## Solar Neutrinos

Over on the Category Theory Community Server, John van de Wetering asked me how many times a typical solar neutrino oscillates on its flight from the Sun to the Earth. I didn’t know, and I thought it would be fun to estimate this.

So let’s do it! Let’s do a rough calculation, and worry about details later. For those too lazy to even jump to the end, here are the results:

• A neutrino takes about 500 seconds to travel from the Sun to the Earth.

• Because a typical solar neutrino moving is moving close to the speed of light, time dilation affects it dramatically, and the time of travel from the Sun to the Earth experienced by the neutrino is much less: very roughly, 1/6 of a millisecond.

• There are different kinds of oscillation. If we keep track only of its slower oscillations, a typical solar neutrino oscillates roughly once for each 1250 meters of its flight through space.

• As it travels from Sun to Earth, this typical neutrino does about 120 million oscillations.

Let’s start at the beginning.

The Sun emits a lot of electron neutrinos. Most are produced from a reaction where two protons collide and one turns into a neutron, emitting a positron and an electron neutrino. The proton and neutron then stick together forming a ‘deuteron’, but let’s not worry about that.

More importantly, the energy of the neutrinos produced from these so-called pp reaction is at most 400 keV. That means 400,000 eV, where an eV or ‘electron volt’ is the energy an electron picks up as it falls through a potential of one volt. If you look at this chart:

you’ll see most solar neutrinos have a somewhat lower energy. Let’s say 300 keV.

By comparison to the rest mass of a neutrino, this is huge. Nobody knows neutrino masses very accurately—as we’ll see, people know more about differences of squares of the three neutrino masses. But a very rough estimate for the rest mass of the lightest neutrino might be 0.1 eV/c2. Here like particle physicists I’m measuring mass in units of energy divided by the speed of light squared. An eV, or electron volt, is the change in energy of an electron as it undergoes a one-volt change in potential.

This mass could be way off, say by a factor of 10 or more. But it’s good enough to show this: solar neutrinos are moving very close to the speed of light!

Remember, the energy of a moving particle, divided by its ‘mass energy’, the energy due to its mass, is

$\displaystyle{ \frac{1}{\sqrt{1 - v^2/c^2}} }$

Our solar neutrino, using our very rough guess about its mass, has

$\displaystyle{ \frac{1}{\sqrt{1 - v^2/c^2}} \approx \frac{300 \textrm{keV}}{0.1 \textrm{eV}} = 3 \cdot 10^6 }$

It has an energy 3 million times its rest energy! That gives

$\displaystyle{ 1 - v^2/c^2 \approx \frac{1}{9 \cdot 10^{12}} }$

or

$\displaystyle{ v^2/c^2 \approx 1 - \frac{1}{9 \cdot 10^{12}} }$

or using a Taylor series trick

$\displaystyle{ v/c \approx 1 - \frac{1}{18 \cdot 10^{12}} }$

or if I didn’t push the wrong button on my calculator

$v \approx 0.99999999999994 \; c$

This is ridiculously close to the speed of light.

It’s more useful to remember that our neutrino’s energy is roughly 3 million times what it would be at rest. And relativity says that due to time dilation, the passage of time experienced by this neutrino is slowed down by the same factor!

It takes 500 seconds for light to go from the Sun to the Earth. Our neutrino will take a tiny bit longer—the difference is not worth worrying about. But because of time dilation, the travel time ‘experienced by the neutrino’ will be

$\displaystyle{ \frac{500 \; \textrm{sec}}{3 \cdot 10^6} \approx 1.67 \cdot 10^{-4} \; \textrm{sec} }$

This figure is very rough, due to how poorly we know the neutrino’s mass, but it’s about a 1/6 of a millisecond.

Now let’s think about how the neutrino oscillates.

To keep things simple, let’s assume our electron neutrino gets out of the Sun without anything happening to it. What happens next?

There are three flavors of neutrino—and as it shoots through space, what started as an electron neutrino will ‘oscillate’ between all three flavors, like this:

Here black means electron neutrino, blue means muon neutrino and red means tau neutrino.

You’ll notice that both high-frequency and low-frequency oscillations are going on. This is because the three flavors of neutrino are nontrivial linear combinations of three ‘mass eigenstates’, each of which has a phase that oscillates at a different rate. Two of the mass eigenstates are very close in mass, and this small mass difference causes a small energy difference which causes the slower oscillation. The third mass eigenstate is farther away from the other two, so we also get a more rapid oscillation. As you can see, this is especially noticeable in how the neutrino flickers back and forth between being a muon and a tau neutrino.

But all this is a bit complicated, so let’s just focus on the slower oscillations. How many of those oscillations happen as our friend the neutrino wings its way from Sun to Earth?

To estimate this, let’s pretend there are only the two mass eigenstates that are very close in mass, and ignore the third. The two masses $m_1$ and $m_2$ are not actually known very accurately. What we know is

$m_2^2 - m_1^2 \approx 0.000074 \; \textrm{eV}^2/c^4$

The reason we know these differences in squares of mass is actually by doing measurements of neutrino oscillations: these differences actually determine the frequency of the neutrino oscillations! Let’s see why.

If something has energy $E,$ quantum mechanics says its phase will oscillate over time like this:

$\exp(-i t E / \hbar)$

where $\hbar$ is Planck’s constant and the minus sign is just an unfortunate convention. But all we detect is the absolute value of this, which is just 1: that doesn’t change. So to actually see oscillations we should think about something that can have two different energies $E_1$ and $E_2$. Then we need to think about things like

$\exp(-i t E_1 / \hbar) - \exp(-i t E_2 / \hbar)$

or other linear combinations of these two functions. But their difference illustrates the point nicely: we have

$\exp(-i t E_1 / \hbar) - \exp(-i t E_2 / \hbar) =$

$\exp(-i t E_1) (1 - \exp(-it (E_2 - E_1)/\hbar)$

and the absolute value of this changes with time! It’s

$| 1 - \exp(-it (E_2 - E_1)/\hbar)|$

and the takeaway message here is that it oscillates at a frequency depending on the energy difference,

$\omega = (E_2 - E_1) / \hbar$

So, if we have two kinds of neutrino, it’s the energy difference of the two mass eigenstates that determines how fast a superposition of these two will oscillate. It’s very similar to how when two piano strings are oscillating at almost but not quite the same frequency, you’ll hear ‘beats’ as they go in and out of phase—and the frequency of these beats depends on the difference of their piano strings’ frequencies.

So energy differences are what we care about. But how is energy related to mass? In units where the speed of light is 1, special relativity tells us this:

$E^2 = m^2 + p^2$

where $m$ is mass and $p$ is momentum. One of the mind-blowing moments of my early physics education was watching someone do a Taylor expansion for low momenta and getting this:

$\displaystyle{ E = \sqrt{m^2 + p^2} \approx m + \frac{p^2}{2m} + \cdots }$

It looks more impressive if you don’t set the speed of light $c$ equal to 1:

$\displaystyle{ E = \sqrt{m^2c^4 + p^2c^2} \approx mc^2 + \frac{p^2}{2m} + \cdots }$

So we see that at low momenta the energy is Einstein’s famous $E = mc^2$ plus the kinetic energy $p^2/2m$ famous from classical mechanics before relativity!

But all this is useless for our solar neutrino, which is ‘ultra-relativistic’: it’s moving almost at the speed of light! Now $p^2$ is much bigger than $m^2,$ not smaller, in units where $c = 1.$ So we should do a different Taylor expansion, where we treat $m^2$ as the small perturbation:

$\displaystyle{ E = \sqrt{p^2 + m^2} \approx p + \frac{m^2}{2p} + \cdots }$

Cute, eh? Everything is backwards from what I learned in school: we just switch $m$ and $p.$

This shows us that if we have a neutrino with some large momentum $p$ and it’s a linear combination of two different mass eigenstates with masses $m_1$ and $m_2,$ it’ll be a blend of two energies:

$\displaystyle{ E_1 = \sqrt{p^2 + m_1^2} \approx p + \frac{m_1^2}{2p} + \cdots }$

and

$\displaystyle{ E_2 = \sqrt{p^2 + m_2^2} \approx p + \frac{m_2^2}{2p} + \cdots }$

So, the energy difference is

$\displaystyle{E_2 - E_1 = \frac{1}{2 p} (m_2^2 - m_1^2) }$

and this is what determines the rate at which the neutrino oscillates.

If we stop working in units where $c = 1$ we get

$\displaystyle{E_2 - E_1 = \frac{c^3}{2 p} (m_2^2 - m_1^2) }$

So, the frequency of oscillations is

$\displaystyle{\omega = (E_2 - E_1) / \hbar = \frac{c^3}{2 \hbar p} (m_2^2 - m_1^2) }$

This frequency says how the relative phase rotates around in radians per second. But it’s more useful to think about radians per distance traveled; let’s call that $k.$ Since our neutrino is moving at almost the speed of light, to get this we just divide by $c.$

$\displaystyle{k = \frac{c^2}{2 \hbar p} (m_2^2 - m_1^2) }$

And because the neutrino is ultra-relativistic, its momentum almost obeys $E = p c.$ Here $E$ could be either $E_1$ or $E_2;$ they’re so close the difference doesn’t matter here. So we get

$\displaystyle{k = \frac{c^3}{2 \hbar E} (m_2^2 - m_1^2) }$

This is why people doing experiments with neutrino oscillations measure differences of squares of neutrino masses, not neutrino masses.

For our solar neutrino we’re assuming

$E = 300 \; \mathrm{keV}$

and remember

$m_2^2 - m_1^2 \approx 0.000074 \; \textrm{eV}^2/c^4$

Plugging these in we get

$\displaystyle{k = \frac{1}{2 \hbar c} \frac{0.000074 \; \textrm{eV}}{300,000} }$

Now it gets annoying, and this is where I usually make mistakes. We use

$c = 3 \cdot 10^8 \; \textrm{meter} / \textrm{second}$

$\hbar = 1.05 \cdot 10^{-34} \; \textrm{kilogram} \, \textrm{meter}^2 / \textrm{second}$

$\textrm{eV} = 1.60 \cdot 10^{-19} \; \textrm{kilogram} \, \textrm{meter}^2 / \textrm{second}^2$

and get

$\displaystyle{ k \approx \frac{1}{1600 \; \textrm{meter}} }$

It’s funny how multiplying and dividing all these large and tiny numbers leaves us with something at the human scale!

But actually my computation was sloppy at one point. I warned you! I think it’s actually off by a factor of two. Wikipedia says right answer is

$\displaystyle{k = \frac{c^3}{4 \hbar E} (m_2^2 - m_1^2) }$

and this gives

$\displaystyle{ k \approx \frac{1}{3200 \; \textrm{meter}} }$

So, the neutrino oscillates at a rate of about one radian every 3200 meters! And to get the wavelength of the oscillation we need to multiply by $2 \pi.$ So our solar neutrino makes a complete oscillation about once every 20 kilometers!

And the distance from the Earth to the Sun is 150 million kilometers. So, our neutrino oscillates about 7.5 million times on its trip here.

You should take all this with a grain of salt since I easily could have made some mistakes. If you find errors please let me know! I leave you with a puzzle:

Puzzle. Where does the missing factor of 2 come from?

I don’t think you need to know fancy physics to solve this. I think the mistake is visible in my calculations.

### 18 Responses to Solar Neutrinos

1. Geoff Clare says:

Since you used the time experienced by the neutrino in your calculation, you also need to use the distance experienced by the neutrino. Applying the same factor of 3 million, that distance is 50 km and the neutrino will oscillate about 40 times.

• John Baez says:

No.

For starters, I didn’t actually use the time experienced by the neutrino in my calculation of how many times the neutrino oscillates as it comes from the Sun to Earth. I computed that time just for fun, as a side-show!

That time could easily be wrong by an order of magnitude, since it’s essentially proportional to the neutrino’s mass, and we don’t know the masses of neutrinos very well. What we know well are the differences in the squares of their masses—because that’s what determines their rate of oscillation.

(For details read my blog article.)

• Geoff Clare says:

Thanks for putting me straight. I admit I glossed over the details of the calculation and just assumed that the reason you worked out the time experienced by the neutrino was because you needed it later on. It might help others avoid making the same mistake if you were to add something like “(Working this out was just for fun – it’s not used in the calculations below.)” after you say “it’s about a 1/6 of a millisecond”.

2. arch1 says:

Eyeballing your “plugging it in” formula it looked like your result was about 30% low, so I started looking for the remaining factor of 5/7 before I remembered that’s not how you physicists think:-)

• John Baez says:

Are you claiming I did that calculation at the end incorrectly? Multiplying and dividing large numbers is an error-prone business with my current setup, so it’s quite possible.

There are no factors of 5/7 in this game. In the overall argument I deliberately made a mistake that involves a factor of 2, and I want to see if anyone can find that. But there could also be mistakes where I simply multiplied numbers wrong, etc. Please help me catch them!

For a while I was forgetting that neutrino masses squared are really measured in units of keV/c4, not keV. The reason is that physicists often set c = 1 and measure masses in keV, when they really mean keV/c2. This led to a final answer that was off by a factor of

$c^4 \approx 8 \cdot 10^{33}$

3. Wyrd Smythe says:

In my old age I’m trying to learn quantum mechanics (and the necessary math). I must be making progress; I was able to follow along with nearly every bit of that. I really enjoyed it, thanks!

• John Baez says:

Great, thanks! I’d always wondered why we know the differences of squares of masses of neutrinos much better than their masses, so it was fun to work through this. It turns out that’s more about special relativity than quantum mechanics, though a little quantum mechanics comes into play.

• Wyrd Smythe says:

That may be why I kept up as well as I did! I feel that I understand SR okay. (Enough to presume to write blog posts about it!)

I’ve read your recent neutrino posts, and the comments, with great interest. I had no idea the flavor and mass eigenstates were so different, let alone about the possibility of the masses being inverted. (I agreed with your comment it would be weird if that were so, and I saw the comments to the effect that we’re now pretty sure it’s not.)

I rather like that the SM still has its own mysteries! ;)

4. amarashiki says:

After many, many years…Your style and wisdom continue to amaze me, John. Great piece. Something to learn always from you. Beyond the given comment about the space and time felt by the neutrino itself, I just to add some of my own knowledge of my favorite particle:

1st. It is true that a photon comes from the sun in 500 seconds, iff you assume it comes from the solar corona. Due to nuclear opacity and the solar density, photons require a much more long time to departure (as surely you know from supernova physics: neutrinos come first, then photons…It is lot of fun to make the numbers you did for the sun, for the SN1987A neutrinos and photons assuming they came from the Magallanic clouds…). About the calculation of the time that the photon needs to cross over the nuclear shells and arrive to the corona, I did the numbers at the end of this post http://www.thespectrumofriemannium.com/2019/08/20/log218-atomic-elephants/

2nd. With respect to neutrinos, my master thesis topic 10 years ago, there is a relatively unknown interaction called coherent neutrino-nuclei scattering, mediated by neutral current interactions (via Z boson interchanges!) whose interaction cross-section is proportional to the Fermi constant squared and the so-called weak charge. It was detected, despite its tininess, by the COHERENT collaboration. It was, again, a SM prediction…SM rules. SM rocks. SM remains even after the end of 2020. Why the coherent neutrino-nuclei interaction is important and interesting? There are two reasons: 1) it mimicks DM interactions (due to neutrality of Z-bosons) and it provides an irreducible background to DM experiments (in a few years, DM detectors will be able to detect coherent neutrino-nuclei interactions coming from the sun! So, we need to understand it in order to not confuse known neutrino interactions with putative DM particles due to the nuclear recoils of the latter), 2) In principle, coherent neutrino nuclei interactions could be sensitive enough to rule out new forces affecting the weak charge and the nuclear form factor or even they could provide a tool to probe the nuclear form factors whose knowledge is important to decide the Majorana character of the neutrinos in different experiments.

Said this, I also have some posts about neutrino and beyond neutrino physics at my blog, but I will allow for eager readers to explore for them. Only one more thing…Even if sterile neutrinos are out there, Cosmology and Majorana neutrino searches will push also for the not less important issue of the spectrum type (normal or inverted). Also, it is expected that if neutrino has INVERTED spectrum, the next decade will provide evidence of it. If the spectrum is of normal hierarchy, we shall expect about two decades from now to see it probed (unless new technology in neutrino baseline experiments shows it clear).

BONUS: Many people bet that neutrinos, at least SM neutrinos, are Dirac particles, despite the fact it introduces as the little hierarchy problem (or why neutrinos masses are so low as compared with any other SM fermion; interestingly, it is not highly highlighted that there is no MASSLESS fermion in the SM due to neutrino oscillations and thus, there are no Weyl fermions in that theory). That neutrino masses and oscillations can be obtained by the seesaw mechanism is not quite the point…I would like to comment that, if you assume the seesaw mechanism as universal, the Planck mass can not be the scale of right-handed neutrinos if it has to explain the possible SM neutrino masses. Therefore, neutrino oscillations with the seesaw, provide a test of beyond Standard Model physics that is NOT the Planck scale (or TOE scale). Surprisingly, right-handed neutrino could be close to GUT scale, and future multimessenger astronomy could give us hints of these huge neutrino energy scale. There is a mechanism, that can be called the ZeVatron (Megatron or Galvatron are boring Transformers here, lol), in which neutrinos interacting with the CMB photons could produce events with ZeV energy at some kind of experiments!

BONUS(II): Cosmological vs. cosmogenic neutrinos. The ZeV neutrinos are called cosmogenic, but there are also cosmological neutrinos out there! Even here! There are about 411 photons/m³ of the CMB recombination event, but also about 411/4 neutrinos/m³ of the CMB time! They have, asumming SM only physics, a current temperature around 2 K (more precisely about 1.945 K if T(CMB) is taken 2.725 K). Of course, this temperature could be lesser if there are extra relativistic neutrino out there (it seems the 4th neutrino hypothesis have not been ruled out yet completely…but I can not be sure about it right now). The detection of cosmological neutrinos is another experimental challenge for the next 1-3 decades. Hopefully, a new technique used in next generation DM detectors will provide a way to search for them!

By the way, I like a lot the count down rule 3 K, 2 K, 1 K to remember the temperature of the cosmological photon, neutrino and graviton backgrounds (assuming SM physics only!).

• John Baez says:

Thanks for the long comment! I got pulled into learning a bit more neutrino physics because I’ve been thinking about octonions and the Standard Model and I realized I should learn more about the trickiest corner of the Standard Model: neutrinos!

With respect to neutrinos, my master thesis topic 10 years ago, there is a relatively unknown interaction called coherent neutrino-nuclei scattering, mediated by neutral current interactions (via Z boson interchanges!) whose interaction cross-section is proportional to the Fermi constant squared and the so-called weak charge. It was detected, despite its tininess, by the COHERENT collaboration.

That’s incredibly cool. I’d vaguely heard about that, which is why I wrote

To keep things simple, let’s assume our electron neutrino gets out of the Sun without anything happening to it.

I also have some posts about neutrino and beyond neutrino physics at my blog, but I will allow for eager readers to explore for them.

I’ll check them out.

Many people bet that neutrinos, at least SM neutrinos, are Dirac particles, despite the fact it introduces the little hierarchy problem (or why neutrinos masses are so low as compared with any other SM fermion)

I guess I bet this too. The miraculous similarity between quarks and leptons seems too good to be just partially true; since quarks don’t seem to have a Majorana mass it seems sad to spoil the similarity by giving neutrinos a Majorana mass, even it helps solve a “hierarchy problem”. Since we don’t really understand anything about why particles have the masses they do, it seems a bit risky to postulate new physics to get them to have masses on the right order of magnitude without fine-tuning.

Anyway, all this reasoning, including mine here, is very hand-wavy. So I don’t feel very certain about any of these things.

There are about 411 photons/m³ of the CMB recombination event, but also about 411/4 neutrinos/m³ of the CMB time! They have, asumming SM only physics, a current temperature around 2 K (more precisely about 1.945 K if T(CMB) is taken 2.725 K).

• Paul Frampton, Stephen D. H. Hsu, Thomas W. Kephart and David Reeb,What is the entropy of the universe?, Classical and Quantum Gravity 26 (2009), 145005.

• Chas A. Egan and Charles H. Lineweaver, A
larger estimate of the entropy of the universe
,
The Astrophysical Journal 710 (2010), 1825.

The first paper estimates the entropy of the observable universe at very roughly 10102 bits. The second estimates it at very roughly 5 × 10104 bits. The difference is due to an increased estimate of the number of supermassive black holes: that is, black holes of masses at least 10 million times that of the Sun, living at the centers of galaxies. These dominate the entropy of the universe!

But neutrinos also play a big role. Measured in bits, here’s how different things contribute to the entropy of the observable universe:

• stars: 1081 bits.

• interstellar and intergalactic gas and dust: 1082 bits.

• gravitons: 1088 bits.

• neutrinos: 1090 bits.

• photons: 1090 bits.

• stellar black holes: 1098 bits.

• supermassive black holes: 5 × 10104 bits.

5. Greg Egan says:

Great post!

I think there might be a factor of 10 error. Wikipedia gives $\delta m_{12}^2 \approx 7.59 \times 10^{-5}$, and though you write 0.000079 in most places (not sure if that should be 0.000076), in the formula for $k$ you have one less zero, and you seem to have used that typo in the subsequent calculations.

Anyway, FWIW I get 1/1556 metres from the same formula for $k$.

On the factor of 2, do you mean the frequency of the absolute value of the difference of exponentials? I guess that boils down to asking something equivalent to whether the frequency of:

$\sqrt{2-2\cos(t)} = 2|\sin(t/2)|$

is 1 or 1/2 … which confuses me a bit, because this isn’t a pure sinusoid, and it does have period $2\pi$ rather than $4\pi$.

• John Baez says:

Greg wrote:

I think there might be a factor of 10 error. Wikipedia gives $\delta m_{12}^2 \approx 7.59 \times 10^{-5}$, and though you write 0.000079 in most places (not sure if that should be 0.000076), in the formula for $k$ you have one less zero, and you seem to have used that typo in the subsequent calculations.

Yikes—thanks for catching that. I did all these calculations while writing the blog article rather than on a separate pad, because I started out planning to write a much shorter article. That made such mistakes almost inevitable. I caught some but not all! I’ll fix this one.

On the factor of 2, do you mean the frequency of the absolute value of the difference of exponentials?

Yes! That’s what I was alluding to. I agree this is confusing, at least as I described it. But we should probably compute either the expected value of an observable or a transition probability, and as a function the answer to those questions has to be a finite linear combination of exponentials (since it always is, for any quantum system with a finite-dimensional state space).

• John Baez says:

The latest, best data from NuFit 5.0 2020 says that $\Delta m_{12}^2$ is $7.42 \cdot 10^{-5} \; \textrm{keV}^2/c^4$, with one-σ error bars of 0.2. NuFit is a website solely devoted to your neutrino data needs. So I’ll redo my calculation with that value.

By the way, the three-σ range for this quantity goes from $6.82$ to $8.04 \cdot 10^{-5} \; \textrm{keV}^2/c^4$, showing how hard it is to measure these things! We’re not even sure of the first decimal, though it’s probably 7.

• Greg Egan says:

Thanks!

BTW, I recently stumbled on this interesting, but possibly dated, paper that seems to raise the possibility that chirality oscillations, in addition to the flavour oscillations, could help explain some observations:

https://arxiv.org/abs/hep-ph/0511232

• John Baez says:

Hmm, my instinctive reaction to that claim is suspicion.

In the last few weeks I’ve taken to browsing the arXiv on selected aspects of particle physics. It’s amazing what a range of stuff there is. Searching for the string “neutrino mass matrix” gets 960 hits. One hopes that the better ideas get amplified, but I keep hoping for an unnoticed gem.

6. amarashiki says:

By the way, some computations of you remember myself doing the slides of my Master Thesis 10 years ago. Available for free here http://www.thespectrumofriemannium.com/talks/masterthesistalk/#dearflip-df_11464/1/
Bonus if you catch: 1) the movie from where one ending slide comes from, 2) TV series from where the Z-neutrino stuff appears.

7. amarashiki says:

By the way, the solution to the riddle of the 2 factor is easy done for 2 neutrino species…
Take: $A=\exp(ix/2)$, $B=\exp(iy/2)$. And suppose only two neutrino especies, mixing as $\Psi'=\sum_{i=1}^2 U_i\Psi$, and assume for simplicity that the PMNS mixing matrix is the unit or Kronecker delta matrix (ironically speaking, it happens to be aproximately true for quarks not for neutrinos or leptons)
The transition matrix would be roughly

$\langle i\vert j\rangle\rightarrow \mathcal{A}(i,j)=\vert \langle i\vert j\rangle \vert^2.$

Then, simple complex calculus provides

$\langle i\vert j\rangle^2\propto 4\cos^2(x-y)/2\sim 4\sin^2(x-y)/2.$

where the last step is true up to a phase…Take $x=m_1^2/2L$, $y=m_2^2/2L$, then you get roughly

$(x-y)/2\sim \Delta m^2 c^3/4E\hbar=1/L_{\nu} =k_\nu$.

In the complete 3 flavor (or N-flavor) case, you get for the probability something like

$\delta_{ij}-4Re (B_{ij})\sin^2(\Delta m_{ij} c^2L/4E\hbar)+2Im (C_{ij})\sin^2(\Delta m_{ij} c^2L/2E\hbar)$

Indeed, to consider $\Delta m^2c^3 L/4E\hbar$ as the basic phase or $\Delta m^2c^3 L/2E\hbar$ is not too important, since the point is phase differences! And the point that makes all this thing important is that massless neutrinos (Weyl fermions) can not explain the neutrino oscillations. I did as undergraduate a work about this for my Astrophysics class…The old SM died the day we discovered neutrino oscillations. They are evidence for new physics beyond the Standard Model! Indeed, not only the spectrum is mysterious, as well it is the number of massive neutrinos! What? Yes, in principle since we do measure mass differences, it could show up that one neutrino flavor could be indeed massless. I think. I have not read about this topic since my graduated ages…In principle, I believe, a quasidegenerated neutrino spectrum is “almost” excluded. And at least, there should be 2 massive neutrino species (there can be 3 as well, of course). People have been trying models of neutrino masses through symmetries of the PMNS matrix (complementary somehow to the CKM quark mixing matrix). Before the measurement of $\theta_{13}$ tribimaximal matrices were preferred for mixing models of neutrinos, I am not sure of what is the state-of-art of this stuff right now…I should read but my desktop computer is apparently dead, and my current laptop is also 7 years old, so I am minimizing workflows until new PC comes in…

In summary, the observation of neutrino oscillations give up Weyl neutrino for the 3 flavor states, since we measure differences between flavor squared masses of flavor states. Moreover, the seesaw mechanism provides a way to explain neutrino masses BUT the upper scale can not be Planck scale or even the GUT scale, …Using seesaw Type I you can guess that arount 1-100 PeV some new physics should arise due to neutrinos (maybe PeV scale Right-Handed superheavy neutrinos / dark matter? No one knows…Cosmic rays and other extreme astronomy is important there for those neutrino/cosmic ray events). Finally, Majorana phases are relevant but could be harder to detect than expected (some recent papers on the arXiv warn us about it…). Hard work is ahead of us…

8. amarashiki says:

Beyond the 2 non parse equations (I did not care and did not close two }), the formula of the probability should have a sin not squared after the Im(part)…Sorry for the typos, but I can not edit it myself now…

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