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 forU(1) andS**U(N) gauge groups, but alsoS**O(N),S**p(2N) 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 likeS**U(5) orS**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 gauginoλ of a gauge groupA 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 lettersA, B to label the gauge groups and small lettera, 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 = ∂_μV_ν^A, a − ∂_νV_μ^A, a + g_Af_A^abcV_μ^A, bV_ν^A, c,

and the covariant derivative is

D_μλ_A^a = ∂_μλ_A^a + g_Af^abcA_μ^bλ^c.

Here,f_A^abc is the structure constant of the gauge groupA. 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, theD-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 scalarsi, j in the model,T_A**r^a are the generators of the gauge groupA for a irreducible representationr. For Abelian groupsT_A**r^a simplify to the chargesQ_ϕ^A of the different fields. In addition, Abelian gauge groups can come also with another feature: a Fayet-IliopoulosD-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 anyU(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 derivativesD_μ ≡ ∂_μ − i**g_AV_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 tensorsF^μ**ν of two different Abelian groupsA,B .κ is in general an × n matrix ifn 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 vectorΥ carrying all gauge fieldsV_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 particleϕ with all gauge fields can be expressed by

Θ_ϕ^TG̃**Υ

Θ_ϕ is a vector containing the chargesQ_ϕ^x ofϕ under allU(1) groupsx andG̃ is an × 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^TG**Υ

with a generaln × n matrixG 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 allU(1) groups, andg_x**y are the entries of the matrixG. Gauge-kinetic mixing is not only included in the interactions with vector bosons, but also in the derivation of theD-terms. Therefore, theD-terms for the Abelian sector in SARAH read

𝔏_D, U(1) = ∑_i**j(ϕ_i^*ϕ_i)(G^TΘ_ϕi)(G**Θ_ϕj)(ϕ_j^*ϕ_j)

while the non-AbelianD-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 bosonsV_μ^A, a. SARAH usesR_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 bosonV_μ^A with massM_A. From the gauge fixing part, the interactions of ghost fieldsη̄_A^a are derived by

𝔏_Ghost = −η̄_A^a(δℱ_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.

See also