Neutrino Mixing and Masses · Elementary Particle Physics

Three angles, two splittings, one phase, and no absolute scale anywhere. That list is the whole of what this chapter can measure, and the missing last item is why two more sections exist.

🎯 Why this matters

Because only differences of squares appear, no oscillation experiment can ever reach the mass scale, however precise it becomes. That is why the last two sections abandon beams entirely and go to cosmology, to the final electronvolt of a beta spectrum, and to a decay nobody has seen.

Chapter 9 built the Standard Model and, in the middle of it, stated plainly that the recipe gives neutrinos no mass at all — the BEH mechanism needs a right-chirality partner and nobody has ever seen one.

Neutrinos change flavour. A neutrino made as $\nu_\mu$ can be detected as $\nu_\tau$ . That single fact requires them to have mass, so this chapter is the first in the book about physics the Standard Model gets wrong.

The good news for you is that the mathematics is not new. It is §7.11’s quark mixing and §8.1’s meson oscillation with the letters changed. What is new is the numbers, and they are startling.

📐 Physics you need first — why changing flavour proves mass

The argument is three lines and it is worth having before anything else.

A neutrino is made in a weak interaction, so it is made in a flavour state: whatever comes out alongside a $\mu^+$ is called $\nu_\mu$ , by definition. But the states that propagate with a definite frequency are the states of definite mass, $\nu_1$ , $\nu_2$ , $\nu_3$ . If those are different states — if the flavour state is a superposition — then each mass component picks up its own phase $e^{-iE_i t}$ as it flies.

Now the punchline. In the ultra-relativistic limit,

E_i \simeq p + \frac{m_i^2}{2E}

so the relative phase between two components grows as $(m_i^2 - m_j^2)\,t/2E$ . If all the masses were equal, that difference would be identically zero, every component would keep step forever, and the superposition would still be exactly $\nu_\mu$ when it arrived. No flavour change, at any distance.

So flavour change ⟹ unequal $m_i^2$ ⟹ at least two neutrinos are massive. It does not tell you how heavy — only that they differ. That is why every number in this section is a difference of squares, and why the absolute scale is still unknown fifty years later.

⚙️ Engineer’s bridge — one change of basis, three chapters

You have now met this object three times, and it is the same object:

	eigenbasis (propagates)	measurement basis (couples)	the matrix
§7.11 quarks	$d, s, b$ (mass)	$d', s', b'$ (weak)	CKM $V$
§8.1 kaons	$K_S, K_L$ (mass, width)	$K^0, \bar K^0$ (strangeness)	a 2×2 rotation
here	$\nu_1, \nu_2, \nu_3$ (mass)	$\nu_e, \nu_\mu, \nu_\tau$ (weak)	PMNS $U$

In signal-processing terms: the Hamiltonian is diagonal in one basis, the detector is diagonal in another, and the unitary that connects them is the whole story. Two normal modes with slightly different eigenfrequencies, excited together, produce a beat — and the beat is the flavour oscillation. You have seen this in coupled LC tanks, in coupled pendulums and in any two-mode system you have ever simulated.

Where the analogy holds: exactly, for the linear algebra. $U$ is unitary, $|U_{\alpha i}|^2$ is a probability, rows and columns sum to 1.

Where it breaks: a beat in a circuit is energy sloshing between two physical oscillators you could probe separately. Here there is only ever one particle, and $|U_{\alpha i}|^2$ is the probability of a measurement outcome, not a share of a divisible thing. And — the subject of §10.3 — matter can change the eigenbasis as the particle flies, which no passive LC network does.

Two mechanisms, and the book is emphatic that they are different

Almost everything you will read elsewhere calls all of this “neutrino oscillation”. Bettini refuses to, and the distinction is worth carrying from the start.

The two ways a neutrino changes flavour. Left: two mass components beat against each other, and the probability returns — this is oscillation, and it happens in vacuum. Right: matter shifts the effective masses, the state rides one branch down to the surface, and there is nothing to come back to — this is adiabatic flavour conversion. The book insists these are not the same phenomenon, and this site keeps them apart. Detail in §10.2 and §10.3.

Oscillation is interference. The beam carries two (or three) mass components at once, their relative phase grows with $L/E$ , and the flavour composition returns to where it started. It happens in vacuum, it is periodic, it is reversible. Discovered in atmospheric $\nu_\mu$ and confirmed in accelerator beams; energies from below 1 GeV to tens of GeV, baselines from hundreds to thousands of kilometres.

Adiabatic flavour conversion is dynamical. Ordinary matter contains electrons but no muons, so $\nu_e$ alone picks up an extra charged-current forward-scattering amplitude — a refractive index for one flavour only. In a varying density the effective mass eigenstates rotate as the neutrino flies, and it can be carried from one to another and left there. Only one state propagates, so nothing interferes, and the process does not undo itself. This is what happens to the $\nu_e$ born in the core of the Sun.

Aside — why the naming matters

You will constantly see the solar effect called “matter oscillation” or “the MSW oscillation”. Bettini writes, in as many words, that the phenomenon is often called oscillation, but it is not — oscillation is an interference phenomenon and AFC is a dynamical one. Keeping them apart pays off twice: it explains why the solar survival probability has a step in it rather than a wiggle (§10.4), and it is why this site gives them two different widgets instead of one with a mode switch.

The idea, and how long it took

7 of 93 entries. Each dot is one advance; stacked dots share a year. Filter by kind and watch the pattern: theory and experiment take turns, and both wait on the machines and the detectors underneath them.

1962
1968
1985
1987
1992
1998
2002

Bruno Pontecorvo proposed in 1957 that a neutrino might oscillate into its own antineutrino, by analogy with $K^0 \bar K^0$ — which, as it turned out, does not happen. At the time exactly one neutrino species was known; it had just been discovered. In 1962, immediately after the second neutrino was found, groups in Kyoto (Katayama and co-workers) and Nagoya (Maki, Nakagawa and Sakata) proposed mixing between $\nu_e$ and $\nu_\mu$ , and the Nagoya paper raised “transmutation” between flavours. Pontecorvo returned to it in 1967, now with flavours rather than particle–antiparticle, and made the observation that decided the next forty years of experiment: the ideal source is the Sun, and if the effect exists we should see about half the expected electron neutrinos.

The matter effect came later: Wolfenstein computed the potential in uniform matter in 1978, and Mikheyev and Smirnov worked out the consequence for a varying density in 1985.

The mixing matrix

The object that connects the two bases is the PMNS matrix $U$ — Pontecorvo–Maki–Nakagawa–Sakata, after the people who first wrote it down.

\begin{pmatrix}\htmlClass{t-f}{\nu_e}\\ \htmlClass{t-f}{\nu_\mu}\\ \htmlClass{t-f}{\nu_\tau}\end{pmatrix} = \begin{pmatrix} \htmlClass{t-u}{U_{e1}} & \htmlClass{t-u}{U_{e2}} & \htmlClass{t-u}{U_{e3}}\\ \htmlClass{t-u}{U_{\mu1}} & \htmlClass{t-u}{U_{\mu2}} & \htmlClass{t-u}{U_{\mu3}}\\ \htmlClass{t-u}{U_{\tau1}} & \htmlClass{t-u}{U_{\tau2}} & \htmlClass{t-u}{U_{\tau3}} \end{pmatrix} \begin{pmatrix}\htmlClass{t-m}{\nu_1}\\ \htmlClass{t-m}{\nu_2}\\ \htmlClass{t-m}{\nu_3}\end{pmatrix}

(10.1)

Bettini p. 437. The definition of the PMNS matrix: flavour states as combinations of mass states.

Every symbol, one at a time

Hover or tap a symbol above — it lights up in the equation and its meaning, units and type appear here.

💡 What this really says — the neutrino you make is not the neutrino that travels

The neutrino you make is not the neutrino that travels. Read the first row: a $\nu_e$ is $U_{e1}\nu_1 + U_{e2}\nu_2 + U_{e3}\nu_3$ — three different particles, with three different masses, in superposition. Read the first column instead and you get the complementary statement: a $\nu_1$ , if you could somehow make one, would be found as $\nu_e$ with probability $|U_{e1}|^2$ .

Compare §7.11, where the same equation is written for quarks. There the matrix is nearly the identity, so a $d$ quark is a $d'$ quark to within a few per cent and mixing is a correction. Here it is not going to be a correction.

Unitarity means the matrix can be written as three rotations and some phases. If neutrinos are Dirac particles, as the Standard Model assumes, all but one phase can be absorbed into the fields and you get exactly the form of (7.91). If they are completely neutral — Majorana — two more phases survive, and they are called the Majorana phases .

U = \begin{pmatrix}1&0&0\\0&\htmlClass{t-23}{c_{23}}&\htmlClass{t-23}{s_{23}}\\0&-\htmlClass{t-23}{s_{23}}&\htmlClass{t-23}{c_{23}}\end{pmatrix} \begin{pmatrix}\htmlClass{t-13}{c_{13}}&0&\htmlClass{t-13}{s_{13}}\htmlClass{t-d}{e^{-i\delta}}\\0&1&0\\-\htmlClass{t-13}{s_{13}}\htmlClass{t-d}{e^{i\delta}}&0&\htmlClass{t-13}{c_{13}}\end{pmatrix} \begin{pmatrix}\htmlClass{t-12}{c_{12}}&\htmlClass{t-12}{s_{12}}&0\\-\htmlClass{t-12}{s_{12}}&\htmlClass{t-12}{c_{12}}&0\\0&0&1\end{pmatrix} \begin{pmatrix}\htmlClass{t-e}{e^{i\eta_1}}&0&0\\0&\htmlClass{t-e}{e^{i\eta_2}}&0\\0&0&1\end{pmatrix}

(10.2)

Bettini p. 437, with the 1–2 block written so that the product reproduces the book's own second form — see the erratum below. Identical in structure to the CKM factorisation of (7.91).

Every symbol, one at a time

Hover or tap a symbol above — it lights up in the equation and its meaning, units and type appear here.

💡 What this really says — nine complex numbers, and only four of them are free

Nine complex numbers, and only four of them are free. Unitarity kills most of the freedom, and rephasing the fields kills most of what is left, leaving three angles and one phase — exactly the count of the CKM matrix, for exactly the same reason.

The two Majorana phases sit in a diagonal matrix on the right, which is the whole reason they are invisible to everything in this chapter until §10.7. Any flavour transition probability is built from $|U_{\alpha i}|^2$ and $U_{\alpha i}U^*_{\beta i}$ products in which each mass index appears once with a bar and once without — so $e^{i\eta_i}$ meets $e^{-i\eta_i}$ and cancels. Only a process where the same neutrino line is used twice without conjugation can see them, and there is exactly one such process in nature that we can look for.

Aside — the same typographic slip as (7.91), and it is still harmless

The book prints the 1–2 block of (10.2) as $\left(\begin{smallmatrix}c_{12}&-s_{12}\\s_{12}&c_{12}\end{smallmatrix}\right)$ , which turns the opposite way from the one its own second form needs. Carrying out the product exactly as printed gives $U_{e2} = -s_{12}c_{13}$ , where the expanded matrix immediately below has $+s_{12}c_{13}$ .

Multiply (10.2) out both ways

import numpy as np

s12, c12 = 0.5514, 0.8343
s23, c23 = 0.7423, 0.6702
s13, c13 = 0.1463, 0.9892
d = np.deg2rad(238.0)

R23 = np.array([[1,0,0],[0,c23,s23],[0,-s23,c23]], complex)
R13 = np.array([[c13,0,s13*np.exp(-1j*d)],[0,1,0],[-s13*np.exp(1j*d),0,c13]])
printed = np.array([[c12,-s12,0],[s12,c12,0],[0,0,1]], complex)   # as (10.2) prints it
needed  = printed.T                                               # transpose

target = s12*c13                        # the (1,2) element of (10.2)'s second form
for name, R12 in (("as printed", printed), ("transposed", needed)):
    U12 = (R23 @ R13 @ R12)[0, 1]
    print(f"{name:11s}: U_e2 = {U12.real:+.4f}   second form wants {target:+.4f}")

prints

as printed : U_e2 = -0.5454   second form wants +0.5454
transposed : U_e2 = +0.5454   second form wants +0.5454

This is the same slip the site found in (7.91) at §7.11, and it is harmless for the same reason: an overall sign on one element is absorbable into a field phase, so no observable moves. It is listed here only because a reader who multiplies the matrices out — which is exactly what this site asks you to do — will get a sign they cannot account for. The version drawn above is the one that reproduces the book’s own expansion.

The angles, and why they are the headline

\begin{aligned} \sin^2\htmlClass{t-12}{\theta_{12}} &= 0.304^{+0.014}_{-0.013}, &\htmlClass{t-12}{\theta_{12}} &\approx 33.5^\circ\\ \sin^2\htmlClass{t-23}{\theta_{23}} &= 0.551^{+0.019}_{-0.080}, &\htmlClass{t-23}{\theta_{23}} &\approx 48^\circ\\ \sin^2\htmlClass{t-13}{\theta_{13}} &= 0.0214^{+0.0009}_{-0.0007}, &\htmlClass{t-13}{\theta_{13}} &\approx 8.4^\circ \end{aligned}

(10.3)

Bettini p. 438, best-fit values from the global fit of Capozzi et al. 2018. Note that the experiments measure sin²θ, not θ — the angle is derived.

Every symbol, one at a time

Hover or tap a symbol above — it lights up in the equation and its meaning, units and type appear here.

💡 What this really says — the quark matrix is nearly the identity — this one is not a perturbation of anything

Read the right-hand column and compare it with the quark angles of (7.95): 13.2°, 2.4°, 0.21°. The neutrino angles are 33.5°, 48°, 8.4°. The quark matrix is a small perturbation of the identity; the neutrino matrix is not a perturbation of anything.

Two families are almost completely mixed ( $\theta_{23} \approx 45^\circ$ is the maximum possible), one is heavily mixed, and even the “small” angle is forty times the quark sector’s smallest. There is no accepted explanation. It is one of the sharpest unexplained facts in physics: the same operation, applied to quarks and to leptons, produces two matrices that look nothing alike.

Bettini Fig. 10.1 redrawn as a direct comparison, because the comparison is the point. Grey ray = the mass basis; coloured ray = the flavour basis. The book draws the neutrino rotations in three dimensions and sends you to (7.95) for the quark numbers; putting them on the same dial makes the sentence “unlike the case of quarks, the neutrino mixing angles are large” into a picture. Quark values from Bettini Eq. (7.95).

🔢 Worked example — the matrix, from three numbers and a phase

Nothing in (10.4) is independent data: the entire matrix follows from the three angles of (10.3) plus $\delta$ . Build it and read off what each mass state is made of.

Build U from the angles, and read the flavour content

import numpy as np

s12s, s23s, s13s = 0.304, 0.551, 0.0214          # (10.3)
d = np.deg2rad(238.0)                             # (10.5)
s12, s23, s13 = np.sqrt([s12s, s23s, s13s])
c12, c23, c13 = np.sqrt(1 - np.array([s12s, s23s, s13s]))

R23 = np.array([[1,0,0],[0,c23,s23],[0,-s23,c23]], complex)
R13 = np.array([[c13,0,s13*np.exp(-1j*d)],[0,1,0],[-s13*np.exp(1j*d),0,c13]])
R12 = np.array([[c12,s12,0],[-s12,c12,0],[0,0,1]], complex)
U = R23 @ R13 @ R12

P = np.abs(U)**2
print("|U|   ", np.round(np.abs(U), 3).tolist())
print("|U|^2 ", np.round(P, 3).tolist())
print("rows  ", np.round(P.sum(1), 12))
print("cols  ", np.round(P.sum(0), 12))
for k, nu in enumerate(("nu1", "nu2", "nu3")):
    e, m, t = P[:, k] * 100
    print(f"{nu}: {e:4.0f}% nu_e  {m:4.0f}% nu_mu  {t:4.0f}% nu_tau")

prints

|U|    [[0.825, 0.545, 0.146], [0.331, 0.593, 0.734], [0.458, 0.592, 0.663]]
|U|^2  [[0.681, 0.297, 0.021], [0.109, 0.352, 0.539], [0.21, 0.351, 0.439]]
rows   [1. 1. 1.]
cols   [1. 1. 1.]
nu1:   68% nu_e    11% nu_mu    21% nu_tau
nu2:   30% nu_e    35% nu_mu    35% nu_tau
nu3:    2% nu_e    54% nu_mu    44% nu_tau

Every one of those nine magnitudes lands inside the measured 3σ range of (10.4), and rows and columns sum to 1 to machine precision — which they must, since the matrix was constructed from rotations.

The book says $\nu_1$ is “about 70 % $\nu_e$ and the rest is half $\nu_\mu$ and half $\nu_\tau$ ”. The 70 % is right. The rest is 11 % and 21 %, not half and half — and the asymmetry is entirely the doing of $\delta$ . Set $\delta = 0$ in the snippet above and the split moves to 5 % / 26 %; set $\delta = 180^\circ$ and it becomes 20 % / 12 %. This one number is the most visible fingerprint of leptonic CP violation in a static picture of the matrix.

|U| = \begin{pmatrix} \htmlClass{t-r1}{0.803-0.845} & \htmlClass{t-r1}{0.514-0.578} & \htmlClass{t-r1}{0.143-0.155}\\ 0.244-0.498 & 0.502-0.693 & 0.632-0.768\\ 0.272-0.517 & 0.473-0.672 & 0.623-0.761 \end{pmatrix}

(10.4)

Bettini p. 438: the 3σ ranges of the magnitudes, from NuFit 5.2. These are ranges, not values with errors — the entries are strongly correlated.

Every symbol, one at a time

Hover or tap a symbol above — it lights up in the equation and its meaning, units and type appear here.

💡 What this really says — no hierarchy at all, where the quarks had three clean orders of magnitude

Put this matrix beside the CKM matrix of (7.93) and the contrast is the physics. In the quark sector the diagonal is near 1, the first off-diagonals are ~0.2 and the corners are ~0.004: three clean orders of magnitude. Here there is no hierarchy at all. Six of the nine entries could be anywhere between 0.24 and 0.77. A neutrino mass state is, to a decent approximation, an even mixture of flavours.

Note also what kind of numbers these are: 3σ ranges from a global fit, not measurements with error bars. The entries are strongly correlated, because they all descend from the same three angles.

The PMNS matrix, and what constrains each element

	1	2	3	Σ\|U\|² − 1
e				−0.0015 ± 0.0164
μ				+0.0006 ± 0.0576
τ				+0.0002 ± 0.0622
Σ\|U\|² − 1	−0.0008 ± 0.0481	−0.0001 ± 0.0557	+0.0002 ± 0.0452

Click any element to see which experiment measures it — they are nine different measurements, not one. Rows and columns must each sum to 1 if the matrix is unitary; the figures on the right and below are how far off, with the uncertainty propagated from the elements.

Central values computed from the angles of (10.3) and δ = 238°, with a 1σ taken as one third of the half-width of (10.4)'s 3σ ranges. The unitarity check here is not the test it was for the CKM matrix — the ranges come from a global fit that assumes unitarity, so it closes by construction. Compare the genuine version at §7.11.

\htmlClass{t-d}{\delta} = \left(238^{+41}_{-33}\right)^\circ

(10.5)

Bettini p. 438, from the global fit of Capozzi et al. 2018. The 3σ range is 149° ≤ δ ≤ 358°.

Every symbol, one at a time

Hover or tap a symbol above — it lights up in the equation and its meaning, units and type appear here.

💡 What this really says — a 3σ hint, in a field that calls 5σ a discovery

If $\delta$ is anything other than $0^\circ$ or $180^\circ$ , neutrinos and antineutrinos oscillate differently, and the lepton sector violates CP. The best fit sits near the value that maximises the effect. But look at the range before getting excited: the CP-conserving value $360^\circ \equiv 0^\circ$ is only just outside it. This is a $\sim3\sigma$ hint from a field that calls $5\sigma$ a discovery, and it is the single most-wanted number in neutrino physics — the target of the experiments in §10.5.

Why anyone cares beyond bookkeeping: CP violation in the quark sector is far too small to explain why the Universe contains matter and not antimatter. Leptonic CP violation is, at present, the most popular candidate for the missing ingredient.

The mass spectrum: two differences and no scale

Both oscillation and AFC depend only on differences of squared masses, never on the masses themselves. So the chapter can pin down two numbers — the solar splitting $\delta m^2$ and the atmospheric splitting $\Delta m^2$ — and remain completely silent about the third.

\htmlClass{t-s}{\delta m^2} \equiv m_2^2 - m_1^2, \qquad\qquad \htmlClass{t-a}{\Delta m^2} \equiv m_3^2 - \frac{m_1^2 + m_2^2}{2}

(10.6, 10.7)

Bettini p. 438. Two definitions, and the second one is not the obvious choice.

Every symbol, one at a time

Hover or tap a symbol above — it lights up in the equation and its meaning, units and type appear here.

💡 What this really says — which side the lone state sits on is the whole ordering question

Two of the three masses are close together and the third is far away. Which side it is on is unknown, and that is the entire content of the mass ordering question: $\Delta m^2 > 0$ is normal ordering (the lone state is heaviest), $\Delta m^2 < 0$ is inverted ordering (it is lightest).

The definition of $\Delta m^2$ using the average of $m_1^2$ and $m_2^2$ looks fussy but earns its keep: with it, the magnitude is the same quantity in both orderings, so an experiment can quote $|\Delta m^2|$ without first deciding which world it lives in.

the zero of m² is unknown — the whole picture can slide up

Bettini Fig. 10.2 redrawn, with the flavour bars computed rather than sketched — each bar is the |U_αi|² column from the worked example above, drawn to scale. Read them and the two-flavour approximations of the next sections explain themselves: ν₃ is almost pure ν_μ/ν_τ, so the fast oscillation barely involves ν_e; ν₁ is mostly ν_e, so the slow one is what the Sun shows you. The dashed axis is the point: the spectrum’s offset is unmeasured.

\htmlClass{t-s}{\delta m^2} = 73.4^{+1.7}_{-1.4}\ \text{meV}^2, \qquad \left|\htmlClass{t-a}{\Delta m^2}\right| = 2455^{+45}_{-32}\ \text{meV}^2, \qquad \htmlClass{t-al}{\alpha} \equiv \left|\delta m^2/\Delta m^2\right| = 0.03

(10.8, 10.9)

Bettini p. 439. Two measured splittings and the dimensionless ratio that decouples them.

Every symbol, one at a time

Hover or tap a symbol above — it lights up in the equation and its meaning, units and type appear here.

💡 What this really says — two frequencies a factor of 33 apart — which is what licenses every two-flavour formula

Two frequencies, a factor of 33 apart, and the hierarchy parameter $\alpha$ is the one number that says so. That separation is what makes neutrino physics tractable: you never have to solve the full three-flavour problem to understand a given experiment, because at any given $L/E$ one of the two oscillations has not started yet and the other has washed out. It is the same convenience you exploit when a system has widely separated poles — you analyse the fast loop and the slow loop independently and only worry about the crossover.

The crossover is not merely a nuisance, though. An experiment placed exactly where both matter can see the interference between them, and that interference carries the sign of $\Delta m^2$ . That is JUNO’s entire design.

🔢 Worked example — how light can the neutrinos be?

The splittings do not give the masses, but they give a floor. Set the lightest state to zero and the other two are then fixed.

The minimum mass spectrum in each ordering

import numpy as np

dm2, Dm2 = 73.4, 2455.0                      # (10.8), meV^2
print(f"alpha = dm2/Dm2 = {dm2/Dm2:.4f}")    # (10.9)

def spectrum(lightest, ordering):
    if ordering == "NO":                     # m1 < m2 < m3
        m1 = lightest
        m2 = np.sqrt(m1**2 + dm2)
        m3 = np.sqrt((m1**2 + m2**2)/2 + Dm2)
    else:                                    # IO: m3 < m1 < m2
        m3 = lightest
        half = m3**2 + Dm2                   # = (m1^2+m2^2)/2
        m1 = np.sqrt(half - dm2/2)
        m2 = np.sqrt(half + dm2/2)
    return np.array([m1, m2, m3])

for o in ("NO", "IO"):
    m = spectrum(0.0, o)
    print(f"{o} with the lightest at 0: m = "
          f"{m[0]:5.1f} {m[1]:5.1f} {m[2]:5.1f} meV,  sum = {m.sum():5.1f} meV")
print(f"cosmology (Abbott 2022) allows sum < 130 meV")

prints

alpha = dm2/Dm2 = 0.0299
NO with the lightest at 0: m =   0.0   8.6  49.9 meV,  sum =  58.5 meV
IO with the lightest at 0: m =  49.2  49.9   0.0 meV,  sum =  99.1 meV
cosmology (Abbott 2022) allows sum < 130 meV

Two results the book does not spell out here, though both are implicit in its Fig. 10.16 five sections later.

At least one neutrino weighs 50 meV or more, whichever ordering holds. The heaviest neutrino is at most $10^{-7}$ times the electron mass, and it is not zero.

Inverted ordering has a floor of 99 meV on $\sum m_i$ ; normal ordering only 58 meV. Cosmology already bounds $\sum m_i < 130\ \text{meV}$ (§10.6). So IO is being squeezed from a completely different direction than oscillation experiments squeeze it — one more reason the global fit prefers NO by over $3\sigma$ .

📏 Reading meV², and why the units look odd

$\delta m^2 = 73.4\ \text{meV}^2$ is a squared mass in natural units ( $\hbar = c = 1$ ), where mass, energy and momentum share a unit. It is $7.34\times10^{-5}\ \text{eV}^2$ — and the oscillation formula of §10.2 wants it in eV², so watch the prefix.

A useful order-of-magnitude anchor: $\sqrt{73.4}\ \text{meV} = 8.6\ \text{meV}$ is about $1.7\times10^{-8}$ of the electron mass, and about $10^{-11}$ of the top quark’s. Whatever gives neutrinos mass is not the mechanism that gives everything else mass — which is one of the open problems the Epilogue surveys.

Which experiment measures what

Two words recur in that census. A disappearance experiment measures the flux of the source’s own flavour and asks whether some has gone missing; an appearance experiment looks for a flavour that was not there to begin with. Only the second can ever see $\delta$ .

The section closes with a census, and it is more than bookkeeping: every experiment in this chapter is one point on the $L/E$ axis, and the field is the coverage of that axis over eleven orders of magnitude.

Where each source sits on the L/E axis

import numpy as np

Dm2 = 2455e-6                      # eV^2, the atmospheric splitting (10.8)
first_max = (np.pi/2) / (1.27*Dm2) # 1.27 Dm2 L/E = pi/2, L in km, E in GeV
print(f"first atmospheric maximum at L/E = {first_max:.0f} km/GeV\n")

rows = [("solar",        1.5e8, 1e-3),   # 1 AU, ~1 MeV
        ("atmospheric",  1.3e4, 1.0),    # up through the Earth, ~1 GeV
        ("reactor MBL",  1.5,   3e-3),   # Daya Bay
        ("reactor LBL",  180.0, 3.6e-3), # KamLAND
        ("accelerator",  295.0, 0.6)]    # T2K
for name, L, E in rows:
    r = L/E
    print(f"{name:12s} L = {L:9.4g} km  E = {E:6.4g} GeV  ->  L/E = {r:10.3g} km/GeV"
          f"  = {r/first_max:9.3g} x the first maximum")

prints

first atmospheric maximum at L/E = 504 km/GeV

solar        L =   1.5e+08 km  E =  0.001 GeV  ->  L/E =    1.5e+11 km/GeV  =  2.98e+08 x the first maximum
atmospheric  L =   1.3e+04 km  E =      1 GeV  ->  L/E =    1.3e+04 km/GeV  =      25.8 x the first maximum
reactor MBL  L =       1.5 km  E =  0.003 GeV  ->  L/E =        500 km/GeV  =     0.992 x the first maximum
reactor LBL  L =       180 km  E = 0.0036 GeV  ->  L/E =      5e+04 km/GeV  =      99.2 x the first maximum
accelerator  L =       295 km  E =    0.6 GeV  ->  L/E =        492 km/GeV  =     0.976 x the first maximum

💡 What this really says — every experiment in the chapter is one point on the L/E axis

Read the last column. Daya Bay and T2K sit at 1.0 — they were built on the first oscillation maximum, on purpose, and that is what “choosing the baseline” means. Atmospheric neutrinos span the maximum, which is why one detector maps the whole curve. Solar neutrinos are $3\times10^8$ maxima past it: the fast oscillation is averaged into a constant and only the slow physics survives, which is why the Sun measures $\theta_{12}$ and not $\theta_{23}$ .

And KamLAND at 99 maxima is the subtle one. The atmospheric oscillation is completely washed out there, so what KamLAND actually watches is the solar oscillation — with reactors, on Earth. It is the experiment that turned the solar result into a laboratory measurement.

Bettini pp. 439–440. Which sources constrain which parameters, and why.

Source↕	Flavour produced↕	Energy↕	Baseline↕	L/E (km/GeV)↕	Constrains↕
	$\nu_e$	MeV scale	1 AU		θ₁₂ dominantly, plus δm² and θ₁₃
Atmospheric		10²–10⁵ MeV		10⁻¹–10⁵	Δm² and θ₂₃ predominantly, plus θ₁₃ and δ
Reactor, medium baseline	$\bar\nu_e$	few MeV		5×10²	Δm² and θ₁₃
	$\bar\nu_e$	few MeV	10–100 km	5×10⁴	Δm², θ₁₃ and θ₁₂
Accelerator, disappearance		1–10 GeV	10²–10³ km	10²–10³	Δm² and θ₂₃
Accelerator, appearance	$\nu_\mu \to \nu_e\ \text{or}\ \nu_\tau$	1–10 GeV	10²–10³ km	10²–10³

L/E values are order-of-magnitude, from the PyBlock above. The atmospheric first maximum is 504 km/GeV.

Erratum — the chapter roadmap on p. 435 is shifted by one section

The chapter introduction promises the mass limits “in Section 10.5”, the completely-neutral-fermion question “In Section 10.6”, and the neutrinoless double-beta test “Finally, in Section 10.7”. The actual sections are:

	printed promise	actually
mass limits	§10.5	§10.6
are neutrinos completely neutral?	§10.6	§10.7
the 0ν2β test	§10.7	inside §10.7, not a section of its own

The cause is visible in the same paragraph: the roadmap never mentions §10.5 at all — the search for CP violation in leptons, which is one of the most active areas in the field. Everything after the omission is off by one. The site’s section links go to the real ones.

Erratum — the Sun is 10⁸ km away, not 10⁷

Describing solar neutrinos, p. 439 says they reach us “travelling a distance of the order of $10^7$ km”. One astronomical unit is $1.496\times10^8\ \text{km}$ — an order of magnitude more.

It matters a little more than a typo, because $L$ is half of the only variable that governs oscillation. With the correct $1.5\times10^8$ km at 1 MeV, the solar $L/E$ is $1.5\times10^{11}$ km/GeV, which is $3\times10^8$ times past the first atmospheric maximum — comfortably in the fully-averaged regime that §10.3 and §10.4 rely on.

🔑 If you remember only three things

Oscillation is blind to mass itself. What it sees is the phase between two components of one particle, so it reports a splitting and can never report a scale.
The lepton mixing matrix is nothing like the quark one. Two angles are large, and even the smallest is forty times its quark counterpart. No theory explains why.
Two of the matrix’s phases are physical and unobservable at once. They exist only if the neutrino is its own antiparticle, cancel out of every oscillation, and surface in exactly one decay nobody has seen.

Where this goes next

You now have the vocabulary and the numbers. What you do not have is any reason to believe them, and the rest of the chapter is that evidence, in the order it arrived.

§10.2 derives the two-flavour oscillation formula from the Schrödinger equation — the derivation you already met for kaons in §8.1 — and then spends it on the atmospheric $\nu_\mu$ deficit, the accelerator beams that confirmed it, and the reactor experiments that finally caught $\theta_{13}$ .

§10.3 does the matter effect properly: the potential, the resonance, the level crossing, and why a small vacuum angle can become maximal mixing inside a star. §10.4 is the thirty-four-year solar neutrino puzzle and how SNO ended it.

§10.5 hunts $\delta$ , and then turns to what oscillation can never tell you — the absolute mass scale — from tritium decay and from cosmology. §10.7 asks the question underneath all of it: is the neutrino its own antiparticle?

✅ Check yourself — neutrino mixing and masses

0/6 answered · 0 correct

1.Neutrino flavour change proves the neutrinos are massive. What exactly does it prove, and what does it not?
2.The two Majorana phases η₁ and η₂ are physical, yet nothing in §10.1 through §10.6 can detect them. Why not?
3.Set δ = 0° in the worked example's snippet instead of 238°. The book says ν₁ is about 70 % ν_e with the rest split half and half between ν_μ and ν_τ. What happens?
4.The hierarchy parameter is α = |δm²/Δm²| = 0.03. Which statements does that one number license? (Select all that apply.)
5.Reactor experiments at ~1 km and accelerator experiments at ~300 km both sit at L/E ≈ 500 km/GeV. Coincidence?
6.In the PMNS widget above, the row and column sums close on 1 within errors. Why is that a much weaker statement than the same check on the CKM matrix in §7.11?