^{3}.

We define the

One-loop Feynman integrals in generic quantum field theories are known to be polylogarithmic (see, e.g., Refs.

We define the geometric “rigidity” of a Feynman integral to be its degree of “nonpolylogarithmicity”. More concretely, after eliminating a maximal number of rational integrations, we imagine all the ways in which a Feynman integral can be expressed as a sum of polylogarithms of weight

In this Letter, we probe the limits of Feynman-integral rigidity. In particular, we prove that a large class of massless Feynman integrals in four dimensions have a rigidity bounded by

All of our results are described in terms of the Symanzik polynomial representation of Feynman integrals. We review this formalism momentarily and use this to motivate the notion of

The examples in which we are presently most interested are easiest to understand using the Symanzik polynomial formalism. (We refer the more interested reader to, e.g., Ref.

Note that

Before moving on, we should describe why we consider marginal integrals especially important for physics: They are capable of having

In a large and growing number of examples, the (nonpolylogarithmic part of the) geometry relevant to individual Feynman integrals has been found to be Calabi-Yau. Examples include the massive

The argument is fairly straightforward (and essentially the same as Brown describes in Ref.

Let us suppose that we have done the above, resulting in a form of the integral which is a polylogarithm integrated over a sum of rational forms whose denominators are each irreducibly quadratic or higher in all

As the sum of the weights in

In two dimensions, the only (tadpole-free) marginal integrals are the so-called “sunrise” (or “banana”) graphs depicted in Fig.

The marginal,

The Symanzik representation of the

Consider now a finite marginal

At this stage we may already conclude that the rigidity of these integrals is

It is worth clarifying the role of condition (ii) above. For an integral satisfying condition (i), performing the third integration results in an integral over the square root of the discriminant of the denominator of

Although criteria (i) and (ii) are fairly stringent, it turns out that many marginal, four-dimensional integrals saturate this bound in rigidity. Explicit two- through five-loop finite integrals with maximal rigidity (and one example of a three-loop integral with

Examples of (a) two-, (b) three-, (c) four-, and (d) five-loop marginal integrals with maximal rigidity; (e) a marginal integral with submaximal rigidity.

Massless, finite four-dimensional Feynman integrals with rigidity

Massless, finite four-dimensional Feynman integrals with rigidity

Notice that each of the infinite families of examples depends on a fixed number of external legs. In particular, those shown in Fig.

Massless, finite four-dimensional Feynman integrals with rigidity

At any fixed loop order and spacetime dimension (or, equivalently, fixed multiplicity), the scope of Feynman integral complexity is bounded by the finiteness in extent of the relevant loop integrands. In this Letter, we have identified several infinite classes of four-dimensional Feynman integrals—relevant to a wide range of quantum field theories—that involve more complicated geometries at each loop order than all previously known examples. Nevertheless, the extent of relevant geometries is

It is not clear whether the geometry of the (CY) manifold relevant to a given Feynman integral is unique—or if it depends on the order of integrations, for example. In the case which is best understood, the elliptic double box

Whether or not the geometry of the hypersurface is uniquely fixed by the graph, our analysis above shows that—at least for the maximally rigid examples discussed—each can be written as a weight-two polylogarithm integrated over some

Having proven an upper bound on rigidity for the class of marginal Feynman integrals, we conjecture that the same bound holds for all

It is natural to wonder if a similar statement to our bounded rigidity holds for higher (even) numbers of dimensions. That is, is the rigidity of any

We have seen that, in two and four dimensions, marginal Feynman integrals exist that saturate the rigidity bounds described above. Beyond one loop for

The two most rigid three-loop integrals in the planar limit: (a) the traintrack with rigidity 2 and (b) a diagram with rigidity 3.

Finally, we should note that the notion of geometric rigidity is rather coarse and in need of considerable refinement and elaboration. For example, an integral over an elliptic curve is clearly simpler than one over a higher-genus surface of the same dimension, and the product of two elliptic curves is clearly less complicated than a generic K3 surface. (Moreover, there are interesting distinctions to be drawn among integrals sharing the same underlying geometry; see, e.g., Refs.

We are grateful to Lara Anderson, Spencer Bloch, Claude Duhr, Paul Oehlmann, Erik Panzer, Pierre Vanhove, and Cristian Vergu for helpful discussions and to Pierre Vanhove for helpful comments on early versions of this draft. Finally, we thank Tristan Hübsch for his uniquely evocative textbook. This project has received funding from the Danish Independent Research Fund under Grant No. DFF-4002-00037 (M. W.), the European Union’s Horizon 2020 research and innovation program under grant agreement No. 793151 (M. v. H.), an ERC starting grant (No. 757978), and a grant from the Villum Fonden (J. L. B., A. J. M., M. v. H., and M. W.).