...  ...  @@ 7,44 +7,80 @@ Supersymmetric RGEs 





### General






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



SARAH calculates the SUSY RGEs at the one and twoloop level. In general, the $`\beta`$function of a parameter $`c`$ is parametrized by






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



```math



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



```






*β*<sub>*c*</sub><sup>(1)</sup>, *β*<sub>*c*</sub><sup>(2)</sup> are the coefficients at one and twoloop level. The results used by SARAH are mainly based on Ref..



$`\beta^{(1)}_c`$ and $`\beta^{(2)}_c`$ are the coefficients at one and twoloop level.



The results used by SARAH are mainly based on [hepph/9311340](https://arxiv.org/abs/hepph/9311340) and the review/corrections (that include wavefunctionren. mixing) in [1809.06797](https://arxiv.org/abs/1809.06797).






### Gauge couplings






For the gauge couplings the generic oneloop expression is rather simple and reads






*β*<sub>*g*<sub>*A*</sub></sub><sup>(1)</sup> = *g*<sub>*A*</sub><sup>3</sup>(*S*(*R*)−3*C*(*G*))



```math



\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.






*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 twoloop expressions are more complicated and are skipped here.



The twoloop correction is



```math



\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 in SARAH are the anomalous dimensions *γ* for all superfields. These can be also parametrized by



The starting point for the calculation of the RGEs for the superpotential terms



```math



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_k



```






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



in SARAH are the anomalous dimensions $`\gamma`$ for all superfields. These can be also parametrized by






One has to stress that*i*, *j* are not generation indices but label the different fields. Generic formula for the one and twoloop coefficients *γ*<sup>(1)</sup>, *γ*<sup>(2)</sup> are given in Ref. as well. SARAH includes the case of an anomalous dimension matrix with offdiagonal entries, i.e. *ϕ̂*<sub>*i*</sub> ≠ *ϕ̂*<sub>*j*</sub>. That’s for instance necessary in models with vector like quarks where the superpotential reads



```math



\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}



```






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



One has to stress that $`i,j`$ are not generation indices but label the different fields. Generic formula for the one and twoloop coefficients $`\gamma^{(1)}`$, $`\gamma^{(2)}`$ are given in [1809.06797](https://arxiv.org/pdf/1809.06797.pdf) and references therein.



SARAH includes the case of an anomalous dimension matrix with offdiagonal entries.



That’s for instance necessary in models with vector like quarks where the superpotential reads






*γ*<sub>*û**Û*</sub> is not vanishing but receives already at oneloop contributions∝*Y*<sub>*u*</sub>*Y*<sub>*U*</sub>.



From the anomalous dimensions it is straightforward to get the*β*functions of the superpotential terms: for a generic superpotential ofthe coefficients*β*<sup>(*x*)</sup> are given by



```math



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}}}



```






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



$`\gamma_{u U}`$ is not vanishing but receives already at oneloop 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






up to constant coefficients.



```math



\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).






### Softbreaking terms






In the softbreaking sector SARAH includes also all standard terms of the form






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



```math



 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 twoloop are given again in the ref. above.









The generic expressions for *B*’s, *T*’s, *m*<sup>2</sup>’s and *M*’s up to twoloop are given again in Ref. which is used by SARAH. The *β*function for the linear softterm *t* is calculated using Ref. . For the quartic softterm *Q* the approach of Ref. is adopted. In this approach *γ̄* is defined by



<details>



<summary markdown="span">Note (needs update of Refs):</summary>






The $`\beta`$function for the linear softterm *t* is calculated using Ref. . For the quartic softterm *Q* the approach of Ref. is adopted. In this approach *γ̄* is defined by






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




...  ...  @@ 53,50 +89,71 @@ The *β*functions for *Q* can then expressed by *γ* and *γ̄*: 


*β*<sub>*Q*<sub>*i**j**k**l*</sub></sub><sup>(*x*)</sup> = \[*Q*<sup>*i**j**k**a*</sup>*γ*<sub>*a**ϕ̂*<sub>*l*</sub></sub><sup>(*x*)</sup>+2*W*<sup>*i**j**k**a*</sup>*γ̄*<sub>*a**ϕ̂*<sub>*l*</sub></sub><sup>(*x*)</sup>\] + (*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.



</details>






### FayetIliopoulos terms






The running of the FayetIliopoulos*D*term*ξ* receives two contributions:



The running of the FayetIliopoulos $`D`$term $`\xi`$ receives two contributions:






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



```math



\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*<sub>*ξ̂*</sub><sup>(*x*)</sup>, is known even to three loops . SARAH has implemented the one and twoloop results which are rather simple:



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 twoloop results which are rather simple:






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



```math



\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}



```






*σ*<sub>1</sub> and *σ*<sub>3</sub> are traces which are also used to express the *β*functions of the softscalar masses at one and twoloop, see for instance Ref. .



$`\simga_{1,2}`$ are traces which are also used to express the $`β`$functions of the softscalar masses at one and twoloop.






### Gaugino mass parameters






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



Finally, the $`\beta`$functions for the gaugino mass parameters are






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



```math



\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*<sub>*M*</sub><sup>(*x*)</sup> are also given in Ref. .*β*<sub>*M*</sub><sup>(1)</sup> has actually a rather simple form similar to the one of the gauge couplings. One finds



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






*β*<sub>*M*<sub>*A*</sub></sub><sup>(1)</sup> = 2*g*<sub>*A*</sub><sup>2</sup>(*S*(*R*)−3*C*(*G*))*M*<sub>*A*</sub>



```math



\beta^{(1)}_{M_A} = 2 g_A^2 \left( S(R)3C(G)\right)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 twoloop 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*<sub>*D*</sub><sup>*ϕ̂**λ*</sup>*Ψ**λ*<sub>*i*</sub> 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:



The changes in the RGEs in the presence of Dirac gaugino mass terms are known today at the twoloop level, see [1206.6697](https://arxiv.org/abs/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:






```math



\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}



```






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



The tadpole term receives two new contributions from FayetIliopoulos terms discussed above and terms mimicking $`B`$insertions






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



```math



\beta_t^{DG}(x) = \beta_t(x) + \beta_{\hat{\xi}}(x) + \beta_D(x)






*β*<sub>*t*, *D**G*</sub><sup>(*x*)</sup> = *β*<sub>*t*</sub><sup>(*x*)</sup> + *β*<sub>*ξ̂*</sub><sup>(*x*)</sup> + *β*<sub>*D*</sub><sup>(*x*)</sup>



```






Thus, the only missing piece is *β*<sub>*D*</sub><sup>(*x*)</sup> which are now also calculated by SARAH up to twoloop.



Thus, the only missing piece is $`\beta_D(x)`$ which is also calculated by SARAH up to twoloop.






### 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



The set of SUSY RGEs is completed by using the results of e.g. [1310.7629](https://arxiv.org/abs/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






*β*<sub>*v*<sub>*ϕ*</sub></sub><sup>(*x*)</sup> = (*γ*<sub>*ϕ**a*</sub><sup>*S*, (*x*)</sup> + *γ̂*<sub>*ϕ**a*</sub><sup>*S*, (*x*)</sup>)*v*<sub>*a*</sub>



```math



\beta_{v_\phi^a}(x) = v_\phi^b\left( \gamma_{\phi_a \phi_b} + \hat{\gamma}_{\phi_a \phi_b} \right)



```






*γ*<sup>*S*</sup> is the anomalous dimension of the scalar*ϕ* which receives the VEV *v*<sub>*ϕ*</sub>. The gauge dependent parts which vanish in Landau gauge are absorbed in *γ̂*<sup>*S*</sup>.



$`\gamma_\phi`$ 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 $`\hat{\gamma}_\phi`$.






NonSupersymmetric RGEs





...  ...  @@ 105,30 +162,33 @@ NonSupersymmetric RGEs 





SARAH calculates also the for a general quantum field theory at the one and twoloop 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*<sub>*c*</sub><sup>(1)</sup>, *β*<sub>*c*</sub><sup>(2)</sup> are the coefficients at one and twoloop level. The results used by SARAH are mainly based on Ref..



```math



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



```



The implementation in SARAH was published in [1309.7223](https://arxiv.org/abs/1309.7223). For details on the calculation of nonSUSY RGEs see the references therein.






### Gauge couplings and potential terms






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



SARAH sticks very close to the generic results of M. E. Machacek and M. T. Vaughn to calculate the betafunctions for all gauge couplings and the parameters of the potential. Therefore, we refer to these papers for many more details: [10.1016/05503213(83)906107](https://doi.org/10.1016/05503213(83)906107), [10.1016/05503213(85)900409](https://doi.org/10.1016/05503213(85)900409) and [10.1016/05503213(84)905339](https://doi.org/10.1016/05503213(84)905339).






### Vacuum expectation values






As in the SUSY case, the gaugedependent parts of the running of the VEVs is taken from Refs. .



As in the SUSY case, the gaugedependent 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, gaugekinetic 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.



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, gaugekinetic 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](https://arxiv.org/abs/1107.2670). These results have been implemented in SARAH.






### Example






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






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



```math



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. \[sec:supported<sub>m</sub>odels\], and I introduced*V*<sub>*ϕ̂*</sub> = *G*<sup>*T*</sup>*Q*<sub>*ϕ̂*</sub>. 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.



Here, $`G`$ and $`M`$ are matrices carrying the gauge couplings and gaugino masses of all $`U(1)`$ groups (see also [supported gauge sectors](/Supported_gauge_sectors "wikilink")) 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.






See also





...  ...  