# Mass Mixing Effect and Oblique Radiative Corrections in ExtendedSU(2)R×SU(2)L×U(1) Effective Theory

Authors: Ying Zhang
Journal: Advances in High Energy Physics
 (2012)

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## Abstract

We analyze the properties of electroweak chiral effective Lagrangian with an extendedSU(2)R gauge group. Right-handed SU(2)R gauge bosons affect electroweak observables by mixing with
electroweak gauge bosons WL,μ and Bμ. We discuss all possible mass mixing terms and calculate
the exact physical mass eigenvalues by diagonalization of mixing matrix without any approximate
assumptions. The contributions to oblique radiative corrections parameters STU from SU(2)R fields are also presented.


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## Body

## 1. Introduction

Although the standard model (SM) has been checked very successfully by more and more high energy physics experiments, the as yet undiscovered Higgs, introduced as a basic scaler field in SM, remains as the only unknown component of the electroweak symmetry-breaking mechanism (EWSBM) unknown. That situation has prompted many extensions to SM [1–3]. A new SU(1)R group, associated with an additional triplet of gauge bosons W′± and Z′, is often considered for different reasons as an extension to the gauge symmetry [4–6]. This extension often appears in superstring-inspired models as well as GUT models [7]. The non-Abelian SU(2)R contains sufficient complexity to incorporate interesting issues related to spontaneous parity violation (SPV) and precise electroweak observables, although remains simple enough that phenomenology can be subjected to analysis. SU(2)R gauge bosons can improve unitarity of not only WW but also WZ scattering processes and delay the breaking scale of unitarity.Many left-right symmetry models with symmetry groupSU(2)R×SU(2)L×U(1) have been used in studying EWSBM. The common feature of these models is the existence of multi-Higgs bosons that then raises phenomenological issues related to multi-Higgs structure dependencies. To obtain an universal physical analysis, we adopt the nonlinear realization of the chiral Lagrangian to describe extended SU(2)R electroweak gauge models given the symmetry breaking pattern SU(2)R×SU(2)L×U(1)→U(1)em. This chiral Lagrangian has already been written down in [8]. The model is a generalization of the conventional linearly realized models with multi-Higgs. Within the extended non-Abelian chiral effective Lagrangian, multi-Higgs effects are parameterized by a set of coefficients that describes all possible interactions among the gauge bosons and provides a model-independent platform to investigate interesting physics [8].In the paper, we focus on mass mixing effects in left-right chiral effective Lagrangian. Mass mixings are main focus in the contribution of the right-handed gauge bosons to electroweak observables at low-energy scales. TheSU(2)R gauge triplet can be regarded as a copy of the SU(2)L gauge triplet of SM, but with heavier masses. Right-handed charged gauge bosons WR± can mix with left-handed WL±, and physical mass eigenstates of W′± and W± are eigenvalues of the charged mass matrix. Similarly, WR3 takes part in WR3-WL3-B three-body mixing to form physical massive neutral bosons Z′, Z,and a massless photon. The nonlinearly realized chiral effective Lagrangian provides us with all possible mass-mixing channels that are allowed by left-right symmetry. Calculating these mixings, we obtain a complete mass mixing contribution to the electroweak observables and a largest parameter space for new physics. Oblique radiative corrections of SU(2)R bosons can be obtained from the mass mixing rotation matrix, which indicates shifts to the SM with new physics.The paper is organized as follows. Section2 reviews SU(2)R×SU(2)L×U(1) effective theory with all possible mass mixing terms in the gauge eigenstates basis. Section 3 presents calculations of the charged and neutral mass eigenvalues to obtain physical boson masses estimates. We improved our diagonalization calculation program for the neutral bosons sector in our paper [8] to yield a set of exact solutions for the rotation matrix and the mass eigenvalues without making any approximating assumptions. Oblique radiative corrections coming from the nonstandard mass mixing beyond SM are studied in Section 4. Furthermore, two kinds of special cases are considered corresponding to condition MWR≫MWL case and left-right symmetry. Finally, we give a short summary in Section 5.


## 2. Left-Right Symmetry Effective Lagrangian

LetWR,μa, WL,μa, Bμ be electroweak gauge fields (a=1,2,3) corresponding to the gauge group SU(2)R, SU(2)L, and U(1), respectively, and UL,R be the two by two unitary unimodular matrices corresponding to left- and right-handed Goldstone boson fields. Under SU(2)R⊗SU(2)L⊗U(1) transformations, the gauge boson fields transform as(2.1)τa2Wi,μa⟶Riτa2Wi,μa(x)Ri†-igiRi∂μRi†,Bμ⟶Bμ-1g∂μθ0,Ui⟶RiUiR0†
with R0=e(i/2)τ3θ0(x) andRi=e(i/2)τaθia(x)for i=R,L. The covariant derivative of the Goldstone fields takes the form(2.2)DμUR=∂μUR+igRτa2WR,μaUR-igULτ32Bμ,DμUL=∂μUL+igLτa2WL,μaUL-igULτ32Bμ.
For convenience in present discussion, we will discard conventional EWCL SU(2) covariant building blocks [9–13] and introduce U(1) invariant building blocks (for i=L,R) (2.3)Xiμ=Ui†(DμUi),W¯i,μν=Ui†giWi,μνUi,Bμν=∂μBν-∂νBμ.
Here, (2.4)Wi,μν=Wi,μνaτa2=∂μWi,νaτa2-∂νWi,μaτa2+igi[Wi,μaτa2,Wi,νbτb2].
With the help of these building blocks, we can write a leading-order chiral Lagrangian as(2.5)LM=-14fL2〈XL,μXLμ〉-14fR2〈XR,μXRμ〉+12κ̃fLfR〈XLμXRμ〉+14βL,1fL2〈τ3XL,μ〉2+14βR,1fL2〈τ3XR,μ〉2+14β̃1fLfR〈τ3XL,μ〉〈τ3XRμ〉.
Here, 〈〉 stands for the trace in flavor space. fL and fR are the scales for spontaneous symmetry breaking in the electroweak sector and parity, respectively. The coefficient βL,R,1 generates extra mass for the left-handed (right-handed) third component in breaking the SU(2)L,R isospin symmetry. The coefficient κ parameterizes the mixing between the left- and right-handed gauge bosons whereas the coefficient β̃1 controls the mixing between left-handed WL3 and right-handed WR3.The neutral current interactions are(2.6)-LNC=WRμ3JRμ+WLμ3JLμ+BμJ0μ
whereas the charged current interactions are(2.7)-LCC=WRμ+JR-,μ+WLμ+JL-,μ+h.c.
Here,(2.8)JL,R±,μ=gL,R2Ψ¯L,Rτ±γμΨL,R.The kinetic part has the simple form(2.9)LK=-14WL,μνaWLμν,a-14WR,μνaWRμν,a-14BμνBμν+iΨ¯iγμDμΨi.Adding Yukawa terms(2.10)LY=Ψ¯LULMUR†ΨR+h.c.,
the total Lagrangian is the sum of all the above terms(2.11)L=LM+LK+LNC+LCC+LY.


## 3. Diagonalization and Mass Eigenstates

In this section, we calculate the mass eigenvalues of the left-right symmetry effective Lagrangian by rotating the mass mixing matrix from the gauge basis to the mass basis.


### 3.1. Charged Gauge Bosons

Taking the unitary gaugeUL=UR=1, the charged gauge boson mass terms can be expressed as(3.1)LCM=14fL2gL2WL,μ+WL,μ-+14fR2gR2WR,μ+WR,μ--14κ̃fLfRgLgR(WL,μ+WR,μ-+WR,μ+WL,μ-).
Here, we have used charged boson definitions Wi,μ1=(Wi,μ++Wi,μ-)/2 and Wi,μ2=i(Wi,μ+-Wi,μ-)/2 for i=L,R.We make an orthogonal rotationV for WL± and WR±(3.2)(WR±WL±)=(cos⁡⁡ξsin⁡⁡ξ-sin⁡⁡ξcos⁡⁡ξ)(W′±W±)≡V(W′±W±)
to eliminate the cross-terms involving WL and WR in (3.1) to keep the kinetic term diagonal. The mixing angle ξ is expressed as(3.3)tan⁡2ξ=2κ̃fLfRgLgRfL2gL2-fR2gR2.After this rotation, the charged boson mass-squared matrix for the charged bosons becomes(3.4)VTMCV=diag⁡(MW′2,MW2),
and the heavy and light charged boson masses are (3.5)MW′2=18[fL2gL2+fR2gR2+(fL2gL2-fR2gR2)2+4κ̃2fL2fR2gL2gR2]≃14fR2gR2{1+κ̃2fL2gL2fR2gR2-fL2gL2},MW2=18[fL2gL2+fR2gR2-(fL2gL2-fR2gR2)2+4κ̃2fL2fR2gL2gR2]≃14fL2gL2{1-κ̃2fR2gR2fR2gR2-fL2gL2}.
We notice that the charged boson mixing angle ξ is controlled by the coefficient κ̃. W-W′ mixing causes W couplings to the right-handed fermion with gRW=gLsinξ/2. gRW can yield the contributions to b→sγ (see paper [14]) and must be restrained so that gRW/gLW<4×10-3, which requires ξ<4×10-3.


### 3.2. Neutral Gauge Bosons

Now, let us discuss the neutral boson sector. The neutral mass terms in our chiral Lagrangian (2.5) can be readily separated out (3.6)LMn=18(1-2βL,1)fL2(gLWL,μ3-gBμ)2+18(1-2βR,1)fR2(gRWR,μ3-gBμ)2-14(κ̃+β̃1)fLfR(gLWL,μ3-gBμ)(gRWR3,μ-gBμ).
It can be written in matrix form(3.7)LMn=12GμTMnGμ
with neutral gauge bosons 𝒢μ≡(WR,μ,WL,μ,Bμ) and mass-squared matrix (3.8)Mn≡(fR2gR24-κfRfLgRgL4fRgRg0(κfL-fR)4-κfRfLgRgL4fL2gL24fLgLg0(κfR-fL)4fRgRg0(κfL-fR)4fLgLg0(κfR-fL)4(fR2+fL2-2κfRfL)g024).
Note that the βL,R,1 do not appear in the above mass-squared matrix because these can be absorbed by a redefinition of VEV fL,R(3.9)fL,R⟶fL,R1-2βL,R,1.
For the sake of convenience, we will retain using the same notation for the redefined VEV fL,R but keep in mind that this redefinition has been made. The new parameter κ in the above formula is a combination of κ̃ and β̃1, namely, κ=κ̃+β̃1. Taking into account the VEVs re-definition, we have (3.10)κ=κ̃+β̃11-2βL,11-2βR,1.
The physical masses of the neutral bosons are the eigenvalues of the matrix ℳn. To obtain the diagonalized eigenvalues, we define the mass eigenstates as 𝒢¯μ=(Z′,Zμ,Aμ)T which are related to 𝒢μ by a special rotation U-1(3.11)G¯μ=(G4gRG1G4-G2G3-G2gLG1G4-G2G3(G2-G4)g0G1G4-G2G3-G3gRG1G4-G2G3G1gLG1G4-G2G3(G3-G1)g0G1G4-G2G3gRG5λ1gLG5λ2g0G5)Gμ(3.12)≡U-1Gμ
with undetermined couplings Gi (i=1,…,5) and parameters λi (i=1,2). This complicated rotation is motivated by the following simple relations: the rotation U relates (3.13)gRWR,μ-g0Bμ=G1Z′¯μ+G2Z¯μ,gLWL,μ-g0Bμ=G3Z′¯μ+G4Z¯μ,gRWR,μ+λ1gLWL,μ+λ2g0Bμ=G5A¯μ
which diagonalizes the B-WL and B-WR mixings automatically while simultaneously keeping the photon massless. To maintain a diagonal kinetic energy matrix, U must satisfy six independent orthogonality conditions (3.14)UUT=1.
Adding one mass diagonalization condition for the WR-WL mass mixing, there are seven independent equations that determine five Gi (i=1,…,5) and two λi (i=1,2). Solving these equations, we obtain(3.15)λ1=gR2gL2,λ2=gR2g02,G1=(κfRC-fL)fL(fRC-κfL)fRG3,G2=CG4,G3=(fRC-κfL)fRg02(1-C)2+gR2+C2gL2fR2C2+fL2-2CκfRfL,G4=1fR2C2+fL2-2CκfRfL((fR2C+fL2)2fR2fL2[gR2+C2gL2+g02(1+C)2]κ2-2fRfL[fR2C(gR2+g02)+fL2C(gL2+g02)+g02(fR2C2+fL2)]κ+gR2fR4C2+gL2fL4+g02(fR2C+fL2)2)1/2,G5=gR21gR2+1gL2+1g02
with a real C that satisfies the quadratic equation(3.16)(κfRfL(gL2+g02)-fR2g02)C2+[fR2(gR2+g02)-fL2(gL2+g02)]C+fL2g02-κfRfL(gR2+g02)=0.
The mass eigenvalues of the physical Z′ and Z then become(3.17)MZ′2=(UTMU)1,1,MZ2=(UTMU)2,2.
Up to now, we have obtained the exact rotation matrix elements without any approximate assumption. The total rotation U in (3.12) can be expressed in terms of (3.11), (3.15), and (3.16).


## 3.1. Charged Gauge Bosons

Taking the unitary gaugeUL=UR=1, the charged gauge boson mass terms can be expressed as(3.1)LCM=14fL2gL2WL,μ+WL,μ-+14fR2gR2WR,μ+WR,μ--14κ̃fLfRgLgR(WL,μ+WR,μ-+WR,μ+WL,μ-).
Here, we have used charged boson definitions Wi,μ1=(Wi,μ++Wi,μ-)/2 and Wi,μ2=i(Wi,μ+-Wi,μ-)/2 for i=L,R.We make an orthogonal rotationV for WL± and WR±(3.2)(WR±WL±)=(cos⁡⁡ξsin⁡⁡ξ-sin⁡⁡ξcos⁡⁡ξ)(W′±W±)≡V(W′±W±)
to eliminate the cross-terms involving WL and WR in (3.1) to keep the kinetic term diagonal. The mixing angle ξ is expressed as(3.3)tan⁡2ξ=2κ̃fLfRgLgRfL2gL2-fR2gR2.After this rotation, the charged boson mass-squared matrix for the charged bosons becomes(3.4)VTMCV=diag⁡(MW′2,MW2),
and the heavy and light charged boson masses are (3.5)MW′2=18[fL2gL2+fR2gR2+(fL2gL2-fR2gR2)2+4κ̃2fL2fR2gL2gR2]≃14fR2gR2{1+κ̃2fL2gL2fR2gR2-fL2gL2},MW2=18[fL2gL2+fR2gR2-(fL2gL2-fR2gR2)2+4κ̃2fL2fR2gL2gR2]≃14fL2gL2{1-κ̃2fR2gR2fR2gR2-fL2gL2}.
We notice that the charged boson mixing angle ξ is controlled by the coefficient κ̃. W-W′ mixing causes W couplings to the right-handed fermion with gRW=gLsinξ/2. gRW can yield the contributions to b→sγ (see paper [14]) and must be restrained so that gRW/gLW<4×10-3, which requires ξ<4×10-3.


## 3.2. Neutral Gauge Bosons

Now, let us discuss the neutral boson sector. The neutral mass terms in our chiral Lagrangian (2.5) can be readily separated out (3.6)LMn=18(1-2βL,1)fL2(gLWL,μ3-gBμ)2+18(1-2βR,1)fR2(gRWR,μ3-gBμ)2-14(κ̃+β̃1)fLfR(gLWL,μ3-gBμ)(gRWR3,μ-gBμ).
It can be written in matrix form(3.7)LMn=12GμTMnGμ
with neutral gauge bosons 𝒢μ≡(WR,μ,WL,μ,Bμ) and mass-squared matrix (3.8)Mn≡(fR2gR24-κfRfLgRgL4fRgRg0(κfL-fR)4-κfRfLgRgL4fL2gL24fLgLg0(κfR-fL)4fRgRg0(κfL-fR)4fLgLg0(κfR-fL)4(fR2+fL2-2κfRfL)g024).
Note that the βL,R,1 do not appear in the above mass-squared matrix because these can be absorbed by a redefinition of VEV fL,R(3.9)fL,R⟶fL,R1-2βL,R,1.
For the sake of convenience, we will retain using the same notation for the redefined VEV fL,R but keep in mind that this redefinition has been made. The new parameter κ in the above formula is a combination of κ̃ and β̃1, namely, κ=κ̃+β̃1. Taking into account the VEVs re-definition, we have (3.10)κ=κ̃+β̃11-2βL,11-2βR,1.
The physical masses of the neutral bosons are the eigenvalues of the matrix ℳn. To obtain the diagonalized eigenvalues, we define the mass eigenstates as 𝒢¯μ=(Z′,Zμ,Aμ)T which are related to 𝒢μ by a special rotation U-1(3.11)G¯μ=(G4gRG1G4-G2G3-G2gLG1G4-G2G3(G2-G4)g0G1G4-G2G3-G3gRG1G4-G2G3G1gLG1G4-G2G3(G3-G1)g0G1G4-G2G3gRG5λ1gLG5λ2g0G5)Gμ(3.12)≡U-1Gμ
with undetermined couplings Gi (i=1,…,5) and parameters λi (i=1,2). This complicated rotation is motivated by the following simple relations: the rotation U relates (3.13)gRWR,μ-g0Bμ=G1Z′¯μ+G2Z¯μ,gLWL,μ-g0Bμ=G3Z′¯μ+G4Z¯μ,gRWR,μ+λ1gLWL,μ+λ2g0Bμ=G5A¯μ
which diagonalizes the B-WL and B-WR mixings automatically while simultaneously keeping the photon massless. To maintain a diagonal kinetic energy matrix, U must satisfy six independent orthogonality conditions (3.14)UUT=1.
Adding one mass diagonalization condition for the WR-WL mass mixing, there are seven independent equations that determine five Gi (i=1,…,5) and two λi (i=1,2). Solving these equations, we obtain(3.15)λ1=gR2gL2,λ2=gR2g02,G1=(κfRC-fL)fL(fRC-κfL)fRG3,G2=CG4,G3=(fRC-κfL)fRg02(1-C)2+gR2+C2gL2fR2C2+fL2-2CκfRfL,G4=1fR2C2+fL2-2CκfRfL((fR2C+fL2)2fR2fL2[gR2+C2gL2+g02(1+C)2]κ2-2fRfL[fR2C(gR2+g02)+fL2C(gL2+g02)+g02(fR2C2+fL2)]κ+gR2fR4C2+gL2fL4+g02(fR2C+fL2)2)1/2,G5=gR21gR2+1gL2+1g02
with a real C that satisfies the quadratic equation(3.16)(κfRfL(gL2+g02)-fR2g02)C2+[fR2(gR2+g02)-fL2(gL2+g02)]C+fL2g02-κfRfL(gR2+g02)=0.
The mass eigenvalues of the physical Z′ and Z then become(3.17)MZ′2=(UTMU)1,1,MZ2=(UTMU)2,2.
Up to now, we have obtained the exact rotation matrix elements without any approximate assumption. The total rotation U in (3.12) can be expressed in terms of (3.11), (3.15), and (3.16).


## 4. Oblique Radiative Corrections

To clearly see the new physics correction, we can separate a standard electroweak rotation from the total rotation in  (3.12)(4.1)U≡U′Uem
with the standard electroweak rotation(4.2)Uem=(1000cθsθ0-sθcθ).
From  (3.12) and (4.1), we can calculate the oblique radiative corrections coming from the right-handed gauge bosons in light of Holdom's work [15](4.3)S=4sθcθα{(sθ2-cθ2)U32′-2cθsθ(U33′-1)+2cθsθ(U22′-1)},T=2α{(U22′-1)-ΔMZ},U=-8sθ2α{cθsθU32′+sθ2(U33′-1)+cθ2(U22′-1)},
where sθ and cθ are the respective sine and cosine of the standard Weinberg angle from SM, and ΔMZ is the new physical shift in the Z mass ΔMZ=MZ-MZ∣SM. Furthermore, we calculate to leading order the results for two special conditions.


### 4.1. Case  1:fR≫fL and gR≫gL/0

This case corresponds to aSU(2)R breaking scale that is much higher than the electroweak breaking scale and MWR≫MWL. It is easy to calculate the U′ rotation from  (4.1), (4.2), and (3.15). We only list leading-order terms (4.4)U11′≃1,U12′≃cθsθr32,U13′≃r,U21′≃-κfRfLsθcθr(1+(3-cθ2)r22),U22′≃1,U23′≃-cθsθr22,U31′≃r+κfRrfL(1+(1-2cθ2)r22),U32′≃cθsθr22,U33′≃1
with coupling ratio r≡g0/gR. Obviously, in the limit of heavy MWR, gR≫gL,0, this new physics rotation matrix U′ becomes a unitary matrix. Indeed, it is a requirement of the SM structure and a good self-checking condition of our calculation.From (3.17), we can calculate the gauge boson mass eigenvalues(4.5)MZ′2=(UTMU)1,1≃fR2gR24(1+r2)(1-κ2),(4.6)MZ2=(UTMU)2,2≃fL2(gL2+g02)4{1+(2κfLfR-sθ2)r2}.From  (4.5), the mass shift can be calculated(4.7)ΔMZ≃-sθ22r2.
Using (4.3), the leading-order terms to the oblique radiative correction parameters are (4.8)αS≃sθ2cθ2(1+2sθ2)r2,αT≃sθ2r2,αU≃4sθ6r2.
Adopting the new physics constraints S<0.11, T<0.14, U<0.16 [16] and taking sθ2=0.2311,α=1/137, we can estimate the coupling ratio r<0.05.


### 4.2. Case  2:fR=fL and gR≫gL/0

The conditions correspond to left-right symmetry.MWR≫MWL requires gR≫gL/0. Hence, the leading-order terms to the matrix elements of U′ are (4.9)U11′≃1,U12′≃-r3cθ(cθ2+cθ2sθ2+1)sθ+rcθ(2+sθ4)sθ,U13′≃r-κr(2+sθ4)2,U21′≃-r3sθcθ(32-cθ2)+sθcθr3κ,U22′≃1U23′≃-sθcθr2+{sθ3cθ2-r2sθ3cθ(1+3cθ2)4}κ,U31′≃r+cθ2sθ3r3κ,U32′≃sθcθr22+{sθ3cθ2-r2cθ4sθ[4+3sθ4(1+cθ2)]}κ,U33′≃1.
When taking r→0 and κ→0, matrix U′ becomes unitary. The leading order terms for the gauge boson masses are (4.10)MZ′2=(UTMU)1,1≃f2gR24(1+r2),MZ2=(UTMU)2,2≃f2(gL2+g02)4(1-sθ2r22).The shift in mass ofZ is (4.11)ΔMZ≃-sθ24r2.
In this case, the leading-order terms of the oblique radiative correction parameters are (4.12)αS≃2r2sθ2(1-2sθ2)cθ2+{6(1-2cθ2)sθ4+r2(4cθ2-3)(3sθ2+4)}cθ2κ,αT≃sθ2r22,αU≃4sθ6r2+2{2sθ2(1-3cθ2sθ2)+r2(2cθ2-1)(3sθ4+4)}sθ2κ.
From T<0.10, we can estimate coupling ratio r<0.09 implying a lower limit for the Z′ mass of about 0.8 TeV.


## 4.1. Case  1:fR≫fL and gR≫gL/0

This case corresponds to aSU(2)R breaking scale that is much higher than the electroweak breaking scale and MWR≫MWL. It is easy to calculate the U′ rotation from  (4.1), (4.2), and (3.15). We only list leading-order terms (4.4)U11′≃1,U12′≃cθsθr32,U13′≃r,U21′≃-κfRfLsθcθr(1+(3-cθ2)r22),U22′≃1,U23′≃-cθsθr22,U31′≃r+κfRrfL(1+(1-2cθ2)r22),U32′≃cθsθr22,U33′≃1
with coupling ratio r≡g0/gR. Obviously, in the limit of heavy MWR, gR≫gL,0, this new physics rotation matrix U′ becomes a unitary matrix. Indeed, it is a requirement of the SM structure and a good self-checking condition of our calculation.From (3.17), we can calculate the gauge boson mass eigenvalues(4.5)MZ′2=(UTMU)1,1≃fR2gR24(1+r2)(1-κ2),(4.6)MZ2=(UTMU)2,2≃fL2(gL2+g02)4{1+(2κfLfR-sθ2)r2}.From  (4.5), the mass shift can be calculated(4.7)ΔMZ≃-sθ22r2.
Using (4.3), the leading-order terms to the oblique radiative correction parameters are (4.8)αS≃sθ2cθ2(1+2sθ2)r2,αT≃sθ2r2,αU≃4sθ6r2.
Adopting the new physics constraints S<0.11, T<0.14, U<0.16 [16] and taking sθ2=0.2311,α=1/137, we can estimate the coupling ratio r<0.05.


## 4.2. Case  2:fR=fL and gR≫gL/0

The conditions correspond to left-right symmetry.MWR≫MWL requires gR≫gL/0. Hence, the leading-order terms to the matrix elements of U′ are (4.9)U11′≃1,U12′≃-r3cθ(cθ2+cθ2sθ2+1)sθ+rcθ(2+sθ4)sθ,U13′≃r-κr(2+sθ4)2,U21′≃-r3sθcθ(32-cθ2)+sθcθr3κ,U22′≃1U23′≃-sθcθr2+{sθ3cθ2-r2sθ3cθ(1+3cθ2)4}κ,U31′≃r+cθ2sθ3r3κ,U32′≃sθcθr22+{sθ3cθ2-r2cθ4sθ[4+3sθ4(1+cθ2)]}κ,U33′≃1.
When taking r→0 and κ→0, matrix U′ becomes unitary. The leading order terms for the gauge boson masses are (4.10)MZ′2=(UTMU)1,1≃f2gR24(1+r2),MZ2=(UTMU)2,2≃f2(gL2+g02)4(1-sθ2r22).The shift in mass ofZ is (4.11)ΔMZ≃-sθ24r2.
In this case, the leading-order terms of the oblique radiative correction parameters are (4.12)αS≃2r2sθ2(1-2sθ2)cθ2+{6(1-2cθ2)sθ4+r2(4cθ2-3)(3sθ2+4)}cθ2κ,αT≃sθ2r22,αU≃4sθ6r2+2{2sθ2(1-3cθ2sθ2)+r2(2cθ2-1)(3sθ4+4)}sθ2κ.
From T<0.10, we can estimate coupling ratio r<0.09 implying a lower limit for the Z′ mass of about 0.8 TeV.


## 5. A Short Summary

To summarize, we have reviewed nonlinearly realized electroweak chiral Lagrangian for the gauge groupSU(2)R×SU(2)L×U(1) and diagonalized gauge eigenstates using all possible mass mixing terms to obtain exact mass eigenstates and the rotation matrix. The oblique radiative corrections from right-handed gauge bosons have been estimated to leading order.



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