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66 lines
3.5 KiB
66 lines
3.5 KiB
\midheading{$W\gamma$ production, processes 290-299, 2941, 2991}
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\label{subsec:wgamma}
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These processes represent the production of a $W$ boson which subsequently
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decays leptonically, in association with a real photon.
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Since this process includes 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 $W \to \ell \nu$ will include
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photon radiation from the lepton, so that a non-zero $R(\gamma,\ell)_{min}$
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({\tt Rgalmin}) should
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also be supplied. This will ensure that the cross section is well-defined.
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Virtual amplitudes are taken from ref.~\cite{Dixon:1998py}.
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The calculation of processes {\tt 290} and {\tt 295} may be performed
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at NLO and NNLO using the Frixione algorithm~\cite{Frixione:1998jh} or standard isolation.
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The phenomenology of these processes at NNLO has been treated in ref.~\cite{Campbell:2021mlr}.
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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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For processes {\tt 290} and {\tt 295} the role of {\tt mtrans34cut} changes to become a cut
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on the transverse mass on the $M_{345}$ system, i.e. the photon is included with the leptons in the cut.
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Processes {\tt 292} and {\tt 297} represent the $W\gamma$+jet production
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processes. They may be computed to NLO.
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Processes {\tt 294} and {\tt 299} represent the photon-induced reactions,
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$p + \gamma \to e \nu \gamma j$ and should be computed at NLO.
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Processes {\tt 2941} and {\tt 2991} represent the photon-induced reactions,
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$p + \gamma \to e \nu \gamma j j$ and should be computed at NLO.
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\bottomheading{Anomalous $WW\gamma$ couplings}
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Processes {\tt 290}-{\tt 297} may also be computed including the effect of anomalous $WW\gamma$ couplings, making
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use of the amplitudes calculated in Ref.~\cite{DeFlorian:2000sg}. Including only dimension 6 operators
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or less and demanding gauge, $C$ and $CP$ invariance gives the general form of the anomalous
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vertex~\cite{DeFlorian:2000sg},
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\begin{eqnarray}
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\Gamma^{\alpha \beta \mu}_{W W \gamma}(q, \bar q, p) &=&
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{\bar q}^\alpha g^{\beta \mu}
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\biggl( 2 + \Delta\kappa^\gamma + \lambda^\gamma {q^2\over M_W^2} \biggr)
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- q^\beta g^{\alpha \mu}
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\biggl( 2 + \Delta\kappa^\gamma + \lambda^\gamma {{\bar q}^2\over M_W^2}
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\biggr) \nonumber \\
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&& \hskip 1 cm
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+ \bigl( {\bar q}^\mu - q^\mu \bigr)
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\Biggl[ - g^{\alpha \beta} \biggl(
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1 + {1\over2} p^2 \frac{\lambda^\gamma}{M_W^2} \biggr)
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+\frac{\lambda^\gamma}{M_W^2} p^\alpha p^\beta \Biggr] \,,
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\end{eqnarray}
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where the overall coupling has been chosen to be $-|e|$. The parameters that
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specify the anomalous couplings, $\Delta\kappa^\gamma$ and $\lambda^\gamma$, are
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specified in the input file as already discussed in Section~\ref{subsec:diboson}.
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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 the 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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\Delta \kappa^{\gamma} \rightarrow
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\frac{\Delta \kappa_1^{\gamma}}{(1+\hat{s}/\Lambda^2)^2}, \qquad
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\lambda^{\gamma} \rightarrow
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\frac{\Delta \lambda^{\gamma}}{(1+\hat{s}/\Lambda^2)^2} \;,
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\end{equation}
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where $\hat{s}$ is the $W\gamma$ pair invariant mass.
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The Standard Model cross section is obtained by setting $\Delta\kappa^\gamma = \lambda^\gamma = 0$.
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