Advances in High Energy Physics
VolumeΒ 2012Β (2012), Article IDΒ 761953, 12 pages
doi:10.1155/2012/761953
Research Article

Mass Mixing Effect and Oblique Radiative Corrections in Extended 𝑆 π‘ˆ ( 2 ) 𝑅 Γ— 𝑆 π‘ˆ ( 2 ) 𝐿 Γ— π‘ˆ ( 1 ) Effective Theory

Ying Zhang

School of Science, Xi'an Jiaotong University, Xi'an 710049, China

Received 29 August 2011; Accepted 15 September 2011

Academic Editor: MarcΒ Sher

Copyright Β© 2012 Ying Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We analyze the properties of electroweak chiral effective Lagrangian with an extended 𝑆 π‘ˆ ( 2 ) 𝑅 gauge group. Right-handed 𝑆 π‘ˆ ( 2 ) 𝑅 gauge bosons affect electroweak observables by mixing with electroweak gauge bosons π‘Š 𝐿 , πœ‡ and 𝐡 πœ‡ . 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 𝑆 π‘ˆ ( 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 𝑆 π‘ˆ ( 1 ) 𝑅 group, associated with an additional triplet of gauge bosons π‘Š β€² Β± and 𝑍 β€² , 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 𝑆 π‘ˆ ( 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. 𝑆 π‘ˆ ( 2 ) 𝑅 gauge bosons can improve unitarity of not only π‘Š π‘Š but also π‘Š 𝑍 scattering processes and delay the breaking scale of unitarity.

Many left-right symmetry models with symmetry group 𝑆 π‘ˆ ( 2 ) 𝑅 Γ— 𝑆 π‘ˆ ( 2 ) 𝐿 Γ— π‘ˆ ( 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 𝑆 π‘ˆ ( 2 ) 𝑅 electroweak gauge models given the symmetry breaking pattern 𝑆 π‘ˆ ( 2 ) 𝑅 Γ— 𝑆 π‘ˆ ( 2 ) 𝐿 Γ— π‘ˆ ( 1 ) β†’ π‘ˆ ( 1 ) e m . 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 𝑆 π‘ˆ ( 2 ) 𝑅 gauge triplet can be regarded as a copy of the 𝑆 π‘ˆ ( 2 ) 𝐿 gauge triplet of SM, but with heavier masses. Right-handed charged gauge bosons π‘Š Β± 𝑅 can mix with left-handed π‘Š Β± 𝐿 , and physical mass eigenstates of π‘Š β€² Β± and π‘Š Β± are eigenvalues of the charged mass matrix. Similarly, π‘Š 3 𝑅 takes part in π‘Š 3 𝑅 βˆ’ π‘Š 3 𝐿 βˆ’ 𝐡 three-body mixing to form physical massive neutral bosons 𝑍 β€² , 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 𝑆 π‘ˆ ( 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 𝑆 π‘ˆ ( 2 ) 𝑅 Γ— 𝑆 π‘ˆ ( 2 ) 𝐿 Γ— π‘ˆ ( 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 π‘Š 𝑅 ≫ 𝑀 π‘Š 𝐿 case and left-right symmetry. Finally, we give a short summary in Section 5.

2. Left-Right Symmetry Effective Lagrangian

Let π‘Š π‘Ž 𝑅 , πœ‡ , π‘Š π‘Ž 𝐿 , πœ‡ , 𝐡 πœ‡ be electroweak gauge fields ( π‘Ž = 1 , 2 , 3 ) corresponding to the gauge group 𝑆 π‘ˆ ( 2 ) 𝑅 , 𝑆 π‘ˆ ( 2 ) 𝐿 , and π‘ˆ ( 1 ) , respectively, and π‘ˆ 𝐿 , 𝑅 be the two by two unitary unimodular matrices corresponding to left- and right-handed Goldstone boson fields. Under 𝑆 π‘ˆ ( 2 ) 𝑅 βŠ— 𝑆 π‘ˆ ( 2 ) 𝐿 βŠ— π‘ˆ ( 1 ) transformations, the gauge boson fields transform as 𝜏 π‘Ž 2 π‘Š π‘Ž 𝑖 , πœ‡ ⟢ 𝑅 𝑖 𝜏 π‘Ž 2 π‘Š π‘Ž 𝑖 , πœ‡ ( π‘₯ ) 𝑅 † 𝑖 βˆ’ 𝑖 𝑔 𝑖 𝑅 𝑖 πœ• πœ‡ 𝑅 † 𝑖 , 𝐡 πœ‡ ⟢ 𝐡 πœ‡ βˆ’ 1 𝑔 πœ• πœ‡ πœƒ 0 , π‘ˆ 𝑖 ⟢ 𝑅 𝑖 π‘ˆ 𝑖 𝑅 † 0 ( 2 . 1 ) with 𝑅 0 = 𝑒 ( 𝑖 / 2 ) 𝜏 3 πœƒ 0 ( π‘₯ ) and 𝑅 𝑖 = 𝑒 ( 𝑖 / 2 ) 𝜏 π‘Ž πœƒ π‘Ž 𝑖 ( π‘₯ ) for 𝑖 = 𝑅 , 𝐿 . The covariant derivative of the Goldstone fields takes the form 𝐷 πœ‡ π‘ˆ 𝑅 = πœ• πœ‡ π‘ˆ 𝑅 + 𝑖 𝑔 𝑅 𝜏 π‘Ž 2 π‘Š π‘Ž 𝑅 , πœ‡ π‘ˆ 𝑅 βˆ’ 𝑖 𝑔 π‘ˆ 𝐿 𝜏 3 2 𝐡 πœ‡ , 𝐷 πœ‡ π‘ˆ 𝐿 = πœ• πœ‡ π‘ˆ 𝐿 + 𝑖 𝑔 𝐿 𝜏 π‘Ž 2 π‘Š π‘Ž 𝐿 , πœ‡ π‘ˆ 𝐿 βˆ’ 𝑖 𝑔 π‘ˆ 𝐿 𝜏 3 2 𝐡 πœ‡ . ( 2 . 2 ) For convenience in present discussion, we will discard conventional EWCL 𝑆 π‘ˆ ( 2 ) covariant building blocks [9–13] and introduce π‘ˆ ( 1 ) invariant building blocks (for 𝑖 = 𝐿 , 𝑅 ) 𝑋 πœ‡ 𝑖 = π‘ˆ † 𝑖 ξ€· 𝐷 πœ‡ π‘ˆ 𝑖 ξ€Έ , π‘Š 𝑖 , πœ‡ 𝜈 = π‘ˆ † 𝑖 𝑔 𝑖 π‘Š 𝑖 , πœ‡ 𝜈 π‘ˆ 𝑖 , 𝐡 πœ‡ 𝜈 = πœ• πœ‡ 𝐡 𝜈 βˆ’ πœ• 𝜈 𝐡 πœ‡ . ( 2 . 3 ) Here, π‘Š 𝑖 , πœ‡ 𝜈 = π‘Š π‘Ž 𝑖 , πœ‡ 𝜈 𝜏 π‘Ž 2 = πœ• πœ‡ π‘Š π‘Ž 𝑖 , 𝜈 𝜏 π‘Ž 2 βˆ’ πœ• 𝜈 π‘Š π‘Ž 𝑖 , πœ‡ 𝜏 π‘Ž 2 + 𝑖 𝑔 𝑖 ξ‚Έ π‘Š π‘Ž 𝑖 , πœ‡ 𝜏 π‘Ž 2 , π‘Š 𝑏 𝑖 , 𝜈 𝜏 𝑏 2 ξ‚Ή . ( 2 . 4 ) With the help of these building blocks, we can write a leading-order chiral Lagrangian as β„’ 𝑀 1 = βˆ’ 4 𝑓 2 𝐿  𝑋 𝐿 , πœ‡ 𝑋 πœ‡ 𝐿  βˆ’ 1 4 𝑓 2 𝑅  𝑋 𝑅 , πœ‡ 𝑋 πœ‡ 𝑅  + 1 2  πœ… 𝑓 𝐿 𝑓 𝑅  𝑋 πœ‡ 𝐿 𝑋 πœ‡ 𝑅  + 1 4 𝛽 𝐿 , 1 𝑓 2 𝐿  𝜏 3 𝑋 𝐿 , πœ‡  2 + 1 4 𝛽 𝑅 , 1 𝑓 2 𝐿  𝜏 3 𝑋 𝑅 , πœ‡  2 + 1 4 Μƒ 𝛽 1 𝑓 𝐿 𝑓 𝑅  𝜏 3 𝑋 𝐿 , πœ‡ 𝜏   3 𝑋 πœ‡ 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 𝑆 π‘ˆ ( 2 ) 𝐿 , 𝑅 isospin symmetry. The coefficient πœ… parameterizes the mixing between the left- and right-handed gauge bosons whereas the coefficient Μƒ 𝛽 1 controls the mixing between left-handed π‘Š 3 𝐿 and right-handed π‘Š 3 𝑅 .

The neutral current interactions are βˆ’ β„’ N C = π‘Š 3 𝑅 πœ‡ 𝐽 πœ‡ 𝑅 + π‘Š 3 𝐿 πœ‡ 𝐽 πœ‡ 𝐿 + 𝐡 πœ‡ 𝐽 πœ‡ 0 ( 2 . 6 ) whereas the charged current interactions are βˆ’ β„’ C C = π‘Š + 𝑅 πœ‡ 𝐽 𝑅 βˆ’ , πœ‡ + π‘Š + 𝐿 πœ‡ 𝐽 𝐿 βˆ’ , πœ‡ + h . c . ( 2 . 7 ) Here, 𝐽 Β± , πœ‡ 𝐿 , 𝑅 = 𝑔 𝐿 , 𝑅 √ 2 Ξ¨ 𝐿 , 𝑅 𝜏 Β± 𝛾 πœ‡ Ξ¨ 𝐿 , 𝑅 . ( 2 . 8 )

The kinetic part has the simple form β„’ 𝐾 1 = βˆ’ 4 π‘Š π‘Ž 𝐿 , πœ‡ 𝜈 π‘Š 𝐿 πœ‡ 𝜈 , π‘Ž βˆ’ 1 4 π‘Š π‘Ž 𝑅 , πœ‡ 𝜈 π‘Š 𝑅 πœ‡ 𝜈 , π‘Ž βˆ’ 1 4 𝐡 πœ‡ 𝜈 𝐡 πœ‡ 𝜈 + 𝑖 Ξ¨ 𝑖 𝛾 πœ‡ 𝐷 πœ‡ Ξ¨ 𝑖 . ( 2 . 9 )

Adding Yukawa terms β„’ π‘Œ = Ξ¨ 𝐿 π‘ˆ 𝐿 𝑀 π‘ˆ † 𝑅 Ξ¨ 𝑅 + h . c . , ( 2 . 1 0 ) the total Lagrangian is the sum of all the above terms β„’ = β„’ 𝑀 + β„’ 𝐾 + β„’ N C + β„’ C C + β„’ π‘Œ . ( 2 . 1 1 )

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 π‘ˆ 𝐿 = π‘ˆ 𝑅 = 1 , the charged gauge boson mass terms can be expressed as β„’ C M = 1 4 𝑓 2 𝐿 𝑔 2 𝐿 π‘Š + 𝐿 , πœ‡ π‘Š βˆ’ 𝐿 , πœ‡ + 1 4 𝑓 2 𝑅 𝑔 2 𝑅 π‘Š + 𝑅 , πœ‡ π‘Š βˆ’ 𝑅 , πœ‡ βˆ’ 1 4  πœ… 𝑓 𝐿 𝑓 𝑅 𝑔 𝐿 𝑔 𝑅 ξ‚€ π‘Š + 𝐿 , πœ‡ π‘Š βˆ’ 𝑅 , πœ‡ + π‘Š + 𝑅 , πœ‡ π‘Š βˆ’ 𝐿 , πœ‡  . ( 3 . 1 ) Here, we have used charged boson definitions π‘Š 1 𝑖 , πœ‡ = ( π‘Š + 𝑖 , πœ‡ + π‘Š βˆ’ 𝑖 , πœ‡ √ ) / 2 and π‘Š 2 𝑖 , πœ‡ = 𝑖 ( π‘Š + 𝑖 , πœ‡ βˆ’ π‘Š βˆ’ 𝑖 , πœ‡ √ ) / 2 for 𝑖 = 𝐿 , 𝑅 .

We make an orthogonal rotation 𝑉 for π‘Š Β± 𝐿 and π‘Š Β± 𝑅 βŽ› ⎜ ⎜ ⎝ π‘Š Β± 𝑅 π‘Š Β± 𝐿 ⎞ ⎟ ⎟ ⎠ = βŽ› ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎠ βŽ› ⎜ ⎜ ⎝ c o s πœ‰ s i n πœ‰ βˆ’ s i n πœ‰ c o s πœ‰ π‘Š β€² Β± π‘Š Β± ⎞ ⎟ ⎟ ⎠ βŽ› ⎜ ⎜ ⎝ ≑ 𝑉 π‘Š β€² Β± π‘Š Β± ⎞ ⎟ ⎟ ⎠ ( 3 . 2 ) to eliminate the cross-terms involving π‘Š 𝐿 and π‘Š 𝑅 in (3.1) to keep the kinetic term diagonal. The mixing angle πœ‰ is expressed as t a n 2 πœ‰ = 2  πœ… 𝑓 𝐿 𝑓 𝑅 𝑔 𝐿 𝑔 𝑅 𝑓 2 𝐿 𝑔 2 𝐿 βˆ’ 𝑓 2 𝑅 𝑔 2 𝑅 . ( 3 . 3 )

After this rotation, the charged boson mass-squared matrix for the charged bosons becomes 𝑉 𝑇 β„³ 𝐢 ξ€· 𝑀 𝑉 = d i a g 2 π‘Š ξ…ž , 𝑀 2 π‘Š ξ€Έ , ( 3 . 4 ) and the heavy and light charged boson masses are 𝑀 2 π‘Š ξ…ž = 1 8 ξ‚Έ 𝑓 2 𝐿 𝑔 2 𝐿 + 𝑓 2 𝑅 𝑔 2 𝑅 +  ξ€· 𝑓 2 𝐿 𝑔 2 𝐿 βˆ’ 𝑓 2 𝑅 𝑔 2 𝑅 ξ€Έ 2 + 4  πœ… 2 𝑓 2 𝐿 𝑓 2 𝑅 𝑔 2 𝐿 𝑔 2 𝑅 ξ‚Ή ≃ 1 4 𝑓 2 𝑅 𝑔 2 𝑅 ξƒ― 1 +  πœ… 2 𝑓 2 𝐿 𝑔 2 𝐿 𝑓 2 𝑅 𝑔 2 𝑅 βˆ’ 𝑓 2 𝐿 𝑔 2 𝐿 ξƒ° , 𝑀 2 π‘Š = 1 8 ξ‚Έ 𝑓 2 𝐿 𝑔 2 𝐿 + 𝑓 2 𝑅 𝑔 2 𝑅 βˆ’  ξ€· 𝑓 2 𝐿 𝑔 2 𝐿 βˆ’ 𝑓 2 𝑅 𝑔 2 𝑅 ξ€Έ 2 + 4  πœ… 2 𝑓 2 𝐿 𝑓 2 𝑅 𝑔 2 𝐿 𝑔 2 𝑅 ξ‚Ή ≃ 1 4 𝑓 2 𝐿 𝑔 2 𝐿 ξƒ― 1 βˆ’  πœ… 2 𝑓 2 𝑅 𝑔 2 𝑅 𝑓 2 𝑅 𝑔 2 𝑅 βˆ’ 𝑓 2 𝐿 𝑔 2 𝐿 ξƒ° . ( 3 . 5 ) We notice that the charged boson mixing angle πœ‰ is controlled by the coefficient  πœ… . π‘Š βˆ’ π‘Š β€² mixing causes π‘Š couplings to the right-handed fermion with 𝑔 π‘Š 𝑅 = 𝑔 𝐿 √ s i n πœ‰ / 2 . 𝑔 π‘Š 𝑅 can yield the contributions to 𝑏 β†’ 𝑠 𝛾 (see paper [14]) and must be restrained so that 𝑔 π‘Š 𝑅 / 𝑔 π‘Š 𝐿 < 4 Γ— 1 0 βˆ’ 3 , which requires πœ‰ < 4 Γ— 1 0 βˆ’ 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 β„’ 𝑀 𝑛 = 1 8 ξ€· 1 βˆ’ 2 𝛽 𝐿 , 1 ξ€Έ 𝑓 2 𝐿 ξ‚€ 𝑔 𝐿 π‘Š 3 𝐿 , πœ‡ βˆ’ 𝑔 𝐡 πœ‡  2 + 1 8 ξ€· 1 βˆ’ 2 𝛽 𝑅 , 1 ξ€Έ 𝑓 2 𝑅 ξ‚€ 𝑔 𝑅 π‘Š 3 𝑅 , πœ‡ βˆ’ 𝑔 𝐡 πœ‡  2 βˆ’ 1 4 ξ€· Μƒ 𝛽  πœ… + 1 ξ€Έ 𝑓 𝐿 𝑓 𝑅 ξ‚€ 𝑔 𝐿 π‘Š 3 𝐿 , πœ‡ βˆ’ 𝑔 𝐡 πœ‡ 𝑔  ξ‚€ 𝑅 π‘Š 𝑅 3 , πœ‡ βˆ’ 𝑔 𝐡 πœ‡  . ( 3 . 6 ) It can be written in matrix form β„’ 𝑀 𝑛 = 1 2 𝒒 𝑇 πœ‡ β„³ 𝑛 𝒒 πœ‡ ( 3 . 7 ) with neutral gauge bosons 𝒒 πœ‡ ≑ ( π‘Š 𝑅 , πœ‡ , π‘Š L , πœ‡ , 𝐡 πœ‡ ) and mass-squared matrix β„³ 𝑛 ≑ βŽ› ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 𝑓 2 𝑅 𝑔 2 𝑅 4 βˆ’ πœ… 𝑓 𝑅 𝑓 𝐿 𝑔 𝑅 𝑔 𝐿 4 𝑓 𝑅 𝑔 𝑅 𝑔 0 ξ€· πœ… 𝑓 𝐿 βˆ’ 𝑓 𝑅 ξ€Έ 4 βˆ’ πœ… 𝑓 𝑅 𝑓 𝐿 𝑔 𝑅 𝑔 𝐿 4 𝑓 2 𝐿 𝑔 2 𝐿 4 𝑓 𝐿 𝑔 𝐿 𝑔 0 ξ€· πœ… 𝑓 𝑅 βˆ’ 𝑓 𝐿 ξ€Έ 4 𝑓 𝑅 𝑔 𝑅 𝑔 0 ξ€· πœ… 𝑓 𝐿 βˆ’ 𝑓 𝑅 ξ€Έ 4 𝑓 𝐿 𝑔 𝐿 𝑔 0 ξ€· πœ… 𝑓 𝑅 βˆ’ 𝑓 𝐿 ξ€Έ 4 ξ€· 𝑓 2 𝑅 + 𝑓 2 𝐿 βˆ’ 2 πœ… 𝑓 𝑅 𝑓 𝐿 ξ€Έ 𝑔 2 0 4 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . ( 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 , 𝑅 𝑓 𝐿 , 𝑅 ⟢ 𝑓 𝐿 , 𝑅 √ 1 βˆ’ 2 𝛽 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 πœ… in the above formula is a combination of  πœ… and Μƒ 𝛽 1 , namely, Μƒ 𝛽 πœ… =  πœ… + 1 . Taking into account the VEVs re-definition, we have Μƒ 𝛽 πœ… =  πœ… + 1 √ 1 βˆ’ 2 𝛽 𝐿 , 1 √ 1 βˆ’ 2 𝛽 R , 1 . ( 3 . 1 0 ) The physical masses of the neutral bosons are the eigenvalues of the matrix β„³ 𝑛 . To obtain the diagonalized eigenvalues, we define the mass eigenstates as 𝒒 πœ‡ = ( 𝑍 ξ…ž , 𝑍 πœ‡ , 𝐴 πœ‡ ) 𝑇 which are related to 𝒒 πœ‡ by a special rotation π‘ˆ βˆ’ 1 𝒒 πœ‡ = βŽ› ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 𝐺 4 𝑔 𝑅 𝐺 1 𝐺 4 βˆ’ 𝐺 2 𝐺 3 βˆ’ 𝐺 2 𝑔 𝐿 𝐺 1 𝐺 4 βˆ’ 𝐺 2 𝐺 3 ξ€· 𝐺 2 βˆ’ 𝐺 4 ξ€Έ 𝑔 0 𝐺 1 𝐺 4 βˆ’ 𝐺 2 𝐺 3 βˆ’ 𝐺 3 𝑔 𝑅 𝐺 1 𝐺 4 βˆ’ 𝐺 2 𝐺 3 𝐺 1 𝑔 𝐿 𝐺 1 𝐺 4 βˆ’ 𝐺 2 𝐺 3 ξ€· 𝐺 3 βˆ’ 𝐺 1 ξ€Έ 𝑔 0 𝐺 1 𝐺 4 βˆ’ 𝐺 2 𝐺 3 𝑔 𝑅 𝐺 5 πœ† 1 𝑔 𝐿 𝐺 5 πœ† 2 𝑔 0 𝐺 5 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 𝒒 πœ‡ ( 3 . 1 1 ) ≑ π‘ˆ βˆ’ 1 𝒒 πœ‡ ( 3 . 1 2 ) with undetermined couplings 𝐺 𝑖 ( 𝑖 = 1 , … , 5 ) and parameters πœ† 𝑖 ( 𝑖 = 1 , 2 ). This complicated rotation is motivated by the following simple relations: the rotation π‘ˆ relates 𝑔 𝑅 π‘Š 𝑅 , πœ‡ βˆ’ 𝑔 0 𝐡 πœ‡ = 𝐺 1 𝑍 β€² πœ‡ + 𝐺 2 𝑍 πœ‡ , 𝑔 𝐿 π‘Š 𝐿 , πœ‡ βˆ’ 𝑔 0 𝐡 πœ‡ = 𝐺 3 𝑍 β€² πœ‡ + 𝐺 4 𝑍 πœ‡ , 𝑔 𝑅 π‘Š 𝑅 , πœ‡ + πœ† 1 𝑔 𝐿 π‘Š 𝐿 , πœ‡ + πœ† 2 𝑔 0 𝐡 πœ‡ = 𝐺 5 𝐴 πœ‡ ( 3 . 1 3 ) which diagonalizes the 𝐡 βˆ’ π‘Š 𝐿 and 𝐡 βˆ’ π‘Š 𝑅 mixings automatically while simultaneously keeping the photon massless. To maintain a diagonal kinetic energy matrix, π‘ˆ must satisfy six independent orthogonality conditions π‘ˆ π‘ˆ 𝑇 = 1 . ( 3 . 1 4 ) Adding one mass diagonalization condition for the π‘Š 𝑅 βˆ’ π‘Š 𝐿 mass mixing, there are seven independent equations that determine five 𝐺 𝑖 ( 𝑖 = 1 , … , 5 ) and two πœ† 𝑖 ( 𝑖 = 1 , 2 ). Solving these equations, we obtain πœ† 1 = 𝑔 2 𝑅 𝑔 2 𝐿 , πœ† 2 = 𝑔 2 𝑅 𝑔 2 0 , 𝐺 1 = ξ€· πœ… 𝑓 𝑅 𝐢 βˆ’ 𝑓 𝐿 ξ€Έ 𝑓 𝐿 ξ€· 𝑓 𝑅 𝐢 βˆ’ πœ… 𝑓 𝐿 ξ€Έ 𝑓 𝑅 𝐺 3 , 𝐺 2 = 𝐢 𝐺 4 , 𝐺 3 = ξ€· 𝑓 𝑅 𝐢 βˆ’ πœ… 𝑓 𝐿 ξ€Έ 𝑓 𝑅  𝑔 2 0 ( 1 βˆ’ 𝐢 ) 2 + 𝑔 2 𝑅 + 𝐢 2 𝑔 2 𝐿 𝑓 2 𝑅 𝐢 2 + 𝑓 2 𝐿 βˆ’ 2 𝐢 πœ… 𝑓 𝑅 𝑓 𝐿 , 𝐺 4 = 1 𝑓 2 𝑅 𝐢 2 + 𝑓 2 𝐿 βˆ’ 2 𝐢 πœ… 𝑓 𝑅 𝑓 𝐿 ξ‚€ 𝑓 2 𝑅 𝑓 2 𝐿 ξ€Ί 𝑔 2 𝑅 + 𝐢 2 𝑔 2 𝐿 + 𝑔 2 0 ( 1 + 𝐢 ) 2 ξ€» πœ… 2 βˆ’ 2 𝑓 𝑅 𝑓 𝐿 ξ€Ί 𝑓 2 𝑅 𝐢 ξ€· 𝑔 2 𝑅 + 𝑔 2 0 ξ€Έ + 𝑓 2 𝐿 𝐢 ξ€· 𝑔 2 𝐿 + 𝑔 2 0 ξ€Έ + 𝑔 2 0 ξ€· 𝑓 2 𝑅 𝐢 2 + 𝑓 2 𝐿 πœ… ξ€Έ ξ€» + 𝑔 2 𝑅 𝑓 4 𝑅 𝐢 2 + 𝑔 2 𝐿 𝑓 4 𝐿 + 𝑔 2 0 ξ€· 𝑓 2 𝑅 𝐢 + 𝑓 2 𝐿 ξ€Έ 2  1 / 2 , 𝐺 5 = 𝑔 2 𝑅 ξƒŽ 1 𝑔 2 𝑅 + 1 𝑔 2 𝐿 + 1 𝑔 2 0 ( 3 . 1 5 ) with a real 𝐢 that satisfies the quadratic equation ξ€· πœ… 𝑓 𝑅 𝑓 𝐿 ξ€· 𝑔 2 𝐿 + 𝑔 2 0 ξ€Έ βˆ’ 𝑓 2 𝑅 𝑔 2 0 ξ€Έ 𝐢 2 + ξ€Ί 𝑓 2 𝑅 ξ€· 𝑔 2 𝑅 + 𝑔 2 0 ξ€Έ βˆ’ 𝑓 2 𝐿 ξ€· 𝑔 2 𝐿 + 𝑔 2 0 ξ€Έ ξ€» 𝐢 + 𝑓 2 𝐿 𝑔 2 0 βˆ’ πœ… 𝑓 𝑅 𝑓 𝐿 ξ€· 𝑔 2 𝑅 + 𝑔 2 0 ξ€Έ = 0 . ( 3 . 1 6 ) The mass eigenvalues of the physical 𝑍 β€² and 𝑍 then become 𝑀 2 𝑍 β€² = ξ€· π‘ˆ 𝑇 ξ€Έ β„³ π‘ˆ 1 , 1 , 𝑀 2 𝑍 = ξ€· π‘ˆ 𝑇 ξ€Έ β„³ π‘ˆ 2 , 2 . ( 3 . 1 7 ) Up to now, we have obtained the exact rotation matrix elements without any approximate assumption. The total rotation π‘ˆ 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) π‘ˆ ≑ π‘ˆ β€² π‘ˆ e m ( 4 . 1 ) with the standard electroweak rotation π‘ˆ e m = βŽ› ⎜ ⎜ ⎜ ⎜ ⎝ 1 0 0 0 𝑐 πœƒ 𝑠 πœƒ 0 βˆ’ 𝑠 πœƒ 𝑐 πœƒ ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ . ( 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 = 4 𝑠 πœƒ 𝑐 πœƒ 𝛼 𝑠 ξ€½ ξ€· 2 πœƒ βˆ’ 𝑐 2 πœƒ ξ€Έ π‘ˆ ξ…ž 3 2 βˆ’ 2 𝑐 πœƒ 𝑠 πœƒ ξ€· π‘ˆ ξ…ž 3 3 ξ€Έ βˆ’ 1 + 2 𝑐 πœƒ 𝑠 πœƒ ξ€· π‘ˆ ξ…ž 2 2 , 2 βˆ’ 1 ξ€Έ ξ€Ύ 𝑇 = 𝛼 π‘ˆ ξ€½ ξ€· ξ…ž 2 2 ξ€Έ βˆ’ 1 βˆ’ Ξ” 𝑀 𝑍 ξ€Ύ , π‘ˆ = βˆ’ 8 𝑠 2 πœƒ 𝛼 ξ€½ 𝑐 πœƒ 𝑠 πœƒ π‘ˆ ξ…ž 3 2 + 𝑠 2 πœƒ ξ€· π‘ˆ ξ…ž 3 3 ξ€Έ βˆ’ 1 + 𝑐 2 πœƒ ξ€· π‘ˆ ξ…ž 2 2 , βˆ’ 1 ξ€Έ ξ€Ύ ( 4 . 3 ) where 𝑠 πœƒ and 𝑐 πœƒ are the respective sine and cosine of the standard Weinberg angle from SM, and Ξ” 𝑀 𝑍 is the new physical shift in the 𝑍 mass Ξ” 𝑀 𝑍 = 𝑀 𝑍 βˆ’ 𝑀 𝑍 ∣ 𝑆 𝑀 . 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 𝑆 π‘ˆ ( 2 ) 𝑅 breaking scale that is much higher than the electroweak breaking scale and 𝑀 π‘Š 𝑅 ≫ 𝑀 π‘Š 𝐿 . It is easy to calculate the π‘ˆ β€² rotation from (4.1), (4.2), and (3.15). We only list leading-order terms π‘ˆ ξ…ž 1 1 π‘ˆ ≃ 1 , ξ…ž 1 2 ≃ 𝑐 πœƒ 𝑠 πœƒ π‘Ÿ 3 2 , π‘ˆ ξ…ž 1 3 π‘ˆ ≃ π‘Ÿ , ξ…ž 2 1 ≃ βˆ’ πœ… 𝑓 𝑅 𝑓 𝐿 𝑠 πœƒ 𝑐 πœƒ π‘Ÿ  ξ€· 1 + 3 βˆ’ 𝑐 2 πœƒ ξ€Έ π‘Ÿ 2 2 ξƒͺ , π‘ˆ ξ…ž 2 2 π‘ˆ ≃ 1 , ξ…ž 2 3 𝑐 ≃ βˆ’ πœƒ 𝑠 πœƒ π‘Ÿ 2 2 , π‘ˆ ξ…ž 3 1 ≃ π‘Ÿ + πœ… 𝑓 𝑅 π‘Ÿ 𝑓 𝐿  ξ€· 1 + 1 βˆ’ 2 𝑐 2 πœƒ ξ€Έ π‘Ÿ 2 2 ξƒͺ , π‘ˆ ξ…ž 3 2 ≃ 𝑐 πœƒ 𝑠 πœƒ π‘Ÿ 2 2 , π‘ˆ ξ…ž 3 3 ≃ 1 ( 4 . 4 ) with coupling ratio π‘Ÿ ≑ 𝑔 0 / 𝑔 𝑅 . Obviously, in the limit of heavy 𝑀 π‘Š 𝑅 , 𝑔 𝑅 ≫ 𝑔 𝐿 , 0 , this new physics rotation matrix π‘ˆ β€² 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 𝑀 2 𝑍 ξ…ž = ξ€· π‘ˆ 𝑇 ξ€Έ β„³ π‘ˆ 1 , 1 ≃ 𝑓 2 𝑅 𝑔 2 𝑅 4 ξ€· 1 + π‘Ÿ 2 ξ€Έ ξ€· 1 βˆ’ πœ… 2 ξ€Έ , ( 4 . 5 ) 𝑀 2 𝑍 = ξ€· π‘ˆ 𝑇 ξ€Έ β„³ π‘ˆ 2 , 2 ≃ 𝑓 2 𝐿 ξ€· 𝑔 2 𝐿 + 𝑔 2 0 ξ€Έ 4 ξ‚» ξ‚΅ 2 1 + πœ… 𝑓 𝐿 𝑓 𝑅 βˆ’ 𝑠 2 πœƒ ξ‚Ά π‘Ÿ 2 ξ‚Ό . ( 4 . 6 )

From (4.5), the mass shift can be calculated Ξ” 𝑀 𝑍 𝑠 ≃ βˆ’ 2 πœƒ 2 π‘Ÿ 2 . ( 4 . 7 ) Using (4.3), the leading-order terms to the oblique radiative correction parameters are 𝛼 𝑆 ≃ 𝑠 2 πœƒ 𝑐 2 πœƒ ξ€· 1 + 2 𝑠 2 πœƒ ξ€Έ π‘Ÿ 2 , 𝛼 𝑇 ≃ 𝑠 2 πœƒ π‘Ÿ 2 , 𝛼 π‘ˆ ≃ 4 𝑠 6 πœƒ π‘Ÿ 2 . ( 4 . 8 ) Adopting the new physics constraints 𝑆 < 0 . 1 1 , 𝑇 < 0 . 1 4 , π‘ˆ < 0 . 1 6 [16] and taking 𝑠 2 πœƒ = 0 . 2 3 1 1 , 𝛼 = 1 / 1 3 7 , we can estimate the coupling ratio π‘Ÿ < 0 . 0 5 .

4.2. Case  2: 𝑓 𝑅 = 𝑓 𝐿 and 𝑔 𝑅 ≫ 𝑔 𝐿 / 0

The conditions correspond to left-right symmetry. 𝑀 π‘Š 𝑅 ≫ 𝑀 π‘Š 𝐿 requires 𝑔 𝑅 ≫ 𝑔 𝐿 / 0 . Hence, the leading-order terms to the matrix elements of π‘ˆ β€² are π‘ˆ ξ…ž 1 1 π‘ˆ ≃ 1 , ξ…ž 1 2 π‘Ÿ ≃ βˆ’ 3 𝑐 πœƒ ξ€· 𝑐 2 πœƒ + 𝑐 2 πœƒ 𝑠 2 πœƒ ξ€Έ + 1 𝑠 πœƒ + π‘Ÿ 𝑐 πœƒ ξ€· 2 + 𝑠 4 πœƒ ξ€Έ 𝑠 πœƒ , π‘ˆ ξ…ž 1 3 ξ€· ≃ π‘Ÿ βˆ’ πœ… π‘Ÿ 2 + 𝑠 4 πœƒ ξ€Έ 2 , π‘ˆ ξ…ž 2 1 ≃ βˆ’ π‘Ÿ 3 𝑠 πœƒ 𝑐 πœƒ ξ‚€ 3 2 βˆ’ 𝑐 2 πœƒ  + 𝑠 πœƒ 𝑐 πœƒ π‘Ÿ 3 π‘ˆ πœ… , ξ…ž 2 2 π‘ˆ ≃ 1 ξ…ž 2 3 ≃ βˆ’ 𝑠 πœƒ 𝑐 πœƒ π‘Ÿ 2 + ξƒ― 𝑠 3 πœƒ 𝑐 πœƒ 2 βˆ’ π‘Ÿ 2 𝑠 3 πœƒ 𝑐 πœƒ ξ€· 1 + 3 𝑐 2 πœƒ ξ€Έ 4 ξƒ° π‘ˆ πœ… , ξ…ž 3 1 ≃ π‘Ÿ + 𝑐 2 πœƒ 𝑠 3 πœƒ π‘Ÿ 3 π‘ˆ πœ… , ξ…ž 3 2 ≃ 𝑠 πœƒ 𝑐 πœƒ π‘Ÿ 2 2 + ξƒ― 𝑠 3 πœƒ 𝑐 πœƒ 2 βˆ’ π‘Ÿ 2 𝑐 πœƒ 4 𝑠 πœƒ ξ€Ί 4 + 3 𝑠 4 πœƒ ξ€· 1 + 𝑐 2 πœƒ ξƒ° π‘ˆ ξ€Έ ξ€» πœ… , ξ…ž 3 3 ≃ 1 . ( 4 . 9 ) When taking π‘Ÿ β†’ 0 and πœ… β†’ 0 , matrix π‘ˆ β€² becomes unitary. The leading order terms for the gauge boson masses are 𝑀 2 𝑍 ξ…ž = ξ€· π‘ˆ 𝑇 ξ€Έ β„³ π‘ˆ 1 , 1 ≃ 𝑓 2 𝑔 2 𝑅 4 ξ€· 1 + π‘Ÿ 2 ξ€Έ , 𝑀 2 𝑍 = ξ€· π‘ˆ 𝑇 ξ€Έ β„³ π‘ˆ 2 , 2 ≃ 𝑓 2 ξ€· 𝑔 2 𝐿 + 𝑔 2 0 ξ€Έ 4  𝑠 1 βˆ’ 2 πœƒ π‘Ÿ 2 2 ξƒͺ . ( 4 . 1 0 )

The shift in mass of 𝑍 is Ξ” 𝑀 𝑍 𝑠 ≃ βˆ’ 2 πœƒ 4 π‘Ÿ 2 . ( 4 . 1 1 ) In this case, the leading-order terms of the oblique radiative correction parameters are 𝛼 𝑆 ≃ 2 π‘Ÿ 2 𝑠 2 πœƒ ξ€· 1 βˆ’ 2 𝑠 2 πœƒ ξ€Έ 𝑐 2 πœƒ + ξ€½ 6 ξ€· 1 βˆ’ 2 𝑐 2 πœƒ ξ€Έ 𝑠 4 πœƒ + π‘Ÿ 2 ξ€· 4 𝑐 2 πœƒ βˆ’ 3 ξ€Έ ξ€· 3 𝑠 2 πœƒ 𝑐 + 4 ξ€Έ ξ€Ύ 2 πœƒ 𝑠 πœ… , 𝛼 𝑇 ≃ 2 πœƒ π‘Ÿ 2 2 , 𝛼 π‘ˆ ≃ 4 𝑠 6 πœƒ π‘Ÿ 2 ξ€½ + 2 2 𝑠 2 πœƒ ξ€· 1 βˆ’ 3 𝑐 2 πœƒ 𝑠 2 πœƒ ξ€Έ + π‘Ÿ 2 ξ€· 2 𝑐 2 πœƒ βˆ’ 1 ξ€Έ ξ€· 3 𝑠 4 πœƒ 𝑠 + 4 ξ€Έ ξ€Ύ 2 πœƒ πœ… . ( 4 . 1 2 ) From 𝑇 < 0 . 1 0 , we can estimate coupling ratio π‘Ÿ < 0 . 0 9 implying a lower limit for the 𝑍 β€² mass of about 0.8 TeV.

5. A Short Summary

To summarize, we have reviewed nonlinearly realized electroweak chiral Lagrangian for the gauge group 𝑆 π‘ˆ ( 2 ) 𝑅 Γ— 𝑆 π‘ˆ ( 2 ) 𝐿 Γ— π‘ˆ ( 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.

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