Twoloop corrections to the \(\rho \) parameter in TwoHiggsDoublet models
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Abstract
Models with two scalar doublets are among the simplest extensions of the Standard Model which fulfill the relation \(\rho = 1\) at lowest order for the \(\rho \) parameter as favored by experimental data for electroweak observables allowing only small deviations from unity. Such small deviations \(\varDelta \rho \) originate exclusively from quantum effects with special sensitivity to mass splittings between different isospin components of fermions and scalars. In this paper the dominant twoloop electroweak corrections to \(\varDelta \rho \) are calculated in the CPconserving THDM, resulting from the topYukawa coupling and the selfcouplings of the Higgs bosons in the gaugeless limit. The onshell renormalization scheme is applied. With the assumption that one of the CPeven neutral scalars represents the scalar boson observed by the LHC experiments, with standard properties, the twoloop nonstandard contributions in \(\varDelta \rho \) can be separated from the standard ones. These contributions are of particular interest since they increase with mass splittings between nonstandard Higgs bosons and can be additionally enhanced by \(\tan \beta \) and \(\lambda _5\), an additional free coefficient of the Higgs potential, and can thus modify the oneloop result substantially. Numerical results are given for the dependence on the various nonstandard parameters, and the influence on the calculation of electroweak precision observables is discussed.
1 Introduction
Highprecision experiments at electron–positron and hadron colliders together with highly accurate measurements at low energies have imposed stringent tests on the Standard Model (SM) and possible extensions. The experimental accuracy in the electroweak observables is sensitive to the quantum effects and requires the highest standards on the theoretical side as well. A sizable amount of theoretical work has contributed over more than two decades to a steadily rising improvement of the SM predictions and also as regards specific new physics scenarios like supersymmetric extensions. The highly accurate measurements and theoretical predictions, at the level of 0.1% precision and better, provide unique tests of the quantum structure of the SM, which has been impressively confirmed by the discovery of a Higgs particle by ATLAS [1] and CMS [2]. Moreover, it opens the possibility to obtain indirect informations on potential heavy new physics beyond the SM, in particular on the not yet sufficiently explored scalar sector.
With the meanwhile very precisely measured Higgsboson mass [3] of \(M_H = 125.09 \pm 0.24 \; \mathrm{GeV}\) the SM input is now completely determined and the SM predictions for the set of precision observables are unique, being in overall good agreement with the data. This improves the sensitivity to physics beyond the SM and makes constraints on the parameters of extended models quite severe.
Models with two scalar doublets in the Higgs sector are among the simplest extensions of the Standard Model (a review on theory and phenomenology can be found in [4]). They fulfill the relation \(\rho = 1\) at lowest order for the \(\rho \) parameter as favored by experimental data for electroweak precision observables allowing only small deviations from unity. Such small deviations \(\varDelta \rho \) naturally originate exclusively from quantum effects in models with Higgs doublets, with special sensitivity on mass splittings between different isospin components of fermions and scalars. \(\varDelta \rho \) can be related to the vectorboson selfenergies and plays the most prominent role in the higherorder calculation of precision observables, constituting the leading process independent loop corrections to accurately measured quantities like the W–Z mass correlation and the effective weak mixing angle \(\sin ^2\theta _\mathrm{eff}\).
The calculation of electroweak precision observables in the general THDM has a long history [5, 6, 7, 8, 9, 10, 11, 12, 13]. Details of the oneloop renormalization of the THDM have been discussed in various papers [14, 15, 16, 17, 18, 19, 20], with the emphasis of the more recent ones on the Higgs sector aiming at loopimproved predictions for Higgsboson observables. The current status of precision observables is given by complete oneloop calculations which can be augmented in the subset of the SM loop contributions by incorporating the known higherorder terms of the SM; the nonstandard contribution is of oneloop order and systematic twoloop calculations have not been done (Ref. [13] contains some higherorder terms by means of effective couplings in the oneloop Higgs contributions). This is different from the supersymmetric version of the THDM, the MSSM, where the nonstandard oneloop corrections to precision observables have been improved by the twoloop contributions to \(\varDelta \rho \) resulting from the strong and Yukawa interactions [21, 22, 23, 24]. In order to achieve a similar quality of the theoretical predictions also in the general THDM, the first step consists in getting the twoloop contributions to \(\varDelta \rho \) from those sectors where the oneloop effects are large, i.e. from the topYukawa interaction and the selfinteraction of the extended Higgs scalars.
In this paper we present the leading twoloop corrections to \(\varDelta \rho \) in the CPconserving THDM which result from the topYukawa coupling and the selfcouplings of the Higgs bosons. Technically they are obtained in the approximation of the gaugeless limit where the electroweak gauge couplings are set to zero (and thus the gaugeboson masses, but keeping \(M_W/M_Z\) fixed). With the assumption that one of the CPeven neutral scalars represents the scalar boson observed by the LHC experiments, with SM properties, the twoloop nonstandard contributions in \(\varDelta \rho \) can be separated from the SM ones. These contributions are of particular interest since they involve corrections proportional to \(m_t^4\), or increase with mass splittings between nonstandard Higgs bosons and can be additionally enhanced by \(\tan \beta \) and \(\lambda _5\), an additional free coefficient of the Higgs potential.
The paper is organized as follows. Section 2 contains the basic features of the THDM and specifies the notations, and Sect. 3 describes the simplifications made for our twoloop calculation. Aspects of custodial symmetry in the context of the THDM are considered in Sect. 4 which provide a deeper understanding of the various higherorder contributions to \(\varDelta \rho \). The renormalization scheme is specified in Sect. 5, and the calculation of \(\varDelta \rho \) is described in Sect. 6. The appendix contains the Feynman rules for the counterterm vertices and the definitions of the one and twoloop scalar integrals. The numerical analysis is the content of Sect. 7, and conclusions are given in Sect. 8.
2 The TwoHiggsDoublet model

typeI: all leptons and quarks couple only to the doublet \(\varPhi _2\);

typeII: the uptype quarks couple to the doublet \(\varPhi _2\), while all the downtype quarks and leptons couple to the doublet \(\varPhi _1\);

typeX or lepton specific model: all quarks couple to \(\varPhi _2\) and all leptons couple to \(\varPhi _1\);

typeY or flipped model: the uptype quarks and leptons couple to the doublet \(\varPhi _2\) while the downtype quarks couple only to \(\varPhi _1\).
Models with a more general structure for the Higgs–fermion interactions are usually referred to as typeIII models [28, 29, 30] and allow couplings of all the SM fermions to both Higgs doublets. The more general Higgs–fermion couplings are then strongly restricted by the absence of FCNCs.
3 Approximations and outline of the calculation
In order to evaluate the leading twoloop contributions from the Yukawa sector and from the Higgs selfinteractions a number of approximations can be made.
3.1 Gaugeless limit and topYukawa approximation
In addition we are using the topYukawa approximation in which all the fermion masses with the exception of the topquark mass are neglected. Especially for the bottom quark, which appears in some of the diagrams for the \(\mathscr {O}\left( \alpha _t^2\right) \) contributions, we set \(m_b=0\).
Differently from the topYukawa coupling, which is universal in all of the four models, the Yukawa coupling of the bottom quark is model specific. In models of typeI and typeX, the bottom and topYukawa interactions have the same structure, and the additional contributions to \(\varDelta \rho \) from the b quark are negligible due to the small value of \(m_b\). In models of typeII or typeY, the bYukawa coupling can be enhanced by \(t_\beta \), and the topYukawa approximation is justified in these models as long as we do not consider large values of \(t_\beta \). For large \(t_\beta \) values additional constraints from flavor physics would have to be taken into account as well.
3.2 The alignment limit
3.3 Outline of the calculation
All needed diagrams and amplitudes are generated with the help of the Mathematica package FeynArts [35]. The evaluation of the oneloop amplitudes and the calculation of the renormalization constants is done with the help of the package FormCalc [36], which is also employed to generate a Fortran expression of the result. In the numerical analysis of the oneloop result the integrals are evaluated with the program LoopTools [36].
The package TwoCalc [37, 38] is applied to deal with the Lorentz and Dirac algebra of the twoloop amplitudes and to reduce the tensor integrals to scalar integrals. In the gaugeless limit the external momenta of all the twoloop diagrams are equal to zero and the result depends only on the oneloop functions \(A_0\) and \(B_0\) (see Appendix B.1) and on the twoloop function \(T_{134}\) (see Appendix B.2) for which analytic expressions are known [39, 40] and Fortran functions are encoded in the program FeynHiggs [41, 42]. For the automation of the calculation and the implementation of the result in Fortran, the techniques from [43] are employed.
4 Custodial symmetry in the SM and the THDM
The custodial symmetry is an approximate global \({SU(2)_L\times SU(2)_R}\) symmetry of the SM which is responsible for the treelevel value of the \(\rho \) parameter [44, 45, 46]. Since the Higgs potential respects the remaining \(SU(2)_{L+R}\) after electroweak symmetry breaking the \(\rho \) parameter is protected from large radiative corrections in the Higgs mass. In the gauge interaction the custodial symmetry is only approximate since it is broken by the hypercharge coupling \(g_1\). Moreover, the custodial symmetry is broken by the Yukawa interaction which leads to large corrections to the \(\rho \) parameter for large mass differences between quarks in the same doublet [47, 48, 49]. A detailed review can be found for example in [50].
4.1 Custodial symmetry in the SM
4.2 Custodial symmetry in the THDM
A scalar potential with two doublets leads to additional terms which can violate the custodial symmetry. A lot of work has been dedicated to investigations of how the custodial symmetry can be restored in the THDM [51, 52, 53, 54, 55, 56], since there are several possibilities to implement the \({SU(2)_L\times SU(2)_R}\) transformations for two doublets. One way is to introduce matrices similar to (57) for the two original doublets in (1). The potential is then custodial invariant for \(m_{H^\pm }=m_{A^0}\) [51, 52]. Different implementations of the custodial transformations were found in [52, 53]; these require \(m_{H^\pm }=m_{H^0}\) in order to obtain a custodialsymmetric potential. However, as shown by [54, 55, 56] these different implementations of the \(SU(2)_L \times SU(2)_R\) transformations are dependent on the selected basis of the two doublets and can be related to each other by a unitary change of the basis. Since the two doublets have the same quantum numbers, such a change of basis maintains the gauge interaction but modifies the form of the potential and the Yukawa interaction.
4.2.1 Custodial symmetry for \(\chi =0\)
4.2.2 Custodial symmetry for \(\chi =\frac{\pi }{2}\)
5 Renormalization scheme
6 Corrections to the \(\rho \) parameter
Vertex and boxdiagram corrections to charged and neutral current processes do not contribute in the gaugeless limit and for vanishing masses of the external fermions, as well as \(\gamma \)–Z mixing in the neutral current interaction.
6.1 Oneloop corrections in the SM and the THDM
6.2 Higherorder corrections in the THDM
With the assumptions from Sect. 3 we have two sources for the twoloop contribution Open image in new window : the topYukawa interaction and the scalar selfinteraction. Due to the alignment limit we can subdivide the topYukawa corrections into two parts. The first one is identical to the twoloop topYukawa contribution in the SM and is discussed in Sect. 6.2.1. The second one originates from the coupling between the top quark and the nonstandard scalars \(H^0\), \(A^0\) and \(H^\pm \) and is described in more detail in Sect. 6.2.2. A similar separation can be made for the additional corrections to the \(\rho \) parameter from the scalar selfinteraction. The part \(V_\mathrm{I}\) of the potential (see (51)), which describes only the interaction between \(h^0\) and the Goldstone bosons \(G^0\), \(G^\pm \), is invariant under the custodial symmetry and the corresponding contributions to the vectorboson selfenergies in \(\varDelta \rho \) cancel each other. The remaining part of the potential gives rise to two finite subsets in Open image in new window . One follows from the interaction between the SMlike scalars \(h^0\), \(G^0\), \(G^\pm \) and the nonstandard scalars \(H^0\), \(A^0\), \(H^\pm \) and is discussed in Sect. 6.2.4. The other one contains only the nonstandard scalars \(H^0\), \(A^0\) and \(H^\pm \) as internal particles in the gaugeboson selfenergies and is described in Sect. 6.2.3.

Open image in new window originates from the coupling between the top quark and the SMlike scalars \(h^0\), \(G^0\) and \(G^\pm \) (see Sect. 6.2.1);

Open image in new window is the part which follows from the topYukawa interaction of the nonstandard scalars \(H^0\), \(A^0\) and \(H^\pm \) (see Sect. 6.2.2);

Open image in new window contains the scalar selfcoupling between the nonstandard scalars (see Sect. 6.2.3);

Open image in new window follows from the interaction between the SMlike scalars and the nonstandard scalars (see Sect. 6.2.4).
For oneloop subrenormalization we need the diagrams shown in Fig. 2 for the selfenergies with the top quarks and in Fig. 3 for the scalar contribution. In the gaugeless limit only two types of renormalization constants survive: the counterterm \(\delta s_W^2\) from the counterterm insertions in the vertices, and the mass counterterms in the propagators of the internal particles. All field counterterms of the internal particles drop out in the calculation, and all other counterterms are zero in the gaugeless limit.

the part Open image in new window contains the correction to the topmass counterterm from the SMlike scalars \(h^0\), \(G^0\), \(G^\pm \) as shown in the selfenergy diagrams in Fig. 5;

the second part Open image in new window contains the part of \(\delta m_t\) which comes from the topquark selfenergy corrections from the nonstandard scalars as depicted in Fig. 6.

Open image in new window contains the nonstandard scalar mass counterterms originating from the topYukawa coupling. The corresponding diagrams are shown in Fig. 7.

Open image in new window labels the part which contains only nonstandard scalars in the calculation of \(\delta m_{H^0}^2\), \(\delta m_{A^0}^2\) and \(\delta m_{H^\pm }^2\). The diagrams are displayed in Fig. 9.

Open image in new window incorporates the contribution to the mass counterterms of \(H^0\), \(A^0\) and \(H^\pm \) which originates from the couplings of the nonstandard scalars to the SMlike scalars. The corresponding selfenergy diagrams are presented in Fig. 11.
6.2.1 Standard model corrections from the topYukawa coupling
6.2.2 Nonstandard corrections from the topYukawa coupling
6.2.3 Scalar corrections from the interaction of the nonstandard scalars
The interaction between the nonstandard scalars gives another finite subset. When inspecting this contribution we found that all the corrections from a coupling between four nonstandard scalars are canceled. The twoloop diagrams which contain such a coupling can be written as a product of two scalar oneloop integrals. The mass counterterms in the subloop renormalization lead to the same product from the corrections to the scalar selfenergies, but with an opposite sign. Consequently the two terms cancel each other.
6.2.4 Scalar corrections from the interaction of the nonstandard scalars with the SM scalars
6.3 The InertHiggsDoublet model

There is no nonstandard correction to \(\varDelta \rho \) from the topYukawa interaction, since the interaction of the fermions with the nonstandard scalars is forbidden by the \(Z_2\) symmetry.

The part \(V^\text {IHDM}_{I}\) has the same structure as the scalar potential of the SM and will not lead to contributions to the \(\rho \) parameter since it is invariant under the custodial symmetry (see Sect. 4).

In the IHDM all the quartic couplings between four nonstandard scalars are proportional to \(\varLambda _2\). However, in our calculation in the aligned THDM we found that all the contributions to \(\varDelta \rho \) from couplings between four nonstandard scalars vanish (see Sect. 6.2.3). The responsible arguments can also be transferred to the IHDM.
 When we identify \(H_1\) with \(\varPhi _{\mathrm{SM}}\) and \(H_2\) with \(\varPhi _{\mathrm{NS}}\) we see that the part \(V_{\mathrm{III}}\) of the potential in the aligned THDM can be obtained by the replacementin \(V^\text {IHDM}_\mathrm{III}\). Consequently for the calculation of the \(\rho \) parameter in the IHDM we get corrections corresponding to Open image in new window and Open image in new window . The oneloop correction Open image in new window is identical in the IHDM since it is independent of \(\lambda _5\). The twoloop part Open image in new window can be written in terms of the IHDM parameter \(\mu _2^2\) by using (159).$$\begin{aligned} \mu _2^2 = \frac{1}{2}\lambda _5 v^2 \frac{ m_{h^0}^2}{2} \end{aligned}$$(159)

As mentioned in Sect. 6.2.3, the correction Open image in new window contains the interaction between three of the nonstandard scalars \(H^0\), \(A^0\) and \(H^\pm \) which follows from the part \(V_{\mathrm{IV}}\) of the potential in (50). In the IHDM couplings between three nonstandard scalars are forbidden because of the exact \(Z_2\) symmetry. As a consequence, corrections to the \(\rho \) parameter which would correspond to Open image in new window are absent in the IHDM.
7 Numerical results
In this part we present the numerical results of the twoloop corrections to the \(\rho \) parameter. We study the dependence on the various parameters of the aligned THDM and compare the nonstandard twoloop contributions with the oneloop result which is part of existing calculations of electroweak precision observables so far. In this way the parameter regions emerge where the oneloop calculations are insufficient and bounds on parameters derived from experimental precision data will be significantly changed when the twoloop terms are taken into account.
7.1 Results for the topYukawa contribution
We start with the analysis of the contribution Open image in new window which is originating from the coupling between the top quark and the nonstandard scalars. As a first test of our result we examine the behavior in the socalled decoupling limit [73], in which the masses of the nonstandard scalars are much larger than \(m_h^0\). In this limit the scalar sector of the THDM can be described by an effective theory which is identical to the SM Higgs sector. Consequently we expect Open image in new window to vanish for large, equal nonstandard Higgs masses. The decoupling scenario is investigated in the upper panel of Fig. 12, where Open image in new window is shown for degenerate masses of the nonstandard scalars. The solid lines represent results for different values of \(t_\beta \). Since the topYukawa coupling breaks the custodial symmetry this contribution is still nonzero, even if the custodial symmetry in the Higgs potential is restored by equal masses of the charged and neutral Higgs states. As expected it approaches zero when the masses increase. Moreover, we can see that larger values of \(t_\beta \) suppress the correction. The reason is that the coupling of the top quark to the scalars \(H^0\), \(A^0\) and \(H^\pm \) scales with \(t_\beta ^{1}\) in the alignment limit (see Sect. 3.2).
The influence of \(t_\beta \) is visualised on the lower panel of Fig. 12 with Open image in new window for the mass configurations as described by the legend, showing the decrease of the contribution with \(t_\beta \). In addition different mass splittings between charged and neutral scalars yield noticable deviations in the result and can even lead to different signs. In general, the topYukawa contribution is of the order of the SM value \(\delta \rho ^{(2)}_\mathrm{t,SM}\) or smaller.
7.2 Results for the nonstandard scalar contribution
We now discuss the numerical results of the contribution Open image in new window which originates from the coupling between three nonstandard scalars as described in Sect. 6.2.3. The influence of a mass splitting between charged and neutral scalars is presented in Fig. 13. The two panels show results for \(m_{H^0}=350 \text { GeV}\), \(m_{A^0}=400\text { GeV}\) and \(\lambda _5=\pm 1\). The variation of \(m_{H^\pm }\) is performed such that it yields similar mass differences for the specified parameter settings. The different lines correspond to different values of \(t_\beta \) as defined in the legend. For comparison the blue dashed line displays the result for the oneloop nonstandard correction Open image in new window . The gray area indicates the bounds from the T parameter in (165).
We see that the contribution Open image in new window can give corrections to the \(\rho \) parameter which are comparable in size or even larger than the oneloop correction. The reason is the new couplings between three nonstandard scalars which enter for the first time in the twoloop contribution. Adding the twoloop corrections to the oneloop result can lead to noticeable modifications of the parameter region allowed by the constraints on T.
However, for \(m_{H^0}=m_{H^\pm }\) we have Open image in new window since in that case \(V_{\mathrm{III}}\) is invariant only under custodial transformations for \(\chi =\frac{\pi }{2}\), but then \(V_{\mathrm{IV}}\) is not invariant and the triple couplings between three nonstandard scalars hence break the custodial symmetry (see Sect. 4.2.2).
The dependence of Open image in new window on \(t_\beta \) is visualized directly in Fig. 14 for different values of \(\lambda _5\), displaying the increase with \(t_\beta \) and the modification by the choice of \(\lambda _5\) according to (168).
7.3 Results for the mixed scalar contribution
In Fig. 15 we analyze the influence of a mass splitting between the charged and neutral scalars. We show two scenarios for different values of \(m_{H^0}\) and \(m_{A^0}\), while the mass of \(m_{H^\pm }\) is varied in such a way that the mass splittings are comparable. The three solid lines present the results for different values of \(\lambda _5\). The blue dashed line gives the oneloop contribution Open image in new window for comparison.
Since the correction Open image in new window is independent of \(t_\beta \) it will be the dominant scalar twoloop correction to the \(\rho \) parameter for \(t_\beta \approx 1\) where Open image in new window is small. However, for \(m_{H^0}=m_{H^\pm }\) both the oneloop correction Open image in new window and Open image in new window vanish independently of \(t_\beta \), and Open image in new window is the only remaining scalar correction to the \(\rho \) parameter (for \(t_\beta \ne 1\)).
For the InertHiggsDoubletModel (IHDM), as explained in Sect. 6.3, the only nonstandard twoloop correction to the \(\rho \) parameter is equivalent to Open image in new window . Conventionally, the parameter \(\mu _2^2\) is often used as a free input parameter. The results in Fig. 15 can easily be interpreted in the IHDM by means of the relation (159) to trade \(\lambda _5\) for \(\mu _2^2\).
7.4 Results for a light pseudoscalar
The purely nonstandard scalar contribution Open image in new window vanishes for \(t_\beta =1\), but otherwise has a strong variation with \(t_\beta \) (and \(\lambda _5\)). It is shown in Fig. 17, the analogous plot to Fig. 13, now with a light \(A^0\). Since the common zero of all curves corresponds to \(m_{A^0}\), the twoloop contribution Open image in new window is always negative for \(m_{H^0, H^\pm } > m_{A^0}\) and thus can diminish Open image in new window substantially for \(m_{H^\pm } > m_{H^0}\) when \(t_\beta \) increases. Again, the situation \(m_{H^\pm } < m_{H^0}\) is disfavored.
For \(m_{A^0}<m_{h^0}/2\), the coupling of \(h^0\) to two pseudoscalars has to be small to suppress the decay channel \(h^0\rightarrow A^0 A^0\) [78]. In the alignment limit this requires one to restrict the value of \(\lambda _5\) to \(\lambda _5v^2 \simeq 2m_{A^0}^2+m_{h^0}^2\) (see (171)).
Scenarios with a light \(A^0\) are especially interesting in the THDM, since Barr–Zee type twoloop diagrams can provide an explanation for the \(3\sigma \) difference between the SM prediction and the measured value of the muon anomalous magnetic moment \(a_\mu \) [79]. An improved agreement between theory and experiment consistent with several theoretical and experimental constraints can be achieved in a typeX model with very large values of \(t_\beta \) (see [80, 81] and references therein). Usually \(m_{H^\pm }=m_{H^0}\) is assumed, to fulfill the constraints from electroweak precision observables. For the \(\rho \) parameter this means vanishing contributions from Open image in new window and Open image in new window . Furthermore, the topYukawa contribution Open image in new window is strongly suppressed. However, for such large values of \(t_\beta \), the nonstandard scalar contribution Open image in new window would completely run out of control unless the scalar selfcoupling is kept small by adjusting \(\lambda _5\) very close to \(\lambda _5 =2m_{H^0}^2/v^2\). An additional aspect of typeX models with very large \(t_\beta \) is the enhanced Yukawa coupling of the \(\tau \) lepton. This could yield a further twoloop contribution to the \(\rho \) parameter, which we did not consider in this work.
8 Conclusions
We have given an overview over the calculation of the twoloop contributions to the \(\rho \)parameter in the CPconserving TwoHiggsDoublet Model where one of the CPeven scalars (\(h^0\)) is identified with the scalar resonance at 125 GeV observed by the LHC experiments ATLAS and CMS. The approximation of the gaugeless limit and massless fermions except the top quark yield the leading contributions from the topYukawa coupling and the selfcouplings of the Higgs bosons, which can be separated into standard and nonstandard contributions. As already at the oneloop level, the nonstandard contributions from the scalar selfinteractions are particularly sensitive to mass splittings between neutral and charged scalars. As a new feature, the twoloop contributions have a significant dependence on the parameters \(\tan \beta \) and \(\lambda _5\), the coefficient of the THDM scalar potential that is not fixed by the masses of the neutral and charged Higgs bosons, and thus can modify the oneloop result substantially. Moreover, this significant dependence on the additional parameters can be exploited to get more indirect information on the Higgs potential from electroweak precision data than with the currently available oneloop calculations.
Notes
Acknowledgements
This work was supported in part by the Deutsche Forschungsgemeinschaft (DFG) under Grant No. EXC153 (Excellence Cluster Structure and Origin of the Universe). We thank Georg Weiglein for useful discussions and Thomas Hahn and Sebastian Paßehr for their helpful support in the installation and handling of the twoloop calculational tools.
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