# Generic RGE calculation

## Supersymmetric RGEs

### General

SARAH calculates the SUSY RGEs at the one- and two-loop level. In general, the*β*-function of a parameter*c* is parametrized by

`\frac{d}{dt} c \equiv \beta_c = \frac{1}{16 \pi^2} \beta^{(1)}_c + \frac{1}{(16 \pi^2)} \beta^{(2)}_c`

*β*_{c}^{(1)}, *β*_{c}^{(2)} are the coefficients at one- and two-loop level. The results used by SARAH are mainly based on Ref..

### Gauge couplings

For the gauge couplings the generic one-loop expression is rather simple and reads

*β*_{gA}^{(1)} = *g*_{A}^{3}(*S*(*R*)−3*C*(*G*))

*S*(*R*) is the Dynkin index for the gauge group summed over all chiral superfields charged under that group, and *C*(*G*) is the Casimir of the adjoint representation of the group. The two-loop expressions are more complicated and are skipped here.

### Superpotential terms

The starting point for the calculation of the RGEs for the superpotential terms in SARAH are the anomalous dimensions *γ* for all superfields. These can be also parametrized by

`\gamma_{\hat \phi_i \hat \phi_j} = \frac{1}{16\pi^2} \gamma^{(1)}_{\hat \phi_i \hat \phi_i} + \frac{1}{(16 \pi^2)^2} \gamma^{(2)}_{\hat \phi_i \hat \phi_j}`

One has to stress that*i*, *j* are not generation indices but label the different fields. Generic formula for the one- and two-loop coefficients *γ*^{(1)}, *γ*^{(2)} are given in Ref. as well. SARAH includes the case of an anomalous dimension matrix with off-diagonal entries, i.e. *ϕ̂*_{i} ≠ *ϕ̂*_{j}. That’s for instance necessary in models with vector like quarks where the superpotential reads

`W \supset Y_u \hat{u} \hat{q} \hat{H}_u + Y_U \hat{U} \hat{q} \hat{H}_u + M_U \hat{U} \hat{\bar{{U}}}`

*γ*_{û**Û} is not vanishing but receives already at one-loop contributions∝*Y*_{u}*Y*_{U}.
From the anomalous dimensions it is straightforward to get the*β*-functions of the superpotential terms: for a generic superpotential ofthe coefficients*β*^{(x)} are given by

`\beta^{(x)}_{L^i} \sim L^a \gamma^{(x)}_{a \hat \phi_i} \\ \beta^{(x)}_{M^{ij}} \sim M^{ia} \gamma^{(x)}_{a \hat \phi_j} + (j\leftrightarrow i) \\ \beta^{(x)}_{Y^{ijk}} \sim Y^{ija} \gamma^{(x)}_{a \hat \phi_k} + (k\leftrightarrow i) + (k\leftrightarrow j) \\ \beta^{(x)}_{W^{ijkl}} \sim W^{ijka} \gamma^{(x)}_{a \hat \phi_l} + (l\leftrightarrow i) + (l\leftrightarrow j) + (l\leftrightarrow k)`

up to constant coefficients.

### Soft-breaking terms

In the soft-breaking sector SARAH includes also all standard terms of the form

`- \mathfrak{L}_{SB} = t^i \phi_i + \frac{1}{2} B^{ij} \phi_i \phi_j + \frac{1}{3!} T^{ijk} \phi_i \phi_j \phi_k + \frac{1}{4!} Q^{ijkl} \phi_i \phi_j \phi_k \phi_l + \frac{1}{2} (m^2)^j_i \phi^{\* i} \phi_j - \frac{1}{2} M \lambda \lambda`

The generic expressions for *B*’s, *T*’s, *m*^{2}’s and *M*’s up to two-loop are given again in Ref. which is used by SARAH. The *β*-function for the linear soft-term *t* is calculated using Ref. . For the quartic soft-term *Q* the approach of Ref. is adopted. In this approach *γ̄* is defined by

`\bar{\gamma}^{(x)}_{\hat \phi_i \hat \phi_j} = \left(M_A g_A^2 \frac{\partial }{\partial g_A^2} - T^{lmn} \frac{\partial}{\partial Y^{lmn}} \right) \gamma^{(x)}_{\hat \phi_i \hat \phi_j}`

The *β*-functions for *Q* can then expressed by *γ* and *γ̄*:

*β*_{Qijk**l}^{(x)} = [*Q*^{ijk**a}*γ*_{a**ϕ̂l}^{(x)}+2*W*^{ijk**a}*γ̄*_{a**ϕ̂l}^{(x)}] + (*l* *i*)+(*l* *j*)+(*l* *k*)

In principle, the same approach can also be used for*B* and*T* terms as long as no gauge singlet exists in the model. Because of this restriction, SARAH uses the more general expressions.

### Fayet-Iliopoulos terms

The running of the Fayet-Iliopoulos*D*-term*ξ* receives two contributions:

`\beta_{\xi_A}^{(x)} = \frac{\beta_{g_A}^{(x)}}{g_A} \xi_A + \beta^{(x)}_{\hat \xi_A}`

The first part is already fixed by the running of the gauge coupling of the Abelian group, the second part, *β*_{ξ̂}^{(x)}, is known even to three loops . SARAH has implemented the one- and two-loop results which are rather simple:

`\beta^{(1)}_{\hat \xi_A} = 2 g_A \sum_i (Q^A_{\phi_i} m_{\phi_i \phi_i}^2) \equiv \sigma_{1,A}\\ \beta^{(1)}_{\hat \xi_A} = - 4 g_A \sum_{ij} (Q^A_{\phi_i}m^2_{\phi_i \phi_j} \gamma^{(1)}_{\hat \phi_j \hat \phi_i}) \equiv \sigma_{3,A}`

*σ*_{1} and *σ*_{3} are traces which are also used to express the *β*-functions of the soft-scalar masses at one- and two-loop, see for instance Ref. .

### Gaugino mass parameters

Finally, the*β*-functions for the gaugino mass parameters are

`\frac{d}{dt} \equiv \beta_M = \frac{1}{16\pi^2} \beta_M^{(1)} + \frac{1}{(16 \pi^2)} \beta_M^{(2)}`

where the expressions for*β*_{M}^{(x)} are also given in Ref. .*β*_{M}^{(1)} has actually a rather simple form similar to the one of the gauge couplings. One finds

*β*_{MA}^{(1)} = 2*g*_{A}^{2}(*S*(*R*)−3*C*(*G*))*M*_{A}

Therefore, the running of the gaugino masses are strongly correlated with the one of the gauge couplings. Thus, for a GUT model the hierarchy of the running gaugino masses is the same as the one for the gauge couplings.

### Dirac gauginos

The changes in the RGEs in the presence of Dirac gaugino mass terms are known today at the two-loop level, see Ref. . SARAH makes use of these results to obtain the*β*-functions for the new mass parameters as well as to include new contribution to the RGEs of tadpole terms in presence of Dirac gauginos. The*β* functions of a Dirac mass terms *m*_{D}^{ϕ̂**λ}*Ψ**λ*_{i} are related to the anomalous dimension of the involved chiral superfield*ϕ̂*, whose fermionic component is *Ψ*, and to the running of the corresponding gauge coupling:

`\beta_{m^{\hat \phi A}_D} = \gamma_{\hat \phi a} m_D^{a A} + \frac{\beta_{g_A}}{g_A} m_D^{\hat \phi A}`

The tadpole term receives two new contributions from Fayet-Iliopoulos terms discussed above and terms mimicking*B* insertions

*β*_{t, D**G}^{(x)} = *β*_{t}^{(x)} + *β*_{ξ̂}^{(x)} + *β*_{D}^{(x)}

Thus, the only missing piece is *β*_{D}^{(x)} which are now also calculated by SARAH up to two-loop.

### Vacuum expectation values

The set of SUSY RGEs is completed by using the results of Refs. to get the gauge dependence in the running of the VEVs. As consequence, the *β*-functions for the VEVs consist of two parts which are calculated independently by SARAH

*β*_{vϕ}^{(x)} = (*γ*_{ϕ**a}^{S, (x)} + *γ̂*_{ϕ**a}^{S, (x)})*v*_{a}

*γ*^{S} is the anomalous dimension of the scalar*ϕ* which receives the VEV *v*_{ϕ}. The gauge dependent parts which vanish in Landau gauge are absorbed in *γ̂*^{S}.

## Non-Supersymmetric RGEs

### General

SARAH calculates also the for a general quantum field theory at the one- and two-loop level. The parameterisation is the same as for a SUSY model:

`\frac{d}{dt} c \equiv \beta_c = \frac{1}{16 \pi^2} \beta^{(1)}_c + \frac{1}{(16 \pi^2)} \beta^{(2)}_c`

*β*_{c}^{(1)}, *β*_{c}^{(2)} are the coefficients at one- and two-loop level. The results used by SARAH are mainly based on Ref..

### Gauge couplings and potential terms

SARAH sticks very close to the generic results of Ref. to calculate the beta-functions for all gauge couplings and the parameters of the potential. Therefore, we refer to this paper for many more details.

### Vacuum expectation values

As in the SUSY case, the gauge-dependent parts of the running of the VEVs is taken from Refs. .

## Gauge kinetic mixing

The expressions presented in literature do usually not cover all possibilities and are not sufficient for any possible SUSY models which can be implemented in SARAH. Therefore, SARAH has implemented also some more results from literature which became available in the last few years. In the case of several *U*(1)’s, gauge-kinetic mixing can arise if the groups are not orthogonal. Substitution rules to translate the results to those including gauge kinetic mixing where presented in Ref. for SUSY and in Ref. for a general quantum field theory. These results have been implemented in SARAH.

### Example

In order to include gauge-kinetic mixing in the running of the gauge couplings and gaugino masses one can use the substitutions

`g_A^3 S(R) \to G \sum_{\hat \phi} V_{\hat \phi} V_{\hat \phi}^T \\ g_A^2 M_A S(R) \to M \sum_{\hat \phi} V_{\hat \phi} V_{\hat \phi}^T + \sum_{\hat \phi} V_{\hat \phi} V_{\hat \phi}^T M`

Here, *G* and *M* are matrices carrying the gauge couplings and gaugino masses of all *U*(1) groups, see also sec. [sec:supported_{m}odels], and I introduced*V*_{ϕ̂} = *G*^{T}*Q*_{ϕ̂}. The sums are running over all chiral superfields *ϕ̂*. Also for all other terms involving gauge couplings and gaugino masses appearing in the *β* functions similar rules are presented in Ref. which are used by SARAH.