Terminology for "one-loop" in position space? So I know that in momentum space the term "one-loop" order means that we have on integral over undetermined momentum. How is the term used when referring to position space diagrams:


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*Integrals involving one integral of undetermined position? - seems unlikely since this would be the same as an expansion in the coupling constant.

*or simply those diagrams that would be "one-loop" if calculated in momentum space? - seems more likely but I can't find a source and I doubt my instinct here. 

 A: Short answer.
Your second option is correct.
Longer answer.
A Feynman diagram is nothing but a graph, in the sense of graph theory. It is mapped to $\mathbb C$ (or, more precisely, the ring of formal power series in the coupling constants) under the Feynman rules of the theory.
It is important to stress that these are two different levels. First, the graph itself, which knows nothing about the underlying theory (or physics at all), and second, its value under the Feynman rules. The number of independent loops is concerned about the first of these levels. This number depends only on the graph -- it is a purely graph-theoretic concept. In fact, it is one of the invariants of the graph (with respect to isomorphisms), and can be related to its size and the degree of its vertices (by means of some well-known topological relations).
In this sense, whether a graph is one-loop or not has nothing to do with the Feynman rules or physics at all; a mathematician would be able to tell you the answer while at the same time being oblivious to what a Feynman diagram is. That being said, it can be shown that the number of independent loops of a graph is precisely the number of momentum integrals that are not fixed by momentum conservation (by the aforementioned topological relations). Thus, your second option is correct. 
