1 Introduction

One of the main goals of the LHC machine is to investigate the individual properties (mass, width, spin, CP quantum numbers) of the Higgs boson as well as its interactions (with both matter and forces) and to look into evidence for new physics. These features have been probed by the ATLAS and CMS collaborations in different production and decay channels, using proton-proton (pp) collision data collected at centre-of-mass energies of 7, 8 [1] and 13 TeV [2, 3]. More recently, there have been observations involving the rare decay of the Higgs boson into a Z-boson and a photon (\(H \rightarrow Z\gamma \)) with a significance of \(3.4\sigma \) [4], which could provide insights into the Higgs boson’s coupling to both the electroweak gauge bosons and photons, contributing to our comprehension of the dynamics and interplays within the Higgs sector. Although the current measurements of the Higgs mass, spin, width and couplings to SM fermions and vector bosons [2, 3] are all indeed in good agreement with the SM theoretical predictions, the small deviations in the SM-like Higgs couplings probed in various production modes for the five key decay channels \(H \rightarrow \gamma \gamma ,~ZZ^{*},~WW^{*},~\tau \tau ~{\text {and}}~b\overline{b}\) [2, 3] still provide signs of possible potential BSM contributions to the total Higgs width and hints of new physics through the invisible and/or undetected decays. The indirect constraints from the current fit of couplings measurements and direct searches for \(H \rightarrow inv\) (i.e., to‘invisible’ final states) performed by ATLAS and CMS collaborations have placed upper limits on the Branching Ratio (BR) of Higgs boson to invisible particles and undetected BSM particles at 95% C.L. (Confidence Level) [5,6,7,8].

The quest to uncover the trilinear couplings also holds a prominent position on the LHC’s research agenda. Such interactions can be probed with sufficient luminosity, although, at present (i.e., at the end of Run 2), they are not determined yet. A measurement of this interaction is one of the highest priority goals during, possibly, Run 3 and, certainly, at the High Luminosity LHC (HL-LHC), both of which would, therefore, start shedding light on the nature of the Higgs boson and the shape of the Higgs potential, which in turn has implications for the vacuum metastability, the hierarchy problem as well as the strength of the Electro-Weak (EW) phase transition. However, probing Higgs’s self-interactions, both trilinear and quartic couplings, in multi-Higgs production is experimentally very challenging. From the theory side on the other hand, many models with an extended scalar sector, like the 2-Higgs Doublet model (2HDM) [9,10,11,12,13], the Next-to-2HDM

(N2HDM) [14,15,16,17] and a variety of both minimal and non-minimal Supersymmetric (SUSY) models [18,19,20,21,22,23,24] motivate additional features of new di-Higgs final states, as they all present with additional CP-even and/or -odd Higgs states, which can be accessible by the LHC experiments in a variety of signatures. Among the prominent processes are searches for exotic Higgs decays to a pair of light scalars or pseudo-scalars, e.g. \(H \rightarrow aa, hh\), which then decay to SM particles.

This paper focuses on the popular 2HDM. After EW Symmetry Breaking (EWSB), the scalar sector of the 2HDM predicts five physical Higgs states, two CP-even Higgs bosons (h,  H, with \(m_h<m_H\)), one CP-odd one (a) and a pair of charged ones (\(H^\pm \)). The rich (pseudo)scalar sector of the 2HDM and the different sets of Yukawa couplings that can be realised then offer a very interesting production and decay phenomenology of neutral and charged Higgs states at the LHC, even after scrutinising the 2HDM parameter space by considering different theoretical (vacuum stability, perturbativity, unitarity, etc.) and experimental (from SM-like Higgs data and null searches for companion states, flavour physics and low energy observables, etc.) constraints. Furthermore, the 2HDM is also attractive because one can impose a simple \(Z_2\) discrete symmetry to the Yukawa sector in order to suppress Flavour Changing Neutral Currents (FCNCs) at tree level [25, 26], which then forces one doublet to couple to a given type of fermions and leading as a result to four Yukawa interactions (termed, Type-I, Type-II, Type-X and Type-Y). In fact, in order to realise EWSB in such a way that the 2HDM is compliant with all experimental data, it is finally customary to allow for a soft breaking of this \(Z_2\) symmetry. Herein, we will use the latter setup with a Type-I Yukawa structure. This particular scenario can accommodate both light neutral (10–100 GeV) and charged scalars (100–200 GeV), while one can not obtain such light Higgs states within the 2HDM Type-II or Type-Y configuration due ot the significant constraints arising from flavour physics, specifically from \(B\rightarrow X_s \gamma \) which enforces a lower bound of 800 GeV on the charged Higgs mass [27].

In the present study, we plan to take advantage of the direct access to some trilinear Higgs couplings that the LHC can access, entering multi-Higgs processes such as \(H \rightarrow hh(aa)\) and \(H^\pm \rightarrow W^\pm a\), to test the sensitivity of the parameter space combination of 2HDM Type-I to light Higgs searches in cascade (or chain) decays. As the analysis progress, we aim to explore the scope of a new search for light Higgs bosons at the LHC Run 3 (with an integrated luminosity of 300 \(\text {fb}^{-1}\)) as well as the HL-LHC (with an integrated luminosity of 3000 \(\text {fb}^{-1}\)), on the basis of the knowledge acquired from the study of the aforementioned signatures. We focus mainly on the configuration of Type-I with inverted scenario, which in turn offers an alternative and new promising signal, in the form of the following cascade decays \(H \rightarrow Z^{*}a \rightarrow Z^{*}Z^{*}h \rightarrow b\overline{b} \mu ^- \mu ^+ jj\). The main Higgs production process is via gluon fusion \(gg \rightarrow H\).

2 2HDM type-I

The 2HDM is one of the simplest well-motivated extensions of the SM. In this section, we briefly review the theoretical structure of this model. The scalar sector of the 2HDM consists of two complex \(SU(2)_L\) doublets, \(\varPhi _1\) and \(\varPhi _2\), with hypercharge \(Y = +1\). The most general \(SU(2)_L\times U(1)_Y\) invariant scalar potential can be written as follows [11,12,13]:

$$\begin{aligned} V(\varPhi _1,\varPhi _2)= & {} m_{11}^2 \varPhi _1^\dagger \varPhi _1+m_{22}^2\varPhi _2^\dagger \varPhi _2-[m_{12}^2\varPhi _1^\dagger \varPhi _2+\mathrm{h.c.}] \nonumber \\{} & {} + \frac{\lambda _1}{2}(\varPhi _1^\dagger \varPhi _1)^2 +\frac{\lambda _2}{2}(\varPhi _2^\dagger \varPhi _2)^2\nonumber \\{} & {} +\lambda _3(\varPhi _1^\dagger \varPhi _1)(\varPhi _2^\dagger \varPhi _2)\nonumber \\{} & {} +\lambda _4(\varPhi _1^\dagger \varPhi _2)(\varPhi _2^\dagger \varPhi _1) +\left\{ \frac{\lambda _5}{2}(\varPhi _1^\dagger \varPhi _2)^2+\mathrm{h.c.}\right\} \nonumber \\{} & {} +\left\{ \big [\lambda _6(\varPhi _1^\dagger \varPhi _1) +\lambda _7(\varPhi _2^\dagger \varPhi _2)\big ] \varPhi _1^\dagger \varPhi _2+\mathrm{h.c.}\right\} . \nonumber \\ \end{aligned}$$
(1)

Assuming CP-conservation in the 2HDM and following the hermiticity of the scalar potential, \(m_{11}^2\), \(m_{22}^2\), \(m_{12}^2\), \(\lambda _{1,2,3,4,5,6}\) are real parameters. Invoking the described \({Z}_2\) symmetry, to avoid tree-level Higgs-mediated FCNCs at tree level, implies that \(\lambda _{6} = \lambda _{7} = 0\). Also notice that the bilinear term proportional to \(m_{12}^2\) breaks the \(Z_2\) symmetry softly. Using the two minimisation conditions of the scalar potential and the combination \(v^2=v_1^2+v_2^2=(2\sqrt{2} G_F)^{-1}\), one can then trade the Lagrangian parameters of the 2HDM for a more convenient set of variables,

\(\alpha \),  \(\tan \beta = \frac{v_2}{v_1}\),  \(m_{h}\),  \(m_{H}\) ,  \(m_a\),  \(m_{H^\pm }\)  and \(m_{12}^2\),

where \(\alpha \) is the CP-even mixing angle, \(v_1\) and \(v_2\) are the Vaccum Expectations Values (VEVs) of the two Higgs doublets \(\varPhi _1\) and \(\varPhi _2\), respectively.

2.1 Yukawa couplings

The general structure of the Yukawa Lagrangian when both Higgs fields couple to all fermions is given by [13]:

$$\begin{aligned} {{\mathcal {L}}}_Y= & {} \bar{Q'}_L ( Y^{u}_1 {{{\tilde{\varPhi }}}_{1}} + Y^{u}_2 {{{\tilde{\varPhi }}}_{2}}) u'_{R} +\bar{Q'}_L (Y^{d}_1 \varPhi _{1} + Y^{d}_2 \varPhi _{2}) d'_{R} \nonumber \\{} & {} + \bar{L'_{L}} (Y^{l}_{1}\varPhi _{1} + Y^{l}_{2}\varPhi _{2}) l'_R + \mathrm {h.c.}, \end{aligned}$$
(2)

where \(Q'_L\) and \(L'_L\) are the weak isospin quark and lepton doublets, \(u'_R\) and \(d'_R\) denote the right-handed quark singlets while \(Y_{1,2}^{u}\), \(Y_{1,2}^{d}\) and \(Y_{1,2}^{l}\) are couplings matrices in flavour space. This form of Yukawa interaction gives rise to large FCNCs at tree level, which is strongly constrained by B-physics observables. Implementing \(Z_2\) symmetry [25, 26] allows only one doublet to couple to a given right-handed fermion field. Depending on the \(Z_2\) assignment, one can have the four types of models previously refered to as Type-I, Type-II, Type-X and Type-Y. In the mass-eigenstate basis, they can be unified and expressed as follows:

$$\begin{aligned} -{{\mathcal {L}}}_Y&=+\sum _{f=u,d,\ell } m_f \bar{f} f+\left( \frac{m_f}{v}\kappa _h^f \bar{f} fh+\frac{m_f}{v}\kappa _H^f\bar{f} fH\right) \nonumber \\&\quad -i\frac{m_f}{v}\kappa _A^f \bar{f} \gamma _5fA \nonumber \\ {}&\quad +\frac{\sqrt{2}}{v}\bar{u} \left( m_u V \kappa _A^u P_L+ V m_d\kappa _A^d P_R \right) d H^+ \nonumber \\&\quad +\frac{\sqrt{2}m_\ell \kappa _A^\ell }{v}{{\bar{\nu }}}_L \ell _R H^+ +\text {h.c.}, \end{aligned}$$
(3)

where \(P_{L,R}=(1\pm \gamma _5)/2\) are the projection operators and V denotes the Cabibbo–Kobayashi–Maskawa (CKM) matrix.

Here, we focus only on Type-I, where only one doublet \(\varPhi _2\) couples to all fermions, and thus the Higgs-fermion couplings are flavour diagonal in the fermion mass basis and depend only on the mixing angles, \(\alpha \) and \(\beta \), as follows:

$$\begin{aligned} \kappa ^{u,~d,~l}_{h}&= c_\alpha / s_\beta ,~\kappa ^{u,~d,~l}_{H} =s_\alpha / s_\beta ,\\ \kappa ^{u}_{A}&= \cot \beta ,~\kappa ^{d,~l}_{A}= -\cot \beta , \end{aligned}$$

where we have used the short-hand notation c and s for \(\cos \) and \(\sin \), respectively.

2.2 Theoretical and experimental constraints

We now describe briefly a set of, in turn, theoretical and experimental constraints that must be satisfied by the parameter space of the 2HDM.

  • Perturbative unitarity constraints require a variety of tree-level 2-to-2 body scatterings processes (e.g. scalar-scalar, gauge boson-gauge boson and scalar-gauge boson) to remain unitary at high energy [28,29,30].

  • Vacuum stability [31] requires the scalar potential to be finite at large field values, leading to the following bounds:

    $$\begin{aligned} \lambda _{1,2}>0, \,\, \lambda _3>- \sqrt{\lambda _1\lambda _2}, \,\, \lambda _3+\lambda _4-|\lambda _5|> - \sqrt{\lambda _1\lambda _2}. \end{aligned}$$

  • Perturbativity requires the quartic couplings to obey \(|\lambda _i| < 4\pi \) (\(i=1,\ldots ,5\)).

Table 1 \((\times )\)/(\(\checkmark \)) indicate searches (not yet)/(already) implemented in HiggsBounds \(-\)5.10.0
Full size table

The above constraints have been implemented in 2HDMC \(-\)1.8.0 [32]. This public code is then used to explore the 2HDM parameter space and to compute the different Higgs BRs in each point of it. (2HDMC also provides an interface to HiggsBounds and HiggsSignals, see below.)

Experimental observations impose the following constraints:

  • EW precision observables, i.e., the oblique parameters, S, T and U [33, 34] are required to be within 95% C.L. of their experimental measurements, the current fit values (with the correlation parameters) are given by [35]:

    $$\begin{aligned} S&= -0.02\pm 0.10,~~T = 0.03\pm 0.12,\ \ \\ U&= 0.01\pm 0.11,~~\rho _{ST} = 0.92,~~\rho _{SU} = -0.80,\ \ \\ \rho _{TU}&= -0.93,~~\chi _{ST,~SU,~TU} < 5.99. \end{aligned}$$

  • Consistency with the Z width measurement \(\varGamma _Z = 2.4952 \pm 0.0023\) GeV from LEP [36, 37] is required.

  • Constraints from LHC, Tevatron and LEP searches which failed to find companion Higgs states are taken into account via HiggsBounds \(-\)5.10.0 [38], which allows to test the exclusion limits at 95% C.L.

  • The code HiggsSignals \(-\)2.6.2 [39] is used to check the signal strength measurements of the SM-like Higgs boson discovered at the LHC in 2012.

  • Constraints from B meson decays are enforced by Superiso-v1.4 [40], using the current measurements:

    • \(\textrm{BR}(\overline{B}\rightarrow X_s\gamma )|_{E_\gamma <1.6\mathrm {~GeV}}= \left( 3.32\pm 0.15\right) \times 10^{-4}\) [41],

    • \(\textrm{BR}(B_s\rightarrow \mu ^+\mu ^-)_{\text {(LHCb)}}\) = \(\left( 3.09^{+0.46}_{-0.43}\right) \times 10^{-9}\) [42, 43],

    • \(\textrm{BR}(B_s\rightarrow \mu ^+\mu ^-)_{\text { (CMS)}}\)=\(\left( 3.83^{+0.38}_{-0.36}\right) \times 10^{-9}\) [44],

    • \(\textrm{BR}(B^0\rightarrow \mu ^+\mu ^-)_{\text {(LHCb)}}\)=\(\left( 1.2^{+0.8}_{-0.7}\right) \times 10^{-10}\) [42, 43],

    • \(\textrm{BR}(B^0\rightarrow \mu ^+\mu ^-)_{\text {(CMS)}}\)=\(\left( 0.37^{+0.75}_{-0.67}\right) \times 10^{-10}\) [44].

  • Constraints from recent searches for light pseudoscalar states in the mass range [15,  62.5] GeV, in proton-proton collision at \(\sqrt{s}=13~\) TeV, in \(\mu ^+\mu ^-b\overline{b}\) [45, 46], \( \mu ^+ \mu ^- \tau ^+\tau ^-\) [47] and \(\tau ^+\tau ^-b\overline{b}\) [48] final states, are included in HiggsBounds. Since no significant excess is observed, upper limits are set on \(\textrm{BR}(H \rightarrow aa \rightarrow \mu ^+ \mu ^- b \overline{b},~\mu ^+ \mu ^- \tau ^+ \tau ^-,~\tau ^+\tau ^-b\overline{b} )\) [45,46,47,48]. However, lately, additional constraints from such Higgs cascade decays have emerged, not included in the numerical tool, so we had to deal with these separately. For example, the CMS group has reported a search for \(H \rightarrow aa \rightarrow 4 \gamma \) [49], using the data collected at \(\sqrt{s} =13\) TeV, with an integrated luminosity of 132 fb\(^{-1}\). Upper limits can then be set on \(\textrm{BR}(H \rightarrow aa \rightarrow 4\gamma )\) at 95% C.L, since no significant deviation is observed.Footnote 1 The ATLAS group [53] has also recently searched for the exotic decay of the Higgs boson into two light pseudoscalars in \(\mu ^+\mu ^-b\overline{b}\) final state at \(\sqrt{s}=13\) TeV with an integrated luminosity of 137 fb\(^{-1}\), in the range of masses varying from 15 to 60 GeV. An upper limit is placed on \(\textrm{BR}(H \rightarrow aa \rightarrow \mu ^+\mu ^-b\overline{b})\)Footnote 2 at 95%C.L. Table 1 summarises several searches for exotic decays of the Higgs bosons in various final states, performed by the two collaborations ATLAS and CMS at Run 2, targeting a different ranges of masses.

3 Numerical analyses

The (pseudo)scalar sector of the 2HDM involves two CP-even Higgs bosons, h and H. One of these scalars can be identified as the 125 GeV state observed at the LHC. As mentioned, in this analysis, we will assume that the heaviest Higgs state H is the SM-like one with a mass of 125 GeV and that h and a are lighter than H. We the perform a scan over the following ranges,

$$\begin{aligned} m_h&\in [10~\text {GeV},~90~\text {GeV}],~m_H = 125~\text {GeV},\nonumber \\ m_a&\in [10~\text {GeV},~90~\text {GeV}],~m_{H^\pm }\in [100~\text {GeV},~160~\text {GeV}],\nonumber \\ \tan \beta&\in [2.5,~25],~\sin (\beta -\alpha ) \in [-0.7,~0.0], \end{aligned}$$
(4)

with \(m_{12}^2\)= \(m_a^2\tan \beta /(1+\tan ^2\beta )\). Assuming \(m_H = 125\) GeV and \(m_{h,a} < 90\) GeV, the decay channels \(H \rightarrow h h,aa,~\) \(aZ^{*}\) could be open, leading to invisible or undetected SM-like Higgs decays that are restricted by the current precision measurements of Higgs couplings. CMS performed a combination of searches, using data collected at \(\sqrt{s} = 7,~8,~13\) TeV [7], for Higgs bosons decaying into invisible particles, which targets the following production channels: Vector Boson Fusion (VBF), Higgs-Strahlung (HS) and gluon-gluon Fusion (ggF) (allowing for initial state radiation). The combination of all the searches, assuming these SM-like production modes, yields an observed (expected) upper limit on \(\textrm{BR}(H \rightarrow inv)\) of 0.19 (0.15) at 95% C.L. The ATLAS group reported a direct search for Higgs bosons produced via VBF with subsequent invisible decays, for 139 \(\text {fb}^{-1}\) of pp collision data at \(\sqrt{s} = 13\) TeV [8]. An observed (expected) upper limit of 0.145 (0.103) is placed on \(\textrm{BR}(H \rightarrow inv)\) at 95% C.L., as a function of the assumed production cross sections. As for now, both ATLAS and CMS have placed, respectively, an upper bound of 0.10 [6] and 0.15 [5] on \(\textrm{BR}(H \rightarrow \text {invisible})\) at 95% CL. In our analysis, we will assume that \(\textrm{BR}(H \rightarrow inv)\) designates the sum of the following decay rates, \(\textrm{BR}(H \rightarrow hh)\), \(\textrm{BR}(H\rightarrow aa)\) and \(\textrm{BR}(H \rightarrow aZ^{*})\).

Fig. 1
figure 1

Allowed parameter space in the 2HDM Type-I at 95% C.L. The solid red and green regions are excluded by the Z width constraint and the LEP search for \(h\rightarrow aa\) [54], respectively. The most sensitive searches for the relevant (\(m_h, m_a\)) regions are shown by coloured dots. Each coloured point in the \((m_h, m_a)\) plane satisfies all theoretical requirements and up-to-date experimental constraints. In the white space, any combination of masses is ruled out by current experimental searches

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Fig. 2
figure 2

Observed and expected upper limits on \(B(H \rightarrow aa (hh) \rightarrow \tau ^+\tau ^- b\overline{b} )\) [51] at 95% C.L. in the 2HDM Type-I. Grey points are allowed by theoretical constraints. As stated above, red colour indicates the combination of the scanned model parameters, which is excluded by existing experimental searches checked by HiggsBounds [38], whereas blue points satisfy both theoretical and experimental constraints

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After performing a random scan over 2HDM Type-I parameters, we show in Fig. 1 the allowed regions by theoretical and experimental constraints. The figure also captures the constraint from the Z width (bottom-left red region), which forbids possible mass combinations \((m_h,~m_a)\) when \(\cos (\beta -\alpha )\rightarrow 1\). Additionally, the constraint from the LEP search for the \(e^+ e^- \rightarrow (h \rightarrow aa)a \rightarrow (b\overline{b}b\overline{b})b\overline{b}\) process [54] excludes the bottom-right region corresponding to \(m_h > 2m_a\), where the decay channel \(h \rightarrow aa\) is kinematically open (bottom-right green region).

Each (coloured) point in the \((m_h, m_a)\) plane implies that there is a combination of the scanned model parameters, which obeys the aforementioned theoretical constraints and evades the current experimental limits in all searched modes, whereas the white space corresponds to the case where any possible mass combination is forbidden by an observed signature(s) in one or more existing experimental searches.

Within this region, the most sensitive channels for the model parameter points, as determined by HiggsBounds, are shown by coloured dots. Note that each coloured point in the parameter space of the 2HDM is allowed by current experimental searches. Obviously, there are two distinct regions in the figure. The one in the top left corner corresponds to low masses of h (\(m_h < m_H/2\)), and high masses of a (\(m_a> m_H/2\)), while the second one corresponds to the \(m_{a,~h}> m_H/2\) scenario. It is interesting to note that there are no acceptable points when \(40~\text {GeV}<m_h<m_H/2\) and \(m_a > m_H/2\). This is due to the fact that this parameter combination is excluded by LEP searches for \(e^+e^- \rightarrow ah \rightarrow b\overline{b}b\overline{b}\) [54] and an ATLAS search for events with at least \(3\gamma \) in \(pp \rightarrow H_\textrm{SM} \rightarrow h h \rightarrow 4\gamma \) [55].

Finally, it is noteworthy that the most sensitives searches for the region with low \(m_h\) and high \(m_a\) are the LEP searches for processes such as \(e^+e^- \rightarrow ah \rightarrow b\overline{b}b\overline{b}\) and \(e^+ e^- \rightarrow (h)Z \rightarrow (b\overline{b})Z\) [54]. Therefore, an update from the LHC during Run 3 is unlikely to rule out this mass combination over the plane \((m_h,~m_a)\) of the 2HDM Type-I. We will be focusing on this region in the second part of our study.

We now turn to the reinterpretation of exotic Higgs decay searches, i.e., \(H \rightarrow aa\) in \(\tau ^+\tau ^-b\overline{b}\), \(\mu ^+\mu ^-b\overline{b}\) and \(\mu ^+\mu ^-\tau ^+\tau ^-\) final states in the framework of the 2HDM Type-I, while taking advantage of the parameter space discussed above. The recasting of \(\tau ^+\tau ^-b\overline{b}\), \(\mu ^+\mu ^-b\overline{b}\) and \(\mu ^+\mu ^-\tau ^+\tau ^-\) searches for \(H\rightarrow hh\) is also possible since these processes share similar kinematics (in the same spirit as in Ref. [56]). It is relevant to note that the constraints from the search for light pseudoscalars in the \(\tau ^+\tau ^-b\overline{b}\) final state are much stronger than the ones from \( \mu ^+\mu ^-b\overline{b}\) and \(\mu ^+\mu ^-\tau ^+\tau ^-\) searches. CMS has set an upper limit, between 1.7% and 7.6%, on \(\textrm{BR}(H \rightarrow aa \rightarrow \tau ^+\tau ^-b\overline{b})\) at 95% C.L. [51], assuming the SM production of primary Higgs boson. We show in Fig. 2 the outcome from reinterpreting the \(H \rightarrow aa (hh) \rightarrow \tau ^+\tau ^-b\overline{b}\) search [51] in the 2HDM Type-I. The yellow and green bands represent the uncertainties at \(\pm 1\sigma \) and \(\pm 2\sigma \) associated with the expected exclusion limits. Grey points satisfy theoretical constraints described in Sect. 2.2, whereas red points are excluded by null searches (i.e., by HiggsBounds [38]). The blue points satisfy both theoretical and experimental constraints. The area of sensitivity to \(H \rightarrow aa (hh) \rightarrow \tau ^+\tau ^-b\overline{b}\) is already excluded by existing experimental searches (red points). In this connection, the BR of Higgs SM-like Higgs state decaying into hh and/or aa is very restricted and cannot exceed \(9\%\) at 95% C.L., again, in the 2HDM Type-I.

Fig. 3
figure 3

Observed and expected upper limits on \(B(H \rightarrow aa (hh) \rightarrow \mu ^+ \mu ^- b\overline{b})\) [51] at 95% C.L. in the 2HDM Type-I

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One can draw a similar conclusion form reinterpreting \(H \rightarrow hh(aa) \rightarrow \mu ^+\mu ^- b\overline{b}\) [51] in our reference framework. Figure 3 shows that the parameter space with sensitivity to this search is excluded. One should keep in mind that the \(\mu ^+\mu ^-b\overline{b}\) final state is well-balanced between large \(\textrm{BR}(h/a \rightarrow b\overline{b})\) and a clean di-muon resonance that is easy to trigger on. This exercise emphasises that the 2HDM Type-I may not be a good framework for reinterpreting searches for exotic Higgs decays into light pseudoscalar in “traditional” final states such as \(\mu ^+\mu ^-b\overline{b}\), \(\tau ^+\tau ^-b\overline{b}\) and \(\mu ^+\mu ^-\tau ^+\tau ^-\).

We also address here light charged Higgs decay in the mass ranges where \(m_{H^\pm } < m_t-m_b\) and \(m_{h,a} < 90~\text {GeV}\). In this configuration, the charged Higgs state can be be produced from top quark decays, i.e., \(t \rightarrow b H^+\), followed by its bosonic decays to \(H^\pm \rightarrow W^\pm h (a)\), instead of the standard fermionic decay modes like \(\tau \nu \) and cs. Many studies motivated these channels as alternative modes to search for light charged Higgs bosons that could dominate over the conventional fermionic channels, because of large BRs when they are kinematically allowed, in models such as our 2HDM Type-I [57,58,59,60]. ATLAS [61] and CMS [62] have considered the ranges \(m_a \in [15,75]\) GeV and \(m_{H^\pm } < m_t-m_b\) to search for light charged Higgs bosons in \(pp \rightarrow t \overline{t} \rightarrow b\overline{b}H^+ W^-\) with \(H^+ \rightarrow W^+ a\) and \(a \rightarrow \mu ^+ \mu ^-\) at \(\sqrt{s}=13~\text {TeV}\), since the \(\mu ^+\mu ^-\) finale state provides the aforementioned experimental advantages, which offset the suppressed rate of \(\textrm{BR}(a \rightarrow \mu ^+ \mu ^-)\). Previously, both CDF and the LEP collaborations have searched for \(H^\pm \rightarrow W^\pm a\) with \(a \rightarrow b\overline{b}\) [63], \(a \rightarrow \tau ^+\tau ^-\) [64] and \(a \rightarrow b\overline{b}\) [65]. In addition, LEP experiments [66] have set a lower bound on the charged Higgs boson mass of \(m_{H^\pm } > 72.5~\text {GeV}\) in the 2HDM Type-I for \(m_a>12~\text {GeV}\) at 95% C.L.

Fig. 4
figure 4

Observed and expected upper limits on \(\textrm{BR}(t \rightarrow H^+b) \times \textrm{BR}(H^+ \rightarrow W^+ a) \times \textrm{BR}(a \rightarrow \mu ^+ \mu ^-)\) [62] at 95% C.L. in the 2HDM Type-I

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Figure 4 shows the CMS observed and expected exclusion limits on the product of the BRs of \(t \rightarrow bH^\pm ,~H^\pm \rightarrow W^\pm a\) and \(a \rightarrow \mu ^+\mu ^-\) [62] as a function of \(m_a\) predicted by the 2HDM Type-I, with respect to several theoretical and experimental constraints. We adopt here \(m_{H^\pm } = m_a + 85\) GeV [62], which enables us to consider \(H^\pm \rightarrow W^{\pm (*)} a\), with \(W^{\pm (*)}\) being on/off shell, by randomly sampling values of the charged Higgs mass between 100 GeV and 160 GeV (see Eq. (4)). A noteworthy observation is that the 2HDM Type-I offers sufficient sensitivity, when the prediction of the model exceeds the expected limit produced at \(\sqrt{s} = 13\) with an integrated luminosity of \(35.9~\text {fb}^{-1}\) (purple stars). Such a signature could be exploited to search for a light \(H^\pm \) at future experiments, Run 3 and/or the HL-LHC, given the available energies and luminosities by then. Therefore, we present in Table 2 some Benchmark Points (BPs) to test the actual sensitivity of these experiments to the 2HDM Type-I parameter space.

Table 2 BPs in the 2HDM Type-I
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We move now to discuss a new analysis, where we deploy the parameter space of the 2HDM Type-I following the outcomes of reinterpreting previous searches for light Higgs bosons, \(pp\rightarrow H_\textrm{SM}\rightarrow hh(aa)\), in different final states, in order to search for a new signature.

Fig. 5
figure 5

\(m_h\) and \(m_a\) vs. \(\textrm{BR}(H \rightarrow Z^{*}a)\) (left) and \(\sigma (gg \rightarrow H \rightarrow Z^{*}a \rightarrow Z^{*}Z^{*}h)\) (right) at 95% C.L. in the 2HDM Type-I

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Figure 5 shows the result of performing a scan over the parameter space of 2HDM Type I, wherein (recall) the heaviest Higgs state is identified as the discovered SM-like one. Each sampled point is required to satisfy the theoretical and experimental constraints described in Sect. 2.2. In the left panel, we illustrate \(m_a\) vs. \(m_h\) with the BR of \(H\rightarrow a Z^{(*)}\) on the colour gauge. Since \( m_H/2< m_a < 125~\text {GeV}\), \(H \rightarrow a Z^{(*)}\) will proceed with Z being off-shell, which explains the suppressed BR (\(<0.2\%\)). In this configuration, \(H \rightarrow aa\) will not be open, thus, \(H \rightarrow hh\) would only contribute significantly to the undetected decays of H. It should be pointed out that the total amount of \(\textrm{BR}( H \rightarrow aa^{*}+ aZ^{*}+ hh)\) should not exceed 12% as required by \(\textrm{BR}(H \rightarrow inv)\) [2]. In the right panel, we show \(m_a\) as a function of \(m_h\) with \(\sigma (H \rightarrow aZ^{*} \rightarrow h Z^{*}Z^{*} \rightarrow Z^{*}Z^{*})\) on the colour gauge. Once the decay chain \(H \rightarrow a Z^{*}\) is open, the subsequent decay of a could lead to \(a \rightarrow Z^{*}h\) with Z being off-shell and h decaying to fermions and/or \(\gamma \gamma \). We use Sushi [67,68,69] to compute the cross section of Higgs production at LO.Footnote 3 It is worth highlighting that Figs. 5, 6 and 14 focus particularly on the region characterized by small \(m_h\) and large \(m_a\) in contrast to Fig. 1, which displays both regions. The cross-section of our signature, i.e., \(gg \rightarrow H \rightarrow aZ^{*} \rightarrow h(\rightarrow b\overline{b}) Z^{*}Z^{*}\), is remarkably negligible in the region with large \(m_h\) and \(m_a\) (\(m_h>60~\text {GeV}\) and \(m_a>40~\text {GeV}\) ), primarily due to the fact that the pseudoscalar decay width (\(\varGamma _a\)) is dominated by \(a \rightarrow b\bar{b}\) with a branching ratio of 85%, rather than \(a \rightarrow Z^{*}h\). Note that the contribution of \(a \rightarrow b\bar{b}\) is negligible in the region with low \(m_h\) and high \(m_a\). Instead, the predominant channel in this region is \(a\rightarrow Z^{*}h\) with Z being off-shell, boasting a branching ratio that can reach 90%. Furthermore, the decay of \(H_{SM} \rightarrow aZ^{*}\) is suppressed in the region with large \(m_h\) and \(m_a\). In fact, \(H_{SM}\) tends to decay to \(aZ^{*}\) in the specific part of the region where the observation of \(a \rightarrow Z^{*}h\) is not possible.

We show in Fig. 6 the \(gg \rightarrow H \rightarrow aZ^{*} \rightarrow h Z^{*}Z^{*}\) cross section, where \(h \rightarrow b\overline{b}\). The process could yield a cross section of 0.006 pb. In the right panel of Fig. 6 we show the BR of \(h \rightarrow b\overline{b}\) in this region of the 2HDM Type-I parameter space. Obviously, the decay width of h is dominated by the decay mode \( h \rightarrow b\overline{b}\). Thus, in what follows, we focus on the case where h decays to \(b\overline{b}\) and \(Z^{(*)}Z^{(*)} \rightarrow \mu ^+\mu ^- jj\). Such a scenario could be an alternative channel to search for light Higgs bosons at Run 3 and the HL-LHC.

Fig. 6
figure 6

\(m_h\) and \(m_a\) vs. \(\textrm{BR}(h \rightarrow b\overline{b})\) (left) and \(\sigma (gg \rightarrow H \rightarrow aZ^{*} \rightarrow h Z^{*}Z^{*} \rightarrow Z^{*}Z^{*}b\overline{b})\) (right) at 95% C.L. in the 2HDM Type-I

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4 Signal vs. background analysis

We describe here the toolbox used to generate and analyse MC events. MadGraph-v.9.2.5 [70] is used to generate parton level configurations of both signal and background processes.Footnote 4 The events are passed then to PYTHIA8 [72] to simulate parton showering, hadronisation and decays. Finally, we use Delphes \(-\)3.5.0 [73] with the standard CMS cardFootnote 5 to perform detector simulation. We resort to MadAnalysis [74] to apply cuts and to conduct the analysis.

The major background processes are top pair production in association with 2 Initial State Radiation (ISR) jets,Footnote 6ZZ production with additional \(b\overline{b}\) quarks, \(ZW + b\overline{b} \rightarrow \mu ^+ \mu ^- jjb\overline{b}\) and Drell-Yan plus jets (DY+jets). We show in Table 3 the corresponding cross sections at \(\sqrt{s}=13\) TeV for the LHC energy. We have generated MC samples of \((10^6)\) events. Unsurprisingly, the irreducible backgrounds \(pp \rightarrow Z^{(*)}Z^{(*)}b\overline{b} \rightarrow b\overline{b}jj\mu ^+ \mu ^-\) (from both QCD and EW interactions) and \(pp \rightarrow ZWb\overline{b} \rightarrow b\overline{b}jj\mu ^+ \mu ^-\) are negligible whereas \(pp \rightarrow gg t\overline{t} \rightarrow gg\mu ^+\mu ^- jj b\overline{b} \nu _\mu {\overline{\nu }}_\mu \) and DY+jets are large.

Table 3 The parton level cross sections of the background processes at LO
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We considered a few BPs for the signal given by \(gg\rightarrow H \rightarrow aZ^{*} \rightarrow h Z^{*}Z^{*} \rightarrow \mu ^+ \mu ^- ~jj~b\overline{b}\) to perform the MC simulation. The input parameters of each BP are given in Table 4. Note that the light Higgs width, \(\varGamma (h)\), is not small enough to lead to a large lifetime and hence, long-lived particles producing displaced vertices inside the detector. The proper decay length \(c\tau _0\) is in fact only a tiny fraction of micrometers.Footnote 7 The different kinematic distributions at parton level in Fig. 7 show that the requirement of central pseudorapidity of the muons is generally satisfied however the \(p_T\) of these can be rather small. To address this, one can invoke the di-muon scouting trigger,Footnote 8 which involves lowering the transverse momentum (\(p_T^\mu \)) of muons to 4.5 GeV [77, 78]. Note that the \(p_T\) threshold of such trigger is reduced to 3 GeV [79] in Run 3. Figure 8 shows the invariant mass distributions of the two b-jets, \(m_{b\overline{b}}\), and that of the full final state, \(m_{jj \mu ^+ \mu ^- b\overline{b}}\), for the signal and the irreducible background processes at parton level, noting that \(m_{b\overline{b}}\) is close to light Higgs mass \(m_h\) and \(m_{jj \mu ^+ \mu ^- b\overline{b}}\) is close to SM-like Higgs mass \(m_H\) (for the signal, unlike the irreducible backgrounds). We will clearly leverage these underlying partonic shapes in our detector levelFootnote 9 analysis, to which we proceed next, in the presence of the following sequence of acceptance cuts:

$$\begin{aligned} p_T^{b}&> 20~\text {GeV},~p_T(j)> 20~\text {GeV},~p_T^{\mu _{1}}>10~\text {GeV}, \\ p_T^{\mu _{2}}&>5~\text {GeV},~|\eta (l,b)|< 2.5,~|\eta (j)|< 5.0,~\varDelta R > 0.4. \end{aligned}$$

Additionally, we only consider events with two oppositely charged muons, two b-jets and two light jets in the final state. Events are further preselected after requiring the invariant mass of the two b-jets (\(m_{b\bar{b}}\)), the two light jets \((m_{jj})\) and the di-muon systemFootnote 10 (\(m_{\mu \mu }\)) to lie within the following mass ranges, as shown in Fig. 9,

$$\begin{aligned} 5~\text {GeV}&<m_{b\bar{b}}<40~\text {GeV},~10~\text {GeV}<m_{\mu \mu }<50~\text {GeV}, \\ 10~\text {GeV}&<m_{jj}<50~\text {GeV} . \end{aligned}$$
Table 4 Selected BPs with parton level cross section and other observables at LO. (All masses and widths are in GeV, with \(m_H = 125~\text {GeV}\).)
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Fig. 7
figure 7

The transverse momentum (left) and pseudorapidity (right) of the hardest muon for the signal (all BPs)

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Fig. 8
figure 8

The invariant mass of the \(b\bar{b}\) (left) and \(\mu ^+ \mu ^- ~jj~b\overline{b}\) (right) system for the signal (BP1) and the irreducible backgrounds \((ZZb\overline{b},~ZWb\overline{b})\) at parton level

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Fig. 9
figure 9

Correlation between and \(m_{b\overline{b}}\) (left panel), \(m_{\mu \mu }\) (middle panel) and \(m_{jj}\) (right panel) for signal (BP3), at detector level

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Fig. 10
figure 10

The and highest \(p_T\) distributions for muons, b- and light-jets (clockwise) for signal (BP3) and background processes, \(ggt\bar{t}\) (blue) and DY+jets (green), at detector level

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Fig. 11
figure 11

\(\varDelta R\) distributions between the two (\(p_T\) ordered) b-jets and muons, from hardest to softest (clockwise) for signal (red) (BP3) and background processes, \(ggt\bar{t}\) (blue) and DY+jets (green), at detector level

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Figure 10 displays the distributions of the missing transverse energy () and the highest \(p_T\)’s of b-jets, light-jets and muons for signal and background processes at detector level. (As mentioned previously, the irreducible backgrounds stemming from \(ZZ b\overline{b}\) and \(ZWb\bar{b}\) processes are negligible, so we have not emulated these at detector level.) The distribution from simulated samples of background events is mainly from di-leptonic decay of \(ggt\overline{t}\), i.e., with \(t \overline{t}\rightarrow W^+bW^-\overline{b} \rightarrow (\mu ^+ \nu _\mu b) (\mu ^- {\overline{\nu }}_\mu \overline{b})\) whereas the in both signal and DY+jets events is due to the semi-leptonic b-meson decays (alongside detector effects). Furthermore, Fig. 11 illustrates the different angular separations between b-quarks and muons for signal and dominant backgrounds, where one can read that background arsing from top pair production has only a minimal component with muons coming from semi-leptonic b-meson decays (as intimated).

To enhance the signals and suppress the backgrounds arising from \(ggt\overline{t}\) and DY+jets, we have adopted several kinematic cuts, which choice is based on comparing different distributions of the signal and background processes at the detector level. Specifically, this has been done through 2D distributions correlating the missing transverse momentum to a series of kinematic variables pertaining to some of the visible objects in the final state as illustrated in Figs. 12 and 13 for signal and background processes mentioned above. One can read that the signal and backgrounds distributions are anti-correlated. In fact, forcing the missing transverse energy to be below 30 GeV will strongly favour the signal over the backgrounds, specifically, \(ggt\bar{t}\).Footnote 11 Additionally, selecting events with \(p_T^{j} < 75\) GeV and \(p_T^{\mu } < 40\) GeV would enhance the signal significance and suppress both background processes. Through similar reasoning, we require the invariant mass of the system to satisfy \(m_H<180~\text {GeV}\). We show in Table 5 the event rates of the backgrounds after applying the cutflow discussed above. Notably, DY+jets events are significantly reduced after considering a final state with exactly 2 b-jets, 2 light jets and 2 muons, while adhering to the invariant masses requirements.Footnote 12 Both backgrounds processes are completely removed after applying the kinematic cuts and restricting the invariant mass of the system to be below 180 GeV to further differentiate the signal from the remaining background processes.

Fig. 12
figure 12

Correlation between \(p_T^j\) and for signal (BP3) (red), \(ggt\bar{t}\) (blue) and DY+jets (green) at detector level

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Fig. 13
figure 13

Correlation between \(p_T^\mu \) and for signal (BP3) (red colour), \(ggt\bar{t}\) (blue colour) and DY+jets (green colour) at detector level

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Table 5 Event rates of the two dominant background processes, \(ggt\overline{t}\) and DY+jets, with \(\sqrt{s} = 13~\text {TeV}\) and \(\mathcal {L}=300~\text {fb}^{-1}\) after applying the cutflow
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We have then computed the significance (for \(\sqrt{s}=13\) TeV and \(\mathcal {L} = 300~ \text {fb}^{-1}\)), defined as \(\varSigma =\frac{S}{\sqrt{B+S}}= \sqrt{S}\),Footnote 13 where S(B) is the signal(background) yield after the discussed cutflow, for not only our three initial BPs (whose \(\varSigma \) rates are 3.05, 3.18 and 3.42 for BP1, BP2 and BP3, respectively), but also those appearing in Table 6. We have done so in order to be able to map the 2HDM Type-I parameter space in detail, so as to acquire a sense of the true portion of it that can be tested by forthcoming experiments. Note that we have kept the same cutflow already illustrated for all such new BPs too. Also, it is at this stage that we take into account the aforementioned QCD K-factor for the signal. Many of the latter can have a significance larger than 3 and up to nearly 4, for Run 3 energy and luminosity. To observe their distribution over the (\(m_h, m_a\)) plane, we have finally produced Fig. 14, indeed, assuming \(\sqrt{s}=13\) TeV and \(\mathcal {L} = 300~ \text {fb}^{-1}\), where both significance \((\varSigma )\) and efficiency \((\epsilon )\) are mapped. Hence, at Run 3, we can conclude that a substantial portion of the 2HDM Type-I parameter space can offer some evidence of the signal we have pursued. Furthermore, we notice that a larger efficiency can be obtained for small \(m_a\): this is because the loss of efficiency with b-tagging is over-compensated by a simultaneous higher efficiency for both j- and \(\mu \)-tagging. Needless to say, at the HL-LHC, where \(\mathcal {L} = 3000~ \text {fb}^{-1}\), most of the sampled parameter space of the 2HDM Type-I would be discoverable.

Table 6 Extended list of BPs used in the MC simulation for the 2HDM Type-I parameter scan, highlighting the h and a masses as well as the signal LO cross section and event rate after the full cutflow, together with its significance \(\varSigma \) and efficiency \(\epsilon \). Recall that NNLO QCD K-factor has been used for Higgs production. Here, \(\sqrt{s}=13\) TeV and \({{\mathcal {L}}}=300\) fb\(^{-1}\)
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Fig. 14
figure 14

Significance (left) and efficiency (right) of each BP produced in our analysis over the (\(m_h, m_a\)) projection of the 2HDM Type-I parameter space, after the full cutflow described in the text

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5 Conclusions

In this paper, we have shown the outcome of performing some recasting over the parameter space of the 2HDM Type I, wherein the heaviest CP-even Higgs state H is identified with the discovered SM-like one, \(H_\textrm{SM}\), while h and a are lighter. After considering the available experimental data from searches for exotic Higgs decay into two light (pseudo)scalars, we have found that the corresponding parameter space for which there is sensitivity via \(H_\textrm{SM}\rightarrow hh(aa)\rightarrow \tau ^+\tau ^- b\overline{b}\) at Run 2 is already excluded by existing constraints from BSM Higgs searches. Furthermore, we have shown that there are regions of the 2HDM Type-I parameter space compliant with theoretical and experimental constraints yielding substantial BR\((H^\pm \rightarrow W^\pm a)\) and BR\((H \rightarrow Z^{*}Z^{*}h)\). The large size of the former has been exploited in other literature. Here, concerning the latter, we have made the case for looking at the process \(pp\rightarrow H_\textrm{SM}\rightarrow Z^{*}A\rightarrow Z^{*} Z^{*}h\) via \( ZZ\rightarrow \mu ^+\mu ^- jj\) and \(h\rightarrow b\overline{b}\) decays, specifically, in the region with large \(m_a\) and small \(m_h\). After performing a full MC analysis down to detector level, we have proven that the overwhelming backgrounds arising from both top-quark pair production in association with 2 ISR jets and DY+jets can be suppressed after applying efficient kinematics cuts, leading to a large significance of this hitherto unexplored light Higgs signature already at Run 3 of the LHC, where evidence of it can be seen, further affording one with clear discovery potential at the HL-LHC.