Masses of physical scalars in two Higgs doublet models
We find bounds on scalar masses resulting from a criterion of naturalness, in a broad class of two Higgs doublet models (2HDMs). Specifically, we assume the cancellation of quadratic divergences in what are called the type I, type II, lepton-specific and flipped 2HDMs, with an additional U(1) symmetry. This results in a set of relations among masses of the physical scalars and coupling constants, a generalization of the Veltman conditions of the Standard Model. Assuming that the lighter -even neutral Higgs particle is the observed scalar particle of mass 125 GeV, and imposing further the constraints from the electroweak T-parameter, stability, and perturbative unitarity, we calculate the range of the mass of each of the remaining physical scalars.
With the discovery of a 125 GeV neutral scalar boson Aad:2012tfa (); Chatrchyan:2012ufa (), the menagerie of fundamental particles in the Standard Model appears to be complete. Some questions still remain unanswered, including the origins of neutrino mass and dark matter, keeping the door open for physics beyond the Standard Model. Among the simplest extensions of the Standard Model are two Higgs doublet models (2HDMs) (for a recent review see Branco:2011iw ()). Originally motivated by supersymmetry, where a second Higgs doublet is essential, 2HDMs have also been studied in several other contexts. Peccei-Quinn symmetry Peccei:1977hh (); Peccei:1977ur () solves the strong CP problem, but must be spontaneously broken. The corresponding Goldstone boson is the axion, which can be a combination of the phases of two Higgs doublets. Models of baryogenesis often involve 2HDMs Turok:1990zg () because their mass spectrum can be adjusted to produce CP violation, both explicit and spontaneous. Another motivation, one that is important to us, is their use in models of dark matter Ma:2006km (); Ma:2008uza (); Barbieri:2006dq (). These models are the inert doublet models, so called because one of the Higgs doublets does not couple to the fermions. Of the 2HDMs we will consider, the Yukawa couplings of one model (type I) approach the inert doublet model for large values of the ratio of the vacuum expectation values (VEVs) of the two Higgs fields. The other models also have small couplings to one or more types of fermions in that limit.
In this paper we consider 2HDMs with a softly broken global U(1) symmetry Peccei:1977hh (); Ferreira:2009jb (), with the parameters chosen so as to make the 2HDM ‘SM-like’. We choose the fermion transformations under this U(1) symmetry, and impose a naturalness condition of vanishing quadratic divergences on the scalar sector of the models. Using additional restrictions coming from partial wave unitarity, vacuum stability, and the parameter measuring ‘new physics’, and assuming that the lighter CP-even Higgs particle in the 2HDMs is the one observed at the Large Hadron Collider (LHC), we find bounds on the masses of the additional scalar particles for each of the 2HDMs.
with real . This potential is invariant under the symmetry except for a soft breaking term Additional dimension-4 terms, including one allowed by a softly broken symmetry Gunion:1992hs () are also set to zero by this U(1) symmetry.
The scalar doublets are parametrized as
where the VEVs may be taken to be real and positive without any loss of generality. Three of these fields get “eaten” by the and gauge bosons; the remaining five are physical scalar (Higgs) fields. There is a pair of charged scalars denoted by , two neutral CP even scalars and , and one CP odd pseudoscalar denoted by . With
these fields are given by the combinations
If we rotated fields by the angle ,
Thus in order for to be the Higgs boson of the Standard Model, we require which has been called the SM-like or alignment limit Ferreira:2014naa (). Accordingly, we will assume in the rest of this paper.
Ii Veltman Conditions
The scalar masses get quadratically divergent contributions which require a fine-tuning of parameters. We thus impose naturalness conditions, a generalization of the Veltman conditions for the Standard Model, that these contributions cancel Veltman:1980mj (). The resulting masses and couplings should not then require fine-tuning.
The Yukawa potential for the 2HDMs is of the form
where are 3-vectors of isodoublets in the space of generations, are 3-vectors of singlets, etc. are complex matrices in generation space containing the Yukawa coupling constants, and
Cancellation of quadratic divergences in the scalar masses gives rise to four mass relations, which we may call the Veltman conditions for the 2HDMs being considered Newton:1993xc (),
where are the and coupling constants. A fourth equation is the complex conjugate of the third one. As we will see below, the last equation vanish identically for all the 2HDMs we consider. The mass relations come from the first two equations above.
When the fermions are in mass eigenstates, the Yukawa matrices are automatically diagonal if there is only one Higgs doublet as in the Standard Model, so there is no FCNC at the tree level. But in the presence of a second scalar doublet, the two Yukawa matrices will not be simultaneously diagonalizable in general, and thus the Yukawa couplings will not be flavor diagonal. Neutral Higgs scalars will mediate FCNCs. The necessary and sufficient condition for the absence of FCNC at tree level is that all fermions of a given charge and helicity transform according to the same irreducible representation of SU(2), corresponding to the same eigenvalue of and that a basis exists in which they receive their contributions in the mass matrix from a single source Glashow:1976nt (); Paschos:1976ay ().
For the fermions of the Standard Model, this theorem implies that all right-handed singlets of a given charge must couple to the same Higgs doublet. We will ensure this using the global U(1) symmetry mentioned earlier, which generalizes a symmetry more commonly employed for this purpose. The left handed fermion doublets remain unchanged under this symmetry, The transformations of right handed fermion singlets determine the type of 2HDM. There are four such possibilities, which may be identified by the right-handed fields which transform under the U(1): type I (none), type II () , lepton specific () , flipped () . We note in passing that another way of avoiding FCNCs at tree level is by aligning the Yukawa and mass matrices in flavor space Pich:2009sp (). However, only these four 2HDMs admit symmetries such as the U(1) Ferreira:2010xe ().
The fermion mass matrix is diagonalized by independent unitary transformations on the left and right-handed fermion fields. In any of the 2HDMs, either or vanish for each fermion type For example, in the Type II model couples to down-type quarks and charged leptons, while couples to up-type quarks, so Thus Eq. (12) is automatically satisfied in each 2HDM. The non-vanishing Yukawa matrices are related to the fermion masses by Newton:1993xc ()
where stands for charged leptons, up-type quarks, or down-type quarks, and the sum is taken over generations.
In order to rewrite the Veltman conditions in terms of the known masses, we first note that in the alignment limit and with the global U(1) symmetry, the independent parameters in the scalar potential may be taken to be the masses the angle the electroweak VEV and the constant The are related to these parameters by Akeroyd:2000wc ()
Inserting Eq. (13) — Eq. (18) into Eq. (10) and Eq. (11), we get the Veltman conditions in terms of the physical particle masses. These are shown in Table 1. The Yukawa matrices which vanish in each model are listed in the second column. We note here that although naturalness conditions in specific 2HDMs have been studied earlier on a few occasions Grzadkowski:2009iz (); Jora:2013opa (), they were not done in the SM-like scenario, nor expressed in terms of the physical masses for the different types as in here.
Iii Bounds on the masses of heavy and charged scalars
We now display our main results, the bounds we have obtained for the masses of the heavy and charged Higgs particles. We will assume that the particle is the one that has been observed at the LHC, so that GeV, and GeV. Let us consider the example of the type II model to explain our derivation of the bounds.
Since we want the bounds on and let us rewrite VC1 and VC2 for the type II model in a convenient form,
On the right hand side of either equation, all but the last term are experimentally known. The symmetry implies that and we impose the restriction of based on the validity of perturbativity. Comparing with Eq. (18), we see that this last puts a restriction GeV.
For a fixed value of we plot both equations on the plane for various values of The point where the two curves cross for a given value of , is an allowed value of the pair
We can restrict the allowed range of the masses even further by imposing constraints coming from stability, perturbative unitarity, and the oblique electroweak -parameter. Conditions for stability, i.e. for the scalar potential being bounded from below, were examined in Sher:1988mj (); Gunion:2002zf (); Branco:2011iw (), and found to provide lower bounds on certain combinations of the quartic couplings On the other hand, the requirement of perturbative unitarity translates into upper limits on combinations of the , which for two-Higgs models have been derived by many authors Maalampi:1991fb (); Kanemura:1993hm (); Akeroyd:2000wc (); Horejsi:2005da (). One condition coming from perturbative unitarity is
Stability provides the inequalities
so that we can write Eq. (21) as with It then follows that
In terms of the scalar masses, this reads
For , this inequality implies that and are almost degenerate, a result also found in Bhattacharyya:2013rya (). In Fig. 1 we have shown this degeneracy by plotting against for different values of It is easy to see from the plots that the degeneracy is more pronounced at higher values of for any value of For these plots we have used the perturbativity condition which restricts 617 GeV.
We will also need another inequality which follows from the condition
The parameter is constrained by the global fit to precision electroweak data to be Baak:2013ppa ()
Our results consist of the pairs for each type of 2HDM, satisfying the two Veltman conditions, and consistent with the constraints from stability, tree-level unitarity and the parameter. For we have plotted the curves corresponding to VC1 and VC2 for several values of These have been superimposed on the bound determined by (24), (26), and (29). The resulting plot is shown in Fig. 2. VC1 produces ellipses, and VC2 gives a narrow band of hyperbolae. Their crossings which fall inside the band representing the bound from the inequalities are the allowed masses. From the plot we can estimate the individual bounds: for all four models, we find approximately 550 GeV 700 GeV, and about 450 GeV 620 GeV, with a higher implying a higher As mentioned earlier, is close to as a result of (24). We also note that direct searches have put a rough lower bound of GeV Beringer:1900zz ().
Some comments are in order for the values of some parameters that we have used in this analysis. We chose so that the 2HDMs are in the alignment limit, in which the lighter CP-even scalar has the couplings of the Higgs particle of the Standard Model. We note that in the decoupling limit Gunion:2002zf () defined by subject to a condition of perturbativity , we also find . (The relation between these and ours may be found in Gunion:2002zf ().) Although we find from our computations in this paper that must be large, we do not require it a priori, so our results are valid for the SM-like alignment limit of the 2HDMs, without going to the decoupling limit. It is worth pointing out that the issue of distinguishing between the decoupling limit and the SM-like scenario was first explored in Ginzburg:2004vp ().
Perturbativity requires that the quartic couplings of the physical Higgs fields are small. Our choice of keeps the models inside the perturbative regime, and this requirement also keeps 617 GeV. Allowing for larger values of would also allow higher values of as well as of and . In that sense, what we have found in this paper are the lower bounds on the masses of and in the SM-like limit of 2HDMs.
The most important parameter in the 2HDMs is There is no consensus on the value of except that it should be larger than unity, based on constraints coming from and mixing Arhrib:2009hc (). Several arguments have been proffered for a large in 2HDMs of different types, using muon in lepton specific 2HDM Cao:2009as (), or using in type I and flipped models Park:2006gk (), which also suppresses the branching ratio to a rough agreement with 95 CL limits from the light charged Higgs searches at the LHC Aad:2012tj (); Chatrchyan:2012vca (). A large value of also makes the heavy Higgs particle difficult to detect Randall:2007as (). We have used a conservative to estimate the scalar masses and — note that is not very far from because of the degeneracy relation (24). A larger makes the degeneracy more pronounced, so the inequality band becomes narrower. This narrows the ranges of and also pushing the region of overlap upwards, making the heavy and charged Higgses more difficult to detect. Recent analyses of LHC data at TeV, as a search for the pseudoscalar Higgs particle, also appear to favor a value of 5 or larger for near the alignment limit Aad:2015wra (); Khachatryan:2015lba () .
Acknowledgements.We thank B. Grzadkowski, B. Swiezewska and X.-F. Han for useful comments, and the anonymous referee for informing us about Ref. Ginzburg:2004vp (). AB wishes to thank D. Das for useful discussions.
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