## EWSCHEME: the electroweak parameters and couplings

For the new version (2.5.0) of VBFNLO, the implementation of the input EWSCHEME has been altered. The manner in which the parameters and couplings are determined for each EWSCHEME choice is described below, as are the changes to EWSCHEME 1 and 4.

Note that the old implementation of EWSCHEME is still present in the code. In the file utilities/parameters.F, the subroutine 'setEWpara' sets the electroweak parameters. The old version of this subroutine is commented out.

#### 1. Determination of the couplings

The couplings are set by VBFNLO in the following manner1). The parameters $g_{2}$ and $e$ are set according to the input EWSCHEME as described in Section 2.

##### 1.1 Higgs couplings

\begin{eqnarray} HWW &=& g_{2} M_{W} \nonumber \\ HZZ &=& \frac{g_{2} M_{W}}{\cos^{2} \theta_{W}} \end{eqnarray}

##### 1.2 Triboson couplings

\begin{eqnarray} \setcounter{equation}{2} WW\gamma &=& -e \nonumber \\ WWZ &=& - \frac{g_{2}}{\cos \theta_{W}} \end{eqnarray}

##### 1.3 Fermion-fermion-boson couplings

Fermion-fermion-photon couplings: \begin{eqnarray} \setcounter{equation}{3} l^{+} l^{-} \gamma &=& e \nonumber \\ u \overline{u} \gamma &=& - \frac{2}{3} e \nonumber \\ d \overline{d} \gamma &=& \frac{1}{3} e \end{eqnarray}

Fermion-fermion-W boson couplings \begin{eqnarray} \setcounter{equation}{4} (f \overline{f} W)_{L} &=& \frac{g_{2}}{\sqrt{2}} \end{eqnarray}

Fermion-fermion-Z boson couplings \begin{eqnarray} \setcounter{equation}{5} (\nu \overline{\nu} Z)_{L} &=& \frac{g_{2}}{2 \cos \theta_{W}} \nonumber \\ (l^{+} l^{-} Z)_{L} &=& - \frac{g_{2}}{2 \cos \theta_{W}} (1 - 2 \sin^{2} \theta_{W}) \nonumber \\ (l^{+} l^{-} Z)_{R} &=& \frac{g_{2} \sin^{2} \theta_{W}}{\cos \theta_{W}} \nonumber \\ (u \overline{u} Z)_{L} &=& \frac{g_{2}}{2 \cos \theta_{W}} \left(1 - \frac{4}{3} \sin^{2} \theta_{W}\right) \nonumber \\ (u \overline{u} Z)_{R} &=& - \frac{2 g_{2} \sin^{2} \theta_{W}}{3 \cos \theta_{W}} \nonumber \\ (d \overline{d} Z)_{L} &=& - \frac{g_{2}}{2 \cos \theta_{W}} \left( 1 - \frac{2}{3} \sin^{2} \theta_{W} \right) \nonumber \\ (d \overline{d} Z)_{R} &=& \frac{g_{2} \sin^{2} \theta_{W}}{3 \cos \theta_{W}} \end{eqnarray}

#### 2. The input EWSCHEME

VBFNLO provides six options for the calculation of the electroweak parameters, controlled by the variable EWSCHEME in vbfnlo.dat.

##### 2.1 EWSCHEME = 1

In this scheme, the Fermi constant, $G_{F}$, the fine structure constant, $\alpha$, and the mass of the $Z$ boson, $M_{Z}$, are used to calculate the mass of the $W$ boson, $M_{W}$, and the sine of the weak mixing angle, $\sin \theta_{W}$. Note that, unless the input values are carefully chosen, this can lead to an unrealistic mass of the $W$ boson – a warning is printed if $M_{W} < 80$ GeV.

The parameters and couplings are determined as follows:

\begin{eqnarray} \setcounter{equation}{6} M_{W}^{2} &=& \frac{1}{2} M_{Z}^{2} + \sqrt{\frac{1}{4} M_{Z}^{4} - \frac{\pi \alpha M_{Z}^{2}}{\sqrt{2} G_{F}}} \nonumber \\ % \sin^{2} \theta_{W} &=& 1 - \frac{M_{W}^{2}}{M_{Z}^{2}} \nonumber \\ % e &=& \sqrt{4 \pi \alpha} \nonumber \\ % g_{2} &=& \frac{e}{\sin \theta_{W}} \end{eqnarray}

##### 2.2 EWSCHEME = 2

In this scheme, the values of $G_{F}$, $\sin^{2} \theta_{W}$ and $M_{Z}$ are read from vbfnlo.dat and the values of $M_{W}$ and $\alpha$ are calculated from them.

\begin{eqnarray} \setcounter{equation}{7} M_{W} &=& M_{Z} \sqrt{1 - \sin^{2} \theta_{W}} \nonumber \\ % e &=& M_{Z} \cos \theta_{W} \sin \theta_{W} \sqrt{\frac{8 G_{F}}{\sqrt{2}}} \nonumber \\ % g_{2} &=& M_{Z} \cos \theta_{W} \sqrt{\frac{8 G_{F}}{\sqrt{2}}} \nonumber \\ % \alpha &=& M_{Z}^{2} \cos^{2} \theta_{W} \sin^{2} \theta_{W} \frac{2 G_{F}}{\pi \sqrt{2}} \end{eqnarray}

##### 2.3 EWSCHEME = 3

In this scheme, $G_{F}$, $M_{W}$ and $M_{Z}$ are taken as inputs from vbfnlo.dat and are used to calculate $\alpha$ and $\sin^{2} \theta_{W}$.

\begin{eqnarray} \setcounter{equation}{8} \sin^{2} \theta_{W} &=& 1 - \frac{M_{W}^{2}}{M_{Z}^{2}} \nonumber \\ % e &=& M_{Z} \cos \theta_{W} \sin \theta_{W} \sqrt{\frac{8 G_{F}}{\sqrt{2}}} \nonumber \\ % g_{2} &=& M_{Z} \cos \theta_{W} \sqrt{\frac{8 G_{F}}{\sqrt{2}}} \nonumber \\ % \alpha &=& M_{Z}^{2} \cos^{2} \theta_{W} \sin^{2} \theta_{W} \frac{2 G_{F}}{\pi \sqrt{2}} \end{eqnarray}

##### 2.4 EWSCHEME = 4

In this scheme, all values in vbfnlo.dat (i.e. $G_{F}$, $\alpha$, $M_{W}$, $M_{Z}$ and $\sin^{2} \theta_{W}$) are taken as input. These parameters are not independent and – if they are not chosen consistently – this may lead to problems with gauge invariance. Inconsistent values are not allowed when electroweak corrections to Higgs production via vector boson fusion are being calculated, as they lead to divergent cross sections.

The couplings are set according to: \begin{eqnarray} \setcounter{equation}{9} e &=& \sqrt{4 \pi \alpha} \nonumber \\ % g_{2} &=& M_{Z} \cos \theta_{W} \sqrt{\frac{8 G_{F}}{\sqrt{2}}} \end{eqnarray}

##### 2.5 EWSCHEME = 5

In this scheme, the input values of $M_{W}$, $M_{Z}$ and $\alpha$ are used to calculate all couplings, as well as $\sin^{2} \theta_{W}$. $G_{F}$ is not used, and the input value of $\alpha$ should be set to $\alpha(M_{Z})$.

\begin{eqnarray} \setcounter{equation}{10} \sin^{2} \theta_{W} &=& 1 - \frac{M_{W}^{2}}{M_{Z}^{2}} \nonumber \\ % e &=& \sqrt{4 \pi \alpha} \nonumber \\ % g_{2} &=& \frac{e}{\sin \theta_{W}} \end{eqnarray}

##### 2.6 EWSCHEME = 6

In this scheme, the input values of $M_{W}$, $M_{Z}$ and $\alpha$ are used to calculate all couplings, as well as $\sin^{2} \theta_{W}$. $G_{F}$ is not used, and the input value of $\alpha$ should be set to $\alpha(0)$.

\begin{eqnarray} \setcounter{equation}{11} \sin^{2} \theta_{W} &=& 1 - \frac{M_{W}^{2}}{M_{Z}^{2}} \nonumber \\ % e &=& \sqrt{4 \pi \alpha} \nonumber \\ % g_{2} &=& \frac{e}{\sin \theta_{W}} \end{eqnarray}

The input value of $\Delta \alpha$ does not affect the couplings – it is used only in the calculation of the charge renormalisation constant if the electroweak corrections are requested (see the notes on electroweak renormalisation).

#### 3. Changes for VBFNLO 2.5.0

Several changes to the calculation of the electroweak parameters have been made for VBFNLO 2.5.0, in order to simplify the implementation. EWSCHEME = 5 and 6 have been added: these are useful mainly when calculating the electroweak corrections to Higgs production via weak boson fusion (see the notes on electroweak renormalisation for further details about the effect of the choice of EWSCHEME on the charge renormalisation procedure).

Note that the old code for determining the electroweak parameters is still present in VBFNLO. In the file utilities/parameters.F, there are two versions of the first subroutine, setEWpara, the second of which is commented out. If you wish to revert to the old implementation of EWSCHEME, simply comment out the first version of the subroutine and uncomment the second.

##### 3.1 EWSCHEME = 1

In previous versions of VBFNLO, if this EWSCHEME was chosen, the manner in which $e$ and $g_{2}$ were calculated was dependent on the value of $\alpha$ that was input by the user, in the manner described below.

##### 3.1.1 EWSCHEME = 1 : $\alpha > \alpha(0)$

In this case, versions of VBFNLO prior to 2.5.0 used the same method as later versions – that described in Section 2.1.

##### 3.1.2 EWSCHEME = 1 : $\alpha = \alpha(0)$

In this case: \begin{eqnarray} \setcounter{equation}{12} \sin^{2} \theta_{W} &=& 0.2312 \nonumber \\ % M_{W} &=& M_{Z} \sqrt{1 - \sin^{2} \theta_{W}} \nonumber \\ % e &=& \sqrt{4 \pi \alpha} \nonumber \\ % g_{2} &=& M_{Z} \cos \theta_{W} \sqrt{\frac{8 G_{F}}{\sqrt{2}}} \end{eqnarray}

Note that in this case the Higgs couplings and the fermion-fermion-photon couplings are inconsistent, which can lead to problems with gauge invariance! If this option is chosen, a warning message is output.

##### 3.1.3 EWSCHEME = 1 : $\alpha < \alpha(0)$

In this case: \begin{eqnarray} \setcounter{equation}{13} \sin^{2} \theta_{W} &=& 0.2312 \nonumber \\ % M_{W} &=& M_{Z} \sqrt{1 - \sin^{2} \theta_{W}} \nonumber \\ % e &=& M_{Z} \cos \theta_{W} \sin \theta_{W} \sqrt{\frac{8 G_{F}}{\sqrt{2}}} \nonumber \\ % g_{2} &=& \frac{2 G_{F}}{\pi \sqrt{2}} M_{Z}^{2} \cos^{2} \theta_{W} \sin^{2} \theta_{W} \end{eqnarray}

Note that in this case the input value of $\alpha$ is not used: the couplings are set according to the input $G_{F}$.

##### 3.2 EWSCHEME = 4

In this scheme all values in vbfnlo.dat (i.e. $G_{F}$, $\alpha$, $M_{W}$, $M_{Z}$ and $\sin^{2} \theta_{W}$) can be taken as input. In versions of VBFNLO prior to 2.5.0, however, for certain values of input $\alpha$, various inputs are re-calculated. As with EWSCHEME = 1, there are three methods by which the couplings are calculated, depending on the value of $\alpha$ in vbfnlo.dat.

##### 3.2.1 EWSCHEME = 4 : $\alpha > \alpha(0)$

In this case:

\begin{eqnarray} \setcounter{equation}{14} e &=& \sqrt{4 \pi \alpha} \nonumber \\ % g_{2} &=& \frac{1}{\sin \theta_{W}} \sqrt{4 \pi \alpha} \end{eqnarray}

Note that, in this case, the input value of $G_{F}$ does not affect the couplings.

##### 3.2.2 EWSCHEME = 4 : $\alpha = \alpha(0)$

In this case, the versions of VBFNLO before 2.5.0 used the method used in current versions, described in Section 2.4.

##### 3.2.3 EWSCHEME = 4 : $\alpha < \alpha(0)$

In this case, the couplings are set according to $G_{F}$

\begin{eqnarray} \setcounter{equation}{15} g_{2} &=& M_{Z} \cos \theta_{W} \sqrt{\frac{8 G_{F}}{\sqrt{2}}} \nonumber \\ % e &=& \cos \theta_{W} \sin \theta_{W} \sqrt{\frac{8 G_{F}}{\sqrt{2}}} \nonumber \\ % \alpha &=& M_{Z}^{2} \cos^{2} \theta_{W} \sin^{2} \theta_{W} \frac{2 G_{F}}{\pi \sqrt{2}} \end{eqnarray}

I.e. the value of $\alpha$ input in vbfnlo.dat is not used, and is re-calculated.

1)
Note that these are the Standard Model couplings. By convention, the sign of the SU(2) covariant derivative is different in the MSSM to the SM. This is dealt with internally by VBFNLO.