Is Higgsium a possibility in 2HDMs?

We investigate the possibility of a Higgs-Higgs bound state in the two Higgs doublet model. Specifically we look for the effect of dimension six operators, generated by new physics at a scale of a few TeV, on the self-couplings of the heavy CP even scalar field in the model. Following the pioneering work of Grinstein and Trott [1], we construct an effective field theory formalism to examine the physics of the Higgs sector. The magnitudes of the attractive and repulsive coupling strengths are compared to estimate the possibility of the formation of the H − H bound state. Another way to check if a bound state is formed or not is from the formation and decay times of the bound state. The possibilities in various types of two Higgs doublet models have been discussed elaborately in the paper. ∗ ani73biswas@gmail.com 1 ar X iv :1 90 1. 05 32 5v 2 [ he pph ] 1 7 Ja n 20 19

In this paper we consider 2HDMs with a softly broken global U(1) symmetry [13][14][15][16][17][18][19], with the parameters so chosen as to make the 2HDM SM-like. An approximate custodial SU (2) C symmetry [20][21][22] has also been imposed on the SM Lagrangian density and its higher dimensional extension. This SU (2) C custodial symmetry must be respected by the total Lagrangian density. Also this custodial symmetry must be respected up to hypercharge and Yukawa coupling violations. Obviously there will be operators that break the custodial symmetry but in order to preserve this approximate custodial symmetry their coefficients are taken to be naturally suppressed.
The introduction of a second Higgs doublet and its higher dimensional extension modify the relation M W = M Z cos θ W , which is commonly parametrized by the ρ parameter. The ρ parameter is defined as ρ = This relationship is expected to be respected as precisely as possible. Since the PDG quotes ρ 0 = 1.00039 ± 0.00019 for the global fit [23] of precision electro-weak observables any physics beyond the Standard Model must keep the ρ parameter within these limits, when the particles of the new physics are integrated out. As we know in order to integrate out any particle from the theory it's mass must be sufficiently higher than the scale of the electroweak symmetry breaking (v ∼ 246 GeV) thus we have to choose the new physics scale well above the electroweak scale. Thus we choose the new scale above a scale of M ∼ 1TeV. At this high energy scale the quanta of the unknown new Physics may be integrated out and we are left with a low energy effective theory. This low energy manifestation is the 2HDM supplemented with non-renormalizable local operators, of dimension D > 4, which are constructed of 2HDM fields and obey the symmetry of the 2HDM. This approach is model independent, but the new physics is parametrized in terms of several arbitrary parameters and nothing is known a priori about these coefficients.
There are experimental constraints on the scale of M. The K 0 −K 0 mixing restrict M to be ≥ 10 4 TeV. Here the fact that flavour changing neutral currents are absent in nature has been taken into account. The Minimal Flavour Violation hypothesis [24][25][26][27][28][29][30][31][32][33] relaxes the bounds on M and restricts the higher dimensional operator basis. Thus a safe choice for the new physics scale would be a few TeV while naturally avoiding flavour changing neutral currents.
In this paper we address the question of whether a bound state of the CP-even heavy Higgs particle can form or not. The repulsive interaction of the quartic coupling and the attractive interaction determined by the cubic coupling compete to form the bound state.
For large enough coupling the exchange interaction is strong enough to produce binding.
We aim to find a necessary condition on the coupling for which a non relativistic (NR) bound state may form. For this purpose we follow the procedure adopted by Grinstein and Trott in [1] where they have formulated a non-relativistic effective theory for Higgs-Higgs interactions to study the Higgs-Higgs bound state of the SM Higgs. We borrow the name 'Higgsium' for the Higgs-Higgs bound state from their work. There the bound state of the SM Higgs particle was given this name, and it was found that this state was not likely to form for the light Higgs particle. In 2HDM the lighter CP-even scalar h is identified with the SM Higgs in the alignment limit. Here we attempt to find the bound state of H , which is the heavier CP-even scalar. Since h is identified with the SM Higgs particle, it is not likely to form a bound state, so we have adopted the same nomenclature for the H − H bound state. Another way to check if a bound state is formed or not is from the comparison of the formation and decay times of the bound state. The rest of the paper has been devoted to study the possibility of the formation of the bound state in various types of two Higgs doublet models. In the N/D method [34,35]  H . The N/D method had been studied to account for the bound state of two particles (not necessarily the Higgs particle) in [36,37]. It was found that for the Standard Model Higgs particle, bound states occur only if m H > 1.3 TeV. Since this is an order of magnitude higher than the observed mass of the Higgs particle, we have to conclude that the Higgs does not form a bound state with itself. Furthermore, even for such heavy Higgs bosons the binding was weak.
Another way to treat the problem of relativistic two-particle bound states in SM involves the variational method within the Hamiltonian formalism of quantum field theory [38][39][40].
This method can be extended to accommodate three-particle systems [41][42][43]. In principle, the variational method does not depend on the coupling strength, in contrast with perturbation theory which becomes increasingly suspect as the coupling becomes stronger. This fact is relevant since the Higgs self-coupling approaches a strong regime as the Higgs boson mass becomes large, rendering perturbation theory results questionable. It has been estimated that the Higgs boson mass at which perturbation theory ceases to act is approximately 700 GeV [2,44] whereupon the theory behaves like a strongly coupled one. Gunion and others treated the possibility of heavier Higgs to be discovered at the LHC and hence applied the variational method rather than the perturbative method.
Leo and Darewych in their work [38][39][40] found that two-Higgs bound states which they called "Higgsonium" would appear only for rather obese minimal Standard Model Higgs particles with mass m H > 894 GeV. This was quite similar to the 810 GeV estimate [45] which was obtained by using a phenomenological Yukawa potential to describe the Higgs-Higgs interaction.
In the approach of Bethe and Salpeter, which we mention for the sake of completeness [46], the relativistic S-matrix formalism of Feynman was applied to the bound-state problem for two interacting Fermi-Dirac particles. The bound state was described by a wave function depending on separate times for each of the two particles forming the bound state. Integral equations for this wave function were derived with kernels in the form of an expansion in powers of g 2 , the dimensionless coupling constant for the interaction. Each term in these expansions gave Lorentz-invariant equations. The validity and physical significance of these equations were discussed. In extreme non-relativistic approximation and to lowest order in g 2 they reduced to the appropriate Schrödinger equation.

II. FORMALISM OF THE TWO HIGGS DOUBLET MODEL
The Lagrangian density of the two Higgs doublet model containing two Higgs doublets φ 1 and φ 2 of hypercharge 1 2 is given by where the covariant derivative is σ I are the Pauli matrices and W I µ and B µ are SU(2) and U(1) gauge boson operators. We will work with the scalar potential [2,47] where the λ i are real parameters. To avoid flavor-changing neutral currents (FCNCs) [48,49], we impose an additional U(1) symmetry. The potential . Additional dimension four terms, including one allowed by a softly broken Z 2 symmetry [50] are also set to zero by this U(1) symmetry. One such term was λ 6 ( 1 We consider the new physics scale at M ∼ 1 TeV. Now the 2HDM Lagrangian density is supplemented with higher dimension operators. The effective Lagrangian density of this extended 2HDM can be written as where the dimension six operators that preserve the symmetries of the 2HDM (here U(1) symmetry) and custodial SU (2) C [51] in the Higgs sector are given by Appendix (A) discusses how the fields transform under custodial symmetry.
We expand the scalar fields about their vacuum expectation values v 1 and v 2 , Here h and H are the CP-even Higgs fields, with h(x) = 0 and H(x) = 0 , and The six fields ξ a i include the three Goldstone bosons which get eaten by the W ± and Z bosons to make them massive, while the other three combine to become the charged Higgs bosons and the CP-odd Higgs boson. The vevs v 1 and v 2 , and therefore v = v 2 1 + v 2 2 , will be taken to be small compared to M , the scale of new physics since v = 246 GeV 1 TeV. We note that degrees of freedom are easier to identify when the doublets are written in terms of real and imaginary parts of the complex scalar fields, but for our calculations in this paper it is more convenient to work in the unitarity gauge, in which the gauge transformation has been used to remove the Goldstone bosons from the Lagrangian.
The charged scalars and the CP-odd scalar will still remain in the Lagrangian, but we can neglect their contribution for the bound state calculations.
In order to normalize the kinetic term to have a coefficient of 1 2 , we redefine the fields as We write the potential in terms of the rescaled fields, focusing on the self couplings of the CP neutral Higgs fields. We call this potential V eff . Though we will be discussing the possibility of bound state formation of the heavy CP-even Higgs field, but still for the sake of completeness we will write the self couplings of the light CP-even Higgs field too. In terms of the rescaled fields, the terms in the effective potential which are of interest to us can be written as The mass terms and the coupling constants are related to the original λ i . Since we are interested in the bound state formation of H therefore we write down the cubic and quartic self couplings and also the mass of H in terms of the original λ i and evaluate their relative strengths.
The mixed cubic and quartic Higgs boson couplings corresponding to the coupling constants λ eff 14 , λ eff 15 and λ eff 16 are needed for the h − H bound state, but since the calculations for that are much more involved, we will leave the study of that bound state for another occasion. As regards the fermionic operators, since they are not needed in the theory for bound state formation we do not explicitly show them here.
We should mention here that for a single Higgs particle, the effective field theory may also be written as a nonlinear realization analogous to the σ and π fields of QCD as described by a chiral Lagrangian [1]. It is not obvious to us how to write a nonlinear realization involving neutral and charged Higgs fields along with the necessary Goldstone bosons, nor is it clear whether that will help in looking for bound states. So we will stick to the linear realization.

III. EFFECTIVE COUPLINGS AT LOW ENERGY
When we discuss physics at the low energy scale, the heavier momentum modes need to be integrated out from the theory. As the top Yukawa coupling is fairly large compared to the other fermions, we need to estimate the effects of the top quark on the possibility of a bound state of H. When the top quark mass is much heavier than the Higgs mass, the effect of integrating out the top quark shows up in the modified coupling constants and masses of the Higgs particle.
We will use this approximation even when the Higgs is slightly heavier than the top.
As mentioned in [1], for the case of a single Higgs field, the approximation is known to work better than one would expect when m h < 2m t . This is because of the absence of any non-analytic dependence on the mass since the Higgs is the pseudo-goldstone boson of spontaneously broken scale invariance [52][53][54]. However, when the mass of the Higgs particle is more than 2m t , we cannot integrate out the top quark. We will work in the Alignment limit and thus we must set m h = 125 GeV. The remaining heavier CP-even Higgs can have any mass above 125 GeV restricted by constraints coming from perturbative unitarity and stability. In [55] its mass was further restricted by use of Naturalness conditions, and the bounds were found to be 450 GeV m H 620 GeV for tan β =5. It is worth mentioning here that though these limits on m H are for Type -II 2HDM but the other types of 2HDMs also exhibit the mass ranges for H in the close vicinity of these limits. Moreover when these mass ranges were evaluated the most recent value of ρ -parameter was used [56].
In this paper we will usually work with these limits, but also consider the possibility that the heavier Higgs has a mass smaller than 2m t . We will not consider the situation where the heavier CP-even Higgs is identified with the Standard Model Higgs (Reverse Alignment limit), for reasons discussed in [57].
Let us also mention here our choice for the parameters used in the calculations. There is no bound on the value of tan β, which is perhaps the most important parameter in the 2HDMs, except that it should be larger than unity. This is based on constraints coming from Z → bb, B qBq mixing [58], muon g − 2 in lepton specific 2HDM [59] or using b → sγ in type I and flipped models [60]. Thus we take tan β =5 which is a reasonable choice, v = 246 GeV, m t = 174 GeV and the new physics scale M to be 1TeV. We broadly categorize the heavier Higgs boson mass as m H < 2m t and m H > 2m t . For m H < 2m t the top quark is integrated out while for m H > 2m t , the effect of top quark is retained in the theory.
Integrating out the top quark: m H 2m t The top mass term and couplings to the Higgs bosons are given by where ξ stands for the Yukawa coupling of H with the fermion indicated in the superscript.
The values of ξ for up-type and down-type quarks are displayed in Table I. Fig. 1 .

(3.2)
For leading order in p 2 /m 2 t → 0, the amplitude is given by The leading order term gives a factor of − where N c stands for the three colors of the top quark. Contributions of other quarks as Higgs-self couplings, We consider the non-relativistic Schrödinger equation to gain some idea about the bound state formation. It reads The above potential has contributions from a Yukawa exchange and a contact interaction, We have put ξ t H ≈ − cot β , which is its value for all types of 2HDMs in the alignment limit. Letting (C 3 φ 2 + 1 4 C 4 φ 2 ) ∼ 1, for v = 246 GeV, m t = 174 GeV, tan β = 5 and keeping the λ's well within the perturbative bounds by choosing λ i ∼ 1 , we have evaluated the strengths of the cubic and quartic couplings from Eqs. (4.5) and (4.6) for m H = 300 GeV. We have found that |λ eff 12 | − |λ eff 13 | = −0.02 , i.e. |λ eff 12 | ≈ |λ eff 13 | at the level of accuracy we are considering. We conclude that in this case of a not too heavy H , an H − H bound state may form, but it is also likely to have a very short lifetime.
Category II: m H ≥ 2m t Naturalness arguments coupled with unitarity, perturbativity and constraints from the T-parameter lead to a heavy H with a mass between 450 GeV and 620 GeV [55] . In this case we cannot integrate out the top quark. For the cubic and quartic couplings we find .
(4.8) For a non-relativistic bound state we can approximate the relative momenta of H by (4.9) The , (4.10) and Now we proceed in two ways. First we fix a value of tan β consistent with observations [58][59][60] and find the range of u H for which a bound state may form. Next we fix a non-relativistic value of u H and find the range for tan β.
Let us fix tan β = 5 and use Eq. (4.12) for various types of two Higgs doublet models in the alignment limit to find the range of u H . Next we fix u H = 0.01c and find the range of tan β. The results are displayed in Table III. In Natural System of Units we set c = 1 and thus we will refrain from writing c from now onwards. We see that when we fix tan β = 5 , The total decay width is then approximately Γ H = Γ H b + Γ H t where the expression for Γ H b is given in Eq. (4.10), and the decay time is the inverse of the total decay width, τ H = (Γ H ) −1 .
The range of the mass of H in the Alignment limit was derived in [55] as m H ∈ [450, 620] GeV. Here we estimate τ H for the two extreme values of m H in this range, namely m H =450 and 620 GeV for various types of 2HDMs in the alignment limit and for tan β = 5 . We also display the lower limits of the relative velocity using the logic that the formation time of the bound state must be shorter than the decay time of the parent particle if the bound state is to form. For all types of 2HDMs in alignment limit the relative Htt coupling is ξ t H = − cot β and as we have seen in the first case ξ b H is type dependent. The results are displayed in the Table IV. 2HDMs  GeV. For this very reason we have studied the bound state formation of the CP even nonstandard Higgs boson whose mass spectra is flexible. Though we have imposed Naturalness conditions and have restricted the mass of H within bounds, still these bounds are much higher on the mass scale and make the probability of bound state formation a possibility.
If the Naturalness criteria is withdrawn and the potential is only subjected to stability and perturbative unitarity constraints then m H is much more flexible and the entire spectrum can be studied for the possibility of the bound state formation.
There are many questions that we have not addressed. The immediate one is the mass of the bound state and its life time. Another important point is the detectability of the bound state. In future works we could attempt to address these questions.
In future we would like to study the possibility of formation of h-H bound state. Further since the λ i s get restricted when expressed in terms of the masses of the physical Higgs bosons of the two Higgs doublet model, we would like to study the variation of effective cubic (attractive) and quartic (repulsive) coupling strengths with λ i s maintaining the perturbative unitarity condition. We also intend to study the formation of bound states by elevating the restrictions imposed by Naturalness. It would also be interesting to solve the bound state equation in the more general, fully relativistic case.
In [62] the two Higgs doublet fields are given as Then φ i are also two SU (2) L doublets with components where φ − i = φ + i . The Higgs bi-doublet fields are given by The SU (2) L × U (1) Y gauge symmetry acts on the Higgs bi-doublets as The Lagrangian has the following global symmetry in the limit where hypercharge vanishes When the Higgs fields acquire their respective vacuum expectation values, both SU (2) L and SU (2) R are broken, however the subgroup SU (2) L=R is unbroken, i.e at φ 0 i = v i , one has When the vacuum expectation values are chosen to be real, v i = v i , Φ i is proportional to the 2 × 2 identity matrix and the vacuum preserves a group SU (2) V (the V stands for "vectorial") corresponding to the identical matrices SU (2) L=R i.e, This remaining group preserved by the vacuum is the custodial-symmetry group.