Geometric Analysis of Feynman-Parameter Integrals

• Geometric framework in Feynman-parameter space is a method that maps integrals onto geometric objects like polytopes to systematically identify dominant scaling regions.
• It employs convex hull construction of exponent vectors in an (n+1)-dimensional space to automate the classification and asymptotic expansion of multi-loop integrals.
• The approach offers significant advantages in automation, frame independence, and exclusion of scaleless regions, thus streamlining comparisons with traditional momentum-space methods.

A geometric framework in Feynman-parameter space refers to a suite of methods that express, analyze, and systematically manipulate Feynman integrals by mapping analytic, combinatorial, or algebraic properties of quantum field theory amplitudes onto geometric objects—typically polytopes, hyperplanes, or convex hulls—in the alpha (Feynman-parameter) representation. Central to this approach is relating the structure of the Feynman-parameter integrand (and its singularities, asymptotics, or regions of dominance) with concrete objects in a multidimensional parametric space, providing a powerful lens for the automation, classification, and computation of multi-loop integrals.

1. Representation of Feynman Integrals in Alpha Space

The geometric framework operates on the alpha (or Feynman-parameter) representation of loop integrals. For an $\ell$-loop, $n$-denominator scalar integral, the standard representation is

$$I(a_1, ..., a_n) = c \int_{0}^{1} dx_1 ... dx_n\ \delta\left(1 - \sum_{k} x_k\right) x_1^{a_1-1} ... x_n^{a_n-1} \mathcal{U}^a\, \mathcal{F}^b$$

where $\mathcal{U}$ and $\mathcal{F}$ are graph-dependent homogeneous polynomials (the first and second Symanzik polynomials). These polynomials encode combinatorial and kinematic information about the graph: $\mathcal{U}$ depends on the spanning tree structure and $\mathcal{F}$ on kinematics (external momenta, masses).

Every monomial in $\mathcal{F}$ or $\mathcal{U}$ is indexed by a vector of exponents:

$\rho^{r_0} x_1^{r_1} x_2^{r_2} \cdots x_n^{r_n}$

with $r_0$ the exponent of the expansion parameter $\rho$ and $r_i$ corresponding to $x_i$.

2. Convex Hulls and the Asymptotic Expansion

The crucial geometric insight of (Pak et al., 2010) is that the relevant contributions to a non-threshold-type asymptotic expansion about a small parameter $\rho$ can be characterized by constructing the convex hull of the exponent vectors of the monomials in $\mathcal{U}$ and $\mathcal{F}$ in an $(n+1)$-dimensional real vector space. All monomials map to points $(r_0, r_1, ..., r_n)$ which, due to the homogeneity constraints, lie on an $n$-dimensional affine hyperplane.

The scaling transform $x_i \sim \rho^{v_i}$ projects the contribution of a term onto the scalar direction $(1, v_1, ..., v_n)$: $\rho^{r_0 + v_1 r_1 + ... + v_n r_n}$ Defining $v = (1, v_1, ..., v_n)$, the dominance of a set of terms (or region) corresponds to the set of points (monomials) whose exponents project maximally in a given direction $v$.

The algorithm consists of:

• Enumerating all exponent vectors for monomials in $\mathcal{U}$ and $\mathcal{F}$.
• Constructing their convex hull in $(n+1)$-dimensional space (using e.g., QuickHull).
• Identifying “bottom” facets: facets whose normal vectors have positive $v_0$ when normalized so the first (the $\rho$ direction) equals one.
• Each such facet’s normal defines a unique scaling regime of parameters $x_i \sim \rho^{v_i}$ contributing to the expansion. Points on a given facet correspond to monomials that survive in that region.

These facets geometrically encode the interplay and balance between the parameters in the leading behavior, separating physical (non-scaleless) regions contributing to the expansion from scaleless (vanishing or power-suppressed) ones.

3. Automation, Covariance, and Comparison with Traditional Region Expansion

The geometric approach provides several practical advantages:

• Automation: The convex hull reduction turns the identification of leading regions into systematic computational geometry, making it possible to automate region-finding for high-multiplicity, multi-loop integrals.
• Frame Independence: The method operates solely on the alpha-representation; it is covariant and independent of momentum routing or frame choice.
• Exclusion of Scaleless Regions: Scaleless (vanishing) expansions manifest as facets that do not correspond to any physical region.
• Comparison with Momentum-Space Regions: While traditional expansion-by-regions in momentum space is labor-intensive and frame-dependent, the geometric approach systematically organizes all contributions, matching and supplementing conventional region analysis.

4. Case Studies and Performance

Two key examples from (Pak et al., 2010):

• One-loop propagator: Three contributing regions (hard, soft, and massless) are identified both in alpha and momentum space. The scaling of $x_1, x_2$ with respect to $\rho=|m^2/p^2|$ arises from the normals to the bottom facets.
• Two-loop vertex integral: For an integral with six denominators, twenty-five points are mapped to $\mathbb{R}^7$. The convex hull yields four bottom facets for $\mathcal{F}$, plus an additional “hard” region from $\mathcal{U}$. The resultant regions match the known momentum regions (hard, collinear, ultrasoft, etc.) and new scaling possibilities are recovered straightforwardly.

The method has been implemented in Mathematica using the open-source QHull library for convex hull computation. Complexity scales approximately as $\mathcal{O}(M^{\lfloor d/2\rfloor})$ for $M$ points in $d$ dimensions, limiting the approach for diagrams with a very large number of lines without further algorithmic optimization.

5. Limitations and Prospects

Several limitations and open directions are highlighted:

• Threshold-type Expansions: The method, as presented, does not handle threshold expansions (where nontrivial cancellations on surfaces occur), and special substitutions or frame choices may still be required.
• Parametric Generalizations: The algorithm assumes conventional Feynman integrals in the alpha-representation; extensions to more complex forms with involved parametrics are indicated as a direction for future development.
• Computational Scalability: Improvement in convex hull algorithms or further exploitation of graph combinatorics would be required for handling highly complex integrals.

The authors plan to generalize the code to more sophisticated parametric integrals, and algorithmic improvements in convex hull computation are likely to further enhance practical scalability.

6. Geometric Framework Summarized

The geometric framework for asymptotic expansion of Feynman integrals consists of:

• Monomial Mapping: Associate each monomial in $\mathcal{U}$ and $\mathcal{F}$ to its exponent vector in $(n+1)$-space.
• Convex Hull Construction: Find the convex hull of all exponent points.
• Facet Identification: Extract bottom facets whose normals define scaling directions.
• Region Assignment: Assign physical regions/subspaces corresponding to each facet by extracting the scaling relation $x_i \sim \rho^{v_i}$ (with normalization $v_0=1$).
• Automated Expansion: Expand the integral in each regime, keeping only terms associated to the facet to identify surviving contributions.

This geometric construction transforms the qualitative and technical process of region expansion into a robust, visual, and computationally tractable method for analyzing asymptotics of multi-scale Feynman integrals. It provides a basis for further extension, automation, and deeper understanding of the analytic structure of perturbative amplitudes.