 Okay, so first I want to thank the organizers for the opportunity to speak at this conference. And I'd like to talk a little bit about the status of naturalness and the hierarchy problem is in particular an approach to it, which goes by the name of neutral naturalness or colorless top partners. So the question is, is the discovery of the Higgs and the fact that we haven't seen anything else at the LHC the end of the idea of natural theories of the electro-wig scale? Are we just giving up on naturalness? And then the question is, so we have already even experimentally seen that we have the wig scale, but then there is a scale determined experimentally by direct and indirect bounds, which is already at the few TV scale here, which is telling us that what we have here in this gap between the wig scale and the few TV is the standard model, but it's tuned to some extent. And so this is what we call the little hierarchy. And then the question is, this is already established experimentally as a problem. We haven't seen anything beyond the standard model. And so there is an extent to which the standard model is already experimentally determined to be a tuned theory to some degree. And then the question is, the quadratic UV sensitivity on the Higgs mass is dominated in particular by the top contribution to the Higgs mass square. And typically what we do in order to cure this in natural theories of the electro-wig scale is to impose a symmetry. And this symmetry will, among other things, relate the top quark to its partner under the symmetry. And this partner will have a one-loop contribution to the Higgs mass square, which will cancel this quadratic sensitivity to the UV scale. Now typically in the theories we consider for the past 20 years, the top partners carry color. And they're easily produced at the LHC, and this results in these stringent bounds for the most part. So the point is, can we build theories where the top partners do not have SU3 color, and therefore they're harder to have been seen at the LHC? So if the symmetry protecting the Higgs mass does not commute with the SU3 color, and it exchanges into another SU3, let's call it SU3 prime, then these top partners will be charged in this new color. The bounds will be less stringent, and the hope is that these theories with colorless top partners will be more natural than the extensions of the standard model we have today. So essentially the idea of these theories of colorless top partners, or neutral natural theories, is to fill this little hierarchy gap with the standard model plus at the minimum colorless top partners, so as to solve at least the hierarchy problem, or the little hierarchy problem here. Eventually most of the theories, all of the theories we have thought of act this way will have a cutoff not far above this scale of the future TV. So how do we do this? So what are the ingredients for a neutral natural theory, or for a theory with colorless top partners? Well first of all, you need a symmetry protecting the Higgs, just like any other theory protecting the UV sensitivity of the Higgs mass. It could be that the Higgs is a pseudonambulgolstone boson, it could be a version of supersymmetry. But you also need to extend the color gauge symmetry to have at least two copies of SU3. So it could be SU6 broken down to SU3 square, or it could be two copies of SU3. We'll see some examples. And then you either impose a discrete symmetry, or somehow you are going to extract the degrees of freedom by some procedure that can be generalized as orbit folding, such that the top partners will actually have the other color once we extract the light degrees of freedom of these theories. So the theories that have been worked on are old to begin with. I'm going to talk about the twin Higgs and folded supersymmetry, but there are also, as you see, any solution to the hierarchy problem can be turned into a colorless top partner type theory by these kinds of procedures. So if you pick your favorite theory, for example, there is a little Higgs version called the Quirky Little Higgs. So as you see, these theories are rather old, 10 years or nine years. But our interest is renewed in them because of the situation of not having seen anything at the LTC. So let's just quickly review the twin Higgs. It's the 10th anniversary of the twin Higgs, I guess. So the Higgs in these theories, it's the Nambu Boston Boston of a spontaneously broken global symmetry. So we start, for example, there are many ways to actually do this, but let's just start with the original way in which the global symmetry is SU4 broken down to SU3. And so there are seven NGBs out of which there will be four that we'll use for the Higgs. And then, so here is the potential for these four of SU4 that I call H, big H. Now, the next step is to gauge a subgroup of this SU4, which is SU2A times SU2B. So now the four can be split in these two doublets. And we'll use this top doublet, HA, as our standard model Higgs doublet. Now, when we gauge this subgroup of SU4, the gauge interactions will break the global symmetry explicitly. And this will lead, generally, you know that this leads to quadratically diverging contributions to the potential of big H. So you compute those contributions from the gauge loops, and they are quadratically divergence. And so here it is, this is GA and GB are the gauge captains of the SU2A, SU2B. And here are the cutoffs, lambda A and lambda B, which in principle, the behavior of the theories in the UV does not have to be the same, so in principle. However, if I impose a Z2 symmetry, such that the behavior in the UV is the same, and then the gauge captains GA and GB are the same, and also the cutoffs of the theory are the same, since the Z2 is good in the UV, then it turns out that the gauge contributions to the, the quadratically diverging gauge contributions are of this form. So if you just get the cutoffs and the gauge captains out, and then it turns out that this is SU4 symmetric. And so if you think of the Higgs as coming from the NGBs in this global breaking, then the Higgs do not enter H dagger A, H dagger H is SU4 symmetric, and so this will not contribute to the Higgs mass square. So this is essentially the philosophy of the Twin Higgs idea, the imposition of this Z2 symmetry in the UV is what protects the Higgs, the Higgs from UV sensitivity. So if you extend this to all standard model interactions, not just the SU2 interactions, then essentially you have a standard model mirror sector, so we have the standard model A, which is us, the standard model B, which is the Twin standard model, and for example, if you look at the top Yukawas, now you can write them in this way in general, there is the A sector Yukawas, and there is the B sector Yukawas, that will generate these quadratically divergent contributions for the A and the B Higgs doublets. But if you impose the Z2, once again the contributions from the Yukawas, the quadratically divergent contributions from the Yukawas are indeed SU4 symmetric and will not affect the Higgs mass square. There will be, yes, logarithmic divergent or logarithmic sensitivity to the cutoff, because, well, I'm not going to the little, but there will be contributions that go like HA to the fourth and HB to the fourth, and those contributions do have logarithmic sensitivity to the cutoff, but the quadratic sensitivity is all we care about here, because we're trying to solve the little hierarchy problem. So if you go to the nonlinear representation, this becomes clear, here are our two doublets, but I can write them as the Nambu-Gauston bosons here, exponentiating, and some VEV-F, and here in the Nambu-Gauston matrix, I'll just will write the four degrees of freedom that I will use for the Higgs doublet, which is right here, and all the other three NGBs are eaten by the SU2B gauge bosons, this formulation, and so when you do that, you can expand the H matrix and rewrite the Yuccaoas and take the first two terms of the expansion, and you will see that the cancellation of our top quadratic sensitivity to the Higgs mass comes from this term, which is normalizable in this nonlinear formulation, and this cancellation, the symmetry is just coming from this fact here, the fact that the Z2 symmetry imposes that the lambda A and lambda B are in the same, and so this cancels the quadratic divergence to the Higgs mass, but the top partner, the B tops are not carrying SU3B color, they are SU3A color, they are carrying SU3B color, so this is a colorless top partner solving the hierarchy problem for you. So up to this point, you have an exact Z2 symmetry, but if you do have an exact Z2 symmetry, what happens is that this VEV, or the VEV that parameterizes the breaking of SU4 will be the same as the weak scale, because essentially all the A sector has to be identical to the B sector, so our VEV and the VEV of the B Higgs, the twin Higgs, will be the same, so you need to break this Z2, but you break it softly by, for example, adding a term like this, a soft term like this, and this allows a hierarchy, a mild hierarchy between the electric scale and this F scale of the twin Higgs VEV, and typically what will happen, for example, for the phenomenology of the twin Higgs to start being explored, is that the couplings of the Higgs to standard model fields will be suppressed by the cosine of V over F, essentially, right? So V over F will be this small hierarchy in order to avoid bounds, essentially that I'm going to show you now. If this was identical to our weak scale, in fact it was the same as the weak scale, then for example the invisible width of the Higgs will be, I don't know, 50 percent, so that's clearly excluded, right? So we need to break the Z2 at least a mildly in a soft way. So there are several incarnations that are used now for the phenomenology at the LHC. The one that is the original is now called the identical twin. It's a complete copy of the standard model. F over V goes somewhere between 3 to 10 where the 3 comes from, essentially the invisible width of the Higgs, and the 10 is just a matter of taste and naturalness. This will force, for example, the twin QCD to have a strong scale which is somewhat above the QCD scale. You will have light twin porcs and leptons, and then the question is whether you will, for the model building, you will keep the twin photon massless or you will have to break it, and this is a question of, for example, cosmology, things like that. So this is the identical twin model. In the identical twin model then, as I said, all Higgs couplings to the standard model states are suppressed by this cosine of V over F. In this way, for example, if rho is the Higgs, the radial state, our Higgs in the twin Higgs model, then this is the production with respect to the standard model and the decay to our sector states, W, Z, B, etc. And then the invisible width will go like the sine square of this angle times the standard model width in equivalent states times some kinematic factor delta which is less than 1 when essentially the masses of the B sector are heavier than our masses. Essentially, so you need this kinematic factor to account for the invisible width. And so as you see when delta as F becomes much bigger than the electric scale, delta becomes much smaller than 1 and you reduce the invisible width. That's the idea. So if you look at the Higgs couplings, so here in the green line you see the invisible width as a function here of the top partner mass, but essentially you can see this as a function of also the scale F. The top partner mass is proportional to some extent to F over V times the top Yukawa couplings times the cosine or the cotangent actually of V over F. So essentially if you just look at the current bound from the invisible width, you probably are around this area over here which still leaves you with a tuning which is not too bad. It's 30% tuning for this theory. On the blue line we see the signal strength for whatever channel of the Higgs that you can imagine so the signal strength and you see that you get to signal strength that are around 5% of the standard model and you still are around 10% tuning. So you could imagine that you will reach high luminosity LHC stages just by measuring the Higgs coupling and then you wonder if you're going to be able to see to tell to exclude these theories just by looking at the Higgs couplings and excluding deviations and if you are with a 5% signal strength you still will have theories that are, you know this theory will still be around 10% natural or just below 10% natural. So it is possible to imagine that the identical twin Higgs may still be around as a not that fine tune even at the end of the high luminosity LHC which is of course terrible but this is the nature of this model. Now there is also another version which is more tailored to having signals which is called the fraternal twin. In the fraternal twin only the minimal fremian content to some hierarchy problem is considered. This is kind of like the twin Higgs version of natural Susie. In here you only have the third generation quarks and leptons and then it turns out that the glubals of this twin sector are the lighter hydronic states and they cannot decay to light mesons because there are no light quarks in this theory and so it turns out that these glubals the Higgs can decay to these glubals with branching ratios that are okay very small 10 to the minus 3, 10 to the minus 4 however these glubals are typically long lived in this model and they can generate highly displaced vertices. So the hope in the twin Higgs models that have this fraternal twin type spectrum is that you will be able to see from this the case of the Higgs to these twin glubals these highly displaced vertices. Okay so for this you really have to go to this fraternal twin setup but other than that you just look at you have to look at the Higgs caplings. Recently there have been a lot of activity in the twin Higgs dark matter idea. Why is this interesting? Well it turns out that the twin strong sector generates a higher strong scale by a little bit as I mentioned earlier which means that in the identical twin case the twin baryons are a factor of a few heavier than our proton and so this is very interesting for asymmetric dark matter model building and in fact there's a recent some recent work by Marco Farina and here you can see assuming just essentially that this factor of a few is between four and six so you get five. This five factor is the mass of the twin baryons is five times our proton mass then you will get the the the the relic density you need for dark matter and so in this case you compute the cross section indirect detection by using either higher dimensional operators or Higgs portal interactions and this is what you get from one TV higher dimensional operators down and so you see that it's possible that this simple and natural idea of having a twin Higgs dark matter may be accessible somehow in the future experiments although if you just look at the at the Higgs interaction is somewhere down here below the neutrino flow but then there is some activity also for the fraternal twin Higgs there are some papers here there is also either a symmetric dark matter which is typically what you want to do for twin bottom baryons in this case only the bottom will be your your baryon because there are no other light works this is a three spin three half dark matter candidate but you can also do a thermal relic in this case mostly the twin tau would be the dark matter so this is this is what the twin Higgs that's for you and so it's a difficult it's a difficult phenomenology if you really stick to the twin to the identical twin Higgs and in principle there could be some deviations in the Higgs coupling but it's hard to see what the path to a smoking gun would be that will definitely get this model now what about supersymmetric can we build a supersymmetric theory that does this thing that has cordless squirks so that you know you wouldn't have produced the squirks at the LHC and the answer is it's not really but yes and no so let's see what does that mean you know so so this this theory goes by then it's not twin supersymmetry it's folded supersymmetry but that's I don't know there's a motivation that hopefully will be clear so the idea is that again exactly the same these squirks in the cancellation we we don't want them to have a suit recolor so how do we do this idea how do we implement this idea that this course canceling the quadratic sensitivity to the of the Higgs mass do not have color how do we do this so in order to to motivate this idea I think this toy example is all we need for technically and this goes goes by the name of bifold protection and essentially this this simple model is like this suppose you have a global un a flavor symmetry which is this un so I have a singlet and I have this these fermions which have this this there are fundamental of the un and then we have this simple coupling between the singlet and the and the and these fermions and of course in this in this setup the singlet is quadratically diverging has a quadratic sensitivity to to uv scales this is this is the obvious thing so the first step to have bifold protection is to supersymmetrize the theory and to think of this as a super potential okay so fine now now you have some protection for supersymmetry in principle if supersymmetry is exact then the second step which is the the by in bifold protection is to duplicate the the flavor index between n to 2n so now I have 2n in the loop that gives me the quadratic divergence so you so far it looks like a useful thing useless thing to do but let's see what we do with that the next step is to define a parity which which is actually an element of this of the global group this global group this u2n now is instead of un I have a u2n flavor symmetry this is one element of this group and I picked this this element with pluses in the diagonal up to from 1 to n and with minuses from n plus 1 to 2n right so this element will be defined as transformation and then it turns out that if you look at that super potential the theory is invariant under the following transformations the singlet going to itself the the q's going to minus gamma q and then the q bars going to minus gamma well this the star but this one is real minus gamma star q bar and also there is the theory is also invariant under this z2r let's call it z2r symmetry where boson under which bosons transform to themselves and fermions to minus themselves okay so now so the theory is invariant under this but now I'm going to ask that the states that are odd and they're the product of these two symmetries are projected out so this is sort of like on or before project projection you can think of this as being an extra dimensional theory and this would be like a boundary condition so I'm going to say this will be like a boundary condition projecting out states that do not that are not even under this this this problem so here is the z2 gamma so here I have the fermions from one to n and from n plus one to two n so gamma it hits them with a minus one here and with a plus one here and the scalars as I said before because of this z2r are are are even here and odd in the fermions okay so here is is so now the product of these two has to be positive and so what happens is now that this mechanism essentially selects the fermions with indices between one and n so these guys will stay in the spectrum at low energies and the scalars with n plus one to two n indices so now I have this theory that is in principle not super symmetric anymore because this q1 to qn are not super partners of the qn plus one to q2n scalars however what I have now is that so this is this is my low energy spectrum I are already folded out or motted out all the rest of the of the of the states and I ended up with this in the low energy theory but still I have in the loop the fermions from one to n and the scalars from n plus one to two n cancelled each other the quadratic divergences and the theory is uh uh UV insensitive at least quadratically to one loop okay so this this is the mechanism by for protection and it's inspired in the large n orbital correspondence that uh uh was it was it was proposed by these people uh years many many years almost 20 years ago so this this allows us to construct then a theory that has at one loop no quadratic sensitivity in in the in this in this singlet in the scalar particle but uh uh it's not supersymmetric there is some sort of accidental supersymmetry in the low energy spectrum but not real Susie okay so in that sense it was the it's no Susie but some leftover supersymmetry is there okay if you think about it this the uh to loop this doesn't work anymore right for example this the scalars are not protected by this by this symmetry the scalars themselves will be quadratically sensitive uh if I close if I open this loop of the scalars and I close the singlet the scalars will the mass of the scalar will diverge no will have a quadratic divergence that means that if I have a two loop contribution to the singlet it's also will have another a quadratic divergence uh to the singlet at two loop but I don't care about two loops I only care about one loop because I want to solve the little hierarchy problem two loops will be important above that scale of few TV so to implement this then uh all I do is now I have to of course do this in the real world so now I'm going to have to have my three copies of SU3 my Z2 that relates the the the SU3 couplings in the UV and then the weak sector I'm going to leave untouched and I'm going to assume that the Orvifold in some extra dimensional theory gives me the fermions of our sector with our color but the scalars with the wrong color and essentially this leads to these interactions in the in the in the UK in the top yukawa which still protects uh Higgs mass square to one loop okay so so this is the setup in extra dimensional theories I have no time to talk about it obviously and so uh uh SU3 is broken in extra dimensional uh setup uh by boundary conditions and then uh essentially I end up uh uh I end up with uh with this spectrum this hyper multiplicates leads to our our fermions and this uh hyper multiplicates will have scalar zero modes with the wrong color okay so this is this is what we do uh uh the interesting thing that I do want to point out is that the scalar masses are actually calculable uh in this theory and this is very old calculation from these people uh because essentially what happens here is that sucy breaking is non-local essentially this setup we have an n equals two supersymmetry in the bulk and then you have n equals one supersymmetry in in this fixed point and in this fixed point but these two n equals ones supersymmetries are different and so the low energy spectrum sees a broken sucy uh but in order for uh the zero modes to to acquire masses I will have to go from this point to this point so I see the different two n equals one supersymmetries and then then I realize sucy is broken and so this gives me this uh sort of uh calculation of the of the masses so the the masses are several hundred gv if I assume that one over r for example is a few tv is my e bounds right that's all this transparency says so this is the setup I have accidental sucy between say a few tv and the weak scale but then there will be a 5d sucy and eventually it's a cutoff because this is a non-normalizable theory and they will have to be a uv completion right all of these theories the twin hicks quick little hicks for the sucy they have a cutoff which is not very far from these these few tv scale so they are really they require uv completions to become actual real theories and that's an important thing I don't have time to talk about signals but there are signals and I'm just going to say that the fact that I have an su3 that confines the su3 prime uh and that the squirks are hundreds of gv it means that they don't hydronize okay so these guys will have quirky type behavior they will generate sort of a global radiation as well as a photon radiation they will have to come back to each other and annihilate and from there you can get bounds which are at right now at the 450 gv level so I'm going to skip this and I'm just going to summarize so we still have natural tiers of electro-exymmetry breaking that are not ruled out by data it's true the signals are collider are different I didn't have time to talk about this but for example in folded sucy globals can be produced both in this in this squirt production they will emit it and also hicks can decay to them and these globals will actually have displaced vertices or interest in the case and so in principle there are interesting signals for these globals I didn't talk about it but I just want to flash the reference for this person that this did this great work and you'll see in this paper that global decays in folded sucy could be an important signal at the LHC in the future so however it's not impossible to imagine that it will end up high luminosity LHC with some natural parameter space in some theory like identical twin hicks and still we haven't seen anything it's not an impossible scenario okay but all of these tiers have low cutoffs below 20 TV so experimentally this would point to higher energies obviously in theory it would point to the fact that we need more reasonable new completions okay and of course I pointed out to the fact that there is interesting dark matter model building okay thank you we have time for one or two questions maybe three if they are quick could you comment give me a comment on the degradation were effective effective number of neutrons I think in this model in this model you you'd have twin neutrons or a twin photons and it may may may conflict with the plant result right right okay so okay I didn't want to comment on this because we are working on it but certainly you you have to get rid of it in the identical twin case you have to get rid of the twin neutrinos somehow this is non-trivial because you have to get rid of them in a way you know you break the z2 somehow and so ideally what you do is you break the z2 because you have to break the z2 anyway at the same time given by a mechanism that also gives masses to the twin neutrinos and to the twin photon that's that would that's very ambitious and that's what we're trying to do and so you really have to do that because you can you can imagine that you can get rid you can get rid of the neutrinos and keep the twin photon but that's still borderline with with structure formation amazingly enough so you can get rid of you know NF is still borderline for one degree of freedom like the twin photon but there's still problems that my collaborators know much more about they have to do with silk dumping and things like that so so I think you have to get rid in these tears you have to get rid of the twin photon also yeah you're right so in the discussion of natural Susie we usually talk about the need at low energies for three states uh Higgs you know stop and then at two loops gluinos but in this context you're really just talking about the color as top partners is the reason why I don't have problems in these scenarios with stuff at at three level you know the analogy of the Higgs you know or why I don't really have to care about two loop effects you know the analogy of the gluino uh so you're talking about uh the twin the fraternal twin case where I only have or or any any of these cases you know is there a mapping between the natural Susie language of what we expect for the low-energy states and again it's Higgs you know stop and gluino is what we say kind of tree level one loop two loop well so for example in the fall that Susie you have the Higgs you know in fact they are they are candidate to dark matter so you have them there uh and then uh and then the gluinos are not that heavy they are they are so they will they are at the at the say one over R right so so in principle uh and now in the twin case um in the twin case in the fraternal twin case um you you have the gluino is still there you know that it's it's even confined it's not broken at all so so it's not yeah it's not quite quite a mapping but it's the gluino is still around it's confined it forms all kinds of uh uh of bound states at low energy you know 5 10 20 g depends on lambda qcd prime so yeah so your point is really to solve the little hierarchy problem i'm not carrying too much about the the big one i do care about the big one but uh if if if the if the idea it turns out that all the natural natural theories are such that they have it cut off and then they they are usually such that you solve only the little hierarchy problem at one you can do that and then this may be uh the first step you need to do but we need to work on uv completions to convince ourselves that this is a sensible low energy theory of course okay so uh you are giving us very good arguments for 100 tv colliders yeah okay so thank you very much and uh thank you again