Precise prediction of the MSSM Higgs boson masses for low MA

Precise predictions for Higgs boson masses in the Minimal Supersymmetric Standard Model can be obtained by combining fixed-order calculations with effective field theory (EFT) methods for the resummation of large logarithms in case of heavy superpartners. This hybrid approach is implemented in the computer code FeynHiggs and has been applied in previous studies for calculating the mass of the lightest CP-even Higgs boson for low, intermediate and high SUSY scales. In these works it was assumed that the non-standard Higgs bosons share a common mass scale with the supersymmetric squark particles, leaving the Standard Model as the low-energy EFT. In this article, we relax this restriction and report on the implemention of a Two-Higgs-Doublet Model (THDM) as effective theory below the SUSY scale into our hybrid approach. We explain in detail the various steps how our EFT calculation is consistently combined with the fixed-order calculation within the code FeynHiggs. In our numerical investigation we find effects on the mass of the lightest CP-even Higgs boson h of up to 9 GeV in scenarios with low $M_A$ , low $\tan\beta$ and high SUSY scales, when compared with previous versions of FeynHiggs. Comparisons to other publicly available pure EFT codes with a THDM show good agreement. Effects on the mass of the second lightest CP-even Higgs boson H are found to be negligible in the phenomenologically interesting parameter regions where H can be traded for h as the experimentally observed Higgs particle.


Introduction
Precise measurements of the properties of the Higgs boson, discovered by the ATLAS and CMS collaborations at the CERN Large Hadron Collider [1,2] in 2012, are not only crucial for testing the Standard Model (SM) but also allow to constrain physics beyond the Standard Model. Supersymmetric extensions of the SM are theoretically well motivated, in particular the Minimial Supersymmetric Standard Model (MSSM) with quite specific predictions for Higgs bosons.
In the MSSM, the Higgs sector consists of two Higgs doublets, with vacuum expectation values (vevs) v 1 and v 2 which can be chosen real and non-negative without loss of generality. After electroweak symmetry breaking, the two Higgs doublets accommodate five physical Higgs states: the light and heavy CP-even h and H bosons, the CP-odd A boson, and the pair H ± of charged Higgs bosons. At the tree level, all Higgs boson masses are determined by two parameters, conventionally chosen to be tan β = v 2 /v 1 and the mass of the A boson, M A . These tree-level relations, however, are affected by large higher-order corrections resulting from the quantum effects of the MSSM.
Since no direct evidence for SUSY particles has been found so far, the range of MSSM parameters can only be constrained indirectly. In addition to the classic set of precision observables (Z and W boson masses, effective electroweak mixing angle, . . . ), the mass of the Higgs boson, determined by ATLAS and CMS [3] to be 125.09 ± 0.24 GeV, can serve as an additional powerful precision observable. When interpreted as the mass of a light Higgs boson within the MSSM spectrum, it is very sensitive especially to the parameters of the top-squark sector and can therefore be used to assess the SUSY scale. In the light of the high level of precision reached by the experiments, an accurate and reliable theoretical prediction is essential.
The advantage of EFT methods is a resummation of logarithms which become large if the relevant scales are widely separated, like in the case of a high SUSY scale M SUSY . Fixed-order calculations become unreliable for such wide scale separations, since the appearance of large logarithms can spoil the perturbative expansion. Fixed-order calculations, on the other hand, capture all terms with inverse powers of M SUSY . Though suppressed in case of a high scale, they can become relevant for lower M SUSY and thus are needed for accurate predictions as well. These terms are missed in EFT calculations, at least when no higherdimensional operators are taken into account (see [40] for a study including higher-dimensional operators).
In order to obtain results as accurate as possible for all SUSY scales, hybrid methods have been developed [41][42][43][44][45][46]. In [41,42,45] the strategy has been pursued to incorporate an EFT calculation on top of a fixed-order calculation. Additional subtraction terms are introduced to avoid double counting of terms contained in both the EFT and the fixed-order calculation. The method has been implemented in the publicly available computer code FeynHiggs [7,13,[47][48][49]. So far, the EFT calculation in that approach was restricted to scenarios in which all non-SM particles share a common mass scale (with the only exception of possibly light electroweakinos and/or a light gluino), leaving the SM as the low-energy EFT.
In this paper, we report on an improvement of this method by introducing a Two-Higgs-Doublet Model (THDM) as the effective theory below the SUSY scale, in replacement of the SM. This setup allows to cover the possibility of light non-standard Higgs bosons in the EFT calculation. Also scenarios with additional light electroweakinos are considered, which are especially interesting in view of the increasingly tight constraints on colored SUSY particles from the LHC. We give a detailed description of the steps needed to combine the THDM EFT calculation with the fixed-order calculation and illustrate the impact of the new hybrid version on the Higgs boson masses by numerical comparisons with previous versions of FeynHiggs. An earlier pure EFT study [38] with an effective THDM found potentially large effects originating from the resummation of logarithms of the SUSY scale over M A , and observed significant differences with respect to FeynHiggs in specific parameter regions. We will clarify this situation by a detailed comparison and explain the current differences between [38] and our new THDM-improved hybrid calculation.
The outline of this paper is as follows: In Section 2, we detail on the EFT calculation. Subsequently, we describe the consistent combination with the fixed-order part in Section 3. In Section 4, we compare our approach to that of other publicly available codes. This is followed by a numerical analysis in Section 5, with conclusions in Section 6. The sections A to E of the Appendix provide additional technical information.

EFT calculation of Higgs-boson masses
If the SUSY particles are significantly heavier than all SM particles, they can be integrated out. In the simplest case when all non-standard particles occur at a common mass scale, the SUSY scale M SUSY , the remaining EFT is the SM, with the Higgs self-coupling determined via matching conditions at M SUSY . This self-coupling and all the other remaining SM couplings are evolved from the SUSY scale down to the electroweak scale by means of renormalization group equations (RGEs); fixing the remaining SM couplings at the electroweak scale by matching to physical observables determines the input quantitites from which the SM Higgs-boson mass can be calculated. This approach has the advantage that large logarithmic contributions are resummed. On the other hand, terms suppressed by M SUSY are missed unless higherdimensional operators are included in the EFT Lagrangian.
The assumption that all non-standard particles have a common mass scale is quite restrictive. For a better flexibility and wider applicabilty, more refined EFTs have to be considered. In our approach, we allow for several independent mass scales where each of them corresponds to the appearance of distinguished new phenomena. To be specific, we take into account five mass scales in our EFT calculation: the SM scale M t , the non-standard Higgs-boson scale M A , the electroweakino scale M χ , the gluino mass scale Mg and the SUSY scale M SUSY . We define the SUSY scale to be the mass scale of sfermions, which we assume to be approximately mass degenerate. Below M SUSY , sleptons and squarks are removed from the EFT; below Mg, we remove the gluino. The electroweakino scale M χ is defined by where M 1,2 are the soft-breaking electroweakino mass parameters and µ is the Higgsino mass parameter. Below M χ , we remove the electroweakinos from the EFT. M A marks the scale at which the heavy Higgs bosons are integrated out. We assume M A and M χ to be smaller or equal to M SUSY . Therefore, we have a set of eight different EFTs: the SM, the SM plus electroweakinos, the THDM and the THDM plus electroweakinos (plus the same with added gluino). This diversity leads to various different hierarchies, as illustrated in Fig. 1.
In our EFT calculation, we take into account full one-loop threshold corrections and full two-loop RGEs. This implies a full LL and NLL resummation. Additionally, we include O(α s α t ) matching conditions for the Higgs self-couplings. O(α 2 t ) threshold corrections for matching the THDM to the MSSM are currently not known. Moreover, three-loop RGEs for the THDM are not yet available. Since the SM three-loop running is negligible, one may believe that this also holds for the three-loop THDM running [38]. Nevertheless, the resummation of NNLL contributions is incomplete.

Relevant EFTs
Here we give a brief overview of the various EFTs appearing in our calculation and specify our notation. We will not describe EFTs with gluino, since the presence of the gluino does not induce any effective couplings that are relevant at the order of the calculation presented in this paper. It, however, does alter the RGEs (see [42]).

The Two-Higgs-Doublet Model
Decoupling all sfermions, gauginos and Higgsinos from the full MSSM leads to a THDM as the remaining effective theory below the SUSY scale. The THDM Higgs potential can be written as follows, where Φ 1,2 denote the two doublets of scalar fields. Since we consider only the real MSSM, all the coefficients can be chosen as real parameters. At the minimum of the potential each Higgs field Φ i acquires a vacuum expectation value (vev), Decomposing the Higgs fields into components according to introducing the quantities and expanding the potential around the minimum yields the mass matrix of the CP-even neutral Higgs bosons, with the entries where the abbreviations for a generic angle γ have been introduced. With the minimum conditions for the Higgs potential, m 2 1 and m 2 2 can be eliminated; the following relations for the masses of the CP-odd neutral A boson and of the charged H ± bosons are obtained, and the CP-even mass matrix M 2 φφ can be cast into the following form, The tree-level mass eigenstates h and H are obtained by a rotation, with the angle α determined by Often, it is useful to work in the Higgs basis instead of the h, H mass eigenstate basis [50]. It is obtained by rotating the original doublets Φ 1,2 by the angle β, In this basis, only H 1 acquires a vev, and the mass matrix (13) is transformed into with To get from the Higgs basis to the mass eigenstate basis, we have to rotate by the angle α − β.
We also need the Yukawa part of the effective THDM Lagrangian, which is given by with the third-generation quark doublet Q L and the Pauli matrix σ 2 . h t and h t are the effective top Yukawa couplings. All other Yukawa couplings are neglected in the EFT calculation; they are, however, fully captured through the diagrammatic calculation at the one-loop level, in case of the bottom Yukawa coupling also at the two-loop level.
As already noted in [38], the effective THDM with the Yukawa texture as given in Eq. (25) is not a type II model where only Φ 2 couples to up-type quarks. Although the tree-level Yukawa sector of the MSSM is that of a THDM of type II, loop corrections induce also a coupling of Φ 1 to the top-quark, which enters through the matching procedure in the effective THDM. Differently from [38], we take this coupling fully into account in all the affected RGEs and threshold corrections. Hence, we have to deal with 12 coupling constants, consisting of three gauge couplings, seven Higgs self-couplings, and two Yukawa couplings. We derived the RGEs for the considered THDM using the Mathematica package SARAH [51]. The corresponding expressions are available from the authors upon request.

The Two Higgs-Doublet Model with Electroweakinos
If in addition to the non-SM Higgs bosons also light electroweak gauginos and Higgsinos (EWinos) are present, the effective Lagrangian below the scale M SUSY is the one of the THDM described above, extended by extra mass and interaction terms with Φ SM being the SM-like Higgs doublet, The scalar potential in the SM part of the Lagrangian is given by g 1u,1d,2u,2d are the effective Higgs-Higgsino-Gaugino couplings of the SM+EWinos, in obvious notation. The number of couplings is reduced to 8. Two-loop RGEs can be found in [35]. Below the electroweakino scale the effective model is eventually the pure SM.

Matching the EFTs
After having specified the various EFTs, we describe how they are matched to each other. To derive the matching conditions, we have to compare physical amplitudes with external light particles computed in the EFT valid below the matching scale and in the full model (or the more complete EFT) valid above the matching scale. The difference between the physical amplitudes has to be absorbed by adapting the effective couplings in the particular EFT that is to be matched.
Terms contributing to the matching conditions arise from different vertex corrections and from different normalizations of the external fields. The part coming from the vertex corrections is obtained by calculating the vertex functions in the high-energy and the low-energy theory. The difference can then directly be absorbed into the effective coupling of the low-energy theory. At least at the one-loop level, at which we mostly work, this procedure is straightforward. Therefore, we will not go into more details.
If all external fields are non-mixed mass eigenstates, the external leg corrections are just given the corresponding LSZ factors, the wave-function renormalization. The difference between the LSZ factors in the high-energy and the low-energy theory has again to be absorbed by the low-energy effective coupling.
In case of mixing in the external fields, a more careful discussion is required. Even when the external fields are diagonal at the tree level, loop contributions to the two-point vertices induce mixing between the mass eigenstates at higher orders. This transition has to be included as further external leg corrections, in addition to the LSZ factors. In the MSSM and the THDM, the mixing between the CP-even Higgs bosons h, H is the important issue. It is ascribed to a non-diagonal self-energy Σ hH .
Conveniently, all external leg corrections can be written in form of a single matrix, the Z-matrix (see [7] for more details). It gives the relation between the external, asymptotical-free physical states and the treelevel mass eigenstates used for the calculation of the vertex correction. At the one-loop level, the MSSM relation reads where we used the symbol to mark MSSM quantities. Σ hh and Σ HH are the diagonal self-energies entering the LSZ factors. The prime denotes the derivative with respect to the external momentum. The corresponding relation in the THDM is written as follows, where we used the symbol to mark THDM quantities. Eqs. (32) and (33) yield the relation between the mass eigenstates of the MSSM and the THDM, where the ∆Σ xy summarize the differences between the self-energies, for x, y ∈ {h, H}, The mass eigenstates are related to the original field components φ 1,2 via Eq. (17), With these relations, Eq. (34) can be transformed into a relation between the component fields φ 1,2 , In the THDM, the mixing angle α is a free parameter. We fix it at lowest order by requiring that the THDM fields φ 1,2 are aligned with the MSSM fields φ 1,2 . Consequently, the two mixing angles α and α are equal at the tree level. At the one-loop level we change the tree-level basis of the THDM slightly allowing for a small misalignment between the THDM and the MSSM fields, Using this shift to replace α by α in Eq. (38) we obtain, expanded up to the one-loop level, Next we expand ∆Σ hH (m 2 All higher order derivatives of the ∆Σ xy are suppressed by M SUSY and therefore negligible in the EFT calculation. For the same reason, we drop the specification of the external momentum in all derivatives of ∆Σ xy in the following (which is always taken at m 2 h ). Using the expansion (41) and partly rewriting the self-energies yields with the notation ∆Σ ij ≡ ∆Σ φiφj for i, j ∈ {1, 2). The second matrix corresponds to the one-loop part of a unitary matrix and thereby to a basis transformation by a rotation. It can be absorbed by adjusting ∆α according to The first matrix in Eq. (42) is not unitary and hence cannot be removed by a basis transformation. Therefore, there is a remaining difference between the normalization of the φ 1,2 fields in the MSSM and the THDM, given by the following relation, which corresponds to the one used in [52]. As noted above, it is only valid at the one-loop level. We have to take care of this relation whenever we match a coupling involving an external Higgs field. This is achieved by rescaling the Higgs doublets of the THDM (or the MSSM) according to Eq. (44).
Since we rescale the whole doublets, a relation similar to Eq. (44) also holds for the vevs, This directly implies or respectively, with H 1,2 being the fields of the Higgs basis defined in Eq. (19). Following this procedure and including vertex corrections, we derived a full set of one-loop threshold corrections for all appearing effective couplings and hierarchies. Below, we list only the tree-level matching conditions and the dominant one-loop corrections, i.e. those proportional to the strong gauge coupling or the top Yukawa couplings. Full one-loop threshold corrections for all effective couplings including electroweak contributions are listed in App. A.
In addition to the calculation of matching conditions, we will also need Eq. (44) for combining the diagrammatic fixed-order calculation and the EFT calculation.

Matching the THDM to the MSSM
The Higgs self-couplings in the THDM scalar potential are fixed at the tree level by [52] where g and g are the electroweak gauge couplings. At one-loop order these relations receive additional contributions [52], A t is the stop trilinear coupling and h t the top Yukawa coupling of the MSSM. 1 The factor k ≡ (4π) −2 is used to mark the loop-order. In addition to these one-loop corrections, we also include O(α s α t ) threshold corrections, listed in App. A.7. Forμ = 1, the effective top Yukawa couplings are given by The full expressions forμ = 1 are given in App. A. The threshold correction for tan β is obtained from Eq. (47) yielding Matching the THDM+EWinos to the MSSM Neglecting the weak gauge couplings, the relations for matching the THDM to the MSSM are also valid when the THDM+EWinos is matched to the MSSM. The additional effective Higgs-Higgsino-Gaugino couplings of the THDM+EWinos fulfill the tree-level relationŝ Matching the THDM to the THDM+EWinos Matching the THDM to the THDM+EWinos, the Higgs self-couplings, the gauge couplings, the top Yukawa couplings and t β are not modified at the tree level. If the weak gauge couplings are neglected, there are also no loop corrections. The full one-loop corrections including the weak gauge couplings are listed in App. A.

Matching the SM to the THDM
In this specific case, the characteristic scale for all the couplings below is the mass M A . In the decoupling , which is assumed when the heavy Higgs bosons are integrated out, the Higgs self-coupling λ of the SM is obtained by with c 11 from Eq. (22), β = β THDM , and the one-loop correction The THDM top Yukawa couplings are related to the SM top Yukawa coupling y t via As stated above, the SM top Yukawa coupling is extracted from the top pole mass at the scale M t . The top Yukawa couplings of the THDM are then determined by numerically solving the system of RGEs with the boundary conditions given in Eqs. (67), (59) and (60) (see also Eqs. (108) and (109) for more general expressions).

Matching the SM+EWinos to the THDM+EWinos
Neglecting the weak gauge couplings, the relations for matching the SM to the THDM are also valid when the SM+EWinos is matched to the THDM+EWinos. At the tree level, the effective Higgs-Higgsino-Gaugino couplings of the SM+EWinos and the THDM+EWinos are related bỹ One-loop corrections proportional to the electroweak gauge couplings can be found in App. A.

Matching the SM to the SM+EWinos or the MSSM
The matching conditions of the SM to the SM+EWinos or to the MSSM are well-known and can be found in [35,37].

Matching EFTs without and with gluino
If the gluino is integrated out, no threshold corrections arise at the one-loop level. At the two-loop level however, the matching conditions of the scalar self-couplings between the THDM and the MSSM are modified if a gluino is added to the THDM. Corresponding expressions are listed in App. A.7.

Calculation of pole masses in the EFT approach
The proper way to calculate the physical masses of the CP-even Higgs bosons in the EFT framework depends on the mass hierarchy. For M A M t , the low-energy theory is the SM (or the SM+EWinos). Therefore, the procedure described e.g. in [45] can be applied. For M A ∼ M t , though, there is no need to integrate out the non-standard Higgs bosons and the low-energy theory is better described by a THDM (or a THDM+EWinos). In this case, the physical masses of the CP-even Higgs bosons are obtained by finding the poles of the propagators, i.e. the zeroes of the determinant of the inverse propagator matrix, depicted here in the Higgs basis as a possible choice, .
The widetilde indicates, as in Section 2.2, that the corresponding quantities are those of the THDM, at the scale M A . The quantitites m 2 HiHj are the entries of the matrix M HH defined in Eq. (21), and the various Σ's denote the corresponding self-energies of the THDM (or the THDM+EWinos) renormalized in the MS scheme.
In situations where M A is larger than M t , but the separation is also not too large, e.g. M A − M t ∼ 100 GeV, it is difficult to decide if the SM should be used as low-energy theory or the THDM might the better choice. Therefore, a smooth transition between the two cases is beneficial. To implement such a transition, we follow a procedure similar to the one introduced in [38]: We include the contribution of the running between M A and M t , (72) In this way both limits, M A M t and M A ∼ M t , are properly recovered. 3

Combination of fixed-order and EFT calculation
The program FeynHiggs already contains a state-of-the-art fixed-order calculation, i.e., it comprises full one- higher-order corrections to the Higgs self-energies [7,13,18,19,21,23,26,29,30,[47][48][49]. For these corrections, a mixed OS/DR scheme is employed (see [7] for more details), with the stop sector renormalized by default using the OS scheme. With version 2.14.0, the possibility of using the DR scheme for the renormalization of the stop sector was introduced [45]. Field renormalization of the Higgs doublets and thereby the renormalization of tan β is always performed in the DR scheme, independent of the renormalization of the stop sector.
Our goal is to combine the result of this diagrammatic fixed-order calculation with the EFT calculation described in Section 2. This combination is done in several steps. First, we have to relate the quantities computed in the EFT approach, namely the entries of the inverse propagator matrix, the two-point vertex function, to those in the fixed-order approach. Second, proper subtraction terms have to be identified and subtracted such that double-counting of terms appearing in the two results is avoided. Finally, differences in input parameters resulting from different renormalization schemes have to be considered by proper conversion of the parameters.
We choose to perform the combination in the gauge eigenstate basis. Therefore, we need to know the relation between the two-point vertex function matrix in the full MSSM, denoted by ∆ −1 φ φ , and in the effective THDM, labeled as ∆ −1 φ φ . Again, as in Section 2.2, the symbol is used to mark quantities in the full MSSM, and to mark quantities in the effective THDM. The two matrices have to be equal in case of Higgs fields with the same normalization in either of the models. In our case, however, the Higgs field normalization is different, as specified by Eq. (44), which leads to the relation As noted in Section 2.2 this formula is valid only in the decoupling limit of M SUSY M t and at the one-loop level. Explicit formulae for the ∆Σ ij are listed in App. B.
In the combination of the EFT and fixed-order results, it is convenient to take account of Eq. (73) by introducing a finite shift in the field renormalization constants of the fixed-order result. Originally, the MSSM Higgs fields are renormalized by the scale transformation (up to two-loop order) The divergent pieces are fixed via the DR prescription in terms of the one-and two-loop self-energies, In FeynHiggs so far, the DR definition of the field renormalization constants is employed. We now add finite pieces to compensate for the different normalization of the MSSM and THDM Higgs doublets, redefining with the proper choice, according to Eq. (73), Since Eq. (73) is valid only at the one-loop level, it cannot be applied for the two-loop field counterterms δ (2) Z ij . These two-loop terms, however, drop out completely (see App. C for more details).
With the additional finite parts introduced in the field renormalization constants, the inverse propagator matrix of the MSSM becomes equal to that of effective THDM (with restriction to the same perturbative order). Hence, the combination of the fixed-order (MSSM) and the EFT (THDM) approach is straightforward, which means that the MSSM inverse propagator matrix is replaced by where ∆ EFT contains the resummed logarithms and corresponding subtraction terms, We checked numerically that the logarithms of the EFT calculation properly recover the logarithmic behavior of the full fixed-order result when restricted to the same perturbative order. For more details on the calculation of the subtraction terms we refer to [42,45].

Redefinition of tan β
As mentioned above, in FeynHiggs by default the DR-scheme is employed for field renormalization of the Higgs doublets and for the renormalization of tan β. Thus, there is a renormalization scale entering the diagrammatic calculation. By default, it is chosen to be equal to the pole mass M t of the top quark. This in particular means that tan β is normally a MSSM DR quantity defined at the scale M t . The redefinition of the field renormalization constants by a finite shift, as described above, has an impact on the renormalization and hence the conceptual definition of tan β. In presence of an off-diagonal field renormalization constant, the counterterm of tan β is given by (assuming still For the corresponding two-loop counterterm, see App. C. With the finite parts of the field renormalization constants in Eq. (79) and switching to the Higgs basis, we find Comparing this result to Eq. (47), we realize that tan β by now is not a MSSM quantity anymore, but instead a quantity of the THDM. Furthermore, the scale is changed to M A , since the THDM part in ∆Σ H1H2 is evaluated at the scale M A . In conclusion, the finite shift in the field normalization constants of the MSSM leads to the conversion Hence, t THDM β (M A ) is the proper input parameter of the fixed-order calculation.

Conversion of input parameters
The diagrammatic calculation implemented in FeynHiggs employs either the OS or the DR scheme for the renormalization of the stop sector. In case of an OS renormalization, this means in particular that the stop masses and the stop mixing angle are renormalized on-shell. For the EFT calculation however, respective DR quantities are needed. Therefore, the parameters have to be converted. As argued in [42], one-loop conversion including only logarithmic terms is sufficient to reproduce the diagrammatic OS expressions from the EFT DR result. Any further terms in the conversion induce higher order contributions which are presently not under control.
It was noticed in [41,42] that the conversion of the stop mass scale does not involve large logarithms; only the stop mixing parameter X t was found to be affected by logarithmic terms. In that previous analysis, a common scale M A = M SUSY was assumed. Here, we extend the conversion formulas to the case of using the abbreviations More details and full one-loop expressions for the parameter conversion are given in App. D.

Comparison to other codes
There are two other publicly available codes for calculating the Higgs pole masses via a THDM matched to the MSSM: the MhEFT package [53], based on [38], and the program FlexibleSUSY [54] in the recent version [46], based on [55]. As pointed out in [46], agreement has been found with the MhEFT results. We therefore restrict ourselves to a comparison of FeynHiggs to MhEFT (version 1.1).
The basis of MhEFT is a pure EFT calculation. Therefore, terms suppressed by heavy scales are absent. Apart from this obvious distinction, there are a few more differences to FeynHiggs: • MhEFT does not employ the DR scheme for renormalization of the SUSY parameters. Instead, MS renormalization is used. Therefore, conversion of the input parameters is needed for the comparison with FeynHiggs. Corresponding conversion formulas can be found in [56].
Although, as argued in [45], this conversion will induce unwanted higher order terms, it is currently the only way to compare the two results, since neither FeynHiggs offers the possibility of a MS renormalization nor MhEFT the possibility of a DR renormalization. In practice it is a viable method since the numerical impact of the conversion is almost negligible, owing to the small numerical difference between MS and DR parameters.
• The EFT calculations entering FeynHiggs and MhEFT differ in various aspects. MhEFT assumes a type II THDM as the effective THDM in the evolution equations. Furthermore, EWino contributions to the various threshold corrections are neglected. Also in the RGEs, EWino contributions are neglected at the two-loop level and only taken into account in approximate form at the one-loop level. In addition the one-loop threshold corrections between the SM and the THDM are neglected for the top Yukawa coupling and approximated for the SM Higgs self-coupling (i.e., the heavy Higgs contribution to the one-loop threshold correction between the SM and the MSSM is used). On the other hand, MhEFT has implemented an approximation for the O(α 2 t ) threshold corrections for the quartic couplings by including the known O(α 2 t ) threshold correction from matching the SM to the MSSM in λ 2 , whereas all other self-couplings receive no O(α 2 t ) threshold correction.
These differences should be kept in mind, when interpreting the numerical results of the comparison presented in Section 5.

Numerical results
In this Section, we investigate the numerical impact of the implementation of an effective THDM into FeynHiggs. This means in practice that we compare the results from the latest version FeynHiggs2.14.1 to those from the calculation presented in this paper, which is implemented in a still private FeynHiggs version based on FeynHiggs2.14.1. In addition, we show results from FeynHiggs2.14.0 to point out the impact of the non-degenerate O(α 2 t ) threshold corrections [40] , which were implemented as a new feature in FeynHiggs2.14.1. The degenerate O(α 2 t ) threshold corrections [39], used in FeynHiggs2.14.0, implicitly assume M A = M SUSY . We furthermore compare the results of the calculation presented in this paper to those of MhEFT.
For illustration of the numerical effects, we investigate simplified scenarios with a common mass scale M S for all sfermions, and M χ for the EWinos, setting (if not stated otherwise) The numerical impact of the effective THDM can also get large for M SUSY ∼ 1 TeV and moderate values of tan β, if µ > M SUSY . This corresponds, however, to a hierarchy which we did not cover in this paper.
For the SUSY parameters, we use the DR-scheme with the corresponding renormalization scale being M SUSY . The DR scheme is also used for X t (except in Fig. 6, where the OS scheme is used). tan β is defined as tan β THDM (M A ), unless stated otherwise.
Aside from the simplified scenarios, we also study a more complicated situation, the "low-tan β-high" scenario proposed by the LHC Higgs Cross Section Working Group in [57].

Shifts from tan β definition
As explained in Section 3, we account for the different normalization of the Higgs doublets in the full MSSM and the effective THDM by introducing a finite shift in the field renormalization constants of the fixed-order calculation. This changes the definition of tan β: from a MSSM quantity to one of the THDM, along with a change of the renormalization scale from M t (the default of FeynHiggs) to M A . In the right panel, the same set of curves is displayed, but now for M A equal to M SUSY . Therefore, the curves using tan β THDM (M A ) and tan β MSSM (M SUSY ) are very close; again, the additional non-logarithmic threshold correction of tan β between the THDM and the MSSM turns out to be negligible. Due to the large scale separation between M t and M SUSY the curve using tan β MSSM (M t ) is shifted downwards by up to 2 GeV between tan β ∼ 1.2 and tan β ∼ 6. For tan β 1.2, a small upwards shift up to 1 GeV is visible.
Note that for the rest of this section, tan β is defined as tan β THDM (M A ) for all displayed results.

Impact of the effective THDM
Having investigated the numerical effect of different definitions of tan β, we now scrutinize the impact of the main result of this paper -the implementation of an effective THDM into the hybrid framework of FeynHiggs. In Fig. 3    So far, all the numerical results refer to the DR scheme for the stop-sector renormalization. As a distinct feature of FeynHiggs, also the OS scheme can be used for renormalizing the stop input parameters. In order to illustrate the use of OS renormalization, we include when turning to the THDM case are similar in size, although slighty more pronounced in the OS scheme.
Here, it is however important to note that the shift between FeynHiggs with and without effective THDM depends sensitively on the Higgsino mass parameter µ when the OS scheme is used 5 . This is due to the needed conversion of X t between the DR and the OS scheme, according to Eq. (85), which involves an extra term that can become large for M A M SUSY , low tan β, µ ∼ M SUSY and X OS t /M SUSY ∼ 2, inducing large differences between X OS t and X DR t . This signals that in those regions the one-loop conversion is insufficient yielding unreliable results for M h , and recommends the use of the DR scheme.
The M SUSY scale dependence of the effect from implementing the THDM is explicitly shown in Fig. 7.

Results for the heavier Higgs bosons
The role of the SM-like Higgs boson can not only be played by the h boson, also the H boson is a potential candidate (see [58,59] for recent studies) and deserves a closer inspection. In the following, we investigate the prediction for the mass of H boson within our hybrid approach.
In this class of scenarios M A is smaller than M t . In consequence, the proper EFT at the electroweak scale is the THDM and not the SM. In the present study, we approximate the values of the SM MS couplings (y t , g 1 , g 2 , g 3 ) at the scale M t computed in [60] as boundary values for the EFT calculation. Thus, the EFT at the scale M t is replaced by the SM, which is then matched to the THDM. This procedure avoids the detailed calculation of the THDM MS couplings at the electroweak scale, but neglects THDM-specific terms (i.e., terms of order O(M t /M A )).  M H can only be significantly decreased by raising tan β. This possibility is analyzed in the right plot of Fig. 9, where M A is set to 80 GeV. The shift between the results with and without effective THDM shrinks for rising tan β, as was the case for M h . To reach the desired value of 125 GeV for M H , tan β has to be at least > 7. In this region, however, the difference between the results with and without the effective THDM is completely negligible. Also the uncertainty induced by not including contributions from non-SM Higgs bosons in the extraction of the low-energy couplings, estimated above, is totally negligible.
In addition, we also investigated the impact of the effective THDM on the prediction of the charged Higgs mass M H ± . For the calculation of M H ± no resummation of large logarithms was available before. Nevertheless, we only find negligible shifts below 1 GeV in the scenarios considered above.
As noted above, the numerical impact of the effective THDM on the heavier Higgs boson masses might be enhanced in case of µ > M SUSY , which is not covered in this work.

The "low-tan β-high" scenario
In the "low-tan β-high" scenario, defined in [57], all soft SUSY-breaking sfermion masses, as well as the gluino mass, are set equal to M SUSY . The value of M SUSY is chosen such that the result for M h is close to the experimentally determined mass and varies between a few TeV (in case of large M A or tan β) and 100 TeV (in case of small M A or tan β). In its original definition, the OS scheme was employed for renormalization, with the OS stop mixing parameter varying with tan β as follows, for tan β ≤ 2 0.0375 tan 2 β − 0.7 tan β + 3.25 for 2 < tan β ≤ 8.6 0 for 8.6 < tan β Owing to the problems with OS parameters in scenarios with low M A mentioned in Section 5.2, we define all parameters as DR quantitites 6 . Accordingly, we modify the values for X t , In this way, X DR t /M SUSY will be close to the value which maximizes M h when tan β = 1 is approached. The remaining parameters are given by The results in [57] were produced using FeynHiggs2.10.4. Since then, many additional improvements were implemented in FeynHiggs (see also the discussions in [42,45] of important changes that have entered the versions 2.13.0 and 2.14.0). To point out the effect of the most recent developments since FeynHiggs2.14.0, we show the difference between the most topical version of FeynHiggs with effective THDM and the non-THDM version 2.14.0 in the right panel of Fig. 10. The diagram shows that for the considered scenario the M h values obtained with an effective THDM are below the values obtained without effective THDM. For tan β 3, the downwards shift is small (below 1 GeV). For smaller tan β, the shift increases to about 4 GeV for M A = 500 GeV. If in addition also M A is small (∼ 200 GeV), the difference amounts to even more than 8 GeV.

Comparison to MhEFT
After investigating the numerical impact of an effective THDM on the hybrid calculation of FeynHiggs, we compare our results to MhEFT (version 1.1).
First, we compare the results for M h in dependence of M A (see left panel of Fig. 11). We choose tan β = 1 to maximize the impact of the effective THDM. For vanishing stop mixing, FeynHiggs and MhEFT are in close agreement. Also for X DR t /M SUSY = √ 6, the two codes agree within ∼ 1 GeV. The remaining deviation is caused by the different parameterization of non-logarithmic terms (see [45] for an extensive discussion). For low M A this constant shift is compensated by terms of O(M t /M A ) originating from the THDM self-energies (see Eq. (70)) which are included in FeynHiggs but not in MhEFT.
In the right panel of Fig. 11, the results are compared as a function of tan β, setting M A = 200 GeV. The overall good agreement is confirmed. Especially around tan β ∼ 3 the two results are very close to each other, whereas the agreement is slightly worse for smaller or higher values of tan β (but still within 1 GeV). Reasons for the disagreement are again the different parameterization of non-logarithmic terms as well as terms of O(M t /M A ).
This behavior is also reflected in the left panel of Fig. 12 showing M h as a function of X DR t . For tan β = 2.5 and tan β = 3.5, FeynHiggs and MhEFT nearly superpose each other. Only for |X DR t /M SUSY | > 2.5, small deviations are visible which originate from the different parameterizations of non-logarithmic terms. These terms become large for large |X DR t /M SUSY |. For tan β = 1, a deviation of 1 GeV is visible for In the right panel of Fig. 12, we have another look at the "low-tan β-high" scenario using the DR scheme, as defined in Section 5.4. In the whole M A -tan β plane the difference between the two codes is smaller than 2 GeV. Especially for low M A or low tan β the two codes agree very well, whereas FeynHiggs yields slightly larger results than MhEFT in the rest of the parameter plane.
Finally, we comment on the comparison between FeynHiggs and MhEFT shown in [38] (see Fig. 10 and 11 therein). The authors of [38] compared the two codes in the low-tan β-high scenario and found deviations of up to 15 GeV. According to their claim, this discrepancy was mainly caused by the missing implementation of an effective THDM in FeynHiggs. In our Fig. 10, right panel, we found, however, the effective THDM to induce shifts of not more than 8 GeV. This raises the question for the origin of the remaining difference  of ∼ 7 GeV. One reason is certainly the fact that FeynHiggs has evolved a lot since version 2.10.2, which was taken for the comparison in [38]. A second more important reason is the parameter conversion used for the comparison, which was done for the "low-tan β-high" scenario defined with OS parameters, Eq. (88). Therefore, the OS stop mixing parameter had to be converted to the MS scheme which is employed in MhEFT.
In this conversion, M A = M SUSY was assumed. Thereby, an important logarithmic contribution was missed (last term in Eq. (85)), which is especially large for low tan β and low M A , thus exactly in the parameter region where the largest deviation between FeynHiggs and MhEFT was observed.

Conclusions
In this paper, we discussed the implementation of an effective THDM into the hybrid framework of FeynHiggs for the calculation of the MSSM Higgs boson mass spectrum. Our new EFT calculation allows to treat the case of light non-SM Higgs bosons as well as of light EWinos and a light gluino. Furthermore, it includes complete one-loop and dominant two-loop threshold corrections and takes all appearing effective couplings fully into account. In this context, we also discussed how the matching between the various EFT versions is performed paying special attention to the different normalization of the Higgs doublets in the MSSM and the THDM. This difference in field normalization plays a crucial role in the combination of the existing fixed-order calculation in FeynHiggs with the new EFT calculation for low M A . Our accounting of the different normalizations is done by introducing finite shifts in the field renormalization constants of the fixed-order calculation, which affects also the conceptual definition of tan β as an input parameter. Moreover, we investigated the effect of a low M A in the scheme conversion of the parameters for the stop sector, which is necessary if OS input parameters are used.
In our numerical study, we compared FeynHiggs2.14.0 and FeynHiggs2.14.1, both with the SM as the EFT, to our new computation with an effective THDM, which is implemented in a still private FeynHiggs version based on 2.14.1. We found the switch to an effective THDM to cause a negative shift in M h of up to 3 GeV with respect to FeynHiggs2.14.1. This maximal value is reached when tan β ∼ 1 and the hierarchy between the SUSY scale and M A is large (M SUSY /M A ∼ 10 3 ). The shift shrinks quickly when tan β is increased. For tan β 7, the effects resulting fom the THDM are almost completely negligible. Similarly, the shift decreases when M A is increased or M SUSY is lowered. Larger shifts, up to 10 GeV, are found when comparing to FeynHiggs2.14.0. In that version, the implemented O(α 2 t ) threshold correction implicitly assumed M A to be equal to M SUSY , leading to an overestimate of M h in scenarios with M A M SUSY . We also investigated predictions for the mass of the second CP-even Higgs boson H. In the phenomenologically most interesting parameter region, where the H boson can play the role of the SM Higgs boson, we found the shift induced by an effective THDM to be negligible. Also the prediction of the charged Higgs boson mass is only marginally affected. In addition, we looked at the "low-tanb-high" benchmark scenario developed by the LHCHXSWG. For this scenario, we found corrections of up to -8 GeV for tan β 3 with the consequence that the updated M h prediction is too low for meeting the experimental Higgs boson mass. Finally, we compared our results with those of the code MhEFT finding good agreement within 1 GeV throughout the considered parameter space.
Our calculation will become publicly available as part of the code FeynHiggs in a future version. We leave possible improvements of the present work, like the implementation of threshold corrections valid for arbitrary masses of the decoupled particles, or O(α 2 t ) threshold corrections, for future work.

A Threshold corrections
In this Appendix one-loop formulas for matching the various EFTs to each other are provided. All expressions are derived under the assumption that all particles that are integrated out have masses equal to the matching scale. The couplings on the right hand side of all following expressions have to be evaluated at the scale given on the left hand side of the corresponding expressions. Couplings not listed do not receive any one-loop contributions to the matching conditions. In addition, two-loop O(α s α t ) corrections for the matching of the THDM quartic couplings to the full MSSM are given.
Expression for matching the SM to the MSSM and the SM to the SM+EWinos are listed e.g. in [37].
See e.g. [61] for more details on the origin of these contributions.

A.2 Matching the SM to the THDM
The SM Higgs self-coupling is obtained in terms of the λ i of the THDM by Plugging in the tree-level expressions for the λ i from the matching of the THDM to the MSSM, we recover the heavy Higgs contribution to the matching condition of the SM Higgs self-coupling to the full MSSM given in Eq. (10) of [37]. The top Yukawa coupling of the SM y t is related to the top Yukawa couplings of the THDM via This correction corresponds to the heavy Higgs contribution to the threshold of the top Yukawa coupling when matching the SM to the MSSM given in Eq. (24) of [37].
whereas the WFR corrections read where theΣ ij = ∂ ∂p 2Σφ iφj | p 2 =0 are given bŷ The scheme change from DR to MS yields the additional contributions ∆ DR→MS λ 5,6,7 = 0, which have already been calculated in [62]. The EWino corrections can be obtained by replacing the effective Higgs-Higgsino-Gaugino couplingŝ g 1uu,1ud,.. in the expression for matching the THDM to the THDM+EWinos given below by their tree-level values.
Due to the wave-function renormalization, also β receives a threshold correction, ∆Σ H1H2 receives corrections from sfermions and EWinos, Only when taking into account this threshold correction, can the well known one-loop matching condition of λ (when matching the SM to the MSSM) be recovered from Eq. (95) considering the limit M A → M SUSY .
The top Yukawa couplings are obtained at the one-loop level via Here, we implicitly assume that M 1 = M 2 = µ.
The appearing functions are given by with as limiting values.

B Difference in field normalization
In this Appendix, we give explicit formulas for the difference of the field normalization between MSSM and THDM fields. The expressions are valid up to terms of O(M t /M SUSY , M A /M SUSY , M t /M A ). The contribution from sfermions is given by The contribution from electroweakinos reads In addition, also all non SUSY particles, i.e. the particles of the THDM, yield a contribution if the renormalization scales of the THDM and the MSSM are not equal, with Q being the renormalization scale of the MSSM and Q the scale of the THDM.

C Dependence on field renormalization constants
Here, we specify in more detail how the renormalized two-loop self-energies are influenced by field renormalization. The discussion is valid in the limit of vanishing electroweak gauge couplings (gaugeless limit), which is the current approximation applied for the two-loop fixed-order corrections implemented in FeynHiggs.
The notation follows closely that of [30], where also more details about the renormalization as well as the applied approximations can be found. Field renormalization is performed by rescaling the original MSSM Higgs fields, introducing loopexpanded renormalization constants up to the two-loop level, φ 1 φ 2 → 1 + 1 2 δ (1) Z 11 + 1 2 ∆ (2) Z 11 1 2 δ (1) Z 12 + 1 2 ∆ (2) Z 12 1 2 δ (1) Z 12 + 1 2 ∆ (2) Z 12 1 + 1 2 δ (1) Z 22 + 1 2 ∆ (2) Z 22 They involve the tadpole counterterms, the counterms for tan β, as well as the renormalization constants of the electric charge, of the W -boson mass, and of sin θ W in the combination With the conditions above all renormalization constants entering the renormalized self-energies in Eq. (143) are determined. The two-loop field renormalization constants appear exclusively in the Z-dependent two-loop counterterms of Eq. (143), either directly or through the two-loop mass, tadpole and tan β counterterms. In the combinations of Eq. (144) they completely drop out and hence are not needed for the renormalized selfenergies (143). This was already noted for the diagonal field counterterms in [30] for the O(α 2 t ) corrections. The one-loop field renormalization constants δ (1) Z ij enter the two-loop renormalized self-energies Eq. (143) both through the counterterms and through the unrenormalized self-energies via one-loop subrenormalization. We extract the following dependence on δ (1) Z ij , The subscript δZ indicates that only terms proportional to any of the field renormalization constants are kept. As a cross-check, we verified that adding a finite part to any δ (1) Z ij does not lead to additional divergencies. This is important for our method in Section 3 to incorporate the different normalization of the THDM fields as a finite shift in the one-loop field renormalization constants of the MSSM.

D Scheme conversion for low M A
In this Appendix, we list the formulas necessary to convert the parameters of the stop sector from the OS to the DR scheme. Building upon the expressions given in [16,36], we extend those to the case of M A = M S . First, we give the expression for calculating the DR top quark mass of the MSSM in terms of the OS top quark mass, and for the conversion of the stop mixing parameter,