 And we are live. Excellent. So hello, everyone. Welcome to another session of the Latin American Webinars on Physics. My name is Joel Jones from the PUCP in Peru, and I will be your host today. And this is webinar number 134. And we are having Robert Ziegler as a speaker. Robert did his PhD at CISA in Trieste, which was unfollowed by postdocs at TUM in Munich, LPTHE in Paris, and CERN. And now he's part of the staff at KIT in Calzru. So today, Robert will tell us about the important facts about axion dark matter and its connection to flavor. So we are, of course, very happy to have him as a speaker. Now, before we begin, let me remind all viewers that you can ask questions and give us comments via the YouTube live chat system. And these questions will be passed on to Robert at the end of his talk. So, well, Robert, please feel free to share your slides and start whenever you're ready. OK, yes, thank you very much for this nice introduction. So can you see my slides? Yes. OK, perfect. So you can also see my pointer, right? Yes. Perfect. OK, yeah, so let me start. So today, I'm going to talk about the possibility that the invisible axion has flavor-violating couplings to standard model fermions. For example, this one here, which is a lepton flavor-violating. And I will discuss the consequences of these scenarios. And I will show you that these couplings then allow for the possibility to produce axions from decays of standard model particles. And this is relevant not only for flavor factories, so laboratory experiments, but also in astrophysical environments, for example, supernova, and finally, also for cosmology, where such decays can produce axions in the early universe, which allows for a very efficient production of axion back-met. OK, yeah, so this is the outline of this talk. I will start with a brief introduction, discussing the strong CP problem at the QCD axion solution. Then I will talk about axion, dark matter, and the phenomenology, or let's say the standard phenomenology of the QCD axion. Before I turn to the main part of this talk, namely, the phenomenology of flavored axions, which means axions with flavor-violating couplings to standard model fermions, which is not very often discussed in the literature. But I think this is a very, very important possibilities, and it allows to test these couplings and look for the QCD axion also with precision flavor experiments in a way that is orthogonal to usual axion searches with axion halo and helioscopes. Interestingly, this will be also important for axion production in supernova 1987A. And finally, this allows also for axion production in the early universe from decays of standard model particles. And here we'll discuss the case where one has decays of standard model leptons, which produce axions via lepton flavor-violating decays. And all of this is based on these couple of papers here that I wrote with several friends and colleagues. OK, so let me start with a strong CP problem. So in general, a gauge and Lorentz symmetries allow for a single term in the Lagrangian, which is called the topological term of QCD. So it's simply a coefficient of a gg till the operator, where g is the field shrinks of QCD. And one can show that this term violates CP and parity. And actually, this term by itself is not physical observable, because one can show that under fermion field with definitions, these term shifts. And actually, the only combination that is invariant under such field with definitions is a combination of these theta term here together with the argument of the determinant of mass, quark mass matrices. But I will simply go to the basis where the mass matrices are already diagonals. So this means that this term here is physical, and as such, it can contribute to physical observables. And as I said, this is a CP violating. So in particular, it can contribute to CP violating observables. For example, the electric dipole moment of the neutron. And if one makes these calculations, one finds that the neutron EDM is then roughly proportional to this angle. And it turns out that this contribution for order 1 theta is way below the experimental upper bound that we have on the neutron. So this experimental upper bound is of the order of 10 to the minus 26 electron centimeters. And this implies that in order to comply with experiment, this theta parameter, which is just a free parameter in the standard model, has actually to be smaller than around 10 to the minus 10. And the reason, I mean, in principle, there is no problem of having such a small parameter, but it would be nice to have a dynamical explanation of this fact. And this is called the strong CP problem. Now, this is very similar to other small number problems that we have in particle physics. So for example, the cosmological constant problem or the flavor problem or the hierarchy problem. But in contrast to those problems, actually, the strong CP problem is somewhat more difficult to explain, I mean, to apply the usual solutions that we have, which are based either on symmetries or on tropic arguments. So one of the problems is that this term, of course, violates CP and parity. But actually, we know that CP is already broken by the weak interactions. While parity is, of course, broken, I mean, very strongly by the v minus a structure of the standard point. So therefore, it is relatively difficult to come up with solutions to the strong CP problem that are based on imposing those symmetries and then breaking it spontaneously, because one has to watch out that the theta term is then not relatively generated, which is a problem, because it is so tight. But nevertheless, you can do this. And this goes under a class of solutions here, where you use either spontaneous P or CP violation. Now, on the other hand, it is interesting to notice that actually, in contrast to, for example, the cosmological constant problem, there is not really an entropic solution to the strong CP problem, because basically, this parameter does not have or does have only little impact on structure formation in the universe. So the theta parameter could be really many orders of magnitude larger than this without having any impact on galaxy formation and so on. So this leaves us with these solutions here, which basically just fall into two classes. So either these symmetry solutions, or which are based on heavy new physics, or a dynamical solution, which I think is the most elegant one, which is based on light new physics. And this is a peculiarity axiom solution. So the essence of this mechanism is the observation that if this theta parameter would be a dynamical field and not just a parameter that we write in the Lagrangian, then one can show that QCD would generate a non-perturbative potential for these fields, which actually has a trivial minimum where the vacuum expectation value of these field vanishes. And so one can calculate starting from these interactions here. One can calculate the contribution to the axiom potential by non-perturbative effects. And what one finds is a potential that roughly has this cosine structure here, which indeed is minimized when theta equal to 0. And theta equal to 0 will be precisely the parameter that will enter the neutron area. I mean, this is the clue to the solution. So in order to explicitly realize this idea, we need a field that couples to gg tilde and has no other potential. Now, a field without any other potential is a Goldstone boson. So essentially, what we need is a new global symmetry of the Lagrangian, which is usually called the Petscher-Quinn symmetry, which then is broken spontaneously. This will give rise to a massless Goldstone boson. And then we only need another ingredient in order to couple this field to gg tilde. And this is achieved provided that the symmetry is anomalous under QCD. So we have a Goldstone boson, which implies that the axiom is this phase here suppressed by the scale where this Petscher-Quinn symmetry is broken. And then this underlying U1 symmetry that is spontaneously broken is also explicitly broken by the chiral anomaly, which means that the corresponding current is not conserved, but is actually proportional to the gluon and its dual. And these two ingredients imply that the Goldstone boson will couple to gg tilde exactly as the tether parameter. And this means that we have precisely realized this first situation here, namely that we have promoted the tether parameter to a dynamical field. And this dynamical field is called axiom. OK, now, although the Goldstone boson is massless, because of the QCD anomaly, one can show that actually QCD generates a tiny axion mass, which then only depends on the Petscher-Quinn breaking scale. And actually, you can see this already from this potential here. If you take this simplified form here, just simply by expanding this to second order in tether, where tether is then identified with the axion divided by the Petscher-Quinn breaking scale, you see that this potential then generates a mass term for the axion, which is given by m pi f pi over lambda Petscher-Quinn. And this means it's inversely proportional to the Petscher-Quinn breaking scale. And actually, using chiral perturbation theory, you can calculate the axion potential, and therefore, the axion mass very precisely. This has been done in this paper here by Giovanni Belladuro and collaborators. And the result is that you'll find this axion mass here, which you can actually quote this for significant digits. You see that it only depends on the Petscher-Quinn breaking scale, FA, which is basically just these combination here. So the Petscher-Quinn breaking square divided by two times the color anomaly coefficient. And you see that for typical values of the axion decay constant, which are of the order of 10 to the 9GV, or so the axion is extremely light with a mass of the order of milli electron. So this relation holds for the QCD axion. In recent years, this notion has been generalized to an axion-like particle. So basically, this is also a particle that has couplings as the QCD axion. But the mass receives additional contributions on top of these anomaly here. So there is simply an explicit breaking of the Petscher-Quinn symmetry, which then simply implies that the mass of these axion-like particles is a free parameter. Now, this at the same time implies that it cannot solve any more of the strong CP problem, at least not in the very simple or standard realizations of the QCD axion, so that the main motivation of having this generalization here comes not from the strong CP problem, but from dark matter and phenomenology. So in particular, the phenomenology of the ALB can even be looked for at the LHC, while for the QCD axion, actually, it would be impossible to find signals of the QCD axion at the LHC. So this generalization, I think, is more to motivate experimentalists to look for such particles, even if they might be not as motivated as originally QCD axion. OK, yeah, so as I said, one of the main motivation for these particles is dark matter. And the reason is that these particles are indeed excellent dark matter candidates, because they are pseudo-goldstone bosons of a Petscher-Quinn asymmetry, which typically is broken at very high scales. The fact that they are pseudo-goldstone bosons imply that these are very light particles. And the fact that the Petscher-Quinn is broken at very high energies implies that they're essentially decoupled, because the Petscher-Quinn breaking scale suppresses all interactions. Now, this implies that not only that their couplings to the standard models are suppressed, but also that the lightness implies that the decay rates are extremely tiny. So in general, these couplings can always decay into two photons, but you see that the corresponding lifetime for typical values of the decay constant and the axion mass easily extends the H of the universe. So in general, we need very large F and a small MA. But if you remember, this situation is precisely realized for the QCD axion, where the axion mass is indeed inversely proportional to the axion decay constant. Yeah, so this means that these kind of axions are stable on cosmological scales, and so we only need to produce them in the early universe. And there are several possibilities to do this, but maybe the most common ones are the so-called misalignment mechanisms, decays of topological defects, terminal freeze and terminal freeze. So let me briefly discuss two of them in somewhat more detail. So the first one is the classic misalignment mechanism that has been proposed first for the QCD axion. And the observation is simply that typically this axion has a very high occupation number, so it essentially behaves like a classical scalar field. And if you write down the equation of motion of a classical scalar field in an expanding universe, you find this equation here, which is precisely the equation of an harmonic oscillator with a time-dependent friction, which is given by the Hubble parameter and a time-dependent mass, which is the QCD axion mass, which in general is temperature-dependent. And essentially, you have then a picture where in the very early universe, at temperatures much above the QCD phase transition, the Hubble friction dominates over the mass term, so the axion is essentially frozen. Then the universe cools down. And close to the QCD phase transition, this potential is switched on. And the Hubble parameter becomes much smaller, such that the axion starts oscillating around the minimum of this potential. And the energy which is stored in this oscillation actually looks exactly like cold dark matter. So therefore, you can calculate the relic abundance in terms of only two parameters. So the first one is the axion and decay concept. And the second is the original misalignment angle that, unfortunately, is just a boundary condition in these very simple scenarios where the petroquin symmetry is broken before in flation. Yeah, but you see that typically you get the right amount of axion dark matter for axion decay constants, which are close to 10 to the 11 G. Now, a second possibility to produce albs in the early universe is to thermal freezing, which always works for particles which have very tiny couplings to the standard model, such that these particles are never in internal equilibrium. Indeed, as I already discussed, this decay constant has to be extremely large, and therefore these couplings to the standard model are extremely small. So indeed, if one starts in such a situation where the original population of the dark matter axions is zero, actually, so this very tiny coupling of the axion with the standard model slowly axion population builds up and basically then remains a constant in a co-moving volume because the standard model particles that have produced the axion either through decays or through scattering then fall out of the, I mean, I mean, they become non-relativistic, so such that their number density becomes exponentially suppressed, and this means that essentially the axion population remains constant. So as an explicit example, one can consider decays of standard model particles into axions and the corresponding contribution to the relic abundance is then given by these two terms here. So the first is the ratio of axion mass by this standard model mass. So if this is of the order of 10 to the minus 3, then the right abundance is reproduced if the decay rate of the standard model particles over their mass is of the order of 10 to the minus 22. So indeed, one needs extremely small couplings to have thermal freeze, and but this is precisely the case for such axion-like particles. So these are the two mechanisms that I will consider for an axion dark matter production, but as I said there, many more. OK, yeah, so I have discussed now the main motivation for the axion, so how can we look for it? Now, in general, we can simply write down the couplings of the axion to the standard model by using an effective Lagrangian, which is well below the petroquin breaking scales, where one writes down simply all couplings of the axion, which respect the shift symmetry, which is the relic of the petroquin symmetry, which implies that, in general, the axion couples duratively to fermion and scalar fields. But actually, because this petroquin symmetry is anomalous, there will be a couplings of the axion to the gauge fields, which are just linear in the axion, so they violate these shifts. And of course, because the axion is a goldstone ball, then all the couplings will be suppressed by the petroquin breaking scale. So the most important terms relevant for phenomenology are then these three terms. So the first one, the axion couplings to gluons, which is proportional to the color anomaly, which, as I discussed already, solves the strong CP problem and generates the axion mass. So this is a mandatory for the QCD axion. But then, in general, we have other model dependent terms. So the first one is this here, the axion coupling now to the photon field strings, which is proportional to the electromagnetic anomaly coefficient, which, of course, contributes to the axion couplings to photons. And then finally, we have axion couplings to fermion fields, which respect the shift symmetry. And in general, they are flavor violating, so they are by far the largest number of terms. But let me first discuss the axion couplings to photons, which is the one that is considered, I mean, almost exclusively in the literature. So as I said, these direct contribution, which comes from the electromagnetic anomaly, contributes, of course, to this coupling. But actually, there are other couplings. The first one is a contribution that is constant in this parametrization, which actually just comes from axion pion mixing. And while these two here are the anomaly terms, there are actually then also other contributions which come from one loop diagrams with standard model fermions, which give contribution to the axion couplings to photons at one loop. But typically, they are suppressed then for very small axion masses. So for the QCD axion there, MA is much smaller than all of the standard model fermion masses. These terms are neglected. And then you have this famous relation here, which gives you the axion couplings to photons, where you see that actually it is difficult to realize a scenario where this coupling is small, because you have to tune this ratio of anomaly coefficients versus this model independent contribution, which comes from axion pion mixing. So for this reason, this coupling is pretty generic. And therefore, it motivates the corresponding axion searches in experiments. And essentially, the main classes of these experiments are the so-called haloscopes. For example, the ADMX experiment, where dark meta-axion is converted into a photon, which is then resonantly detected by a microwave cavity. So this photon has the energy of the axion mass, which, of course, is unknown. So basically, one has to look for the axion by carefully tuning the resonance frequency of these microwave cavities, which, of course, is then also limited by the size of these experiments. Now on the other hand, you have haloscopes, for example, the Cast experiment, which does not rely on axions being dark meta, but simply using the fact that in the sun, the photons which are produced in the stellar plasma are converted into axions due to the presence of very strong electric fields. And the plasma, these axions then escape from the sun and travel to Earth, where they are again converted back into photons using the very strong static magnetic fields, which here is essentially the, shows these dipole magnetizing for lab experiments. So here's the difference is that now the photon has the energy of the typical solar photon, so it's in the x-ray region. So this shows the usual plane that one shows for axion, for the constraints on axion couplings to photons, where here you see the axion, a photon coupling, and here you see the axion mass. And here, essentially, this band here is the QC axion plane. And as you can see, there are several planned experiments and existing constraints populating these region here, so in particular, there will be a lot of new experiments that will cover these most interesting region here, where the axion can be dark matter. So I don't want to go through all of these experiments, but basically just say that the most interesting region of the parameter space, where the axion can be dark matter, for masses roughly below 0.1 electron volt, has only been recently scratched by experiments. And then let me just say that what you see here are the haloscopes, which indeed have, I mean, probe only a quite tiny region in the axion mass parameter space, where the haloscopes, in particular, the cast experiment here, basically can exclude a wide range of axion masses, however only. I mean, however, they don't reach the same sensitivities as the haloscopes. OK. Yeah, so now let me discuss the main topic of this seminar, namely flavor-violating axions. And as I said, usually, this possibility is ignored. And people just write flavor diagonal couplings, but actually, just from an effective field theory point of view, such couplings can in general be flavor-violating. Now, precisely as in the standard model, once you allow for flavor violation, this gives a proliferation of parameters, but it also enriches the phenomenology, because now the axion can couple to all sorts of standard model particles. And in particular, can be produced via the decays of these particles. So for example, in the leptons sector, you have axion production from mu2 EDKs or tau2 EDKs. Then you have production from meson decays like k2 pi A, but even production from hyperron decays or baryons, with a strangeness, for example, lambda hyperron, which decays into a neutron and an axion. And as I'm going to show in the rest of this talk, this allows for efficient production in position-flavor factories, enhancing the direct solution for the axion, supernova 1987A, which is relevant for star cooling, and the early universe from where one can produce axions via the freezing mechanism via these decays of standard model particles. OK, now before I will discuss this, let me briefly motivate the origin of these flavor-violating couplings. So essentially, the couplings to fermions are determined by rotating the petroquin charges to the fermion mass basis by the rotations that diagonalize the eukaryl couplings. And so you see that such off-diagonal entries are present whenever standard model fermions carry petroquin charges, which are not aligned to the eukaryl meaning that these petroquin charges are not diagonal in the same basis as the standard model eukaryls. So usually, it is assumed that the petroquin charges are proportional to the unit matrix. So indeed, in this case, this simply drops out, and then all of these charges will remain proportional to the unity matrix in any basis. But this is an assumption that you don't have to do. And particularly, if you don't do it, this gives an equally good solution to the strong CP problem. And however, this means also that these off-diagonal couplings depend on this misalignment between eukaryls and petroquin charges, which in general is not predicted unless you have a theory of flavor. But this motivates a particular decent scenario where you can predict these flavor-violating couplings, namely when you identify petroquin with a flavor symmetry that explains the hierarchical structure of the eukaryl couplings. And actually, this was proposed already by Frank Wilczek in 82. And in these papers here, we provide an explicit realization of these scenarios where we either identify u1 petroquin with a single frog at Nielsen symmetry, or with a more complicated non-Abean symmetry, where then one can at least parametrically predict these flavor-violating couplings here and show that actually they can be related to CKM elements, which shows that in general, they can be pretty large. OK, but now let me discuss about the phenomenological consequences. So in particular, we can constrain now these flavor-violating couplings by looking for a standard model decays with missing energy. Because of course, this axion in the final state is a dark matter, so it just escapes the detector. And this signature looks exactly like a rare decays of standard model mesons, where one has a final state neutrino pair. Instead, that in contrast to the standard model, this is a two-body decay, not a three-body decay. And now this means that the phenomenology for quarks and leptons is entirely different precisely because in the quark sector, the standard model background is very tiny because decays like K to pi nu nu are extremely strongly suppressed, so of the order of 10 to the minus 10. While for the leptons, of course, these decays are the main decays of the standard model muon. Yeah, nevertheless, one can profit from using polarized muons where then one looks under angers where the standard model looks to zero. But nevertheless, these are the processes that we want to look at and we want to look only in the two-body region of the resulting phase space. Now interestingly, the experimental analysis of such two-body decays are very rare. And this means that one has to take those three-body decays and then re-curse them in the relevant region. Now, I just want to emphasize that actually I find it very interesting to look for the two-body decays instead of looking for small deviations from standard model three-body decays because these two-body decays can probe much larger new physics scales. So the reason is simply that the axiom coupling is a dimension 5 operator while this coupling here, which then has just standard model field. So just a four-pharmium operator with neutrinos is a dimension 6 operator. And so if you imply the constraint from measurements of B2K nu nu, this will probe only scales up to 10 TeV while this one here can probe scales of the order of 10 to the 5 TeV. So therefore, these two-body decays can really test much, much larger scales than what we can do by looking for these three-body decays. And there's another remark. These four-pharmium operators are typically more constrained by mixing of neutral mesons. So for example, giving a bound on 200 TeV if you have a BSPS operator, which indeed for having new physics here is much stronger than looking for this B2K nu nu decay while for the axiom it's exactly the other way round. So this gives the same bound of 200 TeV. But looking for the two-body decay gives much more stringent constraints. So since I don't have so much time left, let me just go quickly over this. So this is basically just shows the two-body recast that we did for available standard model data that we derived in this paper here. Yeah, basically, I mean, this is just a pure, poor, serious recast. So it would be much nicer if the experimenters would directly provide constraints on, for example, D2 pi A, which at present there is no bound in the PDG. Okay, yeah, so this shows then the summary of all the present constraints on flavor violating axiom couplings for quarks and leptons. And I don't wanna go through all of these plot here, but let me just mention that the strongest constraint on the axiom decay constant comes from NA62 experiment which looks for K2 pi A, which provides constraints of the order of 10 to the 12th GEV, which I think is really the largest scale that one can probe with flavor physics. So you'll see now that the other flavor transitions are weaker, but still will scratch into the most interesting region where the axiom can be dark matter. And for example, in the lepton sector, this is already done by present experiments. Also interestingly is that there will be a new M2-3E experiment which will strongly increase these bounds. Unfortunately, there is no proposal by the Mac2 collaboration, but in these papers here, we have proposed modification where one can also look for such decays. There was also this other recent paper here which proposes not to look for M2-EA, but M2-EA gamma, which also then reaches similar sensitivities. Okay, so now let me mention that actually there's an interesting fact here that, namely, if you want to test the axiom couplings to SD quarks, the strongest constraint on this actually comes from astrophysics. Yeah, and the reason is that these coupling contributes to hyperron decays and while laboratory bounds on invisible hyperron decays are extremely weak, actually it turns out that in the very hot proton neutron star which is formed during co-collapse supernova, the temperatures that are reached out of the order of 40 MeV which allows them to have a large population of hyperrons in the star and these hyperrons can decay into axioms that will leave these proton neutron star providing additional cooling to the star which is strongly constrained by observations and these limits the corresponding energy loss rate. You can simply estimate these energy loss rate in terms of these decay rate here of hyperrons to axioms. And what you'll find is that you get really the best, I mean, an extremely strong bound on invisible hyperron decays of the order of 10 to the minus. Yeah, you can do the same for muons, but it turns out that for muons, this is actually much weaker than the laboratory bound. Okay, so now let me turn to constraints from cosmology. So at the very same decays of standard model particles can produce axioms in the early universe. And if these axioms are very light, then they are constrained by bounds on a dark radiation from the CMB. This has been studied in this recent paper here by Francesco Deramo and his collaborator where they overimpose the bounds that you get from CMB data with the laboratory bounds that we derived. And you'll see that while for muons and kions, they are weaker than the laboratory bounds. Actually for the B sector and for the tau sector, these new generation of CMB telescopes will actually compete with the bounds that could be derived by the Bell II experiment. Okay, now let me finally talk about the possibility to actually use the very same production of axioms by flavor violating decays by of standard model particles to produce actually the axion dark matter abundance. So this we have proposed in this very recent paper here which came out a couple of weeks ago with together with Paulo Panci, Diego Redigolo and Thomas Schretz. And the nice thing is that these essentially fixes the decay rate of the standard model particles to axions in terms of the relic abundance. And in turn, this allows it to predict the decay rate of these, the decay rate of these flavor-relating decays such that one get explicit targets for these experimental searches. Yeah, so as I have already discussed the relic abundance from freeze in production of axion dark matter wire such decays is simply proportional to the axion mass times the decay rate, which in turn is given by these flavor relating coefficient over the axion decay constant. So first of all, this implies that you get a window where this scenario is viable because you get an upper limit from the kinematic threshold because of course, the axion has to be light enough in order to be produced by these decay. But then you get a lower limit from laboratory constraints on the flavor-relating decay rate simply because the smaller the axion mass is the larger the decay rate has to be in order to give the same abundance. And of course, there's an upper experimental bound on these decay rates as I've shown in the previous slide. And so there's a compact parameter space in the axion mass interval where actually the lower bound on the axion mass comes from the warm dark matter board. So the main challenge in this scenario is the dark matter stability of the axion because the axion can always decay into photons. And generically, these contributions will be so strong that it is difficult to make the axion sufficiently long-lived because in the relevant parameter regions, they're extremely stringent constraints from X-ray telescopes, which are actually 10 orders of magnitude stronger than just the lifetime constraint by requiring that the axion lives longer than the age of the universe. So this means that one has to suppress these decays here and you can do this by simply requiring that the petroquin has to be anomaly free and there's a standard model which removes these anomaly contributions here and that the axion is sufficiently lighter than the corresponding lepton in which way you additionally suppress these decay rate which allows them to satisfy these constraints. Okay, so yeah, this implies and how you should consider explicit scenarios which realize this idea. So first of all, you can take the leptons which have a traceless petroquin charges that you need in order to make this anomaly free and then you rotate this in the one-two plane and then you have this coupling matrix of electrons and muons which is controlled by a single rotation angle alpha. And then of course these couplings are also suppressed by the axion decay concept and then you have the axion mass. So in total, these are just the three parameters. Now in general, we have a several contribution to the axion relic abundance which depend only on these three parameters here. And the most important one is these infrared freezing contribution from the lepton decays which essentially give the right abundance for a KV axions at decay constants of the order of 10 to the 90. So in principle, there are also other contributions in particular and UV contributions which depend on the reheating temperature. But here we simply assume a relatively small reheating temperature which suppresses these contribution here and actually also the unavoidable misalignment contribution. So then these contribution here is really the dominant one. Okay, so then we only have to calculate the axion lifetime which you see is essentially of the right order in order to satisfy the constraints for similar choices of the parameters as required to give the relic abundance. So in particular, you have also this warm dark metal constraint here so that the axion mass is heavier than around 20 KV. And essentially you are then left with a two dimensional parameter space because one parameter is fixed by reproducing the observed relic abundance. So this means that you can make two dimensional plots here where here you have, for example, the axion mass, here you have the angle and then you plot the various constraints in this plane. So in principle everywhere here in the white space you can have a model model that gives the correct relic abundance. And from below, this is constrained by searches for the, I mean, laboratory searches for these flavor-relating decays. From above, you get constraints looking for decaying dark matter. So in particular in this mass range coming from x-ray telescopes. So essentially here only this parameter space survives. Essentially the diagonal case where alpha goes to pi half is already excluded. So you really need flavor violation in order to open up this parameter space because for flavor violation will suppress these very stringent constraints here. Yeah, so these scenarios will then be tested in the near future, both by the future x-ray telescopes and by future laboratory searches which look for MUTU-A invisible, in particular at the MUTU-3E experiment at the MAC-2 experiment. Yeah, so then in the town MUTU scenario all the constraints from these direct searches disappear because the bounds are extremely weak at the moment. And so you are just left with these future x-ray constraints. So here you also satisfy the stringent constraints from warm dark matter. You get completely fine and easy models for LFE axion dark matter. However, they are also pretty difficult to probe. Okay, yeah, this brings me to my summary. So yeah, it was a discussing scenarios where dark matter axions have flavor-relating couplings which allows them to produce them via the decays of standard model particles. And this is first of all important in position flavor experiments and allows to probe axion decay constants up to 10 to the 12th or 10 to the 10 G E V. It turns out that this is also relevant for extreme astrophysics environments like the core collapse supernova that happened in 1987 A where the temperatures are large enough in order to produce a sizable abundance of moderately heavy flavors, which then decays into axions and contribute to energy loss. And this allows them to put also stringent constraints on hyperron decays. And then finally, these decays will also happen in the early universe and allow to reproduce or observe dark matter abundance via thermal freezing, which gives rise into a very simple class of dark matter models that then can be tested not only with future x-ray telescopes but also this flavor factories such as Mewtwo 3 E and the next two experiments. Okay, that's it. Thank you very much. Super. Thank you very much for this talk. It's been very, very nice, very comprehensive. I've enjoyed it a lot. So let's open the question round. I don't know if the other people in the audience have questions, they certainly have some, but I would, the others begin. Okay, then I'll start. So first basic question is if freezing only works with flavor violation, right? You cannot have freezing production if you do not have flavor violation, right? Well, so I mean that's, I mean, in principle, you, okay, it was a little bit short here. So in principle, you not only have decays but you also have scattering. So actually, so can you see the slide here? Yes. Oh, okay. Yeah, so basically, I mean, the difference is that, I mean, this will be a process that will be additionally suppressed by Alpha Electromagnetic because you need an additional photon. So, okay, I mean, what I wrote here is now a flavor of diagonal contribution. But actually you would also have like mu gamma to mu A, for example. So there will be a diagonal. Oh, right, right. Yes, but the important thing is that here you need an extra photon which gives you an extra Alpha suppression of this process. So the parameter space is a little bit different. And for example, what you see here, yeah, so this is the mu E scenario. So if I take the angle very close to pi half, then actually the axon only couples to electrons and muons. So I reproduce the, I mean, I go to the limit where I just have the, I mean, the flavor violation is switched off. And you see that indeed here, this parameter space is completely closed. Okay, so yeah, I mean, if you want the diagonal case would be somewhere here. So in the mu E scenario, you really need flavor violation, which goes for very small Alpha that basically Alpha equal to zero would mean you don't couple at all to flavor diagonally, which means that basically the gamma gamma would disappear because you would not couple. I mean, the axon will not decay to photons if you'd only couple to mu E, right? Because I mean, there's a flavor symmetry that forbids this process. And therefore you can open parameters. So, sorry, just, I mean, as a short answer, no. So I mean, for the mu E scenario, actually you need flavor violation, but for example, for the tau, say now you don't need. Yeah. But I mean, in general, you would couple basically to all flavors. I mean, in realistic scenario, so. Okay, okay, great, great. Super. So Roberto has got a follow. Yes, I had a very nice talk, Rohit. So I would like to ask you if in this scenario, do you have some contribution for the G minus two of the mu, for instance? Oh, okay. Yes. Yes. So you mean in this particular scenario, yeah? Yes. Yes, and in this case, because I guess the G minus two with the normal action with the flavor diagonal, there are some. Yes, right, right. Yeah, so essentially, I mean, so in this particular scenario here, so this is if you want a decay, I mean, decay constants, which are more closely to the QCD action. So I mean, 10 to the nine, 10 to the 10 GEV. Yeah, so essentially this means that the, okay. Sorry, right, because I mean, because the action is dark matter here, so therefore these decay constants have to be extremely large in order to suppress this coupling and basically this implies that G minus two is completely hopeless because basically your effective operator scale is 10 to the 10 GEV so that you cannot, you can forget, yeah. But you're right. I mean, so, I mean, you can explain the G minus two, but then you need really, I mean, decay constants of the order of a electric scale or lower than this, yeah. But then the action will always decay to something. Okay, thank you. Any other questions? Okay, let me go with another one. So you mentioned the supernova in bounds, right? Regarding the actions escaping the supernova and having that in the cooling. Remember in the case, and I don't know if it is applicable for here, but in the case that you have, for instance, Neutrinos with dipole-effective couplings, you also got constraints from the cooling of red giants, right? Because you have these plasmons here that are like massive photons that can eventually decay, right? In that case, it was into Neutrinos, but I was wondering if that would be applicable here that like a plasmon would decay into actions and in that way you could cool the red giant. You mean in this particular scenario of the, I mean this dark metal scenario? Well, no, in general, it's a constraint since you were mentioning at some point in your talk about the supernova constraints. I was wondering about if you can get constraints from. Ah, yeah, yeah, definitely, definitely. No, right, so essentially, okay, let me see. Okay, so I have to, oops, don't see my own slide. That's a problem of doing such a comprehensive talk that you get many questions. No, it's very good, okay, so I have to. Ah, okay, no, no, it should work, okay. Right, yeah, so actually you see, okay, so basically here I show the usual constraints that you get from other astrophysical sources. So I didn't have time to discuss them, but basically, no, this gives, I mean, also if I take all the one couplings, it gives me a constraint on the axion decay constant. And I mean, this depends then on the coupling. So I mean, normally supernova are most important for like the nucleon couplings. So the axion, no, the coupling to neutrons and protons. And actually, as it was shown recently also to axion couplings to muons because they only exist in supernova course. While for photons and electrons, actually the constraints indeed come from other sources. So in particular, red giants, which I mean, give bounds both on the photon coupling and the electron coupling. Although I think the strongest bounds on electron couplings now come from white dwarf cooling. Although I think the red giant bound is very similar to this. Yeah, so you are right, no? So basically they sit here. Indeed, this is normally cited as the strongest constraint on QCD axions. No, but you see, there are still, I mean, two orders of magnitude weaker than the flavor coupling also. In that sense, I think this is really nice. Exactly, now of course, I mean, yeah, actually for the electrons, I mean, there is also, I mean, there are these stellar cooling anomalies where actually we observe a little bit of additional cooling. So I think in particular, these white dwarfs now, I don't remember in what other astrophysical systems, but actually this implies that actually there's a hint that the couplings to electron is actually quite close to the present bound. Okay. So, yeah, so that would be one possibility, but I mean, of course, the uncertainties are large, but still, I mean, this is something that I mean. So what do you mean that there is a hint of additional cooling? Exactly. Yes. So there is an additional energy loss which is somehow observed in a couple of... And that would be close to the current bound on the one that you have on your pointer. Right? Yeah, exactly, right. Exactly, yes, exactly. Yes, so that's, yeah, I mean, that's of course very interesting, but yeah, I guess it will take some more time until, I mean, this really becomes conclusive, yeah. But that would give a... Would that be in contradiction with the flavor bound? Oh, well, no, I mean, these are independent couplings. Yeah, yeah, these are independent couplings, right? So, no, this, I mean, that is somehow the problem. No, so because in principle, I mean, all of these couplings here, the electron couplings, no, I mean, all of these are independent couplings if you want in the axiom effective theory. So, I mean, if these would turn out to be true, it simply means there would be a hierarchy between electron couplings and flavor violating couplings, which of course, I mean, is natural because you would expect some degree of flavor violation. I mean, suppression by CKM angles and so on. So that would be, yeah, I mean a big issue, I guess, yeah. I see. Very interesting, very interesting. I think we have a talk here regarding, we have a question, sorry, from the audience. It's ER4000 asking if there are collider constraints and if yes, how comparable are the flavor constraints to those of colliders? Yes, yes, so, I mean, colliders really, I mean, the only thing that applies, I mean, let's say for the QCD axiom, are really, I think, I mean, flavor bounds. I mean, there are also colliders, right? So like E plus, E minus, but they're really, I mean, this is really precision physics. So for example, for LHC, I mean, you, I mean, the only bound that I know that comes close to this maybe 10 to the six or so is also looking for flavor violating decay. So actually, I mean, deviations from BS-MU-MU with an additional axiom, but I mean, direct searches basically, I mean, they're not probes such large scales. So, I mean, because the main problem is that here, this is always invisible axioms, yeah? And I mean, LHC is very bad with, I mean, with missing energy. And so, yeah, this I think is the reason why this is, there are no bounds on dark matter axioms, but of course, once you go for, I mean, you give up axiom dark matter and you consider axioms which decay, I mean, promptly to electrons, photons or so, then indeed, I mean, colliders are very important, yeah? And there have been, yeah, many papers which study these. So in particular, by Martin Bauer and collaborators, which look at constraints that you can get from colliders on axiom-like particles, but this will never constrain axiom, I mean, dark matter axioms, which are invisible. Okay, so there's usually a delay between your answer and the presentation in YouTube. So let's give the person who has some second seat. There's any follow-up question. I guess that in the case of colliders, since you have all of these decays of mesons, you could get like axioms in the forward direction, maybe phaser or something like this, could have something to say. I don't know, this is a coupling source or small. Yeah, so, yeah, definitely, absolutely. I mean, so people discuss this, but basically this will always be, I mean, axioms which decay visibly. I mean, no, maybe you have some displaced decays. So maybe, I mean, just decays after 100 meters, but yeah, if you really want dark matter axioms, okay, let's put it like this. So in my opinion, if you look for such particles, I mean, there is basically no motivation left because, I mean, clearly they are not dark matter because I mean, they decay in our laboratory and they don't solve some CP problem. So still you could argue that, I mean, they will be mediators to the dark sector, but I mean, for me, if you want to have- It will disappear completely. Yes, so of course you have very interesting phenomenology to look for this, but somehow the theoretical motivation on the other hand drops. Yeah, so this is why here I just talk about dark matter axioms, which I think is a, I mean, is a natural theoretical motivation for such fields, but this implies that basically you only have a few experiments, in particular flavor experiments, which are sensitive to such large decay constants that you need in order to make the axiom stable on cosmological scales. And at the same time, of course, lies that, I mean, they are just invisible. Yeah, but I really think that it's interesting because basically then you would look really for, I mean, these experiments will look for the same decay that has produced the dark matter and the early universe. So I think that is, I mean, some of the nice possibilities. So normally, I mean, you, yeah, I mean, it's difficult to produce dark matter, in the usual, okay, well, I mean, at colliders, you could, you can produce a dark matter, but here, I mean, it's much easier. They just come from standard model particles that decays. And you just have to look for these decays. So I think it's a very nice new source of dark matter production that actually normally, I mean, it's really discussed, no? So, I mean, no, you talk about, I mean, direct detection, indirect detection and collider, but yeah, this is in a way, and a new source of dark matter production, which actually is very easy because you just wait. Excellent. Okay, so there wasn't a follow-up question. I don't know if anybody else in the audience wants to ask. Seems not. So I have one final question regarding EDMs, right? So usually when, so EDM, you can generate it on a loop, and usually you have it proportional to the mass of the fermion, right? So here you could have, since you have a vector and axial coupling, maybe you could have some CP violation there that could be enhanced by the tau mass. Yes. Yeah. Ah, yeah, yeah. So I think, yes, I mean, I think there has been, there has been a paper recently on this, I mean, talking in particular about those CP violating axion couplings by Pari, the Paradisi, Duccio and Ramona Grulba. And yes, okay, now I don't quite remember, but I suppose, I mean, that some of the situation is similar to the collider. So that essentially, you know, these EDMs are, I mean, secretly, I've mentioned six operators and the relevant scale that you have here that suppresses these operator is FA squared. Yeah, so I mean, no, it's because it's different so if you want, I mean, normally when you have having new physics is a mass that suppresses those operators like it's, but here, of course the axion mass is basically zero, but what replaces then the, I mean, the heavy scale is simply then the coupling. So basically one over FA that appears everywhere. So basically, but the scale that appears in such operators is FA, which, yeah, I mean, it's 10 to the six, 10 to the seven. So I mean, that's typically, I think, too large in order to see something in those experiments. Yeah, that makes sense, that makes sense. Super, super, great. So that's it from the side of the questions, I guess. And yeah, oh my God, it's already 11 of 06. So it's a good time to do this. So thank you very much, Robert, for giving a wonderful talk and answering the questions so clearly. And that's it from us. Please stay tuned for our next webinar happening in like three weeks or so. And well, we'll be seeing you around. Thank you very much, it was a pleasure. But take care, everyone. And I think we are no longer.