The $W \gamma$ decay of the elusive charged Higgs boson in the two-Higgs-doublet model with vectorlike fermions

The LHC search strategy for the charged Higgs boson $H^\pm$ in a two-Higgs-doublet model crucially depends on the top quark physics: for the low mass region, $H^\pm$ is mainly produced from the decay of a top quark; for the high mass region, $H^\pm$ decays into top and bottom quarks. When the charged Higgs boson mass is similar to the top quark mass, the experimental signal is hard to detect because of the accompanying soft particles. For this elusive charged Higgs boson, we suggest the $W \gamma$ decay mode as an alternative search channel. Since the branching ratio of the loop-induced decay in an ordinary two-Higgs-doublet model is very suppressed, below $\mathcal{O}(10^{-4})$, we extend the model by introducing a vectorlike fermion $SU(2)$ doublet and two singlets. In type-I-II model where the SM fermions are assigned in type-I while the vectorlike fermions are in type-II, we show that the branching ratio can be greatly enhanced to $\sim \mathcal{O}(0.01)$ in a large portion of the parameter space allowed by the Higgs precision data, the electroweak oblique parameters, and the direct search bounds at the LHC. The LHC discovery potential is also promising in the channels of $g \bar{b} \to \bar{t} H^+ (H^+ \to W^+ \gamma)$ and $g g \to H/A \to H^+ W^- (H^+ \to W^+ \gamma)$.


Introduction
The current status of new physics study at the LHC is disappointing to many particle physicists as we do not see any hint for new particles. What awaits us in the near future is higher luminosity, which shall be helpful to probe some faint signals, if any, but not higher energy scale particles. Before we are resigned to the prospect of no new signal for a while, however, we must search every hole and corner. And this task requires a special strategy. The common method of finding a new particle is to resort to the main production channel and the main decay modes, which spans the bulk of the parameter space most effectively. This basic process has been performed expeditiously with dedicated efforts, giving us various exclusion plots with a few holes and corners of the allowed region.
Usually, the corners correspond to small signal rate, which will be covered as more data are coming. Problematic is the type of a blind spot or line, a very narrow allowed region, which is mainly from experimental difficulties.
A good example of the hole corresponds to the charged Higgs boson with its mass near the top-quark mass in the two-Higgs-doublet model (2HDM) [1]. Since the H ± -W ∓ -Z vertex in a 2HDM vanishes at the tree level, the charged Higgs boson mostly decays into fermions. The search strategy at the LHC [2][3][4][5][6][7][8] depends on its mass M H ± . If M H ± < m t , the charged Higgs boson is mainly produced from the decay of a top quark in the top quark pair production, and then H ± decays into τ ν. If M H ± > m t , the production channel is gb → H +t , followed by the decay H + → tb. In any case, the top quark plays a key role in searching for the charged Higgs boson at the LHC, either through production or decay. Therefore, it is very difficult to probe the critical case of M H ± ≈ m t since at least one decay product of either t → H + b or H + → tb is very soft [9]. We need alternative channels to target this elusive charged Higgs boson.
Non-fermionic decay channels of H ± into the SM particles are only the radiative decays of H ± → W ± γ and H ± → W ± Z. In the usual 2HDM, the branching ratios are very suppressed, being at most ∼ O(10 −4 ). So we question whether the branching ratios can be meaningfully enhanced if we extend the 2HDM by introducing vectorlike fermions (VLFs) [10]. A heavy VLF with a mass around the electroweak scale appears in many new physics models [11,12]. One of the biggest advantages of VLFs is its consistency with the Higgs precision data unlike heavy chiral fermions [13,14].
Increasing the branching ratios of the radiative decays, significantly enough to ensure LHC sensitivity, is very challenging. Naively raising the Yukawa couplings of the VLFs with the charged Higgs boson shall confront the constraints from the electroweak oblique parameters since the VLF loop corrections to the vertex of H ± -W ∓ -V (V = γ, Z) are usually correlated with those to the vacuum polarization amplitudes of the SM gauge bosons. We need to contrive a viable model which accommodates significantly large loopinduced decays of the elusive charged Higgs boson while satisfying the other direct and indirect constraints. As will be shown, if we assign the SM fermions in type-I and the new VLFs in type-II, the goal is achieved. In a large portion of the parameter space, Br(H ± → W ± γ) is greatly enhanced by one or two orders of magnitude. However, the W Z decay mode does not change much because of the strong correlation with the electroweak oblique parameterT . This is our main result.
The W γ and W Z modes as a new resonance search at the LHC [15,16] have been studied in other new physics models. A representative one is the Georgi-Machacek (GM) model [17] where the custodial-fiveplet (both singly and doubly) charged Higgs boson is fermiphobic, mainly decaying into W Z or W W through the tree level couplings [18][19][20][21].
Below the threshold of W Z or W W , the loop-induced decays into γγ, Zγ, and W γ were studied [22,23]. In a generalized inert doublet model with a broken Z 2 symmetry, called the stealth Higgs doublet model, H ± → W ± γ was also studied [24]. However, our model is distinguished from these models: (i) the charged Higgs boson mainly decays into fermions; (ii) the W Z decay is very suppressed, not by the kinematics, while the W γ decay is enhanced.
The paper is organized in the following way. In Sec. 2, we review our model, the 2HDM with the SM fermions in type-I and the VLFs in type-II. Section 3 deals with indirect and direct constraints such as the Higgs precision data, the direct searches for the charged Higgs boson and the VLFs at the LHC, and the electroweak oblique parameters.
Particularly for the electroweak oblique parameterT , we shall suggest our ansatz for the parameters. In Sec. 4, we first present the one-loop level calculation of the decay rates of H ± → W ± γ/W ± Z via the VLF loops. This is a new calculation. Then, we show that the branching ratio of H ± → W ± γ can be highly enhanced by one or two orders of magnitude, relative to that without the VLF contributions. Section 5 covers the production channels of the charged Higgs boson in our model as well as the 13 TeV LHC sensitivity to the H ± → W ± γ mode. Section 6 contains our conclusions.

2HDM with Vectorlike Fermions
We consider a 2HDM with vectorlike fermions in the alignment limit. The Higgs sector is extended by introducing two complex SU (2) L Higgs doublet scalar fields, Φ 1 and Φ 2 [1]: where i = 1, 2, and v 1,2 are the nonzero vacuum expectation values (VEVs) of Φ 1,2 . We parametrize t β = v 2 /v 1 in the simplified notation of s x = sin x, c x = cos x, and t x = tan x.
The electroweak symmetry breaking occurs by the nonzero VEV of v = v 2 1 + v 2 2 = 246 GeV.
The fermion sector of the SM is also extended by introducing one SU (2) doublet VLF and two SU (2) singlets as follows: Here U ( ) and D ( ) denote the up-type and down-type fermions, respectively. We shall consider various kinds of the VLFs: (X, T ), the vectorlike quark (VLQ) with the electric charges of (5/3, 2/3); (T, B), the VLQ with (2/3, −1/3); (B, Y ), the VLQ with (−1/3, −4/3); (N, E), the vectorlike lepton (VLL) with (0, −1) [12]. In order to avoid the flavor changing neutral currents (FCNC) at tree level, we introduce a discrete Z 2 symmetry under which Φ 1 → Φ 1 and Φ 2 → −Φ 2 [25,26]. The Z 2 parities of Φ 1 and Φ 2 dictate the scalar potential to be where we allow softly broken Z 2 parity but maintain the CP invariance. Five physical Higgs bosons (the light CP -even scalar h at a mass of 125 GeV, the heavy CP -even scalar H, the CP -odd pseudoscalar A, and two charged Higgs bosons H ± ) are related with the weak eigenstates via where z 0 and w ± are the Goldstone bosons that will be eaten by the Z and W bosons, respectively. The rotation matrix R(θ) is The SM Higgs field is a linear combination of h and H, h SM = s β−α h + c β−α H. Because the observed Higgs boson at a mass of 125 GeV is very SM-like, we take the alignment limit of The fermions can have different Z 2 parities. For the SM fermions, we fix Q L → Q L and L L → L L under Z 2 parity transformation. Then, there are four different choices of Z 2 parities for the right-handed SM fermion fields, leading to type-I, type-II, type-X, and type-Y. The VLFs need not to have the same Z 2 parity with the SM fermions. Since our main purpose is to explore the possibility of highly enhancing Br(H ± → W ± γ/W ± Z 0 ), we consider type-I-II, where the SM fermions are assigned in type-I while the VLFs are in type-II (see Table 1). The Lagrangian for the mass and Yukawa terms of the VLFs is then where Φ = iτ 2 Φ * and we take the simplified assumption of The VLF masses are from the Dirac mass parameters as well as from two VEVs of v 1 and v 2 . The mass matrices M D and M U in the basis of (D , D) and (U , U), respectively, In the large t β limit where c β 1 and s β ≈ 1, the off-diagonal terms of M D are suppressed. The VLF mass matrices are diagonalized by the rotation matrices R(θ F ) as Then, the mass eigenstates of the VLFs are obtained as When θ U ,D 1, U 1 and D 1 are SU (2) doublet-like while U 2 and D 2 are singlet-like. In what follows, we use s U = s θ U and c U = c θ U for notational simplicity. The VLF mixing angles satisfy The Yukawa Lagrangian for the SM fermions and the VLFs is where F = U, D, i, j = 1, 2, and φ = h, H. In our type-I-II model, the normalized Yukawa couplings are Additionally, we shall impose the alignment condition of β − α = π/2.
The Yukawa couplings of the VLFs with neutral Higgs bosons are and those with the charged Higgs boson are 14) The gauge interaction Lagrangian in terms of the VLF mass eigenstates is where F = U, D, g Z = g/c W , and c W is the cosine of the electroweak mixing angle. The normalized gauge couplings arê 3 Constraints on the type-I-II 2HDM Before studying how large Br(H ± → W ± γ/W ± Z) can be, we study the allowed parameter space by the current data. First, the FCNC process of b → sγ plays a sensitive probe for H ± . Comparing the Belle result [27] and the SM calculation with NNLO QCD correction [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]  Based on the results, we shall suggest a benchmark scenario for this model.

Constraints from the LHC Higgs precision data
The new VLFs change the loop-induced h-g-g and h-γ-γ vertices which are stringently constrained by the current Higgs precision measurement. New physics effects are usually parametrized by the coupling modifier κ i . Since κ γ is mainly from W ± loop, the most sensitive one is κ g , which the VLFs change into where the loop function A H 1/2 (τ ) is given in Ref. [72], τ f = m 2 h /m 2 f , F = U, D, and i = 1, 2. As explicitly shown in Eq. (2.13), the vectorlike nature of new fermions yields Unless M F 1 is very different from M F 2 , the contribution from F 1 is considerably canceled by that from F 2 . The ATLAS and CMS combined result at 2σ [73], 0.6 < |κ g | < 1.12, is satisfied in most of the parameter space.  [74].
If H ± q mode is additionally open, the VLQ mass bound can be weaker. As for the VLL, multi-leptonic event searches at the LHC lead to M L 300 GeV from the ATLAS data [75] and M L 270 GeV from the CMS data [76]. For the numerical analysis, therefore, we consider two cases of M Q = 600 GeV and M Q = 1.3 TeV for the VLQs, and one case of M L = 300 GeV for the VLLs.
Another important constraint from LHC direct searches is on the charged Higgs boson, the upper bound on its production cross section times branching ratio.
the production channel is pp → tbH ± and the decay channel is H ± → τ ν: for example, σ · Br 4 pb when M H ± = 100 GeV at the 13 TeV LHC [77].

Constraints from the electroweak oblique parameterT
The electroweak precision test puts one of the strongest indirect constraints on new fermions which affect the gauge boson self-energy diagrams through loops, parametrized by the Peskin-Takeuchi oblique parameters S, T , and U [78]. For more general parametrization, Barbieri et al. extended the parameters intoŜ,T , W , and Y [79], which are defined as follows. We begin with Π V V (q 2 ), the g µν term of the transverse vacuum polarization amplitude Π µν V V (q 2 ) of the gauge boson. Expanding Π V V (q 2 ) up to quadratic order as we defineŜ,T , W , and Y aŝ The traditional Peskin-Takeuchi parameters S and T are related withŜ andT as The current experimental constraints are [79,80] We focus on the most sensitive oblique parameterT here.Ŝ, Y , and W are discussed in Appendix A. For the general vector and axial-vector gauge couplings of L = V µψ (g V γ µ + g A γ 5 γ µ )ψ, Π V V (0) from a single diagram mediated by two fermions with masses m a and Here Div = 1/ + ln 4π − γ is the divergence term in the dimensional regularization, = (4 − D)/2, and µ is the renormalization scale. These divergences are properly canceled out and there is no µ dependence onT from the VLF contributions. The vectorlike nature of new fermions makesT depend only onΠ V , defined bỹ Then,T in our model becomeŝ It is generally known that the smallT prefers very degenerate masses for the new fermions in the loop, which is clearly seen from As will be shown, however, the crucial condition for the enhancement of Br(H ± → W ± γ) is sizable mass difference between the up-type and down-type VLFs. It seems that theT constraint excludes the possibility. Here comes the advantage of our model with vectorlike SU (2) L doublet and singlet fermions. The new fermion spectrum includes U 1 , U 2 , D 1 , and D 2 , leading to six terms in Eq. (3.10). Now each term can be sizable whileT is kept small if the first two terms are canceled by the last four terms. We find that this cancellation In Fig. 1, we show the 2σ allowed region of (θ U , θ D ) by the electroweak oblique param- In conclusion, we find the following simple ansatz to satisfyT = 0:

Benchmark scenario for the numerical analysis
Considering all of the constraints described above, we take the following benchmark scenario: where Q F is the electric charges of the particle F. Note that the ansatz in Eq. (3.12) relates the up-type Yukawa coupling Y U with the down-type Yukawa coupling Y D as which can be clearly seen from Eq. (2.10). For large t β , Y D becomes large.
Some brief comments on the decays of the VLQs, especially the ones with exotic electric charges in Eq. (3.13), are in order here. As being colored fermions, the VLQs are copiously produced through the pair production from the gluon fusion. The question is which Lagrangian terms determine their decay into the SM particles in our type-I-II model. For example, the (X, T ) case has the following Yukawa interactions because of the Z 2 parities in Table 1 and the electric charges of (5/3, 2/3): The first term leads to the mixing between T and the SM up-type quarks. The second term yields the vertices of X-u-H ± and T -d-H ± . The X decays into W + u i and H + u i . In our model, the decays of H ± → W ± γ and H ± → W ± Z occur radiatively through the VLFs as well as the SM top and bottom quarks, as shown in Fig. 2. The loop-induced

Loop induced decays of the charged Higgs boson
where N C is the color factor of the fermion in the loop. We further express M µν in terms of three dimensionless form-factors M 1,2,3 as where p 1 and p 2 are the momenta of W ± and V respectively.
q, ijk , for q = 1, 2, 3 . We summarize the indices of i, j, and k for W ± γ and W ± Z in Table 2.  For W + γ decay, the Ward-identity of p ν 2 M µν = 0 from the gauge invariance relates M 1 with M 2 as The partial decay rate for H + → W + Z is quark loops [82] are shown in Appendix B. Our calculation of the VLF contributions is new. We checked that our expressions for the SM contributions are numerically consistent with those in Ref. [82].  The whole behavior of Br(H ± → W ± γ), especially its sensitive dependence on the VLF electric charges, is not easy to understand since it involves the complicated loop effects from various combinations of the VLFs as in Fig. 2. Nevertheless, we find a reason at least when the VLF loop effects are dominant. Since Considering the direct search bound on the VLQ mass at the LHC, we show Br(H + → W γ) for heavy VLQs with M U 1 = M D 1 = 1.31 TeV in Fig. 4. Heavier VLQs with about twice mass yields much smaller Br(H + → W γ) by an order of magnitude. But still Br(H + → W γ) can be an order of magnitude larger than that without the VLFs.

ΔM[GeV]
In Figure 5, we show that the branching ratios of H + → W + Z as a function of ∆M (left panels) and t β (right panels). We take M H ± = 180 GeV, M U 1 = M D 1 = 600 GeV for the VLQs, M U 1 = M D 1 = 300 GeV for the VLLs, and θ U = θ D = 0.2. The VLQ loop contribution to H ± → W ± Z is not as large as that to H ± → W ± γ, typically a few tens of percent for Y D 5. We find that there is a strong correlation between Br(H ± → W ± Z) and the constraint from the electroweak oblique parameter T .
The reader may question whether the large enhancement of Br(H ± → W ± γ) happens only in the benchmark scenario. To answer the question, we scan all of the    parameters in the range of

ΔM[GeV]
Note that we independently span θ U and θ D , not imposing the condition of θ U = θ D . Y U and Y D are determined by Eq. (2.10). Then we select the parameter sets that satisfy the constraints from the Higgs precision data on κ g , the upper bound on It is fair to say that the VLFs in our model greatly enhance the branching ratio of H ± → W ± γ. Concerning the details, the benchmark point for the (X, T ) case does not represent the whole parameter space: even for large |Y D | 5, the (X, T ) contribution to Br(H ± → W ± γ) can be very destructive or very constructive, while the benchmark point always enhances the branching ratio. For heavier VLQ masses (right panel), the range of the scatter plot is not as wide as that for low VLQ masses. The scatter ranges of the (X, T ), (T, B), and (B, Y ) cases are quite separated. Note that all of these production processes occur at tree level: the VLFs do not play a role here.
pp→tt(t→bH + ) In Fig. 7, we show the cross sections of the production channels in Eq. (5.1) as a function of M H ± at the LHC with √ s = 13 TeV. We consider two cases, t β = 1 (dashed line) and t β = 10 (solid line). We use NNPDF [83] for the parton distribution function inside the proton. For M H ± m t , the dominant production channel is from tt production, followed by t → H + b. The result is based on the calculation of σ(pp → tt) × Br(t → H + b) where we use the next-to-next-to-leading order result of σ tt = 831.8 +20 −29 ± 35 pb with a top quark mass m t = 172.5 GeV at the 13 TeV LHC [84]. The cross section quickly falls down by kinematics as M H ± approaches to m t . The t β dependence on σ(gg → tbH + ) is very large. The cross section with t β = 1 is about 100 times that with t β = 10, which is attributed to the t-b-H + vertex being proportional to 1/t β . Note that the t β dependence on the production cross section is opposite to that on Br(H ± → W ± γ).
For M H ± m t , the production process of gb →tH + becomes dominant. The LO analytic expression for the production process can be found in Ref. [85]. The higher order QCD corrections are given in Ref. [86][87][88]. In this work, we only use the LO result. The t β dependence on σ(gb →tH + ) is the same as that on σ(gg → tbH + ): small t β yields much larger production cross section. The pair production qq → H + H − is via s-channel diagrams mediated by γ and Z, which is independent of t β . We adopt the LO analytic result in Ref. [89]. The production cross section is very small in the whole range of M H ± , being O(1) ∼ O(10) fb.
Another way to produce the charged Higgs boson at the LHC is through the resonant decay of other heavy Higgs bosons. The heavy neutral Higgs bosons, H and A, are produced through the gluon fusion, followed by their decay into a charged Higgs boson: Note that the gluon fusion production of H or A is not significantly affected by the VLFs: (i) the scattering amplitudes of gg → A are proportional to the axial-vector coupling of the fermion in the loop, which vanishes for the VLFs; (ii) for gg → H, to which only the H-F 1 -F 1 and H-F 2 -F 2 (F = U, D) vertices contribute, the relation of y H F 1 F 1 = −y H F 2 F 2 yields considerable cancellation of the VLQ contributions. For the resonant decay of H or A, we make use of the W ± boson as a well-defined tagging particle, in order to reduce the model dependence. Then the decays of H → H ± W ∓ and A → H ± W ∓ go through the following Lagrangian terms:  5.2 LHC discovery potential for the H ± → W ± γ mode Next, we consider the potential for the LHC to observe the elusive charged Higgs boson with M H ± m t through the W γ channel. We ask whether the 5σ discovery is possible with the total integrated luminosity 300 fb −1 . The answer largely depends on its production channel. In the previous section, we have studied three production channels, through the top quark pair production of gg → tt(t → H + b), the single top quark production of gb → H +t , and the heavy Higgs boson production of gg → H/A → H + W − . When M H ± m t , two processes of gg → tt(t → H + b) and gb → H +t have similar production cross sections of O(10) ∼ O(10 3 ) fb, depending on the value of t β (see Fig. 7). However, the process gg →tH + b(H + → W + γ) has the final statestW + bγ: the irreducible background is the top quark pair production accompanied with a photon. As a top-quark factory, it shall be very difficult for the LHC to detect H ± → W ± γ via the gg → tt channel. Therefore, we focus on two production channels of gb → H +t and gg → H/A → H + W − . In order to suppress the SM background, we require the invariant mass of W ± and γ to be 1 First, the gb → H +t process, followed by H + → W + γ, has the irreducible background of the single top quark production with a W boson and a photon 2 . The SM cross section of the tW production at NNLO is σ = 71.7 ± 1.8(scale) ± 3.4(PDF) pb [91], which is an order of magnitude smaller than the tt production. Under the additional cut in Eq. (5.4), the SM result for σ(pp →tW + γ) at √ s = 13 TeV is about 5.5 fb by using MadGraph [92]. With a total integrated luminosity of 300 fb −1 , the 5σ discovery through both pp →tW + γ and pp → tW − γ channels demands the new signal more than about 0.48 fb, based on the significance of S/ √ B without including the systematic uncertainties.
In Fig. 9, we show σ(gb →tH + ) × Br(H + → W + γ) as a function of t β at the 13 TeV LHC. We set M H ± = 170 GeV, M U 1 = M D 1 = 600 GeV for the VLQs, M U 1 = M D 1 = 300 GeV for the VLLs, and ∆M = 600 GeV. The opposite t β dependences on σ(gb →tH + ) and Br(H + → W + γ) lead to rather gentle increase of σ × Br about t β , except for the (X, T ) case with t β 7. The most promising case is (B, Y ), which has σ × Br 0.2 fb. The parameter region with t β 10.5 can be probed with 5σ significance. For the (X, T ) case, t β 11 can be discovered at 5σ. Neither the (T, B) nor (N, E) case can be probed in the whole range of t β . Next we present σ(gg → H/A → H + W − ) × Br(H + → W + γ) as a function of t β in Fig. 10, the H mediated one (left panel) and the A mediated one (right panel). We set M H ± = 170 GeV, M H = M A = 2M H ± , M U 1 = M D 1 = 600 GeV for the VLQs, M U 1 = M D 1 = 300 GeV for the VLLs, and ∆M = 600 GeV. Since the suppression of the production cross section by large t β is weak for the H mediation as shown in Fig. 8, the increase of σ × Br with respect to t β is much larger for the gg → H production channel. With the kinematic cut in Eq. (5.4), the SM result for σ(pp → W + W − γ) at √ s = 13 TeV is about 14.4 fb by using MadGraph. Including both H + W − and H − W + , the 5σ discovery with the total integrated luminosity of 300 fb −1 requires σ × Br 0.77 fb. Through the H resonance channel, the (B, Y ) and (X, T ) cases can be probed for t β 5.2 and t β 6.9 at 5σ, respectively. Neither the (T, B) nor (N, E) case has a chance to be detected at the LHC. For the gg → A production channel, we can see the (B, Y ) case in the whole range of t β , and the (X, T ) case for t β 6. And both (N, E) and (T, B) cases for small t β 3 can be also probed.

σ(gb→H + t)·Br(H + →Wγ)[fb]
Although the significance increases if we include both H and A channels, it would be insignificant without knowing the masses of H and A. If M H = M A = 500 GeV, for example, the production cross section of H (A) becomes only 30% (10%) of that with M H/A = 2M H ± . Then, through the gg → H channel, the (B, Y ) and (X, T ) case can be probed at 5σ for t β 8.2 and t β 8.8, respectively. In the A resonance channel, only very large t β region can be probed: the (X, T ) case with t β 15.5 and the (B, Y ) case with t β 16.7.

Conclusions
Targeting the elusive charged Higgs boson H ± with its mass similar to the top-quark mass, we have explored the theoretical possibility that its radiative decays into W ± γ and W ± Z are large enough to detect H ± at the LHC. We considered a two-Higgsdoublet model with a vectorlike fermion (VLF) SU Introducing a VLF doublet and two singlets, necessary for the interaction with the Higgs doublet fields, plays a crucial role. As being vectorlike, one generation of the new fermions has two up-type fermions, U 1 and U 2 , and two down-type fermions, D 1 and D 2 . And these extended fermions allow significant cancellation among the different VLF contributions to the Higgs precision data as well as to the electroweak oblique parameters, especiallyT . Sizable cancellation to the h-g-g vertex occurs naturally because the hF 1 F 1 coupling is opposite to hF 2 F 2 . The cancellation for the electroweak oblique parameterT requires some fine-tuning. We proposed an ansatz to ensureT = 0 such that M U 1 = M D 1 , M U 2 = M D 2 , and θ U = θ D , which is not so artificial. We have also included the constraints from direct search bounds on the VLFs and the charged Higgs boson at the LHC.
We presented the loop-induced amplitudes of H ± → W ± γ and H ± → W ± Z from the VLFs as well as the SM t and b quarks. For the (B, Y ) and (N, E) cases, Br(H ± → W ± γ) is shown to be enhanced in the whole parameter region, particularly for large ∆M (= M U 2 − M U 1 ) and large t β : if ∆M 500 GeV and t β = 10, the enhancement is by two orders of magnitude. For the (X, T ) case, the behavior of Br(H ± → W ± γ) is dynamic: for small ∆M or small t β , destructive interference with the SM contributions occurs; for large ∆M or t β , the (X, T ) contribution becomes dominant, greatly enhancing the branching ratio. The (T, B) case always yields a smaller signal rate than in the 2HDM without the VLFs, because the opposite signs of the electric charges of T and B bring about sizable cancellation among new fermion contributions and the remaining VLF contribution destructively interferes with the SM contributions. On the contrary, Br(H ± → W ± Z) is very moderately affected by the VLFs, because of the strong correlation with the electroweak oblique parameter T . Therefore, we focused on the W γ mode to probe the elusive charged Higgs boson at the LHC.
We have also studied the production of the charged Higgs boson. At the LHC, the main production is through the top quark (a single or pair production) or through the resonant decay of a heavy Higgs boson H or A. Since the tt production, followed by t → H + b and H + → W + γ, has too large SM background of pp →tbW + γ, we considered two processes, gb → tH + and gg → H/A → H + W − . Based on the parton level calculation at the 13 TeV LHC with the total integrated luminosity 300 fb −1 , we showed that the charged Higgs boson via W γ mode can be probed at 5σ in some cases. Through the gb → tH + production, both the (X, T ) and (B, Y ) cases have the 5σ level potential if ∆M 500 GeV and t β 10. Neither the (T, B) nor (N, E) case has enough significance for the discovery. The prospect of the gg → H/A → H ± W ∓ channel sensitively depends on the masses of heavy Higgs bosons, H and A. If their masses are in the desirable range, e.g., M H/A 2M H ± , this channel can also probe the H ± → W ± γ mode for the (X, T ) and (B, Y ) cases with ∆M 500 GeV and t β 5.
In conclusion, the radiative decay mode W γ can serve as an alternative channel to probe the elusive charged Higgs boson. A theoretically viable model in the extended type-I 2HDM with the vectorlike fermions was suggested to allow the great enhancement of the W γ branching ratio. We expect that this study helps the LHC to search for the charged Higgs boson.

A Vacuum-polarization amplitudes of the SM gauge bosons
For the electroweak oblique parametersŜ, Y , and W , we need the first and second derivatives of the transverse vacuum polarization amplitudes of the SM gauge bosons, which are explicitly shown in Ref. [81]. However, we found some typos in their results. The correct ones arẽ

.(A.4)
In our type-I-II 2HDM,Ŝ iŝ For the one loop calculation, we express the result in term of the loop functions of the LoopTools [97]. Two point function defines B l 's as where µ is the renormalization scale, D = 4−2 , and r Γ = Γ 2 (1 − )Γ(1 + )/Γ(1 − 2 ). The tensorial integral for the one-loop three point function is defined by 3) The decompositions of the tensorial integrals up to rank 2 are where k 1 = p 1 and k 2 = p 1 + p 2 . All of the coefficient functions of B i , C i and C ij are numerically computed by LoopTools. Note that C 00 and B i have UV divergence which should be canceled out.

B.2 Decay Form-Factors from the SM quark contributions
We describe the form factors defined in Eq. (4.2) for each single diagram shown in Figure 2. We compute the diagrams in the unitary gauge, and use the dimensional regularization with D = 4−2 in the MS scheme. As for the UV divergence, we show only the 1/ term. Since there is no tree-level coupling for the H + W − V vertex, all of the UV divergences should be canceled out among themselves after summing all the diagrams. This cancellation serves as a validation of the calculation. For notational simplicity, we introduce the normalized gauge couplings and Yukawa couplings aŝ For H + → W + Z, M 1 is not related with M 2 , given by

B.3 Decay Form Factors from the VLQ contributions
We first present the form factors of M 2 and M 3 for H ± → W ± V (V = γ, Z) through the VLQ loop as M (a) 2,ij = 0 , (B.14)