Semiclassical ${\cal S}$-matrix and black hole entropy in dilaton gravity

We use complex semiclassical method to compute scattering amplitudes of a point particle in dilaton gravity with a boundary. This model has nonzero minimal black hole mass $M_{cr}$. We find that at energies below $M_{cr}$ the particle trivially scatters off the boundary with unit probability. At higher energies the scattering amplitude is exponentially suppressed. The corresponding semiclassical solution is interpreted as formation of an intermediate black hole decaying into the final-state particle. Relating the suppression of the scattering probability to the number of the intermediate black hole states, we find an expression for the black hole entropy consistent with thermodynamics. In addition, we fix the constant part of the entropy which is left free by the thermodynamic arguments. We rederive this result by modifying the standard Euclidean entropy calculation.


Introduction
Black hole (BH) information paradox [1,2] has long history ever since the discovery of BH evaporation [3]. Recently there has been a remarkable progress towards its resolution with first semiclassical calculations indicating unitarity of the evaporation process [4][5][6][7][8]. Notably, these calculations use complex saddle points of the gravitational path integral -replica wormholes [9][10][11][12][13] -to derive correct semiclassical expression for the entanglement entropy of BH radiation. The latter quantity is then shown to follow the Page curve [14,15]. It still remains to be understood, however, how quantum correlations are encoded in the Hawking radiation. Only then the information paradox will be completely resolved [16,17].
A direct approach to study unitarity of BH evaporation is to compute the related elements of the gravitational S-matrix [18]. In this case one treats BH formation and its subsequent decay as a scattering process [18,19] mediated by a metastable bound state. On general grounds, consideration of this complete process appears more adequate than its splitting into separate stages of collapse and evaporation. It was argued in [20] that when both initial and final states of the scattering process are semiclassical, the related amplitudes can be evaluated using complex saddle points of the path integral with appropriate boundary conditions, cf. [21,22].
In this paper we further develop complex semiclassical method for gravitational S-matrix. Using this method, we compute the scattering amplitudes and probe the entropy of black holes in (1 + 1)-dimensional dilaton gravity.
We start with an outline of the method. Consider complex quantum transition including collapse of matter in pure initial state Ψ i into a black hole and evaporation of the latter into the state Ψ f . This process interpolates between the free flat-space states Ψ i and Ψ f and therefore defines a gravitational S-matrix [18]. Schematically, one can write a path integral for the transition amplitude as where S [Φ] is the classical action and Φ includes all fields of the model -matter fields, metric, and Faddeev-Popov ghosts. Precise definition of the gravitational path integral (1.1) is a formidable task. One can assume, however, that the initial and final states of the process are semiclassical. In field theory this means that they contain many quanta at high occupation numbers. Then the integral can be evaluated in the saddle-point approximation, giving , where the semiclassical configuration Φ cl extremizes the integrand in Eq. (1.1) i.e. solves the classical field equations.
Importantly, Φ cl does not coincide with the classical collapsing solution: like all configurations in the path integral (1.1) it starts from the flat space in the past and arrives to it in the future. Since real solutions with these properties do not exist, Φ cl is a complex saddle point describing an exponentially suppressed process. This is to be expected: the intermediate black hole mainly emits Hawking radiation with low occupancies, and the probability of producing a semiclassical state Ψ f is exponentially small.
Generically, there may exist many complex saddle points for Eq. (1.1), and one has to select the physical one giving the main contribution into the path integral. To this end, we use the method suggested in [20] (see [23,24] for quantum mechanical applications). The main idea is to enforce the scattering boundary conditions in the path integral (1.1) with a special variant of a constrained instanton method. After that the physical complex solutions are obtained by smooth deformation of the real solutions that describe classical low-energy scattering without black hole production.
Our method reduces construction of the semiclassical gravitational S-matrix to solution of the classical field equations in the complex domain. Though this is in principle managable, applications to four-dimensional field theories with dynamical gravity are challenging. So far this method has been applied only in spherically reduced models with simplified matter content [20].
Below we consider another simplified model based on the two-dimensional Callan-Giddings-Harvey-Strominger (CGHS) [25,26] dilaton gravity. The model describes interaction of a non-dynamical metric g µν (x) and dilaton φ(x) with matter. The vacuum solution in this model has flat g µν and linear dilaton field φ changing from −∞ to +∞. For positive values of φ, gravity becomes strongly coupled precluding the semiclassical analysis. To make the model tractable, we cut off the strongly coupled region by introducing a reflective boundary along the line of constant dilaton φ(x) = φ 0 , where φ 0 is negative and large [27][28][29][30][31]. All fields in the path integral are then restricted to the submanifold φ < φ 0 (the rightmost region in Fig. 1). This model was shown to be equivalent to the flat limit of the Jackiw-Teitelboim gravity [32,33] with a boundary both at the classical [34] and quantum level [35].
We also make the second radical simplification. Instead of a full-fledged field theory, we represent the matter sector with a point particle of mass m moving along the trajectory x µ = x µ * (τ ). One can interpret it as a toy model for the narrow wavepacket in field theory. We find complex semiclassical solutions Φ cl = {g µν (x), φ(x), x µ * (τ )} and compute the transition amplitudes of the particle. At low energies M the particle trivially scatters off the boundary with unit probability, see the dashed line in Fig. 1. However, once the energy exceeds a certain critical value M cr the semiclassical solutions become complex. Initial and final parts of these solutions describe formation of an intermediate BH with mass M from the particle and, after complex evolution, a particle in the final state. The transition probability equals independently of the particle mass m. Here λ is the CGHS energy scale. Notably, M cr coincides with the minimal mass of black holes in the model. It is worth stressing that our semiclassical method provides the phase of the amplitude, in addition to its absolute value. One can interpret the probability (1.2) as follows [22]. The intermediate BH has entropy Σ BH (M ) and an exponentially large number of states exp(Σ BH ). Then it is expected to decay into the single-particle final state with probability P ∝ exp(−Σ BH ). Comparing to Eq. (1.2), we find the entropy of the CGHS black holes, This expression is consistent with the results for BH entropy in similar models [36][37][38][39]. Our result, however, raises a puzzle. A naive extrapolation to our model of the Gibbons-Hawking Euclidean calculation [40] of the BH entropy gives,  [40,41]. The masses of the latter are minimal only among the black holes with given charges and/or angular momenta, whereas the absolute minimum is reached by the neutral BH with the Planckian mass. In this case collision of charged particles may lead to formation of a neutral BH, with charge and angular momentum carried away by bremsstrahlung. Then the corresponding scattering probability is a continuous function of energy [20].
The present paper is organized as follows. In Sec. 2 we introduce our setup. The scattering amplitude is calculated in Sec. 3. In Sec. 4 we discuss the entropic interpretation of the scattering probability and the Euclidean calculation of BH entropy. Section 5 is devoted to conclusions. Several Appendices contain details of the calculations.

The setup 2.1 Dilaton gravity
We consider non-perturbative scattering in two-dimensional dilaton gravity with a boundary [31], see also [25,[27][28][29][30]35]. The gravitational action 1 describes the CGHS model [25] with non-dynamical metric g µν (x) and dilaton φ(x). Besides, it includes the timelike boundary at φ = φ 0 which cuts off the region of strong coupling. Importantly, a regulating boundary should be present in all configurations in the path integral (1.1), otherwise the CGHS fields would become singular at the quantum level [35,42]. In Eq. (2.1) we included the Gibbons-Hawking term [40] at φ = φ 0 with proper time τ 0 , extrinsic curvature K = ∇ µ n µ 0 and outer normal 2 n µ 0 . Parameter λ sets the energy scale of the model.
The semiclassical expansion is controlled by the combination e 2φ 0 . Indeed, a shift φ → φ + φ 0 brings this parameter in front of the classical action, at the place of the 1 We use the metric signature (−, +) and Greek indices µ, ν, . . . = 0, 1. 2 The direction of the normal is fixed by the condition n µ 0 ∇ µ φ > 0. and work to the leading order in this parameter. Without matter, the general solution in the bulk is, where M is the Arnowitt-Deser-Misner (ADM) mass. This constitutes the twodimensional analog on the Birkhoff theorem [43], which we derive in Appendix A. It is not enough, however, to solve the bulk field equations: one should also add the boundary. This amounts to cutting off the spacetime at φ = φ 0 and imposing the boundary condition

Classical scattering
Now we want to consider scattering of a point particle with mass m and action off the boundary. Here the parameter τ is a proper time of the particle. One can find the particle trajectories using the well-known techniques developed for thin shells in multidimensional gravity [44]. We describe the particle trajectory with radius r = −φ/λ as a function of the proper time r = r * (τ ). The two-dimensional Birkhoff theorem guarantees that the empty spacetime regions to the left and to the right of the particle are either Schwarzschild or Minkowski. Then, if the particle starts evolution in Minkowski spacetime, the solution in the "inner" region r < r * remains flat, Note that we introduced the notation T for the time coordinate in the inner region to emphasize its difference from the time t of the distant observer. Similarly, the "outer" region r > r * is described by the Schwarzschild metric (2.3) with conserved gravitational mass M . Since the particle energy-momentum tensor is concentrated at the worldline, the derivatives of the metric and dilaton change discontinuously across it. In Appendix A.2 we derive the Israel junction conditions for the jumps of the extrinsic curvature and normal derivative of the dilaton, Here the square brackets represent difference of the values at r * + 0 and r * − 0; the worldline normal n µ points towards large r. Substituting the inner and outer spacetimes (2.8), (2.3) into Eq. (2.9) one finds equation of motion for the particle, where dot is a derivative with respect to the proper time τ . This equation has an intuitive form of non-relativistic "energy conservation law" with effective potential V eff . The latter is negative everywhere, it monotonically increases from a finite value at the boundary r = r 0 , The details of the derivation are given in Appendix A.2. Now it is clear that the left-moving particle with energy M < M cr always reaches the boundary r = r 0 at some moment of time τ = τ × . Then it reflects back. In Appendix A.3 we demonstrate that reflection of the particle from the boundary simply flips the sign of its radial velocity,ṙ * (τ × + 0) = −ṙ * (τ × − 0). At late times the particle goes to r → +∞.
The classical story changes completely if the particle energy M exceeds M cr . In this case it first crosses the horizon r h > r 0 of the outer metric (2.3) and thus forms a black hole. Whence the particle can be retrieved only quantum mechanically with exponentially small probability.

Semiclassical method
Quantum S-matrix is an operator connecting initial and final Fock states of the process. It is formally defined aŝ whereÛ andÛ 0 are the interacting and free evolution operators, and the limits t i → −∞, t f → +∞ are assumed. In the path integral representation Eq. (3.1) reads, where Φ denotes all fields of the model on the time contour in Fig. 3, while S and S 0 are the interacting and free classical actions 3 on the respective parts of the contour. Note that the fields at the endpoints of the contour t = 0 − and t = 0 + do not coincide. We also introduced the wave functions Ψ i , Ψ f of the free initial and final states. We generalize Eq. (3.2) to gravity in a straightforward way. In this case Φ includes the particle trajectory r * (τ ), metric g µν , dilaton φ and Faddeev-Popov ghosts. The interacting action involves gravitational and matter contributions, as well as the standard Gibbons-Hawking term S GH at infinity; see its definition in Appendix C. Importantly, we assume that configurations Φ = {g µν (x), φ(x), r * (τ )} in the path integral (3.2) have trivial topology expected from the scattering processes. First, they should contain the boundary φ = φ 0 . Second, they should start from flat spacetime at t = t i and arrive to it in the future. This gives a preferred choice of the asymptotic time t changing from t i to t f by the clock of the distant observer. The free actions S 0 = S m describe a particle in flat spacetime and Ψ i,f ∝ e ∓ipr are the momentum eigenstates of this particle with p = √ M 2 − m 2 . In the semiclassical limit e 2φ 0 1 the classical action S becomes large and the integral (3.2) can be evaluated in the saddle-point approximation, is the total action and Φ cl is a complex classical solution extremizing S tot . The Faddeev-Popov ghosts can be neglected at this point as they don't contribute to the leading exponential term.

From low to high energies
It is straightforward to compute the amplitude at M < M cr substituting the real classical solution into Eq. (3.4), see Appendix C. In the overcritical case, however, the task of finding the relevant saddle-point configuration becomes non-trivial. The ordinary collapsing solutions are of no use here, since they describe formation of black holes and therefore violate the requirement of flat spacetime in the asymptotic future.
To enforce this requirement, we introduce a positive-definite and diffeomorphisminvariant functional T int [Φ] estimating the duration of the scattering process from the viewpoint of a distant observer. Namely, T int [Φ] should be finite on any scattering configuration Φ interpolating between flat spacetimes at t → ±∞, and infinite otherwise. Then we constrain the path integral (3.2) to run only over configurations with finite values of T int [Φ]. Technically, this is implemented by inserting the unity into the integrand of Eq. (3.2) and interchanging the order of integration over DΦ and dT 0 dε. To have a specific example, consider the choice where the integration is concentrated on the line φ = φ ε which is far away from the boundary, |φ ε | |φ 0 |. Clearly, T int in Eq. (3.7) is positive-definite for real g µν and φ. Besides, in the asymptotically Schwarzschild spacetime with mass M one finds T int = dt M 2 /4λ. Thus, this functional estimates the asymptotic time spent by the ADM mass M in the "interaction region" to the left of φ = φ ε . We stress that our method is not specific to the choice (3.7) and can exploit any appropriate positive-definite T int . Inserting the unity (3.6) into Eq. (3.2), one finds the path integral with the "regularized" interacting action and the additional integrations over ε and T 0 . The δ-function (3.6) ensures that the configurations Φ leave the "interaction region" in a finite "time" T 0 . Besides, we can use the positive definiteness of T int , to improve convergence of the path integral. To this end, we deform the contour of ε-integration into the region e ε ≥ 0. At fixed T 0 and ε the semiclassical solutions extremize S ε [Φ]. The additional saddle-point integrals with respect to ε and T 0 give ε = 0. We therefore perform calculations at ε > 0 and send ε → +0 in the end, restoring the original saddle-point equations. The "regularized" semiclassical solutions at ε > 0 have three important properties [20,23,24]. First, they leave the "interaction region" φ > φ ε in finite time. Second, the corresponding fields are generically complex-valued. Third, they can be obtained by smooth deformation of the classical reflecting solutions.
To demonstrate these properties, we consider the "shell-like" term (3.7) concentrated at φ = φ ε . Junction at this shell changes the metric to the left of the shell, at r < −φ ε /λ. By Birkhoff theorem, the form of this metric is still Schwarzschild, Eq. (2.3), but with the complex mass In Appendix B we show that ε is positive and proportional to ε. After this replacement the regularized saddle-point configurations change continuously with energy. At M < M cr they are close to the real classical solutions: the particle trajectory r * (τ ) reaches the boundary at r = r 0 and reflects from it, see Fig. 4a. The outer time t changes almost along the real axis (Fig. 4b). Importantly, the horizon of the outer metric now acquires a positive imaginary part, m r h > 0, see Eq. (2.4). Thus, even at M > M cr the particle continues to evolve along the contour C r in Fig. 4c. It bypasses the horizon in complex r-plane, both on the way in and on the way out. But now the outer Schwarzschild time of the particle is essentially complex. Equations (2.3) and (2.10) imply, where the integral runs along the contour C r in Fig. 4c and the minus (plus) sign oḟ r * correspond to motion prior to (after) reflection at r 0 . The integrand in Eq. (3.10) has a pole at the horizon giving an imaginary time change Res Notably, the time contour in Eq. (3.10) is smooth at finite ε > 0, see Fig. 4d. Since the regularized solutions are now connected to the classical ones, we assume that they also represent the physical saddle points of the path integral (3.2). 4 Once the amplitude (3.4) is computed, we send ε → +0.

The result
By construction, the regularized saddle-point configurations have trivial topology, just like the reflective classical solutions at low energies. Their action S tot is computed in a straightforward way, given Eq. (2.10) for the particle trajectory r * (τ ) and the inner and outer metrics (2.3), (2.8). We perform this computation in Appendix C.
Trajectory of the particle in complex planes of the radial and temporal Schwarzschild coordinates for the regularized solutions at M < M cr (top) and M > M cr (bottom). The radial coordinate r * (τ ) varies along the almost real contour C r as the particle's proper time τ changes from −∞ to +∞. The particle bypasses the event horizon which is shifted upwards in the complex plane.
Here is the result, where is the radial momentum of the particle immediately after the collision with the boundary. This result is finite and valid at all energies. It provides the absolute value and the phase of the amplitude (3.4). In the massless case m = 0 the expression (3.12) simplifies, (3.14) The infinitesimal mass shift iε in Eqs. which is independent of the particle mass m. This gives the probability of overcritical scattering P f i = |A f i | 2 in Eq. (1.2). The real and imaginary parts of the expression (3.12) are shown in Fig. 5.
Let us outline where the imaginary part of S tot comes from. We consider 5 the space-time contour in Figs. 4c,d with almost real r * (τ ). Then the particle action (2.7) is almost real as well by Eq. (2.10). In Appendix C we show that the bulk CGHS Lagrangian in Eq. (2.1) is a total derivative. This means that the interacting action S is a sum of integrals along the boundary φ = φ 0 , initial and final Cauchy surfaces t = t i , t f , and the timelike surface at the spatial infinity r = r ∞ , see Fig. 6. We find that the latter term vanishes. The integrals at t = t i , t f combined with the free action S 0 and the wavefunctions Ψ i , Ψ f give complex contribution into S tot . Its imaginary part comes from the residue of the Schwarzschild time at the horizon, This is similar to the results of the previous studies [20,22]. Remarkably, the contribution of the boundary is also complex. One may notice from Fig. 6 that before and after the collision the boundary lives in flat spacetime. These parts do not contribute into S tot . We find, however, that the collision point τ × corresponds to a non-analytic break of the boundary with the extrinsic curvature proportional to a δ-function, where τ 0 is the boundary proper time and τ 0,× is its value at the collision point. The expression (3.17) is complex because f < 0 at r = r 0 < r h . Substituting it into Eq. (2.1), one finds an imaginary term m S There are no imaginary contributions in addition to Eqs. (3.16) and (3.18). Summing up these terms, we arrive to the expression (3.15). the probability P f i of particle reflection is a continuous function of energy M , as it should be. At M > M cr i.e. above the threshold for classical BH production, this probability is exponentially suppressed. The respective transitions are interpreted as two-stage processes. First, the left-moving particle creates the BH of mass M classically. Second, the intermediate BH decays into the final-state particle with exponentially small probability. One expects [22,47] that the probability of the latter stage is suppressed by the number of BH states exp(Σ BH ). This implies the expression (1.3) for the black hole entropy Σ BH . A following puzzle arises. There is an alternative method for calculating BH entropy based on Euclidean path integral [40]. When applied to our model, this method apparently gives a different result (1.4). Let us briefly review the relevant calculation [38,39]. One computes the thermal partition function where S E is the Euclidean CGHS action, see Appendix D for the precise definition. The integral is taken over configurations with period β in Euclidean time t E = it. In the semiclassical limit the integral is saturated by the saddle point. The instanton corresponds to Euclidean continuation of the BH exterior with the metric This spacetime has topology of a half-tube, where t E serves as a periodic coordinate, see Fig. 7a. The black hole horizon corresponds to the tip of the tube. Notably, this tip is a conical singularity if the period β is not equal to the inverse Hawking temperature T −1 H = 2π/λ. As a consequence, the curvature has a δ-function contribution at the tip of the cone, 3) The singular contribution vanishes for β = T −1 H . Now one evaluates the Euclidean action on this solution, and the free energy, Note that the only non-vanishing contribution into the action comes from the δfunction in Eq. (4.3). Then the thermodynamical formula

Experiments with the thermal gas
We now present several physical arguments against Eq. (1.4). To this end, we couple the dilaton gravity to the gas of massless particles -quanta of some massless scalar field. Notice that the BHs cannot form classically from arbitrary configuration of this field, even if its total mass is higher than M cr . Indeed, the gravitational Lagrangian (2.1) is explicitly proportional to the factor e −2φ ≡ e 2λr . This means that the gravitational interaction decreases exponentially at coordinate distance ∆r ∼ λ −1 from the boundary. Then formation of BHs requires the energy M cr to be concentrated within the interval ∆r ∼ λ −1 . A configuration satisfying this condition, however, cannot carry large coarse-grained entropy. Indeed, the entropy reaches maximum in a thermal state providing the bound Here we related the gas temperature to its energy density T gas = 6ρ gas /π and substituted ρ gas ∼ M cr /∆r. On the other hand, the entropy (1.4) is parametrically larger than Eq. (4.7): Σ naive BH = 4πe −2φ 0 at M = M cr . If this expression were correct, it would be puzzling why the critical BH cannot be formed from states with large entropy.
Further, the results on classical subcritical scattering [31] suggest that the entropy of the critical BH is even smaller than (4.7). Namely, near the threshold of critical BH formation reflection of the classical field from the boundary proceeds as follows. A part of the incoming wavepacket reflects immediately, whereas the remaining part forms a long-lived state with mass M ≈ M cr . The latter state decays into a narrow wavepacket carrying a few highly blue-shifted particles. This may be interpreted as formation of a slightly subcritical black hole decaying classically into a low-entropy state. Extrapolating this picture to the critical black hole, we conclude that it should have an order-one entropy. Our semiclassical formula (1.3) is consistent with this picture.
One may also try to form the black hole in an essentially quantum way. Namely, suppose the spacetime is filled with a massless gas of temperature T ∼ λ. Eventually, the black hole of mass M may appear, eating a part of the gas and providing the first order phase transition. According to Eq. (1.3), a part of gas entropy ∆Σ gas = 2M cr /λ ∼ e −2φ 0 disappears in this process even if the black hole is critical. We have argued, however, that the black hole cannot form in classical collapse of the low-temperature gas. Thus, the probability of this process is exponentially suppressed by the CGHS action S gr ∝ e −2φ 0 ∼ ∆Σ gas . Now, we recall that the entropy of a thermal ensemble can decrease with exponentially small probability due to large fluctuations. The above process appears to be one of them.

Correcting the Euclidean calculation
We now suggest a modification of the Euclidean calculation that reproduces the result (1.3) for the entropy. The approach of Sec. 4.1 misses an important property of our model, namely, the presence of the boundary at φ = φ 0 . This boundary is necessary because it shields the singularities of the CGHS fields in the original Lorentzian path integral [27,35,42,[48][49][50]. Our complex scattering solutions satisfy this property, whereas the Euclidean instanton in Fig. 7a does not.
We cure this problem by adding to the Euclidean spacetime a disjoint cap-like portion with a closed boundary φ = φ 0 , see Fig. 7b. By Birkhoff theorem, the geometry of the cap is given by the black hole metric (4.2), possibly with a different mass parameter M . The latter must be larger than M cr for the cap to be compact and satisfy the inequality φ < φ 0 . The radial coordinate on the cap runs in the interval r 0 < r < r h (M ). Importantly, the signature of the metric on the cap is (−, −) instead of (+, +) in the exterior region.
This configuration does not satisfy the boundary condition (2.5) at φ = φ 0 and thus it is not an exact saddle point of the path integral (4.1). Rather, as shown in Appendix D, it should be interpreted as a constrained instanton extremizing the Euclidean action within a subset of geometries with the boundary. Instead of solving the boundary conditions, one minimizes the action with respect to the free parameter M .
The action of the additional Euclidean cap equals (see Appendix D) where the terms proportional to β have cancelled. The only remaining contribution comes from the δ-function of the curvature at the horizon r h . The latter has an opposite sign to that in Eq.

Conclusions
In this paper we further developed complex semiclassical method for calculating Smatrix elements in gravity. We considered a simplified setup where the point-like quantum particle scatters off the boundary in two-dimensional Callan-Giddings-Harvey-Strominger (CGHS) model. The semiclassical method provided the amplitude of a complete transition between the initial particle moving with energy M towards the boundary and an outgoing final particle with the same energy. At low energies this reflection proceeds classically and the transition probability is of order one. However, once the particle energy exceeds the minimal mass of black holes (BHs) in the model, the amplitude becomes exponentially suppressed. Then the respective transition can be interpreted as production of an intermediate BH and its subsequent decay into an outgoing particle. The probability of such transition is naturally identified with exp(−Σ BH ), where Σ BH is the BH entropy. It is important to stress that our analysis provides not only the absolute value of the amplitude, but also its phase.
Our result implies that the entropy of the minimal-mass BH vanishes. This is consistent with the expressions for BH entropy in similar two-dimensional models [36][37][38][39]. We noticed, however, an apparent conflict between this result and the calculation of the BH entropy using the Gibbons-Hawking Euclidean approach. We suggested a natural modification of the Euclidean calculation that takes into account the presence of the boundary and recovers the correct entropy obtained from the scattering probability.
Our results demonstrate that the semiclassical S-matrix provides important insights about black holes, even if the simplified matter content is considered. Let us outline several directions for future research.
The phase of the amplitude is known to contain information about temporal properties of the scattering process [51]. It will be interesting to extract this information from our results and compare it to the characteristic time scales of the BH evaporation, e.g. the scrambling time [15].
Further development of the semiclassical S-matrix approach will be inclusion of full-fledged matter fields. In field theory, the semiclassical amplitudes can be used for studying quantum correlations in the Hawking radiation and for direct tests of unitarity. As an example, consider the identity satisfied in any unitary theory, Another interesting direction of research would be to relate the semiclassical Smatrix to the new "island" method for calculating the entanglement entropy of the Hawking radiation [4][5][6][7][8][9][10]. The latter method indicates purification of the Hawking radiation in the final state. Technically, it makes use of "replica wormholes" [9][10][11][12][13], saddle points of the gravitational path integral for the trace Trρ n , whereρ is the density matrix of the radiation and n is an arbitrary power. If BH is formed from a pure state |Ψ i , the final density matrix equalsρ =Ŝ|Ψ i Ψ i |Ŝ † . Thus, Trρ n can be formally written using the path integral (1.1) for the S-matrix. This suggests that the relevant saddle-point solutions in the two methods may be related to each other by some kind of analytic continuation.
Acknowledgments We thank V. Rubakov, G. Rubtsov and P. Tinyakov for encouraging interest. This work was supported by the grant RSF 16-12-10494.

A Classical solutions
In this Appendix we summarize the field equations and discuss the relevant solutions.

A.1 Birkhoff theorem
Varying the action (2.1), (2.7) with respect to g µν and φ, we find, where ≡ ∇ µ ∇ µ is the covariant d'Alembertian and is the matter energy-momentum tensor. It will be discussed later.
In an empty spacetime region one sets T µν = 0 and arrives to the system where the second equation is a trace of Eq. (A.1). It will be convenient to use the Schwarzschild gauge, where the spatial coordinate r tracks the dilaton field φ and the time t is orthogonal to r. The first of Eqs. (A.4) gives, To summarize, we demonstrated that the static black hole spacetime (2.3) with arbitrary mass M is the only solution in an empty patch of spacetime. This is the analog of the Birkhoff theorem in the present context [43].

A.2 Junction conditions and equation of motion for the particle
It will be convenient to introduce Gaussian normal coordinates (τ, n) near the particle trajectory x µ * (τ ). Here n measures the geodesic distance to the trajectory and τ is orthogonal to n, ds 2 = −a(τ, n) dτ 2 + dn 2 . (A.7) In these coordinates the particle trajectory is n = 0. We choose τ to coincide with the proper time along the trajectory: a(τ, 0) = 1. By construction, a(τ, n) is continuous at n = 0, as opposed to the metric components in the Schwarzschild gauge. Variation of the action (2.7) with respect to g µν gives the particle energymomentum tensor (A.3), T µν = mẋ µ * ẋ ν * δ(n) , where (ẋ τ * ,ẋ n * ) = (1, 0) is the particle velocity. Since Eq. (A.1) has a δ-function in the r.h.s., the normal derivatives of a and φ are discontinuous at n = 0. Equations (A.1), (A.2) take the from ∂ 2 n φ = m e 2φ δ(n)/4 + (regular terms) , ∂ n (∂ n a/a) = 4∂ 2 n φ + (regular terms) , (A.8) where we have kept only the "singular" terms with the second n-derivatives, which are proportional to δ(n). Now, we integrate Eqs. (A.8) from n = −0 to n = +0 and rewrite them in the covariant form using ∂ n φ = n µ ∇ µ φ and ∂ n a/(2a) = K. We arrive to the junction conditions (2.9). Since n µ ∇ µ φ and K are frame-independent, one can compute them in different coordinate systems (T, r) and (t, r) at the two sides of the particle trajectory. The outer trajectory normal in these regions is whereṪ * andṫ * can be expressed from Eqs. (2.8) and (2.3), Substituting the normal into the first of Eqs.

A.3 Boundary condition and reflection law
The saddle-point configurations Φ cl (x) should extremize the action (2.1) with respect to all variables, in particular, the metrics g µν at the boundary φ = φ 0 . Due to the reparametrization invariance, it is enough to consider only the variations preserving the coordinate position of this boundary. Then δφ = 0 and δn 0 µ ∝ n 0 µ at the line φ = φ 0 . We vary Eq. (2.1) with respect to g µν and leave only the boundary terms, where τ µ = dx µ /dτ 0 is the unit vector along the line φ = φ 0 . Requiring the variation to vanish, we obtain Eq. (2.5).
To derive reflection law for the particle from the boundary, we notice that the collision point τ × divides the particle trajectory x µ * (τ ) into two smooth parts, see Fig. 6. Thus, where τ i and τ f are the initial and final times of the process. We vary Eq. (A.11) with respect to x µ * (τ ), again keeping the position of the boundary intact: n 0 µ δx µ * (τ × ) = 0. We obtain, , where again only the boundary terms are shown. We obtain two equations, where the second one follows from the normalizationẋ µ * ẋ * µ = −1. Now, we rewrite Eqs. (A.12) in the coordinates (T, r) of flat spacetime patches immediately prior to the collision and after it, see Fig. 6. This gives the reflection which is used in the main text.

B Regularization method
Let us demonstrate that the regularization (3.7), (3.8) of the classical action is equivalent to the imaginary shift of the Schwarzschild mass M inside the regulating "shell" r * < r < r ε , where r ε = −φ ε /λ. To this end, we solve the field equations at r > r * . Additional term in the regularized action (3.8) produces imaginary energymomentum tensor, give whereM (r) is the coordinate-dependent mass entering the metric as Integrating the second of Eqs. (B.1) one arrives to the matching condition Here M =M (+∞) is the real conserved energy of the particle and M ε =M (r < r ε ) is the Schwarzschild mass parameter inside the regulating "shell" at φ = φ ε . Expressing M ε from Eq. (B.2), one obtains Eq. (3.9) from the main text.

C Computing the action
Let us calculate the total action (3.5) on the semiclassical solution. We work in the limit ε → +0. To start, we note that the bulk Lagrangian in Eq. (2.1) is a total derivative on the field equation (A.2), Thus, the interacting action (3.3) is a sum of one-dimensional contour integrals over the spatial infinity r = r ∞ → +∞, the boundary φ = φ 0 , Cauchy surfaces t = t i , t f , and the particle worldline r = r * (τ ), see Fig. 6, This expression includes the Gibbons-Hawking term, over the spatial infinity r = r ∞ and surfaces t = t i , t f → ∓∞. Here σ is the proper time or the proper distance, K is the outer-normal extrinsic curvature, and κ = n µ n µ equals +1 (−1) at the timelike line r = r ∞ (spacelike curves t = t i , t f ). The parameter K 0 is introduced in Eq. (C.2) to subtract the vacuum contribution; it equals 2λ at r = r ∞ and zero at t = t i , t f . Recall that in addition to the terms in Eq. (C.1), the total action S tot includes the free actions S 0 and the wave functionals Ψ i,f . We now evaluate all the listed contributions one by one. Spatial infinity. It is straightforward to check that the term at r = r ∞ vanishes as e −2λr∞ , where we evaluated n µ ∇ µ φ = −λ √ f and K = f /2 √ f using the metric (2.3), then sent r ∞ → +∞.
The boundary. Adding the boundary term in Eq. (2.1) to the contribution from the bulk action, one obtains, where the term proportional to λ has cancelled due to the boundary condition (2.5).
Note that φ = φ 0 is a straight line in flat spacetime prior to and after the collision, see Fig. 6. In these regions K = 0. Thus, the only non-zero contribution into S φ=φ 0 comes from the singularity of the extrinsic curvature at the collision point.
To evaluate it, we use several technical steps. We regulate the calculation by slightly shifting the line of integration to r δ ≡ r 0 + δr, see Fig. 8a. In contrast to the boundary, the regulating line intersects the particle trajectory twice, going from the flat geometry to Schwarzschild and back. We will see that each of these intersection points gives a δ-functional contribution into the integral (C.4).
Let us focus on the first intersection point A. In its vicinity we introduce the Gaussian normal coordinates (τ, n) which are continuous at the particle worldline. In these coordinates the line r = r δ has a break at A, see Fig. 8b. This is because the normal n µ 0 to this line has a discontinuity, as we now demonstrate. In the Schwarzschild and flat patches it has the components (n t 0,+ , n r 0,+ ) = 0, − f (r δ ) , (n T 0,− , n r 0,− ) = (0, −1) .
At the intersection point A we can decompose n µ 0,± in the basis of the tangential and normal vectors to the particle trajectory. In the two patches the former equals to whereas the latter is given by Eq. (A.9). In this way we find the components of n µ 0 in the Gaussian normal frame which are different on the two sides of the intersection point A, (n τ 0,± , n n 0,± ) = (− sh ψ ± , − ch ψ ± ) , where sh ψ Here all the quantities are evaluated at A. Now we regularize the break approximating the line r = r δ with a smooth curve. Its normal is (n τ 0 , n n 0 ) = (− sh ψ(τ ), − ch ψ(τ )) , (C. 5) where ψ(τ ) interpolates between ψ − and ψ + . The proper time and extrinsic curvature of the curve are readily computed: dτ 0 = dτ / ch ψ, K = − ch ψ ∂ τ ψ. Integrating K in the vicinity of the point A, we obtain, We see that K contains a δ-function at A. Another δ-function with the same coefficient comes from the second intersection point B. Taking the limit r δ → r 0 , we conclude that the boundary extrinsic curvature is proportional to δ(τ 0 − τ 0,× ). Then Eqs. (C.6) and (2.10) yield Eq. (3.17) from the main text. Finally, using the formula for V eff and expressing r 0 in terms of M cr , we arrive to the boundary action, where p × is defined in Eq. (3.13) and the imaginary part of the logarithm is fixed by the regularization procedure from Appendix B. Initial and final Cauchy surfaces. We define them as the lines of constant Schwarzschild time t = t i,f to the right of the initial and final particle positions r i,f continued as T = const to the left, see Fig. 9. The interacting action at the final surface equals One can check that K = 0 on the outer and inner parts of this surface. Thus, the only non-zero contribution comes from the jump of the normal at the particle position r = r f where the two parts of the surface join. The same calculation as before gives, where p = √ M 2 − m 2 and in the second equality we have sent r f → +∞. Computing the initial contribution at t = t i in a similar way, one obtains, where we explicitly added the Gibbons-Hawking term at infinity. The parameter κ = n µ n µ = ±1 discriminates between the signatures (+, +) and (−, −) of the Euclidean spacetime. Note that this parameter is implicitly present 7 in the original Minkowski action, or the latter would be inconsistent.
To warm up, consider the standard Gibbons-Hawking instanton in Fig. 7a. Since the solution (4.2) is stationary, one may naively expect that its Euclidean action is proportional to dt E = β. This would give zero entropy in Eq. (4.6). However, in the vicinity of the horizon r − r h r h the metric (4.2) takes the form where ρ = 2(r − r h )/λ and θ = λt E are the radial and angular coordinates. Since θ changes between 0 and λβ, this metric describes a cone with angle deficit 2π − λβ.
The respective δ-contribution in curvature, Eq. (4.3), is proportional to the angle deficit, not to β. That is why the standard calculation gives non-zero black hole entropy. Now, let us ensure that every single configuration in the Euclidean path integral (4.1) includes a boundary x µ = x µ b (τ 0 ) and φ equals φ 0 at this boundary. The latter condition is enforced by a δ-function in the integration measure, with the boundary function Λ(τ 0 ) playing the role of a Lagrange multiplier. The product on the l.h.s. is taken over all points on the boundary. This adds an extra term to the Euclidean action, where φ b ≡ φ(x b (τ 0 )). Importantly, the term (D.4) changes the boundary conditions at x = x b . Indeed, variations with respect to g µν and φ now give 8 In what follows we find solutions at a fixed Λ and then take the limit φ b → φ 0 . It is clear that the Gibbons-Hawking instanton in Fig. 7a does not satisfy the condition φ b = φ 0 , as it has φ < φ h < φ 0 . Thus, we have to suggest an alternative. Let us assume that the true saddle-point configuration exists and it is real. Besides, we will consider only the solutions with τ 0 -independent φ b = −λr b . This is reasonable because we will eventually send φ b → φ 0 .
With the above assumptions, the only candidate for correct instanton is the disconnected configuration in Fig. 7b. The Birkhoff theorem guarantees that the additional cap-like part in this configuration is described by the Schwarzschild metric The only additional contribution into the Euclidean action which is not proportional to β comes from the conical singularity at its second horizon r h . This time, however, the metric in the vicinity of r h is negative-definite: ds 2 = −dρ 2 − ρ 2 dθ 2 , where ρ = 2(r h − r)/λ and θ = λt E . Thus, the respective δ-term of the curvature has opposite sign, Note that alternatively one can ignore some saddle-point equations and directly minimize the Euclidean action over the free parameters of the solutions. In particular, the discussion of Sec. 4.3 corresponds to neglecting Eq. (D.6b) and minimizing with respect to the single remaining parameter M .