions under study do not actually yield back all the initial information in the physical scattering region, but may carry it away in the small-b, string-dominated region.
Another piece of information comes from the study of the shifts and the corresponding time delay. We first notice, by eq. (139), that the short-distance shift of has a positive singular term -meaning a lot of time spent in the interaction regionwhich is essentially twice the singular contribution to a H in eq. (132). The factor of 2 comes from ρ(0) scaling like R 2 ∼ (G √ s) 2 -similar to what happens for the H-diagram itself. Furthermore, we already know from sec. 4 that the large-distance shift takes contributions from rescattering insertions and produces the additional (negative) shift ∆ H , the one we started with in eq. (103). We thus have We note that the singular term remains positive after subtraction of ∆ H and of the same order as that of the action -apart for the addition of a term proportional to −H(0) > 0. The very large time delay is thus confirmed at all distances. That means that the singular solutions, wherever they are, spend a long time in the interaction region, before exiting, either around b = b c , or at b = O (λ s ). Therefore, they are connected with long-lived states in that region. Finally, let us look at the associated metric (90), whose geodesics -in the "regular" case ρ(0) = 0 -were argued to provide a complementary picture of particle scattering. 17 We have checked that complex saddle-points do not change the picture.  Figure 12: Radial "effective potential" for different values of the test-particle's impact parameter: |β| < r * (solid blue), r * < |β| < √ 2r * (dashed red), |β| = √ 2r * (dotted green), |β| > √ 2r * (dash-dotted brown).
be typical of an angular momentum barrier with |p θ /v ⊥ | = |β| = r * that -by lack of other explanations -we should attribute to the external source δH eq. (135) that we have put in at r = 0 in order to justify the irregular solutions themselves. For v ⊥ = 0 the repulsive potential is absent, and the geodesics (at x − = 0, say) will reach the shock-wave (at x + = 0, say) at the initial distance r, but will thereby acquire a shift and will be deflected according to the formula which takes into account the "refraction index" of the φ-field. Once again r = r * is unreachable: if initially we try to approach it, the geodesic scattering angle approaches π, and the test particle is reflected backwards. We note at this point that the situation does change -unexpectedly -when |β| increases and overcomes the threshold r * : in this case, according to eq. (148), the centrifugal factor at the origin changes sign, so that the potential becomes partly attractive, close to the origin and above r * ( fig. 12). In particular, for |β| = √ 2r * the potential is everywhere attractive! This somewhat surprising feature opens up the possibility that test particles may reach the small-r region, at least at quantum level, because the small barrier has a transmission coefficient of order unity around the value |β| = √ 2r * , and is thus able to populate the small-r region around λ s in an efficient way. That feature is the counterpart of the selection of possible issues of the particles themselves, which was argued before to have sizeable probability for a small-b exit on the basis of the action stationarity condition. Furthermore, the test-particle action variable 1 p ⊥ r * λs p r dr ∼ √ 2r * log r * λs develops a logarithmic behaviour in the small-r region with frequency ( √ 2r * ) −1 ≃ R −1 : is that perhaps a possible interpretation of the time nature of the variable r in that region?
On the whole, we think, the above discussion shows -at a sort of "phenomenological" level -that the irregular solutions may indeed vehicle the particles' information from distances of order R down to the string size, thus suggesting that the unitarity loss of the ρ(0) = 0 model is due to the opening up of the gate to other worlds. However, due to the lack of the theoretical ingredients from string theory mentioned in the beginning, in the present formulation we have been unable to really assess the reliability of the above suggestion and to discuss the physical interpretation of the state(s), living at λ s , which could be responsible for the singular contributions to the action investigated here and for their intriguing consequences.

Conclusions
To sum up, in our study of the improved eikonal model with subleading correction we have found that: • The source particles, after the interaction, besides being deflected, suffer a time delay which can be interpreted as a shift in the light-cone variables (sec. 2); • The shock-wave metric fields h ±± generated by the source particles are also shifted in an analogous way, after the interaction described by the eikonal amplitude (sec. 3); • The subleading contributions to the eikonal, represented by the (multi) H-diagram and by the rescattering terms, provide corrections to the scattering angle of the particles and to the shifts of the metric fields h ±± and Φ (sec.4). Due to rescattering, the large-distance shift of the particle fields agrees with the scattering-angle description based on the action.
In order to find an adequate form for the self-consistent metric which embodies all these features, we have to modify the ACV expression (90). As far as the shock waves are concerned, the improved expressions are obtained from eq. (69) by replacing the leading profile function a 0 (x) with the full profile function a(x) derived in sec. 4.3 and by accordingly shifting the location of the shock waves of the amount ±πR∆(x) as explained in sec. 4.4. In turn, the support Σ of the transverse field Φ is also shifted from the region x + x − > 0 to the two (past and future) disconnected regions where in the region Σ (+) after the interaction we have used the rotated coordinates of eq. (55). Equivalently, the characteristic function of Σ = Σ (−) ∪ Σ (+) reads The ensuing improved metric we propose is thus (1 −ρ)|x + − πRǫ(x − )∆(x)| + dr 2 1 + 2(πR) 2 χ(Σ)φ + dθ 2 r 2 1 + 2(πR) 2 χ(Σ)(φ + 2r 2φ ) .
It takes into account all the physics results following the analysis of the improved eikonal model and the ACV effective action, and is self-consistent, in the sense that the trajectory shifts caused by the shock-wave metric are felt by the metric itself. showing the position of the shock waves 1(blue) and 2 (green) and the support Σ of the metric field Φ (yellow). The (pink) rectangle EF HG actually corresponds to two different regions of space-time, which are accessed either from the left or from the right.
A word of caution has to be spent concerning the form of the metric in eq. (152) in the region between the forward and past light-wedges. In fact, the set |x + | < πR∆, |x − | < πR∆ enclosed by the continuation of the shock waves -represented by a rectangle EF HG for each longitudinal section at fixed x as depicted in fig. 13 -actually represents two distinct physical regions in our coordinate system. Each region is unambiguously described by continuing the metric either from the left or from the right. For instance, by continuing the metric from the left across the boundary EF H, we enter the region at the left of both shocks, where shock 1 travels from E to G, collides with shock 2 in G, and after that shock 2 moves from G to H and goes on in the x − direction, as shown. On the other hand, either continuation leads to the same form of the metric inside the future light-wedge Σ (+) .
We have also investigated here the scattering features of UV-sensitive solutions of the model in the subcritical range λ s < b, r R, without making a real attempt to restore the string degrees of freedom -which are nevertheless called for by the singular shortdistance behaviour of the rescattering current. This preliminary analysis leads to the suggestion that such solutions may indeed carry the information from distances of order R to distances of order λ s , but a dynamical analysis in the string-dominated region is needed to find out about their fate.
In the case of the H-diagram, besides the insertion (a) on external lines (already computed in the text) we have in principle two insertions, one of which is on the internal (eikonal) propagators. We think that this one ( fig. 10.b) is already included by the Lipatov's vertices, which by definition count the external particles' emissions also. Therefore, in order to avoid double-counting we only estimate here the insertion on the rescattering graviton propagator of momentum k in fig. 10.c, as emitted by Lipatov's vertices.
In order to do that, we compute the imaginary part of the q-insertion amplitude, which has the form 2ℑ∆h ++ (x) = 2πGsR 2 d 2 k d 2 q (2π) 2 h * (k)h(k + q) By going through the same steps as before, we perform the q − -integration first (by keeping the leading (k + q) 2 = 0 contribution only) and we have, in addition, a weighted k + , k-integration, as follows: × dk + dq + 4π In the strong ordering region √ s ≫ k + ≫ |k| ∼ 1/b, |q + | we estimate the effect of the (small) q-translation (or k-dependent rotation) by redefiningq = q − k q + /k + and neglecting the translation in the matrix element h(k + q). The integral in d 2 q becomes d 2q 2π 2 e iq·x q 2 e ik·x q + /k + h * (k)h(k +q) by thus factorizing the exponential e ik·x q + /k + which modifies the subsequent k + and q + integrations. By performing the latter in the phase-space −k + < q + < ∞, we obtain in turn the factor We can see here the small shock-wave rotation tan(θ/2) ≃ |k|/k + = O (1/( √ sb)) ≃ e −y which has been neglected in front of the logarithmic phase space. By then collecting the various factors we obtain the result where we have defined This shift of the imaginary part and the corresponding one for the real part add up to the total shift 2a H of both real and imaginary parts due to the external line insertion and thus occurs at large distances only with a relative minus sign, as emphasized in the text.