VBFNLO is capable of studying the effect of anomalous Higgs couplings. The anomalous coupling parameters are input via the file anom_HVV.dat, and can be entered using one of three different parametrisations.

The general form of the coupling [1] between a Higgs and a pair of vector bosons $X$ and $Y$ is given by $T^{\mu\nu}$:

\begin{eqnarray} T^{\mu\nu} &=& a_{1}(x,y) g^{\mu \nu} + a_{2}(x,y) [ x \cdot y g^{\mu \nu} - y^{\mu} x^{\nu} ] + a_{3}(x,y) \varepsilon^{\mu \nu \rho \sigma} x_{\rho} y_{\sigma} \end{eqnarray}

where $x$ and $y$ are the momenta of the gauge bosons $X$ and $Y$ and $a_{1,2,3}$ are Lorentz-invariant formfactors. VBFNLO uses the input of anom_HVV.dat to calculate the formfactors $a_{1,2,3}$ and multiplies them with the appropriate current to evaluate the matrix elements.

The inputs required for this parametrisation (which is used in [1]) are the coefficients $g$ and mass scale $\Lambda_{5}$ of the dimension-5 operators in the effective Lagrangian:

\begin{eqnarray} \setcounter{equation}{2} \mathcal{L} &=& \frac{g_{5e}^{HZZ}}{2 \Lambda_{5}} H Z_{\mu \nu} Z^{\mu \nu} + \frac{g_{5o}^{HZZ}}{2 \Lambda_{5}} H \tilde{Z}_{\mu \nu} Z^{\mu \nu} + \frac{g_{5e}^{HWW}}{\Lambda_{5}} H W^{+}_{\mu \nu} W_{-}^{\mu \nu} + \frac{g_{5o}^{HWW}}{\Lambda_{5}} H \tilde{W}^{+}_{\mu \nu} W_{-}^{\mu \nu} + \nonumber \\ % && \frac{g_{5e}^{HZ\gamma}}{\Lambda_{5}} H Z_{\mu \nu} A^{\mu \nu} + \frac{g_{5o}^{HZ\gamma}}{\Lambda_{5}} H \tilde{Z}_{\mu \nu} A^{\mu \nu} + \frac{g_{5e}^{H\gamma \gamma}}{2 \Lambda_{5}} H A_{\mu \nu} A^{\mu \nu} + \frac{g_{5o}^{H\gamma \gamma}}{2 \Lambda_{5}} H \tilde{A}_{\mu \nu} A^{\mu \nu} \nonumber \\ \end{eqnarray}

where $V^{\mu \nu}$ and $\tilde{V}^{\mu \nu}$ are the field strength and dual field strength tensors respectively: \begin{eqnarray} \setcounter{equation}{3} V^{\mu \nu} &=& \partial^{\mu} V^{\nu} - \partial^{\nu} V^{\mu} \nonumber \\ % \tilde{V}^{\mu \nu} &=& \frac{1}{2} \varepsilon^{\mu \nu \rho \sigma} V_{\rho \sigma} \end{eqnarray}

These are related to the formfactors of Equation 1 in the following fashion: \begin{eqnarray} \setcounter{equation}{4} a^{HXY}_{2} &=& - \frac{2 g_{5e}^{HXY}}{\Lambda_{5}} \nonumber \\ a^{HXY}_{3} &=& \frac{2 g_{5o}^{HXY}}{\Lambda_{5}} \end{eqnarray} where $X$ and $Y$ are gauge bosons.

The second parametrisation is that used by the L3 Collaboration, as described in [2]. The effective Lagrangian used is

\begin{eqnarray} \setcounter{equation}{5} \mathcal{L}_{eff} &=& g_{H \gamma \gamma} H A_{\mu \nu} A^{\mu \nu} + g_{H Z \gamma}^{(1)} A_{\mu \nu} Z^{\mu} \partial^{\nu} H + g_{H Z \gamma}^{(2)} H A_{\mu \nu} Z^{\mu \nu} + \nonumber \\ % && g_{H Z Z}^{(1)} Z_{\mu \nu} Z^{\mu} \partial^{\nu} H + g_{H Z Z}^{(2)} Z_{\mu \nu} Z^{\mu \nu} H + \nonumber \\ % && g_{HWW}^{(1)} \left( W_{\mu \nu}^{+} W_{-}^{\mu} \partial^{\nu} H + W_{\mu \nu}^{-} W_{+}^{\mu} \right) + g_{HWW}^{(2)} H W_{\mu \nu}^{+} W_{-}^{\mu \nu} \nonumber \\ % && + \mbox{ CP-odd part} \end{eqnarray}

The coefficients $g$ in this effective Lagrangian are parametrised in the following way: \begin{eqnarray} \setcounter{equation}{6} g_{H \gamma \gamma} &=& \frac{e}{2 M_{W} \sin \theta_{W}} \left( d \sin^{2} \theta_{W} + d_{B} cos^{2} \theta_{W} \right) \nonumber \\ % g_{H Z \gamma}^{(1)} &=& \frac{e}{M_{W} \sin \theta_{W}} \left( \Delta g_{1}^{Z} \sin 2 \theta_{W} - \Delta \kappa_{\gamma} \tan \theta_{W} \right) \nonumber \\ % g_{H Z \gamma}^{(2)} &=& \frac{e}{2 M_{W} \sin \theta_{W}} \sin 2 \theta_{W} \left(d - d_{B} \right) \nonumber \\ % g_{HZZ}^{(1)} &=& \frac{e}{M_{W} \sin \theta_{W}} \left( \Delta g_{1}^{Z} \cos 2 \theta_{W} + \Delta \kappa_{\gamma} \tan^{2} \theta_{W} \right) \nonumber \\ % g_{HZZ}^{(2)} &=& \frac{e}{2 M_{W} \sin \theta_{W}} \left( d \cos^{2} \theta_{W} + d_{B} sin^{2} \theta_{W} \right)\nonumber \\ % g_{HWW}^{(1)} &=& \frac{e M_{W}}{\sin \theta_{W} M_{Z}^{2}} \Delta g_{1}^{Z} \nonumber \\ % g_{HWW}^{(2)} &=& \frac{e} {\sin \theta_{W} M_{W}} d \end{eqnarray}

Values for $d$, $d_{B}$, $\Delta g_{1}^{Z}$ and $\Delta \kappa_{\gamma}$ can be input via anom_HVV.dat. Note that $\Delta g_{1}^{Z}$ and $\Delta \kappa_{\gamma}$ are also used to parameterise anomalous $WWZ$ and $WW\gamma$ couplings. VBFNLO can also study these couplings for selected processes, but in this case values for $\Delta g_{1}^{Z}$ and $\Delta \kappa_{\gamma}$ need to be input via the file anomV.dat.

The CP-odd part of the effective Lagrangian has the same general forma as the CP-even part. The CP-odd equivalent of the parameters above are signified with a tilde, but there is no $\tilde{\Delta} g_{1}^{Z}$. I.e. \begin{eqnarray} \setcounter{equation}{7} \tilde{g}_{H \gamma \gamma} &=& \frac{e}{2 M_{W} \sin \theta_{W}} \left( \tilde{d} \sin^{2} \theta_{W} +\tilde{d}_{B} cos^{2} \theta_{W} \right) \nonumber \\ % \tilde{g}_{H Z \gamma}^{(1)} &=& \frac{e}{M_{W} \sin \theta_{W}} \left( -\tilde{\kappa}_{\gamma} \tan \theta_{W} \right) \nonumber \\ % \tilde{g}_{H Z \gamma}^{(2)} &=& \frac{e}{2 M_{W} \sin \theta_{W}} \sin 2 \theta_{W} \left(\tilde{d} - \tilde{d}_{B} \right) \nonumber \\ % \tilde{g}_{HZZ}^{(1)} &=& \frac{e}{M_{W} \sin \theta_{W}} \left( \tilde{\kappa}_{\gamma} \tan^{2} \theta_{W} \right) \nonumber \\ % \tilde{g}_{HZZ}^{(2)} &=& \frac{e}{2 M_{W} \sin \theta_{W}} \left( \tilde{d} \cos^{2} \theta_{W} + \tilde{d}_{B} sin^{2} \theta_{W} \right)\nonumber \\ % \tilde{g}_{HWW}^{(1)} &=& 0 \nonumber \\ % \tilde{g}_{HWW}^{(2)} &=& \frac{e} {\sin \theta_{W} M_{W}} \tilde{d} \end{eqnarray}

These are related to the formfactors $a_{1,2,3}$ as follows: \begin{eqnarray} \setcounter{equation}{8} a_{2}^{H \gamma \gamma} &=& -2 \left(2 g_{H \gamma \gamma}\right) = - \frac{2e}{M_{W} \sin \theta_{W}} \left( d \sin^{2} \theta_{W} + d_{B} cos^{2} \theta_{W} \right) \nonumber \\ a_{3}^{H \gamma \gamma} &=& 2 \left(2 \tilde{g}_{H \gamma \gamma}\right) = \frac{2e}{M_{W} \sin \theta_{W}} \left( \tilde{d} \sin^{2} \theta_{W} + \tilde{d}_{B} cos^{2} \theta_{W} \right) \nonumber \\ % a_{2}^{H Z \gamma} &=& -2 \left( g_{HZ\gamma}^{(2)} + \frac{1}{2}g_{HZ\gamma}^{(1)} \right) \nonumber \\ &=& - \frac{2 e}{M_{W}} \left(\cos \theta_{W} \left(d - d_{B} \right) + \Delta g_{1}^{Z} \cos \theta_{W} - \frac{1}{2 \cos \theta_{W}} \Delta \kappa_{\gamma} \right) \nonumber \\ a_{3}^{H Z \gamma} &=& 2 \left( \tilde{g}_{HZ\gamma}^{(2)} + \frac{1}{2}\tilde{g}_{HZ\gamma}^{(1)} \right) = \frac{2 e}{M_{W}} \left(\cos \theta_{W} \left(\tilde{d} - \tilde{d}_{B} \right) - \frac{1}{2 \cos \theta_{W}} \tilde{\kappa}_{\gamma} \right)\nonumber \\ % a_{2}^{HZZ} &=& -2 \left( 2 g_{HZZ}^{(2)} + g_{HZZ}^{(1)} \right) \nonumber \\ &=& - \frac{2e}{M_{W} \sin \theta_{W}} \left( d \cos^{2} \theta_{W} + d_{B} sin^{2} \theta_{W} + \Delta g_{1}^{Z} \cos 2 \theta_{W} + \Delta \kappa_{\gamma} \tan^{2} \theta_{W} \right) \nonumber \\ a_{3}^{HZZ} &=& 2 \left( 2 \tilde{g}_{HZZ}^{(2)} + \tilde{g}_{HZZ}^{(1)}\right) = \frac{2e}{M_{W} \sin \theta_{W}} \left( \tilde{d} \cos^{2} \theta_{W} + \tilde{d}_{B} sin^{2} \theta_{W} + \tilde{\kappa}_{\gamma} \tan^{2} \theta_{W} \right) \nonumber \\ % a_{2}^{HWW} &=& -2 \left( g_{HWW}^{(2)} + g_{HWW}^{(1)} \right) = - \frac{2e} {\sin \theta_{W} M_{W}} \left( d + \cos^{2} \theta_{W} \Delta g_{1}^{Z} \right) \nonumber \\ a_{3}^{HWW} &=& 2 \tilde{g}_{HWW}^{(2)} = \frac{2e} {\sin \theta_{W} M_{W}} \tilde{d} \end{eqnarray}

The third parameterisation of the anomalous Higgs couplings is that described in [3,4]. The effective Lagrangian used in this case is given by:

\begin{eqnarray} \setcounter{equation}{9} \mathcal{L}_{eff} &=& \frac{f_{WW}}{\Lambda^{2}_{6}} \mathcal{O}_{WW} + \frac{f_{BB}}{\Lambda^{2}_{6}} \mathcal{O}_{BB} + \frac{f_{W}}{\Lambda^{2}_{6}} \mathcal{O}_{W}+ \frac{f_{B}}{\Lambda^{2}_{6}} \mathcal{O}_{B} + \mbox{ CP-odd part } \end{eqnarray}

This can be re-written as:

\begin{eqnarray} \setcounter{equation}{10} \mathcal{L}_{eff} = \frac{e M_{W}}{\sin \theta_{W} \Lambda_{6}^{2}} && \left( - \frac{\sin^{2} \theta_{W}}{2} \left( f_{BB} + f_{WW} \right) H A_{\mu \nu} A^{\mu \nu} \right. \nonumber \\ % && + \frac{\sin \theta_{W}}{2 \cos \theta_{W}} \left(f_{W} - f_{B} \right) A_{\mu \nu} Z^{\mu} \partial^{\nu} H \nonumber \\ && + \frac{\sin \theta_{W}}{\cos \theta_{W}} \left(\sin^{2} \theta_{W} f_{BB} - \cos^{2} \theta_{W} f_{WW} \right) H A_{\mu \nu} Z^{\mu \nu} \nonumber \\ % && + \frac{1}{2 \cos^{2} \theta_{W}} \left( \cos^{2} \theta_{W} f_{W} + \sin^{2} \theta_{W} f_{B} \right) Z_{\mu \nu} Z^{\mu} \partial^{\nu} H \nonumber \\ && - \frac{1}{2 \cos^{2} \theta_{W}} \left( \sin^{4} \theta_{W} f_{BB} + \cos^{4} \theta_{W} f_{WW} \right) H Z_{\mu \nu} Z^{\mu \nu} \nonumber \\ % && \left. + \frac{1}{2} f_{W} \left( W_{\mu \nu}^{+} W^{\mu}_{-} + W_{\mu \nu}^{-} W^{\mu}_{+} \right) \partial^{\nu} H - f_{WW} H W_{\mu \nu}^{+} W^{\mu \nu}_{-} \right) \nonumber \\ % && + \mbox{ CP-odd part} \end{eqnarray}

This parametrisation is defined in terms of the coefficients $\frac{f_{i}}{\Lambda_{6}^{2}}$ of the operators $\mathcal{O}_{i}$, and are related to parametrisation 2 according to: \begin{eqnarray} \setcounter{equation}{11} d &=& - \frac{M_{W}^{2}}{\Lambda^{2}_{6}} f_{WW} \nonumber \\ % d_{B} &=& - \frac{M_{W}^{2}}{\Lambda^{2}_{6}} \frac{\sin^{2}\theta_{W}}{\cos^{2}\theta_{W}} f_{BB} \nonumber \\ % \Delta \kappa_{\gamma} &=& \frac{M_{W}^{2}}{2 \Lambda^{2}_{6}} \left( f_{B} + f_{W} \right) \nonumber \\ % \Delta g_{1}^{Z} &=& \frac{M_{Z}^{2}}{2 \Lambda^{2}_{6}} f_{W} \end{eqnarray} for the CP-even parameters, and \begin{eqnarray} \setcounter{equation}{12} \tilde{d} &=& - \frac{M_{W}^{2}}{\Lambda^{2}_{6}} \tilde{f}_{WW} \nonumber \\ % \tilde{d}_{B} &=& - \frac{M_{W}^{2}}{\Lambda^{2}_{6}} \frac{\sin^{2}\theta_{W}}{\cos^{2}\theta_{W}} \tilde{f}_{BB} \nonumber \\ % \tilde{\kappa}_{\gamma} &=& \frac{M_{W}^{2}}{2 \Lambda^{2}_{6}} \tilde{f}_{B} \end{eqnarray} for the CP-odd parameters.

The formfactors $a_{1,2,3}$ are therefore given by the following formulae. Firstly, for the coupling $H\gamma \gamma$: \begin{eqnarray} \setcounter{equation}{13} a_{2}^{H\gamma\gamma} &=& \frac{2 e M_{W} \sin \theta_{W}}{\Lambda^{2}_{6}} \left( f_{WW} + f_{BB} \right) \nonumber \\ % a_{3}^{H\gamma\gamma} &=& - \frac{2 e M_{W} \sin \theta_{W}}{\Lambda^{2}_{6}} \left( \tilde{f}_{WW} + \tilde{f}_{BB} \right) \nonumber \\ \end{eqnarray}

For the coupling $H Z \gamma$: \begin{eqnarray} \setcounter{equation}{14} a_{2}^{H Z \gamma} &=& \frac{2 e M_{W}}{\Lambda^{2}_{6} \cos \theta_{W}} \left( \cos^{2} \theta_{W} f_{WW} - \sin^{2} \theta_{W} f_{BB} + \frac{1}{4} \left(f_{B} - f_{W} \right) \right) \nonumber \\ % a_{3}^{H Z \gamma} &=& - \frac{2 e M_{W}}{\Lambda^{2}_{6} \cos \theta_{W}} \left( \cos^{2} \theta_{W} \tilde{f}_{WW} - \sin^{2} \theta_{W} \tilde{f}_{BB} + \frac{1}{4} \tilde{f}_{B} \right) \end{eqnarray}

The coupling between a Higgs and a pair of $Z$ bosons is parameterised by the formfactors: \begin{eqnarray} \setcounter{equation}{15} a_{2}^{HZZ} &=& \frac{2 e M_{W}}{\sin \theta_{W} \Lambda^{2}_{6}} \left( \cos^{2}\theta_{W} f_{WW} + \left(\frac{\sin^{2} \theta_{W}}{\cos^{2} \theta_{W}} - \sin^{2} \theta_{W} \right) f_{BB} - \frac{1}{2} f_{W} - \frac{\sin^{2} \theta_{W}}{2 \cos^{2} \theta_{W}} f_{B} \right) \nonumber \\ % a_{3}^{HZZ} &=& - \frac{2 e M_{W}}{\sin \theta_{W} \Lambda^{2}_{6}} \left( \cos^{2}\theta_{W} \tilde{f}_{WW} + \left(\frac{\sin^{2} \theta_{W}}{\cos^{2} \theta_{W}} - \sin^{2} \theta_{W} \right) \tilde{f}_{BB} - \frac{\sin^{2} \theta_{W}}{2 \cos^{2} \theta_{W}} \tilde{f}_{B} \right) \end{eqnarray}

Finally, the formfactors for the coupling $HWW$ are \begin{eqnarray} \setcounter{equation}{16} a_{2}^{HWW} &=& \frac{ 2 e M_{W}}{\sin \theta_{W} \Lambda^{2}_{6}} \left( f_{WW} - \frac{1}{2} f_{W} \right) \nonumber \\ % a_{3}^{HWW} &=& - \frac{ 2 e M_{W}}{\sin \theta_{W} \Lambda^{2}_{6}} \tilde{f}_{WW} \end{eqnarray}

[1] T. Figy and D. Zeppenfeld, Phys. Lett. B591, 297 (2004), hep-ph/0403297
[2] L3 Collaboration, P. Achard et al, Phys. Lett. B589, 89 (2004), hep-ex/0403037
[3] K. Hagiwara, R. Szalapski, and D. Zeppenfeld, Phys.Lett. B318, 155 (1993), hep-ph/9308347
[4] K. Hagiwara, S. Ishihara, R. Szalapski, and D. Zeppenfeld, Phys. Rev. D48, 2182 (1993)