Implications of Sigma Models in the Standard Model and beyond * )

After a brief introduction to the sigma model in QCD, we discuss how the sigma model can be relevant in the Standard Model. It is shown to be useful in the analysis of weak processes, such as the study of ∆I = 1/2 rule, and in the analysis of the Higgs sector as well. The sigma model description is also shown to be quite useful in searching for the effects of New Physics via gauge boson self-interactions. §1. Implications of sigma model in the Standard Model 1.1. Relevance of sigma model in QCD The sigma model was originally developed as an effective low energy theory of QCD. Suppose we have just two massless flavours of quarks, (u, d) with m u = m d = 0. Then the QCD lagrangian has chiral symmetry SU(2) L ×SU(2) R , under independent unitary transformations of left-and right-handed quarks by the elements of SU(2), Actually, such global symmetry is spontaneously broken by the presence of the vacuum expectation values (VEV), < uu >=< dd >= O(Λ 3 QCD), where Λ QCD ∼ 200MeV is a unique mass scale in QCD and is comparable to the pion decay constant , f π ∼ 93MeV, roughly speaking. Though the " vacuum condensations " break chiral symmetry, there remains a vectorial (parity conserving) SU(2) V symmetry with U L = U R , as the condensations are symmetric under the rotation between u and d. Such symmetry is often called isospin symmetry. It is a general wisdom that when some symmetry is spontaneously broken there should appear some massless particles, called Nambu-Goldstone (N-G) bosons. The point which makes this statement so powerful is that the appearance of such massless particles is guaranteed just by the symmetry argument. Thus even though we know little about the non-perturbative aspects of QCD, such as the formation of bound states, the appearance of particles is of no doubt. Then a natural guess is that the N-G bosons should be identified with light hadrons with odd parity, i.e. pions. In reality, pions are *) Invited talk at the International Workshop on " Possible existence of the σ-meson and its implications to hadron physics " ,