





title: Rotations in matter sector



permalink: /Rotations_in_matter_sector/










[Category:Model](/Category:Model "wikilink")






General










The field rotations in the matter sector are defined via the list






DEFINITION[$EIGENSTATES][MatterSector] = { List of Rotations }






for each set of eigenstates.



All possibilities to mix matter fields in the Lagrangian are briefly discussed [here](/Supported_particle_mixing "wikilink"). One can see that there are in general two cases:






 The mass matrix is hermitian or symmetric. That's the case for scalars and Majorana fermions



 The mass matrix is not hermitian. That's the case for Dirac fermions






For both cases the entries in `List of Rotations` look slightly different.






Hermitian or symmetric mass matrix










In this case one basis vector of the old eigenstates has to be defined which gets transformed to new mass eigenstates. Therefore, the general syntax is






{{List of Old Eigenstates},{Name of New Eigenstates, Name of Mixing Matrix}}






with






1. A list of the names of old eigenstates



2. The name of the new eigenstates



3. The name of the mixing matrix






Note, while the list of the old eigenstates can be arbitrary long, just one name is defined for the new eigenstates. This eigenstates will appear automatically with the correct number of generations.






### Examples






1. **DownSquarks in the MSSM**: the mixing of the three generations of *d̃*<sub>*L*</sub> (`SdL`) and *d̃*<sub>*R*</sub> (`SdR`) to six generations of *d̃* (`Sd`) via the rotation matrix *Z*<sup>*D*</sup> (`ZD`) is defined by



{{SdL, SdR}, {Sd, ZD}}






This definition is interpreted as



$\\left(\\begin{array}{c} \\tilde{d}_L^i \\\\ \\tilde{d}_R^i \\end{array}\\right)_k = Z^{D,\\dagger}_{kj} \\tilde{d}_j$






2. **CPeven Higgs bosons in the MSSM**: the mixing of *ϕ*<sub>*d*</sub> (`phid`) and *ϕ*<sub>*u*</sub> (`phi_u`) to two generations of *h* (`hh`) via the rotation matrix *Z*<sup>*H*</sup> (`ZH`) is defined by



{{phid, phiu}, {hh, ZH}}






This definition is interpreted as



$\\left(\\begin{array}{c} \\phi_d \\\\ \\phi_u \\end{array}\\right)_k = Z^{H,T}_{kj} h_j$



Since `phid` and `phiu` are real, SARAH automatically defines `hh` as real as well. In addition, it is also possible to introduce a parametrisation for `ZH` via [parameters.m](/parameters.m "wikilink").






3. **Charged Higgs bosons in the MSSM**: the mixing of *H*<sub>*d*</sub><sup>−</sup> (`SHdm`) and *H*<sub>*u*</sub><sup>+</sup> (`SHup`) to two generations of *H*<sup>±</sup> (`Hpm`) via the rotation matrix *Z*<sup>*P*</sup> (`ZP`) is defined by



{{SHdm,conj[SHup]}, {Hpm, ZP}}






This definition is interpreted as



$\\left(\\begin{array}{c} H_d^ \\\\ (H_u^+)^\* \\end{array}\\right)_k = Z^{P,T}_{kj} H^\\pm_j$



Note the usage of `conj` in the above definition.






4. **Neutralinos in the MSSM**: the mixing of *B̃* (`fB`), *W̃*<sup>0</sup> (`fW0`), *H̃*<sub>*d*</sub><sup>0</sup> (`FHd0`) and *H̃*<sub>*u*</sub><sup>0</sup> (`FHu0`) to four generations of *λ*<sup>0</sup> (`L0`) via the rotation matrix *Z*<sup>*N*</sup> (`ZN`) is defined by



{{fB, fW0, FHd0, FHu0}, {L0, ZN}}






This definition is interpreted as



$\\left(\\begin{array}{c} \\tilde B \\\\ \\tilde W^0 \\\\ \\tilde H_d^0 \\\\ \\tilde H_u^0 \\end{array}\\right)_k = Z^{N,\\dagger}_{kj} \\lambda^0_j$



One can form a Majorana spinor from the Weyl spinors*λ*<sup>0</sup> via






DEFINITION[EWSB][DiracSpinors]={



...



Chi >{ L0, conj[L0]}






Nonhermitian mass matrix










In that case two sets of old eigenstates (*O*<sub>1</sub>,*O*<sub>2</sub>) are rotated to two new sets of mass eigenstates (*N*<sub>1</sub>, *N*<sub>2</sub>) via two rotation matrices (*M*<sub>1</sub>, *M*<sub>2</sub>). The general definition in SARAH is therefore






{{{First Basis},{Second Basis}},{{First States,First Matrix},{Second States,Second Matrix}}}






This is interpreted as



$\\begin{aligned}



O_1 &= M_1 N_1 \\\\



O_2 &= M_2 N_2 \\\\



\\end{aligned}$






### Examples






1. **Up Quarks in the SM or MSSM**: the definition



{{{FuL},{conj[FuR]}},{{FUL,ZUL},{FUR,ZUR}}}






is equivalent to



$U_{L,i} = Z^{U_L}_{ij} u_{L,j} \\hspace{1cm} U^\*_{R,i} = Z^{U_R}_{ij} u_{R,j}$



where the gauge eigenstates are *u*<sub>*L*</sub> (`FuL`) and *u*<sub>*R*</sub> (`FuR`).






2. **Charginos in the MSSM**: the definition



{{{fWm, FHdm}, {fWp, FHup}}, {{Lm,Um}, {Lp,Up}}}






is interpreted as






$\\left( \\begin{array}{c} \\tilde{W}^ \\\\ \\tilde{H}_d^ \\end{array} \\right)_i = U^{,\\dagger}_{ij} \\lambda^_j,



\\hspace{1cm}



\\left( \\begin{array}{c} \\tilde{W}^+ \\\\ \\tilde{H}_u^+ \\end{array} \\right)= U^{+,\\dagger}_{ij} \\lambda^+_j$






Rotation without flavour violation










With the above definitions, SARAH assumes always the most case that the new eigenstates can be a general mixture of all old eigenstates. However, when adding the keyword `NoFlavorMixing` only a mixing between the same generations is considered. Thus, in the definition






{{O1,O2},{N, M},NoFlavorMixing}






it is assumed that the ith generation of `O1` can only mix with the ith generation of `O2`. As consequence, the mass eigenstates carry an additional index 'flavour'.






### Example






If one uses the flag in the MSSM for the upsquark mixing






{{SuL, SuR}, {Su, ZU},NoFlavorMixing}






the mixing is taken to be



$\\begin{aligned}



&\\left(\\begin{array}{c} \\tilde{d}_L^1 \\\\ \\tilde{d}_R^1 \\end{array}\\right)_k = Z^{D_1,\\dagger}_{kj} \\tilde{d}_{1j} \\\\



&\\left(\\begin{array}{c} \\tilde{d}_L^2 \\\\ \\tilde{d}_R^2 \\end{array}\\right)_k = Z^{D_2,\\dagger}_{kj} \\tilde{d}_{2j} \\\\



&\\left(\\begin{array}{c} \\tilde{d}_L^3 \\\\ \\tilde{d}_R^3 \\end{array}\\right)_k = Z^{D_3,\\dagger}_{kj} \\tilde{d}_{3j}



\\end{aligned}$



or more compact



$\\left(\\begin{array}{c} \\tilde{d}_L^f \\\\ \\tilde{d}_R^f \\end{array}\\right)_k = Z^{D_f,\\dagger}_{kj} \\tilde{d}_{fj} \\\\$



The consequences are






1. There are three 2 × 2 rotation matrices which get labelled by a flavour index



2. There are three flavours of fields *d̃* which come with two genetations each.






The parameter and particle definitions read therefore






{ZU, {generation, flavor, flavor}, {3, 2, 2}}



{Su, 1, 3, S, {{generation, 3}, {flavor, 2}, {color, 3}}, 2}






While the usage of flavour indices is supported in the [FeynArts](/FeynArts "wikilink"), [CalcHep/CompHep](/CalcHep/CompHep "wikilink"), [UFO](/UFO "wikilink") and [WHIZARD](/WHIZARD "wikilink") output, it is not possible to generate a [SPheno](/SPheno "wikilink") version for such a model.






See also



 


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