Generic RGE calculation
Supersymmetric RGEs
General
SARAH calculates the SUSY RGEs at the one- and two-loop level. In general, the \beta-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\beta^{(1)}_c and \beta^{(2)}_c are the coefficients at one- and two-loop level.
The results used by SARAH are mainly based on hep-ph/9311340 and have been implemented in 1002.0840.
Gauge couplings
For the gauge couplings the generic one-loop expression is rather simple and reads
\beta_{g}^{(1)}= g^3\left(S(R)-3 C(G) \right)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 correction is
\beta_g^{(2)} = g^5\left[ -6 \left[C(G)\right]^2 + 2 C(G)S(R) + 4 S(R)C(R)\right] - g^3 Y_{ijk}Y^{ijk} C(G)/D(G)where Y_{ijk} are trilinear Superpotential couplings defined below and D(G) is the dimension of the adjoint.
Superpotential terms
The starting point for the calculation of the RGEs for the superpotential terms
W(\phi) = L_i \phi_i + \frac{1}{2} \mu_{ij} \phi_i \phi_j + \frac{1}{6} Y_{ijk} \phi_i \phi_j \phi_kin SARAH are the anomalous dimensions \gamma 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 \gamma^{(1)}, \gamma^{(2)} are given in 1002.0840 and references therein.
SARAH includes the case of an anomalous dimension matrix with off-diagonal entries.
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{\overline{{U}}}\gamma_{u U} is not vanishing but receives already at one-loop contributions \propto Y_u Y_U.
From the anomalous dimensions it is straightforward to get the \beta-functions of the superpotential terms: for a generic superpotential the coefficients are given by
\begin{aligned}
\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)
\end{aligned}up to constant coefficients (for exact equations see the refs. above).
Soft-breaking terms
In the soft-breaking sector SARAH includes also all standard terms of the form
- 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 \lambdaThe generic expressions for B’s, T’s, m^2’s and M’s up to two-loop are given again in the ref. above.
Note (needs update of Refs):
The \beta-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:
βQijk**l(x) = [Qijk**aγa**ϕ̂l(x)+2Wijk**aγ̄a**ϕ̂l(x)] + (l
In principle, the same approach can also be used forB andT 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 \xi 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, \beta_{\hat{\xi}}, is known even to three loops . SARAH has implemented the one- and two-loop results which are rather simple:
\begin{aligned}
\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}\simga_{1,2} are traces which are also used to express the β-functions of the soft-scalar masses at one- and two-loop.
Gaugino mass parameters
Finally, the \beta-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 \beta_M (x) are also given in the standard literature.
\beta^{(1)}_M has actually a rather simple form similar to the one of the gauge couplings. One finds
\beta^{(1)}_{M_A} = 2 g_A^2 \left( S(R)-3C(G)\right)M_ATherefore, 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 1206.6697.
SARAH makes use of these results to obtain the \beta-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 \beta functions of a Dirac mass terms m_D^{\hat{\phi}\lambda} \Psi \lambda_i are related to the anomalous dimension of the involved chiral superfield \phi, whose fermionic component is \Psi, 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
\beta_t^{DG}(x) = \beta_t(x) + \beta_{\hat{\xi}}(x) + \beta_D(x)
Thus, the only missing piece is \beta_D(x) which is also calculated by SARAH up to two-loop.
Vacuum expectation values
The set of SUSY RGEs is completed by using the results of e.g. 1310.7629 to get the gauge dependence in the running of the VEVs.
As consequence, the \beta-functions for the VEVs consist of two parts which are calculated independently by SARAH
\beta_{v_\phi^a}(x) = v_\phi^b\left( \gamma_{\phi_a \phi_b} + \hat{\gamma}_{\phi_a \phi_b} \right)\gamma_\phi is the anomalous dimension of the scalar \phi which receives the VEV v_\phi. The gauge dependent parts which vanish in Landau gauge are absorbed in \hat{\gamma}_\phi.
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)}_cThe implementation in SARAH was published in 1309.7223. For details on the calculation of non-SUSY RGEs see the references therein.
Gauge couplings and potential terms
SARAH sticks very close to the generic results of M. E. Machacek and M. T. Vaughn to calculate the beta-functions for all gauge couplings and the parameters of the potential. Therefore, we refer to these papers for many more details: 10.1016/0550-3213(83)90610-7, 10.1016/0550-3213(85)90040-9 and 10.1016/0550-3213(84)90533-9 and the review/corrections (that include wave-function-ren. mixing in non-SUSY case) in 1809.06797.
Vacuum expectation values
As in the SUSY case, the gauge-dependent parts of the running of the VEVs is taken into account.
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 1107.2670. 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 MHere, G and M are matrices carrying the gauge couplings and gaugino masses of all U(1) groups (see also supported gauge sectors) and introduced V_{\hat{\phi}} = G^T Q_{\hat{\phi}}. The sums are running over all chiral superfields \hat{\phi}. Also for all other terms involving gauge couplings and gaugino masses appearing in the \beta functions similar rules are presented in Ref. which are used by SARAH.