 Yeah, thanks. Well, first of all, good afternoon. Welcome to this last lecture today So and thanks to the organizers for the invitation. It's a pleasure to be back here at the ICTP and now some of the thing disappeared. So I Switch a computer on so I I got as a title for these lectures BSM cosmology and You may wonder what that means actually, I didn't know either so on the other hand looking at the program and Contacting the other speakers It was clear that there are several talks here somehow related to this several lectures related to the same theme like on dark matter on axioms and Then by Subya Sarkar the general introduction to cosmology. So I thought it should somehow complement what you had in the other talks and so I Decided to in fact talking to the others to focus on Barogenesis and Inflation I doesn't work. Okay. No problem. That doesn't work either now Just to get started Let me start from this picture Which is something very similar to what you have seen I think already in the lectures of Subya Sarkar it shows the Say some picture of the hot face of the early universe in fact a little bit more and as you know there's a microwave background which is sort of the end of the Hot big bang and that is something which we know very well and which has been beautifully analyzed now by very satellite Experiments most recently a plank in particular and so that we know in great detail The microwave background and really understand it And I think this is the basis then for extrapolating From here to earlier times and then higher temperatures because you know here The temperature was not that high. Well, it was related to the ionization energy of hydrogen. So the temperature was about Say ten electron boils a thousand Kelvin and So this is something we can be certain about but then the question is how do we go on? Now we can go on in this direction Theoretically because we know the structure of matter beyond atomic physics. There's nuclear physics. There is the standard model weak interactions electric phase transition and Maybe eventually grand unification and the question is how far can we extrapolate and That is for that you have to have theoretical bias. So we don't really know and It is possible for instance that biogenesis Takes place here around the electric phase transition It is possible that it takes place much earlier as in the case of leptogenesis or It is also possible to consider some extreme examples of Biogenesis where in fact takes place at temperatures below the temperature for the electric phase transition so Then the question is what is correct and I will discuss the two main examples and some more exotic ones and the following So basically the question here then is when and how was the baryonyl symmetry generated and Then you see here what we have from the end of the hot big bang phase Namely the CMB There will be eventually something similar Also from the beginning namely gravitational waves and so the question is it would be beautiful say if Gravitational waves would be discovered First of all indirectly via the CMB. There's been some very important discussion recently and otherwise Directly by interferometer experiments in the near future So I think the real hope in this field is that it eventually you will also see this and get maybe a picture similar to what we have from here and Then it is generally a belief that Before that there was a phase of inflation This is not a theorem and I think one can still doubt whether or not this is true On the other hand, there is no really serious candidate to this and so the question is really what was the scale? At which inflation took place and then who is inflat on what really caused inflation and how does it fit? into extensions of the standard ball and Then there is an important question, which I come back to that from time to time, but I Don't really have cannot discuss it in too much detail and that is how these things inflation baryogenesis Dark matter are related and The reason is that if you just take these things by itself There's very little quantitative information and so it's very difficult to make progress unless you treat these different things as pieces of one puzzle which you try to put together So then here's the outline of these lectures I will split the thing Roughly 50-50 into baryogenesis and here I'll spend first some time Lecturing baryogenesis and leptogenesis and then a brief discussion of other models and then move on to inflation and Here first discuss the basic picture very elementary still for those who Maybe are not so familiar with that and then discuss some recent developments Particularly related to the new data from Planck and bicep the question whether inflation is large field inflation or small field inflation Are you connected to reheating and possibly to dark matter? So this is the outline so let's then start with baryogenesis So baryogenesis is a question of what is The origin of this one number the ratio of variance in the universe and the number of photons Now you know that is a very small number the number of photons is about 400 per cubic centimeter And so the number of variants averaged over the universe is say nine orders of magnitude Smaller than that it is now with a remarkable accuracy of about five percent that's been measured by Planck and So the question is what is this? and This is somehow in some sense baryogenesis is a very strange topic because it's just one number and If you think how many papers have been written about this one number during Now well, I would say mostly the last 30 years. It is amazing and we are still not Sure what it is. So you still have a chance to find yourself Say the right answer in fact as everybody knows as the first Paper on that was written by Saharoff. There's almost 50 50th anniversary of this and then I Listed a few other papers actually what I try here with references is a following I will for each of the sections list a few references, which I think are sort of the key in the sense not necessarily it's difficult decision what to pick but Something which you should read if you really want to work on that and want to understand it better And then I will give in the other topics for the other topics also some reviews Now here's the first paper is Saharoff. The second is a connection by a laptop number by tuft Then what happens with that at high temperatures Kuzmin Rubakov Shapochnikov? And then an important one also for all the quantitative studies how chemical potentials are related in High-temperature phase So be a soccer has already discussed some of that but we will need a little more So let's then come to these conditions I think you must have seen them all of you must have seen those Necessary conditions for a meta Entire meta asymmetry you clearly need baryon number violation if you don't have that you cannot generate in a symmetry You also need cncp violation. Otherwise you also cannot have a baryon asymmetry. That's pretty obvious What Ricky is a departure from thermal deviation actually I should say these are necessary conditions They are not sufficient in the sense that if you if Sufficient conditions it's much more difficult to formulate. Of course the question is Is then what you get by doing the calculations? Do you get enough baryon asymmetry and this is not at all a trivial and very often an interplay a tricky interplay of Various pieces and we will see some examples of that Now maybe the most tricky thing which in fact was even wrongly treated in some of the very early papers on Baryogenesis is this third condition and let us just check that if it's not fulfilled You will never have a baryon asymmetry that you can easily do you take say suppose we have just thermal equilibrium then If I have any observable like the baryon number density, then that observable is given by sort of the trace of e to the beta h h is the Hamiltonian of the theory beta is one of the temperature Times the baryon number and then I have to take the trace Of this so this describes the system and this is the operator Which measures or which is the operator the observable? corresponding to barrier number Then you can insert here one which you write as theta theta to the minus one But this is the CPT operator and then you use a trace property put that on the other side and you know that CPT Is something which commutes with a Hamiltonian so you can put this through and then Since the baryon operator is odd under CPT you get This is minus to itself. So in thermal equilibrium it's zero so you need a Departure from thermal equilibrium to explain a baryon asymmetry and how is that realized that is realized is in fact very different in the different examples Actually, I should say What is said here is true under the condition that you generate a baryon asymmetry sort of from a thermal bath There are other ways which in the end may be true That is in fact different that the baryon is asymmetry is generated in a different way Related to the dynamics of scalar field a well-known example, which I will not have time really to discuss is a flaked iron baryogenesis There are many papers on the decay of heavy moduli fields, which you may get in strength theory Which also have lots of problems, but in principle, there's also a possibility and so on so there are others There's cold barrier genesis which you do after there are many things which so These conditions are Relevant only for certain class of models So let me then discuss baryon and lepton In this number in the standard ball as you know in the standard ball you have currents for baryon number It's just the sum of the currents for left-handed and quarks and the right-handed up and down quark and similarly for the leptons you have the lepton doublet and the electron Now what is very important is that the divergence of this current is not zero You know for a conserved quantity the divergence of the current has to be zero here the divergence is given by this anomaly in fact in a somewhat fancy for those of you who are more mathematically Inclined this is say the derivative of a churned simons form, which you will see on the next slide so and It is something Which is the total divergence and if we would not have a non-abiding gauge theory Then we wouldn't have to worry about that Then that would be irrelevant. So for QED, but here for SU2 for The electro weak theory we do have to worry about this and these are the fields the W fields and the u1 field in the standard ball now if you Now know that you have this divergence and you can use that and take the integral over all space and over a finite time interval from an initial to a final time and then What you find is that this is proportional to the number of generations which you have in the standard model Which he has three and then you have a difference of Something called Schrodinger and Simon's number, which is this integral which you just get from calculating the anomaly and now this thing Has Important feature that's a jump Is always an integer that's this topological nature and So the num this thing here this difference as you go from some initial time to some final time can jump by units by plus minus one plus minus two and so on and So in field space that means if you say Generically say it takes a W field space in this direction then there is this in this field space as a potential and this potential Has degenerate points which differ by integers of this turn Simon's number and there's an energy barrier associated with that and that Energy barrier is the energy of the object so-called swallow on object now This is What is important those who know QCD? I'm sure they know that the state of parameter and everything associated with that You have here. You have a tunneling between those vac here and the tunneling amplitude is given by instantons So that gives you this and then if you look at this instant on and then you somehow Contained in that you have this father on and this father on you can also find as a saddle point of the electro week Hamiltonian Which has interesting property if you have this it's a saddle point with just one negative eigenvalue and this one negative eigenvalue then gives you The swallow on a decay rate anyway this just it is this and What you now get from this is that these? Instant on interactions or I find a temperature usually calls fall on interactions. They generate An effective interaction between all the left-handed fermions in the theory So and that's a very democratic interaction. You have 12 left-handed fermions in the standard boil starting from the left-handed up walk up to Say it's a town utrino here you have to Make sure that the arrows all go in the same direction and in fact this interaction describes Is described by these operators some here I goes from 1 to 3 and it gives you an effective interaction of all the fermions in the standard Well, now this changes barely an electron number By 3 violates left or the number by 3 So that means that given the fact that you have this vertex you can This instant on interactions in the standard mall or final temperatures these swallow on processes make all processes possible Which you get from this vertex so for example, you can take three of those lines Revert the arrow and take it as an incoming as an out Incoming state in this case and then the rest is outgoing so you can have these three Right-handed anti quark so if you start from the left-handed quark you take the anti party You get a right-handed anti quark goes into this bunch these three go into these nine left-handed fermions Now you may think that's a very complicated process still it's possible. Well, I make a few comments still it's possible in principle to calculate that and It's a very interesting process. In fact There have also been or there are still discussions of how to search for such things also not just a high temperature but at high energy and there are some interesting papers Investigating the possibility that maybe possible say at a large a hard on collided like a hundred TV collided to see something like that Anyway, now the rate for such processes depends crucially on temperature now what top first top In 76 first pointed out that such processes exist at all and then at zero temperature. He made an estimate This rate here has an instant on factor, which is about 10 to the minus one hundred sixty five So from that you conclude you don't have to worry about that so at zero temperature. You are safe and So we don't have to worry about proton decay due to non-perturbative effects in the standard But as you now go to high temperatures the situation changes when you reach temperatures of the order of the electric phase transition then you get a this follow on energy, which I showed you which Proportional to the temperature dependent expectation value of the Higgs field and that gives you this exponentially suppressed rate with certain pre-factor. So that's the rate for B plus L violation per volume and If you now really go above this electric phase transition to high temperatures, then this approaches a certain constant This is the weak Coupling constant at the critical temperature of this electric phase transition You have the force power of the temperature for dimensional reasons and this is a number Which in fact has been calculated on the lattice. It's about 20 plus or minus 2 That's why it's not so important. What is important there is that the product altogether is about 10 to the minus 6 now Actually, there's a Lots of interesting stories related to this number auto-calculated sort of a beautiful topic for the field theory Which however is not the topic of this lecture now Based on this stuff There is a consensus among theorists that B plus L violating processes on thermal equilibrium in a certain temperature range Between say roughly 100 gv for the electric phase transition up to about 10 to the 12 If you go to even higher temperatures then due to this power here These as follow on processes get I mean you have to that you learned I hope in the lecture by Subir Saka that what is very important for all processes is to always ask yourself in the history of the early universe Is this process and thermal equilibrium or not? And in order to check that what you have to do is to calculate the rate and to compare it to the Hubble parameter and You know the Hubble parameter has a temperature dependence which is Goes like at high temperatures like T squared and you see this goes like T to the fourth So once you have different temperature dependencies then which you always have for various processes then these processes At some time I an equilibrium at some other time are not in equilibrium So I say there is a consensus among the theorists, which means that nobody questions this On the other hand working in this field I must say I sometimes worry about the fact that we have no Experimental evidence for this we have it neither for these electro weak instantons nor do we have it really for QCD instant so There were some recent discussions Whether or not one could check this in heavy iron collisions the corresponding QCD effects but otherwise there is no direct experiment evidence and If you have some idea how to check say this for instance in QCD it will be wonderful I think it's still a very important problem Now let me come to the third ingredient which we need for Biogenesis which are these chemical potentials This is in principle an exercise which you can do in First course on thermodynamics you have a certain number of particles you can associate chemical potentials to them They have reactions these reactions give you relations between the chemical potentials and then You can solve that and see what the implications are Now let's look at the standard mall We have one Higgs doublet and we have NF generations and F of course is three However, I keep it as a free parameter in order to Be able to get some formula which where I can which are more transparent then I think what you saw in the Lecture by Sube Saka is that if you now look at the number density of particle and anti particle Then you have this characteristic factor Where G is the number of internal degrees of freedom and then you have This this chemical potential divided by temperature So that means the chemical potential tells you How big the difference between particle number and anti particle number density is fact It's different by factor of two for fermions and bosons Now let's look at the standard model. Then first you have these As you to instantons, which is a swallow on process from that you get the relation because you have That three times the number of chemical potentials from the quark You get the factor of three here because you have three colors plus the number of laptop chemical potentials is zero Then you have QCD instantons also and That relates just the quarks and gives you this relation. Yeah, G G is the number of internal degrees of freedom. So G Excuse me of what of this particular particle for instance, if you take if you take a photon Which we don't have here then G Would be two Because you have two helicity states if you take say An electron a left-handed electron then you would have G for would be two Because with this field you associate the left-handed electron and the right-handed Anti-electron and so on so you can do that so that is accounting which you have to include in all these which you Have to include here for all these number densities and then You get this me. Yeah I was just wondering why do the sum of all these chemical potentials need to vanish I'm not clear on that. You are not clear on that well Okay, you can either do a Calculation for that or you can if you just want an intuitive argument if you look at if you look at this process then If such reactions here take place very quickly Then that means that if you have say an asymmetry in this so this would be a left-handed Mu on your tree know if you have a Difference between this and its anti-particle then by such a process this would equilibrate And so suppose you start from a state where you have an asymmetry here and no asymmetry In all the other particles then due to these interactions this asymmetry would get distributed Around everybody because these processes take place Is that roughly clear? Or suppose you start from a similar. That's a very important point Or suppose you start from a symmetry in this then this distributes among all the other degrees of freedom Now maybe you should do a little exercise for this But that's that's a very important point in fact the whole bariogenesis always works like that. I mean you start For instance in electroweak bariogenesis you generate an asymmetry say maybe first starting With the top clock and then that gets get this gets distributed by this thing the asymmetry the left-handed top clock of all degrees of freedom in leptogenesis you start from an asymmetry in these neutrinos and Leptode symmetry which you generate from have you by run and tree no decays and that Then also gets distributed over all the degrees of freedom because I think that's Probably a typo no no each ah no no no no no sorry This is just one example of course you have to No, no. Yeah. Yeah. I mean you each color counts so so for example you have to From these states you have to form a color singlet Okay, here. I just wrote this product if you write that out You need epsilon tensors which connect the indices in color and then as you too because the whole operator has to be a singlet So that means these are two down quarks with different Color and then you have still an up quark with yet another color so that this forms a singlet the same is true for the others Okay, so really have so one possibility is that you have two down quarks another is you may have two up quarks and so Just it has to be the total thing has to be color neutral. I just wrote this generically and not explicitly with the indices okay There are a few other questions up here before you go on so I just had another question that's I'm still not clear exactly What is Valeron is is it? Tunneling at a non-zero or a non-documentary level or is it an instability at to that local maximum? Yeah, I mean this father on is Technically you take say you take the the SU2 part of Sustainable Lagrangian involving let's switch off the hyper charge coupling So then we have just the W bosons and the Higgs doublet. Okay, and then you neglect say The dependence on the time coordinates. We have a three-dimensional theory and then you can take this and look for a station report and Then you find one and this is the spot so in principle and So this object then has also certain energy density or certain energy and that's a smaller on energy now and You can look if you take now your full Lagrangian now look for fluctuations around this and and then you find that this is not a stable. It's not a stable Extremum, but there is one direction in field space where The mass squared sort of is negative negative curvature and then I think follow on that. I think it's a Greek name which means Essentially, we should have some Greek you explains it to us It's about to decay something like this So I think this one negative eigenvalue is a crucial quantity So when you make the statement that and standard model very on and left and number are violated Do you mean that the standard model of particle physics with only diamonds and four terms? When you when you say that the Standard model violates very on and left a number symmetry By a loop correction or anomalous You mean that in This is the standard model of particle physics with only terms up to diamonds and four I mean you know you do not add Diamonds and five or higher diamonds and terms and you still get very on and left and number violation It's just the standard model is just as we normalize the complex. Yeah, then I have this confusion. Yeah, so We know that standard model has 19 parameters, I mean 18 usual parameters plus the strong CP parameter and CKM matrix has only one phase. I mean I have always this confusion. So when removing the Unphysical phases of the CKM matrix we use Beryon and leptin number symmetry as a part of it now if you say that that this Beryon and leptin number are violated So I can write the full Lagrangian of the standard model including all these corrections Then since very on a leptin number symmetry and no longer a symmetry of the full Lagrangian Then we cannot use that symmetry to remove Unphysical phases of the CKM matrix and there will be at least one extra phase that that will enter in your analysis So that's actually there's a trick which I cannot now not prove on the backboard but for instance there's a difference between The question is whether you can this anomaly I mean if that violates very on a leptin number You can you will have one additional phase then that that we have not seen again in this in the CKM Yeah, no, I don't know. No, I think that phase is not there but I Mean it's true I mean you have to worry about the fact that well first of all these effects will at zero temperature will be negligible But of course another question is very conceptually Yeah, I mean should I should I make this statement the standard model of particle physics has more than 19 parameters? I mean, I don't that's I don't I don't think so I mean that question is has been discussed in the following sense There is a another anomaly They're related to a Kyle symmetry in QCD which there relates To the so-called theta parameter the question is for the similar parameter Which you would have to count for the weak electroweak theory that would be this additional parameter But the statement is that parameter It's not there Now you it's unphysical. I cannot give you another proof, but we can talk about it. Yes, okay Actually, I'm happy that you asked these questions. So I hope we learned something together on the other hand. I also Have to make some progress. Otherwise, we will never make it too I'm sorry. I just have one more quick question about the Svalorans So is it correct that they? act only on the left-handed fields because Corresponds to the SU 2 left symmetry. So once SU 2 left is broken do also the Svalorans disappear then Well, they become irrelevant. Yeah, that means the rate of these the rate of these Transition becomes zero Mm-hmm it gets exponentially suppressed Okay, and then why don't we have Svalorans? Let's say for SU 3 That's also non-nibillion Well, you have You have you have instantons for SU 3 also So in some sense you have similar processes, but I don't Discuss them here because they are not important for the barrier number. I mean you can See that here for instance in this Relation for the chemical potentials you have Here you have the Svaloran effect on the SU 2 instantons which give you this relation among chemical potentials and correspondingly you have a Such a relation also in QCD so principle you have a similar effect. Mm-hmm. Okay. Thank you Okay. Now There are more relations Here which maybe I skip now and you get some rule for which has related to the fact that the total Hypercharge of Your state should be zero the state should be neutral with respect to hypercharge and It is clear if you have in the different particle species asymmetries and if they don't satisfy certain relation then since all these species have different Hypercharges the total hypercharge would not add up to zero so that you have to satisfy and that gives you this relation Then there is an important the important relations also like this so Reactions like this you get from you cut relations like this you get from you cover interactions. So for instance say suppose you have a left-handed u quark which interacts with a Higgs and turns Into a right-handed you quark and then you see also have To make it a proper reaction say you have a gauge pose on here So nose for the gauge pose on there neutral so the chemical potential is zero and Then that relates these three chemical potential and it gives you This cubic relations Now there is a tricky point here what we shall assume in most of the Essentially in all all the stuff which I will discuss is that these you cover interactions are in thermal equilibrium and That is not true actually and the way you can check that as a following clearly these reactions here are Proportional to this you cover coupling And for some a particle this you cover coupling is very small like for the up quark or for the electron and then the rate the corresponding transition rates they are proportional to the temperature and They are out of equilibrium if you compare that with other processes for very very long time For instance this you cover interactions for the electron if I remember correctly comes into thermal equilibrium only at a temperature of about a TV So these are in the very early universe these particles are decoupled and That can mean that There are other ways of Periogenesis in the end or so in fact such models have been discussed in the literature Which don't follow these rules where I say for some reason you generate the symmetry in one of these species Which is decoupled in the beginning which then later when this guy comes into some equilibrium By his as far on process get distributed over all the other degrees of freedom Anyway, this is just a caveat If you now believe those relations then you can solve them You can define a total barrier number in the lepton number like this in terms of the chemical potentials and you find a remarkable relation Namely that both barrier number and lepton number are proportional to the difference of Baryon and lepton number with numbers here this yes in the standard world is about one-third and The important thing here is that they are both proportional to this difference. So the smaller on processes they conserve the difference of Baryon and lepton number, but they violate the Sun and For instance that already shows you that there is a fundamental difference The difference between electroweak barrier genesis and leptogenesis in the electroweak barrier genesis If you are just stick to the standard model B minus L is always conserved. So B minus L will always be zero still by a very tricky and non-perturbative interaction you Which has to do with bubble nucleation you can Never the less generated by an asymmetry present lepton Leptogenesis the story is completely different at the very in the very early universe You have a decay of right-handed neutrinos that generates an asymmetry in B minus L and this asymmetry in B minus L is then no longer Further affected and gives you the B minus L asymmetry today. So now to electroweak biogenesis the first topic Now an equity barrier genesis. I gave two references here one In fact the thermal field theory phase transition started with Weinberg and Dolan Jekief But I think first this paper by Kirsten and Lindon the Abelian Hicks model contains essentially all the ingredients Which you need to understand the properties of the phase transition So therefore this is certainly one of the key papers and then there are many papers on the mechanism of how you then really Dynamically generate the asymmetry one which is closest. I think to The picture which one has today is this by Korn Kaplan and Nelson. I Also listed some reviews here you find some interesting conceptual discussions Morrissey and Ramsey Muzolf. They have Also a number of very thorough Discussion with emphasis on supersymmetry and here you can find in this Report by constantly you can find the discussion for the few theoretic aspects of the non-equilibrium process Now what we now need for the barrier genesis Or for the phase transition is something like this and I think you all must have seen a picture like this. This corresponds to A second-order phase transition this to a third-order phase transition So what the expansion of the universe does it's the following We have essentially a big thermal bath and as the universe expands it cools down the temperature decreases So you move say from such a temperature Temperature down if the term the phase transition will be second order here You would it would be very smooth and you would then just here develop an instability However, if you have a potential like this then again at very high temperatures The minimum of your field value is here, but then as the temperature decreases You would reach a point where these points are Where you have say a broken phase and a symmetric phase which are essentially degenerate and Then close to that so here you get When you cool down you are Stuck say here, but then when this moves a little bit below then you get a first-order phase transition as By bubble nucleation where you tunnel from here to here actually The whole calculation of this if you look into the literature is quite tricky and it is tricky Even in condensed matter where the particles from which the particle physicists learned a lot in doing this calculation goes back originate to a theory by Jim Langer and Which you need if you want to calculate for instance how rain happens because in rain also at some point You start at high temperatures and then you form bubbles Now here what you usually calculate is a formation of one bubble Okay, I do it now of course rain you get many many many bubbles And so the process how you really get this nucleation the percolation theory associated with that how these bubbles expand And how you calculate that is a complicated process, but quite an interesting topic Yeah Yes It's a run. Yeah Well, how you get this potential I will now explain in detail I how you how you get how you obtain such a potential I will now explain in detail That's in fact an important point. So that's a Final temperature field theory how you calculate how to calculate these effective potentials and what you have to do is then to calculate first these potentials in thermal equilibrium so In order to set the basics and just to say what one is really calculating without doing that then For the full standard bullet Let me say how you calculate this final temperature potential to say for a massive scale of fields That's the easiest case and now which you what you may know from your course in quantum field theory there is a very interesting a connection between quantum field theory in Minkowski space and Finer temperature theory first, of course, you know there is the connection you can make a big quotation and Field theory Minkowski space is equivalent to a Euclidean field theory if you know in the Euclidean field theory takes the time interval over finite length from 0 to beta where beta is one of the temperature then you get Statistical mechanics for this in at a temperature T for this Field theory, that's a very important connection and if you Don't know it. I cannot explain it now, but you should look it up when you have some time in your field theory book Now so now so this is this integral of a beta will then just always denotes this Now what we do is we introduce to this Add to this and I take a massive scale of here. So mu is positive mu squared Now we add a source turn Let me a constant source which is essentially a force which pushes the field away from zero and Then you can calculate this the free energy of the system which you get from this generating functional like From the from this from the partition function, which is e to the minus beta again one of the temperatures the volume and Then you have the free energy density here, which depends on this source J and now taking the derivative gives you phi and phi is now The meaning of this is that it's really the expectation value of the field operator In at a temperature Phi and you have the volume average here, so it's a total average of this field now What you do is as a classical mechanics you the You do a Legendre transform and you get the free energy now for the system where you specify This average field, so you then get This potential as a function of phi and if you calculate that you get first Zero temperature potential which we saw in the previous slide then you get a factor pi minus pi squared over 90 Temperature to the fourth that you should know from statistical mechanics because if you take just photons Then you have two degrees of bosonic degrees of freedom and you get here pi squared over 45 times t through the force So this is just a little check. So this is a free energy and then you get higher things you get here this master and So I'm missing here a term t squared Sorry t squared is missing and then you get a cubic a cubic term in the mass times t and so on This is done now as an expansion in powers of this mass of The temperature but this is small and this mass is this which is just The second derivative of the potential with respect to phi This is the potential so if you differentiate with respect twice with respect to phi Then you get this mass and you have here given the mass you have a simple expression for the scalar field now You can now combine These two terms this was a t is missing and is a tree level mass term here and you write this potential as One-half m squared and then you have a term lambda over 4 t to the fourth t squared times phi square plus lambda to the fourth and so on And so you get a temperature correction to this mass so this is a picture of in the end if you pursue that of quasi particles having a Temperature-dependent mass however, you have to be careful with this This is very useful quantity for many things But you also have to be careful for it because engaged here is it's not gaussian variant And you cannot just treat it as a kinematic mass, but this will not be so important for us now Let me Now starting with the simplest example just where I told you what we are just calculating now to An example which is more interesting and which essentially contains the essence of the electric phase transition actually as I said this work by kieschenitz and linda started that to a Large extent and a nice discussion of this is also given in this book by linda. I think inflation and cosmology Now let's now go to this a billion Higgs one then we have now a scalar field which carries charge So we have a cover and derivative here And then we have this thing and now the crucial point is that this mass squared is negative and therefore This field in the ground state acquires a vacuum Expectation way of course as you know one has to be a little bit careful with that one has to say which gauge this expectation where you Is meant and so on I will skip that and Just tell you that if you do a calculation similar Correctly as to what you do for the real-scale a field then you get now a finer temperature potential Which looks like this it is t squared minus some constant Which is minus mu squared or some coefficient a Which you can which is given by the coupling constants then you have a cubic term linear in the Temperature and then you have a Quartic term These coefficients here are if you can get them from this full expression by defining This quality to be the one where the second derivative of the potential at the origin vanishes and the other Coefficient is the one which well that gives you a certain temperature now And it's a barrier temperature so called barrier temperature because they're the barrier disappears So if you look back at this picture that would be the temp the temperature as a potential close to the origin here looks like this So what is here called t1? I? Now call tp. So this is how you get the potential now What is fast even more important now is this critical temperature? That's the one where? The potential at this non-zero X value is equal to the potential at the origin That's a critical temperature, which is some more larger than this barrier temperature and in fact its value the value of this field relative to the critical temperature is given by this ratio and Here you see that this value this ratio is Big if lambda is small and it becomes smaller and smaller when lambda increases So this is the structure of the model and if you now go get to the standard model you have essentially the same Just as a function these coefficients a and b they now depend on your cover couplings the gauge couplings and so on now I should say Calculating this potential is another completely trivial exercise. There was a lot of work done on that and Mostly in the mid 90s and a little bit later which then now these days is used You have to do loop corrections to get that you have to discuss the gauge dependence their infrared divergences You have to worry of how you treat the goldstone bosons You have to do resummations Then there are non-perturbative effects due to the non-abiding gauge interactions and so on And there were lattice calculations done on that so it's I would say quite an interesting topic of Field theory and I think we have now Essentially a complete understanding of this phase transition the standard model Quantitatively and in extensions most of many of them also quantitatively and at least qualitatively Now it turns out we will see that later that what you need for bariogenesis is you need a phase transition where the jump of this The jump of this field the Higgs field at the this critical temperature here this jump Divided by the critical temperature is larger than one That comes out of the study and we'll see a little bit of that but let me first show you what these calculations gave and What in fact the phase diagram of the electrobic theory is this is done for S you do and I think But I would also face you to cross your one, which is not a very big effect The First what you see here is in fact these were Calculations done some time ago as you can see here this young summarize that and they were done by lattice calculations of the full four-dimensional theory and Letters calculations in an effective three-dimensional theory I mean what is nice about these theories is the finite temperature field theory is one as I said where you have in In the time direction a finite interval and you also have to have for these bosonic fields periodic boundary conditions So effectively what you have is a kaluta client theory a kaluta client theory with one compact dimension and for that What you can do is you can if the temperature is big that means the radius is small You can integrate out The kaluta client modes and go to an effective three-dimensional theory That was a nice way of treating that done by at that time the Helsinki group Kayanti Robo-Kain, Shapochnikov and Leine and they got an effective three-dimensional theory, which they then simulated on the lattice Of course, you have some errors and they are which are not completely controllable the other approach was that was mostly done by my lattice colleagues at easy to do a full four-dimensional simulation and What you can see here the stars I think are three-dimensional the other things four-dimensional and this line is in fact a recent perturbation theory so all that works nicely together and And Now there were two problems with that One is when all this work started the Higgs mass Was known to be heavier say than 40 gv So everybody was hoping for say a light Higgs to be found soon a lap say and then But would have a nice strong first-order phase transition and that was a big motivation for all this work Then while this work was carried out the bound on the Higgs mass increased and increased And when the work was finished it was clear that the Higgs was so heavy I think at that time it was about here that This ratio V over T was too small so that it could work for baryogenesis So a number of people just stopped to work on this On the end on the other hand what is interesting is that there's that there's another effect namely At some point this perturbative calculation simply Becomes wrong and that is at about here at about 80 Gv and there it happens that this first-order phase transition which you have in the lecture week theory Becomes a crossover it becomes a smooth transition That that should happen you can argue on general grounds that it happens here you can roughly understand By in the following way in an unavailability engage theory at high temperature You generate a non-perturbative mass for the vector boson so-called magnetic mass and if this magnetic mass becomes roughly equal to the mass which you generate By the vacuum expectation value of the Higgs these non-perturbative effects become important And the phase transition disappears So here we now know that This is the lattice number if the X miles is above 72 Gv There simply is no phase transition at all in the standard model Nevertheless the work on biogenesis vectoring biogenesis continued and of course you can still have First-order phase transition if you go to extensions So you may think of a two-weeks doublet model as maybe the simplest version you can add a singlet You can go to supersymmetry at that time all viable possibilities and you can again Find a first-order phase transition So then You have to calculate for this first-order phase transition first you have to calculate the rate At which these bubbles form the so-called nucleation rate Which is given here by such in such a semi-classical approximation. It's given by field configuration which interpolates between the two vacua the broken in the Symmetric phase and then you can calculate that and see how these bubbles form and expand and Now you get this picture from Electropic biogenesis you have say one big bubble. Let's look at one big bubble which expands very fast With the velocity of about say 0.1 0.01 times the sound velocity, which I think is about one-third of the velocity of light in this relativistic plasma And then as it expands The particles in the plasma Interact with this bubble wall which you have here at the boundary. So inside you have the true vacuum the Higgs phase where the broken phase where this wave is different from zero and where the sphalon rate is Zero actually so no conversion anymore between Particles between billions of leptons But you have also reflections and here outside the symmetric phase where the Higgs wave is zero and Where the sphalon Processes are active now it's sufficient To do this calculation in the end Essentially for a one-dimensional system where you say you take the bubble wall as a plane Which just propagates and you study a transmission and reflection at this wall And now what is important how the fermions behave? Interacting with this wall. So this is say a typical yukava coupling You have the yukava coupling you have the Higgs field right and left-handed fermion and the bubble wall you can imagine as say something here which interpolates between Value different from zero the real part different from zero and zero outside and then what is important you also have a phase which varies and well Now you have to calculate the baryon asymmetry and that's really Calculating this baryon asymmetry is a difficult problem. I think to really develop how that works Maybe talk roughly ten years so that people agree now on the details. It's really very difficult And you can in principle you can start from a rigorous formalism, which is a Schwinger-Kelchish formalism for treating non-equilibrium processes in quantum field theory From that you can go via some approximations to Boltzmann equations still too complicated You can make further approximations go to diffusion equations, which I think well motivated and then You have these diffusion equations with this moving wall and now What now happens is if you look at the situation say in the rest frame of this wall Which moves then in the rest frame you get a stationary solution a stationary configuration so that means these chemical potential which describe the asymmetry between particles and antiparticles depend just on Now one coordinate in this approximation and is the distance from the wall and what we are interested in is We are interested in the chemical potentials of all the left-handed Quarks because the left-handed quarks say in particular the left-handed talk box that has the biggest interaction with this wall this left-handed Top top quark That's all these left-handed quarks. Those are the ones which then in these follow-on processes Where an asymmetry which you generate in these numbers is transmitted Turned into a baryon as a lepton and baryon asymmetry So therefore we are interested in this and then you can convince yourself that now once you have this Profile of chemical potentials in front of the wall that then the change in baryon number Per time is this follow-on rate and Then you have fear is this left-handed chemical potential and then you have to subtract a term Which is converted away by means of this follow-on transition, which is proportional to the asymmetry which you have So imagine you start from zero then you just have a source term Which comes from this interaction with a bubble ball that gives you say an excess in left-handed quarks over right-handed anti quarks and then But once you have that then the follow-on processes are active and convert part of that into a baryon asymmetry just the left of left-handed particles and Then you have to so this gives you the rate of change of this and then you have to integrate that to get the final result now But the question of course is how do you calculate these objects? And that is in fact then these diffusion equations you have to look at all the particles pieces and you get Well really second order differential equations but to a good approximation first all the different equations Differential equations for these chemical potentials where you have the first derivative This is a wall velocity then use a couple to the other chemical potentials and here you have a source term So that you can derive that's a long literature on how you do that which I cannot discuss here and actually even if I had more time I would not be able to because these details are really for those people who work on that and But once you have that that's what people then solve and in the end they calculate from this the baryon asymmetry And you see then you go back what that enters here you have to go back to the frame where the wall moves and So the baryon asymmetry which you generate depends on the wall velocity These are the chemical potentials which you integrate over all the space say on one side of the bubble and what appears in the exponent is again this follow-on rate and This is a coefficient which you have to calculate the wall velocity again appears So this is just to show you what in principle people do who Calculate in this way the baryon asymmetry Yeah, sorry Well, the CP I will come to the CP a symmetry now in certain models But the superior how the CP a symmetry comes in of course is more dependent I mean the simplest example which I have is Now the two X doublet model that will be on the next slide there you have a CP phase Among in the couplings of the Higgs The two X fields and this via a loop effect coupled to the CP a symmetry which In the end makes this phase change. That's a complicated process Well the CP phase is in here and the source term I mean this source term is essentially if you work this out the source term You get from the interaction Of these fermions with a bubble wall this is a bubble wall and there is a phase and This phase you have to calculate how this phase Is of this bubble wall is then related to the say the phases which you introduce in your tree-level Lagrange Okay, and yeah, how to do that is again No, but of course in principle you are right this phase is crucial without the phase There will be no a symmetry now. We know in the standard model doesn't work. So let's go to the next Possibility to X doublets which has been studied in much detail. So you have two Higgs fields here H1 H2 and You see here the phase appears just here in fact it's it appears together with a mass term and Apart from this term in fact the model has a certain symmetry is that to symmetry Where H1 goes to minus H1 and H2 does not change which forbids such mixed terms So this that to symmetry is broken is now why you use this Lagrange is a long story and you have to worry also about flame changing processes and The CP violation which you generate from that and so Now of course I cannot go through the analysis in detail of such a more because all together it's rather complicated. I just Put here one quantity which also work on that introduce because what is important for the consistency It's also how big are the couplings so you cover couplings which you have here and one measure is that you say how big is says the one look correction to such a coupling and you divided To its three level value to see whether says a perturbative treatment is at all correct. I Come back to this number now And I would like to give you a say of two rather recent papers in fact this last one very recent just some numbers the The first is in fact this one This is came out of a detailed study of these people already a couple of years ago But it's a sorrow treatment and what is plotted here as a x-mask At that time we didn't know where it was now. We know it's here 125 And this is the second the mass of the second neutral Hicks how heavy it is and When it gets heavier that means that's your cover couplings which you have in your Lagrangian become bigger and Then These are lines say in this parameter space Which give you a certain barrier to symmetry which is here in units of 10 to the minus 11 Now we know the barrier in asymmetry is now six times 10 to the minus 10 So we have to go along this line or we have to be about here now This is the ratio of this jump in the X parameter over T This is one and going up here. It also means it has to go to one point five two They did not put the stump here, but I think it's roughly one point five which you need up here So it shows you if you choose appropriate parameters you can make it You can Get 426 you can choose say a coupling lambda such that this Hicks mass is so big Which gives you the right by an asymmetry? But you are then close To the region where the model is strongly coupled because this quantity Delta is already point five So you need essentially such a model you need big couplings to make this work This is interesting because the same couplings and the same CP violation which you need in biogenesis They also affect dipole moments and The dipole moments which you get of this hearing of these models are very close to the current experimental bound As far as I know it's still consistent, but It's close now these people had more recently a Detailed studies in view of LHC that did not study the full Biogenesis, but the strengths of the phase transition and then I think it's plausible that biogenesis will work and So what they get is that you know a zero the pseudo scalar mass In the 2x tablet model should be larger than 400 gv in that case and the second neutral Hicks and the charged Hicks is Should be light in mass so you get a definite mass hierarchy Which is important because of course the charged Hicks is maybe more easily observable and also via loop they modify and the Brunching ratio of the Hicks we have the neutral Hicks to two photons and you can get Corrections 5 even up to 10% so that's so what from that you see is if such a picture Will be correct. It's very important to look for charged Hicks's and to have precision measurements of the Hicks branching ratios now I Think two months ago this paper came out and they look at the electric phase transition See whether they can get a strong phase transition in a particular version of the 2x tablet model where In fact one of the Hicks's does not couple to fermions, but Is in fact stable and can give dark matter so they Present three examples, and it's just instructive to look at that. So for instance if The pseudo scalar mass here that is roughly around 300 for a gv The charged Hicks masses is 300 400 like this. So it's pretty much consistent with this So you expect charged Hicks bosons rarely to see at the LHC Then The one which gives a dark matter Can be pretty light could be 200 heavy than the Hicks, but it could be as small as 5g And what is interesting is you predict sizable deviations due to these charged Hicks's in the branching ratio of Hicks to To gammas So if this picture so you have I think what is nice about this is these groups push the 2x tablet model To a level where one can hope to really either see in effect or To falsified now at the LHC and that is related To this rather light charged Hicks's either you can see them directly or By the loop effect they significantly modify The rate of Hicks to gamma gamma Now this brings me to the second example, which where the motivation is different. That's motivated by Composite Hicks molds you heard about Just in the previous lecture and So the idea here is that you have a Hicks Sector with a composites scale of about a TV and then you may have an additional Singlet in addition to the Hicks tablet and then you can design an agrangian Which gives you first order phase transition where now what is important this first order phase transition is not Caused by thermal corrections But as the first order phase transition is due to the tree level potential the interaction between This singlet and the Hicks tablet This is the neutral part H the LH of the Hicks tablet S is a singlet here You see the temperature behavior, which in fact affects both H and The singlet and I think the coefficients are such that in fact first S develops a web the singlet and then H both fields develop an expectation value and here two examples are given That work was done before the Hicks was discovered So now what is relevant for us is 120 gv and then you see that you have a rather light singlet here 80 gv this is a critical jump in the Hicks web and This is the wall Thickness I will come to that and sorry, this is And this is this quantity divided by the critical temperature So you have a light singlet Lighter than the Hicks, which is I think very interesting that is needed for the phase transition on the other hand such a Particle even if it's a light to see it is difficult because it just comes to the Hicks So how to figure that out and what limits can be set from LHC on that? I think is a very interesting question and as far as I know it has not really been worked out in detail So to look at such models and to really work out seriously I think the LHC implications in my opinion is an important problem Now coming again to CP violation Now CP violation in this model you cannot get from the renormalizable Lagrangian But that means from these terms here, but you have to introduce dimension 5 operator In this model which you can argue you have because it's a composite theory So what you need is a coupling for the top right-handed top left-handed top tablet This is the Hicks, and this is the singlet And this a coupling by a two-loop Generates a dipole moment for the electron Maybe looks quite surprising, but you have this big coupling here here via this there as a couple to the top and Then you have the photon you have the photon and this and also for the neutron you get a dipole moment And again these type of moments are close to the current experiment limits So it's interesting at the same time dangerous in that respect the situation is similar As in the two-weeks tablet model and the scale here Which appears here in front of this thing is about a TV. Okay now note Just one note. I did not discuss the MSSN. I mean there were there was a huge investigation on the MSSN biogenesis in the MSSN, but That requires that first of all that squawks are very heavy, which is okay But that's a stop then it's lighter than the top That I don't know whether one can still defend that Possibility sort of as a case where the stop is essentially degenerate with the stop the stop is essentially Must degenerate with the top it cannot be distinguished, but in any case. I think it's very special So I will not discuss it So Finally, I think I should come close to net but let me do as a summary now the following I Think electric biogenesis is a very interesting topic in on equilibrium quantum field theory And there's been a huge activity on this field for the last on this one number for the last 30 years And I think the interest in this is due to the fact that it's so closely related to electric symmetry breaking and the Higgs mechanism So I think it's very important now to really Make progress on this issue at the LHC Now what I did here finally is I asked one of the experts working on this Because reading these papers sometimes not so easy and I wanted to understand where does the order? I mean is a simple formula is there something simple so that you can see where the orders of magnitude come from? finally in the baryonyl symmetry and Apparently what you get out of these diffusion equations and so on you can roughly parametrize like this You have this follow-on rate the baryonyl symmetry here normalized to the entropy density is This follow-on rate divided by t to the fourth, which is about 10 to the minus six Yeah, as we know then you have the wall sickness The wall sickness and the variation of this phase over the sickness gives you the source terms in the diffusion equations And that is roughly say point one Then you have some CP violating phase and say in the normalizable models you get by a loop effect induced CP violation in the top Bubble wall Interaction so you get a loop factor one over four pi. So this is a gain effect about point one This is the fact about possible boils one suppression if the Particle is a little bit heavy Which does the reflections on this wall? Let me put that to one then one has a power Which comes out of soil with the diffusion equations of this critical Veph over as a critical temperature Which is between three and four and then in these diffusion equations? There is some uncertainty or something which gives you another factor between ten to the minus one and ten to the minus two So this chain of factors with ambiguities of course shows you reflect the complication of this whole calculation If you are now on the optimistic side and take everywhere the best numbers you get Say something like ten to the minus nine and we know you have to get six times ten to the minus ten Now as you see it's possible but from what we Learned strong interactions in the couplings and New particles appear to be unavoidable. So for the two extended molds, you need tracks charged exposons You may have a light singlet You have new interactions. So You really Need to see something at the LHC So dedicated searches at the LHC Stronger bounds on the dipole moment and electric precision tests Should really be able I hope with the next couple of years maybe to bring this topic of electric biogenesis to some end Positive or negative we see and that I think is a maybe good point to stop for today Sorry in the page that you Had the example for the standard model extension with a singlet Can you please bring this to page? This one. Yeah. Yeah, I wonder how to determine the VC it should be a function of TC itself, right? I mean the I did not give the impression as the expression here But of course the VC is a function of the parameters here which you have in the Lagrangian Yes, but how we can This is a function of the parameters and then also Say say this ratio is a function of the parameters. I did not give all these Yeah, actually, I need to know what is the functionality? What's the relation of I mean VC with TC? It should be a function of TC itself, right? well, and then I Have the paper in my room. I can't bring it tomorrow. We can't copy. Okay, so You were saying that the CP a symmetry shows up in the source of the left-handed chemical potential Yeah, I mean also you said the DSU 3's tolerance don't play a role But I guess the DSU 3's tolerance will they they they will dilute that Yeah, no, the SEO what's the SEO 3's follow-ons to is if you look at If you look at these these mu eyes, right? These are chemical potential for all the particles. So you get a system of diffusion equations and then The SEO 3's follow-ons give you some relations between them It's like in this Well, I mean you well, I wouldn't well and no I think it's somehow nice to have them There's your 3's follow-ons because it reduces. I mean you can safely. I think assume that they are in thermal equilibrium And therefore that gives you directly some relations between chemical potentials. So it just reduces the number of degrees of freedom But it also dilutes the left-handed. I guess I didn't really see the left-handed Potentials the right-handed guys. So yes, it makes it reduces. I think that you see here in this formula which I Gave here to estimate the effect. I said diffusion is affected by this So I think it's in there. I mean if you look at these equations, they contain thermal averages of various quantities and then That relates the relations which due to that that contains relations which Due to the SEO 3's follow-ons you have between different chemical potentials I'm curious about the next slide Yeah, and no This plot with the predictions of the the mass of the yes, it is. How can it be so predictive? Even with too much parameters It's a good question that I also ask myself, okay, I mean Personally, I've been working on the phase transition in the past, but I never worked on that now How it can be so predictive is If you look at this paper, this is a rather no no it is a I look at it It's a really a rather careful study and then the question is which Couplings in the end are most important. So for instance if you you have these masses here Are functions of the cover couplings in the theory now some of these couplings in the end for Generating the asymmetry are more important than others and in fact in order to make the phase transition Strongly first order. You are typically driven into certain parameter regions where you know certain Couplings plays a dominant role and in fact these couplings then usually are typically pretty big. That's why I I Listed this number here, which is which gives you the change in coupling as you go from Here from three level to the one look correction so for the whole thing to work at all you are driven to Force to go to a regime where some couplings are large and I guess then Of course they Clearly you have a parameter space I think I want to gather maybe five parameters. So you can say how can I get just a line in this play, okay? Now clearly That means that you have fixed the other parameters How they are fixed is given in the paper and I think I expect that that is Reasonable, but that I'm not I was not able to check in that manner for the case of the singlet Yes They made this prediction also. Yes. Now for the singlet the situation is small transparent because as you see your blessed parameters and The the other difference is in the 2x doublet model You have to make the phase transition first order that means to generate the this bump This barrier and the potential from radiative corrections That's difficult and therefore you go to these large Parameters here you get this first order phase transition from a tree level coupling So I think that is essentially in fact these authors that if you're interested in that these papers are Quite good to read actually also. I hope I It seems I forgot to list the authors here on the slide I Don't in the other case. I think I gave the others here I forgot I mean these are papers by this group espinoza constantine And to other bereave No, no, but these people if you're interested I give you the reference Or maybe I put it even back on the slide tomorrow tonight and these papers are quite well to read and There and you can understand how it happens how you make the phase transition sufficiently strong for this order and then So I think that is safe and you don't need much Extreme parameters for that, but then in order to get the right CP violation. You just introduced by hand Thank you. I still not I am still not convinced about the role of Sphalerons in QCD isn't QCD. Yeah, isn't it true that This is for Sphaleron only exists because of some kind of anomaly and the QCD is vector theory or am I mistaken in this No, that's right. I mean, of course the The Instantons in QCD Relevant for the axial anomaly Yeah, so there is no Sphaleron problem. I mean, there's no effect in QCD, right? Well, I mean there is the People believe that The instant on effects are there in QCD in fact in the old days I mean the early days of QCD people even try to calculate Chiral symmetry breaking which I have in QCD as an effect due to instantons So I think there is no doubt among those people who do say letter simulations for QCD that these Feel configurations which have topological charge Are there and play an important role? I think this is not Debated now. It's another question how to see directly the effect Because they the same way he has a Sphaleron so the SU two instantons change can change barrier number these instant on the QCD change chirality and There were recently some interesting papers. I think by Brookhaven by Shoryak And collaborators where they were looking for ways maybe to see such chirality Changes in heavy iron collisions must be difficult but was discussed Thank you. No more questions Okay, let's thank you