Hepp’s bound for Feynman graphs and matroids. (English) Zbl 1520.81076
In the article, the author puts forward a new graph (and matroid) invariant, the Hepp bound. This invariant is a rational number and behaves similarly to the Feynman period. Feynman periods are special cases of Feynman integrals which are of critical importance in perturbative quantum field theory (see [F. Brown, Commun. Number Theory Phys. 11, No. 3, 453–556 (2017; Zbl 1395.81117)] for a detailed account on Feynman periods).
The invariant provides a bound for the value of Feynman periods. This property was previously used for finiteness proofs in quantum field theory [K. Hepp, Commun. Math. Phys. 2, No. 4, 301–326 (1966; Zbl 1222.81219)]. The article establishes that the Hepp bound is left invariant under a highly nontrivial set of graph transformations that also leave the Feynman period invariant (see [O. Schnetz, Commun. Number Theory Phys. 4, No. 1, 1–47 (2010; Zbl 1202.81160)] for details of these transformations). The Hepp bound is interpreted geometrically as the volume of a specific polytope. Effective formulas and algorithms for the computation of the Hepp bound are proven. The author observes empirically that the Hepp bound’s value correlates with the Feynman period’s value for certain interesting sets of graphs.
The definition of the Hepp bound is motivated as a tropicalization of the Feynman period. The fact that the Hepp bound obeys similar combinatorial transformation rules as the Feynman period indicates that the Hepp bound is an appropriate tropical analog of the period.
What follows is a brief introduction of the main characters in this article. To a connected graph \(G\) with \(N\) edges, we can associate the Kirchhoff polynomial in the variables \(x_1,\ldots,x_{N}\), \[ \Psi_G = \sum_{T\subset G} \prod_{e\not \in T} x_e \in \mathbb Z[x_1,\ldots,x_N], \] where we sum over all spanning trees of \(G\). The following integral involving this polynomial is called the Feynman period, \(\mathcal P(G)\). It is a particular case of a Feynman integral in the parametric representation (see [N. Nakanishi, Graph theory and Feynman integrals. Gordon & Breach Science Publishers, New York, NY (1971; Zbl 0212.29203)] for details on (parametric) Feynman integrals). For each connected graph \(G\) and \(D \in \mathbb R\), we write \[ \mathcal{P}(G) = \int_{\mathbb R_{> 0} ^{N-1}} \left. \frac{d x_1 \cdots d x_{N-1}}{\Psi_{G}^{D/2}} \right|_{x_N=1}, \] where we set \(x_N = 1\) in the integrand. The integral only exists if the graph \(G\) and the value of \(D\) fulfill certain restrictions. Feynman periods are interesting transcendental numbers that are difficult to compute in general. To tropicalize, the author replaces the Kirchhof polynomial with the following piece-wise monomial function: \[ \Psi_G^{\mathrm{tr}} = \max_{T\subset G} \prod_{e\not \in T} x_e, \] where, instead of summing over all monomials, we take the one of maximal value. He then defines the Hepp bound, \(\mathcal H(G)\), by \[ \mathcal{H}(G) = \int_{\mathbb R_{> 0} ^{N-1}} \left. \frac{d x_1 \cdots d x_{N-1}}{\left(\Psi^{\mathrm{tr}}_{G}\right)^{D/2}} \right|_{x_N=1}. \]
The invariant provides a bound for the value of Feynman periods. This property was previously used for finiteness proofs in quantum field theory [K. Hepp, Commun. Math. Phys. 2, No. 4, 301–326 (1966; Zbl 1222.81219)]. The article establishes that the Hepp bound is left invariant under a highly nontrivial set of graph transformations that also leave the Feynman period invariant (see [O. Schnetz, Commun. Number Theory Phys. 4, No. 1, 1–47 (2010; Zbl 1202.81160)] for details of these transformations). The Hepp bound is interpreted geometrically as the volume of a specific polytope. Effective formulas and algorithms for the computation of the Hepp bound are proven. The author observes empirically that the Hepp bound’s value correlates with the Feynman period’s value for certain interesting sets of graphs.
The definition of the Hepp bound is motivated as a tropicalization of the Feynman period. The fact that the Hepp bound obeys similar combinatorial transformation rules as the Feynman period indicates that the Hepp bound is an appropriate tropical analog of the period.
What follows is a brief introduction of the main characters in this article. To a connected graph \(G\) with \(N\) edges, we can associate the Kirchhoff polynomial in the variables \(x_1,\ldots,x_{N}\), \[ \Psi_G = \sum_{T\subset G} \prod_{e\not \in T} x_e \in \mathbb Z[x_1,\ldots,x_N], \] where we sum over all spanning trees of \(G\). The following integral involving this polynomial is called the Feynman period, \(\mathcal P(G)\). It is a particular case of a Feynman integral in the parametric representation (see [N. Nakanishi, Graph theory and Feynman integrals. Gordon & Breach Science Publishers, New York, NY (1971; Zbl 0212.29203)] for details on (parametric) Feynman integrals). For each connected graph \(G\) and \(D \in \mathbb R\), we write \[ \mathcal{P}(G) = \int_{\mathbb R_{> 0} ^{N-1}} \left. \frac{d x_1 \cdots d x_{N-1}}{\Psi_{G}^{D/2}} \right|_{x_N=1}, \] where we set \(x_N = 1\) in the integrand. The integral only exists if the graph \(G\) and the value of \(D\) fulfill certain restrictions. Feynman periods are interesting transcendental numbers that are difficult to compute in general. To tropicalize, the author replaces the Kirchhof polynomial with the following piece-wise monomial function: \[ \Psi_G^{\mathrm{tr}} = \max_{T\subset G} \prod_{e\not \in T} x_e, \] where, instead of summing over all monomials, we take the one of maximal value. He then defines the Hepp bound, \(\mathcal H(G)\), by \[ \mathcal{H}(G) = \int_{\mathbb R_{> 0} ^{N-1}} \left. \frac{d x_1 \cdots d x_{N-1}}{\left(\Psi^{\mathrm{tr}}_{G}\right)^{D/2}} \right|_{x_N=1}. \]
Reviewer: Michael Borinsky (Zürich)
MSC:
| 81Q30 | Feynman integrals and graphs; applications of algebraic topology and algebraic geometry |
| 81T18 | Feynman diagrams |
| 81T15 | Perturbative methods of renormalization applied to problems in quantum field theory |
| 14T90 | Applications of tropical geometry |
| 05B35 | Combinatorial aspects of matroids and geometric lattices |
| 52B40 | Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.) |
| 51M20 | Polyhedra and polytopes; regular figures, division of spaces |
Keywords:
Feynman integrals; tropicalization; Mellin transform; matroid invariants; spanning tree polytope; matroid polytope; polytope volumeReferences:
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