Open access peer-reviewed chapter

# Flavor Physics and Charged Particle

By Takaaki Nomura

Submitted: May 23rd 2018Reviewed: September 10th 2018Published: November 5th 2018

DOI: 10.5772/intechopen.81404

## Abstract

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.

### Keywords

• 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],

ΔaμΔaμexpΔaμth=28.8±8.0×1010,E1

where aμ=g2μ/2. This difference reaches to 3.6σdeviation from the prediction. In addition, new charged particles are introduced when we try to explain anomalies in B-meson decay like BKμ+μand 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 SU2Ldoublet Higgs fields are introduced:

H1=H1+v1+ϕ1+iη1/2,H2=H2+v2+ϕ2+iη2/2,E2

where v1,2is the vacuum expectation values (VEVs) of Higgs fields. In general, one can write Yukawa interaction in terms of Higgs doublet fields as

LY=Q¯LY1dDRH1+Q¯LY2dDRH2+Q¯LY1uURH˜1+Q¯LY2uURH˜2+L¯Y1RH1+L¯Y2RH2+H.c.,E3

where all flavor indices are hidden, PRL=1±γ5/2; QLi=uLidLiand LL=νLieLiare the SU2Lquark and lepton doublets with flavor index i, respectively; fR(f=U,D,) denotes the SU2Lsinglet fermion; Y1,2fare the 3×3Yukawa matrices; and H˜i=iτ2Hiwith τ2being 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

h=sαϕ1+cαϕ2,H=cαϕ1+sαϕ2,H±A=sβϕ1±η1+cβϕ2±η2,E4

where ϕiηiand 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 viare the vacuum expectation values (VEVs) of Hiand v=v12+v22246GeV. In our notation, his 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

LYH±=2d¯LV1vtβmu+XusβuRH+2u¯LVtβvmd+XdcβdRH++2ν¯Ltβvm+XcβRH++H.c.,E5

where Vis the CKM matrix and the matrix Xfis defined by original Yukawa coupling and unitary matrix diagonalizing fermion mass

Xu=VLuY1u2VRu,Xd=VLdY2d2VRd,X=VLY22VR.E6

A doubly charged scalar boson also appears from SU2Ltriplet scalar field:

Δ=δ+/2δ++vΔ+δ0+iη0/2δ+/2,E7

where vΔis 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

LY=hijLLiTCiσ2ΔLLj+h.c.,E8

where LLi=νiiLTwith flavor index iand C=iγ2γ0is the Dirac charge conjugation operator. In terms of the components, the Yukawa interaction can be expanded as

LY=hij12iLTCδ+νjLiLTCδ++jL+νiLTCδ0νjL12νiLTCδ+jL+hji12νiLCδjL+iLCδjLνiLCδ0νjL+12iLCδνjLE9

where C=Cis 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 SU2Lsinglet. The Yukawa couplings associated with charged scalar fields are given by

LY=fijL¯Liciσ2LLjh++geee¯RceRk+++gije¯RiceRjk+++h.c.,E10

where fijis antisymmetric under flavor indices. These Yukawa interactions can be used to generate neutrino mass with the nontrivial interaction in scalar potential:

Vμk++hh+c.c..E11

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×U1Ygauge symmetry can be singlet, doublet, and triplet under SU2L. In order to avoid the stringent constraints from rare Zi±jdecays, we here consider the triplet representations 131and 1,3,0with hypercharges Y=1and Y=0, respectively. The new Yukawa couplings thus can be written such that

LY=L¯Y1Ψ1RH+L¯Y2Ψ2RH˜+mΨ1TrΨ¯1LΨ1R+mΨ2TrΨ¯2LΨ2R+H.c.,E12

where we have suppressed the flavor indices; His the SM Higgs doublet field, H˜=iτ2Hand the neutral component of Higgs field is H0=v+h/2. The representations of two VLLs are

Ψ1=Ψ1/2Ψ10Ψ1Ψ1/2,Ψ2=12Ψ20/2Ψ2+Ψ2Ψ20/2,E13

with Ψ2+=CΨ¯2and Ψ20=CΨ¯20. Since Ψ2is a real representation of SU2L, the factor of 1/2in Ψ2is 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 5×5matrices and are expressed by

M=mYv0mΨ,Mν=mνYνv0mΨ,E14

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

Y=12Y11Y21Y12Y22Y13Y23,Yν=2Y11Y21/2Y12Y22/2Y13Y23/2.E15

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

To diagonalize Mand Mν, the unitary matrices VR,Lχwith χ=,νso that Mχdia=VLχMχVRχare introduced. The information of VLχand VRχcan be obtained from MχMχand MχMχ, respectively. According to Eq. (14), it can be found that the flavor mixings between heavy and light leptons in VRχare 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=0in our phenomenological analysis, only τ-related processes have significant contributions among them. Unlike VRχ, the off-diagonal elements in flavor-mixing matrices VLχare associated with Y1,2v/mΨ. In principle, the mixing effects can be of the order of 0.1without 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 Y1,2as

VLχ13×3εLχεLχ/2εLχεLχ13×3εLχεLχ/2,VR13×3εRεR13×3,E16

where VRν1is 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

Lh=h¯LVLm/vY00VRR+H.c.,E17

where T=e,μ,τ,ττis the state of a physical charged lepton in lepton flavor space. We use the notations of τand τto 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

Lhℓℓ=Cijh¯iLjRh+H.c.,Cijh=mjvδij38v2Y1iY1jmΨ12+v2Y2iY2jmΨ22.E18

If one sets me=mμ=0, it is clear that in addition to the coupling hττbeing modified, the tree-level flavor-changing couplings h- τ- μand h- τ- eare also induced, and the couplings are proportional to mτ/v7.2×103. In order to study the VLL contributions to hγγ, the couplings for hττand hττare expressed as

LhΨΨ=viY1i22mΨ1hττ+viY2i22mΨ2hττ.E19

### 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

LVLTQY=Q¯LY1F1RH˜+Q¯LY2F2RH+y˜1TrF¯1LF1RS+y˜2TrF¯2LF2RS+MF1TrF¯1LF1R+MF2TrF¯2LF2R+h.c.,E20

where QLis 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 F12is the 2×2VLTQ with hypercharge 2/31/3. The representations of F1,2in SU2Lare expressed in terms of their components as follows:

F1=U1/2XD1U1/2,F2=D2/2U2YD2/2.E21

The electric charges of U1,2, D1,2, X, and Yare found to be 2/3, 1/3, 5/3, and 4/3, respectively. Therefore, U1,2D1,2could mix with up (down) type SM quarks. Here MF12is 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 5×5mass matrices for up and down type quarks are found by

Mu=mudia3×3vY1/2vY2/202×3mF2×2,Md=mddia3×3vY1/2vY2/202×3mF2×2,E22

where mudia3×3and mddia3×3denote the diagonal mass matrices of SM quarks and diamF2×2=mF1mF2. Notice that a non-vanished vscould 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 D1,2are not physical states; thus one has to diagonalize Muand Mdto 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=11/3under SU2LU1YSM gauge symmetry, where the doublet and triplet representations can be taken as

Φ7/6=ϕ5/3ϕ2/3,Δ1/3=δ1/3/2δ4/3δ2/3δ1/3/2,E23

where the superscripts are the electric charges of the particles. Accordingly, the LQ Yukawa couplings to the SM fermions are expressed as

LLQ=u¯VkPRϕ5/3+d¯kPRϕ2/3+¯k˜PRuϕ5/3+ν¯k˜PRuϕ2/3+uc¯VyPLνδ2/312uc¯VyPLδ1/312dc¯yPLνδ1/3dc¯yPLδ4/3,+uc¯Vy˜PLdc¯y˜PLν+uc¯wPRS1/3+h.c.,E24

where the flavor indices are hidden, VULuULddenotes the Cabibbo-Kobayashi-Maskawa (CKM) matrix, ULu,dare the unitary matrices used to diagonalize the quark mass matrices, and ULdand URuhave 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×U1Yrepresentations 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 h→τ+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 ττmode is the most significant one, and thus we estimate the influence on hτ+τ. Using the values that satisfy BRhμτ104, the deviation of Γhτ+τfrom the SM prediction can be obtained as

κττΓhτ+τΓSMhτ+τ=16v2Y328mΨ220.88.E25

If the SM Higgs production cross section is not changed, the signal strength for pphτ+τin 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 ττchannel are still large, the precision measurement of μττcan test the effect or give strict limits on the parameters.

#### 3.1.2 τ→μγprocess 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 hμτcan provide contribution to Δaμthrough 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 Δaμis thus suppressed by the factor of mμ2mτ/vmh2so that the value of Δaμis two orders of magnitude smaller than current data Δaμ=aμexpaμSM=28.8±8.0×1010[2]. A similar situation happens in τ3μdecay also. Since the couplings are suppressed by mτ/vand mμ/v, the BR for τ3μis of the order of 1014. We also estimate the process τμγvia the h-mediation. The effective interaction for τμγis expressed by

Lτμγ=e16π2mτμ¯σμνCLPL+CRPRτFμν,E26

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

CRC23hC33h2mh2lnmh2mτ243.E27

Accordingly, the BR for τμγis expressed as

BRτμγBRτeν¯eντ=3αe4πGF2CR2.E28

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

### 3.2 B-meson flavor physics with leptoquarks

This section is based on Ref. [32]. Several interesting excesses in semileptonic Bdecays have been observed in experiments such as (i) the angular observable P5of BKμ+μ[7], where a 3σdeviation due to the integrated luminosity of 3.0 fb 1was found at the LHCb [8, 9], and the same measurement with a 2.6σdeviation 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±0.024and 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 2.5σdeviation 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˜i2and y˜3iw2jare SP×SP, and those from y3iy2jand y˜3iy˜2jare SP×S+P, where Sand Pdenote the scalar and pseudoscalar currents, respectively. Taking the Fierz transformations, the Hamiltonian for the bcν¯decays can be expressed as follows:

Hbc=y˜3iw2j2mS2+k3jk˜i22mΦ2c¯PLb¯jPLνi+y˜3iw2j2mS2+k3jk˜i22mΦ214c¯σμνPLb¯jσμνPLνiaV2ayajy3i4mΔ2c¯γμPLb¯jγμPLνi+aV2ay˜ajy˜3i2mS2c¯γμPLb¯jγμPLνi,E30

where the indices i,jare 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×SPstructure 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 RDand RDexcesses. Note that it is difficult to explain RD,Dby only using the doublet or/and triplet LQs when the RKexcess 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 ϕ2/3and δ4/3can be expressed as

Hbs=k3jk2j2mΦ2s¯γμPLb¯jγμPRj,y3jy2j2mΔ2s¯γμPLb¯jγμPLj,E31

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

Hbs=GFαemVtbVts2πH1μLμ+H2μL5μ,E32

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

H1μ=C9s¯γμPLb2mbq2C7s¯iσμνqνPRb,H2μ=C10s¯γμPLb.E33

The effective Wilson coefficients with LQ contributions are expressed as

C910=C910SM+C910LQ,,C9LQ,j=14cSMk3jk2jmΦ2y3jy2jmΔ2,C10LQ,j=14cSMk3jk2jmΦ2+y3jy2jmΔ2,E34

where cSM=VtbVtsαemGF/2πand Vijis the CKM matrix element. From Eq. (34), it can be seen that when the magnitude of C10LQ,jis decreased, C9LQ,jcan be enhanced. That is, the synchrony of the increasing/decreasing Wilson coefficients of C9NPand C10NPfrom 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+μdecay is satisfied, we can have sizable values of C9LQ,μto fit the anomalies of RKand 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 C7is mainly from the SM contributions.

#### 3.2.2 Constraints from ΔF=2, radiative lepton flavor violating, B+→K+νν¯, Bs→μ+μ−, and Bc→τνprocesses

Before we analyze the muon g2, RD, and RKproblems, we examine the possible constraints due to rare decay processes. Firstly, we discuss the strict constraints from the ΔF=2processes, such as FF¯oscillation, where Fdenotes the neutral pseudoscalar meson. Since KK¯, DD¯, and BdB¯dmixings 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˜1w1i0without affecting the analyses of RDand RK. Thus, the relevant ΔF=2process is BsB¯smixing, where ΔmBs=2B¯sHBsis induced from box diagrams, and the LQ contributions can be formulated as

ΔmBsCbox4π254i=13y3iy2imΔ2+i=13k3ik2imΦ2+Cbox4π2i=13y˜3iy˜2imS2+2i=13y3iy˜2ii=13y˜3iy2imS2mΔ2lnmSmΔ,E35

where Cbox=mBsfBs2/3, fBs0.224GeV is the decay constant of Bs-meson [40], and the current measurement is ΔmBsexp=1.17×1011GeV [2]. To satisfy the RKexcess, the rough magnitude of LQ couplings is y3iy2ik3ik2i5×103. Using our parameter values, it can be shown that the resulting ΔmBsagree with the current experimental data. However, ΔmBscan 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<4.2×1013and BRτeμγ<3.34.4×108[2], and they can strictly constrain the LQ couplings. To understand the constraints due to the γdecays, one expresses their branching ratios (BRs) such as

BRbaγ=48π3αemCbaGF2mb2aRab2+aLab2E36

with Cμe1, Cτe0.1784, and Cτμ0.1736. aRabis written as

aRab34π2dXmtFkk˜Fwy˜ab,E37

where dXdxdydzδ1xyz, aLabcan be obtained from aRabby using Fαβabinstead of Fαβab, and the function Fkk˜is given by

Fkk˜ab=Vk3bk˜a353xΔmtmΦab+231xΔmΦmtab,Fwy˜ab=w3bVy˜3a13xΔmtmSab+231xΔmSmtab,Δm1m2abxm12+y+zm22.E38

Note that Vk3bk3band Vy˜3ay˜3aare 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 mdue to the chirality flip in the external lepton legs, and thus they are ignored. Based on Eq. (37), the muon g2can be obtained as

ΔaμmμaL+aRa=b=μ.E39

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

BRB+K+νν¯131r2BRSMB+K+νν¯,E40
r=1CSMνy˜3y˜22mS2+y3y24mΔ2,CSMν=GFVtbVts2αem2πsin2θWXxt,E41

where xt=mt2/mW2and Xxtcan be parameterized as Xxt0.65xt0.575[41]. According to Eq. (31), the LQs also contribute to Bsμ+μprocess, where the BRs measured by LHCb [42] and prediction in the SM [43] are BRBsμ+μexp=3.0±0.60.2+0.3×109and 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].

BRBsμ+μBRBsμ+μSM=10.24C10LQ,μ2.E42

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

BRBcτν¯τ=τBcmBcmτ2fBc2GF2Vcb28π1mτ2mBc221+εL+mBc2mτmb+mcεP2,E43

where fBcis the Bcdecay constant, and the εL,Pin our model is given as

εL=24GFVcbaV2aya3y334mΔ2+aV2ay˜a3y˜332mS2,εP=24GFVcby˜33w232mS2k33k˜322mΦ2.

Using τBc0.507×1012s, mBc6.275GeV, fBc0.434GeV [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¯τdecay by a few factors at most in our analysis.

#### 3.2.3 Observables: RD∗and RK∗

The observables of RDand RKare 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¯Pby

Pp2qγμbB¯p1=F+q2p1+p2μmB2mP2q2qμ+mB2mP2q2qμF0q2,Pp2qσμνbB¯p1=ip1μp2νp1νp2μ2FTq2mB+mP,E44

where Pcan be the Dq=cor Kq=smeson and the momentum transfer is given by q=p1p2. For the BVdecay where Vis a vector meson, the transition form factors associated with the weak currents are parameterized such that

Vp2εq¯γμbB¯p1=iεμνρσενp1ρp2σ2Vq2mB+mV,Vp2εq¯γμγ5bB¯p1=2mVA0q2εqq2qμ+mB+mVA1q2εμεqq2qμA2q2εqmB+mVp1+p2μmB2mV2q2qμ,Vp2εq¯σμνbB¯p1=εμνρσερp1+p2σT1q2+ερqσmB2mV2q2T2q2T1q2+2εqq2p1ρp2σT2q2T1q2+q2mB2mV2T3q2,E45

where V=DKwhen q=cs, ε0123=1, σμνγ5=i/2εμνρσσρσ, and εμis 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¯band iμq¯γμγ5b=mb+mqq¯γ5b. For numerical estimations, the q2-dependent form factors F+, FT, V, A0, and T1are taken as [47]

fq2=f01q2/M21σ1q2/M2+σ2q4/M4,E46

and the other form factors are taken to be

fq2=f01σ1q2/M2+σ2q4/M4.E47

The values of f0, σ1, and σ2for 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 BDform factors were described by [48].

BDBD
F+F0FTVA0A1A2T1T2T3
f(0)0.670.670.690.760.690.660.620.680.680.33
σ10.570.780.560.570.580.781.400.570.641.46
σ20.41
BKBK
f(0)0.360.360.350.440.450.360.320.390.390.27
σ10.430.700.430.450.460.641.230.450.721.31
σ20.270.360.380.620.41

### Table 2.

BP,Vtransition 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 Bdecay processes, which we use for estimating RDand RK. For the B¯Dν¯decay, the differential decay rate as a function of the invariant mass q2can be given by

dΓDdq2=GF2Vcb2λD256π3mB31m2q22232+m2q2X+2+2m2q2X0+q2mXS2+16231+2m2q2XT2mq2XTX0,E48

where the Xαfunctions and LQ contributions are

X+=λD1+CVF+q2,X0=mB2mD21+CVF0q2XS=mB2mD2mbmcCSq2F0q2,XT=q2λDmB+mDCTFTq2CV=28GFVcbaV2ay˜3y˜amS2y3ya2mΔ2,CS=24GFVcby˜3w22mS2k3k˜22mΦ2,CT=216GFVcby˜3w22mS2+k3k˜22mΦ2,λH=mB4+mH4+q42mB2mH2+mH2q2+q2mB2.E49

We note that the effective couplings CSand CTat the mbscale 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=OTeV2.0at the mbscale [49]. The B¯Dν¯decays involve Dpolarizations and more complicated transition form factors, so the differential decay rate determined by summing all of the Dhelicities are

dΓDdq2=h=L,+,dΓDhdq2=GF2Vcb2λD256π3mB31m2q22h=L,+,VDhq2,E50

where λDis found in Eq. (53) and the detailed VDhfunctions are shown in the appendix. According to Eqs. (48) and (50), RM(M=D,D) can be calculated by

RM=mτ2qmax2dq2dΓMτ/dq2m2qmax2dq2dΓM/dq2E51

where qmax2=mBmM2and ΓM=ΓMe+ΓMμ/2. For the BK+decays, the differential decay rate can be expressed as [50].

dΓKℓℓq2dq2cSM2mB3328π31q2mB23/2×C9F+q2+2mbC7mB+mKFTq22+C10F+q22.E52

From Eq. (52), the measured ratio RKin the range q2=qmin2qmax2=16GeV2can be estimated by

RK=qmin2qmax2dq2dΓKμμ/dq2qmin2qmax2dq2dΓKee/dq2.E53

RKis 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 RDand RKanomalies can fit the experimental data. Before presenting the numerical analysis, we summarize the relevant parameters, which are related to the specific measurements as follows:

muong2:k32k˜23,y˜32w32;RK:k3k2,y3y2;RD:k3k˜2,aV2ay3yay˜3y˜a,y˜3w2.E54

The parameters related to the radiative LFV, ΔB=2, and B+K+νν¯processes are defined as

μ:k32k˜13,k˜23k31,y˜32w31,w32y˜31;τaγ:k33k˜a3,k˜33k3a,y˜33w3a,w33y˜3a;B+K+νν¯:y˜3iy˜2i,y3iy2i;Bsμ+μ:k32k22,y32y22;ΔmBs:iz3iz2i2,iy3iy˜2iiy˜3iy2i,E55

where z3iz2i=k3ik2i,y3iy2i,y˜3iy˜2i. From Eqs. (54) and (55), we can see that in order to avoid the μand τγconstraints 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 ΔmBsis 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 yijcan be selected to obtain the correct sign for C9LQ,jand to decrease the value of C10LQ,μso that Bsμ+μcan fit the experimental data. As mentioned above, to avoid the bounds from the K, Bd, and Dsystems, 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

muong2:k32k˜23;RK:k32k22,y32y22;RD:k32k22,y32y22,y˜3w2;Bsμ+μ:k32k22,y32y22;ΔmBs:k32k222,y32y222,E56

where y˜3iy˜2iare ignored due to the constraint from B+K+νν¯. The typical values of these parameters for fitting the anomalies in the bsμ+μdecay are y32k32,y22k220.07, so the resulting ΔmBsis 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 RDand RDexcesses, 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 mLQto denote the mass of all LQs. From Eqs. (37), (39), and (56), we can see that the muon g2depends only on k32k˜23and mΦ. Here we illustrate Δaμas a function of k32k˜23in Figure 2(a), where the solid, dashed, and dotted lines denote the results for mΦ=1.5, 5, and 10TeV, respectively, and the band is the experimental value with 1σerrors. Due to the mtenhancement, k32k˜230.05with mΦ1 TeV can explain the muon g2anomaly.

According to the relationships shown in Eq. (56), RK, Bsμ+μ, and ΔmBsdepend on the same parameters, i.e., k32k22and y32y22. We show the contours for these observables as a function of k32k22and y32y22in Figure 2(b), where the data with 1σerrors and mLQ=1.5TeV are taken for all LQ masses. Based on these results, we see that ΔmBs<ΔmBsexpin the range of k32k22, y32y22<0.05, where RKand BRBsμ+μcan both fit the experimental data simultaneously. In addition, we show C9LQ,μ=1.50.5in 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 k32k22and y32y22where the observed RKand Bsμ+μand the C9LQ,μ=1.50.5overlap are around k32k22y32y220.0010.004and 0.0250.03. The latter values are at the percentage level, but they are still not sufficiently large to explain the tree-dominated RDand RDanomalies.

After studying the muon g2and RKanomalies, we numerically analyze the ratio of BRB¯Dτν¯τto 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 RDand RDas a function of y˜33w23and y˜32w22y˜31w21in 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=1.5TeV is used, and the data with 2σerrors are taken. For simplicity, we take y˜31w21y˜32w22. When considering the limits from BRB¯Dν¯, we obtain the limits y˜3w21.5and 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 RDand RDare 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 RDand RDhave different responses to the tensor operators, where the latter is more sensitive to the tensor interactions. RDand RDcan be explained simultaneously with the tensor couplings. In order to understand the correlation between BRBcτν¯τand RD, we show the contours for BRBcτν¯τand RDas a function of w23y˜33and mSin Figure 4, where y˜32w22y˜31w210are used, and the gray area is excluded by BRBcτν<0.3. We can see that the predicted BRBcτν¯τis 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 S1/3Yukawa couplings y˜, y˜bντ, and wcan be of O1. These couplings affect the S1/3decays but also their production. Therefore, in addition to the S1/3-pair production, based on the O1Yukawa couplings, the single S1/3production becomes interesting. In the ppcollisions, the single S1/3production can be generated via the gbS1/3ν¯τand gcS1/3τ+channels. Using CalcHEP 3.6 [51, 52] with the CTEQ6 parton distribution functions [53], their production cross sections with w23y˜bντ2and mLQ=1000GeV at s=13TeV can be obtained as 3.9 fb and 2.9 fb, respectively, whereas the S1/3-pair production cross section is σppS1/3S1/32.4fb. If we assume that S1/3predominantly decays into , bντ, and with similar BRs, i.e. BRS1/3f1/3, then the single S1/3production cross section σS1/3Xtimes BRS1/3fwith Xand fas the possible final states can be estimated as around 1fb. The LQ coupling w23involves 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×BRat mLQ1000GeV 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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© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Takaaki Nomura (November 5th 2018). Flavor Physics and Charged Particle, Charged Particles, Malek Maaza and Mahmoud Izerrouken, IntechOpen, DOI: 10.5772/intechopen.81404. Available from:

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