# Naturalness in Type II seesaw model and implications for physical scalars

## Abstract

In this paper we consider a minimal extension to the standard model by a scalar triplet field with hypercharge . This model relies on the seesaw mechanism which provides a consistent explication of neutrino mass generation. We show from naturalness considerations that the Veltman condition is modified by virtue of the additional scalar charged states and that quadratic divergencies at one loop can be driven to zero within the allowed space parameter of the model, the latter is severely constrained by unitarity, boundedness from below and is consistent with the di-photon Higgs decay data of LHC. Furthermore, we analyse the naturalness condition effects to the masses of heavy Higgs bosons , , and , providing a drastic reduction of the ranges of variation of and with an upper bounds at and GeV respectively, while predicting an almost degeneracy for the other neutral Higgs bosons , at about GeV.

## I Introduction

After the LHC’s Run 1 and beginning of Run 2, we are now more confident that the observed GeV scalar boson is the long sought Higgs boson of the Standard Model (SM) atlas12 (); cms12 (). However, although its brilliant success in describing particle physics, still many pressing questions are awaiting convincing solutions that cannot be answered within SM. The hierarchy problem and the neutrinos oscillations are the most illustrative ones. In this context, many theoretical frameworks have been proposed and the most popular one is Supersymmetry.

The search for Supersymmetry at Run I of LHC gave a negative result. Therefore the original motivation of Susy to solve hierarchy problem by suppressing quadratic divergencies (QD) is questionnable. In this case, it is legitimate to propose other perspective to interpret and control the QD. It is known that one has to call upon new physics to deal with such problem. More specifically, the new degrees of freedom in a particular model conspire with those of the Standard Model to modify the Veltman Condition and to soften the divergencies drozd (); masina (); kundu13 (); biswas14 ().

In this paper, we aim to investigate the naturalness problem in the context of Type II Seesaw model, dubbed HTM, with emphasis on its effect of the HTM parameter space . More precisely, we will study how to soften the divergencies and how to gain some insight on the allowed masses of the heavy scalars in the Higgs sector. A more recent work of Kundu et al.Kundu14 () has partially discussed this issue. However, unlike the analysis in Kundu14 (), our study use the most general renormalisable Higgs potential of HTM aa11 () and is essentially based on dimensional regularisation approach which complies with unitarity and Lorentz invariance aa11 (). More importantly, the phenomenological analysis takes into account the full set of theoretical constraints, including unitarity aa11 () and the consistent conditions of boundedness from below aa11 (); spanish15 ().

This work is organised as follows. In section , we briefly review the main features of Higgs Triplet Model and present the full set of constraints on the parameters of the Higgs potential. Section is devoted to the derivation of the modified Veltman condition (mVC) in HTM. The analysis and discussion of the results are performed in section , with emphasis on the effects of mVC on the heavy Higgs bosons, particularly on charged Higgs. Conclusion with summary of our results will be drawn in section .

## Ii Seesaw Type II Model: brief review

Type II seesaw mechanism can be implemented in the Standard Model via a scalar field transforming as a triplet under the gauge group with hypercharge . In this case the most general gauge invariant Lagrangian of the HTM scalar sector is given by aa11 (); Perez08 ():

(1) |

The covariant derivatives are defined by,

(2) | |||||

(3) |

where is the Higgs doublet while (, ), and (, ) represent the and gauge fields and couplings respectively. , with () are the Pauli matrices. The potential reads as,

(4) | |||||

where denotes the trace over matrices. The Triplet and doublet Higgs are represented by:

(5) |

with and .

After the spontaneous electroweak symmetry breaking, the Higgs doublet and triplet fields acquire their
vacuum expectation values and respectively, and seven physical
Higgs bosons appear, consisting of: two neutral scalars (, ), one neutral pseudo-scalar and a pair of simply and doubly charged Higgs bosons and . ^{1}

(6) | |||

(7) | |||

(8) | |||

(9) | |||

(10) |

The coefficients and are the entries of the mass matrix defined by,

(11) |

In the remainder of this paper, we assume the light scalar as the observed Higgs with mass about GeV.

## Iii Theoretical and experimental constraints

The HTM Higgs potential parameters are not free but have to obey several constraints originating from theoretical requirements and experimental data. Thus any phenomenological studies are only reliable in the allowed region of HTM parameter space.

parameter:

First, recall that the parameter in HTM at the tree level is given by the formula, which indicates a deviation from unity. Consistency with the current limit on from precision measurements db12 () requires that the limit resulting in an upper limit on about GeV.

Masses of Higgs bosons :

Many experimental mass limits have been found for the Heavy Higgs bosons. From the LEP direct search results, the lower bounds on GeV for models with more than one doublet in the case of the neutral scalars.

As to the singly charged Higgs mass we use the LEP II latest bounds, GeV from direct search results, whereas the indirect limit is slightly higher GeV lep (). Furthermore, the present lower bound from LHC is GeV, where the excluded mass ranges established by ATLAS atlas_charged () and CMS cms_charged () are taken into account. In the case of the doubly charged Higgs masses, the most recent experimental upper limits reported by ATLAS and CMS are respectively GeV atlas_dcharged () and GeV cms_dcharged (). These bounds originate from analysis assuming 100% branching ratio for decay. However, note that one can find realistic scenarios where this decay channel is suppressed with respect to akeroyd08 () invalidating partially the LHC limits. For example, In HTM with moderate triplet’ VEV, GeV, the analysis of decay channel can easily overpasses the two-sign same lepton channel for where the limit decreases up to GeV goran12 ().

As to the theoretical constraints on the parameter space, we should take into account the perturbativity constraints
on the as well as the stability of the electroweak vacuum that ensure that the potential is bounded from below (BFB).
Let us first recall all the constraints obtained in aa11 ()^{2}

BFB:

(12) | |||

(13) | |||

(14) |

Unitarity:

(15) | |||

(16) | |||

(17) | |||

(18) | |||

(19) | |||

(20) | |||

(21) | |||

(22) | |||

(23) | |||

(24) |

We stress here that all values presented in the plots and in our subsequent analysis (section ) are consistent with all theoretical and experimental bounds described in this section.

## Iv The modified Veltman Condition

The method to collect the quadratic divergencies in a framework of dimensional regularisation is due to Veltman Veltman81 () and needs a dimensional gymnastics since the space-time dimension to pick up the quadratic divergencies depends on the number of loops. But the idea to use the quadratic divergencies to get physical insights is much older, and goes back to the pioneering work of Stuckelberg in 1939 Chakraborty39 (). In the context of a renormalisable theory (which is our case), it is not clear that the deductions coming from such a concept are pertinent. Indeed it provides for relations between the masses or/and the couplings constants of the theory, which could give some credit to an underlying and hypothetical symmetry.

From a more basic level, one can imagine that the relations found by this method are indicative and could provide orders of magnitude, bounds or constraints and we will follow this line in this work. This is more or less the point of view of Chakraborty-Kundu Kundu14 () where they apply the cancellation of quadratic divergencies to the HTM, the Standard Model extended by a scalar triplet. But we differ from this work in the way to pick up the quadratic divergencies. They use the cut-off method to regularise the divergent integrals although we will use the Veltman method that we will sketch later on. Both methods have advantages and back-draws that we will not discuss here, since they provide for similar relations. We just mention that the cut-off method is intuitive but neither Lorentz nor gauge invariant. A contrario, the dimensional regularisation method is both Lorentz and gauge invariant, although less intuitive. We refer to previous works such as Peyranere1991 () and references therein for more details and discussions.

To check the Veltman conditions one needs to calculate the quadratic divergencies which show up in the tadpoles of the two CP-even neutral Higgs of our model, namely and . Since vacuum is supposed to be CP-even, the tadpole associated with the neutral pseudoscalar field vanishes. Also, it is noticeable that no QCD contribution appears at one loop level, hence only the electro-weak part of the HTM model is concerned in this procedure. Since there is no derivative couplings, it is easy to figure out how to calculate each diagram (see the appendix for all relevant vertices required for such calculation). Aside the coupling constant, we just need to consider the propagator of the field in the loop, where the contribution of each loop can be written in a simple form in terms of the Passarino-Veltman function pave ().

In a 4 dimensional space-time, , up to some pure numerical factor, where is the pole term. In a 2 dimensional space-time, is no longer mass dimensioned and it is a pure U.V. divergent number; this property is fundamental because it guarantees a gauge invariant result. This pure number will be forgotten everywhere in this section since irrelevant in the equations, together with other common factors like the dimensional dependence of the coupling constants.

For the propagator of a scalar particle in HTM Higgs sector, (), we have leading to a contribution. For a vectorial particle (, we have leading to a contribution, where and are the transverse and the longitudinal projectors and is the space-time dimension. For a true fermionic particle (massive leptons and quarks), we have leading to a contribution.

To get the final results, one just has to sum up all the possible diagrams, taking into account the for the fermionic loops (including Faddeev-Popov ghosts), the symmetry factor of the diagram and possibly the color factor for the quarks that we forget for conciseness. As a matter of fact, for the Higgs particle one gets:

where the couplings , the symmetry factors and the propagator loops are given in the appendix. This formula leads, in a 2 dimensional space time, to a quite large formula depending on many parameters of the model as well as the mixing angles.

Similarly, for the other CP-even Higgs particle , one only has to change into as indicated in the appendix where these coefficients are listed.

To summarise, the Veltman condition implies that the quadratic divergencies of the two possible tadpoles and of the and CP-even neutral scalar fields vanish. Both results are not very tractable, so we do not write them. The linear combination of the fermionic coupling constants is zero; further it turns out that the combination induces simplification and one ends up with the short and nice expression:

(25) |

where and is the square of the Standard Model vacuum expectation value ( GeV).

Of course, it is very tempting to calculate the orthogonal combination, , which also leads to a simple result, where the quarks contributions must be multiplied by the color factor:

(26) |

Here, we do comment both results. The two linear combinations and reproduce the real neutral components of the triplet and the doublet of the HTM, after their VEV shifts. So and represent the quadratic divergencies of the fields and , as can be shown from Eq. (2.26) of aa11 (). Hence, its is straightforward to understand that all mixing angles disappear in and .

The absence of the parameter in is obvious since is the coupling constant of the pure doublet quartic interaction. In the same way, the absences of and are also natural in since these two couplings only concern the triplet.

We are also faced up to two intriguing questions: why the fermionic part is missing in the first equation? why the parameter is missing in both equations ?

The answer to the first question, on one hand, we have already noticed that the cancellation originates from the combination , that is on the form of the trilinear couplings Higgs-fermion-antifermion. On the other hand, the absence of fermionic part lies in the form of the Yukawa coupling with the triplet , given by Eq. (2.5) in aa11 ()). Indeed, a close look shows that the fermionic doublet is no longer associated with but with , that forbids any coupling, insuring the absence of fermionic contribution at one loop order.

About the second question, it is bizarre because is the strength of a trilinear coupling between
doublets and triplet, so a priori able to give contribution, but the interaction is linear in the triplet and a
doublet-triplet-triplet vertex is excluded, that explains the lack of in . As to ,
the solution is again in the form of the interaction where we have the doublet and the transposed
doublet but not : before breaking of the symmetries and the vev shifts, a coupling
such as is forbidden. Then after breaking and shifting the fields, we
end up with two vertices, and , with opposite values ,
where and are the real and imaginary parts of the shifted field . This feature suffices
to cancel these two contributions in a 2 dimensional space-time, when the loop does not depend on the propagator
mass as explained before. In some sense, the quadratic divergencies record the physics before breaking and
shifting, and this is probably not a pure one-loop effect.

Finally, it is worth to notice that Veltman Condition in Standard Model Peyranere1991 () is recovered when, we remove the couplings and in the doublet formula , so discarding any mixing between doublet and triplet.

## V Analysis: Implications for the Physical Scalar masses

In this section we will focus our analysis on the modified Veltman condition (mVC) given in .
Our aim is twofold: first we will show that one loop quadratic divergencies are softened and go to zero within HTM space parameter.
The latter is consistent with theoretical constraints given in Eqs. (), namely unitarity, BFB as well as absence of
tachyon in the potential (inducing a constraint on parameter). It also complies with the observed Higgs at GeV and with
LHC measurements for Higgs decay to several channels (, , , , and ) aa12 ().
Here we only show results with the photon mode that has the best mass resolution. The second aim of our analysis is to gain more
insight on the masses of the heavy Higgs bosons and the effect of naturalness on their allowed range of variations.

We plot in Fig.1, the scatter plot in the (). The figure shows the excluded regions of space parameter
by unitarity (red), by combined set of BFB and unitarity (green). If in addition, we impose consistency withe ATLAS and CMS combined
data within on the diphoton decay mode, (with a signal strength combined15 ()), the blue area is
also discarded. Furthermore, when the Veltman condition for the doublet and triplet field enters into the game, we see a drastic
reduction of the space parameter to a relatively small allowed region marked in brown.

Fig.2 illustrates the doubly charged Higgs mass as a function of resulting from a scan over different values of and for . We find the only relevant parameter region where all of the constraints are imposed, is the area marked in grey which encodes cancellation of quadratic divergencies. This area
is delineated by in the range[] corresponding to varying from to GeV. For these
mass values, is limited to lie in a reduced interval between and .

The remarkable feature of effects of the modified Veltman condition on the doubly charged Higgs masse is clearly indicated in Fig.3 showing the ratio (signal strength) as a function of the doubly charged Higgs mass for different values of . First we consider the case when when mVC are absent (red plot). We see that, small values of less
than favour low varying between and GeV, while for larger , its range of variation is significantly enlarged with an upper limit about GeV. When mVC is turned on (green plot), we show that the intervals get smaller and smaller for getting larger. We also note that unlike the lower limit which is unaffected by mVC,
the upper bound is very sensitive to mVC and experiment a drastic reduction to GeV. Similar analysis is seen in Fig.4 for the simply charged Higgs mass. Here, the same behaviour is reproduced: the lower mass, GeV, is almost insensitive to the constraints including mVC, while the upper limit shows a substantial reduction from GeV to GeV when the conditions mVC are considered. These overall resulting ranges of and are compatible with the LHC exclusion limits on the charged Higgs bosons () .

## Vi Conclusion

We have studied the naturalness problem In the type II seesaw model. We have shown
that the Veltman condition is modified by virtue of the additional scalar charged states and that quadratic divergencies at one
loop can be driven to zero within at the allowed parameter space of the model. Furthermore, when a set of constraints, including unitarity, consistent boundedness from below, is combined with requirement of compatibility with the diphoton Higgs decay data of LHC for the observed Higgs at GeV, the resulting parameter region is severely constrained. If in addition, the modified Veltman condition is also imposed, the effects on the masses of heavy Higgs bosons , , and
are analysed. The analysis has shown a drastic reduction of the mass spectrum of and with an upper bounds at and GeV respectively. These limits are still consistent with LHC measurements . Besides, we have found an almost mass degeneracy of CP odd Higgs and heavy CP even Higgs, with . ^{3}

## Acknowledgements

M. Chabab acknowledges support from the grant . This work is supported in part by the Moroccan Ministry of Higher Education and Scientific Research under contract . M. C. Peyranere would like to thank Gilbert Moultaka for friendly and helpful discussions.

## Appendix : Feynman Rules

In this appendix, we give the couplings used to calculate the tadpoles of the two neutral CP-even Higgs and .
Since we are interested in the one-loop contributions, only three-leg couplings are useful. Further, within this
restricted class, we look for vertices such as or , where stands for any quantum field
of our model: scalar and vectorial bosons, fermions, Goldstone fields and Faddeev-Popov ghost fields .
To be precise, we have used the well-known linear gauge where the gauge fixing Lagrangians are
for the neutral sector and
for the charged sector.

We note () the couplings to the Higgs (). Since the field fixes the propagator, we also give the values () of the loop due to the propagator of the particle which gain a factor in case of charged fields, and the symmetry factor .

(27) |

(28) |

(30) |

(31) |

(32) |

(34) |

(35) |

(36) |

(37) |

(38) |

Unitarity | Unitarity + | |||

BFB | Unitarity + | |||

BFB + | ||||

Unitarity + | ||||

BFB + | ||||

+ | ||||

mVC | ||||

GeV | GeV | GeV | GeV | |

GeV | GeV | GeV | GeV | |

GeV | GeV | GeV | GeV | |

GeV | GeV | GeV | GeV |