Hunting the Higgs Before the LHC · Elementary Particle Physics

This is a section about twenty years of not finding something, and it is worth reading because each failure narrowed where the answer could be.

🎯 Why this matters

An exclusion is a positive result kept in negative form. Every year of not finding it removed possibilities, so when the particle did appear it appeared inside a window that two decades of failure had drawn.

§9.12 predicted a scalar and refused to say what it weighs. This short section is what two decades of searching did about that — and the honest summary is that both machines were built in the wrong place, for reasons that were nobody’s fault and are worth understanding.

Where to look: a band, not a value

The loop corrections of §9.8 do constrain $M_H$ , logarithmically. The global fit gave

M_H = 94^{\,+29}_{\,-24}\;\text{GeV}, \qquad M_H < 152\;\text{GeV at 95\% CL}

But you cannot search only there. The fit assumes the Standard Model, and the search is partly a test of the Standard Model — so the range must be extended to wherever the theory stops making sense on its own terms. That point is computable: the cross-section for $WW\to H\to ZZ$ exceeds the unitarity limit for $M_H$ above about 700 GeV. Taking some margin, the LHC was designed to reach 1000 GeV.

💡 What this really says — a theory that fails without the particle tells you where to stop looking

The upper bound is a nice piece of reasoning and it is the same argument as §9.10’s, run one level up.

There, $e^+e^-\to W^+W^-$ diverged until the $Z$ exchange was included. Here the divergence is in $W_LW_L\to W_LW_L$ — the scattering of longitudinally polarized $W$ s, the very states that §9.12 showed are the eaten Goldstone bosons. Their scattering amplitude grows with energy, and the diagram that cancels the growth is a Higgs exchange.

So the Higgs is not an optional extra bolted onto the theory. Without it the theory is inconsistent above about a TeV — and if the Higgs is too heavy, its cancellation arrives too late and unitarity is violated before it can help.

That converts “we do not know the mass” into “we know it is below roughly 700 GeV, or something else entirely is going on at the TeV scale”. Either way there was something to find, and that guarantee is what justified building a machine to look for it. An engineer will recognise the form: a stability argument that bounds a parameter you cannot measure directly, by asking where the model breaks.

LEP: limited by energy, and by 7 GeV

At LEP energies the Higgs would decay overwhelmingly to $b\bar b$ , and the production channel is Higgsstrahlung — a $Z$ radiating a Higgs:

Fig. 9.42 — Higgs production at an e⁺e⁻ collider

Click a vertex or an internal line.

Bettini Fig. 9.42. The final state is two b jets plus whatever the second Z decays to. The b jets are tagged by their displaced vertices — §9.9's Example 9.6 again.

The whole search is then governed by one kinematic inequality. You must make an $H$ and a $Z$ , so

\sqrt{s} \;\ge\; M_H + M_Z \qquad\Longrightarrow\qquad M_H \;\lesssim\; \sqrt{s} - M_Z

LEP’s superconducting cavities were pushed as far as the tunnel allowed, reaching $\sqrt s = 209$ GeV for a total of about 0.5 fb⁻¹. Nothing was found, giving

M_H > 114.4\;\text{GeV at 95\% CL}

how close LEP came, and why the Tevatron could not finish the job

MZ, MH, MW = 91.1875, 125.25, 80.377
print(f"LEP's reach: e+e- -> H Z needs sqrt(s) >= M_H + M_Z, M_Z = {MZ:.2f}")
for rs in (189.0, 200.0, 209.0):
    tail = "   <- the end of the line" if rs == 209.0 else ""
    print(f"  sqrt(s) = {rs:5.1f} GeV  ->  M_H up to {rs-MZ:6.1f} GeV{tail}")
print(f"\n  LEP's published limit was 114.4 GeV, against a reach of {209-MZ:.1f}:")
print( "  it explored essentially every mass it was able to produce.")
print(f"\nbut the Higgs is at {MH} GeV, so LEP needed")
print(f"  sqrt(s) = {MH} + {MZ:.2f} = {MH+MZ:.1f} GeV")
print(f"  it reached 209.0.   IT MISSED BY {MH+MZ-209:.1f} GeV -- {(MH+MZ-209)/209*100:.1f}%.")

print("\nthe Tevatron had the opposite problem -- reach, but no sensitivity:")
print( "  below ~130 GeV the Higgs decays to b bbar (58%), and QCD makes")
print( "  b bbar pairs vastly more often.  b-tagging is not enough.")
print(f"  above 2 M_W = {2*MW:.1f} GeV the H -> WW channel opens and rapidly")
print( "  reaches BR ~ 100%, and TWO leptonic W decays are a clean signature.")
print( "  so the Tevatron excluded 147 < M_H < 179 GeV -- a window that does")
print(f"  NOT contain {MH}.")

print("\nso the Higgs sat exactly in the gap between the two machines:")
print(f"  LEP:      could produce up to {209-MZ:.0f} GeV        energy-limited")
print( "  Tevatron: sensitive from about 145 upward    background-limited")
print(f"  M_H = {MH} falls between them, and both missed it.")
print("\nand the electroweak fit was right all along:")
print( "  M_H = 94 +29 -24 GeV, upper limit 152 at 95% CL")
print(f"  the true {MH} is {(MH-94)/29:.2f} sigma above centre -- comfortably inside.")

prints

LEP's reach: e+e- -> H Z needs sqrt(s) >= M_H + M_Z, M_Z = 91.19
sqrt(s) = 189.0 GeV  ->  M_H up to   97.8 GeV
sqrt(s) = 200.0 GeV  ->  M_H up to  108.8 GeV
sqrt(s) = 209.0 GeV  ->  M_H up to  117.8 GeV   <- the end of the line

LEP's published limit was 114.4 GeV, against a reach of 117.8:
it explored essentially every mass it was able to produce.

but the Higgs is at 125.25 GeV, so LEP needed
sqrt(s) = 125.25 + 91.19 = 216.4 GeV
it reached 209.0.   IT MISSED BY 7.4 GeV -- 3.6%.

the Tevatron had the opposite problem -- reach, but no sensitivity:
below ~130 GeV the Higgs decays to b bbar (58%), and QCD makes
b bbar pairs vastly more often.  b-tagging is not enough.
above 2 M_W = 160.8 GeV the H -> WW channel opens and rapidly
reaches BR ~ 100%, and TWO leptonic W decays are a clean signature.
so the Tevatron excluded 147 < M_H < 179 GeV -- a window that does
NOT contain 125.25.

so the Higgs sat exactly in the gap between the two machines:
LEP:      could produce up to 118 GeV        energy-limited
Tevatron: sensitive from about 145 upward    background-limited
M_H = 125.25 falls between them, and both missed it.

and the electroweak fit was right all along:
M_H = 94 +29 -24 GeV, upper limit 152 at 95% CL
the true 125.25 is 1.08 sigma above centre -- comfortably inside.

📏 The Higgs sat in the gap between the two machines, and that was structural

Look at the two limits together. LEP could produce a Higgs up to 118 GeV and was stopped by energy. The Tevatron had energy to spare and was blind below about 145 GeV, stopped by background. The Higgs is at 125.

That is not bad luck twice; it is one fact seen twice. The branching ratios of §9.12 make $H\to b\bar b$ dominant below ~130 GeV and $H\to WW$ dominant above ~160, and those two regimes have opposite experimental characters:

$b\bar b$ has a large branching ratio and a hopeless background at a hadron collider, since QCD produces $b\bar b$ pairs constantly. It is only usable at an $e^+e^-$ machine, where there is no QCD background — and there you are limited by the energy you can build.
$WW$ has a clean leptonic signature but does not open until $2M_W = 161$ GeV.

Between the top of the $e^+e^-$ reach and the opening of the $WW$ channel there is a window that neither kind of machine handles well, and the Higgs is in it. Getting at it needed the third combination — a hadron collider with enough luminosity and resolution to use the rare channels, $\gamma\gamma$ at $2\times10^{-3}$ and $ZZ^*\to4\ell$ at $1.3\times10^{-4}$ .

That is what §9.14 is about, and it is why the LHC’s electromagnetic calorimeters were the objects of such effort.

The Tevatron: reach without sensitivity

At a $p\bar p$ collider the Higgs is made three ways, and the book shows all three because the LHC uses the same list:

Bettini p. 402. The same three appear again in Fig. 9.57 for the LHC, with different relative importance because pp is not pp̄.

mechanism↕	how↕	note↕
gluon fusion	two gluons fuse through a virtual top loop	the Higgs has no direct gluon coupling — this works only because the top Yukawa is ≈ 1 (§9.12). Dominant at the LHC (87 %)
Higgs bremsstrahlung ( $VH$ )	a quark–antiquark pair makes a $W$ or $Z$ , which radiates an $H$	the $p\bar p$ analogue of LEP's Higgsstrahlung, and the channel that eventually gave $H\to b\bar b$ a usable signature — the associated lepton tags the event
vector-boson fusion	a quark from each hadron radiates a $W$ or $Z$ ; the two fuse into an $H$	leaves two forward jets as a tag. It is the dominant mode at high $M_H$ , which is what set the LHC energy — see §9.14

The Tevatron collected about 10 fb⁻¹ in Run II and searched everywhere it could. Below ~130 GeV it was defeated by QCD $b\bar b$ production even with vertex tagging. Above 165 GeV, where $H\to W^+W^-$ opens and rapidly reaches a branching ratio near 100 %, both $W$ s can be caught leptonically and the background is manageable. The result by the LHC’s start:

147\;\text{GeV} < M_H < 179\;\text{GeV}\quad\text{excluded}

M_H = 115 GeV
M_H = 300 GeV
M_H = 1000 GeV
the electroweak fit, 68 % CL (Fig. 9.44)
measured today: m_t = 172.7, M_W = 80.377

Figs. 9.44 and 9.52, redrawn as one plot. Each line is a Higgs mass; measuring M_W and m_t locates you between them. The bands are wide because the dependence on M_H is logarithmic — §9.8's point, made visible: a factor of nine in M_H spans about the width of this plot.

⚙️ Engineer’s bridge — two decades of watching one point move on one plot

The plot above is the most-watched figure in particle physics between roughly 1990 and 2012, and it is worth understanding why it took that form.

You have two measurable quantities, $M_W$ and $m_t$ , and one unknown, $M_H$ . The Standard Model relates all three through the loop corrections of §9.8. So each candidate Higgs mass draws a line in the $(m_t, M_W)$ plane, and measuring the two observables places you somewhere among the lines. That is a prediction of $M_H$ from two quantities that have nothing obviously to do with it.

The reason the lines are so far apart in $m_t$ and so close together in $M_H$ is the quadratic-versus-logarithmic asymmetry again: moving $M_H$ by a factor of nine, from 115 to 1000 GeV, shifts $M_W$ by about as much as moving $m_t$ by 20 GeV.

An engineer will recognise the plot as a nomogram — a graphical solution of a constraint among three variables, where you measure two and read the third. It is also, in modern terms, a two-parameter fit displayed as its confidence contour against a family of model curves, and reading it correctly means remembering that the lines carry theory uncertainty too.

What makes it a good story is that the ellipse and the lines moved independently for twenty years — the ellipse as $M_W$ and $m_t$ were measured better, the lines as the calculations improved — and they never stopped overlapping. The Higgs was found at 125 GeV, inside the band the plot had been pointing at since the early 1990s.

Where it breaks: overlapping bands are not independent confirmations. Both the ellipse and the lines come from the same electroweak fit at one loop, so they share the theory calculation and several inputs — agreement between them tests the measurements, not the framework. And the Higgs enters these corrections only logarithmically, which is why the pre-discovery constraint was a broad band rather than a number: the fit favoured a light Higgs and excluded a heavy one, but “pointing at 125 GeV” is a reading available mainly in hindsight. A diagnostic plot whose bands must overlap by construction can only ever catch a gross failure.

🔑 If you remember only three things

The bounds closed from both directions. Consistency of the theory capped it from above while the machines pushed up from below, and the search was that interval shrinking.
Two limits, two different causes. One machine ran out of energy and the other ran out of events, which is why neither could be fixed by running longer.
Every exclusion is conditional on the model assumed. These were redrawn whenever the model moved, which is why a limit carries a date as well as a number.

Where this goes next

Both pre-LHC machines have now done what they can: a lower limit at 114.4 GeV, an excluded window from 147 to 179, and an electroweak fit preferring something light. The gap that remains — roughly 115 to 145 GeV — is precisely where the Higgs is, and reaching it requires the rare decay channels.

§9.14 builds the machine: 14 TeV, $10^{34}$ cm⁻²s⁻¹, and two detectors designed around exactly the two measurements this section shows are needed — a di-photon mass resolution of 1–2 GeV, and four-lepton reconstruction clean enough to see a peak built from $10^{-4}$ of the Higgses produced.

✅ Check yourself — the search before the LHC

0/6 answered · 0 correct

1.The electroweak fit preferred M_H = 94 +29/−24 GeV with a 95 % upper limit of 152. Why did the LHC still have to be designed to reach 1000 GeV?
2.LEP's Higgs reach was governed by √s ≥ M_H + M_Z, and it reached √s = 209 GeV. How close did it come?
3.The Tevatron had far more than enough energy for a 125 GeV Higgs. Why could it not find one?
4.LEP was blind above ~118 GeV and the Tevatron below ~145. The Higgs is at 125. Was that bad luck?
5.In the M_W–m_t plane, why is a factor of nine in M_H spanned by only the width of the plot?
6.Gluon fusion dominates Higgs production at a hadron collider, yet the Higgs has no coupling to gluons. How?