 Let me start with a little bit of personal history. I think my first contact with Kibo must have been in the early 90s, when I got a little bit close to a small gravitational wave community working at CERN around a gravitational wave antenna called Explorer. I remember quite well that Kibo was very, very skeptical about the bars, the hyogenic bars. And while he was very confident that interferometers would, one day, make it possible to detect gravitational waves. And once more, of course, he was right. However, it was not until I spent in Orsay and IHES around 1998 on a Pascal chair that I had the great pleasure of working with him. We first worked with Alessandro Bonanno on string cosmology. I think Alessandro mentioned that. Then we had a paper on black holes and fundamental strings that we taking from the metal strings. And finally, we had a few papers, a couple of papers on violations of the equivalence principle due to a rolling dilaton, some kind of alternative to a mechanism that Kibo and Sasha Polyakov had earlier. So then, I think around 2003, 2004, our work lines kind of diverged for a while. It did great work, meanwhile. And until recently, they started to converge again a few years ago. And this is what this talk will be about. It will be about this recent re-interaction between the two of us. So let me start with a few words of introduction and see why there is something on top of my slide. But I don't want to touch anything. So the recent detection of gravitational waves from binaries, compact binaries, has stimulated a lot of theoretical work on the subject recently. And while the traditional methods for computing the expected wave force, which are crucial to interpret the signals, like numerical relativity, which was reviewed by Benuzzi, post-Newtonian expansions by Blanchet, and effective one-body formalism that Alessandro reviewed, this being entirely classical methods, there have been new approaches based on taking the classical limit of quantum mechanical scattering amplitudes. Now, this brought together two theory communities, which traditionally had been very far apart, the one of classical general relativity and the community of high-energy particles physics. And this generated a lot of synergy. For instance, we organized a workup at the Galileo Institute this year. And it was, I think, very successful in life. But actually, the high-energy community had been interested in the gravitational two-body problem for a long time. Let's say, since the late 80s, for instance, there were papers by Toph, Mati, Cephalonian myself, Muzinich, and Saldi, and quite a few more. This was done for completely different motivations. We wanted to understand, for instance, the emergence of classical and quantum gravity from some thought experiments in flat spacetime. And this turned out to be quite a successful program. But mostly, we would have liked to understand and shed some new light on the information paradox. If you could check the unitarity of the S-metrics, even when the process is expected to lead to black formation that would solve, by definition, the imposition paradox. And this, I must admit, has not been quite as successful. I mean, I don't have time to tell you how far we went. Now, in that context, the context of particle collisions, to go to high or to transplant, can energy is mandatory in order to make gravity relevant if not dominant in the collision of two elementary particles. This is also important to justify a semi-classical approximation. So this is all to say that the ultra-relativistic limit was absolutely necessary in those studies. Now, what do you see in the full text? Because it's fine. Yes, it's OK. No. No, we can't hear you anymore. No, we don't see or don't hear you. We can't hear you. No, you can't hear us. What's happened? Hey. Hello? We don't. We can't hear you. We can't see you. See us. OK. Now, you did it yourself. I think we need, you need to... Hello? Can you do it? Yes, OK. Oh, I don't know how this happened. Yeah. OK. So I repeat myself. OK. What we need at the time was that also massive astrophysical black holes can be thought of as elementary particles. And so if so, then those gedanken experiments become all that gedanken. However, of course, for massive black holes, the non-relativistic, not to be confused with numerical relativistic NR, occasionally, the mildly relativistic regimes are not the relativistic ones are the most relevant ones. Now, in spite of this difference, four years ago, Thibault suggested that useful input for the EOB, the 31 body, could be obtained from high-energy, ultra relativistic regime of gravitational scattering. And this made a lot of sense because the so-called post-Minkowski expansion, which is in particle physics language, the loop expansion and expansion in G Newton at arbitrary velocities, could then be anchored to two limiting cases, the small velocity, post-Newtonian expansion on the one hand and the ultra relativistic limit you can have a handle on that on the other. This reminds me a little bit and no stock about the two regimes of generical activities, most C, infinite C. There are analogies, of course. Now, in result and in the puzzle, in the beginning of 1919, an impressive calculation by Bern and Company, you see at the bottom of the slide, the full set of authors, led to the first complete, quote-unquote complete, because we will see later in what sense it is complete, 3 p.m. or two-loop result for the gravitational scattering of two massive scalars in general relativity. It was checked to be consistent up to 5 p.m. This is still again in quotation mark and later also at 6 p.m. order, but presented quite a puzzle. The puzzle was that the high energy or just the massless limit of the result by Bern et al. Exhibited a logarithmic divergence in energy in contrast with the perfectly finite result by Amati Cipolloni and myself in 1990. Now, note that gravity is supposed to be free from collinear or mass divergences. And so this was quite puzzling, quite some controversy came up together with a lot of confusion. Furthermore, both the massless limit of ACV 90 and the Bern et al log enhancement at large S over M square were checked, double checked, and were even claimed to be universe, so independent of whether you have gravity, supergravity or what. And yet they looked neutrally incompatible and the prevailing attitude for a while was at the limits in which the momentum transfer is much smaller than the mass. And the mass is much smaller than the momentum transfer with both being small compared to the total energy are unrelated. Fortunately, the solution of the puzzle turned out to be different and in my opinion, happier. So this is the plan of the rest of the talk. I will give you a very, very quick reminder of the work of ACV just to stress the basic formula which gave puzzle. Then I will try to sharpen the puzzle and then to solve it. And the solution goes via the so-called radiation reaction and then I will give two shortcuts to computing the radiation reaction, one due to the bond bit. And finally, I will conclude mentioning some new challenges if you want to go one step further to 4 p.m., three loops. So this is a quick reminder, just a few highlights. What we did in those early days was things like restoring elastic unitarity via an Iconal resummation in impact parameter space, three level amplitudes violate partial wave unitarities. But this is recovered after you will be summed. We saw the emergence of a shock wave metric at order G, G Newton, which was also the starting point of the top paper, and we extended the calculation up to order GQ. This is this ACB 90 result. Then we saw also a few other things which I don't want to spend time on, in particular, if you collide strains instead of point like particles. But today we are not going to discuss that. Or I should also mention that we had the first go at gravitational brainstorming in these reactions. We found some kind of energy crisis, which was later cured to some extent, not fully in work by Kruzinov and myself, and worked with Chaffalonian. So the basic point of the impact parameter S matrix is that it takes a semi-classical form in which there is an exponential on the phase where the phase itself can be expanded in powers of a Newton constant. So delta 0, delta 1, delta 2 correspond to the three level, one loop, two loop, and so on, which corresponds to order G to the n minus n plus 1. And the following results were found in ACB 90. And here I only give D equal to 4, general relativity, and the massless limit, as I said. So the leading iconal approximation is given by this formula to delta 0, is Gs over h bar log p square. And it gives the analog of the Einstein deflection angle. And I emphasize that that is a classical contribution because of the one over h bar. Think about WKB approximation. On the other hand, in the massless limit, there is no one loop classical term. It is completely quantum. You see there is no one over h bar in the second line. And at the two loop, you find this result. This will be the main object of our discussion, a perfectly finite, as I said, two loop contribution to the iconal phase and therefore to the deflection angle. There is also a radiation term, which will be of some relevant in the moment. So now let's look at the massive case because that's where the puzzle came about. So we try to sharpen first and then to solve this 3PM puzzle. And this was in work by Dubekka, Heisenberg, Klusse and myself. So to put the old ACV argument on more rigorous and general grounds, for instance, Q less than M, extensions of GR and so on, we use general properties of the scattering amplitudes, real analyticity, the asymptotics in order to write subtracted fixed dispersion relations causing symmetry perturbative unitary. From those we get information on the high energy limit of the ratio between the real and the imaginary part of the certain contribution to the scattering amplitude. This is very much in analogy with what is done in high energy, soft, adrenic physics. It was very popular when I was very young. It's again kind of popular when analyzing soft, like total cross-sectional data at LHC. Now, the main result is this. I will spare you from the proof that if the imaginary part of the amplitude goes like some power of energy times some log of S to the power P, then those model independent constraint allow to express the asymptotic real part of delta 2 in terms of its imaginary part of end of the icon and phases and lower order three level and one loop. And this is the basic formula that we got. Now it contains a non-universal quantum piece. I told you that delta 1 has no classical limit. However, when you do the whole expansion, it comes out to multiply delta zero, which has a classical, which is classical and the product of the two, therefore is of the right order to give the classical contribution. Now for P generic, we are left with a non-universal because this non-universal piece multiplies P minus one. And also since neither in delta one or delta zero have a log S, real delta two from this piece will not have a finite ultra relativistic limit if P is larger than one, because this will go like log S to a power larger than one. And indeed, when you look at the Bernoulli result for this imaginary delta two, it has P equal two. And therefore it's non-universal and also divergent. And this is why we immediately realized Bernthal had this extra log. On the contrary, if and only if P equal one, real delta two does not depend on the non-universal piece and even approaches a finite ultra relativistic limit, the above equation simplifies. There is still an infrared divergence in each one of these two terms, but they nicely cancel each other. And then the end of the day, you get precisely the old result. Now this shows the crucial role played by the imaginary delta two. And imaginary delta two turns out by unitarity, the turbidity unitarity to be directly related to the inelastic three-particle cut of the two-loop ante. What is it? So we recomputed that the kinematics is shown below. By the way, this is called in jargon, the H diagram. Actually we, I'm going to teach a phone and myself gave this name, but we have to be careful with what one means by the H diagram. For instance, in ACV and in Bernthal, it means something different. In ACV, what you see here on the left-hand side through to something similar on the right-hand side, it's supposed to be the full two to three ampute in the semi-classical limit. It's a limit, but the full amplitude. Whereas in Bernthal, it means really this kind of topology. Now, and indeed the full amplitude is less internet singular than the H diagram of Bernthal. It gives a similar enhancement. So P equal one coming from the rapidity integral. And that's why it gives a smooth limit for the deflection angle. Now, we then decided to see really where the problem was because the contribution was getting tougher and tougher. And then we completed the full amplitude. Although for simplicity, because not easy calculation, we did it in N equal eight supergravity instead of general relativity. With massive external states introduced on the sublime and that arbitrary energy and the techniques we use are by now standards, differential equation plus integration by parts. But very crucially, it included contribution from what is called the full soft region, potential plus condition region, whereas Bernthal had only included the former, the potential region. And then we found that at high energy, the sum of the planar, non-planar ladders, which gives also contribution to the leading item, to the exponentiation of the leading icono, but it also has sub-leading contributions, which are of the same order as the ones coming from the H topology. And the magically the log square S terms in imaginary to cancel in the sum, same by the analysis of the plus crossing argument for the log S term in real A2 and everything works fine. Now, I'll give you just the result in N equal eight supergravity because I think it's pretty, it's simple, and it explains the whole thing. So what is real delta two in the end? Real delta two, there is a pre-factor. These are the standard definitions sigma times M1 and two is basically S, you know, the one that stands variable S or E squared. So the R cross of sigma goes like log sigma. So the famous log, which represents the puzzle, is this guy here. Now, this term is the analog of the Bernoulli result in any collet. And you see, if you only have this term and you go to large sigma, this goes to one and this goes to log. And that's how the problem erodes things. But the new term we found are these two indicated by the red arrows. And this nicely cancels. You see, if you go to large sigma, this goes like sigma to the fifth and the sigma to the fifth with precisely the right confusion to cancel. So this log term becomes sub-leading. This one becomes leading and this is the ACB leading. Now, one thing is that the new and the old terms behave very differently in the non-pelativistic limit, which is sigma goes to one or S goes to a special M1 plus and two square. But on the other hand, you see, they go very similarly in the ultra-pelativistic limit so much that they cancel each other. There is a similar rewriting of the same result in terms of the scattering angle. It's just a little more complicated, but you see exactly the same pattern of cancellation. And also the fact that maybe should have been clear already from the previous slide. That these terms, the new terms, from the point of view of the post-Newtonian expansion have happy integer, namely, they contain odd powers of the velocity from B over C. Now, when we presented this result at the Albert Einstein Institute workshop a little over a year ago, to go immediately jumped and grasp the physical meaning of what we had found. He said, just in the discussion, after I gave the talk, right away, he said that our happy integer post-Newtonian terms meant that we had added to the conservative dynamics of Bernadette's calculation the effect of radiation on the iconal face, so this so-called traditional reaction. So he immediately understood that that was the solution we had found to the puzzle. And now, I don't know how much time I have. Can you hear us? Yes. Yes, three more minutes. Oh, okay, fine. So, I will end up with two shortcuts to obtaining the 3PM result, the radiation reaction. One, in fact, I mean, the fact that Thibault immediately understood what we were doing explains why just a couple of months later, he derived the radiation reaction part of the scattering angle. This time, in general relativity, which is more complicated, as you can see, via a smart shortcut. And I would say chapeau because I thought it was really difficult and I was amazed when I saw this paper. Now, he used a previous result with Donato Bini relating radiation reaction to energy and angular momentum loss. He argued that only the angular momentum loss enters being aboard the G-square. And then he computed this J-rad and got the 3PM radiation reaction correction. And again, it nicely canceled the logarithmic divergence in Bernadette Hall and recovered the smooth ultraviolet beam. And his result has been confirmed by another shortcut which I will explain very quickly in a moment and also by full-fledged computations. It raises another puzzle. If you want a little puzzle inside the main puzzle, but I had a couple of slides on this. What is the true J-rad? What is the relevant J-rad? Well, for lack of time, I will have to skip this. There will be hopefully very soon a paper out by Vitoviskin myself dealing with that, but the end of the story is happy again. Everybody is fine. Yeah, okay, so I'll skip this. I will mention this other approach to radiation reaction. This time from soft theorems, this is our group. I will skip the derivation. It uses again analyticity, crossing, and so on. And the main result is this formula, which I find intriguing and interesting. The real part of Delta II, the contribution of radiation reaction, gets directly related to the so-called zero-frequency limit of the gravitational wave spectrum, which is controlled by Weinberg's soft theorem. So this allows, again, for a very straightforward derivation of the radiation reaction from a very simple calculation. And in fact, by proceeding that way, we were able to give the result in GR. You see the result in GR is much more complicated, but this is fully equivalent to what people found. And you can check again the cancellation of the posh terms and the fact that when you sum up the leading terms, this, this, and this, you, this number, adapt to give this universal case. And we check, we recheck this also in any collate. In any collate, you don't emit only gravitons. You also emit the dilaton, two vectors, and two scalars, which come from simplification. You sum the contributions to all this. And in the end, this time it's more complicated, but the end result in any collate is simply the one I showed in the previous test. I end up with just one slide mentioning that there are new challenges if you go to 4 p.m. There is a partial result for the conservative part of the burn metal this year. Unfortunately, it exhibits the same shortcomings of the 3 p.m. conservative results, only worsened. Not only the ultra relativistic or zero mass limit is even more singular than 3 p.m., but even if you go to finite energy, the result is different. And this is expected actually. But adding radiation reaction, which is called the tail contribution, classical generality, is absolutely essential for recovering the finite result. It's a hard problem. It's now understudied by several groups, including ours. Probably Tibo has in his drawer almost a solution. But anyway, he already came up with some partial p.n. and expanded results, which are already available. But we'll see how we go from there. So the conclusion, I think it's quite maybe not worthwhile to spend some time repeating what I told you already. There is this full soft region which allows a smooth behavior of the iconal phase from the deep Newtonian or more relativistic to the ultra relativistic limit. They're smoothly connected to each other. There is some intriguing relation between radiation reaction and the soft theorems. You can see how it's mathematically equivalent to Damou's connection with the abated angular momentum, but the precise physical reason for the equivalence between the two methods remains to be understood. And at 4 p.m., I told you things are still up there. But so far, OK, sorry, I think I didn't correct the slide. But today, I mean, this is a usual situation. Holding one puzzle just brings up new questions. I thought I would say let's enjoy this happy ending for a very, very happy birthday to both. And I really would like to wish for the future more and more strict interactions with him and now that our work lines have converged again, I hope it will go on for a long time, possibly with new collaborations and really the best, very best wishes for you to go. Thank you. Thank you. So in the end, if there's one urgent question, we can take it. If not, we do not need it. Thanks again, W.E.A. Thank you.