Week 8: Lecture 40: Neutrino oscillations

This lecture we will talk about uh oscillations in a different sector in lepttonic sector. uh in the last uh uh or last to last we looked at uh the oscillations in the hydronic sector where we looked at K and neutral B meons. Uh here we'll look at neutrino oscillations. So first of all we'll recap some recapitulate some of the properties of the neutrino. Then we'll look at the experimental data that pointed towards the possibility of neutrinos changing flavor. Uh then we will talk about two generation mixing, three generation mixing. Uh and then finally we'll also see is there a possibility of a fourth generation and what are the some of the open problems in nutrinos in the field of nutrinos. Okay. So uh summarizing some of the properties of nutrinos, it is a spin half entity uh like the electron or muon or the tow particle tow. It has an electric charge which is as far as we know consistent with zero. But experimental limits uh terrestrial experimental limits put the electric charge the upper bound as 4 into 10us4 of a electronic charge and cosmological uh bounds are much more stringent says 2 10us 15 of electron. The magnetic dipole moment uh is less than uh 28 into 10us 10 * a bore magneton. The bore magnet of course is crossed by 2me C. Uh the mean life by the mass is greater than 300 seconds by EV from a reactor experiment and 7 10^ the 9 second by EV from uh looking at solar neutrinos. So of course the solar neutrino bound is much more stringent as compared to the reactor experiment because the distances are much smaller in terrestrial experiments as compared to if you look at nutrinos coming from the sun. The total number of nutrinos with mass less than about half the uh Z0ero mass which means nutrinos which have less than 45 GV by C^² then if you look at the number of such nutrinos uh from the width of the uh Z0 resonance and uh the partial width to unseen particles uh such as the neutrino uh then you can infer this number to be 2.9963 plus -.074. So this is like seven parts in 3,00 which means about two per mill. Uh with two per mill accuracy we can say that is consistent with three the number of neutrinos with masses less than 45 GP by C² is three. the effective masses of uh the electron neutrino is now pushed to less than 0.45 45 EV by C^² mainly because of the Kine experiment. Then the the limits on new mu and new toao are given here. They are much much higher than the they have not changed over so many years because it would be very difficult to decrease that bound. Okay. So what are the sources of nutrinos which is important if you want to study nutrino properties including nutrino oscillations. So the sun is of course a source of nutrinos. The average energy is uh of the order of less than on the average 10 to the 10 me or so. Nuclear reactors similar less than about 10 m. Particle accelerators of course can go from low energies 10 the minus 2 GV to 10 the 2 GV uh atmospheric nutrinos uh you can get greater than about 100 MV or.1 GV onwards and there is a whole spectrum uh of course falling is a power law uh supernova nutrinos are between about 10 to 100 me supernova are exploding stars which many of them many of which land up as neutron star eventually. Uh then there are geonutrinos due to the uh beta decays that occur in the chain of decays starting from uranium 2 38 to thorium 232 also some contribution from potassium 40 and so on and finally there are the nutrinos which are the most difficult to detect which we haven't yet detected. So all the other six types of nutrinos from these various sources of course we have detected and made measurements which tell us something about the nutrino itself. Uh but cosmic big bang nutrinos which is like the cosmic big bang photons. So the such uh uh microwave background radiation we have of course detected but in the case of neutrinos we haven't detected them. They're extremely difficult to detect and they have energies of course of the order of 10 the minus3 GV or something like 170 micro electron volts. Geonetinos are similar in energy range to uh of the order of 2 to 4 me or so. [snorts] uh the so this should be actually 2 to 4 uh me uh okay so these are the average energies or the range of energies of these uh nutrinos from the various sources here listed uh seven such sources or seven groups of sources uh the nutrino fluxes uh also vary uh between big ranges. Uh for instance, the sun is the most prolific source that we know of in our vicinity. Uh namely it produces about 6 10 14 nutrinos per meter squared/s. So these are the units me per meter squared per second. Nuclear reactors give about 210 ^ of 13 at a kilo distance of 1 kilometer. If you have a 1 ghawatt thermal reactor, many reactors these days are about 3 gawatt thermal and there are some even more powerful ones which are 5 gawatt. Particle accelerators uh the fluxes range between 10 the minus2 per meter squared per second to 10 the 2 and these are at large distances of course uh of the order of tens or hundreds of kilometers. uh there of course you use the fact that uh if you have a relativistic particle [snorts] decaying it uh there is a kinematic focusing so that is not mentioned here under what what conditions are these uh numbers holding atmospheric nutrinos are typically of the order of 10 3 per meter squared/snovas 10 -2 to 10 -1 per m²/s uh geo neutrinos of the order of uh 2 10us 3 uh per m²ared/s cosmic big bang neutrinos uh as I said 1.7 10 -3 per m²/s however uh I I think I made a mistake there I think I like to check this number anyway you We can calculate this from the fact that there are about 330 uh these neutrinos uh per uh cm cubed volume in all of space. Okay. And since these are light and they are you can you can for an order of magnitude estimate you can take a that they will move with the velocity of light. In that case uh the uh so this is I think this is should be plus 13 because 3 10^ the 10 cm so this should be plus [snorts] uh so anyway in the final uh transparency so anyway there's a range of fluxes which I wanted to point out uh with solar neutrinos being the most uh with the highest flux and uh nuclear Reactors also are very strong sources. Particle accelerators the fluxes may not look large but they you can you can uh they are focused kinematically. So you can do them at large experience. Now the cosmic big bang nutrinos are large in flux uh but uh they are extremely low energy. So they're extremely hard to detect. Okay this is the energy dependence of the nutrino cross-sections. Maybe when I uh so this as you can see the neutrino energy spans almost uh 20 orders of magnitude 10 the minus2 electron volt to 10 the 10 10 the 18 electron volt so the cross-section of course rises uh initially it rises like E squar and then it rises more like E and then this peak that you see is uh the uh nutrino can interact with the uh background material to actually produce the zero bzon. Okay. So [clears throat] that's why you get this peak here. Uh cross-sections ranging from 10 the minus uh 31 miban to 10 the minus1 miban. Okay. This is taken from a review article in 2012. Uh so any case the the solar neutrinos were first measured by Ray Davis uh and this was a proposal which uh Davis and Ball uh whose picture is here they made in the early 50s. Initially they had some problem getting this approval for such a project in Brook Haven National Lab. uh but the the argument that clinched it I was told is that they said we can measure the PP uh reaction going to dutarium plus E+ plus the neutrino and uh there is no way of measuring that in the lab because the cross-section is smaller than some 10us 52 or something square cm so so they proposed to measure solar nutrinos get a handle on the PP cross-section and that is what uh actually uh won the uh day and then they got approval for making such a measurement. So uh the idea was actually due to Ponte Cororvo in 1946 when he was at Jock River Laboratories in Canada. He proposed that if you have a suitable target of chlorine 37 actually chlorine because uh in natural chlorine you have about uh 25% of 37 chlorine. So then what happens is that the new E interacts with the chlorine to produce argon 37 and an electron. And this being a radiochemical experiment, you basically keep on producing 37 argon to about one lifetime or of about a little more than a month. And then you separate out this argon from the uh target material which has uh chlorine in it. and uh then you count it in a low background setup using a proportional counter for instance. So this was the proposal of ponte carvo uh ponte corvo sorry and uh the solar nutrinos are expected to produce about uh 37 argon atom 37 argon one atom of 37 argon in about 2 days of exposure to the solar neutrinos. uh [clears throat] so if you run it like say for a 40 days then you get about 20 uh atoms of 37 argon and then you separate them out. So this is this number is for a 600 ton carbon C24 organic compound. Of course Davis was a chemist but uh he was a extremely good experimentalist and Ball was an extremely good theoretician. So he actually calculated what you should expect and that changed over a period of time because the uh various cross-sections that enter into such a calculation nuclear cross-section also got refined and the models for the sol sun also got refined. So in any case uh this resulted in a series of measurements from 1970 to about uh 1986 when this experiment uh ended and uh this was the data. Uh so the the average production of uh argon 37 was about half a uh atom per day. So instead of two days it was just one atom per day which they found. And uh this is in terms of uh uh so-called SNE uh the solar neutrino uh unit uh capture rate unit and uh this was about uh 2.5 or so 2.5 snooze uh so the one SNO is defined at 10us 36 captures per target atom per second. Now both these theory and experiment and theory had to go hand in hand. So bal doggedly pursued the calculation of the solar nutrino flux reducing the calculation uncertainty to about 20% or so and uh what was found was that the uh calculated value is about 7 uh and a half snooze whereas the measured value is about 2 and a half snooze so it is about a third of what you calculate. So initially there was a push back from the particle physics community which said that how are you sure about the absolute calculation that you have made it involves so many uncertainties nuclear cross-sections the dynamics in the stellar solar interior and so on. So when he said that there is a problem uh he was not initially believed [snorts] but uh as we will see uh there were other experiments such as the uh super uh initially the kamioandanda experiment and then the super kamocondi experiment which also saw uh which also by lowering the threshold to about 5 me over a period of time they could measure these solar nutrinos and they also found a roughly a ratio of half of what you expect. So this is what you expect and this is the error bar on that and this is what was measured. Uh it is about half of what you expect. Similarly there was another experiment a radiochemical experiment. So this is online realtime experiment because the electron neutrino coming in uh scattered of the electron and you actually counted those such events in which an electron was scattered. uh there's another radiochemical experiment uh which is based on the gallium uh target and there were actually two one as a Russian based collaboration the other was a European collaboration uh and also later it was joined uh both these were combined so it became a sort of worldwide experiment the GNO experiment and what they found was again a ratio which was about u uh a little more than 50% % because of course this had a much lower threshold and they were sensitive to also PP nutrinos. This one was sensitive to nutrinos above 5 MV energy and this was sensitive to nutrinos above about 1 m energy. So in any case all these experiments showed that there was a shortfall of measured nutrinos of the electron type to what was expected. So there comes the Sbury Nutrino observatory experiment and this was a proposal of uh Herb Chen uh who unfortunately passed away before the detector could actually uh collect data. Uh but the proposal was a brilliant proposal. He proposed a 1 kiloton heavy water pure heavy water detector uh to measure nutrinos via both charge current interactions that is given here. Newe interacts with dutarium which basically provides the neutron target. So if you remember how the anti-utrino was discovered, it was discovered by uh its charge current interaction with a proton. But a neutrino the charge current interaction goes o through the neutron target and uh neutron is a loosely bound system of the neutron and the proton. So the new e interacts with the neutron to produce e minus and then two protons. So the spectator proton and the other proton is produced via this charge current interaction. And so this is this is one way in which uh the new new can be detected. But if the newi transforms itself to any other kind of neutrino uh something which bakal was proposing to solve the solar neutrino problem then uh you can any type of neutrino can break up a ne d ne d ne d ne d ne d ne d ne d ne d ne d ne d neutron and that goes through the so-called neutral current interaction where you exchange a zero bzon. Uh, of course they also uh this could also detect uh using the elastic scattering on electrons which is what the SK detector did. And here of course you have a two amplitudes contributing the charge current amplitude where a uh a new E for instance becomes a uh E and vice versa. uh or it can just scatter off uh if it is of a different type can just scatter off the electron. Uh so for the electron type there are two amplitudes that contribute and you have to coherently sum those. Whereas for the new mu and new toao it's just one amplitude that comes about. So okay so this is just elastic scattering. uh so all these three processes are detected in the heavy water detector. So uh the heavy the the proposal was to build a 1 kiloton uh detector of heavy water and then surrounded by pure water as a muon uh veto shield for the inner detector. Uh the snow experiment produced a definitive result in 2002. Of course they went through three phases and they had uh you know published their results but the final result uh was that the electron type of neutrino was had a flux of about 1.8 8 10^ the 6 per square cm/s whereas the uh total flux of all types of neutrinos was about three times higher at about 5 10 6 per square cm/s and this is the uh publication this is the proposal of Chen in 1985 and the final definitive experiment u with all the data that they had was reported in 2002 so almost 17 years later. Uh so the water cherov detectors of course this is just a uh uh schematic of how water cherenov detectors work and this shows that if you have a charged particle such as a muon or electron it produces a cherenov ring of light and that is captured by photo multiplier tubes. If it is an electron then this ring is uh slightly fuzzed uh is fuzzy whereas if it is a muon then that goes in a almost like a straight line. So this is a much sharper ring. U this will uh be important when we talk about atmospheric nutrinos. So atmospheric nutrinos are uh produced uh by uh cosmic ray protons interacting with the upper atmosphere. The atmosphere is about 10 km thick. So the protons interact with the nucleons in the uh nitrogen 14 and 16 oxygen. Remember uh that uh the atmosphere has about 80% of nitrogen, 20% of oxygen uh and of course a little bit of argon 1% of argon. So when the proton interacts it produces pions which then decay to muons of course producing muon type of neutrinos. Uh on the other hand when the muon decays it produces muon type of neutrino but also a electron type of neutrino. So there are two muon type of neutrinos for every electron type of neutrino and this is pretty robust. It doesn't depend on the absolute flux of the protons. So uh you expect to see this ratio whichever way you look up going or downgoing nutrinos. Uh what was observed by this experiment by initially the kamyoka and then with much better statistics by the super kamyoka experiment was that if you looked at downgoing nutrinos then you saw this ratio as two but if you looked at upgoing nutrinos this ratio deviated from two very with great significance and it was only one. Okay. So the fact that the measured upgoing nutrino uh ratio upgoing to downgoing it should be neutrino ratio neutrino ratio is not u sorry is not one uh sorry I should remove this the upgoing neutrino ratio of n mu new mu No, this is this is not correct. Uh this upgoing neutrino ratio not being two is the so-called atmospheric neutrino problem. Okay. So it is two for downgoing but only one for upgoing and this is the atmospheric nutrino problem. As I already said that this ratio doesn't depend on absolute cosmic ray fluxes because after all you're taking a ratio. So even if you're wrong in the uh absolute flux uh you still get the same ratio. So the final experiment was this which showed looked at subgv uh electron-l like uh events subgv muon-l like events and then multigv and uh electron and muon-l like events. So while this doesn't show too much of a structure, this is obvious in uh the uh multigv data and you can see that this uh uh the number of events for multiGV electron type uh goes like this with as a zenith angle and it is larger for theta equal to0 as compared to for theta equal to 180°. degrees. So, and the same thing is manifest in the muon sector as well. Uh, sorry, in the electron sector you you see it going one way. In the muon sector, you see a strong depletion in the upgoing things. So, when theta equal to0, it is actually downgoing and upgoing is theta equal to 180°. Okay. plasma. So uh it shows up that this uh for upgoing things it is uh deviating from what you expect uh this is what you expect in the muon sector and this is what you see whereas in the electron sector it more or less matches what you expect. So this as I said already this was the so-called atmospheric nutrino problem. Okay. So the following this uh these two problems which were uh measured u and the explanation was that it was due to neutrino oscillations uh that uh led to a Nobel prize in physics for uh Kajita son and Art Macdonald uh from the Sudbury neutron observatory. So after Chen passed away uh the baton was taken over by Art McDonald who did a magnificent job with his team at uh SNO and uh they solved the solar neutrino problem. The atmospheric nutrino problem was pointed out and then of course the same explanation that nutrinos oscillate was uh solved the atmospheric nutrino problem also. Uh what is meant by nutrino oscillations? So nutrinos can be described in two bases the flavor basis and the mass basis. So these are two alternate descriptions and these two descriptions are connected by a unitary matrix. So the uh the basis uh where flavor states exist the e mu and the toao uh can be represented by a unitary matrix times the uh the mass basis. So this is described in terms of m1, m2, m3 and this is in terms of e, mu and toao for uh this is depicted here in a cartoon where if you start out with a certain flavor then uh you can oscillate between electron muon type and maybe even the tow type. Okay. So for two flavor mixing you can actually derive a survival probability for let's say a new mu remaining a new mu and that is 1 - sin^ 2 mixing angle 2 theta where theta is the mixing angle and sin^ squ some number time a delta m^ 2 where delta m² is defined as the difference in squares of the masses of these two mass states [snorts] and times l by e okay So as you go along L this uh number oscillates and uh that is of course what is called neutron oscillation. L is the propagation length. Uh if you put in the numbers in meters and me then you can calculate how much this survival probability is. If you want to do it for higher energy neutrinos in GV range then this distance is to be measured in kilometers. And this is a derivation of the uh two- flavor nutrino oscillation. So suppose the electron and you consider only electron and muon type of nutrinos uh in terms of their mass states mu1 and mu2 then there is only one mixing angle theta and you can write this down such that these two are actually orthogonal. Uh so when you have a weak decay or a weak interaction you actually produce a weak igon state. uh and uh at a later time uh this uh new of t evolves uh with in the mass basis as new 1 t * exponential minus i e1 t and new 2t uh exponential minus i e2t okay uh so these are the uh the constant or the time dependent factors that multiply ly this uh the mass uh the electron or the muon type of uh uh state vector which is of course a function of time. So if you calculate the uh the uh probability that a new mu goes to new e then this can be estimated from here this can be calculated from here sorry it go it goes like sin^ squ theta where theta is the mixing angle here and times sin^ square e2us e1 t by 2 okay so this is again the same factor 1.27 27 delta M21^ 2 L by E where M21 squ is just E2 - E1 is just this difference in mass squared by 2E. So for two flavor mixing survival probability goes like this and uh appearance probability where you change from alpha to beta goes like this. Okay sin^ square 2 theta* this. Notice uh again the same kind of units here kilometers and GV or meters and me. Uh two important remarks. If delta M² is zero then of course there's no oscillation because this term then vanishes. Then P alpha alpha is just one and this vanishes. [snorts] If uh the the oscillation length uh also uh this will uh vanish uh so there is no oscillation also if theta is zero. So no oscillation if theta is equal to zero as you can see right if theta is zero this term vanishes. So then this is just one and this is zero. The oscillation length is just given by this 2.5e by delta m^ 2 uh where if delta m² is uh this number then the oscillation length is about 10 3 km for 1 GV and for the smaller case delta m² small then this oscillation length is very large. because these are uh one comes in the numerator the other comes in the denominator. Okay. So there are two descriptions of three families of nutrinos the lepton states and the the mass states and connected by mixing matrix which goes by the name of PMNS mixing matrix named after ponteorvo mar nakagawa and sakata. Okay. So if there are three masses from the solar neutrino experiments we know that there is in the normal hierarchy you have m1 squ m2 squ and m3 squ uh so m3 squar is larger than m2 squ and larger than m1 squ but in the so-called inverted hierarchy because of the way the solar neutrino spectrum is understood and we'll come to that shortly uh the uh we know the ordering of m_sub_2 with respect to m_sub_1. m_sub_2 squar is larger than m1 squared. However, we don't know whether this m3 is larger than these two or is smaller. And so if it is smaller then this is called the inverted hierarchy, inverted mass hierarchy. So in the three flavor oscillation of course this becomes things become more complicated. So the uh the probability that a neutrino of alpha type goes to beta type is connected by is given by this expression where delta j is 1.27 times the corresponding delta m² i j and so on l by e. Uh now the interesting thing is that uh there are also these are all vacuum so-called vacuum oscillations. However you can have interactions and the new can interact with matter electrons through the neutral current uh which is common to all the uh nutrinos and so this can change the mixing angle and mass. This was first pointed out by Wolfenstein and Miky and Smeirnoff. uh Wolfenstein is 78 and Mikab spinoff in Russia independently in 1985. So the mixing angle actually is given by this in matter where a is matter dependent. It is related to the firmmy coupling constant to the number density of the electrons and is proportional to e. Uh so in case of uh delta 21 being greater than zero then uh there can be a cancellation of this and there can be a resonant enhancement. So even as theta m even if theta is small uh in vacuum in the presence of matter this can become almost of order one and this is the so-called resonance enhancement of the angle but also the uh the delta becomes the gets changed you get an effective capital m2 and effective capital m1 and this also under goes the resonance effect. So for instance this uh if 2 theta if this expression is equal to a then you only have delta 21^ 2 sin^ square 2 theta. Okay. So uh in any case the matter affects the propagation of the electron neutrino that you started with in the solar core. And this is a pretty robust effect because uh if uh depending on what the uh theta in vacuum is and what the delta uh in vacuum is uh at some density you can have a crossover and this is uh this can be used to explain the solar neutrino uh deficiency that you have the fact that we see fewer electron type of neutrinos than what is calculated. So uh okay I have just mentioned that atmospheric nutrinos have a large range of energies and propagation distances but the approximate values of these neutrino oscillation parameters are given here that this theta12 which is what comes into play in the solar neutrino problem is about 33° is uh not very close to zero as you can see then delta m21^ 2 is about 7.5 10us 5 ev Right. [snorts] So as I said that the the matter effects actually explain the uh the reduction of the electron type of nutrino uh and uh this is a pretty robust explanation. So the MSW uh interpretation is the one which is nowadays accepted in the community. Some of the open questions I should point out for instance is there a fourth generation of nutrinos which is sterile and this comes about because there is a so-called smoking gun signal from reactor antiutrino experiment which see a small shortage at small L by E as well as the calibration runs of the gallium detector which also saw uh shortfall of uh events from what you expect the ice cube experiment of course is ruled out uh in the delta m^ 2 sin^ square 2 theta space using atmospheric nutrinos and this is the reference for that has ruled out a large uh area of this space which is in the sterile neutrino se sector [snorts] but there are other important questions like is the neutrino a direct particle or a meana particle a direct particle would be represented by spanner with uh with four components uh corresponding to spin up, spin down and particle and antiparticle. Whereas meona is just a s much simpler description. It has only two components and that happens because uh if it's a marona particle the nutrino is its own antiparticle. So you only have handedness that's all. Uh you don't have anything like a neutrino and an anti-utrino. [snorts] Uh so this was uh uh you know this was proposed by Mayorana in the uh in the early 30s but we still don't have an answer to this question and we will probably discuss it in some again in some later lecture. How do we address this question? Uh what is the absolute mass of the neutrino? Now these oscillation experiments give you differences in square masses of the massagon states but what is the absolute scale? they they don't say anything. Trishium is the only experiment which addresses the absolute mass scale. Uh the anti- the of course there even the nutrinless double beta decay addresses it in a slightly different way. So there is a different combination which comes about uh but we have only found uh upper bounds for the absolute mass in some combination of uh amplitude squared times uh the mass of the neutrino and so on. So we have yet to measure this and there is uh the katine experiment has put the tightest bounds on this of about 045 eV by c^ squ but there are other proposals which will improve on that by about an order of magnitude or so and at that level we should be able to find the absolute scale. Uh also we don't have experimental evidence for nutrinos from the big bang. So these are the open questions. So we'll now come to the summary. We uh re we had a recapitulation of some of the nutrino properties and we looked at the evidences that could be explained by nutrino mixing and resulting oscillations. We described two we discussed a little bit about two flavor oscillations where we saw the uh the uh reduction of a particular flavor as it propagated or a production of a new flavor as it propagated. And then of course there is the the realistic uh uh description which involves three flavor oscillations. Uh we ended up with an unanswered question. Is there a fourth generation of sterile neutrinos and other questions as well as whether the nutrino is a marina or a direct particle? How do we look for uh the microwave uh not uh the analog of the microwave cosmic ray background which is the cosmic neutrino background from the big bang. So I think I'll stop here. Thank you. [music] >> [music] [bell]