W and Z Boson Production

W ’s and Z’s are produced via Drell-Yan annihilation

• Parton-level cross section is a weak interaction

• Weak matrix element for W case is

M = −iVqq′ \frac{g}{√2} ϵλ α(P)v̄(p1)\frac{1}{2} γα(1 − γ5)u(p2)

where g2 = 8M2 W GF /√2

• sum over W polarisation and average over quark spins – Longitudinal W ϵλ 0 = (p/MW , 0, 0, E/MW) (helicity 0)

– Transverse W ϵλ ± = (0, 1/√2, ±i√2, 0) (helicity ±1)

∑ |M|2 = |Vqq′|2g2M2 W = |Vqq′|2 \frac{8GF M4 W}{√2}
⇒ σ̂(qq′ → W + ) = (\frac{1}{2})2 \frac{1}{2ŝ} (|Vqq′|2 \frac{8GF M4 W}{√2} ) 2π ∫ d2(P S)

• 2-body phase space is just δ(ŝ − M2 W)

CKM Matrix

Cabbibo-Kobayashi-Maskawa matrix

• represents mixing of mass eigenstates under weak force

• assumed to be unitary

Latest estimates (from J. Hewett at SSI 95)

( \begin{matrix}V ud & Vus & Vub \\ Vcd & Vcs & Vcb \\ Vtd & Vts & Vtb \end{matrix} )

= ( \begin{matrix}0.9747 − 0.9759 & 0.218 − 0.224 & 0.002 − 0.006 \\ 0.218 − 0.224 & 0.9738 − 0.9752 & 0.032 − 0.048 \\ 0.004 − 0.015 & 0.030 − 0.048 & 0.9988 − 0.9995 \end{matrix} )

Total cross section for W^{+} production given by

σ = \frac{K}{3} ∫{1 \atop 0} dx1 ∫{1 \atop 0} dx2 ∑_{q} f1(x1, M2 W)f2(x2, M2 W)σ̂
where K ∼ 1 + \frac{8π}{9} αs(M2 W) and \frac{1}{3} for colour

• Use sx1x2 = M2 W and transform to to ŝ and W rapidity y

dx1dx2 = \frac{dŝdy}{s}

⇒ σ = \frac{K}{3} ∫{s \atop 0}∫{ymax \atop ymin} \frac{dŝdy}{s} ∑_{q} f1f2 \frac{2πGF M4 W |Vqq′|2}{√2ŝ} δ(ŝ − M2 W)
= \frac{K}{3} \frac{2πGF}{√2} ∫{ymax \atop ymin} dy ∑_{q} |Vqq′|2 (x1x2) f1(x1, M2 W)f2(x2, M2 W)

• In this integration,

x1 = \frac{MW}{√s} e+y and x2 = \frac{MW}{√s} e−y

since we transformed to W rapidity

Formula is now almost complete Can use this expression to get differential cross section

• take into account Cabbibo-mixing

\frac{dσ}{dy}(pp̄ → W + ) = \frac{2πKGF}{3√2} ∑_{q} |Vqq′|2(x1x2)

{cos 2 θC[u(x1)d(x2) + d̄(x1)ū(x2)] + sin2 θC[u(x1)s(x2) + s̄(x1)ū(x2)]}.

• PDF’s evaluated at Q2 = M2 W

If we assume “valence” dominated proton, then

\frac{dσ}{dy}(pp̄ → W + ) = \frac{2πKGF}{3√2} (x1x2)u(x1)d(x2)

• u(x) > d(x) ⇒ W + rapidity correlated with u quark

• Charge asymmetry test of structure functions

• Total cross section comes from integrating over y – ymin and ymax are defined by requiring xi ≤ 1 – Can estimate σ if one knows u(x) and d(x)

• Calculation for Z production essentially the same

Observed Cross Section

Both UA2 and CDF have measured σ_{W}

• In excellent agreement with SM prediction • Predict σ ∼ 3 nb at √s = 630 GeV

• Increases with energy due to – Larger rapidity region (50% increase going to Tevatron)

– Structure function growth at low x (×4)

W Decay

Can calculate W^{−} → e^{−}ν̄_{e} using EWK matrix element

M = −i\frac{g}{√2} ϵλ α(P)v̄(p1)\frac{1}{2} γα(1 − γ5)u(p2)

• Average over W + spin and sum over final state spins

\frac{1}{3} |M|2 = \frac{g2}{6}(−gαβ + \frac{P αP β}{M2 W} )T r(lγαkγβ[\frac{(1 − γ5)}{2} ])
= \frac{g2}{3}(l • k + 2\frac{(P • l)(P • k)}{M2 W} ) = \frac{1}{3} g2M2 W .

Use Fermi’s Golden Rule to calculate decay rate

dΓ(W − → e−ν̄e) = \frac{1}{2MW} \frac{1}{3} g2M2 W \frac{1}{4π2} d2(P S)
⇒ Γ(W − → e−ν̄e) = \frac{1}{48π} g2MW ≃ 0.227 GeV

• 3 leptonic channels contribute

• ūd and c̄s contribute with factor ∼ 3|Vqq′|2

• Gives predicted total width of ΓW = 2.09 GeV

Observing W’s

Most accessible channel is W → lν_{l}

• Large cross section (∼ 2 nb per lepton)

• Have charged lepton with pT > ∼ 20 GeV/c

• Have neutrino leaving E̸T > ∼ 20 GeV

• Good fraction of W production is “central”

– Can focus on interval |η| < ∼ 1.1

– Region with CDF’s best lepton identification • Higher rapity region important for asymmetry measurement

Select samples requiring

• Well-identified e± or μ± candidate with pT > 20 GeV/c • Require E̸T > 20 to tag neutrino

• Require charged lepton “isolated” from nearby energy flow

– Rejects backgrounds from ∗ Dijet backgrounds with one jet faking lepton ∗ Heavy quark semileptonic decay b → lνlX

Provides very clean sample of W boson candidates

W Kinematics

In a W → lν_{l} decay

• Don’t know initial state pz or neutrino pz • W often produced with significant pT

– About 1/3 of all W → lνl have jets

Despite all this “missing” information

• Can see clear signal using transverse mass MT ≡ √2ElT E̸T (1 − cos ϕlν)

• Requires good background control – No other sources of real high pT leptons – Need excellent lepton ID capability

Background Issues

Potential source of backgrounds from jet production

• One jet fragments into either:

– high pT π± ∗ charge-exchanges π±N → π∘N′ in calorimeter ∗ Decays via π± → μ±νμ

– high pT π±π∘ pair

∗ “Overlap” in calorimeter fakes e±

• Heavy quark decay also possible background – Most comes in at low MT

• Rejected by lepton ID cuts

• Less than 1% of sample is remnant QCD

W → τνν largest remaining background

• Rejected primarily by kinematics

• About 2% of sample is remnant τ decays

Other backgrounds are relatively small (< ∼ 0.5%)

• Conversions γ → e+ e−

• Missed Z → l+ l− where one lepton not found

Shape of Backgrounds

Distribution shows size and shape of backgrounds for electrons

• Muon channel similar

• Can further reduce backgrounds with tighter MT cut

W Asymmetry

Showed that W^{+} production aligned with u quark

• W − production aligned with ū quark

• Difference in u and d PDF’s result in asymmetry

• Not washed out by V − A decay of W

Measure

A(y) = \frac{dσ^{+} /dy_{l} − dσ^{−}/dy_{l}}{dσ^{+} /dy_{l} + dσ^{−}/dy_{l}}

W Boson Width

The width of the W boson can be indirectly measured

R = \frac{σ(pp̄ → W → eνe)}{(σ(pp̄ → Z → e+ e−)} = \frac{σ(pp̄ → W)}{σ(pp̄ → Z)} • \frac{Γ(W → eνe)}{Γ(Z → e+ e−)} • \frac{ΓZ}{ΓW}

• Can use SM to calculate total cross sections

• Use LEP data for ΓZ

• Use SM to calculate Γ(W → eνe)

Experiment just needs to measure R

• Sensitive to phase space for heavy quark decay

Γ(W → tb̄) = 3|Vtb|2Γ(W → eνe)
×√1 + r2 t + r2 b − 2rt − 2rb − 2rtrb

× [1 − \frac{1}{2} rt − \frac{1}{2} rb − \frac{1}{2} (rt − rb)2]

Z Selection for R Measurement

Z → e^{+} e^{−} decays easily identified

• Use W sample before E̸T cut

• Require second electron candidate

Results on R

Measured value of R is

R = 10.90 ± 0.32(stat ) ± 0.29(syst )

• Can use it – Measure ΓW = 2.063 ± 0.061 ± 0.060 GeV – set a limit on the top quark mass

• Note that it is independent of how t decays

Mtop > 62 GeV/c2 at 95% CL

Trilinear Gauge Couplings in SM

SM predicts self-interactions among bosons

• s-channel and t-channel amplitudes are both large

• Destructive interference neccessary to maintain unitarity

σ(t − channel ) ∝ \frac{G2F s}{3π}

• Can measure these couplings using

pp̄ → W ±γ and Zγ
pp̄ → W + W −, W ±Z, and ZZ

Total W W cross section estimated to be ∼ 9.5 pb

• One can generalise to include “anomolous couplings” – can include both CP-conserving and non-conserving – In general, there is a large number of parameters

• Initial goal is to verify that σV V is consistent with SM

SM Production of WW

The matrix element for W^{+} W^{−} production is

M(qq̄′ → W + W −) = iv̄(q′)T μνu(q)ϵ∗ μ(W + )ϵν(W −) where the tensor current is

Tμν = e2 (\frac{Qq}{s} + DZ \frac{g{q \atop V} − g{q \atop A}γ5}{sin2 θW} )
× [gμν(W + − W −) + γμ(2W + + W −)ν − γν(2W + + W −)μ]
− e2 \frac{(1 + γ5)}{4 sin2 θW} \frac{γν(q − W −)γμ}{(q − W −)2}

• DZ is Z-propogator

DZ = \frac{1}{ŝ − M2 Z + iMZΓZ}

• g{q \atop V} and g{q \atop A} are vector and axial-vector couplings

• Qq is charge of incoming quark

Shows the separate s and t-channel contributions

• Individually grow ∝ ŝ

• Both must exist (and cancel) to avoid violating unitarity

Observation of WZ Production

Anomolous couplings can be interpreted in various ways

• Sensitive to “new” physics

• Non-SM W W γ couplings modify W dipole moments

μW = \frac{e}{2MW} (2 + Δκ + λ)
QeW = \frac{−e}{2M2 W} (1 + Δκ − λ)

assuming that CP is conserved

Results to date have focused on W W γ and ZZγ vertices

• W γ and Zγ production has been measured

• Limited |Δκ| < 3.0 and λ < 0.95 at 95% CL

New CDF data starting to be sensitive to W W Z

• Predicted cross section is 9.5 pb for W W

• Predicted cross section is 2.5 pb for W Z

• Can expect to see a few events in sample

Search for DiVector Boson Events

Search for events where

• One W or Z decays hadronically into two jets – Require jet ET > 30 GeV

– Require dijet mass 60 < Mjj < 110 GeV/c2 • Other boson decays leptonically

• Sensitive to W W and W Z final states

Large backgrounds from W/Z + jet events

• Require dijet PT > 130 for leptonic W candidates • Require dijet PT > 100 for leptonic Z candidates

Results of W W/W Z Search

Expect to see virtually no signal with SM couplings

• Expect 0.13 events with leptonic W

• Expect 0.02 events with dileptonic Z

• Observe:

– one event with leptonic W decay – no events with leptonic Z decay

Allows us to exclude anomalously large W W Z couplings

• Note that if W W Z coupling is zero, we would expect of order 15 events!

– ⇒ W W Z coupling is non-zero • Can set limits on specific choices of κ and λ

• Most sensitive to W W Z couplings

Can set limits making assumptions

• For example, assume SM correct except for W W Z and W W γ • Assume W W Z and W W γ couplings equal

• Conclude −0.11 ≤ κZ ≤ 2.27 and −0.81 ≤ λZ ≤ 0.84