aa r X i v : . [ h e p - ph ] S e p A model of spontaneous
C P breaking at low scale
Gauhar Abbas
Theoretical Physics Division, Physical Research Laboratory, Navrangpura, Ahmedabad380 009, India
Abstract
We introduce a CP symmetric model where masses of fermions are givenby dimensional-5 operators, and CP is spontaneously broken at TeV scale.The unique feature of the model is that new CP symmetric gauge sectorcoexists with new CP symmetric fermions simultaneously at TeV scale. Anultraviolet completion of the model is also proposed. It is observed thatthe fine-tuning of the SM Higgs boson mass in this model is softened in arelatively small amount approximately up to 6 TeV. Other interesting con-sequences are presence of a possible dark matter candidate whose mass maybe bounded from above by the SM Higgs mass. The model may also providean explanation of recently observed flavour anomalies. Keywords: CP symmetry, extension of the standard model CP -violation can distinguish matter and antimatter in an absolute andconvention-independent way. In this sense, CP -violation is more fundamen-tal ingredient of nature in comparison to P or C violation. Furthermore, itis essential to explain matter-antimatter asymmetry of Universe. However,the amount of CP -violation in the standard model (SM) is too small to ex-plain matter-antimatter asymmetry of Universe. The observation of a small CP -violation in the SM may be a sign that CP could be a good symmetryof a larger underlying theory, and only broken by the vacuum of that theory.There are many successful models in the literature where parity is sponta-neously broken at a high scale[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. However,models of spontaneous CP breaking are theoretically very different from themodels based on spontaneous parity breaking in the sense that there is no Email address: [email protected] (Gauhar Abbas)
Preprint submitted to Elsevier October 1, 2018 auge extension of the SM in these models[14, 15]. In these models, sponta-neous CP breaking is achieved through certain terms in the scalar potentialwhich depend on a common non-vanishing vacuum phase[15].The problematic feature of models of spontaneous CP -violation is theappearance of scalar-pseudoscalar mixing[16, 17]. This gives rise to one-loopcontribution to electric dipole moments of fermions which needs a carefulfine-tuning[18].In this paper, we discuss a model of spontaneous CP breaking inspiredby the model proposed in Ref.[19] which avoids scalar-pseudoscalar mixing.In the model discussed in Ref.[19], the SM gauge symmetry is extended to SU (3) ⊗ SU (2) L ⊗ SU (2) ′ L ⊗ U (1) Y ′ where SU (2) ′ L is the CP counter-partof the SM gauge group SU (2) L . The fermionic fields transformation under SU (3) ⊗ SU (2) L ⊗ SU (2) ′ L ⊗ U (1) Y ′ is given as, l L = νe ! L ∼ (1 , , , − , e R ∼ (1 , , , − , ν eR ∼ (1 , , , q L = ud ! L ∼ (3 , , ,
13 ) , u R ∼ (3 , , ,
43 ) , d R ∼ (3 , , , − l ′ L = ν ′ e ′ ! L ∼ (1 , , , − , e ′ R ∼ (1 , , , − , ν ′ eR ∼ (1 , , , q ′ L = u ′ d ′ ! L ∼ (3 , , ,
13 ) , u ′ R ∼ (3 , , ,
43 ) , d ′ R ∼ (1 , , , − l L , q L are the SM doublets of fermions, and e R , ν eR , d R and u R aresinglets under the SM. l ′ L , q ′ L , e ′ R , ν ′ eR , d ′ R and u ′ R are CP counter-parts of theSM fermions. The quantum numbers for second and third family fermionscan be defined exactly as discussed above for the first family.The transformation of fermionic fields under CP can be written as, ψ L CP → γ Cψ ′ LT , ψ R CP → γ Cψ ′ RT , (2)where ψ L is a doublet of the gauge groups SU (2) L , and ψ ′ L is a doublet ofthe gauge group SU (2) ′ L . ψ R and ψ ′ R are singlets under SU (2) L and SU (2) ′ L .In the model discussed in Ref.[19], the Yukawa Lagrangian, for instancefor electron and its CP counter-part, can be written as, L Y = Γ ¯ l L ϕ L e R + Γ ∗ ¯ l ′ L ϕ ′ L e ′ R + H . c ., (3)2here Γ is 3 × ϕ L is the scalar doublet Higgsfield charged under the SM gauge group SU (2) L and singlet under the gaugegroup SU (2) ′ L . Similarly, the scalar doublet Higgs field ϕ ′ L is charged underthe gauge group SU (2) ′ L and singlet under the SM gauge group SU (2) L .The Large Hadron Collider (LHC) has not found these new fermionsat TeV scale yet. Hence, mass of the lightest charged new fermion shouldbe at least at TeV. The mass of CP counter-part of the electron is m e ′ = m e h ϕ ′ L i / h ϕ L i where m e is mass of the electron, h ϕ L i = 246 GeV is thevacuum expectation value (VEV) of the SM Higgs field, and h ϕ ′ L i is VEVof the Higgs field which is CP counter-part of the SM Higgs field. Now, forelectron mass m e = 0 .
511 MeV, and for instance h ϕ ′ L i = 5 × GeV, themass of the e ′ fermion is 1038 .
65 GeV which could be searched at the LHC.Moreover, there are interaction terms between the SM fermions and their CP counter-parts. These interaction terms may bring down scale of h ϕ ′ L i slightly. However, since we need to recover small masses of the SM neutrinos( ≈ − GeV) without fine-tuning of the neutrino Yukawa couplings, weagain require a very high scale of order 10 − GeV or so for h ϕ ′ L i . In Ref.[19], neutrinos are treated as massless.The unusually high value of h ϕ ′ L i increases the masses of gauge bosonscorresponding to the gauge group SU (2) ′ L to 10 − GeV or so, thuscreating a disparity of scale in the gauge sector of the model. Thus, being soheavy, the new gauge sector is practically inaccessible to any experiment innear future.This problem also occurs in models based on mirror fermions and mirrorsymmetries[5, 20, 21], and is elegantly solved in Refs.[6, 22]. The modelpresented in this work is in fact inspired by the models discussed in Refs.[6,19, 22].The spontaneous symmetry breaking (SSB) in the model follows the pat-tern: SU (2) L ⊗ SU (2) ′ L ⊗ U (1) Y ′ → SU (2) L ⊗ U (1) Y → U (1) EM . (4)For achieving the SSB, we introduce two Higgs doublets which transform inthe following way under SU (3) c ⊗ SU (2) L ⊗ SU (2) ′ L ⊗ U (1) Y ′ : ϕ L = ϕ + ϕ ! L ∼ (1 , , , , ϕ ′ L = ϕ ′ + ϕ ′ ! L ∼ (1 , , , , (5)3nd behave under CP as follows: ϕ L ←→ ϕ ′∗ L (6)Since our aim is to have new gauge sector and new fermionic sector at TeVscale simultaneously, we add two real scalar singlets to provide masses tofermions which will be explained in the following discussion. The quantumnumbers of singlet scalar fields under SU (3) c ⊗ SU (2) L ⊗ SU (2) ′ L ⊗ U (1) Y ′ are, χ : (1 , , , , χ ′ : (1 , , , , (7)and they transform under CP as, χ ←→ χ ′ . (8)For lowering down the scale of spontaneous CP breaking such that new gaugebosons and fermions ψ ′ are simultaneously at TeV scale, we impose a pairof discrete symmetries Z and Z ′ on the fermionic fields ψ L , ψ ′ L and scalarsinglets χ , χ ′ as shown in Table 1. All other fields transform trivially underthese symmetries. Fields Z Z ′ ψ L + - χ + - ψ ′ L - + χ ′ - + Table 1: The charges of fermionic and singlet scalar fields under Z and Z ′ symmetries. The Yukawa Lagrangian is forbidden by the discrete symmetries Z and Z ′ , and masses of the first family leptons are given by the dimension-5 op-erator as, L mass = 1Λ h Γ ¯ l L ϕ L χe R + Γ ∗ ¯ l ′ L ϕ ′ L χ ′ e ′ R + Γ ¯ l L ˜ ϕ L χν e R + Γ ∗ ¯ l ′ L ˜ ϕ ′ L χ ′ ν ′ e R i (9)+ 1Λ h ρ ¯ l L ϕ L χe ′ R + ρ ∗ ¯ l L ′ ϕ ′ L χ ′ e R + κ ¯ ℓ L ˜ ϕ L χν ′ eR + κ ∗ ¯ l L ′ ˜ ϕ ′ L χ ′ ν eR i + H . c ., where ˜ ϕ L = iτ ϕ ∗ L , ˜ ϕ ′ L = iτ ϕ ′∗ L , τ is the second Pauli matrix, and Γ i ( i =1 , ρ and κ are 3 × eγe ′ e ′ Sh L Figure 1: The additional new contribution to the process µ → eγ where h L denotes theSM Higgs boson, S is the scalar particle corresponding to the field χ , and blob representsan effective interaction. be written for quarks of the first family and fermions of other families ingeneral.Since ρ is a 3 × µ → eγ . This additional newcontribution to this process, as shown in Fig.1, effectively occurs at twoloops, and hence highly suppressed in this model.The above Lagrangian provides one of the main features of the model.For keeping the mass of the gauge bosons corresponding to the gauge group SU (2) ′ L at TeV scale, the VEV of the field ϕ ′ L should be such that h ϕ ′ L i >> h ϕ L i = 246 GeV. For instance, if VEV of the field ϕ ′ L is such that h ϕ ′ L i = 1TeV which is much larger than the SM Higgs field VEV h ϕ L i , the massesof gauge bosons corresponding to the gauge group SU (2) ′ L should be around 1TeV. Now, mass of the CP counter-part of the electron is m e ′ = m e h ϕ ′ L ih χ ′ i / h ϕ L ih χ i .Thus, for having the lightest new charged lepton e ′ at TeV scale, the patternof the spontaneous breaking of CP should be such that h χ ′ i >> h ϕ ′ L i >> h ϕ L i and h χ ′ i >> h χ i . For illustration, h ϕ L i = 246 GeV, h ϕ ′ L i = 1 TeV, h χ i = 100 GeV, electron mass m e = 0 .
511 MeV and h χ ′ i = 5 × GeV,the mass of the lightest new charged lepton e ′ is 1038 .
62 GeV. The scale ofthe VEV h χ ′ i may be lower when terms having interactions among the SMfermions and their CP counter parts are included in the masses of fermions.5 igure 2: The fine tuning required to recover the 125 GeV Higgs mass in the SM with acut-off scale 6 TeV. The Majorana mass term for neutrinos is given as, L νMajorana = 1Λ h c ¯ l cL ˜ ϕ ∗ L ˜ ϕ † L l L + c ∗ ¯ l ′ cL ˜ ϕ ′∗ L ˜ ϕ ′† L l ′ L i (10)+ 1Λ h c ¯ l cL ˜ ϕ ∗ L ˜ ϕ ′† L l ′ L + c ∗ ¯ l ′ cL ˜ ϕ ′∗ L ˜ ϕ † L l L i + M ν TR C − ν R + M ∗ ν ′ TR C − ν ′ R + c ν TR C − ν ′ R + H . c . We observe that this model has an explanation for small neutrino massesusing type-I see-saw mechanism.The most general scalar potential of the model can be written as, V = − µ L ϕ † L ϕ L − µ ′ L ϕ ′† L ϕ ′ L − µ χ χ − µ χ ′ χ ′ + λ (cid:16) ( ϕ † L ϕ L ) + ( ϕ ′† L ϕ ′ L ) (cid:17) (11)+ λ ϕ † L ϕ L ϕ ′† L ϕ ′ L + λ (cid:16) χ + χ ′ (cid:17) + λ χ χ ′ + λ (cid:16) ϕ † L ϕ L χ + ϕ ′† L ϕ ′ L χ ′ (cid:17) + λ (cid:16) ϕ † L ϕ L χ ′ + ϕ ′† L ϕ ′ L χ (cid:17) . It should be noted that we have introduced soft CP breaking terms in thescalar potential which are essential to provide spontaneous CP breaking suchthat h χ ′ i = ω ′ / √ >> h ϕ ′ L i = v ′ L / √ >> h ϕ L i = v L / √ h χ ′ i >> h χ i = ω/ √
2. As discussed in Ref.[19], this model solves the strong CP problemnaturally. This conclusion still holds even after introducing real singlet scalarfields.This model mitigates the fine-tuning of the SM Higgs boson mass in arelatively small amount approximately up to 6 TeV by avoiding the one-loop6 L Stt tgg
Figure 3: The production of the scalar singlet S with the SM Higgs boson h L in the gluonfusion at the LHC. Here, blob shows an effective interaction. contribution to the SM Higgs boson mass due to the top quark. For being ona concrete ground, let us assume that the SM is valid up to a cutoff scale of 6TeV. Then, the main three quadratic contribution at one loop to the mass ofthe SM Higgs boson at the scale of 6 TeV are − y t Λ / π ≈ . GeV from the top quark loop, g Λ / π ≈ . GeV from the gauge loop,and λ Λ / π ≈ . GeV from the Higgs loop[23]. Thus, the approxi-mate mass of the SM Higgs boson is m h ≈ m tree − (87 . − . − .
76) (125GeV) .This is depicted in Fig.2. In order to recover the 125 GeV SM Higgs mass,approximately 1% fine-tuning is required. In the model discussed in this pa-per, one loop contribution due to top quark is absent. Moreover, the LHCdata is showing that this discovered Higgs boson is behaving like the SMHiggs boson. Hence, one loop contributions to its mass from the scalar fields ϕ ′ L , χ and χ ′ are expected to be small. Therefore, we can safely take the oneloop contribution to the mass of the SM Higgs in this model at the scale of 6TeV to be m h L ≈ m tree − ( − . − .
76) (125GeV) . Thus, we see that thereis relatively less fine-tuning of the SM Higgs boson mass in this model at thescale of 6 TeV. Moreover, it is possible that there is a further cancellationof the above quadratic divergences by the one loop contributions from thescalar fields ϕ ′ L , χ and χ ′ . This will be studied in future.We remark that for cancelling the quadratic divergence of the SM Higgsboson in the supersymmetric framework, one needs a supersymmetric particlein the vicinity of the discovered Higgs boson. However, there is no sign of sucha particle in the LHC run 1 or 2. Besides this, the decay B s → µ + µ − whichis particularly sensitive to supersymmetry, does not provide any evidence ofsuch a particle too[24, 25]. Hence, any alternative idea to deal with the finetuning of the SM Higgs mass is worth exploring.It should be noted that physical particles corresponding to the singletscalar fields χ and χ ′ are a mixture of scalar fields χ , χ ′ , ϕ L and ϕ ′ L after the7SB. The lighter physical scalar singlet particle S which can be mapped ontothe singlet scalar field χ could be a possible dark matter candidate if its massis less than mass of the discovered SM Higgs boson (125 GeV). The reasonis that being singlet under the whole gauge symmetry of the model, it canonly interact with fermions and the SM Higgs field through the Eq.(9), andother scalars of the model through couplings given in the scalar potential.We need to assume that scalar particles corresponding to scalar fields ϕ ′ L and χ ′ are heavier than the scalar particle corresponding to the singlet scalar field χ . Hence, the decay of the scalar particle, S , corresponding to the singletscalar field χ , which is a mixture of scalar fields χ , χ ′ , ϕ L and ϕ ′ L , mayoccur through its interaction with fermions and the SM Higgs boson givenby the Eq.(9). Thus if its mass is lighter than the mass of the discovered SMHiggs boson, its decay to any final states through the Eq.(9) is forbidden bykinematics, and this particle can decay only at loop level. Thus, there is aninbuilt upper bound on the mass of the possible dark matter candidate inthis model. This particle is testable at the LHC in the process of gluon fusion gg → h L S as shown in Fig.3. The physical scalar singlet particle S ′ whichcan be mapped onto the singlet scalar field χ ′ , being the heaviest particle, isexpected to decay into lighter particles. Hence, it is less probable a candidatefor dark matter. However, a thorough investigation is required in future.Besides this, the SM right handed neutrinos and new neutrinos ν ′ may bedark matter candidates. However, these neutrinos also interact with gaugesector. Hence, they are allowed to decay via a loop having a charged gaugeboson and a charged fermion to a neutrino and a photon: ν i → ν j γ where ν i,j is either the SM right handed neutrinos or new neutrinos ν ′ , and sub-scriptshows the conversion of one type neutrino to other type. Because of mixingof the SM fermions and their CP counter-parts, there will be additionalcontribution to this process having the SM and new ψ ′ fermions in the loop.Whether neutrinos are dark matter candidate, will be determined by the rateof the process ν i → ν j γ .The gauge interactions of the scalar fields are given by the following La-grangian: L GS = (cid:0) D µ,L ϕ L (cid:1) † (cid:0) D µL ϕ L (cid:1) + (cid:16) D ′ µ,L ϕ ′ L (cid:17) † (cid:0) D ′ µL ϕ ′ L (cid:1) , (12)where, the covariant derivatives are, D µ,L ( D ′ µ,L ) = ∂ µ + ig τ a W aµ,L ( W ′ aµ,L ) + ig ′ Y ′ B µ , (13)8here, τ a ’s are the Pauli matrices, and g corresponds to the common couplingof the gauge groups SU (2) L and SU (2) ′ L . The coupling constant of the gaugegroup U (1) Y ′ is g ′ .The masses of charged gauge bosons after the SSB are given by, M W ± L = 12 gv L , M W ′± L = 12 gv ′ L . (14)The mass matrix of the neutral gauge bosons in the basis ( W L , W ′ L , B )can be written as, M = 14 g v L − gg ′ v L g v ′ L − gg ′ v ′ L − gg ′ v L − gg ′ v ′ L g ′ ( v L + v ′ L ) . (15)The weak eigenstates of neutral gauge bosons ( W L , W ′ L , B ) can be con-verted into the physical mass eigenstates ( Z L , Z ′ L , γ ) through an orthogonaltransformation T given as, W L W ′ L B = T Z L Z ′ L γ . (16)The masses of the physical neutral gauge bosons are given as, M Z L = 14 v L g g + 2 g ′ g + g ′ " − g ′ ( g + g ′ ) ǫ , M Z ′ L = 14 v ′ L (cid:0) g + g ′ (cid:1) " g ′ ( g + g ′ ) ǫ , (17)where ǫ = v L /v ′ L , and terms of order O ( ǫ ) are ignored since v ′ L >> v L .We can parametrize the orthogonal matrix T in Eq.(16) in terms of themixing angle θ W L given as,cos θ W L = M W L M Z L ! ǫ =0 = g + g ′ g + 2 g ′ . (18)9hus, the transformation matrix T is given as, T = − cos θ W L − p cos2 θ W L tan θ W L cos θ W L ǫ sin θ W L sin θ W L tan θ W L h cos2 θ WL cos θ WL ǫ i − p cos2 θ W L cos θ W L (cid:2) − tan θ W L ǫ (cid:3) sin θ W L sin θ W L p cos2 θ W L cos θ W L h − tan θ WL cos θ WL ǫ i tan θ W L h θ W L cos2 θ WL cos θ WL ǫ i p cos2 θ W L . (19)It should be noted that there are further terms of order ǫ in the transfor-mation matrix T , however the third column of that matrix is unchanged bythose further terms.The following relations are obtained between couplings of the gauge sym-metries of the model and electric charge: g = e sin θ W L , g ′ = e p cos2 θ W L , e = 2 g + 1 g ′ . (20)Eqs.(9) and (10) provide masses of fermions. For instance, we can writethe Lagrangian for the down type quark and its CP counter-part as, L d = 1Λ (cid:0) Γ d ¯ q L ϕ L d R χ + Γ ∗ d ¯ q ′ L ϕ ′ L d ′ R χ ′ (cid:1) + 1Λ (cid:2) ρ d ¯ q L ϕ L χd ′ R + ρ ∗ d ¯ q L ′ ϕ ′ L χ ′ d R (cid:3) + H . c . = (cid:16) ¯ d L ¯ d ′ L (cid:17) Γ d v L ω ρ d v L ω ρ ∗ d v ′ L ω ′
2Λ Γ ∗ d v ′ L ω ′ d R d ′ R ! + H . c .. (21)The mass matrix in above equation is in general 6 × u L u ′ L ! = X u Y u ! u L u ′ L ! and d L d ′ L ! = X d Y d ! d L d ′ L ! , (22)where X u,d and Y u,d are 3 ×
6, and CKM matrices are given by V CKM = X † u X d and V ′ CKM = Y † u Y d .Now we turn our attention towards phenomenological signatures and con-sequences of this model. Due to mixing of the SM and new fermions, thecharged and neutral current Lagrangians allow new fermions ψ ′ to decay into10 g q ′ q ′ q ′ q Z L qZ L gg g q ′ q ′ qqZ L Z L q q g q ′ q ′ qqZ L Z L q ′ Figure 4: The pair production of new quarks at the LHC and their subsequent decay tothe SM Z L boson and a quark. W L , W ′ L W L , W ′ L q, q ′ ¯ q, ¯ q ′ q ¯ q ¯ qq Figure 5: New box diagrams contributing to K and B meson mixing. a SM W L or Z L boson in association with a SM quark. New fermions arecharged under the colour gauge group SU (3) c . Due to this, we can, for in-stance, produce new fermions ψ ′ via gluon-gluon and quark-antiquark initialstates as shown in Fig.4 at the LHC.There are flavour changing neutral current interactions at tree level inthis model. Hence, K and B mesons mixing, as shown in Fig.5, is expectedto place non-trivial constraints on the masses of the new gauge bosons andfermions. For instance, diagrams which may play a non-trivial role are theones which have a W L or W ′ L with new fermions in the box.A simple UV completion of this model can come from vector-like iso-singlet quarks and leptons. We observe that at least two vector-like isosin-glet quarks of up and down type, one iso-singlet vector-like charged lepton,one iso-singlet vector-like neutrino, and their CP -counterparts are sufficientto provide UV completion of this model. Their quantum numbers under SU (3) c ⊗ SU (2) L ⊗ SU (2) ′ L ⊗ U (1) Y ′ and discrete symmetries Z and Z ′ L h L Q q R S q ′ L h ′ L Q ′ q ′ R S ′ Figure 6: The explicit realization of the SM and new fermions ψ ′ in Eq.(9) by the UVcompletion of the model. Here, q L , q R and h L are the SM quarks and Higgs boson, q ′ L , q ′ R and h ′ L are CP counter parts, Q and Q ′ are vector-like iso-singlet quarks, S and S ′ arescalar particles corresponding to the scalar field χ and χ ′ . given as follows: Q = U L,R : (3 , , ,
43 ) + − ; D L,R : (3 , , , − + − , L = E L,R : (1 , , , − + − ; N L,R : (1 , , , + − , (23)and their CP -counter parts are, Q ′ = U ′ L,R : (3 , , ,
43 ) − + ; D ′ L,R : (3 , , , − − + , L ′ = E ′ L,R : (1 , , , − − + ; N ′ L,R : (1 , , , − + , (24)where charges of Z and Z ′ are given in sub-script.The mass terms for vector-like fermions are the following: L V = (cid:0) M U ¯ U L U R + M ∗ U ¯ U ′ L U ′ R (cid:1) + (cid:0) M D ¯ D L D R + M ∗ D ¯ D ′ L D ′ R (cid:1) (25)+ (cid:0) M E ¯ E L E R + M ∗ E ¯ E ′ L E ′ R (cid:1) + (cid:0) M N ¯ N L N R + M ∗ N ¯ N ′ L N ′ R (cid:1) + H . c .. The interactions of the vector-like fermions with the SM and mirror fermionsare given by, L ′ V ff ′ = (cid:2) y ′ ¯ q L ϕ L Q R + y ′∗ ¯ q ′ L ϕ ′ L Q ′ R (cid:3) + (cid:2) c ′ ¯ l L ϕ L L R + c ′∗ ¯ l ′ L ϕ ′ L L ′ R (cid:3) + H . c . (26)The interactions of singlet SM and mirror fermions with vector-like fermionsare described by the following Lagrangian: L ′′ V ff ′ = (cid:2) y ′′ ¯ q R Q L χ + y ′′∗ ¯ q ′ R Q ′ L χ ′ (cid:3) + (cid:2) c ′′ ¯ l R L L χ + c ′′∗ ¯ l ′ R L ′ L χ ′ (cid:3) + H . c . (27)The Eqs.(25), (26) and (27) provide a realization of the masses of the SMand mirror fermions given in Eq.(9) as shown in Fig.6.Now we discuss consequences of this model for recently observed anoma-lies in flavour physics. The first deviation is observed in B → Kll and12 d ¯ b ¯ sZ ′ L µ − µ + B K ∗ ¯ buB + W ′ L τνu ¯ c D ∗ Figure 7: The contribution of new heavy gauge bosons to B → K ∗ µ + µ − and B + → D ∗ τ ν at tree-level in the model discussed in this paper. B → K ∗ ll decays which proceed through b → sl + l − transition. The opti-mised observable P ′ [26] is showing a deviation of 3 . σ from the SM as mea-sured by the LHCb [27]. Moreover, the ratio R K = B B → Kµ + µ − / B B → Ke + e − measured by the LHCb is hinting towards lepton flavour universality (LFU)violation [28]. Furthermore, the ratio R K ∗ = B B → K ∗ µ + µ − / B B → K ∗ e + e − is re-cently measured by the LHCb, and this is also showing significant deviationfrom the SM prediction and lepton-flavour universality [29]. Moreover, thereis one more deviation from the SM expectation in the b → clν transitionhaving different final state leptons leading to the LFU violation in the ob-servable R D ∗ = Γ( B → D ∗ τ ν ) / Γ( B → D ∗ lν ). The most recent average ofthis observable is 4 σ away from the SM expectation [30].In this model, LFU violation is introduced via mixing of the SM fermionswith new fermions, and anomalies may be explained simultaneously due to acontribution coming from a new neutral and a charged gauge boson as shownin Fig.7. Anomalies in B → Kll and B → K ∗ ll decays may be explained viaadditional contribution of the Z ′ L boson. The ratio R D ∗ may be explained byobserving couplings of W ′ L to a charged lepton and a neutrino. This will beinvestigated in a future study.It should be noted that this model is not constructed to explain abovementioned anomalies. The aim of this model is to restore CP symmetry,which is a more fundamental discrete symmetry than C or P in the sensethat it can distinguish matter and antimatter in an absolute and convention-independent way. However, this model may explain above mentioned anoma-lies simultaneously which is not ad hoc at all. The model also alleviates thefine tuning of the SM Higgs boson relatively in a small amount and providepossible dark matter candidates in the form of a real scalar particle (and the13M right-handed neutrinos and new neutrinos) corresponding to scalar field χ which is testable, for instance, in the process of gluon fusion gg → h L S atthe LHC. Acknowledgements
I thank to the anonymous reviewer for several important comments andsuggestions.
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