1 Introduction

The interpretation of the Higgs signal at within the Minimal Supersymmetric Standard Model (MSSM) depends crucially on the predicted properties of the other Higgs states of the model, as the mass of the charged Higgs boson, . This mass is calculated in the Feynman-diagrammatic approach within the MSSM with real parameters. The result includes the complete one-loop contributions and the two-loop contributions of . The one-loop contributions lead to sizable shifts in the prediction, reaching up to for relatively small values of . Even larger effects can occur depending on the sign and size of the  parameter that enters the corrections affecting the relation between the bottom-quark mass and the bottom Yukawa coupling. The two-loop terms can shift by more than . The two-loop contributions amount to typically about 30% of the one-loop corrections for the examples that we have studied. These effects can be relevant for precision analyses of the charged MSSM Higgs boson.


DESY 13–100


The Charged Higgs Boson Mass of the MSSM

in the Feynman-Diagrammatic Approach

M. Frank***email: m@rkusfrank.de, L. Galetaemail: leo@ifca.unican.es, T. Hahnemail: hahn@feynarts.de, S. Heinemeyer§§§email: Sven.Heinemeyer@cern.ch, W. Hollikemail: hollik@mppmu.mpg.de,

[.3em] H. Rzehakemail: hrzehak@ mail.cern.ch******on leave from Albert-Ludwigs-Universität Freiburg, Physikalisches Institut, D–79104 Freiburg, Germany  and G. Weiglein††††††email: Georg.Weiglein@desy.de

Institut für Theoretische Physik, Universität Karlsruhe,

D–76128 Karlsruhe, Germany‡‡‡‡‡‡former address

Instituto de Física de Cantabria (CSIC–UC), Santander, Spain

Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6,

D–80805 München, Germany

CERN, PH-TH, 1211 Geneva 23, Switzerland

DESY, Notkestraße 85, D–22607 Hamburg, Germany

1 Introduction

The ATLAS and CMS experiments at CERN have recently discovered a new boson with a mass around  [1, 2]. Within the presently still rather large experimental uncertainties this new boson behaves like the Higgs boson of the Standard Model (SM) [3]. However, the newly discovered particle can also be interpreted as the Higgs boson of extended models. The Higgs sector of the Minimal Supersymmetric Standard Model (MSSM) [4] with two scalar doublets accommodates five physical Higgs bosons. In lowest order these are the light and heavy -even and , the -odd , and the charged Higgs bosons . It was shown that the newly discovered boson can be interpreted in principle as the light, but also as the heavy -even Higgs boson of the MSSM, see, e.g., Refs. [5, 6, 7, 8, 9]. In the latter case the charged Higgs boson must be rather light, and the search for the charged Higgs boson could be crucial to investigate this scenario [8]. In the former case the charged Higgs boson is bound to be heavier than the top quark [8]. In both cases the discovery of a charged Higgs boson would constitute an unambiguous sign of physics beyond the SM, serving as a good motivation for searches for the charged Higgs boson.

The Higgs sector of the MSSM can be expressed at lowest order in terms of the gauge couplings, the mass of the -odd Higgs boson, , and , the ratio of the two vacuum expectation values. All other masses and mixing angles can therefore be predicted, e.g. the charged Higgs boson mass,


at tree-level. denote the masses of the  and  boson, respectively. Higher-order contributions can give large corrections to the tree-level relations, where the loop corrected charged Higgs-boson mass is denoted as .

Experimental searches for the neutral MSSM Higgs bosons have been performed at LEP [10, 11], placing important restrictions on the parameter space. At Run II of the Tevatron the search was continued, but is now superseeded by the LHC Higgs searches. Besides the discovery of a SM Higgs-like boson the LHC searches place stringent bounds, in particular in the regions of small and large  [12]. At a future linear collider (LC) a precise determination of the Higgs boson properties (either of the light Higgs boson at or heavier MSSM Higgs bosons within the kinematic reach) will be possible [13, 14, 15, 16]. The interplay of the LHC and the LC in the neutral MSSM Higgs sector has been discussed in Refs. [17, 18].

The charged Higgs bosons of the MSSM (or a more general Two Higgs Doublet Model (THDM)) have also been searched for at LEP [19, 20, 21, 22, 23], yielding a bound of  [24, 25]. The LHC places bounds on the charged Higgs mass, as for the neutral heavy MSSM Higgs bosons, at relatively low values of its mass and at large or very small  [26, 27]. For (with denoting the mass of the top quark) the charged Higgs boson is mainly produced from top quarks and decays mainly as . For the charged Higgs boson is mainly produced together with a top quark and the dominant decay channels are , where the latter is the main search channel. At the LC, for a high-precision determination of the charged Higgs boson properties will be possible [13, 14, 15, 16].

For the MSSM111We concentrate here on the case with real parameters. For the case of complex parameters see Refs. [28, 29] and references therein. the status of higher-order corrections to the masses and mixing angles in the neutral Higgs sector is quite advanced. The complete one-loop result within the MSSM is known [30, 31, 32, 33]. The by far dominant one-loop contribution is the term due to top and stop loops (, being the top-quark Yukawa coupling). The computation of the two-loop corrections has meanwhile reached a stage where all the presumably dominant contributions are available [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48]. In particular, the , , , and contributions to the self-energies are known for vanishing external momenta. For the (s)bottom corrections, which are mainly relevant for large values of , an all-order resummation of the -enhanced term of is performed [49, 50, 51]. The remaining theoretical uncertainty on the lightest -even Higgs boson mass has been estimated to be about  [52, 53, 54]. The public codes FeynHiggs [55, 35, 52, 28] (including all of the above corrections) and CPsuperH [56] exist. A full two-loop effective potential calculation (including even the momentum dependence for the leading pieces and the leading three-loop corrections) has been published [57]. However, no computer code is publicly available. Most recently another leading three-loop calculation, depending on the various SUSY mass hierarchies, became available [58], resulting in the code H3m (which adds the three-loop corrections to the FeynHiggs result).

Also the mass of the charged Higgs boson is affected by higher-order corrections. However, the status is somewhat less advanced as compared to the neutral Higgs bosons. First, in Ref. [59] leading corrections to the relation given in Eq. (1) have been evaluated. The one-loop corrections from and  loops have been derived in Refs. [60, 61]. A nearly complete one-loop calculation, neglecting the terms suppressed by higher powers of the SUSY mass scale, was presented in Ref. [62]. The first full one-loop calculation in the Feynman-diagrammatic (FD) approach has been performed in Ref. [63], and re-evaluated more recently in Refs. [64, 28]. At the two-loop level, within the FD approach, the leading two-loop contributions for the three neutral Higgs bosons in the case of complex soft SUSY-breaking parameters have been obtained [29]. Because of the (-violating) mixing between all three neutral Higgs bosons, in the MSSM with complex parameters usually the charged Higgs mass is chosen as independent (on-shell) input parameter, which by construction does not receive any higher-order corrections. The calculation however involves the evaluation of the contributions to the charged self energy. In the -conserving case, on the other hand, where usually instead of is chosen as independent input parameter, the corresponding self-energy contribution can be utilized to obtain corrections of to the mass .

In the present paper we combine the two-loop terms of with the complete one-loop contribution of Ref. [28] to obtain an improved prediction for the mass of the charged Higgs boson. The results are incorporated in the code FeynHiggs (current version: 2.9.4). An overview of the calculation is given in Sect. 2, whereas in Sect. 3 and Sect. 4 we discuss the size and relevance of the one- and two-loop corrections and investigate the impact of the various sectors of the MSSM on the prediction for . Our conclusions are given in Sect. 5.

2 Higher-order contributions for

2.1 From tree-level to higher-orders

In the MSSM (with real parameters) one conventionally chooses the mass of the -odd Higgs boson, , and (, see Eq. (2.1)) as independent input parameters. Thus the mass of the charged Higgs boson can be predicted in terms of the other parameters and receives a shift from the higher-order contributions.

The two Higgs doublets of the MSSM are decomposed in the following way,


with the two vacuum expectation values and . The hermitian 22-matrix of the charged states , , contains the following elements,


, , denote the soft SUSY-breaking parameters in the Higgs sector, and , are the and gauge couplings, respectively. The mass eigenstates in lowest order in the charged sector follow from unitary transformations on the original fields,


This yields the (square of the) mass eigenvalue for the charged Higgs boson, , as given by Eq. (1). Quantum corrections substantially modify the tree-level mass. The charged Higgs-boson pole mass, , including higher-order contributions entering via the renormalized charged Higgs-boson self-energy, , is obtained by solving the equation


This yields as the real part of the complex zero of Eq. (5). The renormalized charged Higgs-boson self-energy, , is composed of the unrenormalized self-energy, , and counterterm contributions as specified below. In perturbation theory, the self-energy is expanded as follows


in terms of the th-order contributions , and analogously for the renormalized quantities. Details for the one-loop self-energies are given below in Sect. 2.2, and for the two-loop contributions in Sect. 2.3.

A possible mixing with the charged Goldstone boson would contribute to the prediction for the charged Higgs-boson mass from two-loop order onwards via terms of the form . The mixing contributions with yield a two-loop contribution that is subleading compared to the leading terms at that we take into account, as described in Sect. 2.3. Consequently, we neglect those two-loop Higgs–Goldstone mixing contributions throughout our analysis.

2.2 One-loop corrections

Here we review the calculation of the full one-loop corrections to , following Refs. [28, 64]. All self-energies and renormalization constants are understood to be one-loop quantities, dropping the order index. Renormalized self-energies, , can be expressed in terms of the corresponding unrenormalized self-energies, , the field renormalization constants, and the mass counterterms. For the charged Higgs-boson self-energy entering Eq. (5) this expression reads


The independent mass parameters are renormalized according to


while the mass counterterm for the charged Higgs boson, arising from , is a dependent quantity. It is given in terms of the counterterms for and by


We renormalize the  boson and the -odd Higgs boson masses on-shell, yielding the mass counterterms


where is the transverse part of the  boson self-energy.

For field renormalization, required for finite self-energies at arbitrary values of the external momentum , we assign one field-renormalization constant for each Higgs doublet,


For the charged Higgs field this implies




For the determination of the field renormalization constants we adopt the  scheme,


i.e. the renormalization constants consist of divergent parts only, see the discussion in Ref. [28]. () denotes the derivative of the unrenormalized self-energies of the neutral -even Higgs bosons, with the mixing angle set to zero. As default value of the renormalization scale we have chosen .

For the self-energies as specified in Eq. (7) we have evaluated the complete one-loop contributions with the help of the programs FeynArts [65] and FormCalc [66]. As regularization scheme we have used constrained differential regularization [67], which has been shown to be equivalent to dimensional reduction [68] at the one-loop level [66], thus preserving supersymmetry [69, 70]. The corresponding Feynman-diagrams for the charged Higgs boson (and similarly for the  boson, where additional diagrams with gauge boson and ghost loops contribute) are shown in Fig. 2.2. The diagrams for the neutral Higgs bosons, entering and , (i.e. the neutral Higgs boson self-energies) , are depicted in Fig. 2.2.

Figure 1. Generic Feynman diagrams for the self-energy ( = {, , }, = {, , }, = {, , }). Similar diagrams for the  boson self-energy are obtained by replacing the external Higgs boson by a  boson; not all combinations of particle insertions exist.

Figure 1. Generic Feynman diagrams for the self-energy ( = {, , }, = {, , }, = {, , }). Similar diagrams for the  boson self-energy are obtained by replacing the external Higgs boson by a  boson; not all combinations of particle insertions exist.

Figure 2. Generic Feynman diagrams for the , , , self-energies ( = {, , , , , , , , } ). Not all combinations of particle insertions exist for all neutral Higgs bosons.

Figure 2. Generic Feynman diagrams for the , , , self-energies ( = {, , , , , , , , } ). Not all combinations of particle insertions exist for all neutral Higgs bosons.

2.3 Two-loop corrections

We now turn to the corrections at the two-loop level. Again, we drop the order-index for all Higgs boson and SM gauge boson self-energies and renormalization constants, which are in this section understood to be of two-loop order. The terms are obtained in the limit of vanishing gauge couplings and neglecting the dependence on the external momentum [35], keeping only terms , with the top Yukawa coupling as defined above. We neglect the bottom Yukawa coupling in the two-loop Higgs-boson self-energies. In this approximation, the counterterm for is determined as follows


while the renormalization constants and do not contribute,


Consequently, the two-loop contribution to the renormalized self-energy can be written in the following way,


From Eq. (5) we get the two-loop correction to the charged Higgs-boson mass,


with the self-energies evaluated at the two-loop level.

We thus have to evaluate the contributions to the and self-energies. Examples of generic Feynman diagrams for the self-energy are depicted in Fig. 2.3, and in Fig. 2.3 for the  boson self-energy. These contributions have been evaluated using the packages FeynArts [65] and TwoCalc [71].

Figure 3. Examples of generic two-loop diagrams and diagrams with counterterm insertion for the charged Higgs-boson self-energy ().

Figure 3. Examples of generic two-loop diagrams and diagrams with counterterm insertion for the charged Higgs-boson self-energy ().

Figure 4. Examples of generic two-loop diagrams and diagrams with counterterm insertion for the  boson self-energy ().

Figure 4. Examples of generic two-loop diagrams and diagrams with counterterm insertion for the  boson self-energy ().

2.4 Subloop renormalization in the scalar top/bottom sector

Besides the computation of the genuine two-loop diagrams at for the self-energies, one-loop renormalization is required for the and  sector providing the counterterms for one-loop subrenormalization. This yields additional diagrams with counterterm insertions; examples are the fourth diagrams in Figs. 2.3, 2.3. The bilinear part of the and Lagrangian,


contains the stop and sbottom mass matrices and , given by


Here , are soft-breaking parameters, where is the same for and (see below), and is the trilinear soft-breaking parameter. The D-terms do not contribute to and therefore have to be neglected in the calculation of the stop mass values entering the contribution of this order [29]. The mass matrix can be diagonalized with the help of a unitary transformation , which can be parametrized by a mixing angle ,


We follow here the renormalization prescription used in Refs. [72, 73]. In the MSSM the  sector is described in terms of four real parameters (where we assume that and are defined via other sectors): the real soft SUSY-breaking parameters and , the trilinear coupling , and the top Yukawa coupling . Instead of the quantities , and , in the on-shell scheme applied in this paper we choose the on-shell squark masses , and the top-quark mass as independent parameters. It should furthermore be noted that the counterterms are evaluated at , such as to yield the desired contributions when combined with the one-loop diagrams with counterterm insertion.

The following renormalization conditions are imposed:

  • The top-quark mass is defined on-shell, yielding the mass counterterm ,


    referring to the Lorentz decomposition of the self energy


    into a left-handed, a right-handed, and a scalar part, , , , respectively.

  • The stop masses are also determined via on-shell conditions [35, 72], yielding

  • The third condition affects the trilinear coupling . Rewriting the squark mass matrix in terms of the mass eigenvalues and the mixing angle using Eq. (22),


    yields the counterterm matrix by introducing counterterms for the masses and for the angle. One obtains the counterterm for the off-diagonal contribution in the stop sector,


    for which the following renormalization condition has been used [73, 72]:


    Finally we derive the relation between the counterterms and . The two counterterms are mutually related via Eq. (20) and Eq. (26). The off-diagonal entries of the corresponding counterterm matrices yield


    As a result, we obtain for


In the sector, we also encounter four real parameters (with and defined via other sectors): the soft-breaking mass parameters and , the trilinear coupling , and the bottom Yukawa coupling or the -quark mass, respectively (which is neglected for the set of two-loop corrections presented in this paper). SU invariance requires the “left-handed” soft-breaking parameters in the stop and the sbottom sector to be identical (denoted as ). With the approximations described above this yields, e.g., . In the evaluation of the contributions to the Higgs-boson self-energies, the counterterms of the sbottom sector appear only in the self-energy of the charged Higgs boson. In our approximation for the two-loop contributions, where the -quark mass is neglected, and do not mix, and decouples and does not contribute. The two-loop contribution to the charged Higgs-boson self-energy thus depends only on a single parameter of the sbottom sector, which can be chosen as the squark mass . By means of SU invariance, the corresponding mass counterterm is already determined:


With the set of renormalization constants determined in Eqs. (23), (25), (30) and (31) the counterterms arising from the one-loop subrenormalization of the stop and sbottom sectors are fully specified.

Finally, at gluinos appear as virtual particles at the two-loop level; hence, no renormalization in the gluino sector is needed. The corresponding soft-breaking gluino mass parameter is denoted . In the case of real MSSM parameters considered here the gluino mass is given as .

2.5 Higher-order corrections in the sector

We furthermore include in our prediction for corrections beyond the one-loop level originating from the bottom/sbottom sector contributions to and . Potentially large higher-order effects proportional to can arise in the relation between the bottom-quark mass and the bottom Yukawa coupling as described in Refs. [50, 51]. The leading -enhanced contribution in the limit of heavy SUSY masses can be expressed in terms of a quantity and resummed to all orders using an effective Lagrangian approach. The relevant part of the effective Lagrangian is given by




denotes a running bottom quark mass at the scale  in the  scheme that incorporates SM QCD corrections (i.e., no SUSY QCD effects are included in the running). The corresponding mass in the  scheme is denoted by . and are the finite parts of the self-energies defined in analogy to Eq. (24). denotes the  element of the CKM matrix. In the numerical evaluations performed with the program FeynHiggs below we use .

The leading -enhanced one-loop contribution in the limit of heavy SUSY masses takes the simple form [49]


where is evaluated at the scale , and the function is given by


The ellipses in Eq. (35) denote subleading terms that we take over from Ref. [74]. Expanded up to one-loop order, the effective mass is close to the  mass (including SUSY contributions in the running), see Refs. [45, 73]. A recent two-loop calculation of can be found in Ref. [75].

3 Approximation for the two-loop corrections

In Sect. 2 we have described the approximations for getting the two-loop terms, which can be written as terms proportional to . It is well known that for the neutral Higgs bosons this procedure indeed yields the dominant part of the one-loop [30, 31, 32, 33] and the two-loop corrections [34, 35].

For the charged Higgs boson mass, , the described procedure provides the analogous contribution to the mass shift as well,


There are, however, other contributions of similar structure at the one-loop level [59, 60, 61, 62, 63],


which are not covered by our approximations for the two-loop terms because they would correspond to (first), non-vanishing gauge-couplings (second), and in the self-energy (third term). This is justified for large scalar-quark mass scales where the second and third type of terms are suppressed. On the other hand, the term (37) extracted by our approximation can in general be large also for large , both at the one-loop and the two-loop level, as we will explain below.

3.1 The one-loop case

Applying the approximations outlined in Sect. 2.3 at the one-loop level yields the counterterms (all quantities in this section are understood to be one-loop quantities),


and thus


From Eq. (5) we get the one-loop corrected value of the charged Higgs-boson mass,




In the following we use the factor to simplify the notation (),