Dynamical Computation on Coefficients of Electroweak Chiral

TUHEP-TH-07157

Dynamical Computation on Coefficients of Electroweak Chiral Lagrangian from One-doublet and Topcolor-assisted Technicolor Models Hong-Hao Zhang∗ , Shao-Zhou Jiang† , and Qing Wang‡§

arXiv:0705.0115v2 [hep-ph] 2 May 2007

Department of Physics, Tsinghua University, Beijing 100084, China¶ Center for High Energy Physics, Tsinghua University, Beijing 100084, China

Abstract Based on previous studies deriving the chiral Lagrangian for pseudo scalar mesons from the first principle of QCD, we derive the electroweak chiral Lagrangian and build up a formulation for computing its coefficients from one-doublet technicolor model and a schematic topcolor-assisted technicolor model. We find that the coefficients of the electroweak chiral Lagrangian for the topcolor-assisted technicolor model are divided into three parts: direct TC2 interaction part, TC1 and TC2 induced effective Z ′ particle contribution part, and ordinary quarks contribution part. The first two parts are computed in this paper and we show that the direct TC2 interaction part is the same as that in the one-doublet technicolor model, while effective Z ′ contributions are at least proportional to the p2 order parameter β1 in the electroweak chiral Lagrangian and typical features of topcolor-assisted technicolor model are that it only allows positive T and U parameters and the T parameter varies in the range 0 ∼ 1/(25α), the upper bound of T parameter will decrease as long as Z ′ mass become large. The S parameter can be either positive or negative depending on whether the Z ′ mass is large or small. The Z ′ mass is also bounded above and the upper bound depend on value of T parameter. We obtain the values for all the coefficients of the electroweak chiral Lagrangian up to order of p4 . PACS numbers: 11.10.Lm, 11.30.Rd, 12.10.Dm, 12.60.Nz

∗ † ‡ § ¶

Email: [email protected]. Email: [email protected]. Email: [email protected]. corresponding author mailing address

1

Contents

3

I. Introduction II. Derivation of the Electroweak Chiral Lagrangian from the One-doublet Technicolor Model

5

A. The Gasser-Leutwyler’s Prescription

7

B. The Dynamical Computation Prescription

8

1. Schwinger-Dyson Equation for Techniquark Propagator

11

2. Effective Action

15

C. Comparison and Discussion on Two Prescriptions

19

III. Derivation of the Electroweak Chiral Lagrangian from a Topcolor-assisted Technicolor Model

20

A. Topcolor Symmetry Breaking: the Contribution of SU(3)TC1

22

1. The Gap Equation

25

2. Low Energy Expansion for iSTC1 [0, BµA , Bµ , Zµ′ ]

28

B. Electroweak Symmetry Breaking: the Contribution of SU(3)TC2

30

C. Integrate out of Colorons

35

D. Integrate out of Z ′

37 43

IV. Conclusion

45

Acknowledgments

46

A. Two Equivalent EWCL Formalisms (4c,B A )

B. STC1

(4c,B)

(4c,Z ′ )

(4c,B A Z ′ )

[B A ], STC1 [B], STC1 , STC1

0 int C. Scoloron [B A , Z ′ ] and Scoloron [B A , Z ′]

(4c,BZ ′ )

[B A , Z ′ ], STC1

[B, Z ′ ]

48 50 51

References

2

I.

INTRODUCTION

The electroweak symmetry breaking mechanism (EWSBM) remains an intriguing puzzle for particle physics, although the Standard Model (SM) provides us with a version of it through introducing a higgs boson into the theory which suffers from triviality and unnaturalness problems. Beyond SM, numerous new physics models are invented which exhibit many alternative EWSBMs. With the present situation that higgs is still not found in experiment, all new physics models at low energy region should be described by a theory which not only must match all present experiment data, but also have no higgs. This theory is the well-known electroweak chiral Lagrangian (EWCL) [1, 2, 3] which offers the most general and economic description of electroweak interaction at low energy region. With EWCL, new physics models at low energies can be parameterized by a set of coefficients, it universally describes all possible electroweak interactions among existing particles and offers a model independent platform for us to investigate various kinds EWSBMs. Starting from this platform, further phenomenological research focus on finding effective physical processes to fix the certain coefficients of EWCL [4, 5, 6], and theoretical studies concentrate on consistency of EWCL itself such as gauge invariance [7] and computing the values of the coefficients for SM with heavy higgs [8]. Systematic theoretical computation of the EWCL coefficients for other new physics models have not been presented in the literature. The possible reasons are that for weakly coupled models, since one can perform perturbative computations, people prefer to directly discuss physics from the model and then are reluctant to pay the extra price to compute EWCL coefficients. While for strongly coupled models, non-perturbative difficulties for a long time prevent people to perform dynamical computations, only for special coefficients such as S parameter, some non-perturbative technique may be applied to perform calculations [9] or for special QCD-like technicolor models, in terms of their similarities with QCD, one can estimate the coefficients of EWCL in terms of their partners fixed by experimental data in corresponding QCD chiral Lagrangian. The estimation of EWCL coefficients for various models is of special importance in the sense that at present we already have some quantitative constraints on them, such as those for the S, T, U parameters and more generally for anomalous triple and quartic couplings [10], along with the experimental progress, more constraints will be obtained. Once we know the values of the coefficients for detailed models, these constraints can directly be used to judge 3

the correctness of the model. It is the purpose of this paper to develop a formulation to systematically compute coefficients of EWCL for strongly coupled new physics models. For simplicity, in this work we only discuss the bosonic part of EWCL and leave the matter part for future investigations. The basis of our formulation is the knowledge and experiences we obtained previously from a series of works deriving the chiral Lagrangian for pseudo scalar mesons from QCD first principles [11] and calculate coefficients in it [12, 13, 14], where we found confidence and reliability in this work. In fact, the formal derivation from a general underlying technicolor model to EWCL was already achieved in Ref.[15] in which except deriving EWCL, coefficients of EWCL are formally expressed in terms of Green’s functions in underlying technicolor model. Once we know how to compute these Green’s functions, we obtain the corresponding EWCL coefficients. The pity is that the computation is nonperturbative, therefore not easy to achieve, it is the aim of this paper to solve this nonperturbative dynamical computation problems. As the first step of performing dynamical computations, we especially care about the reliability of the formulation we will develop. We take one-doublet technicolor model [16, 17, 18, 19] as the prototype to build up our formulation. Although this model as the earliest and simplest dynamical electroweak symmetry breaking model was already denied by experiment in the sense that it results in too large a value for the S parameter, but due to the following reasons, we still start our investigations from here. First, it is similar as QCD in the theory structures which enable us to easily generalize the techniques developed in dealing with the QCD chiral Lagrangian to this model and we call this generalized formulation the dynamical computation prescription. Second due to their similarities with conventional QCD, the coefficients of their EWCLs can be estimated by just scaling-up corresponding coefficients in QCD Gasser-Leutwyler chiral Lagrangian for pseudo scalar mesons [20] and we call this formulation the Gasser-Leutwyler’s prescription which naively is only applicable for those QCD-like models. So for QCD-like models, we have two prescriptions which enable us to compare them with each other to check the correctness and increase the reliability of our formulations. Beyond the traditional one-doublet technicolor model, we choose the topcolorassisted technicolor model as the first real practice model to perform our computations. The reason to take it is that this model is not QCD-like and is active on the market now which does not seriously contradict with experimental data as the case of one-doublet technicolor model and the dynamics responsible for electroweak symmetry breaking is similar to that 4

in the one-doublet technicolor model. We will find that the coefficients for topcolor-assisted technicolor model can be divided into three parts: direct TC2 interaction part,TC1 and TC2 induced effective Z ′ particle contribution part and ordinary quarks contribution part. The first two parts are computed in this paper and we show that direct TC2 interaction part is same as that in one-doublet technicolor model, while TC1 and TC2 induced effective Z ′ particle contributions is at least proportional to p2 order parameter β1 in EWCL and typical features of topcolor-assisted technicolor model are that it only allows positive T and U parameters and the T parameter varies in the range 0 ∼ 1/(25α),the upper bound of T parameter will decrease as long as Z ′ mass become large. The S parameter can be either positive or negative depending on whether the Z ′ mass is large or small. The Z ′ mass is also bounded above and the upper bound depend on value of T parameter. We obtain the values for all the coefficients of the electroweak chiral Lagrangian up to order of p4 . This paper is organized as follows. Section II is the basics of the work in which we discuss one-doublet technicolor model. We first review the Gasser-Leutwyler’s prescription, then build up our dynamical computation prescription, we show how to consistently set in the dynamical computation equation (the Schwinger-Dyson equation) into our formulation. We make comparison between two prescriptions to check validity of the results from our dynamical computation prescription. Section III is the main part of this work in which we apply our formulations developed in one-doublet technicolor model to topcolor-assisted technicolor model. We perform dynamical calculations on technicolor interactions and then integrate out colorons and Z ′ to compute EWCL coefficients. Since this is the first time to systematically perform dynamical computations on the strongly coupled models, we emphasize the technical side more than physics analysis and display the computation procedure a little bit more in detail. Section IV is the conclusion. In the appendices, we list some requisite formulae.

II.

DERIVATION OF THE ELECTROWEAK CHIRAL LAGRANGIAN FROM

THE ONE-DOUBLET TECHNICOLOR MODEL

Consider the one-doublet technicolor (TC) model proposed by Weinberg and Susskind independently [16,

17,

18,

19].

The techniquarks are assigned to

(SU(N)T C , SU(3)C , SU(2)L , U(1)Y ) as ψL ∼ (N, 1, 2, 0), UR = (1/2 + τ 3 /2)ψR ∼ 5

(N, 1, 1, 1/2), DR = (1/2 − τ 3 /2), ψR ∼ (N, 1, 1, −1/2). With these assignments, the techniquarks have electric charges as defined by Q = T3 + Y , of +1/2 for U and −1/2 for D. It can be shown below, by dynamical analysis through the Schwinger-Dyson equation, that ¯ = the SU(N)T C interaction induces the techniquark condensate hψψi 6 0, which will trigger the electroweak symmetry breaking SU(2)L × U(1)Y → U(1)EM . Neglecting ordinary fermions and gluons, we focus on the action of the techniquark, technicolor-gauge-boson and electroweak-gauge-boson sector, i.e. the electroweak symmetry breaking sector (SBS) of this model, SSBS

 1 α α,µν 1 a a,µν 1 = d x − Fµν F − Wµν W − Bµν B µν 4 4 4    a τ3 τ a α α ¯ / PL − g1 BP / R ψ , / − g2 W +ψ i∂/ − gTC t G 2 2 Z

4

(1)

where gTC , g2 and g1 (Gαµ , Wµa and Bµ ) are the coupling constants (gauge fields) of SU(N)T C × SU(2)L × U(1)Y with technicolor index α (α = 1, 2, . . . , N 2 − 1) and weak α a index a (a = 1, 2, 3) respectively; and Fµν , Wµν and Bµν are the corresponding field strength

tensors; tα (α = 1, 2, . . . , N 2 − 1) are the generators for the fundamental representation of SU(N)T C , while τ a (a = 1, 2, 3) are Pauli matrices; and the left and right chirality projection operators PLR = (1 ∓ γ5 )/2. To derive low energy effective electroweak chiral Lagrangian from the one-doublet TC model, we need to integrate out the technigluons and techniquarks above the electroweak scale which can be formulated as   Z   Z  α  α a a ¯ ¯ ψ = Dµ(U) exp iSeff [U, W , Bµ ] , (2) DGµ D ψDψ exp iSSBS Gµ , Wµ , Bµ , ψ, µ where U(x) is a dimensionless unitary unimodular matrix field in the electroweak chiral Lagrangian, and Dµ(U) denotes the corresponding functional integration measure. As mentioned in previous section, there are two different approaches, one is the GasserLeutwyler’s prescription, the other is the dynamical computation prescription. The second approach we developed in this paper is relatively easy to be generalized to more complicated theories. We will compare the results obtained in both approaches.

6

A.

The Gasser-Leutwyler’s Prescription

As we will see, it is easy to relate QCD-like models to chiral Lagrangian using the GasserLeutwyler’s prescription. To begin, we substitute (1) into the left-hand side of Eq.(2) and the result path integral involved technicolor interaction is analogous to QCD and then we can use technique developed by Gasser and Leutwyler relating it with the path integral of chiral Lagrangian for goldstone bosons induced from SBS [20],      R R 4 a 3 α a 1 α τ τ α α,µν α ¯ ¯ / − g2 2 W / PL − g1 2 BP / R ψ DGµ D ψDψ exp i d x − 4 Fµν F + ψ i∂/ − gTC t G R R 3 ¯ ¯ ∂/ − g2 τ a W / a PL − g1 τ2 BP / R ]ψ} D ψDψ exp{i d4 xψ[i 2 Z ˜ exp{iSTC-induced eff [U˜ , W, B]} , = Dµ(U) (3) in which the denominator of the left hand side of above equation is introduced to insure ˜ W, B] normalized as zero when the technicolor-induced chiral effective action STC-induced eff [U, we switch off technicolor interactions by setting gTC = 0 and STC-induced eff [U˜ , W, B] can be written as  1D 2 (F0 ) µ ˜† ˜ + L1D ˜ 2 STC-induced eff [U˜ , W, B] = dx tr[(∇µ U˜ † )(∇µ U)] 1 [tr(∇ U ∇µ U )] 4 ˜ † ∇ν U˜ ] + L1D tr[(∇µ U˜ † ∇µ U˜ )2 ] +L1D tr[∇µ U˜ † ∇ν U˜ ]tr[∇µ U Z

4

2

3

R µ ˜ ν ˜† L µ ˜† ν ˜ 1D ˜ † R ˜ L,µν ] −iL1D 9 tr[Fµν ∇ U∇ U + Fµν ∇ U ∇ U] + L10 tr[U Fµν U F  R R,µν L +H11D tr[Fµν F + Fµν F L,µν ] + O(p6 ) , (4) 1D 1D 1D 1D where the coefficients F01D , L1D arise from SU(N)TC dynamics at the 1 , L2 , L3 , L10 , H1

scale of 250 GeV, and where ˜ − irµ U˜ + iU˜ lµ , ∇µ U˜ ≡ ∂µ U R Fµν ≡ i[∂µ − irµ , ∂ν − irν ]

L Fµν

˜ † = −U˜ † (∇µ U˜ )U ˜ † = ∂µ U˜ † − ilµ U˜ † + iU˜ † rµ , ∇µ U τa τ3 ≡ i[∂µ − ilµ , ∂ν − ilν ] rµ ≡ −g1 Bµ lµ ≡ −g2 Wµa .(5) 2 2

˜ field in Eq.(4) given in second paper of [20] is 3 × 3 unitary Note that conventional U matrix. However, for the SU(2)L × SU(2)R electroweak chiral Lagrangian we considered in 1D this paper, U˜ is a 2 × 2 unitary matrix, and thus the L1D 1 term and the L3 term in present situation are linearly related, µ ˜† ˜ 2 L1D 3 tr[(∇ U ∇µ U ) ]

=

L1D 3 tr



 † µ  ˜ † (∇µ U˜ ) 2 U˜ (∇ U˜ )U 7



   † µ  2 L1D 3 † µ ˜ (∇ U) ˜ tr U˜ (∇ U˜ )U = (6) 2

Comparing covariant derivative for U˜ given in (5) and covariant derivative given in Ref.[2], a 3 we find we must recognize U˜ † = U, ∇µ U˜ † = Dµ U, F R = −g1 τ Bµν and F L = −g2 τ W a . µν

µν

2

2

µν

Substitute them back into Eq.(4), we obtain  Z L1D (F 1D )2 3 4 tr(Xµ X µ ) + (L1D )[tr(Xµ X µ )]2 STC-induced eff [U, W, B] = dx − 0 1 + 4 2 1D L9 µ ν 2 +L1D g1 Bµν tr(τ 3 X µ X ν ) − iL1D 2 [tr(Xµ Xν )] − i 9 tr(W µν X X ) 2  L1D H11D 2 µν µν 10 3 µν 1D + g1 Bµν tr(τ W ) + g Bµν B + H1 tr(W µν W ) . (7) 2 2 1 where Xµ ≡ U † (Dµ U)

W µν ≡ U † g2 Wµν U ,

(8)

We have reformulated the EWCL in terms of Xµ and τ 3 instead of Vµ and T in Ref.[2], the corresponding relations are given in Appendix A. Comparing (7) with the standard electroweak chiral Lagrangian given in Ref.[2], we find f 2 = (F01D )2 , α5 = L1D 1 +

α1 = L1D 10 ,

β1 = 0 ,

L1D 3 , 2

α2 = α3 = −

αi = 0 (i = 6, 7, . . . , 14) .

L1D 9 , 2

α4 = L1D 2 , (9)

Note that in (7), the term with coefficient H11D do not affect the result αi coefficients. Not like the original case of QCD, H11D now is a finite constant. The divergences are from the / a PL − g1 τ2 BP / R ) in (3) which will only contribute tr(Wµν W µν ) and term Tr log(i∂/ − g2 τ2 W a

3

Bµ B µ terms with divergent coefficients due to gauge invariance. These divergent coefficients in combined with H11D will cause wave function renormalization corrections for Wµa and Bµ fields which further lead redefinitions of Wµa and Bµ fields and their gauge couplings g2 and g1 . This renormalization procedure will have no effects on our EWCL, since all electroweak gauge fields appeared in EWCL are as form of g2 Wµa and g1 Bµ which are renormalization invariant quantities. Due to this consideration, in rest of this paper, we just left the wave function corrections to electroweak gauge fields Wµa and Bµ in the theory and skip the corresponding renormalization procedure.

B.

The Dynamical Computation Prescription

Now we develop a dynamical computation program and we will apply it to more complicated model in next section. 8

We first review the derivation process given in Ref.[15] and start with introducing a local 2 × 2 operator O(x) as O(x) ≡ trlc [ψL (x)ψ¯R (x)] with trlc is the trace with respect to Lorentz and technicolor indices. The transformation of O(x) under SU(2)L × U(1)Y is O(x) → VL (x)O(x)VR† (x). Then we decompose O(x) as O(x) = ξL† (x)σ(x)ξR (x) with the σ(x) represented by a hermitian matrix describes the modular degree of freedom; while ξL (x) and ξR (x) represented by unitary matrices describe the phase degree of freedom of SU(2)L and U(1)Y respectively. Their transformation under SU(2)L × U(1)Y are σ(x) → h(x)σ(x)h† (x), ξL (x) → h(x)ξL (x)VL† (x) and ξR (x) → h(x)ξR (x)VR† (x) with h(x) = eiθh (x)τ

3 /2

belongs to an induced hidden local U(1) symmetry group. Now we define a new field U(x) as U(x) ≡ ξL† (x)ξR (x) which is the nonlinear realization of the goldstone boson fields in the electroweak chiral Lagrangian. Subtracting the σ(x) field, we find that the present decomposition results in a constraint ξL (x)O(x)ξR† (x) − ξR (x)O † (x)ξL† (x) = 0, the functional expression of it is Z

Dµ(U)F [O]δ(ξLOξR† − ξR O † ξL† ) = const. ,

(10)

where Dµ(U) is an effective invariant integration measure; F [O] only depends on O, and it compensates the integration to make it to be a constant. It is easy to show that F [O] is invariant under SU(2)L × U(1)Y transformations. Substituting Eq.(10) into the left-hand side of Eq.(2), we have   Z   Z  α  α a a ¯ ¯ DGµ D ψDψ exp iSSBS Gµ , Wµ , Bµ , ψ, ψ = Dµ(U) exp iSeff [U, Wµ , Bµ ] , (11) where Seff [U, Wµa , Bµ ] (12) Z Z 1 a a,µν 1 ¯ W − Bµν B µν ) − i log DGαµ D ψDψ F [O]δ(ξLOξR† − ξR O †ξL† ) = d4 x(− Wµν 4 4  Z     a 3 1 τ τ α a 4 α α,µν α / − g2 W / PL − g1 BP / R ψ × exp i d x − Fµν F + ψ¯ i∂/ − gTC t G 4 2 2 To match the correct normalization, we introduce in the argument of logarithm function the R 3 a R a ¯ ∂−g / a PL −g1 τ2 BP / R )ψ / 2 τ2 W i d4 xψ(i ¯ / a PL − = exp Tr log(i∂/ − g2 τ W normalization factor D ψDψe 2

3 / R) g1 τ2 BP

and then take a special SU(2)L × U(1)Y rotation, as VL (x) = ξL (x) and VR (x) =

9

ξR (x), on both numerator and denominator of the normalization factor Seff [U, Wµa , Bµ ] (13) Z 1 a a,µν 1 τa a τ3 / PL − g1 BP / R) = d4 x(− Wµν W − Bµν B µν ) + Tr log(i∂/ − g2 W 4 4 2 2 R 4 a R τ3 / 1 α α,µν ¯ / TC tα G / α −g2 τ2 W /a ξ PR )ψξ ] ξ PL −g1 2 B DGαµ D ψ¯ξ Dψξ F [Oξ ]δ(Oξ − Oξ†)ei d x[− 4 Fµν F +ψξ (i∂−g , −i log R R a τ3 / 4 ¯ /a / 2 τ2 W ξ PR )ψξ ξ PL −g1 2 B D ψ¯ξ Dψξ ei d xψξ (i∂−g

where rotated fields are denoted as follows

ψξ = PL ξL (x)ψL (x) + PR ξR (x)ψR (x) , Oξ (x) ≡ ξL(x)O(x)ξR† (x) , (14) a a 3 3 τ τ τ τ a g2 Wξ,µ (x) ≡ ξL (x)[g2 Wµa (x) − i∂µ ]ξL† (x) g1 Bξ,µ (x) ≡ ξR (x)[g1 Bµ (x) − i∂µ ]ξR† (x) . 2 2 2 2 In (13), the possible anomalies caused by this special chiral rotation are canceled between the numerator and the denominator. Thus Eq.(13) can be written as Z 1 a a,µν 1 a Seff [U, Wµ , Bµ ] = d4 x(− Wµν W − Bµν B µν ) + Sanom [U, Wµa , Bµ ] + Snorm [U, Wµa , Bµ ],(15a) 4 4 where  Z  1 α α,µν 4 F = −i log F [Oξ ]δ(Oξ − i d x − Fµν 4    τ3 τa a α α ¯ / PL − g 1 B / PR ψξ / − g2 W . (15b) +ψξ i∂/ − gTC t G 2 ξ 2 ξ Z

Snorm [U, Wµa , Bµ ]

DGαµ D ψ¯ξ Dψξ

Oξ† ) exp

and τa a τ3 / PL − g1 BP / R) W 2 2 τ3 τa a / ξ PL − g 1 B / PR ) , −Tr log(i∂/ − g2 W 2 2 ξ

iSanom [U, Wµa , Bµ ] = Tr log(i∂/ − g2

(15c)

It is worthwhile to show the transformations of the rotated fields under SU(2)L × U(1)Y are ψξ (x) → h(x)ψξ (x), Oξ (x) → h(x)Oξ (x)h† (x) with h(x) defined previously describes an invariant hidden local U(1) symmetry. Thus, the chiral symmetry SU(2)L × U(1)Y covariance of the unrotated fields has been transferred totally to the hidden symmetry U(1) covariance of the rotated fields. We can further find combination of electroweak gauge fields a

a

3

3

a a (x) − g1 τ2 Bξ,µ (x) → h(x)[g2 τ2 Wξ,µ (x) − g1 τ2 Bξ,µ (x)]h† (x) transforms covariantly, g2 τ2 Wξ,µ a

3

a

3

a a while alternative combination g2 τ2 Wξ,µ (x)+g1 τ2 Bξ,µ (x) → h(x)[g2 τ2 Wξ,µ (x)+g1 τ2 Bξ,µ (x)−

2i∂µ ]h† (x) transforms as the ”gauge field” of the hidden local U(1) symmetry. 10

With technique used in Ref.[11], the integration over technigulon fields in Eq.(15b) can ...αn be formally integrated out with help of full n-point Green’s function of the Gαµ -field Gµα11...µ , n

thus Eq.(15b) after integration becomes  Z   Z τa a τ3 † iSnorm [U,Wµa ,Bµ ] 4 ¯ ¯ / PL − g 1 B / PR ψξ e = D ψξ Dψξ F [Oξ ]δ(Oξ − Oξ ) exp i d xψξ i∂/ − g2 W 2 ξ 2 ξ  ∞ Z X (−igTC )n α1 ...αn µ1 µn 4 4 + d x1 . . . d xn Gµ1 ...µn (x1 , . . . , xn )Jξ,α1 (x1 ) . . . Jξ,αn (xn ) . (16) n! n=2 µ µ where effective sources Jξ,α (x) are identified as Jξ,α (x) ≡ ψ¯ξ (x)tα γ µ ψξ (x).

1.

Schwinger-Dyson Equation for Techniquark Propagator

¯ = To show that the technicolor interaction indeed induces the condensate hψψi 6 0 which triggers the electroweak symmetry breaking, we investigate the behavior of the techniquark propagator S σρ (x, x′ ) ≡ hψ σ (x)ψ¯ρ (x′ )i in the following. Neglecting the factor F [Oξ ]δ(Oξ − ξ

Oξ† )

ξ

in Eq.(16), the total functional derivative of the integrand with respect to ψ¯ξσ (x) is zero,

(here and henceforth the suffixes σ and ρ are short notations for Lorentz spinor, techniflavor and technicolor indices,) i.e., Z   Z Z a 3 τ τ δ a 4 4 ¯ ξ ) + i d xψ¯ξ i∂/ − g2 W / PL − g 1 B / PR ψξ exp d x(ψ¯ξ I + Iψ 0 = D ψ¯ξ Dψξ ¯σ 2 ξ 2 ξ δ ψξ (x)  ∞ Z X (−igTC )n α1 ...αn µ1 µn 4 4 Gµ1 ...µn (x1 , . . . , xn )Jξ,α1 (x1 ) . . . Jξ,αn (xn ) , (17) + d x1 . . . d xn n! n=2 ¯ where I(x) and I(x) are the external sources for, respectively, ψ¯ξ (x) and ψξ (x); and which leads to    τa a τ3 / (x)PL − g1 B / (x)PR ψξτ (x) 0 = Iσ (x) + i i∂/x − g2 W (18) 2 ξ 2 ξ στ  ∞ Z X (−igTC )n α1 ...αn µ2 µn 4 4 α1 µ1 τ d x2 . . . d xn G (x, x2 , . . . , xn )(t γ )στ ψξ (x)Jξ,α2 (x2 ) . . . Jξ,αn (xn ) + , (n − 1)! µ1 ...µn I n=2 where we have defined the notation hh · · · iiI in this section by Z Z 

¯ ξ) O(x) I ≡ D ψ¯ξ Dψξ O(x) exp d4 x(ψ¯ξ I + Iψ   Z τ3 τa a 4 ¯ / / PL − g 1 B / PR ψξ +i d xψξ i∂ − g2 W 2 ξ 2 ξ  ∞ Z X (−igTC )n α1 ...αn µ1 µn 4 4 Gµ1 ...µn (x1 , . . . , xn )Jξ,α1 (x1 ) . . . Jξ,αn (xn ) (19) . + d x1 . . . d xn n! n=2 11

Taking functional derivative of Eq.(18) with respect to Iρ (y), and subsequently setting I = I¯ = 0, we obtain   τa a τ3 / (x)PL − g1 B / (x)PR hψξτ (x)ψ¯ξρ (y)i 0 = δσρ δ(x − y) + i i∂/x − g2 W 2 ξ 2 ξ στ ∞ Z n X (−igTC ) α1 ...αn − d4 x2 . . . d4 xn Gµ1 ...µn (x, x2 , . . . , xn ) (n − 1)!

n=2  × ψ¯ρ (y)(tα1 γ µ1 )στ ψ τ (x)J µ2 (x2 ) . . . J µn (xn ) , ξ

ξ

(20)

ξ,αn

ξ,α2

 where we have defined vacuum expectation value (VEV) h · · · i by O(x) ≡

 O(x) I /hh 1 iiI |I=I=0 ¯ . If we neglect higher-point Green’s functions,and further taking fac





 torization approximation, i.e., ψ¯ξρ (y)ψξτ (x)ψ¯ξγ (x2 )ψξδ (x2 ) ≈ ψ¯ξρ (y)ψξτ (x) ψ¯ξγ (x2 )ψξδ (x2 ) −

ρ 

 ψ¯ξ (y)ψξδ (x2 ) ψ¯ξγ (x2 )ψξτ (x) , we obtain

  τ3 τa a / (x)PL − g1 B / (x)PR hψξτ (x)ψ¯ξρ (y)i 0 = δσρ δ(x − y) + i i∂/x − g2 W 2 ξ 2 ξ στ Z



 2 d4 x2 Gαµ11µα22 (x, x2 )(tα1 γ µ1 )στ (tα2 γ µ2 )γδ ψ¯ξρ (y)ψξδ (x2 ) ψ¯ξγ (x2 )ψξτ (x) , (21) −gTC

 where we have used ψ¯ξ (x2 )tα2 γ µ2 ψξ (x2 ) = 0, which comes from the Lorentz and gauge invariance of vacuum. We denote the technifermion propagator S σρ (x, x′ ) ≡ hψ σ (x)ψ¯ρ (x′ )i, ξ

ξ

multiplying inverse of technifermion propagator in both sides of Eq.(21), it then is written as the Schwinger-Dyson equation (SDE) for techniquark propagator,   τa a τ3 −1 / (x)PL − g1 B / (x)PR δ(x − y) 0 = Sσρ (x, y) + i i∂/x − g2 W 2 ξ 2 ξ σρ   2 . −gTC Gµα11µα22 (x, y) tα1 γ µ1 S(x, y)tα2 γ µ2

(22)

σρ

By defining techniquark self energy iΣ as   τa a τ3 −1 / (x)PL − g1 B / (x)PR δ(x − y) , iΣσρ (x, y) ≡ Sσρ (x, y) + i i∂/x − g2 W 2 ξ 2 ξ σρ

(23)

the SDE (22) can also be written as iΣσρ (x, y) =

2 gTC Gαµ11µα22 (x, y)

  α1 µ1 α2 µ2 t γ S(x, y)t γ

.

(24)

σρ

Moreover, from the fact that technigluon propagator is diagonal in the adjoint representation αβ space of technicolor group, i.e., Gαβ µν (x, y) = δ Gµν (x, y), and techniquark propagator is

diagonal in the techniquark representation space of technicolor group, and also (tα tα )ab = 12

C2 (N)δab for the fundamental representation of SU(N), Eq.(24) is diagonal in technicolor indices a, b and diagonal part satisfy 2 µ1 µ2 ij iΣij ηζ (x, y) = C2 (N)gTC Gµ1 µ2 (x, y)[γ S(x, y)γ ]ηζ ,

(25)

where {i, j}, and {η, ζ} are, respectively, techniflavor and Lorentz spinor indices; and the Casimir operator C2 (N) = (N 2 − 1)/(2N). a ♣ Bξ,µ = Wξ,µ = 0 Case: the Gap Equation αβ The technigluon propagator in Landau gauge is Gαβ µν (x, y) = δ

with Gµν (p2 ) =

i (g −pµ pν /p2 ). −p2 [1+Π(−p2 )] µν

R

d4 p −ip(x−y) e Gµν (p2 ) (2π)4

And the techniquark self energy and propagator

are respectively Σij ηζ (x, y)

=

Z

d4 p −ip(x−y) ij e Σηζ (−p2 ) (2π)4

ij Sηζ (x, y)

=

Z

d4 p −ip(x−y) ij e Sηζ (p) , (2π)4

(26)

ij with Sηζ (p) = i[1/(p/ − Σ(−p2 )]ij ηζ . Substituting above results into the SDE (25), we have ij   Z 2 (p − q)µ (p − q)ν i −C2 (N)gTC d4 q ij ν µ 2 (27) γ gµν − γ Σηζ (−p ) = (2π)4 (p − q)2 [1 + Π(−(p − q)2 )] (p − q)2 /q − Σ(−q 2 ) ηζ

from which we can see that the solution of the techniquark self energy must be diagonal in techniflavor space, since the integration kernel is independent of techniflavor indices, i.e., 2 ij 2 Σij ηζ (−p ) = δ Σηζ (−p ). With the assumption that the techniquark self energy is diagonal

and equal in Lorentz spinor space, leads to the following two equations Z 4 d q αTC [−(p − q)2 ] Σ(−q 2 ) 2 i Σ(−p ) = 3C2 (N) , 4π 3 (p − q)2 q 2 − Σ2 (−q 2 )   Z 2 (p − q)µ (p − q)ν µ /q 4 αTC [−(p − q) ] g − γ γν . 0= d q µν (p − q)2 (p − q)2 q 2 − Σ2 (−q 2 )

(28a) (28b)

In which we have labeled the integration kernel with running coupling constant αTC (−p2 ) ≡ 2 2 gTC (−p2 )/(4π) = gTC /(4π[1 + Π(−p2 )]). Eq.(28b) is automatically satisfied when taken

the approximation, αTC [(pE − qE )2 ] = αTC (p2E )θ(p2E − qE2 ) + αTC (qE2 )θ(qE2 − p2E ). The other equation (28a) can be written in Euclidean space as Z 4 Σ( qE2 ) d qE αTC [(pE − qE )2 ] 2 Σ(pE ) = 3C2 (N) , 4π 3 (pE − qE )2 qE2 + Σ2 (qE2 )

(29)

If there is nonzero solution for above equation, we will obtain nonzero techniquark condensate hψ¯ξk ψξj i with k and j techniflavor indices, Z 4 Σ(p2E ) d pE j k jk jk , (30) hψ¯ξ (x)ψξ (x)i = −trlc [S (x, x)] = −4Nδ (2π)4 p2E + Σ2 (p2E ) 13

where trlc is the trace with respect to Lorentz, technicolor indices, and where we have used that the techniquark self energy, the solution of the SDE, must be diagonal in techniflavor space. Thus, nonzero techniquark self energy can give a nontrivial diagonal condensate ¯ = hψψi 6 0, which spontaneously breaks SU(2)L × U(1)Y to U(1)em . a ♣ Bξ,µ 6= 0 and Wξ,µ 6= 0 Case: the Lowest-order Approximation

In the following we consider the effects of the nonzero electroweak gauge fields Bξ,µ and a Wξ,µ . The SDE (25) is explicitly 2 Σ(x, y) = C2 (N)gTC Gµν (x, y)   −1 τa a τ3 µ / (x)PL − g1 B / (x)PR δ(x − y) − Σ(x, y) γ ν , (31) i∂/x − g2 W ×γ 2 ξ 2 ξ

where the techniflavor and Lorentz spinor indices of the techniquark self energy are implicitly contained. In this case, the self energy can no longer be written as the function on the derivatives with respect to spacetime, i.e., Σ(x, y) 6= Σ(∂x2 )δ(x − y). a Suppose the function Σ(−p2 ) is a solution of the SDE in the case Bξ,µ = Wξ,µ = 0, that

is, it satisfies the equation 2

Σ(−p ) =

2 C2 (N)gTC

Z

1 d4 q Gµν (q 2 )γ µ γν , 4 (2π) q/ + p/ − Σ[−(q + p)2 ]

(32)

where in the second equality the legality of the integration measure translation comes from the logarithmical divergence of the fermion self energy. Replacing the variable p by p + ∆ in Eq.(32) and subsequently integrating over p with the weight e−ip(x−y) , we obtain, as long as ∆ is commutative with ∂x and Dirac matrices, Eq.(32) imply 2 Σ[(∂x − i∆)2 ]δ(x − y) = C2 (N)gTC Gµν (x, y)γ µ

1 δ(x − y)γ ν . (33) 2 / − Σ[−(i∂x + ∆) ] i∂/x + ∆

Even if ∆ is noncommutative with ∂x and Dirac matrices, the above equation holds as the / ∆] is higher order of momentum than lowest order approximation, for the commutator [∂, a

3

∆ itself. Now if we take ∆ to be −g2 τ2 Wξa PL − g1 τ2 Bξ PR , ignoring its noncommutative property with ∂x and Dirac matrices, Eq.(33) is just the SDE (31) in the case Bξ,µ 6= 0 and a

3

a a PL +ig1 τ2 Bξ,µ PR )2 ]δ(x−y), which is the hidden symmetry Wξ,µ 6= 0. Thus, Σ[(∂µx +ig2 τ2 Wξ,µ

U(1) covariant, can be regarded as the lowest-order solution of Eq.(31). To further simplify the calculations and still keep this hidden-symmetry covariance of the self energy, we can 14

a

3

a PL + ig1 τ2 Bξ,µ PR reduce the covariant derivative inside the self energy ∇µ ≡ ∂µ + ig2 τ2 Wξ,µ

to its minimal-coupling form i τa a τ3 ∇µ ≡ ∂µ + [g2 Wξ,µ (x) + g1 Bξ,µ (x)] . 2 2 2 a

(34)

3

a (x) + g1 τ2 Bξ,µ (x)]/2 transforms as a gauge field In which as we mentioned before [g2 τ2 Wξ,µ

under hidden U(1) symmetry transformations. Thus, if the function Σ(∂x2 )δ(x − y) is the a self-energy solution of the SDE in the case Bξ,µ = Wξ,µ = 0, we can replace its argument 2

∂x by the minimal-coupling covariant derivative ∇x , i.e., Σ(∇x )δ(x − y), as an approximate a solution of the SDE in the case Bξ,µ 6= 0 and Wξ,µ 6= 0.

2.

Effective Action

The exponential terms on the right-hand side of Eq.(16) can be written explicitly as ∞ Z X

d4 x1 . . . d4 xn

n=2

≈

Z

(−igTC )n α1 ...αn µ1 µn Gµ1 ...µn (x1 , . . . , xn )Jξ,α (x1 ) . . . Jξ,α (xn ) n 1 n!

d4 xd4 x′ ψ¯ξσ (x)Πσρ (x, x′ )ψξρ (x′ ) ,

(35)

where we have taken the approximation of replacing the summation over 2n-fermion interactions with parts of them by their vacuum expectation values, that is, ′

Πσρ (x, x ) =

∞ X

′ Π(n) σρ (x, x ) ,

(36)

n=2

(n) Πσρ (x, x′ )

 (−igTC )n α1 ...αn ′ Gµ1 ...µn (x, x2 . . . , xn−1 , x ) (tα1 γ µ1 )σσ1 ψξσ1 (x) = n × d x2 . . . d xn−1 n!  ρn ′ αn µn α2 µ2 αn−1 µn−1 ¯ ¯ ¯ ×ψξ (x2 )t γ ψξ (x2 ) . . . ψξ (xn−1 )t γ ψξ (xn−1 )ψξ (x )(t γ )ρn ρ (37) Z

4

4

where the factor n comes from n different choices of unaveraged ψ¯ξ ψξ , and the lowest term of which is ′ Π(2) σρ (x, x )

=

2 −gTC Gαµ11µα22 (x, x′ )

  α1 µ1 α2 µ2 t γ S(x, y)t γ

.

(38)

σρ

Comparing Eq.(38) with Eq.(24), we have 2

′ ′ iΠ(2) σρ (x, x ) = Σσρ (x, x ) ≈ Σσρ (∇x )δ(x − y) .

15

(39)

By neglecting the factor F [Oξ ]δ(Oξ − Oξ†) in Eq.(16), we have iSnorm [U,Wµa ,Bµ ]

e

 Z   τa a τ3 4 ¯ ¯ / PL − g 1 B / PR ψξ ≈ D ψξ Dψξ exp i d xψξ i∂/ − g2 W 2 ξ 2 ξ  Z ρ ′ 4 4 ′ ¯σ ′ + d xd x ψξ (x)Πσρ (x, x )ψξ (x ) Z

≈ Det[i∂/ − g2

τ3 τa a 2 / / ξ PR − Σ(∇ )] W ξ PL − g 1 B 2 2

(40)

where in the second equality we have taken further approximation of keeping only the lowest (2)

order, i.e. Πσρ (x, x′ ), of Πσρ (x, x′ ). With these three approximations, we have Snorm [U, Wµa , Bµ ] = −iTr log[i∂/ − g2

τ3 τa a 2 / ξ PL − g 1 B / ξ PR − Σ(∇ )] , W 2 2

(41)

As done in [12], we can parameterize the normal part of the effective action as follows 2

Snorm [U, Wµa , Bµ ] = −iTr log[i∂/ + /v + a/γ5 − Σ(∇ )]  Z 4 = d x trf (F01D )2 a2 − K11D (dµ aµ )2 − K21D (dµ aν − dν aµ )2 + K31D (a2 )2 + K41D (aµ aν )2  1D µν 1D µν + O(p6 ) , (42) −K13 Vµν V + iK14 aµ aν V a

3

a + g1 τ2 Bξ,µ) and aµ ≡ where the fields vµ , aµ are identified with vµ ≡ − 12 (g2 τ2 Wξ,µ a 3 1 a (g τ Wξ,µ −g1 τ2 Bξ,µ ) 2 2 2

and dµ aν ≡ ∂µ aν −i[vµ , aν ], Vµν ≡ i[∂µ −ivµ , ∂ν −ivν ]. Ki1D coefficients

with superscript 1D to denote present one doublet model are functions of techniquark self energy Σ(p2 ) and detail expressions are already written down in (36) of Ref.[12] with the replacement of Nc → N. For anomaly part, compare (15c) and (41), we find its U field dependent part can be produced from normal part by vanishing techniquark self energy Σ, i.e. iSanom [U, Wµa , Bµ ] = Tr log(i∂/ − g2

τa a τ3 / PL − g1 BP / R ) − iSnorm [U, Wµa , Bµ ]|Σ=0 W 2 2

(43)

Notice that U field independent part of pure gauge field part is irrelevant to EWCL. Combine with (42), above relation imply iSanom [U, Wµa , Bµ ]

τ3 τa a / PL − g1 BP / R) + i = Tr log(i∂/ − g2 W 2 2 1D,(anom)

(dµ aν − dν aµ )2 + K3

1D,(anom)

Vµν V µν

−K2

−K13

1D,(anom)

Z

4



1D,(anom)

(a2 )2 + K4  1D,(anom) µν + O(p6 ) , + iK14 aµ aν V 16

1D,(anom)

d x trf − K1

(dµ aµ )2

(aµ aν )2 (44)

with 1D,(anom)

Ki

= −Ki1D |Σ=0

i = 1, 2, 3, 4, 13, 14

(45)

where we have used result that F01D |Σ=0 = 0. Combine normal and anomaly part contribution together, with help of (15a), we finally find Z 1 τa a τ3 a a / PL − g1 BP / R) Seff [U, Wµ , Bµ ] = − d4 x(Wµν W a,µν + Bµν B µν ) + Tr log(i∂/ − g2 W 4 2 2  Z 4 +i d x trf (F01D )2 a2 − K11D,Σ6=0 (dµ aµ )2 − K21D,Σ6=0 (dµ aν − dν aµ )2 + K31D,Σ6=0(a2 )2  1D,Σ6=0 1D,Σ6=0 1D,Σ6=0 2 µν µν + O(p6 ) , (46) +K4 (aµ aν ) − K13 Vµν V + iK14 aµ aν V with Ki1D,Σ6=0 be Σ dependent part of Ki Ki1D,Σ6=0 ≡ Ki1D − Ki1D |Σ=0

i = 1, 2, 3, 4, 13, 14

(47)

After some algebras, terms in Eq.(46) can be reexpressed in terms of Xµ and W µν which are just standard EWCL given in Ref.[2] with coefficients 1D,Σ6=0 1D,Σ6=0 1D,Σ6=0 K21D,Σ6=0 − K13 K13 K14 f = , β1 = 0 , α1 = , α2 = α3 = − + , 2 4 16 1D,Σ6=0 1D,Σ6=0 1D,Σ6=0 Σ6=0 K41D,Σ6=0 + 2K13 − K14 K31D,Σ6=0 − K41D,Σ6=0 − 4K13 + 2K14 α4 = , α5 = . (48) 16 32 2

(F01D )2

With formulae of Ki coefficients given in Ref.[12], we can substitute the solution of SD equation (29) into them and then obtain numerical results for those nonzero αi coefficients. In TABLE I, we list down the numerical calculation results for different kind of dynamics. To obtain above numerical result, we have solved Schwinger-Dyson equation (29) with following running coupling which was used as model A in Ref.[12]    7 (11N12π , for ln(p2 /Λ2TC ) ≤ −2;  −2Nf )   12π 4 2 2 2 for − 2 ≤ ln(p2 /Λ2TC ) ≤ 0.5; αTC (p2 ) = {7 − 5 [2 + ln(p /ΛTC )] } (11N −2Nf ) ,   1 12π   , for 0.5 ≤ ln(p2 /Λ2TC )  2 ln(p2 /ΛTC ) (11N − 2Nf )

(49)

In which the fermion number is taken to be Nf = 2 corresponding to present one doublet techniquark. Although there is a dimensional parameter ΛTC appear in αTC (p2 ), except dimensional coefficient F01D , all dimensionless result coefficients αi ,

i = 1, 2, 3, 4, 5 are

independent of this parameter. This can be seen as follows, if we scale up ΛTC as λΛTC , 17

TABLE I. The obtained nonzero values of the O(p4 ) coefficients α1 , α2 = α3 , α4 , α5 for one doublet technicolor model with the conventional strong interaction QCD theory values given in Ref.[12] for model A and experimental values for comparison. ΛTC and ΛQCD are in TeV, they are determined by f = 250GeV and fπ = 93MeV respectively. The coefficients are in units of 10−3 . QCD values are taken by using relation (9). N

ΛTC

α1

α2 = α3

α4

α5

3

1.34

-6.90

-2.43

2.02

-2.69

4

1.15

-9.26

-3.28

2.87

-3.69

5

1.03

-11.6

-4.11

3.60

-4.62

6

0.94

-13.9

-4.93

4.32

-5.54

QCD Theor

ΛQCD =0.484∗10−3

-7.06

-2.54

2.20

-2.81

−6.0 ± 0.7 −2.7 ± 0.4 1.7 ± 0.7 −1.3 ± 1.5

QCD Expt

αTC (p) defined above satisfy αTC (p2 )|λΛTC = αTC (λ−2 p2 )|ΛTC which, by (29), result a scalingup techniquark self energy Σ(p2 )|λΛTC = λΣ(λ−2 p2 )|ΛTC , since an alternative expression of (29) is λΣ(λ−2 p2E )

= 3C2 (N)

Z

d4 qE αTC [λ−2 (pE − qE )2 ] λΣ(λ−2 qE2 ) . 4π 3 (pE − qE )2 qE2 + λ2 Σ2 (λ−2 qE2 )

(50)

Further from (36) of Ref.[12], we find coefficients KiΣ6=0 , i = 1, 2, 3, 4, 13, 14 are invariant and F0 is changed to λF0 under exchanging Σ(p2 ) → λΣ(λ−2 p2 ) if we take cutoff in the formulae Λ → ∞. Due to this invariance for KiΣ6=0 , i = 1, 2, 3, 4, 13, 14, from (51), we can see then αi , i = 1, 2, 3, 4, 5 are independent of ΛTC and F01D scales same as ΛTC . It is this scale dependence for F0 which makes ordinary QCD contribution small to electroweak symmetry breaking and leads necessity for new interactions at higher energy scale. The scale relations above are result of present rough approximations, it will simplify our future computations very much in next section. From TABLE I, we see that within error of our approximation, the numerical result exhibit the scaling-up behavior among different N.

18

C.

Comparison and Discussion on Two Prescriptions

Compare results from Gasser-Leutwyler’s prescription and dynamical computation prescription, (9) and (48), we find results are same as long as we identify 1D,Σ6=0 1D,Σ6=0 K21D,Σ6=0 + K13 K21D,Σ6=0 − K13 1D =− L10 = 4 2 1D,Σ6=0 1D,Σ6=0 1D,Σ6=0 1D,Σ6=0 1D,Σ6=0 K14 K4 + 2K13 − K14 K13 1D 1D − L2 = L9 = 2 8 16 1D,Σ6=0 1D,Σ6=0 1D,Σ6=0 1D,Σ6=0 1D L3 + 2K14 − 4K13 − K4 K L1D = 3 1 + 2 32

H11D

(51)

which is just the result (25) obtained in Ref.[12]. This shows that two prescriptions are equivalent in results. The merit of Gasser-Leutwyler’s prescription is its simplicity and express result coefficients of electroweak chiral lagrangian in terms those in Gasser-Leutwyler chiral lagrangian for pseudo scalar mesons in strong interaction, but we can only apply this prescription to so called QCD-like theories for which the technicolor interaction must be vector-like. On the other hand, dynamical computation prescription, although much more complex but touch the dynamics details, do not limit us in the type of detail interactions. this has very strong potential to be applied to more complicated theories, such as chirallike technicolor models. Since we involve detail dynamical computation in this prescription, not like Gasser-Leutwyler’s prescription the coefficients are expressed in terms of strong interaction experiment fixed values, we can give detail theoretical computation result for all coefficients and it further allow us to test possible effects on the coefficients from variation of the dynamics. The first property qualitatively drew out from (41) and (15c) for their trace operation is that all coefficients are proportional to N. This is the well known scaling-up result for one doublet technicolor model, i.e., present coefficients can be got by (9) but identify the Li with the corresponding Gasser-Leutwyler chiral lagrangian for pseudo scalar mesons by an scaling-up factor N/Nc with Nc = 3 be number of color for strong interaction. In fact, it was this direct correspondence lead to the death of one doublet technicolor model, since 1D negative experiment value for L1D 10 result large positive S = −16πα1 = −16πL10 parameter

which contradict with present electroweak precision measurement data. The second property quantitatively drew out from (41) and (15c) is that except the overall N factor in front of all coefficients, remaining part of coefficients depends on dynamics, 19

so exactly speaking, they are not precisely equivalent to their strong interaction partners. For one doublet technicolor model, when N 6= Nc , not only we will have overall scaling-up factor N/Nc , but also we will have different techniquark self energy Σ due to the difference in running coupling constant (49) appeared in SDE (29). Nf = 2 also cause similar differences. But, since the estimations over the values in original strong interaction suffers large errors either in experiment or theories, this difference caused by dynamics hides in these uncertainties.

III.

DERIVATION OF THE ELECTROWEAK CHIRAL LAGRANGIAN FROM

A TOPCOLOR-ASSISTED TECHNICOLOR MODEL

There are several options in topcolor-assisted technicolor model building: (1) TC breaks both the EW interactions and the TopC interactions; (2) TC breaks EW, and something else breaks TopC; (3) TC breaks only TopC and something else drives EWSB (e.g., a fourth generation condensate driven by TopC). For definiteness, we will focus on a skeletal model in category (1) in the following. Consider a schematic TC2 model proposed by C. T. Hill [21]. The technicolor group is chosen to be GTC = SU(3)TC1 × SU(3)TC2 . The gauge charge assignments of techniquarks in GTC × SU(3)1 × SU(3)2 × SU(2)L × U(1)Y1 × U(1)Y2 are shown as Table II. TABLE II. Gauge charge assignments of techniquarks a schematic Topcolor-assisted Technicolor model. Ordinary quarks and additional fields (such as leptons) required for anomaly cancellation ¯ are not shown. The techniquark condensate hQQi breaks SU (3)1 × SU (3)2 × U (1)Y1 × U (1)Y2 → SU (3) × U (1)Y , while hT¯T i breaks SU (2)L × U (1)Y → U (1)EM . field

SU (3)TC1 SU (3)TC2 SU (3)1 SU (3)2 SU (2)L U (1)Y1 U (1)Y2

QL

3

1

3

1

1

1 2

0

QR

3

1

1

3

1

0

1 2

TL = (T, B)L 1

3

1

1

2

0

1 6

TR = (T, B)R 1

3

1

1

1

0

( 23 , − 13 )

20

The action of the symmetry breaking sector (SBS) then is Z SSBS = d4 x(Lgauge + Ltechniquark ) ,

(52)

with different part of Lagrangian given by 1 α αµν 1 α αµν 1 A Aµν 1 A Aµν 1 a aµν F1 − F2µν F2 − A1µν A1 − A2µν A2 − Wµν W Lgauge = − F1µν 4 4 4 4 4 1 1 µν µν − B1µν B1 − B2µν B2 , (53) 4 4
λA A λA A 1 1 ¯ i∂/ − g31 r1α G / α1 − h1 A / 1 PL − h2 A / 2 PR − q1 B / 1 PL − q2 B / 2 PR Q Ltechniquark = Q 2 2 2 2 3 a   1 1 τ τ / 2 PR T , / PL − q2 B / 2 PL − q2 ( + )B / α2 − g2 W (54) +T¯ i∂/ − g32 r2α G 2 6 6 2

where g31 , g32 , h1 , h2 , g2 , q1 and q2 are the coupling constants of, respectively, SU(3)TC1 , SU(3)TC2 , SU(3)1 , SU(3)2 , SU(2)L , U(1)Y1 and U(1)Y2 ; and the corresponding gauge fields A a α α (field strength tensors) are denoted by Gα1µ , Gα2µ , AA 1µ , A2µ , Wµ , B1µ and B2µ (F1µν , F2µν ,

A a AA 1µν , A2µν , Wµν , B1µν and B2µν ) with the superscripts α and A running from 1 to 8 and

a from 1 to 3; r1α and r2α (α = 1, . . . , 8) are the generators of, respectively, SU(3)TC1 and SU(3)TC2 , while λA (A = 1, . . . , 8) and τ a (a = 1, 2, 3) are, respectively, Gell-Mann and Pauli matrices. We do not consider the ordinary quarks in this work for following considerations, as we mentioned in the Introduction that this paper only discuss bosonic part of EWCL and matter part of EWCL will discussed in future. The matter part of EWCL mainly deals with effective interactions among ordinary fermions which certainly include ordinary quarks. Ignoring discussion of these effective interactions, only concentrate on their contribution to bosonic part EWCL coefficients is not self-consistent and efficient. Further one special feature of topcolor-assisted technicolor model is its arrangements on the interactions among ordinary quarks, especially for top and bottom quark mass splitting and the top pions resulted from top quark condensation through topcolor interactions, therefore dealing with quark interactions is a separated important issue which needs special care, previous formal derivations from underlying gauge theory to low energy chiral Lagrangian, no matter QCD and electroweak theory, all not involve in matter part of chiral Lagrangian and further initial computation shows that we need some special techniques to handle topbottom splitting which are beyond those techniques developed in this paper, to simplify the computations and reduce the lengthy formulae, we will not involve in discussion of ordinary quark in this paper and would rather specially focus our attentions on this issue in future works. 21

The strategy to derive the electroweak chiral Lagrangian from the schematic topcolorassisted technicolor model can be formulated as   Z a ¯ exp iSEW [Wµ , Bµ ] = D QDQD T¯DT DGα1µ DGα2µ DBµA DZµ′   A A a α α ¯ ¯ × exp iSSBS [G1µ , G2µ , A1µ , A2µ , Wµ , B1µ , B2µ , Q, Q, T , T ]   Z a a = N [Wµ , Bµ ] Dµ(U) exp iSeff [U, Wµ , Bµ ] , (55) where U(x) is a dimensionless unitary unimodular matrix field in the electroweak chiral Lagrangian, and Dµ(U) denotes normalized functional integration measure on U. The normalization factor N [Wµa , Bµ ] is determined through requirement that when the gauge coupling g32 is switched off Seff [U, Wµa , Bµ ] vanishes, this leads the electroweak gauge fields Wµa , Bµ dependent part of N [Wµa , Bµ ] is Z a ¯ N [Wµ , Bµ ] = D QDQD T¯ DT DGα1µ DBµA DZµ′   α A A a ¯ ¯ × exp iSSBS [G1µ , 0, A1µ , A2µ , Wµ , B1µ , B2µ , Q, Q, T , T ]

(56)

Since there different interactions in present model, in following several subsections, we discuss them and their contributions to EWCL separately.

A.

Topcolor Symmetry Breaking: the Contribution of SU (3)TC1

It can be shown below, by Schwinger-Dyson analysis, that the SU(3)TC1 interaction ¯ induces the techniquark condensate hQQi = 6 0, which will trigger the topcolor symmetry breaking SU(3)1 × SU(3)2 × U(1)Y1 × U(1)Y2 → SU(3)c × U(1)Y at the scale Λ = 1 TeV. This typically leaves a degenerate, massive color octet of “colorons”, BµA , and a singlet heavy Zµ′ in the coset space [SU(3)1 × SU(3)2 × U(1)Y1 × U(1)Y2 ]/[SU(3)c × U(1)Y ]. The gluon AA µ and coloron BµA (the SM U(1)Y field Bµ and the U(1)′ field Zµ′ ) are defined by orthogonal rotations with mixing angle θ (θ′ ):      cos θ A  = BµA AA AA 1µ A2µ µ sin θ      cos θ′ ′ B1µ B2µ = Zµ Bµ  sin θ′ 22

− sin θ cos θ



(57a)



(57b)

 ,

− sin θ′ cos θ

′

 ,

which lead to λA A λA A λA A λA A / 1 PL − h2 A / 2 PR = −g3 A / − g3 (cot θPL − tan θPR ) B / , A 2 2 2 2 1 1 1 ′ 1 / 1 PL − q2 B / 2 PR = −g1 B / − g1 (cot θ′ PL − tan θ′ PR ) Z / , −q1 B 2 2 2 2 −h1

(58a) (58b)

with g3 ≡ h1 sin θ = h2 cos θ ,

(59a)

g1 ≡ q1 sin θ′ = q2 cos θ′ .

(59b)

As a first step , we formally integrate out the SU(3)TC1 technigluons Gα1µ in Eq.(55) by ...αn introducing full n-point Green’s function of the Gα1µ -field Gαµ11...µ n   exp iSEW [Wµa , Bµ ]  Z Z  Z 1 α αµν 1 a aµν α A ′ 4 D T¯DT DG2µ DBµ DZµ exp i d4 x(− F2µν F2 = exp i d x(− Wµν W ) 4 4 1 1 1 1 Aµν Aµν A − AA − AA − B1µν B1µν − B2µν B2µν ) + iSTC1 [AA 1µ , A2µ , B1µ , B2µ ] 1µν A1 2µν A2 4Z 4 4 4  a 3   τ 1 1 τ α a 4 α / 2 − g2 W / PL − q2 B / PL − q2 ( + )B / PR T , +i d x T¯ i∂/ − g32 r2 G (60) 2 6 2 6 2 2

where   A A exp iSTC1 [A1µ , A2µ , B1µ , B2µ ]  Z Z A
λA A 1 1 ¯ ¯ i∂/ − h1 λ A /A / 2 PR − q1 B / 1 PL − q2 B / 2 PR Q ≡ D QDQ exp i d4 xQ P − h A L 2 1 2 2 2 2  Z ∞ X (−ig31 )n α1 ...αn µ1 µn Gµ1 ...µn (x1 , . . . , xn )J1α (x1 ) . . . J1α (xn ) . (61) + d4 x1 . . . d4 xn n 1 n! n=2 µ α µ ¯ J1α (x) ≡ Q(x)r 1 γ Q(x) is effective source.

¯ σ (x) Since the total functional derivative of the integrand in Eq.(61) with respect to Q is zero, (here and henceforth the suffixes σ and ρ are short notations for Lorentz spinor, techniflavor and technicolor indices,) i.e., Z Z Z δ λA A λA A 4 4 ¯ / ¯ ¯ ¯ / / PR 0 = D QDQ ¯ σ exp d x(QI + IQ) + i d xQ i∂ − h1 A1 PL − h2 A 2 2 2 δ Q (x) ∞ Z X
1 (−ig31 )n α1 ...αn 1 / 1 PL − q2 B / 2 PR Q + d4 x1 . . . d4 xn Gµ1 ...µn (x1 , . . . , xn ) −q1 B 2 2 n! n=2  µ1 µn ×J1α (x1 ) . . . J1α (xn ) , (62) n 1 23

¯ are the external sources for, respectively, Q(x) ¯ where I(x) and I(x) and Q(x), then continue the similar procedure from (18) to (21), by neglecting higher-point Green’s functions and taking factorization approximation, we obtain  λA A λA A 1 / 1 (x)PL − h2 A / 2 (x)PR − q1 B / (x)PL 0 = δσρ δ(x − y) + i i∂/x − h1 A 2 2 2 1  Z 1 τ ρ 2 ¯ / (x)PR hQ (x)Q (y)i − g31 d4 x2 Gαµ11µα22 (x, x2 ) −q2 B 2 2 στ

ρ  γ  ¯ (y)Qδ (x2 ) Q ¯ (x2 )Qτ (x) , ×(r1α1 γ µ1 )στ (r1α2 γ µ2 )γδ Q

(63)



 where hO(x) ≡ O(x) I /hh 1 iiI I=I=0 and we have defined the notation hh · · · iiI in this ¯

section by



 O(x) I ≡

Z

A

¯ i∂/ − h1 λ A /A PL d4 xQ 2 1 ∞ Z X
λA A 1 1 (−ig31 )n / 2 PR − q1 B / 1 PL − q2 B / 2 PR Q + −h2 A d4 x1 . . . d4 xn 2 2 2 n!  n=2 µ1 µn ...αn ×Gµα11...µ (x1 , . . . , xn )J1α (x1 ) . . . J1α (xn ) . (64) n n 1

Z

¯ D QDQ O(x) exp

¯ + IQ) ¯ +i d x(QI 4

Z

¯ ρ (x′ )i, Eq.(63) can be written as Denote the technifermion propagator S σρ (x, x′ ) ≡ hQσ (x)Q SDE for techniquark propagator,   λA A 1 1 λA A −1 / (x)PL − h2 A / (x)PR − q1 B / (x)PL − q2 B / (x)PR 0 = Sσρ (x, y) + i i∂/x − h1 A 2 1 2 2 2 1 2 2 σρ   2 . (65) ×δ(x − y) − g31 Gµα11µα22 (x, y) r1α1 γ µ1 S(x, y)r1α2 γ µ2 σρ

By defining techniquark self energy Σ as  λA A λA A 1 −1 / 1 (x)PL − h2 A / 2 (x)PR − q1 B / (x)PL iΣσρ (x, y) ≡ Sσρ (x, y) + i i∂/x − h1 A 2 2 2 1  1 / (x)PR δ(x − y) , −q2 B (66) 2 2 σρ the SDE (65) can be written as iΣσρ (x, y) =

2 g31 Gαµ11µα22 (x, y)

  α1 µ1 α2 µ2 r1 γ S(x, y)r1 γ

.

(67)

σρ

Moreover, from the fact that technigluon propagator is diagonal in the adjoint representation ¯ is space of SU(3)TC1 , i.e., Gαβ (x, y) = δ αβ Gµν (x, y), and techniquark propagator hQQi µν

24

diagonal in the fundamental representation space of SU(3)TC1 , and also (r1α r1α )ab = C2 (3)δab , Eq.(67) is diagonal in indices a, b and diagonal part becomes 2 µ1 µ2 ij iΣij ηζ (x, y) = C2 (3)g31 Gµ1 µ2 (x, y)[γ S(x, y)γ ]ηζ ,

(68)

where {i, j}, and {η, ζ} are, respectively, techniflavor, and Lorentz spinor indices; and the Casimir operator C2 (3) = (32 − 1)/(2 × 3) = 4/3.

1.

The Gap Equation

A We first consider the case of AA 1µ = A2µ = B1µ = B2µ = 0. The SU(3)TC1 technigluon

propagator in Landau gauge is   Z pµ pν i d4 p −ip(x−y) αβ αβ gµν − 2 e , Gµν (x, y) = δ (2π)4 −p2 [1 + Π(−p2 )] p

(69)

A In the case of AA 1µ = A2µ = B1µ = B2µ = 0, the SU(3)TC1 techniquark self energy and

propagator are respectively Z d4 p −ip(x−y) ij ij Σηζ (x, y) = e Σηζ (−p2 ) (2π)4

ij Sηζ (x, y)

=

Z

d4 p −ip(x−y) ij e Sηζ (p) , (2π)4

(70)

ij with Sηζ (p) = i{1/(/p − Σ(−p2 )]}ij ηζ . Substituting above equations into the SDE (68), we

have 2 Σij ηζ (−p )

=

Z

ij   2 (p − q)µ (p − q)ν i −C2 (3)g31 d4 q ν µ (71) γ gµν − γ (2π)4 (p − q)2 [1 + Π(−(p − q)2 )] (p − q)2 /q − Σ(−q 2 ) ηζ

As discussions of dynamical computation prescription for one-doublet technicolor model, 2 ij 2 2 above equation will lead Σij ηζ (−p ) = δ δηζ ΣTC (pE ) and in Euclidean space ΣTC (pE ) satisfy

ΣTC1 (p2E )

= 3C2 (3)

Z

d4 qE α31 [(pE − qE )2 ] ΣTC1 (qE2 ) . 4π 3 (pE − qE )2 qE2 + Σ2TC1 (qE2 )

(72)

¯ k Qj i with k and j techniflavor indices, The corresponding techniquark condensate hQ Z 4 ΣTC (p2E ) d pE k j jk ¯ , (73) hQ (x)Q (x)i = −12δ (2π)4 p2E + Σ2TC (p2E ) where trlc is the trace with respect to Lorentz, technicolor indices. Nonzero techniquark self ¯ energy can give a nontrivial diagonal condensate hQQi = 6 0, which spontaneously breaks SU(3)1 × SU(3)2 × U(1)Y1 × U(1)Y2 → SU(3)c × U(1)Y . 25

A In the following we consider the effects of the nonzero electroweak gauge fields AA 1µ , A2µ ,

B1µ and B2µ . The SDE (68) is explicitly  λA A λA A 1 2 µ / 1 (x)PL − h2 A / 2 (x)PR − q1 B / (x)PL i∂/x − h1 A Σ(x, y) = C2 (3)g31 Gµν (x, y)γ 2 2 2 1  −1 1 / 2 (x)PR δ(x − y) − Σ(x, y) γ ν , (74) −q2 B 2 where the techniflavor and Lorentz spinor indices of the techniquark self energy are implicitly contained. A Suppose the function ΣTC1 (−p2 ) is a solution of the SDE in the case AA 1µ = A2µ = B1µ =

B2µ = 0, that is, it satisfies the equation Z 1 d4 q 2 2 Gµν (q 2 )γ µ ΣTC1 (−p ) = C2 (3)g31 γν , 4 (2π) /q + /p − ΣTC1 [−(q + p)2 ]

(75)

Replacing the variable p by p + ∆ in Eq.(75) and subsequently integrating over p with the weight e−ip(x−y) , we obtain, as long as ∆ is commutative with ∂x and Dirac matrices, ΣTC1 [(∂x − i∆)2 ]δ(x − y) 2 = C2 (3)g31 Gµν (x, y)γ µ

1 / − ΣTC1 [−(i∂x + ∆)2 ] i∂/x + ∆

δ(x − y)γ ν .

(76)

Even if ∆ is noncommutative with ∂x and Dirac matrices, the above equation holds as / ∆] is higher order of momentum the lowest order approximation, for the commutator [∂, A

A

λ 1 1 A than ∆ itself. Now if we take ∆ to be −h1 λ2 AA 1 PL − h2 2 A2 PR − q1 2 B1 PL − q2 2 B2 PR ,

ignoring its noncommutative property with ∂x and Dirac matrices, Eq.(76) is just the SDE A

λ A A x (74) in the case AA 1µ 6= 0, A2µ 6= 0, B1µ 6= 0, and B2µ 6= 0. Thus, ΣTC1 [(∂µ + ih1 2 A1µ PL + A

1 1 2 ih2 λ2 AA 2µ PR +iq1 2 B1µ PL +iq2 2 B2µ PR ) ]δ(x−y), which is SU(3)1 ×SU(3)2 ×U(1)Y1 ×U(1)Y2

covariant, can be regarded as the lowest-order solution of Eq.(74). From Eqs.(58a, 58b), we can write the covariant derivative of SU(3)1 × SU(3)2 × U(1)Y1 × U(1)Y2 as λA A λA 1 1 A1µ PL + ih2 AA (77) 2µ PR + iq1 B1µ PL + iq2 B2µ PR 2 2 2 2 λA 1 λA A 1 = ∂µ + ig3 AA + ig B + ig (cot θP − tan θP ) Bµ + ig1 (cot θ′ PL − tan θ′ ) Zµ′ , 1 µ 3 L R µ 2 2 2 2

∇µ ≡ ∂µ + ih1

where AA µ and Bµ are the gauge fields of the unbroken symmetry group SU(3)c × U(1)Y . To further simplify the calculations, we can just keep this SU(3)c × U(1)Y covariance of the self energy, that is, we can replace ∇µ by the covariant derivative of SU(3)c × U(1)Y , ∇µ ≡ ∂µ + ig3

λA A 1 Aµ + ig1 Bµ 2 2 26

(78)

inside the techniquark self energy. Thus, if the function Σ(∂x2 )δ(x − y) is the self-energy A solution of the SDE in the case AA 1µ = A2µ = B1µ = B2µ = 0, we can replace its argument 2

∂x by the SU(3)c × U(1)Y covariant derivative ∇x , i.e., Σ(∇x )δ(x − y), as an approximate A solution the SDE in the case AA 1µ 6= 0, A2µ 6= 0, B1µ 6= 0, and B2µ 6= 0. ¯ The exponential terms on Now we are ready to integrate out the techniquarks Q and Q.

the right-hand side of Eq.(61) can be written explicitly as ∞ Z X

d4 x1 . . . d4 xn

n=2

≈

Z

(−ig31 )n α1 ...αn µ1 µn Gµ1 ...µn (x1 , . . . , xn )J1,α (x1 ) . . . J1,α (xn ) n 1 n!

¯ σ (x)Πσρ (x, x′ )Qρ (x′ ) , d4 xd4 x′ Q

(79)

where in the last equality we have taken the approximation of replacing the summation over 2n-fermion interactions with parts of them by their vacuum expectation values, that is, Πσρ (x, x′ ) =

∞ X

′ Π(n) σρ (x, x ) ,

(80)

n=2

(n) Πσρ (x, x′ )

 (−ig31 )n α1 ...αn ′ Gµ1 ...µn (x, x2 . . . , xn−1 , x ) (r1α1 γ µ1 )σσ1 Qσ1 (x) = n × d x2 . . . d xn−1 n!  α α α µ n−1 µ ρ µ ′ ¯ 2 )r1 2 γ 2 Q(x2 ) . . . Q(x ¯ n−1 )r1 γ n−1 Q(xn−1 )Q ¯ n (x )(r1 n γ n )ρn ρ , (81) ×Q(x Z

4

4

¯ and the lowest term where the factor n comes from n different choices of unaveraged QQ, of which is ′ Π(2) σρ (x, x )

  (−ig31 )2 α1 α2 α1 µ1 α2 µ2 ′ σ1 ρ2 ′ ¯ = 2· Gµ1 µ2 (x, x ) (r1 γ )σσ1 Q (x)Q (x )(r1 γ )ρ2 ρ 2!   α1 µ1 α2 µ2 2 α1 α2 ′ . = −g31 Gµ1 µ2 (x, x ) r1 γ S(x, y)r1 γ

(82)

σρ

Comparing Eq.(82) with Eq.(67), we have 2

′ ′ iΠ(2) σρ (x, x ) = Σσρ (x, x ) ≈ Σσρ (∇x )δ(x − y) .

(83)

Substituting Eq.(79) into Eq.(61), we obtain   A A exp iSTC1 [A1µ , A2µ , B1µ , B2µ ]  λA A λA A 1 1 ′ / − g3 (cot θPL − tan θPR ) B / − g1 B / − g1 (cot θ′ PL − tan θ′ PR ) Z / ≈ Det i∂/ − g3 A 2 2 2 2  2 −ΣTC1 (∇ ) , (84) 27

(2)

where we have taken further approximation of keeping only the lowest order, i.e. Πσρ (x, x′ ), of Πσρ (x, x′ ). With all these approximations, we have  λA A 1 λA A A A ′ / − g3 (cot θPL − tan θPR ) B / − g1 B / iSTC1 [Aµ , Bµ , Bµ , Zµ ] = Tr log i∂/ − g3 A 2 2 2  1 ′ 2 ′ ′ / − ΣTC1 (∇ ) , (85) −g1 (cot θ PL − tan θ PR ) Z 2 Since we know QCD-induced condensate is too weak to give sufficiently large masses of W and Z bosons and thus it is negligible when we consider the main cause responsible for the electroweak symmetry breaking which imply that ordinary QCD gluon fields has very little effects on our technicolor and electroweak interactions, therefore for simplicity, we ignore them by just vanishing gluon field AA µ = 0. In next two subsubsections, we perform low energy expansion and explicitly expand above action up to order of p4 .

2.

Low Energy Expansion for iSTC1 [0, BµA , Bµ , Zµ′ ]

We have 2

iSTC1 [0, BµA , Bµ , Zµ′ ] = Tr log[i∂/ + v/1 + a /1 γ5 − ΣTC1 (∇ )] Z (4) = i d4 x(F0TC1 )2 tr[a21 (x)] + STC1 [0, BµA , Bµ , Zµ′ ] + O(p6 ) , (86) where the parameter F0TC1 is depend on the techniquark self energy ΣTC1 . The fields vµ and aµ are identified with g3 λA 1 g1 (cot θ − tan θ) BµA − g1 Bµ − (cot θ′ − tan θ′ )Zµ′ 2 2 2 4 g3 λA A g 1 ≡ − (cot θ + tan θ) Bµ − (cot θ′ + tan θ′ )Zµ′ . 2 2 4

v1,µ ≡ −

(87a)

a1,µ

(87b)

Substituting Eqs.(87) into Eq.(86), we obtain, at the order of p2 , (2)

STC1 [0, BµA , Bµ , Zµ′ ] Z (F0TC1 )2 d4 x[2g32 (cot θ + tan θ)2 BµA B A,,µ + 3g12 (cot θ′ + tan θ′ )2 Z ′2 ] . = 16

(88)

Now we come to consider the p4 order effective action. It can be divided into two parts (4)

(4d)

(4c)

STC1 [0, BµA , Bµ , Zµ′ ] = STC1 [0, BµA , Bµ , Zµ′ ] + STC1 [0, BµA , Bµ , Zµ′ ] (4d)

iSTC1 [0, BµA , Bµ , Zµ′ ] = Tr log[i∂/ + v/1 + a / 1 γ5 ]

(90) 2

(4c)

iSTC1 [0, BµA , Bµ , Zµ′ ] = Tr log[i∂/ + v/1 + a / 1 γ5 ] /1 γ5 − ΣTC1 (∇ )] − Tr log[i∂/ + v/1 + a 28

(89)

(4d)

STC1 [0, BµA , Bµ , Zµ′ ] is divergent part of action, which can calculated by following standard formula Z 1 (91) iTr log[i∂/ + /lPL + /rPR ] = − K d4 x tr[r µν rµν + lµν lµν ] 2 rµν = ∂µ rν − ∂ν rµ − i(rµ rν − rν rµ ) lµν = ∂µ lν − ∂ν lµ − i(lµ lν − lν lµ ) 2 κ 1 (log 2 + γ) (92) K=− 2 48π Λ with K a divergent constant depend on the ratio between ultraviolet cutoff Λ and infrared cutoff κ of the theory. Identify λA A 1 g1 rµ = v1,µ + a1,µ = −g3 cot θ Bµ − g1 Bµ − cot θ′ Zµ′ 2 2 2 λA A 1 g1 lµ = v1,µ − a1,µ = g3 tan θ Bµ − g1 Bµ + tan θ′ Zµ′ 2 2 2

(93) (94)

With these preparations, (4d) ′ STC1 [0, BµA , Bµ , Zc,µ ]

 2 Z g 3g 2 1 4 A A BrA,µν + tan2 θBl,µν BlA,µν ) + 1 Bµν B µν = − K d x 3 (cot2 θBr,µν 2 2 2  2 2 3g1 3g1 2 ′ 2 ′ ′ ′,µν ′ ′ ′,µν (cot θ + tan θ )Zµν Z + (cot θ − tan θ )Bµν Z + 4 2

with A Br,µν = ∂µ BνA − ∂ν BµA − g3 cot θf ABC BµB BνC

A Bl,µν = ∂µ BνA − ∂ν BµA + g3 tan θf ABC BµB BνC

′ Zµν = ∂µ Zν′ − ∂ν Zµ′

Bµν = ∂µ Bν − ∂ν Bµ

(95)

(4c)

STC1 [0, BµA , Bµ , Zµ′ ] is convergent part of action, which can calculated by following standard formula (4c) STC1 [0, BµA , Bµ , Zµ′ ]

=

Z

d4 x tr[−K1TC1,Σ6=0 (dµ aµ1 )2 − K2TC1,Σ6=0 (dµ a1,ν − dν a1,µ )2 + K3TC1,Σ6=0 (a21 )2

TC1,Σ6=0 TC1,Σ6=0 V1,µν aµ1 aν1 ] , V1,µν V1µν + iK14 +K4TC1,Σ6=0 (a1,µ a1,ν )2 − K13

with V1,µν ≡ ∂µ v1,ν − ∂ν v1,µ − i(v1,µ v1,ν − v1,ν v1,µ ) and dµ a1,ν ≡ ∂µ a1,ν − i(v1,µ a1,ν − a1,ν v1,µ ). (4c)

STC1 [0, BµA , Bµ , Zµ′ ] can be further divided into four parts (4c,B A )

(4c)

STC1 [0, BµA , Bµ , Zµ′ ] = STC1

(4c,B A Z ′ )

+STC1 (4c,B A )

The detail form of STC1

(4c,B)

(4c,Z ′ )

(4c,B)

[B A ] + STC1 [B] + STC1 [Z ′ ] (4c,BZ ′ )

[B A , Z ′ ] + STC1

(4c,Z ′ )

(4c,B A Z ′ )

[B A ], STC1 [B], STC1 [Z ′ ], STC1

[B, Z ′ ]

(97) (4c,BZ ′ )

[B A , Z ′ ] and STC1

[B, Z ′ ]

are given in (B1)-(B5) respectively. Since TC1 interaction is SU(3) gauge interaction which 29

(96)

is same as QCD interaction and the quark number Nf are all equal to three 1 , as we discussed before, due to scale invariance we have KiTC1,Σ6=0 , i = 2, 3, 4, 13, 14 are equal to those of QCD values within our approximations KiTC1,Σ6=0 = KiΣ6=0

i = 2, 3, 4, 13, 14

(98)

Use relation given in Ref.[12] 1 1 1 Σ6=0 Σ6=0 Σ6=0 Σ6=0 H1 = − (K2Σ6=0 + K13 ) L10 = (K2Σ6=0 − K13 ) L9 = (4K13 ) (99) − K14 4 2 8 1 1 Σ6=0 Σ6=0 Σ6=0 Σ6=0 ) (100) ) L3 = (K3Σ6=0 − 2K4Σ6=0 − 6K13 + 3K14 − K14 L1 = (K4Σ6=0 + 2K13 32 16 In Table III, we list down original QCD calculation result given in Ref.[12], the value for H1 in the original paper is a divergent constant therefore not given its value, now the divergent (4d)

part is already extracted out by STC1 [0, BµA , Bµ , Zµ′ ], H1 here is a convergent quantity which can be obtained in original formula for H1 by subtracting out its divergent part caused by terms with Σ = 0. TABLE III. The obtained nonzero values of the O(p4 ) coefficients H1 , L10 , L9 , L2 , L1 , L3 for topcolor-assisted technicolor model. The F0TC1 and ΛT C1 are in units of TeV and coefficients are in units of 10−3 . F0TC1 ΛTC1 H1 L10 1

L9

L2

L1

L3

5.21 43.0 -7.04 5.06 2.19 1.10 -7.81

We finally obtain K2TC1,Σ6=0 = L10 − 2H1

K3TC1,Σ6=0 = 64L1 + 16L3 + 8L9 + 2L10 + 4H1

K4TC1,Σ6=0 = 32L1 − 8L9 − 2L10 − 4H1 TC1,Σ6=0 = −L10 − 2H1 K13

B.

TC1,Σ6=0 = −4L10 − 8L9 − 8H1 K14

(101)

Electroweak Symmetry Breaking: the Contribution of SU (3)TC2

Likewise, it is easily check that the SU(3)TC2 interaction does induce the techniquark condensate hT¯ T i = 6 0, which triggers the electroweak symmetry breaking SU(2)L ×U(1)Y → 1

This is in fact an approximation in which we have ignored possible effects on the running of TC1 gauge coupling constant from ordinary color gauge fields and coloron fields.

30

U(1)EM . Integrating out the SU(3)TC2 technigluons Gα2µ and the techniquarks T and T¯, Eq.(60) can be written as   a exp iSEW [Wµ , Bµ ]  Z Z  Z 1 a aµν 1 Aµν 4 A ′ = exp i d x(− Wµν W ) (102) DBµ DZµ exp i d4 x(− AA 1µν A1 4 4  1 1 A Aµν 1 µν µν A A ′ a − A2µν A2 − B1µν B1 − B2µν B2 ) + iSTC1 [Aµ , Bµ , Bµ , Zµ ] + iSTC2 [Wµ , B2µ ] , 4 4 4 A ′ where STC1 [AA µ , Bµ , Bµ , Zµ ] has been given in Eq.(85) for its general form and expanded up

to order of p2 in (88) and p4 in (89), and STC2 [Wµa , B2µ ] is given by   a exp iSTC2 [Wµ , B2µ ] (103)  Z   Z   1 α αµν α α 4 α / 2 + /l2 PL + /r2 PR T = D T¯ DT DG2µ exp i d x − F2µν F2 + T¯ i∂/ − g32 r2 G 4 a

with l2,µ ≡ −g2 τ2 Wµa − q2 61 B2µ and r2,µ ≡ −q2 ( 16 +

τ3 )B2µ . 2

By means of the Gasser-Leutwyler’s prescription presented in section II, the functional integration (103) can be related to the QCD-type chiral Lagrangian by      R R 4 α αµν 1 α α α ¯ ¯ / 2 + /l2 PL + /r2 PR T D T DT DG2µ exp i d x − 4 F2µν F2 + T i∂/ − g32 r2 G R  R D T¯DT exp i d4 xT¯[i∂/ + /l2 PL + /r2 PR ]T Z ˜ exp{iSTC2-induced eff [U˜ , l2,µ , r2,µ ]} , = Dµ(U) (104) with the SU(3)TC2 -induced chiral effective action  TC2 2 Z (F0 ) 4 ˜ + LTC2 ˜ † ∇µ U)] ˜ 2 ˜ tr[(∇µ U˜ † )(∇µ U)] [tr(∇µ U STC2-induced eff [U , l2,µ , r2,µ ] = dx 1 4 ˜ † ∇ν U˜ ] + LTC2 tr[(∇µ U˜ † ∇µ U˜ )2 ] +LTC2 tr[∇µ U˜ † ∇ν U˜ ]tr[∇µ U 2

3

R ˜ νU ˜ † + F L ∇µ U˜ † ∇ν U] ˜ + LTC2 tr[U˜ † F R U˜ F L,µν ] −iLTC2 tr[Fµν ∇µ U∇ 9 µν 10 µν  R R,µν L +H1TC2 tr[Fµν F + Fµν F L,µν ] , (105)

where ˜ ≡ ∂µ U˜ − ir2,µ U˜ + iU˜ l2,µ , ∇µ U R Fµν ≡ i[∂µ − ir2,µ , ∂ν − ir2,ν ] ,

˜ † − il2,µ U˜ † + iU˜ † r2,µ , ∇µ U˜ † = −U˜ † (∇µ U˜ )U˜ † = ∂µ U L Fµν ≡ i[∂µ − il2,µ , ∂ν − il2,ν ] .

31

(106)

TC2 The coefficients F0TC2 , LTC2 , LTC2 , LTC2 , LTC2 arise from SU(3)TC2 dynamics. These 1 2 3 10 , H1

coefficients relates to the KiTC2 coefficients as that appeared in one doublet technicolor model as ′

′

TC2,Σ TC2,Σ6=0 KTC2,Σ + K13 K2TC2,Σ6=0 − K13 =− 2 LTC2 = 10 4 2 TC2,Σ6=0 TC2,Σ6=0 TC2,Σ6=0 TC2,Σ6=0 TC2,Σ6=0 K14 K4 + 2K13 − K14 K13 TC2 − L = LTC2 = 2 9 2 8 16 TC2,Σ6=0 TC2,Σ6=0 TC2,Σ6=0 TC2,Σ6=0 TC2 L K − K4 − 4K13 + 2K14 LTC2 + 3 = 3 (107) 1 2 32

H1TC2

KiTC2 coefficients with superscript TC2 denote present TC2 interaction, they are functions of technifermion T self energy ΣTC2 (p2 ) and detail expressions are already written down in (36) of Ref.[12] with the replacement of Nc → 3 and subtract out their ΣTC2 (p2 ) = 0 parts. Since TC2 interactions among techiquark doublet T is SU(3) which is same as one doublet technicolor model discussed before except ΛTC2 may be different as ΛTC of one doublet technicolor model, but as we discussed before KiTC2 , i = 1, 2, 3, 4, 13, 14 are independent of ΛTC2 , therefore our KiTC2 , i = 1, 2, 3, 4, 13, 14 are same as those obtained in one doublet technicolor model. This results present LTC2 , i = 1, 2, 3, 9, 10 and H1TC2 coefficients are i same as those in one doublet technicolor model,in Table VI, we list down the numerical calculation result in which the method is already mentioned in previous section and except the result for H1TC2, all others are already used in Table I. TABLE VI. The obtained nonzero values of the O(p4 ) coefficients TC2 , LTC2 , LTC2 , LTC2 for topcolor-assisted technicolor model. H1TC2 , LTC2 3 1 2 10 , L9

The F0TC2 and ΛT C2 are in units of TeV and coefficients are in units of 10−3 . 1D TC2 = L1D LTC2 = L1D LTC2 = L1D LTC2 = L1D F0TC2 ΛTC2 H1TC2 = H11D LTC2 3 3 1 1 2 2 9 10 = L10 L9

0.25 1.34

43.0

-6.90

4.87

2.02

1.01

-7.40

˜ is a 2 × 2 unitary matrix, and thus the LTC2 term and Similar as one-doublet case for U 1 the LTC2 term are linearly related, 3 ˜ † ∇µ U˜ )2 ] LTC2 tr[(∇µ U 3

   † µ  2 LTC2 3 † µ ˜ (∇ U) ˜ U˜ (∇ U˜ ) tr U = 2

(108)

Comparing Eqs.(106) with standard covariant derivative given in Ref.[2], we need to recognize U˜ † = U ,

a

3

˜ µ U ≡ ∂µ U + ig2 τ W a U − Uiq2 τ B2µ . ∇µ U˜ † = D 2 µ 2 32

(109)

a

3

L a R = −g2 τ2 Wµν − q2 61 B2µν with B2µν ≡ ∂µ B2ν − ∂ν B2µ is the And Fµν = −q2 ( 61 + τ2 )B2µν , Fµν

U(1)Y2 gauge field strength tensor. Substituting above equations back into Eq.(105), we obtain  Z 1D (F TC2 )2 4 ˜µX ˜ µ ) + (L1D + L3 )[tr(X ˜µX ˜ µ )]2 STC2-induced eff [U, W, B2 ] = dx − 0 tr(X 1 4 2 1D ˜ µX ˜ ν ) − iL1D tr(W µν X ˜ µX ˜ ν) ˜µ X ˜ ν )]2 − i L9 q2 B2µν tr(τ 3 X +L1D [tr( X 9 2 2 L1D 1 µν µν 1D 2 + 10 q2 B2µν tr(τ 3 W ) + (L1D 10 + 11H1 )q2 B2µν B2 2 18  µν

+H11D tr(W µν W ) ,

(110)

˜ µ is defined by where X ˜ µ ≡ U † (D ˜ µ U) . X

(111)

With (111) and from Eqs.(109), (57b) and (59b), we obtain 3

˜ µ = Xµ + ig1 tan θ′ Z ′ τ . X µ 2

(112)

Substituting Eq.(112) into Eq.(110), we obtain, at the order of p2 , (2)

STC2-induced eff [U, W a , B cos θ′ − Z ′ sin θ′ ]   Z (F0TC2 )2 g12 4 µ ′ ′ 3 µ 2 ′ ′2 = d x − tr(Xµ X ) − ig1 tan θ Zµ tr(τ X ) + tan θ Z . 4 2

33

(113)

Similarly detail algebra gives (4)

iSTC2-induced eff [U, W a , B cos θ′ − Z ′ sin θ′ ]  Z L1D 4 1D 2 1D µ ν = i d x (L1 + 3 )[tr(Xµ X µ )]2 + L1D 2 [tr(Xµ Xν )] − iL9 tr(W µν X X ) 2 1D 1 L L1D µν −i 9 g1 Bµν tr(τ 3 X µ X ν ) + 10 g1 Bµν tr(τ 3 W ) + (L1D + 11H11D )g12 Bµν B µν 2 2 18 10 L1D µν 3 )g12 tan2 θ′ Z ′,2 (trX 2 ) (114) + +H11D tr(W µν W ) − (L1D 1 2 L1D 1 3 1D +(L1D + L + )[ g14 tan4 θ′ Z ′,4 − ig13 tan3 θ′ Z ′,2 Z ′,µ tr(Xµ τ 3 )] 1 2 2 4 1 1D 2 − L2 g1 tan2 θ′ [Z ′2 tr(Xν τ 3 )tr(X ν τ 3 ) + Z ′µ Z ′ν tr(Xµ τ 3 )tr(Xν τ 3 )] 2 L1D 3 2 2 ′ ′,µ ′,ν −(L1D + )g12 tan2 θ′ Z ′,µ Z ′,ν tr(Xµ τ 3 )(Xν τ 3 ) − L1D 1 2 g1 tan θ Z Z tr(Xµ Xν ) 2 L1D 3 ′ ′ 3 µ ν +2i(L1D + )g1 tan θ′ Zµ′ tr(X µ τ 3 )(trX 2 ) + 2iL1D 1 2 g1 tan θ Zµ tr(Xν τ )tr(X X ) 2 1 L1D ′ ′ µ 3 ′ 3 ν ′ tr(τ 3 X µ X ν ) + g1 L1D +i 9 g1 tan θ′ Zµν 9 tan θ [tr(W µν X τ )Zν + tr(W µν τ X )Zµ ] 2 2  L1D 1 1D µν 10 ′ ′ 3 1D 2 2 ′ ′ ′,µν 2 ′ ′ µν g1 tan θ Zµν tr(τ W ) + (L10 + 11H1 )[g1 tan θ Zµν Z − 2g1 tan θ Zµν B ] . − 2 18 Thus, from Eqs.(103–111) we obtain   τa a 1 1 τ3 / 2 PR / PL − q2 B / 2 PL − q2 ( + )B iSTC2 [Wµa , B2µ ] = Tr ln i∂/ − g2 W (115) 2 6 6 2   Z (2) (4) × log Dµ(U) exp iSTC2-induced eff [U, W, B2 ] + iSTC2-induced eff [U, W, B2 ] .   3 a / a PL − q2 61 B / 2 PL − q2 ( 16 + τ2 )B / 2 PR , which is at We still left to compute Tr ln i∂/ − g2 τ2 W   3 a / a PL − q2 16 B / 2 PL − q2 ( 61 + τ2 )B / 2 PR as least order of p4 . We can write Tr ln i∂/ − g2 τ2 W

 τa a 1 1 τ3 / 2 PR = Tr log[i∂/ + /l2 PL + /r2 PR ] (116) / / W PL − q2 B 2 PL − q2 ( + )B 2 6 6 2 1 µ τ a a,µ 1 τ a a,µ µ l2 = −g2 W − q2 B2 = −g2 W − (g1 B µ − g1 tan θ′ Z ′,µ ) 2 6 2 6 3 3 τ 1 τ 1 r2µ = −q2 ( + )B2µ = −( + )(g1 B µ − g1 tan θ′ Z ′,µ ) 6 2 6 2 

Tr ln i∂/ − g2

Then with help of (91), computation gives   τa a 1 1 τ3 / 2 PR / PL − q2 B / 2 PL − q2 ( + )B iTr ln i∂/ − g2 W (117) 2 6 6 2   Z 1 11 2 1 2 a a,µν 4 µν 2 ′ ′ ′,µν ′ ′ µν =− K d x g (Bµν B + tan θ Zµν Z − 2 tan θ Zµν B ) + g2 Wµν W 2 18 1 2 34

Substituting Eq.(115) into Eq.(102) and then comparing it with the last line of Eq.(55), we have   a iSeff [U, Wµ , Bµ ] Z  Z  Z 1 1 1 a aµν Aµν A ′ 4 DBµ DZµ exp i d4 x(− AA − AA AAµν = exp i d x(− Wµν W ) 1µν A1 4 4 4 2µν 2   1 1 τa a 1 1 τ3 / 2 PR / PL − q2 B / 2 PL − q2 ( + )B − B1µν B1µν − B2µν B2µν ) + Tr ln i∂/ − g2 W 4 4 2 6 6 2  N [Wµa, Bµ ] exp

A ′ +iSTC1 [AA µ , Bµ , Bµ , Zµ ] + iSTC2-induced eff [U, W, B2 ]

,

(118)

AA µ =0

where we have put the SU(3)c gluon fields AA µ = 0 on the right-hand side, for the QCD effects are small here. The normalization factor from its definition (56) can be calculated similarly as previous procedure, the only difference is that we switch off TC2 interaction by taking g32 = 0 and it will result STC2-induced eff [U, W, B2 ] vanishes, then this leads ignoring term iSTC2-induced eff [U, W, B2 ] in above expression, we get expression for N [Wµa, Bµ ].

C.

Integrate out of Colorons

Now, as shown in Eqs.(118), the next work is to integrate out the SU(3)c octet of colorons, BµA . From Eqs.(57a) and (59a), it is straightforward to get A A AA 1µν = Aµν sin θ + B1,µν cos θ ,

(119a)

A A AA 2µν = Aµν cos θ − B2,µν sin θ ,

(119b)

where A B C B1,µν ≡ ∂µ BνA − ∂ν BµA + g3 f ABC (cot θBµB BνC + BµB AC ν + Aµ Bν )

(120)

A B C B2,µν ≡ ∂µ BνA − ∂ν BµA − g3 f ABC (tan θBµB BνC + BµB AC ν + Aµ Bν )

(121)

then Eqs.(118) become   a a N [Wµ , Bµ ] exp iSeff [U, Wµ , Bµ ]  Z Z  1 a aµν 4 A ′ = exp i d x(− Wµν W ) DBµ DZµ exp iSTC1 [0, BµA , Bµ , Zµ′ ] (122) 4 Z 1 A Aµν 2 1 1 1 A Aµν B1 cos2 θ − B2µν B2 sin θ − B1µν B1µν − B2µν B2µν ) +i d4 x(− B1µν 4 4 4 4    τa a 1 1 τ3 / PR + iSTC2-induced eff [U, W, B2 ] , / PL − q2 B / PL − q2 ( + )B +Tr ln i∂/ − g2 W 2 6 2 6 2 2 35

Ignoring term iSTC2-induced eff [U, W, B2 ] in above expression, we get expression for N [Wµa , Bµ ]. R A ′ In above result, if we denote the coloron involved part be DBµA eiScoloron [B ,Z ] , then  Z 1 A Aµν 4c,B A 4c,B A Z ′ A ′ A A ′ 4 Scoloron [B , Z ] = STC1 [B ] + STC1 [B , Z ] + d x − B1µν B1 cos2 θ 4 (F TC1 )2 2 1 A Aµν 2 B2 sin θ + 0 g3 (cot θ + tan θ)2 BµA B A,,µ − B2µν 4 8  g32 A,µν 2 A A,µν 2 A − K(cot θBr,µν Br + tan θBl,µν Bl ) (123) 4 0 int = Scoloron [B A , Z ′ ] + Scoloron [B A , Z ′ ] (124) 0 int with Scoloron [B A , Z ′ ] be linear and quadratic in coloron fields and Scoloron [B A , Z ′ ] be cubic

and quartic in coloron fields, the detail form of them are given in (C1) and (C2). Now coloron fields is not correctly normalized, since the coefficient in front of kinetic term is not A standard −1/4. We now introduce normalized fields BR,µ as

1 A (125) BµA = BR,µ c 1 TC1,Σ6=0 1 (cot θ − tan θ)2 + K(cot2 θ + tan2 θ)] c2 = 1 + g32 [ K2TC1,Σ6=0 (cot θ + tan θ)2 + K13 2 2 (126) 0 With them, Scoloron [B A , Z ′ ] in terms of normalized coloron fields become Z 1 A −1,µν 0 A ′ A Scoloron [B , Z ] = d4 x BR,µ (x)DB (Z ′ )BR,ν (x) 2

(127)

with −1,µν −1,µν −1,µν 2 DB (Z ′ ) = DB0 + ∆µν (Z ′ ) DB0 = g µν (∂ 2 + Mcoloron ) − (1 + λB )∂ µ ∂ ν (128) 1 5 ′ ∆µν (Z ′ ) = [g µν ( K3TC1,Σ6=0 + K4TC1,Σ6=0 )Zν′ ′ Z ′,ν + (2K3TC1,Σ6=0 + K4TC1,Σ6=0 )Z ′,µ Z ′,ν ] 2 4 2 4 2 ′ ′ 2 g g (cot θ + tan θ) (cot θ + tan θ ) × 1 3 (129) 32c2 1 cot θ + tan θ TC1 g3 F0TC1 Mcoloron = g3 F0 = (130) 2 c 2c sin θ cos θ 1 (cot θ + tan θ)2 TC1,Σ6=0 (131) K1 λB = − g32 4 c2

Here we recover the estimation for coloron mass Mcoloron ∼ g3 Λ/(sin θ cos θ) given in Ref.[17] if we identify Λ = F0TC1 /(2c). We now denote the result action after integration over colorons as Z

DBµA eiScoloron [B

A ,Z ′ ]

36

¯

′

= eiScoloron [Z ]

(132)

µν S¯coloron [Z ′ ] are all vacuum diagrams with propagator DB (Z ′ ) and vertices determined by int Scoloron [B A , Z ′ ]. The loop expansion result is

1 −1 iS¯coloron [Z ′ ] = − Tr log DB (Z ′ ) + two or more loop contributions 2

(133)

The first term in the r.h.s. of above equation is one loop result, if we further perform low energy expansion for it and drop out total derivative terms, we find the contributions from one loop term is quartically divergent up to order of p4 which will be vanish if we take dimensional regularization, then up to order of one loop precision, colorons makes no contributions.

D.

Integrate out of Z ′

From Eqs.(57b) and (59b) imply B1µν = Bµν sin θ′ + (∂µ Zν′ − ∂ν Zµ′ ) cos θ′ ,

(134a)

B2µν = Bµν cos θ′ − (∂µ Zν′ − ∂ν Zµ′ ) sin θ′ .

(134b)

Substituting B1µ = Bµ sin θ′ + Zµ′ cos θ′ , B2µ = Bµ cos θ′ − Zµ′ sin θ′ and Eq.(134) into the right-hand side of Eq.(122), combined with (132), we get   a a N [Wµ , Bµ ] exp iSeff [U, Wµ , Bµ ] Z  Z   Z 1 1 1 a aµν ′ 4 4 DZµ exp i d x − Bµν B µν − (∂µ Zν′ − ∂ν Zµ′ )2 = exp i d x(− Wµν W ) 4 4 4 2 TC1 2 2 2 3g (F ) 1 3g 3g ′ + 1 0 (cot θ′ + tan θ′ )2 Z ′2 − K[ 1 Bµν B µν + 1 (cot2 θ′ + tan2 θ′ )Zµν Z ′,µν 16 2 2 4  3g12 (4c,B) (4c,Z ′ ) ′ ′ ′,µν (cot θ − tan θ )Bµν Z ] + iS¯coloron [Z ′ ] + iSTC1 [B] + iSTC1 [Z ′ ] + 2   1 1 τ3 τa a (4c,BZ ′ ) / PL − q2 B / 2 PL − q2 ( + )B / 2 PR +iSTC1 [B, Z ′ ] + Tr ln i∂/ − g2 W 2 6 6 2  +iSTC2-induced eff [U, W, B2 ] ,

(135)

Ignoring term iSTC2-induced eff [U, W, B2 ] in above expression, we get expression for N [Wµa , Bµ ]. R ′ a In above result, if we denote the Z ′ involved part be DZµ′ eiSZ ′ [Z ,U,W ,B] , then we will find

that Z ′ field in SZ ′ [Z ′ , U, W a , B] is not correctly normalized, since the coefficient in front of

37

′ kinetic term is not standard −1/4. We now introduce normalized fields ZR,µ as

1 ′ Z (136) c′ R,µ 3g 2 3g 2 (137) = 1 + K 1 (cot2 θ′ + tan2 θ′ ) + 1 [K2TC1,Σ6=0 (cot θ′ + tan θ′ )2 2 4 2 11 TC1,Σ6=0 + 11H11D )g12 tan2 θ′ +K13 (cot θ′ − tan θ′ )2 ] − g12 K tan2 θ′ − (L1D 9 9 10

Zµ′ = c′,2

then ′

a

SZ ′ [Z , U, W , B] =

Z

1 ′ g4 µ ′ ′ d4 x [ ZR,µ (x)DZ−1,µν ZR,ν (x) + ZR′,µ JZ,µ + ZR2 ZR,µ J3Z + g4Z ′41 ZR′,4 ](138) 2 c

with DZ−1,µν = g µν (∂ 2 + MZ2 ′ ) − (1 + λZ )∂ µ ∂ ν + ∆µν Z (X) 2 TC2 2 2 TC1 2 g (F ) ) 3g (F (cot θ′ + tan θ′ )2 + 1 0′2 tan2 θ′ MZ2 ′ = 1 0′2 8c 4c 2 TC1,Σ6=0 3g1 (cot θ′ + tan θ′ )2 λZ = −K1 ′2 8c 2 ′ tan θ 2 1D µν 2 1D 1D 3 ∆µν (X) = −g {(2L1D 1 1 + L3 )g (trX ) + (2L1 + L3 )tr(Xµ τ )tr(Xν τ 3) Z ′2 c 1D µν 3 ν 3 3 3 +2L1D 2 tr(Xµ Xν ) + L2 [g tr(Xν τ )tr(X τ ) + tr(Xµ τ )tr(Xν τ )]}

(139) (140) (141)

(142)

and g12 γ ν ∂ Bµν + J˜Zµ (143) c′ (F TC2 )2 µ JZ0 (144) = −ig1 0 ′ tan θ′ tr(τ 3 X µ ) 4c TC1,Σ6=0 1 1D 1 ′ (145) γ = −(3K + K13 ) (cot θ′ − tan θ′ ) − (11K + 2L1D 10 + 22H1 ) tan θ 2 9 2i L1D 2i 3 J˜Z,µ = ′ (L1D + )g1 tan θ′ tr(Xµ τ 3 )(trX 2 ) + ′ L1D g1 tan θ′ tr(Xν τ 3 )(trXµ Xν ) 1 c 2 c 2 1 i 1D ′ 3 3 ν L g1 tan θ′ ∂ ν [trτ 3 (Xµ Xν − Xν Xµ )] + ′ g1 L1D 9 tan θ tr[(W µν τ − τ W µν )X ] + 2c 2c′ 9 1 − ′ L1D g1 tan θ′ ∂ ν tr(τ 3 W µν ) c 10 L1D 1 3 3 1D (cot θ′ + tan θ′ )4 + (L1D + L + ) tan4 θ′ (146) g4Z = (K3TC1,Σ6=0 + K4TC1,Σ6=0 ) 1 2 256 2 4 L1D i µ 3 1D J3Z + L + )g13 tan3 θ′ tr(Xµ τ 3 ) (147) = − ′3 (L1D 1 2 c 2 µ JZµ = JZ0 +

We denote the result action after the integration over Z ′ as Z ′ a a ¯ DZµ′ eiSZ ′ [Z ,U,W ,B] = eiSZ ′ [U,W ,B] 38

(148)

We can use loop expansion to calculate above integration S¯Z ′ [U, W a , B] = SZ ′ [Zc′ , U, W a, B] + loop terms

(149)

with classical field Zc′ satisfy ∂ ′ (x) ∂Zc,µ

  ′ a SZ ′ [Zc , U, W , B] + loop terms = 0

(150)

With (138), the solution is Zc′µ (x) = −DZµν JZ,ν (x) + O(p3 ) + loop terms

(151)

then S¯Z ′ [U, W a , B] =

1 g14 µν µν µ′ ν ′ µν 2 ′ ′ d x [− JZ,µ DZ JZ,ν − J3Z,µ (DZ JZ,ν )(DZ JZ,ν ) + g4Z ′4 (DZ JZ,ν )4 ] 2 c +loop terms (152)

Z

4

where −1 DZ−1,µν DZ,νλ = DZµν DZ,νλ = gλµ

(153)

and it is not difficult to show that if we are accurate up to order of p4 , then p order Zc′ solution is enough, all contributions from p3 order Zc′ are at least belong to order of p6 . With these results, (135) become   a a N [Wµ , Bµ ] exp iSeff [U, Wµ , Bµ ] (154)   Z 1 a aµν 1 3g 2 (4c,B) a 4 = exp iS¯Z ′ [U, W , B] + iSTC1 [B] + i d x − Wµν W − Bµν B µν − K 1 Bµν B µν 4 4 4 1 (F TC2 )2 L1D 1 11 3 a W a,µν ) − 0 tr(Xµ X µ ) + (L1D )[tr(Xµ X µ )]2 + K( g12 Bµν B µν + g22 Wµν 1 + 2 18 2 4 2 1D 1D L L µν 10 9 2 1D 3 µ ν µ ν +L1D [tr(X X )] − iL tr(W g B tr(τ X X ) + g1 Bµν tr(τ 3 W ) X X ) − i µ ν 1 µν µν 2 9 2 2  1 µν 1D 2 µν + H11D tr(W µν W ) , + (L1D 10 + 11H1 )g1 Bµν B 18 Ignoring term with coefficients F0TC2 , L1D and H11D in above expression, we get expression i for N [Wµa , Bµ ], with it we finally obtain Seff [U, Wµa , Bµ ]  Z L1D (F TC2 )2 3 2 a 4 tr(Xµ X µ ) + (L1D )[tr(Xµ X µ )]2 + L1D Seff [U, Wµ , Bµ ] = dx − 0 1 + 2 [tr(Xµ Xν )] 4 2 1D L1D L9 µν 10 3 µ ν µ ν −iL1D tr(W g B tr(τ X X ) + g1 Bµν tr(τ 3 W ) X X ) − i 1 µν µν 9 2 2 1 1D µν + (L10 + 11H11D )g12 Bµν B µν + H11D tr(W µν W ) + ∆Seff [U, Wµa , Bµ ](155) 18 39

i.e. our result EWCL is equal to standard one-doublet technicolor model result plus contributions from Z ′ , we denote this Z ′ contribution part ∆Seff [U, Wµa, Bµ ], 1D . ∆Seff [U, Wµa , Bµ ] = S¯Z ′ [U, W a , B] − S¯Z ′ [U, W a , B]|F0TC2 =0,L1D i =H1 =0

(156)

Correspondingly, EWCL coefficients for topcolor assisted technicolor model can also be divided into two parts f 2 = (F0TC2 )2 in which αi |one

αi = αi |one

β1 = ∆β1

doulet

+ ∆αi i = 1, 2, . . . , 14

(157)

i = 1, 2, . . . , 14 are coefficients from one-doublet technicolor model,

doulet

their values are given in (9) and Table I. ∆β1 and ∆αi i = 1, 2, . . . , 14 are contributions from Z ′ and ordinary quarks, since we do not consider ordinary quarks in this work, so in the next part of this paper, we calculate Z ′ contributions. With help of (152), (139) and (143) Z 1 1 g 2γ a ∆Seff [U, Wµ , Bµ ] = d4 x [− JZ0,µ DZµν JZ0,ν − 2 JZ0,µ (J˜Zµ + 1′ ∂ν B µν ) 2 MZ ′ c 4 g4Z g 4 1 µ 2 ] JZ0 + ′4 81 JZ0 − 6 J3Z,µ JZ0 MZ ′ c MZ ′

(158)

With help of following algebra relations, ∂µ tr[τ 3 X µ ] = 0 tr[τ 3 (∂µ Xν − ∂ν Xµ )] = −2tr(τ 3 Xµ Xν ) + itr(τ 3 W µν ) − ig1 Bµν tr(τ 3 Xµ Xν )tr(τ 3 X µ X ν )

(159)

= [tr(Xµ Xν )]2 − [tr(Xµ X µ )]2 − tr(Xµ Xν )tr(τ 3 X µ )tr(τ 3 X ν ) + tr(Xµ X µ )[tr(τ 3 Xν )]2 tr(T A)tr(T BC) + tr(T B)tr(T CA) + tr(T C)tr(T AB) = 2tr(ABC) where trA = trB = trC = 0 and T 2 = 1. We can show (158) leads to the form of standard EWCL, further combined with (155) and (144), we can read out p2 coefficient β1 =

(F0TC2 )2 g12 (F0TC2 )2 2 ′ tan θ = 8c′2 MZ2 ′ 3(F0TC1 )2 (cot2 θ′ + 1)2 + 2(F0TC2 )2

(160)

which imply a positive β1 which is further bounded above. With the fact that f = F0TC2 = 250GeV and original model requirement F0TC1 = 1TeV, we find 2β1 =

1 . 24(cot θ′ + 1)2 + 1 2

40

(161)

Combine with αT = 2β1 given in Ref.[2], we obtain result that topcolor-assisted technicolor model produce positive and bounded above T parameter! The upper limit of β1 is 1/50 which corresponds to upper limit of T parameter 1/(25α) ∼ 5.1. From (160), we know β1 coefficient is uniquely determined by parameter θ′ , therefore instead of using θ′ as the input parameter of the theory, we can further use β1 or T = 2β1 /α as the parameter of the theory. The p4 order coefficients can be read out from derived EWCL (155) and (158), we list down the results as following, (F0TC2 )2 α1 = (1 − + β1 − 2γβ1 cot θ′ 2 2MZ ′ (F0TC2 )2 1 − β1 − 2γβ1 cot θ′ α2 = − (1 − 2β1 )L1D 9 2 2MZ2 ′ (F0TC2 )2 1 1D α = L + β1 + 2β1 L1D α3 = − (1 − 2β1 )L1D 4 9 2 9 2 2MZ2 ′ L1D (F TC2 )2 3 α5 = L1D − 0 2 β1 − 2β1 L1D 1 + 9 2 2MZ ′ L1D (F TC2 )2 9 1D − 4β (L + ) α6 = − 0 2 β1 + 4β12L1D 1 2 2 2MZ ′ 2 (F TC2 )2 1D 1D α7 = β1 0 2 + 2(β12 − β1 )(2L1D 1 + L3 ) + 2β1 L9 2MZ ′ (F TC2 )2 α8 = −β1 0 2 + β1 L1D 10 2MZ ′ (F TC2 )2 1D α9 = −β1 0 2 + 2β1 (−L1D 9 + L10 ) 2MZ ′ L1D 3 1D ) + 16β14 g4Z cot4 θ′ α10 = (4β12 − 8β13 )(L1D + L + 1 2 2 α11 = α12 = α13 = α14 = 0 2β1 )L1D 10

(162)

Several features of this result are: 1. Except the part of one-doublet technicolor model result, all corrections from Z ′ particle are at least proportional to β1 which vanish if the mixing disappear by θ′ = 0. 2. Since L1D 10 < 0, therefore (162) tells us α8 is negative and then U = −16πα8 is always positive in this model. 3. Except α1 , α2 and α10 , all other coefficients are determined by one-doublet technicolor model coefficients given in Table VI and two other parameter β1 and F0TC2 /MZ ′ .

41

4. α10 further depend on parameter g4Z which from (146) further depend on K3TC1,Σ6=0 + K4TC1,Σ6=0 which are already given by (101) and Table III. 5. α1 and α2 further depend on γ which from (145) further rely on an extra parameter K. We can combine (160) and (137) together to fix K, (F0TC2 )2 3 1 5 3 tan2 θ′ = 2 + K( cot2 θ′ + tan2 θ′ ) + [K2TC1,Σ6=0 (cot θ′ + tan θ′ )2 2 8β1 MZ ′ g1 2 18 4 2 TC1,Σ6=0 1D 2 ′ +K13 (cot θ′ − tan θ′ )2 ] − (L1D 10 + 11H1 ) tan θ 9 Once K is fixed, with help of (92), we can determine the ratio of infrared cutoff κ and ultraviolet cutoff Λ, in Fig.1(a), we draw the κ/Λ as function of T and MZ ′ , we find our calculation do produce very large hierarchy and we further find not all T and MZ ′ region is available if we consider the natural criteria Λ > κ. This criteria leads constraints that as long as Z ′ mass become large, the allowed range for T parameter become smaller and smaller approaching to zero, for example, T < 0.37 for MZ ′ = 0.5TeV, T < 0.0223 for MZ ′ = 1TeV and T < 0.004 for MZ ′ = 2TeV. In Fig.1(b), we draw Z ′ mass as function of T parameter and κ/Λ. The line of κ/Λ = 1 give the upper bound of Z ′ mass. The upper bound of Z ′ mass depend on value of T parameter, the smaller the T , the larger the upper bound of MZ ′ . 6. For fixed MZ ′ , there exists a special θ′ value which maximizes α1 . The parameter, S = −16πα1 , is of special importance in new physics search, in Fig.2, we draw a graph of minimal S parameter with different T parameter. We see that if the Z ′ mass is low enough, say MZ ′ < 0.441 TeV or T > 0.176, S will become negative. Since we already know F0TC2 = 250GeV, therefore all EWCL coefficients depend on two physical parameters β1 and MZ ′ . Combined with αT = 2β1 , we can use the present experimental result for the T parameter to fix β1 . In Fig.3, we draw graphs for the S and U parameters in terms of the T parameter. We take three typical Z ′ masses MZ ′ = 0.5, 1, 2 TeV for references. In Fig.4, we draw graphs for all p4 order nonzero coefficients in terms of the T parameter. Where for α3 and α10 , we only draw one line for each of them, since they are independent of the Z ′ mass.

42

FIG. 1: (a). The ratio of infrared cutoff and ultraviolet cutoff κ/Λ as function of T parameter and Z ′ mass in unit of TeV. (b). Z ′ mass in unit of TeV as function of T parameter and κ/Λ.

800

2.4

κ/Λ=1

2.2 600

Mz’=2

400

2

κ/Λ=10−3

1.8

κ/Λ=10−6

Mz’=1

Mz’

ln(κ/Λ)

1.6 200 Mz’=0.5

κ/Λ=10−10

1.4 1.2

0

1 0.8

−200

0.6 −400 −3 10

−2

−1

10

10

0

10

−2

−1

10

10 T

T

(a) κ/Λ

IV.

0.4 −3 10

(b) MZ ′

CONCLUSION

In this paper, we have set up a formulation to perform the dynamical computation of the bosonic part of EWCL for the one-doublet and topcolor-assisted technicolor models. The one-doublet technicolor model as the earliest and simplest dynamical symmetry breaking model are taken as the trial model to test our formulation. We find our formulation recovers standard scaling-up results. The topcolor-assisted technicolor model is the main model we handle in this paper. We have computed its TC1 dynamics in detail and verify the dynamical symmetry breaking of the theory, TC1 interaction will induce effective interactions among colorons and Z ′ which are characterized by a divergent constant K, a dimensional constant F0TC1 and a series of dimensionless QCD constants L1 , L3 , L9 , L10 , H1 . For TC2 dynamics, it will induce effective interaction for Z ′ , electroweak gauge fields and their goldstone bosons. Due to its similarity with QCD, we use Gasser-Leutwyler prescription to describe its low energy effects in terms of low energy effective Lagrangian with a divergent constant K, dimensional constant F0TC2 and a series of dimensionless constants 1D 1D 1D 1D 1D L1D the same as those in one-doublet technicolor model. Due to 1 , L2 , L3 , L9 , L10 , H1

43

0

10

FIG. 2: The dashed line is the minimal S parameter in topcolor assisted technicolor model for different T . The solid lines are the isolines for different choices of Z ′ mass in units of TeV. 0.4 Mz’=0.8 Mz’=0.7

0.3

Mz’=0.6

Min(S)

0.2

Mz’=0.5

0.1

0

−0.1

Mz’=0.4

−0.2

−2

10

−1

10 T

0

10

dynamics similarities between TC2 and QCD, TC2 interaction make a direct contributions to EWCL coefficients a part which is the same as that of one-doublet technicolor model. Further corrections are from effective interactions among colorons, Z ′ and ordinary quarks induced by TC1 and TC2 interactions. We have shown that colorons make no contributions to EWCL coefficients within the approximations we have made in this paper, while ordinary quark are ignored in this paper for future investigations. In fact, for some special EWCL coefficients, such as S = −16πα1 , αT = β1 and U = −16πα8 parameters, general fermion contributions to them are already calculated [23], S, T, U and triple-gauge-vertices from a heavy non-degenerate fermion doublet has been estimated in ref.[9, 24]. One can based on these general results to estimate possible contributions to some of EWCL coefficients. For topcolor-assisted technicolor model in this paper, the main work is to estimate the effects of Z ′ particle. Our computation shows that contributions from Z ′ particle are at least proportional to β1 and then vanish if β1 is zero. One typical feature of the model is the positivity and bounded above of β1 parameter which means the T parameter must vary in the range 0 ∼ 1/(25α) and the positive U parameter. If we consider the natural criteria Λ > κ which will further constraints the allowed range for T parameter approaching to zero as long as Z ′ mass become large, for example, T < 0.37 for MZ ′ = 0.5TeV, T < 0.0223 for MZ ′ = 1TeV and T < 0.004 for MZ ′ = 2TeV. For S parameter, it can be either positive and negative 44

FIG. 3: The S and U parameters for topcolor assisted technicolor model. F0TC2 = 250 GeV, the T parameter and MZ ′ = {0.5, 1, 2} TeV are as input parameters of the model.

3

0.025

2.5

0.02

2 0.015

Mz’=0.5

U

S

Mz’=2 1.5 Mz’=1

0.01

1 Mz’=1

0.005

Mz’=0.5

0.5

Mz’=2 0 −3 10

−2

−1

10

10

0

10

T

0 −3 10

−2

−1

10

10 T

(a) S

(b) U

depending on the Z ′ mass is large or small. As long as MZ ′ < 0.441 TeV or T > 0.176, we may find negative S. There exist a upper bound for the mass of Z ′ which is depend on value of T parameter, the smaller the T , the larger the upper bound of MZ ′ . Except U(1)Y coupling g1 and coefficients determined in one-doublet technicolor model and QCD, all EWCL coefficients rely on experimental T parameter and coloron mass MZ ′ . We have taken typical values of MZ ′ and vary T parameter to estimate all EWCL coefficients up to order of p4 . Further works on matter part of EWCL and computing EWCL coefficients for other dynamical symmetry breaking new physics models are in progress and will be reported elsewhere.

Acknowledgments

We would like to thank Y. P. Kuang, H. J. He and J. K. Parry for helpful discussions. This work was supported by National Science Foundation of China (NSFC) under Grant No. 10435040 and Specialized Research Fund for the Doctoral Program of High Education of China.

45

0

10

FIG. 4: Nonzero EWCL coefficients αi (i = 1, 2, . . . , 10) for the topcolor-assisted technicolor model up to order of O(p4 ). F0TC2 = 250 GeV, the T parameter and MZ ′ = {0.5, 1, 2} TeV are as input parameters of the model.

0

0.01

Mz’=0.5

−0.01

0 Mz’=0.5

−0.02

−0.01

−0.03

α2

α1

Mz’=1

Mz’=2

−0.04

−0.03

−0.05

−0.04

−0.06 −3 10

−2

−1

10

10

0

10

Mz’=1 Mz’=2

−0.02

−0.05 −3 10

−2

0

10

10

T

(a) α1

(b) α2

−3

−2.415

−1

10

T

−3

x 10

2.6

x 10

2.5

−2.42

2.4

α4

α3

−2.425 Mz’=0.5

2.3

−2.43 2.2 Mz’=1

−2.435

−2.44 −3 10

2.1

−2

−1

10

10

0

10

2 −3 10

Mz’=2

−2

−1

10

T

10 T

(c) α3

(d) α4

APPENDIX A: TWO EQUIVALENT EWCL FORMALISMS

The electroweak chiral Lagrangian is constructed using a dimensionless unitary unimodular 2×2 matrix field U(x), In Ref.[2], the electroweak chiral Lagrangian has been constructed

46

0

10

FIG. 4:

−4

−3

−2.5

x 10

0

x 10

Mz’=2 Mz’=1

−2.6

−1 Mz’=2

−2.7

−2

Mz’=1

Mz’=0.5

α5

α6

−2.8 −3

Mz’=0.5

−2.9

−4 −3 −5

−3.1

−3.2 −3 10

−2

−1

10

−6 −3 10

0

10

10

−2

−1

10

10

T

(a) α5

(b) α6

−4

6

0

10

T

−4

x 10

1

5

x 10

0

4

Mz’=2 Mz’=1

−1

α8

α7

Mz’=0.5 3

2

Mz’=0.5

−2

−3 Mz’=1 Mz’=2

1

0 −3 10

−2

−1

10

10

−4

−5 −3 10

0

10

−2

−1

10

0

10

T

10

T

(c) α7

(d) α8

with the building blocks which are SU(2)L covariant and U(1)Y invariant as: T ≡ Uτ 3 U † ,

Vµ ≡ (Dµ U)U † ,

g1 Bµν ,

g2 Wµν ≡ g2

τa a W . 2 µν

(A1)

Alternatively, we reformulate the electroweak chiral Lagrangian equivalently with SU(2)L invariant and U(1)Y covariant building blocks as: τ3 ,

Xµ ≡ U † (Dµ U) ,

g1 Bµν ,

47

W µν ≡ U † g2 Wµν U ,

(A2)

FIG. 4:

−4

0

−8

x 10

0 Mz’=2 Mz’=1

−1

x 10

−0.5 −1

Mz’=0.5

−1.5 α10

α9

−2

−3

−2 −2.5

−4 −3 −5 −3.5 −6 −3 10

−2

−1

10

0

10

10

−4 −3 10

−2

−1

10

10

T

(a) α9

(b) α10

among which, τ 3 and g1 Bµν are both SU(2)L and U(1)Y invariant, while Xµ and W µν are bilinearly U(1)Y covariant. This second formulation is largely used throughout this paper. We list down the corresponding relations of the two formalisms. (4c,B A )

APPENDIX B: STC1

(4c,B A ) STC1 [B A ]

(4c,B)

(4c,Z ′ )

(4c,B A Z ′ )

[B A ], STC1 [B], STC1 , STC1

(4c,BZ ′ )

[B A , Z ′ ], STC1

[B, Z ′ ]

 g2 = d x − K1TC1,Σ6=0 3 (cot θ + tan θ)2 (∂µ B A,µ )2 8 2 g A −K2TC1,Σ6=0 3 (cot θ + tan θ)2 Ba,µν BaA,µν 8 g4 g4 +K3TC1,Σ6=0 [ 3 (cot θ + tan θ)4 (BµA B A,µ )2 + 3 (cot θ + tan θ)4 (dABC BµB B C,µ )2 ] 192 128 4 g +K4TC1,Σ6=0 { 3 (cot θ + tan θ)4 (BµA BνA )2 192 4 g + 3 (cot θ + tan θ)4 [(if ABC + dABC )BµB BνC ]2 } 128 2 TC1,Σ6=0 g3 A (cot θ − tan θ)2 Bv,µν BvA,µν −K13 8  3 TC1,Σ6=0 g3 2 A,µν ABC B C (B1) (cot θ − tan θ)(cot θ + tan θ) Bv f Bµ Bν , +K14 32 Z

4

48

0

10

T

TABLE I: The Symmetry Breaking Sector of the Electroweak Chiral Lagrangian Formulation I

Formulation II = − 14 f 2 tr(Vµ V µ ) − 14 f 2 tr(Xµ X µ )

L(2)

1 2 † µ 4 f tr[(Dµ U )(D U )]

L(2)′

1 2 2 4 β1 f [tr(T Vµ )]

1 2 3 2 4 β1 f [tr(τ Xµ )]

L1

1 µν 2 α1 g2 g1 Bµν tr(T W )

µν 1 3 2 α1 g1 Bµν tr(τ W )

L2

1 µ ν 2 iα2 g1 Bµν tr(T [V , V ])

iα2 g1 Bµν tr(τ 3 X µ X ν )

L3

iα3 g2 tr(Wµν [V µ , V ν ])

2iα3 tr(W µν X µ X ν )

L4

α4 [tr(Vµ Vν )]2

α4 [tr(Xµ Xν )]2

L5

α5 [tr(Vµ V µ )]2

α5 [tr(Xµ X µ )]2

L6

α6 tr(Vµ Vν )tr(T V µ )tr(T V ν )

α6 tr(Xµ Xν )tr(τ 3 X µ )tr(τ 3 X ν )

L7

α7 tr(Vµ V µ )tr(T Vν )tr(T V ν )

α7 tr(Xµ X µ )tr(τ 3 Xν )tr(τ 3 X ν )

L8

1 2 2 4 α8 g2 [tr(T Wµν )]

1 2 3 4 α8 [tr(τ W µν )]

L9

1 µ ν 2 iα9 g2 tr(T Wµν )tr(T [V , V ])

iα9 tr(τ 3 W µν )tr(τ 3 X µ X ν )

L10

1 2 2 α10 [tr(T Vµ )tr(T Vν )]

1 3 3 2 2 α10 [tr(τ Xµ )tr(τ Xν )]

L11 α11 g2 ǫµνρλ tr(T Vµ )tr(Vν Wρλ )

α11 ǫµνρλ tr(τ 3 Xµ )tr(Xν W ρλ )

L12 α12 g2 tr(T Vµ )tr(Vν W µν )

α12 tr(τ 3 Xµ )tr(Xν W

L13 α13 g2 g1 ǫµνρσ Bµν tr(T Wρσ )

α13 ǫµνρσ g1 Bµν tr(τ 3 W ρσ )

L14 α14 g22 ǫµνρσ tr(T Wµν )tr(T Wρσ )

α14 ǫµνρσ tr(τ 3 W µν )tr(τ 3 W ρσ )

(4c,B) STC1 [B]

(4c,Z ′ ) STC1

=

=

Z

  2 3g TC1,Σ6 = 0 1 µν , Bµν B d4 x − K13 4

µν

)

(B2)

3g12 (cot θ′ + tan θ′ )2 (∂µ Z ′,µ )2 16 3g 4 +(K3TC1,Σ6=0 + K4TC1,Σ6=0 ) 1 (cot θ′ + tan θ′ )4 (Zµ′ Z ′,µ )2 256  2 3g1 TC1,Σ6=0 TC1,Σ6=0 ′ ′ 2 ′ ′ 2 ′ ′,µν , (B3) [K (cot θ + tan θ ) + K13 (cot θ − tan θ ) ]Zµν Z − 16 2

Z

4

dx



− K1TC1,Σ6=0

49

(4c,B A Z ′ ) STC1 [B A , Z ′ ]

 g 2g 2 = d x K3TC1,Σ6=0 [ 1 3 (cot θ + tan θ)2 (cot θ′ + tan θ′ )2 BµA B A,µ Zν′ Z ′,ν 64 2 2 g g + 1 3 (cot θ + tan θ)2 (cot θ′ + tan θ′ )2 (BµA Z ′,µ )2 32 g1 g33 + (cot θ + tan θ)3 (cot θ′ + tan θ′ )dABC BµB B C,µ BµA Z ′,µ ] 32 g 2g 2 +K4TC1,Σ6=0 { 1 3 (cot θ + tan θ)2 (cot θ′ + tan θ′ )2 (BµA Z ′,µ )2 256 g12 g32 (cot θ + tan θ)2 (cot θ′ + tan θ′ )2 [BµA B A,µ Zν′ Z ′,ν + (BµA Z ′,µ )2 ] + 64  g1 g33 3 ′ ′ ABC B C A,µ ′,ν + (cot θ + tan θ) (cot θ + tan θ )d Bµ Bν B Z } , (B4) 32 Z

4

(4c,BZ ′ ) STC1 [B, Z ′ ]

=

A with Bv,µν = ∂µ BνA − ∂ν BµA −

Z

  2 TC1,Σ6=0 g1 ′ ′ ′,µν , (cot θ − tan θ )Bµν Z d x − K13 4 4

g3 (cot θ 2

(B5)

− tan θ)f ABC BµB BνC .

int 0 [B A , Z ′ ] [B A , Z ′ ] AND Scoloron APPENDIX C: Scoloron

0 Scoloron [B A , Z ′ ]  Z g2g2 (F TC1 )2 2 4 g3 (cot θ + tan θ)2 + [K3TC1,Σ6=0 1 3 (cot θ + tan θ)2 (cot θ′ + tan θ′ )2 = d x g µν { 0 8 64 2 2 g g +K4TC1,Σ6=0 1 3 (cot θ + tan θ)2 (cot θ′ + tan θ′ )2 ]Zλ′ Z ′,λ } 64 2 2 TC1,Σ6=0 g1 g3 (cot θ + tan θ)2 (cot θ′ + tan θ′ )2 +[K3 32  2 2 5g TC1,Σ6=0 1 g3 2 ′ ′ 2 ′,µ ′,ν +K4 BµA BA,ν (cot θ + tan θ) (cot θ + tan θ ) ]Z Z 256 g2 −K1TC1,Σ6=0 3 (cot θ + tan θ)2 (∂µ B A,µ )2 8 2 g2 TC1,Σ6=0 g3 (cot θ − tan θ)2 [−K2TC1,Σ6=0 3 (cot θ + tan θ)2 − K13 8 8  1 g32 2 2 A A µ A,ν ν A,µ − − K(cot θ + tan θ)](∂µ Bν − ∂ν Bµ )(∂ B − ∂ B ) (C1) 4 4

50

and int Scoloron [B A , Z ′ ]  Z g2 4 = d x − K2TC1,Σ6=0 3 (cot θ + tan θ)2 [(∂µ BνA − ∂ν BµA )2g3 (− cot θ + tan θ)f ABC B B,µ B C,ν 8 ′

′

′

(C2)

′

+g32(− cot θ + tan θ)2 f ABC BµB BνC f AB C B B ,µ B C ,ν ] g34 g4 (cot θ + tan θ)4 (BµA B A,µ )2 + 3 (cot θ + tan θ)4 (dABC BµB B C,µ )2 ] 192 128 4 g4 g +K4TC1,Σ6=0 { 3 (cot θ + tan θ)4 (BµA BνA )2 + 3 (cot θ + tan θ)4 [(if ABC + dABC )BµB BνC ]2 } 192 128 2 g TC1,Σ6=0 3 (cot θ − tan θ)2 [(∂µ BνA − ∂ν BµA )g3 (− cot θ + tan θ)f ABC B B,µ B C,ν −K13 8 3 1 ′ ′ ′ ′ TC1,Σ6=0 g3 + g32 (− cot θ + tan θ)2 f ABC BµB BνC f AB C B B ,µ B C ,ν ] + K14 (cot θ − tan θ)(cot θ + tan θ)2 4 32 g3 µ A,ν ν A,µ ABC B,µ C,ν ABC B C ×[∂ B − ∂ B + (− cot θ + tan θ)f B B ]f Bµ Bν 2 g1 g 3 +K3TC1,Σ6=0 3 (cot θ + tan θ)3 (cot θ′ + tan θ′ )dABC BµB B C,µ BνA Z ′,ν 32 g1 g 3 +K4TC1,Σ6=0 3 (cot θ + tan θ)3 (cot θ′ + tan θ′ )dABC BµB BνC B A,µ Z ′,ν 32 1 ′ ′ ′ ′ − cos2 θ[2g3 (∂µ BνA − ∂ν BµA ) cot θf ABC B B,µ B C,ν + g32 cot2 θf ABC B B,µ B C,ν f AB C BµB BνC ] 4 1 ′ ′ ′ ′ − sin2 θ[−2g3 (∂µ BνA − ∂ν BµA ) tan θf ABC B B,µ B C,ν + g32 tan2 θf ABC B B,µ B C,ν f AB C BµB BνC ] 4 g32 ′ ′ ′ ′ − K cot2 θ[−2g3 (∂µ BνA − ∂ν BµA ) cot θf ABC B B,µ B C,ν + g32 cot2 θf ABC B B,µ B C,ν f AB C BµB BνC ] 4  g32 2 A A ABC B,µ C,ν 2 2 ABC B,µ C,ν AB ′ C ′ B ′ C ′ B B + g3 tan θf B B f Bµ Bν ] , − K tan θ[2g3 (∂µ Bν − ∂ν Bµ ) tan θf 4 +K3TC1,Σ6=0 [

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