Limits on the charged Higgs parameters in the two Higgs doublet model using CMS $\sqrt{s}=13$ TeV results

Abstract

The latest CMS results on the upper limits on $\sigma _{H^\pm }$BR($H^\pm \rightarrow \tau ^\pm \nu )$ and $\sigma _{H^\pm }$BR($H^+ \rightarrow t{\bar{b}}$) for $\sqrt{s}=13$ TeV at an integrated luminosity of 35.9 $\hbox {fb}^{-1}$ are used to impose constraints on the charged Higgs $H^\pm$ parameters within the Two Higgs Doublet Model (2HDM). The 2HDM is the simplest extension of the Standard Model (SM) under the same gauge symmetry to contain charged Higgs and is relatively little constrained compared to the Minimal Supersymmetric Standard Model (MSSM). The latest results lead to much more stringent constraints on the charged Higgs parameter space than for the earlier 8 TeV results. The CMS collaboration also studied the exotic bosonic decays $H^\pm \rightarrow W^\pm A$ and $A \rightarrow \mu ^+ \mu ^-$ for the first time and put upper limits on the BR($t\rightarrow H^+ b$) for the light charged Higgs boson. These constraints lead to the exclusion of parameter space which is not excluded by the $\tau \nu$ channel. For comparison the exclusion regions from flavor physics constraints are also discussed.

1 Introduction

The Standard Model (SM) of particle physics is the most successful model in explaining nearly all particle physics phenomenology. The discovery of a neutral scalar of mass 125 GeV with properties similar to the Higgs boson in SM [1,2,3,4] makes SM the most acceptable model of particle physics. Despite being successful, the SM fails to explain the existence of dark matter, neutrino oscillations and the matter–antimatter asymmetry. SM also does not explain the mass hierarchy in elementary particles and gravity is not included. Apart from that, there is no fundamental reason to have only one Higgs doublet (i.e. minimal under the SM gauge symmetry) and the discovery of another scalar boson (neutral or charged) would require an extension of the SM. The simplest extension of SM under the same (SM) gauge symmetry is the Two Higgs Doublet Model (2HDM) [5,6,7,8,9,10,11]. So far there is no evidence for any other scalar up to a mass of a few TeV and hence the parameter space of 2HDM is getting significantly constrained by experimental observations [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]. The scalar sector of 2HDM consists of five scalars, two CP even scalars (h and H), one CP odd scalar (or pseudoscalar) A and two charged Higgs $H^{\pm }$. The most general Yukawa sector (Type III) of 2HDM leads to flavor changing neutral currents (FCNCs) at tree level. To avoid the FCNC, Glashow and Weinberg implemented a discrete symmetry in the Yukawa sector which leads to four possible types of Yukawa interactions in 2HDM, i.e., Type I, Type II, Type X (lepton specific) and Type Y (flipped model). A brief review on 2HDM is given in Sect. 2.

The production of a charged Higgs particle, depending on its mass with respect to the top quark, can be divided into light ($M_{H^\pm }\ll M_t$), intermediate ($M_{H^\pm }\sim M_t$) and heavy ($M_{H^\pm }\gg M_t$) scenarios [31,32,33,34]. Throughout the analysis the alignment limit is considered i.e. $\sin (\beta -\alpha )\rightarrow 1$ (where the mixing angles $\beta$ and $\alpha$ are defined in Sect. 2), so that the neutral scalar h behaves like the SM Higgs boson. The precisely measured electroweak parameter T is highly sensitive on the mass splitting of $H^\pm , H$ and A. The alignment limit and minimum mass splitting restricts the charged Higgs decay mostly into the fermionic sector and the experimental constraints put an exclusion bound on the charged Higgs ($M_{H^\pm }$–$\tan \beta$) parameter space. In this paper the 13 TeV CMS results [35,36,37] at an integrated luminosity of 35.9 $\hbox {fb}^{-1}$ are used to restrict the charged Higgs parameter space as discussed in Sect. 4. Throughout the paper, the notation $H^\pm \rightarrow \tau ^\pm \nu$ is used for both $H^+\rightarrow \tau ^+ \nu$ and $H^- \rightarrow \tau ^- {\bar{\nu }}$ (similarly for the tb channel). For the charged Higgs production cross section, $\sigma _{H^\pm }$ denotes the sum of $\sigma _{H^+}$ and $\sigma _{H^-}$. A comparison of exclusion limits on charged Higgs parameter space from 13 TeV and 8 TeV CMS results is presented in Sect. 4 along with the indirect flavor physics constraint coming from $B\rightarrow X_s \gamma$. As mentioned before, the bosonic decays of a charged Higgs boson into $W^\pm h$, $W^\pm H$ and $W^\pm A$ are highly suppressed due to the alignment limit and the limited phase space. But once the bosonic decay channel is open, the bounds coming from the $H^\pm \rightarrow \tau ^\pm \nu$ and $H^+ \rightarrow t{\bar{b}}$ become weak [38, 39]. In this paper, the latest result from the CMS collaboration [37] is used in the Type I scenario in the mass range $M_{H^\pm }\in [100,160]$ GeV, where the mass splitting $M_{H^\pm }-M_A=85$ GeV and $M_{H^\pm }\sim M_H$ is still allowed by the T parameter constraint.

2 Two Higgs doublet model (2HDM) review

For 2HDM, the most general scalar potential [7] is

$$\begin{aligned}&{\mathcal {V}}(\Phi _1,\Phi _2)\nonumber \\&\quad =m_{11}^2\Phi _1^{\dagger }\Phi _1 + m_{22}^2\Phi _2^{\dagger }\Phi _2 - [m^2_{12}\Phi _1^\dagger \Phi _2 +{\text {h.c.}}] \nonumber \\&\qquad +\frac{1}{2}\lambda _1(\Phi _1^\dagger \Phi _1)^2 + \frac{1}{2}\lambda _2(\Phi _2^\dagger \Phi _2)^2 \nonumber \\&\qquad +\lambda _3(\Phi _1^\dagger \Phi _1)(\Phi _2^\dagger \Phi _2) +\lambda _4 (\Phi _1^\dagger \Phi _2)(\Phi _2^\dagger \Phi _1)\nonumber \\&\qquad +\left[ \frac{\lambda _5}{2}(\Phi _1^\dagger \Phi _2)^2 + h.c. \right] \end{aligned}$$

(1)

where $\Phi _{1,2}$ are two isospin doublets with hypercharge $Y=1/2$. To avoid tree level FCNC the ${\mathcal {Z}}_2$ symmetry is imposed under which $\Phi _1\rightarrow \Phi _1$ and $\Phi _2\rightarrow -\Phi _2$. This symmetry is softly broken by the parameter $m_{12}\ne 0$. The parameters $m_{12}$ and $\lambda _5$ are considered real assuming CP invariance. The two Higgs doublets are parameterized as

$$\begin{aligned} \Phi _i = \left( \begin{array}{c}\phi ^+_i\\ \frac{v_i +\rho _i+i\eta _i}{\sqrt{2}} \end{array}\right) \end{aligned}$$

(2)

where $\langle \rho _1 \rangle =v_1$, $\langle \rho _2 \rangle =v_2$, $\tan \beta =v_2/v_1$ and $v=\sqrt{v_1^2 + v_2^2}\approx 246$ GeV.

The physical mass eigenstates are given by

$$\begin{aligned} \left( \begin{array}{c} G^\pm \\ H^\pm \\ \end{array}\right)= & {} R(\beta )\left( \begin{array}{c} \phi _1^\pm \\ \phi _2^\pm \\ \end{array}\right) , \nonumber \\ \left( \begin{array}{c} G\\ A\\ \end{array}\right)= & {} R(\beta )\left( \begin{array}{c} \eta _1\\ \eta _2\\ \end{array}\right) ,\nonumber \\ \left( \begin{array}{c} H\\ h\\ \end{array}\right)= & {} R(\alpha )\left( \begin{array}{c} \rho _1\\ \rho _2\\ \end{array}\right) . \end{aligned}$$

(3)

Here $G^{\pm }$ and G are the Nambu–Goldstone bosons that are eaten as the longitudinal components of the massive gauge bosons. The rotation matrix is given by

$$\begin{aligned} R(\theta )= \left( \begin{array}{cc} \cos \theta &{} \sin \theta \\ -\sin \theta &{} \cos \theta \\ \end{array}\right) . \end{aligned}$$

(4)

Minimization of the scalar potential in Eq. (1) gives

$$\begin{aligned} m_{11}^2= & {} m_{12}^2\tan \beta -\frac{v_1^2 \lambda _1}{2} - \frac{v_2^2 \lambda _{345}}{2} ,\nonumber \\ m_{22}^2= & {} m_{12}^2\tan ^{-1}\beta -\frac{v_2^2 \lambda _2}{2} - \frac{v_1^2 \lambda _{345}}{2}, \end{aligned}$$

(5)

where $\lambda _{345} = \lambda _3 + \lambda _4 + \lambda _5$. The parameters $\lambda _{i}$ written in terms of the physical parameters $M_{H^\pm }, M_A, M_H, M_h, m_{12}$ and the mixing angles $\alpha$ and $\beta$ $(\tan \beta =v_2/v_1)$ are

$$\begin{aligned} \lambda _1= & {} \frac{M_{H}^2 R(\alpha )^2_{11}v_1+M_{h}^2 R(\alpha )^2_{12}v_1-m_{12}^2v_2}{v_1^3} ,\nonumber \\ \lambda _2= & {} \frac{M_{H}^2 R(\alpha )^2_{21}v_2+M_{h}^2 R(\alpha )^2_{22}v_2-m_{12}^2v_1}{v_2^3} ,\nonumber \\ \lambda _3= & {} \frac{M_H^2 R(\alpha )_{11}R(\alpha )_{21}+M_h^2 R(\alpha )_{12}R(\alpha )_{22}}{v_1 v_2} \nonumber \\&+\frac{m_{12}^2 -v_1 v_2\lambda _{45}}{v_1 v_2} ,\nonumber \\ \lambda _4= & {} \frac{m_{12}^2}{v_1 v_2}+\frac{M_A^2}{v^2}-\frac{2M_{H^\pm }^2}{v^2}, \nonumber \\ \lambda _5= & {} \frac{m_{12}^2}{v_1 v_2}-\frac{M_A^2}{v^2}, \end{aligned}$$

(6)

where $\lambda _{45}=\lambda _4 + \lambda _5$.

The most general Yukawa Lagrangian under the ${\mathcal {Z}}_2$ symmetry is

$$\begin{aligned} {\mathcal {L}}^{\text {2HDM}}_{\text {Yukawa}}= & {} -{\bar{Q}}_LY_u{\tilde{\Phi }}_u u_R-{\bar{Q}}_LY_d\Phi _d d_R \nonumber \\&-{\bar{L}}Y_l \Phi _l l_R + h.c. \end{aligned}$$

(7)

where $\Phi _f (f=u,d$ or l) is either $\Phi _1$ or $\Phi _2$, depending on the Yukawa models of 2HDM. The four possible ${\mathcal {Z}}_2$ charge assignments of the quarks and charged leptons are summarized in Table 1.

Table 1 Charge assignment under ${\mathcal {Z}}_2$ symmetry to avoid FCNC at tree level

In Type I 2HDM, the second Higgs doublet $\Phi _2$ couples to the fermions, so all the quarks and charged leptons get their masses from the VEV of $\Phi _2$ (ie. $v_2$). In Type II 2HDM, up-type quarks couple to $\Phi _2$ whereas down-type quarks and charged leptons couple to $\Phi _1$. Hence in Type II up-type quarks get masses from $v_2$ and down-type quarks and charged leptons get masses from $v_1$. The Higgs sector of the Minimal Supersymmetric Standard Model (MSSM) is a special 2HDM whose Yukawa interaction is of Type II. For Type X (also called the Lepton Specific Model), the quark sector is similar to Type I but the charged leptons are coupled to $\Phi _1$ and finally in the Type Y (also called the Flipped Model) the quark sector is similar to Type II but the leptons are coupled to $\Phi _2$. Among them, Type II 2HDM has been most widely investigated because of its resemblance with MSSM.

The Yukawa interactions of $H^\pm$ with quarks and leptons take the form

$$\begin{aligned} {\mathcal {L}}_Y= & {} -\frac{\sqrt{2}}{v}H^+{\bar{u}}[\xi _d V M_d P_R - \xi _u M_u V P_L]d \nonumber \\&- \frac{\sqrt{2}}{v} H^+ \xi _l{\bar{\nu }}M_l P_R l \end{aligned}$$

(8)

where V is the CKM matrix and $P_{R,L}=\frac{1}{2}(1\pm \gamma _5)$ are the chirality projection operators (Table 2).

Table 2 Choices of the couplings $\xi _f$ for the four Yukawa models of 2HDM

3 $H^{\pm }$ production and decay channels

The production cross section of the charged Higgs particle depends on its mass with respect to top quark and can be classified into three categories. The light charged Higgs scenario is defined by the mass of charged Higgs being light enough ($M_{H^\pm }\lesssim 150$ GeV) such that the on-shell decay of the top quark, $t\rightarrow H^+b$, is allowed. The production cross section for the light scenario is simply given by the product of top pair production (double-resonant mode), $p p \rightarrow t {\bar{t}}$, times the branching fraction of the top into charged Higgs, $t\rightarrow H^+b$. The $p p \rightarrow t {\bar{t}}$ cross section has been computed at NNLO in QCD including resummation of NNLL soft gluon terms using the code Top$++2.0$ [40]. The heavy charged Higgs scenario is defined for $M_{H^\pm } > rsim 200$ GeV where the charged Higgs mass is sufficiently large compared to top quark. In this scenario, the dominant charged Higgs production channel is the associated production with a top quark (single-resonant mode) $pp \rightarrow tbH^{\pm }$. The production cross section for the heavy charged Higgs boson is computed in the 4FS and 5FS schemes in Refs. [32, 33], and combined to obtain the total cross section using the Santander matching scheme [41] for different values of $\tan \beta$.¹ The intermediate charged Higgs scenario is considered for $M_{H^\pm }$ close to top quark i.e. $150\lesssim M_{H^\pm }\lesssim 200$ GeV. In this region, the non-resonant top quark production mode also contributes along with the single-resonant and double-resonant modes. Cross sections at NLO QCD accuracy in the 4FS scheme as given in Ref. [31] are considered. Figure 1 shows the leading order (LO) diagrams of charged Higgs production in the three scenarios.² Since the $H^\pm$ interaction to the quark sector of Type I and Type X and similarly that for Type II and Type Y are the same, the production cross sections in different models are related by $\sigma ^{H^\pm }_{\text {Type I}}=\sigma ^{H^\pm }_{\text {Type X}}$ and $\sigma ^{H^\pm }_{\text {Type II}}=\sigma ^{H^\pm }_{\text {Type Y}}$.

Fig. 1

Leading order (LO) diagrams for the charged Higgs particle are shown. The double-resonant top pair production (top diagram) is the dominant process for light $H^\pm .$ The single-resonant top quark production (middle diagram) is dominant for heavy $H^\pm$. In the intermediate $H^\pm$ scenario ($M_{H^\pm }\sim M_t$), both of these production channels along with the non-resonant top quark production (bottom diagram) are taken into account

For the case of charged Higgs fermionic decays, in Type I all the fermionic couplings are proportional to $\cot \beta$ and hence the branching fractions are independent of $\tan \beta$. The $\tau \nu$ channel is the dominant decay channel for light charged Higgs in Type I. However, for heavy charged Higgs scenario in Type I, the Br($H^\pm \rightarrow \tau \nu$) is suppressed by $M_{\tau }^2/M_{t}^2$ over Br($H^+ \rightarrow t{\bar{b}}$), leading to nearly $100\%$ branching fraction in the $t{\bar{b}}$ channel. In Type II and Type X the lepton sector coupling to $H^\pm$ being proportional to $\tan \beta$, the decay into $\tau \nu$ is dominant for light $H^\pm$ and quite sizable for heavy $H^\pm$ for $\tan \beta > rsim 1$. As seen in Fig. 2 for heavy $H^\pm$ scenario in Type X, the $H^\pm$ branching fraction to $\tau \nu$ starts to dominate over the $t{\bar{b}}$ channel for large $\tan \beta$. In Type Y, because of the $\cot \beta$ dependence in the lepton sector the $\tau \nu$ channel gets suppressed compared to the hadronic decay modes (dominantly into $t{\bar{b}}$ for heavy $H^\pm$). The branching fractions computed using the public code Hdecay [42, 43] are shown in Fig. 2 for $M_{H^\pm }=250$ GeV, for all Yukawa types of 2HDM. The code Hdecay also includes the three-body decay of the charged Higgs particle, i.e., $H^+ \rightarrow t^* {\bar{b}} \rightarrow W^+ b {\bar{b}}$ below the two-body decay threshold of $H^+ \rightarrow t {\bar{b}}$ mode [44]. Note that the branching fraction of $H^\pm$ into the fermionic sector is given for situations where there are no $H^\pm$ decays into the neutral scalars.

Apart from the fermionic decays, $H^\pm$ can also decay to $W^\pm$ and neutral scalars h, H or A. The couplings to $W^\pm$ and neutral scalars are (all fields are incoming)

$$\begin{aligned}&H^\mp W^\pm h : \frac{\mp i g}{2} \cos (\beta - \alpha )(p_\mu - p^\mp _\mu ) ,\nonumber \\&H^\mp W^\pm H : \frac{\mp i g}{2} \sin (\beta - \alpha )(p_\mu - p^\mp _\mu ) ,\nonumber \\&H^\mp W^\pm A : \frac{g}{2}(p_\mu - p^\mp _\mu ), \end{aligned}$$

(9)

where $p_\mu$ and $p^\mp _\mu$ are the momenta of the neutral and charged scalars. In the alignment limit $\sin (\beta -\alpha )\rightarrow 1$ (which is considered throughout the paper) $H^\pm$ decay to h is suppressed. The decays into the H and A channels depend on the mass splitting allowed by the T parameter. In the generic 2HDM, there are no mass relations between $H^\pm , H$ and A unlike MSSM and for some parameter choice, the bosonic decays can be more dominant over the fermionic decays once the channels are open.

Fig. 2

Branching fractions of the charged Higgs particle into the dominant fermionic sectors as a function of $\tan \beta$ for $M_{H^\pm } = 250$ GeV. The alignment limit $\sin (\beta - \alpha )\rightarrow 1$ and degenerate $M_{H^\pm }$, $M_H$ and $M_A$ are considered to prevent $H^\pm \rightarrow W^\pm \phi$ ($\phi =h,H,A$) and satisfy the T parameter constraint

4 Experimental constraints

The theoretical constraints of 2HDM consist of vacuum stability [45, 46], perturbative unitarity [47, 48] and tree level unitarity [49,50,51]. The Electro-Weak Precision Observables (EWPOs) $S (0.05\pm 0.11), T (0.09\pm 0.13)$ and $U (0.01\pm 0.11)$ [52, 53], specially the T parameter [54], restrict the mass splitting of $H^\pm$, H and A. In this paper, $M_{H^\pm }=M_H=M_A$ is considered to impose the exclusion limits from the $H^\pm \rightarrow \tau ^\pm \nu$ and $H^+ \rightarrow t {\bar{b}}$ channels over the mass range $M_{H^\pm } \in [80,2000]$ GeV. Perturbative unitarity for a wide region of $\tan \beta$ can be satisfied by a proper choice of the soft ${\mathcal {Z}}_2$ breaking parameter, $m_{12}^2=M_A^2 \sin \beta \cos \beta$. The theoretical constraints are checked using the package 2Hdmc-1.7.0 [55]. The alignment limit $\sin (\beta -\alpha )\rightarrow 1$ is the condition most favored by the experimentalists. In this limit the couplings of the neutral scalar h in 2HDM is similar to the SM Higgs boson and can be identified with the observed 125 GeV Higgs boson. In the alignment limit the other CP even scalar, H, behaves as gauge-phobic i.e. its coupling to the gauge bosons $W^\pm /Z$ is very suppressed. In the context of a charged Higgs analysis for the $H^\pm \rightarrow \tau ^\pm \nu$ and $H^+ \rightarrow t{\bar{b}}$ channels, the alignment limit is useful as it completely suppresses the $H^\pm \rightarrow W^\pm h$ decay. LEP experiments [56] have given limits on the mass of the charged Higgs boson in 2HDM from the charged Higgs searches in Drell–Yan events, $e^+ e^- \rightarrow Z/\gamma \rightarrow H^+ H^-$, excluding $M_{H^\pm } \lesssim 80$ GeV (Type II) and $M_{H^\pm } \lesssim 72.5$ GeV (Type I) at $95\%$ confidence level. Among the constraints from B meson decays (flavor physics constraints), the $B\rightarrow X_s\gamma$ decay [57] puts a very strong constraint on Type II and Type Y 2HDM, excluding $M_{H^\pm } \lesssim 580$ GeV and almost independently of $\tan \beta$. For Type I and Type X, the $B\rightarrow X_s\gamma$ constraint is sensitive only for low $\tan \beta .$ So for $M_H^{\pm } \lesssim 580$ GeV, Type II and Type Y are not considered further.

The LHC experiments have already set limits on the $M_{H^\pm }$–$\tan \beta$ plane using $\sqrt{s}=8$ TeV observations from the $H^\pm \rightarrow \tau ^\pm \nu$ [58, 59] and $H^+ \rightarrow t {\bar{b}}$ [59, 60] channels. For $M_{H^\pm }\in [80,160]$ GeV, the most important constraint comes from the $H^\pm \rightarrow \tau ^\pm \nu$ channel.³ The exclusion regions are shown with green colors in Fig. 3 using the 8 TeV CMS results at an integrated luminosity of 19.7 $\hbox {fb}^{-1}$ for Type I and Type X. In Type X the leptonic coupling being proportional to $\tan \beta$ excludes a slightly larger region of $\tan \beta$. Using the upper bounds on the $\sigma _{H^\pm }$BR$(H^\pm \rightarrow \tau ^\pm \nu )$ from the latest CMS results [35] for $\sqrt{s}=13$ TeV at an integrated luminosity of 35.9 $\hbox {fb}^{-1}$, a much larger region of $\tan \beta$ is excluded as shown in red colors in Fig. 3 for both Type I and Type X. Just above $M_{H^\pm }=160$ GeV $\tan \beta \lesssim 1$ is allowed by this channel in the Type X model. This is because the exclusion in Type X at low $\tan \beta$ is less severe than Type I.

Fig. 3

Exclusion region in Type I and Type X from the upper limits on $\sigma _{H^\pm }BR(H^\pm \rightarrow \tau ^\pm \nu )$ CMS 13 TeV observations are shown in red color. The green color shows the exclusion region from the upper limits on BR$(t \rightarrow H^+ b )$BR($H^\pm \rightarrow \tau ^\pm \nu$) CMS 8 TeV observations. The region below the black dashed line is excluded by the BR($B\rightarrow X_s\gamma$) constraint

For $M_{H^\pm }$ greater than the top quark mass, the constraint coming from the $\tau \nu$ channel does not put any significant bound on the $M_{H^\pm }$–$\tan \beta$ parameter space. Therefore in the higher mass range the $H^+ \rightarrow t{\bar{b}}$ channel has to be studied. The tb channel, unlike the $\tau \nu$ channel, is not clean enough and suffers from various QCD backgrounds, but sophisticated analysis is used to study the tb channel in both 8 TeV and 13 TeV by the CMS collaboration. The CMS 8 TeV upper limit on $\sigma (pp \rightarrow t(b)H^+)$ assuming BR($H^+ \rightarrow t {\bar{b}}) = 100\%$ [59] restricts the parameter space for $200<M_{H^\pm }<600$ GeV. The exclusion region using the 8 TeV results are shown in green colors in Fig. 4 for Type I. A recent paper from the CMS collaboration [36] for $\sqrt{s}=13$ TeV and 35.9 $\hbox {fb}^{-1}$ puts an upper limit at $95\%$ CL on $\sigma _{H^\pm }\text {BR}(H^+ \rightarrow t{\bar{b}})$ with the single-lepton and dilepton final states combined. The resulting exclusion regions in the $M_{H^\pm }$–$\tan \beta$ plane for heavy charged Higgs $M_{H^\pm }\in [200,2000]$ GeV in Type I and $M_{H^\pm }\in [600,2000]$ in Type II are shown in Fig. 4 with red colors. Since the charged Higgs leptonic decay mode in TypeY is much suppressed compared to the $H^+ \rightarrow t{\bar{b}}$ mode and for the Type X scenario, the $H^+ \rightarrow t {\bar{b}}$ mode is dominant for low $\tan \beta$ as shown in Fig. 2 (bottom two plots). The exclusion regions of Type X and Type Y are equivalent to the exclusion regions of Type I and Type II, respectively.

Fig. 4

Exclusion region in Type I and Type II from the upper limits on $\sigma _{H^\pm }BR(H^+ \rightarrow t{\bar{b}})$ CMS 13 TeV observations are shown in red color. The green color (upper plot) shows the exclusion region from the upper limits on $\sigma (pp\rightarrow {\bar{t}}(b)H^+)$ CMS 8 TeV observations assuming BR($H^+ \rightarrow t {\bar{b}})=1$. The region below the black dashed line is excluded by BR($B\rightarrow X_s\gamma$) constraint and the region below the continuous black line in Type II (bottom plot) is excluded by the BR$(B_s \rightarrow \mu ^+ \mu ^-)$ constraint

So far, the charged Higgs decay to the gauge boson and neutral scalars $H^\pm \rightarrow W^\pm H/A$ are not considered by assuming near mass degeneracy of $H^\pm , H$ and A. But once the bosonic decays are kinematically allowed, the charged Higgs boson can significantly decay into these channels. Figure 5 shows the exclusion regions coming from the $H^\pm \rightarrow \tau ^\pm \nu$ channel where the mass difference $M_{H^\pm }-M_A=85$ GeV is considered for $M_{H^\pm }\in [100,160]$ GeV and $M_{H^\pm }\sim M_H$. The red regions are excluded by using the upper limits on $\sigma _{H^\pm }BR(H^\pm \rightarrow \tau ^\pm \nu )$ CMS 13 TeV observations and the green regions are excluded by using the upper limits on BR$(t \rightarrow H^+ b )$BR($H^\pm \rightarrow \tau ^\pm \nu$) CMS 8 TeV observations. For this choice of the mass difference, the exclusion regions are less than for Fig. 3 because of the significant decay of $H^\pm$ into $W^\pm A$. As mentioned above, in Type X the leptonic coupling being proportional to $\tan \beta$ excludes a larger region than for Type I.

Fig. 5

Exclusion region in Type I and Type X from the upper limits on $\sigma _{H^\pm }BR(H^\pm \rightarrow \tau ^\pm \nu )$ CMS 13 TeV observations are shown in red color. The green color shows the exclusion region from the upper limits on BR$(t \rightarrow H^+ b )$BR($H^\pm \rightarrow \tau ^\pm \nu$) CMS 8 TeV observations. The region below the black dashed line is excluded by the BR($B\rightarrow X_s\gamma$) constraint. Here the mass difference $M_{H^\pm }-M_A=85$ GeV for $M_{H^\pm }\in [100,160]$ GeV and $M_{H^\pm }\sim M_{H}$ is considered. The exclusion region is less compared to Fig. 3 where the masses of $H^\pm$, H and A are nearly the same

The CMS collaboration [37] recently studied the scenario where the mass difference of $H^\pm$ and A is $\sim 85$ GeV for $M_{H^\pm }\in [100,160]$ GeV. The charged Higgs produced in pp collision in LHC at an integrated luminosity of 35.9 $\hbox {fb}^{-1}$ decays dominantly into $W^\pm$ and A with final states $e\mu \mu$ or $\mu \mu \mu$. In this analysis, the CMS assumed BR($H^\pm \rightarrow W^\pm A) = 1$ and BR($A\rightarrow \mu ^+ \mu ^-) = 3\times 10^{-4}$. Also this is the first experimental result in the channel $H^\pm \rightarrow W^\pm A$, $A\rightarrow \mu ^+ \mu ^-$ at LHC to put upper limits on BR$(t \rightarrow H^+ b)$. Such a low branching fraction of $A\rightarrow \mu ^+ \mu ^-$ can be realized in Type I 2HDM where BR$(A\rightarrow \mu ^+ \mu ^-) \sim 2.4\times 10^{-4}$ for $A\in [15,75]$ GeV and it goes very well with the CMS assumption. The other assumption, BR($H^\pm \rightarrow W^\pm A)=1$, is satisfied in Type I scenario when $\tan \beta \ge 1$ as seen in Fig. 6. In Type I scenario the charged Higgs coupling to the fermionic sector being proportional to $\cot \beta$, the assumption BR($H^\pm \rightarrow W^\pm A)=1$ starts to fail for $\tan \beta < 1$. The theoretical constraints can be satisfied with a proper choice of $m^2_{12}$ and the oblique parameter T can be satisfied by considering $M_{H^\pm }\cong M_H$. The observed upper limits at 95$\%$ CL on BR$(t \rightarrow H^+ b)$ for $M_H^{\pm }\in [100,160]$ GeV and $M_{H^\pm }-M_A=85$ GeV with the above assumptions are used to find the exclusion region. In Fig. 6 the red region (above $\tan \beta \ge 1$) shows the exclusion region where we have smoothly fitted the observed CMS upper limit on BR$(t \rightarrow H^+ b)$ in the range of 0.63 to 2.9$\%$. Other 2HDMs like Type II and Type Y are not considered, as for this mass range of charged Higgs Type II and Type Y are ruled out by the $B\rightarrow X_s \gamma$ constraint. Unlike Type I where all the fermionic couplings of A are proportional to $\cot \beta$, in Type X the pseudoscalar coupling to the lepton sector is proportional to $\tan \beta$ whereas its coupling to the quark sector is proportional to $\cot \beta$. Thus the BR$(A\rightarrow \mu ^+\mu ^-)$ increases with $\tan \beta$. The CMS assumption is satisfied in Type X scenario only when $\tan \beta$ is close to 1 and for this situation the theoretically estimated BR$(t\rightarrow H^+ b)$ is the same as in Type I (because of the same coupling) and above the upper limit of the CMS observation. Comparing Figs. 5 and 6, the exclusion regions coming from the $\tau \nu$ channel are weak once the $H^\pm \rightarrow W^\pm A$ channel is open. Figure 6 (bottom plot) excludes the regions of parameter space which are not excluded in Fig. 5 (top plot).

Fig. 6

Branching fraction of $H^\pm \rightarrow W^\pm A$ as a function of $\tan \beta$ (upper plot) for $M_{H^\pm } - M_A=85$ GeV and $M_{H^\pm }=150$ GeV. The exclusion region (bottom plot) in Type I is shown in red color from the first upper limits on BR$(t \rightarrow H^+ b$) CMS 13 TeV observations in the charged Higgs decay mode: $H^\pm \rightarrow W^\pm A$ and $A \rightarrow \mu ^+ \mu ^-$ with the assumptions BR$(H^\pm \rightarrow W^\pm A)=1$ and BR$(A\rightarrow \mu ^+\mu ^-)=3\times 10^{-4}$. For the charged Higgs bosonic decay channel the mass difference $M_{H^\pm }-M_A=85$ GeV for $M_{H^\pm }\in [100,160]$ GeV is considered. The region below the black dashed line is excluded by BR($B\rightarrow X_s\gamma$) constraint

For completeness, the indirect constraints from the flavor physics is also considered in the paper as the B meson decay depends strongly on the parameters $M_{H^\pm }$ and $\tan \beta$. The public code SuperIso-3.7 [61] is used for flavor physics computation. As mentioned above, for Type II and Type Y, a charged Higgs boson lighter than $\sim 580$ GeV is completely ruled out for a large region of $\tan \beta$ by the BR($B\rightarrow X_s \gamma$) constraint [62], which is measured to be $(3.32\pm 0.15)\times 10^{-4}$ [63]. For Type I (and similarly for Type X) the BR($B\rightarrow X_s \gamma$) constraint excludes $\tan \beta \lesssim 2$. In Figs. 3, 4, 5 and 6 the regions below the black dashed lines are excluded by the BR($B\rightarrow X_s \gamma$) observation. In Type II (and similarly in Type Y) for $M_{H^\pm }$ above 600 GeV, the rare decay of $B_s \rightarrow \mu ^+ \mu ^-$ (the branching fraction of which is measured to be ($3.0\pm 0.6\pm 0.25)\times 10^{-9}$) as reported by LHCb collaboration excludes a greater region of parameter space than the $B\rightarrow X_s\gamma$ constraint. For the Type II scenario in Fig. 4 the region below the black continuous line is excluded by BR($B_s \rightarrow \mu ^+ \mu ^-$) constraint.

5 Summary and conclusions

The 2HDM is the simplest extension of SM containing charged Higgs. The two most dominant channels, $H^\pm \rightarrow \tau ^\pm \nu$ and $H^+ \rightarrow t {\bar{b}}$, for the search of $H^\pm$ are studied using the latest CMS results for $\sqrt{s}=13$ TeV at an integrated luminosity of 35.9 fb $^{-1}$. The $\tau \nu$ channel excludes a large region of $\tan \beta < {\mathcal {O}}(15)$ for a charged Higgs mass less than 160 GeV both in Type I and Type X. For a heavy charged Higgs boson, the $\tau \nu$ channel does not lead to any significant constraint on the parameter space. However, in this case, the tb channel excludes a significant range of values of $\tan \beta$ in Type I and II and this behavior is carried over to Type X and Type Y. The exclusion regions obtained from the 13 TeV CMS results are compared with the exclusion regions from 8 TeV CMS results. Exclusion bounds from B meson decays are also discussed for all Yukawa types of 2HDM. The fermionic channels are studied for situations where the exotic decays of a charged Higgs boson into a gauge boson and neutral scalars ($H^\pm \rightarrow W^\pm /h/H/A$) are suppressed either by the alignment limit or due to limited phase space. But once the bosonic decay channels are open, they can be the dominant charged Higgs decay channels and the constraints from $H^\pm \rightarrow \tau ^\pm \nu$ and $H^+ \rightarrow t{\bar{b}}$ will be less restrictive. The CMS collaboration for the first time studied the exotic bosonic decay channel $H^\pm \rightarrow W^\pm A$ and $A \rightarrow \mu ^+ \mu ^-$ to put upper limits on BR($t \rightarrow H^+ b$) for $M_{H^\pm } \in [100,160]$ GeV with a mass splitting of $M_{H^\pm }-M_A=85$ GeV. These results are used to exclude a significant parameter space of the charged Higgs boson in Type I 2HDM which is not excluded by the $\tau \nu$ channel. It is expected that a significant parameter space of the charged Higgs boson will be excluded in all Yukawa types of 2HDM (as well as in MSSM) if these exotic bosonic decay channels of charged Higgs are analyzed by CMS or ATLAS collaborations for various charged Higgs mass ranges.