Geometric approach to asymptotic expansion of Feynman integrals

We present an algorithm that reveals relevant contributions in non-threshold-type asymptotic expansion of Feynman integrals about a small parameter. It is shown that the problem reduces to finding a convex hull of a set of points in a multidimensional vector space.

A specific integral is determined by the exponents a 1 and a 2 and depends on the two parameters, m 2 and p 2 . The structure of the expansion does not depend on a 1 and a 2 and we will not mention those exponents in the following discussion.
We consider the asymptotics of I 1 (a 1 , a 2 ; p 2 , m 2 ) in the limit when |p 2 | ≫ m 2 , or ρ = |m 2 /p 2 | ≪ 1. The naive Taylor expansion does not capture the complete asymptotic behaviour since the integration variables (components of k) span all values from −∞ to +∞, and in particular can be as small as m or as large as |p 2 |.
The prescription in this case is to find regions, or scalings of momentum components that after the expansion provide non-zero contributions. In each region, we first Taylor expand the integrand and drop the scaling restrictions. In our example, there are three non-zero regions, summarized in Tab. I. For example, in the region (c) one expands D 2 as follows: In the last line, we dropped the restriction on k and the problem reduced to evaluation of (multiple) Feynman integrals with simpler denominator factors, various denominator exponents and possibly more complex numerators. The non-trivial statement is that the double-counting which could have been introduced disappears in the sum of all regions. For the purpose of the following discussion we assume that the tensor reduction of numerators and the proliferation of terms can be managed; we will focus on the transformations of denominator factors in every region (e.g. (k + p) 2 + m 2 → p 2 in the example above). Some types of asymptotic expansion may require more elaborate choice of regions. For example, k 0 may scale differently from k i or some combination of components may have a separate scale (this happens, e.g., in Sudakov limits). It is thus desirable to have an explicitly covariant formalism to identify regions, independent of the frame choice and the routing of momenta. For that purpose, we may switch to the alpha-representation of integrals. We re-write an integral with n lines (denominator factors) over D-dimensional loop momenta as an integral over n positive parameters x 1 , .., x n . Information about the graph is then encoded in the two homogeneous polynomials, U and F . For example, our integral above is The expansion by regions may also be formulated in the alpha-representation [2,10]. Instead of finding the scaling behaviour of loop momentum components, here we deal with the scaling of each parameter x i that directly corresponds to the scale of the i-th line (denominator factor) of the original integral. During the expansion, only the leading terms remain in the polynomials U and F , and the resulting alpha-representation represents the integrals obtained by the expansion in the momentum space. The last column of Tab. I demonstrates the scaling of alpha-parameters and the polynomials corresponding to each region.
Note that in the language of alpha-parameters, the difference between the threshold-type and non-threshold expansion becomes clear. Let us consider integral of Eq. 1 in the threshold limit y = m 2 + p 2 4 ≪ m 2 (that, of course, implies that p 2 < 0, i.e. this limit is essentially non-Euclidean). Choosing the frame where p = (p 0 , 0) and re-routing the loop momentum, we obtain the denominator factors D 1 = k 2 0 + k 2 + k 0 p 0 + y and D 2 = k 2 0 + k 2 − k 0 p 0 + y. This integral has two non-vanishing regions. The first "hard" region is characterized by k ∼ m. The second "potential" region corresponds to k 0 ∼ y/m, | k| ∼ √ y. In the language of alpha-parameters, in the "hard" region only the second term survives in the polynomial F = y( do not know a general rule to identify such surfaces and find substitutions revealing such regions. Instead, we focus on the "usual" regions that can be determined by examining independently the monomials in U and F . However limited, this problem is still important for many applications.

III. GENERAL FORMALISM
We consider an l-loop Feynman integral which depends on n exponents a 1 , ..., a n , scalar products of e external momenta p 1 , ..., p e and parameters (such as masses) in denominator factors D i . The latter must be quadratic in momenta but other than that may have any form, e.g. correspond to a massive, such as −(k i + p j ) 2 + m 2 k − i0, or a static propagator, such as (−2k i p j ± i0). The alpha-representation for this integral has a general structure where coefficient c and exponents a and b depend only on l, D, and a i . U and F are homogeneous polynomials (of order l and l + 1, respectively) of integration variables x i , and F also depends on the kinematic invariants. If the denominators D i correspond to some graph and have a standard form −k 2 + m 2 − i0 (in Minkowski space), then the functions U and F can be read off the graph in terms of trees and 2-trees [13]. In a more general case, one may obtain U and F with a tool found at http://www-ttp.particle.uni-karlsruhe.de/~asmirnov/Tools-UF.htm. In what follows, we will only discuss the properties of U and F that are independent of specific indices a i . In dimensional regularization, "scaleless" integrals (having no inherent scale) turn to zero. More specifically, an integral is scaleless if it is possible to re-scale some loop momenta or their components so that the result remains proportional to the original integral, or In particular, massless vacuum bubbles vanish: In terms of the alpha-representation Eq. 5, a similar statement applies to homogeneity of U and F with respect to a subset {B} of integration variables x i ({B} should not coincide with the full set of In order to avoid separate treatment of U and F , one may consider the product UF that incorporates the scaling and asymptotic properties of both factors (but may contain many terms).

IV. GEOMETRIC INTERPRETATION OF ASYMPTOTIC EXPANSION
Let us start with some integral in the alpha-representation Eq. 5 with integration variables x 1 , ..., x n and a small expansion parameter ρ. Each of M terms in F corresponds to a vector of n + 1 exponents (we here neglect common factors and numeric coefficients, irrelevant to the non-threshold expansion): and F corresponds to a set {F } of M points in (n + 1)-dimensional vector space. Due to homogeneity of F , all these points belong to an n-dimensional hyperplane r 1 + ... + r n = l + 1, parallel to the 0-th axis (the axis of r 0 ). Terms of U have no explicit powers of ρ in the coefficients. The corresponding set {U } is thus confined to an (n − 2)-dimensional hyperplane r 0 = 0, r 1 + ... + r n = l. In Fig. 1 we present such points corresponding to the example in Eq. 1, where the three terms of {F } are denoted with crossed points and the two terms of {U } with diamonds.
If we fix the scales of alpha-parameters as x i ∼ ρ vi , then the scale of a monomial is ρ r0 x r1 1 ...x rn n ∼ ρ r0+v1r1+...+rnvn ∼ ρ r v with r = (r 0 , ..., r n ) from {F } and v = (1, v 1 , ..., v n ). Graphically, r v represents the length of a projection of the vector r on the direction v.
Some special choices of directions v determine the regions of expansion that we seek. The terms in F that remain after the expansion are all characterized by the same scale in powers of ρ. All points of the corresponding subset {F ′ } then feature the same value of the projection on v, i.e. these points belong to the hyperplane orthogonal to v. The points corresponding to the neglected terms will be located "above" this hyperplane (since v always points "up" with respect to the 0-th axis). In other words, {F ′ } belong to a facet of the envelope, or the "convex hull" of the set {F }, while the corresponding v is the normal vector to that facet. In a similar manner we may define a subset {U ′ } of the remaining terms in U.
Relating the three expansion regions of Tab. I to graphics in Fig. 1, we find the corresponding points and vectors (points as denoted in the figure): Here we exploit the freedom to re-scale all x i by the same power of ρ, i.e. shift v by any vector a = (0, A, ..., A). If v corresponds to a region, then v ′ = v + a determines the same region. For example, v ′ = (1, 2, 0) also corresponds to the region (c) above. It is convenient to choose v parallel to the plane where points {F } are confined, i.e. orthogonal to the vector (0, 1, ..., 1).
Normally, only a few scaling choices produce non-zero regions. In our example, the choice x 1 ∼ ρ 2 , x 2 ∼ ρ 0 , leading to U = x 2 , F = x 2 2 m 2 , or {F ′ } = (A), {U ′ } = (D), corresponds to a scaleless integral. As discussed above, this implies an existence of a scaling leaving both U and F invariant up to a pre-factor (in this case, x 1 → ax 1 ).
The requirement that a region does not vanish can be easily formulated in the geometrical language. Consider the polynomial UF and the corresponding set of points {U F }. After the expansion with the chosen scalings, we are left with its subset {U F ′ }. Numeric coefficients and kinematic invariants are irrelevant to the scalefulness of the region, and we get rid of them by projecting {U F ′ } on the plane r 0 = 0. The thus obtained set of points {U F ′ 0 } belongs to the (n − 1)-dimensional hyperplane.
Scalelessness implies that all terms of the polynomial UF ′ are homogeneous with respect to a certain re-scaling. We thus deduce that the points of {U F ′ 0 } must then belong to an orthogonal space of the corresponding re-scaling vector v h . In other words, {U F ′ 0 } is confined to at most (n − 2)-dimensional subspace, and its (n − 1)-dimensional volume is zero. The latter property can be easily checked (and used to check whether a given integral vanishes). However, it is easy to see that the "bottom" facets of the convex hull for {U F } automatically correspond to scaleful regions: their dimension is (n − 1) by construction (otherwise they become "ridges" or "vertices"), and they (by selection) are not orthogonal to the plane r 0 = 0 (thus the projection has non-zero volume).
Finally, we may formulate the general procedure to determine the expansion regions. We start by building the set of points {U F }. Next, we find the n-dimensional convex hull C of the set {U F } in the n-dimensional plane r 1 + ... + r n = l + 1, using any preferred algorithm. The implementation that we chose, QHull [14], does not allow building hulls of dimensionality lower than the dimension of vector space. Thus, one has to introduce local coordinate system and deal with non-integer coordinate values. However, it is also possible to project {U F } along any of axes r i , i = 0, e.g. consider (n − 1)-dimensional points r = (r 0 , r 1 , ..., r n−1 ). Convex hull C ′ built for this projection will be the projection of the "true" convex hull C. Its dimensions will be stretched but the correspondence of the points to the facets and the vertices will persist. From the (n − 1)-dimensional facets of C we then select the "bottom", i.e. facets with normal vectors v pointing "up", with v 0 > 0. For each of those "bottom" facets, we choose the normal vector v such that v 0 = 1. Its components 1 to n represent the relative scales of alpha-parameters x i and thus uniquely determine an expansion region.

V. LESS TRIVIAL EXAMPLE
Let us consider the integral in Fig. 2, this time defined in the Minkowski space (this example was first considered in [2] and [10], Chapter 10): With S = m 2 + M 2 − s = −2p 1 p 2 , its alpha-representation reads: . Since U is scaleful, we also have to add the "hard" region: v 0 = (1, 0, 0, 0, 0, 0, 0). (The latter would appear automatically, had we used the product UF .) For illustration, let us consider the most non-trivial "ultrasoft-collinear" region corresponding to v 1 , or the scaling of alpha-parameters x 1 ∼ m 0 , x 2 ∼ 1/m 2 , x 3 ∼ 1/m 2 , x 4 ∼ 1/m 2 , x 5 ∼ 1/m 2 , and x 6 ∼ 1/m 4 . First, we may check that v 1 is indeed orthogonal to the plane containing the points 5, 6, 15, 22, 23, 24, and 25 from {F } (in the order as in Eq. 9). Those points correspond to the terms remaining in F ′ after the expansion: M 2 x 2 2 x 3 + M 2 x 2 2 x 5 + . . .. In the momentum space, the interpretation becomes clear only in the special reference frame, where p 1 = (M, 0, 0), n − , and n ± = (1/2, 0, ∓1/2). We also decompose the first loop momentum in plus-and minusparts, k 1 = (k + + k − , k, k + − k − ). To reproduce the "ultrasoft-collinear" region, we should prescribe the following scales to the components of loop momenta: After the expansion, the denominator factors scale as: One can easily see how the powers of m here correspond to the components of − v 1 .

VI. IMPLEMENTATION
We wrote a Mathematica program that determines the expansion regions of a given Feynman integral based on the procedure described above. The general problem of building a convex hull of M points in n dimensions is well-known in computational geometry; we employ the algorithm quickhull [14] that has complexity O M ⌊d/2⌋ . It is sufficient when the number of lines is not too large; for example, finding 11 expansion regions of a 4-loop integral with 10 lines takes about 10 seconds on a laptop PC.
The output is a list of vectors specifying the scales of the alpha-parameters factors, or the non-zero components of vectors v i .

VII. CONCLUSION
We present an algorithm to find the relevant regions of expansion for a Feynman integral in a given limit of momenta and masses. The algorithm is implemented in Wolfram Mathematica language and uses open-source package QHull. The program and examples can be downloaded from our web-page. In the future, we plan to extend the code in order to apply it to more general parametric integrals.