Gravitational bremsstrahlung from transplanckian-energy collisions: a progress report
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Veneziano, Gabriele
Gabriele Veneziano - Gravitational bremsstrahlung from transplanckian-energy collisions: a progress report
 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 Van Sam 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, that energy is 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. 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 whole zone, 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. 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 as 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 the valley Gomez et al. And with Massimo Bianchi and his 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. 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 emerging 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 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, 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 ops of Einstein's formula. So here is the definition of R, the short 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 University, NYU next to Andre 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. And so the classical general activity calculation is based on work I did together with Andre 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 dE d omega. As omega goes to zero, this limit is smooth. Okay? 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 60s to Weinberg's work, for instance. And the result after the result for the process at hand, namely this high energy relativistic collision with small deflection angle is this. Okay? Is this for the d 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, and interested, this is a parenthetical remark with, again, with Adasi and Bianchi, we are trying to see whether the subleading 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 subleading 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 Dierth, Dierth and Payne. They wrote several papers. This is very interesting work. They discuss mainly a regime which is not very interesting is 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 Dierth does. So this is yet another check that we are on the right track. On the other hand, Kipthorn and Kovacs in about the same time, you know, had an ice paper saying that this gives a definitive treatment of classical gravitational Braemstrahlung 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 contact, we, I mean Grosinoff 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-ethvistic and as fast as if they are massless, as they emitted waves themselves. So apparently numerical is very difficult to disentangle the actual Braemstrahlung 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 factors over the three and theta deflection angle a bit bigger than, bigger 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 Braemstrahlung which, okay, has the right pre-factor but has the wrong exponential or rather in this exponential, we like very much this damping, 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, 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 short-shield radius, namely at small deflection angle. And that 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 Ciafalone, but in this 2007 paper we were making some strong approximation trying to get to the critical points for collapse and so on. Was not the main aim of that work to find the spectrum, 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 sorts 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 primstralo. So the recent progress announced in the title, as I said, goes back to, yes. There are no logs. There are soft logs. There are soft divergences, but no collinear divergences. So if you take QED, then there are two kinds of infrared effects. One is the soft divergences, and those are common to gravity and gauge. And one, 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. OK, 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. So 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. But 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 that's 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, okay, 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 of the, we interfere all these waves. But when you do this, you have to be very careful with phases, okay. 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, you know, time is actually this particular combination of space and time, which gets shifted as this particle hits, you know, encounters the other particle at impact parameter B. This is 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. So this is 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. So this is the formula. It's 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, this quant, this observed quantity is related to what I call news functions, okay, which you have, we, you define a, a, a, a sky plus if you want, but very far from this force and so this is the crucial object. It takes the form of a two dimensional integral because you integrate over all these points on this plane and then reconstruct all these, these, these waves which contain, as you can see, the gravitational retardation effects plus pure geometric effects. And what is important to, 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, this object goes to zero at x goes to zero. You see, you know, removes some singular behavior coming from this object. This object is not, nothing but the Riemann tensor associated with the shock wave, you know, 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, you know, 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 CCCV. The same problem had this address 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 and which is needed for gauging variance, for covariance, okay? 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, okay? 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. Now usually you would not include emission from these exchange gravitons, okay, 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, okay? 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. But 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. Do you understand 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 you. Yes. Well, you mean in the same way that I drew my picture? It looks like this on-shell effective. Yeah. Yeah. Yeah. Yeah. Yeah. 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. Yeah. Yeah. Yeah. 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. Same note. Okay. 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, okay? You want to find the barrier. You try to interfere this with the absorption from the last rank, then you find a mismatch of momentum transfer, okay? 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 for omega bigger than r to the minus 1. So this characteristic scale, which is the Hawking scale, the Hawking radiation scale, appears at that point. So and this decoherence causes a break from this flat spectrum. I told you that at omega equals 0, the spectrum is flat. It's omega independent. Well, above r to the minus 1 is no longer flat. It just starts to fall, okay? Which is a nice, very interesting feature. Now, okay, I want to make this story a little bit 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 have, you emit a graviton, say, instead of two bodies, you have three bodies and you have to look at the phase, 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, what 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 is just the function of, 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 the, 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, it 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. Okay. We have analyzed mostly numerically the property of the spectrum and 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 over the b to the minus 1. If you insert that value, you get precisely the zero frequency limit. So what we believe is that the validity of the actual zero 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 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 and scales like 1 over omega, which of course gives a log, 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 this, at a scale, omega star, which is 1 over r divided by the scattering angle to the minus 2. Now, another thing, an 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 cutoff, then of course you can even compute the total efficiency for emitting gravitational waves and it comes out like that. We argued with Kruzinov that above the spectrum should become omega to the minus 2 instead of omega to the minus 1. And then, okay, we later realized that 1 over omega square, I don't know if you know, is also the spectrum of time integrated spectrum of the evaporation of a short-shield black hole according to Hawking. I mean, if you collect all your quanta, then the spectrum ends up being 1 over omega square. 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. Another region 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, so there are some nice pictures. Fiss Refs apparently liked 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, this is, I think it's an, oh, sorry, this is an amusing picture. So this would be the spectrum as a function of omega. It's integrated over azimuth, but still 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, then all of a sudden the plateau ends. And the other steep, you know, this steep curve, 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 t that still. Okay, then, okay, as I said, 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 we'd 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 impact 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 over 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, but claimed by Thalian 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 impact 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 impact parameter of the collision, okay, with Amati and Ciappalloni, we were always discussing different regimes in this plane, now they integrate over impact 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 other 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 some smart techniques about computing scattering amplitudes, how are they called? Scattering equations or string equations, I forgot, okay, string equations, okay. So they come up with this result for very large N, so 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 large factor is square C, or C is not the speed of light, C is the constant of the one, E squared is put for convenience. It's E is E. Or there's no. Or there's no. 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 I 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 the, 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 this value, but that this value, you see they differ by 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, they argue that somehow this final state can saturate unitarity. But you know this final state entropy factor is put by him. So what is the parameter of small S? Small, no, small S is E squared, is the, is the understand variable, is just the center of mass energy square. Yeah, sorry, should have defined everything. So like it was in, in all my previous slides. Yeah. So, okay, this is, is also the mass of the black hole that you supposedly form. And the reason why we take, by the way, transplankian collisions is that we would like to form black holes larger than a plank length. Okay. And that's why you need transplankian energy. Okay. So, what we did with, okay, you can criticize very much. I mean this, this argument because exclusive cross sections have infrared singularities. So of course at three level 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 level, the exclusive cross section should, should blow up because of soft quanta being emitted. And or at fixed multiplicity, the virtual corrections after they are resummed, they make the, 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 the, 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, of my talk was about how to define this inclusive cross sections in the case of gravity where we use the fact that there are no collinear singularities. In a sense the problem is simpler, you know, you, 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, insuitably 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 quant, 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 a, an infrared safe quantity. And by the way, we, we found this could be amusing. We found a nice formula for the, for the gravitational. Well, this is a virtual correction, a virtual gravity on corrections to the 2 to n process, which as Weimberg argued has a smooth massless limit and the massless limit is, is, you never wrote it down even for 2 to 2 process. Has this nice formula, s log s plus t log t plus u log u if you want. I mean, if it was a, 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, and we, we studied this b zero factor, which is the damping factor due to the virtual corrections is always positive and for instance, is, is, if the process is completely collinear 2 to n, but everything goes in the same direction. This b zero goes to zero, but that's the only case in which it is zero. And if for instance, the particles are emitted at 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 one over r appears naturally. In fact, if bar e, e bar is bigger than th, e bar is the energy threshold for calling it jet or a hard graviton. And if you take this threshold too high, higher than the Hawking temperature, then that cross section it will be damped, it 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 the story short and John, so to, 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. You know, 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, Hairwick 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 Hairwick 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 state 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 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. There's 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. And 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. OK, I think that's all I had. Thank you. 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 s? Or it's always equal of the order of magnitude of square root of s? No. OK, 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. Now I think the mass energy loss is equal to square root of s all around this line. Not exactly square root of s. No, no, no. It should be a finite fraction. I think everybody abdu... 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 cutoff 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, OK, this is what Colferai and Ciaffaloni have done in this last paper, but admittedly 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 we... OK, now the result is more comfortable, but before we know exactly where to cut off. In my opinion, it's just that, you know, we make some linear approximation to some equations and we neglect some nonlinear terms and those should fix the problem, but it's not easy. 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 gradually approach 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 the end, well, the end you just get a zero mass black hole. Really? Oh. And then if you have less in back parameter, then you form a bigger one. So I don't know that... No, no, I mean, well, no, I think at least at sufficiently large impact parameter there is simply deflection. That I think is... The critical case. Oh, the critical case. So you have to get the critical case. The critical. 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, just spirals in at some fixed angle. I understand what you mean. Yeah, it could very well be that in the critical case the whole thing is radiated away and you are left with no black hole. The small parameter for the fraction that you emit then at the same time for the one. Well, yeah, I mean, it will be actually a lot of emission. Now, you say there will be a lot of emission over the one. I guess I'm just imagining that it emits some... It's a minute of order, a fraction of the energy it has, each revolution. Yeah, I understand your argument. Yeah. It's because you would like to form an extreme 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 radiate away the... Whereas if it's really central... If it's acid particles then of course the limit you'd have twice the rest mass of the particle. Well, a little bit more. You could very well be true that what signals critical behavior is when you emit... If I knew I had to do numerical calculations it would be an interesting thing because you could have this cell thermal 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 you still... There's no first order transition that all of a sudden you get half of the energy radiated. Before you collapse, certainly everything comes out. Everything that goes in comes out. This is the equality between... Just saying in the limit... Yeah, I understand....of course you only you may have a fraction of the initial energy. But I'm saying in this limiting case... Yeah, yeah, yeah, yeah, which makes sense because it's... 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 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 diagonal momentum. But what is... No, I don't... You mean the class... 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 at 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 when it is collapsed some of it will go away in the location. Okay, if you only... I mean we pick things. Yeah, yeah, yeah. No, you see... Yeah, I suspect that you at some point you need some real non-perturbed physics that you form some long-lived metastable state and that is very hard to see how you could get it from here. You know, some kind of bound states. Non-perturbed. 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 dronic reaction in which the energy of the final state scales with the energy of the initial state. Okay, you have final scaling. You look at single particle distribution in a dronic 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-urativistic regime of LIGO where it did merger the velocities, half the velocity applied. The instantaneous gravitational wave flux is millidison. It's 10 minus 3. So it's a thing where... Millidison. So you could think it's still relativistic. Things are for the unity so you could get very non-protegrativism. 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-urativistic thing, gravity has this property. I agree with the value for this of containing its own cut of softening. I think so too. And then we will not see very small things. Yeah, I agree too. That's my expectation. Somehow generative you should know how to cure this problem on its own without invoking quantum mechanics for this problem. Okay, no further questions. Thank you. Thank you very much. Thank you very much.
 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 Van Sam 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,  that energy is 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.  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 whole zone, 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.  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 as 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 the valley Gomez et al.  And with Massimo Bianchi and his 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.  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 emerging 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 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, 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  ops of Einstein's formula.  So here is the definition of R, the short 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 University, NYU next to Andre 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.  And so the classical general activity calculation is based on work I did together with Andre  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 dE d omega.  As omega goes to zero, this limit is smooth.  Okay?  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 60s to Weinberg's work, for  instance.  And the result after the result for the process at hand, namely this high energy relativistic  collision with small deflection angle is this.  Okay?  Is this for the d 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, and interested, this is a parenthetical remark with, again, with Adasi  and Bianchi, we are trying to see whether the subleading 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 subleading 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  Dierth, Dierth and Payne.  They wrote several papers.  This is very interesting work.  They discuss mainly a regime which is not very interesting is 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 Dierth  does.  So this is yet another check that we are on the right track.  On the other hand, Kipthorn and Kovacs in about the same time, you know, had an ice  paper saying that this gives a definitive treatment of classical gravitational Braemstrahlung  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 contact, we, I mean Grosinoff 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-ethvistic and as  fast as if they are massless, as they emitted waves themselves.  So apparently numerical is very difficult to disentangle the actual Braemstrahlung 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 factors over the three and theta deflection angle a bit bigger than, bigger  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 Braemstrahlung  which, okay, has the right pre-factor but has the wrong exponential or rather in this  exponential, we like very much this damping, 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, 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 short-shield radius, namely at small deflection  angle.  And that 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 Ciafalone, but in this 2007  paper we were making some strong approximation trying to get to the critical points for collapse  and so on.  Was not the main aim of that work to find the spectrum, 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 sorts 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 primstralo.  So the recent progress announced in the title, as I said, goes back to, yes.  There are no logs.  There are soft logs.  There are soft divergences, but no collinear divergences.  So if you take QED, then there are two kinds of infrared effects.  One is the soft divergences, and those are common to gravity and gauge.  And one, 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.  OK, 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.  So 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.  But 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 that's 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, okay, 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 of the, we interfere all these waves.  But when you do this, you have to be very careful with phases, okay.  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, you  know, time is actually this particular combination of space and time, which gets shifted as this  particle hits, you know, encounters the other particle at impact parameter B.  This is 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. So this is 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.  So this is the formula.  It's 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, this quant, this observed quantity is related to what I call news functions, okay,  which you have, we, you define a, a, a, a sky plus if you want, but very far from this  force and so this is the crucial object.  It takes the form of a two dimensional integral because you integrate over all these points  on this plane and then reconstruct all these, these, these waves which contain, as you can  see, the gravitational retardation effects plus pure geometric effects.  And what is important to, 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, this object goes to zero at x goes to  zero.  You see, you know, removes some singular behavior coming from this object.  This object is not, nothing but the Riemann tensor associated with the shock wave, you  know, 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,  you know, 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 CCCV.  The same problem had this address 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 and which is needed for gauging variance, for covariance, okay?  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,  okay?  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.  Now usually you would not include emission from these exchange gravitons, okay, 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, okay?  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.  But 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.  Do you understand 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 you.  Yes.  Well, you mean in the same way that I drew my picture?  It looks like this on-shell effective.  Yeah.  Yeah.  Yeah.  Yeah.  Yeah.  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.  Yeah.  Yeah.  Yeah.  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.  Same note.  Okay.  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, okay?  You want to find the barrier.  You try to interfere this with the absorption from the last rank, then you find a mismatch  of momentum transfer, okay?  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 for omega bigger than r to  the minus 1.  So this characteristic scale, which is the Hawking scale, the Hawking radiation scale,  appears at that point.  So and this decoherence causes a break from this flat spectrum.  I told you that at omega equals 0, the spectrum is flat.  It's omega independent.  Well, above r to the minus 1 is no longer flat.  It just starts to fall, okay?  Which is a nice, very interesting feature.  Now, okay, I want to make this story a little bit 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 have, you emit a graviton, say, instead of two bodies, you have three bodies  and you have to look at the phase, 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, what 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 is just the function of, 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 the, 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, it 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.  Okay.  We have analyzed mostly numerically the property of the spectrum and 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 over the b to the minus 1.  If you insert that value, you get precisely the zero frequency limit.  So what we believe is that the validity of the actual zero 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 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 and scales like 1 over omega, which of course gives a log, 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 this, at a scale, omega star, which is 1 over r divided by the scattering  angle to the minus 2.  Now, another thing, an 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 cutoff, then of course you can even compute the total efficiency  for emitting gravitational waves and it comes out like that.  We argued with Kruzinov that above the spectrum should become omega to the minus 2 instead  of omega to the minus 1.  And then, okay, we later realized that 1 over omega square, I don't know if you know, is  also the spectrum of time integrated spectrum of the evaporation of a short-shield black  hole according to Hawking.  I mean, if you collect all your quanta, then the spectrum ends up being 1 over omega square.  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.  Another region 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, so there are some nice pictures.  Fiss Refs apparently liked 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, this is, I think it's an, oh, sorry, this is an amusing picture.  So this would be the spectrum as a function of omega.  It's integrated over azimuth, but still 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, then all of a sudden the plateau  ends.  And the other steep, you know, this steep curve, 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 t that still.  Okay, then, okay, as I said, 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 we'd 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 impact 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 over 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, but claimed  by Thalian 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 impact 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 impact parameter of the collision, okay, with Amati and Ciappalloni, we were  always discussing different regimes in this plane, now they integrate over impact 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 other 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 some  smart techniques about computing scattering amplitudes, how are they called?  Scattering equations or string equations, I forgot, okay, string equations, okay.  So they come up with this result for very large N, so 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 large factor is square C, or C is not the speed of light, C is the  constant of the one, E squared is put for convenience.  It's E is E.  Or there's no.  Or there's no.  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 I 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 the, 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 this value, but that this value,  you see they differ by 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, they argue that somehow this final state can saturate unitarity.  But you know this final state entropy factor is put by him.  So what is the parameter of small S?  Small, no, small S is E squared, is the, is the understand variable, is just the center  of mass energy square.  Yeah, sorry, should have defined everything.  So like it was in, in all my previous slides.  Yeah.  So, okay, this is, is also the mass of the black hole that you supposedly form.  And the reason why we take, by the way, transplankian collisions is that we would like to form  black holes larger than a plank length.  Okay.  And that's why you need transplankian energy.  Okay.  So, what we did with, okay, you can criticize very much.  I mean this, this argument because exclusive cross sections have infrared singularities.  So of course at three level 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 level, the exclusive cross section should, should blow up because of soft quanta  being emitted.  And or at fixed multiplicity, the virtual corrections after they are resummed, they  make the, 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 the, 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, of my talk was about how to define this inclusive cross sections in the  case of gravity where we use the fact that there are no collinear singularities.  In a sense the problem is simpler, you know, you, 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, insuitably 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 quant, 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 a, an infrared safe quantity.  And by the way, we, we found this could be amusing.  We found a nice formula for the, for the gravitational.  Well, this is a virtual correction, a virtual gravity on corrections to the 2 to n process,  which as Weimberg argued has a smooth massless limit and the massless limit is, is, you never  wrote it down even for 2 to 2 process.  Has this nice formula, s log s plus t log t plus u log u if you want.  I mean, if it was a, 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, and we, we studied this b zero factor, which is the damping factor due to the virtual  corrections is always positive and for instance, is, is, if the process is completely collinear  2 to n, but everything goes in the same direction.  This b zero goes to zero, but that's the only case in which it is zero.  And if for instance, the particles are emitted at 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 one over r appears  naturally.  In fact, if bar e, e bar is bigger than th, e bar is the energy threshold for calling  it jet or a hard graviton.  And if you take this threshold too high, higher than the Hawking temperature, then that cross  section it will be damped, it 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 the story short and John, so to, 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.  You know, 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, Hairwick 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 Hairwick 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 state 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 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.  There's 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.  And 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.  OK, I think that's all I had.  Thank you.  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 s?  Or it's always equal of the order of magnitude of square root of s?  No.  OK, 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.  Now I think the mass energy loss is equal to square root of s all around this line.  Not exactly square root of s.  No, no, no.  It should be a finite fraction.  I think everybody abdu...  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 cutoff 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, OK, this is what Colferai and Ciaffaloni have done in this last paper,  but admittedly 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 we...  OK, now the result is more comfortable, but before we know exactly where to cut off.  In my opinion, it's just that, you know, we make some linear approximation to some equations  and we neglect some nonlinear terms and those should fix the problem, but it's not easy.  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 gradually approach 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 the end, well, the end you just get a zero mass black hole.  Really?  Oh.  And then if you have less in back parameter, then you form a bigger one.  So I don't know that...  No, no, I mean, well, no, I think at least at sufficiently large impact parameter there  is simply deflection.  That I think is...  The critical case.  Oh, the critical case.  So you have to get the critical case.  The critical.  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, just spirals in at some fixed angle.  I understand what you mean.  Yeah, it could very well be that in the critical case the whole thing is radiated away and  you are left with no black hole.  The small parameter for the fraction that you emit then at the same time for the one.  Well, yeah, I mean, it will be actually a lot of emission.  Now, you say there will be a lot of emission over the one.  I guess I'm just imagining that it emits some...  It's a minute of order, a fraction of the energy it has, each revolution.  Yeah, I understand your argument.  Yeah.  It's because you would like to form an extreme 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 radiate away the...  Whereas if it's really central...  If it's acid particles then of course the limit you'd have twice the rest mass of the  particle.  Well, a little bit more.  You could very well be true that what signals critical behavior is when you emit...  If I knew I had to do numerical calculations it would be an interesting thing because you  could have this cell thermal 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 you still...  There's no first order transition that all of a sudden you get half of the energy radiated.  Before you collapse, certainly everything comes out.  Everything that goes in comes out.  This is the equality between...  Just saying in the limit...  Yeah, I understand. ...of course you only you may have a fraction of the initial energy.  But I'm saying in this limiting case...  Yeah, yeah, yeah, yeah, which makes sense because it's...  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 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 diagonal momentum.  But what is...  No, I don't...  You mean the class...  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 at 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 when it is collapsed some of it will  go away in the location.  Okay, if you only...  I mean we pick things.  Yeah, yeah, yeah.  No, you see...  Yeah, I suspect that you at some point you need some real non-perturbed physics that  you form some long-lived metastable state and that is very hard to see how you could  get it from here.  You know, some kind of bound states.  Non-perturbed.  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 dronic reaction in which the energy of the final  state scales with the energy of the initial state.  Okay, you have final scaling.  You look at single particle distribution in a dronic 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-urativistic regime of LIGO where it did merger the velocities,  half the velocity applied.  The instantaneous gravitational wave flux is millidison.  It's 10 minus 3.  So it's a thing where...  Millidison.  So you could think it's still relativistic.  Things are for the unity so you could get very non-protegrativism.  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-urativistic thing, gravity has this property.  I agree with the value for this of containing its own cut of softening.  I think so too.  And then we will not see very small things.  Yeah, I agree too.  That's my expectation.  Somehow generative you should know how to cure this problem on its own without invoking  quantum mechanics for this problem.  Okay, no further questions.  Thank you.  Thank you very much.  Thank you very much.
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