# Implication of Quadratic Divergences Cancellation in the Two Higgs Doublet Model

###### Abstract

Abstract: With the aim of exploring the Higgs sector of the Two Higgs Doublet Model (2HDM), we have chosen the exact and soft symmetry breaking versions of the 2HDM with non-zero vacuum expectation values for both Higgs doublets (Mixed Model). We consider two SM-like scenarios: with 125 GeV and 125 GeV . We have applied the condition for cancellation of quadratic divergences in the type II 2HDM in order to derive masses of the heavy scalars. Solutions of two relevant conditions were found in the considered SM-like scenarios. After applying the current LHC data for the observed 125 GeV Higgs boson, the precision electroweak data test and lower limits on the mass of , the allowed region of parameters shrink strongly.

## I Introduction

The SM describes the physics of elementary particles with a very good accuracy Glashow (). Experiments have confirmed its predictions with remarkable precision. One of the most precise aspects of the model is associated with the Higgs sector. However, the SM is not a completely perfect model, since it is unable to provide adequate explanations for many questions in particle and astrophysics Darvishi:2016tni (); Darvishi:2016fwo (); Bonilla:2014xba (); Darvishi:2016gvm (); Krawczyk:2015xhl (); Darvishi:2017fwr (). One of the problems of the SM is the naturalness of the Higgs mass. From experimental data, we know that the Higgs boson mass (125 GeV) is of the order of the electroweak scale, but from the naturalness perspective, this mass is much larger than the electroweak scale. This is because of the large radiative corrections to the Higgs mass which implies an unnatural tuning between the tree-level Higgs mass and the radiative corrections. These radiative corrections diverge, showing a quadratic sensitivity to the largest scale in the theory weinberg (). Solutions to this hierarchy problem imply new physics beyond the SM, which must be able to compensate these large corrections to the Higgs boson mass. This goal can be obtained with the presence of new symmetries and particles. Veltman suggested that the radiative corrections to the scalar mass vanish (or are kept at a manageable level) Veltman (). This is known as the Veltman condition.

In this paper, we apply Veltman condition to the 2HDM to predict masses of the additional neutral scalars in two possible scenarios, with the SM like- and the SM like- bosons. The reader can find similar discussions in wu (); ma (); Grzadkowski:2010dn (); Chowdhury:2015yja (); Biswas:2014uba (); Chakraborty:2014oma (). In 2HDM Lagrangian, four different types of Yukawa interactions arise with no FCNC at tree level Barger:1989fj (); Grossman:1994jb (). In type I, all the fermions couple with the first doublet and none with the . In type II, the down-type quark and the charged leptons couple to the first doublet, and the up-type quarks to the second doublet. In type III or the flipped models, the down-type quarks couple to the and the up-type quarks and the charged leptons couple to the . In type IV or the lepton-specific models, all quarks couple to the and the charged leptons couple to the . Among fermions, we include only the dominate top and bottom quark contributions and neglect leptons. Therefore type I and type IV become identical. The same statement is true for the type II and the type III. In 2HDM type I, vanishing quadratic divergence are possible due to negative scalar quartic couplings, but this solution is contradict to the condition of positivity of the Higgs potential Chakraborty:2014oma (); Ma:2014cfa (); Barbieri:2006dq (), therefore we concentrate on the 2HDM Model type II .

## Ii Mixed Model with a soft symmetry breaking

The Higgs sector of the 2HDM consists of two SU(2) scalar doublets, and . The 2HDM potential depends on quadratic and quartic parameters, respectively and (i= 1…, 5), from which five Higgs boson masses come up after the spontaneous symmetry breakdown (SSB). The most general invariant Higgs potential for two doublets,

with a soft symmetry () violation is given by:

(1) |

All parameters are assumed to be real, so that CP is conserved in the model.

In order to have a stable minimum, the parameters of the potential need to satisfy the positivity conditions leading to the potential bounded from below. This behaviour is governed by the quadratic terms, which have the following positivity conditions Klimenko:1984qx ():

(2) |

The high energy scattering matrix of the scalar sector at tree level contains only sâwave amplitudes that are described by the quartic part of the potential. The tree level unitarity constraints require that the eigenvalues of this scattering matrix, , be less than the unitarity limitPlehn:2009nd (); Ginzburg:0312374 (). This means, the requirement (or , with being the 0th partial s-wave amplitude for the body scatterings) corresponds to . Finally, one can also impose harder constraints on the parameters of the potential based on arguments of perturbativity, by demanding that the quartic Higgs couplings fulfill Plehn:2009nd (); Ginzburg:0312374 (); Swiezewska:2012ej (); Krauss:2017xpj (); Cacchio:2016qyh (); my2 ().

The Mixed Model is based on the vacuum with nonzero VEV for both doublets, respectively and , with . The minimization conditions are as follows:

(3) | |||

(4) |

where and . It is well known that such minimum is the same a global minimum, i.e. vacuum my2 ().

There are five Higgs particles, with masses as follows:

(5) | |||

(6) |

with the other two mass squared, , being the eigenvalues of the matrix

where . This matrix, written in terms of the mass squared of physical particles , with , and the mixing angle is given by

The ratio of the coupling constant () of the neutral Higgs boson to the corresponding SM coupling , called the relative couplings

(9) |

are summarised in the table 1 (see e.g. reference CP2005 ()).

( and ) | (up-type quarks ) | (down-type quarks ) | |
---|---|---|---|

One sees that all basic couplings can be represented by the couplings to the gauge boson V, for and the parameter.

So, we consider two cases which define our SM-like scenarios:

GeV,
(SM-like scenario)
GeV,

(SM-like scenario).

In both cases, we identify a SM-like Higgs boson with the 125 GeV Higgs particle observed at LHC. Therefore, in the SM-like scenario, the neutral Higgs partner () can only be heavier, while in the SM-like scenario - the partner particle can only be lighter than 125 GeV.

In this analysis, we apply the positivity conditions and perturbative unitarity condition. We keep the mixing angles in the range: and .

## Iii Cancellation of the quadratic divergences

The cancellation of the quadratic divergences at one-loop applied to 2HDM with Model II for Yukawa interaction leads to a set of two conditions wu (), namely:

(10) | |||

It should be noted that the Eqs. (III) and (III) do not depend on the choice of a gauge parameter and the cancellation of the quadratic divergences in the tadpole graphs (Collins:2006ib ()) does not give the additional independent conditions Blumhofer:1994xv (). We include only the dominate top and bottom quarks contributions (, ). Expressing ’s parameters by masses and the mass parameter , we have

(12) |

where

(13) |

(14) |

and

(15) |

(16) |

(17) |

(18) |

In the following section, we solve the Eqs. (12), expressing the condition for cancellation of the quadratic divergences to derive masses of the partner Higgs particles.

## Iv Approximate solution of the cancellation conditions

It is useful to look first at the approximate solution, which can be obtained analytically. In the 2HDM model with the soft symmetry breaking we have for the strict SM-like (alignment) scenarios, with or :

These formula follow directly from equations (II) and (II). The difference of and is given in terms of and the mass of the neutral partner for the SM-like or particle, mining respectively the or the boson. From difference of equations (III) and (III) we found that

(21) |

Combining the equations (LABEL:23), (LABEL:24) and (21), we obtain the following expressions for masses squared of the partner of the SM-like or Higgs particle,

(22) |

Obviously, the above prediction for the mass of the partner Higgs particle has been obtained without additional constraints. We have plotted versus , as given by Eq. (22) for and 100 GeV, in the figure 1.

In the figure the mass limit 127 GeV for the SM-like particle is used. The hachure area (i.e. all masses less than 127 GeV) is the allowed region for in the SM-like scenario and the white area (i.e. all masses higher than 127 GeV) is allowed for in the SM-like scenario. Let us look at the case. It is clear that for the SM-like , solutions exist only for and the mass of should be larger than . The solutions for the SM-like exist for large () with mass of below . For positive , in the intermediate new regions open up e.g. for GeV. For negative , all curves are lying below the reference curves.

## V Solving the cancellation conditions

Here we present the results of numerical solutions of Eqs. (12) for the SM-like scenarios, as described above. We apply the positivity and the perturbative unitarity constraints on parameters of the model. We have performed three scans for three considered SM-like scenarios (SM-like , SM-like H+ and SM-like H-) with the mass window of the SM-like Higgs 124-127 GeV and the relative coupling to gauge bosons between 0.90 and 1.00, in agreement with the newest LHC data, which are presented in table 2.

We assume being bounded to the region and using following regions of the parameters of the model:

Note that very recently the new lower bound on has been derived, much higher than the used by us in the scan 360 GeV oldmisiak (), namely: GeV Misiak:2017bgg (). In the calculations, we use GeV, GeV, GeV, GeV CMS (); Beringer (). Solutions of the Eqs. (12) were found by using and independently by a program, written by us.

Performing our scanning we found no solution for the SM-like scenario in both cases , for mass of the charged Higgs boson larger than GeV oldmisiak ().

For SM-like scenario there are solutions only for positive , in the region 200 - 400 GeV. Figure 2 shows the correlation between and (panel (a)) and the correlation between and (panel (b)). In both panels, the first region from the left (lower region) is obtained for large (above 40), while the right one corresponds to low (below 5). Figure 2(c) shows the correlation of with . In the low the lower limit for is 500 GeV. The correlation between versus and versus are similar to correlation between versus . Also, in parts (d), (e) and (f) of the Figure 2, we have shown the correlations vs , vs and vs , respectively.

Below, we will look closer to the obtained results, by confronting the obtained predictions for observables with experimental data. We propose 5 benchmarks for the SM-like scenario, which will be discussed in section VII, in agreement with theoretical constraints (positivity and perturbative unitarity) and experimental constraints, mainly coming from the measurements of SM-like Higgs boson. In addition, properties of the partner particles () is checked, which is important for future search. In such calculations the program was used 2hdmc ().

## Vi Experimental constraints

We apply experimental limits from LHC for the SM-like Higgs particle and solve the cancellation conditions by scanning over and the mixing parameters (), keeping the values of mass and coupling to the gauge bosons (i.e. ) within the experimental bounds. We confront the resulting solutions with existing data for the 125 GeV Higgs boson, in particular the experimental data on Higgs boson couplings (, and ) and Higgs signal strength (, and ) from ATLAS Aad:2014eha () and CMS Khachatryan:2014ira (), as well as the combined ATLAS+CMS results 201606 (). Also the total Higgs decay width measured at the LHC is an important constraint, see tables 2 and 3.

We also keep in mind other existing limits on additional Higgs particles which appear in 2HDM, as the lower mass limit of the taken to be 360 GeV (based on the earlier analysis oldmisiak ()) and check if the obtained solutions are in agreement with the oblique parameters , , constraints, being sensitive to presence of extra (heavy) Higgses that are contained in the 2HDM.

Below, the following short notation will be used: , and for , and , respectively.

## Vii Benchmarks

Results from scan lead to five benchmark points h1-h5, presented in table 2. The results of the scanning show that for the SM-like scenario, solutions in agreement with existing data only exist for small (0.45 -). Values of observables for the SM-like Higgs particle for five benchmark points h1-h5 are presented in table 3. The large solutions, (above ) exist, however they lead to too large , (above 2). For convenience, we add the experimental data to both tables (with 1 accuracy from the fit assuming and ).

Benchmarks correspond to solutions with masses GeV and GeV and GeV. The newest result of the reference Misiak:2017bgg () with lower bound on GeV can limit our benchmarks to (h3, h4) only. There is a small tension, at the 2 , for all benchmarks for the coupling with the newest combined LHC result 201606 (), which has surprisingly small uncertainty 0.16 (before the individual results were ATLAS Aad:2014eha () and CMS Khachatryan:2014ira (), in perfect agreement with all our benchmarks). Also, benchmarks (h2,h3) correspond to slightly too small mass of the in the light of the new combined CMS and ATLAS value of 125.09 0.24 GeV 201606m ().

We compare our benchmarks to the experimental data on 2HDM (II) on the plot versus , see figure 3. Benchmark h4 is very close to the best fit point found by ATLAS. We would like to point out that our benchmarks results from the cancellation of the quadratic divergences and no fitting procedure has been performed. Note, that the h4 benchmark corresponds to heavy and degenerate and bosons, with mass 650 GeV, while is even heavier with mass 830 GeV.

B mark | |||||||||
---|---|---|---|---|---|---|---|---|---|

exp | - | - | 1.00 (0.92-1.00) | - | - | - | - | - | |

h1 | -1.24627 | 0.451897 | 0.995014 | -0.0997376 | 124.426 | 573.832 | 444.16 | 454.424 | |

h2 | -1.10678 | 0.481736 | 0.999886 | 0.0150858 | 124.082 | 505.298 | 266.59 | 375.488 | |

h3 | -1.00657 | 0.507350 | 0.995518 | 0.0945748 | 124.242 | 736.961 | 567.37 | 598.392 | |

h4 | -0.96384 | 0.589698 | 0.997252 | 0.0740784 | 125.252 | 826.947 | 650.08 | 645.560 | |

h5 | -0.94625 | 1.077030 | 0.980477 | -0.1966340 | 125.771 | 605.931 | 448.48 | 438.628 |

B point | ||||||||
---|---|---|---|---|---|---|---|---|

exp | ||||||||

h1 | 0.77 | 1.04 | 1.03 | 0.98 | 0.96 | -0.00 | -0.02 | -0.00 |

h2 | 1.03 | 0.99 | 1.09 | 0.97 | 0.96 | -0.00 | -0.24 | -0.00 |

h3 | 1.17 | 0.94 | 1.05 | 0.98 | 0.95 | -0.00 | -0.07 | -0.00 |

h4 | 1.12 | 0.95 | 1.11 | 1.08 | 0.97 | -0.00 | 0.01 | -0.00 |

h5 | 0.79 | 1.19 | 1.31 | 1.35 | 0.77 | 0.01 | 0.01 | -0.00 |

It is worth to look for the properties of the heavy neutral Higgs boson , the partner of the SM-like bosons. The corresponding observables are given in table 4, here the relative couplings are calculated in respect to couplings the would-be SM Higgs boson with the same mass as . The coupling to quark is negative, what is easy to understand looking at the table 1. Its absolute value is enhanced, as compared to the SM value, and the corresponding ratio varies from 1.1 to 2.30. One observes a huge enhancement in , it is from 50 to 153 times larger the SM one, at the same time decay channel looks modest (0.31-1.44). Also, the total width is similar to the one predicted by the SM - the corresponding ratio varies from 0.3 to 1.09. For a possible search for such particle, the channel would be the best. The channel is hopeless, but the decays to and , govern by coupling, may be useful. Note, that all heavy Higgs bosons have masses below 850 GeV, in the energy range being currently probed by the LHC.

B point | ||||||
---|---|---|---|---|---|---|

h1 | 573.832 | -2.30 | 0.34 | 69.14 | 0.62 | 0.88 |

h2 | 505.298 | -2.06 | 0.49 | 14.88 | 0.31 | 1.09 |

h3 | 736.961 | -1.86 | 0.59 | 152.71 | 1.21 | 0.78 |

h4 | 826.947 | -1.61 | 0.66 | 49.95 | 1.44 | 0.53 |

h5 | 605.931 | -1.10 | 0.85 | 72.79 | 0.79 | 0.28 |

## Viii Summary and Conclusion

In this paper, we have investigated the cancellation of the quadratic divergences in the 2HDM, applying positivity conditions and perturbativity constraint. We have chosen the soft symmetry breaking version of the 2HDM with non-zero vacuum expectation values for both Higgs doublets (Mixed Model) and considered two SM-like scenarios, with 125 GeV and 125 GeV .

We have chosen 5 benchmarks in agreements with experimental data for the 125 GeV Higgs particle from the LHC and checked that our benchmark points are in agreement with the oblique parameters , and , at the 3 .

We compare our benchmarks to the experimental constraints for 2HDM (II) on the versus plane. All benchmarks are close to or within the allowed 95 CL region, especially our benchmark h4 is very close to the best fit point found by ATLAS. We would like to point out that our benchmarks results from the cancellation of the quadratic divergences and no fitting procedure has been performed. Note, that the h4 benchmark corresponds to heavy and degenerate and bosons, with mass 650 GeV, while is even heavier with mass 830 GeV.

###### Acknowledgements.

This work is supported in part by the National Science Center, Poland, the HARMONIA project under contract UMO-2015/18/M/ST2/00518. ND thanks Prof. M. Misiak and Dr. M.R. Masouminia for their helpful discussions.## References

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