Improved predictions for intermediate and heavy Supersymmetry in the MSSM and beyond

For a long time, the minimal supersymmetric standard model (MSSM) with light masses for the supersymmetric states was considered as the most natural extension of the Standard Model of particle physics. Consequently, a valid approximation was to match the MSSM to the precision measurement directly at the electroweak scale. This approach was also utilized by all dedicated spectrum generators for the MSSM. However, the higher the supersymmetric (SUSY) scale is, the bigger the uncertainties which are introduced by this matching. We point out important consequences of a two-scale matching, where the running parameters within the SM are calculated at $M_Z$ and evaluated up to the SUSY scale, where they are matched to the full model. We show the impact on gauge coupling unification as well as the SUSY mass spectrum. Also the Higgs mass prediction for large supersymmetric masses has been improved by performing the calculation within an effective SM. The approach presented here is now also available in the spectrum generator SPheno. Moreover, also SARAH was extended accordingly and gives the possibility to study these effects now in many different supersymmetric models.


Introduction
The discovery of the Higgs with a mass of about 125 GeV [1,2] is, to date, the biggest success of the large hadron collider (LHC). In contrast, there has not been any evidence for new physics. This puts very strong constraints on the masses of new coloured particles as predicted, for instance, by supersymmetry (SUSY); working with very simplified assumptions, it is possible to exclude gluinos and first/second generation squarks nearly up to 2 TeV [3 -6]. These experimental results raise not only the question if minimal supersymmetry is still a good solution to the fine-tuning or hierarchy problem of the standard model of particle physics (SM), but also gives new challenges to study the MSSM precisely.
In the past many studies for the MSSM were done under the impression that the scale of supersymmetry, M SUSY , should be close to the electroweak scale M Z . Under this assumption it was possible to calculate the gauge couplings in the DR scheme directly from m Z , G F and αem as well as the DR Yukawa couplings from the pole-mass of the top-quark and the running MS lepton and light quark masses given at Q = m Z .
More importantly the Higgs mass(es) has been calculated at fixed order in the full supersymmetric model. However, both calculations became less accurate the larger M SUSY is because potentially large logarithms of form log MSUSY M Z and log MSUSY m h , respectively, appear. Therefore, there are ongoing efforts to improve the calculation in the presence of supersymmetric scales which are well above the electroweak one. The first road is to keep the current set-up in principle but improve it by higher order corrections: for instance, SoftSUSY provides the possibility to include higher order corrections to the threshold corrections at the weak scale and in the renormalisation group equation (RGE) running between the weak and SUSY scale, in order to get a better determination of the DR parameters at the SUSY scale [7]. The first ansatz is to calculate the Higgs mass still in the full MSSM but extends the two-loop fixed order calculation by a resummation of potential large logarithm involving stops. That's for instance done by FeynHiggs since a few years [8][9][10]. The second approach, which becomes more and more popular, is to work in an effective theory below M SUSY : SusyHD [11] and recent versions of FlexibleSUSY [12] as well as FeynHiggs [13] can consider below M SUSY only the degrees of freedom of the SM, and match the SM to the MSSM just at the SUSY scale. Also the Higgs mass calculation is done in the effective SM by obtaining a value of the quartic Higgs coupling λ SM from the matching between the MSSM and SM at M SUSY . The idea to work in an effective SM below M SUSY was already well explored in literature before it became easily available via public tools, see e.g. Refs. [14][15][16][17][18][19][20]. Similarly, also a general Two-Higgs-Doublet-Model was already considered as low energy limit of the MSSM [21][22][23]. Finally, since several years Split-SUSY variants of the MSSM become more and more popular in which the coloured SUSY particles are integrated out [15][16][17][24][25][26].
We have now also extended the stand-alone spectrum generator SPheno [27,28] as well as the Mathematica package SARAH [29][30][31][32][33][34], which gives the possibility to auto-generate a spectrum generator for a given model, to improve the predictions for moderate and heavy SUSY scales. Here, we made use of the second approach: the running DR parameters at the SUSY scale are obtained via a two-scale matching procedure and the Higgs mass calculation can optionally be done within an effective SM. We give in the following not only details of our exact approach but discuss also phenomenological consequences of the improved calculations. We focus not only on the Higgs mass prediction, which has been already discussed to some extent in the recent year, but show also potential important effects on the SUSY mass spectrum. Beside the MSSM we consider also it is minimal extension, the NMSSM. This paper is organised as follows: in sec. 2 we summarize our approach to obtain the DR parameters at the SUSY scale as well as to calculate the mass of the SM-like Higgs. Many details for the matching are given in appendix A, where also the differences between stand-alone SPheno and the SARAH generated version are discussed. In sec. 3 we discuss the numerical impact of the improved calculation on the running parameters, but also on the SUSY and Higgs masses in the MSSM and beyond. We conclude in sec. 4.
2 Matching procedure and effective Higgs mass calculation

The two-scale matching in SARAH
So far, all dedicated MSSM spectrum generators such as SoftSUSY [7,[35][36][37], Suspect [38] or SPheno were adapting the procedure of Ref. [39] to obtain the running gauge and Yukawa couplings at the SUSY scale. All details of the calculations are summarised in Appendix A.1. The principle idea is that all measured SM parameters are already translated at M Z into DR values taking into account the complete MSSM spectrum which are then evaluated to the SUSY scale by using the RGEs of the MSSM. This procedure suffers from increasing uncertainties when the separation of the electroweak and SUSY scale becomes large. In order to reduce the theoretical uncertainty for large SUSY scales, SoftSUSY is able since some time to include the two-loop SUSY thresholds in the calculation of the DR parameters and to perform a three-loop RGE running between M Z and M SUSY . With these additional corrections, potential large effects in the prediction of the Higgsino mass parameter but also for the Higgs mass were found. The drawback of this ansatz is that it is computational very expensive and slows down the evaluation of a given parameter point significantly. Moreover, only the effects of a more precise determination of the top Yukawa coupling on the Higgs mass are caught in this approach up to some extent, while still potential large logarithm in the fixed order Higgs mass calculation can be present. Therefore, we are using another ansatz in SPheno and SARAH 1 which is closer to the setup of NMSSMCalc [40] or specific versions of FlexibleSUSY [12,41]: the matching at the electroweak scale includes only SM thresholds to obtain the MS values of the gauge and Yukawa couplings and the electroweak vacuum expectation value (VEV). These parameters are then evolved up to the SUSY scale using SM RGEs, and the translation from MS to DR scheme and the inclusion of SUSY thresholds is done at the SUSY scale. All details of the calculation are given in Appendix A. The precision to obtain the DR parameters at the SUSY scale via this two-scale matching (2SM) is as follows in SARAH/SPheno 1. The MS parameters at the weak scale are calculated using: -One-loop electroweak corrections to the fermion masses -Two-loop QCD corrections to the top quark mass -One-loop corrections to δ V B as well as one-and two-loop corrections to δρ 2. The SM RGEs are available up to three-loop 3. The MS-DR conversion of the running fermion masses is done at two-loop αs and at one-loop in case of the electroweak gauge couplings Higgs pole mass calculation: oneand two-loop corrections included The DR parameters obtained in that way are then used to calculate the SUSY and Higgs masses at M SUSY . Since both, the matching at the M Z and M SUSY depends on these masses, one needs to iterate the matching procedure. For this reason it is necessary to calculate the quartic self-coupling λ SM (M SUSY ) within the SM which is a function of the SUSY masses and parameters. A handy and very general ansatz to obtain λ SM (M SUSY ) was presented in Ref. [12]: one can match the Higgs pole masses in the full MSSM and the SM at the SUSY scale m SM,pole from what one can derive λ SM as Here, Π h (M SUSY ) are the radiative corrections to the Higgs mass within the SM which are calculated using MS parameters at this scale, while the pole mass calculation in the MSSM involves DR parameters. The equivalence of this ansatz to the matching of four point function as for instance performed in Refs. [19,20] and used also by SUSYHD has been explicitly shown in Ref. [12]. SM RGEs are used are afterwards to run λ SM to M Z , and the MS parameters are recalculated at this scale. This procedure is iterated until the mass spectrum at the SUSY scale has converged.

Differences between SARAH and SPheno in the new matching routines
The above procedure corresponds to the details in SARAH whereas the procedure implemented in the stand-alone SPheno differs in the following details: at Q = m t : the top Yukawa coupling is optionally replaced by the fit formula given by eq. (57) of Ref. [42] at Q = m t : the strong coupling g 3 is optionally replaced by the fit formula given by eq. (60) of Ref. [42] at Q = M SU SY : the thresholds corrections to the gauge and Yukawa interactions are calculated in the electroweak basis assuming an unbroken SU (2) L × U (1) Y . The full formulae are given in AppendixA.
The flags to use/not use the fit formulae of Ref. [42] are given in Appendix A.2. If not indicated otherwise, these fit formulae are used in the following comparisons.

The effective Higgs mass calculation
So far, the mass calculation with SPheno/SARAH would have stopped after the conversion of the mass spectrum at M SUSY . However, this could lead to a large theoretical uncertainty in the Higgs mass prediction for large SUSY masses: the fixed order Higgs mass calculation as performed by SPheno/SARAH would become inaccurate because of the appearance of large logarithms ∼ log(M SUSY /Mew). In order to cure this, one could do a resummation of these large logs. However, in our setup it is much easier to use the value λ SM (M SUSY ), which is already known, and run it to the top mass scale. By this running all large logarithms get re-summed and one can then calculate m h at m t within the SM including radiative corrections. In SARAH/SPheno we include the full SM one-loop corrections as well as the two-loop corrections O(α t (αsα t )) to m h . The schematic procedure for the matching and Higgs mass calculation is summarized in Fig. 1.  In addition, we show the results for SoftSUSY using one-loop (dashed orange) and two-loop SUSY thresholds (full brown), as well as for FlexibleSUSY (green). On the right we give the difference between the results of two calculations as indicated.
All the efforts to disentangle the weak and the SUSY scale in the matching are done to get more accurate values of the running DR parameters at the SUSY scale. Therefore, we want to start the discussion of the impact of the new matching procedure with presenting the changes in the DR parameters at the SUSY scale. The results for the top and bottom Yukawa couplings are shown in Fig. 2 and those for the three gauge couplings g 1 , g 2 and g 3 are depicted in Fig. 3. Since the exact matching procedure using two scales is slightly different between SPheno and SARAH as explained in sec. 2.2 we show the new results for both codes. Since we have turned off here the fit formula of Ref. [42] in the SPheno calculation, the remaining differences appearing here are due to the threshold corrections of the gauge and Yukawa couplings at M SUSY . One sees that in particular the top Yukawa coupling changes significantly compared to the older calculation with only one matching scale (1SM). For M SUSY = 100 TeV, the calculated DR value with SARAH using the two-scale matching is nearly 10% below the one for the one-scale matching. These large changes are in agreement with the results of Ref. [12] where the impact of a 2SM on the top Yukawa coupling has also been analysed analytically. We show for comparison also the calculated couplings in SoftSUSY with and without two-loop SUSY thresholds and three loop RGEs. It is obvious that there was a non-negligible difference between the old results and the one-loop results of SoftSUSY although both calculations were of the same order in perturbation theory. The reason are the matching conditions which can schematically be written as where all loop corrections are summarised in Σ. SPheno usesm t = m DR t while SoftSUSY and other codes set The result obtained with the new two-scale matching agrees now rather well with the SoftSUSY results once the two-loop SUSY corrections in the matching are included up to several TeV. However, for even higher SUSY scales one finds that even the SUSY calculation with two-loop thresholds gives sizeable differences to the RGE re-summed calculation. On the other side, we find an excellent agreement with FlexibleSUSY which performs also a two-scale matching, but uses a different matching procedure at the SUSY scale 2 . A similar, but less pronounced effect can be seen for the bottom Yukawa coupling. Here, the changes between the one and two-scale matching account for a shift of about 6% for a SUSY scale of 100 TeV.
For the gauge couplings, the difference between the one and two-scale matching are in general much smaller than for the Yukawa couplings. The changes are usually well below 1 % even for a SUSY scale of 100 TeV. The only exception is SoftSUSY when turning on the two-loop thresholds to the strong coupling. In that case a significant decrease in g 3 with increasing M SU SY is seen. This effect is not confirmed by the RGE re-summed calculations.

Gauge coupling unification
The shifts in the gauge couplings are rather small even for very large SUSY masses in the multi TeV range. Thus, they play phenomenologically only a sub-dominant role compared to the larger effects in the top Yukawa coupling. However, if one embeds the MSSM into a UV complete framework like supergravity, the running gauge couplings g DR 1 and g DR 2 are usually used as starting point to find the scale of grand unification, M GUT by imposing the condition Also the goodness of complete unification, i.e. the remaining difference between g 3 (M GUT ) compared to the other two couplings is very sensitive to the values of g DR  2 We have adapted the approach of Ref. [39] to a two scale approach: we calculate the DR gauge and Yukawa couplings from the running MS values of αew, sin Θ W , g 3 as well as from the running fermion masses and CKM matrix at the SUSY scale. The calculation is similar to the corresponding matching of the measured values of these parameters to the DR parameters at M Z done before. All details are given in appendix A.2. In contrast, FlexibleSUSY demands the equality of pole masses in the SM and MSSM at the SUSY scale to get the matching conditions for the SM gauge and Yukawa couplings.    Fig. 4. The predicted value for the GUT scale as function of M SUSY changes only slightly when using the new two-scale matching compared to the one-scale matching. In a complete GUT-model, the difference ∆g has to be explained by threshold corrections to heavy GUT-scale particles [43,44] as we are using two-loop RGE running . Therefore, the right plot of this figure indicates the possible size of such corrections due to the GUT-scale spectrum. The prediction for ∆g is different comparing the one-and two-scale matching, but also comparing the new results of SARAH and SPheno. The dominant origin of this difference is the inclusion of the two-loop correction to g 3 in SPheno, i.e. the difference between the two lines can be taken as an estimate for the theoretical uncertainty in ∆g coming from higher order effects: only two-loop SM corrections in the matching of g 3 are included in SPheno, but not the two-loop SUSY thresholds. Also, for consistency three-loop RGEs of g 3 up to M GUT would be necessary. However, for small m 0 also the terms O(v 2 /M 2 SUSY ), which are neglected in SPheno by computing the thresholds in the SU (2) L × U (1) Y limit become important and introduce a difference in the prediction of the GUT scale, which enters logarithmically in the unification condition.

SUSY masses
The changes in the DR parameters at the SUSY scale influence also the mass spectrum. This has very important consequences in particular on the Higgs mass which are discussed in the dedicated section sec. 3.4. For now, we concentrate on the SUSY masses. In that case, the masses do hardly change if all SUSY specific parameters are defined at the SUSY scale because only tiny changes in the F -and D-term contributions as well as in the radiative corrections will appear. Those are found to be hardly in the percent range even for large SUSY scales. Larger effects are present, if on considers unified scenarios in which the SUSY parameters are set via boundary conditions at a scale well above the SUSY scale. The additional RGE running between the high scale, which is often associated with the GUT scale via eq. 4, will then introduce a larger dependence on DR values of SM gauge and Yukawa couplings at M SUSY . As example, we consider again the CMSSM. For simplicity, we fix in the following, if not stated otherwise, A 0 = 0, µ > 0, tan β = 10 and perform a scan over m 0 and For the other masses, the changes in the DR parameters account only for moderate changes of 1% and below. The only exception are fine-tuned region with a Higgsino LSP which we discuss below in more detail. Here, we also display the changes in the bino LSP mass because there small shifts can have sizeable effects in the calculation of the relic density, e.g. in case of Higgs resonances or in case of co-annihiliation. The impact of the DR parameters at M SUSY on the prediction of the light stop mass depends also on the chosen value for A 0 . For non-vanishing A 0 , the changes can become larger as shown in Fig. 6. Setting A 0 = +1.5m 0 we find that the stop mass changes by more than 5% for m 0 > 4 TeV. These changes are still very moderate and have hardly any phenomenological impact at the LHC. However, as mentioned above they can become important for instance in stau or stop co-annihilation to explain the dark matter abundance in the universe [45].
A much more pronounced effect can be observed for the µ parameter in the so called 'Focus-Point'-region [46][47][48][49] from the minimisation conditions of the potential. This result at tree-level in where we have assumed in the last step tan β 1. The special feature of the focus point region is that cancellations in the RGE contributions to m 2 Hu result in moderately small µ which is much smaller than the other SUSY mass parameters. How well these cancellation work depends strongly on the value of the top Yukawa coupling. Hence, we find that in the focus point region, which is usually needs moderate M 1/2 and large m 0 , the value of µ changes by more than 25% as shown in Fig. 7. Thus, also the Higgsino masses vary significantly between the one and two-scale matching calculation.
If one assumes that a large µ-parameter is the main source of fine-tuning in the MSSM, these changes in µ have also an impact on naturalness considerations. Using the approximate formula ∆ the fine-tuning 3 , on sees that the fine-tuning prediction could reduce a factor of 2 and more in the focus point region when going from the one-scale matching to the two-scale matching.

Higgs mass in the MSSM
The impact of heavy SUSY masses on the Higgs mass is nowadays a widely discussed topic. While fixed order calculations suffer from increasing uncertainties, there are two methods to improve the accuracy: (i) resumming the stop contributions as done by FeynHiggs; (ii) working with a EFT ansatz as first done by SusyHD  and later incorporated in FlexibleSUSY as well. The pole mass matching described in sec. 2, which was used so far only in FlexibleSUSY and now also by SPheno/SARAH, has the additional advantage that it includes terms O(v 2 /M 2 SUSY ). This is in contrast to previous calculations to obtain λ SM from the effective potential which are used by SusyHD for instance. Thus, these EFT calculation have a larger uncertainty for not too large M SUSY , while the predictions using a pole mass matching are still reliable for M SUSY of 1 TeV and even below.
We give a comparison of the Higgs mass prediction of the new SARAH and SPheno versions against previous calculations as well as the current versions of FeynHiggs For simplicity, we assume a degeneracy of the SUSY soft masses as well as M A and µ at the SUSY scale: We neglect all trilinear soft-terms but the one involving the stops which is parametrised as usual by The results for the Higgs mass prediction for A t = 0, ±M SUSY and M SUSY up to 100 TeV are summarised in Figs. 8 -10. One can see in Fig. 8   Since the agreement between the different codes becomes impressively good even for very large SUSY masses, we give in Fig. 9 the numerical differences between the Higgs mass predictions of SARAH compared to the other codes. Also the difference between the one-scale matching and the two-scale matching using a one-or two-loop calculation of λ is shown: for M SUSY = 100 TeV the Higgs mass prediction decreases by about 7 GeV when doing it via the EFT approach. The remaining differences to SusyHD and FlexibleSUSY is always better than 1 GeV, most often even better than 0.5 GeV 6 . The increasing difference between SARAH and FlexibleSUSY compared to SPheno and SusyHD comes from the calculation of the top Yukawa coupling in the SM: while SARAH and FlexibleSUSY use two-loop thresholds, SPheno and SusyHD have included even higher order corrections via the fit formula of Ref. [42]. These correction need not to be included because they are of a higher loop level than the Higgs mass calculation is done. Thus, the difference between these two calculations give an impression of the minimal, theoretical uncertainty which is at least present. The differences between the codes also don't grow significantly if we use non-vanishing values for A t as shown in Fig. 10: the overall changes in the Higgs mass between the SARAH calculation in the full MSSM and in the effective SM changes again by 7-8 GeV for very large SUSY scales, while the difference to the other codes is in the range of 1 GeV and less.

Higgs mass beyond the MSSM
With SARAH it is also possible to generate a spectrum generator for models beyond the MSSM which calculates mass spectra, decays and precision observables [52]. Also for these models two-loop Higgs mass calculations are performed by default. All important two-loop corrections stemming from new particles and/or new interactions are covered as discussed in detail in Refs. [53][54][55]. The calculations make use of the generic results of Refs. [56][57][58][59][60] and the only approximations used in the SARAH implementation of the two-loop calculations are (i) the gaugeless limit, i.e. setting g 1 = g 2 = 0, and (ii) neglecting momentum dependence, i.e. p 2 = 0. Thus, SARAH provides for models beyond the MSSM the same precision in the Higgs mass as it does for the MSSM. Moreover,the obtained results with SARAH include already for the next-to-minimal supersymmetric standard model (NMSSM) corrections which are not available otherwise [61,62]. However, there is one additional subtlety when using these two-loop corrections in extended Higgs sector which we need to discuss before coming to the results of the EFT approach: massless states appearing in the two-loop calculations usually cause divergences. Since the calculations are done in Landau gauge, these divergences are often associated with the Goldstone bosons of broken gauge groups what has caused the name 'Goldstone boson catastrophe' [63,64]. For many cases this behaviour was already under control in SARAH by the treatment of the D-terms what induced finite Goldstone masses as explained in Ref. [55]. However, for large SUSY scales, it can still happen that the ratio m S /M SUSY for some scalar mass m S becomes very small and introduces numerical problems. As short-term workaround we have introduced for this reason a regulator R which defines the minimal scalar mass squared as function of the renormalisation scale Q m 2 S,min = RQ 2 All scalar masses which appear in the two-loop integrals which are small than m 2 S,min are then replaced by RQ 2 . We found that numerical dependence on R is usually small for values of R between 0.1 and 0.001. Nevertheless, the results of Ref. [65] shall be included in SARAH in the near future to have a rigorous solution to the Goldstone boson catastrophe which is independent of any regulator [66].  Fig. 10 The same as Fig. 9 for non-vanishing At.
We can turn now to the discussion of the changes in the Higgs mass prediction when using the EFT ansatz. In general, it is possible to use the two-scale matching together with an effective calculation of the Higgs mass within the SM also for non-minimal models. The procedure is exactly the same as for the MSSM. SARAH uses the calculated Higgs mass in the full model to obtain λ SM (M SUSY ) via a pole mass matching. It then evaluates λ SM (m t ) and calculates m h at that scale using SM corrections. We briefly discuss the impact of the new calculation at the example of the NMSSM 7 . For this purpose, we relate the NMSSM specific, dimensionful parameters to the SUSY scale via With this parametrisation we find that the heavy MSSM-like scalars get a tree-level mass of M SUSY while also the scalar singlets are sufficiently heavy to be integrated out at M SUSY . We set in addition Thus, the only free parameters left are λ and M SUSY . The Higgs mass for a variation of M SUSY for λ = 0.1, 0.3, 0.5, 0.7 is shown in Fig. 11. Here, we also show the results with and without regulator R. One can see that the numerical problems associated with small masses, which in this case here are the light Higgs as well as the two Goldstone bosons, show up for increasing M SUSY . The larger λ is, the more pronounced these problems are. However, with a regulator R = 0.01 this behaviour can be prevented for all values of λ and M SUSY shown here for the one-and two-scale matching. We find that the results with regulator masses are in agreement with Ref. [12] within the indicated uncertainties. The impact on the Higgs mass using the new two-scale matching is similar as for the MSSM: for SUSY masses up to 2 TeV, the effects are small and less than 2 GeV, but they quickly increase with increasing M SUSY . For M SUSY = 25 TeV, the difference in the Higgs mass prediction is between 5.5 and 6.5 GeV. For our example we find that the differences depend only weakly on the value of λ. 7 We refer to Ref. [67] for an introduction into the NMSSM and for questions regarding the notation in the following Similarly, one can use now SARAH to study also the Higgs masses for other models in the presence of large SUSY scales more precisely. However, a detailed exploration of these effects in other models is beyond the scope of this paper. Here, we want to stress that one should be careful with models with extended Higgs sector because not all scalar masses become automatically large if M SUSY is large. Examples are for instance models with extended gauge sectors in which a second light scalar can appear because of D-flat directions [68][69][70]. In these cases, a sizeable mixing between the SM-like Higgs and another scalar can be present, i.e. the calculation of m h within an effective SM might now be valid. Therefore, SARAH does not perform this calculation by default, if a second CP-even scalar with a mass below 500 GeV is present.

Perturbativity limit of new interactions
Many models beyond the MSSM are attractive because they give a tree-level enhancement of the Higgs mass. This is quite interesting from the point of view because it reduces the required loop contributions to obtain m h = 125.1. Usually this allows for smaller values of A t which is important for the stability of the scalar potential [71][72][73][74][75]. The best studied example is again the NMSSM which pushes the Higgs mass via new F -term contributions which are proportional to λ 2 . We demonstrate this in Fig. 13 where we compare the dependence of the Higgs mass on the stop mixing parameter X t as defined as In the NMSSM, µ is replaced by µ eff . We see for a SUSY scale of 5 TeV and the chosen value of tan β = 2 and λ = 0.6 even without stop mixing the Higgs mass can be found in the correct mass range of 122-128 GeV. Because of this large impact of λ on the Higgs mass , it is very important to know how big λ can be in order to be still in agreement with gauge couplings unification at M GU T .  In Figure 13 we display the maximal value of λ(M SUSY ) which does not lead to a Landau pole below M GU T for different values of κ(M SUSY ) and for M SUSY up to 25 TeV and tan β = 4, and show the differences between the one-and two-scale matching. Because of the smaller top Yukawa coupling in the two-scale approach, one finds that slightly larger values of λ(M SUSY are allowed that for the one-scale matching.

Conclusion
We have presented the new two-scale matching procedure in SARAH/SPheno to improve the prediction of the running DR gauge and Yukawa couplings at the SUSY scale for large values of M SUSY . Together with the new matching, also the possibility of an EFT Higgs mass calculation is introduced. In the EFT calculation λ SM is obtained via a Higgs pole mass matching at M SUSY and the SM-like Higgs mass is calculated within the SM at the top mass scale. We have shown various consequences of the two-scale matching and the EFT Higgs mass calculation in the MSSM and beyond. In particular, we have compared the Higgs mass prediction for SUSY scales up to 100 TeV and found a good agreement with other EFT codes as SusyHD and FlexibleSUSY. We have also shown that the value of µ in the CMSSM can change significantly because of the changes in the top Yukawa coupling. This has an direct impact on naturalness considerations.

Acknowledgements
We thank Alexander Voigt for helpful discussions concerning the matching procedure in FlexibleSUSY and Eliel Camargo for his contribution in the early stage of this work. W.P. has been supported by the DFG, project nr. PO 1337/7-1.

A.1: One scale matching
Before we present the new two-scale matching which is now performed by SARAH/SPheno, we review the current procedure. The first step is that all DR parameters are calculated already at m Z and two-loop SUSY RGEs are used for the running to M SUSY .

A.1.1: Strong coupling
The strong interaction coupling at the weak scale is matched to the input value α The corrections due to the new coloured states in the MSSM are given by For any other BSM model, ∆ MSSM s is adjusted by SARAH to fit to the particle content.

A.1.2: Electroweak sector
The EW gauge sector of the MSSM is determined by four fundamental parameters. These are usually the gauge couplings for SU (2) L × U (1) Y and the electroweak VEVs for the up-and down-Higgs The relations between the input and DR parameters is as follows: 1. The electroweak coupling constant is calculated from Again, if another model shall be considered, the value of ∆em is calculated by SARAH automatically.
2. The Weinberg angle sin DR Θ W at the scale m Z is obtained iteratively from the above-computed α DR em (m Z ), together with G F and m Z , via where we have introduced Here, Π T V V (p 2 ) (V = Z, W ) are the DR-renormalized transverse parts of the self-energies of the vector bosons, computed at the renormalization scale Q = m Z , and δ (2) r and ∆ρ (2) are two-loop corrections as given in [39,76] and ρ 2 (r) =19 − 16.5r + 43 12 r 2 + 7 120 14) The one-loop vertex and box corrections δ VB implemented into SPheno are hard-coded and taken from literature [77][78][79], while the ones used by SARAH are auto-generated and include therefore all one-loop corrections beyond the MSSM. Also the self-energies Π T are automatically calculated by SARAH at the full one-loop level.
3. The electroweak VEV v used to calculate v d and vu at m Z is obtained from Here, the running mass m DR Z is given by

A.1.3: Yukawa couplings
In order to calculate the value of the DR-renormalized Yukawa coupling at the SUSY scale, SPheno used so far the approach of Ref. [39].
Here, Σ S,R,L are usually the one-loop self-energies without photon and gluon corrections. Only for the top-quark photon and gluon corrections need to be included and in addition one identifies The DR Yukawa matrices fulfilling  24) with δr defined in eq. (A.8). The following one-loop SM contributions are used: 25) and the two-loop corrections δ ( The MS Yukawa matrices are calculated iteratively from the condition that the MS fermion masses are reproduced once the one-loop SM corrections with massive bosons are included: Here,Σ are the self-energies without the photonic and gluonic contributions. The eigenvalues of m For the top Yukawa and strong gauge coupling one can include in SPheno an additional threshold at m t at which higher order corrections are included by using the fit formulae [42] Y t (m t ) =0.9369 + 0.00556

Electroweak sector:
The electroweak gauge coupling is calculated from g MS 1 , g MS 2 and translated into its DR value via where m Z has to be replace by M SUSY in eq. (A.6). In addition, it is helpful to define for later use as well as The DR values of the Weinberg angle and electroweak VEV are now given by where the SUSY corrections are calculated as 44) sin Θ DR W and v DR together with calculated α DR ew (M SUSY ) and the input value for tan β determine g DR 1 (M SUSY ), 3. Yukawa couplings As first step, the running MS Yukawa couplings are translated in DR values via [83] m DR In the next step, the one-loop corrections due to the SUSY particles and the heavy Higgs-doublet H, where H is to the SM-Higgs orthogonal combination of Hu and H d . Here we distinguish between holomorphic and non-holomorphic corrections where the first denotes loop contributions to the existing tree-level coupling and the second the loop-induced ones to the second Higgs-doublet. We give here for simplicity the different contributions for the case of real parameters neglecting flavour mixing. The case with flavour mixing can be easily obtained from appendix A Ref. [84].
-Taking either f = t or f = b we obtain for the gluino contributions -Taking either f = t or f = b we obtain for the single higgsino contributions -For the mixed wino/higgsino contributions we find with L f = Q in case of f = t, b and L f = L in case f = τ .
-For the mixed bino/higgsino contributions we find where f = d, For completeness we note, that the equivalence of the resummation of the two-point function (as done in case of SARAH) with the resummation of the three-point function (as done in SPheno) has been shown in [85].