Searching for LFV Flavon decays at hadron colliders
The search for Flavons with a mass of (1) TeV at current and future colliders might probe low-scale flavor models. We are interested in the simplest model that invokes the Froggatt-Nielsen (FN) mechanism with an Abelian flavor symmetry, which includes a Higgs doublet and a FN complex singlet. Assuming a CP conserving scalar potential, there are a -even and a -odd Flavons with lepton flavor violating (LFV) couplings. The former can mix with the standard-model-like Higgs boson, thereby inducing tree-level LFV Higgs interactions that may be at the reach of the LHC. We study the constraints on the parameter space of the model from low-energy LFV processes, which are then used to evaluate the Flavon decay widths and the () production cross section at hadron colliders. After imposing several kinematic cuts to reduce the SM main background, we find that for about 200-350 GeV, the decay might be at the reach of the LHC for a luminosity in the range 1-3 ab, however, a luminosity of the order of 10 ab would be required to detect the decay. On the other hand a future 100 TeV collider could probe masses as high as (10) TeV if it reaches an integrated luminosity of at least 20 ab. Therefore, the 100 TeV Collider could work as a Flavon factory.
After the discovery of a Higgs-like particle with a mass GeV Aad:2012tfa; Chatrchyan:2012xdj, the search for new physics (NP) has become the one of the next goals of the LHC. Although current measurements of the spin, parity, and couplings of the Higgs boson seem consistent with the standard model (SM) Gunion:1989we, its light mass seems troublesome, i.e. the hierarchy problem, and calls for new physics (NP). The SM has also other open issues, such as the flavor problem, unification, etc. Pomarol:2012sb; Martin:1997ns, which also encourages the study of NP models.
The couplings of the Higgs particle to a pair of massive gauge bosons or fermions have strengths proportional to the masses of such particles. However, the LHC has tested only a few of such Higgs couplings, namely the ones to the gauge bosons and the heaviest fermions. Along these lines, many studies have been devoted to analyze the pattern of Higgs couplings derived from LHC data, for instance Espinosa:2012ir; Giardino:2013bma. However, non-standard Higgs couplings, including the flavor violating (FV) ones, are predicted in many models of physics beyond the SM Branco:2011iw; DiazCruz:2004tr; DiazCruz:2004pj; DiazCruz:2002er. In particular, the observation of neutrino oscillations, which is associated with massive neutrinos, motivates the occurrence of lepton flavor violation (LFV) in nature Ma:2009dk. Within the SM, LFV processes vanish at any order of perturbation theory, which motivates the study of SM extensions that predict sizeable LFV effects that could be at the reach of detection. Apart from decays such as and , particularly interesting is the decay , which was studied first in Refs. Pilaftsis:1992st; DiazCruz:1999xe, with subsequent analyses on the detectability of the signal appearing soon after Han:2000jz; Assamagan:2002kf. This motivated a plethora of calculations in the framework of several SM extensions, such as theories with massive neutrinos, supersymmetric theories, etc. Arganda:2004bz; DiazCruz:2002er; Brignole:2004ah; DiazCruz:2008ry; Chamorro-Solano:2017toq; Chamorro-Solano:2016ugt; Lami:2016mjf. After the Higgs boson discovery, the decay offers a great opportunity to search for NP at the LHC. Although a slight excess of the branching ratio was reported at the LHC run I, with a significance of 2.4 standard deviations Khachatryan:2015kon, a subsequent study CMS:2016qvi ruled out such an excess and put the limit with 95% C.L. In the model we are interested in, LFV effects are induced at the tree-level in the scalar sector, so it is thus worth assessing their phenomenology.
Another open issue in the SM is the flavor problem Isidori:2010kg, which has long been the focus of interest, with several proposals meant to address it, such as textures, GUT-inspired relations, symmetries, radiative generation, etc. In particular, a flavor symmetry approach can be supplemented with the Froggatt-Nielsen (FN) mechanism, which assumes that above some scale , there is a symmetry (perhaps of Abelian type ) that forbids the appearance of Yukawa couplings; SM fermions are charged under this symmetry. However, the Yukawa matrices can arise through non-renormalizable operators. The Higgs spectrum of these models could include a light Flavon , which could mix with the Higgs bosons when the flavor scale is of the order of the TeVs. Quite recently, the phenomenology of Higgs-Flavons at particle colliders has been the focus of attention Bolanos:2016aik; Bauer:2016rxs; Huitu:2016pwk; Berger:2014gga; Diaz-Cruz:2014pla.
Although the states we are interested in arise from the mixing of Higgs bosons and Flavons, we will still call them Flavons for short. Depending on the particular model, there could be several potentially detectable Flavon decays, which would be indistinguishable from the decays of a heavy Higgs boson, therefore, to search for a distinctive signature, we will focus on the one arising from the LFV decay , with . The minimal model that introduces the FN mechanism with an Abelian flavor symmetry includes a scalar sector consisting of a Higgs doublet and a FN complex singlet. From now on we will refer to this model as the FN extension of the Standard Model (FNSM). Such a model predicts a -even Flavon and a -odd one . Also, the couplings of the light SM-like Higgs boson would deviate from the SM ones, such that two possible scenarios are possible: firstly, the mixing of the real part of the doublet with the real component of the FN singlet could induce sizable LFV Higgs couplings of the light physical Higgs boson, which might affect the light Higgs phenomenology; secondly, the -even Flavon could be very heavy, so its mixing with the light Higgs boson would be negligible and unconstrained by LHC Higgs data, though in such a case the -odd state could be the lighter one, thereby giving rise to a potentially detectable LFV signal.
In this paper we are interested in studying the possible detection of both -even and -odd Flavons at the LHC and a future 100 TeV collider via their LFV decays. It has been pointed out that a 100 TeV collider would allow for a detailed study of several topics of interest in particle physics, such as Higgs physics and the electroweak symmetry breaking mechanism Contino:2016spe. It will also be useful to search for possible signals of dark matter, SUSY theories, and other extension models Golling:2016gvc.
We will start our analysis by considering the constraints on the parameter space of the FNSM obtained from Higgs data at the LHC, low energy LFV, and the muon magnetic dipole moment. A set of benchmarks will then be used to estimate the Flavon decay modes, focusing on the LFV ones, as well as their production cross sections by gluon fusion at the LHC and a future 100 TeV collider. We will then explore the possibility that the -even and -odd Flavons could be detected via the decay channel, for which we will make a Monte Carlo analysis of the signal and the SM main background.
The organization of our work is as follows. In Sec. II we describe the realization of the FN mechanism within the simplest model, namely that with one Higgs doublet and one FN singlet. In particular, we present the Higgs potential and Yukawa Lagrangian, from which the Flavon couplings can be extracted. Section III is devoted to the constraints on the parameters space of the model and the benchmarks of parameter values we will be using in our analysis, whereas Sec. IV is focused on the analysis of the decay modes of both a -even and a -odd Flavon as well as their production cross sections via gluon fusion at the LHC and a future 100 TeV collider. We also present the Monte Carlo analysis of the signal and its main background. The conclusions and outlook are presented in Sec. V.
Ii The scalar sector of the minimal FNSM
The scalar sector of the FNSM includes the usual SM Higgs doublet
and a complex singlet
where denotes the SM vacuum expectation value (VEV) and that of the FN singlet, whereas and will be identified with the pseudo-goldstone bosons that become the longitudinal modes of the and gauge bosons.
ii.1 The Higgs potential
We turn now to discuss the minimal -conserving Higgs potential with a softly-broken global symmetry, which is given as follows
We are therefore left with the following -symmetric terms (), and the -soft-breaking term . The latter is required to avoid a massless Goldstone boson when . An extensive analysis of this potential was presented in reference Bonilla:2014xba, where the parameter space that allows a viable model was identified.
Imposing the minimization conditions on renders the following relations:
ii.2 The Scalar Mass Matrix
In a -invariant potential, the -even (real) and -odd (imaginary) components of the mass matrix do not mix. In this case the mass matrix for the real components in the basis is given by:
whereas the mass matrix for the imaginary components, in the basis , reads
We notice that the mass scale for the -odd state arising from the FN singlet is different from the VEV , which is the -breaking scale, and therefore it could be much lighter. As for the mixing of the real components of the doublet and the singlet , the mass eigenstates are obtained through the standard rotation:
In what follows we will identify the mass eigenstate as the SM-like Higgs boson with GeV, while the mass eigenstates and will be assumed to be heavier. Although they arise from Flavon-Higgs mixing, in the present work we will still refer to and as Flavons for short. The properties of the -even Flavon will depend on the size of its mixing with the lightest state. On the other hand, the -odd state, which does not couple to gauge bosons, it does couple to the SM fermions, including both diagonal and non-diagonal interactions.
Our analysis of Flavon decays requires the knowledge of cubic interactions, such as the trilinear vertex , which is given in the minimal model by:
ii.3 Yukawa sector and LFV interactions
The FN Lagrangian of the model includes the terms that become the Yukawa couplings once the flavor symmetry is spontaneously broken. It is given by:
where denote the Abelian charges that reproduce the observed fermion masses, for each fermion type. The Flavon field is assumed to have flavor charge equal to -1, such that is -invariant. Then, the Yukawa couplings arise after the spontaneous breaking of the flavor symmetry, i.e. , where denotes the Flavon VEV, whereas denotes a heavy mass scale, which represents the mass of heavy fields that transmit such symmetry breaking to the quarks and leptons. For specific examples of structures for the Yukawa matrices of fermion type, see Diaz-Cruz:2014pla.
In the unitary gauge we set , thus we can write the neutral component of the Higgs field as follows:
whereas the powers of the Flavon field can be expanded as
Then, after substituting the Flavon mass eigenstates, one gets finally the following interaction Lagrangian for the Higgs-fermion couplings
where we use the usual short-hand notation and . Here, is the diagonal mixing matrix, whereas the information about the size of FV Higgs couplings is contained in the matrices, with given in the flavor basis as
which remains non-diagonal once the fermion mass matrices are diagonalized, thereby giving rise to FV scalar couplings. The diagonal and non-diagonal interactions of the , , and scalar bosons with massive fermions are thus given by:
where the Feynman rule for the vertex includes a Dirac matrix and .
Besides the Yukawa couplings, we also need to specify the scalar-to-gauge-boson couplings. They can be readily extracted from the kinetic terms of the Higgs doublet and the singlet, which transforms trivially under the SM gauge group. Thus after substituting Eq. (8) in the kinetic term, we obtain that the and couplings to gauge boson pairs are SM-like, with the coupling constants given by , for , with . Thus the coupling constants are
We are interested in the possible detection of -even and -odd Flavons with masses of the TeV order at both the LHC and a future collider with a center-of-mass energy of TeV. Depending on the particular model, there could be several potentially detectable decays of such Flavons, but some of them would also arise from heavy Higgs bosons, for instance within multi-Higgs doublet models. Thus, in order to search for a distinctive Flavon signature, we shall focus on the one arising from the LFV decay (). In order to determine the detectability of this decay, we will present a Monte Carlo analysis of the signal and the most relevant SM backgrounds.
Iii Constraints on the parameter space of the FNSM
In order to evaluate the Flavon decays and production modes at a hadron collider we need to analyze the most up-to-date constraints on the model parameters. For the mixing angle we can use the data obtained by the LHC collaborations on the Higgs boson properties, whereas the LFV couplings can be constrained via the experimental data on the muon anomalous magnetic dipole moment (AMDM) , the LFV decays of the tau lepton and , as well as the experimental constraint on the decay. All the necessary formulas to perform our analysis below are presented in Appendix A.
iii.1 Mixing angle
We shall use the universal Higgs fit of Ref. Giardino:2013bma, which presents constraints on the parameters , defined as (small) deviations of the Higgs couplings from the SM values, i.e. . For the and gauge bosons, the corresponding constraints are and . Regarding the fermion couplings, we notice that this universal Higgs fit is valid for the -conserving case, which we are considering here; therefore we can apply them to constrain the properties of the -even SM-like Higgs boson. Furthermore, the constraints derived from the gauge interactions provide the strongest constraints on the mixing angle . Since the lightest scalar boson couples with the SM gauge bosons with a strength that deviates from the SM couplings by the factor , to satisfy the bound on we need to have . We will use a conservative approach and use the benchmark in our analysis below.
iii.2 Diagonal and matrix elements
In this work we will use the 2-family approximation, neglecting FV with the leptons of the first fermion family. We will also assume that there is no -violating phase, which means that we have three free parameters: , , and . To constrain the diagonal matrix element, we refer again to the universal Higgs fit of Ref. Giardino:2013bma and consider the constraint on the deviation of the SM coupling, namely, . We thus show in Fig. 1 the allowed area on the plane for two values of . We observe that in order to agree with the universal Higgs fit constraint when , must be of the order of for TeV and for TeV, but when we must have values of the order in the complete interval. Since we are considering , we will use as benchmark.
As far as is concerned, there is no available data to constraint the coupling but we will assume the hierarchy and set , otherwise the SM coupling would be swamped by the corrections of the FNSM.
iii.3 Non-diagonal matrix element
We will consider the current experimental bounds on Olive:2016xmw, the tau decay Olive:2016xmw, and the Higgs boson decay Aad:2015gha; Khachatryan:2015kon to constrain the matrix elements Arroyo:2013tna. Notice that the current bound on the decay width gives very weak constraints, so we will omit such a process in our analysis. Two scenarios arise when dealing with constraints on LFV couplings:
Scenario I: the FNSM is assumed to be responsible for the discrepancy between the theoretical and experimental values of the muon AMDM.
Scenario II: the FNSM Flavons would not be responsible for the discrepancy. It could happen instead that future calculations of the SM hadronic contribution would settle such a discrepancy without requiring any NP. Otherwise, the ultraviolet completion of the FNSM could give extra contributions, which would solve the puzzle. In any case, by requiring that the new contribution from the Flavons to is smaller than the experimental central value. We find that it is enough to satisfy the LHC constrain from in order to have a viable parameter space.
We will now assess the implications of both scenarios. To avoid large corrections to the diagonal lepton scalar couplings, we will take and . We show in Fig. 2 the area allowed in the plane by the experimental constraints on the muon AMDM along with the and decays for the indicated values of the Higgs Flavon masses and the mixing angle . We observe that scenario I is not consistent with the constraint on the matrix element from the decay, which requires small values of this parameter, of the order of for TeV and for TeV, whereas the muon AMDM requires values of as large as 1 for TeV. Therefore we will assume scenario II and take as benchmark , which is allowed for TeV.
iii.4 and matrix elements
According to the universal Higgs fit Giardino:2013bma, the allowed value for the deviation of the SM coupling is . We will use again the two-family approximation and take the values and .
iii.5 Summary of Benchmarks for the model parameters
In summary, in our study below we will use the following benchmarks:
Mixing angle : As discussed above, to satisfy the fit on the 125 GeV Higgs couplings measured at the LHC, the following constraint must be obeyed . We will thus use the benchmark .
FN singlet VEV : It appears in the Flavon couplings and the LFV SM Higgs coupling. We will consider the value TeV.
matrix: It determines the strength of the LFV scalar couplings. We will use the 2-family approximation and take , , and , which are consistent with the constraint on the LFV decay .
interaction: This vertex depends on a combination of parameters that appear in the Higgs potential. However, these parameters could be traded by an effective coupling , which can take values of the order of . We will thus fix .
A summary of the benchmarks we are going to consider in our analysis is presented in Table 1.
Iv Search for LFV Flavon decays at hadron colliders
As stated above, the aim of this work is to analyze the detectability of the LFV signal arising from the Flavon decays, as predicted by the FNSM, at the LHC and a future 100 TeV collider. Below we will present an analysis concentrating on the main Flavon production mechanism, i.e. gluon fusion, as well as the branching ratios of its dominant decay modes. Then, we will present the Monte Carlo analysis of the and decay signatures, including the study of the potential SM background. We will present a conservative analysis meant to find out whether it is possible to have evidence of our signal at the LHC and the future 100 TeV collider.
iv.1 Production cross-sections of the -even and -odd Flavons
We now turn to analyze the main production mode of both and Flavons at hadronic colliders, namely, by gluon fusion. The High-Luminosity Large Hadron Collider project aims to increase potential discoveries contemplating a luminosity of up to =3 ab about 2025. As far as a 100 TeV collider is concerned the future circular collider (FCC) contemplates an integrated luminosity of until (10 ab). We consider the integrated luminosities shown in tha table 2.
In Fig. 3 we show the production cross section of a -even Flavon as a function of its mass at the LHC and a future 100 TeV collider. We also show the event numbers on the right axis of each plot. As far as the -odd Flavon is concerned, the respective production cross section and event numbers are presented in Fig. 4.
The dominant contribution to gluon fusion arises from loops carrying the top quark as shown in Fig 5. This explains the suppression of the production of the -odd Flavon as compared to that of the -even one, as observed in Figs. 3 and 4, which stems from the appearance of the coupling in the corresponding cross section. For instance, taking into account the parameter values of Table 1, we have , whereas for the -even Flavon .
iv.2 Flavon decays
iv.2.1 Two-body decays
We now analyze the behavior of the branching ratios of the dominant decay modes, including the FV ones, of both the -even and -odd Flavons. Analytical expressions for the partial decay widths are presented in Appendix B. It is worth mentioning that a crosscheck was done by comparing the numerical results obtained via our own C language code implementing the analytic expressions of Appendix B, and those computed with the aid of the CalcHEP package Belyaev:2012qa, for which we used an implementation of the FNSM Feynman rules obtained with LanHEP Semenov:2014rea. In Fig. 6 we present the relevant branching ratios of the decays modes of a -even Flavon as functions of its mass for the benchmarks of Table 1.
We observe in Fig. 6 that in the scenario under study and for ranging between and GeV, the dominant decay mode would be , followed by . Once the channel became open, its branching ratio would be about the same order of magnitude as that of the decay. Other relevant decay modes would be , , and , whereas the one-loop induced decays and would have tiny branching ratios.
As far as the -odd Flavon is concerned, since it does not couple to gauge bosons at the tree-level, its main decay modes are into fermion pairs. The corresponding branching ratios are shown in Fig. 7. We observe that the decay modes could have branching ratios up to two orders of magnitude larger than the analogue branching ratios of the -even Flavon . For the parameter values used here, the main decay channels of the -odd Flavon would be , , , whereas the decay would be very suppressed due to the coupling. The one-loop decays and would also have very small branching ratios of the order of .
iv.2.2 Three-body decays
At the tree-level the -even Flavon also couples with a fermion pair and a SM Higgs boson, thus it is worth analyzing the behavior of the three-body decay modes . The corresponding decay width is presented in Appendix B. In Fig. 8 we show the branching ratios for these three-body decays. We observe that, for a relatively light Flavon with mass around 300 GeV, the decay channel would have a branching ratio as large as . On the other hand, other kinematically allowed decays would reach branching ratios as high as . For a heavier Flavon with GeV, the decay would become open and could be the dominant three-body decay mode for GeV. Although these decay channels seem worth a more detailed study, we will content ourselves with obtaining the event numbers that could be achieved at the LHC and the FCC. We consider values for the luminosities of the table 2. In the Fig. 9 we shown the number of events for the processes with .
iv.3 Search for LFV Flavon decays at the LHC and a future 100 TeV collider
We are interested in the possible detection of the -even and -odd Flavons via their LFV decay into a pair at the LHC and the FCC. We thus show in Fig. 10 the event numbers for the processes and , for TeV and an integrated luminosity for the values displayed in the Table 2. We also use the same set of parameter values of Table 1. We note that for a -even Flavon with a mass about TeV, there would be about (10) signal events at the LHC and events at the FCC. These event numbers would decrease by about two orders of magnitude for a -odd Flavon.
We will now analyze the signature of the LFV Flavon decays and , with , and their potential SM background. We are inspired in the analysis carried out by the CMS collaboration in Refs. Khachatryan:2015kon, ATLAS collaboration Aad:2016blu and the work of the authors of Primulando:2016eod. The ATLAS and CMS collaborations considered the two following tau decay channels: electron decay and the hadron decay . For our analysis we will concentrate instead on the electron decay. As far as our computation scheme is concerned, we first use the LanHEP routines to obtain the FNSM Feynman rules for Madgraph Alwall:2011uj. In this way, the signal and background events can be generated by MadGraph5 and MadGraph5aMCNLO, respectively, interfaced with Pythia 6 Sjostrand:2006za and Delphes 3 deFavereau:2013fsa for detector simulations. The background events were generated at NLO in QCD and the signal at LO, using the CT10 parton distribution functions Gao:2013xoa. The signal and main background events are as follows:
Signal: the signal is with . The electron channel must contain exactly two opposite-charge leptons, one an electron and the other a muon. Then, we search for the final state +miss energy. We consider the specific case for which an integrated luminosity of ab for the LHC is considered and in the interval ab for the FCC.
Background: the main SM background arises from production via the Drell-Yan process, followed by the decay as well as and pair production and jets. In this work we will only consider the main background to assess how our signal could be searched for.
iv.3.1 Analysis at the LHC
We start by analyzing the possible detection of the -even Flavon at the LHC with =14 TeV. For illustrative purpose, we use the following set of values for : 200, 250, 300, and 350 GeV. We generated events for the signal and the SM main background. Afterwards, the kinematic analysis was done via MadAnalysis-5 Conte:2012fm. The cuts applied to both the signal and background are shown in Table 3, where we also show the event numbers of the signal (S) and background (B) after the kinematic cuts are applied, along with the signal significance for GeV. The effect of the cuts on the signal and background event numbers is best illustrated in Fig. 11, where we show how the efficiencies and evolve after each cut is successively applied. One can observe that once the kinematic cuts are applied, the resulting signal efficiency is about , whereas that for the background is around . With a luminosity of ab, the signal significance is about . However, if we take into account an integrated luminosity of ab it increases up to 6.5. The net effect is shown in Fig. 12, where the signal significance is plotted as a function of the luminosity for the chosen values of .
|Cut number||Cuts||Signal (S)||Background (B)|
|Initial (no cuts)||9190||29320240||1.7|
Therefore, it seems troublesome that a -even Flavon with a mass greater than 300 GeV could be detected at the LHC via the decay channel. In such a case, we can turn to the decay modes and , which are the dominant ones. These decay channels seem more promising for the detection of a heavy Flavon. Along this line, to assess the potentiality of the and decay channels for the Flavon detection, we show in Fig. 13 the corresponding event numbers that could be produced at the LHC and a 100 TeV collider. We note that about events would be produced at the LHC for TeV, whereas the number of events would be slightly smaller. This seems more promising for the signal detection, though a more detailed analysis of the background would be required to draw a definitive conclusion.
As for the -odd Flavon, we notice that it has a smaller production rate, therefore, in order to have evidence of the decay, higher luminosities, of the order of , would be required. This seems inaccessible for the LHC, however, we expect that the search for this decay could be possible at the FCC.
iv.3.2 Analysis at the future 100 TeV collider
The building of a 100 TeV collider is under consideration Arkani-Hamed:2015vfh. The luminosity and center-of-mass energy are crucial factors to allow detectable LFV signatures of the FNSM. Luminosity goals for a 100 TeV collider are discussed by the authors of Ref. Arkani-Hamed:2015vfh. Again, we consider the values of the Table 2, although up to 30 ab might be reached. In our analysis we consider the conservative kinematic cuts in order to give a general overview assess how our signal could be searched for. The applied cuts to both the signal and background are shown in Table 4, whereas in Fig. 14 we show the signal significance as a function of the luminosity and the Flavon mass , with . We observe that at a 100 TeV collider with integrated luminosity of 10 ab, it would be possible to probe -even Flavon masses in the multi-TeV range, up to about 4 TeV. We also notice that -odd Flavon masses could be searched until the order 1 TeV.
|Cut number||Cuts||Signal (S)||Background (B)|
|Initial (no cuts)||90720||364075164||4.75|
In this work we have explored the possibility that the LFV decay channel of -even and -odd Higgs Flavons with a mass of a few hundreds of GeVs could be at the reach of detection at the LHC and a future 100 TeV collider, which would serve as a possible probe of low-scale flavor models. For the theoretical framework, we have considered the simplest Froggatt-Nielsen model, with an Abelian flavor symmetry and a -conserving Higgs sector that includes a Higgs doublet and a Froggatt-Nielsen complex singlet. In this model the -even Flavon can mix with the SM-like Higgs boson, thereby inducing tree-level LFV interactions mediated by the latter. In this work we concentrate instead on the LFV couplings of both the -even and -odd Higgs Flavons. After studying the constraints on the parameter space of the model from low-energy LFV processes, we choose a set of benchmarks and estimate the relevant decay modes and the production cross section of the Flavons via gluon fusion at the LHC and a future 100 TeV collider. We then consider a set of kinematic cuts for both the signal and the SM main background. It is found that the LHC has the potential to discover the LFV decay for between 200 and 350 GeVs provided that luminosities, of the order of 1-3 ab, are achieved. In such a case other decay channels would be more appropriate to search for the signal of a Flavon at the LHC. As far as a future 100 TeV collider is concerned, it would be able to probe the LFV decay channel for Flavon masses as heavy as 10 TeVs, as long as an integrated luminosity of at least 20 ab was available, which has been deemed viable in the literature regarding the possible construction of such a collider. Therefore, besides other physics goals, a 100 TeV Collider might also work as a Flavon factory.
Acknowledgements.We acknowledge support from Conacyt and SNI (México). Partial support from VIEP-BUAP is also acknowledged.
Appendix A Flavon contributions to LFV and decays and the muon anomaly
In this Appendix we present the analytical expressions necessary to obtain the constraints on the LFV Flavons couplings shown in Fig. 2. Although these results were meant for -even and -odd scalar bosons, they are also valid for the Flavons.
The CMS collaboration reported a bound on the respective branching ratio: CMS:2016qvi.
As far as the decay is concerned, it arises at the one-loop level and receives contributions of the SM Higgs boson and the Flavons via the Feynman diagram of Fig. 15(a). The respective decay width is
where the and coefficients stand for the contribution of -even and -odd scalar bosons, respectively, which in the limit of and , can be approximated as Harnik:2012pb
Two-loop contributions can be relevant and the respective expressions are reported in Harnik:2012pb. The current experimental limit on the branching ratio is Olive:2016xmw.
As for the decay, it receives contributions from the exchange of a SM-Higgs boson and the Flavons as depicted in the Feynman diagram of Fig. 15(b). The tree-level decay width can be approximated as
where . It has been pointed in Ref. Harnik:2012pb, however, that the one-loop contribution is dominant. We refrain from presenting the corresponding expression as this process, for which the experimental limit on the respective branching ratio is Olive:2016xmw, gives very weak constraints on the FNSM parameters.
Finally, the muon AMDM also receives contributions from the SM Higgs boson and the Flavons, which are induced by a triangle diagram similar to the diagram of Fig. 15(a) but with two external muons. The corresponding contribution can be approximated for as Harnik:2012pb
where one must take into account the NP corrections to the coupling only. If the Flavons are too heavy, the dominant NP contribution would arise from the SM Higgs boson.
The discrepancy between SM theoretical prediction and the experimental value is Olive:2016xmw
Thus, the requirement that this discrepancy is accounted for by Eq. (25) leads to the bound with C.L.
Appendix B Decay widths of -even and -odd scalar bosons
b.1 -even scalar boson decays
The most relevant decays of both -even and -odd scalar bosons have been long studied in the literature. We will present the relevant decay widths for the sake of completeness as they are also valid for the Flavons. We will assume that all the couplings are SM-like, other than the couplings, which stand for the couplings of a -odd scalar boson with the and particles. The tree-level two-body widths are as follows.
The decay width is given by
where and is the color number. From here we easily obtain the flavor conserving decay width.
The decays of a heavy -even scalar boson into pairs of real electroweak gauge bosons can also be kinematically allowed. The corresponding decay width is
with for .
Other relevant decays are those arising at the one-loop level, such as and . The two-photon decay width can be written as
with the subscript standing for the spin of the charged particle circulating into the loop. The function is given by
On the other hand, the two-gluon decay can receive contributions of quarks only and the respective decay width can be obtained from Eq. (29) by summing over quarks only and making the replacements , .
Formulas for other decay channels such as , which has a considerably suppressed decay width, as well as radiative corrections for the above decay widths can be found in the literature Djouadi:2005gi; Djouadi:2005gj.
b.2 -odd scalar boson decays
The decay of a -odd scalar boson into a pair of fermions of distinct flavor is given by
where we now use the definition . The FC decay width follows easily.
There are no decays into electroweak gauge bosons at the tree-level. On the other hand, the two-photon decay proceeds through charged fermion loops only and the corresponding decay width can be obtained from (29) by making the replacement and summing over fermions only, with
whereas the two-gluon decay width can be obtained by summing over quarks only and making the additional replacements and .
b.3 Three-body decay
As far as three-body decays are concerned, the study of the decay channel could be interesting as it can also have a sizeable branching ratio. This decay receives contribution from the four Feynman diagrams shown in Fig. 16. After some algebra, we can write the decay width as follows
where the integration domain is given by
and the average square amplitude is