Symmetries and renormalisation in twoHiggsdoublet models
Abstract
We discuss the classification of symmetries and the corresponding symmetry groups in the twoHiggsdoublet model (THDM). We give an easily useable method how to determine the symmetry class and corresponding symmetry group of a given THDM Higgs potential. One of the symmetry classes corresponds to a Higgs potential with several simultaneous generalised CP symmetries. Extending the CP symmetry of this class to the Yukawa sector in a straightforward way, the socalled maximallyCPsymmetric model (MCPM) is obtained. We study the evolution of the quartic Higgspotential parameters under a change of renormalisation point. Finally we compute the so called oblique parameters , , and , in the MCPM and we identify large regions of viable parameter space with respect to electroweak precision measurements. We present the corresponding allowed regions for the masses of the physical Higgs bosons. Reasonable ranges for these masses, up to several hundred GeV, are obtained which should make the (extra) Higgs bosons detectable in LHC experiments.
I Introduction
In today’s particle physics one of the main hunting grounds of theorists and experimentalists alike are scalars. In the Standard Model (SM) we have as scalar one Higgsboson doublet field, playing an essential role. It is supposed to be responsible for electroweak symmetry breaking thereby giving mass to the and bosons as well as to quarks and leptons. However, more complicated Higgs sectors are by no means excluded experimentally. On the contrary, there are good theoretical reasons for more than one Higgsboson doublet field. Extended Higgs sectors are, for instance, required in supersymmetric models; see for instance Fayet:1974fj (); Fayet:1974pd (); Inoue:1982pi (); Inoue:1983pp (); Flores:1982pr (); Gunion:1984yn (), and in many models trying to solve the so called strong CP problem Peccei:1977hh (); Peccei:1977ur ().
One simple extension of the SM scalar sector has two Higgsboson doublet fields. This twoHiggsdoublet model (THDM) has been studied extensively in the literature; see Lee:1973iz (); Deshpande:1977rw (); Georgi:1978xz (); Haber:1978jt (); Donoghue:1978cj (); Golowich:1978nh (); Hall:1981bc (); Haber:1993an (); Cvetic:1993cy (); Botella:1994cs (); Lavoura:1994fv (); Lavoura:1994yu (); Velhinho:1994vh (); Bernreuther:1998rx (); Davidson:2005cw (); Ginzburg:2004vp (); Barbieri:2005kf (); Nagel:2004sw (); Maniatis:2006fs (); Nishi:2006tg (); Ivanov:2006yq (); Maniatis:2007vn (); Ferreira:2009wh (); Mahmoudi:2009zx (); Grzadkowski:2010dj () and references therein. In our group we have, in particular, emphasised the usefulness of gaugeinvariant bilinears for studying properties of THDMs and we have introduced a special THDM, the maximally CPsymmetric model (MCPM) which may give some understanding of the family structure and the fermion mass hierarchies observed in Nature Maniatis:2007de (). Predictions of the MCPM for highenergy proton–antiproton and proton–proton collisions were presented in Maniatis:2009vp (); Maniatis:2009by (); Maniatis:2010sb ().
THDM’s with additional symmetries were studied in Ivanov:2007de (); Ma:2009ax (); Ferreira:2010hy (). A review of the relation between the usual field formalism and the geometric picture for THDMs working with field bilinears was given in Ferreira:2010yh ().
In the present work we make some remarks concerning symmetries and the corresponding groups for THDMs. We discuss the renormalisation procedure in view of the symmetry constraints on the potential parameters. As an explicit example we treat the renormalisation of the dimensionfour couplings in the MCPM. Finally we calculate the so called oblique parameters , , Peskin:1990zt () for the MCPM. Comparing with electroweak precision data we derive restrictions on the masses of the (extra compared to the SM) Higgs bosons for the MCPM.
Ii The bilinear formalism
We consider models with the particle content as in the SM but with two Higgsboson doublets
(1) 
where . Both doublets are assigned weak hypercharge . We use the conventions for kinematics etc. as in Maniatis:2007de (). The most general gauge invariant and renormalisable potential of the THDM may be written in terms of fields as Haber:1993an ()
(2) 
with , , real, , complex. To study the properties of the Higgs potential, it is convenient to write it in terms of field bilinears Nagel:2004sw (); Maniatis:2006fs (); Nishi:2006tg (); Ivanov:2006yq (). In Nagel:2004sw (); Maniatis:2006fs () a onetoone correspondence of bilinear gaugeinvariant expressions with a Minkowskitype four vector was revealed leading to a simple geometric interpretation. We arrange the fields of (1) in a matrix
(3) 
and define the hermitian, positive semi definite, matrix
(4) 
Its decomposition reads
(5) 
with Pauli matrices . In this way one defines the real bilinears
(6) 
We have
(7) 
In terms of these bilinears the general THDM potential (2) can be written in the simple form
(8) 
with and parameters , , threecomponent vectors , and the matrix . All parameters in (8) are real. The translation from the conventional parameters to the bilinear parameters is
(9) 
We also define the matrix of the parameters corresponding to the potential terms quadratic in the bilinears,
(10) 
Since both Higgs doublets carry the same quantum numbers we may also consider the unitarily mixed fields
(11) 
with . For the bilinears a basis, or Higgsfamily, transformation (11) of the fields corresponds to a SO(3) rotation given by
(12) 
Here is obtained from
(13) 
We note that every proper rotation matrix is a rotation about an axis and can be represented, in a suitable basis, as
(14) 
where is the angle of rotation.
We shall also consider generalized CP (GCP) transformations Lee:1966ik (); Ecker:1981wv (); Ecker:1983hz (); Bernabeu:1986fc (); Ecker:1987qp (); Neufeld:1987wa (); Lavoura:1994fv (); Botella:1994cs (), where
(15) 
with . Note that the ordinary CP transformation is the special case of in (15). In space the generalized CP transformations (15) correspond to the improper rotations Maniatis:2007vn (); Ferreira:2009wh ()
(16) 
Here
(17) 
with the matrix for reflection on the 1–3 plane. We define the matrices () for the reflections on the coordinate planes in space as
(18) 
Here and in the following proper rotation matrices will be denoted by , , etc., improper rotation matrices by , , , etc. By a suitable basis choice we can always arrange that the improper rotation matrix has the form
(19) 
Note that for we get the GCP transformation corresponding to a reflection on the 1–2 plane in space () accompanied by the spacetime transformation . A basis transformation (12) exchanging the 2 and 3 axes in space shows that this is equivalent to the standard CP transformation where in (16). For more details on GCPs in THDMs see Maniatis:2007vn (); Ferreira:2009wh (); Ferreira:2010hy ().
Finally we recall from Maniatis:2006fs () that a transformation (12) in space with always corresponds to a field transformation (11) which is unique up to gauge transformations. Similarly, a space transformation (16) with , , always corresponds to a GCP transformation (15) of the fields which is unique up to gauge transformations.
Iii Symmetry classes and symmetry groups
The general THDM potential has 14 parameters; see (9). Considering only the scalar sector we can make a basis change as in (11), (12) to diagonalise , thereby reducing the number of parameters to 11. One may want to further reduce this number by imposing symmetries. This can be Higgsfamily or GCP symmetries. A Higgsfamily transformation (11), (12) is a symmetry of the potential if and only if the parameters (9) satisfy
(20) 
A GCP transformation (15), (16) is a symmetry if and only if
(21) 
In Ivanov:2007de () the possible symmetry classes of THDMs were derived, however, only potentials which are stable in the strong sense were considered. Here we define, as in Maniatis:2006fs (), a potential to be stable in the strong sense if stability is guaranteed by the quartic field terms alone and in the weak sense if it is guaranteed only after inclusion of the quadratic field terms in (2) respectively (8). A potential being bounded from below but having directions in field space where it does not grow indefinitely for the fields going to infinity has only marginal stability. In all other cases the potential is unstable. In Ferreira:2010hy () the symmetry classes of the THDMs were further studied and also softly broken symmetries were considered.
We give in Table 1 the maximal symmetry group for each symmetry class and the corresponding constraints on the potential (8). Note that in Table 1 the classes are defined to be mutually exclusive, that is, we assign a THDM to a certain class if it has the corresponding group (up to trivial equivalences) as symmetry group and not a bigger one. If the parameters of a THDM potential are not satisfying any of the constraints of Table 1, the theory has no symmetry group except the trivial one, that is, the unit transformation. In appendix A we present a derivation of these symmetry classes and groups where, as mentioned above, we do not use any assumptions on the stability of the potential (2), (8). The methods explained in apppendix A also give an easy practical recipe for finding out if a THDM potential has a symmetry and which one this is. The symmetry relations as given in Table 1 will be used in section IV for the discussion of the renormalisation in specific THDMs.
symmetry class and group  constraints on and  constraints on  
U(1)  
SO(3)  
CP1  
CP2  ,  all different  
CP3  , 
We emphasize that in Table 1 we give the exact conditions for the parameters of the scalar potential to have the symmetry group as listed and not a bigger one. The elements of give the corresponding transformations in space. For proper rotations these are Higgs basis transformations; see eq. (12), for improper rotations these are generalized CP transformations; see eq. (16). Of course, a group of a symmetry class may contain the groups of other classes as subgroups, as is obvious from Table 1. For instance, the group O(3) contains all other groups as subgroups and, clearly, the potential of the SO(3) symmetry class has all other symmetries as well. The numbering of the eigenvalues of in Table 1 is  without loss of generality  chosen conveniently, in order to give the same invariance group and not an equivalent one for all subclasses of one class. For the cases of degenerate eigenvalues of it is understood that a convenient choice of basis in the degenerate subspaces gives the groups as listed. Other choices of bases give equivalent groups.
In Table 1 we have listed subclasses for , U(1), and CP1. These are distinguished by the degeneracies of the eigenvalues and for the CP1 case also by relations for and . These subclasses of a class correspond to the same symmetry group and therefore lead to no new symmetry classes. Under renormalisation only the groups will be preserved. That is, the subclasses of one class will not be invariant under renormalisation but will mix among each other. Considering the theory of the two Higgsboson doublets alone the renormalisation of the potential parameters can not lead from one symmetry class to another one. If we start, for instance, with a theory of the CP2 class where all are different we can not come by renormalisation to the CP3 or SO(3) classes where two, respectively all three, of the ’s are equal. We shall elaborate on this point below in section IV in connection with the renormalisation in the MCPM which is a complete theory including fermions and bosons.
Iv Renormalisation of the dimension four couplings in the MCPM
In this section we consider the renormalisationgroup equations (RGEs) for the dimension four couplings in the maximally CP symmetric model (MCPM) as constructed and studied in Maniatis:2007de (); Maniatis:2009vp (); Maniatis:2009by (); Maniatis:2010sb (). In the MCPM the Higgs potential parameters (9), in a diagonal basis of the matrix , have to fulfill
(22) 
In conventional notation of the Higgs potential (2) this corresponds to the constraints
(23) 
Without loss of generality we can assume
(24) 
From Table 1 we see that the Higgs potential satisfying (22) can be in the symmetry classes CP2, CP3, or SO(3). As shown in Maniatis:2007de (), stability, the correcte electroweak symmetry breaking (EWSB), and absence of zeromass charged Higgs bosons require and are guaranteed by
(25) 
In the MCPM there are five physical Higgs bosons, three neutral ones, , , , and a charged pair, . Their squared masses in terms of the model parameters are, at tree level,
(26) 
Here
(27) 
is the standard vacuumexpectation value. Requiring now also absence of zeromass neutral Higgs bosons and absence of mass degeneracy between and leads to
(28) 
replacing the weaker condition (24). From Table 1 we see that we are dealing now with potentials in the CP2 symmetry class with the corresponding symmetry group as listed there. The main point of the MCPM is that the symmetry group of the CP2 class is required to be respected also by the complete Lagrangian, including the fermions, the gaugeboson, and the Yukawa sectors. It was shown in Maniatis:2007de () that with this requirement a coupling of the two Higgsboson doublets to only one fermion family is not possible with nonvanishing Yukawa couplings. However, with a coupling of the two Higgsboson doublets to two fermion families this is indeed possible with one fermion family acquiring masses and the other remaining massless. With a third fermion family kept uncoupled to the Higgsboson doublets this model gives very roughly what we observe in Nature: two rather light fermion families and one very heavy (the third) fermion family.
The complete Lagrangian of the MCPM is recalled in App. B. The parameters of the MCPM are as follows.

Higgs potential parameters
(29) 
Yukawa sector coupling constants
(30) related to the thirdfermionfamily masses
(31) 
Gauge couplings
(32) of the gauge groups , , and , respectively.
Let us now proceed and consider the oneloop RGEs in this model. The oneloop RGEs for the couplings of the dimensionfour terms in any renormalisable gauge theory are given in Cheng:1973nv (). The RGEs given there apply to the deep Euclidean region where coupling terms of dimension two can be neglected. Also shifts of scalar fields to give them zero vacuum expectation value after EWSB are irrelevant there. For the quartic Higgspotential couplings including the and gauge interactions with couplings and , respectively, taking also the Yukawa couplings (68) into account we find for the MCPM from the results of Cheng:1973nv ()
(33) 
Here with the mass scale of the renormalisation point and a convenient reference scale, for instance, TeV. The RGEs of the ’s can easily be translated to space. For the generic THDM Higgs potential this was done in Ma:2009ax (). In the case of the MCPM we have to extend these RGEs by including the Yukawa interactions (68). From the RGEs for the parameters of the generic Higgs potential as given in Ma:2009ax () we can check that the diagonality of the matrix and , see (22), are preserved under oneloop renormalisation in the MCPM. This must be so, since this is guaranteed by the symmetry group of the CP2 class; see Table 1. Here we find from (9), (10), and (33),
(34) 
As mentioned above these RGEs apply in the deep Euclidean region.
Let us now discuss the evolution of the differences of the eigenvalues of :
(35) 
From (34) we find
(36) 
Suppose now that we start at TeV, corresponding to , with the conditions (28). We have then, in particular,
(37) 
From (36) we can see that the one loop RGEs preserve this property as long as all couplings stay finite. Indeed, suppose that for we have
(38) 
where is a constant. We get then from (36)
(39) 
(40) 
(41) 
Thus, stays positive for . A similar argument applies for the evolution to negative values. Hence, can not change sign as long as the theory parameters stay finite.
The analogous result for can not be derived in the same way from (36). This is due to the terms not proporotional to on the r.h.s of (36). But in the pure scalar theory, that is, if we set and we can again derive the analogue of (41).
We conclude that the one loop RGEs preserve but, in the full theory, not necessarily If now for some value we have we have for the Higgs potential a higher symmetry, here the CP3 symmetry, where two eigenvalues of are equal; see Table 1. But for the full theory this CP3 symmetry is not realised. Thus, in the full theory the RGEs can lead to renormalisation scales where the Higgs potential alone shows a higher symmetry than the full theory. Of course, only the symmetry of the full theory is relevant for physics.
V Oblique parameters in the MCPM
The oblique parameters , , and denote certain combinations of selfenergies of the electroweak gauge bosons with respect to any new contributions compared to the SM Peskin:1990zt (). In any model beyond the SM the oblique parameters can be computed and compared to the electroweak precision data Nakamura:2010zzi () which require:
(42) 
For the case of the general THDM the oblique parameters have been computed in Froggatt:1991qw (); Haber:2010bw ().
We shall now derive the predictions for the oblique parameters in the MCPM. In the MCPM the Yukawa couplings are completely fixed and the only free parameters we encounter in the calculation of the oblique parameters are the Higgsboson masses , , , and . Here and are the CPeven and is the CPodd Higgs boson and denotes the pair of charged Higgs bosons. In Figure 1 we show the contour plots for the 1 (dark) and 2 (bright) deviations of the oblique parameters from the electroweak precision data (42) in the – plane. The mass of the SMlike Higgsboson is fixed to 125 GeV. The chargedHiggsboson mass is set to different values in the range of 125500 GeV in the various plots. In Figure 2 we show analogous plots but for a mass of the SMlike Higgs boson of 170 GeV. Note that we have always in the MCPM which is the reason that there are no allowed regions of parameter space above the diagonal of equal masses in Figures 1 and 2
We see from Figures 1 and 2 that there are large regions for the masses of the Higgs bosons , , and where the electroweak constraints (42) are satisfied. The allowed regions for these masses, up to several hundred GeV, are very reasonable. The CP odd extra Higgs boson could even be below 100 GeV in mass. But then it would be necessary to study all other experimental constraints for such a lowmass boson. Furthermore, we see from Figures 1 and 2 that with increasing masses of and also the allowed domains for the masses of the Higgs bosons and shift to higher mass values.
Vi Conclusions
In this paper we started with briefly reviewing the bilinear formalism which turns out to be quite powerful for the study of the THDM. We have discussed the classification of the possible symmetry classes without any assumption on the stability type of the THDM potential. We have given a practical and easily usable method how to determine the symmetry class of a given THDM Higgs potential. We have defined the symmetry classes to be mutually exclusive; see Table 1. We have also given the symmetry group corresponding to each symmetry class. We have focussed on one of these symmetry classes, denoted by CP2, in some detail. The CP2 symmetric THDM has a number of simultaneous CP invariances. As shown in Maniatis:2007de () the extension of the CP symmetries of the potential to the Yukawa interactions leads in a straightforward way to the socalled maximally CPsymmetric model (MCPM). In this model the Yukawa couplings are completely fixed. We have studied the renormalisationgroup equations of the quartic Higgspotential parameters in this model. We have found that the symmetries of this model are preserved by the RGEs, as it has to be.
The MCPM has a hierarchy of quartic couplings . We have shown that considering the theory of the Higgs bosons alone this hierarchy of quartic couplings turn out to be stable against renormalisation group evolution. However, taking the Yukawa couplings into account is stable but not necessarily . Reaching at a certain renormalisation scale would elevate the CP2 symmetry of the Higgs potential to a CP3 symmetry. But, of course, this does not imply that the full MCPM which includes fermions and gauge bosons has a higher symmetry than CP2 at this renormalisation scale.
Eventually, we have computed the oblique parameters in the MCPM. We find for large parameter space agreement with the electroweak precision measurements. In particular we have presented the 1 and 2 contours of valid regions in the – mass plane for different choices for the chargedHiggsboson mass and for SMlike Higgsboson masses of 125 GeV and 170 GeV, respectively. The allowed regions for the masses of the Higgs bosons are in a reasonable range; see Figures 1 and 2. These Higgs bosons with masses below 500 GeV should therefore be detectable in the LHC experiments. As shown in Maniatis:2009vp (); Maniatis:2009by (); Maniatis:2010sb () in the MCPM these Higgs bosons have characteristic production and decay properties giving clear experimental signatures.
Appendix A Derivation of symmetry classes
In this appendix we give a recipe which allows an easy identification of the symmetry class of any given THDM potential (8).
The first step is to diagonalise by a basis transformation (12). We get then
(43) 
Since is symmetric a diagonalisation is always possible. Therefore we work in the following in the diagonal basis and consider and (9) in this basis. Now we have to distinguish three cases for the ’s.

, , all different.
Then we see from (20) and (21) that only diagonal O(3) matrices or may lead to symmetries, that is, we have to consider
(44) Now we can easily check the conditions for , from (20) and (21). We can have the following cases:

.

Exactly one pair fulfills where .

Exactly two pairs fulfill where .

and .


Exactly two eigenvalues of are equal.
Without loss of generality we set
(48) From (20), (21) we see that allows now as invariances
(49) (50) (51) (52) Note that is included in (52) for : .
Now we consider again all possibilities for and .