... | ... | @@ -13,26 +13,31 @@ It is usually not necessary to define any index structure for terms appearing in |
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### Simple contractions
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The rules to contract indices corresponding to the fundamental (upper) and anti-fundamental (lower) representation of a gauge group are
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The rules to contract indices corresponding to the fundamental (upper) and anti-fundamental (lower) representation of a gauge group are:
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- An upper and a lower index is contracted via a Kronecker Delta
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- N upper or lower indices of a *S**U*(*N*) are contracted with the epsilon Tensor
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- an upper and a lower index is contracted via a Kronecker Delta
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- N upper or lower indices of a SU(N) are contracted with the epsilon Tensor
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#### Example
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1. $`3 \times \bar{3}`$ of *S**U*(3):
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1. $`3 \times \bar{3}`$ of SU(3):
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Delta[col1,col2] T[{col1}].Tc[{col2}]
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2. 2 × 2 of *S**U*(2):
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2. $`2\times 2`$ of SU(2):
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epsTensor[lef1,lef2] Hu[{lef1}].Hd[{lef2}]
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3. 3 × 3 × 3 of *S**U*(3):
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3. $`3\times 3 \times 3`$ of SU(3):
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epsTensor[col1,col2,col3] T[{col1}].T[{col2}].T[{col3}]
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4. 2 × 3 × 2 of *S**U*(2):
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4. $`2\times 3\times 2`$ of SU(2):
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Delta[lef1,lef2] epsTensor[lef2b,lef3] Hu[{lef1}].T[{lef2,lef2b}].Hu[{lef3}]
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5. $`3 \times \bar{3}`$ of *S**U*(3) and 2 × 2 of *S**U*(2):
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5. $`3 \times \bar{3}`$ of SU(3) and $`2\times 2`$ of SU(2):
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Delta[col1,col2] epsTensor[lef1,lef3] Q[{lef1,col1}].u[{col2}].Hu[{lef3}]
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### Clebsch-Gordan Coefficients
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... | ... | @@ -61,11 +66,12 @@ Self-defined contractions |
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In principle, it is also possible that the user can define a contraction of indices which is not the standard one. In particular for the *S**U*(2) it might be necessary to adjust the index contraction: there is some ambiguity because of the relation among the fundamental and anti-fundamental representation in this group. One can see that SARAH used contractions via
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- `ShowSuperpotentialContractions` for SUSY models
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- `` SA`LagrangianContractions `` for non-SUSY models
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- ``SA`LagrangianContractions `` for non-SUSY models
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#### Example
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1. 2 × 3 × 2 of *S**U*(2): depending of the definition of the triplet `t` the demanded contraction might be
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1. $`2 \times 3 \times 2`$ of SU(2): depending of the definition of the triplet `t` the demanded contraction might be
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epsTensor[lef1,lef2] epsTensor[lef2b,lef3] l[{lef1}].t[{lef2,lef2b}].l[{lef3}]
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The default contraction of SARAH would have been instead
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... | ... | @@ -76,7 +82,8 @@ In principle, it is also possible that the user can define a contraction of indi |
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epsTensor[lef1,lef2] epsTensor[lef2b,lef3] l.t.l
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2. |*H*<sub>*u*</sub><sup>†</sup>*σ*<sub>*a*</sub>*H*<sub>*u*</sub> + *H*<sub>*d*</sub><sup>†</sup>*σ*<sub>*a*</sub>*H*<sub>*d*</sub>|<sup>2</sup> can be written by using the completeness relation of the Pauli matrices as
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2. $`|H_u^\dagger \sigma_a H_u + H_d^\dagger \sigma_a H_d|^2`$ can be written by using the completeness relation of the Pauli matrices as
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(2 Delta[lef1,lef4] Delta[lef2,lef3]- Delta[lef1,lef2] Delta[lef3,lef4])*
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(conj[SHd].SHd.SHd.conj[SHd] + conj[SHu].SHu.SHu.conj[SHu] - 2 conj[SHd].SHd.SHu.conj[SHu]) )
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... | ... | @@ -99,6 +106,7 @@ where `COUPLING` is the name of the coupling for which you want to 'hard-code' t |
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ContractionRGE[LT]=Delta[lef1,lef2] Delta[lef3, lef4];
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2. **Triplet-doublet interaction**: Writing the interaction between to scalar doublets `H` and a scalar triplet `Ts` as
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muT conj[H].Ts.H
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One can define an explicit contraction via
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... | ... | @@ -106,7 +114,4 @@ where `COUPLING` is the name of the coupling for which you want to 'hard-code' t |
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ContractionRGE[muT]=InvMat[99][lef1,lef2,lef3];
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Off[Part::pspec];
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InvMat[99][a___Integer]:={{{1/Sqrt[2], 0}, {0, 1/Sqrt[2]}, {0, -(I/Sqrt[2])}}, {{0, -(1/Sqrt[2])}, {1/Sqrt[2], 0}, {I/Sqrt[2], 0}}}[[/a|a]];
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On[Part::pspec];
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See also
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-------- |
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\ No newline at end of file |
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On[Part::pspec]; |
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\ No newline at end of file |