臺灣博碩士論文加值系統

為了實現兩步電弱相變，此碩論探討了將標準模型擴增一個實數單態粒子。我們利用許多不同方策(scheme)去探討且量化模型中的方策以及規範依賴。在考慮第一階圈圖計算時，on-shell (OS)-like方策中的Nambu-Goldstone 波色子需要被重求和以避免紅外發散，而我們量化其重求和後對電弱相變的影響。在OS-like以及MS-bar方策中,兩者所計算的電弱相變之臨界溫度相當一致。在規範依賴的探討中，採用High-temperature以及Patel-Ramsey-Musolf方策來做比較。在某些方策中，分析出的結果對重整化能量尺度有依賴性，此顯示了高階修正是必須的。但無論是對規範有依賴或無依賴的方策，最終資料分析顯示，兩者都在理論誤差以內。

In this thesis, the standard model is extended with a real singlet scalar $S$ to achieve a two-step electroweak phase transition (EWPT). The model is investigated with several schemes to quantify the scheme dependence and the gauge dependence issue. In on-shell(OS)-like scheme, at the one-loop order, Nambu-Goldstone boson contributions are needed to be resummed to circumvent the IR divergence; their effects in the EWPT are studied and quantified. The critical temperatures and critical vacuum expectation values of the EWPT in the OS-like and the MS-bar schemes are highly consistent to each other; we also compare the results with two gauge-independent schemes (the high temperature and the Patel-Ramsey-Musolf schemes). Even though higher order corrections are needed for scale-dependent schemes, the general trend of the results are consistent and the analyses show the differences of gauge-dependent and -independent schemes are within theoretical uncertainties.

口試委員會審定書 . . . . . . . . . . . . . . . . . . . . . . ii
誌謝 . . . . . . . . . . . . . . . . . . . . . . iv
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . v
摘要 . . . . . . . . . . . . . . . . . . . . . . vi
Abstract . . . . . . . . . . . . . . . . . . . . . . vii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . 1
2 Model: SM + Real Singlet Scalar . . . . . . . . . . . . . . . . . . . . . . 8
2.1 On-shell-like Scheme (OS-like) . . . . . . . . . . . . . . . . . . . . . . 10
2.2 MS Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Thermal History and Thermal Potential . . . . . . . . . . . . . . . . . . 13
2.3.1 Standard method of searching T C and critical VEVs . . . . . . . 14
2.4 High Temperature (HT) . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Patel-Ramsey-Musolf (PRM) Scheme . . . . . . . . . . . . . . . . . . . 15
2.5.1 Gauge-independent T C and VEVs . . . . . . . . . . . . . . . . . 16
3 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . 18
3.1 Critical Temperature and Critical VEV . . . . . . . . . . . . . . . . . . . 19
3.2 (non)Thermal Gauge and NG Boson Contribution . . . . . . . . . . . . . 20
3.3 Scheme Dependence Comparison . . . . . . . . . . . . . . . . . . . . . 22
3.4 Scale Dependence of T C . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . 26
4.1 Discussion: Dark Matter, Vacuum Stability and Perturbativity . . . . . . 26
4.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
A Generating Functional of 1(not 1)-PI . . . . . . . . . . . . . . . . . . . . . . 28
B Effective Potential in One-Loop . . . . . . . . . . . . . . . . . . . . . . 31
C Approximate Thermal Function . . . . . . . . . . . . . . . . . . . . . . 36
C.1 Boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
C.1.1 HTEB (a < 0:35) . . . . . . . . . . . . . . . . . . . . . . . . . . 37
C.1.2 PFFB (0:35 a 9:0) . . . . . . . . . . . . . . . . . . . . . . . 37
C.1.3 LTEB (a > 9:0) . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
C.1.4 Bessel Approximation for a 2 i ℜ . . . . . . . . . . . . . . . . . 38
C.2 Fermion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
C.2.1 HTEF (a < 0:32) . . . . . . . . . . . . . . . . . . . . . . . . . . 39
C.2.2 PFFF (0:32 a 9:0) . . . . . . . . . . . . . . . . . . . . . . . 39
C.2.3 LTEF (a > 9:0) . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
C.2.4 a 2 i ℜ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
D Field-Dependent Mass . . . . . . . . . . . . . . . . . . . . . . 41
D.1 Higgs Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
D.2 NG Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
D.3 Gauge Boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
D.4 Top Quark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
E An Illustrative Application PRM . . . . . . . . . . . . . . . . . . . . . . 45
E.1 Gauge-invariant T C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Bibliography . . . . . . . . . . . . . . . . . . . . . . 54

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