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75 lines
4.2 KiB
75 lines
4.2 KiB
\midheading{$Z\gamma$, production, processes 300, 305}
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\label{subsec:zgamma}
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Processes {\tt 300} and {\tt 305} represent the production of a $Z$ boson (or virtual photon for process {\tt 300})
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in association with a real photon based on ref.~\cite{Campbell:2017aul}. The $Z/\gamma^*$ subsequently decays into
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either an $e^+ e^-$ pair ({\tt nproc=300}) or neutrinos ({\tt nproc=305}).
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Since these processes include a real photon, the cross section diverges
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when the photon is very soft or in the direction of the beam.
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Thus in order to produce sensible results, the input file must supply values for both
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{\tt gammptmin} and {\tt gammrapmax}. Moreover, when the parameters {\tt zerowidth}
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and {\tt removebr} are set to {\tt .false.} the decay $Z \to e^- e^+$ ({\tt nproc=300})
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will include photon radiation from both leptons, so that a non-zero $R(\gamma,\ell)_{min}$
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({\tt Rgalmin})
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should also be supplied. This will ensure that the cross section is well-defined.
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The calculation of processes {\tt 300} may be performed
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at NNLO using the Frixione algorithm~\cite{Frixione:1998jh} or standard isolation.
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The one-loop virtual diagrams are taken from \cite{Dixon:1998py} and the two-loop virtual diagrams
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are taken from \cite{Gehrmann:2011ab}.
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%Processes {\tt 302} and {\tt 307} represents the production of a $Z$ boson (or virtual photon)
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%in association with a real photon and an additional jet. These processes are also available at NLO including
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%the full fragmentation processes. Anomalous couplings are not available for these processes.
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%Processes {\tt 304} and {\tt 309} represents the production of a $Z$ boson (or virtual photon)
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%in association with a real photon and and two additional jets. These processes are available at leading order only.
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%When {\tt removebr} is true in process {\tt 300} or {\tt 302} the $Z$ boson does not decay.
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For the process {\tt 300} the role of {\tt mtrans34cut} changes to become a cut
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on the invariant mass on the $M_{345}$ system, i.e. the photon is included with the leptons in the cut.
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\bottomheading{Anomalous $ZZ\gamma$ and $Z\gamma\gamma$ couplings}
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Processes {\tt 300}-{\tt 305} may also be computed including the effect of anomalous couplings between $Z$ bosons and
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photons.
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Note that, at present, the effect of anomalous couplings is not included in the gluon-gluon
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initiated contributions.
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The anomalous $Z\gamma Z$ vertex (not present at all in the Standard Model),
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considering operators up to dimension 8, is given by~\cite{DeFlorian:2000sg},
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\includegraphics[width=0.9\textwidth]{./sections/gobbets/ZZgamma.png}
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%\begin{eqnarray}
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% && \Gamma^{\alpha \beta \mu}_{Z \gamma Z}(q_1, q_2, p) =
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% \frac{i(p^2-q_1^2)}{M_Z^2} \Biggl(
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% h_1^Z \bigl( q_2^\mu g^{\alpha\beta} - q_2^\alpha g^{\mu \beta}
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% \bigr) \nonumber \\
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%&& + \frac{h_2^Z}{M_Z^2} p^\alpha \Bigl( p\cdot q_2\ g^{\mu\beta} -
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% q_2^\mu p^\beta \Bigr)
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% - h_3^Z \varepsilon^{\mu\alpha\beta\nu} q_{2\, \nu}
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% - \frac{h_4^Z}{M_Z^2} \varepsilon^{\mu\beta\nu\sigma} p^\alpha
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%p_\nu q_{2\, \sigma} \Biggl)
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%\end{eqnarray}
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where the overall coupling has been chosen to be $|e|$ (and
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$\epsilon^{0123}=+1$). The non-standard $Z_\alpha(q_1) \gamma_\beta(q_2)
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\gamma_\mu(p)$ momentum-space vertex can be obtained from
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this equation by setting $q_1^2 \to 0$ and replacing $h_i^Z \to
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h_i^\gamma$.
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The parameters that
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specify the anomalous couplings, $h_i^Z$ and $h_i^\gamma$ (for $i=1\ldots 4$), are
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specified in the input file as, e.g. {\tt h1(Z)} and {\tt h1(gamma)}.
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If the input file contains a negative value for the form-factor scale $\Lambda$
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then no suppression factors are applied to these anomalous couplings.
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Otherwise, the couplings are included
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in MCFM only after suppression by dipole form factors,
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\begin{equation}
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h_{1,3}^{Z/\gamma} \rightarrow
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\frac{h_{1,3}^{Z/\gamma}}{(1+\hat{s}/\Lambda^2)^3}, \qquad
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h_{2,4}^{Z/\gamma} \rightarrow
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\frac{h_{2,4}^{Z/\gamma}}{(1+\hat{s}/\Lambda^2)^4}, \qquad
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\end{equation}
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where $\hat{s}$ is the $Z\gamma$ pair invariant mass. Note that these form factors are slightly
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different from those discussed in Sections~\ref{subsec:diboson} and~\ref{subsec:wgamma}. The
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form factors can be modified in {\ttfamily src/Need/set\_anomcoup.f}.
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The Standard Model cross section is obtained by setting $h_i^Z = h_i^\gamma = 0$ for $i=1\ldots 4$.
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