Prospects for charged Higgs searches at the LHC

Prospects for charged Higgs searches at the LHC

Abstract

The goal of this report is to summarize the current situation and discuss possible search strategies for charged scalars, in non-supersymmetric extensions of the Standard Model at the LHC. Such scalars appear in Multi-Higgs-Doublet models (MHDM), in particular in the popular Two-Higgs-Doublet model (2HDM), allowing for charged and additional neutral Higgs bosons. These models have the attractive property that electroweak precision observables are automatically in agreement with the Standard Model at the tree level. For the most popular version of this framework, Model II, a discovery of a charged Higgs boson remains challenging, since the parameter space is becoming very constrained, and the QCD background is very high. We also briefly comment on models with dark matter which constrain the corresponding charged scalars that occur in these models. The stakes of a possible discovery of an extended scalar sector are very high, and these searches should be pursued in all conceivable channels, at the LHC and at future colliders.

A.G. Akeroyd, M. Aoki, A. Arhrib, L. Basso, I.F. Ginzburg, R. Guedes, J. Hernandez-Sanchez, K. Huitu, T. Hurth, M. Kadastik, S. Kanemura, K. Kannike, W. Khater, M. KrawczykaaaCorresponding authors: Maria.Krawczyk@fuw.edu.pl, Per.Osland@uib.no, F. Mahmoudi, S. Moretti, S. Najjari, P. Osland, G.M. Pruna, M. Purmohammadi, A. Racioppi, M. Raidal, R. Santos, P. Sharma, D. Sokołowska, O. Stål, K. Yagyu, E. Yildirim

School of Physics and Astronomy, University of Southampton, Highfield, Southampton SO17 1BJ, United Kingdom,

Institute for Theoretical Physics, Kanazawa University, Kanazawa 920-1192, Japan,

Département de Mathématique, Faculté des Sciences et Techniques, Université Abdelmalek Essaâdi, B. 416, Tangier, Morocco,

LPHEA, Faculté des Sciences-Semlalia, B.P. 2390 Marrakesh, Morocco,

CPPM, Aix-Marseille Université, CNRS-IN2P3, UMR 7346, 163 avenue de Luminy, 13288 Marseille Cedex 9, France,

Sobolev Inst. of Mathematics SB RAS and Novosibirsk University, 630090 Novosibirsk, Russia,

IHC, Instituto de História Contemporanea, FCSH - New University of Lisbon, Portugal,

Facultad de Ciencias de la Electrónica, Benemérita Universidad Autónoma de Puebla,

Apdo. Postal 542, C.P. 72570 Puebla, Puebla, México

and Dual C-P Institute of High Energy Physics, México,

Department of Physics, and Helsinki Institute of Physics, P.O.Box 64 (Gustaf Hällströmin katu 2), FIN-00014 University of Helsinki, Finland,

PRISMA Cluster of Excellence and Institute for Physics (THEP), Johannes Gutenberg University, D-55099 Mainz, Germany,

National Institute of Chemical Physics and Biophysics, Rävala 10, 10143 Tallinn, Estonia,

Department of Physics, University of Toyama, 3190 Gofuku, Toyama 930-8555, Japan,

Department of Physics, Birzeit University, Palestine,

Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland,

Univ Lyon, Univ Lyon 1, ENS de Lyon, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574, F-69230 Saint-Genis-Laval, France,

Theoretical Physics Department, CERN, CH-1211 Geneva 23, Switzerland,

Department of Physics and Technology, University of Bergen, Postboks 7803, N-5020 Bergen, Norway,

Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland,

Centro de Física Teórica e Computacional, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edifício C8 1749-016 Lisboa, Portugal,

Instituto Superior de Engenharia de Lisboa - ISEL, 1959-007 Lisboa, Portugal,

Center of Excellence in Particle Physics (CoEPP), The University of Adelaide, South Australia,

The Oskar Klein Centre, Department of Physics, Stockholm University, SE-106 91 Stockholm, Sweden

1 Introduction

In the summer of 2012 an SM-like Higgs particle () was found at the LHC [Aad:2012tfa, Chatrchyan:2012ufa]. As of today its properties agree with the SM predictions at the 20% level [Khachatryan:2014jba, Aad:2015ona]. Its mass derived from the and channels is [Aad:2015zhl]. However, the SM-like limit exists in various models with extra neutral Higgs scalars. A charged Higgs boson () would be the most striking signal of an extended Higgs sector, for example with more than one Higgs doublet. Such a discovery at the LHC is a distinct possibility, with or without supersymmetry. However, a charged Higgs particle might be rather hard to find, even if it is abundantly produced.

We here survey existing results on charged scalar phenomenology, and discuss possible strategies for further searches at the LHC. Such scalars appear in Multi-Higgs-Doublet models (MHDM), in particular in the popular Two-Higgs-Doublet model (2HDM) [Gunion:1989we, Branco:2011iw], allowing for charged and more neutral Higgs bosons. We focus on these models, since they have the attractive property that electroweak precision observables are automatically in agreement with the Standard Model at the tree level, in particular, [Ross:1975fq, Veltman:1976rt, Veltman:1977kh].

The production rate and the decay pattern would depend on details of the theoretical model [Gunion:1989we], especially the Yukawa interaction. It is useful to distinguish two cases, depending on whether the mass of the charged scalar () is below or above the top mass. Since an extended Higgs sector naturally leads to Flavor-Changing Neutral Currents (FCNC), these would have to be suppressed [Glashow:1976nt, Paige:1977nz]. This is normally achieved by imposing discrete symmetries in modeling the Yukawa interactions. For example, in the 2HDM with Model II Yukawa interactions a symmetry under the transformation , is assumed. In this case, the data constrain the mass of to be above approximately 480 GeV [Misiak:2015xwa]. A recent study concludes that this limit is even higher, in the range 570–800 GeV [Misiak:2017bgg]. Our results can easily be re-interpreted for this new limit. Alternatively, if all fermion masses are generated by only one doublet (, Model I) there is no enhancement in the Yukawa coupling of with down-type quarks and the allowed mass range is less constrained. The same is true for the Model X (also called Model IV or lepton-specific 2HDM) [Akeroyd:1994ga, Logan:2009uf], where the second doublet is responsible for the mass of all quarks, while the first doublet deals with leptons. Charged Higgs mass below has been excluded at LEP [Abbiendi:2013hk]. Low and high values of are excluded by various theoretical and experimental model-dependent constraints.

An extension of the scalar sector also offers an opportunity to introduce additional CP violation [Lee:1973iz], which may facilitate baryogenesis [Riotto:1999yt].

Charged scalars may also appear in models explaining dark matter (DM). These are charged scalars not involved in the spontaneous symmetry breaking, and we will denote them as . Such charged particles will typically be members of an “inert” or “dark” sector, the lightest neutral member of which is the DM particle (). In these scenarios a symmetry will make the scalar DM stable and forbid any charged-scalar Yukawa coupling. Consequently, the phenomenology of the , the charged component of a -odd doublet, is rather different from the one in usual 2HDM models. In particular, may become long-lived and induce observable displaced vertices in its leptonic decays. This is a background-free experimental signature and would allow one to discover the at the LHC.

The SM-like scenario (also referred to as the “alignment limit”) observed at the LHC corresponds to the case when the relative couplings of the 125 GeV Higgs particle to the electroweak gauge bosons with respect to the ones in the SM are close to unity. We will assume that this applies to the lightest neutral, mainly CP-even Higgs particle, denoted . Still there are two distinct options possible—with and without decoupling of other scalars in the model. In the case of decoupling, very high masses of other Higgs particles (both neutral and charged) arise from the soft breaking term in the potential without any conflict with unitarity.

The focus of this paper will be the -softly-broken 2HDM, but we will also briefly discuss models with more doublets. In such models, one pair of charged Higgs-like scalars would occur for each additional doublet. We also briefly describe scalar dark matter models.

This work arose as a continuation of activities around the workshops “Prospects for Charged Higgs Discovery at Colliders”, taking place every two years in Uppsala. The paper is organized as follows. In sections 24 we review the basic theoretical framework. Then, in section 5 we review charged Higgs decays, and in section 6 we review charged-Higgs production at the LHC. Section 7 is devoted to an overview of different experimental constraints. Proposed search channels for the 2HDM are presented in section 8, whereas in sections 9 and 10 we discuss models with several doublets, and models with dark matter, respectively. Section 11 contains a brief summary. Technical details are collected in appendices.

2 Potential and states

The general 2HDM potential allows for various vacua, including CP violating, charge breaking and inert ones, leading to distinct phenomenologies. Here we consider the case when both doublets have non-zero vacuum expectation values. CP violation, explicit or spontaneous, is possible in this case.

2.1 The potential

We limit ourselves to studying the softly -violating 2HDM potential, which reads

(2.1)

Apart from the term , this potential exhibits a symmetry,

(2.2)

The most general potential contains in addition two more quartic terms, with coefficients and , and violates symmetry in a hard way [Gunion:1989we]. The parameters , and are real. There are various bases in which this potential can be written, often they are defined by fixing properties of the vacuum state. The potential (2.1) can lead to CP violation, provided .

2.2 Mass eigenstates

We use the following decomposition of the doublets (see Appendix A):

(2.3)

which corresponds to a basis where both have a non-zero, real and positive, vacuum expectation value (vev). Here , , , with .

We adopt the mixing matrix , between the scalar fields and mass eigenstates (for the CP conserving case CP-even , and CP-odd , respectively) defined by

(2.4)

satisfying

(2.5)

The rotation matrix is parametrized in terms of three rotation angles as [Accomando:2006ga]

(2.6)

with , , and . In Eq. (2.4), is the combination of ’s which is orthogonal to the neutral Nambu–Goldstone boson. In terms of these angles, the limits of CP conservation correspond to [ElKaffas:2007rq]

(2.7)

The charged Higgs bosons are the combination orthogonal to the charged Nambu–Goldstone bosons: , and their mass is given by

(2.8)

where we define a mass parameter by

(2.9)

Note also the following relation arising from the extremum condition:

(2.10)

2.3 Gauge couplings

With all momenta incoming, we have the gauge couplings [ElKaffas:2006nt]:

(2.11)

Specifically, for coupling to the lightest neutral Higgs boson, the -matrix (2.6) gives:

(2.12)

The familiar CP-conserving limit is obtained by evaluating for , , , with the mapping , and . In that limit, we recover the results of [Gunion:1989we]:

(2.13)

The strict SM-like limit corresponds to , however the experimental data from the LHC [Khachatryan:2014jba, Aad:2015ona] allow for a departure from this limit111Note that in the 2HDM, this factor cannot exceed 1. down to approximately 0.7, which we are going to allow in our study.

In the following analysis, the gauge couplings to neutral Higgs bosons are also involved. They differ from the SM coupling by the factor ():

(2.14)

In particular, for , this factor becomes . In the CP-conserving case, we have

(2.15)

Note that the couplings (2.11) and (2.14) are given by unitary matrices, and hence satisfy sum rules. Furthermore, for any , the relative couplings of (2.11) (the expression in the square brackets) and (2.14) satisfy the following relation [Ginzburg:2014pra]:

(2.16)

These relations are valid for both the CP-conserving and the CP-violating cases.

3 Theoretical constraints

The 2HDM is subject to various theoretical constraints. First, it has to have a stable vacuum222Here we perform an analysis at the tree level, for more advanced studies, see [Nie:1998yn, Ferreira:2004yd, Goudelis:2013uca, Swiezewska:2015paa, Khan:2015ipa]., what leads to so-called positivity constraints for the potential [Deshpande:1977rw, Nie:1998yn, Kanemura:1999xf], as . Second, we should be sure to deal with a particular vacuum (a global minimum) as in some cases various minima can coexist [Barroso:2013awa, Ginzburg:2010wa, Swiezewska:2012ej].

Other types of constraints arise from requiring perturbativity of the calculations, tree-level unitarity [Kanemura:1993hm, Akeroyd:2000wc, Arhrib:2000is, Ginzburg:2003fe, Ginzburg:2005dt] and perturbativity of the Yukawa couplings. In general, imposing tree-level unitarity has a significant effect at high values of and , by excluding such values. These constraints limit the absolute values of the parameters as well as , the latter both at very low and very high values. This limit is particularly strong for a symmetric model [WahabElKaffas:2007xd, Gorczyca:2011he, Swiezewska:2012ej]. The dominant one-loop corrections to the perturbative unitarity constraints for the model with softly-broken symmetry are also available [Grinstein:2015rtl].

The electroweak precision data, parametrized in terms of and [Kennedy:1988sn, Peskin:1990zt, Altarelli:1990zd, Peskin:1991sw, Altarelli:1991fk, Grimus:2007if, Grimus:2008nb], also provide important constraints on these models.

4 Yukawa Interaction

There are various models of Yukawa interactions, all of them, except Model III, lead to suppression of FCNCs at the tree level, assuming some vanishing Yukawa matrices. The most popular is Model II, in which up-type quarks couple to one (our choice: ) while down-type quarks and charged leptons couple to the other scalar doublet (). They are presented schematically in Table 1. For a self-contained description of the 2HDM Yukawa sector, see Appendix B.333The absence of tree-level FCNC interactions can also be obtained by imposing flavor space alignment of the Yukawa couplings of the two scalar doublets [Jung:2010ik].

Model
I
II
III
X
Y
Table 1: The most popular models of the Yukawa interactions in the 2HDM (also referred to as “Types”). The symbols , , refer to up- and down-type quarks, and charged leptons of any generation. Here, and refer to the Higgs doublet coupled to the particular fermion. Also other conventions are being used in the literature, see Appendix B.

For Model II, and the third generation, the neutral-sector Yukawa couplings are:

(4.1)

Explicitly, for the charged Higgs bosons in Model II, we have for the coupling to the third generation of quarks [Gunion:1989we]

(4.2)

where is the appropriate element of the CKM matrix. For other Yukawa models the factors and will be substituted according to Table 6 in Appendix B.

As mentioned above, the range in (or ) is , which can be taken as , or . This is different from the MSSM, where only a range of is required [Gunion:1986nh], . The spontaneous breaking of the symmetry and the convention of having a positive value for means that the sign (phase) of the field is relevant. This doubling of the range in the 2HDM as compared with the MSSM is the origin of “wrong-sign” Yukawa couplings.

5 Charged Higgs boson decays

This section presents an overview of the different decay modes, illustrated with branching ratio plots for parameter sets that are chosen to exhibit the most interesting features. Branching ratios required for modes considered in sections 810 are calculated independently.

As discussed in [Gunion:1989we, Moretti:1994ds, Djouadi:1995gv, Djouadi:1997yw, Kanemura:2009mk, Eriksson:2009ws], a charged Higgs boson can decay to a fermion-antifermion pair,

(5.1a)
(5.1b)
(5.1c)
(5.1d)

(note that (5.1b) refers to a mixed-generation final state), to gauge bosons,

(5.2a)
(5.2b)

or to a neutral Higgs boson and a gauge boson:

(5.3)

and their charge conjugates.

Below, we consider branching ratios mainly for the CP-conserving case. For the lightest neutral scalar we take the mass . Neither experimental nor theoretical constraints are here imposed. (They have significant impacts, as will be discussed in subsequent sections.) For the calculation of branching ratios, we use the software 2HDMC [Eriksson:2009ws] and HDECAY [Djouadi:1997yw, Harlander:2013qxa]. As discussed in [Harlander:2013qxa], branching ratios are calculated at leading order in the 2HDM parameters, but include QCD corrections according to [Mendez:1990jr, Li:1990ag, Djouadi:1994gf], and three-body modes via off-shell extensions of , , and . The treatment of three-body decays is according to Ref. [Djouadi:1995gv].

For light charged Higgs bosons, , Model II is excluded by the constraint discussed in section 7. For Model I (which in this region is not excluded by ), the open channels have fermionic couplings proportional to . The gauge couplings (involving decays to a and a neutral Higgs) are proportional to or , whereas the corresponding Yukawa couplings depend on the masses involved, together with .

The CP-violating case for the special channel is presented in section 5.4.

Figure 1: Light charged-Higgs branching ratios vs . Left: Models I and X, right: Models II and Y. The panel on the right is only for illustration, such a light is excluded for the models II and Y.

5.1 Branching ratios vs

Below, we consider branching ratios, assuming for simplicity , in the low and high mass regions.

5.1.1 Light ()

For a light charged Higgs boson, such as might be produced in top decay, the and channels would be closed, and the and channels would dominate. The relevant Yukawa couplings are given by and the fermion masses involved. With scalar masses taken as follows:

(5.4)

we show in Fig. 1 branching ratios for the different Yukawa models.

Since the and couplings for Model I are the same, the branching ratios are independent of , as seen in the left panel. For Models X and II the couplings to and have different dependences on , and consequently the branching ratios will depend on . In the case of Model Y, the channel is for controlled by the term , which dominates over the channel at high .

5.1.2 Heavy ()

Figure 2: Heavy charged-Higgs branching ratios vs for two light neutral Higgs bosons and . Left: Models I and X, right: Models II and Y. Upper two panels: , lower panels: . The dashed vertical lines are for comparison with Figs. 34.

Below, we consider separately the two cases where one more neutral scalar is light, besides , this being either or . For a case where both the channels and are open, whereas is not, exemplified by the masses

(5.5)

we show in Fig. 2 branching ratios for the different Yukawa models. Two values of are considered, 1 and 0.7. For comparison with section 5.2, we have drawn dashed lines at , 3 and 30.

For Model I (left part of Fig. 2), the dominant decay rates are to the heaviest fermion-antifermion pair and to together with or (for the considered parameters, both and are kinematically available). Model X differs in having an enhanced coupling to tau leptons at high , see Table 6 in Appendix B. If the decay to is kinematically not accessible, the mode may be accessible at high .

For Model II (right part of Fig. 2), the dominant decay rates are to the heaviest fermion-antifermion pair at low and high values of , with or dominating at medium (if kinematically available). At high it is the down-type quark that has the dominant coupling. Hence, modulo phase space effects, the rate is only suppressed by the mass ratio . Model Y differs from Model II in not having enhanced coupling to the tau at high values of .

Whereas the couplings and hence the decay rates to and , for fixed values of , are independent of , the branching ratios are not. They will depend on the strengths of the competing Yukawa couplings. The strength of the channel increases with , and is therefore absent in the upper panels where .

It should also be noted that if the channel is not kinematically available, the channel would dominate for all values of . The channel, which may offer less background for experimental searches, is only relevant at higher , and then only in Models II and X.

When is light, such that the channels and are both open, whereas is not, the situation is similar to the previous case, with the mode replaced by the mode. The choice turns off the mode (see Eq. (2.13)), and there is a competition among the and the modes, except for the region of high , where also the mode can be relevant.

Figure 3: Charged-Higgs branching ratios vs , for and . Here, two light neutral Higgs bosons and (125 GeV and 130 GeV) are considered.

5.2 Branching ratios vs

In Figs. 34 we show how the branching ratios change with the charged Higgs mass. Here, we have taken (Fig. 3), 3 and 30 (Fig. 4), together with the neutral-sector masses

(5.6)

(note that here we take ) and consider the two values and 0.7, corresponding to different strengths of the gauge couplings (2.13).

Figure 4: Charged-Higgs branching ratios vs , for and 30, with two light neutral Higgs bosons and (125 GeV and 130 GeV). Left: Models I and X, right: Models II and Y. Top: , bottom:

The picture from Figs. 1 and 2 is confirmed: At low masses, the channel dominates, whereas at higher masses, the channel will compete against and , if these channels are kinematically open, and not suppressed by some particular values of the mixing angles.

Of course, for (Fig. 3), all four Yukawa models give the same result. Qualitatively, the result is simple. At low masses, the and channels dominate, whereas above the threshold, the channel dominates. There is however some competition with the and channels. Similar results hold for , the only difference being that the branching ratio rises faster with mass, and the mode disappears completely in this limit. Even below the threshold, branching ratios for three-body decays via an off-shell can be significant [Djouadi:1995gv]. The strength of the channel is proportional to , and is therefore absent for (not shown).

At higher values of (Fig. 4), the interplay with the and channels becomes more complicated. At high charged-Higgs masses, the rate can be important (if kinematically open). On the other hand, the channel can dominate over , because of the larger phase space. Here, we present the case of . The case of is similar, the main difference is a higher branching ratio, while the channel disappears. It should be noted that three-body channels that proceed via and can be important also below threshold, if the channel is closed.

Figure 5: Product of branching ratios, , for Model I, and three values of , as indicated.

5.3 Top decay to

A light charged Higgs boson may emerge in the decay of the top quark

(5.7)

followed by a model-dependent decay. In Model I possible channels are and , as shown in Fig. 1. For the former case, the product is shown in Fig. 5 for three values of . Note that recent LHC data have already excluded a substantial region of the low- and low- parameter region in Model I, see section 7.2.3.

5.4 The partial width

In this section we consider the decay mode , allowing for the possibility that the lightest Higgs boson, , is not an eigenstate of CP.

The coupling is given by Eq. (2.12). The partial width, relative to its maximum value, is given by the quantity

(5.8)

which is shown in Fig. 6. We note that there is no dependence on the mixing angle . If or , then CP is conserved along the axis with .

Figure 6: Relative partial width for , given by Eq. (5.8), vs and , for and 2. The white “circles” outline the region within which the coupling squared deviates by at most 10% or 30% from the SM value.

In the alignment limit,

(5.9)

which is closely approached by the LHC data on the Higgs-gauge-boson coupling, the coupling actually vanishes.

Hence, the decay crucially depends on some deviation from this limit. We note that the coupling is proportional to . Thus, the deviation of the square of this coupling from unity (which represents the SM-limit), is given by the expression (5.8). Note that the experimental constraint (on the deviation of the coupling squared from unity) is 15–20% at the 95% CL [Khachatryan:2014jba, Aad:2015ona].

For comparison, a recent study of decay modes that explicitly exhibit CP violation in Model II [Fontes:2015xva], compatible with all experimental constraints, considers values in the range 1.3 to 3.3, with parameter points displaced from the alignment limit by ranging from 1.5% to 83.2% (the one furthest away has a negative value of ).

This decay channel is also interesting for Model I [Keus:2015hva].

6 production mechanisms at the LHC

This section describes production and detection channels at the LHC. Since a charged Higgs boson couples to mass, it will predominantly be produced in connection with heavy fermions, , , and , or bosons, or , and likewise for the decays. The cross sections given here, are for illustration only. For the studies presented in sections 810 they are calculated independently.

We shall here split the discussion of possible production mechanisms into two mass regimes, according to whether the charged Higgs boson can be produced (in the on-shell approximation) in a top decay or whether it could decay to a top and a bottom quark. These two mass regimes will be referred to as “low” and “high” mass, respectively.

While discussing such processes in hadron-hadron collisions one should be aware that there are two approaches to the treatment of heavy quarks in the initial state. One may take the heavy flavors as being generated from the gluons, then the relevant number of active quarks is (or sometimes 3). Alternatively, the -quark can be included as a constituent of the hadron, then an parton density should be used in the calculation of the corresponding cross section. These two approaches are referred to as the 4-flavor and 5-flavor schemes, abbreviated 4FS and 5FS. This should be kept in mind when referring to the lists of possible subprocesses initiated by heavy quarks and the corresponding figures in the following discussion. Below, we will use the notation , and to denote quarks which are not -quarks. We only indicate -quarks when they couple to Higgs bosons, thus enhancing the rate.

For some discussions it is useful to distinguish “bosonic” and “fermionic” production mechanisms, since the former, corresponding to final states involving only and , may proceed via an intermediate neutral Higgs, and thus depend strongly on its mass, see e.g., Ref. [Basso:2015dka].

6.1 Production processes

Below, we list all important production processes represented in Figs. 11-14 in the 5FS.444Charge-conjugated processes are not shown separately. Higgs radiation from initial-state quarks are not shown explicitly.

Figure 7: Feynman diagrams for the production processes (6.1) and (6.2).

6.1.1 Single production

A single can be accompanied by a (Fig. 7a, “bosonic”) [Dicus:1989vf, BarrientosBendezu:1998gd, Moretti:1998xq, BarrientosBendezu:1999vd, Brein:2000cv, Hollik:2001hy, Asakawa:2005nx, Eriksson:2006yt, Hashemi:2010ce]:

(6.1a)
(6.1b)

or by a and a jet (Fig. 7b, “fermionic”) [Gunion:1986pe, DiazCruz:1992gg, Moretti:1996ra, Miller:1999bm, Moretti:1999bw, Zhu:2001nt, Plehn:2002vy, Berger:2003sm, Kidonakis:2004ib, Weydert:2009vr, Kidonakis:2010ux, Flechl:2014wfa, Degrande:2015vpa, Kidonakis:2016eeu, Degrande:2016hyf]:555Note that in the 5FS (6.2) can be a tree-level process, whereas (6.1a) can not.

(6.2)

The pioneering study [Dicus:1989vf] of the bosonic process (6.1) already discussed both the triangle and box contributions to the one-loop -initiated production, but considered massless -quarks, i.e., the -quark Yukawa couplings were omitted. This was subsequently restored in a complete one-loop calculation of the -initiated process [BarrientosBendezu:1998gd, BarrientosBendezu:1999vd], and it was realized that there can be a strong cancellation between the triangle- and box diagrams. This interplay of triangle and box diagrams has also been explored in the MSSM [Brein:2000cv].

NLO QCD corrections to the -initiated production process were found to reduce the cross section by [Hollik:2001hy]. On the other hand, possible -channel resonant production via heavier neutral Higgs bosons (see Fig. 7a (i) and (iii)) was seen to provide possible enhancements of up to two orders of magnitude [Asakawa:2005nx]. These authors also pointed out that one should use running-mass Yukawa couplings, an effect which significantly reduced the cross section at high mass [Eriksson:2006yt].

A first comparison of the signal with the background [Moretti:1998xq] (in the context of the MSSM) concluded that the signal could not be extracted from the background. More optimistic conclusions were reached for the channel [Eriksson:2006yt, Hashemi:2010ce], again in the context of the MSSM.

The first study [Gunion:1986pe] of the fermionic process (6.2) pointed out that there is a double counting issue (see sect. 6.1.2). Subsequently, it was realized [DiazCruz:1992gg, Borzumati:1999th] that the process could be described as , where a gluon splits into and one of these is not observed. As mentioned above, this approach is in recent literature referred to as the four-flavor scheme (4FS) whereas in the five-flavor scheme (5FS) one considers -quarks as proton constituents.

NLO QCD corrections to the cross section have been calculated [Zhu:2001nt, Plehn:2002vy, Degrande:2016hyf], and the resulting scale dependence studied [Plehn:2002vy, Berger:2003sm], both in the 5FS and the 4FS. In a series of papers by Kidonakis [Kidonakis:2004ib, Kidonakis:2010ux, Kidonakis:2016eeu], soft-gluon corrections have been included at the “approximate NNLO” order and found to be significant near threshold, i.e., for heavy . A recent study [Degrande:2016hyf] is devoted to total cross sections in the intermediate-mass region, , providing a reliable interpolation between low and high masses.

These fixed-order cross section calculations have been merged with parton showers [Alwall:2004xw, Weydert:2009vr, Flechl:2014wfa, Degrande:2015vpa], both at LO and NLO, in the 4FS and in the 5FS. The 5FS results are found to exhibit less scale dependence [Degrande:2015vpa].

Different background studies [Moretti:1996ra, Miller:1999bm, Moretti:1999bw] compared triple -tagging vs 4--tagging, identifying parameter regions where either is more efficient.

Figure 8: Feynman diagrams for the production processes (6.3).
Figure 9: Feynman diagrams for the production processes (6.4). If the line has no arrow, it represents either a quark or an antiquark.
Figure 10: Feynman diagrams for the production processes (6.5).
Figure 11: Feynman diagrams for the production processes (6.6).

In addition to the importance of the channel at low mass, the following processes containing two accompanying jets (see Fig. 8) are important at high charged-Higgs mass:

(6.3a)
(6.3b)

There are also processes with a single and two jets (see Fig. 9):

(6.4)

In this particular case, with many possible gauge boson couplings, one of the final-state jets could be a .

In addition, single production can be initiated by a -quark,

(6.5)

as illustrated in Fig. 10.

In the 5FS, single production can also take place from and quarks, typically accompanied by a gluon jet [He:1998ie, DiazCruz:2001gf, Slabospitsky:2002gw, Dittmaier:2007uw] (Fig. 11):

(6.6a)
(6.6b)

Similarly, one can consider initial states.

At infinite order the 4FS and the 5FS should only differ by terms of , but the perturbation series of the two schemes are organized differently. Some authors (see, e.g., Ref. [Flechl:2014wfa]) advocate combining the two schemes according to the “Santander matching” [Harlander:2011aa]:

(6.7)

with the relative weight factor

(6.8)

since the difference between the two schemes is logarithmic, and in the limit of the 5FS should be exact.

6.1.2 The double counting and NWA issues

A -quark in the initial state may be seen as a constituent of the proton (5FS), or as resulting from the gluon splitting into (4FS). Adding (with one possibly not detected) and in the 5FS one may therefore commit double counting [Barnett:1987jw, Olness:1987ep]. The resolution lies in subtracting a suitably defined infrared-divergent part of the gluon-initiated amplitude [Alwall:2004xw].666For a complete discussion on the flavour scheme choice in inclusive charged Higgs production associated with fermions see IV.3.2 of [deFlorian:2016spz] and references therein. The problem can largely be circumvented by choosing either the 5FS or the 4FS. For a more pragmatic approach, see Refs. [Belyaev:2001qm, Belyaev:2002eq].

Figure 12: Feynman diagrams for the production processes (6.9).
Figure 13: Feynman diagrams for the pair production processes (6.10a).
Figure 14: Feynman diagrams for the pair production processes (6.10b).

A related issue is the one of low-mass production via -quark decay, followed by (with a spectator), usually treated in the Narrow Width Approximation (NWA). The NWA however fails the closer the top and charged Higgs masses are, in which case the finite top width needs to be accounted for, which in turn implies that the full gauge invariant set of diagrams yielding has to be computed. Considerable effort has been devoted to understanding this implementation, see also Refs. [Guchait:2001pi, Alwall:2003tc, Assamagan:2004gv].

6.1.3 and production

We can have a single production in association with a neutral Higgs boson [Kanemura:2001hz, Akeroyd:2003bt, Akeroyd:2003jp, Cao:2003tr, Belyaev:2006rf, Miao:2010rg]:

(6.9)

as shown in Fig. 12.

For pair production we have [Eichten:1984eu, Willenbrock:1986ry, Glover:1987nx, Dicus:1987ic, Jiang:1997cg, Krause:1997rc, BarrientosBendezu:1999gp, Brein:1999sy, Moretti:2001pp, Moretti:2003px, Alves:2005kr]:

(6.10a)
(6.10b)

as illustrated in Figs. 13 and 14, respectively. These mechanisms would be important for light charged Higgs bosons, as allowed in Models I and X.

6.2 Production cross sections

In this section, predictions for single Higgs production at 14 TeV for the CP-conserving 2HDM, Models I and II (valid also for X and Y) are discussed.

In Fig. 15, cross sections for the main production channels are shown at leading order, sorted by the parton-level mechanism [Basso:2015dka]777In the Feynman diagrams is represented by its dominant decay products .. The relevant partonic channels can be categorized as:

  • “fermionic”: , Fig. 7 b (solid),

  • “fermionic”: , Fig. 8 a, b (dotted),

  • “bosonic”: , Fig. 7 a (i) (dash-dotted).

The charge-conjugated channels are understood to be added unless specified otherwise. No constraints are imposed here, neither from theory (like positivity, unitarity), nor from experiments.

The CTEQ6L (5FS) parton distribution functions [Pumplin:2002vw] are adopted here, with the scale . Three values of are considered, and and are held fixed at . Furthermore, we consider the CP-conserving alignment limit, with . The bosonic cross section is accompanied by a next-to-leading order QCD -factor enhancement [Spira:1995rr].

Figure 15: Charged Higgs production cross sections in the 2HDM, at 14 TeV. Left: Model I (or X). Right: Model II (or Y). Solid and dotted curves refer to “fermionic” channels, whereas dash-dotted refer to “bosonic” ones (see text).

Several points are worth mentioning:

  • To any contribution at fixed order in the perturbative expansion of the gauge coupling, the three cross sections are to be merged with regards to the interpretation in different flavour schemes, as discussed above. In the following, we focus on the first fermionic channel in the 5FS at the tree level.

  • The enhancement exhibited by the dotted curve at low masses is due to resonant production of -quarks which decay to . However, in Model I this mode is essentially excluded by LHC data (see section 7.2.4), and in Model II it is excluded by the -constraint (see section 7.1.2).

  • Model I differs from Model II also for , because of a different relative sign between the Yukawa couplings proportional to and those proportional to , see Table 6.

  • Models X and Y will have the same production cross sections as Models I and II, respectively, but the sensitivity in the -channel would be different.

  • The bumpy structure seen for the bosonic mode is due to resonant production of neutral Higgs bosons, and depends on the values of and . Note that in the MSSM the masses of the heavier neutral Higgs bosons are close to that of the charged one, and this resonant behavior is absent.

While recent studies (see section 6.1.1) provide a more accurate calculation of the cross section than what is given here, they typically leave out the 2HDM model-specific -channel (possibly resonant) contribution to the cross section.

Figure 16: Charged Higgs bosonic production cross sections in the 2HDM, Model II, for 14 TeV, and a fixed value of , plotted vs for , 2, 3 and 4.

In Fig. 16, the bosonic charged-Higgs production cross section vs for a set of CP-conserving parameter points that satisfy the theoretical and experimental constraints [Basso:2015dka] (see also [Basso:2012st, Basso:2013wna]) are presented. These are shown in different colors for different values of . The spread in cross section values for each value of and reflects the range of allowed values of the other parameters scanned over, namely , and .

Low values of are enhanced for the bosonic mode due to the contribution of the -quark in the loop, whereas the modulation is due to resonant production. In the CP-violating case, this modulation is more pronounced [Basso:2015dka].

As summarized by the LHC Top Physics Working Group the cross section has been calculated at next-to-next-to leading order (NNLO) in QCD including resummation of next-to-next-to-leading logarithmic (NNLL) soft gluon terms with the software Top++2.0 [Beneke:2011mq, Cacciari:2011hy, Czakon:2011xx, Baernreuther:2012ws, Czakon:2012zr, Czakon:2012pz, Czakon:2013goa]. The decay width is available at NNLO [Czarnecki:1998qc, Chetyrkin:1999ju, Blokland:2004ye, Blokland:2005vq, Czarnecki:2010gb, Gao:2012ja, Brucherseifer:2013iv], while the decay width is available at NLO [Czarnecki:1992zm].

7 Experimental constraints

Here we review various experimental constraints for charged Higgs bosons derived from different low (mainly -physics) and high (mainly LEP, Tevatron and LHC) energy processes. Also some relevant information on the neutral Higgs sector is presented. Some observables depend solely on exchange, and are thus independent of CP violation in the potential, whereas other constraints depend on the exchange of neutral Higgs bosons, and are sensitive to the CP violation introduced via the mixing discussed in subsection 2.2. Due to the possibility of , in addition to exchange, we are getting constraints from a variety of processes, some at tree and some at the loop level. In addition, we present general constraints coming from electroweak precision measurements, , , the muon magnetic moment and the electric dipole moment of the electron. The experimental constraints listed below are valid only for Model II, if not stated otherwise.888Analyses with general Yukawa couplings can be found in Refs. [Mahmoudi:2009zx] and [Crivellin:2013wna]. Also, some of the constraints are updated, with respect to those used in the studies presented in later sections.

The charged-Higgs contribution may substantially modify the branching ratios for -production in -decays [Krawczyk:1987zj]. An attempt to describe various and anomalies (also ) in the 2HDM, Model III, with a novel ansatz relating up- and down-type Yukawa couplings, can be found in [Cline:2015lqp]. This analysis points towards an mass around 100 GeV, with masses of other neutral Higgs bosons in the range 100–125 GeV. A similar approach to describe various low energy anomalies by introducing additional scalars can be found in [Crivellin:2015hha]. Here, a lepton-specific 2HDM (i.e., of type X) with non-standard Yukawa couplings has been analysed with the second neutral CP-even Higgs boson light (below 100 GeV) and a relatively light , with a mass of the order of 200 GeV.

7.1 Low-energy constraints

As mentioned above, several decays involving heavy-flavor quarks could be affected by in addition to -exchange. Data on such processes provide constraints on the coupling (represented by ) and the mass, . Below, we discuss the most important ones.

7.1.1 Constraints from tree-level exchange

:

The measurement of the branching ratio of the inclusive process [Abbiendi:2001fi] leads to the following constraint, at the CL,

(7.1)

This is in fact a very weak constraint. (A similar result can be obtained from the leptonic tau decays at the tree level [Krawczyk:2004na].) A more recent measurement for the exclusive case gives [Agashe:2014kda]999The error of the measurement, given by HFAG [Amhis:2014hma] and released after the PDG 2014 [Agashe:2014kda], is slightly lower: (.. With a Standard Model prediction of [Charles:2004jd]101010We have added in quadrature symmetrized statistical and systematic errors. , we obtain

(7.2)

Interpreted in the framework of the 2HDM at the tree level, one finds [Hou:1992sy, Grossman:1994ax, Grossman:1995yp]

(7.3)

Two sectors of the ratio are excluded. Note that this exclusion is relevant for high values of .

:

The ratios [Aubert:2007dsa]

(7.4)

are sensitive to -exchange, and lead to constraints similar to the one following from [Nierste:2008qe]. In fact, there has been some tension between BaBar results [Aubert:2007dsa, Lees:2012xj, Lees:2013uzd] and both the 2HDM (II) and the SM. These ratios have also been measured by Belle [Huschle:2015rga, Abdesselam:2016cgx] and LHCb [Aaij:2015yra]. Recent averages [Freytsis:2015qca, Cline:2015lqp] are summarized in Table 2, together with the SM predictions [Fajfer:2012vx, Lattice:2015rga, Na:2015kha]. They are compatible at the level. A comparison with the 2HDM (II) concludes [Huschle:2015rga] that the results are compatible for . However, in view of the high values for required by the constraint, uncomfortably high values of would be required. The studies given for Model II in section 8.3 do not take this constraint into account.

Ratio Experiment SM
[Freytsis:2015qca]
[Cline:2015lqp]
Table 2: Average experimental values [Freytsis:2015qca, Cline:2015lqp] and SM predictions [Fajfer:2012vx, Lattice:2015rga, Na:2015kha, Cline:2015lqp, Crivellin:2015hha].
:

Severe constraints can be obtained, which are competitive with those from [Akeroyd:2009tn].

7.1.2 Constraints from loop-level exchange

Figure 17: BR as a function of for Model I (left) and Model II (right), at two values of . Solid and dashed lines correspond to the NNLO 2HDM and SM predictions, respectively. (Shown are central values with shifts.) Dotted curves represent the experimental average. [Reprinted with kind permission from the authors, Fig. 2 of [Misiak:2017bgg]].
:

The transition may also proceed via charged Higgs boson exchange, which is sensitive to the values of and . The allowed region depends on higher-order QCD effects. A huge effort has been devoted to the calculation of these corrections, the bulk of which are the same as in the SM [Chetyrkin:1996vx, Buras:1997bk, Bauer:1997fe, Bobeth:1999mk, Buras:2002tp, Misiak:2004ew, Neubert:2004dd, Melnikov:2005bx, Misiak:2006zs, Misiak:2006ab, Asatrian:2006rq, Czakon:2006ss, Boughezal:2007ny, Ewerth:2008nv, Misiak:2010sk, Asatrian:2010rq, Ferroglia:2010xe, Misiak:2010tk, Kaminski:2012eb, Czakon:2015exa]. They are now complete up to NNLO order. On top of these, there are 2HDM-specific contributions [Ciafaloni:1997un, Ciuchini:1997xe, Borzumati:1998tg, Bobeth:1999ww, Gambino:2001ew, Misiak:2015xwa] that depend on and . The result is that mass roughly up to is excluded for high values of [Misiak:2015xwa], with even stronger constraints for very low values of . Recently, a new analysis [Trabelsi:2015] of Belle results [Saito:2014das] concludes that the lower limit is 540 GeV. Also note the new result of Misiak and Steinhauser [Misiak:2017bgg] with lower limit in the range 570–800 GeV, see Fig. 17 (right) for high and high masses. We have here adopted the more conservative value of 480 GeV, however our results can easily be re-interpreted for this new limit. Constraints from decay for lower masses are presented in Fig. 19 together with other constraints.

For low values of , the constraint is even more severe. This comes about from the charged-Higgs coupling to and quarks ( and ) containing terms proportional to and (). The product of these two couplings determine the loop contribution, where there is an intermediate state, and leads to terms proportional to (responsible for the constraint at low ) and (responsible for the constraint that is independent of ). For Models I and X, on the other hand, both these couplings are proportional to . Thus, the constraint is in these models only effective at low values of .111111For early studies, see [Grossman:1994jb, Akeroyd:1994ga]. This can be seen in Fig. 17 (left) and Fig. 18, where the new results from the analysis applied to Model I of the 2HDM are shown. We stress that Model I can avoid the constraints and hence it can accommodate a light .

Figure 18: 2HDM 95%C.L. exclusion (lower part) in the plane of and . [Reprinted with kind permission from the authors, Fig. 4 of [Misiak:2017bgg]].
mixing:

Due to the possibility of charged-Higgs exchange, in addition to exchange, the mixing constraint excludes low values of (for ) and low values of [Abbott:1979dt, Inami:1980fz, Athanasiu:1985ie, Glashow:1987qe, Geng:1988bq, Urban:1997gw]. Recent values for the oscillation parameters and are given in Ref. [Deschamps:2009rh], only at very low values of do they add to the constraints coming from .

7.1.3 Other precision constraints

and :

The precisely measured electroweak (oblique) parameters and correspond to radiative corrections, and are (especially ) sensitive to the mass splitting of the additional scalars of the theory. In papers [Grimus:2007if, Grimus:2008nb] general expressions for these quantities are derived for the MHDMs and by confronting them with experimental results, in particular , strong constraints are obtained on the masses of scalars. In general, imposes a constraint on the splitting in the scalar sector, a mass splitting among the neutral scalars gives a negative contribution to , whereas a splitting between the charged and neutral scalars gives a positive contribution. A recent study [Gorbahn:2015gxa] also demonstrates how RGE running may induce contributions to and . Current data on and are given in [Agashe:2014kda].

The muon anomalous magnetic moment:

We are here considering heavy Higgs bosons (), with a focus on the Model II, therefore, according to [Cheung:2003pw, Chang:2000ii, WahabElKaffas:2007xd], the 2HDM contribution to the muon anomalous magnetic moment is negligible even for as high as (see, however, [Krawczyk:2002df]).

The electron electric dipole moment:

The bounds on electric dipole moments constrain the allowed amount of CP violation of the model. For the study of the CP-non-conserving Model II presented in section 8.3, the bound [Regan:2002ta] (see also [Pilaftsis:2002fe]):

(7.5)

was adopted at the level. (More recently, an order-of-magnitude stronger bound has been established [Baron:2013eja].) The contribution due to neutral Higgs exchange, via the two-loop Barr–Zee effect [Barr:1990vd], is given by Eq. (3.2) of [Pilaftsis:2002fe].

7.1.4 Summary of low-energy constraints

A summary of constraints of the 2HDM Model II coming from low-energy physics performed by the “Gfitter” group [Flacher:2008zq] is presented on Fig. 19. The more recent inclusion of higher-order effects pushes the constraint up to around 480 GeV [Misiak:2015xwa] or even higher, as discussed above. See also Refs. [Deschamps:2009rh, Bona:2009cj, Enomoto:2015wbn].

Figure 19: Model II 95% CL exclusion regions in the (, ) plane. [Reprinted with kind permission from EPJC and the authors of “Gfitter” [Flacher:2008zq]]. A new analysis, including the updated bound from , is being prepared by the “Gfitter” group.

7.2 High-energy constraints

Most bounds on charged Higgs bosons are obtained in the low-mass region, where a charged Higgs might be produced in the decay of a top quark, , with the subsequently decaying according to Eqs. (5.1a-c), (5.2) or (5.3). Of special interest are the decays and . For comparison with data, products like are relevant, as presented in section 5.3. At high charged-Higgs masses, the rate can be important (if kinematically open). On the other hand, the channel can dominate over , because of the larger phase space. However, as illustrated in Fig. 4, it vanishes in the alignment limit.

7.2.1 Charged-Higgs constraints from LEP

The branching ratio would be affected by Higgs exchange. Experimentally [Agashe:2014kda]. The contributions from neutral Higgs bosons to are negligible [ElKaffas:2006nt], however, charged Higgs boson contributions, as given by [Denner:1991ie], Eq. (4.2), exclude low values of and low . See also Fig. 19.

LEP and the Tevatron have given limits on the mass and couplings, for charged Higgs bosons in the 2HDM. At LEP a lower mass limit of 80 GeV that refers to the Model II scenario for