Open access peer-reviewed chapter

Flavor Physics and Charged Particle

Written By

Takaaki Nomura

Reviewed: 10 September 2018 Published: 05 November 2018

DOI: 10.5772/intechopen.81404

From the Edited Volume

Charged Particles

Edited by Malek Maaza and Mahmoud Izerrouken

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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
  • leptoquarks
  • charged scalar boson

1. 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, g2μ, which shows a long-standing discrepancy between experimental observations [1, 2] and theoretical predictions [3, 4, 5, 6],


where aμ=g2μ/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 BKμ+μ and BDτν [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

Singly charged scalar appears from two-Higgs doublet model (2HDM) [17, 18] in which two SU2L doublet Higgs fields are introduced:


where v1,2 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, PRL=1±γ5/2; QLi=uLidLi and LL=νLieLi are the SU2L quark and lepton doublets with flavor index i, respectively; fR (f=U,D,) denotes the SU2L singlet fermion; Y1,2f are the 3×3 Yukawa matrices; and H˜i=iτ2Hi with τ2 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 ϕiηi and ηi± denote the real (imaginary) parts of the neutral and charged components of Hi, respectively; cαsα=cosαsinα, cβ=cosβ=v1/v, sβ=sinβ=v2/v, and vi are the vacuum expectation values (VEVs) of Hi and v=v12+v22246 GeV. In our notation, h is the SM-like Higgs, while H, A, and H± 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 Xf is defined by original Yukawa coupling and unitary matrix diagonalizing fermion mass


A doubly charged scalar boson also appears from SU2L triplet scalar field:


where vΔ 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 LLi=νiiLT with flavor index i and C=iγ2γ0 is the Dirac charge conjugation operator. In terms of the components, the Yukawa interaction can be expanded as


where C=C 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±k±± which are SU2L singlet. The Yukawa couplings associated with charged scalar fields are given by


where fij 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. [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 SU2L×U1Y gauge symmetry can be singlet, doublet, and triplet under SU2L. In order to avoid the stringent constraints from rare Zi±j decays, we here consider the triplet representations 131 and 1,3,0 with hypercharges Y=1 and Y=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˜=iτ2H and the neutral component of Higgs field is H0=v+h/2. The representations of two VLLs are


with Ψ2+=CΨ¯2 and Ψ20=CΨ¯20. Since Ψ2 is a real representation of SU2L, the factor of 1/2 in Ψ2 is required to obtain the correct mass term for Majorana fermion Ψ20. Due to the new Yukawa terms of Y1,2, the heavy neutral and charged leptons can mix with the SM leptons, after electroweak symmetry breaking (EWSB). Then the lepton mass matrices become 5×5 matrices and are expressed by


where the basis is chosen such that the SM lepton mass matrices are in diagonalized form, m is the SM charged lepton mass matrix, mΨ= diagmΨ1mΨ2, and


Note that the elements of Yχ should be read as Yij=Yij, where the index i=1,2 distinguishes the Yukawa couplings of the different VLLs and the index j=1,2,3 stands for the flavors of the SM leptons.

To diagonalize M and Mν, the unitary matrices VR,Lχ with χ=,ν so that Mχdia=VLχMχVRχ are introduced. The information of VLχ and VRχ can be obtained from MχMχ and MχMχ, respectively. According to Eq. (14), it can be found that the flavor mixings between heavy and light leptons in VRχ are proportional to the lepton masses. Since the neutrino masses are tiny, it is a good approximation to assume VRν1. If one further sets me=mμ=0 in our phenomenological analysis, only τ-related processes have significant contributions among them. Unlike VRχ, the off-diagonal elements in flavor-mixing matrices VLχ are associated with Y1,2v/mΨ. In principle, the mixing effects can be of the order of 0.1 without conflict. In our example later, we examine these effects on hτμ. To be more specific, we choose parametrization that the unitary matrices in terms of Y1,2 as


where VRν1 is used in our approximation, εLχvYχ/mΨ, and εRvmY/mΨ2. 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 T=e,μ,τ,ττ is the state of a physical charged lepton in lepton flavor space. We use the notations of τ and τ 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 me=mμ=0, it is clear that in addition to the coupling hττ being modified, the tree-level flavor-changing couplings h- τ- μ and h- τ- e are also induced, and the couplings are proportional to mτ/v7.2×103. In order to study the VLL contributions to hγγ, the couplings for hττ and hττ are expressed as


2.3 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 QL 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˜=iτ2H, and F12 is the 2×2 VLTQ with hypercharge 2/31/3. The representations of F1,2 in SU2L are expressed in terms of their components as follows:


The electric charges of U1,2, D1,2, X, and Y are found to be 2/3, 1/3, 5/3, and 4/3, respectively. Therefore, U1,2D1,2 could mix with up (down) type SM quarks. Here MF12 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 5×5 mass matrices for up and down type quarks are found by


where mudia3×3 and mddia3×3 denote the diagonal mass matrices of SM quarks and diamF2×2=mF1mF2. Notice that a non-vanished vs could shift the masses of VLTQs. Since vsv, we neglect the small effects hereafter. Due to the presence of Y1,2, the SM quarks, U1,2, and D1,2 are not physical states; thus one has to diagonalize Mu and Md to get the mass eigenstates in general. If vY1,2imF1,2, we expect that the off-diagonal elements of unitary matrices for diagonalizing the mass matrices should be of order of vY1,2i/mF1,2. By adjusting Y1,2i, 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 Φ7/6=27/6, Δ1/3=31/3, and S1/3=11/3 under SU2LU1Y 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, VULuULd denotes the Cabibbo-Kobayashi-Maskawa (CKM) matrix, ULu,d are the unitary matrices used to diagonalize the quark mass matrices, and ULd and URu have been absorbed into k, k˜, y, y˜, 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, 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].

Particle typeSU3SU2U1YExamples of application
Charged scalar1,3,1, 1,2,1/2Neutrino mass, lepton flavor violation
Vector-like lepton131, 1,3,0Lepton flavor violation
Vector-like quark3,3,2/3, 331/3Quark flavor physics
Scalar leptoquark3,2,7/6, 3,3,1/3, 3,1,1/3Meson decay, lepton flavor violation

Table 1.

List of examples of charged particles from new physics discussed in this review showing SU3×SU2×U1Y representations and applications to phenomenology.


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. [29].

3.1.1 Modification to hτ+τ 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 ττ mode is the most significant one, and thus we estimate the influence on hτ+τ. Using the values that satisfy BRhμτ104, the deviation of Γhτ+τ from the SM prediction can be obtained as


If the SM Higgs production cross section is not changed, the signal strength for pphτ+τ in our estimation is μττ0.88, where the measurements from ATLAS and CMS are 1.440.37+0.42 [33] and 0.91±0.27 [34], respectively. Although the current data errors for the ττ channel are still large, the precision measurement of μττ can test the effect or give strict limits on the parameters.

3.1.2 τμγ 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 g2, denoted by Δaμ. The lepton flavor-changing coupling hμτ can provide contribution to Δaμ through the Higgs-mediated loop diagrams. However, as shown in Eq. (18), the induced couplings are associated with mj/v¯LiRj; only the right-handed tau lepton has a significant contribution. The induced Δaμ is thus suppressed by the factor of mμ2mτ/vmh2 so that the value of Δaμ is two orders of magnitude smaller than current data Δaμ=aμexpaμSM=28.8±8.0×1010 [2]. A similar situation happens in τ3μ decay also. Since the couplings are suppressed by mτ/v and mμ/v, the BR for τ3μ is of the order of 1014. We also estimate the process τμγ via the h-mediation. The effective interaction for τμγ is expressed by


where CL=0 and the Wilson coefficient CR from the one loop is obtained as


Accordingly, the BR for τμγ is expressed as


We present the contours for BRτμγ as a function of coupling Y and mΨ in Figure 1, where the numbers on the plots are in units of 1012. It can be seen that the resultant BRτμγ can be only up to 1012, where the current experimental upper bound is BRτμγ<4.4×108 [2].

Figure 1.

Contours for BRτμγ (dashed) as a function of Y and mΨ, where the constraint from ΓinvZ (solid) is included. (The plot is taken from ref. [29]).

3.2 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 P5 of BKμ+μ [7], where a 3σ deviation due to the integrated luminosity of 3.0 fb 1 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 RD,D, which are defined and measured as follows:

RD=B¯DτνB¯Dlν={ 0.375±0.064±0.026Belle[ 11 ],0.440±0.058±0.042BaBar[ 12,13 ], RD=B¯DτνB¯Dlν={ 0.302±0.030±0.011Belle[ 14 ],0.270±0.035±0.025+0.028Belle[ 15 ],0.332±0.024±0.018BaBar[ 12,13 ],0.336±0.027±0.030LHCb[ 16 ], E29

where =eμ, and these measurements can test the violation of lepton flavor universality. The averaged results from the heavy flavor averaging group are RD=0.403±0.040±0.024 and RD=0.310±0.015±0.008 [35], and the SM predictions are around RD0.3 [36, 37] and RD0.25, respectively. Further tests of lepton flavor universality can be made using the branching fraction ratios: RK=BRBKμ+μ/BRBKe+e. The current LHCb measurements are RK=0.7450.074+0.090±0.036 [38] and RK=0.690.07+0.11±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 g2), where its latest measurement is Δaμ=aμexpaμSM=28.8±8.0×1010 [2]. These anomalies would be explained by introducing LQs and we review possible scenarios in the following.

3.2.1 Effective interactions for semileptonic B-decay

According to the interactions in Eq. (24), we first formulate the four-Fermi interactions for the bcν¯ and bs+ decays. For the bcν¯ processes, the induced current-current interactions from k3jk˜i2 and y˜3iw2j are SP×SP, and those from y3iy2j and y˜3iy˜2j are SP×S+P, where S and P denote the scalar and pseudoscalar currents, respectively. Taking the Fierz transformations, the Hamiltonian for the bcν¯ 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 SP×SP structure as well as a tensor structure. However, the singlet LQ can produce currents of VA×VA, SP×SP, and tensor structures. Nevertheless, we show later that the singlet LQ makes the main contribution to the RD and RD excesses. Note that it is difficult to explain RD,D by only using the doublet or/and triplet LQs when the RK excess and other strict constraints are satisfied at the same time.

With the Yukawa couplings in Eq. (24), the effective Hamiltonian for the bs+ decays mediated by ϕ2/3 and δ4/3 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 bs+ decays is written as


where the leptonic currents are denoted by Lμ5=¯γμγ5, and the related hadronic currents are defined as


The effective Wilson coefficients with LQ contributions are expressed as


where cSM=VtbVtsαemGF/2π and Vij is the CKM matrix element. From Eq. (34), it can be seen that when the magnitude of C10LQ,j is decreased, C9LQ,j can be enhanced. That is, the synchrony of the increasing/decreasing Wilson coefficients of C9NP and C10NP from new physics is diminished in this model. In addition, the sign of C9LQ, can be different from that of C10LQ,. As a result, when the constraint from Bsμ+μ decay is satisfied, we can have sizable values of C9LQ,μ to fit the anomalies of RK and angular observable in BKμ+μ. 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 C7 is mainly from the SM contributions.

3.2.2 Constraints from ΔF=2, radiative lepton flavor violating, B+K+νν¯, Bsμ+μ, and Bcτν processes

Before we analyze the muon g2, RD, and RK problems, we examine the possible constraints due to rare decay processes. Firstly, we discuss the strict constraints from the ΔF=2 processes, such as FF¯ oscillation, where F denotes the neutral pseudoscalar meson. Since KK¯, DD¯, and BdB¯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 k1k˜1y1y˜1w1i0 without affecting the analyses of RD and RK. Thus, the relevant ΔF=2 process is BsB¯s mixing, where ΔmBs=2B¯sHBs is induced from box diagrams, and the LQ contributions can be formulated as


where Cbox=mBsfBs2/3, fBs0.224 GeV is the decay constant of Bs-meson [40], and the current measurement is ΔmBsexp=1.17×1011 GeV [2]. To satisfy the RK excess, the rough magnitude of LQ couplings is y3iy2ik3ik2i5×103. Using our parameter values, it can be shown that the resulting ΔmBs agree with the current experimental data. However, ΔmBs can indeed constrain the parameters involved in the bcν¯ decays.

In addition to the muon g2, the introduced LQs can also contribute to the lepton flavor violating processes γ, where the current upper bounds are BRμ<4.2×1013 and BRτeμγ<3.34.4×108 [2], and they can strictly constrain the LQ couplings. To understand the constraints due to the γ decays, one expresses their branching ratios (BRs) such as


with Cμe1, Cτe0.1784, and Cτμ0.1736. aRab is written as


where dXdxdydzδ1xyz, aLab can be obtained from aRab by using Fαβab instead of Fαβab, and the function Fkk˜ is given by


Note that Vk3bk3b and Vy˜3ay˜3a are due to Vub,cbVtb1. 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 mt. Other LQ contributions are suppressed by m due to the chirality flip in the external lepton legs, and thus they are ignored. Based on Eq. (37), the muon g2 can be obtained as


As mentioned earlier, the singlet LQ does not contribute to bs+ at the tree level, but it can induce the bν¯ process, where the current upper bound is B+K+νν¯<1.6×105, and the SM result is around 4×106. Thus, B+K+νν¯ can bound the parameters of y˜3iy˜2i. 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+K+νν¯ as


where xt=mt2/mW2 and Xxt can be parameterized as Xxt0.65xt0.575 [41]. According to Eq. (31), the LQs also contribute to Bsμ+μ process, where the BRs measured by LHCb [42] and prediction in the SM [43] are BRBsμ+μexp=3.0±0.60.2+0.3×109 and BRBsμ+μSM=3.65±0.23×109, respectively. The experimental data are consistent with the SM prediction, and in order to consider the constraint from Bsμ+μ, we use the expression for the BR as [44].


In addition to the BDτν¯ decay, the induced effective Hamiltonian in Eq. (30) also contributes to the Bcτν¯ process, where the allowed upper limit is BRBcτν¯<30% [45]. According to previous results given by [45], we express the BR for Bcτν¯ as


where fBc is the Bc decay constant, and the εL,P in our model is given as


Using τBc0.507×1012 s, mBc6.275 GeV, fBc0.434 GeV [46], and Vcb0.04, the SM result is BRSMBcτν¯τ2.1%. One can see that the effects of the new physics can enhance the Bcτν¯τ decay by a few factors at most in our analysis.

3.2.3 Observables: RD and RK

The observables of RD and RK are the branching fraction ratios that are insensitive to the hadronic effects giving clearer test of lepton universality in B-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 B¯P by


where P can be the Dq=c or Kq=s meson and the momentum transfer is given by q=p1p2. For the BV decay where V is a vector meson, the transition form factors associated with the weak currents are parameterized such that


where V=DK when q=cs, ε0123=1, σμνγ5=i/2εμνρσσρσ, and εμ 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., iμq¯γμb=mbmqq¯b and iμq¯γμγ5b=mb+mqq¯γ5b. For numerical estimations, the q2-dependent form factors F+, FT, V, A0, and T1 are taken as [47]


and the other form factors are taken to be


The values of f0, σ1, and σ2 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 BD form factors were described by [48].


Table 2.

BP,V transition form factors, as parameterized in Eqs. (46) and (47).

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 RD and RK. For the B¯Dν¯ decay, the differential decay rate as a function of the invariant mass q2 can be given by


where the Xα functions and LQ contributions are


We note that the effective couplings CS and CT at the mb scale can be obtained from the LQ mass scale via the renormalization group (RG) equation. Our numerical analysis considers the RG running effects with CS/CTμ=mb/CS/CTμ=OTeV2.0 at the mb scale [49]. The B¯Dν¯ 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 λD is found in Eq. (53) and the detailed VDh functions are shown in the appendix. According to Eqs. (48) and (50), RM (M=D,D) can be calculated by


where qmax2=mBmM2 and ΓM=ΓMe+ΓMμ/2. For the BK+ decays, the differential decay rate can be expressed as [50].


From Eq. (52), the measured ratio RK in the range q2=qmin2qmax2=16 GeV2 can be estimated by


RK is similar to RK, and thus we only show the result for RK.

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 RD and RK 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, ΔB=2, and B+K+νν¯ processes are defined as


where z3iz2i=k3ik2i,y3iy2i,y˜3iy˜2i. From Eqs. (54) and (55), we can see that in order to avoid the μ and τγ constraints and obtain a sizable and positive Δaμ, we can set (k˜13,33, k31,33, w3i) as a small value. From the upper limit of B+K+νν¯, we obtain y˜3iy˜2i<0.03, and thus the resulting ΔmBs 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 kijk˜jiyij, where the sign of yij can be selected to obtain the correct sign for C9LQ,j and to decrease the value of C10LQ,μ so that Bsμ+μ can fit the experimental data. As mentioned above, to avoid the bounds from the K, Bd, and D systems, we also adopt k1k˜1y1iy˜1iw1i0. When we omit these small coupling constants, the correlations of the parameters in Eqs. (54) and (55) can be further simplified as


where y˜3iy˜2i are ignored due to the constraint from B+K+νν¯. The typical values of these parameters for fitting the anomalies in the bsμ+μ decay are y32k32,y22k220.07, so the resulting ΔmBs is smaller than the current data, but these parameters are too small to explain RD. Thus, we must depend on the singlet LQ to resolve the RD and RD excesses, where the main free parameters are now y˜3w2.

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 mLQ to denote the mass of all LQs. From Eqs. (37), (39), and (56), we can see that the muon g2 depends only on k32k˜23 and mΦ. Here we illustrate Δaμ as a function of k32k˜23 in Figure 2(a), where the solid, dashed, and dotted lines denote the results for mΦ=1.5, 5, and 10 TeV, respectively, and the band is the experimental value with 1σ errors. Due to the mt enhancement, k32k˜230.05 with mΦ 1 TeV can explain the muon g2 anomaly.

Figure 2.

(a) Δaμ as a function of k32k˜23 with mΦ=1.5,5,10 TeV, where the band denotes the experimental data with 1σ errors. (b) Contours for RK, Bsμ+μ, ΔmBs, and C9LQ,μ as a function of k32k22 and y32y22, where the ranges of RK and Bsμ+μ are the experimental values with 1σ errors and mLQ=1.5 TeV. For C9LQ,μ, we show the range for CLQ,μ=1.50.5. (These plots are taken from Ref. [32]).

According to the relationships shown in Eq. (56), RK, Bsμ+μ, and ΔmBs depend on the same parameters, i.e., k32k22 and y32y22. We show the contours for these observables as a function of k32k22 and y32y22 in Figure 2(b), where the data with 1σ errors and mLQ=1.5 TeV are taken for all LQ masses. Based on these results, we see that ΔmBs<ΔmBsexp in the range of k32k22, y32y22<0.05, where RK and BRBsμ+μ can both fit the experimental data simultaneously. In addition, we show C9LQ,μ=1.50.5 in the same plot. We can see that C9LQ,μ1, which is used to explain the angular observable P5, can also be achieved in the same common region. According to Figure 2(b), the preferred values of k32k22 and y32y22 where the observed RK and Bsμ+μ and the C9LQ,μ=1.50.5 overlap are around k32k22y32y220.0010.004 and 0.0250.03. The latter values are at the percentage level, but they are still not sufficiently large to explain the tree-dominated RD and RD anomalies.

After studying the muon g2 and RK anomalies, we numerically analyze the ratio of BRB¯Dτν¯τ to BRB¯Dν¯, i.e., RD(). The introduced doublet and triplet LQs cannot efficiently enhance RD, 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 RD and RD as a function of y˜33w23 and y˜32w22y˜31w21 in Figure 3(a) and (b), where the horizontal dashed and vertical dotted lines in both plots denote BRexpBDν¯τν¯τ=2.27±0.110.77±0.25% and BRexpBDντν¯τ=5.69±0.191.88±0.20%, respectively, and mLQ=1.5 TeV is used, and the data with 2σ errors are taken. For simplicity, we take y˜31w21y˜32w22. When considering the limits from BRB¯Dν¯, we obtain the limits y˜3w21.5 and y˜33w23>0. In order to clearly demonstrate the influence of tensor-type interactions, we also calculate the situation by setting CT=0. The contours obtained for RD and RD are shown in Figure 3(c) and (d), where the solid and dashed lines denote the cases with and without CT, respectively. According to these plots, we can see that RD and RD have different responses to the tensor operators, where the latter is more sensitive to the tensor interactions. RD and RD can be explained simultaneously with the tensor couplings. In order to understand the correlation between BRBcτν¯τ and RD, we show the contours for BRBcτν¯τ and RD as a function of w23y˜33 and mS in Figure 4, where y˜32w22y˜31w210 are used, and the gray area is excluded by BRBcτν<0.3. We can see that the predicted BRBcτν¯τ is much smaller than the experimental bound.

Figure 3.

Contours for (a) RD and (b) RD, where the solid lines denote the data with 1σ and 2σ errors, respectively. The horizontal dashed lines in both plots denote the BRexpB+Dν, whereas the vertical dotted lines are the BRexpB+Dτντ. Contours for (c) RD and (d) RD, where the solid and dashed lines denote the situations with and without tensor operator contributions, respectively. In this case, we take mLQ=1.5 TeV. (These plots are taken from ref. [32]).

Figure 4.

Contours for BRBcτν¯τ and RD as a function of w23y˜23 and mS. (The plot is taken from ref. [32]).

Finally, we make some remarks regarding the constraint due to the LQ search at the LHC. Due to the flavor physics constraints, only the S1/3 Yukawa couplings y˜, y˜bντ, and w can be of O1. These couplings affect the S1/3 decays but also their production. Therefore, in addition to the S1/3-pair production, based on the O1 Yukawa couplings, the single S1/3 production becomes interesting. In the pp collisions, the single S1/3 production can be generated via the gbS1/3ν¯τ and gcS1/3τ+ channels. Using CalcHEP 3.6 [51, 52] with the CTEQ6 parton distribution functions [53], their production cross sections with w23y˜bντ2 and mLQ=1000 GeV at s=13 TeV can be obtained as 3.9 fb and 2.9 fb, respectively, whereas the S1/3-pair production cross section is σppS1/3S1/32.4 fb. If we assume that S1/3 predominantly decays into , bντ, and with similar BRs, i.e. BRS1/3f1/3, then the single S1/3 production cross section σS1/3X times BRS1/3f with X and f as the possible final states can be estimated as around 1 fb. The LQ coupling w23 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 second-generation scalar LQ, we find that the values of σ×BR at mLQ1000 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.


4. 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.


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Written By

Takaaki Nomura

Reviewed: 10 September 2018 Published: 05 November 2018