Higgs boson discovery in 2012

The ATLAS experiment and CMS experiment searched proton-proton collisions at the Large Hadron Collider at collision energies of

7

and

8 TeV

. The ATLAS experiment used silicon tracking, a

2 T

solenoid, liquid-argon electromagnetic calorimeters and an external muon spectrometer. Its July 2012 paper reported a combined local statistical significance of

6.0

standard deviations near

126.5 GeV

; this paper result is distinct from the initial 4 July announcement.^[ref]

The CMS experiment used silicon tracking in a

3.8 T

solenoid, lead-tungstate electromagnetic calorimeters and external muon chambers. Electrons combine tracks and particle showers; muons combine tracking and chamber hits; photons supply shower energies and directions. Selected

H \to γγ

and

H \to Z Z^{*} \to 4 ℓ

candidates, with

ℓ = e, μ

, yielded invariant mass peaks. It checked lepton efficiencies with Z boson decays, estimated fake-lepton backgrounds from control samples with relaxed identification, and fitted the smooth diphoton background from data. Its combined local statistical significance was

5.0

standard deviations near

125 GeV

.^[ref]

The ATLAS experiment compared signal-plus-background and background-only hypotheses with a profile likelihood, accounting for systematic uncertainties and correlations between decay channels. Diphoton and four-lepton candidates gave precise invariant masses;

W W^{*}

decays supplied additional evidence with poorer mass reconstruction because of escaping neutrinos. A local p-value concerns one specified mass; scanning masses requires a look-elsewhere effect correction.^[ref]

Two-photon decay excludes spin one under the usual assumptions, but does not prove spin zero. A mass peak does not determine every Higgs boson coupling. Production rates, branching fractions and angular correlations require additional tests.^[ref]

Table of contents
- Higgs resonance resolution and background model Higgs boson discovery in 2012

Higgs resonance resolution and background model ( $m_{γγ}^{2} = 2 E_{1} E_{2} (1 - cos α)$ )

 0  0

The bridge from detector readout to a particle resonance is four-momentum conservation. In units

c = 1

, sum the measured final-state four-momenta:

m^{2} = (\sum_{i} E_{i})^{2} - ∣ \sum_{i} p_{i} ∣^{2} .

(1)

For two photons separated by angle

α

∣ p_{i} ∣ = E_{i}

gives the displayed diphoton formula. Two back-to-back photons of

60 GeV

each have

m_{γγ} = 120 GeV

. A four-lepton candidate instead uses four four-momenta in the same sum. Collision debris outside the selected decay is not included. Since the partons carry unknown fractions of the incoming proton momenta, reconstructing the decay products is more informative than assigning the full beam energy to each candidate.

Figure 1.
Detector functions, reconstructed mass and a chosen resolution comparison
. Functional schematic followed by an original expectation model: sixty selected signal events at $120 GeV$ , a flat background of eight events per $1 0^{9}$ electronvolts, and chosen Gaussian mass widths $1.5$ and $3 GeV$ . No historical event counts or copied discovery histograms are shown. **Details**.

Suppose the intrinsic decay width is negligible compared with the stipulated detector spread. Let the reconstructed signal mass follow a normal distribution with mean

m_{0} = 120 GeV

and standard deviation

σ_{m}

. For a total expected selected signal

S = 60

and flat background density

b = 8

events per

1 0^{9}

electronvolts, the expected count density is

\frac{d N}{d m} = b + \frac{S}{2 π σ _{m}} exp [- \frac{( m - m _{0} ) ^{2}}{2 σ _{m}^{2}}] .

(2)

This is a deliberately chosen response model, not a fit to either experiment. In the fixed window

m_{0} - a < m < m_{0} + a

with

a = 3 GeV

, direct integration gives

B_{w} = 2 ab = 48, S_{w} = S erf (\frac{a}{2 σ _{m}}) .

(3)

The expected signal counts are

57.2700

for

σ_{m} = 1.5 GeV

and

40.9614

for

σ_{m} = 3 GeV

. Doubling the spread lowers the fraction in the same window without changing the total signal. Detector resolution can therefore change the visibility of a peak even with identical particle production.

As a separate counting example, stipulate an observed integer count

n = 70

and an exactly known background mean

B_{w} = 48

. The Poisson distribution gives a likelihood

L (s) = e^{- (48 + s)} (48 + s)^{70} /70!

for nonnegative signal mean

s

. Its maximum-likelihood estimator is

s = 22

, and the likelihood ratio statistic against

s = 0

q_{0} = - 2 lo g \frac{L ( 0 )}{L ( 22 )} = 2 [70 lo g \frac{70}{48} - 22] ≃ 8.82119.

(4)

This tests a specified window with a specified background, and is not either experiment's discovery significance. If the background mean is allowed to vary freely, background alone can choose

70

and fit that count exactly. Control samples, a defensible mass-shape model, nuisance parameters and corrections for searching many masses are therefore essential to the experimental inference.