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Arithmetic and geometry of a K3 surface emerging from virtual corrections to Drell--Yan scattering

Marco Besier, Dino Festi, Michael Harrison, Bartosz Naskrecki

TL;DR

The paper resolves a key geometric question arising from two-loop EW–QCD corrections to Drell--Yan scattering by showing the associated hypersurface $X_{DY}$ is birational to a K3 surface with a rich Picard lattice: $\rho(\overline{S})=19$, $\operatorname{disc}(\operatorname{Pic}(\overline{S}))=24$, and discriminant group $(\mathbb{Z}/2\mathbb{Z})^2\times(\mathbb{Z}/6\mathbb{Z})$. It provides two independent computations of the Picard lattice, constructs an explicit Shioda--Inose structure connecting the K3 surface to a pair of elliptic curves, and analyzes the elliptic fibrations and their Mordell--Weil groups to obtain a concrete $\mathrm{NS}(\overline{S})$ basis. The work further computes the Brauer group, discusses supersingular reductions, and presents a second, fibrational route to the Picard lattice, including a detailed height-pairing analysis and a discriminant computation that culminates in a transparent lattice decomposition $N= L\oplus U$ with $\operatorname{disc}(L)=-24$ and $\operatorname{disc}(U)=-1$. These results illuminate the interplay between perturbative quantum-field-theory integrals and algebraic geometry, offering a pathway to express Drell--Yan master integrals via elliptic or modular structures. Overall, the paper advances both the mathematical understanding of K3 surfaces arising from physics-inspired square roots and the potential for applying elliptic-fibration methods to high-precision Feynman integral computations.

Abstract

We study a K3 surface, which appears in the two-loop mixed electroweak-quantum chromodynamic virtual corrections to Drell--Yan scattering. A detailed analysis of the geometric Picard lattice is presented, computing its rank and discriminant in two independent ways: first using explicit divisors on the surface and then using an explicit elliptic fibration. We also study in detail the elliptic fibrations of the surface and use them to provide an explicit Shioda--Inose structure. Moreover, we point out the physical relevance of our results.

Arithmetic and geometry of a K3 surface emerging from virtual corrections to Drell--Yan scattering

TL;DR

The paper resolves a key geometric question arising from two-loop EW–QCD corrections to Drell--Yan scattering by showing the associated hypersurface is birational to a K3 surface with a rich Picard lattice: , , and discriminant group . It provides two independent computations of the Picard lattice, constructs an explicit Shioda--Inose structure connecting the K3 surface to a pair of elliptic curves, and analyzes the elliptic fibrations and their Mordell--Weil groups to obtain a concrete basis. The work further computes the Brauer group, discusses supersingular reductions, and presents a second, fibrational route to the Picard lattice, including a detailed height-pairing analysis and a discriminant computation that culminates in a transparent lattice decomposition with and . These results illuminate the interplay between perturbative quantum-field-theory integrals and algebraic geometry, offering a pathway to express Drell--Yan master integrals via elliptic or modular structures. Overall, the paper advances both the mathematical understanding of K3 surfaces arising from physics-inspired square roots and the potential for applying elliptic-fibration methods to high-precision Feynman integral computations.

Abstract

We study a K3 surface, which appears in the two-loop mixed electroweak-quantum chromodynamic virtual corrections to Drell--Yan scattering. A detailed analysis of the geometric Picard lattice is presented, computing its rank and discriminant in two independent ways: first using explicit divisors on the surface and then using an explicit elliptic fibration. We also study in detail the elliptic fibrations of the surface and use them to provide an explicit Shioda--Inose structure. Moreover, we point out the physical relevance of our results.

Paper Structure

This paper contains 24 sections, 28 theorems, 66 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Physical background and motivation
  3. Particle physics and Drell--Yan scattering
  4. Feynman integrals via differential equations
  5. The problem of rationalising square roots
  6. Motivation for a mathematical investigation of the Drell--Yan square root
  7. The Drell--Yan K3 surface and its Picard lattice
  8. The Drell--Yan K3 surface
  9. Computing the geometric Picard lattice
  10. An application: computation of the Brauer group
  11. Elliptic fibrations on the surface
  12. Background on elliptic fibrations
  13. General method and results for the Drell--Yan surface
  14. First elliptic fibration
  15. Second elliptic fibration
  16. ...and 9 more sections

Key Result

Theorem 1.2

The surface $X_{DY}$ defined in eq:DY is birationally equivalent to a K3 surface. Its Picard lattice has rank $19$, discriminant $24$ and discriminant group isomorphic to $\mathbb Z/2\mathbb Z \times \mathbb Z/2\mathbb Z\times \mathbb Z/6\mathbb Z$. The surface $X_{DY}$ admits an explicit Shioda-Ino

Figures (3)

  • Figure 1: Shioda--Inose structure.
  • Figure 2: Left-to-right maps: degree 3; top-to-bottom maps: degree 2; back-to-front maps: degree 7. A star denotes an explicit elliptic curve which can be computed from the given isogeny.
  • Figure 3: The dual graph of the $-2$-curves which generate the Néron--Severi group.

Theorems & Definitions (75)

  • Remark 1.1
  • Theorem 1.2
  • Definition 1.3
  • Remark 1.4
  • Definition 1.5
  • Theorem 1.6
  • Corollary 2.1
  • proof
  • Corollary 2.2
  • proof
  • ...and 65 more