Electroweak production of multiple (pseudo)scalars in the 2HDM

The two-Higgs Doublet Model (2HDM) is the most minimal extension of the Standard Model (SM) containing extra Higgs doublet fields. Given the multiplicity of Higgs states in a 2HDM, its Higgs potential is significantly more involved than the SM one. Importantly, it contains a multitude of Higgs triple self-couplings, unlike the SM, which only has one. These interactions are key to understanding the phenomenology of the 2HDM, as they uniquely determine the form of the potential. Several studies analysing the prospects of measuring these couplings at the Large Hadron Collider (LHC) have found them to be quite low generally. However, such studies have largely concentrated on Higgs pair-production induced by gluon-gluon scattering, either via direct annihilation or followed by their splitting into $b$-(anti)quark pairs, which in turn annihilate leaving behind spectator $b$-(anti)quarks. Both of these channels are therefore governed by QCD dynamics. We compare here the yields of such channels to those initiated by (primarily) valence quarks, which involve Electro-Weak (EW) interactions only, for neutral multi-Higgs final states. We find that EW production can be dominant over QCD production for certain final state combinations. We also illustrate that charged final states, which can only be produced via EW modes, could serve as important probes of some $H^\pm$ triple couplings, that are inaccessible in QCD-induced processes, during Run 2 and 3 of the LHC. Our analysis covers regions of the parameter space of the Type-I 2HDM that have escaped the most up-to-date experimental constraints coming from EW precision data, LHC measurements of the 125 GeV Higgs boson properties, searches for additional Higgs states, and flavour physics.


Introduction
The 2012 discovery of a neutral Higgs boson [1,2], H obs , with a mass near 125 GeV, is strong evidence for gauge boson masses being induced by the Higgs mechanism of Electroweak Symmetry Breaking (EWSB). While the Higgs boson data collected at the LHC is still consistent with the minimal EWSB dynamics of the SM, some other experimental results cannot be reconciled with it. In particular, certain anomalies in the flavour sector [3,4,5,6,7] are far more compatible with an extended Higgs sector [8,9,10,11,12] than with the SM. In view of this, as the H obs state emerges from a Higgs doublet in the SM, the phenomenology of its minimal extension by another Higgs doublet, which results in the two-Higgs Doublet Model, deserves particular attention.
In the 2HDM Higgs sector, five physical states emerge after EWSB: three neutral, of which two are scalars (h and H, with m h < m H ) and one a pseudoscalar (A), plus a charged pair (H ± ). The theory of this scenario is well-understood (see, e.g., [13,14]), but its phenomenological investigation is far from complete at present. In particular, while there exist some indications of what the accessible discovery channels of the additional Higgs bosons of a 2HDM could be at the LHC, little effort has been spent on assessing which are the most suitable channels to pin down the specific nature of the underlying Higgs dynamics. The reason is that there are several incarnations of the 2HDM and, although each of them yields a different phenomenological pattern in general, there exists a significant level of degeneracy among them if only the production and decay channels of a single Higgs state are studied. Indeed, for an unequivocal extraction of a 2HDM scenario involved in EWSB, the various components of the scalar potential ought to be accessed experimentally. This makes the study of multi-Higgs final states mandatory.
In the context of the LHC, several analyses exist in literature, addressing double, or even triple, Higgs production, assuming a 2HDM to be the underlying framework (see, e.g., Ref. [15,16] for a review). However, the majority of such analyses have concentrated on production modes induced by QCD dynamics, notably gluon-gluon (gg) fusion into a (neutral) pair of Higgs states. These pairs emerge either from a primary Higgs state (resonantly or otherwise) or as Higgs-strahlung from a box diagram involving heavy fermion loops. Alternatively, because Higgs couplings to quarks are of Yukawa type (i.e., proportional to the quark mass), the bb scattering channel has also been exploited. It should be noted that b-quarks are not valence partons and are therefore produced from a (double) gluon splitting. Hence this channel is also intrinsically gg-induced.
While these QCD processes clearly afford one the possibility of the direct measurements of a number of terms in the 2HDM Lagrangian, the complete list of these terms is much longer. In order to remedy this, we study here double and triple Higgs boson production in qq ( ) -induced EW interactions, where q represents predominantly a valence u, d, in the Type-I 2HDM. This theoretical scenario has been shown to yield spectacular signals involving light neutral Higgs states, with a mass smaller than that of H obs , that are potentially accessible at the LHC, see Refs. [17,18,19,20]. Here, we assess the complementary portion of the Type-I 2HDM parameter space, wherein the lighter of the two scalar Higgs states has a mass of 125 GeV, along the lines of [21], which considered a similar setup but concentrated exclusively on charged Higgs boson signals. We will argue that the cross sections for the production of some of these double (and triple) Higgs final states could be accessible within the already scheduled LHC Runs. We will in particular show that in certain cases not only can these cross sections be larger for EW processes compared to the QCD-initiated processes, but the former can also possibly provide access to some of the Higgs self-couplings that none of the latter can.
The article is organised as fellows. In Sec. 2 we review in some detail the various types of minimally flavour-violating 2HDM and identify the Higgs-Higgs and Higgs-gauge couplings available in it. In Sec. 3 we discuss parameter space regions of the Type-I 2HDM which are amenable to LHC investigation in multi-Higgs final states, satisfying all the theoretical and experimental constraints of relevance. In Sec. 4 we discuss our results. Finally, we present our conclusions in Sec. 5.

The two-Higgs Doublet Model
The 2HDM contains two Higgs doublet fields, Φ 1 and Φ 2 , and its most general potential can be written as Upon EWSB, Φ 1 and Φ 2 are defined in terms of their respective vacuum expectation values v 1 and v 2 , the physical Higgs states h, H, A and H ± and the Goldstone bosons G and G ± as where α is the mixing angle of the CP-even interaction states and tan β ≡ v 1 /v 2 . Upon minimisation of the Higgs potential in Eq. (1), after rewriting it in terms of these expanded fields, the bare masses m 2 11 and m 2 22 get replaced by v 1,2 . Similarly, the quartic couplings λ 1−5 in Eq. (1) can be traded for the masses of the four physical Higgs bosons as well as the mixing parameter sin(β − α). The free parameters of a 2HDM thus include m h , m H , m A , m H ± , λ 6 , λ 7 , m 2 12 , tan β and sin(β − α). If all the SM fermions couple to both the Higgs fields of a 2HDM, it can lead to dangerous flavour-changing neutral currents (FCNCs). In order to avoid large FCNCs, the most general approach taken is to enforce a Z 2 symmetry on the Lagrangian, so that each type of fermion only couples to one of the doublets [22,23]. This symmetry is softly broken by the m 2 12 term in the Higgs potential above and explicitly broken by the λ 6,7 terms. In the following we restrict ourselves to the CP-conserving case λ 6 = λ 7 = 0.
The Type-I 2HDM is obtained if (conventionally) Φ 1 → −Φ 1 under the Z 2 symmetry, so that all the quarks and charged leptons couple only to Φ 2 . On the other hand, the Type-II 2HDM observes the transformation property so that only these mutually couple, while the up-type quarks couple instead to Φ 2 . The Type-III (or Type Y or flipped) model is built such that Φ 2 couples to the up-type quarks and the leptons and Φ 1 couples to the downtype quarks only while in the Type-IV (or Type X or lepton-specific) model Φ 2 couples to all the quarks and Φ 1 to all the leptons. In this study, we will concentrate on the Type-I 2HDM, for whose allowed parameter space the relevance of the aforementioned EW processes with respect to the QCD-induced ones is most pronounced. (We will defer the study of the other Types to future publications. ) We are in particular interested in the couplings of the (pseudo)scalars to gauge bosons and the triple-Higgs couplings. The (pseudo)scalar-gauge couplings λ HAZ and λ HH + W − are proportional to sin(β − α), and λ hAZ and λ hH + W − to cos(β − α), while λ AH + W − is independent of the 2HDM angles. The LHC data requires at least one of h and H to have a mass near 125 GeV and SM-like couplings. In order for h to satisfy this condition, | sin(β − α)| (| cos(β − α)|) should not be too far from 1 (0). This implies that couplings proportional to sin(β − α) should be larger than those proportional to cos(β − α), which indeed vanishes in the decoupling limit [24]. 1 However, given the current measurements of the properties of the H obs , this limit need not be strictly adhered. For this reason we treat sin(β − α) as a free parameter here.

Parameter space scans and constraints
We numerically scanned the parameters of the Type-I 2HDM using the 2HDM Calculator (2HDMC) [28] in the ranges: • Unitarity (default unitarity limit is 16π), perturbativity (default perturbativity limit is 4π) and Higgs potential stability conditions were enforced with methods provided by 2HDMC.
• The oblique parameters S, T and U were calculated with 2HDMC methods and were required to fall within the 95% Confidence Level (CL) ellipsoid based on 2018 PDG values [29]: with correlations ρ ST = 0.92, ρ SU = −0.66 and ρ T U = −0.86.
• All scalar states in the models satisfied all (95% CL) constraints included in the program HiggsBounds 5.2.0 [30].
• The B-physics observables were calculated with SuperIso 3.4 [31]. They were required to meet the limits from the SuperIso manual (95% CL), except for the three Branching Ratios (BRs) listed in Tab. 1, for which we applied the constraints on (m H ± , tan β) derived in [32].  • The signal strengths for h → γγ, ZZ, W W , τ τ and bb, calculated using HiggsSignals 2.2.0 [33], were required to match within 2σ of the LHC measurements for H obs given in Tab. 1.

Results and discussion
For each scanned point, we calculated tree-level cross sections in pp collisions with √ s = 13 TeV for . These cross sections were calculated using the 2HDMC model [28] with MadGraph5 aMC@NLO [34]. For the neutral 2-Body Final States (2BFSs), we also calculated the cross sections for bb → h i h j in the five flavour scheme using the same methods and for gg → h i h j (gluon-gluon fusion) using MadGraph based codes [16]. From these, we estimated cross sections for the 3-Body Final States (3BFSs) . This was done by multiplying the cross section for a given 2 → 2 process (where available) with the appropriate BR, considering all possible on-shell decays of the heavier (pseudo)scalars.

Charged final states
The charged 2BFSs, each containing the H ± along with one neutral Higgs state, are shown in Fig. 1. These are all necessarily produced by an initial qq state, having no counterpart in gg/bb production, and each shows cross sections of at least 10 fb in some kinematic regions. Whereas the cross sections for HH ± and AH ± states are strongly correlated with their cumulative masses, those of hH ± show greater variation. We find that this variation is correlated with sin(β − α), with maximal cross sections corresponding to minimal sin(β − α). This is consistent with a cross section dominated by an s-channel W ± , whose coupling to hH ± is proportional to cos(β − α), as noted earlier. Because the h is required to have very SM-like properties, the points selected by our scans have |sin(β − α)| close to 1, which means that cos(β − α) may span several orders of magnitude, resulting in large variation in possible hH ± cross sections. Conversely, the HH ± W ∓ coupling varies as sin(β − α) and the AH ± W ∓ coupling has no dependence on sin(β − α), so the cross sections for other charged 2BFSs are also consistent with dominant s-channel W ± production, being determined almost entirely by the final state kinematics.
If we consider the possibility of either the charged or neutral Higgs in a 2BFS decaying, we can have final states containing either three Higgs bosons or two Higgs bosons accompanied by one gauge boson. The cross sections for such 3BFSs, for processes where it exceeds 1 fb for at least one point from the scan, are shown in Fig. 2. The maximal cross sections for all such 3BFSs are summarised in Tab. 2. We note that there are several possible processes which would lead to cross sections of this size, and all of the possible h i → h j + h k /V k decays are represented, excepting one; H → H + H − does not appear, because our scan did not select any points meeting the condition m H > 2m H ± required for this decay. We also note in Fig. 2   this is simply because most points do not have masses which satisfy the kinematic requirements for these decays. However, our broad scan did find some points where the cross sections containing these decays are very substantial and a more comprehensive scanning should find additional candidates.

Neutral final states
The neutral final states may be produced by qq-induced processes as well as via loop-induced processes initiated by a pair of gluons. The cross sections for the 2 → 2 neutral processes are shown in Figs. 3 and 4 as a comparison between qq and gg/bb production. We find that, for H + H − , hA and HA final states, the qq cross sections can all exceed 10 fb and, for some regions of parameter space, dominate the combined gg + bb production, as shown in Fig. 3. While the remaining neutral 2BFSs, namely hh, HH, AA and hH, have EW cross sections unlikely to be relevant at the LHC, their gg/bb production can be significant, as seen in Fig. 4, so these are still   the more useful modes. For the neutral 2BFSs too we consider the possibility of one of the Higgs bosons decaying, and the resulting 3BFSs for which qq cross sections exceed 1 fb are shown in Fig. 5 and Tab. 2. Again, we see some cross sections dominated by qq production. Here too all of the possible Higgs-to-Higgs decays are included, apart from H → H + H − , which is not kinematically available to any of our points. As with the charged 3BFSs, plots involving H ± → W ± A, H → AA, or H → AZ are sparsely populated, since these decays are only allowed for a small number of scanned points.

Higgs boson couplings from multi-Higgs states at the LHC
Evidently, based on our results so far, several different processes and final states could potentially be observed at the LHC and serve as probes of the various couplings appearing in the 2HDM Lagrangian. In Tab. 3 we have listed the ten triple-Higgs couplings (a -h) and the six (pseudo)scalargauge couplings (i -n) that appear in the 2HDM Lagrangian (assuming minimal flavour violation) row-wise and all the possible di-Higgs 2BFS combinations column-wise. If a coupling can potentially enter the secondary vertex of both gg/bb-and qq-initiated s-channel production processes of  a given 2BFS at the LHC, the corresponding cell is checked. A box around a checkmark, however, means that the corresponding 2BFS can only be qq -produced (i.e., when it has a non-zero net electric charge). We also show in parentheses a 3BFS, comprising of at least two Higgs bosons and at most one gauge boson, that can result from the 2BFS at the top of the column.
For a 3BFS, the coupling at the start of the row in the table appears, instead of the secondary vertex in the production process of the 2BFS at the top of the column, in the tertiary vertex between one of the two incoming Higgs bosons and an outgoing Higgs+Higgs/gauge state. A 3BFS is typeset in grey if this incoming Higgs state can only be off-shell, i.e., if its mass, m x , is smaller than the sum of the masses, m j + m k , of the two outgoing particles. In such a case, the cross section for the corresponding process cannot be evaluated in the σ(gg/bb/qq ( ) approach adopted here, and it therefore does not contribute to the cumulative cross section shown for a given 3BFS in the scatter plots in the previous sections. The rightmost graph in Fig. 6 illustrates such a scenario. Again, a 3BFS is typeset in bold if the corresponding 2BFS can result only from qq -initiated processes, otherwise it can be both gg/bb-and qq-produced. Finally, a box around a neutral 2BFS at the top of a column or a 3BFS in parentheses indicates that the cross section for qq production can exceed that for gg/bb production (for certain parameter space configurations), while one around a charged (bold) 3BFS implies that the total cross section can be larger than 1 fb.
There are some important inferences that can be drawn from the   Figure 6: Examples of s-channel diagrams considered for the three-body final states included in Tab. 3 below. Processes like the one on the right are not taken into account in the scatter plots shown above, as two of the three final state particles result from an incoming Higgs state that is necessarily off-shell. Thus the corresponding cross sections cannot be calculated as . Such 3BFSs have therefore been typeset in grey colour in the table.
the statements regarding the 3BFSs are valid only in the parameter space regions that satisfy m x > m j + m k ). One can notice many instances where a coupling appears in more relevant 3BFSs than 2BFSs. While a given 2BFS typically reflects contributions from several diagrams containing different couplings, the 3BFSs often arise from multiple initial 2BFSs, and the decays leading to 3BFSs reflect not only the relevant coupling, but also all other couplings and masses involved in determining the width of the decaying particle. A careful kinematical selection of events might help   disentangle (some of) these couplings from each other, and complementary analyses of the two types of states can greatly enhance the potential of the LHC to probe such couplings.
While only qq ( ) -production is available at leading order for charged 3BFSs, it is clearly the preferred mode also for several neutral 3BFSs, especially those involving the λ hAZ , λ HAZ and λ hH + W − , λ HH + W − couplings. Additionally, we see that all of the charged 3BFSs that include a W ± can have a cross section in excess of 1 fb, which is a consequence of the cross section for the HH ± and AH ± 2BFSs reaching up to 100 fb, as noted in Fig. 1 earlier. As a result, qq ( ) -production of the relevant 3BFSs, if observed, could prove crucial for pinning down the λ hH + W − , λ HH + W − and λ AH + W − couplings at the LHC.

The triple-Higgs couplings
Of particular relevance for disentangling the underlying Higgs dynamics are the triple-Higgs couplings. In Tab. 3, rows b and g, we see that the couplings λ hhH and λ HAA enter, respectively, processes for which EW production dominates for neutral 3BFSs hhA and AAA, and at the same time, also EW processes giving substantial cross sections for charged 3BFSs hhH ± and AAH ± . In order to get an impression of the possible sizes of these couplings, we show in Fig. 7 the correlations of the λ hhH and λ HAA couplings with each other and, for reference, with the SM-like Higgs self-coupling λ hhh . In Fig. 8 we further show these couplings as functions of m H and m A . In each figure we plot all the parameter space points passing the constraints imposed in our numerical scan in light grey. On top of these, a colour heat map depicts the size of the cross section for the EW production, when kinematically allowed, of the corresponding 3BFS, i.e., σ(hhH ± ) for the λ hhH coupling and σ(AAH ± ) for the λ HAA coupling. 2 There are very few points above the kinematic threshold in the upper right plot of Fig. 7 and in the lower two plots in Fig. 8, which can also be seen in the lower middle plot in Fig. 2.
The λ hhH and λ HAA couplings range from essentially zero up to a factor three larger than the coupling in the SM (i.e., 1), while the λ hhh coupling of the SM-like Higgs boson is mostly close to the SM value or smaller, with only a few points in our scan having a larger value. The λ HAA coupling in particular can be sizeable, and may lead to a large σ(AAH ± ), although a relatively small portion of the scanned parameter space lies above the threshold for this process. On the other hand, for the production process of hhH ± , which is sensitive to the λ hhH coupling, a large portion of our parameter space is above threshold. While the cross section σ(hhH ± ) is generally smaller than σ(AAH ± ), parameter space points where it is sufficiently large do exist.

Conclusions
In order to fully establish the EWSB mechanism, whether in the SM theory or beyond it, a full reconstruction of the Higgs potential is required. This entails measuring experimentally the triple-Higgs couplings, which can only be achieved if scattering processes yielding two or more Higgs bosons can be isolated in the detector. Historically, most studies of these couplings have exploited production modes that are enhanced in the hadronic environment of the LHC, primarily gluongluon fusion. Such studies have covered both the SM as well as extended Higgs sectors, chiefly 2HDMs, with and without Supersymmetry. In such beyond-the-SM scenarios, couplings of the Higgs bosons to b-(anti)quarks can be enlarged with respect to the SM case, so that bb-induced production can be relevant in onsetting final states with two or more Higgs bosons. This approach is somewhat limited, though, on two accounts. Firstly, these subchannels cannot lead to electrically charged final states. Hence, they necessarily miss out on some couplings involving a charged Higgs boson, in parameter space regions of the 2HDMs where the neutral final state production processes these couplings might alternatively enter are kinematically unavailable. Secondly, there could exist further production channels (for neutral final states) offering access to many other triple-Higgs couplings, also needed to reconstruct the full EWSB potential.
In this paper, we have therefore concentrated on EW-induced channels, where the initial state constitutes (primarily) of valence quark flavours, which annihilate via both electrically neutral and charged currents into neutral and charged 2-Higgs (and up to 3-Higgs) final states. We have shown that the production cross sections for several charged final states (precluded to the gg and bb production modes) are large enough to be potentially accessible at the LHC, either during the Runs 2 and 3 or at its High Luminosity (HL-LHC) stage (depending on the parameter space configuration). We have also illustrated that such EW-induced channels can often be competitive with, when not overtaking, those induced by gg and bb fusion, other than offering more probes of various triple-Higgs couplings. Finally, as these EW channels are often mediated by weak gauge bosons (i.e., W ± and Z states), they can provide sensitivity to couplings involving one of these and two Higgs bosons.
We have come to these conclusions after studying, as a preliminary step of a long-term investigation that will eventually include a complete detector simulation, the fully inclusive parton-level yield of the aforementioned EW channels. This study tackled the phenomenology of the so-called Type-I 2HDM, as illustrative for conditions which may emerge in other possible non-minimal Higgs constructs, in the presence of standard theoretical constraints as well as the latest experimental limits coming from EW precision data, collider searches for the Higgs boson(s), and measurements of the heavy flavour observables.
In short, we advocate, alongside the time-honoured analyses based on QCD-induced processes, those of EW processes as well, which we have shown to offer improved and expanded sensitivity to both Higgs and gauge-Higgs structure of the underlying EWSB dynamics, which may or may not be the same as the SM ones.