Indirect bounds on heavy scalar masses of the two-Higgs-doublet model in light of recent Higgs boson searches

We study an upper bound on masses of additional scalar bosons from the electroweak precision data and theoretical constraints such as perturbative unitarity and vacuum stability in the two Higgs doublet model taking account of recent Higgs boson search results. If the mass of the Standard-Model-like Higgs boson is rather heavy and is outside the allowed region by the electroweak precision data, such a discrepancy should be compensated by contributions from the additional scalar bosons. We show the upper bound on masses of the additional scalar bosons to be about 2 $(1)$ TeV for the mass of the Standard-Model-like Higgs boson to be 240 $(500)$ GeV.

The Standard Model (SM) for elementary particles based on the SU (3) C × SU (2) L × U (1) Y gauge group has been tested accurately [1]. However, no Higgs boson has been confirmed yet.

Discovery of the Higgs boson is the most important issue at the Fermilab Tevatron and the CERN
Large Hadron Collider (LHC). Direct searches for the Higgs boson at CERN LEP have set a lower mass bound on the SM Higgs boson to be 114.4 GeV [2]. The Tevatron experiment has excluded the mass of the SM Higgs boson around 160 GeV [3]. Recently, the first results from the ATLAS and CMS experiments at the LHC have been reported [4,5]. The Higgs boson mass around 160 GeV and 300-450 GeV has been excluded by the data with the integrated luminosity of about 1 fb −1 .
It is well known that an upper bound on the mass of the Higgs boson is obtained by the tree level unitarity for elastic scattering processes of longitudinally-polarized vector bosons, such as In the SM, since the scattering amplitudes are proportional to the Higgs boson mass in the high energy limit, a large Higgs boson mass leads to a violation of the unitarity.
Consequently the upper bound is obtained on the mass as about 710 GeV [6,7]. On the other hand, if the Higgs boson is absent, the scattering amplitudes grow for high energies. The violation of the tree level unitarity then occurs at √ s ∼ 1.2 TeV, where √ s is the centre-of-mass energy of the W W scattering [8]. The  From the electroweak precision data with the theoretical study for radiative corrections [12], the mass of the Higgs boson in the SM is indicated to be m h = 90 +27 −22 GeV and m h < 161 GeV at the 95% Confidence Level (CL) [13]. Notice that this indirect bound on the mass cannot be applied if new physics exists below the TeV scale and affects the calculation for the radiative correction. In such a case, even if the Higgs boson is heavy, the electroweak precision data can be satisfied by the contribution from the new physics.
In the two Higgs doublet model (THDM), radiative corrections to the electroweak observables have already been calculated, and the possible allowed regions for the parameter space are evaluated under the electroweak precision data [14,15] and theoretical constraints [16][17][18][19][20]. Flavor physics data such as b → sγ [21,22], B → τ ν [23] and tau leptonic decays [24,25] in the THDM can further constrain the parameter space depending on types of Yukawa interactions. In particular, the mass of charged Higgs bosons is bounded from the b → sγ data to be greater than 295 GeV [26] by assuming the Type-II Yukawa interaction.
In this Letter, in light of recent Higgs boson searches, we reanalyze the constraint on the parameters in the THDM by using the electroweak precision data and the theoretical constraints from tree level unitarity and vacuum stability. In particular, we show an upper bound on the masses of the additional heavy scalar bosons can be obtained in the THDM depending on the mass of the SM-like Higgs boson. For a relatively large mass of the SM-like Higgs boson, a relatively large mass difference between the CP-odd Higgs boson and the charged Higgs boson is required in order to satisfy bounds from the electroweak precision data. They are bounded from above by the theoretical constraints. For a SM Higgs boson mass to be 240 (500) GeV, an upper bound on the mass of the CP-odd Higgs boson of about 2 (1) TeV is obtained.
The most general THDM with the doublet fields Φ 1 and Φ 2 are constrained by flavor changing neutral current (FCNC) processes. We here consider the model with the softly-broken discrete Z 2 symmetry under Φ 1 → Φ 1 and Φ 2 → −Φ 2 to avoid FCNC constraints [27]. There are four kind of Yukawa interaction under the discrete symmetry [21]. In this Letter, we do not specify the type of the Yukawa interaction because it does not affect the following discussions. The Higgs potential is then given by The soft-breaking mass parameter m 2 3 and the coupling constant λ 5 are complex in general. We here take them to be real assuming that CP is conserved in the Higgs sector.
The Higgs doublets Φ i (i = 1, 2) can be written in terms of the component fields as where the vacuum expectation values (VEVs) v 1 and v 2 satisfy v 2 1 + v 2 2 = v ≃ 246 GeV. The mass eigenstates are obtained by rotating the component fields as  where ω ± and z are the Nambu-Goldstone bosons, h, H, A and H ± are respectively two CP-even, one CP-odd and charged Higgs bosons, and The eight parameters m 2 1 -m 2 3 and λ 1 -λ 5 are replaced by the VEV v, the mixing angles α and β(= tan −1 v 2 v 1 ), the Higgs boson masses m h , m H , m A and m H ± , and the soft Z 2 breaking parameter M 2 = m 2 3 /(cos β sin β). In particular, the quartic coupling constants are expressed in terms of physical Higgs boson masses, mixing angles and the soft Z 2 breaking mass parameter M 2 as The coupling constants of the CP-even Higgs bosons with the weak boson hW W and HW W are proportional to sin(β − α) and cos(β − α). When sin(β − α) = 1, only h couples to the gauge bosons and behaves as the SM Higgs boson. We call this limit as the SM-like limit.
As discussed in Ref. [28], the masses of the heavy Higgs bosons (H, A and H ± ) are expressed while the mass of h is the SM-like form ∼ λ i v 2 . When M 2 ≫ λ i v 2 the heavier Higgs bosons have the common mass ∼ M . In this case, the effect of these bosons decouples in the large mass limit and the low energy theory becomes the SM with h being at the electroweak scale as the SM Higgs boson. On the contrary, when M 2 ∼ λ i v 2 the effect of these bosons does not decouple, and so-called nondecoupling effects appear in the low energy observables. Notice that the mass difference between the heavy Higgs bosons is independent of M , so that the effect on the low energy observables can be large if the mass differences are not small. We also note that the mass difference between heavy Higgs bosons are related to the violation of the custodial SU (2) symmetry [29], which causes significant deviation in the electroweak rho parameter from the SM prediction in the positive direction. As we see soon below, this positive contribution to the rho parameter (or the T parameter [12]) can be used to compensate the negative contribution of the heavy SM-like Higgs boson.
New physics effects on the electroweak oblique parameters are parameterized by the S, T and U parameters [12]. By fixing U = 0, the central values of S and T are given by [1] S = 0.03 ± 0.09, T = 0.07 ± 0.08, (ρ ST = 0.87), In the THDM, the contributions to the electroweak parameters from the scalar boson loops are given by [14] S where For the case with m 0 ≈ m 1 , we have When all the additional heavy scalar bosons are degenerate m A = m H = m H ± , we obtain S Φ = T Φ = 0. In the SM-like limit sin(β − α) = 1 with the further assumption m H = m A , we have In understood by the approximate formula for m A ∼ m H ± in Eq. (19). Therefore, the deviation of the T parameter is insensitive to M .
In the SM, the mass of the Higgs boson is constrained due to tree level unitarity. It has been studied by considering 6×6 scattering matrix of two body scalar states (hh, hz, zz, ω + ω − , hω + , zω + ) where each eigenvalues of scattering matrices are restricted to be less than a criteria ξ as |a 0 | ≤ ξ [6] where a 0 is the S wave amplitude matrix. For ξ = 1/2, the Higgs boson mass is bounded to be less than about 710 GeV. In the THDM, there are 14 neutral [16], 8 singly charged and a doubly charged two body states [17]. In our numerical analysis, absolute values of all eigenvalues for the S wave amplitude matrix are required to be less than 1/2 as for a criteria to keep perturbativity [30].
For the constraint from vacuum stability, the Higgs potential is required to be positive for a large value of the order parameter. In the SM, this condition is expressed by λ > 0 at the tree level. In the THDM, the condition for vacuum stability is given by [18][19][20] In unitarity and stability bounds do not depend on tan β in the same limit. Regions excluded by the conditions from tree level unitarity and vacuum stability are shown as the green and yellow areas, respectively, while that excluded by the electroweak precision data at the 90% (68%) CL is indicated by the blue (light-blue) area. Although the bounds from perturbative unitarity, vacuum stability and the electroweak precision data with the oblique corrections do not depend on the type of Yukawa interaction, the direct search results for the charged Higgs boson depends on that via the decay process. The region with charged Higgs boson mass below 79.3 GeV [1] is shown as the gray area, which is excluded assuming B(H + → τ + ν) + B(H + → cs)=1. Depending on the type of Yukawa interaction [21,31], we may have additional constraints from the flavor physics data analyses such as B s → X s γ [21,22,26], B + → τ + ν [23] and τ → ℓνν(ℓ = e, µ) [24,25]. We do not consider these constraints in FIG. 2  Finally, we shortly discuss the implication to the collider phenomenology. At the LHC, a heavy SM-like Higgs boson can be found via the gluon fusion gg → h or vector boson fusion V V → h (V = W and Z) with the decays into W W ( * ) and ZZ ( * ) [32]. The additional neutral scalar bosons φ (= H and A) would be produced via gluon fusion gg → φ [33], associated production with heavy quarks pp → ttφ, bbφ [34], pair production pp → W ± * → φH ± [31,35,36] and pp → Z * → AH [31,36], charged Higgs boson production would be via gb → H ± t [37], pp → W ± H ∓ [38] and pp → H + H − [39]. As we discussed above, if the SM-like Higgs boson is heavy, a large mass splitting between additional heavy Higgs bosons H ± and A (or H) is required. In such a case, their decays into a lighter scalar boson associated with a weak gauge boson H ± → φW ± (or φ → H ± W ∓ ) can be significant [40]. These decay modes are kinematically suppressed by the degeneracy of scalar boson masses in most of previous discussions, for example, on Higgs boson decays in the minimal supersymmetric SM [41]. In addition, the one-loop induced decay process H ± → W ± Z can also be enhanced when the mass difference between A and H ± is large [42]. The bosonic decay branching fractions of scalar bosons can dominate over their fermionic decay modes.
Therefore, detailed studies for these decay modes will be important to test the scenario with large mass splitting between additional heavy Higgs bosons [43].
In conclusion, we have analyzed theoretical bounds and experimental constraints in the THDM. For a given Higgs boson mass, the magnitude of the mass difference between additional heavy scalar bosons can be determined to satisfy the electroweak precision data. However, the mass difference requires a large coupling constant in the Higgs potential, and too large coupling constant violates tree level unitarity. Therefore, we have found that an upper bound on the additional heavy Higgs bosons is obtained when the SM Higgs boson is heavy. For example, m A is bounded to be less than around 2 (1)