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---
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title: Rotations in gauge sector
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permalink: /Rotations_in_gauge_sector/
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---
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[Category:Model](/Category:Model "wikilink")
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General
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-------
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The rotations in the gauge sector are defined via
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DEFINITION[$EIGENSTATES][GaugeSector]=
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{ {{Old 1a, Old 2a,...},{New 1a, New 2b,..},MixingMatrix 1},
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{{Old 1b, Old 2b,...},{New 1b, New 2b,..},MixingMatrix 2},
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...
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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: (*N*<sub>1</sub>, *N*<sub>2</sub>, …)<sup>*T*</sup> = *M*(*O*<sub>1</sub>, *O*<sub>2</sub>, …)<sup>2</sup>
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### Remarks
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1. In contrast to [Rotations in matter sector](/Rotations_in_matter_sector "wikilink") the different (charge) components of a field are given explicitly
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2. The new mass eigenstates are taken to appear only with one generation. Therefore, not a single name as for [Rotations in matter sector](/Rotations_in_matter_sector "wikilink") are given but several ones
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3. One can use these definitions to rotate vector bosons and components of gauginos, but not matter fields
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4. In the case of vector-bosons, the names for the new eigenstates must begin with `V`
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5. The ghosts for the rotated vector-bosons are added automatically
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6. By counting the degrees of freedom SARAH checks if the new eigenstates are complex or real.
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7. A parametrisation for the mixing matrix can be given via [parameters.m](/parameters.m "wikilink")
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8. If the rotation is written in terms of mixing angles, it is necessary for the [SPheno](/SPheno "wikilink") output to give relation to calculate the angles from the numerical values of the rotation matrix, see [Rotations angles in SPheno](/Rotations_angles_in_SPheno "wikilink")
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Example
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-------
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1. **Neutral gauge bosons in the SM**: the rotation
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$\\begin{aligned}
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W_3 &=& \\sin\\Theta_W \\gamma + \\cos\\Theta_W Z \\\\
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B &=& \\cos\\Theta_W \\gamma - \\sin\\Theta_W Z \\end{aligned}$
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is defined via
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DEFINITION[EWSB][GaugeSector]=
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{ {{VB,VWB[3]},{VP,VZ},ZZ},
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...
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};
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`VP` and `VZ` are set to real to match the number of degrees of freedom. `ZZ` can be defined in [parameters.m](/parameters.m "wikilink") via
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{ZZ, ...
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Dependence -> {{Cos[ThetaW],-Sin[ThetaW]},
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{Sin[ThetaW],Cos[ThetaW]}}
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};
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to get the standard definition of the Weinberg angle.
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2. **Extra neutral gauge boson**: in the presence of an extra*U*(1) group the mixing
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(*B*, *W*<sub>3</sub>, *B*′) → (*γ*, *Z*, *Z*′)
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can easily be defined based on the above definition via
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DEFINITION[EWSB][GaugeSector]=
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{ {{VB,VWB[3],VBp},{VP,VZ,VZp},ZZ},
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...
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};
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Two important remarks:
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1. SARAH always assumes that the eigenstates are mass-ordered. Thus the above definition is correct for models with a heavy*Z*′. For dark photon models with light extra degrees of freedom for instance it needs to be changed to
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{{VB,VWB[3],VBp},{VP,VZp,VZ},ZZ}
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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`.
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3. **Charged gauge bosons in the SM**: the rotation
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$\\begin{aligned}
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W_1 &=& \\frac{1}{\\sqrt{2}} \\left(W^- + {W^-}^\* \\right) \\\\
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W_2 &=& i \\frac{1}{\\sqrt{2}} \\left({W^-}^\* - W^- \\right) \\\\
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\\end{aligned}$
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is defined via
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DEFINITION[EWSB][GaugeSector]=
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{ {{VWB[1],VWB[2]},{VWm,conj[VWm]},ZW},
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...
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};
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Here, SARAH understands that `VWm` is complex and `ZW` can be set to
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{ZW, ...
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Dependence -> 1/Sqrt[2] {{1, 1},
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{-\[ImaginaryI],\[ImaginaryI]}}
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};
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in [parameters.m](/parameters.m "wikilink").
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4. **Winos in the MSSM**: the relation
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$\\begin{aligned}
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\\tilde W_1 &=& \\frac{1}{\\sqrt{2}} \\left(\\tilde W^- + {\\tilde W^-}^\* \\right) \\\\
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\\tilde W_2 &=& i \\frac{1}{\\sqrt{2}} \\left({\\tilde W^-}^\* - \\tilde W^- \\right) \\\\
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\\tilde W_3 &=& \\tilde W^0 \\\\
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\\end{aligned}$
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is defined via
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DEFINITION[EWSB][GaugeSector] =
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{
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...,
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{{fWB[1],fWB[2],fWB[3]},{fWm,fWp,fW0},ZfW}
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};
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together with
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{ZfW, ...
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Dependence -> 1/Sqrt[2] {{1, 1, 0},
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{-\[ImaginaryI],\[ImaginaryI],0},
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{0,0,Sqrt[2]} }
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};
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See also
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-------- |
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\ No newline at end of file |