Rotations angles in SPheno

General

If a rotation angle is used to parametrise the mixing in the gauge sector a relation must be given how SPheno can calculate this angle from the numerical values of the rotation matrix. For this purpose, the flag DependenceSPheno must be used in parameters.m

Example

1. The Weinberg angle in the SM: the angle can be calculated from the diagonalisation matrix of the neutral gauge bosons Z (called ZZ) via Θ_W = acos|Z₁₁| That's done in parameters.m

   {ZZ, { Description -> "Weinberg-Angle",
                ...
          DependenceSPheno -> ArcCos[Abs[ZZ[1,1]]] }}

2. Models with extra gauge bosons: for models with a Z′ gauge bosons stemming from a U(1) tensored to the SM gauge groups, the rotation matrix reads in general $\left(\begin{array}{ccc} \cos\Theta&-\sin\Theta \cos\Theta'& \sin\Theta \sin\Theta' \\ \sin\Theta&\cos\Theta \cos\Theta'&-\cos\Theta \sin\Theta'\\ 0& \sin\Theta'& \cos\Theta' \end{array}\right)$ and the angles can be calculate from $\begin{aligned} \Theta &=& \text{acos}|Z_{11}|\\ \Theta' &=& \text{acos}|Z_{33}| \end{aligned}$ All of that is defined in parameters.m via {ZZ,{ Dependence -> {{Cos[ThetaW],-Sin[ThetaW] Cos[ThetaWp], Sin[ThetaW] Sin[ThetaWp] }, {Sin[ThetaW],Cos[ThetaW] Cos[ThetaWp],-Cos[ThetaW] Sin[ThetaWp]}, {0, Sin[ThetaWp], Cos[ThetaWp]}}, ...};

   {ThetaW,{
              DependenceSPheno -> ArcCos[Abs[ZZ[1,1]]],
      ...
   };
   
   {ThetaWp, {
              DependenceSPheno -> ArcCos[Abs[ZZ[3,3]]],
      ...
   };

See also