1PR

1PR stands for one-particle-reducible, a term used in quantum field theory and related areas to classify Feynman diagrams by their reducibility. A diagram is 1PR if it can be separated into two disconnected pieces by cutting a single internal propagator line. In other words, the diagram represents a process in which a single intermediate propagator links two sub-processes.

This concept is used to distinguish 1PR diagrams from one-particle-irreducible (1PI) diagrams, which cannot be split by removing just one internal line. The distinction helps organize perturbative expansions and underpins the construction of the Dyson equation and the 1PI effective action. In perturbation theory, 1PR diagrams are generated by successive insertions of the self-energy into propagators and are not considered primitive building blocks of the theory.

In practical calculations, contributions to propagators and vertex functions are often separated into 1PI parts, which define the proper vertices, and 1PR parts, which are built from these 1PI pieces by dressing the propagators. Through Dyson’s equation, the full propagator sums the 1PR insertions, effectively resumming an infinite subset of diagrams.

The 1PR classification is widely used in high-energy physics, statistical field theory, and many-body physics, wherever diagrammatic perturbation theory is employed. It provides a structural language for understanding how complex processes decompose into simpler sub-processes and how different classes of diagrams contribute to physical quantities.