 Good morning. Can you hear me? Okay good. So Giovanni, thank you very much for for this introduction and I should disappoint you because I will not say everything about neutrinos. You see this is very modest title of my lectures but in reality this is a vast field and so what I will do I will just cover some selected topics in neutrino physics and I'm sure that if you invite some other speaker you will get probably substantially different course of the lectures. This is fine so I will speak on the things which I think I understand better than others. Now let me ask you first how many of you are really working in neutrino physics? Can you raise your hands just to see percentage? Because I'm also thinking that how to do these lectures useful for everybody and the point is I have selected topics which actually have kind of general value so you may meet such a situation in some other fields and areas and also I want to stress some open questions. I think for you it is very important to understand what are the most kind of hot subjects in this field and what are open questions and problems because some of you are still looking for topics of your research. So this is my goal I don't know if I will succeed or not. So I will start with just reminding what neutrinos are. Actually I was supposed that it will be first lecture on standard model and by Peskin so it would be some kind of introduction to what I'm going to speak but let me still remind. So what are neutrinos? Neutrinos are neutral and they have zero electric charge and they have zero QCD charge. So they are extremely light so they are much lighter than the mass of the electron much lighter than the proton and of course the first temptation is somehow to connect these properties and the first one is unique for known fermions. Neutrinos is the only fermion which has known fermion which has such properties zero charges zero conserved charges and so many developments in neutrino physics are based on this property of neutrino. Neutrinos have accordingly only weak and gravitational interactions so they are very elusive and essentially they are kind of eternal strangers because what happens once produced especially at low energies neutrinos travel forever essentially with very little chance to interact somewhere. This is bad of course for our detections but this is good because neutrinos bring some very important and unique information about universe about properties of particles about everything. So neutrino is particle with spin one half and so this is the most abundant component of the universe and neutrinos are everywhere. Actually sterile neutrinos exist the number of neutrinos in the universe even bigger than a number of photons. Neutrinos play special role not only in construction of the standard model actually they play it crucial role in this but I also pray a special and not completely understood role in evolution of the universe being probably connected to dark sector dark matter and dark energy. So in this sense neutrino provide a portal to this hidden sector very important and many kind of developments related to this actually is interesting you know so they have many problems in our visible sector but we are trying now to solve it in the hidden one. So this is the status of the field in one glance there are many sources of neutrinos the sun atmosphere the earth then we have detected neutrinos from supernova 87a and expecting some more burst from supernovas. Neutrinos from cosmic rays of cosmic neutrinos this is highlight this is something which has been discovered very recently and a number of developments are related to this discovery. The universe of course is the source of neutrinos. Neutrinos are around us and we more or less know that they are around us. Now there are various artificial sources of neutrino beams accelerators reactors and radiative sources so we have plenty of the data on neutrinos and it seems that they are quite well described in so-called three neutrino paradigm with massive neutrino. So what is this we have standard model and we have three neutrinos with masses and mixing and so with quite peculiar though properties of mixing and very small masses which is something special about neutrino and then the question which we ask is this that's it if there is something else on the top of this up to say plank scale and this question is still open it's interesting to of course to discuss. Now of course one can introduce neutrino masses in very gentle way because neutrino masses are supposed something which is beyond the standard model. However you can produce new introduce neutrino masses generate without strong perturbation of the rest of the theory. This is so-called famous dimension five Weinberg operator which produces the generates a Majorana masses of neutrinos and here is some scale at which such an interaction is generated. Of course it is not present in the standard model but you can introduce on the top of the standard model this introduction. One thing actually tells us that maybe this is not the end of the story is that this lambda is actually should be much smaller than plank scale mass which indicates that there is something still between electroweak scale and the plank scale and I will speak about this in probably the last lecture. So neutrino masses are considered as some physics beyond the standard model actually understanding of masses of other particles other fermions quarks and leptons is also somehow physics beyond the standard model. We do not understand pattern of the masses we don't know how hierarchy is produced of course we can embed in the standard model so we can describe masses of quarks and leptons but we do not understand particular pattern of the masses which we see and mixing also. So this is highlight I mentioned already this is detection of of cosmic neutrinos of high energies and these are two first events detected actually almost three years ago in PEV range who knows what is PEV so TV is you know it's TV is LHC right so this is PEV a thousand times more and there were two events first events which were identified as something which cannot be explained by just the common origins like atmospheric neutrino fluxes and now there are 37 events and there is very interesting development in this area. Now neutrino masses are related to some new physics which we should identify the question what is this new physics you know we were waiting for many many years finally to discover finite mass of neutrino and mixing but unfortunately till now we cannot identify what is really this new physics behind neutrino masses and mixing and the first question you ask is what is the scale of neutrino mass generation and you see now we have so many possibilities which cover 28 orders of magnitude starting from even electron volt or subelectron volt scale and there are kind of interesting developments maybe we are looking in the wrong place what is the origin of neutrino masses maybe very light scale of mass scale energy scale is somehow fundamental and maybe neutrinos and maybe dark energy indicates toward this possibility so this is quite something interesting which actually under development another important scale is of course electrovic and LHC scale and here there's a lot of activity I will cover something in at the end of the course and this is for me something like looking under the lamp so it's just because we know that these energies are available and then we try to test everything what we can test at these energies there is some sense of course you really have small masses and one of the ways to explain this is to assume that they are generated radiatively and if you generate them radiatively so the smallest is somehow related to some small you color couplings or some other couplings and therefore from this point of view there is sense of course to search for some new particles or some new physics at a TV scale associated to neutrino mass generation and actually many things in between can be filled in let me mention the last one as this grant unification and plank scale masses and actually neutrino masses indicate toward the scale because you can generate after electric scale squared over the mass of neutrino some new characteristics of the dimension the mass right so and what is interesting that this is something like 10 to the 14 10 to the 16 gv's and actually if you introduce three right-handed neutrinos in addition to what we have in the standard model and you see some mechanism which I will discuss later then one of these right-handed neutrinos can be easily at precisely grant unification scale so for me it's some hint that neutrino smallest of neutrino mass indicate toward the scale and actually plank scale can also be involved in generation of neutrino masses so 28 orders of magnitude means that we really have no good kind of glue to what happens with neutrino masses and what is the generation of the neutrino mass mechanism now concerning mixing we have quite interesting pattern of left and mixing and the idea is spanned from symmetry down to anarchy and randomness especially after recent discovery of non-zero value of 1 3 mixing so finally for many many years actually the driving force in developments of the field was so-called anomalies so we had solar neutrino normally atmospheric neutrino normally which boiled up eventually to discovery of neutrino masses and mixing so what are anomalies now we are discussing and here you see a few this is the old one some result which I will discuss lsnd then mini boon result and reactor neutrino results gallium calibration results solar neutrino spectrum which indicate that maybe there is something beyond these three neutrino paradigm I will discuss this in details later some anomalies disappear some kind of produce eventually important results and you see it looks like a sterile neutrino is a solution or for all our existing problem sterile neutrino or something from hidden sector because sterile neutrinos are the particles which have no usual interactions and so so they do not interact with photons W bosons gluons and therefore can be considered as something which is in this hidden sector now so keeping all this in mind I have selected the following topics of my lectures I will discuss for the theory of propagation and I think this has a general value and you will see I will use actually these results during all these four lectures and this is something which you can you know it's not just some kind of assumptions or something like in walked something specific some unrelated to reality this we know this exist so you will not waste your time at least listening this you may waste your time listening about models of neutrino mass and mixing and most probably you will waste your time because there are thousands of models and probably only one is true so I can imagine and people are now doing classifications of these models you know there's some kind of zoo of the theories so I would I would say that from theoretical point of view so what what is the theory of neutrino masses so they have ground and this is something which is solid and then there is a lot of speculation it sounds like like like for me it's neutrino physics looks like this so this is solid stuff and these are speculations so this one is solid then I will discuss phenomenology and the way we are determining neutrino parameters and I will also discuss what are kind of hot topics here because now we are thinking how determine neutrino mass hierarchy and CP violation in the tonic sector so there are these are hot topics in and the subject then it will be more speculative toward understanding of neutrino mass and mixing what I will try to do I will here to use some kind of bottom-up approach trying to analyze information which we have and try to see what are hints what are kind of inferences from what we have observed keeping in mind this kind of disastrous situation when we have 28 orders of magnitude unknown scales for new physics and also this kind of big span of possibilities in explanation of mixing and finally I will discuss something which is beyond three neutrino paradigm and I will speak on sterile neutrinos questions okay so masses mixing and the theory of propagation we have three neutrinos electron muon and tau which we are identifying as neutrino flavor states they correspond to certain charge current interactions and they're associated with certain charge current leptons electron neutrino with electron muon neutrino with muon tau neutrino with tau lepton flavor is characteristic of interactions so for instance in neutron decay we produce electron anti-neutrina together with electron in pine decay mostly we are producing muon neutrino together with muon apart from flavor states we introduce mass states the states which have definite masses nu1 nu2 and nu3 so sometimes we call this a mass against states and the essence of mixing is that flavor states do not coincide with mass states flavor states are combinations of mass states and vice versa mass states have kind of composite flavor so in the standard model as we know not much to say that electron neutrino and electron form electric doublet and the same is for two other generations the only what I want to say is that actually phenomenological definition which I gave in the previous slide may differ from this theoretical one and the point is when I'm saying oh if I have in some process appearance of electron and neutrino then neutrino state which accompanies is electron neutrino now imagine that electron neutrino has some mixture of some heavy states which actually in CISO mechanism this is the case and in many recent developments this is also the case this means if you have this heavy state what you produce in weak decays is not this electron neutrino but only part of the state because heavy fermion heavy left on cannot be produced just kinematically so the state which you are producing in at low energies for instance in beta decay may differ from the state which you put in this charge current in this doublet or in charge current so that electron electron neutrino can be the sum of let me put here uai nu i plus for instance u e4 nu 4 and the mass of these four state can be quite large it can be at electric scale for instance and so of course you cannot produce this state in a beta decay what are you are producing is this one only this part there are some interesting effects related to such a possibility for instance the state which you are producing here is not orthogonal to muon neutrino and therefore you may see some left number violating effects related to such a mixture of heavy state of course this mixing is should be quite small actually the bound on this mixing is below 0.1 or maybe even less 0.05 so it's small effect but we are in the era of precision measurements and so what happens that you produce only part of electron neutrino state and it is not orthogonal to muon neutrino this complete combination is actually orthogonal to nu mu but not part and the overlap of nu mu and produce nu e at low energies will be determined by this of mixture I may elaborate on this a little bit later now mixing how we describe mixing what you see here actually what you see here you can use to explain almost everything I mean it's just enough to stay with this diagram and explains all the neutrino physics and masses and mixing using this diagram so you see here the mass spectrum of 3 neutrinos so this is the mass scale and these boxes correspond to different mass states okay now since we have mixing that mass states do not coincide with flavor states and they are combinations of different flavor states and this you see by these colored boxes the red color corresponds to nu e green one to nu mu and blue to nu tau and actually I proposed this type of description long long time ago but now people are using different colors everybody uses its own color sometimes muon flavor is red you know some arguing okay not electron neutrino should be green or something like this so this is a region but other are just interpretations anyway so what is the meaning of this suppose you have beam of neutrino new one this is kind of Gedanken experiment that means this question how can you produce this beam of neutrino one and then you explore how this new one state interacts in your detector and then what you see that in say two third of the cases in the length is a two-third here it interacts as electron neutrino producing collectance in one six of the cases it will interact as muon neutrino and in one six approximately cases as a tau neutrino so the size of these books if everything is just one then the size of this box gives the probability to find in a given mass state a given flavor state so if you have this beam new one then it will produce both electrons and muon and also tau in this proportion this is the essence of of the mixing of course if you have just one neutrino then that gives you the probability however if you have many nutrients flux then these sizes give the portion of the number of muons electrons and tau you will observe in interaction so this be now you can see here quite interesting pattern right it's not just something which is random but first of all you see the mixing is not small small mixing would mean that a given mass state is completely almost like has the same color of the one color right so here you see that the mixture of different here flavors in a given mass states is kind of comparable it's not small the only small is this this is this famous one three mixing which is something like the size of this is something like two percent the rest is quite big now you see here it's almost like half and half which we call maximal mixing so if you have half and half mixing so another pattern here is you have one third one third and one third here it's also interesting that gives you some idea that maybe some symmetry is behind this it looks like claps Gordon coefficients right so it's it's not like in pork sector this is this is something interesting and I will discuss this later I think in the third lecture so what else you can see here apart from so let me say so how we describe this so we have three flavor states and three must states and so this is the metric mixing metrics which connects flavor and mass states and modulize squared of elements of this matrix give precisely the sizes of these boxes okay and I have a written here for instance that the size of these boxes just you e1 modulize square and the rest you can read also so we have this PMNS matrix which is three by three complex matrix and we also parameterize these in terms of three mixing angles so let me define how this mixing angles are introduced we introduce one to mixing angle which actually gives the ratio of this red part so this red part and this one so it gives you relative contributions of electron flavor in the second and the third state one three mixing angle sine square theta one three it's just precisely give this small red part and now two three mixing tangent square theta two three describes distribution of muon and tau flavor in the heaviest state so if amount of muon and tau flavor is the same than the tangent square theta two three is one this is maximum mixing this matrix also has some complex phases actually there's only one complex phase which is feasible which is kind of has a physical meaning in oscillation experiments actually all together there are three phases which have music physical meaning and two exist if neutrinos are Majorana particles so if neutrinos Dirac than only one phase exists this is so-called CP violation phase and it is analogy with what we have in quark sector changes of CP violation phase actually affect distribution of muon and tau flavor here and also here so if you change phase delta and this picture is for phase delta zero then you have slight changes of borders here usually we are parameterizing this mixing matrix in this way so there are three rotations one two one three and two three and here is the matrix of complex phases I will show you in the next slide and it is important this is so-called standard part parametrization it is important this very convenient parametrization so mostly we are using this parametrization and this is very easy to use these when you discuss propagation of neutrinos so that yes so we haven't measured these phases and the measurement of this phase delta actually here is this matrix which is diagonal matrix of this type so this is one of the major issues to measure this the CP violation phase okay so microphone the question is how we measure these phases and what so what are here I'm discussing this is just Dirac so-called Dirac phase now if neutrinos is Majorana then there are two additional phases which we are actually attaching from this side and those will be can be measured in double-beta decay experiment I will discuss this right all three phases enter this this expression yeah in double-beta decay but this we can measure independently in oscillation experiments and not only this because of smallness of one three mixing essentially double beta decay is sensitive to only one of these Majorana phases so this is kind of explicit form of this UP matrix and probably I should not say much apart from you see the phase appears here in this one three element the phases also appear here and therefore you see this these elements are affected by the phase delta another important point is that this phase factor appears always with s13 this is important actually if s13 13 mixing angle would be zero then this phase is also has no physical meaning you can just remove it by the phasing and therefore you two one three non zero mixing angle this phase appear here and it's it has a physical sense but all the effects are proportional also to s13 or s13 squared so there's certain smallness here now let me save you was but how we get this mixing it's just more to fix notations so origin of the mixing is actually inequality of mass matrices of charged leptons and neutrinos and so what you are doing you are diagonalizing these mass matrices and that gives you the mixing and that gives you the mass spectrum when you diagonalize these matrices so here you see expressions for charged leptons mass matrix in terms of diagonal and these are the matrix of eigen values which means the mass of electron moon and tau and these are two mixing matrices you L left rotation and this is rotation of right-handed components now this is mass matrix of neutrino and if it is my orana it can be eaten in such a way again we have here diagonal matrix which is the matrix of mass eigen values and that two rotations but here if neutrino is my orana and my orana master is organized by using just two companies on the left-handed components and conjugate of left-handed components are involved I will discuss this later in details so you have the same matrix here and here on the left here is also a transparent matrix and then if you have charge current and which was written there and you insert in the charge current expressions in terms of mass states you can write your charge current in this way which is this is neutrino mass states they are given here this is electron muon and tau and then you will get here this matrix up mns which has such an expression so what enters here the left-handed rotation from diagonalization of charge left and mass matrix and also left-handed rotation from diagonalization of neutrino mass matrix so this is eventually what is PMS matrix I don't want to make this exercise I think you can write this repeat I recommend to do this by yourself now we use very often flavor basis the basis in which charge left and mass matrix is diagonal so we call this flavor basis and then in this case the up mns matrix is just a left-handed rotation of neutrinos given by diagonalization of neutrino mass matrix in these bases questions to measure what from experiments well it depends what you do I mean it depends on the experiment so you are speaking about mass against states of charged leptons or neutrinos oh so you see it's very drastic difference of the question is what what we are you usually dealing in the experiment as what is interesting the name contrast to quarks where we deal with mass states both up quarks and down quarks and so we can easily just using decays to measure for instance mixing angles in contrast to these neutrinos because they are light they are produced as a coherent state unfortunately or unfortunately this coherent state is just kept for a long time so and therefore here we measure mixing angle in completely different way so in neutrino sector we deal with flavor states which are combinations of mass states and again I was speaking about this gedankin experiment it would be easy in some to some extent if you would have beams of mass against states wait wait wait yeah so while we're talking about the bounds on this you before the mixing between electron and the safe fourth generation neutrinos so I understand that you cannot produce new four via beta-dic interaction and you can make the neutrinos that are produced in the beta decay to interact immediately with matter and see if anyone emits and that's how you can determine that and my question is that does this bound on you if or depends on your measurement of you you want you e2 and e3 so you have to give some inputs for you e1 you e2 and you e3 to extract a bound on you e4 so does the measurement of you you and depends on the presence of you e4 or not yeah so one way of course to measure if you know precisely these elements if you measure then you can use unitarity condition that the modular sum of modular square of these plus these should be one and so they you can just subtract from one modular square of these elements and then you get that but usually we are using different method to measure this this angle and for instance as I mentioned already in this case muon neutrina and electron neutrina are not orthogonal so these type of the schemes produce some effects like lepton number violation like mu e gum for instance or you can search for appearance of this heavy hypothetical so we can just say hypothetical particle directly from you can still produce at LHC for instance and then see if they are decaying in visible particles actually my question is that that does my measurement of you e1 depends on the presence of you e4 or not i mean the experimental extraction of you e1 so if I assume fourth generation or yes it depends slightly that's was my question actually so you cannot use unitarity then so we are doing these measurements very often just in assumption that there is a unitarity and only three but you need to reanalyze again data if you assume this and this also there are some general considerations that also affects the oscillation pattern thanks okay so as I said mixing has kind of a dual meaning you saw the spectrum now suppose you produce electron neutrinos what they are producing you are producing something like what is shown here so you are producing not just one mass state but a combination of mass states and you see mostly it will be new one but one third will be new two and then a little bit new three so when we are saying that we are producing electron neutrina means that we are producing a combination of of mass states are similarly from for new mu and new tau now the question which is quite non-trivial so who mixes this new thing why we have this why we have still the fluxes of electron muon and tau neutrina and the answer is actually not very trivial and it is somehow non-trivial interplay of charge current weak interactions and some kinematical properties of certain reactions for instance in neutron decay or in beta decays of new clay we are producing electron anti-neutrino just because we cannot produce muon and we cannot produce tau but kinematic reasons because energy release in beta decay something like few MEVs and muon and tau leptons have much bigger masses so we just do not produce them otherwise if you would have mass split between neutron and proton very big say GEVs or I don't know 100 GEVs then we would produce all three neutrinos species and then it will be quite non-trivial to figure out what is what is what in the case of pi indicates different reason in pi in decay we mostly produce muon neutrino so so pi in pi in decay we mix other produce other combination of mass state and the reason is because of chirality suppression just angular momentum conservation tells us that it should be a flip of helicity so you have here pi and then you produce muon and you produce muon and you produce neutrino so to satisfy angular momentum conservation you should have a flip here of of helicity and this flip is proportional to the mass of the lepton in this case mass of the muon for the decay into electron and electron neutrino you need to flip the helicity of electron and this mass is much smaller and therefore this mode dominates and the mode of decay into electron and electron neutrino is suppressed by four orders of magnitude also there are some features when we can enhance appearance of tau lepton now what about neutral current so what what is the what is produced in decay in the case of decay of z zero so who tell me so can we do something about this so what is produced if you have z zero decay so what is the state of neutrino and neutrinos which is produced here is it electron neutrino muon neutrino tau neutrino what what say all of them okay so all of them in the same kind of amount right but but you see z zero has no sensitivity to flavor so what is really state which is maybe it's just mass states and one new one new one new one bar new two new two bar new three new three bar because that zero doesn't understand what is flavor so which combination of the states and what is this is this combination which is kind of incoherent some of these three channels or what any guesses okay i'll give you answer later but let us proceed further but it's a kind of interesting physics somehow related to einstein padelsky roson paradox and the answer here that you produce all three neutrino types of neutrinos and the state which you are producing is coherent some of the states of three neutrinos it's not just you are separating you have new e new bar channel independently new new new new bar channel and new tau new tau bar no you are producing some sum of this coherent sum of this then you can wonder how we get correct number of neutrino species three in z zero decay but it turns out that even if you consider this kind of coherent sum it still produces three neutrino species in the decay rate of z zero so what i will discuss in some details and i think it's important to understand this oscillation or theory of oscillations and adiabatic conversion and in fact all all the data which we have for time being can be understood using these two processes so using the data from all these experiments and these two types of the processes we can extract neutrino parameters now oscillations what are oscillations oscillations are periodic process of transformation of one neutrino species into another one so you produce neon neutrino and then periodically in the course of propagation it transforms to electron neutrino and then it's periodic process it's after some time and distance it is going back to elect neon neutrino and this is kind of last for a while oscillations were kind of proposed many years ago actually 50 years 58 years ago by Bruno ponte corva who has this paper mesonium and anti mesonium but and in some last part in one paragraph he mentioned neutrino oscillations and what he was saying actually was trying to find some objects which are knowledge of k zero k zero bar k zero k zero bar were known by that time people discuss oscillations between them and then ponte corva so maybe we there are some other systems which have the same behavior and he discussed this muon muon anti muon system now he said that okay oscillations imply non-zero masses or mass square differences and mixing that also does mention well actually this is very subtle point are we sure that oscillations are required non-zero masses and one should remember one thing actually this is something which you need finally to test very well in experiment what we observe in oscillation but mostly I will speak about masses but I just want to warn you and so you will ask further questions actually what we are testing in oscillations are not directly masses we are testing dispersion relations whatever change dispersion relation that is relation between energy and momentum of your system can influence oscillations and actually can lead to oscillations in oscillations in usual case they have no direct sensitivity to masses and masses something which flip helicity in oscillation there is no flip of helicity so left handed neutrinos or left handed neutrinos are not flipping helicity this is irrelevant and therefore in oscillations we are not testing immediately this nature of mass okay so something which can fake the mass will also produce oscillations it can be just interaction so and actually famous paper by wolfenstein he actually have written this paper trying to get oscillations without masses I will mention this later so this is what I put the question and let me make some some comment what is interesting that proposal of neutrinos relations was motivated by rumors about wrong results actually so this interesting fact in history sometimes even wrong results can produce something which is very beautiful and correct so in that epoch it was davis experiment davis famous by his measurements of solar neutrino fluxes but what he had started he produced this chlorine detector actually it was also suggestion by bruno ponte corva and then what he did he just put it close to atomic reactor and to see if he sees some signal or not what is important that this experiment is sensitive only to electron neutrinos and not electron anti neutrinos sun produces electron neutrinos reactors produce mostly electron anti neutrinos so you do not expect any signal effect from reactor neutrino beam and in the first series of the experiments he saw some excess of events which could be interpreted that something happens and he actually detects electron neutrinos and then bruno ponte corva started thought oh maybe anti neutrinos produced in the reactor actually flip on the way to neutrinos and that caused the effect in davis experiment so this is why he started and the first his suggestion was neutrino anti neutrino oscillations not what we are discussing now flavor oscillations and the first proposal it was neutrino anti neutrino solution so eventually it was realized that the effect is just some background and nothing has been officially published but the rest was this comment in in ponte corva paper it was like you know when some towns were just start to organize it was written so this town has been mentioned you know in 11th century but it was like here so historical it was not much development between this suggestion and eventually this boom of studies of neutrino oscillations so what we have now now we have many observations of oscillations results are well described by standard oscillation formula in the literature there is very naive derivation of this oscillation formula essentially a few lines and probably you have studied also this using textbooks were just few lines and then you realize that something is wrong with this derivation and you immediately realize some paradoxes and then you start to write the papers because you say okay so I know better than so it's absolutely wrong and so I want to derive correct oscillation formula so in steel I'm kind of referring every year few papers with kind of new theory of neutrino oscillations with some corrections to the theory of neutrino oscillations so I hope no one of you will write after my lectures such a papers so the point is that derivation is oversimplified it's over it gives correct result actually it's interesting why it gives correct result even these are people wondering so but it's really oversimplified the point is that things are a bit more complicated and to some extent quite interesting but in most of the situations what we deal with and in experiments actually things are very much simplified and you come to this very simple naive formula but conceptually conceptually it's important to pass through this more correct way to understand what is behind of the final result actually who has derived oscillation formulas for oscillation probability some you are not counted you are so what what I want to do actually I will I will immediately derive kind of in the way I think I mean somebody may object is correct way the simplest also one can use complete field theory to do this derivation but I will I will derive and you please follow I think that that's what I want you to keep home after after these lectures so there's some reference list which I'm my my kind of analysis is based on so in principle it should be no problem right we have Lagrangian we have discharge current interactions right so W boson this is leptin this is a neutrino plus hermitian conjugate and we have mass terms for neutrinos again I'm using here my urana neutrino before this is the same left state and this is a masses mass of of charged leptin and then we know this machinery of quantum field theory so what's the problem just use this Lagrangian and do what you want to do so in principle theory should give you correct answer so we use quantum field theory even quantum mechanics sometimes it's this enough to compute amplitudes probabilities of the process and eventually observables numbers of events in different experiments well actually if you do this then things become not very simple and here what is interesting we deal with quantum mechanics at microscopic distances so for instance you can stay within the wave packet you know and you may have Feynman diagram which is two kilometers long you know you produce neutrino and then you have propagator of neutrino which is two kilometers long or maybe 10 kilometers long so it's kind of interesting and fascinating subject so the point is that even if you know the field theory right still your setup should be adjusted to particular physical problems usually we deal with computation of amplitudes of scattering and usually what we are saying oh at infinity we have three particles which approach each other and they interact in some interaction region and then we see what is going out of this and so we can use then just playing waves describing particles in initial and final state right so usually say interaction region is very small you know this is 10 to the minus a certain centimeters and your particles approaching from from say several meters it's already infinity right many many many orders of magnitude here the situation is a bit different and actually here we deal with two regions of interaction source production region of neutrinos and detector detection region and actually here the oscillation is finite size process and the result of oscillation depends on the distance between source and the detector and that should be embedded already in your formalism if you just do everything go to infinities you will lose in oscillations so this is finite size and finite time process and they have here two regions of interaction in contrast to scattering of course you can probably consider only here external particles and here external particles but then your formalism should know about localization of source and the detector so what is this localization you can actually embed localization of your quantum field here or quantum mechanics formalism in two different ways so in principle you need to use wave functions of particles which participating in production of neutrinos suppose you have a neutron decay then you need to use neutron wave function and not at infinity but in your detector with all these properties of the detector okay and it is this wave function which will localize production region of neutrinos the source the same is detection you need to use again wave functions for instance if scattering occurs on on proton then you need to explicitly write the wave function of proton and find this wave function it depends on our distance and it is this which gives you localization so of course you can use this wave packets for external particles or you can explicitly make integration over finite regions so this is another way somehow approximate but you can do two independent computations here and here so in a sense you can use to apply this s-matrix formalism twice for the region of production or region of detection of neutrinos and then you can if you make this kind of a finite space time integration over production and detection regions and you can use to some approximation plane waves also questions you see the difference right so there's a difference and you need to do with these bloody things with wave functions because if you have plane waves everything is fine you know you just make integration you get that delta functions so delta functions delta functions but here you need to make integration over this you need to find these wave functions of particles accompanying neutrinos and then you can should compute integrals and this is you know things become very complicated again fortunately you can do this in quite general way and in many cases the results do not depend on particular shape of wave functions of your particles so how we treat neutrinos actually we can treat neutrinos again quantum field theory is just propagator between source and detection region but if the distance is quite big then neutrinos are quite quickly become on the mass shell of very very good approximation on the mass shell and so you can actually truncate the process and consider independently production process then propagation of neutrino here you can use neutrinos just using as a free particles not just a propagator but just as particles on the mass shell and then you can independently treat the process of detection questions you know one you can actually do everything even without introducing notional flavor states which actually produce this hassle you need you know to do this mismatch you know various things coherence non-coherence but you can do in the following way you can just work immediately with mass states which are very useful and because this mass states they propagate independently nothing happens with them so you produce mass state and it propagates because this is eigen state of Hamiltonian in vacuum right so it's nice to work with eigen states of Hamiltonian right nice so because they just propagate independently and then the only what you need you need to see how they were produced and how they interact and and you need to compute this amplitudes and sum these amplitudes coherently so to say to sum amplitudes and then compute moduli square I will tell you this more details so you can work again with mass eigen states which are this new i and you have this charge current and therefore you have here these interactions of mass states with charge leptons and you can treat this block which is just your gauge coupling and here is u p m n s matrix as interaction constant so once you know these elements of p m n s matrix you know this coupling constant and so you can compute what are what is the probability or amplitude of probability to produce a given mass state of neutrino I just work with mass states that's it and here is how you produce for instance from pi and again you can compute immediately what is the amplitude of production of mass states and then to see how they they propagate trivially nothing happens and then see how they interact in your detector so for this you need to use wave packets of of neutrinos and usual setup is the following this is in the approximation of factorization if the region of production and detection are quite small and you can neglect oscillations in the production and detection region then you can truncate separate three all the process in three parts one is production then oscillation and propagation of neutrinos and oscillations here and then the detection okay this is how we usually treat these things and now this is some derivation which actually I will go slowly and probably we'll finish at this point and I want you to understand how what's going on I could do this on the black board if you want but let me just to go step by step suppose we produce neutrino the type alpha alpha can be electron muon or tau and they are produced in the source in some which which is centered at x equals zero and time equals zero so after formation of the wave packet inside this production region we get this state which actually is the combination of mass states new k right so what enters in this form this is summation over mass states one two three this is the element of pms matrix right conjugate here and this is the wave function of mass state this is very general right this is kind of coupling of with which you produce a given state mass state agree and then wave function and this is given by by the production process and properties of your source right so this you need to compute in general okay now the wave function of a given mass state can be in general written in such a form so this is wave packet so it's kind of combination of plane waves and this what you see here this is exponent it's just plain wave right and so this is what we call shape factor of you want to just do Fourier transform if you want so f is the amplitude of probability to find plane wave with a given momentum p and p k is the average momentum and a given okay derivation is not very long don't don't don't worry why you tell me if if something is not clear so this is wave function of individual mass state and so we have this sum now here we have the energy and the energy is related to momentum in this way that's just usual dispersion relation right we have mass against state is mass k and then the energy is given by this and I have mentioned already this is the momentum distribution function of the packet and p k is the mean momentum on your way back wave packet is combination of plane waves with different momentum but it's just usual standard quantum mechanics nothing more agree so now we are doing the following let us expand this expression for energy is one around the energy which corresponds to average momentum so we have a wave packet this is p scale and this is this f and this is p k actually wave packet can be more complicated form it can be even like this for plane decay for instance it's not it's not gaussian it's more complicated something so what we will do is just this is Taylor expansion right it's nothing this is the first term we just take this energy at momentum p k then this is the second term and here we have d e k over d p this is the third term let's just do this three or expansion of this expression so this is nothing but expansion of this now what is here here is first term is just energy which corresponds to averaged momentum in the packet the second term is quite interesting because here we see d e over d p and this is nothing but group velocity of your wave packet okay and group velocity is given by if you do this differentiation it's just p over the energy the third term is has quite interesting also nature the third term describes so-called spread of the wave packets i will explain a little bit later so when you produce wave packet what it it does it propagates right but it also spreads because of presence of different momentum in the wave packet and so for instance wave packet from supernova wave packet from supernova has the width 10 to the minus 11 centimeters when it arrives from the center of the galaxy for instance to the earth it has the size five meters can imagine so that's this is inflation no paolo this is something comparable how many folds or not well it's interesting and people haven't discussed this but about but there are kind of conceptual questions what to do with this spread usually we are neglecting the spread which may not be completely correct in all the cases and if you neglect the spread then our expression for energy has such a they have just two terms so the first one is what we have at pk and then on the second one which is proportional to group velocity remember this derivative of energy is just give you the group velocity now in searching this expression for energy into our formula for the wave packet for this for for this wave function so we inserted here okay we will get immediately so so the terms which we have no integration we integrate over dp so for instance if we have here energy of the k state that this term we will go immediately out and some other and so we will have here two factors the first factor is the phase factor so we can actually rewrite it in this way where the phase of the k state is given just pk x minus e k time and it depends on the averaged or mean characteristic of your wave packet this is also important okay now the second term again everything comes from here when you make the substitution can be written in such a way so it's just given by again integral this is this is this is fk right which is what's staying here and this is again phase factor this is what is left here when what you cannot put out of the integral important property here is that the distance coordinate and the time enter in such a combination it's x minus vk this means that this factor describes propagation of your object with velocity vk right and we neglect the change of the shape also shape changes of the wave packet but we neglected for time being so this is what we call shape factor shape factor should be computed using the process of the production and it is process dependent so let me give you the picture and now we come to this picture how we describe our states propagating states using this expression for this what we got before the product of phase factors and shape factors and since our state is combinational here I use example with two mass eigenstates then we have these two terms with two different shape factors with two different group velocities with two different phases and so this is how we describe our state and here is a graphic representation suppose we produce electron neutrino then according to this two neutrino mixing formulas just for simplification this electron neutrino is composed of two wave packet two wave packets which correspond to the state new one and new two okay so the amplitude here is given by just cosine theta and this is by sine theta which enters in this mixing formula each of these eigenstates has composite flavor as we decided right once we have mixing then then it should be composed of two different flavors and you see here red part which is electron flavor green one is muon and both packets contain both muon and electron parts and you can wonder how come this is electron neutrino but it has muon parts and the point is that you see in muon neutrino these two parts should be with opposite sign here so they should actually have opposite phases here they have the same phases this and therefore these green parts which cancel each other so it's destructive interference if you produce electron neutrino although they contain some muon neutrino company but these neutrino muon neutrino companies have opposite sign and they call and they cancel each other now what's going when this system start to propagate what's going on is the following in initial moment the phase difference is zero and therefore these green parts cancel each other so you have just electron neutrino but these wave packets they will have different phase velocities because they have different masses and therefore phase here f1 is not equal f2 in general they have different masses and therefore the phases will be different and so phase difference will change from zero and start to increase and therefore in the already next moment of time you will have no complete constellation of these green parts and so this means that you will see appearance of muon neutrino in your originally produced electron neutrino flag and those are the oscillations so the oscillations are just the process of the phase difference change between these two wave packets I will about to finish this and let me just summarize some things so when you have this propagation of the wave packets few things happen the phase difference changes the phase difference between mass states and that produces oscillation then what happens is that wave packets start to separate because they have also different group velocities because of difference of masses and so if you have first overlap of the wave packet then they start to separate from each other and the third there is a spread of individual wave packets so and I will continue next lecture