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Rotations_in_gauge_sector

Rotations in gauge sector

General

The rotations in the gauge sector are defined via

DEFINITION[$EIGENSTATES][GaugeSector]=
{ {{Old 1a, Old 2a,...},{New 1a, New 2b,..},MixingMatrix 1},
  {{Old 1b, Old 2b,...},{New 1b, New 2b,..},MixingMatrix 2},
  ...

Here, Old Nx is the name of the old and New Nx of rotated eigenstates. MixingMatrix X is the rotation matrix relating the old and new basis. SARAH interprets this definition as matrix multiplication: (N1, N2, …)T = M(O1, O2, …)2

Remarks

  1. In contrast to Rotations in matter sector the different (charge) components of a field are given explicitly
  2. The new mass eigenstates are taken to appear only with one generation. Therefore, not a single name as for Rotations in matter sector are given but several ones
  3. One can use these definitions to rotate vector bosons and components of gauginos, but not matter fields
  4. In the case of vector-bosons, the names for the new eigenstates must begin with V
  5. The ghosts for the rotated vector-bosons are added automatically
  6. By counting the degrees of freedom SARAH checks if the new eigenstates are complex or real.
  7. A parametrisation for the mixing matrix can be given via parameters.m
  8. If the rotation is written in terms of mixing angles, it is necessary for the SPheno output to give relation to calculate the angles from the numerical values of the rotation matrix, see Rotations angles in SPheno

Example

  1. Neutral gauge bosons in the SM: the rotation $\begin{aligned} W_3 &=& \sin\Theta_W \gamma + \cos\Theta_W Z \\ B &=& \cos\Theta_W \gamma - \sin\Theta_W Z \end{aligned}$ is defined via DEFINITION[EWSB][GaugeSector]= { {{VB,VWB[3]},{VP,VZ},ZZ}, ... };

    VP and VZ are set to real to match the number of degrees of freedom. ZZ can be defined in parameters.m via

    {ZZ, ...
     Dependence ->  {{Cos[ThetaW],-Sin[ThetaW]},
                {Sin[ThetaW],Cos[ThetaW]}}
};

    to get the standard definition of the Weinberg angle.

  2. Extra neutral gauge boson: in the presence of an extraU(1) group the mixing (B, W3, B′) → (γ, Z, Z′) can easily be defined based on the above definition via DEFINITION[EWSB][GaugeSector]= { {{VB,VWB[3],VBp},{VP,VZ,VZp},ZZ}, ... };

    Two important remarks:

    1. SARAH always assumes that the eigenstates are mass-ordered. Thus the above definition is correct for models with a heavyZ′. For dark photon models with light extra degrees of freedom for instance it needs to be changed to {{VB,VWB[3],VBp},{VP,VZp,VZ},ZZ}

    2. One can either parametrise ZZ again by two or three rotation angles corresponding to the form of the mass matrix. If not, all expressions are written in terms of ZZ[x,y] with integers x,y.

  3. Charged gauge bosons in the SM: the rotation $\begin{aligned} W_1 &=& \frac{1}{\sqrt{2}} \left(W^- + {W^-}^* \right) \\ W_2 &=& i \frac{1}{\sqrt{2}} \left({W^-}^* - W^- \right) \\ \end{aligned}$ is defined via DEFINITION[EWSB][GaugeSector]= { {{VWB[1],VWB[2]},{VWm,conj[VWm]},ZW}, ... };

    Here, SARAH understands that VWm is complex and ZW can be set to

    {ZW, ...
      Dependence ->   1/Sqrt[2] {{1, 1},
            {-\[ImaginaryI],\[ImaginaryI]}}
};

    in parameters.m.

  4. Winos in the MSSM: the relation $\begin{aligned} \tilde W_1 &=& \frac{1}{\sqrt{2}} \left(\tilde W^- + {\tilde W^-}^* \right) \\ \tilde W_2 &=& i \frac{1}{\sqrt{2}} \left({\tilde W^-}^* - \tilde W^- \right) \\ \tilde W_3 &=& \tilde W^0 \\ \end{aligned}$ is defined via DEFINITION[EWSB][GaugeSector] = { ..., {{fWB[1],fWB[2],fWB[3]},{fWm,fWp,fW0},ZfW} };

    together with

    {ZfW, ...
       Dependence ->   1/Sqrt[2] {{1, 1, 0},
                      {-\[ImaginaryI],\[ImaginaryI],0},
                      {0,0,Sqrt[2]} }
};

See also