Open Access is an initiative that aims to make scientific research freely available to all. To date our community has made over 100 million downloads. It’s based on principles of collaboration, unobstructed discovery, and, most importantly, scientific progression. As PhD students, we found it difficult to access the research we needed, so we decided to create a new Open Access publisher that levels the playing field for scientists across the world. How? By making research easy to access, and puts the academic needs of the researchers before the business interests of publishers.
We are a community of more than 103,000 authors and editors from 3,291 institutions spanning 160 countries, including Nobel Prize winners and some of the world’s most-cited researchers. Publishing on IntechOpen allows authors to earn citations and find new collaborators, meaning more people see your work not only from your own field of study, but from other related fields too.
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
chapter and author info
Author
Takaaki Nomura*
School of Physics, Korea Institute for Advanced Study, Seoul, Republic of Korea
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−2μ, which shows a long-standing discrepancy between experimental observations [1, 2] and theoretical predictions [3, 4, 5, 6],
Δaμ≡Δaμexp−Δaμth=28.8±8.0×10−10,E1
where aμ=g−2μ/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 B→K∗μ+μ−and B→D∗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.
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
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,ℓ) denotes the SU2Lsinglet fermion; Y1,2fare the 3×3Yukawa matrices; and H˜i=iτ2Hi∗with 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+v22≈246GeV. 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
where Vis the CKM matrix and the matrix Xfis defined by original Yukawa coupling and unitary matrix diagonalizing fermion mass
Xu=VLuY1u2VRu†,Xd=VLdY2d2VRd†,Xℓ=VLℓY2ℓ2VRℓ†.E6
A doubly charged scalar boson also appears from SU2Ltriplet scalar field:
Δ=δ+/2δ++vΔ+δ0+iη0/2−δ+/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=νiℓiLTwith flavor index iand C=iγ2γ0is the Dirac charge conjugation operator. In terms of the components, the Yukawa interaction can be expanded as
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
where fijis antisymmetric under flavor indices. These Yukawa interactions can be used to generate neutrino mass with the nontrivial interaction in scalar potential:
V⊃μk++h−h−+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 Z→ℓi±ℓj∓decays, we here consider the triplet representations 13−1and 1,3,0with hypercharges Y=−1and Y=0, respectively. The new Yukawa couplings thus can be written such that
where we have suppressed the flavor indices; His 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
Ψ1=Ψ1−/2Ψ10Ψ1−−−Ψ1−/2,Ψ2=12Ψ20/2Ψ2+Ψ2−−Ψ20/2,E13
with Ψ2+=CΨ¯2−and 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
Mℓ=mℓYℓv0mΨ,Mν=mνYνv0mΨ,E14
where the basis is chosen such that the SM lepton mass matrices are in diagonalized form, mℓis the SM charged lepton mass matrix, mΨ=diagmΨ1mΨ2, and
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 Mℓand Mν, the unitary matrices VR,Lχwith χ=ℓ,νso that Mχdia=VLχMχVRχ†are introduced. The information of VLχand VRχcan be obtained from MχMχ†and Mχ†Mχ, 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 VRν≈1. 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 h→τμ. To be more specific, we choose parametrization that the unitary matrices in terms of Y1,2as
where VRν≈1is used in our approximation, εLχ≈vYχ/mΨ, and εRℓ≈vmℓ†Yℓ/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
−Lhℓ′ℓ′=hℓ¯L′VLℓmℓ/vYℓ00VRℓ†ℓR′+H.c.,E17
where ℓ′T=e,μ,τ,τ′τ′′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 hττbeing modified, the tree-level flavor-changing couplings h- τ- μand h- τ- eare also induced, and the couplings are proportional to mτ/v≈7.2×10−3. In order to study the VLL contributions to h→γγ, the couplings for hτ′τ′and hτ′′τ′′are expressed as
−LhΨΨ=v∑iY1i22mΨ1hτ′τ′+v∑iY2i22mΨ2hτ″τ″.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
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 2/3−1/3. The representations of F1,2in SU2Lare expressed in terms of their components as follows:
F1=U1/2XD1−U1/2,F2=D2/2U2Y−D2/2.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
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 vs≪v, 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 vY1,2i≪mF1,2, 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=δ1/3/2δ4/3δ−2/3−δ1/3/2,E23
where the superscripts are the electric charges of the particles. Accordingly, the LQ Yukawa couplings to the SM fermions are expressed as
where the flavor indices are hidden, V≡ULuULd†denotes 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 type
SU3SU2U1Y
Examples of application
Charged scalar
1,3,1, 1,2,1/2
Neutrino mass, lepton flavor violation
Vector-like lepton
13−1, 1,3,0
Lepton flavor violation
Vector-like quark
3,3,2/3, 33−1/3
Quark flavor physics
Scalar leptoquark
3,2,7/6, 3,3,1/3, 3,1,1/3
Meson 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 h→τ+τ−. Using the values that satisfy BRh→μτ≈10−4, the deviation of Γh→τ+τ−from the SM prediction can be obtained as
κττ≡Γh→τ+τ−ΓSMh→τ+τ−=1−6v2Y328mΨ22≈0.88.E25
If the SM Higgs production cross section is not changed, the signal strength for pp→h→τ+τ−in our estimation is μττ≈0.88, where the measurements from ATLAS and CMS are 1.44−0.37+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 g−2, 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 mℓj/vℓ¯LiℓRj; 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=aμexp−aμSM=28.8±8.0×10−10[2]. A similar situation happens in τ→3μdecay also. Since the couplings are suppressed by mτ/vand mμ/v, the BR for τ→3μis of the order of 10−14. We also estimate the process τ→μγvia the h-mediation. The effective interaction for τ→μγis expressed by
Lτ→μγ=e16π2mτμ¯σμνCLPL+CRPRτFμν,E26
where CL=0and the Wilson coefficient CRfrom the one loop is obtained as
CR≈C23hC33h2mh2lnmh2mτ2−43.E27
Accordingly, the BR for τ→μγis expressed as
BRτ→μγBRτ→eν¯eντ=3αe4πGF2CR2.E28
We present the contours for BRτ→μγas a function of coupling Yand mΨin Figure 1, where the numbers on the plots are in units of 10−12. It can be seen that the resultant BRτ→μγcan be only up to 10−12, where the current experimental upper bound is BRτ→μγ<4.4×10−8[2].
Figure 1.
Contours for BRτ→μγ (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 Bdecays have been observed in experiments such as (i) the angular observable P5′of B→K∗μ+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:
where ℓ=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 RD≈0.3[36, 37] and RD∗≈0.25, respectively. Further tests of lepton flavor universality can be made using the branching fraction ratios: RK∗=BRB→K∗μ+μ−/BRB→K∗e+e−. The current LHCb measurements are RK=0.745−0.074+0.090±0.036[38] and RK∗=0.69−0.07+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 g−2), where its latest measurement is Δaμ=aμexp−aμSM=28.8±8.0×10−10[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 b→cℓ′ν¯ℓ′and b→sℓ′+ℓ′−decays. For the b→cℓ′ν¯ℓ′processes, the induced current-current interactions from k3jk˜i2and y˜3iw2jare S−P×S−P, and those from y3iy2jand y˜3iy˜2jare S−P×S+P, where Sand Pdenote the scalar and pseudoscalar currents, respectively. Taking the Fierz transformations, the Hamiltonian for the b→cℓ′ν¯ℓ′decays can be expressed as follows:
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 S−P×S−Pstructure as well as a tensor structure. However, the singlet LQ can produce currents of V−A×V−A, S−P×S−P, and tensor structures. Nevertheless, we show later that the singlet LQ makes the main contribution to the RDand RD∗excesses. Note that it is difficult to explain RD,D∗by 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 b→sℓ′+ℓ′−decays mediated by ϕ2/3and δ4/3can 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′+ℓ′−decays is written as
Hb→s=GFαemVtbVts∗2πH1μLμ+H2μL5μ,E32
where the leptonic currents are denoted by Lμ5=ℓ¯γμγ5ℓ, and the related hadronic currents are defined as
where cSM=VtbVts∗αemGF/2πand Vijis the CKM matrix element. From Eq. (34), it can be seen that when the magnitude of C10LQ,ℓjis decreased, C9LQ,ℓjcan 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,ℓ′can be different from that of C10LQ,ℓ′. As a result, when the constraint from Bs→μ+μ−decay is satisfied, we can have sizable values of C9LQ,μto fit the anomalies of RKand angular observable in B→K∗μ+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 g−2, RD∗, and RK∗problems, we examine the possible constraints due to rare decay processes. Firstly, we discuss the strict constraints from the ΔF=2processes, such as F−F¯oscillation, where Fdenotes the neutral pseudoscalar meson. Since K−K¯, D−D¯, and Bd−B¯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 k1ℓ′≈k˜ℓ′1≈y1ℓ′≈y˜1ℓ′≈w1i≈0without affecting the analyses of RD∗and RK∗. Thus, the relevant ΔF=2process is Bs−B¯smixing, where ΔmBs=2∣B¯sHBs∣is induced from box diagrams, and the LQ contributions can be formulated as
where Cbox=mBsfBs2/3, fBs≈0.224GeV is the decay constant of Bs-meson [40], and the current measurement is ΔmBsexp=1.17×10−11GeV [2]. To satisfy the RK∗excess, the rough magnitude of LQ couplings is ∣y3iy2i∣∼∣k3ik2i∣∼5×10−3. 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 b→cℓ′ν¯ℓ′decays.
In addition to the muon g−2, the introduced LQs can also contribute to the lepton flavor violating processes ℓ′→ℓγ, where the current upper bounds are BRμ→eγ<4.2×10−13and BRτ→eμγ<3.34.4×10−8[2], and they can strictly constrain the LQ couplings. To understand the constraints due to the ℓ′→ℓγdecays, one expresses their branching ratios (BRs) such as
BRℓb→ℓaγ=48π3αemCbaGF2mℓb2aRab2+aLab2E36
with Cμe≈1, Cτe≈0.1784, and Cτμ≈0.1736. aRabis written as
aRab≈34π2∫dXmtFkk˜−Fwy˜ab,E37
where ∫dX≡∫dxdydzδ1−x−y−z, aLabcan be obtained from aRabby using Fαβ†abinstead of Fαβab, and the function Fkk˜is given by
Note that Vk3b≈k3band Vy˜3a≈y˜3aare due to Vub,cb≪Vtb≈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 mt. Other LQ contributions are suppressed by mℓdue to the chirality flip in the external lepton legs, and thus they are ignored. Based on Eq. (37), the muon g−2can be obtained as
Δaμ≃−mμaL+aRa=b=μ.E39
As mentioned earlier, the singlet LQ does not contribute to b→sℓ′+ℓ−at the tree level, but it can induce the b→sνν¯process, where the current upper bound is B+→K+νν¯<1.6×10−5, and the SM result is around 4×10−6. 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/mW2and Xxtcan be parameterized as Xxt≈0.65xt0.575[41]. According to Eq. (31), the LQs also contribute to Bs→μ+μ−process, where the BRs measured by LHCb [42] and prediction in the SM [43] are BRBs→μ+μ−exp=3.0±0.6−0.2+0.3×10−9and BRBs→μ+μ−SM=3.65±0.23×10−9, respectively. The experimental data are consistent with the SM prediction, and in order to consider the constraint from Bs→μ+μ−, we use the expression for the BR as [44].
BRBs→μ+μ−BRBs→μ+μ−SM=1−0.24C10LQ,μ2.E42
In addition to the B−→D∗τν¯decay, the induced effective Hamiltonian in Eq. (30) also contributes to the Bc→τν¯process, where the allowed upper limit is BRBc−→τν¯<30%[45]. According to previous results given by [45], we express the BR for Bc→τν¯as
Using τBc≈0.507×10−12s, mBc≈6.275GeV, fBc≈0.434GeV [46], and Vcb≈0.04, the SM result is BRSMBc→τν¯τ≈2.1%. One can see that the effects of the new physics can enhance the Bc→τν¯τdecay 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¯→Pby
where Pcan be the Dq=cor Kq=smeson and the momentum transfer is given by q=p1−p2. For the B→Vdecay where Vis a vector meson, the transition form factors associated with the weak currents are parameterized such that
where V=D∗K∗when 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., i∂μq¯γμb=mb−mqq¯band i∂μq¯γμγ5b=−mb+mqq¯γ5b. For numerical estimations, the q2-dependent form factors F+, FT, V, A0, and T1are taken as [47]
fq2=f01−q2/M21−σ1q2/M2+σ2q4/M4,E46
and the other form factors are taken to be
fq2=f01−σ1q2/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 B→Dform factors were described by [48].
B→D
B→D∗
F+
F0
FT
V
A0
A1
A2
T1
T2
T3
f(0)
0.67
0.67
0.69
0.76
0.69
0.66
0.62
0.68
0.68
0.33
σ1
0.57
0.78
0.56
0.57
0.58
0.78
1.40
0.57
0.64
1.46
σ2
0.41
B→K
B→K∗
f(0)
0.36
0.36
0.35
0.44
0.45
0.36
0.32
0.39
0.39
0.27
σ1
0.43
0.70
0.43
0.45
0.46
0.64
1.23
0.45
0.72
1.31
σ2
0.27
0.36
0.38
0.62
0.41
Table 2.
B→P,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 RD∗and RK. For the B¯→Dℓ′ν¯ℓ′decay, the differential decay rate as a function of the invariant mass q2can be given by
We note that the effective couplings CSℓ′and CTℓ′at 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 CSℓ′/CTℓ′μ=mb/CSℓ′/CTℓ′μ=OTeV∼2.0at the mbscale [49]. The B¯→D∗ℓ′ν¯ℓ′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 VD∗ℓ′hfunctions are shown in the appendix. According to Eqs. (48) and (50), RM(M=D,D∗) can be calculated by
RM=∫mτ2qmax2dq2dΓMτ/dq2∫mℓ2qmax2dq2dΓMℓ/dq2E51
where qmax2=mB−mM2and ΓMℓ=ΓMe+ΓMμ/2. For the B→Kℓ+ℓ−decays, the differential decay rate can be expressed as [50].
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:
where z3iz2i=k3ik2i,y3iy2i,y˜3iy˜2i. From Eqs. (54) and (55), we can see that in order to avoid the μ→eγand τ→ℓγconstraints 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 kij≈k˜ji≈∣yij∣, where the sign of yijcan be selected to obtain the correct sign for C9LQ,ℓjand to decrease the value of C10LQ,μso that Bs→μ+μ−can fit the experimental data. As mentioned above, to avoid the bounds from the K, Bd, and Dsystems, we also adopt k1ℓ′≈k˜ℓ′1≈y1i≈y˜1i≈w1i∼0. 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˜2iare ignored due to the constraint from B+→K+νν¯. The typical values of these parameters for fitting the anomalies in the b→sμ+μ−decay are y32k32,y22k22∼0.07, 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 RD∗excesses, where the main free parameters are now y˜3ℓ′w2ℓ′.
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 g−2depends 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˜23∼0.05with mΦ∼1 TeV can explain the muon g−2anomaly.
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, Bs→μ+μ−, ΔmBs, and C9LQ,μ as a function of k32k22 and y32y22, where the ranges of RK and Bs→μ+μ− are the experimental values with 1σ errors and mLQ=1.5 TeV. For C9LQ,μ, we show the range for CLQ,μ=−1.5−0.5. (These plots are taken from Ref. [32]).
According to the relationships shown in Eq. (56), RK, Bs→μ+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 BRBs→μ+μ−can both fit the experimental data simultaneously. In addition, we show C9LQ,μ=−1.5−0.5in the same plot. We can see that C9LQ,μ∼−1, 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 Bs→μ+μ−and the C9LQ,μ=−1.5−0.5overlap are around k32k22y32y22∼−0.0010.004and ∼0.0250.03. The latter values are at the percentage level, but they are still not sufficiently large to explain the tree-dominated RDand RD∗anomalies.
After studying the muon g−2and RKanomalies, we numerically analyze the ratio of BRB¯→D∗τν¯τto BRB¯→D∗ℓν¯ℓ, 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 RD∗as 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 BRexpB−→Dℓν¯ℓτν¯τ=2.27±0.110.77±0.25%and BRexpB−→D∗ℓνℓτν¯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˜31w21≈y˜32w22. When considering the limits from BRB¯→D∗ℓ′ν¯ℓ′, we obtain the limits ∣y˜3ℓw2ℓ∣≤1.5and y˜33w23>0. In order to clearly demonstrate the influence of tensor-type interactions, we also calculate the situation by setting CTℓ′=0. The contours obtained for RDand RD∗are shown in Figure 3(c) and (d), where the solid and dashed lines denote the cases with and without CTℓ′, respectively. According to these plots, we can see that RDand RD∗have different responses to the tensor operators, where the latter is more sensitive to the tensor interactions. RDand RD∗can be explained simultaneously with the tensor couplings. In order to understand the correlation between BRBc→τν¯τand RD∗, we show the contours for BRBc→τν¯τand RD∗as a function of w23y˜33and mSin Figure 4, where y˜32w22≈y˜31w21≈0are used, and the gray area is excluded by BRBc−→τν<0.3. We can see that the predicted BRBc→τν¯τis 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+→D∗ℓνℓ, whereas the vertical dotted lines are the BRexpB+→D∗τντ. 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 BRBc→τν¯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 S1/3Yukawa couplings y˜tτ, y˜bντ, and wcτcan 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 gb→S−1/3ν¯τand gc→S−1/3τ+channels. Using CalcHEP 3.6 [51, 52] with the CTEQ6 parton distribution functions [53], their production cross sections with ∣w23∣∼∣y˜bντ∣∼2and 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 σpp→S−1/3S1/3≈2.4fb. If we assume that S−1/3predominantly decays into tτ, bντ, and cτwith similar BRs, i.e. BRS−1/3→f∼1/3, then the single S1/3production cross section σS−1/3Xtimes BRS−1/3→fwith 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 mLQ∼1000GeV 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.
We are IntechOpen, the world's leading publisher of Open Access books. Built by scientists, for scientists. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals. We share our knowledge and peer-reveiwed research papers with libraries, scientific and engineering societies, and also work with corporate R&D departments and government entities.
