 Well, I wish to thank, of course, the organizers for giving me this opportunity. And I will start with a little introduction, I mean, like Vincent before me, perhaps to apologize, because, well, you can ask, why is this topic of any relevance for this workshop? Well, as I will try to argue, my excuse is that by looking at this gedanken experiments at energies much larger than the Planck scale, and in a regime where collapse is not expected, I emphasize that, of course, we arrive at the next matrix, which is, on one hand, unitary, so information preserving, and on the other hand, it shows the emergence of the Hawking temperature scale, which is something much smaller than Planck. So, and in fact, the farther you go in the incoming energy above Planck, the lower you go in this characteristic energy scale, which will be the characteristic energy scale of what comes out of the process. The outline is this. It's in two parts, rather disconnected, so if time doesn't allow, then I will skip, depending on the, also, how many questions are asked, and so on and so forth. So I don't want to rush, but I have about a little over one minute per slide. I don't know if I can make it in 50 minutes. We don't know how many slides you have. Yeah, well, yeah, about a little less than the number of minutes I have. So, sorry. So in part one, we'll try to solve an unsolved exercise, which should have been a textbook exercise, but it's not. So maybe in a future textbook, it will appear. And this, I will deal with this problem in two ways. One is purely classical GR, and second is a quantum s-metrics approach, and then I will compare the results, finding actually very good agreement or perfect agreement, I would say, of course, by taking a suitable limit of the quantum calculation. And then I will discuss the gravitational wave energy spectrum. We can also look at the angular spectrum, but in particular, the energy spectrum and show that the integral of this energy spectrum still has a logarithmic divergence. Of course, this, we believe, is due to some breakdown of some approximations at high frequency. And then, part two, it will be discussing a claim by Diwali Gomez et al. And with Massimo Bianchi and the student Andrea Adasi, we have tried to reinterpret what they did, and I think the conclusion is quite interesting. And then I'll finish with a question. So the textbook exercise is very simple. Well, but let me start by saying, you know, the problem of computing gravitational wave emission by binary system is almost as old as GR. It has become gradually very relevant for testing general relativity first. So remember the Pulsar's test of general relativity, and later for searches of gravitational waves, first in bars and interferometers, and of course, even more recently after the observation of gravitational waves emitted by the coalescence of two black holes. For instance, there is this effective one-body approach that Thibault and Alessandro Bonanno have pioneered. And they are, of course, the numerical relativity calculations. But most of the time, this process is in the non-relativistic regime, with the exception, perhaps, of the merging itself when high speeds, you know, roughly, say, V over C of the order 0.3 to 0.6 are reached. Now, of course, much less attention has been devoted to a more academic problem, which is the following. Consider the collision of two massless or highly relativistic gravitational interacting particles. We neglect every other interaction. In the regime, which is explained, deformed a little bit later, in which they deflect each other's trajectory by a small angle, theta Einstein. And with theta Einstein, much less than one, so, but still much bigger than the inverse of gamma, which is very small. Now, theta S or theta Einstein for small deflection angle is given by this generalization of Einstein's formula. My finger is too big. So, here is the definition of R, the Schwarzschild radius associated with the center mass energy. Now, the problem is to compute the gravitational wave spectrum, which is associated with this collision, to lowest order in the deflection angle. And I remember I was having lunch at the New York Univet, NYU, next to Andrei Gruzinov, and he asked me, how can it possibly be an unsolved problem? He asked me. Yet, I had asked several experts about whether that problem had been solved, and nobody gave me a solution. So, we decided to work on it. So, the classical general activity calculation is based on work I did together with Andrei Gruzinov. Now, what do we actually know about gravitational wave emission in general? Well, there is a well-known zero-frequency limit. You know, it goes down to the 70s, which gives us a solid prediction for D e d omega. As omega goes to zero, this limit is smooth. It goes to a constant, this ratio. And it can be obtained either by classical or by quantum argument, and the latter uses well-known soft graviton theorems, which go back to the sixties to Weinberg's work, for instance. And the result for the process at hand, namely this high-energy relativistic collision with small deflection angle is this, okay? Is this for the omega gs over pi with precise coefficients? You see, it goes like the square of the deflection angle up to a logarithmic enhancement, and this is the omega going to zero limit. Now, by the way, an interesting, this is a parenthetical remark, again with Adasi and Bianchi, we are trying to see whether the sub-leading corrections to the soft theorems that now people are working on, you know, several people, whether that can give the first correction to the spectrum, it is not quite obvious to us. And then to compare with what I will tell you later, because we pretend to have the spectrum up to a certain frequency, so to check whether the sub-leading soft corrections give, indeed, what we find by other methods would be interesting. It's still work in progress. Now, the other thing we know is there is some work of the, in the 70s, in particular by Peter Diaz, Diaz and Payne. They wrote several papers. This is very interesting work. They discuss mainly a regime which is not very interesting. It's a regime in which the deflection angle is smaller than one over gamma, since our gamma will be infinite. That is not so interesting. But in this paper, they also, he also has interesting remarks about the regime we are interested in. And actually the result I will present can be obtained by suitable limit of what Diaz does. So this is yet another check that we are on the right track. On the other hand, Kip Thorne and Kovacs in about the same time, you know, had a nice paper saying that this gives a definitive treatment of classical gravitational Bremstra-Lung produced by two stars at arbitrary mass flying past each other. So not merging, but just flying past each other. But you see only if the angle of gravitational deflection is much smaller than the inverse gamma to the minus one. So this is precisely what we are not interested in. So unfortunately, this very nice paper doesn't help. Third, there are numerical relativity calculations by the group of Pretorius, Spearhake, and also others. We, I mean Grozinoff and myself contacted them and they admitted they cannot cope with this numerically. And the calculation is challenging because basically the deflected particles, you know, when you have these two massless particle collide and then deflect each other, then they have associated with them two shock waves. And the two shock waves travel as fast, almost as fast if they are ultra-relativistic and as fast as if they are massless, as they emitted waves themselves. So apparently numerical is very difficult to disentangle the actual Bremstra-Alun radiation from the shock waves produced by the final particles. Maybe they will make progress in the future, but for the moment they seem to be limited to Lorentz-Gamma-Fatos of all the three and theta deflection angle a bit bigger than, or less, maybe less than gamma to the minus one. Okay, so now a first attempt to this problem can be found in a paper I wrote with Amati and Ciaffaloni in 2007, which however produced an energy crisis because we got, perhaps as we learn later by a too naive an argument, a spectrum of the gravitational Bremstra-Alun which has the right pre-factor but has the wrong exponential, or rather in this exponential we like very much this damping factor in transverse momentum with respect to the beam axis of one over B. B is the impact parameter of the collision, I should have said. But this damping in frequency is too generous because remember we are at very small angle, so r square over B square is very small. So that means this is a very weak damping. And if you integrate this energy over omega you get very strange results. You find that the energy fraction emitted in gravitational waves is already or the one for impact parameters still much larger than the Schwarzschild radius, namely at small deflection angle. And in particular Slava Ritschkov was making this remark to us smelled very bad, okay? Something looked wrong. On the other hand, okay, it came out of our calculation. Now of course, okay, I don't have time of course to review the work I did with Amati and Ciaffaloni but in this 2007 paper we were making some strong approximation trying to get to the critical points for collapse and so on. It was not the main aim of that work to find the spectrum. It was a byproduct, but this byproduct could have been greatly affected by the approximations we made. So bottom line, we need the gravitational, sorry, the GR answer to the question what is the actual cutoff in omega for the gravitational waves emitted in such a process? And there are related questions. For instance, is the, I mean, this cutoff in omega, is it singular in the massless limit? Second question, is it singular in the classical limit? Okay, because you could invent all sort of scales for this cutoff in omega. I mean, you could say it's 1 over b. You can say it's 1 over r. Or you can say, you know, it's, I forgot. I mean, you can take 1 over b times gamma if the limit m goes to 0 is singular or you can use quantum mechanics and you can say that omega cannot be bigger, of course, than e over the total energy square root of s over h bar, okay? In this case, the massless limit is singular. In the second case, the classical limit is singular. My answer to both questions is no, no. I don't think those limits are singular. As far as we know, gravity has no collinear singularities. In other words, the emission of gravitons is not singular when the emitting particle goes to zero mass, unlike what happens in gauge theories in four dimensions. And also, I don't believe that you really need quantum mechanics to regulate this problem. So the recent progress announced in the title, as I said, goes back to, yes. There are soft logs. There are soft divergences but no collinear. So if you take QED, then there are two kind of infrared effects. One is the soft divergences and those are common to gravity and gauge. And one are the collinear divergences when the emission is collinear, which is precisely what happens if you have massless particles. And Weinberg already noticed that in gravity the second kind of divergences are not there. I'll come back to that. Okay, so as I said in the outline, I will present first the classical calculation that I did with Kruzinov and there is parallel work by Spirin and Tomaras and in regions where we could compare the results, there is also agreement. And then I will come to the quantum calculation. The simple classical treatment that we did, I must say mostly the idea came from Kruzinov, is obtained applying the Huygens principle in the following way. I tried to draw a picture with a keynote, but it's a bit messy so you don't have to look at the whole details. Just to get the feeling and then we'll see the formula. And you'll see in the formula, certain quantities that appear in the picture will be in the formula. So this represents supposedly one shock wave is one particle which travels as z minus equals zero, namely z equals t. z is this direction. This is the other particle in green which travels as z equals minus t and they smash into each other as z equals t equals zero. These two shock waves, then they are deflected and I show only the deflection of the left moving guy, sorry, right moving coming from the left, which is this red thing. So you see the black plane gets a little tilt. That tilt is the deflection angle and then the particle keeps going. This is this blue line. Now I observe the radiation at any given direction in that hemisphere, not necessarily very close to the direction of deflection and we reconstruct the far field by organs, namely by taking every point in this black plane as an emitter of a wave and then with some over all the, we interfere all these waves. But when you do this, you have to be very careful with phases. And they are geometric phases simply due to the fact that, okay, depending on where you look, some points are farther than others from the point where you do your calculation. But on top of this is also two inoptics, but if you are in gravity, you also have to take into account time delays. Okay, so there are so-called shifts that you have to apply. For instance, just to make one well-known example, this Z minus is the time shift. Time is actually this particular combination of space and time, which gets shifted as this particle hits and contrasts the other particle at impact parameter B. It's minus 2R log B. This is the shift. However, okay, the wave which is emitted in the sense of organs by this point X, you know, will suffer a gravitational shift which is not log B, but log of this, of the length of this vector X minus B. This log of B minus X squared. Now, so you have to combine all these waves and you have to subtract the contribution from the shock wave. So you have to look at the metric, but then subtract the contribution to the metric coming from the shock wave because that has nothing to do with radiation. Okay? That would be there anyway. The formula is not very simple, but it's not very complicated either. I can write it in, as you can see in one slide. There is some little bit of notation. Now, this is just general, okay? This observed quantity is related to what I call news functions, okay, which you have, we define a S cry plus if you want, but very far from the source. And so this is a crucial object. It takes the form of a two-dimensional integral because you integrate over all these points on this plane and then you reconstruct all these waves which contain, as you can see, the gravitational retardation effects plus pure geometric effects. And what is important to notice is that when you subtract the deflected shock wave, you get something nicer than if you don't because, for instance, if you go to small x, these two exponents become the same and so this object goes to zero and x goes to zero. You see, it removes some singular behavior coming from this object. This object is nothing but the Riemann tensor associated with the shock wave projected along some direction, okay? The shock wave has no reach but has Riemann associated with it and it's quite simple. Okay, so this is the explicit formula. And now you see you can study the emitted radiation both in terms of omega and in terms of the deflection, of the observation angle and we prefer to write it not in terms of the observation angle but this theta tilde, which is the difference, which is the observation angle with respect to the direction, the deflected shock wave, okay? Because it seems to play a more important role than the angle itself. Okay, now I will give you the property of this pattern but okay, before doing that, let me get to the quantum treatment of the same problem in very simple terms. So in these two papers with my Florentine friends, Ciaffaloni, Colferai and later on a third C, looks like the Soviet Union or CCV. The same problem is addressed at the quantum level and so improving on what I told you before, in 2007 we got this energy crisis. Now, there are important points physically, I mean before going into any details. There is one observation which I think is crucial. In the usual soft-graviton recipe, Weinberg 65, you only take emission from the external legs to leading order, for very small soft-gravitons. You only take the emission from the external legs. Now indeed, when you look at the sub-leading terms, there is also a term which comes from internal emission which is needed for gauging variance, for covariance. But we don't know yet whether that is related to what we are doing here. But certainly we do keep emission from the internal legs as well if the gravitons is not extremely soft. And the reason is that the internal gravitons, I don't know if I have a picture of the ladder. The elastic process before we talk about any emission is determined by some of ladder diagrams. Now the ladder diagrams of course has ranks along the ladder. These are the gravitons you exchange between the two very energetic quanta. So usually you would not include emission from these exchange gravitons because of the usual recipe. But what turns out to be the case is that these are very close to the mass shell. So they are almost on shell. They are almost like the external particles. The technical reason for that is something which has been called by Giddings, I think, fractionation of the exchange transverse momentum. This is perhaps a point worth spending a few words. If you have this high energy collision and the deflection angle is finite, even if small, of course the momentum transfer in the process is huge. If you had a single graviton carry out all this transverse momentum, that graviton would be far off shell. And there would be no reason to include emission of other soft gravitons from that off shell object. But as it turns out, what dominates this process is the exchange of so many gravitons that each one of them is very close to the mass shell. So for this reason the emission from the exchange gravitons cannot be neglected for not so soft gravitons. But these not so soft gravitons are very crucial to compute the energy loss. So if you are only interested in the zero frequency limit, fine, you already know the answer. If you are interested to know the spectrum in some range of omega and integrated to find the efficiency of the process for producing gravitational waves, then you are facing this problem. So in fact I added this question mark. Is this taken care of by this non-leading correction to the soft theorems? This is what we are trying to check. But we don't know the answer yet. Just in terms of diagrams, the fact that you emit from the shock wave which is separated from the particle. Because the particle is still very classical for me. Yes. Well, you mean when I drew my picture? It looks like this on-shell effective. Yeah. I think there is some point to it. Namely that there is a relation between emitting from the full shock wave and from the interior of the graph and the fact that not only the insertion on the external lines count. I think there is a relation. Now, the exact details I don't think are clear but there is certainly a relation. Now, the second important point is and this is where the new scale appears, you know, omega of order 1 over r. So a very small frequency. It appears because of some decoherence effects. The reason is the following. The emitter gravitons can be emitted from any rank in the ladder. However, if omega is smaller than r to the minus 1, this emission is coherent. Namely, it adds up coherently. But if omega is bigger than r to the minus 1, then it turns out that there is a suppression. If you emit the graviton from, say, the first rank and you want to reabsorb it, you know, when you do your S matrix and then you square it. You want to try to interfere this with the absorption from the last rank, then you find a mismatch of momentum transfer. Because simply the territory has bent and so on and so forth. But we can see that in the mathematics, of course, but this is the physical explanation. So this produces a damping, a further damping of the spectrum from omega bigger than r to the minus 1. So this characteristic scale, this is the Hawking scale, the Hawking Radial scale, appears at that point. So, and this decorience causes a break from this flat spectrum. I told you that at omega equals 0, the spectrum is flat, is omega independent. Well, above r to the minus 1 is no longer flat. It just starts to fall, which is a nice, very interesting feature. Now, okay, I want to make this story short because at first, in the first paper, we were taking only this effect into account. We got something which was close, but not quite what we were getting with the classical calculation. The same function, if you, I don't know if you paid attention, you may be too tired. The same function that appears in the classical treatment appears, but there it was exponentiated. Here it comes in the numerator. However, we then discovered that we were missing another effect. And this is simply this iconal phase. And when you meet a graviton, say, instead of two bodies, you have three bodies, and you have to look at the face, at the iconal phase for this three-body system, which consists of three terms, and this is not quite the same as the two-body phase. You take also this into account, and then is the emitted graviton. You look at the single graviton, so you have two particles plus the third. Now, you see, if you have only two, we know the iconal phase. It's just a function with a log of B squared and with a pre-factor. Now we have to do the same for the three pairs of the three particles. It doesn't add up exactly the same thing. And when you take this correction into account, magically, you get exactly the classical result if you take, I don't know, I don't even have it here. You get exactly the same result that we got from the classical calculation. I mean, there is this limit to be taken, but it's really very trivial. I mean, the quantum calculation gives this. Of course, there is an H bar here if omega is a frequency. So you expand this quantity for a small argument and then you get the classical result. So that's the only little step from going from the quantum to the classical result. Okay, so I'm half an hour, slightly late, I guess. So let me describe the spectrum. We have analyzed mostly numerically the properties of the spectrum in the classical limit and let me illustrate it first by saying a few things and then I have some pictures. So the spectrum is like this. For omega between 1 over B and 1 over R, these are two very distinct scales because the ratio is very small. The spectrum is almost flat, okay? It's flat up to a log. That's what we find. However, the approximations really, this Huygens-Franholfer approximations breaks down and we believe that below this scale, this log freezes down, okay? You cannot go just to 0. You can go down up to omega or the B to the minus 1. If you insert that value, you get precisely the 0 frequency limit. So what we believe is that the validity of the actual 0 frequency limit is limited by 1 over B, okay? Up to 1 over B, it's pretty safe to use this. Above 1 over B, but below 1 over R, this is the result we get. However, above R to the minus 1, because of this decoherence effect, and okay, I can explain more detail, the spectrum becomes, if you want scaling variant, I mean in this simple sense, that D, the omega is like E over omega up to this theta square factor. So it's small in scales like 1 over omega, which of course gives a lot, if you pass this up to arbitrarily high omega, gives an efficiency which diverges logarithmically. Whereas previously with this constant spectrum, divergence was linear. Okay, now we don't know exactly, we know that our sum approximations break down and they break down roughly at the scale omega star, which is 1 over R divided by the scattering angle to the minus 2. Now, another amusing thing is that if we use that formula above that scale, then you start to violate something which is not a sacred bound. It's called the Dyson bound. I think this I learned from Gary that is called Dyson bound, that maybe there is a bound on D, D, T which has precisely the dimensions of 1 over G Newton. Now, it's not clear that this is an absolute bound in nature, but it could very well be that it's a true bound for this process. So if we assume that kind of cut-off, then of course you can even compute the total efficiency for emitting gravitational waves and it comes out like that. And we argued with Gruzinoff that above this pattern should become omega to the minus 2 instead of omega to the minus 1. And then, okay, we later realized that 1 over omega squared, I don't know if you know, is also the spectrum of time-integrated spectrum of the evaporation of the Schwarzschild black hole according to Hawking. If you collect all your quanta, then the spectrum ends up being 1 over omega squared. Okay. This is, I will skip it, is an explanation of how you get these different behaviors in these different regions. There are regions in which you can trust Weinberg's approximation. Other regions you have to go over to work by Lipatov that can be considered as an improvement of Weinberg when the frequency is not so small. And, okay, this is a plot, you know, frequency and also the scattering angle. So it depends also. What I gave you, it was integrated over the angle and this more differential description explains better how the various regimes behave. But, okay, you can look it up in the transparencies. And, okay, there are some nice pictures. Fiss revs apparently like them and they put them in some gallery of figures that they have. I don't know that's what my collaborators told me. But, for instance, I think it's an, oh, sorry, this is an amusing picture. So this would be the spectrum as a function of omega is integrated over azimuth, until this theta over theta s. So maybe it's theta tilde. I'm not sure. I think it's theta tilde over theta s. So this is the frequency. This is the angular distribution. And this is the plateau, the zero frequency plateau. But if you go to zero, which means omega r equal one, the plateau ends. And the other steep, you know, these steep curves, one is due to phase space and the lack of collinear divergence and the other is the transverse momentum cutoff. This is simply kinematic boundary. So, basically, the efficiency comes from integrating over this plateau but there is this one over omega tilde that still, okay. Then, okay, as I say, we had a nice picture where we, for different values of omega, we give the full angular distribution in theta and phi. Now, we now want to understand two things. What provides the large frequency cutoff and also would like to extend the reasoning towards large angle of the collapse regime and there have been some first steps done by Ciafalone and Colferi in a recent paper. Now, what I want to stress is that the emerging picture seems to be quite appealing, namely, the transverse momenta are limited by one over b. The longitudinal ones are controlled by the largest scale, one over r, with some leakage at high frequencies. Now, if that behavior persists, as you approach some critical input parameter at which collapse should occur, that would give a rather interesting final state because the distribution will become more and more isotropic. The average number of quanta that you produce is of order of entropy of a black hole and the characteristic energy is of order of the Hawking temperature. So, I think this ends part one and I have 15 or so minutes. So, maybe I will go quickly over the second part because, okay, it's unrelated but the conclusions are similar. So, it adds to the whole picture. So, the second part is about the claim, a recent claim by, well, not so recent by now, a claim by Valiant Company, here is the full list. It's in D equal 4, like also in my previous discussion, no attempt to project on a fixed input parameter or a fixed partial wave. So, if you want, well, I didn't show this before, but in a plane in which you have the energy and the input parameter of the collision, okay, with Amati and Ciappalloni, we're always discussing different regimes in this plane. Now, they integrate over input parameter, okay. So, they look at the whole thing, hopefully also picking up contributions from this right region which is where classically you expect to collapse, okay. This is the, rather standard picture. So, what they have considered is an interesting process. They took two initial, so the initial state is precisely our initial state, two ultra high energy gravitons, okay, and they looked specifically at N final gravitons where N is so large that the energy of the final gravitons is ultra low energy instead of ultra high energy. Now, typically, but this is only typically, they would like to come to this as a prediction, typically the energy of the final particle would be of the order of the Hawking temperature associated with the incoming energy. And they claim that in both in quantum field theory but also in string theory, they can estimate the three level, I should have underlined three level cross section at large N by, you know, some smart techniques about computing scattering amplitudes, how are they called, scattering equations or string equations, I forgot, okay. String equations, okay. So, they came up with this, with this result for very large N, which you can roughly understand. This N factorial is typical in field theory when you have two to N processes for large N. Then this factor can be understood because the average invariant energy of two particles is square root of S over N. So, that's a typical interaction between two of the final gravitons to the N because that's the number of particles. So, by the way, this fudge factors, E squared C, oh, C is not the speed of light, C is the constant over the one. E squared is put for convenience. E is E. Yeah, yeah, it's put for convenience because then if you take the large N limit, E squared disappears and it goes like this, okay. For large N, you can think of it this way. Now, then this for me can be roughly understood, as I said. However, with an argument that I cannot follow, we have been corresponding with them, they picked up a precise value of N and they picked up this value of N. Okay, if it's N equal just the numerator, then you work it out very easily. You get a cross-section which goes like E to the minus N. And N, okay, is the, this N is of the order of course of the entropy, the black hole of mass square root of S. Now, but this N does not correspond to the dominant N in the sum, okay. If you take the N which maximizes this expression, that is not at this value, but at this value, you see they differ by a factor E squared. And there the cross-section is exponentially large, not exponentially small, so it makes a big difference. Now, then they say they add a final state entropy factor E to the plus S to kill this E to the minus S and they argue that somehow this final state can saturate unitarity, but you know, this final state entropy factor is put by hand. What is the parameter of small S? Small, no, small S is E squared. It's the spandestum variable. It's just the center of mass energy squared. Yeah, sorry, should have defined everything. So, like it was in all my previous slides. Okay, this is also the mass of a black hole that you supposedly form. And the reason why we take, by the way, trans-plankian collisions is that we would like to form black holes larger than a plank length. Okay, and that's why you need trans-plankian energy. Okay, so what we did with Okay, you can criticize very much. I mean, this argument because exclusive cross-sections have infrared singularities. So, of course, at three levels, you don't see them. But if you add loops, you expect to have all sorts of divergences in the loop corrections. And at three levels, the exclusive cross-section should blow up because of soft quanta. And, or at fixed multiplicity, the virtual corrections after they are resummed, they make any exclusive cross-section go to zero. So, you only, you have to define suitably inclusive enough cross-section which are free from infrared problems. And so, that's what we did with Adasi Bianchi and Bianchi, and we gave an interpretation of their results that seems to resolve its tension with the other work and even eventually justify qualitatively at least their basic claim. So, the rest of my talk was about how to define these inclusive cross-sections in the case of gravity where we used the fact that there are no collinear singularities. So, in a sense, the problem is simpler. You know, you only have to allow for emission of extra stuff, soft stuff. So, you have virtual corrections and real emission and, as usual, they cancel each other in suitably defined quantities. So, we define something like gravitational jets, but they are not jets because of the absence of collinear singularities. So, you only look at how many hard quanta you produce and then you don't look at the softer quanta and you put some bound on how much energy can go into this unobserved soft quanta. So, this is an infrared safe quantity and, by the way, we found, this could be amusing, we found a nice formula for the gravitational, well, this is the virtual correction, the virtual graviton corrections to the 2-to-n process which, as Weimberg argued, has a smooth massless limit and the massless limit is, you never wrote it down even for 2-to-2 processes, has this nice formula, s log s plus t log t plus u log u, if you want. But, I mean, if it was a 2-to-2 process, it would be s log s plus t log t plus u log u, which, by the way, is the same exponent which appears in the beta function at fixed angle. But this is the generalization to 2-to-n and so we studied this B0 factor, which is the damping factor due to the virtual corrections. It is always positive and, for instance, if the process is completely collinear, 2-to-n, but everything goes in the same direction, this B0 goes to zero, but that's the only case in which it's zero. And if, for instance, the particles are emitted on the transverse plane with respect to the collision, then it maximizes and so on. Anyway, the bottom line I want to jump to it is that, again, the scale 1 over r appears naturally. In fact, if bar e, e bar is bigger than th, e bar is the energy threshold for calling it a jet or a hard graviton. Then if you take this threshold too high, higher than the Hawking temperature, then that cross-section, it will be damped and will be very small. So you have to allow, indeed, for quanta like Hawking quanta and then eventually find single jet cross-sections which have even the Hawking factor. Okay, I want to cut this story short. And John, so if you want a conclusion or even a question, you could say, we would like to say that perhaps we are talking here about pre-collapse, which could be the analogy of QCD's pre-confinement. Many years ago, I had a paper with Daniela Mati pointing out that the perturbative evolution of a QCD jet produces a partonic state which consists of many color singlets of limited mass and resembles energetically the adorned final state. But of course, some non-perturbative physics was still necessary to get down to hadrons. And present codes, for instance, Herwig uses very much this property of QCD jets. It adds to it gluon interference. So you see interference also comes in there. In fact, this IG in Herwig means interfering gluons. Now, in the gravity case, a general pattern seems to emerge where at the quantum level, the transition between the dispersive and collapsed phase is smoothed out. Classically, you either collapse or you don't collapse. And here somehow, as you approach the collapsed region, it seems that things change only gradually and smoothly. So as one approaches a critical value for B, for the impact parameter, the nature of the final state appears to change smoothly. Maybe rapidly, but smoothly, from one characteristic of a dispersive final state to one with high multiplicity and soft quanta, which is reminiscent of Hawking's radiation with high multiplicity and average energies of the order of the Hawking temperature. Of course, it's always up to numbers of order one. But even in this case, I didn't show exactly the spectra, but the spectra are not thermal. The thermal Planck spectrum is quite depressed at small frequency. We get more like a Bremstrahlung spectrum, which has more quanta. It has a divergent number of quanta at low energy, although the energy carried by this low quanta is not divergent. Now, a Planck spectrum is much, much... I cannot call it softer, much less damped at small frequency. So we don't see any sign of thermalization. Again, here, even if we can get to the stage, we need some non-perturbative physics for thermalization. So this is the classical picture, and we think that the quantum picture may be smoother. I couldn't talk about related work in which in this so-called string-gravity regime we can also try to approach this classical boundary between forming or not forming black holes. And here, too, you see this softening of the final state and the emergence of the Hawking temperature scale. Okay, I think that's all I have. Thank you. If I sit along the critical line to form black holes, what do you predict about the variation of the total emitted energy in gravitational waves? Is there a regime where it is small compared to square root of X? Or it's always equal of the order of magnitude of square root of X? Okay, so I think if I understand... So forget about this. This is a string regime which we haven't discussed. So the regime that we have tried to describe is following this arrow. This is really when the skating angle becomes over the one and later you could enter the black hole formation regime. And the mass energy loss is then equal to square root of X all along this line? Not exactly square root of X. No, no, no, it should be a finite fraction. I think everybody... Is it a finite fraction? No, we have no control over... You see, that would depend on two things. On being able to find the true cut-off omega star. One, second, to be able to really approach that line. That line corresponds to skating angles over the 2 pi. The party will start to coalesce. So at that scale... Okay, this is what Colferai and Ciaffaloni have done in this last paper, but admittedly it's a lot of guesswork. So there I think everybody, I don't know your... Expects that there is a finite fraction that goes into radiation and the rest collapses. Now the puzzle was that we were getting the same kind of result even much, much before when the angle is small. Now the result is more comfortable, but before we know exactly where to cut off. In my opinion it's just that we make some linear approximation to some equations and we neglect some nonlinear terms and those should fix the problem. But it's not an easy... Ask me what's wrong with this intuition. I have the intuition if you had two absolutely massless particles well suppose it's classical that the things will come in and they'll just spiral and it'll have gradually approached some thing with a conformal killing. Anyway that'll just become self-similar and it'll just radiate all its energy away before it. So I would have thought in the limit of massless particles that could radiate all of their energy away before at the end you just get a zero mass black hole. Really? And then if you have less in back parameter then you'd form a bigger one. So I don't know that... No, I mean, well no. I think at least at sufficiently large input parameter there is simply deflection. That I think is... The critical case. Oh, the critical case. I'm suggesting at the critical case they just spiral in and so therefore each time over a time scale it's of order the energy remaining then it emits roughly that energy. I mean it just spirals in at some fixed angle. I understand what you mean. It could very well be that in the critical case the whole thing is radiated away and you're left with no black hole. The small parameter for the fraction that you emit at theta for the wall. Well, yeah. I mean it would be actually a lot of emission. Now you say there would be a lot of emission of order one. I guess I'm just imagining that it emits of order a fraction of the energy it has each revolution. Yeah, I understand your argument. Yeah, because you would like to form an extremal black hole but there will be a lot of emission that finally prevents you from forming it at all. That's what they are saying. Yeah, you just radiated away. Whereas if it's really central... No, then it is. Well, a little bit more. You could very well be true that what signals critical behavior is when you are... If I knew I would do numerical calculations it would be an interesting thing because you could have this cell signal solution. In a sense, continuity would also support what you say, right? Because before you collapse you certainly send all the energy out to infinity. Now you say as you approach the collapse there is no first order transition that all of a sudden you get half of the energy radiated. Before you collapse, certainly everything comes out, right? Everything that goes in comes out. This is the equality between... Just saying in the limit, of course... Yeah, I understand. ...if you create each other then of course you may have a fighting fraction of the initial energy. But I'm saying in this limiting case... Yeah, yeah, yeah, yeah. Which makes sense because it's... Yeah, if you go just a little bit above you don't collapse at all and then all the energy also goes to infinity. Yeah, I think you have a good point. Why... How is it possible to obtain a Hawking... something that could be identified as Hawking radiation from a classical calculation? Because of course there is the impact parameter that will give something that will be H-bar, which is a nine-year-old matter, but what is north? Oh, I don't... You mean the classic... Now you are referring to the classical calculation, right? Well, first of all the classical calculation but then how is it possible to control the quantum calculation to make sure that this is Hawking radiation, not some other kind of radiation? I don't know. I mean what I can say is that my collaborators are trying to keep the quantum part of the calculation. Well, of course I think if you do an honest classical calculation you shouldn't get any radiation out. Yes, and there should be a sharp transition between the regime of collapse and the regime of no-collapse. These are these collapse criteria. I mean on this basis I drew this critical line. Yes, so I mean somehow the quantum must be important. I think that for the radiation a small deflection angle is probably okay but as you approach the critical regime who knows? But in the power it will look smooth because even with the collapse the summer heat will go away and the radiation will wear. Okay, if you only... With big things, you know? Yeah, yeah. No, you see, I suspect that you at some point you need some real non-perturbative physics that you form some long-lived metastable state and that is very hard to see how you could get it from here. Some kind of bound states. Okay, non-perturbative. Yeah, I presume. So as I said, okay, it's not about really black holes. That's why I apologize in the beginning but I think the fact that you already see the emergence of some scales it's interesting. You see it's very different from a electronic reaction in which the energy of the final state scales with the energy of the initial state. Okay, you have Feynman's scaling. You look at single particle distribution in adorned physics. They typically scale, you know, the energy of the final particle is proportional to the energy of the incoming particle. Here is the inverse proportion. That's what comes out. And this is because the effective coupling constant is Gs. So it contains two powers of energy and that is also the typical multiplicity. So when you divide square root of S by S you get one square root of S. It's as simple as that. Just a small comment. In the semi-relativistic regime of LIGO where it did merger the velocities, half the velocity of light the instantaneous gravitational wave flux is millidison. It's 10 minus 3. Millidison. So you could think it's still relativistic. Things are for the unity so you could get very non-prolucrative. But this is a regime where there is still a small parameter and one can describe even the transition, merger the transition to buy very simple, I mean, EOB and describe everything. So I would not I would still expect that even in the ultra-relativistic thing, gravity as this property agreed with the value for this of containing its own cut of softening I think so too. And then we would not see very small things. That's my expectation. Somehow general relativity should know how to cure this problem on its own without invoking quantum mechanics for this problem. Thank you very much.