# Constraining parameter space in type-II two-Higgs doublet model

in light of a 126 GeV Higgs boson

###### Abstract

We explore the implications of a 126 GeV Higgs boson indicated by the recent LHC results for two-Higgs doublet model (2HDM). Identifying the 126 GeV Higgs boson as either the lighter or heavier of CP even neutral Higgs bosons in 2HDM, we examine how the masses of Higgs fields and mixing parameters can be constrained by the theoretical conditions and experimental constraints. The theoretical conditions taken into account are the vacuum stability, perturbativity and unitarity required to be satisfied up to a cut-off scale. We also show how bounds on the masses of Higgs bosons and mixing parameters depend on the cut-off scale. In addition, we investigate whether the allowed regions of parameter space can accommodate particularly the enhanced di-photon signals, and decay modes of the Higgs boson, and examine the prediction of the signal strength of decay mode for the allowed regions of the parameter space.

###### pacs:

14.80.Cp, 29.20.-c## I Introduction

Both the ATLAS and CMS experiments have discovered a new particle consistent with the Higgs boson higgs () with a mass of around 126 GeV at about significance ATLAS (); CMS (). A common belief among particle physicists that the SM is not the ultimate theory of fundamental interactions calls for new physics beyond the SM, such as supersymmetry (SUSY) and extra dimension models. Many new physics beyond the SM contain more than one Higgs doublet of the SM Branco (). In this regards, it must deserve to examine whether signals detected at the LHC imply the existence of more Higgs sectors or not.

The purpose of this work is to examine the implications of a 126 GeV Higgs boson indicated by the recent LHC results for two-Higgs doublet model (2HDM). We will focus on how severe the theoretical conditions and experimental results on the Higgs sectors can constrain the masses of Higgs fields and mixing parameters in 2HDM in the light of a 126 GeV Higgs boson. The theoretical conditions taken into account are the vacuum stability, perturbativity and unitarity which are required to be satisfied up to a cut-off scale. Then one can obtain constraints on the couplings of the Higgs potential in 2HDM, which in turn lead to bounds on the masses of scalar bosons as well as mixing parameters. Although there are a few works on the estimation of bounds on the masses of scalar fields in 2HDM by applying the vacuum stability, perturbativity 2HDMvacuum (); Ferreira () and unitarity unitarity (), our new points are to show how the parameter spaces in 2HDM are constrained by those theoretical conditions applied up to a cut-off scale by identifying the 126 GeV Higgs boson as either lighter or heavier of CP even neutral scalar bosons, and to see how bounds on the masses of scalar bosons depend on the cut-off scale. In addition, we will examine how experimental constraints on the parameters of scalar bosons from the LEP can constrain the parameter spaces further. As expected, LEP results can severely constrain the parameter space for scalar bosons in the scenario that the new scalar boson observed at the LHC is the heavier CP even neutral scalar boson in the 2HDM. Finally, we will investigate whether the allowed regions of parameter space can accommodate the enhanced di-photon signals, and decay modes of the Higgs boson observed at the LHC, and examine the prediction of the signal strength of decay mode for the allowed parameter regions.

## Ii Higgs sector in 2HDM, theoretical and experimental constraints

The renormalizable gauge invariant scalar potential of 2HDM with softly broken symmetry we consider is given by 2HDM ()

(1) | |||||

where and are two complex Higgs doublet fields with . We note that the dangerous FCNC does not occur in the form of the scalar potential given by Eq.(1) even if non-zero softly breaking the symmetry is allowed. Depending on how to couple the Higgs doublets to the fermions, 2HDMs are classified into four types Branco (). Among them, the Yukawa couplings of type II 2HDM arises in the minimal supersymmetric standard model which is one of the most promising candidates for the new physics model beyond the SM. In this paper, we focus on the type-II 2HDM, in which the one Higgs doublet couples only to the down type quarks and the charged leptons while the another Higgs doublet couples only to the up-type quarks. We require that the scalar potential conserves the CP symmetry, which is achieved by taking all the parameters in Eq.(1) to be real and the squared mass of pseudo-scalar to be greater than for the absence of explicit and spontaneous CP violation, respectively 2HDM (). After spontaneous symmetry breaking, the Higgs doublets have the vacuum expectation values as follows,

(2) |

where and We take and to be positive, so that is allowed. There are five physical Higgs particles in 2HDMs : two CP-even Higgs and (), a CP-odd Higgs and a charged Higgs pair (). Following 2HDM (), the squared masses for the CP-odd and charged Higgs states are given by

(3) |

and the squared masses for neutral Higgs () are given by

(4) |

where and with , , and . The couplings of the two neutral CP even Higgs bosons to fermions and bosons relative to the SM couplings in type II 2HDM are shown in Table 1.

Light Higgs () | Heavy Higgs () | |
---|---|---|

D,L | ||

U | ||

W or Z | ||

AZ |

The stable vacuum guaranteed when the scalar potential (1) is bounded from below can be obtained only if the following conditions are satisfied Ferreira (); 2HDM (); 2hdz2 (); Ivanov ()

(5) |

Since radiative corrections give rise to the modification of the couplings in the scalar potential, we need to require that the stability conditions (5) are valid for all energy scales up to a cut-off scale . As is known, the stability conditions (5) can lead us to lower bounds on the couplings 2HDM (), which in turn give rise to bounds on the masses of the Higgs fields. In addition, we require the perturbativity for the quartic couplings in the scalar potential at all scales up to the cut-off scale and unitarity at the cut-off scale unitarity (). It is worthwhile to notice that those theoretical conditions can constrain not only Higgs masses but also mixing parameters and via the renormalization group (RG) evolutions. In our numerical analysis, we used RG equations for the parameters , gauge couplings and Yukawa couplings presented in ref.RGrunning (). In particular, we take the top quark pole mass and QCD coupling constant at Z boson mass scale () to be 172 GeV and 0.1185, respectively.

On the other hand, experimental results from the LEP give rise to constraints on the masses of Higgs bosons and the mixing parameters in the case that the masses of light neutral Higgs bosons lie between 10 GeV and 150 GeV LEP (); LEP2 () .
For the charged Higgs bosons, the experimental lower bound on their masses is GeV ALEPH ().
The non-observation of in the LEP experiment
indicates that only the Higgs masses satisfied with are kinematically allowed zdecay ().
In addition, when GeV,
non-observation of the Higgsstrahlung process at the LEP constrains the parameter space of and ^{3}^{3}3The parameter introduced in LEP () is equivalent to
in our model.
.
We also consider the Higgs pair production process, ,
if they are kinematically allowed.
Non-observation of those Higgs pair productions can lead to the constraints on light neutral Higgs masses and mixing parameters
as shown in LEP2 ().

In addition, we take into account the new physics contributions to the electroweak precision parameters and , which are defined by pdg (); Toussaint (); kanemura (); Baak (); Neil ()

(6) | |||||

(7) | |||||

where and the formulae for as well as are given in hunter (); cheung (); Chankowski (), and the function is given by Toussaint (); kanemura (); Baak ()

(8) |

By fixing , the allowed values of and for are given by pdg ()

(9) |

We impose the conditions Eq.(9) in our numerical analysis. On top of the constraints from and , we consider the measurement of pdg () as well as the experimental results of the process bsg (), which give rise to the constraints on the plane. In the Type-II 2HDM, it is known that yields the strictest bound on the plane in the small region rb (); d2hdm (). The measurements of mixing also lead to the constraints on the plane but less severe ones in comparison with that from d2hdm (). Combining the theoretical constraints with the experimental ones, we investigate how the masses of Higgs bosons and mixing parameters can be constrained.

## Iii Allowed regions of parameter spaces

Let us study the implication of the 126 GeV Higgs boson indicated by the recent LHC results by identifying it as the lighter or heavier of the CP even neutral Higgs bosons. Instead of fixing a particular value of the Higgs boson mass, we broaden it to be and then investigate parameter space in consistent with the range. In our numerical analysis, the scanned regions of the parameters, and , are

(10) |

Note that small below 0.3 is ruled out by breaking down of perturbativity of Higgs-top Yukawa coupling H-top (). We observed from our numerical analysis that vacuum stability excludes the region of GeV for the case of GeV and GeV. For the case of GeV, there are allowed regions of the parameter space above GeV, but such large values of lead to large values of scalar masses in the 2HDM leading to so-called decoupling limit, so we cut the size of by 1000 GeV in our analysis. Instead of getting the regions of the parameter space with solid boundaries, we plot allowed data points by randomly scanning the input parameters such as and restricted by Eq.(10) and then picking out the data points satisfying theoretical conditions and experimental constraints.

### iii.1 Case for GeV

Assuming that the mass of the heavier neutral CP even Higgs is around 126 GeV, let us examine how the parameter space of Higgs masses and mixing parameters can be constrained by theoretical conditions and experimental constraints explained in Sec. II. Also, we investigate how the allowed regions of the parameter space depend on the cut-off scale.

Fig. 1 shows how the regions of parameter spaces in the plains (left-hand panels) and (right-hand panels) are constrained by the theoretical conditions and experimental results. The panels in the upper, middle and lower rows correspond to the cases of the cut off scale and TeV, respectively. The territories covered by the red points present the allowed regions by the theoretical conditions. The blue regions survive the constraints on , parameter and . Further imposing constraints coming from the direct searches for Higgs fields via the Higgsstrahlung and Higgs pair productions at the LEP and the bound on from ALEPH, the green data points finally survive. We see that the allowed regions get wider as the cut off scale gets lower. The black horizontal lines correspond to the lower limit of coming from the experimental constraint from btosgam (), and thus the regions below the lines are excluded if no new effects on flavor physics are introduced in 2HDM. We also display the cyan lines corresponding to the lower limit of the charged higgs mass from the ALEPH experiment, GeV ALEPH (). From our numerical analysis, we found that the constraint from excludes all parameter regions survived other constraints for TeV in the case of GeV.

In Fig. 2, the points represent the parameter space in the plain constrained by the theoretical conditions and experimental constraints for 1 (a), 14 (b), 100 (c) TeV, respectively. The brown and red points survive all the constraints we consider, whereas the green and blue points survive all the constraints except for and thus correspond to GeV. In particular, we display the points consistent with SM-like Higgs, , in blue ( GeV) and red ( GeV). As explained above, we do not see any data points for TeV survived the constraint from . We see that the region of is excluded in all cases we consider. It is worthwhile to notice that the mixing parameter can be constrained by not only the LEP experiments but also the theoretical conditions. As the cut off scale increases, the allowed regions get narrowed as shown in Fig. 2.

Let us discuss how the allowed regions obtained above can be confronted with the recent LHC data by considering the channels , , directly searched to probe the SM-like Higgs boson at the LHC. The recent experimental results of the signal strengths for and decay modes are given by ATLAS (); CMS (),

(11) | |||||

(12) |

where with . The results are not incompatible with the SM predictions. As for the measurements for the di-photon channel, the current ATLAS results show a deviation from the SM prediction ATLAS ()

(13) |

whereas the CMS results appears to be compatible with the SM prediction CMS ()

(14) |

In Fig. 3-(a,b), we display plots of vs. for the allowed regions of parameter space shown in Figs. 1 and 2. The left- and right-hand panels correspond to and TeV, respectively. The magenta (green) cross-bars represent the ATLAS (CMS) experimental results. The dashed (solid) cross-bars correspond to the experimental results for signal strengths. In Fig. 3-(c,d), we present the predictions of for the same parameter regions taken in Fig. 3-(a,b). The magenta (green) shaded regions stand for the ATLAS (CMS) results for the di-photon signals. Fig. 4 shows the same as Fig. 3 but for TeV. All blue points in Figs. 3 and 4 correspond to the green ones in Figs. 1 and 2. In particular, we display in Fig. 3 the data points survived the constraint from in cyan. We see from Figs. 3 and 4 that there is no data point compatible with the enhanced di-photon signal measured at ATLAS, whereas there are parameter regions accommodating both the di-photon and vector boson pair signals observed from CMS. To see in detail why the allowed regions of parameter space in this case can not lead to enhancement of di-phton signal, let us consider the formula of enhanced di-photon signal strength given by

(15) |

where is the heavy CP-even neutral higgs () or the light neutral higgs (), and denotes the total decay width of . Because of the convention, , the coupling of heavy neutral higgs () to up-type quarks relative to that of SM, , is smaller than one, whereas the coupling of to down type quarks (or charged leptons), , is larger than one. This indicates that both (A) and (C) in Eq. (15) should be smaller than one because the dominant contribution of gluon fusion is mediated by top quark loop, and yields the most dominant contribution to the branching ratio of the 126 GeV decay. In addition, the dominant contribution (mediated by -loop) to the term (B) is proportional to which can not be larger than one. Thus, the predictions of the signal strength for the di-photon channel can not be enhanced in this case.

### iii.2 Case for GeV

Assuming the lighter CP-even neutral Higgs mass () is around 126 GeV, let us examine how the parameter space of Higgs masses and mixing parameters can be constrained, and how constraining the parameter spaces depends on the cut-off scale. In this case, the experimental results coming from the direct search for the Higgs bosons at the LEP do not further constrain the parameter space survived the theoretical constraints.

In Fig. 5, we show how the regions of parameter spaces in the plains (left panels) and (right panels) are constrained by the theoretical conditions and experimental results. The panels from top to bottom correspond to the cases of the cut off scale 1 TeV, 10 TeV, 40 TeV and 100 TeV, respectively. The territories covered by all the points present the allowed regions by stability, perturbativity and unitarity. The green points survive the constraint on , and . In this case, we scan only the parameter space satisfying the experimental constraint from GeV). The allowed regions appear to get narrowed as the cut-off scale increases.

In Fig. 6, we plot the allowed data points obtained in Fig. 5 in the plain (). The panels (a), (b), (c) and (d) correspond to and 100 TeV, respectively. The points survived all the constraints we consider are displayed in green. Among the points survived all the constraints, the ones corresponding to the SM-like Higgs with and the ones consistent with the measurement of the enhanced di-photon at ATLAS are displayed in blue and red, respectively. It is likely that the allowed regions get narrowed as the cut off scale increases. We see that the region of is excluded in the case of TeV.

In Fig. 7, we show how the predictions of are correlated with those of for the allowed regions of parameter space shown in Fig. 5 for 1 (a), 10 (b), (c) and 100 (d) TeV. In Fig. 8, we plot the predictions of for the same parameter space taken in Fig. 7. All blue points in Figs. 7 and 8 correspond to the green ones in Fig. 5 and 6. The colored cross-bars and shaded regions are the same as in Fig. 3. As can be seen from Figs. 7 and 8, the allowed region of parameter space is so wide that it could be in consistent with the experimental results of the signal strengths from not only CMS but also ATLAS for TeV. Contrary to the case of GeV, in this case, the coupling of lighter neutral higgs () to up-type quarks relative to that of SM, , can be larger than one, whereas the coupling of to down type quarks (or charged leptons), , can be smaller than one, which give rise to enhancements of both (A) and (C) terms in Eq. (15). Those enhancements can be sufficient to enhance the di-photon signal strength after compensating the possible suppression of the term (B) in Eq.(15).

We note that there are several works in the literature d2hdm (); diphoton (); extended diphoton () that study the enhanced di-photon signals in the extended Higgs models, and the authors in diphoton () have obtained the parameter region explaining the enhanced di-photon signal in the case of 2HDM, but we have examined the same problem by taking into account the experimental constraints from the LEP experiments and theoretical conditions valid for all renormalization scales up to given cut-off scale. So we obtain even stronger constraints on and compared with those obtained in diphoton ().

Before concluding, remarks on the implications of type-I 2HDM are in order. Compared with type-II 2HDM, the main difference in type-I 2HDM is the Higgs couplings to the fermions. Those couplings are the same as in the SM but multiplied by and for the Higgses and , respectively. Contrary to type-II model, the Yukawa couplings of down-type quarks can not be enhanced unless is very small. Small values of are excluded or disfavored by perturbativity of Yukawa couplings and constraints from -physics. In particular, the constraints from -physics lead to different implications of type-I model. It is known that type-I model is not severely constrained by Branco (). Thus, contrary to type-II model, light charged Higgs can be allowed in type-I model, which can non-negligibly contribute to the Higgs decays and productions. Thus, the implications of the Higgs signal strengths for the di-photon and are different from those in type-II model. In Fig. 9, we display the allowed points by theoretical and experimental constraints in the plains and for TeV in type-I model. The panels (a) and (b) correspond to the case of GeV and GeV, respectively. Contrary to type-II model, large positive values of are allowed for the case of GeV and most small values of are exluded for the case of GeV in type-I model. For the allowed points, we calculate the signal strengths of the di-photon, gauge boson pairs and , and the results are displayed in Figs. 10 and 11. The left (right) panels correspond to GeV. The colored cross-bars and shaded regions are the same as in the case of type-II model. The predictions for both cases are consistent with the recent results from CMS. While the predictions for the case of GeV in type-II model are so wide that they could cover the enhancement of di-photon signal observed at ATLAS, those in type-I model do not so.

In conclusion, we have examined the implications of 126 GeV Higgs boson indicated by the recent LHC results for type II 2HDM. Identifying the 126 GeV Higgs as either the lighter or heavier of the CP even neutral Higgs, we have obtained the allowed values of Higgs masses and mixing parameters by imposing the theoretical conditions and experimental results on the Higgs sectors. The theoretical conditions taken into account are the vacuum stability, perturbativity and unitarity required to be satisfied up to a cut-off scale. So, the allowed regions are turned out to be strongly dependent of the cut-off scale. We have shown how the experimental constraints on the parameters for Higgs bosons from the LEP as well as B physics, and electroweak precision constraints can constrain the parameter spaces further. Finally, we have found that all the allowed parameter points for the case of are incompatible with the enhanced di-photon signal of ATLAS, whereas there exist parameter regions simultaneously accommodating the di-photon and vector boson pair signals observed at the CMS. On the other hand, in the case of GeV, the allowed region of parameter space is so wide that it could be compatible with not only CMS but also ATLAS experimental results of the signal strengths for TeV. We have also predicted the signal strengths for channel of the Higgs decay for the allowed parameter regions.

###### Acknowledgements.

This work was supported by NRF grant funded by MEST (No.2011-0029758).## References

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