COMPOSITE HIGGS MODEL

In the following section I discuss the Higgs sector of the Composite Higgs model. In this section we consider a general nature of the composite Higgs model where the symmetry in the strong interaction sector is G. We assume that the global symmetry G is dynamically broken to the subgroup H at the strong scale f ∼ TeV. And, we assume the subgroup H0 ⊂ G is gauged by external gauge bosons. In the symmetry breaking pattern G → H, n=dim(G)−dim(H) degrees of freedom become the Goldstone bosons. Among those the Goldstone bosons n₀ = dim(H₀)− dim(H₁) are eaten by the elementary gauge bosons, where H1 = H ∩H0 is the unbroken gauge symmetry after the EWSB. Then, the n−n0

Goldstone bosons become the physical Higgs bosons. In Fig. 11, we represent this condition of the composite sector by a simple cartoon.

For realistic model, we have to impose several conditions on the above idea of the strong dynamics. The first, the subgroup H have to contain the SM electroweak gauge symmetry as H ⊃ GSM = SU(2)L×U(1)Y. The second, the coset space G/H have to contain at least oneSU(2)L doublet Goldstone boson, which become the physical Higgs boson and the eaten Goldstone bosons. The third, the strong sector have to have a custodial symmetry to prevent from large contributions of the Peskin-Takeuchi T parameter and Zb¯b corrections.

When we take into account these conditions, the minimal model is G=SO(5), H =SO(4) and H0 =GSM. ¹

A. Minimal Composite Higgs model

In this section, we consider the minimal symmetry breaking pattern SO(5) → SO(4) ∼ SU(2)L×SU(2)R at the strong scale f ∼ TeV. And, the SM subgroup SU(2)L×U(1)Y

of H is gauged. The coset space SO(5)/SO(4) corresponds to four real Goldstone bosons, which transform as a 4 of the SO(4). Among them, one Goldstone boson is the physical Higgs boson and the others are eaten by the three gauge bosons so that there is no extra physical Higgs bosons. We introduce an extra global U(1)X for the correct hyper charges

1 In the paper [4], the composite Higgs model is considered with the global symmetryG=SU(3)_L. However, in this case, the model does not have the custodial symmetry. Therefore,T parameter andZb¯bcorrections are large and allowed regions are very small.

H¹ H H⁰

G →H n elementary

sector

sector composite

n₀

Higgs

FIG. 11: A general condition of symmetry breaking in the composite sector. In this thesis we consider the minimal model, G=SO(5), H =SO(4) and H₀ =SU(2)×U(1)_Y. In the condition n= 4, n₀ = 3 and one physical Higgs boson.

of the composite fermions. The hypercharges of the composite fermions are obtained by Y =T_R³+X. The composite Higgs models of these set-up are called the Minimal Composite Higgs Model (MCHM).

The four Goldstone bosons can be parametrized by the broken generatorsT^ˆa(ˆa= 1 ∼4) of the coset space SO(5)/SO(4) and the four real scalars h^aˆ. From the nonlinear sigma model, the Goldstone matrix Σis given by

Σ=Σ0e^Π/f, Σ0 = (0,0,0,0,1), Π=−iT^ˆah^ˆa√

2, (68)

where

(T^ˆa)ij =− i

√2(δ_i^ˆaδ_i⁵−δ^a_j^ˆδ⁵_i). (69) Then, the Σ can be written by

Σ= sinh/f

h (h¹, h², h³, h⁴, hcoth/f), h= /"

(h^ˆa)². (70) In the unitary gauge the vacuum expectation value (VEV) can be aligned to the any direction by a SO(4) rotation. If we identify theh4 with the physical Higgs bosonH, theΣ

is given by

Σ=Σ0U (71)

=Σ0

⎛

⎜

⎜

⎜

⎝

1₃ 0 0

0 cosH/f sinH/f 0 −sinH/f cosH/f

⎞

⎟

⎟

⎟

⎠

(72)

= (0,0,0,sinH/f ,cosH/f), (73)

where we define the matrix U for later convenience.

B. The effective Lagrangian of the gauge sector

Before discussing the explicit models, we consider the effective Lagrangian of the MCHM after integrating out the resonances of the composite sector. This procedure is useful to study properties of the composite Higgs model based on the symmetry and to calculate the loop induced Higgs potential in following section. At first, we consider the effective Lagrangian of the gauge sector. For convenience, we assume that the global symmetry SO(5)×U(1)X

is gauged by introducing extra non-dynamical gauge bosons. Then, we can derive a general SO(5)×U(1)X invariant effective action of the SM gauge bosons. After integrating out the particles which are heavy due to the strong dynamics the effective Lagrangian of the gauge bosons at the quadratic level is given by

L^gauge_eff = 1

2P_T^µν[Π^X₀ (p)XµXν +Π0(p)Tr(AµAν) +Π1(p)ΣAµAνΣ^T], (74) whereP_T^µν =η^µν−p^µp^ν/p² andXµis the U(1)X gauge boson andAµ=A^a_µT^a+A^ˆa_µT^ˆa is the SO(5) gauge bosons of the unbroken and the broken generators. The unbroken generators (T_L,R^a )ij (a= 1,2,3, i, j = 1, ...,5) are given by

(T_L^a)ij =−i 2

%1

22^abc(δ_i^bδ^c_j −δ_j^bδ_i^c) +δ_i^aδ_j⁴−δ_i⁴δ_j^a( , (T_R^a)ij =−i

2

%1

22^abc(δ_i^bδ^c_j −δ_j^bδ_i^c)−δ_i^aδ_j⁴+δ_i⁴δ_j^a( ,

(75)

where L, R mean the symmetry SO(4) ∼ SU(2)_L×SU(2)_R. The broken generators are Eq. 69. The form factors Π0,1,Π^X₀ come from the strong dynamics, and we can not pertur-batively calculate within the 4D effective theory. We can perturbatively calculate the form

factors in the 5D Adsmodel or the deconstructed 4D model. The poles of the form factors are gauge resonances. The form factors contain effects of mixings between the elementary and the composite gauge bosons. When we expand on theSO(4) preserving vacuumΣ=Σ0, the effective action can be written by

L^gauge_eff = 1

2P_T^µν[Π^X0 (p)XµXν +Πa(p)A^a_µA^a_ν +Πˆa(p)A^ˆa_µA^ˆa_ν] (76) whereΠa=Π0,Πˆa =Π0+Π1/2 are the form factors of the unbroken and broken generators.

In the large N expansion the Πa, Πˆa are written by an infinite sum of resonances, Πa =p²"

n

fρn

p²+m²_ρn Πˆa =p²"

n

fan

p²+m²_an +1

2f², (77)

where mρ_n, man are masses of the gauge resonances at n-th level of 6-plet and 4-plet of SO(4), while fρn and fan are decay constants of the gauge resonances respectively. Then, the Π0,Π1 at the zero momentum are given by

Π0(0) =Π^X₀ (0) = 0, Π1(0) =f². (78) By using Eq. 73 and 74 and disappearing the non-dynamical gauge bosons, the effective Lagrangian of the elementary gauge bosons are given by

L^gauge_eff = 1 2P_T^µν%#

Π^X₀ (p) +Π^X₀ (p)) + sin²(H/f)

4 Π1(p)$ BµBν

+#

Π0(p) + sin²(H/f)

4 Π1(p)$

A^a_µA^a_ν + 2 sin²(H/f)Π1(p) ˆH^†T^aYHAˆ ^a_µBν

(,

(79)

where A^a_µ, Bµ are SU(2)L×U(1)Y gauge bosons and ˆH is defined as Hˆ = 1

h

⎛

⎝

h¹−ih² h³−ih⁴

⎞

⎠. (80)

The ˆH^t = (0,1) when the Higgs vev is aligned along the h³ direction by a SO(4) rotation.

When we expand the form factors at small momenta and use Eq. 78, the effective Lagrangian is given by

L^gauge_eff = 1 2P_T^µν%1

2

#f²sin²(⟨H⟩/f) 4

$

(BµBν +W_µ³W_ν³−2W_µ³Bν) +#f²sin²(⟨H⟩/f)

4

$W_µ⁺W_ν⁻ + p²

2 [Π^′₀(0)W_µ^aW_ν^a+ (Π^′₀(0) +Π^X₀ ^′(0))BµBν] +...( .

(81)

Therefore, relations among the form factors and the gauge couplings are given by 1

g² =−Π^′₀(0), 1

g^′2 =−(Π^′₀(0) +Π^X₀ ^′(0)). (82) .

The Higgs bosons kinetic terms for canonical normalization of gauge boson given by L^Higgs_kin = f²

2 (DµΣ)(D^µΣ)^T, DµΣ=∂_µΣ+W_µ^aΣT_L^a+BµΣT_R³, (83)

= 1

2∂_µH∂^µH+ g²f²

4 sin(H/f)(W_µ⁺W^µ−+ 1

2 cos²θ_WZ_µZ^µ). (84) When we expand the Higgs boson around the vevH → ⟨H⟩+H, we obtain

f²

4 sin(H/f) = f²[sin²(⟨H⟩/f) + 2 sin(⟨H⟩/f) cos(⟨H⟩/f)(H/f) + (1−2 sin²(⟨H⟩/f)(H/f)²+...]

=v²+ 2v7

1−ξH+ (1−2ξ)H²+...,

(85)

where ξ=v²/f². Therefore, the Higgs vev is given by v =fsin⟨H⟩

f = 246 GeV. (86)

and the Higgs couplings to the gauge bosons V =W, Z are given by gHV V =g_{HV V}^SM 7

1−ξ gHHV V =g_{HHV V}^SM 7

1−2ξ. (87)

Therefore, deviations of the Higgs coupling with the SM massive gauge bosons depend on only the strong breaking scale f.

C. The effective Lagrangian of the fermion sector

Next, we consider the effective Lagrangian of fermions. The effective Lagrangian of fermion depends on representations under the global symmetry G of the composite sector.

Main contributions to the loop induced Higgs potential come from the top sector since the compositeness of the top sector is large. Therefore, the dependence on the representation in the effective Lagrangian of the 3rd generation is important for the EWSB. And, properties of the Higgs couplings different among the representations.

At first, we consider the4(spinor) representation. This choice is minimal in the composite Higgs model of the SO(5)/SO(4) coset space and called MCHM4. Like the gauge boson

case we embed the elementary fermions q_L, u_R, d_R in spinor representations of SO(5) with non-dynamical fields. The spinor representation of SO(5) consists of SU(2)L and SU(2)R

doublets. The spinor representations can be written by

Ψ⁴q =

⎡

⎣ qL

QL

⎤

⎦, Ψ⁴u =

⎡

⎢

⎢

⎢

⎣ q^u_R

⎛

⎝ uR

d^′_R

⎞

⎠

⎤

⎥

⎥

⎥

⎦

, Ψ⁴d=

⎡

⎢

⎢

⎢

⎣ q_R^d

⎛

⎝ u^′_R dR

⎞

⎠

⎤

⎥

⎥

⎥

⎦

, (88)

where the U(1)X charge of these fields are 1/6. Then, the fermion effective Lagrangian of the spinor representation is given by

Lf ermion(4)

ef f = "

r=q,u,d

Ψ¯⁴_r ̸p[Π^r₀(p) +Π^r₁(p)ΓⁱΣi]Ψ⁴_r+ "

r=u,d

Ψ¯⁴_q[M₀^r(p) +M₁^r(p)ΓⁱΣi]Ψ⁴_r, (89) where

ΓⁱΣi =

⎛

⎝

1cos(H/f) σˆsin(H/f) ˆ

σ^†sin(H/f) −1cos(H/f),

⎞

⎠ σˆ ={−→σ,−i1}H^ˆa/H (90) and Π^r0,1, M_0,1^r are form factors of the composite fermions. The form factors contain effects of mixings between the elementary fields and the composite fields. By setting the non-dynamical fields are zero, the effective Lagrangian of the qL, uR, dR is given by

Lf ermion(4)

ef f = ¯qL ̸p(Π^q₀(p) +Π^q₁(p) cos(H/f))qL+ ¯uR ̸p(Π^u₁(p)−Π^u₁(p) cos(H/f))uR

+ ¯dR̸p(Π^u1(p)−Π^u1(p) cos(H/f))dR

+ sin(H/f)M₁^u(p)¯q_LHu_R+ sin(H/f)M₁^d(p)¯q_LHd_R+h.c.

(91)

After normalizing the kinetic term, we approximately obtain fermion masses of the4 repre-sentation m⁴_u,d at the low energy limit p→0

m⁴_u,d ∼ M₁^u,d(0) /Z_q⁴Z_u,d⁴

shch ∼ M₁^u,d(0) /Z_q⁴Z_u,d⁴

v

f, (92)

where Z_q⁴ = Π^q₀(0) + chΠ^q₁(0) and Z_u,d⁴ = Π^u,d₀ (0) − chΠ^u,d₁ (0), sh = sin(H/f) and ch = cos(H/f). By expanding Lf ermion(4)

ef f we obtain approximately deviations of the Higgs couplings to the fermions like the gauge bosoms

gHf f =g_{Hf f}^SM 7

1−ξ. (93)

The allowed parameter regions of the model compatible with current precision measurements is very small from the Zb¯b corrections. Therefore, we don’t consider this model in the following.

Next, we consider the 5(fundamental) representation of SO(5). This model is called the MCHM5. The 5 representation is written by 4 and 1 in the SO(4) language, 5 = 4⊕1.

In the MCHM5 we can not simultaneously generate bottom and top quark masses by one U(1)X charge. We introduce four fundamental multiplets Ψ⁵_q1(X = 2/3),Ψ⁵_q2(X = −1/3), Ψ⁵u(X = 2/3) and Ψ⁵d(X = −1/3) with another U(1)X charge for this purpose. The top quark mass term is generated by mixing between Ψ⁵q1(X = 2/3) and Ψ⁵u(X = 2/3). On the other hand, the bottom quark mass term is generated by mixing between Ψ⁵_q2(X =−1/3) and Ψ⁵d(X = −1/3). We embed the elementary fields into the 5 representation with non-dynamical fields.

Ψ⁵q1 =

⎡

⎢

⎢

⎢

⎣

⎛

⎝ q^′_1L

qL

⎞

⎠

u^′_L

⎤

⎥

⎥

⎥

⎦

Ψ⁵q2 =

⎡

⎢

⎢

⎢

⎣

⎛

⎝ qL

q_2L^′

⎞

⎠

d^′_L

⎤

⎥

⎥

⎥

⎦

Ψ⁵u =

⎡

⎢

⎢

⎢

⎣

⎛

⎝ q_R^u q^′_R^u

⎞

⎠

u_R

⎤

⎥

⎥

⎥

⎦

Ψ⁵d =

⎡

⎢

⎢

⎢

⎣

⎛

⎝ q^′d_R q_R^d

⎞

⎠

d_R

⎤

⎥

⎥

⎥

⎦

, (94)

where the fields except theqL, uR anddRare non-dynamical fields. The effective Lagrangian of the MCHM5 is given by

Lf ermion(5)

ef f = "

r=q1,q2,u,d

Ψ⁵r

i ̸p(δ^ijΠ^r0(p) +ΣⁱΣ^jΠ^r1(p))Ψ⁵r

j + ¯Ψ⁵q1ⁱ(δ^ijM₀^u(p) +ΣⁱΣ^jM₁^u(p))Ψ⁵u^j

+ ¯Ψ⁵_q2ⁱ(δ^ijM₀^d(p) +ΣⁱΣ^jM₁^d(p))Ψ⁵_d^j +h.c.

(95) By expanding the Lagrangian, we obtain the effective Lagrangian of the elementary fields qL, uL and dR,

Lf ermion(5)

ef f = ¯q_L ̸p[Π^q10 (p) +Π^q20 (p) + s²_h

2(Π^q11 (p) +Π^q21 (p))]q_L

+ ¯uR̸p(Π^u₀(p) +c²_hΠ^u₁(p))uR+ ¯dR ̸p(Π^d₀(p) +c²_hΠ^d₁(p))dR

+s_hc_h

√2 (¯qLM₁^u(p)uR+ ¯qLM₁^ddR) +h.c.,

(96)

where the form factors contain the effects of the mixing between the elementary and the composite sector. Then, the SM fermion masses m⁵_u,d is approximately given by

m⁵_u,d ∼ shch

√2

M₁^u,d(0)

/Z_q⁵Z_u,d⁵ ∼ M₁^u,d(0) /2Z_q⁵Z_u,d⁵

v

f, (97)

where Z_q⁵ =Π^q10 (0) +Π^q20 (0) +^s₂^2h(Π^q11 (0) +Π^q21 (0)), Z_u,d⁵ =Π^u,d0 (0) +c²_hΠ^u,d1 (0).

By using the m⁵_u,d we can discuss the yukawa coupling. The form factors can be decom-posed in the SO(4) representations and given by

Π^q1₀ (p) = 1 +Π^q_Q¹₄(p), Π^q2₀ (p) = 1 +Π^q_Q²₄(p), Π^q1₁ (p) =Π^q_Q¹₁(p)−Π^q_Q¹₄(p),Π^q2₁ (p) = Π^q_Q²₁(p)−Π^q_Q²₄(p), Π^u₀(p) = 1 +Π^u_Q4(p), Π^u₁(p) =Π^u_Q1(p)−Π^u_Q4(p), Π^d₀(p) = 1 +Π^d_Q4(p),

Π^d1(p) =Π^dQ1(p)−Π^dQ4(p),

M₁^u(p) =M_Q^u₁(p)−M_Q^u₄(p), M₁^d(p) = M_Q^d₁(p)−M_Q^d₄(p),

(98) where by large N expansion the form factors under theSO(4) symmetry are given by

Π^q_Q¹_4,1^,q2(p) ="

n

|f_Q^q1_4,1^,q2|² p²+ (m^u,d

Q⁽ⁿ⁾_4,1)², Π^u,d_Q4,1(p) ="

n

|f_Q^u,d_4,1|² p²+ (m^u,d

Q⁽ⁿ⁾_4,1)², M_Q^u,d_4,1(p) ="

n

f_Q^q1_4,1^,q2f_Q^u,d_4,1m^u,d

Q⁽ⁿ⁾_4,1

p²+ (m^u,d

Q⁽ⁿ⁾_4,1)² .

(99)

The m^u,d

Q⁽ⁿ⁾_4,1 is fermion resonance masses and the f_Q^q1_4,1^,q2, f_Q^u,d_4,1 corresponds to elementary-composite mixing mass parameters. We take into account only the lightest resonances in each representation of SO(4), m^u,d

Q⁽ⁿ⁾_4,1 = M_u⁵_4,1,d4,1. And, we define f_Q^q1₄ = f_Q^q1₁ = y⁵_uLf, f_Q^q2₄ = f_Q^q2₁ = y_dL⁵ f, f_Q^u₄ = f_Q^u₁ =y_uR⁵ f, f_Q^d₄ = f_Q^d₁ = y⁵_dRf. These notations will be used in the fol-lowing discussions of phenomenologies. Then, deviations of the yukawa couplings are given by

y⁵_u m⁵_u = 1

m⁵_u

∂m⁵_u

∂v = 2

ftan(2v/f) + f

2Z_q⁵Z_u⁵_R sin(2v/f)%, 1

|M_u⁵₄|² − 1

|M_u⁵₁|² .

(|y_uL⁵ |²/2−|y_uR⁵ |²) +

, 1

|M_d⁵₄|² − 1

|M_d⁵₁|² .

|y⁵_dL|²/2− f²|y_uR⁵ |² 2

, 1

|M_u⁵₄|² − 1

|M_u⁵₁|²

. ,|y⁵_uL|²

|M_u⁵₄|² + |y_dL⁵ |²

|M_d⁵₄|² .(

, y_d⁵

m⁵_d = 1 m⁵_d

∂m⁵_d

∂v = 2

ftan(2v/f)+ f

2Z_q⁵Z_d⁵_R sin(2v/f)%, 1

|M_u⁵₄|² − 1

|M_u⁵₁|² .

|y_uL⁵ |²/2 +

, 1

|M_d⁵₄|² − 1

|M_d⁵₁|² .

(|y⁵_dL|²/2−|y⁵_dR|²)− f²|y_dR⁵ |² 2

, 1

|M_d⁵₄|² − 1

|M_d⁵₁|²

. ,|y_uL⁵ |²

|M_u⁵₄|² + |y_dL⁵ |²

|M_d⁵₄|² .(

. (100) The deviations of the yukawa couplings in the first order is given by

2

ftan(2v/f) = 1−2ξ

√1−ξ. (101)

The contributions from the second order can be large in typical relations among the elementary-composite mixing parameters and resonance masses. The deviation from the first order approximation may be measured at the ILC. I discuss about the yukawa cou-plings more precisely in the following sections. In the 5 representation corrections of the EWPT are relatively small. Therefore, in the following section, we discuss phenomenolo-gies of the 5 representation in detail with the explicit SO(4) fermion resonances and the elementary-composite mixings.

Finally, we consider the 10 (anti-symmetric) representation. This model is called the MCHM10. In the SO(4), the 10 representation is written by 4 and 6, 10 =4⊕6. In the MCHM10 the masses of the up-type quark and bottom-type quark can be simultaneously generated by one U(1)X charge X = 2/3. The elementary fields embedded on the 10 representation of SO(5) are given by

Ψ¹⁰_q = 1 2

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

dL

−idL

0

_uL

iuL

dL idL uL −iuL 0

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

, Ψ¹⁰_u = 1 2

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

−uR

uR

u_R 0

−uR

0 0

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠ ,

Ψ¹⁰d = 1 2

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

i^√dR₂ −^d√R₂

d_R

√2 i^d√R₂

−i^√dR₂ −^d√R₂ 0

dR

√2 −i^d√R₂

0 0

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠ .

(102)

where the non-dynamical fields have already been vanished. The effective Lagrangian of the MCHM10 is given by

Lf ermion(10)

ef f = "

r=q,u,d

[Tr( ¯Ψ¹⁰_r ̸pΠ^r₀(p)Ψ¹⁰_r ) +ΣΨ¯¹⁰_r ̸pΠ^r₁(p)Ψ¹⁰_r Σ^T] + "

r=u,d

[Tr( ¯Ψ¹⁰q M^r₀(p)Ψ¹⁰r ) +ΣΨ¯¹⁰q M^r₁(p)Ψ¹⁰r Σ^T] + h.c.

(103)

By expanding, we obtain the effective Lagrangian of the dynamical fields, Lf ermion(10)

ef f = ¯uL ̸p[Π^q₀(p) + (c²_h 2 +s²_h

4)Π^q₁(p)]uL

+ ¯d_L̸p[Π^q0(p) + c²_h

2Π^q1(p)]d_L + ¯uR ̸p(Π^u0(p) + s²_h

4 Π^u1(p))uR+ ¯dR ̸p(Π^d0(p) + s²_h

4Π^d1(p))dR

−shch

4 q¯LM₁^u(p)uR− shch

2√

2q¯LM₁^ddR

(104)

The fermion masses m¹⁰_u,d are approximately given by m¹⁰_u ∼ shch

4

M₁^u(0)

7Z_u¹⁰_LZ_u¹⁰_R, m¹⁰_d ∼ shch

2√ 2

M₁^d(0) /Z_d¹⁰_LZ_d¹⁰_R

(105)

whereZ_u¹⁰_L =Π^q0(0)+(^c₂^2h+^s₄^2h)Π^q1(0),Z_d¹⁰_L =Π^q0(0)+^c₂^2hΠ^q1(0) andZ_u¹⁰_R,dR =Π^u,d0 (0)+^s₄^2hΠ^u,d1 (0).

The form of the top and bottom masses is different from that of the 5 representation.

LIke the 5 we discuss the yukawa couplings in the 10. The form factors can be decom-posed in the SO(4) representations,

Π^q0(p) = 1 +Π^qQ6(p), Π^u0(p) = 1 +Π^uQ6(p), Π^q1(p) = 2(Π^qQ4(p)−Π^qQ6(p)),

Π^u1(p) = 2(Π^uQ4(p)−Π^uQ6(p)), Π^d0(p) = 1 +Π^dQ6(p),Π^d1(p) = 2(Π^dQ4(p)−Π^dQ6(p)), M₁^u(p) = 2(M_Q^u₄(p)−M_Q^u₆(p)), M₁^d(p) = 2(M_Q^d₄(p)−M_Q^d₆(p)),

(106)

where by large N expansion the form factors under theSO(4) symmetry are given by Π^qQ4,6(p) ="

n

|f_Q^q_4,6|² p²+ (m_Q(n)

4,6)², Π^u,dQ4,6(p) ="

n

|f_Q^u,d_4,6|² p²+ (m_Q(n)

4,6)², M_Q^u,d_4,6(p) ="

n

f_Q^q_4,6f_Q^u,d_4,6m_Q⁽ⁿ⁾

4,6

p²+ (m_Q(n) 4,6)² .

(107)

We take into account the lightest modes of them_Q⁽ⁿ⁾

4,6 =M_4,6¹⁰. When we define f_Q^q₄ =f_Q^q₆ = y¹⁰_Lf, f_Q^u₄ = f_Q^u₆ = y¹⁰_uRf, f_Q^d₄ = f_Q^d₆ = y¹⁰_dRf, deviations of the yukawa couplings in the 10

representation are given by y_u¹⁰

m¹⁰_u = 1 m¹⁰_u

∂m¹⁰_u

∂v = 2

ftan(2v/f) + f

4Z_u¹⁰_LZ_u¹⁰_R sin(2v/f)%, 1

M₄¹⁰² − 1 M₆¹⁰²

.

(y_L¹⁰²−y¹⁰_uR²)

− f²y_L¹⁰²y_uR¹⁰² 2

, 1

M₄¹⁰² − 1 M₆¹⁰²

.2

(, y_d¹⁰

m¹⁰_d = 1 m¹⁰_d

∂m¹⁰_d

∂v = 2

ftan(2v/f) + f

2Z_d¹⁰_LZ_d¹⁰_R sin(2v/f)%, 1

M₄¹⁰² − 1 M₆¹⁰²

.

(y_L¹⁰²−y_dR¹⁰²/2)

− f²y_L¹⁰²y_dR¹⁰² 2

, 2

M₄¹⁰⁴ + 1

M₆¹⁰⁴ − 1 M₄¹⁰²M₆¹⁰²

.( ,

(108) where the approximation of the first order is the same as the 5 represantation. We also discuss phenomenologies of the10representation with the explicitSO(4) fermion resonances in the following section.

ドキュメント内 本文 Thesis 総合研究大学院大学学術情報リポジトリ A1751本文 (ページ 36-46)