Conversion to JHU notation

JHU is a parton-level generator which produces LO information about the process $ab \rightarrow X \rightarrow VV$, where the resonance $X$ can be spin-0, spin-1 or spin-2 with generic couplings. In other words, JHU can simulate the particles included in VBFNLO when anomalous Higgs couplings or a spin-2 particle are included. The notation used by JHU, however, is somewhat different to that used by VBFNLO. This page provides the conversion between the two notations.

In VBFNLO, the coupling between an anomalous Higgs boson and a pair of vector bosons is given by

\begin{eqnarray} T^{\mu\nu} &=& a^{V}_{1}(q_{1},q_{2}) g^{\mu \nu} + a^{V}_{2}(q_{1},q_{2}) [ q_{1} \cdot q_{2} g^{\mu \nu} - q_{2}^{\mu} q_{1}^{\nu} ] + a^{V}_{3}(q_{1},q_{2}) \varepsilon^{\mu \nu \rho \sigma} q_{1}_{\rho} q_{2}_{\sigma} \nonumber \end{eqnarray}

where $q_{1}$ and $q_{2}$ are the momenta of the gauge bosons and $a^{V}_{1,2,3}$ are Lorentz-invariant formfactors. The conversion between the formfactors of the coupling and the VBFNLO inputs are given in detail here.

This differs slightly from the notation used by JHU, where the amplitude of a scalar $X$ decaying into two vector bosons is given by

\begin{eqnarray} A(X\rightarrow VV) = \frac{1}{v} \epsilon_{1}^{*\mu} \epsilon_{2}^{*\nu} \left( a^{J}_{1} g_{\mu\nu} m_{X}^{2} + a^{J}_{2} q_{\mu}q_{\nu} + a^{J}_{3} \varepsilon_{\mu\nu\alpha\beta} q_{1}^{\alpha} q_{2}^{\beta} \right) \nonumber \end{eqnarray}

where $v$ is the vacuum expectation value of the scalar and $q = q_{1} + q_{2}$ is the scalar's momentum. Note that for on-shell decays in JHU $q_{1} \cdot q_{2} = \frac{1}{2} \left( m_{X}^{2} - 2 m_{V}^{2} \right)$.

The conversion between JHU and VBFNLO is thus

\begin{eqnarray} a_{1}^{J} &=& \frac{v}{m_{X}^{2}} \left( a_{1}^{V} + \frac{1}{2} \left[m_{X}^{2} - 2 m_{V}^{2} \right] \right) \nonumber \\ a_{2}^{J} &=& - v a_{2}^{V} \nonumber \\ a_{3}^{J} &=& v a_{3}^{V} \nonumber \end{eqnarray}

VBFNLO parametrises a spin-2 singlet $t_{\mu \nu}$ in terms of an effective Lagrangian

\begin{eqnarray} \mathcal{L}_{\text{singlet}} &=& \frac{1}{\Lambda} t_{\mu \nu} \left( f_{1} B^{\alpha \nu} B^{\mu}_{\alpha} + f_{2} W_{i}^{\alpha \nu} W^{i,\mu}_{\alpha} + f_3 \widetilde{B}^{\alpha \nu} B^\mu_{\alpha} +f_4\widetilde{W}_i^{\alpha \nu} W^{i, \mu}_{\alpha} + 2 f_{5} (D^{\mu} \Phi)^{\dagger} (D^{\nu}\Phi) \right) \nonumber \end{eqnarray}

where the field strength and dual field strength tensors are $V^{\mu \nu}$ and $\widetilde{V}^{\mu \nu}$ and the VBFNLO inputs are the scale $\Lambda$ and the coefficients $f_{1..5}$.

JHU gives an amplitude for an on-shell spin-2 particle $S_{2}$ decaying into two identical vector bosons of:

\begin{eqnarray} A(S_{2} \rightarrow VV) &=& \frac{1}{\Lambda} \epsilon^{*\mu}_{1} \epsilon^{*\nu}_{2} \left( c_{1} (q_1 \cdot q_2) t_{\mu \nu} + c_{2} g_{\mu \nu} t_{\alpha \beta} \tilde{q}^{\alpha} \tilde{q}^{\beta} + 2 c_{4} \left[ q_{1\nu} q_{2}^{\alpha} t_{\mu \alpha} + q_{2\mu} q_{1}^{\alpha} t_{\nu \alpha} \right] + \right. \nonumber \\ && \left. c_{6} t^{\alpha \beta} \tilde{q}_{\beta} \epsilon_{\mu \nu \alpha \rho} q^{\rho} + ... \right) \nonumber \end{eqnarray}

where $q_{1}$ and $q_{2}$ are the momenta of the vector bosons and

\begin{eqnarray} \tilde{q} &=& q_{1} - q_{2} \nonumber \\ q &=& q_{1} + q_{2} \nonumber \end{eqnarray}

The $+ ...$ in the amplitude above signifies higher order terms suppressed by a factor of $\frac{1}{\Lambda^{2}}$, which VBFNLO does not include – i.e.

\begin{eqnarray} c_{3} &=& 0 \nonumber \\ c_{5} &=& 0 \nonumber \\ c_{7} &=& 0 \nonumber \end{eqnarray}

When on-shell, the VBFNLO terms $f_{3}$ and $f_{4}$ do not contribute, and as such are not present in JHU. The $c_{6}$ term in the JHU amplitude is not SU(2) gauge invariant and would, if derived from an effective Lagrangian, be a higher order term. It is therefore not included in VBFNLO – i.e. \begin{eqnarray} c_{6} &=& 0 \nonumber \end{eqnarray}

Note that, since VBFNLO uses an effective Lagrangian approach involving the field strength tensors $W_{\mu \nu}^{i}$ and $B_{\mu\nu}$ the couplings of the spin-2 particle to the various gauge bosons are of course related. In JHU, no relations between the $S_{2} VV$ couplings are enforced. Taking each of the $S_{2} VV$ vertices in turn, the conversion between VBFNLO and JHU is as follows, where $g$ and $g'$ are the gauge couplings of $SU(2)_{L}$ and $U(1)_{Y}$ respectively:

• $S_{2} W^{+} W^{-}$

\begin{eqnarray} c_{1} &=& 2 i f_{2} + \frac{i}{2} f_{5} g^{2} \frac{v^{2}}{q_1 \cdot q_2} \nonumber \\ &=& 2 i f_{2} + i f_{5} g^{2} \frac{v^{2}}{m_{S_{2}}^{2} - 2 m_W^2} \nonumber \\ c_{2} &=& - \frac{i f_{2}} {2} \nonumber \\ c_{4} &=& - i f_{2} \nonumber \end{eqnarray}

• $S_{2} ZZ$

\begin{eqnarray} c_{1} &=& 2 i \left( f_{2} \cos^{2} \theta_{W} + f_{1} \sin^{2} \theta_{W} \right) + \frac{i}{2} f_{5} \frac{v^{2}}{q_1 \cdot q_2} \left(g^{2} + g'^{2} \right) \nonumber \\ &=& 2 i \left( f_{2} \cos^{2} \theta_{W} + f_{1} \sin^{2} \theta_{W} \right) + i f_{5} \frac{v^{2}}{m_{S_{2}}^{2} - 2 m_Z^2} \left(g^{2} + g'^{2} \right) \nonumber \\ c_{2} &=& -\frac{i}{2} \left( f_{2} \cos^{2} \theta_{W} + f_{1} \sin^{2} \theta_{W} \right) \nonumber \\ c_{4} &=& -i \left( f_{2} \cos^{2} \theta_{W} + f_{1} \sin^{2} \theta_{W} \right) \nonumber \end{eqnarray}

• $S_{2} \gamma \gamma$

\begin{eqnarray} c_{1} &=& 2i \left( f_{1} \cos^{2} \theta_{W} + f_{2} \sin^{2} \theta_{W} \right) \nonumber \\ c_{2} &=& -\frac{i}{2} \left( f_{1} \cos^{2} \theta_{W} + f_{2} \sin^{2} \theta_{W} \right) \nonumber \\ c_{4} &=& - i \left( f_{1} \cos^{2} \theta_{W} + f_{2} \sin^{2} \theta_{W} \right) \nonumber \end{eqnarray}

• The $S_{2} \gamma Z$ vertex is not included in JHU.