... | ... | @@ -2,31 +2,30 @@ |
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Running the check for a newly implemented model
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-----------------------------------------------
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## Running the check for a newly implemented model
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After the initialization of a model via <span>Start\[“MODEL”\]</span> it can be checked for (self-) consistency using the command
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After the initialization of a model via `Start[“MODELNAME”]` it can be checked for (self-) consistency using the command
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CheckModel;
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Performed Checks
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----------------
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## Performed Checks
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#### Causes the particle content gauge anomalies?
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### Causes the particle content gauge anomalies?
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Gauge anomalies are caused by triangle diagrams with three external gauge bosons and internal fermions . The corresponding conditions for all *S**U*(*N*)<sub>*A*</sub> groups to be anomaly free are
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Gauge anomalies are caused by triangle diagrams with three external gauge bosons and internal fermions . The corresponding conditions for all $`SU(N)_A`$ groups to be anomaly free are
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$` \sum_i \mbox{Tr}\left[T_{Ar}^a(\psi_i) T_{Ar}^a(\psi_i) T_{Ar}^a(\psi_i)\right] = 0 `$
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$`\sum_i \text{Tr}\left[T_{Ar}^a(\psi_i) T_{Ar}^a(\psi_i) T_{Ar}^a(\psi_i)\right] = 0 `$
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Again,*T*<sub>*A**r*</sub><sup>*a*</sup>(*ψ*<sub>*i*</sub>) are the generators for a fermion *ψ*<sub>*i*</sub> transforming as irreducible representation*r* under the gauge group /math>SU(N)_A</math>. The sum is taken over all chiral superfields. In the Abelian sector several conditions have to be fulfilled depending on the number of*U*(1) gauge groups
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The $`
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$`\sum_i \text{Tr}\left[`$ are the generators for a fermion $`\Psi_i`$ transforming as irreducible representation $`r`$ under the gauge group $`SU(N)_A`$. The sum is taken over all chiral superfields. In the Abelian sector several conditions have to be fulfilled depending on the number of U(1) gauge groups
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$` U(1)_A^3 : \sum_i (Q^A_{\psi_i})^3 = 0 \\ U(1)_A\times U(1)_B^2 : \thinspace \sum_i Q^A_{\psi_i} (Q^B_{\psi_i})^2 = 0 \\ U(1)_A\times U(1)_B\times U(1)_C : \thinspace \sum_i Q^A_{\psi_i} Q^B_{\psi_i} Q^C_{\psi_i}= 0`$
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The mixed condition involving Abelian and non-Abelian groups is
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*U*(1)<sub>*A*</sub> × *S**U*(*N*)<sub>*B*</sub><sup>2</sup> : ∑<sub>*i*</sub>*Q*<sub>*ψ*<sub>*i*</sub></sub><sup>*A*</sup> Tr\[*T*<sub>*B**r*</sub><sup>*a*</sup>(*ψ*<sub>*i*</sub>)*T*<sub>*B**r*</sub><sup>*a*</sup>(*ψ*<sub>*i*</sub>)\] = 0
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$`U(1)_A \times SU(N)_B^2: \,\,\,\, \sum_i Q_{\Psi_i}^A \text{Tr}\left[ T_{Br}^a(\Psi_i) T_{Br}^a(\Psi_i)\left] =0 `$
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Finally, conditions involving gravity𝔊 are
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Finally, conditions involving gravityi $`\mathfrak{G}`$ are
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$` \mathfrak{G} \times U(1)_A^2 : \sum_i (Q^A_{\psi_i})^2 = 0 \\ \mathfrak{G} \times U(1)_A\times U(1)_B : \thinspace \sum_i Q^A_{\psi_i} Q^B_{\psi_i} = 0 \\ \mathfrak{G}^2 \times U(1)_A : \sum_i Q^A_{\psi_i} = 0 `$
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... | ... | @@ -34,7 +33,7 @@ If one if these conditions is not-fulfilled a warning is printed by SARAH. If so |
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#### Leads the particle content to the Witten anomaly?
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SARAH checks that there is an even number of *S**U*(2) doublets. This is the necessary for a model in order to be free of the Witten anomaly
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SARAH checks that there is an even number of SU(2) doublets. This is the necessary for a model in order to be free of the Witten anomaly
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#### Are all terms in the (super)potential in agreement with global and local symmetries?
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... | ... | |