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The Masses of the Neutral CP-even Higgs Bosons in the MSSM: Accurate Analysis at the Two-Loop Level

We present detailed results of a diagrammatic calculation of the leading two-loop QCD corrections to the masses of the neutral CP-even Higgs bosons in the Minimal Supersymmetric Standard Model (MSSM). The two-loop corrections are incorporated into the full diagrammatic one-loop result and supplemented with refinement terms that take into account leading electroweak two-loop and higher-order QCD contributions. The dependence of the results for the Higgs-boson masses on the various MSSM parameters is analyzed in detail, with a particular focus on the part of the parameter space accessible at LEP2 and the upgraded Tevatron. For the mass of the lightest Higgs boson, m_h, a parameter scan has been performed, yielding an upper limit on m_h which depends only on tan(beta). The results for the Higgs-boson masses are compared with results obtained by renormalization group methods. Good agreement is found in the case of vanishing mixing in the scalar quark sector, while sizable deviations occur if squark mixing is taken into account.


Introduction
The search for the lightest Higgs boson is a crucial test of Supersymmetry (SUSY) which can be performed with the present and the next generation of accelerators. The prediction of a relatively light Higgs boson is common to all supersymmetric models whose couplings remain in the perturbative regime up to a very high energy scale [1]. A precise prediction for the mass of the lightest Higgs boson in terms of the relevant SUSY parameters is necessary in order to determine the discovery and exclusion potential of LEP2 and the upgraded Tevatron, and for physics at the LHC and future linear colliders, where eventually a highprecision measurement of the mass of this particle might be possible. A precise knowledge of the mass of the heavier CP-even Higgs boson, m H , is important for resolving the mass splitting between the CP-even and -odd Higgs-boson masses.
In the Minimal Supersymmetric Standard Model (MSSM) [2] at the tree level the mass m h of the lightest Higgs boson is restricted to be smaller than the Z-boson mass. However, this bound is strongly affected by the inclusion of radiative corrections: the dominant one-loop corrections arise from the top and scalar-top loops which yield terms of the form G F m 4 t ln(mt 1 mt 2 /m 2 t ) [3]. These results have been improved by performing a complete oneloop calculation in the on-shell scheme, which takes into account the contributions of all sectors of the MSSM [4,5,6]. Beyond one-loop order, renormalization group (RG) methods have been applied in order to include leading logarithmic higher-order contributions [7,8,9,10]. In the effective potential approach diagrammatic results for the dominant two-loop contributions have been obtained in the limiting case of vanishingt-mixing and infinitely large M A and tan β [11]. The calculation of the leading QCD corrections in this approach has recently been generalized to the case of arbitrary tan β and non-vanishingt-mixing [12].
Up to now phenomenological analyses have been based either on the RG results [7,8,9,10], or on the complete one-loop on-shell results [4,5,6]. These results differ by large leading logarithmic higher-order contributions, which are not included in the one-loop onshell results, but also by non-leading one-loop contributions, which are neglected in the RG approach. The numerical difference in the Higgs-mass predictions between the two approaches reaches up to 20 GeV.
Recently a Feynman-diagrammatic calculation of the leading two-loop corrections of O(αα s ) to the masses of the neutral CP-even Higgs bosons has been performed [13,14]. Compared to the leading one-loop result the two-loop contribution was found to give rise to a considerable reduction of the m h value. The leading two-loop corrections have been combined with the full diagrammatic one-loop on-shell result [5] and further refinements have been included concerning the leading two-loop Yukawa corrections of O(G 2 F m 6 t ) [8,15] and leading QCD corrections beyond two-loop order.
In this paper we present in detail the steps of this calculation. The results for the masses of the neutral CP-even Higgs bosons are analyzed in terms of the relevant parameters of the MSSM. A parameter scan for the lightest Higgs-boson mass is performed yielding an upper bound for m h within the MSSM (apart from certain threshold regions which correspond to very specific configurations of the MSSM parameters) given exclusively in terms of tan β . This upper bound is discussed in view of the discovery potential of LEP2 and the upgraded Tevatron. The results for m h are compared with the corresponding results obtained by RG methods. The comparison is performed both in terms of the (unobservable) parameters of the scalar top mass matrix and in terms of the physical stop masses and the stop mixing angle.
The paper is organized as follows: Section 2 contains our notations and a description fo the renormalization procedure as required for the corrections in the MSSM Higgs sector in O(αα s ). The main features of the calculation are discussed in section 3. In section 4 we present a detailed numerical analysis of the results for the neutral CP-even Higgs-boson masses as functions of the different SUSY parameters. We perform a scan for m h over the parameters mg, M A , M, µ and thet-mixing parameter and determine the maximal possible values of m h as a function of tan β . Finally numerical comparisons are shown with results obtained by renormalization group (RG) methods. In section 5 we give our conclusions.

The Higgs sector of the MSSM
The Higgs sector of the MSSM consists of two Higgs doublets H 1 , H 2 with opposite hypercharges Y 1 = −1 and Y 2 = +1 and non-vanishing vacuum expectation values v 1 and v 2 . The Higgs doublets can be decomposed according to The vacuum expectation values define the angle β via The Higgs potential, including all soft SUSY breaking terms reads [16] (ǫ 12 = −1): where m 2 i ≡ |µ| 2 +m 2 i (i = 1, 2);m 1 ,m 2 , m 12 are the soft SUSY breaking terms, and µ denotes the mixing between H 1 and H 2 . The coupling constants of the Higgs self-interaction are, contrary to the SM, determined through the gauge coupling constants g 1 and g 2 . Besides g 1 , g 2 two independent parameters are required to fix the potential (3) at the tree level. Conventionally they are chosen as tan β and M 2 A = −m 2 12 (tan β + cot β ), where M A is the mass of the CP-odd A boson.
The diagonalization of the bilinear part of the Higgs potential, i.e. the Higgs mass matrices, is performed via the orthogonal transformations with β from eq. (2). The mixing angle α is determined through One gets the following Higgs spectrum: The masses of the gauge bosons are given in analogy to the SM: At tree level the mass matrix of the neutral CP-even Higgs bosons is given in the φ 1 -φ 2basis in terms of M Z , M A , and tan β by which by diagonalization according to eq. (4) yields the tree-level Higgs-boson masses In order to slightly simplify the two-loop calculation, we have chosen to perform it in the φ 1 -φ 2 -basis. In this way the angle α does not appear in the calculation of the two-loop self-energies, but enters at the end when the rotation into the physical basis is performed.
In order to deal with the arising divergencies and to establish the meaning of the physical parameters beyond the tree level, one has to renormalize the Higgs and the scalar top sector of the MSSM. For the corrections of O(αα s ) to the Higgs-boson masses, in the focus of this discussion here, renormalization up to the two-loop level is needed. In the following we specify the renormalization for the relevant quantities in this calculation (explicitly listed are only those terms that actually contribute at O(αα s )). The renormalization of the complete one-loop contributions to the neutral CP-even Higgs-boson masses has been performed according to Ref. [5]. We use the following notation: Σ (1) and Σ (2) denote the one-and two-loop part of an unrenormalized self-energy,Σ (1) andΣ (2) denote the one-and two-loop part of a renormalized self-energy, and Σ ′ (k 2 ) = ∂ ∂k 2 Σ(k 2 ). t 1 and t 2 denote the unrenormalized tadpoles; t (1) and t (2) represent the one-and two-loop part of an unrenormalized tadpole, andt 1 ,t 2 denote the renormalized tadpoles.
The renormalization of the masses and fields is performed as follows: This yields for the renormalized two-loop self-energies of φ 1 and φ 2 : where it is understood that the unrenormalized self-energies at two-loop order also contain the contributions arising from the subloop renormalization. The expressions δV φ 2 and δV (2) φ 1 φ 2 are the two-loop counterterm contributions from the Higgs potential: with the electroweak mixing angle s 2 The counterterms are fixed by imposing on-shell renormalization conditions for the renormalized self-energies. For the A boson this reads: The tadpole conditions are:t The conditions for the tadpoles have the consequence that the v i remain the minima of the Higgs potential also at the two-loop level. The resulting expressions for the renormalization constants contributing to the leading two-loop corrections to the neutral CP-even Higgs-boson masses, expressed in terms of unrenormalized self-energies and tadpoles, are given in Sec. 3.1.

The scalar quark sector of the MSSM
Renormalization in the squark sector is needed in the present calculation at the one-loop level, i.e. at O(α s ). As above, we work in the on-shell scheme. In the following the formulas are written for one flavor.
The squark mass term of the MSSM Lagrangian is given by where Z = The renormalization of the fields, the masses, and the mixing angle is then performed viaf In the mass eigenstate basis, the field renormalization reads: with The renormalized diagonal and non-diagonal self-energies in this basis have the following structure:Σf We impose the following on-shell renormalization conditions: which determines the renormalization constants to be The unsymmetric renormalization condition (51) is chosen for convenience since it leads to δZf 2 = δZf 1 and accordingly to δZf 12 = 0, which simplifies the expression for the counterterm of the mixing angle. In eq. (52) we have imposed the condition that the non-diagonal selfenergy vanishes at q 2 = m 2 f 1 . Alternatively one could choose q 2 = m 2 f 2 , instead; the numerical difference arising from these different choices is irrelevant for the results of the Higgs-boson masses, as we have checked explicitly.
Taking into account that neither δµ nor δ tan β are of O(α s ), one obtains from eq. (37): For completeness we also list the expression for the quark mass counterterm in the onshell scheme, where the scalar functions in the decomposition of the fermion self-energy Σ f (p) are defined according to 3 Calculation of the neutral CP-even Higgs-boson masses 3.1 Leading two-loop contributions to the Higgs-boson self-energies The dominant one-loop contributions to the Higgs-boson mass matrix in eq. (10) are given by terms of the form G F m 4 t ln(mt 1 mt 2 /m 2 t ), which arise from t-andt-loops. They can be obtained by evaluating the contribution of the t-t-sector to the φ 1,2 self-energies at zero external momentum from the Yukawa part of the theory (neglecting the gauge couplings). Accordingly, the leading contributions to the one-loop corrected Higgs-boson masses are derived by diagonalizing the matrix where theΣ (1) denote the one-loop Yukawa contributions of the t-t-sector to the renormalized one-loop φ 1,2 self-energies. For completeness, we list here the explicit form of these dominant one-loop corrections (in the numerical results given in Sec. 4 we use the complete one-loop on-shell result as given in Ref. [5]): By comparison with the full one-loop result [4,5,6] it has been shown that these contributions indeed contain the bulk of the one-loop corrections. They typically approximate the full one-loop result within 5 GeV.
In order to derive the leading two-loop contributions to the masses of the neutral CPeven Higgs bosons we have evaluated the QCD corrections to eq. (61) [13,14]. Accordingly, we have calculated the O(αα s ) contribution of the t-t-sector to the φ 1,2 self-energies at zero momentum transfer, neglecting the gauge couplings. Because of the large value of the strong coupling constant these are expected to be the most sizable two-loop corrections (see also Ref. [11]).

Evaluation of the relevant Feynman diagrams
The calculations have been performed using Dimensional Reduction (DRED) [17], which is necessary in order to preserve the relevant SUSY relations. Naive application (without an appropriate shift in the couplings) of Dimensional Regularization (DREG) [18], on the other hand, does not lead to a finite result. The same observation has also been made in Ref. [11]. The Feynman diagrams contributing to the φ 1 , φ 2 and A self-energies are depicted in The gluon-exchange contribution of O(α s ) to the quark mass counterterm reads in DRED: where 2δ = 4 − n with n the space-time dimension, γ E is Euler's constant, and µ is the 't Hooft scale. The explicit form of the other counterterms of the quark and scalar quark sector can be found in Ref. [19]. Some of the diagrams shown in Figs We now briefly describe the evaluation of the two-loop diagrams. As explained above, the calculation involves irreducible two-loop diagrams at zero momentum-transfer and counterterm diagrams. In deriving our results we have made strong use of computer algebra tools: the diagrams were generated with the Mathematica package FeynArts [20]. For this purpose we have implemented a model file which contains the relevant part of the MSSM Lagrangian, i.e. all SUSY propagators (t 1 ,t 2 ,b 1 ,b 2 ,g) needed for the QCD-corrections and the appropriate vertices (Higgs boson-squark vertices, squark-gluon and squark-gluino vertices). The program inserts propagators and vertices into the graphs in all possible ways and creates the amplitudes including all symmetry factors. The evaluation of the two-loop diagrams and counterterms was performed with the Mathematica package TwoCalc [21]. By means of two-loop tensor integral decompositions it reduces the amplitudes to a minimal set of standard scalar integrals, consisting in this case of products of the basic one-loop integrals A 0 , B 0 [22] (the B 0 functions originate from the counterterm contributions only) and the two-loop function T 134 , which is the genuine two-loop scalar integral at zero momentumtransfer (vacuum integral). This integral is known for arbitrary internal masses and admits a compact representation for δ → 0 in terms of logarithms and dilogarithms (see for instance Ref. [23]). It should be noted that from the expansion of the one-loop two-point function B 0 , only the term B fin 0 contributes, while B δ 0 drops out in our final result. From the output generated with TwoCalc a FORTRAN code was created which allows a fast calculation for a given set of parameters. This code has been implemented into the FORTRAN program FeynHiggs [24], see below.
Our results for the two-loop φ 1,2 self-energies are given in terms of the SUSY parameters tan β , M A , µ, mt 1 , mt 2 , θt, and mg. In the general case the results are by far too lengthy to be given here explicitly. In the special case of vanishing mixing in thet-sector, µ = 0, and mt 1 = mt 2 = mt, a relatively compact expression can be derived which is given in Ref. [13]. We have performed an expansion of this result for large values of mg. It yields for the leading termsΣ This shows that the gluino does not decouple from the two-loop result, contrary to the case of the two-loop QCD contributions to the ρ-parameter in the MSSM [19,25]. In Ref. [11] a result for the limiting case has been given. In this limit we obtain

Determination of the Higgs-boson masses
In the Feynman-diagrammatic approach the Higgs-boson masses are derived beyond tree level by determining the poles of the h − H-propagator matrix whose inverse is given by where again theΣ denote the renormalized Higgs-boson self-energies, now in the h−H-basis.
Determining the poles of the matrix ∆ Higgs in eq. (76) is equivalent to solving the equation In our calculation the complete one-loop result for the Higgs-boson self-energies in the on-shell scheme [5] is combined with the leading two-loop contributions, which have been outlined in the previous section. The matrix eq. (76) therefore contains the renormalized Higgs-boson self-energiesΣ where the momentum dependence is neglected only in the two-loop contribution.
Since the two-loop contribution has been calculated in the φ 1 -φ 2 -basis, a rotation into the h-H-basis, according to eq. (4), has to be performed: We have implemented two further corrections beyond O(αα s ) into the prediction for m h , which are illustrated in Figs. 6, 7, 9 and 10. The leading two-loop Yukawa correction of O(G 2 F m 6 t ) is taken over from the result obtained by renormalization group methods. It reads [8,15] The second higher-order contribution which has been implemented concerns leading QCD corrections beyond two-loop order, taken into account by using the MS top mass 2 for the two-loop contributions instead of the pole mass, m t = 175 GeV. In thet mass matrix, however, we continue to use the pole mass as an input parameter. Only when performing the comparison with the RG results we use m t in thet mass matrix for the two-loop result, since in the RG results the running masses appear everywhere. This three-loop effect gives rise to a shift up to 1.5 GeV in the prediction for m h .
The complete one-loop calculation together with the leading two-loop corrections and the other corrections beyond O(αα s ) have been implemented into the FORTRAN code Feyn-Higgs [24]. This code can be linked to existing programs as a subroutine, thus providing an accurate calculation of m h and m H which can be used for further phenomenological analyses. FeynHiggs is available via its WWW page http://www-itp.physik.uni-karlsruhe.de/feynhiggs.
The scalar top masses and the mixing angle are related to the parameters Mt L , Mt R and M LR t of thet mass matrix, which reads with   More relevant for todays' colliders is the mass of the lighter Higgs boson, m h , on which we will focus in the following discussion. In Fig. 9 m h is shown in the two scenarios with tan β = 1.6 and tan β = 40 as a function of mq for no mixing and maximal mixing and for M A = 200, 1000 GeV. The tree-level, the one-loop and the two-loop results with the two corrections beyond O(αα s ) are shown (the values of mq are such that the corresponding t-masses lie within the experimentally allowed region). In all scenarios of Fig. 9 see Fig. 11. However, the location of the maximal Higgs-boson mass, depending on M LR t , is shifted towards smaller values, typically by about 40%. The numerical difference in m h in the two splitting scenarios Mt L /Mt R = 300/1000 and Mt L /Mt R = 1000/300 is small. They differ by up to 2 GeV only in the large tan β scenario when M LR t > 1000 GeV.
The variation of m h with m t is rather strong. The scenarios for no mixing and maximal mixing and for tan β = 1.6 and tan β = 40 are shown in Fig. 12, where m t is varied around    Varying tan β around the value tan β = 1.6 has a relatively large impact on m h (higher values for m h are obtained for larger tan β ), while the effect of varying tan β around tan β = 40 is marginal. This is shown in Fig. 13 for M A = 200, 1000 GeV, mq = 200, 1000 GeV for the no-mixing and the maximal-mixing scenario. For tan β > 15 the variation is less than 1 GeV 3 .
In Fig. 14  The dependence of m h on µ, the Higgs mixing parameter, is depicted in Fig. 15. The parameter µ enters via the non-diagonal Higgs-squark coupling at one-and two-loop order and via the chargino and neutralino sector in the one-loop self-energies. It should be noted that for the plots in Fig. 15 we have set m b = 0 GeV, thus suppressing the contribution of the b −b-sector. The reason is that for large µ and for large tan β some Higgs-sbottom couplings can become rather large, which makes the perturbative calculation questionable in this case. The variation of m h with µ in Fig. 15  Finally we show the dependence on the gluino mass, mg, which enters only at the twoloop level. Fig. 16 depicts m h as a function of mg in the scenarios with mq = 500, 1000 GeV for tan β = 1.6 and tan β = 40 in the no-mixing and the maximal-mixing case. Small variations below 1 GeV occur in the no-mixing scenario, while change in m h up to 4 GeV arises in the maximal-mixing scenario. m h reaches a maximum at about mg ≈ 0.8 mq. Since the parameter mg is absent in the RG approach, a variation of m h with mg can directly be seen as a deviation of the diagrammatic result from the RG result, see Sec. 4.3.
As pointed out in Ref. [14] it is desirable to express the predictions for the observable m h in terms of other physical observables. This provides the possibility to directly compare results obtained by different approaches making use of different renormalization schemes. Therefore we show in Fig. 17 the dependence of m h on the parameters mt 1 , mt 2 and θt, which, since we are working in the on-shell scheme, directly correspond to the physical ones. We show m h as a function of mt 2 for ∆mt = 0 GeV and θt = 0 (no mixing) and for ∆mt = 340 GeV and θt = −π/4 (maximal mixing), where ∆mt ≡ mt 2 − mt 1 . The choice of ∆mt ≈ 340 GeV corresponds to M LR t /mq ≈ 2 in terms of the soft SUSY breaking    parameters. The two-loop results shown here contain also the corrections beyond O(αα s ).
In these plots we have furthermore imposed the ρ -parameter constraint: We have required that the contribution of the third generation of scalar quarks to the ρ-parameter [19,25] does not exceed the value of 1.3 · 10 −3 , which corresponds approximately to the resolution of ∆ρ when it is determined from experimental data [28].

Upper bound for m h as a function of tan β
Since, as shown in Fig. 13, smaller values for m h are obtained for small tan β , this part of the parameter space can to a large extend be covered at todays' colliders. The discovery limit for m h at LEP2 is expected to be slightly above 100 GeV [29].   impression about the maximally possible values for m h . The choice M = µ = 0 GeV is experimentally excluded. We nevertheless use these values since the difference in m h to the case with experimentally not excluded M and µ is very small, typically below 0.5 GeV.
In Fig. 18 we show the maximal Higgs-boson mass value, including also the corrections beyond O(αα s ), as a function of tan β ; the other parameters are chosen according to eqs. (88). For the top-quark mass the most recent experimental value m t = 173.8 GeV [30] is chosen and, since m h grows with increasing m t , the experimental value plus one and plus two standard deviations (m t = 178.8, 183.8 GeV) 5 . The common squark mass parameter is chosen to be mq = 1000 GeV as a high, and mq = 2000 GeV as a very high value. On the left side of Fig. 18 we show the full tan β range (tan β ≤ 50), whereas on the right side we focus on the range especially interesting for LEP2 and the upgraded Tevatron (tan β ≤ 5).
In the tan β In the interesting region around tan β = 1.6 the covered region of the tan β -parameter space depends strongly on the maximally accessible energy of todays' colliders, see Tab. 1. For an exclusion limit of m h > 107 GeV, for instance, LEP2 covers tan β < 1.6 completely only if m t is constrained to its present 1 σ limit. On the other hand, taking a very conservative point of view and choosing m t at the 2 σ bound, no limit on tan β can be set, even for mq = 1000 GeV.
One should keep in mind, however, that the Higgs-boson masses depicted in Fig. 18 are the maximally possible upper values, i.e. for smaller mixing in thet-sector the region tan β < 1.6 can be covered by LEP2 for all other sets of parameters. One can also see that a precise measurement of m t is decisive in order to set stringent bounds on tan β in the MSSM.
In conclusion, our results confirm that for the scenario with tan β = 1.6 the parameter space of the MSSM can be covered to a very large extent. Only for maximal mixing, very large soft SUSY breaking parameters in thet-sector and m t at its upper (1 − 2)σ limit the light Higgs boson can escape the detection at LEP2. For increasing tan β , however, the parameter space in which the Higgs boson is not accessible at LEP2 increases rapidly.
Concerning the large tan β region, LEP2 and the upgraded Tevatron can probe only the region of no mixing in thet-sector. The LHC and a future linear e + e − -collider are needed in order to test the parameter space with larget-mixing.
In an analogous way we have also analyzed the maximal value for m h as a function of the physical parameters, see Fig. 19 and Tab. 2. We have chosen mt 2 = 1000, 2000 GeV, ∆mt = 340 GeV and θt = −π/4. The other MSSM parameters are chosen according to eqs. (88). Fig. 19 shows the result for the maximal range of tan β and for the interesting range for LEP2, tan β ≤ 5. The results in Fig. 19 and Tab. 2 are slightly lower than the values obtained with the unphysical input parameters. This is due to the fact that the values obtained for the squark masses in the first scenario (for all other parameters chosen to be equal) are always larger than for the latter case with physical input parameters. The analysis of the upper bound of m h , however, can be taken over directly from the case with unphysical input parameters.

Numerical comparison with the RG approach
We now turn to the comparison of our diagrammatic results with the predictions obtained via RG methods. For this comparison we made use of the FORTRAN code corresponding to Ref. [9], except for the one-loop results in Figs. 20 and 21, where we used the code described in Ref.
[10] 6 .   We begin with the case of large values of M A , for which the RG approach is most easily applicable and is expected to work most accurately. In order to study different contributions separately, we have first compared the diagrammatic one-loop on-shell result [5] with the one-loop leading log result (without renormalization group improvement) given in Ref. [10]. Since the available code uses the running top mass m t = m t (m t ) ≈ 167.3 GeV we have also used this top mass for the full diagrammatic one-loop calculation. In Fig. 20  In the next step of comparison we analyzed the no-mixing case at the two-loop level: we have compared our diagrammatic result for the no-mixing case, including the Yukawa correction and the running top mass effect, with the RG results obtained in Ref. [9]. We have adopted the scale M (the SU(2) gaugino mass parameter) as M = mq, in order to treat it in the same way as it has been done in the RG approach. As can be seen in Fig. 22, after the inclusion of the corrections beyond O(αα s ) the diagrammatic result for the nomixing case agrees very well with the RG result. For the scenario with M A = 1000 GeV the deviation between the results exceeds 2 GeV only for tan β = 1.6 and mq < 150 GeV. For    Therefore we switch from the set of unphysical parameters to a set of physical parameters: In Fig. 25 we compare the results for the lightest Higgs-boson mass, obtained by the Feynmandiagrammatic method and by the RG method, in terms of this new set of parameters: m h is shown as a function of mt 2 with the mass difference ∆mt ≡ mt 2 − mt 1 and the mixing angle θt as further input parameters. In the context of the RG approach the runningtmasses, derived from thet mass matrix, are considered as an approximation for the physical masses. In our approach, on the other hand, since we are working in the on-shell scheme, thet-masses and the mixing angle directly correspond to physical parameters. In Fig. 25 we have furthermore implemented the same ∆ρ constraints on the range of the third generation scalar quark masses as in Fig. 17.
Similarly to the comparison shown in Fig. 23 and 24, very good agreement is found in Fig. 25 between the results of the two approaches in the case of vanishingt-mixing. The deviation is typically less than 1 GeV and never exceeds 2 GeV. Using the physical parameters as input, the maximal-mixing scenario is realized by setting θt = −π/4 and ∆mt ≈ 340 GeV (i.e. thet-masses obtained for M LR t /mq ≈ 2 have a mass difference of about 340 GeV.) In this scenario again (as in Figs. 23 and 24) the diagrammatic result yields values for m h which are higher by about 5 GeV. The peaks in the plots for M A = 1 TeV and maximal mixing in thet-sector around mt 2 = 660 GeV are again due to the threshold M A = mt 1 + mt 2 in the one-loop contribution, originating from the stop-loop diagram in the A self-energy.

Conclusions
Using the Feynman diagrammatic method we have calculated the leading O(αα s ) corrections to the masses of the neutral CP-even Higgs bosons in the MSSM. The two-loop result has been implemented into the prediction based on the complete diagrammatic one-loop on-shell result. Two further corrections beyond O(αα s ) have been added in order to incorporate leading electroweak two-loop and higher-order QCD contributions. The results have been obtained using the on-shell scheme, which means a renormalization of all sectors of the MSSM at one-loop order and of the Higgs-boson sector at two-loop order. In our two-loop calculation we have imposed no restrictions on the parameters of the Higgs and scalar top sector of the model. Thus the results are valid for arbitrary values of the relevant MSSM parameters. The complete result has been implemented into the FORTRAN program FeynHiggs [24] which is available via its WWW page http://www-itp.physik.uni-karlsruhe.de/feynhiggs . In this way we provide the at present most precise prediction for m h and m H based on Feynman-diagrammatic calculations.
The two-loop corrections lead to a large reduction of the one-loop on-shell result. We have performed a detailed analysis of the dependence of m h on the various MSSM parameters.  Figure 25: Comparison between the Feynman-diagrammatic calculations and the results obtained by renormalization group methods [9] in terms of physical parameters. The mass of the lightest Higgs boson is shown for the two scenarios with tan β = 1.6 and tan β = 40 and for M A = 200, 1000 GeV as a function of the heavier physicalt mass mt 2 . For the curves with θt = 0 a mass difference ∆mt = 0 GeV is taken, whereas for θt = −π/4 we choose ∆mt = 340 GeV, for which the maximal Higgs-boson masses are achieved.
Concerning the scalar top sector the analysis has been carried out in terms of the (unphysical) soft SUSY breaking parameters Mt L , Mt R and M LR t as well as in terms of the physical parameters mt 2 , ∆mt ≡ mt 2 − mt 1 and θt.
A scan over the parameters µ, M, mg, M A and M LR t has been performed in order to determine the maximally possible value for m h as a function of tan β . Our results show that for the scenario with tan β = 1.6 the parameter space of the MSSM can be covered almost completely. Only for maximal mixing, very large soft SUSY breaking parameters in thet-sector and m t at its upper experimental limit the light Higgs boson can escape the detection at LEP2 in this scenario. Concerning the large tan β region, LEP2 and the upgraded Tevatron can probe only the region of no mixing in thet-sector.
We have compared our results, obtained by a Feynman diagrammatic calculation (where also the corrections beyond O(αα s ) have been included), with the results obtained via RG methods. Concerning the one-loop contributions we find very good agreement between these two approaches. The same is valid for the two-loop corrections in the case of vanishing mixing in thet-sector. On the other hand, in the case of non-vanishing mixing sizable deviations between the two approaches occur. For moderate mixing they reach up to 5 GeV, for |M LR t /mq| > ∼ 2.5 they can be very large. In the diagrammatic approach the maximal value for m h is reached for M LR t /mq ≈ ±2, whereas the RG results have a maximum at M LR t /mq ≈ ±2.4, i.e. at the one-loop value. This holds for all combinations of tan β , mq and M A . The fact that the parameter mg is absent in the RG results can give rise to an additional deviation between the two approaches of about ±2 GeV.
We have furthermore discussed the issue of how results obtained via different approaches using different renormalization schemes can be readily compared to each other also when corrections beyond one-loop order are incorporated. For this purpose it is adequate to express the prediction for the Higgs-boson masses in terms of other physical observables, i.e. the physical masses and mixing angles of the model. Accordingly, we have compared the results obtained by our diagrammatic two-loop calculation with those obtained by RG methods in terms of the physical observables mt 2 , ∆mt ≡ mt 2 − mt 1 and θt. As for the comparison in terms of the unphysical parameters, we have found good agreement for the case of vanishing mixing in thet-sector. For large splitting between thet-masses, however, the Higgs-boson masses obtained by the Feynman diagrammatic calculation are about 5 GeV larger than the ones calculated in the RG approach.