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We present a systematic analysis of next-to-next-to-leading-order diagrams that contribute to the pp → ppπ 0 production at threshold. Analytic expressions are given for the effective transition operators, and the relative importance of various types of diagrams is discussed. The vertex-correction-type graphs are found to give only small corrections to lower order graphs in conformity with expectations. By contrast, we find very large contributions from the two-pion exchange graphs which can be interpreted as a part of effective σ-meson exchange diagrams. a. Introduction High-precision measurements [1,2] of neutral pion production in proton-proton collisions pp → ppπ 0 just above the threshold have generated renewed theoretical scrutiny of this reaction [3]-[16]. This reaction is unique among two-nucleon pion production processes in that it is not well described by the single-nucleon process (Born term), Fig.1(a), and the s-wave pion rescattering process, Fig.1(b). The reason is that the " large " Weinberg-Tomozawa term does not contribute to the pp → ppπ 0 reaction, in contrast with e.g. charged-pion production pp → pnπ + , thus rendering pp → ppπ 0 particularly sensitive to and hence a perfect testing ground for the less-well-understood " small " isoscalar s-wave pion rescattering terms. From the calculations done thus far it is clear that the two most basic processes, Fig.1(a) and 1(b), give much smaller pp → ppπ 0 cross sections than the measured ones. Lee and Riska's model calculation [4] suggests that shorter range isoscalar meson-exchange processes, like the σ-and ω-exchanges between the two protons, might be very important for this reaction. In heavy-baryon chiral perturbation theory [HBχPT] [17] one can define the " large " and " small " terms in the Hamiltonian by way of chiral-order counting. 1 In this language the first pion rescattering contributions to pp → ppπ 0 are one chiral-order higher than those of the charged-pion production (e.g., pp → pnπ +). This means that pp → ppπ 0 production 1 Extension of HBχPT to cases with two or more nucleons is not unique, however: Cohen et al. [14] have proposed a modification of Weinberg's chiral counting rules. We do not address this issue here but simply use the standard counting rulesà la Weinberg [18].