Pseudovector components of the pion , π 0 → γγ , and F π ( q 2 )

As a consequence of dynamical chiral symmetry breaking the pion BetheSalpeter amplitude necessarily contains terms proportional to γ5 γ · P and γ5 γ · k, where k is the relative and P the total momentum of the constituents. These terms are essential for the preservation of low energy theorems, such as the Gell-Mann–Oakes-Renner relation and those describing anomalous decays of the pion, and to obtaining an electromagnetic pion form factor that falls as 1/q2 for large q2, up to calculable ln q2-corrections. In a simple model, which correlates lowand high-energy pion observables, we find qFπ(q 2) ∼ 0.12 0.19GeV2 for q2 ∼ 10GeV . Pacs Numbers: 13.40.Gp, 14.40.Aq, 12.38.Lg, 24.85.+p Typeset using REVTEX 1 I. PION AS A BOUND STATE Understanding the pion is a key problem in strong interaction physics. As the lowest mass excitation in the strong interaction spectrum it must provide the long-range attraction in N -N potentials [1]. In QCD, it is a quark-antiquark bound state whose lowand highenergy properties should be understandable in terms of its internal structure, and it is also that nearly-massless, collective excitation which is the realisation of the Goldstone mode associated with dynamical chiral symmetry breaking (DCSB). An explanation of these properties requires a melding of the study of the many body aspects of the QCD vacuum with the analysis of two body bound states. The Dyson-Schwinger equations (DSEs) provide a single, Poincaré invariant framework that is well suited to this problem. The DSEs are a system of coupled integral equations and truncations are employed to define a tractable problem. In truncating the system it is straightforward to preserve the global symmetries of a gauge field theory [2] and, although preserving the local symmetry is more difficult, progress is being made [3]. The approach has been applied extensively [4] to the study of confinement, and to DCSB where the similarity between the ground state of QCD and that of a superconductor can be exploited, with the QCD gap equation realised as the quark DSE. It has also been employed in studying meson-meson and meson-photon interactions [5], heavy meson decays [6], QCD at finite temperature and density [7], and strong interaction contributions to weak interaction phenomena [8,9]. Studying the pion as a bound state requires an understanding of its (fully-amputated) Bethe-Salpeter amplitude, which has the general form Γπ(k;P ) = τ γ5 [ iEπ(k;P ) + γ · PFπ(k;P ) (1) + γ · k k · P Gπ(k;P ) + σμν kμPν Hπ(k;P ) ] , where {τ }j=1...3 are the Pauli matrices. Γπ satisfies the renormalised, homogeneous BetheSalpeter equation