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SARAH

Potential

Last edited by Martin Gabelmann
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Potential

General

The potential of non-supersymmetric models can be defined via the entries of

DEFINITION[GaugeES][LagrangianInput]= {
    {LagPiece1,  {AddHC->True/False}},
    {LagPiece2,  {AddHC->True/False}},
         ...
};

Here, LagPieceX are the parts of the Lagrangian which shall be added. The option defines if SARAH shall also add the hermitian conjugated to the Lagrangian (AddHC->True). If this option is not set, the hermitian conjugated is not added by default. This option is supposed to be used to define the matter interactions of the potential of a model. It could be used also to define new interactions involving vector bosons. However, this option has to be used carefully since it isn't tested as much as the general purpose.

Terms in the Lagrangina

The parts added to the Lagrangian must have mass dimesion 4 and have the general form:

[Coefficient] Parameter  [Contraction] Field1.Field2.[Field3.Field4]

The parts in brackets can be put optionally. The purpose of the different parts is

  1. Coefficient: a numerical coefficient. If it is not put, 1 is taken
  2. Parameter: a name for the coupling
  3. Contraction: one can define a contractions of the charge indices explicitly, see Automatic index contraction for more information
  4. FieldX: a term in the potential can involve up to four fields.

In general, the following interactions are supported (see also Supported matter sector):

  • Two-scalar-Interactions
  • Three-scalar-Interactions
  • Four-scalar-Interactions
  • Two-fermion-Interactions
  • Interactions between two-fermions and one scalar.

Note, a tadpole term for one scalar singlet is not supported! The main reason are that this term can always be shifted away and therefore the generic RGEs are not given in literature.

Example

  1. Scalar potential in the SM: the terms V_H = \mu |H|^2 + \frac{1}{2} \lambda |H|^4 in the SM are defined via

    LagHiggs = Mu2 conj[H].H - 1/2 \[Lambda] conj[H].H.conj[H].H;

DEFINITION[GaugeES][LagrangianInput]= {
    {LagHiggs,{AddHC -> False}},
        ...
};

    SARAH adds automatically the SU(2) indices and contracts them. Therefore, the short input above is interpreted as:

    Mu2 Delta[lef1,lef2] conj[H[{lef1}]].H[{lef2}] - 1/2 \[Lambda] Delta[lef1,lef2] Delta[lef3,lef4] conj[H[{lef1}]].H[{lef2}].conj[H[{lef3}]].H[{lef4}]

  2. Yukawa interactions in the SM: the terms V_Y = Y_d H^\dagger\bar{d} q + Y_e H^\dagger\bar{e} l + Y_u H\bar{u} q in the SM are defined via

    LagYukawa = - (Yd conj[H].d.q + Ye conj[H].e.l + Yu H.u.q);

DEFINITION[GaugeES][LagrangianInput]= {
    {LagYukawa,{AddHC -> True}},
        ...
};

    (note that e, u and d are defined as their complex conjugates in the model file and hence there are nor conj[e] etc but e)

    SARAH adds automatically all indices and contracts them. For instance, Yd conj[H].d.q is interpreted as

    Delta[lef1,lef3] Delta[col2,col3] Yd[gen2,gen3] conj[H[{lef1}]].d[{col2}].q[{lef3,col3}]

See also