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Stream: community: general

Topic: neutrinos

John Baez (Dec 24 2020 at 02:19):

One thing that's somehow underappreciated: nobody knows for sure whether or not if neutrinos are their own antiparticles, and this is a question particle physicists would really love to settle.

John Baez (Dec 24 2020 at 02:20):

But it turns out to be very hard to tell if a nearly undetectable particle that we normally find moving very near the speed of light is its own antiparticle or not!

John Baez (Dec 24 2020 at 02:20):

Physicists usually phrase this puzzle in more jargon: "do neutrinos have a Dirac mass or a Majorana mass?"

John Baez (Dec 24 2020 at 02:21):

I bet they are not their own antiparticles.

John Baez (Dec 24 2020 at 02:26):

If neutrinos are their own antiparticles, then neutrinoless double beta decay should be possible. This is a kind of radioactive decay where two neutrons turn into two protons and two electrons but not antineutrinos.... because they actually emit two antineutrinos but these annihilate!

John Baez (Dec 24 2020 at 02:27):

Physicists have been looking for this and not found it (yet).

Tim Hosgood (Dec 24 2020 at 02:39):

does the abstract theory/model behind particle physics give any predictions about things like this, or is this really a “you have to do experiments” sort of field?

John Baez (Dec 24 2020 at 05:12):

Based on plausible and experimentally supported axioms, quantum field theory puts very strong restrictions on what particles can do - but they don't settle the question of whether neutrinos are their own antiparticles.

In the 1980s, the Standard Model said neutrinos are massless, and not their own antiparticles. Experiments later showed they have nonzero mass.

You can modify the Standard Model and make it more reasonable-looking by making neutrinos massive without making them their own antiparticles. With this "new improved" Standard Model, the neutrinos are just like the other matter particles!

But there's something funny about neutrinos: they're ridiculously light-weight compared to the other matter particles. Also, nobody has ever seen a neutrino spinning clockwise along its axis of motion. In the "new improved" Standard Model these "right-handed" neutrinos exist but don't interact at all with other matter.

There are various alternative theories that try to explain these mysteries. If neutrinos were there own antiparticles, that would give a different way of explaining their mass, which to some extent makes their light mass natural. On the other hand, noninteracting right-handed neutrinos would make the Standard Model incredibly elegant from the viewpoint of group representation theory. Since I want the Universe to be mathematically beautiful, I find this convincing.

In short, there's a big interplay between theory and experiment here, but it seems we need some experiments to settle whether neutrinos are their own antiparticles, because there are reasonable theories that go either way.

David Michael Roberts (Dec 24 2020 at 05:48):

@Tim Hosgood It's like knowing that representations of Lie groups are involved and important, but you don't know which ones are actually observed in nature.

Tim Hosgood (Dec 24 2020 at 11:06):

that makes sense, thanks!

John Baez (Dec 24 2020 at 17:16):

David Michael Roberts said:

Tim Hosgood It's like knowing that representations of Lie groups are involved and important, but you don't know which ones are actually observed in nature.

Yes, it's kinda like that. Of course that's not exactly what's going on here. If we stick to the Standard Model gauge group and assume there are no unobserved particles other than one right-handed neutrino for each of the 3 observed left-handed neutrinos, there are three ways that neutrinos can have nonzero mass. In one way they are their own antiparticles (a "Majorana mass"), in another they're not (a "Dirac mass"), and in the third they are but they get mass from both the Dirac and Majorana mechanisms.

John Baez (Dec 24 2020 at 17:17):

The set of options becomes extraordinarily more diverse if we let ourselves postulate various other not-yet-observed particles, and you can read hundreds of physics papers exploring such scenarios. But still, neutrinos are either their own antiparticles or they're not!

John Baez (Dec 26 2020 at 19:23):

I decided to rewrite the section of this dealing with neutrinos, since it's 13 year old:

• Open questions in physics.

John Baez (Dec 26 2020 at 19:27):

I haven't done it yet, but I solicited help here:

• Neutrino puzzles, Azimuth.

and here:

• Neutrino puzzles and open questions about neutrinos today, Physics Forums.

John Baez (Dec 26 2020 at 19:28):

Physics Forums seems rather dormant lately, but I was happy to see some good comments there... maybe all the experts are sleeping, just waiting for someone to stir up a conversation?

John Baez (Dec 26 2020 at 19:29):

It seems a common mode of interaction there is "post a comment trying to assert superiority by trying to find small mistakes in the previous comment".

John Baez (Dec 26 2020 at 19:31):

It reminds me to rein in my own tendency to make comments like that. Even if one is trying to be helpful by correcting small mistakes, it can be unpleasant. One should at least say some explicitly friendly things while doing it.

John Baez (Dec 26 2020 at 19:33):

Anyway, I think once the conversation got going, most people there became more friendly. It helps to react to criticisms by either pointing out why they were wrong or accepting them and thanking the person who made them, either way not displaying grumpiness. (The last part is exceedingly hard.)

John Baez (Dec 26 2020 at 19:34):

Anyway, I'm catching up on my neutrino physics! They've actually been learning good stuff with all the experiments they've been doing.

Fabrizio Genovese (Dec 27 2020 at 01:25):

John Baez said:

It seems a common mode of interaction there is "post a comment trying to assert superiority by trying to find small mistakes in the previous comment".

Fun fact: One of the best ways to get a response on Stack Exchance or Mathoverflow is to post your question and then post a fake answer from a different account. Nothing motivates (most) mathematicians more than proving someone else is wrong. I find it a truly pitiful thing, but at least I learned to game it. :smile:

Nathanael Arkor (Dec 27 2020 at 01:37):

This has not been my experience of either MSE/MO, or other mathematicians.

John Baez (Dec 27 2020 at 01:42):

MSE and MO are reasonably polite due to the strongly enforced question/answer format, but if you get a bunch of men interacting in any sort of online forum with a specialized focus, you usually see a pecking order emerge - and if the top dogs (mixing my metaphors here) are aggressive, this behavior tends to spread.

John Baez (Dec 27 2020 at 01:59):

Okay, I've updated the section of "open questions in physics" that deals with neutrinos. Here are my new open questions about neutrinos:

• Neutrino puzzles (part 2).

I may continue to fiddle with it as experts weigh in.

Dan Doel (Dec 27 2020 at 05:47):

Having to correct misleading information constantly just sounds like a recipe for burning out the people who actually know the right answer.

Matteo Capucci (he/him) (Dec 27 2020 at 08:54):

I love wikis since you can quietly correct things without hurting anyone's feelings :sparkles:

John van de Wetering (Dec 27 2020 at 10:36):

That might be true in theory, but in practice you still get edit wars by people that disagree what is correct.
In general it is a very nice system though :)

Matteo Capucci (he/him) (Dec 27 2020 at 11:10):

Well, I'm talking about small, inequivocable mistakes.

John van de Wetering (Dec 27 2020 at 11:11):

Just started reading your other page https://math.ucr.edu/home/baez/neutrinos.html.
I think there is a typo in the third displayed equation. It should be $f_i = \sum_j U_{ij} e_j$ if I'm not mistaken

John van de Wetering (Dec 27 2020 at 11:13):

Talking about wanting to fix small mistakes :P

John van de Wetering (Dec 27 2020 at 11:19):

Also, in the inline equation after "The mass eigenstates have bland names" I think there is a latex error.

John van de Wetering (Dec 27 2020 at 11:20):

And now to actually engage with the content of the article: I hadn't realised that the flavour and mass eigenstates where that mixed! I thought it was only a small bit of mixing

John van de Wetering (Dec 27 2020 at 11:20):

Physics is so weird

John Baez (Dec 27 2020 at 21:13):

John van de Wetering said:

Just started reading your other page https://math.ucr.edu/home/baez/neutrinos.html.
I think there is a typo in the third displayed equation. It should be $f_i = \sum_j U_{ij} e_j$ if I'm not mistaken

Yes, that's what it should say. I think that's what it actually says there. Are my eyes deceiving me?

And now to actually engage with the content of the article: I hadn't realised that the flavour and mass eigenstates where that mixed! I thought it was only a small bit of mixing.

I hadn't realized it either until this week! For quarks it's pretty small. For neutrinos the mixing is huge. That's one reason I decided to actually add the mixing matrix for neutrinos, or at least the absolute values of its entries, to my webpage.

Physics is so weird.

Yes, this is not how I would have designed the world. But I was not consulted. And these mysteries suggest that there are some important things we still don't understand!

John van de Wetering (Dec 27 2020 at 23:30):

John Baez said:

John van de Wetering said:

Just started reading your other page https://math.ucr.edu/home/baez/neutrinos.html.
I think there is a typo in the third displayed equation. It should be $f_i = \sum_j U_{ij} e_j$ if I'm not mistaken

Yes, that's what it should say. I think that's what it actually says there. Are my eyes deceiving me?

There is a $f_j$ instead of an $e_j$

John van de Wetering (Dec 27 2020 at 23:33):

With the amount of mixing that occurs for neutrino's, how many times do neutrino's 'rotate' through their states on their way from the sun?
It must be at least one full time around because we only observe 1/3 of the electron-neutrino's we would expect, right? But neutrino's travel at such a high speed that they wouldn't have too much time to flip, or should you only consider this fact from an outside perspective?

John Baez (Dec 27 2020 at 23:41):

John van de Wetering said:

John Baez said:
Are my eyes deceiving me?

There is a $f_j$ instead of an $e_j$

Wow! It's amazing how when I read math I see what should be there instead of what is there. Thanks.

John Baez (Dec 27 2020 at 23:45):

With the amount of mixing that occurs for neutrino's, how many times do neutrino's 'rotate' through their states on their way from the sun?
It must be at least one full time around because we only observe 1/3 of the electron-neutrino's we would expect, right?

Right. They rotate around so much they get effectively randomized.

But you're making me want to estimate the rate at which they rotate, say in radians per million kilometers traveled.

John Baez (Dec 27 2020 at 23:45):

But neutrinos travel at such a high speed that they wouldn't have too much time to flip, or should you only consider this fact from an outside perspective?

Let's figure it out.

John Baez (Dec 27 2020 at 23:47):

Apparently most solar neutrinos have an energy below 400 keV:

https://en.wikipedia.org/wiki/Solar_neutrino#Observed_data

John Baez (Dec 27 2020 at 23:48):

Judging from the blue curve in the graph, a lot of them have an energy of about 300 keV. So let's use that for our estimate.

John Baez (Dec 27 2020 at 23:49):

Now, the mass of the lightest neutrino is not known, which makes it very hard to know how fast a 300-keV neutrino is going!

John Baez (Dec 27 2020 at 23:50):

But the difference in masses between the two lightest neutrinos is about .009 eV:

https://en.wikipedia.org/wiki/Neutrino#Mass

John Baez (Dec 27 2020 at 23:51):

So let's just assume our neutrino has a mass of 0.009 eV. If its energy is 300 keV, how fast is it going? Or better: what amount of time dilation does it experience?

John Baez (Dec 27 2020 at 23:52):

Its energy is 300,000 / 0.009 times its rest energy - that's 33,333,333, but let's say 30,000,000.

John Baez (Dec 27 2020 at 23:53):

So this means it's moving at a speed $v$ such that

$1/\sqrt{1 - v^2/c^2} = 30,000,000$

John Baez (Dec 27 2020 at 23:54):

That's very close to the speed of light - we call it "ultrarelativistic".

John Baez (Dec 27 2020 at 23:54):

But this also means that the time dilation factor is 30,000,000.

John Baez (Dec 27 2020 at 23:55):

Google assures me it takes 8 minutes and 20 seconds for light to go from the Sun to the Earth. That's 500 seconds.

John Baez (Dec 27 2020 at 23:56):

So our solar neutrino will only age 500/30,000,000 seconds during its trip to Earth!

John Baez (Dec 27 2020 at 23:58):

That's 0.0000166... seconds or about 17 microseconds.

John Baez (Dec 27 2020 at 23:58):

That was fun!

John Baez (Dec 27 2020 at 23:59):

But then the question is: does that count as a short time, or a long time, compared to the rate at which the neutrino is oscillating between different flavor states?

John Baez (Dec 28 2020 at 00:03):

I think it must be a long time because I know people can even do experiments here on Earth where they see neutrinos oscillating: they make them in one location, and detect them in another location 500 meters away, and they see that some have changed flavor!

John Baez (Dec 28 2020 at 00:03):

But what's the frequency at which neutrinos oscillate, roughly?

John Baez (Dec 28 2020 at 00:06):

Believe it or not, I've never seen anyone say this in simple terms like "oh, roughly a million times a second".

John Baez (Dec 28 2020 at 00:06):

So I have to figure it out...

John Baez (Dec 28 2020 at 00:08):

Wikipedia has some stuff on this:

https://en.wikipedia.org/wiki/Neutrino_oscillation#Propagation_and_interference

John Baez (Dec 28 2020 at 00:10):

Hmm, I'm gonna get lazy and use their formulas... they say that the number of times the neutrino oscillates when moving a certain distance L is about

$1.27 \times \frac{\Delta m^2}{{\rm eV}^2} \frac{L}{\rm km} \frac{\rm GeV}{E}$

John Baez (Dec 28 2020 at 00:13):

It's not good to just grab formulas like this off the shelf, but I'm getting lazy... here $\Delta m^2$ is the difference in squared masses between two neutrinos, and for the lightest two it's about $0.000079 \mathrm{eV}^2$. I already used this number earlier in my calculation... I took its square root.

John Baez (Dec 28 2020 at 00:15):

$E$ is the energy of our neutrino, and we're assuming that's 300 MeV, and a GeV is 1000 eV, so GeV/E is about 3 for us.

John Baez (Dec 28 2020 at 00:15):

So we get

$1.27 \times 0.000079 \times 3 \frac{L}{\mathrm{km}}$

John Baez (Dec 28 2020 at 00:16):

That's about 0.0003 L/km.

John Baez (Dec 28 2020 at 00:16):

In other words, a typical solar neutrino oscillates roughly 0.0003 times per kilometer of flight!

John Baez (Dec 28 2020 at 00:17):

You should check my arithmetic, I often make mistakes.

John Baez (Dec 28 2020 at 00:17):

But anyway, the Sun is about 150 million kilometers away according to Google!

John Baez (Dec 28 2020 at 00:18):

So I'm getting that our neutrino oscillates about 45,000 times while it goes from the Sun to the Earth!

John van de Wetering (Dec 28 2020 at 11:21):

John Baez said:

$E$ is the energy of our neutrino, and we're assuming that's 300 MeV, and a GeV is 1000 eV, so GeV/E is about 3 for us.

I thought you were taking the energy to be 400keV instead of 300MeV? That would make the oscillation about 1000 times faster.

John van de Wetering (Dec 28 2020 at 11:22):

In any case, that is a lot of oscillation!

John van de Wetering (Dec 28 2020 at 11:25):

Reminds me of this wonderful movie: https://youtu.be/DGf0AHky0Os?t=63

John van de Wetering (Dec 28 2020 at 11:25):

The neutrino's have mutated!

John Baez (Dec 28 2020 at 16:58):

John van de Wetering said:

John Baez said:

$E$ is the energy of our neutrino, and we're assuming that's 300 MeV, and a GeV is 1000 eV, so GeV/E is about 3 for us.

I thought you were taking the energy to be 400keV instead of 300MeV? That would make the oscillation about 1000 times faster.

Whoops!

I was taking it to be 300 MeV; the maximum energy of the neutrinos produced by the most common reaction is 400 MeV but the graph made it look like the most common energy was about 300 MeV.

John Baez (Dec 28 2020 at 17:15):

So yeah, this calculation is off by 1000. Let me redo it; I think I'll post a more polished version on my blog.

Wikipedia says that the number of times the neutrino oscillates when moving a certain distance $L$ is about

$N = 1.27 \times \frac{\Delta m^2}{{\rm eV}^2} \frac{L}{\rm km} \frac{\rm GeV}{E}$

Here $\Delta m^2$ is the difference in squared masses between two neutrinos, and for the lightest two it's the smallest: about $0.000079 \mathrm{eV}^2$. $E$ is the energy of our neutrino, and we're assuming that's 300 keV, and a GeV is 1000 eV, so GeV/E is about 3000.

So we get

$N \approx 1.27 \times 0.000079 \times 3000 \frac{L}{\mathrm{km}} \approx 0.3 \frac{L}{\mathrm{km}}$

In other words, a typical solar neutrino oscillates roughly once for each 3 kilometers of flight!

The Sun is about 150 million kilometers away according to Google. So a typical neutrino - if we pretend it only oscillates between type 1 and type 2 - would oscillate 50 million times as it travels from the Sun to the Earth!

In fact it also has some component of type 3, and that gives a $\Delta m^2$ of $0.0027 \mathrm{eV}^2$, which is 34 times bigger than $0.000079 \mathrm{eV}^2$, so there will also be oscillations that are 34 times faster. (Imagine a superposition of two sine waves of different frequencies.)

Simon Burton (Dec 28 2020 at 19:57):

They travel close to the speed of light, yes? So 50 million / 500 seconds = 100 kHz.

John Baez (Dec 28 2020 at 20:03):

Yes, they move very close to the speed of light. So measured in our frame of reference that's their frequency of oscillation. Nice!

John Baez (Dec 28 2020 at 20:05):

Earlier in this thread I worked out that they are time-dilated by a factor of about 30,000,000. So those 500 seconds feels like about

$500/30,000,000 = 1.67 \cdot 10^{-5}$

seconds to them - let's say 17 microseconds.

John Baez (Dec 28 2020 at 20:06):

(Real-world physicists are absolutely unafraid to talk about what something "feels like" to a neutrino. :upside_down: Avoiding anthropocentrism is important, but to do physics you have to learn to put yourself in the place of a neutrino, and imagine yourself oscillating and whizzing along.)

Simon Burton (Dec 28 2020 at 20:17):

That's pretty wild. Reminds me of the science fiction book "Tau zero". With that much time dilation, the neutrino "experiences" about a day passing through our galaxy: 100000 * 24 * 365 / 30000000 = 29.2 hours.

John Baez (Dec 28 2020 at 20:26):

I once read that, but I don't remember it.

If we do interstellar travel we should pack our information into microscopic pellets, make lots of copies, and shoot them off at ultrarelativistic speeds. Then the hard part is how to stop and turn ourselves back into some useful form. Greg Egan has some nice stories about that, like Glory.

Jason Erbele (Dec 28 2020 at 21:39):

That's a pretty cool set of calculations to get a decent ballpark estimate for the oscillation frequency of neutrinos. I'm curious, though – how good is the approximation $\Delta m \approx \sqrt{\Delta m^2}$ , given that $\Delta m^2$ is the change in the square of the mass?
We should be able to approximate using $\Delta m^2 = (m+\Delta m)^2 - m^2 = 2m\Delta m + (\Delta m)^2$ but that has an annoying dependence on $m$ . Is $m \ll \Delta m$ for neutrinos? That seems like the only way to justify making that approximation.

John Baez (Dec 28 2020 at 22:16):

We don't know the neutrinos masses very well; what we really know is $\Delta m^2$ since that can be measured by studying neutrino oscillations. There are 3 neutrinos, though, so we really know $m_2^2 - m_1^2$ and $m_3^2 - m_2^2$. The first is much smaller than the second. The first is about 0.000079 $\mathrm{eV}^2$, and the second is about 0.0027 $\mathrm{eV}^2$.

($m_3^2 - m_1^2$ is almost the same as $m_3^2 - m_2^2$ since neutrinos 1 and 2 are much closer in mass.)

Luckily we only need these differences of squares to estimate the rate of neutrino oscillations in our frame of reference. But it's not really luck: we know these differences of squares because we can measure them by measuring the rate of neutrino oscillations!

My estimate of the neutrino's actual mass(es) is a complete wild guess, and this infects my estimate for the amount of relativistic time dilation it suffers, and thus the rate of neutrino oscillations in its frame of reference.

We know the lightest neutrino has a mass of less than 2.2 eV, and we know at least one neutrino must have a mass of more than 0.05 eV. That's a pathetically big range of values:

https://en.wikipedia.org/wiki/Neutrino#Mass

Jason Erbele (Dec 28 2020 at 22:28):

That lower bound on the highest neutrino mass makes sense... $m_2 > 0$ , so $m_3 > \sqrt{0.0027 \text{ eV}^2}$ which is close to 0.05 eV.

Jason Erbele (Dec 28 2020 at 22:31):

I suppose one could make a similar series of calculations assuming $m_1 = 2.2\text{ eV}$ to get the other bound on the range of values for how much time dilation neutrinos experience.

John Baez (Dec 28 2020 at 22:34):

Yes. Actually in my blog post for computing this time dilation I'm gonna assume a mass of 0.1 eV. My guess here, 0.01 eV, is probably too low.

John Baez (Dec 28 2020 at 22:35):

But this part only affects the neutrino's time dilation and things that depend on that - not stuff we directly see "at rest" here on Earth.

John Baez (Dec 28 2020 at 22:36):

So it's just a fun sideshow.

Jason Erbele (Dec 28 2020 at 22:36):

Right. It only affects how the neutrinos feel. :upside_down:

John Baez (Dec 28 2020 at 22:36):

Right now with this new assumption I'm getting that the neutrino moves at about

0.99999999999994 c.

John Baez (Dec 28 2020 at 22:36):

So it feels really stressed-out.

Jason Erbele (Dec 28 2020 at 22:46):

Hmmm. At that rate, the neutrinos detected from the famous supernova SN 1987A in the Large Magellanic Cloud took about a third of a second longer to reach Earth than unimpeded light would have taken. No wonder they're stressed out!

John Baez (Dec 28 2020 at 22:49):

But the good news for the neutrinos is that they escape the supernova's core before the light does!

John Baez (Dec 28 2020 at 22:52):

How close to a supernova would you need to be, to get a lethal dose of neutrinos?

The short answer is: much closer than you'd ever want to be to a supernova.

John Baez (Dec 28 2020 at 22:54):

Basically inside the supernova.

John Baez (Dec 29 2020 at 06:46):

I've done the calculations of neutrino oscillations from scratch here:

• Solar neutrinos.

Instead of using the pre-packaged formula on Wikipedia, I derived it all from scratch. I'm still a bit confused about why the numbers are coming out a bit different. (I assumed a different neutrino mass, but that doesn't affect the rate of oscilllation.)

John Baez (Dec 29 2020 at 06:47):

It was fun! It's been a while since I've done a good old fashioned physics calculation.

John van de Wetering (Dec 29 2020 at 11:33):

Nice calculation!

John van de Wetering (Dec 29 2020 at 11:34):

Follow-up question: for different masses the dilation factor will be different. So does the neutrino actually spread out in a spatial superposition as it travels?

John Baez (Dec 29 2020 at 18:10):

I was wondering about that. I guess it must! It's not surprising for a wavefunction to spread out in space, but this case is interesting. Someone must have studied this.

Refurio Anachro (Dec 29 2020 at 18:45):

Uh, so that's not common knowledge?! Disclaimer: I don't know about this stuff, but it's not the first time I think about this.

Since the components spread out with slightly different speeds, and may change back, this might give interesting patterns. Since they start out the same, a compact neutrino source should be called a naser?

I suppose being so light makes them more similar to light.

John Baez (Dec 29 2020 at 18:49):

Refurio Anachro said:

Uh, so that's not common knowledge?!

I've never heard about it, but "common knowledge" is a funny thing: it could be that everyone who works on neutrino physics knows about this, and there could be lots of papers about it.

But maybe it's not very interesting. Since neutrinos are emitted randomly and detected "one at a time", you just see one when you see it: I can't imagine an experiment that detects that its wavefunction has smeared out.

John Baez (Dec 29 2020 at 18:53):

If you could make a pulse of neutrinos all having roughly the same energy, and then detect this pulse a light-year away, you should be able to notice that the lightest neutrinos arrive before the heavier ones. But for some reason nobody does these experiments. :upside_down:

Jason Erbele (Dec 30 2020 at 19:19):

John Baez said:

If you could make a pulse of neutrinos all having roughly the same energy, and then detect this pulse a light-year away, you should be able to notice that the lightest neutrinos arrive before the heavier ones. But for some reason nobody does these experiments. :upside_down:

Since the neutrino flavors oscillate, wouldn't such an experiment merely notice the neutrinos that spent more time in the lighter flavors arrive first, regardless of their flavors at the detection? Supernovae are the best source of neutrino pulses that I can think of, so they could be used and still be detectable at large distances. I don't suppose the neutrinos released in a supernova collapse have roughly the same energy, though... or do they?

John Baez (Dec 30 2020 at 19:58):

Since the neutrino flavors oscillate, wouldn't such an experiment merely notice the neutrinos that spent more time in the lighter flavors arrive first, regardless of their flavors at the detection?

Now you're making me want to actually figure this stuff out. If you could make a pulse of "light neutrinos" - neutrinos in the mass eigenstate $\nu_1$ - they would not oscillate. And if they all had the same energy, they'd go faster than a pulse of "heavy neutrinos" - say, those in the eigenstate $\nu_3$ - that also had the same energy.

John Baez (Dec 30 2020 at 20:00):

The way in which solar neutrinos "oscillate" is just that when they're formed, they're formed in a superposition of mass eigenstates, and the different mass eigenstates have phases that change at different rates.