Phenomenology of Large Mixing for the CP-even Neutral Scalars of the Higgs Triplet Model

# Phenomenology of Large Mixing for the CP-even Neutral Scalars of the Higgs Triplet Model

A.G. Akeroyd111akeroyd@ncu.edu.tw Department of Physics and Center for Mathematics and Theoretical Physics, National Central University, Chungli, Taiwan 320    Cheng-Wei Chiang222chengwei@ncu.edu.tw Department of Physics and Center for Mathematics and Theoretical Physics, National Central University, Chungli, Taiwan 320 Institute of Physics, Academia Sinica, Taipei, Taiwan 115
July 15, 2019
###### Abstract

The Higgs Triplet Model contains two CP-even neutral scalar eigenstates, each having components from an isospin doublet and an isospin triplet scalar field. The mixing angle can be maximal if the masses of the scalar eigenstates are close to degeneracy. We quantify the dependence of the mixing angle on the mass splitting and on the vacuum expectation value of the neutral triplet scalar. We determine the parameter space for maximal mixing, and study the observability of both CP-even Higgs bosons at the CERN LHC.

###### pacs:
12.60.Fr, 14.80.Cp

## I Introduction

The firm evidence that neutrinos oscillate and possess small masses below the eV scale Fukuda:1998mi () necessitates physics beyond the Standard Model (SM), which could manifest itself at the CERN Large Hadron Collider (LHC) and/or in low energy experiments which search for lepton flavour violation (LFV) Kuno:1999jp (). Consequently, models of neutrino mass generation which can be probed at present and forthcoming experiments are of great phenomenological interest.

Neutrinos may obtain masses via the vacuum expectation value (VEV) of a neutral Higgs boson in an isospin triplet representation Konetschny:1977bn (); Mohapatra:1979ia (); Magg:1980ut (); Schechter:1980gr (); Cheng:1980qt (). A particularly simple implementation of this mechanism of neutrino mass generation is the “Higgs Triplet Model” (HTM) in which the SM Lagrangian is augmented solely by an triplet of scalar particles with hypercharge  Konetschny:1977bn (); Schechter:1980gr (); Cheng:1980qt (). In the HTM, neutrinos acquire Majorana masses given by the product of a triplet Yukawa coupling () and a triplet VEV (). Consequently, there is a direct connection between and the neutrino mass matrix, which gives rise to phenomenological predictions for processes which depend on Ma:1998dx (); Chun:2003ej (); Kakizaki:2003jk (); Garayoa:2007fw (); Akeroyd:2007zv (); Kadastik:2007yd (); Perez:2008ha (); delAguila:2008cj (); Akeroyd:2009nu (); Fukuyama:2009xk (); Akeroyd:2009hb (). A distinctive signal of the HTM would be the observation of a doubly charged Higgs boson (), whose mass () may be of the order of the electroweak scale. Such particles can be produced with sizeable rates at hadron colliders in the processes Barger:1982cy (); Gunion:1989in (); Han:2007bk (); Huitu:1996su () and  Barger:1982cy (); Dion:1998pw (); Akeroyd:2005gt (), where is a singly charged Higgs boson in the same triplet representation. Direct searches for have been carried out at the Fermilab Tevatron, assuming the production channel and decays , and mass limits in the range  GeV have been obtained Acosta:2004uj (); Abazov:2004au (); :2008iy (); Aaltonen:2008ip (). The CERN Large Hadron Collider (LHC), using the above production mechanisms, will offer improved sensitivity to Perez:2008ha (); delAguila:2008cj (); Han:2007bk (); Hektor:2007uu (). The phenomenology of the singly charged Higgs boson is also attractive at hadron colliders, with production via followed by the decay Perez:2008ha (); delAguila:2008cj (); Akeroyd:2009hb ().

The phenomenology of the neutral Higgs bosons in the HTM has received much less attention than that of the charged Higgs bosons. There are two CP-even scalars (, where ) and one CP-odd scalar (), which are composed of both isospin doublet and isospin triplet fields. In phenomenological studies of the HTM, it is common to assume that the mass term for the scalar triplet () is considerably larger than the mass of the isospin doublet scalar. This assumption guarantees that the mixing angle for the two CP-even scalars is small, being of the order ( is the VEV of the isospin doublet), where is required to maintain within experimental error. Therefore, is essentially composed of the isospin doublet field and plays the role of the SM Higgs boson, while is essentially composed of the isospin triplet field, and is difficult to detect at hadron colliders.

However, as pointed out explicitly in Ref. Dey:2008jm (), the mixing angle for the CP-even scalars can be maximal in the region of parameter space around degenerate masses for the CP-even scalars. We quantify in detail this parameter space of large mixing in the CP-even sector, and study its phenomenology. When the mixing angle is large, can be produced with observable rates in the standard search channels for the SM Higgs boson. Importantly, in the HTM can be considerably lighter than and . Therefore, might be detected earlier than or , especially if the decay modes and dominate (corresponding to GeV), for which the LHC has sensitivity inferior to that for the leptonic channels and described above.

Our work is organized as follows. The HTM is briefly reviewed in Section II. In Section III, the scalar mass matrices, the mixing angle, and the Higgs potential minimization and stability conditions are presented. The numerical analysis and phenomenology are discussed in Section IV. Our conclusions are given in Section V.

## Ii The Higgs Triplet Model

In the HTM, an complex triplet of scalar fields is added to the SM Lagrangian. Such a model can provide a Majorana mass for the observed neutrinos (without the introduction of additional neutrinos) via the gauge-invariant Yukawa interactions:

 L=hijψTiLCiσ2ΔψjL+h.c. (1)

Here is a complex and symmetric coupling matrix, is the Dirac charge conjugation operator, is a Pauli matrix, is a left-handed lepton doublet, and is a representation of the complex triplet fields:

 Δ=(δ+/√2δ++δ0−δ+/√2) . (2)

A non-zero triplet VEV, , gives rise to the following mass matrix for neutrinos:

 mij=2hij⟨δ0⟩=√2hijvΔ . (3)

This simple expression for tree-level Majorana masses of the observed neutrinos is essentially the main motivation for studying the HTM. Realistic neutrino masses can be obtained with a perturbative provided that eV. The presence of a non-zero gives rise to at tree level. Therefore GeV is necessary in order to comply with the measurement of . We will discuss this bound on in more detail later.

Neutrino oscillation experiments have provided much information on (see, for example, Ref. Schwetz:2008er ()), and so the couplings are already constrained up to an arbitrary scalar factor (the triplet VEV, ). The necessary non-zero arises from the minimization of the most general invariant Higgs potential, which is written as follows Joshipira:1991yy (); Ma:1998dx (); Perez:2008ha ():

 V(H,Δ) = −m2H H†H + λ4(H†H)2 + M2Δ TrΔ†Δ + (μ HT iσ2 Δ†H + h.c.) (4) + λ1 (H†H)TrΔ†Δ + λ2 (TrΔ†Δ)2 + λ3 Tr(Δ†Δ)2 + λ4 H†ΔΔ†H .

Here is the SM Higgs doublet. Variants of the above form for are given in Refs. Chun:2003ej (); Dey:2008jm (); Abada:2007ux (); Gogoladze:2008gf (), which are equivalent to a reparametrization of some .

As in the SM, the term (where ) ensures , which spontaneously breaks to . The mass term for the triplet scalars is given by and usually is taken. One can take values of which are arbitrarily large, but recently much attention has been given to the case of TeV, since this would allow the triplet scalars (especially the distinctive doubly charged scalar, ) to be within the discovery range of the LHC. The main production mechanisms for at hadron colliders are (i) Barger:1982cy (); Gunion:1989in (); Han:2007bk (); Huitu:1996su (), which depends on one unknown parameter, ; and (ii)  Barger:1982cy (); Dion:1998pw (); Akeroyd:2005gt (), which depends on two unknown parameters, and . In the HTM one has if is small (see later). Production mechanisms which depend on the triplet VEV ( and fusion via Huitu:1996su (); Vega:1989tt ()) are not competitive with the above processes at the energies of the Fermilab Tevatron, but can be the dominant source of at the LHC if (1 GeV) and GeV.

The term leads to the triplet VEV, as will be shown explicitly in the next section. In an early version of the HTM Gelmini:1980re (), the term is absent, but can still arise by taking the “wrong sign” choice for (). This leads to spontaneous violation of the lepton number since Majorana mass has come from a Higgs potential which originally conserves the lepton number. The resulting Higgs spectrum then contains a massless triplet scalar (called Majoron, , a Goldstone boson) and another light scalar (). This is a dramatic prediction, and pair production via would give a large contribution to the invisible width of . This model is therefore testable, and it is now excluded because the invisible width for has been measured at the CERN Large Electron Positron Collider (LEP), and its value is in good agreement with the SM prediction (in which the invisible width comes from only).

The inclusion of the term ) explicitly breaks lepton number when is assigned , and eliminates the Majoron. Alternatively, assigning would conserve lepton number in the Higgs potential but break it in the Yukawa interaction of Eq. (1). Therefore, the lepton number is always broken irrespective of the assignment of or because of the presence of both Eq. (1) and ). Thus the above scalar potential together with the triplet Yukawa interactions of Eq. (1) lead to a model of neutrino mass generation which is viable phenomenologically.

One can work in a simplified scalar potential (e.g., Ref. Perez:2008ha ()) by neglecting the quartic couplings (where ) involving the triplet field . The resulting scalar potential then depends on four parameters (, , , ), but only three parameters are independent because the VEV for the doublet field ( GeV) is fixed by the mass of . The three independent parameters are usually chosen as or . The inclusion of generates additional trilinear and quartic couplings among the scalar mass eigenstates, which contribute to the term which mixes the CP-even scalars. The terms with and , which involve both triplet and doublet fields, are of particular interest because they can give sizeable contributions to the masses of the triplet fields (when replacing the fields by ). In this work we will study the HTM with a scalar potential given by Eq. (4).

## Iii Minimization equations and mass matrices for the scalar fields

Following the notation of Ref. Perez:2008ha (), the neutral complex scalar fields are expressed as follows:

 ϕ0=(v0 + h0 +iξ0)/√2 ,  and  δ0=(vΔ + Δ0 +iη0)/√2 . (5)

For non-zero and , the minimization conditions for the global minimum of the potential are:

 −m2H+λ4v20+12(λ1+λ4)v2Δ−√2μvΔ=0 , and (6) M2ΔvΔ+12(λ1+λ4)v20vΔ−1√2μv20+(λ2+λ3)v3Δ=0 . (7)

In the simplified potential which sets , the expression for resulting from the minimization of is:

 vΔ=μv20√2M2Δ . (8)

For , this expression is sometimes referred to as the “Type II seesaw mechanism,” since a small value for arises without requiring a small value of . However, the case of TeV is of phenomenological interest because the triplet scalars would be produced at the LHC, and such a scenario requires a few GeV.

After imposing the above tadpole conditions to eliminate and , one finds that the mass-squared matrix () for the CP-even states is:

 M2even=⎛⎜⎝λv20/2[(λ1+λ4)vΔ−√2μ]v0[(λ1+λ4)vΔ−√2μ]v0(√2μv20+4(λ2+λ3)v3Δ)/2vΔ⎞⎟⎠ . (9)

Note that depends on all seven parameters of the scalar potential. The mass eigenstates are denoted by and , where :

 H1 = cosθ0 h0 + sinθ0 Δ0,H2=−sinθ0 h0 + cosθ0Δ0. (10)

The square of the mass eigenvalues are given by:

 M2H1,M2H2=12[M211+M222±√(M211−M222)2+(4M212)2] (11)

For the case of , the explicit expressions for the squared masses of and expanded to terms linear in are:

 M2H1=12λv20−2√2μv0ϵ+O(ϵ2) , (12) M2H2=μv0√2ϵ+2√2μv0ϵ+O(ϵ2) . (13)

The mass-squared matrix () for the CP-odd states is:

 M2odd=μ(2√2vΔ−√2v0−√2v0v20/(√2vΔ)) , (14)

which is completely independent of the scalar quartic couplings ( and ) and only depends on two parameters ( and ) of the scalar potential. One of the eigenstates is the neutral Goldstone boson which becomes the longitudinal polarization mode of the boson after the breaking of the symmetry. The massive eigenstate is denoted by :

 A0=−sinα ξ0 + cosα η0. (15)

The squared mass of is given by:

 M2A0=μv0√2ϵ+2√2μv0ϵ (16)

Note that also becomes massless (“a triplet Majoron”) in the limit of Gelmini:1980re () and , i.e., the scenario of spontaneous breaking (not explicit breaking) of lepton number caused by . For our choice of and , the sign of and must be the same in order to ensure a positive mass for . We choose and to be positive.

The mass-squared matrix for the singly charged states () is:

 M2±=(μ−λ4vΔ2√2)(√2vΔ−v0−v0v20/(√2vΔ)) , (17)

where one of the eigenstates has a vanishing eigenvalue and serves as the charged Goldstone boson that later becomes the longitudinal polarization mode of the boson. The massive eigenstate is denoted by :

 H±=−sinθ± ϕ± + cosθ± δ± . (18)

Note that only one scalar quartic coupling () appears in , and the mass matrix depends on three parameters. The squared mass of is given by:

 M2H±=μv0√2ϵ−λ44v20+√2μvΔ−λ42v2Δ . (19)

Finally, the squared mass of the doubly-charged state () is given by:

 M2H±±=μv20√2vΔ−λ42v20−λ3v2Δ , (20)

which depends on four parameters of the model.

The above mass matrices for the neutral scalar fields are presented in Ref. Joshipira:1991yy () in the context of an extension of the HTM which includes a singlet scalar field. The mass matrices for the neutral and charged scalars are given in Ref. Ma:1998dx () in the approximation of neglecting one or two of (see also Ref. Frampton:2002rn ()). The scalar mass matrices for the Majoron model with and are given in Ref. Gelmini:1980re (); Georgi:1981pg ().

In Fig. 1, the masses of the scalars of the HTM are presented as a function of for three values of ( and ). Other parameters are fixed as follows: triplet VEV GeV, , , and . The current experimental bounds on the masses of the scalars are respected by choosing above a threshold value. In the plots for and we take , while in the plot for we take . We note that the mass of the doubly charged scalar starts at GeV in the plot with . This is not in conflict with the experimental lower bound because the decay mode has a branching ratio for our choice of GeV, and there has been no explicit search for in this decay channel. As increases (or equivalently, as increases) one can see that remains constant, its magnitude being determined by the entry in the CP-even mass matrix , which is independent of . In contrast, and all increase with . The dominant part of the mass splitting among them is proportional to . For the case of , the latter scalars are approximately degenerate, with very small splittings caused by electromagnetic corrections (see, for example, Ref. Perez:2008ha ()), and from other which have a dependence on the small parameter (see the explicit expressions for the masses of the scalars given in Eqs. (13), (16), (19) and (20)). For one has the mass hierarchy (). In Ref. Dey:2008jm (), the analogous versions of Fig. 1 show a sizeable mass splitting between and the degenerate scalars , , , even when . We cannot reproduce this result.

In Fig. 1, there is a region where and are approximately degenerate in mass. This corresponds to the case of for the CP-even mass matrix, for which the mixing angle in Eq. (10) becomes maximal. The mixing angles () for the mass matrices () are given by the general formula:

 tan2θ=2M212M211−M222 . (21)

Maximal mixing () is achieved at , irrespective of the value of (provided that ). The condition is realized in each of the above mass matrices for the following choice of parameters, respectively:

 M2even: λ=√2μ/vΔ+4(λ2+λ3)v2Δv20 (22) M2odd: 4v2Δ=v20 (23) M2±: 2v2Δ=v20 (24)

Maximal mixing can never be achieved for and because of the constraint . Hence the mixing angles for the CP-odd and singly charged scalars are always small in the HTM, with . However, maximal mixing is possible for the CP-even scalars and has been discussed first in Ref. Dey:2008jm (). In Eq. (22), if one neglects the small term proportional to (which is suppressed by ) one has a simple condition for maximal mixing for , given by . This condition for can be satisfied in the HTM, provided that the masses of the scalars respect their current lower bounds. We will study in detail the phenomenology of the scenario of a large mixing angle for the CP-even scalars, which is possible when and are approximately degenerate.

In general, the mixing angle for the CP-even scalars will be non-zero, apart from fine-tuned choices of parameters for which . In the limit of one has . Notably, even a very small isospin doublet component (e.g., corresponding to values of ) can dominate the branching ratios of , and Perez:2008ha (). This is because (which determines the strength of the decays mediated by the triplet component) and (which determines the mixing angle) are related by Eq. (3).

The lower bounds on the masses of the scalars in the HTM from direct searches depend on their decay modes, and such bounds can be quite different from those in the Two Higgs Doublet Model (2HDM). For example, the decay modes and (decays which are not present at tree level in the 2HDM) can be the dominant channels in the HTM Gunion:1989ci (); Datta:1999nc (). The production processes and have rates which depend on gauge couplings. The most conservative mass limits which can be imposed are , and , which ensure that the scalars do not contribute to the width of the boson (which is measured very precisely). Mass bounds for specific decay channels are stronger than these, and can be applied to the scalars in the relevant regions of the HTM parameter space. We will discuss the mass bounds for , and below.

The parameters of the scalar potential are also constrained by requiring that it is bounded from below, and the electroweak minimum is a global one. The following constraint on can be derived by requiring that the scalar potential is bounded from below:333This constraint was also derived in Ref. Dey:2008jm () for an alternative parametrization of the scalar potential.

 λ1+λ4+2√λ(λ2+λ3)>0 (25)

An upper limit on can be obtained from considering its effect on the parameter. In the SM at tree level, but higher-order contributions (mainly from virtual top- and bottom-quark loops) give rise to a correction , and thus . In the HTM is negative at the tree level:

 ρ≡1+δρ=1−2ϵ21+4ϵ2. (26)

The measurement leads to the bound , or  GeV at CL () Abada:2007ux (); Fukuyama:2009xk (). Experimentally, positive values of are preferred ( Amsler:2008zzb ()). However, the above bound on is not rigorous because it is obtained by comparing the above tree-level expression for in the HTM with the experimentally-allowed value of , in which the dominant SM contribution from virtual top- and bottom-quark loops has already been computed. Clearly this is not a consistent treatment of the HTM and SM contributions to , which are being evaluated at the tree level and the loop level, respectively. A full analysis at the loop level in the HTM requires renormalization of . Explicit analyses have been performed for a model with a real scalar triplet, which has significantly fewer scalar fields and does not contain doubly charged Higgs bosons. The bounds on the triplet vacuum expectation value for the real scalar triplet are found to be similar in magnitude to those derived from the tree-level analysis Blank:1997qa (). We are not aware of an explicit analysis in the HTM, although some studies have been done for other models which contain a complex scalar triplet, as well as additional fields which are not present in the HTM (e.g., Little Higgs models Chen:2003fm () and Left-Right symmetric models with for the triplet of Czakon:1999ga ()). Therefore, in the HTM it seems reasonable to assume a maximum value of the order of a few GeV for the triplet VEV, although the exact bound is not known and will have a dependence on the parameters of the scalar potential.

### iii.1 Mass limits for A0,h1,h2

The best limits on the masses of come from the CERN LEP experiments. For scalar masses probed by LEP, the dominant decay modes for and depend on the value of Perez:2008ha (). For small triplet VEV ( GeV) the dominant decay is to two neutrinos (), while for larger triplet VEV ( GeV) the dominant decay is to two quarks () through the doublet component. The main production mechanisms at LEP are , , and . The relevant couplings are given in Table 1, and they depend on two terms which involve the mixing angles in the CP-even () and CP-odd () sectors. As explained earlier, in the HTM one always has , and if for the CP-even mass matrix. In this scenario of and (which corresponds to most of the parameter space) there will be a SM-like CP-even scalar which can be produced via , and thus the LEP bound GeV can be applied. The mechanisms and would have very small cross sections since and . However, pair production of and is possible via , since the coupling is unsuppressed in this limit. If the decays modes and are dominant then LEP searches can be applied, and the limit GeV can be derived Amsler:2008zzb (). If the decays and are dominant, then one can have the signature , where the photon () originates from bremsstrahlung from or . Some mass limits can be derived from LEP data for the search for “ missing energy” (the mass bound GeV was derived in Ref. Datta:1999nc ()). In the case of a large mixing angle (), all production mechanisms would be relevant. The phenomenology of this scenario is studied in the next section. The couplings and are more relevant for phenomenology at the LHC. Pair production of scalars via the and couplings is not so promising at hadron colliders.

### iii.2 Discovery channels for H1 and H2 at the LHC

The phenomenology of the SM Higgs boson [i.e., a scalar which arises solely from an isospin doublet, in Eq. (5)] at the LHC has been studied in great detail, and is reviewed in Ref. Djouadi:2005gi (). Much of these analyses can be applied to and of the HTM, whose isospin doublet component corresponds to the SM Higgs scalar multiplied by the mixing angle or [see Eq. (9)]. For an isospin doublet scalar field in the mass range GeV, the optimal discovery channels are Djouadi:2005gi ():

• Gluon-gluon fusion, followed by decay of to : ,

• Weak-boson fusion, followed by decay of to : ,

• Weak-boson fusion, followed by decay of to : ,

The statistical significance of the signal depends on the channel and on the mass of , and in channel (iii) it can be as high as for fb. The significances for channels (i) and (iii) increase as increases from GeV to GeV, while the significance for channel (ii) decreases in the same mass range. We will quote specific numbers for the significances later. The channel is important for the case of GeV, although the statistical significance is below for fb.

We will not explicitly consider the search channels , where the decay is mediated by loops involving , charged fermions and charged scalars. The -loop contribution to depends on the couplings , and in the SM it is the dominant contribution. The couplings depend on two terms, one being proportional to and the other being proportional to (see Table 1). Since , the term proportional to is dominant for the case of a large mixing angle of interest to us. In the case of , one has and the coupling is vanishing Perez:2008ha () because of a cancellation between the two terms – see the approximate form of the coupling in Table 1, where the term in brackets is (from Eq. 8). The magnitude of the contributions from the loops involving charged fermions is considerably smaller than that of the loop involving .

Charged scalars ( and ) also contribute to the decays , with the contribution from having a factor of four enhancement at the amplitude level (because of its electric charge) relative to that from . The magnitude of these scalar loops depends on the trilinear couplings and , in which the dominant contribution comes from terms of the form and , and there is also a dependence on the mixing angle . However, the loop function () for such scalar contributions is much smaller than that for the boson () (see eg., Ref. Djouadi:2005gi ()). For the parameter choice in our numerical analysis ( GeV, and GeV), one has and . Hence the loop dominates unless large couplings are considered. Importantly, in our numerical analysis we will focus on the phenomenologically interesting case of and , and the couplings have no contribution from . Therefore, in such a scenario the dominant scalar-loop contribution is from that mediated by , which does not have the aforementioned enhancement factor of 4. Thus, the branching ratios for for the case of in our numerical analysis are essentially the same as that in the SM.

In this work we will focus on the prospects for detection of and in the above channels (i),(ii) and (iii), for the case of a large mixing angle . For the case of ( in the HTM, the eigenstate would be dominantly composed of , and thus the above significances can be applied directly to . In this scenario, would be almost entirely composed of the triplet field , and thus it cannot be produced with an observable rate by the above mechanisms. This can be seen from Table 1, where the and couplings (which are needed for weak-boson fusion) are very small. Moreover, when is essentially composed of the triplet field it only couples very weakly to quarks (through its isospin doublet component), thus rendering the gluon-gluon fusion process ineffective. Although the coupling is unsuppressed in the limit of , the production of scalars via this coupling is not so promising at hadron colliders. That is, followed by decays of and to quarks and/or neutrinos does not have such a large cross section, and its experimental signature would suffer from large backgrounds.

## Iv Numerical Results

In Ref. Dey:2008jm (), the dependence of the mixing angle on the theoretical parameter is studied. In this section we study in detail the parameter space of the HTM where the mixing angle can be sizeable. In Fig. 2, contours of the mixing angle for the CP-even Higgs bosons are plotted in the - plane for . The left panel has GeV and the right panel has GeV. Other parameters are fixed as follows: , , and . This choice of parameters satisfies the constraint in Eq. (25). The ratio is the same (=0.4) in both panels, and it is this ratio which essentially determines the value of [see Eq. (13)]. The choice of gives GeV in both figures, and is the lightest Higgs boson in the spectrum. The direct search limits for the scalars are respected (see Fig. 1). The choice of satisfies the condition for maximal mixing444Note that for fixed the value of which gives maximal mixing is slightly different for GeV and GeV due to the dependence of the second term on in Eq. (22). in Eq. (22) for GeV (left panel) and GeV (right panel), and thus the vertical axis corresponds to the contour of (where ). For GeV in the left panel ( in the right panel), one has , and so the mixing angle decreases away from its maximum value (see Eq. 21). For a given value of , a larger can be obtained with a more negative . Negative values enhance the magnitude of the off-diagonal term in the CP-even mass matrix because is taken to be positive (and ).

In Fig. 3, the mixing angle is plotted as a function of the mass splitting , for three values of . All other parameters are fixed as in Fig. 2. The only parameter which is varied is , starting from GeV ( GeV) for the left (right) panel, and this generates the mass splitting by increasing while maintaining GeV. We emphasize that the mass splitting is potentially an experimental observable, and determines whether and can be observed as separate particles. The left panel has GeV and the right panel has GeV. All the curves start at because the choice of satisfies the condition for maximal mixing in Eq. (22) for GeV (left panel) and GeV (right panel). It is evident that maximal mixing can be obtained for mass splittings GeV and GeV for GeV and GeV respectively. For the case of maximal mixing (i.e., ) the mass splitting is solely caused by the term in Eq. (