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86 lines
4.3 KiB
86 lines
4.3 KiB
\bottomheading{$ZZ$ production, processes 81--84, 90}
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The $Z$'s can either both decay leptonically ({\tt nproc=81}), one can
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decay leptonically while the other decays into neutrinos ({\tt
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nproc=82}) or bottom quarks ({\tt nproc=83}), or one decays into
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neutrinos and the other into a bottom quark pair ({\tt nproc=84}). In
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process {\tt 83} the mass of the $b$-quark is neglected. Note that, in
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processes {\tt 83}--{\tt 84}, the NLO corrections do not include
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radiation from the bottom quarks that are produced by the $Z$ decay.
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In process {\tt 90} the two $Z$ bosons decay to identical charged
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leptons, and interference effects between the decay products of the
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two $Z$ bosons are included. In all cases these processes also
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include the contribution from a virtual photon.
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When {\tt removebr} is true in process {\tt 81}, neither of the $Z$
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bosons decays.
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This process has been treated in several papers,
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\cite{Campbell:1999ah,Campbell:2011bn,Boughezal:2016wmq,Campbell:2022gdq}.
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Processes {\tt 81} and {\tt 82} can be calculated at NNLO.
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The NNLO calculation includes contributions from the process $gg \to ZZ$
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that proceeds through quark loops. The calculation of loops
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containing the third quark generation includes the effect of both the
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top and the bottom quark mass ($m_t,m_b \neq 0$), while the first two
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generations are considered massless. For numerical stability, a small
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cut on the transverse momentum of the $Z$ bosons is applied:
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$p_T(Z)>0.1$~GeV. This typically removes less than $0.1$\% of the
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total cross section. The values of these cutoffs can be changed by
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editing {\tt src/ZZ/getggZZamps.f} and recompiling.
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\bottomheading{Anomalous $WWZ$}
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\label{sec:anomalous}
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It is possible to specify anomalous trilinear
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couplings for the $W^+W^-Z$ and $W^+W^-\gamma$ vertices that are
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relevant for $WW$ and $WZ$ production. To run in this mode, one must
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set {\tt zerowidth} equal to {\tt .true.} and modify the appropriate
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lines for the couplings in {\tt input.ini} (see below). Note that, at
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present, the effect of anomalous couplings is not included in the
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gluon-gluon initiated contributions to the $WW$ process.
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The anomalous couplings appear in the Lagrangian,
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${\cal L} = {\cal L}_{SM} + {\cal L}_{anom}$ as follows
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(where ${\cal L}_{SM}$ represents the usual Standard Model Lagrangian and
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${\cal L}_{anom}$ is taken from Ref.~\cite{Dixon:1999di}):
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\includegraphics[width=\textwidth]{./sections/gobbets/Lagrangian.png}
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%\begin{eqnarray}
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%{\cal L}_{anom} & = & i g_{WWZ} \Biggl[
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% \Delta g_1^Z \left( W^*_{\mu\nu}W^\mu Z^\nu - W_{\mu\nu}W^{*\mu} Z^\nu \right)
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%+\Delta\kappa^Z W^*_\mu W_\nu Z^{\mu\nu} \nonumber \\
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% & &+
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% \frac{\lambda^Z}{M_W^2} W^*_{\rho\mu} W^\mu_\nu Z^{\nu\rho} \Biggr]
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%+i g_{WW\gamma} \Biggl[
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% \Delta\kappa^\gamma W^*_\mu W_\nu \gamma^{\mu\nu}
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%+\frac{\lambda^\gamma}{M_W^2} W^*_{\rho\mu} W^\mu_\nu\gamma^{\nu\rho}
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% \Biggr], \nonumber
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%\end{eqnarray}
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where $X_{\mu\nu} \equiv \partial_\mu X_{\nu} - \partial_\nu X_{\mu}$
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and the overall coupling factors are $g_{WW\gamma}=-e$,
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$g_{WWZ}=-e\cot\theta_w$.
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This is the most general Lagrangian that conserves $C$ and $P$
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separately and electromagnetic gauge invariance requires that there
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is no equivalent of the $\Delta g_1^Z$ term for the photon coupling.
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In order to avoid a violation of unitarity, these couplings are often
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included only after suppression by dipole form factors,
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\includegraphics[width=0.9\textwidth]{./sections/gobbets/Dipole.png}
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%\begin{equation}
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%\Delta g_1^Z \rightarrow \frac{\Delta g_1^Z}{(1+\hat{s}/\Lambda^2)^2}, \qquad
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%\Delta \kappa^{Z/\gamma} \rightarrow \frac{\Delta \kappa_1^{Z/\gamma}}{(1+\hat{s}/\Lambda^2)^2}, \qquad
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%\lambda^{Z/\gamma} \rightarrow \frac{\Delta \lambda^{Z/\gamma}}{(1+\hat{s}/\Lambda^2)^2},
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%\end{equation}
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where $\hat{s}$ is the vector boson pair invariant mass and $\Lambda$
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is an additional parameter giving the scale of new physics, which should
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be in the TeV range.
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These form factors should be produced by the new physics associated
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with the anomalous couplings and this choice is somewhat
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arbitrary. The use of the form factors can be disabled as described
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below. The file {\tt input.ini} contains the values of the $6$
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parameters which specify the anomalous couplings. If the input file
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contains a negative value for the form-factor scale then the
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suppression factors described above are not applied.
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