A neutrino that never interferes with anything can still change flavour completely. The mechanism is a moving eigenbasis rather than a growing phase, and it takes a star to supply one.
🎯 Why this matters
The method works only because the Sun’s density falls slowly enough, which nobody arranged and nobody could check in advance. A star steeper than ours would have sent the same neutrinos out with a different answer, and the measurement would have been read wrong.Everything so far has been oscillation: two mass components interfering, the probability returning periodically, and nothing in the medium mattering at all.
This section is the other mechanism, and the book is emphatic that it is not the same one. Ordinary matter contains electrons and no muons, so alone picks up an extra charged-current amplitude — a refractive index for one flavour. In a medium whose density changes along the path, the effective mass eigenstates rotate as the neutrino flies, and it can be carried from one to the other and left there. Only one state propagates, so nothing interferes, and the process does not undo itself.
That is adiabatic flavour conversion adiabatic flavour conversion AFC: flavour change caused by propagation through matter of varying density, when the effective mixing angle rotates slowly enough that the state stays on one eigenvalue branch. Dynamical and irreversible, and not oscillation — nothing interferes, because only one state propagates. defined in §10.1 — open in glossary , the MSW effect MSW effect Mikheyev–Smirnov–Wolfenstein: adiabatic flavour conversion as it happens in the Sun. Wolfenstein (1978) supplied the matter potential, Mikheyev and Smirnov (1985) the level-crossing consequence. defined in §10.3 — open in glossary , and it is what happens to the born in the core of the Sun.
The Sun as a neutrino source
Solar neutrinos solar neutrinos electron neutrinos only, from the pp chain: the pp continuum below 420 keV (98 % of the flux, and fixed by the solar luminosity alone), the ⁷Be lines at 0.86 MeV, and the ⁸B continuum out to 14 MeV — rare, model-dependent, and the easiest to detect. defined in §10.3 — open in glossary all come from one chain of reactions, and every step of it emits electron neutrinos and only electron neutrinos.
Bettini p. 452. The net reaction that powers the Sun, and the elementary step that starts it.
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 source is pure ν_e, and that is all neutrino physics needs from the Sun
Thermal energies in the core are tens of keV; the Coulomb barrier between two protons is a thousand times that. Fusion happens anyway, by tunnelling, at a rate so low that it takes the whole mass of a star to make a noticeable amount of light — and that is a feature. A reaction that was easy would have burned out long ago.
For neutrino physics only one thing matters: the source is pure , at a known rate. Whatever else is uncertain about the Sun, the flavour is not.
Bettini Fig. 10.11(a). The three boxes that emit neutrinos are outlined in cyan, and each is a different experiment’s target: pp below 420 keV needs gallium, ⁷Be at 0.86 MeV needs chlorine or a very clean scintillator, ⁸B out to 14 MeV is the only one a water Cherenkov can see. The last branch is printed as 0.01 % in the book — see the erratum below.
🔢 Worked example — Example 10.1, and the flux budget it implies
Every helium nucleus releases 26.7 MeV and two neutrinos. About 0.6 MeV of that leaves with the neutrinos, so 26.1 MeV arrives as light. Divide the solar constant by the light-per-neutrino and you have the neutrino flux, with no solar model at all.
The solar constant gives the neutrino flux
# Example 10.1, then the budget it implies
S, J_per_MeV = 1.3e3, 1.602176634e-13 # W m^-2, J/MeV
E_per_nu = 26.1/2 # MeV carried by photons per neutrino
flux = S/(E_per_nu*J_per_MeV)
print(f"energy per neutrino = {E_per_nu:.2f} MeV = {E_per_nu*J_per_MeV:.2e} J")
print(f"total neutrino flux = {flux:.2e} m^-2 s^-1 (the book gives 6.2e14)\n")
ssm = {"pp": 5.98e14, "7Be": 4.86e13, "pep": 1.44e12, "8B": 5.58e10} # m^-2 s^-1
tot = sum(ssm.values())
for k, v in ssm.items():
print(f" {k:4s} {v:9.2e} m^-2 s^-1 {100*v/tot:6.2f} % of the total")
print(f" sum {tot:9.2e} vs {flux:.2e} from the solar constant alone")
print(f"\n8B / 7Be = {ssm['8B']/ssm['7Be']:.4f} -> the 7Be+p branch is "
f"{100*ssm['8B']/ssm['7Be']:.2f} %, not 0.01 %")energy per neutrino = 13.05 MeV = 2.09e-12 J total neutrino flux = 6.22e+14 m^-2 s^-1 (the book gives 6.2e14) pp 5.98e+14 m^-2 s^-1 92.27 % of the total 7Be 4.86e+13 m^-2 s^-1 7.50 % of the total pep 1.44e+12 m^-2 s^-1 0.22 % of the total 8B 5.58e+10 m^-2 s^-1 0.01 % of the total sum 6.48e+14 vs 6.22e+14 from the solar constant alone 8B / 7Be = 0.0011 -> the 7Be+p branch is 0.11 %, not 0.01 %
Sixty billion neutrinos cross every square centimetre of you every second, and the estimate needed nothing but a light meter and the arithmetic above.
The budget underneath is the reason the field is hard. The pp component is 92 % of the flux, is fixed by the luminosity alone, and is almost model-independent — and it is the hardest to detect, because 420 keV is below almost every threshold. The ⁸B component is one part in , depends on the core temperature as , and is the only one the big water Cherenkov detectors can see. Everything in §10.4 follows from that mismatch.
Erratum — the ⁷Be branching in Fig. 10.11(a) is 0.1 %, not 0.01 %
The figure labels the two ⁷Be branches 99.9 % and 0.01 %. A pair of branches must sum to 100 %, and 99.9 + 0.01 = 99.91. The second should be 0.1 %.
The book’s own Fig. 10.11(b) says so too: the ⁸B flux there is about of the ⁷Be flux, which is the branching ratio, because every ⁸B comes from a ⁷Be. The standard solar model gives against m⁻²s⁻¹ — 0.11 %.
A plausible origin, offered as an observation rather than a claim: 0.01 % is the right number for a different quantity — the ⁸B share of the total neutrino flux, as the worked example above prints.
- pp (5.98×10¹⁴ total)
- ⁸B (5.58×10¹⁰ total)
- ⁷Be and pep lines (m⁻² s⁻¹)
What matter does to a neutrino
All three flavours scatter off electrons and quarks by the neutral current, identically. Only can also do it by the charged current, because only can turn into the electron that is already there. That asymmetry is the entire mechanism, and its size is the matter potential matter potential the extra charged-current forward-scattering amplitude that only ν_e has, V = √2 G_F N_e. It is a refractive index for neutrinos, n − 1 ≈ 10⁻¹⁸ in the solar core; being proportional to G_F rather than G_F² is why something this small is observable. defined in §10.3 — open in glossary .
Bettini p. 454. The extra potential seen by an electron neutrino, and by nothing else.
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 refractive index that differs from 1 in the eighteenth decimal place
A neutrino crossing matter behaves exactly like light crossing glass: it does not get absorbed, it gets slowed — and only one polarization, so to speak, is slowed. The refractive index is , and at 10 MeV it is away from 1 in the centre of the Sun and in the Earth.
Those are the two most absurd numbers in the chapter, and neither is negligible. An index that differs from 1 in the eighteenth decimal place changes what comes out of the Sun by a factor of two, because a phase difference accumulates over km of stellar interior. The lesson is the one you already know from interferometry: an unmeasurably small index difference becomes an enormous effect given a long enough arm.
⚙️ Engineer’s bridge — this is a tapered directional coupler
§10.2a mapped vacuum oscillation onto two coupled resonators: off-diagonal element = coupling, diagonal difference = detuning, transfer fraction = . Matter keeps the coupling fixed — the book says so explicitly, “these do not depend on density” — and changes only the detuning.
You have built this. It is a tapered coupler, or an adiabatic mode converter: two waveguides whose propagation constants are swept through each other along the propagation direction. Sweep slowly and the power follows one supermode across the crossing and comes out entirely in the other guide. Sweep fast and it stays put and you get a beat instead. The slow-sweep condition is called adiabatic in both fields, and it is the same condition.
Where it holds: exactly. Same 2×2 Hamiltonian, same avoided crossing, same adiabaticity criterion, same one-way transfer.
Where it breaks: in a coupler you choose the taper. Here the taper is the density profile of a star, nobody designed it, and — as Bettini notes — nature happened to pick the regime in which it works. Also, the coupler’s output is a field amplitude you can measure; here it is the probability of a yes/no outcome on one particle.
Bettini p. 454, WITH THE MATTER TERM HALVED so that it agrees with the book's own (10.37)–(10.39) — see the erratum below. Two flavours, because ν_μ and ν_τ are indistinguishable at solar energies and θ₁₃ is dropped.
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 — matter moves one knob — the detuning, never the coupling
One knob moved. Compare with §10.2a’s vacuum Hamiltonian (10.16): the off-diagonal entries are identical, and matter has added something to the diagonal and to nothing else.
By (10.17) — the innocuous ratio that Bettini flagged as “useful in the following” — mixing is the competition between those two. So matter cannot change how strongly the flavours couple; it can only change how badly they are detuned. And a detuning can be driven through zero.
Erratum — (10.36)‘s matter term is twice what (10.37)–(10.39) need
The book prints on the first diagonal entry and on the second — the full potential difference on each, rather than half of it on each. Then
so feeding (10.36) into (10.17) requires , while (10.38) defines . The two equations, three lines apart, disagree by a factor 2.
Which of the two is the odd one out?
import numpy as np
hbarc, GF = 1.9732698e-7, 1.1663788e-23 # eV m, eV^-2
dm2, th = 73.4e-6, np.deg2rad(33.5) # eV^2, rad
Ne, E = 3e31, 5.0e6 # m^-3, eV
V = np.sqrt(2)*GF*(Ne*hbarc**3) # (10.34)
d = dm2/(4*E)
def H(v): # (10.36) with the matter term v on each diagonal
return np.array([[-d*np.cos(2*th) + v, d*np.sin(2*th)],
[ d*np.sin(2*th), d*np.cos(2*th) - v]])
A = 2*np.sqrt(2)*GF*(Ne*hbarc**3)*E # (10.38)
target = dm2*np.sin(2*th)/(dm2*np.cos(2*th) - A) # (10.37) right-hand side
for name, v in (("as printed, V", V), ("traceless, V/2", V/2)):
h = H(v)
print(f"{name:16s}: (10.17) gives tan2th_m = {2*h[0,1]/(h[1,1]-h[0,0]):+.5f}"
f" (10.37)+(10.38) want {target:+.5f}")
print()
for name, f in (("as printed, V", 4.0), ("traceless, V/2", 2.0)):
Eres = dm2*np.cos(2*th)/(f*np.sqrt(2)*GF*(6e31*hbarc**3))/1e6
print(f"{name:16s}: the Sun's core reaches the resonance for E > {Eres:.2f} MeV"
f" (the book says about 2 MeV)")as printed, V : (10.17) gives tan2th_m = -1.42649 (10.37)+(10.38) want -7.23212 traceless, V/2 : (10.17) gives tan2th_m = -7.23212 (10.37)+(10.38) want -7.23212 as printed, V : the Sun's core reaches the resonance for E > 0.94 MeV (the book says about 2 MeV) traceless, V/2 : the Sun's core reaches the resonance for E > 1.89 MeV (the book says about 2 MeV)
(10.36) is the odd one out. With the term halved, (10.17) reproduces (10.37)+(10.38) exactly, and the resonance threshold comes out at 1.9 MeV, which is the “about 2 MeV” the book states in (10.43). As printed it would be 0.94 MeV, and the whole energy scale of the solar story would move by a factor of two.
The physical reason for the half: only is observable, and putting and on the two entries is what makes the added matrix traceless — an equal shift on both flavours is an overall phase and can never do anything. The equations that follow, and the diagram in Fig. 10.12, all use the correct version.
The resonance
Put (10.36) into (10.17) and out comes the effective mixing angle in matter effective mixing angle in matter θ_m, the vacuum mixing angle rotated by the local electron density. Its tangent diverges at the resonance, so a small vacuum angle can become maximal mixing inside a star. defined in §10.3 — open in glossary — the vacuum angle as the medium has rewritten it.
Bettini p. 455. The vacuum mixing angle, rotated by the medium. This is (10.17) with the matter Hamiltonian in it.
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 angle in matter can be anything, whatever it is in vacuum
The mixing angle in matter can be anything, no matter what it is in vacuum. Set the density so that and the denominator vanishes: exactly, maximal mixing, even if the vacuum angle were one degree.
That is the sentence that resolved a thirty-four-year puzzle. It also means the vacuum angle cannot be read off a solar measurement without knowing the density where the neutrino was born — the Sun is not a passive filter, it is part of the apparatus.
The density at which the denominator vanishes is the MSW resonance MSW resonance the density N_e = δm² cos2θ₁₂ / (2√2 G_F E) at which the effective mixing becomes maximal. Read as a condition on energy instead, it puts a transition at about 2 MeV that splits the solar neutrino spectrum into a low-energy oscillating half and a high-energy converting half. defined in §10.3 — open in glossary .
Bettini p. 455 — the resonance condition, written as a density. Reading it as a condition on ENERGY instead is what splits the solar spectrum in two.
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 Sun is an energy sorter, and the boundary sits mid-spectrum
The Sun is an energy sorter. It has a maximum density, so it has a minimum energy it can resonate. Neutrinos below that threshold walk out as if the Sun were not there; neutrinos above it get converted. One star, two completely different physics regimes, and the boundary sits at about 2 MeV — right in the middle of the solar neutrino spectrum, which is either a coincidence or the luckiest thing in the subject.
Plot the two eigenvalues against density and the picture is Fig. 10.12: a level crossing level crossing the anti-crossing of the two effective mass eigenvalues at the MSW resonance: the ν_e-like branch is heaviest at high density and lightest in vacuum. Borrowed wholesale from atomic and molecular physics. defined in §10.3 — open in glossary that the coupling turns into an avoided one, with the two branches approaching to within δm²sin2θ₁₂ and no closer.
Matter, the level crossing, and what the Sun does to a νₑ
- m̃²₂ — the state a solar νₑ is born in
- m̃²₁
- νₑ diagonal element
- ν_α diagonal element
This neutrino meets the resonance. It is born at θ_m = 63° — well past 45°, so almost pure ν̃₂ — rides the upper branch out through the resonance, and leaves the Sun as ν₂. Nothing interferes with it, so nothing oscillates: the survival probability is just the νₑ content of ν₂, sin²θ = 0.305.
Drag the energy slider through 1.9 MeV and watch the red line cross the grey one. That single crossing splits the solar neutrino spectrum in two, and it is why thirty-four years of experiments at different energies measured three different deficits.
🪜 The journey of a solar νₑ, in five steps
Step 1 of 5 — Born in the core(10.41)
Why you may do this: At the core density A ≫ δm²cos2θ₁₂, so the denominator of (10.37) is large and negative and θ_m → 90°. A ν_e is then, to a good approximation, the single matter eigenstate ν̃₂ — the HEAVIER one. Nothing else is produced.
This is the step that makes the whole phenomenon possible: the neutrino is born in an eigenstate, not a superposition. There is nothing for it to interfere with, which is why what follows is not oscillation.
Bettini pp. 455–456, Eqs. (10.40)–(10.44). This is the argument the widget above draws; here it is in words, with the justification for each step.
Bettini p. 456, with the exponent of N₀ corrected from 10³³ and the units of the constant from eV to eV⁻¹ — see the errata below. Below this energy a neutrino never meets a resonance anywhere in the Sun.
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 lines, one threshold, and all of §10.4 downstream of it
Two lines, one threshold, and the whole of §10.4 is downstream of it. Below 2 MeV — the pp and ⁷Be neutrinos, 99 % of the flux — nothing happens: they oscillate on the way out and arrive with the oscillation averaged. Above 2 MeV — the ⁸B neutrinos, one part in — the resonance is met and they arrive as pure .
The experiments of the 1970s and 1980s each sampled a different part of that divide and each got a different answer. Nobody could see why until this mechanism was on the table.
Erratum — N₀ is 6×10³¹ m⁻³, and the book contradicts its own 10³³ three times
Equation (10.42) prints m⁻³ for the electron density at the centre of the Sun. It is — two orders of magnitude — and the book refutes it three times on its own pages.
- Its own ρ₀. p. 452 gives the central mass density as kg m⁻³. At any plausible electron fraction that is electrons per m³, not .
- Its own constant, in the very next equation. (10.43) evaluates as eV⁻¹. With that constant is ; with it is , a hundred times smaller.
- Its own refractive index. p. 454 states at 10 MeV in the solar core. gives ; would give .
As printed, (10.43) would put the resonance threshold at 19 keV instead of 2 MeV, and there would be no solar neutrino story to tell. Everything else in the section — (10.39), (10.43)‘s numerical answer, Fig. 10.12’s axis — is right.
(The same equation prints the conversion constant in eV where eV⁻¹ is needed. δm²cos2θ₁₂ is in eV², so as written the right-hand side has dimensions of eV³.)
What arrives, at both ends of the spectrum
Bettini pp. 456–457. The two plateaus, and the vacuum oscillation amplitude that produces the lower one.
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 — one number, measured twice, two different ways
The ratio of the two plateaus is , and both come from alone. So a detector that spans the transition measures the same mixing angle twice by two unrelated mechanisms — an averaged vacuum oscillation below, an adiabatic conversion above — and they must agree.
That is a far stronger test than either measurement alone, and the book draws the figure (§10.4’s Fig. 10.14) without pointing it out.
🔢 Worked example — one formula for the whole curve
The book gives the two limits separately and describes the transition in words. They are both cases of a single expression: for an adiabatic journey starting at density ,
Below the threshold and this collapses to (10.46); well above it and it collapses to (10.44).
The transition, and both of the book's limits from one line
import numpy as np
hbarc, GF = 1.9732698e-7, 1.1663788e-23 # eV m, eV^-2
dm2, th = 73.4e-6, np.deg2rad(33.5) # eV^2, theta_12
N0 = 6e31 # m^-3, solar centre (see erratum 1)
def A(Ne, E_MeV): # (10.38), in eV^2
return 2*np.sqrt(2)*GF*(Ne*hbarc**3)*(E_MeV*1e6)
def theta_m(Ne, E): # (10.37)
return 0.5*np.arctan2(dm2*np.sin(2*th), dm2*np.cos(2*th) - A(Ne, E))
def Pee(Ne, E): # adiabatic survival, produced at density Ne
return 0.5 + 0.5*np.cos(2*theta_m(Ne, E))*np.cos(2*th)
print(f"resonance A = dm2 cos2th = {dm2*np.cos(2*th):.3e} eV^2")
E_res = dm2*np.cos(2*th)/(2*np.sqrt(2)*GF*(N0*hbarc**3)*1e6)
print(f"the resonance is reachable in the Sun for E > {E_res:.2f} MeV (10.43)\n")
print(" E (MeV) N_e^res (m^-3) theta_m at the core P_ee")
for E in (0.3, 0.86, 2.0, 5.0, 10.0):
Nres = dm2*np.cos(2*th)/(2*np.sqrt(2)*GF*(E*1e6))/hbarc**3
print(f" {E:6.2f} {Nres:11.3e} {np.rad2deg(theta_m(N0,E)):6.1f} deg"
f" {Pee(N0,E):.3f}")
print(f"\nthe two limits the book gives separately:")
print(f" (10.46) low E: 1 - sin^2(2th)/2 = {1-0.5*np.sin(2*th)**2:.3f}"
f" <- P_ee at 0.1 MeV = {Pee(N0,0.1):.3f}")
print(f" (10.44) high E: sin^2(th) = {np.sin(th)**2:.3f}"
f" <- P_ee at 15 MeV = {Pee(N0,15.0):.3f}")resonance A = dm2 cos2th = 2.868e-05 eV^2 the resonance is reachable in the Sun for E > 1.89 MeV (10.43) E (MeV) N_e^res (m^-3) theta_m at the core P_ee 0.30 3.771e+32 35.2 deg 0.566 0.86 1.316e+32 38.5 deg 0.544 2.00 5.657e+31 45.7 deg 0.495 5.00 2.263e+31 62.5 deg 0.388 10.00 1.131e+31 75.6 deg 0.329 the two limits the book gives separately: (10.46) low E: 1 - sin^2(2th)/2 = 0.576 <- P_ee at 0.1 MeV = 0.573 (10.44) high E: sin^2(th) = 0.305 <- P_ee at 15 MeV = 0.315
Read the middle column: the effective angle at the birthplace runs from 35° for a pp neutrino, through 45.7° at 2 MeV — the resonance, arriving exactly where (10.43) says it should — to 75.6° at 10 MeV.
And read the last: 0.544 for the ⁷Be line at 0.86 MeV, 0.329 at 10 MeV. Those are the numbers Borexino and SNO measured, and §10.4 will put them on a plot.
📏 Why the OTHER resonance does not exist in the Sun
There are two square-mass differences, so in principle two resonances. Bettini says the one would need energies “about 33 times larger” than the 2 MeV limit and therefore does not occur in the Sun — only in supernovae.
The 33 is . Keeping the cosine factors that (10.39) actually carries, the honest ratio is , because while . The conclusion is unchanged and made stronger — the threshold is 155 MeV, not 66 MeV — but the factor is a reminder that each resonance belongs to its own angle as well as its own splitting.
The pendulums again, with somebody pulling the thread
§10.2a built the two coupled pendulums for vacuum oscillation. Bettini reuses them here, with one change that captures the entire difference between the two mechanisms.
Bettini’s variation on Fig. 10.5(f). In §10.2a the two lengths were fixed and the energy sloshed back and forth forever. Here somebody pulls the thread: the left pendulum’s length sweeps slowly past the right one, and the energy crosses once and stays. Same apparatus, one moving part, and the difference between oscillation and conversion.
Do the sweep too fast and the energy stays where it was — that is the non-adiabatic case, and it is why adiabaticity adiabaticity the requirement that the density change slowly compared with the oscillation period at resonance, so the state follows its own eigenvalue branch instead of hopping to the other. The Sun satisfies it comfortably; that is luck, not design. defined in §10.3 — open in glossary is a condition and not a guarantee. It holds in the Sun, and Bettini is careful to say that nobody knew that at the beginning.
🔑 If you remember only three things
-
A refractive index is an amplitude, not a cross-section. It goes as G_F rather than G_F², which is the only reason an interaction this weak reorganises the beam instead of passing through it.
-
Adiabatic means slow compared with the local oscillation. Sweep the density faster than that and the state hops branches instead of following one, and no conversion happens at all.
-
The threshold belongs to the star, not to the neutrino. The Sun has a maximum density, so it has a minimum energy it can convert, and everything below that walks out untouched.
Where this goes next
You now have the mechanism and the number that matters: 2 MeV, above which the Sun converts and below which it merely oscillates.
§10.4 is the thirty-four years it took to work that out. Homestake saw a third of the expected rate in 1968, Kamiokande a half, GALLEX and SAGE about 0.55 — three different deficits from three different experiments at three different energies, each of them believed to be a problem with somebody’s apparatus or with the Sun. They were three points on the curve the widget above draws, and SNO closed the argument in 2002 by counting the neutrinos of all flavours and finding the total exactly where the solar model had always put it.
✅ Check yourself — matter, resonance and conversion
0/6 answered · 0 correct
1.Matter changes the mixing angle. Which part of the Hamiltonian does it change, and why does that let a tiny vacuum angle become 45°?
2.Why is adiabatic flavour conversion not oscillation, in the book's insistent phrasing?
3.In the widget, drag the energy slider from 0.3 MeV to 10 MeV on the eigenvalues view. What crosses what, and what does it decide?
4.Eq. (10.36) as printed disagrees with (10.37)–(10.39) by a factor 2. Which checks settle which one is wrong? (Select all that apply.)
5.The high-energy plateau is P_ee = sin²θ₁₂ = 0.30. Why is there no L or E in that expression?
6.The two solar plateaus, 0.576 and 0.304, both come from θ₁₂ alone. What does that buy an experiment that spans the transition?