## Renormalisation of the electroweak parameters

The electroweak corrections to Higgs production via vector boson fusion have been implemented for version 2.5.0 of VBFNLO, in both the SM and the MSSM.

VBFNLO follows the Standard Model (SM) conventions of [1]. One thing to note is that the MSSM typically uses a different sign convention for the SU(2) covariant derivative to the Standard Model as described in [1]. In practice, this means that any $\sin \theta_{W}$ in the SM is replaced by $(-\sin \theta_{W})$ in the MSSM, and a minus sign is included for every Higgs field in the MSSM couplings. Unless it is specifically stated otherwise, the renormalisation constants given here use the MSSM convention.

In VBFNLO, a renormalisation scheme is used where the electroweak sector is renormalised on-shell, thus ensuring that all renormalised masses of the gauge bosons correspond to the physical, measured masses. The renormalisation constants of the electroweak sector are (where $V$ stands for either a $W$ or $Z$ boson):

\begin{eqnarray} \delta M_{V}^{2} &=& \mbox{Re} \left( \Sigma^{T}_{VV} (M_{V}^{2})\right) \\ % \delta \sin \theta_{W} &=& -\frac{1}{2} \frac {\cos^{2} \theta_{W}}{\sin \theta_{W}} \left( \frac{\delta M_{Z}^{2}} {M_{Z}^{2}} - \frac{\delta M_{W}^{2}} {M_{W}^{2}} \right) \\ % \delta Z_{VV} &=& -\mbox{Re} \left( \Sigma^{'}_{VV} (M_{V}^{2}) \right) \\ % \delta Z_{\gamma Z} &=& - \frac{2}{M_{Z}^{2}} \mbox{Re} \left( \Sigma_{\gamma Z} (M_{Z}^{2}) \right) \\ % \delta Z_{Z \gamma} &=& \frac{2}{M_{Z}^{2}} \Sigma^{T}_{\gamma Z} (0) \end{eqnarray}

Here, $\Sigma_{XY}$ denotes the self energy $X \rightarrow Y$, the prime ($\Sigma^{'}$) signifies that the derivative with respect to momentum $p^{2}$ is taken, and the superscript $T$ denotes the transverse part of the self energy.

The field renormalisation of the fermions is also performed on-shell, with a constant given by: \begin{eqnarray} \setcounter{equation}{6} \delta Z_{f_{X}} &=& -\mbox{Re} \Sigma_{ff}^{X} - m_{f}^{2} \frac {\partial}{\partial p^{2}} \mbox{Re} \left[ \Sigma_{ff}^{L}(p^{2}) + \Sigma_{ff}^{R}(p^{2}) + 2 \Sigma_{ff}^{S}(p^{2}) \right] \Bigl\lvert _{p^{2} = m^{2}_{f}} \end{eqnarray} Here, the superscripts $L, R$ and $S$ denote the left, right and scalar parts, and $X$ is either $L$ or $R$.

#### 1. Higgs renormalisation

We renormalise the Higgs fields using the $\overline{DR}$ scheme, where the superscript $div$ signifies those terms which are proportional to the UV divergence $\Delta$. The correct on-shell properties of the external Higgs are ensured by finite wavefunction normalisation factors. These so-called Z-factors'' are a convenient way of taking higher order corrections (which are, in the Higgs sector, typically large) into account, and are described in more detail in the notes on Higgs propagator corrections. The field renormalisation constants themselves are given by:

\begin{eqnarray} \setcounter{equation}{7} \delta Z_{hh} &=& - \left[\mbox{Re}\Sigma'_{hh}(m_{h}^{2})\right]^{div} \\ % \delta Z_{HH} &=& - \left[\mbox{Re}\Sigma'_{HH}(m_{H}^{2})\right]^{div} \\ % \delta Z_{hH} &=& \delta Z_{Hh} = \frac {\sin\alpha \cos\alpha} {\cos 2 \alpha} \left( \delta Z_{hh} - \delta Z_{HH} \right) \\ % \delta \tan \beta &=& \frac {1} {2 \cos 2 \alpha} \left( \delta Z_{hh} - \delta Z_{HH} \right) \end{eqnarray}

Note that the Higgs masses used in these formulae are the tree level masses1). In the Standard Model, the Higgs field renormalisation is defined in the on-shell scheme:

\begin{eqnarray} \setcounter{equation}{11} \delta Z_{HH} = -\mbox{Re}\left(\Sigma'_{HH}(m_{H}^{2})\right) \end{eqnarray}

In the MSSM with real parameters, the CP odd Higgs boson $A$ is renormalised on-shell, whereas in the complex MSSM the charged Higgs $H^{\pm}$ is required to be on-shell. This follows the decision of which of the two masses is used as the independent parameter input – in the MSSM with complex parameters, $M_{A}$ is not a mass eigenstate at higher orders, due to mixing with the other neutral Higgs bosons, and consequently $M_{H^{\pm}}$ is used as the input.

#### 2. Charge renormalisation

The counterterms needed for the calculation of higher order corrections to weak boson fusion include the charge renormalisation constant, $\delta Z_{e}$. In VBFNLO there are three options for the charge renormalisation, controlled by the choice of EWSCHEME in vbfnlo.dat. This choice can have a large effect on the relative size of the electroweak corrections, although the magnitude of the NLO cross section should not be significantly affected. Note that when discussing charge renormalisation we will work with the Standard Model convention for the SU(2) covariant derivative.

The form of the renormalisation constant $\delta Z_{e}$ is derived by imposing the condition that the loop corrections to the vertex $ee\gamma$ vanish in the Thompson limit. \begin{eqnarray} \setcounter{equation}{12} \delta Z_{e} &=& \frac{1}{2} \Pi_{\gamma}(0) - \frac {\sin\theta_{W}} {\cos\theta_{W}} \frac{\sum^{T}_{\gamma Z} (0)} {M_{Z}^{2}} \nonumber \\ % &=&\frac{1}{2} \frac{\partial}{\partial q^{2}} \sum\nolimits_{\gamma \gamma} \left(q^{2} \right) \vert_{q^{2}=0} - \frac {\sin\theta_{W}} {\cos\theta_{W}} \frac{\sum^{T}_{\gamma Z} (0)} {M_{Z}^{2}} \end{eqnarray}

Difficulties can be encountered, however, when calculating $\Pi^{\gamma}(0)$, as it involves large contributions from a logarithmic term involving the fermion masses. This leads to problems as the masses of the light quarks are not well defined. In order to avoid dependence on the light quark masses, the quantity $\Delta \alpha$ can be used to split the $\Pi_{\gamma}(0)$ in Equation 12 into its light (i.e.\ all leptons and all quarks except the top) and heavy (the top quark, bosons and any SUSY particles) parts and $\Delta \alpha$ can be used to replace the light particles' contribution to $\Pi^{\gamma}(0)$.

\begin{eqnarray} \setcounter{equation}{13} \delta Z_{e} &=& \frac{1}{2} \left( \Delta \alpha + \frac{1}{M_{Z}^{2}} \mbox{Re} \left[\sum\nolimits^{light}_{\gamma \gamma}(M_{Z}^{2}) \right] \right) + \frac{1}{2} \Pi_{\gamma}^{heavy}(0)- \frac {\sin\theta_{W}} {\cos\theta_{W}} \frac{\sum^{T}_{\gamma Z} (0)} {M_{Z}^{2}} \end{eqnarray}

This is the method used when EWSCHEME = 6 is chosen in vbfnlo.dat. The value of $\alpha(0)$ and $\Delta \alpha$ are taken as user input.

We can also choose to absorb the $\Delta \alpha$ contribution into the tree level matrix element and parametrise using $\alpha(M_{Z}^{2})$: \begin{eqnarray} \setcounter{equation}{14} \alpha(M_{Z}^{2}) = \frac {\alpha(0)} {1 - \Delta \alpha} \end{eqnarray}

This means that $\delta Z_{e}$ needs to be replaced by $\delta Z_{e}^{\alpha(M_{Z}^{2})}$ in the following way (up to higher order terms) \begin{eqnarray} \setcounter{equation}{15} \delta Z_{e}^{\alpha(M_{Z}^{2})} &=& \delta Z_{e} - \frac{1}{2} \Delta \alpha \end{eqnarray} In this way, $\Delta \alpha$ drops out of the charge renormalisation constant $\delta Z_{e}^{\alpha(M_{Z}^{2})}$. This is the formula used to determine the charge renormalisation constant when EWSCHEME = 5 is chosen.

A third option is to parametrise the lowest order coupling via the Fermi constant $G_{F}$ – this method is followed when EWSCHEME = 3 is selected. The Fermi constant, determined from muon decay, is related to the fine structure constant $\alpha(0)$ according to \begin{eqnarray} \setcounter{equation}{16} G_{F} = \frac{\alpha(0) \pi}{\sqrt{2} M_{W}^{2} \sin^{2}\theta_{W}} \left(1 + \Delta r \right) \end{eqnarray}

The quantity $\Delta r$ summarises the higher order corrections to muon decay. The reparametrisation of the lowest order coupling in terms of the Fermi constant yields a corresponding shift in the charge renormalisation constant.

\begin{eqnarray} \setcounter{equation}{17} \delta Z_{e}^{G_{F}} = \delta Z_{e} - \frac{1}{2} \Delta r \end{eqnarray}

When only loop contributions from fermions (or fermions and sfermions in the MSSM) are being considered, the simple expression below can be used, which comes directly from Equation 16: \begin{eqnarray} \setcounter{equation}{18} \delta Z_{e}^{G_{F},(s)fermion} = \frac {\delta \sin\theta_{W}} {\sin\theta_{W}} - \frac{1}{2} \left( \frac {\sum^{T}_{WW}(0) - \delta M_{W}^{2}} {M_{W}^{2}} \right) \end{eqnarray}

In the full Standard Model, however, once the full corrections to muon decay are computed, the expression for $\Delta r$ (and hence $\delta Z_{e}^{G_{F}}$) is more complicated, although it is still relatively compact [2-4]. Reparametrisation of the lowest order coupling by the Fermi constant yields the corresponding replacement of the charge renormalisation constant by: \begin{eqnarray} \setcounter{equation}{19} \delta Z_{e}^{G_{F},SM} &=& \frac {\delta \sin \theta_{W}}{\sin \theta_{W}} - \frac { \sum_{WW}(0) - \delta M_{W}^{2}}{M_{W}^{2}} - \frac{1}{\sin\theta_{W} \cos\theta_{W}} \delta Z_{Z \gamma} - \nonumber \\ % &&\frac{\alpha}{8 \pi \sin^{2} \theta_{W}} \left( 6 + \frac{7 - 4 \sin^{2} \theta_{W}}{2 \sin^{2} \theta_{W}} \ln \left(\cos^{2} \theta_{W} \right) \right) \end{eqnarray}

Due to the extra diagrams that need to be considered in the full MSSM (see, for example, [5]), the expression corresponding to Equation 19 in the MSSM has a somewhat more complicated structure. When working in the full MSSM in VBFNLO, we therefore use an approximation whereby the (finite) contributions to $\Delta r$ from the charginos and neutralinos are neglected.

Note that, if EWSCHEME = 1, 2, 4 is selected, VBFNLO selects the charge renormalisation scheme that best reflects the LO Higgs couplings – i.e. if the $\alpha$ used in the LO coupling is closest to $\alpha(0)$ (as opposed to $\alpha(M_{Z}^{2})$ or $\alpha(G_{F})$) then charge renormalisation via $\alpha(0)$ will be used.

##### References

[1] A. Denner, Fortschr. Phys. 41, 307 (1993), arXiv:0709.1075 [hep-ph]
[2] W. J. Marciano and A. Sirlin, Phys. Rev. D22, 2695 (1980)
[3] A. Sirlin, Phys. Rev. D22, 971 (1980)
[4] W. Hollik, Fortschritte der Physik 38, 165 (1990)
[5] P. H. Chankowski, A. Dabelstein, W. Hollik, W. M. Mösle, S. Pokorski, and J. Rosiek, Nuclear Physics B417, 101 (1994)

1)
Note that lower case $m_{Higgs}$ will refer to tree level masses, while upper case $M_{Higgs}$ refers to loop corrected masses.