Two Higgs Doublets and a Complex Singlet:
Disentangling the Decay Topologies and Associated Phenomenology
We present a systematic study of an extension of the Standard Model (SM) with two Higgs doublets and one complex singlet (2HDM+S). In order to gain analytical understanding of the parameter space, we re-parameterize the 27 parameters in the Lagrangian by quantities more closely related to physical observables: physical masses, mixing angles, trilinear and quadratic couplings, and vacuum expectation values. Embedding the 125 GeV SM-like Higgs boson observed at the LHC places stringent constraints on the parameter space. In particular, the mixing of the SM-like interaction state with the remaining states is severely constrained, requiring approximate alignment without decoupling in the region of parameter space where the additional Higgs bosons are light enough to be accessible at the LHC. In contrast to 2HDM models, large branching ratios of the heavy Higgs bosons into two lighter Higgs bosons or a light Higgs and a boson, so-called Higgs cascade decays, are ubiquitous in the 2HDM+S. Using currently available limits, future projections, and our own collider simulations, we show that combining different final states arising from Higgs cascades would allow to probe most of the interesting region of parameter space with Higgs boson masses up to 1 TeV at the LHC with of data.
With the discovery of a Higgs boson at the Large Hadron Collider (LHC) in 2012 Aad:2012tfa (); Chatrchyan:2012xdj () with mass GeV Aad:2015zhl (); Sirunyan:2017exp () and couplings compatible with those of a Standard Model (SM) Higgs Khachatryan:2016vau (); CMS-PAS-HIG-16-042 (); CMS-PAS-HIG-17-031 (); Sirunyan:2017exp (); ATLAS-CONF-2017-047 (), the final ingredient of the SM has been found. However, despite the SM holding up to all laboratory test it has been subjected to so far, it fails to explain the behavior of the Universe at large scales. In particular, the SM does not contain a suitable candidate for the observed Dark Matter and fails to explain the matter-antimatter asymmetry. Beyond such phenomenological problems, the SM also suffers from some issues more theoretical in nature, e.g. the hierarchy problem or the lack of explanation of the SM’s flavor structure.
During the past 50 years, a multitude of Beyond the Standard Model (BSM) particle physics models have been developed to address the aforementioned problems. The vast majority of BSM models, in particular those valid to energy scales much larger than the electroweak scale, feature a scalar sector extended beyond the SM’s one -doublet Higgs. For example, consistent supersymmetric extensions of the SM require a Higgs sector containing at least two Higgs doublets. Furthermore, while the parameters of the Higgs sector are those least well measured in the SM, many of the SM’s shortcomings are intimately related to the scalar sector e.g. the hierarchy problem, the matter-antimatter asymmetry, and the flavor structure.
There are two avenues for studying BSM physics: One can either take a top down approach, starting from well-motivated SM extensions valid at energy scales much larger than the electroweak scale, or, one can take a bottom up approach by parameterizing our ignorance of high-scale physics by writing down the most general form of BSM models at the weak scale. In the case of extensions of the SM’s Higgs sector, this may be exemplified by the well studied case of BSM models containing two Higgs doublets. One can either choose to study more complete models containing such a Higgs sector, e.g. the Minimal Supersymmetric Standard Model (MSSM), or, one can take a more model independent approach and study the most general form of a two Higgs doublet Model (2HDM).
From a bottom up perspective, extensions of 2HDMs with an additional singlet may be motivated from the well known fact that they facilitate baryogenesis Profumo:2007wc (); Barger:2008jx (); Cline:2012hg () to explain the matter-antimatter asymmetry, as well as being useful for constructing Dark Matter models Baum:2017enm (). In addition, well motivated top down BSM models exist containing such a scalar sector, for example the Next-to-Minimal Supersymmetric Standard Model (NMSSM) Maniatis:2009re (); Ellwanger:2009dp (). In this work, we take the bottom up approach to study extensions of the SM’s scalar sector with one additional doublet and one complex SM gauge singlet , which we refer to as 2HDM+S in the following.
While a large body of literature exists on general 2HDM models, see e.g. Refs. Branco:2011iw (); Bernon:2015qea (); Bernon:2015wef (); Basler:2017nzu (); Cherchiglia:2017uwv () for recent works, no systematic study of 2HDM+S models exists in the literature to the best of our knowledge. Some aspects of 2HDM+S models have been discussed in the appendix of Ref. Carena:2015moc (), and the extension of 2HDMs with a real singlet has been discussed in Refs. Chen:2013jvg (); Muhlleitner:2016mzt (); Muhlleitner:2017dkd (). In this work, we provide a first systematic study of the 2HDM+S parameter space. The physical Higgs sector contains 5 real neutral Higgs bosons and one charged Higgs boson after electroweak symmetry breaking. Assuming CP-conservation, the 5 neutral Higgs bosons can be subdivided into 3 CP-even states and 2 CP-odd states. In order to be compatible with the observed phenomenology, one of the CP-even states must have couplings to pairs of SM particles similar to those of a SM Higgs with a mass of GeV, such that it can be identified with the 125 GeV Higgs boson observed at the LHC. As we will see, the presence of this SM-like state has important implications for the behavior of the remaining Higgs bosons.
Extended Higgs sectors such as the 2HDM+S can be tested at the LHC in two complimentary ways: one can study the behavior of the observed approximately SM-like , constraining its possible mixing with additional Higgs bosons, or, one may directly search for the remaining Higgs bosons in the model. The first way has thus far constrained a number of couplings of to SM particles to be within of the SM values. In most cases, future data expected to be collected at the LHC during the next 20 years will allow only for marginal improvements on these bounds Englert:2015hrx (). Note however, that a Higgs factory such as the International Linear Collider would allow for precision tests of the SM-like Higgs boson, probing some of its couplings below the level Asner:2013psa (). This would yield powerful constraints on any BSM Higgs sector.
Such indirect searches are complimented by direct searches for the additional Higgs bosons as carried out at the LHC in a plenitude of final states. Most commonly, one searches for direct production of the additional Higgs bosons which then decay into pairs of SM particles. Such search strategies are similar to those employed in the hunt for the 125 GeV Higgs boson prior to its discovery. For a 2HDM Higgs sector, searches for additional Higgs bosons are complicated by three different issues: 1) While CP-even states are allowed to decay into pairs of SM vector bosons, the SM-like nature of suppresses the couplings of the additional states to pairs of vector bosons. 111Here and in the rest of the article, vector bosons refers specifically to the weak gauge bosons: and . 2) In addition to suppressing the branching ratio into pairs of SM vector bosons, the SM-like nature of also suppresses the coupling of heavy Higgs bosons to pairs of SM-like Higgs boson or a boson and a SM-like Higgs, reducing the power of the corresponding search channels. 3) For large regions of parameter space the decays of any Higgs boson with mass GeV are dominated by decays into pairs of top quarks. Due to the interference of this signal with the QCD background Dicus:1994bm (); Barcelo:2010bm (); Barger:2011pu (); Bai:2014fkl (); Jung:2015gta (); Craig:2015jba (); Gori:2016zto (); Carena:2016npr (), such decays are challenging to use for Higgs searches at the LHC with current search strategies.
All three of these complications are still present in models with enlarged Higgs sectors such as the 2HDM+S. In addition, the presence of the singlet, which can mix with the doublets, complicates conventional searches further since the production cross section of any Higgs state at the LHC is suppressed with a growing singlet fraction. However, the presence of additional Higgs bosons allows for new decay channels of the heavy Higgs bosons, so-called Higgs cascades, where the heavy Higgs bosons decay into two lighter Higgs states or a light Higgs and a boson Kang:2013rj (); King:2014xwa (); Carena:2015moc (); Ellwanger:2015uaz (); Costa:2015llh (); Baum:2017gbj (); Ellwanger:2017skc (). Such decays can also be present in the generic 2HDM without the singlet if the CP-even and CP-odd Higgs bosons have non-degenerate masses, although large mass splittings are difficult to achieve in consistent 2HDMs Krauss:2018thf (). However, in this case the direct production cross section of the lighter state has no suppression from a singlet component, generally making direct searches for such a state more powerful. We briefly discuss the limiting case of decoupling the singlet from the doublets in Sec. 2.3.
As mentioned above, the branching ratios of any of the heavy Higgs bosons to pairs of SM-like ’s, pairs of vector bosons, or a boson and a , are suppressed by the SM-like nature of . Of the remaining decay modes with possibly large, , branching ratios, those consisting of a SM-like Higgs boson or a boson and one additional non SM-like Higgs are most useful for searches at the LHC. This is because one can use the decay products of the SM-like Higgs or the boson with known mass and branching ratios to tag such events. Furthermore, final state search signatures will also contain the decay products of the non SM-like Higgs bosons: if they decay into visible states, e.g. a pair of SM particles, one searches for a resonance in the invariant mass spectrum of the decay products. However, the non SM-like Higgs bosons can also decay into additional new states, for example pairs of new stable neutral particle playing the role of a Dark Matter candidate. In this case, the decay products of the non SM-like Higgs bosons produced in the Higgs cascades would leave the detector without depositing energy, manifesting as missing transverse energy () in the detector, giving rise to so-called mono- and mono-Higgs signatures.
As shown in the context of the NMSSM in Refs. Carena:2015moc (); Baum:2017gbj (); Ellwanger:2017skc (), the branching ratios of Higgs cascades are sizable in large parts of the parameter space, leading to observable signatures at the LHC. As we will see below, this is also the case in the general 2HDM+S.
The remainder of this work is organized as follows: In Sec. 2, we discuss the 2HDM+S. We focus on the mass spectrum, and the implications of the SM-like nature of the 125 GeV Higgs boson in Sec. 2.1, and the couplings between the Higgs bosons and the relevant model parameters for the Higgs cascades in Sec. 2.2. In Sec. 2.3 we discuss the production cross sections and decay modes of the 2HDM+S Higgs bosons, and in Sec. 2.5 we discuss the Higgs cascades in more detail. The more experimentally inclined reader may skip directly to Sec. 3 where we discuss the future reach of the LHC for the full 2HDM+S using Higgs cascades. In Sec. 3.2 we demonstrate that the 2HDM+S gives rise to cross sections within reach of the Higgs cascade searches at the LHC for large regions of parameter space. We reserve Sec. 4 for our conclusions. Appendices A, B, C, and D contain details about the mass matrices, trilinear couplings, quartic couplings, and the collider simulation performed for estimating the reach of the mono- channel, respectively.
The most general 2HDM scalar potential is given by Branco:2011iw ()
where , are doublets with hypercharge . The parameters have dimension mass squared, while the are dimensionless. In this work, we study the CP-conserving case, where all parameters can be chosen manifestly real. Extending the field content by a complex scalar SM gauge singlet , the most general scalar potential for the additional terms is
The parameter has dimensions mass cubed, the parameters have dimension mass, and the are dimensionless. For the CP-conserving case, all parameters can again be chosen to be manifestly real.
The most general scalar potential for a 2HDM+S model is then given by
Out of the 29 parameters only 27 are physical: One can perform a real rotation in space to remove one parameter from the doublet sector Branco:2011iw () and one can perform a real shift of to remove one parameter from the singlet sector. We use these transformations to remove the parameter and the tadpole term from the scalar potential.
One can furthermore use the minimization conditions
to trade three parameters, e.g. , for the vacuum expectation values (vevs)
Per definition, the vevs lie along the neutral direction of the doublets and are real for our choice of the CP-conserving 2HDM+S. In addition to the field transformations discussed above, one can redefine each of the doublets as well as the singlet by a phase. Demanding the potential to remain manifestly CP-conserving under such transformations, i.e. all parameters to remain real, constrains these transformation to in the doublet sector and for the singlet. Although these transformations cannot be employed to absorb additional parameters, one can use them to choose all vevs positive, .
As customary, we define
The observed mass of the boson GeV is obtained for GeV.
It is useful to rotate the Higgs fields to the extended Higgs basis Georgi:1978ri (); Donoghue:1978cj (); gunion2008higgs (); Lavoura:1994fv (); Botella:1994cs (); Branco99 (); Gunion:2002zf (); Carena:2015moc ()222Note that there are different conventions in the literature for the Higgs basis differing by an overall sign of and . Taking these into account, our potential and couplings for the 2HDM+S can be mapped directly to the potential and couplings given in the appendices of Ref. Carena:2015moc ().
where and are the neutral CP-even and CP-odd real Higgs basis interaction states and () is the neutral (charged) Goldstone mode. In this basis, of the states coming from the doublets, only acquires a vev, and it is straightforward to work out the coupling of SM fermions to the Higgs basis states to e.g. study possible flavor changing neutral currents (FCNCs). Potentially dangerous FCNCs can be omitted if the groups of right-handed SM fermions with the same quantum numbers couple to only one of the doublets, respectively, as in the so-called Type I, Type II, flipped, and lepton specific 2HDM versions 333In Type I models, all SM fermions couple to the same doublet. In Type II models, one doublet couples to up-type fermions and the other doublet to the down-type fermions. In flipped models, the up-type quarks and the charged leptons couple to one of the doublets while down-type quarks couple to the other doublet. Finally, in lepton specific models, all quarks couple to one doublet while the charged leptons couple to the other doublet. Branco:2011iw (). Note that couplings of the fermions to the singlet are forbidden by gauge invariance. Restricting ourselves to the case where the values of the Yukawa matrices are chosen such that the observed SM fermion masses are obtained, the couplings of pairs of SM particles to the Higgs basis states can be written as
where “” (“”) stands for SM-fermions coupling to (), “VV” for pairs of or gauge bosons, and is the coupling of a SM Higgs boson to such particles. Note that CP-odd states couple to pseudoscalar fermion bilinears , instead the CP-even states couple to the scalar bilinear .
For concreteness, in the following we assume a Type II Yukawa structure,
where , the are the Yukawa matrices, and the left-handed quarks and leptons as well as the right-handed up-type (down-type) quarks () and the right-handed leptons should be understood as vectors in generation space. Note that our results in the remainder of this paper will in general hold also for a different Yukawa structure. Some quantitative details may change because of the change of the Yukawa enhancement/suppression of the fermion couplings. However, such modifications will be small since we mostly consider the low regime in this work.
2.1 Higgs Mass Eigenstates and Alignment
The mass eigenstates are obtained from the diagonalization of the squared-mass matrix for the Higgs basis states. The neutral and charged Goldstone modes and are massless by construction and do not mix with the other interaction states. In the following, we remove the Goldstone modes from the theory by choosing the unitary gauge. Furthermore, there is no mixing between CP-even and CP-odd states in the CP-conserving 2HDM+S.
We denote the three CP-even mass eigenstates
where is identified with the GeV SM-like state observed at the LHC, and and are ordered by masses, . Each mass eigenstate is an admixture of the extended Higgs basis interaction states,
where with SM, NSM, S denotes the components of the mass eigenstates in terms of the interaction basis. Likewise, we denote the two CP-odd mass eigenstates
where again , and
where the components are similarly denoted by .
The () are obtained from diagonalizing the (symmetric) squared mass matrix for the CP-even (CP-odd) Higgs bosons, (). The values of the entries of the mass matrices and the mass of the charged Higgs boson are recorded in Appendix A.
The observation of a GeV mass eigenstate with couplings to SM particles compatible with that of a SM Higgs boson at the LHC implies that our model must contain a mass eigenstate with
or, in other words, a 125 GeV mass eigenstate approximately aligned with the interaction state.
The observed mass approximately fixes , while the mixing of with is suppressed  for
and the mixing of with is suppressed  for
Hence we see that there are two possibilities to achieve alignment: either the left hand sides of Eqs. (21) and (22) go to zero, or, the right hand sides become large while the left hand sides remains non-zero and sizable. The latter possibility is the so-called decoupling limit, corresponding to
implying . The first option is the so-called alignment without decoupling limit, and is of particular interest for LHC phenomenology. This is because the additional CP-even mass eigenstates and are not necessarily much heavier than and hence may be directly accessible at the LHC Gunion:2002zf (); Carena:2013ooa (); Carena:2014nza (); Bernon:2015qea (); Bernon:2015wef (); Carena:2015moc (). In this case,
must be satisfied in order to ensure approximate alignment. Neglecting radiative corrections, perfect alignment is achieved for
where the first condition ensures and the second condition . Here and in the following we employ a shorthand notation,
2.2 Couplings and Parameters
The trilinear couplings of the Higgs bosons can be obtained from the scalar potential as
We first note that CP-conservation forbids couplings such as or . In the following, we discuss the remaining trilinear couplings in the extended Higgs basis. From these, the couplings for the mass eigenstates can be obtained via
Some of the couplings allowed by CP-conservation, vanish due to gauge invariance. For example, the coupling
is identically zero since it would arise only through terms proportional to , , in the scalar potential which are forbidden by invariance.
A number of the trilinear couplings are proportional to entries in the mass matrices: Categorizing the Higgs basis states as SM-like (), NSM-like (), and singlet-like (), we find that couplings involving only SM-like states are proportional to the corresponding diagonal entry of the CP-even mass matrix
Couplings involving two SM-like states are proportional to the corresponding entry in the CP-even squared mass matrix mixing such states,
Couplings involving one SM-like state, one NSM-like state, and one singlet-like state are proportional to the singlet-doublet mixing entries in one of the mass matrices
This implies that the couplings of two SM-like states to one NSM-like or one singlet-like state are suppressed in the alignment limit. Indeed, for perfect alignment one finds
and hence the couplings of the mass eigenstates,
are suppressed in proximity to alignment, and vanish for perfect alignment.
Besides the trilinear couplings proportional to the entries of the mass matrices, a few of the couplings are identical or can be written as linear combinations of entries of the mass matrices. Accounting for such degeneracies, we find that there are 10 independent trilinear couplings,
For completeness we provide a list of the trilinear couplings and their values not a priori forbidden by CP or charge conservation in Table 1 located in Appendix B, which match the results obtained in Ref. Carena:2015moc ().
It is straightforward to show from the scalar potential that there are 4 independent quartic couplings between the interaction states which cannot be written as linear combinations of the entries of the mass matrices or the trilinear couplings,
We list the values of these couplings in Appendix C.
All together, the 9 entries of the mass matrices, which we parameterize by the 5 physical masses
and 4 mixing angles
the 3 vevs, which we parameterized by
the charged Higgs mass
the 10 independent linear couplings listed in Eq. 38, and the 4 independent quartic couplings given in Eq. 39 yield a set of 27 independent parameters. In the remainder of this work we will describe the 2HDM+S parameter space in terms of these parameters since they are more closely related to physical observables instead of the 27 parameters in the scalar potential. That said, we note that of these 27 parameters, only a subset are relevant for the analysis of the heavy Higgs decay topologies, especially given the constraints due to the SM-like nature of . In particular, the quartics play no role (however, these may be interesting when considering di-Higgs production), and as we shall see, only a few of the trilinear couplings listed in Eq. (38) will be needed.
2.3 Production Cross Section and Partial Decay Widths
For moderate/low values of , the production cross section for non-standard Higgs bosons at the LHC is dominated by gluon fusion. For larger values of , associated production may become relevant. Note that in proximity to the alignment limit the vector boson fusion production cross sections of additional Higgs boson is suppressed since only couples to pairs of vector bosons.
To first approximation, the gluon fusion production cross section can be parameterized by the contribution from top quarks in the loop. At one-loop order, the ratio of the gluon fusion production cross sections of a CP-even Higgs boson and a CP-odd Higgs boson is then given by Spira:1995rr ()
where is the gluon fusion production cross section of a SM Higgs boson with mass , , is the mass of , is the mass of , is the top quark mass, and
As long as the narrow width approximation is valid, the production cross section of any final state arising from gluon fusion production of a Higgs boson is given by the product of the gluon fusion production cross section of the Higgs boson of interest, and its branching ratio into the relevant final state. The relevant branching ratio is given in terms of the partial decay widths by
where the width of the state is obtained by summing over all partial widths. In the remainder of this section, we discuss partial widths of the most relevant decay modes of the additional Higgs bosons.
The partial decay width for a CP-even mass eigenstate into pairs of vector bosons is given by
whereas such decays are forbidden for CP-odd mass eigenstates at tree level.
Decays into pairs of SM fermions are allowed for both CP-even and CP-odd Higgs bosons. The corresponding partial widths can be written
where () for decays into SM quarks (leptons), (), and () for CP-even (CP-odd). Here and in the following, we use to refer to the mixing angles for both the CP-even and the CP-odd mass eigenstates with mass ; for example, or depending on context.
If there are additional Majorana fermions , e.g. Dark Matter candidates, which couple to the Higgs bosons via a coupling 444Interactions with the doublet Higgs fields may be generated via their mixing with the singlet or from higher dimensional operators as would result from integrating out a heavy Dirac fermion doublet Baum:2017enm ().
the corresponding partial width is given by
where () for a CP-even (CP-odd) .
In addition to the above, the extended Higgs sector of the 2HDM+S model allows potentially large decay widths of the heaviest Higgs bosons into pairs of lighter Higgs bosons or a Higgs and a boson. CP-conservation allows decays into pairs of Higgs bosons only if they are of the type (), (), or (), while decays into a and a Higgs boson must be of the type () or (). The corresponding partial widths are given by
where the trilinear couplings between the Higgs mass eigenstates are given in Eq. (29).
2.4 Branching Ratios: SM Fermions, Misalignment and the 2HDM Limit
Considering the decays of the heavy Higgs into pairs of SM particles, it is worth repeating that in the alignment limit only the SM-like mass eigenstate decays into pairs of vector bosons, while the corresponding branching ratio for the non SM-like mass eigenstates such as () vanish. The dominant decay mode into SM particles will thus be into the pair of kinematically accessible SM fermions with the largest suppressed (enhanced) Yukawa coupling
where () for up-type (down-type) fermions accounts for the suppression (enhancement) of the Yukawa couplings. For moderate values of , the dominant decay mode will thus be into pairs of top quarks if . However, decays into bottom quarks will dominate over those into top quarks even for if . In the following, we identify the regions of parameter space where other decay modes may dominate over decays into pairs of SM fermions.
First, we note that if kinematically accessible, any of the Higgs bosons can decay into possible additional singlet fermions . Ignoring kinematic factors, such decays would compete with decays into SM fermions if
where is the coupling of to as defined in Eq. (50).
Departures from perfect alignment allow for decays of the non SM-like CP-even states and into pairs of vector bosons. From Eqs. (47) and (49), we see that for , decays into pairs of () bosons will compete with decays into pairs of SM fermions if
where we have ignored the additional contribution to the effective Yukawa coupling from the SM component of . Since and are only allowed to have a small component from phenomenology, these modes are expected to be very suppressed unless is moderate, rendering both the bottom and top Yukawa couplings small, and , which would significantly reduce the production cross section of such a Higgs boson at the LHC.
Here, is the () component of the daughter CP-even (CP-odd) . Similar to decays into vector bosons, decays of the type are suppressed by the required approximate alignment , making this mode experimentally challenging. On the other hand, decays of the type are not suppressed if kinematically accessible, since is not constrained by the required alignment of . However, if simultaneously becomes large the production cross section of is suppressed.
Here, is the trilinear coupling between the mass eigenstates participating in the cascade decay, and is the () component of the parent CP-even (CP-odd) .
Recalling the form of the trilinear couplings in Sec. 2.2, we observe that decays of the non SM-like Higgs bosons into pairs of SM-like Higgs bosons are suppressed in proximity to the alignment limit, . For example (), to leading order, is governed by . Thus, for decays into pairs of ’s to compete with decays into ,
From the form of the trilinear couplings tabulated in Tab. 1, we see that , and . Hence, for and low to moderate values of , can be of the order of the bottom Yukawa. Above the top threshold, can only compete with the top Yukawa if is large enough to compensate for the small , i.e. if has a substantial singlet component, and if is significantly heavier than . However, in this case the production cross section of would be significantly reduced, making this channel challenging.
The trilinear couplings governing the decays of , for , are not suppressed by alignment. For example, decays of and are governed by