# Supported gauge sectors

#### Gauge groups

The gauge sector of a SUSY model in SARAH is fixed by defining a set of vector superfields. SARAH is not restricted to three vector superfields like in the MSSM, but many more gauge groups can be defined. To improve the power in dealing with gauge groups, SARAH has linked routines from the Mathematica package Susyno. SARAH together with Susyno take care of all group-theoretical calculations: the Dynkin and Casimir invariants are calculated, and the needed representation matrices as well as Clebsch-Gordan coefficients are derived. This is not only done for*U*(1) and*S**U*(*N*) gauge groups, but also*S**O*(*N*),*S**p*(2*N*) and expectational groups can be used. For all Abelian groups also a GUT normalization can be given. This factor comes usually from considerations about the embedding of a model in a greater symmetry group like*S**U*(5) or*S**O*(10). If a GUT normalization is defined for a group, it will be used in the calculation of the RGEs. The soft-breaking terms for a gaugino*λ* of a gauge group*A* are usually included as

` \mathfrak{L}_{SB,\lambda_A} = \frac{1}{2} \lambda_A^a \lambda_A^a M_A + h.c.`

#### Gauge interactions

With the definition of the vector superfields already the self-interactions of vector bosons as well as the interactions between vector bosons and gauginos are fixed. Those are taken to be

` \mathfrak{L}= - \frac{1}{4} F^{A,a}_{\mu\nu} F^{A,\mu\nu a} - i \lambda_A^{\dagger a } \bar{\sigma}^\mu D_\mu \lambda_A^a`

I’m using here and in the following capital letters*A*, *B* to label the gauge groups and small letter*a*, *b*, *c* to label the generators, vector bosons and gauginos of a particular gauge group. The field strength tensor is defined as

*F*_{μ**ν}^{A, a} = ∂_{μ}*V*_{ν}^{A, a} − ∂_{ν}*V*_{μ}^{A, a} + *g*_{A}*f*_{A}^{abc}*V*_{μ}^{A, b}*V*_{ν}^{A, c},

and the covariant derivative is

*D*_{μ}*λ*_{A}^{a} = ∂_{μ}*λ*_{A}^{a} + *g*_{A}*f*^{abc}*A*_{μ}^{b}*λ*^{c}.

Here,*f*_{A}^{abc} is the structure constant of the gauge group*A*. Plugging eq. ([eq:FieldStrength]) in the first term of eq. ([eq:LagVS]) leads to self-interactions of three and four gauge bosons. In general, the procedure to obtain the Lagrangian from the vector and chiral superfields is very similar to Ref. . Interested readers might check this reference for more details.

#### Gauge interactions of matter fields

Vector superfields usually don’t come alone but also matter fields are present. This is discussed below. Here, it is assumed that a number of chiral superfields are present and we want to discuss the gauge interactions which are taken into account for those. First, the*D*-terms stemming from the auxiliary component of the superfield are calculated. These terms cause four scalar interactions and read

` \mathfrak{L}_{D_A} = \frac{1}{2} g^2_A \sum_{i,j} |(\phi_i^\* T_{A r}^a \phi_j)|^2`

Here, the sum is over all scalars*i*, *j* in the model,*T*_{A**r}^{a} are the generators of the gauge group*A* for a irreducible representation*r*. For Abelian groups*T*_{A**r}^{a} simplify to the charges*Q*_{ϕ}^{A} of the different fields. In addition, Abelian gauge groups can come also with another feature: a Fayet-Iliopoulos*D*-term :

\mathfrak{L}*{FI,A} = \xi_A \frac{g_A}{2} \sum*{i} (\phi_i^* Q^A_\phi \phi_i)

This term can optionally be included in SARAH for any*U*(1).
The other gauge–matter interactions are those stemming from the kinetic terms:

\mathfrak{L}*{kin} = - D^\mu \phi^{*i} D*\mu \phi_i - i \psi^{\dagger i } \bar{\sigma}^\mu D_\mu \psi_i

with covariant derivatives*D*_{μ} ≡ ∂_{μ} − *i**g*_{A}*V*_{μ}^{A, a}(*T*_{A**r}^{a}). The SUSY counterparts of these interactions are those between gauginos and matter fermions and scalars:

`\mathfrak{L}_{GFS} = - \sqrt{2} g_A (\phi_i^\* T_{Ar}^a \psi_j) \lambda_A^a + \mbox{h.c.} .`

#### Gauge kinetic mixing

The terms mentioned so far cover all gauge interactions which are possible in the MSSM. These are derived for any other SUSY model in exactly the same way. However, there is another subtlety which arises if more than one Abelian gauge group is present. In that case

` \mathfrak{L}= -\frac{1}{4} \kappa F_{\mu\nu}^A F^{B,\mu\nu} \hspace{1cm} A\neq B`

are allowed for field strength tensors*F*^{μ**ν} of two different Abelian groups*A*,*B* .*κ* is in general a*n* × *n* matrix if*n* Abelian groups are present. SARAH fully includes the effect of kinetic mixing independent of the number of Abelian groups. For this purpose SARAH is not working with field strength interactions like eq. ([eq:off]) but performs a rotation to bring the field strength in a diagonal form. That’s done by a redefinition of the vector*Υ* carrying all gauge fields*V*_{X}^{μ}:

\Upsilon \to \sqrt{\kappa} \Upsilon

This rotation has an impact on the interactions of the gauge bosons with matter fields. In general, the interaction of a particle*ϕ* with all gauge fields can be expressed by

*Θ*_{ϕ}^{T}*G̃**Υ*

*Θ*_{ϕ} is a vector containing the charges*Q*_{ϕ}^{x} of*ϕ* under all*U*(1) groups*x* and*G̃* is a*n* × *n* diagonal matrix carrying the gauge couplings of the different groups. After the rotation according to eq. ([eq:kappaRot]) the interaction part can be expressed by

*Θ*_{ϕ}^{T}*G**Υ*

with a general*n* × *n* matrix*G* which is no longer diagonal. In that way, the effect of gauge kinetic mixing has been absorbed in ’off-diagonal’ gauge couplings. That means the covariant derivative in SARAH reads

`D_\mu \phi = \left(\partial_\mu - i \sum_{x,y} Q_\phi^x g_{xy} V^\mu_y \right)\phi \hspace{1cm}`

*x*, *y* are running over all*U*(1) groups, and*g*_{x**y} are the entries of the matrix*G*. Gauge-kinetic mixing is not only included in the interactions with vector bosons, but also in the derivation of the*D*-terms. Therefore, the*D*-terms for the Abelian sector in SARAH read

𝔏_{D, U(1)} = ∑_{i**j}(*ϕ*_{i}^{*}*ϕ*_{i})(*G*^{T}*Θ*_{ϕi})(*G**Θ*_{ϕj})(*ϕ*_{j}^{*}*ϕ*_{j})

while the non-Abelian*D*-terms keep the standard form eq. ([eq:Dterms]). Finally, also ’off-diagonal’ gaugino masses are introduced. The soft-breaking part of the Lagrangian reads then

` \mathfrak{L}_{SB,\lambda,U(1)} \supset \sum_{xy} \frac{1}{2} \lambda_x \lambda_y M_{xy} + h.c.`

SARAH takes the off-diagonal gaugino masses to be symmetric
*M*_{x**y} = *M*_{y**x}
.

### Gauge fixing sector

All terms written down so far lead to a Lagrangian which is invariant under a general gauge transformation. To break this invariance one can add ’gauge fixing’ terms to the Lagrangian. The general form of these terms is

`\mathfrak{L}_{GF} = - \frac{1}{2} |\mathscr{F}_A^a|^2 .`

Here,ℱ_{A}^{a} is usually a function involving partial derivatives of gauge bosons*V*_{μ}^{A, a}. SARAH uses*R*_{ξ} gauge. That means that for an unbroken gauge symmetry, the gauge fixing terms are

`\mathfrak{L}_{GF} = - \frac{1}{2 R_{\xi_A}} \left|\partial^\mu V_\mu^{A,a} \right|^2 .`

For broken symmetries, the gauge fixings terms are chosen in a way that the mixing terms between vector bosons and scalars disappears from the Lagrangian. This generates usually terms of the form

` \mathfrak{L}_{GF, R_\xi} = - \frac{1}{2 R_{\xi_A}} \left| \partial^\mu V^{A}_\mu + R_{\xi_{A}} M_A G^A \right|^2`

Here,*G*^{A} is the Goldstone boson of the vector boson*V*_{μ}^{A} with mass*M*_{A}. From the gauge fixing part, the interactions of ghost fields*η̄*_{A}^{a} are derived by

𝔏_{Ghost} = −*η̄*_{A}^{a}(*δ*ℱ_{A}^{a}).

Here,*δ* assigns the operator for a BRST transformation. All steps to get the gauge fixing parts and the ghost interactions are completely done automatically by SARAH and adjusted to the gauge groups in the model.