# Feynman diagrams vs Feynman integrals?

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?

• 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? Aug 8, 2015 at 21:14
• 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). Aug 8, 2015 at 23:16
• 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. Aug 14, 2015 at 14:00
• 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 ... Aug 14, 2015 at 15:12