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---
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title: Potential
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permalink: /Potential/
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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 potential of non-supersymmetric models can be defined via the entries of
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DEFINITION[GaugeES][LagrangianInput]= {
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{LagPiece1, {AddHC->True/False}},
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{LagPiece2, {AddHC->True/False}},
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...
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};
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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.
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Terms in the Lagrangina
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-----------------------
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The parts added to the Lagrangian must have mass dimesion 4 and have the general form:
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[Coefficient] Parameter [Contraction] Field1.Field2.[Field3.Field4]
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The parts in brackets can be put optionally. The purpose of the different parts is
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1. `Coefficient`: a numerical coefficient. If it is not put, 1 is taken
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2. `Parameter`: a name for the coupling
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3. `Contraction`: one can define a contractions of the charge indices explicitly, see [Automatic index contraction](/Automatic_index_contraction "wikilink") for more information
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4. `FieldX`: a term in the potential can involve up to four fields.
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In general, the following interactions are supported (see also [Supported matter sector](/Supported_matter_sector "wikilink")):
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- Two-scalar-Interactions
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- Three-scalar-Interactions
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- Four-scalar-Interactions
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- Two-fermion-Interactions
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- Interactions between two-fermions and one scalar.
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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](/RGEs "wikilink") are not given in literature.
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Example
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-------
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1. **Scalar potential in the SM**: the terms
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$V_H = \\mu |H|^2 + \\frac12 \\lambda |H|^4$
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in the SM are defined via
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LagHiggs = Mu2 conj[H].H - 1/2 \[Lambda] conj[H].H.conj[H].H;
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DEFINITION[GaugeES][LagrangianInput]= {
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{LagHiggs,{AddHC -> False}},
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...
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};
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SARAH adds automatically the *S**U*(2) indices and contracts them. Therefore, the short input above is interpreted as:
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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}]
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2. **Yukawa interactions in the SM**: the terms
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*V*<sub>*Y*</sub> = *Y*<sub>*d*</sub>*H*<sup>\*</sup>*d**q* + *Y*<sub>*e*</sub>*H*<sup>\*</sup>*e**l* + *Y*<sub>*u*</sub>*H**u**q*
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in the SM are defined via
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LagYukawa = - (Yd conj[H].d.q + Ye conj[H].e.l + Yu H.u.q);
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DEFINITION[GaugeES][LagrangianInput]= {
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{LagYukawa,{AddHC -> True}},
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...
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};
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SARAH adds automatically all indices and contracts them. For instance, `Yd conj[H].d.q` is interpreted as
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Delta[lef1,lef3] Delta[col2,col3] Yd[gen2,gen3] conj[H[{lef1}]].d[{col2}].q[{lef3,col3}]
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See also
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-------- |
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\ No newline at end of file |