Abstract
Recently, the Higgs boson masses in the Minimal supersymmetric standard model (MSSM) and their mixing have been calculated using the complex twoHiggsdoublet model (cTHDM) as an effective field theory (EFT) of the MSSM. Here, we discuss the implementation of this calculation, which we improve in several aspects, into the hybrid framework of FeynHiggs by combing the cTHDMEFT calculation with the existing fixedorder calculation. This combination allows accurate predictions also in the intermediate regime where some SUSY particles are relatively light, some relatively heavy and some in between. Moreover, the implementation provides precise predictions for the Higgs decay rates and production crosssections.
Introduction
The measurements of an increasing number of SMlike observables by the experiments at the Large Hadron Collider (LHC) approach an accuracy that allows extensions of the Standard Model (SM) to be probed thoroughly beyond the search of yet undiscovered particles. Particularly important in this context are the measurements of the properties of the discovered Higgs boson: with the increased accuracy, these properties are turning into new precision observables that permit tests of many extensions of the Standard Model, namely the ones requiring changes in the Higgs sector. These extensions address questions that the SM cannot answer, such as what dark matter is made up of and why there is more matter than antimatter in the universe.
The results of the measurements performed at the LHC experiments confirm so far the predictions of the SM and therefore constrain the parameter space of possible SM extensions to the parts, which reproduce the SMlike behaviour.
One such model, which can, for example, provide a darkmatter candidate as well as additional sources of \(\mathcal {CP}\) violation needed for the explanation of the matter–antimatter asymmetry, is the Mininmal Supersymmetric Standard Model (MSSM), built upon the concept of Supersymmetry (SUSY). It extends the SM by adding a second Higgs doublet, turning the SM into a TwoHiggsDoublet Model (THDM), as well as a superpartner to each THDM degree of freedom. The two Higgs doublets lead to five physical Higgs bosons: at treelevel, these are the \(\mathcal {CP}\)even h and H bosons, the \(\mathcal {CP}\)odd A boson and the charged \(H^\pm \) bosons. Beyond the treelevel, quantum corrections lead to mixing of the treelevel mass eigenstates. In presence of \(\mathcal {CP}\) violation the true mass eigenstates of the neutral Higgs bosons are labelled by \(h_{1,2,3}\) and are in general mixtures of \(\mathcal{CP}\)even as well as \(\mathcal{CP}\)odd states.
In the context of the MSSM, especially the Higgs boson mass measurement [1,2,3] is a unique opportunity to constrain the MSSM Higgs sector. While the Higgs boson mass is a free parameter in the SM, the mass of the MSSM SMlike Higgs boson can be predicted in terms of the model parameters.
In order to provide a precise prediction, the calculation of higherorder corrections is mandatory: higherorder corrections are essential in the context of the MSSM, since they lead to a circumvention of the upper treelevel mass bound of the lightest MSSM Higgs boson, which is given by the Zboson mass, \(M_Z \approx 91\) GeV, and thereby to an experimentally allowed Higgsboson mass.^{Footnote 1} These higherorder corrections can also change the coupling behaviour of the Higgs bosons, for example by mixing \(\mathcal{CP}\)even and \(\mathcal{CP}\)odd states, and are therefore also important for predicting partial decay widths and production cross sections.
Given the importance of these quantum corrections, much work has been invested in calculating them. These calculations fall into three categories:

In the fixedorder (FO) approach, contributions to the Higgs twopointvertex functions are calculated at a fixedorder of the couplings taking into account the complete model. This results in corrections of the Higgsboson masses and mixings (see [5,6,7,8,9,10,11,12,13,14] for recent fixedorder calculations).

In the effective field theory (EFT) approach, all heavy particles are integrated out and the Higgs mass is then calculated in the lowenergy EFT (see [15,16,17,18,19,20,21,22,23,24,25,26,27] for recent EFT calculations).

While the FO approach is precise for low but not for high SUSYbreaking mass scales (leading to light or heavy superpartner particles, respectively), it is vice versa for the EFT approach. To obtain a precise prediction also for intermediary scale, hybrid approaches combining both approaches have been developed (see [15, 21, 25, 26, 28,29,30,31,32,33,34,35,36]).
In this paper, we will follow the hybrid approach. We combine the results of the recent EFT calculation assuming a THDM as lowenergy theory and allowing for nonvanishing phases of the complex parameters [24, 37,38,39,40] with the fixedorder calculation provided by FeynHiggs [41,42,43,44,45,46,47,48,49,50,51,52,53,54,55]. The nonvanishing phases allow for \(\mathcal {CP}\)violating effects. We also include some further improvements of the EFT calculation. On the one hand, this will increase the validity range of the prediction of the Higgsboson mass as implemented in FeynHiggs. On the other hand, it allows for a better prediction of the mixing of the Higgs bosons, which is important for the calculation of partial decay widths and production cross sections, since the THDM allows for a consistent description of the mixing of all the Higgs bosons. The combination can improve the predictions that have been exploited when studying not only a \(\mathcal {CP}\)odd admixture to the lightest Higgs boson but also the \(\mathcal {CP}\)violating mixing between the heavy Higgs bosons and corresponding changes of exclusion bounds [4].
This paper is organized as follows: in Sect. 2 we discuss the implemented EFT calculation. The combination with the fixedorder calculation is then explained in Sect. 3. We present numerical results in Sect. 4 and conclusions in Sect. 5. Explicit expressions for threshold corrections are given in Appendix A.
EFT calculation
Our EFT calculation closely follows the one published in [24]. At the scale \(M_{\text {SUSY}}\), which we choose to be the geometric mean of the stop SUSY softbreaking mass parameters, we integrate out all sfermions as well as the gauginos and Higgsinos (i.e., we do not take into account separate electroweakino (EWino) and gluino thresholds as in [21] for real parameters). The resulting EFT is the complex THDM (cTHDM) with the Higgs potential,
where \(\Phi _{1,2}\) are the two Higgs doublets. The Higgs mixing parameter \(m_{12}^2\) as well as the quartic couplings \(\lambda _5\), \(\lambda _6\) and \(\lambda _7\) are potentially complex parameters while the mass parameters \(m_{11}^2\), \(m_{22}^2\) and the quartic couplings \(\lambda _1\) to \(\lambda _4\) are real.
The Yukawa part of the Lagrangian reads
with \(Q_L\) being the doublet of the lefthanded quarks and \(t_R\) the righthanded quark singlet. Both topYukawa couplings, \(h_t\) and \(h_t'\), can be complex. In this case, a real topquark mass can be obtained by a redefinition of the righthanded topquark field. We neglect the bottomYukawa couplings (which were taken into account in [24]). In the fixedorder calculation of FeynHiggs and thereby also in in the fixedorder part of our combined hybrid calculation, the bottomYukawa coupling of the MSSM is, however, fully taken into account at the one and twoloop level. These twoloop fixedorder corrections are evaluated in the gaugeless limit and interpolated for nonvanishing phases (see [56] for details). As investigated in detail in [26] for the case of the SM as EFT, the resummation of corrections proportional to the bottomYukawa coupling beyond the order of the fixedorder calculation becomes relevant only for \(\tan \beta > rsim 25\) and negative \(\mu \).^{Footnote 2}
After fixing the cTHDM parameters at \(M_{\text {SUSY}}\) by matching to the full MSSM, we evolve the couplings down to the scale of the nonSM Higgs bosons, which we take to be the charged Higgs mass \(M_{H^\pm }\). For this evolution, we use the renormalization group equations (RGEs) derived in [24] following [57,58,59,60,61,62,63] and compared to the results of [64]. At the scale \(M_{H^\pm }\), we integrate out the heavy Higgs bosons and recover the SM as EFT.
We improve the calculation presented in [24] by the following aspects: we take into account

full oneloop threshold corrections between the SM and the complex THDM (i.e., for the SM Higgs selfcoupling and the topYukawa coupling),

the purely electroweak contributions to the threshold corrections of the THDM Higgs selfcouplings at the SUSY scale (i.e., contributions from electroweakinos and the change from \(\overline{\text {DR}}\), used in the MSSM, to \(\overline{\text {MS}}\), used in the THDM),

\(\mathcal{O}(\alpha _t\alpha _s)\) contributions to the threshold corrections of the THDM Higgs selfcouplings at the SUSY scale,

electroweak contributions to the threshold corrections of the THDM topYukawa couplings at the SUSY scale.
Explicit expressions for these extended threshold corrections are listed in Appendix A.
Moreover, for obtaining pure EFT results, we improve the extraction of the physical SMlike Higgs mass at the electroweak scale. In [24], only the leading \(\mathcal{O}(\alpha _t)\) SM corrections were included. We take into account full oneas well as twoloop corrections [65] for the SMlike Higgs boson.
As in [21, 24], we calculate all threshold corrections in the limit of degenerate sfermion and electroweakino masses. More explicitly, we assume all sfermions and the gluino to have masses equal to \(M_{\text {SUSY}}\). Moreover, we set \(M_1 = M_2 = \mu \) (where \(M_1\) and \(M_2\) are the bino and wino soft SUSYbreaking mass and \(\mu \) is the Higgsino mass parameter). When the EFT calculation is combined with the fixedorder calculation in the hybrid approach, the effect of nondegenerate masses is, however, fully captured at the order of the fixedorder calculation.
Combination with fixedorder calculation
In this Section, we discuss how the EFT calculation, presented in Sect. 2, is combined with the existing fixedorder calculation implemented in FeynHiggs. This fixedorder calculation incorporates full oneloop as well as \(\mathcal{O}(\alpha _t\alpha _s,\alpha _b\alpha _s,\alpha _t^2,\alpha _t\alpha _b,\alpha _b^2)\) twoloop corrections [41,42,43,44,45,46,47,48,49,50,51,52,53,54,55] (\(\alpha _{b,t} = h_{b,t}^2/(4\pi )\) with \(h_{b,t}\) being the bottom and topYukawa couplings in the MSSM; \(\alpha _s = g_3^2/(4\pi )\) with \(g_3\) being the strong gauge coupling). The phases of complex parameters are fully taken into account at the oneloop level as well as at \(\mathcal{O}(\alpha _t\alpha _s,\alpha _t^2)\). The remaining twoloop corrections are interpolated in the case of nonvanishing phases.
For the combination of the EFT calculation with the fixedorder calculation, we largely follow the procedure outlined in [21]: we first redefine the Higgs fields of the fixedorder calculation in order to match the normalization of the Higgs fields in the EFT. The prescription of [21], valid only for the case of real input parameters, was extended to the case of complex input parameters in [33]. As a second step, we add the individual results of the EFT and the fixedorder calculation for each specific element of the Higgs twopointvertexfunction matrix, \(\Gamma _{hHA}\). Additional subtraction terms ensure that terms contained in the EFT as well as the fixedorder calculation are not doublecounted,^{Footnote 3}
with
where the \(\hat{\Sigma }^{\text {FO}}_{ij}\)’s are the fixedorder selfenergies, the \(\Delta _{ij}^\text {EFT}\)’s are the EFT results, i.e. the elements of the corresponding THDM mass matrix, the \(\Delta _{ij}^\text {sub}\)’s are the subtraction terms (including treelevel and higherorder terms), and \(m_h\), \(m_H\), \(m_A\) are the treelevel masses of the light CPeven, the heavy CPeven, and the CPodd Higgs boson, respectively. The pole masses \(M_{h_1}\), \(M_{h_2}\), \(M_{h_3}\) with \(M_{h_1} \le M_{h_2} \le M_{h_3}\) are then determined by finding the poles of this Higgs twopointvertexfunction matrix. In [21], only the logarithms in the \(\mathcal {CP}\)even h, Hsubmatrix have been resummed in the hybrid approach.
In comparison to the real THDM (rTHDM) hybrid calculation presented in [21], we change the treatment of the \(\mathcal{O}(\alpha _t^2)\) fixedorder corrections. The reason for this change is the following: these \(\mathcal{O}(\alpha _t^2)\) fixedorder corrections are by default parameterized in terms of the SM \(\overline{\text {MS}}\) topquark mass \(\overline{m}_t (M_t)\) evaluated at the onshell topquark mass \(M_t\). They include nonlogarithmic SUSY contributions which are constant if \(M_{\text {SUSY}}\) is varied (keeping \(\widehat{X}_t = X_t/M_{\text {SUSY}} \) fixed where \(X_t\) is the top squark mixing parameter). For the case of the THDM as EFT, these fixedorder corrections have no EFT counterpart, since the \(\mathcal{O}(\alpha _t^2)\) threshold corrections between the SM and the THDM, which have been unknown until recently (see [27]), are not implemented in our current setup.^{Footnote 4}
Consequently, in [21] no corresponding nonlogarithmic subtraction term was included. This resulted into a discrepancy between the hybrid results using the THDM and the SM as EFT for high \(M_{\text {SUSY}}\) (\(\sim 100\,\, \mathrm {TeV}\)) and \(M_A = M_{\text {SUSY}} \), where one would expect a good agreement of the two calculations. For such a high \(M_{\text {SUSY}}\), the \(\mathcal{O}(\alpha _t^2)\) threshold correction between the SM and the MSSM are actually negligible, since the topYukawa coupling shrinks with rising \(M_{\text {SUSY}}\) (also the \(\mathcal{O}(\alpha _t^2)\) matching conditions between the THDM and the MSSM will show the same behaviour). For the combined THDM hybrid result of [21], in contrast, the nonlogarithmic \(\mathcal{O}(\alpha _t^2)\) fixedorder terms, employing the SM \(\overline{\text {MS}}\) topquark mass \(\overline{m}_t (M_t)\) instead of the topYukawa coupling at the scale \(M_{\text {SUSY}}\), nonnegligibly contribute to the final result. This shows clearly that for very high \(M_{\text {SUSY}}\), the \(\mathcal{O}(\alpha _t^2)\) fixedorder terms should not be included if the EFT counterpart (the \(\mathcal{O}(\alpha _t^2)\) threshold correction and the corresponding subtraction term) is not available. For low \(M_{\text {SUSY}}\), the inclusion of the fixedorder \(\mathcal{O}(\alpha _t^2)\), however, can improve the result. Since, in this paper, we are mainly interested in the low \(\tan \beta \) region,^{Footnote 5} where a high \(M_{\text {SUSY}}\) is needed to obtain \(M_h \simeq 125\,\, \mathrm {GeV}\), we chose to subtract the SUSY \(\mathcal{O}(\alpha _t^2)\) contribution to the hh Higgs selfenergy.
An additional step has to be considered for the combination of the fixedorder and the EFT result if the input parameters of the fixedorder calculation and the EFT calculation are not renormalized in the same scheme. In this context, the stop mixing parameter, \(X_t\), is especially relevant. While \(X_t\) can be renormalized either in the \(\overline{\text {DR}}\) or the OS scheme in case of the fixedorder calculation, in the EFT calculation \(X_t\) is renormalized in the \(\overline{\text {DR}}\) scheme. In case of an OS input parameter, we need to convert \(X_t\) from the OS to the \(\overline{\text {DR}}\) scheme. As argued in [28, 31], a conversion of \(X_t\) taking into account only large logarithmic terms is sufficient. For the present study, we generalize the formula given in [21], which is valid in case of a large hierarchy between the nonSM Higgs scale and the SUSY scale, to the case of complex input parameters (see also [26]),
where \(\hat{X}_t = X_t/M_{\text {SUSY}} \) and \(\hat{Y}_t = \hat{X}_t + 2\hat{\mu }^*/s_{2\beta }\) (with \(\hat{\mu }= \mu /M_{\text {SUSY}} \) and \(\mu \) being the Higgsino mass parameter) and \(t_\gamma \equiv \tan \gamma \) and \(s_\gamma \equiv \sin \gamma \) are used for abbreviation.
Numerical results
In this section, we numerially investigate the improved EFT calculation as well as the combined hybrid calculation. For most of the numerical results, we consider a simple twoscale scenario. All sfermion SUSY softbreaking mass parameters \(M_{\tilde{f}}\) as well as the gaugino and Higgsino mass parameters, \(M_1\), \(M_2\), \(M_3\), \(\mu \), are set equal to \(M_{\text {SUSY}}\). As second scale, we allow the mass of the charged Higgs boson \(M_{H^\pm }\) to be different from \(M_{\text {SUSY}}\). We set all SUSYsoft trilinear couplings except of \(A_t\) to zero. In summary,
where \(X_t\) is the top squark mixing parameter.
If not stated otherwise, all parameters are fixed in the \(\overline{\text {DR}}\) scheme at the scale \(M_{\text {SUSY}}\) (apart from \(M_{H^\pm }\), which we fix in the \(\overline{\text {MS}}\) scheme at the scale \(M_{H^\pm }\) and in case of a pure EFT calculation and in the OS scheme in case of a hybrid calculation, and \(\tan \beta \), which is always fixed in the \(\overline{\text {MS}}\) scheme at the scale \(M_{H^\pm }\)) and all phases are set to zero.^{Footnote 6}
First, we assess the numerical size of taking into account the \(\mathcal{O}(\alpha _t\alpha _s)\) threshold corrections in the left plot of Fig. 1. It shows the mass of the lightest Higgs boson \(M_{h_1}\) in dependence on \(M_{H^\pm }\). The mass \(M_{h_1}\) is calculated in the pure EFT approach using three different EFTs: SM EFT (blue), cTHDM EFT without \(\mathcal{O}(\alpha _t\alpha _s)\) threshold corrections (green) and cTHDM EFT with \(\mathcal{O}(\alpha _t\alpha _s)\) threshold corrections (red). The SMEFT result is based on the calculation presented in [26] including the full phase dependence in the threshold corrections between the SM and the MSSM.
We choose \(M_{\text {SUSY}} =2\,\, \mathrm {TeV}\), \(t_\beta =8\) and \(\phi _{M_3} = \pi /4\). For smaller \(t_\beta \), higher \(M_{\text {SUSY}}\) values would be required to obtain \(M_{h_1} \sim 125\,\, \mathrm {GeV}\). Since the value of the strong gauge coupling, entering the calculation evaluated at \(M_{\text {SUSY}}\), shrinks with rising \(M_{\text {SUSY}}\), the numerical impact of the \(\mathcal{O}(\alpha _t\alpha _s)\) threshold corrections would be even smaller.
For \(\widehat{X}_t ^{\overline{\text {DR}}} = 0\), the numerical effect of the \(\mathcal{O}(\alpha _t\alpha _s)\) threshold corrections is completely negligible. The THDMEFT results with and without these corrections lie on top of each other. For \(\widehat{X}_t ^{\overline{\text {DR}}} = \sqrt{6}\), there is a small approximately constant difference between both curves (\(\sim 0.2\,\, \mathrm {GeV}\)). The shift induced by the \(\mathcal{O}(\alpha _t\alpha _s)\) threshold corrections is relatively small in comparison to results obtained in the literature for the case of the SM as EFT (see e.g. [16, 17, 28]). This is explained by the fact that we express the \(\mathcal{O}(\alpha _t)\) threshold correction in terms of the MSSM topYukawa coupling. As discussed in [36], this choice absorbs the numerically most significant twoloop terms (i.e., those terms including the highest powers of \(\widehat{A}_t \) and \(\widehat{\mu }\)).
The SMEFT result is in good agreement with the THDMEFT results for \(M_{H^\pm } > rsim 300\,\, \mathrm {GeV}\). The remaining small difference in this region is explained by the fact that we do not include \(\mathcal{O}(\alpha _t^2)\) threshold corrections in the THDM EFT, which have only been calculated very recently [27], while the SM EFT includes the \(\mathcal{O}(\alpha _t^2)\) threshold correction between the SM and the MSSM. For \(M_{H^\pm } \lesssim 300\,\, \mathrm {GeV}\), the SMEFT result lies up to \(\sim 1\,\, \mathrm {GeV}\) higher than the THDMEFT results. This is explained by sizeable Higgsmixing effects which are taken into account in the THDMEFT results but not in the SMEFT results. The numerically effect of the Higgs mixing is stronger for lower \(t_\beta \) values [21].
It should be noted that low values of \(M_{H^\pm } \) below \(M_{H^\pm } \approx 500\) GeV are experimentally disfavoured (see for example the phenomenological studies in [4, 66, 67]). Therefore, our numerical investigation in the left plot of Fig. 1 can be regarded as a conservative estimate of the expected maximal size of the effect.
To assess the sensitivity of \(M_{h_1}\) on the phases, we compare the different EFT predictions in the right plot of Fig. 1 in dependence on \(\phi _{M_3}\) for \(M_{H^\pm } = 200 \,\, \mathrm {GeV}\) and \(\widehat{X}_t ^{\overline{\text {DR}}} = \sqrt{6}\). The line coloring is the same as in the left plot of Fig. 1.
All three curves show a quite similar dependence on \(\phi _{M_3} \). The SM curve is shifted upwards with respect to the cTHDM curves by \(\sim 1\,\, \mathrm {GeV}\), since Higgs mixing effects are not included. Including the \(\mathcal{O}(\alpha _t\alpha _s)\) threshold corrections in the cTHDM calculation leads to an upwards shift of \(\sim 0.2\,\, \mathrm {GeV}\) but does not introduce any sizeable new phase dependence. For smaller values of \(\widehat{X}_t ^{\overline{\text {DR}}} \), the phase dependence is similar but of smaller magnitude.
In Fig. 2, we compare the results of hybrid calculations using the SM (blue), the real THDM interpolated in the case of complex input parameters (green)^{Footnote 7} and the cTHDM (red) as EFTs. For both plots, we set \(M_{\text {SUSY}} = 100\,\, \mathrm {TeV}\) and \(\phi _{M_3} = \pi /4\). For the solid lines, \(\widehat{X}_t ^{\overline{\text {DR}}} = 0\) is chosen, while for the dashed lines \(\widehat{X}_t ^{\overline{\text {DR}}} = \sqrt{6}\) is set.
In the left plot, we show \(M_{h_1}\) in dependence on \(M_{H^\pm }\) for \(t_\beta = 3\). For \(\widehat{X}_t ^{\overline{\text {DR}}} = 0\), the THDM results agree very well (the red curve lies on top of the green curve), since the dependence on \(\phi _{M_3} \) is very small. For \(M_{H^\pm } \sim M_{\text {SUSY}} \), also the SM result agrees very well with the THDM results. In comparison to Fig. 1, the \(\mathcal{O}(\alpha _t^2)\) threshold correction is negligible since the topYukawa coupling entering the threshold corrections is very small for \(M_{\text {SUSY}} = 100 \,\, \mathrm {TeV}\). For \(M_{H^\pm } \ll M_{\text {SUSY}} \) there is a sizeable difference between the SM and the THDM results again caused by Higgsmixing effects which become increasingly important if \(M_{H^\pm }\) is lowered. Note that via the combination of the EFT and the fixedorder calculation in the hybrid approach, Higgsmixing effects are included at the one and twoloop level also for the result using the SM as EFT. Larger differences between using the SM as EFT and the THDM as EFT would be found in the pure EFT approach.
The general behaviour is similar for \(\widehat{X}_t ^{\overline{\text {DR}}} = \sqrt{6}\). The gluino phase, \(\phi _{M_3}\), has, however, a larger numerical impact. This results in an approximately constant shift (\(\sim 0.5\,\, \mathrm {GeV}\)) of the rTHDM curve (using interpolation for handling the complex input parameters) with respect to the cTHDM curve.
The right plot of Fig. 2 is analogeous to the left plot but we show \(M_{h_1}\) in dependence of \(t_\beta \) for \(M_{H^\pm } = 500\,\, \mathrm {GeV}\). The behaviour of the different curves is similar as in the left plot. All results agree very well for \(t_\beta \sim 10\). For lower \(t_\beta \), Higgsmixing effects become relevant leading to a difference between the SM and the THDM results of \(\sim 2\,\, \mathrm {GeV}\) for \(\tan \beta \sim 3\). The nonzero gluino phase is only relevant for \(\widehat{X}_t ^{\overline{\text {DR}}} = \sqrt{6}\) leading to a shift of \(\sim 0.4\,\, \mathrm {GeV}\) between the interpolated rTHDM result and the cTHDM results. For \(\widehat{X}_t ^{\overline{\text {DR}}} = 0\), \(\widehat{A}_t  \le 1\) since \(\widehat{\mu } = 1\) is chosen. Since in this case, the dependence on \(\phi _{M_3} \) is not enhanced by powers of \(\widehat{A}_t \) or \(\widehat{\mu }\) (see explicit expressions in Appendix A) the red and green solid lines lie on top of each other.
We further investigate the dependence on the the phases in Fig. 3. The same hybrid calculations as in Fig. 2 are shown in dependence on \(\phi _{M_3}\) for \(M_{\text {SUSY}} = 100 \,\, \mathrm {TeV}\), \(M_{H^\pm } = 500\,\, \mathrm {GeV}\), \(t_\beta = 3\) and \(\widehat{X}_t ^{\overline{\text {DR}}} = \sqrt{6}\).
In the left plot, we set \(\phi _{A_t} = 0\). The SM and the cTHDM curves are approximately parallel to each other. The offset between both curves is caused by Higgsmixing effects, which are fully taken into account for the cTHDM curve while for the SM EFT results the Higgsmixing corrections are only included via the fixedorder corrections. The interpolated rTDHM curve agrees as expected with the cTHDM curve for \(\phi _{M_3} = 0,\pm \pi \). In between these phase values, the interpolation of the EFT part of the calculation leads to shifts of \(\sim 0.5\,\, \mathrm {GeV}\). The kink at \(\phi _{M_3} = 0\) originates from the combination of the fully phase dependent fixedorder calculation and the interpolated rTHDM EFT calculation (see also [26, 56]).
The difference of the interpolated rTHDM and the cTHDM results is enlarged if a second phase is nonzero. In the right plot of Fig. 3, we set \(\phi _{A_t} = \pi /2\). While the noninterpolated results show only a very mild dependence on \(\phi _{M_3}\) as expected by the fact that the strong gauge coupling entering the threshold corrections at \(M_{\text {SUSY}}\) is small for \(M_{\text {SUSY}} =100\,\, \mathrm {TeV}\). In contrast, the interpolated rTHDM curve shows a strong dependence on \(\phi _{M_3}\) leading to difference in comparison to the cTHDM curve of up to \(3.5\,\, \mathrm {GeV}\).
For the scenarios of Fig. 3, the dependence on \(\phi _{A_t} \) is very similar to the one on \(\phi _{M_3} \).
Next, we study phenomenological applications of our cTHDM calculation. In the left plot of Fig. 4, we try to assess the maximally allowed \(\mathcal {CP}\)odd component of the SMlike Higgs bosons. Following [24], we evaluate the size of the component by the (1, 3) entry of the Higgs mixing matrix squared. We employ a similar scenario as used in [24]. That means we set \(\widehat{A}_t  = 3\), \(\widehat{\mu } = 3\), \(\phi _{A_t} = 2\pi /3\) and \(\phi _{M_3} = \pi /4\). For every point in the \((M_{H^\pm },t_\beta )\) parameter plane we adjust \(M_{\text {SUSY}}\) such that our pure cTHDM EFT calculation yields \(M_{h_1} \sim 125\,\, \mathrm {GeV}\). As upper limit on \(M_{\text {SUSY}}\), we employ \(10^{16}\,\, \mathrm {GeV}\).^{Footnote 8} After fixing \(M_{\text {SUSY}}\), we calculate the \(\mathcal {CP}\)odd component of the SMlike Higgs boson as outlined in [24].
We show three different sets of contours: \(M_{\text {SUSY}} \) contours (red), \(M_{h_1}\) contours (green) and the \(\mathcal {CP}\)odd component of the SMlike Higgs (blue). The minimal value of \(M_{\text {SUSY}}\) is reached for high \(t_\beta \) and \(M_{H^\pm }\) in the upper right corner of the plot. Lowering \(t_\beta \) and \(M_{H^\pm }\) implies the need to increase \(M_{\text {SUSY}}\) in order to guarantee \(M_{h_1} \sim 125\,\, \mathrm {GeV}\). For \(t_\beta \sim 34.5\), the maximum \(M_{\text {SUSY}}\) value of \(10^{16}\,\, \mathrm {GeV}\) is reached. In the same region, \(M_{h_1}\) drops below 125 GeV. The grayshaded region, in which \(M_{h_1}\) is lower than 122 GeV, is excluded by the experimentally measured Higgs mass (assuming a conservatively estimated theoretical uncertainty of \(\sim 3\,\, \mathrm {GeV}\)).
The size of the \(\mathcal {CP}\)odd component of the SMlike Higgs is highest for large \(t_\beta \) and small \(M_{H^\pm }\). This is seemingly in contrast to the findings in [24], where the largest \(\mathcal {CP}\)odd component was found for low \(t_\beta \) and small \(M_{H^\pm }\), where Higgsmixing effects are most important. In [24], however, a fixed SUSY scale of 30 TeV was considered. Correspondingly, \(M_{h_1}\) dropped below 122 GeV already for \(t_\beta \lesssim 810\). Increasing the SUSY scale in this region leads to smaller threshold corrections, since the stronge gauge as well as the topYukawa coupling decrease with rising \(M_{\text {SUSY}}\). Since these threshold corrections induce the \(\mathcal {CP}\)violation, larger \(M_{\text {SUSY}}\) values directly result into a smaller \(\mathcal {CP}\)odd component of the SMlike Higgs boson.
The considered parameter space is already tightly constraints by experimental searches for heavy Higgs bosons as well as coupling measurements of the Higgs boson found at the LHC (see [4, 66] for a detailed discussion of the experimental constraints in similar scenarios). This implies that if a nonzero \(\mathcal {CP}\)odd component of the discovered Higgs boson is established at the LHC, it would probably be very difficult to explain such an observation within the MSSM.^{Footnote 9}
It should be noted that these tight constraints come solely from the Higgs sector. Measurements such as the ones of the electric dipole moments will restrict the allowed values of the \(\mathcal {CP}\)violating phases and might lead to even lower values of an experimentally allowed \(\mathcal {CP}\)odd component of the Higgs boson, see for example [69,70,71,72,73].
As second phenomenological application, we study the \(M_h^{125}(\mathrm {CPV})\) benchmark scenario defined in [4]. In this scenario the SUSY scale is set to 2 TeV, \(X_t^{\text {OS}} = 2.8\,\, \mathrm {TeV}\) and \(\phi _{A_t} = 2\pi /15\). The nonzero \(A_t\) phase leads to mixing between the heavy Higgs bosons, which are almost massdegenerate. Consequently, large interference effects occur in processes involving an schannel exchange of these Higgs bosons. Phenomenologically, the production of the heavy Higgs bosons via bassociated production and their subsequent decay into a pair of tau leptons is very relevant, since it leads to stringent constraints on the MSSM Higgs sector.
In [4], the interference effects in this channel were calculated using FeynHiggs version 2.14.3, employing a hybrid calculation with the SM as EFT,^{Footnote 10} and SusHi [74, 75] with its extension SusHiMi [76]. The interference was quantified factorizing the experimentally measurable coherent crosssection in the form
with
see [77,78,79] for a more details.
Here, we repeat the calculation of the interference factor \(\eta \) with our new hybrid calculation using the cTHDM as EFT (implemented in FeynHiggs and linked to SusHi). Since the release of FeynHiggs version 2.14.3, also the SMEFT calculation has been updated. We show here the results obtained using the SMEFT calculation presented in [26] which includes the full phase dependence in the threshold corrections. In the given scenario, the numerical difference to the interpolated result used in [4] is very small.
In the right plot of Fig. 4, the results using the SM as EFT (blue) and the cTHDM as EFT (red) are compared. As visible in the plot, only small differences between both results are visible. The point of maximal interference is shifted slightly to lower \(M_{H^\pm }\) values if the THDM is used as EFT. This is mainly due to small shifts in the heavy Higgs masses \(M_{h_2}\) and \(M_{h_3}\). The twoloop fixedorder corrections, included in both results, already contain the leading logarithms as well as the leading phase dependence. Due to the small hierarchy between \(M_{H^\pm }\) and \(M_{\text {SUSY}}\), the resummation of higherorder logarithms appearing in the \(\mathcal {CP}\)mixing selfenergy is negligible. Therefore, the hybrid result using the SM as EFT is a good approximation of the full hybrid result using the cTHDM as EFT. This shows the benefits of the hybrid framework combining fixedorder and EFT calculations.
Conclusions
\(\mathcal{CP}\) violation in BSMHiggs sectors is one intriguing possibility to overcome the lack of \(\mathcal{CP}\) violation in the SM; this lack renders an explanation of the matter–antimatter asymmetry of the Universe difficult. In this work, we discussed precision predictions in the \(\mathcal{CP}\)violating MSSM using the cTHDM as EFT.
We improved an existing cTHDMEFT calculation of the MSSM Higgs boson masses by calculating so far missing oneloop threshold corrections as well as by including the leading twoloop \(\mathcal{O}(\alpha _t\alpha _s)\) threshold corrections. We found that the resulting numerical impact on the lightest SMlike Higgs boson mass was small (\(\sim 0.2\,\, \mathrm {GeV}\)), since the dominant pieces are already absorbed into the oneloop threshold correction by parameterizing them in terms of MSSM parameters.
As a next step, we combined this improved EFT calculation with the existing fixedorder calculation implemented in the public code FeynHiggs. In this way, a precise prediction of the MSSM Higgs boson masses for low, intermediary and high SUSY scales is possible. In comparison to the already implemented rTHDMEFT calculation, which is interpolated in the case of nonvanishing \(\mathcal{CP}\)violating phases, we found shifts of several GeV if more than one phase is nonzero.
As a further benefit, the implementation into FeynHiggs allows precise predictions for the Higgs decay rates and production crosssections resumming all large logarithmic contributions in the Higgs selfenergies. As example application, we investigate how large the \(\mathcal{CP}\)odd component of the SMlike Higgs boson can become in the MSSM. Since the \(\mathcal{CP}\)odd component can not exceed a few percent, a discovery of a \(\mathcal{CP}\)odd component at the LHC, assuming the expected experimental reach, would most likely rule out the MSSM. As second application, we studied the interference effect between the two heavier Higgs bosons in the \(M_h^{125}(\mathrm {CPV})\) Higgs benchmark scenario, which was defined in [4]. In comparison to using the SM as EFT, the use of the cTHDM as EFT did not significantly shift the interference pattern indicating that the twoloop fixedorder corrections already account for the bulk of the corrections. This demonstrate the advantages of the hybrid approach combining fixedorder and EFT calculations.
The presented calculation will become part of an upcoming FeynHiggs version.
Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors’ comment: All newly derived expressions are written down in the Appendix.]
Notes
 1.
The scenario where a heavy MSSM Higgs boson plays the role of the discovered one is strongly constrained [4].
 2.
We expect very similar results for the THDM as EFT, since a large hierarchy between \(M_{H^\pm }\) and \(M_{\text {SUSY}}\) is tightly constrained by collider searches for heavy Higgses decaying to tau leptons (see e.g. [4]) for large \(\tan \beta \).
 3.
We neglect mixing with the Goldstone bosons, which is a subleading twoloop correction.
 4.
 5.
In the high \(\tan \beta \) region (i.e. \(\tan \beta > rsim 10\)) a large hierarchy between the SUSY scale and the scale of the heavy Higgs bosons is excluded due to an interplay of the SMlike Higgs mass constraint and tight constraints from heavy Higgs searches (see e.g. [4, 66]). Since in addition Higgs mixing effects are suppressed for large \(\tan \beta \), using the SM as EFT provides a very accurate approximation of the THDMEFT result [21].
 6.
We do not include a band for the theoretical uncertainty, since a reliable uncertainty estimate requires further investigations (for the case of the SM as EFT see, for example, Ref. [35]). Moreover, the main purpose of the shown plots is to investigate the dependencies of \(M_h\) on the model parameters and not to employ \(M_h\) to constrain these parameters.
 7.
 8.
A similar procedure has been applied in [66].
 9.
In the high \(\tan \beta \) region, \(\mathcal {CP}\)violating effects from the sbottom sector can become important. This region is, however, even tighter constrained as the low \(\tan \beta \) region (see e.g. [4]). Another possibility might be scenarios in which the second lightest MSSM Higgs boson is SMlike. But also these scenarios are tightly constraint (see e.g. [4]). Note, however, that a more sizeable \(\mathcal {CP}\)odd component of the SMlike Higgs boson can be obtained in models beyond the MSSM (see e.g. [68]).
 10.
In FeynHiggs2.14.3 the phase dependence of the threshold corrections calculated in [26] was not yet implemented. Instead the EFT calculation was interpolated in case of complex input parameters.
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Acknowledgements
We thank Ivan Sobolev and Georg Weiglein for useful discussions as well as Stefan Liebler for help with SusHi. H.B. is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy  EXC 2121 “Quantum Universe” 390833306. H.R.’s and N.M.’s work was partially funded by the Danish National Research Foundation, grant number DNRF90. H.R. is also supported by the German Federal Ministry for Education and Research (BMBF) under contract no. 05H18VFCA1. Part of this work was supported by a STSM Grant from COST Action CA16201 PARTICLEFACE.
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Threshold corrections: explicit expressions
Threshold corrections: explicit expressions
In this Appendix, we list all threshold corrections taken into account in addition to those presented in [24].
Matching the SM to the cTHDM
The SM Higgs selfcoupling is obtained in terms of the \(\lambda _i\) of the THDM by
with
where \(k\equiv (4\pi )^{2}\). The SM topYukawa coupling, \(y_t\), is given in terms of the cTHDM topYukawa couplings by
The expressions given in Eqs. (9)–(11) are only valid if the matching scale between the SM and the cTHDM is set equal to \(M_{H^\pm }\).
Matching the cTHDM to the MSSM
In addition to the calculation presented in [24], we take into account all oneloop EWino contributions as well as the \(\mathcal{O}(\alpha _t\alpha _s)\) correction for the matching of the THDM Higgs selfcouplings. All threshold corrections between the cTHDM and the MSSM are derived in the limit of all sfermion masses, the gluino mass, and the matching scale Q set equal to \(M_{\text {SUSY}}\). Moreover, \(M_1 = M_2 = \mu \) is assumed.
Oneloop EWino contribution
The oneloop EWino contributions to the matching of THDM Higgs selfcouplings are given by
where \(\phi _{M_1} \), \(\phi _{M_2} \) and \(\phi _{\mu } \) are the phases of the SUSY softbreaking masses \(M_1\), \(M_2\) and \(\mu \), respectively. g is the \(SU(2)_L\) gauge coupling; \(g_y\), the \(U(1)_Y\) gauge coupling. In the limit of vanishing phases, these expressions agree with the results given in [21, 80].
The different contributions to the oneloop threshold corrections of the topYukawa couplings can be categorized in the following way,
where here \(h_t\) is the the MSSM topYukawa coupling. The contributions arising from the change of the regularization scheme, \(\Delta _{{\overline{\text {DR}}} \leftrightarrow {\overline{\text {MS}}}}\), have been derived in [21] and are not modified in the presence of complex phases. The contributions from diagrams involving gluinos, \(\Delta _{\tilde{g}}\), can be found in [24]. In the same work, also the contributions from diagrams involving at least one squark (but no gluino), \(\Delta _{\tilde{q}}\), have been derived in the limit of vanishing electroweak gauge couplings. The full expressions, including electroweak gauge couplings, are given by
The loopfunctions \(\mathcal {G}_i\) are given by
with the limiting values
The loopfunctions \(\mathcal {F}_{i}\) were defined in App. A.3 of Ref. [21] and are given by
Using the identity
the expressions given in [21] are recovered if all phases are set to zero.
The contributions from diagrams involving only gauginos or Higgsinos is given by
Since no vertex diagrams involving only gauginos or Higgsinos exist, these corrections arise purely from externalleg contributions of the Higgs boson.
Twoloop \(\mathcal{O}(\alpha _t\alpha _s)\) contribution
The \(\mathcal{O}(\alpha _t\alpha _s)\) contributions to the matching of THDM Higgs selfcouplings are given by
These expressions are obtained from the \(\mathcal{O}(\alpha _t\alpha _s)\) threshold correction between the SM and the MSSM [26] using the strategy proposed in [19]. Since only the real parts of the \(\lambda _i\)’s enter the treelevel matching condition between the SM and the THDM (see (10a)), it does not allow to obtain information about the imaginary parts of the Higgs selfcouplings (we chose to set these contributions to zero). The complete \(\mathcal{O}(\alpha _t\alpha _s)\) threshold correction have recently been published in [27] but are not taken into account in this work. We expect the differences to the expressions presented here to have only a very small numerical effect.
Similar results have been obtained in [40]. In there, the stop softSUSYbreaking parameters have, however, been defined in the \(\overline{\text {MS}}\) scheme (see [16] for more details).
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Bahl, H., Murphy, N. & Rzehak, H. Hybrid calculation of the MSSM Higgs boson masses using the complex THDM as EFT. Eur. Phys. J. C 81, 128 (2021). https://doi.org/10.1140/epjc/s10052021089397
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