Lepton flavor violating signals of a little Higgs model at the high energy linear colliders
Abstract
Littlest Higgs model predicts the existence of the doubly charged scalars , which generally have large flavor changing couplings to leptons. We calculate the contributions of to the lepton flavor violating processes and , and compare our numerical results with the current experimental upper limits on these processes. We find that some of these processes can give severe constraints on the coupling constant and the mass parameter . Taking into account the constraints on these free parameters, we further discuss the possible lepton flavor violating signals of at the high energy linear collider experiments. Our numerical results show that the possible signals of might be detected via the subprocesses in the future experiments.
I. Introduction
It is well known that the individual lepton numbers , and are automatically conserved and the tree level lepton flavor violating processes are absent in the standard model . However, the neutrino oscillation experiments have made one believe that neutrinos are massive, oscillate in flavors, which presently provide the only experimental hints of new physics and imply that the separated lepton numbers are not conserved[1]. Thus, the requires some modification to account for the pattern of neutrino mixing, in which the processes are allowed. The observation of the signals in present or future high energy experiments would be a clear signature of new physics beyond the .
Some of popular specific models beyond the generally predict the presence of new particles, such as new gauge bosons and new scalars, which can naturally lead to the tree level coupling. In general, these new particles could enhance branching ratios for some processes and perhaps bring them into the observable threshold of the present and next generations of collider experiments. Furthermore, nonobservability of these processes can lead to strong constraints on the free parameters of new physics. Thus, studying the possible signals of new physics in various high energy collider experiments is very interesting and needed.
Little Higgs models[2] employ an extended set of global and gauge symmetries in order to avoid the one-loop quadratic divergences and thus provide a new method to solve the hierarchy between the scale of possible new physics and the electroweak scale . In this kind of models, the Higgs boson is a pseudo-Goldstone boson of a global symmetry which is spontaneously broken at some high scales. Electroweak symmetry breaking is induced by radiative corrections leading to a Coleman-Weinberg type of potential. Quadratic divergence cancellation of radiative corrections to the Higgs boson mass are due to contributions from new particles with the same spin as the particles. This type of models can be regarded as one of the important candidates of the new physics beyond the .
The littlest Higgs model [3] is one of the simplest and phenomenologically viable models, which realizes the little Higgs idea. Recently, using of the fact that the model contains a complex triplet Higgs boson , Refs.[4,5,6] have discussed the possibility to introduce lepton number violating interactions and generation of neutrino mass in the little Higgs scenario. Ref.[5] has shown that most satisfactory way of incorporating neutrino masses is to include a lepton number violating interaction between the triplet scalars and lepton doublets. The tree level neutrino masses are mainly generated by the vacuum expectation value of the complex triplet , which does not affect the cancellation of quadratic divergences in the Higgs mass. The neutrino masses can be given by the term , in which ( are generation indices) is the Yukawa coupling constant. As long as the triplet is restricted to be extremely small, the value of is of natural order one, i.e. 1, which might produce large contributions to some of processes[6,7].
The aim of this paper is to study the contributions of the couplings predicted by the model to the processes and and compare our numerical results with the present experimental bounds on these processes, and see whether the constraints on the free parameter can be obtained. We further calculate the contributions of the model to the processes and ( or ) , and discuss the possibility of detecting the signals of the model via these processes in the future high energy linear collider experiments.
This paper is organized as follows. Section II contains a short summary of the relevant couplings of the scalars (doubly charged scalar , charged scalars , and neutral scalar ) to lepton doublets. The contributions of these couplings to the processes and are calculated in section III. Using the current experimental upper limits on these processes, we try to give the constraints on the coupling constant in this section. Section IV is devoted to the computation of the production cross sections of the processes and induced by the doubly charged scalars . Some phenomenological analyses are also included in this section. Our conclusions are given in section V.
II. The couplings of the triplet scalars
The model[3] consists of a nonlinear model with a global symmetry and a locally gauged symmetry . The global symmetry is broken down to its subgroup at a scale , which results in 14 Goldstone bosons . Four of these are eaten by the gauge bosons , resulting from the breaking of , giving them masses. The Higgs boson remains as a light pseudo Goldstone boson and other give masses to the gauge bosons and form a scalar triplet . The complex triplet offers a chance to introduce lepton number violating interactions in the theory.
In the context of the model, the lepton number violating interaction which is invariant under the full gauge group, can be written as[5,7]:
(1) |
Where and are generation indices, and (= 1, 2) are indices, and is a left handed lepton doublet. is the Yukawa coupling constant and is the charge-conjugation operator. Because of non-linear nature of , this interaction can give rise to a mass matrix for the neutrinos as:
(2) |
One can see from Eq.(2) that, if we would like to stabilize the Higgs mass and at the same time ensure neutrino masses consistent with experimental data[8], the coupling constant must be of order , which is unnaturally small. However, it has been shown[4,5] that the lepton number violating interaction only involving the complex scalar triplet can give a neutrino mass matrix . Considering the current bounds on the neutrino mass[8], there should be:
(3) |
Thus, the coupling constant can naturally be of order one or at least not be unnaturally small provided the of the triplet scalar is restricted to be extremely small.
In this scenario, the triplet scalar has the couplings to the left handed lepton pairs, which can be written as[5]:
(4) |
Considering these couplings, Ref.[5] has investigated the decays of the scalars and , and found that the most striking signature comes from the doubly charged scalars . The constraints on the coupling constant and the triplet scalar mass parameter coming from the muon anomalous magnetic moment and the process are studied in Ref.[7]. In the next section, we will calculate the contributions of the charged scalars and to the processes and .
III. The charged scalars and the processes and
Decay Process | Current limit | Bound |
---|---|---|
[10] | —– | |
[12] | —– | |
[13] | —– | |
[11] | ||
[14] | ||
[14] | ||
[14] | ||
Table 1: The current experimental upper limits on the branching ratios of some processes and the corresponding upper constraints on the free parameters.
The observation of neutrino oscillations[1] implies that the individual lepton numbers are violated, suggesting the appearance of the processes, such as and . The branching ratios of these processes are extremely small in the with right handed neutrinos. For example, Ref.[9] has shown . Such small branching ratio is unobservable.
The present experimental upper limits on the branching ratios [10], [11], [12], [13], and [14] are given in Table 1. Future experiments with increased sensitivity can reduce these current limits by a few orders of magnitude(see, e.g.[15]). In this section, we will use these data to give the constraints on the free parameters and .
The couplings of the charged scalars and given in Eq.(4) can lead to the radiative decays at the one loop level mediated by the exchange of the charged scalars and , as shown in Fig.1. For the doubly charged scalar , the photon can be attached either to the internal lepton line or to the scalar line. For the charged scalar , the photon can be only attached to the scalar line[16].
Using Eq.(4), the expression of the branching ratio can be written as at leading order:
(5) |
Where is the fine structure constant and is the Fermi constant. The factor means that, when the internal lepton is the same as one of the leptons and , the contributions of to is four times those for and . and are the masses of the scalars and , respectively. In the model, the scalar mass is generated through the Coleman-Weinberg mechanism and the scalars , and degenerate at the lowest order[5]. Thus, we can assume and write the branching ratio as:
(6) |
In particular, for the decay process , we obtain the following expression for the branching ratio :
(7) |
From above equations, we can see that the process can not be able to constrain independently. However, if we assume for ( is the flavor-diagonal coupling constant) and for ( is the flavor-mixing coupling constant), then we can obtain the constraints on the combination of the free parameters , and . Observably, the most stringent constraint should come from the current experimental upper limits on the branching ratio . Thus, in Fig.2, we have shown the coupling constant as a function of the mass parameter for , and . From Fig.2, one can see the upper limit on strongly depend on the values of and . For and , there must be .
In the model, the processes can be generated at tree level through the exchange of doubly charged scalar , as depicted in Fig.3.
The expressions of the branching ratios for the processes are given by[16,17]
(8) | |||||
(9) | |||||
(10) | |||||
(11) | |||||
(12) |
Certainly, up to one loop, the processes get additional contributions from the processes . Thus, the charged scalars and have contributions to the processes at one loop. However, compared with the tree level contributions, they are very small, which can be safely neglected.
The processes also can not give the constraints on the coupling constants independently, but would be able to constrain the combination . Our numerical results are given in Table 1.
In the following section, we will take into account these constraints coming from the processes and , estimate the contributions of the doubly charged scalars to the processes and , and discuss the possibility of detecting the signals for the doubly charged scalars at the experiments.
IV. The doubly charged scalars and the
processes
and
In general, the doubly charged scalars can not couple to quarks and their couplings to leptons break the lepton number by two units, leading to a distinct signature, namely a pair of same sign leptons. The discovery of a doubly charged scalar would have important implications for our understanding of the Higgs sector and more importantly, for what lies beyond the . This fact has made one give more elaborate theoretical calculations in the framework of some specific models beyond the and see whether the signatures of this kind of new particles can be detected in the future high energy experiments. For example, the production and decay of the doubly charged scalars and their possible signals at the have been extensively studied in Refs.[18,19]. In this section, we will consider the contributions of the doubly charged scalars predicted by the model to the processes and ( or ). The processes can be seen as the subprocesses of the processes . For example, the doubly charged scalar generates contributes to the process through the subprocess , as shown in Fig.4.
Using Eq.(4), the expression of the cross section for the subprocess can be easily written as:
(13) |
Where is the center-of-mass energy of the subprocess . is the total decay width of the doubly charged scalar , which has been given by Ref.[5] in the case of the triplet scalars and degenerating at lowest order with a common mass :
(14) | |||||
Where is the coupling constant, is the coupling constant. In above equation, the final-state masses have been neglected compared to the mass parameter . It has been shown that, for , the main decay modes of are . Furthermore, the FX coupling constant are subject to very stringent bounds from the process . In this case, the decay width can be approximately written as:
(15) |
Considering the current bounds on the neutrino mass[8], there should be:
(16) |
so leads to , which does not conflict with the most stringent constraint from the process . Thus, in our numerical calculation, we will take Eq.(15) as the total decay width of .
Using the equivalent particle approximation method[20], the effective cross section for the process can be approximately written as[19]:
(17) |
Where and . is the equivalent electron distribution function of the initial positron, which gives the probability that an electron with energy is emitted from a positron beam with energy . The relevant expression can be written as[21]:
(18) |
In Fig.5 and Fig.6, we plot the production cross sections and for the processes and as function of the FD coupling constant , respectively. In these figures, we have assumed and taken and . From Fig.5 and Fig.6 one can see that the values of and are strongly depend on the value of the coupling constant . For and , the values of the subprocess cross section and the effective cross section are larger than fb and fb, respectively.
The signal of the doubly charged scalar given by the process is so distinctive and is the background free, discovery would be signalled by even few events. In Fig.7, we plot the discovery region in the plane at 95% confidence level for seeing 5 events, in which we have assumed the future with the energy and the yearly integrated luminosity of [22]. From this figure, one can see that, in wide range of the parameter space, the signals of should be detected in the future ILC experiments.
The doubly charged scalar can also has contributions to the processes and . However, the experimental upper limits on the processes , , and can give severe constraints on the combination , which makes the production cross sections of these processes very small. For example, even if we take and , the production cross sections , and are smaller than fb, fb, and fb, respectively. Thus, it is very difficult to detect the signals of via the processes in the future experiments.
Certainly, the doubly charged scalar has contributions to the processes and . Similar with above calculation, we can give the values of the production cross sections for these processes. We find that the cross section is equal to the cross section . Thus, the conclusions for the doubly charged scalar are also apply to the doubly charged scalar .
V. Conclusions
To solve the so-called hierarchy or fine tuning problem of the , the little Higgs theory was proposed as a kind of models to accomplished by a naturally light Higgs boson. The model is one of the simplest and phenomenologically viable models. In the model, neutrino masses and mixings can be generated by coupling the scalar triplet to the leptons in a interaction, whose magnitude is proportional to the triplet multiplied by the Yukawa coupling constant without invoking a right handed neutrino. This scenario predicts the existence of doubly charged scalars . For smaller values of i.e. , the doubly charged scalars have large flavor changing coupling to leptons, which can generate significantly contributions to some processes and give characteristic signatures in the future high energy experiments.
In this paper, we first consider the processes and in the context of the model. For the process , it involves all of the FX coupling constants , we can not give the simple constraints about the free parameters and . Thus, for the fixed values of the FX coupling constant , we take into account the current experimental upper limit of the and plot the FD coupling constant as a function of the mass parameter . Our numerical results show that the upper limit on is strongly depend on the free parameters and .
Using the present experimental upper limits on the branching ratios , we obtain the constraints on the combination . We find that the most stringent constraint comes from the process . In all of the parameter space, there must be .
The characteristic signals of the processes is same-sign dileptons or two same-sign different flavor leptons, which is the background free and offers excellent potential for doubly charged scalar discovery. To see whether the doubly charged scalar can be detected in the future experiments, we discuss the contributions of to the processes and . We find that the triplet scalar can give significantly contributions to the processes . In wide range of the parameter space of the model, the possible signals of might be observed in the future experiments. However, the production cross sections of the processes mediated by are very small. The contributions of the triplet scalar to the processes are equal to those of for the processes , Thus, our conclusions are also apply to the doubly charged scalar .
Some popular models beyond the predict the existence of doubly charged scalars, which generally have the lepton number and lepton flavor changing couplings to leptons and might produce distinct experimental signatures in the current or future high energy experiments. Their observation would signal physics outside the current paradigm and further test the new physics models. Search for this kind of new particles has been one of the important goals of the high energy experiments[23]. Thus, the possibly signals of the doubly charged scalars predicted by the little Higgs models should be more studied in the future.
Acknowledgments
This work was supported in part by Program for New Century Excellent Talents in University(NCET-04-0290), the National Natural Science Foundation of China under the Grants No.10475037 and 10675057.
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