We use power-counting arguments as an organizing principle to apply chiral perturbation theory, including an explicit $Delta$, to the $p p rightarrow p p pi^0$ reaction near threshold. There are two lowest-order leading mechanisms expected to contribute to the amplitude with similar magnitudes: an impulse term, and a $Delta$-excitation mechanism. We examine formally sub-leading but potentially large mechanisms, including pion-rescattering and short-ranged contributions. We show that the pion-rescattering contribution is enhanced by off-shell effects and has a sign opposite to that of a recent estimate based on a PCAC pion interpolating field. Our result is that the impulse term interferes destructively with the pion rescattering and $Delta$-excitation terms. In addition, we have modeled the short-ranged interaction using $sigma$ and $omega$ exchange mechanisms. A recoil correction to the impulse approximation is small. The total amplitude obtained including all of these processes is found to yield cross sections substantially smaller than the measured ones.
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