 Thank you. It's a great pleasure to be here. Many of you already pointed out how Samsung influenced you in your scientific careers, learning about his work from his papers. I first encountered Samsung's name and him in some sense through a textbook when I was studying gauge theories. And that kind of brings me to the context which I wanted to kind of emphasize. And that's a development of gauge theories, not only anomalous integrability that is usually emphasized. So if you look historically in the 1970s, gauge theories have been paid lots of attention by some wonderful people, some of them teachers of Samsung and many of you in the audience. And then in the early 80s, little bit more subtle questions were remaining to be solved. Many of them truly to anomalies, but many others as well. And Samsung and his generations, again some of in this audience participated and created that wonderful theory which is going to last and be applicable to nature as well as to our deeper understanding of how field theories work. So it's a great pleasure to be here. Same applies to his later work on integration on the modular space of an instanton. Again understanding gauge theories in a deeper level. So it's a wonderful subject and perhaps worthy of new textbooks now. So with that, I'm going to talk about effective field theory. This theory is not yet as developed as ordinary non-ambilient gauge theories for vector fields. It's a theory of a tensor field and as you know in those sorts of contexts we are encountering all sorts of issues already in the massless generality. And what I'll be talking about is a massive extension of generality and issues that are encountered in that context are specific to that theory and that's what I want to talk about and see some future, whether in future we can resolve them. So let me begin with motivations. I'm going to have two slides for motivations. One is more theoretical and it's the theoretical slide. And another one is cosmological. For the theory we know that if you ask what's the theory of a massless spin-2 in a world in which preserves Lorentz invariance, then the representations of the Poincare group tell you that state has to have two degrees of freedom and we know how to write the Lagrangian and that's a unique Lagrangian which we call general activity. So classically it's a good theory, quantum mechanically. You can quantize this in an effective field theory and what will happen is that it will break down the scale which is at the order of M-plank. But if you are interested in observables below that, it's a great theory in fact. So the question I would like to understand if you can have a similar story for a massive spin-2. In the moment I pronounce these words, massless spin-2 imply underlying Poincare invariance of the world at short scales. Then we have to have five degrees of freedom according to the group representation and hence you have to write down the Lagrangian which propagates five degrees of freedom on an arbitrary background. So that's a theoretical question and then the next question is what is the scale at which that theory breaks down in an effective field theory? Well, we do have examples as you know of massive spin-2 states. Kalutza clings are massive spin-2s. In QCD there are massive spin-2 glugos. I don't know if Shenei will be talking about some of them, but well they are massive spin-2 glugos but also massive spin-2, you know, quark-anti-quark states and things like this. They do exist in nature. Well, regestates have typically also massive spin-2s. But the interesting point is that in all these examples massive spin-2s come together with someone else. It's very hard to arrange for situations in which you separate scale of a massive spin-2 from higher states. It will be fine to separate, let's say, massless states and then just one massive spin-2 that actually will be something that I'll be interested in but decouple it from, in general, infinite number of other states. In particular, if you look at the Kalutza Klein theories, at the linearized level each massive spin-2 has the Lagrangian which is field-powered Lagrangian, the only consistent one. But for the full consistency of the theory, at the non-linear level, the next level, and the next level that you all mix together and contribute and make theory consistent just that way. So there is no parameter by which you can really separate the next Kalutza Klein from the previous one. And same applies to QCD. So in QCD, if you take large n limit to decouple states from each other, then you also lose self-interaction of those globals. For sure, you know, massive spin-2 global can by itself be described by fields-powered Lagrangian. That's fine, but the question is what the non-linear interactions are. And that's what I'll be addressing. So I want to have a theory of a stand-alone massive spin-2. And the question is whether it's possible or not. So now, from a more cosmology application point of view, this sort of theory is an attempt to describe dark energy. I should say that today we have a perfect parameterization of what's seen in the sky. We just write down the Einstein equation, or Friedmann equation, which is a 0,0 component, stick in energies and matter that we believe are existing in the universe, then that includes dark matter, dark energy, and dark energy with typically parameterized by the cosmological constant, and that all feeds data. There is no issue there. Nevertheless, people pursue alternatives to that description first because, well, I don't know if it's first or not, but one reason is because the cosmological constant, as you know, is unstable with respect to quantum loop corrections. There is a tremendous fine tuning. We're setting up that number that's small. But again, one more practical reason is that if you have an alternative scenario in which predicts different type of observables, then perhaps you can measure and distinguish the theories from each other. So both of these lines of thoughts are pursued, so that's why other theories aren't being looked. And massive gravity is one of them. So not only the reason why you would think massive gravity and would give you something like dark energy is the argument is similar to a scholar field theory. If you have a massless scholar, that doesn't have negative pressure. That kind of produces negative pressure. In order to produce negative pressure, you have to have the mass term or the potential term. The mass term being the simplest one that would give you negative pressure. And as you know, the dark energy, the cosmological constant, necessarily has negative pressure. Defining feature of it. So with that argument, you would think that perhaps if you have massive gravity, that mass term would give you something similar to the cosmological constant. And you will see that it really works that way precisely. And physically, what it is, is that this is some kind of cognizant of massive gravity. In fact, the helicity zero states of those massive gravity gives you the accelerated expansion. So another one is, of course, the long standing problem of the old cosmological constant. Whether that can be solved in this context. In fact, historically, you may be aware that when Einstein introduced the cosmological constant, he thought that was a mass term. And actually, that was the reason why he introduced the cosmological constant, because he wanted to have static universe. And as everything else in Einstein, it's very physical. He said, oh, how I make universe static? Well, if I screen gravity, if I have it Yukawa screened as opposed to, you know, one over Newtonian, then that screening somehow should stop the expansion that otherwise he was seeing in his equation in time-dependent equations, as we know now. That's why he introduced the cosmological constant. He thought it was a graviton mass. But we know that the cosmological constant is not a graviton mass simply because it doesn't change the degrees of freedom. It's still a theory with two degrees of freedom. But in any case, the question is whether that logic that Einstein had of screening of things at very large distances perhaps can be used to screen this big cosmological constant. That was another reason why these theories are explored. And the last one, I won't elaborate too much unless the questions are about that. This emerged through the, you know, through work on these theories, not that somebody could predict this. Some of these theories, in particular, what are called bi-gravity theories, these are theories of one massless and one massive graviton that cost us very nicely yesterday. So those have a chance to offer alternative descriptions of very early universe. And again, that's a huge subject that I won't necessarily be talking about. Okay, let me now go to the issue that I'd like to talk about. And that's a question about degrees of freedom and their interactions. So as I mentioned already, the massive spin-2 would have three more degrees of freedom as compared to massless spin-2. And the question is how those additional degrees of freedom are self-interacting or interacting with other stuff. And you can ask the same question for a massive spin-1, non-Aberian massive spin-1, as it was done very long ago, even before Samsung. So take SU2 for simplicity. So local SU2 I'm talking about. And put in, in addition to the Young-Mills term, put in the mass term by hand. That's what actually Glashow did in 1962. That's part of the standard model, in a sense. There is no issue with gauge invariance or something. You can always restore gauge invariance by introducing Schuchelberg fields and so on and so forth. The only question is how that theory differs from the massless one in terms of, let's say, high energy behavior. And it does in a very significant way because of precisely those additional degrees of freedom. So as you know, each massive spin-1 would have extra one degree of freedom. That would be at least a zero state. And because it's SU2, now you have three of them. So this A, this pi A, these are longitudinal degrees of freedom for three massive gauge bosons. A runs from one to three. So that's what pi A's are. And then you can go to a certain limit. It's called the coupling limit. That's essentially physically it's a high energy limit. You're going to get energies which are higher than the mass scale and in a weakly coupling approximation. And you can extract the terms from the Lagrangian which are the strongest in that limit. And these are the ones calculated in that paper for the longitudinal polarizations. And as you see, well, this is some kind of non-linear sigma model. But as you see, it breaks down as an effective field theory if you start quantizing it now perturbatively. Then it breaks down at the scale v. So if you expand this, the leading term will be d pi d pi pi squared over v squared. And if you look at the quartic scattering amplitude of four pi's, that amplitude will be violating perturbative unitarity at the scale of v. And hence, that's the validity of this effective field theory. So something needs to come. So you have two options. Either you declare that that's the theory. Well, perturbative unit weight down. But you can do maybe non-perturbative calculations. You can actually need to do lattice calculations or something like this. But there is no... So clearly what we will be producing is some kind of bound states of those pi's about that scale v. And then it's difficult analytically to address that. But it might be that this theory by itself looks like in the standard model the nature chose... Well, we know that it chose different route. And there is an additional particle. On top of this polarisation, there is an additional particle which we call the Higgs boson. And that Higgs boson comes in and cancels those growing amplitudes so that the resulting theory is still weakly coupled. So you can do perturbative calculations way beyond this scale v, which in the context of conventional Higgs, if I were to write down the Higgs model with S2, this v would have been the mass of the gauge boson divided by the coupling constant. Okay? But still not complete, right? Well, because of the Landau pole, but that's a different story, right? So that's entirely different. Oh, if you embed it in a nanobillion group, then it is complete. So that's a different story, yeah. Right. So, now, ideally that's what you would like to, well, I don't know, ideally or not, that's an easy option that you would expect for polarisation of mass of graviton to interact strongly at some scale and then perhaps if you were to introduce additional degrees of freedom on top of five, then those degrees of freedom somehow softening the behavior. But there are now calculations, very nice calculations by these authors who show that actually with spin zero, spin a half, and spin one, you cannot really do that cancellation. In the massive gauge case, that was happening due to spin zero. Here, even spin one cannot do it. So perhaps then you should be introducing spin two, but then you might be back to conversations such as, you know, infinite towers of Kalutza-Clean modes and so on and so forth. So that's something that we'd like to understand. So to understand that, now, let me go to the action or Lagrangian of the massive theory. So in general relativity, as we mentioned, there are two helicity states of a massless graviton, and we describe them by ten components of the symmetric tensor g mu nu. The reason why we introduce that redundancy is because that's manifestly covariant formulation, and we don't want to deal with non-covariant formulations. So we introduce extra fields, which are obviously constrained. So here, to describe massive theory, you need to introduce yet other extra fields for the, you know, full diff invariant formulation. So I'm going to have g mu nu, and then I'm going to have four scholars. So these phi a's, now a runs from zero, one, two, three, and these are scholars of conventional diffeomorphisms. But internal space metric for these phi a's will be non-compact. It will be, in simplest cases, will be Minkowski. Okay? So that means that one of the signatures will be, what I call it, zero in purpose. It will be negative and the other one's positive. And then from these objects, from these fields, I want to compose the following metrics. And before the metrics, let me introduce this f tilde. This is called a fiducial metric. So it's d phi d phi contracted with internal space Minkowski metric. So this whole thing is also a Minkowski metric. I mean, it doesn't have Riemann-Korwitsch. Nevertheless, this is a Minkowski metric in arbitrary coordination. And so you can think of this as some other metric, as Kostas explained yesterday. You can think of it as some other metric which has its own dynamic and for which you've taken its corresponding M-plank to goes to infinity so that you decouple the dynamics. But you can think in both ways depending what questions you want to address. So there is a trivial transition from one to the other. That's not the whole point of all these conversations. So in any case, you have this metric and then out of this metric and the physical metric you can pose a matrix K which is a unit matrix minus square root of the inverse G times that matrix. So there is the square root of structure. The reason why this has to be the square root of structure is because I already alluded that the target space of the sigma model is non-compact. And typically when you write down conventional Lagrangian for some sigma models like this you're gonna end up with ghosts precisely because of that non-compact sort of signature that comes from the internal space constructions. And this is the Lagrangian which avoids those ghosts because in the end in a very subtle way not all four phi's are physical. They get constrained and only three are physical and the one that typically would have been non-physical actually gets projected out through the form of this Lagrangian. So it took some time to construct that but anyway we have it now. So then the full Lagrangian is the Einstein-Hilbert term and then the master times this which you can consider as the master and these are potential some potential or some interactions of gravitons and everything is written in terms of traces of that matrix k that I defined. This is trace square minus trace of the square which you can also rewrite as that too. That too means epsilon, epsilon times two elements and the other indices contracted between the two epsilons and then there is that three and then there is that four which is a conventional determinant. This is just determinant of this k matrix and so these are just obvious notations. So the point being is that you cannot write anything else if you want to preserve just five degrees of freedom. The moment you write something else you're going to get extra six and six degree of freedom which will be a ghost degree of freedom but five is preserved by this structure. And again going to the bi-gravity making a connection to the bi-gravity so if I were to add here a usual Einstein-Hilbert term for this object without necessarily defining it this way then I would have the bi-gravity theory. So the whole point in those theories is what are the cross terms between one graviton and the other graviton so that was a challenge otherwise these are just Einstein-Hilbert terms. Okay now well again going back to the gauge fields we know how symmetry breaking patterns work and who eats whom. Here there is also symmetry breaking pattern. The global symmetry is the product of two Poincare one is obvious that's what usual you know space time Poincare that in our theory we use the other one is the internal space Poincare of those flies. So you can think of that as some kind of target space and then in the background in which this is a massive graviton let's say forcibly you choose a flat background and you should get a solution and on it massive graviton propagates there is a breaking of this product group down to the diagonal due to the fact that these five a's acquire coordinate dependent webs in spite of the fact that they are coordinate dependent webs it doesn't break full Poincare invariance there is a remnant of Poincare invariance that's the one that we observe in our universe. So this is what's going on in that theory in the similar decoupling limit that I alluded to in the case of massive spin one you can also define that kind of limit here there the group is Galilean this Galilean group down to the Poincare so this is a 15 parameter group which breaks down to Poincare through those Galilean fields so important clarification to this theory is brought by the decoupling limit as it was the case for massive spin one and again physically this is a limit in which energies are a lot higher than your mass so you don't care about those small mass corrections you care about high energy behavior and you need that to answer the question as to when this theory breaks down and remarkably you can calculate that limit exactly and in that limit you only get finite number of terms initially you have infinite number of terms and now you get finite number of terms surviving that limit so there is of course the Einstein-Hilbert part which is the usual but then your tensor fluctuation mixes with the following combination of the tensors which are made of the Helicity-0 state so this little pi now is a Helicity-0 for massive graviton we apply two derivatives that gives you capital pi and out of the capital pi you construct the tensors like so x, y and z and as you see these tensors are kinematically concerned they are automatically concerned you don't need to use equations of motion for them to be concerned so if this were an ordinary scalar for instance right so you would still couple h mu nu times some stuff some t mu nu of the scalar and then the conservation of the t mu nu would necessarily invoke the equations of motion of the phi of the scalar this is not the case here because this you don't require to invoke the equations of motion these are automatically concerned and actually that is very important for the structure of this theory so it's then different variant well linearized diffs are the exact symmetry of this theory in this limit you have exact symmetry of linearized diffs in spite of the fact that this is a nonlinear theory and that tells you that h mu nu will have just two degrees of freedom like in general activity while additional degrees of freedom one being this phi will be entirely encoded in this other part of the Lagrangian which in general you can die well in some cases I'm sorry you should say you can diagonalize in fact but not always okay so alpha and beta are two arbitrary parameters which are not fixed by this construction I already had them in the original Lagrangian so they stay in this approximation too it remains to be seen as to what physical principle fixes them there are some considerations which reduce the parameter space for those alpha and beta but from the fundamental considerations they're not really fixed an interesting fact about them is that they are also not renormalized so if you now think loops in this theory which is a nonlinear interacting theory the loops won't generate any renormalization of alpha and beta so I can explain that in terms of the Galileons or actually maybe in terms of these tensorial structures it has something to do precisely with these vertices that when you have external legs the only ones which have no derivatives on the external legs could potentially generate renormalization of alpha and beta as you see however the tensorial structures are such that you always get two momenta acting on the external leg otherwise everything else is zero and that has something to do with these tensors here so alpha and beta are not renormalized which means that if you go away from the decoupling limit in the full theory the terms that I have written will receive corrections but those corrections will be small they will be proportional to the mass itself and then there will be an additional multiplicator which is the ratio of the mass over and plunge to some power which means that loop corrections don't really ruin the structure itself they give you small corrections and so you can certainly generate some other other things but that part is protected and that precisely has something to do with the structure in the decoupling limit so it's somewhat similar with carer limit of QCD when you take Quark masses to zero you have exact certain exact symmetries and then you built on those exact symmetries and then you get into the into the theory that your corrections due to small up and down masses sometimes even a strange Quark mass so one connection to well something that might interest Samsung is is the the nature of those terms in the context of the cost construction I'm not elaborating on it here done by these people if you ask a question of the following sort let me start with the Galilean group which is a symmetry group in this limit and it breaks down to Poincaré as I was telling you that's a spontaneous breaking so you can construct a corset which is Galilean or Poincaré and you can ask what's the corresponding effective Lagrangian so you can go through the regular construction you will find out that these terms are Galilean terms which are actually diagonalization of this you cannot write them as the terms of the effective Lagrangian so this again just to compare with the in QCD would have written all those terms is just the corset terms conventional corset terms built out of the construction here you cannot just write them so you can mention more and you can write that form and take the projection on the 4-dimensional space in this case and then you get these terms so they are like 2-minute terms now so let me now explain what the strong coupling problem is and what scale it kicks in and the simplest way to see that is to look at the first to look at the linearized part is a linearized part in which I've introduced the Stuckelberg field this Pi so I'm trying to extract the polarization and this is in the leading order they are sub-leading orders in mass in the leading order you got you know kinetic terms for H this is simplified notation all the indices are there and so on and so forth but structurally it's just kinetic for H and then the stress energy tensor and as you see this scale M now is actually irrelevant because you can just rescale Pi by that M or another way of saying same thing is that you can first diagonalize this this kinetic mixing if you diagonalize you're going to get the kinetic term for Pi but then also the interaction of Pi with the stress tensor and then you can rescale the Pi and then you can see that the Pi will couple to the stress tensor with equal strengths as the tensor field okay so that's the essence of so-called Van Damme-Mann-Zacharov discontinuity that doesn't matter how small mass you take the effect is of order one in the linearized theory you have a new effect which is of order one which is independent of the mass and the massive theory differs from mass less one by effects of order one so that's really an issue that needs to be addressed and there is what's called Weinstein mechanism to resolve that but let me put it aside for a moment the related consideration is that you can look now at the nonlinear terms these are the same terms that I wrote in previous slide just in a diagonalized way and when you rescale the Pi so in that scale you can call lambda3 it's a combination and in Planck and the mass squared to one-third power so that's a scale at which let's say 2 by 2 scattering of this Pi will start violating perturbative unitarity and theory will come out of the perturbative control so the conclusion then is that you can have the theory of standalone with just five degrees of freedom they will be preserved in almost all meaningful backgrounds and that theory will have strong interaction at this scale which of course if you use cosmological values of M this is very very low but theoretically we want to understand how that scale may potentially be softened or resolved by new physics or other physics such as the Higgs particle I talk actually the rest of my talk after this cosmology example I will give you quickly it's going to be about that aspect how to go beyond the three scale in the UV so before yes as it was important in the first part that we were saying there are five degrees of freedom for general background you can have a classical background where the square wood business becomes sick so which would be low energy there could be background where the theory becomes what happens in this well it's I wouldn't say the theory becomes those are not would there be six degrees of freedom no those will be just not there is a big class but some of them are not solution some of them are not meaningful solution of this of this theory so what Hibbo is alluding to what's the square root of the matrix and actually it's also not uniquely defined meaningful solutions so but you can you can put it on on a very solid classical theory background which is that I can I can to answer that question I can write this Lagrangian exactly the same way where K is an independent matrix now it's not defined yet in these units it's just independent matrix and then I have a constraint which enforces the quadratic equation either has a real solution it doesn't have real solution depending on the matter and energy in your theory and that that's easier to precisely answer the important question that you are raising for some cases you will have you also depending on boundary conditions will have reasonable solutions in some cases that quadratic equation will have some complex K or what not and those are not solutions your Cauchy problem should be defined in a way that you start with conventional Cauchy data and then you evolve so that you never go into those branches so these are typically done by boundary conditions so yeah so this is a quick slide about you know cosmology I mean this is not doing a fear treatment of it there is a big amount of work actually very good work on that and there is a generic solution in this theory so you can the way you can think about this you didn't mention how to set the fifth force yes so fifth force that's a very important question again and probably the rest of my talk would have taken that so the way that traditionally is solved that is a fine mechanism and what it does is that due to the nonlinear terms your linearized approximation breaks down very very far from your classical object this is somewhat counterintuitive when you think to begin with but then you know there is a second level of intuition which is perfect so even if you take some out of the perturbative regime far away far away if I remember right if you take a solar mass object in isolation the scale at which the perturbative treatment of the pi stops is something like kiloparsec or something like that so astronomical scale so within that kiloparsec you cannot do perturbation theory for that pi it just breaks down right there that's called pi in the interior of that Weinstein radius and then expand around it that's possible or it's the other way around in that region you can do actually small mass expansion so small mass expansion breaks down towards the Weinstein radius from inside and the other expansion breaks down when you come from outside but you can do either but of course there are some cases if you do symmetrically symmetric solution you can do the exact solution for pi which is that doesn't violate solar system test it doesn't yeah it doesn't violate the solar system test that's a remarkable thing it gives you this pi gets so screened by the Weinstein mechanism is that it gives you tiny corrections to the GR and in fact in in this model in the massive gravity those corrections are way too small to be measured by anything that I'm aware of but in other theory which is called there is somewhat similar story there so in that theory those corrections are a little bit stronger and in fact the next generation of the lunar the orbit measurement might be able to pick those corrections so these are tiny, tiny corrections to the GR predictions yeah very important but here now I'm going to address that question differently because I'm going to have an extension of the theory and in that extension it will be solved differently so that's how I emphasize it so the point again I'm trying to make this point is that the Sitter solution is a generic solution when you right now the Einstein equation you have you know conventional Einstein's tensor then on the right-hand side you have all these mass terms and those mass terms give you the existing backgrounds in which they act like dark energy so at the level of the background then this identical to dark energy well you know that physically they will be different because you have more degrees of freedom in fluctuations but they are also challenging here because very often for instance in the simplest cases very often those fluctuations become actually infinitely strongly coupled on this background that's written here literally on this solution they are infinitely strongly coupled so you got to expand that theory already for that purposes if you want to be applying it to cosmology you got to expand it and then this was done in many works notably by people from Japan from Mukoyama's group and so on and so forth but I'm not going to that direction although it's very interesting by itself so the brief summary before I go to the new proposal is that so there are in the linearized theory there are three you know Nambu Goldstone bosons of the spontaneously broken symmetry of course the rank of the group that breaks is a lot higher than Tantradik number three because it's not a compact group counting is different and those three are the ones that get eaten by massless tensor and becomes massive and nonlinear interactions then preserve all those five degrees of freedom on arbitrary background and there are two three parameters but on certain backgrounds you can get instabilities due to the existing five modes so those instabilities are not related to anything new just same five modes and the reason why you get those instabilities is precisely because of these low strongly coupled operators so on some very reasonable backgrounds those strongly coupled operators become of the same order as the as the relevant operators that you started with and they you know deform light con in various possible ways so you can get instabilities through that that's another suggestion as to to the fact that you you need to go about that scale understand what's happening about that scale you see for the massive spin one you wouldn't think of that because there we don't think about the backgrounds we think about scattering and particle physics while for gravity you've got to think of the background and the background is such that it distorts the light con then you should you should address that so that's another indication that you have something something this theory is not is to have some extra degrees of freedom which would change that so here is here is a here is a proposal now how to increase that that scale and it's based on the idea that if you had an ADS background instead of Minkowski let's let's pretend for a second that we are in ADS space so in the ADS space that we this continuity is not there to as was discovered by my colleague I think simultaneously by Jan Kogan and Muslo Polis and Papazoglu at the same time so in ADS background you have an additional scale which is the scale of your curvature and then if you take mass goes to zero that that limit is continuous in the linearized theory and reason reason why that's happening is because in terms of the Helicity zero mode the Helicity in the mass of that mode so in the previous considerations that that gave rise to this kind of strong company behavior there was a implicit assumption that no additional kinetic term for Pi exists and in Minkowski space that's a consequence of the symmetries so there is no way in a Poincare invariant theory you can get anything here because this is not an arbitrary spin tool so that's also different of this theory from all sorts of theories that people call modified gravity in which arbitrarily you put this and that you cannot put arbitrarily anything here everything is related to each other through representations of the Poincare group in fact so there is no way you can do anything here but if it's ADS then that enables you to have the kinetic term for Pi already at the tree level as you if you run the same arguments that I run on that slide gives you additional suppression to the coupling to the matter and it also will give you additional suppression to the higher dimensional operators and in principle if you lived in this space then you would not you would not need to invoke nonlinear Weinstein mechanism although I would argue it's a curvature everywhere in space while if you are discussing the let's say Schwarzschild solution so then that's a coordinate dependent curvature that gives the extra kinetic term for the for the for the Pi so on the other hand we don't live in ADS space so somehow we want to use benefit of this mechanism while still being in Minkowski space or some cosmological the four-dimensional theory into a five-dimensional ADS space I'm five-dimensional I'm saying for simplicity it could be higher-dimensional but let me stick to five-dimensional because that's simpler so you begin with 5D ADS and then in that 5D ADS you necessarily generate a massive theory in 5D ADS you necessarily generate new kinetic term for the Pi which will be and if you manage then for your four-dimensional slice to Minkowski you will necessarily inherit that kinetic term because the volume of that space will be finite if you do it properly and in a finite volume this kinetic term will give you four-dimensional kinetic term the coefficient will be just this multiplied by the size of that extra dimension so you can imagine something you have a brain, you have ADS-5 I'll give you those solutions exactly on the next slides but this is a simplified version of what's going on there then in addition to the standard terms that I talked about this was the Pi kinetic term after the diagonalization you get a new one coming from the fifth dimension involving five-dimensional curvature and that's not in contradiction with anything because your four-dimensional world now is a complex world it's a it's a it's a it's a mixture of four-dimensional gravitation with a tower of Kalutza claim modes which have no mass gap so there is there is this aspect that you've introduced the states way below the strong coupling scale of the theory so in principle that's what you would expect so this Pi to T will now be rescaled by this number and if this number is big then you won't have issues with fifth force in general you won't have this continuity at that level because this has an additional parameter in it which is the curvature and then also it will raise the strong coupling scale precisely of the same argument so as long as this is much bigger than to the fourth which is very easy to justify then the is through this term not through this term and then that reduces all the high-dimensional operators to be suppressed by higher scale so here is a full theory again so the five-dimensional world is Einstein-Hilbert the cosmological constant and the potentials for massive gravity in five-dimensions notice now that because we are in five-dimensions you have one more possible term remember those terms based on epsilon contractions and you saturate number of terms by the number of indices that the epsilon can have so in five-dimensions that's five one more and the definitions are similar to the four-dimensional definitions I use the bars for five-dimensional entities you define this matrix in the same fashion you have Stuckelberg fields in the five-dimension space and you have now a fiducial metric here which I want to make in more general I want to make it to be ADS and again if you were to if I were to write by gravity which I will on the next slide this is very easy to achieve so you have for that second gravity and also a cosmological constant and the Planck scale so that you get this precisely so the matching the bulk boundary matching is a standard one the metric the immune component of the bulk metric at z equals zero is the four-dimension component you can do this in a general-co-orient way but this is in a simplest representation in the boundary values of the capital five bulk Stuckelbergs are what we call the Stuckelbergs in the four-dimensional theory and then this fiducial metric of the five-dimensional massive theory is a fiducial metric of the four-dimensional theory so there is nothing nothing new here but that theory now you write down the equations and it has a solution which is given the ADS fiducial metric which I arranged for by having let's say second-graviton with appropriate dynamics and the coordinate in that second space are fives now I haven't chosen any gauge that's why I keep fives there is a solution for the physical metric which is also ADS just like that and then if you were to introduce Randall-Sundrom-Brain you actually you go to the usual Randall-Sundrom matching condition because what happens was that those case they are on this background those case are zero they don't change any story in the background solution for the Randall-Sundrom in fact so it's identically to that so this is what I already alluded to so I keep in mind that there is a kinetic term in the cosmological term for the second-graviton if you wish to and then this one you can take to be small but the curvature to be the same as the one for the physical ADS space and it's a consistent solution in that by gravity as well so okay then you start working with the theory to try to understand all this strongly coupled behavior and other aspects and remarkably now you so you can take the decoupling limit which is similar to the other one and what I said in a simplest kind of similar type set up precisely carries through you get this extra kinetic term now here I already rescaled this pie there is no mass appearing in front but there is of course mass and then you look at the look at the non-linear terms and because of that rescaling that mass appears here now so these are the typically non-linear terms which gives you strongly coupled behavior in the theory and in addition to the conventional ones that exist in you know flat space flat fiducial space mass of gravity you get now a new ones having something with the with the fact that there is a curvature in extra space and in fact those new ones are worse so you got to take care of them you do rescaling all the calculations and so this is what you'll find that five-dimensional theory has a strong scale which is something that you would expect it's not just five-dimensional plan scale times the power of mass the curvature of the space time also enters higher because I'm imagining this curvature is a lot higher than the mass scale of course so just to give you the roughly you know the scales of magnitude the bulk is like gut you can think of the gut scale bulk and the curvature is a gut scale and things like that which you can naturally do so therefore this will be a lot higher than the flat space scale so this lambda 7 so you can integrate out exactly the dynamics of all the KK modes associated with this pi and so you get the propagator in 4D you get a propagator like this well it looks scary but actually it has lots of interesting physics in it so all these if you look at this propagator which is the inverse of this all the pulse that you will see so this is actually a branch cut just the KK mode with no mask up but in low energy approximation when the momenta are smaller than the curvature this ratio of K1 or K2 these are McDonald's functions it gives you another square root of box 4 and square root of box 4 combined into square root of box so it gives you kinetic term for normal kinetic term for pi weighted by this scale because of that the scale goes high actually 19 orders of magnitude you can win through this process so now your new strong scale is 19 orders magnitude higher than the old one and the Weinstein you don't necessarily need to invoke Weinstein mechanism because the VDVZ discontinuity is not the anymore for the arguments that I emphasize so because I probably have one or two minutes I want to jump onto the holographic interpretation of this so now if you assume that there is a holographic interpretation then you wouldn't expect to have conserved energy momentum tensor in your four-dimensional CFD dual because your gravity in the bulk is massive and the anomalous dimension of your team you knew should be mass squared divided by the curvature squared and then you would ask what it could be so in fact the correlation functions as was done in this paper so what you will uncover is some kind of non-local 1pi action which precisely captures all the all the you know all the calculations of classical gravity theory and what you find is that this theory four-dimensional CFD is like unparticles if you follow that story of unparticles so this is precisely what you find this CFD dual particle theory unparticle theory does have team in you well this little bit of terminology but the ones that I'm talking about it doesn't have team in you you can have unparticle theory which doesn't have team in you yeah then those are the ones sometimes people also call unparticles some conventional CFD stuff which has team in you I mean the one that doesn't and the interesting thing is that if you followed if you pursue that this should be efficient session time so if you if you pursue that you can even write down effective 1PA actions and then think of them coming from five-dimensional flat space construction with lift sheets type scaling you just take a slide you just take a slide exactly for the exactly from for the you lose team in you in full theory has team in you but for the slides doesn't so that's a best interpretation but very quickly actually making connection with cost assess talk so you can make a way to embed this you know bulk massive theory into you know more fundamental theory and you can proceed step by step so there is a there is a construction in which the the bulk mass can be now generated by loop that was done by Porati, Davliu, Aroni, Clark, Karch and Kereces as cost assess cited yesterday if you stick to super gravity approximation that's fine step even in this theory you can address what's going on and then hopefully you can address in the context that cost assess developing with the postdoc here so I want I want to review this because it was very nicely done yesterday in fact but you can precisely repeat that construction in the case with the brain with the Randolph-Sandran brain so you can put in a Randolph-Sandran you find the exact solution and calculate the loop as it was done by Porati because the presence of the Randolph-Sandran brain doesn't change too much changes a little bit but all the all the same arguments go through and then you do get generated Fitzpaul master in the bulk through the loop but this theory is different from the one that I just talked about in which I introduced mass by hand and the difference of course is that is actually the vector that precisely is eaten by the massless stuff and Costa had this group-theory representations for how that works and then there are other bound states right so however the other bound states they don't give you strongly coupled behavior so you know what the strong coupling scale of the theory is it's a curvature of the bulk which is a bulk cosmological constant and until that scale it doesn't go strong at all right so I'll I'll stop here I'm over time so so mass gravity is theoretically interesting QFT it's an effective field theory if you if you just are asking a question can I have just scattering of a massive spin two on flat three and that's that's where it will break down if you are asking more refined questions as to whether you can be using this for cosmology that those questions call for completion of this theory additional degrees of freedom it's somewhat similar in the standard model in which you have to have Higgs boson to go beyond the scale which is V as to what the sector changes it's something more complicated remains to be seen and this exercise that I showed you in which the scale got you know raised due to essentially the Kalutza-Klein without without a gap that tells you that some spin to should be participating in the whole whole solution of that theory which is another indication for the same thing but if you just take very pragmatic approach of magnitude increase of the scale of this lambda-3 if you just introduce the bulk bulk mass and well hopefully this loop-generated gravity on mass is more fundamental way of thinking about this theory in the context of supergravities and perhaps in the context of string theories once Kostas is done with his construction we can also apply that thank you just to understand in the case of the scale of lambda-3 and you have a lambda-3 which is very low energy product of most or whatever so what in a physical sense then what should we see experimentally in low energy gravity there would be new modes I mean you are saying we need a new theory well it depends what you presume about what happens at that scale if you insist that there is no new degrees of freedom what the theory is in that case what can you say in that case it's there is a there is a refinement that you should do which is that there is a Weinstein mechanism if you put this in some cosmological context let's say FRW or even in the context of today's universe let's say in the solar system right so those kinetic terms will get renormalization through the classical backgrounds already those classical backgrounds will be to shorter and shorter scales so they prevent you to see new effects that might potentially be at lambda 3 however if you follow realistic numbers let's say in the solar system still the effective breaking scale would be somewhere in which you can do measurements as to what they are it's hard to say because you cannot do analytic calculations right you have to resort numerics but you know that you should change the theory or you are saying the same theory can be predicted in principle in principle if you were to do to be able to do non-perturbative calculations in a realistic setup you don't have to change the theory you get some predictions out of it question will be are they ruled out or not this was similar to what I was referring to more clear a case of massive spin 1 let's say massive spin 1 you don't adopt Higgs mechanism by hand the Glashow model so you get this strongly coupled behavior of the Nambo Goldstone bosons at V so in that case you know we have a little bit better control because we can think of the bound states of those helices 0s into you know higher mass composite objects similar to composite Ws and so on and so forth and then there is whole theory there is whole effective field theory for those experiment and so on and so forth let's say 10 years ago that was still a viable approach for the standard model you know people made predictions for the LHC except that they all got ruled out by discovering the Higgs mechanism so it's similar here if you are kind of willing to leave strongly coupled behavior and provide the mechanism or tools of calculation let's say lattice calculation or anything else well you can test yes but this will be quantum effects then on the microscopic scale well they might be also classical too also classical too so it's a combination of the two so I'd say quantum in a classical background that's what I would characterize yes true but the point is that fundamentally as a field theory you ask that question what happens about that scale can I resolve that scale one good question is can this this gravity be a bound state for massless gravity you have all kinds of theorems prohibiting to be a bound state this one there's no a priori impediment at the level of no-go theory perhaps it's a bound state if it is then you can resolve it perhaps so those are interesting questions or if you don't resolve then is there generalized Higgs type mechanism and we know that it cannot just be one-spin one-particle-spin-zoonoparticle exchange has to be complex this also indicates in that direction cost us any big questions just to make sure I understood this raising of the break-down scale by 90 or 30 is compared to what scale is this the usual lambda 3 and the M-square implied that's exactly right yes so it's 19 or there's a minus compared to that scale yeah so it's so this lambda star is what you get effectively in 4D so this is what you get in 5D yes which is nature this is the same formula that you had actually but for 5D but this is what you get in 4D yes and as you see instead of one power of mass you have power of curvature so it's compared to this scale that you can still this is the high scale it's compared to okay right yes you probably explained it and I missed it I didn't quite understand how do you update the commander to understand the synergy of arguments about scattering and the phase shift of that delay well so right so this is what I alluded to when I spoke about the parameter range for there is this alpha so in this notation alpha 3 and alpha 4 and in the decoupling limit I call them alpha and beta so there is a parameter range for those for which the superluminal scattering is not happening and that first paper that did that was by Cheng and I cited it somewhere yeah so I had it somewhere cited I am okay I can find it now but anyway when you embed it in ADS yeah so when I embed it in ADS actually if I go by the option in which in which those so those operators are very much suppressed this gives you additional benefit in those conversations so you can you can imagine that you would have again this similar parameter space perhaps even broader parameter space for which you would you would not encounter those issues but if I remember well the problem is where alpha 3 and alpha 4 are driven near 0 by these limits and this point is a point where the range time mechanism does not work so you close in or use 0 in a very dangerous phenomenological region no no but the whole point is that the pi doesn't couple to stress tensor anymore you don't need Weinstein mechanism that's the whole point yeah you raise that but also there is a linear screening of pi now you don't need non-linear screening of pi you see so let me limiting theory where alpha 3 and alpha 4 equal 0 has no problems alpha 3 oh yeah yeah exactly in my notation yeah exactly if you take if you take alpha 3 and alpha 4 to this point if you if you take this times to 0 it won't have that problem that you are referring to I don't remember if it may have the super luminal super luminous yeah probably yes so is this safe theory alpha 3 and alpha 4 equal 0 I believe I believe it may may not be for the for the reasons that Emmanuel was saying I think the parameter range which is not ruled out has non-zero alpha 3 and alpha 4 if I remember we burn exactly to alpha 3 and alpha 4 equals the small regions in this graph yes but it's I'm using different I'm using different coefficients the point is that there is there is two parameters and then there is a parameter space and in some linear combination there is a special point and there is a whole kind of you know circle around that which addresses all those questions okay thank you applause