Flavor Physics and Charged Particle

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.


Introduction
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, g À 2 ð Þ μ , which shows a long-standing discrepancy between experimental observations [1,2] and theoretical predictions [3][4][5][6], where a μ ¼ g À 2 ð Þ μ =2. This difference reaches to 3:6σ deviation from the prediction. In addition, new charged particles are introduced when we try to explain anomalies in B-meson decay like B ! K * ð Þ μ þ μ À and B ! D * ð Þ τν [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. 1 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.

Charged scalar bosons
Singly charged scalar appears from two-Higgs doublet model (2HDM) [17,18] in which two SU 2 ð Þ L doublet Higgs fields are introduced: where v 1, 2 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, are the SU 2 ð Þ L quark and lepton doublets with flavor index i, respectively; f R (f ¼ U, D, ℓ) denotes the SU 2 ð Þ L singlet fermion; Y f 1, 2 are the 3 Â 3 Yukawa matrices; andH i ¼ iτ 2 H * i with τ 2 being the Pauli matrix. There are two CPeven 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 h ¼ Às α ϕ 1 þ c α ϕ 2 , H ¼ c α ϕ 1 þ s α ϕ 2 , where ϕ i η i ð Þ and η AE i denote the real (imaginary) parts of the neutral and charged components of H i , respectively; c α s α ð Þ ¼ cos α sin α ð Þ, c β ¼ cos β ¼ v 1 =v, s β ¼ sin β ¼ v 2 =v, and v i are the vacuum expectation values (VEVs) of H i and v ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi v 2 1 þ v 2 2 p ≈ 246 GeV. In our notation, h is the SM-like Higgs, while H, A, and H AE are new particles which appear in the 2HDM. In particular, Yukawa interactions with charged Higgs are given by where V is the CKM matrix and the matrix X f is defined by original Yukawa coupling and unitary matrix diagonalizing fermion mass A doubly charged scalar boson also appears from SU 2 ð Þ L triplet scalar field: where v Δ 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 L L i ¼ ν i ; ℓ i ð Þ T L with flavor index i and C ¼ iγ 2 γ 0 is the Dirac charge conjugation operator. In terms of the components, the Yukawa interaction can be expanded as where C † ¼ ÀC 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 h AE ; k AEAE À Á which are SU 2 ð Þ L singlet. The Yukawa couplings associated with charged scalar fields are given by where f ij 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.

Vector-like leptons
The vector-like leptons (VLLs) are discussed in Ref. [29]. 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 SU 2 ð Þ L Â U 1 ð Þ Y gauge symmetry can be singlet, doublet, and triplet under SU 2 ð Þ L . In order to avoid the stringent constraints from rare Z ! ℓ AE i ℓ ∓ j decays, we here consider the triplet representations 1; 3; À1 ð Þand 1; 3; 0 ð Þwith hypercharges Y ¼ À1 and Y ¼ 0, respectively. The new Yukawa couplings thus can be written such that where we have suppressed the flavor indices; H is the SM Higgs doublet field, H ¼ iτ 2 H * and the neutral component of Higgs field . The representations of two VLLs are where the basis is chosen such that the SM lepton mass matrices are in diagonalized form, m ℓ is the SM charged lepton mass matrix, Note that the elements of Y χ should be read as Y ij ¼ Y i ð Þ j , where the index i ¼ 1, 2 distinguishes the Yukawa couplings of the different VLLs and the index j ¼ 1; 2; 3 stands for the flavors of the SM leptons.
To diagonalize M ℓ and M ν , the unitary matrices V χ R, L with χ ¼ ℓ, ν so that The information of V χ L and V χ R can be obtained from M χ M † χ and M † χ M χ , respectively. According to Eq. (14), it can be found that the flavor mixings between heavy and light leptons in V χ R are proportional to the lepton masses. Since the neutrino masses are tiny, it is a good approximation to assume V ν R ≈ 1. If one further sets m e ¼ m μ ¼ 0 in our phenomenological analysis, only τrelated processes have significant contributions among them. Unlike V χ R , the offdiagonal elements in flavor-mixing matrices V χ L are associated with Y 1, 2 v=m Ψ . In principle, the mixing effects can be of the order of 0:1 without conflict. In our example later, we examine these effects on h ! τμ. To be more specific, we choose parametrization that the unitary matrices in terms of Y 1, 2 as where V ν R ≈ 1 is used in our approximation, ε χ L ≈ vY χ =m Ψ , and ε ℓ R ≈ vm † ℓ Y ℓ =m 2 Ψ . 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 ℓ 0 T ¼ e,μ,τ,τ 0 ; τ 0 0 À Á is the state of a physical charged lepton in lepton flavor space. We use the notations of τ 0 and τ 0 0 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 m e ¼ m μ ¼ 0, it is clear that in addition to the coupling hττ being modified, the tree-level flavor-changing couplings h-τ-μ and h-τe are also induced, and the couplings are proportional to m τ =v ≈ 7:2 Â 10 À3 . In order to study the VLL contributions to h ! γγ, the couplings for hτ 0 τ 0 and hτ 0 0 τ 0 0 are expressed as

Vector-like quarks
Here we consider vector-like triplet quarks (VLTQs) that are discussed in Ref. [30] 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 Q L 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, H ¼ iτ 2 H * , and F 1 2 ð Þ is the 2 Â 2 VLTQ with hypercharge 2=3 À1=3 ð Þ. The representations of F 1, 2 in SU 2 ð Þ L are expressed in terms of their components as follows: The electric charges of U 1, 2 , D 1, 2 , X, and Y are found to be 2=3, À1=3, 5=3, and À4=3, respectively. Therefore, U 1, 2 D 1, 2 ð Þcould mix with up (down) type SM quarks. Here M F 1 2 ð Þ 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 5 Â 5 mass matrices for up and down type quarks are found by vY 2 = ffiffi ffi 2 p À À À À À | À À ÀÀ À À ÀÀ ÀvY 2 =2 À À À À À | À À ÀÀ À À ÀÀ Since v s ≪ v, we neglect the small effects hereafter. Due to the presence of Y 1, 2 , the SM quarks, U 1, 2 , and D 1, 2 are not physical states; thus one has to diagonalize M u and M d to get the mass eigenstates in general. If vY i 1, 2 ≪ m F 1, 2 , we expect that the off-diagonal elements of unitary matrices for diagonalizing the mass matrices should be of order of vY i 1, 2 =m F 1, 2 . By adjusting Y i 1, 2 , the off-diagonal effects could be enhanced and lead to interesting phenomena in collider physics.

Scalar leptoquarks
In this subsection we consider leptoquarks (LQs) which are discussed for example in Refs. [31,32]. The three LQs are 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, V U u L U d † L denotes the Cabibbo-Kobayashi-Maskawa (CKM) matrix, U u, d L are the unitary matrices used to diagonalize the quark mass matrices, and U d L and U u R have been absorbed into k,k, y,ỹ, and w. 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, Examples of application 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. [31].

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.

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. [29]. (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 ττ mode is the most significant one, and thus we estimate the influence on h ! τ þ τ À . Using the values that satisfy BR h ! μτ ð Þ≈ 10 À4 , the devia- Þfrom the SM prediction can be obtained as If the SM Higgs production cross section is not changed, the signal strength for pp ! h ! τ þ τ À in our estimation is μ ττ ≈ 0:88, where the measurements from ATLAS and CMS are 1:44 þ0:42 À0:37 [33] and 0:91 AE 0:27 [34], respectively. Although the current data errors for the ττ channel are still large, the precision measurement of μ ττ can test the effect or give strict limits on the parameters.

τ ! μγ 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 g À 2, denoted by Δa μ . The lepton flavorchanging coupling hμτ can provide contribution to Δa μ through the Higgs-mediated loop diagrams. However, as shown in Eq. (18), the induced couplings are associated with m ℓj =vℓ Li ℓ Rj ; only the right-handed tau lepton has a significant contribution. The induced Δa μ is thus suppressed by the factor of m 2 μ m τ = vm 2 h À Á so that the value of Δa μ is two orders of magnitude smaller than current data . A similar situation happens in τ ! 3μ decay also. Since the couplings are suppressed by m τ =v and m μ =v, the BR for τ ! 3μ is of the order of 10 À14 . We also estimate the process τ ! μγ via the h-mediation. The effective interaction for τ ! μγ is expressed by where C L ¼ 0 and the Wilson coefficient C R from the one loop is obtained as Accordingly, the BR for τ ! μγ is expressed as We present the contours for BR τ ! μγ ð Þas a function of coupling Y and m Ψ in Figure 1, where the numbers on the plots are in units of 10 À12 . It can be seen that the resultant BR τ ! μγ ð Þcan be only up to 10 À12 , where the current experimental upper bound is BR τ ! μγ ð Þ< 4:4 Â 10 À8 [2].

B-meson flavor physics with leptoquarks
This section is based on Ref. [32]. Several interesting excesses in semileptonic B decays have been observed in experiments such as (i) the angular observable P 0 , where a 3σ deviation due to the integrated luminosity of 3.0 fb À1 was found at the LHCb [8,9], and the same measurement with a 2:6σ deviation was also confirmed by Belle [10] and (ii) the branching fraction ratios R D, D * , which are defined and measured as follows:  where ℓ ¼ e; μ ð Þ, and these measurements can test the violation of lepton flavor universality. The averaged results from the heavy flavor averaging group are R D ¼ 0:403 AE 0:040 AE 0:024 and R D * ¼ 0:310 AE 0:015 AE 0:008 [35], and the SM predictions are around R D ≈ 0:3 [36,37] and R D * ≈ 0:25, respectively. Further tests of lepton flavor universality can be made using the branching fraction ratios: . The current LHCb measurements are R K ¼ 0:745 þ0:090 À0:074 AE 0:036 [38] and R K * ¼ 0:69 þ0:11 À0:07 AE 0:05 [39], which indicate a more than 2:5σ deviation from the SM prediction. Furthermore, a known anomaly is the muon anomalous magnetic dipole moment (muon g À 2), where its latest mea- . These anomalies would be explained by introducing LQs and we review possible scenarios in the following.

Effective interactions for semileptonic B-decay
According to the interactions in Eq. (24), we first formulate the four-Fermi interactions for the b ! cℓ 0 ν ℓ 0 and b ! sℓ 0 þ ℓ 0 À decays. For the b ! cℓ 0 ν ℓ 0 processes, the induced current-current interactions from k 3jk i2 andỹ 3i w 2j are S À P ð ÞÂ S À P ð Þ, and those from y 3i y 2j andỹ 3iỹ2j are S À P ð ÞÂ S þ P ð Þ, where S and P denote the scalar and pseudoscalar currents, respectively. Taking the Fierz transformations, the Hamiltonian for the b ! cℓ 0 ν ℓ 0 decays can be expressed as follows: where the indices i, j 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 W-boson one. The doublet LQ generates an S À P ð ÞÂ S À P ð Þstructure as well as a tensor structure. However, the singlet LQ can produce currents of V À A ð ÞÂ V À A ð Þ, S À P ð ÞÂ S À P ð Þ, and tensor structures. Nevertheless, we show later that the singlet LQ makes the main contribution to the R D and R D * excesses. Note that it is difficult to explain R D, D * by only using the doublet or/and triplet LQs when the R K excess and other strict constraints are satisfied at the same time.
With the Yukawa couplings in Eq. (24), the effective Hamiltonian for the b ! sℓ 0 þ ℓ 0 À decays mediated by ϕ 2=3 and δ 4=3 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 b ! sℓ 0 þ ℓ 0 À decays is written as where the leptonic currents are denoted by L 5 ð Þ μ ¼ ℓγ μ γ 5 ð Þℓ, and the related hadronic currents are defined as The effective Wilson coefficients with LQ contributions are expressed as Eq. (34), it can be seen that when the magnitude of C LQ, ℓ j 10 is decreased, C LQ, ℓ j 9 can be enhanced. That is, the synchrony of the increasing/decreasing Wilson coefficients of C NP 9 and C NP 10 from new physics is diminished in this model. In addition, the sign of C LQ, ℓ 0 9 can be different from that of C LQ, ℓ 0

10
. As a result, when the constraint from B s ! μ þ μ À decay is satisfied, we can have sizable values of C LQ, μ 9 to fit the anomalies of R K and angular observable in B ! K * μ þ μ À . 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 C 7 is mainly from the SM contributions.
3.2.2 Constraints from ΔF ¼ 2, radiative lepton flavor violating, B þ ! K þ νν, B s ! μ þ μ À , and B c ! τν processes Before we analyze the muon g À 2, R D * ð Þ , and R K * ð Þ problems, we examine the possible constraints due to rare decay processes. Firstly, we discuss the strict constraints from the ΔF ¼ 2 processes, such as F À F oscillation, where F denotes the neutral pseudoscalar meson. Since K À K, D À D, and B d À B d 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 k 1ℓ 0 ≈k ℓ 0 1 ≈ y 1ℓ 0 ≈ỹ 1ℓ 0 ≈ w 1i ≈ 0 without affecting the analyses of R D * ð Þ and R K * ð Þ. Thus, the relevant ΔF ¼ 2 process is B s À B s mixing, where Δm B s ¼ 2| B s jHjB s | is induced from box diagrams, and the LQ contributions can be formulated as where C box ¼ m B s f 2 B s =3, f B s ≈ 0:224 GeV is the decay constant of B s -meson [40], and the current measurement is Δm exp B s ¼ 1:17 Â 10 À11 GeV [2]. To satisfy the R K * ð Þ excess, the rough magnitude of LQ couplings is |y 3i y 2i | $ |k 3i k 2i | $ 5 Â 10 À3 . Using our parameter values, it can be shown that the resulting Δm B s agree with the current experimental data. However, Δm B s can indeed constrain the parameters involved in the b ! cℓ 0 ν ℓ 0 decays.
In addition to the muon g À 2, the introduced LQs can also contribute to the lepton flavor violating processes ℓ 0 ! ℓγ, where the current upper bounds are BR μ ! eγ ð Þ< 4:2 Â 10 À13 and BR τ ! e μ ð Þγ ð Þ< 3:3 4:4 ð ÞÂ10 À8 [2], and they can strictly constrain the LQ couplings. To understand the constraints due to the ℓ 0 ! ℓγ decays, one expresses their branching ratios (BRs) such as with C μe ≈ 1, C τe ≈ 0:1784, and C τμ ≈ 0:1736. a R ð Þ ab is written as where Ð dX ½ Ð dxdydz 1 À x À y À z ð Þ , a L ð Þ ab can be obtained from a R ð Þ ab by using F † αβ ab instead of F αβ À Á ab , and the function F kk is given by Note that Vk 3b ≈ k 3b and Vỹ 3a ≈ỹ 3a are due to V ub, cb ≪V tb ≈ 1. 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 m t . Other LQ contributions are suppressed by m ℓ due to the chirality flip in the external lepton legs, and thus they are ignored. Based on Eq. (37), the muon g À 2 can be obtained as As mentioned earlier, the singlet LQ does not contribute to b ! sℓ 0 þ ℓ À at the tree level, but it can induce the b ! sνν process, where the current upper bound is B þ ! K þ νν < 1:6 Â 10 À5 , and the SM result is around 4 Â 10 À6 . Thus, B þ ! K þ νν can bound the parameters ofỹ 3iỹ2i . The four-Fermi interaction structure, which is induced by the LQ, is the same as that induced by the W-boson, so we can formulate the BR for B þ ! K þ νν as where x t ¼ m 2 t =m 2 W and X x t ð Þ can be parameterized as X x t ð Þ≈ 0:65x 0:575 t [41]. According to Eq. (31), the LQs also contribute to B s ! μ þ μ À process, where the BRs measured by LHCb [42] and prediction in the SM [43] are BR B s ! μ þ μ À ð Þ exp ¼ 3:0 AE 0:6 þ0:3 À0:2 À Á Â 10 À9 and BR B s ! μ þ μ À ð Þ SM ¼ 3:65 AE 0:23 ð ÞÂ10 À9 , respectively. The experimental data are consistent with the SM prediction, and in order to consider the constraint from B s ! μ þ μ À , we use the expression for the BR as [44].
In addition to the B À ! D * ð Þ τν decay, the induced effective Hamiltonian in Eq. (30) also contributes to the B c ! τν process, where the allowed upper limit is BR B À c ! τν À Á < 30% [45]. According to previous results given by [45], we express the BR for B c ! τν as where f B c is the B c decay constant, and the ε L, P in our model is given as Using τ B c ≈ 0:507 Â 10 À12 s, m B c ≈ 6:275 GeV, f B c ≈ 0:434 GeV [46], and V cb ≈ 0:04, the SM result is BR SM B c ! τν τ ð Þ≈ 2:1%. One can see that the effects of the new physics can enhance the B c ! τν τ decay by a few factors at most in our analysis.

ð Þ
The observables of R D * ð Þ and R K * ð Þ are the branching fraction ratios that are insensitive to the hadronic effects giving clearer test of lepton universality in Bmeson 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 B ! P by where P can be the D q ¼ c ð Þor K q ¼ s ð Þmeson and the momentum transfer is given by q ¼ p 1 À p 2 . For the B ! V decay where V is a vector meson, the transition form factors associated with the weak currents are parameterized such that ð Þε μνρσ σ ρσ , and ε μ 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., i∂ μ qγ For numerical estimations, the q 2 -dependent form factors F þ , F T , V, A 0 , and T 1 are taken as [47] and the other form factors are taken to be The values of f 0 ð Þ, σ 1 , and σ 2 for each form factor are summarized in Table 2. A detailed discussion of the form factors can be referred to [47]. The next-to-next-leading (NNL) effects obtained with the LCQCD Some Rule approach for the B ! D form factors were described by [48].
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 B decay processes, which we use for estimating R D * ð Þ and R K . For the B ! Dℓ 0 ν ℓ 0 decay, the differential decay rate as a function of the invariant mass q 2 can be given by  (46) and (47).
where the X ℓ 0 α n o functions and LQ contributions are We note that the effective couplings C ℓ 0 S and C ℓ 0 T at the m b scale can be obtained from the LQ mass scale via the renormalization group (RG) equation. Our numerical analysis considers the RG running effects with $ 2:0 at the m b scale [49]. The B ! D * ℓ 0 ν ℓ 0 decays involve D * polarizations and more complicated transition form factors, so the differential decay rate determined by summing all of the D * helicities are where λ D * is found in Eq. (53) and the detailed V For the B ! Kℓ þ ℓ À decays, the differential decay rate can be expressed as [50].
From Eq. (52), the measured ratio R K in the range q 2 ¼ q 2 min ; q 2 max Â Ã ¼ 1; 6 ½ GeV 2 can be estimated by (53) R K * is similar to R K , and thus we only show the result for R K .

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 R D * ð Þ and R K * ð Þ 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: muon g À 2 : k 32k23 ,ỹ 32 w 32 ; R K : k 3ℓ k 2ℓ , y 3ℓ y 2ℓ ; The parameters related to the radiative LFV, ΔB ¼ 2, and B þ ! K þ νν processes are defined as μ ! eγ : k 32k13 ,k 23 k 31 ,ỹ 32 w 31 , w 32ỹ 31 ; τ ! ℓ a γ : k 33ka3 ,k 33 k 3a ,ỹ 33 w 3a , w 33ỹ 3a ; B þ ! K þ νν :ỹ 3iỹ2i , y 3i y 2i ; B s ! μ þ μ À : k 32 k 22 , y 32 y 22 ; where z 3i z 2i ¼ k 3i k 2i , y 3i y 2i ,ỹ 3iỹ2i . From Eqs. (54) and (55), we can see that in order to avoid the μ ! eγ and τ ! ℓγ constraints and obtain a sizable and positive Δa μ , we can set (k 13, 33 , k 31, 33 , w 3i ) as a small value. From the upper limit of B þ ! K þ νν, we obtainỹ 3iỹ2i < 0:03, and thus the resulting Δm B s 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 k ij ≈k ji ≈ |y ij |, where the sign of y ij can be selected to obtain the correct sign for C LQ, ℓ j 9 and to decrease the value of C LQ, μ 10 so that B s ! μ þ μ À can fit the experimental data. As mentioned above, to avoid the bounds from the K, B d , and D systems, we also adopt k 1ℓ 0 ≈k ℓ 0 1 ≈ y 1i ≈ỹ 1i ≈ w 1i $ 0. When we omit these small coupling constants, the correlations of the parameters in Eqs. (54) and (55) can be further simplified as muon g À 2 : k 32k23 ; R K : k 32 k 22 , y 32 y 22 ; R D * ð Þ : k 32 k 22 , y 32 y 22 ,ỹ 3ℓ 0 w 2ℓ 0 ; B s ! μ þ μ À : k 32 k 22 , y 32 y 22 ; Δm B s : k 32 k 22 ð Þ 2 , y 32 y 22 whereỹ 3iỹ2i are ignored due to the constraint from B þ ! K þ νν. The typical values of these parameters for fitting the anomalies in the b ! sμ þ μ À decay are y 32 k 32 ð Þ, y 22 k 22 ð Þ $ 0:07, so the resulting Δm B s is smaller than the current data, but these parameters are too small to explain R D * ð Þ . Thus, we must depend on the singlet LQ to resolve the R D and R D * excesses, where the main free parameters are now y 3ℓ 0 w 2ℓ 0 .
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 m LQ to denote the mass of all LQs. From Eqs. (37), (39), and (56), we can see that the muon g À 2 depends only on k 32k 23 and m Φ . Here we illustrate Δa μ as a function of k 32k 23 in Figure 2(a), where the solid, dashed, and dotted lines denote the results for m Φ ¼ 1:5, 5, and 10 TeV, respectively, and the band is the experimental value with 1σ errors. Due to the m t enhancement, k 32k 23 $ 0:05 with m Φ $ 1 TeV can explain the muon g À 2 anomaly.
According to the relationships shown in Eq. (56), R K , B s ! μ þ μ À , and Δm B s depend on the same parameters, i.e., k 32 k 22 and y 32 y 22 . We show the contours for these observables as a function of k 32 k 22 and y 32 y 22 in Figure 2(b), where the data with 1σ errors and m LQ ¼ 1:5 TeV are taken for all LQ masses. Based on these results, we see that Δm B s < Δm exp B s in the range of |k 32 k 22 |, |y 32 y 22 | < 0:05, where R K and BR B s ! μ þ μ À ð Þcan both fit the experimental data simultaneously. In addition, we show C LQ , μ 9 ¼ À1:5; À0:5 ½ in the same plot. We can see that C LQ , μ 9 $ À1, which is used to explain the angular observable P 0 5 , can also be achieved in the same common region. According to Figure 2(b), the preferred values of k 32 k 22 and y 32 y 22 where the observed R K and B s ! μ þ μ À and the C LQ, μ 9 ¼ À1:5; À0:5 ½ overlap are around k 32 k 22 ; y 32 y 22 À Á $ À0:001; 0:004 ð Þ and $ 0:025; 0:03 ð Þ . The latter values are at the percentage level, but they are still not sufficiently large to explain the treedominated R D and R D * anomalies.
After studying the muon g À 2 and R K anomalies, we numerically analyze the ratio of BR B ! D * ð Þ τν τ À Á to BR B ! D * ð Þ ℓν ℓ À Á , i.e., R D * ð Þ. The introduced doublet and triplet LQs cannot efficiently enhance R D * ð Þ , 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 R D and R D * as a function of y 33 w 23 andỹ 32 w 22ỹ 31 w 21 Þ À in Figure 3(a) and (b), where the horizontal dashed and vertical dotted lines in both plots denote BR exp B À ! D ℓν ℓ ; τν τ ½ ð Þ ¼2:27AE ½ 0:11; 0:77 AE 0:25% and BR exp B À ! D * ℓν ℓ ; τν τ ½ ð Þ ¼5:69 AE 0:19; 1:88 AE 0:20 ½ %, respectively, and m LQ ¼ 1:5 TeV is used, and the data with 2σ errors are taken. For simplicity, we takeỹ 31 w 21 ≈ỹ 32 w 22 . When considering the limits from BR B ! D * ð Þ ℓ 0 ν ℓ 0 À Á , we obtain the limits |ỹ 3ℓ w 2ℓ |≤1:5 andỹ 33 w 23 >0. In order to clearly demonstrate the influence of tensor-type interactions, we also calculate the situation by setting C ℓ 0 T ¼ 0. The contours obtained for R D and R D * are shown in Figure 3(c) and (d), where the solid and dashed lines denote the cases with and without C ℓ 0 T , respectively. According to these plots, we can see that R D and R D * have different responses to the tensor operators, where the latter is more sensitive to the tensor interactions. R D and R D * can be explained simultaneously with the tensor couplings. In order to understand the correlation between BR B c ! τν τ ð Þand R D * ð Þ, we show the contours for BR B c ! τν τ ð Þand R D * ð Þ as a function of w 23ỹ 33 and m S in Figure 4, whereỹ 32 w 22 ≈ỹ 31 w 21 ≈ 0 are used, and the gray area is excluded by BR B À c ! τν À Á < 0:3. We can see that the predicted BR B c ! τν τ ð Þis 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 S 1=3 Yukawa couplingsỹ tτ , y bν τ , and w cτ can be of O 1 ð Þ. These couplings affect the S 1=3 decays but also their production. Therefore, in addition to the S 1=3 -pair production, based on the O 1 ð Þ Yukawa couplings, the single S 1=3 production becomes interesting. In the pp collisions, the single S 1=3 production can be generated via the gb ! S À1=3 ν τ and gc ! S À1=3 τ þ channels. Using CalcHEP 3.6 [51,52] with the CTEQ6 parton distribution functions [53], their production cross sections with |w 23 | $ |ỹ bν τ | $ ffiffi ffi 2 p and m LQ ¼ 1000 GeV at ffiffi s p ¼ 13 TeV can be obtained as 3.9 fb and 2.9 fb, respectively, whereas the S 1=3 -pair production cross section is σ pp ! S À1=3 S 1=3 ≈ 2:4 fb. If we assume that S À1=3 predominantly decays into tτ, bν τ , and cτ with similar BRs, i.e.
BR S À1=3 ! f $ 1=3, then the single S 1=3 production cross section σ S À1=3 X times BR S À1=3 ! f with X and f as the possible final states can be estimated as around 1 fb. The LQ coupling w 23 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 [54] based on a single production of the secondgeneration scalar LQ, we find that the values of σ Â BR at m LQ $ 1000 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.

Conclusions
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 B-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.