 Let me remind where we stopped yesterday and I will repeat some important things which we actually discussed and try to explain a little bit more probably. I want you to follow all the things. So remember we discussed flavor states and mass states. When we are saying that we produce electron neutrino, this means that we are producing a combination of mass states. And then, so we describe this mass states using wave packets so that this electron neutrino evolves into some state which can be written as the sum over mass states. If we consider evolution in vacuum, in matter it will be eigenstate of neutrinos in matter. Then we have these elements of mixing matrix which tells you, if you produce electron neutrino, what is amount of eigenstate i in the electron neutrino. And then we introduce wave function of the eigenstate and then we have this new i. And we are working here actually in this factorization approximation which means that we consider all the process, production, evolution and detection of neutrinos. We split it in three parts, production independently assuming that there is not much evolution in the production region. And so we are focusing now on evolution of neutrino which is the second phase and then it will be detection. We also have split this wave function of the eigenstates in two parts. One is the phase factor and another one is the shape factor. And an approximation when we use just two terms in expansion of energy over momenta or the average momenta in the wave packet. We get this shape factor which depends only on such a combination of distance, time and vi is the group velocity. And this is approximately one minus mi square over two energies taking into account that square. Taking into account that neutrinos are ultra-latvistic and so you just make here expansion of p or e if you want and so you get this expression for the group velocity. Neutrinos have small masses and therefore this velocity is very close to one. But what is important that this group velocity is different for different mass eigenstates. Which means that different mass eigenstates or wave packets of different mass eigenstates propagate with different velocity. So if you take into account other terms in our expansion then this g will depend on x or t independently also apart from this combination. And this means that wave packet will change the shape in particular due to spread due to presence of different momenta. So this is an interesting combination which tells you that this factor indeed describes propagation. If you change t then x changes according to this law. Suppose you take the maximum of the wave packet and then coordinate of maximum of wave packet changes with time as vi times t. Which means that this shape factor describes propagation of the wave packet. This is the propagation term. And the picture you see here precisely reflects these expressions. So I have shown these shape factors probably I should say another also at the following. For simplicity I will consider just two neutrina case not three but two neutrina case. For several reasons first of all it's easy just to see the results. And the second one it also has a practical meaning because in many situations three neutrina problem, three neutrina mixing reduces to two neutrinos. So we consider two neutrina case for simplicity. And in this case the mixing matrix becomes just two by two matrix. And in two by two case we have just one mixing angle. So this matrix becomes just cosine theta, sine theta, minus sine theta, and cosine theta. And this matrix connects my flavor states in this case for instance new e and new mu. Sometime I will use some other one. It connects this with my states new one and new two. So in many cases I will consider just two neutrina situation. So then this sum will be just sum of two terms, right? Which correspond to new one and new two. And you see here precisely this combination. So it will be cosine which is just new e one in the two neutrina case. Then shape factor, then the phase factor and this is new one. And this is for the second state, sine theta, then gain shape factor, phase factor and new two. And what you see here precisely corresponds to this expression. So we have this shape factor and then this periodic structure is inscribed. We just follow from this phase. Well actually what one needs to write not just up but it should be symmetrically down. Because you have this e and even in complex space. But for simplicity I'm just using this type and many people are using just this type of picture for the wave packets. Questions. I will use this picture several times so it's better if you understand and ask me questions now. Please give the microphone. Or you can give me, I will run. I just have some exercise. To just say that the three neutrina mixing is reducing to two neutrina mixing but I didn't actually understand why you say that. So we know that there are three neutrinos, right? Now I'm saying that in various cases and you should consider of course evolution of a whole system which includes certain neutrina. But for instance in the first approximation since one three mixing is small in the first approximation you just can take it to be zero. What we will see later that mass differences and you probably saw already in the one of the first pictures are quite different. So there are three states and the mass split is different. There are hierarchy of splits of the masses. You will see that in this case you will have two kind of scales of oscillation phenomena. One is short scale relatively. It depends on the mass. Another one is big one. So in some experiments which we have short baseline then you can neglect long baseline effects and therefore the problem again reduces effectively to two neutrinos. More questions? Now we see here also that each of these wave packets has two companies, red one and green one, right? This is because we have electron neutrina and mule neutrina and you see this expression which just corresponds to such a relation in this mixing matrix but you can invert these expressions and express new one and new two in terms of new E and new mule, right? Which means that each of these eigen states, mass eigen states is again it has a composite flavor, not pure flavor and this is the essence of mixing and we also saw this picture last lecture when I showed the spectrum of neutrinos, okay? So this is complete description of electron neutrina. Now you see here green part here and green part here and so you can wonder again why this mule neutrina appears into electron neutrina and the point is that in the production point the phase difference is zero so this is the phase difference. Then these green parts which will cancel each other and so what you will see is just electron neutrina so they will have destructive interference and you can just check that if you write explicitly expressions that this constellation occurs and this is because you see in electron neutrina new one and new two are in phase. This is sine plus and here the minus sign so if in one case you have constructive interference the red parts here sum up then green parts cancel it. So this is in the moment when the phase is zero. Now what happens if you produce this state? So this system of two wave packets starts to propagate according to what is written here according to this it will start to propagate. The shape factor doesn't change the form doesn't change if you neglect a spread of the wave packets, okay? And also the size of the wave packets doesn't change the size just is determined by this cosine and sine. So in the process of propagation the only what happens the only what happens is phase difference changes between this wave packet and this wave packet and this change of the phase difference is again due to difference of the masses you see we have here different group velocities since we have different group velocities here the velocity of this packet and this one is different so actually if they travel a long time they start to shift with respect to each other so upper and down wave packets and at some point stopped overlap and this what is called the loss of propagation coherence I will come to this question a bit later Now what is this oscillation phase? This is the difference of the phases of these two wave packets So each of these phases is given by this expression so this is the energy, this is momentum this is energy and momentum, average energy and momentum in each packet, right? So this is the way we derived such a formula remember we introduced average momentum of particle and then defined also the energy which corresponds to this momentum So this average energy and average momentum is determined by the production process so you need to compute actually so you consider the process of production of neutrino and you compute what is the shape of these wave packets and you compute what is the average energy and average momentum so we are using this dispersion relation Now how we can write this phase difference? This is due to difference of the energies and difference of momentum so this is phase difference in general and I'm not assuming equality of energies and momentum which is confusion point in many textbook derivations So suppose I take a given delta E actually gain it is given by difference of these energies average energies in each wave packet then delta P is given by this expression so it's just delta Dp over De delta E this is Taylor expansion Dp over Dm squared delta M squared just Taylor expansion because P depends both on energy and it depends on the mass squared Now what is staying here is nothing but inverse of group velocity we already defined this here and so this means that this expression can be written in such a way and then you can easily compute of course this derivative just using this and then you get such an expression for delta P which corresponds to a given delta E So then we insert this delta P in such an expression and finally we get this formula for the phase difference for oscillation phase There are two terms here one term is given by delta M squared over two energies and x is the distance neutrino traveled and this is kind of standard term standard oscillation phase which you see in all the textbooks but there is another piece here but it has very interesting feature so first of all it's given by this group velocity time minus x and this expression is actually varies from zero up to the size of the wave packet because the tails are exponentially die so the typical value of this expression in brackets is just the size of the wave packet the width of the wave packet and here is delta E difference of average energies in the wave packets over the group velocity so you can put here just one and then this delta E should be proportional to delta M squared because if mass differences are zero then you should not expect any difference of energy so it should be proportional to delta M squared so typically this delta E is delta M squared over two energies and now things which you see here is nothing but the phase which is acquired on the size of the wave packet here what is staying x and here is sigma x and if the size of the wave packet is very small then you can usually neglect this term and so you get the standard expression for the oscillation phase so this is important issue because there are many discussions about the oscillation phase some people are finding factor of two now it's kind of more quiet period but this is important to understand so this is what we are getting here for the oscillation phase to a good approximation it's given by just this expression but there is some small correction and actually this small correction also reflects a possible oscillations within the source because source of size of the source determines for you the sigma x the size of the wave packet and then to some extent that reflects the oscillation phase which is acquired on this small distance of the size of the wave packet so the crucial point that this phase is proportional to delta M squared and inversely proportional to the energy of nutrient and now let me show come back to this pictorial stuff and see what happens so what happens the phase difference changes so oscillation phase is proportional to x nutrient or time so nutrient propagates and the phase difference changes and what happens is you have in initial moment something like this then in the next moment you may have so after a while you will have this one when the phase difference becomes pi so that's phase zero in the case of stop of oscillations stop to oscillate Giovanni okay it works oscillation stop to oscillate nutrient stop to oscillate anyway so that's in the case of phase of pi you see there is a shift of the phase relative shift of the phase and now in such a situation these green parts will not interfere destructively so they will interfere constructively and so this means that you will have appearance of nutrient in your nutrient flux so that's in fact you see I just put relatively I mean that's that what happens for two different phases but it's smooth process so that the phases are phase difference slowly increases then after a while you will get again the phase you will get the phase 2 pi so which means that you are coming back to the original state and so that gives you the periodic process of nutrient oscillations now I didn't mention detection actually detection plays very crucial role it's actually quite symmetric production and the detection and what do you see in the detector actually depends on the properties of your detector in particular that kind of interesting phenomenon that you can even restore coherence even if it lost on the way of nutrient traveling if you ask me I will put it on this more and actually you can introduce some kind of generalized wave packet which takes into account properties of production and properties of the detection and in this factorization approximation it works well now let's come to oscillation probability and so what we are computing we have created this state which is written here and in two neutrino case it was written before and now we can compute what is the probability that we will in detector find an electron neutrino so we compute this matrix element right so which means that we are computing c now I am writing this in terms of mass against this nu1 plus sin nu2 very often I will use c and s instead of cos sin and sin and so that should be understood that this is cos sin and then I have here this expression which I had before let me show you it was in the previous slide just not to repeat again so this one so I need to insert this expression and here you see cos here in front of nu1 sin here in front of nu2 and when I insert here such an expression what I will get using also that nui nuj they are orthogonal so it's delta ij so which means that nu1 nu1 will give me 1 and nu1 nu2 give me 0 and so I just need to pick up the terms which are with the same labelling of eigenstate so what I will get cos sin2 it comes when I put this one and then g factor g1 plus sin2 g2 I will meet here these arguments and then I will write here i to the phase I don't know if you see well here do you see this or not I should ok so that is expression which is what I am getting for the amplitude and I put here the phase so I put out the factor i phi1 and I can always do this because eventually I am computing moduli square so the probability is given by moduli square of this expression and probably I should not write because it's written already in this so that is expression moduli square this and make integration and now integration depends on how I am organizing my experiment and so what I did here I just integrated over dx so I have considered the probability in a given moment of time but you can do the following you can just fix x but integrate over time because usually you are not measuring time you get the same result so this is expression which you get eventually for the probability this is integral over moduli square of this expression ok and now what you get the following I didn't mention this but I should now to stress that of course this way function should be normalized right they should be normalized in a way that psi x t ok moduli square and if it fix t and make integration over dx it should give me one right and therefore from here I am getting that if I can do the same things with this factor g again integrate over dx of g factors then I should get again one because that differs from psi just by this exponent right so that moduli square of shape factor integrated over x again should give me one and this simplifies a lot our expression because here you see these two terms and they come when we square this and then make integration then take into account that the integral of this is one the second term this one comes when I square this again make integration this produces sine for the force theta and there is no shape factor here and there is interference term which is kind of cross term here so it's proportional to 2 sine square theta cos sine square theta then the phase appear here cos sine phi right the oscillation phase and this additional factor which is the integral over the product of two shape factors ok now what is this product of these shape factors if they overlap if there is no separation of the wave packets then again you get one and so in this case your expression the probability is given by such a thing so I just put instead of this force powers I express this as one minus two sine square theta multiplied by cos sine square theta and I get this expression out of this immediately if the integral is one and then I can further transform this and I'm getting such an expression which is standard oscillation formula so the point is that of course you can get this oscillation formula standard oscillation formula using plane waves or point like presentation of neutrinos but then you met this conceptual questions you can ask also questions why we should take equal moment or equal energies there is nothing here no assumptions here but you see then how you get what is behind this standard oscillation formula for instance behind this is assumption that you have overlap complete overlap now if your wave packets separated completely then this integral is zero and then you will get expression which is just cos sine to the force theta and sine to the force theta and this is nothing but expression of averaged oscillation probability but of course there are intermediate steps when you have partial overlap so let me make also comment you see you get this expressions which actually do not depend on precise shape of this G in two cases either you have complete overlap then it doesn't matter what is the shaping you need do not need to make computations of your production process or when you have complete separation of the wave packet again things are simple in most of the cases fortunately we have such a situation so either you have no substantial shift of the wave packets and then you get this expression your things are averaged if you want to pick up kind of intermediate situation then you are becoming sensitive to the shape factor questions oh let me make final comment so this oscillation phase I just reproduced again the same expression you saw before and one can introduce the oscillation length which is the distance over which the phase becomes 2 pi which means that the system is back to initial state and this oscillation length is given by this expression it's just you equate this so this is the phase so you equate this to 2 pi and so you get this expression oscillation length is given by 4 pi energy over delta m squared okay questions that phase why it's disapear now so again that is the assumption that I can neglect oscillation effects on the distance of the size of the wave packet assuming that wave packet is short enough and usually we deal with sizes of the wave packet which is maybe 10 to the minus 6 centimeters maybe 10 to the minus 11 centimeters and the oscillation lengths are kilometers or meters kilometers which is much much bigger which means that with very high accuracy you can actually neglect however in some very specific situations you may still take into account this additional term so this is neglecting this additional term actually you see explicitly what are really assumptions behind this usual oscillation formula let's see so which one the oscillation length yes because this is m squared this is inverse of the mass and this is energy here so it's one over energy which is the length I don't get it to what energy it corresponds to that energy we use there and in previous in phase and so on and so forth so look you are right because in this system there are even two average energies right even more energies because your wave packet is actually collection of the energies so even I have no difference of energies I just put one energy if there are some differences energies important in some then I write this explicitly because this difference of the energy is extremely small it's much much much smaller than the energy itself ok so it's just to avoid this many indices I just put E which means that kind of one of the energies in this wave packet and the situation is the same for other energies in the wave packet ok in the oscillation phase we can see the energy dependence yes it seems to be that it is useful to actually observe the sinusoidal pattern of oscillation by changing the energy of nitrino beam was it done actually in the experiment absolutely yeah so I will discuss this at length this is kind of next topic of what I'm going to speak you can observe oscillations in two different ways either you can fix the energy or some energy interval and look how things change with the distance but more often we are doing precisely what you are saying fix distance because you have source and you have detector and fix distance and then you study what happens for different energies and then you will see oscillatory pattern but in energy scale yeah my question is just in continuation to the question asked by another participant you were mentioning that most of the problems in the nitrino oscillations you said that it can be reduced to two nitrino problems the main motivation behind this reduction is just that one three mixing angle you mentioned that it's but as far as I understand the hierarchy in the nitrino mixing angles is not that small yeah it's not that large right so I just don't want you know I'm lazy and I don't want to bother too much with long formulas so actually what you need to write you need to write three terms in the expression here right so it's three terms in the expression here and then you will have how many at least three interference terms just to avoid this is for simply I want to kind of to tell you what are conceptual issues but then I will show you the results and analysis is done in three nitrinos you are right in many cases you cannot neglect this third mixing and it is important and especially with present day accuracy of measurements so this is crucial to make how to take into account all three all three oscillation modes yeah but in principle conceptually everything is here so what I was telling you you will have three wave packets then you will have interference of three wave packets then each of these wave packets will have three colors and then you need to consider all possible interferences that's it but it's just lengthy but the same thing then you will have two different oscillation lengths because you have two different delta M squares shall we continue so what I have shown you before was survival probability so we have produced electron neutrino and then I was asking what is the probability to find again electron neutrinos in my detector but you can also find what is the transition probability or appearance sorry and then it's just one minus expression which you saw before and so this is expression for the transition probability it's given by sine square theta two theta and then there is oscillatory factor and you see here that the depth of oscillation is determined by mixing angle by sine square to theta so that was if mixing is small then you will have oscillations with very small depths if it's maximal then you have maximal oscillation depth so there are some repetitions of what I have said already you see all the complications are absorbed in this derivation in normalization or reduced to partial averaging of the oscillation pattern now let me repeat again oscillation is the effect of phase difference change you will see some other effects which have some other degrees of freedom but here the only degree of freedom which is operating is the phase and phase difference so oscillation is the effect of phase difference increase on monotonous phase difference increase with distance and the phase here is just proportional to the distance over time of propagation at mixtures of mass eigen states in a given propagating state do not change so the sizes of the wave packs and the shape factors do not change here at least in our approximation and also flavor composition of individual eigen states is fixed by just mixing angle so this is inversion of formals which I have shown before now there are some slides with stars here which I don't want to go into discuss but if you ask me later I can answer I can come back to this so one is this explanation of this loss of coherence in the propagation and what is interesting you can actually restore in spite of the fact that your wave packets arrived already separated in principle the detection process can again restore this coherence and you may see interference picture but for this you need to have very good energy resolution very long time of detection process coherent time and you see it's like the case of the pendulum suppose you have pendulum okay and then one wave packet arrives so it stays here and knock it so it starts to you know to vibrate now if you have friction and when the second wave packet arrives it stays again it again knock it and vibration will be independent on what you had before but suppose you have very small friction very good impedance so the first wave packet came knock your system it start to vibrate then when second arrive it still continue to vibrate and then effect of the second heat will depend on what happened before so this is more or less how coherence can be restored in the detector so you need to have very long coherence time of detection for this another important thing which I want to just mention is the equivalence of the pictures in momentum representation space and configuration space so what I have discussed is the pattern in configuration space right how things developed in the X in coordinate in usual coordinates but in quantum mechanics we have this equivalence but you can consider all the things in just an energy momentum space and there is kind of one-to-one equivalence of what I have discussed here and what you can get in this energy momentum consideration and in this connection I can just mention this famous Tadolski theorem who was saying that actually if you want to compute observational effects in your detector you can forget about wave packets and just to make integration over energy of the produced spectrum and energy resolution of your detector and you will get equivalent result which is true if you ask me I can elaborate more and finally it's another way stars this is what is called the spread of the wave packets and the spread is given by such an expression it's mass square over energy of neutrino cube this is the size of the wave packet and this is the energy space and this is the length of propagation so what happens is you had such a narrow wave packet and it becomes very wide what's interesting that when this wave packet becomes very wide actually there is a loss of coherence even within the wave packet so it becomes more like a classical and the effect is nothing but related to the fact that higher energies will arrive first so neutrinos with higher energies will arrive first and then lower energies later in the wave packet you have the spectrum of energies some interval of energies and when your system travels long then the first what is arriving high energy pieces of your wave packet so suppose we kind of solve all these problems with wave packet with coherence with everything and then the problem is reduced to some extent like problem forget all these complications and now let's do the following so let's take this vector with three neutrinos so it's electron, neutrino, muon and tau and evolution of these neutrinos, at least in flavor space can be described by such a question which is just a usual Schrodinger question where Hamiltonian is given by mass matrix multiplied by mass matrix dagger over two energies approximately and plus I have written already in general some potential so let me explain this expression because we will use this equation so this part is vacuum part what we call and actually this is nothing but generalization of such an expression for energy in ultra-relativistic case you will have energy equals to P plus M squared over two energies just for one particle now we have three particles and generalization is straightforward it is just then instead of M squared you will have mass matrix squared is three by three matrix multiplied by mass matrix dagger and these two energies okay and actually this part can be written in terms of mixing matrices and diagonal matrix you can describe this as our CKM matrix this is CKM matrix dagger and these are masses of squared of neutrinos and this is in flavor basis now in many situations I will just assume that mass matrix of charge is diagonal so we are working in flavor basis this is already reflected by the fact that I am using here electron, mu and tau neutrino functions now if neutrinos are propagating in medium and they are interacting with medium then you will have some additional term in your Hamiltonian and this is V which is staying here and I will discuss this later okay so essentially now we need to solve these equations for different situations different situations means that different distances and different shape of the potential etc now very important thing and I again want you to really to pick up this because things become very simple if you use this graphic representation it is very easy but you see many things will be understood in a very simple way if you understand this so we use wave functions however we can introduce polarization vectors so now I am again back to two neutrinos, just for simplicity of course you can generalize this for three neutrinos so we introduce polarization vector which is just this wave function the 2 by 2 this is sigma matrix, Pauli matrix and again this psi this is definition if you want right now explicitly the components and now we have three components because we have here three Pauli matrices so these three explicitly components are given here so that it is a real part of the wave function of electron neutrino put just a neutrino sign without psi so that's meant that this is wave function of a given neutrino so three components are here written here is a real part of this product new E dagger new here, tau then imaginary part and then that component is just given by wave function of E dagger, wave function of E minus one half evolution question which I have written already before is given by such an expression and in two neutrinos case it can be rewritten in such a form it's just nothing more complicated but when you express your Hamiltonian in terms of mass eigenvalues and mixing angles and in this case it's just one mixing angle then you get expression for this B which is written here it has three components this is one, two pi over oscillation lengths and again in general I'm writing here the oscillation lengths in medium it is valid in vacuum or it also valid in medium I want to have more general immediately so three components this is the first sine two theta mixing angle zero and this cosine theta it's just expression for the Hamiltonian in two neutrinos case rewritten in terms of mixing parameters and mass eigenvalues so let me so that's I'm using for this mass metric such an expression so these are mixing matrices and in two neutrinos it's just expressed in terms of one mixing angle now what I'm doing is the following I take this P and I want to find now evolution equation for this polarization vector what I'm doing I'm just differentiating this equation and I will get two terms so why I'm differentiating this side dagger and the second one where I'm differentiating this one side and then I use this evolution equation to express these derivatives and I'm coming to such an expression now see this and I'm asking question but if you're really interested I mean that's 5 minutes derivation it's nothing complicated so do I understand how it was derived so this is the definition I differentiate this P over T so I get expression which actually is derivatives of psi and then psi then here's psi and here's derivative of psi and I use this evolution equation for psi yeah sorry oh yeah so in terms of just wave functions right yes so in many cases we are doing this but let me tell you the following so you will see how things becomes very visible and simple if I use this polarization vector so you can do this of course actually this is the way to solve using density matrix actually components of this P are the elements of density matrix but in some cases like collective effects in supernova which when you have also scattering of neutrinos or anti-neutrinos I cannot imagine it's so complicated to solve in terms of wave functions using this formalism it becomes much much easier and you will see these examples so what is this equation tell me so if it resembles for you something so what is this yeah so this expression for spin precession of electron magnetic field if I identifying B with magnetic field so this is the expression for B it's 2 pi this is oscillation length and it's sine cosine then my vector P according to this equation we will just precess around this vector B so this is beauty of physics when you have absorbed the same phenomena sorry when you have different phenomena and actually there is the same physics behind and here we see this analogy so we developed this to describe oscillations but this is nothing but precession of spin of the electron and here you see this picture so you have this P vector and this is the space in which this vector is moving now precession occurs around the vector B and actually vector B if you saw this expression for B it's 2 theta angle inclined with respect to this axis Z now this direction of axis Z correspond to pure electron neutrino state you see what is here is PEE probability to find electron neutrino minus 1 hull so if this PEE is 1 which means that you have electron neutrino then you will have this axis you have this vector here it's just 1 hull ok so if you have PEE 0 which means that your electron neutrino completely converted to another neutrino species then your vector is down so that's notation so up means electron neutrino and down means millon neutrino or whatever it is and then magnetic field vector is inclined here it has angle 2 theta m so this axis is determined by mixing angle and what happens is the following you produce electron neutrino and so your neutrino vector in initial state is just like here it's directed in this way but then the evolution will be just precession of this neutrino vector around this axis on the surface of the cone and the cone angle is just 2 theta m or 2 theta in vacuum so then if you have this evolution of course projection of our vector on the axis z changes periodically and this is just analogy of oscillation because the probability of PEE is just given by this projection on the axis plus one half so if you want to find what is the probability to find electron neutrino you need to project your vector on to this axis z and add one half questions yeah that's while defining this polarization vector we are using the two flavors that is new E and new tau that is probably if you are considering the atmospheric neutrino oscillations again I will tell you how to apply this for specific systems but for me it can be new new tau I can do this and then it will be for atmospheric neutrino the main channel but many others so I will apply this type of pattern for different channels and different situations but you can consider this as a kind of toy model if you want to explain to you this graphic representation and now what's going on oscillation is just this process our neutrino vector I will just show you and then you can ask again so that's the procedure that's how things so the vector is moving around this axis so please just go back to slides where you generalize the Hamiltonian for matter effect I will discuss matter effect next so that will be in details that's one no yeah this one so when you write the Hamiltonian you just say E is P plus M squared by Y YC so I understand that this M squared by YC is generalized to MM dagger by YC and this interaction potential comes due to the matter effects but why there is no P in matter effect good question yeah I have forgotten to tell you good thanks here is the difference some terms here which are different for different neutrino species so if you have some contribution to this Hamiltonian which is the same for all three neutrinos it will not produce any effect so here we simplified many things in particular here we are considering that P is the same for all three neutrinos and then we can remove this term but we can do the same with energies or even more complicated case but again the rule is that if you have something which is the same for all three neutrinos species you can remove it from your Hamiltonian it doesn't produce any phase difference at least in this situation because if you have no inelastic processes that can matter but not here so I just want to finish this part by telling what is the experimental setup for the source then usually we have some near detector to control the flux which is coming out of production region and then we have some far detector where we detect the neutrinos and there are different experiments which aimed at detection of the same type of neutrino which is produced so we call this disappearance experiment or the experiments where detector detects neutrinos of other type and so then we can actually determine what is disappearance probability or disappearance channel that we have already discussed the phase has such an expression and therefore we can search for oscillations or oscillatory part either in distance changing the distance changing x or changing the energy so there are two ways to search for oscillatory effects so now I'm starting a part on the matter effects and I will use what I have already developed for y-cam oscillation so it will be much much faster if there are still some questions for the first part please ask me because I will be using what we have discussed already not repeating various things good so tell me, suppose you have light and light is propagating in transparent medium so what happens we can just say spherical ball of glass and light is propagating so what happens which kyber phenomena you expect to see this is refraction phenomena so you have refraction index in transparent and you may have some reflections you may have internal reflection inside your ball or if your system is such that the refraction index is different for different polarization even have rotation of polarization of the light in your medium now imagine that the earth the sun and many other things are just this glass balls for neutrinos neutrinos have no negligible inelastic interactions and only what happens is just refraction phenomena the earth is some this transparent ball for neutrinos at low energies right and so the only what happens is this refraction phenomena and so you can estimate significance of kind of refraction phenomena and inelastic processes by just comparing imaginary and real parts of the amplitudes so you can consider scattering amplitudes and then imaginary part is responsible for inelastic processes and the real part of the amplitude is precisely giving you refraction phenomena refraction phenomena and so I don't know if this formula exists here no it's not but the ratio of this imaginary and real parts is something like s total energy in the center of mass over the mass of W boson squared so if you have lower energies then you can neglect inelastic part and the only phenomena you see is just this refraction phenomena so we will discuss now refraction phenomena of neutrinos and the first person who started to do this was Lincoln Wolfenstein who unfortunately died just two months ago so one of the person who actually has driven all these developments now elastic forward scattering this is what actually is responsible for refraction phenomena can be described by potential so you can imagine that neutrino light also neutrino propagation medium and then it just suffers from forward scattering and that can be described by just potential so the medium creates potential in which a neutrinos propagate and actually refraction index its deviation from one is just given by this potential over the momentum and for instance for 10 MeV neutrinos inside the earth this N minus 1 is 10 to the minus 20 and inside the sun is 10 to the minus 18 and what is important here again the difference of the potentials as always the difference of the potentials is important not just absolutely well and here we are speaking about difference of the potentials for different neutrinos species in particular for electron neutrino and for muon neutrino, electron neutrino and tau neutrino and so we need actually to look at the interactions for electron neutrino and muon neutrino and the only diagram the only process which in standard model gives this difference is this one scattering of electron neutrinos on electrons due to charge current all other diagrams are common they are the same for muon neutrino and tau neutrino so this is the only process which gives the difference and we can compute what is the potential which is produced by this type of scattering and so this is given by G Fermi we are at low energies, we just use four fermionic Hamiltonian the square root of 2 and of course it should be proportional to the density of the electrons you can actually even use this dimensional arguments to derive such a formula because you know the bigger amount of density of the electrons the stronger refraction phenomena at least in some interval it's just linear and G Fermi is giving you the strengths of interaction if you are clear and you can use dimensions nothing more can be essentially written I have here more detailed derivation but I will not focus much on this if you want you can go through these slides or ask me, essentially you need to compute effective Hamiltonian when you make kind of averaging over integration of the wave functions of the medium and that produces for you the density of the electrons now suppose we have now neutrino propagating a medium and in this case you need to add term to your Hamiltonian which corresponds to interaction of neutrino is medium so we add to our Hamiltonian this V term right so we have neutrino and they interact in medium there is some energy associated to this interaction so we need to add to Hamiltonian which is the energy right operator so we have the zero part and V now let us compare how we move from vacuum case to the case in matter so in vacuum we have this H0 now we have this additional term we have different Hamiltonian the eigen states in vacuum are new one and new two eigen states are the mass states the eigen states of Hamiltonian are mass states mass states propagate independently so if you produce mass states it will propagate it will not transform to anything now in matter because of the presence of additional term you will have some other set of the eigen states which differ from mass states so in medium some other states propagate independently if medium has a constant density and of course eigen values change so in vacuum we have this eigen values which is determined by the mass of neutrino and energy and here we will have something else I will show you expressions for this now how we define the mixing in medium in vacuum mixing connects flavor states with mass states in medium mixing in medium connects again flavor states with eigen states in medium well actually what is interesting and I will discuss this later that mass eigen states start oscillating in medium so if you have just a new one state with definite mass and then it enters medium it starts to oscillate because it's not eigen state in medium and recently super k collaboration has published kind of about 3 sigma effect of so day night effect on solar neutrinos and it is related to oscillations of mass states inside the earth so it's not just abstract things we actually observe now this effect now this is again evolution question with the parts which I have explained already the matrix of potentials is given here again in 2 by 2 case so it's diagonal and this v e is precisely what you saw before square root of 2 g fermi and the density of the electrons so it's important that this matrix is diagonal and again the difference of the potentials is the matrix here in general you have here some v electron general here is v mu but the difference is this v e so you can subtract common parts this is for oscillation phenomenon however if you want to discuss some kind of internal refraction and many others in reflection then you need to take total potential for oscillations the difference matters evolution question in terms of mixing angles delta m square and here you see this v e now you can use this equation and you can diagonalize this and find what is the mixing angle so you find this relation between flavor states and eigenstates diagonalizing Hamiltonian and this is expression for mixing angle in a medium so what is important here sine square 2 theta m is proportional to vacuum and you have here denominator kind of interesting dependence on parameters here you see this 2 energy potential over delta m square square and then sine square 2 theta so if potential is 0 then denominator is just 1 and you get come back to vacuum expression for mixing angle now here is interesting factor because it's some energies and potentials it can be 0 right and this size of the potential is given here this is what is called resonance condition if this condition is satisfied then your mixing angle in medium becomes just 1 because this term vanishes and you have sine square 2 theta here and here and so it's just give you 1 so this is the resonance this is resonance condition this is resonance in your 3 in the mixing ok now you can also find immediately what are eigenvalues when you make diagonalization of your Hamiltonian and the difference of the eigenvalues which is important here again is given by such an expression which is essentially square root of what is staying here this is delta m square over 2 energies this is what you would have in vacuum and these are additional terms and again this term becomes 1 when potential is 0 now let me show you some pictures so here you see dependence of mixing angle in medium or on density or energy are actually in this combination this is 0 and here are for 2 mixing angles now since we are now generalizing for 3 in the case for small mixing angle like 1, 3 you will have this sharp peak resonance peak for large mixing which is realized for 1 to mixing very wide so this is what we call resonance what is the meaning of the resonance you can rewrite this condition for mixing angle to be 1 in matter in such a form and it gives you clarifies the physical meaning of resonance the oscillation lengths in vacuum equals in the resonance so called refraction lengths multiplied by cosine sorry I have forgotten to introduce this refraction lengths it was in the first slide here so let me come back so we introduce potential and then we can introduce refraction lengths which is just 2 pi over v and the meaning of this refraction lengths is this is the distance over which the phase, additional phase associated with interactions will be 2 pi now 2 contributions to phase one is from difference of delta m squares which we have discussed but also it will be contribution to phase due to interaction with medium and the distance refraction lengths is the distance over which we get this additional 2 pi phase so the meaning of resonance and if mixing is small is that refraction lengths is approximately equal to oscillation lengths in medium so the sense is very clear and physically simple so you have medium and it is characterized by frequency which is 1 over L0 so it's essentially frequency and you have the system which is characterized by its own frequency which is 1 over L new in vacuum so resonance when these two frequencies coincide so that's common meaning of the resonance or vacuum oscillation lengths is approximately refraction lengths this when resonance happens questions? so you go back to the resonance condition that you said that yeah so here is my confusion so the left hand side this V is completely determined right we know the electron density or something but the right hand side there is some ambiguity what we choose for delta m square and cos 2 theta so do you guarantee that such a thing in the case of for instance solar we precisely use this question to determine what is delta m square solar but then you have to choose something for cos 2 theta also right so this is the way and I will discuss this how we actually we use these relations to determine the parameters a priori we don't know so when it was kind of introduced we didn't know what is delta m square we didn't know what is this theta angle actually for a long time in for solar neutrinos people discussed that maybe mixing angle is very small or maybe large and different ranges of delta m squares have been considered yes okay inside the supernova there are even two resonances which are related to delta m square and delta m3 are realized it's even more complicated situation now this is the picture which is complementary this is dependence of eigen values of Hamiltonian for large mixing and for small mixing and near the resonance the splitting becomes very small and you can find this from expression which I have written for you the difference of two eigen values so this is nothing but level crossing phenomena which is very common in physics and appears in many places and so it's also very simple way to see how transitions occur again this is dependence of eigen values of our system on density here is a density or energy nobody ask me what is this negative part what is negative part so you see even in the previous it was negative value here you see it's negative it's zero and what's interesting that this is continuous continuation to this negative part so what is this negative part of this diagrams so this are the scales is n over e or v over e if you want I don't hear you sorry if you have holes in the Dirac's negative electron density it's simply just for neutrino channel so what is interesting if you have resonance in neutrino channel there is no resonance in anti-neutrino channel and actually this plane corresponds to negative value of v I put here n but more correctly I need to put potential potential might have different signs so it can be positive and negative it has opposite sign for neutrinos and anti-neutrinos so this part of the plane corresponds to anti-neutrinos to negative value of potential so this is this level crossing scheme for three neutrinos and let me just this last topic which I want to discuss what was going on with with oscillations actually if the density is constant if you have uniform medium then everything is exactly like in the case of vacuum oscillations you have precisely the same expression for oscillation probability with the only difference that now you need to put here mixing angle in medium and here you need to put oscillation lengths in medium and oscillation lengths of medium is 2 pi I don't think it's us it's here 2 pi over difference of eigenvalues this is general expression and this is reduced to our usual expression if you are in vacuum and a maximal effect again when this sine squared to theta and this is the resonance so this is how the oscillation lengths in matter depends on energy so it increases with energy at lower energies then it becomes very big in resonance and then it approaches to refraction lengths so oscillation lengths also depends on energy here this is again expression for refraction lengths and this is oscillation lengths and so now it's just immediate consequence of what we have discussed resonance enhancement of oscillation suppose we have the source which produces some spectrum of neutrino neutrinos propagate in medium with constant density and then detected here then what you will observe in your detector so you will observe then the ratio of fluxes suppose you have electroneutrina in your detector detects electroneutrina then the ratio of absorbed flux or initial one will have this oscillatory picture which is in green line so this is in energy scale and at energies which are close to resonance energy you have oscillations with maximum depth so this is oscillation in energy scale so this is for two different widths of the layer so you have different patterns now if if mixing is small then the resonance resonance is narrow and you get your ratio of probabilities of this type or this type depending on the width of your layer so this is actually what is realized in the case of propagation of atmospheric neutrinos or accelerator neutrinos inside the earth and so this is graphic representation of this effect when you have resonance then the mixing angle is pi over 4 right so then two mixing angles is just give you pi over 2 and then what happens your cone becomes completely open so the axis two mixing angles which determine the axis of the cone this is the b is pi over 2 and then the oscillation will proceed just like a rotational precession along this axis and of course then you have maximal change of the projection on the axis z so if you have the spectrum then for different energies neutrinos will evolve around move over the surfaces of different cones this is for energy one so you will have such an evolution in resonance it will be evolution in this way then it will be like this and precisely in resonance evolution will be all this type and probably I stop here thank you