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
-
The Weinberg angle in the SM: the angle can be calculated from the diagonalisation matrix of the neutral gauge bosons Z (called
ZZ
) via ΘW = acos|Z11| That's done in parameters.m{ZZ, { Description -> "Weinberg-Angle", ... DependenceSPheno -> ArcCos[Abs[ZZ[1,1]]] }}
-
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]]], ... };