Left-Right Symmetric Models with Peccei-Quinn Symmetry

Transcription

1 Left-Right Symmetric Models with Peccei-Quinn Symmetry Pei-Hong Gu Max-Planck-Institut für Kernphysik, Heidelberg PHG, ; PHG, Manfred Lindner, Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200

2 Outline Introduction Left-Right Symmetric Models Double and Linear Seesaw from PQ Symmetry Breaking Inflation and Flavor Mixing from PQ Symmetry Breaking Summary Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 2

3 Introduction The SU(3) c SU(2) L U() Y standard model standard model (SM) has been tested to a very high accuracy. However, there are several theoretical motivations to consider theories beyond the SM. For example, the SM lefthanded fermions transform as doublets while the SM right-handed fermions transform as singlets. This unequal treatment of the left- and right-handed fermions is dictated by parity violation at low energy. It is thus natural to start with a theory with left-right symmetry (Pati, Salam, 74 ; Mohapatra, Pati, 75 ; Mohapatra, Senjanović, 75 ) at high energy, where both the left- and righthanded fermions are treated on equal footing and parity is conserved. At some high energy the left-right symmetry is spontaneously broken, which also induces a spontaneous parity violation as observed at low energy. This inspires us to consider a left-right symmetric extension of the SM. Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 3

4 The strong CP problem is another big challenge to the SM. The Peccei- Quinn (Peccei, Quinn, 77 ) (PQ) symmetry predicts the existence of an axion (Peccei, Quinn, 77 ; Weinberg, 78 ; Wilczek, 78 ) which would solve the strong CP problem. The original axion model (Peccei, Quinn, 77 ; Weinberg, 78 ; Wilczek, 78 ) has been ruled out. Alternatively, we can consider the Kim-Shifman-Vainshtein-Zakharov (Kim, 79 ; Shifman, Vainshtein, Zakharov, 80 ) (KSVZ) model or the Dine-Fischler-Srednicki-Zhitnitsky (Dine, Fischler, Srednicki, 8 ; Zhitnitsky, 80 ) (DFSZ) model for the invisible axion. The axion could also be a dark matter candidate (Preskill, Wise, Wilczek, 83 ; Abbott, Sikivie, 83 ; Dine, 83 ). The PQ symmetry could have various other interesting implications on particle physics and cosmology. For example, we can relate the PQ symmetry breaking to the neutrino mass-generation (Shin, 87 ). Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 4

5 Recently we proposed some left-right symmetric models (PHG, ; PHG, Lindner, ), where the fermion singlets for double (Mohapatra, 86 ) and linear (Barr, 04 ) seesaw can naturally obtain their masses through the PQ symmetry breaking. The PQ symmetry breaking thus is tightly related to the neutrino mass-generation. The PQ symmetry breaking even can simultaneously realize the natural inflation (Freese, Frieman, Olinto, 90 ; Adams et al., 93 ) and generate the lepton and quark mixing. Our models can be embedded into the SO(0) grand unification theories (GUT). Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 5

6 Left-Right Symmetric Models The left-right symmetric models (Pati, Salam, 74 ; Mohapatra, Pati, 75 ; Mohapatra, Senjanović, 75 ) are based on the gauge group SU(3) c SU(2) L SU(2) R U() B L. In the left-right symmetric context, the left-handed fermions are SU(2) L doublets as they are in the standard model while the right-handed fermions (the standard model right-handed fermions plus the right-handed neutrinos) are placed in SU(2) R doublets, q L (3, 2,, [ ul 3 ) = d L ], q R (3,, 2, 3 ) = [ ur d R ] ; l L (, 2,, ) = [ ν L e L ], l R (, 2,, ) = [ ν R e R ]. Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 6

7 In order to generate the fermion masses, we need some Higgs scalars to construct the Yukawa interactions. For example, in the most popular leftright symmetric model with two triplet and one bi-doublet Higgs scalars, the charged fermions and the neutral neutrinos can obtain their Dirac masses though the Yukawa interactions involving the Higgs bi-doublet. Since the right-handed neutrinos obtain their heavy Majorana masses through the Yukawa interactions between the right-handed lepton doublets and Higgs triplet, we can naturally realize the type-i seesaw (Minkowski, 77 ; Yanagida, 79 ; Gell- Mann, Ramond, Slansky, 79 ; Glashow, 80 ; Mohapatra, Senjanović, 80 ). At the same time, the left-handed Higgs triplet, which have the Yukawa interactions with the left-handed lepton doublets, can acquire a small vacuum expectation value for the type-ii seesaw (Magg, Wetterich, 80 ; Schechter, Valle, 80 ; Cheng, Li, 80 ; Lazarides, Shafi, Wetterich, 8 ; Mohapatra, Senjanović, 8 ). Therefore, the small neutrino masses can be understood in a natural way. This scenario also accommodates leptogenesis (Fukugita, Yanagida, 86 ) one of the most popular mechanisms to explain the cosmological baryon asymmetry. Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 7

8 In the original left-right symmetric model, the Higgs sector contains two doublet and one bi-doublet Higgs scalars, χ L (, 2,, ) = χ0 L χ L, χ R (,, 2, ) = χ0 R χ R, φ(, 2, 2, 0) = The allowed Yukawa interactions are φ0 φ+ 2 φ φ 0 2. L Y = y q q L φq R ỹ q q L φq R y l l L φl R ỹ l l L φl R + H.c.. Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 8

9 The discrete left-right symmetry can be parity P or charge-conjugation C. The fields should transform as or χ L χ L P χ R, φ C χ R, φ P φ, q L P q R, l L C φ T, q L C q c R, l L P l R, C l c R. Once the discrete left-right symmetry is determined, we can constrain the Yukawa couplings and other parameters in the lagrangian. For example, in the case of charge-conjugation, the Yukawa couplings should be y q = y T q, ỹ q = ỹ T q, y l = y T l, ỹ l = ỹ T l. Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 9

10 Through the Yukawa interactions with the Higgs bi-doublet, the charged fermions and the neutral neutrinos can obtain their Dirac masses, m d = y q φ ỹ q φ 0, m u = y q φ 0 + ỹ q φ 0 2, m e = y l φ ỹ l φ0, m ν = y l φ 0 + ỹ l φ0 2. It is difficult to generate the small neutrino masses unless we fine tune the Yukawa couplings. Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 0

11 We can revive the original left-right symmetric model by adding some fermion singlets, S R (,,, 0) P SR c (,,, 0) or S C R (,,, 0) S R (,,, 0). The fermion singlets with a Majorana mass term can have the Yukawa couplings with the lepton and Higgs doublets, L h L l L χ L S R h R l c R χ R S R 2 M S S R Sc R + H.c. with h L = h R or h L = h R. The full mass terms involving the left- and right-handed neutrinos and the fermion singlets should be L 2 [ νl ν c R Sc R] 0 m ν h L χ 0 L m T ν 0 h R χ 0 R ν c L ν R + H.c.. h T L χ0 L ht R χ0 R M S S R Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200

12 When m ν and h χ L are much smaller than h χ R and/or M S, we can get the neutrino masses by the seesaw formula, i.e. L 2 m ν ν L ν c L + H.c. with m ν = m ν h T χ 0 R M S h χ 0 R mt ν ( m ν + mt ν ) χ0 L χ 0 R. ν L ν R S R S R ν R ν L ν L S R ν R ν L ν L ν R S R ν L FIG. : Double/Inverse (top) and linear (bottom) seesaw. In the double/inverse (Majorana, 86 ; Majorana, Valle, 86 ) and linear (Barr, 03 ) seesaw scenario, we can realize the leptogenesis in two ways (PHG, Sarkar, 0 ): the usual leptogenesis with double seesaw, the resonant leptogenesis with inverse seesaw. Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 2

13 Double and Linear Seesaw from PQ Symmetry Breaking The fermion singlets play an essential role to realize the double and linear seesaw in the left-right symmetric model. Their Majorana masses can be simply input by hand as they are allowed by the gauge symmetry. A more attractive possibility is to introduce certain spontaneous symmetry breaking. We introduce a complex scalar singlet, σ(,,, 0). The one Higgs bi-doublet is extended to be two Higgs bi-doublet, φ (, 2, 2, 0) = φ0 φ+ 2 φ φ0 2, φ 2 (, 2, 2, 0) = φ0 2 φ+ 22 φ 2 φ0 22. Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 3

14 We take the discrete left-right symmetry to be the charge-conjugation. The transformation of the complex scalar singlet should be σ C σ. We then impose a global symmetry, under which the fields carry the quantum numbers as below, for q L, q c R, l L, lc R, S R ; 2 for φ, φ 2, σ ; 0 for χ L, χ R. Clearly, the above global symmetry is consistent with the discrete left-right symmetry. Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 4

15 The full scalar potential then should be V = µ 2 σ σ 2 + µ 2 χ( χ L 2 + χ R 2 ) + µ 2 ij Tr(φ i φ j ) + λ σ σ 4 +λ χ ( χ L 4 + χ R 4 ) + λ χ χ L 2 χ R 2 + λ ijkl Tr(φ i φ j )Tr(φ k φ l ) +λ ijkl Tr(φ i φ j )Tr( φ k φ l ) + κ σχ σ 2 ( χ L 2 + χ R 2 ) + ρ ij σ 2 Tr(φ i φ j ) +ξ ij ( χ L 2 + χ R 2 )Tr(φ i φ j ) + ξ ij (χ L φ i φ j χ L + χt R φt i φ j χ R ) +[α ij σ 2 Tr( φ i φ j ) + β i σχ L φ i χ R + γ i σ χ φ L i χ R + H.c.]. The allowed Yukawa interactions should be L Y = y i q q L φ i q R y i l l L φ i l R h( l L χ L S R + l c R χ R S R ) 2 gσs R Sc R + H.c.. Clearly, the lepton and quark doublets, the Higgs doublets, the Higgs bidoublets, the fermion singlets and the scalar singlet can, respectively, belong to the 6 Fi, 6 H, 0 H,2, Fi and H representation in the SO(0) GUTs. Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 5

16 After the complex scalar singlet σ develops a vacuum expectation value to spontaneously break the global symmetry, it can be described by σ = 2 (f + ρ) exp ( i a f ). In the presence of the α, β, γ-terms in the potential, like the structure of the DFSZ model, the Nambu-Goldstone boson a can couple to the quarks, L 2f ( µa)( q L γ µ q L q R γ µ q R ) = 2f ( µa)(ūγ µ γ 5 u + dγ µ γ 5 d). Therefore, the global symmetry is the PQ symmetry and the Nambu-Goldstone boson is an axion. Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 6

17 The charged fermions and the neutral neutrinos have the Dirac masses as below, m d = yq φ y2 q φ 0 22, m u = yq φ 0 + y2 q φ 0 2, m e = yl φ0 2 + y2 l φ0 22, m ν = yl φ0 + y2 l φ0 2, Clearly, we can get the desired mass spectrum of the charged fermions. The fermion singlets also obtain their Majorana masses, M S = 2 gf. Therefore, the double and linear seesaw is ready. Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 7

18 Inflation and Flavor Mixing from PQ Symmetry Breaking We now consider a more attractive scenario where the PQ symmetry breaking can simultaneously lead to inflation (Guth, 8; Linde, 82; Albrecht, Steinhardt, 82 ) and flavor mixing. At the left-right level our model contains one Higgs bi-doublet for each family, two Higgs doublets, two leptoquark doublets and six complex singlets in the scalar sector while three neutral singlets and three generations of lepton and quark doublets in the fermion sector. The six scalar singlets are responsible for a U() 6 global symmetry breaking at the Planck scale. Because of the Yukawa interactions between the scalar and fermion singlets, the U() 6 symmetry is explicitly broken down to a U() 3 symmetry (Barbieri et al., 05 ; PHG, He, Sarkar, 07 ). Three Nambu- Goldstone bosons will obtain heavy masses through the Coleman-Weinberg (Coleman, Weinberg, 73 ) potential while the other three will pick up tiny masses through the color anomaly (Adler, 69 ; Bell, Jackiw, 69 ; Bardeen, 69 ). Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 8

19 The heavy and light pseudo Nambu-Goldstone bosons can act as the inflaton and the axion, respectively. This inflationary scenario can also avoid the cosmological domain wall problem (Sikivie, 82 ). In the absence of any off-diagonal Yukawa couplings involving the lepton and quark doublets, we can make use of the mixed fermion singlets to induce the lepton mixing by tree-level seesaw and the quark mixing by one-loop diagrams. Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 9

20 Besides three generations of fermion doublets and two Higgs doublets, our model contains three Higgs bi-doublets, two leptoquark doublets, η L (3, 2,, 3 ) = six complex singlets, φ i (, 2, 2, 0) = η +2/3 L η /3 L and three neutral fermion singlets, φ0 i φ i φ+ i2 φ0 i2,, η R (3,, 2, 3 ) = σ ij (,,, 0) = σ ji (,,, 0), S Ri (,,, 0). η +2/3 R η /3 R, Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 20

21 We assume a discrete left-right symmetry which is connected to chargeconjugation and the fields thus will transform as φ i φ T i, χ L χ R, η L η R, σ ij σ ij, q Li q c R i, l Li l c R i, S Ri S Ri. In the presence of the above left-right symmetry, we can impose a family symmetry U() F = U() 3, under which the left- and right-handed fermion doublets carry an equal but opposite charge for each family, i.e. ( δ i, δ i2, δ i3 ) for l Li l c R i and q Li q c R i. We also assign the U() F charges for other fields, ( δ i, δ i2, δ i3 ) for S Ri, (δ i + δ j, δ i2 + δ j2, δ i3 + δ j3 ) for σ ij, ( 2δ i, 2δ i2, 2δ i3 ) for φ i, (0, 0, 0) for χ L,R, η L,R. Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 2

22 We further introduce a global symmetry U() G, under which the fields carry the following quantum numbers, 2 for σ ij, for χ L χ R, η L η R, Sc R i, 0 for l Li l c R i, q Li q c R i, φ i. The allowed Yukawa interactions should be, L Y = y qi q Li φ i q Ri y li l Li φ i l Ri f i ( l Li χ L S Ri + l c R i χ R S R i ) h i ( q Li η L S Ri + q c R i η R S R i ) 2 g ij σ ij Sc R i S Rj + H.c.. Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 22

23 For simplicity, we do not write down the full scalar potential where α ij σ ii σ jj Tr φ i φ j + β ij σ2 ij Tr φ i φ j + H.c. should be absent due to the global symmetry U() G. Instead, we only give the part relevant for generating the mixing between the left- and right-handed leptoquarks, V λ i σ ii η L φ i η R + κη L χ L χ R η R + H.c.. Clearly, the lepton and quark doublets, the Higgs and leptoquark doublets, the Higgs bi-doublets, the fermion singlets and the scalar singlets can, respectively, belong to the 6 Fi, 6 H, 0 Hi, Fi and Hij representation in SO(0) GUTs. Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 23

24 Each scalar singlet σ ij has an independent phase transformation to perform a U() 6 symmetry. However, the six scalar singlets have the Yukawa interactions with the three fermion singlets so that the U() 6 symmetry should be explicitly broken down to a U() 3 symmetry. After the six scalar singlets develop their vacuum expectation values, there will be six Nambu-Goldstone bosons, i.e. σ ij = 2 (f ij + ξ ij )e iϕ ij f ij. The fermion singlets then obtain their Majorana masses M ij = 2 g ij f ij e iϕ ij f ij = M ij e iϕ ij f ij, which will result in a Coleman-Weinberg potential, V = 32π 2Tr ( M M M M ) ln with Λ being the ultraviolet cutoff. [ ( Λ 2 M M )] Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 24

25 Only three Nambu-Goldstone bosons can exist in the Coleman-Weinberg potential while the other three can be absorbed by the three fermion singlets. For example, we can take Clearly, we have ϕ ij the logarithm S Ri e i ϕ ii i ϕ ij 2f f ii S Ri and then M ij = M ij e ij ln with ϕ ij f ij = ϕ ij ϕ ii ϕ jj. f ij 2f ii 2f jj = 0 for i = j. By taking a reasonable simplification on ( Λ 2 M M ) constant = O(), the Coleman-Weinberg potential can be expanded by Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 25

26 V 32π 2 { (M 2 + M M 2 3 )2 + (M M M 2 23 )2 +(M M M 2 33 )2 + 2(M 2 M M 2 2 M M 2 3 M 2 23 ) +2(M 2 M M 2 2 M M 2 3 M 2 33 ) +2(M 2 2 M M 2 22 M M 2 23 M 2 33 ) +4M M 22 M 2 2 cos ( 2ϕ 2 f 2 +4M 22 M 2 33 M 2 23 cos ( 2ϕ 23 [ M cos ( ϕ 2 f 2 +M 33 cos ( ϕ 2 f 2 + ϕ 3 f 3 ϕ 3 f 3 ) ) f 23 ) ϕ 23 f 23 )]} ϕ 23 f M M 33 M 2 3 cos ( 2ϕ 3 f 3 + 8M 2 M 3 M 23 + M 22 cos ( ϕ 2. f 2 ϕ 3 f 3 ) ) + ϕ 23 f 23 Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 26

27 A combination of ϕ 2, ϕ 3 and ϕ 23 below, can have a potential of the form as V = µ 4 ( ± cos ϕ f ) with µ = O(M ij ), f = O(f ij ). The pseudo Nambu-Goldstone boson ϕ can realize the natural inflation for µ = O(0 5 GeV) and f = O(M Pl ). Note that the Majorana masses M ij should be determined by the Yukawa couplings g ij = O(0 4 ) for the given symmetry breaking scales f ij = O(M Pl ). Benefited from the inflation, our model can escape from the cosmological domain wall problem. Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 27

28 On the other hand, the three Nambu-Goldstone bosons absorbed in the three fermion singlets can have derivative couplings with the quarks, L 2f ii ( µ ϕ ii )( q Li γ µ q Li q Ri γ µ q Ri ) = 2f ii ( µ ϕ ii )(ū i γ µ γ 5 u i + d i γ µ γ 5 d i ). Therefore, the Nambu-Goldstone bosons ϕ ii can obtain their tiny masses through the color anomaly. Clearly, the pseudo Nambu-Goldstone bosons ϕ ii play the role of the invisible axion while the family symmetry U() F is identified with the PQ symmetry. Note for the PQ symmetry breaking at the Planck scale, we can (Pi, 84 ; Linde, 88 ) choose the initial value of the axion by the anthropic argument (Weinberg, 87 ) to give a desired dark matter relic density (Visinelli, Gondolo, 0 ). Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 28

29 The charged leptons have a diagonal 3 3 mass matrix, i.e. L m e ē L e R + H.c. with ( m e ) i = y li φ 0 i2. The mass terms involving the neutral leptons are given by L m ν ν L ν R f χ 0 L ν L S R f χ0 R νc R S R 2 MS c R S R + H.c. with ( m ν ) i = y li φ 0 i. For f χ 0 R and/or M much bigger than m ν and f χ 0 L, we can obtain the double/inverse and linear seesaw, i.e. L 2 m ν ν L ν c L + H.c. with m ν = m ν f T χ 0 R M f χ 0 R mt ν ( m ν + m T ν ) χ0 L χ 0 R. ν Lj ν Rj S Rj S Ri ν Ri ν Li ν Li ν S Ri ν Li Ri ν Li ν Ri S Ri ν Li FIG. 2: Tree-level seesaw for lepton masses and mixing. Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 29

30 Since the charged lepton mass matrix has a diagonal structure, the neutrino mass matrix should account for the lepton mixing described by the PMNS matrix. With the six elements in the Majorana mass matrix M, we have enough flexibilities to fit the known masses and mixing (Smirnov, 93 ). Note that with a left-right symmetry breaking scale χ 0 R = O(03 GeV), the double/inverse seesaw term should be the double seesaw for M = O(0 5 GeV). In this scenario, the inflaton should decay into the righthanded neutrinos through the off-shell fermion singlets. Subsequently, the decays of the right-handed neutrinos can realize the non-thermal or thermal leptogenesis. Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 30

31 At tree level the mass matrices of the down- and up-type quarks are both diagonal, L m 0 d d L d R m 0 uū L u R + H.c. with ( m 0 d ) i = y q i φ 0 i2 and ( m0 u) i = y qi φ 0 i. At one-loop order the leptoquarks can mediate the mixing of the fermion singlets to the quark sector. η /3 R (η 2/3 R ) η /3 L (η 2/3 L ) d Rj (u Rj ) S Rj S Ri d Li (u Li ) FIG. 3: One-loop diagrams for quark masses and mixing. Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 3

32 To calculate the loop-induced quark masses, we define the mass eigenstates of the leptoquarks, η 2/3 = η 2/3 L cos θ 2/3 η2/3 R sin θ 2/3, η 2/3 2 = η 2/3 L sin θ 2/3 + η2/3 R cos θ 2/3, η /3 = η /3 L cos θ /3 η/3 R sin θ /3, η /3 2 = η /3 L sin θ /3 + η/3 R cos θ /3, with the rotation angles, and the masses, θ 2/3 [0, π] for δ 2 2/3 = 0 or θ 2/3 = π 4 for δ2 2/3 0, θ /3 [0, π] for δ 2 /3 = 0 or θ /3 = π 4 for δ2 /3 0, M 2 η 2/3 2 M 2 η /3 2 M 2 η 2/3 M 2 η /3 = 2δ 2 2/3 sin 2θ 2/3, = 2δ 2 /3 sin 2θ /3. Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 32

33 Here we have defined δ 2 2/3 = 2 λ i f ii φ 0 i + κ χ0 L χ0 R, δ 2 /3 = 2 λ i f ii φ 0 i2. We further rotate the fermion singlets to diagonalize their Majorana mass matrix, U MU = diag{m S, M S2, M S3 }. Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 33

34 The loop-induced quark masses can be calculated by ( m d ) ij = sin 2θ /3 32π 2 h i h j U ik U kj T M S k M 2 η /3 M 2 η /3 M 2 S k ln M 2 η /3 M 2 S k ( m u) ij = sin 2θ 2/3 32π 2 h i h j U ik U kj T M S k M 2 η 2/3 M 2 η 2/3 M 2 S k ln M 2 η 2/3 M 2 S k M 2 η /3 2 M 2 η /3 2 M 2 S k ln M 2 η /3 2 M 2 S k sin2 2θ /3 h i h j M ij δ2 /3 6π 2 M 2 η 2/3 2 M 2 η 2/3 2 M 2 S k ln M 2 η /3 M 2 η 2/3 2 M 2 S k sin2 2θ 2/3 h i h j M ij δ2 2/3 6π 2 M 2 η 2/3,. Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 34

35 Therefore we have ( m d ) ij = ( m u) ij = sin 2θ /3 0 6 GeV M η /3 sin 2θ 2/3 0 6 GeV M η 2/3 ( ) hi h 2 j GeV, ( ) hi h 2 j GeV. M ij 0 5 GeV M ij 0 5 GeV δ 2 /3 0 9 GeV 00 GeV δ 2 2/3 0 9 GeV 00 GeV Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 35

36 We now check that the tree-level and loop-order quark mass matrices definitely can induce the desired quark masses and mixing for a proper parameter choice. Since the discrete left-right symmetry is charge-conjugation, the quark mass matrices can be diagonalized by m d = m 0 d + m d = V d diag{m d, m s, m b }V d, m u = m 0 u + m u = V u diag{m u, m c, m t }V u. The CKM matrix can be defined by V CKM = V u V d = V ud V us V ub V cd V cs V cb V td V ts V tb. Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 36

37 For demonstration, let s consider a limiting case where V d = V CKM, V u =. Now the off-diagonal quark masses can be given by ( m d ) 2 = m d V ud V cd + m sv us V cs + m b V ub V cb = ( m d ) 2, ( m d ) 3 = m d V ud V td + m sv us V ts + m b V ub V tb = ( m d ) 3, ( m d ) 23 = m d V cd V td + m sv cs V ts + m b V cb V tb = ( m d ) 32. Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 37

38 We then read ( m d ) 2 = ( m d ) 2 = MeV, ( m d ) 3 = ( m d ) 3 = MeV, ( m d ) 23 = ( m d ) 32 = MeV, by taking the CKM matrix (PDG 08 ) and the down-type quark masses (Fusaoka, Koide, 98 ; Xing, Zhang, Zhou, 08 ) at µ = m Z, V CKM = ± ± ± ± ± ± m d = MeV, m s = MeV, m b = 2.89 ± 0.09 GeV., Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 38

39 Summary The PQ symmetry can be embedded into the left-right symmetric models and then into the SO(0) GUTs. The PQ symmetry breaking can naturally lead to the double and linear seesaw for generating the small neutrino masses. The PQ symmetry breaking can simultaneously realize the natural inflation and the lepton and quark mixing. Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 39

40 Thanks! Pei-Hong Gu Institute of Theoretical Physics, Chinese Academy of Sciences, 20/2/200 40