---
title: 'Top-W: Charged-Current Top–W Interactions'
url: https://www.emergentmind.com/topics/top-w
type: topic
---

# Top-W: Charged-Current Top–W Interactions

Top–\(W\) denotes the set of interactions and observables governed by the charged-current coupling between the top quark and the \(W\) boson, most directly through the \(tWb\) vertex, the decay \(t\to Wb\), associated \(tW\) production, and processes in which top loops modify \(W\)-boson scattering. Because the top quark decays before hadronizing, its decay products retain spin information, so \(W\)-polarization observables, off-shell line shapes, and associated-production rates furnish unusually direct probes of chirality structure, \(|V_{tb}|\), anomalous couplings, and the top width \(\Gamma_t\). In collider phenomenology, this sector therefore links precision top decay measurements, single-top production, multivariate reconstruction methods, and higher-order QCD and electroweak calculations [1510.01084] [2408.13870].

## 1. The \(tWb\) vertex and its standard parameterizations

In the Standard Model, the interaction of the top and bottom quarks with the charged \(W\) boson is commonly written as
\[
\mathcal L_{tbW}
=
-\,\frac{g}{\sqrt2}\,W^-_\mu\,\bar b\,\gamma^\mu\bigl(V_L P_L+V_R P_R\bigr)t
+\mathrm{h.c.},
\]
with \(P_{L,R}=(1\mp\gamma^5)/2\). At tree level one has \(V_L\equiv V_{tb}\simeq1\) and \(V_R=0\), so the coupling is purely left-chiral [1510.01084].

A more general, gauge-invariant, dimension-six extension introduces tensor terms,
\[
\mathcal{L}_{Wtb}
=
-\frac{g}{\sqrt2}\,\bar b\,\gamma^\mu\bigl(V_LP_L + V_RP_R\bigr)\,t\,W_\mu^-
-\frac{g}{\sqrt2}\,\bar b\,\frac{i\sigma^{\mu\nu}q_\nu}{M_W}\bigl(g_LP_L + g_RP_R\bigr)\,t\,W_\mu^-
+\mathrm{h.c.},
\]
where nonzero \(\{V_R,g_L,g_R\}\) modify the decay amplitude and the polarization pattern of the emitted \(W\) boson [1005.5382].

The polarization fractions are defined either as \((f_-,f_0,f_+)\) or \((F_L,F_0,F_R)\), with the unitarity relation \(f_-+f_0+f_+=1\) or equivalently \(F_L+F_0+F_R=1\). In the Standard Model, representative predictions quoted for \(m_t\simeq172.5\) GeV and \(m_W=80.4\) GeV are \(f_0^{\rm SM}\simeq0.696\), \(f_-^{\rm SM}\simeq0.303\), and \(f_+^{\rm SM}\simeq3.8\times10^{-4}\), while another NLO prediction gives \(f_0^{\rm SM}=0.698\pm0.012\), \(f_-^{\rm SM}=0.301\pm0.012\), and \(f_+^{\rm SM}=4.1\times10^{-4}\) [1211.4523] [1110.2954]. At leading order and neglecting \(m_b\), one also finds
\[
F_0=\frac{m_t^2}{m_t^2+2m_W^2}\approx0.70,\qquad
F_L=\frac{2m_W^2}{m_t^2+2m_W^2}\approx0.30,\qquad
F_R=0,
\]
which is the standard \(V\!-\!A\) limit [1510.01084].

## 2. Polarization observables in top decay

The conventional observable is the helicity angle \(\theta^*\), defined in the \(W\) rest frame as the angle between the charged lepton, or down-type quark, and the direction of the parent top quark or of the \(W\) flight direction in the top rest frame, depending on the convention used in the measurement. The normalized decay distribution is
\[
\frac{1}{\Gamma}\frac{d\Gamma}{d\cos\theta^*}
=
\frac{3}{8}(1-\cos\theta^*)^2 f_-
+
\frac{3}{4}(1-\cos^2\theta^*) f_0
+
\frac{3}{8}(1+\cos\theta^*)^2 f_+,
\]
or equivalently with \((F_L,F_0,F_R)\) in place of \((f_-,f_0,f_+)\). This distribution underlies essentially all direct determinations of \(W\)-helicity fractions in top decay [1211.4523] [1003.0224].

For polarized tops, the polarization analysis can be extended beyond helicity fractions. The decay density matrix in the top rest frame contains diagonal terms associated with the usual helicity widths and off-diagonal terms sensitive to additional spin structure. On that basis one can define transverse and normal \(W\)-polarization fractions, \(F_i^T\) and \(F_i^N\), along axes orthogonal to the \(W\) momentum. The sum rules
\[
F^T_0=F^N_0=\tfrac12(F_++F_-)
\]
and, for real couplings,
\[
F^N_+=F^N_-=\tfrac12-\tfrac14(F_++F_-)
\]
follow directly from the density-matrix decomposition [1005.5382].

A particularly distinctive observable is the forward–backward asymmetry in the normal direction,
\[
A_N
=
\frac{N(\cos\theta_N>0)-N(\cos\theta_N<0)}
{N(\cos\theta_N>0)+N(\cos\theta_N<0)}
=
\tfrac34\,P\bigl(F^N_+-F^N_-\bigr),
\]
with \(P\) the top polarization. In the general \(Wtb\) vertex this asymmetry is proportional to \(\Im(V_Lg_R^*)+\cdots\), and for small \(g_R\) with \(V_L\simeq1\) one has \(A_N\simeq0.64\,P\,\Im g_R\). This makes normal polarization directly sensitive to complex phases in the \(Wtb\) interaction that are not isolated by helicity fractions alone [1005.5382].

The same generalized treatment also modifies the spin-analyzing powers in
\[
\frac{1}{\Gamma}\frac{d\Gamma}{d\cos\theta_X}
=
\tfrac12\bigl(1+\alpha_X P\cos\theta_X\bigr).
\]
In the Standard Model, the quoted values are \(\alpha_{\ell^+}\approx1\), \(\alpha_b\approx-0.41\), and \(\alpha_\nu\approx-0.32\), so the charged lepton remains the optimal spin analyzer [1005.5382].

## 3. Experimental measurements of \(W\)-boson polarization in top decay

Tevatron and LHC measurements have used matrix-element or template-likelihood methods to extract the helicity fractions from lepton+jets, dilepton, and single-top samples. The central measurements quoted in the supplied literature are summarized below.

| Measurement | Dataset | Result |
|---|---:|---|
| CDF, lepton+jets | \(2.7\ \mathrm{fb}^{-1}\) | \(f_0=0.88\pm0.11\pm0.06,\ f_+=-0.15\pm0.07\pm0.06\) |
| D0, \(\ell+\)jets and dilepton | \(5.4\ \mathrm{fb}^{-1}\) | \(f_0=0.669\pm0.078\pm0.065,\ f_+=0.023\pm0.041\pm0.034\) |
| CDF, full Run II lepton+jets | \(8.7\ \mathrm{fb}^{-1}\) | \(f_0=0.726\pm0.066\pm0.067,\ f_+=-0.045\pm0.044\pm0.058\) |
| CMS, single-top at \(8\) TeV | \(19.7\ \mathrm{fb}^{-1}\) | \(F_L=0.298\pm0.028\pm0.032,\ F_0=0.720\pm0.039\pm0.037,\ F_R=-0.018\pm0.019\pm0.011\) |

The earlier CDF lepton+jets analysis reconstructed both leptonic and hadronic \(W\) decays, used a per-event likelihood based on leading-order matrix elements and detector transfer functions, and obtained a statistical correlation \(\rho=-0.59\) in the simultaneous \((f_0,f_+)\) fit. Constraining \(f_+=0\) gave \(f_0=0.70\pm0.07\;(\mathrm{stat})\pm0.04\;(\mathrm{syst})\), while fixing \(f_0=0.70\) yielded \(f_+=-0.01\pm0.02\;(\mathrm{stat})\pm0.05\;(\mathrm{syst})\) and \(f_+<0.12\) at \(95\%\) C.L. [1003.0224].

The D0 measurement combined \(\ell+\)jets and dilepton channels and used reweighted \(V\!-\!A\) and \(V\!+\!A\) Monte Carlo templates in a binned Poisson likelihood. It reported \(f_0=0.669\pm0.078\;(\mathrm{stat.})\pm0.065\;(\mathrm{syst.})\) and \(f_+=0.023\pm0.041\;(\mathrm{stat.})\pm0.034\;(\mathrm{syst.})\), with statistical correlation \(\rho(f_0,f_+)\approx-0.83\). Fixing \(f_0\equiv0.698\) gave \(f_+=0.010\pm0.022\;(\mathrm{stat})\pm0.030\;(\mathrm{syst.})\); the result was stated to be consistent at the \(98\%\) level with the Standard Model [1110.2954].

CDF’s full Run II result used 2 574 lepton+jets candidates with \(\ge1\) \(b\)-tag and a matrix-element-based unbinned likelihood \(L(f_0,f_+)\). It measured \(f_0=0.726\pm0.066\;(\mathrm{stat})\pm0.067\;(\mathrm{syst})\) and \(f_+=-0.045\pm0.044\;(\mathrm{stat})\pm0.058\;(\mathrm{syst})\), with \(\mathrm{Corr}(f_0,f_+)=-0.69\). The dominant systematic uncertainties in the simultaneous fit were background modeling, ISR/FSR variation, MC-generator choice, color reconnection, method/calibration, jet-energy scale, PDF uncertainty, and pileup or multiple interactions [1211.4523].

The LHC measurements quoted in the review are numerically compatible with the Tevatron results and with the Standard Model expectations \(F_0\approx0.70\), \(F_L\approx0.30\), \(F_R\approx0\). A plausible implication is that the observed chirality pattern remains dominated by the Standard Model \(V\!-\!A\) structure at the present precision level [1510.01084].

## 4. Associated production channels: \(tW\) and \(t\bar tW\)

Associated single-top production with a \(W\) boson, usually called the \(tW\) mode, is one of the three electroweak single-top production mechanisms. At leading order it proceeds through partonic channels written in the supplied sources as \(g b\to tW^-\) or \(b g\to tW\), and it yields a final state with two \(W\) bosons and a \(b\) quark after the top decay. Because the process is directly sensitive to the \(W\!-\!t\!-\!b\) vertex, it provides an experimental handle on \(|V_{tb}|\) [1205.5764].

At NLO, \(tW\) interferes with \(t\bar t\) production, so signal definition requires a prescription. The two standard schemes are diagram removal, which omits doubly resonant \(t\bar t\) graphs, and diagram subtraction, which cancels them with a local counterterm. CMS explicitly used the diagram removal scheme in its 8 TeV observation and retained the DR–DS difference as a systematic uncertainty [1401.2942].

The first LHC evidence from ATLAS at \(\sqrt s=7\) TeV used \(2.05\ \mathrm{fb}^{-1}\) in dilepton final states with two isolated leptons, significant missing transverse momentum, and at least one jet. A boosted decision tree in the TMVA framework, trained in the one-jet signal region with 22 kinematic and topological variables, was fitted simultaneously across the 1-, 2-, and \(\ge3\)-jet bins. The observed significance was \(3.3\sigma\) with expected sensitivity \(3.4\sigma\), and the measured cross section was
\[
\sigma_{Wt}=16.8\pm2.9\;(\mathrm{stat})\pm4.9\;(\mathrm{syst})\ \mathrm{pb},
\]
from which \(|V_{tb}|=1.03^{+0.16}_{-0.19}\) was derived assuming \(|V_{ts}|\) and \(|V_{td}|\) are small [1205.5764].

CMS performed an early 7 TeV search with \(2.1\ \mathrm{fb}^{-1}\), using dilepton triggers, categories \(1j1t\), \(2j1t\), and \(2j2t\), a data-driven Drell–Yan estimate, and a simultaneous Poisson-likelihood fit to nine channels. The extracted cross section was
\[
\sigma_{tW}=22^{+9}_{-7}\ \mathrm{pb},
\]
with an observed excess corresponding to \(2.7\sigma\) and an expected significance of \(1.8\pm0.9\,\sigma\) [1201.4997].

CMS subsequently reported the first observation at \(\sqrt s=8\) TeV with \(12.2\ \mathrm{fb}^{-1}\). Events with two leptons and a \(b\)-tagged jet were analyzed with boosted decision trees using 13 kinematic and topological variables, and a simultaneous binned maximum-likelihood fit was performed in the 1j1t signal region and in the 2j1t and 2j2t control regions. The measured cross section was
\[
\sigma_{tW}=23.4\pm5.4\ \mathrm{pb},
\]
with an observed significance of \(6.1\sigma\). Using \(\sigma_{tW}^{\rm meas}=\lvert V_{tb}\rvert^2\,\sigma_{tW}^{\rm th}\), CMS extracted \(|V_{tb}|=1.03\pm0.12\;(\mathrm{exp})\pm0.04\;(\mathrm{th})\) and set the lower limit \(|V_{tb}|>0.78\) at \(95\%\) C.L. [1401.2942].

The same review that summarized \(W\)-helicity measurements also quoted associated \(t\bar tW\) production at 8 TeV. ATLAS reported
\[
\sigma_{t\bar tW}=300^{+120}_{-100}\,(\mathrm{stat})\,^{+70}_{-40}\,(\mathrm{syst})\ \mathrm{fb},
\]
for \(20.3\ \mathrm{fb}^{-1}\), while CMS reported
\[
\sigma_{t\bar tW}=382^{+117}_{-102}\,(\mathrm{tot})\ \mathrm{fb}
\]
for \(19.5\ \mathrm{fb}^{-1}\). These channels were extracted from multilepton categories and were stated to agree with SM NLO predictions within combined uncertainties of approximately \(15\)–\(25\%\) [1510.01084].

On the theory side, higher-order corrections to \(tW\) production have been pushed through approximate \(N^3\)LO in QCD with NLO electroweak corrections included. Using MSHT20 NNLO PDFs and \(\mu_F=\mu_R=m_t\), the 13 TeV total cross sections quoted are \(66.0\) pb at LO, \(81.0\) pb at NLO QCD, \(87.5\) pb at aNNLO QCD, \(89.5\) pb at a\(N^3\)LO QCD, and \(86.8\) pb at a\(N^3\)LO QCD+EW, with \(\delta_{\rm EW}\approx-3\%\). The residual scale dependence is quoted as shrinking from \(+20/-18\%\) at LO to \(+3/-2.5\%\) at a\(N^3\)LO [2408.13870].

## 5. Off-shell \(W^+W^-b\bar b\) production and the top width

A distinct top–\(W\) observable is the extraction of the top width from off-shell regions of
\[
e^+e^-\to W^+W^-b\bar b.
\]
The method defines reconstructed masses
\[
M_+\equiv M(W^+,J_b),\qquad M_-\equiv M(W^-,J_{\bar b}),
\]
selects a double-resonant on-shell window \(M_{\min}\le M_\pm\le M_{\max}\), and compares it with a single-resonant off-shell region in which one of \(M_\pm\) lies in \([M_{\min},M_{\max}]\) while the other is above \(M_{\max}\) [1511.02350].

Near the top pole, the double-resonant contribution scales as \(g_{tbW}^4/\Gamma_t^2\), while the single-resonant contribution scales as \(g_{tbW}^4/\Gamma_t\). Their ratio
\[
R\equiv \frac{N_{\rm off}}{N_{\rm on}}
\simeq
\frac{\sigma_{\rm SR_1}}{\sigma_{\rm DR}}
\]
is therefore linear in \(\Gamma_t\) and independent of \(g_{tbW}\) at leading order up to subleading terms in \(\Gamma_t/m_t\) [1511.02350].

At NLO QCD, the differential cross sections acquire process-dependent one-loop plus real-emission \(K\)-factors, denoted \(K_{\rm DR}(M_+,M_-)\) and \(K_{\rm SR_1}(M_+,M_-)\), but the ratio-based strategy remains viable after integrating over the selected windows. For the benchmark \(\sqrt s=500\) GeV, \(R_{\rm jet}=0.5\), and \([M_{\min},M_{\max}]=[165,180]\) GeV, one quoted result is
\[
R(\Gamma_t)\simeq (0.07\ \mathrm{GeV}^{-1})\,\Gamma_t \pm (\text{scale uncertainty }0.0007),
\]
while benchmark sensitivities summarized for an assumed experimental fractional error \(\delta R\approx5\%\) give:

| \(\sqrt s\) | Polarization | \(dR/d\Gamma_t\ [\mathrm{GeV}^{-1}]\) | \(\Delta\Gamma_t^{\rm total}\) |
|---:|---|---:|---:|
| 500 GeV | unpolarized | 0.11 | 0.19 GeV |
| 500 GeV | \((P_{e^+},P_{e^-})=(-1,+1)\) | 0.14 | 0.14 GeV |
| 600 GeV | unpolarized | 0.15 | 0.18 GeV |

These benchmarks were summarized as giving \(\delta\Gamma_t\) of order \(100\)–\(200\) MeV at an \(e^+e^-\) collider [1511.02350].

The same ratio idea can be applied to \(pp\to W^+W^-b\bar b\), since the single- to double-resonant ratio is again proportional to \(\Gamma_t\) at LO. However, the hadron-collider implementation was explicitly judged challenging because the LO cross section scales as \(\alpha_s^2\), factorization and renormalization scale uncertainties are large, and exclusive resonance regions are affected by additional QCD radiation and matching issues. A plausible implication is that the \(e^+e^-\) environment is intrinsically cleaner for width extraction by off-shell line-shape ratios [1511.02350].

## 6. Event-level polarization control and top-induced \(W^+W^-\) dynamics

Modern studies of the top–\(W\) sector also rely on event-level control of polarization in Monte Carlo samples. The custom-angle-replacement method modifies decay angular distributions in a pre-existing sample so that a desired polarization state is reproduced while all production kinematics are unchanged. For semileptonic \(t\to Wb\to \ell\nu b\), the method operates in a factorized production-times-decay approximation, reconstructs the original event in the top and \(W\) rest frames, resamples the four angles \((\theta,\phi,\theta^*,\phi^*)\) from the fully differential decay distribution, and then boosts the reconstructed decay products back to the lab frame [2208.00424].

The fully differential distribution is four-dimensional and involves three Lorentz frames. In the implementation summarized in the source, the top density matrix is \(\rho_t=\frac12[1+\mathbf P\cdot\sigma]\), the helicity amplitudes are \(a_{\lambda_W,\lambda_b}\), and the resampling leaves momentum magnitudes fixed. Because the phase-space Jacobian is unchanged, all events in the new sample carry unit weight and no extra Jacobian or weight correction is needed under the stated conditions. Validation tests reported perfect agreement in one-dimensional angular distributions and agreement at the \(10^{-3}\)–\(10^{-2}\) level in template-fit tests of polarization and spin-correlation coefficients [2208.00424].

Beyond on-shell top production and decay, top quarks also affect \(W^+W^-\) scattering through loop corrections. In the Higgs–Electroweak Chiral Lagrangian, the left-handed \(Wtb\) interaction arises from the covariant derivative term
\[
\mathcal L_{Wtb}^{\rm kin}
=
-\frac{g}{\sqrt2}\,\bar t\,\gamma^\mu P_L\,b\,W^+_\mu
-\frac{g}{\sqrt2}\,\bar b\,\gamma^\mu P_L\,t\,W^-_\mu,
\]
while the top Yukawa sector is parameterized by Higgs functions \(\mathcal G(h)=1+c_1\,h/v+c_2\,h^2/v^2+\cdots\) and \(\mathcal F(h)=1+2a\,h/v+b\,h^2/v^2+\cdots\). In the Standard Model limit, \(a=b=c_1=d_3=1\) and \(c_2=0\) [2012.12242].

The impact of fermion loops can be quantified through
\[
R_J(s)=
\frac{\Im a_J^{(\rm ferm)}(s)}
{\Im a_J^{(\rm bos)}(s)+\Im a_J^{(\rm ferm)}(s)},
\]
for partial waves \(J=0,1\). In scans with \(a=b=d_3=1\), \(c_1\in[0.9,1.1]\), and \(\sqrt s=0.5\)–\(3\) TeV, the quoted values are \(R_0(0.5\,\mathrm{TeV})\approx10\%\) to \(80\%\), falling to a few percent by \(3\) TeV. For the \(J=1\) channel, scanning \(a\in[0.9,1.1]\) gives \(R_1(0.5\,\mathrm{TeV})\simeq80\%\)–\(90\%\), and even at \(3\) TeV the range remains \(10\%\)–\(90\%\). In the Minimal Composite Higgs benchmark with \(a=c_1=d_3=\sqrt{1-\xi}\) and \(b=1-2\xi\), one still finds \(R_0\lesssim45\%\) with a peak around \(\sqrt s\sim0.6\) TeV and \(R_1\sim20\%\) to \(90\%\) [2012.12242].

This suggests that the top–\(W\) sector is not exhausted by direct decay or associated-production observables. In effective-field-theory descriptions of electroweak symmetry breaking, top-induced loop effects can be comparable to bosonic contributions in specific channels, especially in the \(J=1\) partial wave, and therefore enter the interpretation of longitudinal \(WW\) scattering and vector-boson-scattering phenomenology [2012.12242].

Source: https://www.emergentmind.com/topics/top-w