Loop suppressed light fermion masses with U(1)_R gauge symmetry
aa r X i v : . [ h e p - ph ] J un KIAS-P17023
Loop suppressed light fermion masses with U (1) R gauge symmetry Takaaki Nomura ∗ and Hiroshi Okada † School of Physics, KIAS, Seoul 02455, Korea Physics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan 300 (Dated: July 24, 2018)
Abstract
We propose a model with two Higgs doublet where quark and charged-lepton masses in the firstand second families are induced at one-loop level, and neutrino masses are induced at the two-loop level. In our model we introduce an extra U (1) R gauge symmetry that plays a crucial rolein achieving desired terms in no conflict with anomaly cancellation. We show the mechanism togenerate fermion masses, the resultant mass matrices and Yukawa interactions in mass eigenstates,and discuss several interesting phenomenologies such as muon anomalous magnetic dipole momentand dark matter candidate that are arisen from this model. ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION Radiatively induced mass scenarios have widely been applied to various models and suc-cessfully been achieved as theories at low energy scale ( ∼ TeV) that induce masses of lightfermions such as neutrinos, include dark matter (DM) candidate, and explain muon anoma-lous magnetic dipole moment (muon g −
2) without conflicts with various constraints suchas flavor changing neutral currents (FCNCs), lepton flavor violations (LFVs), and quarkand lepton masses and their mixings. Thus a lot of authors have historically been workingalong this ideas. Here we classify such radiative models as the number of the loops, i.e. ,refs. [1–90] mainly focusses on the scenarios at one-loop level, and refs. [91–125] at two-looplevel. Moreover, refs. [126–128] discuss the systematic analysis of (Dirac) neutrino oscilla-tion, charged lepton flavor violation, and collider physics in the framework of neutrinophilicand inert two Higgs doublet model (THDM), respectively.One of the mysteries in the standard model (SM) is the hierarchical structure of fermionmasses in both quark and lepton sectors, which indicates large hierarchy of the Yukawacoupling constants. In particular, masses of the SM neutrinos are very small compared tothe other fermion masses. It is thus challenging to understand the hierarchical structure offermion masses applying a scenario of radiatively induced mass; some attempts to resolveflavor hierarchies in THDM are found, for example, in Refs. [129–132].In this paper, we propose a new type of THDM scenario that can explain the small fermionmasses in the SM, i.e. , the first and second families in the quark and charged lepton sectors,and the tiny masses of active neutrinos, by applying a radiatively induced mass mechanism.Here the second isospin doublet Higgs has small vacuum expectation value (VEV), whichprovides such lighter fermion masses in the first and second families, while the SM-like Higgsprovides the mass of third family fermions in the SM; top quark, bottom quark, and tauon.To realize such a small VEV and family dependence, we impose a U (1) R gauge symmetry infamily dependent way and introduce extra scalar fields with U (1) R charges. Then the VEVof second Higgs doublet is induced at the one-loop level, which could be an appropriatereason of the smallness due to the loop suppression. In addition, active neutrino masses areinduced at two-loop level with the canonical seesaw mechanism. As a bonus of introducingthe extra scalars, we can also explain the muon g −
2, and obtain a dark matter candidate,as is often the case with radiatively induced mass models.2 uarks LeptonsFermions Q αL u iR d iR t R b R L αL e iR N iR τ R SU (3) C SU (2) L U (1) Y
16 23 −
13 23 − − − − U (1) R x − x − x x Z + + + + + + + + +TABLE I: Field contents of fermions and their charge assignments under SU (2) L × U (1) Y × U (1) R × Z , where each of the flavor index is defined as α ≡ − i = 1 , = 0 InertBosons Φ Φ ϕ ϕ η S χSU (2) L U (1) Y
12 12 U (1) R x x x x x Z + + + + − − − TABLE II: Boson sector, where all the bosons are SU (3) C singlet. This paper is organized as follows. In Sec. II, we show our model, and establish thequark and lepton sector, and derive the analytical forms of FCNCs, LFVs, muon anomalousmagnetic dipole moment. We conclude and discuss in Sec. III.
II. MODEL SETUP
In this section, we introduce our model, analyze mass matrices in quark and leptonsector and discuss some phenomenologies. First of all we impose an additional U (1) R gaugesymmetry, where only the first and second families of right-handed SM fermions and N R have nonzero charge x , where N R constitutes Majorana field after the spontaneous U (1) R gauge symmetry breaking. All of the fermion contents and their assignments are summarizedin Table I, in which i = 1 , α = 1 − N R is two, since the anomaly arising from U (1) R gauge symmetrycancels out in each of one generation [73, 121].For the scalar sector with nonzero VEVs, we introduce two SU (2) L doublet scalars Φ and Φ , and two SU (2) L singlet scalars ϕ and ϕ which are charged under U (1) R . HereΦ is supposed to be the SM-like Higgs doublet, while Φ is the additional Higgs doubletwith tiny VEV and has non-zero U (1) R charge. For SM singlet scalars, ϕ plays a role ininducing the tiny VEV of Φ at the one-loop level, and ϕ provides the Majorana fermions N R after the spontaneous U (1) R breaking. On the other hand, SU (2) L singlet scalars S , χ ,and doublet scalar η are inert scalars because of odd parity under the Z , and they play arole in generating the tiny VEV of Φ by running inside a loop diagram. In addition, thelightest state of these neutral scalars can be a dark matter candidate [28]. All of the scalarcontents and their assignments are summarized in Table II, where we assume S to be areal field for simplicity. We also note that massive Z ′ boson appears after U (1) R symmetrybreaking. In this paper, we omit detailed analysis for phenomenology of Z ′ and just assumemass of Z ′ is sufficiently heavy to avoid constraints from collider experiments. A. Yukawa interactions and scalar sector
Yukawa Lagrangian : Under our fields and symmetries, the renormalizable Lagrangiansfor quark and lepton sector are given by −L Q = ( y u ) αj ¯ Q L α u R j ˜Φ + ( y d ) αj ¯ Q L α Φ d R j + ( y t ) α ¯ Q L α t R ˜Φ + ( y b ) α ¯ Q L α Φ b R + c . c ., (II.1) −L L = ( y ν ) αj ¯ L L α N R j ˜Φ + ( y ℓ ) αj ¯ L L α Φ e R j + ( y τ ) α ¯ L L α e R ˜Φ + ( y N ) ii ¯ N R i N CR i ϕ + c . c ., (II.2)where ˜Φ , ≡ ( iσ )Φ ∗ , with σ being the second Pauli matrix. Here we note that the SM-likeHiggs doublet Φ only couples to third family right-handed fermions while Φ couples firstand second families right-handed fermions because of the gauge invariance under U (1) R .4 calar potential : In our model, scalar potential is given by V = X a =1 − ( µ ϕ a | ϕ a | ) + µ S S + µ χ | χ | + µ η | η | + λ h (Φ † η ) χϕ + c . c . i + λ ′ h (Φ † η ) Sϕ ∗ + c . c . i + µ ( χSϕ ∗ + c . c . ) + X a=1 − (cid:0) λ ϕ a | ϕ a | + λ ϕ a S | ϕ a | S + λ ϕ a χ | ϕ a | | χ | + λ ϕ a η | ϕ a | | η | (cid:1) + λ S S + λ χ | χ | + λ η | η | + λ Sχ S | χ | + λ Sη S | η | + λ χη | χ | | η | + X i =1 , " X a =1 , ( λ ϕ a Φ i | ϕ a | | Φ i | ) + λ S Φ i S | Φ i | + λ χ Φ i | χ | | Φ i | + λ Φ i η | Φ i | | η | + λ ′ Φ i η | Φ † i η | + µ | Φ | + µ | Φ | + λ | Φ | + λ | Φ | + λ | Φ | | Φ | + λ | Φ † Φ | , (II.3)where we choose some parameters in the potential so that h Φ i ≡ v / √ U (1) R symmetry breaking, effective mass term µ Φ † Φ isgiven via Eq.(II.3), and µ is given by µ = − λ λ ′ µv ϕ √ π ) m χ m S ln h m χ m S i + m η m S ln h m S m η i + m χ m η ln h m η m χ i ( m χ − m S )( m χ − m η )( m S − m η ) , (II.4) m χ = µ χ + λ χ Φ v + λ χ Φ v + λ ϕ χ v ϕ + λ ϕ χ v ϕ , (II.5) m S = µ S + λ S Φ v + λ S Φ v + λ ϕ S v ϕ + λ ϕ S v ϕ , (II.6) m η = µ η + λ Φ η v + λ Φ η v + λ ′ Φ η v + λ ′ Φ η v + λ ϕ η v ϕ + λ ϕ η v ϕ , (II.7)where h ϕ i i ≡ v ϕ i / √ i = 1 −
2) and v = 0. The resultant scalar potential in the THDMsector is given by V T HDM = µ (Φ † Φ + c . c . ) + µ | Φ | + µ | Φ | + λ | Φ | + λ | Φ | + λ | Φ | | Φ | + λ | Φ † Φ | , (II.8)where h Φ i i ≡ v i / √ i = 1 −
2) and we chose that µ is negative and µ is positive. Taking v /v << v /v S << we finally obtain the formula of Φ VEV as v ≈ v µ µ + v ( λ + λ ) + v ϕ λ ϕ Φ + v ϕ λ ϕ Φ . (II.9) To achieve it, one has to assume to be: 0 < µ + λ v − ( λ + λ ) v , arising from the tadpole condition: ∂V THDM ∂ Φ (cid:12)(cid:12)(cid:12) v ,v = 0. Z symmetry and no λ [(Φ † Φ ) + h.c. ] term [133].Including their VEVs, the scalar fields are parameterized asΦ i = h v + h + ia √ , Φ = h +2 v + h + ia √ , η = η + η R + iη I √ , (II.10) ϕ a = v ϕ a + ϕ R a + iϕ I a √ , ( a = 1 , , χ = χ R + iχ I √ , S = s R √ , (II.11)where ϕ I a does not have nonzero mass eigenvalues, and either of them is absorbed by thelongitudinal degrees of freedom of Z ′ gauge boson. After the spontaneous symmetrybreaking, neutral bosons mix each other and their mass eigenstates and eigenvalues aredefined by: Diag.( m H , m H , m H , m H ) = O H m ( ϕ R , ϕ R , h , h ) O TH , Diag.( m G , m A ) = O C m ( a , a ) O TC , Diag.( m ω ± , m H ± ) = O C m ( h ± , h ± ) O TC , Diag.( m η R , m η R , m η R ) = O R m ( η R , s R , χ R ) O TR , Diag.( m η I , m η I ) = O I m ( η I , χ I ) O TI , (II.12)where O H,C,R,I denotes the mixing matrices which diagonalize the mass matrices accord-ingly. Here G and ω ± do not have nonzero mass eigenvalue, and they are absorbed by thelongitudinal degrees of freedom of neutral SM gauge boson Z and charged gauge boson W ± respectively as Numbu-Goldstone (NG) bosons. The mass matrices in the right-hand side ofEq. (II.12) are given by the parameters in scalar potential. For neutral CP-even components A physical massless boson at the tree level seems to underlie our model. And it will be severely constrainedby non-Newtonian forces, if its mass is extremely tiny compared to 1 eV [134]. However since its vanishingmass originates from an accidental global symmetry after all the gauge symmetry breaking, it can alwaysbe massive at higher dimensional operators [135]. In our case, for example, five dimensional operators; M pl (Φ † Φ ) ϕ and M pl (Φ † Φ ) ϕ that retain all the gauge symmetries, violate the accidental symmetry.Then one finds it nonzero mass with Planck scale suppression. Nevertheless, the mass scale can begenerated up to 1 MeV in case v << v ϕ . Thus we can evade this constraint via this effect.
6e obtain m ( ϕ R , ϕ R , h , h ) = v ϕ λ ϕ v ϕ λ ϕ v v ϕ λ ϕ Φ v v ϕ λ ϕ Φ v λ − v µ v v v ϕ λ ϕ Φ v v ϕ λ ϕ Φ v v ( λ + λ ) + µ v λ − v µ v , (II.13)where the matrix has symmetric structure. We also obtain the mass matrices for CP-oddand charged components as m ( a , a ) = − v µ v µ µ − v µ v , m A = − ( v + v ) µ v v , (II.14) m ( h ± , h ± ) = − v ( v v λ +2 µ )2 v v v λ + µ v v λ + µ − v ( v v λ +2 µ )2 v , m H ± = − ( v + v )( v v λ + 2 µ )2 v v . (II.15)The mass matrices for inert scalar sector are given by m ( m η R , m η R , m η R ) = ( m η R ) v v ϕ λ ′ ( m η R ) v v ϕ λ m η R ) , ( m η R ) = m η , ( m η R ) = m S , ( m η R ) = m χ , (II.16) m ( η I , χ I ) = ( m η R ) − v v ϕ λ − v v ϕ λ ( m η R ) ,m η I = ( m η R ) + ( m η R ) − q [( m η R ) − ( m η R ) ] + v v ϕ λ ,m η I = ( m η R ) + ( m η R ) + q [( m η R ) − ( m η R ) ] + v v ϕ λ . (II.17)Here we explicitly show the 2 by 2 matrices; O C and O I , as O C ≡ c β s β − s β c β , s β = v p v + v , (II.18) O I ≡ c a s a − s a c a , s a = − v v ϕ λ m η I − m η I , (II.19)7 IG. 1: The one loop diagram which induces masses first and second families of quarks and chargedleptons. where c a ≡ cos a and s a ≡ sin a , and we define v ≡ p v + v and tan β ≡ v v which lead v = v cos β and v = v sin β as in the other THDMs. The mass eigenvalues m H a ( a = 1 − H ( ≡ h SM ) is the SM-like Higgs andthe other three neutral bosons are the additional(heavier) Higgs bosons. Here η R i ( i = 1 − η I i ( i = 1 −
2) is the masseigenstate of the imaginary part of inert neutral boson. All of the mass eigenvalues andmixings are written in terms of VEVs, and quartic couplings in the Higgs potential afterinserting the tadpole conditions: ∂V /∂φ | v ,v ,v ϕ ,v ϕ = 0 and ∂V /∂ϕ R | v ,v ,v ϕ ,v ϕ = 0. Alsothe mass of η ± is given by m η ± = v ϕ λ ϕ η + v λ Φ η + v ϕ λ ϕ η + v λ Φ η + 2 µ η . (II.20)We note that in THD sector SM-like couplings are preferred for gauge interactions of h SM by the current Higgs data [136]. Note also that a mixing between Higgs and extra scalarsinglet modifies the SM Higgs couplings which is tested by the Higgs measurements at theLHC. The mixing angle is constrained as sin θ . . . Quark sector In this subsection, we will analyze the quark sector. First of all, let us focus on theYukawa sector, in which the measured SM quark masses and their mixings are induced.Up and down quark mass matrices are diagonalized by D u ≡ ( M diag.u ) = V u L M u V † u R , and D d ( ≡ M diag.d ) = V d L M d V † d R , where V ′ s are unitary matrix to give their diagonalizationmatrices. Then CKM matrix is defined by V CKM ≡ V † d L V u L , where it can be parametrizedby three mixings with one phase as follows: V u ( d ) L ≡ c u ( d ) c u ( d ) c u ( d ) s u ( d ) s u ( d ) − c u ( d ) s u ( d ) − s s u ( d ) c u ( d ) c u ( d ) c u ( d ) − s u ( d ) s u ( d ) s u ( d ) s u ( d ) c u ( d ) s u ( d ) s u ( d ) − c u ( d ) s u ( d ) c u ( d ) − s u ( d ) c u ( d ) − c u ( d ) s u ( d ) s u ( d ) c u ( d ) c u ( d ) , (II.21)The mass matrix in our form is written in terms of the dominant contribution ( M (1) t ( b ) ) that isproportional to v and the sub-dominant one ( M (2) u ( d ) ) that is proportional to v . Also we canwrite the left-handed mixing matrix in terms of linear combination as V u ( d ) L ≡ V (1) t ( b ) L + V (2) u ( d ) L ,where V (1) t ( b ) L ( V (2) u ( d ) L ) corresponds to M (1) t ( b ) ( M (2) u ( d ) ). Then we consider the product of the massmatrix given by (cid:16) M u ( d ) M † u ( d ) (cid:17) αβ = (cid:16) ( M (1) t ( b ) )( M (1) t ( b ) ) † (cid:17) αβ + (cid:16) ( M (2) u ( d ) )( M (2) u ( d ) ) † (cid:17) αβ = v (( y t ( b ) ) ) ( y t ( b ) ) ( y t ( b ) ) ( y t ( b ) ) ( y t ( b ) ) ( y t ( b ) ) ( y t ( b ) ) (( y t ( b ) ) ) ( y t ( b ) ) ( y t ( b ) ) ( y t ( b ) ) ( y t ( b ) ) ( y t ( b ) ) ( y t ( b ) ) ( y t ( b ) ) ) + v ( y u ( d ) ) + ( y u ( d ) ) ( y u ( d ) ) ( y u ( d ) ) + ( y u ( d ) ) ( y u ( d ) ) ( y u ( d ) ) + ( y u ( d ) ) ( y u ( d ) ) ( y u ( d ) ) + ( y u ( d ) ) ( y u ( d ) ) ( y u ( d ) ) ( y u ( d ) ) + ( y u ( d ) ) ( y u ( d ) ) ( y u ( d ) ) + ( y u ( d ) ) , (II.22)which is diagonalized by V u ( d ) L . When we redefine a t ( b ) ≡ ( y t ( b ) ) ( y t ( b ) ) and b t ( b ) ≡ ( y t ( b ) ) ( y t ( b ) ) in( M (1) t ( b ) ) αβ , we can rewrite the leading term as (cid:16) ( M (1) t ( b ) )( M (1) t ( b ) ) † (cid:17) αβ = ( v ( y t ( b ) ) ) a t ( b ) a t ( b ) b t ( b ) a t ( b ) a t ( b ) b t ( b ) b t ( b ) b t ( b ) a t ( b ) b t ( b ) . (II.23)9ts resulting mass eigenvalues and mixing matrix are given by | D (1) t ( b ) | = Diag. (cid:18) , , ( v ( y t ( b ) ) ) a t ( b ) + b t ( b ) ) (cid:19) ≡ (0 , , | m t ( b ) | ) , (II.24) V (1) t ( b ) L = − q a t ( b ) a t ( b ) q a t ( b ) − a t ( b ) b t ( b ) q a t ( b ) q a t ( b ) + b t ( b ) q a t ( b ) q a t ( b ) + b t ( b ) − b t ( b ) q a t ( b ) q a t ( b ) + b t ( b ) a t ( b ) q a t ( b ) + b t ( b ) b t ( b ) q a t ( b ) + b t ( b ) q a t ( b ) + b t ( b ) . (II.25)It suggests that the leading term provides the top and bottom masses only. Thus the firstand second masses are generated via subleading matrix ( M (2) u ( d ) ), where it is arisen at theone-loop level as can be seen in fig. 1.The first and second quark mass eigenvalues are calculated by solving the secular equation δm q δm q δm q δm q ≡ ( V (1) q L ) i (cid:16) ( M (2) u ( d ) )( M (2) u ( d ) ) † (cid:17) ij ( V (1) † q L ) j ( V (1) q L ) i (cid:16) ( M (2) u ( d ) )( M (2) u ( d ) ) † (cid:17) ij ( V (1) † q L ) j ( V (1) q L ) i (cid:16) ( M (2) u ( d ) )( M (2) u ( d ) ) † (cid:17) ij ( V (1) † q L ) j ( V (1) q L ) i (cid:16) ( M (2) u ( d ) )( M (2) u ( d ) ) † (cid:17) ij ( V (1) † q L ) j , (II.26)where δm qij ( i, j = 1 ,
2) is written in terms of bi-linear combinations of a ( b ) t ( b ) and( y u ( d ) ) k,ℓ ( k = 1 − , ( ℓ = 1 , | D (2) u ( d ) | ≡ Diag.( | m u ( d ) | , | m c ( s ) | ,
0) = Diag.( δm q − δm q , δm q + δm q , , (II.27) V (2) u ( d ) L = − δm q √ δm q + δm δm q √ δm q + δm q δm √ δm q + δm q δm q √ δm q + δm q O (cid:16) v v (cid:17) O (cid:16) v v (cid:17) O (cid:16) v v (cid:17) ≈ − δm √ δm q + δm q δm q √ δm q + δm q δm √ δm q + δm q δm q √ δm q + δm q
00 0 0 , (II.28)where δm q ≡ ( q ( δm q − δm q ) + 4 δm q δm q − δm q + δm q ) /
2, and δm ij implies10 m u ij or δm d ij . Totally one finds | D u ( d ) | = Diag. (cid:18) δm q − δm q , δm q + δm q , ( v ( y t ( b ) ) ) a t ( b ) + b t ( b ) ) (cid:19) , (II.29) V u ( d ) L ≈ − q a t ( b ) − δm q √ δm q + δm q δm √ δm + δm a t ( b ) q a t ( b ) − a t ( b ) b t ( b ) q a t ( b ) q a t ( b ) + b t ( b ) + δm √ δm + δm q a t ( b ) q a t ( b ) + b t ( b ) + δm q √ δm q + δm q − b t ( b ) q a t ( b ) q a t ( b ) + b t ( b ) a t ( b ) q a t ( b ) + b t ( b ) b t ( b ) q a t ( b ) + b t ( b ) q a t ( b ) + b t ( b ) . (II.30)Comparing Eq. (II.21) and Eq. (II.30), one finds the following relations: s u ( d ) ≈ q a u ( d ) ( V u ( d ) L ) , s u ( d ) ≈ − b u ( d ) q a u ( d ) + b u ( d ) , s u ( d ) ≈ − a u ( d ) q a u ( d ) . (II.31)Since V CKM is close to the unit matrix, one approximately finds to be V CKM ≈ V u L ≈ V d L .Here we take v ≈
10 GeV to explain the charm mass ∼ . FCNCs : Now that all the mass eigenstates have been derived in the quark sector, werewrite the interacting Lagrangian in terms of the mass eigenstate as follows: −L Q i nt = − ( V d L ) βα [( y u ) αj c β − ( y t ) α s β ] ¯ d L β u R γ H − + ( V u L ) βα [( y d ) αj c β − ( y b ) α s β ] ¯ u L β d R γ H + + ( V u L ) βα √ (cid:2) ( y u ) αj ( O TH ) a − ( y t ) α ( O TH ) a (cid:3) ¯ u L β u R γ H a − i ( V u L ) βα √ y u ) αj c β − ( y t ) α s β ] ¯ u L β u R γ A + ( V d L ) βα √ (cid:2) ( y d ) αj ( O TH ) a + ( y b ) α ( O TH ) a (cid:3) ¯ d L β d R γ H a − i ( V d L ) βα √ y d ) αj c β − ( y b ) α s β ] ¯ d L β d R γ A + c.c. ≡ − ( Y u ) βγ ¯ d L β u R γ H − + ( Y d ) βγ ¯ u L β d R γ H + + ( Y ′ u ) aβγ ¯ u L β u R γ H a − i ( Y ′′ u ) βγ ¯ u L β u R γ A + ( Y ′ d ) aβγ ¯ d L β d R γ H a − i ( Y ′′ d ) βγ ¯ d L β d R γ A + c .c., (II.32)where a = 1 − M − ¯ M mixing : It is given in terms of the above Lagrangian, where the leading contri-bution of Y is induced at the one-loop level, which are found in Appendix. While the one11 eson ( a, b, c, d ) m M [GeV] f M [GeV] ∆ m exp M [GeV] D ( c, u, ¯ u, ¯ c ) 1.865 0.212 6.25 × − B ( d, b, ¯ b, ¯ d ) 5.280 0.191 3.36 × − B s ( s, b, ¯ b, ¯ s ) 5.367 0.200 1.17 × − K ( d, s, ¯ s, ¯ d ) 0.488 0.160 3.48 × − TABLE III: The experimental values for M − ¯ M mixing. of Y ′ and Y ′′ is done at the tree level. Then its resulting form is found to be∆ m M ( d a ¯ d c → ¯ d b d d )( Y ′ d , Y ′′ d ) ≈ (cid:18) m M m d a + m d c (cid:19) m M f M × R e X i h ( Y ′ id ) ca ( Y ′ id ) bd + ( Y ′ i † d ) ca ( Y ′ i † d ) bd i m H i − h ( Y ′′ d ) ca Y ′′ d ) bd + ( Y ′′ † d ) ca ( Y ′′ † d ) bd i m A −
124 + 14 (cid:18) m M m d a + m d c (cid:19) ! m M f M × R e X i h ( Y ′ id ) ca ( Y ′ i † d ) bd + ( Y ′ i † d ) ca ( Y ′ id ) bd i m H i + h ( Y ′′ d ) ca ( Y ′′ † d ) bd + ( Y ′′ † d ) ca ( Y ′′ d ) bd i m A , (II.33)where ∆ m M ( u a ¯ u c → ¯ u b u d ) = ∆ m M ( d a ¯ d c → ¯ d b d d )( Y ′ u , Y ′′ d )( u ↔ d ) and x ab ≡ m a m b . Theexperimental values for the mixing are given in Table III and we apply phenomenologicalconstraint ∆ m M ≤ ∆ m exp M . From the above current bounds, severe constraints are found.Here we conservatively discuss the order of the Yukawa couplings and masses of scalarbosons allowed by the constraints. The flavor violating components of ( Y ′ u ( d ) ) a is stronglyconstrained to be less than O (10 − ) when corresponding H a is the SM Higgs boson . Onthe other hand one can take Y ′′ u ( d ) = O (10 − ) if m A = O (100) GeV. Y u ( d ) contribute M − ¯ M mixing at the one-loop level as shown in Appendix, and Y ′ u ( d ) = O (0 .
1) if m H ± = O (1)TeV. Notice here that above estimations are that for Yukawa couplings which violate flavorsand flavor conserving couplings are less constrained. The masses of extra bosons are thuspreferred to be heavier than SM Higgs to avoid the constraints. In case where H a is not the SM Higgs, one can take the same order as Y ′′ u ( d ) and m A . b → sµ − µ + and b → cℓ − i ¯ ν j . The lepton universality violating decay b → sµ − µ + is measured as the ratio R K ≡ B ( B → Kµµ ) B ( B → Kee ) = 0 . +0 . − . ± .
036 by LHCb [148],which has deviation from the SM prediction. This process is found by the following effectiveHamiltonian in our model: H eff. = − √ " X i ( Y ′ id ) βα ( Y iL ) cd m H i + ( Y ′′ d ) βα ( Y ′ ℓ ) cd m A ! ( ¯ d β P R d α )(¯ ℓ c P R ℓ d )+ X i ( Y ′ id ) βα ( Y i † L ) cd m H i + ( Y ′′ d ) βα ( Y ′ † ℓ ) cd m A ! ( ¯ d β P R d α )(¯ ℓ c P L ℓ d )+ X i ( Y ′ i † d ) βα ( Y iL ) cd m H i + ( Y ′′ † d ) βα ( Y ′ ℓ ) cd m A ! ( ¯ d β P L d α )(¯ ℓ c P R ℓ d )+ X i ( Y ′ i † d ) βα ( Y i † L ) cd m H i + ( Y ′′ † d ) βα ( Y ′ † ℓ ) cd m A ! ( ¯ d β P L d α )(¯ ℓ c P L ℓ d ) . The semi-leptonic decay b → cℓ − i ¯ ν j is measured as the ratio R D ≡ B ( ¯ B → Dτν ) B ( ¯ B → Dℓν ) = 0 . ± . ± .
024 by flavor averaging group (HFAG) [149], which also has deviation from the SMprediction. This process is also found by the following effective Hamiltonian in our model: H eff. = ( Y † νℓ ) ij m H ± (cid:2) ( Y † u ) ba (¯ u b P L d a )(¯ ℓ i P L ν j ) − ( Y d ) ba (¯ u b P R d a )(¯ ℓ i P L ν j ) (cid:3) . However since all the effective Hamiltonians discussed above depend on the Y ( ′ , ′′ ) u ( d ) and 1 /m A ,1 /m H i , 1 /m H ± , which are severely restricted by the bounds of M − ¯ M mixings. Hence itcould be difficult to explain such anomalies in our order estimations. C. Lepton sector
In this subsection, we will discuss the lepton sector, where neutrinos are canonical seesawtype. Thus the process to induce the mass matrix in the charged-lepton sector is the sameas the down-quark sector, by changing b → τ and d → ℓ in the quark sector. The massmatrix is diagonalized by D ℓ ( ≡ M diag.ℓ ) = V ℓ L M ℓ V † ℓ R , while the neutrino mass matrix isdiagonalized by D ν ( ≡ M diag.ν ) = U ν M ν U Tν , where V ℓ L and U ν are unitary matrix to givetheir diagonalization matrices. Then MNS matrix is defined by V MNS ≡ V † ℓ L U ν .13hen the charged-lepton mass matrix is arisen at the one-loop level as can be seen infig. 1, and the resulting form is straightforwardly written as (cid:0) ( M ℓ )( M ℓ ) † (cid:1) αβ = (cid:16) ( M (1) ℓ )( M (1) ℓ ) † (cid:17) αβ + (cid:16) ( M (2) ℓ )( M (2) ℓ ) † (cid:17) αβ = v ( y τ ) ( y τ ) ( y τ ) ( y τ ) ( y τ ) ( y τ ) ( y τ ) ( y τ ) ( y τ ) ( y τ ) ( y τ ) ( y τ ) ( y τ ) ( y τ ) ( y τ ) + v ( y ℓ ) + ( y ℓ ) ( y ℓ ) ( y ℓ ) + ( y ℓ ) ( y ℓ ) ( y ℓ ) + ( y ℓ ) ( y ℓ ) ( y ℓ ) + ( y ℓ ) ( y ℓ ) ( y ℓ ) ( y ℓ ) + ( y ℓ ) ( y ℓ ) ( y ℓ ) + ( y ℓ ) . (II.34)Following the quark sector, the mass eigenvalues D ℓ ≡ Diag.( m e , m µ , m τ ) and eigenstate arerespectively given by | D ℓ | = Diag. (cid:18) δm ℓ − δm ℓ , δm ℓ + δm ℓ , ( v ( y t ( b ) ) ) a τ + b τ ) (cid:19) , (II.35) V ℓ L ≈ − √ a τ − δm ℓ √ δm + δm ℓ δm √ δm ℓ + δm ℓ a τ √ a τ − a τ b τ √ a τ √ a τ + b τ + δm ℓ √ δm ℓ + δm ℓ q a t ( b ) √ a τ + b τ + δm ℓ √ δm ℓ + δm ℓ − b τ √ a τ √ a τ + b τ a τ √ a τ + b τ b τ √ a τ + b τ √ a τ + b τ , (II.36)where a τ ≡ ( y τ ) ( y τ ) , b τ ≡ ( y τ ) ( y τ ) , δm ℓ , and δm ℓij , ( i, j ) = 1 , s ℓ ≈ p a τ ( V ℓ L ) , s ℓ ≈ − b τ p a τ + b τ , s ℓ ≈ − a τ p a τ . (II.37)The neutrino mass matrix is arisen at the two-loop level as can be seen in fig. 2, and theresulting form is given by ( M ν ) αβ = v X i =1 − ( y ν ) αi ( M − N ) ii ( y ν ) Tiβ , (II.38)where M N ≡ y N v ϕ / √
2. We apply Casas-Ibarra parametrization [143] to reproduce neutrinooscillation data, then one finds the following relation: y ν = 2 v V † MNS V † ℓ L p D ν O p M N , (II.39)14 IG. 2: The two loop diagram which induces masses of active neutrinos.Process ( α, β ) Experimental bounds (90% CL) References µ − → e − γ (2 , BR ( µ → eγ ) < . × − [141] τ − → e − γ (3 , Br ( τ → eγ ) < . × − [142] τ − → µ − γ (3 , BR ( τ → µγ ) < . × − [142]TABLE IV: Summary of ℓ α → ℓ β γ process and the lower bound of experimental data. where O (= OO T = 1) is an arbitrary orthogonal matrix with complex values. LFVs : Now that all the mass eigenstates have been derived in the lepton sector, werewrite the interacting Lagrangian in terms of the mass eigenstate as follows: −L L i nt = − c β ( V ℓ L ) βα ( y ν ) αj ¯ ℓ L β N R j H − + ( U ν ) βα [( y ℓ ) αj c β − ( y τ ) α s β ] ¯ ν L β e R γ H + + 1 √ (cid:2) ( V ℓ L ) βα ( y ℓ ) αj ( O TH ) a + ( V ℓ L ) βα ( y τ ) α ( O TH ) a (cid:3) ¯ ℓ L β e R γ H a + i √ V ℓ L ) βα ( y ℓ ) αj c β − ( V ℓ L ) βα ( y τ ) α s β ] ¯ ℓ L β e R γ A + c .c. (II.40) ≡ − ( Y ν ) βj ¯ ℓ L β N R j H − + ( Y νℓ ) βγ ¯ ν L β e R γ H + + ( Y L ) aβγ ¯ ℓ L β e R γ H a + i ( Y ′ ℓ ) βγ ¯ ℓ L β e R γ A + c .c., (II.41)where a = 1 − Y ν , Y νℓ , Y ν , and Y ′ ℓ can respectively be arbitralscale by controlling the parameters O , U ν , O H and V ℓ L . ℓ α → ℓ β γ : The lepton flavor (LFVs) violation processes give the constraints on ourparameters. The experimental bounds are found in Table. IV. The most known processes15re ℓ α → ℓ β γ , and its branching ratio is given by BR ( ℓ α → ℓ β γ ) ≈ π α em C αβ G F m ℓ α (cid:0) | a R + a R + a R | + | a L + a L + a L | (cid:1) αβ (II.42)where α em ≈ /
128 is the fine-structure constant, C αβ = (1 , . , . α, β ) =((2 , , (3 , , (3 , G F ≈ . × − GeV − is the Fermi constant, and a R αβ and a L αβ arecomputed as( a R ) αβ = ( Y ν ) βj ( Y † ν ) jα m ℓ α π ) M N j + 3 M N j m H ± − M N j m H ± + m H ± + 12 M N j m H ± ln h m H ± M Nj i ( M N j − m H ± ) , ( a L ) αβ = ( Y ν ) βj ( Y † ν ) jα m ℓ β π ) M N j + 3 M N j m H ± − M N j m H ± + m H ± + 12 M N j m H ± ln h m H ± M Nj i ( M N j − m H ± ) , ( a R ) αβ = − ( Y L ) aβγ ( Y † L ) aγα m ℓ α π ) m H a , ( a L ) αβ = − ( Y L ) aβγ ( Y † L ) aγα m ℓ β π ) m H a , ( a R ) αβ = − ( Y ′ ℓ ) βγ ( Y ′ † ℓ ) γα m ℓ α π ) m A , ( a L ) αβ = − ( Y ′ ℓ ) βγ ( Y ′ † ℓ ) γα m ℓ β π ) m A . (II.43) Muon anomalous magnetic dipole moment ( g − µ : Through the same process from theabove LFVs, there exists the contribution to ( g − µ , and its form ∆ a µ is simply given by∆ a µ ≈ − m µ ( a R + a R + a R + a L + a L + a L ) µµ . (II.44)This value can be tested by current experiments ∆ a µ = (28 . ± . × − [144]. As can beseen in Eq. (II.43), one finds that the first two forms a R ( L ) give negative contribution, whilethe others provide positive contribution. Note that from the flavor violation in quark sector,extra scalar bosons are preferred to be heavier than SM Higgs. Thus we here assume thedominant contribution to the muon g − µ → eγ , the stringent constraint BR ( µ → eγ ),are approximately given by SM Higgs as∆ a µ ∼ − m µ ( a R + a L ) µµ = X γ =1 ( Y L ) γ ( Y † L ) γ π ) m µ m H , (II.45) BR ( µ → eγ ) ∼ π α em G F m µ | ( a R ) µe | = | P γ =1 ( Y L ) γ ( Y † L ) γ | πG F m H , (II.46)where m H ( ≈
125 GeV) is the mass of the SM Higgs. As can be seen in Eqs. (II.45) and(II.46), one can satisfy the constraint of LFV due to the independent parameters. Thus16e show the allowed range of the current measurement of muon g − Y L ) γ ( Y † L ) γ : 2 . . X γ =1 ( Y L ) γ ( Y † L ) γ . . . (II.47) D. Dark matter
In our scenario, real scalar S is considered as a DM candidate, where we assume to beno mixing between S and η R that is natural assumption because of v << v .Our DM candidate S can interact via a Higgs portal coupling S - S - h SM . However theHiggs portal coupling is strongly constrained by the direct detection search at the LUXexperiment [145]. We then assume the SM Higgs portal coupling is negligibly small bychoosing some parameters in the scalar potential to avoid the constraint from the directdetection. We then consider that S dominantly interacts with one of the extra scalar singlets H ≃ ϕ , assuming small mixing among CP-even scalars. Then the dominant annihilationprocess is 2 S → H via four point coupling of S - S - H - H taking mass relation m H SSH H , (II.48)where the coupling λ SSHH is given by combination of couplings in the potential Eq. (II.3).In case of small mixing limit, it is λ SSHH ∼ λ ϕ S . The relic density of DM is then givenby [146] Ω h ≈ . × p g ∗ ( x f ) M P l J ( x f )[GeV] , (II.49)where g ∗ ( x f ≈ ≈ M P l ≈ . × , and J ( x f )( ≡ R ∞ x f dx h σv rel i x ) is given by J ( x f ) = Z ∞ x f dx R ∞ m S ds p s − m S s ( σv rel ) K (cid:16) √ sm S x (cid:17) m S x [ K ( x )] , ( σv rel ) = | λ SSHH | π s s − m H s . (II.50) Here we assume DM pair annihilate into H pair but annihilation mode into H pair is also possible ifwe chose H is lighter than DM. SSHH = Λ SSHH = Λ SSHH = W h =0.12 100 200 3001500.020.050.100.200.501.00 m S @ GeV D W h FIG. 3: Relic density of DM in terms of the DM mass, where λ SSHH = (1 . , . , . 0) represent thelines of red, blue, and magenta, respectively. Here we fixed m H = 100 GeV for simplicity. Here s is a Mandelstam variable, and K , are the modified Bessel functions of the secondkind of order 1 and 2, respectively. The observed relic density is Ω h ≈ . 12 [147]. We showthe relic density in terms of the DM mass in Fig. 3 for several values of the coupling constantfixing m H = 100 GeV, which suggests that the order one quartic coupling is needed. III. CONCLUSIONS AND DISCUSSIONS We have proposed a model with two Higgs doublet Φ , in which quark and charged-leptonmasses in the first and second families are induced at one-loop level and neutrino massesare induced at the two-loop level. In the model we have introduced an extra U (1) R gaugesymmetry in family dependent way that plays a crucial role in achieving desired interactionterms in no conflict with anomaly cancellation. The second Higgs doublet Φ is also chargedunder U (1) R and couples to only the first and second families of right-handed fermions. Wehave then considered the scenario in which vacuum expectation value of Φ is absent at treelevel and induced at one-loop level via spontaneous symmetry breaking of gauge symmetries.After the gauge symmetry breaking, we have obtained the scalar potential of THDM withsoftly broken Z symmetry where Φ † Φ term is suppressed by loop effect and λ [(Φ † Φ ) + h.c. ] term is absent at tree level. We have shown the fermion masses where first and secondfamilies are loop suppressed and discussed structure of the mass matrices. Here we emphasize18hat our original Yukawa couplings could be less hierarchical compared to the SM or generalTHDM because of the loop suppression effect for the first and second families. The Yukawacouplings with mass eigenstates are also derived and we discussed several phenomenologiessuch as flavor changing neutral current in the quark sector, lepton flavor violations, muon g − 2. In addition, we have analyzed relic density for the dark matter candidate in thismodel which can be accommodated with observed data.In the model, rich phenomenologies can be considered such as flavor violating SM Higgsdecay and collider physics although we have not discussed. It will be also interesting toinvestigate difference from other THDMs in detail since we have specific structure of Yukawacouplings where one Higgs doublet couples to third family right-handed fermions and thesecond doublet couples to other families of right-handed fermion. In addition, we can discussphysics of extra Z ′ gauge boson which comes from our U (1) R . More detailed analysis of themodel will be done elsewhere. Appendix M − ¯ M mixing: The one-loop contribution that is proportional to Y ′ and Y ′′ is found tobe ∆ m (2) d ( Y u , Y d ) = m u α m u β m M f M π ) m H ± (cid:18) m M m d a + m d c (cid:19) (III.1) × R e h ( Y u Y d ) ba ( Y u Y d ) cd + ( Y † d Y † u ) ba ( Y † d Y † u ) cd i F II ( x u α H ± , x u β H ± ) ,F II ( x , x ) = Z dadbdc δ ( a + b + c − a ( a + bx + cx ) , (III.2)where ∆ m (2) u ( Y d , Y u ) = ∆ m (2) d ( Y u , Y d )( u ↔ d ), x ab ≡ m a m b , and( a, b, c, d ) = ( c, u, ¯ u, ¯ c ) , for D , (III.3)( a, b, c, d ) = ( d, b, ¯ b, ¯ d ) , for B , (III.4)( a, b, c, d ) = ( s, b, ¯ b, ¯ s, ) , for B S , (III.5)( a, b, c, d ) = ( d, s, ¯ s, ¯ d ) , for K . (III.6)In the order estimation of M − ¯ M mixing, it satisfies if we take Y u ( d ) . O (1) and m H ± = O (1)TeV. 19 cknowledgments H. O. is sincerely grateful for all the KIAS members, Korean cordial persons, foods,culture, weather, and all the other things. [1] A. Zee, Phys. Lett. B , 389 (1980) [Erratum-ibid. 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