ar X iv : h ep - p h / 02 11 27 5 v 1 1 8 N ov 2 00 2 Meson Form Factors and the Quark - Level Linear σ Model
چکیده
The quark-level linear σ model (LσM) is employed to compute a variety of electromagnetic and weak observables of light mesons, including pion and kaon form factors and charge radii, charged-pion polarizabilities, semileptonic weak Kl3 decay, semileptonic weak radiative pion and kaon form factors, radiative decays of vector mesons, and nonleptonic weak K2π decay. The agreement of all these predicted observables with experiment is striking. In passing, the tight link between the LσM and vector-meson dominance is shown. Some conclusions are drawn on the LσM in connection with lattice and renormalization-group approaches to QCD. 1 Survey of LσM and chiral Goldberger–Treiman relations For the past eight years, there has been much experimental [1] and theoretical [2] activity, as well as combined workshops [3], concerning isoscalar scalar mesons in general, and the σ meson in particular. Very recently, we have employed electromagnetic (e.m.) and weak processes to conclude that the mostly nonstrange n̄n resonances are the f0(600) and the f0(1370), while the f0(980) and the f0(1500) are mainly s̄s [4]. In the present paper, we study meson (π,K) form factors in general, and specialize at a later stage to a specific scheme, namely the quark-level linear σ model (LσM). Nonperturbatively solving [5] the strong-interaction Nambu-type gap equations δfπ = fπ and δm̂ = m (where fπ is the pion decay constant and m̂ is the nonstrange constituent quark mass) in 1 quark-loop order, regularization schemes lead to the NJL [6] and Z = 0 compositeness [7] relations mσ = 2m̂ , g = 2π √ Nc , (1) with m̂ ∼ MN/3 and meson-quark coupling g = 2π/ √ 3 = 3.628. For a more detailed description of the quark-level LσM, see the Appendix. Here, we survey instead meson form factors and related data in a LσM context for strong, e.m., and weak interactions. This chiral LσM is based on the quark-level pion and kaon Goldberger–Treiman relations (GTRs) fπ g = m̂ = 1 2 (mu +md) , fK g = 1 2 (ms + m̂) , (2) for fπ ≈ 93 MeV (fπ ≈ 90 MeV in the chiral limit (CL) [8]), fK/fπ ≈ 1.22, and ms ≈ 1.44 m̂ (from Eq. (2)). We begin in Sec. 2 by studying meson vector form factors and their measured charge radii. In Sec. 3 we survey charged-pion polarizabilities for γγ → ππ, and compare the results with the LσM predictions. In Sec. 4 we study the semileptonic weak Kl3 decays and the form factor f+(k ) evaluated at k = 0. Then in Sec. 5 we examine the radiative semileptonic weak form factors for π → eνγ and K → eνγ decays, with the observed pion second axial-vector form factor implying a pion charge radius rπ ∼ 0.6 fm, also found in Sec. 2 from data [9] and from the theoretical LσM. In Sec. 6 we return to the LσM and its link with vector-meson dominance (VMD). Finally, in Sec. 7 we begin by studying the ∆I=1/2 rule for two-pion decays of the kaon in connection with the σ as the pion’s chiral partner, and end by showing that the mass of the now experimentally confirmed scalar κ meson is consistent with the observed K → 2π decay rate. We summarize our results and draw our conclusions in Sec. 8. 2 Meson vector form factors and charge radii The charged-pion and kaon e.m. vector currents are defined as