Enhanced Charged Higgs Production through W-Higgs Fusion in W-b Scattering

We study the associated production of a charged Higgs boson with a bottom quark and a light quark at the LHC via p p \to H^\pm\,b\,j in the Two Higgs Doublet Models (2HDMs). Using the effective W approximation, we show that there is exact cancellation among various Feynman diagrams in high energy limit. This may imply that the production of charged Higgs can be significantly enhanced in the presence of large mass differences among the neutral Higgs bosons via W^\pm-Higgs fusion in the p p \to H^\pm\,b\,j process. Particularly, we emphasize the potential enhancement due to a light pseudoscalar boson $A$, which is still allowed by the current data by which we explicitly calculate the allowed regions in (M_A,\,\tan\beta) plane, and show that the production cross section can be as large as 0.1 pb for large $\tan\beta$. We also show that the transverse momentum distribution of the b quark can potentially distinguish the W^\pm-A fusion diagram from the top diagram. Finally, we point out further enhancement when we go beyond the 2HDMs.


I. INTRODUCTION
A new scalar boson h was discovered in the run I of LHC with 7 ⊕ 8 TeV energies in 2012 [1,2]. The combined measurement of the mass of the boson performed by the ATLAS and CMS collaborations based on the data from h → γγ and h → ZZ → 4l channels is m h = 125.09 ± 0.21 (stat.) ± 0.11 (syst.) GeV [3]. Furthermore, the measured properties of the new particle are best described by the standard-model (SM) Higgs boson [4,5].
The mission of the new LHC run at 13 TeV (and later upgraded to 14 TeV) is two folds: the first task is the improvement of the scalar boson mass and scalar boson coupling measurements and the second one would be to find a clear hint of new physics. By performing accurate measurements of the scalar boson couplings to the SM particles would be helpful to determine if the Higgs-like particle is indeed the SM Higgs boson or a Higgs boson that belongs to a higher representation, such as models with extra Higgs doublets, extra triplets, or singlets. Most of higher Higgs representations with extra doublet or triplet Higgs fields predict in their spectrum one or more singly-or doubly-charged Higgs bosons. A discovery of such charged Higgs bosons would be an indisputable signal of new physics.
In the two-Higgs-doublet models (2HDMs) or the minimal supersymmetric standard model (MSSM), the charged Higgs boson can be abundantly produced both at hadron and e + e − colliders. At hadron colliders, the charged Higgs boson can be produced through several channels: • Production from top decay. If the mass of the charged Higgs boson is smaller than m t − m b , the production of tt pairs provides an excellent source of the charged Higgs bosons. If kinematically allowed, one of the top and anti-top quarks, say the antitop quark can decay into H −b , competing with the SM decay oft → W −b . This mechanism pp → tt → tbH − can provide an important source of light charged Higgs bosons and offers a much cleaner signature than that of direct production.
• Single charged Higgs production. The most important ones are gb → tH − and gg → tbH − [6]. These are QCD processes, and thus the cross sections are expected to be large. We can also have a single charged Higgs boson produced in association with a W ± gauge boson via the loop process gg → W ± H ∓ or the tree level process bb → W ± H ∓ [7]. Similarly, the single charged Higgs boson can be produced in association with a Higgs boson: qq → W ± * → φH ± where φ denotes one the the three neutral MSSM Higgs bosons [8]. Most of these processes are of the Drell-Yan type, they are expected to give substantial cross sections only for the charged Higgs mass below about 200 GeV.
• Single charged Higgs boson production associated with a bottom quark and a light quark qb → q H + b in the MSSM framework in which the neutral heavier Higgs bosons are almost degenerate [9].
At the Tevatron and LHC, detection of light charged Higgs boson with M H ± < m t − m b is straightforward from tt production followed by the decayt →bH − or t → bH + . Such a light charged Higgs boson can be detected for any value of tan β in the τ ν decay which is indeed the dominant decay mode. The ATLAS and CMS have already had an exclusion on B(t → bH + ) × B(H ± → τ ν) based on this decay channel [12,13].
In the MSSM and 2HDMs, the heavy charged Higgs boson with M H ± > ∼ m t would decay predominantly into tb. The experimental search is rather difficult due to large irreducible and reducible backgrounds associated with H + → tb decay. However, in Refs. [14] it has been demonstrated that the H + → tb signature can lead to a visible signal at the LHC provided that the charged Higgs mass is below 600 GeV and tan β is either below < ∼ 1.5 or above > ∼ 40. An alternative decay mode to detect a heavy charged Higgs boson is H ± → τ ν [15], even if such a decay is suppressed for heavy charged Higgs bosons, it has the advantage of being much cleaner than H + → tb. Recently, a new technique using the jet substructure for the heavy charged Higgs boson decaying to tb has been proposed in [16].
In the MSSM, the branching ratio of the decay mode B(H ± → W ± h) could at best be at the level of 10% for low tan β while in the 2HDM-I * it could dominate over B(H + → tb).
Therefore, H ± → W ± h could be an alternative channel to discover the heavy charged Higgs boson at the LHC [17]. Similarly, when the CP-odd Higgs boson A is light enough, the decay of H ± → W ± A could be the dominant one in the 2HDMs and could also be used to search for heavy charged Higgs bosons. Finally, in the models with higher Higgs representations * See Section II for classification of 2HDMs.
such as the triplet representation of the Higgs, the charged Higgs boson could decay into W ± Z with a significant branching fraction [18]. This decay channel could lead to isolated leptons in the final state and could be used to distinguish between models with charged Higgs bosons.
The aim of this work is to study singly-charged Higgs boson production in association with a bottom quark and a jet q with the subprocess qb → q H + b. Such a process had been studied for the first time in Ref. [9] which showed that the rate is rather small in the MSSM due to a huge cancellation between the top-and Higgs-mediated diagrams as we will show. In the present study, we discuss the production rate of this process and its sensitivity to tan β in the 2HDMs where the masses of the heavier Higgs bosons are not fixed by one mass parameter as in the MSSM. Specifically, we demonstrate that the process possesses destructive interference between the s-and t-channel diagrams, which significantly reduces the cross section. Especially, when the two heavier neutral Higgs bosons are decoupled from the lightest one and they are degenerate, the cross section is canceled to a large extent. In addition, we show that with a relatively light CP-odd Higgs boson, which is still allowed by the current data, the production cross section of the charged Higgs boson via W ± -Higgs fusion in the pp → H ± b j process can be significantly enhanced at the LHC.
The organization of the work is as follows. In the next section, we write down the framework for the 2HDMs, provide analytic understanding of the process in terms of the 2 → 2 subprocess, and also describe the full 2 → 3 process in detail. We present the numerical results in Sec. III. Some cases beyond the 2HDMs are considered in Sec. IV and we conclude in Sec. V.

II.
qb → q H + b IN TWO HIGGS DOUBLET MODELS

A. Brief review of two-Higgs-doublet models
In 2HDMs the electroweak symmetry breaking is performed by two scalar fields Φ 1 and Φ 2 which are parameterized by † : (1) † For an overview, see Ref. [19].
We denote v 1 = v cos β = vc β and v 2 = v sin β = vs β . The parameterization of the general scalar potential which is gauge invariant and possesses a general CP structure can be found in [20]. In the present study we are mainly interested in Higgs coupling to fermions and gauge couplings to be listed slightly later.
The general structure for Yukawa couplings is given in the following interactions where Q T = (u L , d L ), L T = (ν L , l L ), and Φ i = iτ 2 Φ * i with We note that there is a freedom to redefine the two linear combinations of Φ 2 and Φ 1 to eliminate the coupling of the up-type quarks to Φ 1 [21]. The 2HDMs are classified according to the values of η l 1,2 and η d 1,2 as in Table I. To define the Higgs mass eigenstates, we first rotate the imaginary components a i and the charged ones φ + 1 and φ + 2 in order to obtain the would-be-goldstones G 0 and G ± that would be eaten by the longitudinal components of the Z and W ± bosons. These rotations result in an CP-odd state a = A = −s β a 1 + c β a 2 and a pair of charged Higgs bosons In the most general case with CP violation, the mass eigenstates of the neutral Higgs bosons are obtained by diagonalizing the 3 × 3 mass matrix M 2 0 by an orthogonal 3 × 3 mixing matrix O that relates the interaction eigenstates to the mass eigenstates as follow: Here the states H i do not have to carry any definite CP-parity and they have both CP-even and CP-odd components.
After identifying the Yukawa couplings by one can easily obtain, from the above Lagrangian, the following Higgs-fermion-fermion in- and where P L,R = (1 ∓ γ 5 )/2.
Before moving to the next subsection, we present the mixing matrix O in the CPconserving case in terms of the mixing angle α. In our numerical study, to deliver our findings more clearly, we focus on the CP-conserving case. In this case the matrix O takes the following form: assuming H 3 is the pure CP-odd state or H 3 = A. In this notation, the decoupling limit of the 2HDM [23], which seems to be favored by the current LHC data, is β − α → π/2: and (d).
B. Subprocess W + b → H + b and unitarity In this subsection, we present the amplitude of the process q b → q H ± b in the effective W approximation. In this process, the dominant contribution comes from the region where the W boson emitted from the incoming quark q is close to on shell and one can approximately represent the process by the W boson scattering with the incoming b quark or anti-b quark to give H ± b or H ±b in the final state: The process W + b → H + b receives contributions from Fig. 1 interactions needed for these two subprocesses can be obtained from the Yukawa interactions given by Eqs. (6) and (7) and from the covariant derivatives: , where and in types II and IV and in types I and III.
The amplitude of each diagram for where s = (p 1 + q 1 ) 2 = (p 2 + q 2 ) 2 , t = (p 1 − p 2 ) 2 = (q 2 − q 1 ) 2 , and u = (p 1 − q 2 ) 2 = (p 2 − q 1 ) 2 and µ (q 1 ) denotes the polarization vector of W + boson. The amplitudes for the (c) and (d) diagrams in Fig. 1 can be obtained by replacing u(p 1,2 ) with v(p 1,2 ) and (s − m 2 t ) with (u − m 2 t ). In the high-energy limit, s, |t|, |u| where we have taken the longitudinally polarized W or µ ( denoting the four-momenta of the exchanging top quark with p t . Incidentally, the square of the 4-momenta of the internal neutral Higgs is p 2 We note that the c R term, which is suppressed by m t / √ s, is neglected here. In types II and IV, the c R term could be important when tan β < ∼ m t /m b ∼ 7. As shall be seen, the total cross section takes its smallest value at tan β ∼ 7. When tan β > ∼ 7, compared to the c L term, the c R term could be safely neglected when On the other hand, in types I and III, the c R term can be neglected only if √ s/m t m t /m b . Therefore, the high-enegy limit should be applied with more cautions at the LHC for types I and III. But, for the 2HDM types I and III, the production cross sections are suppressed by 1/ tan 2 β with increasing tan β and the largest value with tan β = 1 is only ∼ 30 fb, as shall be shown. The amplitude M (a)+(b) for the b-initiated processes consists of the contributions from the t-channel Higgs-exchange diagrams (a) and the s-channel top-exchange diagram (b) .
The c L term in the second line is from the s-channel diagram and all the others from the t-channel ones. Therefore, the high-energy limit has been obtained by taking On the other hand, the high-energy limit of the amplitude M (c)+(d) for thē b-initiated processes can be obtained by replacing u(p 1,2 ) with v(p 1,2 ) in Eq. (16) and taking . Otherwise, the expression given by Eq. (16) can be applicable for both the bandb-initiated processes.
The high-energy limit expression Eq. (16) contains two non-interfering terms both of which grow as √ −t and therefore the absence of these unitarity-breaking terms require the following three types of sum rules: The first one gives the relation between the charged Higgs coupling to t and b quarks (c L ) and the sum over the Higgs states of the scalar and pseudoscalar products (g S i S i + g P i P i ) of the neutral Higgs couplings to b quarks and those to the charged Higgs and W . The second relation shows the sum over the Higgs states of the scalar products should be the same as that of the pseudoscalar ones. And the third relation implies that there is no CP violation if the scalar-pseudoscalar products are summed over the three Higgs states.
These interesting sum rules can be explicitly checked in each 2HDM. In types II and IV, using the orthogonality of the mixing matrix O, we find that With c L = tan β, the unitarity conditions are satisfied automatically. On the other hand, in types I and III, we find that With c L = −1/ tan β, the unitarity conditions are again satisfied automatically.
This is the proof for the unitarity of the subprocess W + b → bH + in the high energy limit in the general 2HDMs with or without CP violation. The same proof also applies to the case ofb initiated subprocess W +b →bH + .
C. The full process qb → q H + b After discussing the essence of the physics involved in the 2 → 2 subprocess, we shall describe the full 2 → 3 process ‡ . We shall consider the CP-conserving case for simplicity, unless stated otherwise. In this case, without loss of generality, we identify H 1 = h, H 2 = H, and H 3 = A, where h and H denote the lighter and heavier CP-even Higgs bosons, ‡ For a full consideration of NLO corrections, one may need to take account of the 2 → 4 process: qg → qH + bb. We leave this part for further work. is that the former has a s-channel exchange top propagator while the latter has a u-channel one. Similarly, the fermion-line direction of the q can be reversed to includeq →q transition.
Therefore, we have a number of initial states for production of H + : (u, c,d,s) ⊗ (b,b). We can then take the charge conjugate to obtain the H − processes.
The diagram in Fig. 2 Table II up to some normalizations. Incidentally, the non-vanishing neutral Higgs couplings to charged Higgs and W are given by using the form of O given by Eq. (8). Abb  Fig. 2(a) and one s-channel diagram mediated by the top quark in Fig. 2(b). Similarly, for Wb → H +b subprocess we have three t-channel diagrams with H i = h, H, A in Fig. 2(c) and one u-channel diagram mediated by the top quark in Fig. 2(d). We have shown in the previous subsection using the effective W approximation that there is strong cancellation among the diagrams, and indeed all four diagrams will exactly cancel one another in the high energy limit. Therefore, if we employ a much lighter CP-odd Higgs boson, which is still allowed by the current data, we expect a strong enhancement to the production cross section of this process. Experimentally, one can use this process to search for the charged Higgs boson and investigate the effects of light CP-odd Higgs boson. Perhaps, a negative search would close out the entire window of light CP-odd Higgs boson.

III. NUMERICAL RESULTS
In this section, we first present some numerical results for the subprocesses W + b → bH + and W +b →bH + for a given value of center-of-mass energy √ S and then consider the full process pp → H + bj in the 2HDM of type I (III) and II (IV).
A. The 2 → 2 subprocess in the effective W approximation We shall limit ourself to the CP conserving case taking H 1 = h, H 2 = H, and H 3 = A.
And the couplings of the neutral Higgs bosons to the charged Higgs and W are: S h = cos(β − α), S H = − sin(β − α), and P A = 1. Neglecting the contribution from the lightest Higgs boson h as in the decoupling limit cos(β − α) → 0, we observe that in the high-energy limit the cross section of the subprocess behaves like We note that the cross section suffers a huge cancellation between the top-and Higgsmediated diagrams and a further cancellation between the Higgs-mediated diagrams. Taking the type II model as an example, we find with c L = −S H g S H = −P A g P A = tan β. Note, for the W +b → H +b process, while the high-energy behavior of the t-only, (t + H)-only, and (t + A)-only amplitudes remains the same.
Furthermore, independent of the type of 2HDMs we note that for between u-channel top diagram and t-channel W -A and W -H fusion diagrams. We stress that the destructive interference in theb-initiated process is less severe than the b-initiated one, such that the total cross section for W +b → H +b is about one order of magnitude larger than that for W + b → H + b. This is because the cancellation between the u-and t-channel

B. For full process pp → H ± bj
In the previous subsection we have shown analytically and illustrated numerically the cancellation in the subprocesses W + b → bH + and W +b → H +b between the top diagram and W -A and W -H fusion diagrams using the effective W approximation. In Fig. 4, we show the cross sections for the full 2 → 3 processes qb → q H + b (upper panels), qb → q H +b It is clear from the left panels that the cross sections are enhanced for both small tan β ≈ 1 and large tan β, the latter of which is associated with enhanced bottom Yukawa couplings.
A dip indeed occurs around tan β ≈ 6 in the tan β plots, which corresponds to where the top and bottom Yukawa couplings become similar m b tan β ≈ m t / tan β. We stress that our results are in good agreement with Ref. [9]. ¶ Our numerical calculations of the several cross sections for the b-andb-initiated full 2 → 3 processes presented here are carried out by use of the Helicity Amplitude Method [24]. We compare our results for the total cross sections with those obtained using MadGraph [25] and find excellent agreements.  Since we have observed that in order to enhance pp → bH + j cross sections one needs both non-degenerate A and H and also large tan β, here we attempt to find what would be the largest possible value for tan β such that it is still consistent with τ + τ − data for 100 ≤ M A ≤ 340 GeV and assuming that the heavy CP-even Higgs boson is rather heavy. A similar study with the 7 TeV data had been done in [28] for 2HDM. Because of CP invariance the CP-odd Higgs boson A does not couple to W W or ZZ, and the partial decay widths into loop mediated gg, γγ, and γZ channels are highly suppressed. The decay channel A → hZ, which is proportional to cos(β − α), will also be severely suppressed if we assume that β − α is close to the decoupling limit. Therefore, the CP-odd Higgs boson predominantly decays into fermion pairs: qq, q = b, s, d, c, u and l + l − l = τ, µ, e.
In 2HDM-I and -IV, the coupling Aτ + τ − is proportional to 1/ tan β while in 2HDM-II and -III it is proportional to tan β. On the other hand, from Table I the coupling of Abb is proportional to tan β in 2HDM-II and -IV but 1/ tan β in 2HDM-I and -III. Thus, it is clear that in 2HDM-I (resp. II) both the production rate gg → A and the decay A → τ + τ − are suppressed (resp. enhanced) for large tan β. We then expect a strong exclusion for large tan β in type II but not in type I. In all four 2HDM types we expect some enhancement for small 0.5 ≤ tan β ≤ 1 because Att is proportional to m t / tan β.
For a given CP-odd Higgs mass M A and tan β, the cross section of gg → A, which only depends on these 2 parameters, is computed with help of SUSHI public code [29].
In the decoupling limit A → Zh is vanishing, and if A → {W ± H ∓ , ZH, tt} are closed, the branching ratio of A → τ + τ depends on tan β and M A only, and there is no sin α dependence. Therefore, the cross section gg → A → τ + τ − will depend only on tan β and M A . Hence, our exclusion from τ + τ − data can be given in the plane of (M A , tan β). After computing the cross section gg → A times the branching fraction of A → τ + τ − , we compare our theoretical predictions with the ATLAS data [26]. Note that the ATLAS data were given for M A = 90, 100, ...340 GeV with steps of 10 GeV or even larger in some cases. Therefore, we have used linear interpolations for M A values in-between the data.
We draw in Fig. 7

IV. BEYOND TWO-HIGGS-DOUBLET MODELS
Another interesting possibility to enhance the production of charged Higgs boson might be the case in which a 2HDM is not an ultraviolet (UV) complete theory and the UV cutoff locates far above the mass scale of heavy Higgs bosons * * . In this case, taking one of the 2HDMs as a low-energy reference model, the relevant interactions may be parameterized as In the MSSM, for example, including the tan β-enhanced SUSY threshold corrections to the down-type Yukawa couplings, we have with [30] where g and H are the contributions from the sbottom-gluino exchange diagram and from stop-Higgsino diagram, respectively. Their explicit expressions are where M 3 is the gluino mass parameter, h t and A t are the top-quark Yukawa and trilinear couplings, respectively.
Without loss of generality we choose the 2HDM type II as the reference model, and show the change in production cross sections with the variations in the couplings ξ S,P i and ξ L .
We used Eq. (25) as the guidance. We first show the ratio of the cross sections for varying ξ S i = ξ P i = ξ L between −2 and +2 to the cross section at the 2HDM type II values, i.e., ξ S i = ξ P i = ξ L = 1 in Fig. 8, for tan β = 1, 30. The tan β = 1 curves show almost no * * We note that this kind of enhancement arising from unitarity violation might be dangerous and should be taken with caution, because a UV-complete model might contain new interactions to restore the unitarity.  sensitivity to ξ S i = ξ P i = ξ L because the process is dominated by the top-Yukawa term. On the other hand, the tan β = 30 curves are dominated by the bottom-Yukawa term. It is obvious that the ratio is close to zero for ξ S i = ξ P i = ξ L = 0, and is one for ξ S i = ξ P i = ξ L = 1. The ratio grows as the square of the couplings around 0 to almost 4 at ξ S i = ξ P i = ξ L = ±2. If for some higher scale dynamics such that ξ S i and ξ P i do not change in the same manner, we show the effects on the cross sections in Fig. 9. On the left panel, we show the ratio of cross sections with varying ξ P A between −2 and +2 to the cross section at the 2HDM value (i.e. ξ P A = 1) by keeping all other parameters at their 2HDM type II values. Again, the sensitivity at tan β = 1 is negligible, while it becomes quite nontrivial for large tan β = 30.
As we have shown in Sec. III that the W -A diagram interferes destructively with the top diagram, we can now turn the destructive interference into constructive one by reversing the sign of ξ P A . Furthermore, when ξ P A is negative the second term in Eq. (21) would not vanish. It is then very clear that the ratio becomes quite large at negative ξ P A . At ξ P A = 0, the ratio is already larger than 1 because no interference comes from the W -A diagram.
Similar behavior occurs for ξ S H as shown on the right panel in Fig. 9. The ratios that ξ S H can attain are very similar to those of ξ P A .

V. CONCLUSIONS
We have performed the study of b-andb-initiated processes of pp → jH ± b/b in the 2HDM framework at the LHC-14 in the decoupling limit (sin α = − cos β), which is favored by the current Higgs data. We have identified strong cancellations between the top diagram and the W -H i (H i = h, H, A) diagrams which rendered the process very suppressed. The cancellation is indeed the strongest when the A and H are degenerate and in the decoupling limit. We pointed out that if the pseudoscalar Higgs boson A is much lighter than the CP-even Higgs boson H, the cross section of charged-Higgs production can be substantially enhanced, because the cancellation is no longer complete. We have explicitly obtained the exclusion in parameter space of (M A , tan β) for 2HDM types I to IV based on the LHC data on σ(g → Φ) × B(Φ → τ + τ − ). In the allowed paramete space, the size of production cross section can be as large as O(50) fb for M A = 100 GeV and tan β = 30 for types II and IV.
This is the main result of the work.
We offer the following comments on the findings of this work as follows.
1. The b-initiated process for production of H + in qb → qH + b suffers from a very strong cancellation between the top diagram and the W -H i diagrams. However, thebinitiated process for production of H + in qb → qH +b suffers a less severe cancellation, mainly because of the u-channel top-exchange instead of s-channel.
2. The strong cancellation is dictated by the absence of the unitarity-breaking terms and we find the sum rules expressed by the relevant Higgs couplings, see Eq. (17).
of this work has been done.