It is well-known how one can translate a (physical) Feynman diagram into integrals of kind: $$I(p_1, \dots, p_n) = \idotsint \prod_{l=1}^{L} \frac{d^D k_l}{(2\pi)^D} \frac{\text{scalar products}}{\text{propagator factors}},$$ where the propagators are of kind $D(q, m) = q^2 - m^2 + i0$ for $q$ a linear combination of loop and external momenta. However, is the converse true, i.e. that an arbitrary Feynman integral can be depicted via Feynman diagrams?
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$\begingroup$ Well, what definition for "Feynman integral" do you use that's not "integral obtained from a Feynman diagram", so that this question isn't tautological? $\endgroup$– ACuriousMind ♦Aug 8, 2015 at 21:14
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$\begingroup$ I've seen many definitions. For example, in a book about Feynman integrals by Smirnov they are defined as integrals, whose integrand is a product of propagator-like factors, i.e. $E(\{k\},\{p\}, m) = \sum A_{ij} \, (q_i \cdot q_j) - m^2$, where $\{q\} = \{k\} \cup \{p\}$ denote all loop and external momenta (and $A_ij$ are parameters). $\endgroup$– NewbieAug 8, 2015 at 23:16
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$\begingroup$ Sorry for offtop, but could you maybe explain a little why this question is important to you? Because it is much like asking if any hat can be used as a boomerang. $\endgroup$– Prof. LegolasovAug 14, 2015 at 14:00
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1$\begingroup$ Mainly, because diagramatic Feynman integrals are better studied and have properties, which the general integrals may not have. For example, for the diagramatic ones is known that they admit Hopf algebra structure, as hinted in arxiv.org/abs/hep-th/0202110v3 or arxiv.org/abs/1504.00206 ... $\endgroup$– NewbieAug 14, 2015 at 15:12
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There are Feynman integrals that don't come from diagrams. These integrals occur naturally when trying to use tensor reduction to remove the numerator in some Feynman integrals. The generalization of a graph/diagram is called a Matroid and in Feynman integrals (as you described them) where the momentum flow in the denominators can not come from a graph, it can come from a matroid.
For more details, see:
Kreimer, Dirk, and Karen Yeats. "Tensor structure from scalar Feynman matroids." Physics Letters B 698.5 (2011): 443-450. [link]
