Week 9: Lecture 41: Measuring neutrino mixing matrix parameters

So in a previous uh lecture we saw how nutrinos mix. Uh there are three flavors of nutrinos and so when there are weak interaction processes they respect the lepton number. However, when they propagate, they propagate as mass states and again when they get detected again they have to respect their electron number. So uh so this phenomenon is called nutrino oscillation that it propagates as uh according to their masses and therefore uh at some point where you to detect them you you don't get back the same uh leptton number that you had originally. It's a mix of all three flavors of the nutrinos. Anyway, so this lecture we will uh talk about how you measure some of these nutrino mixing matrix parameters and uh we'll talk about uh you know the mixing matrix which has three angles uh akin to the uh three oiler angles when you rotate a frame of reference. uh so a vector in the unrotated and rotated frames are connected by 3x3 matrix called the rotation matrix in this this is analogous to that so you get three angles however you also get a CP phase so the the numbers are complex unlike in the case of uh rotation matrix and then there are of course three masses so uh we'll talk about over the next uh two lectures on uh how you use solar neutrinos, atmospheric neutrinos and accelerator based neutrino experiments and of course also reactive nutrinos to get a handle on these parameters of the mixing matrix. Okay. So let's uh uh recap what we did in an earlier lecture. So uh in the three f three flavor nutrino oscillation picture of course the nutrinos come in three flavors new mu and new toao. they are created and interact with matter via their flavor states. Uh so if they have a non-zero mass then the mass igon states may not be the same as the flavor state. So you can express one in terms of the other and the uh connection is made through this matrix which has nine elements and this complex. So the number of parameters of course because this is a unitary matrix there are seven parameters if the neutrino is a direct particle and as I said the three parameters corresponding to the three angles theta 1 3 theta12 and theta 2 3 and then uh the three masses m1 m2 m3 uh and a uh cp phase delta cp. So the oscillation probability is given by this p alpha is a rather complex looking uh you know expression. However is is not that difficult to evaluate once you have these parameters and the delta here is just 1.27 * delta m²ig in where this is in terms of e squ l is in kilometers and e is in gv. Of course, if E is in meters, then E has to be in me. And this delta MIJ squared is just mi^ 2 minus mj squ. Okay, so the difference of the square of the masses. Now there is something called the matter effect which was first worked out by Wolfenstein and there was a small error in that and Mikav Smeirnov independently worked on this. So what they found is that if you have matter then the new E interacts with matter electrons uh through the neutral current as well as the uh the charge uh changing current uh but the neutral current is common to all. So ultimately what ends up uh affecting this is that the the charge current of new E with electrons matter electrons. So that is what changes the effective angle of mixing as well as the delta which is the difference in mass squared in matter. So these are the two expressions and if delta 21 is positive then a is positive for neutrinos and a is negative for anti-utrinos. Uh so any case neutrinos coming out from the sun are neutrinos and not anti-utrinos of the electron type and uh therefore this is having a a is having a positive sign. So delta 21 cos theta uh when it hits a when it is equal to a then you have a resonance and that is the so-called MSW resonance. So this is a very uh general way of mixing and it is rather robust. It doesn't depend on particular values of theta. So at some density corresponding to this delta 21 you will and the energy. So the a is dependent of course on the energy is of course dependent on the coupling fmy coupling constant and the density of electrons. So if uh E is high this resonance will occur at a smaller NE and similarly if E is low this will occur at a higher NE. Higher NE means deeper inside the sun. Uh so uh this is a robust way of getting mixing and that uh basically accounted for the uh shortage of neutrinos uh that Davis saw and that also other experiments. So we will come to that uh uh apart from the Davis experiment which we have already talked about we'll talk about some of the other experiments which measured solar neutrinos. Okay. So the uh so-called Ponte Corvo uh M&S I forgot Sakata is one of the names and uh Maki I think is one of them and the other is I think Nakayama or something. So this that was explained in the earlier uh talk. So anyway this PMs neutral mass mixing matrix is written down like this where Sig and CI correspond to the uh sine of that theta corresponding theta and cos of that corresponding theta where these thetas change from 0 to uh they lie between 0 and pi by 2. So 0 and 90° whereas the delta CP goes from 0 to 360° and that enters into some of these matrix elements. So uh five of them here. Okay. [snorts] Uh now the best values for PDG from experiments for normal ordering is that M21 is uh 7.59 plus some error bar 10us 5. So roughly 7.5 10 - 5 UV ^ 2. Delta M31 is 2.5 roughly 10us 3 ev^ 2. So this is about 30 times larger uh 30 35 times larger than delta m21^ squ. Theta 2 is about 34°. Theta13 is small but not zero 8.5° or so. Theta 23 is about close to 45 from whatever best fit values they have. This is on the other side of 45. And this in fact was uh was one of the goals of some of these experiments to find out on which side of uh 45 this is the lower or higher side. The delta CP 1 177 plus - 20° of course this is uh I mean this this is not at the five sigma level yet. Uh so this is I think at the one sigma level. So this is of course something that will be confirmed by future experiments. Okay. So what are the various sources and what are the experiments? What are the parameters that they want to measure and what are the secondary parameters? So the main goal is here and the secondary goals uh will be a byproduct of that experiment. So for instance the solar home stake as well as the sage galaxy and go the SNO and the super kamioande they the main parameter was theta12 and the secondary parameters were this delta m21 squar and theta13 the reactor experiment the long baseline reactor experiment kland in Japan there delta m21 squar was the main goal and then theta12 and theta13 were the secondary parameters similarly the reactor medium baseline. Uh there's the Diab experiment, the Renault and uh double show. Uh they were aiming at theta13 and also modulus of delta m^ 2 31 and 32. Okay. Uh so uh the atmospheric nutrinos uh and the so this is one experiment and then the super kamioande uh this aim to measure theta 23 and uh delta again the same parameter here and also the sign of delta uh squared 3132. So this is the so-called ordering of the mass ordering of the nutrinos whether uh so from solar neutrino experiments we know that m_sub_2 is greater than m1 because otherwise you won't hit the uh msw resonance but we don't know whether uh m3 is larger than these two and that of course that difference as we saw is small um is larger or smaller and that's the hierarchy problem or the mass ordering problem and these accelerator experiments hope to do that to to measure the mass hierarchy especially uh this one DUNE is not mentioned here but DUNE is will measure that and also to some extent T2K uh also of course they have a much bigger goal namely that they want to measure the uh CP violation in the neutrino sector and again the secondary parameter is the theta 13 okay so again recall The solar neutrino spectrum u the solar neutrinos the mainly the bulk of the solar neutrinos of course PP neutrinos but they have a small end point that's about 420 keV or so and berillium 7 nutrinos uh the when berillium 7 decays to the ground state then that gives a a gamma ray of the order of 800 keV or uh u of course it also goes to an excited state that branching ratio is smaller by a factor of I think four or something and then there are other sources of nutrinos uh these 13 nitrogen 15 oxygen 17 florine etc they come about electron capture on nitrogen 13 they all come about from the CNO cycle and we'll see how that was measured for the first time uh the highest energy nutrinos are actually the HP nutrinos but they are so scarce uh that they would they have not yet been detected as far as we know. uh on the other hand the boronate nutrinos are have been detected they're a very small fraction again of the order of you know few times 10 the minus4 of the whole spectrum but since they are the highest energy and since the neutrino interaction cross-section goes like eu squared so they can be uh measured and indeed they have been they are the bulk of the signal in uh in the super kamio quantity detector also the bulk of the signal in the Davis radiochemical experiment and so on. Okay. So where are these nutrinos produced? So this is a is a graph graphical representation of that. The eight boron of course is this uh uh you know this dark line dark black. So where is that? That is somewhere here. And that uh purple line uh comes about from uh of the order of 5% or thereabouts of the uh core of the sun. So five the radius from where it comes out that is peaked at about 5% of the total of the full radius of the sun. On the other hand, the PP which is this uh orange one that uh extends beyond this more than 10% also it's a broader distribution because of course uh this can occur at lower temperatures whereas the boron uh which is produced by berillium 7 plus proton uh since it has to cross the berillium 7 coolum barrier then of course it has to be more energetic the proton has to be more energetic and that happens at the hotter temperatures in the core of the sun. Okay. So the reference is given here. It's a 2025 uh review article on solar neutrinos. Now we have already talked about the Davis experiment uh the radiochemical experiment which started this whole business of measuring solar neutrinos and then there was a solar neutral problem. It was a very difficult experiment. It is not a online experiment. But there is another uh radiochemical experiment based on gallium 71. Now why another radiochemical experiment? Well, as I told you the uh Davis experiment which involved chlorine 37 was actually more sensitive to the boron uh nutrinos, boron 8 nutrinos. So they are the higher energy nutrinos whereas uh gallium 71 has the uh beauty that it has a very low threshold for the new EE reaction and uh so it is sensitive. So the threshold is only 233 KV and as as is shown here the pputinos extend up to about 420 KV. So uh things below the threshold of course cannot be detected by this uh through this reaction but things above can be and therefore this gallium is sensitive to the most abundant nutrinos as well. Of course because the cross-section goes like E squ it it's uh I mean the production of 71 germanium is sensitive to the PP nutrinos as well as the berillium 7 as well as the boron nutrinos. Okay, there is a weightage for each of these of course dependent on the energy. Uh so in any case this was uh you know conceived uh and then two experiments were carried out. One is the Soviet American gallium experiment so-called sage. it uh the experiment was done in uh the uh Soviet era which means in Russia with about 50 to 57 tons the various stages of the experiment uh of liquid gallium metal and this was done at a depth of about 2 km in the Caucus mountains in Russia. What they did is to uh add a carrier like natural germanmanium only 700 microgram and then you extract the 71 germanmanium in the midst of this 50 odd tons of gallium and then you count it making a suitable gas I think it was GF4 gas in a very small proportional counter with carefully chosen materials and so on to reduce the radioactivity. U now uh this was done. So the low-lying levels uh that can be excited by solar neutrinos are shown also here. So this is natural gallium gallium. You can excite the half minus 5 half minus and three half minus. So of course when you do that uh when you extract 71 germanium say let's say this is excited here this will decay by gamma rays and come to the ground state and similarly this state and then you count the 71 germanmaniums. So when you want to translate this uh into a solar neutrino flux you have to of course know the matrix elements cons. So this is easy relatively because this beta decays to gallium 71 the other states you have to do experiments uh to see what those uh matrix elements are and these have been indeed carried out. Uh this uh is taken from a reference given here. It's a talk uh sometime by M. Jaitler. Okay. So uh then you of course need to calibrate this uh you know radiochemical detectors. This was done by using extremely strong sources of chromium 51 and 37 argon. And uh if you go back to the earlier slide you will see the uh the kind of uh neutrino energies that you get. These are uh electron capture sources. So they have a moneretic uh electron type of nutrino coming out. So this uh can calibrate uh this detector by uh through this same reaction. Um and argon is slightly higher energy neutrino. Uh so anyway so this can also excite five half minus and half minus and so on. So in any case uh uh the sorry I think I made a mistake in when you do a new E reaction you produce germanium 71 so yeah that's true you can excite these states from the argon 37 this is just about the energy so you can only go to the ground state okay so using these sources they found that the experimental theory was about 95 with some error bar and also 79 for the argon 37 case. So you know this was this is not one. So that led to this uh problem that maybe there is a sterile neutrino and so on. Um so in any case they used this in uh extracting the uh you know capture rate and this capture rate was found to be in solar neutrino units uh 64.7 plus - 2.4 four this was the apparatus that was used in the uh galax experiment galaxy and later became the go experiment and uh if you go back this was the kind of apparatus that was used in the sage experiment. So there were actually uh seven such modules uh and they were used to uh this is representative of one of them. You can see the size as compared to a human being and this shows the proportional counter. So these have stirrers which mix the gallium. So then when you extract the germanium you get uh all the or with some efficiency close to uh 80% 90% you get the germanium 71s and then you can count them. Okay. So this is as far as the uh sage experiment is concerned. So you can see that this was less than half of what was expected on the basis of the standard solar model SSM. This is on 138 snooze and they got uh about 65. The galax experiment was done with a smaller amount of gallium. This was basically a European collaboration Germany, UK, Italy and so on. And they did this experiment in the Grand Saso lab between 91 and 97. The final result was 78 plus - 8 snow. So you can see that the accuracy is poorer because of the mass of the gallium and also because it was done over a period of 6 years whereas the stage was done over a longer period of something like 26 years. So an improved version of uh galax was the GN O which ran for 5 years 1998 to 2003 and these two experiment gave results which was consistent which were consistent with each other and the final combined rate was 67.5 plus - 5.1 snooze their calibration was done with much stronger sources almost two times stronger than the uh two times more than three times stronger than the ones used in sage. Uh so the 37 argon was about four four and a half times stronger. So it was extremely strong sources to calibrate the detector. But there also they found a shortage uh in the sense that experiment measured experimental theory was less than one uh and uh with the argon it was 81. So it is almost like 20% but of course the error bar is also like 10%. and uh so uh 10 to even more slightly 12 13%. Uh so these results were consistent with new e mixing taken together the sage and the galax they were consistent with this mixing but uh the main goal was of course to measure the total capture rate and to see if it fits within the you know paradigm of nutrino oscillations. uh the sage uh experiment later morphed into the so-called bakan experiment on sterile nutrinos and what they did is they had a chamber with a inner volume and an outer volume and they uh they had a source inside so the smaller one was actually closer to this and the bigger one was slightly further away. So in principle you have two distances average distances and they were looking for differences between these two. Uh what they found is that again for the experimental theory for the inner and outer uh targets they found this ratio was consistent with each other but it was different from one and within you know this was like uh a 4 and a half sigma effect okay uh but it was lower by about 22%. So again these results confirmed uh that there seems to be an indication for something I mean something strange that is happening and this is consistent with new e to new s oscillations provided delta m squ is larger than about.5 e squ where this delta m squ corresponds to the standard neutrino the electron neutrino and the sterile neutrino. Uh now you might ask okay why are we bringing in the fourth generation of uh nutrinos? Well we know from the Z0ero uh profile that there can be only three nutrinos with masses less than 45 GV. So this sterile neutrino is not supposed to really be part of that family but it only comes about through mixing. So there are theories which try to uh you know predict such a thing or explain such a thing but we still don't have a definitive answer as to the whether sterile neutrons exist or not and I'll come to that uh in in the subsequent lecture. Uh so any case this experiment was consistent with the delta m² being greater than.5 e^² and sin^ square 2 theta being about 04. So pretty large mixing and this is a reference fractor paper in 2022 where these results are summarized. Okay. Uh solar neutrals are also have also been measured in super kamio which is the 50 kiloton water cherankov detector with a uh active volume or a so which I mean a shielded volume of something like 22 kiloton. The complete data set they had four major runs and the complete data set shows a flux which is mainly the eight boron flux of about 2.3 with these error bars 10^ the 6 nutrinos electron neutrinos per cm squared per second. Uh okay the best fit parameters are given here sin^ square theta12 solar is.3 or so and delta m^ 2 is about 6.1 with some fairly large error bars 10 the minus 5 e^ squ so you can see that uh this error bar for the theta12 is smaller than the error bar that you get from the delta m^ 2 uh the e distribution is given here So this is the solar and then the solar plus the kamland. Uh these are shown here in the dark and the blue uh or green lines. And uh this is the typical theta distribution for a particular energy bin 6.9 to 7.5 me or so. And you can see that it is uh peaked in the direction of the sun. Uh so uh the the vector from the earth to the sun centers of these two objects uh that is your uh theta equal to zero and so cos theta is one. So what is plotted here is cos theta versus uh events per bin and it peaks in a direction towards the sun. This is uh this is a final paper from 1 to four uh solar energy measurements at super K. A cute thing uh if you were to take the picture of the sun in not in photons but in nutrinos then you will find that it is the center of it is the actually main source of nutrinos. Of course because of the uh way the detector is and because of finite resolutions and so on that come in this width as large as 20° whereas we know that the sun subends an angle in our let's say in an image of the order of.53°. However, this is dominated by the statistics because nutrinos are interactions are very few relative to photons and they so it depends on the detection method. However, this could be in principle improved to about.12° in which case you will get a picture of this sun uh in the neutrino sector. So this is the super kameo quande data at 1500 days and as we saw it was peaked at cos theta equal to 1. So it is basically taken from there that we get this uh uh image. Okay, there is a another detector which is again a real-time detector like super kamio not radiochemical which uh is this borax detector. This is a 300 ton liquid cintillator at Grand Saso uh again under a mountain and this is a schematic of that. So there are there's a veto around this detector which and which is looked at by photo multiplier tubes. So you can veto cosmic muons the residual cosmic muons that still come in uh even below the mountain and uh then of course the main cintilator is of the order of 300 tons. Now this is one of the purest cintilators in the world. Of course now this is dismantled but uh this had a purity such that the uh I mean it went the cintilator of course goes in a closed loop through a purifying setup and to remove any impurities. uh but towards the end the cintilator was so pure that it had only a uh a background of uranium and thorium of the order of 10 the minus 20 or 10us 21 atoms of uranium for every atom for every molecule of cintilator extremely pure so uh this is the uh nature paper that showed evidence for pp neutrinos and So these PP neutrinos are this here. This is of course carbon 14 and so you have to subtract. So this is measured very carefully and subtracted out and uh so the contribution from PP is this red line here from berilium is this so so to speak compton edge. Now, of course, this is a it's a little hard to see, but you uh you can always uh I mean, they have done a very careful analysis and so this was one of the first evidences for PP nutrinos in in a real-time detector. But the evidence for of course berillium 7 was stronger. In fact, this orexino was built for uh looking at berillium 7 capture nutrinos. But they show Compton spectrum of course because uh the nutrino can scatter off the electron and deposit uh energy from a maximum to zero. So this is the CNO contribution. Uh so it's even harder to measure and then there are these boronate nutrinos which of course you see the evidence for here the background is much less. Okay. So uh they they have measure a flux of pp nutrinos 6.1 with an error of 12%. They measure the berillium 7 with much better accuracy. Five into 10 the power of let me see this is I think 10 ^ of 10 probably this is n yeah this this 10 is this n is this 10 and bium 7 is this five plus - 3% into 10 ^ of 9 cm per cm squared per second the boronate nutrinos are of the order of 6 into 10^ the 6 uh nutrinos per square centimeter per second and CNO in this paper they just put a limit at 9 95% confidence level uh one year later they uh one year it's let me see was it one year later no it was uh uh about 6 years later they published a result in nature which showed the CNO contributions the red contribution the CNO contribution now of uh they have done an extremely careful analysis of the distribution of these neutrinos with regard with respect to where in the detector they have come about and so on where the interaction has taken place and so on. So they uh they say the absence of CNO neutrinos is ruled out at 5 sigma whereas the presence is shown to be at the 3.5 sigma level. Okay, this is from the boron. However, of course, you can see that this is still uh you know in background which is about five to 10 times higher uh depending on where you are in the spectrum. Uh however, we do believe their careful analysis and so this was the first evidence of CNO neutrinos which actually constitute only about 1% of the total neutrino flux. Okay. Now we come to reactor antiutrinos. So there is this commland experiment with a 1 kiloton liquid cintilator that's shown here uh in the same area as the super kamio detector and uh in fact it I think occupied the space of the uh first kamoka 3 kiloton water chenov detector and it so happens that Japan produces a lot of I mean power through their nuclear reactors and uh it so happens that at this kamyoka site the you know the kind of uh effective baseline that they get is uh of the order of about uh 175 km plus - 35 okay and 70 gaw of total power. So this means that there is effectively a you know reasonably uh you know distance which is reasonably unique 175 plus - 35 km and they could do this experiment to actually repeat what was done with the solar neutrinos but now this time on the earth using nuclear reactors. Of course remember that this is done with anti-utrinos and the solar neutrinos are neutrinos. So uh so that's but if if uh uh CPT holds then of course this is would be probing the same things. Okay. So Kamland has these final spectra that they have. So this dark line shows what you expect if there are no oscillations and this is what you see in the experiment. And uh they have also evidence for geonutrinos. So this is the first experiment to find evidence for geonutrinos. So that is shown here. After you subtract the reactor contribution, you get this and this actually corresponds to about I think 40 terowatt of power in the in the earth which comes about because of radioactive decay. Uh the parameter extractions are like this. the s tan square 2 theta12 and delta m^ 2 1 this uh this is the plot. So the values the real values are somewhere in between in in this circle or in this ellipoid ellipse uh or a deformed ellipse you might say. And these are the various confidence levels. And so while it measures delta m21 squared fairly accurately it it's the error bar on the tan square theta13 is somewhat more and is also shown here. delta m squ is more accurate than but if you combine these of course this and the super k then of course you get a uh you get both of these together with some error bars. Okay. Uh it improves the accuracy of either experiment. Okay. So then there is the final reactor experiment the dab experiment which was specifically designed to look for uh theta 13. It could have been zero. But it turns out that sin square 2 theta13 from this experiment which is done at uh with different detectors between about 3 and 1.5 kilmters. So the uh the probability that a new e bar remains a new e bar is just given by 1 minus sin^ square 2 theta 1 3. It doesn't depend on delta cp etc. and u also on the other uh neutrino parameters. So if you know these distances you can extract theta13 and that is what they did to remarkable accuracy uh to something like let's say two parts in 85. So it's about a 2 and a half% accurate result and of course they also got delta m 32 with uh similar kind of accuracy and uh these are the latest results from the ab. So in summary, we did a recap of the nutrino mass matrix and experiments uh to address the mixing parameters. We also looked at some of the solar and reactor experiments that bear on these parameters. In the next uh lecture we will talk about uh accelerator mainly accelerator based experiments which uh address these mixing parameters. Thank you. >> [music] [music] [music and bell] [music]