Supersymmetry and the Minimal Supersymmetric Standard Model
One reason that physicists explore supersymmetry (SUSY) is because it offers an extension to the more familiar space–time symmetries of quantum field theory. These symmetries are grouped into the Poincaré group and internal symmetries. The Coleman–Mandula
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Supersymmetry and the Minimal Supersymmetric Standard Model
3.1 Why Supersymmetry? 1. One reason that physicists explore supersymmetry (SUSY) is because it offers an extension to the more familiar space–time symmetries of quantum field theory. These symmetries are grouped into the Poincaré group and internal symmetries. The Coleman–Mandula theorem (Coleman and Mandula 1967) showed that under certain assumptions, the symmetries of the S-matrix must be a direct product of the Poincaré group with a compact internal symmetry group or, if there is no mass gap, the conformal group with a compact internal symmetry group. In 1975, the Haag–Lopuszanski–Sohnius theorem (Haag et al. 1975) showed that considering symmetry generators that satisfy anticommutation relations allows for such nontrivial extensions of space–time symmetry. This extension of the Coleman–Mandula theorem prompted some physicists to study this wider class of theories. 2. One of the main motivations for SUSY comes from the quadratically divergent contributions to the Higgs mass squared. The quantum mechanical interactions of the Higgs boson causes a large renormalization of the Higgs mass and, unless there is an accidental cancelation or fine tuning, the natural size of the Higgs mass is the highest scale possible. However, taking all precision measurements together in a global fit, the current experiment infers that the Standard Model Higgs boson mass must be lighter than around 200 GeV (95% c.l.) (2006). This problem is known as the hierarchy problem (Weinberg 1976; Gildener 1976; Susskind 1979; t Hooft 1979). Consider a massive fermion loop correction to the propagator for the Higgs field, as shown in Fig. 3.1. If the Higgs field couples to a Dirac fermion f with a term in the Lagrangian kf Hf f , the leading correction is then given by
B. Liu, Muonium–Antimuonium Oscillations in an Extended Minimal Supersymmetric Standard Model, Springer Theses, DOI: 10.1007/978-1-4419-8330-5_3, Ó Springer Science+Business Media, LLC 2011
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3 Supersymmetry and the Minimal Supersymmetric Standard Model
Fig. 3.1 Fermion loop contribution to the selfenergy of the Higgs boson
Dm2H ¼
jkf j2 2 ½K 2m2f lnðKUV =mf Þ þ 8p2 UV
ð3:1Þ
Here, KUV is an ultraviolet momentum cutoff used to regulate the loop integral. If the cutoff, KUV , is replaced by the Planck mass, Mplanck , the resulting correction would be some thirty degrees of magnitude larger than the experimental bound on the Higgs. Supersymmetry provides automatic cancelations between fermionic and bosonic Higgs interactions. For example, consider a scalar field S, which couples to Higgs with a Lagrangian term kS jHj2 jSj2 . Then the Feynman graph in Fig. 3.2 gives a correction Dm2H ¼
kS ½K2 2m2S lnðKUV =ms Þ þ 16p2 UV
ð3:2Þ
If each of the quarks and leptons of the Standard Model is accompanied by two complex scalars with kS ¼ jkf j2 , then the KUV contributions of Figs. 3.1, 3.2 will cancel. Supersymmetry relates the fermion and boson couplings in just this manner. Each Standard Mod
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