FCC-ee · Z(ℓℓ/νν/qq) H(jj) · an interactive dissection

Two thirds of the Higgs, and nobody has seen most of it

B(H→b̄b) + B(H→gg) + B(H→c̄c) = 69.3%. The LHC has one of those three at ~15% precision, one at the edge of existence, and one not at all. This is a working dissection of the FCC-ee note that claims 0.21% / 1.6% / 0.8% — with the kinematics, the tagger, and the likelihood put back in your hands so you can find out where those numbers actually come from, and where they are softer than they look.

Fig. 01

01 / 09 Why

Beam

Beam energy spread per beam 222 MeV Track momentum term a (σ_p/p = a·p) 2.0e-5 ISR strength nominal

Lepton flavour

Background level nominal

m_recoil — Z(ℓℓ)H

Peak σ (core)—

From beam spread—

From tracking—

S/B in ±2σ—

m_{rec}^{2} = {| P_{i} - P_{ℓℓ} |}^{2} = s - 2 \sqrt{s} \cdot E_{ℓℓ} + m_{ℓℓ}^{2}

The Higgs mass is measured without touching the Higgs. Differentiate and you get the two error budgets that matter:

\frac{\partial m_{rec}}{\partial E_{ℓℓ}} = - \frac{\sqrt{s}}{m_{rec}}

→ ×1.92 at 240 GeV, ×2.92 at 365 GeV

\frac{\partial m_{rec}}{\partial \sqrt{s}} = \frac{\sqrt{s} - E_{ℓℓ}}{m_{rec}}

→ ×1.08 at 240 GeV, ×2.16 at 365 GeV

Both amplify. Both amplify worse at 365. This is one of several reasons the ZH programme lives at 240 and merely visits 365.

Check yourself — the irreducible floor

The note smears each beam by 222 MeV at 240 GeV. Turn the tracking term to zero and read off the peak width. Why is it not 222 MeV, and what does that number imply about how much effort is worth spending on the tracker?

√s fluctuates by √2 × 222 ≈ 314 MeV (two independent beams), and that propagates with the coefficient (√s−E_ℓℓ)/m_rec ≈ 1.08, giving ≈ 340 MeV of recoil-mass width before a single detector element is switched on. Since E_ℓℓ for a ~50 GeV Z carries roughly 105 GeV and the tracker delivers σ_p/p ≈ 0.2–0.5% at those momenta, the tracking contribution is comparable, not dominant. The practical consequence: doubling tracker performance buys you well under a factor √2 on the recoil peak, whereas monochromatisation of the beams would move the floor directly. It also means the recoil mass resolution is not what limits B(H→b̄b) — the tagging is.

Jet energy scale bias

Applied to both jets 1.000 Jet energy resolution at 50 GeV 5.0% Jet angular resolution 10 mrad

Probe point (drag on the map)

m_vis over the (E_jj, m_jj) plane

raw m_jj rescaled m_vis = α·m_jj

require:

(\sqrt{s}, \vec{0}) - α (E_{jj}, {\vec{p}}_{jj})

has invariant mass

m_{Z}

\Rightarrow α^{2} m_{jj}^{2} - 2 α E_{jj} \sqrt{s} + (s - m_{Z}^{2}) = 0

\Rightarrow α = \frac{E_{jj} \sqrt{s} - \sqrt{Δ}}{m_{jj}^{2}}, Δ = E_{jj}^{2} s - m_{jj}^{2} (s - m_{Z}^{2})

Both roots are real and positive (Δ ≥ 0 whenever m_jj ≤ E_jj√(s/(s−m_Z²)), which for a physical jet pair is always). The minus root is the physical one; the plus root sits near m_vis ≈ 400 GeV and is discarded.

Check yourself — why this is not just a cosmetic rescaling

Drag the jet energy scale slider to 0.90 and watch the two histograms. The raw m_jj peak walks off to 112 GeV. The m_vis peak does not move at all. Prove that algebraically, then say what the residual systematic on m_vis actually is.

Let both jets be mismeasured by a common factor k: E_jj→kE_jj, m_jj→km_jj. Then Δ → k²Δ, and

α → (kE√s − k√Δ)/(k²m²) = α/k, so m_vis = α m_jj → (α/k)(k m_jj) = α m_jj, exactly invariant.

The constraint has absorbed the entire jet energy scale into α. What survives is everything that is not a common multiplicative factor: the jet energy resolution (which makes m_jj/E_jj fluctuate), angular resolution, out-of-cone and neutrino losses that differ between b-jets and gluon jets, and — crucially — the assumption that the recoiling object really is an on-shell Z. For H→WW*→qqℓν or any final state with genuine missing energy the constraint is wrong, which is exactly why m_vis and m_miss are fitted together in two dimensions rather than either alone.

Detector

Jet energy resolution at 50 GeV 5.0% Jet direction resolution 5 mrad χ² cut 30

Pairing

Z(qq)H(b̄b) at √s = 240 — constrained fit

σ(m_H) raw—

σ(m_H) fitted—

Mispairing rate—

χ² cut efficiency—

E_{1} + E_{2} + E_{3} + E_{4} = \sqrt{s}, \sum_{i} {\vec{p}}_{i} = \vec{0}, {\vec{p}}_{i} = E_{i} {\vec{β}}_{i}

[1 1 1 1; β^{x}; β^{y}; β^{z}] \cdot \vec{E} = {(\sqrt{s}, 0, 0, 0)}^{T}

Four equations, four unknowns. The jet directions — measured to a few milliradians — completely determine the jet energies, which are measured to several percent. You are throwing away the calorimeter's energy measurement and replacing it with geometry plus the beam energy. That is the single biggest lever in the fully hadronic channel, and it is only available at a lepton collider.

Check yourself — when does the 4×4 solve fail?

Push the angular resolution to 60 mrad and watch the fitted resolution overtake the raw one. Separately: name three physical configurations in which this fit is systematically biased even with a perfect detector, and connect them to the note's <1% "inversion failed or unphysical" rate.

Conditioning. The matrix rows are the four unit vectors plus a row of ones. If the four jets are close to coplanar or two are nearly collinear, the determinant collapses and small angular errors are amplified without bound — the classic near-degenerate linear system. That is the <1% the note keeps at nominal values rather than discarding.

Physical biases: (i) semileptonic b and c decays put real neutrinos in the jets, so Σ E < √s and the fit compensates by inflating all energies — a flavour-dependent bias that is worst in exactly the H→b̄b sample the analysis cares most about; (ii) ISR photons down the beampipe remove longitudinal momentum and energy, breaking both constraint rows (this is why the effective √s is not the nominal √s); (iii) any final state with more than four partons — H→WW*→4q plus the Z gives six — is forced into a four-jet hypothesis, so the constraint is solved with the wrong topology and produces a plausible but meaningless χ². The note's χ²<30 cut removes some of (iii), which is visible in its efficiency table as the ~40% efficiency for H→WW versus ~80% for H→qq.

Tagger separation power A_f

b 6.0 c 4.4 s 2.6 g 3.4

Categorisation

Integrated luminosity 10.8 ab⁻¹

Process → category migration (row-normalised, %)

B-like purity—

C-like purity—

S-like purity—

G-like purity—

bc sg light / other

per jet j:

K_{j} = (b_{j}, c_{j}, s_{j}, g_{j}, τ_{j}, u_{j}, d_{j}), \sum_{f} K_{j, f} = 1

event score:

K = K_{1} + K_{2} \in {[0, 2]}^{7}, category = {argmax}_{f} K_{f}

The toy above models each jet's seven-vector as a softmax over logits A_f·Sim[f][·] plus Gumbel noise — i.e. a multinomial-logit response with a hand-set flavour-confusion structure (b↔c, s↔g↔u/d). It is not the note's GNN, but it has the same two degrees of freedom that matter: per-flavour separation, and which flavours bleed into which.

Three things to try. (1) Slide s down to zero: the S-like category fills with gluon and light-quark jets and the H→s̄s row goes flat — this is the whole story of why B(H→s̄s) is a limit and not a measurement. (2) Slide b up to maximum: B-like purity saturates but δμ_b̄b barely moves, because b-tagging was never the bottleneck. (3) Switch on L/M/H purity subdivision and watch the highest-purity bins carve out a nearly background-free b̄b region — that single move is worth more than any amount of extra tagger performance.

Check yourself — why summing the two jets' scores is not obviously right

The note forms K = K₁ + K₂ and takes the argmax. An alternative is the product (a likelihood ratio for "both jets are flavour f"), or a genuine 2-jet joint classifier. Under what conditions does the sum lose information relative to the product, and why might the note's choice still be defensible?

The sum is dominated by whichever jet is more confident: an event with K₁ = (0.95 b) and K₂ = (0.20 b, 0.45 g) still lands with B = 1.15 > G = 0.5 and is called B-like, even though the second jet actively prefers gluon. The product 0.95×0.20 = 0.19 versus 0.0×0.45 would flag the inconsistency. So the sum is insensitive to disagreement between the jets, which is precisely the signature of the dangerous backgrounds — Z(bb̄) recoiling against something else, or g→bb̄ splitting giving one b-like jet.

The defence is threefold. First, for a genuine H→f f̄ decay the two jets are the same flavour, so the product's extra discriminating power applies mostly to backgrounds you are already suppressing by mass. Second, the product is numerically brutal: one jet with a score of 10⁻⁴ kills the event, and tagger scores that far into the tail are not calibrated. Third, and most importantly, the note does not use the categorisation to measure — it uses it to partition, and then fits templates within each partition. Any orthogonal partition gives an unbiased answer; a better one just gives a smaller variance. That is why you can play with the sliders above and never introduce a bias, only lose sensitivity.

Category yields

Scaling

Integrated luminosity 10.8 ab⁻¹ MC events per data event ∞ (off) Background normalisation systematic 0.0%

Parameters of interest

δμ per decay — Fisher vs the note

δμ b̄b—

δμ c̄c—

δμ gg—

δμ s̄s—

Poisson, Gaussian limit, nuisance-free:

F_{ab} = \sum_{i} \frac{S_{ia} S_{ib}}{V_{i}}, {δ μ}_{a} = \sqrt{{(F^{- 1})}_{aa}}

V_{i} = (\sum_{a} S_{ia} + B_{i}) + \frac{B_{i}^{2}}{N_{i}^{MC}} + {(σ_{B} B_{i})}^{2}

i: one category — a single bin of the fit. Here the 21 rows of the note's Table 4, transcribed into this page: b̄b low/mid/high, c̄c low/mid/high, and so on. The sum Σ_i runs over all of them.
a, b: one of the seven decays being measured (b̄b, c̄c, gg, s̄s, WW, ZZ, ττ) — the columns of Table 4, and the buttons in the POI list.
μ_a: the signal strength of decay a: how many of those Higgs decays you see, divided by how many the Standard Model predicts. μ = 1 means "exactly as predicted". δμ_a is the uncertainty on it — the percentage the panel reports, and the number the whole paper is about.
S_ia: expected signal events of decay a landing in category i, assuming μ = 1. Read straight off Table 4.
B_i: expected background events in category i — everything that is not a Higgs decay (ZZ, WW, Z/γ*).
V_i: the variance of the event count in that bin: how much the number would jump around if you repeated the experiment. Square root of it is the error bar on the bin.
N_i^MC: how many simulated events the background estimate was built from. Fewer simulated events ⇒ a fuzzier prediction ⇒ larger V.
σ_B: a systematic on the background normalisation, as a fraction (0.05 = "we know the background rate to 5%").
F_ab: the Fisher information matrix: how sharply the data pin down the μ's. Big F = small errors. Inverting it turns information into a covariance matrix, and (F⁻¹)_aa is the variance of μ_a with the other six left free to float.

The three terms of V are the whole argument of the paper's results section: data statistics, Monte-Carlo statistics (the Barlow–Beeston penalty, why 0.26% becomes 0.33%), and normalisation systematics (which the note largely does not include). Toggle a decay out of the POI list and watch the others tighten — that difference is the price of measuring seven signal strengths at once rather than one.

Do it yourself — δμ(b̄b) on the back of an envelope

Every number in the blue bars comes out of three lines of arithmetic. Here is the b̄b one, in full, using only the numbers printed in the note's Table 4. You need a calculator, not a computer.

Why the formula looks like that. The expected count in bin i is T_i = Σ_a μ_aS_ia + B_i. Ask how fast that count moves when you turn one signal strength: ∂T_i/∂μ_a = S_ia. A bin is informative about μ_a when it responds strongly (large S_ia) and is quiet (small V_i) — so each bin contributes S_iaS_ib/V_i, and you add up the bins. That is the entire content of the first equation.

The one-parameter shortcut. If you measured b̄b alone, there is no matrix to invert: δμ = 1/√F_bb = 1/√(Σ_i S_i,b̄b²/V_i). The panel says the seven-parameter answer is almost the same — blue and grey bars coincide — so the shortcut is good to about a percent here. Take the simplest case: no MC statistics, no systematics, so V_i is just the total count T_i.

Three rows of Table 4. For each, add all seven signal columns and the three background columns to get T, then square the b̄b column and divide:

category	S (b̄b)	T (all)	S²/T
b̄b low	3 702.9	5 392.4	2 542.7
b̄b mid	6 885.3	8 586.1	5 521.4
b̄b high	48 179.9	50 911.4	45 595.0
sum			53 659

δμ = 1/√53 659 = 0.00432 = 0.43% — the number in the readout above. The other 18 categories contribute a few hundred more (b̄b leaks into the ZZ and HZZ bins), which is why the panel is a shade tighter than this hand estimate.

What each of the three variance terms does to that number. Slide MC events per data event to 1×: the panel adds T_i of simulation variance to every bin — signal and background templates both come from simulation — so V doubles, δμ grows by exactly √2, and 0.43% becomes 0.61%. That single factor is the whole gap to the note's published 0.60%. Now slide background systematic up instead: nothing much happens to b̄b, because its high-purity bin is 95% signal and (σ_BB)² is negligible against T — but watch s̄s, which is almost pure background, blow up immediately. That contrast is the physics: b̄b is limited by how much you simulated, s̄s by how well you know your background.

Try the same three lines on c̄c (rows 4–6 of Table 4, S = 557.8 / 635.2 / 1 768.8 on totals 3 596.9 / 1 910.1 / 2 736.9). You should land near 2.6%, and you will see why. The three b̄b bins are 69% / 80% / 95% signal; the c̄c bins are 16% / 33% / 65%. Everything else in those bins is count that fluctuates without carrying any information about μ_c̄c — and there are twenty times fewer c̄c events to begin with (2 962 against 58 768, which is just the branching-ratio ratio 2.891 / 58.24).

Check yourself — the missing 40% on b̄b

With the note's own Table 4 numbers, no MC statistics and no systematics, the counting Fisher gives δμ = 0.43% / 2.64% / 1.34% for b̄b / c̄c / gg against the note's 0.60% / 3.47% / 1.93%. Turning off the correlations changes nothing at all. Find the single knob that closes the gap, and say what it means physically.

First, rule out the obvious. Set any subset of the seven POIs to fixed: the remaining uncertainties barely move, and the correlation matrix is nearly diagonal. That is not an artefact of the toy — the note says the same thing, that its results are "extremely close to independent in terms of number of signal strengths we simultaneously measure". So the simultaneous fit is not the cost.

Now slide MC events per data event to 1×. The Fisher estimate becomes 0.61% / 3.73% / 1.90% — the note's numbers, to the precision this toy deserves. The gap is Monte-Carlo statistics, which the note explicitly implements and which nothing else in the analysis can remove.

Look at the note's own Table 1 to see why. The equivalent luminosity ratio L_gen/L is 0.177 for Z/γ*(qq̄), 0.952 for Z/γ*(ℓℓ), 2.10 for WW and 3.83 for ZZ. The dominant backgrounds are simulated with fewer events than the experiment will collect. A background bin with B expected data events built from N_MC weighted simulated events carries an extra variance B²/N_MC, which at L_gen ≈ L simply doubles the effective variance everywhere.

The consequence is a genuinely encouraging one, and the note says it in one sentence that is easy to miss: "for the future FCC-ee experiments it should not be a problem to generate significantly larger samples than those used in this study". Every published number here is degraded by roughly 40% purely by the size of a Monte-Carlo sample produced in 2023. The physics reach is the ungapped number. What is not recoverable by buying CPU is the experimental systematic budget, which is absent entirely.

One exception is worth noticing: s̄s goes the other way (counting gives 232%, the note 223%). Being background-dominated, its uncertainty is set by √B rather than by the MC replication of B, so MC statistics barely touch it — and the note's finer sub-categorisation buys back a few percent that pure counting throws away.

Coupling modifiers

κ_Z 1.000 κ_W 1.000

Run

Separation power of m_miss nominal

68% / 95% contours in the (κ_W, κ_Z) plane from ν̄νH(b̄b)

σ(ZH→νν̄H)—

σ(WW→H)—

δκ_Z—

δκ_W—

ZH, m_miss peaks at m_Z W-fusion + interference

ν̄ ν H final state = κ_{Z}^{2} \cdot [e^{+} e^{-} \to ZH, Z \to ν ν̄] + κ_{W}^{2} \cdot [e^{+} e^{-} \to ν_{e} {ν̄}_{e} H via WW fusion] + κ_{Z} κ_{W} \cdot [interference]

240 GeV: 40.1 fb ZH vs 6.1 fb fusion — a 15% contamination. 365 GeV: 29.2 fb vs 29.2 fb — they are equal, and the note stops trying to treat the sample inclusively. This is why Table 18 splits the production modes at 365 but Table 14 does not at 240.

The part that makes it model-independent

Everything on this page measures σ×B, not B. The recoil-mass technique in the Z(ℓℓ) channel is what converts one into the other: counting Z(ℓℓ) events with a recoil mass near 125 GeV measures σ_ZH ∝ κ_Z² without looking at the Higgs decay at all. Divide, and you get an absolute branching ratio. Then the chain closes:

01 · Recoil

N(Z→ℓℓ, m_rec≈125) → σ_ZH ∝ κ_Z². Decay-blind, so genuinely absolute.

02 · Divide

(σ_ZH×B_XX)/σ_ZH → B_XX. The 0.21% here is the numerator only.

03 · Close

Γ_H = Γ(H→ZZ)/B_ZZ ∝ κ_Z²/B_ZZ — the total width, no assumption about invisible decays.

04 · Couple

κ_f² ∝ Γ_H·B_ff. Every hadronic BR in this note feeds this last step.

Check yourself — the 365 GeV inversion

At 240 GeV the note gets ±0.38% on ZH×B(b̄b) and ±1.99% on ν_eν̄_eH×B(b̄b). At 365 GeV the two are ±0.73% and ±0.68% — the fusion channel has overtaken Higgsstrahlung. Reconstruct the argument, and then say why this makes the 365 GeV run scientifically necessary rather than merely nice.

σ_ZH falls as roughly 1/s above threshold, while W-fusion is a t-channel process whose cross-section grows logarithmically: 6.1 → 29.2 fb between 240 and 365, against 40.1 → 29.2 fb for ZH→νν̄H. Combined with 3.0 ab⁻¹ instead of 10.8, the ZH sample shrinks by ~13× and the fusion sample only by ~2×, so the fusion measurement wins.

Why it matters: ZH and Z-fusion amplitudes both scale with κ_Z, so a 240-only programme measures κ_Z superbly and κ_W only through H→WW* decay — which is degenerate with the total width. W-fusion gives a production-side handle on κ_W, breaking that degeneracy directly. Slide κ_W in the panel with the 240-only run selected and watch the contour stretch along κ_W; add 365 and it closes. The 365 GeV run is not just about tt̄ and the top Yukawa.

Everything the note concludes, in one table

Expected precision (%) on σ(ZH)×B(H→jj) at 68% CL, √s = 240 GeV, L = 10.8 ab⁻¹ (four detectors). Note 23 of the paper. Two variants of the ν̄ν analysis are quoted because they were done independently.

Channel	b̄b	c̄c	gg	s̄s	ττ	ZZ	WW
Z → ℓℓ	0.60	3.47	1.93	220	2.54	7.65	1.49
Z → qq	0.32	3.52	3.07	410	110	50	8.74
Z → νν̄ (I)	0.35	2.06	1.01	100	10.6	11.4	1.28
Z → νν̄ (II)	0.33	2.27	0.94	140	21.8	19.8	1.89
Combined (I)	0.21	1.56	0.85	89	2.46	6.24	0.95
Combined (II)	0.21	1.66	0.80	105	3.97	10.1	1.16

Split by production mode and energy

Relative uncertainty (%) on σ_ZH×B and σ_{ν_eν̄_e}×B, 68% CL. Table 24. Note the ordering flip in b̄b between the two energies.

	240 · ZH	240 · ν_eν̄_eH	365 · ZH	365 · ν_eν̄_eH
H → b̄b	0.21	1.89	0.41	0.67
H → c̄c	1.61	19.4	3.13	3.49
H → s̄s	120	990	360	290
H → gg	0.80	5.50	2.21	2.66
H → WW	1.17	15.6	3.18	5.36
H → ZZ	9.94	130	26.0	37.1
H → ττ	3.67	∞	11.0	24.2

Inputs you will want at hand

	240 GeV	365 GeV
Integrated lumi	10.8 ab⁻¹ (4 IPs, 3 years)	3.0 ab⁻¹ (5 years)
σ(ZH)	≈ 200 fb — near the L·σ maximum, not the σ maximum (which is at ~255 GeV)	≈ 125 fb
σ by Z decay	qq̄H 136.35 · νν̄H 46.2 · eeH 7.17 · μμH 6.76 fb	qq̄H 84.36 · νν̄H 53.94 · eeH 7.39 · μμH 4.19 fb
W-fusion	≈ 6.1 fb (15% of ZH→νν̄H)	≈ 29.2 fb (117% of ZH→νν̄H)
Z-fusion (eeH)	0.40 fb vs 6.76 fb Z(ee)H	3.20 fb vs 4.19 fb
Beam energy spread	222 MeV per beam	—
Beamspot	σ_z 0.64 mm · σ_x 9.8 μm · σ_y 25.4 nm	—

Detector	Modified IDEA as simulated
Tracking	Si vertex (innermost layer R = 1.2 cm) + low-mass drift chamber (1.6% X₀ transverse) + Si micro-strips, in 2 T. Beam pipe R = 1 cm, 0.67% X₀. 100% efficiency assumed above 100 MeV, \|η\| < 2.56.
ECAL	σ_E/E = 3%/√E ⊕ 0.2%/E ⊕ 0.5% — dual-readout crystals, solenoid moved behind the ECAL so it does not degrade this
HCAL	σ_E/E = 30%/√E ⊕ 5%/E ⊕ 1% — dual-readout lead/fibre
ℓ / γ ID	99% above 2 GeV, \|η\| < 3; mis-identification neglected entirely
Jets	Durham exclusive k_T, N = 2 (or 4), E-scheme; flavour tagging by GNN with 7 outputs summing to unity, trained on Z(νν̄)H

Higgs branching ratios assumed

Decay	B (%)	Status at the LHC	FCC-ee combined
b̄b	58.24	observed, HL-LHC proj. 4.4% (ATLAS+CMS)	0.21%
WW	21.37	observed	0.95%
gg	8.187	no attempt	0.85%
ττ	6.272	observed	2.46%
c̄c	2.891	searches only, sensitivity ≈ SM	1.56%
ZZ	2.619	observed	6.24%
s̄s	0.024	unfeasible	limit at 1.6 × SM

Where this note is softer than its abstract

01 · No experimental systematics anywhere in the leading channels

The ℓℓH and ν̄νH(I) fits state explicitly that there are no systematic uncertainties, only Monte-Carlo statistics. The ν̄νH(II) analysis adds 0.1% (signal) / 5% (background) log-normal normalisations and then reports that the sidebands constrain the backgrounds to "better than a percent level". Every headline number is therefore a statistical floor. Flavour-tagging efficiency calibration, fragmentation modelling, and the jet energy response for b- versus gluon-jets are all absent, and each of them is plausibly at the 0.3–1% level for b̄b. Use the correlated-floor slider on the combination tab to see what that does.

02 · Table 1 contains unphysical placeholder samples

The rows ν̄νH(bd̄), ν̄νH(bs̄), ν̄νH(sd̄), ν̄νH(cū) — and their eeH/μμH counterparts — carry σ = 1000 fb, five times the entire ZH cross-section, for flavour-violating Higgs decays that are forbidden at any observable rate. These are clearly generator bookkeeping placeholders that were never renormalised, and they appear in the figure legends (llH(bd), llH(bs), llH(sd), llH(cu)). If any of them entered a yield table at face value it would swamp the analysis. Nothing in the results suggests they did, but their presence means the sample bookkeeping was not audited.

03 · Table 18 quotes the wrong luminosity

The 365 GeV ν̄νjj results are headed "√s 365 GeV, integrated luminosity 10.8 ab⁻¹" while the run plan, Table 2 and every other table use 3.0 ab⁻¹ at that energy. Either the table header is a copy-paste error, or the numbers are optimistic by √(10.8/3.0) = 1.9. Since Table 24's 365 GeV column (ZH b̄b ±0.41%) is better than Table 18's ±0.73%, the inconsistency is not merely cosmetic.

04 · The s-tagger is doing work no experiment has ever demonstrated

Strange tagging relies on identified high-momentum kaons — in IDEA, on dE/dx or cluster counting in the drift chamber plus time-of-flight. The Delphes parameterisation encodes an assumed performance; there is no fragmentation systematic, no assessment of how the s-tag responds to a gluon jet that fragments into strange hadrons, and no variation of the strangeness-suppression parameter in the hadronisation model. The resulting limit — B(H→s̄s) < 1.6 × SM — is also being placed on a target whose SM value carries a large uncertainty from m_s, which the note itself flags but does not propagate.

05 · "gg" is not a clean parton label

A gluon-tagged category is contaminated by g→b̄b and g→c̄c splittings inside jets, and conversely H→b̄b events radiate hard gluons. The seven-output tagger was trained with the true flavour "based on the generated decay of the Higgs boson" — i.e. the label is the decay mode, not the parton reaching the detector. That is the right label for this measurement, but it means the tagger has partly learned to identify the Higgs decay via its global event shape, and its performance will not transfer to any other sample. It also makes the confusion matrices in Figures 7 and 14 optimistic if the parton shower model is wrong.

06 · S/√(S+B) is quoted as "significance" in the yield tables

The per-category numbers in parentheses in Tables 4 and 9 use S/√(S+B), which is neither the asymptotic discovery significance nor the quantity entering the profile likelihood. It systematically overstates categories where B ≫ S and understates highly pure ones. It is a sorting aid, not a result — but it is presented adjacent to the yields as if it were one.

07 · Four identical detectors is an assumption, not a design

10.8 ab⁻¹ presumes four interaction points each with IDEA-like performance. The note tells you the scaling honestly (halving the IPs costs a factor 1.7 in luminosity, not 2, because of the machine layout), but the headline 0.21% becomes ~0.28% for two IPs before any systematic is added — and the current FCC-ee baseline has fluctuated between two and four IPs during the design study.

08 · Internal inconsistency in the ν̄ν selection table

Table 8's selection row reads "70 < m_vis < 150 GeV, 60 < m_miss < 220 GeV" while the text specifies m_miss in 50–140 GeV and the fit is performed over 60–140 GeV. The efficiencies quoted (86–92%) are only reproducible with one of these. Small, but the reader cannot reconstruct the cutflow.

Harder problems

Problem 1 — the ℓℓ efficiency dip on b̄b

In the ℓℓH cutflow, the "no additional leptons with p > 25 GeV" cut removes 5% of H→b̄b events but essentially none of H→gg or H→s̄s. The note remarks it "might be considered to veto additional isolated leptons instead". Quantify the effect from first principles and say what the right fix costs.

B hadrons decay semileptonically with B(b→ℓν X) ≈ 10.5% per lepton species, and with the b→c→ℓ cascade the inclusive rate of a lepton from a b-jet is ~20%. With two b-jets and a spectrum that puts a few percent of those leptons above 25 GeV (the b-jets carry ~60 GeV each, and the lepton takes a soft fraction), you land at a few percent per event. The observed 5% is entirely consistent.

The proposed fix — vetoing only isolated leptons — recovers most of it, because leptons from heavy-flavour decay sit inside the jet cone. The cost is that the veto's original job was to suppress H→ττ with a leptonic τ and ZZ→ℓℓℓℓ, whose leptons are isolated, so isolation is exactly the discriminant you want. In practice the fix is nearly free, worth ~5% in b̄b yield, i.e. ~2.5% in δμ — small compared to the 40% gap identified in the Fisher exercise, which is why nobody bothered. Recognising which inefficiencies are worth fixing is the skill; this one is not.

Problem 2 — the exclusive-2 clustering choice

Every analysis here forces the event into exactly N jets with the exclusive Durham algorithm and uses d_2,3, d_3,4 as discriminants. Why is exclusive clustering the natural choice at an e⁺e⁻ collider and pathological at a hadron collider, and what specifically do d_2,3 and d_3,4 buy in this analysis?

Exclusive clustering asserts that every reconstructed particle belongs to one of N jets. At an e⁺e⁻ collider that is nearly true: there is no underlying event, no pile-up, no beam remnant, and the total visible energy is √s by construction. At the LHC the same statement is false at the level of tens of GeV per event, which is why hadron colliders use inclusive algorithms with a fixed R.

d_i,i+1 is the k_T distance at which the event transitions from i+1 to i jets, d = 2 min(E_i², E_j²)(1−cos θ_ij). It is therefore a continuous measure of "how badly does this event want to be more than two jets". H→gg and H→WW*→4q produce large d_2,3; H→b̄b produces small d_2,3 but non-negligible d_3,4 from b-decay substructure; a leptonic τ leaves d_2,3 = 0 exactly, which is why the note's cut is literally d_2,3 > 0 — a topology veto disguised as a kinematic one. Feeding log(d) rather than d to the network is not cosmetic either: the distribution is roughly log-uniform over three decades.

Problem 3 — reconstruct the ν̄ν sensitivity from the yield table

In the ν̄νH categories, the three b̄b bins hold S = 61 092 / 55 486 / 101 039 signal on totals of 199 630 / 88 233 / 111 892. Estimate δμ_b̄b by hand and compare with the note's 0.35%. Then explain why the b̄b_high bin, with 90% purity, contributes only about half the information.

Σ S²/T = 61092²/199630 + 55486²/88233 + 101039²/111892 = 18 695 + 34 891 + 91 249 ≈ 144 800, plus a few hundred from the leakage bins. δμ = 1/√144 800 ≈ 0.26%, against the note's 0.35%, the difference again being MC statistics and the seven-parameter fit (compare Table 14: 0.26% without MC stat, 0.33% with — the counting estimate reproduces the no-MC-stat number almost exactly).

The high-purity bin contributes 91 249 of the 145 000, i.e. 63%. Its information density is S²/T = S·(S/T) = S × purity, so information scales as yield times purity, not purity alone. The low-purity bin has 61 092 events at 31% purity → 19 000; the high bin has 101 039 at 90% → 91 000. Doubling purity in a bin is worth exactly as much as doubling its yield. That symmetry is the reason the note's L/M/H subdivision (which changes neither total yield nor total purity, only their correlation across bins) improves the result at all: it moves yield into the high-purity end of the distribution.

Problem 4 — the s̄s limit, done properly

The note reports a 95% CL upper limit of 1.6 × SM on B(H→s̄s) from the combination, with δμ ≈ 89–105%. Show that these two statements are consistent, and identify the statistical subtlety that makes "1.6 × SM" a slightly odd thing to quote.

For a Gaussian likelihood centred on μ = 1 with σ_μ ≈ 0.9, the one-sided 95% upper limit is μ < 1 + 1.64σ ≈ 2.5. To land at 1.6 you need σ_μ ≈ 0.37, or an expected limit computed on the background-only Asimov set (μ = 0), where the limit is 1.64σ ≈ 1.5. The 1.6 is therefore a background-only expected limit, while the 89–105% are signal-injected uncertainties on μ. Both are legitimate; they answer different questions and should not be read as the same number in different units.

The oddity: quoting a limit in units of the SM prediction when that prediction is itself poorly known. B(H→s̄s) ∝ m_s(m_H)², and m_s carries a several-percent uncertainty which is amplified by the running and squaring. The note flags this in the introduction ("a large uncertainty related to the limited knowledge of the mass of the strange quark") and then quotes a limit in exactly those units. The physically meaningful statement is a limit on κ_s·y_s^SM, or better, on the absolute partial width.

Problem 5 — design the measurement you would actually do

You have been asked to improve B(H→c̄c) beyond 1.6%. You may spend effort on exactly one of: (a) charm tagging, (b) jet energy resolution, (c) more luminosity, (d) the categorisation scheme. Argue for one using the numbers on this page.

Work out what limits it. In the ℓℓ channel the c̄c categories hold S ≈ 2 960 on T ≈ 8 244 — purity 36%, and the contamination is dominated by ZZ background (4 587 events) rather than by other Higgs decays (≈ 100 events). So c̄c is background-limited, not tag-limited.

(c) more luminosity scales as 1/√L and is the least efficient use of anything. (b) jet energy resolution is nearly irrelevant, because the discriminant is m_recoil from the leptons, which never touches the jets. (a) charm tagging helps but you are fighting Z→cc̄ inside ZZ, which has the same flavour content as your signal — better c-tagging does not separate them at all; only the recoil mass does. That leaves (d): the ZZ background is separable from ZH by the Z polar-angle distribution (1/σ dσ/dcos θ rises steeply toward |cos θ| = 1 for ZZ and is flat for ZH — the note plots exactly this in Fig. 3 and feeds it to the network) and by the recoil lineshape. Finer categorisation in cos θ_ℓℓ and tighter recoil binning attack the actual limitation. In the ν̄ν channel the same reasoning points instead at the 2D (m_vis, m_miss) template granularity.

The general lesson: look at what is in the denominator of S²/(S+B) before you improve anything.

Two thirds of the Higgs, and nobody has seen most of it

Beam

Lepton flavour

Jet energy scale bias

Probe point (drag on the map)

Detector

Pairing

Tagger separation power A_f

Categorisation

Category yields

Scaling

Parameters of interest

Channels in the combination

Correlated tagging systematic

Coupling modifiers

Run

The part that makes it model-independent

Everything the note concludes, in one table

Split by production mode and energy

Inputs you will want at hand

Higgs branching ratios assumed

Where this note is softer than its abstract

Harder problems

Beam

Lepton flavour

Jet energy scale bias

Probe point (drag on the map)

Detector

Pairing

Tagger separation power Af

Categorisation

Category yields

Scaling

Parameters of interest

Channels in the combination

Correlated tagging systematic

Coupling modifiers

Run

The part that makes it model-independent

Everything the note concludes, in one table

Split by production mode and energy

Inputs you will want at hand

Higgs branching ratios assumed

Where this note is softer than its abstract

Harder problems

Tagger separation power A_f