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The form factor for γπ + → π + π 0 was calculated in a simple–minded constituent model with a constant quark mass parameter, as well as in the Schwinger-Dyson approach. The comparative discussion of these and various other theoretical results on this anomalous process, as well as the scarce already available data (hopefully to be supplemented by more accurate CEBAF data soon), seem to favor Schwinger– Dyson modeling which would yield relatively small low–momentum values of the constituent (dynamically dressed) quark mass function. The form factor for the anomalous process γ(q) → π + (p 1)π 0 (p 2)π − (p 3) was calculated as the quark " box "-amplitude in two related approaches 1,2. In our Ref. 1 , the intermediate fermion " box " loop is the one of constituent quarks with the constant quark mass parameter M. The predictions of this quark loop model 1 are given in Fig. 1 by the long-dashed curve for M = 330 MeV, by the line of empty boxes for M = 400 MeV, and by the line of crosses for the large value M = 580 MeV. (In the lowest order in pion interactions, they are also the form factors of the σ-model and of the chiral quark model.) Our second Ref. 2 employs the Schwinger-Dyson (SD) approach 3 , where the box loop amplitude is evaluated with the dressed quark propagator S(k) = 1 ik / A(k 2) + m + B(k 2) ≡ Z(k 2) i k / + M(k 2) (1) containing the momentum-dependent, mostly dynamically generated quark mass function M(k 2), while m is the small explicit chiral symmetry breaking. In Fig. 1, the solid curve gives our γ3π form factor obtained in the SD approach for the empirical pion mass, m π = 138.5 MeV, while the dashed curve gives it in the chiral limit, m π = 0 = m. To understand the relationship between the predictions of these two approaches, one should, besides the curves in Fig. 1, compare also the analytic expressions we derived for the form factors [esp. Eqs. (20)–(21) in Ref. 2 and analogous formulas in Ref. 1 ]. This way, one can see, first, why the constant, momentum-independent term is smaller in the SD case, causing the downward shift of the SD form factors with respect to those 1

THE γ → 3π FORM FACTOR AS A CONSTRAINT ON SCHWINGER–DYSON MODELING OF LIGHT QUARKS