Rotations_in_matter_sector

Rotations in matter sector

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. 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

A list of the names of old eigenstates
The name of the new eigenstates
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

Down-Squarks in the MSSM: the mixing of the three generations of d̃_L (SdL) and d̃_R (SdR) to six generations of d̃ (Sd) via the rotation matrix Z^D (ZD) is defined by {{SdL, SdR}, {Sd, ZD}}

This definition is interpreted as $\left(\begin{array}{c} \tilde{d}_L^i \\ \tilde{d}_R^i \end{array}\right)k = Z^{D,\dagger}{kj} \tilde{d}_j$
CP-even Higgs bosons in the MSSM: the mixing of ϕ_d (phid) and ϕ_u (phi_u) to two generations of h (hh) via the rotation matrix Z^H (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.
Charged Higgs bosons in the MSSM: the mixing of H_d⁻ (SHdm) and H_u⁺ (SHup) to two generations of H^± (Hpm) via the rotation matrix Z^P (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.
Neutralinos in the MSSM: the mixing of B̃ (fB), W̃⁰ (fW0), H̃_d⁰ (FHd0) and H̃_u⁰ (FHu0) to four generations of λ⁰ (L0) via the rotation matrix Z^N (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 spinorsλ⁰ via
```
DEFINITION[EWSB][DiracSpinors]={
...
Chi ->{ L0, conj[L0]}
```

Non-hermitian mass matrix

In that case two sets of old eigenstates (O₁,O₂) are rotated to two new sets of mass eigenstates (N₁, N₂) via two rotation matrices (M₁, M₂). The general definition in SARAH is therefore

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

This is interpreted as O_1 = M_1 N_1 \\ O_2 = M_2 N_2 \\

Examples

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_L (FuL) and u_R (FuR).
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 i-th generation of O1 can only mix with the i-th generation of O2. As consequence, the mass eigenstates carry an additional index 'flavour'.

Example

If one uses the flag in the MSSM for the up-squark mixing

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

the mixing is taken to be \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} 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

There are three 2 × 2 rotation matrices which get labelled by a flavour index
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, CalcHep/CompHep, UFO and WHIZARD output, it is not possible to generate a SPheno version for such a model.

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