In version 2.5.0 of VBFNLO, the implementation of the input EWSCHEME was 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 manner. The parameters \(g_2\) and \(e\) are set according to the input EWSCHEME.

    1. Higgs couplings $$HWW = g_2 M_W$$ $$HZZ = \frac{g_2 M_W}{\cos^2{\theta_W}}$$
    2. Triboson couplings $$WW\gamma = -e$$ $$WWZ = -\frac{g_2}{\cos{\theta_W}}$$
    3. Fermion-fermion-boson couplings
      • Fermion-fermion-photon couplings: $$\ell^+ \ell^- \gamma = e$$ $$u\bar{u}\gamma = -\frac{2}{3}e$$ $$d\bar{d}\gamma = \frac{1}{3}e$$
      • Fermion-fermion-W couplings: $$(f\bar{f}W)_L = \frac{g_2}{\sqrt{2}}$$
      • Fermion-fermion-Z couplings: $$(\nu \bar{\nu} Z)_L = \frac{g_2}{2\cos{\theta_W}}$$ $$(\ell^+ \ell^- Z)_L = -\frac{g_2}{2\cos{\theta_W}(1-2\sin^2{\theta_W})}$$ $$(\ell^+ \ell^- Z)_R = \frac{g_2\sin^2{\theta_W}}{\cos{\theta_W}}$$ $$(u \bar{u} Z)_L = \frac{g_2}{2\cos{\theta_W}} \left(1 - \frac{4}{3}\sin^2{\theta_W}\right)$$ $$(u \bar{u} Z)_R = -\frac{2g_2 \sin^2{\theta_W}}{3\cos{\theta_W}}$$ $$(d \bar{d} Z)_L = -\frac{g_2}{2\cos{\theta_W}}\left(1 - \frac{2}{3}\sin^2{\theta_W}\right)$$ $$(d \bar{d} Z)_R = -\frac{g_2 \sin^2{\theta_W}}{3\cos{\theta_W}}$$
  2. The input EWSCHEME

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

    1. EWSCHEME = 1

      In this scheme, the Fermi constant, GF, 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 \(M_W < 80\; \rm{GeV}\).

      The parameters and couplings are determined as follows:

      $$M_W^2 = \frac{1}{2}M_Z^2 + \sqrt{\frac{1}{4}M_Z^4 - \frac{\pi \alpha M_Z^2}{\sqrt{2}G_F}}$$ $$\sin^2{\theta_W} = 1 -\frac{M_W^2}{M_Z^2}$$ $$e = \sqrt{4\pi \alpha}$$ $$g_2 = \frac{e}{\sin{\theta_W}}$$
    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.

      $$M_W = M_Z\sqrt{1-\sin^2{\theta_W}}$$ $$e=M_Z\cos{\theta_W}\sin{\theta_W}\sqrt{\frac{8G_F}{\sqrt{2}}}$$ $$g_2 = M_Z\cos{\theta_W}\sqrt{\frac{8G_F}{\sqrt{2}}}$$ $$\alpha = M_Z^2\cos^2{\theta_W}\sin^2{\theta_W}\frac{2G_F}{\pi \sqrt{2}}$$
    3. EWSCHEME = 3

      In this scheme, the values of \(G_F\), \(M_W\) and \(M_Z\) are read from vbfnlo.dat and the values of \(\alpha\) and \(\sin^2{\theta_W}\) are calculated from them.

      $$\sin^2{\theta_W} = 1 - \frac{M_W^2}{M_Z^2}$$ $$e=M_Z\cos{\theta_W}\sin{\theta_W}\sqrt{\frac{8G_F}{\sqrt{2}}}$$ $$g_2 = M_Z\cos{\theta_W}\sqrt{\frac{8G_F}{\sqrt{2}}}$$ $$\alpha = M_Z^2\cos^2{\theta_W}\sin^2{\theta_W}\frac{2G_F}{\pi \sqrt{2}}$$
    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: $$e = \sqrt{4\pi\alpha}$$ $$g_2 = M_Z\cos{\theta_W}\sqrt{\frac{8G_F}{\sqrt{2}}}$$

    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 \(\alpha (M_Z)\).

      $$\sin^2{\theta_W} = 1 - \frac{M_W^2}{M_Z^2}$$ $$e = \sqrt{4\pi\alpha}$$ $$g_2 = \frac{e}{\sin{\theta_W}}$$
    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 \(\alpha (0)\).

      $$\sin^2{\theta_W} = 1 - \frac{M_W^2}{M_Z^2}$$ $$e = \sqrt{4\pi\alpha}$$ $$g_2 = \frac{e}{\sin{\theta_W}}$$

      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).