The W Mass

In the SM, the W and Z Mass satisfy the relationship

sin2 θW = 1 − \frac{M2 W}{M2 Z}

• Defines θW based on the physical IVB masses

• but EWK theory is renormalisable – Coupling constants are a function of Q2

– Have to take “running” into account (introduce Δr)

M2 W = \frac{1}{2} \{1 + [1 − \frac{4πα(1 + Δr)}{√2GF M2 Z} ]1/2 \} M2 Z

SM predicts M_{W} based on low-energy data

• Existence of split heavy fermions doublet (t, b) affects MW /MZ • effects of top quark and Higgs mass different

• Introduces – quadratic dependence on Mtop

– logarithmic dependence on MHiggs

Prediction for M_{W} vs M_{top}

Can use all LEP data and low-energy data

• Have to assume MHiggs

• Clearly sensitive test! – point represents CDF accuracy after Run II

W Mass Measurement

W production cross section is ∼ 20 nb at Tevatron

• Can readily observe W → eνe and W → μνμ

• Select high ET electron or muon

• Require significant E̸T to tag neutrino

Can infer M_{W} from transverse mass distribution

MT = √2ET E̸T (1 − cos ϕlν)

• Not totally Lorentz invariant – Some sensitivity to W momentum

• Sensitive to structure functions

How Do We Accurately Measure M_{W}?

We don’t know initial state p_{z} or neutrino p_{z}

• Cannot reconstruct an invariant mass

• Mass information exists in lepton pT and E̸T

Suppose W decays at rest with l^{±} angle θ^{∗}

• Assume decay angular distribution f(cos θ∗)

• Transform to pT = (M sin θ∗)/2

\frac{dN}{dpT} = \frac{1}{1 − 4p2T /M2} f (±√1 − 4p2T /M2 )

• Has singularity at pT = M/2, moderated by – Physical width and pT of W

– Detector resolution

Reducing Systematics

Can reduce sensitivity to these effects by using M_{T}

• Much less sensitive to W pT distribution

• More resolution than pT (uses more information)

Doesn’t get rid of all systematics

• Still residual W pT dependence – Get rid of it by looking only at W ’s near rest

• Now depend on E̸T measurement – Involves entire calorimeter response

• Have to worry about effect of PDF’s

• Still have to deal with calibration

Data Sample from Run 1A

Select events with clean electron or muon

• For muons, require – pT > 25 GeV/c

– satisfies standard muon ID cuts • For electrons, require

– ET > 25 GeV

– satisfies standard electron ID cuts • Require E̸T > 25 GeV

• Require no “significant” recoil system – |u⃗| < 20 GeV

– No jets with ET > 30 GeV

• Left with 4663 μν and 8067 eνe candidates

Fit 65 < M_{T} < 100 GeV to predicted lineshape

• 5718 electron events

• 3268 muon events

Systematic Uncertainties on W Mass

Control of systematics most challenging aspect

1. Energy and momentum scale (120 and 45 MeV/c2) • Determined using Υ/ψ → μ+ μ− and Z → μ+ μ−/e+ e− • calorimeter scale set by tracking system

2. Resolution function of “detector” (∼ 80 MeV/c2) • Constrained by studies of Z events

3. Uncertainies in “recoil” system physics (∼ 60 MeV/c2) • Limited by statistics on Z sample

4. Structure functions (50 MeV/c2)

• Use MRS D ′ as default

• Check variation caused by other choices

Accounting for correlations in these uncertainties

• Overall systematic uncertainty is 130 MeV/c2

Momentum and Calorimeter Calibration

CTC was aligned and calibrated in situ

• Aligned with zillions of charged particles

• All material accounted for

• Used J/ψ → μ+ μ− as calibration

• Applied a correction factor of 0.99984 ± 0.00054

• Causes a ±45 MeV/c2 uncertainty on MW

Calorimeter was calibrated by comparing E_{cal} with p

• Use E/p distribution for electrons

• Take into account material in detector (0.09 X0)

• Limited by statistics, uncertainty in material – Statistics gives 0.08% uncertainty

– Material correction has 0.09% uncertainty • Overall uncertainty is 0.136% on calorimeter scale

Overall uncertainties on W mass from calibration are

• Momentum scale: ±45 MeV/c2

• Calorimeter scale: ±110 MeV/c2

E/p Comparison

9

Checks of CTC Calibrations

Z mass from electrons a good check

• Use tracks to determine absolute calorimeter ET

Dimuon mass distribution absolute check of pT

Effect of Underlying Event

Have to understand effect of underlying event on M_{T}

• Enters in two ways:

1. Modifies u⃗ (and therefore E̸T)

2. Modifies electron energy measurement

Can study u⃗ effects by looking at Z events

• Z momentum well measured by dilepton final state

• Can determine calorimeter response to given recoil

Combining Channels

To combine channels, have to understand uncertainties

• Some uncertainties unique to each channel, others correlated • Uncorrelated ones are reduced by averaging

• Correlated uncertainties effect average result

Uncertainty  ΔM_{W}^{e} ΔM_{W}^{μ} Common
Statistical  145  205
Energy Scale  120  45  45
Resolution  80  60  0
p_{T} model  45  45  25
Recoil model  60  60  60
PDF  50  50  50
Total  230  240  100

Fit to Muon Data

Results of fit to muon data

• Fit data to predicted shape – function of MW, ΓW

• Use only most sensitive MT range

MW = 80.310 ± 0.205 ± 0.130 GeV/c2

Fit to Electron Data

Results of fit to electron data

MW = 80.490 ± 0.145 ± 0.175 GeV/c2

Summary of W Mass

Combined mass from both channels is M_{W} = 80.41 ± 0.18 GeV/c^{2}

How Well Can We Do?

The statistical errors will decrease ×2

• Already have ×5 more data

• Will come done to < ∼ 100 MeV/channel

Systematic uncertainties are the challenge

• Experience has shown them to also scale with sample size – All determined by other data

• Should hope to get to < ∼ 50 MeV/c2