Epilogue: Seven Things the Standard Model Does Not Do

None of these seven is a wrong prediction. They are things the Standard Model does not address, accommodates without explaining, or explains by orders of magnitude too little.

🎯 Why this matters

Because the seven overlap, an experiment aimed at one can settle another. A neutralino detection would answer §12.2 and §12.4 at once, and an axion would answer §12.2 and §12.5 — which is why the field’s effort is not divided seven ways.

Eleven chapters have been about what is known. This one is about what is not.

The book’s Epilogue is four pages of prose with no equations, no figures and no data — a list of seven open problems, a paragraph each. That is the right length for a closing chapter and the wrong length for a reader who now knows enough to check things, so every equation, figure and number on this page is one this site supplies. Where the book says “quadratically divergent”, you get the number. Where it says supernovae “are dimmer than expected”, you get the slider.

Aside — how to read a chapter with no results in it

Every previous chapter obeyed the book’s own rule: state nothing that is not experimentally confirmed. This one suspends it, and it is worth knowing which sentences are which.

Measured: the 5 / 27 / 68 budget, the supernova dimming, the SUSY exclusion limits, the bound on the strong CP parameter, the absence of antimatter, the parameter count. These are facts, and most of them are facts about absences.

Hypothesis: WIMPs, supersymmetry, the axion, leptogenesis. Not one has been observed. They are here because each would explain something, and because each makes a prediction someone is currently testing.

The distinction matters because the two are interleaved in the book’s prose. A reader who does not hold them apart will finish this chapter believing that dark matter is made of WIMPs, which is not known and may well be false.

12.1 Gravity

The Standard Model describes three interactions. The fourth has no quantum theory, and ch11 showed the only experimental handle anyone has on it. What ch11 did not need is the second constant gravity turns out to require — the cosmological constant .

This section adds one idea that is not in ch11, and it is a good one: gravity needs two fundamental constants where every other interaction needs one.

\underbrace{\htmlClass{t-g}{G_{\mu\nu}}}_{[L^{-2}]} + \underbrace{\htmlClass{t-l}{\Lambda}\,g_{\mu\nu}}_{[L^{-2}]\cdot[1]} = \underbrace{\frac{8\pi G}{c^4}\,T_{\mu\nu}}_{[L^{-2}]}

(12.1 — supplied by this site; the book states the argument in words)

The Einstein field equations, with the cosmological constant. The book gives only the dimensional argument for why Λ has to be there; the equation it is an argument about is worth writing down.

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 — why two constants and not one

Every other interaction in this book has exactly one dimensionful thing to measure — a coupling. Gravity has $G$ , which sets how strongly matter curves space-time, and it also has $\Lambda$ , which is not a coupling at all: it is a curvature the Universe has when nothing is there.

The reason is the one §11.1 laboured: the gravitational field is the metric, and the metric is dimensionless. Curvature is not. Relating the two requires a constant carrying $[L^{-2}]$ , and there is nowhere else in the theory for it to come from.

The book adds one more thing about it, and it is not decoration: like $c$ , Λ must be universal and frame-independent. It has to be, for the same reason $c$ does — it appears in an equation required to hold in every frame, multiplying a tensor, so a Λ that differed between observers or between places would break the covariance the whole theory is built on. That puts a number nobody can derive in the same category as the one constant this book has leaned on since §1.1.

Λ is small — about $10^{-52}\ \mathrm{m^{-2}}$ , a radius of curvature comparable to the size of the observable Universe — which is why nobody needed it for eighty years. §12.3 is about the discovery that it is not zero.

⚠️ Λ is the fourth thing in this book called Λ

Three of them have appeared already, and none is related to the others:

symbol	meaning	where
$\Lambda$	the Λ baryon, uds	§4.4
$\Lambda_{\rm QCD}$	the QCD scale, ~200 MeV, where the strong coupling blows up	§6.4
$\Lambda$	a generic cut-off energy in a loop integral	§5.8, and again in §12.4 below
$\Lambda$	the cosmological constant, $[L^{-2}]$	here

The last two appear on this page, four paragraphs apart. Physics ran out of capital lambdas some time ago.

12.2 Dark Matter

Of the mass–energy in the Universe, about 5 % is the matter this book has spent eleven chapters on. 27 % is matter that has never been seen except through its gravity. The remaining 68 % is §12.3’s problem.

The mass–energy budget, drawn from the book’s own three numbers. The sliver on the left is the subject of this entire book. The book gives the percentages in prose across two sections and never puts them side by side; side by side is the only way the proportion registers.

Dark matter neither emits nor absorbs light, so every measurement of it is gravitational: galaxy rotation curves, the velocity dispersion of clusters, gravitational lensing, and the pattern of the microwave background. Neutrinos are dark, but there are not enough of them — they contribute at most 1 % — and they are far too fast to clump into the structures we see. So dark matter is made of particles that are not in this book.

The standard candidate is a WIMP : mass tens to hundreds of GeV, interacting about as feebly as a neutrino. That combination is what makes the search so hard, and it is worth doing the arithmetic the book leaves out.

🔢 Worked example — what a WIMP actually does when it hits something

The energy budget of a dark-matter detector

import numpy as np
c = 2.99792458e8
v = 230e3 / c                      # halo speed, in units of c

print("=== a WIMP arriving from the halo ===")
print(f"  the Galactic halo speed is about 230 km/s = {v:.2e} c\n")
print("     m_WIMP        kinetic energy    recoil off Xe-131 (max)")
mN = 131 * 0.9315                  # xenon-131 mass in GeV
for m in (10., 100., 1000.):
    T = 0.5 * m * 1e6 * v**2                       # keV
    mu = m * mN / (m + mN)                         # reduced mass
    Er = 2 * mu**2 * v**2 / mN * 1e6               # max recoil, keV
    print(f"    {m:6.0f} GeV      {T:9.1f} keV      {Er:9.1f} keV")
print("\n  the book says 'their energy is small, of several keV' -- see the erratum")

muXe = 100. * mN / (100. + mN)
E100 = 2 * muXe**2 * v**2 / mN * 1e6
print(f"\n  however heavy the WIMP, the recoil cannot exceed 2 m_N v^2 = "
      f"{2*mN*v**2*1e6:.0f} keV")

print("\n=== why the experiment lives 1.4 km underground ===")
dEdx = 2.0                                          # MeV/cm in xenon
print(f"  a cosmic-ray muon deposits about {dEdx:.0f} MeV/cm in xenon")
print(f"  = {dEdx*1e3:.1e} keV, which is {dEdx*1e3/E100:.0f} times a 100 GeV recoil")
print(f"  rock overburden of 1.4 km cuts the muon flux by about 1e-06")

prints

=== a WIMP arriving from the halo ===
the Galactic halo speed is about 230 km/s = 7.67e-04 c

   m_WIMP        kinetic energy    recoil off Xe-131 (max)
      10 GeV            2.9 keV            0.8 keV
     100 GeV           29.4 keV           29.1 keV
    1000 GeV          294.3 keV          114.1 keV

the book says 'their energy is small, of several keV' -- see the erratum

however heavy the WIMP, the recoil cannot exceed 2 m_N v^2 = 144 keV

=== why the experiment lives 1.4 km underground ===
a cosmic-ray muon deposits about 2 MeV/cm in xenon
= 2.0e+03 keV, which is 69 times a 100 GeV recoil
rock overburden of 1.4 km cuts the muon flux by about 1e-06

Two numbers explain the entire design. The signal is tens of keV deposited in one nucleus, once, and the ambient noise — a cosmic ray, a decay of natural uranium in the steel of the vessel, a speck of radon — is thousands of times larger and constant. So the detector goes a kilometre underground, is built of materials assayed for radioactivity at the part-per-trillion level, and is its own target: there is no beam to switch off and no coincidence to demand, only a very quiet room and a long wait.

The strategy has a name — direct detection — and it is one of three, the others being production at a collider and the search for annihilation products in the sky.

Look at the third column against the second. The recoil is not proportional to the WIMP’s energy: it climbs towards a ceiling of $2m_Nv^2 = 144$ keV and stops, because a projectile much heavier than its target can only transfer so much. And at the light end it collapses — a 10 GeV WIMP leaves 0.8 keV in a xenon nucleus, below most thresholds. That is the whole reason the field uses several target materials: xenon is deaf to light WIMPs and saturating for heavy ones, so germanium, argon and silicon carry the ends of the mass range.

Erratum — a WIMP’s energy is not “several keV”

p. 491 says WIMPs “should be around us in very large numbers, but their energy is small, of several keV”. Two lines earlier the same paragraph puts their mass at “tens or hundreds of GeV”.

Those two statements are consistent only at the very bottom of the stated range. With the halo speed of about 230 km/s, $\tfrac12 mv^2$ is 2.9 keV at 10 GeV, 29 keV at 100 GeV and 294 keV at 1 TeV — the block above computes all three.

The number the sentence probably wants is the nuclear recoil energy, which is a few to a few tens of keV and is what direct-detection experiments actually quote as their threshold. But the sentence says “their energy”, meaning the WIMP’s own, and for the masses it proposes that is tens to hundreds of keV.

12.3 Dark Energy

The remaining 68 % is not matter. Dark energy has negative pressure, its density does not fall as the Universe expands, and it makes the expansion accelerate. That last property is a claim about the scale factor $a(t)$ — the one number saying how much the Universe has grown — and specifically about its second derivative.

The evidence is one measurement. A type Ia supernova is the explosion of a white dwarf pushed over the Chandrasekhar mass, and because that threshold is always the same, so is the intrinsic brightness: it is a standard candle visible across gigaparsecs. Compare how bright it looks against how bright it is and you have its distance; compare that against its redshift and you have the expansion history.

The book states the result in a form that is easy to nod along to and hard to check: if the Universe held only matter, distant supernovae would look brighter and nearer than their redshifts suggest; instead they are dimmer and further. That is an argument about a sign. Here is the slider.

What the Universe is made of, and what that does to its history

a(t)

matter only
your universe

Ω_m — matter = 0.320Ω_Λ — dark energy = 0.680

keep it flat (Ω_m + Ω_Λ = 1, which the CMB says it is)

Ω_k — curvature

-0.000

q₀ = Ω_m/2 − Ω_Λ

-0.520

age

no Big Bang

ä turns positive

z = 0.62, 6.0 Gyr ago

The expansion is accelerating today. Positive on the right-hand plot means dimmer and further than an empty universe — which is what the supernovae showed. A matter-only universe is the dashed line, on the wrong side of zero.

Bettini §12.3, supplied. Drag Ω_m to 1 and Ω_Λ to 0 — the universe everyone expected before 1998 — and the right-hand curve goes below zero: supernovae brighter than an empty universe. The measured Universe puts it above. That sign is the whole discovery, and the book makes the argument in prose without ever plotting it. The left panel is the same physics as a history: a matter-only universe is only 9.6 Gyr old, younger than the oldest stars in it.

⚠️ Four symbols this section borrows, all of them already taken

symbol	here it means	where you last saw it
$\Omega_m$ , $\Omega_\Lambda$ , $\Omega_k$	density parameters — the fraction of the critical density in matter, in Λ and in curvature, so that they sum to 1 for a flat universe. §10.6 defines the critical density	Ω is also §11.1’s orbital angular frequency and the Ω⁻ baryon of §4.8
$z$	redshift — the fractional stretching of a wavelength by the expansion. $1 + z$ is how much the Universe has grown since the light left	not a coordinate, and not ch08’s $\Delta z$ vertex displacement
$a(t)$	the scale factor, normalised to 1 today, with $a = 1/(1+z)$	not an amplitude, and not the $a = (g-2)/2$ of §5.9
Gpc	gigaparsec = $3.086\times10^{25}$ m = 1000 Mpc	§10.6 and ch11 measured in Mpc; this chapter jumps three orders of magnitude without saying so

The last row is the one that bites. A supernova at 3.5 Gpc is 3500 Mpc — well beyond anything the gravitational-wave standard sirens of §11.3 reach, and roughly a quarter of the way to the edge of the observable Universe.

💡 What this really says — three ways to see the same fact

As a sign. Matter decelerates the expansion, so in a matter-only universe light from a given redshift has had less time to travel and the source is nearer, hence brighter. Every supernova beyond about $z = 0.3$ came in fainter than that.

As a dilution law. Matter density falls as $a^{-3}$ — the same stuff in a bigger box. Dark energy’s density does not fall at all. So the two swap dominance exactly once in the history of the Universe, and we happen to live shortly after the swap.

As an age. A matter-only universe with the measured $H_0$ is 9.6 Gyr old. The oldest globular clusters are about 12.5 Gyr old. The stars were older than the Universe, and that contradiction was on the books for a decade before the supernovae resolved it.

🔢 Worked example — the book’s own numbers, derived from its own percentages

§12.3 states that supernovae “at distances between about 1 Gpc and 3.5 Gpc … exploded between 2.8 and 8 Gyr in the past” and never says where those numbers come from. They come from the percentages in §12.2 and §12.3.

From 5 / 27 / 68 to the book's distances and times

import numpy as np
H0 = 67.7
T_H = 977.79 / H0                      # Hubble time, Gyr
D_H = 299792.458 / H0                  # Hubble distance, Mpc
Om, OL = 0.05 + 0.27, 0.68
Ok = 1 - Om - OL
E = lambda z: np.sqrt(Om*(1+z)**3 + Ok*(1+z)**2 + OL)
def integ(f, a, b, n=200000):
    x = np.linspace(a, b, n)
    return np.trapezoid(f(x), x)

print("=== the book's own budget ===")
print(f"  Om = 0.05 + 0.27 = {Om:.2f}     OL = {OL:.2f}     Ok = 1 - Om - OL = {abs(Ok):.2f}")

print("\n=== its supernova window, recomputed ===")
zs = np.linspace(1e-5, 3, 60000)
Dc = np.concatenate([[0], np.cumsum(np.diff(zs)/E(zs[:-1]))]) * D_H
for D, book in ((1000., 2.8), (3500., 8)):
    z = float(np.interp(D, Dc, zs))
    t = integ(lambda x: 1/((1+x)*E(x)), 0, z) * T_H
    print(f"  comoving {D/1000:.1f} Gpc -> z = {z:.3f} -> exploded {t:5.2f} Gyr ago   (book: {book})")

print("\n=== when did the expansion start accelerating? ===")
zt = (2*OL/Om)**(1/3) - 1
print(f"  a-double-dot = 0 at 1+z = (2 OL/Om)^(1/3):  z_t = {zt:.3f}, "
      f"{integ(lambda x: 1/((1+x)*E(x)), 0, zt)*T_H:.2f} Gyr ago")
age = lambda om, ol: integ(lambda a: 1/(a*np.sqrt(om/a**3 + (1-om-ol)/a**2 + ol)), 1e-7, 1) * T_H
print(f"  age of the Universe: {age(Om, OL):.2f} Gyr     (matter only: {age(1.0, 0.0):.2f} Gyr)")

print("\n=== the sign the whole section is about, in magnitudes vs an empty universe ===")
def dmu(z, om, ol):
    ok = 1 - om - ol
    e = lambda x: np.sqrt(om*(1+x)**3 + ok*(1+x)**2 + ol)
    dc = integ(lambda x: 1/e(x), 1e-9, z) * D_H
    dm = dc if abs(ok) < 1e-4 else (D_H/np.sqrt(abs(ok)))*(np.sinh if ok > 0 else np.sin)(np.sqrt(abs(ok))*dc/D_H)
    return 5*np.log10((1+z)*dm / (D_H*z*(1+z/2)))
for z in (0.5, 1.0):
    print(f"  z = {z:.1f}:  measured {dmu(z, Om, OL):+.3f}   matter-only {dmu(z, 1.0, 0.0):+.3f}")
print("  positive = dimmer and further. Only one of these matches the data.")

prints

=== the book's own budget ===
Om = 0.05 + 0.27 = 0.32     OL = 0.68     Ok = 1 - Om - OL = 0.00

=== its supernova window, recomputed ===
comoving 1.0 Gpc -> z = 0.240 -> exploded  2.93 Gyr ago   (book: 2.8)
comoving 3.5 Gpc -> z = 1.051 -> exploded  8.09 Gyr ago   (book: 8)

=== when did the expansion start accelerating? ===
a-double-dot = 0 at 1+z = (2 OL/Om)^(1/3):  z_t = 0.620, 5.99 Gyr ago
age of the Universe: 13.67 Gyr     (matter only: 9.63 Gyr)

=== the sign the whole section is about, in magnitudes vs an empty universe ===
z = 0.5:  measured +0.108   matter-only -0.276
z = 1.0:  measured +0.036   matter-only -0.537
positive = dimmer and further. Only one of these matches the data.

2.93 and 8.09 Gyr, against the book’s 2.8 and 8. The Epilogue’s two numbers are not decoration — they follow from its own percentages, and reproducing them is the best evidence available that this four-page chapter hangs together.

One thing worth noticing, because a careful reader will: the switch from deceleration to acceleration happens at $z_t = 0.62$ , 6.0 Gyr ago — which is inside the 2.8–8 Gyr window the book describes as showing acceleration. There is no contradiction. A supernova’s brightness measures the integrated expansion along the whole path, not $\ddot a$ at the moment it exploded, so the window straddling the transition is expected. But the book’s “between 8 and 10 Gyr, the opposite effect appears” is a statement about where the integrated effect changes sign, not about where the acceleration began.

⚙️ Engineer’s bridge — this is a system identification problem

Strip the astronomy and §12.3 is something you have done: you have a system whose step response you can measure, and you want its transfer function.

The measurement is $\mu(z)$ — brightness against redshift — which is an integral of $1/E(z)$ . The model has two free parameters, $\Omega_m$ and $\Omega_\Lambda$ , and each enters $E$ with a different power of $(1+z)$ : matter as the cube, curvature as the square, Λ as a constant. That is exactly a mixture-of-exponentials fit, and the identifiability question is the same one: with data over a limited range of $z$ , how well can you separate two terms whose ratio changes slowly?

The answer is what makes the plot in the widget worth staring at. Over $z < 0.3$ the two models are nearly indistinguishable, which is why the measurement needed supernovae at $z \approx 1$ and why it could not have been done in 1980. The information is in the curvature of the residual, not its value.

Where it breaks: a transfer function is a property of a fixed system, and you can repeat the experiment. Here there is one Universe, one realisation, and no control. Every systematic — dust reddening, evolution of the supernova population with epoch, selection of brighter objects at greater distance — mimics exactly the signal being looked for. That is why the 1998 result needed two independent teams and why “we measured a residual of 0.2 mag” was a much harder claim than it looks.

12.4 SUSY

Supersymmetry is the one item on this list motivated by no observation at all. It fixes a problem of taste — the hierarchy problem , which is the question of why the Higgs mass is as small as it is.

\delta m_H^2 \;\sim\; \frac{\htmlClass{t-y}{\lambda^2}}{16\pi^2}\,\htmlClass{t-c}{\Lambda^2} \qquad\Longrightarrow\qquad \frac{\delta m_H^2}{m_H^2}\Big|_{\Lambda = M_{\rm Pl}} \approx \htmlClass{t-n}{6\times10^{31}}

(12.2 — supplied; the book says only 'quadratic divergences')

The hierarchy problem, as a number. A scalar's mass gets a correction proportional to the square of whatever scale the theory is cut off at — unlike a fermion's, which is protected by chiral symmetry, or a gauge boson's, which is protected by gauge symmetry. The Higgs is the only elementary scalar in the Standard Model, and it is the only particle with this problem.

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 — fine-tuning is not a contradiction

Nothing above is a paradox. A theory is allowed to have a bare parameter that nearly cancels a large correction; the observable comes out right and every prediction still works.

The objection is structural, and it is the same objection you would raise about a circuit whose output depended on two resistors matching to thirty-two digits. It would work. You would not believe it was designed that way, and you would go looking for the feedback loop that enforces the match.

Supersymmetry is that feedback loop. Every fermion loop contributing to $\delta m_H^2$ gets a boson partner loop with the opposite sign, and the two cancel exactly if the partners are degenerate — and approximately if the partners are close in mass. Which is why the argument has an expiry date: push the sparticles far above a TeV and the cancellation becomes as fine-tuned as the thing it was invented to explain.

Bettini p. 492. What the LHC has excluded, with the third-family limit corrected — see the erratum below.

Sparticle↕	partner of↕	excluded below↕	why the limit sits there↕
gluino g̃	the gluon		strongly produced, so the cross-section is large and the reach is the best of any sparticle
squark q̃ (1st, 2nd family)	u, d, s, c	1–2 TeV	also strongly produced, but a squark is made in pairs and each carries away less of the collision energy
squark q̃ (3rd family)	t, b
slepton ℓ̃	e, μ, τ	~600 GeV	produced only electroweakly, so the cross-section is orders of magnitude smaller than for coloured sparticles
neutralino χ̃⁰	γ, Z, H (mixed)	model-dependent

These are lower limits: the particle is excluded BELOW the quoted mass, under the model assumptions each search makes. Click a cell.

Erratum — the LHC cannot exclude a 12 TeV squark

p. 492 gives the exclusion limits as “2–3 TeV for the gluinos, 1–2 TeV for the 1st and 2nd family squarks, 0.6–12 TeV for the 3rd family squarks and around 600 GeV for sleptons”.

It should be 0.6–1.2 TeV — a lost decimal point. Three ways to see it without consulting anything outside the book:

The energy is not there. The sentence says the search is done “at LHC”, whose proton–proton collision energy is 13 TeV. Squarks are pair-produced, so a 12 TeV squark needs at least 24 TeV. No LHC search can set that limit.
It inverts the book’s own ordering. The list runs downwards in strength — gluinos 2–3, first two families 1–2, sleptons 0.6 — and third-family squarks always give the weakest squark limits, which is why their range starts at 0.6. Ending it at 12 puts them four times above the gluinos.
The lower end is right. 0.6 is exactly where the third-family limits do start, which is what you would expect if one character were lost from the upper end and not from the lower.

None of this kills supersymmetry, and the book is careful about why: the models have so many free parameters that no prediction of the mass spectrum is possible at all, so every limit in the table excludes a corner of a parameter space rather than the idea. What the book will commit to is that “several versions of the theory appear to be under strong pressure” — which is as close as this author comes to an opinion in 493 pages.

💡 What this really says — the cure has more free parameters than the disease

Hold that sentence next to §12.7, which is about to argue that the Standard Model’s 19 unpredicted numbers are the strongest evidence it is not final.

The minimal supersymmetric extension of that model adds, on the standard count, about 105 more — soft-breaking masses, phases and mixing angles, none of them predicted either. Supersymmetry answers a fine-tuning objection by introducing six times as many free parameters as the theory it is fixing.

That is not a refutation; the parameters are the price of not knowing how SUSY is broken, and the hierarchy argument stands on its own. But it is why “no prediction on the mass spectrum is possible” is in the book’s sentence, and it is why an exclusion limit on a sparticle mass is a much weaker statement than an exclusion limit on, say, the Higgs mass was. You cannot falsify a theory that can be re-fitted. That, and not the absence of a signal, is the thing the searches are really up against.

12.5 Strong CP

The strong CP problem is made entirely of an absence.

\mathcal{L}_{\rm QCD} \supset \htmlClass{t-t}{\bar\theta}\,\frac{g_s^2}{32\pi^2}\, \htmlClass{t-o}{G^{\mu\nu}_a \tilde G_{a\,\mu\nu}}, \qquad \htmlClass{t-o}{\tilde G^{\mu\nu}} = \tfrac12\epsilon^{\mu\nu\alpha\beta}G_{\alpha\beta}, \qquad \htmlClass{t-t}{|\bar\theta|} < 10^{-10}

(12.3 — supplied; the book gives the operator and the bound in words)

Bettini p. 492. The book names the operator and quotes the bound; the coefficient it is a bound on has a name, θ̄, and naming it is what turns 'the strong CP problem' into a sentence you can state.

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.

⚠️ The tilde changed meaning one section ago

In §12.4 a tilde marked a supersymmetric partner: $\tilde q$ a squark, $\tilde g$ a gluino, $\tilde\chi^0$ a neutralino. One section later, $\tilde G$ is the dual field-strength tensor, $\tilde G^{\mu\nu} = \tfrac12 \epsilon^{\mu\nu\alpha\beta}G_{\alpha\beta}$ — an ordinary QCD object with nothing supersymmetric about it, and it would have existed had SUSY never been proposed.

Two unrelated conventions, four paragraphs apart, on the same page. The way to keep them apart is that the SUSY tilde sits over a particle symbol and the dual tilde over a tensor: $\tilde g$ is a gluino, $\tilde G$ is not a “gravitino” or a partner of anything — it is the gluon field strength with its indices contracted against ε.

💡 What this really says — the opposite of every other problem here

Every other open question in this chapter is about something missing: no quantum gravity, no dark-matter particle, no supersymmetric partners. This one is about something present and inexplicably switched off.

QCD is allowed a term that violates CP. It is gauge invariant, it has the right dimension, and by the principle the section opens with — a Lagrangian contains every operator its symmetry permits — it should be there with a coefficient of order one. Instead the neutron’s electric dipole moment says the coefficient is below $10^{-10}$ .

Compare the weak interaction, which violates CP happily and whose CP-violating phase is not small (§8.9 measured $\gamma \approx 65°$ ). The puzzle is not that CP violation is rare. It is that the strong interaction, alone, has an available knob turned to zero to ten decimal places.

The axion is the standard resolution: promote $\bar\theta$ from a constant to a field, and the dynamics drive it to zero. The field’s quantum is a very light pseudoscalar which would also, conveniently, be dark matter.

the QCD axion band — mass and coupling are not independent
g = 0.7×10⁻¹⁰ (m/eV) GeV⁻¹, the benchmark line
the region haloscopes have reached

Supplied — the book says only that 'we know, within an order of magnitude, the relation between its mass and its coupling to photons' and that experiments 'are now reaching the sensitivity to explore the very small cross-section region for some values of the mass'. That relation is the whole reason the search is possible: an unknown particle with two free parameters could hide anywhere on this plane, but the QCD axion is confined to a diagonal band, so a search only has to sweep along a line. The dashed box is roughly where microwave-cavity haloscopes have looked — a few μeV of the several decades that are open.

⚙️ Engineer’s bridge — the axion search is a swept-frequency lock-in

The experiment is one you could describe to an RF engineer without mentioning particle physics. Put a high-Q microwave cavity in a strong magnetic field. If axions of mass $m$ are streaming through it, the field converts a few of them into photons of frequency $f = mc^2/h$ — and if the cavity is tuned to that frequency, the conversion is resonantly enhanced by the cavity Q.

So: sweep the cavity’s resonance, and look for excess power at the tuned frequency. It is a lock-in amplifier hunting a carrier of unknown frequency, with a quality factor of $10^5$ and a signal power around $10^{-23}$ W.

The mass–coupling relation is what makes it tractable: knowing $g \propto m$ turns a two-dimensional search into a one-dimensional scan. And the scan rate is the whole design problem — time spent at each frequency buys signal-to-noise as $\sqrt{t}$ , so covering a decade of mass at a useful sensitivity takes years.

Where it breaks: a lock-in needs a reference to mix against and here there is none — nobody knows the frequency, which is precisely what is being searched for. So the analysis is a blind scan with a look-elsewhere penalty over millions of independent frequency bins, and the threshold for a claim has to be set accordingly. It is the same statistics as the Higgs search of §9.15, applied to a spectrum rather than an invariant mass.

Erratum — the axion is not a meson

p. 492 introduces it as “the ‘axion’, a very light pseudoscalar meson”.

This book defines a meson in ch04 as a quark–antiquark bound state — composite, hadronic, strongly interacting, with a mass of order the QCD scale. The axion is none of those: it is an elementary particle with no colour and no constituents, and the same sentence puts its mass in the μeV range, fifteen orders of magnitude below the lightest actual meson.

It is a pseudoscalar boson. The slip is worth correcting here rather than letting it pass, because this site has spent a chapter teaching that “meson” is a statement about what something is made of.

12.6 Matter and antimatter

The Universe is made of matter. PAMELA and AMS, magnetic spectrometers in orbit, have looked for antinuclei in the cosmic rays and set strong limits: there is no antimatter region out there.

That is a problem, because the Big Bang should have made equal amounts. Getting from a symmetric start to the Universe we observe is called baryogenesis , and Sakharov showed in 1967 that it takes three things at once — the Sakharov conditions , of which §12.6 names one.

🪜 Sakharov's three conditions — the book names one of them

Step 1 of 5 — 1. Baryon number must not be conserved

Why you may do this: Obvious once stated, and the one the book omits: if B is exactly conserved and starts at zero, it stays at zero forever, no matter what else happens. Something must be able to change it.

The Standard Model does have such a process — an electroweak "sphaleron", non-perturbative and utterly negligible today, but fast in the early Universe. So this condition is met. [§12.7](#127-structural-problems) asks whether B is conserved at all; this is why the question matters.

Sakharov (1967). A Universe that starts matter–antimatter symmetric can only end up asymmetric if all three hold. §12.6 names only the second, which makes the argument look weaker than it is.

💡 What this really says — the number that connects two chapters

Read §12.6 next to §10.5 and the point of ten years of neutrino-beam physics changes shape.

$\delta_{CP}$ is not just a parameter of the PMNS matrix. It is the last unmeasured quantity in the Standard Model that could plausibly explain why there is any matter at all — and the current best fit, near $-\pi/2$ , is close to the maximum possible violation.

That is either the most important hint in particle physics or a $2\sigma$ fluctuation. Hyper-Kamiokande and DUNE exist to find out.

Erratum — “the product of the square sines” is not the quantity

p. 493 states the condition for leptonic CP violation as: “the product of the square sines of three neutrino mixing angles and of the Dirac phase δ is large enough”.

Read literally that is $\sin^2\theta_{12}\sin^2\theta_{23}\sin^2\theta_{13} \sin^2\delta$ , and it cannot be the right quantity for a reason that needs nothing beyond ch03: a CP-violating observable must change sign under CP, which sends $\delta \to -\delta$ . Any function of $\sin^2\delta$ is even in $\delta$ and therefore identical for the process and its CP-conjugate — it can only ever predict no asymmetry.

The quantity that is meant is the Jarlskog invariant, the same object §8.8 used for the quarks, written for the PMNS matrix:

J = \underbrace{\htmlClass{t-a}{c_{12}s_{12}\,c_{23}s_{23}}}_{\text{sines and cosines}} \;\underbrace{\htmlClass{t-b}{c_{13}^2\,s_{13}}}_{\text{the only square}} \;\underbrace{\htmlClass{t-d}{\sin\delta}}_{\text{carries the sign}}

(12.4 — supplied; the book describes this quantity in words and describes it wrongly)

The Jarlskog invariant for the PMNS matrix, with c_ij = cos θ_ij and s_ij = sin θ_ij. It is the unique rephasing-invariant measure of CP violation in a 3 × 3 unitary mixing matrix: change the arbitrary phases of the fields however you like and every element of the matrix moves, but J does not.

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.

Sines and cosines, only $\cos\theta_{13}$ squared, and $\sin\delta$ to the first power — which is what carries the sign. The block in §12.7 evaluates it for both sectors: $J_{\rm CKM} = 2.9\times10^{-5}$ against $|J_{\rm PMNS}| = 3.3\times10^{-2}$ at $\delta = -\pi/2$ . The lepton sector can supply about 1100 times more CP violation than the quark sector, and that one ratio is the whole reason §12.6 ends where it does.

The book’s intent is clear enough — it is arguing that no angle is small and that δ is not near 0 or π, which is exactly the condition for $J$ to be large — and the phrasing is probably a compression of the $\sin^2 2\theta_{ij}$ that ch10’s oscillation formulas use throughout. But the compressed version is not checkable and $J$ is.

12.7 Structural problems

The last section is not about a missing phenomenon. It is about the shape of the theory: the Standard Model has too many free parameters to be a final answer — numbers it cannot predict and must be told.

Bettini p. 493, counted. The book says 'too many free parameters' and lists them in prose; here is the bill.

Group↕	count↕	what they are↕
gauge couplings	3
charged fermion masses		6 quarks + 3 charged leptons — equivalently, 9 Yukawa couplings
CKM quark mixing	4
Higgs sector	2	the vacuum expectation value and the Higgs mass
strong CP angle θ̄	1	measured to be < 10⁻¹⁰ and unexplained — §12.5
— subtotal, the Standard Model as written —	19	nineteen numbers with no theory behind any of them
neutrino masses	3	not in the model at all; ch10 measured that they exist
PMNS lepton mixing		3 angles + 1 Dirac phase, plus 2 Majorana phases if the neutrino is its own antiparticle
— total, with neutrinos —	26 or 28	and every one of them is measured, not derived

A free parameter is a number the theory cannot predict and must be told. Click a row for what it is and why nothing fixes it.

Then there is the pattern of those numbers, which is stranger than their count.

neutrinos (mass scale, from ch10)
charged leptons
quarks

§12.7's 'the masses of the fermions … span 13 orders of magnitude', drawn. The gap that should stop you is between the neutrinos and the electron: seven empty decades, with nothing in them. Every mass here is a free parameter — the Standard Model predicts not one of them, and it does not explain the gap either. That is what 'why such big differences?' means.

🔢 Worked example — the span, and the two questions the book ends on

13 orders of magnitude, and one part in 10²¹

import numpy as np
print("=== the fermion mass range (12.7) ===")
mnu, mtop = 0.05, 172.69e9
print(f"  lightest:  neutrino mass scale ~ {mnu:.2e} eV")
print(f"  heaviest:  top quark            {mtop:.2e} eV")
print(f"  span: {np.log10(mtop/mnu):.1f} orders of magnitude   (book says 13)")
print(f"  and if the lightest neutrino is 0.01 eV, {np.log10(mtop/0.01):.1f} -- so 13 is right")
print(f"\n  the gap between the heaviest neutrino and the electron alone: "
      f"{np.log10(0.511e6/mnu):.1f} decades")

print("\n=== quark mixing is small, lepton mixing is large (12.7) ===")
ckm  = {'12': 13.0, '23': 2.4, '13': 0.2}
pmns = {'12': 33.4, '23': 49.0, '13': 8.6}
for k in ('12', '23', '13'):
    print(f"  CKM  theta_{k} = {ckm[k]:4.1f} deg    PMNS theta_{k} = {pmns[k]:4.1f} deg")
print("  the same physics, two sectors, and nothing explains the difference")

print("\n=== how much CP violation each sector can supply (12.6) ===")
def jarlskog(a, delta_deg):
    t12, t23, t13 = (np.radians(a[k]) for k in ('12', '23', '13'))
    return (np.cos(t12)*np.sin(t12) * np.cos(t23)*np.sin(t23)
            * np.cos(t13)**2 * np.sin(t13) * np.sin(np.radians(delta_deg)))
Jq = jarlskog(ckm, 65.0)          # CKM, delta = gamma = 65 deg
Jl = jarlskog(pmns, -90.0)        # PMNS, delta near -pi/2
print(f"  J_CKM  (delta = 65 deg)   = {Jq:+.2e}")
print(f"  J_PMNS (delta = -90 deg)  = {Jl:+.2e}")
print(f"  the lepton sector can supply {abs(Jl/Jq):.0f} times more CP violation")
print("  J changes sign with delta -- which sin^2(delta) could never do")

print("\n=== the neutrality of matter (12.7) ===")
print("  |q_p + q_e| / e < 1e-21")
print(f"  for comparison, the best mass measurement is about 1 part in {1e11:.0e}")
print("  charge cancels 10 orders of magnitude better than anything is weighed")

prints

=== the fermion mass range (12.7) ===
lightest:  neutrino mass scale ~ 5.00e-02 eV
heaviest:  top quark            1.73e+11 eV
span: 12.5 orders of magnitude   (book says 13)
and if the lightest neutrino is 0.01 eV, 13.2 -- so 13 is right

the gap between the heaviest neutrino and the electron alone: 7.0 decades

=== quark mixing is small, lepton mixing is large (12.7) ===
CKM  theta_12 = 13.0 deg    PMNS theta_12 = 33.4 deg
CKM  theta_23 =  2.4 deg    PMNS theta_23 = 49.0 deg
CKM  theta_13 =  0.2 deg    PMNS theta_13 =  8.6 deg
the same physics, two sectors, and nothing explains the difference

=== how much CP violation each sector can supply (12.6) ===
J_CKM  (delta = 65 deg)   = +2.90e-05
J_PMNS (delta = -90 deg)  = -3.33e-02
the lepton sector can supply 1147 times more CP violation
J changes sign with delta -- which sin^2(delta) could never do

=== the neutrality of matter (12.7) ===
|q_p + q_e| / e < 1e-21
for comparison, the best mass measurement is about 1 part in 1e+11
charge cancels 10 orders of magnitude better than anything is weighed

The third block is the one to sit with. The proton is a bound state of three quarks with charges $+\tfrac23, +\tfrac23, -\tfrac13$ ; the electron is elementary and unrelated to any of them. Nothing in the Standard Model requires their charges to cancel — charge quantisation is an input, not a result. And yet the neutrality of matter holds to better than one part in $10^{21}$ , which is ten orders of magnitude better than the most precise mass measurement anyone has made.

That is not a small coincidence. It is the strongest experimental hint that quarks and leptons are related by something the Standard Model does not contain.

The five questions the book ends on

Having counted the parameters, §12.7 closes the volume with what it calls “even more ambitious questions”. They are worth listing carefully, because they are not rhetorical: every one of them is a running experiment with a published limit, and that is exactly the point the book’s final sentence makes.

Bettini p. 493, the closing list, with the experiment attached to each question. The book asks them and stops; the limits are what turn them from philosophy into a programme.

The question↕	why it is asked↕	how it is being answered↕	where it stands↕
Are baryon and lepton number conserved?		searches for B-violating decays, and for neutrinoless double beta decay, which violates L by two units
Is the proton really stable?			τ > 10³⁴ yr
Why do the proton and electron charges cancel exactly?	charge quantisation is put into the SM by hand: nothing requires a composite of three quarks to have exactly the opposite charge to an elementary lepton	test the neutrality of bulk matter — a macroscopic sample would carry a measurable charge if the cancellation were imperfect
Why are there just three families?			N_ν = 2.984 ± 0.008
Are there more than three spatial dimensions?		look for deviations from 1/r² gravity at sub-millimetre distances, and for missing energy at the LHC as a graviton escapes into the extra dimension	no deviation above ~30 μm

Every row is an experiment currently taking data or under construction. Click a cell for what is actually measured.

The second row is the one to hold on to. Proton decay has never been observed, and the limit — a lifetime beyond $10^{34}$ years — is not a measurement of patience but of scale: nobody waits for one proton, they watch $10^{34}$ of them for a year. That substitution of quantity for time is the same move as the luminosity argument of §1.8, and it is what makes a question about $10^{24}$ times the age of the Universe answerable at all.

Erratum — four parameters, not four elements

p. 493: “The values of the four elements of the CKM matrix are also not predicted.”

The CKM matrix has nine elements. Four is the number of independent parameters — three mixing angles and one CP-violating phase — which is what §7.11 derives from unitarity and the freedom to rephase the quark fields.

The refutation is in the very next clause of the same sentence, which counts neutrino mixing as “four or six quantities depending on the neutrinos being Dirac or Majorana”. That is the parameter count done correctly for the PMNS matrix, one line later.

Erratum — the budget is in §10.6, not §10.4

p. 490 opens §12.2 with “As we mentioned in Section 10.4, overwhelming evidence from astrophysical observations shows that the matter we know represents only a small fraction…”.

§10.4 is The Shorter Period Oscillation — solar neutrinos, SNO’s three bands, KamLAND — and contains no cosmology whatsoever. The mass–energy budget, the density parameters and the ΛCDM digression are in §10.6, Limits on Neutrino Mass.

The book’s own section titles settle it. It is the sixth cross-reference slip this site has found, five of them in ch10 — a reminder that in a book revised across three editions, a section number is the least stable thing on the page.

🔑 If you remember only three things

Five per cent of the Universe took eleven chapters. The other ninety-five has no identified particle in it, and two of the seven problems are about that alone.
A free parameter is a measurement the theory could not make. Nineteen of them is not a contradiction, and saying so is the whole of §12.7.
The absences bound theories more tightly than the detections would. A dipole moment that is not there, an antinucleus never seen, a proton that has not decayed — each is a limit with a number on it.

Where this goes next

Seven problems, and they are not seven independent things.

💡 What this really says — three of the seven are entangled

The book never says this, and it is the most interesting structural fact in the chapter.

The neutralino of §12.4 is a WIMP. If supersymmetry exists at the TeV scale, its lightest particle is stable, massive and weakly interacting — which means §12.4’s solution is also §12.2’s.

The axion of §12.5 is also a dark-matter candidate. A light, feebly coupled, cosmologically abundant boson is exactly what §12.2 needs, and it arrives as a by-product of fixing the strong CP problem.

And $\delta_{CP}$ ties §12.6 to ch10. The phase measured in a neutrino beam is the input to leptogenesis .

So the seven are really: gravity (alone), the parameter structure (alone), and a knot of four in which any of three hypothetical particles would resolve two problems at once. That is why those three searches get the money.

⚙️ Engineer’s bridge — what “the model is incomplete” actually means

A working engineer meets the Standard Model’s situation regularly: a model that fits every measurement, has no internal contradiction, and is obviously not the final story.

The tells are the ones you would list in a design review. Too many magic numbers — 26 constants, none derived. Unexplained scale separation — 13 orders of magnitude in the masses, with a seven-decade hole in the middle. A parameter pinned to zero with no mechanism — θ̄ < 10⁻¹⁰. A quantity that only works if two unrelated terms cancel to 32 digits — the Higgs mass. And a subsystem that is not modelled at all — gravity.

None of these is a bug report. Every one is the kind of observation that makes you say the abstraction is leaking, and go looking for the layer below.

Where it breaks: in engineering you can usually open the box. Here the layer below sits at $10^{19}$ GeV, fifteen orders of magnitude beyond the largest machine ever built, and the only probes are the ones in this chapter — rare processes, cosmological leftovers, and precision on quantities the model does predict. That is why the seven problems are attacked from the top down, and why progress is measured in exclusion limits rather than discoveries.

You now have the whole of the book: the kinematics and detectors of ch01, the particle zoo and the quark model, the symmetries that organise them, the three interactions and their unification, the neutrinos that broke the model’s last assumption, and the gravitational waves that gave the fourth interaction its first data.

The book closes with a sentence worth keeping:

“These questions are not metaphysical, but physical, as they can be and are addressed experimentally.”

Every problem on this page has an experiment attached — a detector under a mountain, a cavity in a magnet, a beam across Japan, a collider at CERN, an interferometer in Louisiana. None of them may find anything. That is what makes them experiments.

✅ Check yourself — what the Standard Model does not do

0/6 answered · 0 correct

1.Why does gravity need two fundamental constants when the other three interactions each need one?
2.In the widget, set Ω_m = 1 and Ω_Λ = 0. The supernova residual curve goes below zero. What does that mean, and why did it settle the question?
3.The hierarchy problem is described as a fine-tuning, not a contradiction. What exactly is the objection? (Select all that apply.)
4.What makes the strong CP problem different in kind from the other six problems in this chapter?
5.The book says CP violation is 'a necessary ingredient' for the matter–antimatter asymmetry. What does that leave out, and why does it matter?
6.The book says the SUSY exclusion limits reach '0.6–12 TeV for the 3rd family squarks'. How do you know from inside the book that this is wrong?