Abstract
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 Feynmandiagrammatic approach within the MSSM with real parameters. The result includes the complete oneloop contributions and the twoloop contributions of . The oneloop 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 bottomquark mass and the bottom Yukawa coupling. The twoloop terms can shift by more than . The twoloop contributions amount to typically about 30% of the oneloop corrections for the examples that we have studied. These effects can be relevant for precision analyses of the charged MSSM Higgs boson.
CERN–PH–TH/2013–114
DESY 13–100
MPP–2013–147
The Charged Higgs Boson Mass of the MSSM
in the FeynmanDiagrammatic Approach
M. Frank^{*}^{*}*email: m@rkusfrank.de, L. Galeta^{†}^{†}†email: leo@ifca.unican.es, T. Hahn^{‡}^{‡}‡email: hahn@feynarts.de, S. Heinemeyer^{§}^{§}§email: Sven.Heinemeyer@cern.ch, W. Hollik^{¶}^{¶}¶email: hollik@mppmu.mpg.de,
[.3em] H. Rzehak^{∥}^{∥}∥email: hrzehak@ mail.cern.ch^{**}^{**}**on leave from AlbertLudwigsUniversität 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
MaxPlanckInstitut für Physik (WernerHeisenbergInstitut), Föhringer Ring 6,
D–80805 München, Germany
CERN, PHTH, 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,
(1) 
at treelevel. denote the masses of the and boson, respectively. Higherorder contributions can give large corrections to the treelevel relations, where the loop corrected charged Higgsboson 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 Higgslike 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 highprecision determination of the charged Higgs boson properties will be possible [13, 14, 15, 16].
For the MSSM^{1}^{1}1We concentrate here on the case with real parameters. For the case of complex parameters see Refs. [28, 29] and references therein. the status of higherorder corrections to the masses and mixing angles in the neutral Higgs sector is quite advanced. The complete oneloop result within the MSSM is known [30, 31, 32, 33]. The by far dominant oneloop contribution is the term due to top and stop loops (, being the topquark Yukawa coupling). The computation of the twoloop 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 selfenergies are known for vanishing external momenta. For the (s)bottom corrections, which are mainly relevant for large values of , an allorder 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 twoloop effective potential calculation (including even the momentum dependence for the leading pieces and the leading threeloop corrections) has been published [57]. However, no computer code is publicly available. Most recently another leading threeloop calculation, depending on the various SUSY mass hierarchies, became available [58], resulting in the code H3m (which adds the threeloop corrections to the FeynHiggs result).
Also the mass of the charged Higgs boson is affected by higherorder 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 oneloop corrections from and loops have been derived in Refs. [60, 61]. A nearly complete oneloop calculation, neglecting the terms suppressed by higher powers of the SUSY mass scale, was presented in Ref. [62]. The first full oneloop calculation in the Feynmandiagrammatic (FD) approach has been performed in Ref. [63], and reevaluated more recently in Refs. [64, 28]. At the twoloop level, within the FD approach, the leading twoloop contributions for the three neutral Higgs bosons in the case of complex soft SUSYbreaking 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 (onshell) input parameter, which by construction does not receive any higherorder 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 selfenergy contribution can be utilized to obtain corrections of to the mass .
In the present paper we combine the twoloop terms of with the complete oneloop 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 twoloop corrections and investigate the impact of the various sectors of the MSSM on the prediction for . Our conclusions are given in Sect. 5.
2 Higherorder contributions for
2.1 From treelevel to higherorders
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 higherorder contributions.
The two Higgs doublets of the MSSM are decomposed in the following way,
(2) 
with the two vacuum expectation values and . The hermitian 22matrix of the charged states , , contains the following elements,
(3) 
, , denote the soft SUSYbreaking 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,
(4) 
This yields the (square of the) mass eigenvalue for the charged Higgs boson, , as given by Eq. (1). Quantum corrections substantially modify the treelevel mass. The charged Higgsboson pole mass, , including higherorder contributions entering via the renormalized charged Higgsboson selfenergy, , is obtained by solving the equation
(5) 
This yields as the real part of the complex zero of Eq. (5). The renormalized charged Higgsboson selfenergy, , is composed of the unrenormalized selfenergy, , and counterterm contributions as specified below. In perturbation theory, the selfenergy is expanded as follows
(6) 
in terms of the thorder contributions , and analogously for the renormalized quantities. Details for the oneloop selfenergies are given below in Sect. 2.2, and for the twoloop contributions in Sect. 2.3.
A possible mixing with the charged Goldstone boson would contribute to the prediction for the charged Higgsboson mass from twoloop order onwards via terms of the form . The mixing contributions with yield a twoloop 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 twoloop Higgs–Goldstone mixing contributions throughout our analysis.
2.2 Oneloop corrections
Here we review the calculation of the full oneloop corrections to , following Refs. [28, 64]. All selfenergies and renormalization constants are understood to be oneloop quantities, dropping the order index. Renormalized selfenergies, , can be expressed in terms of the corresponding unrenormalized selfenergies, , the field renormalization constants, and the mass counterterms. For the charged Higgsboson selfenergy entering Eq. (5) this expression reads
(7) 
The independent mass parameters are renormalized according to
(8) 
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
(9) 
We renormalize the boson and the odd Higgs boson masses onshell, yielding the mass counterterms
(10) 
where is the transverse part of the boson selfenergy.
For field renormalization, required for finite selfenergies at arbitrary values of the external momentum , we assign one fieldrenormalization constant for each Higgs doublet,
(11) 
For the charged Higgs field this implies
(12) 
with
(13) 
For the determination of the field renormalization constants we adopt the scheme,
(14) 
i.e. the renormalization constants consist of divergent parts only, see the discussion in Ref. [28]. () denotes the derivative of the unrenormalized selfenergies 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 selfenergies as specified in Eq. (7) we have evaluated the complete oneloop 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 oneloop level [66], thus preserving supersymmetry [69, 70]. The corresponding Feynmandiagrams 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 selfenergies) , are depicted in Fig. 2.2.
2.3 Twoloop corrections
We now turn to the corrections at the twoloop level. Again, we drop the orderindex for all Higgs boson and SM gauge boson selfenergies and renormalization constants, which are in this section understood to be of twoloop 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 twoloop Higgsboson selfenergies. In this approximation, the counterterm for is determined as follows
(15) 
while the renormalization constants and do not contribute,
(16) 
Consequently, the twoloop contribution to the renormalized selfenergy can be written in the following way,
(17) 
From Eq. (5) we get the twoloop correction to the charged Higgsboson mass,
(18) 
with the selfenergies evaluated at the twoloop level.
2.4 Subloop renormalization in the scalar top/bottom sector
Besides the computation of the genuine twoloop diagrams at for the selfenergies, oneloop renormalization is required for the and sector providing the counterterms for oneloop 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,
(19) 
contains the stop and sbottom mass matrices and , given by
(20)  
(21) 
Here , are softbreaking parameters, where is the same for and (see below), and is the trilinear softbreaking parameter. The Dterms 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 ,
(22) 
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 SUSYbreaking parameters and , the trilinear coupling , and the top Yukawa coupling . Instead of the quantities , and , in the onshell scheme applied in this paper we choose the onshell squark masses , and the topquark 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 oneloop diagrams with counterterm insertion.
The following renormalization conditions are imposed:

The topquark mass is defined onshell, yielding the mass counterterm ,
(23) referring to the Lorentz decomposition of the self energy
(24) into a lefthanded, a righthanded, and a scalar part, , , , respectively.

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),
(26) yields the counterterm matrix by introducing counterterms for the masses and for the angle. One obtains the counterterm for the offdiagonal contribution in the stop sector,
(27) for which the following renormalization condition has been used [73, 72]:
(28) Finally we derive the relation between the counterterms and . The two counterterms are mutually related via Eq. (20) and Eq. (26). The offdiagonal entries of the corresponding counterterm matrices yield
(29) As a result, we obtain for
(30)
In the sector, we also encounter four real parameters (with and defined via other sectors): the softbreaking mass parameters and , the trilinear coupling , and the bottom Yukawa coupling or the quark mass, respectively (which is neglected for the set of twoloop corrections presented in this paper). SU invariance requires the “lefthanded” softbreaking 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 Higgsboson selfenergies, the counterterms of the sbottom sector appear only in the selfenergy of the charged Higgs boson. In our approximation for the twoloop contributions, where the quark mass is neglected, and do not mix, and decouples and does not contribute. The twoloop contribution to the charged Higgsboson selfenergy 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:
(31) 
With the set of renormalization constants determined in Eqs. (23), (25), (30) and (31) the counterterms arising from the oneloop subrenormalization of the stop and sbottom sectors are fully specified.
Finally, at gluinos appear as virtual particles at the twoloop level; hence, no renormalization in the gluino sector is needed. The corresponding softbreaking gluino mass parameter is denoted . In the case of real MSSM parameters considered here the gluino mass is given as .
2.5 Higherorder corrections in the sector
We furthermore include in our prediction for corrections beyond the oneloop level originating from the bottom/sbottom sector contributions to and . Potentially large higherorder effects proportional to can arise in the relation between the bottomquark 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
(32) 
Here
(33)  
(34) 
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 selfenergies 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 oneloop contribution in the limit of heavy SUSY masses takes the simple form [49]
(35) 
where is evaluated at the scale , and the function is given by
(36)  
The ellipses in Eq. (35) denote subleading terms that we take over from Ref. [74]. Expanded up to oneloop order, the effective mass is close to the mass (including SUSY contributions in the running), see Refs. [45, 73]. A recent twoloop calculation of can be found in Ref. [75].
3 Approximation for the twoloop corrections
In Sect. 2 we have described the approximations for getting the twoloop 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 oneloop [30, 31, 32, 33] and the twoloop corrections [34, 35].
For the charged Higgs boson mass, , the described procedure provides the analogous contribution to the mass shift as well,
(37) 
There are, however, other contributions of similar structure at the oneloop level [59, 60, 61, 62, 63],
(38) 
which are not covered by our approximations for the twoloop terms because they would correspond to (first), nonvanishing gaugecouplings (second), and in the selfenergy (third term). This is justified for large scalarquark 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 oneloop and the twoloop level, as we will explain below.
3.1 The oneloop case
Applying the approximations outlined in Sect. 2.3 at the oneloop level yields the counterterms (all quantities in this section are understood to be oneloop quantities),
(39) 
and thus
(40) 
From Eq. (5) we get the oneloop corrected value of the charged Higgsboson mass,
(41) 
with
(42) 
In the following we use the factor to simplify the notation (),