Simple Standard Model Extension by Heavy Charged Scalar

We consider a Standard Model extension by a heavy charged scalar gauged only under the $U_{Y}(1)$ weak hypercharge gauge group. Such an extension, being gauge invariant with respect to the SM gauge group, is a simple special case of the well known Zee model. Since the interactions of the charged scalar with the Standard Model fermions turn out to be significantly suppressed compared to the Standard Model interactions, the charged scalar provides an example of a long-lived charged particle being interesting to search for at the LHC. We present the pair and single production cross sections of the charged scalar at different colliders and the possible decay widths for various boson masses. It is shown that the current ATLAS and CMS searches at 8 and 13 TeV collision energy lead to the bounds on the scalar boson mass of about 300--320 GeV. The limits are expected to be much larger for higher collision energies and, assuming $15~ab^{-1}$ integrated luminosity, reach about 2.7 TeV at future 27 TeV LHC thus covering the most interesting mass region.


Introduction
With the discovery of the Higgs boson at the LHC, the Standard Model (SM) was completed in the sense that all the predicted particles have been found and all the interaction structures have been fixed. However, not all the interactions in the gauge and Higgs sectors are confirmed experimentally. The Standard Model is based on the fundamental principles such as gauge invariance, the absence of chiral anomalies, unitarity and renormalizability. It is a common knowledge that the SM works extremely well explaining an enormous amount of experimental facts and results. However, because of a number of theoretical problems such as the hierarchy problem and the inability to explain the presence of Dark Matter or the nature of CP violation, the SM is considered as a sort of effective theory describing phenomena up to the electroweak or TeV energy scale. A large number of various experimentally allowed beyond the SM models and scenarios are proposed motivating intensive searches for new physics in the terrestrial and space experiments, in particular, at the LHC. However, up to now no convincing results confirming any concrete BSM direction have been obtained.
Among various objects predicted by new physics models a special attention has been recently paid to the so-called HSCP (heavy stable charged particles) or LLP (long-lived particles). Various SM extensions predict the existence of such particles [1]- [13]. A number of searches for LLP and HSCP have been performed at the Tevatron and the LHC [14]- [19].
In this paper we discuss shortly a very simple SM extension by a charged scalar boson interacting with the U Y (1) weak hypercharge gauge boson and potentially giving an example of a long-lived charged particle. Such a model from rather different perspectives has been considered in paper [20] and quite recently in paper [21]. This SM extension by the extra charged scalar can be naturally called csSM.
Generic SM extensions by an arbitrary number of Higgs singlets and doublets were considered by P. Langacker in his famous review paper [22]. We consider in more detail one particular case with an extra complex scalar field S interacting in a gauge invariant manner only with the U Y (1) weak hypercharge gauge field and with the Higgs field. The scalar field potential of the model coincides with that of the SM extension by singlet complex scalar with U(1) symmetry discussed in paper [23], where this scalar field couples only to the Higgs field and is shown to give a reliable explanation of the cold dark matter. In our model we identify this U(1) symmetry with the weak hypercharge U Y (1) symmetry, which makes the complex scalar electrically charged and forbids its interpretation as a dark matter particle. The model (csSM) can be viewed as a simplified variant of the Zee model [24]. The original Zee model includes an extra scalar SU(2) doublet and gives rise to a number of intriguing interactions in the lepton sector, which lead to processes with lepton number violation [24] (see a recent discussion in [25]), and to radiatively induced Majorana neutrino masses [26,27]. The parameter space of the Zee model allowed by the experimental data has been recently worked out [28] showing that the masses of the additional scalars in the range of a few hundreds GeV are possible, but they have to lie in the range below a few TeV.

The Minimal Model
The minimal part of the SM Lagrangian extended by the scalar field carrying a non-trivial representation of the U Y (1) weak hypercharge group includes the terms of dimension not greater than four. If one requires, in addition, lepton number conservation, as it takes place in the SM, the simplest model Lagrangian contains the kinetic term and the mass and self-coupling terms of the charged scalar boson field: where the covariant derivative is given by being the SM weak hypercharge gauge field, g 1 is the SM U Y (1) coupling and Y S is the weak hypercharge of the new scalar field S.
The potential V (S) may have, in general, the following gauge invariant form where |µ S | 2 is a mass parameter, λ S is the S-boson quartic self-coupling, λ ΦS is the coupling of the scalar S to the Higgs field. The last term has been included into the potential, because it contributes to the mass term after spontaneous symmetry breaking. Let us stress a few points here: • The S-field is a charged field, so it cannot have a nontrivial vacuum expectation value. Therefore, it cannot influence the value the SM ρparameter.
• Since the gauge boson B is expressed in the SM as a linear combination of the photon and the Z-boson fields, B ν = A ν cosθ W − Z ν sinθ W , the S-scalar couples to the photon with the constant e Y S 2 , where the electromagnetic constant e is equal to g 1 cosθ W , as it is usual in the SM. The S-scalar is an electrically charged field. As will be shown later, the hypercharge of the S-field is equal to two with the electric charge being equal to one (Q S = Y S /2). Thus, we denote the S-field as S − and the complex conjugate field S * as S + .
• The mass term parameter µ S could be equal to zero. In this case the mass of the S-boson comes from the interaction with the Higgs field in a similar way as for the other SM particles. In this case the mass of the S-boson is equal to M 2 S = λ ΦS v 2 /2 and its natural value is of the order of hundred GeV.
• If only the dimension 4 or less operators are included, there are no gauge invariant operators containing the charged scalar and the quark fields. We did not include into the Lagrangian the gauge invariant operators of dimension four, which describe the interaction of the S-scalar with the SM lepton fields giving lepton number violating vertices, they will be discussed shortly later. As a result, in this approximation the S-scalar is a stable particle.
In a simplest variant of the model the last property leads to the prediction of a stable charged scalar boson. Obviously, if the mass of the boson is of the order of a few hundreds GeV, the existence of the boson will not contradict the limits from precision electroweak measurements, in particular, the limits on S and T-parameters [28].

Pair Production Cross Sections
Charged scalars can be produced at the LHC in pairs via the Drell-Yan process in collisions of quark-antiquark pairs as well as in the gluon-gluon fusion. The production cross section as a function of the charged scalar mass is shown in Fig.1 for three different proton-proton collision energies √ s = 13, 14, 27 TeV. 1 One can see from Fig The production in quark-antiquark pair collisions was also discussed in paper [21]. However, there is an additional contribution to the pair production cross section, which comes from the gluon-gluon fusion mechanism and which was not mentioned in [21]. Two gluons produce a virtual SM Higgs boson via the top loop triangle diagram and the virtual Higgs boson decays to a pair of the charged scalars. The production cross section for the case, in which the mass of the S-boson comes from the interaction with the Higgs field corresponding to the coupling constant chosen as λ ΦS = 2M 2 S /v 2 , is shown in Fig.1b for the collision energies √ s = 13, 14, 27 TeV. The gluon-gluon fusion cross section also decreases with the grows of the mass. The level of the cross section is comparable or even higher than that for the Drell-Yan quark-antiquark annihilation process in Fig.1a. The cross section in Fig.1b includes the NNLO K-factors as given in [34]. The cross section for 1 TeV scalar boson mass is about 3.5×10 −1 fb at the collision energy 27 TeV adding to the production rate about 5200 events at the high luminosity 15 ab −1 . The pair production cross sections from the gluon-gluon fusion mechanism presented in Fig.1b refer to the case, where the total S-boson mass is generated by the Higgs mechanism. However, as it was noted after formula (2), there can exist a proper mass term. In this case the cross section should be multiplied by the factor ξ 2 , ξ = λ ΦS v 2 /2M 2 S < 1 denoting the part of the S-boson mass squared coming from the Higgs mechanism. Thus, the pair production cross sections presented in Fig.1b should be considered as the maximal possible for given S-boson masses. If the S-boson is discovered, measurements of its Drell-Yan production cross section will allow one to determine the value of the parameter ξ and thus to find out, whether there exists another mass generation mechanism besides the SM Higgs mechanism.
Searches for stable charged particles presented in [19] at the LHC energy 13 TeV give the lowest bound on the production cross section of about 4 f b -2 f b for the luminosity 2.5 f b −1 . This corresponds to 10 -5 events expected for the stable charged particle production. Therefore, one gets an upper bound on the charged scalar mass of about 300 GeV and 390 GeV corresponding to 10 and 5 expected events respectively summing up the mentioned contributions from the qq and gluon-gluon sub-processes to the charged scalar production cross section.
Assuming the same lowest number of expected events from 10 to 5 one can estimate from the computed cross sections the expected upper limits on the boson mass for various cases of collision energies and luminosities. So, for the proton-proton collision energy 14 TeV and the luminosity 300 f b −1 the expected mass limits are calculated to be about 1000 GeV and 1150 GeV respectively. For the benchmark energy 27 TeV and the luminosity 15 ab −1 the limits on the charged scalar mass are expected to be 3.1 TeV and 3.4 TeV.
For completeness the production cross section in e + e − collisions is shown in Fig.2 as a function of collision energy for the scalar mass 100 TeV, 200 TeV, and 300 GeV. The level of the cross section in Fig.2 is large enough giving good prospects to study the charged scalars in detail, if its mass is in the kinematically accessible range. Surely, if the scalar in that mass range having the specified production cross sections (Fig.1) had existed, it would have been already discovered at the LHC.

Interactions with Leptons and Quarks
If only the above discussed terms (operators) of dimension 4 had been present in the extended SM Lagrangian, the charged scalar boson would not have had interactions leading to its decay and/or single production, and therefore the boson would have been stable. However, gauge invariant operators of dimension four and five involving the charged scalar field can be constructed, which lead to decays of the boson. We will first discuss the gauge invariant terms of dimension four involving the lepton fields.
The transformation properties of the S-scalar field under the gauge group of the SM allow the existence of the following dimension 4 terms describing the coupling of the S-scalar to leptons [24]: where ν c denotes the charge conjugate neutrino field. Obviously, these interactions lead to lepton number violation in the S-scalar decay processes. However, it turns out that at low energies this lepton number violation is very small due to the large S-scalar mass. Moreover, one can show with the help of Fierz identities that the S-scalar mediated interactions of leptons conserve lepton number and can be brought to the standard form of Fermi's four fermion interaction, which imposes constraints on the coupling constants f ik [35,36]. The results of these papers with the present day values of the Fermi constant [37,38] and the probabilities of the decays τ → µν µ ν τ , τ → eν e ν τ , µ → eγ [38] give |f 12 | 2 < 3 × 10 −6 G F M 2 S , |f 13 | 2 , |f 23 | 2 < 2.8 × 10 −2 G F M 2 S . A full parameter scan of the Zee model carried out in paper [28] and including a fit of the neutrino mixing angles and mass differences gives the constraints on the coupling constants f ik , which turn out to be much more stringent: |f 12 |, |f 13 |, |f 23 | < 10 −6 . For these values of the coupling constants the partial widths of the S-scalar decays to leptons are less than 0.5 eV for the S-scalar mass up to 5 TeV.
The interaction of the S-scalar with the quark fields can take place only due to gauge invariant terms of dimension five or larger. Here we will discuss the gauge invariant terms of dimension five involving the quark fields. To introduce the notations let us first recall the well-known fact that, in the SM, the most general interaction Lagrangian of the Higgs field and the quark fields includes a mixing of the fermion fields from various generations: where Γ u,d are generically possible mixing coefficients with up-and downtype quark fields. The Higgs and the conjugate Higgs SU L (2) doublet fields in the unitary gauge are After spontaneous symmetry breaking Lagrangian (4) in the unitary gauge takes the following form where M ij = Γ ij v/ √ 2 is a generic mass mixing matrix. In order to obtain the physical mass eigenstates of quarks, the matrices M ij should be diagonalized by unitary transformations of the left-and righthanded quark fields: The matrices U are chosen such that As it is well known, the SM neutral currents remain the same after the above unitary transformation providing the absence of the FCNC at three level. However, after the transformation to the physical degrees of freedom the charged currents get a unitary matrix in front of the down quark fields, called the Cabbibo-Kobayashi-Mascawa (CKM) mixing matrix. Similarly, after the unitary transformation of the lepton fields, one gets the Pontecorvo-Maki-Nakagawa-Sakata neutrino mixing matrix (PMNS) in front of the massive neutrino fields in the charged leptonic currents.
In a similar manner one can write a gauge invariant Lagrangian for the interaction of the SM fermions with the charged scalar boson: where λ u,d are dimensionless matrices and Λ is the scale of "new physics". After the substitution of the Higgs field and the transformation (6) of the quark fields to the mass eigenstates, one gets the following interaction Lagrangian in the unitary gauge The elements of matrices µ d,u have the dimension of mass, the matrices are not diagonal in general, they may contain complex phases leading to CP violation. Here we do not discuss such a general case.
Let us consider a simple particular case, where matrices µ d,u are proportional (or equal) to the mass matrices M ij . In this case matrices V d,u contain the products of the CKM matrix or its hermitian conjugated matrix and the diagonal mass matrices for the up-and down-type quarks. The interactions of the two first quark generations are therefore naturally suppressed by the corresponding quark masses allowing to overcome the FCNC constrains [28]. The dominating part is the interaction of the charged scalar with the top-bottom quark charged current. In fact, the interaction structure is very similar to that of the charged Higgs in the 2HDM or MSSM taken at tan β=1 (see [39,40,41,42,43]). However, in comparison with the 2HDM or MSSM the interaction vertices are suppressed by the factor of the order of v/Λ.
It is worth noting that interactions similar to those described by formulas (8) and (9) can exist also in the lepton sector. If the neutrinos are considered to be massless, the corresponding formulas will include only the terms similar to the second ones in formulas (8), (9). If the neutrinos are considered to be massive, they will be absolutely similar to formulas (8), (9). However, it is natural to expect the entries of the corresponding mass matrices µ ν,e = λ ν,e v/ √ 2 to be of the order of neutrino and charged lepton masses, and in this case the contribution of these terms to the S-scalar decay processes is negligible compared with the decay to t-quark.
The dominating production channel pp → t + S − + X in the case of the scalar boson being heavier than the top quark is similar to the charged Higgs case with the suppression factor (v/Λ) 2 . If the scale is not very large, the production cross section could be large enough to be interesting for searches at the LHC as shown in Fig.3. The NLO corrections make the result much more stable with respect to the factorization/renormalization scale variation while the NLO K-factor is found to vary in the range of 1.4 or less [44]. The single production cross section decreases quadratically with the scale and becomes smaller than the considered above pair production at the scale greater than a few tens TeV.
As was mentioned, the mass of the S-scalar in the csSM may arise from the decay width varies from 10 −1 GeV to 10 −4 GeV (left plot). However if the scale is in the GUT range (right plot) the width becomes very small 10 −24 GeV -10 −27 GeV. In this case the life time of the scalar might be 0.1 sec or more leading to a microscopic travel distance before the decay. For the case of large scales Λ the single boson production cross section becomes negligible at colliders, and the charged scalars may be produced only in pairs. This corresponds to the case of long-lived charged particles with the discussed above current and expected limits on the charged scalar boson mass.
For rather small scales in TeV range the charged scalar may be produced either singly or in pairs with subsequent decays into top and bottom quarks. However both production cross sections are significantly smaller than the top pair and the single top cross sections. In this case a careful analysis is needed in order to estimate, whether or not a small signal of the charged scalar could be extracted from much larger backgrounds at the LHC.

Concluding Remarks
A simple gauge invariant extension of the SM considered in this study may provide an example of a heavy stable charged (HSCP) or long-lived (LLP) particle. The model contains, in addition to the SM fields, only the charged scalar field gauged only under the U Y (1) weak hypercharge gauge group. The model can be considered as a simple special case of the well known Zee model. In the simplest case, assuming the presence of only dimension 4 operators and lepton number conservation, the gauge invariant Lagrangian of the model contains only the gauge interaction of the charged scalar and its interaction with the SM Higgs field. Since in this case one cannot construct gauge invariant interactions of the scalar with the SM fermions, the charged scalar boson is a stable particle. The main production mode is the charged scalar Drell-Yan pair production via the photon and Z-boson exchange in quark-antiquark and via the SM Higgs exchange in gluon-gluon collisions. From the computed cross sections and the results of searches for HSCP at the LHC one can estimate the current bounds on the charged scalar boson mass to be about 400 GeV and the expected bounds at higher collision energies and larger luminosity. In particular, at future 27 GeV LHC with the luminosity of 15 ab −1 the bound is expected to reach 3.4 TeV covering the most interesting mass regions following from the overall parameter space analysis for the Zee model as found in [28]. Allowing higher dimensional operators and violation of the lepton number one can add to the Lagrangian the interactions of charged scalar field with the SM fermions leading to decays of the scalar boson. The dimension 4 operators containing lepton fields violate lepton number conservation, and the corresponding coupling strengths are significantly constrained by the muon decay, the neutrino mass measurements and oscillation data. The dimension 5 operators in the quark sector are naturally proportional to the fermion masses and the CKM matrix elements. The dominating decay mode of the charged scalar boson is, therefore, the decay to the top and the bottom quarks and the dominating single boson production channel is the associated production with the top quark. This is rather similar to the charged Higgs production and decay in 2HDM or MSSM at tan β = 1, although with an additional suppression by the factor v 2 Λ 2 . The single production cross section varies from 10 f b to 10 −5 f b in the mass range between 200 GeV and 4 TeV and in the range of the scale Λ from 2 TeV to 30 TeV. The decay width depends strongly on the scalar boson mass and the scale Λ and for the TeV scale regions takes values from 100 MeV to 0.1 MeV or so. If the scale is much larger, say, in the GUT range, the decay width to the top and bottom quarks becomes very small. In this case the width could be dominated by lepton number violating decays, but this obviously depends on the small lepton violating coupling strengths.
We did not discuss cosmology issues of the csSM model. This is planned to be presented in a separate study.