Abstract
1989 marks an important year for high energy community.In August of that year, LEP at CERN started taking unprecedentedly precise data at Z-polewhich can probe the electroweak quantum correction.The accuracy of $Z$- mass measurement achieved to $0.1\%$ at that year.About the same time, SLC and the Mark II detector were switched on at SLAC, and FNALbegan the precision studies of the W mass.%The experiments which measure SM observables, the accuracy often reach $0.1\%$ and sometimesmuch better, usually be called precision measurements.Through measuring the EW loop quantum correction to some observables, people knew top massshould range in $150-190$GeV before the discovery of top . The estimate was not far fromthe observed value. That's an impressive success of SM.The precision measurements are not only used to test SM but also have been used to probethe effects of physics beyond SM.%In Chapter 2, firstly, I want to give some examples to show how the renormalization(RG)program of SM works and how the quantum correction modifies the observables. But the subjectis vest and well-known, I would only provide some `flavor' of it.Secondly, I would like to emphasize that there are many ways to do RG and the observablesare different from scheme to scheme. One has to be careful when comparing the theoreticalpredictions and experiment data.Thirdly, I would like to include the projects we have done[4,5] which was motivated bythe deviation($\sim 3\sigma$) of $Zb\bar{b}$ coupling between the SM prediction and precisionmeasurements.%Following the line, in Chapter 3, I present the project[8] we were doing whilecomposing this note. We calculated the 2-loop Barr-Zee type diagrams\footnote{\prl 65 21 1990 . }involving exotic Higgs sector contributing to muon $(g-2)$.The electron $g-2$ is the most accurate experiment, to $10^{-12}$, human being has ever done.But the quantum loop effects, except the photonic loops, are severely suppressed by thetiny mass of electron. So it serves mainly as a rigid proof of QED but left very littleinformation of EW quantum correction let alone the new physics.In the case of muon, since the mass of muon is about 200 times bigger then electron's,the EW quantum correction, although small, must be included. With the on going improvementof experiment E821 at BNL\footnote{ see for example, B. Lee Roberts, hep-ex/0002005 (2000).}, which was planned to achieve the accuracy of $10^{-10}$, we arevery likely to test the 2-loop EW quantum correction. If lucky, we can also see the effectsof new physics. On the other hand, by comparing with muon $g-2$ experiment one can putstringent limit on physics beyond SM.%%%% Discuss why the Barr-Zee diagram is important.Before the project on muon $(g-2)$, we had applied the same technique to study the 2-loop EDM ofelectron and neutron due to the charged Higgs[6]( Chapter 4) and constraint on R-parityviolation parameter by neutron EDM[7](Chapter 5).%In general, the two loop calculation is very tedious and involving too many diagrams.And two loop diagrams involving Higgs are most suppressed by at least two powers of light fermionmass. The only 2-loop diagram involving Higgs sector which escape the light fermion masssuppression is Barr-Zee type diagram. For having a glance at how it looks like, turn toChapter 3.Also we have a trick to do it. In short, the leading one-loop effective vertex ofvector-vector-scalar or vector-vector-pseudoscalar coupling can be expressed as angauge invariant form. Say, let the Lorentz index of the two vector be $\mu$ and $\nu$,the momentums they carry are $k$ and $q$, then the effective vertex must be in the form of$S[k^\nu q^\mu-k\cdot q g^{\mu\nu}]+i\epsilon^{\mu\nu\alpha\beta}k_\alpha q_\beta P$, wherethe $S$ and $P$ are functions of $k$ and $q$.%%Basically, the calculations of $(g-2)$ and EDM are the same but just being applied to differenttopics. I will only present the detail of the calculations in the Appendix of Chapter 3 which dealswith muon $(g-2)$.The other two projects on EDM will be presented like a journal paper, in facts, I just copy andpaste from our paper version with little modification. The needed information and somecalculation details are arranged in the Appendix. The other technical points are scattered inthe explanation boxes in the text.