The Global Fit, and What Is Still Untouched · Elementary Particle Physics

Twenty-one numbers go in and nothing in the framework explains any of them, which is the precise sense in which this is a model rather than a theory.

🎯 Why this matters

Twenty-one free numbers do not make it unfalsifiable. Each is measured once and everything else — hundreds of quantities across a dozen unrelated experiments — follows with nothing left to adjust.

With the Higgs found, every field the Standard Model contains has been observed: three families of spin-½ fermions, twelve spin-1 gauge fields — one electromagnetic, three weak, eight strong — and one scalar. This last section asks the only question left. Do all the constraints hold at once?

Why it is a model and not a theory

The book makes a distinction worth taking seriously, in a parenthesis:

the observables … in terms of the couplings and the masses of the fermions, quantities that the model does not predict (which is why it is called a ‘model’ and not a ‘theory’)

the parameters the Standard Model does not predict

rows = [('gauge couplings g, g-prime, g_s', 3),
        ('charged-fermion masses (Yukawa couplings)', 9),
        ('CKM: three angles and one phase', 4),
        ('the Higgs sector: mu^2 and lambda (or v, M_H)', 2),
        ('the QCD theta parameter', 1)]
print("the free parameters, with massless neutrinos:\n")
for nm, n in rows:
    print(f"  {nm:46s} {n:2d}")
print(f"  {'':46s} --")
print(f"  {'total':46s} {sum(n for _, n in rows):2d}")
print("\n  (neutrino masses and the PMNS matrix add at least 7 more -- Ch. 10)")

print("\nso what makes the global fit a TEST rather than a parameterisation?\n")
print("  the electroweak fit takes about FIVE of these as inputs -- alpha,")
print("  G_F, M_Z, m_t, M_H (with alpha_s and the hadronic vacuum polarisation)")
print("  -- and then PREDICTS the remaining sixteen observables.\n")
print("  21 measurements, 5 inputs, 16 predictions.  the fit has 16 chances")
print("  to fail and does not take any of them.")

flavour = 9 + 4
print("\nwhat the 19 parameters actually are, by fraction:")
print(f"  9 of 19 are Yukawa couplings -- masses put in by hand")
print(f"  4 of 19 are the CKM matrix")
print(f"  so {flavour} of 19 -- more than two thirds -- describe FLAVOUR, and the")
print( "  Standard Model explains none of it.  the gauge sector, which is the")
print( "  part this chapter has been celebrating, needs only three.")

prints

the free parameters, with massless neutrinos:

gauge couplings g, g-prime, g_s                 3
charged-fermion masses (Yukawa couplings)       9
CKM: three angles and one phase                 4
the Higgs sector: mu^2 and lambda (or v, M_H)   2
the QCD theta parameter                         1
                                               --
total                                          19

(neutrino masses and the PMNS matrix add at least 7 more -- Ch. 10)

so what makes the global fit a TEST rather than a parameterisation?

the electroweak fit takes about FIVE of these as inputs -- alpha,
G_F, M_Z, m_t, M_H (with alpha_s and the hadronic vacuum polarization)
-- and then PREDICTS the remaining sixteen observables.

21 measurements, 5 inputs, 16 predictions.  the fit has 16 chances
to fail and does not take any of them.

what the 19 parameters actually are, by fraction:
9 of 19 are Yukawa couplings -- masses put in by hand
4 of 19 are the CKM matrix
so 13 of 19 -- more than two thirds -- describe FLAVOUR, and the
Standard Model explains none of it.  the gauge sector, which is the
part this chapter has been celebrating, needs only three.

The 21 inputs, decoded

The book spends a page saying what the labels on Fig. 9.65 mean, which is exactly the right thing to do — the plot is unreadable otherwise.

Bettini pp. 428–429. Every row was measured somewhere in Chapters 5–9, and the last column says where.

input↕	what it is↕	measured in↕
$M_Z$ , $\Gamma_Z$	the mass and total width of the Z	§9.9 — 23 ppm and 0.09 %
$\sigma^0_{\text{had}}$	the hadronic cross-section at the Z peak	§9.9
$R^0_{\text{lep}}$	the ratio of leptonic to hadronic cross-sections at the peak	§9.9
$R^0_c$ , $R^0_b$	the partial rates into charm and beauty, as fractions of the hadronic width	§9.9 — possible only because $c$ and $b$ leave displaced vertices
$A_\ell$ (LEP), $A_\ell$ (SLD)	the asymmetry between right- and left-polarized beams — the SLC measurement LEP could not make, since only a linear collider could polarize	§9.9
$A_c$ , $A_b$	the same for the heavy quarks	§9.9
$A^{0,\ell}_{FB}$ , $A^{0,c}_{FB}$ , $A^{0,b}_{FB}$	forward–backward asymmetries — the cross-section integrated over the forward hemisphere against the backward one	§9.4 listed this as a way to get the angle
$\sin^2\theta^{\ell}_{\text{eff}}$	the effective mixing angle, defined per fermion by $\sin^2\theta_{\text{eff}} = \tfrac{1}{4\|Q_f\|}\left(1 - c_V^f/c_A^f\right)$ — a second definition, linked to the first by the theory	LEP and the Tevatron
$\Delta\alpha^{(5)}_{\text{had}}(M_Z^2)$	the hadronic contribution to the running of α up to $M_Z$ — the one input that cannot be computed perturbatively	§5.8, from $e^+e^-\to$ hadrons data
$\alpha_s(M_Z^2)$	the strong coupling at the Z mass	§6.5
$m_t$	the top mass	§9.11 — and still theory-limited

the 21 pulls (Gfitter 2018)
perfect agreement

Fig. 9.65 redrawn schematically. A pull is (measured − fitted)/σ_experimental, so with 21 inputs you EXPECT about seven beyond ±1σ and one beyond ±2σ. That is what is seen. The famous outlier near 2.5σ is A^0,b_FB, the beauty forward–backward asymmetry, which has sat there for two decades.

💡 What this really says — a perfect pull plot would be evidence that something is wrong

The instinct on seeing a plot of deviations is to want them all at zero. That instinct is exactly backwards, and the pull plot is the clearest place to say so.

A pull is a deviation divided by its own uncertainty. If the model is right and the errors are honestly estimated, the pulls are drawn from a standard normal distribution — so out of 21 you should find about seven outside ±1σ, about one outside ±2σ, and roughly a 5 % chance of one outside ±2.6σ.

Fig. 9.65 shows precisely that population. The single ~2.5σ entry — the forward–backward asymmetry into beauty, $A^{0,b}_{FB}$ , which has been the electroweak fit’s most persistent outlier for twenty years — is expected in a sample of 21. It is interesting, it has been examined exhaustively, and it is not evidence of anything.

All pulls near zero would mean the errors are overestimated, which is a defect rather than a triumph: it means the measurements are less informative than they claim. An engineer knows the same signature — residuals that are too small mean you have over-fitted or over-quoted your tolerances, and a χ² per degree of freedom well below 1 is a reason to check your error bars, not to celebrate.

So the right way to read Fig. 9.65 is: the scatter is the right size. Not “the model agrees”, but “the model agrees and the disagreements are the size they should be”, which is a much stronger statement and the one the fit actually supports.

⚙️ Engineer’s bridge — one theory, and experiments with nothing else in common

What makes this plot remarkable is not the agreement but the heterogeneity of what is being made to agree. The book lists it and the list deserves reading slowly.

The measurements were made on neutrino beams (§9.4), at $e^+e^-$ colliders (§9.9), at $ep$ , $p\bar p$ (§9.7, §9.11) and $pp$ machines (§9.14), with heavy ions, and in precision atomic physics (§9.4 again, at the ppm level). The theory needed to interpret them spans perturbative expansions, resummations and lattice calculations (§6.10).

Between an atomic parity-violation experiment at $|Q| \sim$ keV and a top-mass measurement at a TeV there is nothing in common — not the apparatus, not the technique, not the systematic uncertainties, not the community, and in several cases not the century. The only thing they share is the theory being tested.

That is what an over-constrained system buys you, and an engineer will recognise the design principle from any safety-critical measurement: make the redundant measurements as unlike each other as possible. Two thermometers of the same model agreeing tells you little; a thermometer agreeing with a thermocouple and an infrared camera tells you the temperature. Correlated methods share correlated failures, and the whole value of the global fit is that its inputs cannot fail together.

Where it breaks: “their inputs cannot fail together” is the claim, and it is only partly true. The 21 inputs share α, share the same one-loop calculations, and several share the same hadronic and lattice inputs — $\Delta\alpha^{(5)}_{\rm had}$ enters almost everything. A common theory error moves the whole fit and shows up as agreement, not as tension. That is the failure mode a global fit is blind to by construction: it is superb at catching one bad measurement and structurally incapable of catching one bad input shared by all of them. The standing 2.5σ pull on $A^{0,b}_{FB}$ has survived two decades precisely because no amount of extra correlated data resolves it.

What has not been tested

The book ends the chapter on the honest note, and it is worth taking as seriously as the agreement:

Bettini p. 430. Two of these are the subject of the next decade; the third is the subject of the next two chapters.

untested↕	why it matters↕	what will do it↕
Higgs couplings to the first two generations	the Yukawa couplings of the light fermions are 9 of the theory's 19 free parameters, and almost none of them has been checked. Only $H\to\mu\mu$ has been seen at all, at 2–3σ (§9.19)	Run 3 and the HL-LHC, by statistics
the shape of the BEH potential	completely untested. The self-couplings of three and four Higgs bosons depend on the single constant λ, which the Standard Model fixes from $M_H$ and $G_F$ — so measuring $HHH$ tests the mechanism of §9.12, not just its consequences	the HL-LHC, and it is hard: di-Higgs production is orders of magnitude rarer than single
anything to do with neutrino mass	the BEH recipe gives neutrinos nothing, because it needs a right-chirality partner (§9.12). But neutrinos oscillate, so they have mass	Chapter 10 — and it is physics beyond this chapter

📏 λ is the last free parameter of the mechanism, and nothing has touched it

It is worth being precise about what “the structure of the BEH potential remains completely untested” means, because everything else in the chapter has been tested.

§9.12’s potential is $V = \tfrac{\lambda}{4}(\Phi^2 - \upsilon^2)^2$ — two parameters. One of them, $\upsilon = 246$ GeV, has been known since the 1930s through $G_F$ . The other, $\lambda$ , enters only two places:

the Higgs mass, $M_H = \upsilon\sqrt{2\lambda}$ — measured, giving $\lambda = M_H^2/2\upsilon^2 = 0.129$ ;
and the Higgs self-couplings, $g_{HHH} = 3M_H^2/\upsilon$ and $g_{HHHH} = 3M_H^2/\upsilon^2$ — never measured.

So $\lambda$ has been extracted from one observable and never checked against a second. Every coupling measurement of §9.19 tests how the Higgs talks to other particles; none tests how it talks to itself, and the self-coupling is what encodes the shape of the potential — the very thing that makes the symmetry break.

Put another way: we have confirmed that a scalar field with a non-zero vacuum value gives mass to everything in the predicted proportions. We have not confirmed that its potential is a Mexican hat rather than something else with the same minimum. That measurement is the HL-LHC’s headline goal.

Erratum — “up to 4000 fb⁻¹ per year”

The chapter closes by saying the High Luminosity LHC, starting in 2029, will be “capable of collecting up to 4000 fb⁻¹ per year”.

4000 fb⁻¹ is the integrated total of the whole HL-LHC programme, accumulated over about a decade — which is what the same sentence says two lines earlier (“explored in the coming decennium”). It is not an annual rate.

The book’s own numbers show why. A running year is $10^7$ s (§9.6’s mnemonic), so 4000 fb⁻¹ per year requires

\mathcal{L} = \frac{4\times10^{42}\;\text{cm}^{-2}}{10^{7}\;\text{s}} = 4\times10^{35}\;\text{cm}^{-2}\text{s}^{-1}

— twenty times the $2\times10^{34}$ that §9.14 says the LHC is currently limited to by the heat load on the focusing quadrupoles, and eight times the HL-LHC’s own design value. At the realistic $5\times10^{34}$ the annual figure is a few hundred fb⁻¹, and 4000 takes the decade.

Confirmed on the render of PDF p. 448.

Nine Yukawa couplings, one line — and five of them have never been checked

1 MeV10 MeV100 MeV1 GeV10 GeV100 GeV10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹fermion mass (log)coupling y = m/υy = m / υ — the prediction, slope 1never measuredtbτcμsduemeasured2–3σ only (H→μμ)untested

Supplied — the table above says the first two generations are untested and the scale of that gap only lands as a picture. The line is not a fit: the Standard Model says a fermion’s coupling to the Higgs field is its mass divided by υ = 246 GeV, so all nine points are forced onto slope 1 with nothing to adjust. What has been confirmed covers the top three decades and stops. The electron’s Yukawa is 3×10⁻⁵ of the top’s, and no experiment has touched it — which matters because the mechanism of §9.12 is only tested where a point sits on that line.

Nine of the model’s nineteen free parameters live on this plot; three are established, a fourth is at 2–3σ, and five have never been probed at all.

🔑 If you remember only three things

Most of the inputs are masses nobody predicts. The count is dominated by the Yukawa sector, which is the part of the model with no structure in it at all.
Observed is not understood. Every field the model contains has now been seen, and the numbers describing them remain inputs rather than results.
The part that has never been tested is named here. The shape of the potential itself is untouched, and everything confirmed so far is a consequence of the mechanism rather than its form.

Where this goes next

Chapter 9 ends with the Standard Model verified in every sector it describes — and with two doors held open.

The first is inside the model. The Higgs potential’s shape is unmeasured, and so are most of the Yukawa couplings. Neither is a crack; both are simply not yet done.

The second is outside it, and the book is blunt about this. “Sectors of particle physics, specifically with neutrinos, and in cosmology show that physics beyond the SM exists.” §9.12 showed why: the BEH mechanism gives neutrinos no mass at all, because it requires a right-chirality partner that has never been seen. And neutrinos oscillate.

Chapter 10 is that story — Chapter 8’s formalism run again with large mixing angles instead of small ones, and with a mass whose origin nothing in Chapters 7–9 supplies.

✅ Check yourself — the global fit

0/6 answered · 0 correct

1.Why is the Standard Model called a model rather than a theory?
2.The fit uses 21 measured inputs. What makes that a test rather than a parameterisation?
3.One of the 21 pulls sits near 2.5σ. Is that a problem?
4.Which part of the BEH mechanism remains completely untested?
5.The fit's inputs come from neutrino beams, e⁺e⁻, ep, pp̄, pp, heavy ions and atomic physics. Why does that heterogeneity matter?
6.The book says the HL-LHC will be 'capable of collecting up to 4000 fb⁻¹ per year'. Check it.