List of examples of charged particles from new physics discussed in this review showing representations and applications to phenomenology.
We have new charged particles in many scenarios of physics beyond the Standard Model where these particles are sometimes motivated to explain experimental anomalies. Furthermore, such new charged particles are important target at the collider experiments such as the Large Hadron Collider in searching for a signature of new physics. If these new particles interact with known particles in the Standard Model, they would induce interesting phenomenology of flavor physics in both lepton and quark sectors. Then, we review some candidate of new charged particles and its applications to flavor physics. In particular, vector-like lepton and leptoquarks are discussed for lepton flavor physics and B-meson physics.
- flavor physics
- charged particle from beyond the standard model
- B-meson decay
- vector-like lepton/quark
- charged scalar boson
Charged particles are often considered in the physics beyond the Standard Model (BSM) of particle physics as new heavy particles which are not observed at the experiments. Such charged particles can have rich phenomenology since it would interact with particles in the Standard Model (SM). Furthermore they are motivated to explain some experimental anomalies indicating deviation from predictions in the SM. For example, some charged particle interaction can accommodate with the anomalous magnetic moment of the muon, , which shows a long-standing discrepancy between experimental observations [1, 2] and theoretical predictions [3, 4, 5, 6],
where . This difference reaches to deviation from the prediction. In addition, new charged particles are introduced when we try to explain anomalies in -meson decay like and [7, 8, 9, 10, 11, 12, 13, 14, 15, 16].
In this chapter, we review some candidates of new charged particles from BSM physics. After listing some examples of them, the applications to some flavor physics will be discussed focusing on some specific cases. We find it interesting to consider new charged particles which are related to flavor physics in both lepton and quark sectors.
2. Some charged particles from beyond the standard model physics
In this section we review some examples of charged particles which are induced from BSM physics.
2.1 Charged scalar bosons
where is the vacuum expectation values (VEVs) of Higgs fields. In general, one can write Yukawa interaction in terms of Higgs doublet fields as
where all flavor indices are hidden, ; and are the quark and lepton doublets with flavor index , respectively; () denotes the singlet fermion; are the Yukawa matrices; and with being the Pauli matrix. There are two CP-even scalars, one CP-odd pseudoscalar, and two charged Higgs particles in the 2HDM, and the relations between physical and weak eigenstates can be given by
where and denote the real (imaginary) parts of the neutral and charged components of , respectively; , , , and are the vacuum expectation values (VEVs) of and GeV. In our notation, is the SM-like Higgs, while , , and are new particles which appear in the 2HDM. In particular, Yukawa interactions with charged Higgs are given by
where is the CKM matrix and the matrix is defined by original Yukawa coupling and unitary matrix diagonalizing fermion mass
A doubly charged scalar boson also appears from triplet scalar field:
where is the VEV of the triplet scalar. Such a triplet scalar is motivated to generate neutrino mass known as Higgs triplet model or type-II seesaw mechanism [19, 20, 21, 22, 23, 24, 25, 26]. We can write Yukawa interaction of triplet scalar and lepton doublets by
where with flavor index and is the Dirac charge conjugation operator. In terms of the components, the Yukawa interaction can be expanded as
where is used. Another example of model including doubly charged scalar is Zee-Babu type model [27, 28] for neutrino mass generation at two-loop level. In such a type of model, one introduces singly and doubly charged scalars which are singlet. The Yukawa couplings associated with charged scalar fields are given by
where is antisymmetric under flavor indices. These Yukawa interactions can be used to generate neutrino mass with the nontrivial interaction in scalar potential:
Note that these charged scalars also contribute to lepton flavor violation processes.
2.2 Vector-like leptons
The vector-like leptons (VLLs) are discussed in Ref. . They are new charged particles without conflict of gauge anomaly problem and induce rich lepton flavor physics. To obtain mixing with the SM leptons, the representations of VLL under gauge symmetry can be singlet, doublet, and triplet under . In order to avoid the stringent constraints from rare decays, we here consider the triplet representations and with hypercharges and , respectively. The new Yukawa couplings thus can be written such that
where we have suppressed the flavor indices; is the SM Higgs doublet field, and the neutral component of Higgs field is . The representations of two VLLs are
with and . Since is a real representation of , the factor of in is required to obtain the correct mass term for Majorana fermion . Due to the new Yukawa terms of , the heavy neutral and charged leptons can mix with the SM leptons, after electroweak symmetry breaking (EWSB). Then the lepton mass matrices become matrices and are expressed by
where the basis is chosen such that the SM lepton mass matrices are in diagonalized form, is the SM charged lepton mass matrix, diag, and
Note that the elements of should be read as , where the index distinguishes the Yukawa couplings of the different VLLs and the index stands for the flavors of the SM leptons.
To diagonalize and , the unitary matrices with so that are introduced. The information of and can be obtained from and , respectively. According to Eq. (14), it can be found that the flavor mixings between heavy and light leptons in are proportional to the lepton masses. Since the neutrino masses are tiny, it is a good approximation to assume . If one further sets in our phenomenological analysis, only -related processes have significant contributions among them. Unlike , the off-diagonal elements in flavor-mixing matrices are associated with . In principle, the mixing effects can be of the order of without conflict. In our example later, we examine these effects on . To be more specific, we choose parametrization that the unitary matrices in terms of as
where is used in our approximation, , and . Combining the SM Higgs couplings and new Yukawa couplings of Eq. (12), the Higgs couplings to all singly charged leptons are obtained such as
where is the state of a physical charged lepton in lepton flavor space. We use the notations of and to denote the heavy-charged VLLs in mass basis. Using the parametrization of Eq. (16), the Higgs couplings to the SM-charged leptons can be formulated by
If one sets , it is clear that in addition to the coupling being modified, the tree-level flavor-changing couplings - - and - - are also induced, and the couplings are proportional to . In order to study the VLL contributions to , the couplings for and are expressed as
2.3 Vector-like quarks
Here we consider vector-like triplet quarks (VLTQs) that are discussed in Ref.  The gauge invariant Yukawa couplings of VLTQs to the SM quarks, to the SM Higgs doublet and to the new Higgs singlet field are written as
where is the left-handed SM quark doublet and it could be regarded as mass eigenstate before VLTQs are introduced; here all flavor indices are hidden, , and is the VLTQ with hypercharge . The representations of in are expressed in terms of their components as follows:
The electric charges of , , , and are found to be , , , and , respectively. Therefore, could mix with up (down) type SM quarks. Here is the mass of VLTQ, and due to the gauge symmetry, the VLTQs in the same multiplet state are degenerate. By the Yukawa couplings of Eq. (20), the mass matrices for up and down type quarks are found by
where and denote the diagonal mass matrices of SM quarks and . Notice that a non-vanished could shift the masses of VLTQs. Since , we neglect the small effects hereafter. Due to the presence of , the SM quarks, , and are not physical states; thus one has to diagonalize and to get the mass eigenstates in general. If , we expect that the off-diagonal elements of unitary matrices for diagonalizing the mass matrices should be of order of . By adjusting , the off-diagonal effects could be enhanced and lead to interesting phenomena in collider physics.
2.4 Scalar leptoquarks
In this subsection we consider leptoquarks (LQs) which are discussed for example in Refs. [31, 32]. The three LQs are , , and under SM gauge symmetry, where the doublet and triplet representations can be taken as
where the superscripts are the electric charges of the particles. Accordingly, the LQ Yukawa couplings to the SM fermions are expressed as
where the flavor indices are hidden, denotes the Cabibbo-Kobayashi-Maskawa (CKM) matrix, are the unitary matrices used to diagonalize the quark mass matrices, and and have been absorbed into , , , , and . In this setup, we treat the neutrinos as massless particles and their flavor mixing effects are rotated away as an approximation. There is no evidence for any new CP violation, so in the following, we treat the Yukawa couplings as real numbers for simplicity.
The scalar LQs can also couple to the SM Higgs field via the scalar potential, and the cross section for the Higgs to diphoton can be modified in principle. However, the couplings of the LQs to the Higgs are independent parameters and irrelevant to the flavors, so by taking proper values for the parameters, the signal strength parameter for the Higgs to diphoton can fit the LHC data. For detailed analysis see Table 1 in Ref. .
|Particle type||Examples of application|
|Charged scalar||,||Neutrino mass, lepton flavor violation|
|Vector-like lepton||,||Lepton flavor violation|
|Vector-like quark||,||Quark flavor physics|
|Scalar leptoquark||, ,||Meson decay, lepton flavor violation|
3. Examples of applying charged particles to flavor physics
In this section, we review applications of charged particles to flavor physics by considering VLLs and LQs as examples.
3.1 Flavor physics from vector-like lepton
Introduction of VLLs contributes to lepton flavor physics via Yukawa interactions discussed in previous section. Here we review the leptonic decay of the SM Higgs and LFV decay of charged lepton as an illustration based on Ref. .
3.1.1 Modification to branching ratio
From Eq. (18), it can be seen that the modified Higgs couplings to the SM leptons are proportional lepton masses. By comparison with other lepton channels, it can be seen that the mode is the most significant one, and thus we estimate the influence on . Using the values that satisfy , the deviation of from the SM prediction can be obtained as
If the SM Higgs production cross section is not changed, the signal strength for in our estimation is , where the measurements from ATLAS and CMS are  and , respectively. Although the current data errors for the channel are still large, the precision measurement of can test the effect or give strict limits on the parameters.
3.1.2 process in vector-like lepton model
In the following, we investigate the contributions of new couplings in Eq. (18) to the rare tau decays and to the flavor-conserving muon anomalous magnetic moment. We first investigate the muon , denoted by . The lepton flavor-changing coupling can provide contribution to through the Higgs-mediated loop diagrams. However, as shown in Eq. (18), the induced couplings are associated with ; only the right-handed tau lepton has a significant contribution. The induced is thus suppressed by the factor of so that the value of is two orders of magnitude smaller than current data . A similar situation happens in decay also. Since the couplings are suppressed by and , the BR for is of the order of . We also estimate the process via the -mediation. The effective interaction for is expressed by
where and the Wilson coefficient from the one loop is obtained as
Accordingly, the BR for is expressed as
We present the contours for as a function of coupling and in Figure 1, where the numbers on the plots are in units of . It can be seen that the resultant can be only up to , where the current experimental upper bound is .
3.2 -meson flavor physics with leptoquarks
This section is based on Ref. . Several interesting excesses in semileptonic decays have been observed in experiments such as (i) the angular observable of , where a deviation due to the integrated luminosity of 3.0 fb was found at the LHCb [8, 9], and the same measurement with a deviation was also confirmed by Belle  and (ii) the branching fraction ratios , which are defined and measured as follows:
where , and these measurements can test the violation of lepton flavor universality. The averaged results from the heavy flavor averaging group are and , and the SM predictions are around [36, 37] and , respectively. Further tests of lepton flavor universality can be made using the branching fraction ratios: . The current LHCb measurements are  and , which indicate a more than deviation from the SM prediction. Furthermore, a known anomaly is the muon anomalous magnetic dipole moment (muon ), where its latest measurement is . These anomalies would be explained by introducing LQs and we review possible scenarios in the following.
3.2.1 Effective interactions for semileptonic -decay
According to the interactions in Eq. (24), we first formulate the four-Fermi interactions for the and decays. For the processes, the induced current-current interactions from and are , and those from and are , where and denote the scalar and pseudoscalar currents, respectively. Taking the Fierz transformations, the Hamiltonian for the decays can be expressed as follows:
where the indices are the lepton flavors and the LQs in the same representation are taken as degenerate particles in mass. It can be seen that the interaction structure obtained from the triplet LQ is the same as that from the -boson one. The doublet LQ generates an structure as well as a tensor structure. However, the singlet LQ can produce currents of , , and tensor structures. Nevertheless, we show later that the singlet LQ makes the main contribution to the and excesses. Note that it is difficult to explain by only using the doublet or/and triplet LQs when the excess and other strict constraints are satisfied at the same time.
With the Yukawa couplings in Eq. (24), the effective Hamiltonian for the decays mediated by and can be expressed as
where the Fierz transformations have been applied. By Eq. (31), it can be clearly seen that the quark currents from both the doublet and triplet LQs are left-handed; however, the lepton current from the doublet (triplet) LQ is right(left)-handed. When one includes Eq. (31) in the SM contributions, the effective Hamiltonian for the decays is written as
where the leptonic currents are denoted by , and the related hadronic currents are defined as
The effective Wilson coefficients with LQ contributions are expressed as
where and is the CKM matrix element. From Eq. (34), it can be seen that when the magnitude of is decreased, can be enhanced. That is, the synchrony of the increasing/decreasing Wilson coefficients of and from new physics is diminished in this model. In addition, the sign of can be different from that of . As a result, when the constraint from decay is satisfied, we can have sizable values of to fit the anomalies of and angular observable in . Although the LQs can contribute to the electromagnetic dipole operators, since the effects are through one-loop diagrams and are also small, the associated Wilson coefficient is mainly from the SM contributions.
3.2.2 Constraints from , radiative lepton flavor violating, , , and processes
Before we analyze the muon , , and problems, we examine the possible constraints due to rare decay processes. Firstly, we discuss the strict constraints from the processes, such as oscillation, where denotes the neutral pseudoscalar meson. Since , , and mixings are involved, the first-generation quarks and the anomalies mentioned earlier are associated with the second- and third-generation quarks. Therefore, we can avoid the constraints by assuming that without affecting the analyses of and . Thus, the relevant process is mixing, where is induced from box diagrams, and the LQ contributions can be formulated as
where , GeV is the decay constant of -meson , and the current measurement is GeV . To satisfy the excess, the rough magnitude of LQ couplings is . Using our parameter values, it can be shown that the resulting agree with the current experimental data. However, can indeed constrain the parameters involved in the decays.
In addition to the muon , the introduced LQs can also contribute to the lepton flavor violating processes , where the current upper bounds are and , and they can strictly constrain the LQ couplings. To understand the constraints due to the decays, one expresses their branching ratios (BRs) such as
with , , and . is written as
where , can be obtained from by using instead of , and the function is given by
Note that and are due to . From Eq. (24), we can see that the doublet and singlet LQs can simultaneously couple to both left- and right-handed charged leptons, and the results are enhanced by . Other LQ contributions are suppressed by due to the chirality flip in the external lepton legs, and thus they are ignored. Based on Eq. (37), the muon can be obtained as
As mentioned earlier, the singlet LQ does not contribute to at the tree level, but it can induce the process, where the current upper bound is , and the SM result is around . Thus, can bound the parameters of . The four-Fermi interaction structure, which is induced by the LQ, is the same as that induced by the -boson, so we can formulate the BR for as
where and can be parameterized as . According to Eq. (31), the LQs also contribute to process, where the BRs measured by LHCb  and prediction in the SM  are and , respectively. The experimental data are consistent with the SM prediction, and in order to consider the constraint from , we use the expression for the BR as .
In addition to the decay, the induced effective Hamiltonian in Eq. (30) also contributes to the process, where the allowed upper limit is . According to previous results given by , we express the BR for as
where is the decay constant, and the in our model is given as
Using s, GeV, GeV , and , the SM result is . One can see that the effects of the new physics can enhance the decay by a few factors at most in our analysis.
3.2.3 Observables: and
The observables of and are the branching fraction ratios that are insensitive to the hadronic effects giving clearer test of lepton universality in -meson decay, but the associated BRs still depend on the transition form factors. In order to calculate the BR for each semileptonic decay process, we parameterize the transition form factors for by
where can be the or meson and the momentum transfer is given by . For the decay where is a vector meson, the transition form factors associated with the weak currents are parameterized such that
where when , , , and is the polarization vector of the vector meson. Here we note that the form factors associated with the weak scalar/pseudoscalar currents can be obtained through the equations of motion, i.e., and . For numerical estimations, the -dependent form factors , , , , and are taken as 
and the other form factors are taken to be
The values of , , and for each form factor are summarized in Table 2. A detailed discussion of the form factors can be referred to . The next-to-next-leading (NNL) effects obtained with the LCQCD Some Rule approach for the form factors were described by .
According to the form factors in Eqs. (44) and (45), and the interactions in Eqs. (30) and (32), we briefly summarize the differential decay rates for the semileptonic decay processes, which we use for estimating and . For the decay, the differential decay rate as a function of the invariant mass can be given by
where the functions and LQ contributions are
We note that the effective couplings and at the scale can be obtained from the LQ mass scale via the renormalization group (RG) equation. Our numerical analysis considers the RG running effects with at the scale . The decays involve polarizations and more complicated transition form factors, so the differential decay rate determined by summing all of the helicities are
where and . For the decays, the differential decay rate can be expressed as .
From Eq. (52), the measured ratio in the range GeV can be estimated by
is similar to , and thus we only show the result for .
3.2.4 Numerical analysis
After discussing the possible constraints and observables of interest, we now present the numerical analysis to determine the common parameter region where the and anomalies can fit the experimental data. Before presenting the numerical analysis, we summarize the relevant parameters, which are related to the specific measurements as follows:
The parameters related to the radiative LFV, , and processes are defined as
where . From Eqs. (54) and (55), we can see that in order to avoid the and constraints and obtain a sizable and positive , we can set (, , ) as a small value. From the upper limit of , we obtain , and thus the resulting is smaller than the current data. In order to further suppress the number of free parameters and avoid large fine-tuning of couplings, we employ the scheme with , where the sign of can be selected to obtain the correct sign for and to decrease the value of so that can fit the experimental data. As mentioned above, to avoid the bounds from the , , and systems, we also adopt . When we omit these small coupling constants, the correlations of the parameters in Eqs. (54) and (55) can be further simplified as
where are ignored due to the constraint from . The typical values of these parameters for fitting the anomalies in the decay are , so the resulting is smaller than the current data, but these parameters are too small to explain . Thus, we must depend on the singlet LQ to resolve the and excesses, where the main free parameters are now .
After discussing the constraints and the correlations among various processes, we present the numerical analysis. There are several LQs in this scenario, but we use to denote the mass of all LQs. From Eqs. (37), (39), and (56), we can see that the muon depends only on and . Here we illustrate as a function of in Figure 2(a), where the solid, dashed, and dotted lines denote the results for , , and TeV, respectively, and the band is the experimental value with errors. Due to the enhancement, with 1 TeV can explain the muon anomaly.
According to the relationships shown in Eq. (56), , , and depend on the same parameters, i.e., and . We show the contours for these observables as a function of and in Figure 2(b), where the data with errors and TeV are taken for all LQ masses. Based on these results, we see that in the range of , , where and can both fit the experimental data simultaneously. In addition, we show in the same plot. We can see that , which is used to explain the angular observable , can also be achieved in the same common region. According to Figure 2(b), the preferred values of and where the observed and and the overlap are around and . The latter values are at the percentage level, but they are still not sufficiently large to explain the tree-dominated and anomalies.
After studying the muon and anomalies, we numerically analyze the ratio of to , i.e., . The introduced doublet and triplet LQs cannot efficiently enhance , so in the following estimations, we only focus on the singlet LQ contributions, where the four-Fermi interactions shown in Eq. (30) come mainly from the scalar- and tensor-type interaction structures. Based on Eqs. (48), (50), and (51), we show the contours for and as a function of and in Figure 3(a) and (b), where the horizontal dashed and vertical dotted lines in both plots denote and , respectively, and TeV is used, and the data with errors are taken. For simplicity, we take . When considering the limits from , we obtain the limits and . In order to clearly demonstrate the influence of tensor-type interactions, we also calculate the situation by setting . The contours obtained for and are shown in Figure 3(c) and (d), where the solid and dashed lines denote the cases with and without , respectively. According to these plots, we can see that and have different responses to the tensor operators, where the latter is more sensitive to the tensor interactions. and can be explained simultaneously with the tensor couplings. In order to understand the correlation between and , we show the contours for and as a function of and in Figure 4, where are used, and the gray area is excluded by . We can see that the predicted is much smaller than the experimental bound.
Finally, we make some remarks regarding the constraint due to the LQ search at the LHC. Due to the flavor physics constraints, only the Yukawa couplings , , and can be of . These couplings affect the decays but also their production. Therefore, in addition to the -pair production, based on the Yukawa couplings, the single production becomes interesting. In the collisions, the single production can be generated via the and channels. Using CalcHEP 3.6 [51, 52] with the CTEQ6 parton distribution functions , their production cross sections with and GeV at TeV can be obtained as 3.9 fb and 2.9 fb, respectively, whereas the -pair production cross section is fb. If we assume that predominantly decays into , , and with similar BRs, i.e. , then the single production cross section times with and as the possible final states can be estimated as around fb. The LQ coupling involves different generations, so the constraints due to the collider measurements may not be applied directly. However, if we compare this with the CMS experiment  based on a single production of the second-generation scalar LQ, we find that the values of at GeV are still lower than the CMS upper limit with few fb. The significance of this discovery depends on the kinematic cuts and event selection conditions, but this discussion is beyond the scope of this study, and we leave the detailed analysis for future research.
We have reviewed some charged particles which appear from physics beyond the Standard Model of particle physics. Some possible candidates of them are listed such as charged scalar boson, vector-like leptons, vector-like quarks, and leptoquarks. After showing some properties and interactions of these particles, we reviewed some applications to flavor physics in which lepton flavor physics with vector-like lepton and -meson physics with leptoquarks are focused on as an illumination. We have seen rich phenomenology that would be induced from such new charged particles, and they will be also tested in the future experiments.
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