Probing Charged Higgs Bosons in the 2-Higgs Doublet Model Type-II with Vector-Like Quarks

We study the phenomenology of charged Higgs bosons ( H ± ) and Vector-Like Quarks (VLQs), denoted as T , the latter possessing a charge identical to the top quark one, within the framework of the Two Higgs Doublet Model Type-II (2HDM-II). Upon examining two scenarios, one featuring a singlet ( T ) (2HDM-II+( T )) and another a doublet ( T B ) (2HDM-II+( T B )), we discover that the presence of VLQs has a significant effect on the (pseudo)scalar sector of the 2HDM-II. In particular, this leads to a reduction in the strict constraint on the mass of the charged Higgs boson, which is imposed by B -physics observables, specifically B → X s γ . The observed reduction stems from modifications in the charged Higgs couplings to the Standard Model (SM) top and bottom quarks. Notably, the degree of this reduction varies distinctly between the singlet 2HDM+( T ) and doublet 2HDM+( T B ) scenarios. Additionally, our investigation extends to constraints imposed by the oblique parameters S and T on the VLQ mixing angles. Furthermore, to facilitate efficient exploration of the ’2HDM-II+VLQ’ parameter space, we present results on pair production of VLQs T ( pp → T ¯ T ), followed by T → H ± b and H ± → tb decays, yielding a distinctive 2 t 4 b final state. This investigation thus provides valuable insights guiding the search for extended Higgs and quark


Introduction
Evidence of either a light or heavy charged spin-0 boson, H ± , with a fundamental (i.e., pointlike) structure would be a crucial piece of evidence suggesting physics Beyond the Standard Model (BSM), as such a state would not replicate any of those existing within the SM.The latter, in fact, includes charged bosons, but these are spin-1.Conversely, it includes a spin-0 boson, but this is chargeless.As a consequence, the search for charged Higgs bosons in recent years has received a great boost at existing accelerator machines like the Large Hadron Collider (LHC) at CERN.However, despite all the efforts being made, still no H ± signals have been seen.Thus, data from direct H ± searches from both the ATLAS and CMS experiments have excluded large areas of the parameter spaces pertaining to BSM scenarios incorporating these new states.In particular, this has been done for a 2-Higgs Doublet Model Type-II (2HDM-II), including its Supersymmetric incarnation, the Minimal Supersymmetric Standard Model (MSSM).However, while the latter is able to escape LHCb indirect limits on H ± states via, chiefly, b → sγ driven decays of B mesons (thanks to new (s)particles in the corresponding loops cancelling the H ± contributions), this is not possible for the former.Therefore, in the 2HDM-II, a rather stringent lower limit applies on its mass, m H ± , of 580 GeV or so, quite irrespectively of the other model parameters.
Such a constraint can, however, be lifted if the 2HDM-II is supplemented with additional objects flowing inside the loops, enabling the b → sγ transitions, similarly to what happens in the MSSM.The purpose of this paper is to investigate whether a similar phenomenon can be triggered by the presence of Vector-Like Quarks (VLQs) alongside the particle states of the 2HDM-II.VLQs are (heavy) spin-1/2 states that transform as triplets under colour, but, differently from the SM quarks, their Left-and Right-Handed (LH and RH) couplings have the same Electro-Weak (EW) quantum numbers.They are predicted by several theoretical constructs: e.g., models with a gauged flavour group [1][2][3][4], non-minimal Supersymmetric scenarios [5][6][7][8][9][10], Grand Unified Theories [11,12]), little Higgs [13,14] and Composite Higgs [15][16][17][18][19][20][21][22] models, to name but a few.In most of these frameworks, VLQs appear as partners of the third generation of quarks, mixing with top and bottom quarks.Further, if their mass is not exceedingly large for the LHC kinematic reach, they can be accessible in its detectors in a variety of final states [23][24][25][26][27], so that some of these have already been explored by the ATLAS and CMS collaborations .
Here, rather than invoking a complete theoretical scenario embedding VLQs (like, e.g., Compositeness), we adopt a simplified approach by manually extending the 2HDM-II with two VLQ representations (singlet and doublet), in the spirit of Refs.[27,[54][55][56][57][58], i.e., for the purpose of assessing the phenomenological consequences of such a modifications in general, irrespectively of the underlying theory.Indeed, we will be able to prove that the VLQ loops entering b → sγ transitions can cancel the large contributions due to H ± ones, thereby reducing the aforementioned limit on m H ± down to approximately 200 GeV for the doublet scenario (2HDM+T B) and to around 500 GeV in the singlet scenario(2HDM+T ).However, we shall see that achieving this reduction relies on large mixing angles, a condition largely contradicted by constraints imposed by the oblique parameters S and T .The focus then will shift to the production of H ± states through pp → T T and T → H ± b, with H ± → tb, resulting in a final state characterised by two top and four bottom quarks (2t4b).
The plan of this paper is as follows: The next section is devoted to introduce our 2HDM-II+VLQ framework.Then we proceed to discuss both theoretical and experimental bounds on it.Sect. 4 presents our numerical results for the two chosen VLQ representations in turn, including providing Benchmark Points (BPs) amenable to further phenomenological investigation aimed at extracting signatures of the 2HDM-II+VLQ scenario at the LHC.Finally, we conclude.(We also have some appendices.)

Model Descriptions
In this section, we only provide a brief overview of the 2HDM-II+VLQ realisations that are relevant to our work.Let us begin with recalling the well known CP-conserving 2HDM scalar potential for two doublet fields (Φ 1 , Φ 2 ) with a discrete Z 2 symmetry, Φ 1 → −Φ 1 , that is only violated softly by dimension-2 terms [59,60]: Here, all parameters are real.The two complex scalar doublets Φ 1,2 may be rotated into a basis, H 1,2 , where only one obtains a Vacuum Expectation Value (VEV).Using the minimisation conditions of the potential for the implementation of EW Symmetry Breaking (EWSB), the 2HDM can be fully described in terms of seven independent parameters: , sin(β − α) and the soft breaking parameter m 2 12 .When we impose that no (significant) tree-level Flavour Changing Neutral Currents (FCNCs) are present in the theory, four Yukawa versions of the 2HDM can then be realised, depending on how the Z 2 symmetry is implemented into the fermion sector.These are: Type-I, where only Φ 2 couples to all fermions; Type-II, where Φ 2 couples to up-type quarks and Φ 1 couples to charged leptons and down-type quarks; Type-Y (or Flipped), where Φ 2 couples to charged leptons and up-type quarks and Φ 1 couples to down-type quarks; Type-X (or Lepton Specific), where Φ 2 couples to quarks and Φ 1 couples to charged leptons 1 .
We now move on to discuss the VLQ side of the model and we start by listing the representations of two gauge-covariant multiplets ((T ) and (T B)) in Tab. 2, where the fields T and B have electric charges 2/3 and −1/3, respectively.Specifically, the T is chosen as triplet under the colour group SU (3) C and singlet under the EW group SU (2) L ×U (1) Y .Furthermore, the RH and LH components T 0 L,R have the same EW and colour quantum numbers.The mixing of VLQs with the first and second generations of SM quarks is heavily constrained by low energy physics measurement constraints.One such example of such constraints originates from the EW Precision Observables (EWPOs), including oblique parameter corrections, these being radiative corrections to quantities such as the W ± boson mass (m W ± ) and to the effective mixing angle (sin 2 θ W ) at high orders [61,62].Such corrections may be sensitive to VLQs and may impose restrictions on their masses and couplings.Additionally, as mentioned, VLQs, especially the third generation, may also affect the properties of top and bottom quarks through the mixing of fermions [63].This may have implications for processes like the decay of the Z boson into bottom quarks, which were measured with high precision at the LEP e + e − collider at energies near the Z resonance [64,65].Deviations from SM predictions in such measurements have the potential to provide strong constraints on the properties of VLQs.We thus focus on scenarios in which the VLQs interact solely to third-generation SM quarks.
Component fields Table 1: Singlet and doublet VLQ representations under the SM gauge group.
In the Higgs basis, the Yukawa Lagrangian can be written as: Here, u R actually runs over (u R , c R , t R , T R ) and d R actually runs over (d R , s R , b R , B R ) while y u,d are 3×4 Yukawa matrices.When only the top quark "mixes" with T , the relation between mass eigenstates (T L,R ) and weak eigenstates (T 0 L,R ) can be factored into two 2 × 2 unitary matrices U L,R , such that where θ is the mixing angle between mass and weak eigenstates and ϕ is a possible CP-violating phase which will be ignored in our work.In the weak eigenstate basis, the diagonalisation of the mass matrices makes the Lagrangian of the third generation and heavy quark mass terms such that where M 0 is a bare mass and the y ij 's are Yukawa couplings.For the singlet case y 43 = 0, while for the doublet one has y 34 = 0. Using standard techniques of diagonalisation, the mixing matrices are obtained by with M q the two mass matrices in Eq. ( 4) and M q diag the diagonalised ones.The mixing angles in the LH and RH sectors are not independent parameters.Using Eq. ( 5) and depending on the VLQs representation, one can find: with (q, m q , m Q ) = (u, m t , m T ) and (d, m b , m B ).

Theoretical and Experimental Bounds
In this section, we list the constraints that we have used to check the validity of our results.
From the theoretical side, we have the following requirements: • Unitarity constraints require the S-wave component of the various (pseudo)scalar-(pseudo)scalar, (pseudo)scalar-gauge boson, and gauge-gauge bosons scatterings to be unitary at high energy [66].
• Vacuum stability constraints require the potential to be bounded from below and positive in any arbitrary direction in the field space, as a consequence, the λ i parameters should satisfy the conditions as [67,68]: • Constraints from EWPOs, implemented through the oblique parameters2 , S and T [71], require that, for a parameter point of our model to be allowed, the corresponding χ 2 (S 2HDM-II + S VLQ , T 2HDM-II + T VLQ ) is within 95% Confidence Level (CL) in matching the global fit results [72]: S = 0.05 ± 0.08, T = 0.09 ± 0.07, ρ S,T = 0.92 ± 0.11.
Note that unitarity, perturbativity, vacuum stability, as well as S and T constraints, are enforced through the public code 2HDMC-1.8.03 [73].
• Constraints from flavour physics are taken at 95% CL from experimental measurements are accounted for via the public code SuperIso v4.1 [81].Specifically, we have used the following measurements for the most relevant Branching Ratios BRs: 4 Numerical Results In this section, we present our numerical results by first describing the Lagrangian terms relevant to b → sγ transitions in the 2HDM-II+VLQ scenarios considered, then by charactering the implications of the latter in the case of the (T ) and (T B) representations chosen here in turn, and, finally, by presenting some BPs amenable to experimental investigation.

Charged Higgs Boson Contributions to Flavour Observables
In the following, we shall discuss the results of the most relevant B-physics constraints, from b → sγ transitions, onto the 2HDM-II+VLQ scenarios considered.
The Lagrange density, which defines the interactions of the charged Higgs boson with third generation fermions can be written as: where P L/R = (1 ± γ 5 )/2 are the chiral projection operators.For the 2HDM-II, 2HDM-II+(T ), and 2HDM-II+(T B) cases, the couplings κ t and κ b take the values presented in Tab. 2. We emphasise here the role of the Yukawa couplings, as it is the changes of the latter occurring in presence of VLQs which are responsible for the forthcoming results, as opposed to the role of the VLQs in the loop observables that we will be describing, owing to the far too large values of their masses (of order 750 GeV or more).
Table 2: Yukawa couplings of the charged Higgs bosons H ± to the third generation of quarks in the 2HDM-II and 2HDM-II+VLQ representations studied here ((T ) and (T B)).Here The table summarises the Yukawa couplings of charged Higgs bosons to quarks of the third generation for three distinct representations: 2HDM-II, 2HDM-II+(T ), and 2HDM-II+(T B), providing insights into the modification patterns in different scenarios.As indicated, the couplings of the charged Higgs boson to SM quarks, specifically to the top (t) and bottom (b), undergo modifications in both VLQ representations 4 .These alterations lead to significant changes in observables related to B-physics processes, such as b → sγ and B s/d → µ + µ − .The contributions to the corresponding Wilson coefficients (C 7,8 ) are proportional to κ i κ * j , where the terms can be further decomposed into two parts.Detailed expressions of C t i,κ b κ * t can be found in Ref. [86].
The contributions5 to C t,model i take the form: The outcomes of these modifications will be presented in the following subsections.

2HDM-II+(T )
We start with Fig. 1, where we present explicitly the excluded parts of the (m orange, which is hardly visible in the plots), and B u → τ ν (blue).This figure illustrates that higher values of s L lead to a less stringent constraint from B → X s γ, driving the exclusion regions towards smaller values of the charged Higgs boson mass, reaching approximately m H ± ≃ 500 GeV for s L = 0.45.Conversely, the limits from B s → µ + µ − become more restrictive as s L increases, excluding tan β ≲ 7 for s L = 0.25 and tan β ≲ 5 for s L = 0.45.Here, it is important to note that the new term introduced in the relevant H ± tb coupling depends solely on the mixing angle s L , which thus controls the inclusion of VLQs in the 2HDM-II, as discussed previously.However, the potential for this reduction is moderated by the oblique parameters S and T , as will be discussed subsequently.These parameters restrict the range of the mixing angle s L , thereby playing a critical role in preventing a substantial decrease in the charged Higgs mass limit, particularly by constraining the possibility of larger mixing angles.∆χ 2 (S, T ) Figure 2: Allowed points by all constraints (HiggsBounds, HiggsSignals, SuperIso and theoretical ones) superimposed onto the fit limits in the (S, T ) plane from EWPO data at 95% CL (with a correlation of 92%), with the colour code indicating ∆χ 2 (S, T ).Results are presented for the 2HDM-II+(T ).

Scanned ranges
Table 3: 2HDM and VLQs parameters with their scanned ranges for both VLQ representations considered here.Masses are in GeV.(Note that we have taken the lightest CP-even neutral Higgs boson, h, to be the one observed at ∼ 125 GeV.) In Fig. 2, we show the result of the analysis for our scan over the (S, T ) plane.The colour coding indicates the difference with respect to the χ 2 (S, T ) values, and the black lines show the 68% CL (solid) and 95% CL (dashed) contours.The scanned 2HDM-II+VLQ parameters 6are given in Tab. 3. In the remainder of our analysis, we will present results from the 2HDM-II+(T ) regions within the 2σ band also compliant with flavour constraints.It is important to highlight that the shape of these points is a result of the cancellation between the contributions from the 2HDM and VLQ states to the oblique parameters S and T .In Fig. 4, we illustrate the BR(T → H + b) (left) and the ratio Γ T /m T (right) as a function of m T and tan β.It is clear from the left plot that the 2HDM-II+(T ) scenario cannot ultimately predict dominant production rates for a H ± emerging from a heavy T quark, no matter its final decay pattern.Specifically, the decay rate of T → H ± b reaches a maximum value of 23% for tan β ≤ 1.5 and m T ≥ 1500 GeV and becomes negligible for tan β ≥ 4, essentially due to the dependence of the BR on the mixing angle s L .Recall, in fact, that the latter is constrained to be rather small by the oblique parameters S and T .One can also read from the right panel of the figure that the heavy quark total width can be at most 10% of its mass at low tan β (indicated by dark blue points), so that this state is generally rather narrow and then can (tentatively, depending on the H ± decays) be reconstructed in LHC analysis.
We then present in Fig. 5

2HDM-II+(T B)
In this section, we discuss the case of the 2HDM-II+(T B).In the SM extended with such a VLQ multiplet, both mixing angles in the up-and down-type quark sectors enter the phenomenology of the model.For a given θ b R , θ t R , and m T mass, the relationship between the mass eigenstates and the mixing angles is given by [27]: Using the above relation, one can then compute m B for a given m T and mixing angles θ b R and θ t R .Additionally, the left mixings θ b L , θ t L may be calculated using Eq. ( 6).In Fig. 6, we once again present the excluded regions over the (m H ± , tan β) plane at a 95% CL.The exclusions are derived from B → X s γ (depicted as hatched areas) as well as from B 0 d → µ + µ − (depicted in green), B 0 s → µ + µ − (depicted in orange), and B u → τ ν (depicted in blue).Notably, in this representation, the limit imposed by B → X s γ can be further diminished compared to the singlet scenario.The exclusions shift the regions for B → X s γ towards smaller values of the charged Higgs boson mass, particularly reaching m H ± ∼ 400 GeV for s d R = 0.2 and m H ± ≃ 180 GeV for s d R = 0.45, maintaining a fixed value of s u R at 0.05 in both cases.Similarly to the previous scenario, the constraints from B s → µ + µ − become more stringent than in the 2HDM case in this representation, excluding all values above tan β ≃ 5.
Following the scan described in Tab. 3, we again present in Fig. 7 our 2HDM-II+(T B) surviving points in the (S, T ) plane, wherein the colour illustrates the usual difference in χ 2 (S, T ) values.As previously, we will define the data set to be used for additional analysis as the one lying within the illustrated 2σ band.As evident, in this representation, the shape differs from the one presented in the previous representation, and this is again attributed to the cancellation of contributions from 2HDM-II and VLQ states to the oblique parameters S and T .∆χ 2 (S, T ) Figure 7: Allowed points by all constraints (HiggsBounds, HiggsSignals, SuperIso and theoretical ones) superimposed onto the fit limits in the (S, T ) plane from EWPO data at 95% CL (with a correlation of 92%), with the colour code indicating ∆χ 2 (S, T ).Results are presented for the 2HDM-II+(T B).Fig. 8 illustrates the exclusion region derived from the oblique parameters S and T at 2σ.It is evident from the plot that EWPOs impose a stringent limit on the mixing angles.Specifically, as the mixing angle s u R increases, the exclusion region expands, covering the entire mass range of m T .Similarly, for the mixing angle s d R , an increase in the VLQ mass m T results in a larger exclusion region.From Fig. 9 (left), in contrast to the previous VLQ realisation, it is clear that charged Higgs boson production from T decays can reach more than 90% in the corresponding BR for medium tan β.In the right plot of the figure, we display Γ T /m T as a function of m T and tan β, where it can be seen that the total decay width of the T state can reach 30% of m T .Hence, unlike the singlet case, here we are normally in presence of a rather sizeable T → H ± b rate, which is affected by phase space effects only for light T states.Conversely, though, the T state can be quite wide, thereby rendering attempts at reconstructing its mass from kinematic analysis potentially more difficult than in the singlet case.
In Fig. 10, we present in the (m T , tan β) plane the aforementioned distribution of points mapped against BR 2 (T → H + b) × BR 2 (H + → tb) (left panel) and σ T T × BR 2 (T → H + b) × BR 2 (H + → tb) (right panel).One can see from these plots that the signal 2t4b could reach values up to 100 fb for medium tan β and for m T ≤ 1000 GeV.Therefore, owing to the enhanced coupling T H + b in this scenario, a much larger cross section is observed for this signature compared to the 2HDM-II+(T ) case.

BPs
Before concluding, in order to encourage experimental analyses of these scenarios at the LHC, we propose three BPs in Tab. 4 for the 2HDM-II+(T B) representation, which exhibits a more substantial cross-section compared to the 2HDM-II+(T ) one.The chosen BPs feature T masses within the range of approximately 800 GeV to 1300 GeV.Furthermore, the H ± masses are set to be heavy, enabling attempts to extract charged Higgs boson signatures in its tb decay, presenting a significant signal in the form of 2t4b final states originating from the pair production of the new VLQ T , followed bt T → H + b and H + → tb decays.

Conclusions
This paper goes into great detail about how charged Higgs bosons are made with a new top partner T in two different forms: 2HDM-II+(T ) (singlet) and 2HDM-II+(T B) (doublet).Our findings reveal a compelling capability of VLQs to alleviate the stringent limit imposed by B → X s γ on the charged Higgs mass (m H ± ).Specifically, in the singlet scenario, the mass limit can be reduced to around 500 GeV, while in the doublet case, it can decrease to approximately 200 GeV,particularly when the mixing angles are large.However, the inclusion of constraints from the oblique parameters S and T implies additional limitations, allowing only small VLQ mixing angles, in turn somewhat restraining the reduction in the aforementioned limit on m H ± .Consequently, in the 2HDM-II+(T ) scenario, the mass limit decreases to about 567 GeV, and in the 2HDM-II+(T B) scenario, it reduces to approximately 360 GeV, compared to the typical 580 GeV in the 2HDM-II.
Furthermore, we observed a remarkable distinction between the 2HDM-II+(T B) and 2HDM-II+(T ) representations.The former exhibits a substantial production rate of charged Higgs bosons, approaching nearly 100%, while the latter achieves only 25%, in terms of the BR(T → H + b).Our study thus further delved into the pair production of T (pp → T T ), followed by the decays T → H ± b of both VLQs.In both scenarios, the charged Higgs boson subsequently undergoes decay into tb, yielding a distinctive final state characterised by two top and four bottom quarks (2t4b).Notably, our results indicate that this signal can reach 100 fb in the 2HDM-II+(T B) whereas for the 2HDM-II+(T ) case, the corresponding rates are significantly smaller, so as to be of little relevance for the forthcoming LHC runs.Hence, we finally produced three BPs in the 2HDM-II+(T B) amenable to experimental investigation.
While a comprehensive analysis of H ± decays has not been performed in this study, we are confident that the H ± → tb channel (2t4b final state) from T T production and decay is a promising avenue for further exploration.This endeavour, most likely during Run 3 and certainly at the HL-LHC, awaits a future publication where a detailed analysis of H ± decay channels will be performed.

A Charged Higgs Boson Couplings
In this section, we provide further elaboration on the charged Higgs couplings within the 2HDM-II+(T ) and 2HDM-II+(T B) frameworks.
Assuming that the new VLQs predominantly mix with the third generation (i, j = 3), we can express this as: Let's focus on the charged Higgs boson H ± : Given our assumption that the new VLQs primarily mix with the third generation, we can write: 9 Note that in the 2HDM-II: Now, let's express ( * ) in the form of the mass matrix M u : Consequently, Eq. ( 14) can be formulated as: Subsequently, we obtain: Hence: BR(H ± → XY ) In Fig. 11, the BRs of T and H ± in the 2HDM-II+(T ) scenario are illustrated as functions of their respective masses.The left panel indicates that the BR(T → H + b) can reach a maximum of 25% for m T > 2000 GeV.This limitation is due to the coupling depicted earlier, which is proportional to the mixing angle s u L , a parameter constrained to be very small by the oblique parameters S and T .Shifting to the right panel, where clearly we can see that the charged Higgs boson predominantly decays through the fermionic channel H + → t b achieving approximately 100% across the entire range of charged Higgs masses.

A.2 2HDM-II+(T B)
The Yukawa Lagrangian for the doublet (T B) case can be expressed as: Following the same steps as in the previous scenario, we can express the charged Higgs Yukawa Lagrangian as: Now, let's express ( * ) and ( * * ) in the form of the mass matrices M u and M d : Then Eq. ( 22) can be written as: For the coupling of H + tb: we use : we get: then we get: For the coupling of H ± T b : we use Eq. ( 27) we get: Finally we get: BR(H ± → XY ) Similar to Fig. 11, we illustrate in Fig. 12 the BRs of T H ± as functions of their respective masses, but within the context of the 2HDM-II+(T B) scenario.Beginning with the left panel, in contrast to the previous scenario (2HDM-II+(T )), the production of charged Higgs from the new top quark T dominates for m T > 1000 GeV, reaching nearly 100%.Shifting to the right panel, and as anticipated for heavy charged Higgs masses, the primary decay channel remains the fermionic one H + → t b, achieving approximately 100% BR across the entire range of charged Higgs masses.

B LHC Limits on VLQs
In this section, we evaluate the compatibility of our results with the latest LHC constraints.Figure 13 presents our data as orange points, juxtaposed with the ATLAS limits, represented by blue [90] and green [91] lines , on the (m T , κ) plane.The coupling κ, explained further in [26,92], takes the form of s u L c u L in the singlet scenario (2HDM+T ) and s u R c u R in the doublet scenario (2HDM+T B).Importantly, , as discussed previously, the oblique parameters S and T impose stringent constraints on the mixing angles, limiting them to small values.This confinement is pivotal, ensuring our scenarios comply with the existing LHC limits and simultaneously allowing exploration of a broader VLQ mass (m T ) range.Blue lines represent ATLAS limits [91] (solid for observed, dashed for expected), while green lines represent [90].Notably, our analysis includes an extrapolation of these limits to expand the mass range downward to 750 GeV, considering that the original dataset commenced at ∼1000 GeV.

C VLQ Contributions to the Oblique Parameters
Herein, we derive and present analytical and general expressions for the two VLQ representations10 , T and T B. These expressions are formulated as functions of the scalar Passarino-Veltman functions, which are a standard in this area of study.The oblique parameters are expressed in terms of the two-point functions of the gauge bosons, as detailed below: Here, α denotes the fine-structure constant, while c W and s W represent the cosine and sine of the Weinberg angle, respectively.
For the analytical calculations of the self-energy of gauge bosons, as depicted in equations ( 33) and (34), we employ the FeynArts/FormCalc public codes.These calculations specifically address scenarios where VLQs interact exclusively with third-generation SM quarks.We concentrate on the contributions from these heavy quarks, namely the SM top and bottom quarks, along with the newly introduced VLQs, since the other contributions remain the same as in the SM.With this focus, the expressions for the electroweak gauge boson self-energies can be formulated as follows: −Π ZZ (q 2 ) = 2 −Π W W (q 2 ) = Here, c V f f represents the couplings between the gauge boson Z or W and two fermions (f = t, b, T, B), detailed in the next subsection C.1.The functions g ab , including g V V and g Zγ , are defined as:

C.1 Gauge Interactions of VLQs
The introduction of VLQs modifies the neutral and charged current interactions.The couplings between these exotic quarks the third-generation SM quarks, as well as the electroweak massive gauge bosons, are described as follows: Zq ′ q = e 2s w c w γ µ (κ L Zq ′ q L + κ R Zq ′ q R), Where κ L,R V q ′ q are the components for both left-and right-handed couplings of the Z and W bosons.Note that these couplings depend on the chosen representations of VLQs as follows: In the SU (2) L vector-like singlet scenario with T , the charged current couplings κ L,R W q ′ q are: The neutral current couplings κ L,R Zq ′ q are defined as:

C.1.2 (T B) Doublet
In the (T B) doublet scenario, the couplings of T , B, and the third-generation SM quarks with the W -boson are as follows: The neutral current couplings to the Z boson are expressed as:

Figure 1 :
Figure 1: Excluded regions of the (m H ± , tan β) parameter space by flavour constraints at 95% CL.Plots are presented for the 2HDM-II+(T ) singlet with s u L = 0 (left), s u L = 0.2 (middle) and s u L = 0.45 (right).

Figure 3 :
Figure 3: Exclusion regions by (S, T ) in the (m T , s d R ) plane with fixed parameters: m H = 532.80GeV, m A = 524.26GeV and m H ± = 588.02GeV, with tan β = 5.19.

Fig. 3
Fig. 3 illustrates the excluded regions at 95% CL by S and T in the (m T , s u L ) plane, considering the fixed parameters m H = 532.80GeV, m A = 524.26GeV, and m H ± = 588.02,with tan β = 5.19.From this figure, it is evident that the imposed constraints only permit low mixing angles, specifically s u L ≲ 0.1.Additionally, the stringency of the constraint increases with rising VLQ mass, m T .

Figure 6 :
Figure 6: Excluded regions of the (m H ± , tan β) parameter space by flavour constraints at 95% CL.Plots are presented for the 2HDM-II+(T B) doublet with s d R = 0 (left), s d R = 0.2 (middle) and s d R = 0.45 (right), with fixed s u R = 0.05.

Figure 8 :
Figure 8: Exclusion regions by (S, T ) in the (m T , s d R ) plane for various values of s u R , with fixed parameters: m H = 532.80,m A = 524.26and m H ± = 588.02,with tan β = 5.19.

Figure 11 :
Figure 11: → XY ) as a function of m T , with XY= H + b blue, ht red, W + b yellow and Zt green.Right: BR(H ± → XY ) as a function of m H ± , with XY= W + A blue, τ ν yellow, and tb green.For 2HDM-II+(T ).

Figure 13 :
Figure13: points in the (m T , κ) plane for (T ) singlet left and (T B) doublet right.Blue lines represent ATLAS limits[91] (solid for observed, dashed for expected), while green lines represent[90].Notably, our analysis includes an extrapolation of these limits to expand the mass range downward to 750 GeV, considering that the original dataset commenced at ∼1000 GeV.

Table 4 :
The full description of our BPs.Masses are in GeV.