Gauge group constants

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SARAH supports not only chiral superfields in the fundamental representation but in any irreducible representation of $SU(N)$. In most cases, it is possible to fix the transformation properties of the chiral superfield by stating the corresponding dimension $D$. If the dimension is not unique, also the Dynkin labels are needed. For calculating kinetic terms and D-terms, it is necessary to derive from representation the corresponding generators. Furthermore, the eigenvalues $C_2$ of the quadratic Casimir for any irreducible representation $r$

$T^a T^a \phi(r) = C_2(r) \phi(r)$

as well as the Dynkin index $I$

$Tr(T^a T^b) \phi(r) = I \delta_{a b} \phi(r)$

are needed for the calculation of the RGEs. All of that is derived by SARAH due to the technique of Young tableaux. The following steps are evolved:

1. The corresponding Young tableaux fitting to the dimension $D$ is calculated using the hook formula:

$D = \Pi_i \frac{N + d_i}{h_i}$

$d_i$ is the distance of the $i.$ box to the left upper corner and $h_i$ is the hook of that box.

2. The vector for the highest weight $\Lambda$ in Dynkin basis is extracted from the tableaux.

3. The fundamental weights for the given gauge group are calculated.

4. The value of $C_2(r)$ is calculated using the Weyl formula

$C_2(r) = ( \Lambda, \Lambda + \rho).$

$\rho$ is the Weyl vector.

5. The Dynkin index $I(r)$ is calculated from $C_2(r)$. For this step, the value for the fundamental representation is normalized to $\frac{1}{2}$.

$I(r) = C_2(r) \frac{D(r)}{D(G)}$

With $D(G)$ as dimension of the adjoint representation.

6. The number of co- and contra-variant indices is extracted from the Young tableaux. With this information, the generators are written as tensor product.

The user can calculate this information independently from the model using the new command

CheckIrrepSUN[Dim,N]

Dim is the dimension of the irreducible representation and N is the dimension of the $SU(N)$ gauge group. The result is a vector containing the following information: (i) repeating the dimension of the field, (ii) number of covariant indices, (iii) number of contra-variant indices, (iv) value of the quadratic Casimir $C_2(r)$, (v) value of the Dynkin index $I(r)$, (vi) Dynkin labels for the highest weight.

Examples
1. Fundamental representation The properties of a particle, transforming under the fundamental representation of $SU(3)$ are calculated via CheckIrrepSUN[3,3]. The output is the well known result

{3, 1, 0, 4/3, 1/2, {1, 0}}
2. Adjoint representation The properties of a field transforming as 24 of $SU(5)$ are calculated by CheckIrrepSUN[24,5] . The output will be

{24, 1, 1, 5, 5, {1, 0, 0, 1}}
3. Different representations with the same dimension The 70 under $SU(5)$ is not unique. Therefore, CheckIrrepSUN[{70, {0, 0, 0, 4}}, 5] returns

{70, 0, 4, 72/5, 42, {0, 0, 0, 4}}

while CheckIrrepSUN[{70, {2, 0, 0, 1}}, 5] leads to

{70, 2, 1, 42/5, 49/2, {2, 0, 0, 1}}