O ct 1 99 2 Modelling the Off - Shell Dependence of π o − η Mixing with Quark Loops

It is shown that including form factors for the quark-pseudoscalar meson couplings of the Georgi-Manohar chiral quark model allows one to obtain the leading off-shell dependence of π − η mixing (as predicted by chiral perturbation theory) from the effect of quark loops on the meson propagators. Implications for ρ − ω mixing and for the effects on meson mixing contributions to few body charge symmetry breaking observables are also discussed. INTRODUCTION It is commonly held that the bulk of non-electromagnetic charge symmetry breaking (CSB) in few-nucleon systems is due to isoscalar-isovector mixing in the intermediate meson propagators of one-boson-exchange graphs. The dominant contributions of this type are associated with ρo−ω and, to a lesser extent, πo−η mixing [1-4]. Recently, however, it has been pointed out that some problems exist with the standard evaluations of these contributions. First, recall that the ρ −ω mixing matrix element is obtained from experimental data on ee → ππ in the ρ− ω interference region and therefore corresponds to q ≃ mω. The standard analysis assumes this matrix element is q-independent and, therefore, uses the experimental value unchanged in the NN scattering region, for which q < 0. Not only has the validity of this ansatz never been tested, but, in Ref. 5, using a model in which ρo−ω mixing is generated by an intermediate quark loop as a consequence of the inequality of up and down quark masses, Goldman, Henderson, and Thomas (GHT) raise the possibility of significant q-dependence of the ρ − ω mixing matrix element. Second, a recent evaluation [6] of πo− η mixing to one loop in chiral perturbation theory (ChPT), shows that 1) the mixing matrix element is, indeed, q-dependent (varying by ∼ 20% over a range from q ≃ mη to q ≃ −mη, comparable to that involved in the extrapolation of the ρ −ω matrix element from q ≃ mω to the scattering region) and 2) the magnitude of the mixing, even without this q-dependence, is less than obtained by the model analysis of Ref. 3c. The results of Ref. 6, being at one loop in ChPT, provide only the linear-in-q dependence of π − η mixing; the higher order dependence is unconstrained but will be small in the region of validity of the chiral expansion (|q2| < ∼ mη). Since the convergence of the chiral expansion to one loop in this region has been extensively tested [7,8], and the framework of ChPT is a rigorous consequence of QCD, the results of Ref. 6 should be considered very reliable (up to an overall ∼20% scale uncertainty associated with the treatment of electromagnetic corrections of pseudoscalar masses only to leading order in the chiral