O(\alpha_s^2) corrections to the running top-Yukawa coupling and the mass of the lightest Higgs boson in the MSSM

In this paper we propose a method to compute the running top-Yukawa coupling in supersymmetric models with heavy mass spectrum based on the"running"and"decoupling"procedure. In order to enable this approach we compute the two-loop SUSY-QCD radiative corrections required in the decoupling process. The method has the advantage that large logarithmic corrections are automatically resummed through the Renormalization Group Equations. As phenomenological application we study the effects of this approach on the prediction of the lightest Higgs boson mass at three-loop accuracy. We observe a significant reduction of the renormalization scale dependence as compared to the direct method, that is based on the conversion relation between the running and pole mass for the top quark. The effect of resummation of large logarithmic contributions consists in an increased prediction for the Higgs boson mass, an observation in agreement with the previous analyses.


Introduction
The recent discovery of the Higgs boson at the Large Hadron Collider (LHC) was a major milestone not just in particle physics but in the history of science. It bears the knowledge how the mass comes about at quantum level by the Higgs mechanism. With the Higgs boson discovery, particle physics entered a new era of tremendous intensity of detailed and careful study of its properties. Hopefully, accurate understanding of the Higgs phenomenology together with new information from experiments at the LHC will provide us a tool for exploring new physics. Great interest is currently devoted to the study of the Higgs boson couplings to the electroweak gauge bosons W and Z and to the top-and bottom-quarks or tau-leptons. Deviations in these couplings could possibly be observed once the currently large uncertainties will be improved as part of the LHC program and at a future Higgs factory. It has been shown that the Higgs couplings will be sensitive to new physics at the multi-TeV scale once percent level precision on coupling measurements will be achieved (for a recent review see [1]).
The aim of this paper is to propose a method for the computation of the top-Yukawa coupling within the Minimal Supersymmetric Standard Model (MSSM) taking into account radiative corrections at O(α 2 s ) accuracy. In general, the relationships between the running couplings and the physical observables like particle masses are affected by large radiative corrections [2,3]. Within the SM, the relationship between the running top-Yukawa coupling and the physical top-quark pole mass receives three type of radiative corrections: i) higher order corrections to the running Yukawa coupling, ii) contribution to the fermion pole mass, and iii) corrections to the relation between the Fermi constant and the SM parameters. The first contribution is known at three-loop accuracy [4,5] taking into account corrections from all sectors of the SM. The radiative corrections to the top-quark pole mass are known in QCD up to three-loop order [6][7][8][9] and in the electroweak sector up to two loops [10][11][12][13]. The third contribution is known in the SM with two-and three-loop accuracy for the genuine electroweak [10,14,15] and mixed QCDelectroweak [16] sectors, respectively. As was explicitly shown there are two sources for the large radiative corrections: the pure QCD contributions to the top-quark pole mass of about 10 GeV, 2 GeV and 0.5 GeV [6] at one-, two-and three-loop order, respectively; and the tadpole diagrams when they are taken into account for a gauge-independent definition of the running-mass [12]. Their magnitude is comparable with that from QCD sector at the one-loop order and amounts to about 0.5 GeV at two loops. As can be concluded from the numerical values cited above the radiative corrections are very important for the QCD sector, and even the third-order terms in the perturbative series are necessary in order to cope with the current experimental accuracy on the top-quark mass [17]. However, the situation is going to change if the International Linear Collider (ILC) is built, where a precision of O(100) MeV is expected. When physics beyond the SM is considered, the radiative corrections to the top-quark pole mass might receive much larger contributions than in the SM and even diagrams beyond the three-loop order have to be taken into account to reach the current experimental accuracy. By now, the radiative corrections to the fermion pole mass are known at two-loop order for a general theory with massless gauge bosons [18]. The numerical evaluation of the two-loop self-energy diagrams has been implemented in the code TSIL [19]. For supersymmetric models with masses at the few TeV scale 1 the radiative corrections to the top-quark pole mass increase by about a factor of four as compared to the SM results as we show in section 3.2. Thus, the two-loop contributions might become one order of magnitude larger than the experimental uncertainties and the effects of higher order corrections have to be considered. The computation of on-shell self-energy diagrams with several mass scales at the three-loop order is, for the time being, feasible only using asymptotic expansion techniques and is computationally very involved. In this paper, we propose an alternative method that can be applied as long as the top-quark mass is much smaller than the masses of supersymmetric particles. In the present paper, we focus on the dominant SUSY-QCD corrections to the running Yukawa coupling in the MSSM and postpone the study of the contributions originating from Yukawa and electroweak interactions for a later publication. The SUSY-QCD corrections to the relation between the running Yukawa coupling and the top-quark pole mass reduce to the corrections to the ratio between the pole and the running masses. Explicitly, the running top-quark mass is determined in the SM with the highest available precision from the experimentally measured pole mass 2 . Then, the running mass is evolved up to the SUSY scale using the Renormalization Group Equations (RGEs) of the SM. Afterwards, the running top-quark mass in the SM is converted to its value in the MSSM. In this step, threshold corrections at the SUSY-scale are required. In the last step, the running top-quark mass in the MSSM is evolved at the desired energy scale with the help of MSSM RGEs. As the RGEs in the SM [4,5] and the MSSM [22,23] are known to three-loop order, the threshold corrections are required at the two-loop order. They are known for light quark masses (e.g. bottom quark) in the SM to three-loop accuracy [24,25] and in the MSSM to two loops [26,27]. Great interest was devoted to the determination of the effective bottom-Yukawa coupling in SUSY-models with large tan β parameter [28][29][30]. For these models the resummation of tan β enhanced contributions is considered on top of the two-loop order calculation.
One goal of this paper is to present the computation of the two-loop SUSY-QCD threshold corrections to the running top-quark mass. The calculation is similar with the ones performed in [26,27]. The advantage of the method presented here as compared to the direct calculation of on-shell self-energy diagrams in the MSSM is that the occurring large logarithms of the form ln(M top /M SUSY ) are automatically resummed by the use of RGEs. The result is a much better convergence of the perturbative series as will be explicitly shown in the next sections. The second aim is to study the phenomenological effects of the above calculation. Obviously, the top-Yukawa coupling is an essential ingredient in all processes involving interactions between the Higgs boson and top-quark and top-squarks. However, the most prominent example is probably the effects on the lightest Higgs boson mass, that receives radiative corrections enhanced by the fourth power in top-Yukawa coupling. A detailed numerical analysis of these effects will be discussed in section 4.
The paper is structured as follows: in section 2 we present our computational framework; explicit analytical results are discussed in section 3 together with their numerical implementation; in section 4 we perform a phenomenological analysis of the effects of the two-loop SUSY-QCD corrections determined in the previous section on the prediction of the lightest Higgs boson mass with three-loop accuracy; in section 5 we summarise our conclusions.

Framework
An elegant approach to get rid of unwanted large logarithms occurring in the predictions for observables in multi-scale processes is to formulate an effective theory (ET) (for a review see Ref. [31]). The parameters of the ET must be modified in order to take into account the effects of the heavy fields. The ET parameters are related to the parameters of the full theory by the so-called matching or decoupling relations. These take into account threshold corrections generated by the heavy degrees of freedom that are integrated out when the ET is constructed. In the following, we concentrate on the calculation of the decoupling coefficients for the strong coupling and the top-quark mass within SUSY-QCD. They are defined through the following relations between the bare quantities where the primes label the quantities in the effective theory. The decoupling coefficients have been computed in QCD including corrections up to the four-loop order for the strong coupling [32] and three-loop order for quark masses [24,25]. In the MSSM the two-loop SUSY-QCD [26,27,33] and SUSY-EW [26,30] expressions are known. Very recently, even the three-loop SUSY-QCD corrections to decoupling coefficient of the strong coupling were computed [34]. We consider MSSM with n f = 6 active quark and squark flavours and ng = 1 gluinos. Furthermore, we assume that all SUSY-particles including squarks and the gluino are much heavier than the SM particles. Integrating out the heavy fields from the full Lagrange density, we obtain the Lagrange density corresponding to the "effective" SM with n f quarks plus non-renormalizable interactions. The latter are suppressed by negative powers of the heavy masses and will be neglected here. Since the decoupling coefficients are universal quantities, they are independent of the momenta carried by the incoming and outgoing particles. The authors of Refs. [24] showed that the bare decoupling coefficients for the quark mass ζ 0 m and for the strong coupling constant ζ 0 s can be derived via the relations where Σ s (p 2 ), Σ v (p 2 ) and Σ A (p 2 ) are the scalar, the vector and the axial-vector components of the top-quark self-energy defined through Note that the axial-vector component starts contributing at the two-loop order. Π 0,h c (p 2 ) and Π 0,h g (p 2 ) are the ghost and gluon vacuum polarizations and Γ 0,h gcc (p, q) denotes the amputated Green function contributing to the gluon-ghost-ghost vertex. The superscript h indicates that in the framework of DREG or DRED only diagrams containing at least one heavy particle inside the loops contribute and that only the hard regions in the asymptotic expansion of the diagrams are taken into account. In Fig. 1 some sample Feynman diagrams contributing to the decoupling coefficients for the top-quark mass are shown.
The Feynman diagrams were generated with QGRAF [36] and further processed with q2e and exp [37,38]. The reduction of various vacuum integrals to the master integral was performed with a self written FORM [39] routine [35]. The reduction of topologies with two different massive and one massless lines requires a careful treatment. The related master integral can be easily derived from its general expression valid for massive lines, given in Ref. [40].
The finite decoupling coefficients are obtained upon the renormalization of the bare parameters. They are given by where Z ′ s and Z ′ m correspond to the renormalization constants in the effective theory, and Z s and Z m denote the same quantities in the full theory. Since we are interested in the two-loop results for ζ i , i = s, m, the corresponding renormalization constants for SUSY-QCD and QCD have to be implemented with the same accuracy. Analytical results for the latter up to the three-loop order can be found in e.g. Refs. [31,41,42].
Apart from the renormalization constants of the external fields, also parameter renormalization is required. For the renormalization of the gluino and squark masses and the squark mixing angle we choose the on-shell scheme. This scheme allows us to directly use the physical parameters in the running analyses making the implementation very simple. The explicit formulae at the one-loop order can be found in Refs. [43,44]. The two-loop counterterms are known analytically only for specific mass hierarchies [45] and numerically for arbitrary masses [19]. For the calculation of ζ αs the simultaneous renormalization of up-and down-type squarks is required. We follow the prescription proposed in Ref. [46] and fixed the counterterms for the up-squarks and the heavier down-squark to be on-shell and derive the counterterm for the lighter down-squark accordingly. In our limit of neglecting the top-quark mass as compared to the SUSY mass scale it holds and where δmd 1 stands for the mass counterterm of the light down-squark and δm os u 1 denotes the on-shell mass counterterm of the light up-squark. Explicit formulae at the one-loop level can be found in Refs. [43,44].
For the computation of the decoupling coefficient of the top-quark mass at order O(α 2 s ) one needs to renormalize in addition the top-quark mass and the ǫ-scalar mass. As the top-quark mass is neglected w.r.t. SUSY particle masses, an explicit dependence of the radiative corrections on m t can occur only through top-Yukawa coupling. In order to avoid the occurrence of large logarithms of the form α 2 s log(µ 2 /m 2 t ) with µ ≃M , wherẽ M stands for the SUSY scale, one has to renormalize the top-Yukawa coupling in the DR scheme. In this way, the large logarithms are absorbed into the running mass and the higher order corrections are maintained small. As a consequence, the top-squark mixing parameter X t = A t − µ SUSY / tan β will be renormalized in a mixed scheme. Its counterterm is derived from the relation between the top-quark and squark masses and the mixing angle and mixing parameter.
Finally, the last parameter to be renormalized is the ǫ-scalar mass 3 . To obtain decoupling coefficients independent of the unphysical parameter m ǫ , one has to modify the top squark masses by finite quantities [48,49]. We adopt in the present calculation the method proposed in Ref. [26] to choose the ǫ-scalars massive and integrate them out together with the SUSY particles. In this way we achieve conversion from MS to DR schemes and decoupling of the heavy masses in a single step.

Two-loop threshold corrections
The two-loop results for the decoupling coefficient for the strong coupling is very compact and we reproduce them below.
For a review on their role in multi-loop calculation see [47].
In the formula above the sum q runs over all quark flavours and gen over the number of generations. C A , C F are the quadratic Casimir invariants for the adjoint and fundamental representations, T = 1/2 is the Dynkin index and L x = ln(µ 2 /m 2 x ), with x =g,q. B 0,fin (p 2 , m 1 , m 2 ) denotes the fine part of the B 0 -function [50]. The asymmetry w.r.t. up-and down-type quarks originate from the special renormalization scheme of downtype squarks relative to the up-type squarks. Here, α s = α (SQCD) s denotes the strong coupling constant in the full theory.

Limits
The final results for two-loop threshold corrections for ζ mt are too lengthy to be displayed here.
They are available in Mathematica and Fortran format from http://www.ttp.kit.edu/Progdata/ttp14/ttp14-025. Instead, we present them for two special mass hierarchies.

Scenario A
We consider first the case of all supersymmetric particles having masses of the same order of magnitude and being much heavier than the top-quark.
ζM s , ζM mt are functions of the supersymmetric massM , the soft SUSY breaking parameters X t = A t − µ SUSY / tan β, the strong coupling constant in the full theory α (SQCD) s , and the decoupling scale µ. The superscript (6) indicates that the parameters are defined in QCD with n f = 6 quarks.

Scenario B
The second scenario we consider is the so called "split-SUSY" one, with squarks much heavier than all the other particles: The result reads:

Numerical results
In this section we present the phenomenological effects of the two-loop SUSY-QCD threshold corrections on the prediction of the running top-quark mass at the SUSY mass scale. We also present the comparison with the direct prediction obtained from the ratio between the running and the pole mass within SUSY-QCD as described in the code TSIL [19].
Our method can be summarised in the following sequence: where M OS t denotes the top-quark pole mass and m MS t and m SQCD,DR t stand for the running top-quark mass in the SM and SUSY-QCD in the MS and DR schemes, respectively. µ dec is the scale at which the decoupling is performed and it is usually chosen comparable with SUSY masses. We fix it to be the arithmetic average over the squarks and gluino on-shell masses: But of course, µ dec can be chosen arbitrarily and the dependence of the running topquark on it is a measure of the theoretical uncertainties. Also in the numerical setup, we implemented it as a free parameter that can be varied. In the step (i) the relation between the top-quark pole and running masses in the SM is required. We implemented the three-loop results [6][7][8][9]. The RGEs for the SM and the MSSM that are necessary in the steps (ii) and (iv) are known to three-loop accuracy [4,5,22,23] as well. Let us mention, that in QCD the quark anomalous dimension was recently computed even at the five loop order [51]. For consistency, the threshold corrections evaluated in the step (iii) are necessary at two loops. For the explicit numerical evaluation we use for the SM parameters their values cited in Ref. [52]. For the MSSM parameters we employ two scenarios that we call "heavy Higgs" and "heavy sfermions", respectively. We obtain the numerical values with the help of the spectrum generator SOFTSUSY [53]. The input parameters for the spectrum generator are as follows: i) In the "heavy sfermions" scenario we define all DR breaking parameters at the input scale following the Supersymmetry Les Houches Accord (SLHA) [54,55] EXTPAR 0 Q in =m t . Wherem t is an alias for the right handed top squark mass breaking parameter EXTPAR 46 and kept as free parameter. Further we identify the third generation doublet mass breakingm Q 3 alias EXTPAR 43 withm t . All other sfermion mass breaking parameters have a common valuem f =m t + 1TeV. The trilinear couplings (EXTPAR 11-13) are given by A t = 20GeV and A τ = A b = 4TeV. The gaugino mass parameters (EXTPAR 1-3 ii) In the "heavy Higgs" scenario we define all DR breaking parameters at the input Here M SUSY = 1TeV is a common breaking mass parameter for all sfermions exceptm t (EXTPAR 46), which we keep as free parameter. The remaining input parameters are given by: Here s w and c w are the sine and cosine of the Weinberg mixing angle θ W . In contrast to the first scenario we do have light Higgs masses for sub TeV values ofm t due to the stop mixing. Moreover one can have a very light stop of order 300GeV form t values having nearly same size. Please note that for the pure SUSY-QCD analysis done in this paper, the effect of changing breaking parameters of particles transforming as QCD singlets is very weak with respect to SUSY-QCD decoupling effects. We provide the SLHA input files for the two scenarios considered here in electronic format on the web page http://www.ttp.kit.edu/Progdata/ttp14/ttp14-025.
In the following, we denote as leading order (LO) value for the running top-quark mass in the decoupling method (DEC), the value obtained with one-loop RGEs and without threshold corrections. The next-to-leading (NLO) and the next-to-next-to-leading order (NNLO) values in the decoupling method are calculated employing two-and three-loop RGes and one-and two-loop threshold corrections, respectively. The NLO top-quark mass computed in the direct method ( with the code TSIL) takes into account the oneloop relation between the running-and pole-quark mass, whereas the NNLO prediction is based on the two-loop relation.
In Fig. 2 we present the running top-quark mass in the full theory as a function of the scale µ ren at which it is evaluated. The black (middle) curves display the results obtained with the method proposed in this paper (that we refer at as the decoupling method and is denoted as "DEC" in the legend of the plot) at one-(dotted line) two-(dashed line) and three-loops (full line). The blue curves show the predictions obtained directly via the ratio between the top-quark pole and the running masses in the SUSY-QCD, using the code TSIL. The dashed line corresponds to the one-loop results and the full line displays the two-loop contributions. As can be seen from the figure, the radiative corrections for the decoupling method are very small (tenth of MeV between one-and two-loop order contributions and negligible at the three loops) as compared with the current experimental uncertainty on the top quark pole mass of about 1 GeV. The perturbative series is very well converging and the contributions from the unknown higher order corrections are negligible for all renormalization scales. The radiative corrections obtained via the direct method are much larger than the experimental uncertainty. Even at the two-loop order, they amount to 10 GeV at the electro-weak scale and increase further at renormalization scales comparable with the squark masses. In this case higher order contributions are necessary to bring the theoretical precision at the same level as the experimental one. One observes also that the predictions obtained in the two methods at the two-loop order agree very well for small renormalization scales below 400 GeV. This can be explained by the fact that in this case the logarithmic contributions (of the form ln(M top /µ)) are small and the resummation gives only minor corrections. When the running top-quark mass is evaluated at high energy scales the resummation of the large logarithms becomes important and the difference between the two predictions can reach about 10 GeV. Let us also point out that in the domain where the resummation is expected to bring only small effects the differences between the two methods decrease considerably when going from one to two loops as is expected in perturbation theory.
In Fig. 3 the running top-quark mass is shown as a function of the squark mass breaking parameterm t , that can be interpreted as a scale for the SUSY mass spectrum. In this case the renormalization scale is chosen as geometric mean value of the top-squark masses. One notices that the radiative corrections calculated with the decoupling approach are very small for all SUSY scales. The direct computation deliver even at the two-loop order radiative corrections of about O(10) GeV. As expected, the two methods provide results in very good agreement at the two-loop order for low SUSY mass scales, where no large logarithmic corrections are present. Fig. 4 shows the running top-quark mass as a function of the renormalization scale in the "heavy Higgs" scenario. For the chosen input parameters, the predictions of the two methods differ at the one-loop level by more than 10 GeV. At the two-loop level the predictions of the two approaches agree well within the present experimental uncertainty on the top-quark pole mass for low-energy scales, whereas for large renormalization scales the difference amounts to few GeV. Let us also point out that the differences between one-and two-loop contributions within the decoupling method amounts to about 4 GeV, whereas the genuine three-loop contributions are very small, below 100 MeV. This observation proves the good convergence of the perturbative methods in the decoupling approach. The direct calculation based on the code TSIL provides similarly large radiative corrections at the two-loop level. However, in order to reduce the theoretical uncertainty at a similar level with the experimental one, we need higher order radiative corrections that are currently not available.   Figure 3: Dependence of the running top-quark mass on the SUSY scalem t for the "heavy sfermion scenario". The convention for the curves is the same as in Fig. 2.
In summary, we conclude that the two methods provide results in good agreement for low SUSY mass scales or renormalization scales, but they differ significantly when the SUSY particles become heavy, in the multi TeV range. Also, the rapid convergence of the perturbative series for the decoupling method allows us to reduce the genuine theoretical uncertainties due to unknown higher order corrections well below the present experimental uncertainty on the top-quark pole mass. The discrepancies between the predictions obtained within the two methods can have important phenomenological implications, depending on the process and observables under consideration.

4
The mass of the lightest Higgs boson  Figure 4: Dependence of the running top quark mass on the renormalization scale within the "heavy Higgs" scenario. The convention for the curves is the same as in Fig. 2.
Since the dependence of the Higgs boson mass on the SUSY masses is logarithmic, highprecision measurements and theoretical predictions are required. The genuine theoretical uncertainties due to unknown higher order corrections are expected to grow with the SUSY mass scale. For top-squark masses in the multi TeV range they were estimated to be of about 5 − 7 GeV [59,60]. By now, the complete one-loop [61,62] and dominant twoloop [63,64] corrections are implemented in the numerical programs FeynHiggs [65] and CPsuperH [66] using on-shell particle masses, and in SOFTSUSY [53] and SPheno [67] using running parameters. The dominant SUSY-QCD three-loop corrections are taken into account in the code H3m [45], for which a mixed renormalization scheme was employed. However, the three-loop contributions are known only for specific mass hierarchies between the SUSY particles. The dominant contributions to the leading (LL) and next-to-leading (NLL) logarithmic terms in the ratio between the top quark mass and the typical scale of SUSY particle masses ln(M top /M SUSY ) have been obtained in Ref [68]. Very recently, the generalization of the LL and NLL approximation has been derived [59] and implemented in the code FeynHiggs, up to the seventh loop-order. Furthermore, in Ref. [60] the recent calculations of the three-loop beta-functions for the SM coupling and the two-loop corrections to the Higgs boson mass in the SM have been used to derive (presumably) the dominant NNLL corrections at the four-loop order.
In this section we focus on the numerical effects that the new prescription for the determination of the running top-quark mass and top-Yukawa coupling has on the predictions of the lightest Higgs boson mass in the MSSM. We implemented the resummation of the large logarithms of the form ln(M top /M SUSY ) contained in the running of the top-Yukawa coupling (as discussed in the previous section) on top of the three-loop SUSY-QCD corrections to the lightest Higgs boson mass encoded in the program H3m. As will be shown, this type of resummation is necessary for SUSY masses in the multi TeV range and enable us to reduce the effects of the unknown higher order contributions. In the following, we evaluate the running top-quark mass and coupling with the highest accuracy both for the decoupling and the direct methods and use them further for the calculation of the lightest Higgs boson mass at one-, two-and three-loop accuracy.
In order to allow a convenient evaluation of general MSSM scenarios including the stated low scale ones, the Mathematica package SLAM [69] has been implemented in H3m. It provides an easy way to use interface for calling and reading SUSY spectrum generator output full automatically using the SLHA. Moreover it enables the ability to save and recall SUSY spectra in and from a data base. Besides reading in user provided SLHA spectrum generator output files, it is now possible to enter the SLHA input file for the spectrum generators defining the SUSY scenario directly in Mathematica relaxing the restriction to predefined scenarios in earlier versions of H3m. The MSSM parameters derived in this way are stored and further used to compute the running top-quark mass, the running top-Yukawa coupling and the strong coupling constant in the MSSM using the decoupling method described in section 2. This step is realized through stand-alone routines as is also explained in the flowchart plot in Fig. 5. Afterwards, the stored values for the input MSSM parameters together with the DR couplings just derived are delivered to the code H3m. The computation of the lightest Higgs boson mass follows then the steps described in Ref. [45]. The user has also the possibility to choose the way the running top-quark mass is computed. The command SetOptions[H3GetSLHA, calcmt->{''MtTSIL''}] allows to use at this stage the code TSIL, whereas the decoupling method is implemented as default option. A direct comparison between the predictions obtained with the two methods will be presented below. The new version of the code H3m together with few simple example programs are available from https://www.ttp.kit.edu/Progdata/ttp14/ttp14-025/. Apart from the implementation of the interface program SLAM and the routine for the computation of the running top-quark mass and the strong coupling constant through the decoupling method, we improved on the determination of the mixing angle and reduced oscillations in the Higgs mass by setting the W-boson mass fixed to is on-shell value as given by the PDG [52].
In Fig. 6 it is shown the dependence of the lightest Higgs boson mass M h in the "heavy sfermion" scenario on the renormalization scale, taking into account beyond one-loop only SUSY-QCD radiative corrections. The black curves display the one-(dotted line), two-(dashed line) and three-loop (full line) contributions obtained with the running top-quark mass in the decoupling method. The blue curves present the same predictions using the code TSIL for the calculation of the running top-quark mass. The explicit value of the running-top quark mass can be read from Fig. 2. It is known that the renormalization scale dependence of an observable gives an estimation for the magnitude of unknown higher order corrections. This enables us to use it for the determination of the theoretical uncertainty. As expected the renormalization scale dependence is reduced when going from one-to two-and to three-loop order corrections in both schemes. However, the direct determination of running top-quark mass (blue curves) is affected by large logarithmic corrections that in turn induces large radiative corrections to the Higgs boson mass. Even at the three-loop order, the scale variation of M h amounts to about 5 GeV, more than an order of magnitude larger than the current experimental accuracy on M h and few times bigger than the parametric uncertainties. In contrast, the resummation of the logarithmic corrections to the running top-Yukawa coupling through the decoupling method renders the scale dependence of M h at three-loop order very mild about tens of MeV. One can also observe, that low values for the renormalization scale around the top-quark pole mass are not well suited for the present scenario, especially when the three-loop order contributions are not taken into account. In this case, radiative corrections even beyond the three-loop order are required in order to cope with the experimental precision. The difference between the predictions for M h obtained with the two methods is sizeable and can amount to few GeV for large values of the renormalization scale.
The dependence of M h on the SUSY breaking parameterm t is shown in Fig. 7. As described in section 3.2,m t can be interpreted as an estimation of the SUSY mass pa- rameter. The convention for the line style is the same as in the previous figure. The renormalization scale is fixed as the geometric mean value of the top-squark masses, thus in the TeV range. The radiative corrections to the Higgs boson mass increase with the SUSY mass scale as expected. The predictions obtained using the two methods for the derivation of the top-Yukawa coupling are in good agreement for low SUSY scales of about 500 GeV, but they differ significantly for heavy SUSY spectrum in the multi TeV range. The Higgs boson mass predicted through the decoupling method is always heavier and has a steeper dependence on the SUSY spectrum as compared to the one obtained through the direct method. This difference can be explained by the effects of resumming large logarithms within the first approach. Since we use the RGEs at the three-loop order the next-to-next-to-leading-logarithms are resummed. A similar behaviour of predictions for M h based on resummation was observed in the previous works [59,60]. Let us point out the big impact of the resummation procedure for constraining the SUSY parameter space. While the prediction of M h through the decoupling method allows SUSY mass scales of about 4 TeV, the present scenario is already excluded when the direct method is employed. This observation explains also the need for very precise theoretical predictions for the Higgs boson mass.
In Fig. 8 the dependence of M h on the renormalization scale is shown for the "heavy Higgs" scenario. As can be read from the figure, the scale dependence is reduced when  higher order radiative corrections are taken into account. However, even at the three-loop order the variation of M h with the renormalization scale can amount to few GeV for both methods. It is also important to notice that low values of the renormalization scale are characterized by large radiative contributions. A better alternative is the choice of the renormalization scales above 500 GeV. In the decoupling method this choice will reduce the scale variation to about 200 MeV. As for the previous scenario, we observe a milder scale variation when the decoupling method is employed, that can be associated with smaller higher order corrections. The difference between the predictions obtained in the two frameworks amounts to few GeV for renormalization scales in the TeV range. The two methods provide same values for M h for renormalization scale of about 400 MeV, for which also the predictions for the running top-quark mass coincide.

Conclusions
In this paper, we consider the calculation of O(α 2 s ) radiative corrections to the running top-quark mass and Yukawa coupling with the MSSM. Our method is based on the "running and decoupling" technique, that has the advantage to resumm the large logarithms by the use of RGEs. Our numerical analysis performed in section 3.2 showed that the method is very stable upon higher order radiative corrections. The remaining theoretical uncertainty is estimated by half of the magnitude of genuine three-loop order contributions and amounts to about 100 MeV. The method proposed here provides results for the running top-quark mass in good agreement with the predictions of the code TSIL for light SUSY spectra and for low-energy scales at which the top-quark mass is computed. For heavy SUSY particles and for high renormalization scales the differences between the two methods can easily reach few GeV. However, the relation between the running and the on-shell top-quark mass in the MSSM, on which the code TSIL rely, is affected by large radiative corrections in the range of GeV. One also observes that the two methods agree better when going from LO to NLO and NNLO. The exact analytical results of our method are available in electronic format. In addition, we provide in the paper the results for two specific mass hierarchies.
The second part of the paper presents the effects that the new determination of the running top-Yukawa coupling has on the prediction of the lightest Higgs boson mass in the MSSM. We observe a much milder dependence on the renormalization scale of the Higgs boson mass predicted with three-loop accuracy. The renormalization scale dependence is usually interpreted as a measure for the missing higher order corrections. Thus, this improvement is very welcomed given the present difficulty to achieve radiative corrections to the Higgs boson mass beyond the two-loop level. We also notice that the predictions obtained through the decoupling method are higher than the one derived in the direct method. The difference can amount to few GeV for SUSY masses in the TeV range. This behaviour can be explained by the effect of resumming the large logarithms through the use of RGEs in the determination of the running top-Yukawa coupling. Furthermore, we implemented the decoupling method described above together with the code SLAM, that provides an interface to spectrum generators, in the existing code H3m. In this way, the code H3m computes the three-loop SUSY-QCD corrections to the Higgs boson mass, taking into account the resummation of the large logarithms of the form ln(M top /M SUSY ).