Mass Mixing Effect and Oblique Radiative Corrections in Extended 𝑆𝑈(2)_𝑅×𝑆𝑈(2)_𝐿×𝑈(1) Effective Theory

Abstract

We analyze the properties of electroweak chiral effective Lagrangian with an extended 𝑆𝑈(2)_𝑅 gauge group. Right-handed 𝑆𝑈(2)_𝑅 gauge bosons affect electroweak observables by mixing with electroweak gauge bosons 𝑊_𝐿,𝜇 and 𝐵_𝜇. 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.

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 group 𝑆𝑈(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. The 𝑆𝑈(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 𝑊^±_𝑅 can mix with left-handed 𝑊^±_𝐿, and physical mass eigenstates of 𝑊′^± and 𝑊^± are eigenvalues of the charged mass matrix. Similarly, 𝑊³_𝑅 takes part in 𝑊³_𝑅−𝑊³_𝐿−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. Section 2 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 𝑀_𝑊𝑅≫𝑀_𝑊𝐿 case and left-right symmetry. Finally, we give a short summary in Section 5.

2. Left-Right Symmetry Effective Lagrangian

Let 𝑊^𝑎_𝑅,𝜇, 𝑊^𝑎_𝐿,𝜇, 𝐵_𝜇 be electroweak gauge fields (𝑎=1,2,3) corresponding to the gauge group 𝑆𝑈(2)_𝑅, 𝑆𝑈(2)_𝐿, and 𝑈(1), respectively, and 𝑈_𝐿,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 as𝜏^𝑎2𝑊^𝑎_𝑖,𝜇⟶𝑅_𝑖𝜏^𝑎2𝑊^𝑎_𝑖,𝜇(𝑥)𝑅^†_𝑖−𝑖𝑔_𝑖𝑅_𝑖𝜕_𝜇𝑅^†_𝑖,𝐵_𝜇⟶𝐵_𝜇−1𝑔𝜕_𝜇𝜃⁰,𝑈_𝑖⟶𝑅_𝑖𝑈_𝑖𝑅^†₀ (2.1) with 𝑅₀=𝑒^{(𝑖/2)𝜏3𝜃0(𝑥)} and𝑅_𝑖=𝑒^{(𝑖/2)𝜏𝑎𝜃𝑎𝑖(𝑥)}for 𝑖=𝑅,𝐿. The covariant derivative of the Goldstone fields takes the form𝐷_𝜇𝑈_𝑅=𝜕_𝜇𝑈_𝑅+𝑖𝑔_𝑅𝜏^𝑎2𝑊^𝑎_𝑅,𝜇𝑈_𝑅−𝑖𝑔𝑈_𝐿𝜏₃2𝐵_𝜇,𝐷_𝜇𝑈_𝐿=𝜕_𝜇𝑈_𝐿+𝑖𝑔_𝐿𝜏^𝑎2𝑊^𝑎_𝐿,𝜇𝑈_𝐿−𝑖𝑔𝑈_𝐿𝜏₃2𝐵_𝜇. (2.2) 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) 𝑋^𝜇_𝑖=𝑈^†_𝑖𝐷^𝜇𝑈_𝑖,𝑊_{𝑖,𝜇𝜈}=𝑈^†_𝑖𝑔_𝑖𝑊_{𝑖,𝜇𝜈}𝑈_𝑖,𝐵_𝜇𝜈=𝜕_𝜇𝐵_𝜈−𝜕_𝜈𝐵_𝜇. (2.3) Here, 𝑊_{𝑖,𝜇𝜈}=𝑊^𝑎_{𝑖,𝜇𝜈}𝜏^𝑎2=𝜕_𝜇𝑊^𝑎_𝑖,𝜈𝜏^𝑎2−𝜕_𝜈𝑊^𝑎_𝑖,𝜇𝜏^𝑎2+𝑖𝑔_𝑖𝑊^𝑎_𝑖,𝜇𝜏^𝑎2,𝑊^𝑏_𝑖,𝜈𝜏^𝑏2. (2.4) With the help of these building blocks, we can write a leading-order chiral Lagrangian asℒ_𝑀1=−4𝑓²_𝐿𝑋_𝐿,𝜇𝑋^𝜇_𝐿−14𝑓²_𝑅𝑋_𝑅,𝜇𝑋^𝜇_𝑅+12𝜅𝑓_𝐿𝑓_𝑅𝑋^𝜇_𝐿𝑋^𝜇_𝑅+14𝛽_𝐿,1𝑓²_𝐿𝜏³𝑋_𝐿,𝜇²+14𝛽_𝑅,1𝑓²_𝐿𝜏³𝑋_𝑅,𝜇²+14𝛽₁𝑓_𝐿𝑓_𝑅𝜏³𝑋_𝐿,𝜇𝜏³𝑋^𝜇_𝑅. (2.5) Here, ⟨⟩ stands for the trace in flavor space. 𝑓_𝐿 and 𝑓_𝑅 are the scales for spontaneous symmetry breaking in the electroweak sector and parity, respectively. The coefficient 𝛽_𝐿,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 𝛽₁ controls the mixing between left-handed 𝑊³_𝐿 and right-handed 𝑊³_𝑅.

The neutral current interactions are−ℒ_NC=𝑊³_𝑅𝜇𝐽^𝜇_𝑅+𝑊³_𝐿𝜇𝐽^𝜇_𝐿+𝐵_𝜇𝐽^𝜇₀ (2.6) whereas the charged current interactions are−ℒ_CC=𝑊⁺_𝑅𝜇𝐽_𝑅^−,𝜇+𝑊⁺_𝐿𝜇𝐽_𝐿^−,𝜇+h.c. (2.7) Here,𝐽^±,𝜇_𝐿,𝑅=𝑔_𝐿,𝑅√2Ψ_𝐿,𝑅𝜏^±𝛾^𝜇Ψ_𝐿,𝑅. (2.8)

The kinetic part has the simple formℒ_𝐾1=−4𝑊^𝑎_{𝐿,𝜇𝜈}𝑊_𝐿^{𝜇𝜈,𝑎}−14𝑊^𝑎_{𝑅,𝜇𝜈}𝑊_𝑅^{𝜇𝜈,𝑎}−14𝐵_𝜇𝜈𝐵^𝜇𝜈+𝑖Ψ_𝑖𝛾_𝜇𝐷_𝜇Ψ_𝑖. (2.9)

Adding Yukawa termsℒ_𝑌=Ψ_𝐿𝑈_𝐿𝑀𝑈^†_𝑅Ψ_𝑅+h.c., (2.10) the total Lagrangian is the sum of all the above termsℒ=ℒ_𝑀+ℒ_𝐾+ℒ_NC+ℒ_CC+ℒ_𝑌. (2.11)

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 gauge 𝑈_𝐿=𝑈_𝑅=1, the charged gauge boson mass terms can be expressed asℒ_CM=14𝑓²_𝐿𝑔²_𝐿𝑊⁺_𝐿,𝜇𝑊⁻_𝐿,𝜇+14𝑓²_𝑅𝑔²_𝑅𝑊⁺_𝑅,𝜇𝑊⁻_𝑅,𝜇−14𝜅𝑓_𝐿𝑓_𝑅𝑔_𝐿𝑔_𝑅𝑊⁺_𝐿,𝜇𝑊⁻_𝑅,𝜇+𝑊⁺_𝑅,𝜇𝑊⁻_𝐿,𝜇. (3.1) Here, we have used charged boson definitions 𝑊¹_𝑖,𝜇=(𝑊⁺_𝑖,𝜇+𝑊⁻_𝑖,𝜇)/√2 and 𝑊²_𝑖,𝜇=𝑖(𝑊⁺_𝑖,𝜇−𝑊⁻_𝑖,𝜇)/√2 for 𝑖=𝐿,𝑅.

We make an orthogonal rotation 𝑉 for 𝑊^±_𝐿 and 𝑊^±_𝑅𝑊^±_𝑅𝑊^±_𝐿=cos𝜉sin𝜉−sin𝜉cos𝜉𝑊′^±𝑊^±≡𝑉𝑊′^±𝑊^± (3.2) to eliminate the cross-terms involving 𝑊_𝐿 and 𝑊_𝑅 in (3.1) to keep the kinetic term diagonal. The mixing angle 𝜉 is expressed astan2𝜉=2𝜅𝑓_𝐿𝑓_𝑅𝑔_𝐿𝑔_𝑅𝑓²_𝐿𝑔²_𝐿−𝑓²_𝑅𝑔²_𝑅. (3.3)

After this rotation, the charged boson mass-squared matrix for the charged bosons becomes 𝑉^𝑇ℳ_𝐶𝑉=diag𝑀²_𝑊′,𝑀²_𝑊, (3.4) and the heavy and light charged boson masses are 𝑀²_𝑊′=18𝑓²_𝐿𝑔²_𝐿+𝑓²_𝑅𝑔²_𝑅+𝑓²_𝐿𝑔²_𝐿−𝑓²_𝑅𝑔²_𝑅²+4𝜅²𝑓²_𝐿𝑓²_𝑅𝑔²_𝐿𝑔²_𝑅≃14𝑓²_𝑅𝑔²_𝑅1+𝜅²𝑓²_𝐿𝑔²_𝐿𝑓²_𝑅𝑔²_𝑅−𝑓²_𝐿𝑔²_𝐿,𝑀²_𝑊=18𝑓²_𝐿𝑔²_𝐿+𝑓²_𝑅𝑔²_𝑅−𝑓²_𝐿𝑔²_𝐿−𝑓²_𝑅𝑔²_𝑅²+4𝜅²𝑓²_𝐿𝑓²_𝑅𝑔²_𝐿𝑔²_𝑅≃14𝑓²_𝐿𝑔²_𝐿1−𝜅²𝑓²_𝑅𝑔²_𝑅𝑓²_𝑅𝑔²_𝑅−𝑓²_𝐿𝑔²_𝐿. (3.5) We notice that the charged boson mixing angle 𝜉 is controlled by the coefficient 𝜅. 𝑊−𝑊′ mixing causes 𝑊 couplings to the right-handed fermion with 𝑔^𝑊_𝑅=𝑔_𝐿sin𝜉/√2. 𝑔^𝑊_𝑅 can yield the contributions to 𝑏→𝑠𝛾 (see paper [14]) and must be restrained so that 𝑔^𝑊_𝑅/𝑔^𝑊_𝐿<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 ℒ_𝑀𝑛=181−2𝛽_𝐿,1𝑓²_𝐿𝑔_𝐿𝑊³_𝐿,𝜇−𝑔𝐵_𝜇²+181−2𝛽_𝑅,1𝑓²_𝑅𝑔_𝑅𝑊³_𝑅,𝜇−𝑔𝐵_𝜇²−14𝜅+𝛽₁𝑓_𝐿𝑓_𝑅𝑔_𝐿𝑊³_𝐿,𝜇−𝑔𝐵_𝜇𝑔_𝑅𝑊_𝑅^3,𝜇−𝑔𝐵_𝜇. (3.6) It can be written in matrix formℒ_𝑀𝑛=12𝒢^𝑇_𝜇ℳ_𝑛𝒢^𝜇 (3.7) with neutral gauge bosons 𝒢_𝜇≡(𝑊_𝑅,𝜇,𝑊_L,𝜇,𝐵_𝜇) and mass-squared matrix ℳ_𝑛≡𝑓²_𝑅𝑔²_𝑅4−𝜅𝑓_𝑅𝑓_𝐿𝑔_𝑅𝑔_𝐿4𝑓_𝑅𝑔_𝑅𝑔₀𝜅𝑓_𝐿−𝑓_𝑅4−𝜅𝑓_𝑅𝑓_𝐿𝑔_𝑅𝑔_𝐿4𝑓²_𝐿𝑔²_𝐿4𝑓_𝐿𝑔_𝐿𝑔₀𝜅𝑓_𝑅−𝑓_𝐿4𝑓_𝑅𝑔_𝑅𝑔₀𝜅𝑓_𝐿−𝑓_𝑅4𝑓_𝐿𝑔_𝐿𝑔₀𝜅𝑓_𝑅−𝑓_𝐿4𝑓²_𝑅+𝑓²_𝐿−2𝜅𝑓_𝑅𝑓_𝐿𝑔²₀4. (3.8) Note that the 𝛽_𝐿,𝑅,1 do not appear in the above mass-squared matrix because these can be absorbed by a redefinition of VEV 𝑓_𝐿,𝑅𝑓_𝐿,𝑅⟶𝑓_𝐿,𝑅1−2𝛽_𝐿,𝑅,1. (3.9) For the sake of convenience, we will retain using the same notation for the redefined VEV 𝑓_𝐿,𝑅 but keep in mind that this redefinition has been made. The new parameter 𝜅 in the above formula is a combination of 𝜅 and 𝛽₁, namely, 𝜅=𝜅+𝛽₁. Taking into account the VEVs re-definition, we have 𝛽𝜅=𝜅+₁1−2𝛽_𝐿,11−2𝛽_R,1. (3.10) The physical masses of the neutral bosons are the eigenvalues of the matrix ℳ_𝑛. To obtain the diagonalized eigenvalues, we define the mass eigenstates as 𝒢_𝜇=(𝑍^′,𝑍_𝜇,𝐴_𝜇)^𝑇 which are related to 𝒢_𝜇 by a special rotation 𝑈⁻¹𝒢_𝜇=𝐺₄𝑔_𝑅𝐺₁𝐺₄−𝐺₂𝐺₃−𝐺₂𝑔_𝐿𝐺₁𝐺₄−𝐺₂𝐺₃𝐺₂−𝐺₄𝑔₀𝐺₁𝐺₄−𝐺₂𝐺₃−𝐺₃𝑔_𝑅𝐺₁𝐺₄−𝐺₂𝐺₃𝐺₁𝑔_𝐿𝐺₁𝐺₄−𝐺₂𝐺₃𝐺₃−𝐺₁𝑔₀𝐺₁𝐺₄−𝐺₂𝐺₃𝑔_𝑅𝐺₅𝜆₁𝑔_𝐿𝐺₅𝜆₂𝑔₀𝐺₅𝒢_𝜇≡𝑈⁻¹𝒢_𝜇 (3.11)(3.12) with undetermined couplings 𝐺_𝑖 (𝑖=1,…,5) and parameters 𝜆_𝑖 (𝑖=1,2). This complicated rotation is motivated by the following simple relations: the rotation 𝑈 relates 𝑔_𝑅𝑊_𝑅,𝜇−𝑔₀𝐵_𝜇=𝐺₁𝑍′_𝜇+𝐺₂𝑍_𝜇,𝑔_𝐿𝑊_𝐿,𝜇−𝑔₀𝐵_𝜇=𝐺₃𝑍′_𝜇+𝐺₄𝑍_𝜇,𝑔_𝑅𝑊_𝑅,𝜇+𝜆₁𝑔_𝐿𝑊_𝐿,𝜇+𝜆₂𝑔₀𝐵_𝜇=𝐺₅𝐴_𝜇 (3.13) which diagonalizes the 𝐵−𝑊_𝐿 and 𝐵−𝑊_𝑅 mixings automatically while simultaneously keeping the photon massless. To maintain a diagonal kinetic energy matrix, 𝑈 must satisfy six independent orthogonality conditions 𝑈𝑈^𝑇=1. (3.14) Adding one mass diagonalization condition for the 𝑊_𝑅−𝑊_𝐿 mass mixing, there are seven independent equations that determine five 𝐺_𝑖 (𝑖=1,…,5) and two 𝜆_𝑖 (𝑖=1,2). Solving these equations, we obtain𝜆₁=𝑔²_𝑅𝑔²_𝐿,𝜆₂=𝑔²_𝑅𝑔²₀,𝐺₁=𝜅𝑓_𝑅𝐶−𝑓_𝐿𝑓_𝐿𝑓_𝑅𝐶−𝜅𝑓_𝐿𝑓_𝑅𝐺₃,𝐺₂=𝐶𝐺₄,𝐺₃=𝑓_𝑅𝐶−𝜅𝑓_𝐿𝑓_𝑅𝑔²₀(1−𝐶)²+𝑔²_𝑅+𝐶²𝑔²_𝐿𝑓²_𝑅𝐶²+𝑓²_𝐿−2𝐶𝜅𝑓_𝑅𝑓_𝐿,𝐺₄=1𝑓²_𝑅𝐶²+𝑓²_𝐿−2𝐶𝜅𝑓_𝑅𝑓_𝐿𝑓²_𝑅𝑓²_𝐿𝑔²_𝑅+𝐶²𝑔²_𝐿+𝑔²₀(1+𝐶)²𝜅²−2𝑓_𝑅𝑓_𝐿𝑓²_𝑅𝐶𝑔²_𝑅+𝑔²₀+𝑓²_𝐿𝐶𝑔²_𝐿+𝑔²₀+𝑔²₀𝑓²_𝑅𝐶²+𝑓²_𝐿𝜅+𝑔²_𝑅𝑓⁴_𝑅𝐶²+𝑔²_𝐿𝑓⁴_𝐿+𝑔²₀𝑓²_𝑅𝐶+𝑓²_𝐿^21/2,𝐺₅=𝑔²_𝑅1𝑔²_𝑅+1𝑔²_𝐿+1𝑔²₀ (3.15) with a real 𝐶 that satisfies the quadratic equation𝜅𝑓_𝑅𝑓_𝐿𝑔²_𝐿+𝑔²₀−𝑓²_𝑅𝑔²₀𝐶²+𝑓²_𝑅𝑔²_𝑅+𝑔²₀−𝑓²_𝐿𝑔²_𝐿+𝑔²₀𝐶+𝑓²_𝐿𝑔²₀−𝜅𝑓_𝑅𝑓_𝐿𝑔²_𝑅+𝑔²₀=0. (3.16) The mass eigenvalues of the physical 𝑍′ and 𝑍 then become𝑀²_𝑍′=𝑈^𝑇ℳ𝑈_1,1,𝑀²_𝑍=𝑈^𝑇ℳ𝑈_2,2. (3.17) 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)𝑈≡𝑈′𝑈_em (4.1) with the standard electroweak rotation𝑈_em=1000𝑐_𝜃𝑠_𝜃0−𝑠_𝜃𝑐_𝜃. (4.2) 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]𝑆=4𝑠_𝜃𝑐_𝜃𝛼𝑠²_𝜃−𝑐²_𝜃𝑈^′₃₂−2𝑐_𝜃𝑠_𝜃𝑈^′₃₃−1+2𝑐_𝜃𝑠_𝜃𝑈^′₂₂,2−1𝑇=𝛼𝑈^′₂₂−1−Δ𝑀_𝑍,𝑈=−8𝑠²_𝜃𝛼𝑐_𝜃𝑠_𝜃𝑈^′₃₂+𝑠²_𝜃𝑈^′₃₃−1+𝑐²_𝜃𝑈^′₂₂−1, (4.3) where 𝑠_𝜃 and 𝑐_𝜃 are the respective sine and cosine of the standard Weinberg angle from SM, and Δ𝑀_𝑍 is the new physical shift in the 𝑍 mass Δ𝑀_𝑍=𝑀_𝑍−𝑀_𝑍∣_𝑆𝑀. Furthermore, we calculate to leading order the results for two special conditions.

4.1. Case  1: 𝑓_𝑅≫𝑓_𝐿 and 𝑔_𝑅≫𝑔_𝐿/0

This case corresponds to a 𝑆𝑈(2)_𝑅 breaking scale that is much higher than the electroweak breaking scale and 𝑀_𝑊𝑅≫𝑀_𝑊𝐿. It is easy to calculate the 𝑈′ rotation from (4.1), (4.2), and (3.15). We only list leading-order terms 𝑈^′₁₁𝑈≃1,_′12≃𝑐_𝜃𝑠_𝜃𝑟³2,𝑈^′₁₃𝑈≃𝑟,_′21≃−𝜅𝑓_𝑅𝑓_𝐿𝑠_𝜃𝑐_𝜃𝑟1+3−𝑐²_𝜃𝑟²2,𝑈^′₂₂𝑈≃1,_′23𝑐≃−^𝜃𝑠_𝜃𝑟²2,𝑈^′₃₁≃𝑟+𝜅𝑓_𝑅𝑟𝑓_𝐿1+1−2𝑐²_𝜃𝑟²2,𝑈^′₃₂≃𝑐_𝜃𝑠_𝜃𝑟²2,𝑈^′₃₃≃1 (4.4) with coupling ratio 𝑟≡𝑔₀/𝑔_𝑅. Obviously, in the limit of heavy 𝑀_𝑊𝑅, 𝑔_𝑅≫𝑔_𝐿,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 eigenvalues𝑀²_𝑍′=𝑈^𝑇ℳ𝑈_1,1≃𝑓²_𝑅𝑔²_𝑅41+𝑟²1−𝜅²,𝑀²_𝑍=𝑈^𝑇ℳ𝑈_2,2≃𝑓²_𝐿𝑔²_𝐿+𝑔²₀41+2𝜅𝑓_𝐿𝑓_𝑅−𝑠²_𝜃𝑟². (4.5)(4.6)

From (4.5), the mass shift can be calculatedΔ𝑀_𝑍𝑠≃−^2𝜃2𝑟². (4.7) Using (4.3), the leading-order terms to the oblique radiative correction parameters are 𝛼𝑆≃𝑠²_𝜃𝑐²_𝜃1+2𝑠²_𝜃𝑟²,𝛼𝑇≃𝑠²_𝜃𝑟²,𝛼𝑈≃4𝑠⁶_𝜃𝑟². (4.8) Adopting the new physics constraints 𝑆<0.11, 𝑇<0.14, 𝑈<0.16 [16] and taking 𝑠²_𝜃=0.2311,𝛼=1/137, we can estimate the coupling ratio 𝑟<0.05.

4.2. Case  2: 𝑓_𝑅=𝑓_𝐿 and 𝑔_𝑅≫𝑔_𝐿/0

The conditions correspond to left-right symmetry. 𝑀_𝑊𝑅≫𝑀_𝑊𝐿 requires 𝑔_𝑅≫𝑔_𝐿/0. Hence, the leading-order terms to the matrix elements of 𝑈′ are 𝑈^′₁₁𝑈≃1,_′12𝑟≃−³𝑐_𝜃𝑐²_𝜃+𝑐²_𝜃𝑠²_𝜃+1𝑠_𝜃+𝑟𝑐_𝜃2+𝑠⁴_𝜃𝑠_𝜃,𝑈^′₁₃≃𝑟−𝜅𝑟2+𝑠⁴_𝜃2,𝑈^′₂₁≃−𝑟³𝑠_𝜃𝑐_𝜃32−𝑐²_𝜃+𝑠_𝜃𝑐_𝜃𝑟³𝑈𝜅,_′22𝑈≃1_′23≃−𝑠_𝜃𝑐_𝜃𝑟²+𝑠³_𝜃𝑐_𝜃2−𝑟²𝑠³_𝜃𝑐_𝜃1+3𝑐²_𝜃4𝑈𝜅,_′31≃𝑟+𝑐²_𝜃𝑠³_𝜃𝑟³𝑈𝜅,_′32≃𝑠_𝜃𝑐_𝜃𝑟²2+𝑠³_𝜃𝑐_𝜃2−𝑟²𝑐_𝜃4𝑠_𝜃4+3𝑠⁴_𝜃1+𝑐²_𝜃𝑈𝜅,_′33≃1. (4.9) When taking 𝑟→0 and 𝜅→0, matrix 𝑈′ becomes unitary. The leading order terms for the gauge boson masses are 𝑀²_𝑍′=𝑈^𝑇ℳ𝑈_1,1≃𝑓²𝑔²_𝑅41+𝑟²,𝑀²_𝑍=𝑈^𝑇ℳ𝑈_2,2≃𝑓²𝑔²_𝐿+𝑔²₀4𝑠1−^2𝜃𝑟²2. (4.10)

The shift in mass of 𝑍 is Δ𝑀_𝑍𝑠≃−^2𝜃4𝑟². (4.11) In this case, the leading-order terms of the oblique radiative correction parameters are 𝛼𝑆≃2𝑟²𝑠²_𝜃1−2𝑠²_𝜃𝑐²_𝜃+61−2𝑐²_𝜃𝑠⁴_𝜃+𝑟²4𝑐²_𝜃−33𝑠²_𝜃+4𝑐²_𝜃𝑠𝜅,𝛼𝑇≃^2𝜃𝑟²2,𝛼𝑈≃4𝑠⁶_𝜃𝑟²+22𝑠²_𝜃1−3𝑐²_𝜃𝑠²_𝜃+𝑟²2𝑐²_𝜃−13𝑠⁴_𝜃+4𝑠²_𝜃𝜅. (4.12) 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 group 𝑆𝑈(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.

Acknowledgments

This work was supported by National Science Foundation of China (NSFC) under no. 11005084 and no. 10947152 and partly by the Fundamental Research Funds for the Central University.